Document
NASA/TM-2002-210733
String Stability of a Linear Formation Flight
Control System
Michael J. Allen, Jack Ryan, and Curtis E. Hanson NASA Dryden Flight Research Center Edwards, California James E Parle University of Southern California Los, Angeles, California
August 2002
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NASA/TM-2002-210733
String Stability of a Linear Formation
Flight Control System
Michael J. Allen, Jack Ryan, and Curtis E. Hanson NASA Dryden Flight Research Center Edwards, California James E Parle University of Southern California Los Angeles, California National Aeronautics and Space Administration Dryden Flight Research Center Edwards, California 93523-0273
August 2002
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Available from the following: NASA Center for AeroSpace Information (CASI) National Technical Information Service (NTIS) 7121 Standard Drive 5285 Port Royal Road Hanover, MD 21076-1320 Springfield, VA 22161-2171 (301) 621-0390 (703) 487-4650
STRING STABILITY OF A LINEAR FORMATION FLIGHT CONTROL
SYSTEM
Michael J. Allen,* Jack Ryan, _ Curtis E. Hanson_ NASA Dryden Flight Research Center Edwards, California James F. Parle § University of Southern California Los Angeles, California
Abstract Nomenclature
Acronyms String stability analysis of an autonomous formation 6-DOF six degrees-of-freedom flight system was performed using linear and nonlinear Autonomous Formation Flight simulations. String stability is a measure of how position AFF errors propagate from one vehicle to another in a BIBO bounded input, bounded output cascaded system. In the formation flight system Dryden Flight Research Center (Edwards considered here, each ith aircraft uses information from DFRC Air Force Base, California) itself and the preceding ((i-1) th) aircraft to track a commanded relative position. A possible solution for F/A-18 twin-engine jet fighter aircraft (Boeing, meeting performance requirements with such a system USA) is to allow string instability. This paper explores two GPS Global Positioning System results of string instability and outlines analysis International Organization for techniques for string unstable systems. The three ISO Standardization analysis techniques presented here are: linear, nonlinear formation performance, and ride quality. The linear MSDV motion sickness dose value, (m/sec) 15 technique was developed from a worst-case scenario and National Aeronautics and Space could be applied to the design of a string unstable NASA Administration (Washington, D. C.)
controller. The nonlinear formation performance and ride quality analysis techniques both use nonlinear PID proportional-plus-integral-plus-derivative formation simulation. Three of the four PSD Power Spectral Density formation-controller gain-sets analyzed in this paper SISO were limited more by ride quality than by performance. single input, single output Formations of up to seven aircraft in a cascaded Symbols formation could be used in the presence of light gusts with this string unstable system. amp amplitude of oscillation (peak value) frequency-weighted acceleration, m/s 2 a w *Aerospace Engineer dB decibels +Aerospace Engineer E _Aerospace Engineer East direction in Earth tangent reference §Aerospace Engineer frame Copyright © 2002 by the American Institute of Aeronautics and absolute-position error of the ith aircraft, ft Astronautics, Inc. No copyright is asserted in the United States under Ei, abs relative-position error of the ith aircraft, ft Title 17, U.S. Code. The U.S. Government has a royalty-free license El, tel to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by the copyright G 1 (s) example of a string unstable closed-loop owner.
transfer function Note that use of trade names or names of manufacturers in this document does not constitute an official endorsement of such products G 2(s) example of a string stable closed-loop or manufacturers, either expressed or implied, by the National transfer function Aeronautics and Space Administration.
American Institute of Aeronautics and Astronautics
G_ closed-loop transfer function lateral relative-position command, ft
AP Y cm d Hz hertz, cycles/sec lateral relative-position error, ft AP y _ aircraft formation index i AP z vertical relative position, ft imaginary number, _/(-1)
J vertical relative-position command, ft
AP Z_m d bank angle gain
x, vertical relative-position error, ft
AP z _ lateral position integral gain
Avy lateral relative velocity, ft/sec
KIy vertical position integral gain lateral relative-velocity error, ft/sec KI z A Vy_ motion sickness constant
K_
Avz vertical relative velocity, ft/sec
normal acceleration gain KN z vertical relative-velocity error, ft/sec A Vz_ lateral velocity gain (Y Kvy real part of complex number s, S (5 + jo) vertical velocity gain
Kv_
bank angle, deg Ky lateral position gain O3 input frequency, rad/sec
Kz vertical position gain
max maximum frequency of Mm, rad/sec 03 m Mm peak magnitude of the closed-loop transfer
Introduction
function n total number of aircraft in series formation Loosely speaking, string stability is a measure of how errors propagate through a series of interconnected N North direction in earth tangent reference systems. A formation of aircraft is considered string frame stable if, for instance, a position error between the first
Xz vertical acceleration, g and second aircraft results in a smaller position error
between the second and third aircraft.
position of the ith aircraft, ft Pi Pin single position input, ft An example of string instability can be found by considering a formation of piloted aircraft in initial position of the aircraft, ft Po, i low-visibility conditions such as clouds. With this Pout single position output, ft limited visibility, each pilot can only see the aircraft directly ahead and attempts to track a position relative to Y-axis leading aircraft position, ft PYleading that aircraft. Typically, any position changes to the first Y-axis trailing aircraft position, ft lOYtrailing aircraft are reacted to by the second aircraft with slight overshoot. Each aircraft overshoots the motion of the Z-axis leading aircraft position, ft lOZleading previous aircraft. This can cause unacceptable motion of Z-axis trailing aircraft position, ft lOZtrailing the last aircraft in the string.
Rn ratio of absolute-position error, given by There has been much work done investigating string equation (6) instability, most of which is directed toward automotive S Laplace transform variable, s cy+j¢o technology such as adaptive cruise control 1 and the t automated highway system. 2' 3 The vast majority of this time, seconds work has been directed toward investigating varying X longitudinal axes of formation reference strategies to avoid or correct string instability. The frame common conclusions are that either large amounts of Y lateral axes of formation reference frame intervehicle communication are needed, or trajectory following or headway guidance approaches are Z vertical axes of formation reference frame needed.4, 5 input direction, deg 5'1 Most studies are constrained by the requirement that a 5'2 output direction, deg formation of infinite size must be string stable. The APy lateral relative position, ft system described in this paper is limited in formation American Institute of Aeronantics and Astronautics size and is therefore free to explore string unstable The design objectives were to have the trailing aircraft, systems.
in a two-ship formation, maintain relative position in the lateral and vertical directions. Response to commands Autonomous Formation Flight was to be brisk and smooth without adversely affecting pilot comfort. The design was limited to a two-ship This paper focuses its investigation on the string-stability properties of the Autonomous Formation formation and therefore string stability issues were not Flight (AFF) control system. The AFF program was considered in the design. A proportional-plus- developed to try to obtain drag reduction, and hence integral-plus-derivative (PID) controller with state improve fuel efficiency, through formation flight. 6 feedback was used in this system (fig 3 and 4). Limited by hardware constraints, the control system outputs are In a manner similar to birds in a V-shaped formation, longitudinal-stick and lateral-stick commands. The each individual aircraft can reduce its induced drag formation controller inputs are lateral-position errors, through wingtip-vortex interaction.6, 7, 8, 9 A full vertical-position errors, lateral-velocity errors, description of the AFF program can be found in references 6 and 10. Two F/A-18 aircraft in close vertical-velocity errors, local normal acceleration, and formation flight are shown in figure 1.
local bank angle. Longitudinal position was controlled by the pilot with the throttle.
The guidance and navigation algorithms use a formation reference-frame aligned with a fixed formation-heading (fig 5). Details of the guidance and navigation can be found in reference 6.
Four different gain-sets were developed and flight-tested. They are referred to as A-gains, B-gains, C-gains, and D-gains. The A-gains were designed with high stability margins and were used for initial tests of the system in flight. The B-gains were designed to meet performance goals with adequate stability margins. The C-gains were designed with lower stability margins to EC01-0328-03 allow better performance. The D-gains were designed to Figure 1. Two F/A-18 aircraft in formation flight.
give zero steady-state error in the presence of sensor biases. The D-gains are the only gains with a nonzero The AFF system shown in figure 2 was used to position a trailing aircraft relative to a leading aircraft. position integral term.
Ap, rcmd Trailing aircraft formation control system p )" I PYtrailing Leading PZleading- I Formation F/A-18 Trailing Ztrailing aircraft
inner- I._1 aircraft
I gu _ c, IAVY
aero- loop I-_ aero- dynamics nav_g_ion _ controller I [ dynamics VYtrailing VZtrailing
VY'sa"'n'
020186 Figure 2. Formation flight control system.
American Institute of Aeronantics and Astronautics
Linear String Stability
This paper investigates the effects of extending the AFF two-ship formation controller to an n-ship formation. It was expected that an n-ship formation .Yerr a.. , would be string-unstable; however, the authors were interested in attempting to quantify the degree to which the performance of the system and ride quality degraded with such a control system.
AVYerr _U - Two types of stability are used in this paper; 1) 020187 bounded-input bounded-output (BIBO) stability and string stability. The BIBO stability of a single linear Figure 3. Lateral formation control system (individual stability) is met if the real part of system.
each pole of the system is negative. 11 A formation of aircraft is BIBO-stable if bounded motion of the 1st aircraft results in bounded motion of the n th aircraft.
String stability only describes the growth or decay of errors in the formation.
The cascaded system shown in figure 6 is used to approximate n aircraft flying under formation control.
Each identical system, G(s9, attempts to follow the
"Zerr
position of the preceding system. Here G(s) is a linear single-input single-output (SISO) system that is assumed to be individually stable. Sheikholeslam and Desoer 12 show that a cascaded system of identical vehicles (G(s)) will be string stable if AVZerr ;U - NZ O2Ol 88 IG(j¢o)[ < 1 for all ¢o > 0 (1) Figure 4. Vertical formation control system. where jco is substituted for s because cy 0 for constant oscillatory motion. A cascaded system that satisfies equation (1) will attenuate formation errors.
The n th aircraft of a string stable system will attenuate motion of the 1st aircraft. If the system is string unstable according to equation (1), then errors will be amplified Formation X by the formation.
ing _ Aircraft 2 Aircraft 3 Aircraft n Trailing 4' 020190 _i:¢raft /"
// V _\
\ Figure 6. Cascaded formation flight system.
\ \ N If the formation size is finite and known, then the formation can be BIBO-stable even if the system is string unstable by eq (1). The transfer function of the S f// formation shown in figure 6 with input P] and output Pn is
//1
020189
P.(s) 1
(G(s)) n (2) Figure 5. Formation reference frame.
Pl(s)
American Institute of Aeronantics and Astronautics where n is the number of aircraft connected in series Figure 8 shows the step responses of the two systems formation and Pi is the position of the i th aircraft. The when connected in series. The response of G](s) shows formation transfer function (G(s))n 1 has n-1 multiple the exponential amplification of errors caused by string poles and n-1 multiple zeros of G(s). If G(s) is instability. Also note that since G](s) and G2(s ) are both individually stable then the formation will be BIBO individually stable, errors go to zero when the input is stable.
held steady. A string unstable formation of systems such as Gl(S ) will have an unbounded output only if the formation size is infinite.
Consider the two second-order systems: GI(S) 2 -- Command s +6s+ 100 ...... S ,stem 1 ..... S ,stem 2 ...... S ,stem 3 .... S ,stem 4 G2(s) 2 s + 16s + 100 ............... S ,stem 5
s.o _ /_ _
with transfer function magnitudes shown in figure 7.
G](s) is string unstable according to equation (1) and a
2.0 #q_i l iii !_ _
formation of these systems would amplify a 10 rad/sec E .................... _.k2. i-::_ \ _.M::',, __.:--.. :: ....
sinusoidal input. If the input to G](s) were removed and W O the formation size was finite, all systems in the . 5 . . . . . . . . . . . . . . . . . . . / _ , _ _ 1 1 _ 1 _ 1 1 _ 1 1 1 1 1 1 _ 1 1 / 1 1 1 1 1 _ 1 1 1 : _ : . . . . . . . . . . . . . _ . . . . . . . . . . . . . . . . . . .
a.
I?. _:L...:_i.li I i "
formation would return to zero because each system is o ...... i 'i............ i...............................................
individually stable. For comparison, G2(s ) is string stable and would not amplify errors when connected in -1.0 series.
0 2 3 4 5 6 Time, sec 020192 (a) Gffs).
-- Gl(S) string unstable -- Command -- ---- G2(s ) string stable 6 .....
...... System 1 ..... System 2 ........................ ......... ........ .......... ii ................... .......... ......
...... System 3 4 ......... ',--',-+---i- ................ .......... ',-;,--.-', ...............i......... .......
.... System 4 ............... S ,stem 5 3.0 " 2 ......... i_.._L.i.......k. ................ .......... i L.......ii..i.i ............. i......... .......
"O i i i i i i i i i 2.5 • • • . . . . . . . . . _ 1 1 1 1 1 1 _ 1 1 1 1 1 _ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 _ . . . . . . . . . . . . . . . . . . . . . . . . . . _ 1 1 1 _ 1 1 1 1 1 1 1 1 _ 1 1 1 _ . . . . . . . . _ . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.0 • • 1.5 , • • i- ¢0 i i i i i i M. i i i 101 I i i i i i 4,,. i i i t 1 " O . . . . . . . . . . . . . . . . . . .
" I I,;//
-2 ......... i------ -----i----------- ---- ................. ......... i------i----- ------i\------i .................... ......
_2 .5 li j UZ" i i
O I ) _ 0 _ : : : _ 1 1 1 1 1 1 1 . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
-3 i i i i _i
-1.0 0 2 3 4 5 6 100 101 Time, sec Frequency, rad/sec 020193 020191 (b) G2(s ).
Figure 7. Magnitude of closed-loop transfer functions Figure 8. Step response of a string unstable system and from second-order example. G 1 is string unstable, G 3 is a string stable system with five aircraft.
string stable.
American Institute of Aeronantics and Astronautics Forstring stability analysis anabsolute-position error affected by the phase lag of G(s) or the time delay of the
wasdefined as the difference between the aircraft
interaircraft communications. These effects degrade tracking performance but do not determine the peak position (Pi)and itsinitialposition (Po,i).
motion of the n th aircraft described by equation (6).
Ei,abs Pi Po, i (3) Conversion of the AFF System to a Single-Input Single-Output System The absolute-position error (Ei,abs) in equation (3) is different than the relative-position error used by the The AFF system (fig 2) was converted to a linear controller.
SISO model so that equation (6) could be used. The system was not broken into separate lateral and vertical El,tel Pi Pi 1 (4) SISO models because of the possibility that a combination of lateral and vertical inputs could be the Each aircraft is assumed to have an initial relative-position error of zero. worst-case input. First, it was assumed that the system input only consists of leading aircraft motion. All other In a worst-case scenario, the formation will be excited inputs and disturbances were assumed to be zero. The by constant oscillatory motion. The absolute-position second assumption was that the leading aircraft error amplitude ratio of the n th aircraft to the first velocities could be exactly calculated from the leading aircraft for sinusoidal input is given in equation (5). For this analysis, the excitation is restricted to be only on the aircraft positions. With these assumptions, the system 1st aircraft.
was reduced to a two-input two-output system. The two inputs were leading aircraft Y position, PYleadin,' and iG(jo_)ln 1 amp(En,abs) for any ¢o > 0 (5) leading aircraft Z position, Pzzeading. The two outputs amp(El,abs) were trailing aircraft Y position, PYtr_Z_ng and trailing aircraft Z position, Pztr_Z_ng.
Equation (5) can be used to predict the largest amplitude of oscillation possible for the n th aircraft due An independent variable (71) was created for the to motion of the 1st aircraft at a given frequency.
purpose of combining the lateral and vertical axes of the system in order that a worst-case combination of inputs The peak magnitude of G(/'¢o), referred to as Mm, could be used. The two inputs were parameterized with defines the maximum value of equation (5). Substituting input direction 5'1 using equation (7) and equation (8).
the frequency at which M m occurs, tom, into equation (5) produces equation (6).
PYl*_ding = Pin" C0S(5'1) (7) (Mm)n 1 F(Enabs)_ = max .v,....
(6) Pzz,_d_n * = Pin" sin(5'l) (8)
L(El,, bs)j =
The further assumption of a constant 5'1 reduces the Equation (6) can be used as a bound on the system for system input to one parameter, Pin. The two system string stability during the design of the controller. This outputs were combined in a similar fashion by defining gives the designer an initial look at the degree of string 5'2 and Pout" instability of the system and a tool for linear control design. Equation (6) does not replace nonlinear formation simulation or ride quality analysis described (9) later in this paper. System excitation at a different frequency, or with non-sinusoidal motion, will result in a response that is less than the response predicted by equation (6). Equation (6) is only valid if the communication parameters between the aircraft are [py2 + pz 2 (10) Pout = '_1 trailing trailing single-input single-output (SISO).
An important observation from equation (6) is that C0S(5'2 5'1) the amplitude of the absolute-position error is not American Institute of Aeronautics and Astronautics Equations (7) and (8) force the leading aircraft to Figure 12 shows the magnitude of the closed-loop move along a line defined by the fixed angle 5'1 and transfer function of a single formation control system equations (9) and (10) project the two trailing aircraft with each of the four gain-sets. Linear models of the positions onto the same line to form a single trailing system were used for this analysis. Each gain-set results aircraft position, Pout. Figure 9 shows pictorially how 5'1 in a system with a peak that is greater than 1.0 (0 dB) was used to create Pin and Pout. The resulting SISO and is therefore string unstable according to system is shown in figure 10.
equation (6). The data in figure 12 show a trend of higher peak magnitudes for higher bandwidths, indicating a tradeoff between performance and string stability for this system in agreement with previous PZleading work. 13 Nonlinear Formation Performance Formation performance is based on the relative-position error described by equation (4). String
[ %r.,,n,)
instability can cause the relative-position tracking performance of the n th aircraft to degrade beyond acceptable limits. Formation performance can be Po "- 020194 analyzed with linear system models connected in series Figure 9. Parameterization of lateral and or with nonlinear formation simulation. A 6-DOF vertical positions into Pin and Pout. nonlinear formation flight simulation was used in this analysis. 14 Formations were simulated by performing multiple runs of a single aircraft and formation autopilot Linear Results simulation, l° The ith aircraft tracked the motion of a previously recorded (i-1) th aircraft simulation. This The string stability analysis was performed at the process was repeated for each aircraft in formation. The input direction (5'1) that yielded the highest peak of the combinations of lateral and vertical inputs to the closed-loop transfer function (Mm) for each gain-set.
formation were determined using equations (7) and (8).
The effect of input direction was found by plotting M m with varying values of 5'1 as shown in figure 11. The Two test cases were chosen for nonlinear formation input direction that yielded the highest values of M m performance analysis to represent the set of all inputs was found to be 5'1 90 deg for all gain-sets. This expected to occur during flight. The first test case was a input direction was used in the linear string stability step input on the relative-position command of the first analysis. Longitudinal string stability was ignored aircraft in formation. The second test case used only during the linear analysis because longitudinal position was controlled by the pilot. light turbulence to excite the formation.
Plant Pin i Pout 020195 Figure 10. Single-inpu_single-output representation of the formation control system.
American Institute of Aeronantics and Astronautics before the maneuver began. The exponential mA magnification of errors in this formation is a result of .... m --'--C string instability. Formations using the B-, C-, and .......... D D-gains were also simulated. Results of these simulations show that the B-, C-, and D- gains all have higher step responses than the A-gains.
7 ............................................................... i............. /:_":: ...........................
m 6 -- Aircraft I Aircraft 5 ........ Aircraft 2 ................... Aircraft 6
i:s
....... Aircraft 3 ......................... Aircraft 7 ........ _,,,_,_ ..............................................................................................................
IE ....... Aircraft 4 4 ____2_22_2/_ ........................................................................
fi!
1121i2 .................. i..............................................................
2530 ................................... [i i [i ....................
0 45 90 _'I' deg 020196 ¢1 10 I- Figure 11. Input direction survey. 5 W o 0 a.
-5 -10 mA -15 ..... C i1|1|1|11 D -20 -25
e : M ..... i i i ,_._'i',_
0 20 40 60 80 100 120
6 ........................ (A:O_nsI_'N;_<z''VV
Time, sac 020198 4 iiiiiiiiiiiii iiiiiiiiiiiiiiiiii_Xiil;_.i i Figure 13. Six degrees-of-freedom nonlinear m 0 i _.'_ formation simulation using A-gains with step command excitation at the first aircraft.
=m e- (U -10 '-5 _" The results from the nonlinear formation simulation with step input excitation were compared to the linear -15 results from equation (6). The absolute error ratio of the 3rd aircraft to the 1st aircraft for each gain-set is plotted 10-2 10-| in figure 14. These comparisons show that the linear analysis technique is very conservative for predicting Frequency, Hz 020197 the response of the formation to step inputs. This result Figure 12. Magnitude of closed-loop transfer functions is expected because equation (6) used the assumption of for gain-sets A, B, C, and D.
constant sinusoidal input at a frequency chosen to give the highest absolute-position errors. Step inputs give a good indication of the system dynamics, but do not Nonlinear Formation Performance Results excite string stability dynamics as well as constant oscillatory input.
The absolute positions of each aircraft in a seven-aircraft formation, due to step excitation, are plotted in figure 13. The A-gains were used for this test. The simulation of a seven-aircraft formation in light Step commands on the relative-position command of turbulence using A-gains yielded less severe aircraft ±10 ft were given to the 1st aircraft to excite the motion. Relative-position commands were set to zero formation. The nonzero initial conditions of some of the for these tests. Figure 15 shows the positions of a aircraft are a result of error magnification that occurred seven-aircraft formation in light turbulence using the American Institute of Aeronantics and Astronautics Each aircraft in the seven-aircraft formation using the Nonlinear simulation A-gains meets the AFF phase-0 goal of a standard NN Rn deviation of relative-position error that is less than 9 ft in light gusts. 6 The AFF goal was intended for use with W two-aircraft formations but is used here as a J=7 m semiarbitrary limit of relative-position tracking ul ',-'6 performance. Figure 16 shows the relative-position error E 5 of each aircraft in formation in light gusts. Comparison
_4 of figure 15 to figure 16 illustrates the difference
I/I between absolute aircraft motion predicted by
_a
IJLI equation (6) and relative tracking performance given by
"1'2
equation (4). The absolute-position errors are larger and E 1 grow quicker than the relative-position errors. This is because the relative-position errors are reduced when A B C D Gain-set the aircraft are in phase with each other.
020"199 Figure 14. Comparison of nonlinear and linear string E2, i'el stability results for the third aircraft E3, rel E4, rel E5, rel A-gains. The motion of the first three aircraft is E6, tel dominated by gust response, but the remaining aircraft ............ E7, tel in formation seem to be driven more by string instability. The seventh aircraft has constant oscillatory motion with a peak of approximately 20 ft and a frequency of approximately 0.063 Hz. Note that com for the A-gains is 0.086 Hz. Although excited by random gusts, the aircraft near the back of the formation became oscillatory with a frequency of oscillation close to co m.
Aircraft 1 ..... Aircraft 5 ........ Aircraft 2 ................... Aircraft 6 Aircraft 3 .......................... Aircraft 7 I_ -10 ....... ii Aircraft 4 -15 ........................................................................................................ ;'-" - ..............
-200 10 20 30 40 50 60 70 80
15 I ... jk
Time, sec 020201 10 Figure 16. Aircraft relative position from 6-DOF nonlinear formation simulation with light turbulence O using A-gains.
W O Q.
N 0 ... I_.._ . ....,...,,,...._ ........... ,i_:..., ........... _. ,...:...,,: .............
The standard deviation of relative-position error for -5 each of the four gain-sets is plotted in figure 17. This figure compares the relative-position tracking -10 performance of each aircraft to the 9 ft standard deviation limit used for this study. The C-gains are -15 0 10 20 30 40 50 60 70 80 limited by performance at the 5th aircraft. The A-gains Time, sec have sufficient tracking performance with up to seven 020200 aircraft in formation. The design of the A-gains with Figure 15. Aircraft position from 6-DOF nonlinear increased stability margins resulted in a gain-set with formation simulation with light turbulence using low bandwidth but improved string stability A-gains.
characteristics when compared to other gain-sets.
American Institute of Aeronantics and Astronautics The limit for acceptable ride quality was found by limiting the percentage of the general population that would vomit after an hour of flight in these conditions.
A limit of 10 percent was chosen because it has been used in previous ride-quality analysis with military aircraft. ]6 A K m factor of one-third was used in this analysis. ]5 Ride-quality limits and K m factors will vary with aircraft type, mission, and passenger characteristics.
Ride Quality Results Figure 18 shows the ride quality of each aircraft in formation for each gain-set. An unexpected result of this .=.
U) analysis is that gain-sets B, C, and D are all constrained more by ride quality than by relative-position tracking performance. This is because the aircraft motion resulting from string instability determined by the M m 0 _ _ values of the B-, C- and D-gains occur at a frequency 2 3 4 5 6 7 that is conducive to motion sickness. This can be seen Aircraft number, I 020202 by comparing the magnitude plots of the closed-loop Figure 17. Relative-position error standard transfer functions to the frequency-weighting curve deviations for each gain-set.
given in ISO-2631-1 for motion sickness (figure 19).
The A-gains have reduced MSDV z values because they Ride Quality Analysis have better string stability and because the ISO frequency-weighting curve at the value of co m for the String instability also adversely affects aircraft ride A-gains is not weighted as heavily.
quality. ""Ride quality" typically describes the passenger's level of comfort. Measurements made using the nonlinear formation simulations were used to 4O assess the ride quality at the i th aircraft. The
o . i i i
International Organization for Standardization (ISO) standard for motion sickness 15 was used to translate the
.oi i i
accelerations experienced into a measure of pilot comfort. More specifically, motion sickness dose values (MSDV) were calculated for each of the n aircraft in the formation. The MSDV has been defined such that higher values correspond to greater likelihood of motion sickness. The MSDV for vertical acceleration is defined as MSDVz = aw(t ) dt (11) lO here aw is a frequency-weighted acceleration in the z-direction and T is the total period during which motion could occur. Data used in the ride-quality 2 3 4 6 7 analysis was taken from time histories from the Aircraft number, i 020203 nonlinear formation simulation lasting 200 seconds.
Lateral ride-quality analysis was not performed because Figure 18. Aircraft ride quality from lateral acceleration weightings are not given in the ISO nonlinear formation simulation data with standard for motion sickness and because lateral light gusts. High MDSV indicate poor ride accelerations were found to be significantly lower than quality.
vertical accelerations during these tests.
American Institute of Aeronantics and Astronautics equal to the frequency of the peak magnitude of the ............. ISO motion sickness closed-loop transfer function of the system. Adequate weighting curve -- A-gains performance with formations of up to seven aircraft .... B-gains could be obtained using the A-gains in light turbulence.
----- C-gains .......... D-gains The ride quality of each aircraft in formation was evaluated for four gain-sets using the nonlinear simulation and the ISO-2631 standard. 15 Three of the gain-sets are limited more by ride quality than by performance. The exception is the A-gains, because the m 0 "0 magnifications of position errors caused by string "0 instability occur at a lower frequency. Each aircraft in a ._= -S seven-aircraft formation using A-gains would meet a performance specifications and have adequate ride :Z -10 quality.
-15 A string unstable formation control system can be considered when the formation size is limited and the -20 guidance and communication are similar to the system 10-2 10-1 100 presented in this paper. It is recommended that a string Frequency, Hz 020204 stability constraint is included in the design of such Figure 19. Comparison of linear closed-loop transfer systems.
function magnitudes to the ISO-2631 motion sickness References frequency weighting curve.
]Liang, Chi-Ying and Huei Peng, "Optimal Adaptive Cruise Control with Guaranteed String Stability," Concluding Remarks Vehicle System Dynamics, Vol. 31 (1999), pp. 313-330.
Analysis techniques in this paper can be used to 2Gehring, Ottmar and Hans Fritz, "Practical Results determine the effect of string instability on an aircraft of a Longitudinal Control Concept for Truck Platooning under autonomous formation control. Linear analysis, with Vehicle to Vehicle Communication," IEEE nonlinear performance analysis, and ride-quality Co@rence on Intelligent Transportation Systems, ITSC analysis were used to analyze a formation flight system '97, November 1997, Boston, MA for string stability. The three analysis methods were applied to a formation control system using F/A-18 3Wang, Yibing and Zengjin Han, "Stability of an aircraft to demonstrate the results of string instability.
Automated Vehicle Platoon," Proceedings' of the American Control Conference, Philadelphia, The linear string stability equations provided in this Pennsylvania, June 1998.
paper can be used to determine a relative measure of string stability. The equations show that the amplitude 4Kaminer, Isaac, Antonio Pascoal, Eric Hallberg, and of oscillation of the aircraft in a cascaded formation is Carlos Silvestre, "Trajectory Tracking for Autonomous not affected by the phase lag of each closed-loop Vehicles: An Integrated Approach to Guidance and formation system or by the interaircraft communication Control," Journal of Guidance, Control, and Dynamics', delay. The linear technique discussed in this paper could Vol. 21, No. 1, Jan-Feb 1998.
be applied to the design of a formation control system to conservatively limit a SISO system for string stability.
5Swaroop, D., J.K. Hedrick, C. C. Chien, and R Ioannou, "A Comparison of Spacing and Headway Performance analysis was executed with a nonlinear Control Laws for Automatically Controlled Vehicles," simulation of the system. Nonlinear formation Vehicle System Dynamics, Vol. 23 (1994), pp. 597-625.
simulation also demonstrated the conservativeness of 6Hanson, Curtis E., Jack Ryan, Michael J. Allen, and the linear analysis. Formation simulation performed with excitation consisting of only light turbulence Steven R. Jacobson, "An Overview of Flight Test produced oscillatory motion of the aircraft in the back of Results for a Formation Flight Autopilot", American the formation. The frequency of oscillation was nearly Institute of Aeronautics and Astronautics, Guidance, American Institute of Aeronantics and Astronautics Navigation, and Control Conference, Monterey, California, August 2002. (publishing concurrently) 7Beukenberg, Markus and Dietrich Hummel, "Aerodynamics, Performance and Control of Airplanes in Formation Flight," Proceedings' of the International Council of the Aeronautical Sciences, Stockholm, Sweden, Sept. 9-14, 1990.
8Hummel, Dietrich, "Formation Flight as an Energy-Saving Mechanism", Israel Journal of Zoology, Vol. 41, March, 1995. pp. 261-278.
9Weimerskirch, Henri, Julien Martin, Yannick Clerquin, Peggy Alexandre, and Sarka Jiraskova "Energy Saving in Flight Formation," Nature, Vol. 413, 18 Oct. 2001, pp. 697 and 698.
l°Ryan, Jack, Curtis E. Hanson, Ken A Norlin, and Michael J. Allen, "Data Synchronization Discrepancies in a Formation Flight Control System," Proceedings' of SFTE 32nd Annual International Symposium, 10-14 Sept. 2001, Seattle, Washington.
lichen, Chi-Tsong, Linear System Theory and Design, Oxford University Press, New York, 1999.
12Sheikholeslam, Shahab and Charles A. Desoer, "Longitudinal Control of a Platoon of Vehicles with no Communication of Lead Vehicle Information: A System Level Study," IEEE Transactions on Vehicular Technology, Vol. 42, No. 4 Nov. 1993.
13Swaroop, D. and J. K. Hedrick, "String Stability of Interconnected Systems," IEEE Transactions on Automatic Control, Vol. 41, No. 3, March 1996.
14Norlin, Ken A. Flight Simulation Software at NASA Dryden Flight Research Center. NASA TM 104315, Oct. 1995.
15International Organization for Standardization, Mechanical Vibration and Shock Evaluation of Human Exposure to Whole-Body Vibration. Part 1, General Requirements, ISO 2631-1 :1997(E).
16jacobson, S. and Moynes, J. "Ride Qualities Criteria for the B-2 Bomber," AIAA 90-3256, AIAA/AHSVASEE Aircraft Design, Systems and Operations Conference, Sept 17-19, 1990.
AmericanInstituteof Aeronantics andAstronautics
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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORTTYPE AND DATES COVERED August 2002 Technical Memorandum 4. TITLE AND SUBTITLE 5. FUNDING NUMBERS String Stability of a Linear Formation Flight Control System WU 706-35-00-E8-20-00-AFF 6. AUTHOR(S) Michael J. Allen, Jack Ryan, James F. Parle, and Curtis E. Hanson 8. PERFORMING ORGANIZATION 7. PERFORMING ORGANIZATION NAME(S)AND ADDRESS(ES) REPORT NUMBER NASA Dryden Flight Research Center RO. Box 273 H-2504 Edwards, California 93523-0273 10. SPONSORING/MONITORING 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) AGENCY REPORTNUMBER National Aeronautics and Space Administration NASA/TM-2002-210733 Washington, DC 20546-0001 11. SUPPLEMENTARY NOTES Control Conference, Also presented at the AIAA Guidance, Navigation, and Monterey, California, August 5-8, 2002.
12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE Unclassified--Unlimited Subject Category -- 08 This report is available at http://www.dfrc.nasa.gov/DTRS/ 13. ABSTRACT (Maximum 200 words) String stability analysis of an autonomous formation flight system was performed using linear and nonlinear simulations. String stability is a measure of how position errors propagate from one vehicle to another in a cascaded system. In the formation flight system considered here, each i'" aircraft uses information from itself and the preceding ((i-1) th) aircraft to track a commanded relative position. A possible solution for meeting performance requirements with such a system is to allow string instability. This paper explores two results of string instability and outlines analysis techniques for string unstable systems. The three analysis techniques presented here are: linear, nonlinear formation performance, and ride quality. The linear technique was developed from a worst-case scenario and could be applied to the design of a string unstable controller. The nonlinear formation performance and ride quality analysis techniques both use nonlinear formation simulation.
Three of the four formation-controller gain-sets analyzed in this paper were limited more by ride quality than by performance. Formations of up to seven aircraft in a cascaded formation could be used in the presence of light gusts with this string unstable system.
14. SUBJECTTERMS 15. NUMBER OF PAGES Aircraft, Formation, Leader-follower, Stability, String 16. PRICE CODE 17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT OF REPORT OF THIS PAGE OF ABSTRACT Unclassified Unclassified Unclassified Unlimited NSN 7840-01-280-8800 Standard Form 298 (Rev. 2-89) Prescribed byANSI Std Z39 18 298 102