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Dynamics and control of evolving space platforms : an approach with application Suleman, Afzal 1992

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DYNAMICS AND CONTROL OF EVOLVING SPACEPLATFORMS: AN APPROACH WITH APPLICATIONAFZAL SULEMANB.Sc. (Honours), Imperial College, University of London, 1984M.Sc., Imperial College, University of London, 1986A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate StudiesDepartment of Mechanical EngineeringWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAAugust 1992© Afzal Suleman, 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 my department or by his or her representatives. It is understood that copying or publication ofthis thesis for financial gain shall not be allowed without my written permission.Department of Mechanical EngineeringThe University of British ColumbiaVancouver, B.C. CANADADate: August 1992ABSTRACTA relatively general formulation for studying dynamics and control of a large classof space systems is developed. The formulation has the following distinctive features:(a) it is applicable to an arbitrary number of beam. plate, membrane and rigidbody members, in any desired orbit, interconnected to form an open branch-type topology:(b) joints between the flexible members are considered rigid permitting arbitrarylarge angle rotation and linear translation between the structural components;(c) symbolic manipulation is used to synthesize the nonlinear, nonautonomousand coupled equations of motion thus providing an efficient modelling capability with optimum allocation of computer resources:(d) the governing equations are programmed in a modular fashion to isolate theeffects of appendage slewing and translation, libra.tional dynamics, structuralflexibility and orbita.l parameters;(e) both the nonlinear and linear forms of the equations of motion have beenformulated and programmed to help assess relative performance of variouscontrol strategies with reference to linear as well as nonlinear dynamics.The above multibody dynamics formalism involves representing structural deformation in terms of system modes. This feature has several advantages: the formulation effort and derivation time are dramatically reduced; the complexity of thegoverning equations of motion is considerably simplified: the terms representing structural flexibility contributions are decoupled due to orthogonality of the normal modeswith respect to the mass and stiffness matrices: and the physical interpretation of theresults becomes more meaningful, since the modal frequencies represent resonance11conditions for the overall structure. For geometrically time varying systems, themodes are updated at user specified intervals, thus maintaining a faithful representation of structural fiexiblity throughout the simulation sequence. Furthermore, thefinite element method used in the calculation of system modes makes the present algorithm ideal for visualization of the spacecraft dynamics and control through computeranimation. A video depicting modal interactions of the evolving Space Station hasbeen produced in collaboration with the University Computer Services VisualizationGroup.Applicability and versatility of the general formulation are illustrated throughthe analysis of two evolutionary stages of the Space Station: the Eirst Milestoneonfiguration and the ssembly Qomplete Configuration. Effects of the number ofsystem modes, and operational disturbances (solar panel sun tracking, Orbiter docking, crew motion and manipulator tasks) are investigated. Control strategies usingboth linear and nonlinear dynamics have been implemented and their relative performance compared. It is shown that the controller imparts the Space Station, which hasa gravitationally unstable orientation, a desired degree of stability. The simulationresults represent important information and may help in defining the design loads forthe Space Station’s main truss structure, solar arrays, modules and other secondarycomponents.Summarizing, the unique feature of this study is evident in the developmentof an interdisciplinary integrated algorithm synthesizing multihody dynamics, finiteelement method for modal discretization, symbolic manipulation, application of linearand nonlinear control approaches, and computer animation.111TABLE OF CONTENTSABSTRACTLIST OF SYMBOLSLIST OF ACRONYMSLIST OF FIGURESLIST OF TABLESACKNOWLEDGEMENT2. FLEXIBILITY MODELLING ISSUES .. 242.1 Modal Discretization2.1.1 The assumed modes method2.1.2 The finite element method2.1.3 The transformation matrix C2.1.4 Illustrative example2.2 Model Reduction2.2.1 Model reduction algorithms2.2.2 Illustrative example2.3 Dynamic Stiffness2.3.1 Illustrative examples . .2.4 Summary 613. MULTIBODY FORMULATION. . . .. 62Uiiixiiivxiiiiv681114INTRODUCTION1.1 Background1.1.1 Multibody dynamics1.1.2 Structural deformation representation1.1.3 Attitude and vibration control . .1.2 Scope of the Present Investigation.120242ö•. 2628• • 3043434448483.1 Formulation of the Problem3.1.1 System geometry .6263iv3.2.3 Joint rotation3.2.4 Joint translation3.2.5 Shift in the cent.er of mass3.3 Kinetics3.3.1 Kinetic energy3.3.2 Potential energy3.4 Discretization3.5 Equations of Motion3.5.1 Mass matrix3.6 Disturbance Environment3.6.1 Solar array sun tracking .3.6.2 Aerodynamic torque3.6.3 Operation of the Mobile Servicing3.6.5 Orbiter docking3.6.5 Intra-Vehicular Activity (WA)3.6.6 Extra-Vehicular Activity (EVA)3.7 Summary9090911021071081091101111183.2 Kinematics3.2.1 Reference frames and position vectors3.2.2 Position and orientation in space6565656870707272747777798181828284848789System (MSS)4. MODAL FUNCTIONS AND INTEGRALS4.1 Finite Element Formulation . .4.1.1 Beam element4.1.2 Plate element4.1.3 Geometric nonlinearities .4.1.4 Lumped mass element4.1.5 Modal analysis4.2 Space Station Model Description .4.2.1 Finite element representa.t ion4.3 System Modes for the Evolving Space StationV4.3.1 Nominal configurations.4.3.2 Solar array sun tracking4.3.3 MSS operational maneuvers4.4 Modal Integrals4.4.1 Integral evaluation procedure4.5 Summary 139140141141model . . 143144145146147148149150158165165166169169177181185186189189190• 118125131135• 1355. CONTROL METHODOLOGY5.1 Linear Control5.1.1 Linearization5.1.2 State space representation of the mathematical5.1.3 Momentum management5.1.4 The Linear Quadratic Regulator (LQR) .5.1.5 Controllability and observability5.1.6 The torque equilibrium attitude5.1.7 Effect of structural flexibility5.2 LQR Simulation Results and Discussion5.2.1 The gravity-gradient orientation5.2.2 The Local Vertical-Local Horizontal (LVLH) orientation5.3 Nonlinear Control5.3.1 Feedback Linearization Technique (FLT)5.3.2 Extension of the FLT to flexible systems5.4 FLT Simulation Results and Discussion5.4.1 The gravity-gradient orientation5.4.2 The Local Vertical-Local Horizontal (LVLH) orientation5.5 Summary6. MULTIBODY IMPLEMENTATION6.1 Program Architecture6.1.1 Structural Dynamics Module (SDM)6.1.2 Multibodv Dynamics Module (MDM)6.1.3 System Control Module (SCM)vi6.2 Program Flowchart .6.2.1 Subroutine MODEL6.2.2 Subroutine MODES6.2.3 Subroutine FCN6.3 Computational Considerations6.4 Code Validation6.4.1 Total energy6.4.2 Test case studies6.5 SummaryAPPENDICES192192194194195196197198204205205206207214239239242256257257261263A EQUATIONS OF MOTIONB FINITE ELEMENT MATRICESC SIMULATION PROGRAM FILE STRUCTURERESPONSE TO OPERATIONAL DISTURBANCES7.1 Space Station Mission Requirements. .7.2 First Milestone Configuration (FMC) . .7.2.1 Modal convergence7.2.2 Response to operational disturbances7.3 Assembly Complete Configuration (ACC)7.3.1 Modal convergence7.3.2 Dynamic response to disturbances7.4 Summary8. CONCLUDING REMARKS8.1 Original Contribution8.2 Future WorkBIBLIOGRAPHY273291298viiLIST OF SYMBOLSdt nodal displacement vector for the i-th finite elementposition vector from O to O, Fig. 3-2dma. drn mass elements in body B and B, respectivelyf system structural frequencyh angular momentum per unit mass of spacecraftk, Jk kk unit vectors in the directions of Xk, Yk, and Zk axes, respectively; k = p, c, iI direction cosines of Rcm with respect to axesrn. m mass of the bodies B and B, respectivelytotal number of finite elements used to discretize the structuret timecross-sectional area of the i-th finite elementB matrix with elements which are derivatives of elements of Nttransformation matrices defining orientation of F relative toFD fiexural rigidity of the plateYoung’s modulus of the i-th finite elementE unit matrixreference frame for coordinate axes Xk,Yk,Zk; k = o,c,iG shear modulus of the i-th finite elementangular momentum of spacecraft with respect to XC,YC,ZCviiiaxesI inertia matrix of spacecraft with respect to the axes(12)i, (I3)i second moment of area for the i-tb beam element in the localy and z directions, respectivelyK, K displacement and velocity gain matriceslength of ith beam finite elementM total mass of spacecraftM mass matrixN total number of B bodiesN finite element local shape functions matrixO, O origins of the coordinate axes for bodies B and B, respectivelyOm Oim centres of mass of the undeformed and deformed configurations of spacecraft, respectivelyQ, r control effort vectors for flexible and rigid coordinates respectivelyQ,. Q. Q> control effort for pitch, roll and yaw degrees of freedom, respectively-cm position vector from the centre of force to the instantaneouscentre of mass of spacecraftR, R1 position vectors of the mass elements dine and dm, respectively as measured from the centre of forceRcrn. R. R magnitudes of R. R and R7, respectivelyCm cm cmixposition vector from 0cm to the instantaneous centre of massof spacecraft;position vector from O to the centre of mass of undeformedspacecraftT total kinetic energyT system kinetic energy due to various coupling effects where i= s (slew), v (vibration). t (translation)U potential energy of spacecraft; Ue + UgUe strain energy of spacecraftUg gravitational potential energy of spacecraftXk.Yk.Zk body coordinate axes associated with B and B, respectively;k = c, i. or i.jXQ,YQ,ZQ inertial coordinate system located at the earth’s centercoordinate axes with origin at C and parallel to XC,YC,ZC,respectivelyXS,YS,ZS orbital frame with X in the direction of the orbit-normal, Y3along the local vertical, and Z8 towards the local horizontalshear deformation coefficient in the y and z directions, respectively6, 6 vectors representing transverse vibration of drn and dm,respectivelyx nonlinear controller damping ratioe eccentricityfinite element generalized strain vectorxrotation about the local horizontal axis, Z1, of the intermediate frameX1.Y,Zrotation about the local vertical axis, Y2, of the intermediateframeX2,YZgravitational constante true anomaly,vectors denoting positions of drn and dm, respectively, inthe undeformed configuration of the spacecraft, Figure 3-2librational velocity vector with respect to axesrotation about the orbit normal. X8structural damping ratioxiLIST OF ACRONYMSACC ssembly complete configurationCMG Control Momentum yroCOPS COntrol of Flexible StructuresESA uropean space AgencyEVA Rxtra vehicular ctivityFMC First Milestone onfigurationFEM Finite Element MethodFLT feedback Linearization lechniqueJEM Japanese Experiment ModuleLQR Linear Quadratic RegulatorLSS Large space StructuresLVLH Local Vertical - Local HorizontalMSS Mobile Servicing SystemMTC Man Tended ConfigurationPMC Permanentl Manned onfigurationPV PhotoVoltaicQCLC Quasi-losed-Loop ControlQOLC Qua.si-Qpen-oop ControlRCS Reaction Control SystemTEA Iorque Equilibrium AttitudeTDRSS Tracking and ata Relay Satellite SystemxiiSCOLE Spacecraft ntro1 Laboratory xperimentSPDM pecia1 urpose .extrous ManipulatorSSM Substructure Synthesis MethodDot ( ) and prime ( ‘ ) represent differentiations with respect to time t andtrue anomaly G, respectively. Subscripts o and e indicate initial and equilibriumconditions, respectively. Unless stated otherwise, overbar (- ) represents a vector;and boldfaced symbol refers to a matrix.XliiLIST OF FIGURES1-1 The configuration of the Space Station Freedom as proposed in 1988.It was revised in early 1991. More changes are anticipated before thespace platform becomes a reality by the turn of the century1-2 The Space Station’s assembly sequence showing a schematic of thefour milestone configurations 52-1 General flexible multihody model 292-2 L-shape cantilever beam 312-3 Forcing function profiles applied at the free tip of the L-shape beam:(a) impulse function ; (b) ramp function 322-4 System modes for the L-shape beam: (a) Case 1: (b) Case 2; (c) Case 3 332-5 Component modes used to represent the body B1 as a cantilever beamwith tip mass 352-6 Component modes used to represent the body B2 as a cantilever beam 352-7 Dynamical response for the structure in Case 1 for the three solutionprocedures in the presence of: (a) ramp function;(b) impulse function 402-8 A comparison of transverse tip response in Case 2 for the three solutionprocedures in the presence of: (a) ramp function; (b) impulse function 412-9 Displacement response for the structure in Case 3 for the three solutionprocedures in the presence of: (a) ramp function; (h) impulse function 422-10 Reduced order model for the L-shape structure with load appliedin the transverse direction and the output is the longitudinal tipdisplacement of the beam B1 46Xiv2-11 Evaluation of the reduced order model through a comparison ofthe dynamic response to a 5 Ns impulse using modes 3, 4, 5 and 9and the full order model (first 10 modes) 462-12 Reduced order model for the L-shape structure with load applied inthe transverse direction to B2 and the output of interest is thetransverse tip displacement of beam B1 472-13 Evaluation of the reduced order model is demonstrated by carryingout the dynamic simulation for a 5 Ns impulse with modes 1 and 2,and compared to the full order model (first 10 system modes) 472-14 Two degree of freedom lumped mass model [72] 502-15 Spin-up maneuver profile for the 2-d.o.f. system 532-16 Tube angular deflection for spin-up to: (a) 3 rad/s;(b) 1 rad/s; (c) 0.1 rad/s 532-17 Development of stiffness force during the spin-up 552-18 Cantilever beam model [72] 562-19 Spin maneuver profile 582-20 Time history of the beam tip displacement: (a) =6 rad/s;(b) Q =3 rad/s 593-1 A schematic diagram of the spacecraft model 643-2 Position vectors 663-3 Reference frames used in the formulation 673-4 Pitch. Roll and Yaw rotation sequence to determine spacecraftattitude motion 683-5 Aerodynamic torque history 833-6 Effect of eccentricity on orbital time and maneuvering profiles:(a) displacement-true anomaly history for a sine-on-ramp maneuverxvcompleted in 1800; (b) corresponding velocity profile; and(c) resulting acceleration history 843-7 Translational maneuver profile for the Mobile Servicing System. . . 853-8 Orbiter docking simulation model 863-9 Crew motion simulation models:(a) longitudinal IVA; (b) transverse IVA;(c) treadmill walk; (d) treadmill jog;(e) coughing and sneezing; (f) EVA 884-1 Beam-type finite element with twelve degrees of freedom 934-2 Plate-type finite element with 24 degrees of freedom 1034-3 Lumped mass-type finite element 1094-4 A finite element model of: (a) the main truss a.t the ACC stagewith attached modules and payloads; (h) solar array;(c) PV radiator; (d) station radiator; (e) resistojet andstinger assembly; (f) RCS boom assembly 1124-5 Frequency spectrum and representative mode shapes for: (a) the FMCshowing its closely spaced and overlapping character; (b) the MTC;(c) the PMC. Note, the spectrum shows further shrinkage; (d) the ACC.Now, the first forty modes are compressed in the frequency rangeof 0.1-0.8 Hz 1214-6 Solar array a-joint rotation for Sun tracking 1274-7 Frequency spectrum for the FMC as a function of the a-joint rotationthrough 90° 1274-8 Progressive change in the dominant character of the 27 system mode,from torsion to bending, during 90° rotation of the a-joint 1284-9 Solar array ,8-joint rotation for Sun tracking 129xvi4-10 Frequency spectrum for the FMC as a function of the solar arraya-joint maneuver 1294-11 Character of the 40th mode showing local changes during 900 rotationof the solar panels about the ,8-joint 1304-12 MSS payload positioning operation consisting of slewing and translationalmaneuvers 1324-13 Time-steps during inpiane maneuver of the MSS 1334-14 Frequency spectrum for the FMC of the Space Station as a functionof the MSS inpiane slewing and translational maneuvers 1334-15 25th modal function, at three discrete instants, as affected bythe inplaie maneuver of the MSS 1344-16 A schematic diagram showing time-steps during the out-of-planemaneuver of the MSS 1364-17 Frequency excursions for the FMC as function of the MSSout-of-plane maneuver 1364-18 Variation of the 2O system mode during the out-of-plane maneuver 1374-19 Discrete finite element modal functions for beam and plate typemembers 1385-1 Attitude and momentum management controller architecture 1505-2 The Space Station in the gravity gradient orientation 1525-3 The attitude response of the uncontrolled Space Stationfor an initial disturbance of 50 in pitch, roll and yaw,applied simultaneously 1525-4 Librational response of the Space Station to an independentexcitation in pitch, yaw and roll. Note, the pitch and yawdisturbances lead to essentially uncoupled motions. The systemxviiappears to become unstable in yaw through its coupling with roll. . . 1535-5 The controlled angular displacement and rates of the SpaceStation in the gravity-gradient mode of operation for an initialdisturbance of 5° in pitch, roll and yaw, applied simultaneously 1565-6 The control effort and required change in the angular momentumfor an initial disturbance of 5° in pitch, roll and yaw,for three different gain settings 1575-7 The effect of structural flexibility on the attitude controlfor the Space Station in the gravity-gradient orientation 1595-8 A schematic diagram showing the FMC in the Local Vertical-LocalHorizontal mode 1605-9 The free attitude dynamics response of the Space Station for aninitial disturbance of 5° in pitch, roll and yaw,applied separately 1625-10 The controlled response and the associated effort time historiesfor the Space Station in the L\TH mode of operation 1635-11 The effect of structural flexibility on the attitude controlfor the Space Station in the LVLH orientation 1645-12 Block diagram for the nonlinear control:(a) quasi open loop technique; (h) quasi-closed loop technique 1705-13 Controlled librational response of the rigid FMC, in thegravity-gradient orientation, for three different sets of gainswith an initial angular disturbance of 5° in pitch. roll and ya.w 1735-14 Controlled librational response of the rigid FMC. in thegravity-gradient orientation, for three different dampingratios with an initial angular displacement of 5°in pitch, roll and yaw 174xviii5-15 Controlled damped response of the flexible FMC in thegravity-gradient orientation, with preset controller frequencyx = lO2rad/s, in the presence of librationaldisturbance of 50, imparted to the system in pitch,roll and yaw: (a) critical damping ratio; (b) = 0.1 1765-16 Controlled librational response of the rigid FMC in the LVLHorientation for three different sets of gains with an initiala librational displacement of 5° imparted to the system 1795-17 Controlled librational response of the rigid FMC in the LVLHorientation for three different damping ratios. Initial angulardisplacement corresponds to 50 in pitch. roll and yaw 1805-18 Controlled damped librational response of the flexible FMCin the LVLH orientation, with preset controller frequencyx = lO2rad/s, with an initial angular displacementof 50 imparted to the system in the librational degrees offreedom:(a) critical damping ratio; (b) = 0.1 1826-1 Flowchart showing the interfacing between the three modules:structural dynamics, multihody dynamics, and control 1886-2 Six particular topological cases. The REDUCE based formulationarrives at the explicit equat.ions of motion with optimum allocationof computer resources for the case under consideration 1916-3 Multibody dynamics flowchart depicting the main subroutinescomprising the program and corresponding data flow . . 1936-4 Response of the Space Shuttle to an initial disturbanceof 50 in pitch, roll and yaw, applied simultaneously 1996-5 Librational response of the rigid one-link manipulator duringa translational maneuver of 2Dm completed at several speeds 200xix6-6 Librational response of the rigid one-link manipulator during aslewing maneuver through 1800 completed at several speeds 2016-7 Effect of the payloa.d mass on the librational response of aone-link rigid manipulator executing a simultaneous translationaland slewing maneuver 2026-8 Effect of the arm flexibility on pitch response and pointingaccuracy of a single link manipulator executing the combinedtranslational and slewing maneuvers 2037-1 Allowable microgravity level envelopes for the proposed microgravitylaboratory experiments 2067-2 The coordinate system, appendage numbering scheme and attituderepresentation for the FMC of the proposed Space Station 2081-3 (a.) Librational displacement and angular velocity response of the FMCto an initial displacement of 50 in pitch, roll and yaw, appliedsimultaneously. Note. the results for 1, 5, 10 and 20 mode are identical;(b) Comparison of the nonlinear control effort required to drive thesystem to its design orientation (LVLH) with an increase in the numberof system modes included in the simulation. The system is subjectedto an initial attitude displacement of 5° in pitch, roll and yaw;(c) The solar panel tip displacement and acceleration profilesshowing the effect of number of system modes in the dynamicsimulation. The system has been given an initial attitudedisplacement of 50 in pitch, roll and yaw;(d) The main truss z-direction tip displacement and accelerationprofiles showing the effect of number of system modes in thedynamic simulation. The system has been given an initial attitudedisplacement of 5° in pitch, roll and yaw 209xx7-4 (a) The Space Station response to the solar panels rotationto track the sun for optimum power output. Note that the modalinformation is upda.ted every 15° to maintain a faithfulmodel of the system geometry and structural deformation;(b) The shift in the center of mass due to structural deformationduring the solar panels tracking maneuver:(c) The Space Station response to the solar panels rotation,to track the sun for optimum power output, in the presence ofaerodynamic torque 2167-5 (a) A schematic diagram of the MSS motion profilecomposed of slewing and translational maneuvers;(b) The MSS slewing and translational maneuvers time-historyused in the simulation;(c) The dynamics rsponse of the Space Station in presenceof the MSS payload positioning maneuver 2207-6 The Space Station controlled response to:(a) the Shutle docking along the local vertical;(b) the Shuttle docking along the local horizontal;(c) the Shuttle docking along the orbit normal;(d) the Shuttle docking along the local verticalwith the structural damping ratio = 0.01 2257-7 The Space Station response to the EVA tethered motionalong the loca.1 vertical, (a) in the absence of structural damping;(b) in the presence of damping, (= 0.01 2317-8 The Space Station response:(a) in the presence of treadmill jog along the local vertical;(b) in the presence of treadmill walk along the local vertical 233Xxi7-9 The Space Station response to the induced IVA disturbancealong the local vertical:(a) along the module: (b) across the module 2357-10 The coordinate system, appendage numbering scheme andattitude orientation proposed for the ACC Space Station 2407-11 Effect of increasing the number of modes on the system responsefor an intial disturbance of 50 in pitch, roll and yaw:(a) librational displacement and angular velocity of the Space Stationin the presence of the number of modes (1,5.10,20) in the presence ofnonlinear FLT controller with identical gains;(b) torque effort for the FLT controller;(c) solar panel tip deflection and acceleration time-histories;(d) station radiator tip displacement and acceleration time histories;(e) the main truss tip displacement and acceleration 2427-12 Response of the Space Station (ACC) to the Shuttle dockingalong the local horizontal 2497-13 Response of the ACC to the disturbance introduced by anastronaut treadmill jogging along the local horizontal 2507-14 Damped response of the ACC during a tet.hered EVA sortiealong the local horizontal 2517-15 The ACC response in the presence of:(a) IVA transverse motion along the local horizontal;(h) IVA longitudinal motion along the local horizontal 2527-16 The Space Station response in the presence of astronautcoughing and sneezing with a force transmitted along thelocal horizontal 254xxiiLIST OF TABLES1-1 Summary of milestone flight configurations showing progressive evolutionof the Space Station 42-1 System mode frequencies for the L-shape cantilever structure 342-2 Component mode frequencies for the beam B1 modelled as a cantileverwith a tip mass 362-3 Component mode frequencies for the body B2 modelledas a cantilevered beam 364-1 The Space Station component data 1114-2 Mass and inertia propoerties of the Space Station modules 1174-3 The Space Station payload characteristics 1175-1 Closed-loop eigenvalues for the gravity-gradient FMC controlleddynamics with linear attitude and momentum controller 55-2 Closed-loop eigenvalues for the LVLH FMC controlleddynamics with linear attitude and momentum controller 1615-3 Control torque demand for the LQR and FLT approaches 1847-1 Peak nonlinear attitude control efforts, displacement and accelerationlevels at various FMC Space Station locations for the Shuttle dockingand crew motion disturbailces 2387-2 Peak nonlinear attitude control efforts, displacement and accelerationlevels at various ACC Space Station locations for the Shuttle dockingand crew motion disturbances 255xxiiiACKNOWLEDGEMENTSI wish to express my gratitude to Professor V. J. Modi, who gave unsparingly ofhis time in the guidance of my research. He has made a substantial contribution tobot.h my understanding and appreciation of dynamics.I also wish to acknowledge the insights, suggestions, and encouragement of friendsand colleagues Said Marandi, Alfred Ng and Fakhredine Karra.y. I am also indebted toProfessor N. Nathan, who as a member of the supervisory committee, offered valuableadvice and suggestions during the course of this thesis.Special mention must be made of my parents, Mr. Suleman Allimahomed andMrs. Zuleka Ismail, whose constant love and faith made this accomplishment possible,and to them I dedicate this work.This study was supported by the University Graduate Fellowship. Further assistance was provided by the Natural Sciences and Engineering Research Councilof Canada, Grant No. A-2181, and the IRIS Center of Excellence Program GrantNo. IRIS/C-8, 5-55380, both held by Professor V. J. Modi.xxivCHAPTERONEINTRODUCTION“When the President of the United States in 198. directed NASA to build a spacestation, he invited the friends and allies of the U.S. to join in the challenge of creating a machine that could be manned and opera.ted well into the 21st century. Thatinvitation came after almost five years of cooperation in manned Shuttle operationsand almost 25 years of cooperation in unmanned space activity. Extending that cooperation to a manned space station ensured that development of the space frontierbeyond the year 2000 would be a global undertaking. And rio matter how we measureit, Space Station Freedom will be the biggest international technological project everundertaken” --Aerospace America, September 1990.The U.S, Japan, Canada, and the 10 member nations of the uropea.n pa.cegency (ESA) have embarked upon a project that might have an impact. on globalaffairs far beyond the realm of space exploration. Figure 1-1 illustrates one of theproposed Space Station configurations at. the completion of the assembly sequence.This permanently manned platform will require about 30 Shuttle flights during itsassembly. It will operate at an altitude of 400 kilometers. in a near circular orbit atan inclination of 28.5° with respect to the earth’s eciuatorial plane[1].The station consists of contributions by the various international partners, anda U.S. provided infrastructure that includes truss assembi , electric power, and crewliving quarters. The U.S will also build a laboratory Ino(lule and a polar orbitingplatform: the ESA will provide a co-orbiting Free-Flying Laboratory, and a secondIThe configuration of the Space Station Freedom as proposed in 1988.It was revised in early 1991. More changes are anticipated before thespace platform becomes a reality by the t.urn of the century.polar platform; and Japan will supply the Experiment Module (JEM) consisting ofan exposed facility together with a logistics module. Canada’s contribution will bethe Mobile servicing ystem (MSS), a complex robotic machine, which will he usedto assemble, service, and maintain the entire station.Gimbaled solar arrays will track the Sun and provide power at any relative alignment of the Space Station with respect to the Sun. Heat rejection is achieved bynonrotating radiators located on the main truss. The baseline configuration involvesfour pairs of solar arrays and two pairs of radiators positioned on the main truss asshown. The pressurized modules (Habitation, Laboratory and Logistics) are locatednear the geometric center of the main truss.So’ar Pai]Figure 1-12Primary advantages of the configuration include: favourable view afforded toall payloads; safe accomodation of tethered systems and communication antennas;provision of wide clearances for orbiter rendezvous and berthing, construction andservicing; and convenient transition to evolutionary phases.Details of the individual missions are still being worked out; however, NASA [2]has established four milestone flight configurations as summarized in Table 1-1 , andschematically shown in Figure 1-2. The first stage. referred in the acronym form asthe £irst Milestone configuration (FMC), will deliver 11 bays of the main truss. Twoof the bays are outboard of the articulating a-joint which, together with the a-jointon the panel axis, allows the solar arrays to track the sun. Hardware delivered willalso include a pair of solar arrays providing 18.15 kW of power, a panel radiator,2 Reaction Control System (RCS) modules, fuel storage tanks for flight control andreboost., and limited avionics and communication equipment. Once assembled, theFMC will be a fully functional spa.cecraft awaiting the return of the Shuttle to progressto the next stage of the assembly sequence.Subsequent flights will deploy a station radiator, six Control Momentum yros(CMG) for additional attitude control with a complete communications coverage provided by a Iracking and a.ta elay satellite ystemn (TDR.SS). The Shuttle will alsodeliver a t.elerobotic servicer to extend support to xtra Vehicular ctivity (EVA)of the crew during assembly and maintenance, a pressurized docking adapter anda Laboratory Module, thus accomplishing the second landmark in the evolutionaryprocess, namely the Man lended Configuration (MTC).These elements will provide a base for deployment for the various modules, payloads and crew for a Permanently Manned Configuration (PMC). with the main trussextending over 23 hay lengths. Finally, the Assembly Complete Configuration (ACC)3Table 1-1 Summary of milestone flight, configurations showing progressive evolution of the Space Station.MILESTONE FLIGHTSFIRST MILESTONE MASS (Kgs)Main Truss (60m), PV arrays (2),(FMC) PV Radiator (1), Stinger/Resistojet. 21,580MAN-TENDED Main Truss (6Dm), PV arrays (2),Pv Radiator (1), Station Radiator (1), 73,779(MTC) U.S. Lab. Module, Press. Dock. AdapterPERMANENTLY Main Truss (115m), PV Arrays (4),MANNED PV Radiator (2), Station Radiator (2), 150,490(PMC) Lab. , Logistics arid Hab. Modules.ASSEMBLY Main truss (155m), PV Arrays (8),COMPLETE PV Rad. (4), Station Rad. (2), Lab., 273,647(ACC) Logistics, Hab, Jap. and ESA Modules.will mark the end of the first phase of development with the deployment of the variouscontributing elements from Canada, Europe and Japan.Canada’s contribution pert.ains to the development and operation of the MobileServicing System (MSS) which will assemble, maintain and service the station. Therobotic MSS comprises the Mobile Servicing Center (MSC), Maintenance Depot., Special Purpose Dextrous Manipulator (SPDM) and, on Earth, ground support facilities.The MSC is Freedom’s core external assembly, maintenance, and servicing system. Itconsists of a base structure mounted on a transport.er, with accomodations for manipulators, payloads on orbit replacement units, servicing devices, tools, utilities, anda thermal controller. Included is a large remote manipulator system to capture andberth the fully loaded Orbiter and undertake a variety of posit.ioning and handlingtasks. The MSC will be controlled by the crew from work areas inside Freedom’s4Figure 1-2 The Space Station’s assembly sequence showing a schematic of thefour milestone configurations.PV ArrayStation RacatoLabOratory ModuleMan Tended ConfigurationSnger/ResatoetLthoratory ModuleLogistics ModuleHthitalion ModuleJanese ModuleESA ModuleLaboratory Module1_ogistics ModuleHabitation ModulePermanently Manned Configuration Assembly Complete Configuration5pressurized cupolas through direct visual control, supplemented by television cameras. The Maintenance Depot will he attached to the truss structure for storage ofspare parts and serve as a base for assembling large structures.The SPDM, when attached to the station manipulator, will give the MSS an additional dextrous capabilityfor tasks requiring a higher degree of precision.Maiy advanced technologies will be built into the design of the robotic devicesdeveloped by Canada. These include vision; force and tactile feedback: automatedsystem health checks; diagnostic and failure management; automated path trajectory:and collision avoidance management. Through its participation in the Space StationProgram, Canada has reiterated its committment to excel and maintain the edge inspace technology.1.1 BackgroundIn the early days of space exploration, spacecraft were relatively small, compact, and mechanically simple. They were modelled as rigid bodies for purposesof motion simulation, stability determination, and active control design. Even thenthis approximation, introduced by neglecting flexibility, was sometimes innacurate asdemonstrated by the instability of the Explorer I spacecraft in 1958 [3]. The abnormalbehaviour of this satellite was attributed to energy dissipation induced by vibrationof the long wire turnstile antennas, which protruded from the cylindrical housing ofthe satellite. Similar anomalous behaviour due to structural flexibility and energydissipation was also exhibited by Alouette I and Explorer XX spacecraft.These and subsequent experiences with missions such as the Orbiting GeophysicalObservatory III[4] in 1966 (in this case excessive oscillations were induced by controlsystem interactions with flexible beams) led to a vigorous program of research inmultibody systems with flexible components. The approximate analytic and numeri6cal techniques developed in the course of this research proved to be quite successful indesigning spacecraft with modest size and flexibility. Indeed, the existing literatureon dynamics of flexible spacecraft is so voluminous that a 1974 review by Modi[5] onrigid bodies with flexible appendages contained more than two hundred references.However, a new generation of spacecraft with large flexible components (radararrays, masts, solar collectors, communications platforms. etc.) are slowly emerging,presenting new challenges to accurately model their dynamics and develop controlprocedures. In general, these structures are characterized by interconnected flexiblebodies having small structural damping and low, closely spaced frequency spectra.The tasks of controlling the rigid body rotations (librations) for pointing accuracyand stabilizing the flexible structure vibrations pose dynamical and control designproblems, never encountered before.A question arises: why not conduct ground based experiments before deploying astructure or its subassemblies in space? During the past decade, NASA has been actively involved in a study such as 5tructural control Laboratory Experiment (SCOLE,6), a part of the more general program referred to as the COntrol of Flexible Structures(COPS). Lately, it has been renamed as Control-Structure Interaction (CSI) study,however the ultimate objective remains essentially the same: through scale modelstudies on ground and their prototype counterpart investigations in space, to developdependable mathematical models for predicting dynamical response and its control forflexible structures operating in the space environment. Unfortunately, ground-basedexperiments have their limitations as accurate representation of the gravitational,magnetic, solar radiation, free molecular and other fields has been found elusive.Thus, refined mathematical models and comprehensive control simulation techniqueswill be necessary to accurately and reliably predict complex dynamical interactionsin large space structures. Moreover, as spacecraft become more complex and architecturallv metarnorphical, development of precise dynamical models and derivation ofthe corresponding equations of motion for transient and evolutionary stages becomeoverwhelming. Hence, considerable a.ttention has been directed towards developmentof computer algorithms to automate the dynamical simulation process for complexsystems. In an effort to make these programs applicable to a large class of systems,the number of structural members constituting a system is considered a variable, i.e.left arbitrary. The phrase “multibody computer program” has been coined to denoteapplicability of the code to a. system with an arbitrary topology.1.1.1 Multibody dynamicsIn the field of spacecraft. dynamics, the first paper describing a general multibodydynamics formulation was published by Hooker and Margulies[7] in 1965. This workwas based on Newton-Euler equations and is applicable to a point connected setof rigid bodies in a topological tree, where the constraint torques are obtained viaLagrange multipliers. At about the same time, Roberson and Wittenburg[8] treateda system with n-rigid bodies independently and derived the dynamical equations inthe matrix form.Ever since, a number of multibody formalisms have been reported in the literature. Ness and Farrenkopf [9], and Ho amid Gluck [10] extended the above modelsto the flexible n-body system. Ness and Farrenkopf chose the unified approach todeal with the nonlinear equations for the total motion of the system, while Ho andGluck opted for the perturbation approach to deal with the flexibility dynamics. Theunified approach is based upon an algorithm utilizing a digital computer to synthesize the dynamic and kinematic equations of motion. In the perturbation method,deformation is represented as a small perturbation of the nominal motion, obtained8by treating the system as a collection of rigid bodies.Kane and Levinson [11] employed D’Alembert’s principle and the concept of angular momentum for derivation of the equations of motion for a flexible tree typetopological system. This approach, commonly referred to as Kane’s method, is a. vectorial formulation based on the projection of forces and torques along the directionsdefined by the “partial velocity” and “partial angular velocity” vectors.Vii-Quoc and Simo [12] have proposed multibody formulations for both openchain and closed-loop structures. The equations of motion for the individual bodiesare expressed with respect to a translational frame positioned parallel t.o the inertia.lreference, whose origin is placed at the instantaneous center of mass of the system.This approach uses geometrically exact structura.l theories.Modi and Lips [13] have presented a general Lagrangian formulation for the iibrational dynamics of cluster type spacecraft with an arbitrary number of deployingflexible appendages. Modi and Ibrahim [14] extended the above model to include shiftin the center of mass, changing central rigid body inertia and offset of the appendageattachment point. Modi and Ng[15] furt.her extended the multibody formulation toinclude an all flexible two-tier tree type configuration, incorporating thermal deformations and appendage deployment maneuvers.Russel [16] developed a. set of transformed equations for an arbitrary tree configuration using the Newton-Euler formulation where the dynamic state variables arethe free components of the transformed momentum; however, the constrained components of momentum and the transformed velocities a.re retained as intermediatevariables, so that. the final equations have a simple form.Other multihody formalisms documented in the literature include the studies byKurdila (Maggi&s approach) [17], Keat (velocity transform method) [18], Ho (direct9path technique) [19] and Meirovitch (perturbation approach) [20]. Jerkovsky [21]presents an excellent overview of the relative merits and disadvantages of selectedmomentum (Newton-Euler) aild velocity (Lagrange) formulations mentioned above.The earlier multihody derivations were based on Newton-Euler approaches. Themethods of Newton and Euler, which involve physically visualizable quantities represented by Gibbsian vectors and dyadics, are generally recognized as useful in understanding behaviour of relatively simple systems, such as particles in space or gyroscopes. However, it is believed that, in providing the transition from the physicalworld of vectorial mechanics to the abstract analytical realm of generalized scalarformulation found in analytical mechanics, Lagrange gave us superior procedures forderiving equations of motion for complex mechanical systems.A comment concerning the Lagrangian approach to the problem may be appropriate. It has not been popular in multibody dynamics because the kinetic energyexpression can become extremely lengthy and perhaps unmanageable (before the advent of high speed computers), as indicated by Hooker[7]. On the other hand, itseffectiveness has been tested by a variety of problems in analytical dynamics for morethan 200 years. More specifically, the approach automatically satisfies holonomicconstraints. It provides expressions for useful functions such as Lagrangian, Hamiltonian, conjugate momenta, etc., and the form of the governing equations conveysa clear physical meaning in terms of contributing forces. Equally important is thefact that the equations are readily amenable to stability study and well suited forcontrol design. Furthermore. the method of Lagrange yields the governing equationsof motion whose structure is independent of the system geometry. Finally, if equations are to be derived by symbolic processing, the primary criterion for selecting aderivation procedure would he amenability to automation. which encourages reduced10dependence on engineering judgement. These considerations favour the Lagrangianapproach for studying dynamics of large multibody flexible systems.The pioneering research in rnultibodv dynamics was driven largely by the allureof the equations themselves, and not by the need of computer programs to simulatedynamics of the spacecraft being designed at the time. For Hooker and Margulies,and for Roherson and Wittenhurg, the equations themselves were the goal, and theirelegance was more valued than their utility. Nowadays, the situation is quite different;the research efforts are governed by the need to develop tools for design and testingof spacecraft committed for development. Several general purpose computer codesaimed at studying dynamics of multibody systems have been commercially available for sometime. They include DISCOS[22], ALLFLEX[23], TREETOPS[24] andSD/FAST[25J. These are primarily suitable for systems with large rigid body motionwith flexible members uidergoing small deformations .As expected, each asserts avariety of distinctive features and they have been used extensively with a varyingdegree of success often governed by the nature of the system, experience of the userand computational tools available. In general, they present a scope for improvementin representation of axial foreshortening leading to dynamic stiffness effects, rotaryinertia and shear deformation; computationa.1 strategies, in particular with referencet.o symbolic manipulation; selection of admissible functions; and user-friendliness.1.1.2 Structural deformation representationFlexible multihody simulation algorithms employ discretization of the continuumbased on the classical assumed modes method [26]. It proposes that the deformationfield for each flexible component in the multibody chain can be expressed as a seriesof spatial and temporal functions. In general, the spatial functions can be any admissihle function satisfying geometric boundary conditions and they are often referred to11as mode shapes while the corresponding temporal functions are termed generalizedcoordinates. A daunting task facing dynamicists and control engineers is the choiceof modes in discretizing structural deformations. In particular, the focus is on selecting the modes which adequately capture the interaction dynamics involving systemparameters, control characteristics and intial disturbances.To establish a framework for selection of modal functions, consider a typical spacecraft with a rigid huh and elastic appendages. A hierarchy of modes would need tohe selected in order to faithfully represent deformation history of the appendages.Either component modes may be used in which the appendages vibrate with respectto the central body but independently of each other; or system modal representationmay be performed where the entire structure vibrates accounting for dynamical interactions throughout the spacecraft. The component modes’ method was pioneered byHurty[27] in the early 1960s. It involves determination of the appendage admissiblefunctions with enforced geometric compatibility between the adjacent elements of thesystem. Since temporal generalized coordinates are associated with each mode fora given component, the size of the problem is directly dependent on the number ofappendages and modes. Another drawback of this method is in the development of adynamically faithful set of admissible functions for geometrically complex structureswith interconnected flexible components. Further investigations on modal selectionby Craig and Bampton[28], Benfleld and Hruda[29] and Hughes[30] were aimed atimproving modal convergence by more precise specification of geometric conditionsat internal boundaries between the substructures. On the other hand, with ‘systemmodes’, frequently obtained by the finite element method, the size of the problemis independent of the geometric complexity of the structure. The study by Hablani[31j suggests that, for a given order of discretization, prediction of the spacecraft’sdynamics improves as one migrates from the component to the system modes. Fur12thermore, system modes are physically more meaningful, since a modal frequencyrepresents resonance of the the entire struct.ure.Another important issue is to model accurately a flexible lattice structure. Forexample, in the case of the Space Station, should the truss structure be consideredhomogeneous, and if so, how? Design of the lattice structure, which constitutes themain truss, must he highly reliable since it cannot be tested full scale in its operationalenvironment prior to the flight. On the other hand, a detailed finite element analysisof the structure using truss bar elements would involve a large number of elementsand nodes thus becoming uneconomical. especially at this initial design phase whenthe structure and its associated systems are subject to modifications.Procedures presently used to study space lattice structures fall into three categories [32.33]: (i) matrix methods, (ii) field techniques, and (iii) continuum modelling.The first consists of numerically solving a set of algebraic equations in a direct manner.The other two are analytical approaches.Numerical methods, such as the finite element and finite difference procedures,use matrix approach based on discrete element idealization. Field techniques attemptto describe lattice structures or a pattern of elements analytically. The popularityof this approach is due to the fact that the elemental nature of the lattice bay ispreserved in the governing equilibrium equations. In comparison with the numericalmethods, a field analysis does not increase the problem size as the number of bays inthe truss structure is increased.The last method consists of approximating a repetitive lattice by an equivalentcontinuum. This ensures that the continuum model exhibit equivalent energy levelsto the actual discrete lattice. Here qualitative decisions ieduce the dirnensionalitvof the mathematical model and physically identify the nature of the deformation131(e.g. warping, bending, shear). Furthermore, as the number of repeating modules isincreased, the accuracy of the response improves although the model size does notincrease.This energy equivalence concept has been demostrated in a variety of investiga.tions. For instance, continuous systems involving particular types of beam andplate type lattice structures have been developed by Berry et al.[34]. and Juang andSun [35]. Their studies suggest. the necessity to model large truss structures by thegeometrically nonlinear Timoshenko beams. While shear and rotary inertia lead tosmall corrections to the Bernoulli- Euler theory for the lower modes of long and thinbeams, significant errors may be introduced if they are neglected when dealing withthicker beams, or for the higher modes of any beam.1.1.3 Attitude and vibration controlThe subject. of attitude and vibration control in iarge space Structures (LSS) hasreceived considerable att.ent ion and has evolved quite rapidly in the last. thirty years.Balas[36] and Meirovitch[37] have presented an excellent overview of approaches to thecontrol of LSS. Unlike the rigid spacecraft design, LSS control is an interdisciplinarysubject drawing on structural mechanics, continuum representation, optimization andidentificat.ion. Issues in modelling and control design include control/structure interactions, actuator and sensor selection and placement, controllability and observability,control and observation spillover, sensitivity and robustness, modelling uncertaintiesand errors, to name a few.In application of linear control theory to flexible orbiting systems, three procedures have been commonly used t.o develop control laws for large flexible st.ructures:(a) decoupling techniques; (b) pole placement; and (c) optimal linear regulator theory.11The decoupling technique can be applied in two situations where: (i) the linearstate equations are decoupled using state variable feedback; and (ii) the open iooplinear equations are first transformed into a decoupled set in the modal coordinatesand then control laws are developed independently for each mode. Thus it becomesnecessary to transform the control laws as expressed in modal coordinates to theactual control in the original coordinates.In the pole placement method, the overall transient requirements of the system areconsidered instead of concentrating on the behaviour of the individual coordinates.The linearized equations of motion are recast in the state space form[38j and thefeedback control law selected.The linear regulator theory allows one to set, a priori, distinct penalty weightingfunctions on the control effort as well as the state variables. The feedback controllaw is selected such that a quadratic performance criterion is minimized[39].Both the linear regulator problem and the pole clustering method can result insome of the closed loop frequencies being orders of magnitude greater than those of theuncontrolled system. These higher frequencies may also correspond to the frequenciesof higher modes not included in the truncated model. To account for such effects,the order of the original system model will have to be increased in order to avoid theeffects of spillover[40]. On the other hand, these methods have the advantage of beingapplicable even in situations where the number of actuators is less than the numberof modes in the mathematical model, in contrast to the the decoupling methods.The primary requirement of the flight control system is to maintain the SpaceStation attitude within ±5° with reference to the orbital frame. Control MomentumGyros (CMGs) will be utilized as the primary actuating devices for most of the assembly sequence due to their greater torque capability for given weight and power con1.3sumption, as opposed to momentum wheels. However, they have limited momentumstorage capabilities before reaching saturation. Therefore. a scheme for desaturationof the CMGs will have to be developed to remove secular momentum build-up. Desaturation methods include the use of magnetic torquer bars, aerodynamic torques,fluid desaturation, reaction control systems, and gravity gradient torques[41].Although the subject of attitude control of spacecraft has received considerableattention and has evolved rapidly in the past three decades[42,43], studies dealingspecifically with attitude control of the Space Station are relatively recent and few.Sham and Spector[44] developed a continuous feedback controller architecture thatintegrated the functions of attitude as well as momentum control using gravity gradient torques. They also provided a compensation scheme t.o decouple roll and yawdynamics so that roll, pitch and yaw controllers can he designed independently.Tie et al.[45] incorporated “cyclic-disturbance rejection filter” in an integratedattitude/momentum controller. Inclusion of the filters is motivated by the fact thatthe spacecraft with articulated solar arrays in near earth orbits are subjected to torquedisturbances at multiples of the orbit frequency. The filters are used to suppressthe effects of disturbances on roll momentum, yaw attitude, and pitch attitude ormomentum. However, they do not suppress roll attitude or yaw momentum at orbitalfrequency, nor do they suppress constant attitude offsets.Sunkel and Shieh[46J. using the controller architecture in [45], have proposed thematrix sign scheme for placing the closed-loop poles in a prescribed sector of the left-half plane. This method has an advantage over the Linear Quadratic Regulator (LQR)procedure since, for nonautonomous systems, it does not require time consumingiterations to determine the optimal gain matrices for the cost-function.Kumar et al.[47] have proposed a method which utilizes an approximate knowl16edge of the Torque Equilibrium Attitude (TEA) in roll, pitch and yaw directions. Incase of large deviations, the method reduces amplitudes of attitude oscillation andpeak momentum requirements, thus decreasing the time required to reach the steadystate.Warren et al.[48] present a method for “asymptotic momentum management” thatlets the spacecraft assume a periodic attitude motion for which no control torqueis required. i.e. a dynamic TEA. The controller is configured to suppress CMGs’momentum in all axes at all frequencies using disturbance rejection filters. For fixedperiodic disturbances, the result is a periodic spacecraft motion that (for a linearmodel) requires no CMG input even for unstable spacecraft dynamics.The contributions mentioned above have been based on linearized equations obtained by assuming small deviations from the designed or operational orientation andthe products of inertia have been neglected. However, due to the triaxia.l nature ofthe Space Station, the discrepancy between the TEA and the designed orientationis expected to be significantly large. Following on this poilit, Harduvel[49] opted foran approach to momentum management of the Space Station that permits three-axiscoupled dynamics that automatically generates gravity gradient and aerodynamictorques required for momentum control by maneuvering the spacecraft.When suppressing structural vibrations, the closely spaced modal frequencies,coupled with the uncertainties in structural modelling, place stringent robustness requirements on the control system. A frequently used approach for ensuring robustnessis to colocate the sensors and actuators[36]. However, because of physical limitationson hardware placement, this approach is often not feasible, resulting in undesirablecontrol/structure interactions.NASA was once involved in two experiments to study these phenomena. The17COPS Mast Flight System experiment [50] represents a typical sensor/actuator placement problem, where control of the primary modes requires consideration of theinteractions with the secondary modes. Wie[50] has proposed two control strategiesfor this experiment: a rapid transient control of the tip deflection to an impulsetype disturbance; aid a steady-state vibration suppression using noncolocated actuator/sensor pairs. Ozguner et al.[51] have studied the feasibility of decentralizedcontrol applied the COPS configuration.The SCOLE[6] was designed to provide a physical test bed for assessment ofcontrol procedures applicable to complex flexible spacecraft. The SCOLE problemis defined as two design challenges. The first challenge is to design control laws fora mathematical model of a large antenna attached to the Space Shuttle by a longflexible mast. The second challenge is to design and implement the control scheme ona laboratory model and assess its effectiveness in achieving desired pointing accuracy.Modi and Morita[52] have developed three different control strategies for the SCOLEconfiguration in which the optimal linear quadratic regulat:or theory and the directvelocity feedback principle are employed.Bainum et al.[53-54] have provided considerable insight into the behaviour ofcomplex large space systems by modelling basic systems such as flexible beams andplates in orbit. Numerous studies aimed at control of flexible spacecraft have beendocumented in the literature. They include contributions by Denman and Jeon (eigenvalue relocation,55), Ih et al. (adaptive control,56), Meirovitch et al. (perturbationapproach,57), Young (decentralized control,58) and Williams and Juang (pole/zerocancellation,59), to name a few. Ng[60] has reviewed this literature at considerablelength.The linear control procedures, based either on t.he Bellman principle of optimalitv18or the Pontryagin maximum principle, have served efficiently in the design of a largeclass of dynamical systems. However, in reality, modelling uncertainties may renderthe nonlinear character of the system important enough to make the linearizationunsuitable. The mathematical theory of bilinear systems and the general nonlinearcontrollability and observability theory have been investigated, via certain aspectsof the differential geometry such as Lie algebra[61j, to yield some understanding ofnonlinear control systems. Dwyer and Batten[62] have proposed an attitude motioncontroller for a rigid spacecraft based on the inversion of a. nonlinear input-outputmap. Singh and Bossart[63j attempted a linear representation of the nonlinear dynamics of the rigid Space Station using feedback linearization theory[64}. More recently, Karray and Modi[65] extended the application of the feedback linearizationtechnique to flexible systems. The nonlinear control strategy based on this techniquehas several advantages over controller designs based on linearized dynamics. Firstly,a linearized model is only approximate and does not embody the more complete information contained in a nonlinear model. Also, linearized models are valid only nearoperating points, thus restricting the authority of the designed controller to smallscale maneuvers. Obviously, any attempt at overcoming this limitation would requiregain scheduling. More recently, the design of robust control strategies based on theH approach[66] has received some attention.This brief literature survey provides a synopsis on current research activities inthe field of dynamics and control of LSS. with particular application to the proposedSpace Station. It revealed that more realistic and accurate mathematical modellingtechniques are needed in order to design and succesfully deploy large flexible systemsin space with an acceptable degree of confidence.191.2 Scope of the Present InvestigationThe Space Station program, as it stands today, consists of design, construction,operation, maintenance and evolutionary phases extending over two decades. It willbe relatively compact and rigid during the Space Shuttle transportation, but later,upon integration, will emerge like a butterfly from its cocoon, extending antennasand booms and unfurling solar arrays until the structure has undergone completemetamorphosis. Accordingly, this dissertation primarily concentrates on understanding of complex dynamics and control issues associated with this synergetic process.The emphasis throughout is on the development of a methodology applicable to alarge class of systems leading to physical appreciation of the system behaviour in thepresence of structural flexihlitv and operational disturbances.Flexibility, both in the structural and modelling sense, is the key issue governingthe Space Station dynamics and control simulation study. Chapter 2 focuses on structural flexibility modelling aspects during simulation of large overall motions of systems containing deformable bodies. In this context, moda.1 representation techniques,model reduction methods and dynamic stiffening phenomena are inspected. Throughsimple examples, it clearly brings to light, rather convincingly for the first time, limitations of the component mode representation. Chapter 3 develops a comprehensivegeneral formulation for studying dynamics of large flexible orbiting systems. Theformulation is applicable to space systems with an arbitrary number of beam, plate,membrane and rigid members, in any desired orbit, interconnected to form an openbranch-type topology. It allows for appendage slewing and translation and accountsfor transient system inertias, shift in the centre of mass, geometric nonlinearity, sheardeformation and rotary inertia effects. Lagrangian formulation of flexible multibodysystems accounting for simultaneous joint translation and rotation represents an im20portant feature seldom encountered in existing multibody formulations.The multibody dynamics formalism involves the flexibility discretization in termsof modal functions. Chapter 4 outlines the finite element formulation used in obtaming the system modes. For geometrically time varying systems, the modes areupdated at user specified intervals, thus maintaining a faithful representation of structural fiexibhity throughout the simulation sequence. This feature constitutes an original contribution to the field of multibody dynamics of large flexible space systems.By employing structural deformation representation in terms of system modes, theflexibility vector is directly expressed in terms of the system frame rather than withrespect to the local appendage base coordinates. This has several distinct advantages:the formulation effort and derivation time are dramatically reduced; the complexityof the nonlinear, nonautonomous and coupled differential equations of motion is considerably simplified : the expressions representing structural flexibility are decoupleddue to the orthogonality of the normal modes with respect to the mass and stiffnessmatrices; and the physical interpretation of the results becomes more meaningful,since the modal frequencies represent resonance conditions for the overall structure.Next, a general purpose computer program has been developed based on theformulation presented in Chapter 3. The implementation has been carried out inthree stages, and these constitute the major building blocks of the simulation program.The Structural Dynamics Module defines the geometric and flexibility characteristicsof the system under study. The spacecraft physical data. in terms of nodal meshand structural parameters are specified by the user, and the finite element modalanalysis program generates the dicretized modal functions, the centroidal vector, andinertia properties. This information is supplied to the Modal Integrator for processingof modal momentum coefficients and subsequently transferred to the next stage of21simulation, the Multibody Dynamics Module, where the equations of motion areobtained by a judicious application of symbolic manipulation.The principal disturbances acting on t.he Space Station are the gyroscopic andgravity-gradient torques associated with the orbital motion, and those contributedby the operational environment. The gyroscopic and gravity-gradient torques areinherently present in the formulation of the systeni dynamics. Operational disturbances are expected to be the key factor affecting microgravity environment of theStation and the pointing performance of attached payloads. They are incorporatedthrough the generalized force vector. The transient disturbances include Shuttle docking, treadmill activities, crew member kickoffs and nominal MSS operations. Steadystate disturbance sources considered include solar array sun tracking and aerodynamicforces. Results on multibody system response to such a wide spectrum of disturbancesrepresent an important contribution to a rather scarce literature on the subject.The Space Station will be operating in a nominally unstable attitude orientationin a disturbance rich environment. Thus, the primary requirement of the SpaceStation flight control system will be to maintain attitude, microgravity level andpayload pointing excursions within accept able limits. With these points in mind,Chapter 5 proposes linear and nonlinear control strategies by employing the LinearQuadratic egulator (LQR) theory and the feedback inearization technique (FLT),respectively. To that end, the differential equations of motion have been linearizedabout the time dependent equilibrium position and subsequently manipulated to putin the form suitable for integration. In other words, the Control Module arranges thelinearized equations of motion in state space form for interfacing with user-definedcontrol strategies. Nonlinear control of large flexible structures, such as the SpaceStation, represents a field wide open to innovative contributions.22Finally, the finite element analysis used in the representation of structural deformation makes the present algorithm ideal for visualization of the spacecraft dynamicsand control through computer animation. A video depicting modal interactions ofthe evolving Space Station has been produced in collaboration with the UniversityComputer Services Visualization Group.The present formulation, associated algorithm and computer implementation constitute a comprehensive and versatile simulation tool for studying dynamics and control of a large class of complex space systems. Its applicability is demonstrated bystudying various realistic situations for the evolutionary Space Station. The uniquefeature of this study is evident in the development of an interdisciplinary integratedalgorithm synthesizing multibody dynamics, finite element method for modal discretizat.ion, symbolic manipulation, application of linear and nonlinear control theories, and computer animation.23CHAPTERTWOFLEXIBILITY MODELLING ISSUESThis chapter brings to light flexibility modelling aspects in connection with themultihody dynamics simulation. It examines fundamental issues that arise duringformulation of the equations of motion for dynamics and control simulation of largeflexible orbiting structures. Modal discretization and model reduction techniques areassessed; and the effect of dynamic stiffening in rotating flexible structures investigated. The section concludes with some remarks on the relevance of results withrespect to modelling of the flexible Space Station type configurations.2.1 Modal DiscretizationThe class of dynamical systems, generally referred to as flexible multibody systems, are characterized by assemblage of rigid and elastic bodies. The motion ofsuch systems is described by a set of hybrid, nonlinear, noiiautonomous and coupleddifferential equations. In order to extract meaningful information from this complexmathematical representation, the partial differential equations are first discretizedinto a finite set of ordinary differential equations through introduction of admissible functions. satisfying geometric and natural boundary conditions, together withtime dependent generalized coordinates. Thus the procedure approximates elasticdeformations of the system by a series of spatial and temporal functions.6 =T(x)(t), (2.1)24where is a matrix of spatial admissible functions; and is a vector of temporalgeneralized coordinates. Focus here is on the choice of the spatial functions. Themore commonly used approaches are: (i) the assumed modes method; and (ii) thefinite element method. These are briefly touched upon here.2.1.1 The assumed modes methodFor discretizatiori a.t the component level, the individual structural member (body)is allowed to vibrate with assumed boundary conditions (fixed or free) while the restof the system is held stationary. Based on this assumption. the use of a consistentkinematic procedure ensures geometric compatibility, i.e. the same displacement andslope at a point common to any two substructures. Provisions can be made formatching natural boundary commditions (bending moment and shearing force) to improve convergence. The component mode set may also he enlarged until t.he solutionof the system equations converges, with an increase in the solution time.Another avenue to represent structural deformation is to discretize using systemmodes, where the entire structure vibrates in unison such that dynamic interactionsbetween the various components of the spacecraft take place. For a given order of discretization, accuracy of the system dynamics is likely to improve as one migrates fromthe component to system modes. The system representation is physically meaningful, since the modal frequencies represent resonance character of the entire structure;the flexibility contributions are uncoupled, implying reduced computational effort;and modes can he chosen in a frequency range of interest. Nevertheless, the systemmodes approach has not yet been characterized, assessed, and documented in theopen literature, in the context of a general flexible multibody dynamics formalism.The system modes can he calculated in their entirety. However, if the model contains a large number of degrees of freedom so as to make the modelling of the complete2.5system intractable and computationally inefficient, one can resort to the componentMode synthesis (CMS), first proposed by Hurty[27]. The technique consists of modelling the motion of the individual substructures. Three types of modes are selected torepresent their motion: rigid-body modes, constraint modes and normal modes. Theconstraint modes are defined as those producing unit displacement on each redundantconstraint in turn, with all other constraints fixed. The normal modes represent themodes of vibration of the substructures with all constraints fixed. Craig and Bampton[28] subsequently suggested two types of substructure modes: boundary modesproviding for displacements and rotations at points along the substructure boundaries, which are related to the constraint modes of Hurty: and modes correspondingto completely restrained boundaries. The substructures modelled individually aremade to act together as a single unit. This is achieved by eliminating the redundantgeneralized coordinates, arising from the fact that displacements at points commonto two adjacents substructures are included twice in the overall formulation, once foreach substructure. The elimination process is based on the use of constraint equationsresulting from the enforcement of compatibility conditions, i.e. the displacement of aboundary point shared by two substructures is the same. In yet another approach tothe problem, proposed by Benfield and Hriida [29], the effect of adjacent substructuresis accounted for by subjecting a given substructure to inertial and stiffness loadingsat the boundaries.2.1.2 The finite element methodDiscretization of a structure using the hierarchical finite element method is closeto the assumed component modes method. Iii the component modes approach, theadmissible functions are defined over substructures and in the finite element method,the local functions are polynomials defined over finite elements. Thus, in the finite26element approach, the system is regarded as an assembly of many discrete elements,where each element locally models a portion of the domain. The family of piecewisecontinuous elements constitutes a finite element model. The nature of the local interpolation functions, the size of the finite elements, and the degree of inter-elementcontinuity affect the precision of this approach.The entire structure is required to move in unison by imposing rotational, translational, and continuity constraints at the joints and boundaries between contiguouselements. In addition, the internal forces are required to balance at the joints. Consequently, the finite element method expresses the displacement of any point in astructure as a function of a finite number of displacements, usually at the boundariesof the elements.The flexibility of the method lies in the fact that the analyst is free to specify andvary the degree of modelling complexity in the various subsystems that comprise thesystem.The major drawback of the finite element approach is that the degrees of freedomrequired for accurate representation of deformations may be large. In fact, the SpaceStation structural model will require several thousand degrees of freedom to modelit precisely. Fortunately, the nllmber of system modes required is relatively small. Itis of interest to note that the assumed modes method for complex structures usuallyconverges with fewer degrees of freedom than the finite element representation ofthe same structure. However, the convergence behaviour is rather dependent on theparticular structure and its complexity.For a typical spacecraft modelling problem, the system natural frequencies andcorresponding eigenvectors or modes are among the most important final results of afinite element analysis. These in turn, can be used in the assumed modes method, by27associating each system mode with a generalized coordinate, as will he demonstratedin the ilustra.tive example to follow.2.1.3 The transformation matrix CConsider a flexible multibody system as shown in Figure 2-1. The equations ofmotion can be derived by adopting a consistent kinematic procedure to describe themotion. To this end, introduce an inertial frame F. with the origin at O, a set ofbody axes X, Y, Z (frame F)with the origin at O and attached to the centralbody B at O; and a frame Fa with the origin at Oa and attached to the body Baat °a To begin with, consider the formulation procedure aimed at the central bodyB and appendage Ba. Extension to other appendages follows logically. The positionvectors to typical deformed mass elemeits in bodies B and Ba with respect to theinertial frame F can be written as:(2.2)aRc+ca+ca+Ca+6a); (2.3)wherevector from O to O:position vector from O to an undeformed elemental mass in B;elastic deformation vector of drn w.r.t.. F;ca vector evaluated at °aca vector evaluated at °atransformation matrix to define orientation of Fa w.r.t. F.The transformation matrix C, givell in Eq. (2.3), is used to project vectorsdefined in the local coordinate frame onto the system frame attached to body B.28BaGeneral flexible multibodv modelThis is done to guarantee displacement and slope compatibility between the varioussubstructures, as required by the consistent kinematic procedure.A comment concetning C in the derivation of the equations of motion would beappropriate. In the component modes method, C is a function of the joint flexibilityrotation, which translates into dependence on the generalized coordinates. Thus= C(p), (2.4)which, upon substitution into the Lagrangian equations, yields terms like d/dt(6C/6),8c/8p and 0C/8. These terms considerably complicate the formulation process.On the other hand, using system modes, C is independent of the generalized coBFigure 2-129ordinates, since the structural deformations are automatically expressed in terms ofthe coordinate frame attached to the central body B. This results in equations ofmotion which are relatively simple, and of manageable size. resulting in an efficientformulation and computer implementation.2.1.4 Illustrative exampleIn order to illustrate the application of discretization methods for elastic systems, consider an L-shaped beam as shown in Figure 2-2. Though geometricallysimple, it contains one important fea.ture of large space structures - interconnectionbetween flexible bodies. The proposed Space Station with a large number of flexibleinterconnected bodies presents a challenge in dynamical analysis and control seldomencountered before. The simple L-shaped beam may reveal some useful characteristicbehaviour.The structural parameters used in this analysis are: the beam length L1 = =10 m; the cross-sectional area A1 = A2 = 3.5 m; the moment of cross-sectional area= ‘2 = 1 m4; and the beam masses m1 = rn2 = 130 kgs.It is proposed to study the dynamics of the structure by assigning three differentcombinations of bending stiffness to the two beams:Case 1 Case 2 Case 3Body E. Nm2 E. Nrn2 E. Nrn2B1 2x107 2x107 2x107B2 2x103 2x105 2x107The dynamic response of the beam B2 at the tip in the horizontal direction wasmonitored for two different types of forcing functions: an impulse, and a ramp profile(Figure 2-3).30F2Figure 2-2 L-shape cantilever beam.The three different structural deformation modelling strategies are compared:(i) The system modes method, where each generalized coordinate is associated witha system mode, calculated by the finite element method. The first six systemmodes are employed in the discretization process for each case study as shown inFigures 2-4(a), (b) and (c). The associated frequency spectra for the three casesare presented in Table 2-1.(ii) The component modes method, where each generalized coordinate is associatedwith an admissible function in the set used to discretize the body. Three termsin the summation are employed to represent the flexibility of each of the bodies.For body B1, the first three modes corresponding to a. cantilevered beam with anF13150 50Figure 2-3 Forcing function profiles applied at t.he free tip of the L-shape beam:(a) impulse; (b) ramp.equivalent tip mass (mass of body B2 plus the inertia of B2 w.r.t the frame atthe joint), to simulate the presence of beam B2, are considered. For body B2, thefirst three modes corresponding to a cantilevered beam are employed (Figures 2-5and 2-6). The corresponding frequency spectra are given in Tables 2-2 and 2-3,respectively.(iii) The direct. int.egration method, available in the general purpose finite element program ANSYS[67]. Here, the equations of motion are integrated using a numericalstep-by-step procedure. The term direct implies that prior to the numericalintegration no transformation of the equations int.o a different form is carriedout[68].za)C)0LL0za,C.)I0LI00Time (s) 0.5 0 Time (s) 532Figure 2-4 System modes for the L-shape beam: (a) Case 1.Mode I Mode 2 Mode 3I-Mode 5 Mode 6—-Figure 2-4 System modes for the L-shape beam: (b) Case 2.Mode 433Table 2-1 System mode frequencies for the L-shape cantilever structure.System Mode Frequencies (Hz)Casel Case2 Case3fi 0.07 0.67 2.31f2 0.42 2.82 6.26f 0.57 4.18 28.84f4 1.11 10.22 40.42f 1.75 11.20 56.18f6 2.07 19.93 93.62The simulation results, comparing the three methods of solution, are presentedFigure 2-4 System modes for the L-shape beam: (c) Case 3.34Mode 1 Mode 2 Mode 3extension_______,_EE1/.Mode 4 Mode 5 Mode 6extensionFigure 2-5 Component modes used to represent the body B1 as a cantileverbeam with tip mass.Mode 1 Mode 2 Mode 3. extension—----. ------...Mode4 Mode 5 Mode 6extension______•%_.___/Figure 2-6 Component modes used to represent the body B2 as a cantileverbeam.33Table 2-2 Component mode frequencies for the beam B1 modelled as a cantilever with a tip mass.Component Mode Frequencies (Hz)Case 1 Case 2 Case 3fi 2.38 2.38 2.38f2 8.30 8.30 8.30f3 31.72 31.72 31.72f 4996 4996 4996f. 11863 11863 11863f 126.91 126.91 126.91Table 2-3 Component mode frequencies for the body B2 modelled as a cantilevered beam.Component Mode Frequencies (Hz)Case 1 Case 2 Case 3fi 0.07 0.69 6.92f2 0.42 4.20 41.76f 0.58 5.80 58.00f 1.11 11.11 111.19f 1.75 17.53 175.32• f6 2.03 20.28 202.75in Figures 2-7, 2-8 and 2-9. For Case 1, where the frequency spectra of both thebeams are quite distinct, there is an excellent correlation between the three methods.The dynamics is dominated by the more flexible beam (body B2): and the frequencytables reveal that the overall system frequencies and the beam B2 frequencies are36comparable. The dynamic simulation shows that the maximum tip displacement inthe y-direction reaches around 0.3 m when the system is subjected to a. 5 Ns impulse.With a. ramp forcing function of 50 N at the end of 2 seconds, the beam transversedisplacement reaches 5 m.In the second case, when the frequencies of the individual beams are closer, butstill distinct, a. slight discrepancy in the results is observed. The direct integrationsolution and the system modes superposition method agreed very well, while thecomponent modes solution slightly underestimated the magnitude of t.he response.Nevertheless, the frequency content of all three solutions continue t.o exhibit. an excellent correlation. The table of frequencies for this case shows that there is still agood match between t.he system frequencies and beam B2 frequencies. The responseof t.he struct.ure when subjected to the impulse and ramp forcing functions shows themaximum tip transverse displacements of 0.04 in and 0.12 m, respectively.In Case 3, the two beams have identical structural characteristics. Note, thecomponent modes method, with three modes used to discretize each beam as before,fails to exhibit the correct dynamic behaviour (Figure 2-9). In fact, while the directintegration and the system modes superposition results agree quite well, the component modes solution shows significant discrepancy in the response, both in magnitudeand frequency content. This anomaly was further probed. An attempt was made toimprove the component modes solution by increasing the number of modes in thediscretization process, since one would expect the solution to converge to the rightvalue in the limit (as the number of modes tends to cc). With six modes representing each beam, there was a small improvement, in the results, indicating the correcttrend, however, it still underestimated the maximum tip displacement by approximately 50% (ramp case). This suggests that treating B2 as a cantilever beam does37not simulate the physical reality when the two beams have comparable elastic characteristics. In other words, the assumed boundary conditions were not valid. Thus thecomponent modes solution appears to he dependent on the physical characteristicsof the system, and that the boundary conditions have to be judiciously selected foreach set of design parameters, to converge to the right solution. In the present case,body B2 may have to be modelled as a beam with springs at the point of attachmentin order to properly account for inertia and stiffness characteristics of beam B1 atthe point of attachment. This is an important piece of information in the context oflarge multibody flexible systems, such as the proposed Space Station. The component modes method is prone to errors if proper steps are not taken to incorporate theappropriate boundary conditions between the various structural members. Nevertheless, this can be circumvented by using component mode synthesis [27,28,29]. Thisapproach entails loading a component. with mass and stiffness contributions from theremaining parts of the structure in an attempt at capturing the modes of the systemthat the component is a part of. This requires the evaluation of the connectivitymatrix; and the component mode set may need to be recomputed if the geometry,mass, or stiffness properties of any component change during the design process. Tosummarize:(1) The component modes method is unable to accurately model the system dynamicswhen the boundary conditions are not. properly defined. In the context of largeflexible multibody systems, such as the proposed Space Station, this representsa serious limitation of the approach. Moreover, since the transformation matrixC is dependent on the generalized coordinates, the equations of motions arerelatively more complicated and lengthy. This reflects in increased formulationand computational efforts by at least an order of magnitude compared to thesystem modes approach.38(ii) The finite element method. though it offers an accurate and versatile modellingtechnique, requires a large number of degrees of freedom. This makes its application impractical in the implementation of flexible multibody dynamics andcontrol simulations.(iii) The system modes approach offers an alternative, which synthesizes desirable features of the two methods described above. The generalized coordinates are nowassociated with the system modes of the structure; the system representation isphysically meaningful, since the modal frequencies represent resonance characterof the entire structure; the contributions of the structural deformations are uncoupled, implying reduced computational effort: the formulation is considerablysimplified as the transformation matrix C is no longer a function of the generalized coordinates; and the boundary conditions are automatically taken intoaccount by the finite element method. while evaluating the system modes. However, the modal set needs to be updated for geometrically time varying systems tomaintain a faithful structural deormation representation during the simulation.395.0xIc’JE-5.0cta)Figure 2-7 Dynamical response for the structure in Case 1 for the three solution procedures in the presence of: (a) ramp function; (b) impulsefunction.0.0 Time (s) 10.0- System Reponse due to a 5 N S Impulse0.50Ea)C-)Cl)0.00><.50(b’)Direct Integration [ANSYS] -System ModesComponent Modes0.0 Time (s) 10.0400.00><ciI—c’.J44:Ect50.0 Time (s) 10.00.05Ea)C.)ciCl)0.00><0-0.05Figure 2-80.0 Time(s) 10.0A comparison of transverse tip response in Case 2 for the three solution procedures in the presence of: (a) ramp function; (b) impulsefunction.41Ic0.00C’)><I—Ea)crI—c.’JEFigure 2- response for the structure in Case 3 for the three solution procedures in the presence of: (a) ramp function; (b) impulsefunction.Time (s) 5.0Time (s)422.2 Model ReductionIn order to perform dynamical simulation of a system using the discretizationmethods presented earlier, one has to specify the number of modes to be employed.Of course, in general, an increase in the number of admissible functions leads toan improved accuracy. On the other hand, the computational cost also rises. Thismotivates the question: How to choose the number of assumed modal functions inthe dynamic simulation to insure a desired accuracy with a minimum cost? Despitesignificant past research efforts. this question remains a difficult one to resolve ingeneral, except for relatively simple systems. In order to shed some light on thisissue, results for the L-shaped cantilever beam are presented and discussed in thissection.2.2.1 Model reduction algorithmsHaving established, in the preceding section, the advantages in modelling structural deformation at the system level, consider discretization of the physical structureusing system modes. Assuming that the original modal set is an accurate representation of the physical structure, the challenge is to choose a reduced order modal setwhich represents the full order model as closely as possible.Blelloch and Carney[69] have documented two modal selection algorithms. Thefirst algorithm arranges modes in terms of their contribution to the peak magnitude(L error) of the transfer function. which is identical to the approximate balancedsingular value proposed by Ioore [70]:Iii”I 1IIG(s)II = max- 2)2 + 4(2w2 (2.5)• where:43ith modal frequency;damping ratio;‘I’ modal values at locations of force inputs;modal values at locations of displacement outputs;modal values at locations of velocity outputs;modal values at locations of acceleration outputs;The second algorithm, Skeltons modal cost[71j. orders modes in terms of theircontribution to the RMS value (L2 error) of the transfer function, and it can beWrjtteri as:v — +‘‘2i1 (264(L2.2.2 Illustrative exampleThe planar L-shaped cantilever beam with small modal damping is used to demonstrate applicability of the model reduction methods reviewed above. The dampingratio is taken to be 1 for all the modes. It was assumed that a force actuator anddisplacement sensor are located at ‘F’ and ‘u’, respectively. The input/ouput transferfunction for an impulse response is considered and the modal truncation is performedby retaining modes with large singular values. In other words, modal truncation forthe open-loop performance preserves modes with large peak magnitude in the transferfunction.When an impulse (5 N’s) is applied in the transverse direction at the tip of thebeam B2, it is necessary to include modes 3, 4, 5 and 9 in the modal set for accurateprediction of the longitudinal tip deflection of beam B1 (Figure 2-10). Figure 2-11substantiates this finding. Here, the tip response with only modes 3, 4, 5 and 9correlates quite well with the full order model (first ten modes). However, the response44given by the most dominant mode (mode 5) does not represent. an accurate solution.If the sensor is designed to estimate the transverse displacement of beam B1, themodel reduction algorithm picks modes 1 and 2 only (Figure 2-12). Response resultsin Figure 2-13 show that the first two modes are sufficient to obtain an accurateestimate of the displacement. solution.From the results presented, it can be concluded that the modal selection criteriapicks modes which have a high modal energy for the displacement output of interest.By inspecting Fig. 2-4, it can be observed that beam B1 experiences a. large longitudinal displacement in modes 3, 4, 5, and 9, while modes 1 and 2 exhibit a largetransverse displacement. However, in any realistic situation, engineering judgementwill need to be combined with the results of modal selection algorithms. For a complex system such as the Space Station, although modal selection may eliminate a fewmodes which do not. couple strongly with any input or output, the number of modesretained is still likely to be large.4.3-200-400•11.50101 102 (rad/s) 10Va)VDC0x‘VU,0C.)•00Figure 2-100IFigure 2-11Reduced order model for the L-shape structure with load appliedin the transverse direction and the output is the longitudinal tipdisplacement of the beam B1.1.50-1.50.0 Time (s) 0.1Evaluation of the reduced order model through a comparison of thedynamic response to a 5 Ns impulse using modes 3, 4, 5 and 9 andthe full order model (first 10 modes).46C0xC.0Ct102 (0 (rad!s) io0101 102 (0 (rad/s) ioCto -200VC0)Ct-4001.5Figure 2-12 Reduced order model for the L-shape structure with load appliedin the transverse direction to B2 and the output of interest is thetransverse tip displacement of beam B1.Ct0x25a -2.5FrECua,Figure 2-130.0 Time (s) 1.0Evaluation of the reduced order model is demonstrated by carryingout the dynamic simulation for a 5 Ns impulse with modes 1 and 2,and compared to the full order model (first 10 system modes).472.3 Dynamic StiffnessIn the assumed modes method presented in Section 2.1, the structural flexibilityinformation is contained in 6, the deformation vector. Let6 = öiei + 62e + 63ë, (2.7)where ë1, ë2 and ë3 are mutually perpendicular unit vectors fixed in the local coordinate frame of the appendage. In this ‘conventional’ approach, the ö, (i = 1, 2, 3) areusually expressed as i.e. 6 are uncoupled. Thus the velocity expression in theLagrangian is assembled using the assumption that each component of 6, resolved inthe directions of basic vectors è1, ë2 and e3. can be described in terms of a sum of theproducts of independent modal functions and generalized temporal coordinates. Inother words, the deformations are implicitely assumed linear; and displacements inthree orthogonal directions are taken to be independent, which is clearly not true. Forexample, when a beam bends, the axial deformation is a function of the transversedefiections. When the modal functions are assumed to he kinematically independentthrough this “premature linearization”, as introduced in the conventional analysis,the structural component will fail to exhibit the true dynamical behaviour during theoverall system response.The inherent coupling between the axial and bending deformations may howeverbe captured ‘dynamically’ in the elastic forces acting on the differential structural elements. These terms are commonly referred to as arising from geometric nonlinearities.This point is illustrated below by means of two examples.2.3.1 Illustrative examplesConsider a 2-degree of freedom spring-mass system illustrated in Figure 2-14[72j.48It consists of a spinning disk on which a hinged hollow massless tube rests. A torsionalspring of stiffness K8 is collocated at the hinge point which in turn is located atdistance d from the center of the disk. A particle P of mass n slides along the wallsof the tube while it is connected to an axial spring of stiffness K attached to thetube at distance L from the hinge. The disk rotates at an angular rate of (t).A typical assumption made in the dynamic analysis of structures undergoing largeoverall motions is that the structural deformations remain small. In order to capturethis assumption in the simple 2-d.o.f. model, it is required that the rotation ofthe tube as well as the displacement, s, he small. Dynamical equations of motionfor the system can now be written in terms of the natural set of coordinates (s,6).By studying this system, one can gain insight into deleterious effects of “prematurelinearization” in the study of dynamics of large flexible structures.The position vector of particle P is given by:- 1d+(L+s)cos(L+s)sinO J (2.8)The formulation of the equations of motion by the Lagrangian method requires anexpression for the velocity of the particle P. Differentiating Eq. (2.8) w.r.t. time gives.cose— (L+s)8sinO— cl(L+ s)siii8 29.siii9+(L+s)&cos+{d+ (L+s)cos]Substituting Eq. (2.9) into the kinetic energy expression T = , expressing thestrain energy as U = Ks2 + K9&2,and applying the Lagrangian procedure, thenonlinear equations of motion can he written as:rn — in (L + s) (Ô + )2 — ç2 d cos + in d l sin 0+ K3s = 0; (2.10)m(L+s)0+2m(L+s)(0+!2)+rnd(L+s)sin0+md(L+s) cos8+rnL2+2rnLsQ+ins2Q+K6= 0. (2.11)49PFigure 2-14 Two degree of freedom lumped mass model [72].The linearized equations of motion are arrived at in two ways: first using the properway where the full nonlinear equations are obtained first and then linearized in s and8. In the second procedure, the expression for the velocity of P in s and 6 is linearized.Using the linearized velocity expression, the system kinetic energy is obtained and thecorresponding governing equations of motion obtained. The linearization is performedonly with respect to the reference state of the system. The linearized equations ofmotion given by the two approaches are as follows:(i) Correctly Linearized Equations of Motionmass matrix gyroscopic matrixmL2 0 1 + 0 2mL 50 m —2mL 0S50stiffness matrix circulatory matrixK9 + mLd2 rnQ(L + d) 5 + 0 m L 5 &+m(L+d) Ks—mc22 1sf —m2L 0— f_rnL(L+dfl (219)— mQ2(L+d) J’(ii) Incorrectly Linearized Equations of Motionmass matrix gyroscopic matrixmL2 0 0 2mLQ J&0 m sJ —2m 0stiffness matrix circulatory matrix+K9mL22 m(L+d) 5 & + 0 mcLf&m(L + d) Ks — ml2 .s J —mQL 0 s— 5 —m L(L + (2 13)1 mc22(L+d)These two sets of equations have a similar form except for one crucial difference.The stiffness term appearing in the properly linearized set of equations Eq. (2.12)increases in magnitude with an increase in 2, whereas the stiffness term in Eq. (2.13)decreases in magnitude with an increase in Q.Consider three simulated spin-ups: 0 — 3 rad/s; 0 —* 1 rad/s; and 0 —* 0.3 rad/s.The spin-up profiles are shown in Figure 2-15. The following system parameters areselected:K9 = 1.444 Nm.;Ks= 576,081 N/rn;d = 1 in;51L = 10 in;m = 12 kg.For each spin-up case, the solutions to three different sets of equations of are monitored: (i) the full nonlinear equations; (ii) the correctly linearized equations; and (iii)the prematurely linearized equations. For all the cases, variation of 0 with time isplotted as shown Figures 2-16(a), (b) and (c).The prematurely linearized equations of motion can lead to innacurate resultswith instability present for a spin rate of 3 rad/s (higher than the system’s transversenatural frequency). Substantial amplitude and frequency errors are present in the1 rad/s case. Only for the 0.3 rad/s (which is approximately an order of magnitudelower than the transverse natural frequency) do the results from the two equationsets agree with the full nonlinear equations of motion. Figure 2-17 presents plots ofthe induced stiffness forces through the spin-up maneuver for the three profiles. Thedependence of structural stiffness on the Spin rate is apparent, with a large associatedwith the 3 rad/s case. The results clearly indicate that the “conventional theory”, acounterpart of the premature linearization, used in deriving the equations of motion,can lead to erroneous results.Next, consider a continuous, uniform, cantilever beam attached to a rotating baseas shown in Figure 2-18. The differential mass element din experiences a centrifugalforce F induced by the base spin motion. The position vector = + S locates P inframe {a1,a2}, where ? represents the undeformed position of P whileS = 1a+6ãis the deformation. In addition, din feels a spring force I’ arising from structuralstiffness.Accounting for the fact that the beam transverse aiid axial displacements arecoupled, leads to terms referred to here as “dynamic stiffness” since they only appear523 rad/sU)VG)4-CCl)0.0CuC04.-C.)‘I—ciCu-0.6C1 racils0.3 rad/sFigure 2-150.00.0 Time (s)Spin-up maneuver profile for the 2-d.o.f. system.30.0SPIN-UP to 30Figure 2-1610 20Time (s) 30Tube angular deflection for spin-up to: (a) 3 rad/s.53I.0a)a)(tDCFigure 2-16•0IC0.4-C.)ci)D0)0.00Figure 2-160Tube angular deflection for spin-up to: (b) 1 rad/s.30Tube angular deflection for spin-up to: (c) 0.3 rad/s.SPIN-UP to 1 rad/sIncorrectly LinearizedEquations of Motion0.30.0-0.30 10 20Time (s)30SPIN-UP to10 20Time (s)541000a)D-500Figure 2-17 Development of stiffness force during the spin-up.in the stiffness matrix and are always preceded by a quantity (e.g., Q) characterizingthe dynamic behaviour of the base.In the “conventional formulation”, the deformations 6 and 62 are independentlyexpanded in modal sums. In addition, linear strain-displacement relation is assumedin evaluating the elastic potential energy. This is equivalent to regarding the component of the spring force, F, along a2 as zero. However, the component of thecentrifugal force, F, along a2 is positive, thus producing a destabilizing effect. Itshould be noted that working with the coordinates (6, 62) is equivalent to using thenatural set of coordinates (s. 6) as described previously in the spring-mass example.The spin destabilizing effect can be corrected by accounting for the fact that P’ actually acts along b1 rather than a1. As a result, the component. of F along a2, althoughnonlinear, becomes negative thus producing a stablizing effect. This nonlinear term(geometric stiffening) can be captured in the energy based Lagrangian formulationTime (s) 3055b2 —b1Figure 2-18 Cantilever beam model [72].by using nonlinear strain-displacement relations in evaluation of the elastic potentialenergy. Clearly this approach retains all the conventional features and can be appliedto more complex structures. Thus, the equations of motion for the spinning cantileverwith the nonlinear geometric stiffness terms included can be written as:inertia coriolis centripetalF—-‘ F Th Frn L f ( d + m - m L L’( iqi)linearst iffnessEl f’ d2(q) 4± dd2 d2geometric stiffness+ 2L f’ (1 2) [d(Eii)] [d] Fg (2.14)F p Fpa156The modal functions chosen were the first three eigenfunctions of an identical uniform cantilever beam. The numerica.l integration was carried out by theIMSL:DGEAR[73j. Simulation of the spinning cantilever beam was conducted utilizing Eq. (2.14) for: (a) the conventional; and (b) the geometrically nonlinear formulations. The conventional approach neglects the geometric stiffness term in Eq. (2.14).The following numerical values for system parameters were used in the simulation:m = 12 kg;L=lOrn:A = 4 x i0 m2,E = 7 x 1O° N/rn2,G = 3 x 1010 N/rn2I = 2 x 10 rn4.The base was spun-up to 6 rad/s with time-histories as shown in Figure 2-19 andthe inplane tip displacements were recorded (Figure 2-20), which indicates that theconventional formulation predicts instability while inclusion of the geometric stiffningterm properly corrects the solution. In Figure 2-20, with the spin rate of 3 rad/s,the conventional formulation shows a stable but drastically different result from thatusing the nonlinear approach. Thus the conventional modelling method may show aperfectly reasonable but completely incorrect result when simulating spinning systemsundergoing large overall motions and small structural deforniations. It may be pointedout that the conventional modal approach yields accurate results (i.e identical tothose predicted by the nonlinear formulation) when the base spin rate is an order ofmagnitude smaller than the fundamental bending frequency of the beam.The geometric stiffening effect. can also be included using the modal interdepen576.03.00.0Figure 2-19 Spin maneuver profile.dence approach as proposed by Kane et al.[74], which brings the spin induced stiffness terms into the equations of motion through the generalized inertia terms. Thismethod is computationally efficient, however, generalization to an arbitrary systemmay be difficult. Other related studies include the development of a nonlinear strainmodal beam theory by Laskin, Likins, and Longman [75];, and the flexible mechanism model of Turcic, Midha, and Bosnik [76], which uses standard finite elementgeometric stiffness matrices to include spin-induced dynamic stiffness effect.Realizing that something may be amiss in the conventional approach, the naturalquestion seem to be: In what practical situations can one safely apply the conventionalapproach? What situations are likely to require inclusion of geometric nonlinearitiesin the context of large space structures in general, and in particular, with referenceto the Space Station dynamics simulation? To answer these questions, let us considerthe Space Station mission requirements:0 10 20 30Time (s)58_. 0.0C0SPIN-UP to 6 rad/s10 20Time (s)0 30Figure 2-20 Time history of the beam tip displacement: (a) =6 rad/s.0.0-0.3EC)30Figure 2-20 Time history of the beam tip displacement: (b) ! =3 rad/s.0 10 20Time (s)59(i) Stationkeeping In the Stationkeeping case, the overall motions of the componentsare negligible, and small free vibrations are of importance. Thus, it is safe to inferthat the conventional approach will provide accurate simulation results.(ii) MSS Operation The rate of slewing maneuvers is expected to be orders of magnitude lower than the lowest fundamental bending frequency of the appendage.Moreover, the maneuvers are over short durations, with ranges involving fractionsof a revolution. Thus, it seems that the terms missing from the conventional formulation are likely to be small if the corresponding deflections of the flexiblemembes are kept within limits of the linear elasticity;(iii) Spinning Motion There are no high speed rotational motions associated with theSpace Station. Tracking of the sun by the solar panels is at an extremely slowra.te. Nevertheless, for a spinning flexible body, the conventional formulation willyield incorrect results. If the spin speed is very slow in relation to the flexibilityof the body, error will not be drastic, although any parametric studiy undertakenmay prove to be misleading.602.4 SummaryFlexibility modelling aspects concerning discretization, model reduction and thedynamic stiffness effects on the response of flexible structures have been investigated.Based on the analysis, the following general remarks can be made:(i) Discretization of complex systems using system modes is quite attractive in termsof accuracy as well as reduced degrees of freedom. On the other hand, the useof component modes, though convenient, may lead to ina.ccuracies under certain combinations of system parameters. The situation can be improved throughaccurate representation of the boundary conditions, however, this is not alwaysapparent.(ii) ‘When reducing the model of the structure, a weighting factor can be defined interms of modal contribution to t.he peak error (L) or to the RMS error (L2norm) in the transfer function.(iii) The ‘premature linearization” in the “conventional theory” gives equations of motion which can lead to erroneous results. The situation can be corrected by usingnonlinear strain-displacement relations in the calculation of the strain energy. Itis important to note that the conventional theory is not fundamentally flawed; itis the introduction of the linearization process that leads to the discrepancy.61CHAPTERTHREEMULTIBODY FORMULATIONGiven the evolving nature of the Space Station Freedo7n, and having acknowledged the importance of structural flexibility in the dynamics and control study oflarge space structures, an effective methodology that takes into account transient inertia, changing geometry and structural flexibility is highly desirable. This chapterpresents the development of a relatively general formulation particularly suitable forstudying dynamics and control of a large class of flexible space systems comprised ofinterconnected flexible members.3.1 Formulation of the ProblemThis section presents a relatively general Lagrangian formulation of the nonlinear,nonautonomous and coupled differential equations of motion governing the dynamicsof a. system of interconnected flexible members. The formulation has the followingdistinctive features:(a) it is applicable to an arbitrary number of beam, plate, membrane and rigid bodymembers, in any desired orbit, interconnected to form an open branch-type topology;(b) joints between the flexible members permit large angle rotation aimd linear translation between the structural components;(c) the formulation accounts for the gravity gradient potential, the effects of transientsystem inertias and shift in the centre of mass:62(d) the flexible character of the system is described by three-dimensional systemmodal functions obtained using the finite element method;(e) geometric nonlinearity, shear deformation, rotary inertia, variable mass density,flexural rigidity and cross sectional area effects have been included in a consistentmanner, both at the multibody dynamics formulation level as well as in the modaldiscretization procedure;(f) symbolic manipulation is used to synthesize the nonlinear, nonautonornous differential equations of motion thus providing a general and efficient modellingcapability with optimum allocation of computer resources;(g) the governing equations are programmed in a modular fashion to isolate the effectsof appendage slewing and translation, librationa.l dynamics, structural flexibilityand orbital parameters;(h) operational disturbances (Space Shuttle docking, crew motion and maintenanceoperation maneuvers) have been implemented in this dynamic simulation tool.Other disturbances can easily be incorporated through generalized forces andinitial conditions;(1) both the nonlinear and linear forms of the equations of motion have been formulated to permit assessment of a wide variety of control strategies, both linear andnonlinear.3.1.1 System geometryThe system model selected for study consists of an arbitrary number of flexiblebodies connected to form a branched geometry such that B (i = 1, 2,... , N) bodiesare connected to a central body B (Figure 3-1). Thus the total number of interconnected structural members amounts to 1 + N. It should be noted that the number63TrajectoryFigure 3-1 A schematic diagram of the spacecraft model.and locations of the flexible appendages are kept arbitrary so that the configurationcan be used to study a large and varied class of future spacecraft.For instance, the European Space Agency’s L-SAT (Large SATellite system,Olympus) represents a new generation of communications satellite. It has two solar panels connected to a central body. The satellite’s central rigid body and twosolar panels can be simulated by bodies B, B1 and B2, respectively. In the case ofthe Space Station, all the evolutionary stages can be studied by modelling the maintruss as central body B while the modules, radiators, stinger, and solar panels canbe represented by bodies B.EahPerigee643.2 Kinematics3.2.1 Reference frames and position vectorsConsider the spacecraft model in Figure 3-2. The centres of mass of the undeformed and deformed configurations of the spacecraft are taken at °m and Om,respectively. Let X0, Y0, Z0 be the inertial coordinate system (F0) located at theEarth’s centre (Figure 3-3). Attached to each member of the model is a body coordinate system helpful in defining relative motion between the members. Thus referenceframe F is attached to body B at an arbitrary point O. In addition, there is aframe Ff (Xf, Yf, Zf) at °im so oriented as to have corresponding axes parallelto those of frame F. Location of the elemental mass dm of the central body Brelative to O is defined by two vectors: ,o indicates the undeformed position of themass element: and the transverse vibration vector leads to the final orientation ofthe deflected mass element.Frame F is attached to body B at the joint. O between the bodies B and B.An arbitrary mass element drn on body B can be reached through the direct pathfrom O via O. Thus motion of dm, caused by librational and vibrational motionsof B, can be expressed in terms of the system coordinate frame in the followingform: the relative position of O with respect to O is denoted by the vector d. Thevectorial quatities Ci (rigid body position) and 6j (structural deformation) definethe orientation and position of the elemental mass dm in body B, relative to O,where C is the matrix representing the orientation of body B relative to body B.3.2.2 Position and orientation in spaceLet the spacecraft. with its instantaneous centre of mass at he negotiatingan arbitrary trajectory about the centre of force coinciding with the homogeneous,GoFigure 3-2 Position vectors.spherical Earth’s centre. At any instant, the position of Om is determined by theorbital elements.As the spacecraft has finite dimensions, i.e. it has mass as well as inertia. Thesystem is free to undergo librational motion about its center of mass. Let X8, Y8,Z represent a moving coordinate frame along the orbit normal, local vertical, andlocal horizontal, respectively. Any spatial orientation of Xf, Yf, Zf with respect tothe system frame F can be described by three modified Eulerian rotations: a pitchmotion. ‘, about theX8-axis; a roll motion, , about the Z1-axis; and a yaw motion,A, about the Y2axis, where the Z1 and Y2 axes are associated with intermediateTrajectoryAppendageB1Perigee66zoFigure 3-3 Reference frames used in the formulation.coordinate frames (Figure 3-4)It should be pointed out that the resulting rotation matrix has a singularity whenthe angle of rotation approaches 00 or 900. Since there are 12 possible ways to definethe Euler angles, the singularity problem can be avoided by choosing the appropriateset [43] so that it is outside of the range of spacecraft maneuvers. From Figure 3-4.it can be seen that the librational velocity vector, , is given by=(3.1)vi00Earthxo Perigee67Figure 3-4 Pitch, Roll and Yaw rotation sequence to determine spacecraft attitude motion.where represents the spacecraft’s orbital rate.3.2.3 Joint rotationThe appendages are permitted to rotate about the hinge position; and the rotationmatrix which transforms the position vector to an elemental mass onto the coordinatesystem F is given by(3.2)ZIRollZ 2YawY2 ,YYlTrajectoryZr %% 0 I%_ __SEDX1 \EarthSPerigeeSSS.68where C. and C represent the initial orientation of the appendage and the rotationdue to the prescribed slewing motion with respect to the central body, respectively.Note, as opposed to the component modes method, the transformation matrix dueto the flexibility of the central body at the point of attachment is not present in theequation. With discretization of the structure usillg the system modes, the deformations are automatically expressed with respect to the system frame attached at thecentral body, thus symplifving the derivation of the equations of motion, as explainedin Chapter 2.R.otation MatricesThe modified Eulerian rotation sequence which determines orientation of the appendages with respect to the coordinate frame attached to the central body B canbe written asRot(. /3,7) = Rot(X, a)Rot(Y, /3)Rot(Z, ‘rny), (3.3)which corresponds to rotations about X by an angle , follwed by a rotation /3 aboutY, and finally a rotation about Z axis by an angle :1 0 0Rot(X. c) = 0 cos o — sin o ; (3.4)ocos/3 0 Sin/3Rot(Y.B) = 0 1 0 : (3.5)—sinf3 0 cos/3COS7 —Sin7 0Rot(Z.’y) = sin7 COS7 0. (3.6)0 0 169Carrying the multiplication, one obtainscos/3cos-y sinasinj3cos7 — cosasin7Rot(7,3, a) = cos 3 sin sin a sin 3 sin’-,’ + cos a cos— sin,13 sinacos/3cos a sin /3 COS 7 + sin a sin’-,’cosasin/3sin7— sinacos7 . (3.7)cos a cos /3Equation (3.7) represents the general form of the transformation matrix used to represent vectorial quantities in terms of the reference frame attached to the central bodyB.3.2.4 Joint translationThe appendages are permitted to undergo prescribed translation maneuvers withrespect to the central body B. Thus, the joint position vector d (Figure 3-2) hascontributions from two sources: the initial rigid body position denoted by C(Oj) andthe user prescribed translational maneuver Thus, d can be written as=(O) + Z(O). (3.8)3.2.5 Shift in the center of massThe vector‘im’ which denotes the position of Oim relative to Om, representst.he shift in the instantaneous centre of mass of the spacecraft due to its deformation(Figure 3-2). It plays an important role in evaluation of kinetic and potential energiesof the system and hence may affect its dynamics significantly under certain conditions.From Figure 3-3, with reference to X, Y and Z axes, R and R can be writtenas:RcRcrn,SirnSm+,Oc+6c: (3.9)70=—— Scm + i + c +.(3.10)Taking moment about the centre of force givescm=(3.11)where M is the mass of the spacecraft. Substituting Eqs. (3.9) and (3.10) intoEq. (3.11) gives—f LQ— cm crn= {fmc+ drn + + cj + dmi] }. (3.12)where:m =position vector of °m the centre of massof the undeformed spacecraft, relative to O;N =number of B bodies;M =m +Since= {fmcdrn+[{mi [c(Oi) + tc(Oi)]+ fmidmi}] }(3.13)can be simplified asirn{fmjdm+Z I dmi}. (3.14)713.3 Kinetics3.3.1 Kinetic energyThe kinetic energy, T, of the system is given byT {f c.Ldmc+[f iidmi]} (3.15)mc mwhere and R are obtained by differentiating Eqs. (3.9) and (3.10) with respectto time. The kinetic energy expression can be written in the formTTorb+Tcm+Tt+Ts+Tv+Tt,v+Ts.t+Ts,v+TI+T, (3.16)where ‘ is the librational velocity vector; I, the inertia matrix; H, the angular momentum with respect to the F frame; the kinetic energy due to pure rotation;and ,TR. the kinetic energy contribution due to coupling between rotational motionand transverse vibrations. The subscripts involved are defined below:orb orbital motion:cm centre of mass motion;s rotational motion due to joint slew;t joint translation between body B and B;v structural vibration.Here contributions from the above sources take the form:1 TTorb = M Rcm Rcm; (3.17a)1 TTcm = — M Scm Scm (3.17b)T =if T dm; (3J7c)i=1 m72f j dm; (3.17d)mTvf cTcdmc+f jTjdmj; (3.17e)mc i=1 m7=d 6+ 6 d1 dm; (3.17f)ii m i=1 mT3, = f T. din + f (C)T d dm; (3.17g)i=1 rn m7T5, = f (O)’ j drn + f Tà15j drn; (3.17h)i=1 mand the angular momentum vector can he written a.sHHcm+Ht+Hs+Hv+Ht.s+Ht.v+Hs,v (3.18)with:1crn _M(cm X scm); (3.19a)f x drn; (3.19b)i=1 m= f (C x à) d7n: (3.19c)i—i m.=x dm; (3.19d)i=1 m?= f (Cj x j + x á) dm; (3.19e)i=1 m=x x j)dmj; (3.19f)i=1 mj= f (6j x + Cpj x thn. (3.19g)i=1 m73The inertia matrix has contributions from the following sources:I = icm+It+Is +Iv+It,s+It,v +Is.v, (3.20)where:‘cm = —M [S Scm E — Scm S]; (3.21a)It=[d7E—d7]dnj,; (3.21b)i=1 mjL = f [(C)T (cC) E — (C) (C)TJ dm.; (3.21c)i=1 mjI = f j E — } dm; (3.21d)i=I m= f [2 dT (C) E — j (Cipi)T — (C) d7} drn; (3.21€)i=1 mIiv=f [2jE-JT_jd]dmj: (3.21!)i=1 m= f [2 (C) E — 6 (C)T — (C) ] drn. (3.21g)i=1 m23.3.2 Potential energyThe potential energy, U, of the spacecraft has contribution from two sources:gravitational potential energy (Ug) and strain energy (Ue) due to elastic deformations,U = Ug + Ue. (3.22)Gravitational Potential EnergyAny nonsymmetrical object of finite dimensions in orbit is subject to a gravitational torque because of its position in the Earths gravitational field. A gravity-74gradient torque results from the inverse square force field with the correspondingpotential energy given by [77](3.23)Substituting the expressions for R and R2 from Eqs. (3.9) and (3.10), and ignoringterms of order 1/Rm and higher, the gravitational potential energy expression canbe rewritten asUg=___— 2 trl+ 2R3 l’ii. (3.24)CTh cm cm.where 1 represents the direction cosine vector of R, with reference to Xf,Yf,Zf axes.From Figure 3-4, 1 is given by= (cos ‘ sin sin + sin ‘ sin ) I+ (cos ,L’ cos )k. (3.25)Elastic Potential EnergyThe spacecraft is assumed to be composed of beam. plate and membrane typemembers. The strain energy of a beam element has contributions from several sourcesand can 1)e written as[78]:uam= Uberiding + Utorsion + Uaxiai + Ushear + Ugeo.noniin. ; (3.26)Ubending= EI 1L 2 dx +1L 2dx; (3.27a)Ushear= GA1L(Du+ 62) dx+— ) dx: (3.27b)o Ux1)EAfI’thLl2Uaxjai= —) dx: (3.27c)2GJIL68i2U00 =— (-b---) dx: (3.27d)20EA 1L 0u1 (0722 2 EA fL Ou 9723 2Ugeo.noniin. =—— dx +—J -h--— (_) dx; (3.27e)Ox Ox? 22 Jowhere u and O (i=1,2,3), represent the translational and rotational structural deformations, respectively.For a plate type component, the elastic energy contributions arises from the following sources[79]:Uiening + Utorsion + Umembrac (3.28)D b a 1(02723 2 02723 02723+2v(Ubending 2 L L [ Ox2 I \ Ox2 ) ( Oy2)02723 21+ (02) j dxdy: (3.29a)b a O2u3 2) dxdy; (3.29b)U5ion = Df L (1— (OxOy6D b 0u1”2+0u3 2 Ou2 2Urnembrane = i J J (- Ox (--) + ()/ 0722 iOu3 2 1 Ou3 2 / On3+.[(_) +--_)]r Oui)(Ou2 1 iOu2 /OU32 1 Ou 0u3\21+2v1(——)+—,--— +(---)(--- jL\Ox 0+ — +2(—(—,+(—-(1— ii) [(Oul)2 On1 OU2’ iOu222 Oy \OyJ\OxJ \Ox’+2 vdx(— dxdyOn1(Ou3)(Oud OU3)(0u3)]}\ Oy) Ox \ Oy I \ Ox 0 ; (3.29c)76where D and ii represent the flexural rigidity and the Poisson’s ratio of the plate,respectively.3.4 DiscretizationThe spacecraft model has been discretized by expressing elastic deformations interms of admissible functions. Thus linear displacements and angular rotations of aflexible structure in three orthogonal directions can be expressed asp(t)y3(x,y,z) p(t)U3 ‘i qz(x,y.z) p(t) — fT15 330— 81 v(x,y,z) p(t) — ( .82 v(x.y,z) p(t)83 v(x.yz) p(t)where and I are discrete eigenvector matrices corresponding to translational androtational deformations, respectively, and is the corresponding generalized coordinates vector.The discretization of the physical system is carried out using system modes, wherethe number of generalized coordinates depend on the number of modes considered.For example. if one accounts for the m system modes to represent the flexibility ofthe model, the total number of generalized coordinate is given byNq5+m. (3.31)3.5 Equations of MotionUsing the Lagrangian procedure, the governing equations of motion can be obtained fromdDT UT DU -- + Qq (3.32)where and Qq represent the generalized coordinates and associated forces, respectively. The above equations can he rewritten in a vector form asIF9M() “ = - (3.33)lFpJwhere:- d &‘ d - T dcZ’ d f0,T\ -dt \ 1 dt dt dt \ IT dH 0,T- (3.34a)- d16T’ d(8HTo )ã) )___)DT 1 T0’ OH+ - + -- + (—) + . (3.34b)Op 2 Op Op OpHere M() represents the nonlinear mass matrix, while F9 and F correspond tothe nonlinea.r stiffness, gyroscopic and forcing terms for the librational and vibrational degrees of freedom, respectively. is comprised of two vectors, and,where= {, , .} for the librational degrees of freedom and = {pl,p2, p.} forthe vibrational generalized coordinates. The nonlinear, nonautonomous and coupledequations of motion are presented in their entirety in Appendix A.In general, the effect of librational and vibrational motions on the orbital dynamics is small unless the system dimension are comparable to Rcm[80.81]. Hence, theorbit can he represented by the classical Keplerian relations:h2Rcm = (3.35)i-i(l +ecosG)78p2 361cm’-’ — ‘where h is the angular momentum per unit mass of the system; e, the eccentricity ofthe orbit; and G. the true anomaly.In space dynamics, it is convenient to express the independent variable (t, time) interms of the true anomaly (0). This transformation facilitates physical interpretationof results in terms of the time scale that is easy to comprehend. The dynamical studyis performed with respect to the orbital unit of time (time taken for one orbitalrevolution). The transformation between the variables is readily accomplished using:(3.37)c12— 2( d2 2esiri0 d 3 38dt2 dO2 1+ ecosO dO3.5.1 Mass matrixAs depicted in the equations of motion, the mass matrix has to be expressedexplicit.ely so that it can be evaluated and subsequently inverted. The matrix can bepartioned in the following form,M9,0M=Mp.e Iwhere:cosq5cos\ —siriq cosçbsin\ cosbcos) —sin, 0M,6 = — .sin \ 0 cos I — sin 0 10 1 0 cosq5sin\ cos\ 0= M8r — i= I—jn 0 cos\ I[0 1 0 jF(fmc — fmc 5czq5yd7nc) 1{ (fmc zx - fmc cdrnc)L (fmc — fmc cdmc) j[(fmc ybdm— fmczcbdrnc) 1+ I (fmc zdrn— fmcxcdmc)L(fmc xdm — fmc ycdmc) jN F(f — fm 1+ I (frnj- fmj xdm) I‘ L(im — fmj1drn) jN F (2 fmj pbdm — fmj pc5d7n) 1+ ( fmj pjdrn — fmjL (1 fmj pdm — 2 fmj jN F (d2 I T dm — d3 fmj thm) 1‘T7i Z+ (3 f dm — d1 fmj drni)1-ni ‘ix‘ L (d1 f T dm — d2 fmj dmi) jrn iF (cm2 fmc — Scm3 I dmc) 1mc V— I (scm3 fmc — Scm1 fmc czdmc)L (semi fmc — 5cm2 fmc cxdmc) JN(scm2 fm — s3 f, izdrni)+ (scm3 fm — 8cm1 fmj xdrn)N]};[(scm.i frnj — 5cm2 fmj iydmi)= f 4dm + f drnJmc i=1 mjN N1’_f cdmc+f d1n)(f d7flc+f drni).Jmc i=1 m me i=1 m803.6 Disturbance EnvironmentMany complex experiments and observations are planned using the Space Stationas a platform. One importait research activity pertains to material as well as physiological behaviour in the microgravity environrnent, to be conducted mostly withinthe laboratory module. As can be expected, the microgravity level would be quitesensitive to the Space Station dynamical response and associated acceleration environment. Thus it would be necessary to evaluate the acceleration field at criticallocations under various likely disturbances to ascertain that the operation remainswithin specified limits, and if required. to introduce appropriate control measures.Even the routine activity of astronauts and the orbit maintenance may result in disturbances of concern. Thus, an accurate assessment of operational disturbances isnecessary to predict if the design requirements are likely to be satisfied.To carry out. such assessment. models have been proposed for a spectrum ofdisturbances likely to he encountered by the Space Station[82j. The disturbancesources can be classified into two categories: steady state and transient. Steady statedisturbances are solar array rotations for tracking t:he sun and payload articulations.The transient disturbances include shuttle docking and the crew activities.3.6.1 Solar array sun trackingThe rotation of t.he solar panels to track the sun for optimum exposure is dependent. on the position of the spacecraft in the orbit, i.e. on the true anomaly e.For a circular orbit, this results in a maneuver at an uniform rate 9. For ellipticaltrajectories, it is convenient to use eccentric anomaly (E) which relates to the trueanomaly[83j:(sinE.cosE) (1 — e)siflO e+cosO ): (3.39)1+ecosG 1+ecosOSi1t = (3.40)p n(E—esinE)n=/ (3.41)where a is the semi-major axis of the ellipse and GMe is the Earth’s gravitationalconstant.3.6.2 Aerodynamic torqueThe aerodynamic disturbances are modelled as a bias plus periodic terms and aregiven by[48]:Qaero = 4 + 1.2 sin Ot + 3.5 sin. 2Gt (3.42a)= 1 + sin et + 0.5 sin 2Gt (3.42b)Qaero = 1 + sin Gt + 0.5 sin 2t (3.42c)The aerodynamic torque frequencies at the orbital rate are caused by the rotationof the solar panels to track the sun: and the effect. of the the earth’s bulge accountsfor disturbances at twice the orbital rate. (Figure 3-5).3.6.3 Operation of the Mobile Servicing System (MSS)The Mobile servicing vstem (MSS) operation consists in moving payloads andassisting in the construction of the Space Station in orbit. Thus, the MSS maneuversinclude rotational of the arm and translational motion of the base.Payload SlewingThe sine-on-ramp profile has been selected to simulate slewing of the MSS arms.82EZ6CC0E4z0a,c00.0 5.0Figure 3-5 Aerodynamic torque history.This maneuver profile provides smooth velocity and acceleration time histories, withzero values at the beginning and end of the maneuver. Figure 3-6 shows the displacement, velocity and acceleration profiles associated with this type of maneuveras functions of true anomaly and orbital eccentricity.Payload TranslationOperation of the MSS can impose transient stress field on the Space Station. Atypical prescribed translational profile of the MSS along the X-axis of the Station,shown in Figure 3-7 for a maneuver over one bay, is discussed here.Each prescribed translation is divided into ‘bay maneuvers’, where one bay consists of a 5-meter truss. Each individual maneuver is identical in the sense that thedisplacement profile, and hence the velocity and acceleration profiles, are prescribedin the same way. It is assumed that one bay maneuver takes 300 seconds.Orbits83Figure 3-6 Effect of eccentricity on orbital time and maneuvering profiles: (a)displacement-true anomaly history for a sine-on-ramp maneuver completed in 180°; (b) corresponding velocity profile; and (c) resultingacceleration history.3.6.4 Orbiter dockingAs the Space construction would require more than 30 flights of the Orbiter, itsdocking maneuver represents a major disturbance. The Shuttle will rendezvous withthe Space Station at regular intervals for resupply, exchange of crew, and return ofmanufactured items as well as waste products. The contact forces from berthingor docking will be an order of magnitude higher than any of the operational forcesapplied to the station. The disturbance is modelled as a 2,225 N force acting alongthe berthing axis for a period of 1 second. (Figure 3-8)3.6.5 Intra-Vehicular Activity (IVA)Although individual crew movements inside a spacecraft may be of a random(a) (b)I-.it/20(c)0xC,,VI-.10500(%J0ct:it 0 e,rad 7t0.5-0.50 e,rad 0 e,. rad It845E0c.’J0xU)E 0.4-01><U)0 500Figure 3-7 Translational maneuver profile for the Mobile Servicing System.nature, they follow statistical patterns. The most common crew induced disturbancesinclude treadmill walking and jogging, coughing and sneezing, and intra-vehicularmotion along and across the modules.Onboard Crew MotionThe model for the disturbance due to onboard crew movements attempt to simulate the forces caused by a crew member pushing off one wall of the habitation moduleand coming to rest at the opposite wall. The first crew motion model is a end to endtraverse of the habitation module, called an axial crew kick-off. The force generatedTime,s832225.0za)C-)I00.0___________________________________0 Time,s 3Figure 3-8 Orbiter docking simulation model.by the crew member increases linearly to a maximum of 111.25 N over an intervalof of 1 second and decreases linearly to zero as wall contact is lost. At the opposingwall, the force increases instantly t.o 111.25 N as the wall is contacted and decreaseslinearly to zero over a 1 second interval. The time between force pulses, 13 seconds,was calculated from a free-flight distance of 9 meters (Figure 3-9a). The second crewmotion model is a side to side traverse of the habitation module called a transversecrew kick-off. The force time history is essentially the same as for the axial motionexcept that the time between the pulses, three seconds, was calculated from a freeflight distance of 3 meters (Figure 3-9b).Treadmill ExerciseCrew will be required to maintain physical fitness by exercising daily on facilitiesavailable on board. Treadmill equiment will be a part of such a setup, and two simulation models for walking and jogging are available for an 80 Kg-astronaut exercise86routine. (Figures 3-9c and 3-9d).Coughing and SneezingThe coughing and sneezing model shown in Figure 3-9(e) is essentially a randomsignal obtained experimentally during a Skylab mission.3.6.6 Extra-Vehicular Activity (EVA)The Space St.ation will require Extra-Vehicular Activity (EVA) by the astronautsfor its assembly, operation and maintenance. Figure 3-9(f) presents a simulationmodel for a tethered sortie by an astronaut during a Station maintenance mission[33j.It consists of a short triangular wave followed by a square wave of a slightly longerduration. The peak force is about 1100 N in the initial phase of the tethered maneuver.Due to time limitations and aailabi1ity of data, the following additional disturbances were identified but not included in this study:• Solar Pressure;• Rotating Machinery Dynamics:• Shuttle RCS Plume Impingement:• Construction Dynamics:• Manufacturing Dynamics.Of course, this is not a complete list. However, taken together with the oneswhich have been implemented in this study, it does constitute the set of disturbanceswhich is considered relevant for dynamics and control study of the Space Station.8725.0 2 50zC)C.,0zC)C-,0LLza)C.,0U-0 Time,s Time,sza)C)zC)C.)za)C)0LL100.0- Time,S 10 00.0-25.015 0250.00.0-250.010005000Crew motion simulation models:(a) longitudinal IVA; (h) transverse fl/A;(c) treadmill walk; (d) treadmill jog:(e) coughing and sneezing: (f) EVA.88Astronaut’s Coughing and SneezingTime,S 10(e)Time,s0Figure 3-9Time,S 53.7 SummaryThe multibody dynamics formulation presented here constitutes the core of thesimulation program developed to study dynamics and control of flexible, multibodysystems with particular application to the proposed Space Station. The multibodyformulation approach has the following distintive features:• Derivation of the equations of motion by matrix manipulation and symbolic implementation makes explicit linearization about an arbitrary orientation possible.Furthemore, it is well suited to the assessment of the system’s sensitivity to governing parameters, and it permits determination of the system’s observability,controllability and stability properties.• Vectorial designation is used throughout the derivation of the equations of motion,which is convenient for identification of the various contributing terms (inertia,gyroscopic, stiffness, and forcing terms) to the dynamics of the problem.• The component eigenfunctions (for each contituent flexible body) are replacedby more general system admissible functions, which yield relatively simple andmanageable equations of motion.The Space Station has been modelled as a system of interconnected flexible bodies.The interface between bodies consists of three rotational degrees of freedom at thejoints. In addition. the formulation provides for user-defined, prescribed translationalmotion of the appendages with respect to the central body. A detailed discussion ofthe computer implementation of this general multihody formulation is presented inChapter 6.89CHAPTERFOURMODAL FUNCTIONS AND INTEGRALSThe formulation presented in Chapter 3 employs spatial functions, referred toas assumed modal functions, in conjunction with generalized coordinates to describeelastic deformations of the flexible system. In the governing dynamical equations ofmotion, the modal functions appear, in a variety of comhiiia.tions with other systemparameters as integrands of certain definite integrals (referred to as modal integrals).These integrals have to be evaluated prior to, or during numerical simulation. Sincethe modal integrals play such a significant role in the simulation algorithm, their accurate evaluation is of utmost. importance. This chapter presents a. detailed formulationand solution of the associated eigenproblem by the finite element method, togetherwith the necessary steps to utilize discret.e eigenvectors in the integral evaluationprocess.4.1 Finite Element FormulationWhen the structure to be analyzed is not uniform, or when effects such as sheardeformations, rotary inertia, warping, or coupled bending and torsion cannot be neglected, it is usually impossible to obtain an analytical, closed-form eigensolution.This makes it necessary to resort to numerical techniques to obtain eigenvalues andcorresponding eigenvectors. Furthermore, if the system under consideration is of atransient and evolutionary character, the mathematical model will also require freedom and ease with which system parameters such as geometry, flexibility and inertiacan be varied. To this end, the finite element approach appears quite attractive90because of the following reasons:(i) The admissible functions are quite simple (low degree polynomials) and computationally desirable. Integrals involving such polynomials can be evaluated inclosed-forms, thus eliminating errors that may result from numerical integration.(ii) Even for a complex flexible structure like the Space Station, system modes can heobtained quite readily. Moreover, the nodal feature of the finite element methodmakes handling of the boundary conditions straightforward.(iii) The formulation is not affected by tl1e evolving design. Once characteristics ofthe system are identified, a finite element that properly accounts for the physicalproperties is developed. Contribution of this element in the system’s dynamicalanalysis is automatically accounted for in the assembly process. When a different structural configuration is encountered, say during the integration of a solarpanel with the power boom, a new mesh layout and subsequent reassembly of thestructure is all that is required.The Space Station is a large flexible structure which consists of beam, plate andlumped mass type components. This section presents the finite element formulation ofsuch members with emphasis on the procedural steps to extract the system eigenvaluesand eigenvectors.4.1.1 Beam elementProcedures for modelling large truss structures like the Space Station’s powerboom by means of equivalent continuous systems have been developed by Berry etal.[34J, and Juang and Sun[35]. Their studies suggest the necessity to model largetruss structures by the geometrically nonlinear Timoshenko beams. While shear androtary inertia lead to small corrections to the Bernoulli- Euler theory for the lower91modes of long, thin beams, significant errors may be introduced if they are neglectedwhen dealing with thicker beams, or for the higher modes of any beam.A number of Timoshenko beam-type finite elements have been proposed in theliterature [84— 88]. They can be divided into two classes: simple and complex. Asimple element has a total of four degrees of freedom, two at each of the two nodes,for unidirectional bending in a principal plane. A complex element has more thanfour degrees of freedom, i.e. more than two degrees of freedom at a node or morethan two nodes.The simple beam element has been selected in the present study. It is taken to be astraight bar of uniform cross-section capable of resisting axial forces, bending momentsabout the the two principal axes in the plane of its cros-sect ion, and twisting momentabout its centroidal axis. The following displacements are present on the beam:axial displacements d1 and d7; transverse displacements d2. d3, d8 and dg; rotational(bending) displacements d5, d6, d11 and d12; and rotational (twisting) displacementsd4 and d10. The location and positive directions of these displacements are shown inFigure 4-1.Mass and Stiffness MatricesConsider the beam discretized into e1 elements. Let the jth element be describedby a length L, material properties E, G, p, and cross-sectional properties A, I,I, , and t. The expressions for the elemental stiffness and consistent massmatrices were determined by writing the kinetic a.nd potential energy expressionsfor a single element and Lagrange equations were subsequently employed to developelemental equations of motion in matrix form. The kinetic energy, Tt, of the ith92d9d12d7-’/Jd3 d11d10d6d4d1 d5Figure 4-1 Beam-type finit.e element with twelve degrees of freedom.element can be written as[89J,T= 1dx (i 1..., flel), (4.1)where is a 12 x 1 vector whose elements are the time derivatives of dZ, defined.civi(4.2)d_li U3iOU,11 1d12 i+1i+11J393Here and O symbolize the extension, two transverse displacements,and three cross-section rotations, respectively, of the element i, expressed in a localcoordinate s stein fixed in the element i. N’ represents the local shape functionmatrix of size 6 x 12 such that.t’f1,(x)d(t)12J’f,(x)d(t)ç Jv,(x)d(t) 43,(x)d(t)‘_‘ A1j(x) d(t)A/j(x) d(t)and R is a 6 x 6 diagonal matrix of mass and inertia properties given by100 0 0 00100 0 00 0 1 0 0 0—p 0 0 0 0 0 (4.4)0 0 0 0 ‘22 00 0 0 0 0 133heie I = 1jA0 (z = 1 2 3)The potential energ, P for element i is obtained b subtracting from the strainenergy the ork done b3 f the geneialized nodal forces vector Thus= U’ — W. (4.5)where the symbol W denoting the work done by the nodal forces, is given byWt (d f, (4.6)and the strain energy Ui stored in the th beam element isU =— J (i)T(ai)dx (4.7)294The symbol E represents the elemental generalized strain vector, defined asdö d1_1 -(EI3)i 073 0 0 (3EOI2)l 0 0 0 _c;1(4.8)-0::;0 0 0 0 0 1dx2Denoting the square matrix by E and the right-most column matrix by ‘, Eq.(4.8) can be rewritten more concisely as= E6’. (4.9)The vector ö appearing in Eq. (4.7) pertains to the generalized stresses and is givenby= S6’. (4.10)where S is defined asEAQ 0 0 0 0 00 EI 0 0 0 00 0 EI 0 0 00 0 0 0 00 0 0 0 EI 00 0 0 0 0 EISince the vector is the product of the shape function and the nodal displacement,it follows that ‘ can be written similarly as= BJ’, (4.12)where B1 is a 6 x 12 matrix obtained through differentatioii of N1. In particular, for93= 1, • • flel:= (j = 1,... , 12); (4.13)= d3.Ai (4.14)Z3 = dZ3ii; (4.15)dA[.= dx(4.16)=d:,i; (4.17)= d2J (4.18)If a. constitutive matrix D is defined as(4.19)the strain energy expression becomes= f dxi. (4.20)Inserting the expressions for W and U from Eqs. (4.6) and (4.20) into Eq. (4.5)and substituting the resulting expression for P along with the expression for T fromEq. (4.1) into t.he Lagrange equation,doT2a(TiPi)0 (421)dt 0d — ‘96yields elemental dynamical equations of the form MJ + K1 = , where theelemental mass and stiffness matrices are given by:Mt= J (Ni)TEi(Ni) dxi: (4.22)0= f (B)TD1(Bi) dxi. (4.23)The shape functions can be derived in their entirety as follows. Assuming that 6 (x, t)can be represented in terms of a linear function of the two nodal extensions d1 (t) andd7(t), as well as in terms of a linear polynomial in x, leads to= 1 — (4.24)= x/L. (4.25)Let 6(x, t) he represented both as a linear function of the four time dependent.quantities d2(t), d6(t), d8(t) and d12(t), and as a cubic polynomial in x. Then, fori1=!f,2(x)dt)+f,6(x)dt)+ JV28(s)dt) + A/,12 (x)d12(t); (4.26)Z2,1(t) +Z2,(t)x +Z2.3(t)x +Z2,4(t); (4.27)where Z2,(i = 1.2,3. 4) are unknown coefficients in the cubic polynomial representing. To determineA.241,6.,12’ it is also necessary to develop an equationfor the slope 6 in terms of Z2 (i = 1,... 4), so that Eq. (4.26) can be evaluatedalong with Eq. (4.27) at x = 0 and x = Lt in order to provide four equations infour unknowlls. According to the Timoshenko-beam theory:=—(4.28)97— D8 2V.xi 0—2 j(8JVI3— Dx — A0G’ ‘ Dx— D6 o2EoI3DA0G Dx’___GA0 —GA0 )i2Dx 2EI32 DxiGA0 -GA0— 2E0I3=(2EOI3)2,2+2Z2,3x+ 3Z2,4x).The solution of the resulting linear, ordinary differential equation appearing in Eq.(4.33) will be expressed as the sum of a complementary (homogeneous) solution[O(x,t)] and a. particular solution [8(x.t)]. Taking the complementary solutionas a cubic polynomial in x,[(xi.t)] = Z6,1(t) +Z6,2(t)x +Z6,3(t)x +Z6,4x, (4.34)the unknowns Z6 (i = 1.2,3.4) can be found by substituting [8(x, t)j intoEq. (4.33) with the right-hand side of the equation set to zero. ThusGA02 Z6.3 + 6Z.4 x— ( )t(Z + Z5,2x + Z53x +Z6,4x) = 0.2E013 (4.35)(4.29)(4.30)(4.31)where 4 is the shear angle as a function of x, for element i, while V2 and M3 arethe cross sectional shear force and bending moment, respectively. The superscripti appearing on the parentheses indicates quantities pertaining to the th element.Equation (4.31) represents a second order. linear, partial differential equation thatcan be written in the formor, equivalently,(4.32)(4.33)98from which it follows thatZ6,1 = Z6,2 = = Z6,4 = 0. (4.36)Hence, the homogeneous or complementary solution vanishes when expressed in termsof a cubic polynomial in x. A particular solution is found by assuming a form[8(xt)] = z2, + 2Z,3x+ 3Z2,4x+. (437)where is an unknown constant. Substituting t)] from Eq. (4.37) into Eq.(4.33) leads to6 Z2,4—GA0 )i [z2. + 2Z,3x+ 3Z2,4x+ ] =2E013(A ) [Z22 + 2Z,3x+ 3Z24x] (4.38)which gives= (62EOI3)iz (4.39)Thus, the total solution for O(x,t) is= [(x,t)] += + 6(’3)tZ24+ 2Z,3x+ 3Z24x. (4.40)Evaluating Eqs. (4.27) and (4.40) at x = 0 and P gives the following four relations:d2 = 1Z2, + 0Z2, + 0Z2,3 + 0Z24; (4.41)d5 = 0Z2 + 1Z22 + 0Z2,3 + (Li)2Z2,4; (4.42)d8 = 1Z2,i + (L)Z2,+ (L)2Z.3+ (L)3Z2,4; (4.43)d12 = 0Z2.1 + 1Z2, + 2(L)Z,3+ (3 + )(Lt)Z24; (4.44)where the symbol 2 denotes t.he term [12aEoI/GAo(L)]. Eqs. (4.41)-(4.44),99when solved for Z2, (i = 1,2,3,4), giver 1 r 1 0I Z2,L I — i 2/(1 + 2) (2 + 2)/[2(1 + 2)]I3(L) I— I —3/(1 + 2) —(2 + 2/2)/(1 + 2)LZ2,4(L)i L 2/(1 + 2) 1/(1 + 2)0 0 1 d]2/(1 + 2) 2/2(1 + 2) I d6L.(4.45)3/(1 + 2) —(1— 2/2)/(1 + 2) I I d8 i—2/(1+2) 1/(1+2) i Ld12JFrom Eqs. (4.26), (4.27) and (4.45), it follows that, for (i = 1,...,nj)X2= [i + 2 — — 3(r) + 2()3] (1 + 2); (4.46)2 Xj 2j2 2 X26(xi) = + - 2(u) -+ (.)3] (1 + 2)’4.47)xi 3xi,8(xi) = [2()- 3()2+ 2(p) 1(1 + 2)’; (4.48)I 2 X x 2 2 X 2 X7 3= L_T() - () + + () 1(1 + 2); (4.49)andAl,(x) = 0, (j = 1,3,4,5,7,9.10.11). (4.50)The shape functions for AIJ (j = 1, . . . , 12) can he developed in a similar way.They are, for (i = 1,...X2,3(x) = + 3— 3() — 3(r) + 2()3] (i + 3); (4.51)X 2 X 3]xi 3x5(Xi) = Li[_()-+2()2+ - () j (i + 3).52)x 31= [3()+ 3()2- 2(p) j (1 + 3); (4.53)3 x x2 3X2 X3= Lt + + _--() - (i;) ](1 + 3); (4.54)and= 0, (j = 7,8, 10,12): (4.55)100where the symbol denotes the term [12a3EoI2/GAo(L)].Let &(xj,t) be represented as=J’f41’,(x)dt)+ A’,10do(t) (4.56)and8 (xi, t) = Z4,1(t) + Z4,2(t)xj. (4.57)Evaluating Eq. (4.56) at x = 0. L gives:Z4,l = d4; (4.58)d10 — d4=4.59Thus:= 1 — (4.60)A’,10(x) = (4.61)and= 0. (j = 1.2.3,5,6,7.8.9,11,12). (4.62)Finally consider (x.t) written in terms of Z3. (i = 1.2,3,4) and assumed in theformt) =Jf3(x)dt) +f,5(x)dt)+A19(x)dg(t) + (x)d1i(t), (4.63)then:= [6(p) - 6()2] (1 + 3); (4.64)= [i + - 4() - + 3(Xz)2] (1 + 3)’; (4.65)[_6() + 6()2] (i + ) (4.66)101= [_2() + + (Xi)2] (1 + (4.67)and= 0, (j = 1,2,4,6,7,8,10,12). (4.68)Similarly:2(xi) = + 6(] (1 + (4.69)f6,(x) = [i + 2 — 4() — + 3()2] (1 +2y’; (4.70)8(xi) - 6(.)2] (1+2y’; (4.71)A12(x) = [_2() + + 3()2] (1 + (4.72)andAi,(x) = 0. (j = 1, 3,4. 5. 7,9, 10, 11). (4.73)These shape function were introduced into Eqs. (4.22) and (4.23), and usingthe symbolic manipulation code REDUCE [90], the elemental stiffness and consistentmass matrices were obtained as given in Appendix B. This represents a summary ofthe procedure involved in the derivation of a compatible beam type finite elementformulation using CUbIC shape functions.4.1.2 Plate elementThe 24 degree of freedom plate element has been selected in the present study.Przemieniecki[91] and Zienkiewicz[92] have presented a comprehensive displacementformulation for this element. For completeness and continuity, only the essentialfeatures of the stiffness and mass matrices derivation are touched upon here.Consider deformation of a plate element subject to ‘in-plane’ and ‘lateraf forces,102d22d19d15Figure 4-2 Plate-type finite element. with 24 degrees of freedom.d8as shown in Figure 4-2. The nodal displacements are comprised of: membrane displacements d1. d2, d7, d8, d13, d14, d19 and d20; transverse displacements d3, d9, d15and d21; rotational (bending) displacements d4, d5, d10, d11, d16, d17, d22 and d23;and rotational (in-plane twisting) displacements d6, d12, d18 and d24.Mass and Stiffness MatricesConsider the plate discretized into eL elements. Let the th element. be describedby a length a, width b and thickness t with the material properties E, G, pt, andv. The nondimensional coordinates and i are defined asd18d16d9d12d21dd113d3d1 d5xi•‘ ati77 (4.74)103Expressions for the element stiffness and consistent mass matrices were given earlierin Eqs. (4.22) and (4.23), respectively, which are valid for the plate element as well.The displacements are defined in terms of the nodal parameters using the appropriateshape functions matrix of size 3 x 24 such that24.A/,(x) d(t)5= 6 = iV2,(x) d(t) . (4.75)j=1 A13(x)d(t)The generalized strain matrix for the plate element can be defined asElx662Ely ‘E = . (4.76)2 (3axayThus from Eqs. (4.75) and (4.76), the strain-displacement relationship becomes= (4.77)where B2 is a 6x24 matrix representing derivatives of elements in N. In particular,a.= 8x’ (j = 1 ,24); (4.78)aAr•Z3i 2,j•2j—______2,3J 8y Ux’‘3i — 3,j4j Dx2 ‘=(4.82)104= (4.83)For linear elastic behaviour, the constitutive matrix D is defined asD= [ ]. (4.84)where:(1_v2) 0 (1_v)/2](4.85)1 ii 0D= 12 1 2\i-i 1 0 . (4.86)‘ 0 0 (1—v)/2The shape functions are taken to he the same as in reference [91]. For membranedeformation in the x-direction:f2(xi.y) = (1— ); (4.87)= (4.88)A14(xi,y) = (1—(4.89)A12o(xui.yz)= (1— )(1—ii); (4.90)andAq(, y2) = 0, (j = 1,3—7,9— 13,15 — 19,21 —24). (4.91)In the v-direction:A1(xty) = (1—j); (4.92)A/7(x.yt)= (4.93)1054,13(x,y) = (1—= (1 — )(1 — 7]);!417(z,y)A,21 (xt,y)N3.22 (xi, y)(4.94)(4.95)= 0, (j = 1,2,6 — 8, 12 — 14, 18 — 20,24). (4.109)The deflection functions represented by Eqs (4.97)-(4.109) ensure that the boundarydeflections at the adjacent plate elements are compatible; however, rotations of theelement edges at a common boundary are not compatible, and consequently disconandy). = 0, (j = 2 — 6,8 — 12.14 — 18.20 — 24).For bending deformations:.43(xi. y)A’,4(x y)J’*.f5(x,y).,A9(xi y).A,10(xy).A11 (xi, y)= (3- 22(1 -7]) + 7](1 - ?])(1 -27]);= (1- 2( -= 7](1 —= (3_227]_7](1_7])(1_27]);= (1 —= —(1 —(4.96)(4.97)(4.98)(4.99)(4.100)(4.101)(4.102)(4.103)(4.104)(4.105)(4.106)(4.107)(4.108)N3,15(x.yt) = (1- )(3 - 27])7]2 + (1 - )(1 - 2)7];= —(1 —= —(1 — )(1 —=1_7]_(3_22(1_7])_(1_)(3_27])7]2;=- )2(-and= (1 — )7](1 —106tinuities in slope exist across boundaries. The stiffness and mass matrices based onthe above mterpolation functions are presented in Appendix B.4.1.3 Geometric nonlinearitiesWhen the defiections of the beam or plate are large enough to cause significantchanges in the geometry of the structure, nonlinearities in the strain-displacementrelations need to be include in the formulation. The matrix analysis method developedabove can be extended to account for such nonlinear behaviour. For example, thenonlinear strain-displacement relationship may be taken in the form as suggested byZienkiewicz [92]:DtiDxEx 032E Dy— Dz—I 1 Dz DyI7zxI Ox ‘ Dz\7xyDy Dx(2 (22 Dx) kOx) Dx)1 +(2+2 ‘Dy) Dy) \Dy)1 (.i2 + +2 \ Oz I \ Dz I \ Dz I+ (4.110)Dz Dy Dz Dy Dz DyDx Oz Ox Dz Dx Dz1+a.a+a.aDy Dx Dy Dx Dy DxSpecializing the above relation to the one-dimensional beam problem and neglecting10the contributions from the shear strains,= +[U622 + ()2]. (4.111)Substituting the above expression into f01 E dx, the nonlinear geometric stiffnessmatrix for the beam element can be calculated as given in Appendix B, using theprocedure described before.4.1.4 Lumped mass elementThe simplest form of mathematical model for inertia properties of structural elements is the lumped-mass representation. In this idealization, concentrated massesare placed at the node points.Mass MatrixThe generalized mass element has six degrees of freedom: three in translation(d1.d2,d3) and three in rotation (d4.d5.d5) about the noda.l x, y and z axes, respectively (Figure 4-3). ThusmO 00 0 0OmO 0 0 00 Om 0 0 0Mium.p= 0 0 0 j 0 0 (4.112)0 0 0 0 00 0 0 0 0The element. defined by a single nodal point. contributes only to the structuralmass matrix and has no associated displacement function.108Figure 4-3d1d3Lumped in ass-type finite elementd24.1.5 Modal analysisFree vibration of an elastic system is governed byTakingM d+K = 0.= cbet,(4.113)(4.114)for harmonic response, where is a column matrix of amplitude and w is the circularfrequency of oscillatiops, gives= M4’A. (4.115)Here, the columns in are the eigenvectors and A is a diagonal matrix listing thecorresponding eigenvalues. The solution procedure used to solve Eq. (4.115) is basedon the subspace iteration method [68]. The eigenvalue solver was implemented numerically using EISPACI< [93], in the public domain library. The results were checkedwith benchmark cases available in the Examples Manual of ANSYS [67].d6d4d51094.2 Space Station Model DescriptionThe principal model components constituting the proposed space station freedomare the main truss, solar arrays, PV and station radiators, the stinger/resistojetassembly, the RCS boom module. modules cluster and attached payloads.The First Milestone Configuration (FMC) consists of a 60 m long truss, a 11.5 x1.15 x 0.03 meters array radiator, a pair of solar panels (33 x 6 x 0.25) meters,the 12.7-meter long Reaction Control System (RCS) boom and the 26.7-meter longStinger/Resistojet assembly.In the Man Tended Configuration, the main truss length remains the same, however, the system’s weight. increases through addition of the laboratory module andstation radiator.The Permanently Manned Configuration (PMC) has both the length as well asthe mass of the Space Station virtually doubled. The number of solar panels increasesto four to meet additional demand on power. This requires further rejection of energyto maintain the desired thermal environment through addition of photovoltaic andstation radiators.Finally, the assembly process is complete ( Assembly Complete Configuration,ACC) with the station reaching its fina.l length of 155 m, addition of two pairs of solararrays and a pair of PV radiators, thus attaining a total weight of around 274,000 Kg.The system properties used correspond to those specified in the NASA’s EngineeringData Book[2} and these are summarized in Table 4-1. As pointed out before, thespace station configuration has been revised and, perhaps, will undergo modificationsin future. However, the procedure developed here is general, and the associatednumerical algorithms are capable of incorporating such changes quite readily. In anycase, it does not affect the thrust of the present discussion.110Table 4-1 The Space Station component data.SPACE STATION FINITE DIMENSIONS MASS BENDING TORSIONALCOMPONENTS ELEMENT STIFFNESS STIFFNESSTYPE rn kg Nm2 x 106 IVm2 x 106Main Truss Beam 155 x 1.86 x 1.86 24,161 1500 1125Solar Array Plate 33 x 6 x 0.25 444 70 53PV radiator Plate 11.5 x 1.15 x 0.03 450 10,000 7520ST. Radiator Plate 11.5 x 9 x 0.1 1.393 200 150RCS Boom Beam 12.7 x 1.86 x 1.86 164 2 1.5Stinger Beam 26.7 x 1.86 x 1.86 270 12.5 Finite element representationMain Truss StructureThe main truss structure is the backbone of the Space Station. All the majorhardware is supported by the truss which consists of individual bays each 5 m. Thestruts are made of graphite epoxy material with a modulus of elasticity Etrut =90.0 x Pa.The truss was discietized into continuous beam finite elements Equnalent aluesof El, GJ and EA for the finite elements were calculated using the properties of theindividual members.For equivalent continuous beams representing trusses with cubic repeating bays,stiffness parameters are given by [94]:El = CEIh2(EA)stru; (4.116)111PV Radiator Support Modules PV RadiatorFigure 4-4 A finite element model of: (a) the main truss at the ACC stage withattached modules and payloads.GJ = CGJk2(EA)struj; (4.117)EA = CEA(EA)8trut; (4.118)where h is the dimension of the cubic repeating truss-bay: (EA)3trut is the axialstiffness of the individual strut member; and CEJ, CGJ, CEA are dimensionless coefficients whose values depend only on the geometrical arrangement of the memberscomprising a repeating truss-bay. For the Space Station, these coefficients take thevalues of 1.07, 0.26 and 5.41 for CEI, CQJ and CEA, respectively. The finite elementmodel of the main truss in the ACC stage is shown in Figure 4-4(a).Solar ArraysEach solar array mast is 33 meters long and supports an array blanket. Perhapsan accurate mathematical representation would involve modelling the array mastTank FarmAntenna—PV Array SupportG,N&C KitStation Radiator Support— w\/PV Array Support112Rigid Support Solar Array_______‘I__IF 11 II ii ii 111t IIIF ii i 11 ii itFigure 4-4 A finite element model of: (b) solar array.by the geometrically nonlinear beam element with a superposed membrane elementrepresenting the blanket. However, for the purposes of a preliminary analysis, itwas decided to model the mast/blanket arrangement as plate elements, as shown inFigure 4-4(b), with the design constraint of making the solar array exhibit a frequencyof 0.1 Hz in fundamental bending when cantilevered to a fixed base.PV and Station RadiatorsThe PV radiators are the heat rejecting elements for the solar array batteries andassociated control electronics. There is one radiator for every pair of solar arrays. Thefinite element model for a PV radiator is shown in Figure 4-4(c), where the idealizedplate elements are used, with the fundamental cantilever frequency of 0.1 Hz.The Station radiators reject. the heat generated from the Space Station and payloads. The finite element model for the Station radiation is shown in Figure 4-4(d).113rjaq CD CD CD CD CD 0 CD 0TII-”‘-1 CD CD CD CD CD 0 CD 0 C-i 0ICo a. Cl)c 0_____________0LILLJILZ1L1L.IL.10.EnCl)C 0En LILIStinger and Resisto jet AssemblyThe resistojet, located on the stinger, is the main waste disposal system for theSpace Station. The waste products are first incinerated and the resulting gases ventedthrough the resistojet. The stinger (26.7 meters) is a deployable boom designed tohave a minimum frequency of 0.5 Hz and idealized as a cantilevered beam, with theresistojet modelled as a tip mass (Figure 4-4e).Reaction Control System (R CS) AssemblyThe Space Station attitude control is accomplished in part using reaction controljets mounted at the tip of the deployable booms. Each RCS boom, 12.7 m long andsupporting a tip mass of 190 kg, is located 5 meters inboard of the alpha joint. Theelectronics support weight of each boom was lumped at the base of the boom. TheRCS booms are similar in design to the resistojet/stinger assembly, and they alsohave a fundamental frequency of 0.5 Hz iii bending (Figure 4-4f).Modules and PayloadsThe modules are ring and stinger stiffened cylinders with hemispherical caps.There are, in all, five modules: one habitation, three laboratory, and one logistics.The mass and inertia properties of the modules are summarized in Table 4-2. Themasses are modelled as rigid bodies and placed at the geometric center of the maintruss.Table 4-3 summarizes weights of miscellaneous payloads and support systems.All the loads, e.g. the antenna assemblies; and the Guidance,Navigation and Control(GNC) kit are modelled as lumped masses with al)propriate inertias and their locationalong the main truss as given in Table 4-4.115Stinger______\_Res istojetFigure 4-4 A finite element model of: (e) resistojet and stinger assembly.RCS Boom\Jet ModuleFigure 4-4 A finite element model of: (f) RCS boom assembly.//116Table 4-2 Mass and inertia propoerties of the Space Station modules.MODULES MASS I I Ikg kgm2 x 108 kgrn2 x kgin2 x i0Laboratory 31,060 2.46 1.55 1.55Logistics 8,154 1.05 1.05 1.05Habitation 25,898 2.05 1.29 1.29Japanese 23,380 1.80 1.02 0.97European 23,323 1.51 1.02 1.02Resource Node 27,853 0.73 0.05 0.07Airlock 8,635 0.24 0.02 0.02Table 4-3 The Space Station payload characteristics.SUPPORT MASS I J, IzzLOADS kg kgin2 kgm2 kgrn2Solar Array Electronics 793 20 20 20PV Radiator Electronics 3.913 98 98 98St. Radiator Electronics 568 14 14 14Antenna Module 285 7 7 7RCS Jet Module 190 5 5 5Resistojet 200 5 5 5G,N & C Kit 1,132 28 28 281174.3 System Modes for the Evolving Space StationThe first forty system modes (including the six rigid body modes) for the fourmilestone configurations were obtained to discretize the continuous system.The frequency spectra provide the free vibration frequencies and associated system modes. The mode characterization help appreciate relative contributions of different parts of the Station.In general, modal displacements fall into the following three categories:(a.) Solar Array Deformation Modes: these are the modes in which the solar arraysdeform significantly in and out of the X-Y plane as cantilever plates and theremainder of the Station responds only slightly so as to maintain the dynamicequilibrium. Modes which are dominated by the twisting motion of the arrayplates are also included in this category.(b) Radiator Modes: These are associated with the PV and Station radiator deformations. Both are modelled to have a fundamental bending frequency of 0.1Hz.(c) Stinger/RCS Boom Coupled Modes: Since both these components are designedto have a fundamental bending frequency of 0.5 Hz, they appear in combinationin the system modes.(c) Overall System Modes: in general, these modes involve an overall motion of theStation, with solar array and radiator deformations coupled with response of themain truss in and out of the X-Y and X-Z planes.4.3.1 Nominal configurationsFor the FMC, the appendage response in bending dominated the first six elasticmodes (f — f12) with frequencies in the range of 0.1 - 0.5 Hz (Figure 4-5a). Of these,118the first three modes (fr, fs. f9) pertain to the PV array and radiator while fio — f12correspond t.o the RCS boom and stinger assembly. It is of interest to recognize thatthe torsional motion of the main truss is represented by f9 while the correspondingbending in Z and Y directions correspond to f21 = 2.30 Hz and f22 2.35 Hz,respectively. Note, the stinger and R.CS boom motions are coupled as both have afundamental frequency of 0.5 Hz. On the other hand, the PV arrays and radiatorshave their fundamental component frequency of 0.1 Hz as cantilevers. The solar arraydeformation modes display pure torsional motion in simmetric and asymmetric modesat fiG and f17 (1.14 Hz) with higher harmonics represented by f24 f25 (2.4 Hz) andf31,f32 (5.97 Hz).In the Man Tended Configuration, the system mass increases by around 70%,primarily due to addition of the U.S. Laboratory Module near the end of the truss andthe station radiator halfway along its length. As expected, the appendage responsedoes not exhibit any change except for the additional station radiatofs fundamentalbending at fio (Figure 4-5b). Presence of the module lowers the truss fundamentalfrequencies in Y and Z directions to 1.71 Hz (f21) and 1.74 Hz (f22), respectively.Of particular interest is the fact tha.t the Space Station in its FMC and MTC stagesexhibits main truss torsional characteristics below the bending modes, a feature notobserved with the PMC and ACC stages. This is consistent with the bending stiffnessvarying inversely as the cube of the truss length.In the PMC stage, the main truss has grown to 115 in. This results in the trussfundamental frequency to be lower than that in torsion as pointed out. Furthermore,due to the geometrical symmetry, there are more possible combinations of symmetricand asymmetric modes. Note. now the first forty modes are clustered in 0 - 2 Hz range(Figure 4-Sc)! Among them only six were found to be associated with the overall119system motion while the rest involved individual or coupled component motions.The main truss fundamental frequency in bending corresponds to f15 = 0.34 Hz (Ydirection, local vertical), and is coupled with the PV array bending. Because of theincreased mass along the main truss and at its geometric centre due to the modules,these frequencies were considerably lower compared to those observed in the MTCcase. Furthermore, presence of the modules significantly reduced the deflection at. thetruss center (origin of the x,y,z-frame). Hence the main truss behaved more like twocantilevered beams fixed at the modules, as can be expected.The Assembly Complete Configuration (AC C) exhibits essentially the same trendsas those observed for the PMC (Figure 4-Sd). With an increase in the truss lengthto 155 m, and addition of solar panels, radiators, modules, etc. the first. forty modesnow cover the frequency range of 0 - 0.8 Hz. Note. the extreme clustering of modesand overlapping will lead to complex interactiois between degrees of freedom, andtheir control a challenging task.Summarizing, it can he inferred that the appendages dominate the flexibility dynamics; however, the modes involving their pure motion are not significantly affectedby the evolutionary character of the Space Station. Furthermore, the Station hasmany closely spaced modes, partly due to the presence of identical, multiple systemcomponents (8 solar arrays, 4 radiators, 2 station radiators). However, growth ofthe main truss and its interaction with the cluster of modules significantly affect thesystem modes as observed through the changes in the truss bending frequencies.120Frequency Spectrum for FMJ1315LHHHArray bendingPV rad. bendingMain truss torsionArray bendingRCS boom bendingStinger bendingArray bendingPV rad. bendingArray torsionArray bendingPV rad. bendingMain truss bendingArray torsionStinger bendingRCS boom bendingMain truss bending[ PV array symmetric bendJ [v array unsymmetric torsionjFigure 4-5 Frequency spectrum and mode shapes for: (a) the FMC showing itsclosely spaced and overlapping character.-1.0—log f -0I__1.0-[17 =O.1O8HT] [!_=1136J121Frequency Spectrum for MTC-1.0 Array bending_____PV rad. bending[og Main truss torsionArray bendingStation rad. bendingStinger bendingRCS boom bendingArray bendingPV rad. bending0 Station rad. bendingArray torsionMain truss bendingArray bendingPV rad. bendingArray torsionStation rad. bendingStinger bendingMain truss bending1.0• RCS boom bendingFigure 4-5 Frequency spectrum and representative mode shapes for: (b) theMTC.( Main truss Y-bendingI PV array bendingnger bending= 1.7Oi][f io = 0.161 HzlLstation radiator bendingy122Frequency Spectrum for PMCArray bendingPV rad. bendingStation rad. bendingMain truss bendingRCS boom bendingStinger bendingArray bendingPV rad. bendingStation rad. bendingArray torsionMain truss bendingArray bendingFrequency spectrum and representative mode shapes for: (c) thePMC. Note, the spectrum shows further shrinkage.-1.0gfJ01.0yL Pv array bending][f7 =0.106Hz)LMain truss Y-bendingPV array bendingP 15 = 0.336HJFigure 4-5123I Frequency Spectrum for ACC-1.0Figure 4-5 Frequency spectrum and representative mode shapes for: (d) theACC. Note. now the first forty modes are compressed in the frequency range of 0.1- 0.8 Hz.Array bendingPV rad. bendingStation rad. bendingIogzfl0Main truss bendingRCS boom bendingStinger bendingArray bendingPVrad. bending1rMain truss Y-bendi’lLPV array bending JO.256 HzyyTPV array bending ][17 =O.lO5Hj1244.3.2 Solar array sun trackingAs pointed out before, the Space Station will be operating with its main trussalong the local horizontal and the solar arrays perpendicular to the orbital plane. Thearrays are provided with the rotational capability, about the alpha and beta joints,in order to track the sun for optimum exposure. Another design objective, which willrequire rotation of the solar panels, is to maintain a “feathered” flight configurationin order to reduce the aerodynamic drag. Obviously, different orienta.tions of the solarpanels due to these maneuvers will affect. to some extent, overall structural flexibilityof the Space Station and the associated frequency spectrum. This is the focus ofthe study in the present section with reference to the First Milestone Configuration(FMC). The rotational rates of the solar panels are relatively slow, such that a quasi-static condition prevails during the maneuver.For a 90° rotation of the solar panels about the -joiiit (Figure 4-6), the . frequency spectrum undergoes significaiit changes, particularly at modes 16, 18, 19, 25,27, 29 and 33, with variations as large as 35% in mode 16 (Figure 4-7). On closeinspection of the modal displacements, it is of interest to note that mode 27 startswith the solar arrays undergoing torsional motion, and as the maneuver progresses,the deformations become predominant in bending (Figure 4-8). Similar changes inthe behaviour were observed in other modes as well. For instance, mode 18 exhibitedthe main truss bending about the Y- axis coupled with the solar array bending atthe start of the maneuver,and by the end, the structural response was characterizedentirely by the torsional motion of the arrays. Also of interest is the interchange ofenergy among the modes and between the components in the same mode. Considerfor example mode 29. At the beginning of the maneuver, large bending displacements of the solar panels were observed, coupled with slight bending of the radiator,125stinger and RCS boom. By the end of the maneuver, the radiator exhibits largedisplacements with a small motion of the solar arrays.The second case study consists of rotating the arrays 900 about the 8-joint (Figure 4-9). Now, the flexible character of the system is hardly influenced (Figure 4-10).It is apparent that only modes 25 and 40 undergo any perceptible changes in frequency, however, they are quite small compared to the changes noted with the a-jointrotations. Figure 4-11 shows typical variations in t.he character of the 40th mode withthe rotation about the ,3-joint. Although the overall modal character of the systemdoes not change, local variations can be discerned, particularly in the progressiveincrease iii displacements of tile main truss and PV radiator. A similar behaviourwas observed in mode 25.The results suggest that significant effects of the solar panels rotation is confinedto a few modesThis information is utilized in the multibodv dynamics simulations such thatthe modes are updated, so as to maintain an accurate representation of the systemflexibility during the maneuver.12690OFigure 4-6N>%C.)a)Da)LLa)Cl)>C,)2.00.0Figure 4-7a-jointSolar array a-joint rotation for Sun tracking.450 900Solar Panel Slewing ManeuverFrequency spectrum for the FMC as a function of the a-joint rotationthrough 90°.0°12727=3468Hza =450/1’ 3.751 Hz3.773 Hz/. IFigure 4-8 Progre55j change in the dominant character of the 27th Systemmode, from torsion to bending during gOO rotation of the a-joj128C D1\) bI-.’t\)I.-.oqSystemFrequency(Hz)9)CCCl,C -S -S -S C -S C C -S U)-S cqoq-“CDo,.o CDp CDCDCD -S-S E -S CD C) p 0 C CD C,, 0 p -S p pC CC,) 0 D CD Cr) CD (oO;,D CD CD -SCo 0CCaC,)o0)000(0 0 00 -I-ITJ (DII ci aii4•O)cxo0NCl)1.0CDoq I—•0 C) p C) p CD-4.C!)0II-0)(0OiCD0x0NoZC 0)0, CI)Oc0N4.3.3 MSS operational maneuversThe Mobile Servicing System (MSS), a two-arm robotic manipulator, will assistin the contruction, operation and maintenance of the Space Station. Modi et al.[—971 have studied at length the dynamics and control of the MSS itself. However,it would of considerable interest to assess dynamical response of the Station duringthe manipulator slewing and translational maneuvers. As the focus here is on theSpace Station response, and not the manipulator dynamics, the MSS is modelled asa single flexible link.Consider a manipulator system, 15 meters in length, and uniformly distributedmass of 3200 kgs, carrying a 3,200 Kgs payload at the end. As an illustration, thetask consists of moving the payload from the module attachment area to a depot nearthe root of the solar arrays. Two cases are considered:Inpiane ManeuverThe path consists of an initial slewing maneuver through 900, followed by a 22.5-meter translation along the main truss, and terminating with a 90° slew, as indicatedin Figure 4-12. In this case the manipulator motion describes a trajectory in the planeformed by the solar arra.y plates (Figure 4-13). The maneuver has been discretizedinto ten time-steps (t1 —* t10). During this task, the frequency spectrum undergoessignificant excursions in modes 11, 20, 23, 24 and 25, with changes in frequency aslarge as 30 in mode 20 (Figure 4-14). Furthermore, the associated modal displacements also exhibited considerable changes during the maneuver. For example, mode25 (Figure 4-15) displays a transfer of energy, in this case from the manipulator armto the PV radiator. Mode 11 started by having the strain energy stored in bendingof the solar panels and PV radiator; at the time-step t. the RCS boom and stinger131Figure 4-12 MSS payload positioning operation consisting of slewing and translational maneuvers.displayed predominant modal motion; and by the end of the maneuver (t10) the elastic energy reverted back to the PV radiator and solar arrays. Mode 20 displayed avery interesting behaviour: at the beginning of the maneuver, the motion of the maintruss, at its free end, was suppressed by the presence of the robot arm, which actseffectively as an added inertia on an anti-node of a free-free beam.Out-of-Plane ManeuverHere the manipulator describes a trajectory in the plane formed by the PV radiator plate (Figure 4-16). A similar task profile as in the previous case has been implemented for this maneuver. As depicted in the frequency spectrum (Figure 4-17), asimilar pattern as before emerges, with the exception of mode 22, where an excursionSlewing TranslationSlewing132t5 t4 t3Figure 4-13 Time-steps during inpiane maneuver of the MSS.tbFigure 4-14 Frequency spectrum for the FMC of the Space Station as a functionof the MSS inpiane slewing and translational maneuvers.t2tiN>C.)c2.Oa)Da)I.UE>1•(I)ooti t5MSS Maneuver133= 2.089 Hzf25=2.212HzH \ I- - - -----1-1• 9I f25=2.124HzI YXz4 IFigure 4-15 25t4 modal function, at three discrete instants, as affected by theinpiane maneuver of the MSS.134of 11% takes place in frequency. The modal displacements also exhibit a similar behaviour to that observed in the previous case (Figure 4-18). Note the energy exchangebetween the various constituents in the 25th system mode during the maneuver. Atthe beginning of the operation, the mode is governed by the displacements of thesolar arrays, stinger and the MSS arm. As the maneuver unfolds, the strain energyis progressively transferred to the PV radiator.Summarizing, it can be inferred that the MSS maneuvers may significantly affectthe modal character of the structure as demonstrated through the two examples.Obviously, for a different maneuvering trajectory. a different structural behaviour willbe observed during the simulation. The necessary structural flexibility informationis delivered to the Multibody Dynamics module via the Modal Integral Processorinterface, in terms of modal coefficient matrices, as discussed in the next section.4.4 Modal IntegralsIn the governing dynamical equatioiis of motion, the modal functions appear, in avariety of combinations with other system parameters as integrands of certain definiteintegrals (referred to as modal integrals). The integrals have to be evaluated prior to,or during numerical simulation. These modal identities represent a measure of thestructural flexibility of the system.4.4.1 Integral evaluation procedureThe “flexiblity influence coefficients” take the following form:fmT dm Modal Linear Momentum Coefficient Matrix;f dm Modal Angular Momentum Coefficient. Matrix.The finite element method calculates the integrands of the above identities in discrete135Figure 4-16 A schematic diagram showing time-steps during t.he out-of-plane maneuver of the MSS.N>0c2.OG)DG)I—UE>C,)0.0MSS ManeuverFigure 4-17 Frequency excursions for the FMC as function of the MSS out-ofplane maneuver.t43 t2 titi t5 tb136II 1) z N-I.0‘I-L (7’C)NxBeam Modal Function Plate Modal Function0n,mA:Figure 4-19 Discrete finite element modal functions for beam and plate typemembers.form. As an example, for beam or plate appendages, the modal functions are givenin Figure 4-19. The above integrals were evaluated using the code FITPACK [98],a set of subroutines in the public domain. It is specifically designed to perform lineand area integration of discrete functions.A few of the integrals, however, can be evaluated more readily by taking advantageof properties possessed by the modal functions. For instance, due to the eigenvectororthogonality:Im T dm = (4.119)Im dm = A: (4.120)where E is the rn x m unit matrix and A is the m x rn diagonal matrix having )j =in the jth row and column.This concludes the discussion concerning numerical implementation of the finiteelement procedures in the calculation of the modal functions and integrals.01384.5 SummaryStructural dynamic models for the four milestone configurations were developedfor the flexible dynamics and control analysis. The main truss, stinger and RCS boomassemblies were represented as beams. The solar arrays and radiators were modelledas plates with the fundamental cantilevered frequency in bending of 0.1 Hz. Payloadsand modules were treated as lumped masses. The system modes were of four basictypes: modes dominated by array mast deformation; modes with significant radiatordeformations; those characterized by stinger and RCS boom deformations; and theones displaying coupled main truss/appendage deformations.The operational maneuvers of the solar panels and MSS may significantly affectthe flexibility description of the system and hence should be accounted for during thesimulation process. This can be achieved by updating the system modes during themaneuver.139CHAPTERFIVECONTROL METHODOLOGYThe primary purpose of the Space Station controller is to maintain attitude displacements and rates within design limits, by means of Control Momentum yros(CMGs), thus ensuring payload pointing and desirable microgravity environment.In this study, two attitude control techniques are proposed. First, a linear control approach is adopted. To this end. the equations of motion are linearized aboutan arbitrary equilibrium position and subsequently written in the state space matrixform. The Linear Quadratic Regulator (LQR.) is applied and the optimal controllergains for the state variable feedback are established by solving the steady state algebraic Ricatti equation. The performance index, defined by the controller energyand deviation of the state from the equilibrium, is minimized. The matrix equationsare solved numerically to obtain the controlled response of the system as well as theeffort required. Furthermore, in order to avoid saturation of the CMGs’ momentum, the equations of motion are augmented by including the CMG dynamics in thesimulation.Next, the nonlinear control based Feedback Linearization lechnique (FLT) isapplied. Using the original nonlinear dynamics of the multibody system, the controllerfirst determines the effort to effectively linearize the system and introduces a linearcompensator to achieve the desired system output.1405.1 Linear ControlApplication of a linear control theory would require, to begin with, linearizationof the equations of motion about an operating point. Feedback is subsequently introduced, where a real time comparison of the available output with the desired stateserves as a measure of error. This is used to determine inputs that will subsequentlydecrease the error. An important aspect of the control procedure is to achieve it inan optimal fashion, consistent with some specified performance index, over a periodof time.The basis of linear control laws is the linearized plant model associated withthe nonlinear multibody dynamics formulation. The equations of motion derived inChapter 3, and presented in detail in Appendix A. are nonlinear, nonautonomousand coupled. The nonlinearities are due to coupling between the attitude motion andstructural flexibility. A linear model for the multibody dynamics has been developed,with linearization performed about a planned attitude trajectory.5.1.1 LinearizationThe complete nonlinear, nonautonomous and coupled differential equations ofmotion for the rigid system can be written as- d 8Z,T - 6.z’T d’ -M33&=—=-Iw————---Iw—--—---I———=- Il, (.1)ao dt dt (1 + ecosO) ô9where:(5.2), -,T i -T j -,T ,U 1.1W U (.1W U UW U (1W=—---s- + +——-- ; (5.3)dt dt dt dt c3 o141=- + —s- - +—-(5.4)ftz,T-- =+ +-- ; (5.5)d& d’ d8T dôT— =— + + ——n- +——- ; (5.6)dt dt dt dt dt(5.7)alT alT alT=-- +--(5.8)Eq. (5.1) can be linearized about an arbitrary equilibrium position and cast inthe classical formMM+H+KM=F, (5.9)where:M=d (5.10)dtd&z’T d8cZ,T a.T dzH =—I Dü + — I ‘ + —n- I —dt dt d+ [—j ILZo + I5; (5.11)i i —T —TU (J U UU (JU’ UL uK=—-—- I,o+ —-—s- I+ —- I— + —s- I —dt dt c98 j dt dta-T alT alT+ —- I + —=— ILz,+—=-ITO + -- iT; (5.12)a& U6O a&142C d &z,T dz’1 18(,T -I—— ID0+ -—-r— I —I + I’o+ — ito; (5.13)[dt dtj L o owhere the mass matrix 1 is symmetric and positive definite. Coriolis and other yelocitv related forces due to attitude motion are included in H. Stiffness forces arisingfrom the gravitational potential are considered in matrix K. The vector F consistsof forces attributed to the orbital eccentricity and other parameters independent ofthe generalized coordinates.5.1.2 State space representation of the mathematical modelThe linearized equations of motion (5.9) can be put iii the form= A11 + B1ã+ F1, (5.14)where:Co E ..A1== system characteristic matrix; (5.15)B1= [] = control influence matrix; (5.16)== forcing function; (5.17)o = null matrix; (5.18)E = 3 x 3 identity matrix. (5.19)1435.1.3 Momentum managementThe Space Station will employ CMGs as a primary actuating device during normalflight operation. Since the CMGs are momentum exchange devices, external controltorques must be used to desaturate the CMGs, that is, to bring the momentum backto the nominal value. One approach to the CMG momentum management is tointegrate the momentum vector [45].The equations of motion were augmented to account for CMGs’ momentum control,x =ü, (5.20)where the vector i represents deviation of the CMGs’ momentum from equilibriumcondition; and LD* is the angular velocity vector linearized about the torque equilibrium position. As both the momentum and its integral will he used in the feedback,the following state vector is defined for control:X2 ={hi,h3fhidt,fdtfh}. (5.21)Thus, the standard state-space form for the CMGs’ dynamics becomes= A22 + B2ü, (5.22)where the matrices A2 and B2 are:0 W: W 0 0 0_w3* 0 cô’ 0 0 0A— LL)’ 0 000 5232— 1 0 0 0000 1 0 0000 0 1 000B2 = —B1. (5.24)144The system of Eqs. (5.14) and (5.22) can now be transformed into the followingaugmented state-space form= A + Bü + P. (5.25)where the 12th order state vector is{}, (5.26)and the system matrices A (12 x 12) and B (12 x 3) are:A_Al 0 5270A2’B=[]; (5.28)P= [‘]. (5.29)5.1.4 The Linear Quadratic Regulator (LQR)For large complex space structures, the conventional procedures applicable tosingle-input single-output systems for pole placement by gain selection cannot beused to obtain a unique control law. In order to develop a control strategy consistent with proper speed (bandwidth). damping characteristics and the available powerlimitation, one has to resort to an optimal control theory. The linear control theoryadopted is of the classical Linear uadratic Regulator (LQR) type with the performance index for minimum tracking error and energy given byJ = f (TQ+TRu) dt, (5.30)where Q is the symmetric state penalty matrix; and R is the symmetric controlpenalty matrix. The matrix R is required to he positive definite while Q can bepositive semi-definite.145Thus, the optimal control force ü that minimizes this criterion is given byu = _R_1BTYR± = —G±, (5.31)where YR is the solution to the Ricatti equation, which for the infinite time casebecomes-YRA - ATYR + YRBR’BT- Q =0. (5.32)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 design of a suitable observer.Substituting the feedback law into the system Eq. (3.14) yields(5.33)representing the response of the closed loop system. The characteristic equation forthe system is given bydet[sE— AG] = 0. (5.34)The closed ioop poles of the system are given by eigenvalues of A— BG. Thecontrol 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 he either real or complex conjugate.5.1.5 Controllability and observabilityFor a nonautonomous system, the check on controllability and observability arenot as straight forward as the time invariant case. They are verified by checkingthe ranks of the controllability and observahilitv matrices, at different instants. Thisprovides information concerning the controllable and observable character of the sys146tern at distinct instants, and by the continuity assumption, at all points along thereference trajectory.In the present. case, controllability of the system was established by the numericaltechnique that utilizes Householder’s transformation [99]. In Eq. (5.14), A and Bare the system characteristic and control influence matrices, of dimensions 6 x 6 and6 x 3, respectively. The system is controllable if and only if the rank of C = 6. HereC is the 6 x 9 matrix given byC = [BABIA2BI...1].It is not easy to check linear dependence of large order matrices, particularly inthe presence of numerical round-off error. Instead, the matrix is triaiigularized usingHouseholder’s approach t.hat transforms CT t.o an upper triangular mat.rix C-”. Ifthere a.re any zeroes in t.he diagonal of the upper triangular submatrix C’’, thenthe matrix is singular and C is not of rank 6 suggesting an uncontrollable system.The system was considered to he observable.5.1.6 The torque equilibrium attitudeThe equilibrium configuration for the Spa.ce Station is not at ± = 0. This isbecause of the triaxial nature of the Space Station inertia and moments contributedby the environment. For some evolving Space Station configurations, large deviationsfrom the 0 position are observed, thus making it necessary to consider the fullcoupling of the pit.ch roll and yaw equations of motion, through contributions of theproducts of inert.ia terms. Linearization of the equations of motion is performedaround an arbitrary operating point, which for the librating spacecraft is the TorqueEquilibrium Attitude (TEA). Under nominal operation, the Space Station’s TEA will147change due to its evolving character and environmental inputs. Thus, from Eq. (5.14),0 = + P, (5.35)which gives= —A’F. (5.36)Equation (5.35), represents a linear time varying system, where A and P arefunctions of time. They change due to the maneuvers of the various Space Stationappendages and orbital eccentricity effects. The system is considered to have a quasistatic equilibrium configuration, as given by Eq. (5.36), during each time-step ofintegration.In general, the state of a system can be cast in the form=€ + x (5.37)where Lx represents deviation from the equilibrium state.5.1.7 Effect of structural flexibilityThe controller design presented above is based on the attitude dynamics of arigid spacecraft. However, the Space Station is a highly flexible system with a verylow frequency spectrum as demonstrated in Chapter 4. Thus, the interaction of theattitude control system with vibration modes can become a matter of concern.The equations of motion are now augmented to include the effects of structuralflexibility on the system dynamics. To account for structural flexibility, the generalized coordinate vector, rearranged into the rigid and flexible degrees of freedom, canbe written as= (“..p1.p2 pfl) = {}. (5.38)148where 8 and depict attitude and vibrational generalized coordinate vectors, respectively. The equations of motion can be written asM09- 39p.e 1J I pJ I Jwhere the generalized control vector Q is based on the linearized attitude equationsof motion for the rigid system. Here the matrix M is partioned into four submat rices:M6,;M6,; M,6 and with Me. and corresponding to coefficients of thesubvectors 8 and , respectively. Details of the matrix M were given in Chapter 3.The control law = a = G, is based on the rigid spacecraft model, with thestructural flexibility acting as a disturbance to the rigid system. Thus, the controller,based on the linearized equations of motion, is applied to the complete nonlinearequations of motion.5.2 LQR Simulation Results and DiscussionThe overall architecture of the at.titude control and momentum management system is shown in Figure 5-1. The input signals to the attitude and momentum management controller are the attitude angles and angular rates, the CMG momentumand the momentum integral, and an estimate of the TEA. The static calculation ofthe TEA can be implemented off-line using tabulated estimates of the system inertia.To illustrate application of the proposed controller, consider the First MilestoneConfiguration (FMC). The inertia matrix for the FMC is taken as0.56 x 106 0 —0.19 x 106‘FMC = 0 0.78 x i0 0 kgrn2. (5.40)—0.19 x 106 0 0.79 xTwo spatial orientations in a 400 Km altitude. circular orbit are considered: thegravity gradient orientation, with the main truss along the local vertical and the149hd 0hFigure 5-1 Attitude and momentum management controller architecture.solar panels aligned with the orbit normal; and the Local Vertical-Local Horizontal(LVLH) mode, with the main truss along the local horizontal, and the solar panelsparallel to the orbit normal. Noise-free estimates of all the states and the body-axiscomponents of the CMGs’ momentum are considered to he available. The controltorque and angular momentum saturation values are set at. 200 Nm and 27,000 Nms,respectively [45].5.2.1 The gravity-gradient orientationThe gravity gradient orientation consists of deploying a. spacecraft in orbit withits minimum moment of inertia axis along the local vertical, which in the present caseentails aligning the main truss with the local vertical. The solar panels are deployedalong the orbit normal in order to optimally track the sun. as shown in Figure 5-2.°TEA150This orientation demands minimum control for its stabilization (Figure 5-3) due tothe bounded character of pitch aid roll response. However, it has the disadvantageof not exposing the various payloads on the main truss with an unobstructed view ofthe earth.To attain appreciation of the system dynamics during transition to instability,the uncontrolled system was subjected to large pitch, yaw and roll disturbances insequence (Figure 5-4). With a pitch disturbance of 5°, the roll and yaw degrees offreedom remain unexcited, while the pitch response shows stable oscillations aboutthe equilibrium point (‘e = —1.47°). However, with a roll and yaw disturbance of 5°,applied independently, the yaw exhibits a divergent behaviour in both the cases, thusindicating a strong coupling between the roll and yaw. This suggests that roll controlcould provide a key to ensure stability of the Space Station in the gravity-gradientconfiguration.Consider the linearized equations of motion for a single rigid body , with principa.linertias (I.I).I),in a circular orbit:+ 3e2(I— IA)’I’ = 0; (5.41)I>\ — €q5) + (Iv, — I)(G2.) + O) = 0; (5.42)Ib + + (I — J)(4e2— e) = 0. (5.43)The gravity gradient restoring torques T (i = i.. ). in a 400 km circular orbit,can he expressed as:T = 392 (IA — I) 30.0 Nm: (5.44)TA = O2 (Id,— Iii,) 0.2 Nm; (5.45)Tc1, = 492 (IA — ib) 40.0 Nm. (5.46)151C)0)-Da)-cDFigure 5-3 The attitude response of the uncontrolled Space Station for an initialdisturbance of 5° in pitch, roll and yaw, applied simultaneously.OrbitNormalxFigure 5-2 The Space Station in the gravity gradient orientation.200-200.0 Orbits 1.0152Figure 5-4 Librational response of the Space Station to an independent excitation in pitch yaw and roll. Note, the pitch and yaw disturbanceslead to essentially uncoupled motions. The system appears to become unstable in yaw through its coupling with roll.153Gravity-Gradient FreeLibratioriWe 147°Pe‘4J0=5;klo=oopo=oo0 Orbit 1C)-DC)-Da)VD4-,C,a)-D-oD4-,4-,50-550-550-5o=oo0 Orbit 10 Orbit 1NOWeo=ooThis suggests that the deviation about the yaw axis is likely to he significantlygreater than that about the pitch and roll axes, which is clearly demonstrated in thesimulation results shown in Figure 5-3. The corresponding librational frequencies areapproximately:= 1.65; = 0.50; = 1.95. (5.47)0 0 0indicating that the yaw period is relatively long (about two orbital revolutions). Thetrends indicated by this simplified analysis are confirmed through more general simulation results presented in Figs. 5-3 and 5-4.Next, the attitude controller was activated. Three different gain settings wereused to investigate the closed-loop system response with an initial disturbance of5° in pitch, roll and yaw, applied simultaneously. The controller performance ismeasured with respect to parameters such as settling time, control effort and angularmomentum requirements (Figures 5-5 and 5-6). The closed loop eigenvalues for Cases1, 2 and 3 are given in Table 5-1, where the gains have been selected iteratively. Asexpected, in all the three cases, the pitch response is driven to the equilibrium orientation (,L’e 1.5°, e = = 0°). The settling times vary from 0.05 orbit (5 minutes)for Case 1 to 0.2 orbit (20 minutes) for Case 3. The angular rates are also keptwithin the allowable limits of 0.2 deg/s. The control effort reveals that Case 3 with asettling time of 5 minutes, imposes the highest control torque (150 Nm in pitch) andangular momentum demands (5.000 Nins) on the CMGs. The CMGs’ momentumchange is the input, to the pitch axis disturbance rejection and it approacheszero after reaching a maximum value of around 5, 000 Nms. This high demand onimpulse is due to the large angle maneuver required to return the spacecraft fromthe imposed initial disturbance of 5° to the pitch TEA of —1.5°. The yaw degree offreedom requires minimal control, with Q. = 50 Nm and h, = 1, 000 Nms in Case154Table 5-1 Closed-loop eigenvalues for the gravity-gradient FMC controlled dynamics with linear attitude and momentum controller.Closed-Loop ElgenvaluesAttitude MomentumCase 1 —4.7 ± 4.4 —0.5 ± 0.7—16.9 ± 16.7 —0.3 ± 1.2—4.6 ± 4.4 —0.6, —0.005Case 2 —9.6 ± 9.3 —0.6 + 1.2—3.0 ± 2.4 —0.4 ± 1.6—2.9 ± 2.6 —0.9, —0.015Case 3 —29.9 ± 29.9 —0.3 ± 0.3—8.1 ± 8.0 —0.1 ± 1.0—8.0 ± 7.9 —0.3, —0.0013. This is expected since the minimum axis of inertia is aligned with the local vertical, an inherent characteristic of gravity-gradient stabilized systems. The controllerprovides satisfactory closed-loop transient response, with the CMG momentum peakand torque demand well below the specified limits. In addition, the energy requirements about the yaw axis are lower than the pitch axis since the deviation from theequilibrium is small: )c—= 1.5°; ‘o — = 6.5°Next, the performance of the controller in the presence of structural flexibilitywas tested. In this simulation, the controller is required to drive the system towardsthe design orientation, i.e. (‘ = .X = = 0°), thus aligning the spacecraft axes withthe orbital frame. The quantities of interest include the angular attitude excursions,the angular velocity with respect to the inertial frame, the solar panel tip deflectionand acceleration histories, and the control torque demanded by the maneuver. Thefundamental system mode has been used to represent. the deformation history of theSpace Station (f = 0.68 rad/s). The matrices Q and R are assumed diagonal1550xcrC)C).C,)01><U)C)C)Gravity-Gradient ControHedLQR•• Rigid ModelOrbit 0.2C’)0-Ds—TEA03N’°po0.050-550-550-50.0 Orbit i.o—a-— Case 1—0— Case 2Case 3J0.0 Orbit 0.2 0.0 Orbit 0.20.0 OrbitFigure 5-51.0 0.0 Orbit 0.2The controlled angular displacement and rates of the Space Stationin the gravity-gradient mode of operation for an initial disturbanceof 50 in pitch, roll and yaw, applied simultaneously.156Gravity-Gradient ControlledLQR•• Rigid Mode)50><C/)Ez-(.0 Orbit 0.2150Ez0-15050EZ 0O5OE 100z0-10010—10.0 Orbit 0.2Case 30.0 Orbit 0.10.0Figure 5-6C?0><C’)Ez-cC0>(U)Ez.0 Orbit 0.1‘-I0-5Orbit 0.2 0.0 Orbit 0.2The control effort and required change in the angular momentum foran initial disturbance of 50 in pitch, roll and yaw, for three differentgain settings.157and similar weightings as in Case 2 are used for the three librational degrees offreedom. The yaw degree of freedom displays the shortest settling time (t8 = 5minutes) although it experiences the highest angular velocity (w, = 2.5 x 10—2 deg/s),as shown in Figure 5-7. The solar array tip displacement and acceleration profilesshow that the vibration is coupled to the attitude motion, with the displacementsteady-state value reaching 0.2 x io— rn.These simulations show that the proposed control scheme tunes the open-loopunstable Space Station to a stable, oscillatory motion which minimizes the controleffort during steady state operations. Note that. it is not intended here to fine-tunethe gains for best performance and/or robustness. The objective is to demonstratethe applicability of the linear attitude controller to control the Space Station.5.2.2 The Local Vertical-Local Horizontal (LVLH) orientationIn the LVLH orientation, the Space Station has its main truss along the localhorizontal and the solar panels are aligned with the orbit normal for optimum exposure to the sun (Figure 5-8). As depicted in Figure 5-9, this mode exhibits instabilityin all the three librational degrees of freedom. However, with control, the missionrequirements are best fulfilled by this orientation, because it provides an excellentview of the earth to the observation equipment located along the main truss.The LQR controller was applied to the rigid Space Station, and the controlledresponse is shown in Figure 5-10 for three different sets of gains. The resulting closedloop eigenvalues for Cases 1. 2 and 3 are given in Table 5-2.The controller performssatisfactorily in all the three cases, with the settling time varying from t = 5 minutes(Case 3), to t = 20 minutes (Case 2). The control efforts display typical torquevalues of Q, = 50 Nm (Case 1) to Q, = 100 Nm (Case 3), which are well withinthe hardware specifications. The roll degree of freedom demands the least effort of158Figure 5-7 The effect of structural flexibility on the attitude control for theSpace Station in the gravity-gradient orientation.Gravity-Gradient ControlledLQR•• Flexibility• = 0.68 rad/sc..JCU)C)VSOrbitC)VC)VDCE010Cd,C1><c’JU)EOrbit 0.50.0 0.5 0.0____0___ ________ ___E0.0 nt 0.5— 0-200.050.00 Orbit 0.00 Orbit 0.25159OrbftNormalxA schematic diagram showing the FMC in the local vertical-localhorizontal mode.Q 70 Nm (Case 3), since the Space Station’s minimum moment of inertia axis isnow aligned with local horizontal.Results showing effectiveness of the controller in the presence of flexibility are presented in Figure 5-11. The trends are similar to those observed for the gravity-gradientcase (Fig. 5-7). The LVLH orientation requires less energy in roll, as expected. Thegains considered in this simulation correspond to Case 1, of the rigid body study.The Station experiences a maximum angular velocity in yaw (0.03 deg/s), while thepitch and roll rates display similar profiles with lower peak values. The solar panelsundergo higher tip displacements than those observed in the gravity-gradient case(Fig. 5-7), probably due to higher control effort needed to stabilize the spacecraft.Maximum control efforts in pitch, roll and yaw are = 8 Nm, QA = 12 Nm andFigure 5-8160Table 5-2 Closed-loop eigenvalues for the LVLH FMC controlled dynamics withlinear attitude and momentum controller.Closed-Loop EigenvaluesAttitude MomentumCase 1 —16.9 ± 16.7 —0.7 ± 0.5—5.0 ± 4.2 —0.1 ± 1.0—4.8 ± 4.3 —0.23. —0.02Case 2 —9.6 ± 9.3 —1.2 ± 0.6—3.5 ± 2.3 —0.1 ± 0.9—3.2 ± 2.6 —0.6. —0.06Case 3 —30.0 + 29.9 —0.4 ± 0.3—8.3 ± 7.8 —0.02 ± 1.0—8.1 + 7.8 —0.006, —0.07Q 15 Nm, as compared to Qj, = 8 Nm; Q. = 10 Nm: and Q = 8 Nm, in thegravity gradient case.161‘VFigure 5-9 The free attitude dynamics response of the Space Station for aninitial disturbance of 50 in pitch, roll and yaw, applied separately.LVLH FreeLibratione = -1.47°e0(Pe0°N0°A0=O°po=oo50.00 Orbit 0.25C)a)-Da)DC)a)-Da)DWoWe-5 po=oo0.00 Orbit 0.25D-\ 0:-50.00 Orbit 0.251625 50ElJo00.0 0.5 0.0 Orbits 0.25 50Eoo_____Z 0-5 O5Q0.0 Orbit 0.5 0.0 Orbit o.i5 50Ep°0 Z 05 O5Q0.0 Orbit 0.2 0.0 0.1Figure 5-10 The controlled response and the associated effort time histories forthe Space Station in the LVLH mode of operation.LVLH N’ ControlledLQR•• Rigid ModelOrbit6—0--- Case 1Case 2Case 3pOrb it163Figure 5-11 The effect of structural fjexibilitv on the attitude control for theSpace Station in the LVLH orientation.LVLH ControlledLQR.. Flexibility• f = 0.68 rad/sC”CU)C)VS0.0 Orbitci)VC)Vci101><C”U)EOrbit 0.50.0 Orbit0.5 0.00E0.5 Z—2 00.00 Orbit 0.05 0.00 Orbit 0.251645.3 Nonlinear ControlNonlinear control has received considerable attention in the past decade, particularly in the robotics applications. Linear control techniques based on either theBellman’s principle of optimality or on the Pontrya.gin’s maximum principle, fail toprovide reliable and accurate results, particularly when the nonlinearities of the system become important. To overcome this limitation, Freund[100] proposed the use ofthe state feedback to decouple the nonlinear system in such a way that an arbitraryplacement of poles becomes possible.Inverse control, based on the Feedback Linearization echnique (FLT), was firstinvestigated by Beijczy[1O1] and used by Singh and Schy[1021 for control of a rigid armrobot. Spong and Vidyasagar[103] also used the FLT to formulate a control procedurefor rigid manipulators. Given a dynamical model of the system, the controller firstutilizes the feedback to linearize the system followed by a linear compensator toachieve the desired output.Advantages of the FLT approach include the relatively simple implementation ofthe controller as compared to the LQR technique; and the straighforward compensatordesign. Here, the FLT is applied to the FMC of the Space Station to achieve attitudecontrol in the presence of structural flexibility.5.3.1 Feedback linearization techniqueThis procedure has been applied with success in many control problems dealingwith rigid systems. A particular application is in the trajectory tracking of a givenstructure where the dynamics involves only the rigid modes. For example, consider asystem described by a set. of equations in the formM(,t) + P’(, ,t) = (&. 6.t), (5.48)1 6owhere the generalized coordinate vector accounts only for the rigid degrees of freedom. The objective is to seek a nonlinear feedback control L t), which whensubstituted iii the above equation leads to a linear closed ioop system. It has beenshown [64.103] that, the resulting system becomes asymptotically stable around thenominal trajectory if the driving control efforts are given by= M(. t) + P’(, 6. t). (549)whered + K(d— ) + K(8 — 8) = (5.50)and 8d, 6d and d correspond to the desired trajectory characteristics. Here K andare the 3 x 3 matrices of position and velocity feedback gains, respectively. Theyare so chosen as to insure stable behaviour of the tracking error. e = 8— 8d, given by+K+Ke= 0. (5.51)A suitable choice for K and K is= diag{,...,x};K = (5.52)where Xi and represent the controller frequency and damping ratio, respectively.This results in a globally decoupled system with each generalized coordinate responding as a second-order damped oscillator. The natural frequencies j determine thespeed of response of the corresponding generalized coordinates. A larger value of Xigives rise to a faster response of of the it/i degree of freedom.5.3.2 Extension of the FLT to flexible systemsConsider the system given in Eq. (5.39). Recently, Karray and Modi[65] have extended the FLT to include structural flexibility for a model of an orbiting manipulator166system studied by Chan[97j. The basic idea here is to design a controller capable oftransforming the rigid part of the dynamics into a canonical, decoupled state spacemodel. This, obviously, implies a completely controllable system. Note, here the stateof the system is not transformed through a diffeornorphic mapping; rather it is thecontrol effort that makes the rigid part of the system behave as if it were completelylinear. Also it is important. to notice t.hat if the system were not in the form similarto that in Eq. (5.48), then a diffeomorphic transformation and a special form of thecontrol effort are needed for reducing the system to the canonical form. This can beachieved only if necessary and sufficient conditions pertaining to controllability andinvolutivity of nonlinear systems are met[104j.Now, if the observable states are chosen to be the components of the rigid modesubvector 8, then by selecting a suitable control vector Qo, the linearized equationsof motion become‘, , ) = (5.53)where:(5.54)withM = —M’8,M];=[F6—M6P’]; (5.55)and ‘iY takes the form given in Eq. (5.50). The control effort can be expressed as thesum of two parts, Q (primary) and Q62 (secondary):Q01 = Med + F; Q82 = M(KV + Kdë). (5.56)The primary controller is so designed as to compensate for the nonlinear effects corresponding to rigid part of the system. In practice, the system properties and the167dynamical model are usually not precisely known. To account for modelling uncertainties, i.e. to impart robust character, a secondary controller is introduced. Thefunction of the primary controller is to offset the nonlinear effects inherent in the attitude degrees of freedom; whereas the secondary controller ensures robust behaviourof the error. The question of flexible modes which interact with the rigid ones throughM6, still remains as they are needed for computation of the control effort Q. Twodifferent control schemes are proposed to that end: one leads to a Quasi-Qpen Loopontrol (QOLC) procedure; while the other is termed the Quasi-Closed Loop Controlscheme.Quasi-Open Loop ControlThe central idea here is to evaluate flexibility generalized coordinates through anoff-line procedure, i.e., dynamics of the p is computed independent of . However, itis still governed by the desired trajectory specified for the rigid degrees of freedom ascharacterized by 8d and 6d. Thus the dynamics of evolves according to= —M{M.8+ F(8d8dp,p)}. (5.57)Integration of this set of equations, which can be carried out off-line, permits thedesigner to assess the evolving behaviour of p and , and compute the control effortQe with the tracking error vector governed by Eq. (5.50). Of course, this implies thedynamics of the flexible generalized coordinates to be stable for the control study.It is important to recognize that the choice of as in Eq. (5.50), instead of beingsimply 0d, gives the system a more robust behaviour, similar to that attained with aproportional plus derivative controller.Quasi-Closed Loop Control168Here, responses in the rigid and flexible degrees of freedom are computed simultaneously according to the following dynamical relations:(5.58)= —M,{M.6+ (5.59)Now P’ is a. function of and 0 instead of being governed by 0d and 0d. Thedisadvantage of the scheme is the relatively large computational effort as comparedto QOLC. However, the QCLC is less sensitive to system uncertainties.The two control strategies discussed above are presented as a block diagram inFigures 5-12(a) and (b), respectively. It is mainly composed of two control elements,t.he plant a.nd the controller. The controller itself is made up of two sub-controllers,primary and secondary. In the case of QOLC,the primary controller receives informa.tion concerning the desired trajectory and the vibrational dynamics excited by thecommanded profile through an off-line procedure. On the other hand, in the case ofQ CLC, t.he primary controller continues to receive the desired trajectory commandsas before, however, now it is supplied with the actual dynamic state of the system. Inboth t.he cases, the secondary controller has, as main inputs, the desired trajectorycharacteristics a.nd the actual trajectory output, which are considered available.5.4 FLT Simulation Results and DiscussionThe FLT is applied t.o the F\lC of the Space. Station for the two spatial orientations described earlier: the gravity-gradient; and the LVLH.5.4.1 The gravity-gradient orientationConsider the rigid Space Station. Three different cases of controller frequency fora critically damped response (damping ratio = 1) are considered and the perfor169Figure 5-12 Block diagram for the nonlinear control: (a) quasi open loop technique;Secondary ControllerTorqueEquilibriumAttitude170Figure 5-12 Block diagram for nonlinear control: (b) quasi-closed loop technique.171TorqueEquilibriumAttitudemance is compared: (i) = 0.5 x 10—2 rad/s, (ii) x = 1.0 x 102 rad/s, and (iii)x = 1.5 x 10—2 rad/s. For simplicity, the resulting gains K = x2E and K = 2Eare chosen, where E is the identity matrix. The plotted quantities of interest includethe angular displacement and the control effort histories. As shown in Figure 5-13,even with a large initial angular displacement of 50 to the system in pitch, roll andyaw, the controller is able to regain the TEA (L’e = 1.5°, ‘e = 0°) in around0.02 orbit. 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 criticallv damped behaviour, which depends on the assigned controller frequency x anddamping (. With the same set of gains in the three directions, the spacecraft displays similar dynamic response in pitch, roll and aw. The control torques involvedin accomplishing the proposed task are well within the allowable limits, with a peaktorque in pitch of 100 Nm, 60 Nm and 20 Nm for Cases 1, 2 and 3, respectively. Alsonote that the control torque time histories for pitch and roll are essentially the same,because the Space Station inertias in pitch and roll differe by less than 1%.Next, the effects of controller damping on the system response is studied. In thissimulation, the controller frequency is set to = 1.0 x 10_2 rad/s, and the dampingratio effect is monitored for three cases: (a.) ( = 0.05, (1)) ( = 0.10, and (c) ( = 1.0(critical). As expected, a lower damping ratio leads t.o a larger settling time and ahigher control effort compared to the critical case. For Case (a), the settling timeincreases to 80 minutes (0.8 orbit.) with the corresponding control effort in pitch of= 95 Nm, compared to 60 Nm needed for the critical case (Figure 5-14). ForCase (b), the system takes 60 minutes to settle within 10% of the initial disturbance,with a corresponding torque in pitch Q = 75 Nm.The effect of structural flexibility on the nonlinear attitude controller is investi172Figure 5-13 Controlled librational response of the rigid FMC, in the gravity-gradient orientation, for three different sets of gains with an initialangular disturbance of 50 in pitch, roll and yaw.Gravity-Gradient ControlledFLT•• Rigid Model5IJo0-550E2:0O5QOrbit 0.20.0 Orbit 0.5o : C C C 0 0 0 •C 0 0—D--- Case—0-— Case 2-5______0.0 Orbit 0.5eq eq eq0.0 Orbit 0.50.05EzoO50.050E200 -50Orbit 0.10.0 Orbit 0.2173Ez0Gravity-Gradient ControlledFLT•• Rigid Model50Ez o0-50Orbit 1.0 0.0 Orbit 1.0Orbit5‘lJo0-50.05o0-50.05 50Epo0 z 0.5 0-500.0 1.0 0.0 1.0Figure 5-14 Controlled librational response of the rigid FMC, in the gravity-gradient orientation, for three different damping ratios with an initialangular displacement of 5° in pitch, roll and yaw.1.0 0.0 Orbit 1.0Orbit Orb it174gated next. To that end, the fundamental system mode for the FMC (w1 = 0.68 rad/s,Chapter 4), is used to represent the structural deformation. To keep the controllerbandwidth well below the structural frequency spectrum, the controller frequencyhas been set at x = 10—2 ra.d/s, which is one order of magnitude lower than thatof the fundamental system frequency. The plotted quantities include the libra.tionof the spacecraft, the angular velocity with respect to the orbital frame andw, the solar array tip deflection(6solar) and acceleration(6solar) histories , and thecontrol effort (Figure 5-15a). The settling time for the librational motion is around10 minutes (( = 1). The angular velocities history also presents a similar profile, withpeak values of 0.02 deg/s. The solar arrays are excited only slightly by the control torque. The structural deformation history is modulated by the control torqueprofile in pitch and it reaches a limit cycle type response once the system attainsthe required attitude orientation. The maximum solar tip displacement was of theorder of 4.0 x 10 m, with the tip acceleration reaching 1.5 x 10 m/s2, which isapproximately 1.5 g. A comparison of results for the QOLC and QCLC approachesshowed virtually no difference in the control effort (inset.). The pitch torque shows aresidual value in the steady-state regime, due to the flexibility of the system.The controller damping is now set. to = 0.1 while the frequency x remainsthe same. The corresponding response is shown in Figure 5-15(b). The trends arethe same as in Fig. (5-15a), however, the settling time rises to 40 minutes and thecontroller torque in pitch and roll increases to 80 Nm. The librational motion hardlyexcites the flexible a.ppendages(6solar = 6 x 10 ‘rn and 6sølar = 2.0 X i0 mis2).The control effort in pit.ch shows modulation due to the structural deformation (inset).It may be emphasized that these modulations were confined to the pitch degree offreedom only, because the dominant motion of the solar panels induces a torque a.boutthe orbit normal. As the control effort. diminishes with attitude, so does the amplitude175Figure 5-15 Controlled damped response of the flexible FMC in the gravity-gradient orientation, with preset controller frequency x lO2rad/s,in the presence of librational disturbance of 5°: (a) critical dampingratio.Gravity-Gradient ControlledFLT••• Flexibility0.00 Orbit0c’JCCl)a)V•0-S-20.00 Orbit 0.2500.00Eo0COrbit0.25NvVWMM1WCx005 Ez-0.050.00 Orbit 0.00 Orbit 0.251 ‘76of vibration, though t.he rate of decay is minimal.5.4.2 The Local Vertical-Local Horizontal (LVLH) orientationNow, consider the Space Station oriented in an unstable LVLH configuration. Thesame set of gains as in the previous Cases 1, 2 and 3 are considered with an initialangular displacement of 5° imparted to the system about the three librational axes.The duration of the simulation is typically for 0.5 orbit., however in some cases alonger period is necessary to establish trends. The angular displacement and controltorque simulation profiles reveal a similar trend as in the gravity-gradient case, butnow the minimum control effort is associated with the roll, since the minimum inertiaaxis is aligned with the local horizontal. The settling time varies from t. = 5 minutesto t 20 minutes and the control torques ta.kes the approximat.e value of ç., = 100Nm in pitch a.nd yaw, while the roll control t.orque is very small (Q = 1.0 Nm), asshown in Figure 5-16.Next. the effect of the damping on the controller’s performance is investigatedusing the same damping ratios as before: ( = 0.05: = 0.10; and = 1.0. Asexpected, a lower damping ratio results in a higher control effort and a longer settlingtime. Typical settling times var from 10 - 80 minutes with the control torque rangingfrom Q, = 55 Nm to Q = 10 Nm (Figure 5-17). Of course, a shorter settling timecorresponds to a larger control effort.When the flexible Space Sta.tion is considered. with preset controller frequencyx = 10_2 ra.d/s and damping ratio = 1.0, the response profiles exhibit analogousbehaviour in all three axes (Figure 5-18a). As before, the solar panels exhibit smalltip displacement (0.2 x iO in, amplitude) at the fundamental system frequency(approximately 650 cycles/orbit). The minimum control torque is associated with theroll axis. Due to essentially the same iiiertias in pitch arid yaw, the control torques177Figure 5-15 Controlled damped response of the flexible FMC in the gravity-gradient orientation with preset controller frequency x = lO2rad/s,and with an initial angular displacement of 5° imparted to the systemin pitch, roll and yaw: (b) ( = 0.1.178Gravity-Gradient ControlledFLT•• x=l 0; =0.i• Flexibflityc’J0xLI)C,G)V90.0 Orbit 1.0 0.05CC-51010><CM0EOrbit 1.00.0 OrbitC..’0><E1.0 Z-—ww—-20.05 0.00.00 Orbit Orbit 1.0LVLH ControlledFLT•• Rigid Model5woo-5E50zI •6 10 P6 •6—TEA 00-50Orbit0.02500.2o.o Orbit 0.5o : eq cc eq •O CO-0-- Case I—0- Case2-5 Case3o.o Orbit 0.5P0 CO P0-50.0Figure 5-16Ez-25Ez00.0 Orbit 0.210—1Orbit 0.5 0.0 Orbit 0.2Controlled librational response of the rigid FMC in the LVLH orientation for three different sets of gains with an initial librationaldisplacement of 50 imparted to the system.179LVLH ControlledFLT• Vooo5°• Rigid Modelw500-50.0EzOrbit51.000Orbit-51.00.0 Orbit51.0500-500.0E50z0-50E10z-101.0 0.0 1.0Controlled librational response of the rigid FMC in the LVLH orientation for three different damping ratios. Initial angular displacementcorresponds to 5° in pitch, roll and yaw.0.000Orbit 1.0o.o OrbitFigure 5-17Orbit180about these axes exhibit similar trends. Decreasing the damping ratio to = 0.1results in a longer settling time (t3 = 40 minutes) and control torque (Q, = 80 Nm).Furthermore. the pitch control t.orque shows modulation at the system vibrationalfrequency as before (Figure 5-18b).5.5 SummaryPerformance of two distinctly attitude control strategies is studied, with referenceto a. specific configuration of the proposed Space Station. In LQR approach, theequa.tions of motion are linearized about a desired trajectory, and the optimal gainsdetermined so as t.o minimize a preformance criterion. The equations of motion wereaugmented t.o 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 Space Station, which has a gravitationally unstableinertia configuration, to a stable, small amplitude oscillatory motion in the steady-state.The nonlinear control technique (FLT) was also found to be effective in controllingthe attitude motion of the FMC. The controller performance for three sets of frequencyand damping was compared for the rigid Space Station. Even when the satellite isinitially disturbed by 5° in pitch, roll and yaw simultaneously, the resulting gains areadequate to reduce the attitude error to within design limits in less than 5 minutes. Ingeneral, the rate of response is directly proportional to the magnitude of the controlgains. With the inclusion of flexibility in the Space Station model, both the QOLCand QCLC were found to be effective.The peak control torque demands for the LQR and FLT approaches are present.edin Table 5-3 to help assess their relative performance.181C00(I,0xc’.JU)Ce=c’J0U)C)C)V0.25 0.0000005Figure 5-18 Controlled damped librational response of the flexible FMC in theLVLH orientation, with preset controller frequency x = lO2rad/s,with an initial angular displacement of 50 imparted to the system inthe librational degrees of freedom: (a) critical damping ratio.182LVLH ControlledFLT.• x=1O; C=1.O. Flexibility-50.00 Orbit Orbit 0.250.00 OrbitWVWWVVWWVW\.iy(_-N0.00 Orbit 0.05 0.00 Orbit 0.25Figure 5-18 Controlled damped librational response of the flexible FMC in theLVLH orientation, with preset controller frequency x = 102rad/swith an initial angular disturbance of 5° introduced about the pitch,roll and yaw axes: (b) = 0.1.ControlledFLT•• x=i 0; 01• FlexibilitycJC‘—5><0C)V•0-SOrbit 0.05C-eC-50xE000><cJCl)EOrbit 1.00.0 1.0o.o Orbit Orbit 0.0 Orbit 1.0183Table 5-3 Control torque demand for the LQR and FLT approaches.CONTROL TORQUE, NmGravity-Gradient LVLHRigid Flexible Rigid FlexibleLQR FLT LQR FLT LQR FLT LQR FLTib 11 17 8 19 17 20 12 15) 10 1 9 2 10 14 9 188 13 8 22 10 0.3 10 0.5In general, the LQR provided stabilization with very lower control torques compared to the FLT, for a given settling time. In simulations with identical initialconditions, it was observed that for the gravity-gradient orientation, typical controltorque values in pitch were around 10 Nm, while for the FLT the control torquesapproached 20 Nm, for a settling time of 20 minutes. The objective here is not toobtain vast amount of response results for uncontrolled and controlled dynamics, butillustrate application of the methodology discussed before with reference to differentcontrol strategies. The gains and the desired trajectory were arbitrarily selected inthe present investigation .A more systematic approach in making these choices isrequired. To establish effectiveness of a specific control procedure and its relativemerit, further study is necessary with a variety of distuhances, and a.ccounting forsystem uncertainties.184CHAPTERSIXMULTIBODY IMPLEMENTATIONA general purpose computer program has been developed based on the formulation procedures presented in Chapters 3, 4 and ö. The implementation of themultibody dynamics and control study has been carried out in t.hree stages, whichconstitute the major building blocks of the present simulation tool. The StructuralDynamics Module defines the geometric and flexibility characteristics of the systemunder study. The spacecraft physical data, in terms of nodal mesh and structuralparameters are specified by the user, and the finite element modal analysis programgenerates the modal functions. the centroidal vector, and inertia properties. Thisinformation is supplied to the Modal Integrator Interface for processing of modal momentum coefficients and subsequently transferred to the next stage of simulation, theMultibody Dynamics Module. At t.his stage, the equations of motion are assembledand coded, without user interaction, by symbolic manipulation with optimum allocation of computer resources. Validity of the governing equations were establishedthrough checks against benchmark case studies documented in the literature.At user’s request, the Control Module makes available the linear a.nd nonlinearequations of motion for application of control strategies based on the Linear QuadraticRegulator and Feedback Linearization techniques.Finally, a graphical interface program has been developed in order to make the system dynamics output available in a suitable format for interfacing with Dat aView[105].This procedure allows the user to animate and visualize the simulation sequence. As185a.n example, a video depicting modal interactions of the evolving Space Station hasbeen produced in collaboration with the University Computer Services VisualizationGroup.6.1 Program ArchitectureSeveral general purpose computer codes aimed at studying dynamics of multihodysystems have been commercially available for sometime. They include DISCOS[22],ALLFLEX[23], TREETOPS [24] and SD/FAST[251. Primarily, they are suitable forsystems with large rigid body motion with flexible members undergoing small deformations. As expected, each asserts a variety of distintive and desirable features andthey have been used extensively with a varying degree of success often governed bythe nature of the system, experience of the user and computational tools available.In general, they present a scope for improvement in representation of axial foreshortening leading to dynamic stiffness effects, rotary inertia and shear deformation.Furthermore, it is widely recognized that[106j:(i) rendering the program operational often demands enormous time and effort;(ii) user-end modification and adaptability of the program is often cumbersome if notimpossible;(iii) contribution of the various forces and moments to the governing equations ofmotion is usually not available explicitely, making the analysis of results andphysical appreciation of the system dynamics difficult;(iv) general character of the formulation and computational methodology appliedleads to non-optimal numerical performance when dealing with relatively simple system configurations.The computer program developed in this study contains several features which186attempt to alleviate some of the drawbacks mentioned above and thus result in anefficient code for studying multibody dynamics. The program provides for directelimination (by command) of higher order terms in the explicit equations of motionwhen they are found to have negligible contribution to the overall dynamics. Inaddition to providing an accurate reflection of the analyst’s intentions in the derivationof equations of motion, this procedure is virtually free of wasteful operations such as,additions of zeros, multiplications by unity and taking dot product of orthogonalvectors.The procedure may be considered to he optimal in the sense that it leaves to theanalyst the tasks he is best trained to perform. while transferring to the computer themanually prohibitive algebraic manipulation and lengthy operations associated withderivations. This feature is introduced by incorporating symbolic manipulation in thederivation of equations of motion. The symbolic manipulation, i.e., the nonnumericalcomputation with a. digital computer, yields scala.r equations of motion specificallytailored to complex dynamical systems, where the analyst has the freedom and insightto incorporate any required degree of fidelity in the model. Furthermore, the outputof the symbolic manipulation is a. completely portable FORTRAN or C code in theformat M = F(, , t), which can be delivered via a file for processing withoutrequiring any programming. The procedure considerably reduces the developmentcost compared to that of a special purpose implicit code; and helps in providing somephysical insight into the behaviour of the system.As pointed out before, the implementation of the algorithm involves three stagesas shown in Figure 6-1: structural ynamics Module (SDM), Multihody ynamicsModule (MDM); and System Control Module (SCM). These modules constitute thet.hree principal blocks of the present multibody dynamics simulation program.187USERINTERFACEFigure6-1Flowchart showingtheinterfacingbetweenthethreemodules:structuraldynamics,multibodydynamics,andcontrol.j SYSTEMPHYSICALIPROPERTIESIFHOperationalandenvironmentaldisturbancesFLEXIBLEMULTII300YSTRUCTURAL4 DYNAMICSMODULEFiniteElementModalAnalysisProgramDYNAMICSMODULE00 00SYSTEMCONTROLMODULESymbolicManipulation(REDUCE)ControlDesign•Linear(LQR)•Nonlinear(FLT)SystemDynamicNonlinearNonautonomousandCoupledEquationsofMotion6.1.1 Structural Dynamics Module (SDM)The SDM defines the geometric and flexibility characteristics of the spacecraftunder study and it consists of the code relevant to the theory presented in Chapter 4.The spacecraft physical data, in terms of nodal mesh and structural parameters arespecified by the user, and the finite element modal analysis program generates thesystem modes, the centroidal vector for the undeformed structure, and inertia dyadicsfor the overall structure about the central body coordinate frame, as well as for theappendages about the hinge position. This information is supplied to the ModalIntegrator for processing and calculation of modal coefficients which are subsequentlytransferred to the next stage of the simulation, the MDM.6.1.2 Multibody Dynamics Module (MDM)An important aspect, when considering integration and data management, is theapproach to managing the software itself. Although software management seems tohave lower visibility than data management, it is more important from the development and validation point of view. Efficiency of the software development has beeninsured by the modular and vectorized character of the formulation itself. To help validate the code, it is important to be able to identify and isolate the various dynamicalcharacteristics of a system and study their mutually exclusive interaction dynamics.For example, the Space Station constitutes a dynamical system of evolutionary andhighy complex character. When analysing such a structure, it is easy to get entangledin the never ending maze of cause-and-effect dynamics between the flexibility of theoverall structure, slewing of the solar panels to track the sun for optimal exposure,translation of the Mobile Servicing System, crew motion and shuttle docking, to namea few possibilities.189Furthermore, the debugging procedure during the program development and computational efficiency can be dramatically expedited by extracting specific equationsof motion for the particular problem under consideration. To this end, the symbolicmanipulation program REDUCE [90] has been incorporated in the simulation loop.By identifying the classification in Figure 6-2 that best represents the system understudy, the program arrives at appropriate equations of motion with optimum allocation of memory thus avoiding unnecessary utilization of computer resources. Forexample, in Case 1, the Space Station would be represented as a rigid body, thus permitting identification of attitude dynamic characteristics and control requirements.If one wishes to study the effect of solar panels or robotic mobile manipulator slewingmaneuvers on the attitude motion, Case 3 would be the most suitable one. However,if the user decides that the effects of solar panels flexibility are important as well,then the algorithm with the choice of configuration in Case 4 automatically yieldsthe appropriate governing equations. Of course, the configuration in Case 6 woulddefinitely constitute a more truthful representation of the real structure, however atthe expense of higher computational demands.6.1.3 System Control Module (SCM)The governing equations of motion have been programmed in both their nonlinearas well as linearized versions. This provides considerable versatility in assessing relative merit of various linear and nonlinear control strategies. In the present study, thefocus is on the LQR using the linearized equations of motion and the FLT employingtheir nonlinear representation.190CDSixparticulartopologicalcases.TheREDUCEbasedformulationarrivesattheexplicitequationsofmotionwithoptimumallocationof computer resourcesforthecaseunderconsideration.Figure6-26.2 Program FlowchartFigure 6-3 shows the main subroutines and data flow in the program. A detaileddescription of each subroutine follows:6.2.1 Subroutine MODELThis subroutine reads in the the following simulation parameters:• Geometric: the spacecraft geometric data calculated by the finite element structural analysis block such as the inertia matrix (IMAT) and centroidal vector(GMI) for the undeformed system, the mass values (MASSI) and inertia matricesfor the appendages IMATI with reference to the coordinate system at the hingeposition, the initial hinge position vector (DIO) for the appendage, the first moment of area (RHOIMI) and the initial orientation of the appendages (ROTINJ);• Orbital: orbital period of the spacecraft (PERIOD), orbital eccentricity (EGG)and the initial spacecraft orientation with respect to the orbital axes (ORIENT);• Slew: the slewing maneuver profile (ROTSLW), the initial angular positionso, ‘Yo, the final angular positions aj, ,Bj’, 7f and the duration of the maneuvertf — t0;• Translation: the translational maneuver profile (TRASLW, the initial hingepositions x0, yo. z0, the final hinge positions Xf, Yf Zf. and the duration of themaneuver tj — to;• Simulation Time: the number of orbits for the duration of simulation (NOR).and the number of points per orbit in the output file for data processing (NPTO);• Initial Conditions: the initial conditions associated with the attitude motion inpitch, roll and yaw. with angular positions o. A0 and angular rates ‘o. o. .Xo;and initial conditions associated with the structural vibration, with deformation192ASSEMBLY OF EQUATIONS OF MOTIONI (Subroutine FCN)INERTIA MATRIX (I)(Subroutine IASSMB)ANGULAR VELOCITY (w)(Subroutine ANGVEL)MASS MATRIX (M)(Subroutine MASSMB)(Ug)ELASTIC P.E. (Ue)(Subroutine STRAIN)JOINT SLEWING (Ci)(Subroutine JSLEW)4KINETIC ENERGY (T)(Subroutine TASSMB)ANGULAR MOMENTUM (H)(Subroutine HASSMB)FINITE ELEMENTSTRUCTURALANALYSISMODEL GENERATIONMODAL DATA(Subroutine MODEL)MODAL INTEGRALPROCESSOR(Subroutine MODES)0Ui-JD0 GRAVITATIONAL P.E.(Subroutine GRAVIT)COMMONBLOCKSJOINT TRANSLATION (d)(Subroutine JTRANS)CENTER OF MASS SHIFT (S)(Subroutine SHIFT)Figure 6-3 Multibody dynamics flowchart depicting the main subroutines comprising the program and corresponding data flow.193displacernentsp10.2,3•••.and deformation velocities10.2,3.,po;• Integration: the parameters associated with the subroutine (IMSL:DGEAR) forintegration of the equations of motion, the initial step size (HE), the tolerance(TOL), the method of iteration (ME TI-I), the method of iteration (MITER), theindex for the integration (INDEX:Appendix C shows a sample of a complete input specification file for the FirstMilestone Configuration (FMC) of the Space Station. This information is writteninto a file using any available text editor, and its format is as given in the examplefile. The bold words have special meaning to the program, while the italic words arejust names provided by the user. The central body is the distinguished base body ofspecial importance, and the program computes orientation and translation relativeto the coordinate frame attached to it.6.2.2 Subroutine MODESThe MODES subroutine reads in the modal displacement values from the finiteelement free vibration analysis, and subsequently processes this information to calculate the modal coefficients required for the multibody simulation. The FITPACK[98]library of subroutines has been used to cla.culate the modal integral coefficients shownin Appendix A.6.2.3 Subroutine FCNThe subroutine FCN is the nerve center of this computer program. It callsdifferent subprograms to calculate the kinetic energy associated with deformation(TASSMB), angular momentum vector and derivatives (HASSMB), the transient inertia of the system (IASSMB), angular velocity of the system associated with theattitude motion (ANGVEL), the mass matrix (MASSMB), the gravitational poten194tial energy (GRAVIT), the elastic potential energy (STRAIN), the joint transformation matrix due to the initial orientation of the appendages and slewing maneuvers (JSLET4I, the joint translation vector (JTRANS), and the shift in the center ofmass (SHIFT). These subprograms are repeatedly called by the integration packageIMSL:DGEAR.As mentioned before, the equations of motion are programmed in their nonlinearas well as linearized forms. Thus the response time history can be obtained usingeither one of them. This helps in assessment of the effect of nonlinearities in a givensituation and, if negligible, can lead to a significant reduction in the computationalcost. Of course, they also provide the infrastructure for the control study.6.3 Computational ConsiderationsThe equations of motion were analyzed using a SUN SPARCstation. In order touse the IMSL:DGEAR integration subroutine, the Lagrangian equations of motionare put in the following form suitable for integration,= —M1()[H(. , 9) + K(. 9) — R(9)]=Letting= {. }, gives .‘ = {, ‘(a, , 9)}, which represents a set of 2N firstorder nonlinear differential equations. Thus, the integration subroutine finds approximations to the solution of a system of first order ordinary differential equations of theform ‘=f(9. ) with initial conditions. There are two built-in approaches availableto the user: the implicit Adams method and the backward differentiation procedure,also referred to as Gear’s stiff method. For each of the procedures, six different iteration schemes are available. The choice is based on the nature of the problem, storage195needed, etc. and therefore requires some experimentation. The prime considerationis the stiffness of the problem. For the reasons of efficiency and speed, the Adamsmethod is used for nonst.iff problems; while in the case of deployment and/or flexibility, the stiff routine is used. An analytic Jacobian is supplied or calculated internallyby the finite difference approach in case of linear and nonlinear problems, respectively.For every step, the DGEAR tests for the possibility that the step-size is too largeto pass the error test (based on the specified tolerance), and if so it automaticallyadjusts it. The step-size of integration specified by the user is employed only as astarting value, and is adjusted automatically by the subroutine. Values in the rangeof 106 to 10 were used in this study.The error, which is controlled by way of the specified tolerance, is an estimateof the local discrepancy due to truncation, t.hat is, the error in a single step withthe starting data regarded as exact. This should be distinguished from the globaltruncation error due to all steps taken to obtain y(x). The later is neither estimatednor controlled by the routine. The user manual for the subroutine advices that, if theproblem is mathematically stable, the global error at a given x should vary smoothlywith the chosen tolerance limit in a monotonically increasing manner. Different tolerance levels are used here ranging from iO to 10_b, depending on the problem.6.4 Code ValidationOne of the major challenges of flexible multihody dynamics simulation codes isto demonstrate model validation and establish credibility. Obviously resolution of acomplex nonlinear problem is undertaken primarily because the solution is not available. Thus assessment of the accuracy associated with the formulation and relevantcomputer code does present a challenging task. Several avenues are available to thisend. To start with, simulations of a given system with different methods of integration196routine and error tolerances should give comparable results. Any major discrepancywould point to some numerical sensitivities. Moreover, in most situations, the designer knows about several inherent properties of the system, such as the total energyand angular momentum. In the present case, validity of the governing equations wereestablished by monitoring the total energy of the system, and through checks againsttest cases documented in the literature.6.4.1 Total energyFor a conservative system, total energy, a scalar quantity, must remain constant[107], i.e.7lAD__________LI— 1V1 I cm £ tcm——p3‘cm cm+ + 2R3 = constant. (6.1)cmwhere:= cosAcos —sin +sin\cos. (6.2)The total energy of the system was monitored for the rigid FMC Space Station.The spacecraft is assumed to be in a Lagrangian attitude orientation, where theminimum axis of inertia is aligned with the local vertical, while the maximum axis ofinertia is aligned with the orbit normal. The total energy of the system was monitoredby imparting librational inertial condition of 10 in pitch, roll and yaw, separately. Thecalculated total energy of the system was constant, with a value of around of 2.9 Jfor the prescribed initial conditions. The change in total energy was zero, with errorsof 108, which are within the tolerance limit set up by the integration routine. Thus,the check for the conservation of energy also served as a. tool for validation of theintegration routine IMSL:DGEAR used in the present simulation program.1976.4.2 Test case studiesThe present dynamics and control simulation tool was further tested by comparingthe results with those documented in the literature for particular simplified models.Attempt was made to test each of the six possible configurations (Figure 6-2) with acorresponding representative case that is reported.(1) Rigid Space Shuttle dynamicsThis case study is representative of Configuration 1 (Figure 6-2), where the systemis taken as a single rigid body with the focus on the attitude motion. Modi andIbrahim[108] have carried out a study of the rigid Space Shuttle dynamics, and theobjective here is to compare with the results obtained.The orbiter is assumed to negotiate a circular orbit with a period of 90.3 minutes,which corresponds to an altitude of 330 km from the earth’s surface. The inertiamatrix for the Shuttle was taken to he the same as that used in [108].8286760 27116 —8135‘shuttle = 8646050 328108 kg’rn2.sym. 1091430Figure 6-4 shows the librational response of the Orbiter when subjected to a smalldist.urbance of 5° in pitch, roll and yaw. The Lagrange configuration representing theminimum moment of inertia axis along the local vertical and the nmximum momentof inertia axis aligned with the orbit normal is stable as expected, and the dynamicbehaviour matched precisely with the results of Modi and Ibrahim.(2)Rigid one-link manipulator systemThe rigid one-link manipulator system studied by Modi and Chan[109] is repre198200)0U)000U)C)C0-200.0Figure 6-4 Response of the Space Shuttle to an initial disturbance of 50 in pitch,roll and yaw. applied simultaneously.sentative of Configuration 3. This case requires modelling of a rigid spacecraft withslewing and translating appendages. The tailored equa.tions of motion were derivedby REDUCE and used to verify the code for the correct implementation of appendageslew (ROTSLW) and translation (TRASLW) subroutines.In the simulation, period of the circular orbit is taken to be 100 minutes, theplatform mass as 214,000 Kg, and the manipulator has a mass of 3200 Kg. Thelength of the platform is 11.5 rn, compared to a 15 m arm for the manipulator. Thedynamic response for several combinations of slewing and translation maneuvers wassimulated.Figure 6-5 aims to verify the program code with regard to the translational maneuver subroutine (TRASLW). Using the sinusoidal on ramp profile for the specifiedOrbits 5.01990.0“Jo-0.40Figure 6-5 Librational response of the rigid one-link manipulator during a translational maneuver of 20rn completed at several speeds.coordinate, the maneuvers are specified by their magnitude and duration. A manipulator translation of 20 meters along the Station is completed for a duration rangingfrom 0.01 to 0.8 of an orbit (1 to 80 minutes). Note that a short duration requiresmore energy to execute the maneuver, leading to a larger disturbance. The resultantplatform librations are not symmetric with respect to ,I’ 0 since it no longer represents the equilibrium state after the maneuver due to a change in the system inertia.The results agreed with the simulations reported in Chan’s thesis.Next, the attention is directed towards verification of the part. of the code relatedto the slewing maneuvers subroutine (ROTSLW). With the mobile base fixed at thecenter of the station, the manipulator arm is slewed through 1800 with the beginningand end position of the arm perpendicular to the local vertical. Several slewing duraOrbits 42000.2‘V0-0.0-0.2Figure 6-6 Librational response of the rigid one-link manipulator during a slewing maneuver through 1800 completed at several speeds.tions (r = 0.01, 0.1, 0.5, 0.8 of an orbit) are used to assess effect of the maneuveringspeed on the response. The results in Figure 6-6 show essentially that faster slewmaneuvers induce higher librational amplitudes. It is interesting to recognize that, asthe arm swings in the counterclockwise sense, the station rotates clockwise (negative‘) to conserve the angular momentum.Finally, both translation and slewing maneuvers are applied together. Here, theone-link manipulator translates a distance of 30m from the Space Station’s centerand the arm simultaneously slews through 90° in 0.01 orbit, with varying payloadmass (Figure 6-7). As expected, and confirming the results obtained by Chan[109], alarger payload leads to a higher pitch amplitude. A payload of 32,000 kg, ten timesthe manipulator mass, causes the platform to pitch by 8.4° within 0.5 orbit while the0 Orbits 4201Figure 6-7 Effect of the payload mass on the librational response of a one-linkrigid manipulator executing a simultaneous translational and slewingmaneuver.corresponding amplitude in the absence of the payload is merely 0.6°.(3 )Flexible one-link manipulator systemThe space based one-link flexible manipulator system represents the spacecraftConfiguration 4, where the central body is rigid and the appendages are flexible.Chan[109] has also conducted a planar case study of such a system. The modelis similar to the rigid link case with a 32,000 kg payload to simulate extreme effects.Figure 6-8 shows the librational response for several values of the arm stiffness givenin frequency (as a cantilever beam in the first mode): 1.2, 1.5, 2.0, and 5.0 rad/s.Note, including flexibility in the manipulator link only modulates the rigid arm pitch10.01,00.0-10.0//arm:i!Stationm=0—— m=2—A--— m=100.0 Orbits 1.22025.0e(m)0.0-5.0Figure 6-8 Effect of the arm flexibility on pitch response and pointing accuracyof a single link manipulator executing the combined translationaland slewing maneuvers.response with superposition of a low amplitude, high frequency component.As the link becomes more flexible, the increased deformations lead to a greatererror () in the payload position, i.e. the link’s tip deviation w.r.t. the reference condition. The results check with the solutions obtained by Chan leading to confidencein the multibody dynamics formulation and corresponding code developed here. Orbits 0.10.0 Orbits 0.12036.5 SummaryA general purpose flexible multibody dynamics simulation program has been developed. The state-of-the-art computer program developed in this study has severalfeatures which makes it highly efficient and versatile:(i) the symbolic manipulation procedure has been included in the simulation loop inorder to extract desired specialized equations of motion for the particular systemconfiguration under study. This has the added advantage of transferring thetedious task of formulation of equations of motion to the computer, thus allowingthe structural dynamicist to devote time and effort towards understanding andinterpreting the physics of the problem.(ii) The program provides for direct elimination of higher order terms in the equationsof motion when these terms are found to have negligible contribition to the overalldynamics.(iii) The program structure is highly modular and vectorized, which contributes tothe relative ease with which it can be debugged.(iv) Finally, validity of the governing equations and the computer code was establishedthrough checks against benchmark case studies documented in the literature.204CHAPTERSEVENRESPONSE TO OPERATIONAL DISTURBANCESThe multibody dynamics and control program developed in this thesis constitutesa comprehensive simulation tool in the study of a large class of spacecraft. Thefeatures incorporated combine to tackle a variety of operational disturbances andenvironmental conditions encountered in practice.The Space Station represents an evolving platform with time varying geometry,inertia, flexibility, damping and other structural properties during its constructionalphase. Each stage in the development represents a challenging dynamical problem indesign, dynamics, stability and control.Here, applicability and versatility of the simulation tool is demonstrated by studying two of the Space Station’s evolutionary milestone stages: the First Milestoneconfiguration (FMC); a.nd the Assembly Complete configuration (ACC). Dynamicresponse of the Space Station in the presence of operational disturbances is studiedduring nominal mission. The results provide important information which my proveto be useful in defining design loads for the Space Station’s main truss, solar arrays, and secondary components. It also provides useful information concerning themicrogravity environment and control effect during various operational phases.7,1 Space Station Mission RequirementsThe attitude control system for the Space Station must he capable of providing:(i) sun pointing attitude to the solar arrays to maximize their power output; (ii) a20öFigure 7-1 Allowable microgravity level envelopes for the proposed microgravitylaboratory experiments.LVLH orientation; and maintain control during (iii) Space Shuttle docking: (iv) MSSmaneuvers; (v) crew induced forces. Furthermore, microgravity experiments plannedaboard the Space Station will impose acceleration limitations as depicted in Figure 7-1.7.2 First Milestone Configuration (FMC)The Space Station is projected to be assembled in space using, approximately,30 Space Shuttle flights. The FMC represents the evolutionary stage at the end ofthe second flight. Its main truss structure will be 60 meters long with a total massof 21,580 kgs. Major equipment installed on the FMC includes two solar arrays, aPV radiator, a stinger and a RCS boom. Various support loads are located along themain truss. The design attitude orientation is LVLH. Figure 7-2 shows the coordinateframes, librational angles, and appendage numbering scheme used in the simulation.Considering the first m system modes to represent structural deformations, the gen206era.lized coordinates and the degrees of freedom are:The numerical data used in the simulation were presented in Chapter Modal convergenceIn order to simillate the system dynamics using the assumed modes approach, itis necessary to specify the number of modes to he used. In theory, as one increasesthe number of assumed modes, the results should approach t.he true response. On theother hand, this leads to an increase in the computational cost. So it is important tostudy the character of modal convergence to strike some balance between accuracyand cost.The dynamic behaviour was monitored in the presence of an initial disturbanceof 5° of the system in pitch, roll and aw, applied simultaneously. The structuraldeformation was represented by considering 1, 5, 10 and 20 lowest modes. Plottedquantities of interest include the lihrational displacements and corresponding angularvelocities, the nonlinear control effort to maintain the Station in the LVLH orientation; modal displacements as well as accelerations at the tip of the upper solar paneland at the modules location on the main truss (i.e. end of the truss). Displacements and accelerations at. the tip of the PV radiator, stinger and RCS boom are alsomonitored.Figure 7-3(a) shows controlled a.ttitude response of the system. Nonlinear controller gains are based on x = 102 rad/s and ( = 1 (Chapter 5). The maximumangular velocity in pitch rea.ches 0.02 deg/s for a settling time of 0.1 orbit (10 mmutes). The corresponding control effort plots (Figure 7-3b) show that contributionof the higher modes manifests as high frequency local variations of ±2Nm in the207Appendage Appendage Space StationNumber Type ComponentCentral Body Beam Main Truss1 Plate Upper Solar Array2 Plate Lower Solar Array3 Beam PV Radiator4 Beam Stinger5 Beam RCS BoomOrbitNormalFigure 7-2 The coordinate system appendage numbering scheme and attituderepresentation for the FMC of the proposed Space Station.208Figure 7-3 (a) Librational displacement and angular velocity response of theFMC to an initial displacement of 5° in pitch, roll and yaw, appliedsimultaneously. Note, the results for 1, 5, 10 and 20 modes areidentical.characteristic control effort profile. The control requirements in pitch and yaw peakat 75 Nm, while the roll degree of freedom requires only 2.5 Nm. It is interestingto note that, for the single mode case, the roll and yaw degree of freedom do notshow a significant contribution from the structural flexibility of the system. Here, thestructural deformations were confined to the solar panels vibrating in a symmetricfashion, while other components of the Station remained relatively stationary. However, with the second and higher modes included, the radiator and other secondarymembers displayed a more active contribution to the system deformation, leading tointeractions with the roll and yaw degrees of freedom.The solar panels do not exhibit a noticeable improvement in modal amplitudeSpacecraft Configuration: FMCOrbital Orientation: LVLHInitial Conditions: 141= =Attitude, deg5020-20 Orbit 0.25 0 Orbit 0.25209Cl)0)0If)Cl)0)0Figure 7-3 (b) Comparison of the nonlinear control effort required to drive thesystem to its design orientation (LVLH) with an increase in the number of system modes included in the simulation. The system is subjected to an initial attitude displacement of 5° in pitch, roll and yaw.210Spacecraft Configuration: FMCOrbital Orientation: LVLHInitial Conditions:= == 50C)V0Q,Nm I Q,NmQA,Nm100-10100-10100-10100-10-• Cl)C)V000 Orbit 0.2 0 Orbit 0.2 0 Orbit 0.2prediction as the number of modes is increased (Figure 7-3c). During the transientdynamics, i.e. while the controller is driving the Station to the LVLH orientation, thetransverse tip displacement reaches 5 x i0 rn, while in the steady state, it settlesat 3 x i0 m. However, the higher frequency modes induce local variations in thefrequency for the acceleration time histories at the tip of the panel. with peaks of3 x i0 m/s2( 30kg). The PV radiator tip also exhibited similar behaviour (notshown). Here, the maximum transverse tip displacement and acceleration values inthe steady state were 3.5 x 10 rn and 4 x i0 rn/s2 ( 40kg), respectively.The main truss, stinger and RCS boom are treated here as beam type structuralmembers. The possible tip displacement and acceleration directions include the extensional motion (x), and transverse vibrations (y,z). In this particular simulation,the z-displacement at the module location on the main truss is plotted. Figure 7-3(d)shows that while the amplitude and frequency content of the tip displacernent profileare not affected, the acceleration prediction at the modules location experiences adrastic change. particularly with 20 modes. This is due to the fact that, at the highermodes, the main truss plays a more dominant role in the structural dynamics of thesystem (main truss bending occurs at system mode f22, which is accounted for in the20-modes solution). The acceleration level at the moclilles location was found to hearound 5kg.Summarizing, it can be inferred that the system displacement can be predictedwith accuracy using 5 system modes (less than lVc error in the solar panel tip displacement compared to that given by the 20 modes solution). However, the accelerationprofiles showed that the higher modes contribute to local variations in the acceleration time histories (±2jig). although the general character of the response remainsessentially the same. On the other hand, dynamics of the main truss showed that it211Spacecraft Configuration: FMCOrbital Orientation: LVLHInitial Conditions: N’0 = == 50Figure 7-3 (c) The solar panel tip displacement and acceleration profiles showingthe effect of number of system modes in the dynamic simulation. Thesystem has been given an initial attitude displacement of 50 in pitch,roll and yaw.soTar’ m xl soIar’ m/s2xl50-5—0)0C)V00)V1Cl,C)00c’J50-550-550-550-55-5C)-D0U)0)V0L()Cl)0)V00U)C)V00C%.j50-550-50 Orbit 0.05 0 Orbit 0.01222Cl)V0L()U)C)V0CC’,C)V00C’.’Figure 7-3Cl)C)V0U)C)V001U)a)000cJ(d) The main truss z-direction tip displacement and acceleration profiles showing the effect of number of system modes in the dynamicsimulation. The system has been given an initial attitude displacernent of 5° in pitch, roll and yaw.213Spacecraft Configuration: FMCOrbital Orientation: LVLHInitial Conditions: N0 = == 50mxlO6truss’C)V01•9 —,m/sxl QOtruss’C)V050-5550-550-550of\J\f\J\/\r\-550-550ViW10-50Orbit 0.05 0 Orbit 0.01is necessary to include the higher system modes in order to reasonably predict theacceleration loads at the modules location with sufficuent accuracy. Thus, dependingon the use, determination of accurate information would require judicious selectionof modal functions with significant contributions to the variable of interest.7.2.2 Response to operational disturbancesSolar Array Sun TrackingThe solar panels will he required to track the sun for optimum power output.Consider the FMC of the Space Station, in a a near-circular orbit (e0.02), with theLVLH orientation as shown in Figure 7-2. The sun tracking maneuver consists ofrotating the panels about the local axis parallel to the orbit normal, at the orbitalrate 0, so as to maintain the the array facing the sun. The simulation is carried outfor 0.25 orbit (25 minutes), i.e the solar panels are required to rotate through 900during this time. Ten flexible system modes (beyond the first six rigid body modes,f “ fin) are considered to represent structural deformation of the Station. Sincethe spacecraft experiences changes in geometry during the tracking maneuver, theassociated frequency spectrum changes accordingly, as explained in Chapter 4. Thus,in order to maintain a faithful representation of the system geometry and structuralflexibility, the modes need to be updated at a suitable interval. Here, the interval istaken to be 15°, thus giving six updating steps during the maneuver, Of course, onecan use any desired intrval for updating the information depending on the problemin hand, accuracy required and acceptable computational cost.Figure 7-4(a) shows response of the FMC during the solar panels sun trackingmaneuver. The control effort to maintain the Station in the LVLH orientation isminimal. with the peak Q, = 1.4 Nm and Q,, = Q = 0.5 Nm. Since the controller214is commanded to drive the system to the LVLH orientation, which is not a torqueequilibrium position ( = 1.5°, ? = = 00, the control effort in the pitch degreeof freedom continues to persist at an average level of 1 Nm.Of interest is the transfer of energy between the solar panels and the PV radiatorduring the maneuver. It can he seen that a transfer of energy is taking place. Atperigee, the beginning of the maneuver, the solar panel tip deflection is larger thanthe PV radiator (3 x io— and 3 x iO rn. respectively. At the end of the maneuver,when the spacecraft has completed 0.25 orbits, the PV radiator appears to containmost of the modal energy, with t.he tip displacement considerably higher t.han thatof the solar panel. The main truss displacement at the modules location increasesduring the maneuver, while the microgravity levels stay well within the alowahie limitof 1.0kg.Also of particular interest is the shift in the center of mass of t.he system, dueto the slewing maneuver and flexibility, which is accounted for in the model. Sincethe rotation of the solar panels takes place about the longitudinal axis, there is nonoticeable change in the shift of the c.m. vector due to the maneuver. The contribution, primarily from the structural deformation as indicated by high frequencymodulations, is indeed very small , of the order of 10—6 in (Figure 7-4b).Next, the effect of aerodynamic torque was explored during the sun trackingmaneuver. The torque model accounts for the diurnal bulge at twice the orbital rateand was presented earlier. (Fig. 3-25). Figure 7-4depicts the corresponding dynamicalresponse. The TEA shifts from ?e = l.5°,Ae = 0°. to = 13°,) = 90 ande= 70 This change in equilibrium reflected in an increase in the control effort(from Q = 1 Nm, Q> = Q = 0 Nm to Q = 4.5 Nm, Q>, = 1.5 Nm, and Q = 1.4Nm). The solar panels and secondary members do not exhibit any significant change215Figure 7-4 (a) The Space Station response to the solar panels rotation to trackthe sun for optimum power output. Note that the modal informationis updated every 15° to maintain a faithful model of the systemgeometry and Structural deformation.216Spacecraft Configuration: FMCOrbital Orientation: LVLHDisturbance: Array Sun TrackingModal Updating: f7— 1j52IPrntL0 Orbit 0.01Ez10__—10 Orbit 0.2511Ez-0—1Ez00E0xE0xENLo0 Orbit 0.2530-330-350-5nF—’_1C1x0x0)000)N20-220-210—1AUlL..- “1111111i,ti,itII IjijJijiuni’rflri. XrI..flIIV —0 Orbft 0.25 0 Orbit 0.252030-20.25Figure 7-4 (b) The shift in the center of mass due to structural deformationduring the solar panels tracking maneuver.in behaviour, with similar responses as before in displacement and acceleration.To summarize, rotation of the solar panels for tracking the sun, even in thepresence of aerodynamic drag, is not likely to affect the microgravity experiments.Furthermore, the control effort required to maintain the spacecraft in the LVLHorientation is rather minimal.MSS Payload PositioninThe MSS manipulator arm, among other tasks, will be used to position payloadsalong the Space Station’s main truss. Here it is proposed to investigate a maneuverdesigned specifically for this purpose. Consider the case where a disabled satellite hasbeen retrieved by the Space Shuttle and delivered to the Space Station docking bay,0 Orbit217Figure 7-4 (c) The Space Station response to the solar panels rotation, to trackthe sun for optimum power output, in the presence of aerodynamictorque.n eaeroSpacecraft Configuration: FMCOrbital Orientation: LVLHDisturbance: Array Sun TrackingAerodynamic TorqueModal Updating: f7—.. f.6Ez-1310 Orbit 0.250 Orbit 0.253o____-3__ig””’_.-Ez01xE0xE0.(00xEj0 Orbit 0.25::-J$$,30-350.5o1C)00CxC)>0.0xC)20-220-250.5.r I itiri I0 Orbit 0.25“!‘‘0 Orbit 0.25218and it is to be transferred to the maintenance depot for repair. To accomplish thistask, the manipulator is commanded to perform a series of slewing and translationalmaneuvers as depicted in Figure 7-5(a). The maneuver consists of three distinct steps:(i) a 900 slewing motion in the plane of the solar panels, divided into four 22.5° increments. Each increment follows a sine-on-ramp profile, presented in Chapter 3; (ii)a translation of 22.5 m along the main truss, divided into five steps. Each incrementfollows the MSS maneuver profile proposed by the Space Station handbook and presented in Fig. 3-20; and finally (iii) a 90° rotation to position the satellite at the theroot of the solar panels, with the slewing motion composed of 4 x 22.5° steps. Themaneuver time history is presented in Figure 7-5(b). The manipulator data used inthe analysis were presented earlier in Chapter 4, and the payload mass is assumed tobe 3,200 kgs.During this MSS task, the system flexibility characteristic is changing, as discussed in Chapter 4. Thus, during the simulation, the system modes are updated atpredefined intervals. A total of 13 modal updating steps are considered, each coinciding with the completion of the maneuver interval (Fig. 7-Sb). Thus. at the endof 22.50 rotation of the MSS arm, the modal information is updated to accuratelyreflect the change in geometry a.nd flexibility characteristics.The ensuing dynamic response is presented in Figure 7-.5. The various maneuversare well demarked in the plots. It can be observed that the slewing maneuvers exertconsiderable disturbance to the Station environment compared to the translationmaneuvers. The complete positioning task lasts 0.225 orbit (22.5 minutes). Theinplane slewing of the MSS arm exerts a moment about the local vertical; this torqueis transmitted to the Space Station, which in turn is counteracted by the CMGs witha corresponding peak control effort in the yaw degree of freedom (Q>, = 1014.5 Nm).219t9toTIME-STEP MANEUVER SPAN DURATION (s)0- 1 slewing 22.5 ° 751 - 2 slewing 22.50 752 - 3 slewing 22.5 753 - 4 slewing 22.5 0 754 - 5 translation 2.5 m 1505 -6 translation 5.0 m 1506-7 translation 5.0 m 1507-8 translation 5.0 m 1508 - 9 translation 5.0 m 1509. 10 slewing 22.5 ° 7510 - 11 slewing 22.5 011 - 12 slewing 22.5 0 7512- 13 slewing 22.50 75Figure 7-5 (a) A schematic diagram of the MSS motion profile composed ofslewing and translational maneuvers.t4TranslationSlewing220Figure 7-5 (b) The MSS slewing and translational maneuvers time-history usedin the simulation.Spacecraft Configuration: ACCOrbital Orientation: LVLHDisturbance: MSS ManeuverTask: 3200Kgs Payload PositionirSpan: 900; 22.5 m; 9Q0Translation‘ifSlewing/42(10><C’,2x0>(U)2xMW_0-2510—110—10Itit/2010030.3VctU)-Dc’J0xU)VJ__AMirOrbit 0.25 0 Orbit 0.25221The MSS arm displays a maximum transverse tip displacement of 3 x in withcorresponding acceleration of 5kg. The acceleration levels around the modules on themain truss were found to be quite high (10 fig).Simulations were also carried out for a 1,000 kgs payload. It was observed thatthe control efforts and acceleration levels at various Station location decreased considerably (50).Orbiter DockingDocking of the Space Shuttle to the Space Station is simulated by a 2,225 N pulseof one second duration as described in Section 3.6. A few simple calculations weremade in an attempt to develop insight into the effects of the docking action on themomentum storage and torque requirements for control of the Station. Consider theShuttle (mo =100,000 kgs) approa.ching the Space Station with an impact velocityof V0 = 0.075 m/s, and aiming to dock at the module attachment point on the maintruss, which is at a distance of r = 34 m from the center of mass of the Station. Theangular momentum requirement is likely to be H = rn x V0 x r = 225 x Nms,while the peak torque demand is anticipated at Q = 75 x i0 Nm. It is improbablethat a CMG system alone would he designed to absorb such torque and momentumdemands. In this study, three possible approach directions are investigated: (i) theShuttle assumed to approach the Space Station along the local vertical axis; (ii) theclosure along the local horizontal axis, in the direction of the trajectory; and (iii) theapproach considered in the orbit normal direction. In each of the cases, the berthingtakes place at the docking module, approximate1 located 34 m fiom the center ofmass of the StationThe platform responses of interest are the displacement and acceleration levels222r-.Figure 7-5 (c) The dynamics rsponse of the Space Station in presence of theMSS payload positioning maneuver.223Spacecraft Configuration: ACCOrbital Orientation: LVLHDisturbance: MSS ManeuverTask: 3200Kgs Payload PositioninSpan: 90°; 22.5 m; 90030-3/30-3><EzcC>(Ez00>(Ej01E0(.00xE(.00 Orbit 0.25) 5- 0E-530-310—150-5•00x0):e.050-510—10x0):(.0Orbit 0.25 0 Orbit 0.25at the module location, the tip of the appendages (solar panel, radiator, stinger andRCS boom), and the control effort required to maintain the Station in the LVLHorientation (‘ = A = = 0°).Figure 7-6(a) shows the control effort for Case (i). Note a significantly largedemand on the pitch controller ( 75. 578 Nm) compared to the efforts in yaw androll which have peak values of 1937 Nm and 1207 Nm. respectively.Time histories of displacement and acceleration at selected locations on the maintruss and various appendages show that the most severe effect is at the tip of thesolar panels, as expected. The displacement and acceleration levels peaked at. 0.24 mand 28 x g, respectively. Relatively large tip displacements for the solar panels,33 m in length, may suggest. a need for damping or improved control. The maintruss exhibited a maximum displacement and acceleration values of approximately13 x i0 m and 1.6 x i0 g, respectively, at the modules location. The stingerand RCS boom experienced significant displacements in the local y-direction, withdisplacement and acceleration profiles comparable to those at the tip of the radiator.Figure 7-6(b) shows the system response in Case (ii). Here, the overall dockingeffects on the FMC space platform appear t.o be less severe than in Case (i). In general,t.he t.ip displacements and acceleration levels are lower. For example, the solar panelsexhibit a peak displacement of 0.14 rn, compared to 0.24 ni encountered in Case (i).The peak acceleration levels reach only 564.tg. Of interest is the high frequency (0.5Hz) response with significant axial accelarations at the tip of the stinger and RCSboom (3 x i03tg).When the berthing takes place along the orbit normal, Case (iii), the FMC issubjected to a high disturbance about the local vertical, requiring a. correspondinglarge control effort (Q.x = 75. 650 Nm). On the other hand. the control torques in224Spacecraft Configuration: FMCOrbital Orientation: LVLHDisturbance: Shuttle DockingDirection: Local VerticalPosition: Docking Module0Ez020-2I-50 Orbit 0.0130-30 Orbit 0.01C’,200 Orbit 0.010xE0xEc.J0xED0xC)>.:O.0-)<C)- 0. -3:O—10—10.01 00 Orbit0 Orbit 0.01 :20-20>(C):COrbit 0.01i0 Orbit 0.010 Orbit 0.01Figure 7-6 The Space Station controlled response to: (a) the Shutle dockingalong the local vertical.225Figure 7-6 The Space Station controlled response to: (b) the Shuttle dockingalong the local horizontal.Spacecraft Configuration: FMCOrbital Orientation: LVLHDisturbance: Shuttle Docking%Direction: Local HorizontalPosition: Docking Module0 Orbit 0.01Ez030-30 Orbit 0.01cJ0xEz00xE0C.,0>(Ej0 Orbit 0.0130-350-500x0N30-310—10 Orbit 0.010 51>(C) 0--5:c. 0 Orbit0 Orbit 0.010..—1•••C) ) 0——.0.01 0 Orbit 0.01 :O 0 Orbit 0.01226pitch and roll are mainly required to counteract the flexibility effects on attitudemotion (Figure 7-6c). The displacement and acceleration profiles for the componentsagain show that both the lower system modes (0.1 Hz), for the solar panels andradiator, and higher system modes (0.5Hz), in the case of the stinger and RCS boom,are excited. The solar panels react to this disturbance direction less severely than inthe previous cases, however, the responses at the tip of the stinger and RCS boomwere comparable.Next. the effect of structural damping is investigated. Since the approach alongthe local vertical resulted in the highest acceleration loads at several location on theStation, a modal damping (( = 0.01) was incorpora.ted in the analysis. Figure 7-6(d) shows that at the end of 0.01 orbit, an attenuation of 507o in displacement andacceleration levels is achieved .A similar decrease in response was observed at otherSpace Station locations as well.Crew MotionThe crew induced disturbances investigated here consist of six models presentedin Section 3.6: transverse and longitudinal motions of the astronaut inside the laboratory/habitation modules, the walking and jogging exercise routines on the treadmill,the extra-vehicular tethered motion during the Station construction and maintenance,and the astronauts coughing and sneezing.The EVA tethered motion was found to be the highest crew disturbance. Duringthis operation, the acceleration levels at the modules exceeded the allowable gravitylevel of 1kg, with a peak value of 170 jLg when the force was applied at the habitationmodule along the local vertical. The maximum displacement of 1.6 x 10—2 m occurs atthe tip of the upper solar panel (Figure 7-7a). The effect of structural damping ((=227Spacecraft Configuration: FMCOrbital Orientation: LVLHDisturbance: Shuttle DockingDirection: Orbit NormalPosition: Docking Module0><Ez050-500 Orbit 0.011c%J0xEz(.J01xEC><EN0-10 Orbit 0.0110—110—10OrbitcJ0xa0x0)aN0.0150-510—1Orbit 0.01 0 Orbit 0.015 [ [\ 1\ 1\ [\ f\0 / \i \I \I \I\I \ 0:.V J V \j V—3 !FlihH!lTWW!fhT!!IT:(c 0 Orbit 0.01 0 Orbit 0.01C0><0)aU‘C30-3Figure 7-60 Orbit 0.01The Space Station controlled response to: (c) the Shuttle dockingalong the orbit normal.228Figure 7-6 The Space Station controlled response to: (d) the Shuttle dockingalong the local vertical with the structural damping ratio = 0.01.Spacecraft Configuration: FMCOrbital Orientation: LVLHDisturbance: Shuttle DockingDirection: Local VerticalPosition: Docking Module5nFdocking2c)0><Ez0-50 Orbit0-20.0130-3c=o.o10xEz00E(Vc,J0xE00C)>:0xC)- 0. -30CxC)20-2Orbit-w10—1-20 Orbit 0.01 :0200 Orbit0.0120-20.01,.02 -2:C.0 0 Orbit 0.010 Orbit 0,012290.01) results in, as before, around 50% attenuation in displacement and accelerationprofiles within 1 minute (0.01 orbit, Figure 7-7b).The next disturbance, in terms of severety, was found to be associated with thetreadmill jogging. Required control torques exceed the CMG specifications, whichmay point to the fact that the treadmill exercise may have to wait until the permanently manned configuration milestone is achieved. Effect of the jogging motion,simulated for 10 seconds, is clearly indicated in the simulation plots. During thetransient period, the solar panels exhibit a maximum tip acceleration of 3315 gig,while the stinger shows a y-acceleration of 3508 g(Figure 7-8a). As can be expected,the walking action on the treadmill represents a less severe disturbance with controleffort demands reduce by around 80% (Figure i-Sb).The transverse and longitudinal IVA astronaut. motion responses show that carewill have to be exercised by the astronauts during the microgravity experiments.Acceleration levels exceeding 25 g were observed around the modules location area,which are beyond the recommended limits (Figures 7-9a and 7-9b), with the effect ofkicking action clearly captured in the simulation plots.The simulation is able to capture response of the Station to even minor disturbances such as coughing and sneezing, which are not expected to induce any noticeableeffects on the Space Station environment, with a minimal demand on the control effort(Q, = 32 Nm, Q, = 0.9 Nm, Q = 0.5 Nm).The peak displacements and accelerations at several Space Station locations, andthe control effort required to maintain the platform in a LVLH orientation, for thevarious disturbances are summarized in Table 7-1. Note in general, the local verticalrepresents the most sensitive direction for application of a disturbance. As expected,docking of the Space Shuttle is by far the most demanding disturbance the Space230Figure 7-7 The Space Station response to the EVA tethered motion along thelocal vertical. (a) in the absence of structural damping.Spacecraft Configuration: FMCOrbital Orientation: LVLHDisturbance: Tethered EVADirection: Local VerticalPosition: Habitation ModuleFtethoj.50 Orbit 0.01eqxEzaeq2C0><Ezeq0>cE0><E0 Orbit 0.01—0.010-20 Orbit—30-310—1C.,0c.oeq0xU)U)>.30-320-20 Orbit 0.01 0 OrbitC? C.)0 2 0-.0 WRN-2 I>->:c.o 0 Orbit 0.01 :c0.0130-3C?C1:O20-20 Orbit 0.01 0 Orbit 0.01231Figure 7-7 The Space Station response to the EVA tethered motion along thelocal vertical, (b) in the presence of damping, ( = 0.01.Spacecraft Configuration: FMCOrbital Orientation: LVLHDisturbance: Tethered EVADirection: Local VerticalPosition: Laboratory ModuleFtether5020C.J0><Ez0-5-20 Orbit 0.01 0C01Ez0C’,0xE0xEc=0.o1 30-3Orbit 0.01 0 Orbit 0.010x:20-210—1010wC;J0xC)20-2Orbit 0.01 0 OrbitC? C.,Ii.i____:c.o 0 Orbit 0.01 :C0.01C?01C)0)C)0 Orbit 0.01 0 Orbit 0.01232Spacecraft Configuration: FMCOrbital Orientation: LVLHDisturbance: Treadmill JogDirection: Local VerticalPosition: Habitation Module tread-jogFigure 7-8 The Space Station response: (a) in the presence of treadmill jogalong the local vertical.50-50C’J0xEzc20-2Orbit 0.01 0 Orbit 0.0150-550-50 Orbit 0.01C?0>cEz00EU)0><EU)C0><C)U)0.:C?0xC)U):CxC)U)>-30-320-2Orbit 0.01020-200C?0C)CU):QOrbit 0.01C?0><C)U)C):OOrbit 0.01- —0 Orbit 0.010 Orbit 0.01233Figure 7-8 The Space Station response: (b) in the presence of treadmill walkalong the local vertical.Spacecraft Configuration: FMCOrbital Orientation: LVLHDisturbance: Treadmill WalkDirection: Local VerticalPosition: Habitation Module tread-walk0 Orbit 0.01 0 Orbit. 0.01 0 Orbit 0.01—0xEza0xE0xEzCAcJ0xC)C.10—130-30cJ01>(C)0:C.O1C)0:O50-530-3—Orbit 0.01 0 Orbit 0.010xC)C0:50-50 Orbit 0.01cJ01C)a)C.)0 Orbit 0.01 0 Orbit 0.01234Figure 7-9 The Space Station response to the induced WA disturbance alongthe local vertical: (a) along the module.Spacecraft Configuration: FMCOrbital Orientation: LVLHDisturbance: Longitudinal IVADirection: Local VerticalPosition: Habitation Module FivaiongT10—100><Ez20-2LA0 Orbit 0.01Orbit 0.010Ezci020Ez0E0xEcJ0C)>-20 Orbit 0.0130-31010200xC):C0C):O30-320-2Orbit 0 Orbit0.0130-3C%J00 Orbit 0.01 :C.Oc.J0C)0.01—20 Orbit 0.010 Orbit 0.01235cJ0C):cJ0C)Spacecraft Configuration: FMCOrbital Orientation: LVLHDisturbance: Transverse IVADirection: Local VerticalPosition: Habitation ModuleFivatrans0 Orbit 0.0120><EzcJ0xE0E10—130-310-10-20 Orbit0 Orbit 0.01-c.J0‘—3xC)-0—.3:LC0.010 Orbit 0.01o2*:j2Orbit 0.0130-300 Orbit 0.01cJ0><C)C>0 Orbit 0.01 :Figure 7-90 Orbit 0.01The Space Station response to the induced WA disturbance alongthe local vertical: (a) across the module.236Station will be subjected to. The information not only provides some insight into thesystem dynamics of such a flexible system, but should also prove useful in designingan appropriat.e control procedure to meet the operational specifications.237I’.) 00Table7-1PeaknonlinearattitudecontroleffortsandmaximumdisplacementandaccelerationlevelsatvariousFMCSpaceStationlocationsforShuttledockingandcrewmotiondisturbances.FirstMilestoneConfiguration(FMC)LocationDisturbance(LocalVertical)DockingIVALong.IVATrans.Tread.WalkTread.JogEVATetherCoughDisp.,mm13.,pg15982225341861700.7Disp.,mm2401.,pg279693864536013315301612Disp.,mm1200.,pg13363163192255141212646Disp.,mm2301.,pg322604314476403508316413Disp.,mm1200.,pg2244618625319089113749r/,Nm7557884884915277644847832Control4’,Nm19372630402242030.9),Nm12071518241291180.57.3 Assembly Complete Configuration (ACC)The ACC represents the evolutionary Space Station at the end of the projected30 Shuttle flights. Its main truss structure will be 155 meters long and the Station’stotal mass is expected to be 210,580 kgs. Major components of the ACC include eightsolar arrays, four PV radiator, two station radiators, the stinger and two RCS booms.Various support loads are located along the main truss, as described in Chapter 4.The baseline configuration has gone through several changes. Numbers quoted herecorrespond to one of the earlier configurations. Figure 7-10 shows the coordinatesystem, the librational orientation, and appendage numbering scheme used in thenumerical simulation.7.3.1 Modal convergenceThe modal convergence investigation was carried out for the ACC. With an initialangular displacement of 5° in pitch, yaw and roll, and the structural deformation wasrepresented by considering 1, 5, 10 and 20 system modes. As before, the variablesof interest include: the librational displacement and angular velocity; the nonlinearcontrol effort. as calculated by the FLT scheme; the modal displacement and accelera.tion profiles at the modules location on the main trus ( taken to be the midpoint ofthe main truss) as well as the tip of the solar panel and station radiator.Figure 7-11(a) shows the controlled attitude response of the system. The same setof controller gains as in the FMC case is used here. The librational response displaysa critically damped behaviour, as commanded by the controller, with the angularvelocity reaching a maximum of 0.02 deg/s in pitch. Note that the attitude responseis identical to the FMC case, since the attitude behaviour is preset by the controllergains specified in the FLT nonlinear technique. The corresponding control effort plots239Appendage Appendage Space StationNumber Type ComponentCentral Body Beam Main TrussI Plate Solar Array2 Plate Solar Array3 Plate Solar Array4 Plate Solar Array5 Plate Solar Array6 Plate Solar Array7 Plate Solar Array8 Plate Solar Array9 Plate PV Radiator10 Plate PV Radiator11 Plate PV Radiator12 Plate PV Radiator13 Plate Station Radiator14 Plate Station Radiator15 Beam RCS Boom16 Beam RCS Boom17 Beam StingerFigure 7-10 The coordinate system, appendage numbering scheme and attitudeorientation proposed for the ACC Space Station.240(Figure 7-10) show that it is necessary to take into account at least 10 modes in orderto attain a reasonably accurate response in amplitude and frequency. The demandon the controller is large in pitch and yaw (Q, = 1100 Nm and Q>. = 1050 Nm),while the roll degree of freedom requires Q = 500 Nm. Note, these represent aconsiderable increase in control effort compared to the FMC case. The 20-modessolution exhibits local high frequency modulations in the control profile, however, themagnitude and general character of the response remain essentially the same.The solar panel tip dynamics confirms that 10 modes are necessary in orderto accurately represent the flexible character of the system. Contributions of thehigher modes are apparent in the markedly different responses of the 1, 5 a.nd 10-mode solutions. Predict.ion of the peak solar panel tip displacement improves from0.2 x 10 m to 1.0 x i0 m through the use of 10 modes in the solution. Thecorresponding acceleration values increase from 100 g in the 1-mode solution, to 400,ug in the 10 modes-solution (Figure 7-10). The presence of beat in the response isattributed to the closely spaced frequencies in the spectrum (Chapter 4). The 1-modesolution fails to predict the beat character which is clearly visible in the 10-modesolution. The station radiator show a rather dramatic improvement in the results(Figure 7-10). Obviously, significant contributions come from the higher modes and itis necessary to use at least 20 modes (perhaps more) to properly assess the dynamicalbehaviour of the station radiator. The maximum displacement and acceleration valuesare 1.5 x 10 meters and 140 ,ug in the 20-mode solution, compared to 3 x iO mand 1.öjg, respectively for the 10-mode. In a sense, the behaviour substantiates theearlier obervation made for the FMC case, that in order to accurately predict thedynamic response of a particular component of interest, it should exhibit high energyin the modes considered in the simulation.241Spacecraft Configuration: ACCOrbital Orientation: LVLHInitial Conditions:= == 50Figure 7-11 Effect of increasing the number of modes on the system response foran intial disturbance of 50 in pitch, roll and yaw: (a) librational displacement and angular velocity of the Space Station in the presenceof the number of modes (1,5,10,20) in the presence of nonlinear FLTcontroller with identical gains.The main truss module location (Figure 7-10), as well as the stinger and the RCSboom tips exhibit modal excursions in the local z-direction. Contribution from thehigher modes essentially affect the frequency of the response at all the three locations.7.3.2 Dynamic response to disturbancesOrbiter DockingAs in the FMC analysis, three berthing directions were considered in the presentstudy: the local vertical, local horizontal and orbit normal . The center of massof the system is located at {O, —0.17, —O.09} m from the geometric centre of theAttitude, deg wdeg/s xl 025020-20 Orbit 0.25 0 Orbit 0.25242U,a)0Li)I Q,Nmx1O2Vr iLJLt1.k._L Ii’. .1Figure 7-11 Effect of increasing the number of modes on the system response foran intial disturbance of 50 in pitch, roll and yaw: (b) torque effortfor the FLT controller.Spacecraft Configuration: ACCOrbital Orientation: LVLHInitial Conditions:= == 505Q, Nm x102a)V0 0-5Ct)G)V001Cl,a)-D00c’JOrbit 0.2 0 Orbit 0 Orbit 0.2243Figure 7-11 Effect of increasing the number of modes on the system response foran intial disturbance of 5° in pitch, roll and yaw: (c) solar panel tipdeflection and acceleration time-histories.Spacecraft Configuration: ACCOrbital Orientation: LVLHInitial Conditions: i4r0 = =ösoIar m x104510—110—10a)a)01U)V0Lr)U,C)V00Cl)a)000c\JU)V050-51 Cl)C)V00I0—15U,C)V000 Orbit 0 Orbit 0.250.25244Figure 7-11 Effect of increasing the number of modes on the system response foran intial disturbance of 50 in pitch, roll and yaw: (d) solar radiatortip displacement and acceleration time histories.Spacecraft Configuration: ACCOrbital Orientation: LVLHInitial Conditions: w = 0050C)0-58C)010It)U)C)00U)C)V00c’J50-550-550-51500-150C,,C)V0It)C’,a)V00Cl)a)V00c’J50-550-51000-100-0 Orbit10.25 0 Orbit 0.25245Lbci)V0toCl)0V00Lb0V0toCOci)-o00rtruss’nana awtLmD’mflWflmL.naIFigure 7-11 Effect of increasing the number of modes on the system response foran intial disturbance of 50 in pitch, roll and yaw: (e) the main trusstip displacement and acceleration.Spacecraft Configuration: ACCOrbital Orientation: LVLHInitial Conditions: xji,= == 505z mxlO5truss’a)V0r50-5ci)V0r-50-550-550-5550-550-550-5Lb0V00c’.JSn**Ø$*$**..*d$WLb0V00c%J00Orbit-5W wv”fl’W’fl’w nwwJflWf_________a—a —a-a0.25 0 Orbit 0.25246spacecraft. The docking effect and the control effort requirements are considerablylower since the disturbance is applied closer to the center of mass than in the FMCcase (Figure 7-12). The docking action along the local horizontal was found to bethe most severe, with the peak control efforts as Q = 2194 Nm, QA = 42 Nm andQ = 3055 Nm. The higher demand on the controller in the pitch and roll degreesof freedom is due to large structural deformations and accelerations at the tip of thesolar panels (6 = 3.5 x 10—2 m and 6 = 2330kg), resulting in large moments aboutthe center of mass of the system. At the location of the modules (approximatelyat the midpoint of the truss). displacement and acceleration in the local z-direction,though small were perceptible, 6 = 2.6 x iO m and SZ = 18jg. The PV radiator,the Station radiator, the stinger and the RCS boom tip acceleration profiles were alsostudied, for completeness. The acceleration levels were found to he comparable tothe main truss profile, with the PV radiator exhibiting a beat-type response. Otherdocking approaches along the local vertical and orbit normal resulted in much lowercontrol effort and acceleration loads.Crew MotionThe treadmill jogging motion presents a considerable disturbance to the Stationenvironment. The forcing function is applied for 10 seconds along the local horizontal, and the resulting dynamics is shown in Figure 7-13. Although the accelerationlevels are quite low (2.1ig), they are still beyond the recommended microgravitylimits around the laboratory module (lJLg). Thus the microgravity experiments andthe physical exercise will not be able to proceed simult.aneouly, unless a. dampingmechanism is introduced to isolate the disturbance. Recently, NASA-Lockheed haveproposed a device that supports the exercise equipment yet cancels out the vibrations,247thus allowing astronauts to work out. strenuously without interfering with science experiments. The device is called the “Isolated/Stabilized Exercise Platform’, and itsdebut is planned for June 1992, in the mid-deck of the Space Shuttle Columbia [110].Effect of the jogging motion is clearly indicated by the simulation plots, with thesystem following a free vibration response in the steady state regime, i.e. when thedisturbance has ceased. The treadmill walking action induces less severe accelerationsloads on the Space Station. The peak acceleration level of 59jig occured at the tip ofthe solar panels, which is 75% lower than that for the jogging routine case.A tethered EVA sortie represents a considerable disturbance to the system, withrelatively high displacement a.nd acceleration values at the solar panel tip. Figure 7-14shows the response in the presence of structural damping ( = 0.01). It attenuatest.he response by around 30% at the end of 1 minute of simulation. The cont.rol effortrequired to maintain the Station in the LVLH orientation is comparable to the joggingcase, with the maximum control effort in roll of 333 Nm.The IVA activity generat.ed low acceleration loads around the laboratory module,with 0.23 ig level caused by the transverse motion. The longitudinal kicking actioncaused even lower loads (0.05 fig). The resulting dynamics when the loading historiesare applied along the local horizontal is shown in Figures 7-15(a) and (h). Again theeffect. of the kicking action is clearly visible in both the cases.Finally, t.he cough and sneeze force history reveals tha.t minimal efforts are required and resulting acceleration levels hardly affect the Station (Figure 7-16).The peak displacements and accelerations at several Space Station locations, andthe control effort required to maintain the platform in the LVLH orientation, forthe Shuttle docking and crew disturbances applied in the local vertical direction aresummarized in Table 7-2.248Cx0)0)U)0____________________________________Figure 7-12 Response of the Space Station (ACC) to the Shuttle docking alongthe local horizontal.ipacecraft Configuration: ACCOrbital Orientation: LVLHDisturbance: Shuttle DockingDirection: Local HorizontalPosition: Docking ModuleF20-2c’J0><Eza10-102Orbit 0.01C0xEzaC?0>i:Eza0xE0eoC1EU)N0-20 Orbit 0.0130-330-30 Orbit 0.0122x-2:C.O22N-20 OrbitH0 Orbit 0.010.01 0 Orbit 0.01\JW\/H0 Orbit 0.01 0 Orbit 0.01 0 Orbit 0.01249:c.O 0 Orbits 0.012 °21 I‘— I )>< Io__________-2 ‘-10 Orbits 0.01 0 Orbits 0.01 0 Orbits 0.01Figure 7-13 Response of the ACC to the disturbance introduced by an astronauttreadmill jogging along the local horizontal.Spacecraft Configuration ACCOrbital Orientation: LVLHDisturbance: Treadmill JoggingDirection: Local HorizontalPosition Habitation Module20-20 Orbits 0.010EzC0><E0xE/\j=50-510—100 Orbits 0.01 0 Orbits 0.0103. 3:0xC)10—1OrbitsWWW\0.01 0 Orbits 0.01250:c 0 Orbit 0.012:HH0 Orbit 0.01Figure 7-14 Damped response of the ACC during a tethered EVA sortie alongthe local horizontal.Spacecraft Configuration: ACCOrbital Orientation: LVLHDisturbance: Tethered EVADirection: Local HorizontalPosition: Laboratory Modu’eEza50-5CxEzc0xE0U,0xELOc’Ja0 Orbit 0.01 0 Orbit 0.01 0 Orbit 0.0120-250-510—10C=0.oi0x0):0)N:20-220-2Orbit 0.01 0 Orbit 0.010 Orbit 0.01 0 Orbit 0.01251Figure 7-15CxC)0001xC)ci,>:C%J0-><C)The ACC response in the presence of (a) IVA transverse motion alongthe local horizontal.Spacecraft Configuration: ACCOrbital Orientation: LVLHDisturbance: Transverse IVADirection: Local HorizontalPosition: Habitation Moduleiva-transb><Ez00>cE0xEOrbitx0):50-530-330-300):20-20 Orbit:LO 0 Orbit 0.01Orbit0.010 Orbit 0.01 0 Orbit 0.010.01H0 Orbit 0.0125203 02:3HH:H____ ____:c 0 Orbit 0.01 0 Orbit 0.01Figure 7-15 The ACC response in the presence of (b) IVA longitudinal motionalong the local horizontal.Spacecraft Configuration: ACCOrbital Orientation: LVLHDisturbance: Longitudinal IVA F.iva.iongDirection: Local HorizontalPosition: Habitation Module30-30 Orbit 0.010xEz050.50xEzxEz00xECD01xEN30-30 Orbit0 Orbit 0,011C)C50-530-30.01200C) 0-2Orbit 0.01 Orbit:V0 Orbit 0.01c’.JC>(C)0.01H0 Orbit 0.012530><ZLV______________C.,__________________________________________:Figure 7-16 The Space Station response in the presence of astronaut coughingand sneezing with a force transmitted along the local horizontal.Spacecraft Configuration: ACCOrbital Orientation: LVLHDisturbance: Cough and SneezeDirection: Local HorizontalPosition: Habitation ModuleFi0 Orbit 0.0101><Ezc03>(E 0Z3Ez0>cECD01EU)co0 Orbit 0.0150-530-30 Orbit 0.0100U):30-320-20 Orbit 0.01 00xC)U)00)>.U):C.Q 0 Orbit 0.01Orbit 0.01FH.0 Orbit 0.01 0 Orbit 0.01H0 Orbit 0.01254AssemblyCompleteConfiguration(ACC)LocationDisturbance(LocalHorizontal)DockingIVALong.IVATrans.Tread.WalkTread.JogEVATetherCoughDisp.,pm2603.52.92.310132.1ModulesAcc.,pg180.,pm350008205805708101942540SolarAcc.,pg233046355926226713Disp.,pm500210210210220213210PVRad.Acc.,pg18666666Disp.,pm30043.62.311134StingerAcc.,pg180.220.230.41220.1Disp.,pm200211.6571.5RCSBoomAcc.,pg100.,pm24031.71.1811.40.8St.Rad.Acc.,pg160.220.20.35222.2‘,Nm21943029502522442.4Control4,,Nm420.50.51.3540.5),Nm305533407637133314PeaknonlinearattitudecontroleffortsandmaximumdisplacementandaccelerationlevelsatvariousACCSpaceStationlocationsforShuttledockingandcrewmotiondisturbances.L’30’C-)’Table7-27.4 SummaryThe attitude control system successfully regulated the effects of induced disturbances, such as crew motion and Orbiter berthing. on the librational motion of theStation.In this investigation, the dynamic response of the Space Station’s FMC and ACCmilestone configurations was determined for Shuttle docking, crew motions, MSSoperations, solar panel tracking and aerodynamic disturbances. Acceleration responsetime histories at several station locations were presented for each case and the peakresponse values tabulated for comparison.The results indicate that t.he Orbiter docking leads to the highest response of theSpace Station, i.e. it is the most severe one. The peak translational accelerations forthis case were of the order of iOg at several locations. Crew motion caused peakresponse of the order of 104g. The sola.r array sun tracking maneuver led to thepeak accelerations of only 105g.256CHAPTEREIGHTCONCLUDING REMARKS8.1 Original ContributionThe computer age has given the dynamicists and control engineers enormous capability to explore complex problems which were either beyond their reach or were approached with highly approximate models. Now they can develop efficient and highlyfaithful models of systems such as the flexible orbiting space platforms using themethodology established by Euler, Lagrange, Hamilton and others, who themselvesmay have not envisioned such end applications of their elegant formalisms. As a consequence, the discipline of multibody dynamics and control has undergone significanttheoretical and experimental advances, primarily motivated by the unprecedentedchallenges in the design of large space structures. In spite of significant efforts aimed atdevelopment of mathematical and numerical tools, there is considerable scope for lasting contribution to improve upon the present state-of-the-art, with particular emphasis on the development of general and efficient methodologies applicable to a largeclass of space systems leading to physical appreciation of dynamical interactionsbetween flexibility. librational motion and disturhaices, as well as their control.Flexibility is the key parameter governing the dynamics and control of large spacestructures. Through a simple example of the L-shaped cantilever structure, the limitation of representing deformations using component modes has been clearly demonstrated.257The use of Lagrangian procedure ill conjunction with structural representationusing system modes is applied to the evolving Space Station for the first time. Theapproach leads to nonlinear, nonautonomous and coupled equa.tions of motion of relatively reduced complexity, with a considerable saving in time and effort. Equallyimportant is the better physical appreciation of the contributing generalized forcesand the response results, as now the modal frequencies corespond to resonance conditions of the overall structure. Structural flexibility terms are decoupled due to theorthogonality of the normal modes with respect to the mass and stiffness matrices.For geometrically time varying systems, the modes can be updated at user specified intervals, thus maintaining faithful representation of the structura.l flexibilitythroughout t.he simulation sequence.To summarize, a relatively general Lagrangian formulation of the nonlinear, nonautonomous and coupled differential equations of motion, governing the dynamics of asystem of interconnected flexible members with the following distinctive features hasbeen developed:(a) it is applicable to an arbitrary number of beam, plate, membrane and rigid bodymembers, in any desired orbit, interconnected to form an open branch-type topology:(b) flexible appendages are permitted to have large angle rotations and linear translations with respect to the central body;(c) the formulation accounts for the gravity potential, the effects of transient systeminertias and shift in the centre of mass;(d) the flexible character of the system is described by means of three-dimensionalsystem modal functions obtained by the finite element method;(e) geometric nonlinearities, shear deformations, rotary inertias, variable mass den258sity, flexural rigidity aid cross sectional area effects have been included in aconsistent manner, both at the multibody dynamics formulation level and at themodal discretization level;(f) symbolic manipulation is used to synthesize the equations of motion thus providing a general and efficient modelling capability with optimum allocation ofcomputer resources;(g) the governing equations are programmed in a modular fashion to isolate the effectsof appendage slewing and translation. librational dynamics, structural flexibilityand orbital parameters:(h) operational disturbances (Space Shuttle docking, crew motion and MSS operationmaneuvers) have been implemented in this dynamic simulation tool and additionaldisturbance models can easily be incorporated through generalized forces andinitial conditions;(i) both the nonlinear and linear forms of the equations of motion have been formulated to permit assessment of a variety of control strategies.A general purpose flexible multihody dynamics and control simulation programhas been developed. The implementation has been carried out in three stages:(a) the Structural Dynamics Module defines the geometric and flexibility characteristics of the system under study. The spacecraft physical data, in terms of nodalmesh and structural parameters are specified by the user, and the finite elementmodal analysis program generates the modal functions, the centroidal vector, andinertia properties;(b) this information is supplied to the modal integral interface for processing of modalmomentum coefficients and subsequently transferred to the next stage of the simulation, the Multibosy Dynamics Module. At this stage, the tailored equations of259motions, for the selected configuration (1-6) which best represents the spacecraft,are assembled and coded, by symbolic manipulation;(c) at user’s request, the Control Module makes available the nonlinear and nonlinear equations of motion for application of control strategies based on the LinearQuadratic Regulator and Feedback Linearization techniques.Versatility of this simulation tool is illustrated through the analysis of two evolutionary Space Station configurations: the FMC and the ACC. Effect of the number ofmodes on convergence to the true solution is investigated. Prediction of the dynamicresponse of the Space Station to disturbances encountered during normal operationrepresents an important step in defining the design loads for the main truss, as wellas for the modules and secondary components. The control effort profile to maintainthe LVL•H attitude of the Station, the displacement and acceleration response timehistories for several locations are presented in a tabular form to permit assessmentof their relative importance. Results indicate that the Orbiter docking representsthe most severe disturbance for the Space Station. The peak translational accelerations for this case, of the order 103g, violate the desigi criterion. Crew motioncaused peak responses in the range of 104g, while the array rotation to track thesun represents a rather weak disturbance leading to around 105g.The finite element procedure used in the representation of structural deformationmakes the present algorithm ideal to carry out visualization of the spacecraft dynamicsand control simulations through animation. A video depicting the modal interactionsbetween the various Space Station flexible appendages throughout its evolutionarystages has been produced in collaboration with the University Computer ServicesVisualization Group.Summarizing, the unique features of this study are evident in the development260of an interdisciplinary integrated algorithm synthesizing multibody dynamics, finiteelement method for modal discretization, symbolic manipulation, application of linearand nonlinear control strategies, and computer animation.8.2 Future WorkThe innovative efforts discussed here represent only the first steps towards understanding the dynamical and control of evolving space platforms. Some thoughtsconcerning future work, which are likely to prove useful, are indicated below:(i) The two modelling techniques (system modes vs. component modes) with regardto structural deformation representation should be further investigated. The ultimate objective is to establish criteria which will help assess the suitability ofone method over the other in the analysis of flexible multibody systems.(ii) The present multibody formulation should be extended to include shells, in a manner similar to beams and plates. With regard to the system modes obtained forthe Space Station Freedom, a more detailed analysis of the solar panels is necessary considering their important role. Modelling the panels by a beam/membraneassembly may provide a more accurate representation.(iii) The multibody dynamics program needs to be automated in order to transferinformation from one module to another without user intervention (Figure 6-1). In this respect, the structural dynamics module should form an integralpart of the multibody dynamics module, such that modal updating is performedautomatically. The applicability of parallel processing should be invest igat ed,such that the various tasks of calculating the kinetic energy, angular momentumvector, inertia matrix could be concurrently performed. This is essential for realt.ime dynamical and control simulation:261(iv) The FMC and ACC responses predicted in this study only serve as general guidelines. The disturbance forcing functions will have to be better defined beforeextensive and accurate analyses are performed.(v) The simulation results obtained from the multibody dynamics program shouldbe in a form compatible with existing animation packages, such as DataView, sothat the dynamic and control simulation histories can be visually processed;(vi) structural vibration control will have to be implemented in order to completethe present simulation algorithm. 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Proceedings of IEEE Conference on Decision and Control, Fort Lauderdale, florida, December 1985. pp. 1767-1772.[104] Isidori, A., Nonlinear Control Systems: an Introduction, Springer Verlag, Berlin,Heidelberg, 1986, pp. 178-215.[105] Hogg, J., “DataView- A Computer Animation Packag&’, The University Computer Services. The University of British Columbia, 1991.[106] Likins, P\V. “Multibody Dynamics- An Historical Perspective”, Proceedings ofthe Workshop on Multibody Simulation, JPL D-5190, Vol. 1, l988,pp. 10-24.[107] Marandi, S.R., and Modi, V.J., “An Alternate Transition from the Lagrangian ofa Satellite to Equations of MOtion”, Celest.tial Mechanics, Vol. 45, 1988. pp. 333-386.[108] Ibrahim, A.E.M., Mathematical Modelling of Flexible Multibody Dynamics withApplication to Orbiting Systems. Ph.D. Thesis, The University of British Columbia,April 1988, pp. 135-141.[109] Chan, J.K., Dynamics and Control of an Orbiting Space Platform Based MobileFlexible Manipulator, M.A.Sc. Thesis, The University of British Columbia, April1990, pp. 135-160.[110] SPAC’E, Vol. 8, No. 2, April-May 1992, A Shephard Press Publication, Bucks.U.K., pp. 38-44.272APPENDIXAEQUATIONS OF MOTIONUsing the Lagrangian procedure, the governing equations of motion can be obtained fromdOT OT 0(1 (A.1)where and Fq represent the generalized coordinates and associated forces, respectivelv. The above equations can be rewritten in the vector form as(Q& ‘1M”= - (A.2)1Q Jwhere:- d &DT - &‘ d - DL,T d d -= —{——}Iw — —.-—Iw —— —{---—}Hdt dt 88 dt dt o01,T dH 0,T 01,T- (A.3)- d OT* d OFIT 8ITdOT*T01 OH 0(1+--+ ‘+{--}+. (A.4)Here M represents the nonlinear mass matrix, while Q and,correspond to the nonlinear stiffness, gyroscopic and forcing terms for the libratiorial and vibrational degreesof freedom, respectively. is comprised of two vectors, and p, where 8 = {‘, , }273for the librational degrees of freedom and= {p1’p2 ...,pn} for the vibrationaldegrees of freedom.The various terms which contribute to the equations of motion given by (A.1) -(A.3) are next presented in matrix form, ready to he programmed:o 1fl (cosqcosA’)1 —sin O:o IA) (cossinAJo —?1b sinq5 cosA —e sinq cosAo —‘ cosq5— 9 cosço sin sin A — 9 sin g5 sin A—\cosA 00 0;—5 0cosq5cosA—sinq5 0cosq5sinA cosAcos cos Aw= —sznq5cosqS sinA—sin A0cos A98— cos sinA — cosA— 9 cos sinA0iJ cosb cosA — sinA + 9 cosq cosAcosq5 cosA — sinA 0Sifl 0 1cosq5 sinA cosA 0—q5 sin cos A — cos sin A= —cos—5 sinb sinA+A cosq cosAdtdzdt10274— sin q cos A — cos q sin A+ —cosq 0—sirzq5sinA cosq5cosAo —singcosA+ 0 —cosqo sinq sinAA0.AAs explained in Chapter 3, the kinetic energy expression can be written in the formT=Torb+Tcm+Tt+Ts+Tv+Tt.v+Ts,t+Tt.s+TI+TL (A.5)= T* + + TL (A.6)where i is the librational velocity vector: I, the inertia matrix: H, the angular momentum with respect t.o the F frame: L,T I ‘. the kinetic energy due to pure rotation;and ,T B, the kinetic energy contribution due to coupling between rotational motionand transverse vibrations. The subscripts involved are defined below:orb orbital motion:cm centre of mass motion;s rotational motion due to joint slew;t joint translation between body B and B;v structural vibration.The contributions from the various sources to T* take the form:—cosA0—sinA—cosA0—cosq5cosA+2751N T.T*=tfcdrnc+fdi didmi+f d+ yTf + fi dm+ cxT drn CiT + fc5P dm Cx2T + f dm C3T]+Tfidrnil;n xl+ dm+dm + fiT dm CHere the following identities apply:fl x 3[cx Cy Cz1nx3_-= [ix IN &13x1 ( ‘I i Id = d2 ; [d]x = d?3 0 —dixld3 J —d,2 dixl 0C C C21,1 1.2 1.3 C21Ci = Cr21 C222 C223 =ci31 c32 c33276The angular momentum and its derivatives can be written as:[ f Zc d772 + f yc drnc] J3x1H= [— J zq dm + f xcc dma][- f Yc dm + f xc dmc]T (_ f CZ dm + f cycz drnc)+ pT f CZ dm + f cCX dine) PT (_ f Cy dine + f dmc)[dj X mC3 f /5j / drn7 •T + f iT d7n -T+ C3 fj ‘ d?n T—d2 f 75j /ff dm •T + drn _T_pT J + p’’ f dm+ pT f _pT_pT f +T f qqçdrn—e3 (f jdrn) d2 + C2 (f d?n) d3e3 (f jdrn) d1 — e1 (f dm) d3(f drn) d1 + e1 (f din) d12—d3 ëj2 (f dm) + d2 ëj (f ,5dm)+Nd3 ë (f drn) — d1 ëj (f dm)—d2 (f d7n) + d1 ëj2 (f dm)277—f q5T dint p d2 + f qdm p d3fdmpd1_fdrnpd3—fdmpdj1— fdrnpd2—d3 fqdm.+d2fbdrnj+ d3 f Tdmi — d1 f Tdmi—d2 f dinj+d1f’dmj2 f p + C%3 f p+ f pdm p—e f p— f ödrn P + f pCt3 f P + f Oqdrn 5+ c3 f dm — ë f drnCi2 f + e f çbdm= [(_ f CZ drn + f cy drnc)( f CZ dm + f C2 drn)(- f drn + f CxCy drnc) ](--f cbjq’dm + f j5dm)+(f idm- f jdrn)(—f jbdm + f jdrn)+[-f drn d2 + f d32 8f g5drn d1 — f q5din d3— f cb’drnj d1 + f çbd?n d2]+ [_f dm •T + j •T- -T T - -T Tf 1)p dm c1— f p dm c3-T •T - -T Ti— f dm c1 + f dint c2 jnx3= [[ f zc dm + f dmc][— f zq5 drn + f xb dma][— f Yc dm + f X dma] I+ [( I Cz dm + f CYCZ dmc)( f CZ d’in + f drnc) 5( f Cy dm + f drnc) ](f & thi + I jqdrnj)T p+ N- f jdm)T(f cb dm + I d7n. ) T p+ [_f’drnjd3+fdmd2f q5drn d3 — f bdm d1279— f dìn d2 + f qdrn d1]+ [—f q,”dmj + f q5ö”dm Tf ë — f q’dm e;— f q OTdrnj ç + j ;]nX3d oH= [(_ f CZ dm + f Cy dmc)(— f q5CZb dm + f drn)( f Cy drn + f dmc) ](—f bdrn + fN+i=1 (-f dm + f din)Tj+ [—f bc1m d3 + f 1’dmj d2f q5drn d3 — f bdrn d1— qT din d2 + f d1]J tx+ [—f bj’dm •T + •Tf bj’drn — f qjdrn— f q5j dm C2 + f .T]280[- f dm + f Yc d7n]{— f Yc dm + f dmc]T(_ Czcy dm + f CyCz drnc)+ T_j dm + f dn)T ( j CyCX drn + f drn)pT (- f Cz% dm + f CyCZ dine)+ pT (_ f bCZ drn + f &c drn)pT (_ f Cy dm + f drn)+ + [djX } rnC3 f 5j dm •T + C2 f ij drn TC3 fPt ,oT drn e:T — e fo dmjë3TC2 f 75’ dm •T + f drn •T(—f q5jçb,drn + f(f idrn — f q5jçbdrn)(-f jdm + f jdm)P (f + f çbj,,,q5dm) 3:+ p (f- f jdm)p (—f q5q5drn + f drn)281(f drn) 2 + C2 (f drn) d3e (fpdm)d1—e (fdrn)d3(f dm) + e1 (f pdrn) d2—d3 2 (f drn) + d2 e (f1dm)+ d3 (f dm) — d1 e (f dm)—d2 e (f dm) + d1 2 (f dm)—f dm d2 + f q51’drn+ fdinpd1—fdrnjd3—f bTdm 1 + f ‘dni d2—d3 f dm1j3 + d2 f q5?d7n j5fTdrn—d72 fdrn—d1fbPdmji2 f P d7m J + C3 f+ ë f jdm—CZ3 f— f ,oçbdrn + i f jçb’dmf ,oq’drn + ë f1oq5’drn+ ëj f — ë f jdm2 f15drn + e f j5jq5’drn282The inertia matrix and its derivatives take the form:32’ (f + J qqdm)pTpT f6qdmp— pT f cbcdrni PpT ftcdmiP_pT_pT f ppT ++ f ( [E] — ) dm+ —(fy+fx)p-(fz+fx)p— (fx+fy)p2(fx+fz)p-(fz+fy)p283-- (Iz +fy)2(fx+fy)p+ f ( c [EJ — c ) dm+cicrT (f qçdrn + f ‘dm) 5_T fdmp_pT fdmp_pTT (f q5q5’d?n. + f drn)_T_T fdmpT fdmp:T(fTdm. + fçbçbTdmj)+2dTCi(fMrni)[E] _i(f dm)C — C(fdln)d7284[2 d7 (f dm) p [E] - p (f dm) - (f dm) p dT](f,oqdinj)p + CZ3 (fdrn)]p(f ‘drn) P— 2 (f qdm) p(f drnj) j3 — ë3 (f ,öqdm)— ë1 (f P — (f Tdmj) p2 [e (f oq5T din1) P + C?3 (f1drn) P1— Ct2 (I1qdrn)p — e3 (f1dm)p— e (f/51dm1)p — ej3 (fjTdrn1)p— ë1 (f o!’din) — ci (f,ö2q5dm) p2 [ëj (f j drn1)p + j2 (fo1drn)p]dIN—2(fxc+fzc’).—2852 [e12— C11- —C11—— (fz +fyc).Ø--jT (f q5bdrn + f q5drn)+2 _pT_T fq5q5’dmjjp1’ fdmjpT (f + f bcb’dm)pT fzdmi_pT f’dmj_T f q’dm(f bqdm + f din) P+ {2 T (fidmi) + 2 j Oj (f dm)} [E]drn)C[ - i(fdrni)O286— Oi(f — Cj(fpjdmj)djT+[(2T(fdrn) + 2 d7 (f dm) ) [E]—T (f dm) — (f drn) - (f d7n) - (f dm) p2[ë (f dm)p + Ci3 (f dm)p]— (f ob,drnj)p — Ci2 (f oq5drn)p_1 (f ,ñd7n) — CZ3 (f ‘dm)p— e (f5’drnj)p— 2 (fdm)p2 [e (f p + C3 (f jbTdmj) 1(föq”thn)p— Ci3 (fi3din)p— e (f 1o drn)p—e (fs3cdm)—(fdrn)p—C3 (födm)p2 [e1 (f j5dm)p + 2 (fi3dm)pj2 [e2 + Cj3— (f oiq5,dm)— 2 (f Oicdrn)— (f ,äjdm) —‘(f ojbdrnj)287— ej1 (fp’drn7)j— ë2 (fiTdmi)2 [e1 (f1oq” dm) + äi3 (f ãjdm?) }— Cj2 (f öcbdm) — ci (f ô’dm)— 1 (fTdrn.) — e3_,j1— C3 (fTd7nj)J(f iTdmi)i + Ci2 (f,ö,dm)ØjUT—2(fc’tfZc)i— (fx+fz)ëj2 (fx + fy,) j288eT(f + f dm5+2—e f ,dmn1p—ëTf’drnj—ëTf4dmji3ë (f din + f ‘drn) :P— fçdmp—eT fè’ (f + f din)2 [ëj2 (f icTdrn) e + e (f1oçb’din) ej—(f qdm) e — c2 (f ej—(f ö’dm) e — e (f öjTdrnj) e— (f oq5Pdm) e — 2 (f ë2 [e1 (f jTdmj) e + (f dm) e}—ëj2 (f ,ödm) — ëj (f ,öJ,dm) e— (f öjc7’dm) ë — ë3 (f ôj’dm) e— (f öjcffdm) e — e3 (f oqdm) e2 (f ,ö4Tdmj) e2 + (f e]289[2 d7 (f Tdrnj) dm e [Ej — i e (f drn)— (f dm) ë d7]The gravitational potential energy contribution, Ug, can be rewritten asUg = JLeM — trl+ 3Ie (A.7)cm cm cmwhere 1 represents the direction cosine vector of Rcm with reference to XC,YC,ZC axes.From Chapter 3:= (cos ib sin ç sin \ + sin ‘ sin )) I+ (cos b cos+ (cossinbcos — sin?/’cos\) 1cc: (A.8)DUg 3,Ue Dlj88 2Rc3m88DUg /e tr-- fTDIfR— 2R3 2R3 8F cm 1-’ cm290APPENDIXBFINITE ELEMENT MATRICESB.1. Beam ElementGEOMETRIC STIFFNESS MATRIX KgKg= i = 1,... , 12, j = 1,... . 12; gi,j = 9j.j.— n. — 6+2+522 — 6+1O3+532gi.i — , g2,2— 5(1+2)2— 5(1+F3)2——(2++ji. — (++_)l294,4 ,— (1+’) g6.6— (1+2)2—— 6+1O2+5 — 6+1O3+532g77 — g8.8—999—go,o = 0; 911,11 = 912.12=1 _6_1O2_522__________926 1O(1+2) 92,8 = 5(1+2)2 92,12 = lO(1+2)2—1 _6_1O3_532—l935 1O(1+3) : 939 = 5(1+3)2 g3,1’ = 1O(1+a)2—1 4 4’ 2_________-1= 1O(1+4)2 ‘ 9s,ii= (1+3) g6,8 = 1O(1+F2)2—1 ‘2 ‘‘2 2_ ___-l-1g6.12= (1+2) 98,12 = 1O(1+2) 9.ii = 1O(1+3)All other gj,j are equal to zero.291??r?r??00V’CI—iC)CIIIIIIIIIIIti9rzI!L00lC.+I0 NCOC.boC’IIIIIIIIIIMASS MATRIX M?n,=m,i;A— B- pAl— (1 + 2) — (1 +pAl (ri). pAl(1+2)2 1 ‘ (1+3)2 1pAl 13 72 2 613 73 32 pAll1 pAlrn3,= ( + + ——)B + rn4,= 3A =1 21 2 2 2 2 2 2 C;11 112 2 1 2rn2,6= + 120 +----)lA + (- — ——)iC; rn1212 = rn6,9 3 2 6rn88=rn2213 32 2 1 2rn1010=rn44,11 11 2 1 pAlrn3,5= (-j- + 120 + ---)lB + (- - _j)lD; =9rn3,9= ( + —— + ——)B — rn11, =13 3 2 1rn99=m33;13 32 2 1 pAll1m410== 6A1 2 1 2 2 D: m10,=rn4.1 2 2 1 2 22C;rn8.12 = —rn2,6 ; rn9.11 = rn5,9 ; rn6,8 = —n2, 12All other are equal to zero.293B.2. Plate ElementSTIFFNESS MATRIX KK = C k,3, i=1, 24,j=1.....24; k,=k,2;ic3, = 4(32 + ,8_2) + (14 — 4v);ic3,s = 23 + -(1 + 4v)}a;ic3,10 = [2/3—a + (1 — zi)]b:ic3,15 = 2(2,8 — /3_2) — (14 — 4zi):ic3.17 = [_132 + -(1 — zi)}a:ic3,22 = [/3—2 — (1 + 4v)]b:=.p—2 + (1 — zi)jb2= [2/3 + k(1 — v)Jb;k4,11 = 0= [113—2 + (1 — v)Jb2;ic4,21 = [p2 — (1 + 4v)]b:ic4.23 = 0;= [.p2 + — zi)Ja2ic5,10 = 0ic5,15 = [_,32 + (1— v)la;k5,17 = 1/32 + (1 — ii)Ja2;ic5,22 = 0ic9, = 4(132 + j3_2 + (14 — 4zi);= [2/3 + (1 + 4v)}a;ic9.16 = [_p2 + (1 + 4zi)]b:ic9,21 = 2(/3 + /3_2) + (14—ic9,23 = [/32 — (1 —ic10, = [p_Z + (1— v)1b2ic10.5 = [_/3— + (i + 4v)Jb;ic10.7 = 0ic10,22 = [.p2 + (1 — z)]b2k3,4 = [2/3—2 + (1 + 4v)]b;k3,9 = 2(132 — 2/3_2)—(14 — 4zi):ic3,11 = [2 — 1(j + 4zi)}a;ic3,16 = [p2—(1 — zi)]b;k3,21 = _2(2/32 — /3_2)— (14 — 4zi)k3,23 = [2/32 + (1 —ic4,5 = vab;ic4,10 = [3_2— — v)]b2ic4,15 = [_p_2 + (1 —= 0ic4,22 = [p_2—(1— z)Jb2ic5,9 = [/32—‘(1 + 4zi)ja;ic5,11 = [2p2— (1 — v)]a2;ic5,16 = 0ic5,2’ic5,23k9,,0ic9,15k9,,7kg,22= _[2/32 + (1— v)la;=[2!321(1v)1a2= [2j3— + (1 + 4v)}b;=2(/32_2/3_2)-(14 - 4v);= [2/32 + (1 — v)Ja;= [_p-2 + (1 - v)jb;Bending StiffnessEt3C= 12(1 — v2)abk10,,k,0,,6ic10,2ic,0.23= [2/32 + (1 + 4v)}a;= p_2 — 4(—= [/3_2— (1 — v)jb;=0;294k,,,,, t/32 + (1 v)Ja2;k,,,6 0k,,,2, = {_/32 + (1 v)1a;k,,,23 = 1/32 + 1ff(1—v)1a2;k15,,5 = 4(32 + /3_2 + (14 4v);k,5,,7 = _[2/92 + (1 +4zi)Ja;k,522 = f23—2 + — zi)Jb;k,6,, [3_2 + (1 v)1b2;k,62, {2/3_2+(1—v)1b;k,623 ok,717 = + (1 — v)Ja2:k,7,22 = 0k21, = 4(/32 +/32) + (14 — 4v),= _f2/32 + (1 +4v)ja;k2222 = [3_2 + — v)]b2;= + (1 — v)Ja2;Membrane St iffness= {_2 + 1(1 +4v)ja;k,7,23 = {/32—(1—= f2/3_2 + (1 + 4v)Jb;k2223—vab:4/3 + 2(1—k,,7 23—2(1 —k,,,3 —2/3 — (1—k,,9 = —4/3 + (1—k2, =4/3’+2(i —)1/3;k2,8= —4/3’+(i v)J/3;k2,,4 —2/3’—(1—k220 (1 — 3zi);k7,8 (1 + v):k714 = —(1 — 3ii);k7,20 = —(1 + v);k8,3 (1 — 3zí);= 2j3’ — 2(1—k8,20 —2/3’ —(1 —v)3;k,3,4 =-1(1 + v);k,3,20 = —(1 — 3z);k,4,,9 = (1 — 3v)k,919 4/3+2(1—zi)J/31;k’20, = 4/3—’ + 2(1 — i’)J3.Ft== —(1 + v);—iik,,4 1(1 + v)—1_I.__3r •—J.—Yjk2,3 = (1 + Li);I..—3i-“2.19 — — oYjk77 =4/3+2(1 —i’)J,3’;k7,,3—4/3 + (1—k7,,9 = —2/3 — (1 —k’88 4/3’+2(i )1/3;k819= —(1 +v);k,313 4/3 + 2(1—k,319 = 2/3 2(1k,414 4/3—’ + 2(1k,4,20—4/3’ + (1—k,920 = (1 + v);k,,,5k,,,7k,,22k,52,k,6,,7k,6,22= _[2/32 + (1—i/)la;=1!32_(1_v)1a2;= {2/3+(1+4Li)1b;2(/3 2/32) (14 4v);= f_/32+ (1 + 4zi)Ja;= vab;=/3_2_(1_’1)1b2;295MASS MATRIX MBendingM = i= 1,...,24. j = 1....,24; =rn,;pab— 176,400rn3, = 24, 178;rn3,g = 8, 582;m315 = 2, 758;rn3,21 = 8, 582;rn4, = 560b2m4,io = —420b2rn4,16 = —210b27fl4.22 280brn5, = 560a2= 280arn5,17 = —210a2= —420a2rn99 = 24, 178;rn9,15 = 8,582;m921 = 2, 758;77210,10 = 560b2rn1016 = 280bm1022 = —210brn11, = 560a2= —420a2rnll,23 = —210a2rn15, = 24, 178rn1521 = 8.582;rn16, = 560b2rn1622 = —420b2Tfl17,1 = 560a22i7.23 280arn21, = 24,178rn22.2 = 560b2rn233 = 560a2rn3,4 = 3,227b;rn310 = —1. 918b;7723,16 = —812b;rn3,22 = L 393b:rn4,5 = 441ab;m411 = 294ab;in4,17 = —196ab;rn4,23 = —294ab;rn5,9 = 1,393a;7725,15 = 812a;m521 = 1, 918ain3,5 = 3, 227a;rn3,11 = 1, 393a;7723,17 = —812arn3,23 = —1,918arn4,9 = 1918b;rn4,15 = 812b;rn4,21 = 1, 393brn5,107725.16?ns ,22in9,rn9,17rn9,2377210,1577210,21rn11,6rn11,22rn15,777215,23rn16,21= —294ab;= —196ab;= 294ab;= 3,227a;= —1,918a;= —812a;= —1,393b;= —812b;= —294ab;= 196ab;= —3,227a;= —1,393a;= —1.,918b;772910m9 15in9,22rn10,11rn10rn10,23m11 1577211,2177215,16rnls.277116,1777116,23m17,21rn21 ,227fl22,23= —3,227b;= —1,393b;= 812b;= —441ab;= 294ab;= 196ab;= 1,918a;= 812a:= —3,227b;= 1,918b;= 441ab;= 294ab;= —1,393a;= 3,227b;= 3,227b;rn17,22 = —294ab;rn21,3 = —3, 227a;296Membranern1, = 4;rn1,9 = 2rn214 = 1rn7,13 = 2rn8,14 = 2;rn13,9 = 2rn19, = 4pV36’rn1,7 = 2;rn2, = 4;rn2,0 = 2rn7,19 = 1rn8,20 = 1;rn14, = 4;20.20 = 4rn1,3 = 1;rn2,8 = 2;m7,7 = 4;rn8, = 4;rn13, = 4rn14,20 = 2i=1,...,6,j=1,...,6; rn,3=rn,;rn1, = rn1rn4, =rn2, = in1rn5,= 12jrn3, = rn1rn6, = I3B.3. Lumped Mass ElementMASS MATRIX MM =297APPENDIXCSIMULATION PROGRAM STRUCTUREThis appendix provides a simplified user’s guide to the REDUCE/FORTRAN-77computer program which has been developed in the present study. The simulationprocedure has two stages. The first stage pertains to the REDUCE symbolic manipulation program. The implementation of the equations of motion in FORTRAN-77code is carried out by speciflying the configuration under study. The user may specifyany of the six configurations discussed in Chapter 6 that best describes the proposedsystem. The output from the REDUCE consists of the code which will perform theactual dynamic simulation. The program may operate in an interactive or batchenvironment and all information must be specified via input files.298C.1 REDUCE Sample FileThe following pages show sample listings of three REDUCE files. The T-VIBRA.RDC,H-VIBRA.RDC and I-VIBRA.RDC files pertain to the structural vibration contributions to the kinetic energy, angular momentum and inertia matrix, respectively.Similar files are available for the contribution from the libration, appendage slewingand translation, and gravitational and potential energy terms.The output from these files is in FORTRAN-77 programming language, which isready for compilation and execution.299SPACE STATION DYNAMICS AND CONTROL SIMULATIONREDUCE FILE - T-VIBRA.RDC;begin;gentranliteralC” ,cr!*,tab!*, SUBROUTINE TVIBRA(THETA,Y,RTV)” ,cr!*,C” ,cr!*;>>$gentrandeclarea-h, o-z implicit real*S;>>>>$gentran<<declarey(eval 2*nmodes+6)) :real*8;q(eval nmodes)) :real*8;qdot(eval( nmodes)) :real*8;RTv1 eva.l(nmodes ) :real*8;RTv2 eval(nrnodes ) :real*8;RTv(eval(nmodes)) :real*8;RTv2a(eval(nrnodes) , 1 .eval(ni)) :real*8;>>>>$gentranliteralC” ,cr!*,tab!*,”INCLUDE ‘CBLOCK.F’,cr!*,C” ,cr!*;>>$off period;gentrannmodes := eval(nmodes);ni := eval(ni);>$on period;genstmtnum!* : 1;gentran300EC := 2.0*ecc*dsin(theta)/(1 .0Iecc*dcos(theta)):>$Comment: Change of variables from to, ;gentran<<for k := 1 : nrnodes doq(k) := y(k+3);qdot.(k) := y(k+nmodes+6);>>$>>$gentranfor k := 1 : nmodes doRTv1(k) := for 1 := 1 : nmodes sum EC*(phxxrnc(k.l)+phyyrnc(k,1)+phzzrnc(k,1) )*qdot( U:>>$gent ranfor j := 1 : ni dofor k := 1 : nmodes doRtv2a(k1J) := for 1 := 1: nmodes sum EC*(phxxmi(k.l,j)lphyymi(kJ,j )+phzzmi(kJ,j ))*qdot (U:for k := 1 : nmodes doRTv2(k) := for j := 1: ni sum RTv2a(k,1,j);>$>>$gentranfor k := 1 : nmodes doRTv(k) := RTv1(k) + RTv2(k);>>$;end:returnend;end;301REDUCE FILE- H-VIBRA.RDC;begin;gent ranliteralC” ,cr!*,tab!*, SUBROUTINE HVIBRA(THETAY,HVV,QHVV,PHVV,RHVV,SHVV)” ,cr!*,“C” ,cr!*;>>$gentran<<declarea-h, o-z : implicit real*8;>>$>$gent randeclare<<y eval 2*nmodes+6)) :real*8;q eval nmodes)) :real*8;qdot(eval(nmodes)) :real*8;Hvla(eval(nmodes),3)) :real*8;Hvl(3) :real*8:Hv2a( eval(nmodes) ,3 ,eval(ni)) :real*8;Hv2b(3,eval(ni)) :rea.l*8;Hv2 3) :real*8;Hvv 3) :real*8;QHv2a(eval(nmodes) ,3,eval( ni)) :real*8;Q Hv2(eval(nmodes),3 :real*8;QHvv(eval(nmodes).3 :real*8;PHvv(eva.1(nmodes),3) :real*8;RHvv( eval(nmodes) 3) :real*8;SHvv(3) :real*8;>>$>$gentranliteralC” ,cr!*,tab!*,”INCLUDE ‘CBLOCK.F’’,cr!*,“C” ,cr!*;>>$off period;gentraniimodes := eval(nmodes):302ni := eval(ni);>>$on period;genstmtnum!* : 1;gentranEC := 2.O*ecc*dsin(theta)/(1 .O+ecc*dcos(theta));>$Comment: Change of variables from ‘ to, ;gentranfor k := 1 : nmodes do<<q(k) := y(k+3);qdot(k) := y(k+nmodes+6);>>$>$Comment: Find ;gentranfor k := 1 : nmodes doHv 1 a(k, 1Hvla(k,2Hvla(k,3>>$1 : urnodes sum q(k *Hvla k,11 : nmodes sum q(k *HT1a k,21 : nmodes sum q(k *H;rla k,3for j := 1 : ni dofor k := 1 : nmodes do:= for 1 := 1 : iimodes sum(phyzmi(k,l,j )-phzymi(k,l,j ))*qdot(l);for 1 := 1 : nmodes sum(phzxmi(k,1,j )-phxzmi( k,l,j ))*qdot(l);for 1 := 1 : nmodes sum(phxyrni(k,1,j )-phyxmi(k,l,j ) )*qdot(l);:= for 1 := 1 : nmodes sum(phyzmc(k,l)_phzymc k,l))*qdot 1);for 1 := 1 : umodes sum(phzxmc(k,l)-phxzmc k,1))*qdot 1);:= for 1 := 1: nmodes sum(phxymc(k,1)phyxmc(k,l))*qdot(l);Hvl 1Hvl 2Hvl(3)for k :=for kfor k :=Hv2a(k,1,j)Hv2a(k.2,j)Hv2a(k,3,j)>>$>>;for k := 1: 3 doHv2b(kJ) := for 1 := 1: nmodes sum q(l)*Hv2a(l,k,j);for k := 1: 3 doHv2(k) := for j := 1: ni sum Hv2h(kJ);303for k := 1: 3 doHvv(k) := Hvl(k) + Hv2(k);>$Comment: Findgentranfor j := 1 : ni dofor 1 := 1: 3 dofor k := 1 : nrnodes doQHv2a(k,1,j) := Hv2a(k,1,j);for 1 := 1: 3 dofor k := 1 : nmodes doQHv2(k,1) := forj := 1: ni sum QHv2a(k,1,j);for 1 := 1: 3 dofor k := 1 : nmodes doQHvv(k,1) := Hvla(kJ) + QHv2(k,1);>$Comment: Findgentranfor k := 1 : nmodes doHvla k,1 := for 1 := 1: nmodes sum(phzymc(k,1 - phyzmc(k,1))*q(1);Hvla k,2 : for 1 := 1 : nmodes sum(phxzmc(k,1 - phzxmc(k.1))*q(1);Hvla k,3 := for 1 := 1: nmodes sum(phyxrnc(k,1 - phxymc(k,1))*q(1);>>$>>$gentranfor j := 1 : ni dofor k := 1 : nmodes doHv2a(k,1,j) := for 1 := 1 : nmodes sum(phyzmi(1 ,k,j )—phzyrni(1,k,j ) )q(1);Hv2a(k.2J) := for 1 := 1 : nmodes sum(phzxmi(1k,j )-phxzmi(1,k,j ))*q(1);Hv2a(k,3,j) := for 1 := 1 : iimodes sum(phxymi(1,k,j)—phyxmi(1,k,j) )*q(1);>>$>>$for 1 := 1: 3 dofor k := 1 : nmodes doQHv2(k.1) := for j := 1: ni sum Hv2a(k,1,j);forl:=1:3dofor k := 1 : nmodes do304PHvv(k,1) := Hvla(k.1) + QHv2(kj);>$Comment: Find --:dt Oqgentran<<for k := 1 : nmodes do<<Hvla k.1) := for 1 := 1: nmodes sum(phzymc k,1 _phyzmc(k,1))*qdot(1Hvla k,2 := for 1 := 1 : nmodes sum(phxzmc k,1 _phzxrnc(k,1))*qdot(1Hvla(k,3 := for 1 := 1: nmodes sum(phyxmc k,1 _phxymc(k,1))*qdot(1);>>$>>$gentranfor j := 1 : ni dofor k := 1 : nmodes doHv2a(k,1J) := for 1 := 1 : nrnodes sum(phyzmi (1 k,j )—phzymi (1,k,j ) ) *qdot (1);Hv2a(k,2.j) := for 1 := 1 : nmodes sum(phzxmi(1,k,j )_phxzmi(1,k,j))*qdot.(1);Hv2a(k,3,j) := for 1 := 1 : nmodes sum(phxymi(1 .k,j )_phyxrni(1k,j))*qdot(1);>>$>$for 1 := 1: 3 dofor k := 1 : nmodes doQHv2(k,1) := for j := 1: ni sum Hv2a(k1,j);for 1 := 1: 3 dofor k := 1 : nmodes doRHvv(k.1) := Hvla(k.1) + QHv2(k.1);>$Comment: Findgentranfor k := 1 : nmodes doHvla(k.1) := for 1 := 1: nmodes sum(phyzmc(k,1)_phzymc(k,1))*qdot(1):Hvla(k,2) := for 1 := 1: nmodes sum(phzxmc(k,1)_phxzmc(k,1))*qdot(1);Hv1a(k3) := for 1 := 1: nmodes sum(phxymc(k,1)_phyxmc(k,1))*qdot(1):>>$Hvl(1) := for k := 1: nmodes sum EC*qdot k)+q(k *Hvla(k,1Hvl(2) := for k := 1: nmodes sum EC*qdot k)+q(k *Hvla(k.2Hvl(3) := for k := 1: nmodes sum EC*qdot.(k)+q(k *Hvla(k,3>>$305gentranfor j := 1 : ni dofor k := 1 : nmodes doHv2a(k,1,j) := for 1 := 1 nmodes sum(phyzmi(k.1,j )_phzymi(k,1,j))*qdot(1);Hv2a(k,2,j) := for 1 := 1 nmodes sum(phzxmi(k,1J )-phxzmi(k,1J ))*qdot(1);Hv2a(k,3,j) := for 1 := 1 : nrnodes sum(phxymi(k,1,j )-phyxmi(k,1,j ) ) *qOt(];>$Hv2b 1,j := for k := 1: nmodes sum EC*qdot k +q(k *Hv2a(k,1JHv2b 2,j := for k := 1: umodes sum EC*qdot k +q(k *Hv2a(k.2,jHv2h(3,j := for k := 1: nmodes sum (EC*qdot(k)+q(k))*Hv2a(k,3,j>$for k := 1: 3 do Hv2(k) := for j := 1: ni sum Hv2b(k,j);>>$gentranfor k := 1: 3 doSHvv(k) := Hvl(k) + Hv2(k);>>$;end;returnend;end;306REDUCE FILE - I-VIBRA.RDC;begin;gentranliteralC” ,cr!*,tab!*, SUBROUTINE IVIBRA(Y,IV,QIV,SIV)” ,cr!*,C” ,cr!*;>>gentrandeclare<<a-h, o-z implicit real*8;>>$>>$gentrandeclare<<y(eval(2*nmodes+6)) :real*8;q(eval(nmodes)) :real*8;qdot(eval(nrnodes)) :real*8;Tmxxc(eval(nmodes)) :real*8;Tmvxc(eval(nmodes)) :real*8;Tmzxc(eval(nmodes)) :real*8;Tmyyc(eval(nmodes)) :real*8;Tmzyc ( eval (nmodes)) :real*8;Tmzzc(eval( nmodes)) :real*8;Tmxxi(eval(nmodes).eval(ni)) :real*8;Tnvxi(eval(nmodes).eval(ni)) :real*8;Trnzxi(eval(nmodes ) .eval(ni) :real*8;Tmyyi( eval(nmodes) ,eval ( ni) :real* 8;Tmzvi( eval(nmodes) .eval(ni) :real*8;Trnzzi(eval(nmocles) .eval(ni)) :real*8;Ivc(3,3) :real*8;Ivil(3.3.eval(ni)) :real*8;Ivi(3,3) :real*8;Iv(3.3) :real*8;QIvc(33.eval(nmodes)) :real*8;QIvil (3.3.eval(nmodes) .eval(ni)) :real*8;QIvi(3.3 ,eval(nmodes)) :real*8;QIv(3,3 ,eval(nmodes)) :real*8;SIv(3,3) :real*8;>>$>>$gent ran307literalC” .cr!*,tab!*,’IXCLUDE ‘C-BLOCK.F’” cr!*,“C” ,cr!*;>>$off period;gentrannmodes := eval(nmodes);ni := eval(ni);>$on period;genstmtnum!* : 1:gentranEC := _2.O*ecc*dsin(theta)/(1.O+ecc*dcos(theta));>$Comment: Change of variables from to q, ;gentranfor k := 1 : nmodes doq(k) := y(k+3);qdot(k) := y(k+nmodes+6);>>$>$Comment : Find vibration inertia for central body B;gentran<<for k := 1 : umodes doTmxxc(k := for I := 1 : nmode.s sum phxxrnc(k,1)*q(I);Tmvxc(k := for 1 := 1 : nmodes sum phyxmc(k,l)*q(1);Tmzxc(k := for I := 1 : nmodes sum phzxmc(k,1)*q(l);Tmyyc(k := for 1 := 1: nmodes sum phyymc(k,1)*q(l);Tmzyc(k := for 1 := 1 : umodes sum phzymc(k,l)*q(1);Tmzzc(k) := for 1 := 1 : nmodes sum phzzmc(k,1)*q(1);>$Ivc(1,1) := for k := 1: nmodes sum q(k)*(Tmyyc(k)+Tmzzc(k));Ivc(2.1) := for k := 1 : nmodes sum - q(k)*Tmyxc(k);Ivc(3,1) := for k := 1 : nmodes sum - q(k)*Tmzxc(k);Ivc(1,2 := Ivc(2.1);Ivc 2,2 := for k := 1: nmodes sum q(k)*(Tmxxc(k)+Tmzzc(k));lye 3,2 : for k := 1 : nmodes sum - q(k)*Tmzyc(k);Ivc 13 := Ivc(3.1lye 2,3 := Ivc(3,2Ivc(3.3 := for k := 1: nmodes sum q(k)*(Tmxxc(k)+Tmyyc(k));308>$Comment: Find vibration inertia for appendages B;gent ranfor j := 1 : ni do<<for k := 1 : nmodes doTmxxi(k,j) := for 1 := 1: nmodes sum phxxmi(k,1,j)*q(1);Tmyxi(k,j) := for 1 := 1: nmodes sum phyxmi(k,1,j)*q(1);Tmzxi(k,j) := for I := 1: nrnodes sum phzxmi(k,1J)*q(1);Tmyyi(k,j) := for 1 := 1: nrnodes sum phyymi(kJJ)*q(1);Tmzyi(k,j) := for I := 1: nmodes sum phzymi(k,1,j)*q(1);Tmzzi(kJ) := for 1 := 1: nmodes sum phzzmi(k,1,j)*q(1);>$Ivil(1,1,j) := for k := 1: nmodes sum q(k)*(Tmyyi(kJ)+Tmzzi(k,j));Ivil(2,1,j) := for k := 1: nmodes sum- q k *Tnxi(k,j);Ivil(3,1,j := for k := 1: nmodes sum- q k *Tmzxi(kJ);Ivil(1,2J := Iv(2,i);Ivii(2,2,j := for k := 1: nmodes sum q(k)*(Tmxxi(k,j)+Tmzzi(k,j));lvii 3,2,j := for k := 1: nmodes sum - q(k)*Tmzyi(k,j);lvii i,3,j := Iv(3,1Ivil(2,3,j := Iv(3,2Ivii(33,j := for k := 1: nmodes sum q(k)*(Tmxxi(k,j)+Tmyyi(k,j));>$for k := 1: 3 dofor 1 := 1: 3 doIvi(1,k) := for j := 1: ni sum Ivil(1,k,j);>>$gentranfor k := 1: 3 dofor 1 := 1: 3 doIv(l,k) := Ivc(1,k) + Ivi(1,k);>$Comment: Findgentranfor k := 1 : nmodes doQIvc( 1,1 ,k) := 2.O*(Tmyyc(k)+Tmzzc(k));QIvc(2,i ,k) := 2.O*Tmyxc(k);QIvc(3.i,k := _2.O*Tmzxc(k);QIvc i,2.k := QIvc(2,i,k);QIvc 2,2.k := 2.O*(Tmxxc(k)+Tmzzc(k));QIvc 3,2.k := _2.O*Tmzyc(k);QIvc i,3,k := QIvc(3,i,kQIvc(23,k := QIvc(3,2,k309>>$for j := 1 : ni do<<>>$for k 1 : nrnodes do<<for k := 1 : nmodes dofor 11 := 1: 3 dofor 12 := 1: 3 doQIvi(12,1i.k) := forj := 1: ni sum QIvil(12,11,k,j);for k := 1 : nmodes dofor 11 := 1: 3 do>>$Comment: Findfor 12 := 1: 3 doQIv(12,11 ,k) := QIvc(12J1 ,k) + QIvi(12,11 ,k);gentranfor k := 1 : nrnodes do>>$Ivc LiIvc 2,1Ivc(3J)Ivc( 1,2)Ivc(2.2)Ivc (3.2)lye (1,3)Tmxxc(k) := for 1 := 1Tmyxe(k) := for 1 := 1Tmzxc(k) := for 1 := 1Trnyyc(k) := for 1 := 1Tmzvc(k) := for 1 := 1Tmzzc(k) := for 1 := 1nmodes sum phxxmc(k,1) *qdot (1);umodes sum phyxmc(k,1) *qdot (1);nmodes sum phzxme(k.1)*qdot(1);nmodes sum phyyrnc( k,1)*qdot,( 1);nmodes sum phzymc(k,1 )*qdot (1);nmodes sum phzzmc(k,1)*qdot(1);>>$gentran<<QIvc(3,3,k) := 2.O*(Tmxxc(k)+Tmyyc(k)):QIvii(i,1,k,j)Qlvii(2,i,k,j)Qlvii(3,i,k,j)QIvil(1 .2,k,j’QIvil (2.2,k,jQIvil(3.2.k,j)QIvil(1.3,kJQIvil (2,3.k,jQIvil(3,3,kJ: 2.O*(Tnwyi(kj)+Tmzzi(kJ));:= _2.O*Tmyxi(k,j);:= 2.O*Tmzxi(k,j);:= QIvil(2,1,k.j)::= 2.O*(Tmxxi(k,j)+Tmzzi(k,j));:= _2.O*Tmzyi(kJ);:= QIvil(3,1,k,j);:= QIvil(3,2,k,j)::= 2.O*(Tmxxi(k.j)+Tnwvi(k.j));>>$:= for k := 1:= for k := 1for k := 1:= Iv(2,1);for k :=:= for k :=:= Iv(3.1);nmodes sum q(k)*(Tmyyc(k)+Tmzzc(k)):nmodes sum- q(k *Tmyxc(k);nmodes sum- q(k *Tmzxc(k);1: nrnodes sum q(k)*(Tmxxc(k)+Tmzzc(k));1 : nmodes sum - q(k)*Trnzyc(k):310>>$gentranIvc(2,3) := Iv(3,2);Ivc(3,3) := for k := 1: nrnodes sum q(k)*(Tmxxc(k)+Tmyyc(k));for j := 1 : ni dofor k := 1 : nrnodes do>>$gentran>>$Tmxxi(k,j)Tmyxi(k,j)Tmzxi(kJ)Tmvyi(k,j)Tmzvi(k,j)Tmzzi(k,j)for k := 1 : 3 dofor 1 1: 3 doIvi(1,k) := for jnrnodes sum phxxrni(kJ,j )*qdot(1);nmodes sum phyxrni(k,1,j)*qdot(1);nmodes sum phzxmi(k,1,j)*qdot(1);nmodes sum phyymi(k,1,j ) *qdot (1);nrnodes sum phzymi(k,1J ) *qd(];nmodes sum phzzmi (k,1 ,j ) *qdot (1);1: nmodes sum q(k) * (Tmxxi(k,j ) +Tmzzi(k,j));1: nmodes sum - q(k)Tmzyi(k,j);1: nmodes sum q(k)*(Tmxxi(k,j)+Tmyyi(kj));:= 1: ni sum Ivii(1,k,j);:= for 1:= for 1for 1:= for 1:= for 1:=i=1:=1:= for 1 := 1>>$Ivil(i,1,jlvi 1(2,1,jIvil(3,iJIvii(i,2J)lvi 1(2,2,j)lvii (3,2 ,j)lvii 1,3Jlvii 2,3,jlvii 3,3,jnmodes sumnmodes sumnmodes sumfor k := 1:= for k := i:= for k i:= Iv(2,i);:= for k :=:= for k :=:= Iv(3,i);:= Iv(3,2)::= for k :=q(k)*(myyi(kj)+Trnzzi(k,j));- q(k)*Tmyxi(k,j);- q(k)*Tmzxi(k,j);;end;;end:for k := i 3 dofor 1 := i : 3 doSIv(1,k) := 2O*Ivc(1,k) + 2.O*Ivi(1.k);>$ret urnend311C.2 Input Data FileThe following pages show the format of the input file as required by the FMCMAIN.F simulation program for the FMC of the Space Station. This informationis put into a file using any available text editor, and is completely free format. Thebold words have special meaning to the program, while the italic words are includeto provide a description of the variable.312SPACE STATION DYNAMICS AND CONTROL SIMULATIONFIRST MILESTONE CONFIGURATION - FMCINPUT FILE TO FMC-MAINY. IMATXC =‘ Inertia Matrix For Central Body8.8D6 O.ODO O.ODOO.ODO 2.9D7 O.ODOO.ODO O.ODO 2.9D7• CCMAT = Initial Orientation of spacecraftO.ODO O.ODO O.ODO. ECCENT = Eccentricity of OrbitO.ODO. PERIOD = Period of Orbit (Minutes)l.0D2. ROTSLW Rotational Maneuver ParametersB1 t0 = O.DO c = O.DO /3 = O.DO o = O.DOtf = O.DO = O.DO /3f = O.DO 7f = O.DOB2 to = O.DO co = O.DO j3 = O.DO 7o = O.DOtf = O.DO = O.DO 13f = O.DO‘‘= O.DOB3 to = O.DO a = O.DO !3o = O.DO 7o = O.DOtf = O.DO O.DO /3f = O.DO 7f = O.DOB4 to = O.DO co = O.DO = O.DO ‘yo = O.DOtf = O.DO c = O.DO /3 = O.DO 7f = O.DOB5 t0 = O.DO ao = O.DO /3o = O.DO 7o = O.DOtf = O.DO = O.DO,I3 = O.DO 7f = O.DO. TRASLW Translational Maneuver ParametersB1 tçj = O.DO x = O.DO Yo = O.DO z0 = O.DOtf = O.DO Xf = O.DO 1Jf = O.DO Zf = O.DOB2 t0 = O.DO x0 = O.DO Yo = O.DO z0 = O.DOtf = O.DO Xf = O.DO 7Jf = O.DO Zf = O.DOB3 t0 = O.DO x0 = O.DO Yo = O.DO z0 = O.DOtf = O.DO Xf = O.DO 7Jf = O.DO Zf = O.DOB4 t0 = O.DO x0 = O.DO Yo = O.DO z0 = O.DOtf = O.DO Xf = O.DO Yf = O.DO Zf = O.DOB5 t0 = O.DO x0 = O.DO Yo = O.DO z0 = O.DOtf = O.DO Xf = O.DO = O.DO Z = O.DO313• IMATXI Inertia Matrix for AppendagesB1 1334.3D0 0.D0 0.D00.ODO 161174.3D0 0.D00.ODO 0.D0 162504.ODOB2 1334.3D0 0.D0 0.D00.ODO 161174.3D0 0.D00.ODO 0.D0 162504.ODOB3 49.6D0 0.D0 0.D00.ODO 19837.6D0 0.D00.ODO 0.D0 19887.2D0B4 94.4D0 0.D0 0.D00.ODO 47ö63.1DO 0.D00.ODO 0.D0 47563.1DOB5 155.5DO 0.D0 0.D00.ODO 149719.1DO 0.D00.ODO 0.D0 149719.1DO• RHOIMI First Moment of AreaB -658604.ODO 0.ODO 0.ODOB1 7326.ODO 0.ODO 0.ODOB2 7326.ODO 0.ODO 0.ODOB3 2587.5DO 0.ODD 0.ODDB4 4089.4D0 0.ODO 0.ODOB5 6808.5D0 0.ODO 0.ODO• DIO = Initial Hinge PositionB1 -52.5D0 LDO 0.ODOB2 -52.5D0 -1.D0 0.ODOB3 -47.5D0 0.D0 0..5D0B4 -2.öDO 0.D0 0.00B5- 2.5D0 0.D0 0.D0• EMASS = Mass Val’uesB 19426.3D0B1 444.0D0B2 444.ODOB3 450.ODOB4 404.ODOB5 390.ODO• ROTINI Initial Rigid Orientation of AppendagesB1 0.ODO 0.ODO -90.ODOB2 0.ODO 0.ODO 90.ODOB3 -90.ODO 0.ODO -90.ODO314B4 0.ODO 90.ODO 0.00B5 0.ODO 90.ODO 0.D0. IMSL = Integration Routine ParametersHe l.OD-8Tol l.OD-8Meth 2Miter 0Index 1. SIMUL Simulation Run TimeNOR l.ODONPTO 5.0D2. INICO Initial ConditionsDISPO 5.D0 5.D0 5.dO 0.dO 0.dO 0.dO0.dO 0.dO 0.D0 0.D0 0.D0 0.D0VELO 0.D0 0.D0 0.D0 0.D0 0.D0 0.D00.D0 0.D0 0.D0 0.D0 0.D0 0.D0. MODAL Modal FunctionsfelO 7felO8felO9fellOfellifell 2fell 3fell 4fell5fell6. OPTION = Simulation and Output optionsICTRL 1 0=UNCONTROLLEl=CONTROLLEDIQUASI 2 l=OPEN LOOP , 2=CLOSE LOOPISOLAR 1 0=PRINT , l=DON’T PRINTIPVRAD 1 0=PRINT , l=DON’T PRINTIRCSYS 0 0=PRINT , l=DON’T PRINTISTING 0 0=PRINT l=DON’T PRINTITRUSS 0 0=PRINT , l=DON’T PRINTIUPDAT 0 0=NO UPDATE , l=YESIGFOPT 0 0=NO DISTURB. , l=YES•EOF315


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