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Dynamics and control of flexible spacecraft : a case study Muneer, Khan Mohd 1992

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DYNAMICS AND CONTROL OF FLEXIBLE SPACECRAFT :A CASE STUDYKHAN MOHD MUNEERB. Tech, Indian Institute of Technology, Kanpur, India, 1989A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate StudiesDepartment of Mechanical EngineeringWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIADecember 1992© Khan Mohd Muneer, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of ki2_(.1\01, \ The University of British ColumbiaVancouver, CanadaDate 1U )I\ 'PJZ cDE-6 (2/88)ABSTRACTThe First Milestone Configuration (FMC) of the proposed Space Station Freedomis considered for investigating the dynamics and control aspects of a space based or-biting flexible structure. The system dynamics is governed by an extremely lengthy(even in the matrix form), highly nonlinear, nonautonomous and coupled set of equa-tions amenable only to a numerical solution. Two distinctly different discretizationprocedures, using the system modes and the component modes, are applied to thedesign as well as the Lagrange orientations of the FMC. The system is found to beinherently unstable suggesting the need for control. To this end two control tech-niques are employed: the classical Linear Quadratic Regulator (LQR) is applied tothe linearized set of governing equations of motion; and the Feedback LinearisationTechnique (FLT) for the complete nonlinear set. The amount of information obtainedthrough a planned parametric study of the system dynamics and control is indeedenormous.The results clearly show that under critical combinations of parameters thesystem can become unstable . However, both the LQR and FLT procedures, togetherwith the on board Control Momentum Gyros (CMG's), can restore the system to theequilibrium state. To gain better insight into the physics of the system behaviour,the uncontrolled as well as the controlled responses of the FMC are animated on anIRIS graphics workstation. The thesis ends with a few recommendations for futurestudy.iiTABLE OF CONTENTSABSTRACT ^  iiLIST OF SYMBOLS ^  viiLIST OF FIGURES  xiiLIST OF TABLES ^ xviiiACKNOWLEDGEMENT  xix1. INTRODUCTION ^  1^1.1^Preliminary Remarks  11.2^A Brief Review of the Relevant Literature ^  31.3^Scope of the Present Investigation  72. FORMULATION OF THE PROBLEM ^ 152.1 Preliminary Remarks ^ 152.2 Kinematics of the Problem 152.2.1^First Milestone Configuration (FMC) ^ 152.2.2^Coordinate system ^ 162.2.3^Position of spacecraft in space ^ 202.2.4^Shift in the center of mass 222.3 Elastic Deformations ^ 232.3.1^Background 232.3.2^Transverse vibrations ^ 24iii2.4^Kinetics ^  252.4.1 Rotation matrices ^  262.4.2 Kinetic energy  272.4.3 Potential energy ^  292.5^Lagrangian Formulation  303. DYNAMICAL STUDIES ^  333.1^Preliminary Remarks  333.2^First Milestone Configuration ^  333.3^System and Component Modes  343.3.1 Component mode discretization ^  363.3.2 System mode discretization  39^3.3.3 Comparision for the component and system modes   393.4^Flexibility Effects ^  403.5^System Mode Simulation Study ^  573.5.1 Lagrange orientation  613.5.2 Summary ^  703.6^Response to Operational Disturbances ^  704. CONTROL STUDY ^  724.1^Linear Control  724.1.1 Linearization of the equations of motion   73iv^4.1.2^State space equivalent ^^4.1.3^The Linear Quadratic Regulator (LQR) ^4.1.4^Effect of flexibility ^7475764.2 LQR Simulation Results and Discussion ^ 774.3 Nonlinear Control ^ 844.4 Feedback Linearization Technique (FLT) ^ 874.5 Implementation Of the Control for a Flexible System ^ 884.5.1^Quasi-open loop control ^ 904.5.2^Quasi-closed loop control 904.6 FLT Simulation Results and Discussion ^ 904.7 Summary ^ 985. ANIMATION  1025.1^Methodology of Animation ^  1025.1.1 Object definition  1025.1.2 Scene composition ^  1045.1.3 Transformation management and synchronisation . . . ^ 1066. CONCLUDING REMARKS ^ 1106.1^Concluding Comments  1106.2^Recommendations for Future Work ^  112BIBLIOGRAPHY^  113viLIST OF SYMBOLSdi^ position vectors from 0, to Oidme , dmi^elemental mass in body Be , and Bi , respectivelyfc, fi fundamental frequency of bodies Be , Bih^ angular momentum per unit mass of spacecrafti orbit inclination with respect to the ecliptic planedirection cosines of Rcm with respect to Xp ,Yp ,Zp axes1b, 1p^beam and plate lengths, respectivelylc , li length of bodies Be , and Bi, respectivelymb^ beam mass per unit lengthme, rni mass of the bodies Be , and Bi, respectivelyt^ timet, plate thicknessfLk^ unit vectors {i k , jk , kk}Tb ,v , w transverse vibration of a beam in its Y and Z directions,respectivelyCi^ transformation matrix defining orientation of Fi relative toF,, respectivelyCCM,01,^position vector from C i to the instantaneous centre of massof spacecraftCcm^ position vector from 0, to the centre of mass of undeformedviiC i ,CfDHz ,tH 31. ' tflaysJaysspacecraftcentres of mass of the undeformed and deformed configura-tions of spacecraft, respectivelyflexural rigidity of the plategeneralized coordinate associated with the s ih and tth modesin its X and Y directions, respectively, for a plate undergoingtransverse vibrationsdimensionless generalized coordinate; 1-1 3k ' t = Hy:' t //kangular momentum of spacecraft with respect to X,,Y,,Z,axesinertia matrix of spacecraft with respect to the XC ,YC ,ZC axes(Ixx)k, (Iyy)k, (Izz)k^principal inertia of Body Bk about Xk, Yk, and 2k axes,respectivelyKp , Kv^displacement and velocity gain matricesM total mass of spacecraftN^total number of Bi0,, Oi origins of the coordinate axes for bodies 13,, and Bi respec-tivelyPT, Q rk^generalized coordinates associated with the r th transverse vi-bration mode of a beam in its Y and Z directions, respectivelydimensionless generalized coordinates; Pk^Pk Ilk , Q rk =Q rkl lkQ 1b, Qq5, Qi^control effort for pitch, roll and yaw degrees of freedom, re-spectivelyDr Q T`.°e kviiiRAD^tip deflection of the radiatorRcm position vector from the centre of force to the instantaneouscentre of mass of spacecraftRc ,R,^position vectors of the mass elements dm c ,anddmi, respec-tively as measured from the centre of forceRcm , R , Rz^magnitudes of Rim , Rc , and Ri, respectivelySP^ tip deflection of the solar panelsSTp tip deflection of the stinger in the Y transverse directionSTQ^tip deflection of the stinger in the Z transverse direction di-rectionT^ total kinetic energy of spacecraftTRp tip deflection of the truss in the Y transverse directionT Rc2^tip deflection of the truss in the Z transverse directionTn., system kinetic energy due to various coupling effectsU potential energy of spacecraft; Ue + UgU unit matrixUe^strain energy of spacecraftUg^gravitational potential energy of the spacecraftXk,Yk,Zk^body coordinate axes associated with Bc , Bi, respectivelyX0,Y0,Z0^inertial coordinate system located at the earth's centerXp ,Yp ,Zp^coordinate axes with origin at Cf and parallel to x e ,y c ,z,,respectivelyX,,Y,,Z,^orbital frame with X, in the direction of the orbit-normal, Y8ixePc, Pi,wwalong the local vertical, and Z, towards the local horizontalvectors representing transverse vibration of dm c , and dmi,respectivelyeccentricityrotation about the local horizontal axis, Z1, of the interme-diate frame Xi ,Y1,Z1 {cos^cos 01, cos 01}Trotation about the local vertical axis, Y2, of the intermediateframe X2,Y2, Z2gravitational constanttrue anomalylongitude of the ascending nodevectors denoting positions of dm c , dmi, respectively, in theundeformed configuration of the spacecraftargument of the perigee point;librational velocity vector with respect to Xp ,Y1,,Zp axesrotation about the orbit normal, X,ABBREVIATIONS c.111 .DODOFFEMFLTFMCcenter of massDesign Orientation of the FMCDegrees Of FreedomFinite Element MethodFeedback Linearization TechniqueFirst Milestone ConfigurationLO^Lagrange Orientation of the FMCPVA PhotoVoltaic ArrayQCLC^Quasi-Closed-Loop ControlQOLC Quasi-Open-Loop ControlSSM^Substructure Synthesis MethodDot (^) and prime ( ) represent differentiations with respect to time tand true anomaly 8, respectively. Subscripts o and e indicate initial and equilibriumconditions, respectively. Unless stated otherwise, overbar ( ) represents a vector;boldfaced symbol, a matrix; and underbar ( _ ) refers to a dimensionless quantity.xiLIST OF FIGURES1-1^The European Space Agency's L—SAT (Olympus) launched in 1989.^. . 21-2^A schematic diagram of the the proposed Space Station Freedom asof 1988. It is characterised by a truss extending to 155m with 66mlong solar panels (tip to tip), essentially rigid modules at the center,and a mobile flexible manipulator.^ 41-3^The four milestone configurations of the evolving space stationFreedom. ^  51-4^Four different orientations of the Space Station's First MilestoneConfiguration (FMC). Though unstable, NASA's choice of thedesign orientation is primarily governed by the power productionthrough the photovoltaic conversion ^  91-5^Schematic of the design orientation in orbit. ^  121-6^Schematic of the Lagrange orientation in orbit.Note, the alignmentof the axes of the maximum and the minimum moments of inertia. . . 131-7^A schematic diagram showing the aspects studied in the thesis.^. 142-1^The main truss and the appendages constitute a branched geometrywith one bifurcation. ^  172-2^A schematic diagram of the First Milestone Configuration showingthe interconnected flexible members and the reference coordinatesystems.   182-3^Schematic diagram of the FMC showing the instantaneous center ofmass and direct path to a mass element with respect to the inertialxiiframe F0. ^  192-4^Modified Eulerian rotations 0, 0, and A defining an arbitrary spatialorientation of spacecraft.   213-1^Configuration of the FMC used in the numerical simulation: (a)coordinate systems; (b) design configuration.   353-2^Librational response of the rigid FMC showing the inherent unstablecharacter of the design orientation.Note it is not in the equilibriumstate. ^  363-3^Librational behavior of the rigid FMC(DO) showing the unstableresponse in the presence of an external disturbance in pitch (0 = 10°).^373-4^Libration behaviour of the rigid FMC in Lagrange orientation inabsence of the external disturbances. The pitch response is dueto deviation from the equilibrium configuration.   383-5^Librational response using two distinct programs, employing systemand component mode discretizations, in the limiting case of zeroflexibility, for the design orientation.   413-6^Librational response using two distinct programs, employing systemand component mode discretizations, in the limiting case of zeroflexibility, for the Lagrange orientation. ^  423-7^Dynamical response for a disturbance of 0 = = A = 2° in thedesign orientation of the FMC , using the system modes forflexibility discretization^  433-8^Dynamical response for a disturbance of 0 = zG = A = 2° in thedesign orientation of the FMC, using the component modes forflexibility discretization^  443-9^The effect of flexible appendages on the librational response of theFMC with one of the appendages (Solar panel by 0.001cm tip)disturbed slightly, in the Lagrange orientation. ^  453-10 The effect of flexible appendages on the librational response of theFMC with one of the appendages (Solar panel by 0.001cm tip)disturbed slightly, in the design orientation. ^  463-11 Dynamical response of the FMC in design orientation,with solar panelsreduced by 90 percent of their lengths and a tip deflection of 1 cm.Note that the high frequency modulations are absent from thelibrational response ^  483-12 Dynamical response of the FMC for a small power boom disturbancein the local horizontal direction (TRp = 0.0001652), in the designorientation.   503-13 Dynamical response of the FMC for a small power boom disturbancein the local vertical direction,(TRQ = 0.0001652), in the designorientation.   513-14 Dynamical response of the FMC for a small disturbance to the stingerin the local horizontal direction, (STp = 0.000094), in the designorientation.   523-15 Dynamical response of the FMC for a small disturbance to the stingerin the orbit normal direction, (STQ = 0.000094), in the designorientation.   533-16 FMC dynamical response to a tip deflection of 1 cm in the solarpanels (SP=1 cm), in the design orientation^  553-17 Dynamical response of the FMC for a deflection (RAD=0.00049), inxivthe PV radiator, for design orientation. ^  563-18 Frequency spectrum for the FMC. Note the frequencies are relativelysmall and quite closely spaced .The first forty modes are within10 Hz !. Such packed character with overlap makes determinationof the system dynamics and control extremely difficult. ^ 593-19 Response of the FMC in design orientation with zero initial condition,using system modes for flexibility discretization. ^  623-20 System mode simulation of the dynamical response for a tipdisturbance of lcm in the PV Array in the design orientationof the FMC    633-21 Librational responses of the FMC in Lagrange orientation forincreasing roll disturbances of = 4° and 12°. ^  643-22 Dynamical response of the FMC in Lagrange Orientation for apitch disturbance of 10°.   663-23 Dynamical response of the Lagrange Orientation for a disturbanceof 10° in pitch, and solar panels deflected by lcm at the tip. ^ 673-24 Dynamical response of the FMC in Lagrange Orientation for 0.2 cmdeflection of the radiator. ^  683-25 Dynamical response of the FMC in Lagrange Orientation for a tipdeflection of lcm for the truss. ^  69^4-1^Schematic of LQR control strategy  784-2^Librational response of the FMC in Lagrange orientation to a pitchdisturbance of 10°   804-3^Librational response of the FMC in Lagrange Orientation forxvdisturbances of 10° in yaw and roll independently. ^ 81^4-4^The controlled and uncontrolled libration responses , as well as thecontrol efforts for the application of LQR to the Lagrange Orientationof the FMC = = q = 10°) ^  844-5^The controlled and uncontrolled libration responses , as well as theassociated control efforts for the application of LQR to the designorientation of the FMC ( = 10° ).   854-6^The controlled response, and control effort for the FMC in LagrangeOrientation for application of FLT with initial disturbance of 10°in pitch, yaw and roll.   924-7^The controlled response, and control effort for the FMC in LagrangeOrientation for application of FLT with initial disturbance of 10°in pitch, yaw and roll, with flexibility included and differentfrequencies. ^  934-8^The controlled response of the FMC in design orientation for theapplication of FLT, with a disturbance of 10° in pitch yaw androll with flexibility included. ^  954-9^Generalized vibrational coordinates after the application of FLT tothe design configuration of the FMC, demonstrating the stabilityin flexible response.   964-10 Effect of damping on the controlled response and control effort for theapplication of FLT on the design orientation of the FMC. ^ 974-11 FLT response of the FMC (rigid) in design configuration with initialconditions of 10° in pitch , and a preset frequency Of 0.01 rad/sec.^. . 994-12 Effect of frequency on the controlled response and effort for thexviapplication of FLT on the design orientation of the FMC withinitial librational disturbance of 10° in pitch yawand roll simultaneously ^  100^5-1^Flowchart showing the animation methodology employed ^ 1075-2^Scene hierarchy for the FMC animation^  108xviiLIST OF TABLES^1-1^Properties of the FMC. Here: 13,, central body, main mast; B1, stinger;B2, panel radiator; B3, B4, solar panels. Moments of Inertia are withreference to the local body frames. ^^4-1^Typical numerical parameters for the control study (LQR). ^xviiiACKNOWLEDGEMENTSI would like to thank Prof. V.J. Modi for giving me the opportunity to workunder him.In ways too numerous to enumerate, have I benefited from his insight,wisdom and erudition. Were it not for his able guidance, throughout it's preparation,this thesis may never have seen the light of the day. The thesis has benefited from thehelp received from my seniors. In particular I would like to thank Dr. Afzal Sulemanand Dr. Alfred Ng for explaining a lot of concepts to me.I also take this opportunityto thank Mr Allan Steeves for helping out with the problems connected with the soft-ware. The project was supported by the National Sciences and Engineering ResearchCouncil of Canada, Grant No. A-2181; and the IRIS Center of Excellence, Grant No.C-8/5-55380.xix1. INTRODUCTION1.1 Preliminary Remarks"Space", as Gene Rodenberry, the creator of famous teleserial Startrek onceaptly said, "is the final frontier for mankind's advancement and continues to be soin our times". Beginning with the launch of the first satellite Sputnik in 1957, theexploration of space has continued unabated, with ever increasing tempo, as thedemand on spacecraft capabilities in terms of telecommunications, weather forcastingand remote sensing have increased .With the passage of time, spacecraft , have tended to become larger in size andcomplex in design. On the other hand, man made space structures, no matter howcomplex they may be , are still subservient to an all-embracing elegance and sim-plicity of Newtonian and Keplerian mechanics. The rigid and simple structures ofyesteryears, have evolved from their rigid pupal stage into breathtakingly beautifulbutterflies of today, that are necessarily large and flexible. The following two examplesof contemporary spacecraft illustrate this point:(i) The European Space Agency's L—SAT (Large SATellite system, Olympus,Figure 1-1 ), launched in 1989, represents a new generation of communicationssatellite. It has two solar panels, each 25 m in length, connected to a centralbody.(ii) The U.S. led Space Station Freedom, scheduled to be completed around 1998,will have a main truss of 155 m in length*. Equipment attached to the trussincludes habitat, command and laboratory modules, power generation equip-ment and system control assembly, stinger and resistojet, photovoltaic (PV)* The configuration has been revised recently.1Figure 1-1^The European Space Agency's LSAT (Olympus) launched in 1989.2arrays, PV array and station radiators, etc.(Figure 1-2 ). Of course such agigantic structure cannot be carried in it's entirety to the operational altitudeof around 500 km, but will have to be constructed, through around seventeenflights of the Space Shuttle. Thus here we have a space platform , that willbe evolving with it's inertia, flexibility , damping and other characteristicschanging with time. NASA has identified four milestone configurations asshown in Figure 1-3 .1.2 A Brief Review of the Relevant LiteratureOver the past thirty years, the amount of literature accumulated on the subjectof spacecraft dynamics is literally enormous. It can be classified into four broad cat-egories: formulation; dynamics and control; environmental effects; and experimentalvalidation.Hughes [1] derived the equations of motion for a chain of flexible bodies withterminal members rigid. The Newton-Euler approach was used, and the resultingequations were linear in the angular rates as well as elastic deformations. The equa-tions were tailor made for control system design.Using the Lagrangian procedure, Modi and Ibrahim [2] presented the generalequations of motion for studying librational and vibrational dynamics of a large classof spacecraft with deployable flexible members. The equations accounted for thegravitational effects, shifting center of mass, changing rigid body inertia, appendageoffset and transverse oscillation. Appendages with variable mass density, flexuralrigidity, and cross-sectional area along its length can also be accommodated.The literature on spacecraft control is primarily concerned with two aspects: at-titude control and vibration control. Balas [3] as well as Meirovitch and Oz [4] have34,Figure 1 -2^A schematic diagram of the the proposed Space Station Freedom as of 1988. It ischaracterised by a truss extending to 155m with 66m long solar panels (tip to tip),essentially rigid modules at the center, and a mobile flexible manipulator.Figure 1 -3^The four milestone configurations of the evolving space station Freedom.provided overviews of the vibration control problems. A few studies aimed at controlalgorithms for flexible spacecraft are briefly touched upon here.A Flexible spacecraft, being a distributed parameter system, needs to be discretizedto study the associated dynamics and its control. A continuos system can be trans-formed into the modal-space using modal functions for discretization. Meirovitch andOz [5] have found that the control as applied to the transformed system (modal-spacecontrol) is more efficient than that executed on the discrete system ("actual space"control). Subsequent contributions by the authors [6-8] study application of themodal-space control to different spacecraft configurations.Wie and Bryson [9] modeled flexible space structures using single-input single-output transcendental transfer functions. The models were simple enoughtfor polesand zeroes to be determined analytically. The results were then used in the pre-liminary controller design. Wie [10] applied this approach to Control Of FlexibleStructures (COFS—I) mast flight system . Chu et al. [11] employed the same ideain modeling and designing the Space Station attitude controller. Using multi-inputmulti-output transfer functions and numerical algorithms, Kida et al. [12] designedthe controller for flexible spacecraft with constrained and unconstrained modes.Goh and Caughey [13] have explored the idea of stiffness modification in vibrationsuppression of flexible structures. The control scheme guaranteed global stability byvirtue of the positive definite rate of energy decay. The implementation required notthe conventional actuators but rather transducers which converted strain into controlsignals and then into electronic damping. This was considered to be a favourablefeature.Modi and Brereton [14] studied the planar librational stability of a slender flex-ible satellite under the influence of solar heating.Modi and Ng [15] gave a general6formulation for the dynamical simulation of a large class of spacecrafts with attachedflexible appendages using the component mode approach. Modi and Suleman [16]also gave a general formulation for the dynamics and control study of a large class offlexible spacecrafts using the system mode discretization. The present study employsthe formulations as given by Modi and Ng, and Modi and Suleman to investigatethe dynamics and control of the First Milestone Configuration (FMC) of the evolvingspace station Freedom.1.3 Scope of the Present InvestigationIn the present study the emphasis is on the applicability of the above mentionedformulations [15,16] to explore complex dynamics and control of large space struc-tures. The focus is on the use of relatively simple mathematical models to gain betterphysical understanding of interactions between librational dynamics and flexibility.Of course, the final objective is to develop suitable control strategies that would leadto an acceptable system performance. The thesis can be divided into four parts: for-mulation of the problem; dynamical study; control implementation ; and animation.The problem formulation begins with the study of kinematics of a spacecraftcomprised of interconnected flexible bodies. The discretization of elastic deflectionfollows, leading to the expressions for kinetic and potential energies. Using the La-grangian procedure, equations of motion, applicable to First Milestone Configuration(FMC), are obtained.Next the attention is directed towards dynamics of the FMC. During integrationof the proposed space station, the evolving structure can present a variety of possibleorientations each offering some attractive features. Several of these orientations areshown in Figure 1-4 . Stability is likely to be one of the major considerations inselecting an orientation during assembly.The present study focuses on two important7orientations of contemporary interest:(i) the design orientation as suggested by NASA (Figure 1-5 );(ii) the more stable Lagrange configuration (Figure 1-6 ). Here the minimum momentof inertia axis is aligned with the local vertical and the maximum moment ofinertia axis lies along the orbit normal.Note, that for the design orientation the main truss is along the orbit normal.Table 1-1 summarizes geometric, stiffness and inertia properties of the FMC withrespect to the local body fixed frames. It, may be noted that for the Lagrangeorientation the axis of minimum moment inertia is along the local vertical and theaxis of maximum moment of inertia is along the orbit normal. The solar panels arein the plane of the orbit. The objective here is to study complex interactions betweenlibrational and vibrational dynamics, flexibility and initial disturbances. The amountof information obtained is literally enormous; however, for brevity, only some typicalresults useful in establishing trends are presented here.Results suggest that the designorientation is inherently unstable while the Lagrange orientation can become unstableunder critical combination of parameters and initial conditions.The next logical step is to develop control strategies, applicable to such a formidableclass of problems in general and the FMC in particular. To achieve this, both lin-ear and nonlinear control strategies are explored. The time-tested Linear QuadraticRegulator (LQG) is chosen for its simplicity and hence ease of implementation. Onthe other hand, the Feedback Linearization Technique (FLT) represents the latestdevelopment in the field and accounts for the complete nonlinear dynamics.Towards the end the issue related to the visualisation of the free and controlled re-sponses of the FMC, for better appreciation of the physics of the system, is addressed.The VERTIGO animation software is employed to render, in three dimensions, ani-8Orbit NormalLocalVerticalCOmome°p LocaHorizontalLocalVerticalDesign OrientationLocalVertical1111111111111■11111111101Orbit NormalLocalHorizontal1111111111111^111111111110Gravity Gradient OrientationLocalHorizontalLocaHorizontalOrbit NormalOrbit NormalLVLH Orientation1111 111111111LocalVertical11111 111111 11Lagarange OrientationFigure 1-4^Four different orientations of the Space Station's First Milestone Configuration (FMC).Though unstable, NASA's choice of the design orientation is primarily governed by the powerproduction through the photovoltaic conversionmated librational response of the rigid FMC.The concluding chapter summarizes more important results and presents rec-ommendations for future studies. An overview of the thesis layout is presented inFigure 1-7 .1 0Table 1-1^Properties of the FMC. Here: Bc , central body, main mast; B1, stinger;B2, panel radiator; B3, B4, solar panels. Moments of Inertia are with referenceto the local body frames.BODY LENGTH,mMASS,KgFREQ. ,HzIxx 9kg-m2IYY 9kg-m2izz ,kg-m2Bc 60 15,840 1.9 15x104 4.37x106 4.28x106Bl 26.7 270 0.5 0.0 64,160 64,160B2 11.5 450 0.1 50 19,837 19,837B3 , B4 33 444 0.1 1,332 161,172 162,504PeFigure 1-5^Schematic of the design orientation in orbit.12Figure 1 -6^Schematic of the Lagrange orientation in orbit.Note, the alignmentof the axes of the maximum and the minimum moment of inertias.13Figure 1 -7^A schematic diagram showing the aspects studied in the thesis.2. FORMULATION OF THE PROBLEM2.1 Preliminary RemarksThe Newton-Euler method and the Lagrangian approach are commonly used inthe dynamical formulation of multibody systems. The Newton-Euler method is basedon the principle of angular momentum whereas the Lagrangian approach relies on thesystem energy. In the present case, the governing equations of motion are expectedto be highly nonlinear, nonautonomous, and coupled; hence, a closed-form solution isnot expected to exist. Furthermore, in general, the center of mass is not stationaryand geometric nonlinearity effects can be significant. The Lagrangian procedure istherefore selected to assure accuracy of the governing equations.This chapter is divided into three sections: kinematics, kinetics, and the La-grangian formulation. The kinematics begins with a discussion of the system geom-etry and reference coordinate systems used to identify the deformed configuration.The spatial orientation of the system as described by a set of orbital elements andmodified Eulerian rotations is presented next. Finally, the shift in the center of massdue to deformations, and the associated rotation matrices, are discussed. The ki-netics of the problem deals with evaluation of the kinetic and potential energies andthe governing equations of motion obtained through the Lagrangian approach. TheLagarangian method for obtaining the equations of motion is outlined.2.2 Kinematics of the Problem2.2.1 First Milestone Configuration (FMC)The FMC consists of a central truss with solar panels ,stinger and radiator con-15nected as appendages. The system model selected for this study consists of flexiblebodies interconnected to form a branched geometry: central body B, is connected tobodies Bi (B i , , B4). as shown in Figure 2-1 . When applied to the First Mile-stone Configuration of the proposed space station Freedom, the central body B, maysimulate the main truss with the modules, power generation equipment and systemcontrol assembly treated as lumped masses. The stinger, PV arrays and radiators arerepresented by bodies Bi (i=1,2,3,4).2.2.2 Coordinate systemConsider the FMC model in Figure 2-2 . Attached to each member of the modelis a body coordinate system helpful in defining relative motion between the mem-bers.Therefore there are four coordinate systems attached to the appendages. Thecentres of mass of the undeformed and deformed configurations of the system arelocated at Ci and Cf, respectively (Figure 2-3 ). Let XO ,YO ,ZQ be the inertial co-ordinate system located at the earth's centre. Thus reference frame F, is attachedto body B, at an arbitrary point O. Frame Fi , with origin at 0i, is attached tobody Bi at the joint between body Bi and Bo . In addition, for defining attitude andsolar radiation incidence angles, a reference frame is located at Cf such that the axesXP , Yp , and Zr are parallel to X,, Ye , and Z,, respectively. Note, an arbitrary masselement dmi on body Bi can be reached through a direct path from 0, via Oi (Fig-ure 2-3). 0,, in turn, is located with respect to the instantaneous center of mass Cfand the inertial reference frame, Fo . Thus motion of dmi caused by librational andvibrational motions of B, and Bi can be expressed in terms of the inertial coordinatesystem. The relative position of Oi with respect to 0, is denoted by the vector di .The location of the elemental mass of the central body, dm,, relative to 0, isdefined by a series of vectors. pc indicates the undeformed position of the element.16Figure 2 - 1^The main truss and the appendages constitute a branched geometrywith one bifurcation.17Figure 2 - 2^A schematic diagram of the First Milestone Configuration showingthe interconnected flexible members and the reference coordinatesystems.18Figure 2 -3^Schematic diagram of the FMC showing the instantaneous center ofmass and direct path to a mass element with respect to the inertialframe F0.19The transverse vibration of the element, S c , shifts the element to the final position.Similarly, pi , and Si define the location of the elemental mass dmi, in body Bi, relativeto frameOrientation of the coordinate axes Xi,Yi,Zi relative to X,,Y,,Z, is defined by thematrices Cf. It should be noted that the transverse vibration of 13, and Bi result inthe time-varying characteristics of Cf .2.2.3 Position of spacecraft in spaceConsider the First Milestone Configuration, with its instantaneous centre of massat Cf negotiating an arbitrary trajectory about the centre of force coinciding withthe homogeneous, spherical earth's centre. As the FMC has finite dimensions, i.e.it has mass as well as inertia, in addition to negotiating the trajectory, it is free toundergo librational motion about its center of mass. Let X,, Y,, Z, represent movingcoordinates along the orbit normal, local vertical, and local horizontal, respectively(Figure 2-4 ). Any spatial orientation of Xp ,Yp ,Zp with respect to X„,Y,,Z, canbe described by three modified Eulerian rotations in the following sequence: a pitchmotion, V), about the X 3-axis giving rise to the first set of intermediate axes X 1 ,Y1 ,Z1 ;a roll motion about the Z1 -axis generating the intermediate axes X2,Y2,Z2; andfinally, a yaw motion, A, about the Y2 -axis yielding Xp ,Yp ,Zp . It can be shown thatthe librational velocity vector, (1,, is given by,= sin A + (e ij)) cos cos Ai ip — (é + /,b) sin 0]:jpcos A +(B+ 1k) cos q5 sin ALicp ,^ (2.1)where B represents the orbital rate of the spacecraft.20Figure 2-4^Modified Eulerian rotations V), 0, and A defining an arbitrary spatialorientation of spacecraft.212.2.4 Shift in the center of massThe centre of mass of the First Milestone Configuration is the reference point todescribe the spacecraft's librational and orbital motions. For a rigid system, where thecentre of mass remains stationary with respect to a body fixed axis, can be determinedeasily; however, this is no longer true for a flexible system. The general expressiondescribing a shift in the centre of mass due to elastic deformations is derived below.Consider the spacecraft in Figure 2-3. Here, C i and Cf represent the centres ofmass of the undeformed and deformed configurations of the system, respectively. Thevector which denotes the position of Cf relative to C i , represents the shift inthe instantaneous centre of mass of the spacecraft due to its deformation. This vectorwill be necessary in evaluation of the kinetic and potential energies of the system.With reference to Xp ,Yp ,Zp axes, the vectors from the center of the earth to masselements dme and dmi, as represented by fic and lei , respectively, can be written as:•Rc = Rcm —^+ Pc + Sc ;Ri =^— C —^+ CT (Pi + Si) ; •Taking moment about the centre of force givesR,m^dm, + ELI kidmill ,M{ fme^i=1^771iwhere M is the mass of the spacecraft. Substituting Eq. (2.2) into Eq. (2.3) yieldsC = 1 fl {P, 6,1 dm,mc+E{fm {C11 + {Pi + Ed} dmimiwhere:(2.2)(2. 3)(2.4)22dcm = dim + Cf!sm ;= position vector of Ci, the centre of mass of the undeformedspacecraft, relative to Oc;Cm = position vector of Cf relative to C i ;N = number of Bi bodies;M = total mass of the spacecraft,^E [mi] .i=1SinceC'Crri ={if Pc dmc E [{7nipc(00 + fm Ci Pi &nil]^c^,^(2.5 a)m i=1^ iCim can be simplified as(CCf,, L^m^= — {1^dm ^ [fmisc(oi) + f C; Si &nil] ,^(2.5b)mc i=1^miwhere Sk(Ol) (k = c,i; / = i ) represents Sk, evaluated at the coordinates of On andU is the unit matrix. Equation (2.4) is the general expression valid for both rigid aswell as flexible systems; however, for a rigid system, Eq. (2.5a) is adequate.2.3 Elastic Deformations2.3.1 BackgroundEvaluation of the kinetic and potential energies requires expressions for the systemdeformation. For a multibody system, two distinct approaches have been popular toestimate elastic deformations: the Finite Element Method (FEM); and SubstructureSynthesis Method (SSM). In the FEM, the system is first subdivided into finite ele-ments with degrees of freedom at the nodes. Using the local degrees of freedom asgeneralized coordinates, the mass and stiffness matrices of the element can be de-23rived readily. Applying the boundary conditions for the system and compatibilityrequirements between adjacent elements, the system mass and stiffness matrices canbe assembled from the corresponding matrices of the elements. The system modescan then be evaluated numerically using finite element subroutines.In the SSM, the system's flexural motion is represented in terms of the compo-nents' dynamics. The first step is to obtain the series of admissible functions, bysolving the eigenvalue problem for each component, representing its elastic defor-mation. These functions are referred to as "component modes" by Hurty [17], whopioneered the SSM approach. It should be pointed out that the FEM can be usedin deriving the component modes. The objective here is to transform the state-spaceform of the system's equations of motion into the balanced form such that the control-lability and observability grammians are equal and diagonal. Generalized coordinateswhich have small diagonal elements are least controllable or observable; hence, theyare often discarded.2.3.2 Transverse vibrationsThe transverse vibration of the substructures can be obtained by the SSM. How-ever, convergence of any set of admissible functions to the actual solution is guaran-teed by Rayleigh-Ritz procedure provided the admissible functions satisfy the kine-matic boundary conditions and form a complete set. With this in mind, the vibra-tional displacements of beam-type elements are represented in terms of modes of theEuler-Bernoulli beams. For body Bc , the admissible functions used are similar tothose of a free-free beam [18],O r (x) = cosh(Pr 4x ) cos(f3r — -yr [sinh(Or sin(/3' )] ,r = 1, 2, . . .^(2.6)24where Or is the solution of the equationcosh(13r) cos(/3T) — 1 = 0 ,and -yr is given byT = sin(3r) + sinh(f3r)-Y — cos(Or) + cosh(3r) •For Bi cantilevered modes are selected,Or(x) = cosh(f3r ) — cos(fr• ) — yT {sinh(f3r ) — sin(Or-T-d] ,lb^lb^lb^r = 1, 2, ...^(2.7)where ,fir is the solution of the equationcosh( IF) cos(f3r) + 1 = 0 ,and -yr is given by, _ sin(OT) — sinh(8r),ycos(f3r) — cosh(f3r) •Using Eqs. (2.6) and (2.7), the transverse vibrations of a beam-type substructure inits reference Y and Z directions, v b and w b , respectively, can be written as:n= E pr(07,b7(x) ;r=1n= E Qr(t)or(x) ;r=i(2.8)where Pr (t) and QT (t) are the generalized coordinates associated with vibrations inthe Y and Z directions, respectively; and Or(x) are the corresponding admissiblefunctions.2.4 Kineticsv bw b252.4.1 Rotation matricesMatrix^denotes the orientation of the frame Fi relative to the frame Fe . Tworotation sequences are needed to determine C?: the first one, Cr, defines the rigidbody orientation of with respect to F e whereas the second one, C jc 'f , defines therotation of frame Fi relative to Fe due to elastic deformations of the body Bc . Amodified Eulerian rotation of the following sequence is selected: a rotation 0; c aboutXe-axis, followed by 0;c about Ye-axis, and finally 0;c about Ze -axis, i.e.,cos 0;cCF' r = sin Orzc0— sin Orzc 0 cos 0;c 0 sin Bye^-ccos 0:c 0 1 00 1 — sin Or 0 cos ByeYc_ 1^0^00 cos 0; c — sin 0; c0 sin 0; c^cos 0; cSimilarly,cos^cos 0;c sin 0; c sin 0;c cos 0;c — cos 0; c sin 0;c^= cos 0;c sin 0;c sin 9 sin Bye ^0;c + cos 0; c cos 0;c[— sin 0;c^ycsin 0; c cos 0;ccos 9 sin 0;c cos 0;c + sin 0; c sin Orzccos Or sin Or sin V — sin O cos Orxc^ye^xc^zc^•cos 0; c cos 0;c(2.9a)Cic, f =[cos etc cos 01ccos 0‘c sin 0f— sin Bysin 0 -le sin By cos 01c — cos 01.c sin 0 -zfcsin 0-1c sin of sin 61c + cos 0‘c cos 01csin 0.1c cos 0‘ccos 9 sin Oifc cos 0ifc + sin 01c sin 0-zfccos 0 -1c sin 0t.c sin 0-zfc — sin 01, cos 0ifccos Otc cos 0‘,(2.9b)Nowcf = c ie'f x c ic'r.The choice of the rotation sequence is somewhat arbitrary because the multiplication269 f =^Rcr t ) d'tk c.zc c dxl izc=cizc^cos^.T=1(2.10b)of rotation matrices is commutative for small rotations; however, as pointed out byHughes [19], the concern is the location of the singularity. For the sequence selected,the singularity is located at 9 yr = 7r/2 or Oyfc = 7r/2. Ofc , O‘c , and 0ife are functions ofthe elastic deformations of Be . Consider the FMC model , with a beam type centralbody attached to another body at coordinates (4 c , dyc , dzc ). Initially, in absence ofany deformations:0;,it2 'eyrc = 0 ;0:c = 0 .^ (2.10a)With the inclusion of elastic deformations, O zfc , 0 -‘c , and Oirc can be evaluated using:Of = •me^2 ,Of^— E Qrc(0 d0;^c4c —yc lc dx c izc=dz eT=1cos Of ;2.4.2 Kinetic energyThe kinetic energy, T, of the spacecraft is given byT = 2 m{Ie^kc dmc^[f lei^dmil ,i.1 miwhere Rc , Ri , are obtained by differentiating Eq. (2.2) with respect to time:(2.11)= Rcf^•— C — C +^—m^cm^cm c +co x^—ki .kcni — ac fm —6icm +4i+cfCso+df(si)+w x Holm _ oLi + + Cf(pi Si)] ;+ +Se] ;27x^— Cln.„ + di] .^ (2.12)Substituting Eq. (2.12) into Eq. (2.11), the kinetic energy expression can be writtenin the formT = Torb + Ton+ Th Tjr + Tv + Thjr + Th,t, + Tjr,v1 T_+ —2w Lys w + (DT ILys ,1 _T,YTorb + Taya 2 la a CO wT Hsys ,Tay, = Torb Tem + Th Tj + Tv,(2.13)where (.7) is the libration velocity vector; Isys , the inertia matrix; 'Lys the angularmomentum with respect to the Fic frame; . c,DTIsysc.D, the kinetic energy due to purerotation; and CDTrisys , the kinetic energy due to coupling between rotational motionand transverse vibration. Tsy, represents the kinetic energy contributions due tovarious effects with the subscripts involved defined below:orb^orbital motion;cm^centre of mass motion;h^hinge position between body Bc and Bi or between body Bi and Bi,j;j^joint rotation due to elastic deformation;v^transverse vibration.For instance, Tv refers to the contribution of kinetic energy due to the transversevibration velocity. The evaluation of the integrals require a priori knowledge of p-kEk, and 4 (k = c, i,); hence, the configuration and location of each body must bespecified before the evaluation can proceed. The matrix I.", which represents inertiaof the system, is time-dependent and consists of several components,Lys =^ih + Jr + Tv + ih,r^Ir,v^ (2.14)28where the subscripts cm, h, and v have the same meaning as before. Subscript rdenotes contribution from the rigid body component.The angular momentum vector, Riy,, can be written asHag, =^Hh+^Hv+11h,jr^flh,v^(2.15)with the subscripts defined as before. Similar to L ys , Ray, is a time-dependentquantity.2.4.3 Potential energyThe potential energy, U, of the spacecraft has contribution from two sources:gravitational potential energy, Ug , and strain energy due to transverse vibration andthermal deformation, Ue,U = Ue UgThe potential energy due to gravity gradient is given byN{fmc drRnc, z^dmi igi}U =Substituting the expressions for R e , Ri, from Eq. (2.2), and ignoring the terms oforder 1/R,47n and higher, Ug can be written asUy^ tleM^31-1e --T^= Rcn,^2.„^7,^tr [Lye] + 241 I 3Y, 14 (2.16)where p e is the gravitational constant and T represents the direction cosine vector ofRe, with reference to Xp ,Yp ,Zp axes. From Figure 2-4, T is given by1 = (cos 1,b sin 0 cos A + sin ti) sin A)i e cos cos Oic+ (cos '0 sin 0 sin A — sin //) cos A)kc •^ (2.17)29The strain energy expression for a beam and a plate are [2]:1 f rEI (a 2 w b V^a2vb)2} dib2 jib i. ^ aX2 )^Eizz( ax2D^(52wn2 + 2v (02wP (52w" ) (52w "  ) 22 hp t aX 2 )^ ax2 ) ay 2 )^ ay2+ 2(1 — (a2wp 2 \ dA •PaXayUe,beamUe,plate =(2.18)where D and v are the flexural rigidity and Poisson's ratio of the plate, respectively;and EIyy and Elzz are the bending stiffness of the beam about Y and Z axes,respectively. By specifying the configuration of each body constituting the system,the strain energy can be evaluated.2.5 Lagrangian FormulationUsing the Lagrangian procedure, the governing equations of motion can be ob-tained fromd f aT, aT au r,`qwhere q and Q q are the generalized coordinates and generalized forces, respectively.The number of generalized coordinates depends on the system configuration, i.e., thedegrees of freedom involved.In general, the effect of librational and vibrational motion on the orbital motionis small unless the system dimension is comparable to Rcni, [20,21]. Hence, the orbitcan be represented by the classical Keplerian relations:h2Rcm = Pe(1 e cos 19)Rc271,9 = h;^ (2.19)where h is the angular momentum per unit mass of the system, p e is the gravitational301k/I4) 11A req1qnvK.0KKAKq1QQQA. . .Q q1Q qnvMlib^Mlib,vibmTlib,vib Mvib _constant, and e is the eccentricity of the orbit. Therefore, q and Ng would reduce to:q =^A, Pry,  Q rc -E: i , Q iri ;Ng = 3 + (2 x n„) + (2 x (N x nri )).Here n„ and nri denote the number of modes used to represent the transverse vibra-tion of B, and Bi, respectively; and P", Q:'', Pri, and Q:i , are the nondimensionalvalues Pic/lc , Cfc c//c , Piri//i, and Q iri//i, respectively. In parametric studies of space-craft dynamics, dimensionless parameters and independent variable are desirable. Forinstance, simulation results using time as the independent variable from t = Tinitial tot = T final are applicable for a particular orbit only. In contrast, using true anomalyas the independent variable, simulations for 0 = °initial to 0 = ° final are valid forsimilar orbits at different altitudes. Variables with a dimension of time, such as fre-quencies, are nondimensionalized by the orbital rate at the perigee point, 0,,. As forthe time derivatives, their true anomaly counterparts are obtained using the followingrelations:d^• ddt = 070 "2e sind2^• [ d2= 62c/(9 2^1 + e cos dd8dt 2(2.20a)(2.20b)where Eq. (2.20b) is derived from Eqs. (2.19) and (2.20a).The general equations of motion can now be written as:31Or^M(q)q" C (q, q', 0) K (q, 0) = Q(0) ;^ (2.21)where primes denote differentiation with respect to the true anomaly; n, is the totalnumber of vibrational degrees of freedom such that Nq = 3 -I- nv . Here M is a non-singular symmetric matrix of dimension Nq x Nq . The entries in M come from secondorder terms of dlc10(aT N I ). C is a Nq x 1 vector representing the gyroscopic termsof the system. They come from two sources: the Coriolis terms of dl c10(87115q 1 ) andfrom aTlaq. K, also a Nq x 1 vector, denotes the stiffness of the system. Its solecontribution is from (aU/aq). Q, the generalized force vector of dimension Nq x 1, isevaluated using the virtual work principle.323.^DYNAMICAL STUDIES3.1 Preliminary RemarksThe objective of this chapter is to study the dynamical responses of the FMC asaffected by orientations and initial conditions. To this end, dynamic simulations ,oftwo configurations of the FMC , of contemporary interest, were undertaken.3.2 First Milestone ConfigurationThe proposed Space Station Freedom which is currently in the design and devel-opment phase, will be assembled with thirty flights of the space shuttle. The firstthree flights would result in the construction of the First Milestone Configuration(FMC) . It will have an overall length of 60 m and a mass of 17,680 kg. Majorequipment installed in the FMC includes two PV arrays, an alpha joint, fuel storagetanks, stinger and resistojet, avionics, and RCS. The design configuration of the FMCwould be such that the axial directions of the power boom and PV arrays are parallelto the orbit normal and local vertical, respectively. In general, the design configu-ration is not identical to the equilibrium configuration. The FMC is simulated hereby modelling the power boom, as a free-free beam, with lumped masses representingthe alpha joint, fuel storage tanks, avionics, and RCS. The stinger and the resistojetare treated as a cantilever beam and point mass, respectively. The PV arrays andPV array radiator are represented as cantilevered plates.Taking the component modeapproach for discretization and considering only the first mode of vibration for theflexible elements, the generalized coordinates and the degrees of freedom are:libration^stinger^SolarPanelsr--d■--„, ,---...--.„ ,----"—.--,,/, „ At,^Di n i^Di n i^"1,1^"1,1q = '7' , w ,^, --c , 'c‘ , 4=1 , L 1 , IL2^, 173 , andlf!' l •„_..., -,.....-c ---. ,.—..„,..—,truss^radiator33Ng =10 .The numerical data and the configuration used in the simulations are obtained orestimated from the NASA reports [22,23] and are summarized in Table 1-1. Figure 3-1shows the system under study in the design orientation. The normalised inertia matrixof the system is found to be,1.1715 0 —0.1088Lys =^0[ 9.2108 0—0.1088 0 9.5538Consider first the rigid body dynamics of the FMC. i.e Here, the power boomas well as the attached appendages are assumed to be rigid. The equilibrium con-figuration of the FMC corresponds to: &e = 0 ° , qe = 0.74°, and .\ e = 0°, whichis slightly different from the design orientation. Note, the equilibrium configurationdeviates from design configuration due to small roll rotation of 0.74 degree, due to thepresence of the radiator and the stinger. Hence the system, even in absence of anyexternal or internal disturbance, would exhibit some motion. In the present case , itis apparent that the system as well as the equilibrium configuration are unstable (Fig-ure 3-2). Introduction of an initial disturbance of 10° in pitch only accentuates theunstable response and the system starts to tumble in less than one orbit as shown in(Figure 3-3 ). Obviously , there is a need to develop a suitable control. Next considerthe rigid body dynamics of the FMC for the Lagrange configuration (Figure 3-4 ).It's equilibrium position corresponds to 0, = —1.47°, and A e = 0 ° Abe = 0°. Thus inthe classical Lagrange orientation the system experiences a small angular disturbanceof —1.47° in pitch. Note ,the librational response, to this disturbance is stable in allthe three degrees of freedom : pitch, yaw and roll.3.3 System and Component Modes34Figure 3 - 1^Configuration of the FMC used in the numerical simulation: (a) co-ordinate systems; (b) design configuration.35Figure 3-2^Librational response of the rigid FMC showing the inherent unstablecharacter of the design orientation.Note it is not in the equilibriumstate.3.3.1 Component mode discretizationThe dynamical analysis of the flexibility for the FMC can be accomplished in oneof the following two ways. In the component mode method, the number of generalizedco-ordinates depends on the system configuration, i.e, the degrees of freedom involved(for holonomic systems).For instance , consider a beam type central body Be , under-going general libration motion with N beams (Body Bi) attached to it. Assuming mto represent the number of flexural deformation modes for each of the flexible mem-bers the total number of generalized coordinates is given by for each of the flexiblemembers, the total number of generalized co-ordinates is given byNg = 5 + m + (N x m)36Figure 3 - 3^Librational behavior of the rigid FMC (DO) showing the unstableresponse in the presence of an external disturbance in pitch (0 =10°.)37Figure 3 -4^Libration behaviour of the rigid FMC in Lagrange orientation inabsence of the external disturbances. The pitch response is due todeviation from the equilibrium configuration.38. Also , each flexible member is modelled seperately (plate ,or beam etc.) and thedisplacements due to flexibility are referred to the local frame of reference.3.3.2 System mode discretizationIn the system mode method, the number of generalized coordinates depend onthe number of modes considered in the discretization of the continuum motion andnot on the complexity of the system configuration.For instance, if one accounts forthe m system modes to represent the flexibility of the model, the total number ofgeneralized coordinates is given byNg = 5 + mIn the system mode discretization the whole system is considered as one flexiblemember and boundary conditions, between members, are evaluated using FEM.3.3.3 Comparision for the component and system modesThe responses for the two modes of discretization, namely the system mode andthe component mode cannot be compared directly, however if the effect of flexibilityis decreased, then in principle the two responses should converge to the rigid case.Figure 3-5 and Figure 3-6 show librational responses as obtained by two distinctly dif-ferent programs:the one based on component modes while the other using the systemmodes. Of course in the present case we are checking their prediction in the limitingcase of zero flexibility, i.e the rigid case. Ideally they should give the identical results.As expected the results exhibit similarity of trends in their librational responses. Theminor discrepancies may be attributed to differences in the computational method-ology associated with the programs and to the disparity in the boundary conditionswhich cannot be matched precisely, for the two methods of discretizations employed.39Next, the flexible deformations were represented by the first mode for each of theflexible members (2 solar panels, truss , stinger and radiator) in the component modediscretization. With three librational degrees of freedom, this would give ten gen-eralised co-ordinates. Correspondingly,seven system modes were selected to give thesame number of degrees of freedom. The system is subjected to an initial disturbanceof 2° in pitch roll and yaw simultaneously.Note the predicted responses ( Figure 3-7and Figure 3-8 ). This is expected as the component mode synthesis is unable tosatisfy all the boundary conditions of the continuum system.3.4 Flexibility EffectsThe effect of flexibility on the system response , for the configurations understudy is shown in in Figure 3-9 and Figure 3-10 . Since the solar panels have largelengths and flexibility they dominate the response in the flexible degrees of freedom.Note the flexible response is coupled with the librational response, though the extentof interaction is small. Even with the solar panels disturbed by a small amount(0.001 cm ) the coupling becomes all too apparent. For better appreciation of theresponse, the vibration degrees of freedom are plotted in terms of the tip deflectionof the centerline. Using Eq. (2.8), the tip deflections of a beam element in Yk and Zkdirections, respectively, are given by:ns, = E P,:covkvo ;r=1n.51= E (21(t)/Paik) •^ (3.1)r=1For a plate element , the displacement is a function of both the Xk and Yk coordinates.Figure 3-2 shows the librational response of the system initially in the designorientation. This serves as a reference for other cases with initial disturbances. Here,40Figure 3- 5^Librational response using two distinct programs, employing systemand component mode discretizations, in the limiting case of zeroflexibility, for the design orientation41PIFigure 3-6^Librational response using two distinct programs, employing systemand component mode discretizations, in the limiting case of zeroflexibility, for the Lagrange orientation42Figure 3-7^Dynamical response for a disturbance of 0 = '0 = A = 2° in thedesign orientation of the FMC , using the system modes for flexibilitydiscretization43Figure 3-8^Dynamical response for a disturbance of 0 = 0 = A = 2° in thedesign orientation of the FMC, using the component modes for flex-ibility discretization.44Figure 3 - 9^The effect of flexible appendages on the librational response of theFMC with one of the appendages (Solar panel by 0.001cm tip) dis-turbed slightly, in the Lagrange orientation.45Figure 3 -10 The effect of flexible appendages on the librational response of theFMC with one of the appendages (Solar panel by 0.001cm tip) dis-turbed slightly, in the design orientation.46deviation from the equilibrium configuration serves as a disturbance to the system.Even with small disturbances, the system gradually undergoes unstable librationalmotion. The periodic oscillations of the flexible degrees of freedom in the direction ofthe local horizontal, or the orbit normal indicate their non-zero equilibrium startingposition. Due to the coupling effect between the librational and vibrational degreesof freedom, their response amplitudes gradually increase with time. It is difficultto completely isolate the flexibility effects since the equations are coupled nonau-tonomous and highly nonlinear.However, some insight can be obtained if the flexiblecontributions are reduced .One way to accomplish this is to reduce the length of theflexible appendages. Figure 3-11 shows the effect of reducing the solar panel lengthsby 90% (i.e from 33m to 3.3m ). The response shows that the high frequency modu-lations of the the libration degrees of freedom are reduced when the mass and inertiaof the solar panels is reduced by ninety percent.The magnitude of all the flexible re-sponses decreases by almost two orders of magnitude .The beat phenomena displayedby stinger response under flexibility effects is also absent.The effect of the power boom disturbance, in the local horizontal direction isshown in Figure 3-12 . Here , the power boom is initially deformed in the firstmode with a tip deflection of 2 cm in the local horizontal direction ((P 1 ) 0 = 1.652 x10 -4 ). Even with this small disturbance, the pitch and roll responses show highfrequency modulations of 0.03° in amplitude.(in pitch). Similar trend is observed inthe roll response. Under this initial condition, the power boom is expected to bevibrating symmetrically about the local vertical; hence, it is not surprising to seethat the transverse motion of the truss (T R p) induces corresponding modulationsin the roll. Of some interest is the beat response of the stinger. Modi and Ng [24]have investigated the beat response character of a gravity-gradient oriented satellitewith two appendages. The flexible element experiencing the largest deflection is the47Figure 3-11 Dynamical response of the FMC in design orientation,with solar pan-els reduced by 90 percent of their lengths and a tip deflection of 1cm. Note that the high frequency modulations are absent from thelibrational response48stinger. Its maximum amplitude is about 1 mm in the local horizontal direction.In Figure 3-13 , the power boom is initially disturbed in the local vertical direc-tion, (Q 1 ) 0 = 1.652 x 10 -4 . Since this initial condition gives rise to predominantlysymmetrical motion about the local horizontal, the yaw motion is excited with highfrequency harmonics. However, as can be expected, both pitch and roll remain es-sentially free of the flexibility induced modulations. The beat response of the stingeris modified . This shows the directional characteristics of the FMC power boom:for a known response in direction X when disturbed in direction Y, one cannot pre-dict response in the direction Y when disturbed in the direction X.Of Course, theflexible members are excited by a small amount, however , the numerical computa-tions are able to capture their trends. Exception is the solar panel response which issignificantly larger.Response of the system with the stinger subjected to an initial disturbance isshown in Figure 3-14 . The stinger is deformed initially in the first mode of a cantileverbeam with a tip deflection of 0.5 cm in the local horizontal direction, (Pi )° = 0.9360 x10 -4 , Similar to the power boom disturbance case, both pitch and roll responses havehigh frequency harmonics although the modulations in the roll motion are hardlynoticeable. Also, the initial condition has no effect on the yaw response at all. As forthe power boom, it is slightly excited in the local vertical direction. The excitationof other flexible members are also small.Figure 3-15 shows that a disturbanceto the stinger, along the orbit normal, bythe same amount as before,induces high frequency modulations in the pitch and abeat phenomena in the TRp degree of freedom.Figure 3-16 shows the influence of PV array on the system response. This isin addition to the initial 10° disturbance in pitch . Hence to assess the effect of49Figure 3-12 Dynamical response of the FMC for a small power boom disturbancein the local horizontal direction (T Rp = 0.0001652), in the designorientation.50Figure 3 -13 Dynamical response of the FMC for a small power boom disturbancein the local vertical direction,(TRQ = 0.0001652), in the design ori-entation.51Figure 3- 14 Dynamical response of the FMC for a small disturbance to the stingerin the local horizontal direction, (STp = 0.000094), in the designorientation.520.0^Orbit 0.10.0 0.10.0500.000-0.050-0.1000.0-0.0-2.0-!)1,11111 i1,!1 1,1,11111!3.00.0- io-4 m0.1TR Q0.1Orbit^0.1PerigeeSPFigure 3-15 Dynamical response of the FMC for a small disturbance to the stingerin the orbit normal direction, (STQ = 0.000094), in the design ori-entation.53flexibility, it should be compared with Figure 3-11. The major effect , as far asthe librational response is concerned, are the high frequency modulations in the yawdegree of freedom. The main truss and stinger show relatively small motion at theirends, however the radiator motion is large and diverging at least in this early phase.Larger duration computation showed it to be a beat phenomenon with the peakamplitude of around 0.2 mm.In Figure 3-17 , the PV array radiator is initially deflected in the first mode of acantilever plate with a tip deflection of 1.0 cm [(I-4 1 ) 0 = 0.4970 x 10 -3 ]. The pitchresponse shows low frequency harmonics of a small amplitude. The excitation of thepower boom and the stinger is rather small. The only flexible member that showsnoticeable deformation is the PV array . Similar coupling between the radiator andsolar panel motions was observed before (Figure 3-16). These results also point tothe coupling of power boom vibration in the local horizontal direction with the PVarray deformation, through their similarity in the beat response. As can be expected,reduction in the solar panel lengths (which decreases it's flexibility ) significantlydiminished it's coupling with the radiator.In summary, Figures 3-9 to 3-17 demonstrate that flexibility effects on the FMCresponse cannot be overlooked. A small disturbance applied to any flexible membercan significantly affect the rigid body motion through it's high frequency modulations.The power boom disturbance in the local vertical direction is the most critical one asthe resulting high frequency modulated pitch and roll responses would require highbandwidth controllers. Furthermore, with the power boom or the stinger subjectedto a given magnitude of disturbance, its direction is critical in predicting the rigidbody as well as vibratory responses of other components.54Figure 3-16 FMC dynamical response to a tip deflection of 1 cm in the SolarPanels (SP=lcm), in the design orientation.55Figure 3-17 Dynamical response of the FMC for a deflection (RAD=0.00049), inthe PV radiator, for design orientation.563.5 System Mode Simulation StudyConsidering the first m system modes to represent structural deformations, thegeneralized coordinates and the degrees of freedom are:4 - {ii, A) 0)P1) P2) •••)P7r11The numerical data used in the simulation were presented earlier. The first forty sys-tem modes (including the six rigid body modes) for the First Milestone Configurationwere computed using a seperate finite element program, to discretize the continuoussystem. Figure 3-18 shows the frequency spectrum of the First Milestone Orientation.Of course, use of such a large number of modes would increase the computational timeand cost. Objective here is to demonstrate the applicability of the numerical codeto a specific configuration. Hence, in the present study only first two system modesare used for actual simulation, to save the computational effort.The main objective isto emphasize the point that component modes have inherent limitations in correctlypredicting the response, unless the boundary conditions , on the global scale, areaccurately accounted for. This is seldom possible. They may predict correct trendsbut one can expect accurate answers only in isolated simple cases. The frequencyspectrum provides the free vibration frequencies and associated system modes. Themode characterization help appreciate relative contributions of different parts of theStation.In general, modal displacements fall into the following three categories:(a) Solar Array Deformation Modes: these are the modes in which the solar ar-rays deform significantly in and out of the X-Y plane as cantilever platesand the remainder of the Station responds only slightly so as to maintain thedynamic equilibrium.57(b) Radiator Modes: These are associated with the PV solar panels and Stationradiator deformations. Both are modelled to have a fundamental bendingfrequency of 0.1 Hz.(c) Stinger/RCS Boom Coupled Modes: Since both these components are de-signed to have a fundamental bending frequency of 0.5 Hz, they appear incombination in the system modes.(d) Overall System Modes: in general, these modes involve an overall motion ofthe Station, with solar array and radiator deformations coupled with responseof the main truss in and out of the X-Y and X-Z planes.For the FMC, the appendage response in bending dominate the first six elasticmodes (3'7 — f12) with frequencies in the range of 0.1 - 0.5 Hz . Of these, the first threemodes (f7, f8, f9) pertain to the PV array and radiator while fim — f12 correspondto the RCS boom and stinger assembly. The torsional motion of the main truss isrepresented by f9 while the corresponding bending in Z and Y directions is representedby f21 = 2.30 Hz and f22 = 2.35 Hz, respectively. The stinger and RCS boommotions are coupled as both have a fundamental frequency of 0.5 Hz. The PV arraysand radiators, which are modelled as cantilevers have their fundamental componentfrequency as 0.1 Hz . The solar array deformation modes display pure torsionalmotion in symmetric and asymmetric modes at f16 and fi7 (1.14 Hz) with higherharmonics represented by 1.24, i25 (2.4 Hz) and f31, f32 (5.97 Hz).The arrays are provided with the rotational capability, about the alpha and betajoints, in order to track the sun for optimum exposure. Although the maneuversare slow and quasi-static conditions prevail, different orientations of the solar panelsdue to these maneuvers will affect, to some extent, overall structural flexibility of theFMC and the associated frequency spectrum. A rotation of the solar panels about the58Figure 3-18 Frequency spectrum for the FMC. Note the frequencies are relativelysmall and quite closely spaced .The first forty modes are within 10Hz !. Such packed character with overlap makes determination ofthe system dynamics and control extremely difficult.59a-joint affects, modes 16, 18, 19, 25, 27, 29 and 33 . Similar changes in the behaviourwere observed in other modes as well. Of particular, interest was the interchange ofenergy among the modes and between the components in the same mode. In orderto simulate the system dynamics using the assumed modes approach, it is necessaryto specify the number of modes to be used. In theory, as one increases the number ofassumed modes, the results should approach the true response. On the other hand,this leads to an increase in the computational cost. So it is important to study thecharacter of modal convergence to strike some balance between accuracy and cost.Suleman[ 25 ] has also shown in his dissertation that the significant effects of the solarpanels rotation is confined to a few modes.He also found that system response canbe predicted with an accuracy of 1 percent by using only five modes. On the otherhand , acceleration profiles were affected by higher modes (above five), specially thetruss bending corresponding to the 22n d mode.The dynamic behaviour was monitored in the presence of an initial disturbanceof 10° in pitch, roll and yaw, applied simultaneously. The structural deformation wasrepresented by 1, 5, 10 or 20 lowest modes. Plotted quantities of interest include thelibrational displacements , modal displacements at the tip of the upper solar panel,and that at the modules location on the main truss (i.e. at the end of the truss).It was interesting to note that, for the single mode case, the roll and yaw degree offreedom do not show a significant contribution from the structural flexibility of thesystem. Here, the structural deformations were confined to the solar panels vibratingin a symmetric fashion, while other components of the Station remained relatively sta-tionary. However, with the second and higher modes included, the radiator and othersecondary members displayed a more active contribution to the system deformation,leading to interactions with the roll and yaw degrees of freedom.60Figure 3-19 shows the system mode simulation of the flexible free response of theFMC in the design orientation. The effect of flexibility on the librational responsehas a very small magnitude.The solar panels exhibit highest magnitude of all flexibleappendages, with regards to the flexible response.3.5.1 Lagrange orientationThe component mode solution has an inherent limitation in that it is unableto take care of the boundary conditions. To illustrate limitation of the componentmodes solution, response of the previously studied design orientation of the FMC wasobtained using, the first two system modes corresponding to the peak displacementof 1 cm in the PV Array (Figure 3-20 ). It should be pointed out, to be precise,that such comparision is not quite accurate. For preciseness, one should match thedeflection as well as the energy input during a specified disturbance using system andcomponent modes.The system mode solution being more precise , and the Lagrange orientationbeing stable, it was thought approriate to focus attention on it's response analysisusing the first two system modes.Figure 3-21 shows the librational response of the FMC in the Lagrange orientationfor disturbances of 0 = 4° and 12°. Note the sensitivity of the Librational responseto disturbances in the roll degree of freedom. The pitch response which is stable forsmall roll disturbances becomes unstable for a large roll disturbance of 12°. Henceroll is a critical parameter as far as the librational response of the FMC in LagrangeOrientation is concerned.Figure 3-22 shows the Lagrange Orientation response for a ib = 10°. A rela-tively large disturbance of 10° does not cause any significant variation in the yaw61Figure 3-19 Response of the FMC in design orientation with zero initial condi-tion, using system modes for flexibility discretization62Figure 3-20 System mode simulation of the dynamical response for a tip distur-bance of 1cm in the PV Array in the design orientation of the FMC.63Figure 3-21^Librational responses of the FMC in Lagrange orientation for in-creasing roll disturbances of 0 = 4° and 12°.64and roll responses. Their magnitudes remain relatively small over one orbit. Thisdemonstrates the stability of the Lagrange orientation with respect to relatively largedisturbances in pitch.Figure 3-23 shows the Lagrange Orientation flexible response for a tip deflection oflcm in the solar panels.This is in addition to a pitch disturbance of ten degrees. As canbe seen the yaw and roll degrees of freedom show some high frequency modulations,although their magnitudes are pretty small. Also note the magnitudes of the trussresponse in the two directions (local vertical and local horizontal) differ by six ordersof magnitude. This is in keeping with the directional nature of the truss response asfound out earlier in the case of design orientation.Figure 3-24 shows the Lagrange Orientation response for the disturbance of 0.2cm in the cantilever plate mode of the radiator. Once again the yaw and roll have lowmagnitude , high frequency modulations.Also note that the magnitude of the solarpanel excitation is higher than that for other appendages. This once again confirmsthe domination of the solar panels on the flexible response.Figure 3-25 shows the response of the FMC in LO for a truss disturbance of1.0 cm .All the flexible members are excited. Note the difference in the frequency ofthe response for the truss and the solar panels. The directional nature of the trussresponse is also amply demonstrated. However the magnitudes of the solar panelresponse is higher. Since only two system modes are used hence the high frequencycoupling of the librational response has a low magnitude.The solar panels do not exhibit a noticeable improvement in modal amplitudeprediction as the number of modes is increased . However the coupling of the flexibleresponse with the librational response becomes more apparent with the inclusion ofhigher nodes. The main truss, stinger and RCS boom are treated here as beam type65Figure 3-22 Dynamical response of the FMC in Lagrange Orientation for a pitchdisturbance of 10°.66Perigee0.0^0.40^ 0.0^ 0.05\8.00.0-8.0-6x 10 mI I/1TR PII II I8.00.0-8.00.0x10 9 mI//STPIIi0.058.00.0-8.04.0II0.0^A^f-4.00.0 Orbit^0.05 0.0^Orbit^0.05x 10-9 m 2.00.0-2.0SPitFigure 3-23 Dynamical response of the Lagrange Orientation for a disturbanceof 10° in pitch, and solar panels deflected by lcm at the tip67Figure 3-24 Dynamical response of the FMC in Lagrange Orientation for 0.2 cmdeflection of the radiator68Figure 3-25 Dynamical response of the FMC in Lagrange Orientation for a tipdeflection of lcm for the truss.69structural members as before. The major items of interest are the tip displacementsand accelerations of the flexible members together with the librational motion. Thedirection of the tip displacement profiles are generally not affected by the number ofmodes usedr.However acceleration prediction at the modules location experiences adrastic change, with the accounting of higher modes. Susceptibility of the accelerationprofiles to higher modes, is due to the fact that , the main truss plays a more dominantrole in the structural dynamics of the system and it's bending occurs at the systemmode f22 .(Figure 3-18)3.5.2 SummarySummarizing, it can be inferred that the system displacement can be predictedwith considerable accuracy using 5 system modes (less than 1% error in the solarpanel tip displacement compared to that given by the 20 modes solution) as found bySuleman [25]. However, the higher modes contribute to local variations in the accel-eration time histories (±2ps), although the general character of the response remainsessentially the same. On the other hand, dynamics of the main truss suggests that itis necessary to include the higher system modes in order to predict the accelerationloads at the modules location with sufficuent accuracy. Thus, depending on the infor-mation needed judicious selection of modal functions would be required. As expected, the component mode solutions only suggest trends, unless suitable procedure fortheir synthesis to account for effective boundary conditions globally is found. Onthe other hand, system modes are quite attractive as they do not suffer from suchlimitation.3.6 Response to Operational DisturbancesThe solar panels will be required to track the sun for optimum power output.70Consider the FMC of the Space Station, in a a near-circular orbit (e=0.02). Thesun tracking maneuver consists of rotating the panel and the truss axes (a and /3respectively.) at the orbital rate 6, so as to maintain the the array facing the sun.Since the spacecraft experiences changes in geometry during the tracking maneuver,the associated frequency spectrum changes with time . Thus, in order to maintaina faithful representation of the system geometry and structural flexibility, the modesneed to be updated at a suitable interval. As pointed out by Suleman [25], of interestis the transfer of energy between the solar panels and the PV radiator during themaneuver. Furthermore, there will be a shift in the center of mass of the system, dueto the slewing maneuver and flexibility, which is accounted for in the model. For therotation of the solar panels about the longitudinal axis, there is no noticeable shift ofthe c.m. A contribution, from the structural deformations appears as high frequencymodulations, which are indeed very small , of the order of 10 -6 m . The solar panelsand secondary members do not exhibit any significant change in behaviour, withsimilar responses as before in displacement and acceleration. Hence it appears that,rotation of the solar panels for tracking the sun, even in the presence of aerodynamicdrag, is not likely to affect the microgravity environment significantly. Furthermore,as found later , the control effort required to maintain the spacecraft in the desiredorientation rather minimal.714. CONTROL STUDYAs shown by the dynamical response study, the FMC is inherently unstable inthe design orientation, and hence there is a need for control. Even the Lagrangeorientation becomes unstable with a small excitation in roll. In any case,for preciseorientation of the space station, to facilitate it's construction and operation , activecontrol is necessary. The main purpose of the controller on board the FMC is tomaintain attitude displacements and rates within design limits, by means of ControlMomentum Gyros (CMG), thus ensuring desirable accuracy in microgravity environ-ment. Two attitude control techniques are studied here, to that end. First, a linearcontrol approach, the Linear Quadratic Regulator is adopted. The equations of mo-tion are linearized about an arbitrary equilibrium position and subsequently writtenin the state space matrix form. The optimal controller gains for the state variablefeedback are established by solving the steady state algebraic Ricatti equation. Theperformance index, defined by the controller energy and deviation of the state fromthe equilibrium, is minimized [26]. The matrix equations are solved numerically toobtain the controlled response of the system as well as the effort required. Subse-quently, a nonlinear control based on the Feedback Linearization Technique (FLT)is applied. Utilizing , the original nonlinear dynamics of the multibody system, thecontroller first determines the effort to effectively linearize the system and introducesa linear compensator to achieve the desired system output.4.1 Linear ControlThe use of a linear control strategy requires linearization of the equations ofmotion about an operating point. Feedback is subsequently introduced, where anonline comparision of the available output with the desired state serves as a measure72of error. This is used to determine inputs that will subsequently decrease the error.An important aspect of the control procedure is to achieve it in an optimal fashion,consistent with some specified performance index, over a period of time.Thus the basis of linear control strategy is the linearized plant model associatedwith the nonlinear multibody dynamics formulation. The equations of motion derivedearlier, are nonlinear, nonautonomous and coupled. The nonlinearities are due toa number of factors, including coupling between the attitude motion and structuralflexibility. A linear model for the multibody dynamics has been developed, here withlinearization performed about a planned attitude trajectory.4.1.1^Linearization of the equations of motionThe complete nonlinear, nonautonomous and coupled differential equations ofmotion for the rigid system can be written as^acDT Ica dt ac-DT^acDT dc,-).^ 362 ^[all i^(4.1)dt 3 ,4^3,4 dt^(1 + ecosO) aewhere:6Z) -= CD^-F^(D ;AO (4.2)Eq. (4.1) can be linearized about an arbitrary equilibrium position and cast in theclassical form [26](4.3)where:Mm =H =rag i rczi3 ,4 J 0 Ldt (4.4)[ d &DT I^(.4) + d acDT 1 T Woekit^°^kit ae- 0a,T i[ (9 J 0^Ldt i173aLD T^3T• [ ae-]^hp()^a8[ -^^A;AO 0(4.5)k= [ d ^+[d aCD1L^°^dt 3,4^woe +t + ^ 1[—1ae a dt 0^3b-^dt oe▪^[3,T^ -TnnT ^Iwo [ -^ I woe [ -a1T-^I TO [ —a^11,a;^(4.6)C/t/ Ao 59 0^ae A-#^ao 0^1) ^3„7, 1^raLDT1 I rczi^racDT 1+ [aril 1 /0^(4 . 7)[dt 3 ,1^ J 0^°^(9 :# J 0^dt 0 + a -d J 0 1 w°^a6 0Here the mass matrix Mm is symmetric and positive definite. H includes Coriolosand other forces due to velocity effects.. Stiffness forces arising from the gravitationalpotential are reflected in matrix K. The vector P comprises of forces attributed to theorbital eccentricity and other parameters independent of the generalized coordinates.^4.1.2^State space equivalentThe linearized equations of motion (4.3) can be put in the formwhere:Aaxa + Bau + Pa,^ (4.8)0A a = [^ system characteristic matrix;^(4.9)13„ = = control influence matrix;^ (4.10)0Pa - = forcing function;^ (4.11)f\411)740 = null matrix;^ (4.12)I = 3 x 3 identity matrix.^ (4.13)4.1.3 The Linear Quadratic Regulator (LQR)The linear control theory employed is of the Linear Quadratic Regulator (LQR)type with the performance index for minimum tracking error and energy given byJ = 1 oo^(isiT Qii + 'ET RU) dt,^ (4.14)owhere Q is the symmetric state penalty matrix; and R is the symmetric controlpenalty matrix. The matrix R is required to be positive definite while Q can bepositive semi-definite.The optimal control force ii that minimizes this criterion is given byu = —R-1 BTYR" = —G- - , (4.15)where YR is the solution to the Ricatti equation, which for the infinite time case hasthe form.—YRA — ATYR + YRBR -1 BTYR — Q = 0.^(4.16)The requirements for a unique control law are that the pair [A, B] be controllableand that the entire system be observable. The system states are assumed availableeither through direct measurement or by a design of the suitable observers.Substituting the feedback law into the system Eq. (4.8) yields= A. -± — BG^(4.17)75representing the response of the closed loop system. The characteristic equation forthe system is represented bydet[sE — AG] = 0. (4.18)The closed loop poles of the system are given by the eigenvalues of A — BG = 0.The control law design consists of choosing appropriate state penalties (Q) and controlpenalties (R) to obtain desired locations of the poles. For a real system, one wouldexpect the poles to be either real or complex conjugate.4.1.4 Effect of flexibilityThe controller design described above is based on the attitude dynamics of arigid spacecraft. It can be extended to the First Milestone Configuration which is ahighly flexible system with a very low frequency spectrum . Thus, the interaction ofthe attitude control system with vibration modes, is a matter of significance to beneglected.The equations of motion are augmented to include the effects of structural flex-ibility on the system dynamics. To account for the deformations , the generalizedcoordinate vector, rearranged into the rigid and flexible degrees of freedom, can bewritten ase = (I, b ) 0, A, Pi, P2, ..., pn) = {1, p}, (4.19)where g and 73 depict attitude and vibrational generalized coordinate vectors, respec-tively. The equations of motion can be written as[m9,9 A49,1 V. 1 + ff:0 1 = I gc9Mp , 0 Mp,p p Fp f lof (4.20)where the generalized control vector Q j is based on the linearized attitude equations76of motion for the rigid system. Here the matrix M is partioned into four submatrices:M9 , 9 ; Mod, ; M,,9 and Mp ,p ; with M9,9 and Mp,p corresponding to coefficients ofthe subvectors B and 75- , respectively. The control law Q 19 = u = Gd, is based onthe rigid spacecraft model, with the structural flexibility acting as a disturbance tothe rigid system. Thus, the controller, based on the linearized equations of motion,is applied to the complete nonlinear equations of motion ,and its validity assessed.Figure 4-1 shows schematically the LQR strategy.4.2 LQR Simulation Results and DiscussionTo illustrate application of the proposed controller, consider the First MilestoneConfiguration (FMC). Two spatial orientations in a 400 Km altitude, circular orbitare considered: the Lagarange orientation, with the main truss along the local verticaland the solar panels aligned with the orbit normal; and the design configuration , withthe main truss along the local horizontal, and the solar panels parallel to the orbitnormal. Noise-free estimates of all the states are considered to be available. Thecontrol torque and angular momentum saturation values for the CMG's are set atpredetermined levels as specified by NASA[22, 23].The Lagrange orientation consists of deploying a spacecraft in orbit with its min-imum moment of inertia axis along the local vertical, and the maximum moment ofinertia along the orbit normal,this in the present case entails aligning of the maintruss with the local vertical , and deployment of the solar panels in the plane of theorbit. This orientation demands minimum control effort for its stabilization , due tothe bounded character of pitch and roll response. However, it has the disadvantageof not exposing the various payloads on the main truss with an unobstructed view ofthe earth.With a pitch disturbance of 10°, the roll and yaw degrees of freedom remain77Figure 4- 1^Schematic of LQR control strategyunexcited, while the pitch response shows stable oscillations about the equilibriumpoint (Ike = —1.47°). (Figure 4-2 ). Figure 3-2lshowed the librational response ofthe FMC for a disturbance of 4 and 12 degrees applied to the roll degree of freedom.Figure 4-3 shows the response of FMC for disturbances of 10° in yaw and roll one byone. However, with a roll and yaw disturbance of 10°, applied independently, the yawexhibits a divergent behaviour in both the cases, thus indicating a strong couplingbetween the roll and yaw degrees of freedom. This suggests that roll control couldprovide a key to ensure stability of the FMC in the Lagrange Orientation.As has been shown by Modi and Suleman [16], the deviation about the yaw axisis likely to be significantly greater than that about the pitch and roll axes, which in-dicates that the yaw period is relatively long (about two orbital revolutions). Threedifferent gain settings are used to investigate the closed-loop system response with alarge initial disturbance of 10° in pitch, roll and yaw, applied simultaneously. Thecontroller performance is measured with respect to parameters such as settling timeand control effort requirements. Results were obtained for different initial distur-bances and imposed settling times (0 = = A = 5°,10°15° , and settling times of5 min , 10 min and 20 min). Only a typical set of results for 10° and a settlingtime of 20 min is presented here for conciseness. Figure 4-4 shows the controlled anduncontrolled libration responses , as well as the control efforts for the application ofLQR to the LO of the FMC. The closed loop eigenvalues used for the case of 0.2 orbit(20 min.) settling time are given in Table 4-1, where the gains have been selectediteratively.As, expected the pitch response is driven to equilibrium orientation.The angular rates are also kept within the allowable limits of 0.2 deg/s.As ex-pected, the control effort reveals that the case with a settling time of 5 minutes,imposes the highest control torque (150 Nm in pitch) and angular momentum de-79Figure 4-2^Librational response of the FMC in Lagrange orientation to a pitchdisturbance of 10°80Figure 4-3^Librational response of the FMC in Lagrange Orientation fordisturbances of 10° in yaw and roll independently.81Table 4 -1 Typical numerical parameters for the control study (LQR).TYPICAL CLOSED LOOP GAINS FOR LAGRANGE ORIENTATION-------------..„...a -.802761E+08 -.388231E-07 -.192604E-08-.695312E-07 -.100103E+09 -.313927E+07-.4145502E-07 0.195826E+07 -.751463E+08-.353745E+08 -.133646E-07 -.530870E-09-.590766E-08 -.105975E+08 0.607783E+060.458071E-07 0.505759E+06 -.345348E+08TYPICAL CLOSED LOOP GAINS FOR DESIGN^ORIENTATION-.995574E+08 -.985051E-02 0.541533E-02.911092E-02 -.994216E+08 -.397904E+080.314057E-02 0.413105E+08 -.124493E+09-.105556E+08 -.217679E-03 0.834481E-02-.163591E-03 -.396923E+08 0.871128E+060.780127E-02 0.887655E+06 -.441554E+08TYPICAL CLOSED LOOP POLES^FROM RICATTI EQUATION-9.433 -9.433 -2.645-2.645 -2.645 -2.6819.474 -9.474 3.358-3.358 1.428 -1.42882mands (5,000 Nms) on the CMGs. The yaw degree of freedom required minimalcontrol, with QA = 50 Nm and ILA = 1, 000 Nms . This is expected since the mini-mum axis of inertia is aligned with the local vertical, an inherent characteristic of thegravity-gradient stabilized systems. The controller provides satisfactory closed-looptransient response, with the CMG momentum peak and torque demand well below thespecified limits. Note, the torque requirement about the yaw axis is lower than thatabout the pitch axis since the deviation from the equilibrium is small: Ao — A e = 10';— =11.47°Figure 4-5 shows the controlled and uncontrolled libration responses , as wellas the control efforts for the application of LQR to the design orientation of theFMC.(z,b = 10°) in the presence of flexibility. The three librational responses arequickly driven to the equilibrium position.Here , the controller is required to drive the system towards the design orientation,i.e. (0 = A = q5 = 0°), thus aligning the spacecraft axes with the orbital frame. Theparameters of interest include the angular attitude excursions, and the control torquedemanded. The fundamental system mode has been used to represent the deformationhistory of the FMC (fpi = 0.68 rad/s). The matrices Q and R are assumed diagonaland similar weightings as in the rigid case, are used for the three librational degreesof freedom. The yaw degree of freedom displays the shortest settling time (t, = 5minutes). also note that the, torque requirement for the yaw degree of freedom is theleast.The results show that the LQR control scheme tunes the open-loop unstable ,oscillatory motion which minimizes the control effort during the steady state opera-tions.4.3 Nonlinear Control83Figure 4 -4^The controlled and uncontrolled libration responses , as well as thecontrol efforts for the application of LQR to the Lagrange Orientationof the FMC^= A = = 100).84Figure 4-5^The controlled and uncontrolled libration responses , as well as theassociated control efforts for the application of LQR to the designorientation of the FMC.( 0 = 10°).85Next, the attention was turned towards accounting of the highly nonlinear sys-tem dynamics. Nonlinear control has received considerable attention in the roboticsresearch, particularly during the past decade. Control strategies based on linearizedsystem models have been found to be inadequate. One possible solution was put for-ward by Freund [27]. The idea is to use the state feedback to decouple the nonlinearsystem in such a way that an arbitrary placement of poles is possible. The tech-nique, however, was found to be difficult to apply to systems with more than threedegrees of freedom. Slotine and Sastry [28] applied the sliding mode theory to thecontrol of robot manipulators. However, unmodelled dynamics usually results in highfrequency oscillations of the manipulator. Slotine [29,30] improved the performanceby using a filtering process with a high bandwidth for the sliding variable. Slotineand Li [31] also incorporated the sliding mode control in an adaptive PD feedbackcontroller. Inverse control, based on the Feedback Linearization Technique (FLT),was first investigated by Beijczy [32] and used by Singh and Schy [33] for rigid armcontrol. Spong and Vidyasagar [26,34] also used the FLT to formulate a robust con-trol procedure for rigid manipulators. Using the FLT and given the dynamical modelof the manipulator, the controller first utilizes the feedback to linearize the systemfollowed by a linear compensator to achieve the desired system output. Spong [35]later extended the method to the control of robots with elastic joints. Advantages ofthis approach are twofold: (i) the control algorithm based on the FLT is simple; and(ii) the compensator design, based on a feedback linearized model, is straightforward.Recently, Modi et al. [36,37] extended the technique to include structural flexibilityfor a model of an orbiting manipulator system studied by Chan [38]. The procedureis found to provide adequate control for both rigid as well as flexible manipulators.4.4 Feedback Linearization Technique (FLT)86Consider a rigid system given bym(q,,t)4r, ^+ F(qf,4,,t)= Q(qr,4,,t) ,^(4.21)where q,. denotes rigid generalized coordinates and Q (q, , 4r , t) is the nonlinear control.Let the control effort have the formQ (qr , 4,,. , t) = M(q, , t)f) + F (q , , 4, , t) ,^ (4.22a)where f) = (4;-)d + Kv[(dr)d — dr] + Icp[(gr)d — dr] ,^(4.22b)With CM cl) (dr)d) and (4;)d representing the desired displacement, velocity and accel-eration, respectively. The nonlinear control when substituted into (4.21) results in alinear closed-loop system,4; = 13 ;^ (4.23a)Or [(6',)d — d;.] + Kv[(dr)d — dr} + Kp[(4r)d — 47.] = 0 .^(4.23b)Since E = (4r)el — 4r denotes the displacement error, Eq. (4.23) can be rewritten as+ Kv e + K p e = 0 .^ (4.24)The role of Kv and Kp is now apparent; they are position and velocity gains to insureasymptotic behaviour of the closed-loop system. A suitable candidate for Kp and Kvwould be diagonal matrices of the formrKp = w2n[ 2CD1Kv = (4.25)leading to a globally decoupled system with each generalized coordinate behaving asa critically damped oscillator. For attitude control of a rigid spacecraft, Kp and K vare 3 x 3 matrices for pitch, roll, and yaw degrees of freedom. In general, a larger value87Mrf 1{.qr 1 {Fr 1 {Q r[ MrrMfr Mff^of^Ff^Q f(4.26)of C.,,,, gives rise to a faster response of the n-th generalized coordinate. In general,the larger the gain, the faster the response; however, this does not necessarily implyhigher control effort.4.5 Implementation Of the Control for a Flexible SystemIn general, dynamics of a flexible spacecraft with 4, and 41 corresponding tolibrational and vibrational generalized coordinates, respectively, is governed byHere Mrr (qr ) is a 3 x 3 submatrix for the librational degrees of freedom; M rf(qT , qf), ofdimension 3 x Nq —3, represents the coupling between the rigid and flexible generalizedcoordinates; Mfr = MrfT ; and Mff(qf) is a Nq — 3 x Ng -3 submatrix for the flexibledegrees of freedom only. Pr and P.f are 3 x 1 and Ng — 3 x 1 vectors, respectively,representing first and second order coupling terms. Assuming only the librationaldegrees of freedom to be observable, the control force (2 f is not applicable and henceset to zero. The objective is to determine Q T such that the closed-loop system islinearized. Rewriting Eq. (4.26) into two sets of equations and with C2 f = 0 gives:1\4,r4, + Mrf -if + Pr = Q, ;^(4.27a)Mfg?. + Mff4:f + Pf = 0 j^(4.27b)which can be solved for qr and 4.f:AU + f = QT. ;^(4.28a)(if. = _mff—i mfr 4; — mff — 1 pf^(4.28b)where:^M = Mrr - Mrf Mff 1 Mfr88f = Pr - MrfMff 1 Ff •As in the case of Eq. (4.22), a suitable choice of Q,. would beQ T(4T) 4f)4T 74 f, t) = 4(4r ) q f ,^+ F(4T) qf , 4r , 4f , t))with v = (47 )d + Kv[(tr)d — 6.] Kp[(6.)d q'r]Now the controlled equations of motion become:4T = ;= — Mff 1 MfrV — Mff(4.29a)(4.29b)Note thatQT =M^ + + M(K,6 + Kp e) ,which can be visualized as a combination of two controllers: primary (Q r ,p ) andsecondary (Q T,,), whereQr,p = M(iir)d^; (4.30a)Q T,, = M(Kve Kp e) .^ (4.30b)The function of the primary controller is to offset the nonlinear effects inherent in therigid degrees of freedom; whereas the secondary controller ensures robust behaviouragainst the error.To evaluate the control effort Q T required, a priori information of M and F isneeded. In turn, calculation of M and F requires the knowledge of q f and (y .f. Tothis end, two schemes, Quasi-Open Loop Control (QOLC) and Quasi-Closed LoopControl (QCLC), are suggested by Modi et al. [36].894.5.1 Quasi-open loop controlHere , the flexible coordinates are evaluated off-line, i.e. integration of Eq. (4.29b)is performed independently and with 4, substituted with (47,) d . The main advantageof the scheme is a reduced computation effort. However, under this scheme, discrep-ancies between the calculated and actual flexible coordinates would exist. Hence, thesuccess of the scheme depends on the robustness of the controller.4.5.2 Quasi-closed loop controlUnder this scheme, both the rigid and flexible coordinates are evaluated simul-taneously. The disadvantage of the scheme is a relatively large computational effortas compared to the QOLC. On the other hand, the QCLC is less sensitive to systemuncertainties. In general, the rate of response was found to be essentially proportionalto the magnitude of the control gains. Advantages of the FLT approach include therelatively simple implementation of the controller as compared to the LQR technique;and the straighforward compensator design. Here, the FLT is applied to the FMC ofthe Space Station to achieve attitude control in the presence of structural flexibility.The basic idea here is to design a controller capable of transforming the rigidpart of the dynamics into a canonical, decoupled state space model. This, obviously,implies a completely controllable system.4.6 FLT Simulation Results and DiscussionThe FLT is applied to the FMC of the Space Station for the two spatial orienta-tions described earlier: the Lagrange ; and the Design orientation. Consider the rigidFMC. Three different cases of the controller frequency (x) for a critically dampedresponse (damping ratio C = 1) are considered and the performance is compared: (i)90x = 0.2 x 10 -2 rad/s, (ii) x = 5.0 x 10 -2 rad/s, and (iii) x = 2.0 x 10 -1 rad/s.For simplicity, the resulting gains Kp = x 2 E and K t, = 2CxE are chosen, where Eis the identity matrix. The quantities of interest include the angular displacementand the control effort histories. As shown in Figure 4-6 , even with a large initialangular disturbance of 10° to the system in pitch, roll and yaw, the controller is ableto regain the equilibrium attitude(O, = 1.5 ° , = 4', = 0°) in around 0.10 orbit(for x =0.05). Note, the character of the librational response is similar for the threedegrees of freedom. The controller's objective is to make the system exhibit a criti-cally damped behaviour, which depends on the assigned controller frequency x anddamping (. With the same set of gains in the three librational directions, the space-craft displays similar dynamic response in pitch, roll and yaw. The control torquesinvolved in accomplishing the proposed task are well within the allowable limits [22,23]. Also note that the control torque time histories for pitch and roll are essentiallythe same, because the inertias in pitch and roll differ by less than 1%.The effect of structural flexibility on the nonlinear attitude controller is investi-gated next. To that end, the fundamental system mode of the FMC (wi = 0.68 rad/s,), is used to represent the structural deformation. To keep the controller band-width well below the structural frequency , the controller frequency has been set atx = 10 -2 rad/s, which is one order of magnitude lower than that of the fundamentalsystem frequency. The parameters of interest include the libration of the spacecraft;and the control effort. (Figure 4-7 ). The settling time for the librational motionis around 10 minutes (C = 1). The angular velocities history also presents a similarprofile. The solar arrays are excited only slightly by the control torque. The controltorque profile in yaw is modulated by flexible deformation history. The control effortsin pitch and roll show a residual value in the steady-state regime, due to the flexibilityof the system.911111(Figure 4-6^The controlled response, and control effort for the FMC in LagrangeOrientation for application of FLT with initial disturbance Of 10° inpitch, yaw and roll,and different frequencies.92-^0.20.0Perigee15.05.05.0Orbit^0.5-^.......... . • • .,0Figure 4-7^The controlled response, and control effort for the FMC in LagrangeOrientation for application of FLT with initial disturbance of 10° inpitch, yaw and roll, with flexibility included .93Next the design orientation is considered with an initial disturbance of 10° inpitch yaw and roll. The controller damping is now set to ( = 0.1 while the frequencyx remains the same. The corresponding response is shown in Figure 4-8 The trendsare similar to as in Figure 4-7. however, the settling time rises to 15 minutes and thecontroller torque in pitch decreases. The control effort in pitch shows modulation,due to the structural deformation . It may be emphasized that these modulationswere confined to the pitch degree of freedom only, because the dominant motionof the solar panels induces a torque about the orbit normal. As the control effortdiminishes with attitude, so does the amplitude of vibration, though the rate ofdecay is minimal. Figure 4-9 shows the generalized vibrational coordinates after theapplication of FLT.As is evident the application of FLT does not destabilize theflexible response.Next, the effect of the damping on the controller's performance is investigatedusing the same damping ratios as before: C = 0.05; ( = 0.10; and C = 1.0. Asexpected, a lower damping ratio results in a higher control effort and a longer settlingtime. Typical settling times vary from 10 - 30 minutes with the control torque rangingfrom Qo = 3000 Nm to Q, = 1 Nm (Figure 4-10 ). Of course, a shorter settling timecorresponds to a larger control effort.When the rigid FMC is considered, with preset controller frequency x = 10 -2 rad/sand damping ratio C = 1.0, the response profiles exhibit analogous behaviour in allthree axes . Figure 4-11 shows FLT response of the FMC (rigid) in design configu-ration with initial conditions of 10° in pitch yaw and roll, and a preset frequency Of0.01 rad/sec. The minimum control torque is associated with the pitch axis. Due tosimilar inertias in pitch and roll, the control torques about these axes exhibit similartrends. Decreasing the damping ratio to C = 0.1 results in a longer settling time94Figure 4-8^The controlled response of the FMC in design orientation for theapplication of FLT, with a disturbance of 10° in pitch yaw and rollwith flexibility included.95Figure 4 -9^Generalized vibrational coordinates after the application of FLT tothe design configuration of the FMC, demonstrating the stability inflexible response.96Figure 4-10 Effect of damping on the controlled response and control effort forthe application of FLT on the design orientation of the FMC.97(ts = 40 minutes) and control torque (Qv, = .05 Nm). The low value of torques isexpected because of the low speed of response. Also note that the yaw degree offreedom requires the maximum torque.Figure 4-12 shows the response for two different frequencies for the design orien-tation. As can be expected, higher frequency means faster response, which impliesshorter settling time hence more control effort.4.7 SummaryPerformance of two distinct attitude control strategies is studied, with referenceto a specific configuration of the proposed Space Station. In the LQR approach, theequations of motion are linearized about a desired trajectory, and the optimal gainsdetermined so as to minimize a preformance criterion. The equations of motion wereaugmented to accounted for proper management of the angular momentum. Theeffect of structural flexibility was included in the simulation loop. The results showthat the controller tunes the FMC, which has a gravitationally unstable design inertiaorientation, to a stable, small amplitude oscillatory motion in the steady-state.The nonlinear control technique (FLT) was also found to be effective in control-ling the attitude motion of the FMC. The controller performance for three sets offrequency and damping was compared for the FMC. Even when the spacecraft is ini-tially disturbed by 10° in pitch, roll and yaw simultaneously, the resulting gains areadequate to bring the attitude error to within design limits in less than 5 minutes.In general, the rate of response is directly proportional to the magnitude of the con-trol gains. With the inclusion of flexibility in the FMC model, both the QOLC andQCLC were found to be effective. In general, the LQR provided stabilization withmuch lower control torques compared to the FLT, for a given settling time. it was alsoobserved that faster the rise time higher the control torque. It should be emphasised98Figure 4-11^FLT response of the FMC (rigid) in design configuration with initialconditions of 10° in pitch , and a preset frequency Of 0.01 rad/sec.99Figure 4 -12 Effect of frequency on the controlled response and effort for the ap-plication of FLT on the design orientation of the FMC with initiallibrational disturbance of 10° in pitch yaw and roll simultaneously100again that the objective here is not to obtain vast amount of response results foruncontrolled and controlled dynamics, but illustrate application of the methodologydiscussed before with reference to different orientations and control strategies. Forestablishing the effectiveness of a specific control procedure and its relative merit,further study is necessary with a variety of distubances, and accounting for systemparameter uncertainties.1015. ANIMATIONTo gain better insight into the physics of the problem and judge the efficacy ofthe controller, the controlled and the uncontrolled responses were animated on anSGI IRIS graphics workstation using VERTIGO animation package.5.1 Methodology of AnimationThree dimensional animation of a rigid object requires six variables as a functionof time,: three rotations and three components of translation. However, if the globalaxis is not the local axis then three extra translations are needed to fix the positionof the local frame with respect to the global frame.Each body that has a coordinateframe assigned to it is called an entity.A number of entities may be grouped togetherto form a group.The relation between entities in a group may be established in twoways :(i) by defining the motion of each of the entities with respect to a global frame.(ii) by specifying the relationship of entities in a group amongst themselves.5.1.1^Object definitionAn object or entity can be defined using the contours.The contours representthe basic cross-sectional shape of the object.Each object may be defined in termsof one or more contours.However, there must be a contour at the first level. Thisimplies that there must be a seed contour to start with.Each contour comprises ofat least a perimeter and an arbitrary number of holes (including none). Perimetersand holes are always closed polygons.When the shape of an object is defined by morethan one contours, each contour must have the same number of holes. Furthermore, in all contours, corresponding perimeters and holes must be defined by the same102number of vertices.To the anchor point (origin of the frame in which the contour isdefined) of each contour is attached a spine.The anchor point need not be containedin the contour.The spine is a three dimensional curve representing the "backbone"of the extruded object.A bend in spine is closely followed by the object.The spine isa mandatory component of any object. When the object is extruded , contours areattached to the object at location "levels". When an object has a closed spine, thecontours at the first and the last extrusion levels must be identical.Every contour inan object must comply with the following rules:(a) All contours must have the same number of perimeters and holes.(b) All corresponding perimeters and holes must have the same number of ver-tices. However for corresponding vertices, continuity, tension and bias canvary from level to level.(c) All corresponding curve segments must have the same number of steps.A path lime marks the beginning and the end of a spine. The path shape is subjectto the following restrictions:(i) The path is allowed to have only one value along the spine.This implies thatthe path is not allowed to loop back on itself .(ii) The path values may be defined only at levels on the spine.(iii) Vertices may not coincide anywhere in the extrusion.(iv) If the spine is closed , scale bevel and wobble points at the spine ends mustbe equal. However, twist values are allowed to differ by a multiple of 360degrees.A bevel path changes the size of the contour , along the spine, by moving all pointson the contour in or out by the same amount .However, the scale path changes thesize of the contour as it is obtained through scaling.The scale path must always be103greater than zero.Once the object is designed ,a number of features are defined for theobject under consideration.These include , rotational and transational parameters.5.1.2 Scene compositionOnce the number of entities in a scene is fixed, the object lighting is decided . Theobject lights are nothing but point sources of light placed at appropriate locations tolight up the scene.They are governed by the same set of rules as any other object.A time dependent transformation for each object is needed to display it's positionand orientation as a function of time. Here the transformation is defined as a setof operations , carried out on an object, light or group (set of objects) to changeit's position size and orientation. Each of the members (children) of the group hasit's own, associated set of transformations, as does the group itself. Since each childof a group has it's own transformations, it can be scaled , positioned and orientedindependent of the other children in that group.The children inherit the group leveltransformations, so that when the group is scaled or translated it affects the childrenaccordingly.Scene hierarchy is defined as the organization of all the entities, comprising thescene.The hierarchy is the overall organization of the entities in the scene being an-imated. The root or the highest level entity always contains the scene, directly orindirectly. The global group may contain any (including zero) number of entities,including lights, objects and other groups. However, groups may not contain them-selves. The relationship between entities to one another in a group is described usinggynealogical terms. An entity may or may not have siblings, where siblings are enti-ties with the same parent. An entity may or may not have children. If it does, it is agroup. If not, it is either an object or a light.There is a set of six fundamental operations which take care of all the translations104and rotations in a scene.These are :(i) Centroid The center of an entity is initially determined when it is created.The centroid transformation changes the position of the center by shifting theorigin of the local space of the entity. However, because rotation, scale andorbit take place about the centroid, these are the parameters affected duringthe centroid transformation .(ii) Scale changes the size of each entity, as well as the group in the three direc-tions. The scaling occurs about the centroid.(iii) Rotate causes motion of the entity about either of the x, y, z axes. Therotation also occurs about the centroid.(iv) Translate moves the entity along either of x,y or z axes. In this case as wellthe translation is measured from the centroid.(v) Orbit transformation is identical to the function rotate, but it is applied aftertranslate. The orbit transformation is applied first to the x-axis, then to they-axis and then to the z-axis.(vi) Visibility , as the term suggests, governs visual manifestation of an object.Visibility is inherited by children of the entity. An object on the screenis visible only if all of the ancestors in it's hierarchy are also visible. Thegeneral principle is that if a group is not visible then it's children are also notvisible; the status of the visibility transformation of the children in that caseis unimportant.Transformations in general follow the following two principals:(a) Children of a group, inherit the transformation of the group.(b) An entity may have a maximum of five transformations associated with it.1055.1.3 Transformation management and synchronisationTransformations applied to entities can be constant over time, or they may vary. If an entity does not change it's position, size, orientation or visibility , over theduration of the animation, it's transformations are constant. However if the entityis to move, rotate or change size in time then it is imperative to assign channels tothe transformations. A channel is a set of values representing the transformationvalues, one for each frame of the animation, and each potentially different from theothers. More than one transformation can be associated with a given channel, sincechannels are merely sets of numbers, they vary from frame to frame; different trans-formations can access these numbers and interpret them differently. For instance, atranslate channel could interpret them as distance units, while a rotate channel couldinterpret them as angles. The x, y, z components of each transformation are nor-mally described by seperate channels. The values to be associated with a channel canbe interpolated through smooth curves (splines) , straight lines (linear), or specifiedmathmatical functions. If the channel curve is described by a mathematical function,there are no keyframes ; the channel value for each frame is calculated directly fromthe mathmatical function. The time dependence of each of the variables associatedwith an entity is described as a function in a channel file.Once the number of framesis fixed , the number of time-steps is also fixed. The values for all the parametersare specified in the channel file associated with the particular parameter. Figure 5-1shows the animation methodology employed. The entities comprising the FMC areshown in Figure 5-2 .Since there are 5 bodies in the FMC , forty five channel files are required forthe whole configuration. However since only the rigid configuration is consideredhere, the relative orientation of the bodies is fixed. The use of this fact reduces the106Trans.! Trans.5Trans.2i Solar`Panel 1ChannelStingerChannelRadiatorChannelVertigoScene ScriptDynamics and Control Simulation ProgramGeneralized CoordinatesFigure 5 - 1^Flowchart showing the animation methodology employed107PV ArrayIIPV ArrayITransformationGroupFMCStinger PVARadiatorFirstMilestoneConfigurationFigure 5-2^Scene hierarchy for the FMC animation.108number of time dependent parameters to nine.Therefore nine channel files are neededto completely render the rigid system animation. The output from the dynamicssimulation program is piped into the channel files , once preprocessed by a suitableinterface. In the present case this is achieved through a C-program which transformsthe output into a suitable form usable by the animation software.1096. CONCLUDING REMARKS6.1 Concluding CommentsThe First Milestone Configuration of the proposed space station Freedom is takenas an example to study the dynamics and control of flexible spacecraft in orbit.Thegoverning equations are highly nonlinear, nonautonomous, and coupled. Two generalformulations developed by Ng[15] and Suleman[25] for the simulation of multibodydynamics of flexible spacecraft structures, with the system discretization, using thesystem and component modes, respectively, is employed in the present study. Basedon the analysis the following general remarks can be made:(i) The operational orientation of the FMC is slightly different from its equilibriumorientation. Furthermore, the equilibrium orientation itself is unstable. Hence inthe operational (i.e design) condition, the system begins to tumble in less thanone orbit, if uncontrolled. Inclusion of flexibility results in the coupling of theelastic response with the rigid modes.(ii) The response of the power boom is sensitive to the direction of the disturbance. Itleads to large amplitude pitch and roll responses with high frequency modulationswhich are undesirable. The truss inertia being comparable to the overall inertia ofthe spacecraft, any disturbance to the truss is likely to render the overall systemunstable.(iii)The flexible appendages in general (as against the main truss) and the solar panelsin particular dominate the flexible response.(iv) Stinger, like the power boom, is sensitive to disturbance.(v) Two control techniques, one linear and the other nonlinear, have been imple-mented. The Linear Quadratic Regulator (LQR) though designed around the110linearised equations of motion is found to be effective even with the flexibilityaugmented equations of motion. The Feedback Linearisation Technique (FLT) isimplemented as nonlinear control technique and it's effectiveness demonstratedin controlling the libration modes of the FMC. The FLT for the control of highlynonlinear, coupled systems seems promising. The method is rather straightfor-ward to implement and the control algorithm is simple. For the control of rigidFMC in the design orientation, using the FLT, the control torque required isminimal. In general the flexibility of the boom and the associated appendagesdo not influence the control effort significantly.It only means that a controllerwith a higher bandwith is required .In general the control effort with the FLT isindependent of the settling time.(vi) The Lagrange configuration, though nominally stable, becomes unstable for rathersmall disturbances in roll. In FLT Control for the flexible case, the control effortincreases for a decrease in the settling time, i.e for an increased frequency.(vii) In the case of the LQR for the FMC ( design as well as Lagrange orientations) thecontroller designed around the linearised equations of motions is able to handleboth,the nonlinearities and the flexibility quite well, without significantly affectingthe flexible response.(viii) The dynamical response of the rigid FMC is animated is animated, to gainbetter insight into the physics of the problem, for both the Lagrange as well asthe design orientations. The controlled responses are also animated using the SGIIRIS graphics workstation and the VERTIGO animation software. (This clearlydemonstrates effectiveness of the controller for both the orientations under study)1116.2 Recommendations for Future Work(a) Attention should be directed towards getting around the high frequency controleffort to make the controller more practical and realisable.(b) The effect of slewing maneuvers of the solar panels on the FMC is likely to be ofimportance and needs to be investigated.(c) Effect of the system orientation on the control effort and performance of thespacecraft should be studied more systematically in order to arrive at betterdesign for the operational configuration.(d) Animation of the flexible response and effect of the controller on the overall re-sponse needs to be rendered to gain better physical insight into the system, per-formance.112BIBLIOGRAPHY[1] Hughes, P.C., "Dynamics of a Chain of Flexible Bodies," The Journal of theAstronautical Sciences, Vol. 27, No. 4, October — December 1979, pp. 359-380.[2] Modi, V.J., and Ibrahim, A.M., "A General Formulation for Librational Dy-namics of Spacecraft with Deploying Appendages," Journal of Guidance, Con-trol and Dynamics, Vol. 7, No. 5, September — October 1984, pp. 563-569.[3] Balas, M.J., "Some Trends in Large Space Structure Control Theory: FondestHopes; Wildest Dreams," Proceedings of the 1979 Joint Automatic ControlsConference, Denver, Colorado, June 1979, pp. 42-53.[4] Meirovitch, L., and Oz, H., "An Assessment of Methods for the Control ofLarge Space Structures," Proceedings of the 1979 Joint Automatic ControlsConference, Denver, Colorado, June 1979, pp. 34-41.[5] Meirovitch, L., and Oz, H., "Modal-Space Control of Distributed GyroscopicSystems," Ignorable Coordinates," Journal of Guidance and Control, Vol. 3,No. 2, March — April 1980, pp. 140-150.[6] Meirovitch, L., and Oz, H., "Modal-Space Control of Large Flexible SpacecraftPossessing Ignorable Coordinates," Journal of Guidance and Control, Vol. 3,No. 6, November — December 1980, pp. 569-577.[7] Meirovitch, L., Baruh, H., and Oz, H., "A Comparison of Control Techniquesfor Large Flexible Systems," Journal of Guidance, Control and Dynamics,Vol. 6, No. 4, July — August 1983, pp. 302-310.[8] Oz, H., and Meirovitch, L., "Optimal Modal-Space Control of Flexible Gyro-scopic Systems," Journal of Guidance and Control, Vol. 3, No. 3, May —June1980, pp. 218-226.[9] Wie, B., and Bryson, A.E. Jr., "Pole-Zero Modeling of Flexible Space Struc-tures," Journal of Guidance, Control and Dynamics, Vol. 11, No. 6, Novem-ber — December 1988, pp. 554-561.[10] Wie, B., "Active Vibration Control Synthesis for the Control of FlexibleStructures Mast Flight System," Journal of Guidance, Control and Dynamics,Vol. 11, No. 3, May — June 1988, pp. 271-277.[11] Chu, P.Y., Wie, B., Gretz, B., and Plescia, C., "Space Station Attitude Con-113trol: Modeling and Design," AIAA Guidance, Navigation and Control Con-ference, August 1988, Minneapolis, Minnesota, Paper No. AIAA-88-4133.[12] Kida, T., Ohkami, Y., and Sambongi, S., "Poles and Transmission Zeros ofFlexible Spacecraft Control Systems," Journal of Guidance, Control and Dy-namics, Vol. 8, No. 2, March — April 1985, pp. 208-213.[13] Goh, C.J., and Caughey, T.K., "A Quasi-Linear Vibration Suppression Tech-nique for Large Space Structures via Stiffness Modification," InternationalJournal of Control, Vol. 41, No. 3, 1985, pp. 803-811.[14] Modi, V.J., and Brereton, R.C., "Planar Librational Stability of a FlexibleSatellites," AIAA Journal, Vol. 6, No. 3, March 1968, pp. 511-517.[15] Ng, C. ,Dynamics and control of Orbiting flexible systems, PhD dissertationApril 1992. University of British Columbia.[16] Modi, V.J. and Suleman, A., "System Modes and Dynamics of the ProposedSpace Station Type Configurations", Third Conference on Nonlinear Vibra-tions, Stability, and Dynamics of Structures and Mechanisms , June 25-27,1990, Blackburg, VA.[17] Hurty, W.C., "Dynamic Analysis of Structural Systems Using ComponentModes," AIAA Journal, Vol. 3, No. 4, 1965, pp. 678-685.[18] Blevins, R.D., Formulas for Natural Frequencies and Mode Shapes, Van Nos-trand Reinhold Co., New York, N.Y., 1979, pp. 108-109.[19] Hughes, P.C., Spacecraft Attitude Dynamics, John Wiley & Sons, New York,N.Y., 1986, pp. 18-22.[20] Moran, J.P., "Effects of Planar Libration on the Orbital Motion of a DumbbellSatellite," ARS Journal, Vol. 31, No. 8, August 1961, pp. 1089-1096.[21] Yu, E.Y., "Long-term Coupling Effects Between the Librational and OrbitalMotions of a Satellite," AIAA Journal, Vol. 2, No. 3, March 1964, pp. 553-555.[22] Space Station Engineering Data Book, NASA SSE-E-87-R1, NASA Space Sta-tion Program Office, November 1987.[23] Modal Analysis of Selected Space Station Configurations, NASA SSE-E-88-R8,NASA Space Station Program Office, Washington D.C., June 1988.114[24] Modi, V.J., and Ng, A.C., "Dynamics of Axisymmetric Satellite Under theInfluence of Solar Radiation: Analytical and Numerical Approaches," TheJournal of the Astronautical Sciences, Vol. 37, No. 1, January - March 1989,pp. 17-40.[25] Suleman, Afzal, Dynamics and Control of evolving Space Structures : An ap-proach with Application , PhD dissertation, August 1992. University of BritishColumbia.[26] Spong, M.W., and Vidyasagar, M., "Robust Linear Compensator Design forNonlinear Robotic Control," Proceedings of IEEE Conference on Robotics andAutomation, St. Louis, Missouri, March 1985, pp. 954-959.[27] Freund, E., "The Structure of Decoupled Nonlinear Systems," InternationalJournal of Control, Vol. 21, No. 3, 1975, pp. 443-450.[28] Slotine, J.E. and Sastry, S.S., "Tracking Control of Non-linear Systems us-ing Sliding Surfaces with Application to Robot Manipulators," InternationalJournal of Control, Vol. 38, No. 2, 1983, pp. 465-492.[29] Slotine, J.E., "Sliding Controller Design for Non-linear Systems," InternationalJournal of Control, Vol. 40, No. 2, 1984, pp. 421-434.[30] Slotine, J.E., "The Robust Control of Robot Manipulators," InternationalJournal of Robotics Research, Vol. 4, No. 2, 1985, pp. 49-64.[31] Slotine, J.E., and Li, W., "On the Adaptive Control of Robot Manipulators,"International Journal of Robotics Research, Vol. 6, No. 3, 1987, pp. 49-59.[32] Beijczy, A.K., Robot Arm Dynamics and Control, JPL TM 33-669, CaliforniaInstitute of Technology, Pasadena, California, 1974.[33] Singh, S.N., and Schy, A.A., "Invertibility and Robust Nonlinear Control ofRobotic Systems," Proceedings of 23rd Conference on Decision and Control,Las Vegas, Nevada, December 1984, pp. 1058-1063.[34] Spong, M.W., and Vidyasagar, M., "Robust Nonlinear Control of Robot Ma-nipulator," Proceedings of the 24th IEEE Conference on Decision and Control,Fort Lauderdale, Florida, December 1985, pp. 1767-1772.[35] Spong, M.W., "Modelling and Control of Elastic Joint Robots," Journal of Dy-namic Systems, Measurement and Control, Vol. 109, December 1987, pp. 310-319.115[36] Karray, F., Modi, V.J., and Chan, J.K., "Inverse Control of Flexible Orbit-ing Manipulators," Proceedings of the American Control Conference, Boston,Mass., June 1991, Editor: A.G. Ulsoy, pp. 1909-1912.[37] Modi, V.J., Karray, F., and Chan, J.K., "On the Control of a Class of Flexi-ble Manipulators Using Feedback Linearization Approach," 42nd Congress ofthe International Astronautical Federation, October, 1991, Montreal, Canada,Paper No. IAF-91-324, also Acta Astranautica, in press.[38] Chan, J.K., Dynamics and Control of an Orbiting Space Platform Based MobileFlexible Manipulator, M.A.Sc. Thesis, The University of British Columbia,April 1990.116


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