AAS/GSFC 13th International Symposium on Space Flight Dynamics

AAS/GSFC 13th International S

Space Flight Dynamics

Torn Stengle, Editor Proceedings o f a conference held at Goddurd Space Flight Center; Greenbelt, Maryland May 11-15,1998 National Aeronautics and Space Administration Goddard Space Flight Center Greenbelt, Maryland 2077 1 ay 1998

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* al ierr rick This paper gives the results of a study which was conducted on the SkyBridge constellation station keeping within an Alcatel and CNES partnership framework. The first step consisted in proving the feasibility of the station keeping on the candidate orbits taking into account a relatively new and stringent station keeping criterion. Then, once the mission orbit choice had been made, the study enabled us to detail the perturbations on the real orbit and to define the type of station keeping maneuver to be performed.

The last part consisted in imagining the best possible station keeping strategy for a constellation in terms of robustness and ground workload.

The SkyBridge station keeping definition is not complete; the results presented here are not final and may still change according to choice of system or satellite.

INTRODUCTION The SkyBridge project, designed and promoted by Alcatel Telecom is an ambitious satellite Telecommunications program. Its goal is to offer continuous interactive multimedia services to millions of users around the world by means of a constellation of Low E a r t h Orbit satellites.

The baseline SkyBridge constellation consisted of a nominal @-satellite constellation.

In a partnership framework with Alcatel the CNES flight dynamics division is in charge of studying station keeping strategy.

first, to assess The major problems raised by constellation station keeping analysis are, the feasibility of a strategy in relation with the mission requirements and, then, to define a strategy which limits the operational workload generated by the orbit control of a large number of satellites.

Space Mathematics Division, Mission Analysis Department, Toutouse Space Center, CNES, 18 avenue Edouard Belin. F- 31401 Toulouse cede& France. Emails : Pascal.Brousse@cnes.fr,Pierre.RozanesOcnes.fr tt Alcatel, 26 avenue J.F. Champdim, 3P 1187,31W7 Toulouse &ex I, France. Emails : erick.lansard.alcateI@~mail.com, vincent.martindalcateI0einail.com.

lowed radiation level.

lunisolar and solar radi ssure perturbations.

Those perturbations and their effects in tenns of orbital behavior must be carefully analyzed, particularly with realistic assumptions on the size, mass and shape of the satellite to precisely determine the long tern orbit evolution caused by non-gravitational forces which are the main input data for the station keeping defintion.

Moreover, such a telecommunication system requires a station keeping criterion which is not commonly used. The SkyBridge system is designed to provide wide band communication over orld to a huge set of ideptical terminals which have the same pointing capability. This terminal feature determines the maximum deviation allowed between the so-called reference orbit and the actual orbit and then defines the station keeping constraints which must be fulfilled over the complete orbit. As for the reference orbit it must be simple enough to be loaded once for all into the user terminal and to provide all the orbital data needed to keep the terminal in good working order: visibility dates of each of the 64 satellites, antenna pointing data for each visibility, etc..

Another key point of a constellation station keeping strategy design is its efficiency to limit the operational workload with regards to the large number of satellites. Even if the orbit surveillance and control is supposed to be automatic with human intervention only in contingency cases, the number of maneuvers should be reduced to a minimum to increase the reliability of the station keeping operation and, then, to decrease the operational workload.

And this reliability should be obtained without adding, as far as possible, constraints on the telecommunication mission such as attitude maneuver before orbit maneuver which could require, in the best case, on board antenna pointing reprogramming and, in the worst case, telecommunication mission interruption.

This paper describes the SkyBridge station keeping strategy analysis and shows the feasibility of the envisaged strategies . " - l _ - - " " " . _ - I " " I U I . - " ~ " . " - " ~ . . . - " " " " . - . " - " " . . - . . " . . - - . . " - . " - " - - - - . . " - " . " " - ~ I i SkyBridge 2 CEO satellite SkyBridge I forbidden to user terminal I f a 5 Q,, User terminal Figure 1 : Frequency Sharing Constraint Any ground point ( user terminal ) in visibility with a SkyBridge satellite with an elevation angle above 10 degrees, cannot establish a l i n k with this satellite if the angle between the line of sight of the user terminal to the satellite and the line of sight of the user tenninal to any point of the geostationary ring is less than 10 degrees.

this constraint, the system is designed to provide to any user terminal located Despite between 70" S and 70"N a continuous single coverage.

It has been decided that the orbital data needed to schedule the terminal antenna pointing will be loaded once for d l into the terminal. This solution was supposed to be simpler t h a n a regular broadcast of updated ephemeris of all the satellites to all the terminals.

But, in order to store limited orbital data in the terminals, the reference orbit should be phased in relation to the E a r t h with a cycle duration of several days, to obtain a repetitive geometry.

The station keeping constraint is given by the maximum authorized deviation between the actual orbit and the reference orbit tracked in open loop by the terminals. For the time being, the system specification defmes a maximum deviation of 0.5 degrees between the line of sight of the user terminal to the reference position and the line of sight of the user terminal to the actual position.

The SkyBridge station keeping requirements, phased reference orbit with a maximum angular deviation of 0.5 deg, leads inevitably to absolute station keeping.

At the early stages of the project, two altitudes were envisaged : one 24 hour resonant orbit at an altitude of 1630 lun and a non resonant orbit at an altitude of 1457 km.

ION KEEPING FEASI AT AN ALTITUDE OF 1630 KM The nominal SkyBridge station keeping orbit is a phased circular orbit and the fmt step of the station keeping definition was to choose the cycle duration and the nominal altitude.

The inclination of the orbit w a s already fmed at 55 deg.

Taking into account the J2 term of the E a r t h potential, the parameters of the phased orbit at these altitude and inclination were as follows: semi major a x i s : 8008.000 km number of node periods during the cycle : 12 cycle duration : 0.9902 days eccentricity : 0 inclination : 55 deg.

With this orbit, the geometry of the constellation is periodical with a period duration of 1 day; this notably limits the amount of data corresponding to the visibility schedule and pointing data stored in terminals.

But this very short cycle duration raised the problem of the strong resonance of the orbit in relation with the E a r t h potential perturbation. The a i m of the preliminary study was to assess the feasibility of the station keeping taking into account this strong resonance and the stringent constraint of 0.5 deg maximum deviation.

This resonance due to the earth potential causes a disturbance whose amplitude is much higher than the amplitude of the other perturbations; the station keeping feasibility study must therefore identify the maximum effect of this disturbance.

The resonant earth potential terms can be easily identified by using the Kaula development of the disturbing potential.

Slm if E - m is even - S l m i f l - m i s 0 C l m i f l - m i s o

Ylmp = ( E -2p + 4)M + ( E - 2p)w + m( - 8 ) where 8 is the sideral time and ae

the Earth radius.

As the orbit is circular, we only consider term q = 0 as functions Glpq(e) are null for e=Q and q#O.

The perturbations occur for y- = o which occurs for 1-2p=1 m=12 and 1-2p= 2 and m=24.

The quadruplets (l,m,p,q) corresponding to the main resonant terms are therefore:

(13,12,6,0),(15,12,7,0),(17,12,8,0) ....... (23,12,11,0),(24,24,11,0)

The effect of these resonant terms on the semi-major axis and the inclination is a periodic effect with a very long period equivalent over a shorter time horimn to a secular effect. An analytical development allows the maximum amplitude of the phenomena to be calculated on the semi-major axis and on the inclination.

this orbit at an altitude of 1630 km, we obtain the following values: For 1.29 d d a y on a and 0.0237 deglyear on i.

This effect is the maximum effect on this orbit but the actual value, in fact, depends on the initial phase and longitude of the ascending node of the orbit.

A station keeping simulation by converting the maximum angular deviation of 0.5 deg into a maximum phase deviation and a maximum node longitude deviation leads to following station keeping costs: T o t a l Av: 20.5 m /s for a life of 8 years inter-maneuver time: 39.5 days with combined semi-major axis and inclination maneuvers.

The station keeping cost, although high, is not necessarily very design-critical in relation to the one generated by the positioning. This result demonstrated that the use of this orbit was feasible from a station keeping point of view.

Yet, additional radiation and space debris considerations led the constellation designers to envisage a lower altitude in order to improve the life time of the satellite. An altitude of 1457 km w a s then selected as the nominal station keeping altitude.

parameters : the inclination was still fuced at 55 deg, the number of nodal periods per day was chosen equal to 12+11/28 which gives a semi major axis equal to 7834.983 km and a cycle duration of 27.709970 days Because the purpose of the reference orbit is to have constant parameters, it was decided to choose a frozen eccentricity. Indeed it is unthinkable to have a reference orbit with an eccentricity vector which is time variable, it is simpler to have a constant eccentricity vector

which is achieved by freezing the eccentricity ( : : : ) -

Taking into account the zonal terms of the earth potential from 52 to J16 and the nominal values of the semi major axis and the inclination, the components of the reference eccentricity vector are as follows : e = 0.94064 0 = 9 o d e g The initial values of the other orbital parameters s2 and ~ F - D ~ M are not specified because they could range from 0 deg to 360 deg depending on date and on satellite considered in the constellation.

In fact, a l l the right ascension of the ascending node and phase values will be deduced, at a given date, fkom the constellation architecture and the actual values reached by the first 4 satellites launched in the plan which will become the fmt plan of the constellation. Today there is no requirement on the initial value of the ascending node longitude of this first plan.

The nominal mean parameters of the reference orbit are s m a r i z e d below : a = 7834.983 &m.

i = 55 deg e = 0.94064 w = 90 deg The SkyBridge constellation architecture is given by the Figure 2.

Figure 2 : SkyBridgeConstellation Architecture Orbit perturbations The hypotheses on the satellite surface and mass taken into account to compute the orbit perturbations are the following : solar panel surface : 47 mz average satellite body surface : 4 m2 satellite m a s s : lo00 kg The perturbations to be analyzed are the perturbations leading to deviation of the actual orbit with respect to the reference orbit. The long t e r m deviation between the two orbits will be controlled by the station keeping strategy to fulfdl the maximum deviation constraint.

The relative perturbations between two different orbital planes of the constellation due to the relative geometry of each plane with respect to the sun, the moon and the earth are not analyzed because they have no impact on the s t a t i o n keeping strategy definition, they just produce a natural shift between the maneuver calendar of each satellite.

The main results of our perturbation analysis are presented below.

SkyBridge Station Keeping: Drag Figure 3 : Drag Perturbation Between 1980 And 1990 Perturbations due to the luni-solar potential As shown i n Figure 4, the luni solar potential leads to a periodical evolution of the inclination but generates a secular drift on i2 which has to be controlled .

Two periods : I30 days due to the moon i 48 &ys due to the sun I I I Lunisolar p e r t u r b a t i o n s on SkyBridge ss.001 Inclination ?is.oDa 15.m1

-

= $5.000 3d.YSY j U.998 2 u.,n

1 54.0)s

- 54.0)s

11.994 lOM0 11IW 1 Z w O l a m 13200 13800 11400 2 - e tcmss ns%an r a t a : 10Y3141/ol/1¶110) IZOOD 1zMO 13200 1 4 0 0 2 1 1 1 . t m l B %lira at. I M Y 3 1 4 I / o l / l Y ( I O I Figure 4 : L u n i s o l a r Perturbations Perturbation due to the solar radiation pressure The m a i n effect of the solar radiation pressure is a periodic evolution of the eccentricity around the frozen eccentricity SkyBridge frozen e c c e n t r i c i t y , r t 4., s 4 . 8 1 2 4 . 5 u d 0.8 a t . . s r? I . . a? 1.5 I? 2.8 r?

Figure 5 : Solar Radiation Pressure Perturbation STATION KEEPING STRATEGIES deviation, 6, compatible with the terminal link budget is fixed to 0.5 deg whatever the relative geometry between the user and the s a t e l l i t e .

To define the station keeping strategy, the maximum value of 6 reached over a complete orbit has to be written as a function of the orbital parameter deviation between the reference orbit and the actual orbit.

To obtain this equation, the first step is to write the maximum angular deviation, &, seen form the earth center between two orbits :

6 , 2 = (sa +cosi my +8i2 + (silli

where : 0 6a is the phase deviation 0 82 is the RAAN deviation e 6i is the inclination deviation and considering small angles.

Then &can be easily l i n k e d to 6 by the following formula : a sin 6, tgs = acoss, -ae The following drawing (Figure 6) shows the optimal evolution over time of 6 considering only an initial phase deviation and the atmospheric drag as disturbance.

0.5 deg time Figure 6 : Sketch Of The Station Keeping Criterion Evolution Figure 7 : Effects Of Perturbations On The Station Keeping Criterion Evolution It appears that : e the atmospheric drag is still the main perturbation for the station keeping criterion, an initial inclination offset of 0 . 0 0 3 deg on the actual orbit allows the secular d r i f t on

X 2 to be decreased. As there is no secular evolution on the inclination deviation, this

enables us to write roughly 6 as the sum of a term which is a h c t i o n of 6a and a quasi constant term

62 = 6oc2 +2 ~a ssz cos i + m2 + 6i2

So the m a i n purpose of the SkyBridge strategy is to maintain the semi-major a x i s around its reference value taking into account the satellite attitude law used to maintain the solar panels pointed towards the sun for power optimization.

ar panels towards s of a rotation around the Z s which is the n is, and a rotation around the Y axis which is the rotation axis of the solar panels.

The optimal combination of these two rotations leads to a continuous yaw steering of the spacecraft over its orbit. The theoretical law is given with the notation of Figure 8 by the following formula tgB tgQi =- sin(a -ao) where p is the angle between the sun line and the orbital plane, @ is the spacecraft yaw axis.

and a is the in orbit position in reference with which is the canonical position.

figure 8 : Angle Definition Even if a simplified law is loaded into the satellite, the yaw angle follows a periodic evolution whose amplitude is a function of p.

The yaw steering strategy is used continuously except when (3 <PO. When (3 is near this

angular limit a futed yaw attitude is utilized to avoid excessive yaw rates. In this case, the satellite is positioned at a yaw angle of 0 or 180 deg.

t u e o ategies oss t ffe 1 .

With this strategy, we only correct the semi-major axis when yaw steering is disable. In this mode, the yaw angle is equal to 0 deg or 180 deg, the nozzle is then oriented according to the speed of the satellite or opposing the speed of the satellite. This configuration enables the semi- major axis to be increased or decreased.

The yaw steering mode is disable periodically every 47 days during a period lasting several days.

This strategy does not allow correction of Q or correction of i.

Strategy 2: In this strategy, whatever the attitude of the satellite, an attitude maneuvre is performed before the orbit maneuver to obtain optimum maneuver attitude. This strategy enables all optimum orbit corrections but leads to strong coupling between the attitude and the orbital maneuvers.

This strategy leads to a depointing of the solar panels during orbital corrections and can lead to an interruption in the telecommunication service.

Strategy 3: In this strategy, we will make semi-major axis maneuvers whatever the yaw angle value ensuring that the off-plane thrusts induced do not lead to evolutions of 6n and 6i incompatible with the station keeping strategy. To avoid the off-plane component from being too high, the maneuvre are only performed when the yaw angle is lower than 60 deg.

This strategy enables semi major axis corrections to be made practically at a l l times. It also enables Q and i corrections to be made if required.

Station keeping strategy trade-off The station keeping of a constellation must be as flexible as possible to, on the one hand, generate minimum possible constraints on the mission and, on the other hand, to limit the operational constraints.

A strategy which is not robust may lead to extra operational work in case of unexpected dispersion. Although this extra work maybe possible for a single satellite system, it is certainly very difficult to absorb when controlling a constellation.

y c o n s t r ~ ~ n g conce Strategy 2, although optimum for orbit control, generates too high a constraint on the payload which must take into account the scheduled maneuvers to repoint its onboard antennas.

Strategy 3 leads to negligible extra costs in terms of consumption while offering all the of types of maneuvers which makes the strategy robust to dispersion. Also, it possibility generates a number of maneuvers near to the optimum number of maneuvers.

The table below summarizes the performances of each of the strategies over a life time of 8 years and taking into account a medium solar activity.

I 3 I 0.38 I 0 . 2 4 I 1 1 I 78 I

This study conducted by CNES in partnership with Alcatel has enabled us to prove the feasibility of the SkyBriclge station keeping on two different orbits, one being highly resonant.

As, fmally, a non-resonant orbit was chosen, the perturbations were analyzed to evidence the perturbations of the real orbit in relation to the reference orbit.

Lastly, a strategy robust to dispersions and without additional constraints was proposed.

It seems today the most suitable for controlling this 64-satellite constellation.

Of course, the Skybridge station keeping is not yet frozen and still subject to refinements. Bu the current paper shows that there exists efficient feasible solutions and the trade-offs between the possible options are well understood.

1 4 ACKNOWLEDGEMENTS The authors would like to thank Luc Lefebvre and Laurent Lagarde from CNES for their contribution to this work.

1. JL Palmade T h e SkyBridge constellation design" International Workshop on Mission Design & Implementation of Satellite Constellations,November 1997.

2. RB Frauenholz The Role of Anomalous Satellite-Fixed Accelerations in TopexPoseidon Orbit Maintenance"AASIAIAA Astrodynam'ks Specialist Conference,August 1993.

3. L. Lefebvre. "Relative Station Keeping Optimization For The Starsys Constellation" 12'h InternationalSymposium on Space Flight Dynamics. June 1997.

Marco M. Cas~onuovo', Carlo Ulivierf and iovanni Laneve$

Uniform Homogeneous Constellations (UHC) o f small satellites on multi-sun- synchronous (MSS) orbits, which allow to revisit any given location every m nodal days with an integer number n/m o f different illumination geometries repeated every n days, have been considered. These constellations could offer a good means of continuous surveiilance for the tropical regions where natural disasters, such as floodings and droughts, are most common.

A study has been d e d out to ascertain how they allow to obtain efficient revisit merages and repeat cycles, deploying satellites on one or more orbital planes whose inclination has been chosen equal to tropical latitudes. The dynamical behaviour of a satellite constellation has been simulated, so to analyze the configuation and coverage evolution under the effects of the major perhubatiOIlS.

INTRODUCTION Satellite remote sensing Hers the only means of obtaining synoptic coverage of large geographical regions; sunsynchronous quasi-polar orbits d o w to observe from low altitudes a wide latitudinal range, excluding the polar caps.

However longitudinal coverage is strictly related to the swath width of the onboard instrument and to the required revisit frequency of observation; the spatial gaps occurring between two consecutive satellite ground traces is uniformly reduced if a suitable value m o f nodal days is accepted. The gap i s so much wider how lower is the latitude; for this reason the observation of low latitude areas is critical since here a high spatial resolution and fast response to the detection of natural and human-caused catastrophes is particularly required. The best solution is obtained when the gap (either temporal or spatial) is reduced by using a constellation with satellites located on circular Multi-SunSynchronous ( M S S ) orbits'. A study has been camed o u t to obtain efficient Post Doc Fellow of the Universiti degli Studi di Roma "La Sapienza', Scuola d'lngegneria Aerospaziale, Via Eudossiana 16, Rome, Italy. Phone: ++39 6 8124629, FAX: ++39 6 8104SSI. e-mail: castronu@crasan.psm.uniromal .it Full Professor of the Universiti degli Sudi di Roma "La Sapienza', affiliated with the Centro di Ricerca Progetto San Marco, Via Salaria 851,00138 Rome, Italy.

*

Assistant Professor of the Universiti degli Sudi di Roma "La Sapienza", Scuola dlngegneria Aerospaziale, Via Eudossiana 76. Rome, Italy.

ora one or more orbital planes whose revisit coverages an ~ n ~ l i ~ a t i o ~ has been The dynamical behavior of a test-case constellation has been simulated nmerkally by means of evohtion under the , so to analyze the effects of the major ~ e ~ r b a t i o n s .

LLlT IN TE A srngle Circular orbit satellite will perform a continuous ground track pattern; the geographical coordinates of its nadir trace for a unifody rotating spherical earth are given as:

4 = l a t i d e = sin-'(sini sinu)

=Iongitude=A,+tan-'(cosi tanu) These relations are obtained by applying Napier's rules to the spherical triangle represented is the angular earth rotation, S2* the orbital nodal precession rate and u the in Figure 1; os argument o f latitude.

Figure 1 Geometry of satellite ground track The westward longitudinal separation between w n s d v e equatorial crossing ( S , ) is given by: where T, is the orbital nodal period, whose expression is where: K = - J~ R; J;; Udess using a sensor with a quite wide swath, the spatial gap between the consecutive revolutions of the track pattern does not guarantee a complete coverage in a period of the order of few days; on the other hand the condition for periodic coverage is that the nadir trace is duplicated after some d a t e period of time after which the previous pattern is retraced and so on indefhitely.

The possibility to have the required Coincidence of the li"h orbital node with the nodal earth rotation is accomplished by adjusting the satellite nodal period so as to produce exactly R revolutions in m nodal days (Dn ): mD, = RT, (3) with 2 z 0, = (4)

0, -!5

When the apparent motion of the Sun (a$ ) is equal to R' (sunsynchronous condition) 0, = solar day.

Since m and R are restricted to integral values Eq. (3) tends to limit T , (and then the correspondmg altitudes)to a series of discrete values, t a k i n g also into account the necessity of considering periods of reasonable durations i n order to locate satellites in low orbits (avoidmg, however, excessive air drag). Substituting E q s . (2) and (4) i n t o Eq. (3) and expressing R' as a function of a and i, we obtain the following polynomial in a for the periodicity condition:

CI a5.5 + Cz a ' + C3 a3.5 + C4 a2 + C, = o

(5) with The coverage pattern also can be considered discrete (intennittent). The m-fold equatorial arc is evenly divided in two different ways: in m equal parts and also in R equal parts; the simplest single division mode which includes both the m and the R-fold divisions is the least common multiple of R and m or Zcm(R,m). This solution is unique; in practical applications the Uniqueness requirement is met simply by assuming that only relatively prime m, R pairs are considered.

The number of orbits perfomed in one day is the repetition factor Q which can be split in two tenns accordmg to the relation k

Q = Ni + Nf = Ni +-

m where Ni is the integer number of orbits per€ormed daily and N- is its fractional part. Then the abovesaid possible solutions @airs of R and m) are given by the values of k, prime integer with m and I S k Sm-I; it defines the ground track spacing amrdmg to the relationship: ;2,=;2,+Sr mod d - (7)

( 3

where A, represmts the crossinglongitude on day d, 5 is the eo-rotating crossing longitude at the initial time (d = 0) and the operator

mod(x) = fiac(x) - intl fkac(x)+i I

allows to have subsequent crossings w i t h i n 5 -t S t / 2, accordmg to Hopkins’ notation’. M e r m days, the longitudinal increment Sr will be divided in m equal increments of longitude S m = Sr / m; during the pattern development between two consecutive nodes, each day’s nodal co-rotating crossing occurs east (or west) of the previous day7snode by f k S m where the sign is positive or negative if k < m / 2 or k > m / 2 respectively.

uences the choice of a uniformly S m i s related to the swath w i d t h of the und tracks at the end of the repeat interval. In fact, coverage of the earth's equatorial region from a satellite i s accomplished if the swath w i d t h on the earth's surface required for coverage is equal to the minimum longitudinal interval S m b earth traces. If Nf = 0 then the number of orbits performed must be an integer and the repeat 1 rn i s only one day.

Our interest is limited to constellations consisting of satellites evenly deployed on circular orbits with the same altitude and inclination (Uniform Homogeneous Constellations - UHC).

Previous works demonstrated that the addition o f the number o f satellites N on the same orbital plane and ofthe number of orbital planes P to a constellation has the potential of improving both revisit and spatial coverage3*'. Table 1 summarizes all the possible situations for smgle and multi- plane UHG.

Table 1 SINGLE AND MULTI-PLANE UHC

r (nodal daw) S, (km) - m/r

P lcrn S , P .m .N - (highest revisit P .N lcrn lcrn frequency) P I ? C ? T 2 S. m - N '

-

(minimum ground N P -1cm lcrn track spacing) where r is the repeat cycle of the constellation (nodal days) and Zcm = Icm (1v,m) is the least common multiple between Nand m and S, is the minimum ground track spacing.

ORBIT SELECTION In order to provide a service of continuous and global surveillance of the tropical regions the mission design must take into account several requirements in the orbit selection. The orbit periodicity (repetitivity), and the same geometry of illumination for remote sensing systems operating in the visible are the main ones.

The condition of multisunsychronism (MSS) is obtained when the difference between the

apparent solar m o t i o n a'

is a submultiple of the and the orbital nodal precession rate $2' difference between the angular e a r t h rotation wE and $2' . Since J ? ' depends on both the inclination i and the semi-major axis a of the orbit, the MSS condition can be expressed by the relation: where n is the number of nodal days ry to reencounter the same geometry of illumination.

The upper sign is valid for 0' < a' , the lower one corresponds to $2' > a' . In our specific case orbits w i t h inclination around the tropical latitudes have been considered so that L ? ' < 0, therefore the upper sign should be considered. Eq. (8) is represented in Figure 2 for inclinations between 20" and 3 5 ' and satellite altitudes between 400 km and 1000 km for various values ofn.

I000 n E W 20 22 24 26 28 30 32 34 orbital in c I i n ation (deg) Egure 2 Multi-suuspchronoussolutions from E q . (8) Among all these solutions only periodic orbits should be considered. This can be obtained by solving simultaneously Eq. (5) and Eq. (8). Explicitkg cos i f r o m Eq. (8) and substituting into Eq. (5) we obtain the following polynomial in a whose solutions are both periodic and MSS, provided that m is a submultiple of n and t h a t R and m are prime.

C1a7 + C 2 a 2 +C,a0*' +C4 = O (9) with To obtain reasonable values for the orbital altitude (say between 400 km and 1100 km) R should be given a value around (12-;.14)x rn. Examples of solutions of Eq. (9) are shown in Table 2.

The selection among all possible solutions has been done in accordance with the following criteria.

The value of the illumination cycle r n has been chosen to be the lowest possible, since a UHC constituted by an equal number of satellites has a revisit freequency of 1 nodal day; the satellite altitude h has been l i m i t e d between 400 km and 1100 km and the orbit inclination i between 23.45' and 35'. Then, for each solution found, the d u m instantanwus field of view (6. ifov) necessary to obtain the global coverage of the earth has been computed. The last column of Table 2 contains the distance between the ground tracks at the equator.

It should be noted that since the orbital inclination i does not appear in Eq. (9), its value will be computed by Eq. (8) once Eq. (9) has been solved for the value of the semi-major axis a. This means that aIso the orbital inclination can assume only discrete values in accordance with the integer numbers rn, n and R that are used as input. Obviously appropriatetriples of rn, n and R can produce orbital inclination values in the desired range.

Accordmg to Table 1 the revisit frequency can be reduced to values below 1 n o & Z day deploying a number N of satellites (equal to m) on each of P orbital planes. So, for example, i n the cases with rn = 2 a revisit frequency of '/z nodaI day can be obtained with a UHC composed by 4 satellites, deployed on 2 orbital planes (2 satellites on each orbital plane) with ascending nodes at 9 0 O apart. Of course the choice of a particular MSS orbit as reference trajectory for a surveillance constellation is the result of a trade-off process among several Connieting parameters such as the illumination cycle duration (that determjnes the number of satellites required to have a specific revisit frequency), the height ofthe orbit (and therefore the required instarrtaneOus field of view to obtain global coverage), the inclination (dete-g the extension of the area to be covered) and the number of different illumination geometries. The requirements of each particular mission will detemine, time by time, the conditions to be m e t by the constellation.

i (da) - k min. ifov (d@ Sm &m) m - n hlkm) 29 1 51.29 1381.90 2 50 644.41 26.60 1381.90 648.18 35.14 29 1 63.06 2 54 37.36 1484.26 2 58 999.34 27.08 27 1 42.06 1484.26 2 60 1000.74 3 1 . 2 2 27 1 44 35.22 910.80 3 48 588.20 2 4 . 2 1 2 700.59 43 32.60 931.98 3 51 26.08 1 931.98 3 54 703.31 3 2 . 8 1 43 1 39.50 27.5 1 977.44 3 57 937.63 28.01 41 2 41 32.29 977.44 3 60 939.82 33.83 2 60 1062.96 40 25.17 1001.88 3 28.28 1 679.24 4 48 561-48 25.85 59 3 29.55 679.24 4 52 565.56 34.86 59 3 37.88 24.74 703.07 4 52 729.52 27.07 57 1 55 20.88 728.64 4 56 906.85 27.30 3 4 60 1094.57 26.64 53 1 1 7 . 6 1 756.13 527.30 5 45 418.09 25.59 76 1 30.47 29.04 541.55 5 50 547.68 3 1 . 5 9 74 4 5 677.42 19.44 556.60 50 24.64 72 2 5 55 749.08 32.76 7 1 23.04 564.44 580.80 5 55 888.20 25.86 69 4 1 6 . 2 3 60 16.15 598.13 5 1037.91 29.50 67 2 6 48 431 . 7 3 91 30.74 440.38 3 2 . 6 1 1 6 48 535.03 27.38 89 5 21.91 450.28 54 759.56 17.76 471.47 6 3 0 . 2 3 85 1 6 54 875.54 83 12.75 482.83 23.9 1 5 NUMERICAL SIMULATION AND ORBIT MAINTENANCE To evaluate the influence of the main perturbations on the orbital parameters of a multi- sunsynchzonous UHC, a particular test case has been umsidered for numerical integration. In order to guarantee a global coverage of the tropical regions with a revisit time of % ylodal day (around 12 hours), a multi-sunsynchronous orbit with an altitude of 648.18 km and an inclination of 35.14’ has been chosen as reference trajectory for the intended satellite constellation. Such an orbit gives rise to repetitive ground track with a revisit interval of 2 no&1 days and a separation between adjacent tracks of 1382 km at the equator. In addition, every 54 m & 1 atzys the same place on the ground is flown over in the same illumination conditions. The relatively high inclination of this orbit implies the Continuow coverage of a quite vast area. If, as in this case, this has to be obtained with a very limited number of satellites (only 4), this results in a large number of illumination and, being the ground tracks &,the equator considerably far apart from each other (1380 km), a very wide instantaneous field of view of the sensor. The pattern of ground tracks for such an orbit during a m-day cycle is shown in Figure 3.

complete Figure 3 Ground track pattern ofthe test case orbit during the m - % cycle The exact repeat of the satellite ground track pattern may be altered as a consequence of the variation of the nodal period, of the nodal precession rate and of the orbital inclination.

The major perturbations affectingthe ground track repeatability are the atmospheric drag and the luni-solar gravitational potential. Luni-solar disturbances cause secular and long-period variations in the nodal period, the nodal precession rate and the inchtion, while the atmospheric drag reduces systematicallythe semi-major axis, affecting both the nodal period (first order effect) and the nodal precession rate (second order effect).

To simulate the behavior of the constellation under the influence of the major perturbations a numerical integration of the mation of the satellites has been carried out by means of GEODYN LT softwarefor a time span of about 110 days. All perturbations suitable for propagation of low earth orbits have been selected.

We have considered 4 satellites (mass = 300 kg, cross section area = 3 Irk and CD = 2.0) uniformly deployed on 2 orbital planes with nodes equally spaced (90O apart) The evolution of the semi-major axis for the 4 satellites considered is represented in Figure 4. As expected we can notice a decrease i n the semi-major axis at an almost constant rate due to the atmospheric drag. However, it is evident that the two planes of the constellation are affected in a different way by the aerodynamic resistance. Such a difference can be explained by the effect of the atmospheric diurnal bulge due to the heating of the atmosphere by direct illumination of the Sun. Nevertheless, because of the multi-sunsynchronicity of these orbits, the differential ef€ect between the two planes i s smoothed out in comparison.with the case of correspondmg sun-synchronous orbits that have constant geometry of illumination.

7829.44 7829.43 ,.

I Y u W829.41 . - # x L 0 -9

+ 7889.4

I 0 UI 7829.39 7819.38 Figure 4 Semi-major axis evolution for the 4 satellites of the constellation The maintenance of the operational orbit considered has been analyzed evaluating the maneuver requirements, taking i n t o account t h a t , a t the altitude considered, the driving natural effect is the reduction of the nodal period due to the atmospheric drag. Tn fact, because of the constant reduction in the semi-major axis induced by the aerodynamic resistance, the satellite ground track slowly drifts to East. The knowledge of the nodal crossing drift is hdamental for planning the orbital corrections needed in order to keep the correct confi,auration of the constellation.

A possible strategy that maximizes the time interval between the maneuvers consists in s t a r t i n g the control cycle by placing each satellite on its eastern boundary of the longitude deadband, with a semi-major a x i s augmented by a suitable amount da with respect to the n o d value. If da i s computed accordmg to the expected semi-major a x i s decay, the sub-satellite ground track will drift westward, u n t i l will reach the westem boundary of the deadband exactly when the semimajor axis is back to its nominal value. Afterwards, the further reduction in the semi-major axis will cause an inversion i n the ground track drift eastward, and, as far as the drag force may be considered constant during a fidl control cycle, the eastern boundary ofthe deadband will be reached when the semimajor axis assumes a value that is lower than the nominal one by da. A t that point the semi-major axis must be increased by 2Aa by means of the propulsion system to repeat the control cycle and maintain the ground track within the required tolerance.

2 6 or circular orbits & & e d by a constant force, the velocity increment dv, required to 1 cycte, the rime interval between maneuvers restore the initial conditions at the end of each and the corresp da can be easily computed': B p Y 3 A AVc = 3 W E R E B

A a = - 2 p Jr a Tc

where p is the air density, B the ballistic coefficient and the earth equatorial radius.

C o n s i d e r i n g a tolerance on the ground track A = f Z km, the Jacchia atmospheric model' with an exospheric temperature of 1000 K, and the assumption that monopropellant hydrazine would be used as fuel for the orbit control system the following values can be obtained for our test case from E q s . (lo), (11) and ( 1 2 ) .

Table 3 CONTROL CYCLE RELEVANT FIGURES

Altitude m-) TC.h.!E& -C AV ( d s ~ e r w ) Fuel M w ) Aa (m)

648.18 19.1 0.4 0 . 1 18 The luni-solar disturbances aEect the track repeatibility, but can be taken into account by means of very small corrections to the semi-major axis control cycle. Nevertheless, due to the resonance existing between the sun motion and the nodal precession rate of multi-sunsynchronous orbits, long period and secular variations of the inclination could be expected.

An inchtion variation involves a displacement of the sub-satellitetrack at higher or lower latitude.

Therefore, an inclination control cycle should be envisaged as well if the track repeatability requirement has to be m e t eveyhere along the orbital path.

rbits for bservation”, Advances i n the Astronautical Sciences, Vol. 76, Univek Jnc, San it Coverage Using -Satellite Constellations”, er N. 88-4276-CP7 A s t r o d p h c s Conference, oh, USA, 19 3. C. Ulivieri, 6. Laneve and S. M. Hejazi Moghaddam, “UPH Constellations for Continuous R e g i o n a l Surveillance”, AAS 97-622, AAS/AIAA Astrodynamics Codmence, Sun Valley, USA, 1997.

4. C. Uiivieri, 6. Laneve and S. M. Hejazi Moghaddam, “Orbit Design Analysis for Remote Sensing SateIlite constellation^'^, Paper presented at the IAF Workshop on Mission Design & hplementation of SateIlite Constellations, Todouse, France, November 1997.

5. L. G. Jacchia, Revised Staric M o d e l s of the Thermosphere and Ecosphere with Empirical Temperature Profiles, Smithsonian Astrophysical Observatory, Special report 332,197 1.

C . Brochet, J.M. Garcia, J. . Enjalbert, T . C601int

During the mission of a constellation, maneuvers must be introduced periodi- cally to reset the drifted satellites. Moreover some satellites may fail during the life of the constellation, and maneuvers have to be done to ensure the desired coverage.

In this paper, we propose several optimization models for this problem. For each model we present the most efficient resolution algorithm. Each model consists in minimizing the total consumption due to maneuvers. It takes into account the trajectory of each satellite and constraints on their relative posi- tions. An additional constraint is introduced to limit the number of satellites that can be simultaneously controlled. Such an optimization problem is a Mixed Integer Non Linear Programming (MINLP). It contains boolean and real variables. Boolean variables determine which satellites can be thrusted, and real variables correspond to the value of maneuvers.

The global problem is splitted on the basis of the generalized Bender’s decomposition method (projection on the boolean variables space).

The first model is linear and differential (relative satellite positions). The sub- problem (calculation of the impulsive thrusts) is solved by a dual approach that finds the solution in a finite number of steps. It provides the global opti- mum in a very short computing time. This model is interesting in the case where the phasing of the constellation is not far from nominal conditions.

The second model is nonlinear and non differential. It represents the real problem without simplifications. The resolution of the sub-problem is done using a direct search approach (Hooke and Jeeves algorithm) to determine real variables in the sub-problem. This model is used to solve the station keeping problem and to determine optimal maneuvers to replace satellites in case of failure.

Numerical experiments and comparison between the two approaches are pre- sented for various constellation configuration parameters.

INTRODUCTION Spatial projects are more and more numerous, and regularly, satellites are launched in orbit in order to begin a mission for many years. The objective of a constellation is for example to ensure a coverage that allows datas communication between two satellites or between a satellite and a ground station. To avoid gaps in the required coverage, relative positions of these satellites must not exceed a fixed threshold. But many perturbations make the trajectory of satellites drifted, and it is necessary to regularly maneuver some of them. This process is called station keeping. These maneuvers have to be calculated in order to minimize the consumption of each satellites, since the mass of ergo1 is limited.

t LAAS-CNRS, 7 avenue du Colonel Roche, 3 1077 Toulouse cedex 04, France.

reover, since the number multaneous maneuvers is limited, additional constraints ve to be taken into account ame model can be used to calculate replacement rnaneu- evious researches have been done on this subject. A linear model of the station keeping optimization problem has been stated, and a method to solve it has been proposed (Ref. 8 and 9). In this paper, we present new robust and efficient methods that can solve this linear problem. We also present a nonlinear station keeping optimization problem and an algorithm to solve it exactly.

In the first section, we present the problem modelling. The second section is devoted to the resolution of the mixed-variables problem. The method is decomposed into two lev- els: the master problem and the sub-problem. It needs successive resolutions of the sub- problem (real variables problem with boolean variables fixed). The two following sections present the resolution of the real variables problem, assuming that constraints can be lin- ear or nonlinear. In the last section, numerical results and comparisons are presented.

STATEMENT OF THE PROBLEM

Station keeping problem In this paper circular Walker constellations with N satellites are considered. The life of the constellation is decomposed into successive station keeping cycles. At each cycle, the station keeping optimization problem must be solved. It consists in finding the lowest values of maneuvers, that maintain the desired coverage, respecting operational con- straints. These constraints mean that all satellites cannot be thrusted simultaneously, but just M among N. To take this into account, each station keeping cycle contains K steps.

Only M maneuvers at the beginning of each step are allowed. That’s why boolean vari-

ables p are introduced in the optimization problem. If is equal to 1 then the corre-

sponding satellite i is allowed to maneuver at the beginning of the step k, otherwise it cannot maneuver.

cycle = T cycle = T cycle = T A eK- 1 ..... .....

T/K T/K F time ei: step i P TStm= start of the mission

1 0 ... satellite 2

P ....

Figure 1 Station keeping cycles Hence, solving the optimization station keeping problem, is to find for a cycle:

0 Which satellite can maneuver at each step? Values of boolean variables: Pik

0 What are the values of these maneuvers? Values of real variables: &Vt Therefore, the problem is a mixed-variables problem. It contains K*N boolean variables and K*N real variables.

The general expression of,this problem is the following: niingv, gJ(SV, P) IAdfI < $ V i , k i denotes satellites whereas k refers to steps K - 1

1 6 $ 1 I Consi Consiis the maximal consumption authorized for satellite i

Vi (1) k = O N P f l M Vk i = 1

( 1 - P f ) . svf = 0 Vi, k

The objective is to minimize the total consumption of each satellite. The problem has two sorts of constraints. The first one is a coverage constraint (distance between two satellites @) and the three remaining constraints are operational IAdl must not exceed a threshold constraints.

Description of maneuvers As it has been mentioned, we treat circular Wglker constellations. Orbital parame-

ters taken into account are [a, i, a, a] respectively semi-major axis, inclination, right

ascension of the ascending node (RAAN) and mean anomaly.

The correction of the semi major a & and the mean anomaly can be done through in-plane maneuvers 6Vt, whereas the correction of the inclination and the M A N have to be done through a combination of in-plane and out of plane maneuvers SVt and 6Vw (Ref. 1 and 2).

Criterion In all station keeping models proposed, the objective is to minimize the total con- sumption of satellite maneuvers.

Possible objective functions are: K - 1 N

minp, GVJl(P, s u J,(P, SV> = I : PfJzp7Gp

k = O i = 1 The first criterion minimizes the s u m of complete maneuvers.

The two others criteria minimize the sum of each component of each maneuvers.

In practice the choice of the criterion will depend on the physical possibility of doing maneuvers in both directions simultaneously or not.

Resolution The global optimization problem Eq. (1) can generally be written as follows: In this paper, this problem Eq. (3) will be called the mixed-variables problem. It belongs to the class of P @&xed Integer Non Linear Programming). This kind of problem is well studied in the literature (Ref. 3,4,5 and 6). All methods proposed, need successive resolutions of the real variables problem Eq. ( 4 ) , that will be called the sub- problem (with fixed boolean variables).

p i n 6 V W ) g ( 6 V ) IO (4) h(6V) = 0

i 6 V € 3IKN

The next section is devoted to the description of an exact method solving the mixed- variables problem Eq. (3). The resolution of the sub-problem Eq. (4) will depend on its characteristics. These methods will be described in the two following sections.

THE RESOLUTION OF TNE MULED-VARIABLES PROBLEM The resolution of the mixed-variables problem Eq. (3) consists in finding values of boolean variables and real one’s. A natural method to solve this problem is to enumerate all boolean variables combinations. For each acceptable combination, the sub-problem is solved. Solutions of sub-problems, for each acceptable boolean combinations, are com- pared each others in order to determine the global optimum of the mixed-variables prob- lem.

Such a method is too long to compute, that’s why, we propose a more efficient method.

Description T h i s method uses the generalized Bender’s decomposition that Geoffrion extended to the non-linear case (Ref. 4 ) . The mixed-variables problem is projected on the boolean vari- ables space. It can be rewritten: Eq. (5) is called the master problem.

To have more details on this method, refers to (Ref. 4,5 and 6).

The objective of the algorithm is to make iteratively the lower bound (of the global mini- mum) increase and the up er bound decrease.

1.Select one acceptable combination of boolean variables p ; it=l (it is the current iteration) 2.Solve the sub-problem: m i n p g q g > f ( ~ ~ p f i ) + p j g ( ~ ~ j , B ) + A ~ ~ ( G V ~ ? V j = I...it pmin

B E vn{o, I}KN -i qmin

11 v = {B/(g@V,B)50)1

LINEAR AND DIFFERENTIAL OPTIMIZATION PROBLEM The linear and differential station-keeping problem In this section we consider a differentialand linear model of orbital parameters. The model is called differential since we consider the evolution of distance between a couple of satellites and not the evolution of each satellite (absolute model will be treated in the next section). Moreover the model is linear since constraints on distances between a cou- ple of satellites are linear. In this case, the model Eq. (1) can be rewritten as follows: The criterion can be one of the ones presented in &. (2).

Coverage constraints are the following: k k 'IAaf +- '1 = I A a ~ + Ca k + Cg - Ca. [(k+ I - l ) A V i ] + Ch C i - [(k+ 1 - l ) A V t i l Vi, k eoa 1 1 k k lAsLf+ '1 = /nnp +- 4 + Cfz,. Cu. [(k -I- 1 - I ) A V i ] + C k . C,. . [(k+ 1 - l ) A V t i 1 e+sL K-1 N lBVf]<Consi Vi and B f l M Vk

[ k = O i = 1

(7) This mixed-variables problem can be solved with the previous method, but the resolution of the sub-problem Eq. (4) is required. The next sub-section is devoted to the resolution of the sub-problem Eq. (4) that contains linear constraints, like for example Eqs. (6) and ( 7 ) .

Resolution The sub-problem Eq. (4) with linear constraints can be solved by the linear simplex algorithm (Ref. 8). However such a method implies the use of a software like Xpress and doesn't solve nonlinear problem with, for example, the criterion Jlor J4. So we developed a specific approach based on analytical calculations. T h i s method is also suitable for non- linear criteria.

An analytical approach. According to the theory of duality, Eq. (4) is similar to: T max > 0, Amin~vL(Gv, p, A) = f ( B V ) + p g ( 6 V ) + A T . h ( 6 V ) (8) P - The resolution of Eq. (8) provides the global optimum of the sub-problem Eq. (4).

The disadvantage of the equivalent problem Eq. (8) is that it contains two optimization problems and more variables than the sub-problem Eq. (4). Indeed, the problem Eq. (8) contains a maximization problem in 6V variables (2KN variables) and a minimization one in p and h (Lagrange multipliers) variables (X variables, if X is the number of con- straints). However, we found solutions to reduce the complexity of the equivalent problem Eq. (8).

1. How to suppress the minimization problem?

In fact, the minimization problem can be solved using analytical expressions. Indeed, if we write stationnarity conditions Eq. (9) we can deduce relations Eq. (10) between 6V optimal variables and other variables (Kuhn Tucker and Lagrange multipliers) of the prob- lem.

6V = sv(p,h) (10) The calculation of the Hessian matrix shows that the expression of 6V found with previ- ous calculations corresponds to the global minimum of the problem.

So the solution of the minimization problem is analytical.

Note that:

- Case of criterion J1 or J2, the relation Eq. (10) is not found directly. The way to

obtain this kind of relation is to use the criterion J1+J3 or J2+J3 instead of J1 or J2.

This is not a problem since values of lSVl are about ( M s ) , so values of (6V)' are about ( W s ) and we can conclude that (6V)2 < < ISVl. That's why criteria J1 or 52 are similar to J1+J3 or J*+J3.

After having substituted 6V by the expression found with Eq. (10) in the Lagrangian func- tion, the new expression of the problem Eq. (8) is the maximization problem with Kuhn- Tucker and Lagrange variables:

Es. (8) =ap, A) (11)

2. How to reduce the number of variables of the problem Eq. (1 l)?

It is stated that a Kuhn Tucker multiplier is positive if the constraint is saturated and null otherwise. So there is no need to solve the problem with all Kuhn Tucker variables but just the ones that correspond to a constraint that will certainly be saturated. Others Kuhn Tucker parameters will be fixed to zero.

The way to find the variables of the problem is to plan what are the studied couples of sat- ellites such that their relatives positions will have to be equal to the threshold.

When variables of the problem are determined, we can solve the problem EQ. (1 1).

If the criterion is J1 or J3 we can write stationnarity conditions: = 0 if p+O =L(p,A) = 0 'dh else, we must use a direct search algorithm (see next section) to solve the problem Eq.

(1 1).

Algorithm. Case where all the saturated constraints are known before the resolution of the sub-problem,just one resolution of the problem Eq. (1 1) is needed. However, if we do not know which are the saturated constraints, we propose the following algorithm: it=I ( m e variable it denotes the current iteration); h=O and p=O I.

with these values calculate 6V=SV(p,h) what is the most violated constraint?

2.

pit of the problem Eq. ( I I).

The corresponding Lagrange multiplier becomes a variable The problem Eq. ( I 1 ) contains now, i t Kuhn-Tuckervariables + all Lagrange variables.

esolution of the equivalent problem Eq. (IZ) that provides optimal values of 1 . 1 , (i=Z..it) andh with these values calculate ~ ~ = ~ V ( 1 . 1 , A ) . qthe solution is feasible then it=it+I and retuna to 2,else stop.

ear s Example. In this paper we just develop calculations, case when the sub-problem is the fol- lowing: K - l N

c B;(sV;P

6 V ; k = O i = l The equivalent problem to solve is: k

-A$ - c", - C$ . CQ 2 [ ( k + 1 - Z)AV$] -$a + 31; { ( 1 - $) - *V;}

l = O The gradient and the hessian matrix of the lagrmgian function are: K - 1

-h( 6 V , p , h)= 2SVf + B c [( j + 1 - k)( pli- - pl{ - p2i- + p2i)l + A ; . ( 1 - $) = 0

a s v: j = k a 2 L(6V, p, h) = 2 > 0 B = B(k, C ; , C , ) as$

B K - 1 k ! . ( 1 -f?)

The new expression of 6V is Svf = --i [ ( j + 1 - k ) ( p i { - -pi{ - p2{- + p2$] + j = k The new problem to solve is: max > 0, A w l y P2, A) P 1 Y 1 . 1 2 - The gradient of this function is: K - 1 k

a

- B . ( f + 1 - k ) . [-A;. (1 -$) + A : + .( 1 ‘f$+ J] - L ( p l , p 2 , h ) = f A ( l , f + 1)-++ aplf k = 0 ; k l t

a a

By the same way, - L ( ~ I , p2, A) and - L ( ~ I , 1.12, A) are calculated and the initial optimiza- aP2; ah; tion problem has the same solution than the following linear system of equations:

a

-L(pl,p2,A) = 0 vpl;#o (apl;

a

v p 2 p o -L(pl, p2, h) = 0 Conclwion As a conclusion, the method presented in this section is very efficient since it can provide the global minimum (optimal maneuvers needed to ensure a good coverage) just resolving a linear equations system. This method can be applied to every station keeping problems such that the criterion is one of the ones quoted in Eq. (2), and with linear con- straints in 6V.

NONLINEAR AND NON DIFFERENTIAL OPTIMIZATION PROBLEM It can happen that perturbations on orbital parameters, makes the linear model not precise enough. That’s why we propose a nonlinear model. This model is also absolute: all orbital parameters of each satellites are independently considered. This will allow not only to replace evenly satellites around the earth in station keeping, but also to correct some satellites too far from the nominal constellation.

In this section we present this new model and methods that can solve this nonlinear station keeping problem.

Nonlinear and non differential station keeping problem Criterion can be one of the ones of Eq. (2). The following criterion that minimizes the mass of consumption of ergo1 of all the N satellites of the constellation, can also be used: *ma is the mass of ergol left of the satellite i at the beginning of the station keeping cycle considered, 0 g is the gravitational constant, 0 Isp is the specific impulsion.

Coverage constraints are the following: p = 3.9860064d4m3/s2 3/2

( a i ) qi 2

with 1 / 2 ae= 6378.140krn (a;, = i f + c o s a . - - G V k wi & J2= 1.08266268 with a = 0 or n; to control i assuming that a = n;/2 or 3n/2 to controlP Operational constraints are the same as in the previous model Eq. (7).

Constraints on absolute position of all satellites: satellites must not be too high or too low and must be close enough to the desired orbit inclination.

aminla. k Samax imin I zi k I imax To balance the consumption of satellites, we can use either a new criterion (Js) or new constraints that will penalize the use of satellites for which the consumption of ergol is more important than for the others.

K - 1 N K-1

J 5 ( & 6 V ) = 2 ( P i . p f - / m Pi = Consprevi.i + 2 ( ! 3 f . / r n )

k = O i = 1 k = O Consp,,i.iis the previous consumption of the satellite i his sub-problem can be solved with a direct search algorithm.

oke and Jeeves) does not need the calculation of the gradient, but only a direct evaluation of the criterion on different points. To take into account con- straints, the use of exact penalty functions is required:

(SP) = m i n 6 V f ( 6 V ) + c1 * max(0, g ( 6 V ) ) + c2 - max(O,lh(6V)I)

(16) Such a method provides the global optimum of the sub-problem Eq. (4) if the optimal val- ues of penalty coefficients C1 and C2 are well evaluated. It is difficult to get analytically exact value of penalty coefficients. However, there exists a way to get acceptable values for coefficients C 1 and q, by solving in a first step the problem in the linear case.

Assuming f(x) is the objective function to minimize, the Hooke and Jeeves algorithm is the following (Ref. 7): * x : is the initial point k-1 k

I

I I

YES NO

I

k

pf f(xE)>f(xo) Then Ai = yAi i = l,n -h

Stop criteria

-

Phase For i = l , n k calculate x = xi - + diei k k then xi = x If f(x) <f(xi- Else k calculate x = xi- - Aiei k k k If f ( x ) < f ( x i - l ) then x: = x else xi = x i - l Figure 4 Hooke and Jeeves algorithm There exist t’wo kinds offailure for the satellite.

1. There is no ergo1 left, to make maneuver, but the satellite is able to work.

hile its trajectory is not too far from the nominal trajectory, the satellite can be consid- ered for the coverage study, but not for station keeping maneuvers.

2. The failure is a mechanic or an electronic’s one: the satellite is thrusted to a higher orbit, and cannot be considered any more for the study of coverage of the constellation.

This kind of failure is seldom predictable. When it happens, there are 3 solutions: 1.The redundancy makes the coverage of the constellation good enough, not to use another satellite, but maneuvers can be done to replace satellites evenly around the earth.

2.The satellite can be replaced by a stand-by satellite located on orbit. In this case, the spare satellite can be in the same orbital plane but at a lower altitude than the failed satel- lite (the altitude, and the mean anomaly have to be corrected), or in another orbital plane (the altitude, the mean anomaly, the RAAN and perhaps the inclination have to be cor- rected).

3.The satellite can be replaced by a satellite stored on the ground.

Case the satellite has to be replaced, if the failure has not been predicted, there’s a constraint on the time length of the replacement, in order to minimize the time of damaged coverage (with gap in coverage).

Optimization problem to replace a satellite.

The solution consists in using the drift of the spare satellite in order to minimize the total maneuvers. Case when the spare satellite is located in the same orbital plane but a t a lower altitude than the failed satellite, just the altitude, and the mean anomaly have to be cor- rected. That’s why only, in-plane maneuvers and drifts are necessary to be controlled.

The optimization problem that consists in minimizing the total maneuvers has two sorts of variables: in-plane maneuvers (either just maneuvers of the spare satellite, or maneuvers of this satellite and all others satellites of the constellation) and the drifts time length.

Constraints are the objective location of the spare satellite, and the maximum time length replacement of the failed satellite. Equations of evolution are non linear, since the spare satellite is first too far from its objective location in the constellation.

This model can be extended to the case when the spare satellite is not in the same orbit plane than the failed satellite.

Conclusion This model can be used to determine optimal maneuvers needed to replace a failed satel- lite. However the way to choose the spare satellite to replace the failed one, has to be mod- eled. This will certainly be useful to the study of the design of a constellation, €or choosing the location of spare satellites.

Conclusion As a conclusion, we can say that this new model is very interesting since it can solve the most complicated casp: nonlinear mixed-variables problem. It’s more precise than the linear one. It allows to take into account various constraints: on the position (relative or absolute) of satellites, OR the consumption that can be limited and OR the balanced COR- his model can also be used to calculate optimal maneuvers needed to llite. A first approach to find optimal maneuvers has been presented.

the global problem, including the choice of the spare satellite, has to be stated.

These numerical results concern constellations such that random perturbations have of orbital parameters of all the N satellites.

been added to initial values Comparison of both methods to solve the mixed-variables problem Methods described (enumeration and exact method splitted on the basis of the gen- eralized Bender’s decomposition) have been used to solve the mixed-variables station keeping problem.

Parameters of the example are the following: N=6 satellites, K=2 steps, M=3. In this example just the mean anomaly is corrected (SV,=O), the criterion is J3, and constraints are linear. The sub-problem is solved using the analytical method Eq. (12).

The value of the global optimum, and the computing time of each method are pre- sented in next table. The boolean combination found is obviously the same with both methods.

Table 1 COMPARISON BETWEEN ENUMERATION AND EXACT METHOD.

Type of result Exact method Enumeration Cost (meter/secl2 2.66 2.66 10-~ Computing time (Sparc 5 ) 6 min 26 12 min 49 This example and many others confirm that the exact method provides the global optimum in a shorter computing time than the enumeration does.

Comparison of analytical method and direct search algorithm In this example, the mean anomaly of a constellation (such that N=16 satellites, K=2 steps), is corrected (SV,=O). The criterion is J3, and constraints are linear.

We solved this station keeping problem using respectively these three methods: Algorithm 1 finds which constraints are saturated, Figure 3. The problem Eq. (11) is solved by the analytical method, i.e. the resolution of the system of equations: Eq. (12).

Algorithm 2 also finds which constraints are saturated, Figure 3. The problem Eq. (11) is solved by the direct search approach, i.e. the Hooke and Jeeves algorithm.

Algorithm 3 solves the problem Eq. (11) with all p and h variables, using the Hooke and Jeeves algorithm.

EA S Type of result AIEorithm 1 Algorithm 2 Algorithm 3

Cost meter/sec - meter2/sec2 2,214 - 4.082 2,214 - 4.082 2,214 - 4,082

Computing time (UltraSparc) 6 sec 1 h05' 1 h40' Number of optimization variables 11 11 96 Values of cost presented in table 2, proves that the direct search (Hooke & Jeeves) algorithm can provide the global optimal solution of a station keeping problem, even for an important number of variables. All maneuvers found, with the three methods, have the same value with both methods about d s .

Comparison of linear and nonlinear models

Let us consider two examples with basic parameters N=20 satellites, K=3 steps, alti- tude=103 meters, inclination=53", Number of plane=5. The mean anomaly and the incli- nation are corrected, since we consider J2 effects. So in and out of plane maneuvers have to be calculated.

Example 1: parameters of the constellation are perturbed: random perturbations have been added on orbital parameters of each satellite such that 0,=103 meters, oi=8.104", ~ ~ = 1 0 - ~ " , C F Q = ~ O - ~ ~ .

Example 2: parameters of the constellation are more perturbed: oa=2. io3 meters, oi=1,2. 0,=2. C F ~ = ~ O - ~ ~ .

We solved these two examples with the linear model and using the analytical method. Optimal and feasible maneuvers found for each example have then been intro- duced in the nonlinear model. The first column of the next table presents thresholds that respectively ]Ai{ and 1 ~ ~ x 1 must not exceed. The two remaining columns present the greatest values of [Ail and IAal, provided when optimal and feasible maneuvers found with the lin- ear model have been introduced in the nonlinear model (for initial orbital parameters of examples 1 and 2).

Table 3 COMPARISON BETWEEN LINEAR AND NONLINEAR MODEL, EXAMPLES 1 & 2.

Threshold examde 1 example 2 A i maximum degrees 0.001 0.001 0.001 ha maximum degrees 0.1 0.134 0.168 These results show that the linear model can be used and provides good results.

However, when constellation parameters are far from the nominal positions, the linear model is not precise enough and the optimal linear solution can violate constraints.

n this paper, a new robust and efficient method that can solve the linear station keeping optimization problem has been presented. provides the global opti- mum of the problem in a very short computing ti sented a nonlinear station keeping optimization problem and an algorithm to solve it. These models and methods are able to solve a large range of station keeping optimi problems: linear or nonlinear evolution model of orbital parameters, corrections of e or absolute positions of sat- ellites, operational constraints, constraints on the consumption of satellites, constraints on the consumption balancing of all the satellites, choice of the objective function that can minimize the sum of complete maneuvers or just the sum of components of maneuvers, ...

Several researches will be devoted to the following items: A hybrid method mixing both analytical approach and direct search method can be inves- tigated in the case where the problem has both linear and nonlinear constraints.

of a stand-by satellite to replace a failed satellite has to be stated, Moreover, the choice and an optimization method has to be developed to solve this problem.

Researches on constellation design are also developed and will certainly provide informa- tions on constraints of the station keeping problem.

REFERENCES 1. O.Zarrouati, “Trajectoires spatiales”, Cepadues Editions, 1987.

2. “MCcanique spatiale”, Tome 1 Cepadues Editions, 1995.

3. Michel Minoux, “Programmation mathCmatique”, Tomes 1&2, Dunod, 1983.

4. A. M. Geoffrion, ‘%eneralized Benders decomposition”, Journal of Optimization The- ory and Applications, 10(4):237, 1972.

5. C. A. Floudas and V. Visweswaran, “A primal-relaxed dual global optimization approach”, Journal of Optimization Theory and Applications, 78(2):187,1993.

6. C. A, Floudas, “Nonlinear and mixed-integer Optimization”, Oxford University Press, 1995.

7. R. Hooke and T. A. Jeeves, “Direct Search Solution of Numerical and Statistical Prob- lems”, Journal ACM, Vol. 8, pp. 212-229, 1961.

8. E. Lasserre, F.Dufour, J.L. Calvet, D.Arzelier, J.Foliard and M.Vincent, “Optimal approach to station acquisition and station keeping of satellite constellations”, 2 0 t h International Symposium on Space Technology and Science, Gifu, Japan, May 19-25, 1996.

9. L. Lefebvre, A. Lamy, P. Brousse, M. Vincent, J. Foliard, F. Dufour, E. Lassene and J.

Bernussou, “Relative station keeping optimization for the starsys constellation”, 12th International Symposium on Space Flight Dynamics, ESOC, Darmstadt, Germany, 2-6 June 1997.

Jean-Claude Agnhse", Pascal Browse+ Many conceptionor scheduling proMems of space systems are based on combinatorial optimization techniques. In this paper, we describe the application of these techniques to the resolution of the scheduling problem appearing in the choice of visibilii windows of satellites of a constellation.

The problem we try to solve is described by: - given a set of trading antennas,

- one antenna can only follow one satellite a t a time and needs a certain

delay to allow reconfigurationbefore being able to track another satellite, - satellitesmust all be tracked more than a certain time every day, - as much as possible the load of the antennas must be equal.

Among all the visibility windows, the problem consist in choosing one set that satisfies theses constraints a t best.

We desaibe several methods initially developed in the framework of the scheduling problem of imaging for the future Spot-5 satellite:

- exact methods like the so called ~ R u s s i a n dolisw based on a Depth

First Branch and Bound algorithm to find an optimal solution a t the pnke of a sometimesvery large computation time,

- approximatemethods like "greedy searchw (iterative or random) to find

a good solution with a very short computationtime.

INTRODUCTION When designing a constellation of satellites, one of the many problems to solve consists in minimizing the ground station network taking into account the great number of satellites to track.

The needs can be summarized as follows:

* A l l satellites must be regularly tracked with a minimum duration for telemetry

(for instance 5 minutes every 36 hours).

9 This regular control must be compatible with a more important control on satellites in contingency (for instance a visibility on every orbit).

* Space Mathematics Division. N u w i c a l Analysis and Applied Mathematics Department, Toulouse Space Center, CNES, 18 avenue Edward Belii, F-31401 Toulouse &ex, France. Email: Jean-Claude.Agnese@cnes.fr Space Mathematics Division, Mission Analysis Department, Toulolse Space Center, CNES, 18 avenue Edward Belin. F-31401 Touiouse cedex, France. Email: Pascal.Brousse@cnes.fr T O these constraints induced by exploitation and station-k~ping can be added needs in visibility due to operations during the positioning phase following multiple launches and to de-orbiting operations.

Dimensioning the ground stations network for the control of a constellation requires then the use of efficient scheduling techniques to reach an optimized result.

PROBLEM MODELING The scheduling problem The visibility scheduIing problem can be informally described as follows: 0 Given a set of satellites of the constellation to be t r a c w 0 Given a set of antennas achieving tracking operations on these satellites; 0 Given a reference time interval; Given a set S of visibility windows corresponding to the different ways to track a satellite by a particular antenna on the reference time interva each window is assumed to meet the requirements (RF visibility, minimum duration.. .);

* Given a weight associated to each window which can be the result of an

aggregation of several criteria like the importance of the satellite.. . typically for

the standard problem of f i n d i n g a visibility window for each satellite, the weight wiU be uniformly 1; 0 Given a set of hard constraints which must be satisfied: - Only one v i s i b i l i t y window needed for each satellite; - Non overlapping (one satellite tracked at a t h e ) and respect of a minimal transition time (reconfiguration delay) between two successive tracking on the same antenna; 0 The problem is to find a subset S’ of S which is admissible (hard constraints met) and which maximizes the s u m of the weights of the windows in S’ (Le. the number of satellites tracked). In addition, the best between two solutions, provided the fact they reach the same maximum will be the one which leads to the most equal load of the antennas and the most uniform repattition in time.

This problem belongs to the class of the Discrete Constrained Optimization Problems and more precisely is a Valued Constraint Satisfaction Problem’”.

VCSP is an extension of the CSP framework where each problem can be characterized by: traint links a subset of the variables and defines forbidden Combinations of values for the variables in V’; A valuation set (to valuate constraints and assignments) with a t o t a l order (to compare two valuations), a minimal element I (to represent constraint satisfaction) and a maximal one T (to represent violation of a hard constraint); 0 A valuation function associating to each constraint c in C an element in E which represents the importance of the satisfaction of c; e An aggregation operator 8 (to aggregate constraints valuations) which respects commutativity and associativity, monotonicity relatively to the order, and for which I is the identity element and T the absorbing one.

Given an assignment A of all the problem variables, the valuation of A is the aggregation by the operator 8 of the valuations of all the constraints not satisfied by A.

The standard objective is to produce an assignment with a minimal valuation. It is an NP-hard problem according to the complexity theory and then its worst-case complexity grows at least exponentially with the problem size.

Modeling as a Valued Constraint Satisfaction Problem The modeling of the visibility scheduling problem within t h e VCSP framework consists then in: 0 Associating a variable v to each visibility window w which represents the possibility to track a specific satellite with a specific antenna; this window is defined by a time interval during which the satellite is in visibility of the antenna.

e Associating to v a domain d of values: d={0,1) corresponding to the two possibilities to achieve (1) or not achieve (0) the tracking of the corresponding satellite on the associated antenna during this particular visibility window; the special value 0 corresponds to the possibility of not selecting w in the schedule; 0 Associating to v a unary constraint forbidding the special value 0 with a valuation equal to the weight of w ( t h e penalty for not selecting w); e Translating as n-ary constraints with the maximal valuation T the requirement of tracking each satellite only once; 0 Translating as binary constraints with the maximal valuation T the constraints of non overlapping and respect of the minimal transition time between two successive tracking on the same antenna (recodigwation delay); 0 Using as valuation set the set of integers between 0 (for I) and an integer greater than the number of satellites to track (for T); s or rder o rs as usual c operator.

valuation of an assignment sum of the weights of the rejected windows it is always possible to produce an assignment where a l l the hard constraints are satisfied (for example by rejecting all the visibility windows), finding an assignment of minimal valuation is equivalent to finding an assignment satisfying all the hard constraints and minimizing the sum of the weights of the rejected windows i.e. the number of non tracked satellites in our case.

Except the unary constraints associated to each variable (the only ones which can be violated), all the other constraints are hard (valuation equal to T). The valuation set and the aggregation operator induce an additive VCSP which is among the most difficult ones to solve.

EXACT METHODS E x a c t methods are systematic tree search procedures. The root of the tree, starting point for the search, is the empty assignment. At each node, the set of variables is partitioned into a set of instanciated variables and a set of uninstanciated variables. The children of a no& corresponds to a l l possible extensions of the current assignment by instantiating a new variable. The leaves of the tree correspond to all the possible assignments. Variable instantiation ordering and value ordering can be used to guide the search. These methods are called exact because they are able to find an optimal solution provided that no running time limit is set. To avoid producing and evaluating all the possible assignments, optimistic evaluations of the partial assignments are used.

is to use c0rh.mercia.l software.

0-1 numbers. So the mode On a typical problem with a set of 870 initial windows, it has led to more than 380000 constraints. The only preprocessing took more t h a n 1 hour on a S U N SS30 workstation and was very long to solve.

Thanks to previous studies on the scheduling of an earth observation satellite, specific methods developed in this context have been adapted to the visibility problem.

They are described below.

Depth First Branch and Bound The most fiequently used algorithm is the Depth First Branch and Bound which can be viewed as an extension to the VCSP framework of the backtrack algorithm widely used within the standard CSP fiamework.

Let us assume that the problem is to find an assignment with a minimal valuation

less than ~6 and greater or equal to 8 (we suppose that it is known by other means that no

assignment with valuation less than f3 exists). By default a=T and f3=L The mechanism

consists in performing a depth first search to find a complete assignment with a valuation less than a. This bound i n i t i a l i z e d to Q strictly decreases during search. Each time a than or equal to the current bound is complete assignment with a valuation greater produced, a backtrack occurs. The algorithm stops when a complete assignment of valuation equal to is found or when no complete assignment of valuation less than the current bound can be found.

This algorithm presents the following advantages: 0 It only requires a limited space linear with respect to the number of variables;

* As soon as a first assignment with a valuation less than C G - J is found, the

algorithm behaves like an anytime algorithm: if interrupted, the best solution found can be returned and its quality cannot but improve over time.

The main problem is that a depth first search can easily be stuck into a portion of the search space where no optimal assignment exists because of the first choices made during the search.

Russian Dolls This algorithm can be seen as an hybridization of Dynamic Programming and Branch and Bound. As it sequentially solves nested problems, it has been called Russian Dolls.

Given a problem with n variables, the method, which assumes a static variable ordering, consists in perfor&g n searches, each one solving with the standard Depth First Branch and Bound algorithm a subproblem limited to a subset of the variables. The words, on the sub-tree issued when solving the i+l" problem This method which can be surprising since it multiplies by n the number of searches has proved to be very efficient. The main explanation is the quality of the valuation of the partial assignments provided by previous searches.

APPROXIMATE METHODS This section presents methods which aim at providing good sohtions but cannot prove optjrnality. The counterpart is their efficiency in terms of computation time which becomes polynomial in the problem size.

Greedy search Visibility windows are first heuristically sorted. Then a solution is built by trying to insert each window in the current solution in the order of the sort and rejecting it i f it is impossible. The algorithm is a one pass process and never comes back on i t s choices.

quality of the solution found greatly depends on the s o r t performed at the beginning. In the visibility problem, the best heuristic lies on a chronological order.

Iterative Greedy Search A way to improve the solution provided by t h i s algorithm is to work in two phases: 0 The first phase deals with the computation of a kasible solution using a greedy algorithm, 0 The solution (result of the first phase) is then improved by a perturbation method based on an iterative inhibition of the selected Windows. For each selected window, it consists in rejecting it and computing a new schedule from t h i s point (the portion of schedule fiom the beginning up to that window being unchanged). If a better solution is found, the window is definitively rejected and the current solution updated else it is definitively selected.

This algorithm is a combination of greedy search (first phase) and limited local search (second phase).

Random Greedy Search Another way to more widely explore the search space is implemented in the Random Greedy Search 0 A upper loop modifies the set of windows in input, inhibiting some of them in a random way. "M inhibition consists in randomly suppressing some of the visibility windows associated to a same satellite.

In order to standardize the computation process of the Merent algorithms, a set of basic functions as been implemented as shown on Figure 1. This functions deal with the basic manipulations necessary to build a solution and verify it. So scheduling algorithms can be easily interchanged.

They are based on two main data structutes representing : 0 The current assignment on which algorithms work and try to i n s e r t the different visibility *daws; The best assignment which describes the best solution found.

Vibility Visibility window visibility window Figure 1 : Basic functions ofthe scheduling a I g o r i t h ~ ~ ~ RESULTS On the next figures we give an example of the output of the scheduling process for 64 satellites (2 in contingency) tracked by 8 antennas in 4 stations on a period of 4 hours.

Label C i s for satellites i n contingency which must be tracked as much as possible without holes of visibility longer than 2 hours. Label N is for specific satellites which must be viewed within the 2 first hours.

The algorithm has f a d a visibility for a l l satellites (included the one in contingency). The load of the different s t a t i o n s is almost equal. The entire process did not took more than a few minutes on the SUN SS30 workstation.

Figure 2: Set of possible visibility windows I I 0 1 2 3 -e, Figure 3: Output of the scheduling algorithm etween the different algorithms.

I otes Often non optimal but very low CPU time I

* I **

Iterative Greedy Search

** I **

Random Greedy Search Best tradeoff Linear Programming in Optimal solution, very long CPU

*** I -

large memory requirements Integer Numbers time, Russian dolls Optimal solution but often prohibitive CPU time I CONCLUSION Exact methods like Russian D o l l s or L i n e a r Programming have the advantage to provide 'optimal solutions and to prove this optimality. Nevertheless they ofken fail on large size problems or in presence of high arity constraints in the sense that they cannot reach a solution in a reasonable computation time. When they fad, the systematic order they use to explore the search space prevents them to produce good quality solutions.

Approximate methods, like Random Greedy Search have the advantage to provide, within a limited time, good quality solutions thanks to their opportunistic way to explore the search space. But they have the drawbacks to provide no guarantee about this quality when some satellites remain untracked after a search for it is impossible to say that it is because the problem is unfeasible or because the algorithm has not found the solution.

In practice, these algorithms are intensively used for mission analysis in order to dimension the ground station network. When the constellation is operational, they will be integrated in the ground segment to plan the satellites tracking operations, both in nominal case and to reconfigure the constellation when some of them fall in contingency.

REFERENCES 1. T. Schiex, "Pr&&ences et incertitudes dans les problernes de satisfaction de contraintes", Technical Report 2/7899 DEW, CERT, 1994.

2. T. S c h i e x , H. Fargier & G. Verfaillie, "Valued constraint satisfaction problems : hard and easy problems", in Proc. Of IJCAI-95. Montrhl, Canada, 1995.

3. JC Agn-, yLogiael ORVISK - Ordonnancement des VisibiliteS SkyBridge", CNES Infernal

Technical Report, D G i W f T ' V N I S / M N , 1997.

Navigation, Guidance, and Control Center Goddard Space Flight Center, NASA The direction to develop small low cost spacecraft has led many scientists to recognize the advantage of flying spacecraft in constellations and formations to achieve the correlated instrument measurements formerly possible only by flying many instruments on a single large platform. Yet, constellations and formation flying impose additional complications on orbit selection and orbit maintenance, especially when each spacecra€t has its own orbit or science requirements.

t h i s paper is to develop an operational control method for The purpose of maintenance of these missions. Examples will be taken from the Earth Observing-1 (EO-1) spacecraft that is part of the New Millennium Progam (NMP) and from proposed Earth System Science Program Office (ESSPO) constellations. R e s u l t s can be used to determine the appropriateness of constellationsand formation flying for a particular case as well as the operational impacts. Applications to the ESSPO and NMP are highly considered in analysis and applications.

After constellation and formation analysis is completed, implementation of a maneuver maintenance strategy becomes the driver. Advances in technology and automation by GSFC's Guidance, Navigation, and Control Center allow more of the burden of the orbit selection and maneuver maintenance to be automated and ultimately placed onboard the spacecraft, mitigating most of the associated operational concerns. This paper presents the GSFC closed-loop control method to fly in either constellations or formations through the use of an autonomous closed loop three-axis navigation control and innovative orbit maintenance support. Simulation results using AutoConm and FreeFlyerm with various fidelity levels of modeling and algorithms are presented.

o Aerospace Engineer, Formation Flying Technology Lead, System Engineering Branch, NASA Goddard Space Flight Center, Greenbelt, Mary'land, 20771.

* Aerospace Engineer, EOS AM-1 Flight Dynamics Lead, Flight D y n a m i c s Analysis Branch, NASA Goddard Space Flight Center, Greenbelt, Maryland, 2077 1.

+ Aerospace Engineer, EO-1 GPS Lead, Flight Dynamics Analysis Branch, NASA Goddard Space Flight / Center, Greenbelt, Maryland, 20771.

Missions such as those'of the Earth System Science Program Office (ESSPO) and New emphasize the use of multiple spacecraft to collect Earth- sts of multiple spacecraft in various orbits which include the System's EOS AM-1, EOS PM, EOS CHEM, and the EOS Laser Altimetry (ICESATs) missions. Other related spacecraft such as the next generation of Landsats are also considered part of this initiative. The EO- 1 spacecraft of the NMP also is using the ESSPO requirements to promote technologies and correlated measurements. The orbit characteristics of several of these missions presented in the following table seem very similar in orbital mechanics terms, however the science goals are varied to achieve a wide range of E a r t h observations in the areas of ground imaging, atmospheric research, and ice sciences. These various spacecraft form a constellation of related spacecraft, potentially taking coincident or sequential measurements of the same location on the E a r t h ' s surface, or correlating measurements of related atmospheric phenomena. The reasons for these temporal measurements range fkom cross-calibration of the instruments as follow-on spacecraft are launched into the same orbit to sequential measurements made by instruments on spacecraft in different polar orbits.

Table - 1 Mission Characteristics As these programs mature, the maintenance of a constellation or formations of spacecraft drives the need for further analysis regarding the design of the spacecraft orbits. Analysis regarding the impacts of a design on subsequent missions and their requirements becomes more important and has highlighted challenges in determining the feasibility of proposed solutions to scientific questions, in accounting for monetary constraints, and in accommodating new technologies which have also posed challenges in the areas of orbit control and temporal observations. Extended analysis has also been driven by the imposition of constellation requirements on future low E a r t h orbiting spacecraft.

FORMATION AND CONSTELLATION DESIGN DRIVERS D e s i g n drivers for formations and constellations come from both scientific and technological disciplines, and in~lude:~"' Small Spacecraft flown as virtual platforms or ESSPO mission segments to meet instrument or scientific requirements.

0 Navigation and communications requirements.

e Spacecraft and instrument operational considerations.

Constellations and formations offer the advantages of reduced launch risk per instrument, the separation of ins ment and spacecraft bus schedules, and the i~p~~mentation of new technology.

However, the use of several spacecraft instead of one large spacecraft bus also has some disadvantages when coincident or sequential observations or calibration of instruments are required. The use of one instnunent's imaging data by another for planning, near-real time operations, or ground data processing can become a significant driver.

The proposed use of ground stations instead of the space network for communication support is another consideration in constellation and formation design.' ESSPO spacecraft are considering the use of X-band direct downlink for scientific data return. In order to assure that direct downlinking of data from numerous spacecraft will be possible without overlap in viewing from the ground station, an analysis was performed of the separation in a constellation which would minimize science data collection concerns5.

Therefore, in considering the maintenance of a constellation or formation, fuel budgets must be analyzed.

The goal is to minimize the required fuel for constellation maintenance by combining this maneuver with other maneuvers already planned to meet other mission requirements such as ground track control.

Navigation system selection also will impact the choice and design of constellations and formations not to mention the impact to the available onboard computer hardware and Attitude Control Systems (ACS). Recently, GPS has come to the forefront for real-time onboard navigation, but other technologies exist which may compliment the spacecrafthardware and provide a robust real-time navigation system. The technology of cross-links between spacecraft for both data communication and relative navigation has yet to be fully explored, but for a true closed-loop design, a real-time cross l i n k must be available.

Orbit mechanics and the need to meet all mission orbit requirements place a great burden on the selection of the constellation and its maintenance. For example, most EOS missions have both ground track and mean local time 0 of node crossing control requirements. These orbital requirements must be met in order to successfully collect scientific data. Also, physical impossibilities will inhibit wishful thinking in the selection of some constellations or the achievement of the formations directly from the launch vehicle.

Some constellations may take a long duration to establish and can impose increased constraints on the launch vehicle to meet injection targets. The operations associated with these maneuvers may also become a driver if the instruments are required to physically change their modes, such as covering up optics during maneuvers to protect against contamination or sun impingement.

Formation And Constellation Defiitions While often used together, achieving and maintaining a constellation are independent concepts from that of formation flying?' A constellation is defined as two or more spacecraft in similar orbits that perform separate control of their orbits. They may provide global or localized science data, but mostly i n a post-processing sense. They do not provide real-time communications between spacecraft. In general, a constellation could contain spacecraft t h a t have no hard requirement concerning maintenance of a relative position. For a large difference in orbital anomalistic angles, relative cross track separations vary over the orbit since the spacecraft are really in different orbit planes. This orbit plane difference in nodal crossing is used as an advantage for constellation maintenance to meet sequential observations by accounting for the Earth rotation. The concern is that the result of relative drift in the along-track direction between two spacecraft yields a different sub-satellite point, thereby impeding the coincident observation requirement on every orbit. However, for the NMF' problem, in order to achieve a higher percentage of coincident observations, the spacecraft have the additional requirement to maintain a formation within the constellation.

Formation flying is an orbital operations concept design in which a spacecraft maintains a predetermined trajectory relative to a reference position without making a physical attachment. ' This reference position may be occupied by another spacecraft if desired. Consider two spacecraft placed in the same orbital plane and at the same altitude, with an initial anomaly separation angle small enough that atmospheric density and gravitational perturbations can be considered constant. These spacecraft will be similarly affected by atmospheric drag and by the gravitational potential field of the Earth provided that they have identical ballistic properties. Ballistic properties are defined here as the ratio of mass to the product of frontal area and coefficient of drag. If the spacecraft are separated in the radial direction, and the respective ballistic properties are di n t , their orbit velocities are also different, and one spacecraft (the formation flyer) will appear to drifi relative to the other (the reference flyer). The drift is most apparent in the along-track (orbital velocity) direction. The approach for determining the formation flying maintenance w a s formulated using basic orbital mechanics and formation flying concepts which are derived from H i l l ' s or Clohessy-Wiltshire Equations of motion.

ESIG GY To consider the methodologies of maintaining constellations and formations, an example from each is discussed in detail. The first methodology discussed is the constellation.

Constellation D e s i g n and Maintenance The mean anomaly separation between spacecraftis used as the basis for our analysis. While some separations may seem exceedingly large, it is determined by the science temporal requirement for coincidentkquential observations and by communication requirements. Also, for spacecraft to observe the same location, their orbit planes must be oriented to account for the rotation rate of the Earth during the time lag between one spacecraft seeing the location and the other spacecraft passing over the same location.

To characterize the definition of lcication, it is assumed that the sequential instrument fields of view are large enough to have an imaging expectancy of at least 80%: A first order approximation to analyze the constellation was completed based on orbital mechanics found in any textbook. W e high order Geopotential and third body effects can be ignored in the analytical results, they should be considered when verifying results. These .values were verified in high order simulations using A I Solutions' AutoConm, or FreeFlyHm. lo The analysis of constellations w a s based on information in Table-1 and on the following assumptions and requirements: * The spacecraft must maintain a minimum true anomaly separation.

* All spacecraft must meet their groundtrack requirements, therefore, meuvers must be performed at intervals defined by the atmospheric conditions and not the constellation maintenance.

The range of spacecraft ballistic coefficient differences are no larger than 15% with a baseline of 50 ks/m2.

a Atmospheric conditions are considered to be relatively uniform over the separation in the orbit planes and between spacecraft.

The maximum separation in radial altitude to meet the maximum ground track requirement is 2 km (+/- 1 km about a reference altitude).

Other mission orbit requirements place additional constraints on the constellation maintenance.

These are ground track control, frozen orbit control, inclination control, mean local time control, and repeating orbits. The principal driver of these is ground track maintenance, which has the most stringent orbit requirements. To meet science requirements for Earth observing instruments, the repeating groundtrack of the sub-satellite must be controlled. Ground track maintenance is performed by varying elements of the orbit to ensure that the orbit repeat cycle is met and reference points at the equator are over- flown each orbit. The ground track accuracy is maintained by changing the orbital nodal period with respect to the fixed E a r t h rotation rate. The nodal period is adjusted by changes to the semi-major axis.

The number and times of the maneuvers to accomplish this are determined by atmospheric conditions. For ESSPO spacecraft, this maneuver frequency varies between one month and six months. Frozen orbit control can be accompliihed through strategic placement of the ground maintenance maneuvers a t no additional fuel cost. The other orbit parameters are rarely adjusted and are not considered here.

Consteuation Targets To maintain the constellation, maneuvers must be performed to control the drifting between spacecraft due to the differential decay rates. The targets used for constellation maintenance are dependent upon the individual requirements of the science goals, operations, and constraints. An example of the targets used most o f t e n for polar orbiting ESSPO type missions are semi-major axis (sma) and eccentricity.

One can maintain an ESSPO constellation by adjusting these parameters to control the individual orbit or to maintain the constellation separation. A change to the s m a will adjust the ng in an orbit period between the spacecraft while the ~ e n ~ c i ~ can adjust the orientation of the relative orbit elements such as argument of periapsis. The sma can be targeted to meet the ground track requirements and to maintain the constellation.

ce Using If one follows the ground track control theme then constellation maintenance is reduced to meeting the mission requirements. The ground track control is realized by a change to the sma and the adjustments made to this parameter will result in a differential drift in the relative mean anomaly. There is no control of the magnitude of the d r i f t between the spacecraft as the drift distance is dependent upon when the maneuvers are performed for the ground track control. The targeted s m a is the required sma to maintain the mission ground track which can be computed via differential correction methods in FreeFlyerm.

Maintenance Using Mean AnomaIy Controt I€ one follows the differentid mean anomaly rate theme, one can adjust the time it will take to transverse a delta mean anomaly between the spacecraft. The selection of the sma of the maneuvering spacecraft can be used as a target to bring about a controlled drife over a given delta anomaly in a given as &e. The derivation of this s b target E simply an algebraic expansion of the mean anomaly rates shown below.

The mean anomaly difference over time can be computed as, where with a , = mean sma, Q = initial mean sma, a,,d = s m a decay rate, pgravitational constant, n = mean motion, and t= time.

Using a desired angular difference and time, this can be expanded to, Solving for the target semi-major axis, am, and using an assumption that the decay rates are subject only to the differential ballistic coefficients yields, where, To consider a sample scenario, this analysis assumes that ground track mainte spacecraft half-way through the ground track maintenance cycle to account for the maximum radial separation (and therefore maximum in-track velocity difference) over time. Orbital decay rates were calculated a t the solar flux maximum, based on +2 sigma predictions. If the ballistic coefficient ( B , ) is 50 kg/m2, the decay rate at 705 km at the beginning of the mission (June 1998) is approximately 0.0028 km/day. If the B, equals 40 kg/m2, the decay rate is approximately 0.0034 W d a y . Decay rates for the -2 sigma solar flux values can be orders of magnitude less (e.g. B,p50kg/m2, decay rate - 0.0003 M d a y four years later) and could give significantly different results. The ground t r a c k maneuvers periodically change the relative semi-major axes of the spacecraft which results in a switching of the sign of the delta mean motion.

The maintenance of the ground track results in a repeating and somewhat uniform increasing and decreasing of the mean anomaly (along-track distance) between the spacecraft as maneuvers change the direction of the differential mean motion. The observed difference in the mean anomaly of each spacecraft varied by approximately +/- 15" over a several month. This difference suggest that ESSPO type separation angle requirements of 40° can easily be met. Furthermore, results suggest that multiple spacecraft can be initially 'stationed' at intervals of 60" to allow for drift. These spacecraft do not need to be in co-planar orbits, since the above sequential observations and station coverage must be met. More importantly, the ground track control results of t h i s analysis suggest that no additional propellant is required to maintain a constellation separation if the coincident observations can be reduced to occurring a t smaller time intervals.

In Figure 1, a mean anomaly separation angle is shown for spacecraft with the same B, but w i t h different ground track requirements of +/- 20 km and +/- 5 km. Figure 2 presents the separation angle for spacecraft that have the same ground t r a c k requirements, but the B , of the formation flyer is 15% that of the reference spacecraft (40 kg/m2 vs 50 kg/mz).

Figure 1 - Constellation Drift of S/C with Different

Figure 2 - Constellation Drift of S/C with Same

Goundtracks Groundtracks Mean Anomaly Control Results The results of using the equations derived above for the sma targets are shown in Figure 3 and 4.

Figure 3 presents the required initial s m a to drift a desired distance in a fixed time and the sma to d r i f t a fixed distance in a desired time. Two examples in the figure show the effects of changing the fixed parameter. The results of this spreadsheet were numerically verified using the FreFFlyerm system and the verified points are noted by the circles and squares. The initial reference sma was 7077 km, which represents a typical mean element of the sma of ESSPO orbits.

The mean anomaly control results of this analysis, while similar to the ground track results, suggest that any given constellation separation magnitude can be controlled. The separations and time can also be used as an input into the ground track control to minimize the separation drift distances and thereby increase the number of sequential instrument observations. The results suggest that ESSPO type separation angle requirements of 4 0 ’ can easily be met. Furthermore, results suggest that ‘stationed’ at smaller separation angles. As with the ground track results, these spacecraft do not need to be i n co-planar orbits, since the above sequential observationsand station coverage must be met.

Figure 3 - Constellation Drift Figure 4 - Initial Return Drift Conditions

A n a l y t i c a l Drift Equations Figure 4 presents a general analytical method to compute the initial radial separation for maintaining a constellations given a desire to control the along track separation. The equations for t h i s plot are analytical and only a meant to given a representative case. The point is that a controlled drift in the along track direction, both away f r o m and back toward a reference spacecraft cannot be achieved by using the generic d r i f t equations previously described. The radial separation required for a controlled d r i f t is an order of magnitude smaller than that for general drift over a given time period. Since the orbit decay is inversely proportional to the ballistic coefficient, the chase spacecraft will decay at an average decay rate similar to that of the reference and is given by,

r2 = ii (BC, p c , )

and the differential orbital decay rate will be

Ai = r; - i2

the initial radial separation can then be given by The maximum downrange drift rate can then be given by substitution into the differential angular rates and the maximum drift is then D , , = D , , -0.5-t The equations are presented here as a general guideline and do not hold up under a high fidelity modeling which includes higher order Geopotential terms and differential orbital perturbations due to large angular separations. Figure 4 presents the drift and initial radial separation only for the decay rates used in the / constellation analysis and need to be modified for each individual case.

6 1 In order to meet the coincident observation requirement without a large variation in the anomaly as previously presented, a formation strategy must be developed and followed. Assuming that the fields of view of the instruments are circular (on the order of one kilometer in diameter) and nadir pointed, a control box can be determined to ensure that the FOVs will overlap to a given percentageg. It is assumed here that this control box is 50 kilometers in the along track direction, given the assumption that the ground error is equal to the along-track error for a sma of 7077 km. Therefore, to meet this two kilometer requirement, an initial altitude displacement for the formation flying spacecraft with respect to the reference is required to affect the formation flying theory.

Formation flying involves position maintenance of multiple spacecraft relative to measured separation errors. It involves the use of an active control scheme to maintain the relative positions of the spacecraft. Optimally, this process will be performed autonomously onboard the spacecraft and is called Enhanced Formation Flying, such as that which will be implemented by GSFC for the New Millennium EO- 1 mission. A complete description of the fundamental of formation flying was previously p~blished'~'~. An example of the orbit dynamics of formation flying is shown in Figure 5.

FIGURE 5. Formation Flying Example Formation flying techniques can be used to meet a variety of mission separation requirements.

When the mission requirements call for a tightly controlled separation (kilometer range), whether the overall separation is small or large, frequent control becomes necessary. Formations of spacecraft are identified using tight or loose control methods. While some separations may seem exceedingly large, they are determined by the science requirement to view coincident sites or a communication requirement of a ground station to view only one spacecraft at a time. For large separations, one must consider the rotation of the Earth if the formation is used to meet concurrent or sequential imaging of the same locations on the ground. Therefore, relative crosstrack separations are used to follow the reference ground track for any temporal requirement. A patent rights application w a s submitted to the GSFC patent counsel by two of the authors for the application of Autonomous Closed Loop 3-Axis Navigation Control Of Spacecraft.12 Formation Flying and Targeting Algorithm Description The algorithm enables the spacecraft to execute complex 3-axis orbital maneuvers autonomously.

Figure 6 illustrates the basic sets of information required for formation targeting as it is incorporated into AutoConRn. The algorithm is suited for multiple burn scenarios but is explained here in a two-bum approach for clarity. The simplest formation flying problem involves two spacecraft orbiting the Earth. One spacecraft, referred to as the control spacecraft, orbits without performing any formation flying maneuvers.

The second spacecraft is the chase spacecraft. It monitors the control spacecraft, and performs maneuvers to maintain the desired formation phasing. The goal of the formation flying algorithm is to perform / e Project (R,,V,) through -At to determine ~ r w v ~ ) (where you should be ut timev.

e Compute(6rw6vJ (dffwence between where YOU are und where you wunt be atto).

FIGURE 6. Formation Orbital Parameters This god is accomplished by finding the state the spacecraft would have at the current time in order to achieve the target state a t the target epoch without maneuvering. This new state is d e d the desired state Sd = (rd, vd); it is the target state propagated backwards in time from the target epoch to the epoch of the initial state. The difference between the initial state and the desired state is:

6 s -(6":)=( - l p 0 - r d )

vo - V d

Then, following the derivation of the state transition matrix given i n Battin 13, the relevant state transition matrix submatrices are: The expressions for F, G, and C are derived from the universal variable. From these submatrices, the C* matrix is computed as follows: The expression for the impulsive maneuver follows immediately:

Av = C*(to)6r - 6v

Keplerian and Non-Keplerian Transfer Orbits The transfer trajectory for constellations and formations does not need to be of a Hohmann type.

Having established both actual and desired states of a spacecraft's location using any navigation filter, all that is needed is a means of autonomously zeroing the difference between the two states. Given two Keplerian trajectories and a chronologically defined maneuver window, a reference non-Keplerian trajectory may be determined which will smoothly transport the spacecraft from its position on the first Keplerian path at the beginning of the maneuver window to a desired position on th&second Keplerian path at the conclusion maneuver window. Control points on the reference trajectory in Figure 7 are calculated at regu ability of the spacecraft to receive and process intervals consistent with position data, fue its thrusters, and account for the effects of each firing. At each step next control point on the reference path is examined and back-computed along a determine small differences between spacecraft position and velocity on the mfmence path and determine which Keplerian path would intersect the reference path at the next control point. These differences are then fed into a system of linearized state transition matrices to determine the incremental AV required to get the spacecraft to the next control position on the reference trajectory. At the conclusion of t h e maneuver window, a final bum is required to match the velocity required to maintain the new Keplerian trajectory.

One can use single or multiple maneuvers to achieve the target condition Keplerian State &$)

,*

* Keplerian State Q Initial Keplerian State w t M a n e u v e r 4 Window FIGURE 7. Non-Keplerian Reference T r a j e c t o r y During Maneuver Algorithm Targets For the formation, the orbit target is described as a location relative to the reference so that the drifting due to ballistic coefficient difference can be utilized. For example, the EO-1 relative position has a three dimensional target that is 450 km behind the reference spacecraft in the along-track direction, a sub- kilometer altitude above the reference, and also a cross track differential to account for the rotation of the E a r t h to meet the observation requirements. This target can easily be misinterpreted as a simple rotation in true anomaly and altitude for the along-track direction and altitude and a node displacement for the cross track requirement. If a true anomaly is used to compute the along-track difference, a completely different orbit will be designed as the change on keplerian elements doesn't take into consideration the true orbit with the perturbations included. It can be shown that propagating an orbital element set with a delta true anomaly either before or after the change in altitude will not give the desired results. Therefore, if one wants the orbit or the formation flying spacecraft to 'fly' a predetermined trajectory, the following method can be used. This method will account for the cross track component as well.

Initially use the reference orbit Cartesian state Offset the altitude of the formation flyer by the desired amount Propagate (numerical methods suggested) the initial state backwardforward by the required time delta, e.g. plus or minus one minute.

Change the Epoch of the f i n a l propagated state of the formation flyer to the original time to effect a change in the cross track to meet coincident observations Change the coordinat&system into ECI from ECEF Fornation Flying Results The following results are taken directly from the AutoConnY' ground system which utilizes the GSFC algorithm. The results are divided into two formation flying scenarios of two spacecraft which maintain either a close or a dynamic formation? The initial conditions were derived from the orbit elements for the Landsat-7 mission which has a s u n - s ~ c ~ n o u s orbit with a descending node and a ground track repeat of 233 orbits in 16 days. The results show formation flying evolution and the effect on the mission groundtrack requirements. Evolution Figures are presented in a control spacecraft rotating coordinate system with the radial direction being the difference in radius magnitude and the alongtrack direction being the arc between the position vectors.

C The first two figures present the maintenance of a formation that has a 10 meters radial separation only. Figure 8 presents the formation evolution in radial and separation distances for a period of 90 days.

To re-initialize this orbit, two maneuvers are used in a Hohmann-like transfer. The first DV to re-establish the 10 m radial position separation by using the algorithm targeting method with a H orbit period and the second DV by using the same method with a . 0 1 orbit period to adjust the velocity components. Figure 9 presents the ground track of these orbits. The initial orbital condition placed the ground track at the "0" error location for convenience.

0.010 0.w5 , 0 . 0 4 0 E .z = 3 . W 5 4 . 1 1 0 r - 1 1 I I I I I I I 1 1 0 . 0 0.5 1.0 1.5 2.0 2 . 5 3 . 0 3 5 1 . 0 4.5 5.1 55 LrgTnctStpla (KaJ

Figure 8 - Close Formation Radial and

Figure 9 - Close Formation Relative Groundtrack

Alongtrack Senaration Figures 10 and 11 present results of starting w i t h an initial along track sepiation of 0 m and an initial radial separation of 20 m and then targeting to a 10 m radial and 0 along track separation whenever either spacecraft performs a maneuver. Therefore, the first maneuver of the formation flyer is to adjust to both the groundtrack of the control spacecraft after its maneuver and to re-establish the initial formation parameters. Figure 10 presents the results when ground track maneuvers have occurred for the control spacecraft. As seen in Figure 11, a ground track maneuver takes place slightly before the time when the along track separation is near zero. The smaller parabola represent the maintenance of the formation to the 10 m radial separation. The formation evolution in radial and separation distances is presented for a period of 90 days.

The next simulation consists of maintaining a dynamic formation where the formation flying spacecraft w a s in a different orbit plane with an along track separation on the order of 450 km. To simulate this, the initial state of the control spacecraft was propagated backward for 1 minute ( 450 km at 7.5 lads) and to maintain the ground track requirement the right ascension of ascending no& was adjusted to account for a one minute Earth rotation. Figures 12 and 13 present the formation evolution in the radial versus along track and cross track versus along track separation for several days. The effect of the pernubations on the orbit elements has an immediate effect in the osculating orbital elements. This results in a very large radial separation approaching +/- lkm. A cross track of +/- 3 Okm was anticipated since that is the effect of the node difference. As the formation evolved, a maneuver was required to re-established the formation at the initial separation of 0 m alongtrack and 30 km cross t r a c k at a radial separation of 10 m. Figure 12 presents the trajectory of the formation flyer. The figure shows the radial separation change fiom approximately 500 m to +10 m and an along track separation fiom 450 km to 0 km. After this state was targeted, a &euver was performed to maintain a fo-rmation similar to the close formation. Figure 13 presents formation evolution after the maneuver.- I I I I I I C t l I I I I I I 0.015 0.010 0.695 - g 0 . 0 0 0 E 0.0 c gj .P E 3d.m t B 0 i P =.a010 QS 4.015

Figure 13 - Post Maneuver Trajectory and

Figure 12 - Dynamic Formation Evolution

Evolution EFT TECHNOLOGY DESCRIPTION The control of the constellations and formations mentioned above use an algorithm that is part of a new technology called AutoConm, which features flight software that is capable of autonomously planning, executing, and calibrating routine spacecraft orbital maneuver^'"'^. The autonomous EO-1 formation flying control software AutoConm builds on this existing capability for the maneuver planning, calibration, and evaluation t a s k s . A fuzzy control engine i s ideal for this application because it can easily handle conflicting constraints between spacecraft subsystems.

The AutoConm flight control system will need data fiom additional sensors and spacecraft subsystems such as propulsion, groundtrack, navigation, and attitude data. It will then be possible to autonomously generate, analyze, and execute the maneuvers required to initialize and maintain the formation between Landsat-7 and EO-1. Figure 14 shows a functional diagram of the AutoConm system.

Because these calculations and decisions are performed onboard the spacecraft) the lengthy period of ground-based planning currently required prior to maneuver execution will be eliminated. The system is general and modular so that it can be easily extended to future missions. Furthermore, the AutoConm flight control system is designed to be compatible with various onboard navigation systems (Le. GPS, or an uploaded ground-based ephemeris). This formation flying technology will demonstrate the capability of EO-1 to fly over the same groundtrack as Landsat-7 within +/-3 kilometers at the equator while autonomously maintaining the formation for extended periods to enable paired scene comparisons between the two satellites.

The Enhanced Formation Flying (EFF) system for the EO-1 application is designed by GSFC, AI Solutions, Inc., and the Hammers Company, who has responsibility for the EO-1 attitude control system (ACS). The flight software, AutoCon-Flight TM will serve as the overall architecture and execute the Goddard developed control algorithm for maneuver decision, design, and execution. This control algorithm will provide a delta-velocity magnitude, burn epoch, and duration to the ACS for execution.

Maneuver implementation is the responsibility of the ACS. Maneuver calibration will be performed autonomously within AutoConm. Integration testing and system verification will be performed with the ACS flight software prior to the mission to demonstrate technology readiness. Ground simulation equipment will be used for system integration, testing, and performance evaluation. Verification of the flight system performance during the operational phase will be conducted according to a validation plan.

This validation will occur in several incremental steps starting with ground verification of the maneuver parameters that have been computed onboard, and will culminate with full onboard autonomous maneuver prediction, planning, and closed-loop onboard maneuver execution. A subsystem interface is shown in Figure 15.

I EFF Subsystem Interfaces

I'

.&caR&- i 1 ...................... J.& I i I I ...................

I

1 i I * ) :...-.......-.I I lr--Ji ................................

Figure 14 - AutoCon Functional Diagram Figure 15 - Autocon Sub-system Diagram

CONCLUSIONS In considering the use of constellations to meet scientific objectives, one must take into account the physical limitations and restrictions imposed. Our results suggest that no additional propellant is required to maintain a large constellation separation if the coincident observations can be reduced to occurring at small time intervals when the orbit mechanics would naturally provide this event. The maintenance of a constellation for constant coincident viewing can be quite complicated but is feasible if one manages and plans for this endeavor. This planning should assess emerging technology and the system engineering aspects of the spacecraft development. It should assess the spacecraft ballistic coefficient in particular as well as the amount of fuel required. The amount of coincident observations that are required to meet mission objectives versus the amount desired should be addressed. The formation can be established to provide coincident observationson a timed schedule, but may miss targets of opportunity for calibration or extra coverage.

This paper shows that the formation flying algorithm presented is a feasible technology that can be used in a closed-loop design to meet science and mission requirements of Low Earth Orbiting missions in the NMP and ESSPO. The algorithm is very robust in that it supports not only benign,ground track control, but demanding 3-D control for inclination and non-Keplerian transfers. To best meet the NMP EO-1 requirements, this innovative technology will be flown onboard the spacecraft which launches The algorithms are being ~n~egrated into AutoConm for both ground support validation onboard autonomy. This system will be implemented as a close-loop flight code onboard Orbiter-1 (EO-I) spacecraft thereby yielding the name of Enhanced Formation Flying system will be implemente the ground for verification. The application of this algorithm and AutoConm system to other or ESSPO programs is unlimited and can be used to fully explore the NASA mandate of faster, better, cheaper spacecraft.

1. G. Asrar, NASA’s Mission To Planet Earth, Earth Observing System, PAM-552, NASA Publication, 2. R. Ticker, “New Millennium P r o g r a m ’ s Earth Orbiter One Flight (NMPEO-1) Opportunity for Industry Partnership”, NASA, GSFC, Greenbelt, MD.

3. R. Carter, Internal memo of EO-1 Requirements, (1997), NASA GSFC Greenbelt, MD.

4. D. Folta, L. Newman, “Establishment And MaintenanceOf A Constellation O f Multiple Spacecraft”, Proceedings of the Flight Mechanics Estimation Theory Symposium,May, 1994 5. L. Newman, ‘%OS-PM Ground Station-to-Multiple S p a c e c r a f t Contact Study”, memorandum to Mr. Wu, EOS-PM Project, May 1994.

6. D. Folta, F. Bordi, C. Scolese, “Considerations on Formation Flying SeparationsFor E a r t h Observing Satellite Missions”, Proceedings of the W A A S SpaceflightMechanics Meeting, February 1992, Colorado Springs, Co.

7. Folta, D.C. and L. Newman (1996) “Foundations of Formation Flying For Mission To Planet Earth and New Millennium, AIAA 96-3645, AIAA Guidance, Navigation, and Control Conference and Exhibit, San Diego, CA 8. Clohessy and Wiltshire, Terminal Guidance System for Satellite Rendezvous, Journal of the Aerospace Sciences, Sept. 1960.

C. Scolese, D. Folta, and F. Bordi, ‘‘Field of View Location and Formation Flying For Polar Orbiting 9.

Missions”, AIAA/AAS Spaceflight Mechanics Meeting, February, 1991, Houston, T x .

10. R. Sperling, AutoCun / FreeFlyer System Description and User’s Guide, (1997) AI Solutions, Inc., Greenbelt, MD. 20770 11. D. Folta and D. Quim, “A 3-D Method for Autonomously Controlling Multiple Spacecraft Orbits”, IEEE Aerospace Conference, Aspen CO., March 21,1998 12. D. Quinn,. and D. Folta (1996) Patent Rights Application and Derivations of Autonomous Closed Loop 3-Axis Navigation Control Of EO-I.

13. Battin, R., An Introduction to the Mathemtics and Methods of Astrodynamics, ALkA Education Series, (1987) Chapters 9 and 11.

14. F. Bauer, D. Quinn*, K. Haamah’, D. Folta’, and John Bristow, “Enhanced Formation Flying Experiments For The New Millennium Program Earth Orbiter (Eo)- 1 Mission - Challenging Technology P r o g r a m Management”, AIAA, 5/97

SAT CONSTEL o[IssIor DESIGN

Marco Concha and Robert DeFazio NASA Goddard Space Flight Center ABSTRAGT The NanoSat constellation concept mission proposes simultaneous operation of multiple swarms of as many as 22 identical 10 kg spacecraft per swarm. The various orbits in a NanoSat swarm vary from 3x5 t o 3x42 R, i n geometry. In this report the unique flight dynamics issues of this constellation satellite mission design are addressed. Studies include orbit design, orbit determination, and error analysis. A preliminary survey determined the orbital parameters that would yield a 100 minute shadow condition maximum while providing adequate ground station access for three ground stations.

Lauri Kraft Newman NASA GSFC, Code 572 Mark E. Hametz, Darrel J. Conway AI Solutions, Inc.

From a flight dynamics perspective, the EOS AM-1 mission design and maneuver operations present a number of interesting challenges. The mission design itself is relatively complex for a low Earth mission, requiring a frozen, Sun-synchronous, polar orbit with a repeating ground track. Beyond the need to design an orbit that meets these requirements, the recent focus on low-cost, “lights out” operations has encouraged a shift to more automated ground support. Flight dynamics activities previously performed in special facilities created solely for that purpose and staffed by personnel with years of design experience are now being shifted to the mission operations centers (MOCs) staffed by flight operations team POT) operators. These operators’ responsibilities include flight dynamics as a small subset of their work; therefore, FOT personnel often do not have the experience to make critical maneuver design decisions. Thus, streamlining the analysis and planning work required for such a complicated orbit design and preparing FOT personnel to take on the routine operation of such a spacecraft both necessitated increasing the automation level of the flight dynamics functionality.

The FreeFZyem software developed by AI Solutions provides a means to achieve both of these goals. The graphic interface enables users to interactively perform analyses that previously required many parametric studies and much data reduction to achieve the same result. In addition, the fuzzy logic engine enables the simultaneous evaluation of multiple conflicting constraints, removing the analyst from the loop and allowing the FOT to perform more of the operations without much background in orbit design.

Modernized techniques were implemented for EOS AM-1 flight dynamics support in several areas, including launch window determination, orbit maintenance maneuver control strategies, and maneuver design and calibration automation. The benefits of implementing these techniques include increased fuel available for on-orbit maneuvering, a simplified orbit maintenance process to minimize science data downtime, and an automated routine maneuver planning process. This paper provides an examination of the modernized techniques implemented for EOS AM- 1 to achieve these benefits.

The challenge in dete ing how best to support each mission is to not only look to the past to learn from suc s and failures of previous missions, but to also look to the future to take advantage of new technologies. S AM-1 is no exc similar fashion to the Landsat mission series, EOS will fly in a Sun- frozen orbit with a 16-day repeat cycle. This orbit necessitates frequent orbit maintenance maneuvers over the life of the mission. Modernized techniques were implemented for EOS AM-1 flight dynamics support in several areas, including launch window determination, orbit inaintenance maneuver control strategies, and maneuver design and calibration automation. The benefits of implementing these techniques include increased fuel available for on-orbit maneuvering, a simplified orbit maintenance process to minimize science data downtime, and an automated routine maneuver planning process.

A cooperative effort with Lockheed Martin (the launch vehicle manufacturer) has resulted in an optimal use of the launch vehicle’s capabilities that has enabled a greater than instantaneous launch window and has minimized the amount of corrective maneuvering required by the spacecraft. Once on orbit, analysis has shown that routine stationkeeping maneuvers executed as single burn maneuvers do not compromise orbital constraints. Additionally, in keeping with NASA’s direction to reduce operations costs, the maneuver design process has been automated through the use of FreePZyerW. This paper details these modemized approaches to meeting the AM-1 requirements described above, including updated analysis methods, simplified ‘maneuver alternatives, and automated operations.

MISSION OVERVIEVV The Earth Observing System AM-1 (EOS AM-1) spacecraft is an E a r t h Systems Science Program Office (ESSPO) initiative to explore global change and the Earth’s environment. EOS AM-1 will be launched no earlier than October 6, 1998 aboard an Atlas IIAS expendable launch vehicle (ELV) from the Western Range of Vandenberg A i r Force Base. After the ascent maneuvers are executed to place EOS AM-1 in its mission orbit at 705 km mean equatorial altitude, the five science instruments aboard the spacecraft will begin taking measurements of the Earth’s environment. These data will later be correlated with data from related instruments on other spacecraft to provide scientists with a more in-depth view of the phenomena under study.

The EOS AM-1 mission orbit is both frozen and Sun-synchronous yith a 16-day repeating ground track. The ground track must be maintained to k20 km of the World Reference System ( W R S ) . The frozen orbit condition must be maintained such that the altitude over a given latitude is within +lo/-5 km of the nominal value at all times. In addition, the Mean Local Time (MLT) of descending node must remain between 10:15 am and 10:45 am throughout the duration of the mission to maintain constant lighting over the E a r t h ’ s surface. Passive control of the 98.2 degree nominal Sun-synchronous inclination prevents the need for out-of-plane maneuvers while maintaining the constant lighting within these mission tolerances. The inclination will be biased above the nominal value t at beginning of li clination drift with time will cause over the course o towards 1 0 : 4 5 and back to 10: 15, requiring no maneuver to adjust the inclination actively. For more details on this technique, see Ref. 1.

TS In preparing to support the launch and operation of EOS AM-1, flight dynamics analysts have incorporated several techniques into the mission plan that, while not new, have not previously been used in an optimized manner. The first of these techniques is the use of guided targeting to achieve optimal inclination targets determined on board using a polynomial to widen the launch window. The second technique involves using one burn instead of the traditional Hohmann transfer to accomplish the combined ground track and frozen orbit maintenance. Performing one burn instead of two minimizes instrument down times and periods of less-accurate data while simplifying operations.

The paragraphs below describe the benefits and concern associated with the use of these techniques and provides analysis verifying the accuracy and reliability of these methods.

Launch Window Widening To achieve a Sun-synchronous orbit, the spacecraft must be launched at the time that the desired orbit plane passes through the launch site longitude. This time occurs once per day in the appropriate (ascending or descending) direction. Ideally, this constraint would imply an infinitely small launch window to accurately achieve the desired MLT. The length of the window may be widened around the exact launch time by making use of the permissible error range on the MLT requirement. However, this error box is often better used to eliminate inclination maintenance maneuvers. The MLT drift throughout the mission may be kept to within the MLT limits by choosing the optimum inclination for a given MLT. This strategy eliminates the need for out-of-plane inclination maintenance maneuvers. Figure 1 (Ref. 2) shows the effects on MLT d r i f t of the optimum inclination choices for EOS AM-1 for 10:20 am and 1 0 : 4 0 am beginning of life MLTs. This maintenance method for Sun-synchronous orbits is described more fully in Ref. 1 . When these considerations are incorporated into the analysis, the desired launch target still requires achieving the optimum combination of inclination and MLT.

Therefore, a virtually instantaneouslaunch window is once again required.

A second option is to make use of guided targeting when available from the ELV for widening the launch window by altering the target orbit parameters during powered flight. Guided targeting is a feature often used by a vehicle to accommodate needs of various payloads, such as azimuth targeting for deep space missions and minimum parking orbit inclination targeting for geosynchronous spacecraft. The Atlas IIAS vehicle that will be used to launch EOS AM-1 is capable of guided targeting implemented though the use of a polynomial in the flight code. EQS AM-1 is taking advantage of this capability to change the inclination and MLT targets depending on the actual minute within the launch window that the ELV lifts o f f . Using this method, the EOS AM-1 window was widened from instantaneous to a 20 minute launch opportunity. Although the EOS AM-1 MLT linqits are between 1015 and 1 0 : 4 5 am indicating that a 30 minute its orbits to those will to prevent aPP s due to vehicle dispersions on the inclination, exceeding the scien at beginning of life. This high drift rate may cause an would cause a high immediate violation of the 1 1 I i ( 0 .

t 2 3 4 5 8 m#ty.bnLI.

Figure 1: Optimum MLT and Inclination Targets for EOS AM-1 The A t l a s fight code computes the inclination target in the following manner (Ref. 3). After launch occurs, the actual l i f t o f f time is used to calculate the desired Greenwich M e a n Time (GMT) of the descending node of the injection orbit from the equation:

GMTDN = G M T , + Atl + At2

where: GMTw is the GMT of l i f t o f f in seconds.

At is the nominal time of launch vehicle flight from l i f t o f f to sgcecraft separation (seconds) At2 is the nominal time from spacecraft separation to the descending node (seconds) Because the exact actual powered flight t i m e s are not known before completion of that flight segment, values must be used for AtLand At2 that are determined pre-launch.

In addition, the flight code cannot accept multiple values for these variables based on launch time, so the same constant values of Atl and At2must be hard coded for use at all points in the launch window. Since the launch will most likely occur at the beginning of the launch window, the 1 0 : 2 0 values for these t@es were computed based on simulated powered flight trajectories by Lockheed Martin and set as constants in the flight code.

Then, the GMTDFcomputed in (1) may be used to compute MLTDN, assuming a constant value for the longtude of descending node (LBN):

GMTDN = { MLTDh - [LDN * (86400 sed360 deg)] }modulo 24

(2) where: the time of the descending node crossing.

Finally, the target inclination may be computed from a polynomial of the form: where the h4LTDN is measured in hours computed using (2), INCTARG is the target inclination in degrees, and the coefficients are computed before launch by fitting a curve to the optimal inclination and MLT targets for each minute of the launch window as shown in Figure 2 (Ref. 2). For the data in Figure 2, CQ = 80.987296, C 1 = 3.460392, and C, = -0.172758. These values of inclination and m T D N computed during flight augment the specified altitude target to fully determine the ELV target orbit.

9 0 . 3 .

98.29 ..

-.

1 m27..

i t i BB26..

9025 -.

98.24 .I One issue raised about this technique is thatwhen using a constant value for the At’s and the LDN in the above equations, the error caused by using one value over the whole launch window might outweigh the benefits of using the guided targeting. This concern arose because the inclination targets varied only 0.05O over the 20 minute launch window. It was eventually decided that holding the values constant did not cause a problem based on the analysis described below. Note that although the ELV is only contractually obligated to provide inclination accuracy w i t h i n 0.1”, historical data has shown that Atlas vehicles routinely achieve inclination targets to w i t h i n hundredths of a degree of the targeted value. Therefore, the targeted values are a reasonable goal for the mission. The closer the vehicle can place the spacecraft to its targeted inclination, the less fuel the spacecraft will have to spend on operationally complex out-of-plane inclination maneuvers to return to the optimum initial state.

Lockheed Martin supplied optimized trajectory runs for 10:20, 1030, and 10:40 optimum MLTDN and inclination targets. Flight Dynamics used these data and held the LDN and At’s from the 10:20 nominal trajectory constant over the whole launch window to compute the inclination target, as will be done in the flight code. The target

inclination was then co sing equations (1) - (3)

Values for the beginning, middle, and end of the launch window are shown in Table 1.

Table 1 TAIXGETS USING BE G OF OW LAUNCH CONSTANTS MLT GMTm At, At2 GMTDN LDN LMST INC Targeted (hours) Target 10:20 67065.8 820.3867 126.6371 68012.82 231.5716 10.33333384 98.29827 10:30 67665.8 820.3867 126.6371 68612.82 231.5716 10.50000051 98.27531 10:40 68265.8 820.3867 126.6371 69212.82 231.5716 10.66666717 98.24275 The worst case encountered in this analysis is a launch a t the end of the window while achieving the 30 LDN dispersions on the target state. If the 9 8 . 2 4 2 7 5 ' target 10:40 state above i s compared with the optimum inclination of 98.2413", the difference is -0.001446309° in inclination, well w i t h i n the inclination dispersion allowance of 0 . 1 " . If non-constant values of LDN and At had been used, the 1 0 : 4 0 target would be computed using the optimal 10:40 LDN and At's, yielding 98.241 15", a difference of 0.00015° from the target computed using 10:20 constants.

The worst-case LDN dispersions computed by Lockheed Martin 'are +/-0.060525" for a l ' 0 : 3 0 am target orbit. Applying these dispersions to the 1020 data above (assuming similar dispersions regardless of when in the launch window l i f t o f f occurs), the targets may be recomputed using the maximum (LDN1m+0.0605250) and minimum (LDN,m-0.0605250) LDNs. Results indicate that the maximum LDN dispersion case causes the 1 0 : 4 0 target to exceed the MLT box by 10.66792605-10.66666666 = 0.00125939 min, well w i t h i n the 0.5 min allowable dispersion. For the minimum LDN dispersion case, the 1020 target exceeds the MLT box by 10.33333333-10.32652272 = 0.00681061 min, also well within the 0.5 min allowable dispersion.

These analyses show that guided targeting as is used for the EOS AM-1 launch is a technique that works well for Sun-synchronous orbits. Thus a technique that is applied regularly to deep space and geosynchronous orbits has been shown to produce significant improvements in the low-Earth regime.

Frozen Orbit Control Having thus improved the launch and ascent process, the maintenance of the mission orbit was also examined for possible improvements. In addition to maintaining the ground track control grid, the science instruments on EOS AM-1 dictate that only small altitude changes can occur over any given latitude. Consequently, a frozen orbit is implemented to constrain the mean argument of perigee near 90 degrees. Freezing the orbit requires a mean eccentricity of 0.00116 for the mission altitude and inclination. If no ground track control maneuvers were required, some infrequent frozen orbit maintenance maneuvers would be required over the life of the mission to reshape the orbit. However, since the mission requires frequent altitude maneuvers, these maneuvers can be used to simultaneously restore the frozen orbit while performing ground track control. The key to utilizing the altitude maneuvers to simultaneously meet the frozen orbit c o n s ~ ~ n t s i restore the orbit orbit should be d argument of perigee as close as spectively). The maneuvers are s, separated by about 180' anomaly, that the Hohmann transfer maneuvers would normally be placed to restore the frozen orbit.

Because it is desirable for operational reasons (less downtime for instruments, less data interruption, less maneuver planning) to perform the least number of maneuvers possible and still maintain the orbit within the science requirements, the technique of using only one maneuver instead of the traditional Hohmann transfer pair to maintain the AM-1 ground track was designed into the AM-1 support. A single maneuver has the same total AV as the pair and is placed appropriately to maintain the frozen orbit condition. This technique was used with success to maintain the semi-frozen orbit of Landsat-5. AM-1 has a more stringent requirement to maintain its frozen altitude to within +lo/-5 km mean altitude and B O " mean argument of perigee. Therefore, analysis was required to investigate the effects of the location of single ground track correction maneuvers w i t h i n the orbit on the frozen orbit condition.

An algorithm w a s developed for the night Dynamics Analysis Branch that determines the best location to perform a single burn to drive the orbit back to the optimal frozen conditions. This algorithm was easily integrated into AI Solutions' object-oriented FreeFZyerm product and is used as part of the automated ground track maintenance maneuver planning process. The algorithm (Ref. 5 ) requires as input the initial mean semi-major axis, argument of perigee, and eccentricity, as well as the values of the AV and burn duration required for the ground track maintenance maneuver that will be accomplished in combination with the frozen orbit maintenance. First, the average orbit velocity is computed from the mean semi-major axis (a) as: Dividing the total desired AV for the ground track maneuver by the burn duration yields a AV per second, 6v. Then the standard variations of Keplerian elements under this AV are given by:

2&(e + cos MA)

Ae = v a g Applying these equations iteratively over the duration of the maneuver allows the initial Keplerian elements to be'coarsely propagated. After each iteration, the mean anomaly (IvfA) is calculated using: may then be varied parametrically to determine the value that best achieves the desired frozen orbit conditions. Figure 3 shows a scan over one orbit at 2" mean anomaly increments. The AV is applied at each step and the post-bum eccentricity and argument of perigee computed. The area highlighted by the circle in Figure 3 shows where the set of post-burn solution points most closely intersects the target point of 0.00116 and 90". This intersection point corresponds to a mean a n o d y of 49", where 9 0 . 2 2 " and 0 . 0 0 1 1689, respectively. The point of 0" MA and the the computed values are direction of increasing MA are both indicated in the figure.

Scan of Burn Location to Restore Frozen Conditions 2 Degree Stepsizt 0 . ~ 0 7 ~ 0.000to 0.000~5 o.ooopo 0.0000s 0.00ioo o.oitos o.oiiio o . o o ~ o.&rzo 0.0012s o.oorao Postbum M e a n E c c e n t r i c i t y Figure 3: Optimum M e a n Anomaly to Achieve EOS AM-1 Frozen Orbit The fromn orbit evolution for an l8-month span with maneuvers determined using this algorithm is shown in Figure 4. The figure indicates that performing the single maneuver ground track corrections at the optimum frozen orbit restoration location achieved the maintenance of the mean argument of perigee to within the 520" allowed by the mission requirement. The straight line portions of the plot indicate places a t which the maneuvers were performed.

The radial position constraint of +lo/-5 km in mean altitude is then met by default, since the argument of perigee requirement is the more stringent of the two as described in Reference 4. This result may be easily seen when examining Figures 5 and 6 from Reference 4, which show the frozen orbit evolution for eccentricities that are increments of 0 . 0 0 2 higher (l3gure 5 ) and lower (Figure 6 ) than the nominal 0.00116 value. The center ellipse in each figure is the nominal eccentricity, and ellipses moving out from the center are incrementally higher or lower, respectively. Based on both figures, the eccentricity must not deviate more than k0.W from the nominal value. For an argument of perigee deviation of SO", keeping the eccentricity deviation within these bounds requires constraining the altitude to within approximately +3.7 W - 2 . 3 km of the 705 km mean nominal, as shown in Figure 7 (Ref. 4). Since these altitude restrictions are tighter 3. ts, ~ a i n ~ n i n g the ~ ~ ~ n t of ensure ti0 is not violated.

0.0008 0.0009 0.0010 0.0011 0.0012 0.0013 0.0014 0.0015 0.0 I6 Eccellricity Figure 4: 18-MonthFrozen Orbit Evolution 0.0025 0.002 c .

.-

-

9 0.0015

; 0.001

tQ a , 0.0005 20 40 60 80 100 120 140 160 Mean Argument o f Perigee (deg) Figure 5: Frozen Orbit Evolution for Eccentricities of 0.0002 Increments Above Nominal 0.0025 0 . 0 0 2 0 . 0 0 1 5 0 . 0 0 1 0.0005 40 60 80 100 120 1 40 Mean Argument of Perigee (deg)

I

I Figure 6: Frozen Orbit Evolution for Eccentricities of 0.0002 Increments Below 0 . 0 0 0 5 0 . 0 0 1 0.001 5 0 . 0 0 2 0.0025 Ecc8rmici!y Figure 7: EOS AM-1 Altitude Variation vs. Eccentricity ADVANCEMENTS IN OPERATIONS TECHNIQUES Flying a spacecraft with multiple orbital and operational constraints such as EOS AM-1 traditionally requires experienced personnel to design, plan, and execute maneuver control strategies. Current directions in NASA are driving towards more streamlined, “lights-out” environments in which spacecraft operators are only present during the day shift. This change of approach forces operators to perform a variety of functions more efficiently. The FreeFZyerm mission design and operations software, a commercial off- the-shelf (COTS) product developed by A I Solutions, Inc. under contract to NASA GSFC, provides the analyst with a l l the functionality required to design and test various control strategies. More importantly, this same strategy is then easily automated in the operations environment.

There are two factors that must be addressed in the mission design process. The first is examining the orbit mechanics to determine the best way to achieve and maintain an orbit that will meet the science requirements. The second, equally impoaant factor is to address the real-world operational issues that must be included in any maneuver plan.

For example, the basic physics behind the ground track control problem is to adjust the orbit period using altitude control. The operational constraints can include ground station viewing requirements and lighting constraints. FreeFZyerm is designed to include both types of considerations in the design process.

The Physics - Ground Track Control

As described earlier, the ground track pattern for EOS AM-1 must remain within S O km of the WRS grid. In order to use the full 220 km ground track control box, the orbit must be raised above the nominal altitude, causing the period to be greater than that of the nominal altitude. In that case, the spacecraft takes longer than nominal to reach the descending node, the Earth turns farther under the orbit plane, and the ground track W t s westward. When the nominal altitude is reached, there is no d r i f t . As the period of the orbit continues to decrease, the spacecraft to reaches the descending node earlier each orbit and the ground track error drifts eastward. The d r i f t continues eastward to the edge of the control box. Consequefitly, periodic altitude raising maneuvers are required prior to reaching the eastern boundary to reset the ground track to the eastern edge of the box.

After the maneuver, drag will again act on the orbit and will slow the westward drift rate until it begins to drift ground track control problem appears as a s here maneuvers are executed at the peaks and nominal altitude is reached in the troughs.

Since the ground track drZt is due largely to atmospheric drag effects and since the EOS AM-1 mission will span periods of both low and high solar flux, the frequency of the maneuvers will vary greatly over the mission lifetime. Also, the magnitude of the maneuvers can vary by factors of up to four between the solar maximum and the solar minimum. This variability has been handled historically by sizing the ground track maneuvers by hand to see what size bum will turn the drift westward while not overshooting the westward boundary.

Stated differently, the analyst would test burn sizes until the turnaround point was at an acceptable l i t near the western edge. This requires analyst knowledge of acceptable limits, d r i f t rates, and flux predictions.

In FreeFZyerm, this process has been automated by numerically implementing the same strategy. The bum size is determined using an internal targeting algorithm based on a differential corrector incorporated into FreeFZyerm. Each iteration is evaluated by checking the longitude error at the turnaround point. This point is numerically defined as the location where the derivative equals zero. Since the longitude error data points contain small oscillations as shown in Figure 5, a running average of the data is first computed to smooth the curve so that a derivative may be calculated accurately.

Figure 9 shows the results of a maneuver targeted with the method described above. In this figure, an initial guess is tested in the curve labeled (l), a perturbation is applied along (2), and then the first iteration (3) is computed, tested, and accepted. This strategy minimizes the analysts’ time pre-launch and allows the FOT to perform functions operationally without prior understanding of the problem.

-5 -10 -15 0 10 20 30 40 50 60 10 80 90 EOSAM1ElapsedOsysFmmEpoch (Days) Figure 9: Ground Track Maneuver Targeting In addition to designing and maintainiig the orbit to meet science requirements, These the operations environment places restrictions on the orbit design as well.

restrictions are often arbitrary and not related to the mechanics of the design itself.

FreeFZyerm is designed to automate operations by addressing both the physics and these operational requirements. The control language in the program allows the user to require any number of conditions to be met before performing an action. Therefore the user can state that if the need for a maneuver is detected and the spacecraft is in view of a ground station and the spacecraft is not in shadow, then the software should plan and execute the required maneuver.

However, since a maneuver plan will not always be comprised of -/false conditions, FreeFZyerm contains a fuzzy logic engine to resolve conflicting constraints or to allow constraint weighting. For example, if a s o f t boundary is reached, there may be time to wait for an ideal maneuver location. However, if the hard boundary is reached an immediate burn may be required. Some examples of these principles as they are being used for the EOS AM-1 mission are described in the remainder of this section.

Operational Considerations - CalculableParameters

A key component to automated maneuver planning is to include operational considerations, such as lighting conditions or maximum thruster on-times. While some constraints are either true or false, others may be approximate constraints. FreeFZyerW provides a mechanism that allows mission constraints to be defined and evaluated in terms of approximations. For instance, a basic maneuver to raise perigee would not necessarily need to occur exactly at apogee, but rather near apogee to allow other constraints (such as acquiring a ground station) to be satisfied.

FreeFZyerTM contains a mechanism that allows combinations of constraints to be evaluated simultaneously and resolved into acceptable actions, even when these constraints appear to conflict. This mechanism is fuzzy logic. Fuzzy logic has been used for control systems in cameras, subways, and automobiles to resolve conflicting control goals. is technique and applies it in the orbit control regime.

Perhaps the least glamorous and yet most valuable of an operational ith the staffing oft constraint utilizing fuzzy logic is time. operations support shift during normal business hours, the scheduling of maneuver times is a key component of the EOS AM-1 control strategy. For AM-1 the FOT desired to restrict the maneuvers to occur mid-week during the late afternoon, allowing sufficient time to plan and execute the burn in a single shift. In FreeFZyerm, the day of the week and time of the day for maneuvers can be added easily into the control logic of the maneuver plan.

Figure 10: Fuzzy Set Utility in li2eeFlyerYM A fuzzy set representing the time of day is shown in Figure 10. The set is defined over a domain ranging from 13 to 17, representing the Greenwich Mean Time (GMT) of a day measured in hours. This domain was chosen to correspond to midday local time/EST. The shape of the set is used to weight the importance of the maneuver time in the control logic. Higher values (Le. higher “degrees of membership” in the fuzzy logic sense) represent more acceptable solutions. This fuzzy set can then be used in FreeFZyerm as a component of the decision algorithm configured by the user. More specifically, the user controls the maneuver plan using the following syntax: If (AM1.LongitudeError > 18 and *AMl.TimeOfDay is atprimeshift) then Maneuver EOSAMl This command line (taken literally from a FreeFZyerm control script, with slight modification for clarity) evaluates the error in the EOS AM-1 ground track and the time of day at the operations center for a modeled spacecraft epoch, and plans a maneuver if the error in the ground track is approaching the control boundary at a time of day that is acceptable for maneuver execution.

Operational Considerations - Non-CalculableParameters

e are events maneuvers is that A contact schedule for EOS AM-1 is delivered electronically on a weekly basis.

This schedule contains the allotted contact opportunities with the TDRS system, a subset

of the geometrically possible contacts computed in FreeFZyerm . The contacts are

approximately 10-minutes in duration and occur approximately twice per orbit. To ensure that the maneuver is planned within these scheduled passes, an ASCII file containing the weekly schedule is read by FreeFZyerm, and is converted to fuzzy sets based on the spacecraft epoch. These fuzzy sets are then incorporated into the control script in a manner similar to that discussed above for the t i m e of day constraint. The new control logic takes the form (Ref. 6 ) : Load InTDRSContact from TDRS-Schedule using AM1.Epoch; If (AM1-LongitudeError > 15 and AM1.Epoch is atprimeshift and AM1.Epoch is InTDRSContact) then Maneuver EOSAMl The Galyst literally sets the control logic using this kind of near-natural language technique. The shapes of the fuzzy sets can easily be modified using extensions to the control language.

The flexibility provided by FreeFZyerm for orbit control makes it an extremely powerful tool for mission analysis, planning and operations. The tool addresses needs in the user community that have been identified for a number of years, and moves the satellite control regime much closer to autonomous operations.

CONCLUSION EOS AM-1 has been able to realize cost savings in several areas. F i r s t , the expert flight dynamics personnel will only be required to support the mission post-launch in a consultation standing. FOT personnel will be able to include the routine flight dynamics activities into their daily schedule with a minimum of impact due to the high level of automation. Maneuvers will be restricted to the nominal work hours of the prime shift.

In addition, time spent by the flight dynamics experts in planning special maneuvers, l i k e the ascent sequence, has been drastically reduced by eliminating the need for running multiple pieces of software in parametric m s .

REFERENCES 1. F o l t a , David C. and Lauri Kr&, “Methodology for the Passive Control of Orbital Inclination and Mean Local Time to Meet Sun-Synchronous Orbit Requirements”, pace olorado Springs, C 2.

Computer Sciences Corporati Interface Control Document,” LMCA- ICD-95-004, Rev. Cy 11 Nov. 1997.

4. Computer Sciences Cogoration, CSC568 16-02, “EOS AM- 1 Ground-Track Control and Frozen Orbit Analysis“ (memorandum), P. Noonan, September 9,1996.

5. A I Solutions, Inc., “EOS AM-1 Mathematical Specifications,” D. Conway and C.

Schiff, March 31,1998.

6. A I Solutions, Inc., “EOS AM-1 FreeFZyerm tutorial,^ Version 1.1, M. Hametz and P . Noonan, January, 1998.

ite Si Roberto Alonso Pablo Anigstein Ricardo Siunehez Pena (CoNAE) ABSTRACT The SAC-A is a Low Cost - Short Schedule - Small Bus dedicated to test equipment and new technologies which may be used in operational or scientific missions with more immunity to failures. T h i s satellite is planned to be launched in July, 1998 as part of the STS 88 mission.

The opportunityto fly in a low orbit for a reasonable period of time (at least 1 year), allows the characterization of the behavior of this new instrumentation in real world applications and also to compute performance.

The 68 kg satellite will have an almost octagonal configuration to be fitted within the Hitchhiker Motorized Door Canister with Hitchhiker Ejection System (HES) envelope. This volume is approximatelya cylinder of 19 inches diameter by 20.5 inches maximum height.

The orbit will be circular @ 200 run altitude with an inclination of 5 1.6 deg, the expected lifetime is about one year.

The experiments on board are: Differential Global Positioning System Receiver (DGPS) 0 CCDCamera 0 Tri Axial Magnetometer (TAM) 0 Argentinian Si Cells 0 UHF ReceiverNHF Transmitter.

From the mission design point of view, the on board power consumption plays a central role for the attitude strategy design: to supply the essential loads at least three solar arrays should be pointed most of the time to the sun. In addition, the DGPS receiver and the CCD camera should be pointed to the zenith-nadir direction. The other experiments do not impose any attitude constraints on the mission.

To fulfill all of the requirements several control modes are implemented. The paper explains in details the strategy adopted.

The sensors to accomplish the mission are: Six coarse sun sensors. Six Argentinian Cells, placed on the satellite to cover the total sphere, although the configuration has some blind holes behind the solar panels.

Tri-axial magnetometer. It is the same fluxgate magnetometer used for the experiment, but the readings are taken only with 10 bits.

The actuators are: . An in house made momentum wheel, located along the direction perpendicular to the sun line.

agnetic Torque Coils. Three circular a i r core coils, one on each axis.

There are four identified operational modes along the lifetime of the satellite, listed below.

Sun Pointing Mode. The solar arrays are placed toward the sun to maximize the power generation. In this mode the whale tracker and the Argentinian Si Cells are tested. During eclipse, the sun signal is not available and the ACS enters in to eclipse submode.

Picture Mode. This mode allows to place the momentum bias normal to the nadir and s u n vector to take E a r t h pictures, after a rotation around the wheel a x i s . The CCD camera is tested with this mode.

DGPS Control Mode. This mode allows to rotate the body Y axis around the sun line in order to point the antenna’s boresight vector as close as possible to the zenith axis, required by the DGPS receiver to lock on the DGPS satellites. The DGPS and the triaxial magnetometer (because a very precise time and orbital position is needed to correlate the Earth magnetic field) are tested during this mode.

Spinning Mode. The satellite is spun along the momentum bias to check the DGPS performance in spinning satellites.

The Safe Hold Mode (with momentum wheel ON) is in effect the Sun Pointing Mode (fiom the power consumption and thermal equilibrium point of view). In case of momentum wheel failure, the satellite will enter in a “safe” spin mode along the sun h e to preserve the necessary power generation.

The on board software has the capability to compute ephemeris, the magnetic field and sun vectors in inertial fi-ame using theoretical models, attitude determination in real time and also detects failures which change the mode to Safe Hold. The software design is flexible enough to allow changes or patches on variables and modify areas of memory.

The proposed configuration is better than other possibiliiies considered in terms of safety and stability, because the precession of the momentum bias for E a r t h observation or to take pictures or to acquire the DGPS satellite is around the sun.

In this paper we will derive control l a w s to perform attitude maneuvers which automaticdy avoid the pointing of a speci- fied spacecraft axis into forbidden directions. These laws will be found by a control-theoretical approach using methods from differential geometry.

We will start by deriving a particularly simple control law ma- neuvehg a spacecraft from rest to rest between prescribed attitudes in the absence of pointing constraints. Subsequently, we will show how maneuvers of this type can be concatenated in such a way that prescribed forbidden directions are guaran- teed to be avoided. The control law obtained in this way does not take recourse to numerical methods and hence can be easily implemented in an on-board attitude control system, possibly for performing maneuvers in an emergency mode. Finally, we will propose an iterative scheme to optimize the solution.

INTRODUCTION The attitude or orientation of a spacecraft (modelled as a rigid body) is the matrix g E SO(3) whose rows are the directions of the body's principal axes with respect to some reference coordinate system. Let us denote by 11,12, I3 the mo- ments of inertia, by w1 , W 2 , 0 3 the angular velocities and by T',T', T3 the exerted torques about the principal axes. Then the attitude kinematics of the spacecraft are described by the equation

j ( t ) = (@l(t)El + W 2 ( t ) E 2 + W 3 ( t ) E 3 ) 9 ( t ) (1)

where .

0 1 0 0 0 0 0

E l = [ H 0 ' I , E2= [o 0 -81, E3= [-1 0 01

(2) 0 0 0 -1 0 1 0 t Fachhochschule Wiesbaden, Fachbereich Mathematik, Naturwissenschaften, Datenver- arbeitung und Umwelttechnik, Kurt-Schumacher-Ring 18, D - 65197 Wiesbaden, Germany.

FAX: (49) 611-9495-382. Electronic mail: spindler@r5.mnd.fh- Phone: (49) 611-9495-377.'

wiesbaden.de.

?

ere s e S In this paper, we will study the problem of steering a spacecraft between given g(t0) = go and g(t1) = 91 while minimizing a cost functional which both attitudes measures the overall angular velocity and penalizes undesired attitudes during the maneuver. By choosing this cost functional judiciously, we will ensure that the angular velocities at the beginning and at the end of the maneuver take prescribed values and also that forbidden pointing directions are avoided during the maneuver.

MATHEMATICAL BACKGROUND Our approach is based on the observation that equation (1) can be considered as a control problem on the Lie group SO(3) (with the angular velocities wi treated as control inputs). T h i s observation w i l l enable us to use differential geometric tech- niques in control The characteristic features of the situation are captured in the following problem formulation.

Basic Control Problem. Let G be a Lie group with Lie algebra g and let

(El, ... ,En) be a vector space basis of 8. Consider a right-invariant dynamical

system g ( t ) = U ( t ) g ( t ) evolving o n G where U ( t ) is the sum of a controlled t e r m

Cbl ui(t)Ei and a drift t e r m CEmil UiEi with given constants um+l,. . . , un. W e

wdl be interested in finding controls t I+ ui(t) (where 1 5 i 5 m) steering the s y s t e m E.m g ( t 0 ) = go t o g(t1) = 91.

To h d suitable control laws which solve this motion planning problem, we propose to introduce a minimization condition which the controls are required to satisfy. This condition should be “reasonable”, but is not at all unique. The idea is not so much that the minimization of a specific cost functional is a strict require- ment, but rather that introducing such a cost functional makes available methods from optimal control theory (specifically, Pontryagin’s Principle) as tools to fmd solutions to our problem. We first consider the special case that the cost functional on time.

does not explicitly depend Theorem 1 . In the control problem described above, let the controls uf be chosen in such a way that a cost functional st’,’ @ ( g ( t ) , u ( t ) ) d t i s minimized. Let t H g * ( t ) be the resulting state trajectory in G so that g n ( t ) = U n ( t ) g * ( t ) . T h e n

there exist a curve t I+ p * ( t ) in g* \ (0) and a number e E Iw such that

j = -E-(g*(t),u*(t)) a (I 5 i 5 m) -

h i m = n (Le., i f the system s f i l l y actuated) then necessarily e f: 0 (absence of abnormal minimizers).

roof. The Hamiltonian H : G x g* x W n 3 R of the system is H ( g , p ; u ) := e@(g,u) + C y = l u i p ( E i ) where E is either zero or an arbitraq nonzero number.

Pontryagin's Principle shows that there is a curve t c3 p * ( t ) in g* \ (0) with

aH

-(g*(t),p*(d);ol*(t)) = 0 (1 5 i 5 rn)

h i for almost all t E {to,tl]. This establishes equations (4) and (5). Ifm = n then E = 0 would imply p * ( t ) = 0 ( f o r almost d t ) according to equation ( 5 ) , contradicting

the choice o f p*. Hence # 0 in this case. m

We now consider a cost functional depending explicitly on time. The asso- ciated optimal control problem will be reformulated as one with a timeinvariant cost functional by f o d y introducing time as an additional state variable. This reformulated control problem will automatically contain a drift term (even if the original control problem does not); hence the full generality of Theorem 1 is needed in obtaining our result.

Theorem 2. In the control problem described above, let the controls uf be chosen in such a way that a cost functional st',' @ ( g ( t ) , u ( t ) , t ) d t is minimized. Let t c3 g * ( t ) be the resulting state trajectory in G so that g*(t) = U*(t)g*(t). Moreover,

let ql -: g x W + g and Po ; g x W + B be the canonical projections of the direct

product g x R. Then there exist a curve t H n*(t) in (g x R)* \ ( ( 0 , O ) ) and a number

e E W such that

a@

T*(t)Z = -e-(g*(t),u*(t),t) (1 5 i 5 m).

(7) h i If m = n t h e n E can be chosen t o be nonzero.

does not explicitly depend on time any more. The claim then follows immediately by applying Theorem 1 to the control problem on F of steering 7 from (gO,e*o) to (91, e t l ) while mhimkhg L : 5(7(t), u(t)) dt.

If m = n we can choose E # 0. Otherwise an optimal control would be neces-

sarily independent of the cost functional, which implies that we could replace the time-dependent cost functional by a time-invariant one. But for a time-inva.riant cost functional the absence of abnormal minimizers was established in Theorem 1 already. rn ATTITUDE MANEUVERS We will now apply the above results to attitude control.' To keep things sim- ple, let us consider the case that a spacecraft shall be steered from rest to rest between prescribed attitudes. (The method to be presented can, with a little ex- tra effort, also be applied to maneuvers between arbitrarily prescribed rotational states.) Moreover, let us assume that certain state constraints have. to be taken into account (such as the need to avoid the pointing of specified spacecraft axes into forbidden directions or the requirement to stay close to a desirable reference trajectory). We will formally treat the angular velocities as control inputs and try to plan a maneuver which is "as smooth as possible" in the sense that a cost functional measuring the overall mgular velocity during the maneuver is minimized. To deal with the state constraints we introduce a term in the cost functional which penalizes undesirable attitudes during the maneuver. (This is reminiscent of a Lyapunov ap- proach using artificial potential functions with peaks about undesired attitudes.') Moreover, in order to gumantee that a rest-to-rest maneuver is being planned, we impose infinite penalties on nonzero angular velocities at the beginning and at the end of the maneuver. Thus we are led to the optimal control problem of determining the angular velocities t w wf(t) which steer the spacecraft between the prescribed attitudes while mbhizing a cost functional of the form where F is chosen according to the nature of the constraints to be considered and where q is a positive weighting function with singularities at the start time t o and at the end time t l . (A typical choice for Q is depicted in Figure 1 below.)

Figure 1 : Typical choice f o r the weighting function Q .

We will now apply Theorem 2 to solve this optimal control problem. Once this is done, the torques required to implement the desired maneuver are found by simply plugging the angular velocities t H ~ r ( t ) into Euler's equations (3).

Theorem 3. Consider the problem of steen'ng a three-axis controlled spacecraft between two specified attitudes g(t0) = go and g(t1) = g1 such that the cost finctional ( 8 ) i s minimized. Denote the optimal angular velocities by t H w f ( t ) and let t c-) g*(t) be the resulting attitude evolution. Then the functions S&(t) := q(t)wr(t) satisjy the diflerential equations Proof. Applying Theorem 2 with E := -1, we obtain a curve t w ~ * ( t ) in

(so(3) x R)* \ ((0,O)) such that for 1 5 i ,< 3 we have n*(t)Z = S2;(t) on the one

hand and = O on the other hand; here the last term vanishes because of the bracket relations [&,E21 = 4 3 3 , [&,&I = -El, [E3,Ell = 4 3 2 .

T h i s yields the claim.

nctio ctions 0 203 -202 -203 0 201 -w1 0 w2 3, we can succinctly formdate our result y stating that if the a n , c s u l a r velocities t H of(t) are optimally chosen with respect to the cost functional (8) and if t H g * ( t ) is the corresponding attitude evolution, then the fun ons t H g * ( t ) and t c3 sl*(t) are solutions of the following system of coupled di tid equations: BOUNDARY CONDITIONS In addition to the differential equation ( l l ) , the two boundary conditions g(to) = g o and g(t1) = g1 have to be satisfied. T h i s is accomplished by follow- ing a shooting procedure whi& we now describe. For s E W3 let t I + (ga(t),na(t)) be the unique solution of the initial value problem s(t) = d ~ ) - ' q w ) g ( ~ ) , g(t0) = go, (12) Q ( t 0 ) = s .

Qt) = @ ( g ( t ) , t ) 9 We want to adjust s in such a way that g a ( t l ) = g1. To do so, we have to investi- gate how the functions t I+ (ga(t),sls(t)) -vary in dependence of the parameter s .

Therefore, we introduce the functions g 8 ( t ) a j ( t ~ s)T = -a;(ty ~ ) ~ ~ ( ~ ) * * s last equation is, a i . The evaluation of the right-hand side of (15) is simplified by noting that (17) can be rewritten in the fonn The desired value of s is such that g e ( t 1 ) = 91. The strategy is to first find a reasonable initial guess do) and then apply a Newton-type algorithm to produce improved values d l ) , d2), .. . until the condition g 8 ( t l ) = g1 i s satisfied within the desired accuracy. Inserting the approximation into the target equation gs+&(tl) = 9 1 , we see that the update equation in the Newton-type iteration is given by In the next paragraph we will show how to find a reasonable initial estimate do).

This paragraph is interesting in its own right, because it yields a very simple al- gorithm to perform a rest-to-rest maneuver between prescribed attitudes in the absence of state constraints .

CONSTRAINT-FREE MANEUVERS Let us for a moment ignore the presence of state constraints. Then we can choose F E 0 in the cost functional, and the equations (9) simply state that the functions s2i remain constant during the maneuver. The contents of the following theorem is that the constants can be explicitly written down as functions of the initial attitude go and the target attitude g1.

Theorem 4. Suppose we want to steer the spacecTaft from the attitude g ( t 0 ) = go t o the attitude g ( t 1 ) = g1 while minimizing the functional

T h e n the optimal cont are given by wZ(t) = cr/q(t) for 1 5 i 5 3, and the

torques which lead t o t angular velocities are given by .r&Q(t) + (12 - I 3 ) C ; C ;

- I i ) C ; C : 0 (23)

& C $ Q ( t ) + ( 1 3

I3cfQ(t) + (I1 - I2)C;c; T h e maneuver is such that any body-fixed axis b E R3 rotates about the azis W a where T T (24) L ( 4 = go 91 -91 90 0 Proof. From (9) we know that the functions = q u r are constants, say ~ r ( t ) = ci/q(t). Let C := clE1 +czEz +c&; then the optimal trajectory t c-) g*(t) in SO(3) satisfies g*(t) = q(t)-lCg*(t). This equation can be explicitly integrated.

In fact, Q being an antiderivative of l / q , we have

g*(t) = -P( (QW - Q(t0))C) go

(25)

The constants ci must be such that g(t1) = g1 which means that exp( [Q(tl) -

&(to)]C ) = gig;' = 7. Rodrimes' formula then shows that ci = cr for 1 5 i 5

3. (Note that the equation cosa = (tr[7] - 1)/2 does not determine a ! uniquely.

However, since our optimization criterion requires cq + cz + ci and hence cy: to be as

s d as possible, we really have a ! = arccos((tr[7] - 1)/2); whence the claim.) The torques T i are then obtained by plugging in o = wr in Euler's equations (3).

If b E R3 is a body-fixed direction (i.e., if b l , b2, b3 are the body-coordinates of a unit vector rigidly attached' to the spacecraft) then the motion o f this axis in space between u := gTb and v := gTb is given by

g*(t)Tb = SoTeXp (-(Q(t) - Q ( t o ) ) L(c*)) b

= SOT- (-(Q(9 - & ( t o ) ) L(c*)) g o g P

(26)

= ~ X P (-(Q(t) - Q ( t o ) ) sOTL(C*>SO) g0Tb

= ~ X P (-(Q(t) - Q ( t o ) ) L(gOTc*)) u

where we used the fact that C = L(c*). This shows that the body-axis rotates about the axis spanned by 2 := grc'. Note that if X := 2 ( Q ( t l ) - Q ( t o ) ) ( s i n a ) / a > 0 and if a := A6 then

L(a) = x * L(gFc*) = x * gTL(c")go = gF(7 - yT)gO

(27) T T

= ' goT(g19oT - g0gT)go = 9 0 91 - 91 9 0 *

m functions t I--) inertial reference system) are specified in such a s is forbidden to coincide with the j-th pointing direction at any t h e ; in fact, the angle between these two axes may be required to exceed a minimum angle (psde. Now it is easy to verify whether or not a maneuver as in Theorem 3 meets this requirement. In fact, as was shown before, each body-axis bj rotates about an axis Wa (uniquely determined by the initial attitude and the target attitude) as shown in Figure 2, and the closest possible angle between the rotating axis and a fixed space direction dj is given by

lej -6jl where Oj := L ( u j , u ) = L(vj,u) and Sj := L(a,dj) (see Figure 3); hence

the maneuver given in Theorem 2 is safe if lej - Sjl > (psde for all j .

a a 'igure 2: Motion of a body-axis dur- Figure 3 : Checking the pointing con- ing the maneuver.

st raint s.

If the maneuver is not safe, then we can simply concatenate several maneuvers of the same type each of which avoids the forbidden directions. We will explicitly do so in the case that there is one axis f o r which there are two forbidden directions, having mind a cryogenically cooled space telescope for which the telescope di- rection-is forbidden to coincide with either the sun or the moon direction. If the circle C along which a maneuver as in Theorem 4 would carry the telescope axis is found to be not safe due to one of the forbidden directions, say d l , then the other forbidden direction dz can make only one of the two regions bounded by C unsafe, but not both; hence (after replacing a by -a if necessary) we can assume that the region containing a is safe for operations. (This is the case if (d2, a) < 0.) The idea is now to slew the telescope axis from its initial position u to the axis direction a and from there to the target direction v . The next theorem shows how to find a safe intermediate attitude 9;.

avoids the forbidden directions.

Proof. The idea is to slew the telescope from u = gTb to a (which requires a = grb, i-e., gia = b) along great-circle through u and a (which requires

* L(u x a) = grgi - $go with a constant c # 0). Then necessarily

and hence

which impEa c = 2. Consequently, giu = 2(u,a)b - goa. Since gi E SO(3) we then

have

gi(a x U ) = gia x giu = b x (2(u,a)b - goa)

= b x (-Sou) = g o a x b = goaxgou = g o ( a x u ) .

Hence gi necessarily maps a to b, u to 2(u,a)b - goa and a x u to go(a x u); this

deterxnines gi Uniquely. Since A maps el to b, e2 to u and e3 to a x u and since B

maps el to b, e2 to 2(u,a)b - goa and e3 to go(a x u), this implies gi = BA-l. B

MANEUVER OPTIMIZATION The maneuver proposed in the previous paragraph is a safe option (and possi- bly useful in an on-board emergency mode), but does not yield an overall smooth motion, as the spacecraft is artificially brought to rest at the maneuver midpoint.

Therefore, we propose to find a better solution by using a shooting procedure as de- scribed above, taking as a starting point the maneuver determined in the constraint- free case. As before, let bj be the spacecraft directions which are not allowed to coincide with the forbidden directions dj. If t H g ( t ) denotes the spacecraft attitude evolution then the angle p j between bj and d j is given by cos p j ( t ) = ( g ( t ) T b j ,d j ( t ) ) .

Hence we will introduce a term of the form C j x ( ( g ( t ) * b j , d j ( t ) ) ) in the cost func- tional where : [-1,1] -+ [O, 00) is an increasing function taking relatively large attitude evolution found for the unconstrained case, where the penalty is low close to the forbidden attitudes and high in safe regions. This penalty term may explic- itly depend on time, imposing higher penalties on deviations towards the end of the maneuver. Thus in the cost functional (8) we will use a control term of the form

where the matrix norm 1 1 . 1 1 is the one derived &om the inner product ((A,B)) :=

t~! ATB = C;,j aijbij.

SIMULATION RESULTS As an example, let us take the initial attitude go = 1 (identity matrix) and the target attitude 0.03154 -0.11772 0.99255 0.25689 -0.95873 -0.12187 .

0.96593 0.25882 0.00000

I

Choosing the normalized time i n t e d [ t o , t4 = [O, I] and the weighting function

q ( t ) := l / ( t - t'), Theorem 4 yields the maneuver whose mgular velocity evolutions

are given in Figure 4.

I I Figure 4: Angular velocities during one-leg maneuver.

I I Figure 5: Angula,r velocities during two-leg maneuver.

The pointing requirement which is violated during the one-leg maneuver (see Figure 6) is now met during the two-leg maneuver (see Figure 7).

150 1751 2 . 2 '3.4 0.6 0 . 8 Figure 6: Angle between telescope Ggure 7: Angle between telescope axis and forbidden direction during axis and forbidden direction during one-leg maneuver. , two-leg maneuver.

0.2 0 . 4 0 . 6 0 . 8 1 I t Figure 8: Deviation between current Figure 9: Deviation between current attitude and target attitude during attitude and target attitude during one-leg maneuver. two-leg maneuver.

Finally, we introduce a penalty term o f the form (28) in the cost functional by

choosing x(z) := 100 e x p ( z - 0.8) and p(z,t) := 1/z. The reference trajectory

t H grer(t) is the one obtained for the one-leg maneuver described above, and the initial values o f the functions t H q(t)wf(t) f o r the first iteration are taken from this maneuver. The shooting method described before converges fast;- the angular velocities of the resulting solution are given in Figure 10 below.

V

I Figure 1 0 : . Angular velocities during optimized maneuver.

0 . 2 0 . 4 0.6 0.8 1 Figure 11: Deviation between cur- Figure 12: Angle between telescope rent attitude and target attitude dur- axis and forbidden direction during ing optimized maneuver. optimized maneuver.

To understand the scale in the deviation plots, we note that the maximum deviation between two elements of SO(3) is given by REFERENCES 1. Roger W. Brockett, R. S. Milman, He.ctor J. Sussmann, Difjrerential Geometric Control Theory, Birkhguser 1983 2. Velimir Jurdjevic, Geometric Control Theory, Cambridge University Press 1996 3. Colin R. McInnes, Large Angle Slew Maneuvers with Autonomous Sun Vector Avoidance, Journal of Guidance, Control, and Dynamics, Vol. 1 7 , No. 4, Jdy-Aupst 1994, pp. 875-877 James R. O'Donnell, Jr., Ph.D.

David J. Mangus Goddard Space Flight Center, Code 572 Greenbelt, Maryland USA 20771 Over 39 years and a long list of missions, the guidance, navigation, and control (GN&C) groups at the Goddard Space Flight Center have gradually developed approaches to the design and implementation of successful spacecraft attitude control systems. With the recent creation of the Guidance, Navigation, and Control Center at Goddard, there is a desire to document some of these design practices to help to ensure their consistent application in the future.

In this paper, we will discuss the beginnings of this effort, drawing primarily on the experience of one of the past attitude control system (ACS) groups at Goddard (what was formerly known as Code 712, the Guidance, Navigation, and Control Branch). We will discuss the analysis and design methods and criteria used, including guidelines for linear and nonlinear analysis, as well as the use of low- and high-fidelity simulation for system design and verification of performance. Descriptions of typical ACS sensor and actuator hardware will be shown, and typical sensor/actuator suites for a variety of mission types detailed. A description of the software and hardware test effort will be given, along with an attempt to make some qualitative estimates on how much effort is involved. The spacecraft and GN&C subsystem review cycles will be discussed, giving an outline of what design reviews are typically held and what informationshould be presented at each stage. Finally, we will point out some of the lessons learned at Goddard.

INTRODUCTION Throughout its history, the Goddard Space Flight Center has had a number of attitude control system (ACS) branches and organizations devoted to the design, development, testing, and operation of the attitude control and determination subsystems of spacecraft. During the many years and many projects with which these groups have been involved, a number of practices, approaches, and lessons learned have been developed that have led to a great many successful missions.

With the recent reorganization of engineering groups at Goddard, a number of attitude control and navigation groups have been merged and combined within the new Guidance, Navigation, and Control Center (GNCC). The approaches and material discussed in this paper primarily reflect the heritage of only one of the antecedent groups to the GNCC, what was formerly Code 712, the Guidance, Navigation, and Control Branch. Code 712 had primary responsibility for the mid-range and larger spacecraft designed and 1 0 3 built at Goddard. Other groups, also part of the new GNCC, designed the ACS subsystems for the smaller missions or were involved in ACS on-orbit operations. A paper written from the points of view of these groups would reflect many of the same general approaches, though it would likely differ in emphasis.

The goal of this paper is to begin the work of documenting the different aspects of the ACS subsystem design process. The paper will cover many topics very broadly in an attempt to give an overview of our work and some feeling for our design approach. It is hoped that this effort will continue in a much more complete and rigorous way, so that the ACS design heritage and expertise of the Goddard ACS groups can be preserved for the future.

UPPORT FOR SPACECRAFT PROJECT PHASES A typical spacecraft project goes through at least five general phases, four of which require the direct, full participation of ACS engineers. These first four phases are the spacecraft conceptual design, development, integration and test, and launch and early operations. Once a spacecraft is on-orbit and operating, ACS involvement usually is limited to support for special events such as orbit maneuvers, anomaly resolution, and participation in end-of-life and other engineering tests.

Following is a list that summarizes many of the tasks that are performed by members of the ACS subsystem throughout these phases of a spacecraft's life.

1. Spacecraft ACS Conceptual D e s i g n Support GN&C systems engineering in defining high-level analysis support Support GN&C systems engineering in determining contractor support Review project-level requirements with scientists Develop GN&C mission-level requirements from project-level requirements 0 Define ACS hardware requirements 0 Define ACS software requirements Define attitude knowledge and control requirements Define trajectory requirements 0 Design GN&C mission scenario Develop control mode block diagrams Develop control mode error budgets 2. System Development Verify control mode rigid body stability margins Develop a low-fidelity time domain simulation (LoFi) Verify control mode pointing performance using LoFi Perform structure modal reduction analysis Perform control mode flexible body stability margins analysis Develop high-fidelity models of selected sensor and actuator hardware 0 a nonlinear high-fidelity time domain simulation (Hiii) Develop Verify control mode error budget using HiFi Verify control mode transitions using HiFi Present at the Critical Design Review Develop the ACS Algorithm Document Support the definition of the flight software test facility Write software test procedures that verify control mode requirements 0 Perform and evaluate results of control mode software testing Support hardware procurement and hardware and software design reviews 3. Integration and T e s t (I&T) Develop and suppoh spacecraft-level ACS hardware aliveness tests Develop and support spacecraft-level ACS hardware functionality test procedures Develop and support spacecraft-level ACS hardware phasing test procedures 0 1 04 vel ACS end-to-end tests eve1 Comprehensive Performance Tests tor calibration requirements Develop mission operation center ACS telemetry page layouts Support pre-launch mission operations Support launch and early on-orbit activities Support on-orbit sensor and actuator calibrations The remainder of this paper will discuss these phases, and the different work that occurs in each, in some more detail. The main concentration, again reflecting the experience of the authors and Code 712, will be on the development and I&T phases, with a little discussion on the systems engineering and conceptual design phase.

SYSTEMS ENGINEERING AND DESIGN ACS support for spacecraft development begins in the initial systems engineering and design effort, which very often begins a number of years before a project is even approved. There are usually a variety of study and “pre-Phase A” efforts involved in the very early spacecraft design; once a project is approved, it enters a much more formal period of development and test.

During the early systems efforts, systems engineers and those who work directly with the ACS subsystem begin the initial design task, taking the science and other mission requirements and using them to design ACS requirements and a subsystem concept that w i l l allow the spacecraft to successfully achieve its science objectives. This early ACS design comprises the following steps: 1. Mission Concept D e s i g n : In this design phase, the object is to develop a concept that will allow for the successful completion of the mission’s science objectives, within the imposed budgetary, time, and other constraints. This mission concept encompasses all phases of the project, from initial design and development, through test, launch, and operations. Specific to the ACS, an initial design of the subsystem is created-including control modes and preliminary hardware sensor and actuator complement-that will allow the mission concept to be successfully implemented.

2. ACS Level Requirements: Once the mission concept is developed, the mission-level requirements must be used to generate subsidiary requirements for each subsystem. For the ACS, this will typically mean requirements for pointing accuracy and stability, momentum management, and orbit maneuvering and maintenance; as these requirements are developed and refined, they can have a direct impact on the subsystem design, both algorithmic and hardware.

3. Error Budgets: With numerical ACS subsystem requirements in hand, it is necessary to develop error budgets that parcel out the required performance goals and allowable errors to the different parts of the subsystem. These budgets will define the type and quality of ACS hardware required, as well as imposing performance and stability requirements on the control algorithms developed in the design process.

4. Trade Studies: It is usually necessary to iterate on all of the above steps, generating a number of options and possibilities for the different aspects of the design. Decisions made at the subsystem level, both in the ACS subsystem and others, can have potential impacts in other subsystems and on the system as a whole. The object of the early stages of the design is for the systems engineers, aided by specialists in the different subsystem areas, to generate a viable preliminary design and attainable set of requirements so that the design and development effort can continue with an expectation of success.

In the ACS analysis and design phase, the requirements and mission concept generated in the early systems engineering effort is turned into reality-at least a mathematical an rithmic reality, at this at can implement point. The first step is to define the control modes-and relations the mission concept defined in the conceptual design of the spacecraft ACS. Then, for each mode, a linear controller design and analysis must be performed.

For the majority of spacecraft attitude control systems, initial design is performed assuming a PD or PID controller. Because of the relatively benign disturbance environment for most space missions, this simple controller usually proves to be sufficient. Using a PD or P I D controller and a linear J / . plant to model the spacecraft, beginning the design of controller gains and analysis of ACS performance is very straightforward. As the physical design of the entire spacecraft becomes more mature, and mathematical models of it more complete, a flexible mode analysis is done to ensure that the spacecraft ACS will not excite any uncontrolled oscillations during operation. Throughout the design process, simulations are developed and run to verify the time-domain performance of the spacecraft.

A final aspect of the ACS algorithmicdesign does not apply specifically to the spacecraft performance within a specific mode, but looks at the management of angular momentum across all modes. Depending on the mission, all spacecraft will encounter a variety of disturbance inputs-aerodynamic, gravity gradient, solar pressure, and magnetic, to name the most common-that, over time, can cause a buildup of system angular momentum within the spacecraft. A system for managing and off-loading this momentum must be designed, as it will eventually affect the performance of the spacecraft ACS.

Control Mode Definition Figure 1 shows an example control mode diagram from the MAP spacecraft. The six modes, five in the spacecraft main ACS processor and one in the separate attitude control electronics (ACE) box, were designed to meet the requirements derived from the mission requirements and concept for the MAP ACS subsystem.

Linear System Design and Stability It is during the linear ACS design and analysis phase of a project that the work most commonly associated with the ACS anaZyst is done. For the large majority of spacecraft, the control laws used as the basis for ACS design remain the same as those used 20 years ago. The initial design and stability analysis is mainly concerned with using the available tools to decide what control law is needed, determine the controller gains and other parameters needed to implement that control law, and then to verify its performance and stability. As more information-such as system inertia matrices and flexible mode analyses-becomes available, further analysis is performed to verify that sufficient performance and stability margins still exist.

The bulk of the engineering judgement and expertise of the ACS analyst with respect to the linear design come into play in two main areas. First, the initial design must include sufficient performance and stability margins so that, as more information about the'system becomes known and more fidelity is included in the design models, the design continues to satisfy all performance and stability criteria. Second, the control law and linear ACS design must be created and translated in such as a way so that it can be turned into physical hardware and software on the spacecraft.

DeltaVMode 1

aulonomous ( DeltaHMode ) ACE A l l Modes CSSnRU-based, RWA-conmlled Figure 1 MAP Control Modes Design and Analysis Tools. Many of the design and analysis concepts used by today's ACS analyst haven't changed from those used his or her predecessor, 20 or 30 years ago. The same collection of methods used to characterizethe performance and stability of systems modeled with linear equations has been used in the aerospace industry for as long as it has been around. Of course, the way these tools are implemented has changed quite a bit with the advent and increasing power of the computer. Today's analyst may be creating root locus, Bode, and Nichols plots much l i e analysts of the past, but he or she is generating them a lot more quickly.

Figure 2 shows a collection of plots that represent some of the typical means of designing and analyzing attitude control systems modeled with linear equations. Root Locus plots allow the designer to work directly with the open- and closed-loop poles and zeros of the linear system; well-known relationships can then be used to derive frequency- and time-domain characteristics, such as natural frequency and damping ratio, rise time and settling time, for the resulting system. Bode and Nichols plots (as well as Nyquist plots, an example of which is not shown) are alternate methods of displaying the same frequency- domain information and characteristics of a system. The most important of these characteristics, as relates to system stability, are the system gain and phase margins. These margins give a measure of how much more o r less gain and how much more phase lag a system can handle before it goes unstable. Adequate margin is needed when designing a system modeled with linear equations to ensure that the real system will remain stable and have acceptable performance in actual operation.

The f i n a l design tool used by the ACS analyst is the time-domain simulation. Through a variety of simulationsof varying degrees of fidelity, the performance of the control system is analyzed and verified.

As mentioned above, many of the design techniques used by ACS designers has remained the same.

The way in which the designer generates and makes use of these tools has changed, and is beginning to change a lot more quickly, as computers become faster. For instance, using a software tool such as the Interactive Control Design Module from Integrated Systems, the linear equations for a system and a baseline controller can be set up, and the controller parameters can be changed and the resulting system performance viewed in real-time.

STEPRESPONSE Figure 2 Linear Analysis and Design Techniques Rigid Body Stability. The initial ACS design and analysis is usually done with a simple line5qr model of the spacecraft, using only rigid body dynamics. At this stage of a project, it is usually not possible to acmtely model any better than this. It is not until the physical design of spacecraft bus and instrument become more mature that details of the flexible body characteristics of the system become available.

There are a number of design criteria that can be applied to a rigid body design. Of course, most of the time-domain criteria are dictated by the requirements of the mission; these include things like slew rate and pointing performance. Controller designs must first satisfy these requirements. However, there are a number of criteria that determine the stability of a system that also must be satisfied. The criteria used are the gain and phase margins of the system'. The margins that we typically use, which must be found with respect to any commandable gain within the control system (i.e., controller gains, reaction wheel or other actuator scale factors, etc.), are a gain margin of 12 dB and a phase margin of 40".

Flexible Body Stability. Flexible body analysis is generally performed starting with a NASTRAN (or other such) model of a system that gives the flexible mode frequencies and modal gains of a system. This list of modes is reduced, based on the modal frequencies and gains compared to the bandwidth of the control system, into a number of modes used for linear analysis. These modes are then included in the linear model of the system as a series of second-order systems of the form: The parameters Ki and q in Eq. (1) are the modal gain and frequency of mode i, respectively. The parameter 5 is the damping ratio of the flexible modes, which is conservatively set to 0.001 (this translates to an impulse response that takes approximately 5.c = 5/sq = 5000/q seconds to die out‘). With the flexible modes in place, stability analysis is performed using the s h e desired gain and phase margins as used above.

Along with determining the stability of a system, generally using frequency-domain techniques such as Bode or Nichols plots, the performance of a system must be determined using time-domain simulations of the system. Generally, three types of simulations are created and used during a project for analysis and design. The first of these is a simple linear simulation, using the same linear model as that used for frequency-domain analysis. Timedomain performance of such a model can be calculated analytically by the same design package that is used to create Bode or Nichols plots, and is typically used as a first cut during the early stages of the design.

Aside from simple linear models of a system, nonlinear models and simulation tools are used with both low-fidelity (LoFi) and high-fidelity (HiFi) models of a system Low Fidelify Simulations. Low-Fidelity (LoFi) simulations, like simple linear models, are typically used during the early stages of an ACS design and analysis effort to broadly characterize the performance of the system. LoFi simulations differ from linear models in that they typically include some nonlinear elements, particularly mixed discrete and continuous systems to model the physical environment and dynamics of a spacecraft along with the discrete controller. ’ Typically, the LoFi simulation is used during the early design stages when there is likely to be a lot of iteration back and forth between design and performance verification, and so a quick turnaround is highly desirable. The intent of the LoFi is to show broad performance characteristics, enough to verify that the design can meet the pointing, slew rate, and other criteria.

H i g h Fide& Simulations. Whereas a LoFi simulation is primarily a design tool, a high-fidelity (mi) simulation is mainly used for design verification. HiFi simulations are created by including as much detail into the simulation as possible, including such things as sensor and actuator performance and noise models, environmental disturbances,quantization error, and even the ability to model non-ACS specific events such as ACS processor warm and cold restarts. Also, as will be discussed in the ACS Flight Software section later in this paper, automatic code generation tools can even allow the HiFi to be used to generate actual flight software.

Figure 3 shows an example of a LoFi and HiFi spacecraft simulation. Notice that the LoFi shows a fairly accurate portrayal of the “macro” performance of the control system. The HiFi, which is plotted on a much tighter scale, gives a better view on the “micro” level, showing the details of the system response i n the presence of noise and other disturbances.

Momentum Management Design and analysis of the momentum management component of the ACS begins in the initial study phase of a spacecraft design, and continues as the spacecraft design matures and the mathematical models used for it continue to be developed. The following considerations and rules of thumb are used in its design: 1. Identify the external torques that contribute to a build-up of system momentum. Typically, these torques are gravity gradient, aerodynamic, and solar pressure. In low-Earth orbits, gravity gradient and aerodynamic torques are the most significant; at higher orbits, the most significant external torque is usually from solar pressure.

Observing Mode Sunline Angle 0 100 300 500 Time (sed Figure 3 Low- and High-Fidelity Simulations 2. Select a momentum unloading control law. The two control laws generally used are referred to as “B dot” and “ H x B . B dot does not require any hardware other than magnetic torquer bars and a magnetometer, but leaves a residual spacecraft body rate equivalent to one or two revolutions per E a r t h orbit, and will not dump momentum stored in the wheels. HxB, on the other hand does not leave the spacecraft with a residual r a t e , but it also requires system momentum information (typically from gyros and reaction wheel tachometers).

3. Using past missions in a similar orbit, analysis of worst-case environmental disturbance torques, and LoFi simulations as a guide, make a first cut at magnetic torquer bar sizing.

4. Select torquer bar sizes including 100% margin. Inertia ratios, and other parameters that influence the spacecraft response to environmental torques, will typically rise by more than 30% from the initial sizing studies to launch.

5. Test the final design using the HiFi simulation. It is best to use a 36-hour test, so that a full 24 hours is covered to allow the E a r t h magnetic dipole to rotate in inertial space, with an additional six hours of overlap on each side.

A similar approach, using similar margins, is taken when magnetic momentum unloading impractical or impossible. Two other possible approaches to momentum unloading are using thrusters (which is very quick and efficient, but requires expendable fuel) or, in higher orbits where the dominant disturbance torque is solar pressure, by trimming the orientation of the solar panels with respect to the sunline.

ACS HARDWARE As the ACS is being designed by analysts, other ACS subsystem engineers are selecting and procuring the hardware to be used to implement that design on the spacecraft. The hardware needed for the ACS 1 1 0 subsystem can be divided up into three categories: flight hardware shared by the ACS and other spacecraft subsystems, the ACS sensor, actuator, and other electronic hardware, and the hardware needed on the ground to integrate and test the ACS system.

Depending on the design of a given system, the ACS will usually be implemented within the main spacecraft processor, within a separate attitude control electronics (ACE) box, or sometimes both. The main processor, along with the hardware and software that comprise the command and data handling (C&DH) subsystem, are integral parts necessary for the successful operation of the ACS. Similarly, the attitude control electronics, used as an interface between the control algorithms implemented in either the main processor or a dedicated ACE processor and the sensor and actuator hardware. Also, instrument data is sometimes used for fine position sensing within the ACS.

Flight Sensor Hardware There is a wide variety of ACS sensor and actuator hardware available for spacecraft missions-each piece of hardware has different characteristicsof performance, cost, lifetime, and other criteria, that make it applicable to a subset of the possible missions. In this section of the paper, we will give examples of some of the major types of sensors and actuators. At the end of the section we will show a table containing some representative mass, power, and performance numbers for some of the sensor types discussed.

Earth Sensor: Earth sensors detect the Earth’s horizons (actually, an infrared radiation band from the CO, layer slightly above each horizon) as seen from space to provide two axes of attitude information with respect to the geodetic nadir vector from the spacecraft to the Earth. Earth sensors cannot measure the yaw angle about this vector. These sensors are very often used as the primary attitude sensors for Earth-pointing spacecraft; even when the pointing requirements of the mission exceed the capabilities of an Earth sensor, they are often used for Earth acquisition after launch and other maneuvers.

There are two general classes of Earth sensors. The first type, scanning Earth sensors, use a moving optical head to detect where the horizon is. Static Earth sensors, on the other hand, are built to operate at a given altitude range and have a fixed field-of-view designed to intersect the Earth horizon at one or more points. In general, scanning sensors are heavier, use more power, and cost more (in addition to causing attitude disturbances that may affect the science payload), but also have a larger range of operation and greater pointing performance.

D i g i t a l Sun Sensor: Digital sun sensors are generally used to detect the orientation of a spacecraft with respect to the vector from the spacecraft to the sun. Like an Earth sensor, sun sensors cannot measure the orientation of the spacecraft about this vector. Digital sun sensors can be used to give two axes of information, and are generally used with other attitude measurements as part of a general attitude determination algorithm, or to provide attitude updates to a Mman filter.

Inertial Reference Unit: Inertial reference units are gyro/accelerometer packages that can be used as an angular rate and acceleration sensor (to save cost, weight, and power, the accelerometers are often left out, unless there is a structural resonance issue). They can also be used to propagate an attitude estimate between measurements from an attitude sensor. Gyros are available in a wide range units, varying considerably in mass, power, and capabilities.

Star Trucker: Star trackers are use to detect and track stars. By correlating measurements of the line of sight vectors to multiple stars, the spacecraft’s current orientation with respect to an inertial reference frame can be determined (the particular inertial frame used depends on that used to specify the star positions).

Most star trackers currently operate by providing information to the spacecraft processor about line-of- sight vectors and magnitudes to detected stars. Algorithms in the processor, using an onboard star catalog, identify the stars detected and use their defined positions to calculate the spacecraft attitude. In the event of a spacecraft where this information is only needed for attitude knowledge, and not attitude control, this processing may also be done on the ground. However, some of the newest star trackers now becoming available are so-called “quaternion output” trackers-they have their own processor and built-in star catalog, and can directly output a quaternion expressing the orientation of the star tracker boresight with respect to an inertial reference frame.

0.25” accuracy 45” field of view 0.017” accuracy 0.0039” resolution Inertial Reference Unit 1.6-5.8 7.5-12 0.1-1 arcsec/pulse (two-axis unit) Star Tracker 8.1 12 8x8” field of view 3 arcsec accuracy Other Sensors: In addition to the sensors discussed above, which are the typical sensors usually used by the ACS mission mode controller, there are a number of other sensors that are generally included or can be used on a spacecraft. These sensors fall into two general categories: First, there are sensors used for a special purpose on a spacecraft, other than a normal mission mode control. Two examples of this would be the course sun sensor, usually used as a part of a minimum hardware sun acquisition control mode, and the three-axis magnetometer, usually used along with magnetic torquer bars for momentum unloading. (It is interesting to note that algorithms, such as the “Contingency Mode” developed for the TRMM spacecraft, have been developed and implemented that use magnetometer measurements, along with measurements from other sensors, such as digital sun sensors, to generate a mission mode attitude estimate.)

Second, there are a number of new technology attitude sensors that are in development and/or a t the experimental stage. These include the use of GPS signals in a differential mode to generate attitude information, and a number of integrated star trackerhnertial reference unit sensors that generate both attitude and attitude rate information.

Flight Actuator Hardware Most spacecraft use one of two types of actuators for most of their flight actuator requirements.

Propulsion systems and thrusters are included on a spacecraft to perform orbit maneuvering and stationkeeping, as well as for momentum management on spacecraft where magfietic torquer rods are not sufficient (or usable, for non-Earth orbiting or high-Earth orbiting spacecraft). The ACS groups work closely with the propulsion groups at Goddard to design a thruster system, where needed, that will satisfy all spacecraft requirements for attitude, orbit, and momentum management. In general, though, most spacecraftrequire reaction or momentum wheels as their primary actuator.

Reaction Wheel: A reaction wheel is used to apply or remove a rate from a spacecraft by making use of the conservation of angular momentum. When torque is applied to a reaction wheel to get it to change its rotation rate about a given axis, a corresponding torque and angular rate change is generated in the spacecraft in the opposite direction. Two of the most important ways that reaction wheels are characterized is through their momentum capacity and their torque authority-in general, larger wheels have more capacity and can generate a larger torque, at the expense of more mass, power, and cost.

Power (W) Performance Mass (k) Reaction Wheel 2.55 5.5-9 (orbit average) 0.012-0.02 Nm torque authority “Type A’’ 25 (peak) 4 Nms momentum capacity Reaction Wheel 10.5 15-40 (orbit average) 0.3 Nm torque authority “Type E” 280 (peak) 50 Nms momentum capacity to reaction wheels; generally, they provide lower torque, but have a omentum wheels are similar higher efficiency. Lastly, in addition to thrusters and wheels, magnetic torquer bars are often used by spacecraft in low-Earth orbit for momentum management.

In addition to the flight hardware that must be acquired to implement an ACS design on a spacecraft, there is a large amount of additional hardware that is required for ground test purposes. Figure 4 shows an example of the hardware used for a hybrid dynamic simulator (HDS), used to provide ground-test capability for a spacecraft, both software algorithms and flight hardware, from low-level testing through final integration and test.

ACS Figure 4 HDS Hardware for ACS Testing Multiple hardware test setups are typically required for many missions, allowing the ability to concurrently test the main spacecraft processor, the independent attitude control electronics (if present), as well as the abiiity to stimulate all flight hardware and hardware interfaces. The production of breadboard and engineering test units @TU) for many of the flight components adds to the cost and effort involved.

Depending on how the test effort is scheduled, very often completely different test "smngs" are required to allow for testing of ACS and C&DH functionalityseparately. When all of this additional hardware required for the testing effort is considered, it becomes a significant fraction of the cost and effort associated with the flight hardware.

ACS FLIGHT SOFTWARE There are three major phases in the production of ACS flight software used to implement the control laws developed to support the spacecraft mission. The fmt phase bridges the gap from the ACS design described above to the flight software developers, usually through the means of an algorithm document.

Next, the ACS algorithms, along with all of the other necessary flight software functions, are developed for the specific processors and hardware architecture selected for the spacecraft. Finally, perhaps most importantly, software (and hardware, as discussed above) is developed and run to fully test the functions of the flight software.

For the TRMM project, ACS analysts converted the algorithms needed to implement the controller into was then provided FORTRAN-like syntax used in the ACS Algorithm Document3. Thi routines necessary to the ACS flight software develop used it as a guide to develop the to implement the algorithms on the spacecraft main and ACE processors.

One of the lessons learned from the TRMM experience is that the FORTRAN vs C syntax used for the algorithm document and flight software, respectively, hampered the ability of the developers to correctly interpret the algorithm document,. as well as making it more difficult for the ACS designers to verify the correctness of the code. Using similar syntax-which in this case would have meant using C-like syntax in the algorithm document-would have made the process a lot easier.

Taking advantage of some of the advances in the technology of the control system design tools, a slightly different approach to that used by TRMM is now possible. Products such as DocumentIt fiom IS1 allow documentation to be generated automatically fiom a system model, such as a HiFi simulation. The documentation generated for the MAP ACS uses the software tool’s ability to create output meant for display on the Web.

The ability to automatically generate documentation is only a small part of what is becoming possible with the newest generation of ACS design tools. As will be discussed in the next section, there are several tools available now that will automatically generate flight software fiom a system model, thus skipping the algorithm to algorithm document to flight software translation process altogether.

Automatically-Generated Code Figure 5 shows an example block diagram from the MAP HiFi simulation, depicting the Observing Mode controller. Instead of translating this controller into a written algorithm, which would then be coded by the ACS flight software developers, by using ISI’s AutoCode ACS development tool, flight code for this controller can be automaticallygenerated.

There are a number of potential benefits to automatic code generation. primarily, once confidence in the code-generation tool is established, it should strengthen the testing effort. Because it is possible to generate some of the flight code much more quickly using this method, it is possible to begin testing earlier, and thus test more thoroughly. Also, because the algorithms are proven within the simulation environment first, before the code is generated, a great deal of lower-level testing can be avoided.

The MAP program is one of the first at Goddard to use automatically generated flight software.

Because of this, the scope of what parts of the ACS were chosen to be Autocoded was limited; even so, approximately 1/3 of the MAP ACS flight software was automatically generated. The consulting group at ISI, which has a lot more experience using automatic code generation, and the IS1 tools in particular, has reported even greater percentages of automatically generated code and improvements of the flight software design cycle for the MSTI-1, MSTI-2, and MSTI-3 programs4.

Development vs Testing Based on the experiences that we have had across many missions, it is safe to say that the flight software testing effort is always underestimated. Just as software is developed to implement ACS algorithms for flight, software test procedures are needed to test that software throughout the development and integration and test phases of the project. The relative amount of effort needed for ACS flight software development vs testing is roughly a 50/50 split between the two. This includes the manpower and time to develop the software, either flight or test, and to run the tests; it doesn’t include ACS analyst support, but that also tends to be pretty evenly split between the two activities.

The relative effort assessments discussed above do not include that involved for configuration management and maintenance of a system for tracking ACS parameters and for filing discrepancy reports (DRs) for the different ACS subsystem components. One of the lessons learned from past programs that is currently being applied is the use of a central database for maintaining the DRs and parameters used throughout the ACS subsystem: HiFi simulation, flight software, Because of this, it is fair to consider this effort as a general overhead important to both the deve nd testing effort.

Each spacecraft goes through a number of design reviews, each of which has a different emphasis and reflects a different stage in the design process. While the specific names of the reviews for each project may vary, the general topics covered at each stage tend to be the same. Some of the typical reviews and topics covered, as they relate to the ACS subsystem, over the course of a project are discussed in this section.

There is some overlap from one review to another, and a number of topics that will be covered in each.

In particular, every review will discuss design changes (and their impacts) and action items (and their resolutions) since the last review, as well as outstanding issues and concerns.

Figure 5 MAP Observing Mode Controller to be AutoCoded Preliminary Design Review (PDR) The first formal review is usually the Preliminary Design Review, or PDR. This is typically conducted after enough time has passed for a first cut at an ACS subsystem.design has been done. The purpose of the initial design and review is to develop a preliminary design for the complete ACS subsystem and to determine the viability of the mission concept. If the mission has any “show stoppers”-mission requirements that cannot be met within the budgetary, time, or other constraints imposed on the project- they should be identified by the PDR. Also, major design issues should be identified. A list of topics typically covered in a PDR is as follows: 1. Requirements 2. Subsystem Analysis Anal rformed ACS

- Descriptions

- Stability

- Performance

Other Topics (Attitude Determination/Error Budgets, Momentum Management) 3. Hardware 0 Procured Components 0 In-House Development 4. Software 5. Operations 6. Testing Critical Design Review (CDR) The Critical Design Review (CDR) is the last subsystem-specific review conducted. By the time it is conducted, the ACS design should be completed, analyzed, and tested to the point that it can be said and demonstrated with confidence that the design will meet all requirements. Issues identified at the PDR should be addressed and closed at the CDR. The design of other parts of the spacecraft should have matured by subsystem CDR so that flexible mode analysis is possible. The CDR will cover much of the same material as the PDR, though the design should be more mature and finalized.

1. Requirements 2. Subsystem Analysis status

- Stability Analyses

- Flexible Mode Analyses

-

ACS Mode Description and Performance Summaries Other Topics (Attitude DeterminationErrorBudgets, Momentum Management) 3. Noncompliance SummarylOpen Issues 0 Hardware 0 Software 4. Failure Detection and Correction 5. Operations 6. Testing Once all subsystems have had their CDRs, there is a spacecraft-level CDR that covers the complete spacecraft, at a higher level. It is through this review, either on its own or with a separate confirmation review, that it is decided whether or not the spacecraft design is complete and good enough to justify continuing with the integration and test process.

Pre-Environmental Review (PER) A Pre-Environmental Review (PER) is conducted immediately prior to the full spacecraft going into environmental testing. The purpose of the review is to ensure that all outstanding issues from previous reviews and I&T have been closed, and that a plan is in place for successfully testing spacecraft aliveness and functionality as it goes through the different environments. Items typically covered in a PER include: 1. Hardware Qualification and Testing 0 For Each Component:

-

Hardware Description and Part Number

- Component Specification Document

-

Manufacturer Environmental Testing and Total Running Time

- Delivery Date

Fun Electrical Integration Procedure Performance Tests pacecraft Level Testing Aliveness, Functional, and Phasing Tests Comprehensive Performance Testing CPT) 0 Dynamic SimulatorTesting 3. Is the Spacecraft Ready for Environmental Testing?

Pre-Ship Review (PSR) A review is conducted before a spacecraft is shipped to its launch site to determine if it is ready for launch. This Pre-Ship Review would cover the following types of topics: 1. Events since PER * Spacecraft Tests w/Dates

- Results

- Completion Status

0 Right Hardware Operating Hours 0 Anomalies Since PER

- Problem

- Cause of problem

- Current status

2. Is the SpacecraftReady for Launch?

CONCLUSION This paper has attempted to cover a lot of material very broadly, to give an idea of what is involved in one approach to a successful spacecraft design. It is by no means the only approach to how to design a spacecraft, nor is this paper meant to be a detailed blueprint of the design process. It is, instead, a summary of some of the experiences that we'have had at Goddard over the years, and is hopefully the first step in a larger effort to document the design expertise currently present in the Guidance, Navigation, and Control Center.

ACKNOWLEDGEMENTS The authors would like to gratefully acknowledge Robert Spagnuolo, Otilia Rodriguez-Alvarez, Debbie Henretty, Maureen Bartholomew, Nelson Rubin, and Martin Frederick, a l l members of Goddard's Guidance, Navigation, and Control Center, for their assistance in the preparation of this paper.

REFERENCES 1. G. E. Mosier and J. W. Croft, "XTE Linear Analysis Objectives Plan", Goddard Space Flight Center, Code 71 2, Internal Memo.

2. R. J. Richards, "An Introduction to Dynamics and Control", Longman Group Limited, London, 1979.

3. J. Bracken, J. San, J. DAgostino, and K. Barnes, "TRMM ACS Algorithm Documenr, Goddard Space Flight Center, Code 712, Build 5.5, September 16, 1997.

4. F. Tubb, R. McEwen, J. Farazfan, and A; Waddell, "MSTIS Spacecraft Attitude Control Software Development using Automatic Code Generation", Integrated Systems, Inc.

ce Accommo~ating §l~ding ode Controller

for Spacecraft Attitude Maneuvers

Jongrae Kim*, Jinho Kim+, John L . Crassidid I n the absence of an external disturbance and uncertainty, sliding mode (variable structure) control is guaranteed to aspptotically stabilize a system, which is provided by using a correction control input dculated using a Lyapunov-type condition Le., sliding mode existence condition. When bounded unmodeled external torques are added, the closed-loop system is no longer globally asymptotically sta- ble since steady-state errors are present. T h e error can be mhimhed by increasing the correction control gain or decreasing the thickness of boundary layer of sliding mode control. But for limited actuator capabfity the maximum control gain and the minimum thickness of boundary layer being allowed may be restricted.

Disturbance accommodating control is a signal synthesis adaptive control. For a short time interval the disturbance is assumed to be modeled by a linear combination of previously selected basis func- tions. A disturbance accommodating observer can be used to iden- tify unmeasurable internal and external disturbances. In this paper, sliding mode control is combined with disturbance accommodating control (i.e.; Disturbce Accommodating Sliding Mode Control) in terms of modified Rodrigues parameters for a spacecraft attitude reg- ulation and tracking maneuvers. The presented disturbance accom- modating sliding mode control has the following advantages: 1) the design procedure is more effective than the traditional sliding sur- face stabilizing problem since steady-state errors are reduced, 2) the designed disturbance accommodating observer is linear, and 3) the robustness of sliding mode is guaranteed in the range of actuator capability. Simulation results are shown that use the disturbance ac- commodating sliding mode control to reduce steady-state errors in the case of applied external disturbances.

INTRODUCTION Spacecraft attitude control for large-angle slewing maneuver poses a difficult problem, including the nonlinear characteristics of the governing equation, modeling *Graduate Student, Dept. Mechanical Eng., The Catholic University of America, 73gimOp1uto.ee.cua.edu t A d a t e Professor, Dept. Aerospace Eng., Inha University, Inchon, Korea, jhkimOdragon.inha.ac.kr %Assistant Professor, Dept. Mechanical Eng., The Catholic University of America, jlcQpluto.ee.cua.edu 119, (gibbs vector). The Rodrigues parameters provide a minimal (Le., three-dimensional) parameterization. However, the Rodrigues parameters have a singularity for 180 deg rotations. Vadali presented an optimal sliding manifold using error quaternions.2 For large angle maneuvers, quaternion feedback was presented by Wie and B a r b z ~ ~ A quaternion feedback regulator was a l s o presented by Wie, Weiss and Arapostathis.4 Quaternions are nonsingular for any rotation, however, the use of quaternions re- quires an extra parameter that leads to a nonminimal parameterization. Crassidis m d Markley developed a sliding mode controller for regulation and tracking problems of spacecraft attitude control based on the modified Rodrigues parameter^.^ The ad- vantages of using modified Rodrigues parameters include the following: 1) rotations up to 360 deg are possible, and 2) the parameters form a minimal parameterization?

Therefore, in this paper, sliding mode control based on modified Rodrigues parame- ters is adopted. All of the above control laws are robust with respect to variations in the moment of inertia tensor on the order of 10 - 20 %.6 One of the drawbacks of sliding mode control is the chattering problem due to dis- turbance and modeling imprecision. For spacecraft attitude control, chattering may be excite the higher frequencies of spacecraft and cause structural failure. Chattering can be settled by smoothing the control input using boundary layer or bandwidth- limited sliding mode control, which was presented by Dwyer and Kim7. However, a globally suitable boundary layer thickness cannot be easily determined. Moreover, for spacecraft attitude control it may be difiicult to predict the external disturbances acting on body. When bounded unmodeled external torques are added, the closed- loop system is no longer globally asymptotically stable since a steady-state error is present. The error can be minimized by increasing the correction control gain or decreasing the thickness of boundary layer of sliding mode control. In this paper we derive this relation using a Lyapunov function. But for limited actuator capability the maximum correction control gain and the minimum thickness of boundary layer being allowed may be restricted. Though the steady-state errors are usually small, in a high-precision attitude pointing or tracking systems, these errors may not tolerable for satisfying a mission requirement.

In this paper, we adopt disturbance accommodating control to minimize steady- state errors in sliding mode control. The disturbance accommodating control concept was first proposed by Johnson?Jo External disturbances w(t) are a s s u e d to sat- isfy P+'w(t)/dtm+' = 0 differential- equation where the external disturbam& are represented as mth-degree polynomials i n time t with unknown coefficients.1o Design procedures and existence of the disturbance observer are presented in (Ref. 11, 1 2 ) .

disturbance accommodating observer es linear behaviors in the responses e accommodating observer include the following: 1) it is linear, and 2) it also compensates the error due to modeling uncertainty.

Combining sliding mode control with a disturbance accommodatingobserver (i.e., Disturbance Accommodating Sliding Mode Control) was presented by Kim, and was applied to a robot manipulator for reducing the upper bound of bandwidth of slid- ing mode contr01.l~ In this paper sliding mode control based on modified Rodrigues parameters is adopted for spacecraft attitude control. Also, a disturbance accom- modating observer is combined with sliding mode control for reducing steady-state errors due to external disturbances. Simulation results that use the disturbance ac- commodating sliding mode control to reduce the steady-state error are shown for the case of regulation and tracking maneuvers.

The organization of this paper proceeds as follows. First, a brief summary of the kinematics and dynamics of a spacecraft is presented. Then, a brief overview of the sliding mode control based on modified Rodrigues parameters is shown. Next, a robust analysis of the sliding mode control with respect to external disturbances is accomplished using a Lyapunov function. A disturbance accommodating observer is derived for reducing the steady-state error. Also, sliding mode control and disturbance accommodating observer are combined. Finally, simulation results are shown for regulation and tracking of a spacecraft.

PROBLEM FORMULATION In this section, a brief review of the kinematic equations of motion using modified Rodrigues parameters, the rigid body dynamics, and sliding mode control based on the kinematics is shown.

Attitude Kinematics and Dynamics The modified Rodrigues parameters are defined by5 p li tan (814) where p is a 3 x 1 vector, i i is a unit vector corresponding to the axis of rotation and 6 is the angle of rotation. The kinematic equations of spacecraft attitude motion described in modified Rodrigues parameters are derived by using the spacecraft's angular velocity (o), given by5

P = 1/4 { ( 1 - P ' P ) 1 3 x 3 + 2 [px] + 2 P P ' } o (2)

The dynamic equation of motion for a rigid body with external disturbance (w) is given by Euler's equation, defined by

Lj = J-' [Jwx] w + J-'U + J-'w

(4) where, J is the spacecraft's inertia (3 x 3) matrix, J-l is the inverse matrix of J, and u is the control input torque (3 x 1) vector.

Sliding Mode Control In this paper it is assumed that measurements of both the spacecraft, attitude and angular rate are available and the dynamics of actuator is neglected. The nonlinear model for spacecraft motion is summasized by5

W = f (0) + J-'U + J-lw

(6) where

1/4 { ( 1 - PTP) 13x3 + 2 bxl+ 2 PP*}

(7) F (PI f (0) E J-' [ J w x ] ~ (8) Sliding mode control introduces velocity vector fields directed toward the sliding sur- face or madold ( s = 0) in its immediate vicinity, where s is given by' s=w-m(p) (9) The quantity m(p) is defined using a desired vector field from the kinematic equation, given by1 4 P ) = F-'(P)d(P) (10) where

F-'(p) = 4 (1 + P*P) { (1 - PTP) 13x3 + 2 bxl+ 2 PP*}

(11) The quantity d(p) is formed by allowing a linear behavior in the sliding motion, given by5

d(P) = (P - Pa) (12)

where pa is the desired reference trajectory and A is a diagonal matrix with negative elements. The input by sliding mode control is divided into two parts. The first is the equivalent control ueq for satisfying the ideal sliding mode conditions (Le., invariant e dm

f(o) - -F(p) [m(p) + s]

dp u , = - - J K S U t ( S , E ) (15) where K is a 3 x 3 positive definite diagonal matrix. The saturation function is used to minimize chattering in the control torques. The function is defined by The detail descriptions of the quantities m(p) and dm/& for the regulation and the tracking problems can be found in (Ref. 5).

CONTROL DESIGN In this section a robust analysis of the sliding mode control with respect to a external disturbance is accomplished using a Lyapunov function. A disturbance accommodating observer is also derived for reducing the steady-state error. Finally sliding mode control and disturbance accommodating observer are combined.

Robust Analysis of Sliding M o d e Control We use the following candidate Lyapunov function V to study global stability of the motion by sliding mode control.8 V = -sTJs

Define an error torque Aw using an estimated external disturbance + and the actual

external disturbance through*

AW = + - w

The first time derivative of the candidate Lyapunov function with the control input reduces to8

V =: - s ~ J K s u ~ ( s , E ) - sTAw

(19) this system is Note that in the absence of an external disturbance estimation error, guaranteed to be globally &ymptoticaUy stable. If bounded unmodeled disturbances are added, but not compensated for in the control law, the system is no longer asyrnp totically stable. If K is large enough so that sTJKsut(s, E) is larger than sTAw, then at the thickness of boundary layer E is sufficiently small and the cor- rection control gain K is sufficiently large to keep the time derivative of Lyapunov function negative-definite with bounded external disturbances in the region of the outer boundary layer. In the boundary layer the dynamics of sliding function is given bv .I K s=-- S - J - ~ A ~ E If the estimation error of external disturbance settles to a d u e and the sliding func- tion s must settle to a finite constant steady-state value sss. Setting the derivative in the dynamics of sliding function to zero we obtain8 K

o = ---sSs - J - ~ A ~

E Therefore the steady-state value of sliding function (Le., tracking error) will converge to the following finite offset8 € sSs = --J-~Aw K The tracking error will not converge to zero but to a finite ofbet. T h i s o h t can be reduced to fall within acceptable limits by decreasing the boundary layer thickness 6 or increasing the correction control gain K. However, decreasing the boundary layer or increasing the correction control gain will limit the error recovery performance by saturating the actuator or will cause high frequency chattering in the actuator! For high-precision attitude tracking, this small error offset or the high gain may not be acceptable. The steady-state error can also be reduced by making Aw smaller.

Disturbance Accommodating Observer The uncertainty associated with some internal and external disturbances w(t) is represented by a semideterministic waveform-model description of the generalized splinefunction type, given by16

where the basis functions fl(t), f&), - fm(t) are completely known and the con-

s t a t weighting coefficientvectors c1, c2, . * * cm are totally unknown and may jump

in value &om time to time. Without loss of generality, it is further assumed that the basis functions f&) satisfy a linear differential equation. As a consequence, there exists a linear dynamical “state model” representation as follows:16 w(t) = W(t)Z 1 24 1 . coulomb and other complex forms of nonlinear damping 2 . uncertain external input disturbances 3 . plant parameter model errors 4. coupling effects in reduced-order state models The basis functions can be chosen a s power series in time t or as orthogonal poly- nomials commonly used in approximation theory.16 The design procedure and the existence problem of the appropriate observer with the stabilizing gain was shown in (Ref. 1 2 ) .

Disturbance Accommodating Sliding M o d e Control In this paper we divide the control input into the equivalent control input uq and the correction control input u , of the sliding mode control and the disturbance ac- commodating control input uhc for canceling the effects of external d i ~ t u r b a n c e s . ~ ~ , ~ ~ After applying the control input to the dynamics of the sliding function, the dynamics and the disturbance model can be written in the following state-space f ~ r m : ' ~ ~ ' ~

S = J - l u , + J - l u d , + J-'w

2 = D ( t ) z + a ( t ) w = H ( t ) z The appropriate disturbance accommodating observer is given by12 4 = D(t)ii-K0(2-4) f = H(t)P where, KO is the observer gain ( 9 x 9) matrix w h i c h provides sufficient time constants in the observer. We adopt the three basis functions a s 1, t, t2 for each body axis (i.e., i = 1, 2 , 3 ) .

~ w & ) = e1 + et + c3t2 (33)

A l l the matrices in the observer are constant, however, the observer in Eq. (31) cannot be directly implemented due to the unmeasurable state z. Define a new state variable Q a s follows:17

Q = 4 - K ~ s

(36) where K1 is a gain matrix (9 x 3 ) . The gain K1 can be tuned to satisfy the following condition:17

K O + KIH = 0

(37) Finally, the modified observer composed by the measurable or known states is derived as follows:18 where the initial condition is given by Q(0) = -Kls(O). Then, the estimation error dynamics becomes18

AQ - (D + KO) AQ = -a($)

(39) where

AQ = (2 - K ~ s ) - (Z - K ~ s )

I f the gain KO is large enough so that the error dynamics is stable and converges fast, then the tracking error offset is reduced. The designed observer is linear and it can be easily implemented in digital software. One of drawbacks of the observer is that the sensor noise is amplified by the gain at the output of the observer. In this case we cannot use the reduced observer form, and have to implement a observer to estimate s.

the state A brief description of the control and system is shown in Figure 1. The estimated states I, % and u & are calculated by the following r e l a t i ~ n : ~ ~ , ' ~ 1 2 6 Figure 1 System Block Diagram SIMULATION The inertia matrix of the simulated spacecraft is given bf

J = diag [ 114 86 87 ] [kg m2]

(44) The initial conditions for the angular velocity are set to zero. The bpundary layer thickness e: in the saturation controller is set to 0 . 0 1 : Also, the control torques are limited to 1.0 N-m? The simulations are performed by Runge-Kutta 5 method in simulink in MATLAB with a maximum step size of 1 sec, minimum step size of 0.0001 sec and a tolerance 1.0 x The external disturbances applied to each body axis are set to 0.3sin(t/lO) N-m. The observer gain Koi for each body axis (i.e., i = 1, 2, 3) is calculated using a pole-placement method as the following: -30.0 0 0

KOi = [ -300.0 0 0 ]

(45) -1000.0 0 0 Simulation cases for the regulation and tracking problems are given by 1. Case A: Sliding Mode Control without the external disturbances B: Sliding Mode Control with the external disturbances 2. Case 3. Case C: Disturbance Accommodating Sliding Mode Control with the external disturbances Regulation The initial conditions for the modified Rodrigues parameters are given by T

.p(O) = [ - 0 . 1 0.5 1 . 0 ]

(46) The rotation for the initial conditions is approximately 206 deg. The diagonal el- ements of the correction control gain K are all set to 0.0015 and the constant X 1 27 maxim= estimation errors for each body axes are smaller thm 1.7 % with respect to the maximurn external torques.

0 1 2 3 4 5 6 time [min] Figure 2 Regulation: Sliding Function Trajectories

$ -1 0 1 2 3 4 5 6

5 x loJ

V I 1

I -5' 1 2 3 4 5 6 0 ~~ 1 2 3 4 5 6 * O time [min] Figure 3 Regulation: Estimation Errors Aw pdl = 0.05 sin(0.005t) pd2 = 0.05 sin(0.006t) p k = -0.05 ~in(0.007t) The sliding function trajectories for each case are shown in Figure 4. As shown in Figure 4, the trajectory of Case B oscillates up and down through the trajectory of Case A. In Case C, when disturbance accommodating sliding mode control applied the trajectory is almost the same its the one of Case A. The estimation errors, Aw, are shown in Figure 5. The maximum estimation errors for each body axes are smaller than 1 7 % with respect to the maximum external torques.

Figure 4 Tracking: Sliding Function Trajectories CONCLUSION A method for compensating the steady-state error of sliding mode control due to external disturbance was presented and applied to spacecraft attitude maneuvers.

The presented disturbance accommodating sliding mode control include the follow- ing advantages: 1) the des& procedure is more dective than the traditional sliding surface stabilizing problem since steady-state errors are reduced, 2) the designed disturbance accommodating observer is linear allowing the use of many design and 0.Wf

- I

d o~ 0 2 4 6 8 10 12 14 16 18 20

-0.05 Figure 5 Tracking: Estimation Errors Aw REFERSNCES 1. Dwyer, T. A. W., and Ramirez, H. S., “Variable-Structure Control of Spacecraft Attitude Maneuvers”, Journal of Guidance Control & Dynamics, Vol. 11, No. 3 , 1988, pp. 262 - 269.

2 . Vadali, S. R., “Variable-Structure Control of Spacecraft Large-Angle Maneuvers”, Journal of Guidance Control & Dynamics, Vol. 9, No. 2, 1986, pp. 235 - 239.

3 . Wie, B., and Barba, P. M., “Quaternion Feedback for Spacecraft Large Angle Maneuvers”, Journal of Guidance Control & Dynamics, Vol. 8, No. 3 , 1985, pp.

360 - 365.

4. Wie, B., W e b , H., and Arapostathis, A., “Quaternion Feedback Regulator for Spacecraft Eigenaxis Rotations”, Journal of Guidance Control & Dynamics, Vol.

12, NO. 3, 1989, pp. 375 - 380.

5. Crassidis, J. L., and Marlcley, F. L., “Sliding Mode Control Using Modified F b drigues Parameters”, Journal of Guidance Control & Dyna;mics, Vol. 19, No. 6 Engineering Notes, 1996, pp. 1381 - 1383.

McDuffie, J. H., and Shtessel, Y. B., “A Sliding Mode Controller and Observer for 6 .

Satellite Attitude Control”, AIAA Conference 1997,A.IAA-97-3755,1997, pp. 1613 - 1619.

130, nautical Sciences, Vol. 44, No. 3, Jul. - Sep., 1996, pp. 373 - 387.

9. Johnson, C. D., “Optimal Control of the Linear Regulator with Constant Distur- bance~’~, IEEE Transactions on Automatic Control, Aug. 1968, pp. 416 - 421.

10. Johnson, C. D., “’Further Study of the Linear Regulator with Disturbances -

The Case of Vector Disturbances Satisfying a Linear Differential Equation”, IEEE Transactions on Automatic Control, Apr. 1970, pp. 222 - 228.

11. Johnson, C. D., “Accommodation of Ekternal Disturbances in Linear Regulator and Servomechanism Problems”, IEEE Transactions on Automatic Control, Vol.

Ac-16, NO. 6, Dec. 1971, pp. 635 - 644.

12. Johnson, C. D., “On Observer for Systems with unknown and inaccessible In- puts”, International Journal of Control, Vol. 21, No. 5, 1975, pp. 825 - 831.

13. Johnson, C. D., “Adaptive Controller Design using Disturbance Accommodation Technique”, International Journal of Control, 1985, Vol. 42, No. 1, pp. 193 - 210.

14. Johnson, C. D., “Application of Linear Adaptive Control to Some Advanced

Benchmark Examples”, Proc. of American Control Conference, 1989, pp. 460 -

465.

15. Addington, S. I., and Johnson, C. D., “Dual-Mode Disturbance Accommodating Pointing Controller for Hubble Space Telescope”, Journal of Guidance Control and Dynamics, Vol. 18, No. 2, M a r . - Apr. 1995, pp. 200 - 207.

16. Johnson, C. D., “A New Approach to Adaptive Control”, Control and Dynamic Systems, Academic Press Inc., 1988, pp. 1 - 69.

17. Kim, J., “Disturbance Accommodating Sliding Mode Control ’), Proc. of Ameri-

can Control Conference, 1992, pp. 888 - 890.

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Dr. Hans Seywald*, r. Renjith Kurnar", Dr. Min Qu* ABSTRACT This paper introduces a nonlinear reorientation and attitude controller based on feedback linearization, Besides user-prescribed rnaximum and minimum gain values, the controller requires no tuning, and is applicable to arbitrary rigid spacecraft configurationS. The user needs to input only data pertaining to the physical problem setup, such as the spacecraft's inertia matrix, initial conditions, target orientation, target angular velocity, control constraint, and slew rate limit. Global asymptotic stability is guaranteed. The controller is computationally inexpensive, and numerical tests show excellent performance in terms of transient behavior and overall maneuver time.

* Analytical Mechanics Associates

IMPROVEMENT OF ORBIT DETERMINATIONAND PREDICTION Mina Ogawa', Maki Maeda', Mikio Sawabe', Masao Hirota', and Yousuke Yamamotott The Advanced Earth Observing Satellite (hereafter ADEOS) was was terminated on the launched on August 17th, 1996 and its Operation end of June 1997. TheADEOS carries a large aperture laser-refledor, referred as "ReBorenec~r-lnSpa"(RIS). In order to hit laser properly onto the RIS, we need a trajectory prediction with accuracy of about 100 m. The flightdynamics team at the NASDmACC (Tracking and Control Center) ordinarily derives a satellite trajectory with Range and Range Rate (RARR) measurements using S-band radio wave.

However,. the trajectory prediction is expectedto be only as accurate as 1 km for the ADEOS. This uncertainty is not acceptable for the RIS.

Although the operation of the ADEOS was terminated, NASDAkeeps on making every efort to improve the accuracy of trajectory determination and propagation of the ADEOS/RIS with Satellite Laser Ranging (SLR) data obtained at world-wide SLR ground stations. As a result we achieved a hundred-fdd and a ten-fold improvement on accuracy of orbit determination and prediction respectively.

This paper presents the summary of the experiment and the latest results. A brief discussion of the post-ADEOS mission plan will be found in this paper as well.

INTRODUCTION Satellite laser ranging (SLR) measurement with a pulse laser beam is one of means for measuring a distance between a ground station and a satellite. SLR measurement provides more accurate trajectory than that derived from the Range and Range Rate (RARR) measurement with radio wave.

The Advanced Earth Observing Satellite (ADEOS), launched into space on August 17, 1996, by National Space Development Agency of Japan (NASDA), and its operation was terminated on the called end of June 1997. One of the sensors on board the ADEOS is a hollow laser reflector Retroreflector-In-Space ( RIS ) (see Figure 1) provided by the Japanese Environment Agency ( EA 1 .

t National Space DevelopmentAgency of Japan (NASDA), Sengen 2-l-I,Tsukuba, Ibaraki, 305-8505, Japan t t Fujitsu Limited, Nakase 1-9-3, Mihama-ku, Chiba-shi,Chiba, 261-8588, Japan The effective diameter of the RIS is 0.5 rn (see Figure 1) and the large surface area enables it to reflect back a large fraction of energy throughput of a laser beam from a ground station without being suffered from satellite jittering. The ADEOG/RIS is originally designed to measure absorption spectrum in a reflected laser beam due to a small amount of Ozone, Methane, and other compounds found in the atmosphere. With the RIS on board the ADEOS, we plan to develop a prototype scheme of advanced trajectory determination system for future NASDA satellite missions.

The finalized scheme, called “Global and high accuracy Trajectory determination System” (GUTS) will be designed to utilize both SLR and differential GPS (Global Positioning System) measurement, and would be able to derive a satellite orbit within a accuracy of 25 ern when a satellite is in orbit at altitude of about 800 km. The finalized scheme will be in operation by the year of 2003.

Final goal of our system is a accuracy of 20 cm using SLR short arc data obtained multi-stations’ simultaneously.

Figure 1 RetrwflectaSIn-Space In this paper we discuss the followingitems: -- Data delivery system for the ADEOS/RIS.

-- Observations with the ADEOS/RIS.

-- Accuracy of trajectory determination using long arc data.

-- Accuracy of trajectory prediction.

-- Effects on accuracy of trajectory determination by each observation and force model.

-- Comparison of trajectory determination accuracy using SLR data and using RARR data.

In order to point a laser beam accurately to the RIS. we must provide high precision orbital parameters to a designated SLR station. Currently uncertainty of a satellite's position must be as small as 100 m for a successful SLR observation, and the orbital solution is also expected to be valid as long as a time interval of 3 days.

As a mean to deliver such a high precision trajectory ephemeris, the NASDA and the Communication Research Laboratory (CRL) have established a data delivery link for the ADEOS/RIS experiment (Figure 2). The CRL is responsible for distributing and archiving of every SLR data taken with the ADEOS/RIS, while the NASDA develops a scheme for estimating accurate trajectory of the ADEOS satellite and predicting its position prior to the next RIS observation.

First, SLR dataobtained at SLR stations around the world (Figure 3) are collected at the CRL through E-mail and/or anonymous file transfer protocol (FTP) via Internet Second, all of the SLR data are automatically transmitted to the Tsukuba Space Center (TKSC), NASDA, via the Earth Observation Center (EOC). NASDA, by using also E-mail and FTP through a network dedicated for this purpose (the data transfer rate is about 1.5 Mbps).

Then, after accumulating sufficient amount of the SLR data, the NASDA analyzes all of the collected data and make available estimated satellite's orbital parameters with high precision prior to the next SLR measurement.

These parameters are provided in a standard format for transferring Tuned Interrange Vector (TIRV) and time bias function. TIRV contains orbital elements estimated at a certain epoch, and predicted orbital elements derived at every 0 UTC. Time bias function is defined as a difference expressed in terms of time between the latest estimated orbital element and the latest observation. Using these information, SLR station is tracking the spacecrafts.

. The predicted orbital positions are weekly transmitted back from the NASDA to each SLR per week and station in a reverse order of the passage mentioned above.

TIRV is derived once time bias function is derived two times per week'.

If the SLR data collected in one week are not enough to determine the ADEOS orbit, TIRV should be derived from RARR measurement with S-band radio wave or 2-line elements received from the GODDARD Space Flight Center, NASA.

Communications Research Earth Laboliltory Tsukuba ^.

uoservauon Space Center (CRL) Center(E0C)

u - -

SLR Stations DATA Internet Circuit * Marithuesafety Agency CRL statlons (@hima, Koganei, Miura, Inushi) * International stations Orbit determination result report time bias function Figure 2 ADEOS/RIS Delivery Netwmk System A HALE.4LAl.A Figure 3 ADEOSMS SLR Netwcrk OBSERVATION WITH THE ADEOSlRlS The collected We have been collecting the ADEOS/RIS SLR data since October 30, 1996.

data number as of April 1.1g98, are 762 passes from 28 SLR stations, including 5 passes obtained after the end of ADEOS's operation. Some of the data were gathered from NASA Crustal mics Data Information System (CDDIS). Those data were not used for tracking ADEQSIRIS mly for analyses. Table 1 shows the summary of the ADEOG/RIS data obtained at each SLR on, from October 30. 1996 to June 30, 1997 and after the termination of the ADEOS operation.

1864 25 MAIDANAK 192 138.0 1868 15 KOMSOMOLSK 61 58.1 1870 MENDELEEVO 44 279 1 4 75.3 1873 SIMEIS 7 37 28.0 1893 KATSIVELY 2 9 12.3 FORT DAVIS 17 85 1 7 . 5 . .

7090 YARAGADEE 8 50 8.6 7105 WASHINGTON 22 142 1 12 10.1- 7109 QUINCY 4 23 7.3- 7110 47 MONUMENTPEAK 242 9.0 7210 MAUI 13 66 10.1 7236 WOHAN 1 4 8.9 7237 CHANGCHUN 52 414 58.6 7249 BWING 38 27 1 462 7308 TOKYO 7 68 53.1 7403 AREQurpA 13 53 8.7 7404 TLRS-2(SANTIAGo) 1 6 33.3 7548 CAGLIARI 11 55 14.7 7805 METSAHOVI 52 390 1 3 33.1 7810 ZflvIMERWALD I 15 1 .o 7824 S A N FERNANDO 43 447 1 6 50.5 7831 HELWAN 1 10 25.3 SHANGHAI 7837 5 18 36.5 7838 SlMOSATO 48 578 33.9 7840 126 579 1 1 9.9 7843 CANBERRA 84 564 8.4 7939 MATERA 30 172 84.6 A 0 8834 WETTZELL .- 360 2.6 TOTAL 737 5190 5 26 * Average of RMS of normal point data ** after data rejection TRAJECTORY DETERMINATIONAND PREDICTION Trajectory Accuracy Improvement The NASDA keeps on improving the accuracy of orbit determination and prediction of the ADEOS/RIS by updating the software models of the ADEOS/RIS analysis system. At first, we applied Tropospheric refraction correction2,center of mass correction, and GEM-T3 which is more accurate geopotential model than that for the ADEQS routine operation ( Ref. 3 . 4 ), and achieved a ten-fold improvement on accuracy of trajectory determination. For further improvement, we applied much more accurate geopotential model JGM-3 and earth radiation pressure, and reduced the uncertainly of SLR station position.

In this chapter, the latest models and results of the ADEOS/RIS trajectory determination and prediction will be reported.

to atio We determined the ADEOS/RIS trajectory using a few days data arc with about one day of During the ADEOWRIS operation, the NASDA used 3 days overlap between data arcs (Figure 4 ) .

Estimated parameters are an orbital element, solar radiation data arc and 0.5 days of overlap.

pressure modification rate rl, and air drag modification rate p 1.

p a : actual atmosphere density pm: model atmosphere density p a = p,(l + p

ra : actual solar radiation pressure r,: model solar radiation pressure

ra = r,( 1 f r,)

The NASDA used the NASDA Orbit Computing System ( NOCS on a main frame or NOCS2 on EWSs ) for the ADEOS routine operation, and uses proto-type GUTS for the ADEQS/RIS analysis.

Parameters and software models considered in the ADEOS operation and our analysis are shown in Appendix A. The conditions of trajectory determination are mostly accordingto the ADEOSIRIS Tracking Standards and the IERS standards 19925.

Figure 4 Data Span fcr Trajectory Detgmination (long arc) Accuracy of Trajectory Determination using LongArc Data We investigate accuracy of trajectory determination by comparing orbits during time span where one data arc and the next data arc overlap, since a trajectory of a spacecraft must be continuous. In order to get enough pass in the period of overlap between data arcs, in this section we use 4 days dataarc with 1 day of overlap. Table 2 shows sample of SLR data taken with the ADEOS/RIS, its 0-C and the estimated value of p and rl scale factor parameters using the latest and full software models. Table 3 shows the accuracy of trajectory determination, and Figure 5 shows the difference between the trajectory derived from arc No. 1 and that from arc No. 2 .

The uncertainty for a trajectory position in the along track is determined to be about 0.4 to 2 3). A similar analysis conducted using 3 . 5 days data arc with 0.5 meters for the ADEOS (Table days of overlap resulted in 20 to 75 meters as uncertainty in the along track (Ref. 4 ) . So, we can say that we achieved a ten-fold improvement on accuracy of trajectory determination from’former analysis in Reference 4 .

Arc No. Start (UTC)

r l

EndWC) .I__.____-- Pass Site Data O-ClRMS(m) __.__I 15 8 1 1996/10/30 3h 1996/11/02 1 2 h , 101 0.303 0.301 0.449 2 1996/11/02 3h 1996/11/05 Ilh 10 7 61 0.210 0.298 0.408 3 1996/11/05 8h 1996/11/08 21h 14 8 83 0.366 0.350 0.440 4 1996/11/08 2 h 1996/11/11 19h 16 8 100 0.232 0.410 0.492 Table 3 ACCURACY OF TRAJECTORY DETERMINATION Overlap span (UTC) Difference in Position(m) Along Cross RSS Arc Start(UTC) End(UTC) Pass Site Radial Trac~ T r a c k No.

-- -

1-2 1996/11/02 Oh - 1996/11/03 Oh 3 3 0.130 0.446 0.275 0.540 2-3 1996/11/05 Oh - 1996/11/06 Oh 2 2 0.725 2.022 1.323 2.523 3-4 1996/11/08 Oh - 1996/11/09 Oh 5 5 0.126 0.402 1.392 1.454 1996/11/2 000 Z O O 400 600 800 10:OO 12:OO 14:00 16:00 1800 20:00 22:OO 000 Time(UTC) Figure 5 Comparing arbits during the overlapped period Accuracy o f Trajectory Prediction We use orbital parameters determined with 4 days arc data (Table 2) in order to investigate an error of a trajectory prediction. For an example, difference between orbital parameters in the data arcs No. 1 and No. 2 is defined as an error of trajectory prediction for the next three days, and difference between No. 1 and No. 3 is defined as an error for the next six days (see Figure 6).

This scheme provides predictioxi errors for a total of 9-days period.

epoch generated TIRVs No.1 0 1 2 3 4 5 6 7 8 9 10 Figure 6 Scheme fcr estimating an error of trajectay prediction In Table 4, one of examples for the error estimate is shown. nEpochn is the point which too.

evaluate two orbital element and that is the epoch of determined orbital element, The orbit predicted from the orbital element No. 1 differs about 2.6 m from the orbital element No. 2.

Although the In other words, the prediction accuracy is 2 . 6 m in three days.

other data show different values in the accuracy. the values are constantly under 30 meters that satisfies our main requirement for a successful SLR measurement However, we must note that this result is only accurate for a period of low solar activity.

Table 4 DIFFERENCE BETWEEN PREDICTED ORBIT AND REFERENCE ORBIT (m RMS) predicted orbit Predicted orbit predicted orbit Epoch Reference orbit from No. 1 from N0.2 from No.3 1996/11/02 Oh No. 1 - 0.152 8.117 N0.2 2.55 1 199611 1/05 Oh - 2.223

26.595 -

1996/11/08 Oh N0.3 9.416 199611 1/11 Oh No.4 83.425 98.300 12.713 Effects on Accuracy of Trajectory Determination by Each Observation and Force Model In order to confirm effects of both various force and observation models, we choose several case studies (see Table 5). For the other models, for example, solar radiation pressure model, air drag model and tidal effect comparing, please see Reference -4. From 0 4 and difference of position and velocity of each case, we evaluate a degree of effects on trajectory determination by each model.

Table 5 SELECTED CASES FOR STUDYING EFFECTS OF VARIOUS MODELS A JGM-3 B : GEM-T3 0. considered X: not considered All of the significant observation and force models are considered in the case a. In cases from b to e, one of the models is omitted from modeling for trajectory determination. Since, the case a should define the best 0-C RMS value (provided that all the selected model is defined accurately),exclusion of one of the selected models should result in a larger value for the 0-C RMS and a degree of its derivation of the 0-C RMS from the case a may be used as a sensitive indicator for determining the importance of the excluded model.

Table 6 shows 0 - C RMS and estimated parameters p , and r,, of data arc No. 1 and arc No. 2 Table 7 and table 8 show the accuracy of trajectory determination and prediction for each case.

respectively.

Table 6 0-c RMS, ESTIMATED rl AND p .I Pass Site Data O-CRMS(m) p 1 I-1 &No. CaseNo. Start(UTQ

--. -- --

1 a 1 9 9 6 / 1 0 / 3 0 3 h - 1 9 9 6 / 1 1 / 0 2 1 2 h 1 5 8 1 0 1 0 . 3 0 3 0 . 3 0 1 0 . 4 4 9 1996/10/303 h - 1996/11/02 1 2 h 1 5 8 1 0 1 2 . 0 9 0 0 . 2 0 9 0 . 3 7 0 1 9 9 6 / 1 0 / 3 0 3h - 1 9 9 6 / 1 1 / 0 2 1 2 h 15 8 1 0 1 0 . 4 3 9 0 . 3 0 4 0 . 4 3 8 1996/10/30 3h - 1996/11/02 1 2 h 15 1 0 1 0.304 0.296 0 . 3 5 9 8 1 9 9 6 1 1 0 1 3 0 3h - 1996/11/02 1 2 h 15 8 1 0 1 1 . 8 4 5 0.287 0 . 5 3 4 1996/11/02 3h - 1 9 9 6 / 1 1 / 0 5 llh 1 0 7 . 6 1 0 . 2 1 0 0.298 0 . 4 0 8 1996/11/023 h - 1 9 9 6 / 1 1 / 0 5 llh 1 0 7 6 1 0.730 0.362 0 . 4 3 1 1996/11/02 3h - 1996/11/05 llh 10 7 6 1 0 . 3 6 0 0 . 2 9 5 0 . 3 9 1 1 9 9 6 / 1 1 / 0 2 3h - 1 9 9 6 / 1 1 / 0 5 llh 1 0 7 6 1 0 . 2 0 9 0 . 2 9 3 0 . 3 1 4 1996/11/023 h - 1996/11/05 llh 1 0 7 61 2 . 0 3 7 0 . 2 9 8 0 . 4 2 1 Table 7 DIFFERENCES IN TRAJECTORY POSITION DURING THE OVERLAPPED PERIOD CaseNo. Radial AlongTrack CrossTrack RSS a 0.130 0.446 0.275 0.540 b 0.939 2.152 0.672 2.442 C 0.117 0.654 0.121 0.675 d 0.223 0.495 0.095 . 0.551 e 0.370 1.846 0.375 1.920 Table 8 POSITION ERROR OF TRAJECTORYPREDICTION FOR THREE DAYS Case No. Radial Along Track Cross Track RSS 2.325 0.476 2.55 1 a 0.937 b -1.145 -38.503 -1.107 38.536 C 0.745 5.034 0.241 5.094 d 0.902 2.726 0.194 2.878 e 2.684 -14.868 0.171 15.109 Figure 7 shows the difference in position between the trajectory derived being corrected the center of mass and that not being corrected. According to the figure 7, the difference in the along track is about 2 meters that is corresponding to the center of mass offset.

(ml 0.5 0 . 0 -0.5 - 1 . 0 -1.5 - 2 . 0 -2.5 - 3 . 0 -3.5 I I I I I I I I I 4 . 0 1996/10/30 1996/10/30 1996/10/31 1996/10/31 1996/11/1 1996/11/1 1996/11/2 1996/11/2 1996/11/3 000 1200 000 moo 000 1200 000 1200 000 T i m e (UTC) Figure 7 Difference in position ( Center of rnass cwrected VS Center of mass not ccrrected) Uncertainty for the earth radiation pressure model has only a minor impact to 0 - C RMS.

The acceleration by the earth radiation pressure model is so small that the effect may included in the accuracy of trajectory determination by this ADEOS/RIS analysis system.

When all the models considered, 0 - 4 2 RMS is 0.5m to 2.5m. therefore, the position accuracy may be 10 meters or better.

Comparison o f Trajectoty Determination Accuracy using SLR Data and using RARR Data In this section, we discuss accuracy of determined and predicted trajectories using SLR data or RARR data The solution obtained by the RARR measurement with S-band radio wave were provided to all the ADEOS users together with the ADEOS mission data during the ADEOS operation period. Using the SLR data and the RARR data in table 9, we estimated the ADEOWRIS trajectory with the conditions showed in Appendix. Table 10 and table 11 illustrate difference in position between those solutions. The difference grows at a alarming rate as each arc passes, which implies that a larger error may propagate through the RARR method.

1 s 1996/10/30 3h 1996/11/02 12h 1R 1996/10/31 Oh 1996/11/02 18h 2 s 1996/11/02 3h 1996/11/05 I l h 2R 1996/11/03 Ih 1996/1’1/05 18h 3R 1996/11/05 Ih 1996/11/07 20h 3 s 19%/11/05 8h 1996/11/08 21h 4R 1996/11/08 2h 1996/11/10 19h 4 s 19%/11/08 2h 1996/11/11 19h Table 10 DIFFERENCE IN TRAJECTORYPOSITIONS ( DERIVED FROM THE RARR VS DERIVED FROM SLR) Time Span (UTC) Difference in Position (m) ArcNo. Start(UTC) a d o Radial AlongTrack CrossTrack RSS

1s vs 1R 1996/10/31 Oh - 1996/11/03 Oh 6.166 22.259 12.274 26.156

25 vs 2R 1996111/03 Oh - 1996/11/06 Oh 7.753 19.385 . 14.346 25.332 3s vs 3R 199W11/05 Oh - 1996/11/08 O h 6.629 16.054 16.301 23.82

4s vs 4R 1996/11/08 Oh - 1996/11/11 Oh 5.647 28.679 21.716 36.413

Table 11 DIFFERENCE BETWEEN PREDICTED ORBIT AND REFERENCE ORBIT (m RMS) Predicted orbit Predictedorbit Predicted orbit Epoch(UTC) Reference orbit from 1R from 2R f r o m 3R 1996/11/02 O h 1s 18.549 47.51 188.202 1996/11/05 Oh 2 s 156.699 16.78 18.446 Oh 3s 423.923 208.242 32.609 1996/11/08 1996/11/11 Oh 4 s 822.053 640.769 150.481 POST-ADEOS MISSION The NASDA plans to launch the ADEOS-II, which succeeds the ADECXS, in the summer of 1999. The ADECS-II will also carry Laser Reflector which has 9 individual cube-comers arranged to provide a quasi-hemispherical array on the pole nadir orientation. A center cube- corner will be oriented toward the nadir normal, and the 8 remaining cube corners will be positioned radially. This configuration will enable us to make an SLR observation in nearly all visible ranges (except a shade part of bus equipment). Its visible ranges is expected to be larger than that of the ADEOS.

Table 1 2 shows the NASDA spacecrafts that are planning to be launched with carrying Laser Reflector in future.

ADEOS-11 2000 LEO 800 prism array ETS-VIII 2002 GEO 36500 TBD ALOS 2003 LEO 700 prism array LEO : Low Earth orbit GEO : Geostationary Orbit We provide an estimate of accuracy for trajectory determination and prediction during the low solar activity period.

Accuracy of predicting a trajectory is less than 30 meters and the accuracy is valid for a period of three days after the calculation. Although the accuracy changes quite randomly from 3 to 30 meters, we achieve our main purpose that accuracy of trajectory prediction must be under 100 m for the defined period. For tracking the RIS for its original scientific experiment, our derived TIRVs are sufficient for a successful run.

For accuracy of trajectory determination, we achieve accuracy of about 3 meters in the along track. Uncertainty for a trajectory position of low earth orbiter results mainly from an uncertainty in the along track direction. Therefore, the accuracy of 3 meters in along track direction provides a good measure for the overall accuracy in position. Since accuracy of trajectory determination prior to this experiment was about 150 meters, the accuracy for the ADEOS is improved significantly.

However, approximately from the year of 1999, the solar activities will become more significant. Then it will become difficult to satisfy the requirement for accuracy of trajectory prediction since an effect of air drag onto trajectory determination is poorly understood.

Unfortunately, we have no means to verify the above assumption for accuracy of trajectory determination and prediction owing to ADEOS operation termination. It would be necessary to determine and predict the trajectory more frequently.

ACKNOWLEDGMENT We wish to thank all of personnels, particularly those who are involved at SLR stations and SLR networks both supporting the ADEOURIS experiment and the ADEOS operation.

REFERENCES 1. ADEOS/RIS Tracking Support Coordination Group, “ADEOS/RIS Tracking Standards”, 1996 2. Marini J. W., and Murray C. W., “Correction of Laser Range Tracking Data for Atmospheric Refraction at Elevations Above 10 Degrees”. NASA GSFC X-591-73-351.1973 3. M. Maeda, M. Ogawa, M. Sawabe , M. Hirota and H. Kunimori, “ADEOS/RIS LASER TRACKING EXPERIMENTS”, 1 Zth International Symposium on Space Fligbt Dynamics, ESA SP403.1997, pp. 131-136 4. M. Maeda. M. Ogawa, M. Sawabe, M. Hirota and H. Kunimori, “ACCURACY OF TRAJECTORY DETERMINATION AND PREDICTION OF ADEOS WITH RIS EXPERIMENT”, Laser Radar Rangingand Atmospheric Lidar Techniques, SPIE Vol. 3218,1997, pp. 31-39 5. McCarthy, Dennis. D., “IERS Standards (1992)”, IERS Technical Note 13.1992 ADEOSRIS experiment by proto-type GUTS ISOFTWARE MODEL ADEOS operation by NOCSiNOCS2 ASTRONOMICAL CONSTANTS Speed of Light 2.99792458 X108 m/s 2.99792458 X IO8 mls 1.4959787066 X lo8 km Astronomical unit 1.4959787 X 1 O8 km 6378.137 km(WGS-84) Equatorial radius of the Earth 6378.138 km (GEM-IOB) lR98.257(GEM- 10B) Flattening of the Earth lR98.257 (GEM-IOB) 7.2921 15 X l d j rad/s 7.2921 15 X md/s Mean spin rate of Earth TIME SYSTEMS Inner TAI TAI Input/Output UTC UTC from IERS Bulletin B UTI - TAI f r o m IERS Bulletin B COORDINATE SYSTEMS Mean equator and equinox of J2000.0 I n e r t i a l Mean equator and equinox of J2000.0 True equator and equinox of date True equator and equinox of date, Input/Output Tuned Interange V e c t o r (TIRV) Precession Newcomb theory Newcomb theory developed by W o o l a r d (EL-DUO0 used) Nutation developed by Woolard (JPL-DE200 used ) from IERS Bulletin B ' from IERS Bulletin B Polar motion WGS-84, C-7 WGS-84, C-7 Geodetic coordinate s y s t e m FORCE MODELS Geopotential GEM- 1 OB to degree and order 36 Geoptential model GEM-T3 to degree and order 50 GEM-T3 to degreeand order 50 JGM-3 ( C,,, S , , rate and J2 rate is not included)' GM: 398600.44 h3/9 (GEM-1OB) GM: 398600.436 km3/s2 1 only solid Earth tide by sun and moon (not IERS) only solid Earth tide by sun and moon (not IERS: Tidal effect hi-Solar gravity JPL-DE2OO luni-Solar ephemeris JPL-DE200 Moon-Earth mass ratio: 0.01230002 0.0 123OOO34(DE2OO) Sun-Earth mass ratio: 3.329460 X lo5 3.32946045 X lo5 (DUOO) Atmospheric drag Jacchia-Nicolet (geomagnetism not considered), Atmospheric density model Jacchia-Nicolet (geomagnetismnot considered) I ~ h h - R ~ b e r t s , Modified Harris-Priester, MSIS86 Radiation pressure Solarconstant =4.560 X loa N/m2 at IAU Solar constant 4 . 5 7 X l o a N/m2 at IAU Solar Radiation Cylindorical model for Earth and Moon s h a d o w Cylindorical model for Earth and Moon shadow N/A Seconddegreezonal model Earth Ratiation singleplate for Solar, Earth radiation and single plate for Solar, Earth radiation and Reflection model atmospheric drag atmospheric drag

ADEOS operation by NOCSNOCSZ I ADEOS/RIS experiment by proto-type GUTS

;OFTWARE MODEL MEASUREMENT MODELS ta use ( Light time equation is solved by [n u s e ( Light time equation is solved by w 3 e instantaneous rangedifferencemethod) instantaneous range difference method) h g e rate [n use none [n use none interma angles (AZ,EL) [n use none o ( , y 1 -way Doppler [n use none lata correction Marini and M u r r a y model Tropospheric refraction STDStAlgonthms NIA Ionospheric refraction rsuchiya Model Considered Center of mass NIA iite displacement NIA only Solid Earth tide by Sun and Moon 1 W M E R I U INTEGMTION Adams-Cowell method Adams-Cowell method dethod first order Adams-Bashfo~th predictor predictor- corrector f i r s t order Adams-Bashforth predictor Adams-Moulton corrector Adams-Moulton corrector iecond order: Stomer predictor second ordec Stomer predictor Cowell corrector Cowell corrector W I M T I O N METHOD Weighted least squares estimation 3asic m e t h o d Weighted least squares estimation [teration of cholesky method equations Iteration of cholesky method equations terative procedure WIM TION PARAMETERS Orbital elements %tesian Keplerian Scale factor of amospheric drag p Scale factor of amospheric drag p 1 Force model Parameters Scale factor of solar radiation I ' 1 Scale factor of solar radiation I ?

Range bias and so on Range bias and so on Observation biases Station Location biases (xb, Yb, Zb) TGTDS : Goddard Trajectory Determination System * Difference between the ADEOS operation system and the ADUlSlRIS experiment system USING DORIS Jean-Paul Berthiast, Sabine H o u v Until now, TOPEWPoseidon precise orbits were needed only for the production of Geophysical Data Record files, and thus were not required until about 5 weeks after data acquisition. Recent developments in operational oceanography now require the rapid delivery of precise altimeter data within days, and possibly hours, of data acquisition. The processing of the altimeter measurements can be accomplished according to this schedule, and the only difficulty rests with the production of the precise orbit ephemerides.

The long delay involved in the current production scheme results from the necessity to collect laser tracking data from ground stations and also from the need to wait for the final and most accurate values of the solar activity and Earth orientation parameters. A reduction in the o r b i t production delay forces the processing to deal with DORIS data only and with predicted values for the parameters. In addition, this reduces the amount of validation that can be performed before delivery.

Fortunately, the spatial and temporal coverage of the DORIS tracking system is such that the DORIS data by itself is sufficient to produce a precise orbit. Also, predictions of solar activity and Earth orientation parameters have improved considerably over the last few years, so that using them instead of actual data does not significantly degrade the orbit accuracy.

Using this strategy, DORIS orbits have been computed on a daily basis within 24 to 48 hours of data acquisition. And since the beginning of October 1997, these o r b i t s have been included on the Poseidon interim Geophysical Data Record files for all cycles when this altimeter is on.

Evaluations of these daily orbits reveal that their radial accuracy is very close to that of the standard precise orbits ephemerides.

INTRODUCTION The DORIS (Doppler Orbitography and Radiopositioning Integrated by Satellite) tracking system w a s designed and developed by the Centre National $Etudes Spatiales (CNES), i n collaboration with the Institut aographique National (IGN) and the Groupe de Recherches en Geodesic Spatiale (GRGS), to achieve the very high level of orbit Manager, Orbit Metrology Group, Centre National d'Etudes Spatiales, I8 avenue E . Belin, 31401 Toulouse Cedex 4, France ' Member of the technical s t a f f , Orbit Metrology Group, Centre National &Etudes Spatiales, 18 avenue E . Belin, 31401 Toulouse Cedex 4, France stance between the spacecraft and the ocean rder to derive the absolute sea surface height de of the spacecraft has to be known with the same level of precision. To reach this goal a major effort was launched as part of the T/P Precise Orbit Determination (POD) activities. It included, among others things, improvements to geopotential models2, the development of sophisticated surface force models3, and the installation of the DORIS tracking s ~ s t e m ~ ’ ~ .

These efforts have resulted in an orbit precision never achieved before for a large satellite in low Earth orbit. The error level of the T/P Precise Orbit Ephemerides (POE) that are routinely produced by NASA and CNES does not exceed 2 to 3 cm RMS in the radial direction as demonstrated by various tests: tracking data residual analysis, especially high elevation laser ranging residuals, comparisons of orbits computed with different data sets 6 7 8 9 (in particular DORIS and laser versus GPS), and altimeter crossover residual analysis ’ ’ ’ .

DORIS is a one-way, ascending Doppler system which utilizes a set of ground beacons that broadcast continuously and omnidirectionally on two frequencies of 2036.25 and 401.25 MHz. Each beacon contains an ultrastable quartz oscillator (USO), as well as sensors for monitoring the temperature, pressure and humidity. The broadcast message, which is transmitted every 10 seconds, consists of the meteorological data, the beacon identification number, a short status report and a synchronization signal. The receiver on- board the satellite receives the dual frequency signal and computes the integrated Doppler count over intervals of 7 or 10 seconds. The receiver is programmed in advance to multiplex the signals from several commonly viewed beacons.

The current DORIS network consists of about 50 beacons covering the entire sudace of the Earth, with the exception of the Southern Pacific ocean. It is routinely used to track t h r e e satellites, SPOT 2 (since 1990), T/P (since 1992) and the recently launched SPOT 4, and produces more than 250 passes of data per satellite and per day. With this network, the DORIS system provides on a daily basis a uniquely spatially and temporally dense set of high precision ground-based tracking data.

DORIS is a centralized tracking system, in which a l l the data are collected on-board of the spacecraft. This makes it possible to compute the orbit either on-board in real-time, or on the ground in near real-time. The real-time capability is now operational on SPOT 4, while the near real-time processing is routinely used to compute daily 1-day orbits for TP.

The key accomplishment was to improve the precision of these orbits to a level comparable to that of the POE.

RATIONALE FOR FAST PRECISE ORBIT PRODUCTION Until now, precise orbits were only needed for the production of Geophysical Data Record (GDR) files, which are designed for the scientific community, and are distributed with a two month delay. Thus POEs were not required until about 5 weeks after data acquisition. The Interim GDR (IGDR) files, which were provided to users requesting fast service, were generated using lower precision operational orbits.

rucial to the quality of the result.

project in this field. Its goal is the implementation within five to seven years of a pre- operational high resolution global Ocean model which assimilates satellite and in situ data.

One of the by-products should be the demonstration of the operational need for space based oceanography data, with real-time availability as one of the key factors.

Looking to the future, near real-time altimeter data should become one key component of the global Ocean observation system. Quick assimilation of this data into global meteorological models could improve weather forecast, both on a short term and seasonal basis (e.g. El Niiio). It could also help predictions of near-surface conditions for the open ocean, which would in turn benefit fishing or transportion industries while improving safety. Even local forecast of coastal currents would be improved through a better knowledge of the deep ocean boundary conditions.

CHALLENGES OF FAST PRECISE ORBIT PRODUCTION These new requirements for fast precise orbit production creates new challenges. The long delay involved in the current production scheme makes it possible to collect laser ranging data from ground stations, and also to benefit from the f i n a l and most accurate values for the solar activity and Earth orientation parameters. A reduction in this delay forces the precise orbit production system to deal with DORIS data only and with predicted values for the parameters. In addition, this reduces the amount of validation which can be performed before delivery.

Fortunately, when available, the DORIS data by itself is sufficient to produce precise orbits. In addition, atmospheric drag is very smaU at the altitude of T/P, and short term predictions of solar fluxes and geomagnetic indices have improved considerably over the last few years, so that using them instead of actual data does not significantly degrade the orbit accuracy.

Similarly, the quality of predictions for Earth orientation parameters has also improved, but it is still not suEcient? at the level of accuracy that we deal with. However, the DORIS data is powerful enough to accurately determine the orbit in the Earth based frame in which the station locations are known, even though it is not known how to precisely relate this ftame to the inertial frame in which the equations of motion are integrated.

There are two elements that contribute to this success. The first one has to do with the ability to recover the Earth orientation parameters while computing the orbit. However, the quality of the result is not sufficient to ensure centimeter level accuracy. The second one is the fact that when the T/P orbit and the DORIS station coordinates are expressed in the The factor of two improvement in the quality of the IERS Bulletin A short term predictions", introduced in early March, might change this conclusion. It is currently under investigation. This change was implemented t o improve the quality of GPS orbit predictions.

approximate E a r t h orientation parameters is of second order. But, in addition, the reduced dynamics stochastic correction technique12 (Em) used in the CNES precise orbit production strategy corrects the dynamical errors using measurements.

ORBIT PRODUCTION STRATEGY Description The near real-time precise orbits are produced at CNES using the tools that were developed for the POE. The ZOOM software is used for a l l the computations, and the Voyager user interface and procedures are used to activate and monitor the various steps of 13.14 theprocessing .

A short s u ~ ~ l s ~ f y of the strategy is as follows: Perform a standard orbit determination using a complete dynamical model. Corrections to the IERS predictions for polar motion parameters are added to the standard state vector (initial conditions, multiplicative coefficients for solar radiation pressure and atmospheric drag, frequency and troposphere bias per pass, constant along-track and once-per-revolution along-track and cross-track empirical forces). The result is a fully dynamical orbit ephemeris expressed in the true-of-date reference frame.

Apply the reduced dynamics stochastic correction technique to adjust a piecewise constant empirical force that follows a first order Markovian evolution scheme. The result is a corrected orbit ephemeris, still expressed in the true-of-date reference frame.

Convert both dynamical and corrected ephemeris to the IERS Terrestrial Reference Frame (ITRF) using the estimated value of the E a r t h orientation parameters. It is only in this fhme that the orbit is accurate, so these are the products which are delivered and archived.

Validate the orbit using data residual analysis, comparison of two successive orbits in the overlapping region, and comparison with the previous day’s extrapolated orbit.

Other verifications, including comparisons with respect to the POE and altimeter crossover residual statistics, are conducted routinely, as soon as these products become available.

Products T/P DORIS data is received at CNES in daily batches in the morning. On day D the orbit is actually computed using data f r o m days D-2 and D-1, and covers 30 hours. This includes a two hour margin at both ends, where the stochastic correction degrades the orbit rather than improving it: this is due to the lack of past information on the correlated stochastic process a t the beginning, and to the lack of future information at the end. This leaves 26 hours of usable ephemeris, covering the period from 8 p.m. on day D-2 to 10 p.m.

rs coming from the purely dynamical solution of day D, from 10 p.m. to midnight, and 24 hours of extrapolation.

These composite orbits are delivered to the Centre de Traitements DORISPoseTdon (CTDP) which reformats them before providing them to AVISO. They are then added to the fast IGDR products and sent to the Service Hydrographique et Oc6anographique de la Marine, and to its US counterpart, NAVOCEANO. During the periods when the PoseTdon altimeter is on, these orbits are also used to produce the standard IGDR.

Operations The processing chains are activated automatically on a daily basis by Unix cron processes. The activation takes place at the same time every day; however, there are sometimes delays in the reception of the DORIS data from NASA's Jet Propulsion Laboratory, or in the preprocessing performed by the CTDP. In this case, the procedure switches to a sleep state and checks for data arxival at regular intervals. Once the data are available the entire processing takes less than one hour on the SUN Ultrasparc Enterprise E5000 of the CNES central computing facilities.

At the end of the processing an E-mail message is sent to the supervising engineer. It contains the status of the individual steps, as well as the results of the various verification tests. Simplified global status messages are available for display on the terminal of an operator. However, during the current development phase, no operator is available to monitor the orbit determination processes during weekends. As a temporary solution, a copy of the final mail message is sent to the private mailbox of one of us (S.H.) who can thus remotely monitor the status of the system, and take appropriate action.

Many of the internal validation tests are associated with expected ranges for results.

Whenever values %e out of k g e , processing is stopped. This prevents delivery of incorrect orbits to the end user. Using the Voyager monitor, the supervising engineer can intervene manually in the system, and resume the processing at any point. He can skip steps, override options, and perform step by step processing, or chained operations.

Many parameters are generated during the daily processing. They are archived for long term monitoring of the orbit quality. In the near future, an automated quality assessment report will be generated based on these parameters and provided to users along with the orbits.

ACCURACYASSESSMENT Near real-time DORIS orbits have been computed over the last few months, using various configurations. Hence, results vary slightly as a function of the configuration used at the time. However, this has nd significant impact on the analysis which is presented here.

g 0.83

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8 0.51 2z 0.45 = 2 u I w e 7 23m6197 22lows7 2 i n m 2 M m 1WOZYW lamugg Figure 1 DORIS residuals RMS value The statistical analysis of the data residual is a good tool to evaluate the quality of the processing. In the case of DORIS, the data appear noise limited around the level of 0.55 d s . RMS values of the DORIS data residuals are plotted on Figure 1 . Daily solution residuals (squares) are at least as good, if not better, than residuals of the POE (triangles).

In particular, daily residuals are lower during fixed yaw periods, when cross-track and along-track directions remain constant relative to the satellite body. In this case, the adjusted once-per-revolution parameters absorb poorly modeled surface forces more easily.

Successive orbits overlap over a 2-hour period. Comparison of the two solutions over this period provides a good estimate of the orbit error. RMS values of these differences are Figure 2.

plotted on 6.

21nm 2 M m 18/021BB 18104199 Figure 2 Orbit overlap differences RMS values in the along-track direction, are a consequence of th 2.6 and 2.2 milliarcseconds for the u and v , which correspond respectively to 7.8 cm and 6.6 cm. These values are consistent with the observed level of cross-track and along-track error.

Figure 3 RMS differences between daily orbits and the POE R M S values of the radial differences between daily orbits and the corresponding POE are also about 2 cm (Figure 3). This is comparable to the radial error level in the POE itself, thus, i n the radial direction, daily orbits appear to be about as accurate as the POE. Cross- track and along-track differences are significantly larger, respectively about 10 cm and 7 cm RMS. Here again, these differences are mostly the result of inaccuracies in the E a r t h orientation parameters.

The larger differences observed since December 1997 are due to a change in the reference system used to generate the near real-time orbits. Originally, these orbits and the POE used the same reference system, based on a set of station coordinates computed at CNES. In December of last year, the reference system of the daily orbits was switched to

the ITTIRF 96 ~olution'~. The almost 5 cm offset along the Z direction between the two

fiames adds to the difference between the orbits in the radial and along-track directions.

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170 l?!i 180 CYCLENUMBER 190 195 200 Figure 4 Altimeter crossover residual RMS value The computation of altimeter crossover residuals provides the best external test of the quality of the orbits as this data is not used to compute the orbit. Figure 4 presents the result of these evaluations. Crossover residuals computed using a combination of 10 daily orbits to produce a full repeat cycle are represented by squares, while crossover results for the POE are represented by triangles. Full cycle crossover residuals for daily orbits and precise orbits are roughly the same. This confirms that the radial orbit error level in the daily orbits and in the POE are nearly identical. However, this test does not provide any reliable estimate of the error level, as the residual signal is dominated by ocean variability, altimeter data noise and tide model errors.

PERSPECTIVES FOR REAL-TIME PRECISE ORBIT DETERMINATION In 1991, CNES started the development of a DORIS based space borne orbit determination system for the SPOT 4 satellite16. The core of the system is a standard DORIS receiver to which a new function has been added to process measurements in real- time and produce an orbit. Positions and velocities are then added to the image data in the spacecraft telemetry and downloaded to the ground image processing centers.

This on-board orbit determination system, DIODE (D6termination Imm6diate d'Orbite par DORIS Embarque), is now fully operational. Figure 5 shows a comparison of the SPOT 4 positions computed on-board and received in the telemetry with a ground based reference orbit. This plot corresponds to the fmt week of operations.

The current results of DIODE are an excellent proof of its quality m reliability. The accuracy of the results, a few meters 3D at 1 sigma, is well within the reqwiements of most space missions, including SPOT.

However, technically, this version of the orbit determination software is obsolete in terms of precision. It w a s delivered to the project in mid 1995, and since then major improvements have been brought to the software to support the development of the new DORIS receivers. These instruments will offer an order of magnitude improvement in precision over the current Work is on-going to t r y to retrofit the SPOT 4 software to benefit from this improvement in accuracy, as well as other new functions such as the ability to initialize autonomously.

DIODE/SPOT$ First three days on orbit 5 0 , t Figure 5 First comparisons of the on-board orbit with a ground-based reference One key factor for this progress is the fact that both the real-time on-board orbit determination s o h a r e and the operational full precision ground-based orbit determination program, ZOOM, are developed within the same group. Thus, challenges arising from the requirements of operational oceanography have stimulated the development of new and more precise real-time processing strategies.

The latest version of DIODE was designed for the European ENVISAT satellite.

When adapted to take into account the rather complex attitude control of the T/P spacecraft, it provides orbits with a radial error level of between 10 and 20 cm RMS. This is not sufficient for the very precise operational applications, but can be used to produce auxiliary wind and wave products.

However, improved modeling techniques and a better tuning of the fdter have led to significantlybetter results in some test caseslg. These results still need to be confiied. If they can be generalized, there is hope that, in the future, real-time precise orbits can be produced.

CONCLUSION Evaluations of near real-tipe daily orbits computed for T/P using DORIS data reveal that their radial accuracy is roughly identical to that of the POE. Their introduction into the fast IGDR products, and into the Pose'idon IGDR, hence their availability to the mission, will suppo onal oceanography by its, and with an even ) based on the real- time on-board orbits. It is hoped that these new developments will help secure the future of operational oceanography, so that we can a l l benefit from its extraordinary potential contribution to weather and marine state forecasting.

ACKNOWL~DGMENT The authors wish to thank Eric Julien and V6ronique Lefr5re of CISI, Aeronautics and Space Branch, Toulouse, France, for their implementation of the complex computer procedures required to produce the near real-time orbits. They also thank Christian Jayles and Patrick Broca for providing them with the fust results of the DIODE experiment on- board of SPOT 4.

REFERENCES 1. L.L. Fu, E.J. Christensen, C. A. Yamarone, M. Lefevre, Y. Menard, M. Dorrer, P. Escudier, “TOPEXROSEIDONmission overview”, Journal of Geophysical Research, Vol. 99, No. C12,1994, pp 24369-24381 B.D. Tapley, et al, “The JGM-3 geopotential model”, Journal of Geophysical Research, Vol. 101, B12, 2.

1996, pp 28029-28049 3. J.A. Marshall, S.B. Luthcke: ‘Modeling radiation forces acting on TOPEX/POSEIDON for precision orbit determination”, Joumal of Spacecraft and Rockets, Vol. 31, No. 1, 1994, pp 89-105 4. F. Nouel, J. Bardina, C Jayles, Y. Labrune, B. Truong, ‘DORIS: a precise satellite positioning doppler system”, Astrodynamics 1987, Advances i n the Astronautical Sciences, J.K. Solder et al. ( e d s ) , Vol. 65, 1988, pp. 311-320 5. F. Nouel, J.-P. Berthias, P. Broca, A. Comps, M. Deleuze, A. Guitart, C. Jayles, P. Laudet, C. Pierret, A. Piuzzi, D. Pradines, D. Taburiau, C. Valorge, “Precise orbit determination with the DORIS/SPOT 2 system: first r e s u l t s ” , Proceedings of the ESA Symposium of Spacecraft Flight Dynamics, Darmstadt, Germany, September-October 1991, pp. 91-95 6. F. Nouel, J.P. Berthias, M. Deleuze, A. Guitart, P. Laudet, A. Piuzzi, D. Pradines, C. Valorge, C. Dejoie, M.F. Susini, D. Taburiau: “Precise CNES orbits for TOPEXPOSEIDON: Is reaching 2 cm still a challenge?”, Journal of Geophysical Research, Vol. 99, No. C12,1994, pp. 24405-24419 7. B.D. Tapley, J.C. Ries, G.W. Davis, R.J. Eanes, B.E. Schutz, C . K . Shum, M.M. Watkins, J.A. Marshall, R.S. Nerem, B.H. htney, S.M. Klosko, S.B. Luthcke, D. Pavlis, R . G . Williamson, N . P . Zelesnki, “Precision orbit determination for TOPEX/POSEIDON”, Journal of Geophysical Research, Vol. 99, No.

CP2,1994, pp. 24383-24404 8. B.D. Tapley, J.C. Ries, “Recent advances in precision orbit determination”, Proceedings ofrhe 1997 A I M GNC, AFM, and MST Conference Md Exhibit, New-Orleans, Louisiana, August 1997 for precise orbit determination?”, , Toulouse, France, April 1998 10. P. Courtier, “The ATOR project and the future of operational oceanography”, Proceedings of the International Symposium “Monitoring the oceans i n the 2000s: an integrated approach”, Biarritz, France, October 1997 11. J. Ray and B. Luzum, Tmproved polar motion predictions”, IGS Electronic Mail, Message no 1816, February 1998 12. B. Barotto and J.-P. Berthias, “First results of Reduced Dynamics w i t h DORIS on TOPEX/Poseidon and SPOT”, Journal of Guidance and Control, Vol. 19, No. 6, pp. 1296-1302 13. F. Nouel, M. Deleuze, P. Laudet, C. Valorge: “Operational aspects of precise orbit determination of SPOT and TOPEX/Poseidon satellites w i t h DORIS”, Spaceflight Dynumics, Advances in the Astronautical Sciences, J. Teles et al. ( e d s ) , Vol. 84,1993, pp. 279-290 14. E. Julien, J.P. Berthias, P. Broca, A. Comps, M. Deleuze, A. Guitart, S . Houry, C. Jayles, P. Laudet, E Nouel, A. pi&, D. Pradines: “The DORIS orbit computation service”, Proceedings of the 11” InternationalAstrodynamics Symposium, Gifu, Japan, May 1996 15. C. Boucher, Z. Altamimi, P. Sillard, “The lTRF!26 realization of the International Terrestrial Reference System”, Proceedings of the IAG Scienec Assembly, Rio de Janeiro, Brazil, 1997 16. J.-P. Berthias, C. Jayles, D. Pradines, “DIODE a DORIS based real-time on-board orbit determination system for SPOT 4”, Spacefzght Dynamics, Advances in the Astronautical Sciences, J. Teles et al. ( e d s ) , Vol. 84,1993, pp. 125-138 17. J.-P. Berthias, P. Broca, J. Fourcade, C. Jayles, D. Pradines, D. Laurichesse, “General characteristicsof real-time on-board orbit determinati~n’~, Proceedings of the 12” Intemational Symposium on Space Flight Dynamics, Darmstadt, Germany, June 1997, pp. 267-274 18. C. Jayles, P. Broca, J.P. Berthias, D. Laurichesse, ‘Navigation autonome temps r & l ” , Proceedings of the DORIS Days Meeting, Toulouse, France, April 1998 19. D. Laurichesse, private communication F. M. Martinez Fadrique, GMV R. C. A. Zandbergen, Logica D. Navarro Reyes, GMV ESA / European Space Operations Centre Robert Bosch Strasse 5,64293 Darmstadt, Germany The Earth observation satellite Envisat-1 will be controlled by the European Space Operations Centre (ESOC). This paper addresses ESOC's orbit determination activities for Envisat, and more particularly the possibility of obtaining high- precision altimetry products for near-real-time ocean surface topography monitoring.

First, this paper presents the current ESOC capabilities in the area of Precise Orbit Determination (POD) and ocean surface model computation, based on the most recent ERS-2 data. A detailed analysis of the models used for ERS and the possibilitiesof implementing newly developed ones is discussed in order to identify potential improvements. Since the precise tracking devices on board Envisat to ERS, this paper also presents provide further sources of improvement compared the advantages which may be expected from Envisat for obtaining better orbits and models.

Finally, the new Navigation Package for Earth Observation Satellites (Napeos), which will perform both the operational and precise orbit determination, will be described in a few words INTRODUCTION Continuing its Earth observation programme, ESA will launch Envisat-1 in November 1999. Envisat, with its improved instruments and tracking devices, will still resemble in many aspects the remote sensing satellites currently in operation. The satellite will be controlled by ESOC, which will also be responsible for all Flight Dynamics activities. These activities include routine operational orbit determination, orbit prediction and orbit maintenance (manoeuvre planning) but also very precise orbit determination resulting i n the available of high-precision altimetry products.

The aim of ESOC Flight Dynamics is not to produce scientific data. Rather, its requirements on orbit determination, prediction and control stem from the need to control the spacecraftand its instruments, and to maintain the ground track of the satellite within a narrow deadband. In addition to these near-real time activities, precise orbit determination (POD) is performed for the evaluation of the routine orbit determination and the performance of the altimeter instrument. POD is aided by the presence of the laser retro-reflector (LRR) and, as a new instrument compared to ERS, the DORIS system (Doppler Orbitography and Radiolocation Integrated by Satellite). These tracking systems shall provide the possibility of obtaining an accuracy of around 5 cm in the radial direction. The altimeter height measurements, when properly processed, also provide a source of tracking data, and the ocean surface topography models obtained in this processing are a valuable spin-off of the Envisat POD.

As the result of several years of experience with the POD of ERS and Topefloseidon, the Envisat mission can build on the knowledge of very accurate models and processing techniques. Of special interest is the processing of altimeter data, while the nearly complete coverage by precise DORIS tracking will allow the computation of more accurate orbits in zones of major interest like the Pacific ocean, where no simultaneous altimetry and precise tracking data were available for ERS. Unfortunately, at the same time Envisat will suffer the consequences of flying during the solar maximum, which will make air drag modelling much more challenging.

ESOC's current capabilitiesin the area of POD and ocean surface model computation will fvst be evaluated using the most recent ERS-2 data. Subsequently, the accuracy which may be expected from Envisat will be explored, with a view to establishing the accuracy of the altimetry by-products of the ESOC Envisat POD.

CURRENTLY IMPLEMENTED MODELS FOR ERS Two parallel activities w i l l exist in the Envisat orbit support operational and high-precision orbit determination.

The operational orbit determination is based on requirements of near-real time orbit determination with maximum stability and reliability and relatively modest accuracy. The dynamic models and trackjng data processing models currently used for ERS yield the required accuracy specified for orbit determination and prediction, both for ERS and Envisat, even in the period of increased solar activity. This orbit determination scenario is not subject to dramatic improvements, mainly due to the accuracy limitations of the S-band tracking system itself. This activity includes the estimation of optimised manoeuvre sequences in order to maintain the ground track within one km from the reference ground track throughout the mission.

Precise orbit determination follows a completely different approach. The most accurate models available are used in order to obtain orbits as accurate as possible. The tracking data used for this purpose are also the most accurate available (satellite laser ranging (SLR) and Precise Range and Range-rate Equipment (PRARE)) with state of the art data processing models. This scenario is subject t o continuous improvement, both in the environmental modelling and in the processing of the tracking data. Of particular interest is the analysis and implementation of models for the correction of altimeter observations.

Although the main objective of the ESOC POD is the processing of ERS data and the generation of ERS high-precision orbits, data from other satellites equipped with differenttracking devices are processed. This allows the verification of algorithms and models which may be used for ERS and the generation of auxiliary data (e.g. station coordinate solutions) for which the ERS orbit configuration is not optimal.

Dynamics and tracking data processing models The following setup, currently used for POD of ERS, Topefloseidon, SPOT and Lageos also forms the basis for the processing of other (future) satellite and tracking configurations: Reference frame Mean equator and equinox of J2000.0 o Nutation (Wahr model) Earth rotation (EOP IERS Bulletin A) o SLR station coordinates from an ERS and Lageos multiarc solution aligned to the lTRF * DORIS station coordinates from ITRF Dynamics o JGM-3 ( 7 0 , 7 0 ) gravity model SISE-90 air densit). model. Detailed drag modelling based on spacecraft geometry and aerodynamic flow; scale factor estimated every twelve hours.

e Luni-solar gravity e Frequency-dependent solid Earth tides (Wahr model) Ocean tides (Schwiderski) o Direct solar radiation pressure model. Detailed modelling based on spacecraft geometry e Albedo and infrared radiation perturbations e Manoeuvre modelling with scale factor estimation e One cycle per revolution (cpr) along-track and cross-track empirid acceleration. One set of coefficients per arc.

Tracking data processing o Murray- Marini tropospheric correction (laser) q Centre of mass correction o Tropospheric, ionospheric and centre of mass corrections from dataset for DORIS o Tropospheric, ionospheric, centre of mass, antenna phase, station mechanical and external corrections for the Precise Range and Range-rate Equipment (pRARE) tracking data.

o SLR range station bias o PRARF. range station bias and pass atmospheric scale factor Altimeter data processing Ocean surface: Solid E a r t h tide correction, Schwiderski model Permanent tide correction, Wahr model o Ocean tide correction, Schwiderski NSWC model Ocean loading, Schwiderski model o Mean Sea Surface, OSU-91A plus ERS-IESOC correction model o Dynamic Sea Surface Topography, ERSESOC model Propagation: 0 Dry troposphericcorrection, Saastamoinen 1972 and ECMWF pressure field Wet troposphericcorrection, ESOC model o Ionospheric correction, Rawer-Bent model Other: Electromagnetic bias correction from Fast-Delivery products Centreofmass 0 Altimeter instrument bias Some of the models used in the processing of altimetry were developed in-house (ERS-2 Altimeter Calibration at ESOC, Romay-Merino et al.); the altimeter data generated in the geodetic phase of ERS-1 gave the possibility of generating a global solution for the mean sea surface with resolution of 0.3 degrees and accuracy better than 10 c m . A spherical harmonic expansion for the altimetry wet tropospheric correction was also computed based on meteorological data from the ECMWF.

Precise Orbit Determination Implementation Satellite laser ranging observations and altimeter normal points are the tracking data types used in the ERS- 2 POD. A parallel POD activity for evaluation purposes uses PRARE as precise tracking data. Both orbit determination activities use five-day arcs overlapping the previous and following arcs by one day each (two- day effective overlap). The three days in the middle of each arc are kept as the precise solution (see Figure 1). With this strategy one can eliminate boundary effects from ill-determhed parameters like aerodynamic coefficients. Comparisons of the one-day arcs centred in each of the two-day overlap periods are made in order to verify internal consistency between successive arcs.

Figure 1 : Precise Orbit Determination Strategy ERS-2 precise orbit determination results Since the launch of ERS-1 in 1991, and subsequently for ERS-2, the ESOC precise orbit determination has seen a gradual improvement in the dynamics and data processing models with a corresponding improvement in the accuracy of the orbit solutions for these satellites. A major step was achieved at the occasion of the relative calibration of the ERS-1 and ERS-2 altimeter instruments during the ERS-2 commissioning phase (ERS-2 Altimeter Calibration at ESOC, Romay-Merim et d.).

The first step in the generation of precise altimetry products is the generation of a high-precision orbit solution. This is accomplished by the simultaneous processing of SLR quick-look data from the EUROLAS and CDDIS data servers and altimetry. The SLR station coordinates used for ERS are based on multi-arc solutions incorporating ERS and Lageos data, where the scale and orientation of these solutions have been made to match the current ITRF solution. The latest solution is based on two years of Lageos and ERS data and coincides in scale and orientation with lTRF-96.

Unfortunately, the ERS-1 PRARE tracking device failed soon after its activation. For ERS-2, the processing of PRARE data has received a lot of attention from the international POD community, and after two years solutions with an accuracy comparable to those based on satellite laser ranging (SLR) and altimetry, but independent of the altimeter data, have become possible (cf. e.g. Incorporation o f PRARE data i n ERS-2 orbit computation, Visser et al.).

Typical one-way rms values of ERS-2 SLR residuals are shown in Figure 2. They are an indication of the total satellite position accuracy during the periods of visibility by a laser station.

Figure 2 : ERS-2 POD SLR Residuals The altimetry residuals, of which rms statistics are shown in Figure 3, are an indication o f the combined radial accuracy o f the orbits and the models used in the processing of the altimeter data.

Epoch Figure 3: ERS-2 POD Altimetry Residuals Another indication o f the orbit accuracy is the consistency between consecutive orbit determinations, The rms difference in the overlapping arcs give an idea of the consistency of the solutions. For the SWaItimetry combination these values are below 5 cm.

Orbits based on (revision five) PRARE data are yielding fits of the tracking data o f about 7 cm RMS in range (one-way) and 0.8 mm/s RMS in range-rate (see Figure 4).

10.0 2.0 9.0 1.8 8.0 1 . 6 7.0 1.4 6 . Q 1.2 cn c n z 1.0 0: 5.0 P E 4 . 0 0.8 a 3.0 0.6 2.0 0 . 4 0.2 1.0 0 . 0 0 . 0 Epcch Figure 4: ERS-2 POD P U R E Residuals ERS-2 altimeter products from PRARE/SLR orbits are being generated at ESOC and shown at its Internet web site ( h t t p : //nng . esoc . esa . d e / ) . A similar future activity will be based on DORISISLR orbits for Envisat The extension of the PRARE MEX station network at the beginning of this year improved the accuracy and consistency of orbits independent from altimetry. The radial internal consistency goes f r o m 5.7 cm for the SLR solution t o 1.9 cm for the P W S L R combination. m e combination SLWaltimetry stays in between with 3.3 cm (see Figure 5). The problem related with the SLR solution is due to the lack of data in the first pare of the analysed period, which is typical for SLR in seasons of bad weather in the northern hemisphere.

20.0 18.0 16.0 14.0 i 5

- 12.0

u1 r a 10.0 a $ 8.0 8 6.0 4.0 2.0 0.0 c c c c c c c c r - c c c c c c c c c r - e m e %s%spp?%ss?%ps>?s?pp d d n ~ n ~ o o o d d d d n ~ n n o o ~ w Figure 5: ERS-2 Orbit Radial Internal Consistency for Various Tracking Scenarios The radial orbit comparison between the altimetry/SLR and PRARE/SLR solutions has an nns of 3 . 0 cm.

Given that the maximum accuracy that can be obtained with the JGM3 model has been estimated to lie around 8 cm, the accuracy of the PRARE/SLR solution should be sufficientfor obtaining altimeter products.

10.0 9.0 8 . 0 7 . 0

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6.0 u1 g 5 . 0 2.0 1.0 0.0 Epoch Figure 6: ERS-2 Radial Orbit Comparison: AltimetryBLR v s . PRAREBLR ENVISAT ORBIT DETERMINATION ESOG Right Dynamics is now preparing for the support of the Envisat mission. The similarities between the two spacecraft and their orbits make it possible to reuse most of the existing ERS systems for Envisat. It is important, however, also to identify the differences between the two missions in order to make the appropriate adjustments in order to maintain the existing performance and improve it where possible.

Four tracking devices are available on Envisat for the purpose of operational and precise orbit determination. These are: S-band transponder: This system provides 2-way range and range difference data from ESA's Multi- Purpose Tracking System (MF'TS). This is the main source for the operational orbit determination. Its relative low accuracy (of the order of 1 meter after pre-processing) makes it unsuitable for POD.

e Laser Retro-Reflectox This is a passive device which provides a capability for high-precision 2-way ranging from the SLR network. The coverage of this network is limited to populated areas of the Earth (Europe, N o r t h America, etc.) and it is very sensitive to meteorological conditions.

e Doppler Orbitography and Radiolocation Integrated by Satellite (DORIS): This system replaces the PRARE system used on ERS. DORIS provides high precision one-way range rate observations from a very uniformly distributed network with a nearly global coverage. The device mounted on-board Envisat incorporates the second generation of the DORIS tracking system.

0 Radar altimetex Although not a tracking device in the first place, it provides height measurements above the instantaneous sea surface, which can be used to improve the operational orbit determination.

The main advantage of Envisat with respect to ERS is the global tracking data coverage provided by DORIS. Also the DORIS station cosrdinates are computed as part of the ITRF from Spot and Topefloseidon solutions.

Precise Orbit Determination Prospect To demonstrate the accuracy achievable in Envisat POD it is necessary to simulate a scenario With a similar satellite, orbit and tracking data. This is most easily achieved by using an existing mission whose characteristics are close to those of Envisat Missions carrying a DORIS instrument are SPOT and Topefloseidon. Although Topefloseidon is more attractive because it also carries two radar altimeters and a laser retro-reflector, the SPOT orbital height is much closer to that of Envisat, and this will be the deciding factor in the achievable orbit determination accuracy. For the analysis, six arbitrarily selected months of SPOT-2 DORIS data were selected. The models used in the analysis are basically those from taken ERS except for the variable area table for drag and radiation pressure. The station coordinates set was directly from the ITRF94.

The potential accuracy of the orbits is to certain extent represented by the level of residuals in the orbit fits.

For the analysed data a value of 0.56 mm/s was computed, which is equivalent with a noise in 20-second one way range normal points of 2 c m This means that the achievable accuracy is of the same order of an orbit computed with SLR, limited by the JGM-3 geopotential to 7-8 cm in the radial direction.

Epoch Figure 7: SPOT-2 DORIS Residuals The internal consistency of the solutions in overlapping arcs, for the analysed period, yielded a value of 1.7 values of 2.68 cm for altimenylSLR cm, which compares very favourably with the aforementioned solutions and 1.9 cm for PRARE/SLR solutions for ERS.

5 . 0 4.5 Epoca Figure 8: SPOT-2 DORIS Orbit Internal Consistency New Models for Envisat The high solar activity expected during the Envisat mission will have two major consequences. Most importantly, the increased error in air density prediction leads to a larger error in the along-track position and velocity restitution. Secondly, the higher solar activity increases the ionospheric delay effects on the tracking data, which are not easy to model properly. The fmt problem is partially compensated by the higher ballistic coefficient of Envisat compared to ERS. The ionospheric modelling problem does not affect the SLR data, and can be mostly eliminated from the DORIS and altimeter instruments because both are - dual-frequency systems. Still, it will be of some interest t o compare the performance of the old Rawer-Bent model with that of the International Reference Ionosphere ( I R I ) , which incorporates data f r o m the latest period and a prediction for the next few months. Another possibility is to use GPS derived ionospheric products from the ESOC GPS analysis facilities. This has the major advantage that the very latest state of the ionosphere can be used, taking then i n t o account any abrupt fluctuation of solar flux and geomagnetic index.

Other important areas with a potential for improvement are the gravity model, the ocean tide model and the mean and dynamic sea surface topography models.

The gravity model currently used for ERS is JGM-3, which is complete to degree and order 70. This model places a Emit on the achievable radial orbit accuracy of around 7-8 cm. More recent models with acclaimed accuracies of 5 cm in the radial direction for ERS are the general-purpose TEG-3 model from the University of Texas in Austin and the ERS-tailored DGM-E04 from the Delft University in the Netherlands. These three models may be compared using ERS-2 and SPOT-2 orbit determination. Table 1 shows a summary of the results, clearly indicating that tracking data residuals are reduced significantly and so are the orbit consistency value.

Table 1: Gravity Models Comparison Table 1 : Gravity Models Comparison (contd.)

Figure 9 shows that the radial orbit consistency is especially improved using the more recent gravity model(s) if the tracking data is sparse.

20.0 1 8 . 0 16.0 14.0

- 12.0

! 2 10.0 a 2 8.0 Q) g 6.0 4.0 2.0 0.0 Epoch Figure 9: ERS-2 SLR Orbit R a d i a l Internal Consistency Geopotential Models Comparison For Envisat it is of interest to predict what the effect of using these gravity models is going to be. The radial comparisons between the orbits computed w i t h JGM-3, TEG-3 (CST/UT) and DGM-EO4 (DEOS) for different tracking scenarios (see Figures loa, 10b and 1Oc) show that differences are driven by the change in model and not by the tracking d a t a . The worst scenario (SLR only) is not much different f r o m the best scenario (SLR + PRARE). It is also noteworthy that the differences between TEG-3 and DGM-E04 orbits computed at ESOC are not far from the differences between the CSR and DEOS computed orbits.

14.0 12.0 10.0 L

-

8 . 0 = . 6 . 0 r( 4 O" 4 . 0 2 . 0 0 . 0 Figure loa: Orbit Comparisons : JGM3 vs. TEG3 Figure loa: Radial Orbit Comparisons: JGM-3 vs. TEE3 Epoch Figure lob: Radial Orbit Comparisons: JGM-3 vs. DGM-E04 Epoch Figure 1Oc: Radial Orbit Comparisons: DGM-E04 v s . TEG-3 er important effect of the use of these improved gravity models is the reduction in geographically correlated error. One can see this computing the differences between Sea Surface Topographies (SST) calculated with JGM-3 and the more recent models. Figures 1 la and 11 b show that this effect is essentially independent of the t y p e of tracking used in the computation.

Figure lla: SST Differences (PRAWSLR) Figure llb: SST Differences (AItimetry/SLR) The next important area of improvement when computing precise altimetry products is the processing of the altimetry data itself. Two significant areas for improvement are the ocean tides model and the ionospheric model. For ocean tides, the CSR 3 . 0 model was compared with Schwiderski using ERS-2 data. For the ionosphere, IEU-95 was tested against the Rawer-Bent model.

Altimeter residuals drop from 16.7 cm RMS using Schwiderski to 16.0 cm R M S using CSR 3.0. The altimeter bias increases by 0.3 cm although no clearly defined trend can be observed in the different arcs.

20.0 19.0 18.0 Epoch Figure 12: ERS-2 Altimeter Residuals: Schwiderski vs. CSR 3.0 Ocean Tide Model 20.0 18.0 16.0 14.0 1 2 . 0 8 10.0 8.0 6.0 4 . 0 2.0 0 . 0 Epoch Feure 13: ERS-2 Altimeter Bias: Schwiderski v s . CSR 3.0 Ocean Tide Model The impact of the detail in the ocean tide modelling in the evaluation of sea surface topographies can add up locally to several centimetres as can be seen in Figure 14.

Figure 1 4 : Schwiderski v s . CSR 3 . 0 Ocean Tide Model The impact of the ionospheric correction is practically negligible, most likely due to the very low current solar activity that makes this correction very small. In periods of high solar activity, l i i for Envisat, in which this correction should become larger, the use of more accurate models than Bent should bring noticeable improvements.

Altimetry Products from ERS and Envisat In order to obtain precise altimetry products, the orbit determination should ideally not use the altimeter data as tracking data, such that aliasing of the models used in the processing i n t o these products is avoided. A typical example is the ESOC mean sea surface model used in the processing of the ERS POD solutions, whose errors risk W i g propagated into the monthly dynamic topography models which are estimated and published on the WWW.

For ERS routine operational orbit determination, altimeter data are an extremely valuable addition to the set of tracking data. For this purpose, a mean sea surface model like the one derived at ESOC is ideally suited, as long as it is used with a consistent dynamic SST model. This model computed on a grid with size of 0.3" is based on ERS-1 altimetry data from the geodetic phase and produces much better residuals than the existing geoid models. Similarly, for ERS-2 POD using laser data, the processing of the altimeter data in the same way results in the best possible precise for verification of the routine operational products. The ESOC mean sea surface model was also ideally suited for the relative altimeter calibration between ERS-1 and ERS-2.

The dynamic SST models computed monthly from ERS-2 POD are affected by the accuracy of the ESOC mean sea surface model. The latter agrees with state-of-the-art geoid models up to degree and order 1 7 , and no significanterrors must be expected here. The SST models are computed to degree and order 23, and a constant (with time) error in these higher-degree terms will be aliased into each of the monthly solutions.

This will have no discernible impact on the variations which are observed between the different months, and which have very clearly shown the effects of the well-known recent El Nino event It has been demonstrated that with the ERS-2 PRARE dam, a precise solution independent from altimetry, but with a very similar accuracy, can nowadays be obtained and it is expected that this will be even more the case for Envisat, thanks to the almost complete global coverage from DORIS. The ESOC mean sea surface model will again be a valuable tool in the relative calibration of the ERS-2 and Envisat altimeter instruments, after which a continued production of monthly dynamic topography models with a delay of less than a few weeks will be possible.'If more detailed geoid models from dedicated gravity missions become available during the lifetime of Envisat, absolute dynamic topography maps data can be obtained in near- real time f r o m the Envisat POD carriedout at ESOC.

Aiming to support the Envisat mission with the highest level of accuracy both in data processing and environmental modelling, ESOC is developing the Navigation Package for Earth Observation Satellites (Napeos). This package, based on several years of experience in precise orbit determination and the processing of tracking data, shall be responsible all orbit and manoeuvre related activities for Envisat, from the retrieval of the data to the dissemination of both operational and precise products.

Napeos shall inherit from the ERS flight dynamics software the knowledge in precise orbit determination and orbit control, including the experience in automatic spacecraft operations and real time product generation. Based on this experience Napeos has been designed as a self-contained package capable of performing all activities required for spacecraft operations, from the data acquisition and pre-processing to the orbit determination (operational and precise), precise product generation and multi-arc physical parameter estimation. On top o f this, the most advanced software engineering techniques and the extensive use of standardisation should make of Napeos a product easy to maintain and enhance.

CONCLUSIONS Based on the experience accumulated during the ERS missions, ESOC is capable of computing orbits with an accuracy equalling those of the world leaders in this field. This experience, combined w i t h the use of the latest available models, will ensure the availability of high-precision orbits and altimetry products for Envisat, within days from data take.

The various analyses and comparisons shown in this paper further demonstrate the capability at ESOC to accommodate new models and tracking data types as they become available. This will be further improved by the use of the Napeos package, for which the capability to implement new models was one of the design- driving feams.

REFERENCES 1. Tapley, C. K. Shum, J. C. Ries, S . R. Poole, P. A. M. Abusali, S. V. Bettadpur, R. J. Eanes, M. C. Kim, H. 3. Rim, B. E. Schutz.

The TEG-3 GeopotenxialModel Center for Space Research, The University of Texas at Austin, Austin, TX 78712 USA 2. M. M. Romay Merino, R. Piriz, R. Zandbergen, J. M. Dow ERS-2 Altheter Calibration at ESOC ESA/European Space Operations Centre, Robert-Bosch Strasse 5,64293 Darmstadt, Germany 3. P. N. A. M. Visser, R. Scharroo, R. Floberhagen, B. A. C. Ambrosius Impact of PRARE on ERS-2 Orbit Determination Delft Institute for Earth-Oriented Space Research (DEOS), Iuuyverweg 1, 2629 HS, D e l f t , The Netherlands 4. P. N. A. M. Visser, R. Scharroo, B. A. C. Ambrosius, Incorporation of PRARE on ERS-2 Orbit Computation AGU 1997 Spring Meeting, May 1997, Baltimore, Maryland Delft Institute for Earth-Oriented Space Research (DEOS), Iuuyverweg 1, 2629 HS, Delft, The Netherlands 5. P. N. A. M. Visser, R. Scharroo, B. A. C. Ambrosius, R. Noomen Incorporation of PRARE on ERS-2 Orbit Computation EGS XXII General Assembly, April 1997, Vienna, Ausma Delft Institute for Earth-Oriented Space Research (DEOS), Kluyverweg 1, 2629 HS, Delft, The Netherlands 6 .

rgen, R. Piriz, J. Dow e d Assembly, April 1997, Vienna, Austria ESAiEuropean Space Operations Cenfxe, Robert-Bosch Strasse 5, Dmstadt, Germany 7. R. Scharroo, P. N . A. M. Visser Field ~ m F r o v e ~ e n t for the ERS Satellites Delft Institute for Earth-Oriented Space Research (DEOS), Kluyverweg 1, 2629 HS, Delft, The Netherlands 8. R. J. Eanes, R. and S. Bettadpur The CSR 3.0 Global Ocean Tide Model Technical Memorandum, CSR-TM-95-06, December 1995 Center for Space Research, The University of Texas at A u s t i n , Austin, TX 78712 USA 9. M. Rutkowska, M. M. Romay Merino Estimation of the Borowiec S t a t i o n Position OAD Working Paper 546 ESAiEuropean Space Operations Centre, Robert-Bosch Strasse 5,64293 Darmstadt, Germany 10. IERS Technical Note 21: “IERS Convention 1996” P a r i s , July 1996 11. PRARE WeeklyReports GeoForschungsZemtmm, Potsdam, Germany 12. DORIS System Description www-projec~cst.cnes.fc8060 CNFS, 18 Avenue Edouard BELIN, 31055 Toulouse Cedex, France F. Delhaise*, 0. Mikkelsent and S. PaUaschke WA/ESOG, Robert-Bosch-Strasse 5, D-64293 Darmstadt9Germany The orbit detemination programs often apply a least squares estimator which provides good statistical infomation on the quality of the ob- tained solution, but does not provide adequate means for the identifica- tion o f error sources. Isolation of error sources may be achieved with certain difficulty, e.g., by selecting specific sets o f the tracking data and varying the solve-for parameters in successive runs.

However, the obtained results of this trial-and-error process are often ambiguous and unhelpful in.the identification of the source for the de- graded orbit determination result. This paper summarizes an investiga- tion into additional functionality for the detection of unmodeled orbit determination errors such as station or transponder delay biases or an unexpectedly high noise level for a particular station. The respective merits of two different basic approaches involving extensions to the common least squares method or, conversely, alternatives to the least squares estimator, were studied.

INTRODUCTION The trajectory determination problem can be defrned as the estimation of a set of p parame- ters denoted by a p-dimensional $ector 2 given an m-dimensional observation vector, 3 (with m D p ), the equations of motion f , and the statistical properties of the random noise E : 3 = ?(e>+: The orbit determination software used in the Flight Dynamics division at ESOC uses a Gauss-Newton iterative procedure based on a weighted least squares estimator. After lineariza- tion around an initial estimate k, , the estimate o f the differential correction is given by: -1 T

A$ = ( F ~ W F ) F w A$

+

where F is the matrix o f partial derivatives o f f with respect to 2 evaluated at 2 = 2,, , A 3 is the vector difference between the observations vector and the computed measurements vector *. EDS Industrien @eut.sdhd) GmbE based at ESOC i. TERMA, Copenhagen,Denmark error sowws loss functionis a useful measure of the degree to w tion, the diagonal elements of the error covariance matrix indicate the dispersion of the estima- ted parameters while the off-diagonal elements represent the interdependence (correlation) among errors in the solve-for parameters.

These statistics done are however somefimes insufficient to identify sowces of error in the of t h i s paper is to investigate additional functionality for orbit estimation process. The purpose detection o f m o d e l e d orbit determination errors. Two different basic approaches were inves- tigated (see Ref. 5 for M e r details).

0 Extensions to the common least squares method 0 Alternative to the least squares estimator EXTENSIONS TO THE LEAST SQUARES METHOD The Gauss-Newton algorithm based on the least squares estimator (also called the Z2 estima- tor) is widely used for solving the orbit determination as it constitutes a good compromise be- tween efficiency and complexity. The main justification for using a least-squares estimator results fiom the Gauss-Markov theorem. It states t h a t under the following hypothesis of the measurement noise distribution: 0 the matrix of the partial derivatives F has full rank, the measurement noise has zero mean, 0 the measurement noise has a covariance matrix positive definite known up to a multiplica- tive factor, then the weighted least-squares estimator is an unbiased linear estimator. If furthermore the weight matrix is the inverse of the noise covariance matrix then the least-squares method cons- titutes the m i n i m m variance estimator within the class of a l l unbiased linear estimators. A fur- ther mathematical justification for the least squares criterion is that it is the maximum likelihood estimator corresponding to a Gaussian distribution o f the random noise.

In practice, however these necessary conditions are hardly ever completely Nfilled. In the context of this study, further investigations have been perfoimed to find a way to diagnose the potential following problems: 0 poor choice for the weight matrix (the covariance of the measurement noise is usually not perfectly known), 0 poor observability of one or more solve-for parameters, nsn-zero mean value of noise, 0 outliers and bias in the observations data, 0 convergence problems in’the strongly non-linear case.

Before investigating possible solutions to the above problems, the so-called “hat” matrix is introduced. This is particularly useful because it yields a measwe o f the observations quality.

= f -&)

(3) where I is the identity rn projection matrix of the measure- is the so-called “hat” matrix and is ments space to the range (F) of the formulated as: -1 T

v = F ( F ~ W F ) F w

(4) The matrix V is a (m x m) idempotent matrix with trace and rank p. It is an important matrix which occurs repeatedly in regression work. I t s diagonal elements play a key role in determi- ning the variance of the residuals. It can be shown that (see Ref. 8): c.

var(ri) = ( 1 - V i i ) a”/ wi

where vii is the i-th diagonal element of the hat matrix, wi is the weight associated to the i- th measurement and CT is the variance of the measurement noise.

The diagonal elements of the hat matrix (called here the leverage values) are a good indicator of the geometry quality of the measurements combined with their given weight value. The high- er the value of the diagonal element vii , the higher the impact of the corresponding measure- ment on the estimated solution. The measurements whose diagonal elements of the hat matrix exceed 3p/m are called leverage measurements.

A s an example, the leverage values vii are pMted in Table 1 for a typical Ariane Geosta- tionary Transfer Orbit (GTO) simulated over a tracking interval of about 11 hours. This exam- ple consists of 2-ways ranges fiom Malindi (Kenya) and Vill&anca (Spain) and antenna angles from P e r t h (Australia). The corresponding weighted partial derivatives of each measurement with respect to each solve-for-parameter evaluated at the initial estimate is also given in the de- fined range ffom l to 9. The maximum value of the partial derivatives for all measurements with respect to a particular solve-for parameter is set to “9” and its minimum value is set to “1”.

The highest quality in terms o f geometry and weight is reached by the measurements num- bered 1 (Malindi), 52 (Malindi) and 55 (Villafianca), 2-ways ranges made at the start and at the end of the tracking interval. These are the computed observations which are the most sensitive to a slight change in the initial estimate. This is confiied by the corresponding large values of the weighted Jacobian matrix.

Table 1 also shows that the leverage values of the angular data are very small, demonstrating will not really influence the obtained solution. This is explained by the that these observations fact that the angular data are much less accurate measurements that the 2-ways ranges.

The following additionalinformation can easily be retrieved from the printed weighted Jaco- bian matrix. The semi-major axis is essentially estimated by the range measurements which are at the end of the tracking interval. The reason is that the semi-major axis is directly correlated to the mean motion and the longer the tracking interval, the better the mean motion estimate.

is poorly estimated by the station M a l i n d i which lies close to the equator (the The inclination iai

- ___11__1 F j_l__ - P -

-

Lime V o b m t . raw res Jacobian n0 stat.

stand student lever.

type hh:m (km. d e d 0 deg

-r TZ33- m

ERK 2 23:32 MAL RNG 120 1 2 2 9 9 7 32167.450 0.0064 0250 0287 0.238 3 23:33 PER AZI 120 323.118 0.0147 -0.573 u s 7 3 O.Oo0 111111 23:33 PER ELE 120 45.718 0.0108 -0.420 -0.420 0.Ooo 111111 5 2352 MAL RNG 131 41097270 O.Ooo8 5.031 -0.033 0.137 1 1 2 9 9 6 11 00.32 MAL RNG 0.0152 -0.590 145 54534S38 -0.616 0.080 1 4 2 9 9 5 16 01:32 MAL RNG 67443.624 0.0195 -0.756 2 5 1 9 9 3 159 -0.789 0.081 17 0152 MAL RNG 162 70316.844 0.0017 0.067 ’ 0.070 0.084 2 6 1 9 9 3 20 02:12 MAL RNG 166 72590.097 0.0131 -0.509 -0333 0.086 2 6 1 9 9 3 21 M:32 MAL RNG 169 74307.738 0.0078 -0.303 -0.317 0.086 2 7 1 9 9 2 25 03:12 MAL RNG 175 76203.585 0.0141 -0.548 -0.571 0.079 3 7 1 8 8 2 27 0352 MAL RNG 180 76186.637 0.0180 0.698 0.723 0 . 0 6 6 3 8 1 8 8 1 28 0353 PER AZI 180 320.816 0.0149 -0.580 -0.580 111111 O.OO0 29 0353 PER ELE 180 45.421 0.0202 -0.785 111111 -0.785 O.OO0 3 0 #12 MAL RNG 75491.652 -0.223 4 8 1 8 8 1 183 0.0057 -0230 0.061 31 W:13 PER AZI 318223 3.628 111111 183 0.0936 3.629 O.OO0 04:13 PER ELE 111111 32 183 43.706 0.0088 0.343 0.343 O.OO0 0452 MAL 72747.763 34 RNG 189 0.0163 -0.634 -0.653 0.058 4 8 1 8 8 2 0532 MAL 38 RNG 195 68170.470 0.0661 2.562 2.666 0.076 5 9 1 7 7 2 42 m12 MAL RNG 202 61659.378 0.0160 -0.622 -0.662 0.116 6 9 1 7 7 3 44 0652 MAL RNG 210 53040.764 0.0066 -0.258 -0281 0.160 7 9 1 7 7 3 47 07: 12 MAL RNG 216 47879.329 0.0008 0.031 0.034 0.169 8 9 1 7 7 4 49 MAL 35915.199 0752 RNG 230 0.0179 0.697 0.803 0 . 2 4 6 9 7 2 8 8 4 0832 MAL 242 52 RNG 29556.943 0.0081 5.315 -0.626 0.746 9 4 2 8 9 4 1011 VLL 55 RNG 125 37818.158 0.0135 -0.525 -2.042 0.933 9 6 9 6 7 4

- - - - - - -

a. column5 lists the true anamaly (deg) cmreqmnk toeachmeamement; columns 8 and 9 list the standardized and studentized residuals respectively; column 10 lists the leverage val- ues and the l a s $ column gives the scaled partial derivatives of all meamemem wzt. each of the solve-for parameters: A: semi-major axis. E eccentricity, I : inclination. N right ascen- sion ofthe ascending node, W: mgument of perigee and V: true anomaly.

Weighting Errors If the number of measurements i s large enough for each measurement type and ground-sta- tion, a possible weighting problem can be diagnosed by comparing the quadratic mean of the standardized residuals block by block. Measurements are defined to belong to the same block if they are of identical type and from the same s t a t i o n . A standardized residual si i s defined as: where wi is the weight value of the i-th measurement. The ratios of the quadratic means should be close to 1 if the weighting correctly reflects the measurement noise.

. A reasonable weighting value of each block can be deduced from the obtained residuals. A suggested weight value for block “k” is: (7) e Perth which is ide a random noise franca and 1.05 for Perth which corresponds to the exact correction to bring to the assumed noise level of the three stations.

The Problem of Observability Multicollinearities among the solve-for parameters (i.e. they are highly correlated) result in much larger variances and correlations for the least squares estimators. This implies a much greater likelihood of a poor estimate of their respective parameters. Knowledge of multicolline- arities and their attendant problems is the first step in correcting its deleterious effects. Nume- rical comparison of the magnitudes of the estimated solve-for parameters and their variance and covariance must be made with standardized variables in order to remove the d i s t o r t i o n s due to different scales. In the normalized space, a measure of the inflation of the variance of least squares estimators due to multicollinearities is done via the so-called V a r i a n c e Mation Factor (V.I.F.). These elements are the diagonal elements of the inverse matrix of R which is a (p x p) matrix defined as (see Ref. 4, Ref. 8): -1 T R = K (F WF) 3C-l ( 8 ) where K is the diagonal matrix formed by the square root of the W-norm of each column of the Jacobian matrix F.

Another fundamental diagnosis of lack of observability is based on the spectral analysis of R and especially of the smallest eigenvalue of R ( p & ) . It can be demonstrated that the large components in the (standardized) eigenvectors corresponding to eigenvalues near zero identifv the solve-for parameters that are involved in the multicollinearity (see Ref. 4, Ref. 8).

The factor by which the magnitude of the solve-for parameters vector i s increased due to multicollinearity can be deduced from the following formula (see Ref. 10): N N

where 3 is the estimated solve-for parameters vector in the normalized space, 2~

is the exact solution in the normalized space, G is the variance of the measurement noise and pj are the eigenvalues of the matrix R.

As an illustration, the variance inflation factors, the eigenvalues and eigenvectors of the ma- t r i x R and the “expansion” factor of formula (9) are printed for three simulated orbit determina- tion runs applied to: 0 a GTO orbit deteimination with coverage from four ground-stations, 0 a geostationary (GEO) orbit determination with a coverage from the four ground-stations, ~- ~ -~ Table 3: Eigenvalues of the matrix R Table 2 shows a clear lack of observability of the right ascension o f the ascending node Q , the argument of perigee o and the true anomaly ffor the geostationary orbits which is well ex- plained by its s m a l l inclination and eccentricity values. For the first GEO for which the coverage is perfomed by 4 stations, only one eigenvalue is close to zero while the GEO covered by a single station shows two direction of lack of observability. These directions given by the comes- ponding eigenvectors of R are the following: ic

GEO-1: el = - 0 . 6 i + 0.6Q - 0 . 4 f

GEO-2: 81 = 0.6Q + 0.70 - 0.3f (10)

e2 = 0.7Q - 0.60 + 0.3f

The expansion factor of equation (9) is also a relevant indicator of an eventual lack of obser- vability of certain parameters. The value of this factor is much larger for the GEO orbit types (it equals 1 and 6, respectively) than for the GTO (= 2 . 1 0 - 4 ) . In conclusion, it is clear that the three angles Q , o and f are very poorly observable, only the s u m of the three angles can be determined with accuracy. This is a well known result for the orbit determination of geostation- ary orbits but this illustrates the utility of these additional parameters in diagnosing the attaina- ble degree of accuracy.

Accommodation to multicollinearities Hoerl (Ref. 9) fust suggested using a ridge-regression like algorithm to control the inflation This algorithm minimizes the and general instability associated with the least squares estimates.

s u m of squares of residuals with the constraint: where a is a positive real number and K is the matrix defined for the equation (8). The rela- tionship of a ridge regression estimate to an ordinary estimate is given by a form of the follo- wing kind (see Ref. 10 for further details): rors).

our software, an approach like the one given by equation (1 1) was selected. The operator has the possibility to give an initial estimate of the covariance matrix of the initial estimate pax different from infinity. In t h i s case, the least squares differential correction is computed at each iteration as: This solution accounts for the fact that the initial estimate Io is known to be accurate to a confidence level given by p . Therefore, any solution is constrained to satisfy the a-priory realization 2, to within the & t s of its uncertainty so that equation (11) is guaranteed to be ve- rified.

This algorithm i s especially useful when the orbit has to be estimated based on only a few measurements, eg. shorter after separation or a€ter a long-duration manoeuvre.

A comparison between the two methods (12) and (13) has been performed. Solution (13) has certainly the advantage that it does not depend on a tuning parameter IC. However, (13) only ensures convergence to a solution close to the i n i t i a l estimate. It will not ensure that the obtained estimate is more accurate t h a n the least squares estimator. Therefore (13) is suitable only when the least squares estimator cannot yield any solution at all. In other situations of real multicol- linearities, a ridge estimate given by (12) is more favourable in order to ensure a more accurate solution than the usual least squares estimate.

Outliers-Leverage Measurements An outlier among residuals is one that is far greater than the rest in absolute value. We must distinguish caremy between a large residual caused by an inadequate model and a large resi- dual caused by poor data (like incorrect operations of measuring means). The former can be remedied by improving the model; the latter has no remedy. A criterion such as that of least squares is very sensitive to large residuals. A few large outliers can transform a potentially use- ful solution to nonsense (e.g. a negative airdrag coefficient). It is therefore important in least squares estimation to detect outliers and attempt either to eliminate them or to improve the mo- del so that they do not influence the estimation.

In our software, this problem is tackled in two ways. On one side it may automatically reject an outlier and on the other side it may use a so-called robust Z2 estimator to proeeed to a gentle elimination of large residuals.

The automatic rejection of outliers is based on the absolute values of the standardized resi- duals. The measurement is rejected if this absolute value is larger than a given threshold. Alter- natively the studentized residuals could also be used as rejection criteria. These are the raw residuals ri scaled by their estimated standard deviation: s t u d e n ~ e d cri ased on residuals ore closely residuals. But acc not always an acceptable rejection criterion. The reason is that the leverage measurements have, by definition, a large value of the studentized residual. Therefore they are easily rejected even if they are just slightly off which is easily the case for leverage measurements which are very sensitive to the initial estimate. See, for example, the high value of the studentized residual of measurement 55 of Table 1 which is not an outlier.

Automatic rejection is not always advisable. Sometimes large residuals provide information which other data points cannot. They may indicate physical effects which are not included in the mathematical model of the orbit propagation.

A s a compromise, we may consider that a large residual indicates that the observation is im- probable and it should be assigned small weight (no cutoff is used). Such a robust estimator which i s by definition less sensitive to large e r r o r s has been implemented as a possible option.

The basic idea of t h i s algoiithm is to define the weight value of each measurement as function of the value of its standardized residual from the previous iteration (see Ref. 3, Ref. 4). This then yields to a reduction of the weight value for the measurements that have large residuals at the previous iteration. The reweighting at each iteration is defined a s : ifri = 0 where s is also calculated from the residuals. It is taken as the median deviation of the resi- duals (Ref. 4). The HAMPEL function y~ is defined as follow: with the values a=1.7, b=3.4 and d . 5 .

Being less sensitive to measurements with large residuals, t h i s algorithm might in some cases point out better than the common least squares algorithm the error sources such as a station bias or an abnormal large measurement noise for a particular station. As an illustration, a GTO orbit was simulakd over one orbital revolution with 2-ways range from four stations: Perth (66 meas.), Kourou (60 meas.), Wllafranca (78 meas.) and Malindi (58 meas.). A station range bias of 600 meters was htrduced for Kourou. Table 4 l i s t s the mean and r.m.s. values obtained by Table 4 shows that the Kourou bias is detectable in a nearly exact way by the robust Z2 me- thod via the mean and rms. values. This method consequently yields a solution of a higher de- gree of accuracy than the common Z2 method. This is a distinct advantage of this robust estima- tor.

However, in order to be able to pinpoint error sources such as individual station biases the following requirements must be fulfilled. The quality in terms of geometry and weight (which can be expressed as the s u m of the leverage values) of the error-free measurements should be sufficiently large for the algorithm to converge to the solution p r i m d y defmed by these error- free measur&ents. Thus, the residuals of the faulty measurements will become large. They are subsequently automatically downweighted by the robust estimator and will then no longer in- fluence the solution.

Convergence When starting the iterative process defmed by equation (2) with an initial estimate far from the solution, the actual contours o f the non-linear loss function are not well approximated by the linearization. The Gauss-Newton procedure may converge very slowly, it may oscillate widely around the solution or it may even fail to converge altogether. An alternative to Gauss-Newton linearizationprocedure is the so-called steepest descent method. The basic idea is to move f r o m an initial estimate into the direction of the “steepest” descent of the loss function. This direction changes continuously as the path is followed. Although this method initially converges rapidly, it slows down when the solution approaches the vicinity of the minimum. Marquardt (Ref. 13) proposed an algorithm which performs an optimal interpolation between the two techniques.

The implementation of the Marquardt algorithm has been performed as a possible option, by adding a constant hk to the diagonal elements o f the normal matrix. The sequence of positive red number hk has to decrease rapidly enough after each iteration to ensure convergence. The following sequence was chosen (Ref. 4): yk k is the iteration number hk = (17) ( k + 1)2 ’ with yk the diagonal elements of the normal matrix at iteration “k”.

Il O ~ t i ~ a t o r One major drawback of the least squares method is its lack of robustness (i.e. its sensitivity to large errors). The I , and I, norms are two possible alternatives for minimizing the vector of residuals of the regression equation (1) (other choice for the norm are possible but seldom used): For the orbit determination with eventual large errors w i t h i n the tracking data, the Z , crite- rion must be rejected because it assigns high weight to large residuals (since the I , criterion is to minimize the largest residual).

In the opposite, the Zl criterion aims to minimize the s u m of the absolute value of the resi- duals. It is therefore much less influenced by large residuals than I , or even Z2. I , solution is the maximum likelihood estimator corresponding to a distribution of the error which goes to zero far more slowly than the normal distribution. It, therefore, encompass large residuals that would be extremely improbable with the least squares Criterion.

In our software, two weighted I , algorithms may be used as a possible alternative to the usu- al least squares method: either a modificatioa of the simplex method as implemented by Barro- dale and Roberts (Ref. 2) or a modification of the dual simplex algorithm as proposed by Abdelmalek (Ref. 1). These two algorithms produced results w i t h a comparable degree of ac- curacy and computing time.

The weighted I, method was tested on numerous example cases w i t h conditions such as a bias on the station or transponder delay, a bad initial estimate or a high noise level for a parti- cular station. These examples showed that a degree of accuracy similar to the common least squares method is attained by the I, solution The rate of convergence is only slightly lower, especially in the proximity of the approaching the solution. Like the iteratively reweighted least squares method (see above) the solution is not influenced by a high noise level of a particular station. Furthermore, and contrary to the least squares method, the exact noise level of each sta- tion is exactly reflected after convergence by each station’s rm.s values. The I , method can also be used to indicate the exact value of a previously unknown station bias (see Table 4), where least squares would tend to even out the residuals for all stations.

ACKNOWLEDGMENT The authors wish to acknowledge the invaluable assistance of Wolfgang Peterhiinsel (EDS Industxien Deutschland GmbH, based at DOC) in setting up the test environment and simula- ting tracking data.

1.

3. ction to Overdete Springer-Verlag New-York Inc, 1990.

R4. Garrou J.P. “Spatial Mechanics”, Vol 1, CNES, 1995 R5. Delhaise E et al. “Study of Fault Detection, Isolation and Recovery Based on Tracking Data“, ESOC Final Report, to be issued, 1998.

R6. Deutch R. “”EstimationTheory”, Prentice-Hall Inc, 1965 R7. Draper N . R . and Smith H. “Applied Regression Analysis”, John Wdey & Sons, Inc, 1966 R8. Gunst R.and Mason R. “Regression Analysis and its Application”, Dekker, Inc, 1980 R9. Hoerl A.E. “Application of Ridge Analysis to Regression Problems” Chemical Enginee- ring Progress 58, p. 5459,1962 RlO. Hoerl A.E. and Kennard R. W. “Ridge Regression: Biased Estimation for Nonorthogonal Probleins”, Technometrics, Vol 12, No.1, 1970.

R11. Hubert M. and Rousseeuw P . “Robust regression with both continuous and binary regres- sor~~’, 1998 R12. Koch K. “Parameter Estimation and Hypothesis Testing in L i n e a r Models”, Springer- Verlag, 1987. ~- R13. Marquardt D. W. “An Algorithm for Least-Squares Estimation of Non-linear Parame- ters”, SIAM, Vol. 11, p. 431,1963.

R14. Meyer S.L. “Data Analysis for scientists and Engineers”, John Wiley & Sons, Inc, 1975 R15. Robison. E.A. “Least Squares Analysis in Terms of Linear Algebra”, Goose Pond press, 198 1 R16. Soop E.M. “Handbook of Geostationary Orbits”, Kluwer Ac. Publishers, 1994.

M S. Kawase.

A concise accuracy model is derived for orbit determination and prediction. Optical and radar tracking, in the form of single-site single- pass obsewation, are evaluated. Orbit prediction errors are formulated analytically as functions of tracking arc length and predictiontime length.

The relative merits of the optics and radars are clarified through the analysis. These results offer a basis for discussing short-term strategies of spacecraft near-miss avoidance in low earth orbits.

INTRODUCTION Suppose a manned space s t a t i o n in a low e a r t h orbit receives a warning of close approach by another orbiting object, probably a satellite no longer in service. Urgent tracking and orbit d e t d o n of the object will be needed, to analyze the approaching geometry and to work out an avoidance maneuver. It will be essential to assess the accuracyof the tracking and orbit determination, because the planning of the avoidance maneuver depends entirely on that accuracy.

It would be best if we could model the accuracy of orbit determination for every possible form of trackmg, but this would be a difficult task if the tracking may involve multiple observation passes using multiple sensors at multiple sites. A viable, concise model of the orbit detemhation accuracy does n o t seem likely. However, if the orbit determinaton is "Urgenf7 as mentioned above, then its traclung period must be short, probably with single-site single-pass observation, and this may change the situation. The present paper will show that the orbit determination accuracy can be modeled in concise formulations for that kind of short arc tracking.

In the present paper we first derive the accuracy model for optical tracking. The derivation is analytic, with its geometrical meaning clarified. The accuracy model is next modified to cover radar tracking, and finally the a n a l y t i c a l results are checked against numerical evaluations.

OPTICAL OBSERVATION AND ORBIT DETERMINATION Assume that the earth is spherical with radius R and the tracked object is in a near-circular orbit of altitude h , as illustrated in Figure 1. The orbital path is assumed, for ease of analysis, to pass near the zenith of the tracking station, T. (Tius restrictive assumption will be relaxed later.) Tracking observationsare made when the objTt is at Pz, which is directly above T, and when it is at PI and P 3 , which are at angle 8 from Pz. Thus we have three observation points, with 8 spec- the length of the observation arc. Although actual tracking will try to acquire as many data points as possible . Space Systems Section, Communications Research Laboratory; Kashima, Ibaraki 3 14-0012 Japan.

/ Phone: +I31299 84 7149, Fax: +81 299 84 7160, e-mail: kawase@crl.go.jp.

ion pas assume only points because the srn effect of is out 0 of our interest.

Figure 1 Tracking Geometry optically observed are two angles u and v as i l l u s t r a t e d in Figure 2, where u measures the object’s angular position dong the flight path, and v measures that across the path. A positive v in Figure 1 would point toward the back of the page.

Any orbit determination needs an a-priori orbit of the object and t h i s is assumed to come from cataloged orbits with a typical accuracy of several kilometers.’ We assume here, for ease of analysis, that the observations made at PI, P 2 , and P 3 are so accurate t h a t these observations newly determine the orbital elements. The sole exception is the semi major axis (SMA), which cannot be determined &om single-pass tracking. The S M A has to be improved by comparing t h e predicted time of arrival into the observer’s field of view against the actual time. So the determination of S M A is left out of our discussion and the other five orbital elements are assessed of the accuracy of determination. The reference time of the orbit determination is set at the t i m e the object passes the zenith point Pz.

I Flight path .

U I

Figure 2 Optical Observations Figure 3 Center-Shift Components a constant displacement A 4 along its flight path, so we have three orbital elements ( D, , D, ,A4 ) to define the in-plane all zero. If each varies f r o m zero slightly, then orbital motion. Suppose these elements were initially variations arise in the tracking observations; this is examined in detail below.

Figure 4 Observation Variation due to Dl Figure 5 Observation Variation due to 0, An orbital arc visible to the tracking station is short ifthe orbital altitude is low. Our observation points PI , PZ , and P 3 then lie nearly in a line as illustrated in Figure 4, and a small fictitious center- s h i f t Dl makes these points move down by Dl to PI7, PZ7, and P 3 ’ . Strictly speaking, 4’4’ exceeds 4P3 because displacements arise along the path -see Eq. (A2) of the Appendix- but this excess is small so that the change fiom (PI , PZ , P3) to (PI’, Pz’, P 3 ’ ) may be regarded as a parallel displacement. Displacement 44’ is detected at T through its transversal component Dlsin6; dividing this by the distance Pl = h / cose makes the following variation arising in the observed u of P I :

6ul = -(q /h)sinewse (1)

The observed u of Pz has no variation, so 624, = o (2) while P3P3’ causes the same variation as 45’ but with a change of sign: 6 2 4 , =(Dl /h)sin6cose (3) Orbital perturbation and the earth‘s rotation are neglected because the objk~t’s time of flight f i o m PI to P 3 is short. Consider next the center-sbift D, and assume that this shift is a “modified center-shift” as defined in the Appendix. That is to say, the orbital circle is regarded as rotating slightly, as illustrated in Figure 3, around fixed Pz. This causes the flight path near P z to incline f r o m being horizontal, as i l l u s t r a t e d in Figure 5 . The linear arrangement of PI, P z , and P 3 then inclines by 0, / ( R + h ) , thus causing a displacement in PI by 44’= 0, / ( R + h ) ehtan6 , and its contribution is to the observation variation 624, = -- D2 sin2 8 R + h e PZ does not move, so that 6u2 = 0 6u3 = -- D2 sin28 R + h Figure 6 Observation Variation due to M Finally consider M , the along-path displacement (see Figure 6). Its transversa,, component causes 1 ie following variations to the observed u of PI Pz , and P3: 6ul =(Mlh)cos28 ( 7 ) 6u2 = M l h (8) 6u3 = ( M I h)cos2 0 (9) Eqs. (1) through (9) are combined to form -sc - s 2 / A c2 where s=sin0, c=cos0,and A=(R+h)lh.Notethat 4 , D 2 , a n d M donotcausevariations in the observed v ; this allows us to invert the relationship, thereby obtaining f 1 ' -- 0 - 2sc A Ac2 (10) \ 0 1 ' 0 ) which ascribes the orbital element variations to the observation variations. (The left-hand side in this context should be SO,, 6 0 , and 6M while we omit '' 6 " s for simple notation.) Now regard 6q, 6u2, and 6u3 as denoting observation errors; then Eq. (10) evaluates the error in the in-plane orbit / determination. 194 orbits. Suppose the object has revolved in its orbit as much as anomaly or true anomaly without diffamce since the orbit y circular.) The orbital element errors Dl and 0, of Eq. (IO) give rise to an along-path error.

s error equals the s u m of the 61s of Eqs. (A2) and (A4) of the Appendix, which makes 2 4 sin y + 2 0 , (cos tp - 1) , and this is rewritten, by using Eq. (IO), as (h/sc)(-6ul+6~3) ~ i n v +(Ah/s2)(-6u1 +2c26u2 - 6 ~ 3 ) ( ~ 0 ~ t p - l ) .

Evaluating this quantity in standard deviation is our f i n a l step. Assume that 6ul, 6u,, and 62.4, are independent of each other and have identical n o d distributions with standard deviation 0,; we then have t h e e r r o r evaluation of Next, for t h e radial error component, sum up the Sr s of Eqs. (A3) and (As) of the Appendix, to make -D1cosyl +D,sin\y. Rewritethis byusing Eq. (1O)tohave h Ah -(6q -62.4,)cos y +~(--62.41 +2c26u, -62.43) sinyl, 2sc 2s and evaluate this in standard deviation in the same manner as the above, to obtain the evaluation of

, 1 + 2 ~ 0 s ~ e

sin2 y radial:

&sine cos2.e sin2 e

Recall that Eq. (10) Contains the relationship A 4 = h 6u,, indicating that a bias error exists along the path. Evaluating this in standard deviation makes another error evaluation of along-path, bias: h 0, (13) Figure 7 Basic Transversal Resolution Evaluations ( l l ) , (12), and (13) have the following physical meaning: If we look upward from station T and project the angle observation error to the flight path (see Figure 7 ) : then the projection factor 0.1 revolution angle 9 [degl Figure 8 Magnification Factor for Optical Tracking.

From top to b o t t o m in each group are for 8 = 20,40,60 deg fdls onto the width of 5 h 0, along the path. We call t h i s width the “basic transversal resolution” and it is qual to the along-path bias (13). Evaluations (1 1) and (12) both have the form of “basic transversal resolution x magmfieation fhetor7’,and the fixtors vary with \y and 6. At the altitude of the space station (h = 435 k m ) , A is 15.7 so that the terms with A become dominant in t h e square roots of evaluations (1 1) and (12). The along-path error thus takes its maximum at M a revolution after the orbit determination, and the radial takes its maximum at 1/4 and 314 of a revolution. T h e along-path error maximum is four times larger in magnitude t h a n the radial maximum. By recalliqg the dominance of the A -terms and looking at Eq. (1 0) once more, we see that the orbit prediction error originates mainly fiom the D, determination error. That is to say, determining the flight path’s i n c l i n a t i o n against the horizontal plane (refer to Figure 5 ) is t h e major difliculty in determining the orbit. The observation arc 6 affects the magmfication factors as shown in Figure 8, where the radial for 180 deg through 360 deg i s omitted because it is periodic. A smaller 0 may cause a problem in that the newly determined and predicted orbits become less accurate, in some region of y , t h a n the a-priori orbit. This problem could be avoided by weighting the a-priori orbit, but at the cost of discarding our accuracy model. The along-path factor in Figure 8 becomes less than one near y = 0 and y = 360 deg, at which points the bias error (13) becomes dominant.

OUT-OF-PLANE ERROR EVALUATION T h e inclination of the orbital plane has two degrees of fieedom. Consider first an inclination such that the orbit-normal vector leans towards T (see Figure 9) by a small angle i , . This causes the pohts PI , P z , and P 3 to move uniformly by (R+h)i, to PI’, P2), and P3’, all in parallel to the local horizontal plane of T. Accordmgly, the observed v of P, varies by ss’/TP, , and the same occurs for the observed v of P 3 ’ , so that we have 6v, = 6v, = A c 0 ~ e . i ~ (14) / I T

b

\\

i2 Figure 9 Observation Variation due to i, Figure 10 Observation Variation due to i2 The other inclination i s illustrated in Figure 10, where the orbital plane rotates around the axis OT by iz , and this makes the hear arrangement of PI , P2 , and P3 change its orientation horizontally by iz with P2 being fixed. The displacement 45 '= h tan 0 - iz then causes a variation in v1 as Sv, =-sine- i2 (15) and the same, with an inverted sign, in v3 as 6v3 =sin& iz (16) Only two observation points PI and P3 are considered because we are determining two orbital elements. Note that il and i2 cause no variations in the observed u . EQuations (14), (15), and (16) are combined to form

( E ; ) =y Ac - s ) ( ; ) s

This can be inverted in order to evaluate the out-of-plane orbit determination errors: i, = (6vl+8v3)/ (2Acose) (17) i2 =(+ +6v3)l(2sin0) (18) The orbital element errors il and i2 cause an orbit prediction error t h a t points out of the plane. This error equals, at a revolution angle w fiom P2, (R + h ) ( il cos y + i2 sin y ) , and this is rewritten, by * apply@g Eqs. (1 7) and (1 8), as SV, +6v3 -6v,+Sv, WS\v + s h y ) .

( R + h ) ( 2 A c o s 8 2sine Evaluating this in standard deviation assumes the same statistics for Sv, and 6v3 as those for 6u s , resulting in the evaluation of / where 0, isthe deviation of the v -observation error. The out-of-plane malplification&&or has the same periodicity as the radial. It is shown, with its first period omitted, in Figure 8. If cr, =a, , the out-of-plane error is always smaller than the radial at the same revolution angle w. Evaluations (1 l), (12), (13), and (19) thus model the accuracy of the opticaUy tracked orbit determination and prediction.

T h e orbital accuracy may be affected by inaccurate modeling of the orbital dynamics, and such inaccuracy originates in m o s t cases from the atmospherical drag for low altitude objects. This kind of inaccuracy however, gives rise f i r s t to a change in SMA, and through t h i s then to a change in the along-path motion. S i n c e the determination of S M A was left out of our present discussion, we analyzed the error relationdip only between observations and orbital elements.

RADAR OBSERVATION Radar provides observations of range p , azimuth a , and elevation E . Variations arising in these observations are examined here, and the tracluag geometry of Figure 1 and Figures 4,5, and 6 are again referred to. We need only two observation points, PI and P 3 in Figure 1, because the radar observation (p, a, E) has one more degree of f r e e d o m than the optics. The other assumptions made so firr do not change.

Consider f i r s t the in-plane orbital elements. Figure 4 shows that v a r i a t i o n s in the elevations sl and s3 of PI and P3 are written in terms of u simply as 6z1 = 6u, and 6e3 = -624, , so that we have the following: 6 ~ 1 = -(Dl / h)sinecose (20) 6 ~ 3 = -(Dl / h)sineWse (21) D2 sin20

= -R+h

6E3 = - D2 sin2e

, R + h as1 = (M/h)cos20 (24) 6 ~ 3 = -(M/ h)cos2 0 (25) N e x t refer to Figures 4,5, and 6 and see t h a t the displacement 44' creates, through its line-of-sight component, variations in the range p1 of PI, as

ap, = -D, case

6p1= -R+h D2h sine

6pl = -Msin0 and similarly to that of Ps, w i t h some sign changes, as .

6p3 = -D1 COS^

6p, = - D2h sine

R + h 6p3 = Msin0 positions of the fixed stars seen in opti its elevation measurement may be observational field of view.

the radar actually does is thought to be the following. Let affected by an unknown bias error.

o elevations, make our tracking observation. The bias error Q = - e 3 , the difference between an accurate observation of can be made.

Accordingly, Eqs. (20) through (25) reduce to the following three equations: G = 0 . 4 (32) = ( 2 M l h)cos2 8 (34) Combine Eqs. (26) through (34) and denote s = sin0 , c = cos0, and A = ( R + h) / h to make -.VIA s l A 0 -2s21(Ah) and invert it as - 1 l c - 1 l c 0 -- -Ac21s A c 2 / s (35) S h

[ ” I - : ( M -S

in order to evaluate the in-plane orbit determination error due to the observation errors 6p, , 6p3 , and &E.

Evaluating the orbit prediction errors f r o m Q. (35) proceeds in the same way as that for the optics.

Set the error standard deviations of range and elevation to crp and Q Z , and the resulting evaluations are These now have the form of “range resolution x magmiication fkctor.” The constant k = ha; I a , , is the ratio of the elevation’s basic transversal resolution to the range resolution, which depends on each radar and on the orbital altitude. Except for this dependency on k , the way t h a t errors (36) and (37) vary with the revolution angle y is similar to errors (1 1) and (12), eliminating the need to plot them.

Particular to the radar is the absence of 11 s i n 0 a t the head and t h i s makes the radar highly accurate when t h e observation arc is short.

The along-path bias error from Eq. (35) is M = (-sin0.6p,+sin0.6p3+h62)/2, which is ’ evaluated in standard deviation as

along-path, bias: up /= sin20 k2

(38) Finally, there i s the out-of-plane error evaluation. In Figure 9, the displaceryent 45’ causes a y angle Lp,p,P,', which is larger than 6v1 the optically observed variation.

sm, and the az observations are thus ~ h y s i ~ l y equivalent to v1 and v3 . Therefore the out-of-plane evaluation can be based on (19), a s

outofplane: - ha, / - -

(39)

45 cos 0 sin e

where Q, is the azimuth error standard deviation. Evaluations (36) through (39) thus model the accuracy of the radar-tracked orbit determination and prediction.

NUMERICAL TEST Our analysis is based on a number of assumptions and approximations. The analytical results were therefbre checked against more exact numerical error-evaluations.

The assumptions of two-body orbits along with two or three observation points is the same. Again, the earth's r o t a t i o n i s neglected. The orbital element variations propagate, without approximations, to the observation variatiofls through the numerical processing of satellite motions and tracking observations. This provides the normal equation of the least square method, fkom which an exact error covariance matrix of 5 x 5 can be used to evaluate the orbital element errors. These errors are then Kepler-propagated in order to evaluate t h e orbit prediction error; this makes our check-reference.

The error standard deviations of optical and radar angles are a l l assumed to be 10 arcsec and that of radar range to be 21 m, and the orbital altitude to be 435 km; this makes constant k =l.

error [ k m l error CkmJ 10 j L 0.1 0.01

0 90 i a o 270 360 0 90 180 210 360

revolution angle v.' [degl revolution angle \y [degl a. Optical b. Radar Figure 11 Numerical Test

- is analytical; o (along-path), + (radial), x (out-of-plane) are numerical; 0 =28 deg

/ vtical evaluations were 0thm compared at the error-maximums (i.e., at \v =180 deg for along-path and at \v =90 or 270 deg for radial and out-of-plane) while the observation arc 6 was being varied. The results, shown in Figure 12, indicate the practical validity of our analytical model.

error-maximum [km] error-maximum [kml 0 . 1 1 .

0 20 40 3 observation arc 0 [degl observation arc 0 [degl a . Optical b . Radar Fignre 12 Numerical Test

- is analytical, o + x (same as in Figure 11) are numerical

The “zenith passing path” has been t h e most restrictive of our assumptions. L e t the orbital path be off the zenith by angle p as viewed from T. This causes the numerical evaluation for the optics to differ fiom what appears in Figure ll(a). This differenceywhich was seen only in the out-of-plane error, caused the evaluation plot to elongate and have an increased maximum. ( I t s minimum changed little.) This increase i n the error-maximum, which is shown in Figure 13(a), depends on f3 and 6 .

The same thing also occurs with the radar, where the along-path error suffers the increase in error- maximum as shown in Figure 13(b). The results suggest that, practically speakingyour analytical model still works for non-zenith passing paths, except for optical tracking with a short observation arc and a large off-zenith angle.

, error-maximum, out-of-plane [kml error-max imum. along-pa t h [ km] 3 3 0 10 20 30 40 0 10 20 30 40 I off-zenith angle B Cdegl off-zenith angle ) 3 [degl

50 i

a . optical b . Radar Figure 13 Effect of O f f - Z e n i t h Path (Numerical Evaluation) From top to bottom are for 0 =28,42,52,60 deg SUMMARY Although restricted to a particular case of single-site single-pass tracking, we have established optical and radar observations. The principal difference orbit determination accuracy models for between the optics and the radar, in tern of our analysis was the radar's superior accuracy of i n - plane orbit determination when the observation arc is short. Otherwise, the orbit prediction errors that arose were common to optics and radars, w i t h major error sources commonly being the dif6culty in determining the f l i g h t path's inclination.

That the orbital prediction e r r o r s t a k e their maximums and minimums at particular points of a revolution will be worth notice when near-miss avoidance strategy is discussed. M o r e precise m o d e l i n g obviously needs to consider the error in SMA. Its effects, however, will stay small in short- t e r m orbital predictions.

The present accuracy modeling Will offer a basis for discussing short-term Strategies of near-miss warning and avoidance in l o w earth orbits.

REFERENCES 1. S. A. Chamberlain, T. A. Slauenwhite, " U n i t e d States Space Command Space Surveillance Network Overview", Proceedings of the F i r s t European Conference on Space Debris, Dannstadt, Germany, 5-7 April 1993.

IX h a p Of Near-Circular 0 Consider an elliptical orbit with semi major axis a and small i n e , Figure Al and perigee at P. Let S be a satellite at true anomaly ae to the opposite of P. Find then the distance fiom 0’ to S.

The formula for the satellite radius is r = a(1-e2)/(1+ecos f ), which is approximated for small e by r=a-aecosf (All The distance in question is then written as 0’s = Jr2 + (ae)2 - 2raecos(?t - f) , which reduces, with e2 being neglected, to 0’s = a . The shape of our orbit is therefore a circle with radius a , centered at 0’, which is shifted f i o m 0 by D = ae .

Position Variation due to Center-Shift Suppose we have a satellite in a circular orbit, and this satellite is a t mean anomaly m . And suppose that the eccentricity changes f i o m its initial zero to a small e resulting in a center-shift by D = ae . Find how much the satellite’s position varies.

Satellite true anomaly f is related to the mean anomaly m by f - m = 2e sin m . The position thus varies, along t h e orbital path, by 61 = a( f - m) = 2aesin m = 2Dsinm As for radial variation, see Eq. (Al) and set 6r = r -a = -aecos f . Approximate ems f G ecosm [ I - (f - m)sin m] z ecosm, since e v - m) is small to the order of e’. The variation is then writtenas 6 r = - a e c o s m = - D c o s m We regard this D in Eqs. (A2) and (A3) as causing the satellite position variation.

Modified Center-Shift Suppose we have a satellite in a circular orbit, to which the following two events occur: a) The center shifts by D .

b) The satellite displaces along the path by - 2 0 .

This combined set of events is referred to as a “modified center-shift,” and it makes E & (AZ) change to 61 = 2D(sin m - 1 ) , while the 6r in Eq. (A3) remains valid with little error. L e t m’= m - 90 deg andusethis m’ to rewrite 61 and 6r as 61 = 2 D(cos m’- 1 ) (A41 Sr = Dsinm’ (As) What we are doing is better understood by looking at Figure A2 as follows: O u r initial orbit is centered at 0, with Q being a quarter revolution point. S is a satellite at revolution angle m’ fiom Q. I Now event a) occurs so t h a t our new orbital circle becomes centered at 0’’ while t h e new orbit passes Q because event b) assures 61 = 6r = 0 for a satellite at Q. Satellite S then experiences a position variation that obeys Eqs. (A4) and (As). Note that O’Q is a radius of the new orbital circle, while the length O’Q is virtually equal to OQ for a small D .

Figure A1 Shape of the Orbit Figure A2 Modified Center-Shift ORBITAL PERTURBATIONS USING GEOPOTENTIAL COEFFICIENTS UP TO €UGH DEGREE AND ORDER: EIGHLY ECCENTRIC ORBXI'S Rodolpho Vilhena de Morae* and Edwin Wmkt Several methads have been proposed for calculations o f the deity hction for a bigh value o f the eaxmicity, however they cannot be used when the high degree and order coefl[icients o f gravity fields are taken into account. The methoaproposedbywnnk' i s numeria& stable in tbis case, bot when i s used, a large number ofterms occurs in formulas for geopotential p3tUTbatiOllS.

Inthispaperweproposeanapplicationofexpansionsofsarnefunctionsofthe eccentric anomaly Eas well as Hansen d c i e n t s i n power series of@ - e), where e* is a fixed value of the eccentricity derived by Da Silva F d 4 .

These series are convergentfor all e < 1.

Recent applications of artificial satellites needs the description of the orbital motion under a precision of centimeters. For example, the radii component of position of altimetric satellites such as ERS-I, ERS-2 and TOPEXPOSEIDON m u s t be determined with a p r e c i s i o n of a few centimeters in order that the alhetric measurements can be conveniently used.

I i Also, in order to avoid c o l l i s i o n s of important and big spacecraftysuch as fbture space stationsy the position of objects (active satellites and space debris) must be computed w i t h a precision of the order of meters.

T a k i n g into account the perturbations due to the geopotential, theories of motion of satellites must be developed, as well as models for the potential, to attain the expected level of description of the sateUite's motion.

Within this aim, the well-known Kaula's geopotentiai perturbations thee$ can be

slightly modified (Wnuk6) introducing the lumped coefficients which group terms w i t h the same frequency. Lumped coefficients shpliiies the derivation of the expressions for the perturbations and enable us to consider a great number of geopotential coefficients.

* orup0 de Dhiimica orbitat e Plamtoiogia, DMA-FEGUNESP, 12SOO-ooO, chm&@M& SP, Bnlz& & . r o d o ~ u n e s p . b r t - . ~ o f t & e A d a a l ~ c z u n i v ~ , P c n n a m , P ~ o m a i l : ~ a m u & p i

- 0.Q besides near circular

are placed in orbits z 0.6 -0.7, geostation eccentricity eo mission).

ations of motion is applied in calculation of orbits in some applications one needs an analytical description of the satellite motion.

Analytical theories of an artificial satellite motion give formulas for perturbations that are in a closed form for the eccentricity only in a case of perturbations due to some zonal harmonic coefficients. In the general case, when an arbitrary degree and order spherical harmonic coefficients have to be taken into account, series of expansions in the eccentricity have to be used in formulas for perturbations. Some dif€icuIties occur when the analytical theory of a satellite motion is used in calculations of a precise position on a highly eccentric orbit. The source of these difficulties is the calculation of the eccentricity functions (Hansen's coefficients) and their derivatives. The Kaula's formula for the eccentricity h c t i o n is not numerically stable for a large values of eccentricity and simultaneously large values of indices.

a Several methods were proposed for calculations of the eccentricity function for high value of the eccentricity ( e.g. Szeto and Lambeck7, Gooding and King-Hele', Rosborough and Lemoineg,), however they cannot be used when the high degree and order co&cients of gravity fields are taken into account. The method proposed by Wnuk' is numerically stable in this case, but when is used, a large number of terms occurs in formulas for geopotential perturbations.

In this paper we propose an application of expansions of some functions of the eccentric anomaly E as well as Hansen coefficients in power series of (e - e), where e* is a fixed value of the eccentricity derived by Da Silva Fernande~~". These series are convergent for all e < 1.

PERTURBATIONS DUE TO THE GEOPOTENTIAL The geopotential Y expressed in orbital elements can be put in the following form (Wmk6): where J=J1 N J=J1

are the generalized lumped coefficients, c,m and SI,,, are normalized geopotential

coefficients, are fbnctions of the normalized inclination h c t i o n ~ ~ ( I ) = ~ , m ~ l - k l 1 2 (I) and of the eccentricity function G , (e) (Kada5), R y*(o, $2, M y 0) = k o + (k + q)M + m(S2 - 0) + (k - m)- , (4) j , = max [klkl + 2E((m - kJ + 1) / Z ) ] , kl = l k l + 2(5,, + &)) , [(k-m+1)/2] Y km = (1)E > N= maxZ , m f q l , E(x) is the Entier h c t i o n and the symbol * stands for summation with step 2.

The general form of the formulas for the f i r s t order geopotential perturbations in the quantity E (an orbital element or a component of the radius vector) is the following (Wnuk'910, Wnuk and Breiter"): where the amplitudes 4 4 ( a 7 e y l ) , BA4(aYe,l)are hctions of generalized lumped coefiicients C : , S," .

As an example, Fig. 1 shows the perturbations due to the geopotential for the Brazilian satellite SCDl. The perturbations obtained are given as knction of the order m of the harmonics. The 70x70 JGM-3 (Nerem at al.",) geopotential model has been used in calculations. The spectrum of the perturbations shows that coefficients of high de&ee.and order must be taken into account if it is necessary to attain precision of centimeters.

- Estimated values of perturbations in components of the radius vector (the radial, eory with numerical integration.

3\

a=7139!zm h e = 0.00457 I = 24.972 deg (D d

P

I U V\

1.OE-3 ' , I ~ l ~ , , I J , ~ l

0 I O 10 30 4 0 5 0 0 8 0 m Harmonic order m Fig. 1. Total perturbations due to the geopotential for the Brazilian satellite SCDl .

1 . W satcflita: SCDl l.OE+3 a=?l39km I m 3 2 e = 0.00457 E I = 24.972 deg Q 1.0€*1 E Y .g I.@€* 0 radii1 -

-

l.OS1

+ tlamvcrse

1&2 1 . 0 E - 3 1 . O H 0 4 0 5 0 8 0 7 0 10 20 30 Harmonic order m Fig.2. Perturbations in components of the radial vector for the Brazilian satellite SCD 1.

ram 1.00 a satellite: SCDl a = 7139kn 0.10 e = Q W 1=24972deg 0.01 0 IOW ZQY) 3om 4om 5ooo T i m (minutes) Fig.3. Differences between numerical integration and the analytical method for the low eccentricity orbit of the Bkzilian satellite SCD 1.

la's e c c e n t ~ c i ~ on G~,~,(+ is relate X t l n (4 by the following equation: The Ti~serand'~ (1889) definition of the Hansen's coefficients X;"(~)as well as the Kaula's' (1966) formula for the h c t i o n are sums of terms, which include factorials and binomial coefficients. This formulation is not numerically stable at the higher eccentricities. In the case of high eccentricities W a g n e r ' " and Gooding and King-Hele' replaced Kaula's formula by an integral formula, which was next used by Rosborough and ~ e m o i n e ~ in the sensitivity studies of%s orbiters for M ~ S gravity recovery. The integraI formula works very well for high eccentricities, however because of computation time and some numerical instabilities, its practical application is limited when indices I , p, q reach high values.

Following Da Silva Fernande~~'~ we propose to calculate the Hansen's Coefficients X:"(e) fiom the following power series of (e - e): where e* is a fixed value of the eccentricity. This series is convergent for a l l values of the eccentricity e < 1 such that ]e - e*l< p(e*). The values of the convergence radius p(e*) are given by Da Silva Fernandes3..The values of Hansen's coefficient of the larger eccentricity e are calculated from the power series with coefficients of derivatives of Hansen's coefficients of the smaller eccentricity e'.

The derivatives of the Hansen's CoefEcients are calculated with the use of the following formulas (~iacaglial'):

jPZS t P =

n - m + l ifn- n + m + l - s i f n + m + l 2 0 and S , ' 00 i f n - m + l < O jl={ oo i f n + m + l l Q The functionf(P) may be expressed as the following power series of the eccentricity e (JWf&Il16): The derivatives v y than be easy obtained, if we use the following relations for derivatives of the Bessel functions: Using the above formulas one may calculate Hansen's coefficients for an arbitrary values of the eccentricity and arbitrary values of indices. This method of calculation of the eccentricity hnction enables to obtain geopotentiat perturbations for orbits with high values of the eccentricily and high order and degree geopotential coefficients. As an example, fig.4 shows geopotential perturbations for the orbit with the eccentricity of 0.73.

One can see that even fQf the h o n k order of 40 perturbations are on a level of meters, and resonance effects have to be taken into account for the higher orders. Because of the resonance effects the comparison w i t h numerical integration (fig.5) is a bit worse then in the case o f near circular orbits, but still is on a level of a few m e t e r s .

l.OE*l ' V 1.0E-3 ' I I l l j l l l , l l l 0 10 20 30 w o s o e o m Harmonic order rn Fig.4. Total perturbations for the high eccentricity satellite orbit.

satellite: testl a = 24470 ktn e = 0.723

J - l - 7 - m IOW 2ooo 3ooo 4om

Time (rnin) Fig5 Differences between numerical integrationand the analytical theory for the high eccentricity satellite orbit.

satellite: testl a = 24470 Ian 0 radial e = 0.723

I = 61.0 deg + transversal

1.0€+3 normal p!

p? l.OE+Z a g l.oE+l

9 I.oE*

L 1.oE-I a IdE-2 I A W 1.0- 0 10 20 30 w 5 o e o m Harmonic order (rn) Fig.6. Perturbations in components of the radius vector for the high eccentricity orbit.

21 1 ions in the radial, transverse an has some finctions of the eccentric anomaly resence of the eccentric-and mean anomalies i n these formulas some difficulties occure. In order to overcome this difficulty let us consider the H(E) functions introduced by Da Silva Fernandes3given as where H ( E ) is an analytical h c t i o n of the eccentric anomaly E, e* is a given value for

the eccentricity e and Az = (e - e*). This expression is convergent for every M as long as

le - e*l c &e*), being h e ' ) a positive real number @a Silva Femndes').

Using the H(E) functions, and after a lenghty calculation, the formulas for the radial component Ar , the transverse component Ail and the binormal components Abcan be expressed by the following series, convergent for high eccentricities: N N O I L L {r2 ms(\v + sM) + r3 cos(y, - sM) + r4sin(\v + sM) + r,sin(v - S M ) ) >,

-Z

2 s=l t=l where A =

yh{i2, cosy + A , sin y +

m=l k=-N q=-Q

{A, COS(W + sM) + &[cos(Y-- sM) + A , sin(y + M) + A, sin(y - SM)))

where 2, = -3e- N N Q an

y,{bl cos(ty + w ) + b2 cos(y/ -a)+ b, s i n ( y + w ) + b, sin(y -a)

Ab = -c

m=lk=-Np-Q L L

+ C'' {b, COS(^/ + o + a) + b6 COS(Y/ -I- o - SM) + b, c d Y - w - M)

b sf. t=1

+ b, cos(ry - w - sM) + b, sin(ty + w + sM) + b,,, sin@ + w - M)

+ b,l s i n ( y - w + sM) + bI2 sin( y - w - M ) ) ) , here b4 = -e(3T& - 41 - e ' T& + 3T&), b,, = -T&d, - 41 - e2 T&(c, + cSL+,) - T&d, + T & m b s , e

4 1 - e2

T&(c, + csL+,) - G d , - T& J s b , . b12 = -T&d, i-

e Here, t y stands for y * . The coefficients R h q and T h q , i=17...,9, as defined i n (Wnuk and Breiter"), are knctions of the lumped coefficients and it is worthwhile to mention that their expressions have ~ i n the denominator. The terms a~t,b,,c,,c', and dg are functions of the eccentricity e dehed as follows @a Silva Fernandes4): C.

where Jn(x) are the Bessel kctions of order n for the variable x.

-”+ CONCLUSIONS New formulas for computation of the orbital perturbations due to the geopotential i n radial,-transverse-and binormal components, valid for highly eccentric orbits, were cos pi as it is usual in

derived. The formulas were transformed to the form z&

sin perturbation theory.

ACKNOWLEDGMENTS T h i s work was supported by FAPESP (96/0612-9,98/00791).

REFERENCES 1. Wnuk, E. Highly Eccentric Satellite Orbits,”Adv.Space Res., V o 1 . 1 9 ,No.11,1997, pp 1735.

2 . S . D a Silva Fernandes, “Some Expansions of the Elliptic Motion to High Eccentricities,” Celestial Mechanics and Dynamical Astronomy, 1994, Vol. 58, pp 297-308, . 3 . S. D a S i l v a Fernandes, “Some Expansions of the Elliptic Motion to High Eccentricities,” Celestial Mechanics and Dynamical Astronomy, 1995, Vol. 62, pp 305-32 1.

. S. Da Silva ~ e ~ d e s , Celestial Mechanics and Dynanlical Astronomy, 1996, Vol. 63, pp 375.

5. a, Theory ofSatellite Geodesy, s s . , 1966.

Harmonic Perturbations for Wigh Order and Degree Harmonics,” Celest. Mech.

nuk, “Tesseral 7.K. Szeto and K. t s ” , Celest. Mech., vo tricity F ~ ~ o n for 27,1982, pp. 325-338 8. R N . Gooding and D. 6. King-Hele,” Explicit Forms of Some FunctionS Arising i n the Analysis of Resonant Satellite Orbits,” RA-E. Technical Report, No. 99035,1988.

9 . 6 . W. Rosborough, and F.G. Lemoine, “Sensitivity Studies of Mars Orbiters for Mars Gravity” Recovery, J. Astronautical Sciences, vo1.39, No.3, pp. 327-357,1991 10. E. Wnuk, Tesseral Harmonic Pertdm ‘om in the keplerian orbital Elements”, Acta Astronomica, Vol. 40,1990 pp 191.

11. E. Wnuk,E. and S . Breiter: Tesseral Harmonic Perturbations in Radial, Transverse and Binormal Components”, CeZestiuZMechunics, Vol. 48,1990, pp. 375.

12. R S. Nerem and 21 others, “Gravity Model Development for TOPEx/poseidn: J o i n t Gravity M o d e l s 1 and 2”, J. Gephys. Res., vol. 99 No. C12, pp. 24,421-24,447 (1994).

13. F. Tisserand, Traite de Mechanipe Celeste, Gauthier - Villars e t F i l s , Paris (1889).

14. C. A Wagner S p e c t r a From the Tracking of Planetary Orbiters7’, J. Gephys. Res.

vo1.84, No.BI2, pp. 68916908 (1979).

15.G.E.0.Giacaglia, “A N o t e on Hanse~’s Coefficients in Satellite Theory”, Celestial Mechanics, Vol 14, 1976, pp. 515.

16 M. P. Jarnagin Jr. 1965, Expansions ixrElIiptic Motion, Astron.papersAm Eph.nautAlmanac, Vo1.18, 1965, XXXVI.

ne Floberghagen Pieter Visser Frank Weischede Massimiliano Vasile ABSTRACT A force model for the lunar albedo effect on low lunar orbiters is developed on the basis of Clementine imagery and absolute albedo measurements. The model, named the Dew Lunar Albedo Model I @LAM-I), is a 15 x 15 spherical harmonics expansion, and is intended to improve force modeling for low satellite orbits, and moreover to help avoid aliasing of non-gravitational force model defects in future lunar gravity solutions from satellite tracking data.

The development of the model from the available lunar albedo data sources is described, followed by a discussion on its calibration using absolute albedo measurements. Further interpretation of the model is based on a comparison with main selenological features. Next, the implementation of DLAM-1 in satellite force computations is outlined, with emphasis on computation costs.

DLAM-1 is also applied in low lunar orbit determination, and results for typical orbits of current and prospective satellite missions are presented.

Finally, the effect of lunar albedo on future solutions for the gravitational potential of the Moon is presented. In this regard, particular interest is on gravity mapping from global data sets, e.g. satellite-to-satellite tracking, which is expected to be one of the experiments of coming lunar missions. It is shown that albedo-induced orbit perturbations have a magnitude and frequency signature which are non-negligible for precise orbit and gravity modeling.

Radial orbit errors are in the order of 1-2 m for one week arcs.

OF SELENE LUNAR LANDER CONSIDERING ORBIT DETER~INAT~ON ERROR Hayato Oonot Shinich Ishikawat Ken Nakajimaff Kentaro Hayashiff Rie Odakau Generally, to achive the high accurate navigation, both the radar altimeter and the inertial measurement sensors consisting of the accelerometers and the gyros are used during the powered decent to the lunar surface. In the current conceptual design, the SELENE is considered to have no (SELenological ENgineering Explorer) lander way to use the radar altimeter during the most part of the powered decent from the restriction of cost,weight,etc.. Therefore, the initial navigation error of the lander at the start of the powered decent brings about an important influence on a safe landing. This initial state error corresponds to the accuracy of the orbit determination in the ground s stem. To relax the effect of this initial state error, we carried out t TI e accuracy analysis of the orbit determination by using the satelJite-to-satellite tracking via a lunar relay satellite. The obtained results are then used in the navigation and guidance error analysis. It can be shown that there is the adequate possibility of the safe landing by the navigation with the only inertial measurement sensors during the braking phase. In this paper, the nominal trajectory, the guidance method and the results of error analyses mentioned above are reported.

INTRODUCTION The SELENE (SELenological ENgineering Explorer) mission will be carried out by a probe combination in the beginning of the next century. The probe is composed of a Lunar relay satellite, a Lunar observation satellite and a Lunar lander. They will be launched by a n H-IIA rocket in the summer of 2003 year. The lander will be on a Lunar circular orbit with the lunar observation satellite and will be operated for a period of about one year, the end of the program of the Lunar surface scientific observation. After that, the lander will be released f r o m the lunar satellite and after the de-orbit maneuver initiated the powered decent from its perilune point at a n altitude of about 15 kilometers. We call a phase from a n altitude of about 15km to reaching an altitude of about 4km BRAKING phase.

Guidance and Propulusion Technology Laboratory. Office of Research and Development. National Space t Development Agency of Japan. Tsukuba Space Center. Sengm 2-1-1. Tsukuba-city. ibaraki, 305 Japan Mitsubishi Space Software Co..Ltd.

In this phase, the navigation of the lander is based on the data from an ine ~ e a s u r e ~ e n t unit. This unit is consisted of acceleroineters and gyros. Therefore, the initial state error at the Powered Decent I iation (F)DI) brings about a great influence on its terminal point state. The cause of this initial state error at PDI is the error of the ground base orbit determination. Especially, the uncertainty of the lunar gravity potential influences the accuracy of the orbit determination seriously.

We performed the orbit determination analysis considering the uncertainty of the lunar gravity potential and the navigation and guidance error analysis based on to confirm it’s feasibility as the primary analysis of SELENE lunar the above result lander mission . Figure 1 shows the flow of these analyses we performed.

PRIMARY ANALYSIS NEXT PHASEANALYSIS Figure I Flow of the Primary Analyses First, we performed the trade off study about the powered decent guidance, and selected a candidate one from this result. To clarify the tolerance of position and velocity error at the powered decent initiation (PDI), using the selected guidance method, we carried out the sensitivity analysis of initial state error with respect to the state at the powered flight terminal point.

Next, we tried the covariance analysis of orbit determination accuracy for the lander including the tracking system with a lunar relay satellite. In this covariance analysis, especially, we considered the error of Lunar potential model Lun6OD (Ref.

1 ) . Then we performed the total error analysis considering the initial state error, the thruster error and the error of onboard inertial measurement sensors. Finally, we redesigned the nominal trajectory to land safely from the results of the above analyses.

NOMINAL TRAJECTORY Table 1 shows the lander characteristics, In Figure 2, we depict the pre- liminary nominal trajectory of the braking phase in powered decent we assumed.

Table 1 LANDER CHARACTERISTICS Initial Mass (ka) Thrust Force (N) Exhaust Velocitv (m/s) 856.000 1700.000 3098.901 p 1 5 ?f, 5 12 START POINT % 9 0 6 12 18 24 SWEPT CENTER ANGLE (deg) Figure 2 Preliminary Nominal Trajectory This trajectory was designed using the bilinear tangent law t o minimize the time transfer under the restriction of along track is freewand the thrust magnitude is fixed (Ref. 2).

POWERED GUIDANCE As the candidates of guidance method for the braking phase, we prepared for the following three types of guidance method.

The first is Proportional Guidance (Gl), this guidance scheme is represented by the following equation.

F D ( t ) = r, + r , t

( 1 ) Here, rD is desired selenocentric radius. ro and rI are constants.

The second guidance is based on Linear Tangent law (62). In this guidance scheme, the desired thrust direction vector XD is where XG ,YG indicate the along track direction vector, radial direction vector respectively, and p , q are constants. The minus sign before XG in Eq.(2) means this guidance is the braking of velocity.

The final guidance (G3) desires the thrust direction X D as

x D ( f ) = x G c o s ( @ + m e ) + YGsin(8+ m f )

(3) where 8 and o are constants. We obtain this Eq.(3) by the parameter transformation of the desired thrust direction vector used in the Space Shuttle Powered Explicit Guidance @ef. 3).

In the guidance scheme G2 and G3, we must predict the increment of position and velocity due to the thrust acceleration aT. These prediction are represented by the following integrals.

The thrust acceleration 8 T under the assumption that the thrust force and the weight flow rate are constant value is 6" a d t ) = r, - t (6) where ck is the effective exhaust velocity, 2 , is the mass zero time.

In 6 2 scheme, Eq.(4) and Eq.(5) can be expressed by some elementary functions. On the other hand, these equations in G3 scheme can not be expressed by any elementary functions, however we can calculate them with sufficient accuracy using the series expansion etc..

In Table 2 and Figure 3, we show the comparison results of each guidance method.

Table 2 CAPABILITY OF EACH GUIDANCE METHOD Time of Fliahtlsec) Fuel ConsumDtionlkq) Sweer, Analeldeq) NOMINAL 693.9 380.6 20.4 G1 696.2 381.9 20.7 G2 694.0 380.7 20.5 G3 693.9 380.6 20.4 We thought the adaptability for the nominal trajectory was important and selected G3 method as the guidance during the braking phase. Though G2 also has the adaptability, 6 3 has the applicable wide range for the thrust direction and the swept center angle in comparison with G2.

E r, 15 k

p 12

I 0 3 6 9 12 15 18 21 24 SWEPT CENTER ANGLE (deg) Figure 3 Comparison of Nominal Trajectory / In the preliminary nominal trajectory, the terminal point o f the br is 3 km altitude from the lunar surface. If the initial state (onboard navigation) error is very large, there is possibility that the height at the braking phase termination is under the lunar surface. To make clear this situation, we analyzed the sensitivity of each state at the terminal point with respect to each initial state error using G3 guidance method. Table 3 shows results of the sensitivity analysis. He=, H,C and L indicate radial, cross and along track direction respectively, and DH,DC and DL indicate the time derivative of H,C and L. The evaluation of sensitivities in the cross track direction are excluded from Table 3, this is because we do not consider the control for the out of a trajectory plane in this analysis.

Table 3 RESULTS OF SENSITIVITY AT TERMINAL POINT * .

nm Ftrq H (kml I (kml D H h W DUm/s) H f l ( k m ) -/+1.351 M.134 -/+1.399 M.014 k 1 (km) -/+0.383 3.058 -/+0.233 3.557 DHf 1 (Ws) -/+0.713 M.025 -/+1.397 M.320 DLfl (Ws) J+0.246 M.000 J+0.464 M.748 In the present investigation, we assume that the recoverable error after the braking phase is the position ermr in H of about 2km, therefore we obtain the error tolerance at PDI as shown in Table 4 under the condition that each initial state error occurs independently.

Table 4 ERROR TOLERANCE AT PDI H(km) !&@ DH(mls) DL(m/s) f l . 4 8 S . 2 2 A2.80 B.13 ORBIT DETERMI In SELENE lander, the state at PDI is uplinked fmm the ground station, and this uplinked state is based on the result of orbit determination. Therefore the state error at PDI corresponds t o the error of orbit determination Here we describe the results of analysis about the orbit determination and propagation accuracy.

The accuracy of orbit determination is computed with the pseudo epoch estimator using Kalman filter algorithm. Then we must consider the e&ct of uncertainty parameters, because the degradation of the accuracy is mainly dominated by these uncertainty parameters. In particular, the degraded e%ct due to the uncertainty of Lunar potential is well-known. We treat this Uncertainty as the considered (unadjusted) parameters. These algorithm @e€ 4) are

K = PA;[ A ~ A ; + PJ-'

S = S- K ( A $ + A,)

(9) is the satellite state (position and velocity) at epoch which is the estimate is the satellite state at time t, red parameters that are treated as systematic error sources Cy corresponds to the error of potential coefficients mentioned above), z is the modeled measurements, P is the error covariance matrix, P, is the measurement variances, K is the Kalman gain and S is the sensitivity matrix in y on the estimated parameters. The propagation of error covariance including the effect of the considered parameters are

P( t ) = - p axct, ( - & ( t ) IT

ax0 ax, where Pr is the diagonal covariance matrix fior the considered parameters.

The lander will be released from an altitude of about lOOkm circular polar orbit ofthe lunar satellite and after de-orbit maneuver, initiated the powered decent from its perilune point at an altitude of about 15 kilometers. Considering these sequences and landing to Mare Serentitatis, for covariance analysis, we set up the schedule of measurements and operations as shown in Figure 4.

PROPAGATIONBASEDON THE ORBITDETERMINATION ___) TRACKINGFOR ORBIT DETERMINATION EPOCH ORBIT DETERMINATION UPLINK OF STATEAND MANEUVERINFORMATION DE-ORBITMANEUVER * START POINT OF REV : DECENDINGNODE POWEREDDECENT Figure 4 Schedule of Measurementand Operation Here we describe the scenarios of measurements and operations. The tracking station that is one of the domestic stations tracks the lunar orbiter and measures range and range rate in the duration of three passes of the lunar orbiter. The orbit determination process is carried out from the final Loss-Of-Sight(LOS) of tracking to the next Acquisition-OfSightdAOS). Timing, orientation and velocity increments for the de-ohit maneuver, and timing, initial orientation and initial state at PDI are uplinked to the lander.

Therefore, to evaluate the initial state error at PDI, we need to propagate the of orbit determination from epoch to PDI for about 9 hours, of error covariance course, then we must take into account the effect of the considered parameters.

Table 5 shows the error sources used in covariance analysis, and Figure 5 shows the error analysis results under the above scenarios.

/ Parameter 1Siama A oriori Uncertaintv Lunar orbiter state vectors at Epoch 100 km in each positioncomponent 100 m/s in each velocity component Lunar potential full coefficients error in L u n G O D Lunar gravity constant 0.004 km3/sed Earthgravity constant 0.004 km3/sed Solar gravity constant 450000 km3/sec2 Earthposition 10 m i n each of X,Y, 20m in Z Solar position 2000 m ineach of X, Y,Z Rangebias 10 m Rangenoise 10 m Range-rate noise 0.5 cmls The wsults in Figure 5 indicate that the position accuracy in the cross track direction is too bad. Because the measurements have no sensiti~ty of the cmss track direction in order that the orbit plane is parallel to the Earth-Moon line in the duration oftracking just before landing. Accuracy of the other elements a w also not good.

1 o2 P Y 2 4 6 10 12 Hours past Epoch

- 10-1

E E El

t

F

a a 10” a, > 1 lo4 ZD v 1 o - ~ a 4 6 8 1 0 12 H o u r s past Epoch Figure 5 Propagation Error of Orbit Determination Accuracy 225 / To improve the propagation accuracy a t PDI, we tried the covariance analysis it determination using satellite-to-satellite tracking. The SELENE mission consists ofa relay, an orbiter and a lander satellite. The initial trajectory ofthe relay satellite is the same inclination and the perilune height as the orbiter and the apolune height of about 2500km. Figure 6 shows the results of the error propagation using the satellite-to-satellite tracking. A priori uncertainty of relay satellite state uses the results obtained by the another analysis about the relay satellite. The other conditions are the same as the above case (see Table 5).

The time span from epoch to PDI i n Fig.6 i s diffenent from it in Fig.5, but the time span from the end of tracking to PDI that is the most important point in this analysis is the same as both cases.

In the case shown in Fig.6, t o be the difference of ascending node between the relay and the orbiter satellite of about 27 deg, the position and velocity error in cross track dimtion are improved incomparison with the case shown in Fig. 5 .

Table 7 shows the propagation results of orbit determination accuracy at PDI of the above two cases.

1 o2 P d 1 o-2

- 10-1

f

d 0 l o 5

i

i

(D lo4 i?

c 10” 0 a 4 6 8 10 12 Hours past Epoch Figure 6 Propagation Error using Satellite-to-Satellite Tracking Station-Orbiter 1.182 111.861 9.41 1 8.764 14.469 0.736 0.600 Station-Relay-Orbiter 0.855 0.526 8.150 7.422 0.331 It is clear that the error of position in L and velocity in H at PDI shown in Table 7 are not satisfied with the error tolerance at PDI shown in Table 4. However, by reasons that the correlation coefficient between position error in L and velocity error in H is about minus one, and the error of same sign in L and DH at PDI occurs the position error of same sign in H at terminal point (see Table 3 ) , there is possibility that these two error factors offset each other. In next section, we describe the total error analysis considering this cancelled effect.

NAVIGATION AND GUIDANCE ERROR ANALYSIS As mentioned above, we selected the guidance method of the braking phase in powered decent, and defined a priori state error at PDI. And we performed the error analyses using these results. Figure 7 shows the outline of this analysis.

NOTE : not consider the potential error during powered decent thrust acc. direction Figure 7 Outline of Navigation and Guidance Error Analysis In this simulation, the terms integrated numerically are the acceleration due to the thrust force and the lunar gravity constant. The perturbation due to the lunar potential is not considered. And the time of flight during the braking phase is in short of about 700sec, the error sources on the environment influenced the lander 5 are disregarded.

motion shown in Table / 3-Siama A oriori Uncertaintv

-

Initial positionin H k855.0 (m) Initial positionin Land velocity in H k8150.0 (m) and -/+7.422 (m/s) Initial velocity in L 39.600 (mls) initial attitude 39,8406 (deg) in each Roll, Pitch and yaw Thrust variation B.0 (%) Isp variation 3 9 . 0 (%) Ta ilaff impulse k450.0 (N sec) Initial nav. pos. in H k855.0 (m) Initial nav. pos in L and vel. in H 28150.0 (m) and -/+7.422 (m/s) Initial nav. vel. in L 39.600 (mls) initial nav. attitude S.0406 (deg) in each Roll, Pitch and yaw Accelerometer bias flOO.O (pg) in each axis Accelerometer scale 2300.0 (ppm) in each axis Gym bias M.015 (deghour) in each a x i s Table 8 shows the main error sources used the navigation and guidance error analysis of the braking phase. In Table 8, we used the propagation results obtained by satellite-to-satellitetracking, because we can not evaluate the effect of large error in cross track direction appropriately to be the lack of the control for the out of plane.

In Table 8, initial position or velocity indicates the error sensed in the onboard navigation system, and initial nav. position or velocity error indicates the error non- sensed in the onboard navigation system. And the position error in L and the velocity error in H are treated as a composite error case by reason of the strong correlation.

Actually, we executed the analyses for total 43 error (86 cases considering the sign) sources such as the inertial measurement sensors, the thruster characteristics,

the initial state and the onboard navigation state . Table 9 shows the results of the

navigation and guidance error analysis including the effect due to the error sources besides those shown in Table 7 .

Table 9 3SIGMA ERROR AT TERMINAL POINT (ALTr3KM) Parameter 3-Siama RSS Main Error Source Time of Flight 80.4 Thrust force variation Fuel Consumption 25.9 Isp variation Position in H 2.8 Initial nav. L pos. & DH vel.

Position in C 0.7 Initial C pos.

Position in L 67.7 Thrust force variation Velocity in H Initial nav. attitude in pitch 8.8 Velocity in C 1.5 Initial nav. attitude in yaw Velocity in L 2.5 Initial nav. L pos. & DH vel.

Considering 3-sigma error sources, Table 9 indicates that the altitude at terminal point of the braking phase is 3;t2.8km and this result is not satisfied with the altitude requirement over lkm at the terminal point. Accordingly, we should modify the altitude of about 3km into 4km at the terminal point to secure from impacting on the surface. By this modification, the range of the altitude at terminal to 6.7km. We tried to point considering 3-sigma error is the value of about 1.2km analyze again using new terminal altitude and codinned to be satisfied with the requirements for the fuel consumption etc..

/ From the above analyses, if some conditions as using -to-satellite tracking data and modifying the altitude into 4km at the terminal point of the braking phase are met, we obtained the conclusion there is the adequate possibility of the safe landing by the navigation with the only inertial measurement sensors during the braking phase.

Lunar landing is the first experiment €or us, we must still study about many things to accomplish the SELENE mission.

In the next phase analysis, we are going to perform more detailed analysis about the orbit determination accuracy with the lunar potential model Glgm2 or improved by the Lunar Prospector Mission.

REFERENCES 1 . A. S . Konopliv, W. L. Sjogrerr, R. N. Wimberly, R. A. Cook and A. Vijayaraghavan, "A High Resolution Lunar Gravity Field and Predicted Orbit Behavior," AAS 93-622,1993 2 . A. E . Bryson,Jr and Yu-chi Ho, "Applied Optimal Control,"Hemisphere Publishing Co., New York, 1975 R. F. Jaggers, "An Explicit Solution to the Exoatmospheric Powered Flight Guidance and 3 .

Trajectory Optimization Problem for Rocket Propelled Vehicles," AIAA 77-1051,1977 4 . G. J. Bierman, "FactorizationMethods for Discrete Sequential Estimation,"Academic Press, New York, Yaswhiro Kawakatsw* A Ywtaka Kaneko* Yos h isada Takizawa* This paper focuses on the three topics related to the trajectory design of SELENE. The first is the orbit maneuver of the orbiter.

The altitude of the orbiter is l O O k m and the orbit is strongly perturbed by high order term of the gravity potential. In order to satisfy the mission requirements, ten maneuvers are scheduled during one year mission. The second topic is the orbit design of relay satellite. The relay satellite has no propulsion system and has no orbit maneuver capability. The orbit of the relay satellite is perturbed mainly by earth's gravity and the shape of the orbit changes through the one year mission. The initial orbit is selected carefully to meet the mission requirements through the mission considering the effect of perturbation. The third topic is the trajectory design of the landing mission. The navigation error in the landing phase is expected to be large value. M a i n reason of the is orbit determination enor and long duration of inertial error navigation. The landing trajectory is designed to permit this navigation error and assure the safe landing.

INTRODUCTION SEUNE (SELenological and Engineering Explorer), the first ISAS & NASDA joint H-IIA rocket in 2003. SEENE is a lunar polar mission to the Moon will be launched by orbiter of 1OOkm altitude with a relay satellite for far-side tracking coverage. The orbiter is composed of two modules, the Mission Module (MM) and the Propulsion Module (PM). PM works for attitude and orbit control of the orbiter system for one year during the global observation. After this observation period, PM separates from the MM and demonstrates soft-landing on the surface, namely it is operated as Lunar Lander in the mission.

*Advanced Mission Research Center, National Space Development Agency of Japan.

Sengen 2-1-1, Tsukubacity, Ibaraki, 305-8505, Japan 23 I cuses on the three topics relate the orbit maneuver of the orbiter to maintain its altitude within the required econd is the orbit design of relay satellite and the third is the trajectory design of the landing mission. etails of these topics are described from the next section.

RBlT RBlTE Orbit Perturbation Analysis Lunar Polar orbiter i s orbited on the 1OOkm circular orbit with inclination of 95 degree. It is required from the science mission group to maintain the altitude of the orbiter within the range of lOOkm f 30km. In other words, the eccentricity of the orbit is required to keep under the value of 0.016.

The orbit of SELENE orbiter in the low lunar orbit is perturbed by several factors.

Those are irregulamess of lunar gravity potential, the gravity of the earth or the sun, solar pressure, and so on. However, the most dominant factor is the irregularness of lunar gravity potential and the other effect of the other factors is relatively low. In this section, is solely taken into the perturbation caused by the irregularness of lunar gravity potential account.

As a lunar gravity potential model, Lun60d(1993)' is adopted in this paper. It is composed of 60 degrees of zonal and non-zonal coefficients with their estimation error.

Data source for the gravity potential estimation are the orbit determination data of Lunar Orbiter and Apllo subsatellites.

One of the mission objectives of Lunar Prospector which now activates is the precise determination of the lunar gravity potential model. New gravity potential model SELENE design no sooner based on the Lunar Prospector data will be reflected to the than it will be published.

Here we want to take up the movement of the eccentricity vector of the orbit under the perturbation of the lunar gravity potential high order term. The eccentricity vector mentioned here has the magnitude of the orbit eccentricity and the direction of the orbit perilune direction. The movement of the tip of the eccentricity vector for the duration of two month is described in Figure 1. As is shown in the figure, the movement of the eccentricity vector is highly complicated in case the whole gravity potential coefficients are taken into account. It is difficult to analyze systematically with the whole potential coefficients taking into account. In the following analysis, we take the effect of the zonal term and non-zonal term separately in order to make the systematic analysis. Considering the non-linear effect between the zonal term and non-zonal term, this analysis, which talce them into account separately, does not give the precise results. However, the following results indicates that the effect of nonlinearity between the zonal and non-zonal term is not so significant.

If the zonal term is only taken into account, the time variance of the eccentricity vector can be calculated anal'yticallf.

Figure 2 shows the drift of the eccentricity vector under the effect of zonal term. The requirements from the science mission, that is to keep the altitude within the range of 100 0.02' 0.015 8.01 0.005 -0.81 -0.01s -0.02 -0.02 -0.015 -0.01 -0.665 0 0 . W 0.B1 0.015 0.02 - - e ~ C O S ( w ) Figure 1 Drift of the Eccentricity Vector under the Perturbation e Xsin( W )

O"f

-0.04 1

Figure 2 Drift of Eccentricity Vector in the Effect of Zonal Term e drift of the eccentricity vector under the effect of non- in the form of the fundamental To analyze the re 3 shows the e numerical simulation is inevita lunar month ( 2 7 . 3 days). The shape of the each 1 month’s drift trajectory resembles well. Additionally, the center of the trajectory drifts along the ellipse shown in figure 1, that is the drift trajectory drawn under the effect of the zonal term only.

The First Month The Second Month ? . . .

-ems -e.eix -0.m e e.w -ems -0.889 e e . - e.eia excos(w) .

e Xcos( w) Figure 3 Drift of Eccentricity Vector of Each Month If the effect of the non-zonal tern is represented by its enclosing circle, the drift of the eccentricity vector can be estimated as a circular region drifting along the perturbation ellipse by the zonal coefficient. In case of the orbit altitude 1OOkm and orbit inclination 95 degree, the radius of the enclosing circle is about 0.008.

In the gravitational potential model, the error of the coefficient has been written additionally. For the system design, it is desirable to consider the worst case in range of the coefficient error. In this perturbation analysis, it is necessary to rightly estimate the effect of the gravitational potential coefficient error on the eccentricity vector drift.

As to the zonal coefficient, it is possible to calculate analytically the sensitivity of the zonal coefficient error against the feature of the eccentricity vector perturbation. In the altitude maintenance controf analysis mentioned after, the worst case f r o m the point of maneuver interval and maneuver quantity is estimated. Figure 4 shows the d r i f t trajectory of the eccentricity vector i n the worst case considering the zonal coefficient error. “The circle in the worst case.

In Large Scale 0. o* 0. 02

-

3 -0.02 b

-2 -o.o*

x u -a OL -a 08 -0. 1 -a I1 0 a005 0.08 0.081 0.02 -0.08 -LOk -0.04 -0.01 0 0.02 0.04 0.06 0.01 e x c O s ( 0) e Xcos( 0) Figure 4 Drift of Eccentricity Vector Considering Zonal Coefficient Error Since the sensitivity of the non-zonal coefficient error cannot be calculated analytically, the worst case of the eccentricity vector perturbation cannot be specified. In the altitude maintenance maneuver analysis mentioned below, the non-zonal coefficient error is not taken into account when the eccentricity vector perturbation is analyzed.

Instead, in order to cope w i t h the non-zonal coefficient error,20% margin for the whole orbit maneuver A V is appropriated.

Altitude Maintenance Control Analysis

The orbit maneuver AV necessary for maintaining pemission altitude which is a

demand from the observation mission is discussed here. It is considered in the worst case including the zonal coefficient error.

The approach of the altitude maintenance control is shown in figure 5. The violated if appropriate maneuver is executed when the enclosing circle touch the limitation circle. In case of the figure, the enclosing circle touches the tolerance limits in the right side. Then the maneuver is carried out to move the eccentricity vector as left as possible in order to gain the maximum maneuver interval. The objective point is the center of the enclosing circle which touches the tolerance limits at the left. By this way, the tolerance limits will never be violated even if the eccentricity vector exists anywhere in the enclosing circle. A V necessary for moving the eccentricity vector in the way mentioned above becomes a controlled quantity per. 1 maneuver. Non-zonal coefficient error is not yet considered i n this stage.

a nm I

<

transition of eckentricity vector e Xcos( w) e xcos(w) (the effect of zonal item) Figure 5 Maneuver Interval and A V in Altiiude Maintenance Control The result of the altitude maintenance control analysis is shown in table 1. It is a result of the worst case considering the zonal term coefficient error. 20% margin €or the Non-Zonal term coefficient error is contained in the AV for 1 year mission.

Table 1 RESULT OF ALTITUDE MAINTENANCE CONTROL ANALYSIS

Maneuver A V for one AV for 1 year

Used Model Interval maneuver maneuver LUN60d 66 days 19.6mls 118m/s e result of the altitude aintenance control analysis is sho result of the worst case considering the zonal term coefficient error. 20% margin for the Non-Zonal term coefficient error is contained in the AV for 1 year mission.

A The relay satellite does not have the orbit maneuver capability, and its orbit through whole mission duration (1 year and 2 months) is decided by the condition at the relay In the initial orbital elements of the relay satellite, ascending node Q, satellite separation.

argument of perilune W , and inclination i (that means the perilune direction and the orbital plane) equals to those of the orbiterwhen the lunar orbit injection sequence of the relay satellite is considered. Additionally, perilune altitude equals to lOOkm which is the altitude of the orbiter. As a result, free design parameter in relay satellite orbit elements is only semimajor axis a (or eccentricity e).

Orbit Settings of Relay Satellite It is necessary to take in mind two roles of the relay satellite, when setting the relay satellites orbit.

The first is to relay signals to the orbiter at the far side of the moon. The signals include command, telemetry, Doppler shift measurements and ranging Four way Doppler shift measurements relayed by the orbiter and the relay satellite will determine the gravitational mapping of the far side of the moon with high resolution and sensitivity. The coverage of lunar gravimetry for the far side is principally restricted by the limited opportunity of four way communication link. From this point of view, the orbit of the relay satellite is required to be visible for long duration from the orbiter at the far side of the moon. In other words, the apolune altitude of the relay satellite’s orbit is required to be sufficiently high.

The second role of the relay satellite is the radio sources for Delta VLBI observation.

The radio sources will be also loaded on the propulsion module of the SEENE orbiter which will be separated and land on the near side of the moon surface after the polar orbital mission period for one year. Delta VU31 observation using radio sources loaded on the relay satellite and the propulsion module on the moon surface will be made for two months after the propulsion module will be landed. The data will be analyzed to derive the precise orbits of the relay satellite and contribute to determine precisely the gravimetry and the libration of the moon. From this point of view, the average altitude of the relay satellite is required to be sufficiently low so that the effect of gravity can be observed clearly.

The semimajor axis of the relay satellite was set to 3000km considering the conditions mentioned above.

elta mission will be carried out for propulsion module on the lunar su year to 1 year and 2 after from the orbit injection of the relay satellite. It i s required from the delta mission to keep the perilune altitude of the relay satellite at Okm-6OOkm during the mission period. In this section, the result of the perilune altitude analysis during the delta VLBI mission period is shown.

There is no orbit maneuver executed for the relay satellite after the separation f r o m the orbiter. Therefore, the orbit through the whole mission duration of the relay satellite is determined by the initial state in the orbit injection.

Firstly, the date of the relay satellite injection is an important factor. Since the main perturbation factor to the relay satellite orbit is the earth‘s gravity, and the positional relation of the earth and relay satellite orbit changes by the injection date. Next, for the orbital element, a, e are set to the above-mentioned value and i , Q are set to the value equal to those of the orbiter. For the remaining element, 0 , it is also decided f r o m the closest approach condition of the translunar trajectory, when the injection date is designated. In short, the free parameter as an initial state is only the injection date, and the orbit of the relay satellite through the whole mission duration will be determined for the injection date.

As a result, the perilune altitude during the delta VLBI mission period will be also determined for the injection date. Figure 6 plots the perilune altitude at the 360th and the 420th day from the relay satellite orbit injection. Horizontal axis indicates the assumed launch date in the summer 2003. “360” and “420” is the day when the delta VLBI mission starts and ends. It indicates that, for some launch date, the perilune altitude at the 420th day becomes negative, and the relay satellite collides with the lunar surface during the delta VLBI mission period. From this result, it can be concluded that, there are only 17 days which can satisfy the requirements of the delta VLBI mission.

Perilune Altitude(km) 6oo Launch Date Figure 6 Perilune Altitude of the Relay Satellite is also constraine ~nstraints suc V in the lunar orbit injection, the launch condition of the rocket, the eclipse on g the launch in summer, 2003, launch windo RY M e r the one year observation, the payload module separates from the mission module and demonstrates soft-landing on the lunar surface, namely it is operated as lunar lander in the mission.

Landing sequence of the propulsion module is shown in figure 7 . The propulsion module separates from mission module on a circular orbit of altitude 1OOkm. The propulsion module executes de-orbit maneuver and it is injected into the elliptical orbit of whose apolune altitude 1OOkm and perilune altitude 15km. After the coasting phase for half way of elliptical orbit, the powered descent phase starts. In the powered descent phase, the maneuver which mainly cancels the horizontal velocity is executed and terminal condition of velocity Om/s and altitude 4km is achieved. Vertical attitude is established within short seconds in the beginning, and the vertical descent maneuver is carried out to achieve soft landing on the lunar surface.

Propulsion Module 15km /--.-m ....................................

15000~1 Transfer \ Power Descent Phase 4050111 Attitude Stabilization Phase 4000m Vertical Descent Phase ' Circular Orbit Figure 7 Landing Sequence Precise sequence of each phase are described in the following.

Before Separation Orbit and attitude deteiination are carried out before the propulsion module separation. The summary of the orbit determination sequence is shown in figure 8.5 or 6 passes are visible in one day from the tracking station in Japan. The visible duration in ass i s about one s to say, the satellite passes the near side of the for 5 or 6 times in the visible duration from Japan. The orbit determination is carried out using the tracking data of first 3 passes, and the orbit determination value is transmitted in the fourth pass. e landing is carried out in the fifth pass. The sixth pass is unavailable for the landing, because the transmission time after landing is sending image data in this case. The attitude determination is carried out using the star sensor mounted on the mission module. The orbit determination value sent from the ground and the attitude determination value from star sensor are used as an initial value of the inertial navigation system of the landing phase.

Figure 8 Orbit Determination Sequence For the reduction of navigation error, the orbit detellnination had better be carried out after the propulsion module is injected into the elliptical orbit. However, the power of the propulsion module is supplied from its battery, and its capacity is not enough for long hours operation. For the sufficient orbit determination accuracy, at least 3 visible pass, that is, about 6 hours of orbit determination period is necessary. This duration is too long to supply power only by the battery on the propulsion module. This is the reason why the orbit determination is carried out before separation. It is about 1 hour after the attitude determination in the time to the landing, and the d r i f t error o f the gyro is negligible. There is no problem in carrying out the attitude determination in this stage.

Separation and De-orbit of Propulsion Module “Mare Serentitatis” in the near side of the moon is planned to be landing point (about 20 de@ and 25 degN). As mentioned above, the propulsion module coast half way of the elliptical orbit before the powered descent phase. Therefore, the injection to the elliptical orbit is carried out in the back side of the moon. The separation of the propulsion module is carried out just before the elliptical orbit injection in order to shorten the operation time of the propulsion module as possible.

The separation velocity is about 1Ocm/s, and the navigation error produced in this

phase is negligible. AV necessary for the elliptical orbit injection is about 2Ws. At this

an output of the accelerometer is referred, and the engine is cut off at the maneuver, timing when the required velocity increment is acquired. As a result, the elliptical orbit injection is achieved without producing large AV error in spite of thrust error, etc..

The propulsion module is injected into the elliptical orbit whose apolune altitude 00km and perilune altitude 15km, and coasts for half w orbit. Its position and

velocity are propagate on the onboard inertial navigation . The powered descent is

started when the vertical velocity becomes zero, in other words, at the perilune passage point.

owered Descent Phase The navigation during the powered descent phase is also done only according to the information of inertial measuring unit (IMU). Though there is a radio altimeter/ velocitymeter installed in the propulsion module underside, it is not yet available in this phase. The reason is that the attitude of propulsion module is almost horizontal, and measurement direction is out of range. As a guidance system, the explicit guidance method to optimize the propellant consumption is used. The descent trajectory is almost reverse of the ascent trajectory of the rocket, and various guidance method used in the rocket launch is applicable for the guidance method in this phase. There is no significant difference in the guidance performance among each method. Only the altitude (4b) and velocity (horizontaVvertid direction W s ) are designated as a terminal condition. The horizontal position is not designated as a guidance target. From the result of the guidance error analysis, the guidance error has fallen within the enough small value. This fact shows that the propulsion module can be guided to the goal point in its own navigation system, even in the existence of the thrust error, etc.. However, the navigation error is a different problem. Since the M U is the only available sensor after the orbit determination, the navigation error considerably increases., For example, the altitude error at the end of the powered descent phase is estimated as 3.6km for 3 (7. In short, even if the guidance system works well and the propulsion module is guided accurately to the target on the of the propulsion module disperses in the inertial navigation system, the actual altitude range of 0.4km-7.6km for 3 (7 at the end of the powered descent phase.

Vertical Descent Phase The attitude of the propulsion module at the end of the powered descent phase leans 70 degrees from the vertical direction. Vertical attitude establishment phase for about 7 seconds is set in the beginning of the vertical descent. It falls about 5Qm in this phase.

The radio altimeter/speedometer is available at this moment, and actual altitude can be used in the navigation system for the first time. Combined navigation information system which uses the information of the IMU and the radio altimeter/velocitymeter is used in this phase. From the result of the navigation error analysis, there is about 2m/s ( for 3 0) horizontal velocity error in the beginning of the vertical descent phase. The horizontal velocity error cannot be removed only by the altimeter information. Therefore, in order to satisfy the landing condition, the radar velocitymeter is necessary. The vertical descent phase is basically a sequence using free fall and the maximum thrust deceleration.

A breaking line which is set at the beginning of the vertical descent phase considering the . .

mass an result of these sequence, the soft landing is achieved.

Three topics related to the trajectory design of SELENE are introduced. The feasibility of the mission is verified through the conceptual design study. The SEIENE project is phased up to the preliminary design phase this spring and more detailed studies, tests and simulations will be done to confirm the system feasibility.

ACKNOWLEDGEMENT We thank the SELENE project team members for their contribution to the work described in this paper.

REFERENCES 1. A. S. Konopliv, W. L . Sjogren, R. N. Wimberly, R. A. Cook and A. Vijayarslghavan, “A High Resolution Lunar Gravity Field and Predicted Orbit Behavior, “ AAS 93-622, 2. R. A. Cook, ‘The Long-Term Behavior of Near-Circular Orbits in a Zonal Gravity Field,” AAS 91-463,1991 p .

ILIANO VASILE' , UNE FLOBERGHAGEN' Abstract In this paper a temporal finite element method is developed for dynamics and optimal control problems. The approach is here proposed in order to determine the optimal initial conditions, and the optimal control law to perform the desired soft landing on the South Pole of the Moon. The spacecraft is modeled as a point mass subject to gravity forces due to the Moon, the Earth and the Sun, and controlled by an ideal throttable thrust. An estimation of total propellant consumption is presented taking into account possible errors arising from a poor knowledge of the present force model.

Finally an optimal control strategy for obstacles avoidance is proposed. Some sample problems show the effectiveness of the proposed approach.

INTRODUCTION Various space agencies are presently studying or approving several new missions to the Moon. The demonstration of landing technologies is an essential feature of all these new mission plans (Lunar-A, SELENE').

In particular the mission target is to perform a precise soft landing in a morphological complex area on the South Pole of the Moon. In order to achieve this mission goal and to lead the project to success it is necessary to minimize risks of a hard landing, reducing costs in term of weight budget.

As demonstrated in previous works' this is a challenging task due to the strongly non uniform and yet not well known gravity field of the Moon which will give great dispersion in final results. Thus the consequences of a wrong force field prediction on the design of a landing trajectory should be analyzed.

The landing problem can be handled as a typical boundary values problem in which the final position and a series of constraints on the trajectory must be met.

In this paper a Spectral Elements in Time (SET) approach is proposed in order to determine the optimal initial conditions, and the optimal control law to perform the desired soft landing on the South Pole of the Moon. The spectral elements approach is based on a variational method coupled with a spectral elements discretization of the solution which leads to a fast and compact memory algorithm and allows the introduction of several constraints conditions in order to fulfill all the requirements of this kind of mission.

The landing mission is here divided in two phases: a coast phase and a homing phase. The analysis of the coast phase is performed considering the GLGM-2 gravity Dipartimento di Ingegneria Aerospaziale Politecnico di Milanno. via Golgi 40, 301 33 Milano. ltuly 'DEOS, Delft Institute for Earth-Oriented Space Research, Kluyverweg 1.2629 +IS, DelTt. The Netherlands / model and the perturbing effects of the Earth and the Sun; however, as stated before, the uncertainty on present gravity models could lead to major errors in coasting trajectory predictions. hus different possible coasting orbits with different geometry have been studied.

A wrong prediction in the coasting trajectory endpoint propagates through a variation of the homing trajectory which could lead to a hard landing. The extra mass and thrust requirements to correct the final descent are analyzed here imposing an error in the gravity model and computing a new optimal homing descent.

The error is estimated considering both the uncertainty on the GLGM-2 coefficients - obtained from the covariance analysis - and a completely different gravity model.

The second phase is characterized by a poor knowledge of the landing site, therefore altimeter data and images should be processed during the mission in order to determine the landing area correctly. Thus an optimal control law is introduced in order to correct the descending trajectory on the basis of real time data acquisition.

Finally some results proving the effectiveness of the proposed approach will conclude this paper and possible initial conditions and control law, to conduct a soft landing, are presented.

SPACECRAFT AND FORCE MODEL The basic set of equations of motion of the spacecraft subject to a generalised force function f and to a control u can be expressed in the following general form: x = f (x, u) (1) The state vector x, in a Cartesian reference frame (see Figure I), can be expressed in terms of displacements 4 and momenta p as follows 7’ (2) x = {Px 7 P y P z , 4 , 7 4 y 7 4,) ; while the function f(x,u) is defined as where u, u,, ,u, are the three components of the control along the Cartesian axis X,Y,Z.

The mass ratio mR is defined as mdm, , where mp is the propellant mass and m, is the dry m a s s of the spacecraft. Furthermore a linear mass flow equation is added to the differential system in order to take into account the dependency of the mass on the thrust modulus Ilull:

a, = -IlU / c (4)

where c is the exhaust gas velocity c=Ispgo, with Isp the specific impulse and go the gravity acceleration on Each surface.

The function U is the .potential due to the gravity forces acting on the spacecraft, namely the gravity field of the moon and third body perturbations.

In a selenocentric reference frame (see Figure I ) , the potential of the lunar gravity field given as expansion into spherical harmonics is a sum of the potential of a sphere and the perturbation accounting for all the deviations of a real body from a spher2: / (6) where p,bf is the gravity parameter of the Moon, R,+f is the mean equatorial radius and 8 is the phase of lunar rotation, namely the angle between some body fixed direction along the equator and some inertial direction along the equator.

The perturbing function due to the presence of the Earth and the Sun can be developed in terms of the displacements q and of the position of the planet relative to the local co- ordinate frame as follows3: Figure.1. Selenocentric reference frame where pg is the gravity parameter of the third body, namely the Earth or the Sun, p g is the selenocentric position radius of the third body and d~ is defined as follows: d , = ( P ~ f q 2 - 2 P B q C O S 6 ) " 2 (8) where 6 is the angle between p and q.

OPTIMAL CONTROL FORMULATION Let consider a performance index of the form:

J = V,(x(t)7t> 1" i-1 L[x(t),u(t),t]dt

(9) where <p(i,r) is a discrete function of the states and time at the final time and L(x,u,t) is an integrand performance index.

The problem is to find a state function x:[t~,tf]+W and a control function ~ : [ t ~ , ~ ] + U c % ~ , that minimize (or maximize) the cost function (9), subject to conditions (1),(4) and to the following conditions on final boundaries: @(X,f) = 0 (10) Adjoin4 the system differential equations ( 1 ) and (4) and boundary conditions (10) to J , respectively with multiplier functions h ( . and v : fl

J =[<pp(x,t)+~~@(x,t)] I r ' +I [L+hT(f -x)+&,,(m,< +IIullc)]dt

( 1 1) [ I For convenience, define a scalar function H a s follows: ~[x(t),u(t),il(t),ti = ~[x(t),u(t),ti+ ; l " ( t ) f ~ x ( t ) , ~ ~ , t i + ~ l w l l ~ ~ & (12) Taking the first variation of J , considering also differential changes in the terminal time f j , , the following is obtained: (13)

+ hT6f + 6h, ( mR + CII 11) + k,l16mR + h,,c = 0

)]dt In order to have boundary conditions of the weak type the following termss are adjoined to expression ( 13):

6hT (x - xb) + hb ax 1 ; + 61: (mR - mi + k:,,6mR If

(14) with the costates boundary values at final time hf defined as follows: W X , O + v T &X,t) OX E$ (15) OX The resulting equation takes the form:

6~ =hb6x 1" +6vT@ 1" +[6hT(x-xb)+6h,(mR -rn;)+kb,6mR] 1'' +

4 t, After an integration by parts of the term 6hTx, the expression (1 6 ) reduces to the form: The controls, both at boundary and at internal nodes, have to satisfy necessary condition H&, h, u, t ) =O and Legendre-Clebsch condition & , ( x , A, u, 02U.

THE SET APPROACH The finite elements in time method (FET) has been successfully applied to a large number of problems in computational mechanics, spacing from rigid body dynamics to structural mechanics, wave propagation, fluid dynamics and optimal c o n t r 0 1 ~ ~ ~ ~ ~ .

In this paper we propose a slightly different approach using, instead of E T , Spectral Elements in Time (SET) a high-order finite element technique that combines the geometric flexibility of finite elements with the high accuracy of spectral methods.

Spectral method, pioneered in the mid 1980's by Anthony Pater2 at MIT for fluid dynamics problems, is here applied to the integration of ODES in the time domain, being spectral elements in time more accurate and efficient in finding the solution for our problems involving less memory space and less computational cost.

Both E T and Spectral Elements methods offer some interesting features that make them attractive in automated numerical procedures: Through the use of spectral basis for shape functions, high order methods can be constructed, therefore allowing the development of automated p and hp adaptive procedures.

Using a time assembly process, they allow the solution of general boundary- value problems. Besides the computation of the system response, this technique provides at a negligible extra computational cost an approximation of the transition matrix that allows to perform a linearised stability analysis of the solution7.

e The variational framework is an ideal context for developing constrained formulations for mechanics8, leading to schemes characterized by robust numerical behavior.

The variational principle (17) is the governing equation for the weak Hamiltonian method for optimal control problems. This formulation provides the base for the development of the SET discretization for general boundary problems.

Now let the time domain D(t&5R be decomposed into N finite time elements: = Uy=1 Dj ( t i 9 ti+, (18) The parametric approximations of the trial functions (x,h,u,rn~) and test functions (6x,6h,6u,6m~) are developed within the space of the polynomials of order k-1 and k respectively: where the functionsfand g are defined as follows: f E Pk-*(Dj) ; g E P k ( D .I ) (22) and the quantities xS,& us and m ~ , are internal node values.

In a more general way we could decompose the domain D as a union of smooth images of the reference time interval [-1,1] where we define a reference parameterv: The basis functions fand g can be constructed by using Lagrangian interpolants

associated with the internal Gauss-Lobatto nodeg. Thus if (5, }k,,, are the set of Gauss-

Lobatto points on the reference interval [-1,1], J(v) will be the Lagrangian interpolating polynomial vanishing at all the Gauss-Lobatto nodes except at where it equals one. Each integral of the continuous form (17) is then replaced by a Gauss quadrature sum: where O i is the weight associated with ti.

By the discrete form (24) two distinct procedures can be derived: 0 An implicit time marching self-starting integration obtained for initial value problems 0 An assembled process developed for boundary value problems, obtained by matching the final boundary state of each element with the initial state of the subsequent element.

Both approaches are taken into account in this effort: while the assembled system is used to solve the optimum problems, the time marching integrator is used to propagate the initial condition founds, forward and backward in time.

By the SET discretization the differential problem is transformed into a system of non-linear algebraic equations. Then a numeric optimization process can be used to minimize the objective function satisfying the differential constraints. All the additional constraints on the state and on the control are discretized and directly implemented into the optimization process. Here a Newton algorithm with line search is adopted to solve the non-linear system arising from discretization. Linearised algebraic equations for a single spectral element yield:

JCC) *AY(C, = -qC) (25)

where is the elemental tangent matrix, qe) the elemental residual vector, Ay(e~=(~,A$Au,h&~ are increments to nodal states, costates and controls, while the subscript refers to elemental quantities. The global matrix formulation can then be obtained through the standard finite element assembly process performed on the corresponding elemental matrices: J *Ay=-R (26) The matrix J is highly sparse" and can be conveniently stored in a compressed format and solved by an iterative sparse solver.

CONVERGENCE ANALYSIS The validation of the optimization algorithm has been performed on several problems taken from Ref.4. In particular we compare our results to two sample problems presented in Ref.5 where a finite elements techniques is developed for / optimal dynamics problems. The first optimal control problem is a transfer of a particle on a rectilinear path with fixed time. The thrust angle is the control and the mass and constant acceleration modulus: particle has constant I

cos(u),asin(u), p , . p, }

(27) 1 E+O The objective function is the final horizontal component of velocity that should 1E-1 be maximized:

E

W 0 1E-2 J = cp(x(t),t) 1" = p: I' (28)

> Initial conditions are fixed and there are 1E-3 .-4 also two terminal constraints on state: fixed

2 1E-4

final height and final vertical component of velocity, which must be zero: 1 E-5 7 . f

10 100 W x J ) 1' ={07(qy-h),P.ryO} I. (29)

Number of Nodes In Figure 2 a comparison in final solution Figure.2. Relative error on final thrust inclination, versus accuracy is made between the solution obtained using a spectral discretization with computational cost polynomials of the 2nd order, both for the control and the state, and the results reported in Ref.5 in which finite elements of the first order for the state and of order zero for the control have been used.

As can be seen increasing the order of the polynomials, especially for the control, reduces the over all cost. Spectral basis for the generation of polynomials of high order guarantees numerical stability of the integration algorithm allowing p adaptivity", where the solution is continuous, and flexible h adaptivity where it's not.

THE LANDING PROBLEM In order to make the right choice for the best landing maneuver two main drivers have been identified the overall cost, in terms of weight budget, and the reliability.

Thus two are the main problems analyzed here: find a reliable orbit, for the coast phase, which can be used as safety path in case of landing abort evaluate the propellant consumption and the maximum thrust needed to perform the desired homing trajectory, taking into account errors arising from non correct modeling of the present force field.

Coast Phase A stable periodic orbit for the coast phase would reduce the risk of a hard landing even without a huge amount of propellant, and could be a reliable parking orbit in case of a landing abort. However the lack of data on the real gravity field of the moon gives a meaningful uncertainty on the frozen/periodic orbit location. This can be clearly seen

in Figures 3-7 where three different solutions, generated' ' using respectively

Lemoine's GLGM-2 model, Konopliv's LunGOd and GLGM-2 plus uncertainties (sigmas) on harmonics coefficients, are represented. In particular the huge variation of 180" in frozen periselenium anomaly should be noted.

Thus different possible orbits with different orbital elements have then been analyzed and three specific cases are discussed here: a direct descent from a lOOx 100 km parking orbit down to a 100x20 km orbit with the periseleniurn over the South ole; a direct descent from a lOOxl00 krn parking orbit down to a 70x20 km frozen/periodic orbit with the periselenium over the South Pole; a two-steps maneuver which exploits an intermediate 50x20 frozen/periodic orbit with the periselenium over the North Pole. Exact orbital elements are reported in Table 1, while in Figure 9 orbit geometry for the three coasting trajectories are represented and, for each maneuver, ignition time and thrust modulus u (constant) reported. Table 2 summarizes the propellant consumption budget for the coast phase, taking into account extra propellant necessary to come back on a stable orbit in case of landing abort.

- 91.0 0.02 S 90.5 2 0.015 a3 M

Q z 5

E d W 2 U 90.0 E 0.01 cd .H m r : a .- e Q U 89.5

2 0.005

89.0 0.00

-0.2 0.2 0.6 1 .o

- 1 .OE-2 O.OE+O 1 .OE-2 RAAN (deg) k=e*cos(omega) Figure 3. Periodic solution, osculating h-k Figure 4. Periodic solution, osculating i-SZ plane: a=1788 km, T=2 months, Lemoine’s plane: a=1788 lun, T=2 months, Lemoine’s GLGM-2.

GLGM-2.

-0.028 r

r

n -0.03 M

i

-0.036 .-.( m a I 1 c -0.04 -0.044 89.0 -5.OE-3 O.OE+O 5.OE-3

-0.5 0.0 0.5 1 .o

k=e*cos(ornega) RAAN (deg) Figure 5. Periodic solutiofi, oscutating h-k Figure 6. Periodic solution, osculating i-SZ plane: a=1788 km, T=2 months, Konopliv’s plane: a=1788 km, T=2 months, Konopliv’s Lun60d. Lun60d.

91.0 r-

- O e o o 4 r I

-0.024 , 89.0

- 1 .OE-2 O.OE+O 1 .OE-2 0.0 0.8 1.5

k=e*cos(omega) M A N (deg) Figure 7. Periodic solution, h-k plane: Figure 8. Periodic solution, i-S2 plane: a=1788 km, T=2 months, Lemoine’s a=1788 km, T=2 months, Lemoine’s GLGM-2+sigmas GLGM-2+sigmas After designing the desired coasting orbit using an estimated force model, computed initial conditions are propagated forward in time using a different gravity model, taken to be true. As estimated force model we take at first LunGOd, using Lemoine’s as true one, and then we take Lemoine’s as estimated and GLGM-2 plus sigmas as true one. Errors in final state vector .computed in this phase will represent uncertainties in initial state vector for the following homing phase.

Table 1. Coasting Orbit Keplerian Elements Ignition Second Ignition Second Ignition u=3.858 d s 2 .

u=3.828 m/s’

T=10.13~ 1

T=10.13 s T=10.26 s 7

Figure 9. Coasting Trajectories Geometry and Orbital Maneuvers 25 1 0.0 1953 ~ o m i n g Phase An optimum criterion aimed to minimize the thrust is adopted for the homing phase: a low maximum thrust will reduce engines dimension and weight and the overall propellant consumption. An optimal trajectory is designed for an estimated force model taking into account a number of constraints characterizing the homing phase.

A landing site placed on a circle of 86" of latitude south on the near side has been selected and different approaching directions have been analyzed in order to identify the best moment to begin the homing maneuver. Final altitude is fixed at 2 m above the ground, where thrusters are cut off and the residual vertical velocity of the space- craft is 3 m/s. Initial conditions are constrained to be on a given coasting trajectory with given orbital keplerian parameters: where v=v(p) is the velocity vector, r=r(q) the radius, a,e,o and h the semi-major axis, the eccentricity, the periapsis and the angular velocity of the coasting orbit respectively. A specific impulse 1,=3 17s is here adopted. From 10 km to 0.002 km an alignment constraint is imposed on the state vector in order to reduce the tangential velocity to zero and to direct the thrust to the ground before the final break. This constraint is implemented by means of an additional weighted objective function w,,, of the form: L =-uz +w,(p) From 5 km to 0.002 km an obstacles avoidance strategy is turned on in order to guide the spacecraft to a suitable place. Results for the homing trajectory, computed 8 elements with polynomials of the 6th order both for state and control vector, using are shown in Figures 10,11,12 and 13. The velocity modulus and the velocity angle a relative to the ground are represented in Figures 14 and 15. The maneuver begins 559.637 km before the landing site and lasts 856 s. It should be noted how,

approaching to the ground, a turns to 270". The thrust modulus and the thrust angle p

/ relative to the ground are represented in Figures 16 and 17 respectively, while in re 18 the propellant consumption history is plotted.

always under 3.912 m/s' and reaches its he specific thrust modulus stays maximum value when the alignment constraint on velocity vector is turned on. At this point two subsequent maneuvers reduce the velocity component tangential to the ground to zero. A final maneuver is performed to reduce the velocity to 3 d s . In order to evaluate the effects of force model errors, initial conditions are forced to be equal to the final conditions taken from the coast phase analysis and a new optimal solution is computed taking into account different gravity models. The result in term of extra thrust and extra propellant mass for the homing phase is summarized in Table 3.

-1 500 -1 600 h E -1700 Y N

-===-=-"A I

-1 800 -1900 - * -2000 I-

I x (km)

Figure.10. Homing trajectory relative to the Moon surface: z-x plane

Oa5 c I - l a ' r

-1680 n h E E -1720 Y

s

N

-1760 r

i I I 4r- Y

2 2-5 t

0 300 600 900 0 300 600 900 Time (s) Time (s) Figure.13. Altitude vs. time Figure.16. Thrust modulus vs. time

2-o r

80 I-- i h a, cd U -20 , 0 300 600 900 0 300 600 900 Time (s) Time (s) Figure.17. Thrust inclination relative Figure.14. Homing: velocity modulus to the ground vs. time

1.0 r

300 r

L h

I 1 I I I

150 0.0 I 1

0 300 600 900 0 300 600 900 Time (s) Time (s) Figure.15. Velocity inclination Figure.18. Propellant mass relative to the ground. consumption vs. time.

he strategy for obstacles avoidance is implemented directly into the optimization process: the objective function L is modified introducing an additional term which is function of the distance do from obstacles. In this way the new optimal solution will maximize the distance from obstacles, minimizing the thrust needed to perform the maneuver. The new objective function is defined as follows: d L* = - t12 -t k,d,) (9) where k, is a weight parameter which characterizes the priority of maneuvers on thrust minimization: the higher k, is the more the maneuvers are fast and expensive.

The distance do is evaluated superimposing a characteristic shape function on as shown in Figure 19.

obstacles I Figure 19. Example of shape function for obstacles avoidance maneuver: shape function <p is superimposed on obstacles 0, and 02. Distance dl and dz are evaluated from the surface of the tangent cone n.

In order to provide a real time control, the differential problem and the additional objective functions are linearized in a neighbour of an initial optimal solution. In the linearized model the shape function is substituted by a sequence of cones tangent (see Figure 19) to the original shape function. In this way at each step Af the optimization algorithm has to evaluate just one sparse matrix inversion to reach an accuracy on the solution of le-6. The weight parameter S is a function of the amplitude of cones, scaled to the dimension of the thrust modulus.

As test cases, different scenarios have been generated with different random distribution of obstacles of several dimensions, ranging from mountains of few kilometers to rocks of some meters. The cone parameter k, becomes a function of obstacles position, altitude and width, and of the estimation of the landing site position, which represents the cone vertex location. Every At seconds, during the descent, the cone geometry is updated on the basis of new data on landing site and obstacles nature. To prove the robustness of the algorithm, different temporization for data acquisition have been simulated, taking into account possible delays. In Figure 20 three possible obstacles avoidance maneuvers are shown, each starting with different initial conditions and characterized by a temporization At=3.55 s. These solutions are integrated using 10 elements with polynomials of the 2nd order both for state and control vector. In Figures 21 and 22 specific thrust modulus and inclination as to the ground are plotted.

On trajectory number 3, the spacecraft detects obstacle 3 at 3.5 km and obstacle 2 only at 1 km from the ground, thus, as can be seen in Figure 20, two subsequent fast maneuvers are necessary to avoid them both. On the contrary, on trajectory number 2 the spacecraft detects obstacle 3 at 3.5 km and obstacle 2 at 2.5 km allowing a more smooth maneuver. On trajectory number 1 obstacle 1 is detected immediately while obstacle 2 is detected below I km, requiring a fast maneuver with a thrust angle of 51".

4 - 3.0 .

-8 2 -

a

.&

1.0 - 1 I I I 120.8 121.5 122.2 122.9 123.t x (km) Figure.20. Example of obstacle avoidance maneuvers

Q

0 30 60 9(

I l o b ' 3 b 1 6 * ' d (

Time (s) Time (s) Figure.21. Thrust modulus Figure. 22. Inclination relative to the ground Table 3. Overall Homing Costs In this paper a numerical approach based on the spectral elements in time technique is presented for optimal control problems. An optimal set of initial conditions and an optimal control law has been derived to perform a soft landing on the South oon. For the final descent a control strategy fo proven to be effective, leading to a fast and simple control in real time. Taking into consideration errors arising from gravity perturbations and uncertainties on landing site location, a total mass ratio of 0.951k0.003137 of propellant is estimated to be necessary with a maximum specific thrust of 3.858 rn/s20.02445 m/s2. From this preliminary analysis a 70x20 frozen/periodic orbit seems to be the optimal coasting orbit, both for cost and reliability. However this aspect must be investigated further on the basis of more reliable gravity data: if the actual periselenium was at 90°, a mission aimed to a soft landing on the North Pole would be recommended.

Future developments aimed to enhancing the model of the spacecraft are at present being studied, in particularly to define a pulsed thrust and to introduce the attitude control in the optimization process.

REFERENCES 1. Kinoshita T.,Itagaki H., Moriuma H., Namura E. Outline o f the experimental Lunar Lander in SELENE Project. ESA SP-403 ISBN 92-9092-295-8 August 1997.

Floberghagen R., Visser P., Vasile M.. Low Lunar Orbits Analysis, determination 2.

and selection for soft landing on the lunar South Pole. ESA SP-403 ISBN 92- 9092-295-8 August 1997.

3. Sans0 F.,Rmummel R. Theory o f Satellite Geodesy and Gravity Field Determination. Lecture Notes in Earth Sciences. Vo1.25, Springer-Verlag, 1988.

Bryson A.E. Jr., Ho Yu-Chi. Applied Optimal Control. Blaisdell Publishing 4.

Company, Waltham, Massachusetts, 1969.

5. Hodges D.H., Bless R.R.. A Weak Hamiltonian Form.for Optimal Control. Jan 01,1989, NASA-CR-185336, NAS 1.26:185336.

Bulirsch R., Miele A., Stoer J., Well K.H.. Optimal Control, Calculus o f 6.

Variation, Optimal Control Theory and Numerical Methods. ISNM Vol. 1 1 1, Birkhauser Verlag 1993 7. Borri M. Helicopter Rotor Dynamics by Finite Elements Time Approximation.

Comp.&Maths. with Appls., vol. 12A, No. 1,1986,~~. 149-160 Basic Feutures o f the Time Finite 8. Borri M.,Bottasso C.L., Mantegazza P.

Elements Approach for Dynamics. Meccanica, Vol.27, 1992, pp. 1 19- 130.

Karniadakis G.E., Bullister E.T. and Patera A.T.. A Spectral Elements Method f o r 9.

Solution of Two- and Three-dimensional Time-dependent Navier-Stokes Equations. Proc. Europe-U.S. Conf. on Finite Element Methods for Nonlinear Problems, Springer-Vedag, p.803, 1985.

L.F.. Preconditioned Mixed Spectral Elements Methods for Elasticity 10. Pavarino and Stokes Problems. AMS(M0S) subject classifications65f 10.

11. Finzi A.E., Vasile M.. Numerical Solution f o r Luntrr Orbits. IAF-97-A.5.08, 48th International Astronautical Congress, October 6- 10, 1997, Torino. Italy.

/

ER ST OR UPROAC NG

OF LUNAR OR P ~ ~ T ~ Y SPACECRAFT"

Shuji ONOf

This paper presents a new algorithm for the independent estimation of the high-order state vector of attitude, angular rate, thrust, and dynamic parameters, needed for the spacecraft control i n the approach and landing to the moon or a planet. Without aid of the inner informa- tion of a gyroscope and an accelerometer,the constructed algorithm ( range & range rate ) of the tracking uses only the outer information station measurement or the topographical rneasurement b y the reflection wave, for the multi-beam emitted from several antennas equipped on a vehicle. The simulation results well verify the effectiveness of this algoritkeven in place of an attitude sensor.The new excellent result i s especially obtained by the topographical assumption of a ground surface model. This paper gives one of fundamental guidelines not only for the new design of the navigation system and the ground control system of an aircraft, a future space-plane, or a planet-surveyor, but also for the identification of a planetary surface model.

INTRODUCTION

The approach and landing on the moon or a planet presently depends upon the on-board control system of an un-manned vehicle, due to an uncertain topographical information and a communication delay upto the Earth. Therefore the right or wrong, namely,the landing rocket and the on-board control system normally operated or not, the landed point was nominal or not, there were any damages in the vehicle or 'not, must be waited until the end of the postflight data analyses for several hours or some days. In the near future,if the interplanetary communication and tracking network and the detailed topographical data will be arranged, the unmanned on-board system will be reliable and safe. Moreover the flight situation will be independently seized and judged in realtime if the manned station will be constructed nearby.

This paper, on the premise of the above arrangements in the near future, aims the independent estimation of the high-order state vector, separated from the vehicle inner information, that is, for the purpose of evaluating the Inertial % Prepared for 13 t b International Symposium on Space Flight Dynamics at Goddar Space Flight Center,W 1998.

Doctor of Engineering, Tsukuba Carrplter Center, Tsukuba Spacecenter, National Space Development A g e w of

t

Japan ( N A S N 1, 2-1-1 Sengen, TsuicUba city, Ibaragi-kea J A P A N , Tel. (81129852-2488, Fax. (811298-52-2394.

easurement Unit ( I M U 1 system of a gyroscope,an accelometer,and an attitude sensor, The high-order state vector of attitude, angular rate, thrust,and dynamic parameters, i s neccesary for the vehicle control in an approach and landing on the moon or a planet. The new algorithm is constructed only with use of the outer information ( range & range rate 1, measured by the tracking station of the known position and velocity, or measured b y the reflection wave from the known topographical model, for the multi-beams emitted from the several antennas on a vehicle.

The first analyses by the tracking station use the previous algorithms for

the stochastic estimation and the geometrical determination . The stochastic

algorithm adopts the extended Kalman filter commonly used in the on-board system.

The geometrical algorithm determinates the high-order state vector by the relative position and velocity amongst the spacecraft antennas and the tracking stations.

Moreover the introduction of the relation of several antennas locat ions on the vehicle ( Fixed Frame method ) realizes a high accurate determination. The dynamic model of 19 dimensional state vector considers the vehicle dynamics of powered flight. The atomospheric effect i s ignored for simplicity, The measurement model of 24 dimensional vector i s formulated in order to discriminate each antenna datum of range and range rate, The measurement by several antennas is assumed to be sequential or simultaneous. A s the target of analysis i s for the Mars, the imaginary tracking stations are supposed’to be G T S ( Geostationary Tracking Satellite 1 on a geostationary orbit of 17000 km altitude, T S ( Tracking Satellite ) on a circular orbit of 10000 km altitude around the Mars.

In the second analyses by the reflection wave from the ground,two topographical algorithms for the stochastic estimation and the geometrical determination were newly constructed with use of the map data with a mountain and a river for example, developed from the assumption of a planetary surface model of sphere. And the Mars i s also the target. The stochastic algorithm was constructed b y the replace of the new measurement model of topographics from the previous algorithm. The geometrical algorithm was also newly constructed under the condition of the optimum evaluation function to be minimum.

In both simulations, each stochastic estimation results that the estimated error variances ( 6 ) of state vector have the excellently converging tendency, and each geometrical determination also shows that the obtained state very narrowly varies around the Nominal state change. The simulation indicates the independent and high accurate est imat ion of high-order f 1 ight state only by assuming the tracking network or only by the premise of the topographical ground surface model. The effectiveness of these algorithms with several antennas equipment, i s well verified,even in place of an attitude sensor. Moreover the application of this technique to the Earth mission i s very easy ,because of the conditions already arranged in the tracking network and also about the detailed topographical model of the Earth.

This paper gives one of fundamental guidelines not only for the new navigation system design of an airplane,a helicopter,moreover a future space-plane, or a planet- surveyor in the re-entry o r the approach and landing phase, which demand the independent attitude estimation without use of the ordinary inertial navigation system, but also for the identification of a planetary surface model.

-0 G 0

This chapter presents the assumption and the result of the analyses.

The detailed algorithms for the stochastic estimation or the geometrical determination only by the outer information measured at the tracking station of the known position and velocity, are shown in the previous papersc2-41 ~ Figure 1 i s the relation between the Mars and a spacecraft in an boost phase.

The spacecraft body coordinate and the Mars centered inertial coordinate are also defined. Four antennas are located in front and rear of the fuselagesand also on both wingtips. The imaginary tracking stations GTS,TS are assumed around the Mars.

The configuration of the vehicle i s only imaginary and used only from the relation to the previous papersc1-4J .

Table 1 i s the apriori parameters. The simulation i s performed for the de-orbit phase from an initial circular orbit around the Mars. The simulation time i s 80 seconds. The de-orbit motor is assumed to be ignited at 5 seconds after the calculation start. The motor burning time i s 70 seconds. The initial values of ( roll/ pitch/ yaw ) the attitude Euler angles and of their rates ( W I , 02, 0 3 1, defined by the nominal attitude of de-orbit direction, are all zero. The initial motor thrust parameter is 7 k = O = 100 ( % ), The initial thrust off-set is ( g 2, 3 ) k = O = 0 ( mm ). The diagonal elements of the initial error co-variance matrix P k = O adopt the values of Table 1 . The initial non-diagonal elements are all zero.

Mars Geostationary K YB

P3

Z B XB w- *-s

I Fig, 1 Spacecraft I

J

and Mars

X E I T ;Vernal Equinox Table 2 depicts the locations of

four antennas ( A , B, C, D ) and

four liquid fuel tanks in the instant- aneous body coordinates.

Body Scale 1 i s the standard antenna layout for the assumed vehicle. Each location of Scale 2 is twice larger than that of Scale 1.

Scale 0 means all antennas locations on the origin of the body coordinates.

The measurement data are generated by adding the normal random numbers Initial Covariance Matrix ( P ~ = o ) : to the Nominal data of range and - range rate, between each antenna on Position CJ-x, Y , Z = 1.0 km

Velocity 6% 9, 2 = 0 . 1 w s

a vehicle and each tracking station

of G T S , T S and the Mars station. Euler Angle 6 # . 6 d = 1 0 . 0 deg

of measurement is The time span h l a r Rate 6 a , l , e ~ = 1.0 deg/s Thrust Offset % 2 , E 3 = 1 0 . 0 ram 0.1 seconds.

The normal random numbers were M o m e n t of Inertia Q I ~ . ~ , ~ ~ = 1 . 0 kgm2 calculated with the random errors Mass 6, = 1 0 . 0 kg in T a b l e 1. Thrust Parameter CT o = 1 . 0 %

Table 2 Body Scale

: Tanks Location

Body Scale

Location of Liquid Fuel Tanks

A 10 m B -5 m

7--$

C 5 m D -5 m e 2 i s the simulation results of the stochastic estimation by the assumed station. The Kalman filter starts to operate at T= 5 seconds after the simulation start, The estimated error variances( o- ) of state vector have the excellently converging tendency. Otherwise i n the geometrical determination, the obtained state i s shown very narrowly t o vary around the Nominal state change.

Both simulations by each algorithm indicate the high accurate estimation and the effectiveness of the several antennas equipment.

One Sigma Change of Velocir 2 o o - Thrust Offset 6, I J $ i ' + u r ' + u i a a d

- 1.0

Moment o f lnertia

. 11l

2 540 s 0.5 5 Irr

.. 520

=s?

2 - 500

'y

: - 0 . 5

D i: -1.0

- 1.0 -

Moment of inertia . Y a w Rate g,, m - J 0 . 5 0 - 2 i f 196

-

= -0.5 - ..

-1.0 -

I92 -I5 t , , , , 2 5 2 - Moment o f Inertia Thrust Parameter of Thrust g - J 1000 3 250 a .

- 5 3 248 * - - 2 4 6 4 - - 2 ," 244 e

2 4 2 -

0 20 4 0 60 80 0 20 40 60 80 0 20 4 0 60 80 T t a e ( s e d Time ( s e d Time ( s e d Fig. 2 Stochastic Estimation b y Tracking Station ( Mars Approach Phase ; case &BOA ) re 3 is the effects of Body Scale to the precision of estimation.

ure (a) is the results of the stochastic estimation, and shows the changes of the error variances at T = 80 seconds in accordance with Body Scale.

The figure (b) i s the geometrical result, The change of the standard deviation is also shown in accordance with processed over the whole time of calculation, Body Scale.

Both results indicate that the precision remarkably ameliorates as the vehicle scale becomes bigger.

( a ) Stochastic Estimation ( b ) Geometrical Determination

g.004} - \ Roil Rate

Yaw Rate

\ Yaw Rate

G. 04 2.02 3 I I 1 0 1 2 3 1 2 3 0 1 2 3 BO& Scale (times) ' Body Scale (times) Body Scale (times) Body Scale (times) Fig. 3 Effects of Body Scale ( Mars Approach Phase 1 C T I O ( Topographical Algorithm ) Fig,4 shows a spacecraft during a motor burn on a planetary ground surface.

Three beams of each different direction are assumed to be enunitted from 4 antennas on a vehicle. The algorithm to estimate the flight state vector by the measurement data ( range and range rate ).of the reflection wave from the ground, i s described a s follows.

Fig.5 shows the spacecraft flight path and the locus of antenna beam on the assumed topograpical model of a planetary ground surface. The spacecraft state vector,considering a boost phase dynamics, i s position,velocity,Euler angle, anguler velocity,thrust,and dynamic parameter,same as the previous chapter. Each reflection wave i s assumed as the same time measurement.

The measurement model i s constructed so as to distinguish each antenna beam. The range is obtained to calculate the intersecting point of beam direction within a planer element of the topographical surface. The range rate i s the time derivative of the range.

Thrust F\\ rn 1

. ’ , <

.- ,-.*

.=‘* .-

Intersection Fig. 4 Spacecraft and Beam Reflect ion The stochastic algorithm as constructed from that of the previous chapter by the ecomposition of the measur ent model considering the topographical information^ Table 3 i s the top0 raphical algorithm f o r the geometrical determi~ation~ Step 1 i s the measurement data generation of range and range rate by addition the measurement errors (the random numbers) to the nominal data of antenna beam reflection, Step 2 i s the generation of the spacecraft flight state with errors by the addition of the random numbers. This state value becomes the initial condition in Step 3 .

Step 3 i s the state determination of position, velocity, Euler angle, and angular rate ( 12 dimensions ) by the Newton method under the condition of the evaluationnal function( S ) minimum. The other high-state vector can be estimated by an iterration of state propagat ion.

Fig. 5 Flight Path and Beam Locus on Topographical Model

* Z W

Table 3 Outline of Determination Algorithm

Generation of Measurement Data with Random Numbers ; step Topographical Reflection Waves for 4 Antennas / 3 Beams.

Addition of Random Numbers to Nominal State Change of Spacecraft.

Step 2

Determination of Flight State ( Position,Velocity,Euler Angle, Angular Rate 1 b y Newton Method with the following condition :

Step 3

where p? j*; Range & Range Rate i n Step 1, a=l,2,3,4 ; Antenna Number, : Beam Number.

p , i : Range & Range Rate i n Step 3, b=l, 2,3 ( Topographical Est imation Resu 1 t s 1 The simulation of topographical estimation by this new algorithm was performed for the same case of de-orbit from a circular orbit of altitude 20 km around the Mars as the previous chapter.

Fig.6 i s the stochastic results of the estimated position,velocity,attitude, attitude rate,and so on. The Kalman filter starts to operate at T=5seconds after the simulation start. From these figures,the error variances ( c7 1 of the state j 10 Thrust Offset ..

w - 5 I I e o I

: -5

e - -10

15 -

.r: 1 . 0 0.

Roll Rate 6,, Y

-3 10 - 3

-

- 0.5 8 5 -

- B 0 -

d - 0 a - 5 - oc c = -0.5

-

-

- -10 Is

Is -15 -1.0 -, LJ - I

- 15

- 1 . 0

-

Pltch Angle 0'8 9 Pitch Rate g , , 5 1 0 - m Xoneat of Inertia 2 -r 5 4 0 1 6 Ixr 3 0.5

- 5 - I

2 . . s j o

-: 0 -

I

= - 5 - 2

E 480 P -0.5 - U 2 -10 I

0- -

-15 - -1.0 I 440 * 202 - Yaw Rate 0 ' , , Moment of Inertia 3 2 % = 200 - E 2 -

- - 198

I- - i 2 196 - e 194 One Signa Change of

252 -

Thrust Paraaeter Moment of Inertia B, I^ 250 - % a

i

e 3 248 -

- -

O 246 - I" i -" 244 -

242 -

0 20 40 60 80 0 20 40 60 80 0 20 40 60 BO Time (sec) Tine ( s e d Time ( s e d Fig. 6 Stochastic Estimation by Topographical Wave ( Mars Approach Phase ; Case Mars-1 ) vector, except the moment of inertia, have the excel lent coverging tendencies.

- 7 i s the geometrical results. The algorithm of come t r ical de t ermina t ion also starts to operate at T= 5seconds. The simulation shows that the determined state varies very narrowly arround the Nominal change.

Table 4,5 are the simulation conditions of both algorithms and the results.

The converged error variances ( CT ) at the end of simulation T = 80 seconds, and the standard deviations calculated from the differences between the Nominal state and the determined value over the simulation time, are both very small.

The high accurate estimation only by the topographical imformation and the effectiveness of several anntennas equipment, are also indicated.

- P +

2 , 1 Position A R = I B-RN.-I.I

0 ' I I I I I Roll Rate dm~ = O I - P ) I N * . I .

a a0..04 - Pitch Angled8 = 8 - 8 ~ . . 1 .

Pitch Rate A m a =ma- ma N . ~ I .

8 0 . 4 g o . 06 $ 0 . 0 4

t

8 0 . 0 4 ;-. 0 4 -0.4 '-. 06 1 t I 0 20 40 60 80 0 20 40 60 80 , Time (sec) Tine (sec) Fig. 7 Geometrical Determination by Topographical Wave ( Mars Approach Phase : Case Mars ) eneral,it has been thought that the estimation of yaw angle only from the ave i s difficult. But this newly constructed algorithm was shown to e able to estimate the high-order flight state especially including the yaw angle ith high precision.

ge of range and range rate This analysis uses the rough ground mode1,and the c is discontinuous. Therefore in the initial operation of Kalman filter,the estimation of position and velocity is not smooth. But the estimation becomes smooth with time and very accurate at the end of operation. Although the next analysis targets for the continuous ground model, the results of the detailed model are the same as the fundamental results and tendencies of this paper.

From the above, the new algorithms are both verified effectively to operate.

Table 4 Stochastic Estimation Results Table 5 Geometrical Determination Results b y Reflect ion Wave b y Reflect ion Wave ( Mars Approach Phase ) ( Mars Approach Phase 1 Mars-1 Run Condition & R - Case Mars Mats Approach Flight Phase Mars Approach Dynamic Model ( dimension 1 1 9 Measurement Model ( dimension ) Measurement Model ( dimension 1 24 (combination) R&RR (Range & RRate) (dination) R&RR Equipments : Doppler Radar,etc . 4 Esuipments : Doppler Radar,etc 4 Generation of Random Noise for R&R : RandGNoise of Measumnt : Range Noise c r a ( m 1 1 . 0 Range Noise C T R ( m 4 1 . 0 RRate Noise 0 RR ( a d s ) 1 . 0 Range Rate Noise CT R R (cm/s) 1 . 0 Data Smoothing No &nothing Location of 4 antennas Each Point Location of 4 antennas Each Point & BIeasurenent by antennas Same Time & Measurement b y antennas Same Time & Bodykale ( Times 1 1 & BodyScale (Times 1 Time Interval DT (sed 0 . 1 heration of Random Noise for X i : Results of Estimated aError ; Position Nois cr X . Y , Z ( m 1 . 0 Position a R ( a 1 2 . 6 0 Velocity Nois a i. 3, i (ads1 1 . 0 Velocity a V (cads1 6 . 9 4 0 Euler Angle Nois a &e,@ (deg 1 0 . 1 Roll Angle ff $ (deg 1 0 . 0 0 0 9 3 Anguler R a t e Nois a ( ~ , . a ~ p b ( */s 0.01.

Pitch Angle a 8 (deg 1 0.00064 Time Interval DT (set) 0.1 Yaw Angle a @ (deg) 0 . 0 1 2 2 9 Result of Estimated E Errors : Roll Rate a w l ( Y S ) 0 . 0 0 0 0 2 Position E R ( m ) I. 6528 Pitch Rate ff a J 2 (O/S) 0.00001 Velocity E v (Cm/S) 1.6613 Yaw Rate c r w 3 0.00016 Roll Angle E 4 (deg 1 0 . 0 9 5 1 5 Thrust Off-set a E ? ( w 1 0 . 0 0 0 0 2 Pitch Angle E 6 (deg 1 0 . 0 9 4 7 9 Thrust O f f - t a ~3 ( I U I I 1 0.00001 Yaw Angle e # (deg 1 0 . 0 9 8 7 7 Thrust Para 0 7 ( 9 6 1 0 . 8 2 1 Roll Rate & 0 1 ( O h ) 0.00972 Mass a M ( k g 1 * 5 . 5 9 Pitch Rate & 0 2 ( o/s) 0 . 0 0 9 5 9 Yaw Rate & w 3 ( o/sl 0 . 0 0 9 4 7 I

co

This paper concludes that the high-order state vector of spacecraft attitude, angular rate, thrust, and dynamic parameters,besides position and velocity, can be independently estimated with high precision by several anntennas equipments on a vehicle, only from the outer measurement data of tracking stations or topographical ( IMU 1.

reflection wave, separated from the vehicle inner information The new topographical algorithms was also verified effectively to operate in place of an attitude sensor. The next target is the estimation for the atomospheric flight on the detailed ground model. Although the detailed analysis is expected to become higher accurate, the results of the detailed model are the same as the fundamental results and tendencies of this paper.

This paper gives one of newly technical guidelines not only for the design of the new navigation system and the ground control system,where the future aircrafts and the spacecraf ts demand the high-order state estimation separated from the vehicle inner information of the ordinary navigation system and I M U data, but also for the identification of a planetary surface model.

Moreover the application of this technique to the Earth around mission is very easy,because of the conditions already arranged in the tracking network and also about the detailed topographical model of the Earth.

REFERENCES 1 ) ON0 S . , ” High-Order State Estimation for Space-Plane with Several Antennas”, Proceedings of the International Symposium on Spacecraft Ground Control and Flight Dynamics ( SCDl 1 , Journal of the Brazilian Society of Mechanical Sciences,Vol.XVI, Special Issue,

ISSN 0100-7386, P . 513-520, (1994)

2) ON0 S . , ” Stochastic Algorithm for High-Order State Estimation of Space-Plane without INRI Data , Transactions of The Japan Society for Aeronautical and Space Sciences, V o l . 38, N O . 1 2 1 , P P . 207-227, ( 1 9 9 5 ) .

3) ON0 S . , Geometrical Algorithm for High-Order State Determination of Space-Plane without I N Data,Transactions o f The Japan Society for Aeronautical and Space Sciences, V o l . 3 8 , N O . 121, P P . 228-242, (1995).

4) ON0 S,,” A Study of Unsteady Dynamics and of High-Order State Estimation for Space Flight Vehicle System ”, Thesis for Doctorate in Engineering, Nagoya University, N O . 1326, P P . 1-175, (1997). , SSI ESIG T SUPPORT David Lozier & Ken Gala1 Space Projects Division Ames Research Center, NASA David Folta & Mark Beckman Guidance, Navigation, and Control Center Goddard Space Flight Center, NASA The Lunar Prospector mission is the first dedicated NASA lunar mapping mission since the Ap0110 Orbiter program which was flown over 25 years ago.

Competitively selected under the NASA Discovery Program, Lunar Prospector was launched on January 7,1998 on the new Lockheed Martin Athena I1 launch vehicle.

The mission design of Lunar Prospector is characterized by a direct minimum energy transfer trajectory to the moon with three scheduled orbit correction maneuvers to remove launch and cislunar injection errors prior to lunar insertion. At lunar encounter, a series of three lunar orbit insertion maneuvers.

and a small circularization burn were executed to achieve a 100 kin altitude polar mapping orbit.

This paper will present the design of the Lunar Prospector transfer, lunar insertion and mapping orbits, including maneuver and orbit determination strategies in the context of mission goals and constraints. Contingency plans for handling transfer orbit injection and lunar orbit insertion anomalies are also summarized. Actual flight operations results are discussed and compared to pre- launch support analysis.

INTRODUCTION Mission Overview The January 7, 1998 launch of the Lunar Prospector spacecraft marked the return of America's space program to the moon, picking up where the Apollo program left off with a IQW altitude polar orbiting mission to map the entire surface of the moon. In contrast to the Apollo program, however, Lunar Bospector was a modest spacecraft funded at a cost of $63 million (including the launch vehicle) by NASA's Discovery Program'. Six science experiments were flown to map the composition of the lunar surface, study the moon's gravity and magnetic fields, investigate levels of tectonic and volcanic activity, and search for evidence of water ice at the lunar poles.

27 1 esc The Lunar Prospector spacecraft (Figure 1) is a spin-stabilized graphite-epoxy drum, 1.4 meters in diameter by 1.22 meters in height, with three radial instrument booms located 120 degrees apart. Power is provided by solar arrays mounted on the outside of the drum. Attitude, spm rate, and velocity control are provided by a blowdown monopropellant hydrazine propulsion subsystem using six 22 N thrusters.

Attitude and spin rate determination are provided by a sun sensor and an Earth/Moon limb sensor.

Telemetry and command functions are provided by redundant S-band transponders through either a medium gain or an omni-directional low gain antenna mounted on a mast aligned the spacecraft spin axis.

Figure 1: Lunar Prospector Spacecraft with Instrument Booms Deployed The nominal Lunar Prospector telemetry rate is 3.6 kbps real-time with no onboard tape recorder.

However, a 53.3 minute delayed transmit capability at the spacecraft permits ground capture of telemetry data taken from the backside of the Moon. The total spacecraft mass at launch was 296.4 kg, including 137.7 kg of hydrazine propellant.

NOMINAL MISSION PROFILE Table 1 provides a summary of the nominal mission profile designed for the Lunar Prospector mission.

The Lunar Prospector spacecraft w a s launched on January 7,1998 at 02:28:44 GMT from the Eastern Test Range. A Lockheed Martin Athena II vehicle placed the payload (spacecraft and injection stage) in a nominal 100 nautical mile circular parking orbit after a 13-minute flight. Following a 42-minute coast to a translunar injection point over North-Western Australia, the payload w a s released and a S t a r 37FM motor was used to apply a nominal 3 142 m/s delta-V over the course of a 64 second burn. The design flight time to the moon was 105 hrs from injection. A series of three lunar insertion burns were designed to place the spacecraft into its 100 km polar mapping orbit about the moon.

Launch Date Selection and Transfer Orbit Design The approach taken by the Lunar Prospector project for establishing launch dates was to limit launch opportunities to a single set of consecutive prime and backup days each month. This approach was motivated m part by a desire to limit launch vehicle preparation costs, as well as a desire to only select launch dates which provided the best geometry in terms of minimizing both operational risk to the mission and overall propellant consumption. Monthly launch dates were identified beginning in September of 1997. Possible earlier launch dates were rejected in order to avoid the September 16, 1997 lunar eclipse by the E a r t h , which would have lasted several hours and necessitated a latger battery.

ECT ESIG In the Fall of 1997, two sets of prime and backup launch dates consisting of January 6/7 and February 4/5 were selected as candidate launch opportunitiesfor the Lunar Prospector mission. These launch dates and the associated transfer trajectories were selected on the basis of the following factors: Sun Angle Geometry: A favorable sun angle of close to 90 degrees relative to the spacecraft z-axis was desired during translunar injection (TLI) and lunar orbit insertion (LOI) for power, thermal and attitude determination considerations.

e Shadowing: In order to minimize risk to the spacecraft, it was desired that the spacecraft be in sunlight following TLI, and for the duration of the 2-day LO1 sequence to place the spacecraft into its low altitude mapping orbit. Additionally, transfer orbit geometries (for both nominal and contingency orbits) whereby the dc-Earth-sun angle approached 180 degrees were to be avoided due to the possibility of long shadow periods (up to 9 hours in some cases) while in the transfer orbit.

Transfer Orbit Inclination: A low transfer orbit inclination with respect to the EarthMoon plane was desired in order to minimize LO1 costs.

: Lunar orbit inse~ion with the moon close to apogee yields a slightly lower DV cost ~ v e r s ~ To minimize propellant consumption and risk to the spacecraft, transfer orbit trajectories that required small turn angles to get from the transfer orbit cruise attitude to the LO1 attitu f r o m the LO1 attitude to the mapping orbit attitude were chosen.

Post ges rapidly near Earth for a given ation acquisition^ While the station look angle inertia1 TLI attitude and antenna configuration, certain trajectories enable post-TLI station coverage sooner than others. Minimizing the time to acquire telemetry and command capability was a goal in order to minimize the time to correct launch and transfer orbit dispersions.

LO1 Station Coverage: As a goal, the periselene arrival time was to be maintained such that dual station coverage w a s available during LOI# 1. Furthermore, it w a s desired that all subsequent LO1 bums be conducted in view of a station (a 64 minute delayed command timer was available for doing bums in the blind when necessary).

I 1

1/12 9 - T -3 Figure 2: Lunar Prospector Transfer Orbit Geometry for Prime and Backup Launch Dates From a given launch s i t e , a launch to the moon is possible on each day of the month, with two launch times (roughly 12 hours apart) available on each launch datez. The launch date establishes the sun/Earth/moon geometry for the -fer orbit. The selection of the launch time, for a given launch date, establishes the inclination of the transfer orbit plane relative to the Earth-moon plane, and influences lighting and station coverage conditions at injection. On a given day, the transfer orbits corresponding to the two possible launch times are typically distinguished by the length of the coast time in the parking orbit (Le. short coast or long coast), or alternatively,by the proximity of th tion point to the ascending vs.

descending node. As a final option in transfer orbit design, for each 1 datehime, two lunar insertion conditions are available over either the northern or southern lunar hemisphere. The selection of the lunar approach geometry affects the required LO1 thrust direction and hence the required as part of LO1 operations. Through a careful lunar approach geometry, transfer orbits for prime and backup launch dates (Figure 2) were arrived at that minimized both the risk to the Lunar Prospector spacecraft and the propellant required to get into orbit about the moon. The resulting transfer orbits selected were low heliocentric inclination orbits with injections near ascending node and lunar insertion in the northern lunar hemisphere.

Transfer Orbit Maneuver Strategy The nominal Lunar Prospector transfer orbit maneuver plan called for a total of three trajectory correction maneuvers (TCMs) with TCM#1 planned at 4.5 hours after separation, TCM#2 at 24 hours after TCM# 1, and finally, TCM#3 at 24 hours prior to lunar orbit heition. A total of 80 m / s was nominally allocated to cover corrections to possible launch and TLI dispersions.

Key to this maneuver strategywas a desire to execute TCM#l as soon as possible after TLI in order to minimize losses associated with buming away f r o m perigee. It was felt that given all the spacecraft events that had to occur before a nominal burn could take place (e.g. 90 deg reorientation to the cruise attitude, boom deployment, orbidattitude determination, maneuver planning, command load generation/execution), a bum 4.5 hour into the mission was an achievable goal, assuming a nominal post-separation timeline of events. Ideally, for this maneuver time, the required correction for any 1aunchlTLI dispersionswould have grown by a factor of 3 . 5 by the time of the maneuver. However, given Lunar Prospectors attitude and thruster firing mode (where both axial and tangential thrusters are fred in a vector mode), and a desire to maintain a fixed time of flight to the moon, it was expected that a 1 m/s launch/TLI error would require 4.5 m/s of equivalent propellant to correct for an orbit maneuver a t 4.5 hours after TLI.

TCM#2 was nominally scheduled 24 hours after TCM#l in order to provide sufficient time to collect tracking data for an orbit solution and plan the maneuver, as well as to pennit the prime operations shift to rest between bums. TCM#3 was scheduled 24 hours prior to lunar insertion to provide any required final corrections to the approach trajectory.

Lunar Orbit Insertion Maneuver Strategy Three LO1 maneuvers were designed to capture the spacecraft into lunar orbit and lower apoapsis most ofthe way into the nominal 100 km polar mapping altitude. Each LO1 bum was to be performed from an inertial attitude using 2 axial jets along the aft side of the spacecraft (axial jets on the antenna end of the spacecraft could not be used for long bums due to antenna heating concerns). The LO1 maneuver sequence was designed with the following goals/constraints in mind S/C Pointing: There was a desire to maintain a single attitude throughout the LO1 maneuver sequence in order to minimize propellant use and operational complexity. By maintaining a fixed argument of periapsis for each intermediate orbit in the LO1 sequence, no attitude maneuvers would be required.

Maneuver Duration and Frequency: While the there was no hardware limitation on the maximum time to bum during each LO1 maneuver, bum efficiency considerationssuggested keeping the bums as small as possible in order to minimize thrust losses over the bum arc. On the other hand, operational considerations suggested keeping the overall number of bums at a manageable number.

e Intermediate Orbit Perturbations: It was recognized that smaller LOI#I bums would result in an initial capture orbit that was more susceptible to third body perturbations (largely from the Earth, and affecting mostly orbit inclination). Such perturbations can be nullified to some degree by biasing the lunar orbit insertion conditions to counteract the anticipated evolution of the orbit. However, the possibility always exists that unforeseen delays might result in an extended stay in the initial capture orbit, whereby excessive orbit errors could accumulate which would require subsequent correction.

ased on the above guidelines, a three-bum LOI sequence was designed to first capture the spacecraft into a 12 hour period orbit, lower it into a 3.5 hour period orbit, and finally drop apoapsis most of the way into the 100 km nominal mapping altitude. This sequence resulted in e LO1 maneuvers of roughly equal size (approximately 30 minutes each), with an initial orbit that experienced relatively small perturbations. The lunar arrival conditions were biased slightly from the nominal 90 degree inclination and 1838 km periselene radius (to 89.8 deg and 1829.7 km, respectively) to allow for inclination growth due to third body perturbations while in the 12 hour capture orbit, and to allow for expected growth in periapsis altitude resulting from LO1 finite bum losses. A constant argument of periapsis was targeted for the first two burns to avoid the need for altitude maneuvers between bums.

Orbit DeterminationStrategy The orbit determination strategy for Lunar Prospector w a s broken into two phases: cislunar phase and mapping phase. Both phases were analyzed pre-mission using covariance analysis and simulated tracking data. The simulations assumed tracking by the Deep Space Network (DSN) stations in California, Australia, and Spain. The DSN stations were expected to provide range, Doppler, and XY angle @SN 26 m stations only) measurements.

There were two primary goals during the transfer/LOI phase: (1) provide predicted ephemerides for mission planning and trajectory design, and (2) provide near-real-time assessments of orbit maneuver performance. During the transferLO1phase, there were seven maneuvers nominally planned: four deterministic (TLI and LOI#l-3) and three corrective (TCM#l-3). During this phase, it was expected that the spacecraft would be continuously tracked by the DSN. After each maneuver, range, Doppler, and XY angles (only for DSN 26 m stations) would be collected and processed to determine the new trajectory.

Due to reduced dynamics as the spacecraft moved away &om perigee, the time required to obtain an accurate converged solution increased with each maneuver in the transfer orbit. Once captured in lunar orbit, the required convergence time was mostly a function of the orbit period. Table 2 shows the expected tracking arc required after each maneuver to obtain a full state batch orbit estimation.

Table 2 LUNAR PROSPECTOR ORBIT DETERMINATION TURN-AROUND TIMES Maneuver TLI TCM-1 TCM-2 TCM-3 LOI-1 LOI-2 LO13 Planned Post- ManeuverODTime 30min" 6hrs 8hrs 12hrs 4hrs 3hrs 2hrs Following each LP orbit maneuver, an updated orbit state was computed in support of preliminary maneuver planning of subsequent burns. The state would be updated several hours prior to the upcoming maneuver as an input to the final maneuver plan. In each case, the predicted velocity uncertainty of the solution at the time of a maneuver was at least an order of magnitude less than the planned delta-V for that maneuver. This ensured that the maneuver plan was not corrupted by trajectory uncertainties.

The capability for near-real-time maneuver assessments was required in support of Lunar Prospector contingency plans --- particularly in support of the critical TLJ and LOI-I bums. The near-real-time * Use of combined tracking data from TDRSS (available at TLI and lasting approximately45 minutes) and the DSN (available starting at 19 minutes after TLI) would have enabled a preliminary solution to be computed within 30 minutes of TLI and a final solution 2 hrs after TLI.

assessment would be made by monitoring DSN Doppler residuals. Once a final maneuver plan was ephemeris was used to available several hours before an upcoming maneuver, the predicted generate simulated nominal Doppler measurements. These Doppler nts were processed using orbit estimation software and compared to the pre-maneuver state in erate a baseline plot of the expected Doppler residual signature over time. Next, simulated fin assuming a hot or cold maneuver, and corresponding Doppler residuals from these off-nominal cases were also plotted. Once the maneuver began, Doppler residuals were compared in near-real-time against the previously generated plots to enable a quick assessment of the maneuver performance. Pre-mission analysis had indicated that for each of the planned deterministic maneuvers, the difference between residual signatures for a nominal and a 5% off-nominal maneuver was greater than the expected residuals associated with statelmeasurement uncertaintiesduring that maneuver. Thus, any deviation in maneuver performance of 5% or greater would be observable.

During the mapping phase of the mission, the main challenge to orbit determination support was that of meeting accuracy requirements. The mission requlrement for post-processed solutions (using LP derived lunar gravity models) was 1 km 1-sigma position accuracy in each of radial, cross-track, and along-track directions. A pre-launch covariance analysis indicated that the lunar potential model was the leading source of orbit estimation error. The lunar potential model to be used initially was GLGM-2 developed at Goddard by F. Lemoine using tracking data from the 1994 Clementine mission3. The covariance analysis indicated that the mission requirements could only be partially met using this model, and only with extensive post-processing. For certain geometries(e.g. when orbit normal was perpendicular to the Earth- Moon line and lunar occultation occurred) the mission requirements would not likely be met4.

As part of one of the experiments to be conducted by Lunar Prospector, tracking measurements were to be used by A. Konopliv at JPL to develop a new lunar potential model. It was planned that a switch to the new potential model would be made when the model became available (approximately two months into the mapping mission) and that LP defmitive orbit data would be regenerated using the new model to ensure that orbit accuracy requirements could be met. The actual orbit accuracy attainable using the new model would not be known until it became available.

Contingency Planning Several orbit contingency plans were devised to handle possible off-nominal performance of the spacecrafl, ground system, and launchhjection sequence. Naturally, Lunar Prospector obit contingency plans centered around critical mission phases consisting of transfer orbit injection and lunar orbit insertion and included the following: Emergency Post-TLI Correction: In the event of a TLI overbum by greater than 20 m/s, or a TLI underbum of between 20 and 35 d s , an emergency correction burn w a s to be executed approximately 40 minutes following spacecraft separation from the TLI stage.

Contingency Phasing Orbit: In the event of a TLI under-performance by more than 35 m/s, a plan was devised whereby the spacecraftwould be initially left in its anomalous orbit about the Earth, then placed into a phasing orbit for several revolutions, with an attempt to capture into lunar orbit approximately one lunar sidereal month (27 days) beyond the nominal capture date.

LO1 #1 Under-PerformanceContingency: In the event that a near-real-time assessment of the LOI#1 maneuver indicated a significant under-performance, plans were in place to extend the bum via command upon completion of the nominal LO1 burn duration.

Delayed LO1 #1 Contingency (3 hours or l e s s ) : For delays in the nominal start time of LOI#l of 3 hours or less, it was planned that a maneuver with a AV essentially along the negative velocity vector of the outgoing hyperbola would be executed to capture into a 2-day period (or less) lunar orbit. For a 3 hour delay in LOI#l start time, it was estimated that a penalty on the order of 400 m/s would be incurred, which would severely jeopardize the mission.

(greater t ): For delays in the nominal start time of nd that a d w a s not the most efficient method to get into lunar orbit. Instead, for the nominal Lunar Prospector January 7 efficient to delay any orbit correction until 3 days past periselene and days after periselene 1 . The propellant penalty associated with this contingency strategy is approximately the same as that of a direct capture after a 3-hour delay in LOI#l start time (400 m / s ) .

Key to these plans was the early detection of anomalous maneuver conditions through near-real - t'm e orbit assessment using the Doppler residual method described in the previous section. In preparation for these contingencies, procedures and data bases of required attitude and burn conditions as a function of time were prepared for quick implementation in the event of a contingency.

LP MANEWER RESULTS Table 3 contains a history of orbit conditions following each maneuver in the transfer and lunar orbit It should be noted that target maneuver AV values listed in the table may insertion phases of the mission.

in some cases be slightly different from ideal AV values expected from a propagation of the pre-maneuver state with nominal maneuver end states targeted. This is due to the existence of slightly different Lunar Prospector propulsion models (all consistent to within a few percent) that were available during the mission. As a result, in some cases the selected burn time was based on an average of the models in an effort to provide an added measure of safety (e.g. a slightly longer bum was used during LOI#l). The estimated performance values in Table 3 are based on the trajectory team's baseline propulsion model and are computed relative to actual commanded maneuver times and the best available thrust calibration estimates going into each bum. Possible calibration error sources include attitude uncertainties and propulsion system adiabatic cooling effects. All post-maneuver states are represented in terms of Mean-of- 52000 Keplerian elements relative to the equatorial plane of the central body indicated.

Table 3 LUNAR PROSPECTOR MANEUVER SUMMARY

-

LOI#l Target 364.4 m/s Maneuver AV Estimated 99.3% Performance Post-AV State 1/ 7/98 1/ 7/98 1/ 8/98 111 1/98 Epoch (GMT): 03:30:00 12:30:00 8:45:00 12:20:00 182799 197202 19623 1 6014.6 a o a n ) : e : 0.96403 0.96727 0.96886 0.69714 i (deg): 29.20 29.27 29.26 89.72 S2 (deg): 318.58 318.20 318.16 192.59 w (deg): 318.09 318.70 3 18.32 150.37 MA (deg): 0.14 13.34 43.74 10.63 Period (hrs): 216.1 242.1 240.3 11.63 359023 .387949 386351 RApaa.psb - 1 : 10207.6 Central Body Earth Earth Earth Moon Lift-off of the Lunar Prospector mission occurred on time -approximately 2 seconds into its 4-minute launch window. Following a nominal launch and parking orbit ins the Athena I1 launch vehicle, 64 seconds to place the spacecraft int sfer orbit to the moon. The the STAR37 motor burned for computed post-TLI orbit state vector is presented in Table 3, while injection state (impulsive) is shown in Table 4. The TLI bum occurred over the North-Western coast of Australia (LON = 125 E, LAT = 18 S ) . The ground track from launch to TLI +18 hours is shown in Figure 3. The solid arcjust east of Florida consists of the 13 minute boost phase. The ground track for the 42 minute coast in the parking orbit (represented by the dashed line beginning at the end of the boost phase) crosses over southern Africa and ends at the TLI point on the NW coast of Australia. The post TLI ground track is labeled with tic marks every hour and is shown for the first 18 hours of the transfer orbit. The combination of Lunar Prospector launch and injection errors is estimated at -9.1 m/s. Planar orbit dispersions (i.e., inclination, ascending node, and argument of perigee) were very small -- on the order of a few hundredth of a degree.

Table 4 LUNAR PROSPECTOR NOMINAL TRANSFER ORBIT INJECTION STATE ' .

0 Q.

0 0 X 0 ..I rn n

ii B . 2

a 0 0 a . - I 0 0 a .

-.bo 30' B 60' E 90' h 120' B 150; B 140b B 150' W110' W 90; W 60' W 30; W 0 '

LONGITUDE Figure 3: Lunar Prospector Launch and Transfer Orbit Injection Ground Track rs Table 5 lists the timeline of key events in the transfer orbit. TCM#I was nominally planned for TLI + 4 . 5 hours, however, a number of factors conspired to delay the maneuver until 8.5 hours after TLI. The nominal post-TLI timeline called for immediate acquisition of the spacecr mode using NASA's Tracking and Data Relay Satellite (TDRS) - West spacecraft and for use of the telemetry and Doppler to verify the health of the Lunar Prospector d c and the performance of the TLI bum. The TDRS support was planned on a best effort basis as part of an on-going effort to establish the potential use of TDRS to support non-TDRS (Le. not equipped with a TDRS-compatible transponder) missions. Unfortunately, due to limited test data on the LP transponder frequency characteristics, and the limited sweep capability of the TDRS spacecraf%, TDRS was initially unable to lock up on the LP telemetry stream. At TLI + 21 minutes, the Deep Space Network's Goldstone station locked on the LP transmit signal in two-way mode, and within a few minutes, data from the 300 bits per second (bps) telemetry stream was received. However, the telemetry data observed was noisy (due to a combination of the low bit rate and marginal LP antenna aspect angle geometry.) and commanding was suspended until the antenna geometry improved and the health of the spacecraft could be ascertained.

Table 5 TRANSFER ORBIT TIMELINE OF KEY EVENTS After the successful DSN acquisition of LP in two-way mode, TDRS was able to lock up on the telemetry stream. However, as TDRS visibility was expected to end a short time later, mission controllers to r i s k re-configuring the ground system to accept the data, and instead decided to were reluctant concentrateon the Goldstone coverage.

About an hour later, with the antenna geometry improving and engineers reassured that the LP attitude, orbit and subsystems were nominal, spacecraft commanding was resumed. At TLI + 02:27, the spacecraft * The marginal antenna angle geometry was expected, but it was hoped that data quality would hold up until the spacecraft could be commanded into its cruise attitude. It is suspected that poor geometry was only partially to blame for the noisy data, and that another factor w a s the low transmit bit rate in use initially to improve chances for TDRS acquisition. The LP transponder was an off-the-shelf item with a 1024 kHz subcarrier frequency designed for higher bit rates. At the 300 bps telemetry rate the LP subcarrierto data r a t e ratio was not optimal for ensuring ground station acquisition, despite pre-launch efforts to configure ground equipment in such a way as to maximize performance using this signal.

was re-oriented 90 degrees toward its cruise attitude and commanded to its nomina1 3600 bps telemetry rate. As a result of the delays and concerns over cooling of the space om deployment mechanisms, a decision was made to alter the target cruise attitude by approximate grees in order to provide additional solar heating of those areas. The science booms were successfully deployed at TLI + 03:44. A 21 degree attitude trim maneuver was performed at TLI + 0558 for thermal reasons.

As a result of these delays in the timeline, the first trajectory correction maneuver was not performed until TLI + 8.5 hours. This delay raised the required AV magnitude of the correction from 38 d s e c (for a TCM#l at TLI + 4.5 hours) to 50 dsec. However, this was still well within the budgeted 80 d s e c allocated for transfer orbit maintenance. TCM#l consisted of a vector burn with a 13 m/s axial component and 48.5 d s tangential component.

TCM#2 took place on schedule and consisted O f a7.4 m / s vector bum (1.2 m/s axial and 7.3 m/s tangential). A final TCM#3 was scheduled to take place at LO1 - 24 hours, but was called off when a propagation of the post-TCM#2 state yielded a projected periselene condition within 10 km of the target .

radius and . 1 deg of the target inclination.

Lunar Orbit Insertion Maneuvers Lunar Prospector LO1 maneuvers occurred according to plan, with the propulsion system performing to between 1 and 3 percent repeatability. LO1 burns #1 and #2 placed the spacecraft into an 1 1.63 hour, then a 3 . 5 2 hour orbit, and LOI#3 dropped apoapsis down to within 50 km of the target mapping orbit altitude.

LOB3 was purposely targeted 3% short as an extra margin of safety, since it was predicted that a 9% hot LOH3 bum would have dropped apoapsis down to lunar radius. A final mapping orbit correction (MOC- 1) maneuver on January 15,1998 circularized the orbit at a 100 km altitude. This maneuver consisted of two axial bums to lower apoapsis and raise periapsis to a target radius of 1838 km. The second bum was executed slightly off-apses to permit DSN coverage of the burn.

Mapping Orbit Maintenance The strategy for maintaining the LP 100 km altitude polar orbit was developed with the following goals: 1. Maintain an altitude band of 100 km +/- 20 km 2. Conduct maneuvers in view of a ground station 3. Minimize the number of maneuvers 4. Use axial maneuvers instead of vector bums if possible The last goal was established for reasons of operational simplicity, since LP vector burns cannot be performed readily during shadow periods for lack of a reference sun pulse. Since the nominal LP spin-axis attitude is within a few degrees of the ecliptic normal (and therefore almost normal to the lunar equator), this goal required that the argument of periapsis be close to zero degrees to allow axial maneuvers to take place parallel to the velocity direction at penapsidapoapsis. Furthermore, as LP maneuvers consist of a two-bum Hohmann sequence, the second goal requires that maneuvers be conducted when the orbit plane is normal to the Earth/moon line - a condition that occurs approximately every 14 days.

Figure 4 shows a history of the LP orbit periapsis/apoapsis altitude and argument of periapsis through the first orbit maintenance maneuver in the mapping orbit, followed by a prediction of the orbit evolution assuming no further maneuvers are conducted. The dashed vertical lines reflect the actual date of the first LP MOC maneuver (MOC#2) and the planned date of the second (56 days apart and coincident with periods of full station coverage). Furthermore, as the plot of orbit argument of periapsis shows, the maneuver dates occur when the line of apsides is within I5 degrees of the equator, allowing axial 28 1 maneuvers to cake place with only minor losses in efficiency. ithout maneuve~, the orbit could be expected to impact the moon within approximately 150 days.

Q U

- 3 8 0

M a Figure 4: LP Orbit Evolution of ApoapsisPeriapsis Altitude and Argument of Periapsis Beyond the First Mapping Orbit Maintenance Maneuver (LP75D lunar potential model) number of maneuvers performed, A 56 day interval between maneuvers was chosen to m d minimizing perturbations to science from the standpoint of reducing operational r i s k to the mis on was to adjust the orbit eccentricity data collection. Therefore, the goal of each mapping orbit and argument of periapsis in order to maintain an altitude variation within +/- 20 km over the next 56-days.

city discussed in References 5 and The phase space approach of plotting argument of p is a usefid tool in understanding LP orbit evolution. Figu tains such a polar plot with eccentricity plotted along the radial direction and argument of periapsis lar direction. In this plot, LP orbit evolution is plotted for a 1 -year duration, starting after MOC#l on January 15,1998. Figure 5 describes how city grows with time, as argument of periapsis gravitates toward a value of 270 degrees. excursions beyond the 4- 20 km range, initial values of 180 deg for argument of periapsis 1 12 km orbit) were targeted as maneuver end conditions to evolve through the zero eccentricity point and p expected that such a strategy,will figure even more p ng the extended mission phase when frequent maneuvers will be required to maintain the 900.09 Figure 5: Lunar Prospector Mapping Orbit Eccentricity and Argument of Periapsis Evolution .

[Eccentricity along radial direction and argument of periapsis along angular direction] LP ORBIT DETERMINATION RESULTS Upon launch of the LP spacecraft, the first task of the orbit team was to assess the performance of the TLI maneuver. The failure to acquire TDRSS data delayed that assessment. when coherent DSN Doppler w a s received, approximately 23 minutes after TLI, the residuals indicated a slightly cold burn. The expected residuals for several off-nominal cases, along with the actual residuals obtained, are shown in Figure 6. The off-nominal cases modeled are -20 and -35 m / s TLI magnitude error and +/- 0.8 and -2.4 deg in argument of perigee error.

The frrst full state estimate w a s not obtained until 2.5 hrs after TLI due to the failure to acquire TDRSS tracking data and due to dropouts in the DSN data. The calibrated TLI magnitude error w a s estimated at -9.1 ds, which matched the near-real-time Doppler assessment data fairly well. TCM#l was not 8.5 hours after TLI.

preformed until After each of the TCMs, a Doppler assessment was made in near-real-time. In each case, the assessment indicated that the maneuver w a s slightly cold. The final calibrated efficiencies of these maneuvers (Table 3) are consistent with these initial Doppler-based assessments. The first orbit solution after TCM#l was obtained after seven hours. The goal was to obtain this solution within six hours of the maneuver, however since TCM#l was performed much farther away from perigee than planned, additional data were needed for a solution. After TCM#2, the first orbit estimate was available eight hours later - exactly as expected.

TCM#3 was cancelled, which meant the orbit trajectory would be well determined for the LOI#l maneuver.

During the 30 minute LOHI maneuver, the DSN lost coherent lock on the spacecraft. As a result, no Doppler data were obtained until immediately after the maneuver. A Doppler assessment at that point indicated a successful lunar capture with only a slightly cold bum. A full orbit state was obtained 1.5 hrs after LOI#l, which was slightly over 2 hours earlier than expected.

4ooo

I

Figure 6: Lunar Prospector Doppler Residual Signatures for Post-TLI Maneuver Assessment LOH2 and LOI#3 were nominal. Doppler assessments were used during each maneuver to estimate maneuver efficiency. In particular a near-real-time assessment of LOI#3 was performed out of concern for an over-bum beyond the 100 km target apoapsis altitude. Full state estimates were available 2 hrs after LOI#2 and 3.5 hrs after LOI#3. The trend for the amount of tracking data needed to converge after LOI#l, 2 & 3 was exactly opposite of what was expected. This has been attributed to the inadequacy of the GLGM-2 potential model at lower altitudes. This effect was not seen in the covariance analysis.

Once the mapping orbit was achieved, different batch arc lengths were attempted with the goal of extending them as long as possible to reduce the amount of processing time (since it was expected that definitive ephemerides would be regenerated with the new potential model at a later date). A 14 hr tracking arc w a s chosen with a 2 hr overlap between two consecutive tracking arcs. Thus two 12 hr definitive ephemeridesper day were placed on the Goddard Lunar Prospector web site (http://fdd.gsfc.nasa.gov/lp/) for use by the LP mission control center and science community.

The first updated LP potential model was available after just two weeks in the mapping orbit. The new model, LP75A, was developed by A. Konopliv of the Jet Propulsion Laboratory. A final model, LP75D7, was available after one month and was used to generate updated ephemerides. A comparison of the orbit accuracy achievable with each of these three models is shown in Table 6. The orbit accuracy is measured as the difference between two consecutive definitive ephemerides over the period of overlapping tracking data. The LP75A solutions consisted of 26 hr arcs with a 2 hr overlap. The LP75D solutions consisted of 55 h r arcs with a 7 hr overlap.

Clearly the LP75D solutions meet the LP mission requirements. As of February 23, updated definitive ephemerides were being generated using the LP75D model. The regeneration of the first five weeks of definitive ephemerides will be completed by mid-April. The entire lunar mapping orbit definitive ephemeris history is available on the Goddard Lunar Prospector web site.

T ~ b l ~ 6 UN APP~NG ORBIT CONCLUSION Lunar Prospector orbit operations occurred largely according to plan and resulted in the successful attainment of the 100 km altitude polar mapping orbit within budgeted orbit maneuver propellant allocations. A large share of the credit for the success of the Lunar hospector mission can be attributed to the straightforward design of the spacecraft and the overall mission, and to a well-build and well-tested spacecraft that performed flawlessly. In particular, the spacecraft propulsion system was well modeled and provided excellent repeatability. Finally, a robust orbit determination strategy, characterized by accurate solutions and fast-turnaround, was also an important factor that contributed to maneuver targeting accuracy and low propellant consumption.

ACKNOWLEDGMENTS Special thanks to Greg M a r r , Karen Richon, Marc0 Concha, Osvaldo Cuevas, Pat Johnson and Joe Toth of NASAIGoddard, and to Marcie Smith of NASAIAmes, Dan Swanson of Lockheed Martin, Mark Ryne and Alex Konopliv of JPL and Dr. Alan Binder of the Lunar Research Institute for their excellent contributions to the success of Lunar Prospector trajectory operations.

REFERENCES 1. Hubbard, G. S., et al., "The Lunar Prospector Discovery Mission: A New Approach to Planetary Science", IAF-97-Q.4.01, October 1997.

2. Clarke, V. C., Jr., "Design of Lunar and Interplanetary Ascent Trajectories," AIAA Journal, Vol. 1, No. 7, July 1963.

3. Lemoine, F. G., et al., "A 70th Degree Lunar Gravity Model (GLGM-2) from Clementine and Other Tracking Data," Journal of Geophysical Research Vol. 102, No. E7, Pgs 16339-16,359 July 25, 1997 4 . Folta, D., et al., "Lunar Prospector Mission Design," Flight Mechanics Symposium, Greenbelt, MD, May 19-21, 1997.

Cook, R. A. and Sweeter, T. H., "Orbit Maintenance for Low Altitude Near-Circular Lunar Orbits", 5.

AAS Paper 92- 185, February 1992.

6. Konopliv, A. S., et al., "A High Resolution Lunar Gravity Field and Predicted Orbit Behavior", AAS Paper 93-622, August 1993.

7. Konopliv, A. S . , "Lunar Prospector Gravity Results", to be presented at AGU Spring Meeting, May 26-29, 1998, Boston, MA.

Werner Enderle, Michael Schmidhuber, Eberhard Gill, Oliver Montenbruck, Armin Braun', Bernd Eisfeller, Oliver Balbacht The succedhlly usage of spaceborne GPS receivers for orbit- and attitude determination has in the past mainly been applied for spacecraft in near-circular Low Earth Orbits (LEO), e.g. TOPEX@OSEJDON, RADCAL,.

However, no experience exists of using GPS signals for spacecraft navigation in altitudes above the GPS (20.000 k m ) , such as the Geostationary W i t (GEO) or the Geo Transfer Orbit (GTO). In this contexf EQUATOR-S w i t h it's highly elliptical orbit offers a unique possibility in order to increases signiscantly the experience concerning GPS visibility, signal reception and GPS based navigation.

The application of GPS based spacecraft navigation in GEO's and HEO's will bring advantages in many aspects. Besides orbitlattitude detennination, GPS offers a potential for increasing command capabilities and decreasing ground station support. This will lead to a reduction of mission cost especially for LEOP phase of geostationary satellites.

This paper describes first experiences in operating a GPS receiver in a highly elliptical orbit and presents fist results of GPS visibility and signal reception conditions within t h i s orbit.

INTRODUCTION EQUATOR-S (see Figure 1) is a scientific satellite for the investigation of the magnetosphere of the sun under the responsibility of the Max-Planck-Instit,. her Extraterrestrische Physik (MPE) in Garching, Germany. EQUATOR-S was launched on December 2, 1997 into the Geostationary Transfer Orbit (200 km x 36000 km) together with a Japanese Communication Satellite JC-SAT 5 on an Ariane 4 (44P) fiom Kourou in French Guyana. EQUATOR-S is under the control of DLR's German Space Operations Center (GSOC). Following a nine day LEOP phase the spin stabilized satellite was boosted (see Figure 2, Table 1) into the final low inclination HE0 (500 km x 67000 k m ) .

Besides several scientific payloads, EQUATOR-S carries also a GPS reeeiver (incl. two antennas, cables and interface'box ) as a technological experiment.

German Space Operations Center (GSOC), DLR, D-82230 Wessling. Germany.

Phone: 4 9 (8153)28 17 52, F a +49 (8153)28 14 60, E-mail: wemer.enderle@dlr.de Instime of Geodesy and Navigation @EN), University FAF Munich, Germany.

igure T :E ita1 E ts S e ~ ~ e ~ m e n t on-board o f her field of interest cone ch can be subdivided i (pseudo range, carrier phas emination of the spin axis orientation, verification d special emphasis is given to near-geostationary altitudes in view of potential applications for autonomous orbit/attitude d e t e ~ ~ n a t i o n and control of fUture g e o s t a t ~ o n ~ satellites.

The GPS system is designed for applications on Earth, above Earth surface in very low altitudes (e.g. airplanes) and satellites in LEO. For this reason the GPS antennas directed towards the Earth with an half cone angle of 21.3 [deg] (Ll). Application of GPS in GEO’s and HEO’s are therefore outside the original GPS specifications. The main problems of using GPS in this orbits are given by the visibility limitations of GPS satellites at high altitudes (geometry), signal reception at high altitudes caused by poor signal strength (link budget) and a wide range of Doppler due to highly variable velocities particularly for HEO’s around perigee.

GPS signals in 3 3 5 0 ’ s and GEO (see Figure 3) will only be The reception and using of possible if the following criteria will be filfilled at the same time 0 the line of sight f r o m host satellite (EQUATOR-S) to the GPS satellite is not blocked by the Earth 0 the line of sight from host satellite (EQUATOR-S) to the GPS satellite is not blocked by a spacecraft component e the signal strength of the received GPS signal is strong enough, so that the receiver can track the signal (the Signal to Noise Ratio (SNR) is the key factor) Figure 3 : Visibility of GPS satellites in HEO’s and GEO 289 ’ tennas. For visi reasons (covering the entire s -§ and one at the bottom ( , constrains apply for it’s operation on / Apart fiom the exciting scientific aspects, the GPS experiment firthemore faces a challenging operations concept due to the given constrains relating the operations. Some of the constrains are EQUATOR-S specific e.g. no Earth pointing attitude at high altitudes (poor GPS visibility), during the GTO the bottom antenna was obstructed by the kick motor, this results in a restricted field of view and limited TM/TC system capabilities. Considering the constrains given by the receiver, the expected Doppler shift between the GPS satellites and the receiver is internally computed f i o m a set of orbital elements using a simple orbit model. Therefore, new reference orbit elements must be uploaded fkom ground after each perigee transit. Since there is a high risk to the receiver electronics fiom the hard radiation in the higher altitudes, the receiver shall be switched off above 40.000 km. The automatic selection of GPS satellites is based on their elevation above the user satellite’s local horizontal plane. For high altitudes and an inertial fixed antenna lobe, the selection algorithm fails and has to be replaced by a manual assignment of GPS satellites to the various receiver channels.

Despite these restrictions, which could at least partially be compensated by appropriate operations procedures, the receiver was suecessfblly switched-on and initialized soon after the separation.

WSIBILITY PREDICTION - VISDBILITY TOOL ‘HEOVIS~

Considering that the automatic search algorithm fails in high altitudes, the necessity for a highly reliable GPS visibility predict tool was essentially for successfidly operations in high altitudes and also for analysis. For this reason a GPS visibility predict tool has been developed. The calculation of the GPS visibility includes the geometric visibility conditions and a link budget [3], [4] computation: Implemented in this software is besides the orbit dynamic of the GPS satellites also the complete orbit and attitude dynamic of EQUATOR-S [ 2 ] . The link budget calculation applies all relevant aspects including the antenna pattern of the GPS satellites emitting antennas and the receiver antenna (see Figure 6 and 7). Detailed description are given in [I]. Some visibility predict results are given in the Figures 8-1 1.

29 1 Figure 6: Emitting antenna pattern Figure 7: receiver antenna pattern Vlslbirty of OPS Satelltss In GTO Bottom Antenna mfJ P I Begin: DOY 337 1997/12103 00:OO.OO Figure 8: Predict number of visible GPS Figure 9: Predict visible PRN Predicted Range between EQUATOR-S Predicted Signal t o Nohe Ratio and th :Ps Satelliter 50 _ , . , . , . , . , . , . , . , . , _ , . , _ 40000

. . i40000

7m0r------ ,*---\ /' P44 /' -rp I 1 3 4 5 8 7 8 9 10 11 11 - 5 7 a 9 10 11 11 Them1 .

Begin: DOY 337 1997112103 Figure 10: Predict of SNR Figure 11: Predict of range The first result was the first GPS satellite which has been tracked (PRN 24) by the bottom antenna (two channels simultaneously) for a duration of 0 3 5 2 min in an altitude of about 34.000 km. As far as we know, this was the first time that a GPS satellite has been tracked around the GEO altitude. The precise information of this event is given in Figure 12. The most Critical parameter for the GPS signal acquisition and tracking in high altitudes is the SNR. Unfortunately the SNR value is not given by the Viceroy receiver in a physically unit. The output of this information is a so called SS parameter (Figure 13 a) ). By using a formulae, given by Motorola for Earth application this SS parameter was converted into the S N R with the physically unit [dBHz] (Figure 13 b)).

Comparing this value with fiom HEOVIS predicted S N R value (38 dBHz) shows, that the converted SNR lies ca. 4dBHz below the predicted SNR. The predicted S N R corresponds with information from Motorola for expected initial acquisition of the GPS signal S N R numbers. Figure 13 c) shows the integrated carrier phase measurements and Figure 13 d) displays the pseudo range obtained fiom the receiver. The values of the pseudo range measurements shows that the tracked GPS PRN 24 must have been in the opposite of the E a r t h , seen from EQUATOR-S. The predicted pseudo range for the tracked GPS PRN24 fit very well with the pseudo range measurements.

The second result presented here is the tracking a GPS satellite (PRN04) within a the first side lobe of the GPS emitting antenna. The ability of signal reception fiom the first side lobe increases in HEO’s the number of visible GPS satellites especially in altitudes > 20.000 km significantly. The Figure 14 displays the tracking of a GPS satellite in the first side lobe.

The last result presented here shall be the GPS tracking with the longest duration (40 min). In Figure 15 a) - 15 f ) are the displayed the main parameters of this measurement.

The GPS PRN29 was tracked for about 40 min from an altitude of 15.000 km up to an altitude of 22.000 km (Figure b)).. The total number of visible GPS satellites was between four and five (Figure a)). In Figure d) one could see that the converted S N R decrease with increasing altitude. The values at the end of the curves indicate a loss of signal. The parameter values of PRN03 were also plotted in all Figures, this was done in order to compare the values obtained by a different tracking mode status. While PRN29 would have been used four a position solution algorithm (mode status 8), PRN03 would not have been used (mode status 7).

It should also mentioned that the maximum number of simultaneously tracked GPS satellites was three and therefore no receiver position solution is available.

Orbit determination results based on C/A-code and carrier phase measurements are expected to be available very soon.

Visibility of GPS Satellites in GTO Bottom Antenna 34300 34250 34200 I 2.

m

z

34150 .-..

34100 34050 05:35 OS36 0531 05:38 0538 OS40 05.41 05:42 0543 0544 0545 Time [hh:mm] DOY 337 1997112lQ3 Figure 12: First tracking of a GPS satellite in GEO altitude 85790- 65705: d) - 65780: 65775- i f 85770: f 65785- 0 m s 65780- c 65755- 85750; . . . , . , 1 Figure 1 1 3 : PRN24 measurements GPS Satellite Visibility 18 nna l i I M Time [h] Figure 14: Tracking of PRNO4 in the side lobe Figure 15: Tracking of PRN29, longest tracking period / CONCLUSION The results presented in this paper shows, that it is generally possible to acquire and track GPS signals in near GEO altitudes. The GPS experiment on-board EQUATOR-S gives first answers about the SNR values in altitudes near GEO. It also could be proved, that it is possible to track GPS satellites in the first side lobe. The fact of GPS signal reception fiom the side lobe increases the number of visible GPS satellites in high altitudes significantly. In spite of all constrains concerning the operations, it could be shown that it was possible to operate the GPS receiver under dif33cult circumstances and receive good measurements. In this context the developed GPS visibility predict SW ‘HEOVIS’ was important and delivered very good results.

The experiment is still running and we are looking forward to reach the next two goals, 1. tracking of four or more GPS satellites at the same time and 2. tracking of GPS satellites above the GEO altitude, REFERENCES 1. W. Enderle, Lagebestimmung von Satelliten in hochexzentrischen Orbits, basierend auf GPS Messungen, Ph.D. Thesis, Technical University of Berlin, will be published.

2. U. Feucht, W. Enderle, D. Moormann, EQUATOR-S Attitude Dynamics Simulation, 12* International Symposium on Space Flight Dynamics, Darmstadt 1997.

3. Motorola: Data Sheets for Viceroy GPS Receiver and Antenna 4. F.M. Czopek, S. Shollenberger, Description and performance of the GPS Block I and II L-Band Antenna and Link-Budget. ION GPS-93 Three g r o u n d ~ b r e ~ n g experiments involving simultaneous collection of QPS data by spacecraft took place between November 1996 and October 1997. These experiments had the god of demonstrating the feasibility of using GPS for relative navigation of spacecraft. The experiments took place w i t h i n ESA's ATV (Automated Transfer Vehicle) Rendezvous Pre-development (ARP) program. This program aims to validate rendezvous technologies that will be used in ATV for its proximity operations around the International Space Station.

The spacecraft involved were NASA's Shuttle and the US/German retrievable Astrospe s a t e l l i t e for the first flight demonstration t h a t took place during the STS-80 flight at the end of 1996 and the Shuttle and the Russian MIR space station during the STS84 and 86 flights. GPS receivers were installed in the spacecraft involved and GPS data were collected for several intervals during the rendezvous and separation phases.

ESOC Flight Dynamics was entrusted with the challenge of providing the most accurate trajectofi8sthat could be computed using the on-board collected GPS data in combination with ground collected data. These precise trajectories were required in order to validate the algorithms to be used in the ATV for relative navigation. The baseline for ATV k that the O F % data collected on-board the A W and the I S S H/il be processed in real time on-board the ATV to obtain a relative navigation solution, ESOC routinely produces precise orbits and clocks for the GPS satellites as part o f its involvement as Analysis Centre in the International GPS Service for Qeodynamics.

With those precise products it is possible t o correct in post-processing for the errors introduced by Selective Availabili. For these experiments the GPS measurements are corrected and then used to calculate the absolute position of each of the two spacecraft. The biggest sources of error that have to be deal with are the ionospheric delay, because only L 1 frequency data are being collected, !he pseudo range noise and the cycle slips in the carrier phase.

So far the data for the first Flight Demonstration have been processed, with results that compare well with those derived f r o m the Shuttle TCS laser ranging system and we are currently waiting for the availability of the data for Flight Demos 2 and 3 . For these last two Flight Demos there is a delay in the retrieval of data because some of them have to be down-linked from the MIR space station, but they should be available soon and it i s expected that data processing will be completed before the end of the year.

T h i s paper will present the strategy used to obtain the best estimated trajectories, the problems found during the analysis of the data and the results obtained, including comparisons with trajectories obtained using other tracking systems or algorithms.

/ESTIMAT Karl Hans Neumayer and Rolf Konig* In the context of scientific investigations related to the CHAMP project and other upcoming LEO missions, the GeoForschungsZen- trum Potsdam (GFZ) initiated the development of an efficient real- time on-board orbit estimation and propagation software called POPE. The methods specific to POPE are in fact adaptions of orbit prediction algorithms that are being in operational use for the PRARE system on-board METEOR 3/7 and ERS-2 for several years.

Simulation studies show that orbit elements generated by POPE from on-board GPS measurements suffice t o bridge a gap of up to one week without GPS data with position errors of 10 t o 20 km on the average for LEO orbits at 300 km altitude.

INTRODUCTION CHAMP1 is a mini-satellite mission for geopotential and atmospheric research that will be launched by the end of 1999 into a near-circular, near-polar orbit. It will cover 5 years, with the orbital altitude decaying from initial 450 km to 300 km towards the end of the mission. The payload instruments include a TurboRogue space receiver providing GPS navigation solutions at 0.1 Hz frequency with an accuracy of approximately 55 m standard deviation in position.

The attitude and orbit control system has to keep CHAMP earth-oriented within 2 degrees during normal mode operations. For this task, orbital position is needed at 1 Hz frequency with errors not exceeding 240 km in along-track direction. From the GPS navigation solutions, accurate positions can easily be interpolated. In case of GPS navigation downtimes, an orbit propagator shall extrapolate positions and velocities for the missing time intervals.

Within this context, we peresent the newly developed propagation software POPE (ERARE elements on-board Orbit Eropagator/Estimator).

"GeoForschungsZentrum Potsdam (GFZ), Div.1: Kinematics and Dynamics of the Earth, c/o DLR, D-82230 OberpfaEenhofen, Germany. Contact: Dr. Karl Hans Neumayer, Phone (+49) 8153 28-1330, FAX (+49) 8153 28- 1585, e-mail hans.neumayerQdlr.de The orbit parametrization in POPE is based on the ephemeris initially designed for the PR-ARE system on-board ERS-13 . PRARE was sucessfully tested on-board the METEOR-3/7 satellite during the years 1992 until 1995. In 1995 ERS-2 overtook

the space segment function4 . The PRAW orbital elements are ge rated on ground

from laser tracking data and uploaded once a week during nominal operation.

Upcoming LEO missions like CHAMP that carry on-board GPS receivers have on- board data processing capabilities. For the orbit prediction chain, off-line processing is no longer necessary. Orbit parameters can be generated from GPS data on-board and on-line on the side in order to be utilized when the GPS receiver is down.

The above-mentioned PRARE orbit parametrization is based on a series expansion of regularized Kepler elements that are used instead of the common Kepler elements a, e, i, w, R, N (2) for near-circular orbits in order to avoid singularities. Here a is the semimajor axis, e the eccentricity, i the inclination, w argument of perigee, R longitude of ascending node, M the mean anomaly and u the argument of latitude. Both sets of elements are connected by the equations where u is the true anomaly.

The series expansion is a result given by classical perturbation theory2 . The overall perturbations of the elements are composed of periodic effects with frequencies that are multiples of the Earth rotation, mean motion of the satellite, and secular rates of longitude of node and argument of perigee. Thus for the PRARE system, orbital elements are represented by linear combinations of common time-dependent and trigonometric polynoms. A core coefficient set of the polynomial expansion has been chosen for the CHAMP mission enabling the design of safe algorithms and providing fast computing performance at the same time. The selection is the result of a significance analysis; it is not specific to CHAMP, indeed all LEO orbits can be approximated in the same way.

For the update of the expansion coefficients, the GPS coordinates and velocities provided by the on-board GPS receiver in the terrestrial reference frame WGS84 are not used directly. In a first step, they are transported into the pseudo-inertial true- of-date system by a removal of the Earth rotation. They are then transformed into osculating regularized Kepler elements (1) that, instead of the original positions and velocities, are considered as the raw measurements.

-4s the above-mentioned series expansion in fact establishes a linear model for the orbit elements in (l), the whole update can be taken care of within the framework of linear estimation theory. That would not be possible if the the GPS coordinates and velocities were used as raw measurements, as the elements in (1) depend on the state vector in Cartesian coordinates in a complicated non-linear manner.

In order to further faciliate the procedures involved we assume that the six mea- sured Kepler elements are mutually uncorrelated a-priori. Thus we have to deal only with six independent noisy measurement channels instead of a fully occupied measure- ment noise matrix, and we need not establish relative weights between the different observed orbit elements.

REVOLUTION-DEPENDENT PERTURBATIONS

From the different kinds of perturbations given by analytic orbit theory2 , the

following ones have been considered important for the orbit propagator POPE: All the elements have perturbations that are periodic with the orbit frequency of the spacecraft. The sixth element, the argument of latitude u, exhibits a linear drift with revolution-dependent components superimposed. The length of the ascending node R has an additional small drift due to the influence of the oblateness of the earth.

If these periodic terms are removed, among the resulting smoothed elements the first four are constants and the last two are linear functions of the elapsed time.

Those constants and the slopes of the linearily varying terms can be conveniently estimated from the smoothed measurements (4).

The indicated smoothing operation is performed by subtracting a trigonometric polynom from each one of the elements listed in (1). The polynomial coefficients Am, B , are given by a numerical approximation of the integrals 1 2=

Am = -1 f(u) dsin(mu)

nm u=o where f(u) symbolically stands for the periodic part of one of the dements in (1).

For the argument of latitude u(t) e.g. we have with

u ( t ) = a(t) + Up@); (8)

Le. u(t) is composed of a purely linear varying part E ( t ) and a purely periodic part U P ( t > .

The smoothed signal which is taken care of by the rest of the filter is M (A, cos(mu) + Bm sin(mu)} .

m= 1 In order to faciliate the book-keeping inside the orbit propagator algorithm, the co- efficients A , , Bm are re-computed during every revolution. However, as revolution- dependent gravity perturbations of orbit elements are known to be extremely stable phenomena, in theory the trigonometric coefficients could be refreshed only sporadi- cally, say once a week.

At this point, some remarks are in order. From the purely logical viewpoint, the argument of latitude u should be represented by an expression of the form M u = ti + E {Am cos(m%) + Bm sin(m.ii)} , m=1 i.e. in (9) the trigonometric quantities cos(mu), sin(mu) should be replaced by cos(mti), sin(mG). The formula (10) is indeed the one which is used in the pre- diction part of the orbit propagator: First, the smoothed argument of latitude G ( t ) is predicted, then the result is inserted in (10) in order to predict u(t). But apart form the fact that, from the numerical viewpoint, cos(mG), sin(mG) are not perceptibly different from cos(mu), sin(mu) needed in (5), (6) and (9), they have a big advantage at the same time: They can be derived from cos(u), sin(u) by well-known trigonomet- ric formulas, and the latter quantities are computed on the side if an update is due by the transform of the state of the spacecraft in Cartesian coordinates into osculating Kepler elements without the necessity of actually invoking trigonometric functions.

It is also clear that the filter algorithm has to work in a revolution-wise interleaving manner: For the removal of the periodic part uP(t),we need at least an approximative value of the linear part ti(t), and for the subsequent filtering of the linear part, the periodic terms have to be removed. This leads to a dedicated startup procedure, for during the first revolution, there is no preceeding revolution interval. Here the drift

term of the argument of latitude is set equal to the orbit frequency 4 - of a

circular orbit, only the second revolution uses the correct drift term, and consequently only during the second revolution the Fourier expansion coefficients of the periodic part are computed correctly. As these coefficients can be used only from the third revolution onwards, and allowing for a stabilizing period for the filter of the smoothed Kepler elements, it is clear that the whole algorithm needs at least three revolutions of the spacecraft to attain its maximal accuracy.

Figure 1 illustrates the transient part for an example LEO: After some five hours, the along-track error has shrunk from the order of several kilometres to several hun- dred metres. The above-mentioned startup procedure covers the first five instead of the first three revolutions in order to stay on the safe side.

..................... .....................

1 2 3 4 5 6 elapsed time (hours) Figure 1 Example for the transient phase of the filter.

FILTERING THE SMOOTHED ELEMENTS The smoothed elements of equation (4) in theory are either constants, or they have a linear trend. They are separately modelled by linear stochastic systems whose state dimensions are either one or two.

Without going into details, we wish to make only two remarks: Due to the small state dimension of the models for the individual elements of equation (4), and as the individual observation noise channels have been assumed to of the filters are extremely simple; be mutually uncorrelated, the update equations

in case of the nominal constants si, f , fj, 5 they can be represented by trivial convex

combinations of predicted and measured value.

The method of filtering chosen is described in the literature as exponential age weighting, also: fading memory fiZte$ without system noise. Due to the stationary character of the filter involved however, only the constant filter feedback matrices ex- plicitly appear in the update part of the filter algorithm, and these matrices have been designed as telecommand parameters. Thus the overall concept (classical Kalman, Kalman without system noise, fading memory filter etc.) the filtering equations are derived from has no effect on the explicit coding, as long as the filters are considered stationary. After some real-world experience with the satellite.in orbit, the filtering algorithm could be redesigned and the filter feedback matrices could be overloaded remotely by the ground segment.

It has already been mentioned in the introduction that linear drift terms and revolution-dependent perturbations do not explain all deviations from an ideal Kepler orbit. There are still the effects to consider that are periodic with the frequency of the Earth rotation. Figure 2 shows the along-track error between a simulated LEO satellite trajectory that is the output of a high-precision orbit integrator and its representation by our ,on-line estimator after the transient phase of Figure 1 with regular updates every 60 seconds between hours 12 through 48. Apart from residual contamination of orbit frequency we see an offset of some 400 m and a superimposed diurnal oscillation of some 600 m amplitude.

I I I ~ 1000 ............................... : ..........................

C"" -500 ...............................................................

t

-750 ............................... ;.. ............................. .............................................................................................. .............................

I I 1 I 12 18 24 30 36 42 48 elapsed time (hours) Figure 2 Diurnal along-track deviations in steady-state.

However, as those daily perturbations do mainly concern the along-track error and therefore, among all the elements of eq. (l), the argument of latitude u(t), and as the revolution-dependent perturbations are known to have an amplitude of 20-30 km at orbit altitudes between 350-450 km and thus surpass the observed diurnal effect by far, we choose to neglect it.

Up to now, we have a kind of cascaded filter: First, the revolution-dependent terms are removed by Fourier techniques, then the remaining smoothed elements are treated with a steady-state Kalman filter. Diurnal effects are neglected.

If trial runs are conducted with simulated orbits that are the output of a high precision orbit integrator, then mainly the argument of latitude u(t) exhibits a run- away phenomenon if a data gap occurs and the orbit propagator is not updated: The difference between the true and the predicted value increases in a roughly quadratic manner, and the corresponding along-track error in a LEO environment may surpass 200 km within two to three days. The reason for this effect is the air drag which has not been accounted for.

Also affected is the semimajor axis a(t) of the spacecraft orbit, but not in such a pronounced manner, and the deviation is more like a linear shrinking.

A first and straightforward attempt to replace the linear model

E @ ) = Bo + e1 (t - t o )

(11) for the smoothed argument of latitude in the Kalman filter with a quadratic model 1 2

a(t) = iio + ti1 (t - t o ) + p ( t - t o )

fails due to numerical reasons: Whereas i i l is fairly large - in fact, it is roughly given

by 2n/90minq1 x 10-3s-1 - it can be seen by a rule-of-thumb estimation from the

perceived runaway phenomenon that i i 2 is of a magnitude of 10-l2s-l. As the input G of (9) into the Kalman filter is still contaminated with a residuum of imperfect removal of revolution-dependent terms and neglected daily effects, fi2 cannot be estimated in a reliable manner, with catastrophic consequences for an attempted prediction.

Thus, for the estimation of the airdrag correction, a brute-force method was cho- sen: Parallel with the actual filter for (11) which is updated with every incoming measurement (9), a second identical dummy filter runs whose state is periodically (e.g. once a week, this time parameter can be adjusted) reset to the state of the first filter, and which receives no update from measurements at all. From the difference between the actual filter and the dummy, the coefficient ii2 of (12) is estimated over the time span between two adjacent resets with a recursive least-squares procedure.

Basically, the dummy filter conducts an attempted prediction over the latter time span neglecting the airdrag, and we simply look up the error we commit and count on it that this error varies only slightly over a few days.

With respect to the airdrag our method of parametrizing the orbit with regularized Kepler elements has one big advantage over other orbit propagation schemes that are based on integrating the dynamic equations of the spacecraft in Cartesian coordinates directly. Those dynamic equations are i: = -grad U ( t , r ) (13) where r and T J = .i. are the three-dimensional vectors of position and velocity and U ( t , r ) is a model function for the gravity potential of the Earth. At a first glance, the latter treatment of (13) saves computer time, as no trigonometric functions have to be evaluated, contrary to ou?'method. But, as the air drag mainly causes an along-track error that, roughly spoken, appears in the argument of latitude u alone, we have to correct only G ( t ) in the simple manner indicated by eqs. (ll), (12) without caring for the underlying physics. In (13) however, the neglection of air friction in Cartesian coordinates does not affect only one, but all entries of the vectors r and in a non-separable manner. Accounting for the airdrag in this framework furthermore means a tricky non-linear on-line estimation of a non-negative scaling parameter y in the extended dynamical model whereas we can rely on linear theory alone as can be seen from eqs. ( l l ) , (12).

SIMULATION STUDIES Tests with simulated orbits generated by a high-precision orbit integrator indicate that a prediction accuracy of 10 to 20 km on the average over a dataless gap of one week at an orbit altitude between 300 to 460 km can be expected from an airdrag correction estimated from the data of the preceeding week. In one case, an along- track error of 80 km occurred. The outcome depends on the actual variability of the atmosphere-mainly influenced by solar and geomagnetic activity-during the time when no measurements are available.

Figure 3 shows the error for a 300 km altitude orbit produced by the filter for a slightly different setting where the air drag effect was estimated from data over two days rather than from a whole week. The coordinate system is the earth-fixed WGS84. During the first two days, the filter received a Cartesian reference frame regular update of vehicle position and velocity at sampling intervals of 60 seconds.

The following data gap of five days (i.e. no update at all) leads to a position error in every coordinate of 12 km at most for all three coordinate directions.

1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 Figure 3 Example orbit error for the three coordinate directions; abscissa: elapsed time in days, ordinate: deviation in km.

CONCLUSION The method for the off-line estimation of PRARE orbit elements on ground that LEO satellites for years has been has proved its reliability by being operational on two successfully transformed into an on-line on-board orbit propagator. The techniques employed as e.g. numerical Fourier expansion and Kalrnan filtering of constants and drift terms, are conservative, straightforward and robust. The use of regularized Kepler elements in the framework of semi-analytic orbit theory guarantees that the air drag effect is mainly restricted to the quadratic runoff of one orbit element alone.

It can be taken care of by an extremely simple estimation procedure. The quality of orbit parameters obtained from a few (2 to 7) days with complete data coverage suffices to restrict the orbit error oyer a data gap of 5 to 7 days to some 10 to 20 km.

1 Reigber, Ch., ock, R., Forste, Ch., Grunwaldt, L., Jakowski, N., Liihr, Schwintzer, P., Tilgner, C.: CHAMP Phase € 3 , Executive Summary, Scientific Tech- nical Report STR96/13, GeoForschungsZentrum Potsdam, 1996 2 W. M. Kaula Theory of Satellite Geodesy, Blaisdell Publishing Company, 1966 3 R. Konig, Predictions for ERS-I, in: Veillet, C., 7th International Workshop on Laser Ranging Instrumentation, Matera 1989, pp.385-392, proceedings, Grasse 1989 4 F. Flechtner, S. Bedrich, F, H. Massmann, PRAR€/ERS-2: System Status and Re- sults, CSTG Bulletin 13, 1997, pp. 67-71 5 Peter S. Maybeck, Stochastic Models, Estimation and Control, Volume 2, Academic Press, 1982 3 10 ATI DIFFERENTIA^ SING INTEGRATED-R Yaakov Oshman* and F. Landis Markleyt A sequential filtering algorithm is presented for attitude and attitude-rate estima- tion from Global Positioning System (GPS) differential carrier phase measurements.

A third-order, minimal-parameter method for solving the attitude matrix kinematic equation is used to parameterize the filter’s state, which renders the resulting estima- tor computationally efficient. Borrowing from tracking theory concepts, the angular acceleration is modeled as an exponentially autocorrelated stochastic process, thus avoiding the use of the uncertain spacecraft dynamic model. The new formulation facilitates the use of aiding vector observations in a unified filtering algorithm, which can enhance the method’s robustness and accuracy. Numerical examples are used to‘ demonstrate the performance of the method.

INTRODUCTION Attitude determination methods using Global Positioning System (GPS) signals have been inten- sively investigated in recent years. In general, these methods can be classified into two main classes.

Point estimation algorithms (also called “deterministic” algorithms), in which the GPS measure ments at each time point are utilized to obtain an attitude solution independently of the solutions at other time points, were introduced, among others, in Refs. 1 , 2 and 3. Stochastic filtering algorithms, which process the measurements sequentially and retain the information content of past measure- ments, can produce better attitude solutions by more effectively filtering the noisy measurements.

Such algorithms were recently introduced in Refs. 4 and 5, both of which utilized extended Kalman filtering to sequentially estimate the attitude from GPS carrier phase difference measurements. Both attitude and attituderate were estimated, and the filters used the nonlinear Euler equations of mo- tion for attitude propagation. While avoiding the traditional usage o f the costly and unreliable gyro package, this approach rendered the resulting filters computationally burdensome and sensitive to inevitable modeling errors.6 In Ref. 4 an attempt was made to robusta the dynamics-based filter by estimating the unknown disturbance torques, modeled as unknown constants.

Although GPS-based attitude estimation methods should enjoy, in principle, the low price and low power consumption of state-of-the-art GPS receivers, and the general availability and robustness of the global positioning system, these methods are very sensitive to multipath effects and to the geometry of the antennae baseline configuration, and they inherently rely on precise knowledge of the antennae baselines in the spacecraft body frame. On the other hand, methods based on *National Research Council Research e c i a t e , NASA/Goddard Space Flight Center, Guidance, Navigation and Control Center/Code 571, Greenbelt, MD 20771; currently on sabbatical from Department of Aerospace Engineering, Technion - Israel Institute of Technology, Haifa 32000, Israel. Email: oshmanQ~.t.technion.ac.~/.

$Staff Engineer, NASA/Goddard Space Flight Center, Guidance, Navigation and Control Center/Code 571. Greenbelt, MD 20771. Email: landis.markley(0gsfc.nasa.gov.

vector observations have reached maturity and popularity in the last three decades. ver, as is well known, they too suffer from disadvantages, that can be attributed to the particular attitude sensors on which they are based. Thus, while their readings are relatively noiseless, Sun sensors are very sensitive to Earth radiation effects, and are rendered completely useless during Eclipse. Star trackers can provide accuracy on the order of a few arc-sec, but are usually extremely expensive.

Magnetometers always provide measurements of the Earth magnetic field in spacecraft flying in low Earth orbits, but they are sensitive to unmodeled residual magnetic fields in the spacecraft and to magnetic field model imperfections and variations.

The method presented herein is a sequential estimator for both the spacecraft attitude matrix and attituderate, which mainly uses differential GPS carrier phase measurements, but can a l s o process aiding vector observations (such as low accuracy coarse Sun sensor measurements, or magnetic field measurements). Conceptually similar to the principle of complementary filtering, the idea underlying this estimator is that, due to the different nature of these signals, the combination of both in a d e d data processing algorithm can benefit from the relative advantages of both sensor systems, while alleviating the disadvantages of both.

The new estimator is based on a third-order &mal-parameter method for solving the attitude matrix evolution equation using integrated-rate parameters (IRp).7 Similarly to Refs. 5 and 4 , the new estimator is a sequential filtering algorithm and not a deterministic (point estimation) algorithm. However, the new algorithm differs from other works addressing the same problem in two main respects. F i r s t , the estimator's propagation model does not utilize the nonlinear Euler equations. Instead, employing an approach borrowed from h e a r tracking theory: the uncertain dynamic model of the spacecraft is abandoned, and the angular acceleration is modeled as a zero- mean stochastic' process with exponential autocorrelation. Combined with the extremely simple evolution equation of the integrated-rate parameters, this results in a simple, linear propagation model. Second, in contrast with other methods relying mainly on the attitude quaternion, the algorithm presented herein directly estimates the attitude matrix, a natural, nonsingular attitude representation. Building upon the minimal, third-order integrated-rate parametrization, the new estimator assigns just three state variables for the parametrization of the nineparameter attitude matrix, which is at the heart of its computational efficiency.

INTEGRATED-RATE PARAMETERS Consider the matrix differential equation V ( t ) = W(t)V(t), V(t0) = & (1) where V ( t ) E Rain, W(t) = - v ( t ) for a l l t 2 t o , &Vz = I and the overdot indicates the temporal derivative. Defining t

A(t,to) P J W ( r ) d r

t o Wo(t) 4 i W(t) - (t - to).cir(t) it can be shown that the following matrix-valued function is a third-order approximation of V(t): Moreover, P is a third-order approximation of an orthogonal matrix, i.e., P(t, tO)VTT(t, to) = I + O ( ( t - to)*) where O ( x ) denotes a function of 2 that has the property that O(z)/z is bounded as x 3 0.

In the 3-D case, the off-diagonal entries of A(t,to), termed integrated-rate parameters, have a simple geometric interpretation: they are the angles resulting from a temporal-integration of the wz(d) w3(t)lT, where wi is the three components of the angular velocity vector w ( t ) d [ w l ( t ) i-axis of the initial coo system, and i = 1,2,3 for angular velocity component along the 2, y, I, respectively. The orthogonal matrix differential equation (1) is rewritten, in this case, as b(t) = n(t)D(t), D(t0) = Do ( 5 ) where D(t) is the attitude matrix, or the direction cosine matrix (DCM), n(t) = - [w(t) x ] , and [ w ( t ) x ] is the usual cross product m a t e corresponding to u(t). In this case, the matrix A(t,to) takes the form where the parameter vector 6 ( t ) is defined as and

Let the sampling period be denoted by T d t k + l - t k . Using the notation B(k)

e(&), the

at time t k is e ( k ) = [e,(k) e2(k) e3(k)jT and Eq. (8) implies parameter vector t k i = i , 2 , 3

e i ( k ) = I I , wi(T) d7, (9)

From Eq. (9) we have

8(k + 1) = e ( k ) f Itk+' ~ ( 7 ) dT

t k Define A(k + 1, k) to be the discretetime analog of A(t, t o ) , i.e.,

~ ( k + 1, k) 4 - [ ( e ( k + 1) - e&)) X ]

(11)

Also, let @(k + 1) 4 - [+(k + 1) x ] , where

Then, the corresponding discretetime equivalent of Eq. (4) is

D(k + 1) = { I + A(k + 1, k) + s A 2 ( k 1 + 1, k) + ;A3(k 1

+ 1, k)

which, using Eqs. (11) and (12), can be written as To avoid using the uncertain spacecraft dynamic model, the spacecraft angular acceleration is mod- eled as a zero-mean stochastic process with exponential autocorrelation function. The acceleration is, therefore, the following first-order Markov process, dynamic model

G(t) = -A&(t) + C ( t )

(15) For simplicity, a decoupled kinematic model is chosen for the three angular rate components, i.e., A diag{7F1, 7F1, 7T1), where { ~ i } f = ~ are the acceleration decorrelation times associated with the corresponding body axes. The driving noise is a zero-mean white process, with power spectral density (PSD) matrix Q(t) = 2AC2, C diag{al,a2, as} (16) The noise variances in Eq. (16) were chosen according to the Singer angular acceleration probabilis- tic model: in which the angular acceleration components, {wi}bl, can be 1) equal to L ~ M ~ with probability p ~ ~ , 2) equal to - W M ~ with probability p ~ ~ , 3) equal to zero with probability poi, or 4 ) uniformly distributed over the interval [ - L ; I M ~ , & M ~ ] with the remaining probability mass. U s i n g this model, it follows that The parameters W M ~ , P M & and poi are considered as filter tuning parameters. As customarily done, they are selected by experience with real and simulated data, so as to optimally adapt the filter to the characteristics of the problem at hand.

[OT(t) ~ * ( t ) bT(t)IT7 then the state Now let the system's state vector be defined as z(t) equation is O I 0

k(t) = F z ( t ) +.ii(t) -= 0 0 I Z ( t ) +

[o 0 -ni [ C ~ t ~

with obvious definitions of F and G ( t ) . Corresponding to the sampling interval T, %hediscrete-time state equation is

z ( k + 1) = cp(T)z(k) + v(k)

(19) where the transition matrix is I TI A-2(e-AT - I + T A )

Q(T) = eFT = ~ A - l ( I -e-AT)

e-AT and v(k) is a zero-mean, white noise sequence, with covariance matrix T eF(T-t)diag{O, 0, Q(t)}eFT(T-t) dt Q ( k ) A E{v(k)vT(k)} = MEASUREMENT PROCESSING GPS Differential Phase Measurements Consider the basic GPS antenna array, depicted in Fig. 1. The array consists of the master antenna, Figure 1 . GPS Phase Difference Measurement Geometry A,, and the slave antenna, Aj. These antennas are located on the satellite’s surface, such that the baseline vector between them, resolved in a body-fixed coordinate system, is Z j . It is assumed that the entire system consists of m g antennas, in addition to the master antenna, so that there exist mb independent baselines. It is also assumed that at time &+I, m, GPS satellites are in view.

Consider the ith satellite, and denote the sightline (unit) direction vector to that satellite, resolved in an inertial coordinate system, by si. Let D(k + 1) be the attitude matrix transforming vectors in the inertial coordinate system to their body-fixed system representations at time &+I. Let Nij(k+l) and A&j(k+ 1) denote the integer and fractional parts, respectively, of the phase difference between the two carrier signals, corresponding to the ith satellite, as acquired by the antennas A , and Aj.

Denoting by X the GPS carrier wavelength, the true (noiseless) signals satisfy

[A&(k + 1) + N i j ( k + l ) ] X = ZTD(k + 1 ) s i

(22) The standard GPS carrier wavelength is 19.03 cm. In this work, it is assumed that the integer part of the phase difference between the two receivers is known from a previous s o l ~ t i o n . ~ ~ ~ In practice, the phase measurements will be contaminated by noise, the primary source of which . is due to the multipath effect.l Denoting the noise corresponding to the baseline 6j and the sightline

si by iiij(k + 1 ) , the real measurement equation is

[A&(k + 1) + Nij(k + 1)]X = ZrD(k + 1 ) s i + iiij(k + 1)

(23) where it is assumed that &j(k + 1) - N(0,Cij( IC + 1 ) ) . Typically it can be assumed that the noise standard deviation is on the order of 5 mm.l From Eq. (23) we obtain the normalized measurement equation

A&(k + 1) + N.j(k + 1) = bTD(k + 1 ) s ; + nij(k + 1)

(24) where we have defined bj

&/A and nij(k + 1) d iiij(k + l}/A The normalized measurement noise

satisfies nij(k + 1) - N ( O , a ~ j ( k + l ) ) , where a & - + 1) = &j(k + l)/X.

GPS Measurement Linearization At t k + l the minimum mean square error (MMSE) predicted vector is f(k+llk), and its corresponding prediction error covariance matrix is P(k + Ilk) 2 E{Y(k + l ( k ) 5 T ( k + I l k ) ) , where the estimation error is 5 ( j l k ) z(j) - f(j1k). Using Eq. ( 1 4 ) , Eq. (24) is rewritten as N i j ( k + 1) + A&(k + 1) = b;D[O(k + 1) - O(k),w(k + l ) , & ( k + l),D(k)]si + nij(k + 1) (25) e*, we linearize the non1inea.r measurement equation (25) about the most recent estimate at tk+l, t.e.,

~ ( k + llk)] [ ~ ~ ~ ~ r (26)

z(k + 1) = i?(k + Ilk) + 6z(k + 1) E d ( k + 1111.) + 6w(k + 1)

&(k + Ilk)

where 6O(k + l ) , 6w(k + 1) and 6G(k + 1) are the perturbations of the state components about the nominal (Le., predicted) state. Let Ij*(klk) denote the a posteriori, orthogonalized estimate of the attitude matrix at time tk, t? be discussed in the next section. Using now the most recent estimates for D(k) and z(k), namely D*(kjk) and P(klk), respectively, in Eq. (25), it follows that A&j(k + 1) + Nij(k + 1) = ?$D[&k + Ilk) + 6O(k + 1) - &k/k),&(k + Ilk) + dw(k + I),

&(k + Ilk) + 6G(k + l),&(klk)]Si + Wj(k + 1)

(27) As discussed in the sequel, the a posteriori IRP estimate is zeroed after each measurement update (due to full reset control of the IRP state). We will, therefore, use the reset value of the IRP estimate, &(klk) = 0, in Eq. (27). Now expand D about the nominal state using a first-order Taylor series expansion, i.e., D[@ + ilk) + 6e(k + i),cj(k + ilk) + 6w(k + i ) , S ( k + ilk) + 6 q k + i ) , P ( k l k ) ] all[@+ l l k ) , w ( k + l ) , & ( k + l/k),Ij*(kIk)]

I bWi(k+l)

+x i=l awi G(k+llk)

aD[@ + llk),&(k + l(k),rj(k + l ) , B * ( k l k ) ]

+c

I&(k+llk) 6&(k + 1)

i=l where (.)I denotes ‘evaluated at C’ and b ( k + l J k ) 4 ll[~(k+llk),rj(k+llk),~(k+llk),Ij*(k(k)] C Differentiating Eq. (13), the sensitivity matrices appearing in Eq. (28) are computed as a

-D[O(k + l),Lj(k + llk),Z(k + llk),B*(klk)] = Gi[O(k + l),$(k + llk)]P(klk)

aei

a 1

-D[d(k + llk),w(k + 1 ) , 3 ( k + llk),P(klk)] = -pi[@ + ljk)]P(klk)

awi a 1

-D[@ + llk),G(k + llk),G(k + l ) , P ( k l k ) ] = -,T2Fi[6(k + lIk)]P(klk)

a W i

for i = 1,2,3, where $(k + Ilk)

&(k + Ilk) - T&(k + Ilk), and

1 1 1 1

Gi(O,+) = ,z(Oe? + e#) - O i l - (1 - - 110112) [eix] + gT(+eT - + SOi [Ox]

Fi(e) = e,@” - OeT

where ei is the unit vector on the ith axis, i = 1,2,3.

Using Eqs. (28), (29) and (30) in Eq. (27) yields

A&(k + 1) + Nij(k + 1) - b;TB(k + 1lk)si = hz(k + l ) d ~ ( k + I.) + nij(k + I)

where the observation vector hij(k + 1) E Rg is defined as

h,j(k + 1) [heT;.(k + 1) h,Tj(k + 1) hrjT;.(k + l)] T

and the elements of the vectors hezj(k + 1) E R3, hUij(k + 1) E R3 and hhij(k + 1) E R3 are p = 1,2,3 (33a) hegp(k + 1) = $Gp[k(k + llk),$(k + l I k ) ] p ( k l k ) s i , p = 1,2,3 (33b) hwijp(k + 1) = ~ Z " b ~ F , [ k ( k + llk)],ri*(klk)si, p = 1,2,3 ( 3 3 ~ ) hhijp(k + 1) = -Th,jjp(k + l), Define now the effective GPS measurement to be

y$(k + 1) 2 ~ + i j ( k + 1) + Nij - b:B(k + 11k)si

(34) Then, using this definition in Eq. (31) yields the following scalar measurement equation:

y$(k + 1) = hijT(k + 1)6z(k + 1) + nij(k + 1)

(35) For the mb baselines and m, sightlines, there exi$ m, x m b scalar measurements like Eq. (35).

We next aggregate d of these equations into a single vector equation, such that the measurement associated with the baseline bj and sightline s i corresponds to the pth component of the vector

measurement equation, where p = (j - l)m, +i. This yields

yqlc + 1) = H"k + l)bz(k + 1) + n q k + 1)

(36) wherethepthrowofthematrixH&(k+l) ish,jT(k+l), n@(k+l)~ N ( 0 , R d ( k + l ) ) , andR&(k+l) is a diagonal matrix whose diagonal elements are R&(k + 1) = uij.

Vector Observation Aiding If the sole source of attitude information is the GPS carrier phase signals, then Eq. (36) should serve as the basis for the development of the measurement update algorithm (in the next section). In the case that vector observations are available, this information structure needs to be augmented.

This Assume that a new pair of corresponding noisy vector measurements is acquired at pair consists of the unit vectors u(k + 1) and v(k + l ) , which represent the values of the same vector r(k+l), as modeled in the reference coordinate system and measured in the body coordinate system, respectively. The direction-cosine matrix D(k + 1) transforms the true vector representation uo into its corresponding true representation vo according t o

vo(k + 1) = D ( k + l)uo(k + 1)

(37) Assuming no constraint on the measurement noise direction, the body-frame measured unit vector,

v(k + l), is related to the true vector according to

where the white sensor measurement noise is nk(k + 1) - N(0, Rk(k + 1)). Since both vo(k + 1) and

v(k + 1) are unit vectors, it follows,from Eq. (38) that

v ( k + 1) = vo(k + 1) + n,(k + 1)

(39) where n,,(k + 1 ) k ?&(k + l ) n : ( k + 1 ) and ?&(IC + 1) 4 1 - vo(k + l ) t $ ( k + 1 ) . ‘Eo a good approx- imation, the effective measurement noise is a zero mean, white Gaussian sequence with covariance

&(k + 1) = ?&(k 4 - l ) R ; ( k + l)P&(k + 1 )

(40) T o account for non-ideal effects (e.g., star catalog errors), it is assumed that the modeled reference vector is related to the true vector according to

u(k + 1 ) = uo(k + 1 ) + n , ( k + 1)

(41) where n , I uo is a zero mean, white Gaussian noise, that is uncorrelated with n , and has a known covariance matrix &(k).

Vector Measurement Linearization Using Eqs. ( 1 1 ) ) ( 1 2 ) and ( 1 3 ) , Eq. (37) can be rewritten as vo(k+1)=D[B(k+1) -e(k),w(k+l),Lj(k+l),D(k)]~(k+l) (42) Lmearizimg about the predicted estimates and using Eqs. (26)) (39) and ( 4 1 ) ) it follows that ~ ( k + 1 ) - n , ( k + 1 ) = D[@+ i l k ) + be(k + i),qk + i l k ) + a ~ ( k + I ) ,

Z ( k + l l k ) + bG(k + 1),3*(kIk)] [u(k + 1 ) - n,(k + l ) ]

(43) where the reset d u e o f the IRF’ estimate, &(klk) = 0, has been used. Expanding D about the nominal state using the first-order Taylor series ( 2 8 ) yields

v(k + 1 ) - B ( k + llk)u(k + 1) =

[Gi[6(k + Ilk), $(k + l ) k ) ] b B i ( k + 1 )

i-1 1 1 + gTF;[i(k + l I k ) ] b w i ( k + 1 ) - zT2Fi[8(k+ l ( k ) ] b & ( k + l ) ] B * ( k ( k ) u ( k + 1) - B ( k + l / k ) n , ( k + 1 ) f %(k + 1 ) = H”(k + l ) S Z ( k + 1 ) - B ( k + l ] k ) & ( k 3.1) + &(k + 1 ) (44)

where the observation matrix H”(k + 1) is written in block matrix form as

H”(k + 1 ) = [Hl(k + 1 ) Hz(k + 1 ) H3(k + l ) ] E R3V9

(45)

and the columns o f the submatrices H i ( k + 1) E R3p3, i = 1,2,3 are

H l j ( k + 1 ) = Gj [d(k + l l k ) , 4 ( k + I l k ) ] l j * ( k l k ) ~ ( k + 1 )

(464

H q ( k + 1 ) = -TF’[J(k + l I k ) ] B ’ ( k / k ) u ( k + 1 )

(46b)

H s j ( k + 1 ) = -TH2j(k + 1 )

(464 for j = 1,2,3. Define now the effective measurement and measurement noise to be, respectively, Then, using these definitions in Eq. (44) yields the following measurement equation:

y”(k + 1 ) = H”(k t l ) S z ( k + 1 ) + n”(k + 1 )

where nY(k + 1 ) - N(0, R”(k + 1 ) ) is the white measurement noise, and

RY(k + 1 ) kk &(k + 1 ) + b ( k + l l k ) % ( k + l ) l j T ( k + i l k )

To process the measurements, define now where n - N(0,R) and R diag{Rb,RY). Since 6z(k + 1 ) = z(k + 1) - f ( k + 1Jk) = f ( k + 1Jk) and f ( k + I l k ) i s an unbiased, MMSE predictor, we have E{Sz(k + 1)) = E { f ( k + I l k ) } = 0 and cov{dz(k + 1 ) ) = cov{Z(k + I l k ) ) = P(k + Ilk), thus Sz(k + 1 ) N N(O,P(k + I l k ) ) . Using the line& measurement equation and the statistical properties of the measurement and prediction

errors, the MMSE estimator of 6z(k + 1) is

h

6z(k + l l k + 1 ) = K ( k + l)y(k + 1 )

(52)

where K ( k + l), the estimator gain matrix, is computed as

K ( k + 1 ) = P(k + l l k ) H T ( k + 1 ) [ H ( k + 1)P(k + l l k ) H T ( k + 1 ) + R(k + 1 ) l - I ( 5 3 ) Also, S(k+llk+l) = f ( k + l l k + l ) - 3 i . ( k + l l k ) which, used in Eq. (52), yields the state mequrement update equation

2 ( k + llk + 1 ) = 2(k + I l k ) + K(k + l)y(k + 1 )

(54)

Subtracting z ( k + 1 ) from both sides of the last equation yields

f ( k + I l k + 1 ) = [ I - K(k + 1 ) H ( k + l ) ] Z ( k + I l k ) - K ( k + l ) n ( k + 1 ) (55) from which the resulting covariance update equation is P(k + I l k + 1 ) = [I - K ( k + l ) H ( k + 1 ) ] P ( k + I l k ) [I - K ( k + l ) H ( k + 1 ) I T

+ K(k + l ) R ( k + 1)KT(k + 1 )

(56) where the filtering error covariance is P(k + I l k + 1 ) E(Z(k + I l k + l ) Z T ( k + I l k + 1 ) ) .

To compute the measurement-updated attitude matrix at time t k + . l , we use the m o s t recent estimate 2 ( k + l l k + 1 ) and the estimated attitude matrix corresponding to time t k in Eq. ( 1 3 ) . This yields

l j ( k + I l k + 1 ) = {I + A(k + 1, k) + $ P ( k + 1, k ) + $43(k + 1, I C )

+ &i(k 1 + 1, k ) @ ( k + llk + 1) - @(k + l l k + l ) & k + 1, k ) ] } b ’ ( k l k )

(57) where the a posteriori estimates of A(k + 1, k) and !P(k + 1) are defined, respectively, as

A(k + 1,k) p - [ i ( k + Ilk + l)x], @(k + Ilk + 1) c -[&k + l ( k + l ) x ]

(58) where $(k+llk+l) cj(k+llk+l)-T&(k+llk+l), and b ( k l k ) is the a posteriori, orthogonalized estimate of the attitude matrix at time t k , to be discussed in the next section.

Finally, since the a posteriori attitude matrix, & , k + Ilk + l ) , is computed based on the a posteriori estimate, d(k + Ilk + l), this implies a full reset control of the parameter vector, i.e., P ( k + 1) = 6(k + 1) - &k + Ilk + l), where @(k + 1) is the reset state vector at &+I, and a corresponding reset of the state estimate, &(k + l / k + 1) = 0, which is then used in the ensuing time propagation step. Since the reset control is applied to both the state vector and its estimate, no changes are necessary in the estimation error covariance matrix.

ATTlTUDE MATRIX ORTHOGONALIZATION To improve the algorithm's accuracy and enhance its stability, an additional orthogonalization pro- cedure is introduced into the estimator, following the measurement update stage. In this procedure, the orthogonal matrix closest to the filtered attitude matrix is computed.

Given the filtered attitude matrix b ( k + llk + 1)7 the matrix orthogonalization problem is to find the matrix subject to DTD = I

P ( k + l ( k + 1)

arg min IIfi(k + Ilk + 1) - Dll, (59)

DERS1s Being a special case of the orthogonal Procrustes problem, the matrix orthogonalization problem can be easily solved using the singular value decomposition (SVD). In cases where the excessive computational burden associated with the SVD might render its use prohibitive, e.g., in real-time attitude determination and control, the following approximate orthogonalization method, based on the iterative method introduced in Ref. 10, can be utilized:

P ( k + l l k + 1) = N ( k + l)B(k + llk + 1)

(60) where 3 1

N ( k + 1) si -I - -d(k + Ilk + l)fiT(k + l l k + 1)

(61) 2 2 Remark 1. Using an approach similar to that used in Ref. 11, it can be shown that, to first-order accuracy, the orthogonalization procedure does not affect the statistical properties of the estimator and, therefore, does not necessitate any adjustments in the algorithm.

PREDl CTl ON In the prediction step at t k , the reset a posteriori estimate at time t k , ?(kllc) (computed with the reset I W estimate) and its corresponding error covariance matrix, P(lclk), are propagated to time tk+l- Using Eq. (19), we have

q k + Ilk) = cP(T)P(klk)

Using this result with E q . (19) 'yields the covariance propagation equation 320 .

To propagate the attitude matrix to t k + l we use the most recent IRP, attitude-rate and angular acceleration estimates, and the orthogonalized DCM estimate corresponding to t k , in Eq. (13). This yields 1 1

B ( k + Ilk) = I + A ( k + 1, IC) + ,A2(k + 1, k) + ,A3(k + 1, k)

{

+ i T [ A ( k + l,k)&(k + Ilk) - &(k + l]k)A(k + l,k)]}b*(klk)

(64)

where the a priori estimates of A(k + 1, k) and %(lc + 1) are defined, respectively, as

A(k+l,k) -[&(k+llk)x], &(k+llk) ii -[&k+llk)x] (65) NUMERICAL STUDY Example I T In this example, three non-orthogonal baselines were used: 61 = [1.0, 1.0, O.OIT, $2 = [O.O, 1 . 0 , 0 . 0 1 , T 63 = [ O . O , 0.0, l.O]*. T w o fixed sightlines were observed at all times, s1 = -$[1.0, 1.0, 1.01 and s2 = &[O.O, 1.0, L O I T . The non-normalized GPS signal noise standard deviation was 5.0 1 1 1 1 1 1 .

When vector measurements were used, the noise equivalent angle of the inertially-referenced obser- vations was set to 5.0 arc-s, while the body-referenced vector measurements were simulated to be acquired by a low accuracy attitude sensor with a noise equivalent angle of 0.1 deg. These mea- surements corresponded to a randomly selected vector, which was kept constant throughout the run.

The angular rates of the satellite satisfied w i ( t ) = Aisin(F-t + $i), where Ai = 0.02,0.05,0.03 deg/s, Qi = 7r/4,7r/2,37r/4 rad, and = 85,45,65 s for i = l , i , 3 , respectively. The initial angular rate estimates were all set to zero. The true initial attitude corresponded to Euler angles of 30 deg, 20 deg and 10 deg in roll, pitch and yaw, respectively, while the filter's initial state corresponded to Euler angles of 25 deg, 15 deg and 5 deg, respectively. The filter was run at a rate of 20 Hz, and the measurement processing rate was 10 H z . The Singer angular acceleration model was used with parameters set to T = 10 s, WM = 1 0 - ~ rad/s2, p M = po = . O O ~ for all three axes.

In Fig. 2, the true and estimated yaw angle time histories, and their corresponding estimation errors, are shown for a typical run, with and without vector measurement aiding. (The estimated yaw angle was computed using the estimated attitude matrix, assuming a 3-2-1 Euler angle sequence).

Using only GPS measurements, the average yaw estimation error was 7.15 x deg, with a standard deviation of 0.095 deg. When vector measurements were used in combination with the GPS signals, the average estimation error was 9.87 x deg, and the estimation error standard deviation reduced to 0.022 deg. In Fig. 3, the third component of the angular velocity vector, its estimates and corresponding estimation errors are shown for the same run. Using GPS only measurements, the steady state estimation error standard deviation was 0.015 deg/s. When vector measurements were used in combination with the GPS signals, the estimation error standard deviation reduced to 0.0065 deg/s (the average rate estimation errors were on the order of deg/s in both cases).

Example II In this example, the same parameters were used as in Example I, except for the following. The three baselines used were now 61 = EO.1, 1.0, O . l l T , 62 = C O . 0 , 1.0, O.OIT, 63 = [O.O, 0.0, 1.0IT. As can be observed, the first two baselines are almost colinear. The angular rates of the satellite were T w = [0,236, O] deg/hr. The Singer angular acceleration model parameters were set to T = 10 s, 3~ = lo-' rad/s2, p~ = po = .001 for all three axes. As in the first example, vector measurements, when available, corresponded to a randomly selected, constant vector.

i

o 20 10 Bo Bo 1w 120 140 160 180 200 Tim (SI (a)Yaw angle (b) Yaw angle estimation error (c) Yaw angle (d) Yaw angle estimation error Figure 2. Yaw Angle Estimation: (a) and (b) GPS Only Measurements, ( c ) and (d) With Vector Measurement Aiding.

In Fig. 4, the true yaw angle time history is shown, along with the estimation error time histories corresponding to the cases where only GPS measurements were used and where vector observations were used along with the GPS measurements. (The estimated yaw angle w a s computed using the estimated attitude matrix, assuming a 3-2-1 Euler angle sequence). As can be observed from Fig. 4, the effect of aiding the GPS measurements with vector observations is very substantial i n this ill- conditioned case. Using only GPS measurements, the average yaw angle steady-state estimation error in this run w a s 7.72 x deg, with an estimation error standard deviation of 0.087 deg.

When the GPS measurements were aided by vector measurements, the average Euler angle steady- state estimation error reduced to 4.6 x deg, with an estimation error standard deviation of 0.022 deg. In Fig. 5, the estimation error of the third rate component is shown, with and without vector observation aiding. Using GPS only measurements, the steady-state rate estimation error standard deviation was 9.34~ deg/s. When vector measurements were used in combination with the GPS signals, the standard deviation reduced to 3 . 5 1 x deg/s (the average rate estimation error was on the order of deg/s in both cases).

0.1 0 20 40 80 80 1w 120 140 180 180 200 (8) (b) w3 estimation error r m (5) (d) w3 estimation error Figure 3 . 0 3 Estimation: (a) and (b) GPS Only Measurements, (c) and (d) With Vector Measurement Aiding.

CONCLUSIONS A nonlinear sequential estimator has been presented, that uses differential GPS carrier phase mea- surements to estimate both the attitude matrix and the angular velocity of a spacecraft. The algorithm is based on the IRP third-order minimal parametrization of the attitude matrix, which is at the heart of its computational efficiency. Avoiding the use of the typically uncertain (and frequently unknown) spacecraft dynamic model, the filter uses a polynomial state space model, in which the spacecraft angular acceleration is modeled as an exponentially autocorrelated stochastic process. When vector observations are available (e.g., from low accuracy Sun sensors or magnetome ters), the estimator’s structure can be easily modified to exploit this additional information and, thereby, significantly enhance the algorithm’s robustness and accuracy. Numerical examples have been presented, that demonstrate the performance of the proposed algorithm and the advantages of aiding the GPS carrier phase signals with vector observations, even when the vector measurements are of relatively low accuracy (a) Yaw angle (b) Yaw angle estimation error (e) Yaw angle estimation error Figure 4. Yaw Angle Estimation: (a) True Angle, (b) GPS Only Measurements, (c) With Vector Measurement Aiding.

ACKNOWLEDGEMENT T h i s work was performed while the f i r s t author held a National R e s e a r c h Council-NASA Goddard Space Flight Center Research Associateship.

REFERENCES 1 . Cohen, C. E . , “Attitude Determination,” Global Positioning System: Theory and Applications, Vol. 1 1 , edited by B. W. Parkinson and J. J. Spilker, Progress in Astronautics and Aeronautics, AIM, Washington, D.C., 1996.

2. Crassidis, J. L. and Markley, F. L., “Attitude Determination Using Global Positioning System Signals,” Proceedings of the AIAA Guidance, Navigation and Control Conference, New Orleans, LA, Aug. 1997, pp. 23-31.

Figure 5. w3 Estimation Error: (a) GPS Only Measurements, (b) With Vector Mea- surement Aiding.

3. Bar-Itzhack, I. Y., Montgomery, P. Y., and Garrick, J. C., “Algorithms for Attitude Determina- tion Using GPS,” Proceedings of the AIAA Guidance, Navigation and Control Conference, New Orleans, LA, Aug. 1997, pp. 841-851.

4 . Fujikawa, S. J. and Zimbelman, D. F., “Spacecraft Attitude Determination by Kalman Filtering of Global Positioning System Signals,” Journal of Guidance, Control, and Dynamics, Vol. 18, NO. 6, Nov.-D~c. 1995, pp. 1365-1371.

5. Axelrad, P. and Ward, L. M., “Spacecraft Attitude Estimation Using the Global Positioning System: Methodology and Results for RADCAL,” Journal of Guidance, Control, and Dynamics, Vol. 19, No. 6, Nw.-Dec. 1996, pp. 1201-1209.

6. Lefferts, E. 3 . and Markley, F. L., “Dynamic Modeling for Attitude Determination,” Proceedings of the AIAA Guidance and Control Conference, San Diego, California, Aug. 1976, Paper No.

761910.

7 . Ronen, M. and Oshman, Y., “A Third-Order, Minimal-Parameter Solution of the Orthogonal Matrix Differential Equation,” Journal of Guidance, Control, and Dynamics, Vol. 20, No. 3, May-June 1997, pp. 516-521.

8. Singer, R. A., “Estimating Optimal Tracking Filter Performance for Manned Maneuvering Tar- gets,,, IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-6, No. 4, Jul. 1970, pp. 473-483.

9. Crassidii, J. L., Markley, F. L., and Lightsey, E. G., “Optimal Integer Resolution for Attitude Determination Using Global Positioning System Signals,” Proceedings of the AAS/GSFC 13th International Symposium on Space Flight Dynamics, NASA Gosddard Space Flight Center, Greenbelt, MD, May 1998.

10. Bar-Itzhack, I. Y. and Meyer, J., “On the Convergence of Iterative Orthogonalization Pro- cesses,” IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-12, No. 2, Mar.

1976, pp. 146-151.

11. Oshman, Y. and Markley, F. L., “Minimal-Parameter Attitude Matrix Estimation from Vector Observations,” Proceedings of the A I A A Guidance, Navigation and Control Conference, New Orleans, LA, Aug. 1997, pp. 12-22, AIAA Paper 97-3451.

ve Mark S. Asher The Johns Hopkins University Applied Physics Laboratory Johns Hopkins Road Laurel, MD 20723-6099 (410) 792-5327 asherms 1 @aplcomm.jhuapl.edu ABSTRACT The existing paradigm for GPS attitude determination is to track the signals in a conventional tracking loop and use the phase observables as input to the attitude determination process. The dominant error sources limiting the accuracy of this process are the differential phase pattern of the antennas and multipath These error sources are essentially indistinguishable and currently limit the accuracy of the GPS attitude determination process to about 0.1 degree for a 1 m baseline. For higher accuracies one needs to include a high quality Inertial Measurement Unit for accelerating platforms or a star camera for space pla$orms. This paper describes improvements to GPS attitude determination which may make it competitive with star cameras down to the 0.01 degree level.

The essential elements of the technique are: 1. Use predetect GPS data to do the attitude estimation, instead of the output of a phase tracker.

2. Use redundant antennas to reject signals inconsistent w i t h a plane wave arriving from infinity.

3. Take advantage of gyro data to reject the multipath signal on the basis of its temporal signature.

The main advantages of the approach are: 1. It tends to reject signals that are not plane waves and that do not have temporal signatures consistent with the direct path signal. Self-multipath signals are re-radiated from reflecting surfaces a few meters away and have significant curvature to their wave fronts. They also do not evolve in time the same way that the direct-path signal does.

2. It is robust since it does not depend on knowledge of the specific form of the multipath.

3. It is not restricted to space platforms, where the multipath is very repeatable and where vehicle motion is low. In fact, the relatively high angular motion of aircraft and terrestrial vehicles would tend to assist in rejection of multipath by the difference in temporal signature.

A practical for processing the predetect data which does not require the accumulation of large amounts of data is presented. The key to this technique is that the GPS spectrum is despread by using the local code and message data bits generated by an auxiliary conventional tracking loop tracking the same GPS satellite. The resultant (narrow band) signal is sampled at a low rate bufFered and optimally combined with similar signals fiom the other antennas as well as the gyro data.

We present computer simulations of the new techniques in a scenario containing extremely strong multipath and examine the parameter sensitivities. It is shown that the use of predetect data can attenuate the multipath error by a factor of three or more. The total improvement from all of the techniques, redundant antennas and gyros, can be a factor of 10.

John L. Crassidis,* F. Landis Markley; E. Glenn Lightseyt In this paper, a new motion-based algorithm for GPS integer ambiguity resolution is derived. The flust step of this algorithm converts the reference sightline vectors into body m e vectors.

This is accomplished by an optimal vectorized transformation of the phase difference measurements. The result of this transformation leads to the conversion of the integer ambiguities to vectorized biases; This essentially converts the problem to the familiar magnetometer-bias determination problem, for which an optimal and efficient solution exists. Also, the formulation in this paper is re-derived to provide a sequential estimate, so that a suitable stopping condition can be found during the vehicle motion. The advantages of the new algorithm include: it does not require an a-priori estimate of the vehicle’s attitude; it provides an inherent integrity check using a covariance-type expression; and it can sequentially estimate the ambiguities during the vehicle only disadvantage of the new algorithm is that it motion. The requires at least three non-coplanar baselines. The performance of the new algorithm is tested on a dynamic hardware simulator.

INTRODUCTION The utilization of phase difference measurements from Global Positioning System (GPS) receivers provides a novel approach for three-axis attitude determination andor estimation. These measurements have been successfully used to determine the attitude of air-based,’ ~pace-based;~~ and sea-based4 vehicles. Since phase differences are used, the correct number of integer wavelengths between a given pair of antennas must be found.

The determination of the integer ambiguities can either be accomplished by using “static” (motionless) or “dynamic” (motion-based) techniques. The ambiguities essentially act as integer biases to the phase difference measurements. Once the integer ambiguities are resolved, then the attitude determination problem can be ~olved.~ The static method fmds a solution that minimizes the error residual at a specific time by searching through an exhaustive list of all possible integers and rejecting classes of * Assistant Professor. The Catholic University of America, Department of Mechanical Engineering, Washington, DC 20064.

’ Engineer. NASA-Goddard Space Flight Center, Guidance, Navigation, and Control Center, Greenbelt, MD 20771.

l becomes too large.

space with knowle set of Diophantine equations?

“instantaneous” attitude solution, for the short b does not guarantee a correct solution in the wrong solution bias, is incorrect. This lack of integrity can cause significant problems if the sensor output is used to control a high bandwidth actuator, such as gas j on a spacecraft.

Another consideration is that static methods sometime require that the antenna array must be w i t h i n a defined angle (typically 30 degrees) of a reference attitude, which is often true for ground-based applications, but is less likely for space-based applications. Also, structural flexibility in the baselines may lead to erroneous solutions. All of the aforementioned limitations imply that static methods, while attractive because of their fast solutions, are not totally acceptable for general purpose applications.

The other techdque for resolving integer ambiguities involves collecting data for a given period of time and performing a batch solution, in which the integer terms remain constant over the collection period. This technique relies on the fact that a certain amount of motion has occurred during the data collection, either fi-om vehicle body rotation or GPS line of sight motion. The main disadvantage of this technique, compared to static approaches, is that it takes time for the motion to occur, which may be on the order of several minutes. Another consideration is that a potentially significant amount of memory is required for the storage of the batch data collection. But, motion-based techniques also have significant advantages over static methods. Most importantly, motion-based techniques are inherently high integrity methods because there are numerous checks that can be implemented into the solution before it is accepted. These include using statistical checks applied to error residuals, matrix condition number checks, and using the closeness of the computed floating-point “integers” to actual integers as a check. The probability of an erroneous solution being reported as valid can be made as small as desired by appropriately setting the thresholds on these integrity on- checks. For these reasons, motion-based techniques have been more widely used for board applications.

Traditional motion-based techniques of integer ambiguity resolution rely on the fact that either GPS line of sight motion or vehicle motion dominates the changes i n differential carrier phase measurements. Coheng developed an algorithm, known as “quasi-static” integer resolution, that can be used when the GPS line of sight motion and the vehicle rotation both account approximately evenly for the differential carrier phase measurement changes. This algorithm can be adapted to almost any vehicle motion, slow or fast, simply by varying the sample rate and the data collection time. The quasi-static method solves a collection ,of differential phase measurements for a single attitude estimate and then considers perturbations to the initial estimate at each measurement epoch to produce a time varying batch solution to the data. Although this is a widely used algorithm, there are certain disadvantages. First, an a-priori attitude estimate must be itude estimate. Finally, if a large observe the motion, large-order e data collection ired. Another method (Ref. 10) performs a parameters independent of each other, determining the integers. This approach has been shown to provide better convergence than Cohen’s method and works well for non-coplanar baselines; however, singular conditions can exist at various attitude rotations and a significant amount of vehicle motion may be necessary for a solution.

In t h i s paper, a new motion-based algorithm is derived. The main advantages of the new algorithm over the prior methods include: (i) it resolves the integer ambiguities without any a-priori attitude knowledge, (ii) it requires less computational effort, since large matrix inverses are not needed, and (iii) it is non-iterative. The only disadvantage The of the new algorithm is that it requires at least three non-coplanar baselines.

algorithm is first shown as a batch solution, and then shown as a sequential solution. A covariance expression is also derived which can be used to bound the integer solution so This is extremely that a sufficient integrity check for convergence can be developed.

i n the sequential formulation, since the solution can be found as the motion occurs, useful rather than taking a batch solution at a specific data collection interval. For these reasons, the new algorithm provides an attractive method for real-time ambiguity resolution.

The organization of this paper proceeds as follows. First, the concept of the GPS phase difference measurement is introduced. Then, a brief review of Cohen’s quasi-static method is shown, and limitations and computational aspects of this algorithm are discussed. Next, the new motion-based algorithm is derived. The conversion of the GPS sightline vector into the body fkme is first reviewed. Then, the batch solution used to resolve the integer ambiguities is derived, followed by the sequential solution. Finally, the new algorithm is validated by using an actual GPS receiver with a hybrid dynamic simulator to simulate the vehicle motions of a low-altitude Earth-orbiting spacecraft.

GPS SENSOR MODEL In this section, a brief background of the GPS phase difference measurement is shown. The main measurement used for attitude determination is the phase difference of the GPS signal received fiom two antennas separated by a baseline. The wavefront angle and wavelength are used to develop a phase difference, as shown in Figure 1. The phase difference measurement is obtained by

br COS 8 = A ( A 4 - n)

(1) where bl is the baseline length (in cm), 8 is the angle between the baseline and the line of sight to the GPS spacecraft, n is the number of integer wavelengths between two

receivers, A 4 is the phase difference (in cycles), and A is the wavelength (in cm) of the

GPS signal. The two GPS fiequency carriers are L1 at 1575.42 MHz and L2 at 1227.6 MHz. As of this writing, non-military applications generally use the Ll frequency. The phase difference can be expressed by f?om one receiver to another, and A E R ~ ~ ~ is the attitude

A&j - T =bi A s . +ng +wg

(3) -1 where A& denotes the phase difference measurement for the i * baseline andf’ sightline, and wg represents a zero-mean Gaussian measurement error with standard deviation mo which is 0.5 cm/A = 0.026 wavelengths for typical phase noise?

To GPS

+

Figure 1 GPS Wavelength and Wavefront Angle QUASI-STATIC APPROACH Cohen’s quasi-static methodg is a motion-based technique that begins by taking measurements for k = 1 to L rmeasurement epochs”) to which a single attitude solution will be determined. At each epoch it is assumed that A 4 baselines exist and N sightlines. The measurement model is linearized by assuming a small perturbation about

a nominal attitude 4 and an assumed set of integer phases (no)g , so that

where 13x3 is a 3 x 3 identity matrix, & is assumed to be a small angle rotation, and

[@XI is a cross product matrix with

Substituting Equation (4) into Equation (3) for all available measurements yields

where 7 is a quasi-identity matrix with possible zeros along the diagonal where states

have been removed at various measurement epochs, and Equation (6) is a set of M N equations for (3 + M N ) states. Allowing perturbations at all epochs leads to This compact representation has LMN rows and (3L+ M N ) states. In principle, the integers and the attitude of the vehicle of the vehicle at each measurement epoch may be found by applying an iterative linearized least-squares approach using Equation (8) and updating the nominal attitude using Equation (4).

The quasi-static method has been successfully implemented to resolve the integer ambiguities on an actual system (Ref. l), and works extremely well when a fairly accurate a-priori attitude is known, and significant vehicle motion is present. However, this approach has a number of disadvantages. First, if the a-priori attitude estimate is poorly known, then the solution may never converge (even if the integers are known exactly).

Second, not only does this algorithm require a good a-priori guess, but requires fairly as time accurate attitude estimates at all measurement times. The reason for this is that increases, the perturbations to the a-priori attitude guess may become too large for the solution to converge. This may be overcome by augmenting the state equations to

unknown, body rate that is also estimated. Finally, a (3L + M N )

include a constant, but matrix inverse is required, which may cause computational problems. Many of these at is i s section a new algorithm to resolve the integer ambiguities is shown. The main advantage of this algorithm is that it is attitude independent. First, a conversion of the sightline vectors into the body w e is shown. This converts the problem into the familiar magnetometer-bias problem. Then, a batch solution for this problem is shown, followed by a sequential approach.

The new algorithm begins by determining the sightline vector in the body fiame,

denoted by S = A s . This is accomplished by minimizing the following loss function"

If at least three non-coplanar baselines exist, the minimization of Equation (9) is straightforward and leads to . s ' - j - A - g j - c .

(loa> -J The computed sightline in the body .frame is related to the sightline vector in the reference fiame by -J s^ - = A s -I . + C -J . + g j (11) where g j is a constant bias since the baselines are assumed constant, and gj is a zero- mean Gaussian process with covariance Rj = BY'. Again, the inverse in Equation (10) exists only if three non-coplanar baseline vectors exist.

The next step is to use an attitude-independent method to find the phase-bias vector cj. Doing this for each sightline gives us all the sightlines in both the body fiame and the reference fiame. The explicit integer phases are not needed for this solution, but it is as mentioned in the Introduction.

important to check that they are close to integer values, In the general case, the explicit integer phases can be found fiorn the attitude solution.

nu =bi T g j

(12) With more th selines, however, Equation (1Oc) does not have a unique solution for g j , so the phases for sightline gj cannot be found fi-om g j alone. We will consider the three-baseline case, which is the most common in practice. If more baselines are available, we are always fi-ee to select a three-baseline subset. Then, after the integer phases have been determined, a refined attitude estimate can be computed using all baselines (Le., three baselines are sufficient to determine an attitude, which may then be used to resolve the integers correspondingto the other baselines).

To eliminate the dependence on the attitude, the square of Equation (1 1) is computed, so that Next, the following effective measurement and noise are defined (14b) Then, the effective measurement can be written as (15)

zj = 2 $ . -1 .g. 1 - I ~ j I f + v j

Alonso and Shuster (Ref. 12) showed that vj is approximatelyGaussian for small g j with mean given by pj E{ vj} = -trace{ R j } (16) and variance given by

4 E E { $ } - $ = 4 ( i j - g j ) T R j ( i j - g j )

(17) The negative-log-likelihood function for the bias is given by The symbol k denotes the variable at time tk . The maximum-likelihood estimate for g j , denoted by g ; , minimizes the negative-log-likelihoodfunction, and satisfies zation of Equation ( 18) is not s ~ i ~ t f o ~ a r d since the likelihood function is quartic in g j . A number of algorithms have been proposed for estimating the bias (see Ref. 12 for a survey). The simplest solution is obtained by scoring, which involves a Newton-Raphson iterative approach. Another approach avoids the minimization of a quartic loss function by using a “centered” estimate. A statistically correct centered estimate is also derived in Ref. 12. Furthermore, Alonso and Shuster show a complete solution of the statistically correct centered estimate that determines the exact maximum likelihood estimate s ; . This involves using the statistically correct centered estimate as an initial estimate, and iterating on a correction term using a Gauss-Newton method.

Although this extension to the statistically correct centered estimate can provide some improvements, this part is not deemed necessary for the GPS problem since the estimated quantity for nG is rounded to the nearest integer.

Batch Solution In this section the statistically correct centered estimate algorithm (see Ref. 12 for First, the details) and its application to the integer ambiguity problem are shown.

following weighted averages are defined where Next, the following variables are defined z j ( k ) zj(k)-Tj, Ej(k) E -1 2 -(k)-F* -1’ Cj(k) vj(k)-Tj, p j ( k ) E p j ( k ) - j i j (22) The statistically correct centered estimate now minimizes the followilig loss function

which is now a quadratic function in c j . The minimization leads directly to

r covariance is given by

I ’

The ambiguity for the i * baseline and]& sightline can be resolved by rounding the following to the nearest integer %j = -1 bTG? J (26) The integer error covariance, denoted by Qj, can be shown to be given by

Qj = k r q k i (27)

Equation (27) can be used to develop an integrity check for the algoritbm. For example, a suitable criterion can be developed ffom a three-sigma bound using 3&.

Sequential Formulation This section expands upon the batch solution so that a sequential estimate of the integers can be found. The main advantage of a sequential formulation is that the convergence (integrity) check can be made on-the-fly @e., in real-time). The covariance

in Equation (25) to be expanded to the L + 1 time point, so that

From the matrix inversion lemma,13 the following sequential formation for the covariance is developed

q ( k + 1) = Kj(k)Pj(k) (29)

where Kj(k)~ I - f ) ( k ) g j ( k + l ) F T ( k + l ) q ( k ) x j ( k + 1 ) + - 4 ( k + l ) 4 EF(k+l) (30)

1 [-J

In order to derive sequential formulas for the quantities in Equation (20), first consider the following identity 0 S r 1 and so <(L)Zj(L- 1 > + 3 (L- l)Zj(L) Zj(L)= (33)

4 ( L ) + 5; (L - 1)

Therefore, the following sequential expressions for the quantities in Equation (20) are given

q k + 1) =

[ 6 (k + 1) Zj (k) + 3 (k) z j (k + l)]

< (k + 1) + 3 ( k )

j ( k + 1 ) = 1 [4 ( k + l ) E j ( k ) + + (k)gj(k+l)] (34b)

4 ( k + l ) + 8 (k)

;lii(k + 1) =

(34c) [ crf (k + l);iii(k) + 3 (k) bj(k + l)]

C++l)+Zf(k) where 1 1 1

=- +

(3 5 )

z ; ( k + 1) Zf(k) crf(k + 1)

The estimated bias in Equation (24) can also be found in a similar manner, so that 1 [ T j ( k + l ) - j i j ( k + 1 ) ] 2 P j ( k + l ) l j s ( k + 1 ) (36)

c?(k + 1 ) = Kj(k)C?(k) +

-J c$(k + 1)

Since the baselines are constant, Equations (26) and (27) can be used directly to determine the sequential integer value and error covariance, given by nG(k) = &Tg;(k) (37a) Qj (k) = @pi (k) (37b) The complete solution proceeds as follows. First, use Equations (lob) and (1Od) to convert the sightline vectors into the body m e . Then, perform an initial batch solution wing Equations (20)-(25) in order to initialize the sequential routine (an accurate initial estimate is not required as will be seen in the results section). Then, perform a sequential estimate for the integers using Equations (29), (30), and (34)-(37). Finally, continue until the covariance in Equation (3%) is below a pre-specified value.

First, the algorithm is fully There are many advantages of the new algorithm.

autonomous (Le., it requires no a-priori information such as an a-priori attitude guess).

Second, the largest matrix inverse is of a 3 x 3 matrix, which makes the algorithm es it s u i ~ ~ l e for ergence can be tual motion in the vehicle. Finally, the integers for other sightlines can be easily resolved by calling the same subroutine. Therefore, the alg be implemented using all available sightlines, and attitude determination can begin once the integers corresponding to two sightlines have been resolved. For these reasons, the new algorithm provides an attractive approach to resolve the integers.

SIMULATION SULTS A hardware simulation of a typical spacecraft attitude determination application was undertaken to demonstrate the performance of the new algorithm. For this simulation, a Northern Telecom 40 channel, 4 RF output STR 2760 unit was used to generate the GPS signals that would be received at a user specified location and velocity. The signals are then provided directly (i.e., they are not actually radiated) to a GPS receiver that has been equipped with software tracking algorithms that allow it operate in space (see Figure 2 ) .

Inputs Visible Satellites GPS Constellation Doppler Shift Simulation Parameters Carrier Phase TCXO Simulated synthesizer

’ computer ’

Performance Position Velocity Attitude Figure 2 Hardware Simulation Block Diagram The receiver that was used was a Trimble TANS Vector; which is a 6 channel, 4 RF input multiplexing receiver that performs 3-axis attitude determination using GPS carrier phase and line of sight measurements. This receiver s o h a r e was modified at Stanford University and NASA-Goddard to allow it to operate in space. This receiver model has been flown and operated successfully on several spacecraft, including: REX-II, OAST- Flyer, GANE, Orbcomm, Microlab, and others.

launch, this motion profile is nonetheless very repres determination applications. The orbit parameters and pointing profile used for the simulation are given in Table 1.

Table 1 §§TI Lewis Orbit parame :rs Semimajor axis (a) 6901.137 km Inclination (i) 97.45 deg Right Ascension of Ascending Node (RAAN) -157.1 deg Eccentricity (e) 0.0001 Pointing profile Earth pointed Launch date August 22,1997 The simulated SSTI Lewis spacecraft has four GPS antennas that form three baselines. The antenna separation distances are 0.61 my 1.12 m, and 1.07 m, respectively.

One antenna (in baseline 3) is located 0.23 m out of plane (below) the other three antennas. On the spacecraft, the antennas are mounted on pedestals with ground planes to minimize signal reflections and multipath. For the simulation, the signal was provided to the GPS receiver without multipath noise. The baseline vectors in wavelengths are given by 2.75 0.00 -3.93

-1 b - -[ 1.641, b2 =[ ,.28] b3 =[ - 3.931 1.23 (38)

-0.12 -0.17 Line biases are first determined before the new algorithm is tested to resolve the integer ambiguities. The GPS raw measurements are processed at 1 Hz over a forty minute simulation. During the simulated run, a minimum of three visible GPS are given at all times. Also, there are a number of eight minute spans when two of the same (in-time) sightlines are available for the ambiguity resolution algorithm. Again, in practice, all available sightlines should be processed simultaneously, but with three baseline vectors only two simultaneously available sightlines are required to determine the attitude of the vehicle.

As mentioned previously, the first step in the algoritbm involves using the baselines and phase difference measurements to convert the sightline vector into the body-frame, using Equations (lob) and (1Od). Then, a small batch run is used to initialize the sequential routine. For this case, only 5 seconds of data was required to perform the initialization. Again, only two sightlines are required to determine the attitude.

Sequential error results (i-e., actual integer minus the computed values without rounding) for the first sightline are shown in Figure 3. The integer error can be found be rounding eck shows that the integers may be resolved well before 5 minutes, which is seen in this case). A plot of the errors for the second sightline is shown in Figure 5. For this case, all of the ambiguities have been resolved within 30 seconds (the error value corresponding to the second baseline goes below -0.5 before 30 seconds). A plot of integrity check for the second sightline is shown in Figure 6. The integrity check shows that the ambiguities are resolved w i t h i n 7 minutes. This hardware simulation of a spacecraft clearly demonstrates that the new algorithm presented in this paper provides an accurate method to resolve the integer ambiguities with even slight vehicle motion.

1 3 4 5 1 7 8 . TmW) Figure 3 E r r o r s for First Sightline Figure 4 Integrity Check for First Sightline Figure 6 Integrity Check for Second Sightline Figure 5 Errors for Second Sightline 34 1 is paper, a new algorithm was develo ed for GPS integer ambiguity resolution.

e new algorithm has several adv over previously existing algo is attitude independent s a-priori attitude estimate ( motion) is required. Second, the algorithm is sequential so that it may be implemented in real-time. Also, a suitable integrity check can be used to determine when the determined values have converged to the correct values. Finally, the algorithm is computationally efficient since only a 3 x 3 matrix inverse is required, and the same subroutine can be used on different sightlines. The only disadvantage of the new algorithm is that it requires at least three non-coplanar baselines. The algorithm was tested using a GPS hardware simulator to simulate the m o t i o n s of a typical low-altitude Earth-orbiting spacecraft. Results indicated that the new algorithm provides a viable and attractive means to effectively resolve the integer ambiguities.

ACKNOWLEDGMENT The first author’s work was partially supported by a NASNASEE Summer Faculty Fellowship, under the supervision of Mr. Frank Bauer at NASA-Goddard Space Flight Center. The author greatly appreciates this support. Also, this author wishes to thank Dr.

Malcolm Shuster of the University of Florida for introducing him to the concept of magnetometer-bias estimation.

REFERENCES 1. C.E. Cohen, “Attitude Determination Using GPS,” PbD. Dissertation, Stanford University, Dec. 1992.

2. E.G. Lightsey, E. Ketchum, T.W. Flatley, J.L. Crassidis, D. Freesland, K. Reiss, and D. Young, “Flight Results of GPS-Based Attitude Control on the REX-XI Spacecraft,” Proceedings of the 1996 ION-GPS, Kansas City, MO, Sept. 1996.

3. P.J. Melvin, L.M. Ward, and P. Axelrad, “The Analysis of GPS Attitude Data from a Slowly Rotating, Symmetrical Gravity Gradient Satellite,” Advances in the Astronautical Sciences, Vol. 89, Part 1, AAS Paper #95-113, pp. 539-558.

Cannon, and B. Loncarevic, “Shipborne GPS Attitude 4. G.Lachapelle, M.E.

Determination During MMST-93,” IEEE Journal o f Oceanic Engineering, Vol. 21, NO. 1, Jan. 1996, pp. 100-105.

5 . J.L. Crassidis, E.G. Lightsey, and F.L. Markley, “Efficient and Optimal Attitude Determination Using Recursive Global Positioning System Signal Operations,” to appear in the Proceedings o f the Guidance, Nawgation, and Control Conference9 Boston, MA, Aug. 1998, AIAA Paper #98-4496.

6. P.G. Quinn, “Instantaneous GPS Attitude Determination,” Proceedings o f the I993 ION-GPS, Salt Lake City, UT, Sept. 1993, pp. 603-615.

7 . C.D. Hill and L. AG, “An Optimal Ambiguity Resolution Technique for Attitude Determination,” Proceedings o f the IEEE Position, Location, and Navigation Symposium, Atlanta, GA, April 1996, pp. 262-269.

son, “A New Motion- Based Algorithm for GPS Attitude Integer Resolution,” Navigation, Vol. 43, No. 2 , S~mrner 1996, pp. 179-190.

11. J . L . Crassidis and F . L . Markley, ‘New Algorithm for Attitude Determination Using Global Positioning System Signals,” Journal of Guidance, Control and Dynamics, Vol. 20, No. 5, Sept.-Oct. 1997, pp. 891-896.

12.R. Alonso and M.D. Shuster, “A New Algorithm for Attitude-Independent Magnetometer Calibration,” Proceedings o f the Flight Mechanics/Estimation Theory Symposium, NASA-Goddard Space Flight Center, Greenbelt, MD, 1994, pp. 5 13-527.

13. R . F . Stengle, Optimal Control and Estimation, Dover Publications, New York, 1994, pg. 62.

Stephen F. ~ n d r e w s * Wendy M. Morgenstern” The Tropical Rainfall Measuring Mission (TRMM) spacecraft is a nadir pointing spacecraft that nominally controls attitude based on the E a r t h Sensor Assembly @SA) output. After a potential single point failure in the ESA was identified, the contingency attitude determination method chosen to backup the ESA-based system w a s a sixth-order extended Kalman filter that uses magnetometer and d i g i t a l sun sensor measurements. A brief description of the TRMM Kalman filter will be given, including some implementation issues and algorithm heritage. Operational aspects of the Kalman filter and some failure detection and correction will be described. The Kalman filter was t e s t e d in a sun pointing attitude and in a nadir pointing attitude during the in-orbit checkout period, and results from those tests will be presented. This paper will describe some lessons learned from the experience of the TRMM team.

INTRODUCTION TRMM Spacecraft The Tropicid Rainfall Measuring Mission (TRMM) spacecraft, seen in Figure 1, is a joint NASANASDA mission that was launched on November 27,1997 from Tanegashima Space Center, Japan. The spacecraft is three-axis stabilized, in a near circular 350 km orbit at a 35 degree inclination. The Mission Mode is nadir pointing, and due to Sun constraints, the spacecraft must be rotated 180 degree about nadir (yaw) every few weeks. The sensor complement includes a static Earth Sensor Assembly (ESA), two two-axis Digital Sun Sensors @SS), a redundant three-axis Inertial Rate Unit (IRU), eight Coarse Sun Sensors (CSS), and two Three-Axis Magnetometers (TAM). The spacecraft is controlled with four Reaction Wheels (RW), twelve thrusters (Reaction Engine Modules, REM), and momentum is unloaded with three Magnetic Torquer B a r s (MTB). In Mission Mode, which is the nominal science configuration, attitude determination is done with the ESA for roll and pitch, and integrated IRU rate for yaw.

Problem Description A potential single point failure of the ESA was first identified at Goddard Space Flight Center (GSFC) in 1992, with the discovery of a “fogging” effect of the ESA lenses’.

This problem could cause the ESA to fail the Mission Mode attitude determination requirement. A backup attitude detennination method was needed to satisfy the system redundancy requirements. Buying another ESA or a star tracker (ST) was not a realistic option, given the TRMM budget and schedule. A software backup using the available sensor measurements added redundancy without requiring additional hardware or affecting other subsystems such as power or structures.

*Aerospace Engineer, Guidance, Navigation, and Control Center, NASA’s GSFC Figure 1 Spacecraft A six state extended results of a trade study of se filter was adapted from the rithrn were the r e addition of a second subroutines, and new approach to the new algorithm.

main portion of the algorithm is a discrete, extended s sake, only a few re oints will be mentioned here.

term residual as used in this paper refers to the difference between the vector y the sensor and the vector predicted by the model. The scalar implementation used in the flight software introduces another tern that is called the adjusted residual. As each measurement component is processed, the vector predicted by the model is u t residual components have to be corrected for referred to as adjusted residual so corrections are usually fthe’residualand the adjusted res1 almost identical.

The algorithm also includes checks on the data in the filter.

first check is made on the availability and quality of the sensor data. For example, if the sun is in the DSS field of view but the measurement is not valid, the filter will not use that DSS measurement. In addition, there is a residual tolerance test that rejects any measurements that create residuals larger than a set tolerance. These checks prevent the estimation from using bad sensor data. This is not an algorithm failure, so no corrective action is taken.

There are three Failure Detection and Correction (FDC) checks designed specifically to monitor the Kalman filter algorithm. Two checks monitor the covariance matrix for divergence and positive semidefiniteness. The third test ensures that the adjusted residual remains within 30 of the expected value of the residual. For a l l three tests, the ACS software autonomously performs the same actions. First, the software stops updating the attitude quaternion and the gyro d r i f t with the failed sensor, and then it commands the spacecraft to a power and thermal safe attitude after a specified amount of tirne.

GROUND TESTING Using software to add redundancy is not a trivial task. For TRMM, the most difficult issue w a s adding the new Kalman fdter code into tested flight software without altering the existing ESA-based controllers. In addition, there were concerns about processor speed. Since it was unclear if the hardware could run fast enough to simultaneously process the ESA information and run the Kalman fdter algorithm, the attitude control system (ACS) team decided to run one algorithm at a time.

A more detailed description of the software implementation and testing can be found in Andrews and D’Agostino’. Tests were run to verify the nominal performance of the Kalman filter, and to ensure t h a t the existing control modes were not affected by the addition of the new algorithm. All test results were nominal, except one. In that test, the filter rejected DSS measurements after an eclipse. The DSS residuals passed the initial tolerance test, but failed the adjusted residual test, causing the filter to reject the DSS data.

This indicates that either the DSS tolerance was set too tight or that the covariance did not grow large enough during eclipse, leaving the filter knowledge of the TAM noise smaller than it should have been. At the time, the ACS team believed the failure was the result of a mismatch between the ‘true’ ephemeris and the ‘modeled’ ephemeris in the test setup.

FLIGHT RESULTS Sun Acquisition Mode Test On the second day of the mission, TRMM was still in Sun Acquisition Mode holding the spacecraft x-axis 16.5 degrees from the sunline. In this mode, the spacecraft is controlled directly off the CSSs and the IRUs; the Kalman filter output is not used in the control bop. The Kalman filter was run for a total of 13000 seconds. After converging for 9640 seconds, the filter was reinitialized during eclipse to study the TAM-only filter performance. The TAM residugs for the entire test are shown in Figure 2. The flat line portions in this figure are periods of loss of signal (LOS), when TRn/lM was not in contact with the ground. It is obvious that the TAM residuals are not the zero mean, white noise processes modeled by the filter equations. The magnitude of this modeling error has oise parameters lter that w i ~ 0.6 i Figure 2 Sun Acquisition Mode Test: TAM Residuals The standard deviations of the attitude estimate, shown i n Figure 3, converged to [ 0 . 0 2 , 0 . 0 0 6 , 0 . 0 0 2 ] degrees within 8000 seconds. In Sun Acquisition Mode, the sun is held in the same location i n the body frame, perpendicular to the z axis and primarily along the x axis. This reduces observability i n the x axis, a phenomenon that is reflected in the relative size of the attitude standard deviations.

Figure 3 Sun Acquisition Mode Test: Standard Deviation of the Attitude Estimate ro bias e s ~ ~ a t e s are shown sizes of the b i e of the gyro bias estimate s reset can be seen clearly, Figure 4 Sun Acquisition Mode Test: Standard Deviation of the Gyro Bias Estimate Figure 5 Sun Acquisition Mode Test: Estimated Biases filter i s first enable after the filter was again, and the test was ended.

: w i -2 I O M I I i 1 Figure 6 Sun Acquisition Mode Test: DSSl Residuals This adjusted residual test failed because the actual DSS adjusted residual was the expected adjusted residual that is calculated from the 30 tolerance, the state larger than covariance, and the sensor noise. Review of the data showed that the state covariance was too small and the 30 tolerance was too tight. The state covariance w a s too small because the Kalman filter was overweighing the TAM measurement, and converging too quickly.

This weighting factor is a function of the TAM measurement noise covariance matrix that was set to model sensor noise on a zero mean process. The actual TAM measurement residual has a nonzero mean due to modeling errors. The 30 tolerance was set too tight because the filter should be allowed to accept 50 DSS d a t a since the DSS w a s performing better than expected after GSFC's Flight Dynamics Facility had Other problems were identified later, lime to analyze several days of flight data. It was found that the DSS heads were misaligned by as much as 0.3 degrees, which caused biased DSS residuals, leading to biased estimates. Also, the influence of the MTBs on the TAM measurements had not been accurately compensated for, and that increased the TAM residuals. In addition, the IRU calibration maneuvers had not been done yet, and the alignment matrices on-board did not properly account for the true IRU alignments. Finally, it w a s found that the magnetic field model on board w a s not inQmally consistent. The coefficients were from a 1995 model, but the epoch time for computation of the secular variations was set to 1990. This means that the residuals between the magnetic field model and the TAM measurements had a much larger bias and variance than expected.

test, the filter parame were retuned by modifying measurement noise covariance matrix to account for the model errors. Second, the 30 to 50. Third, the magnetic field model residual tolerance was increased from coefficients were set to the 1990 values to match the epoch time. Fourth, the DSS parameters were updated to account for some of the misalignment errors.

The on-board software precluded compensating for the DSS misalignments completely, so there were still unmodeled DSS misalignments of up to 0.08 degrees.

Once the changes were made, the Mission Mode test began.

The TAM residuals are shown in Figure 7 , and, as in the Sun Acquisition Mode test, they are neither zero mean nor Gaussian distributed. The high frequency component of the signal is due to the unmodeled 0.5 Hz rotation of one of the payload instruments, and the low frequency variation may be due to the effects of the IvrTl3s on the TAM measurements. The sharp spikes on the plot are caused by the on-board magnetic field model.

I! I -1 Figure 7 Mission Mode Test: TAM Residuals The x/z The standard deviation of the attitude estimate is shown in Figure 8.

(rolYyaw) quarter-orbit coupling is due to the one revolution per orbit rotation of the spacecraft about the y (pitch) axis. The spacecraft y axis is generally perpendicular to the During eclipse, the covariance sunline, and thus shows the greatest estimated accuracy.

increases because the less accurate TAM is the only update sensor available. Figure 9 shows that the gyro bias estimate is also affected by the availability of the DSS measurement. The periods when the bias covariance is increasing or holding steady are periods of eclipse.

35 1 Figure 8 Mission Mode Test: Standard Deviation of the Attitude Estimate Tim (set Figure 9 Mission Mode Test: Standard Deviation of the Gyro Bias Estimate The Kalman filter estimated gyro biases are shown in Figure 10. Upon initiaIization and reinitialization,the initial attitude transient lasts about 2000 seconds. The filter bias estimate settles to the same values before and after the reinitialhation; this shows that the gyro drift rate is steady on a time scale of hours. This result is expected because of the high quality and d r i f t stability of the TRMM RUs.

Time (sec) Figure 10 Mission Mode Test: Estimated Biases The DSS residuals are plotted in Figures 11 and 12. Between measurements, the filter simply stores the last value of the residual, and the data goes static. The effect of the DSS misalignments can be seen in the large initial values of the residuals when the sun first enters the DSS field of view.

Figure 11 Mission Mode Test: DSSl Residuals Figure 12 Mission Mode Test: DSS2 Residuals Figure 13 Mission Mode Test: ESA Attitude The best measure of the performance of the Kalman fdter is the attitude derived from the ESA. Although the ESA data is not processed on board when the Kalman filter is running, the unprocessed data is available in telemetry. With this information, the ESA attitude was calculated on the ground and is shown in Figure 13. The initial attitude : transients are on the order of 1 * in roll, and 0.2" in pitch.

best performance of the filter is the period from 5000 to 10000 seconds, 0.12".

the largest attitude error is about The possible improvements in the Kalman filter fall into two categories: operational issues and performance issues. Operational issues include transitions to the backup mode, testing, software design, and data flow. Performance issues pertain mostly to properly tuning a Urnan filter so that it functions effectively with real sensors.

Operations issues Since the TRMM Kalman filter was adcled late in the testing cycle, the concern about onboard processing power forced the software design to an either/or mindset. Either the ESA processing could be run or the Kalman filter could be run, but not both. The filter was designed under the assumptions that the ESA had failed and that once the Kalman filter was turned on, it would never be turned off. It was also assumed that the Kalman filter would have to replace only the ESA functions, such as the earth acquisition maneuvers, maintaining nadir pointing, and inertial slews and holds.

Assuming the filter only had to replace the ESA functions meant that the fdter's performance during other cases, such as thruster maneuvers, was not thoroughly considered. This led to several oversights in the filter design. First, in all of the ACS control modes, except during thruster maneuvers, the ACS software runs at a 2 Hz cycle.

During thruster maneuvers, the controller runs at 8 Hz. However, since the Kalman filter was only coded as a replacement for ESA functions, the 2 Hz duty cycle was hardcoded into the K a l m filter algorithm, and the filter cannot run during the thruster maneuvers.

Second, the software propagates the attitude estimate during thruster maneuvers, but it does not propagate the filter covariance. Thus, after completing the thruster burn, the covariance gives an incorrect indication of the accuracy of the attitude estimate. A solution is to reinitialize the filter and allow it to reconverge, a process the flight tests show takes several hours, which reduces the quality of the science d a t a for that period of time. Finally, since normal operations, such as the Delta-V maneuver, require the filter to be reinitialized periodically, testing all possible reset conditions should be included in both ground and flight testing, a luxury the TRMM schedule did not allow. More thorough testing might have revealed more of these problems in t i m e for the development team to modify the design, rather than forcing the operations team to resort to work-arounds.

Since it was assumed the Kalman filter would never be turned off, the ground verification did not test the transition between the fdter and ESA processing. Again, this led to several oversights. The first problem concerns the gyro biases. The filter is continually estimating the gyro biases for all three axes. When the filter stops running, these bias estimates are stored in memory. As the spacecraft's orientation changes during nadir pointing, misalignment errors map differently into the gyro drift bias error. If the filter is running, these changes will be compensated for on board. If the fdter is not running, the estimate that is in memory may actually introduce a small error in the d r i f t biases. Thus, it is necessary to reset the gyro drift biases after exiting the filter. However, the initial estimated bias is set in the gyro initialization subroutine, not in the Kalman filter initializatiodresetsubroutine. Commanding a filter reset only reruns the filter initialization.

To zero the estimated bias, the gyro initialization subroutine must be rerun. This can only be accomplished by rebooting the ACS software, which is extremely risky to do in flight.

e ~ ~ ~ n a t e software a few m filter initialization subroutine would have been a much cleaner solution.

These issues make the transition between ESA attitude determination and Kalman a t t i t u d e de the filter The availability is another operational problem that resulted from adding a backup algorithm to a mature software design.

Currently, there are problems getting flight data from the Kalman fdter because the telemetry packet is available only by special request, or asynchronously. This means that the operations team must send a new command to the spacecraft every nine hours to keep the Kalrnan filter data in the telemetry stream. In addition, the filter packet is only issued every eight samples, which is insufficient for a thorough perfomance evaluation if the filter ever becomes the primary attitude determination method. Since it is in an asynchronous packet, the flight recorder does not store the fdter data, so real-time playbacks must be used to regenerate the data on the ground. This is inefficient, and requires a large effort from the Flight Operations team. The resulting data is full of gaps, since the might Operations team only records the telemetry stream during real time passes. Fortunately, the flight software allows the team to modify the data storage operations, so it is possible to record a continuous d a t a stream for the fdter information. Unfortunately, the data rate is still one sample in every eight, and it involves yet another operational workaround. Many of these data problems could have been avoided if the asynchronous packet had been redefined as a synchronous packet that is always available in the telemetry stream and is always sent to the flight recorder, and issued at a higher data rate.

Performance issues The Kalman fdter models assume zero-mean white noise measurement residuals, which is mostly true of the DSS residuals but is not true of the TAM residuals. The filter has no knowledge of biased sensor readings unless they are included in the state equations, so the DSS misalignment has a large impact on the accuracy of the filter. To characterbe the estimation errors caused by instruments, the sensors and relevant instruments must be accurately modeled in the simulation including biases, scale factor errors, and misalignments. In particular, an accurate gyro model is essential if gyro biases are included in the filter states. Also, the full effects of the Earth's magnetic field on the Kalman fdter cannot be properly seen in simulation because the low frequency variations of the E a r t h ' s magnetic field are hard to model accurately. Ideally, the simulation should model all of the errors that will be seen on orbit, but that is not always easy to achieve.

The on-orbit test needs to be run for many hours to adequately test the backup algorithm. Ideally, the filter should be tested under all the conditions where it is expected to be used. As with all on-orbit tests however, this requirement has to be balanced with other subsystem tests and the science schedule.

As mentioned previously, F'DC is designed to capture certain problems, and to keep the spacecraft safe. The three F'DC tests discussed will not indicate if the filter is trying to estimate an attitude error or a gyro bias error larger than it was told to expect. If the true error is larger than the error indicated by state covariance, the filter may converge to an incorrect attitude. This type of error is indicated when the filter's estimates do not match the true attitude. Outside of computer simulation, however, there is no truth model to use for comparison. If this type of problem is suspected, the best option is to compare the estimates from the filter to ground estimates ’ust, or tune, the filter based on the comparison, a luxury that is no s available to the operations team. To properly tune a filter, the designers must be of the largest possible state error, and choose initial state covariance valu convergence time, it estimation errors. Th ~ p ~ e m e n t ~ t ~ o n features The flight software developers should rarely hardcode a number; a table design that allows parameters to be changed with a simple uplink rather than a software patch should be used instead. Software patches require a significant development and testing effort from the software maintenance team, and risk the safety of the spacecraft. For example, some of the DSS misalignment error w a s calibrated out of the data by changing some table values, but updating the magnetic field model to a 1995 epoch will require a software change.

One good feature of the flight software design is that it allowed the flight operations team to safely verify the filter’s performance. There should be a control mode available to check out the Kalman filter performance before controlling with the filter’s attitude estimates. On TRMM, the Sun Acquisition control mode uses the CSSs and IRUs for attitude determination, which allowed the filter to be tested in-flight without affecting the safety of the spacecraft. If the processing power had been sufficient to run both the Kalman fdter and the ESA processing, much of the awkward testing done on TRMM would have been unnecessary. Running both algorithms simultaneously would provide two attitude estimates a t a l l times, allowing ground personnel the luxury of evaluating the long term performance of the filter without affecting nominal mission operations.

A vital safety feature of the TlRMM design is the FDC logic The three FDC tests pertaining to the Kalman fdter allow the on-board algorithm to determine when it is inappropriate to use the filter results. The TRMM design stops updating the filter and autonomously places the spacecraft in a power safe mode if a bad attitude estimate is computed.

With approximately six months to go from a trade study to completely tested flight software, time constraints made it impossible to test the Kalman fdter under every possible flight condition. Better system engineering should be able to identify possible failures and available backups early in the design phase. Flight results and trend data from each component should be reviewed early in the design to identify potential failures. Decisions to incorporate backups for hardware failures, whether using redundant hardware or implemented software backups, should be made early in the program. For TRMM it would have been best if the backup mode had been included in the earlier design, so that the additional processing needs would be reflected in the processor requirements and design.

Once backups algorithms are selected, good subsystem engineering should help identify all possible uses of these algorithms so those conditions can be tested. ‘Expected‘ usage tests do not cover all reasonable situations. The fundamental lesson learned is that software designed for one spacecraft can be reused on a different spacecraft if the software design is modular enough and the reused software is well tested, even though it is easy to underestimate the difficulty of the conversion.

CONCLUSION ln the unlikely event of a complete ESA failure, TRMM can meet pointing n error sources, and because some unexpected errors showed up in flight tests. However, flight testing showed pointing performance better than the required 0.7” and approaching the 0.2” performance of the primary attitude control system.

REFERENCES D. K. Ward, internal memo, “TRMM ACS Peer Review Action Items Regarding Earth Sensor Assembly Concerns”, August 18, 1992.

J. L. Crassidis, S . F. Andrews, F. L. Markley, K. H a , “Contingency Designs for Attitude Determination of TRMM”, Flight MechanicsEstimation Theory Symposium, 1995 pp.

41 9-433.

J. W. Murrell, “Precision Attitude Determination for Multimission Spacecraft, “AIAA Paper 78-1248, Aug. 1978.

E. J. Lefferts, F. L. Markley, and M. D. Shuster, “Kalman Filtering for Spacecraft Attitude Journal o f Guidance, Control and Dynamics, Vol. 5 , No. 5, Sept-Oct. 1982, Estimation,” pp 417-429.

S. F. Andrews, J. M. D’Agostino, “Development, Implementation, and Testing of the TRMM Kalman Filter,” Flight Mechanics Symposium 1997, pp. 457-47 1.

J. S. Wertz, (ed.), Spacecraft Attitude Determination and Control, D. Reidel Publishing Co., Dordrecht, The Netherlands, 1984.

A. Gelb (ed), Applied Optimal Estimation, The M . I . T . Press, Cambridge, Massachusetts, 1974.

AN EFFICIENT ALG FT ATTtTUDE WITH OPTICAL SENSORS Malcolm D. Shuster An earlier algorithm for multiple sensors is extended to provide three axis attitude from multiple linesf-sight observations with a single optical sen- sor. The algorithm, called SCAD, is simpler computationally than either the QUESI' or FOAM algorithmsand, although suboptimal, suf€ers o n l y imper- ceptiile lass of accuracy for typical star cameras with limited fields of view.

An approximate covariance analysis of the algorithm is presented.

INTRODUCTION A central problem in Spacecraft Attitude Determination has been that of determining the three-axis attitudewhich minimizes the cost function where A is the directioncosinematrix' , W k , k = 1, ... , N , are directions (lines of sight, observation vectors) observed i n the spacecraft body frame, ek, k = 1, ... , N, are the eonesponding directions known in an inertial frame (the reference v e c t o r s ) and uk, k = 1, ... , N, are a set of positive weights.

A caret in this work will be used t o denote a unit vector. This cost function was first proposed by G.

Wahba2 in l965 and has been the starting point of many algorithms, of which the most popular has been the QUEST algorithm3, although other attractivealgorithms exist?-*.

Of particular importance is the fact that the Wahba cost function can be derived f r o m maximum- likelihood estimation' provided one assumes the following measurement model1o, which has been called the QUEST model, because it was used in an early amracy study of the QUEST algorithm3, w k = A*, -k A w k , (2) with the measurement error AWk having first and second momentst *Professor, Department of Aerospace Engineering, Mechanics and Engineering science, University of F l o r i d a , Gainesville, Florida 326116250. Phone (352) 392-7164, FAX (352) 392-7303, email: mshuster@ieee.org t In fact, because of the unity constraint on the norm of iVk, the mean of A ~ V ~ will have a small nonvanishing pado equal to -~p,,. This may be safely neglected in our dmsion.

ts to wit The common constant in the numerators of Eq. (5) is arbitrary, of course, but the choice of Eq. (6) makes N One defines the attitude covariance matrix Pee @efk 3,lO) as the covariance of the attitude error vector, which is the rotation vector1 of the small rotation carrying the true attitude into the estimated attitude. Assumhg the QUEST model for the measurements, t h i s leads to the followingexpression for the attitude covariance matrix In actual computations we must replace W k ~ e by W k , because the former is not b o w n in general.

since we will be interested in calcuhting quantities only to lowest nonvanishing order in AWk this replacement will not lead to important errors in general.

In a previousworkff a method was presented which simplilied the attitude estimation process Using data from an Earth albedo sensor. In that work, an approximate measurement for the direction of the Earth albedo centroid was determined by taking an average of the centroid of the directions of individualelements of the Earth albedosensorweighted by the measured intensity, which was compared with a simulated model centroid. The effectivevector measurementwas combined with a measurement of the Sun direction and used as input to the TRIAD algorithm3. It could equally well have been used as input to the QUEST algorithm, but the minuscUle imprmment in accuracy was not justified by the additional computational burden. Brozenec and BenderI2 used a similar averaging of multiple star diredons i n a star camera to generate a reduced set of measurements for the QUEST algorithm. In the present work we present a method for retaining full three-axis attitude information from multiple data from a single optical sensor, t y p i c a l l y a star camera. In addition, rather than relying on heuristic arguments, we will develop the algorithm in a rigorous manner. We call the algorithm SCAD (Star Camera Attitude Determination).

CONSTRUCTION OF A SUBOPTIMAL COST FUNCTION Let us reexamine the Whba cost function, which we write in the form of the datadependent part of the negative-log-likelihoodfunctbng8 lo, assuming that the measurement model of Eqs. (2) through (4) is valid, namely, uce nd the cost f ~ n ~ o n to ob

If now w a n d v are chosen to have the values

with given by Eq. (S), then the second line of Eq. (12) Will knish identically, and the third line will be a minimum (for given A) leaving For a focal-plane sensor with a field of view of f O . l rad per axis (roughly f6 deg per axis), we anticipate that the effective contribution of the second summation in Fq. (14) will be roughly (0.1)2 or one per cent of the first. Thus, the estimation of the spacecraft attitude will be “dominated” by the first term. The second term, which could be discarded if another vector sensor were presentl1*l2, is not unimportant, however, if data from this sensor alone must be used to construct the threeaxis attitude*.

Minimizing only the first term is not sufiicient to determine the spacecraftattitude. If A, rninimizeS the n h first term, then so does R(w, +)A,, where R(W, $) denotes the direction-cosine matrix for a rotation h through an arbitrary angle $ about the direction w, It i s the second term of Eq. (14) which provides the information on 4.

Sice the overall weight of the first summation i n Fq. (14) will be so much greater than that of the second term, we can determine an approximatevalue for the optimal attitude by writing *In hct, i n the illustrative example of (Ref. 12), the second sensor is an Earth horizon scanner, whose data is sutlicientiypoor that attitude a m q . a b o u t the star camera boresight is worsened l y an order of magnitude by the averaging procedure. ”he algorithm of (Ref. 12) will not lead to a loss of attitude estimation accuracy, however, if applied t o the case of two noncoIIinear star ameras, or if the second sensor is a precise Sun sensor.

Given these A; and we anticipate that h A” = R(w, $*) A : , (19) will be a good approximation for the optimal directionashe matrixwhich m i n i m i z e s the cost function of Eq. (10) .

SIMPLIFICATION OF THE COST FUNCTlONS

We can simplify the two cost f u n c t i o n s , V ( A , ) and L”($), without loss of accuracy. Examine first Lt(Ao). Defining I W I - IVI € E (20) IWl ’ we write I V l = (1 - E ) I W l , and we can recast L‘(A,) accordingly in the form h h The optimizing value of A, will cause A,V to be parallel to w independently of the value of E , Thus, we will achieve the identical value of A*, if we discard E in E q . (22).

Ekewise, substituting E q . ( 2 1 ) into 3. (18) leads to h Separating the terms in the argument of the vector norm which are parallel and perpendicular to w leads further to ere fore^ cost L"(qj) = - 1 - R(@, +) .

Note that the simplification of Eqs. (17) and (18) to obtain Eqs. (25) did not rely on any approximation for the value of E . Note also that we have discarded an uninteresting factor in E q . (Wa).

We determine the suboptimal attitude by minimizing the two cost functions of E q . (Z), L'(A,) and L"(+), in sequence.

CONSTRUCTION OF THE SUBOPTIMAL ATTITUDE The Gost function of Eq. (Ea) can be made to vanish exactly for a continuum of solutions A;. Except A

- -

= -V, for which an A; may be found trivially, a suitable A; is given by (Ref. 13) for the special case corresponding to the quaternion' and Rodriguesvector1 - G i i x ?

Ptt = 1+*.F' The particular A; that we chose is of no consequence,provided that it satisfy It remains only to find the angle qj* which minimizes the Cost function o f Eq. (25b).

'Ib determine p we rewrite LN(+), using techniques developed by Davenport14which have become part of the development of the QUEST algorithm3, as ing Euler's formulaf in the form where we have

LN($) = - 1 - tr [B] - sin$tr [B' 1 2 ] ] ] - cos$tr [gT (I - m*)]

4 o t

-

---

2 ~ ~ [ B ] - s s ~ ~ $ - c c o s $ , dtot with A X e , S E ( Z T f ? ) , and C E (tr[B]-WBW) . . .

Muunuzation of L"($) leads straightforwardlyto -s cos$* + c sin$* = 0 , or $* = arctan2(s1 c ) .

Here arctan2(sl c) i s the function which r e m the arc tangent of S/C in the correct quadrant. In the FORTRAN language this function is called ATAN2 The angle $* will be indeterminate if both s and c vanish. Thls i s possible, however, only if all of the W k are identical.

The parallelism of the calculation of t,Y i n the present algorithm with that of g in the QUEST algorithm is apparent. However, these methods are applied to a single angle variable and not to the four components of the quaternion of rotation. The computational burden is therefore much smaller, particularly since the need to compute the overlap eigenvalue has been eliminated.

COVARIANCE ANALYSIS OF THE ALGORITHM

A simple approximate expression for the covariance matrix of the SCAD algorithm can be obtained if we neglect correlationsbetween the t w o steps. In that case, we effectively treat the estimation of A, and 11 as separate maximum liklihood estimation problems and can obtain an approximate estimate error covariance matrix from the Fisher information matrix associated with each of the estimationsteps.

Clearly, the direction-cosinematrix of Eqs. (26) causes the cost function L'(A;) to vanish identically.

Therefore, the coefficient i n E q . (=a) has no direct connection to the covariance mat* o f the subop timal attitude estimate, once we have embarked on our two-step optimization sequence. 'Ib compute / cter

om the definition of w and v we may write

where N ( 3 9 ) k=l A W will have vanishing expectation (to order a-t) and covariance matrix From Eq. (38) it follows that h h

-

G = AV+ AW, h Thus, A w also has vanishing expectation and covariance matrix The datadependent part of the negative-log-likelihoodg corresponding to the measurement model of E q . ( 4 1 ) i s , therefore, (Refs. 9,lO) - T J ( A ) = - 1 - (W-AV) R Z ( % - A + ) ,

(a)

2 W R$ (Ub) -true -true 2 ]]A6 R Z W-Vv +[[W ]]A6

)*

where A - -true W zAV, (45) and A6 i s the attitude error vector. The Hessian matrix (ie., the matrix of second-order partial deriva- tives) of the negative-log-likelihood function of E q . ( 4 4 ) with respect to A6 is then the contribution of h

the effective measurement w to P G ' . Thus

and the single prime denotes that t h i s i s the contribution to the attitude covariance arising from w.

compared to the errors associated with V , because of the geometric dilution of precision associated with the limited field of view of the star tracker, we expect 6, to be very close to WkNe. Thus, we now seek the value of the infinitesimal All, which m i n i m i z e s The information for All, is just the second derivative of this quantity with respect to A+, and since this n angle is about the direction wits conm%ution to the attitude information is just Thus, the inverse of the covariance matrix of our new algorithm is In anticipation of practical application we have made the replacements h The geometricdilution of precision is manifest in the actors IW x w k 1 ' .

The computation of the pseudo-inverse i s easily accomplished. Let Q and 9 be any two unit vectors h such that { Q, +, w} form a right-handed orthonormal set. Since the singularity of I+ i s solely the n

manifestation that w i s a null vector of the covariance m a t r i x , it follows that

where F [a i e ] n - T

-

F F ~ = I - (53) e 2 x 2 matrix R ‘ , R ‘ E F ~ % F , 1 b e nonsingubr if the attitude is observable. Hence, in the only cases of interest The verification that R ‘ is nonsingular would be a routine step in the computation of the attitude to verify observability. A similar step occufs in implementation the QUEST algorithm in which the rank of the QUEST information matrix is tested15.

We may Simplify the expression for the SCAD inverse covariance matrix further by noting that h [[W]] F 3 G = [-9 i GI.

(56) Hence, we have finally Note that the second term in Eq. (57) may be written as Note that the QUEST covariance matrix will be a good approximationfor the SCAD covariance matrix when the field of view of the sensor has a small diameter. The comparison of these expressions will be camed out in the next section.

h If p i s the root-mean-square arc length of the individual star observations from ‘pv, then we anticipate that the first term in Eq. (57) will differ from the corresponding term of Eq. (49) by fractional errors of order p2. By this same token, we observe that the contribution of second term in Eq. (49) or (57) to the total inverse covariance matrix will be small& than that of the first by a factor of p2.

MODEL COVARIANCE ANALYSIS It follows from the Cram&-Rao Theoremg that The important question is how large is the difference between the two attitude covariance matrices. lb answer this question, we examine the two covariances (rather the two inverse covariance matrices) in a simple model, in which the star camera iS assumed to have a circular field of view of angular radius p, and the stars are assumed to be distributed uniformly over the field of view of the sensor. We will assume for convenience that the star camera has its boresight along the spacecraft z-axis. We assume in addition that the covariance matrix of every line-of-sight observation is characterized by the same standard deviation u2.

With S2 the solid angle subtended by the star camera field of view, 52 = 2 r ( l - c a p ) , (60) and W i t h these substitutions the inverse covariance matrix for each star camera using the QUEsT algorithm f o r computing the attitude is N (PgPgUEST)-l= - U2 diag(a, a, b) , (62) where

(a)

and a = (4 + cos p + cos2 p ) / 6 , b = (2 - cosp - cos2 p y 3 .

Note that a s p -+ 0 we have that a -+ 1 and b 4 0.

F o r the SCAD algorithm, we note first that

- w = ( l + c w p ) %

U2 % - = diag(a, a, b) , R=.= ." ( ) diag(a, a, 0).

W N l + c o s p From these results we may compute the inverse covariance m a t r i x f o r the SCAD algorithm given i n Eq. (49) t o obtain

SCAD -' = { ('

+imp) diag(l/a, l/a, 0) + diag(0, 0, b)

(PO@ 1 (9

The two covariance matrices are both diagonal in'the model case examined.

We note first that the variance about the boresight is identical for this example for both the QUEST and the SCAD algorithms U p D

--

QUEST - ' 9

=b f 6deg 1.

1.000000 f 12deg 1.oooO7 1.000000 f 30deg 1.053 1.000000 f 60deg 1 . 0 4 1.000000 f 90deg 1.33 1.000000 where the subscript b stands for "boresight." Thus, not only do we recover the information on the attitude about the boresight, we recover it completely.

The ratio of the standard deviation of the SCAD algorithm to that of the QUEST algorithm for h attitude errors about axes normal to FV is where t stands for "transverse." Since we are interested in this algorithm primarily for a sensor of limited field of view, we define 5 ~ l - c c o s p . (69) Then Thus, for this simple example, the standard deviation of attitude errors for axes perpendicular to the bore sight is approximately- U?CA" a 1 + p4 /24 + O(p6 /96).

@JEST For limited fields of view, the relative loss in accuracy compared to the QUEST algorithm is &per- ceptible. Bble 1 gives the relative loss of accuracy for several fields of view. Note that because of the rotation symmetry of our example about the star camera boresight, the cross covariance between A$ and A6 wiIl vanish. Thus, the errors introduced by our approximatetreatment of the attitude estimate covariance are completelysuppressed.

REFERENCES 1 SHUSTER, M. D. " A Survey of Atthde Representations," Journal oftheAstronautica1 Sciences, Vol. 41, No. 4,Oct.-Dec. 1993, pp. 439-517.

2 WAHBA, G., "Problem 65-1: A Least Squares Estimate of Spacecraft Attitude," Siam RaieW, Vol. 7, No. 3, July 1%5, p. 4 0 9 .

3 SHUSTER, M. D., and OH, S. D. "ThreeAxis Attitude Determination f r o m Vector Observa- tions," Journal of Guidance and+Control, Vol. 4, No. 1, Jan.-Feb. 1981, pp. 70-77.

4 MARKLEY, E L "Attitude Determination using Vector Observations and the Singular Vglue Decomposition," Journal of rhe Astronautical Sciences, Vol. 36, No. 3, July-Sept. 1988, pp. 245- 258.

mitted to the Journal o f Guidance, Control and Dynamics.

8 MORTARI, D., “A Closed-Form Solution to the M h b a Problem,” submitted to the Journal ofthe Astronautical Sciences.

9 SORENSON, H. W., Parameter Estimation, Marcel Dekker, New York, 1980.

10 SHUSTER, M. D., “Maximum Likelihood Estimation of Spacecraft Attitude,” Journal o f t h e h - tronautical Sciences, Vol. 37, No. 1, Jan.-March, 1989, pp. 79-88.

11 FISHER, H. L, MUSSER, I C L, and SHUSTER, M. D., “Coarse Attitude Determination from Earth Albedo Measurements,” IEEE li-an.vactions on Aerospace and Electronic Systems, Vol. 29, No. 1, Jan.-Feb. 1993, pp. 22-26.

BROZENEC, THOMAS E, and BENDER, DOUGLAS J., “A Simple Suboptimal Least-Squares Algorithm for Attitude Determination with Multiple Sensors,” Proceedings, Flight Mechanicsf Estimation Theoty Sympaium, NASA Goddard Space Flight Center, Greenbelt, Maryland, May 17-19,1994, pp. 529-743.

13 SHUSTER, M. D., “Attitude Estimation from the Measurement of a Direction and an Angle,” Revista Brasileka de CZncias MecGnicas, Vol. 16, Special Issue, 1994, pp. 19-23.

14 DAVENPORT, R B. (unpublished).

15 SHUSTER, M. D., (unpublished).

16 SHUSTER, M. D., and NATANSON, 6. A, “Quaternion Computation from a Geometric Point of V i e w , ” Journal ofthe Astronautical Sciences, Vol. 41, No. 4,OcL-Dec. 1993, pp. 545-556.

17 SHUSTER, M. D., ‘Kalman Filtering of Spacecraft Attitude and the QUEST Model,” Journal o f thehtronauticul Sciences Vol. 38, No. 3, July-Sept. 1990, pp. 377-393.

Appendix: Implementation of SCAD The following are the steps for computing the optimal attitude using the SCAD algorithm: e From the input data, w k , k = 1, . .. , N, the corresponding reference vectoxs, i T k , k = 1, . . . , N, and the sensor variances, a : , k = 1, .. . , N, compute: (1) c$,,~ according to Eq. (6); (2) the weights ak, k = 1, . . . , N, according to Eq. (5); (3) W and ‘iil according to Eq. (13); and (4) the matrix C according to Eq. (31).

h h h e From these quantities compute the unit vectors w and ’iil according to q. (15) for W a n d similarly A for 77.

e Compute li+ according to Eqs. (40) and (43).

e Calculate the matrices F and G according to Eqs. (54) and (56) R ' Is ran IfR' nk, pute A; accor~ing to the foll n - - -

- If MT - V > 1 - E for some predetermhed value of E, set A; = I. (The value of E will be

a function of the machine precision and the accuracy of the data.)

- A

- -

- If W a V < -1 + E for some predetermined value of E, set A : = R($ T), where 6 is the

n

representation of the sensor coordinate axis for which Ifi x w l is largest

- Otherwise, use any of Eqs. (26a) through (28) to generate A; either directly or via the

quaternion or Rodrigues vector.

0 Compute B according to Eq. ( 3 1 ) , and 2 , s , and c according to Eq. ( 3 5 ) .

o Compute 8* according to Eq. ( 3 7 ) and A" according to E q . (19).

0 Compute PzFAD according to E q . (57).

This completes the SCAD algorithm.

The above implementation was given with a mind to generating the direction-cosine m a t r i x as final output. If it is desired to generate instead either the quaternion or the Rodrigues vector as final output, one requires the formulae: and combining these directlywith and p: according to the prescriptions i n (Ref. 1).

37 I AVENPORT GYR CALIBRATION SCHEME* G. A. Natanson' The in-flight gyro calibration scheme commonly used by NASA Goddard Space Flight Center (GSFC) attitude ground support teams closely follows an original version of the Davenport algorithm developed in the late seventies. Its basic idea is to minimii the least-squares differences between attitudes gyro- propagated over the course of a maneuver and those determined using post- manuever sensor measurements. The paper representsthe scheme in a recursive form by combining necessary partials into a rectangular matrix, which is propagated in exactly the same way as a Kalman filter's square transition matrix.

The nontrivial structure of the propagation matrix arises fiom the fact that attitude errors are not included in the state vector, and therefore their derivatives with respect to estimated gyro parameters do not appear in the transition matrix defined in the conventional way.

In cases when the required accuracy can be achieved by a single iteration, representation of the Davenport gyro calibration scheme in a recursive form allows one to discard each gyro measurement immediately after it was used to propagate the attitude and state transition matrix. Another advantage of the new approach is that it utilizes the same expression for the error sensitivity matrix as that used by the Kalman filter. As a result the suggested modification of the Davenport algorithm made it possible to reuse software modules implemented in the Kalman filter estimator, where both attitude errors and gyro calibration parameters are included in the state vector.

The new approach has been implemented in the ground calibration utilities used to support the Tropical Rainfall Measuring Mission (TRMM). The paper analyzes some preliminary results of gyro calibration performed by the TRMM ground attitude support team. It is demonstrated that an effect of the second iteration on estimated values of calibration parameters is negligibly small, and therefore there is no need to store processed gyro data. This opens a promising opportunity for onboard implementation of the suggested recursive procedure by combining it with the Kalman filter used to obtain necessary attitude solutions at the beginning and end of each maneuver.

* This work was supported by the National Aeronautics and Space Administration (NASA) I Goddard Space Flight Center (GSFC), Greenbelt, Maryland, Contract GS-35F-4381 G, Task Order No. 5-03365-Y.

Computer Sciences Corporation (CSC), 101 10 Aerospace Rd., Seabrook, MD, USA 20706 Propagation of a state vector from one measurement time to another is usually done' by ~ntroduc~ng a transition matrix formed by partial derivatives of the current state with respect to the state at an epoch time. A well-known techniquez3 has been developed to propagate the transition matrix between sequential measurements using gyro data. The paper extends this propagation technique to the Davenport gyro calibration scheme.&' The main obstacle to such an extension comes from the fact that the cited gyro calibration scheme treats an a priori given change in the spacecraft attitude within a specified time interval as a pseudo-measurement, and therefore attitude errors are not included in the state vector anymore; as a result, their derivatives with respect to estimated gyro parameters (such as misalignments, biases, and scale factors) do not appear in the transition matrix defined in the conventional way. To overcome this complication, the new approach combines necessary partials into a rec&mgular matrix, which can be propagated in exactly the same way as the conventional (square by definition) transition matrix.f3 Assuming that the first iteration eliminates bulk errors, representation of the least-squares gyro calibration scheme in a recursive form allows one to discard each gyro measurement immediately after it is used to propagate the attitude and state transition matrix. Due to a significant decrease in required storage size, this feature of the new approach seems especially promising for onboard applications.

The next Section presents a simplified derivation of the original version of the Davenport algorithm."s Its final result is an explicit expression of the vector attitude residual in terms of the error sensitivity matrices y~ k utilized by Kalman filter estimator? It is shown that the derived expression turns into the conventional one7 if only linear terms are kept in the expansion of each matrix y k as a Taylor series in the duration At k ofthe kth propagation interval.

Section I11 introduces a rectungular matrix which gyro-governed evolution is performed via the same recurrence relations as those used for gyro propagation of the conventional state transition matrix. The derivation is accomplished in Section IV, which outlines main steps of the suggested recursive procedure.

The new algorithm has been implementeds and successfully used to calibrate gyrosg for the Tropical Rainfall Measuring Mission (TRMM). One of the advantages of the suggested modification of the Davenport algorithm is that it allowed a reuse of software modules implemented in the Kalman filter estimator: where both attitude errors and gyro calibration parameters are included in the solve-for state vector. Section V discusses some preliminary of the TRMM gyro calibration. It is shown that the required accuracy of gyro calibration results can be indeed achieved by a single iteration, and therefore each gyro measurement can be indeed discarded immediately after it was used. The paper also studies a possibility to reduce a volume of processed gyro data without jeopardizing the accuracy of calibration.

The Davenport method' is a two-step procedure. The first step is to determine attitudes at the specially selected calibration intervals. For successful calibration, the selected time usually cover a series of maneuvers associated with significant changes in body rates.

It is essential that, regardless of maneuver specifics, each calibration interval must both start and end in a constant-rate mode. To determine the spacecraft attitude at the ends of each interval, sensor measurements are then collected only during time periods within constant-rate modes, when unknown errors in gyro misalignments and scale factors are compensated by additional gyro biases estimated simultaneously with the spacecraft attitude. As a result, one can assume that gyro propagation from one sensor measurement time to another is done accurately enough, despite the fact that gyros have not been properly calibrated yet.

The second step is quaternion propagation starting from the predetermined attitude quaternion at the beginning of each calibration interval and stopping at its end. The resultant propagated quaternion is then compared with the second of two attitude quaternions predetermined for this calibration interval. The comparison is done by multiplying one of the two quaternions at the ending time by the inverse of other. The vector part of the product is then treated as a vector residual, with the total number of these attitude quaternion vector residuals (AQVRs) always equal to the number of the calibration intervals.

Davenport's principal result is an approximate expression for the AQVRs in terms of vector deviations of the observed angular velocity vectors from the true rates. The outline of the Davenport method presented here mainly follows Keat's5 interpretation of Davenport's original work." To simplify the notation, the discussion will be limited only to a single maneuver so that the index labeling different maneuvers can be omitted. An extension of the final expression for an AQVR to a series of sequential maneuvers is performed in a trivial way by attaching an additional index to both residual and all angular velocity vectors.

Let (3 be an observed angular velocity vector obtained by adjusting measured rates with some estimated parameters, where subscript k refers to the k-th available gyro measurement within the maneuver in question. The observed vector differs from a true vector (3 by a rate error 6 5 k, that is, Both vectors (3 f j and 6 k are assumed to remain constant during a time interval Atk so that the quaternion propagated over n intervals (starting from the known quaternion ijinit) can be represented as 6 sin(+ /2), COS($ /2)], is defined via the relation:* where qfin is the given attitude quaternion in the end of the maneuver.

It is assumed that the attitude quaternion qfin can be obtained fiom anit by propagating the latter with the true constant angular velocity vectors 6 k over the time intervals A t k , so that To express the AQVR 2 in terms of errors in gyro parameters, one f i r s t needs to linearize the --1 -

quaternion product gfin gprop in 66 k . At this point one has to deal with unnormalized

quaternions, which form the so-called 'associative algebra'. Note that both attitude and propagation quaternions discussed above are normalized quaternions, which cannot be either summed up or multiplied by a scalar, in contrast with unnormalized quaternions. On the other hand, the inverse operation q-' is well defined just for normalized quaternions. Only the multiplication law given by Eq. (D-8) in Ref. 10 is common for both normalized and unnormalized quaternions. It is essential that, by analogy with orthogonal matrices, the

---

multiplication law is associative, i. e., q(q'q") =(qZj')Ij'' for any three unnormalized

quaternions, q,q', and q". Another important features of the multiplication law are that

q(q'+q'') =qq'"q'' and that q(kq) = ( k q )q' for a scalar multiplication. After the

mentioned features of the multiplication law are established, unnormalized quaternions can be formally treated in the same way as square matrices, with the norm of quaternion given by Eq.

0 - 9 ) in Ref. 10 used instead of the matrix determinant. In particular, all Taylor expansions look very similarly, except that each product should be computed based on the quaternion multiplication law.

Substituting Eqs. (2) and ( 4 ) into Eq. (3) and keeping only terms linear in 6 6 , , one can represent the latter expression as where

*

Note that our definition of the AQVR differs by the factor (-1) f r o m that used by Keat.'

and Note that the sum in the right-hand side of Eq. (5) is formed by the products of normalized quaternions, and hence, to simplify each product, one can take advantage of the existent 3 x 3 orthogonal matrices. M a k i n g use of isomorphism between normalized quaternions and Eq. (1 2-7b) in Ref. 10, one can easily verify that R A( I)RT = A( Rg) , (8) where R is an arbitrary 3x 3 orthogonal matrix, whereas the rotation matrix A( g) associated

with the Gibbs vector is given by Eq. (12-7b) in Ref. 10, with g = 2 tan(W2). Representing

Eq. (8) in the quaternion form and substituting the resultant expression in the sum in the right-

hand side of Eq. (9, one can finally represent the AQVR 2 as

where Rk+kt is the rotation matrix associated with the propagation quaternion q k + K .

As discussed in detail i n Ref. 11, an explicitly expression of the vector in terms of the rate error 6 6 k has the form: where w k is the error sensitivity matrix used by the Kalman filter estimator: that is, withv(cp)=s~cp/cp,q(cp)~~.5 (1-sincp)/cp2, and (bk"d' =lGTIAtk. Note that a slightly different representation for error sensitivity matrix (1 l), compared with Ref. 3, makes it possible + 0.

to compute this matrix using expansions expIicitly stable at the limit 1 6 Finally, AQVRs (9) are represented as linear combinations of errors Axi (i=l,...,p) in gyro parameters: by substituting the P - 6G,= i=l into Eq. (IO). Computation of necessary partials is then performed in a trivial way.

Note that Keat’s formula’ for the AQVR (utilized in the conventional version’ of the Davenport algorithm4) is obtained from Eq. (13) by keeping only the first term in the right-hand side of Eq.

(1 l), which seems to be a sufficiently accurate approximation in most cases (see comments made in the end of Section V). Another minor modification comes from a slightly different choice of the state vector AZ , which is formed by three bias errors Abi (i=1,2,3), three scale factor errors Aki (i=1,2,3), and two misalignment angle errors &k (k=1,2) for each of three gyros (i=1,2,3).

Such a choice of gyro calibration parameterss makes it possible to calibrate each gyro separately, which is convenient in case of a spacecraft having only one gyro, such as Solar and Heliospheric Observatory (SOHO).lz 111. PROPAGATION OF ATTITUDE MATRIX VECTOR RESIDUALS The main purpose of this Section is to show that an attitude residual can be represented in the general form: 6 = fissmAX ,

-

where 8, AT, B, and 6-are usually referred to as a measurement residual, a state error

vector, a sensitivity matrix, and a state transition matrix, respectively. The crucial point is that

u -

the transition matrix &state can be computed as the last term <lp,t,t,=.cD, in a sequence of recurrence relations: where the (p +3) x (p+3) matrix @k-&,k is an incremental transition matrix conventionally used in Kalman filter applications to propagate the attitude state vector (see, for example. Eq. (F8-26) in Ref. 3). A certain complication, however, comes from the fact that the calibration scheme in question estimates only gyro calibration parameters, and therefore attitude errors are not AZ . As a result &k turn out to be rectangular matrices having included into the state vector p+3 rows but only p columns.

As mentioned above, the initial and final attitudes in the Davenport method are determined using sensor measurements in constqnt-rate modes, when solved-for biases compensate for errors in gyro misalignments and scale factors. The attitude matrix vector residual (AMVR), 6 , is then defined via the relation: are the attitude matrices associated with the attitude quaternions in the previous Section.* One thus finds Since we are interested only in terms linear in gyro errors, one can simply put

and make use of Eq. (12) to represent the AMVR G as the last term 6 in the sequence

Eq. (19) immediately leads to the conventional Kalman filter expression3 for propagation of the combined attitude error / gyro calibration parameter state vector:

""1. A 5 3

where r with By initializing sequence (15) via the relation * The author is thankful to J. Sedlak for pointing to a misprint in the definition of the AMVR'ij in Ref. 11 leading to a sign error in Eqs. (IIM), (111-6), (III-8), (111-14), and (111-17) there.

- 0 = one finds that and hence, making use of Eq. (20) at k=l , with a+, o,,,].

By applying mathematical induction to Eq. (20) and making use of the fact that the first three f i t three rows ofthe transition matrix 6 k for any rows ofthe matrix coincide with the k , one can easily veri@ that Substituting the state transition matrix 6srate for sn then immediately leads to Eq. (14), which constitutes the main result of this work.

IV. REPRESENTATION OF THE DAVENPORT GYRO CALIBRATION SCHEME IN A RECURSIVE FORM The suggested recursive procedure has been implemented in the following way.8 Estimation starts by setting elements of the so-called ‘measurement accumulation’ vector AX to zero. One also initializes elements of the covariance matrix P with some a priori values. The attitude matrix is then propagated from one gyro measurement to another:

starting from the given observed attitude A, associated with the quaternion ‘Tiinit at the

At the same step one also computes error sensitivity matrix beginning of the first maneuver.

(11) and rate-dependent partials in the right-hand side of Eq. (22) which are then substituted, e ~otat~on matrix -l+k used to propagate on matrix via recurrence sequence (1 5).

R is computed by linearizing Eqs. (1 6):

6 = -[6Az -6A3,, 6A3, -6A,, , 6A,, -6A,,IT

where &Afi are elements of the orthogonal matrix

After the AE/NR 6 is computed, one updates the measurement accumulation vector and the

inverse covariance matrix W = P” according to the standard equations: where C is a 3 x 3 measurement covariance matrix. The state transition matrix is then reset to 6o and the processing continues starting f r o m the beginning of the next maneuver.

After the last maneuver is processed, one obtains the covariance matrix P by inverting the resultant W matrix and computing the state error vector AT from the measurement accumulation vector Aii : A Z = P A i i . ( 3 3 ) The magnitude of the state error vector A 3 is then compared with the given tolerance to proceed with iterations if necessary.

V. TRMM IN-FLIGHT GYRO CALIBRATION The TRMM is an Earth-pointing three-axis stabilized spacecraft. Its body z-axis is nominally pointed along the geodetic nadir.I3 It can be in ‘+x forward’ or ‘-x forward’ nominal mode, with its body x-axis being approximately either parallel or anti-parallel to the spacecraft veIocity. For power and thermal protection of science instruments from direct exposure to the Sun, yaw maneuvers from one nominal mode to another (similar to those depicted in Figs. la and lb) are periodically performed. Since the body y-axis is parallel (anti-parallel) to the orbit normal in the ‘-x forward‘ (‘+x forward’) mode, the only nonzero component of the spacecraft angular velocity vector is the pitch rate, nominally equal to + I revolution per orbit (RF’O) in the ‘-x forward’ at portions of the dashed n nearly constant in both nominal modes, and therefore the spacecraft attitude can be determined with a sufficient accuracy without a complete gyro calibration, provided that gyro biases are included in the state vector to be solved for. (Unknown errors in scale factors and gyro misalignments manifest themselves as , some additional biases, which differ for different modes.)

0.3

-

% 0.2

-5

0.1 u) aa L s o -0. I 0 500 1000 1500 2000 2500 0.1 Figure 1. TRMM body rates during +X to -X (upper) and -X to +X (lower) yaw maneuvers.

On December 14, 1997 the TRMM was also placed in the ‘-y forward’ mode to calibrate scientific instruments, with the y body axis being anti-parallel to the spacecraft velocity vector.

As seen from gyro rate profiles depicted in Fig. 2, the spacecraft was rotating for about one hour around its body x-axis with the rate of +1 RPO, before coming back to the ‘-x forward’ mode.

0.3 0.2 dashed = y dashdot = z 0.1 h u a3 cs, a3 Z O LD a3 c CII o z -0.1 -0.2 -0.3 0 1000 2000 3000 4000 5000 6000 Time (seconds) Figure 2. TRMM body rates in the -Y forward mode.

addition to maneuvers between the three E h-pointing modes mentioned above, the T anuary 4, 1998 to stay for one orbit in the inertial hold mode used to calibrate a science instrument by pointing it toward cold space. Fig. 3 presents the corresponding x and y body rates. (The 2 : body rate is omitted since its deviations from zero would be practically invisible at the figure scale.)

solid = x dashed = y I -0.01 0 1 000 2000 3000 4000 5000 6000 7000 Time <seconds) Figure 3. TRMM body rates during inertial hold.

Table 1 lists time intervals selected for calibration of the TRMM inertia1 reference unit (IRU)?

The reference attitudes were determined at the beginning and at the end of each maneuver using Digital Sun Sensor (DSS) and Barnes Static E a r t h Sensor Assembly (SESA) measurement^.'^

Table 1 - Intervals of Gyro Data Used for IRU Calibration

Three residuals per maneuver were then obtained by propagating the spacecraft attitude with gyro rates from the beginning of each maneuver and comparing the result with the predetermined reference attitude at the end of the maneuver. Table 2 presents roll, pitch, and yaw attitude residuals obtained by gyro propagation with pre-launch and calibrated gyro biases, scale factors, and misalignment matrix. The corresponding values of calibration parameters for each gyro (i. e., a bias, a scale factor, and misalignment angles* relative to body axes) are listed in Table 3.

* A deviation of the gyro axis f r o m its nominal direction can be derived from two other misalignments; however, it is included in Table 2 just to simplify the notation, with zeros standing for some rather small numbers completely irrelevant to our discussion.

Tab ! 3 - Calibration parameters

Biases (deglsec) Scale factors Misalignment angles (deg) relative body axis x Misalignment angles (deg) relative body axis y Misalignment angles (deg) relative body axis z vestigate the possibility of red~cing the amount of processed gyro data (about 60 000 points At = 0.5 s, 1 s, 2s. for all six ~ a n e u v e s), propagation was p using several ti can be successfully Inspection o f Table 2 and Table 4 shows t other gyro mea of every skipped without any noticeable effect on the accuracy of estimation. Skipping four gyro measurements still gives reasonably good results, though some deg n in the accuracy can be clearly seen.

0.0892 0.0960 0.0019 0.0024 It is essential that the calibration be accomplished by a single iteration. This can be easily seen by comparing corrections to gyro biases and elements of the G-matrix due to the first and second iterations, presented in Table 5. Contributions to attitude residuals from the second iteration are so small that they would have no effect on the values t in Tables 2 and 4 (to the precision shown).

Table 5- Bias and G-Matrix Corrections

I 1 st iteration I 2nd iteration

2-gyro x-gyro y-gyro 2-gyro x-gyro YWrO B i a s -1.74 xl0” 4.15 xIO” -1.15 XIO-’ -3.89 x104 4.18 x10-’ 0.1 1 x104 To study the significance o f higher-order powers of Atk in the error sensitivity, calibration was repeated using only the linear terms in Eq. (1 I), which is equivalent to the use of the Davenport method in its conventional implementation.’ It was found that neglecting higher- order terms does not practically affect the calibration results, so that the linear approximation seems to be sufficiently accurate for calibration purposes.

A new approach has been implemented in the ground calibration utilities’ and successfully used for the accuracy can be achieved M IRU calibration. It has been shown that the requi by a single iteration. As a result the new approach seems to be ecially useful for onboard applications by allowing one to discard each gyro measurement immediately update the state vector and covariance matrix.

Recently the Rossi X-Ray Timing ExpIorer (RXTE) ground launch support team has reportedt4 some problems in Kalman filter estimation of gyro scale factors and gyro misalignments, and the new least-squares approach to gyro calibration makes it possible to extend advantages of the Davenport algorithm to onboard applications. To avoid memory-consuming batch attitude determination, one can use the Kalman filter to determine the spacecraft attitudes before and afier each of the selected maneuvers. The Kalman filter state vector is composed only of attitude and gyro bias errors, so that gyro biases will change with each new gyro measurement, in contrast with those used in the Davenport method. For this reason one has to propagate in parallel two separate transition matrices, namely, propagation rates for the spacecraft attitude and the transition matrix used by the suggested recursive algorithm are obtained by adjusting raw measurements with a priori biases which remain the same for all the selected maneuvers. On the other hand, propagation rates for attitude errors and the transition matrix utilized by the Kalman filter are obtained by adjusting raw measurements with the solved-for biases (and the same scale factors and misalignments as in the former case). Feasibility of this approach is currently investigated using Submillimeter Wave Astronomy Satellite (S WAS) simulated data.

ACKNOWLEDGEMENTS The author would like to thank J. Sedlak for very thorough editing of the manuscript and numerous critical remarks, J. Hashmall for providing preliminary calibration results, as well as the necessary attitude history file and gyro measurements, and J. Glickman for a comprehensive overview of TRMM mission requirements and some useful comments on the paper.

References 1. L. Fallon, 111, and P. V. Rigterink, “Introduction to Estimation Theory” in Spacecraft Attitude Determination and Control, J. Wertz, editor, D. Reidel, Dordrecht, Holland, 1978 2. M. Nicholson, F. Markley, and E. Seidewitz, “Attitude Determination Error Analysis System (ADEAS) Mathematical Specifications Document,” CSC/TM-88/600 1, prepared by Computer Sciences Corporation, October 1988 3. J. Landis et al, “Multimission Three-Axis Stabilized Spacecraft (MTASS) Flight Dynamics Support System,” Section 2.4, CSCITR-9 1/6071R1 UDO, prepared by Computer Sciences Corporation, Sept. 1995 I 4 . P. Davenport, “In-flight Calibration of Gyros,” Goddard Space Flight Center, Spring 1976 5 . ‘Gyro Calibration nalysis for the Energy Astrono y o b s e r v a t o ~ - ~ -77/6082, prepared by Computer Sciences Corpo~atio~, June I for In-flight Gyroscope Calibration,” Flight 7. J. Landis et al, “Multimission Three-Axis Stabilized Spacecrafi (MTASS) Flight Dynamics Support System,” Section 4.2, CSC/TR-9 1/6071R1 U D O , prepared by Computer Sciences Corporation, Sept. 1995 8. G. Klitsch, M. Lambertson, G. Natanson, R. Strang, et al., “Flight Dynamics Distributed Systems (FDDS) Generalized Support Software (GSS) Functional Specification,” Revision 1, Update 4, CSC/TR-92/6023RlUD4, prepared by Computer Sciences Corporation, September 1996 9. J. Hashmall, ““RMM Inertial Reference Unit Calibration,” unpublished memo, Febr. 1998.

10. F. Markley, “Three-Axis Attitude Determination Methods” in Spacecrafr Attitude Determinationand Control, J. Wertz, editor, D. Reidel, Dordrecht, Holland, 1978 11. G. Natanson, “A Transition Matrix Approach to the Davenport Gyro Calibration Scheme ”, Memo CSC-27434-62, Nov. 1997.

12. T. Becher et al., “International Solar-Terrestial Physics (ISTP) / Collaborative Solar- Terrestial Research (COSTR) Initiative: Solar and Heliospheric Observatory (SOHO) Mission, Flight Dynamics Support System (FDSS) Functional Specifications,” CSC/TR-92/6 102ROUD0, prepared by Computer Sciences Corporation, February 1993 13. J. Glickman et al., “Tropical Rainfall Measuring Mission (TRMM) Flight Dynamics Support System (FDSS) Functional Specifications, ” CSC/TR-94/6045ROUDO, prepared by Computer Sciences Corporation, October 1994 14. D. Fink, W. Davis et al., “Rossi X-Ray Timing Explorer (RXTE) Postlaunch Report,” CSC 10032526, prepared by Computer Sciences Corporation, June 1996 M. Challa' and G. Natansont ABSTRACT Two different algorithms-a deterministic magnetic-field-only algorithm and a Kalman filter for gyroless spacecraft-are used to estimate the attitude and rates of the Rossi X-Ray Timing Explorer (RXTE) using only measurements from a bee-axis magnetometer. The performance of these algorithms is examined using in-flight data from various scenarios. In particular, significant enhancements in accuracies are observed when the telemetered magnetometer data are accurately calibrated using a recently developed calibration algorithm.

Interesting features observed in these studies of the inertial-pointing RXTE include a remarkable sensitivity of the filter to the numerical values of the noise parameters and relatively long convergence time spans. By analogy, the accuracy of the deterministic scheme is noticeably lower as a result of reduced rates of change of the body-fixed geomagnetic field. Preliinary results show the filter- per-axis attitude accuracies ranging between 0.1 and 0.5 deg and rate accuracies between 0.001 deg/sec and 0.005 deg./sec, whereas the deterministic method needs a more sophisticated techniques for smoothing time derivatives of the measured geomagnetic field to clearly distinguish both attitude and rate solutions from the numerical noise. Also included is a new theoretical development in the deterministic algoritfm the transformation of a transcendental equation in the original theory into an 8*-order polynomial equation. It is shown that this 8 ' - order polynomial reduces to quadratic equations in the two limiting cases- infinitely high wheel momentum, and constant rates-discussed in previous publications.

INTRODUCTION It has been demonstrated'-'' that the attitude and rates of low-Earth orbiting spacecraft can be simultaneously estimated using measurements of the E a r t h ' s magnetic field, g , using only a three-axis magnetometer (TAM) and no a priori information. The feasibility of this "TAM-Only'' scheme essentially * This work was supported by the Nation+ Aeronautics and Space Administration (NASA) I Goddard Space Flight Center (GSFC), Greenbelt, Maryland, Contract GS -35F4381G, T a s k Order No. S-03365-Y.

Computer Sciences Corporation, 101 10 Aerospace Road, Lanham-Seabrook, MD 20706

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changing direction rapidly enough in the spacecraft body frame to make computation of its time derivative possible, and these changes during the course of an orbit are sufficiently large to enable determination of all three Euler angles using only TAM data.

Our approach consists of using two independent algorithms-deterministic attitude determination from magnetometer-only data (DADMOD) and the Real Time Sequential Filter (RTSF). The DADMOD''3 is a TAM-only algorithm that relates the time derivatives of in inertial and spacecraft body coordinates to determine the attitude and the body rates. The RTSP' is a robust Kalman filter that estimates, in addition to the attitude, errors in rates propagated via Euler's equation. Note that the RTSF is a general algorithm for gyroless spacecraft, however, its sensitivity to rate errors as small as 0 . 0 0 0 3 deg/sec makes it a robust and accurate real-time algorithm even in TAM-only situations with no a priori spacecraft information.

The highlights of our past applications to in-flight data from the Solar, Anomalous, and Magnetospheric Explorer (SAMPEX) and the Earth Radiation Budget Satellite (ERBS) have shown that with a TAM-Only approach: (1) SAMPEX attitude and rate requirements can be met even when the on- board Sun sensor failsM, (2) using partially calibrated magnetometer data from ERBS nominal mission mode the RTSF yields' accuracies,within 0.4 deg and rate accuracies within 0.005 degkec, and (3) ERBS attitude and rates could be reliably determined" during its 1987 control anomaly" when the spacecraft tumbled at approximately 2 deg/sec. Another useful aspect of the past work is the combination of the strengths of these algorithms in an automated scheme7 wherein the deterministic algorithm is used to initialize the more accurate Kalman filter to within a few degrees of the correct spacecraft attitude.

In the present work, we examine the performances of these algorithms during some important scenarios of the RXTE: (1) calibrated and uncalibrated TAMS and (2) during maneuvers. While the fist scenario does not require further explanation, the motivation for the second is the possible application of the RTSF to extend aging missions. For example, ERBS (launch 1984) needs monthly hster-based maneuvers for solar-power purposes, and these are currently conducted using rate information from the one remaining gyro channel. This paper demonstrates that, by providing magnetometer-only rate solutions, the filter can be a useful tool during such maneuvers, especially when the last gyro channel also fails.

The present work concentrates on results for RXTE that are interesting in their own right, because, in contrast to SAMPEX and ERBS: * RXTE is inertial pointing so that $changes very slowly in the body frame, and this leads to observability and convergence issues when only short data spans are used (as is the case here).

RXTE is a zero-momentum spacecraft, whereas SAMPEX and ERBS are momentum-biased about the pitch axis. We believe this leads to the RTSF results being very sensitive to the numerical values of its propagation noise parameters and their relationship to the weightage of the TAM data.

For a similar reason, the accuracy of the deterministic scheme becomes noticeably lower.

The rest of the paper is organized as follows. Section 2 briefly describes the algorithms; included here are recent developments in DADMOD and a novel TAM calibration algorithm'2 that is used to calibrate the RXTE TAM data. Sections 3 and 4 analyze the performances for RXTE and ERBS respectively, and Section 5 summarizes the conclusions.

ALGORITHMS Deterministic Attitude and Rate Determination Using Magnetameter-Only Data (DADMOD) As discussed in detail in,previous publications, determination of the spacecra: attitude and rates based on magnetometer measurements and their first and second time derivatives can be cast in the form of the following vector equation: where the angle @ and the body rate wl around the body-fixed geomagnetic field vector unknown variables. It is essential that A X Bl(@), A 2 5 a2 x BA, where the vectors a, (@) and a are given by Eqs. (5) and (6) in Ref. 10, and BA=BA/ I aA I .

The third vector A o(@) is also perpendicular to BA at any value of the angle @ so that two nontrivial equations to determine both @ and Q are obtained by projecting vector equation (1) on two directions perpendicular to the geomagnetic field. As a new development, we show that these two directions can be chosen in such a way that C(@) = tan(@/2) becomes one of roots of an 8&-0rderpolynomial P & ] .

In fact, by projecting Eq. (1) onto the vector BA x A one finds

Substituting the latter expression into the projection of Eq. (1) onto the vector (@) then gives It can be shown that the vectors A o(@) and a, (a) have the form:

-

where the vectors Qn[C] are formed by polynomials of the n&order in and c(@) = cos ( W 2 ) , and therefore the solution sought for is given by one of roots of the 8*-order polynomial: Note that all coefficients of polynomial (6) are equal to zero when the geomagnetic field is

a and A , vanish. By

directed along one of the spacecraft principal axes of inertia since both vectors

-

A , becomes perpendicular to the vector Hl (Q1) for

analogy these coefficients are nullified if the vector the sought -for root C(@I). However, it can be shown that the ratio (p4[C] )2 / P & ] tends to zero in both cases so that the solution sought for can be found among real roots of the polynomial P & ] . Because Eq.

(3) is no longer applicable, one has to~solve a quadratic equation to find 01 . To avoid instabilities, a special algorithm was developed to select the direction associated with the maximum of the discriminant.

As a result, the resultant solution remains stable as the coefficient of the quadratic term tends to zero.

-

tends to -, while the The ratio ( P 4 [ 5 1 l2 / Pz[<] also vanishes as the wheel momen~um polynomial P6[<] takes the form: A s a result, we come to the quadratic equation 2fSJ = 0 discussed in Ref. 6. A similar decomposition of the polynomial P,j[<] takes place in case of constant body rates after one drops all terms associated with the time derivative of the angular velocity vector a. The solution solved for can be found from the requirement for the vectors B, (a) and A o(@) to be perpendicular to each other, which is equivalent to the condition P & ( @ ) ] = 0. The resultant quadratic equation n",c] = 0 has been studied in detail i n Refs. 1 and 2.

Real Time Sequential Filter (RTSF) In view of space considerations,only details relevant to the tuning of the RTSF are presented here.

A full mathematical description of the RTSF has been provided elsewhere (References 4 and 5).

The RTSF's state vector 2 is comprised of the four components of the attitude quaternion, g , and the corrections, 8 , to the spacecraft's rates, 6:

2 = [g' "'I'

(Note that the components of b' and & are resolved along the spacecraft's x, y, z axes.)

The RTSF uses sensor data to estimate 4' as well as 5, with b ' being estimated kinematically in the same manner as gyro biases for a gyro-based spacecraft; i.e., by attributing differences between the measured and propagated attitudes to errors in 6. The 8 estimates are then used to correct G , and these corrected rates are used as initial conditions to propagate Euler's equation to the next measurement time.

The propagation of b' is modeled via a first-order Markov model: a % $ - = - - + j j b (9) dt z where q, is a white noise term, and T is a finite time constant. A suitable value for T is the time between measurements.

The rates are assumed to contain a white noise component, qa and are propagated using Euler's equation after accounting for the angular momentum contributed by the wheels, and for the total external torques acting on the spacecraft. TAMONLY currently models the gravity-gradient torque and the magnetic control torque acting on the spacecraft. (The aerodynamic drag torque and the radiation pressure torque have been intentionally omitted to reduce the amount of spacecraft modeling required. The RTSF relies on the rate-corrections, b' , to compensate for the small effects of these torques.)

The covariance matrix, P, is propagated by numerically integrating the following equation: - dP = ~ ( 6 ) P + P F~ (GI+ Q dt ( G ) is described in Reference 5 ; the quantity of interest is the 6 x 6 matrix is of the following diagonal form: quantifies the propagation noise and [ Qa, Qa, Qai Qb, Q b , Qb I (1 1) Here Qa is related to the noise term e, and contributes to the growth of the attitude error covariances about the body X-, Y-, and Z-axes during propagation. Similarly, QI, is related to noise term 7 j b , and contributes to the growth of the error covariances of b' during propagation. Another quantity that we must consider during tuning is 0, the strength of the white noise in the TAM measurements.

The filter can be initialized in one of the following two ways before processing a span of telemetry data.

e Inertial initial conditions (IIC), where the spacecraft is assumed at rest in the Geocentric Inertial Coordinates (GCI) with its axes coincidingwith the GCI axes; this results in large initial errors.

e Deterministic initial conditions (DIC) where the filter makes short (2 to 5 min) runs and determines which of the DADMOD solutions is a good u priori solution. This results in small initial errors.

The TAM Calibration Algorithm The effects of TAM calibration were determined using a recently-developed algorithm'2 where the following set of 21 time-independent parameters are used to "adjust" the magnetic field vector measured by the TAM whose axes nominally coincide with the spacecraft body axes.

E ............ 3x3 scale factor/misalignment matrix nominally equal to the identity matrix, 13x3 '

G ............ 3x3 TAM-torquer coupling matrix nominally equal to the null matrix, 03x3

3: ............ 3x1 bias vector nominally equal to the null vector, If at any instant z is the magnetic field vector measured by the T W , fi is the 3x1 dipole moment vector of the magnetic torquer bars, A is the known GCI -to-spacecraft body fiame attitude matrix, and 5 ; is the corresponding Intemationl Geomagnetic Reference Field (IGRF)I3 vector in the inertial frame, the calibration model assumes Bi = I : (E,Bi" - G G D j ) - J + V i , i= 1,2,3 j=1 where g R = A@ is the predicted field in the spacecraft body frame, and G is a white-noise term of root- mean-square (r-m-s) value 0. The goal of the calibration then is to estimate E, G, and j, by applying statistical methods to a span of TAM measurements, {a, ..., gN) and the corresponding predictions {z;,...,z: /.

Resolving E and G into the vectors El, &, E3, el, e2, and e3 as follows, three independent loss functions are now formulated as: where the subscript n denotes the measurement time, and Q ; . i The following notation for the statistical quantities formed from vectors is followed here.

N - Means: (+ p i x , r=l

Covariances: (iif) = (HT)-(i?)(fr)

where the superscript T denotes matrix transpose.

Minimizing Li ( Ei, ai , A ) yields

(19a) i (19b) E:p"16)i -a:( 6 1 6) = kip),

A =E:('i"), -qqi-(13i)i (19c)

where subscript i indicates that the averages are only over the i-th set. Equations (19a) and (19b) can be readily solved yielding: fi: is then obtained by using Equations (20) in (19c).

RESULTS USING THE ROSS1 X-RAY TIMING EXPLORER IN-FLIGHT DATA Overview of the Mission and Data The RXTE is an inertial-pointing spacecraft and was launched in December 1995 into a near- circular orbit of altitude 580 Inn and inclination 23 deg. The primary attitude sensors on board are charge- coupled device star trackers that provide accurate sensordetermined attitudes during inertial periods. The attitude during maneuvers (as many as eight each day) is obtained from accurately calibrated gyros. The predicted field values B' were generated using a 10' order IGRF model for the reference field values.

Three sets of data from 1/4/96,7/4/96, and 11/6/97 were used in the present study. Of these, the first two contain spacecraft slews (primarily about the z axis), while the last is wholly inertial. The telemetry data received at the FDF are nominally 2 sec apart, but various samples at a slower rate were generated to increase observability of the magnetic field variations. Thus, data were generated with pseudo-periods ranging f r o m 4 sec to 40 sec, and several different telemetry periods were used for each set of data.

However, the results presented here used 40 sec sampling for the 7/4/96 data and 8 sec sampling for the other two sets.

Terminology Some notes about the figures and tables presented here are in order. The "truth" models used to evaluate the attitude and rate accuracies of the algorithms are the on-board computer (OBC) determined attitudes and rates computed from their time derivatives. GCI-to-Body attitude results are presented in the form of 1-2-3 Euler angles, and these angles are respectively referred to Bs "Angle- ", " Angle-2", and " Angle-3". The body-frame components of the spacecraft rates are depicted in the figures as "wx", "wy", and "wz". "Raw" and "adjusted" refer to the quality of the TAM data, and denote pre- and post-calibration values for the TAM measurements. "Residuals" are the differences between TAM measure predicted using the RTSF attitude estimates. "TAM angle" is the angle between the measured and predicted fields. It is a convenient scalar parametrization of the separate TAM residuals along the three body axes and, as will be seen, is useful when evaluating the filter in the absence of truth models. Only TAM-1 measurements have been used throughout the paper although TAM-2 measurements are also available for the RXTE. The TAM-2 measurements and the residual statistics are not very different from the TAM-1 measurements, although significant differences do exist in the calibration parameters. "RTSF rate-errors " are the corrections, b" , estimated by the RTSF (see Equation (9)) as part of its state vector and are different from a term such as "error in wz" that refers to the differences between the RTSF rates and gyro rates. Thus b' may be viewed as "rate residuals" since convergence of the RTSF implies small b" .

TAM-1 Calibration Results Excellent residual statistics were obtained after calibration of the data and the results are shown in Table 1 for each axis separately. For example, the root-mean-square residuals are of the order of 0.5 mG.

The mean residuals are most impressive: of the order of (i. e. of the order of "e-14'' in the notation of the Table).

Table 1 RXTE RESIDUAL STATISTICS FOR TAM-1 Pre-Calibration (Raw) Data I Post-Calibration (Adjusted) Data ..

-

Mean Residuals (X,Y,Z) ~ 1/4/96 4.105, -0.481, -1.792 Max: 9.290 Min: -6.949 7/4/96 3.055, -4.510,3.413 Max: 7.984 Min: -3.196 -1.561, -0.579,8.221 Max: 15.670 Min: -6.635 RTSF Tuning The RTSF was tuned as follows. The largest of the r-m-s residuals results for a given dataset of Table 1 was used as the RTSF tuning parameter CT during the TAM-only runs; for example, this value would be 4.72 mG for the raw data of 1/4/96. At the outset of the TAM-only runs approximate numerical values for the filter propagation noise parameters, Qa and Qb of Equation (1I), were obtained by analyzing the errors in the angular momentum of the spacecraft and wheels and the effectsthese errors would have on the RTSF rate and attitude while propagating between measurements. The uncertainties in the system net wheel angular momentum was determined to be about 0.025 N-m, which resulted in rate uncertainties of 9.4x1U6 rad/sec. This implied Q, was of the order of 1U''Lsr rad2/sec where Ar is the telemetry period.

analysis using the convergence properties of the Markov model resulted i n Q b of the order of rad2/sec3.Tuning was then accomplished by: (1) choosing a constant o from Table 1 as stated above, (2) setting Q, equal to Qb during all of the rum, and (3) varying this adjustable single adju about the numerical value of until the attitude errors were minimized. The accuracy of the tuning parameters was verified later by studying the performance of the filter over several orders of magnitude of Qb. For each dataset a few runs were also made using different o but none yielded better performance.

Each dataset was also studied using different telemetry periods. A l l i n all a few hundred runs were made for each dataset, and only a small portion of the results are shown below.

A striking difference between the RTSF performance for RXTE and past experience with SAMPEX and ERBS data is the sensitivity to the numerical values of the numerical parameters, which in turn were somewhat dependent on the telemetry period. Thus, whereas it was sufficient for a b to be accurate to one significant figure for S A M P E X and ERBS, it turns out that the tuning parameters have to be accurate to three to four significantfigures for RXTE. As an example, for the 11/6/97 data w i t h 8 sec telemetry period, the t o t a l RTSF attitude error was 15.6 deg when a b = 1 . 1 ~ 1 0 ~ ' ~ whereas this error dropped to 5.1 deg when Q b = 1.01~10~'~.

DADMOD and RTSF (using IIC) attitude results for the 7/4/96 RXTE data with a telemetry period of 40 sec are presented i n Fig. 1, which shows a spacecraft maneuver about the z - a x i s between 1500 sec and

I I I I 0 , I . \ / I I I

/\ 0 500 1000 1500 2000 2500 30w3 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 TIME (sec) Figure 1. GCI-to-Body 1-2-3 Attitude Euler Angle Results for Adjusted 7/4/96 D a t a ( s o l i d =true, dashed = RTSF, crosses = DADMOD 1st root, circles = DADMOD 2nd root) 2500 sec. In this figure the lines represent the truth and the RTSF sotutions (solid and dashed respectively) while the symbols represent the DAD correct and spurious solutions (crosses and circles respectively).

The filter was started with IIC (large errors) and converged within 100 sec to a metastable spurious state that also shows up in the DAJ3MOD solutions. The RTSF converges to the correct solution only about 1000 sec later-towards the start of the maneuver. This slow convergence of the filter is a direct result of the inertial-pointing nature of the spacecraft, which results in the orbital motion being the sole cause for ," is approximately constant over the maneuver period.) Note that: (I) this is the first independent confirmationof the DADMOD spurious solutions, and (2) such ambiguities will not arise if gyros provided the rate information and a TAM is used solely for attitude information. Note also a relatively large spread of the physical deterministic solution as a result of relatively l o w rate of change of ii.

The slow convergence severely l i m i t s any rating of the accuracy of the filter: statistics for the last 15 points in Fig. 1 reveal r-m-s attitude errors of (0.43,0.39,0.17) deg about the three body axes. For more reliability, the filter was studied using data from the 1/4/96 dataset where an inertial span of nearly 4500 sec duration precedes the maneuver. The results are shown in Figs. 2 and 3.

Fig. 2 presents sample attitude and rate results for the 1/4/96 data of 8 sec telemetry period with the RTSF using IIC. These results were obtained with the numerkal values of Q, = Qb = 2.23x10-" and were deemed the optimal parameters after examining the error statistics. (See Table 2 below). We see that the filter converges by about 4000 see even though the initial errors ranged from about 65 deg in Angle-3 to about 113 in Angle-1. Additional residual results from the same run are presented in Fig. 3. The RTSF state vector evolves so as to minimize all these quantities, and we see that all are small only after 4000 sec.

The convergence is slow here also, and it is instructive to examine the RTSF errors after 3200 sec separated into before, during, and after the maneuver. These are presented in the first and second columns of Table 2. Table 2 also compares these error statistics with the ones obtained using raw TAM data and a different set of tuning parameters (Qa = Qb = l.l2~lO-~) separately determined to be optimal for the raw data. Some clear inferences can be draw from examining Table 2.

0 The attitude errors are significant before the maneuver but noticeably decrease during the maneuver, which we attribute to the increased observability of changes in 2.

In contrast to the attitude errors, the errors in the rates increase during the maneuver, which we attribute to (small) dynamical modeling errors of the spacecraft.

TAM calibration significantly improves TAM-only accuracies.

The above information is also seen qualitatively in the bottom plot of Fig. 2 and the middle and bottom plots of Fig. 3. Thus, Fig. 3 clearly shows us that the rates have converged well before the attitude; this is in accord with past experiences with the RTSF. It should be noted, though, that there will always be differences in the convergence times of the RTSF attitude and rate estimates because the rates are corrected based on the TAM residuals. Nevertheless, it would be interesting to examine in the future if better accuracies result from tuning Q and Q b to yield the same convergence times for both attitude and rates.

The RTSF performance using IIC was further examined using the wholly inertial span of 11/6/97, I I C , and some of the results are shown in Fig. 4. It is clear (especially from the TAM angle plot) that the convergence time is of the order of 4400 sec. The error statistics from 4400 sec to the end of the data span are as follows: r-m-s attitude errors = (0.54,0.13,0.33) deg r-m-s errors in rates = (0.0049,0.0010,0.0024) deg/sec I I 3 I I I , I 0 1000 2OOO 3000 4000 5000 6000 7000 8000 g 4 . 0 2 ' , I 1 I Lu 0 1000 2OOO 3000 4000 5000 6000 7000 8000 Time in sec since 19960104.203059 Figure 2. RTSF Attitude and Rate Results for Adjusted l/4/96 Data (circles = RTSF and solid = truth in top two plots) Q) Q)

s -20 I I

0 1000 2000 3000 4000 5000 6000 7000 8000 0 1000 2000 3000 4M30 5000 6000 7000 8000

z ~~~1 ?3

N s -2 0 1000 MOO 3000 4000 5000 6000 7000 8000 Time in sec since 19960104.M3059 Figure 3. Additional RTSF R&ults for Adjusted Y4/96 Data Showing TAM-1 Residuals (top two plots) and RTSF Rate-Error Estimates (bottom plot) After (0.38,1.33,0.65) (0.0015,0.0038, (0.68,2.06,0.57) (0.0017,0.0043, 0.0034) 0.0033)

I I I I I

n Ln (u e.

N O a 3 - m -3 -1 M 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 c4 u

-

/ (u

s

I I I I I t 1 I 8 5 5 4 (u

=

0) - E2 4 .

z $ 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Time in sec since 19971106.010002 Figure 4. RTSF R e s u l t s for Adjusted 11/6/97 Data (circles = filter in top plot) CONCLUSIONS In contrast to our past experiences with SAMPEX and ERBS, which were momentum-biased spacecraft spinning at the orbit rate, the RTSF performance for the zero-momentum, inertial-pointing RXTE is characterized by extreme sensitivity to filter tuning and long convergence times (about 4000 s e c ) . Thus, while past SAMPEX and ERBS resdts demanded accuracies of only 1 significant figure in the tuning parameters, it was clear that accuracies of three to four significant figures were needed for success application to RXTE. The performance of both DADMOD and RTSF improved significantly once the telemetry period was increased from the nominal value of 2 sec to between 8 and 40 sec. Presently we attribute this sensitivity to a combination of telemetry period and the zero-momentum nature of m.

Careful tuning of the RTSF demonstrated per-axis attitude accuracies between 0.13 and 0.54 deg and rate accuracies between 0.0010 degsec 0.0049 deg./sec when were used. The

corresponding values during the maneuver of 1/4/96 were 0.3 1 - 0.70

- 0.0041 deg/sec.

These results are similar to our past results for ERBS’: attitude within 0.4 deg and rate accuracies within 0 . 0 0 5 degsec We consider these accuracies preliminary in view of the long convergence times and the limited number of post-convergence results. More definitive accuracy studies are needed using long spans of inertial data. It would also be useful to study the relationship between the performance, the tuning, and the telemetry period.

Accurate TAM calibration was performed using a recently developed algorithm, and the subsequent RTSF TAM-only results show a significant improvement in the attitude accuracies upon using calibratea TAM data. This conclusion is in general agreement with studies using a TAM-gyro c~mbination’~.

An important theoretical development presented here is the transformation of a DADMOD transcendental equation into an 8th order polynomial. The first independent evidence of the accuracy of the DADMOD spurious solutions w a s obtained in the form of initial convergence of the RTSF to the metastable solution using the 4/7/96 data. To improve the DADMOD performance for inertial-pointing modes, a more sophisticated technique for smoothing time derivatives of the measured geomagnetic field should be applied to separate the physical root from the numerical noise.

REFERENCES 1. G. Natanson, S. McLaughlin, and R. NicMas, “A Method of Determining Attitude From Magnetometer Data Only,” Proceedings of the Flight MechanicsA?stimation Theory Symposium 1990, NASA Conference Publication 3102, Goddard Space Flight Center, Greenbelt, MD, May 1990 G. Natanson, J. Keat, and S . McLaughlin, Sensor and Advanced Attitude Studies: Detenninistic 2.

Attitude Computation Using Only Magnetometer Data, Goddard Space Flight Center, Flight Dynamics Division, 554-FDD-91/010, prepared by Computer Sciences Corporation, March 199 1 G. Natanson, “A Deterministic Method for Estimating Attitude From Magnetometer Data Only,” Paper 3.

No. IAF-92-0036, Proceedings o f the World Space Congress, Washington, DC, September 1992 M. Challa, Solar, Anomalous, and Magnetospheric Particle Explorer (SMPEX)Real Time Sequential 4.

Filter (RTSF): Evaluation Report, Goddard Space Flight Center, Flight Dynamics Division, 553-FDD- 93/024ROUDO, prepared by Computer Sciences Corporation, April 1993 5. M. Challa, G. Natanson, D. Baker, and J. Deutschmann, “Advantages of Estimating Rate Corrections During Dynamic Propagation of Spacecraft Rates-Applications to Real-Time Attitude Determination of SAMPEX,” Proceedings o f the Flight MechanicsLEstimation Theoj, Symposium 1994, NASA Conference Publication No. 3265, Goddard Space Flight Center, Greenbelt, MD, May 1994 G. Natanson, M. Challa, J. Deutschmann, and D. Baker, “Magnetometer-Only Attitude and Rate 6.

Determination for a Gyroless Spacecdt,“ Proceedings o f the Third Internutional Symposium on Space Mission Operations and Ground Data Systems, NASA Conference Publication 328 1, Greenbelt, MD, November 1994, pp. 791-798.

M. Challa and G. Natanson, “A PC-Based Magnetometer-Only Attitude-and-Rate Determination 7.

System for Gyroless Spacecrafk” Proceedings o f the Flight Mechanics and Estimation Theory Sym- posium, NASA Conference Publication No. 3299, Goddard Space Flight Center, Greenbelt, MD, May M. Challa and C. Wheeler, !‘Accuracy Studies of a Magnetometer-Only Attitude-and-Rate Determi- 8.

nation System,” Proceedings of the Flight Mechanics and Estimation Theory Symposium, NASA Conference Publication No. 3333, Goddard Space Flight Center, Greenbelt, MD, May 1996 . Challa, G. Natanson, and C. Wheeler, "Simultaneous Determination of Spacecraft Attitude and ales Using Only a ~ a g n e ~ o m e t e r ~ Proceedings o f the American lnstitute of Aeronautics and Astronautics (AlAA)/AmericanAstronautical Society (AAS) Astrodynamics Specialist Conference, San Diego, CA, July 1996 10. M. Challa, S. Kotaru, and 6, Natanson, "Magnetometer-Only and Rate Estimates During the Earth Radiation Budget Satellite 1987 Control Anomaly," A dance, Navigation and Control Conference, New Orleans, August 11-13,1997 1 1 . J. Kronenwetter, M. Phenneger, and W. Weaver, "Attitude Analysis of the E a r t h Radiation Budget Satellite (ERBS) Yaw Tum Anomaly", Proceedings o f the Flight MechanicdEshtion Theory Symposium, NASA Conference Publication No. 301 1, Goddard Space Flight Center, Greenbelt, MD, May 1988 1 2 . M. Challa and R. Harman, "A New Magnetometer Calibration Algorithm and Application to the X-Ray Timing Explorer", Proceedings of the AiAA Guidance, Navigation and Conrrol Conference, Boston, 10-12, 1998 ( t o be published) August 1 3 . See, for example, the IGRF World Wide Web page at: http://ftp.ngdc.noaa.gov/Solid,Eartb/Mai 1 4 . J. Hashmall and J. Deutschmann, "An Evaluation of Attitude-Independent Magnetometer-Bias Determination Methods", Proceedings of the Flight MechanicdEstimation Theory Symposium 1996, NASA Conference Publication 3333, Goddard Space night Center, Greenbelt, MD, May 1996 *t Joseph A. Hashmall, Joseph Sedlak,* Daniel Andrews,* and Richard Luquette* A major limitation on the use of infrared horizon sensors for attitude determination is the variability of the height of the infrared Earth horizon. This variation includes a climatological component and a stochastic component of approximately equal importance. The climatological component shows regular variation with season and latitude. Models based on historical measurements have been used to compensate for these systematic changes. The stochastic It can cause component is analogous to tropospheric weather.

extreme, localized changes that for a period of days, overwhelm the climatologicalvariation.

An algorithm has been developed to compensate partially for the climatological variation of horizon height and at least to mitigate the stochastic variation. This method uses attitude and horizon sensor data from spacecraft to update a horizon height history as a function of latitude. For spacecraft that depend on horizon sensors for their attitudes (such as the Total Ozone Mapping Spectrometer-Earth Probe-TOMS-EP) a batch least squares attitude determination system i s used. It is assumed that minimizing the average sensor residual throughout a full orbit of data results in attitudes that are nearly independent of local horizon height variations. The method depends on the additional assumption that the mean horizon height over all latitudes is approximately independent of season. Using these assumptions, the method yields the latitude dependent portion of local horizon height variations.

This paper describes the algorithm used to generate an empirical horizon height. Ideally, an international horizon height database could be established that would rapidly merge data from various spacecraft to provide timely corrections that could be used by all.

*

Computer Sciences Corporation, I O 1 10 Aerospace Rd. LanhdSeabrook, MD 20706, USA $ Phone (301)-794-1279,e-mail joseph.hashmall@gsfc.nasagov NASA Goddard Space Right Center, Greenbelt MD, 20771, USA horizon sensors have recently regained some of their early popularity as attitude cost, their accuracy ough they have the advantage of reliabili for near-Earth orbit missions i s limited by the vari t of the layer of the stratosphere that they detect as the surface of the Earth’s infrared spheroid. Most modem horizon sensors limit their sensitivity to radiation in the 14-16y band to eliminate diurnal effects and because at these wavelengths the Earth spheroid is most stable and homogeneous.

Scanning horizon sensors rotate the field-of-view (FOV) of an infrared telescope around a circular path. The sensor FOV points towards space over part of the circle and points towards Earth during the rest. The angle at which a sudden change between the low radiance of space and the high radiance of the Earth occurs is interpreted as the horizoncrossing angle.

Differences and mean values of Earth-in and Earth-out horizon crossing angles can be used to provide estimates of the spacecraft pitch and roll.

Static horizon sensors have detectors that point towards the Earth horizon when the is near its nominal attitude. The level of output from these detectors represents the spacecraft portion of their FOVs that contains the Earth. Differences and means of the output from detectors viewing different portions of the horizon can be used to provide estimates of the spacecraft pitch and roll.

For both types of sensor, several effects can alter the detector output and result in attitude errors. These effects include the variation of the atmosphere’s radiance in the wavelength range (roughly 14-16p) in which the detectors are sensitive. This paper analyzes those horizon height variations caused by those changes in radiance that result from stratospheric temperature variation. Other phenomena can also cause significant horizon sensor errors. These include the effects Earth’s oblateness, of high, cold clouds, and of the proximity of the Sun or Moon image to the horizon crossing point. Some of these can easily be compensated (e.g. Earth oblateness) while others must be predicted or detected and the contaminated data removed from the processing stream. None of these other effects will be extensively discussed in this paper.

Changes in stratospheric temperatures can be interpreted as changes in the height of the infrared Earth horizon and often are the single largest uncompensated contributor to horizon sensor attitude error. These temperature changes can be classified as climatic or stochastic. In this paper the term “climatic” will be used to designate those effects that depend on latitude and season and which are repeatable from year to year. Similarly, “stochastic” will be used to designate those effects that change rapidly (over a period of days), are usually localized (over a range of a few thousand k m ) , and are correlated in time. Stochastic effects are similar to weather and cannot be accurately predicted long in advance. For attitude determination they can best be modeled as colored noise.

Attempts have been made to analyze and mitigate the effect of the climatic variations of horizon radiance on horizon sensor attitude estimates. Extensive historical measurements of stratospheric temperatures from balloon and rocket probes have been combined with atmospheric models and electronic models of sensor triggering to produce horizon radiance models.’ These models have been incorporated in software to compensate for the climatic variation of the infrared horizon height?3*4 Studies of the stochastic variations of horizon radiance have shown that errors due to these variations can be as large as those due to uncompensated climatic horizon radiance variations. Sudden stratospheric warming events’ can ov the climatic effects in regions of the winter hemisphere a large portion of the winter ing the horizon heigh hemisphere higher than that over the summer hemisphere! vents can last for as little as a few days to as much as a few weeks. Observations of the Earth in GOES-8 and -9 sounders have been correlated with errors in Earth sens Results show that stratospheric radiance changes can arise suddenly and can result in significant attitude errors?

In preparation for the launch of the Total Ozone Mapping Spectrometer-Earth Probe (TOMS-EP), an empirical radiance modeling utility’ (ERMU) was developed by NASA’s Goddard Space Flight Center (GSFC). This utility was to be used in an attempt to minimize the effects of horizon radiance variation on TOMS-EP attitude. This paper is a description of the results of measurements made by the TOMS-EP horizon sensors and their interpretation as changes in the Earth’s infrared horizon height.

TOMS-EP was launched in July 1996 into a Sun synchronous orbit at an altitude of approximately 500 km and an inclination of 97.4 deg. After 16 months of data collection, in December 1997 it was boosted to an altitude of about 740 kin and the inclination adjusted to 98.4 deg to maintain Sun synchrony. The nominal TOMS-EP attitude is Earth pointing with the Z- axis (yaw) pointing towards the Earth, the Y-axis (pitch) along the negative orbit normal, and the X-axis (roll) pointing in the general direction of the spacecraft velocity vector.

The principal attitude sensors used by TOMS-EP are two Ithaco T-scanwheel horizon sensors (HS), two fine Sun sensors (FSS), and a set of accurate rate determining gyros. The two horizon sensors are mounted with their scan axes in the Y-2 plane, canted 20 deg towards nadir from the +Y and -Y-axes respectively. Their half cone angles are 46 deg. The HSs are insensitive to Yaw so the primary Yaw information is obtained from the two Sun sensors. The Sun sensors’ FOVs are directed in the +X and -X directions allowing these sensors to measure the Sun direction while the satellite is near the polar regions.

METHOD The TOMS-EP ERMU was developed to reduce the effect of horizon radiance variation on horizon sensor performance. It was to do this by using horizon sensor data to determine horizon height variations weekly and apply the corrections needed to compensate for these variations in the following week. Although the ERMU was tested in the early phases of the TOMS-EP mission, it was never used in normal operations because its use was found to be unnecessary to meet mission requirements (0.25 deg per axis, 30). The approximately one year span during which horizon height variations were measured serves as an excellent reference for horizon height behavior and mitigation studies.

The basis of the ERMU horizon height measurements is the determination of the This algorithm is implemented in spacecraft attitude using a batch least-squares algorithm.

GSFC’s multimission three-axis stabilized spacecraft (MTASS) coarse and fine-attitude determination system (CFADS). CFADS has been used with data from many missions and has been shown to be flexible, reliable, and accurate.

gorithm minimizes the loss function fo ba9 problem given by: L with respect to a state vector including the attitude at an epoch time and gyro biases. In Eq. (I), is the relative weight of the sensor with a measurement at some time t, to is the reference time, Atois the attitude at the reference time, PRoisthe transition matrix transforming the attitude at epoch to the attitude at time t (obtained by integration of gyro observations with added biases from to to t), and ht and st are reference and observed sensor measurements at time t. The sum is performed over all valid sensor measurements in a batch, including both Sun sensor and horizon sensor measurements. The primary C F A D S output is the attitude of the spacecraft at times throughout the batch obtained by propagating the epoch attitude solution using gyro data modified by the determined gyro biases.

As long as the gyro biases are nearly constant (as assumed in Eq. (1) and as expected from mission experience), the attitudes obtained from CFADS will be far more accurate than single frame attitudes, and in cases with correlated errors in sensor measurements or correlated errors in reference vectors, more accurate than ordinary Kalman filters. Note that errors in horizon sensor measurements due to horizon height variations are highly correlated in time. This is because horizon height changes occur over finite areas of the E a r t h surface (they are correlated in space) and, therefore, measurement deviations persist while the spacecraft traverses the affected region. These correlations strongly affect the accuracy of Kalman filter derived attitudes but have little effect on the batch least-squares attitudes used in this study.

The accuracy of CFADS attitudes arises from the use of all of the sensor data in the batch. Errors arising from sensor deviations at one time can be compensated by opposing errors at other times because of gyro propagation. This compensation does not require uncorrelated errors, but increases accuracy as mean errors over the batch approach zero. In this study, an integral number of orbits were used in all CFADS batches, minimizing (due to North-South mirror symmetry) the effect of climatic horizon radiance variation.

TOMS-EP Earth sensor data is preprocessed onboard the spacecraft and reported as pitch and roll estimates for each of the two horizon sensors. They are reported every 32 seconds.

Once CFADS attitudes have been computed, sensor residuals are obtained by first computing reference roll and pitch angles. These reference angles are computed by converting the negative spacecraft position vectors to body coordinates using the corresponding calculated attitudes, The converted vectors are expressed in terms of roll and pitch angles in an nadir referenced coordinate system. Differences between these reference pitch and roll angles and the observed angles, compensated for sensor misalignment, constitute the sensor residuals.

The spacecraft orbit was divided into equal bins of orbit phase and the roll and pitch errors within each bin were averaged to give mean residuals: Ain and A& for bin n. These are converted into mean horizon height deviations by the following steps:' 1. Compute a mean angular radius of the subtended Earth, p, from the nominal semimajor axis, E, the equatorial Earth radius, Re, and the nominal horizon height , hnom (hnom= 37.9 km), by:

p=sin-( 1 Re + E horn )

2. Compute the ominal Earth chord width, -2 cos p - cos q cosy n = cos (3)

( sinqsiny

where q is the complement of the cant angle and v i s the half cone angle.

3. Geometry coefficients for roll, K, and pitch, K ’ , are computed from: tanp Kr = (4) 2€(sinqcosy - cosq sinycosS2) and tanp ( 5 ) Kp = ~ ( s i n o sinly) 4. Finally; the horizon height deviations for the Earth-in and Earth-out transitions are found from: and where ATn is the mean roll residual and Apn the mean pitch residual in bin n.

A minimum of four orbits of data (approximately 200 Earth observations per sensor per orbit) in each of four days were used to produce weekly averages of Earth-in and Earth-out horizon height deviations for each sensor and for each orbit phase measured from the ascending node. Horizon heights that fell within orbit phase bins having a width of 2 deg were averaged.

This empirical horizon height database was produced for approximately 1 year of TOMS-EP data.

The tables of horizon heights stored by TOMS-EP orbit phase angle and week of year were designed for this single mission. Corrections to the nominal height of any of the horizon crossings were to be obtained by interpolation of phase angle using the previous weeks results.

Once on orbit, it was found that this correction was not needed to attain the mission’s modest attitude accuracy requirements. If one of the TOMS-EP Earth sensors fails it may be necessary to use of the ERMU to improve attitude accuracy.

To make tables of horizon heights more generally useable, and to allow compatible input from spacecraft other than the one making horizon height measurements, it is convenient to express the results in terms of the latitude of the horizon crossing rather than the phase of the spacecraft orbit. This transformation is accomplished by determining the declination angle, p, as: Equation (8) is virtually identical to Eq. (2) except that the oblate Earth radius beneath the (-in or -out) horizon crossing, R,, and the true Earth to spacecraft distance, En, are used in place of nominal values, and the horizon height is corrected by values from the ERMU table. The geometry defining p is shown in Fig. 1. The location of the tangent point is obtained by finding the intersection of the nadir pointing cone with a half-cone angle of p and the scan cone. This ossible scan vectors, S . fter selecting the correct scan vecto h center to this point is simply the sum of the spacecraft position vector, and the scan vector adjuste to the correct length: R n = E n + S n where The latitude, A * of the tangent point can then be computed as: Figure 1. Geometry of Horizon Sensor Tangent Height Strictly speaking, because of oblateness, the Earth radius is a function of the latitude of the horizon crossing, but an iterative algorithm compensating for oblateness changes latitude from the values calculated using Eq. (1 1) only by a negligible amount.

RESULTS AND DISCUSSION For a period of approximately 1 year (from Summer 1996 to Summer 1997) TOMS-EP horizon sensor measurements were used to determine weekly horizon height variation tables.

For each week's entries at least one orbit of data per day for at least 4 days in the week were processed. In each batch of data, observations at 32-second intervals from each of the Earth sensors were processed. A total of about 190 observations per orbit or at least 760 observations each week were used to determine Earth-in and Earth-out horizon heights for each sensor using Eqs (6) and ( 7 ) . Horizon heights were collected in bins of 2 deg phase angle and averaged for each sensor and each horizon crossing (-in or -out).

The latitude corresponding to the Earth-in and Earth-out horizon crossing for the center of each phase angle bin was computed. For each week of a, horizon heights falling in the same 2-deg latitud bin were averaged even if they arose from different horizon crossings or different sensors. lot showing these mean hori ights as a function of season latitude is presen Individual horizon heights range from a minimum of 24.6 km to a maximum of 56.8 km.

The horizon height variations with latitude and season clearly show both regular, climatic trends as well as more chaotic, stochastic effects.

n m Q)

z

Q)

=

CI

-

c June sept. Dec. March Figure 2. Measured Average E a r t h Sensor Horizon Heights (km) From TOMS-EP Data The climatic effects for the study year show expected features as well as some surprising ones: Near the equator, the horizon height varied least throughout the year.

In each hemisphere, there was a regular change of horizon height with season.

Near the North Pole, the minimum horizon heights occurred in the northern autumn.

Near the South Pole, the minimum horizon heights occurred in late in the northern spring.

e At about 50-deg south latitude, a local minimum horizon height was found in all seasons.

e stochastic effects are best demonstrated nce of numerous regions of a ion and 10 or less eg of latitude extent with average horizon heig nt from the surro To better describe the variation of horizon heights with time and position, we values were converted to tangent angles, p in . (8) and Fig. 1, and the differences in these values were computed as a function of latitude for different weeks. These plots of Ap against latitude for adjacent weeks, for a separation of half a year, and for a separation of approximately 1 year are presented in Fig. 3. Again, each set of weekly mean values were obtained by computed average heights (eqs. 6 and 7) using data collected during several arbitrary orbits on each of at least 4 days in each week and converting the average heights to average scanner rotation angles. The plots in Fig. 3. show differences between average scanner angles for typical weeks separated by 1 week, 26 weeks, and 1 year.

Figure 3. Tangent Angle Differences for 1 W e e k , 26 Weeks, and 1 Year Time Difference In these plots, several important features of the mean tangent angles are clear: e Values separated by either 1 week or 1 year have smaller differences than those separated by half a year.

e Values separated by half a year show the largest differences indicating clear seasonal changes.

e Values near the equator are similar, regardless of time difference.

Values separated by 1 week are similar but large differences (= 0.2 deg) occur in some regions.

In order to evaluate the effect of the horizon radiance variance on attitude, T data for early in 1998 were examined. During this period, the spacecraft altitude had been increased to 740 km. At this altitude, horizon height variations subtend a smaller angle at the spacecraft, so the resulting measurement errors are somewhat smaller that they were earlier in the mission.

For each of 3 weeks (Feb 1-7, Feb 8-14, and March 1-7, 1998), data were processed using CFADS for 3 orbits per day on 4 or 5 days in each week. The pitch and roll residuals of each HS were accumulated and averaged in bins of 5 deg of phase angle. Data from any single day contributed an average of 8 values to each bin average. These average residuals for one day (Feb 4, 1998) are plotted against orbit phase in Fig. 4. Error bars for each bin, corresponding to fla deviations from the mean, are also included in this figure.

I 1 I 1

I t : T : ' ;HS2(

........... .. ........ c ... ....... , ...

............................ - 0 . 5 f

1 c c

1 Feb 4, 3998 : : I

....................... ... -, :I ;

i 1

0 60 120 188 240 300 0 60 120 180 240 300 36( Orbit phase (deg) Orbit phase (deg) Figure 4. Mean TOMS-EP Roll and pitch E r r o r s for Feb 4,1998 Several interesting features are apparent in these plots: Single sensor roll and pitch errors can have residuals that are as large as 0.5 deg.

Residuals show systematic variations with orbit phase.

The standard deviations of the residuals in any orbit phase bin may differ by large amounts from those in other bins.

41 1 Noisy bins (those with large standard deviations) seem to be clustered near each other, as are bins with small standard deviations.

The noisiest bins are clustered in the same general region of the orbit (phase near 200 deg) €or both of the sensors.

These observations can be explained by the existence of regions of unusual horizon height. At phase angles in an orbit where one or the other of the horizon crossings €or a HS passes through such a region large residuals are produced. If the region is localized in longitude, the horizon crossing on subsequent orbits will not be affected, producing large standard variations of the mean. At other phase angles, where no anomalous horizon height regions are observed, the standard deviations will be smaller.

Were this explanation true, the plots in Fig. 4 would change in an unusual manner as more data from other days is added. In cases with normal distributions of measurements, adding data should systematically reduce the standard deviations. Assuming n o d distribution, adding 4 additional days of data to the one day shown in Fig. 4 should decrease the standard deviations to less than half their previous values. The actual changes are shown in Fig. 5. In the plots shown in this figure, each bin represents an average of about 40 observations.

. . ..- * ..-. . . ... . . .

I -~

: :w: : -1

0 60 120 180 240 300 0 60 120 180 240 300 36( Orbit phase (deg) Orbit phase (deg) Figure 5. Mean TOMS-EP Roll and Pitch Errors for Feb. 1-7,1998 It is clear from this figure that adding more data increases the standard deviations at most orbit phases. This result is consistent with the growth and decay of anomaly regions within the week period covered by each plot. As a new anomaly region grows it changes the residuals uring the portion of some orbits. This increases the standard viations and shifts the mean.

The same effect occurs as the region disappears.

Is0 for most values of phase angle, adding data from different days reduces the s in standard deviations among adjacent bins. Even w regions (e.g. near 200-deg phase angle in the week of Feb. 1) consistently large, while for others (e.g. between 240 and 300 deg phase) the standard deviations are consistently low.

Figures 6 and 7 show similar averages obtained from data in the following week (Feb. 8- 14, 1998) and three weeks after that (March 1-7, 1998), respectively.

In the plots representing HS1 residuals in the week of Feb. 8-15, there are exceptionally large values of the mean residual as well as standard deviations near 200 deg phase angles. Most of these large values can be traced to short periods (1-2 minutes in duration) during which HS1 measured anomalous values. These roll and pitch values were more than 1 deg from nominal, lasted for several measurement periods, and occurred on at least 2 days. They were not observed in every orbit and, although their approximate position remained the same, moved by several deg of orbit phase from day to day.

--..... ...

0 60 120 180 240 300 0 60 120 180 240 300 36( Orbit phase (deg) Orbit phase (deg) Figure 6. Mean TOMS-EP Roll and Pitch Errors for Feb. 8-14,1998 There are many possible causes for the anomalous behavior near 200 deg phase angle. It is well known that the presence of the Sun or Moon near the position where a HS detects the horizon can cause large errors in attitude measurements." Although Sun and Moon interference and roll errors of this size, examination of stan limited to a 0.1-0.2 deg. Other possible causes for this anomaly, including Sun glint, shading, etc. were not investigated.

These anomalies probably are not an artifact of telemetry processing because the moderately large standard deviations in HS2 for the same period and phase angle are probably due to a response to HSI anomalies by the OBC attitude control system. The facts that they recur at approximately the same phase angle on adjacent days, and that they contain more than a single observation each, makes it likely that these anomalies are caused by a physical phenomenon.

Except for the anomalous portions of the data in the week of Feb. 8-14, the mean values of horizon sensor residuals are quite similar from week to week. Differences of about 0.1 deg can be seen (for example near 125 to 175 deg of phase angle in the plot of pitch residuals). The size of these differences is consistent with the results shown in Fig. 3.

- - a ...... - - - . - * -

0 60 120 180 240 300 0 60 120 180 240 300 3 6 C Orbit phase (deg) Orbit phase (deg) Figure 7. Mean TOMS-EP RoU and Pitch Errors for March 1-7,1998 The standard deviations are consistently larger in some portions of the orbit and smaller in others. The mean of the standard deviations over all of the phase angle bins, both of the sensors, and both attitude directions is about 0.15 deg. This value reveals more about the fraction of the time during which a HS will have relatively large errors than it does about either the random noise the sensor will experience or the reliability of a horizon height model.

he relative size of the residuals throughout any rbit also shows a consistent pattern.

uch of this pattern can be exp~ained by the positions o S horizon crossing points as a nction of orbit phase. The latitudes a nominal orbit and nominal orizon heights) are shown in Fig. 8 When the -in and -out horizon c th HSs fall at nearly the same latitude (e.g.

near orbit phases of 90- and 27O-deg), the standard deviations are smaller than in regions where the horizon crossings fall at different latitudes. The largest standard deviations in all of the weeks occur between orbit phases of 180 and 240 deg. where one of the horizon crossings in each sensor is approaching the summer pole and the other remains at much lower latitudes.

I Latitudes of Earth Sensor Horizon Crossing Points I

Figure 8. Horizon Crossing Latitudes v s . Phase Angles for a Nominal TOMS-EP Orbit CONCLUSIONS The data presented here as well as that cited from earlier studies is consistent with a horizon height model that includes climatic and stochastic effects of approximately the same importance. The climatic effects are, by definition, predictable and can therefore be reduced through the proper use of models. The stochastic effects can not be predicted and their effects can not therefore be completely removed.

Stochastic effects are localized in both time and space and introduce colored noise into the horizon radiance. Treating these effects as uncorrelated noise with normal distribution roduces sensor err0 models that, although convenient to use, are not optimal since they do not curately reflect the system’s statistics.

en evaluating the accuracy of a sensor in the design phase of a mission, that sensor’s measurement error statistics should be used explicitly to e level of attitude error will occur.

Because the horizon height variation contains a significant contribution from sequentially correlated errors, the assumption of white noise and use of the corresponding simple standard deviation as a measure of sensor error yields a misleading description of attitude errors.

Given the observed error distribution on TOMS-EP, it is probable that in any orbit there will be regions in which the sensor error will be much higher than the standard deviation obtained from statistics on measurements over many orbits, many days, and even many years.

Using more observations to determine the standard deviation may refine this value but will not help to determine the fraction of each orbit that the sensor (and the spacecraft attitude) can be expected to have large errors.

The effect of correlated sensor errors cannot easily be removed by the typical Kalman f i l t e r s used for spacecraft attitude determination. These fdters give optimal results assuming uncorrelated, gaussiandistributed noise. Although the inherent sensor noise may indeed be random and white, the sensor and attitude residuals contain contributions from the colored errors in the Earth horizon height model. In a similar case, that of correlated errors in magnetic field models, a filter that explicitly accounts for correlated noise has been developed and has improved attitude determination markedly.” Several approaches are currently being pursued to evaluate and mitigate horizon radiance errors on Earth sensor derived attitudes. These include: development and testing of a correlated noise filter evaluation of the attitude accuracy improvements that can be expected in Kalman filters and through use of a system similar to the ERMU evaluation of the attitude accuracy improvements that can be expected in batch-least squares methods using a system similar to the ERMU evaluation of the attitude accuracy improvements that can be expected in Kalman filters by ignoring HS measurements with high latitude horizon crossing.

Many factors not discussed here affect the height at which a horizon sensor detects the E a r t h . These include the trigger logic and rotation rate for scanning HSs as well as the size and shape of the instantaneous FOV and the sensor mounting geometry. Differences from nominal triggering height at a particular location on the Earth’s surface should be more nearly equivalent for different missions than are the heights themselves.

If a system like the ERMU proves to be successful in reducing attitude errors to a level lower than attainable with climate models, input to horizon height correction tables from many spacecraft might be desirable. In this case, weekly horizon radiance correction tables might be generated and used by each contributing spacecraft to correct its horizon height model and improve its attitude accuracy.

This work was supported by the National Aeronautics and Space Administration (NASA)/ Goddard Space Flight Center (GSFC) under contracts NAS 5-31000 and GS-35F- 43816, task order No. 5-03365-Y.

RENCE 1.

M. C. Phenneger, S . P. Singhal, T. H. Lee, and T. H. Stengle, “Infrared Horizon Sensor Modeling for Attitude Determination and Control: Analysis and Mission Experience,” NASA, TM 86181, March 1985.

2. W. Nutt and M. C. Phenneger, “Horizon Radiance Modeling Utility System Description and User’s Guide,” Computer Sciences Corporation, CSC/SD-78/6032, March 1978.

3. E. J. Burgess, “Earth Radiation Budget Satellite (ERBS) Horizon Radiance Modeling Utility (HRMU) User’s Guide and System Description,” Computer Sciences Corporation, CSC/SD-84/6007, June 1984.

4. R. Shendock, “DE-B Horizon Radiance Modeling Utility (HRpcllU) System Description and User’s Guide,” Computer Sciences Corporation, CSC/SD-80/6096,1980.

5. S . Fritz and S. D. Soules, “Planetary Variations of Stratospheric Temperatures,” Monthly Weather Reviews, Vol. 110, no. 7, July 1972.

6. E. Harvie, 0. Filla, and D. Baker, “In-Flight Measurement of the National Oceanic and Atmospheric Administration (NOM)-1 0 Static Earth Sensor Error,” AAUAZAA Spaceflight Mechanics Meeting, Pasadena, CA. AAS 93-101, Feb. 1993.

7. M. C. Phenneger, W. C. Bryant, J. Baucom, and K. Woodham, “Evaluating the Goes-8 & -9 Earth W Horizon Sensor Error Response Using the Goes-9 Sounder IR Channel 1 thru 4 Data,” Proc. SPIE, Vol. 2812, Oct. 1996.

8. S . Shulman, et al., “Total Ozone Mapping Spectrometer-Earth Probe (TOMS-EP), Flight Dynamics Support System (FDSS), Functional Specifications,” (revised version) Computer Sciences Corporation, CSC/TR-92/6008R 1 UDO, Nov. 1993.

9. G. Wahba, “A Least-Squares Estimate of Spacecraft Attitude,” S Z A M Rev. Vol. 7, No. 3, July 1965.

10. M. C. Phenneger, S. P. Singhal, T. H. Lee, and T. H. Stengle, op. cit., Sec. 4.

11. ibid.

12. J. Sedlak, “Improved Spacecraft Attitude Filter Using a Sequentially Correlated Magnetometer Noise Model, ” Proceedings of the I 6’h Digital Avionics Subsystems Conference, Irvine CA, October 1997.

I.

alla auc he r-Lag racie The accurate orientation of a spacecraft with respect to the Earth is generally achieved through a passive infrared electro-optical sensor as a part of attitude control subsystem : an Earth sensor. The spatial and temporal Earth radiance non-uniformities make up the main limitation of LEO Earth sensor accuracy. Then recent investigations have been performed with CNES (the French National Space Agency) collaboration on a better understanding and on a characterization of E a r t h radiance fluctuations, in order to update compensation laws used to improve the satellite pointing accuracy.

In fact several analyses have been engaged on in-flight Earth sensor radiance telemetry data from French LEO satellites (SPOT 1,2, 3) oriented by Earth sensors and rate gyro systems.

From these in-flight data a first step has consisted of a radiance variations modelling versus Earth coordinates and seasons, and has led to separate deterministic variations from random fluctuations and attitude measurements. However without absolute measurement reference the software modelling accuracy is limited by the gyro one. Now, thanks to the presence of both accurate SED 12 star-trackers and scanning infrared Earth sensors STD 16 on the French satellite HELIOS 1, it is possible to compare the Earth sensor in-flight output data to almost absolute angular references. It is then possible, by analysing these data over a long period of time, to deduce the Earth sensor error and how this error changes versus the satellite motion and the season.

The purpose of this paper is to present the analyses performed with CNES specialists fiom the STD16 sensor transitions and radiance telemetries on board HELIOS lin order to improve the knowledge of the infrared Earth radiance mapping and consequently to significantly decrease the radiance error terms of infrared earth sensors. The opportunity to work on one and a half year period telemetry data has allowed at one hand to optimize the deterministic radiance fluctuations modelling and at the other hand to update our software model of Earth sensor radiance errors. After a brief recall of the sensor operating principle and telemetry characteristics, radiance variations according to spatial and temporal references are analyzed on about 8000 observations.

Existing seasonal compensations laws have been updated from Earth radiance errors modelling and results in terms of performances are discussed. So with these new compensation laws the radiance error can be largely decreased by a factor 50 up to 100 % (mainly depending on the season) compared to the previous laws by which the error was only reduced by a maximum factor of 50%. This new model should improve the prediction of the infrared sensor radiance error and could be used by the AOCS of LEO satellites to compensate the in-flight radiance error whatever the altitude and the orbit parameters.

* SociCtC Anonyme d’Etudes et RCalisations NuclCaires (Sodem), 20 Avenue Descartes, F-9945 1 Limeil-Brevannes Cedex, France.

’ Centre National d’Etudes Spatiales (CNES), 18 avenue Edouard Belin, F- 3 I055 Toulouse Cedex, France.

Attitude determination from InfraRed Earth Sensors (I radiometric Earth observations : the useful information delivered by the sensor is the measurement of the thermal discontinuity between the Earth and the Space radiances.

6 (fig.1) is an Earth rizon crossing sensor designed for ‘Low ) missions. It delive transitions between Earth and Space from pitch attitude angles are determined. To achieve good accuracy performances Earth sensors generally ne to operate in the infrared spectral band of 14-16.5 pm corresponding to the absorption band which provides the more uniform and stable Earth radiance distribution than other bands. However they present non negligible seasonal and spatial variations creating transition shifts.

Today, to withstand the on-market accuracy requirements, it has become necessary to compensate attitude errors due to the non uniformity of Earth radiance.

In fact a long term study based on telemetry data f r o m French remote sensing satellites has been engaged with the CNES (French Space Agency) for few years (Ref.l) to progress in the knowledge of Earth radiance fluctuations in the 15 pm spectral band.

The analysis of telemetry data f r o m SPOT satellites, oriented by SODERN Earth sensors (Ref.2) and rate gyro systems, has allowed to separate the deterministic radiance variations from the random ones for finally deducing a simple E a r t h mapping model. However without absolute measurement reference the accuracy of this s o h a r e modelling was limited by the gyro’s one.

Then this paper, for the first time, presents analysis results from Earth sensor outputs compared to absolute angular references given by star-tracker systems (Ref.3). This new data bank from HELIOS 1 satellite has allowed, with CNES collaboration, on the one hand to update existing infkared Earth radiance model and on the other hand to develop compensation laws of LEO Earth sensors errors due to E a r t h radiance fluctuations.

This study is aimed at the improvement of LEO platforms pointing accuracy.

Figure 1 : The STD16 Earth sensor I axis stabilized spacecrafts The STD16 infiared Earth sensor is designed orbiting at low altitudes. A full desc-ription of STD given in reference (Ref.2).

riefly, STD16 sensor is an Earth infrared Horizon scanner. Its function is to provide the angular position, with respect to some reference, of transitions which occur on the scanned traces between Space and Earth : the knowledge of these transitions enables the AOCS to compute the angular position of the satellite, with respect to the Earth, in terms of pitch and roll.

The main feature of STD16 is that two opposite scan cones are generated by scanning a single bolometer image thanks to a 1 rps rotating head and two fixed mirrors then inducing 4 transitions between Earth and Space (fig.2). Bolometer signal is then fed through an electronic unit which processes it and drives the scanning mechanism. The processing circuit consists in a band pass filter coupled to sign discriminators and proportional threshold detectors. The two Space to E a r t h transition signals are stored in analog memories. The peak values of the four transitions are directly proportional to the magnitude of the radiance peak values.

Figure 2 : Sensor operating principle DATA CHARACTERISTICS Telemetry data description HELIOS 1 satellite, in a circular sun synchronous high inclination (98"), orbits at an altitude of 700 km which results in an orbit period of approximately 90 minutes.

Therefore, the satellite orbits around the Earth about 14 times each day. It is oriented thanks to 5 SODERN equipments : 3 SED12 star-trackers and 2 STDl6 Earth sensors.

42 1 LIOS 1 has been launc analysed over 1 provided by the the L,, L2, L,, L, Earth radiance telemetries measured at each transition between Earth and Space, according to a parameter, (a = satellite position on orbit, a = 0 at the ascending node) (fig.3).

I 3.5

I 1 i ' J i n 2 . 5

- 2

SC?

O N

5 1 . 5

g, I

E" 1 northern hemisphere I southern hemisphere r I I

5 0 . 5

W l I

, I

0 60 120 180 240 300 360 Satellite position on orbit (degrees) Figure 3 : Earth radiance telemetry data (L,) : August 1995.

0 the Earth radiance error, deduced from attitude telemetry data, according to a parameter, (fig.4).

I i 0 60 120 180 240 300 360 i Satellite position on orbit (degrees) I I Figure 4 : STD16 Pitch and Roll errors due to Earth radiance fluctuations (August 95).

The Earth radiance error is defined by the difference between attitude angles directly deduced from Earth sensor outputs and attitude reference angles delivered by the star-tracker system : where : constants = mechanical bias and mission characteristics constants.

With a rate of one day per fifteen days, with 10 to 14 orbits per day, each orbit was containing about from 20 to 50 measurement points (sensor telemetries sampled at 1/80 Hz). The day of the fortnight, for which telemetries are available, has been assumed as pretty representative for the all over fifteen days. However, for confidentiality reasons, March, April and September telemetry data have not been available.

With this data bank, study investigations about Earth radiance characterization have been engaged to improve STD 16 accuracy.

Data accuracy e Earth radiance telemetries : The relationship between Earth radiance in the 15 pm band and the STDl6 telemetry data was calibrated with on-ground Earth simulator. The accuracy of measured values are estimated to be about z k 4 % taking into account thermal effects, quantization noise and memory variations, e Earth radiance errors : of the very high measurement accuracy of the star-tracker (a few 10” Because degrees), we can assume in this way that we have a very representative Earth radiance error.

EARTH RADIANCE FLUCTUATIONS CHARACTERIZATION As it has been previously observed with SPOT 1,2,3 data bank (Ref.l), Earth radiance fluctuations present a deterministicpart and a random one.

Deterministic fluctuations correspond to seasonal variations and random fluctuations are revealed by irregular and unexpected variations.

The seasonal variations correspond to two deterministic phenomena :

- a seasonal radiance gradient on Earth surface, with an important latitude

dependence, - a fluctuation of the infrared horizon depth.

These deterministic fluctuations have been modelled by analytical hctions giving Earth radiance variations versus latitude and seasonal parameters.

ith the aim of estimating attitude errors due to Earth radiance fluctuations before spacecraft launch rather than trying to compensate them in flight, a sof3ware errors model has been developed by SODERN.

A software simulation model has been developed to simulate the Earth sensor operation in front of a thermal scene. This software delivers the Earth sensor output data for all possible input parameters of the modelled radiance configurations. It allows then to estimate STD16 attitude errors due to E a r t h radiance fluctuations according to mission characteristics.

The modelled scene, deduced from previous analyses (Ref.l), represents an infrared Earth radiance mapping according to latitude and seasonal parameters.

In fact seasonal radiance gradients on Earth surface are expressed by monthly function versus latitude and time parameters and the fluctuations of the infrared horizon depth are represented by analytical hctions deduced from NASA studies according to altitude, latitude and season parameters (Ref.5 and Ref.6).

This software developed from STD16 radihnce telemetry data (L,, L,, L,, L4) has been validated and improved using HELIOS 1 very accurate and representative telemetry data.

Val ida ti on The software has been validated in two steps. The comparison between software model and telemetry data has concerned, at first, the thermal scene observed by the sensor ( E a r t h radiance fluctuations) and secondly the Earth sensor errors (errors induced by the Earth radiance).

For available periods, the mean fluctuations of Earth radiance and standard deviations have been compared to the modeled functions. It appears that for the whole year analytical functions give a first order quite good estimation of Earth radiance fluctuations. However the comparison gave also prominence to differences particularly in southern hemisphere for winter periods and less differences in the both hemisphere for spring summer and autumn periods (fig.5).

For attitude errors induced by deterministic radiance fluctuations, the main conclusions of comparison are on the one hand a rather good synchronization of phases according to a parameter but a larger error amplitude from telemetry data and on the other hand a pretty good estimation for pitch angle (fig. 6).

August 1995 : I 2 5 I ‘ A I -90 -60 -30 0 30 60 90 Latitude (degrees) * December 1996 : A

southern herrisphere I northern hemtiphere

1 -90 -60 -30 0 30 60 90

Latitude (degrees) !

May i996 :

I

RFT southern hemisphere I northem hemisphere I 1.5 - I I

-90 -60 - -30 0 30 60 90 !

!

Latitude (degrees) Figure 5 : E a r t h radiance profiles versus latitude parameter.

0.15 A rn 0.1 9) s ?

0.05 9) 2 -0.05

.-

"CI E = -0.1

.-

n -0.15 Position on orbit (degrees) 0 . 0 4 . I

- telemtry data

Fl

- 0 . 0 5 J Position on orbit (degrees) Figure 6 : Radiance errors profdes versus satellite position on orbit (end of December 1996) Updating o f software model The software model improvement has concerned the Earth radiance mapping according to HELIOS available telemetry data.

The approximate and simple formulas for radiance variations, versus latitude and time, deduced from SPOT telemetry data analysis have been modified in the way of a better f i t t i n g of radiance variations including a seasonal dependence.

Spring :

for latitude < 20" S : L = 2.4 + 1.28 * sin (2 z t - to)) (latitude + 20)

for 20" S < latitude < 35" N : L = 2.4 in [W/mZ.sr.p]

( 2 E;- to)) ( latitude . . 35)

for latitude > 35" N : L = 2.4 + 1.0 * sin

0 Summer:

(2 z:-to)) (latitude + 20)

for latitude < 20" S : L = 2.4 + 1.28 * sin in [W / m2.sr.p]

for 20" S < latitude < 20" N : L = 2.4 in [W/mZ.sr.,um]

(2 : : - t o ) ) (latitude - 20)

for latitude > 20" N : L = 2.4 + 0.83 * sin

in [W / mZ.sr.p] 0 Autumn: (2 E:- to)) (latitude + 35) in [W/m2.sr.,um]

for latitude < 35" S : L = 2.4 + 1.32 * sin

for 35" S < latitude < 20" N : L = 2.4 in [W/m2.sr.p]

(2 E:- to)) ( latitude - 20)

for latitude > 20" N : L = 2.4 + 0.83 * sin

in [W/ mz.sr.p] 0 Winter: (2 z : - to)) (latitude + 35)

for latitude < 35" S : L = 2.4 + 1.2 * sin

in [W / m2.sr.pm] for 35" S < latitude < 20" N : L = 2.4 in [W/m2.sr.p]

(2 %(:-to)) (latitude - 20) in [W / m2.sr.,um]

for latitude > 20" N : L = 2.4 + 0.85 * sin

With : t-t, : day of the year dated from April 10' T = 365.25 days latitude : in degrees.

N : northern hemisphere S : southern hemisphere Performances Although more complex functions could have resulted in a better approximation, these simple laws give satisfactory results : the residual errors compared to the measured values are always less than the standard deviation.

een evaluated with respect to the co cations have influence leading to 'a significant improvement in the attitude error estimation. Maximum amplitude error differences between software results and telemetry data are given hereafter (illustration on table1 and figure 8): Table 1 ITUDE ERRORS DIFFERENCES BETWEEN UPDATED SOFTWARE AND TELEMETRY DATA Spring Summer Autumn Winter Attitude angles (degrees) (degrees) (degrees) (degrees) Pitch 0.015" 0.025" 0.010" 0.020" Roll 0.0 10" 0.020" 0.020" 0.015' e Pitch 1.ooE-01 3 1 n u) Q) 5.00E-02 Q) s 0.00E+00 6.00E-02

. - ,- I

.c

.---.- - ELTsimc

u Y -1.00E-01 J L1

- ELTsim

satellite position on orbit (degrees) ._._.._ ELl2495 Roll position on orbit (degrees) satellite Figure 8 : Radiance errors profiles (beginning of December 1996) telemetry data (EL2495), software model (ELsim), software model updated (ELsimc).

These results, given for a maximum pitch radiance error of about 0 . 0 8 0 ' and a imum roll radiance error of about 0.050°, show that a better mode radiance gradie t mapping allows a better esti ion of STDl6 errors but that some residual errors are still remaining. This is mai to the Earth horizon depth rough model and to the fact that longitude effects on Earth radiance are not taken into account. In fact HELIOS telemetry analysis showed a ly longitude influence on attitude errors (particularly for roll errors) ; works are still carried out on this point.

After updating Earth radiance mapping model, the SODERN software model allows to evaluate 75% of pitch radiance errors and 40% of roll radiance errors due to deterministic Earth radiance fluctuations. These performances could be yet improved on the one hand with more telemetry data (yearly repeatability) and on the other hand with the updating of Earth horizon depth model and an analysis of longitude influence on seasonal Earth radiance fluctuations. Therefore, for LEO missions, SODERN software model allows a first order pretty good estimation of STD16 in-flight errors, according to geographical and time parameters, thanks to a simple and representative Earth radiance mapping model. Then consequently this model allows to determine compensation laws to improve the Earth sensor accuracy budget.

CORRECTION LAWS OF ATTITUDE ERRORS DEDUCED FROM TELEMETRY DATA With the aim of improving significantly STD16 accuracy for LEO platforms orbiting at 800 h, compensation laws of radiance errors have been directly deduced from HELIOS telemetry data. This study has been performed in collaboration with the CNES.

Compensation laws of attitude radiance errors are analytical functions depending on a parameter (satellite position on orbit) and on seasonal coefficients.

Correction laws evaluation method The first step was to deduce deterministic error laws from a large dispersion of Earth radiance telemetry data. The best fittings of radiance error variations in time have been obtained by using analytical and statistical methods, such as the least square algorithms method according to polynomial sinus and cosine formulas. This method allows to set free from sensor noise and telemetry accuracy. Nevertheless, for a few periods, telemetry data were not available and error fbnctions have then been extrapolated for these periods.

With this data processing, random errors have also been reducing by an averaging process and statistical analysis.

The deduced laws are then representative of seasonal radiance variations.

’ - (Apl . cos a + A,, . cos 3a + B, . sin 2a + B, sin 4a)

with : R, P : corrected roll and pitch values, R’, P’ : measured roll and pitch values, a : angle from the spacecraft ascending node, A,, ,B, ,A, ,BE : seasonal coefficients (to be changed every 15 days).

Validation on SP The compensation laws for STD16 system are deduced from very representative Earth radiance errors data, thanks to star-tracker systems giving absolute references in term of roll and pitch angles. These compensation laws have been validated by CNES on SPOT telemetry measurements.

Telemetry data collected over 5 years from satellites SPOT 1, 2, 3 represent a large data bank which has been used to deduce compensation laws of attitude radiance errors. In fact with the knowledge of ((measurement / radiance errors )) transfer hction, attitude radiance errors have been reconstituted orbit by orbit.

HELIOS 1 compensation laws have then been statistically validated by applying them on the SPOT 1, 2, 3 radiance error profiles giving very satisfying results. In fact, in comparison with previous compensation laws (deduced from SPOT data bank telemetries) the HELIOS compensation laws improve with an average of 50% the on- board compensation level. The performance in terms of platform pointing accuracy is confidential but HELIOS compensation laws will be used on SPOT 4 satellite.

Performances With HELIOS compensation laws the roll and pitch radiance errors can be largely decreased by a factor 50 up to 100 % (mainly depending on the season) compared to the previous laws by which errors were only reduced by a maximum factor of 50%.

Then it has allowed to determine the following realistic Earth radiance residual error 2): budget (see table Table 2 RESIDUAL EARTH RADIANCE ERRORS Attitude angles Comuensation levels Residual error after compensation worst to best min to max

(%I (degrees)

Roll 50 to 100 0.003 to 0.035 Pitch 70 to 100 0.002 to 0.035 The m ~ ~ m ~ value of residual error takes into account a 0.015” term due to the correction ~ e ~ ~ o d i c i t y (eve Figures 9 and 10 allow to compare roll and pitch radiance errors deduced from telemetry data (‘telemetries’ curves) with radiance errors compensated by HELIOS updated laws (‘new law’ curves).

‘Previous law’ curves correspond to the compensation laws deduced from SPOT 1,2, 3 data bank : so the compensation level did not exceed 50% for pitch and 20% for roll with sometimes some error increase.

It shows that roll compensation is not as good as pitch compensation, but absolute error is smallest. These uncompensated residual errors are mainly due to longitcide influence and software extrapolation accuracy.

I R O U . FEBRUIRY I R O U : YAY 4.W-02 I I 3.-I ROU: DECEMBER o.mE.02 3.00642 2 . 6 . 0 2 1 .we2 0 . 6 4 -1.OOE.02 -2.WE.02 3.mE-02 -4.ooE-02 Figure 9 : Roll radiance error profiles 4 3 1 Y

0.05 , 1

PITCH: MAY

1 4 . 0 2

-0.06 Figure 10 : Pitch radiance error profiles A new data bank of very accurate telemetry data from allowed to progress in the knowledge of Earth radiance.

In fact for the first time Earth sensor telemetry data have been compared to an absolute attitude reference given by a star-trackers system on the same spacecraft. The opportunity to work on 1.5 years period telemetry data has allowed on the one hand to update the SODERN software model of Earth radiance mapping and on the other hand to improve sensor accuracy by computing new errors compensation laws. The estimation level of SODERN soRware to evaluate attitude errors due to radiance fluctuations is 2 times better and is now used for in-flight errors estimation and compensation. And thanks to new compensation laws the correction percentage of the total deterministic Earth radiance error varies from 50 to 100 % with a residual error after compensation not exceeding 0 . 0 3 5 ’ .

From this data bank, the longitude influence on seasonal radiance errors and the random radiance fluctuations could be analysed to still improve the accuracy of LEO platforms.

REFERENCES 1. 0. Brunel, F. Haignere, JP. Krebs, JG. Peltier and M. Burello, A 15 pm Infrared Earth mapping through the STD12 sensor telemetries on board SPOT satellites, Ref. AAS 93-322.

2. 0. Brunel, J. Jamet, JP. Krebs, A versatile Earth Sensor for LEO Satellites, 20’ ISTS Japan, Ref.96-c-72 p. 1996.

3. 0. Brunel, P. Dureux, Y. Kocher, JP. Krebs, Les capteurs d’attitude des plates-formes Envisat et Helios, International Symposium Optronics and Defense, Nato Confidential, Montigny le Bretonneux, France, 3-5 December 1996.

4.

P. Faucher, Calcul de compensations d’erreurs de luminance pour SPOT 4, CNES report 1997.

5 .

Whitman, Ruth I., Thomas B. Mc Kee, and Richard E. Davis, Infiared Horizon Profiles for Winter Conditions from Project scanner, NASA TN D-49.5, Dec. 1968.

6. Thomas, John R., Ernis E. Jones, Rober O’B. Carpenter, and George Ohring, The Analysis of 15pm Infiared Horizon Radiance Variations Over a Range of Meteorological, Geographical, and Seasonal conditions, NASA CR-725, April 1967.

ALYSIS OF EARTH ALBEDO FFECT ON SUN SENS

~ € A S U R E ~ ~ N ~ S BASED ON THEORETICAL MODEL

AND MISSION EXPERIENCE* Dan Brasoveand and Joseph Sedlakt Analysis of flight data from previous missions indicates that anomalous Sun sensor readings could be caused by Earth albedo interference. A previous Sun sensor study presented a detailed mathematical model of this effect. The model can be used to study the effect of both diffusive and specular reflections and to improve Sun angle determination based on perturbed Sun sensor measurements, satellite position, and an approximate knowledge of attitude. The model predicts that diffuse re- flected light can cause errors of up to 10 degrees in Coarse Sun Sensor (CSS) measurements and 5 to 10 arc sec in Fine Sun Sensor (FSS) measurements, depending on spacecraft orbit and attitude. The accuracy of these sensors is affected as long as part of the illuminated Earth surface is present in the sensor field of view. Digital Sun Sensors (DSS) respond in a different manner to the Earth albedo interference.

Most of the time DSS measurements are not affected, but for brief periods of time the Earth albedo can cause errors which are a multiple of the sensor least significant bit and may exceed one degree.

This paper compares model predictions with Tropical Rainfall Measur- ing Mission (TRMM) CSS measurements in order to validate and refine the model. Methods of reducing and mitigating the impact of Earth albedo are discussed. The CSS sensor errors are roughly proportional to the Earth albedo coefficient. Photocells that are sensitive only to ultraviolet emissions would reduce the effective Earth albedo by up to a thousand times, virtually eliminating all errors caused by Earth albedo interference.

* This work was supported by the National Aeronautics and Space Administration (NASA) / Goddard Space Flight Center (GSFC), Greenbelt, MD, USA under Contract GS-35F-4381G, Task Order No. S- 03365-Y .

Computer Sciences Corporation (CSC), 10110 Aerospace Rd., Seabrook, MD, USA 20706.

All Sun sensors are designed based on the assumption that only one bright object, Le., the Sun, is present within the sensor field of view (FOV). Current Sun sensors cannot distinguish the effect of a single light source if several bright objects are simultaneously visible (discount- ing pattern recognition schemes that would not be practical for a relatively simple sensor).

Therefore, it is expected that Earth albedo interference will degrade the accuracy of Sun Sensors. An analysis of Solar Maximum Mission flight data' provided indications that Fine Sun Sensor (FSS) measurements were affected by Earth albedo, but not a definite proof. Many other questions were also left unanswered. Is the accuracy of all Sun Sensors reduced by the Earth albedo interference? Is it possible to model and accurately quantify the effect of illuminated Earth on Sun Sensor measurements? A subsequent study by one of the authors2 attempted to answer these questions by providing a detailed theoretical model of the Earth albedo effect on Coarse Sun Sensors (CSS), Digital Sun Sensors (DSS), and FSS. That study shows that all types of Sun sensors are adversely affected by the Earth 'albedo interference and predicts the accuracy degradation based on spacecraft, Earth and Sun positions, sensor boresight orientation, and sensor design data. For Coarse Sun Sensors (CSS), which are affected most, the theoretical model predicts measurement errors of up to 10 degrees.

The model has been tested before only for a few Sun, Earth, and spacecraft geometries.

The goals of this study are to thoroughly test the model using Tropical Rainfall Measuring Mission (TRMM) flight data and then to determine whether the model could be used to increase the accuracy of CSS measurements. Coarse Sun Sensors were chosen as a benchmark due to their simplicity (their behavior can be predicted without a detailed knowledge of proprietary sensor design data) and significant response to Earth albedo interference. These CSS charac- teristics facilitate the testing of the model and the establishing of a procedure for improving the accuracy of CSS measurements. After being validated by application to the CSS, the procedure can be modified to include other Sun sensor types.

MODELING THE EARTH ALBEDO EFFECT The theoretical model of Earth albedo effect on CSS is discussed briefly here. For more details and for modeling other Sun sensors see Reference 2. The Earth albedo effect has to be determined numerically for each individual CSS eye. The Earth surface is divided into a set of area elements using a map-like grid (see Figure 1). An Earth surface element increases the in- tensity of the electric current produced by a CSS eye whenever the element is located on the illuminated side of the Earth, within the sensor FOV, and-not beyond the spacecraft horizon.

Surface elements with these characteristics will be called active (see Figure 2).

Define the model frame of reference as the Earth centered frame with axes parallel to the CSS eye axes. Mathematically, the three conditions that define an active element (the f h element) can be expressed as follows: [surface element is illuminated)

(xj-xc)2+(q-~)2-(ij-zc)2tan2 <o (surface element within the FOV)

( 2 )

Xi.X, + 5 * (surface element is not beyond horizon)

+ Zi - Z c 2 RLnh

Figure 1. Surface Grid

T

/.-

Figure 2. Spacecraft, Earth, and Sun geometry where (Xc, Y,, 2,) is the CSS eye position, (Xsm, Usun, Zsm) is the Sun vector, and (Xj, 5, Z,) is the position of the$ element, all in the model frame. e CSS FOV is a Neglecting specular reflections, the light flux reflected by the j" active element, &by), is given by

&? = Ai$, Si cosuj (1)

where Ai and 4 are the albedo coefficient and surface area of the active element, respectively, Ips is the incident solar flux, and u is the angle between the n o d to the surface element and the Sun direction. This light is reflected within a solid angle of 2.n steradians (half-sphere). The perturbation flux due to thefh active element, @j, is given by In general, the electric current, I, produced by a bright object in the sensor FOV is I = K$cosa (3) where K is a sensor constant, $ , is the light flux detected by the solar sensor and a is the angle between the sensor boresight and the bright object direction. So, the perturbation flux produces a perturbation current, 4, given by lj = K&Pj C O S ~ ~ (4) where, q is the angle between the eye boresight and the area element direction. Therefore, the maximum current expected from the CSS eye photocell, IO, is I , =K@s (5) Based on Eq. (3), the current due to the Sun can be expressed as I , = K$, c o s a , (6) where as is the Sun angle @e., the angle of interest between the boresight and Sun direction).

Due to the Earth albedo effect, the total current provided by the CSS eye photocell, Itoral, is where n is the total number of active elements. Equations (4), (6), and (7) show that n

Qs COSCC, +zwj C O S ~ ~

'mal j=1 Assuming the only bright object present within the sensor FOV is the Sun, show that the Sun angle is given by a, = Unfortunately, Eq. (9) is always used to calculate the Sun angle, even when other bright objects are present within the field of view. Therefore, the angle a i calculated using Zmtd is not the true Sun angle but a perturbed value: The difference, 6as, between the true Sun angle and the CSS eye measurement thus is given by According to the above theoretical model, Eq. (10) should accurately predict the Earth- induced current whenever the albedo coefficient of each active element is known. However, the local albedo coefficients are strongly dependent on weather conditions over large regions (but not so large that a global average is sufficient). Even using advanced weather monitoring and prediction system, creating and maintaining a database of local albedo coefficients would be a formidable undertaking. The simplest approach is to replace the local albedo coefficient every- where with a constant value, a, and replace Eq. (10) with

ai = .cos( k)

(12) n s j COS U j C O S a !

j = l 2 A [ ( X j - X,)’ + ( Y j - Y,)’ + (Zi - zc,’]

Then the SURangle error is approximately n Sj cosuj cosaj 6a,=a,-a;

j = I 2 + X j - x c , ’ + ( q -q)? +(Z, -z,,*]

The average albedo coefficient of active areas and the average albedo coefficient of the entire Earth3 (Le., 0.30) can be quite different. Therefore, model predictions based on the average Earth albedo can be quite inaccurate. Nonetheless, the model was tested by comparing TRMM flight data with predictions based on Eq. (12) using the average Earth albedo coefficient. The following sections present more detail.

THE TROPICAL RAINFALL MEA~URING MISSION TRMM is one of a series of National Aeronautics and Space Administration (NASA) missions designed for the study of the Earth as a dynamical system. TRMM is a joint project between NASA and the National Space Development Agency (NASDA) of Japan. The TRMM instruments will determine the rate and total amount of rainfall occurring over the tropics and subtropics (from latitude 35 S to 35 N).

The TRMM spacecraft was launched on November 27, 1997 onboard a NASDA H - 1 1 launch vehicle. The nominal orbit is circular with an altitude of 350 lan and an inclination of 35 deg. The attitude is three-axis stabilized and Earth-pointing. Primary attitude sensors include a Barnes static Earth sensor, a Kearfott inertial reference unit, two Adcole digital Sun sensors, and two three-axis magnetometers. The nominal attitude determination accuracy is 0.2 deg per axis (36). The required control accuracy is 0 . 4 deg (30) with stability of 0.1 deglsec.

TRMM is equipped with eight CSS eyes.4 Two eyes (numbers 1 and 2) are located on solar panel 1; another two (numbers 5 and 6) are on solar panel 2. The boresight directions of eyes 1, 2,5, and 6 in the solar panel frame of reference are given in Table 1. The other eyes are located on the body and their boresights in the body frame are given in Table 2.

CSS eye number 1 2 5 6 0.773372 -0.2820903 -0.2796423 0.7697018 Boresight 0.3757137 -0.168597 -0.6468678 0.8598846 unit vector - 0 . 6 1 1 1227 -0.70851 05 -0.427081 66 - 0 . 5 1 6 1380 CSS eye number 3 4 7 8 . 0.7625562 -0.9601 3227 0.3369986 -0.9601 3227 Boresight 0.27954134 0.5507738 0.27954134 -0.8775453 unit vector -0.3383467 - 0 . 0 0 1 63112 - 0 . 3 4 1 0957 - 0 . 0 0 1 63112 measured and predicted CSS outp t currents, some account must be rmined for each tion errors. In particular, separate bi CSS eye. Systematic sensor errors can arise from a number of physical causes. A bias in the measured current shifts the cosine of all angles by the same amount; whereas, an error in the maximum current, IO, is a scale error that changes the cosines all by the same fraction. A mis- alignment will show up as a shift in the angles that depends on the location of the Sun in the field of view.

A simplification occurs for the CSS eyes that are mounted on the solar array panels. For these eyes, the Sun remains at nearly a constant angle throughout the sunlit part of the orbit. In this case, all the calibration parameters can be absorbed into a single bias; separate bias, scale factor, and misalignment parameters cannot be distinguished without observing the Sun over a range of angles in the CSS frame, This study analyzes these solar panel mounted eyes only.

For each eye, a bias was determined using only data from that part of the orbit where the predicted E a r t h albedo interference w a s less than 0.02 deg. These measurements have essen- tially no Earth interference so the difference between the measured and the expected CSS current can be attributed to sensor bias. Table 3 shows the biases obtained by averaging this difference over all points where the Earth interference is negligible. The fourth columri in Table 3 indicates the scatter of observations. This scatter contributes an angular uncertainty propor- tional to the bias standard deviation divided by sin 0. (Angular sensitivity is worst when obser- ving near the boresight.) At 8 = 45 deg, an error of 0.006 corresponds to an angular uncertainty of about 0.5 deg.

Table 3 shows bias values obtained using a data set consisting of one orbit from Feb. 22, 1998. Biases recalculated using another data set from M a r . 10,1998 differ by less than 0.02 lo.

Table 3. TRMM CSS Biases for Selected Solar Array Mounted Eyes EVALUATION OF ERRORS Tests with different grid sizes (see Figure 1) were performed. The total number of grid cells ranged from 7200 to 7,372,800 in these tests. The grid selected for the TRMM analyses had 115,200 cells. This discretization leads to numerical errors of no more than 0.00004 lo.

The measurement residual, I; which is the sensor error after compensating for bias and predicted Earth albedo effect, is r=ai-amvc where c & . , is the measured sensor output corrected only for bias, and a i is the predicted output corrected for albedo interference.

Uncompensated for Earth albedo, the sensor error is where O + is the reference Sun angle, uncorrected for bias or Earth interference. The residual r and error e are displayed in the plots presented below as sensor errors either corrected or uncorrected for Earth albedo interference; the measured angle in both cases is o & so both r and e are corrected for sensor bias.

NUMERICAL APPROACH Based on TRMM telemetry, Sun and spacecraft position vectors in the geocentric inertial reference frame (GCI) were calculated every other second using a set of MATLAFj 4 . 2 scripts.

These vectors were rotated from GCI into the model frame using another MATLAB script. A FORTRAN code was then used to determine the reference Sun angle and predicted CSS output every other second. CSS biases and resulting statistics were calculated using MathCAD 6 . 0 .

RESULTS AND DISCUSSION The CSS eyes that were analyzed are those subject to Earth interference for extended periods of time. These were eyes 1, 5 , and 6 . Figure 3 shows the reference G~~ (dotted line), predicted a; (solid line), and measured sensor output G,, (dashed line) for eye 1 based on the Feb. 22, 1998 data set. The predicted sensor output was calculated using the average Earth albedo coefficient. The value of ar# is nearly constant since the CSS eye is mounted on the solar array which follows the Sun.

A large discrepancy between the measured and predicted angles is apparent in Figure 3 from t = 300 to 600 seconds. This occurs because the model in this prototype version of the code does not take into account Sun occultations by the Earth.

Differences between the reference Sun angle and the measured sensor output seen in Figure 3 are due to Earth albedo interference. The sensor output and the reference angle agree within the measurement uncertainty due to bias scatter (roughly 1 deg) except from 0 to 250 sec and from 3500 to 5500 see. These are the time intervals when the Earth interference is significant and are accurately predicted by the model. For these intervals, the predicted curve is smooth while the measured curve shows abrupt changes. This qualitative difference is due to using Eq.

(12) instead of (lo), i.e., assuming a constant albedo coefficient. In reality this coefficient varies from one Earth area to another; the local albedo coefficient can vary from 0 . 0 5 to 0 . 6 : As a consequence, the predicted and measured output differ by up to 5 deg (at t = 4700 sec).

Nevertheless, the predicted and measured angles show similar overall trends. According to both measured and predicted CSS output, Earth albedo interference causes errors of up to 10 deg (from 4500 to 5500 sec).

time (sec) Figure 3. Reference, predicted, and measured Sun angles for CSS eye 1; Feb. 22,1998 (albedo coefficient = 0.30) The eye 1 errors before and after compensating for predicted Earth albedo effect are shown in Figure 4. For this orbit, model predictions based on the average Earth albedo coefficient pro- vide a significantly better CSS accuracy. After compensating for the Earth albedo effect, the average CSS error is 0.5 deg with a RMS of 2.1 deg. Without compensation, the average error is 2 . 6 deg with a RMS of 3.9 deg. The maximum error is reduced from 9 deg to 5 deg.

T R M M C S S Eye 1 S u n Angle Errors; F e b . 2 2 , 1 9 9 8 -4 I -6 I 2500 3000 3500 4000 4500 5000 5500 6000 time (see) Figure 4. CSS eye 1 errors for Feb. 22,1998, corrected and uncorrected for E a r t h interference (albedo coefficient = 0.30) re 5 shows the reference, measured Sun angles February 22 data set was used here as for Figures 3 and 4 . Again, the model accurately predicts the time intervals when the CSS eye i s exposed to Earth albedo interference (i.e., roughly from 0 to 100 seconds and from 2600 to 5500 seconds); the overall shape of the predicted sensor output also is approximately correct. The predicted and measured outputs differ by up to 2 deg.

TRMM C S S Eye 5 Sun Angles; Feb. 2 2 , 1 9 9 8

I I

5 4 '

0 1000 2000 3000 4000 5000 6000 time (sec) Figure 5. Reference, predicted, and measured Sun angles for CSS eye 5; Feb. 22,1998 (albedo coefficient = 0.30) The eye 5 errors before and after applying the model corrections are shown in Figure 6.

Using the average E a r t h albedo, the model reduces the CSS average error from 2.0 deg to 0 . 4 9 deg and the RMS error from 2.3 to 1 .O deg. The maximum error is reduced from 5 to 2 deg.

TRMM C S S Eye 5 Sun Angle Errors; Feb. 2 2 , 1 9 9 8 i n , I

\ , , Uncorrected

C o rrected

i

10 3000 3500 4000 4500 5 0 0 0 5500 6000 time (sec) Figure 6. CSS eye 5 errors for Feb. 22,1998, corrected and uncorrected for Earth interference (albedo coefficient = 0.30) or eye 6, the re icted, and measure before, the model and reference angles differ at the start of the Sun occultation period. Eye 6 is affected by the Earth albedo from 0 to about 100 seconds and from 3800 to 5500 seconds. The Earth interference periods are accurately predicted. The predicted sensor response again is only qualitatively correct because the average Earth albedo coefficient is used. The eye 6 errors before and after accounting for the Earth albedo interference are shown in Figure 8. The model reduces the maximum error from 7 to 4.0 deg. The average error decreases from 1.7 deg to 0.67, and the RMS decreases from 3.0 to 1.8 deg.

TRMM C S S €ye 6 Sun Angles: Feb. 2 2 , 1 9 9 8 5 8

-

0) (D v) (D - p 5 4 (0 C 5 2 - ' 5 0 \# M eas ure d 4 8 0 1000 2000 3000 4000 5 0 0 0 6000 time (sec) Figure 7. Reference, predicted, and measured Sun angles for CSS eye 6; Feb. 22,1998 (albedo coefficient = 0.30) TRMM C S S €ye 6 Sun Angle Errors; Feb. 2 2 , 1 9 9 8 -2 -4 -6 2500 3000 3 5 0 0 4000 4 5 0 0 5000 5 5 0 0 6000 time (sec) Figure 8. CSS eye 6 errors for Feb. 22,1998, corrected and nncorrected for Earth interference (albedo coefficient = 0.30) 1 ' other times, as shown by analyzing a 0, 1998 data set, del predictions based average Earth albedo coefficient may o orrect the CSS, actu increasing the sensor nce time for eye 1. The error. Figure 9 shows reference, pre spans are accurately predicted, as the pre- n 0 and loo0 dicted and measured sensor output differ by more than 7 deg, while the average uncorrected sensor error is less than 4 deg. Overall, the model correction based on the average Earth albedo coefficient increases the average eye 1 error from 1.0 to 2.0 deg and the RMS from 1.8 to 3.0 deg.

TRMM C S S Eye 1 Sun Angles; Mar. 10,1998 -1 / ' I Measured / Q c 4 5 40 I 0 5 0 0 1000 1500 2000 2500 time (sec) Figure 9. Reference, predicted, and measured Sun angles for CSS eye 1; Mar. 10,1998 (albedo coefficient = 0.30) This increased error is not due to an error in the model itself. Rather, the adverse effect is due the assumption that the average albedo of active areas and the average Earth albedo are equal. The altitude of TRMM is about 350 km, which means that during an entire orbit, a swath of about 16% of the entire Earth area is visible. This is a large area and therefore great albedo variations should be expected. The average error can be reduced to 0 by taking the average albedo for the active areas to be 0.105 instead of 0.30 (see Figure 10). The maximum error then is 1.5 deg and the RMS is 0.94 deg. Finding the optimum coefficient, Le., the albedo coefficient that provides an average corrected error of 0, improves the maximum error and the RMS as compared to the correction based on a global average value of 0.30. This optimum value im- proves the CSS measurements at all times that were analyzed. The optimum albedo coefficient can be easily determined a posteriori. Unfortunately, a cost-effective and general method of de- termining it a priori, i.e., before the CSS measurements are made, is not available. For the most accurate results, Eq. (10) should be used. This method requires a detailed database containing the albedo coefficients of thousands of Earth surface elements that is frequently updated using accurate weather input. Such an approach would be difficult and expensive.

CONCLUSIONS The model presented here accurately predicts the time intervals when Earth albedo affects Sun sensor measurements. Regardless of the Earth albedo coefficient that is used, the model Uncorrected 0 5 0 0 1000 1500 2000 time (sec) Figure 10. CSS eye 1 errors for March 10,1998, corrected and uncorrected for Earth interference (albedo coefficient = 0.105) provides a good qualitative prediction of Earth interference. In general, the model predictions based on the average Earth albedo coefficient increase the CSS accuracy. Nevertheless, there are times when this method severely over-corrects the sensor and reduces accuracy. The opt- imum albedo coefficient improves the accuracy of CSS measurements at all times, but it is not clear how to determine it a priori. The best Earth interference predictions could be made by maintaining a detailed database of local albedo values.

This study shows that the straightforward method of predicting Earth interference using a global mean albedo coefficient is insufficiently accurate. An albedo database adequate to im- prove matters is not readily available. A far more reasonable approach, as mentioned in prev- ious studies,' is to reduce the effect of Earth interference in the first place by using a filter.

Earth albedo is very low for several ranges of ultraviolet and infrared radiation. If the filter restricts the sensor sensitivity to such a range, the Earth albedo becomes negligible relative to the Sun. All types of Sun sensors could benefit from such a design modification.

REFERENCES 1. D. Kulp, Solar Maximum Mission Fine Pointing Sun Sensor Dawn and Dusk Errors, Flight Data and Model Analysis, Computer Sciences Corporation, CSC/TM-87/6700, January 1987.

2. D. Brasoveanu, H. Arabshahi, and M. Phenneger, Study of Earth Albedo Intelference on Sun Sensors, Computer Sciences Corporation, CSC/TM-90/6 103, September 1990.

J. R. Wertz, ed., Spacecrafr Attitude Determination and Control, D. Reidel Publishing Co., 3.

Dordrecht, The Netherlands, 1978.

4. M. Lambertson and J. Glicknkn, Flight Dynamics Distributed Systems, Generalized Sup- port Software, Tropical Rainfall Measuring Mission Application Program Specifications, Vol. 3: Measurement Processing System, Computer Sciences Corporation, November 1997.

c- .Yu.Beliaev, V.M.Stazhkov, N.I.Efimov', V.V.Sazonov+, H.Frank, M.Sehnel1el.t The adjustment procedure of the MOMS-2P German hardware accommodated on the Priroda module integrated with the Mir orbital station is described. This hardware consists of three separate cameras installed in a special manner for surveying the Earth's surface, and angular velocity sensors for attitude determination. The proper interpretation of the observation results requires precise reference of the MOMS-2P coordinate system with respect to the Astro-1 optical star sensor coordinate system. Adjustment is made as a result of the joint processing of measurements of the star sensor and angular velocity sensors.

Measurements are made while maintaining the fured attitude of the station in the orbital coordinate system. Measurements made when maintaining only one attitude state are not sufficient for the adjustment as by these measurements one can determine only two of three angles characterizing mutual arrangement of the MOMS-2P and star sensor coordinate systems. "he joint processing of the measurement data obtained when maintaining two and more different attitudes of the station allows to solve this problem. Examples of actual data processing and results of mathematical modeling are given.

INTRODUCTION The MOMS-2P German equipment has been installed on board the Priroda module of the Mir orbital station. This equipment has special cameras envisaged for the Earth surface observation while maintaining the fixed attitude of the station relative to the orbital coordinate system. For the proper interpretation of the observation results precise reference of the MOMS-2P coordinate system with respect to the Earth-fixed coordinate system is required. Reference is provided by knowledge of the station center of mass position in the Greenwich coordinate system and knowledge of the station attitude in the absolute space. The Greenwich coordinates of the station are determined by a GPS- receiver being a part of the MQMS-2P equipment; the station attitude is defined by readings of the Astro-1 optical star sensor. The latter is not related to MQMS-2P. In order to obtain precise attitude information of the MQMS-2P equipment in the Earth-fixed coordinate system it is necessary to know the MOMS-2P coordinate system relative to the coordinate system of the star sensor. As a result of this preceding adjustment the following is known: orientation of the coordinate systems with reference to each of the three optical units of the star sensor in the coordinate system of the MIR station (Ref.l).

Therefore, it is convenient to defrne the MOMS-2P coordinate system orientation in relation to the construction system indicated. The adjustment of this orientation is * Korolev Rocket-Cosmic Corporation Energia, Russia Phone 007 (095) 513-51-37, Fax 007 (095) 513-61-38, E-mail DNR@MCC.R.SA.RU.

Keldysh Institute of Applied Mathematics, Russia .t DLR, GSOC, Munich, Gemany.

provided as a result of the joint processing of the measurement data of the optical star a part of MO nt. Measurements sensor and angular velocity sensors being for the adjustment are made in nominal operation mode of M n maintaining the fixed attitude of the station in the orbital coordinate system.

ORDINATE SYSTEMS During the adjustment three right-handed Cartesian systems are used:

OX1Xfi3 - inertial coordinate system referenced to mean equator and equinox of

epoch 1950.0.

0 ~ 1 x 2 ~ 3 - construction coordinate system of the station core module.

Oy1yv3- coordinate system referenced to MOMPS-2P angular velocity sensors.

Positions of the cameras in this system are known accurately enough.

The systems introduced are used only to specify components of free vectors, therefore their origins are taken as coinciding.

Let us designate: ay , , - transition matrix fiom coordinate system 0 X 1 ~ 2 ~ 3 to system O X I X ~ X ~ ,

A = II 111,,=1

B = ~ ~ b g ~ ~ ~ , j ~ l - transition matrix fiom coordinate system Oy1yw3 to system O&&&, u = ~ ~ u y ~ ~ . . - transition matrix from system 0yLy2y3 to system ChcIx2x3 with a9 - cosine of I J = l the angle between axes 0 x 0 and Oxj, bo- cosine of the angle between axes Ox. and @ j , UQ is cosine of the angle between axes Oxj and @ J The matrices introduced are related as B = A U. Matrix U is constant and in general, matrices A and B are time-dependent.

Below we will express matrix U elements by Krylov angles t p , 8 andp. These angles are defined in the following way. System 0 X 1 ~ 2 ~ 3 can be converted into system Oylyu3 by three successive rotations: 1) through angle y around axis 0 x 2 , 2) through

angle e around new axis 0x1, 3 ) through anglep around new axis 0 x 3 coinciding with

axis O y 3 . The elements of the transition matrix in terms of Krylov angles are: In an analogous way we will present matrix B elements. Let us designate Krylov

angles 8, w andp used to ‘prescribe U p1, p2 and p3, respectively; analogous let us

designate the Krylov angles used to prescribe B at some fixed instant of time as 41, q2 and 93.

Angles PI, p2 and p3 are known only approximately. Description of the refinement procedure of their values is the aim of this paper.

/ The Astro-1 optical star sensor is installed on board the Kvant-2 module and allows to define the station attitude-coordinate system Ox1xp3 orientation relative to system OX&& at some discrete instants of time. The measurement data of this sensor and methods of its processing are given in (Ref. 1). Based on the star sensor measurement data processing one can define values of matrix A at some instants of time at intervals of several seconds.

The angular velocity sensors being a part of the MOMS-2P equipment allow to measure components a , (a: = 1, 2, 3) of the absolute angular velocity of the station in coordinate system Oy?yzs3. There are two sensor packages and each package measures

four velocity components. For component a there are two sets of measurements

(redundancy), components q and are measured once in each package. Measurements of all the components are made at discrete time steps h = 8.30078 ms. The certificate is 0.0001 deg/s. Examples of the station angular rate accuracy of measurements measurements made by the sensors are given in Figure. 1. In this Figure points conforming to measurements neighbouring in time are connected with straight lines. The broken lines characterize the angular velocity components behavior in time.

-0.034 -0.056 -0.077 -0.oss -0.121 -0.010 -0.042 -0.073 -0.01 1 -0.045 - 0 . 0 1 1 -0-02.5 -0.0-71 0.0 0 - 2 0 . -7 0 . G 0.0 1.0 Figure 1 Examples of the station angular velocity measurements 45 1 / As shown in the Figure, the angular velocity components demonstrate cyclical trend. The measurement data spectral analysis allowed to find a frequency of this trend. It is 24 H z . If such variations took place in reality, then due to the large moments of inertia of the station they would require a very large torque. Rough estimates show that the moment required cannot be provided with any onboard devices. Therefore, the variations indicated are caused by properties of the angular velocity sensors. Using the measurement

data of these sensors, the variations indicated must by eliminated - suppressed using low-

fiequeney filtering.

Such filtering is provided in the following way. Assume that measurements of any component of the angular velocity are given. Let us designate these measurements as xi (i

= 0, 1, ..., mn), where m and n are natural numbers. Measurement xi is made at the instant

of time ti = ih, h > 0. A low-fiequency component contained in this data will be taken as n-1 ?(t) = a + + ak sin(zkt/mnh).

k=l Here a; , E l and a k are coefficients. We will select values of these coefficients fiom the fhctional minimum condition.

m @ = x [ 2 ( i h ) - x i l 2 .

i=O This approximation method is a slight modification of one of the methods considered in (Ref.2). To calculate coefficients there are simple design formulae. The function prescribed by Eq. (1) often experiences noticeable comparatively high-frequency

oscillations. In order to get rid of them this expression is corrected using Lanczos cr -

factors (Ref.2): n-1 sin(& / n) sin(&/ mnh) ? ( t ) = a + p + x a k k=l n k l n One can use other analogous factors as well.

Examples of the low-frequency component extraction from the measurement data of angular velocity components w, (a = 1,2, 3) are given in Figure 2. Filtering is provided with m = 3000. Instant of time t = 0 in this Figure corresponds to instant of time -0-064sa -0.06505 -0.0651 2

/

/ 0.00037 0.000 1 7 - 0 . 0 0 0 0 4 -0.00025 2 , s -0.00046 0 4 6 92 138 2 3 0 deg-’s 0.00075- 0.00063- 0.0 005 1..

0.00039 0.00027-- 8 , s 0.000 15 0 4 6 92 138 230 Figure 2 Examples of the low-frequency component extraction from the measurement data t = 0 in Figure 1. Two plots obtained by measurements of redundant sensors are given for

component e. These plots almost coincide. For the adjustment the approximation of a

low-fiequency component in 9 obtained by readings of only one sensor w a s used.

ADJUSTMENT PROBLEM ANALYSIS Assume that values of matrix A at instants of time tl<t2< ...<t are got as a result of the star sensor measurement data processing. Let us denote a value relating to instant ti as A i .

Matrix B satisfies the following relation: B = B f 2 , (2) where the time differentiation is designated by a point.

-w, w,

a =

w, 0 -w, 0 1 0 -w2 Let us specify angular velocity components w, (a = 1 2,3) with functions constructed by the measurement data of the angular velocity sensors using the previous section method

and now consider relation (2) in which 0 = $2 (0 as a differential equation defining

matrix B. Interval t l I t I tn must lie deeply enough within the interval defining matrix function L ? ( o to exclude the boundary effects related to its construction.

We will represent the solution of Eq. (2) as B O = BoX(t), where BO = B(to), y definition of matrices X ( t ) is a unit matrix, instant to is selected near interval ti I t I tn.

A , B and Uthey are linked with relation AU = B, However, for matrices Ai only relations AiU xB&i, where 4- = X(t$ (i = 1 ,..., n) are valid.

Matrices U and BO are known beforehand only approximately but they can be refined as a result of statistical processing of the measurement data. Following the least- squares method we will take values of these matrices from the functional minimum condition 1 " S = - C t r ( A i U - BoXi)*(AiU - BOXi).

2 i l In minimization we will use explicit expressions for U and BO using two sets of Krylov angles Pa and ql (tr; A = 1, 2, 3), respectively. In this case the problem consists in minimizing S by pa and ql and is reduced to the solution of the following equations: Eqs. (4) are solved numerically by Gauss-Newton iteration method (Ref.3). This method is a version of the Newton method with simplified calculation of the matrix of the linearized system occurring during each iteration. Let us cite the basic formulae used.

First, we will give formulae for partial derivatives of matrices U and BO by angles Pa and 42. We have the following: Values 4al, 4d and 4d are determined by differential identities.

where 8 , w , Q) are ordinary designations for Krylov anglespa. Doubly recurring of Greek indices means the summation from 1 to 3. Values rynl, rynz and ~3 are determined in an analogous way. It is convenient to introduce vectors corresponding to matrices The following formula is true: n B @a i=l i=l is the derivation which is based on the relation trOa= 0 and the condition of orthogonality of matrices A i and U. Let us introduce matrix and using its elements, specify row vector f= (223 - 232, z3l - 213,212 - 221).

Then = - f p a .

In a similar way we will get us go to the second derivatives of S. Using orthogonality properties of Now let matrices A b X b V, Bo and usual simplifications of Gauss-Newton method, Le. assuming

that AiU = BOX;: (i = 1, ..., n> we will get

The system of the linearized equations solved during each iteration of Gauss?

Newton method and defining corrections App and Aq, to available estimates of unknowns pp and q, is the following This system is named normal. Its matrix is s fthis matrix has non zero eigenvalue then this normal system has a unique solution and Gauss- Newton method converges with suficiently exact initial estimates of the unknowns.

However, unfortunately, this zero eigenvalue exists.

The proof will be given assuming that the station orbit is Keplerian and the station maintains a fixed attitude in the orbital coordinate system. It is an idealized case.

Obviously, the degeneration will not take place for the actual orbit, however, the problem of the concurrent refinement of U and Bo will be ill-conditioned. With assumptions made matrix X(t) is defined by Eq. (3) in which D(t) is specified by the orbital coordinate system angular velocity. In this system we will direct axis 3 dong the radius vector of the station and axis 2 along the normal to the orbital plane. Then the angular velocity of this system in its proper basis will be (0, 3 , 0), where vis the station true anomaly. Denote by D the transition matrix fiom coordinate system Oyvw3 to the orbital system. Then cosp 0 sinp 0 1 0

X(t) = DTn2[v(t) - v(to)]D, I72 (p)=

-sinp 0 cosp e2 = (0, 1, 0)', a = D'e2. As Assume that p)e2 = e2 and ern*( p ) = e l , we have X(t) a = a and aTX(t) = a'. Hence, Ma = M ' a = n a.

Let us consider normal Eqs. ( 5 ) with zero right-hand sides

Here 4 p and 4, are unknown values. Let us show that this system has a nontrivial

solution. It is this solution that will be a eigenvector conforming to the zero eigenvalue of the matrix of system (5). Rewrite the latter equations as

Let us determine 4 p and 4, fiom the following systems of equations p d p p = a and

v&, = a. These systems have unique solutions if both sets of Kxylov angles are non-

degenerate. Having substituted such solutions in normal equations we will get obvious identities: Thus, the matrix of normal equations has a zero eigenvalue. The problem of this zero eigenvalue of the matrix of system ( 5 ) is related to non-uniqueness of matrices U and Bo / providing minimum of S. In order to state this fact it will suffice to prove non-uniqueness of these matrices in relation ~ ( t ) = ~&(t) U? Using the on above for X(t) one can

prove that A ( t ) = i o X ( t ) c T , where io = BODT I I 2 = UDrn2(s)D, s is any

number.

COMPUTATIONAL ALGORITHMS Due to system ( 5 ) degeneracy one interval with the angular velocity sensors and optical star sensor measurements is not enough to provide the adjustment. However, by measurements on one interval one can refine two of three angles pa. The case of one interval can be considered more generally. One can specify values of all three angles but in this case it will be required to provide an additional scalar relationship between them.

Let us take such an additional relationship as linear We will add this relationship as an additional equation to system (4). With the appropriate selection of coefficients al, a2, a3 system (4), (6) will have the unique solution.

The numerical search for this solution is performed using the obvious modification of Gauss-Newton method. To the system of normal Eqs. (5) occurring during each iteration of this method we add linearized Eq. (6) and get a system of 7 linear equations relative to 6 unknowns. However, the rank of this new system does not exceed 6; it will equal to 6 with an apt choice of coefficients in Eq.

(6). The solution of this system is found using the singular decomposition of its matrix (Ref.4). We consider the Gauss-Newton iteration process to be convergent when norm of the correction vector w i t h components Ap, dqn is less than the given positive number.

The calculation of the matrix singular decomposition of system (5), (7) allows to find out if the selection of coefficients a, in Eq. (6) was suitable. This matrix must be well-posed enough, Le. the ratio between its maximum and minimum singular values must be not very high.

In order to specify values of all angles pa it is necessary to provide simultaneous processing of two and more intervals with measurements for different attitudes of the station. In this case the following will be specified: matrix Uthe same for all intervals and matrices BO for each interval. If these matrices are given parametrically then during the simultaneous processing of I intervals the parameter 3(1+1) values will be specified. The refinement can be provided by Gauss-Newton method. The calculation of the matrix and right-hand side of the corresponding system of normal equations is made in the following way. First the matrix and vector of the right-hand side of system (5) are calculated for each interval, then all these matrices and vectors are divided into blocks of the following sizes: 3 x 3 and 3 x 1, respectively and the required matrix and vector of the right-hand side of this new system of normal equations of the order of 3(1+1) are formed of the blocks obtained.

As an example we will consider generation of a system of normal equations during the joint processing of two intervals which will be numbered as 1 and 2. Let us specify matrix U by angles pa; matrix B y ) , presented as matrix Bo for the j-th interval, is specified by angles 4 : " . Write system (5) for thej-th interval as Here A p = (klpl, 4 2 , Apj)? 4@ = ( A q ~ ) , A q ~ ) , d q ~ ) r , [ C ~ ) p ' = C : ; ) . Then the system of normal equations occuning during the joint processing of two intervals will be: The covariance estimate matrix of parameters pa and 4 : " obtained as a result of the

w i t h i n usual assumptions of the least - squares method (Ref.4). In

adjustment is calculated this case it is considered (comp. Ref.1) that mismatches of matrices Ai and B&iUT (i = l,..., n) can be described by infinitesimal rotation vectors which components in any Cartesian system are independent random variables with zero mean value and identical variance.

ADJUSTMENT RESULTS Presently there is data of the angular velocity sensors and Astro-1 only for one time interval adequate for adjustment. This interval falls on gh OOm - gh 1 2 " ' , March 14, 1997.

It contains 148 points with the station attitude measurements. As indicated above, one interval is not enough to provide adjustment. However, if an additional scalar constraint is pa then some refinement of these angles is feasible. For example, imposed upon angles one can fix a value of one angle and find values of two other angles. Assume that in Eq.

(6) a 1 = a 3 = 0, a2 = 1, a 0 = 3d2. Such coefficient selection means fixing of angle p2 = 162. The adjustment result is the following: p1 = 0.O62Oyp3 = 90.105". Standard deviations of the given estimates are: c r , , = 0.013O, c r , , = 0.013O.

Let us give some characteristics of the system of normal equations at the minimum point of S. The standardized eigenvector codorming to a zero eigenvalue of the matrix of

system (5) is: (0.004, -0.639. - 0.004, 0.354, 0.612, 0.302). Nonzero eigenvalues of this

matrix are: 1.037, I .38 1 , 324.3, 43 1.1, 538.2. The matrix singular values of system (9, (7) are: 0.6295, 1.050, 1.384,324.3,431.1,538.2.

To check the adjustment methods described above two time intervals with the artificial me~urementdata were used. These intervals were obtained in the following way. The actual measurement data of all sensors was taken and artificially combined with each other. The first artificial interval was of the length of 9 min and there were 275 instants of time with the star sensor measurements on it. The second artificial interval was of the length of 4 min and there were 12 1 instants of time with this sensor measurements on it. The joint processing results of the actual interval considered above and the first artificial interval allowed to get the following results: pi = 0.061°, p2 = 90.299", p3 = 90.057", crp, = 0.013", crp2 = 0.0066", crp, = 0.0059". The maximum and minimum eigenvalues of the matrix of the system of normal equations in the case given are 0.5888 and 1045.

Now let us give the joint processing results of the actual interval and two artificial intervals: p1 = 0.099", p2 = 90.175", p3 = 90.062, crp, = 0.013", op2 = 0.0070", c r , = 0.0062". In the case given the maximum and minimum proper values of the matrix of the system of normal equations are 0.4908 and 1355.

Of the results given the eigenvalues and singular values of matrices of the system of linear equations solved are of the most interest. These.values show a capability of using the proposed adjustment method. Note that the use of the artificial measurement data in calculations has no effect on the conclusion drawn. This conclusion is based only on the selection of the station positions during the adjustment and on the selection of instants of time with measurements. The accuracy rating of specified angles pa that are obtained during the adjustment is also of interest. Though this accuracy rating depends on the artificial data selection the data origin allows to hope for impartiality of the rating indicated. One will not attach great significance to the estimated discrepancy of angles pa and standard deviations opa in case of adjustments provided in two and three intervals.

This discrepancy can be explained by the artificial data preparation method.

CONCLUSION It is shown that the optical star sensor and angular velocity sensors measurements made when maintaining only one f=ed attitude of the station are not sufficient for the adjustment; by these measurements one can determine only two of three angles characterizing the mutual arrangement of the MOMS-2P coordinate system and construction coordinate system of the station. The joint processing of the measurement data obtained when maintaining two and more different attitudes of the station allows to solve this problem. These conclusions are validated by examples of the actual data processing and mathematical modeling results.

REFERENCES 1. M.Yu. Beliaev, N.I.Efirnov, V.V.Sazonov. Determination of the attitude of the Mir orbital complex from indications of an optical star sensor. Cosmic Research, 1995, v.

33, N 4, p.357-364.

2. C. Lanczos. Applied analysis. M., Prentice Hall, Inc., 1956.

3. Y Nonlinear parameter estimation. Academic ress, New York, §an Fr , London, 1974.

4. G ub, CReinsch. Singular values decomposition and least-squares solution.

Nw. Math. 1970, V. 14, N. 5, p. 403-420.

PA A YNA IM R ichmann* and Joseph Sedlak' There are a number of approaches one can take to modeling the dynamics of a flexible body. While one can attempt to capture the full dynamical behavior subject to disturbances from actuators and environmental torques, such a detailed description often is unnecessary. Simplification is possible either by limiting the amplitude of motion to permit linearization of the dynamics equations or by restricting the types of allowed motion. In this work, we study the nonlinear dynamics of bending deformations of wire booms on spinning spacecraft. The theory allows for large amplitude excursions from equilibrium while enforcing constraints on the dynamics to prohibit those modes that are physically less relevant or are expected to damp out fast. These constraints explicitly remove the acoustic modes (i.e., longitudinal sound waves and shear waves) while allowing for arbitrary bending and twisting motions which typically are of lower frequency.

As a test case, a spin axis reorientation maneuver by the Polar Plasma Laboratory (POLAR) spacecraft has been simulated. POLAR was chosen as a representative spacecraft because it has flexible wire antennas that extend to a length of 65 meters. Bending deformations in these antennas could be quite large and have a significant effect on the attitude dynamics of the spacecraft body.

Summary results from the simulation are presented along with a comparison with POLAR flight data.

INTRODUCTION This work describes the design and application of a flexible spacecraft dynamics simulator?named Cartwheel? to model a spin axis reorientation maneuver by the Polar Plasma Laboratory (POLAR) spacecraft. It presents a summary of the simulation results and a comparison with flight data. Overviews of the theory and the software are also given.

The general purpose of this research was to study the applicability of a particular flexible dynamics method to spacecraft attitude dynamics. As described below, the Cart- wheel simulator captures the most important parts of the nonlinear dynamics but leaves out a number of smaller effects. As a prototype analysis tool, Cartwheel has been used for basic theoretical tests but is not intended for operations support.

* The Aerospace Corporation, 2350 El Segundo Blvd., El Segundo, CA, USA 90245.

Computer Sciences Corporation (CSC), 101 10 Aerospace Rd., Seabrook, MD, USA 20706.

ose as a representative spacecraft because it has flexible wire anten- 65 m length, so the wire antennas contribute a large fraction of nas that extend up to the total moment of inertia. e to the great length of these thin wire booms, bending de- formations can have a significant effect on the attitude dynamics of the hub of the spacecraft body. Flexure of the antennas can cause large changes in the total moment of inertia, which, in turn, can affect the spacecraft rotation rate. The motions of the antennas couple through the rigid hub. In addition, the antennas have some intrinsic stiffness, that is, a tendency to spring back to their default orientation (normally straight). The resulting highly nonlinear dynamics leads to complex perturbations to the attitude.

The POLAR spacecraft periodically executes a 180 degree spin reorientation man- euver to prevent the Sun from coming into view of certain instruments. We anticipated that this type of maneuver would induce vibration in the wire booms that could meas- urably influence the attitude dynamics of the hub. This paper describes the application of the Cartwheel simulator to such a maneuver.

The next sections contain overviews of the dynamics theory and the Cartwheel soft- ware. This is followed by a description of the POLAR spacecraft, orbit, attitude, and the spin axis reorientation maneuver. Finally, a summary of the simulation results and com- parison with POLAR flight data is given.

THEORY The flexible spacecraft dynamics simulation program, Cartwheel, was designed to of large bending deformations of wire booms on space- model the nonlinear dynamics craft. In these simulations, we enforce constraints on the dynamics to prohibit the gener- ally lower amplitude, higher frequency acoustic modes (i.e., longitudinal sound waves and shear waves) but to allow the larger, lower frequency bending and twisting motions.

While it is possible to model the full dynamical problem, this is made difficult by the presence of both very slow and very fast characteristic appendage motions. Describing all the characteristic frequencies accurately leads to difficulties and inefficiencies when integrating the governing equations. Dynamical systems with this problem are called s t i f f .

There are two broad kinds of simplifications that make solution of the governing equa- tions more tractable. One can assume the deviations of the appendages from their equil- ibrium positions are of low amplitude and then discard all terms in the equations that are of second or higher order in this amplitude. The resulting linear system of equations can be analyzed in terms of its normal modes of vibration. Alternatively, one can disallow types of motion that are physically less important for the system under consideration. The approach taken in this work is to discard the generally high frequency motions associated with stretching and shearing of the appendages (acoustic modes) while keeping the full nonlinear description of the rest of the motion (flexing and twisting).

Constraining the dynamics to disallow acoustic modes makes numerical solutions easier to obtain without appreciably limiting their validity. The only motions of interest here are those that affect the spacecraft attitude; these are likely to be combinations of simple oscillations and possibly “whiplash” motions of the booms. These motions are not y to excite or couple strongly o the acoustic modes. ition, acoustic in the multi-stranded boom material. Eliminating these modes likely to damp out fast from the start is very nearly the same as describing the full dynamics of a system with highly damped acoustic modes.

The method chosen for enforcing the dynamical constraints is called the impetus- striction method (Ref. 1). The method has proven valuable in theoretical analyses of stability conditions (Ref. 2) as well as in numerical simulations to rod dynamics (Ref. 3).

The key features of the impetus-striction method are as follows: 0 The method applies to systems having a Lagrangian formulation subject to h o b nomic constraints. It transforms constrained Lagrangian dynamics into an uncon- strained Hamiltonian formulation where constraints appear as constants of the motion.

0 Each constraint equation is replaced by its time-derivative; the Lagrange multipliers associated with these time-differentiated constraints are called the strictions and have dimensions of momentum.

0 One derives the conjugate variables to be used in the Hamiltonian formulation from . the constrained Lagrangian (the classical Lagrangian plus constraint terms). Thus, the conjugate variables are not quite the same as the usual momentum variables that arise from the classical Lagrangian alone; these new variables are the impetuses.

- Roughly speaking, the striction describes that part of the momentum arising from the constraint forces, and the impetus describes that part due to the remaining forces. Note that if there are no external forces, it is still the total momentum that is conserved, not the impetus.

0 One constructs a “pre-Hamiltonian” fiom the constrained Lagrangian; this differs as yet undetermined from the usual Hamiltonian only in that it still depends on the strictions.

0 The Hamiltonian is obtained by minimizing the pre-Hamiltonian with respect to the strictions for fixed values of the state variables. This minimization determines the strictions and can usually be reduced to the solution of a system of linear algebraic equations.

0 The resulting unconstrained Hamiltonian system is integrated in time, solving for the strictions at each time step.

OVERVIEW OF CARTWHEEL The physical model of the boom dynamics yields a system of nonlinear partial dif- ferential equations in which the independent variables are time and arc l e n , @ h along each boom, both of which must be discretized for the numerical implementation.

For the discretization in arc Iength (often called the semi-discretization), we model each boom as a sequence of N, elastically connected rigid segments. As implemented, the integration errors of order h2 as the segment length, h, goes to zero, vanish for any segment length when the spacecraft rotates uniformly with zero deflection in the wires.

There is a position and orientation associated with each boom se,oment. Orientations are represented by quaternions, so there are seven configuration variables for each seg- ment, along with their conjugates (the impetuses). Note that the quaternion normaliza- tions are not imposed by brute force but are maintained by the integration method. There are Na antennas (Nn can be different for each antenna). There also is a position and orien- tation for the spacecraft hub and a position for the center of mass of the entire system.

For each time step, the strictions are calculated as the solution of a linear system of equations; the number of strictions is 3NnNa representing the stretch and shear constraints for each segment, plus 3Na to constrain the position of the antenna attachments to the hub, plus 3 more to separate relative position coordinates from the system center of mass.

(The striction equations decouple into N, banded, linear systems, each with 3Nn un- knowns, together with the solution of one system with 3Na unknowns, and one with 3 unknowns for the center of mass.)

For the discretization in time, the Hamiltonian system, consisting of ( 14NnNa + 20)

first-order ordinary differential equations for the coordinates and conjugate impetuses, is stepped forward using a midpoint method. That is, if zk is the state at time t k and F(z) is its time derivative, then This is an implicit integration scheme wherein the time derivative evaluation requires knowledge of the new state. Implicit methods are numerically more stable than explicit methods but require solution of a possibly nonlinear system of equations for z k + ' at each time step. Equation (1) is solved by a futed-point iteration method but a Newton method could also be used.

The integrator was chosen not only for its inherent stability but because it is one of the simplest of the symplectic integration methods. One consequence of this is that it pre- serves quadratic invariants (see, for example, Ref. 4). This means that any theoretical constant of the motion that is quadratic in the state variables will not be subject to any numerical error beyond machine roundoff. In practice, the error level is determined by the tolerance allowed when iterating for a solution to Eq. (1).

Some examples of quadratic invariants are the quaternion norms and the amount of stretch and shear between se,gnents of the appendages. Other invariants may depend on the boundary conditions of a particular example. For this problem, total linear momentum and total angular momentum are quadratic invariants. On the other hand, although the total energy is a conserved quantity (there is no damping in the model), the Hamiltonian is not quadratic in the state variables. Consequently, the code is not expected to conserve the energy as we11 as the quadratic invariants.

enforces the constraints by solving for and imposing the onian system is unconstrained. Thus, requisite forces, and the resulting augmented , they are not set to zero ad hoc. They while the stretch and shear are constrained to on any value but remain small only because of the impose point here is that the finite step size in the time-integration conceivably cause growing errors in the constraints. The use of a symplectic integrator guarantees that this will not happen.

The first test of the code was to examine the invariants. The errors in the quadratic invariants should remain small (close to the tolerance of the solution of Eq. (l)), and the error in the total energy should grow with a slope proportional to the time step squared.

This has been demonstrated remarkably well for both single rod and closed ring simulations (Ref. 3). (Interest in these early rod and ring tests goes beyond just verifying the impetus-striction method. They are well suited for modeling some aspects of biomolecules such as bacterial DNA (Ref. 3.)

Later tests verified that the invariants are properly conserved also for the full space- craft model (multiple booms coupled through a rigid hub) and that accurate results are ob- tained for small perturbations with the booms discretized into relatively few segments.

THE POLAR SPACECRAFT The mission of the POLAR spacecraft is to observe the electromagnetic field of the Earths polar regions. The hub of the spacecraft is about 1.58 m in radius. The mass is roughly 1000 kg. Nominally, the spacecraft spins at a rate of 10 revolutions per minute (rpm) with the spin axis oriented near the orbit normal vector. Spin axis direction is deter- mined using a Sun sensor and an Earth sensor. The spacecraft has two axial booms exten- ding along the positive and negative spin axes. The axial booms are effectively rigid and are rigidly attached to the spacecraft hub. The spacecraft also has two radial booms, hold- ing magnetometers, that are effectively rigid and rigidly attached to the spacecraft hub.

The spacecraft has four flexible wire booms extending radially from the hub and arranged symmetrically. Each wire boom can be deployed to a maximum length of 65 m.

Each wire has a radius of 1 mm. The wire booms serve as antennas to observe the ambient electromagnetic field and also to help stabilize the spacecraft rotation. During the phase of the mission treated in this study, one pair of wire booms w a s deployed to 50 m while the other was deployed to 65 m.

The POLAR spacecraft orbits the Earth in a near polar, elliptical orbit (roughly 2x9 Earth radii). The pre-maneuver orbit had semimajor axis equal to 34251 km, the eccentricity was 0.662, the inclination was 86.01 deg, and the right ascension of the ascending node was 26.809 deg. The orbital period is approximately 12 hours.

During most of the mission, the spacecraft spin axis is maintained along the positive or negative orbit normal. As the Sun-orbit plane geometry changes, it is necessary about every 6 months to reorient the spin axis for solar thermal constraints and to keep the Sun of view of sensitive instruments. At those times, the spin axis is reoriented out of the field ositive to negative orbit normal, or vice versa.

keeping the Sun vector nearly perpendicular to the spin axis. Due to current thruster limitations, this maneuver is performed in two separate 90 degree slews, each lasting roughly 3.5 hours.

~ POLAR spacecraftand insbunents CEPPADISEPS &

' EFI

TlMAS and CAMMICE, notshown, areon the far side of the spacecraR Figure 1 . The POLAR spacecraft and instruments.

This study discusses one such maneuver that was performed on April 16-17, 1996.

We anticipated that the maneuver, although performed slowly, would induce small vibrations in the flexible wire booms. There are no instruments onboard the spacecraft to measure directly the deformations of the booms. However, due to their length during this phase of the mission, the booms contribute about 50% of the total moment of inertia about the spacecraft's spin axis. It was therefore expected that if there were any flexure of the booms, this would have a measurable effect on the attitude of the spacecraft hub.

SIMULATION RESULTS In this section, we describe the results of numerical simulations of the POLAR spin reorientation maneuver and compare results from flexible and rigid body models. Our simulations focused upon the effects of boom flexibility on the attitude dynamics of the spacecraft hub. Because of their length, the booms could exhibit large bending motions under the appropriate circumstances. However, because the spacecraft spins at 10 rpm, each boom experiences a stiffening centrifugal force that tends to inhibit deformations. In addition, the rate of the spin reorientation maneuver is small (90 degrees in 3 . 5 hours), so g forces and torques acting on t e booms are smd ations of the wire booms were found.

o display the deformations, the position of the end of each segment is represented in a rotating body coordinate system attached to the spacecraft hub. If a boom continued to point radially outward from the hub, as it does during a steady spin, then the boom would exhibit no motion in this body frame. During the simulation of the spin reorientation maneuver, each boom oscillated primarily out of the spin plane and parallel to the hub’s spin axis, designated as the z-axis. This motion is shown in Figure 2, where the tip’s displacement is plotted as a function of time each 0.5 second. The spacecraft rotates 200 times during the 20 minute period shown. In Figure 2, the displacement is displayed as a fraction of the boom length, which is 50 meters. The amplitude of the oscillation is about 0.014 of the total length, or about 0.7 m.

I

-0.01 5 ’

1000 1200 0 200 400 600 800 time from maneuver (seconds) Figure 2. Out-of-plane displacement of boom tip for model with two unequal length pairs of wire booms (peak amplitude corresponds t o 0.7 m)- The complex motion in Figure 2 can be approximated well by a superposition of three distinct oscillations:

Ai cos(2m / q ) + Bj sin(2nt / q )

j = I where the periods are T I = 6.0 sec, T2 = 5.778 sec, and T 3 = 5.872 sec. The amplitudes are AI = -5.25~10-~ , BI = 2.8~10-~, A2 = 6 . 9 1 ~ 1 0 ~ ~ , B2 = -3.6~10-~, A3 = - 1 . 8 5 ~ 1 0 - ~ , and B3 = 1 .3x10a.

The oscillation with 6 seo period is associated with the spin rate of the spacecraft.

The other two frequencies appear to be associated with driven oscillations of the two pairs of booms of length 50 m and 65 m, respectively. To test this, we ran a simulation ~ ~ e t e r s the same.

tip of each boom was less complex and decomposed into the sum of oscil only two distinct periods of 6.00 sec and 5.79 sec as shown in Figure 3.

0.015 m

c 1

- W

-

I I

5 0.01

n

-

0.005 - c

2 - - 0

CI c W

5 -0.005

a

-

a v) -0.01 N -0.01 5 0 200 400 600 800 1000 1200 time from maneuver (seconds) Figure 3. Out-of-plane displacement of boom tip for model with four equal length booms (peak amplitude corresponds to 0.5 m).

Although the motion of the tip of each boom is fairly complex, each boom moves nearly as a rigid rod. To demonstrate this, Figure 4 shows the maximum deviation of a boom from the straight line between its base and its tip. Thus, the oscillations shown in Figure 2 are subtracted out, leaving only the deviation caused by any curvature of the h

= X l O d

$ 7 ,

- I

time from maneuver (seconds) Figure 4. Maximum deviation of boom from straight line (peak amplitude corresponds to 3 mm).

boo n, the displaceme asured as a fraction of the boom leng maximum displacement from the straight configuration is seen to be only about 6x10'' times the 50 HI length, or 3 mm.

Because each boom behaves essentially as a rigid body, the most significant flexibility effects can be captured by representing each boom as a rigid rod attached to the hub by a flexible hinge. Simulations with various, reasonable amounts of spring force in the hinge at the base show that even this elasticity has only a small effect. Thus, one can set the deformation restoring force to zero at the base.

In this simulation, the initial conditions were taken from the April 1996 reorien- tation maneuver. The spin axis was initially pointing toward right ascension 116.3 deg and declination -7.7 deg. The final attitude following the first 90 deg maneuver segment was estimated to be right ascension 27.2 deg and declination -80.0 deg (Ref. 6). Figure 5 shows the right ascension and declination of the hub for the simulated maneuver. For comparison, the simulation for a totally rigid spacecraft with the same mass properties is also shown. (Note that the initial attitude differs somewhat from values given in Ref. 6.

This small offset is caused by a different choice of sensor biases solved for together with the attitude and does not affect the analysis.)

~ 1 1 7 Q, E .

5 116

.- E a ,

5 : 115

a E 'g 114 0 200 400 600 800 1000 1200 time from start of maneuver (sec) -8 m a , 3-10 c .o -12 rigid model: solid line

.- I ! -14

flexjble model: dashed line

8 -16

0 200 400 600 800 1000 1200 time from start of maneuver (sec) Figure 5. Rigid body and flexible model simulations showing right ascension and declination of spin axis for the first' 1200 seconds of spin reorientation maneuver.

parison of the two simulations ( exible and rigid body) allows the effects of to be seen clearly. The flexible dy simulation shows sinusoidal oscillations absent from the rigid body simulation. The period of the oscillation is about 2.6 minutes.

rn difference in right ascension between rigid and e model is about maximum difference in declination is about 0.1 higher frequency oscillations in the rigid body attitude due to nutation that are too low in amplitude to show clearly on these plots.)

It should be noted that the flexible dynamics simulator neglects several perturbing forces that could influence the spacecraft motion. It is expected that most of these distur- bances are negligible for the case under consideration. For example, gravity-gradient torques are not included. The spacecraft is near apogee during the maneuver, so the large distance from the Earth (9 Earth radii) greatly reduces the influence of gravity-gradient torque.

Internal dissipation within the wire booms could have a more significant effect on the dynamics. In particular, internal dissipation tends to damp out oscillations in the wires. Thus the current model would not yield physically comct results over long periods of time, but should be accurate for times shorter than the decay time. This is observed to be on the order of tens of minutes (see results below). An accurate.mode1 of internal dissipation could be difficult, due to the complex composite structure of the wire, and would require a more detailed knowledge of the material parameters.

The driving torque was modeled as continuous and constant. In reality, thrusters are used to supply the torque. They are fired in pulses, timed with the spin period, to perform the maneuver. The periodic pulsing of the control torque is expected to excite higher frequency vibrations of the wires that are not excited in the model.

Thermal effects are known to influence the dynamics on other spacecraft. However, these tend to be small and brief disturbances. Furthermore, for the orbital geometry during this reorientation inaneuver, the POLAR spacecraft does not enter the Earth’s shadow so the wire temperature is not expected to vary significantly.

Some other effects one might consider are perturbations due to atmospheric drag and fuel slosh. However, the spacecraft is at a sufficiently high altitude that torques from is not a factor because the maneuver is atmospheric drag are negligible, and fuel slosh performed sufficiently slowly compared to the spin rate.

COMPARISON OF SIMULATION WITH FLIGHT DATA To estimate the true attitude of the POLAR spacecraft during the maneuver, the spacecraft telemetry was processed using an Attitude Ground Support System (Ref. 7) designed for spinning spacecraft. Attitude estimation was based upon Earth horizon sen- sor and spinning Sun sensor readings. The Earth and Sun angle measurements later were in reprocessed using MATLAB where it was possible to insert sensor biases overlooked the initial processing. In particular, this reprocessing corrected timing errors and rejected spurious solutions.

owi reference vectors, the measure yields the angle s to the nadir vector spacecraft spins, ured as a time o converted into an angle using the spin period determin pulses.) The spin axis then is found at the intersecti nadir and Sun angles (another twofold ambiguity). Knowing the path of the planned maneuver made it simple to select the correct solution. (The initial spin direction is unambiguously estimated from a large batch of data obtained before the slew.) Reference 8 gives details on methods for spacecraft attitude determination.

Figure 6 shows the estimated spin axis right ascension and declination angles for POLAR during the interval 03:16:46 GMT to 03:40:06 GMT, on April 16, 1996. The spin reorientation maneuver begins at time 03:20:06 GMT. The results of the flexible dynamics simulation from Figure 5 are also overlaid on Figure 6.

-200 0 200 400 600 800 1000 1200 time from start o f maneuver (sec)

- -8

m 0, 3 - I O c .o .I- -12 observations: solid line

L= 2 -14

-16 -200 0 200 400 600 800 1000 1200 time from start o f maneuver (sec) Figure 6. Observed and simulated spin axis attitude for the first 1200 seconds of spin reorientation maneuver (observations are from POLAR spacecraft maneuver on April 16,1996).

As the maneuver progresses, the Earth horizon sensor scans across the Earth along various chords. The scan cone moves off the Earth at 03:42:58 GMT (time = 1372 sec, off scale on Figure 6). After that time, only Sun data is available and there is insufficient information to compute an attitude until the Earth again comes into view near the end of the maneuver. The estimated attitude also is less reliable due to larger Earth sensor un- certainty during the minutes immediately before the Earth horizon signal is lost. Thus, 47 1 although this 90 deg segment of the maneuver takes 3.5 hrs, the attitude is observable only for the first 23 min while both the Sun and Earth are detectable, and the final 3 min are discarded.

As seen in Figure 6, the flexible body model exhibits oscillations that are qualitativ- ely similar to the oscillations in the observed flight data. These slow oscillations are not present in the rigid body model in Figure 5. The flexible model oscillations match up exactly with the envelope of the boom tip displacements shown in Figure 2 and probahly correspond to a beat frequency between two out-of-plane modes.

The quantitative features of the slow oscillations depend primarily on the material parameters of the wire booms, in particular the mass density and the bending stiffnesses.

The periods are approximately 2.6 min for the flexible simulation and 2.2 min for the observed attitude. This 15% difference might be attributed to uncertainties in the material parameters available to us.

There also is fairly good agreement between the simulated and observed amplitudes, however the observed oscillation decays with time while the amplitude of the simulation does not. This reflects the absence of dissipative terms in the dynamics model. As ex- pected, there also are higher frequency oscillations (dominated by a mode with a 30 sec period) that are not found in the simulation results. The simulation would likely have shown a much richer spectrum if the model had included more realistic pulsed thrusters rather than a continuous control torque.

CONCLUSIONS In this study we have examined the attitude of the POLAR spacecraft during a reorientation maneuver of the spin axis. The attitude computed using flight telemetry was compared with simulated data obtained by modeling the spacecraft as a rigid hub with four flexible wire booms. A comparison with a totally rigid body model also was given.

The flexible model simulation captures the most important qualitative features of the flight data and is in reasonable quantitative agreement, as well.

The Cartwheel simulator employs an implicit midpoint time-stepping scheme to perform the numerical time integration and to preserve the dynamics constraints to a high accwacy. If high numerical accuracy is not maintained, the simulation will diverge after only a few minutes of simulated dynamics. The price for high accuracy is relatively slow computational speed. For example, with each boom represented as a single rigid rod, twenty minutes of simulated data required about ten minutes of CPU time on a DEC Alpha workstation. When each boom was subdivided into five-meter segments, yielding 46 segments in all, then twenty minutes of simulated data required about ten hours of CPU time.

A number of features could be added to Cartwheel to improve its accuracy. The most important additional feature would be a more complete actuator model, particularly pulsed thrusters. Following this, one could improve the modeled material properties of the wire booms, including dissipation and more accurate mass and stiffness parameter nviron~ental pe rbatio~s could contribute small corrections, and sophisticate control laws would allow testing a wider variety of scenarios. These improvements could be implemented thout making major changes to t flexible dynamics model. With these additions, Cartwheel would be usefu s tool to study the nonlinear dynamics of extended systems such as long-boom spinners or tethered satel- lites. Its strength would be most apparent for applications to systems under contingency conditions where the deformations are too extreme to be described using linearized or modal dynamics.

ACKNOWLEDG~ENTS This work was performed under a cooperative agreement among the National Science Foundation (Industrial Postdoctoral Fellowship award number 9505450), the National Aeronautics and Space AdministratiodGoddard Space night Center (contracts 5-3 lOOO), Computer Sciences Corporation, and the University GS-35F-4381G and NAS of Maryland Institute for Physical Science and Technology. Additional support came from the Air Force Office of Scientific Research. The authors would particularly like to thank Professor John Maddocks, currently at the Swiss Federal Institute of Technology, who provided guidance for much of the theoretical work that preceded the simulator development.

REFERENCES 1. D. J. Dichmann, Hamiltonian Dynamics o f a Spatial Elastica and the Stability o f Solitary Waves, Ph. D. thesis, University of Maryland, 1994.

2. D. J. Dichmann, J. H. Maddocks, and R. L. Pego, “Hamiltonian Dynamics of an Elastica and the Stability of Solitary Waves,” Arch. o f Rational Mech., to appear.

3. D. J. Dichmann and J. H. Maddocks, “An Impetus-Striction Simulation of the Dynamics of an Elastica,” J. Nonlinear Sci., Vol. 6, 1996, pp. 271-292.

4. J. M. Sanz-Sema and M. P. Calvo, Numerical Hamiltonian Problems, Chapman and Hall, 1994.

5 . M. Tabor and I. Mapper, “Dynamics of Twist and Writhe and the Modeling of Bacterial Fibers,” in Mathematical Approaches to Biomolecular Structure and Its Applications, 82, Springer, New York, 1996, pp. 139-160.

6. J. Dibble and S. Good, “Interplanetary Physics Laboratory (WIND) and Polar Plasma Laboratory (POLAR) Postlaunch Report,” Goddard Space Flight Center, Flight Dynamics Division, 553-FDD-96/006ROUDO, July 1994.

7. A. Calder, “Multimission Spin-Axis Stabilized Spacecraft (MSASS) Flight Dynamics Support System User’s Guide, Update 1,” Goddard Space Flight Center, Flight Dynamics Division, 552-FDD-9 1/019ROUD1, March 1993.

8. J. S . Wertz, ed., Spacecraft Attitude Determination and Control, D. Reidel Publishing Co., Dordrecht, The Netherlands, 1978.

During slew manoeuvres between scientific observations, the X and Slew (IPS) mode control laws compute profiled momentum demands to the reaction wheels in order to achieve the desired change in three-axis attitude. These momentum profiles are computed on-board, together with the demanded Sun position (FSS) field-of-view, such that the change in attitude is about in the Fine Sun Sensor the eigenaxis. The outputs of the FSS along with the demanded sun position are used to provide closed-loop attitude control about the roll and pitch axes during the slew, with the yaw axis being open-loop.

This implies that a number or a combination of system model parameters must be calibrated, in order to limit the size of the attitude error with respect to the planned target attitude due to the lack of yaw control.

A description of the IPS mode control law and the principal contributors to slew errors is presented and an algorithm based on an Extended Kalman Filter is used to estimate 15 states (3 spacecraft rates, 3 external torques, and 9 components of the ‘effective’ normalised inertia matrix), during a series of small offset manoeuvres separated by stable pointing phases used to estimate the environmental disturbance torques.

The perfomance of the algorithm has been assessed ‘by a complete simulation of the XMM dynamics, kinematics, sensors (Star Tracker and Fine Sun Sensor), control laws and actuators (Reaction Wheel Unit) and the results are presented.

Finally, a series of tests on ISO, which uses the same Star Tracker (STR) and FSS, are planned at the end of it’s operational life. This data will be processed on-ground using the proposed algorithm.

INTRODUCTION ESA’s X-ray Multi Mirror observatory, XMM, is planned for launch on August 2nd 1999 by an Mane 5 launch vehicle.

During the Launch and Early Orbit Phase, GEOP), about seven hours after separation from the launcher, an initial calibration of the yaw principal inertia will be performed to remove the largest source of attitude error for open-loop slews. The LEOP will last up until 10 days after launch whereby it will be followed by a commissioning phase.

During the commissioning phase, a complete calibration of the spacecraft moments and products of inertia, Reaction Wheel Unit (RWU) alignments and reaction wheel moments of inertia will be performed. These uncertainties in the RWU model parameters being the second most important source of attitude errors for open-loop slews.

The Flight Dynamics mission planning functions, during the XMM’ routine scientific phase, should benefit from these improved open-loop slew accuracies.

t Science Systems Space Ltd., 23 Clothier Rd., Brislington, Bristol, BS4 5PS, England. Currently based at the European Space Operations Centre, Robert Bosch Str. 5,64293 Darmstadt, Germany.

The IPS provides two distinct functions which are described as follows.

This is used to maintain a stable fine pointing inertial attitude throughout the mission, including scientific observation phases. During nominal sunlit operations, the Fine Sun Sensor (FSS) is used to control the roll axis.

The pitch and yaw axes are controlled using the Star Tracker (STR).

In order to maintain a true inertial attitude, a sun steering law is implemented on-board. This is used to adapt the roll reference demand input to the control law, in order to take into account the motion of the Sun in the FSS field-of-view.

, During the stable pointing phase, the yaw control law computes a filtered estimate of the yaw disturbance, which the on-board software uses to initialise the controller as soon as the slew phase has been commanded.

This is necessary to compensate for the yaw disturbancetorque throughout the slew phase. Any changes of the disturbance torque throughout the slew will not be compensated. At the same time, the proposed calibration algorithm estimates the disturbancetorques prior to executing the calibration slews.

In addition, it is also possible to make small adjustments to the spacecraft attitude by r e d e f ~ g the demanded position of the guide star and the sun. These changes can be commanded using references generated by a profiled offset steering law (small offset manoeuvre) or in the case of attitude changes less than 1 arcmin, a step change to the demanded positions can be commanded. In both cases, the attitude control references are exactly the same as for the stable pointing phases with three-axis attitude control.

The profiled offset steering law (small offset manoeuvre) is used to execute the slew performance calibration manoeuvres.

Slew Phase Slew manoeuvres are performed open-loop about the yaw axis and closed-loop about the roll and pitch axes, using FSS outputs. The control law inputs are the predicted sun positions and wheel momenta as a function of time, to give the required slew about the eigenaxis.

The computation of the wheel momentum profiles assumes a knowledge of the following system model parameters:- * The spacecraft moments and products of inertia * The moments of inertia of the reaction wheels * The alignments of the reaction wheels with respect to the spacecraft functional coordinate system It is for this reason and the fact that the yaw axis is open-loop, that the accuracy of slew manoeuvres is significantly more dependant upon the accuracy of these on-board parameters, in comparison to attitude control systems where gyroscopes are employed. The build up of attitude error is shown to be about the instantaneous sun-~ine~.~.

It is possible to update these on-board parameters by ground command once they have been calibrated in-orbit.

The calibration algorithm is used to estimate nine parameters that can be expressed as a function of the following unknown system model parameters- Spacecraft Inertia matrix (rigid body) Reaction wheel moment of inertia uncertainties Reaction wheel unit alignment uncertainties Then by selectingthe unknown system model parameters such that the estimationresults are preserved, then in the case of perfect estimates, the effects due to the above uncertainties can be compensated and will therefore not introduce errors during open-loop slew manoeuvres.

The system dynamic model used to propagate the system state vector between measurement updates and the associated measurement equation as required in the formulation of the estimation algorithms, are presented as follows. These algorithms are based on a 15 state Extended Kalman Filter, which is used to estimate 3 spacecraft body rates, 3 external environmental disturbance torques and 9 parameters that are a function of the unknown system model parameters described above.

System Dynamic Model We assume that the spacecraft is a rigid body with a reaction wheel u n i t . The model of the system dynamics is therefore given by:- -I (1) 8 = J ( ~ , , - c o ~ ( J ~ + h ) - h ) where o is the spacecraft body rate vector at time t J is the spacecraft inertia matrix (rigid body) res are the externally applied torques (environmental disturbance torques) h is the wheel momentum vector in spacecraft axes at time t Then the derivative of the wheel momentum vector is given by the equation:- where A , is the transformation from wheel axes to the spacecraft functional frame, J , is a diagonal matrix of wheel moments of inertia, and a , are the reaction wheel speeds which are available in telemetry.

The reaction wheel unit alignment matrix and moments of inertia are in general known only to within specified uncertainties:- = a b w + U b w (3)

J w = Sw i- AJw

where AAh is a 3x3 matrix of the reaction wheel unit alignment uncertainties, and A J , , , is a diagonal 3x3 matrix of reaction wheel moment of inertia uncertainties.

Substituting Eq(3) into Eq(2) yields:- where AAb, is a 3x3 matrix of the reaction wheel unit alignment uncertainties, Alw is a diagonal 3x3 matrix of reaction wheel moment of inertia uncertainties, Ab, is the transformation from wheel axes to the spacecraft functional frame, Awb is the inverse of A b , and h is the wheel momentum vector in the spacecraft functional frame.

* denotes the nominal value of the quantity. In the case of the wheel alignments and moments of inertia, this data is based on pre-flight measurements.

Then substituting Eq(4) into Eq(1) yields:- -I d =

- o A (JO + oil)) - J D-#)

where on neglecting 2nd order error terms:- I - 1 The state vector chosen for estimation is then given by:- where the 9-vector, d, is made up of the rows of the matrix D, (dl, d2, d3).

Eq(5) foxms the basis for propagation of the system state vector between measurement updates.

Linearisation of this equation will be required to propagate between measurement updates: the system state transition matrix, using E q (8), and the error state covariance matrix, using Eq(9). This linearisation is done numerically using the current estimate of the system state vector.

T

P - ( t k ) = W k , t k - l ) p f ( t k - 1 ) @ (fk, f k - 1 ) + w, t k - 1 )

where F(r) is the linearised system dynamics matrix, cf, is the system state transition matrix, P is the error state covariance matrix, and U is the state noise covariance matrix.

The measurements available during the short calibration slews, performed using the IPS small offset manoeuvre law, are the sun vector construceed from the fine sun sensor outputs and the star vector constructed from the star tracker outputs. For the estimation algorithm, equations are required to be a function of the system state vector, which was previously defined by E q (7). During slew manoeuvres the observation vectors, (&-I, Sk), constructed from two successive samples of the sensor measurements, are perpendicular to the instantaneous spacecraft body rate vector. The rate vector can therefore be written a s : - = ' k A ' k - 1 oorlhog AT where AT is the sampling period, and it follows that:- Applying the vector triple product identity yields:- The measurement equation is then a time-varying linear function of the system state vector and can be expressed as a function of the current estimate of the state vector a s : - z k H k x k (13) where it follows from Eq(12) that:- The measurementsprovided by Eq(12) will be used to derive state corrections, using J2q( 15), in order to refine the current estimates of the state vector. Also, the measurement geometry matrix given by Eiq( 14) will be used to compute the update gains, using Eq( 16) and to perForm measurement updates to the error state covariance matrix, using a numerically stable version of E q ( 17).

2 ; = 2 ; + K k ( Z k - H f i i )

(15)

K k = p-( t k ) H i [ H k P - ( f k ) H i + R ]-I (16)

P + ( t k ) = ( I 3 - K k H k ) p - ( t k ) where K is the update gain matrix, P is the error state covariance matrix, and R is the measurement noise covariance matrix.

The estimation algorithm provides an accurate estimate of the following matrix Rearranging, we have Now for the operational wheelset, the transformation from wheel reference axes to the spacecraft functional coordinate system is given by

= r.11 Bzl B~ a b w M b w (20)

where the alignment for the i~ wheel in the spacecraft functional coordinate system is given by

S h ( 6 i + Aei)

which, after neglecting 2nd order terms, becomes Bi = &+ABi Using Eq(22), the i" column of the following matrix is given as follows J W i J W i AJWi ARi = I : - j W i j W i Then, the spacecraft inertia matrix given by Eq( 19) can be written as Defining the error in the spacecraft inertia matrix as

AJ = J - I

Then, Eq(26) can be re-written as AJE = [DI.II D2v21 D3vJ + F where From Eq(28) above and using the fact that Al is a symmetric matrix, then

- -

AJII A J 1 2 -AJ33- Then from Eqs (28), (29), and (: , we can express these equations as a system of linear equations of the form G I -D1 0 0 G2 0 -D, 0

~] f 3

V 2 i G3 0 0 -D,

"'- v3

where the coefficient matrix is rank-deficient and the 15 unknowns are the errors in the system model parameters.

This problem possesses an infnite number of solutions,but exactly one with a minimal 2-norm. A minimisation of the estimation errors is the desired solution. Weights which are a function of the worst case uncertainties in the system model parameters are applied to the state vector, such that the errors are equally weighted.

48 1 The procedures consist of executing 3 independent yaw, pitch and roll slews using the profiled offset steering law. The procedures for each calibration slew are essentially the same for each axis but with different initial conditions. In all cases, a guide star must remain tracked continuously throughout the slew manoeuvre.

Due to the operational time constraints (1 hour) during the LEOP, only the calibration of the yaw principal inertia will be performed.

During the subsequent commissioning phase, after the perigee raising bums and the opening of the mirror doors, the full calibration sequence will be performed.

The following data are read at regular intervals from the local flight dynamics telemetry history files and are converted, as specified earlier, into a suitable format as required by the algorithms: e STR position coordinates, magnitude and star data status.

e Raw FSS roll and pitch angle data.

* Raw wheel speed data.

Calibration Procedure The following describes the operational procedures required to execute the yaw axis calibration small offset manoeuvre:- Within the Inertial Pointing and Slew (IPS) Mode, slew the spacecraft to boresight the Sun on the +Z-axis (FSS axis) such that measured FSS a = = 0 (to within 1 FSS output quantum) and the k X axes are in the ecliptic plane*. This will be done using the open-loop slew mode so that a star map can be processed at the end of the slew. Select a high quality' guide star from one of the stars in the map near the centre of the STR FOV. If necessary, perform a small offset manoeuvre to place the guide star close to the centre of the STR fieId-of-view.

Transition to Thruster Control Mode (TCM). Command the desired wheel speeds in order to avoid operating the wheels in low speed regions ( a , > 300 RPM) and overspeed regions ( w , e 3000 RPM).

These constraints should be taken into account during the planning of the small offset manoeuvre as well as the start and end points.

Transition back to IPS Mode. Slew back to a Sun boresighted attitude with the guide star near the edge of the STR field-of-view, so that the whole yaw width of the STR field-of-view can be used for the calibration small offset manoeuvre (- 4 degrees).

After 1800 seconds without any motion in IPSs, command the calibration small offset manoeuvre to execute a pure yaw slew. The maximum yaw rate during the slew is limited by the on-board control laws to about 32 arcsedsec.

After completion of the calibration slew, and 1800 seconds without any motion in IPS, command another small offset manoeuvre back to the initial attitude. The maximum yaw rate during the slew is limited by the on-board control laws to about 32 arcsedsec.

The pitch and roll axis calibration slews are very similar to the yaw procedure described previously and are described in4.

*. This ensures that there 'is no attitude drift due to the motion of the Sun during the stable pointing phase and the small offset manoeuvre phase, when the sun steering law is not active I-. This star should be a bright star, (8.5 <,m < 2), to reduce the effects of STR noise and biases.

$. This time is allocated for the on-board control law and the on-ground disturbance torque estimation processes to converge.

The algorithm has been validated using a complete simulation of the dynamics, sensors, XNM IPS mode control laws, and actuators. These cases have been selected to demonstrate the correct performance of the estimation algorithm and its sensitivity to various conditions. The algorithm performance and sensitivity has been assessed4 with respect e o the following conditions: 1. Sensitivity to extreme variations of the spacecraft moments and pr f inertia.

2. Sensitivity to extreme variations in reaction wheel moment of inertia and alignment uncertainties.

3. Sensitivity to FSS misalignment and biases.

4.

Sensitivity to environmental disturbance torques.

In all cases, the worst case sensor and actuator noise and quantisations are used4. Also, the reaction wheels were biased to the desired values prior to the calibration small offset manoeuvre.

These simulations consist of independentlyexecuted yaw, pitch and roll slews carried out using the small offset manoeuvre law required to estimate all parameters, (Le. the column vectors of the matrix in Eq(6)). From these column vectors, the components of the spacecraft inertia matrix, ( J I I , J12, J13,522, J23, J33), three reaction wheel unit alignment angles, (Ae1, A92, Ae3), and wheel moments of inertia, (JwI, Jw2, Jw3) are computed by solving Eq(31). The estimation results for two examples are presented in Tables 1 and 2 below.

Table 1 : Uncertaintiesin wheel moments of inertia and RWU alignments (Case 1) Table 2 : Uncertainties in wheel moments of inertia and RWU alignments (Case 2) In order to show clearly the accuracy of open-loop slews, before and after the slew performance calibrations have been performed, three separate simulation cases are presented. The open-loop slew is about the yaw axis with an amplitude of 20' and a slew rate of 20"hour. These are listed as follows:- * Without any compensation applied.

* Compensation only for the yaw principal inertia applied.

e Complete compensation of the spacecraft inertia matrix, reaction wheel moments of inertia and RWU alignments applied.

The simulationresults are presented in the same order as the slew calibration results are presented in Tables 1 and 2, thereby showing the effects of the above three scenarios for each example.

The main conclusion that can be drawn from these simulationresults, is the justification of the statement made earlier about selecting the unknown system model parameters in order to preserve the estimation results of the matrix given in Eq(6). This is clearly seen by the low slew errors shown in the last of the three plots in Figures 1 and 2, where the complete compensation is applied.

also interesting to note that with the introduction of constant FSS misalignments and biases, that the It is estimated parameters from the slew calibrations are referred to the optical references and not the spacecraft functional reference frame. This has the effect of taking into account the alignments of the reaction wheels relative to the optical references. This is necessary so that FSS misalignments and biases do not contribute to slew errors3. Simulation results for this case are given in4.

Finally, several other simulations have been performed the results of which have not been included in this paper, and consist of the following: = Increased slew rates (2Whour up to 90"hour).

* More general slews where the Sun is not boresighted and the eigenaxis is in different directions.

It is confiied that by commanding higher slew rates, the slew accuracy is improved even more, due to the fact that errors are inversely proportional to the slew rate.

Also, more general slews can be worse in terms of accuracy than the pure yaw slew results that are presented in Figures 1 and 2. However, it is assumed in general that they are still si,dicantly more accurate than the corresponding cases where only the yaw principal inertia has been calibrated.

IN-FLIGHT TESTING USING IS0

The aim of these tests is to have an apriori in-orbit evaluation of the following points:- = Rate estimation accuracy from FSS and STR outputs throughout the short calibration slews5. Derived rates6 will be compared with accurate rate signals from the gyropackage to determine the effects of slew rates on the quality of the STR outputs.

* Partial validation of the estimation algorithms. Only a partial verification of the algorithms to estimate the unknown system parameters can be performed due to the fact that the IS0 AOCS multiple gyro failure on-board software does not have a completely gyroless slew mode. The design of this new on-board software assumes that there is at least one gyro left out of the four for use during slew manoeuvres.

Although it is possible to load on-board, different values of the spacecraftinertia matrix, it is unlikely that these effects would produce slew errors as large as those experienced during the Xh4M slew phase, where no gyros are used.

No compensationapplied No compensationapplied Time (semnds) Time Ccsmnds) Compensationfor error in spacecraft Compensation for error in spacecraft yaw principal inertia yaw principal inertia Time (seconds) Time (ramnds) I L Complete compensationapplied Completecompensationapplied Figure 1 : Uncertainties in wheel moments of Figure 2 : Uncertainties in wheel moments of inertia and RWU alignments ( C a s e 1) inertia and RWU alignments(Case 2) A method lo calibrate the spacecraft moments and products of inertia, reaction wheel moments of inertia and alignments for the Inertial Pointing and Slew mode attitude control system has been proposed.

Simulation results show that an order of magnitude improvement in the accuracy open-loop slews is achieved, when these parameters have been loaded on-board in the AOCS , compared with the same slews where only the yaw principal inertia has been calibrated. This should be enough to ensure that for most observational slews that the guide star intended for use at the final target attitude, will lie within the 3 ” x 4 field-of-view of the STR. A small offset manoeuvre can then be used to reposition the guide star to the desired location following every open-loop slew.

ACKNOWLEDGEMENTS The author would like to acknowledge various members of the IS0 and XMM flight dynamics teams for their contributions. In particular, Mr C. Stephenson for setting up the IPS mode calibration tests on IS0 and providing the flight data for these tests and to Mr G. Gienger for his help and suggestions.

REFERENCES 1.

G. Gienger, A. McDonald, and J. Palmer, “Flight Dynamics Support for the Gyroless Operations of XMM,” Proceedings of the International Symposium on Space Flight Dynamics, Dannstadt, June 1997.

2 . H a r r i s R. S. (Matra Marconi Space), “XMM AOCS Design Report, “ XM-RP-MMB-0018, Issue 3, September 1997.

3. Chapman P. D. (Matra Marconi Space), “XMM AOCS Per€ormance Review Document Volume 5, Inertial Pointing and Slew (IPS) Mode” XM-RP-MMB-0026, Issue 2.

4. M. J. Tuttlebee, “Calibration of the Inertial Pointing and Slew Mode for XMM and INTEGFWL, “ XMM- MOC-TN-O1140AD, Issue 1, December 1997.

5. IS0 Flight Control Team, “Mared Space Observatory (ISO) Technology Test Plan, “ TOS-OFC-ISO- ”FVJF, Issue 1 Rev 0, January 1998.

6 . M. J. Tbttlebee, F. Dreger, “Feasibility of FSS?STR based slew attitude determination, “ INT-MOC-TN- 0002-0AD, Issue 1, March 1997.

David Sonnabend’ Abstract For the Rosetta mission to orbit a comet, due to launch in 2003, and spend about a decade en route, the European Space Agency contemplates the use of a series of “hibernation” periods.

T h i s is both to conserve resources, and to reduce expensive ground operations. The general idea is to point the solar arrays at the sun, spin the spacecraft at a low rate about the sun line, suspend communications with the ground, and turn o f f most spacecraft equipment. A factor tending to limit the value of this idea is that, as Rosetta moves in its orbit, the direction to the sun changes, reducing the available power from the array. The worst case from this standpoint is a fhal hibernation period between a 2nd asteroid flyby and the comet approach phase; because the spacecraft is then most distant from the sun. T h i s hibernation could be as long as 3 years, and the sun direction (true anomaly) could change as much as 70 deg. Even biasing the initial direction of the spin axis to favor the later, more distant part of the orbit, would still lead to a maximum array offset o f around 30 deg, when the loss of power would be about 13%, worse if various asymmetries and external disturbances are taken into account. T h i s paper advances a passive technique for using radiation pressure to cause the spin axis to track the sun. A fairly complete analysis is presented, along with calculations of the performance.

1 DISCUSSION

To discuss the main idea, and various possibilities for disturbance, the spacecraft and the body axes need to be defined. When this study was done by the DASA - Aerospatiale proposal team, the spacecraft had a pair of solar wings, on separate array drives. The design details are given below. In x axis, the axis of maximum moment spin mode, the direction to the sun is close to the spacecraft of inertia. The y axis is the solar array shaft axis, the axis of minimum moment of inertia; and z completes a right handed system.

of mass. If there is an x lst, suppose the center of pressure doesn’t coincide with the center displacement, there is clearly no effect. A y displacement causes a torque along z; but this rotates with spin, and averages out. Similarly, a z displacement leads to a y torque, which also averages out.

Clearly, we are not concerned with center of pressure migration. Another possibility is a shape or reflective properties variation between wings. In either case, we again have a rotating disturbance, which averages out. Finally, if the array shaft axis is rotated, either from bias or deliberate control, there will be a force component along z. Either an asymmetry between the wings, or a differential control of the 2 array shaft angles, yields a torque along x. This torque is unaffected by rotation, and thus tends to change the magnitude of the angular momentum. A formula for this “propeller” torque is derived in Section 5, followed by a discussion of possible methods for controlling spin rate.

There is one other substantial source of solar torque. If both wings are bent back, away from the sun, we have what aircraft designers call “positive dihedral”. If the x axis is pointed at the sun, as desired, the effect is balanced, and there is no torque. However, if x is offset from the sun by some angle, then one wing receives greater illumination than the other, yielding a z torque. This torque varies during spin; but, after a half rotation, it’s the other wing that is more favorably placed.

President, Analytical Engineering, 303-530-9641, &mail dsonnabend@lworldnet.att.net Moreover, the torque has the same direction in space; so, the torque history looks more or less like a rectified sine wave; and on average, there is a torque along z , tending to cause the spin axis to precess about the sun line. It will be shown below that the precession cone naturally tracks the sun direction. A similar idea was analyzed in the Reference, and put forward as a method of passive control of spin axis direction. A simplified analysis of the dihedral effect is presented in Section 3, and then applied to the present design.

The configuration explored in the Reference kept the solar panels parallel, but added fixed dihedral vanes at the ends of the panels. Relative to the design proposed here, vanes pose some disadvantages.

1) the vanes add some mass; 2) there are 2 extra deployment joints, with their associated mass and deployment commands; and 3) the most natural accordion folding would cause the vanes to cover the outer array panels, thus eliminating any solar power before deployment. The disadvantage of the proposed design is that dihedral causes some loss of power; however, even with 3 deg of dihedral, the loss is only 0.137%. A few years earlier, a Russian proposal for a series of satellites called Regatta employed essentially this same idea for passive attitude control; but the program died for lack of support; and I haven’t seen their analysis.

2 FLAT PANEL ANALYSIS

The forces and torques discussed in this paper all depend on the force on a flat panel at some angle to the sun. If an object absorbs sunlight, and reemits this energy isotropically, then the pressure is given by Is/c, where I , is the solar irradiance, and c = 2.99776 x lo8 m/s = the speed of light.

Applying this to a flat panel of area A, whose normal is pointed toward the sun, the force i s : where Ise = 1367.5 w/m2, the mean solar irradiance at 1 AU; and T is the solar distance, expressed in AW. In the current design, each wing is 14.142 m long and 2 . 2 1 6 m wide, from which A = 31.339 m2. Thus, at 1 AW, F o = 1.4296 x N on each wing. To stray a bit, counting both wings, plus something for the antenna and spacecraft body, the total solar force would be about 3 x N at 1 AW; so with a nearly dry spacecraft at 1200 kg, the solar acceleration would be about 2 . 5 x m/s2. This would not be directly detectable by any accelerometer Rosetta would likely carry, but possibly would show up in a long term average.

Now suppose the spacecraft remains correctly pointed, but the panel is somehow tipped about the y axis through an angle 0 5 4 < ~ / 2 . Further, suppose that the incoming photons are divided into a fraction Up that’s specularly reflected, a kaction U d that’s diffusely reflected, and a remaining fraction that’s absorbed and isotropically reradiated. Then these fractions obey For the absorbed fraction, the force is along the sun line, and is Fo, reduced by c q 5 (c and s stand for the cosine and sine respectively). The specularly reflected fraction gives twice the force, similarly reduced, whose resultant is along the panel normal. Finally, for the diffusely reflected fraction, the incoming part behaves like the absorbed rays; but the reflected photons are spread into a cosine distribution, whose resultant is along the panel normal. These features are shown in Fig. 1. A well known result is that the average of the cosine over a hemisphere is 2/3. From this, the normal and transverse force components are readily shown to be: FT = - F o (1 - ar)sq5c4 (4) In the latter formula, the transverse direction is taken a s in the panel, and closest to the sun line.

These formulas will be applied below to obtain particular torqtie components. As an aside, the Reference assumed that a , . = 1, when these equations reduce to FN = -2F0c24 ; FT = 0 However, solar cells are dark blue to the eye in sunlight, so we must have a , > a , .

c

Figure I - Flat panel

3 DIHEDRAL TORQUES

To analyze this 3 dimensional problem, it’s helpful to introduce some coordinate systems. lst, a more or less inertial system is defined by choosing a unit vector E 1 along the spacecraft spin axis, followed by an orthogonal unit vector E 2 in the plane containing the sun, and a 3rd unit vector E3, completing an orthonormal system. If the sun is an angle 8 forward from El, then its direction is E, = [ce, se, o]?

( 5 ) where the subscript indicates that the resolution is in the inertial system, and the superscript signifies transpose.

b

Figure 2 - Dihedral forces

The spacecraft system; with base vectors E,, E y , and E,; is obtained by rotating the inertial coordinates about E 1 through the angle a, which advances at spin rate. To analyze dihedral solar panels, we need some definitions. ferring to Fig. 2, shown with negative dihedral, suppose the solar array shaft axes are aligned with Ey, and that these axes are forward along Ez by a distance f. Also suppose the joint connecting the mounting yoke to the beginning of the array is a distance 9 from the center line. We then assume that the arrays are each bent back by a dihedral angle $.

If the array length is 2 4 and if the shafts are displaced by a distance h along z , the locations of the centers of pressure of the 2 wings are

p+/- = [f - &$7 f(g + bc$)7 h]; E [ P I , f p2, h]g (6)

where this time the resolution is in body coordinates. We also need the directions of the outward normals at these points. These are readily worked out from Fig. 2. In body and inertial coordinates we have EN+/- = IC$, fs$, 0 1 ; = I C $ , fs+ca, fs?)sa]T (7) These relations are sufficient to compute the solar offsets used in Section 2, for each wing: c$+/- = E, * EN+/- = cOC.?~, f S~S$CCU (8) It remains to determine the direction of the transverse component of the solar force. This lies along the panel surface, and in the plane defined by the panel normal and the sun direction. It's not hard to show that this obeys:

E, = EN+&+ + ET+s$+ = EN-c$- + ET-$-

(9) These lad relations are only useful if the sun illuminates the front of both wings, a condition met if 1 4 < (n/2) - $* We're now ready to compute the combined torque. Since torque is the cross product of the center of pressure vector by the applied force, we have here:

T = P+ x F+ + P- x F-

(10) where the total force on each wing i s given by F+/- = F N + / - EN+/- + FT+/-%+/-

From (3) and (4), the + wing is

and using (9) to eliminate ET+, this reduces to and similarly for F-. Next, we construct the cross products: p+/- x EN+/- = [Ths$, hc$, Tf'& = [Ths$, hc$ca f P~SCY, hc$sa 'f P ~ C C Y ] ; (14) where

P3 E P~c$ - PIS$ = b + g ~ + - f ~ ? )

On substituting these into (13), and the corresponding F, relation, and averaging over cy for one spin cycle, we find - (16) TN+/- = FOG, and on combining the 2 wings: So the normal components lead only to a torque along E3, vanishing if either 6 or $ is zero.

A somewhat different procedure is needed for the Es terms. This time, the total torque is: 7s = -Fo(l - .~)(p+C(h+ + P-C(h-> X E, (18) The 1st vector is easily found in body axes, transformed to inertial axes, and averaged, when

(P+c(h+ + P - C $ - ) ~ ~ = [2Plc6c+, P2sBs$, 0 1 : (19)

so that Ts = - F o (1 - 0,) P4 E~s~cB where

2P1~$ - P~s$ = 2 f C$ - gs$ - 3bs$c$

P 4 Note that if $ = 0, i.e. there is no dihedral, this reduces to 7 s = -2Fof(l- UT)E3S6C6 ($ = 0) So there is a torque along E 3 even without dihedral. It may be seen that it arises from the transverse components of the force, caused by the absorbed and diffusely reflected fractions of the incident sunlight. Moreover, it's proportional to the distance f , the forward displacement of the solar array drive. axis along x. Finally, we may combine these components and get: For small 6 , this last expression reduces to The scalar 7 ' introduced here is a measure of the control authority available from the dihedral idea; i.e., it's the precession torque per unit solar offset. It will be computed for the proposal design parameters in the next section, and used to determine the dynamics of solar tracking.

In passing, it's been suggested that another source of control torque might come from solar pressure on the high gain antenna, especially as the antenna drive angles could be fixed in any desired positions during hibernation. The calculation of the effect closely follows that of the dihedral arrangement.

To use the same nomenclature, suppose an az-el configuration is adopted, and the azimuth axis is fixed in the position where the elevation axis is normal to the spin axis. Then, if we approximate the antenna reflector as a flat circular plate of radius b = 1.1 m, we have an area of 3.8 m2; when from (1)) r2Fo = 1.734 x lo-' N. Now, suppose the elevation axis grips the plate on an edge at the body coordinates [f, g, 0 1 , and the plate is bent back through an angle $. Here, g is actually the radial coordinate from the spin axis; so the h coordinate used in the dihedral analysis is here unnecessary.

With these definitions, the center of'pressure is at 49 1 corresponding to P+ in (6). The analysis proceeds in the same way, and the spin averaged torque can be read off from (16): 7~ = Fop3 (20rC@C8 4- (26) Now in body axes: P = [PI, P2coI, P2SoII; so that (pC4iav = klc+ce, z ~ 2 s w ,

O I T

and, similar to (20): i - T, = - A F ~ ( ~ - O-,.)P~E~SBCB Thus, if 8 is small, r’ becomes independent of 8: This expression closely resembles (24); it will also be computed in the next section.

4 CONTROLLING SPIN DIRECTION

We may now consider what happens when the sun is in motion, as seen from the spacecraft. For coordinates, take a system whose origin is at the spacecraft, and whose orientation is fixed relative to J2000 (i.e., the stars). Then arrange things such that E3 is the spacecraft orbit normal, and that the sun is initially in the direction El. These axes aren’t the same as in the last section. Then the sun will appear to move forward toward E 2 at a rate n, the current rate of change of the spacecraft true anomaly. Assuming n to be fairly constant (to be refined below), we may express this as

E, = [c(nt), s(nt), oIT (31)

We also need to express the spacecraft angular momentum L in this system. Choosing spin axis latitude p and longitude X as the descriptive variables, we have: L = I,w,[c~cX, c ~ s X , z I=w,EL (32) where I, is the spin moment of inertia of the spacecraft, and w, is the spin angular velocity. In these terms, the angle 8 between EL and E,, as used in the last section is

c8 = EL - E, = c(X - nt)cp z cycp

(33) and the differential longitude 7 must be small.

Clearly, for 8 to be small, both Now, in the last section, the solar torque due to dihedral wings was found to have the form - T r = r’E,s8 = #EL x E, = r’[-s(nt)sp, c(nt)sp, -srcp] (34) where from (24) and small 8 , r’ has the constant value (35) The stage is now set for the dynamics. From L = 7, and the above expressions, we find

ACPSX + BSPCX = wps(nt)sp

(36) Bcp = -wpsrcp where

wp = r’/(Izws) (39)

.I Strictly speaking, this assumes that the spacecraft is spinning on a principal axis, not necessarily the case. However, with good design, and some form of passive damping, the assumption should be good enough. There are actually only 2 independent relations here - the 3rd relation, and from a combination of the 1st 2: Now, the small angle hypothesis may be applied to P and 7, when That this linear system has a sinusoidal solution is immediately apparent: n

P = Ac(wpt + p) + - = As(upt + p)

; WP a s is readily verified by differentiation. Thus, the angular momentum L precesses circularly, but displaced in latitude from the sun by an angle n/wp. Moreover, wp is the precession frequency given By summing the squares, we in (39). The precession is in the opposite direction from the spin.

obtain a constant of the motion, revealing further insight:

(P-$) + y 2 = A 2 (43)

Some reflection shows that, to keep 8 as small as possible, we should point the spin axis initially to the coordinates when A = 0, and the spin axis follows the sun at a fixed separation 8 = n/lwpl. On the other hand, if we initially pointed at the sun ( P = y = 0), the later maximum deviation of 8 would be twice this.

This behavior is illustrated in Fig. 3.

A physical argument may help to make this precession behavior more understandable. Suppose we initialIy point the spin axis to the optimal offset position. Then, from the analysis of the last section, the resulting torque will be orthogonal to both the spin a x i s , and the projected direction of the sun.

In the coordinate system used here, this torque is in the longitude direction, which is just what is needed to follow the sun. As for the magnitude, the required torque is given by r / L = n. But we also have r = f 8, so e = r/# = nL/# = n/wp (45) and we may conclude that just this offset is required to produce the torque needed to cause L to precess at the desired rate n.

8 a E, E,, Figure 3 -Apparent solar precession In following this strategy, there are 2 requirements that limit the choice of w,. One of these arises in the control of spin, and will be discussed in Section 6. The other is because we need to make lwpl >> n in order to keep the offset within reason, say A6 = . 0 1 or . 0 2 rad. This may be stated a s wp > n/A0 (46) &om ( 3 9 ) , the corresponding limitation on w, is w, < r’AO/(nIz) (47) It’s useful to see some numbers. In the design examined here, we have f = .024 m, g = 2 . 8 2 m, and b = 7 . 0 7 1 m. A l s o , the reflection coefficients are a , = 0.69, = 0.31, and a d = 0. Now from (23), for small 6 and $, we have = Fo(39.072$ - -03312) (49)

For a few values of + and T this is

@ - r a d 0 .01 .02 .05 0 . 1 TI at 1 A U - pN-m/rad -4.73 5 1 . 1 107 275 554 rt at 5 AU - pN-m/rad 2.04 4.28 11 -0.189 2 2 . 2 Note that the contribution of the constant term is small; but, in a different design, the distance f , from the center of mass forward to the solar array drive axis, could be a good deal larger. When this analysis was 1st undertaken, it was believed that positive dihedral was necessary for stability.

However, it’s now clear that a negative dihedral would merely reverse the direction of precession and the latitude of the offset. Moreover, from (49), it’s clear that the sign of $ should be opposite from f; so it would be better to make + < 0 here. In this case, f is so small that this change would improve 7’ by only about 2 % .

To compute wp, an average value of 1, is about 2 x lo4 kg/m2. Thus, at 1 AU, for say w, = 1 deg/s, and $J = .05 rad, we get wp = 7 . 8 8 x rad/s, yielding a precession period of 7 . 9 8 x lo6 s or 9 2 . 3 days. Alternatively, for 5 AU, we might lower w, to 0 . 1 deg/s, when wp = 3 . 1 5 x rad/s, for a period of 231 days. As for the precession, if we take A8 = .02 rad, and consider the final hibernation period, about 1 rad is traversed in 3 years, giving n x l o ’ * rad/s, so we require w, < .0011 rad/s = -063 deg/s.

This whole behavior may become clearer by considering what happens with increasing w,. This causes L to increase, increasing the “resistance” to the torque 7, when the tendency to follow E, weakens. Geometrically, wp is decreasing, which increases the latitude offset of the precession circle from the sun. If the spin axis is initially pointed at the sun, the precession circle is tangent to the y axis, and L heads backward. If w, --f 00, L merely moves backward at the rate -n; i.e., it is inertially fixed.

A similar calculation may be made for the high gain antenna. As the position, and elevation drive details weren’t settled at the time of the study, we took f = g = 1 . 5 m as representative nominal dimensions, along with the established b = 1 . 1 m. f and g were varied, to see if T’ could be substantially changed. Also, except f o r the last case, suppose the antenna is painted with thermal white, for which a, = 0 and a d = 0 . 8 . This time we’re free to choose $ to be anywhere in the range 1$1 5 7 r / 2 . So, for each choice of the parameters, $J was varied to find the maximum 1 7 ’ 1 . The results are in the table below.

Case T2Tf f g a, ad $opt m m rad pN-m/rad 1 1 . 5 1 . 5 0 0 . 8 -0.79838 -52.435 2 -1.5 1 . 5 0 0 . 8 0.79845 -52.435 3 0 1 . 5 0 0 . 8 f0.93823 f31.541 4 3 1 . 5 0 0 . 8 -0.70032 -74.714 5 1 . 5 3 0 0 . 8 -0.86885 -65.530 6 1 . 5 1 . 5 1 0 -0.78540 -55.858 Case 1 is the nominal case cited above. For comparison the previous table shows that similar performance is available from the solar arrays with a dihedral bend of about .01 rad; and for 0.1 rad, the arrays yield an order of magnitude greater authority. Also note that $opt < 0, so that bending the array forward is preferable. This preference is much stronger than for the array, as the value of f is much larger here.

Case 2 differs from Case 1 in that the sign of f has been reversed. While such a change would be impractical in the above design, it shows the expected behavior - that only the signs of $opt and r ‘ have reversed. Case 3 supports this conclusion by examining f = 0. As expected, the variation with respect to $ is antisymmetric; and equal performance may be achieved by bending the antenna either forward or backward.

4 and 5 inquire into what happens if the elevation joint can be pushed further out, either Cases forward of the center of mass, o r radially from the spin axis. There are improvements in r’, but not as large as might be expected, as both f and g increase the authority. Finally, Case 6 returns to the nominal dimensions, but replaces the white paint with a purely specular reflector (polished aluminum, say). Since the results vary only slightly, it would appear that the surface properties make little difference.

We’re now in a position to bring all this theory together to calculate the optimal $J. One design constraint is that lAel 5 20 deg during hibernation, to avoid excessive heating of the side panels. If no stronger constraint is active, and $ is given, then the maximum w, during a particular hibernation may be calculated from (49) and (47). However, during the final hibernation, there could be a very tight requirement that the average power loss not exceed 0 . 5 % due to mispointing of the array. This amounts to a limitation on the average values of 4+ and &. From (8), this means we must enforce c9C$ 5 e#+/- = 0.995 (50) Since all these angles must be small, this may be written as where 4~ = 0.10004 rad = 5.731 deg. Our optimization of $ now comes from the need to relax (47) as much as possible, consistent with the constraint ( 5 1 ) . Since we’re combining the contributions of the arrays and the antenna, it’s best to rewrite (49) as

r2rt = -1.4296 x 10-4(39.072$ + .03312)

while the nominal antenna has r2rt = -5.2435 x N-m/rad; so the combined authority is r2rt = -.0055857$ - 5.717 x

= -.0055857($ + 6)

where 6 = .010235 rad. To get optimal performame, we must maximize the function The solution is readily shown to be

$ = 1 4 (Ja - 5) = .068226 rad = 3.9091 deg

(53) when 9 = .073166 rad = 4.1921 deg, and the combined authority is r2rt = -4.3826 x N-m/rad.

For the final hibernation, the average T = 3 . 6 5 AU, and the average n = 1 . 4 x rad/s. Thus, with this design, the upper limit on spin rate is: (4.3826 x 10-4)(4.1921) = 0.49252 deg/s ws 5 (3.65)2(1.4 x 10-8)(2 x lo4) and at this spin rate, the precession rate is wp = 1 . 4 x 10-8/.073166 = 1.9135 x lo-’ rad/s for a precession period of 380.06 days. Rather long, but with accurate initial pointing, it shouldn’t make much difference.

For contrast, we may ask how these results would vary if the average power loss requirement were relaxed. When this study was completed it was believed that a 2% loss could be tolerated, in contrast to the 0 . 5 % used above. In this case we find ( 6 ~ = 11.478 deg, $ = -7.9711 deg, 9 = 8.2588 deg, r2rt = -8.3427 x rad/s, N-m/rad, and w, 5 0.9703 deg/s, at which speed, wp = 9.7128 x for a precession period of 748.73 days.

5 PROPELLOR TORQUES

Up to this point, it’s been assumed that the solar array drive angles have both been set to zero; so that, with no dihedral, both array normals are along the body 2 axis; i.e., EN+/- = fl, 0, 0 1 ; .

More generally, suppose the drives are controlled to the angles q+ and q- for the 2 wings. A pure propeller setting would nominally have q- = -q+; but in order to examine errors, this assumption won’t be made for the main derivation.

From the layout of the spacecraft, the array drive angles must be inserted before the dihedral bends.

As the drives rotate about spacecraft y, this causes the normals to shift to T EN+/- = [cq+/-, 0, -sq+/-]B Now putting in the dihedral bend about the rotated z axis gives (54) E N + = [c$cq+i s'5oc7]+, -s7]+]: EN- = [c$cq-, - s $ q - , -sq-]B T (55) The new rotations also modify the locations of the centers of pressure. Some consideration of the diagram shows that (6) is generalized to [P+i, P z , f ' + 3 ] 5 (56)

P+ = [f - b s $ ~ ) ) , , 9 + h$, h + bs'zos77+1~

when the essential cross products become (in body axes)

b(sZ$cq- + c$)sq- + gsq- + hs$cq-

P- x EN- = fsq- + hc$q- - b(1- c$)s+sT~L~-] E]

(59)

[ [b(s2$JC7)- + c2$> - fs$ + gc$] ell-

To analyze this more general arrangement, we may again make use of the inertial coordinates intro- duced in Section 3. Thus, E, is again given by ( 5 ) ; and again the body axes are rotated forward from inertial axes about E1 by an angle cy = w,t. Thus, after switching the cross products to inertial coordmates we have:

P+ x EN+ = [Vi, UZCC~ - U3=, UZSQ + U~CCY]T (60)

P- x EN- = [Vi, &CCX - KSQ,

+ I$CQ]? (61)

The angles between the sun and the array normals are again important; but instead of (8) we now have

~ 4 - = E, EN- = C$CT@ + Sq-sOsQ - S $ ~ - S ~ C Q

(63) The analysis leading to (10) and (13) is unchanged; so, after averaging over a spin cycle, the inertial components from the EN terms are:

U I (or [(2 - 3s2e)C2$cZ7]+ + s%] + $adc$cq+ce}

(64)

(Uzs$cq+ - U3sq+) (2arc$Jq+cB + $ a d ) SO

(Uzsq+ + U ~ S + C ~ + ) (2rrc$~7)+ce + zed) SO

[(2 - 3s2e)c2$c2q- + s2e] + $adc$q-ce}

ki -

(65) TN- = -Fo -(%s+cq- + fiq-) (2arc+q-ce + s8

[ (&ST]- - &s$c~]-) ( 2 a r c ~ q - ~ e + +gd) se

We also need the torque due to the Es term in the force equations. As in Section 3, these are best worked out directly in the inertial frame. To this end, starting from (56) and (57):

P+ = [P+1, P2ca - P+3sa, P,sa + P+3ca]T (66)

P- = [P-l, - P2ca - P-ssa), P-3ca - P2sCylT (67) On taking the cross products with E,, substituting the results into (13) and the similar equation for F-, and averaging over a, we're led to:

- (P2sq+ + p+3s+crl+

- 1 7,+ = -5Fo(l- ar) (68)

(PZST+ + P+ss+crl+)c4