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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. HansonNASA Dryden Flight Research CenterEdwards, California

James F. ParleUniversity of Southern CaliforniaLos, Angeles, California

August 2002

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NASA/TM-2002-210733


Michael J. Allen, Jack Ryan, and Curtis E. HansonNASA Dryden Flight Research CenterEdwards, California

James F. ParleUniversity of Southern CaliforniaLos Angeles, California

August 2002

National Aeronautics andSpace Administration

Dryden Flight Research CenterEdwards, California 93523-0273

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STRING STABILITY OF A LINEAR FORMATION FLIGHT CONTROL SYSTEM

Michael J. Allen,* Jack Ryan,† Curtis E. Hanson‡

NASA Dryden Flight Research CenterEdwards, California

James F. Parle§

University of Southern CaliforniaLos Angeles, California

Abstract

String stability analysis of an autonomous formationflight system was performed using linear and nonlinearsimulations. String stability is a measure of how positionerrors propagate from one vehicle to another in acascaded system. In the formation flight systemconsidered here, each ith aircraft uses information fromitself and the preceding ((i-1)th) aircraft to track acommanded relative position. A possible solution formeeting performance requirements with such a systemis to allow string instability. This paper explores tworesults of string instability and outlines analysistechniques for string unstable systems. The threeanalysis techniques presented here are: linear, nonlinearformation performance, and ride quality. The lineartechnique was developed from a worst-case scenario andcould be applied to the design of a string unstablecontroller. The nonlinear formation performance andride quality analysis techniques both use nonlinearformation simulation. Three of the fourformation-controller gain-sets analyzed in this paperwere limited more by ride quality than by performance.Formations of up to seven aircraft in a cascadedformation could be used in the presence of light gustswith this string unstable system.

1American Institute of Aero

*Aerospace Engineer†Aerospace Engineer‡Aerospace Engineer§Aerospace Engineer

Copyright 2002 by the American Institute of Aeronautics andAstronautics, Inc. No copyright is asserted in the United States underTitle 17, U.S. Code. The U.S. Government has a royalty-free licenseto exercise all rights under the copyright claimed herein forGovernmental purposes. All other rights are reserved by the copyrightowner.

Note that use of trade names or names of manufacturers in thisdocument does not constitute an official endorsement of such productsor manufacturers, either expressed or implied, by the NationalAeronautics and Space Administration.

Nomenclature

Acronyms

6-DOF six degrees-of-freedom

AFF Autonomous Formation Flight

BIBO bounded input, bounded output

DFRC Dryden Flight Research Center (Edwards Air Force Base, California)

F/A-18 twin-engine jet fighter aircraft (Boeing, USA)

GPS Global Positioning System

ISO International Organization for Standardization

MSDV motion sickness dose value, (m/sec)1.5

NASA National Aeronautics and Space Administration (Washington, D. C.)

PID proportional-plus-integral-plus-derivative

PSD Power Spectral Density

SISO single input, single output

Symbols

amp amplitude of oscillation (peak value)

aw frequency-weighted acceleration, m/s2

dB decibels

E East direction in Earth tangent reference frame

Ei,abs absolute-position error of the ith aircraft, ft

Ei,rel relative-position error of the ith aircraft, ft

G1(s) example of a string unstable closed-loop transfer function

G2(s) example of a string stable closed-loop transfer function

nautics and Astronautics

G(s) closed-loop transfer function

Hz hertz, cycles/sec

i aircraft formation index

j imaginary number, √(-1)

Kφ bank angle gain

lateral position integral gain

vertical position integral gain

Km motion sickness constant

normal acceleration gain

lateral velocity gain

vertical velocity gain

KY lateral position gain

KZ vertical position gain

max maximum

Mm peak magnitude of the closed-loop transfer function

n total number of aircraft in series formation

N North direction in earth tangent reference frame

NZ vertical acceleration, g

Pi position of the ith aircraft, ft

Pin single position input, ft

Po,i initial position of the aircraft, ft

Pout single position output, ft

Y-axis leading aircraft position, ft

Y-axis trailing aircraft position, ft

Z-axis leading aircraft position, ft

Z-axis trailing aircraft position, ft

Rn ratio of absolute-position error, given by equation (6)

s Laplace transform variable, s=σ+jω

t time, seconds

X longitudinal axes of formation reference frame

Y lateral axes of formation reference frame

Z vertical axes of formation reference frame

γ1 input direction, deg

γ2 output direction, deg

∆PY lateral relative position, ft

lateral relative-position command, ft

lateral relative-position error, ft

∆PZ vertical relative position, ft

vertical relative-position command, ft

vertical relative-position error, ft

∆VY lateral relative velocity, ft/sec

lateral relative-velocity error, ft/sec

∆VZ vertical relative velocity, ft/sec

vertical relative-velocity error, ft/sec

σ real part of complex number s,

φ bank angle, deg

ω input frequency, rad/sec

ωm frequency of Mm, rad/sec

Introduction

Loosely speaking, string stability is a measure of howerrors propagate through a series of interconnectedsystems. A formation of aircraft is considered stringstable if, for instance, a position error between the firstand second aircraft results in a smaller position errorbetween the second and third aircraft.

An example of string instability can be found byconsidering a formation of piloted aircraft inlow-visibility conditions such as clouds. With thislimited visibility, each pilot can only see the aircraftdirectly ahead and attempts to track a position relative tothat aircraft. Typically, any position changes to the firstaircraft are reacted to by the second aircraft with slightovershoot. Each aircraft overshoots the motion of theprevious aircraft. This can cause unacceptable motion ofthe last aircraft in the string.

There has been much work done investigating stringinstability, most of which is directed toward automotivetechnology such as adaptive cruise control1 and theautomated highway system.2, 3 The vast majority of thiswork has been directed toward investigating varyingstrategies to avoid or correct string instability. Thecommon conclusions are that either large amounts ofintervehicle communication are needed, or trajectoryfollowing or headway guidance approaches areneeded.4, 5

Most studies are constrained by the requirement that aformation of infinite size must be string stable. Thesystem described in this paper is limited in formation

KIY

KIZ

KNZ

KV Y

KV Z

PY leading

PY trailing

PZleading

PZtrailing

∆PY cmd

∆PY err

∆PZcmd

∆PZerr

∆V Y err

∆V Zerr

s σ jω+=

2American Institute of Aeronautics and Astronautics

size and is therefore free to explore string unstablesystems.

Autonomous Formation Flight

This paper focuses its investigation on thestring-stability properties of the Autonomous FormationFlight (AFF) control system. The AFF program wasdeveloped to try to obtain drag reduction, and henceimprove fuel efficiency, through formation flight.6

In a manner similar to birds in a V-shaped formation,each individual aircraft can reduce its induced dragthrough wingtip-vortex interaction.6, 7, 8, 9 A fulldescription of the AFF program can be found inreferences 6 and 10. Two F/A-18 aircraft in closeformation flight are shown in figure 1.

The AFF system shown in figure 2 was used toposition a trailing aircraft relative to a leading aircraft.

The design objectives were to have the trailing aircraft,in a two-ship formation, maintain relative position in thelateral and vertical directions. Response to commandswas to be brisk and smooth without adversely affectingpilot comfort. The design was limited to a two-shipformation and therefore string stability issues were notconsidered in the design. A proportional-plus-integral-plus-derivative (PID) controller with statefeedback was used in this system (fig 3 and 4). Limitedby hardware constraints, the control system outputs arelongitudinal-stick and lateral-stick commands. Theformation controller inputs are lateral-position errors,vertical-position errors, lateral-velocity errors,vertical-velocity errors, local normal acceleration, andlocal bank angle. Longitudinal position was controlledby the pilot with the throttle.

The guidance and navigation algorithms use aformation reference-frame aligned with a fixedformation-heading (fig 5). Details of the guidance andnavigation can be found in reference 6.

Four different gain-sets were developed andflight-tested. They are referred to as A-gains, B-gains,C-gains, and D-gains. The A-gains were designed withhigh stability margins and were used for initial tests ofthe system in flight. The B-gains were designed to meetperformance goals with adequate stability margins. TheC-gains were designed with lower stability margins toallow better performance. The D-gains were designed togive zero steady-state error in the presence of sensorbiases. The D-gains are the only gains with a nonzeroposition integral term.

EC01-0328-03

Figure 1. Two F/A-18 aircraft in formation flight.


Figure 2. Formation flight control system.

Linear String Stability

This paper investigates the effects of extending theAFF two-ship formation controller to an n-shipformation. It was expected that an n-ship formationwould be string-unstable; however, the authors wereinterested in attempting to quantify the degree to whichthe performance of the system and ride quality degradedwith such a control system.

Two types of stability are used in this paper;bounded-input–bounded-output (BIBO) stability andstring stability. The BIBO stability of a single linearsystem (individual stability) is met if the real part ofeach pole of the system is negative.11 A formation ofaircraft is BIBO-stable if bounded motion of the 1st

aircraft results in bounded motion of the nth aircraft.String stability only describes the growth or decay oferrors in the formation.

The cascaded system shown in figure 6 is used toapproximate n aircraft flying under formation control.Each identical system, G(s), attempts to follow theposition of the preceding system. Here G(s) is a linearsingle-input–single-output (SISO) system that isassumed to be individually stable. Sheikholeslam andDesoer12 show that a cascaded system of identicalvehicles (G(s)) will be string stable if

(1)

where jω is substituted for s because forconstant oscillatory motion. A cascaded system thatsatisfies equation (1) will attenuate formation errors.The nth aircraft of a string stable system will attenuatemotion of the 1st aircraft. If the system is string unstableaccording to equation (1), then errors will be amplifiedby the formation.

If the formation size is finite and known, then theformation can be BIBO-stable even if the system isstring unstable by eq (1). The transfer function of theformation shown in figure 6 with input P1 and output Pnis

(2)

Figure 3. Lateral formation controlsystem.

Figure 4. Vertical formation controlsystem.

Figure 5. Formation reference frame.

G jω( ) 1 for all ω 0><

σ 0=

Figure 6. Cascaded formation flight system.

Pn s( )P1 s( )------------- G s( )( )n 1–

=


where n is the number of aircraft connected in seriesformation and Pi is the position of the ith aircraft. Theformation transfer function has n-1 multiplepoles and n-1 multiple zeros of G(s). If G(s) isindividually stable then the formation will be BIBOstable.

Consider the two second-order systems:

with transfer function magnitudes shown in figure 7.G1(s) is string unstable according to equation (1) and aformation of these systems would amplify a 10 rad/secsinusoidal input. If the input to G1(s) were removed andthe formation size was finite, all systems in theformation would return to zero because each system isindividually stable. For comparison, G2(s) is stringstable and would not amplify errors when connected inseries.

Figure 8 shows the step responses of the two systemswhen connected in series. The response of G1(s) showsthe exponential amplification of errors caused by stringinstability. Also note that since G1(s) and G2(s) are bothindividually stable, errors go to zero when the input isheld steady. A string unstable formation of systems suchas G1(s) will have an unbounded output only if theformation size is infinite.

G s( )( )n 1–

G1 s( ) 100

s2

6s 100+ +--------------------------------=

G2 s( ) 100

s2

16s 100+ +-----------------------------------=

Figure 7. Magnitude of closed-loop transfer functionsfrom second-order example. G1 is string unstable, G2 isstring stable.

(a) G1(s).

(b) G2(s).

Figure 8. Step response of a string unstable system anda string stable system with five aircraft.


For string stability analysis an absolute-position errorwas defined as the difference between the aircraftposition (Pi) and its initial position (Po,i).

(3)

The absolute-position error (Ei,abs) in equation (3) isdifferent than the relative-position error used by thecontroller.

(4)

Each aircraft is assumed to have an initialrelative-position error of zero.

In a worst-case scenario, the formation will be excitedby constant oscillatory motion. The absolute-positionerror amplitude ratio of the nth aircraft to the firstaircraft for sinusoidal input is given in equation (5). Forthis analysis, the excitation is restricted to be only on the1st aircraft.

(5)

Equation (5) can be used to predict the largestamplitude of oscillation possible for the nth aircraft dueto motion of the 1st aircraft at a given frequency.

The peak magnitude of G(jω), referred to as Mm,defines the maximum value of equation (5). Substitutingthe frequency at which Mm occurs, ωm, intoequation (5) produces equation (6).

(6)

Equation (6) can be used as a bound on the system forstring stability during the design of the controller. Thisgives the designer an initial look at the degree of stringinstability of the system and a tool for linear controldesign. Equation (6) does not replace nonlinearformation simulation or ride quality analysis describedlater in this paper. System excitation at a differentfrequency, or with non-sinusoidal motion, will result ina response that is less than the response predicted byequation (6). Equation (6) is only valid if thecommunication parameters between the aircraft aresingle-input–single-output (SISO).

An important observation from equation (6) is thatthe amplitude of the absolute-position error is not

affected by the phase lag of G(s) or the time delay of theinteraircraft communications. These effects degradetracking performance but do not determine the peakmotion of the nth aircraft described by equation (6).

Conversion of the AFF System to a Single-Input–Single-Output System

The AFF system (fig 2) was converted to a linear

SISO model so that equation (6) could be used. The

system was not broken into separate lateral and vertical

SISO models because of the possibility that a

combination of lateral and vertical inputs could be the

worst-case input. First, it was assumed that the system

input only consists of leading aircraft motion. All other

inputs and disturbances were assumed to be zero. The

second assumption was that the leading aircraft

velocities could be exactly calculated from the leading

aircraft positions. With these assumptions, the system

was reduced to a two-input two-output system. The two

inputs were leading aircraft Y position, , and

leading aircraft Z position, . The two outputs

were trailing aircraft Y position, and trailing

aircraft Z position, .

An independent variable (γ1) was created for thepurpose of combining the lateral and vertical axes of thesystem in order that a worst-case combination of inputscould be used. The two inputs were parameterized withinput direction γ1 using equation (7) and equation (8).

(7)

(8)

The further assumption of a constant γ1 reduces thesystem input to one parameter, Pin. The two systemoutputs were combined in a similar fashion by definingγ2 and Pout.

(9)

(10)

Ei abs, Pi Po i,–=

Ei rel, Pi Pi 1––=

G jω( ) n 1– amp En abs,( )amp E1 abs,( )------------------------------- for any ω 0>=

Mm( )n 1–max

En abs,( )E1 abs,( )

-------------------- Rn= =

PY leadingPZleading

PY trailingPZtrailing

PY leadingPin γ1( )cos⋅=

PZleadingPin γ1( )sin⋅=

γ2

PZtrailing

PY trailing

------------------- 1–tan=

Pout PY trailing

2PZtrailing

2+

γ2 γ1–( )cos⋅

=


Equations (7) and (8) force the leading aircraft tomove along a line defined by the fixed angle γ1 andequations (9) and (10) project the two trailing aircraftpositions onto the same line to form a single trailingaircraft position, Pout. Figure 9 shows pictorially how γ1was used to create Pin and Pout. The resulting SISOsystem is shown in figure 10.

Linear Results

The string stability analysis was performed at theinput direction (γ1) that yielded the highest peak of theclosed-loop transfer function (Mm) for each gain-set.The effect of input direction was found by plotting Mmwith varying values of γ1 as shown in figure 11. Theinput direction that yielded the highest values of Mmwas found to be for all gain-sets. Thisinput direction was used in the linear string stabilityanalysis. Longitudinal string stability was ignoredduring the linear analysis because longitudinal positionwas controlled by the pilot.

Figure 12 shows the magnitude of the closed-looptransfer function of a single formation control systemwith each of the four gain-sets. Linear models of thesystem were used for this analysis. Each gain-set resultsin a system with a peak that is greater than 1.0 (0 dB)and is therefore string unstable according toequation (6). The data in figure 12 show a trend ofhigher peak magnitudes for higher bandwidths,indicating a tradeoff between performance and stringstability for this system in agreement with previouswork.13

Nonlinear Formation Performance

Formation performance is based on therelative-position error described by equation (4). Stringinstability can cause the relative-position trackingperformance of the nth aircraft to degrade beyondacceptable limits. Formation performance can beanalyzed with linear system models connected in seriesor with nonlinear formation simulation. A 6-DOFnonlinear formation flight simulation was used in thisanalysis.14 Formations were simulated by performingmultiple runs of a single aircraft and formation autopilotsimulation.10 The ith aircraft tracked the motion of apreviously recorded (i-1)th aircraft simulation. Thisprocess was repeated for each aircraft in formation. Thecombinations of lateral and vertical inputs to theformation were determined using equations (7) and (8).

Two test cases were chosen for nonlinear formationperformance analysis to represent the set of all inputsexpected to occur during flight. The first test case was astep input on the relative-position command of the firstaircraft in formation. The second test case used onlylight turbulence to excite the formation.

Figure 9. Parameterization of lateral andvertical positions into Pin and Pout.

γ1 90 deg=


Figure 10. Single-input–single-output representation of the formationcontrol system.

Nonlinear Formation Performance Results

The absolute positions of each aircraft in aseven-aircraft formation, due to step excitation, areplotted in figure 13. The A-gains were used for this test.Step commands on the relative-position command of±10 ft were given to the 1st aircraft to excite theformation. The nonzero initial conditions of some of theaircraft are a result of error magnification that occurred

before the maneuver began. The exponentialmagnification of errors in this formation is a result ofstring instability. Formations using the B-, C-, andD-gains were also simulated. Results of thesesimulations show that the B-, C-, and D- gains all havehigher step responses than the A-gains.

The results from the nonlinear formation simulationwith step input excitation were compared to the linearresults from equation (6). The absolute error ratio of the3rd aircraft to the 1st aircraft for each gain-set is plottedin figure 14. These comparisons show that the linearanalysis technique is very conservative for predictingthe response of the formation to step inputs. This resultis expected because equation (6) used the assumption ofconstant sinusoidal input at a frequency chosen to givethe highest absolute-position errors. Step inputs give agood indication of the system dynamics, but do notexcite string stability dynamics as well as constantoscillatory input.

The simulation of a seven-aircraft formation in lightturbulence using A-gains yielded less severe aircraftmotion. Relative-position commands were set to zerofor these tests. Figure 15 shows the positions of aseven-aircraft formation in light turbulence using the

Figure 11. Input direction survey.

Figure 12. Magnitude of closed-loop transfer functionsfor gain-sets A, B, C, and D.

Figure 13. Six degrees-of-freedom nonlinearformation simulation using A-gains with stepcommand excitation at the first aircraft.


A-gains. The motion of the first three aircraft isdominated by gust response, but the remaining aircraftin formation seem to be driven more by stringinstability. The seventh aircraft has constant oscillatorymotion with a peak of approximately 20 ft and afrequency of approximately 0.063 Hz. Note that ωm forthe A-gains is 0.086 Hz. Although excited by randomgusts, the aircraft near the back of the formation becameoscillatory with a frequency of oscillation close to ωm.

Each aircraft in the seven-aircraft formation using theA-gains meets the AFF phase-0 goal of a standarddeviation of relative-position error that is less than 9 ftin light gusts.6 The AFF goal was intended for use withtwo-aircraft formations, but is used here as asemiarbitrary limit of relative-position trackingperformance. Figure 16 shows the relative-position errorof each aircraft in formation in light gusts. Comparisonof figure 15 to figure 16 illustrates the differencebetween absolute aircraft motion predicted byequation (6) and relative tracking performance given byequation (4). The absolute-position errors are larger andgrow quicker than the relative-position errors. This isbecause the relative-position errors are reduced whenthe aircraft are in phase with each other.

The standard deviation of relative-position error foreach of the four gain-sets is plotted in figure 17. Thisfigure compares the relative-position trackingperformance of each aircraft to the 9 ft standarddeviation limit used for this study. The C-gains arelimited by performance at the 5th aircraft. The A-gainshave sufficient tracking performance with up to sevenaircraft in formation. The design of the A-gains withincreased stability margins resulted in a gain-set withlow bandwidth but improved string stabilitycharacteristics when compared to other gain-sets.

Figure 14. Comparison of nonlinear and linear stringstability results for the third aircraft

Figure 15. Aircraft position from 6-DOF nonlinearformation simulation with light turbulence usingA-gains.

Figure 16. Aircraft relative position from 6-DOFnonlinear formation simulation with light turbulenceusing A-gains.


Ride Quality Analysis

String instability also adversely affects aircraft ridequality. “Ride quality” typically describes thepassenger’s level of comfort. Measurements madeusing the nonlinear formation simulations were used toassess the ride quality at the ith aircraft. TheInternational Organization for Standardization (ISO)standard for motion sickness15 was used to translate theaccelerations experienced into a measure of pilotcomfort. More specifically, motion sickness dose values(MSDV) were calculated for each of the n aircraft in theformation. The MSDV has been defined such that highervalues correspond to greater likelihood of motionsickness. The MSDV for vertical acceleration is definedas

(11)

here aw is a frequency-weighted acceleration in thez-direction and T is the total period during whichmotion could occur. Data used in the ride-qualityanalysis was taken from time histories from thenonlinear formation simulation lasting 200 seconds.Lateral ride-quality analysis was not performed becauselateral acceleration weightings are not given in the ISOstandard for motion sickness and because lateralaccelerations were found to be significantly lower thanvertical accelerations during these tests.

The limit for acceptable ride quality was found bylimiting the percentage of the general population thatwould vomit after an hour of flight in these conditions.A limit of 10 percent was chosen because it has beenused in previous ride-quality analysis with militaryaircraft.16 A Km factor of one-third was used in thisanalysis.15 Ride-quality limits and Km factors will varywith aircraft type, mission, and passengercharacteristics.

Ride Quality Results

Figure 18 shows the ride quality of each aircraft information for each gain-set. An unexpected result of thisanalysis is that gain-sets B, C, and D are all constrainedmore by ride quality than by relative-position trackingperformance. This is because the aircraft motionresulting from string instability determined by the Mmvalues of the B-, C- and D-gains occur at a frequencythat is conducive to motion sickness. This can be seenby comparing the magnitude plots of the closed-looptransfer functions to the frequency-weighting curvegiven in ISO-2631-1 for motion sickness (figure 19).The A-gains have reduced MSDVz values because theyhave better string stability and because the ISOfrequency-weighting curve at the value of ωm for theA-gains is not weighted as heavily.

Figure 17. Relative-position error standarddeviations for each gain-set.

MSDV z aw t( )2

td0

T

∫ 1 2⁄

=

Figure 18. Aircraft ride quality fromnonlinear formation simulation data withlight gusts. High MDSV indicate poor ridequality.


Concluding Remarks

Analysis techniques in this paper can be used todetermine the effect of string instability on an aircraftunder autonomous formation control. Linear analysis,nonlinear performance analysis, and ride-qualityanalysis were used to analyze a formation flight systemfor string stability. The three analysis methods wereapplied to a formation control system using F/A-18aircraft to demonstrate the results of string instability.

The linear string stability equations provided in thispaper can be used to determine a relative measure ofstring stability. The equations show that the amplitudeof oscillation of the aircraft in a cascaded formation isnot affected by the phase lag of each closed-loopformation system or by the interaircraft communicationdelay. The linear technique discussed in this paper couldbe applied to the design of a formation control system toconservatively limit a SISO system for string stability.

Performance analysis was executed with a nonlinearsimulation of the system. Nonlinear formationsimulation also demonstrated the conservativeness ofthe linear analysis. Formation simulation performedwith excitation consisting of only light turbulenceproduced oscillatory motion of the aircraft in the back ofthe formation. The frequency of oscillation was nearly

equal to the frequency of the peak magnitude of theclosed-loop transfer function of the system. Adequateperformance with formations of up to seven aircraftcould be obtained using the A-gains in light turbulence.

The ride quality of each aircraft in formation wasevaluated for four gain-sets using the nonlinearsimulation and the ISO-2631 standard.15 Three of thegain-sets are limited more by ride quality than byperformance. The exception is the A-gains, because themagnifications of position errors caused by stringinstability occur at a lower frequency. Each aircraft in aseven-aircraft formation using A-gains would meetperformance specifications and have adequate ridequality.

A string unstable formation control system can beconsidered when the formation size is limited and theguidance and communication are similar to the systempresented in this paper. It is recommended that a stringstability constraint is included in the design of suchsystems.

References

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Figure 19. Comparison of linear closed-loop transferfunction magnitudes to the ISO-2631 motion sicknessfrequency weighting curve.


Navigation, and Control Conference, Monterey,California, August 2002. (publishing concurrently)

7Beukenberg, Markus and Dietrich Hummel,“Aerodynamics, Performance and Control of Airplanesin Formation Flight,” Proceedings of the InternationalCouncil of the Aeronautical Sciences, Stockholm,Sweden, Sept. 9-14, 1990.

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16Jacobson, S. and Moynes, J. “Ride QualitiesCriteria for the B-2 Bomber,” AIAA 90-3256,AIAA/AHS/ASEE Aircraft Design, Systems andOperations Conference, Sept 17-19, 1990.


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WU 706-35-00-E8-20-00-AFF

Michael J. Allen, Jack Ryan, James F. Parle, and Curtis E. Hanson

NASA Dryden Flight Research CenterP.O. Box 273Edwards, California 93523-0273

H-2504

National Aeronautics and Space AdministrationWashington, DC 20546-0001 NASA/TM-2002-210733

String stability analysis of an autonomous formation flight system was performed using linear and nonlinearsimulations. String stability is a measure of how position errors propagate from one vehicle to another in acascaded system. In the formation flight system considered here, each

ith

aircraft uses information from itselfand the preceding (

(i-1)th

) aircraft to track a commanded relative position. A possible solution for meetingperformance requirements with such a system is to allow string instability. This paper explores two results ofstring instability and outlines analysis techniques for string unstable systems. The three analysis techniquespresented here are: linear, nonlinear formation performance, and ride quality. The linear technique wasdeveloped from a worst-case scenario and could be applied to the design of a string unstable controller. Thenonlinear 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 thanby performance. Formations of up to seven aircraft in a cascaded formation could be used in the presence oflight gusts with this string unstable system.

Aircraft, Formation, Leader-follower, Stability, String 16

Unclassified Unclassified Unclassified Unlimited

August 2002 Technical Memorandum

Also presented at the AIAA Guidance, Navigation, and Control Conference, Monterey, California,August 5-8, 2002.

Unclassified—UnlimitedSubject Category — 08This report is available at http://www.dfrc.nasa.gov/DTRS/

string stability of a linear formation flight control · pdf filestring stability of a linear...

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