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An approach to dynamics and control of flexible space structure Chen, Yuan 1993

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AN APPROACH TO DYNAMICS AND CONTROL OF FLEXIBLESPACE STRUCTUREByYUAN CHENB. A. Sc. (Electrical Engineering), Shanghai Jiao Tong University,1982A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESMECHANICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1993C) YUAN CHEN, 1993In 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 ILl^cLoj n eo The University of British ColumbiaVancouver, CanadaDateDE-6 (2/88)AbstractA relatively general formulation for studying dynamics and control of a large class ofsystems, characterized by interconnected rigid bodies, beam and plate type structuralmembers forming a tree topology with two level of branching, is developed using theLagrangian procedure. The governing equations are discretized using two fundamentallydifferent approachs of system and component modes synthesis. Versatility of the formu-lation is demonstrated, through its application to two systems of contemporary interest,in the presence of nonlinear control: (i) Space station based, two-arm mobile, flexiblemanipulator; (ii) NASA proposed configuration involving a flexible slewing arm abode aflexible truss with a rigid antenna. Relative merit of the two discretization proceduresis assessed over a range of system parameters through comparison of the controlled anduncontrolled responses. Results suggest that the component mode synthesis, thoughrelatively easy to implement, can lead to inaccurate response unless the boundary con-ditions are modeled precisely. Unfortunately, this is seldom possible, particularly withcomplex systems of current interest. On the other hand, discretization through systemmodes, though precise, would require frequent updating leading to an increase in thecomputational time. The investigation represents an original contribution of far-reachingconsequence to the field. Such a planned approach to assess discretization methodologieswith reference to space based system has not be reported in open literature.Table of ContentsAbstractList of FiguresList of Tables^ viiiList of Symbols ixAcknowledgement^ xiii1 INTRODUCTION 11.1 Preliminary Remarks ^ 11.2 Background to the Problem ^ 21.3 Scope of the Present Investigation 62 KINEMATICS OF THE SYSTEM 82.1 Coordinate System ^ 102.2 Position of Spacecraft in Space ^ 132.3 Shift in the Center of Mass 162.4 Elastic and Thermal Deformations ^ 172.4.1^Background ^ 172.4.2^Deformation Expression for Beam-type Substructure ^ 192.4.3^Thermal Deformation ^ 202.4.4^Transverse vibration 212.5 Rotation Matrices  ^223 KINETICS OF THE SYSTEM^ 243.1 Kinetic Energy ^  243.2 Potential Energy  263.3 Equations of Motion ^  274 AN APPROACH TO CONTROL^ 314.1 Feedback Linearization Technique (FLT) ^  314.2 Control Implementation Procedures  355 COMPUTATIONAL CONSIDERATIONS^ 385.1 Program Structure of the Component Modes Method (CMM) ^ 385.2 Program Structure for the System Modes Method (SMM) ^ 406 RESULTS AND DISCUSSION^ 456.1 Two Link Mobile Servicing System  456.2 NASA's Cotrol-Structure Interaction Model ^  597 CONCLUDING REMARKS AND RECOMMENDATIONS FOR FU-TURE WORK^ 64Bibliography^ 66A DETAILS OF Tsys, In, AND Hn,^ 70B INPUT FILES FOR THE CMM 79C INPUT DATA FILES FOR THE SMM ^ 81ivList of Figures1.1 One of the earlier configurations of the proposed space stationFreedom . ^31.2 Schematic diagrams of the four milestone Configurations of the evolv-ing space station Freedom: (a) First Milestone Configuration (FMC) (b)Man Tended Configuration (MTC) (c) Permanently Manned Configura-tion (PMC) (d) Assembly Complete Configuration (ACC)   42.3 A system of interconnected bodies, forming two levels of branching, con-sidered for study. It can represent a vast variety of systems including theproposed space station ^92.4 Body fixed coordinate system Fp(Xp, Yp, Zp) and the solar radiation inci-dent angles.  ^112.5 Coordinate systems used to identify position of a mass element undergoingrigid body as well as vibrational motions and thermal deformations. . . . 122.6 Orbital parameters for a spacecraft defining position of its center of mass. 142.7 Modified Eulerian rotations specifying arbitrary orientation of the systemin space ^154.8 Block diagram of a feedback linearized control system ^ 344.9 Block diagram of the Quasi-closed loop control system  375.10 Program flow-chart for the CMM simulation ^  395.11 Flow-chart showing the numerical integration procedure using the CMM ^415.12 Program flow-chart for the system modes determination. ^ 425.13 Program flow-chart for the SMM simulation ^  436.14 Geometry of the two-link manipulator supported by a rigid space platform. 476.15 Effect of increasing the amplitude as well as speed of the slewing maneuver,in the plane of the orbit, with the MSS located at the center of the station:(a) 45° in 10 min; (b) 90° in 10 min; (c) 180° in 10 min.   506.16 Effect of the slewing speed on the system response with the manipulatorat the center of the main truss. The slewing maneuver of 180°(inplane) iscompleted in: (a) 10 min; (b) 5 min; (c) 2 min.   516.17 Response plots showing the effect of the manipulator's location: (a) Ma-nipulator at the center of the station; (b) manipulator at the tip of thestation. The slewing rate is 180° in 10 min.   526.18 Response plots showing the effect of the manipulator's location during theslewing maneuver of 180° in 5 min. (a) manipulator at the center of thestation (b) manipulator at the tip of the station   536.19 Controlled response using the FLT applied to the librational degrees offreedom with the manipulator at the center of the station: slewing ma-neuver of 45° in 10 minutes.   546.20 Controlled response using the FLT applied to the librational degrees offreedom with the manipulator at the center of the station: slewing ma-neuver of 90° in 10 minutes  556.21 Controlled response using the FLT applied to the librational degrees offreedom with the manipulator at the center of the station: slewing ma-neuver of 180° in 10 minutes.   566.22 System response in the presence of FLT control with the manipulatorlocated at the tip of the station: slewing maneuver of 180° in 10 minutes. 57vi6.23 System response in the presence of FLT control with the manipulatorlocated at the tip of the station: slewing maneuver of 1800 in 5 minutes. ^586.24 Geometry of the model proposed by NASA for the Control-Structure-Interaction (CSI) study.   616.25 Response of the NASA's CSI model during a 180° maneuver in 10 minutes:(a) discretization using the CMM; (b) discretization using the SMM. Themanipulator is located at the center of the truss.   626.26 Effect of the manipulator location on the system response during the 180°slewing maneuver in 10 minutes: (a) discretization using the CMM; (b)discretization using the SMM. The manipulator is situated at the tip ofthe truss.   63viiList of Tables6.1 Numerical values used in simulation of the two-link manipulator ^ 466.2 Numerical values used during simulation of the CSI configuration dynamics 60viiiList of Symbolsab^ beam radiusdi,j position vectors from 0 c to Oi and Oi to 0,respectively&Tic, drn„ dmij^elemental mass in body Bc, Bi, and Bi,j, respectivelyorbit inclination with respect to the ecliptic planekb^ thermal conductivity of an appendagedirection cosines of R,,, with respect to Xi,, Yp, Zpaxes1b^ beam lengththermal reference length of the beamlc, /i, li,j^ length of bodies Bc, 132 and B2,3, respectivelymb beam mass per unit lengthmc, mi, rn2,3^mass of bodies Bc, Bi and B2,3, respectivelynumber of 132,3 bodies attached to body Bivector representing flexible and rigid generalizedcoordinates(4r)d^^ vector representing the desired value of the rigid generalizedcoordinatessolar radiation intensity; W/m2timebV 7 w^ transverse vibration of a beam in its Y and Z directions,respectivelyC2`, Cfj^ transformation matrices defining orientation ofrelative to Fc, respectivelyixricmEIZZCcm Cf -Fecni—fCcni position vector from C2 to the instantaneous cenetr ofmass of the spacecraftposition vector from 0, to the center of mass of the undeformedspacecraftcenter of mass of the undeformed and deformedconfigurations of the spacecraft, respectivelyflexural rigidity of a beam about its Y and Z axes,respectivelyangular momentum of the spacecraft with respect to theXc,Y , Z, axes, respectivelyinertia matrix of spacecraft with respect to theZ, axes, respectivelyIxr, I,izz^principal inertia of body Bk about Xk,Yk, Zk axes,respectively; k = c,i or i,total mass of spacecraftN, Nj^ total number of B, and .132,,3 bodies, respectively0,, 0i, 0,,j origins of the coordinate axes for bodies Bc , BR: and B2,3respectively(2f,(2,^control effort vectors for flexible and rigidcoordinates, respectivelyposition vector from the center of force to theinstantaneous center of mass of the spacecraftRc, Hz, R.^^position vectors of the mass elements dm,, dm, and dm,,i,respectively, as measured from the center of forcepotential energy of spacecraft; Ue Ugunit matrixET,^ strain energy of the spacecraftUg gravitational potential energy of the spacecraftX0, Yo, Zo^ inertial coordinate system located at the center ofthe earthX p , Yp , Z^ coordinate axes with origin at Cf and parallel toxc, yo, oz,, respectivelyX8, Y3, Z,^ orbital frame with X, in the direction of the orbit-normal,Y. along the local vertical, and Z, towards the local horizontal8c, 8i, 6i,i^ vectors representing transverse vibration of the mass elementsdm,,dmi, dmi,j, respectivelyrotation about the local horizontal axis, Z1 , ofthe intermediate frame Xi,A^ rotation about the local horizontal axis, Y2, ofthe intermediate frame X2, Y2, Z2gravitational constantmatrix representing slewing motion of the body .132,39^ true anomaly03, On, slew angle and maximum slew angle,respectivelylongitude of the ascending nodePc, Pi, Pi,i^ vectors denoting positions of dm,, dmi and dmij, respectively,in the undeformed configuration of the spacecraft'is^ slewing periodTc, Ti, vectors denoting thermal" xideformations of dm, , dmi and dmi,j, respectivelyargument of the perigee pointlibrational velocity vector with respect to Xp, Yp,axesrotation about the orbit normal, X,Abrivations cm : center of massCMM : Component Modes MethodFLT : Feedback Linearization TechniqueMSS : Mobile Servicing SystemSMM : System Modes MethodDot (.) and (') represent differentiations with respect to time t and true anomaly 0,respectively.xiiAcknowledgementI would like to thank Prof. V. J. Modi for his guidance throughout my M.A.Sc programand the preparation of the thesis as well as many other academic and nonacademicaspects.I would like to express my appreciation to Dr. A. Ng, Dr. A. Suleman and Dr. H. Mahfor their help in understanding the previous related work.I would like to thank my wife Chao-ying for her unreserved support and understand-ing. Without it the completion of the thesis would have been impossible.This project was suported by the Natural Sciences and Engineering Research Councilof Canada, Grant No. A-2181; and Networks of Centers of Excellence Program, GrantNo. IRIS/C-8, 5-55380. Both the grants were held by Dr. Modi.Chapter 1INTRODUCTION1.1 Preliminary RemarksThe Unite States, together with Canada, Japan and the European Space Agency, hascommitted itself to the establishment of a space station by the turn of this century. It willbe used for scientific exploration, satellite launch and maintenance, manufacture of prod-ucts in the favourable microgravity environment, the Earth-oriented applied technologiesand numerous other applications.The primary design requirement for the space station is to provide a versatile, ex-pandable, permanent, manned facility for the tasks mentioned above. It will containlaboratories for a wide range of fundamental investigations. Furthermore, the space sta-tion will serve as a platform for satellite launch and repair; as well as assembly of spacestructures which may be too large, in terms of size and weight, to transport by the spaceshuttle or other launch vehicles available today.One of the space station configurations considered by NASA is shown in Figure ( 1.1).It had a 150m long main truss (power boom), aligned with the orbit normal with eightphotovoltaic arrays parallel to the local vertical, each extending to thirty-three metersand together generating 75 kW of power. The gimbaled solar array blankets providepower at any relative alignment of the space station with respect to the Sun, and heatrejection is achieved by nonrotating radiators. Habitation, laboratory and logistics mod-ules are located near the geometric center of the power-boom. The geometry of the space1Chapter 1. INTRODUCTION^ 2station has already gone through several modifications and, at present, its configurationis undergoing further revision. However, one thing is clear: the space station will be agigantic and highly flexible structure. It will be the largest platform (more than 100min length) ever assembled in space.Such a gigantic, massive structure, however, cannot be carried in its entirety to theoperational orbit. It will have to be constructed in space by integration of hardwaredelivered by a number of flights of the space shuttle. For instance, the the first spaceshuttle flight will deliver four truss bays of the power-boom, two of which are outboardof the articulating alpha joint, which allows the photovoltaic solar arrays to track theSun. The hardware delivered will also include a radiator, two Reaction Control System(RCS) modules, fuel storage tanks for flight control and reboost, and limited avionicand communication equipment. Once assembled, it will be a fully functional spacecraft,awaiting the second shuttle flight to progress to the next stage of the assembly sequence.NASA has identified, for such an evolving structure, four milestone configurations asshown in Figure ( 1.2).Given the large size of this orbiting system and the expected growth from the initialconfiguration, the structural flexibility will be a key parameter governing the space stationdynamics and control. The presence of environmental and operational disturbances willonly add to the complexity of the problem. Hence, thorough understanding of interactionsbetween librational and flexible body dynamics is of importance, for the appropriatecontrol system design, to attain the desired performance.1.2 Background to the ProblemHistorically, experiments with scale models have been routinely carried out to obtainuseful information for the prototype design. One is, therefore, tempted to use similarSPACE STATION 'FREEDOM'Figure 1.1: One of the earlier configurations of the proposed space station FreedomPV ArrayV*4PV RadiatorStation RadiatorLaboratory ModuleStinger/Resistojet(b)PV ArrayLaboratory ModuleLogistic ModuleHabitation ModuleJapanese ModuleESA ModulePV RadiatorStation RadiatorStinger/Resistojet(d)PV ArrayLaboratory ModuleLogistic ModuleHabitation ModulePV RadiatorStation RadiatorStinger/Resistojet(c)Chapter 1. INTRODUCTION^ 4Figure 1.2: Schematic diagrams of the four milestone Configurations of the evolvingspace station Freedom: (a) First Milestone Configuration (FMC) (b) Man Tended Con-figuration (MTC) (c) Permanently Manned Configuration (PMC) (d) Assembly CompleteConfiguration (ACC).Chapter 1. INTRODUCTION^ 5approach for the design of space-based structures. In fact, facilities attempting to simu-late some aspects of space environment have proved useful in testing of relatively smalland essentially rigid spacecraft. However, ground-based experiments with large flexi-ble space structures have been found to be of limited value due to practical difficultyin simulating environmental effects such as gravity gradient, magnetic, free molecular,microgravity, solar radiation, etc. This has led to increased dependence on numericalmethods, particularly with larger, extremely elastic and complex space configurations. Ageneral formulation applicable to a large class of systems is always attractive. Once thegoverning equations of motion are established and the associated integration program isoperational, it becomes a powerful versatile tool.Over the past three decades, Modi et al. as well as number of other investigators haveattempted to obtain relatively general formulations for progressively complex systems tostudy their dynamical behaviour [1]- [8]. this has also helped in development of severallinear as well as nonlinear control strategies. Of course, a number of other investigatorshave also approached this class of problems in a variety of different ways [9]. This has ledto the generation of a vast body of literature which has been reviewed quite effectivelyby Modi [10] [11], Ng [12], Suleman [13], and Mah [14].In the above mentioned developments, focus has been on the systems characterizedby a large number of interconnected flexible bodies forming a tree topology. In general,dynamics of such systems is governed by a set of 'hybrid', nonlinear, nonautonomousand coupled equations of motion. Here 'hybrid' refers to the set containing both partial(elastic motion) as well as ordinary (rigid body motion) differential equations. To helpobtain useful information with relative ease, it is conventional to discretize the partialdifferential equations into an ordinary set. This is achieved by representing the elasticdeflections through a series of time dependent generalized coordinates and spatially vary-ing admissible functions, ideally satisfying geometric and natural boundary conditions.Chapter 1. INTRODUCTION^ 6There are two fundamentally different choices for the admissible functions: componentmodes, which try to simulate the system behaviour through synthesis of the local com-ponent behavior; and system modes which attempt to accomplish the same but throughthe consideration of more global behaviour. Although component modes and their variedforms of synthesis have been used in practice quite widely, as shown by Suleman [13],through a simple example, there is a serious doubt as to the validity of the approach. Themajor problem is associated with the difficulty in satisfying complex boundary conditionsin multibody systems. As future space structures are going to be highly flexible and canonly be represented as interconnected multibody systems, the choice of modal functionsfor discretization has become an issue of enormous importance. This is understandableas validity of the dynamical response and control results depend on the accuracy of thediscretization approach.1.3 Scope of the Present InvestigationWith this as background, the thesis aims at analyzing, understanding and hopefully arriv-ing at better appreciation of the problem of discretization through a systematic study at afundamental level. To begin with kinematics and kinetics of a flexible, multibody systemwith two levels of branching is described and the governing equations of motion obtainedusing the Lagrangian procedure. A computer code capable of accommodating discretiza-tion through both Component Mode Method (CMM) and System Mode Method(SMM)is developed as an extension of the contributions by Ng [12], Suleman [13] and Mah [14].Two systems of contemporary interest are considered to asses comparative response:(i) two-link Mobile Flexible Manipulator(MFM) operating on a space platform;(ii) configuration proposed by the Control-Structure Interaction (CSI) program at NASALangley Research Center [15].Chapter 1. INTRODUCTION^ 7Performance of the above two configurations is studied over a range of importantsystem parameters to have some appreciation of the conditions leading to unacceptableperformance. Finally, a control strategy, based on the Feedback Linearization Technique(FLT) accounting for the complete nonlinear dynamics of the system, is developed and itseffectiveness assessed for the libration/vibration control of the above two configurationsusing both the discretization procedures. The thesis ends with concluding comments andrecommendations for future work in this area.Such a comparative study of discretization procedure, as applied to multibody systemswith tree topology and in the presence of nonlinear control, has never been reportedbefore.Chapter 2KINEMATICS OF THE SYSTEMThe system model is identified first and reference coordinate systems explained. Thisis followed by a kinematic study aimed at establishing position and velocity of an ar-bitrary element of the system under consideration in terms of specified and generalizedcoordinates.To help study a wide variety of systems, a relatively general model of interconnectedbeam and plate type members, interconnected to form a tree topology, is considered inFigure ( 2.3). The system is in an arbitrary orbit about the center force, which is alsothe origin of the inertial reference from F0 comprised of coordinates X0, Yo, Z0. Cf is theinstantaneous center of mass of the system. It consists of the central body B, to whichare connected structural members Bi referred to as B1, B2, • • • , BN. This may be lookedupon as the first level of branching from the trunk of a tree (main body). In turn eachBi body (i = 1, 2,       , N) is connected to Bi,3 bodies (j = 1, 2,     ,ni), the second levelof branching. Thus the number of bodies forming the system are 1 N E n3.j=1The number of members N and ni as well as their positions are kept arbitrary. Fur-thermore the members are free to undergo arbitrary translational and rotational maneu-vers at the joints. This permits simulation of a vast variety of systems of contemporaryand future interest. For instance, to simulate the proposed space station Freedom (presentconfiguration which is undergoing changes), the central body Bc may simulate the maintruss with the modules, stinger, radiators, photovoltaic arrays and manipulator repre-sented by Bi and B2 bodies as shown inFigure ( 2.3)8-NFlexibleBody B FlexibleBody B 1*\ OrbitNFlexible \\Body B1,n1 \\\FlexibleBody B2CentralBody BcYo/ 0R cmFlexibleBody BNFlexibleBody B11Chapter 2. KINEMATICS OF THE SYSTEM ^ 9PerigeeFigure 2.3: A system of interconnected bodies, forming two levels of branching, consideredfor study. It can represent a vast variety of systems including the proposed space station.Chapter 2. KINEMATICS OF THE SYSTEM^ 102.1 Coordinate SystemConsider the system model as described before. Let X0, Yo, Zo be the inertial coordinatereference located at the center of the earth. The centers of mass of undeformed anddeformed configurations of the system are located at C2 and c, respectively. Thereis a body coordinate system attached to each member of the model which is helpful indefining relative motion between the members. Thus, reference frame 11 is attached toBc at an arbitrary point O. Frame Fi, with origin at 02, is attached to bodies B, at theconnecting point between body Bi and B. In addition, for defining attitude and solarradiation incidence angles, a reference frame is located at Cf such that the axes Xi,, Ypand Zp are parallel to Xc, Y and Zc, respectively as shown in Figure ( 2.4).Now, an arbitrary mass element dmi on body Bi can be reached through a direct pathfrom 0, via Oi. Oc, in turn, is located with respect to the instantaneous center of mass Cfand the inertial reference frame Fo. Thus, the motion of dmi, caused by librational andvibrational motions of Bc and Bi, can be expressed in terms of the inertial coordinatesystem. Similarly, frame Fi,i is attached to body Bi,i and has its origin at the jointbetween Bi and Bi The relative position of Oi with respect to 0, is denoted by thevector di, while c12,3 defines the position of O relative to O.The location of the elemental mass of the central body, dmc, relative to Oc is definedby pc +77--, +Sc. Here pc indicates the undeformed position of the element; Tc, the thermaldeformation; and finally Sc expresses the transverse vibration of the element. Similarly,T-i and S. define the location of the elemental mass dm„ in body B,, relative to Oi. For the elemental mass dm,,i of body B,,j, its position relative to 0,,j is defined byand Si J. The coordinate systems are shown in Figure (2.5).Orientation of the coordinate axes Xi, Y„ Zi and X, Y ,3, Z relative to Xc, Y, Z, isdefined by the matrices CT and CZ,pi,j, respectively such thatChapter 2. KINEMATICS OF THE SYSTEM^ 11Figure 2.4: Body fixed coordinate system Fp(Xp, Yp, Zp) and the solar radiation incidentangles.Chapter 2. KINEMATICS OF THE SYSTEM ^ 12Figure 2.5: Coordinate systems used to identify position of a mass element undergoingrigid body as well as vibrational motions and thermal deformations.Chapter 2. KINEMATICS OF THE SYSTEM ^ 13= TL, = CiciLi =^ (2.1)where^is the matrix denoting the motion of body B2 ,3 relative to body Bi; 774 (k =p, c,i, or i , j) is the unit column vector in the corresponding coordinate axes. For instance,= {;:cjc, n4criT. It should be noted that the thermal deformation and transverse vibra-tion of B, and Bi result in the time varying characteristics of C. and C, respectively.2.2 Position of Spacecraft in SpaceConsider an arbitrary spacecraft in orbit, as shown in Figure ( 2.6), with its instanta-neous center of mass at Cf . At any instant, the position of Cf is determined by theorbital elements p,i,w, c, R77, and O. Here 8 is the longitude of the ascending node; i,the inclination of orbit with respect to the ecliptic plane; co, the argument of the perigeepoint; c, the eccentricity of the orbit; R, the distance from the center of the earth toCf ; and 9, the true anomaly of the orbit. In general, p, co and c are fixed whileand 19 are considered, approximately, functions of time.As the spacecraft has finite dimensions, i.e. it has mass as well as inertia, it is alsofree to undergo librational motion about its center of mass (in addition to the orbitalmotion). Let X3, Y5, Z, represent the orbital frame with coordinates aligned along theorbit normal , local vertical, and local horizontal, respectively. Any spatial orientation ofXp, Yp, Zp with respect to X8, Y3, Z, can be described by three modified Eulerian rotationsin the following sequence: a pitch motion, 7,b, about the X5-axis giving the first set ofintermediate axes X1, Y, Z1; a roll motion, 0, about the Z1-axis generating the second setof intermediate axes X2, Y2, Z2; and finally a yaw motion, A, about the Y2-axis yieldingXp, n and Zr,, as shown in Figure ( 2.7).Chapter 2. KINEMATICS OF THE SYSTEM^ 14Figure 2.6: Orbital parameters for a spacecraft defining position of its center of mass.Z , Z2Z sxs x 1Local HorizontalYlz^Local VerticalNChapter 2. KINEMATICS OF THE SYSTEM ^ 15Figure 2.7: Modified Eulerian rotations specifying arbitrary orientation of the system inspace.Chapter 2. KINEMATICS OF THE SYSTEM^ 16From the system geometry and Eulerian rotations, the librational velocity vectoris given byc7) = [--4. sin A + (e -F1j)) cos cos Arip + R — (è -c.b) sin Orip-F[cos A + (à +1.k)cos Osin A]rcp,^ (2.2)where 0 represents the orbital rate of the spacecraft.2.3 Shift in the Center of MassInstantaneous position of the center of mass Cf serves as an important reference point inidentifying position of the system in orbit. However, it is affected the system flexibilityas well as translational and slewing motions of the appendages. Hence, determination ofits position, at a given instant, is important [16] [17].In Figure (2.5), C2 and Cf represent the centers of mass of the system during un-deformed and deformed conditions, respectively. The vector Cf , denotes the positionof Cf relative to Ct. Thus it represents the shift in the instantaneous center of mass ofthe spacecraft due to its deformation. This information is necessary in evaluation of thekinetic and potential energies of the system.The vector from the origin of inertial coordinates, the center of the earth, to the masselements dmc, dmi and dmij are denoted by Rc,Ri,Ri,j, respectively. These vectors canbe written as:R,,i — Cf —^+ To, + "Tc + 8c;^ (2.3)Ri^Rcm — Cf^+ +^+ + Si); (2.4)= R — Cf^+ d +^ Tij^ (2.5)Chapter 2. KINEMATICS OF THE SYSTEM ^ 17Taking moment about the center of force gives1R = — {I Rcdmc+ E^Ridmi E fidmi,j1}.^(2.6)cm M^j=1 midSubstituting Eqs.(2.3)-(2.5) into Eq.(2.6) results in1Can=wherecicm =^+ offinc c^c^c i=1(7)- +77-- + ) dm + E {f^+ Cf(Pi^Si)dmini+ E f + C:di,j  ^ ,J=1 mid(2.7)Ccm- = position vector to Ci, the center of mass of the undeformed spacecraft, withrespect to Oc;—Ccnif = position vector to Cf, relative to Ci;ni = number of Bi,i bodies attached to body Bi;N = number of Bi bodies;M = total mass.2.4 Elastic and Thermal Deformations2.4.1 BackgroundIn recent years, greater emphasis has been placed on the design of high-speed, lightweight,precision mechanical systems. These systems, in general, incorporate various types ofChapter 2. KINEMATICS OF THE SYSTEM ^ 18driving, sensing, and controlling devices working together to achieve specified perfor-mance requirements under different loading conditions.In many of these applications, systems cannot be treated as collection of rigid bodies;i.e. flexible character of the system must be accounted for. In such cases, a mechanicalsystem can be modeled as a collection of both rigid and flexible bodies, or an entirelyflexible system depending on the situation. The flexible members may be representedas beams, plates, shells, membranes or their combinations. The design and performanceanalysis of such systems through dynamic simulation can be achieved provided the de-formation effect is incorporated in the mathematical model.The motion of a rigid body in the multibody system can be described by six co-ordinates defining its translation and rotation. They lead to a set of six independentsecond-order differential equations of motion.The exact configuration of a deformable body, however, can be identified only byinfinite number of coordinates. Dynamics of such continuous systems leads to a set ofpartial differential equations of motion which are both time- and space-dependent. To getmeaningful information about the complex system behaviour, it is convenient to convertthe mathematical representation into a set of ordinary differential equations by specifyingdeformations in terms of admissible functions of space and time dependent generalizedcoordinates. Admissible functions should satisfy as many natural and geometric bound-ary conditions as possible, and often structural modes are used to that end. However,the choice of structural modes which will converge to the right answer is still a subject ofconsiderable controversy particularly for complex, highly flexible systems with ill-definedboundary conditions. Broadly speaking, the choice of modes can be classified into twocategories leading to two different methods for studying system dynamics and control(i) modes defining motion of the entire system vibrating in unision, normally obtainedChapter 2. KINEMATICS OF THE SYSTEM^ 19through finite element approach [13], referred to here as the System Mode Method(SMM);(ii) simpler, individual component modes appropriately synthesized to represent the sys-tem behaviour [12] called the Component Mode Method (CMM).In the SMM, the structure is first subdivided into finite elements with degrees offreedom at the nodes. Using the local degrees of freedom as generalized coordinates, themass and stiffness matrices of the element can be derived readily. Applying the boundaryconditions for the system and compatibility requirements between adjacent elements, thesystem mass and stiffness matrices can be assembled from the corresponding matrices ofthe elements. The system modes can be evaluated numerically by using the conventionalfinite element method.In the CMM, the system's flexural motion is represented in terms of the components'dynamics. The first step is to obtain the series of admissible functions, by solving theeigenvalue problem for each component, representing its elastic deformation. Next, theadmissible functions are assembled so that the individual member dynamics becomeshomogeneous constituent of the total system response.2.4.2 Deformation Expression for Beam-type SubstructureThe governing equation for transverse vibration of a thermally fixed beam is given by(942 52mb^522Eh, ^+ m ^t + Mb = 0,^(2.8)ax4 ax2^aowith the appropriate boundary conditions. As an example, for a cantilever beam fixedat x = 0 and free at x = lb the boundary conditions are:Chapter 2. KINEMATICS OF THE SYSTEM^ 20^Wb -49C4ib = 0,^at x = 0;^(2.9)axa2wb^a3wb a AlbEIb^ .114 = E ^ t 0 at x = lb.^(2.10)ax2 ax3^axHere E/b is the bending stiffness of the beam; mb, the mass per unit length; and /14 thethermal bending moment given by.11/4 =EatT(x,y, z)zdA;.Area(2.11)where T(x, y, z) is the difference between the ambient temperature and the temperatureat a point on the substructure with coordinate (x, y, z); and af is the thermal expansioncoefficient of the beam material. The integral is over the cross-sectional area of thesubstructure.In general, a closed-form solution for the system is not available. The problem is over-come by assuming the thermal and elastic deformations to be uncoupled, i.e. the solutionfor thermal displacements can be obtained independent of the elastic displacements.2.4.3 Thermal DeformationThe effect of thermal deformation on transient dynamics of large space structures canbe significant, however, the associated analysis, in general, would be quite formidable.Fortunately, as shown by Modi and Brereton[1] the time constant of the heat balanceequation is quite small, i.e. the system attains the deformed state almost instantaneously.The steady state solution shows that the shape function for a thermally flexed free-freebeam is give by:8 = —ln[cos (1)] cos^ (2.12)Chapter 2. KINEMATICS OF THE SYSTEM ^ 218 = —in icoslt, (\n )lcos z; (2.13)where[s^al^ + zicbc-t (qsas )3/4 (8^eb .)1aaqq, 7rebab)^4 — Eb )2ab^kbbb (2.14)Here:Si,; biz = deflections in the transverse Y and Z directions, respectively;Oy; q = solar radiation incidence angles with respect to appendage reference Y and Zdirections, respectively;the distance along the appendage axis from the center point;/g = the thermal reference length of the appendage. It is a function of the solar radia-tion intensity(q,), Stefa-Boltzman constant(crt), appendage dimensions (ab, bb), andappendage physical properties (as, at, eb, kb).Ng and Modi [18] showed that the solution to Eqs.(2.12) and (2.13) can be approxi-mated with an error of 3% for niit < 0.6:2ay^1 (770— — —^s y;it,^2^lt,) (2.15)2Sz^1 (ri—) cos Os.^ (2.16)/t,^2^/gThus, for instance, the thermal deformation of a beam-type body Bi can be simulated as2^x•-= {0, --() COS^(-12 COS Oz,i}T.2 1:^"1 2 /7(2.17)Chapter 2. KINEMATICS OF THE SYSTEM^ 222.4.4 Transverse vibrationThere are numerous possibilities for selection of the admissible functions. Consider, forexample, the body .13c, a free-free beam, the admissible functions may be given by:Or(x) = cosh (f3r32-) + cos (T) — -yr [sinh (f3r5r) + sin (r-/31th^ib^lb(2.18)r =1,2, --;where Or/b is the solution of the equationcosh (3r) COS (13r) —1 = 0;^ (2.19)and -yr is given byr^sin (,8r)^sinh (13r) (2.20)7^— COS (f3r) + cosh (Or)  For B. and Bi j treated as cantilever beams:Or(x) = cosh (Prl — cos (Or^— -yr {sinh (f3r1 — sin (or-x—tb(2.21)r = 1,2,—  ;where Or is the solution of the equationcosh (Or) cos (Or) + 1 = 0;^ (2.22)and -yris given by= sin Pr) sinh (3")) „yr COS (Or) — cosh (0*)(2.23)2.5 Rotation MatricesMatrix CT appearing in Eq.(2.1) denotes orientation of the frame Fi relative to the frameF. Two rotation sequences are needed to determine C. The first one, Cic'r, definesChapter 2. KINEMATICS OF THE SYSTEM^ 23the rigid body orientation of Fi with respect to Fc; while the second one, CT's, specifiesrotation of the frame Fi relative to Fl due to elastic and thermal deformations of thebody B. The Eulerian rotations have the following sequence: rotation 07.' about theXcaxis, followed by Our about 1''-axis; and finally Orz about Z'-axis; i.e.cos In — sin Orzsin g cos go0 cos Byr 0-sin 19; 1 0 00 0 1 0 0 COS Orr - sin 8;1 — sin g 0 cos OrY _ 0 sin 8: cos Br.cos 9; cos erz sin g sin g cos erz — cos 9: sin trz cos In cos trz sin 9; + sin 8; sin Brzcos 9; sin 19; sin 8; sin g sin g + cos g cos erz cos In sin 9; sin Brz — sin 8; cos Orz— sin Of sin en cos 8; cos in cos 0;(2.24)Similarlyc:" =cos 9t cos 0:: sin 01 sin^cos 9-zf — cos O sin 19:7ze cos 9 cos 0-zf sin 9-;` + sin 0.1 sin 19-Icos 19:: sin 81 sin 01 sin q sin 0:: + cos 0-1 cos t91 cos 9 sin 19L` sin 19-zf — sin 8-1 cos 0-zf— sin 19f^sin q cos eifi^ cos 81 cos Oil; (2.25)Finally,= c7,f x (2.26)Chapter 3KINETICS OF THE SYSTEMWith the kinematics of the system established, the next logical step is to obtain thekinetic and potential energies of the system model under consideration. Application of theLagrangian principle will then provide the equations of motion, one for each generalizedcoordinate, governing the system dynamics.3.1 Kinetic EnergyThe kinetic energy, T, of the system is given by1^ niT = — Rclicdmc E [f Rdmi E I Rijjdmi,]} ,2^nic^ i=1(3.27)where Rc,^and Ri are obtained by differentiating Eqs.(2.3)—(2.5) with respect totime:=^cm—af —Ci +.t-c-1-6.c-Fox(—Cf —C:,„+pcd-T-c-1-8c); (3.28)=^-R1,3 =-Fc7) x [—Cf — Ci + + C^+^82)];^ (3.29)+(azikti,i +^+^+^+^+ si,J)x [-Cc _ cii +^CZidi CZikti,;(A^T-i,i 82,3)].^(3.30)24Chapter 3. KINETICS OF THE SYSTEM^ 25Substituting Eqs.(3.28-3.30) into Eq.(3.27), the kinetic energy expression can written inthe formT = Tort) + [T,„ Th T jr Tt^+ Th,jr Th,t Th,v T,r,t Tint, Tt,e]1+—ZiiT Ysc7.; 4.4771181,8, (3.31)2 Or1 ,T = Torb Tay, + —C7i' 1580 + H sys ,^ (3.32)2^Ywhere (47 is the librational velocity vector; Ins, the inertia matrix; Hsu, , the angularimomeutum with respect to the I', frame ; orLy.0 ,the kinetic energy due to pure ro-2tation ; and C.-vT1/88 , the kinetic energy due to coupling between the rotational motion,transverse vibraion, and thermal deformation. Tsys represents the kinetic energy contri-butions due to various other effects contributed by flexibility, shift in the center of massand assoiated coupling terms. The subscripts involved are as follows:orb = orbital motion;cm = centre of mass motion;h = hinge position between body Bc and Bi or between body Bi and Bi,j;jr = joint rotation due to elastic and thermal deformation;t = thermal defomation;v = transverse vibration.Chapter 3. KINETICS OF THE SYSTEM^ 26For instance, Tt,,, refers to the contribution of kinetic energy due to the rate of thermaldeformation and transverse vibration velocity. Details of the kinetic energy expressionare given in Appendix A.isy, and Hsys represent the inertia and angular momentum vectors of the system,respectively. They are time dependent and can be given by:Lys^Ian + + 1r + +^+ Ih,t +^+ + (3.33) /Lys -,-- Hem+ H h + H^Ht+ He+ 1 kir + 1 h,,. H h,t H+H,.,t^H,.,„^Ht,„,. (3.34)Here subspript r denotes contribution from the rigid body components. The details ofLys and Hn, are also given in Appendix A.3.2 Potential EnergyThe potential energy , U, of the system has two sources: gravitational potential energy,Ug; and strain energy, Ue, due to transverse vibration and thermal deformation,U = Ue Ug.^ (3.35)The potential energy due to the gravity gradient is given by= { f dmc rn dmi^dmijil^Rc^i=1^i^j=1 mij 14JSubstituting the expressions for Rc, Ri and Rij from Eqs.(2.3)-(2.5) and neglecting theterms of order 1/R4 and higher, Ug can be written as(3.36)M Pe Ug= ficm^2R3 trans[IeY e] +  314 -IT I 12R3^ssY (3.37)h2R=/4(1^c cos 8)' (3.41)Chapter 3. KINETICS OF THE SYSTEM ^ 27where tte is the gravitational constant and 7 reprensents the direction cosine vector ofltm with respect to the Xp, Yp, zp axis,1= (cos 7/) sin 4. cos A + sin sin A)i, + cos '17b cos g — c^(cos "1k sin 4' sin ). — sing') cos AP-cc.^ (3. 38)The strain energy expression for a beam isUe,beam = 1 I EIy a2 wb) 2 ETI a2 vb 2vzz  ^dlb ,^(3.39)—2^dx2^ax2where EIvy and EIzz are the bending stiffnesses of the beam about the Y and Z axes,respectively.3.3 Equations of MotionUsing the Lagrangian procedure, the equation of motion can be obtained fromd aT aT au— —aq + —aq Fq,^ (3.40)where q and Fq are the generalized coordinates and generalized forces , respectively.Normally, the effect of librational and vibrational motions on the orbital trajectory issmall. Hence , the orbit can be expressed by the classical Keplerian relations:R9 = h; (3.42)where h is the angular momentum per unit mass of the system; ice is the gravitationalconstant; and E is the eccentricity of the orbit. The general form of the equations ofmotion can be written asChapter 3. KINETICS OF THE SYSTEMMr^Mr, fof,01/AilCbCA KA QA28171, Cg, Kg, Qq,Mf,r^ mf//qn, Cg,„ Kg„.. Qq.,(3.43)OrM(q)re C(q, q', 0) K(q, 0) = Q(0),^(3.44)where primes denote defferentiation with respect to the true normaly 0. Note, n, rep-resents the number of vibrational degrees of freedom. Hence, the total number of gen-eralized coordinates Ng equals the three librational coordinates IP, q5, A plus nv. M is anonsingular asymmetric matrix of dimension Ng x Ng. The entries in M come from thesecond order terms of dId0(OT I aqi). C is a Ng x 1 vector. The terms here are derivedfrom two sources: Coriolis contribution of dId0(37115q9; and from 871/aq' . K, alsoa Ng x 1 vector, denotes the stiffness of the system. It is composed of the terms fromÔU/öq.Q, the generalized force vector of dimension Ng x 1, is evaluated using the virtualwork principle. Nonlinear entries in M together with nonlinear and time varying compo-nents of C, K and Q result in a set of coupled, nonlinear and nonautonomous equationsof motion.As pointed out before, the equations of motion are applicable to a wide variety ofsystems because of the relatively general character of the model. They range from com-munication satellites to the evolving space station Freedom and robotic manipulators,Chapter 3. KINETICS OF THE SYSTEM^ 29to mention a few configurations of current interest. More important features of theformulation may be summarized as follows:(i) It is applicable to an arbitrary number of beam, plate and rigid body members, inany desired orbit, interconnected to form an open branch-type topology.(ii) Joints are provided with translational and rotational degrees of freedom. For ex-ample, in the Space Station configuration, the solar panels can undergo predefinedslewing and translational maneuvers with respect to the central body, to track theSun for optimal power production.(iii) The formulation accounts for the effects of transient system inertias, shift in thecenter of mass, geometric nonlinearities, shear deformations and rotary inertias. Italso considers solar radiation induced thermal effects. Thus it is possible to studythe complex system dynamics involving interactions between librational motion,transverse vibrations and thermal deformations.(vi) Environmental forces (aerodynamic, magnetic, etc.) and operational disturbances(Orbiter docking, EVA activity, etc.) can be incorporated readily through general-ized forces and initial conditions.(v) The governing equations are programmed in a modular fashion to isolate the effectsof slewing , librational dynamics, flexibility, orbital parameters, etc.(vi) The equations are amenable to discretization using both component as well as systemmodes thus facilitating comparison between their relative merit.(vii) The equations of motion can be cast, quite readily, into a form suitable for bothlinear and nonlinear control studies.Chapter 4AN APPROACH TO CONTROLA system is usually designed according to prescribed specifications to achieve a desiredlevel of performance. Besides performing the required tasks efficiently, it must also exhibitseveral desirable properties such as stability, quick response, disturbance rejection, etc.For flexible space based systems, governed by highly nonlinear, nonautonomous andcoupled equations of motion, this is possible only in the presence of active control. Misawaand Hedrick [19] have reviewed the pertinent literature at some length.As can be expected, the control strategy can vary rather widely. There are literallythousands of control algorithms written to meet a variety of situations and performancerequirements. More recent approaches include extended linerization [20],[21]; feedbacklinearization [22]-[24] or, in simplified cases, the inverse control [25] and several others[26]- [32].In the present study, the Feedback Linearization Technique (FLT), which completelyaccounts for the system's nonlinear dynamics, is used to control large, flexible, multibodystructures in space.4.1 Feedback Linearization Technique (FLT)Feedback linearization is an approach for the nonlinear control system design . The basicidea is to algebraically transform the nonlinear system dynamics into an equivalent un-coupled canonical form (partly or fully ) so that linear control techniques can be applied.30Chapter 4. AN APPROACH TO CONTROL^ 31It also can be used as a model-simplifying method in the development of robust or adap-tive nonlinear controllers. In its simplest form, the FLT amounts to cancellation of thenonlinearities so that the closed-loop dynamics is in a linear form. It is applicable to aclass of nonlinear systems described by the so-called companion form, or the controlla-bility canonical form. A system is said to be in the companion form if its dynamics isrepresented byx(n) = f(x)+ b(x)u,^ (4.45)where u is the scalar control input; x is the scalar output of interest with = [x,±, • • • ,x(n)1Tas the state vector; and f(x), a nonlinear function of the states . Note, in the state-spacerepresentation, Eq.(4.45) can be written as xl X2di^ (4.46)xn^f(x) b(x)uUsing the control input (assuming b to be non-zero) for the systems which can beexpressed in the controllability canonical form,U =^— f].^ (4.47)One can cancel the nonlinearities and obtain the simple input-outout relation (multiple-integrator form )Xn-1 Xn(4.48)with the control lawChapter 4. AN APPROACH TO CONTROL ^ 32V = -- k1 X1^ —kn_1xn-1,^ ( 4. 49 )where the ki's are chosen so that Pn + kn-iP12-1 +2 .^+ki is a stable polynomial. Atthe same time, one can place the poles at the desired locations, i.e. reaching the pole-placement target. This leads to an exponentially stable dynamics,(4.50)which implies that x(t) —> 0.For the tasks involving tracking of a desired output xd(x), the control law(n)^ (n-1)U = Xd — kie — k—, • • • ,^, (4.51)with e = x(t)— xd(t) as the tracking error, leads to the exponentially convergent tracking. Note, similar results would be obtained if the scalar x was replaced by a vector and thescalar b by an invertible square matrix.When the nonlinear dynamics is not in the controllability canonical form, one mayhave to use transformations to arrive the form before using the above mentioned feedbacklinearization approach. At times, only partial linearization of the original dynamics ispossible. Consider a system as= f (x , u). (4.52)Let us apply the technique referred to as input-state linearization. There are two stepsinvolved. First, one finds a state transformation Z = w(x) and a input transforma-tion U g (x , v) so that the nonlinear system dynamics is transformed into that of anequivalent linear system. Next, one may use a standard linear technique (such as theChapter 4. AN APPROACH TO CONTROL^ 33—IP- 0^lib  "V- KT Z U=g (x, v) x=f (x, u)Linearization LoopPole Placement LoopZ—w (x)Figure 4.8: Block diagram of a feedback linearized control systempole placement) to design the equivalent input v. The block diagram for the procedureis presented in Figure (4.8).Followings remarks can be made about the method:(i) The result is valid in a large region of the state space except for the singularities.(ii) The technique involves combination of the state transformation as well as the inputtransformation, with the state feedback in both. Thus, it is a linearization byfeedback and hence the designation Feedback Linearization Technique. This isfundamentally different from the Jacobian linearization for a small range operationwhich is the basis for linear control.(iii) In order to implement the control law, the new state components (Z) must beavailable. They must be physically meaningful, measurable and amenable to de-termination from Z=w(x) .(iv) In general, one relies on the system model for both the controller design and theChapter 4. AN APPROACH TO CONTROL ^ 34computation . If there is an uncertainty in the model, it will lead to errors in thecomputation of both the new state Z and the control input U.(v) Tracking control can also be considered. However, the desired motion then needs tobe expressed in terms of the complete new state vector. Complex computationsmay be involved in transforming the desired motion specifications ( of physicaloutput variables ) into specifications in terms of the new state.4.2 Control Implementation ProceduresAs shown in Chapter 3 , dynamics of a flexible spacecraft with and 4f correspondingto librational and vibrational generalized coordinates, respectively, is given by[ Mr,r Mr,1 { r + {Fr^{QrMf,r Mf,f^f^Ff^Q f(4.53)Here Mr,,. is a 3 x 3 matrix for the librational degrees of freedom; Mrj is a 3 x (Ng —3) matrix, representing coupling between the rigid and flexible generalized coordinates;Mf,r = Mr,f; and Mfj is a (Nq — 3) x (Ng — 3) submatrix for the flexible degrees offreedom. Fr and Qr are 3 x 1 vectors representing the first and second order couplingterms and the generalized force for the rigid part of the system, respectively. Similarly, Ffand Q f are (Ng —3) x 1 vectors corresponding to the first and second order coupling termsand generatized force for the flexible part of the system, respectively. Assuming only thegeneralized coordinates of the librational degrees of freedom to be observable, the controlforce Q f is not applicable and hence set to zero. The objective is to determine the controlinput Q,. such that the closed-loop system is linearized and has desired pole-positions.Consistent with the assumption:Chapter 4. AN APPROACH TO CONTROL ^ 35mfrrir+ mrf4f + Fr = Qr;^ (4.54)Mfr^Mf f^F f^; (4.55)which can be solved for "4, and^as:^Mir F = Qr;^ (4.56)-74f^ MI) f; (4.57)where:M = Mrr MrfAlf; Mfr;^ (4.58)F = FR— Mr.fMhieF f• (4.59)Fortunately, the system has the controllability canonical form. A suitable choicefor Q,. would be^C2r = 71177+P;^ (4.60)with"r) = (47.)d + Ko[(r)d — 4r] + ki2R4r)d 4r]-^(4.61)Now the controlled equations of motion become:= "I);^ (4.62)f _^mfru^F f.^ (4.63)Chapter 4. AN APPROACH TO CONTROL ^ 36Here: Q,. , M(Md + F + M(Kv6 + IC136); and e = (q7.)d— qr is the error between desiredoutput and the system output . The controller is composed of two parts: the primarycontroller Qra, = M(Md + F; and the secondary controler: QT,s = M(kve + kpe).The quasi-closed loop control block diagram is shown in Figure (4.9). Note, thecontroller solves the tracking, stability and linearization problems at the same time. Asshown in Chapter 6, the control strategy improves the system behaviour significantly.Chapter 4. AN APPROACH TO CONTROL ^ 37Figure 4.9: Block diagram of the Quasi-closed loop control systemChapter 5COMPUTATIONAL CONSIDERATIONSAs pointed out before, formulation of the problem for dynamical and control studiesof a multibody system, comprised of rigid bodies, beam and plate type members is achallenging task. The governing equations were found to be extremely lengthy (even inthe matrix form), highly nonlinear, nonautonomous, and coupled. Equally formidable isthe development of an efficient numerical code for their integration. This chapter brieflyindicates approach used to this end. Modular nature to help isolate influence of varioussystem parameters as well as user friendliness were the two characteristic features whichguided the process. Depending on the discretization process used, two separate programswere required.5.1 Program Structure of the Component Modes Method (CMM)The overall program is composed of several stages or phases. The first one is the in-put phase, followed by numerical integration, output part and completion test. It isschematically shown in Figure (5.10). The input phase includes:(i) orbital parameters p,i ,(4.; and eccentricity e;(ii) program control parameters, simulation period, numerical integration tolerance, ini-tial step-size, and output data;(iii) initial conditions of the state variables.38Chapter 5. COMPUTATIONAL CONSIDERATIONS^ 39Figure 5.10: Program flow-chart for the CMM simulationChapter 5. COMPUTATIONAL CONSIDERATIONS^ 40Two data files used are referred to as MODEL and MODE. They are described in theAppendix B.The method DGEAR was chosen to perform the numerical integration of the equationsof motion. This is a mature method. In most cases, it is robust. Also it can automaticallyadjust the integration step-size and select appropriate iteration procedure.FCN block calculates the right function values, e.g. value of the the state vector atthe time of T + AT. Here AT is the time step-size used by the DGEAR at the instant.Output and completion test-blocks are determined by the input data file. They areused to output the data in a proper format and stop the simulation at a chosen time,respectively. The numerical integration flowchart is presented in Figure (5.11).5.2 Program Structure for the System Modes Method (SMM)The first step is to obtain system modes using the finite element method. Thus theprogram has two major parts: the finite element module; and the dynamic analysismodule. They are presented in Figure (5.12) and (5.13), respectively. Updating of thesystem modes for spacecraft with time dependent geometries, such as rotating solarpanels or slewing robotic arms, cannot be overemphasized. System modes reflect elasticcharacter of the entire spacecraft, hence their variation with the geometry can affectthe system response significantly. There is an important decision to be made: howoften to update the system modes? Obviously one has to strike a balance between theavailable computational resources, need for real-time information and accuracy. Longupdating time may lead to discontinuity in response, which may not be acceptable. Ofcourse, by frequent updating one can avoid such problem, however, only at a considerablecomputational cost.The input I block in Figure (5.12) introduces material properties of the structureChapter 5. COMPUTATIONAL CONSIDERATIONS ^ 41Figure 5.11: Flow-chart showing the numerical integration procedure using the CMMChapter 5. COMPUTATIONAL CONSIDERATIONS ^ 42Figure 5.12: Program flow-chart for the system modes determination.Chapter 5. COMPUTATIONAL CONSIDERATIONS^ 43Figure 5.13: Program flow-chart for the SMM simulationChapter 5. COMPUTATIONAL CONSIDERATIONS^ 44necessary for calculation of the system modes. Input II, in Figure (5.13), enters thesimulation period, numerical integration and program control parameters, mode updateinformation, etc. Details of the input files are presented in the Appendix C.Chapter 6RESULTS AND DISCUSSION6.1 Two Link Mobile Servicing SystemWith the formulation applicable to a class of multibody systems in hand and the associ-ated computer code operational, the next logical step is to illustrate its effectiveness. Tothat end two different configurations of contemporary interest are considered:(i) a two-arm, mobile, flexible manipulator operating on a rigid platform;(ii) NASA proposed system for Control-Structure-Interaction(CSI) study.There are three basic objectives:(i) potential to undertake parametric studies if desired;(ii) effectiveness of the FLT for control accounting for the complete nonlinear systemdynamics;(iii) relative merit of the system and component modes discretization.It is not intended here to generate a vast body of results for each of the systems underconsideration. Of course the code can provide the information as desired by spacecraftdesigners. The objective is primarily to show the potential of this powerful versatile tool.The proposed space station Freedom, expected to be operational by the turn of thecentury, will support a Mobile Servicing System (MSS). It is essentially a two link manip-ulator on a mobile base which can traverse along the main truss of the station treated as a45Chapter 6. RESULTS AND DISCUSSION^ 46Table 6.1: Numerical values used in simulation of the two-link mani ulatorM (kg) L (m) w(rad/s) .Ix(kgm2) Iyy(kgm2) Izz(kgm2)Center Body 240,120 115.35 rigid 8 x 105 2.67 x 108 2.67 x 108Upper Link 1,800 7.5 0.167 101 33,750 33,750Lower Link 1,800 7.5 0.167 101 33,750 33,750free-free beam. The system is expected to be the workhorse for the station's construction,maintenance, operation, and future evolution.The objective of the simulation here is to assess:(a) effect of the more important system parameters on the response;(b) effectiveness of the nonlinear feedback linearization control.The system geometry is shown in Figure (6.14) with the numerical values used in thesimulation presented in Table (6.1). The slewing maneuver of the manipulator is taken tobe a sine-ramp function thus resulting in zero velocity and acceleration at the beginningand end of the operation:Os = { Om   (r/T8) — (0,,/27) sin(277-/7-8), r; < TmOm^ T > rn, .^(6.64)Here: 08, slew angle at an instant 0 (true anomaly); 19,, maximum slew angle; and-r8, duration of the slew. To help isolate the effect, only the lower arm was consideredto undergo the specified slewing motion in the plane of the orbit. Also only the firstcantilever mode is used to represent the link flexibility. The space station is taken to berigid although the formulation and the program accounts for its flexibility.Figure (6.15) shows the effect of increasing the magnitude as well as speed of theslewing maneuver. Three cases are considered corresponding to the slewing magnitudeof 45°, 90° and 180° completed in 10 minutes with the manipulator located at the centerof the station. Note, the inplane maneuver excites only the pitch librational motion,Chapter 6. RESULTS AND DISCUSSION ^ 47Figure 6.14: Geometry of the two-link manipulator supported by a rigid space platform.Chapter 6. RESULTS AND DISCUSSION^ 48i.e. the out-of plane librations, roll and yaw, remain zero. The response follows theexpected trend. With the increase in amplitude and speed of the maneuver, from 450at 4.5°/min to 1800 at 18°/min, the peak pitch response increases from 0.01° to 0.035°.Corresponding vibratory response at the tip of the lower arm (611) in the 1714 directionis also shown for the three cases. Note., the maximum 611 increases significantly, from0.3mm to 1.2mm .The effect of speed is isolated in the response results given in Figure (6.16). Now, thefixed inplane maneuver of 180° is completed at increasingly faster rates (18° /min, 36°/min,and 90°/min). As can be expected, the librational response is modulated at the vibra-tional frequency with the pick çb exceeding 0.050. Note, the tip response at the lowerarm exceeds 10cm for fastest maneuver considered ! This clearly suggests that activecontrol would be necessary during large , fast maneuvers to maintain the response withinan acceptable level.Figure (6.17) assesses the effect of manipulator location on the system response fora 180° maneuver completed in 10 minutes. Two cases are considered : (a) manipulatorlocated at the center of the station as before (same as Case (a) in Figure (6.16)); (b)manipulator located near the tip of the station. Note, besides the librational and manip-ulator tip response, deflection time history of the upper arm end (81) is also plotted. Ascan be expected, several observations of interest can be made:(i) As expected, the pitch amplitude increases significantly due to a large reaction mo-ment about the center of mass. In the present case it increases almost by a factorof 10 !(ii) The inplane deflection 81 of the upper arm is large than that of the lower arm 611.(iii) Effect of the MSS location on the vibratory response of its arms, though present, isrelatively small.Chapter 6. RESULTS AND DISCUSSION ^ 49Figure (6.18) studies the effect MSS location during a faster maneuver of 1800 in 5minutes (twice the previous slewing rate). Except for the frequency modulation in thepitch, the response character remains essentially the same as in Figure (6.17).Next the attention was turned to the system response with the librational controlusing the FLT. Typical results are given for the influence of the slewing speed with themanipulator located at the center of the system in Figures (6.19)-(6.21) and at the tip ofthe station Figures (6.22)-(6.23). Note, the librational motion is controlled, as expected,quite well depending upon the choice of gains. The vibratory motion being uncontrolledcontinues to persist as no damping is considered. There are several options available tocontrol the manipulator vibrations using the conventional proportional control strategies,linear or nonlinear, with or without damping. These options are not pursued here asthe main objective is to assess applicability of the FLT to this class of system. Theextension to a hybrid control strategy with the FLT applied to the librational degrees andproportional (or other conventional control) control procedure applied to the vibrationaldegrees is rather routine.0.025V °0.000-0.0250.0250.000-0.0250.050^0.100(b)0.000 0.150^0.200 0.250 0.000 0.050 0.100 0.150 0.200 0.250-0.00100-0.000000.00100-0.00000-0.001000.0000.001000.050 0.100 0.150 0.200 0.250(a)0.050^0.100 0.150^0.200 0.2500.0250.000-0.0250.000^0.050^0.100^0.150^0.200^0.250 0.000^0.050 0.100^0.150 0.200^0.250-0.00000_0.00100 - 8^(m)Chapter 6. RESULTS AND DISCUSSION ^ 50TIME, Obit TIME, ObitFigure 6.15: Effect of increasing the amplitude as well as speed of the slewing maneuver,in the plane of the orbit, with the MSS located at the center of the station: (a) 45° in10 min; (b) 90° in 10 min; (c) 180° in 10 min.0.0500.025°-0. 000-0.025-0.0500.000 0.050 0.100 0.150 0.200 0.2500.0500.0250-0.000-0.025-0.0500.000 0.050 0.100 0.150 0.200 0.2500.025iv 0-0.0000.050-0.025-0.0500.000 0.050 0.100 0.150 0.200 0.250-0.1%0000^.000 0.050 0.100^0.150 0.200^0.2500.1000.0500.000-0.0500.100 0.2500.050 0.2000.1500.1000.0500.000-0.050(m)TIME, OrbitChapter 6. RESULTS AND DISCUSSION ^ 51TIME, OrbitFigure 6.16: Effect of the slewing speed on the system response with the manipulator atthe center of the main truss. The slewing maneuver of 1800(inplane) is completed in: (a)10 min; (b) 5 min; (c) 2 min.0.100TIME, Orbit-0.00750.0000.00200 ^0.000^0.050 0.150^0.200520.2500.000 0.050 0.100 0.150TIME, Orbit0.200 0.250-0.00500.0000.00150-0.0025-0.0250.0000.00500.050^0.1000.050 0.1000.150^0.2000.150 0.2000.2500.2500.0010080.00050(TO1,1-0.00000-0.00050-0.001000.0025(m)-0.00000.025-0.0000.001008 1,1 (m)-0.00000-0.001001II0.000 0.1000.050 0.2000.150 0.2500.2000.050 0.1500.100 0.250Chapter 6. RESULTS AND DISCUSSION(a)0.00250.00008 1 (m)-0.0025-0.0050-0.00-0.10-0.20r 01 -0.30-0.40-0.50-0.60(b)Figure 6.17: Response plots showing the effect of the manipulator's location: (a) Manip-ulator at the center of the station; (b) manipulator at the tip of the station. The slewingrate is 1800 in 10 min.Chapter 6. RESULTS AND DISCUSSION 530.025N' ° .000-0.0250.1500.100 0.2000.000 0.050 0.250(a)0.0258 (r.)0.000-0.025-0.050^0.100 0.150^0.200 0.2500.0000.0100 -0.0050 -0.0000-0.0050-0.0100 --0.01y000^ 0.050^0.100 0.150^0.200 0.250-0.20oN'-0.40-0.00-0.10-0.50-0.600.000 0.050 0.100 0.150 0.200 0.2500.0500.0258 (m)0.000-0.025 --0.0%0000^ 0.100^0.150 0.200^0.250(b)0.010000.005000.00000-0.00500-0.01000-0.0159,0000 0.050^0.100 0.150^0.200 0.250TIME, Orbit TIME, OrbitFigure 6.18: Response plots showing the effect of the manipulator's location during theslewing maneuver of 1800 in 5 min. (a) manipulator at the center of the station (b)manipulator at the tip of the station.8 i (m)_0.00000.00100.0005-0.00050.150 0.2000.100 0.2500.000 0.0508^,m,-0.00000.00100.0005-0.00050.050 0.100 0.2000.150 0.2500.000TIME, Orbit-°.if0000.500.25V °-0.00-0.250.2000.050 0.100 0.150 0.250TIME, OrbitChapter 6. RESULTS AND DISCUSSION ^ 54Controlled^ Uncontrolled Figure 6.19: Controlled response using the FLT applied to the librational degrees offreedom with the manipulator at the center of the station: slewing maneuver of 45° in10 minutes..000 0.150 0.200 0.2500.1000.0505 (.)-0.000000.002000.00100-0.001000.100 0.2000.000 0.050 0.150 0.2500.00100.00055^(In)-0.0000-0.00050.2000.100 0.150 0.2500.000 0.0500.02000.0100w 0-0.0000-0.01000.000 0.050 0.100 0.150 0.200 0.2508 i (m)_0.000000.002000.00100-0.001000.000 0.050 0.100 0.150 0.200 0.250ControlledTIME, OrbitUncontrolled0.500.25w 0-0.00-0.25TIME, OrbitChapter 6. RESULTS AND DISCUSSION ^ 55Figure 6.20: Controlled response using the FLT applied to the librational degrees offreedom with the manipulator at the center of the station: slewing maneuver of 900 in10 minutes.0.500.25v 0-0.00-0.25-0.P000 0.050 0.100 0.150 0.200 0.250-0.0250.0000.00500.00255 (m)-0.0000-0.0025-0.00500.0000.002000.001005 1,1 (m)-0.00000-0.00100iir 0-0.0000.0250.0500.0500.1000.1000.1500.1500.2000.2000.2500.250Chapter 6. RESULTS AND DISCUSSION^ 56Controlled Uncontrolled 0.00500.00255 1 (m)-0.0000-0.0025-0.00500.000^0.050^0.100^0.150^0.200^0.2500.002000.001005 1,1 (m)-0.00000-0.001000.000^0.050^0.100^0.150^0.200^0.250^0.000^0.050^0.100^0.150^0.200^0.250^TIME, Orbit TIME, OrbitFigure 6.21: Controlled response using the FLT applied to the librational degrees offreedom with the manipulator at the center of the station: slewing maneuver of 1800 in10 minutes.Chapter 6. RESULTS AND DISCUSSION 570.0000.00300.00200.0010-0.0000-0.0010-0.0020111 11m1,10.050^0.100^0.150^0.000.000 0.050 0.100 0.150 0.200 0.2500.000 0.050 0.100 0.150 0.200 0.250NI -0.00-0.250.500.25-0.5&000^ 0.050 0.100 0.150 0.200 0.2500.01000.00508 1 (m)-0.0000-0.0050-0.00-0.10-0.20y -0.30-0.40-0.50-0.600.000 0.050 0.100 0.150 0.200 0.2508 1 (m)-0.00000.01000.0050-0.0050UncontrolledControlled0.003000.002000.00100-0.00000-0.00100-0.002000.1000.250 0.2508 1,1 (m)0.000^0.050 0.150^0.00TIME, Orbit TIME, OrbitFigure 6.22: System response in the presence of FLT control with the manipulator locatedat the tip of the station: slewing maneuver of 180° in 10 minutes.Chapter 6. RESULTS AND DISCUSSION 58Controlled Uncontrolled-0.600.000 0.050 0.100 0.1500.050 0.150 0.2000.100 0.200 0.2500.2500.500.250-0.00-0.25-0.10.000-0.00-0.10-0.20NI 0-0.30-0.40-0.500.050^0.100 0.150^0.200 0.2500.250 .000-0.0V ^.0008^(m)0.150^0.2000.050^0.100 0.2500.050^0.100 0.150^0.2000.010000.00500-0.00000-0.00500-0.010000.0000.2500.050^0.100^0.150^0.2000.0258 1 (m)0.000-0.0250.0258 (m)0.000-0.0250.01000.0050-0.0000-0.0050-0.0100 -0.000TIME, Orbit TIME, OrbitFigure 6.23: System response in the presence of FLT control with the manipulator locatedat the tip of the station: slewing maneuver of 1800 in 5 minutes.Chapter 6. RESULTS AND DISCUSSION ^ 596.2 NASA's Cotrol-Structure Interaction ModelAs pointed out before, with increase in size of the space structures, flexibility effectshave progressively become more important in ensuring successful completion of a givenmission. The frequency spectrum for a flexible space based system often shows extremelylow fundamental frequency with closely spaced and/or overlapping higher modes. Thisraises the possibility of the control system bandwidth interfering with the system dynam-ics making understanding of the Control-Structure-Interaction(CSI) problems and theirresolution extremely important. This has led to the establishment of a major projectby NASA, involving ground as well as space based experiments, to better appreciate theCSI oriented issues [9]. The CSI project has established an evolutionary structural modelfor dynamics and control study as shown in Figure (6.24). It consists of a uniform beamsupporting a one arm flexible manipulator and a fixed rigid antenna. The manipulatorcan be located at any desired position on the beam. Two such positions are shown in thefigure: manipulator located at the center or the tip of the beam. Objective is to controlthe manipulator's tip position during slewing maneuvers. In the present study, this isattempted using the FLT with the flexibility discretization accomplished using system aswell as component modes. Thus the response results would help assess:(i) effectiveness of the control strategy using the FLT;(ii) comparative response using the two discretization procedures.The numerical values used in the simulation are presented in Table (6.2)The manipulator, initially aligned with the main truss, executes a slewing maneuverof 108° in 10 minutes when located at: (i) the midpoint of the space station (beam);(ii) the tip of the truss, i.e. 57.5 m from the center. Corresponding results are given inFigures (6.25) and (6.26), respectively.Chapter 6. RESULTS AND DISCUSSION^ 60Table 6.2: Numerical values used during simulation of the CSI configuration dynamicsM (kg) L (m) f(Hz) Izx(kgm2) Iyy(kgm2) Izz(kgm2)Main truss 18,000 115 0.193 1.9733 x 107 1.01 x 104 1.9733 x 107Link 3,200 15 0.015 1,805 25.500 25.500Tip Point Mass 1,000 - - - -The results shows that the nonlinear control using the FLT continues to be quiteeffective. Both the manipulator and the truss tip motions are rather small. Furthermore,the general trend of the response time history is essentially the same except for localdetails. The main difference seems to be the chattering response with the SMM whichis attributed to the updating of the system mode (fundamental) during the slewingmaneuver. In the present case the new system mode was calculated every 2.78 minutes(100 min./36 steps), to save the computational effort. Of course, by more frequentupdating, the chattering can be eliminated.Chapter 6. RESULTS AND DISCUSSION ^ 61Figure 6.24:^Geometry of the model proposed by NASA for the Con-trol-Structure-Interaction (CSI) study.1.0000E-45.0000E-5-4.3656E-11-5.0000E-50.000 0.050 0.100 0.150 0.200Chapter 6. RESULTS AND DISCUSSION^ 622.5000E-73.2685E-13-2.5000E-7-5.0000E-7-7.5000E-7-1.0000E-60.0005.0000E-64.0927E-12-5.0000E-6-1.0000E-58 (m)0.050^0.100^0.150^0.200^0.250 -1.5000E-5   c^  '^  '^'^ 0.000^0.050^0.100^0.150^0.2000.001000.00050-0.00000-0.000508 ( )-0.001000.000 0.050 0.100 0.150 0.200 0.250(a) (b)Figure 6.25: Response of the NASA's CSI model during a 1800 maneuver in 10 minutes:(a) discretization using the CMM; (b) discretization using the SMM. The manipulator islocated at the center of the truss.0.000^0.050^0.100^0.1500.000200.000100.00000-0.00010-0.000206.156^6.00(b)0.200Chapter 6. RESULTS AND DISCUSSION^ 630.000 0.050 0.100 0.150 0.200 0.2500.00100.00050.0000-0.0005-0.00100.000 0.050 0.100 0.150 0.200 0.250( a )1.0000E-65.0000E-7-2.6290E-13-5.0000E-72.0000E-5-3.6380E-12-2.0000E-5Figure 6.26: Effect of the manipulator location on the system response during the 1800slewing maneuver in 10 minutes: (a) discretization using the CMM; (b) discretizationusing the SMM. The manipulator is situated at the tip of the truss.Chapter 7CONCLUDING REMARKS AND RECOMMENDATIONS FOR FUTUREWORKAdvent of the computer revolution has made it possible for dynamicists and control engi-neers to analyze complex space based systems. Dynamics of a flexible orbiting platform,supporting a mobile flexible manipulator, represents a system never envisaged by pio-neers of the classical mechanics including Newton, Euler, Lagrange and Hamilton. Yet,their elegant methodologies help us cast such formidable systems into into mathematicalmodels, and computers assist in their analyses.The present thesis has attempted to tackle a class of such challenging problems ofconsiderable practical importance. More important aspects of the study and associatedresults may be summarized as follows:(i) An approach to a relatively general formulation for studying dynamics and control ofsystems, characterized by interconnected flexible bodies, has been explained.(ii) A computer code for the above model, leading to extremely lengthy, highly nonlinear,nonautonomous and coupled equations of motion, has been developed.(iii) The formulation together with the operational integration program represent pow-erful tools for design and parametric evaluation of the system performance.(iv) Application of the above mentioned development to a specific system of the spacestation based flexible manipulator, in the presence of nonlinear FLT control, sug-gests versatility of the computer code. Results show that large and rapid maneuvers64Chapter 7. CONCLUDING REMARKS AND RECOMMENDATIONS FOR FUTURE WORK65can lead to unacceptable response. The information is fundamental to the designof the system as well as the controller.(v) The study using the NASA's control-structure interaction model shows the responsetrends to be essentially the same with the component or the system mode dis-cretization. The differences are primarily local in character, at least for the caseconsidered.The study presents a first step in approaching even the preliminary level of under-standing for the dynamics and control of complex flexible systems. The thesis has es-tablished a methodology, however, systematic parametric studies are necessary with avariety of configurations, of practical importance, to develop a data bank with perfor-mance and design charts. This would demand considerable time, effort and computercost, but should lead to important design tools so urgently needed for the next genera-tion of spacecraft. It will also help in planning of the proposed space based experimentsaimed at dynamics and control of flexible structures, analysis of results when such exper-iments are conducted, with the assessment and improvement of the models and controlalgorithms.Bibliography[1] Modi,V.J., and Brereton,R.C., " Planar Librational Stability of a Flexible Satel-lites",A/AA Journal, Vol.6, No.3, March 1968, pp.511-517.[2] Modi, V.J., and Flanagan, R.C., " Effect of Environmental Forces on the Atti-tude Dynamics of Gravity Oriented Satellites, Part I: High Attitude Orbits", TheAeronautical Journal of the Royal Aeronautical Society, Vol.75, November 1971,pp. 783-793.[3] Modi, V.J., and Flanagan, R.C., "Effect of Environmental Forces on the AttitudeDynamics of Gravity Oriented Satellites, Part II: Intermediate Altitude Orbits Ac-counting for Earth Radiation", The Aeronautical Journal of the Royal AeronauticalSociety, Vol.75, December 1971, pp.846-849.[4] Flanagan, R.C., and Modi, V.J., " Effect of Environmental Forces on the AttitudeDynamics of Gravity Oriented Satellites, Part III: Close Earth Orbits Accounting forAerodynamic Forces", The Aeronautical Journal of the Royal Aeronautrical Society,Vol.76, January 1972, pp.34-40.[5] Modi, V.J., and Flanagan, R.C., " Librational Damping of a Gravity Oriented Sys-tem Using Solar Radiation Pressure", The Aeronautical Journal of the Royal Aero-nautical Society, Vol.75, August 1971, pp.560-564.[6] Modi, V.J., and Kumar,K., " Librational Dynamics of a Satellite with ThermallyFlexed Appendages", The Journal of the Astronautical Sciences, Vol.25, No.1,January-March 1977, pp.3-20.[7] Ibrahim, A.M., Mathematical Modelling of Flexible Multibody Dynamics with Ap-plications to Orbiting Systems, Ph.D. Thesis, The University of British Columbia,April 1988.[8] Chan,J.K., Dynamics and Control of an Orbiting Space Platform Based Mobile Flex-ible Manipulator, M.A.Sc. Thesis, The University of British Columbia, Vancouver,Canada, April 1990.[9] Special Issue on Computer Methods in Flexible Multibody Dynamics, The Interna-tional Journal of Numerical Methods in Engineering, Vol.32, No.8, December 1991.[10] Modi,V.J., "Attitude Dynamics of Satellites with Flexible Appendages- A Brief Re-view", Journal of Spacecraft and Rocket, Vol.58, No. 11, November 1974, pp.743-751.66Bibliography^ 67[11] Modi,V.J., "Spacecraft attitude Dynamics : Evolution and Current Challenges",Invited Address to the NATO/AGARD Symposium on the Space Vehicle FlightMechanics, Luxembourg, Novermber 1989; also Proceedings of the Symposium,AGARD-CP-489, pp.15-1 to 15-26; also Acta Astronautica, Vol.21 No.10, 1990,pp.689-718.[12] Ng,A.C., Dynamics and Control of Orbiting Flexible Systems:A Formulation withApplications, Ph.D.Thesis, The University of British Columbia, Vancouver, Canada,April 1992.[13] Suleman,A., Dynamics and Control of Evolving Space Platforms: an Approach withApplication, Ph.D. Thesis, The University of British Columbia, Vancouver,Canada,Augest 1992.[14] Mah,H.W., On the Nonlinear Dynamics of a Space Platform Based Mobile FlexibleManipulator, Ph.D. Thesis, The University of British Columbia, Vancouver, Canada.October 1992.[15] NASA Langley Material on the Space Station Freedom as a Transportation Nodeand the NASA CSI Program, NASA Langley Space Station Office and CSI ProgramOffice.[16] Mah, H.W., and Modi, V.J., " A Relatively General Formulation for Studying Dy-namics of the Space Station Based MRMS with Applications", AIAA 26th AerospaceSciences Meeting, January 1988 Reno,Nevada,U.S.A., paper No.AIAA-88-0674.[17] Modi, V.J.,and Suleman,A., " An Approach to Dynamics of Flexible Orbiting Sys-tems with Application to the Proposed Space Station", 42nd Congress of the Inter-national Astronautical Federation, October 1991, Montreal, Canada, Paper No.IAF-91-293.[18] Ng, A.C, and Modi, V.J., "Dynamics of Gravity Oriented satellites with ThermallyFlexed Appendages", AAS/AIAA Astrodynamics Specialist Conference, Kalispell,Montana,U.S.A., 1987, Paper No. AAS-87-432.[19] Misawa, E.A., and Hedrick, J.K., "Nonlinear Observers-A State-of-the-Art Survey",ASME Journal of Dynamic Systems, Measurement, and Control, Vol.111, September1989, pp.344-352.[20] Baumann, W.T., and Rugh, W.J., " Feedback Control of Nonlinear Systems ByExtended Linearization", IEEE Journal of Automatic Control, Vol.31, No.1, January1986, pp.40-46.Bibliography^ 68[21] Baumann, W.T., "Feedback Control of Multi-input Nonlinear Systems by ExtendedLinearization", IEEE Journal of Automatic Control, Vol.33, No.2, February 1988,pp.193-197.[22] Wang, D., and Vidyasagar, M., " Control of a Class of Manipulators with a SingleFlexible Link-Part I: Feedback Linearization", ASME Journal of Dynamic Systems,Measurement, and Control, Vol.11, December 1991, pp.655-661.[23] Wang, D., and Vidyasagar, M., " Control of a Class of Manipulators with a Sin-gle Flexible Link-Part II: Observer-Controller Stabilization", ASME Journal of Dy-namic Systems, Measurement, and Control, Vol. 11, December 1991, pp.662-668.[24] Modi, V.J., Karray, F., and Chan, J.K., " On the Control of a Class of FlexibleManipulators Using Feedback Linearization Approach", .42nd Congress of the In-ternational Astronautical Federation, October 1991, Montreal, Canada, Paper No.IAF-91-324.[25] Karray,F., Modi, V.J., and Chan, J.K., " Inverse Control of Flexible Orbiting Ma-nipulators", Proceedings of the American Control Conference, Boston. Mass. U.S.A.,June 1991, Editor: A.G. Ulsoy, pp. 1909-1912.[26] Sharan, A.M., Jain, J., and Kalra, P., "Efficient Methods for Solving Dynamic Prob-lems of Flexible Manipulators", ASME Journal of Dynamic Systems, Measurement,and Control, Vol.114, March 1992, pp.78-88.[27] Tosunoglu, S., Lin, S.H., and Tesar, D., "Accessibility and Controllability of FlexibleRobotic Manipulators", ASME Journal of Dynamic Systems, Measurement, andControl, Vol.114, March 1992, pp.50-58.[28] Fuliu,V., Rattan, K.S, and Brown Jr.,H.B., "Modeling and Control of Single-LinkFlexible Arms with Lumped Masses", ASME Journal of Dynamic Systems, Mea-surement, and Control, Vol.114, March 1992, pp.59-69.[29] Spong, M.W., and Vidyasagar, M., " Robust Linear Compensator Design for Non-linear Robotic Control", Proceedings of the IEEE Conference on Robotics and Au-tomation, St. Louis, Missouri,U.S.A., March 1985, pp.954-959.[30] Spong, M.W., and Vidyasagar, M., " Robust Nonlinear Control of Robot Manip-ulator", Proceedings of the 24th IEEE Conference on Decision and Control, FortLauderdale, Florida,U.S.A., December 1985, pp. 1767-1772.[31] De Silva, C.W., "Trajectory Design for Robotic Manipulator in Space Application",J.Guidance, Control and Dynamics, Vol.14, No.3, 1991, pp.670-674.Bibliography^ 69[32] Xu, J., Bainum, P.M., and Li, F., "Reaction Rejection Techniques for Control ofSpace-Robotic Structures", IMACS/SICE International Symposium on Robotics,Mechantronics and Manufacturing Systems, Kobe, Japan, September 1992.Appendix ADETAILS OF Lys, Ins AND HThe details of the components that make up T of Eq.(3.31) are as follows:1^•^•Torb = — M R,„ • Rcni;21= --MC,   C„;21 NTh E2 i=1[I;1 -2d.dm• + E f (d. i + C7dij) • (di + C7dii)dmij ;mi^2^2j=-1 mijTjr 67(pi +^Si) • dic(Pi Ti Si)dmi+ E p {6Tdii +^+^+ + sii)}j=1 mi.a{67dii (67,pai,./^T-ii 8ii)} dmi,ii1^•^•^1^•Tt — I 'Tc • T-cdmc + —2 2 {.1 (cT.T )   (cT2)dm,2 mcmi70Appendix A. DETAILS OF Tsy,, I ^1151,8 71ni+ E^(CZillio';" ij) • (CZi No+ ij)drnio ;j=1 Mij1 NT = —1•Scdnt, + — E^i) • (CfS. i)dmi2 .17.,^2 is=1ni+ E f^  (azip,i,jeii)dmi,ij=1ni c—^—Thor — E Idi • fci^Ti + 61i)drnt. E^+i=1^mi j=-1{Of cli; +^+ CZpai,j )(Ti; +^+ S2i ildm2,j1niTh,t E + E f^+ cTdii)   (Cf,ilLi,.frii)dmi,ii=1 frnini=^  (Cf8i)dmi E f (di + cfdij)   (cfjp,i,isii)dmii=1 j=1 mijTir,t = E Ef^+ + Si)(67-i)dmi+mi{6L^7dij  ^ Sij)}j=1 mij (CZjiii,j7fij)drni^;Appendix A. DETAILS OF Tn,, Ins AND I I n,^ 72Tjr,t, = E [f + + si)(676-..i)dmi+i=i miEf fafdii +^+^+ + Sij)}nij=1;=r^I• Scdmc+ E [f (Cf'*i) (C)dmi i=i mini+ E f (czipi,J7-ii)   ( CZipilSij)dmi,j .1 rniThe system inertia, Ins, given by Eq.(3.33) is the sum of the following components:'cm —m- recni^— UcmCY,,i ;= E I {di • diU — didi}dmi+ E J {(di+ CNi,;)   ( di +i=1 mi^ j=1—(di Cfc/i,j)(di Cfdadmiji=^{-,ac   Tcu — Tc-gcldincAppendix A. DETAILS OF Tn., In, AND H.y.^ 73N „+ E 1 j {(c)   (CTA)U (C7Ti)(Cfiii)}dmii=1 mini r^+ E^{(cf,ith,3T)z,)   (cf,3itigt.i)uj=1 rn'd;It = f {7-, • T,U Tcrc}dMc+ E {fm {(c7T-i)   (C)U — (C7T-i)(CT-iildmi^i=1^ ini+ E f {(czjiL,,,,Tij)   (cfjp,,,t-ii)u—(Cf,j14,iTi3)(Cf,314,3"Tii)}dm.,i1;^Iv = f {6, •^— 6,S,}dm,mc+ E [f {(cfsi)   (C76i)U — (Cf6i)(Cf6iildmimini+ E I^  (cz,/,,,,xxj=1Appendix A. DETAILS OF Tn,, In, AND Hsy,^ 74Ih,r = E f2di   (cf)u - di(Cfpi) — (Cf-)dildmii=1 mi+ E f {2(di C7dii) (CZjp,i,jp-ii)U — (di +j=1CTdii)}dmi,j1 ;= E [f {2di   (C)U — di(CT-f-i) —+ EJ {2(di C7dij) • (CZitti,iTii)U — (di + Cfdij)(CfjpidTii)—(CZiktidrii)(di C7dij)}dmiIh,t, = E {f {2di   (Cf8j)U —4(CfSi)— (C:8i)dildmii=1 mi+ E {2(di + cTdii)   (c,,,Aijaii)u - (di +j=1 midC7dii)}dmij]{25c .7-cu — pc.rc — Tcpc}dmcmc+ E {2(cfAi)   (c)u - (Cfpi)(Cf7r-i) — (CTi-i)(Caii)}dmii=i miAppendix A. DETAILS OF Tsy,, I ^Hn, 75ni+ Ef^  (cf,paz,ifii)u -mj—(6 3 ,3T=2)(cT,J kiz,31)i, )}dm,31= f {2Tc   ScU --Pc& — 6cPc}drricmc+ E {f {2(C)   (Cf8i)U — (C)(CfSi) — (Cf5i)(CT)i)}dm,ii=1 mini+ E^(C27,iiti,iTzi)(CC:pai,36zi)—( G7,3/.1/43623 )(67,,A2,315, )1drni,3] ;h,,, = f {2T-,   8,U —^—rn„+ E {f {2(cf,T-i)   (CT8i)U — (CTT-i)(Cf8j) — (C78i)(CfT-i)}dmii=i mini+ E{2(czip,i,,,r-ii)   (c,)u -J=1 mid-^)(CZ jiLidrij )1^.Here, U is a unit matrix.Appendix A. DETAILS OF Tsy,, Ins AND Hsy,^ 76The components of Hn,, the system angular momentum vector in Eq.(3.34) , are asgiven below:niHh = E [I (di x di)dmi E / (di + CNi;) x (di + CT.dij)dmidi=1 rni^ j=1 Mij= E [f Cf(Ti+ 77-^Si) x^Si)dmini+ EIcifdii+ ^ii)}rnijx { Ofdii^+^ri;^;Ht = fmc(Tic X't)dnic E (9777--i x (C)dmimini+ E f^x^;mijHy = Inic(Sc X .8c)dmc E [j (c78, x (C)dmini+ E(czjizi,j8i; xj=1 mi,iAppendix A. DETAILS OF T55, In, AND Hn,^ 77H ki, E^x 6T(Toi^+ 8i)dmii=1niE I^x {67c/i; (dZitti,j^+ Sii)}dmi,jx^+ Cfclij)}dmid ;niII kr = E I (qpi x di)drni + E f {Cf,paigiji=1^ j=1 mijH kt = E [f (cT,Ti x + 1, xi=1 mi+E I^x (C/ij=1H kt, = E (c:si x + x CTSi)dmii=1 mini+ EJ {cjaii x (di +j=1+(di + Ci7clij) x (CLILi,j8. ii)}dmi,;];11,,t = I (I), X i-c)dmc + E [f (qpi x cfT-i)dmimc i=i miAppendix A. DETAILS OF Tns, Ins AND H58 ^ 78ni+E f xj=1 midf (T) c X Sc)dnic E [f^x CfSi)dmimc^i=1Htm = f^x 8c -F3c x c)thnc E [1^x^C78i xmc i=i mini+Ef^xj=1XAppendix BINPUT FILES FOR THE CMMInput Data File for the Model:* MSS data based on Julius's two arms model as of June 20, 1990* Read in Ni, Nj*1 1 (Ni** Modeb -- Modes for beam; Modepx -- Modes for plate in X* Modepy — Modes for plate in Y1 1 1 (Modeb, Modepx, Modepy** Nqc - central body g.c.*0 (Nqc133.4d0 1538d0 .0001d0 .00d0 .0d0 .3d0 (Rmi, RLi, 'WLi, TLi2629.59d0 .0030d00 1.d0 0.0d0 .00d0 .00d0 (Rloi, Rh-.5d0 -.5d0 0.0d0^(Ai** Nqi(1) - 1st body g.c.*2^(Nqi1.d0^1.d0 .0001d0 .10d0 .453562d2 .3d0 (Rmc, RLc, WLc, TLc3333d0 .7511d-3^1.d0 0.0d0^.00d0 .00d0 (Rloc, Ric0.d0^0.0d0^.0d0 (Bi=.5 or .06650.d0^0.0d0^.5d0 (Ac** Nqj(1) - 1st j body g.c.*1^2^(Nqj1.d0^1.d0 .0001d0 .10d0 .453562d2 .3d0 (Rmi, RLi, WLi, TLi3333d0 .7511d-3^1.d0 0.0d0^.00d0 .00d0 atIoi, RR1.0d0^.060^.0d0 (Bi.060^.060^+.5d0 (Al79Appendix B. INPUT FILES FOR THE CMM ^ 80Input Data File for the Mode:* MAIN19DThis is MSS model case* long -longitude; inc - inclination;* argu - argument; Ecc - eccentricity1.570796 0.d0 1.570796 0.d0* TOL - tolerance; HE - initial step size1.d-6 1.d-6* METH - method; MITER -* INDEX -1^0^1* Nor - No. of Orbit; Npto - No. of points/Orbit* Nstop - Termination of output; Nout -- Termination of vib. output1 1800 450 450* Ncm - shift in cm; Nshade - Thermal Effect; Nengy - Evaluate energy* Nang - use of alt. w; Nslew - Axis of slew1^0^0^1^3* Shadei - Shadow angle; Tslew - Period of slew* Aslew - angle of slew110.d0 36.d0 -180.d0* Ncon -- nonlinear control0* (Gp(i), Gv(i)) i=1 to 30.64 1.6 0.64 1.6 0.64 1.6* Y(iq), iq=1 to 70.0000d0 0.0000d0 0.0000d0 0.d0 0.d0 0.d0 0.d0* Y(Nq-t-iq), iq=1 to 70.000d0 0.000d0 0.0000d0 0.d0 0.d0 0.d0 0.d0Appendix CINPUT DATA FILES FOR THE SMMData File for the INPUT I:PREP7/OUTPUT,t1a_out/COM 27TH DECEMBER 1990/TITLE, MODAL ANALYSIS OF SPACE STATION (FEL - FIRST ELEMENT LAUNCH)KAN,2KAY,2,40KAY,3/COM DEFINE ELEMENT TYPES, 63=PLATE ELEMENT/COM^4=BEAM ELEMENT/COM 21=GENERALIZED MASSET,1,4ET,2,21/COM/COM DEFINE MATERIAL PROPERTIES FOR CENTRAL TRUSS(BEAM)EX,1,1.5E9NUXY,1,0.333R,1,3.464,1,1,1.86,1.86DENS,1,45/COM DEFINE MATERIAL PROPERTIES FOR MANIPULATOR (BEAM)EX,2,1.E7NUXY,2,0.333R,2,3.464,1,1,1.86,1.86DENS,2,61.66IOM DEFINE MASS PROPERTIES FOR MSS PAYLOADR,3,1000,1000,1000,1,1,181Appendix C. INPUT DATA FILES FOR THE SMM^ 82/COM^ ########/COM DEFINE NODES FOR MAIN TRUSSN,1,0.,57.5,0.N,21,0.,-57.5,0.FILL/COM DEFINE NODES FOR MANIPULATORN,22,-0.1307,56.0,0.N,31,-1.307,42.56,0.FILL/COMNLIST,ALL/COM ###########################W###############/COM DEFINE ELEMENTS FOR MAIN TRUSSTYPE,1REAL,1MAT,1E,1,2E,2,3E,3,4E,4,5E,5,6E,6,7E,7,8E,8,9E,9,10E,10,11E,11,12E,12,13E,13,14E,14,15E,15,16E,16,17E,17,18E,18,19E,19,20E,20,21Appendix C. INPUT DATA FILES FOR THE SMM^ 83/COM DEFINE ELEMENTS FOR MANIPULATORTYPE,1MAT,2REAL,2E,1,22E,22,23E,23,24E,24,25E,25,26E,26,27E,27,28E,28,29E,29,30E,30,31/COM DEFINE ELEMENT FOR MANIPULATOR PAYLOADTYPE,2REAL,3E,31/COM ###########################################WAVESM,31,ALLtota1,12/SHOW,x11/VIEW,1,1,1,1ENUMEPLOTITER,I,1AFWRITEFINISH/EXEC/INPUT,27FINISHAppendix C. INPUT DATA FILES FOR THE SMM^ 84Data File for the INPUT II:##############################################*################## *SPACECRAFT CONFIGURATION DATA*# (FIRST MILESTONE CONFIGURATION)# INPUT FILE TO FMC SPACE STATION# PROGRAM NAMES: FEL GF.F => FLT NONLINEAR CONTROL #################################################################IMATXC => INERTIA MATRIX FOR CENTRAL BODY^19733353.13d0^0.D0^0.DO0.0D0^10103.48d0^0.D00.01)0^0.DO^19733353.13d0KEPLERIAN ORBITAL PARAMETERSCCMAT => INITIAL 1-2-3 ORIENTATION OF SPACECECRAFT0.D0 90.D0 O.DOECCENT => ECCENTRICITY OF ORBIT0.0D0HEIGHT => HEIGHT AT PERIGEE (KM)4.132PERIOD => PERIOD OF ORBIT (MINUTES)1.D2SLEWING MANEUVRE PARAMETERS FOR APPENDAGESROTSLW => 1-2-3 MANEUVRE ANGLES FOR APPENDAGES#1 0.01)0 0.60D0 (THETAO,THETAF IN ORBITS)#1 0.0D0 0.00D0 (ALPHAO,ALPHAF IN ORBITS)#1 0.01)0 0.0D0 (BETAO,BETAF)#1 0.0D0 0.60130 (GAMMAO,GAMMAF)TRANSLATION MANEUVRE PARAMETERS FOR APPENDAGESTRASLW => TRANSLATION VECTORS FOR APPENDAGES#1 0.0D0 0.0130 (TRANSO,TRANSF)#1 0.0D0 0.0D0 (XO.XF)#1 0.0130 0.0D0 (YO,YF)#1 0.0D0 0.0D0 (Z022)Appendix C. INPUT DATA FILES FOR THE SMM^ 859'0%%%%%%9:9%%%%9'0%%%%%%%%9'0%%%%%%%%%%%%%%%%%%%%%%%%%9'0%%%%e709'0%%%%%STRUCTURAL PROPERTIES FOR APPENDAGESIMATXI => INERTIA MATRIX FOR APPENDAGESAPPENDAGE #11805.74D0^O.DO^O.DO^0.0D0^255000.00D0^O.DO0.0D0^O.DO^255000.00d0RHOLNI => FIRST MOMENT OF AREABc O.DO^0.0D0^0.0D0#1 39000.0D0^0.0D0^0.0D0DIO => INITIAL HINGE POSITION VECTOR FOR APPENDAGES#1 ODO^0.D0^0.D0ENIASS => MASS PROPERTIESBc 17905.5D0#1 4200.D0ROT1NI => 1-2-3 INITIAL RIGID ORIENTATION OF APPENDAGES#1 O.DO^O.DO^O.DOIMSL => IMSL:DGEAR PARAMETERSHE => 1.D-8TOL => 1.D-6METH => 2MITER => 0INDEX => 1SIMUL => SIMULATION RUN TIMENOR => 0.20D0NPTO => 10.d3INICO INTTIAL CONDMONS (IN DEGREES FOR ATTITUDE MOTION)DLSPO => O.DO^0.D0^040^0.0d0^0.d0^0.d0^0.d0^0.d0^0.130 0.D0 0.D0 0.D0 0.D0VELO => 0.D0^0.130^0.130^0.130^0.D0^0.D0^0.130^0.D0^0.D0 0.130 0.D0 0.D0 0.130Appendix C. INPUT DATA FILES FOR THE SMM^ 86%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%MODAL => MODAL EIGENVECTOR FILENAMESSTEP! => MODAL EIGENVECTOR FILENAMESmass0701mass0801mass0901mass1001mass1101STEP2 => MODAL EIGENVECTOR FILENAMESmass0702mass0802mass0902mass 1002mass1102STEP35 => MODAL EIGENVECTOR FILENAMESmass0'735mass0835mass0935mass1035mass1135STEP36 => MODAL EIGENVECTOR FILENAMESmass0736mass0836mass0936mass1036mass 1136STEP37 => MODAL EIGENVECTOR FILENAMESmass0'737mass0837mass0937mass 1037mass1137Appendix C. INPUT DATA FILES FOR THE SMM ^ 87OPTION => SIMULATION AND OUTPUT OPTIONSICTRL =>^1 (0=UNCONTROLLED, 1=CONTROLLED )IQUASI =>^1 (1=OPEN LOOP   2=CLOSE LOOP)IGFOFT =>^0 {1=INCLUDE G.F , 0= NO G.F. )ISOLAR =>^1 (0DONT PRINT, 1= PRINT )IPVRAD =>^0 (0-=DON'T PRINT, 1= PRINT )IRCS =>^0 (0=DONT PRINT, 1= PRINT )ISTING =>^0 (0=PRINT , 1= PRINT )ITRUSS =>^1 (0=DON'T PRINT, 1= PRINT )IAERO => 0 (1=INCLUDE AERO, 0= DO NOT )IUPDAT =>^1EOF

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