Multibody dynamics and robust control of flexible spacecraft

Multibody Dynamics andRobust Control of FlexibleSpacecraft

A. GREWALInstitute for Aerospace Research

V. J. MODIUniversity of British Columbia

The paper focuses on an approach to the study of the

dynamics and control of large flexible space structures, comprised

of subassemblies, a subject of considerable contemporary interest.

To begin with, a relatively general Lagrangian formulation

of the problem is presented. The governing equations are

nonlinear, nonautonomous, coupled, and extremely lengthy

even in matrix notation. Next, an efficient computer code

is developed and the versatility of the program illustrated

through a dynamical study of the first element launch (FEL)

configuration of the Space Station Freedom, now superseded

by the International Space Station. Finally, robust control of

the rigid body motion of the FEL configuration using both the

linear-quadratic-Gaussian/loop transfer recovery (LQG/LTR) and

H1 procedures is demonstrated. The controllers designed using

the simplified linear models, prove to be effective in regulating

librational disturbances. Such a global approach–formulation,

numerical code, dynamics, and control–is indeed rare. It can

serve as a powerful tool to gain comprehensive understanding

of dynamical interactions and thus aid in the development of an

effective and efficient control system.

Manuscript received January 5, 1999; revised September 4 andNovember 29, 1999; ready for publication December 1, 1999.

IEEE Log No. T-AES/36/2/05223.

Refereeing of this contribution was handled by T. E. Busch.

Authors’ addresses: A. Grewal, Institute for Aerospace Research,National Research Council, Montreal Rd., Ottawa, Ontario, Canada,K1A 0R6; V. J. Modi, Dept. of Mechanical Engineering, Universityof British Columbia, Vancouver, British Columbia, Canada,V6T 1Z4.

0018-9251/00/$10.00 c° 2000 IEEE

I. INTRODUCTION

Over the past few decades, there has been atrend towards more flexible satellites and spacestructures. This has presented numerous challengesto investigators in the field of spacecraft dynamicsand control. These include the study of flexible,multibody, orbiting structures with deploying,slewing, and translating members, in the presence ofenvironmental disturbances.From the point of view of applicability to the

study of a variety of spacecraft and space platforms,the development of a general multibody dynamicalformulation and associated computer code is desirable.A wide variety of computer codes aimed at dynamicsand control of flexible multibody orbiting systemssuch as TREETOPS [1], are available commercially.As can be expected, each asserts a variety ofdistinctive and desirable features and have been usedin practice with a varying degree of success oftengoverned by the nature of the system, experience ofthe user, and computational tools available. In general,they all display limitations and present a scope forimprovement. Furthermore, it is widely recognizedthat:

1) rendering the program operational oftendemands enormous time and effort;2) user end modification and adaptability of the

program is often very cumbersome if not impossible;3) contribution of the various forces and moments

to the governing equations of motion is usually notavailable explicitly, making the analysis of results andphysical appreciation of the system dynamics difficult.

The difficulty, and in some cases impossibility, ofmodifying these codes for the unique requirementsof a particular study would necessitate their in-housedevelopment, particularly in the research-orientedenvironment. To that end, a multibody formulationand code, which considers an assemblage of beamsand plates in a tree-type topology, has been developed.The governing equations of motion are derived usingthe Lagrangian procedure, and the formulation allowsfor the simultaneous slewing and translation of theappendages. Also, the code computes the linearizedequations of motion which are vital for the synthesisof linear control systems.An assessment of techniques and issues pertaining

to the control of large, flexible space structureshas been addressed in a number of survey papers(e.g., [2]). In a more recent study, Van Woerkom [3]touches upon a number of newly developed robusttechniques for the control of this class of systems.The subject of attitude control of spacecraft with

momentum management has received some attentionin recent years [4]. Sundararajan, et al. [5] appliedthe linear-quadratic-Gaussian/loop transfer recovery(LQG/LTR) design procedure to the problem of

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 36, NO. 2 APRIL 2000 491

Fig. 1. Schematic diagram of multibody model used in the study.

the pointing control of a flexible space antenna. Anapplication of H1 control theory to the design of anattitude and momentum control system of the SpaceStation was explored by Byun, et al. [6]. A numberof investigations [7] have used a two-level controlstrategy to address the spillover problem whereby alow authority controller (LAC) employing colocatedrate feedback is initially used to enhance the dampingof the critical flexure modes. Then a high authoritycontroller (HAC) is employed to provide the necessaryperformance characteristics. An alternate two-levelstrategy whereby robust control synthesis is used inboth the stages (i.e., control of flexible motion andcontrol of libration) has also been studied by Grewaland Modi [8].This study examines the application of two

popular robust control techniques, LQG/LTR andH1, to the problem of attitude control of a flexiblespace platform comprised of subassemblies. Theresulting controllers are implemented on a flexiblenonlinear plant model, and are verified by thenumerical simulation of the entire closed-loop system.It is important to point out that even commerciallydeveloped codes have not tackled the configurationwith control procedures studied here.

II. MULTIBODY FORMULATION

The model considered (Fig. 1) consists of acentral body Bc connected to Bi bodies (i= 1, : : : ,N).Each Bi body is, in turn, connected to Bi,j bodies(j = 1, : : : ,Ni). The number of Bi and Bi,j bodiesis kept completely arbitrary in order to facilitatethe simulation of a large class of flexible, orbitingstructures. This model has 3 levels of bodies, sincemost spacecraft can be adequately represented by a 2or 3 level geometry. For example, the Space StationFreedom, which has since been superseded, can berepresented by 2 levels of bodies, with the powerboom or central truss represented by the central body(Bc). All remaining bodies, which consist of the solararrays, solar and station radiators, stinger and theremote manipulator, can be represented by Bi bodies.

For the present study, six types of reference framesare employed (Fig. 1). These are the inertial referenceframe F0 taken to be fixed at the Earth’s center; theorbital frame Fs; the system frame Fp; and threeframes for the central (Bc), Bi, and Bi,j bodies. Thelatter three frames are denoted by, Fc, Fi and, Fi,j ,respectively.The Lagrangian procedure for developing

the governing equations of motion requires thedetermination of the system kinetic and potentialenergies, as well as the Rayleigh dissipationfunction. The first step involves the determination ofexpressions for the position vectors of mass elementsin bodies Bc, Bi, and Bi,j with respect to F0. This isachieved by establishing a path to the central body,and then extending it to the secondary (Bi) and tertiary(Bi,j) bodies. With respect to the system referenceframe Fp, the position vectors can be expressed as

~Rc = ~Rcm¡ ~Ccm+ ~½c+~±c~Ri = ~Rcm¡ ~Ccm+~di+Cci (~½i+~±i)~Ri,j = ~Rcm¡ ~Ccm+~di+Cci~di,j +Cci Cii,j(~½i,j +~±i,j):

(1)

The ½ and ± terms in the above equations referto position and deformation vectors of masselements in the subscripted bodies. Cf is thesystem center-of-mass, while Cci and C

ii,j refer to

transformation matrices representing the rotation of Bibodies with respect to Bc, and Bi,j bodies with respectto Bi, respectively. The transformation matrices are,in general, functions of the flexibility generalizedcoordinates, as well as the specified angles associatedwith slewing bodies.Next, expressions for system kinetic T and

gravitational potential Ug energies are obtained

T =12

(Zmc

( _~Rc ¢ _~Rc)dmc

+NXi=1

"Zmi

( _~Ri ¢ _~Ri)dmi+NiXj=1

Zmi,j

( _~Ri,j ¢ _~Ri,j)dmi,j#)

;

(2)

Ug =¡¹eM

Rcm¡ ¹e2R3cm

tr[I] +3¹e2R3cm

~lT[I]~l (3)

where ~l is the direction cosine vector of ~Rcm withrespect to the coordinate frame Fp. Also ¹e is thegravitational constant for Earth, and I is the systeminertia matrix. The strain energies for beam and plateelements are given in [10]. From these, expressionsfor the strain energy of an interconnected system ofbeam and plate-like bodies can be developed [9].The presence of structural damping is accountedfor in the governing equations by Rayleigh’sdissipation function R. The flexible character ofthe beam and plate elements are discretized using

492 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 36, NO. 2 APRIL 2000

TABLE IPhysical Characteristics of Major Components of FEL

Length Mass !¤1 Ixx Iyy IzzBody (m) (kg) (rad/s) (kg-m2) (kg-m2) (kg-m2)

Bc 60 15,840 12.16 1:5£ 105 4:37£ 106 4:28£ 106B1 26.7 270 3.14 10 64,160 64,160B2 7.5 800† 6.4 45 60,000 60,000B3 11.5 450 0.628 50 19,837 19,837B4,B5 33 444 0.628 1,332 161,172 162,504

Note: *!1 is the fundamental bending natural frequency, † excluding payload mass of 800 kg.

admissible functions (method of assumed modes)or quasi-comparison functions, which can lead tosignificant improvement in certain cases [11]. Thegoverning equations of motion can now be obtainedthrough application of the Lagrangian principle,

d

dt

µ@T

@ _qi

¶¡ @T

@qi+@U

@qi+@R@ _qi

=Qi, i= 1, : : : ,ng

(4)where qi are the system generalized coordinates,which may be due to orbital, librational, or flexiblemotion, and Qi the nonconservative generalizedforces due to environmental effects and actuators.The nonlinear governing equations are evaluated andprogrammed into the multibody code. They can bewritten as

M(~q, t)~̈q= ~f( _~q,~q,~u, t) (5)

where M(~q, t) is the system mass matrix, and ~uis the vector of control inputs. All the stiffness,damping, gyroscopic, and generalized force terms,which are in general nonlinear and time varying,are incorporated in the vector ~f( _~q,~q,~u, t). A set oflinearized equations may be obtained by expressing(5) in the first-order form, and then evaluating theJacobian of the nonlinear function with respect to thestate and control vectors, respectively. Given the sizeof the equations of motion, the analytical derivationof the Jacobian matrices would be a prohibitive task.However, their numerical computation is possible byapproximating the partial derivatives of the nonlinearvector function by finite differences equations. Thedetails are given in [8].

III. ILLUSTRATIVE EXAMPLE

The model considered for numerical analysisis the first element launch (FEL) configurationonce proposed for the Space Station. The principalcomponent of the FEL is the central truss-likestructure called the power boom. Attached to thepower boom, at approximately its center, are thevarious modules required for experimentation,habitation, logistics, etc. (Fig. 2). Other prominentelements of the FEL are a pair of solar arrays situatedapproximately at one end of the power boom, and a

Fig. 2. Schematic diagram of FEL showing its majorcomponents.

solar radiator which is located inboard of the solararrays. A stinger is located at the other end of thepower boom. The power boom is modeled as afree-free beam, the stinger as a cantilever beam, whilethe solar arrays and radiator are treated as cantileverplates. The principal physical characteristics of theFEL are given in Table I. The FEL, which is orientedin a librationally unstable configuration, is placed ina circular orbit at an altitude of 400 km, and has anorbital period of approximately 92 min.

IV. UNCONTROLLED RESPONSE

The Space Station will be subjected to numerousdisturbances, both of the natural and operationalvariety. Operational disturbances include crew activity,docking of the space shuttle, and maneuvers ofthe remote manipulator system. In this section, theresponse of the system to manipulator maneuver isinvestigated.The physical characteristics of the single-link

manipulator considered in the simulation are alsogiven in Table I. The payload is represented by apoint mass of 800 kg at its tip. The effect of a typicalmaneuver of the manipulator on the dynamics ofthe Station is investigated by simulating a combinedtranslation and rotation maneuver shown in Fig. 2.The base of the manipulator is initially situated 15 m

GREWAL & MODI: MULTIBODY DYNAMICS AND ROBUST CONTROL OF FLEXIBLE SPACECRAFT 493

Fig. 3. Librational response of FEL to manipulator maneuver.

from the center of the power boom (towards the solarpanels), with the long axis of the manipulator locatedalong the orbit normal. The manipulator slews in theplane defined by the orbit normal and local horizontalthrough 180±, while its base translates by 30 m alongthe power boom towards the stinger end. Both thetranslation and rotation profiles are taken to be of thesine-on ramp type, and the maneuver is completedin 60 s. Three admissible functions are employedfor discretizing the flexible nature of the powerboom, manipulator and stinger in each transversedirection. As before, the PV arrays and radiator aremodeled as cantilever plates represented by a singleshape function in the longitudinal and transversedirections. The librational responses to the maneuverare shown in Fig. 3 over 0.02 orbit which correspondsto approximately 110 s. The major librational degreeof freedom (DOF) excited by the maneuver is theyaw angle, which is perturbed by almost 7±. Thepitch degree of freedom is the least affected, with amaximum deviation of less than 0:2± over the periodshown. The roll response starts to grow considerablyin the negative direction, especially at about 20 s afterthe start of the maneuver. The growth is essentiallylinear, reaching a value of 0:7± by 0.02 orbit.The vibrational responses of the power boom

and stinger are rather small and hence not shown.The manipulator vibrational response is presented inFig. 4, along with the responses of the PV radiatorand arrays. The tip deflection of the manipulatoris significantly higher (approximately 50 times)in the z2 direction than in the y2 direction. Thisis due to the fact that the z2 direction lies in theplane of the maneuver. Moreover, at the start ofthe maneuver, the deflection is in the ¡z2 directionbecause the torque applied at the base of the arm

Fig. 4. Vibration response of manipulator, PV radiator, PV arrayto manipulator maneuver.

results in the tip lagging behind. The maximum tipdeflection of the manipulator is approximately 2 cm.The post-maneuver response for all the members isrelatively small. The largest deflections in the systemoccur at the tips of the solar panels (maximum of3 cm). Recognizing that the array vibration lies in theplane of the maneuver, this result is not surprising.

V. CONTROL SYSTEM SYNTHESIS

For operational considerations, at times spacecrafthave to be placed in unstable orientations. For


example, the power boom of the FEL will be alignedwith the orbit normal resulting in an unstablelibrational motion. In many cases, even a stableorientation results in unacceptable librational responsedue to the various environmentally induced andoperational disturbances. Attitude control systems arean integral part of almost all present-day spacecraft.Results from the previous section demonstrated

that the FEL is unstable in libration and is proneto excitation from operational disturbances. Interms of dynamic performance, attitude stabilityand the response characteristics are of primaryimportance. To that end, an attitude control systemis designed employing two different techniques:the LQG/LTR and H1 methods. Vibrations are notcontrolled directly, however, the effect of flexibility isincorporated as modeling error. Thus, the controllerdesign can be made robust against the unmodeledflexible dynamics, ensuring that instability in theclosed-loop system will not arise. In practice the FELis provided with control moment gyros (CMGs) forattitude stabilization and control. A set of three CMGsis assumed to be present, one for each direction.Colocated at the CMG position is a set of threeattitude sensors which measure the pitch, roll, andyaw angles. Two linearized models of the FEL areobtained from the flexible multibody program: 1) thedesign model, which considers the system rigid, and2) the truth model, which accounts for the flexibilitywith discretization using a single shape function foreach flexible member. The design model consistsof a set of three second-order ordinary differentialequations (i.e., 6 states), while the truth model isdescribed by a set of ten second-order ordinarydifferential equations (20 states). Initially, the onlynon-zero state of the system is the yaw angle, whichhas an equilibrium value of ¸e = 0:74

±. Both themodels have the following mathematical form:

_x=Ax+Bu

y=Cx:(6)

Here x, y, and u are the state, output, and controlinput vectors of the system, respectively, while A, B,and C are time-invariant matrices. The objectives ofthe controller design are as follows.

1) stabilization of the attitude motion;2) sufficiently high bandwidth for the system,

resulting in a relatively fast response;3) adequate damping of the attitude motion to

avoid excessive overshoot;4) acceptable control requirement for expected

disturbances to avoid saturation of the CMGs.

As is the case with most engineering designproblems, some of the goals are not entirelycompatible, and hence one has to introduce somerational compromises. The maximum levels for the

CMG torque and its total momentum capacity aretaken as 270 N-m, and 27 kN-m-s, respectively [4].The station is required to maintain an attitude within§1± of the nominal orientation.A common design procedure for multivariable

control systems involves the representation ofthe performance specification as a low frequencybarrier, and identifying a model uncertainty as ahigh frequency bound. Then, the robustness of thecontroller design against both model uncertaintyand the specified performance can be ensured byemploying various tests. In the present case, thestability robustness test for model uncertainty isemployed. The system response to disturbances issimulated, and the resulting control input requirementsare determined to ensure that CMG saturation isavoided. Furthermore, a premultiplicative modeluncertainty or error is employed. The error, denotedby Em(s) is defined as

Gf(s) = [I+Em(s)]Gr(s) (7)

where Gf(s) and Gr(s) are the transfer functionmatrices of the truth and design models, respectively.The criterion for a multi-input, multi-output (MIMO)system to be robust, from the stability point of view,to model uncertainty is obtained from a generalizationof the Nyquist criterion for single-input, single-output(SISO) systems [12]. The robustness condition formultiplicative model error or uncertainty is

¾[I+(Gr(j!)Kc(j!))¡1]> ¾[Em(j!)] (8)

where ¾[¢] and ¾[¢] are the maximum and minimumsingular values, respectively.

A. LQG/LTR Control

The LQG/LTR procedure is described in detail inthe literature (e.g. [13]). For this study the optimallinear regulator gain matrix Klqr is designed first.This is an iterative procedure, accomplished bymanipulating the state weighting matrix (Q) and thecontrol weighting matrix (R) until the performanceand stability robustness criteria are satisfied. TheLQR design is recovered using an optimal stateestimator with specific characteristics, namely thatthe input noise influence matrix ¡ be equal to thecontrol influence matrix B. The input and outputnoise covariance matrices of the state estimator areused as tuning parameters. They are manipulated untilthe loop transfer function of the plant/compensatorpair, broken at the plant input, converges to the looptransfer function of the plant with state feedback (i.e.,the LQR problem), also broken at the input to theplant.Details of the resulting 6-state compensator

model are not shown here. The LQG/LTR designis robust against the modeling error. The test ispresented in Fig. 5, along with the results for


Fig. 5. Stability robustness test, LQG/LTR controller.

the LQR state feedback design. The closeness ofthe two results demonstrates the high degree ofrecovery achieved. The LQG/LTR compensator isimplemented in the nonlinear simulation code byincorporating the controller dynamics in the overallsystem model. A comprehensive set of controlledand uncontrolled nonlinear simulation results wereobtained. For brevity, only the system response toan initial disturbance in roll of 5± is presented. Theroll response converges to equilibrium with a smallovershoot. The pitch and notably the yaw anglesundergo some transients while the roll is beingregulated by the controller (Fig. 6). In around 0.36orbit (2000 s), all the librational DOF settle down totheir steady state values. The vibratory response ofthe PV radiator and array are shown in Fig. 7. Theseprovide a measure of the spillover introduced by the

Fig. 6. Libration response of FEL, LQG/LTR control, Á(0) = 5±.

Fig. 7. Vibration response of PV array and radiator, LQG/LTRcontrol, Á(0) = 5±.

controller. The largest disturbance occurs in the arraywhich has a maximum tip deflection value of 0.7 cm,while the radiator has a maximum deflection value ofless than 0.3 cm. The time histories for the initial 2 sof the CMG torques (see Fig. 8) reveal large values inall three directions, particularly in the pitch and rolldirections. After 2 s the requirements are very modest.

B. H1 Control

In the LQG/LTR control problem, the synthesiswas accomplished by employing an optimal observerdesigned in a particular way to recover the propertiesof the LQR state feedback design. Moreover, theLQR design procedure involves the minimization ofa quadratic performance index. This, indirectly, leadsto the minimization of the H2 norm of the systemtransfer function matrix M(s) [13]. The indirect natureof the optimization results in only a moderate control


Fig. 8. Control torque requirement for FEL, LQG/LTR control,Á(0) = 5±.

over the loop-shaping through the use of the quadraticweights in the performance index. H1 synthesisrefers to an approach to linear control system designwhich entails the minimization of the H1 norm ofthe system transfer function(s). The use of the H1norm allows for precise shaping of the system transferfunction matrices in order to simultaneously satisfyperformance, stability, disturbance rejection, etc.criteria through the use of dynamic weights. Thisrepresents a more direct procedure to the designof a controller than the LQG/LTR approach, whichiteratively satisfies the design criteria. Details of theH1 algorithm can be found in the control literature(e.g. [14]).The H1 control design procedure is applied

to the same plant model as the one employed forthe LQG/LTR controller design. As before, themultiplicative error was computed assuming thetrue system to consist of a 20 state flexible modeldescribed earlier. The mixed-sensitivity problemformulation was adopted, whereby the rigid designmodel was augmented with two weighting functions,W1(j!) and W3(j!), which penalize the sensitivityS(j!) and complementary sensitivity I¡S(j!),transfer functions of the unaugmented plant-controllersystem. The procedure used for the H1 designis referred to as the Glover-Doyle or two-Riccatialgorithm. It was implemented with the RobustControl Toolbox [15] used in conjunction with theMATLAB software. The weighting function W1(j!)penalizes the error between the plant output andthe desired output (defining the performance orbandwidth specification), while W3(j!) penalizes theplant output (defining the bound required for stabilityrobustness). For the present design, W3(j!) is taken

Fig. 9. Stability robustness test, H1 controller.

to be an upper limit of the multiplicative model errorto ensure robustness against unmodeled dynamics,while W1(j!) is used as a design parameter. Thecriteria used to establish whether the final designis acceptable include the speed of the response andthe control effort (CMG torque) required. For thepresent case, W3(j!) is chosen to be equal to 50 s

2.The performance weighting function W1 is taken asa ratio of two quadratic functions, the parametersof which are varied. The resulting augmented plantmodel has 12 states, as does the compensator returnedby the H1 algorithm. The stability robustness test ofthe design is presented in Fig. 9. The system is clearlymore robust to the unmodeled plant dynamics thenin the case of the LQG/LTR compensator, mainlydue to the greater slope of ¾[I+(Gr(s)Kc(s))

¡1](i.e., 40 dB per decade) for ! > 0:1 rad/s. Thisis due to the second-order dependence of W3on s.As with the LQG/LTR design, the H1 controller

was implemented on the nonlinear plant model. Theobjective was to assess its effectiveness over a widerange of librational and vibrational disturbances.However, only the case investigated for the LQG/LTRcompensator is examined here. Fig. 10 shows thecontrolled librational response of the system to a 5±

disturbance in roll. The improved performance withthe H1 controller is apparent. While the overshootin the pitch response is similar to the previous case(maximum of ¼ 0:14±), the overshoot in the yawresponse is considerably better (maximum of ¼ 0:2±for H1, compared with ¼ 1:5± for LQG/LTR).Furthermore, the roll disturbance attenuates muchfaster (less than 400 s), compared with more than2000 s for the LQG/LTR controller. The PV radiatorand array vibrational responses for the H1 controlcase are shown in Fig. 11. The peak vibrationalresponse of the array is considerably lower inmagnitude for this case due to the lower control


Fig. 10. Libration response of FEL, H1 control, Á(0) = 5±.

Fig. 11. Vibration response of PV array and radiator, H1control, Á(0) = 5±.

torques in the pitch and yaw directions (see Fig. 12).In the case of the PV radiator, the peak magnitude ofthe vibration is approximately equal to the LQG/LTRcase. The nature of the responses for both the arrayand radiator, however, is considerably different, withthe vibrations in the H1 case exhibiting a slowlyvarying envelope, which reflects the time history ofthe control torques. In the LQG/LTR case, the controltorques were essentially impulsive in nature, and thiswas manifested in the character of the vibrationalresponses.The control effort is also lower, especially for the

pitch and yaw CMGs. For example, in the LQG/LTRcase, peak torques of 270 N-m and 200 N-m wererequired for the pitch and yaw CMGs, respectively.For the H1 controller these values are 5 N-m and25 N-m, respectively (see Fig. 12). Furthermore, thecontrol torque demand for the roll CMG is relativelysmooth. It does not exhibit any fast “switching” as

Fig. 12. Control torque requirement for FEL, H1 control,Á(0) = 5±.

in the case of the LQG/LTR control (¡270 N-m to+270 N-m, see Fig. 8).

VI. CONCLUDING REMARKS

A rather general formulation applicable to a largeclass of systems, characterized by interconnectedbeam and plate-type members, is presented. Thecorresponding numerical code is also developed. Theyrepresent versatile tools for studying the dynamicsand controls of space-based multibody systems ofcontemporary interest. Their application is illustratedthrough the study of a specific space platformconfiguration.The results suggest that the H1 controller is

superior to the LQG/LTR design in many respects. Itis more robust to the unmodeled (flexible) dynamics.This is borne out not only in the robustness test, butalso in the simulation results. In terms of performance,the H1 design leads to smaller settling times forlibrational disturbances. The control torques requiredin the directions normal to the disturbance aresignificantly lower. Also, because of the ability totailor the closed-loop characteristics of the design,the H1 procedure requires significantly less iterationeffort compared with the LQG/LTR case, to achievethe desired loop shape.It must be emphasized that the H1 controller is

relatively more complex, resulting in a 12th-ordercontroller, while the LQG/LTR procedure produced a6th-order one. Therefore, in terms of implementation,the H1 controller would demand more computingeffort. However, reducing the order of the resultingcompensator, which was not pursued here, may resultin a more attractive design.


REFERENCES

[1] Singh, R. P., et al. (1985)Dynamics of flexible bodies in tree topology–Acomputer-oriented approach.Journal of Guidance, Control, and Dynamics, 8, 5(Sept.—Oct. 1985), 584—590.

[2] Balas, M. J. (1982)Trends in large space strctures control theoory: Fondesthopes, wildest dreams.IEEE Transactions on Automatic Control, AC-27, 3 (June1982), 522—535.

[3] Van Woerkom, P. Th. L. M. (1993)Synthesis and survey of control laws for large flexiblespacecraft.Control-Theory and Advanced Technology, 9, 3 (Sept.1993), 639—669.

[4] Chu, P. Y., et al. (1988)Space Station attitude control: Modeling and design.Presented at the AIAA Guidance, Navigation andControl Conference, Aug. 1988, Minneapolis, MN, PaperAIAA-88-4133.

[5] Sundararajan, et al. (1987)Robust controller synthesis for a large flexible spaceantenna.Journal of Guidance, Control, and Dynamics, 10, 2(Mar.—Apr. 1987), 201—208.

[6] Byun, K.-W., et al. (1991)Robust H1 control design for the Space Station withstructured parameter uncertainty.Journal of Guidance, Control, and Dynamics, 14, 6(Nov.—Dec. 1991), 1115—1122.

[7] Safonov, et al. (1991)H1 robust control synthesis for a large space structure.Journal of Guidance, Control, and Dynamics, 14, 3(May—June 1991), 513—520.

[8] Grewal, A., and Modi, V. J. (1996)Robust attitude and vibration control of the Space Station.Acta Astronautica, 38, 3 (Feb. 1996), 139—160.

[9] Grewal, A. (1994)A study of flexible space structures: Dynamics andcontrol.Ph.D. dissertation, The University of British Columbia,Vancouver, B.C., Sept. 1994.

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[14] McFarlane, D. C., and Glover, K. (1990)Robust controller designs using normalized coprimefactor plant descriptions.Lecture Notes in Control and Information Sciences, Vol.138.New York: Springer-Verlag, 1990.

[15] Chiang, R. Y., and Safonov, M. G. (1992)Robust-Control Toolbox.Natick, MA: The MathWorks, 1992.


Anant Grewal received the B.A.Sc. (Hon.) degree in mechanical engineeringfrom the University of British Columbia, Vancouver, in 1985, the M.S. degreein aerospace engineering from the University of Minnesota, Minneapolis, in 1987,and the Ph.D. degree in mechanical engineering from the University of BritishColumbia, Vancouver, in 1994.He was a Defence Scientist at Defence Research Establishment Valcartier,

Quebec, Canada from 1988 to 1990. Since 1994, he has been a Research Officerwith the Institute for Aerospace Research, National Research Council in Ottawa,Canada. His current research interests include the application of control theory toproblems of noise and vibration reduction.Dr. Grewal is a member of AIAA and ASME.

Vinod J. Modi received a bachelor’s degree in mechanical and electricalengineering from Bombay University, Bombay, India, in 1953. He obtainedthe M.S. at the University of Washington, Seattle, in 1956, and the Ph.D. fromPurdue University, Lafayette, IN, 1959, both in aerospace engineering.He is currently a professor at the University of British Columbia, Canada.His contributions are recognized by many awards including: the CANCAM

Award (1985); B.C. Science Council’s Gold Medal (1986); University of BritishColumbia’s 75th Anniversary Commemorative Medal (1991); Dirk BrouverAward, American Astronautical Society (AAS, 1991); the McCurdy Award,Canadian Aeronautics and Space Institute (CASI, 1993); Distinguished AlumnusAward (1995); the Best Paper Award, IEEE (1995); the Best Paper Award,International Society of Offshore and Polar Engineers (ISOPE, 1996); Mechanicsand Control of Flight Award, American Institute of Aeronautics and Astronautics(AIAA, 1996); the John V. Breakwell Memorial Lecture and Plaque, InternationalAstronautical Federation (IAF, 1996); and the Pendary Aerospace LiteratureAward (AIAA, 1999).He has been a member of the Spaceflight Mechanics Committee (AAS,

1981—1995), Astrodynamics Technical Committee (AIAA, 1980—1984,1989—1993, 1995—1998), and the Astrodynamics Committee of InternationalAstronautical Federation (1984—1994, chairman 1991—1994). He is a fellow ofthe AAS, AIAA, ASME, CASI, British Interplanetary Society, Royal Society ofCanada, as well as a member of the International Academy of Astronautics.


Documents

Multibody dynamics and robust control of flexible spacecraft