Document
NASAKM- .-_/_ 207527
AIAA 79-1709R
Fuel-Conservative GuidanceSystem
for Powered-LiftAircraft
H. Erzberger and J.D. McLean
Reprinted from
Journal ol 6uidance
This paper is declared a work of the U.S. Governmentand therefore is in the and Conlrol Volume 4, Number 3, May-June 1981, Page253.
public domain.
AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS • 1290 AVENUE OF THE AMERICAS • NEW YORK, NEW YORK, N.Y. 10104 VOL. 4, NO. 3, MAY-JUNE 1981 J. GUIDANCE AND CONTROL 253 AIAA 79-1709R
Fuel-Conservative Guidance System for Powered-Lift Aircraft
Heinz Erzberger* and John D. McLean* NASA Ames Research Center, Moffett Field, Calif.
A technique is described for the design of fuel-conservative guidance systems and is applied to a system that was flight tested on board NASA's augmentor wing jet STOL research aircraft. An important operational feature of the system is its ability to rapidly synthesize fuel-efficient trajectories for a large set of initial aircraft positions, altitudes, and headings. This feature allows the aircraft to be flown efficiently under conditions of changing winds and air traffic control vectors. Rapid synthesis of fuel-efficient trajectories is accomplished in the airborne computer by fast-time trajectory integration using a simplified dynamic performance model of the aircraft. This technique also ensures optimum flap deployment and, for powered-lift STOL aircraft, optimum transition to low-speed flight. Also included In the design is accurate prediction of touchdown time for use in four-dimensional guidance applications. Flight test results have demonstrated that the automatically synthesized trajectories produce significant fuel savings relative to manually flown conventional approaches.
Nomenclature = crosstrack error, ft Ay = crosstrack error rate, ft/s Ay = drag force, lb = flap angle, deg = distance of backward and forward = maximum flap angle, deg integration, respectively = fraction of energy rate used for changing = cruise distance, ft d c speed dh = length of ground track from initial to final 0 c = command pitch angle, deg position of aircraft, ft V = vectored thrust angle, in degrees of nozzle E = energy, ft angle
e o = energy rate, ft/s
T = throttle setting, in percent rpm Enmax ,_'nmin = maximum and minimum available energy = fraction of available energy rate rate, respectively, ft/s =commanded and reference bank angles, F,G = perturbation state and control distribution respectively, deg matrices, respectively = acceleration of gravity, ft/s 2 g Hf, n_ = final and initial ground headings of Introduction aircraft, deg N the past, terminal area guidance system design for h = altitude, ft aircraft has concentrated primarily on automatic glide- = final and initial altitudes of aircraft, hf, h, slope tracking, flare, and touchdown. During recent years, respectively, ft designs have been developed to provide automatic guidance K = feedback gain matrix along curved and decelerating approach paths. _ This in- = lateral error and error rate feedback gains k _y,k ,y, creased capability was made possible through the integration L = lift force, lb of digital computers into the flight guidance system.
= speed along ground track, ft/s However, even in the more advanced designs, automatic T = thrust force, lb guidance is limited to a few prestored three-dimensional (3-D) t = time, s flight paths, as in Ref. 1. While the ability to fly complex U = perturbation control vector prestored trajectories is essential, it cannot give optimum = two-dimensional unit vectors 1_ 1 , 112 , U 3 , il 4 performance under actual terminal area operating conditions, = aircraft control vector Uc as shall be explained.
v, = airspeed, ft/s or knots
First, a prestored trajectory cannot optimize fuel con- = final and initial airspeeds of aircraft, vo, sumption or a similar performance measure under actual respectively, ft/s or knots operating conditions. Optimum trajectories depend = wind speed in direction of ground track, V w significantly on aircraft gross weight, wind and temperature knots profiles, and the initial state of the aircraft. These variables W = aircraft weight, Ib cannot be predicted with the required precision prior to X = perturbation state vector takeoff. To prestore optimum trajectories for each of the = final and initial X coordinates of aircraft, gf, X i conditions likely to be encountered would result in an im- respectively, ft possible large memory requirement. Therefore, prestored = final and initial Y coordinates of aircraft, YI' Y_ trajectories must necessarily represent a compromise in respectively, ft performance.
IX = angle of attack, deg Second, in existing systems, the pilot must fly the aircraft = ineritial flight-path angle, deg manually from its current position to the starting point of the = aerodynamic flight-path angle, rad or deg "Ya.
trajectory. This flight segment is known as the capturing = airspeed rate correction due to wind shear, maneuver. Three-dimensional curved trajectories can be ft/s 2 difficult to capture manually, and, if the trajectory also in- cludes a specification of landing time, as is the case in four- Presented as Paper 79-1709 at the AIAA Guidance and Control dimensional (4-D) guidance, the capturing maneuver cannot Conference, Boulder, Colo., Aug. 6-8, 1979; submitted March 7, be done by the pilot without computer assistance. Therefore, 1980; revision received Aug. 14, 1980. This paper is declared a work of the capturing maneuver, because of its variability, can only be the U.S. Government and therefore is in the public domain.
• Research Scientist. Member AIAA. generated by onboard trajectory synthesis.
254 H. ERZBERGER AND J.D. McLEAN J. GUIDANCE AND CONTROL
Third, aircraft in high-density airspace are usually con- Equation (4) specifies the energy rate as a function of the difference between thrust and drag, subject to the constraint trolled by air traffic control vectors and during this period that lift equals weight. Thrust and drag are, in turn, functions cannot follow a prestored flight path. Synthesis of a of the controls producing forces in the flight-path direction; trajectory can only begin after the aircraft has received its final vector and has been cleared for approach. But the initial namely, throttle I", flap angle 8I, nozzle angle v (vectored thrust), and angle of attack or. Equation (5) determines the position of the aircraft at that time varies between ap- relationship between flight-path angle and deceleration for proaches; thus, trajectories require onboard synthesis.
the energy rate calculated from Eq. (4). Equation (5) indicates An initial design of a 4-D guidance system embodying the that, in particular, a given energy rate may be utilized to fly at concept of onboard trajectory synthesis, including an ad- flight-path angle 7o with constant airspeed, or to fly at zero vanced capture law, was previously developed and flight flight-path angle with acceleration dVo/dt. An infinity of tested onboard a conventional aircraft equipped with ad- other combinations of 7o and dV,/dt can also be chosen to vanced avionics. 2 In the design described here, a new yield the same energy rate. This makes possible a simplifying algorithm for generating horizontal capture trajectories has dichotomy in the trajectory synthesis; namely, at any time the been implemented and vertical and speed profiles are syn- desired energy rate is selected first by choice of appropriate thesized using simplified aero/propulsion performance controls and then the linearly related quantities of 7, and models of the aircraft. This method results in profiles that are d II,/dt are selected to generate the specifics of the flight path.
more fuel efficient than those of earlier design. Design of the Since the STOL aircraft studied in this paper has four control law for tracking the synthesized trajectory is based on controls to achieve a specified energy rate and to maintain lift the linearized perturbation guidance approach. Since the equal to weight, there is an excess of two controls over the perturbation equations are aircraft configuration dependent, minimum number needed for a simultaneous solution to Eq.
gain scheduling is used in the feedback law.
(4) and the constraint L= W. These two extra degrees of The Augmentor Wing Jet STOL Research Aircraft freedom in the controls are exploited to minimize power (AWJSRA) was chosen as the test vehicle for this concept.
setting and, therefore, fuel flow at every energy rate. This This type of powered-lift aircraft is highly cost-sensitive to optimization problem is restated in equivalent form as the operational procedures in the terminal area. It also exem- maximization of energy rate for a given power setting: plifies particularly well the unique problems of powered-lift aircraft, namely, high fuel consumption in the STOL mode; E,, (x) = max (T-D)/W (6) dependence of both lift and drag on thrust; and an excess of I,, ill, _f controls over the minimum number needed to determine path and speed. These factors suggest that trajectory optimization Constraint: L(a',r, ct,61) = W could greatly increase the operational efficiency of the air- craft. Implementation of this concept was facilitated by the The maximization must obey various inequality constraints existing installation of an advanced avionics system onboard on the controls: the aircraft.
- 10.5 deg< c_ 19.5 deg Energy Rate Model and Selection of Reference Controls 6 deg< v< 100 deg An energy rate model of aircraft performance has been found to yield a compact and sufficiently accurate 5.6deg<6l<Sf_,,_(V o) [flapplacard] representation of performance for terminal area trajectory synthesis. In this section, a performance model based on In addition, a lift or maneuver margin must be satisfied at energy rate is derived and then applied to determine the every point to guarantee sufficient normal force for changing optimum reference controls for synthesizing trajectories.
the flight path. Pilots familiar with this aircraft specify that at Consider the standard expression for energy rate written as least 0.4 g of normal acceleration must be attainable at any time by an increase in the angle of attack alone.
dE (T-D) Vo
(1) The use of Eq. (6) results in the selection of the controls that
dt W yield the maximum attainable energy rate at each thrust where .3-- E=h + (1/2g) V_ (2) with constraint L = W (Ref. 3). It is assumed throughout this .2 paper that flight-path angles are small such that cos_° =, 1 and sin % =%. Furthermore, it is assumed that flight-path angle rates are so small that their effect on lift is negligible. Dif- .1 ferentiation of Eq. (2) with respect to time gives an equivalent expression for energy rate:
dE dh 1 dVo
a_
d-7= d-7+ _ V° -_7 (3)
-.I z Equations (1) and (3) can be nondimensionalized by dividing them both by Vo. The resulting quantity on the left side, -.2 (I/V°) (dE/dO, is defined as the normalized energy rate/_,, or energy rate for short. By using the relation (dh/dt) ,_ Vo7 °, -.3 the two relations for/_, become E.. = (T-D)/W (4) I I l I I -.4 80 80 100 120 140 160
_: = _o+ I_ dVo (5) INDICATED AIRSPEED. knots
g dt Fig. 1 Energy rate diagram for STOL aircraft: W= 38,000 Ib, sea level 59" F.
withconstraint L = W.
MAY-JUNE 1981 FUEL-CONSERVATIVE GUIDANCE FOR POWERED-LIFT AIRCRAFT 255 setting. This insures the efficient use of thrust at any energy maximum value of 100 deg as the energy rate decreases rate that requires more than the minimum thrust. But energy toward its negative limit of -0.3.
rates more negative than those attainable by Eq. (6) are also In the flight implementation of the algorithm, four of interest. Such negative energy rates must occur at the diagrams, as shown in Fig. !, are utilized: two for sea-level greater of the minimum or idle thrusts required by the altitude at weights of 38,000 and 48,000 lb, and two others for maneuver margin. At a particular airspeed, a decrease in the 5000-ft altitude at similar weights. Experience indicates that energy rate below the minimum attained through Eq. (6) can these provide sufficient data to adequately interpolate the be effected by increasing the vectored thrust angle v and/or controls. Each diagram requires 124 words of memory in the the flap angle 6y. The third control, angle of attack a, is airborne computer. The small circles in Fig. 1 indicate the needed to satisfy the constraint L = W. The two degrees of locations of points that are stored. The energy rate data are freedom in the controls can be exploited to minimize noise also corrected for deviations from the standard temperature exposure along the ground track. Noise under the aircraft is profile. Correction is done by computing a thrust setting known to increase as the nozzles producing the vectored thrust corrected for temperature deviations.
are turned downward. Therefore, a further decrease in energy rate is achieved by first increasing flap angle until it reaches its Synthesis of Complete Profiles limit or placard value and only then by increasing nozzle In the preceding section the criteria of fuel conservation angle.
and noise reduction were used to determine the four reference The result of applying these procedures to the AWJSRA is controls of throttle, nozzle angle, flap angle, and angle of shown in Fig. 1 for a weight of 38,000 lb, sea-level altitude, attack as functions of the energy rate. This approach replaced and standard temperature. The figure gives the envelope of the problem of selecting four control variables with the energy rate vs indicated airspeed with throttle, flaps, and simpler problem of selecting a single, equivalent variable, vectoring nozzle as parameters. Angle of attack is not plotted namely, the energy rate. In this section, we make use of the to avoid cluttering the figure. At any airspeed, the J_nrnm and energy-rate variable in generating efficient terminal area kT,min curves define the range of permissible energy rates. The trajectories.
optimum controls for a given airspeed and energy rate are The problem of terminal area trajectory synthesis can be determined by interpolation between contours of constant stated as the specification of rules for flying an aircraft with controls. For example, at an airspeed of 105 knots and initial state vector [Xi, Yi, hi, H_, Vo,] to a final state vector -P, =-0.17, the optimum controls are found to be _5.f=26 IX/, Y/, hi, /'Is, V,f]. To be of practical interest, such rules deg, _ = 6 deg, and x= 84°70 (point A, Fig. 1). Angle of attack must generate efficient and flyable trajectories connecting (not shown) is 8.4 deg. Maximum energy rate with minimum various initial and final state vectors. By specifying a per- thrust occurs at 112 knots (point B) and corresponds ap- formance criterion such as fuel consumption, we can fit this proximately to (L/D) m_ = 10.
problem into the framework of optimal control theory.
It should be noted that the force-producing controls in this However, the difficulty of solving an optimal control problem experimental STOL aircraft have unusual characteristics that characterized by a five-element state vector makes this ap- account for the relative complexity of Fig. 1. Throttle affects proach computationally impractical for in-flight im- both lift and drag at all speeds, but the effect on lift is greatest plementation. Following Ref. 4, we have adopted the sim- in the STOL regime below about 80 knots. The thrust plifying procedure of separating the synthesis problem into magnitude produced by the vectoring nozzle, referred to as two essentially independent problems.
the hot thrust, is also controlled by the throttle and accounts The first problem consists of synthesizing the horizontal or for about 60% of the total thrust produced by the two 2-D trajectory. References 4 and 5 give algorithms for engines. The remaining 40070 of the thrust, which is the cold computing near-minimum-distance 2-D trajectories as a thrust produced by the fans, energizes the augmentor wing to sequence of an initial constant radius turn, straight flight, and increase lift at STOL speeds.
a final constant radius turn, where the turn radii are chosen so The relationship between the controls and the energy rate is as to avoid exceeding a specified maximum bank angle at the revealed more clearly in Fig. 2 at the example airspeed of 105 maximum groundspeed encountered in each turn. The knots. Many such plots at various airspeeds would be required algorithm used in this onboard computer implementation is to illustrate the complete dependence of the controls on based on a simplified derivation which resulted in a energy rate. As the energy rate decreases below its maximum significant reduction in computing time compared with the value of 0.28, throttle decreases nearly linearly until idle methods given in the references. The derivation can be found throttle is reached. In this interval, flaps increase only in the Appendix. Figure 3 illustrates two of the four types of slightly, while nozzle angle remains at minimum and angle of horizontal trajectories that can occur; the two not shown attack increases. At more negative energy rates, flaps become differ only in the orientation of the turns. The algorithm the dominant control until they reach the placard value of 40 deg at this airspeed. Angle of attack decreases sharply as flap angle increases. Finally, nozzle angle increases toward its IX i, Yi ) , INITIAL INITIAL X POSITION POSITION RUNWAY RUNWAY i . --_ ...f _ INITIAL (XiYil X CENTERED CENTERED _,/ RlUl _1/URN f _"_'" --_ ]COORDINATE
'°- '°°r.'°°r I °7"
COORDINATE --HI/ _" _ f _ 1Ul _ _ SYSTEM
("- 'oi- li/
-_ 92 60 6 l- .,_L_ ;IL ROTTLE V4 _(XC2, YC2) 'V4
_ 2-;84 _ 20 _L
/ _= / I I _"",,----__..._jLAPS
_ JT_i_)-'_lXf, Yf) (xt.Y,)/V.
F,NAL _ _ R_=
/" / IMAX,,
I" 0- eol-_ o/ tl I I I t F--I
POSITION R204 FINA'L " (XC2 Yc2) "R204 _;-;' POSIT ON -2 - 76 L_ -20 [ & I I 1 I I I -.4 -.3 -.2 -.1 0 .1 .2 .3 a) TURN b) ENERGY RATE, EN Fig. 3 Examples of minimum distance, constant turn radius, Fig. 2 Optimum controls as function of energy rate at 105 knots: horizontal capture trajectories to a capture point P.,, on final np- 14"=38,000 Ib, sea level 59°F.
proach: a) turns in same direction; b) turns in opposite direction.
256 H. ERZBERGER AND J.D. McLEAN J. GUIDANCE AND CONTROL computes all feasible trajectories, of which there are at least two and at most four, and chooses the one with the shortest path length. Note that the terminal point lies on an extension of the runway centerline and that the final heading angle is equal to the runway heading. The final point should be chosen d as close as possible to the touchdown point consistent with u.i safe operational practice. For a STOL aircraft, the minimum distance is about l n.mi. u: 100 .... LANDING e=0 The second problem, solved after the horizontal trajectory SPEED has been computed, consists of synthesizing efficient speed and altitude profiles which match the initial and final speeds (a) I I I I and altitudes Vi, h i, and V/,ht, respectively.
The horizontal distance of the trajectorY d,, a known quantity computed in the previous step, adds a third boundary condition to be satisfied by the profiles. While this three-state optimal control problem is much simpler to solve than the original five-state problem, it is still too complex for onboard-
3°°° V
computer implementation. A simpler algorithm was, therefore, developed that generates near-optimum speed- altitude profiles by matching the general characteristics of I- e=O optimum fuel and noise trajectories studied in Refs. 6 and 7, _' 1000 respectively. We briefly explain the rationale for this algorithm with reference to descent, which is the most dif- ficult case.
It was found in Ref. 6 that the descent portion of a 0 (bl _ = 1 I I minimum-fuel descent trajectory is characterized by a delay in 20000 10000 5000 0 the start of the energy decrease as long as possible, consistent DISTANCE FROM TOUCHDOWN with meeting end constraints of speed and altitude. Fur- = -0.13.
Fig. 4 Effect of _ on speed and altitude profiles, with/_n thermore, the energy change consists initially of descent to the final altitude at near-constant indicated airspeed followed by deceleration in level flight. Most of the energy change takes dependent of speed, and let the airspeed to be achieved at place at minimum throttle, as one might expect for minimum touchdown be 100 ft/s. To achieve the desired boundary fuel flight. Minimum-noise descent profiles computed in Ref.
conditions, Eqs. (8), (10), and (11)are integrated in backward 7 are similar in that they also delay the start of energy time starting with the speed and altitude at touchdown. The decrease as long as possible, but approach the final altitude in resulting airspeed and altitude profiles are plotted as a func- a steep descent to maximize the aircraft's altitude above the tion of distance to touchdown in Fig. 4 for e = 1, 0.5, 0.0. The ground near the runway. This means that the deceleration to profile for _ = 1 is seen to approximate the minimum fuel, for the final airspeed takes place before the start of descent or _=0, the minimum noise descent and for e=0.5, a com- during the early portion of the descent. Thus, the two types of promise between fuel and noise minimization.
descent profiles differ primarily in the way they proportion To minimize fuel consumption or noise, changes in energy the use of available energy rate to decrease altitude and should be made at maximum rate when the aircraft enters the airspeed.
powered-lift region of 90 knots and below. This is ac- To facilitate the synthesis of such profiles, a family of complished by setting o to unity and thereby following the decreasing (and by extension, increasing) energy profiles, _7,mi, contour during descent and deceleration. However, for which include the two types described as special cases, is the aircraft under study this can yield energy rates too defined by two parameters, a and _. The first parameter, o, negative for safe operation. A limit less than one is also selects the fraction of minimum/maximum available energy necessary to reserve energy rate for perturbation control. A rate, _',mi,, (J_nmcx) t O be used for decreasing/increasing practical upper limit on a is about 0.9 for the AWJSRA. In energy. The values of Enmin and/_,m_ can be read from Fig. 1 the flight implementation, the two profile parameters are at each indicated airspeed. The second parameter, e, deter- keyboard entries that allow the pilot to choose values ap- mines the fraction of the selected energy rate to be used for propriate for each landing approach. In addition, the pilot deceleration/acceleration. Then, for particular choices of o can specify the maximum deceleration and descent angles via and ¢, the energy rate, airspeed, flight-path angle, altitude, keyboard entry. The maximum safe deceleration for this and horizontal distance are computed as follows: aircraft is limited to about 0.06 g by the maximum rate of which flaps can be extended. The synthesis algorithm is _l_'n = aEnmin (0_ O'_ l) (7) configured to decrease o below its limit if that is necessary to satisfy these constraints.
The backward time integration described above generates an increasing (in backward time) energy profile starting at the
3'. = (I-e)E, (9) desired final speed and altitude. To complete the synthesis of
the descent trajectory, we still need rules for matching this
J/= vo-vo (1o)
profile to the initial speed and altitude of the aircraft. The freedom of the aircraft to maneuver in altitude is restricted by g= VoCOS_,o + V. (11) air traffic control as well as passenger comfort con- siderations. Thus, as an aircraft approaches a terminal area, it is generally not allowed to climb above its initial approach where V,, is the along-track component of windspeed. Note that Eqs. (7-9) are consistent with Eqs. (4) and (5) for all altitude for the purpose of optimizing the approach trajec- values of a and _. Decreasing/increasing energy profiles are tory. The aircraft must hold this altitude until starting the final descent. However, while flying at altitude h_, it may generated by integrating Eqs. (8), (10), and (11) for particular choices of a and _.
change to a new airspeed, V=t, called the terminal area speed, which can be higher or lower than the initial speed Vo,. Unless To illustrate the effect of the parameter _ on the specified by the pilot via keyboard entry, it is chosen to descent/deceleration profiles, assume E, = -0.13, in-
MAY-JUNE 1981 FUEL-CONSERVATIVE GUIDANCE FOR POWERED-LIFT AIRCRAFT 257
corrected for the effect of the bank angle used in flying a turn minimize fuel use per unit distance, and is 140 knots for this by interpolating the controls with lift equal to an aircraft aircraft (it would be 220-250 knots for conventional jet weight multiplied by the load factor l/cos_. Integration step transports).
size varies during synthesis. During decelerations or ac- The various rules contained in the preceding two celerations it is l s, while during altitude changes at fixed paragraphs can now be combined to yield the complete speed it is 5 s. Total time for synthesizing a complete algorithm. The synthesis begins with the backward time in- trajectory consisting of a horizontal trajectory similar to the tegration from final conditions h/, Vaf using the specified a ones shown in Fig. 3 and a speed/altitude profile similar to and E. If the altitude reaches its target value of h_ before the the one in Fig. 5 is about 2 s on the particular airborne airspeed reaches its target value of V,_, we set _ = 1 and then computer used in the flight tests. When the trajectory syn- continue the backward time integration until the airspeed has thesis is time-shared with navigation and other necessary also achieved its target value. When setting _ = l, the flight- computations, the computing time increases to about 6 s.
path angle is forced to zero and the energy rate is used entirely for accelerating (in backward time) toward Vat. On the other Perturbation Guidance Law hand, if the airspeed reaches its target value before the altitude does, we set _ = 0. This stops the airspeed change and Perturbations of the aircraft states from the reference states uses the energy rate entirely for increasing the altitude toward are used in the guidance law to generate perturbation controls its target value of h r. When the second and last variable which are added to the reference controls in order to null reaches its target value, we set a = 0, i.e., E_ = 0, thus com- errors in airspeed, altitude, and crosstrack position. The pleting the backward time integration. Next, we begin a feedback states in the guidance law also include crosstrack forward time integration to get the distance required to error rate and flight-path angle, as well as the integrals of change speed from Vai to Va_ with _ = 1. Let the distances for airspeed and altitude errors. The latter two are used to reduce the backward and forward integrations be d 0 and dr, speed and altitude bias errors caused by inaccuracies in the respectively. A valid trajectory has been generated if the stored energy rate data and errors in the estimates of wind and cruise distance d c, computed from temperature profiles.
The controls are throttle, nozzle, pitch, and roll angles.
dc =d0 -rib -d/ (12) Flaps are not used as perturbation controls because of their relatively low rate limit and an operational constraint that is nonnegative, i.e., de >0. Ifd_ is negative, the synthesis has flap motion be monotonic during an approach. The flap failed because the aircraft is too close to the capture point P/.
command is simply the reference value at each ground track Figure 5 illustrates the various segments of an approach position limited to the placard value at the current airspeed.
trajectory synthesized by the algorithm. As before, we assume Lateral perturbation control is essentially uncoupled from for simplicity that E_ = - 0.13, a constant. Other parameters the longitudinal mode and is accomplished through a roll- defining the problem are indicated in the figure. Note that the angle command to the roll-command autopilot. This com- initial descent at 7° = - 7.5 flattens to 3', = - 3.75 to allow the mand is of the form aircraft to decelerate. The reference controls for this trajectory can be interpolated from Fig. 1.
The airspeed deceleration is corrected for known wind shears, which are computed from a knowledge of V w (h), if where ¢_, is the reference roll angle, and Ay and A._ are the available. The wind shear correction factor is crosstrack error and error rate, respectively. The two gains were chosen to provide a well-damped response and control A(,',= - (dVw/dh) Va% activity compatible with the noise characteristics of the navigation system.
and is added to the right side of Eq. (8) to obtain _he corrected Longitudinal perturbation control for correcting airspeed airspeed rate. Furthermore, the reference controls are and altitude errors is difficult because the reference controls generated by the energy rate schedule of Fig. 1 often lie on a constraint boundary and therefore cannot be perturbed freely 3oo- L I 1 I [ in both directions. The two controls that are often constraint- '_,,_ai : 265 ft/mc I limited during a fuel-conservative approach are throttle a', --- _='_i I I._i_ - - - v, T = 23o ft/_ - 0.065 's DECELERATION ' .3-- 0.065 g's DECELERATION _ [ I o'p ,," 100 J ___Vaf - I l I_ I I I I I I I I I (a) o I t I t I I I .1 f DECELERATION t I DESCENT/DECELERATION 0=0.5 I I o=1,_=o.5 I REGION I1: INCREASE x TO INCR'EASE EN; = e-1 / ] I \ ] INCREASE v TO DECREASE E'N _, MCNIMUM REF NOZZLE
e2 °
VBOUNOARY /
[CRUISE_ DESCENT I -- I d io=1 _=0l / _.Pl FINAL I t l MINIMUM REF THROTTLE _l_ c -l- ' --I_ :._PROACjH I \X / /BOUNDARY REGION IV: _ ¢m -,1 I [ I I I It, AND v ARE --_ / / _. 2000 i I_.----h i I SLOPE I I
PREE --,, _ *
___.v2 o_ / \ I I -.2 "_1 -3,75° \ I I F- p- I -N I -.3 (b) I I NOTE: PITCH ANGLE 8 IS FREE IN ALL REGIONS f MAX NOZZLE BOUNDARY -20000 - 10000 0 5000 I I [ I [ -.4 Pf TOUCH- 60 80 100 120 140 160 DOWN INDICATED AIRSPEED. knots Fig. 5 Example of synthesized STOL approach trajectory. Fig. 6 Constraint boundaries for perturbation controls.
258 H. ERZBERGER AND J.D. McLEAN J. GUIDANCE AND CONTROL and nozzle angle v. Some insight into this problem can be obtained using data from the energy rate schedules. Figure 6 AND LANDING SPEED ENTER COORDINATES I OF FINAL POSITION shows the energy rate envelope from Fig. 1 with the minimum reference nozzle and throttle constraint boundaries. These boundaries divide the envelope into four regions: I, where v _SYNTHESIZE HORIZONTAL L RATE cannot be reduced; II, where neither x nor v can be reduced; TRAJECTORY AND AIRSPEED/ I _ DIAGRAMS III, where x cannot be reduced; and IV, where x and v are free i ENERGY ALTITUDE PROFILE Jel (FIG. 1) to move in either direction. The combinations of controls available for increasing and decreasing En in each region are J -G E'-NERATE indicated in the figure. Note that in region I, the nozzle could FEED- FORWARD t NAVIGATION be used as a additional control variable for decreasing energy STATES SYSTEM: rate. However, this variable is not used because throttle and AND TACAN, pitch provide adequate control of flight-path errors in this MODILS; CONTROLS FROM STATE region. In region IV, the minimum reference throttle is above ESTIMATES idle and is determined by the maneuver margin constraint. At
TR O gRV
each airspeed in this region the negative throttle perturbation STORE PARAMETERS| TRAJECTORY that can be added to the reference throttle to yield the com- PARAMETERS manded throttle is limited to -2°70 for safety reasons.
PERTUR- I AND DISPLAY BATION | Positive and negative throttle perturbations are further TRAJECTORY FEEDBACK limited so that the commanded throttle xc falls into the engine LAW; ROLL, | operating range, 84070 < a'_ 96070.
PITCH l AUTOPILOT ] The perturbation equations and the perturbation control law can be written in state vector notation as AIRC_RAFT 1 dx/dt=Fx+Gu u=Kx Fig. 7 Simplified flow diagram of guidance system.
where where Va, is in units of feet per second. Extensive computer x= (AV, AT, Ah,_AVdt,_Ahdt) r calculations have verified that the closed-loop eigenvalues of this system have damping factors of 0.707 or greater and real u= (Ax, A0,Av) r parts less than -0.05/s at all operating points. These characteristics provide adequate tracking performance. When The delta quantities are the perturbations from reference operating in region I of Fig. 6, the last row of K is set to zero values, i.e., AV= V a - Va,, etc., where Vo is the aircraft and since nozzle angle is not used for control. In regions II and Vat the reference true airspeed, respectively. The commanded III, throttle perturbations are limited to positive values, while controls are the sum of reference and perturbation controls: in region II, nozzle perturbations are limited to positive values. In region IV, each control moves freely but negative uc= ( xr + Ax, O, + AO, v, + Av) throttle perturbations are limited to -2070 rpm, as previously explained. Control limiting can reduce the effectiveness of integral feedback of speed and altitude. Some design con- For a powered-lift STOL aircraft, such as the one used for siderations for these integral feedback loops are given in these flight tests, the values ofFand G are strongly dependent Ref. 9.
upon airspeed and energy rate and are thus time-varying along The throttle and nozzle angle perturbations generated by a trajectory. Quadratic optimal synthesis s would therefore the control law will generally be of opposite sign, because the yield time-varying gain matrices that are also functions of the elements of the first row of K all have opposite sign of the reference trajectory. But it is neither practical nor necessary to third-row elements. Thus, even in region If, where throttle implement a complex, reference-trajectory-dependent gain and nozzle perturbations are each limited to move only in the matrix in order to achieve adequate control system per- formance in this case. positive direction, they are not generally limited simultaneously. This implies that two controls, either throttle The design procedure employed here began by first com- and pitch or nozzle and pitch, are free to move. Transient puting optimum gain matrices at various operating points in response studies using a nonlinear simulation of the aircraft the control region diagram (Fig. 1) using fixed values of Fand and guidance system have shown that the control power is G. The analysis of these gain matrices showed the strongest adequate to provide rapid and well-damped airspeed and dependence on airspeed, reference nozzle angle, and reference altitude error responses in region II.
flaps. Sensitivity of the closed-loop eigenvalues to changes in several of the gains was low, allowing those to be set to zero or held constant throughout the operating region. It was possible Guidance Algorithm Overview to fit the variable gains with relatively simple functions of reference airspeed, nozzle angle, and flap angle. This method A flow chart illustrating the integration of major functions resulted in the following gain matrix: within the guidance system is shown in Fig. 7. The pilot enters -40 -40 -8 -0.6 -_osv r
liar Vor Vor
-14 -0.2 K _ -0.4 0 Vo, 2 0.6 0.3 0.05 max[ 0'8/- 4520 deg I MAY-JUNE 1981 FUEL-CONSERVATIVE GUIDANCE FOR POWERED-LIFT AIRCRAFT 259 BEGIN into the guidance computer the coordinates, altitude, and DECELERATION landing airspeed to be achieved at the desired final position on A B C the approach path (see Fig. 3) or, alternatively, he selects a 160[- 25 T I BEGIN END == ERROR OECE,.E, AT,O,., capture waypoint with prestored coordinates. Trajectory synthesis can begin after the navigation system has computed : RE F ER ENC"E ..........
60 -25 r -t the current position and velocity components of the aircraft.
i F The first step in the synthesis process involves computing the horizontal trajectory parameters using the technique < OZ.._ ..... __ _ _ _..=._ , ERROR, J ,,,Y.- .
r-- _ _":'_o,,_ - ........ _ --' described in the Appendix. This step is always successful. In ¢0 t_.250J : I_-_- t REFERENCE ........... ', the second step, the altitude and speed profile are synthesized i l I00 I using the energy rate diagram of Fig. 1 in conjunction with the _'E logic described in the section on Synthesis of Complete COMMAND Profiles. This step is not always successful. For example, if I- the horizontal path computed in the first step is very short and 105 : REFERENCE : the differences in speed and altitude between initial and final aircraft positions are large, a flyable trajectory along that path may not exist. During synthesis, such a failure is detected as a negative cruise distance d c in Eq. (12). A failure to 100 i , synthesize is an unlikely event in landing approaches initiated ui -----I several miles from the final point, the usual situation; if the failure occurs, however, the synthesis is repeated using up- dated position and velocity vectors, which aircraft motion has changed during the time the synthesis was in progress. The pilot can also fly the aircraft manually to a more favorable U uJ" .,,REFERENCE location (path stretching) for a successful synthesis.
AIRCRAFT I After a trajectory has been successfully synthesized, its <t -25 J parameters are stored and the horizontal path is displayed to Fig. 8 Flight test results, straight-in approach, runway at 140 ft the pilot on a map-like cathode ray display (see Ref. 2 for above sea level.
details of this device). The appearance of the trajectory on the display is also a cue to the pilot that a valid trajectory has been exceed 35 ft and decreased to about 15 ft near the end. Speed successfully synthesized. The stored and displayed trajectory errors during deceleration were less than 10 ft/s, and is refreshed by repeating the synthesis every few seconds until decreased to about 1 ft/s at the end. If allowance is made for the pilot engages the auto track mode. At that time, the last the presence of turbulence, winds, and navigation system synthesized trajectory is frozen and its reference states and noise during the flight, these errors agree reasonably well controls are regenerated in real time. These reference values with simulation results and are acceptably low. The control are fed forward to the perturbation feedback law, which perturbation biases, evidently caused by modeling errors and causes the aircraft to track the synthesized trajectory. Three- the unmodeled wind, are larger than those seen in simulation, dimensional navigation data for computing errors between though they are not excessive. Nozzle bias during the middle the aircraft and the reference trajectory are obtained from of the deceleration averages about 25 deg. While this seems TACAN (TACtical Air Navigation System) or MODILS large, it should be noted that during this interval the throttle is (MODular Instrument Landing System, an experimental at flight idle, where the effect of nozzle on energy rate is a Microwave Landing System), with automatic switching to the minimum. On the whole, the control biases represent fairly most accurate signal.
small errors in the energy rate model. The flight test results It is important to note that once automatic tracking of a can, of course, be used to improve the accuracy of the energy trajectory has begun, the stored trajectory is not refreshed, rate model of the aircraft.
though this may be desirable if unmodeled wind or transients The crosstrack error at the end of deceleration, point C, 0.9 in navigation introduce large tracking errors. The real-time n.mi. from touchdown, was measured by precision radar as operating system has been configured to add this capability in 80 ft. This error is an important criterion for determining how future flight experiments. Also, a technique is incorporated in close point C can be placed to touchdown when navigating the flight software that compensates the trajectory for the with TACAN. Pilots judged the action of the automatic changes in aircraft position occurring during the time (up to 6 control law as smooth and the trajectory synthesis technique s) the trajectory is synthesized.
as a convenient and effective tool for optimizing approach trajectories.
Flight Test Results The fuel consumption of this automatically flown Figure 8 shows the major portion of various time histories trajectory was compared with that of a trajectory flown by a for a straight-in flight test approach starting 7 n.mi. from test pilot under simulated instrument fligh rule conditions. In touchdown, at 3000 ft altitude and 140 knots. The order to provide a basis for comparison, the manually flown deceleration at 0.03 g begins in level flight at point A, 4.7 trajectory began from the same initial distance-to-touchdown, n.mi. from touchdown. The descent begins at point B, 3.7 airspeed, and altitude as the automatically flown trajectory.
n.mi. from touchdown. The entire approach trajectory was The approach was made with the aid of a flight director flown using TACAN for navigation. The TACAN station was system which displayed to the pilot lateral and longitudinal located at the airport a few hundred feet from the runway deviations from a straight-in 7.5 deg approach path. The fuel centerline. Because of its favorable location, the TACAN used for the automatic approach was 381 lb, while that for the station provided sufficient navigation accuracy for flying the manually flown one was 500 lb. Further simulations and approach automatically to within a half-mile of touchdown flight tests are in progress to compare the fuel consumption without switching to the higher precision MODILS as would for various approach trajectories, flight director designs, and normally be required. There was light-to-moderate turbulence wind conditions.
and an average headwind of about 15 knots below 4000 ft as Conclusions measured by a radar tracked weather balloon just prior to takeoff. However, the wind profile was not entered into the The automatic guidance system described in this paper synthesis logic and thus constituted an unmodeled wind.
achieves the dual goal of fully automatic flight and near- Altitude errors, except near the pitchdown point, did not optimal fuel conservation through the technique of fast-time 260 H. ERZBERGER AND J.D. McLEAN J. GUIDANCE AND CONTROL onboard trajectory synthesis. This technique overcomes the and therefore, since/5 and t_z are perpendicular, performance limitations inherent in a stored precalculated trajectory by adapting the trajectory to the unique conditions D- IDI =x/Q e - (ReS e-RtS t ) e (A2) encountered in each landing approach. The ability to adapt is crucial in the terminal area since the initial conditions for where, by definition, starting the approach and the wind and temperature profiles are not predictable with sufficient accuracy prior to takeoff. Q- 101 =x/(xce-xct)2+ (YCe-YCt) z (A3) Synthesized profiles always delay the start of the descent and It can be seen from Eq. (A2) that no real solution exists if deceleration points as much as possible. Flight evaluation using a powered-lift STOL aircraft showed that an Q< IReSe-RtS t I. When the turns are in opposite direc- automatically synthesized approach saves approximately 120 tions, S t=-S 2, and there is no real solution for lb of fuel during the last 7 miles of the approach relative to Q < (R t + Re ), i.e., if the circles intersect. On the other hand, one flown manually with only conventional flight director for rotations in the same directions, S t =$2 and a real guidance. The design procedure described herein for a STOL solution exists unless Q< IR 2 -R t I, i.e., unless one circle lies entirely within the other. From geometric construction it can aircraft is applicable with lesser computer complexity to be shown that there always exist at least two real solutions.
conventional aircraft. The algorithm is also suitable for in- From the definition of the radius vectors, one can write for corporating in advanced flight management systems currently the real solutions: under development by industry.
Appendix = (-RtStsinHi'_ (A4) This appendix derives the expressions for synthesizing Rtf'ztSt \ RtStcosH _ / horizontal capture trajectories for flying an aircraft from a given initial position and heading to a specified final position and and heading. Figure 3 is used to explain the problem and define the variables. The turns are arcs of the circles shown in the figure and the straight portion of the trajectory must be a XC t -X_ Rt_tS t = \yct _ Yi I / (A5) line tangent to both circles. Since the initial and final turns may be either clockwise or counterclockwise, there are four possible combinations of turning directions--two with the Equating Eqs. (A4) and (AS) gives initial and final turns in the same direction, and two in op- posite directions. Figure 3 illustrates one solution of each type. If a given pair of circles is entirely separate, i.e., no part XC t = X t - R t St sinHi YCI = Yt +RtStcosHi (A6) of one circle lies within the other, it is possible to draw four tangent lines between the pair. However, vector b along the Similarly, tangent line from the initial to the final circle coincides with the direction of rotation at both tangent points for only one of XC 2 = Xf- R eSesinH f YC 2 = Y/-RzS2cosH / (A7) the four tangent lines as shown in the figure.
The radius vectors at the tangent points can be used in the In the figure, the final position and the origin of the same manner to compute the components of X2 and .,_'j coordinate system are located on the runway centerline.
However, the derivation is for arbitrary locations. Fur- X 2 = XC t + R t St sin//2 (A8a) thermore, all variables are defined so that the derivation applies to all possible combinations of turning directions.
Figure 3a is for the case where both turns are in the same Ye = YCt - R t St cosHe (A8b) direction and the tangent vector/) does not cross Q, while in X 3 = XC 2 + R2 $2 sinH2 (A9a) Fig. 3b the turns are in opposite directions and/) crosses Q.
Initially, the aircraft is at (X i, Y_) in some inertial Cartesian Yj = YC 2 - R 2 S 2cosH e (A9b) coordinate system with heading H i defined as positive clockwise from the X axis, and 0j is a unit vector in the Subtracting Eq. (A8a) from (A9a) and Eq. (A8b) from (A9b) direction of the velocity. The vector distance from (X i, Yi) to gives the components of/): the center of the turn is given by fitRj, where R t is the radius of turn and ti I a unit vector normal to t_t and positive to the right of t31. Therefore, the vector from (X i, Yi) to the center Xj - X_ = XC 2 - XC t + ( R 2S 2 - R tS t ) sinH 2
(XCI, YCt) is Rtti t for a right turn and -Rtti t for a left (A10)
turn. The directions of turn are accounted for by writing the Y3 - Ye = YC2 - YC t - (R 2S: - R tS t ) cosH2 radius vector as RtSt_ t, where S t = + 1.0 for right turns and S t = - 1.0 for left turns. Similarly, the direction of the final Another expression for the components of/) is: turn is denoted by S 2.
The aircraft moves along the circle from (X i, Yi) to the Xj-X2=DcosH2 Y3 - Y2=DsinH2 (All) tangent point (X 2, Y2) which has a radius vector RtS t ft 2. The tangent vector from (X 2, Y2) at the end of the initial turn to Equating the corresponding pairs in Eqs. (AI0) and (All) (X 3, Yj) at the beginning of the final turn is D. The radius gives vector at (_Xj, Y3) is R2S2ft_, but since fit and ti_ must be normal to D, _2 = :is. Likewise, the headings H 2 and H 3 at DcosH 2 = (XC e - XC t ) + (R 2Se - R t St ) si nH 2 the two tangent points are equal. The final turn ends at (AI2) (Xf, YZ) with heading Hf and radius vector R2Szft_.
DsinH z = ( YC 2 - YC t ) - ( R eS 2 -RtS t )cosHe Using this notation, we can write Equations (A12) can be solved for the tangent ofH e: D+ R2_2S e =R I gteS t + (YC 2 - YC t )D- (R2S 2 -RtS t ) (XC 2 -XC t) or tanH e = (XC e -XC t )D+ (ReS e -RtS t ) ( YC e - YC t ) ()= D +fte (R2S 2 - R I St ) (A1) (Ai3)
MAY-JUNE 1981 FUEL-CONSERVATIVE GUIDANCE FOR POWERED-LIFT AIRCRAFT 261
References Equations (A6-A9) and (AI3) completely specify a capture trajectory for any combination of S I and S 2. However, the I Neuman, F., Watson, D.M., and Bradbury, P., "Operational length of the trajectory is also needed in order to determine Description of an Experimental Digital Avionics System for STOL which of the feasible trajectories gives the minimum distance.
Airplanes," NASA TM X-62,448, Dec. 1975.
The first turn angle is 2 Lee, H.Q., Neuman, F., and Hardy, G.G., "4-D Area Navigation System Description and Flight Tests," NASA TN D-7874, Aug. 1975.
TR t = (H 2 -Hi) + 2rCIS I 3Byrson, A.E. Jr., Desai, M.N., and Hoffman, W.C., "Energy where State Approximation in Performance Optimization of Supersonic Aircraft," Journal of Aircraft, Vol. 6, Nov.-Dec. 1969, pp. 481-487.
4Erzberger, H. and Lee, H.Q., "Terminal-Area Guidance C;=[_ if S,(H2-Hi)_,O if SI(H 2-H i ) <0 (AI4) Algorithms for Automated Air Traffic Control," NASA TN D-6773, April 1972.
and the second turn angle is 5pecsvaradi, T., "Four-Dimensional Guidance Algorithms for Aircraft in an Air Traffic Control Environment," NASA TN D-7829, TR 2 = (H I -1-12) +2rC2S March 1975.
where 6Erzberger, H., McLean, J.D., and Barman, J.F., "Fixed-Range Optimum Trajectories for Short-Haul Aircraft," NASA TN D-8115, [_ if S2(Hf-H2)_O Dec. 1975.
C2= if S2(Hf-H2)<O (AIf) L 7Jakob, H., "An Engineering Optimization Method with Ap- plication to STOL Aircraft Approach and Landing Trajectories," Finally, the total length of the capture path is NASA TN D-6978, Sept. 1972.
d h = I/)1 +R 1 ITRII +R 2 I TR_I (AI6) SBryson, A.E. Jr. and Ho, Y.C., Applied Optimal Control, Blaisdell Publishing Co., Waltham, Mass., 1969, pp. 148-176.
The algorithm computes the length of trajectories for all 9Slater, G., "Analysis of Integral Controls in Linear Quadratic feasible pairs of S I and S 2 and then picks the shortest length Regulator Design," AIAA Paper 79-1743, Proceedings of the AIAA trajectory. A FORTRAN listing and additional details of this Guidance and Control Conference, Boulder, Colo, Aug. 6-8, 1979.
algorithm, including captures with three turns, can be found l°McLean, J.D., "A New Algorithm for Horizontal Capture in Ref. 10.
Trajectories," NASA TM-81186, March 1980.