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On the nonlinear dynamics of a space platform based mobile flexible manipulator Mah, Harry Wayne 1992

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ON THE NONLINEAR DYNAMICS OF ASPACE PLATFORM BASEDMOBILE FLEXIBLE MANIPULATORHARRY WAYNE MAHB.Eng., Technical University of Nova Scotia, 1988M.A.Sc., University of British Columbia, 1986A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate StudiesDepartment of Mechanica’ EngineeringWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAOctober 1992© Harry Wayne Mah, 1992Signature(s) removed to protect privacyIn presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of ZJ\.:t cv-The University of British ColumbiaVancouver, CanadaDate °,(SignatureDE-6 (2/88)Signature(s) removed to protect privacyABSTRACTThe thesis aims at development of a rather versatile tool for studying dynamicsand control of an orbiting flexible manipulator. It is motivated by the Canadiancontribution, in the form of the Mobile Servicing System (MSS), to the U.S. led SpaceStation (Freedom) program, scheduled to be operational by the turn of the century.To begin, a relatively general dynamical formulation is developed for a large classof systems characterized by interconnected beam and/or rigid articulating membersforming a chain-type geometry.As can be expected, the governing nonlinear, nonautonomous and coupled equations of motion, extremely long even in matrix notation, are not amenable to anyknown closed form solution. Hence the attention is focused towards development ofan efficient numerical code, in a modular format to help assess relative importance ofthe various system parameters. Validity of the formulation and the computer code areassessed and their operational aspects demonstrated through a parametric responseanalysis. Emphasis throughout is on methodology and general approach leading tounderstanding of the multibody dynamics problem at the fundamental level.11TABLE OF CONTENTSABSTRACTLIST OF SYMBOLSLIST OF FIGURESACKNOWLEDGEMENT1. INTRODUCTION1.1 Preliminary Remarks1.2 The Space Station Remote Manipulator System1.3 A Brief Review of the Literature1.4 Scope of the Investigation2. FORMULATION OF THE PROBLEM2.1 Preliminary Remarks2.2 Lagrangian Approach2.2.1 Reference frames2.2.2 Specified and generalized coordinates11vuxixv11351014141416162.2.3 From the inertial frame to the jth body frame . . .. 192.2.4 Position and deformation vectors for theelement mass dm1 212.2.5 Assumptions concerning mode shapes asadmissible functions 25in2.2.6 System shifting center of mass 262.2.7 Kinematic expression for position and velocity 262.3 Kinetic Energy 292.4 Potential Energy 322.4.1 Gravitational Potential Energy 322.4.2 Joint Potential Energy 322.4.3 Beam Strain Energy 332.5 Equations of Motion 392.6 Computational Considerations 413. RESULTS AND DISCUSSION . . . 443.1 Preliminary Remarks . . 443.2 Verification of the Code . . 443.3 Free Response . . 473.4 Effect of Slewing Maneuvers 533.4.1 Rigid manipulator and payload with wrist maneuver . . 533.4.2 Flexible manipulator and rigid payloadwith wrist maneuver 593.4.3 Slewing maneuvers of the flexible arms 653.5 Effect of Translational Maneuvers 713.5.1 Inplane maneuver (flexible manipulator) 79iv3.5.2 Out-of-plane maneuver (rigid manipulator) 793.6 Effect of Payload Flexibility 843.7 Further Generalization of the Model 883.7.1 Single branch configuration 923.7.2 Extended ‘flower petal-type’ configuration . 923.7.3 Chain geometry forming single and multipletether-like configurations 994. CONCLUDLNG REMARKS 1074.1 Summary of Conclusions 1074.2 Recommendations for Future Work 109BIBLIOGRAPHY 111APPENDICESA: ROTATION AND PROJECTION MATRICES 117B: SPECIFIED TIME HISTORY 119C: MATRICES AND VECTORS ASSOCIATED WITHDEFORMATION 120D: NATURAL FREQUENCIES AND MODE SHAPES 124E: SHIFT IN CENTER OF MASS 128F: TSF - ENERGY RELATIVE TO SYSTEM FRAME 129G: TQQ - ENERGY IN TRANSLATION W.R.T. SYSTEM FRAME 132H: Tww - ENERGY IN ROTATION W.R.T. SYSTEM FRAME . 134VI: TQW - ENERGY IN COUPLING OF TRANSLATIONAL ANDROTATIONAL DEGREES OF FREEDOMW.R.T. SYSTEM FRAME 137J: TLSF - ENERGY IN COUPLING 139K: - SYSTEM ANGULAR MOMENTUM DUE TOTRANSLATION W.R.T. SYSTEM FRAME 142L: - SYSTEM ANGULAR MOMENTUM DUE TOROTATION W.R.T SYSTEM FRAME 143M: - SYSTEM ANGULAR MOMENTUM DUE TO INTEGRAL 145N: TL - LIBRATIONAL ENERGY OF SYSTEM 1460: TROT - ENERGY CONTRIBUTION FROM ROTARY INERTIAAND TORSION 147P: UL - GRAVITATIONAL POTENTIAL ENERGY 151Q: [M] - MASS MATRIX 155R: VECTOR{Rq} 157vLIST OF SYMBOLSvector locating frame 1 w.r.t. the system framevector locating frame i + 1 w.r.t. the framelocation of the c.rn. of the th body w.r.t. the th framee eccentricity of orbitlocation of elemental mass dm1 on the jth body w.r.t. frame 1location of frame i + 1 (on undeformed body) w.r.t. frame iprojection of along axis xqtorsional spring constant for the ith joint about the fth direction1 index for number of bodies in systeml, dl length and element length of the ith body, respectivelydirection cosines for unit vector along w.r.t. the system frame79 vector projecting 6 on the system framem1, dm: mass and elemental mass of the th body, respectivelyn1 sum of masses from the th to the last bodynj , number of admissible functions used to represent deflectionand rotation of body i in the th direction, respectively{ q } vector of generalized coordinatesth generalized coordinater13 angle of torsion for the tip of the ith bodyvector locating the system c.m. w.r.t. the inertial framelocation of elemental mass dm1 on the body w.r.t. thesystem frametimetip deflection of the th body in direction sjjtip position of the ith body along the axis of the orbital frametip position of the ith body along the th axis of frame 1x12,x13 coordinates for the i’ frame in directions 1,2,3, respectivelyx,x0,x3 coordinates for the orbital frame in directions 1, 2, 3, respectively.vii[Ag], [D J matrices of functions of mode shapes associated with transverseand axial deformation of the th body[B, 1 [C, } matrices of functions of mode shapes associated withforeshortening of the i’ body[C] rotation matrix associated with relative rotation ofbody i + 1 w.r.t. the th bodyYoung’s modulus of elasticity for the th bodyC, shear modulus of elasticity for the ith bodyHSF system angular momentum[I] system inertia diadic w.r.t. the system frame[Iei. j inertia diadic per unit length of elemental mass drn.jsecond moment of area for beam cross-sectional areaabout the pth direction.1, polar moment of inertia for cross-section of the ith beamM, dM mass and elemental mass of the system, respectively[M] mass matrix of the system[F,] projection matrix associated with Zi[P,,, } matrix projecting onto the system frame{ Q } vector of generalized forcesth generalized forceposition of elemental mass w.r.t. the inertial frame{ Rq } vector associated with equation of motionT total kinetic energy of the system[T,] rotation matrix projecting vector from frame i onto the systemframekinetic energy associated with librationTLSF kinetic energy due to coupling of libration with system angularmomentumT0 kinetic energy associated with orbital motionTROT kinetic energy attributed to rotation of elemental masses dm,w.r.t. frame iviiiTSF kinetic energy associated with motion relative to the system frameTTRAN kinetic energy attributed to translation of elemental massesw.r.t. the inertial frameU total potential energy of the system[U] unit matrixU strain energy in axial deformation for the i” bodyU strain energy in bending for the th bodyU strain energy in torsion for the jth bodyUj potential energy attributed to joint complianceUL potential energy attributed to gravitational force fieldUs potential energy attributed to beam strain energyrotation of elemental mass dm1 w.r.t. frame idue to rotary inertia and torsionset of angles describing rotation of the ith body at x3 = h13-due to rotary inertia and torsionrotation of elemental mass dmj about the th directionvector of angles describing specified orientation of frame i + 1w.r.t. the th framespecified angle of rotation about axisvector of joint d.o.f. angles describing orientation of frame i + 1w.r.t. its specified orientation ()generalized coordinate describing angle of rotation about axis rctjvector of generalized coordinates associated withtransverse and axial deformations[S J matrix of generalized coordinates associated with beamforeshorteninggeneralized coordinate for the kth admissible functionassociated with the deflection ,position of element mass dmj w.r.t. the frame[in] matrix of mode shapes associated with rotary inertia andtorsional motions of the bodyix7ijk kth admissible function associated with the rotation ajS true anomaly• vector of generalized coordinates associated withrotary inertia and torsional motionsgeneralized coordinate for the kth admissible functionassociated with the rotation auniversal gravitational constantmass per unit length of the th bodydeflection of elemental mass dm w.r.t. frame idue to transverse and axia’l deformationsdeflection of elemental mass dm1 in the th directionaxial deflection of the ith beam due to foreshortening effect7i position of elemental mass dm: w.r.t. thejth framefor the undeformed th body[a J matrix of mode shape functions associated with beamforeshorteningJ matrix of mode shapes associated with transverse and axialdeformations of the bodykt admissible function for the deflection jvector of librational generalized coordinatesangular velocity at hinge due to , , orangular velocity of elemental mass drn w.r.t. the frameWjk natural frequency of the ith beam associated withthe kthl mode in/about the th directionpartial derivative operatorlibrational angular velocityDots and primes represent differentiation w.r.t. time t and the true anomaly 8,respectively, unless otherwise defined. Tilde [] represents an operation on a generic(3 x 1) vector to yield its skew-symmetric matrix [j.xLIST OF FIGURES1-1 A schematic diagram showing one of the earlier configurations of theproposed space station Freedom. It has gone through severalmodifications and is likely to evolve further 21-2 A schematic diagram of the flexible platform with SSRMS and MT . 41-3 System description with reference frames and position vectors 122-1 The position vector locates a system elemental mass dM w.r.t. theinertial frame 152-2 Planar view of the undeformed body 182-3 Link (beam) in transverse planar deformation 202-4 Position vector locates element mass drn w.r.t. the frame. . 232-5 A link undergoing planar transverse as well as axial deformations. 242-6 Flow chart of computer code 433-1 Data used in comparison with Chan’s study of planar maneuver,a particular case 453-2 A comparison of response results as obtained by the present methodwith those reported by Chan. The correlation is indeed excellent. 463-3 Data used in comparison with Ng’s study 483-4 Comparison of response results, as obtained by Ng and the presentmethod, for an out-of-plane slew maneuver of the upper armthrough 180°. Note, the two sets of results are virtually identical. 493-5 Data for initial planar disturbance 503-6 Response to a planar disturbance:xi(a) time histories of generalized coordinates 51(b) tip deflections w.r.t. local and platform frames 523-7 System data for initial out-of-plane disturbance 543-8 Response to initial out-of-plane disturbance:(a) librational response as well as platform tip and joint defiectionsw.r.t. local frames 55(b) tip and joint 3,4 deflections w.r.t. local frames 56(c) joint 4 rotations as well as tip positions in the orbital frame. 57(d) tip positions in the orbital frame 583-9 Data for wrist slew maneuver of rigid manipulator (3 or 6s maneuver) 603-10 System response to wrist yaw maneuver with the rigid manipulator:(a) libration and platform deflections, 3s maneuver, local frame 1. 61(b) payload tip position in the orbital frame, 3s maneuver 62(c) libration and platform deflections, 6s maneuver, local frame 1. 63(d) payload tip position in the orbital frame, 6s maneuver 643-11 Data for wrist slew maneuver of flexible manipulator with a rigidpayload, 6s maneuver 663-12 System response to wrist yaw maneuver with the flexible manipulator:(a) librational response as well as platform tip and joint deflectionsw.r.t. local frames 67(b) tip and joint 3,4 deflections w.r.t. local frames 68(c) joint 4 rotations as well as tip positions in the orbital frame. 69(d) tip positions in the orbital frame 70xii3-13 Data for simultaneous slew of the flexible arms (6s or 18s maneuver) 723-14 Typical system response to a simultaneous 6s slew maneuver of theflexible arms:(a) joints and upper arm deformations 73(b) lower arm deformations and the tip position 743-15 Typical system response to a simultaneous 18s slew maneuver of theflexible arms:(a) joints and upper arm deformations 75(b) lower arm deformations and the tip position 763-16 System response during simultaneous 6s slew maneuver of the armswith joints treated as rigid:(a) upper and lower arm deformations 77(b) the lower arm tip position 783-17 System data for the manipulator translational maneuver of 5mcompleted in either lOs or 60s 803-18 System response during the translational maneuver of 5m:(a) time-histories of the generalized coordinates for the lOs maneuver 81(b) time-histories of the tip deflections and c.m. position for thelOs maneuver 82(c) time-histories for the 60s maneuver 833-19 System data for an out-of-plane, translational maneuver of therigid manipulator 853-20 Time histories of the system response during an out-of-planetranslational maneuver:xl”(a) librational and platform tip motions. 86(b) payload tip position. 873-21 System data. used in the simulation study with rigid andflexible payloads 893-22 Effect of payload during a translational maneuver:(a) rigid payload. 90(b) flexible payload. 913-23 Illustration of single branch configuration 933-24 Illustration of extended ‘flower petal-type’ configuration. 943-25 System data for single branch configuration having rigid platformwith an array of flexible appendages 953-26 Typical response of the single branch configuration to an appendagetip disturbance:(a) librational motion and tip planar deflections for the four appendages.The out-of-plane response was zero 96(b) time histories of the tip positions w.r.t. the platform frame. . . . 97(c) extended plot of libration and the appendage tip deflectionsshowing the beat response 983-27 System data for the ‘flower petal-type’ configuration 1003-28 Libration and the appendage tip responses for the ‘flower petal-type’configuration. 1013-29 Illustration of multi-tether configuration. 1023-30 Response of rigid tether with point mass payload. 1043-31 Response of multi-tethered system with point mass payloads. . . . . 106xivACKNOWLEDGEMENTSI would like to thank Prof. V.J. Modi for his guidance throughout the preparationof the thesis.Special thanks to colleagues Alfred Ng and Julius Chan for their invaluable cooperation during the debugging phase.The research project is supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. A—2181 and the Center of Excellence Program,Grant No. IRIS/C—8, 5—55380, both held by Prof. Modi.xv1. INTRODUCTION1.1 Preliminary RemarksTo establish man’s presence in space over a relatively long duration, the U.S. hasconimitted itself to the development of an infrastructure for the initial explorationand occupation of the inner solar system. The first indispensable building block ofthis infrastructure is already in place. The Space Transportation System includesa fleet of four manned, orbiting space shuttles designed for low altitude orbits andatmospheric reentry. The second building block is in the planning stage. The Earthorbiting Space Station Freedom, scheduled to be operational by the end of the century, will facilitate undertaking of a variety of missions including satellite launch andmaintenance, space exploration, earth observation, etc. It will enhance Earth orientedtechnologies such as communications, meteorology, navigation etc., and provide a microgravity environment for scientific and commercial utilization. About the size of afootball field ( hOrn length), the base design of the Space Station (Fig. 1-1) suggeststhat it will be an extremely flexible structure with the fundamental natural frequencyin bending less than 0.1 Hz.Various modules will house crew quarters, scientific laboratories, maintenanceand operations units, etc., all supported by the main truss which will also sportearth and space pointing devices such as antennas, telescopes, and other scientificinstruments. The Station’s power requirement will be supplied by an array of highlyflexible solar panels together with the development of solar dynamic generators infuture. Thermal control will be through heat balance with the excess heat dissipatedby radiators. A docking facility will secure the shuttle to the Station, and cargo willbe handled by a highly flexible Space Station Remote Manipulator System (SSRMS).1[vRadiatorl.SPACESTATION‘FREEDOM’IFigure1-1AschematicdiagramshowingoneoftheearlierconfigurationsoftheproposedspacestationFreedom.Ithasgonethroughseveralmodificationsandislikelytoevolvefurther.RCSBoom]SolarPanel2 -IMainTrussiAlso, the SSRMS will assist in the construction, operation and maintenance of theSpace Station.1.2 The Space Station Remote Manipulator SystemThe SSRMS consists of a highly flexible, robotic, manipulator arm supportedby an essentially rigid Mobile Transporter (MT,Fig. 1-2) which is able to traversethe Station through translational and rotational maneuvers. Thus, the manipulatorarm can be transported to any desired location on the Station. The manipulatorconsists of two highly flexible links forming the upper and lower arms, while the endeffector can be a conventional grip (essentially rigid) or a smaller twin arm dextei:ousmanipulator.This Mobile Servicing System (MSS, Canadian contribution to the Space Station) will perform tasks through translational maneuvers of the U.S. supplied MTand rotational maneuvers of the SSRMS. Slewing maneuvers of the SSRMS will becarried out through specification of rotations of the joints at the shoulder, elbow,and wrist of the manipulator arm. As mentioned before, the SSRMS will be utilized for handling and ferrying of payloads, retrieval of disabled satellites for repair,maintenance of the Space Station and other various missions. Performance of suchtasks may require complex maneuvering time histories of the SSRMS and can leadto dynarriical, stability and control problems, especially in light of the high degree offlexibility of the manipulator system and the Station itself. The MSS’ slewing andtranslational maneuvers may lead to undesirable librational as well as vibrationalmotion of the Station, thus adversely affecting its performance. For example, thelarge scale and varied articulation activity onboard the Space Station may present a.challenging multi-payload pointing control problem. For successful completion of themissions, attitude control as well as vibration suppression of the Station and MSS3®flexibleJpayIoadlowerarwithspec.rot.andlastic,dissipatived.o.f.rot.©upperarplatformyaworbitalframeatspacestationc.m..\Figure1-2AschematicdiagramoftheflexibleplatformwithSSRMSandMT.would be necessary prerequisites.In brief, a sound understanding of complex interactions between the rigid bodydynamics, flexibility, MSS maneuvers and external/internal disturbances, is necessaryto undertake safe design of the Space Station and the manipulator system as well ashelp develop appropriate control strategies so that the Space Station can play itsintended role.1.3 A Brief Review of the LiteratureThe field of satellite dynamics and control has evolved significantly during the last30 years and some of the methodologies developed there may assist us in planning anapproach to tackle this formidable problem.In the early stages of space exploration, satellites tended to be small in size, simplein design, and essentially rigid. A vast body of literature dealing with librationaldynamics of satellites has been reviewed quite effectively by Shrivastava et al’.A flexible spacecraft under external disturbances will exhibit not only attituderesponse but also vibrational dynamics, which could lead to coupling displaying resonance, beat or other interesting characteristics. Literature dealing with this class ofproblems pertaining to flexible satellites has been reviewed by Modi2.Environmental effects through interactions with gravity gradient, magnetic, freemolecular, solar radiation and plasma fields further complicate the problem. Fortunately, influence of the environmental forces on librational dynamics of rigid satellitesis quite well understood and a variety of control strategies have proved to be effective. A review of the extensive literature in the field has been presented by Modi andShrivastava3.The three review papers by Modi et al.’3 which cite around 560 papers cover5the ground with reference to satellite dynamics and control quite well, with twoexceptions:(i) Control of flexible satellites is a relatively recent field (Modi’s review paperon flexible satellites was written in 1974), and the available literature is stillrather limited4’9.A relatively recent issue of the Journal of Guidance, Control, and Dynamics, published by the American Institute of Aeronautics andAstronautics, attempts to evaluate the current state in the field20. Broadlyspeaking, the significant conclusions may be listed as follows:• It is recognized that gravitational, free molecular, magnetic, solar and plasmafields cannot be simulated with acceptable accuracy to study dynamics andcontrol of flexible structures. Thus the ground-based simulation facilities thathave been so heavily relied on during the design of essentially rigid spacecraftwould be of limited value in the development of flexible systems.• It is generally agreed by NASA, other space agencies around the world, andthe community of space scientists that mathematical modeling of dynamicsand control will be relied upon more heavily in future in the design of flexiblesystems.• A variety of proposed control strategies in the papers cited above will have tobe checked, modified, and associated algorithms calibrated through carefullyplanned on-orbit experiments.(ii) There is virtually no literature on the control of flexible satellites in the presence of environmental forces.It can be concluded from the above that although dynamics and control of rigidspacecraft is quite well understood, our appreciation of even simple flexible systemdynamics and control is, at best, cursory.6One would expect to find a considerable amount of relevant literature in thefield to facilitate the formulation of equations of motion for a flexible manipulator.After all, ground based robots have been in operation for years. However, there is afundamental difference. Earthbound manipulators for factory automation have been,generally, small in size and essentially rigid. Extensive amount of literature pertainingto ground based robots is thus limited by the constraint of rigid structure2125.An emerging trend toward faster, lighter industrial robots has spawned some recent attempts at exploring the effect of fiexibility2629 using simplified models. Theflexibility generally arises from link elasticity and compliance of the motor/transmissionunits at the joint. The resulting oscillations of the manipulator’s end-effector will adversely influence its ability to track a target trajectory. The analysis of a flexiblemanipulator is further complicated by the coupling between the nonlinear rigid bodymotion and the essentially linear deformations of the elastic links30.Approaches to the derivation of equations of motion for a flexible manipulatorhave been varied. Fresonke et al.3’ as well as Hughes32 employed the Newton-Eulermethod to generate the equations for a flexible serial manipulator, while Low andVidyasagar33 utilized Hamilton’s Principle. Book34 combined a recursive Lagrangianapproach with homogeneous transformation matrices to obtain equations of motionfor a multi-link flexible manipulator. Cetinkunt and Book35 proposed the use ofsymbolic manipulation programs to bypass the algebraic complexities associated withflexible manipulator systems.An important characteristic of a flexible manipulator system, which has beenrarely addressed in the literature, is the aspect of torsional behaviour. Kane et al.36incorporated the effect of torsion, rotary inertia, and shear deformation in their analysis of a 3 link flexible manipulator, and asserted that conventional approaches (lack7ing this provision) were woefully inadequate for successful simulation. Lemak andBanerjee37 have incorporated this concept in a multi-body program, presently limited to arbitrary rigid bodies and symmetric cantilever beams. Poelaert38 utilized theconcept of dynamic stiffness matrix to develop a distributed element program capableof analyzing arbitrary configurations composed of a variety of structural elements —including beams undergoing axial, bending, and torsional vibration.Placing a robot in space introduces Keplerian mechanics considerations: operation in essentially zero gravity environment. However, this is not entirely new. Wedo have considerable experience with the now famous Canadarm, and the associatedliterature3945 would be of some value in a limited way. On the other hand, striking differences between the Canadarm and the proposed SSRMS should be clearlyrecognized. They may be summarized as follows:(i) Arms of the SSRMS are significantly longer than those of the Space Shuttlebased Remote Manipulator System (RMS). Differences in flexibility could besignificant, subject to design considerations.(ii) The Space Shuttle, which supports the RMS, is essentially rigid while theSpace Station, as pointed out before, is a highly flexible structure.(iii) Shoulder of the Canadarm is fixed to the Shuttle while the SSRMS is free totranslate as well as rotate with respect to the Space Station.Thus here we have a flexible manipulator with a. mobile base negotiating a flexibleplatform. Obviously, space robotics presents us with problems far more challengingthan those ever encountered on ground.It is, therefore, understandable why there is virtually no published literature directly pertaining to the SSRMS configuration presented in Figure 2, except for a fewsimple studies aimed at understanding the basic concepts. Misra and Cyril46 studied8the planar dynamics of a 3 link (2 flexible, 1 rigid) manipulator, however, the effectsof attitude motion and gravity-gradient were neglected. Yarnada et al.47, applyingsimilar simplifications in their study of a 5 link rigid manipulator, concluded thatmotion coupling between the spacecraft and manipulator system should not be neglected as the resultant motion of the spacecraft could lead to the end effector missingthe target. A study by Longinan et al.48 addressed the problem of reaction moment,imposed by the Space Shuttle librational control, through development of kinematicequations that adjusted the joint angle commands of the rigid remote manipulatorsystem. As well, Umetani and Yoshida49 derived a newJacobian matrix to controlattitude motion caused by the slewing of a single rigid link. Yamada and Tsuchiya5°optimized the trajectory of a 6 d.o.f. rigid manipulator while simultaneously suppressing the attitude variation.There has been attempts to model the SSRMS in an approximate manner. Chan5’assumed the Station platform to be rigid and reduced the Mobile Transporter andpayload to point masses. The manipulator links were treated as Euler beams and thejoint compliance modeled as torsional springs. However, the system was restricted toplanar motion.In an attempt to attain insight into the interaction of the flexible Station with theMSS, Morita52 developed a Lagrangian formulation for studying dynamics of a systemof interconnected Eulerian beam-type members forming a chain geometry. The systemwas considered to be in an arbitrary orbit with the inplane and out-of-plane slewingmotions permitted at the joints, which were considered rigid. Translational motionat the first joint was permitted to simulate the mobile character of the Space Stationbased manipulator. However, application of the formulation was limited to the SpaceShuttle based S COLE (Structural COntrol aboratory Experiment) configuration9involving slewing maneuver of the flexible mast supported antenna system and itscontrol.A word concerning contribution of Spar Aerospace, the prime contractor of theMSS, would be appropriate. Obviously considerable amount of simulation tools arelikely to have been developed by the organization directly or through contracts. However, as can be expected, the information is considered proprietary in nature, and isnot available in open literature.The review clearly suggests that the dynamics of a flexible manipulator, traversinga flexible space platform and carrying an elastic payload, belongs to the class ofproblems never encountered before. Newton, Euler, Lagrange and Hamilton wouldhave never imagined that the innovative and elegant tools they developed would, infuture, assist in resolving problems of space platforms that never appeared in theirwildest of dreams.1.4 Scope of the InvestigationThe enormous complexity of the dynamics associated with the space station basedMSS cannot be understated. The dynamics of the flexible Station and its intricatecoupling with the flexible manipulator presents a formidable task in mathematicalmodeling. Orbital effects, imposed disturbances, and the stringent performance criteria set for the Station-SSRMS system, make evaluation of its response and controla challenge of daunting proportion. This investigation aims at taking a modest stepin that direction.The nature of this study is necessarily theoretical. Even a modest facility aimedat simulation of the environment can be afforded only at the national level. Moe importantly, as pointed out before, it is generally agreed that flexibility effects and their10control cannot be evaluated with confidence on ground. Validity of numerous paperson dynamics and control of flexible satellites can only be assessed through carefullyplanned on-orbit experiments. A modest beginning to that end took place in Septernber 1985 through the NASA/Lockheed Solar Array Flight Experiment (SAFE). Vibration frequencies, modes and structural damping of a lOift (33m) solar array, deployedfrom the Space Shuttle were measured using two video cameras. A more elaborate experiment by NASA called CSI (Control Structure Interaction) aims at dynamics andcontrol of a flexible beam-type structures during slewing maneuvers. It is scheduledto take place onboard the Shuttle in the mid nineties.The ultimate goal is to provide insight into the dynamics and control of a largeclass of orbiting, mobile, flexible manipulators supported by highly flexible platforms.A major hurdle (but necessary stepping stone) to this end is the formulation of theproblem, i.e. derivation of the equations of motion governing dynamics of this class ofsystems. The main thrust of this thesis then, is the development of a relatively generalLagrangian formulation of the equations of motion for a dynamical system comprisedof an arbitrary number of flexible bodies connected in a chain-link configuration(Fig. 1-3).Salient features of the formulation may be summarized as follows:• The formulation accounts for an arbitrary number of flexible bodies in achain-link configuration.• The translation and rotation (as well as position and orientation) of eachbody is specified w.r.t. the preceding body.• Joint degrees of freedom at body hinges (superimposed on specified rotations)enable modeling of the joint compliance and dissipation, and permit application of a variety of manipulator control strategies.111th hingeFigure 1-3 System description with reference frames and position vectors.2body 11(•e///inertialframecenter of force\of massbody iorbitaJ frame\\.cbody—I lb— I-12• The flexible bodies are modeled as uniform Timoshenko beams (restricted toEuler-Bernoulli beams in the computer code).• The formulation accounts for the transverse, axial, and torsional deformationsof the constituent members comprising the system.• All deformations are discretized using an arbitrary number of assumed modesin conjunction with time dependent generalized coordinates.• The formulation accounts for the effect of gravity gradient, eccentricity of theorbit, transient system inertias, and shift in the system center of mass.• External forces attributed to environmental effects (solar radiation pressure,aerodynamic drag, Earth’s magnetic field), surface contact between bodies,structural damping, etc., can be incorporated through generalized forces.The governing equations of motion are highly nonlinear, nonautonomous andcoupled — necessitating a numerical approach to their solution. The equations arecoded for numerical integration, and some typical results, indicating influence of thesystem parameters, are presented to help establish trends. The emphasis throughoutis on the methodology of approach and understanding at the fundamental level ratherthan compilation of response data useful in the system design. Of course, with theformulation in hand and the computer code operational, such compilation of usefulinformation through a judiciously planned parametric study can be obtained quitereadily. The thesis ends with some concluding remarks and recommendations forfuture study.132. FORMULATION OF THE PROBLEM2.1 Preliminary RemarksIn this chapter, the methodology of approach to the problem is described. First,the system geometry is considered; frames of reference and degrees of freedom aredefined. Next, the kinematics of the system are established, providing a basis fordevelopment of the kinetic energy. Contributions to the potential energy due tothe gravity gradient field and the elastic character of the system are assessed, andgoverning equations of motion are derived using the classical Lagrangian procedure.2.2 Lagrangian ApproachThe energy based Lagrangian approach requires evaluation of the kinetic andpotential energies of the system. The kinetic energy can be obtained in a straightforward manner provided the position and velocity of an arbitrary mass element can beexpressed with respect to an inertial reference.If the position of a mass element dM w.r.t. the inertial frame is denoted by andits velocity by (Fig. 2-1), then the kinetic energy attributed to the translationalmotion of the system mass M w.r.t. the inertial frame can be written asTTRAN2’M. dMHowever, for a system of interconnected flexible bodies in orbit undergoing translational and slewing maneuvers, identification of the spatial orientation of a masselement, forming a part of an arbitrary body, is a bit involved. Before the positionof an elemental mass w.r.t. the inertial frame can be determined, it is necessary toestablish intermediate frames of reference and system degrees of freedom. To that14Figure 2-1 The position vector locates a system elemental mass dM w.r.t.the inertial frame.Rinertial frame15end, a procedure for arriving at the th body frame from the inertial frame must beoutlined. Finally, location of the element mass of the ith body w.r.t. the framehas to be detailed.2.2.1 Reference framesConsider a system of elastic bodies connected arbitrarily to form a chain-typegeometry as shown in Fig. 1-3.There are four distinct reference frames utilized for this problem: inertial, orbital,system, and body.The inertial frame is located at the center of force.The orbital frame has its origin locatçd at the system c.m. and is oriented suchthat its axes (x01, x02, are directed along the local horizontal, orbit normal andlocal vertical axes, respectively.The system frame also has its origin situated at the system center of mass (c.m.).The orientation of the system, frame w.r.t. the orbital frame is determined by apitch, roll, yaw rotational sequence, associated with the libration angles. Thus, thesystem frame, constrained to remain parallel to body frame 1, describes the librationalresponse of the system.The body frame, as the name suggests, refers to a reference frame fixed to a body.The ith body frame (with axes x1, x13) is attached to body i at an arbitrarylocation. It serves as a reference frame for measurement of elastic deformations ofthe body.2.2.2 Specified and generalized coordinatesThe radius vector and true anomaly 9 are treated as specified coordinates. In16general, orbital motion of the system is essentially unaffected by libration, vibrationand maneuvers of the system, unless the system dimensions are comparable to theposition vector R.The librational generalized coordinate vector () describes orientation of the systern frame w.r.t. the orbital frame. Here, pitch, roll, and yaw represent rotations ofthe system frame about the orbit normal, local horizontal, and local vertical axes,respectively.(ibj’) (pitch’)= rollI, yaw)Body i + 1 is attached to the ith body at a hinge point (Fig. 2-2).The location of the hinge point on the undeformed th body is specified by (),where the hinge distance along axis xj is given by h:,,I h11 ‘1i h13jOrientation of body i + 1 w.r.t. the body is specified by a set of angles ,where /3, is a specified angle of rotation about axis x,,I,/33)To model joint compliance (freeplay) at the jth hinge, joint generalized coordinate angles (v,) are superimposed on the specified slew position . Here, y,, is ageneralized coordinate representing the free angle of rotation taken about axis x,,,I ‘77i2I.The flexible bodies are modelled as Timoshenko beams. Transverse and axial17xi’/Yi2(generalized)framei1’//////di001312(specified)SNhhingeFigure2-2Planarviewoftheundeformedbody.beam displacements of the th body are denoted by (xj, t) whilst deformationsattributed to rotary inertia and torsional effects are given by t) . The deformations are discretized using assumed modes in conjunction with the time dependentgeneralized coordinates (t) and X(t’), respectively:(z3,t) = fm.(())(z3,t) = fm.(X(t))2.2.3 From the inertial frame to the i’ body frameThe inertial frame is located at the center of force (Fig. 1-3). The radius vectorand true anomaly 8 define the location of the system center of mass w.r.t. the inertialframe. The system is free to negotiate an arbitrary, specified trajectory. The orbitaland system frames are located at the system c.m. The orientation of the orbital frameis as described earlier. A modified Euler rotation sequence (2-1-3) of the librationangles (otherwise described as pitch-roll-yaw) identifies the orientation of the systemframe w.r.t. the orbital frame.The location of body frame 1 w.r.t. the system c.m. is given by the vector.Frame 1 is constrained to remain parallel to the system frame. Consider a set ofbody frames as indicated in the figure. Frame 1 is attached to body 1 at an arbitrarylocation while the remaining frames are placed at the joints and attached to thecorresponding bodies. For example, the frame is fixed to the body and has itsorigin located at the hinge connecting body i — 1 to the jth body.Position vector locates frame i + 1 w.r.t the frame and is a superpositionof , the specified location of the th hinge on the rigid th body, and (h), thedeflection of the jth hinge due to deformation of the body (Fig. 2-3),19xi’h)(&i(h):i+1=:0(712frameia1+i(x13=h3)Figure2-3Link(beam)intransverseplanardeformation.— + i(J.The modified Euler angles describe orientation of frame i + 1 w.r.t. the ith frame.The matrices associated with rotations of body i+ 1 w.r.t. the th body are designatedby [C] . Here [C3_2]represents a rotation matrix at the ith hinge due to therotation of the th body w.r.t. the frame (i.e. rotary inertia and torsion) evaluatedat the th hinge. Rotation matrices [C3_1]and [C3:] are, respectively, due to specifiedslew and joint d.o.f motion 7 at the ith hinge. Appendix A contains details ofrotation matrices [C].It is convenient to introduce a rotation matrix [T:] to facilitate projection of avector from the th body frame onto the system frame ([T1] = unit matrix),3(i—1)[T]= H [C3].j= 1The mass of the jt1 body is denoted by m, and an elemental mass of this bodyis represented by dm. The sum of masses from the ith body onward to the last (ith)body is given by= m3j=iThe total mass of the system is denoted byM=n1.The time histories for the maneuvers specified for 7iE and are detailed in Appendix B.2.2.4 Position and deformation vectors for the element mass dmFig. 2-4 illustrates the deformation of the i beam. The position vector locates21mass element dm w.r.t. the jth frame. It is a superposition of , the position ofdm when body i is undeformed, and , the deflection of drn., due to transverse andaxial deformations,(x3,t) = +(x3,t).Fig. 2-5 illustrates the planar and axial deformations of the i’ beam. Deflectionsof the mass element dm1 w.r.t. frame i from its nominal equilibrium position inthe j direction are represented as series of admissible functions (assumed modes inparticular) and generalized coordinates:flu.,(x, t) = (x, )S,k (t) , (j = 1,2);k=1fluu,(xu3, t) = &jk(X:3)Sijk(t) + &3fa , (i = 3);where 3fs is an axial deflection in direction 3 accounting for the foreshorteningeffect,1 f3f(d2 (d2’\23fa= J + _) fdZSimilarly, rotations of dm: w.r.t. frame i about the th direction (i.e. deformationdue to rotary inertia and torsion) are given by01:cz(x13,t) = 77uk(zu3).)i,k(t) , (j = 1,2,3)Note that n13 and o1 are the number of admissible functions taken to representthe deformation of the jth body in the j’ direction. The deflection and rotation ofthe th body are denoted by and ,, respectively, and can be written in vector form22Positionvectorlocateselementmassdw.r.t.thethframe.(‘ Jframeibodyidrr—————————Figure2-4xil(i1 0144fN3+i3bodyiframeidm1ci___-.-z—I1-----1:--————--P13a12(Xis,Figure2-5Alinkundergoingplanartransverseaswellasaxialdeformations.as follows:1==[jj +I.. E13 )(cz )= a2=1. cz3)The matrix [J contains the admissible functions used to describe the lateral andaxial vibration of the th body. Similarly, the matrix [j is a collection of admissiblefunctions describing the rotary inertia and torsional motion of the i’ body. Detailsof [jj, [rjj, , and 5 are listed in Appendix C.The matrix [ci] accounts for the foreshortening effect of the th beam. Literaturereview suggests that, in general, for most situations the effect of foreshortening onthe system dynamics is insignificant and hence this effect is not explored in thepresent study. The focus is on assessing influence of more dominant parameters andtheir interaction including librational and vibrational degrees of freedom, slewing aridtranslational maneuvers, initial conditions, etc.2.2.5 Assumptions concerning mode shapes as admissible functionsA daunting task awaits the intrepid engineer intent on obtaining the exact modeshapes for the flexible beams. Slewing and translational maneuvers of the MSS aswell as flexibility of the interconnected bodies necessitates consideration of transientboundary conditions. For example, the lower arm of the MSS, modelled as a beam,would have a slewing, translating base whilst its payload would consist of a timedependent configuration of flexible bodies. Determination of the exact mode shapes,if not impossible, would most certainly be difficult, and quite probably demand considerable effort and CPU time.Faced with the dire prospects of this scenario, alternatives to the exact modes25must be weighed. A major consideration is the CPU cost and one way to avoidrecalculating exact mode shapes for every time step is to remove the time dependencyof the mode shapes.To achieve this end, the space station and manipulator link deformations aremodelled using free-free and cantilever beam modes, respectively. The mode shapesare listed in Appendix D.2.2.6 System shifting center of massVibration of flexible components of the station, joint freeplay, or maneuvers ofthe SSRMS will cause a shift in the location of the system’s center of mass. Thelocation of frame 1 w.r.t. the system c.m. is given by the vector(-).The vector 5, in the preceding expression denotes location of the c.m. of body jw.r.t. frame j, and is defined by the following expression,f , dm, = J (• + ) dra,= m,bj.It should be noted that inclusion of the shift in the system center of mass complicates the kinematics enormously. Development of is detailed in Appendix E.2.2.7 Kinematic expression for position and velocityThe mathematical expression for the position of an arbitrarily located mass element in the inertial frame is given by (Fig. 1-3)26where locates the system center of mass w.r.t. the inertial frame, and F representsthe position vector to the mass element on the th body with respect to the systemcenter of mass.The vector can be written as= + ( [Tj+1)+The vector locates the mass element on the body w.r.t. frame 1The use of Lagrange’s equations of motion requires expression of the velocity ofan arbitrary mass element,—where: = orbital velocity of the system;= librational velocity;8.i= time rate change of w.r.t. the system frame.The librational velocity can be written as= {P]+T8á,where [P,,bj is a matrix projecting on the system frame and is a vector projecting8 on the system frame:[c2s3 c3 0] — I c23 ‘1[Pm] = i C23 —S3 0 ; 19 = c23L—s2 0 1J 2 )Now,27—8t — 8twhere:= ()(l + [Tj)m+[7*+i)n÷i);= [7})+ + [TjIt should be noted that:3(i—1) ,(—[T])= [ (fJ[C])j x [T],3(i—1) j(-[TJ)+1= [ (H [Ck])jj x [T]i;j=1 k=13(i—1) j([T1}), = { (fT [C])j x [T].j=1 k=1The angular velocities about the th hinge are given by:= [P3i_2ji(7:) ;= [P3:_i I= [P3]1where [P3_2j, [P31_1]and [.P3] represent projection matrices (Appendix A).The time derivatives of and are:+ [S][a1}28+i i + [&()] + [6][o()jhence,mJ=Ji+1where: [A} = fm[& 1dm: ; [B:] fm[0u}dmi[Dj= [)] ; [G] =The need may arise for a skew symmetric matrix [] of a generic vector Y. ForV2V3)we obtain [0 V3 V21[v]=1 v3 0 —v1L—v2 V1 02.3 Kinetic EnergyThe velocity of a mass element translating w.r.t. the inertial frame is given by .The total kinetic energy in translation can be obtained by integrating over the entiresystem,TTRAN2’M• dM.Substituting for 1?, the kinetic expression takes the form1 f . — c9.t —TTN 21M8a= 21M F).(x )+ (.)29Recalling that F locates a mass element w.r.t. the system center of mass,Taking the time rate of change of the above equation (w.r.t. the system frame)results in--J dM=J ZdM=O.atM MötConsequently, the kinetic energy expression reduces toTTRAN= 2’M*)dM + .J(x ) ( x )dMThe component kinetic energies can now be identified. The energy associatedwith the orbital motion of the system about, the center of force is= 2’M•)dM=The librational component of the kinetic energy can be rewritten as= 2’M=The energy associated with motion relative to the system frame (MSS maneuvers/vibration, platform vibration) is given byTSF2fM(atat1 v’ I (0, 8=--) dm1,30and the energy due to coupling of libration with the system angular momentumcan be rewritten asTLsF=cz.J (xj.!)dmrn=•HsF.The development of energies TSF, TLSF, and TL are detailed in Appendices F, J,and N respectively.Contribution to the kinetic energy from rotational motions of an element massdrn, w.r.t. frame i (i.e. rotary inertia and torsion) can be written asTROT = J . [Iç ] ZT dli=1 1iwhere: angular velocity of mass element dmwith respect to the body frame;[Ia. } inertia diadic per unit length ofmass element dm:.The development of TROT is detailed in Appendix 0.The total kinetic energy of the system, being equal to the sum of the energiesattributed to translational and rotational motions of the element masses, is now givenbyT = TTRAN + TROT2.4 Potential Energy31• Contribution to the system potential energy arises from three sources: gravitational field; elasticity of the joints; and strain energy due to flexibility of the bodies;U=UL+UJ+Us.2.4.1 Gravitational Potential EnergyThe gravitational contribution (IL is given by+4i[Ij7rc,where: p. = universal gravitational constant;= direction cosines for unit vector along F w.r.t. the system frame.Details of the development of UL are given in Appendix P.2.4.2 Joint Potential EnergyTo model joint compliance at the ith hinge about the jt direction, torsionalspring constants k, are considered. As the potential energy associated with a linearelastic spring is conventionally described in a quadratic form, the joint energy storedat the tfr hinge can be written ask,Hence, the potential energy associated with joint compliance for a system of 1bodies (1 — 1 joints) is given by1=1 jzl32with: k:, = torsional spring constant for the th joint about the th direction;= rotational d.o.f. for the jth joint about the th direction.2.4.3 Beam Strain EnergyThe flexible bodies are modeled as Euler-Bernoulli beams. The beams’ materialis assumed to be linear elastic, isotropic and homogeneous. The beams are assumedto be slender with uniform cross-section. The strain energy arises from beam deformation attributed to bending, torsion, and axial displacement,U = (U + U + Us).Strain energy in bendingThe partial differential equation for an isolated Euler-Bernoulli beam of constantcross-section with no external forces, can be written as52 52 d2•5 + dt2=where mass per unit length of the body, and the index p denotes the directionperpendicular to the th direction (e.g. if j = 1, then p = 2).Beam deflections are written as a series of assumed modes and generalized coordinatesfluEu,(x, t) = &jk (zu3)82 (t)where 8:jJ and 1’u, satisfy the relations5i3k + Wk 6ijk = 0;3382 F ô2tjk 1 22 i — 1LiWjk &jk = 0.LJX3 IJX:3The strain energy for the i body can be written1 ‘EITT I_____Jbi— J0 :jwhere r, is the radius of curvature of the deformed geometry.For small defiections, an approximation can be made that1(52E)2So now,ii 8222Ub.= j E:I:p( 8x3 dx3Integrating, one obtains_____1i 8“1i2 = E1I, 2 — E,‘ 2 1uX13 (JX13 0 i.iX13 (1X13 0l 82 82g..+j jj dx3.For a cantilever beam, shear and bending moment are zero at the free end, whiledeflection and slope are zero at the fixed end. For a free-free beam, the shear andbending moment are zero at both ends. In either case,1 1: 82 82=2 8X3 (EI s;’) dX.Substituting for,in the above expression,34=3J1iqijm ätjkäi;m dx3flu fiji.= I ijk) ijm &ijk Sijm dx13.k1 m1 0Applying the orthogonality condition:J jk ijm dx3 = (k = m)= 0; (k#m).The strain energy associated with transverse deformation in the th direction forthe th beam can be finally written asa..U. = jk j ijk dx3 Skfljj= 1 Lujk ii 3ijlc’flj=k1Wijk 6ijkwhere:= mass of th body;=kth nondimensional natural frequency in transverse deformation.The partial derivative of the strain energy in bending w.r.t. the generalized35coordinate associated with the kth mode is given byau1 27fl Li) 6ijkStrain energy in torsionThe differential equation for a beam in torsion can be written as82a3(x1,t) 823(x,t)8x3 — c2 8t2where c2 = , and p. = mass per unit volume for the body.Discretizing the beam deflection with a series of assumed modes and generalizedcoordinates0i3a3(x,t) = 773k(X:3) .)3k(t)k= 1and substituting for a, the differential equation becomes1(v 7i3k’ (L A3’‘i3k =‘) ?7Rearranging the equation, and equating both sides to a constant,(O2i73k) 1 — 1 1 (82 •k — (W3k 27k — c2 3k at2 / — “ c /results in two ordinary differential equations:82Th3k / Wj3k .. 2a 2 + k—) 73k = 0x13 CA,3k 2+ 3k’3k =36Now, the torsional strain energy for the th beam can be written asGJ 2U1 = f dx32Evaluating the integral results inI 5a13 2 Iti f ö2czdx13 = 3I — j j3 ( 8xA free-free beam has zero torque at each free end. A cantilever beam has zerodeflection at the fixed end and zero torque at the free end. In both cases,G1J 82= J _a( 2 jdx32 o 8z3Substituting for c3 in U results in0i31 17:3m( —_____Ut.= 2 o k=1 m=1 8x2j dx i3k i3mi3Q3 2— 1 i3m (wk) 7i3k dz3 3ki3m—2 k=lm=1 CFrom orthogonality of the assumed mode shapes:J 73k1i m dx3 = -, (k = m);(km);hence,,-‘0:3u.= (Wi)2c i3k1Thus the strain energy in torsion can be written asi3= mtJ: j W‘‘3k4Ak=137where:m: = mass of beam;J = polar moment of inertia for beam cross-section;A1 = cross sectional area;Wi3k = kt nondimensional natural frequency in torsion.The partial derivative of the strain energy in torsion w.r.t. the generalized coordinaterepresenting the beam rotation ‘j3k, isau1 m1J 2=c A W13k “3kUA13kStrain energy in axial deformationThe axial strain energy is given byE1A f’ia432U01= 2 Jodx13and using the procedure similar to that for torsion, one obtainsfl’3U0 = 3k 3j3kwhere:m, mass of the th body;= ktJnondimensional natural frequency in axial deformation.The partial derivative of the strain energy in axial deformation of body i w.r.t. the38generalized coordinate 3k is8U0.__m 2— Wj‘:3kVO:3k2.5 Equations of MotionThe Lagrangian procedure in conjunction with the kinetic and potential energiesgivesdr0T 67’ 8(1v—) ——-‘----=1where q, and Q, represent the ith generalized coordinate and force, respectively. Herethe generalized coordinates are the degrees of freedom associated with the librationalmotion; free joint motion; transverse and axial vibration; rotary inertia and torsion.For a system of 1 bodies, the generalized coordinate vector takes the formwhere:= libration vector;= joint d.o.f. vector for the th hinge;= deflection d.o.f. vector for the i’ body;= rotational d.o.f. vector for the jth body.Substituting for the kinetic and potential energies, and isolating the second derivative terms results in the following form for the equations of motion39where:{R } T = (, ‘ •‘ 1’ ‘‘•• ,‘ 1 ‘2’ ,‘A1);{ Q} = vector of generalized forces.For a system of 1 bodies, the mass matrix [M] takes on the form:m,1 ... rn,,1,711 m,,1 ... m,,bs1 ... rn,1m7171 . . . m7171 m71s1 . . . m7151 m,1, . . .m711 j1 m711 . . . m711 rn711 A1 •mE1 rn mE1 flL,mS1E ms1A . . . m1)mA1A . . . rn),1).rnAAThe vector components of {Rq} are defined as follows:T=,rem —L(T — U);R7. = R7.—£7 (T — U);=,rem —L(T — U);R). = ,rem — LA ( T — U).The partial derivative operator 1q is defined below:I 1’i i I 87t1A A= ‘I 2 I ‘ = 1 87i2I I8‘ 8l,&3) ‘ 873The partial derivative operators L and are defined in Appendix C. It maybe pointed out that partial derivatives of the energies, Lq(T — U), were obtainedthrough computer coding (mere application of the product rule and chain rule), and40hence are not explicitly listed. Elements of [M] and {Rq} are respectively detailedin Appendices Q and R.The equations presented in this thesis are in the dimensional form (i.e. units ofmass, length, and time). Should one desire to nondimensionalize these equations (andconvert the independent variable from time to true anomaly), the following relationswould prove useful:= q’ Ô;= q” Ô2 + q’ ö;6 — —2e sin682 — 1 + e cos61r82 — 1 + ecos8A word concerning the damping would be appropriate here. It is a complex parameter for a space structure with contributions arising from several sources: viscous,structural, free molecular and others. Accurate model of the structural damping hasbeen a challenge that still remains unresolved. The present formulation can easilyincorporate damping as a generalized force when the desired model is made available.In light of the uncertainty, the damping is purposely not included here.2.6 Computational ConsiderationsAs stated earlier, the governing equations of motion for the system are highlynonlinear, nonautonomous and coupled. As can be expected, they are not amenableto any known closed-form solution. Therefore, one is forced to turn to numericalmeans. An I.M.S.L. routine called DGEAR, which is based on the forward differencemethod (Adam’s method5354),was used to that end. The package requires that the41equations of motion be formatted as a set of first order differential equations,=With the initial state , as input, the routine numerically integrates the equations.As the governing equations of motion for the present system under study are secondorder differential equations (for the state ), a simple transformation of the variablessatisfies this requirement:iqjA flow chart outlining the important aspects of the computer code is presentedin Figure 2-6.The program was implemented using a Sun Sparc Station 2. A typical set ofresponse results for given initial conditions took anywhere from 1 hour to 1 week torun, depending upon the configuration complexity and the duration of the simulation.Of course, in the industrial setting, availability of a supercomputer may alleviate theproblem of computational time. However, there is always a scope for refining the codeto make it more efficient.42equationsblock1outputstate\fincrementtime tyes,stopprogramFigure 2-6 Flow chart of computer code.initial block initial blockintegrationsubroutine-input i.c. for state-input parametersfor maneuvers-input body masses,freq., geometries-baic. mode shapefunctionsequations blockno- caic. spec. coord.-calc. systemconfiguration-caic. partial deny.of energies T &-caic. time deny.-assemble massmatrix [MJ-assemble vector Rq-arrange equationsin the form= f(x)433. RESULTS AND DISCUSSION3.1 Preliminary RemarksThe objective here is not to obtain results aimed at design of a system but toillustrate versatility of the formulation. To reduce computational effort, responseplots are often limited to a fraction of an orbit. In such cases, conclusions cannot bedrawn because trends are not yet fully established. The simulation runs were carriedout for circular orbit (orbital period of 100 minutes) with the platform positionedin the gravity gradient orientation. All maneuvers were specified with a cubic timehistory and a 1-2-3 modified Eulerian rotation sequence was employed to describerelative body rotations, unless stated otherwise. All rigid members were modeledas solid cylinders. Flexible bodies were treated as Euler-Bernoulli beams made ofisotropic and homogeneous material (Poisson’s ratio of 0.3). The beams are assumedto be slender with uniform circular cross-section. Free-free mode shapes were usedfor the platform, whilst cantilever modes were utilized for all other flexible members.3.2 Verification of the CodeTo assess validity of the code, particular cases studied by Chan5’ and Ng57 wereselected for comparison. The data used for comparison with Chan’s planar studyare shown in Fig. 3-1, with response results presented in Fig. 3-2. The arms areinitially normal to the main truss and go through a rather severe, combined, andrapid maneuver of 30m in translation and 90° rotation in 60s. The results correlaterather well.The data for the out-of-plane study conducted by Ng are presented in Fig. 3-3.To be consistent, here a 2-1-3 rotation sequence is employed for the relative rotations44payload (body 4)lower arm (body 3)elbow (joint 2)upper arm (body 2)shoulder (joint 1)platform (body 1)initial conditionsall i.c. zeromaneuversh13= 0 to 3Cm in 60 Sp12= 90 to Odeg in 60 s(sine-ramp profiTefor both maneuvers)m l d {h}T {}T(kg) (m) (m) (m) (deg)1 213,440 115.4 5.162 (0,0,) (0,*,0)2 1600 7.5 0.336 (0,0,7.5) (0,0,0)3 1600 7.5 0.336 (0,0,7.5) (0,0,0)4 3200 0.1 0.100fundamental freq. inlateral vibrationnumber of modeslateral11axial00torsion00body1234rigid body0.3183 Hz0.31 83 Hzrigid body• joint stiffness (kg.m2/s) coordinate_presentk2 Yji Y12 Y131 00 120,000 00 no yes no2 00 120,000 0 no yes no3 00 00 00 no no noFigure 3-1 Data used in comparisonparticular case.with Chan’s study of planar maneuver, aI local verticaldirect ion final arm positionlocal horizontaldirect ion12x43x2145pitch14111.00.0-1.00.050.00-0.05upperarmpresentChan0.000.02Figure3-2AcomparisonofresponseresultsasobtainedbythepresentmethodwiththosereportedbyChan.Thecorrelationisindeedexcellent.deg0.005.00.02shoulder0.0-5.0 0.005.00.02‘Y220.0-5.0 0.000.050.00-0.05orbits0.02lowerarm0.00orbits0.02of the body frames. A comparison of responses to an out-of-plane maneuver as givenby the two codes is depicted in Fig. 3-4. Observation reveals excellent agreementin both librational and vibrational degrees of freedom. It may be pointed out thatChan as well as Ng checked their own codes with other available data, conservationof energy, etc. Hence the correlation between their results and those obtained by thepresent approach helped establish a high level of confidence in its accuracy.3.3 Free ResponseFigure 3-5 contains the data used in a free planar disturbance study with theresponse results presented in Figures 3-6(a) and 3-6(b). The joint stiffness valueswere arbitrarily chosen such that the resultant frequency of oscillation of a singlemanipulator link about its end hinge coincides with the link’s fundamental frequencyof 0.551 Hz., i.e.(m2_ 2=3 )(2cf)= (500 (8.8)2) (2(0.551))2154, 780 kgm2/sThe tip of the upper arm is subjected to a nondimensional, initial disturbance of0.01 (541 = 0.01, i.e. 2% of the arm length). Note, this induces pitch libration (z,bi)of the platform, as well as inpiane platform vibration (5), thus suggesting a strongcoupling between the platform response and the manipulator vibration. The presenceof a payload also adds to the coupling (Figure 3-6a). Note, both the beam and jointvibrations show, in general, response at a characteristic low frequency ( 0.43 Hz)with high frequency modulations superimposed on it.47final positionof lower armlocal verticaldirect ionlocal horizontaidirect ionrn d {h1}T {13}T(kg) (m) (m) (m) (deg)1 240,120 115.4 5.162 (0,0,0) (0,90,0)2 1800 7.5 0.336 (0,0,7.5) (*9QØ)3 1800 7.5 0.336fundamental freq. inlateral vibrationnumber of modeslateral1Iaxial00torsion00body. joint stiffness (kg.m2/s) coordinate_presentk2 k3 y11 Y21 oo 00 no no no2 —- 00 no no no123rigid body0.0266 Hz0.0266 HzFigure 3-3 Data used in comparison with Ng’s study.initial positionof lower arm L21 slew maneuverswings arm out of plane(about axis x31)initial conditionsall i.c. zeromaneuvers21 0 to -1 8odeg in 300 S(sine-ramp profilefor rñaneuver)platform (body 1)48dogpitch0.20‘I’i0.050.00-0.05‘V20.050.00-0.05I mnvr.j,ond0.00orbits0.250.00+0.20mFigure3-4Comparisonofresponseresults,asobtainedbyNgandthepresentmethod,foranout-of-planeslewmaneuveroftheupperarmthrough18O.Note,thetwosetsofresultsarevirtuallyidentical.0.00-0.200.000.25dogroll0.000.25t21 t22 +0.20L310.00-0.200.000.25‘I’35.00.0-5.00.000.25mlowerarm0.25lowerarm0.00-0.200.00orbits0.25c’j0x535local verticaldirectionlocal horizontal initial conditionsdirectionmobile transporter(body 2)platform (body 1)641_0.01maneuversnone. rn I d {h}T {13}T(kg) (m) (m) (m) (deg)1 154,500 115.0 5.0 (2.5,0,0) (090,0)2 2000 2.0 6.0 (0,0,2) (0,0,0)3 500 8.8 0.5 (0,0,8.8) (0,0,0)4 500 8.8 0,5 (0,0,8.8) (0,0,0)5 940 5.1 1.0fundamental freq. inlateral vibrationbody123450.193 Hzrigid body0.551 Hz0.551 Hzrigid bodynumber of modeslateral111axial000torsion000. joint stiffness (kg.m2/s) coordinate_presentk1 k3 1 ‘Y2 ‘V31 oo cc no no no2 00 154,780 cc no yes no3 cc 154,780 cc no yes no4 cc 154,780 no yes noFigure 3-5 Data for initial planar disturbance.500.004dogpitchdogelbowWi1321.0-0.002 0.0000.0050.0000.005platformlowerarm::A1VA,\AAAAA/v\,-2E-7-0.0.0000.0050.0000.005dogshoulderdogwrist7221.0142tO0.0000.0050.000orbits0.005upperarm0.01-0.00-0.01 0.000orbits0.005Figure3-6Responsetoaplanardisturbance:(a)timehistoriesofgeneralizedcoordinates.mplatformupperarmm5E-52E-7-5E-5t310.1 0.0-0.1t410.1 0.0-0.115310.20.0-0.2t33,10.20.0-0.2CJ1L.30.0000.005mupperarmt43,10.0000.0050.0000.005mlower arm0.0000.0000.005orbits0.0050.000orbits0.005Figure3-6Responsetoaplanardisturbance:(b)tipdeflectionsw.r.t.localandplatformframes.Figure 3-6(b) shows corresponding tip deflections at the platform and the arms.Two sets of results using different reference frames are presented here. t11, t31, t41show the response with respect to local body frames 1,3 and 4 attached to the platform, upper arm and lower arm, respectively. On the other hand, three plots on theright refer to the tip positions with respect to the platform based observer. As expected, the platform motion due to its large inertia is negligible. However, the armsand the payload display significant deflections.Corresponding data for the same out-of-plane disturbance are presented in Figure 3-7. Note, initially the lower arm is no longer aligned with the upper arm as inthe previous case but is inclined at an angle of 450 System response is illustrated inFigure 3-8.The initial out-of-plane deflection of the lower arm(S42 = 0.01) gives rise to threedimensional librations and vibrations of the platform. Note, the inplane vibrations ofthe links are also excited (31, t41). Furthermore, the configuration induces torsionalvibration (r33, r43) of the manipulator links. It may be pointed out that higherfrequency modulations for the generalized coordinates would have been encounteredin absence of the payload. It is of interest to recognize that the platform motionat around 0.27 Hz is reasonably close to its fundamental frequency of 0.193 Hz. Ofcourse, the high frequency modulations, when present, are due to flexibility of thelinks as mentioned before.3.4 Effect of Slewing Maneuvers3.4.1 Rigid manipulator and payload with wrist maneuverThe question may arise — Is the platform motion influenced by the flexible natureof the manipulator?. Can the platform response to the manipulator maneuvers be53local verticaldirect ionlocal horizontaidirect ionplatform (body 1)initial conditions342_0°1maneuversnone. m1 I d {h1}T {13}T(kg) (m) (m) (m) (deg)1 154,500 115.0 5.0 (2.5,0,0) (0,90,0)2 2000 2.0 6.0 (0,0,2) (0,0,0)3 500 8.8 0.5 (0,0,8.8) (0,45,0)4 500 8.8 0.5 (0,0,8.8) (0,0,0)5 940 5.1 1.0body2345fundamental freq. inlateral vibration0.193 Hzrigid body0.551 Hz0.551 Hzrigid bodynumber of modeslateral111axial000torsion111. joint stiffness (kg.m2/s) coordinate_presentI k1 k2 Y11 1i2 ‘Y31 00 00 Co no no no2 154,780 154,780 00 yes yes no3 00 154,780 00 no yes no4 154,780 154,780 154,780 yes yes yesFigure 3-7 System data for initial out-of-plane disturbance.;. .— _oC —.5 Eii. -lower arm(body 4)C0000payload(body 5)(m.t)54platform2E-41E-li2E-4 0.5“30.0-0.5.5E-4L11OEO-5E- 4platformFigure3-8Responsetoinitialout-of-planedisturbance:(a)librationalresponseaswellasplatformtipandjointdeflectionsw.r.t.local frames.degpitch2 E-3t12OEO-2E-3m0.00000.0008roll5E-4‘V2OEO-5E-40’ 01r130.00080.00001E-16degplatform-2 E-23-1E-16 0.00000.00000.00080.00080.00000.00080.00000.00000.00081.07220.50.0-0.5-1.0orbitsdegshoulder0.00080.0000orbits0.0008upperarmlowerarmt310.10.0-0.1t320.10.0-0.1m0.10.0-0.1t410.00000.0008upperarm0.00000.0008C,’0)m0.00000.00080.10.0-0.1r33lowerarm-1.00.00000.0008doglowerarm0.00000.0008dogelbow1.00.0-1.0r431.00.0-1.01.00.0-1.00.00080.0000 dogwrist0.0000orbits0.00080.0000orbits0.0008Figure3-8Responsetoinitialout-of-planedisturbance:(b)tipandjoint3,4deflectionsw.r.t.localframes.wristplatform1.01420.0 1.01430.0-1.0mupperarm-1.0dog57.5579ti3,057.557557.5571mdog0.00000.00080.00000.00080’wrist13.06131013.0513.04 0.0000upperarm0.00000.0008mplatform0.2t32,0-0.2503L110-0.2506-0.2509•0.000L12,0 -0.002-0.0040.0008inupperarm0.00080.0000 mplatform0.00000.00000.00080.1t33,0-0.0-0.1orbits0.00080.0000orbits0.0008Figure3-8Responsetoinitialout-of-planedisturbance:(c)joint4rotationsaswell astippositionsintheorbitalframe.lowerarmpayload19.28m22.89mt410:::.“°::::0.00000.00080.00000.00080.30mlowerarmmpayloadt42,o::.t52,00.30000.00000.00080.00000.00084.16mlowerarmmpayloadt43,.t53Q9770.0000orbits0.00080.0000orbits0.0008Figure3-8Responsetoinitialout-of-planedisturbance:(d)tippositionsintheorbitalframe.attributed primarily to the rigid body motion of the manipulator system? If so, thenrigid modeling of the manipulator arms may suffice in the study of platform libration() and vibration (t11, t12,r13).To help assess the effect of the manipulator flexibility, it was thought appropriateto conduct a few simulation studies treating the manipulator and the payload rigid.Two different durations of 3s and 6s were considered for a 900 slewing maneuver atthe wrist about an axis parallel to the local vertical (yaw maneuver for the wrist). Thecorresponding system data are summarized in Figure 3-9. The manipulator is locatednear the tip of the platform, around 50 m from its center. As can be expected, thesignificant librational response is confined to yaw (b3 30, Figure 3-lOaj. The tipof the station goes through libratory excursion of around 3 cm with approximately0.05° torsional amplitude (r13). Though apparently small, these responses can affect performance of the platform based payloads. Figure 3-10(b) shows payload tiplocation in the orbital frame. This provides integral effect of the system librationand platform flexibility as observed from the orbital frame. Note, the tip undergoessignificant motion in the pitch, roll-axes plane, of around 4-5 m.Response results were also obtained for a slower 6s maneuver (Figure 3-lOc).Note, the librational response remains virtually unchanged. However, as expected, thevibratory response does show reduction, particularly in the torsional degree of freedom(r 0.02°). The payload tip position plots (Figure 3-lOd) remain essentially thesame as the dominant librational motion remains unaffected.3.4.2 Flexible manipulator and rigid payload with wrist maneuverFigure 3-11 contains the data for an out of plane slew maneuver at the manipulator wrist. Here, a payload is rotated 90 degrees in 6 seconds. System responseis illustrated in Figure 3-12. As can be expected, the out-of-plane slew maneuver,59body12345c -0 om1 I d {h}T {}T(kg) (m) (m) (m) (deg)1 154,500 115.0 5.0 (2.5,0,50) (0,90,0)2 2000 2.0 6.0 (00,2) (0,0,0)3 500 8.8 0.5 (0,0,8.8) (0,0,0)4 500 8.8 0.5 (0,0,8.8) (*,0,0)5 940 5.1 1.0fundamental freq. inlateral vibration0.193 Hzrigid bodyrigid bodyrigid bodyrigid bodyat eral1number of modesaxial0torsionj oint stiffness (kg.m2/s) coordinate p resentk3 Y1 Y2 Y31 00 00 no no no2 0O 00 00 no no no3 00 00 no no no4 00 00 00 no no noFigure 3-9 Data for wrist slew maneuver of rigid manipulator (3 or 6s maneuver).60E.2>. 0 0.2X5(=51lb. dW.52local verticaldirectionlocal horizontaldirectionplatform (body 1)side view of arm illustratingwrist slew maneuver(payload is rotated 90°)initial conditionsall i.c. zeromaneuversp41= 0 to 9odeg in 3 or 6sx13xlipitchplatformdog005m‘V1.0.05t12000-0.00-0.050.0000.0050.0000.005dorollplatform-0.000geg‘V2-0.005r130.1-0.010-0.0000.0050.0000.005yaw4dog0.0mplatform‘V3-0.1_0.0000.0050.0000.005platform005mplatformt110.1t120-0.0-0.00-01I•end-0.050.000orbits0.005OMOOorbits0.005Figure3-10Systemresponsetowristyawmaneuverwiththerigidmanipulator:(a)librationandplatformdeflections,3smaneuver,localframe1.payload53,050 45m0.000orbitspayload0.005locationofpayloadtipinorbitalframeFigure3-10Systemresponsetowristyawmaneuverwiththerigidmanipulator:(b)payloadtippositionintheorbitalframe,3smaneuver.27 22 0.0000mpayload52,0.4 0.0000.0050.005E c’Ja x G) 4-. ( C D 0 0 00-4222527coordinatex01,mj mnvr.j,ond-0.000-0.005-0.010 0.0000.05L12-0.00-0.050.005yawdegt 11,0pitch‘l’i0.050.00mplatform0.0000.0050.0000.005rolldegr130.10.0-0.1degplatform0)4C..)11133 2 0 0.0000.10.0-0.10.005mplatformmnvr.end0.0000.005platform-0.1 0.0000.0050.05mplatform12,0-0.00-0.050.000orbits0.0050.000orbits0.005Figure3-10Systemresponsetowristyawmaneuverwiththerigidmanipulator:(c)librationandplatformdeflections,6smaneuver,localframe1.Figure3-10Systemresponsetowristyawmaneuverwiththerigidmanipulator:(d)payloadtippositionintheorbitalframe,6smaneuver.mpayloadlocationofpayloadtipinorbitalframet51.o t52,o t53,o0.005payloadE 0 x a) C” C D 0 0 027 22 0.000oz:-4 0.000 m50 45 0.0000-40.005payload222527mnvr.endcoordinatex01,morbits0.005induces relatively large out-of-plane motion of the arms and joints (t32, t42, 7,The 8.8 metre links’ maximum defiections are approximately 0.5 metre (5.7% linklength; Figures 3-12a,b) which would result in a significant deviation at the elbow(t32,0 Figure 3-12c) and the grip (t42,0 Figure 3-12d). Note, the librational responseis now oscillatory in character with an amplitude of around 50 (Fig. 3-12a) in contrastto a steady state value of 3° in the rigid manipulator case.Of course, the net result of the manipulator and the platform flexibility is betterreflected in the orbital frame reference (Fig. 3-12d) where the payload tip positionshows significant deviations in the local horizontal (t51,0) and orbit normal (t52,0)directions. This clearly suggests the need for a control strategy.To summarize, the comparison of the librational responses for rigid and flexiblemanipulator cases suggests that rigid modelling of the manipulator system eliminatesoscillatory modulations of the ]ibration. Obviously, the elastic nature of the manipulator arm would exert considerable influence over the librational response. So faras the platform vibrations are concerned, the inpiane response (t11) in the two casesremain quite similar, however, there is some difference in the out-of-plane deflections(t12). Of course, as expected, the torsional resonse of the platform (r13) changesdrastically. It appears that the manipulator must be modelled as a flexible system inorder to obtain a credible response for the platform.3.4.3 Slewing maneuvers of the flexible armsFigure 3-13 presents the data for two simultaneous slew maneuvers (/3ii, /322)for a simplified model — the mobile base and payload are removed and the librationof the rigid platform is considered to be controlled, i.e. specified to be zero. Theresults for a 6s maneuver are shown in Figure 3-14. Deformation of the upper arm(body 2, t22) is rather significant — a lm tip deflection for an 8.8m beam. As well,65. m1 d {3}T(kg) (m) (m) (m) (deg)1 154,500 115.0 5.0 (2.5,0,50) (0,90,0)2 2000 2.0 6.0 (0,0,2) (0,0,0)3 500 8.8 0.5 (0,0,8.8) (0,0,0)4 500 8.8 0.5 (0,0,8.8) (*00)5 .940 5.1 1.0fundamental freq. in number of modesbody lateral vibration lateral axial torsion1 0.193Hz 1 0 12 rigid body ---..3 0.551Hz 1 0 14 0.551Hz 1 0 15 rigid body. joint stiffness (kg.m’/s’) coordinate_presentk3 ‘y11- Y12 Y131 00 00 no no no2 154,780 154,780 yes yes no3 00 154,780 00 no yes no4 154,780 154,780 154,780 yes yes yesFigure 3-11 Data for wrist slew maneuver of flexible manipulator with a rigidpayload, 6s maneuver.E E0 ,. -..2> 00 0.c. •—-x43 Ix41I.h.r.side view of arm illustratingwrist slew maneuver(payload is rotated 90°)xlilocal verticaldirectionlocal horizontal initial conditionsdirect ionall i.c. zeromaneuvers41 0 to 90deg in 6 splatform (body 1)x1366platform14110.04l20.00dogpitch0.0000.0030.03dogroll0.00-0.0314135 00.0000.00300000.0037210.03m0.00-0.03 0.0000.0030.03dogplatform-0.03 0.0000.003-3dogshoulder0.0000.003platformmnvr.endFigure3-12Systemresponsetowristyawmaneuverwiththeflexiblemanipulator:(a)librationalresponseaswellasplatformtipandjointdeflectionsw.r.t.local frames.t12 r130)0.00 3 010.030.00-0.030.047220.020.00-0.02-0.04shoulderdog0.000orbits0.0030.000orbits0.003lowerarm0.030.00-0.030.003upperarmdogdogelbowmnvr.endmupperarmm0.03t310.00-0.03t320.0000.0000.0030.000000.003lowerarm0.03r330.00-0.030.031410.00-0.030.50t420.00-0.500.03r430.00-0.03 1.00:50.0-0.5-1.00.0000.003doglowerarm0.0000.0030.0000.0000.003orbits0.0030.000orbitsFigure3-12Systemresponsetowristyawmaneuverwiththeflexiblemanipulator:(b)tipandjoint3,4deflectionsw.r.t.localframes.0.003wrist0.03dog56.50mplatform142t1300.00—-—_---.-———------.56.25-0.03_________________________________56.00__________________________0.0000.0030.0000.0030.03dogwrist13.25mupperarm143t3100.00______13.00-0.03____________________________12.75____________________________0.0000.0030.0000.003platformupperarm-0.20m1.00mtilo0.50-0.30-0.0______________________________0.0000.0030.0000.003platform49.00mupperarm0.05mt120t330-0.0048.75mnvr.-0.05end448.50____________________________0.000orbits0.0030.000orbits0.003Figure3-12Systemresponsetowristyawmaneuverwiththeflexiblemanipulator:(c)joint4rotationsaswellastippositionsintheorbitalframe.22.00t41,o21.7521.5049.00143048.7548.50lowerarmm51,00.00027 24 2142,0payloadlowerarm1.000.0030.0000.003-0.00 0.000mpayloadlowerarm0.003mmnvr.end052,0-2-449.00153048.7548.500.0000.0000.003mpayloadorbits0.0030.000orbits0.003Figure3-12Systemresponsetowristyawmaneuverwiththeflexiblemanipulator:(d) tippositionsintheorbitalframe.torsion of the arm (r23) reaches an angle of 10 degrees at the tip. To assess the effectsof arm flexibility, a similar simulation has been carried out for a rigid manipulator,and the tip location of the rigid arm (at 6=0.002 orbit) is shown in Figure 3.14(b).Observation reveals that arm flexibility results in the tip wandering from the rigidtip location by as much as 2in. It is clear that flexibility of the manipulator can haveconsiderable influence on the response, and should not be neglected.To reduce the deformation of the flexible manipulator arm, the maneuver periodwas tripled from 6 to 18 seconds (Fig. 3.13). From observation of the results (Figure 3-15), it can beseen that a dramatic decrease in deformation has been obtained— the maximum tip deflection (t22) is now approximately 0.lm and the torsional deformation (r23) reduces to around 2 degrees. The slower maneuver causes the armtip to hover much closer about the rigid tip location (Fig. 3-15b), as compared withthe previous case.To assess the effect of manipulator joint flexibility, the same simplified modelwas used as before but with the joint degrees of freedom excluded. The response tothe same maneuver as before is presented in Figure 3-16. A comparison with theearlier results (Fig. 3.14) revea]s a slight increase in the response frequencies withinsignificant difference in the magnitude. This seems reasonable as removal of thejoint degrees of freedom results in an effectively stiffer arm. Again, the effect of armflexibility is quite evident as shown by the discrepancy between the tip positions forthe flexible and rigid manipulators (Fig. 3-16b). Of course, as seen before during thewrist maneuver, the slewing of the arms would lead to librational response irrespectiveof the joint stiffness magnitude.3.5 Effect of Translational Maneuvers71upper arm (body 2)‘22tower arm(body---1Tr---initial arrn/..-‘position ‘-‘final armpositionside view of arm illust ratingslew maneuverlocal horizontaldirectionlocal verticaldirectionplatform (body 1)initial conditionsall i.c. zeromaneuvers0 to 3odeg in Gs or I 8sp22= 45 to 9odeg in 6s or lBs(note: rot.seq. is 2-1-3)m d, {h}T {13}T(kg) (m) (m) (m) (deg)1 154,500 115.0 5.0 (2.5,0,0) (*,90,0)2 500 8.8 0.5 (0,0,8.8) (0,*,0)3 500 8.8 0.5• joint stiffness (kg.m2/s) coordinate_presentk3 -y1 Y2 1131 154,780 154,780 154,780 yes yes yes2 oc 154,780 no yes nobody123fundamental freq. inlateral vibrationrigid body0.551 Hz0.551 Hznumber of modeslateral11axial00-torsion11Figure 3-13 Data for simultaneous slew of the flexible arms (6s or 18s maneuver).l.h. dir.72-5.0dogshoulder10.0 5.00.0-5.05.00.0-5.0 0.000Figure3-14Typicalsystemresponsetoasimultaneous6sslewmaneuveroftheflexiblearms:(a)jointsandupperarmdeformations.shoulderYii1.0t210.0-1.05.00.0-5.0 0.0005.0dogshoulder122-5.0upperarmm0.0020.0000.0020.0005.00.00.0020.0000.0020.002-10. 0.0000.000 dogelbowmnvr.endorbits0.002orbits0.002lowerarm1.0lowerarmm20mt31311-1.050.0000.0020.0000.002lowerarm1.0mlowerarmmt32t321-1.0______________________________-100.0000.0020.0000.0020.001doglowerarmmlowerarmt33,1-5-0.000—--—--------10mnvrI0.001ondJj-150.000orbits0.0020.000orbits0.002*tippositionforrigidmanipulatorFigure3-14Typical systemresponsetoasimultaneous6sslewmaneuveroftheflexiblearms:(b)lowerarmdeformationsandthetipposition.5.0dogshoulder1.0mupperarmIiit210.00.0-5.0-1.0_______________________0.0000.0060.0000.0065.0degshoulder1.0mupperarmt220.0—.-—————------—0.0-5.0-i.o_____________________________0.0000.006.0.0000.0065.0dogshoulder10.0dogupperarm(31235.00.00.0-5.0-5.0-10.____________________________0.0000.0060.000orbits0.0065.0dogelbow0.0mnvr.-5.0end0.000orbits0.006Figure3-15Typicalsystemresponsetoasimultaneous18sslewmaneuveroftheflexiblearms:(a)jointsandupperarmdeformations.1.0mlowerarmlowerarm2031,115 10 5t532,10-5-10lowerarm0.0060.001dogr33-0.000-0.001 0.000mlowerarm-5-10-15 0.000orbits*tippositionforrigidmanipulatorFigure3-15Typical systemresponsetoasimultaneous18sslewmaneuveroftheflexiblearms:(b)lowerarmdeformationsandthetipposition.t310.0—-1.0 0.0001•0mt320.0-1.00)0.000lowerarm0.0060.000 m0.006lowerarm0.0060.000mnvr.endt33,1orbits0.0060.006upperarmlower armr33-0.000-0.001lowerarmlowerarmFigure3-16Systemresponseduringsimultaneous6sslewmaneuverofthearmswithjointstreatedasrigid:(a)upperandlowerarmdeformations.1.00.0-1.0 0.0000.0021.0m0.0-1.0t21 t22 r23-4-.4t31upperarm0.0000.0000.002,0.00210 5 0 .5-100.0000.001deg0.0020.000orbits0.0020.000orbits0.00220mlowerarmt3110.0000.002mlowerarmt32,l0.0000.002000mlowerarmt33,l-10mnvr.-15end0.000orbits0.002*tippositionforrigidmanipulatorFigure3-16Systemresponseduringsimultaneous6sslewmaneuverofthearmswithjointstreatedasrigid:(b)thelowerarmtipposition.3.5.1 Inplane maneuver (flexible manipulator)As emphasized earlier, the objective in pursuing the parametric analysis is notto accumulate data for design purposes. It is primarily aimed at demonstratingeffectiveness and versatility of the computer code in tackling a variety of situations.Figure 3-17 presents the system data for a. maneuver involving translation of thepayload.Response of the system to a. translational maneuver, in the plane of the orbit, of5 m at two different speeds were considered: the maneuver completed in lOs and 60s.The results are presented in Figure 3-18.At the outset it is apparent that the reaction moment caused by the maneuverleads to the librational response with modulation at a characteristic system frequencyof the arm ( 0.048 Hz, Fig. 3-iSa). Note, the joint and arm responses are in phaseat the fundamental system frequency (0.048 Hz) with tip deflection at the lower arm(grip) of around ±3.5 m and at the payload of approximately ±5 m (Fig. 3-18b).Also shown is the time-history of the location of the system center of mass w.r.t. theplatform (frame one). This reflects as perturbation of the orbital motion which willhave to be corrected through an appropriate control procedure.Figure 3-18(c) assesses effect of increasing the duration of the maneuver, fromlOs to 60s. As expected, the results showed a. sharp decrease in amplitudes of oscillations for all the degrees of freedom. Note, the payload tip now oscillates with anamplitude of 0.3m, a considerable reduction from the 5m oscillation incurred for the‘fast’ maneuver.3.5.2 Out-of-plane maneuver (rigid manipulator)Figure 3-19 presents the system data for a case-study investigating the effect of an79C’) —U,C%.J•‘ .0 — .0 0—.9, E o—‘-2 -.E-g—,F_Vt _•Vt—initial arm positionh13 mnvr.ri51final arm positionmaneuversplatform (body 1)rn-d {h}T {13}T(kg) (m) (m) (m) (deg)1 154,500 115.0 5.0 (2.5,O,) (0,90,0)2 2000 2.0 6.0 (0,0,2) (0,0,0)3 500 8.8 0.5 (0,0,8.8) (0,0,0)4 500 8.8 0.5 (0,0,8.8) (0,0,0)5 940 5.1 1.0fundamental freq. inbody12345lateral vibration0.193 Hzrigid body0.551 Hz0.551 Hzrigid bodylateralnumber of modes111axial000torsion000. joint stiffness (kg.m2/s) coordinate_presentk1 k2 k3 Yi Y2 Y131 oo 00 0° no no no2 00 154,780 00 no yes r3 154,780 00 no yes no4 00 154,780 00 no yes noFigure 3-17 System data for the manipulator translational maneuver of 5m completed in either lOs or 60s.local horizontaldirect ionlocal verticaldi rect ionmobile transporter(body 2)initial conditionsall Ic. zeroxl-Jxli33 L.x31h13=5 to 0 m in 1 Os or 60s80dogpitchdegelbow141i0.l5.00.00.0-5.00.0000.0140.0000.0141E5platformlowerarm0.05-3E-120.00-1E-5-0.050.0000.0140.0000.014oodogshoulderdogwrist5.0050.0-5.0050.0000.0140.0000.014orbitsupperarm10.050.00mnvr-0.05,end0.0000.014orbitsFigure3-18Systemresponseduringthetranslationalmaneuverof5m:(a)time-historiesofthegeneralizedcoordinatesforthelOsmaneuver.upperarmFigure3-18(I) C 0 Cl) 0 0.0 4- Systemresponseduringthetranslationalmaneuverof5m:(b) time-historiesofthetipdeflectionsandc.m.positionforthelOsmaneuver.m0.50.0-0.5 0.0000.0140.50.0-0.5 0.0005.00.0-5.0 0.00000 t.’z0.014.220.28tih.Ooo0.270 0 .-E0.26xO______0.0000.014010 0.000.0.014orbitsm0.10.(ti C-systemc.m.location0.05.w.r.t.frameone0 0 C.,><‘0.00m0.014orbits0.2600.2700.280x11coordinateelbowq1dogpitch2E-110.05-2E-50.00-4E-5-0.051220.51420.050.00.00-0.5-0.050.050.00-0.050.000.04dog0.50.0-0.5platform0.00000.04mlowerarm0.00.0.04dogshoulder0.00 dog0.040.000.04wrist0.000.04U) C 0 (1) 0 0. 00.005.0 0.0 0.00orbits0.04orbits0.04Figure3-18Systemresponseduringthetranslationalmaneuverof 5m:(c)time-historiesforthe60smaneuver.out-of-plane translation maneuver. As against the previous case where the maneuvertook place in the orbital plane, now the plane of the manipulator (considered rigid)is inclined at 45 degrees to the orbital plane during the translation. The resulting response is presented in Figure 3-20. Note, the significant libration motion, particularlyin yaw (Figure 3-20a). This would affect the trajectory tracking of the manipulator,a challenging task in practice, especially in the presence of the arms’ flexibility. Vibration of the 115m platform (t11, t12) at first glance seems to be minimal (0.002mamplitude for tip oscillation). However, excitation of the platform motion cannot beignored as it may interfere with the desired microgravity environment for controlledexperiments.Figure 3-20(b) presents variation of the payload tip position w.r.t. the systemframe. In the absence of libration, the payload tip should trace a straight line in theorbital frame maintaining a constant coordinate position of (t51,0t52,0) (l0.6m,-6.2m). Observation of the payload tip location plot reveals this to be not the casedue to the platform librations (Figure 3-20a). The tip location is shown to be (ll.5m,-5.4m) after 0.02 orbit.3.6 Effect of Payload FlexibilityAn area of interest concerning maneuvers of the MSS is the handling of flexiblepayloads. There may exist the possibility of arm resonance or platform beat responsefor a particular combination of inertias, flexibility parameters, and the MSS maneuvers. To investigate the effect of payload flexibility on the manipulator response,numerical simulation runs were carried out for a translation maneuver consideringboth rigid and flexible payloads. Figure 3-21 presents the system data used duringthis part of the study. Observation of Figure 3-22(a) reveals that during the maneuver(0 to 0.003 orbit), the manipulator’s joint degrees of freedom and beam defections84local verticaldirectionlocal horizontaldirect ion. m1 I d, {h}T {}T(kg) (m) (m) (m) (deg)1 154,500 115.0 5.0 (2.5,0,) (0,90,0)2 2000 2.0 6.0 (0,0,2) (45,0,0)3 500 8.8 0.5 (0,0,8.8) (0,90,0)4 500 8.8 0.5 (0,0,8.8) (0,0,0)5 940 5.1 1.0fundamental freq. inlateral vibrationbody23450.193 Hzrigid bodyrigid bodyrigid bodyrigid bodylateral1number of modesaxial0torsion1. joint stiffness (kg.m2/s) coordinate_presentk1 k2 Yi 1i2 Y0cc cc cc no no no2 00 00 cc no no no3 cc 00 cc no no no4 cc cc no no noFigure 3-19 System data for an out-of-plane, translational maneuver of the rigidmanipulator.the arm is inclined 45°out of the orbital planeinitial arm positionfinal arm positioninitial conditionsall i.c. zeromaneuversh13=-i0 to -40m in 30splatform (body 1)85degpitchmplatform14(1000.005-0.5______0.000.020.000.0202degrolldegplatform•r135E-500-0.2-5E-50.000.020.000.02ciyawm400.00platform0.000.020.000.02platform•mplatformt0.005t0201112,0-0.005mnvr.I0.00end4_____________________________________0.00orbits0.020.00orbits0.02Figure3-20Timehistoriesofthesystemresponseduringanout-of-planetranslationalmaneuver:(a)librationalandplatformtipmotions.12mpayloadlocationofpayloadtipt51,oEinorbitalframe-5010_____________________a)0.000.02Cmpayload0-60-6I•Imnvr.00ond1011120.000.02coordinatex01,m-20mpayload-30-40-50_________________________0.00orbits0.02Figure3-20Timehistoriesofthesystemresponseduringanout-of-planetranslational maneuver:(b)payloadtipposition.are negative, implying that the rigid payload “lags behind” and is being pulled alongby the manipulator arm. After the translation maneuver is over (at 6=0.003 orbits)and the Mobile Transporter has stopped moving, the inertia effect causes the payloadto continue motion, forcing positive displacement at all the three joints (712, 722, 732)and the arms’ tips (t21, t31). The progress of the payload is gradually slowed as themanipulator stiffness force increases until finally it reaches the peak value (manipulator grip location t33,0 = -7.5m). Now the rigid payload begins the reverse journey(t33,0 increases positively) as the elastic manipulator arm attempts to ‘snap back’into its undeformed state.Corresponding response results accounting for the payload flexibility are presentedin Figure 3-22(b). It is apparent that the manipulator response remains essentiallyunaffected by the flexible nature of the payload.A word of caution is appropriate here. Depending on the orientation of thepayload and the type of maneuvers (translational, slewing, combination of the two,planar, out-of-plane, etc.), the system response can change significantly and maydisplay resonance or beat type phenomenon.3.7 Further Generalization of the ModelAs pointed out in the beginning, the formulation for studying dynamics and control of the manipulator with chain-type geometry is indeed quite versatile. In fact, itis one of the most general formulations reported in the open literature. The formulation methodology permits it to adapt the formulation to several different geometries.Although they suggest generalization away from the standard chain geometry, it mayintroduce some constraints. In the first case, there is a departure from the chain geometry (Figure 3-23), however, the central body has to be rigid. On the other hand,now it can have arbitrary number of cantilever beam appendages with any specified88x-JXIIlocal verticaldirectionplatform(body 1)Z rn d {h1}T {13}T(kg) (m) (m) (m) (deg)1 154,500 115.0 5.0 (2.5,0,*) (0,90,0)2 500 8.8 0.5 (0,0,8.8) (0,45,0)3 500 8.8 0.5 (0,0,8.8) (0-45,0)4 500 8.8 0.5 (0,0,8.8) (0,0,0)5 940 5.1 1.0fundamental freq. rilateral vibrationbody2345rigid body0.551 Hz0.551 Hzrigid or 0.2 75 Hzrigid bodylateral1I1number of modesaxial000torsion000. joint stiffness (kg.m2/s) coordinate_presentk3 ‘Y1 ‘Y21 154,780 no yes no2 00 154,780 00 rio yes no3 co 154,780 oo no yes no4 co 00 00 no no noFigure 3-21 System data used in thepayloads.simulation study with rigid and flexiblelocal horizontaldirectionshoulder (joint 1)elbow \ ‘—‘C(joint 2)-..--.-z--z- 5:wrist /‘ upper arm(joint 3)N ,‘‘,‘‘:.12final arm position. XX33payload consists of body 5 (rigid)attached to body 4 (rigid or flexible)3initial conditionsall i.c. zeromaneuversh13=5 toO m in 18s89shoulderwrist21121 0t210.0-0.5mnvr.end2321deg0.0000.006perarm0.5m-0.50 0.00018.03io17.517.016.516.00.0061elbow0.0000.0060.0000.0060—1122t31t330-6.0-9:0mlowerarm0.0000.0060.0000.0060.5mlowerarm0.000orbits0.006Figure3-22Effectofpayloadduringatranslationalmaneuver:(a)rigidpayload.shoulderlowerarm I mnvr.-Jjond-orbits0.00627321mlowerarm0.0000.006mlowerarm0.0000.0062 1 0-1t2100.006dogwrist0.0000.0060.00005upperarm0.0000.0062dogelbow7220.0000.0060.5mt310.0-0.5 0.00018.017.517.016.516.0 0.0-3.0-6.0-9.00.51410.0-0.5 0.000orbits0.006Figure3-22Effectofpayloadduringatranslationalmaneuver:(b)flexiblepayload.orientation. The second case represents further generalization of the original model(Figure 3-24). Here, additional beam type members, of arbitrary inertia and stiffnessproperties, can be added at the hinges resulting in a flower petal-type arrangement.Note, translational or slewing maneuvers at a hinge will result in motion of all succeeding bodies. Two particular examples representing the above mentioned cases areanalyzed here to illustrate these features of the formulation.3.7.1 Single branch configurationObjective here is to model a satellite with central rigid body and an arbitrarynumber of flexible, beam-type appendages attached to it. A simple configuration,with four identical appendages in the orbital plane and normal to the satellite surfaceas indicated in Figure 3-25, is considered here for analysis. One of the appendagesis subjected to an initial planar tip displacement of 10% (lm) of its length as adisturbance. The appendage displacement initiates platform libration, which in turninduces vibration of the other appendages (Figure 3-26a). As clear from the earlierdiscussion, the inpiane disturbance does not excite the out-of-plane motion (t23,1t31,1t431, t,1; Figure 3-26b). An extended plot (Figure 3-26c) clearly reveals abeat response for the appendages.3.7.2 Extended ‘flower petal-type’ configurationHere additional appendages (bodies 2 and 4) were introduced at the shoulderand elbow, respectively as shown in Figure 3-27. Note, all the appendages are in theplane of the orbit, with the system subjected to an initial appendage tip disturbance(planar, 551) of 10% of its length. It can be seen from Figure 3-28 that vibrationof the fifth body results in excitation of the third body (t31) which in turn causesthe platform libration. This leads the second body to vibrate at its fundamental92Figure 3-23 Illustration of single branch configuration.• arbitrary rigid central bodywith array of bodies consisting offlexible beams or rigid members93all bodies consideredas flexible beamsor rigid membersFigure 3-24 illustration of extended ‘flower petal-type’ configuration.cluster of bodiesat each hingecentral bodyin orbit94body 4body 5local verticaldirectionlocal horizontaldirectioninitial conditions21 0.05maneuversnone. m1 I d {h}T {13}T(kg) (m) (m) (m) (deg)1 154,500 115 5.0 (0,0,57.5) (0,0,0)2 500 10 0.5 (2.5,0,-iC) (0,90,0)3 500 10 0.5 105,O,-2.5 (0,90,0)4 500 10 0.5 (2.5,0,-ic) (0,90,0)5 500 10 0.5fundamental freq. inlateral vibrationbody2345rigid body0.551 Hz0.551 Hz0.551 Hz0.551 Hzlateral1111number of modesaxial0000. joint stiffness (kg.m2/s) coordinate_presentk1 k.2 k3 “Y1 Y2 Y131 00 no no no2 00 00 00 no no no3 00 00 00 no no no4 00 00. 00 no no notorsion0000Figure 3-25 System data for single branch configuration having rigid platformwith an array of flexible appendages.body 2body 3x3.(body 1) x51x4195degpitchFigure3-26Typicalresponseofthesinglebranchconfigurationtoanappendagetipdisturbance:(a)librationalmotionandtipplanardeflectionsforthefourappendages.Theout-of-planeresponsewaszero.—0.0000.005N’ 10.010.00+1.0‘210.0-1.0t31body2Co0.3mm0.0-0.3body4t41 t510.0000.0050.3body3o.0-1AJvV\r\fJ-0.30.0000.0050.3mbody50.0wWiW\fJ-0.30.000orbits0.0050.000orbits0.005body467.8mbody267.567.2 0.00012.8mbody312.5—12.2 0.00047.8t33,147.547.2-12.2mbody5-12.5—-12.8 0.000Figure3-26Typicalresponseofthesinglebranchconfigurationtoanappendagetipdisturbance:(b) timehistoriesofthetippositionsw.r.t.theplatformframe.body2+1.00.0-1.0 0.0000.3L4110.0-0.30.005t23,1 t31,10.0000.005-67.2mbody4t43,1-67.5—-67.8 0.0000.005t51,10.0050.005mbody30.005-47.2-47.5-47.80.000orbits0.005mbody50.000orbits0.005j0.4131-0.0-0.41.00.0-1.0t0.4151-0.0-0.4ciegpitch0.02m I m0.20.2Figure3-26Typicalresponseofthesinglebranchconfigurationtoanappendagetipdisturbance:(c)extendedplotoflibrationandtheappendagetipdeflectionsshowingthebeatresponse.,,0.00‘I’1-0.05-0.10-0.15 1.0210.0-1.0body2000.00.2mbody3-.-0.00.00.2mbody4‘:.0.0body5orbitsfrequency of 0.551 Hz.3.7.3 Chain geometry forming single and multiple tether-like configurationsThe formulation, developed here for interconnected, flexible beam-type membersforming a chain geometry, can be applied to a variety of tether-like configurations.They range from: (i) a rigid tether connected to a central body through a pin joint, oran elastic beam-type member with the fixed boundary condition; (ii) interconnectedrigid/flexible members forming a chain-type geometry approaching multi-tether configurations with very low flexural rigidity (Figure 3-29).Two configurations arising as particular cases from the general formulation wereconsidered.Rigid tether with payloadEmployed for study is a single rigid tether with a point mass payload attached tothe free end. This configuration is often considered in the study of tethered satellites,where the tether is assumed to be rigid. The station libration is assumed controlledto emphasize the tether dynamics. The tether hinge is placed at a distance fromthe center of mass of the system (the orbital frame origin). The degrees of freedomfor the tether are joint angles and 712, which respectively represent out-of-planeand inplane rotations of the tether. The data for this run are as follows:m2= 10kg m3= 100kg12= lOOm 13= O.Olmd2 = 0.01 rn d3 = 0.01 m(2Om’ ( 0 )h1z 20m h2= 0j lOm) I.. lOOm)99body 5x53x51body3 >1X3Jjbodyxxlx4lbody 4—x43x33body 1local horizontaldirect ionlocal verticaldirection Iinitial conditions51 0.05maneuversnone. rn I d {h}T {13}T(kg) (m) (m) (m) (deg)1 154,500 115 5.0 (0,0,57.5) (0,90,0)2 500 10 0.5 (0,0,0) (0,-90,0)3 500 10 0.5 (0,0,10) (0-90,0)4 500 10 0.5 (0,0,0) (0,90,0)5 500 10 0.5bodyrigid body0.551 Hz0.551 Hz0.551 Hz0.551 Hzfundamental freq. inlateral vibrationnumber of modes12345lateral1I11axial0000torsion0000joint stiffness (kg.m21s) coordinate_present[____ k2 Y1 i21 oc 00 00 flO no no2 00 00 00 no no no3 00 00 no no no4 00 00 00 no no noFigure 3-27 System data for the ‘flower petal-type’ configuration.100pitchI—.ED I.1i0.010.00 0.0000.0050.3mbody20.0-7\Jf\JW\AAJ\J-0.3t21 t311•1.0L410.0-1.0body4body30.0000.0050.0000.005t51body50.000orbits0.0050.000orbits0.005Figure3-28Librationandtheappendagetipresponsesforthe‘flowerpetal-type’configuration.arbitrary rigid body orflexible beam type member in orbitFigure 3-29 Illustration of multi-tether coafiguration.rigid body or flexible beamjoint with mass permitting3-dimensional generalized rotationsrigid body orflexible beam102Here, a 100 kg point mass payload is attached to the end of a 100 m rigid tether.An initial planar displacement of the tether is:A7ii — U, 712 ‘) , 7ii — 712 —The response results are presented in Figure 3-30.Note, the system is subjected to an inplane disturbance only, of 5 degrees. However, due to coupling, a significant amount of the out-of-plane motion (‘y) results.Projection of the tip motion in the local horizontal, orbit normal-plane (x1,x12 coordinates) is also presented. The energy transfers from inplane (712) to the out-of-plane(711) motion, resulting in the x12 projection.Multi-tethers with payloadsA multiple tether configuration, with a point mass payload attached to the endof each of the two spans, is considered for study. The first tether mass (m2) is hingedat the orbital frame origin (system center of mass), and the second tether mass (rn4)is connected to the end of the first tether. Joint angles 7ii, 712 describe orientationof the first tether with respect to the local vertical, and 731, 732 identify orientationof the second tether with respect to the first tether. The data for the simulation areas follows:rn2 = m4 = 10 kg; m3 = m5 = 100 kg;1214 lOOm; 1315 O.Olm;d2 = d4 = 0.01 m; d3 = d5 = 0.01 m;( 0 ( o “h1- 0; h2=h4’ 0 ; h3= 010) lOOm) I 0.Olm)103dogoutofplane-i----E1I. 0 0 C.) c’Jxparallelwithlocalverticalidhinge7iibody1712I-.C)‘yll0-5‘Y120-5pointmasspayloadorbitnormal0 dogplanar0orbits30rigidtetherlocaltippositionhorizontalinframe120 10 0hingelocation0102030x11coord.,mFigure3-30Responseofrigidtetherwithpointmasspayload.51 = /32 =/33 = /34= (o,o,0)T.The initial state for this response study is:7ii = 712 = 731 = 732= 50;7ii = 712 = 731 = 732 = 0.As against the previous case where the tether attachment was offset from thesystem center of mass, in the present situation the system is subjected to both inpiane(712, 732) and out-of-plane (yi, 731) disturbances. The coupled response results withthe transfer of energy between inpiane and out-of-plane degrees of freedom as shownin the time-histories of the tether end projections (Figure 3-31).105dogoutofplane0 dogplanardogoutofplaneIo.n73110 0-10x230 dog732710 0-1071210 0-10 40E20-o I— o00 0-20x-40I-’ oorbits10orbits1tippositionof40tippositionoffirstrigidtetherEsecondrigidtetherinframe120inframe1LN0 0 ()-20x.-40hingelocation-40-2002040x11coord.,m-40-2002040x11coord.,mFigure3-31Responseofmulti-tetheredsystemwithpointmasspayloads.4. CONCLUDING REMARKS4.1 Summary of ConclusionsA relatively general formulation for studying dynamics of an orbiting, arbitrarychain of translating, slewing flexible bodies has been presented.The formulation accounts for shear deformations and rotary inertia effects associated with Timoshenko beam members forming the manipulator’s arms. Accountingfor joint flexibility in three dimensions as well as specified and generalized coordinates at the joints, with freedom to tranverse over a flexible platform free to librate,and carrying a flexible payload represents a model never reported before in the openliterature. Its versatility would permit dynamical analysis and nonlinear control ofa wide class of space and ground based manipulators. The effect of environmentalforces, such as the solar radiation pressure and free molecular interactions, can beintroduced quite readily as generalized forces.The model not only accounts for an arbitrary number of flexible bodies forminga chain geometry but can tackle a cluster of flexible bodies at joints forming ‘flowerpetal-type’ configurations; rigid central body based single branch geometry applicableto a large class of scientific and communications satellites; and a variety of tethermodels including:(a) rigid tether connected to a central body through a pin joint, or a flexiblebeam with the fixed boundary condition;(b) interconnected rigid/flexible members forming a chain-type geometry approaching multi-tether configurations through appropriate choice of stiffness;(c) as a special case of (b), representing rigid multi-tether systems which havebeen reported in the literature.107The governing equations of motion are extremely lengthy, highly nonlinear, nonautonomous and coupled. A computer code for such a versatile formulation is also notavailable in the open domain. Modular character of the code makes parametric analysis of the system dynamics relatively efficient.Based on the simulation results for two-arm mobile manipulators with payloadsand other spacecraft configurations, the following general conclusions can be made:(i) Translational and slewing maneuvers of the manipulator can induce platformlibrations as well as transverse and torsional vibration of the platform.(ii) Speed of the maneuvers has considerable influence on the platform as wellas the manipulator responses. In general, slower maneuvers induce decreasedlevels of vibration. However, depending on the system parameters, initialconditions and maneuver time histories, the resulting response can be unacceptable requiring active control. Even with small and acceptable level ofdeformations, acceleration levels can be large enough to affect microgravityexperiments onboard the platform.(iii) Platform librations and the system flexibility affect, quite significantly, trajectory tracking of the tip actuator in the orbital frame. This suggests thatcapture of an orbiting object (say a disabled satellite), transfer of a payloadfrom one location to another on the platform, and similar maneuvers wouldrequire careful planning of control strategies.(iv) Vibration of the manipulator arm can influence platform motion. Hence,rigid modeling of the manipulator arm is inadequate for assessing the inducedplatform motion.(v) Torsional deformations of the manipulator links are significant and must beincluded in the model.108(vi) The formulation, developed here for interconnected flexible bodies forming achain-type geometry, can be applied to several other configurations. They include scientic and communications satellites with flexible beam-type membersas well as systems similar to single and multiple tethers.4.2 Recommendations for Future WorkWith a rather general formulation of the problem established and the associatedcomputer code operational, it can be used to advantage in assessing relative importance of a wide variety of system parameters as well as generating design informationfor a class of present and future manipulator designs. In particular, initially theattention may be directed towards the following:(i) Effort should be directed to incorporate generalized displacements at thejoints in addition to rotations covered in the present formulation.(ii) Assessment of the system response accounting for orbital eccentricity, numberof modes, axial deformation, environmental forces, etc., would be a logicalextension to the present study.(iii) Modeling of the Space Station may be improved as the proposed configurationinvolves a vast array of flexible beams, power booms, solar panels, attenae,habitat, experiment and command modules.(iv) Effect of structural damping of the Space Station, manipulator links and jointson the system response should be assessed.(v) Removal and placement of a. payload would result in a closed chain geometrynot dealt with in the present study.(vi) Capture or release of a payload would result in variation of the number ofbodies involved in the simulation. This could be quite challenging if the109payload is flexible or has interconnected bodies, as the number of systemdegrees of freedom will change during the simulation’s integration process.(vii) Perhaps the major effort should be directed towards development of linear aswell as robust, nonlinear control strategies to bring vibrational and librationalresponses within acceptable limits. Accounting for contact forces and pathplanning with collision avoidance represents an extremely important problembut of a higher order of complexity.110BIBLIOGRAPHY1. Shrivastava, S.K., Tschann, C., and Modi, V.J., “Librational Dynamics ofEarth Orbiting Satellites — A Brief Review,” Proceedings of the XIVth Congressof Theoretical and Applied Mechanics, Kharagpur, India, Dec. 1969, pp.284-306.2. Modi, V.J., “Attitude Dynamics of Satellites with Flexible Appendages — ABrief Review,” Journal of Spacecrrift and Rockets, Vol. 11, No. 11, Nov. 1974,pp.743-51.3. Shrivastava, S.K., and Modi, V.J., “Satellite Attitude Dynamics and Controlin the Presence of Environmental Torques — A Brief Survey,” Journal ofGuidance and Control, Vol. 6, No. 6, Nov.-Dec. 1983, pp.461-471.4. Woo, H.H., and Alnianza, J.D., ‘Preliminary Evaluation of an Attitude Control System for the Space Station,” A Collection of Technical Papers of theAIAA Guidance, Navigation and Control Conference, Snowmass, Colorado,August 1985, pp.696-708. -5. Tseng, G.T., and Mahn, R.H., Jr., “Flexible Spacecraft Control Design UsingPole Allocation Technique,” Journal of Guidance and Control, Vol. 1, No. 4,July-Aug. 1978, pp.279-81.6. Hughes, P.C., and Abdel-Rahrnn, T.M., “Stability of Proportional-Plus-Derivative-Plus-Integral Control of Flexible Spacecraft,” Journal of Guidanceand Control, Vol. 2, No. 6, Nov.-Dec. 1979, pp.499-503.7. Martin, G.D., and Bryson, A.E., Jr., “Attitude Control of a Flexible Spacecraft,” Journal of Guidance and Control, Vol. 3, No.1, Jan.-Feb. 1980, pp.37-41.8. 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(Ed)., Proceedings of the 5th VPI £4 SU/AIAA Symposium onDynamics and Control of Large Structures, Blacksburg, Virginia, June 1985.15. Proceedings of the Fourth International Federation of Automatic Control Symposium on Control of Distributed Parameter Systems, Los Angeles, California,June 30 - July 2, 1986.16. Rodriguez, G. (Ed)., Proceedings of the Workshop on Applications of Di..stributed System Theory to the Control of Large Space Structures, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California,July 14-16, 1985.17. A Collection of Technical Papers of the AIAA Guidance, Navigation and Control Conference, Snowmass, Colorado, August 1985.18. Govin, B., and Claudinon, B., “Adaptive Control of Flexible Space Structures,” ESA Journal, Vol. 6, No. 1, 1982, pp.35-51.19. Balas, M.J., “Direct Velocity Feedback Control of Large Space Structures,”Journal of Guidance and Control, Vol. 2, No. 3, May-June 1979, pp.252-253.20. 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Morita, Y., and Modi, V.J., “A Formulation for Sti.idying Dynamics andControl of the Space Station Based MRMS and its Application”, AIAA/AASAstrodynarnics Conference, Minneapolis, Minnesota, U.S.A., August 1988,Paper No. AIAA—88--4269 CP; also Proceedings of the Conference, Editors:R. Holdaway, and B. Kaufman, AIAA Publisher, pp.401-4O953 Gear, C.W., Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1971.54 Hindmarsh, A.C., “Gear: Ordinary Differential Equation System Solver,”Lawrence Livermore Laboratory, Report UCID-30001, Revision 3, Dec. 1974.55 Blevins, R.D., Formulas for Natural Frequencies and Mode Shapes, Van Nostrand Reinhold Co., New York, N.Y., 1979.56 Brereton, R.C., “A Stability Study of Gravity Oriented Satellites,” Ph.D.Thesis, University of British Columbia, Nov. 1967.57 Ng, C.K.A., “Dynamics and Control of Orbiting Flexible Systems: A Formulation with Applications ,“ Ph.D. Thesis, University of British Columbia,April 1992.116Appendix A: ROTATION AND PROJECTION MATRICESLet [C(U)} represent a rotation matrix associated with a set of rotations U.For a 1-2-3 rotation sequence:Fl 0 0 1 F c2 0 32] Fc3 —33 01[C(U)]=0 c1 —3 0 1 0 I 33 C3 0Lo c1 iL—s2 0 c2J Lo 0 1For a 2-1-3 rotation sequence:F c2 0 s2] Fl 0 0 1 Fca 33 01[C(U)J=0 1 0 0 c1 —‘ I 33 C3 0L—s2 0 c2J LU si c1 J Lo 0 1So for the jth hinge:= [C(())][C31_1j = [C()][C3:] = [C(j]Let [P(U)] represent a projection matrix associated with a set of rotations U.For a 1-2-3 rotation sequence:F c23 s3 0][F(v)]=—C2S3 C3 0L 2 0 1117For a 2-1-3 rotation sequence:C3 C1S3 01[P()J=—33 C13 0L 0—3 1So for the jth hinge:[P31_2I =[P31_1J = [P(j][P3] = [P()].118Appendix B: SPECIFIED TIME HISTORYThe location and orientation of body i + 1 w.r.t. the th body are respectively specifiedby vectors and . The elements of the vectors are specified coordinates. Eachspecified coordinate must be looped through a time history routine.If a translational or slewing maneuver is desired for a generic specified coordinate c,the following must be specified:c1 = initial value for specified coordinatec1 = final value for specified coordinateT = duration of maneuverThen the current value of the specified coordinate during the maneuver (0 t T)is given byc = c + c(t)For a cubic time history (initial and terminal velocities are zero),í 3t2 2t3For a sine-ramp time history (initial and terminal velocities as well as accelerationsare zero),Lc(t) = (c — c) [ —119Appendix C: MATRICES AND VECTORS ASSOCIATEDWITH DEFORMATIONIn absence of the foreshortening effect, [oJ = 0 The deflection and rotation,,of body i can be written as:= [i1dirneri3ion (3 x 1) (3xn1) (n1xl)= [771]where:dimension. (3 x 1) (3xo1) (o1xl)ni=j= 130 =j=1Here n, and o are the number of modes used to represent deformation in the jdirection of the i body. The generalized coordinate vector 5, is detailed as follows:(1 ‘=I. 813 )where: ( 8111—= silnhl1—5,2= I1 8—S1513= IIt should be noted that j is a (n x 1) vector associated with deformation of the120th body in the th direction; lateral deformation for directions j = 1, 2 , and axialdeformation for j = 3.The generalized coordinate vector 3 is detailed as follows:15:i113)where:( ( :21 31—— :22 — i32As before, X is an (o x 1) vector associated with deformation of the jth body aboutthe th direction; rotary inertia motion about directions j = 1, 2 , and torsion aboutdirection j 3.The partial derivative operators of the vectors S1, and ) are as follows:(8 (8I ôA1ii8 886 i12 8A t12a 888A i218 8A 8S22 A 8A22L5. = L. =886j212 8A120128 886i31 8A 1318 88632 8)328 886j3n1121The matrix [j(x)] takes the form11 i12 4’,1n 0 0 ... 0 0 0 ... 0 1I 0 0 ... 0 21 t2 •.. 2ns2 0 0 ... 0L 0 0 ... 0 0 0 ... 0 32 1’3ri, Jwhere1k(x,3) is the ktF mode used to represent deformation in the th direction ofthe th body; lateral beam deformation for j = 1,2; axial displacement for j = 3. Themode shapes assumed for lateral and axial deformation are detailed in Appendix D.Similarly, the matrix [(x)] takes the form[lull ?7i12 0 0 ... 0 0 0 ... 0 10 0 ... 0 ii 27i22•••lu2o2 0 0 ... 0L 0 0 ... 0 0 0 ... 0 77131 7i32 •.. 771333 ‘where (x) is the kth mode used to represent deformation about the th direction of the body; rotary inertia displacement about directions j 1,2; torsionaldeformation about directon j = 3.Time limitations did not permit the investigation of the Timoshenko beam. Forthe Euler-Bernoulli beam, the rotary inertia motion of a cross-section of the beamis related to the beam’s slope:d2a(xj t) = —3._(z,t)u.xi3c2 (xe, t) = -————-(x, t)ax13Consequently, the mode shapes chosen to describe rotary inertia motion are:d771k(Zi3) = —----—(2k(x3))aX13d7i2k (x13)=.—(qS, (x13))axi3122Furthermore, as the rotary inertia motion is dependent on the beam slope, the vectorsX and 2 are no longer generalized coordinates. The equations of motion can beadjusted accordingly by combining the equations with the S:2 equations, andthe equations with the equations. The mode shapes selected for torsionaldeformation are described in Appendix D.123Appendix D: NATURAL FREQUENCIES AND MODESHAPESThe formulae, for single span beams, in this appendix were obtained from Blevins55.Notation: x = distance along span of beam;rn = mass per unit length of beam;= mass density of beam materialI = mass moment of inertia of beamabout neutral axisL = span of beam;E = modulus of elasticity;G = shear modulus of elasticity,{G = E/[2 (1 + v)], ii Poisson’s ratio}LATERAL DEFORMATIONnatural frequency (hertz), fk = k () 2; k = 1,2,3...27rL mFree-Free( . )kxmode shape, k(X) = cosh_E_ + co.s—— — Jkl3tflhr + szn.Z_124natural frequency parameters, k and 0kk1 4.73004074 0.9825022152 7.85320462 1.0007773123 10.9956078 0.9999664504 14.1371655 1.0000014505 17.2787597 0.999999937k >5 0.5 ir(2k + 1) 1.0Clamped-Free4kX .)kX ( . .mode shape, 4k(x) cosh_L_ — cos_E_ — k sznh_L_ — sin—i—natural frequency parameters, .&k and ok1 1.87510407 0.7340955142 4.69409113 1.0184673193 7.85475744 0.9992244974 10.99554073 1.0000335535 14.13716839 0.999998550k >5 0.5 r(2k — 1) 1.0For the aforementioned lateral modes, the following integral holds,= LAXIAL DEFORMATIONnatural frequency (hertz), fk=-() ; k = 1,2,3...125Free-Freemode shape, k(X) = cos?-4L_; k = 1,2, 3.nat. freq. par., = k’r; k = 1,2, 3...Clamped-Freemode shape, k(z) sin; k=1,2,3...nat. freq. par., ‘k = (2k — 1); k 1,2, 3..For the aforementioned axial modes, the following integral holds,JL =TORSIONAL DEFORMATIONnatural frequency (hertz), fk= 2L)2; k = 1,2,3...Free-Freemode shape, ?7k(z) = k = 1,2,3...nat. freq. par., = kir; k = 1, 2, 3...Clamped-Freemode shape, (x) = sin4; k = 1,2,3...126nat. freq. par., = .(2k — 1); k = 1, 2, 3...For the aforementioned torsional modes, the following integral holds,JL =127Appendix E: SHIFT IN SYSTEM CENTER OF MASSThe location of the center of mass of the jth body w.r.t. the system frame can bewritten as=The location of the system center of mass w.r.t. the system frame is given by ,DM =m1i=1= (+( [Tj1)+ [T]Sj)m1M + + E[TjJm.Since the system frame has its origin at the system center of mass, must identicallybe zero. To satisfy the above equation, , which locates frame 1 w.r.t. the systemframe must appear as(j).128Appendix F: TSF - ENERGY RELATIVE TO SYSTEMFRAME( “I — 8a1‘ + 2 +“ at at ‘ — at at at at a at‘ I (8 8F’ “8 8:’drn =-a-)+2fdm +J(77)dmiJ 7L drn.=(-[TJm, + [Tfm)+mt(((1—[T,})j+i -i-[T],+1))= (-[1jm + [TJm)+ j+1 + []+)ButJ7tdrni =7’(_M),i=1hence,(811)M+28181(M)12981ö1(M)+j’lt.7t)dmi.Now,a a —öt([j), + [Tj [T] + [T]+2(([TJ)+ [T])i—i i—i([Tk])k+l + ([T]÷1)j1 k=1.[Tk]#k÷l],therefore>[J([T:1) (-[7])dm+J dm1+2J (-[T]). [Tdm]+{2(([T1j)’m-i- [T1]m)•(([‘,j)+i +1 >:—i —imj ([Tk})k+l + ([Tk}+l)i=1 y=1 k=1+2([2J)+ .[Tk]Ik+l].a1—M —at at130() [ (([Tk])k+[Tk]k)mmkj=1 k=1I i—i+2. (([Tkj)k+l +[Tk]k+l)mjnk+lj=1 k=1i—i i—iE ((i)+ +[T]+1). (([Tkj)k+l + [Tkjk+l)n+lnk+lj.j=1 k=1yjf idmm(_[T]). (—[Tj)dm1+ f dm + 2J ([7]) [Tdmj+ (([7J) + [2]) ((-[Tk1)k + [Tk]k)mjmk()+ (((-[Tk])k+l +[Tk]k+l)* (flk+1 () +f))]where:f = 0, (Ic> i—i)f=1,(ki—1)i—i i—i (()÷ + (([Tkj)k+l + [Tk]k+l)fj=1 k=1where: f = ni (i + k+i (i)), j Icf=nk+l(1+n+i()), k>j= 2 * TSF, where TSF = TQQ + Tww + TQwFurther development of energies TQQ, Tww, TQW are given in Appendices G, H, andI, respectively.131Appendix G: TQQ - ENERGY IN TRANSLATION W.R.T.THE SYSTEM FRAME2*TQQ= Jj1 rn j=1 k=1+ ( [Tj+i (nk÷i () + f))]where:f= 0, (k > i—i)f = 1, (k i — 1)1—1 1—1(+) ([TkJk+l)fj=1 k=1where: f = n÷1 (i + k+i (i)), j kf=nk+l(1+n+l()), k>jNoting that i’m. dm = [Ij , the above expression can be written as2*TQQ=Z51 [‘Q:]6i+ b [Mb1bj=1 j=1—1 1—i+ ä’ [Maj÷1a]áj+ii=1 •;=i1T+ b1 [M&a.1 Ii=1 j=1where:132[‘i =1 ([]T[] ++[}T[S}T[]+ [}T[5}{}) dm;[Mb.] = [TjT [Ti] mm () * f f = 1, i ==2,ij;= [T]T [}n1 (i +n+1()) f f= 1, i=j,= 2, i[M&jaj1]= [TjT [Tj]m (f+nj+i()) *2 f= 0, j > i—i,= 1, ji—1.133Appendix H: - ENERGY IN ROTATION W.R.T.THE SYSTEM FRAME2*Tww= J ([Tj). ([T])dm1=1 m: V(-j). (-[Tk])Ekmjmk(-)j=1 k=1+ [2(-[Tj)m. (([Tkj)k+l (nk+i () +f))]where:f=O, (k>i—1)f=1,(ki—1)i—i i—i(‘.,i)+ ([TkJ)k+ifj=i k=iwhere: f = n+ (i + k+i (i)), j kk>j3(1—1) 3(1—i)= foriji=1 j=i= [(U [Cii])T[Ihm+in+i 1( [C1])ii=1 ji=12 3(k—i) 3(k—1)V + ( II [Cil])[Imkl( H [c1j)] *fkn+1 ii+1 ;1j-j-1where: f = 1, i = jf=2, ijHere, [Ii,,] is the inertia diadic associated with the angular velocities and Note134the similarity of this inertia formula with the parallel-axis theorem.Notes on— m, n are indices for hinges associated with angular velocities of the indicesi, j.— m is the index for hinge associated with angular velocity index i, m = 1 fori = 1,2,3; m = 2 for i = 4,5,6; etc.— n is the index for hinge associated with angular velocity index j, n = 1 forj = 1,2,3; n = 2 for j = 4,5,6; etc.— if (i + 1) > 3(k — 1), then fl [C1] = [U] (unit matrix)— if (j + 1) > 3(k — 1), then LI [C1] = [U] (unit matrix)[1m]= J(. ..)[U] -dm.Here, [Imj] represents the inertia diadic for the th body w.r.t. the jth frame. Thereis no restriction on the placement of the body frame.[Ih,] = E {[T}. [T] [U] — [T]T[T]T](_l):m jn1—1 1—1[r1 [U] — fi=m j=nwhere: f = n+1 (i + ni (v)), i i+ [U] - [T T[T}T] (f+ (_1))i=m y=nwhere:f=O, (j > i—i)f=1, (ji—1)135+ [T [U] — [jS+1T[T]T] mj(f + ())where:f=O, (i>j—1)fzl, (ij—1)Here, [Ihmn] is the contribution of masses to the inertia diadic [Ii,] attributed tothe distances of frames m to 1 and n to 1 from frames m or n.136Appendix I: TQW - ENERGY IN COUPLING OF TRANSLATION AND ROTATION W.R.T. THE SYSTEM FRAME2 * TQW = YZ 2J (-[Ti]) [T]dm+ (([j)j. [Tj + ([Tk])k)mmk()=1 k=1+[2(1-[Tij) m• ([Tk]k+l (flk+i () +f))]where:f=zO, (k>i—1)f 1, (k i —1)[[Tim. ( ([Tk])k+l (flk+1 () +f))]where:f=O, (k>i—1)f = 1, (k i—i)1—1 1—1(([])j+i [T]1 +[T]a+i (-[Tk])k+l)fj=lk=1where: f n+1 ( + flk+i()), j kfflk+l(1+fl+l()), k>jThis can be written as:3(1—1)2*TQW WTHQW;HQW = [Nt,. j i + [Nw.a. I 1j+1+ [K.5.÷1] +i137= (fi[ckl)T [7] *2;= (II [Ck}) [Nhma.+i J [Tj *2;= ([ck])T[] [Kj * 2;{Nhmb.} = [T]mmj()+iTTh+1f = 0, i > j;= 1, i j;{Nhmaj}=[Ti]im(f+nj()), f=0,j>i;= 1, ii;+i=m+1f=n(i+n()), ij;i<j;[K}= J [}([] + [S][])dm;where indices m and i are associated with hm and u,, respectively (e.g. m = 2 for i= 1,2,3; m = 3 for i = 4,5,6: etc.).138Appendix J: TLSF - ENERGY IN COUPLINGTLSF represents the energy in coupling between the librational degrees of freedomand the system angular momentum. Now,aj’ iaX--)i+)X---+-),therefore,f _j rix-ä—}dmi=x (M)+1x fm+ x + J xBut:ZJ dm=i(—M);i=1 in1=therefore:i=1 771i i=1 i139Here:a1x --(—M) =() x mk(([Tkj)k + {Tk]k)+ (J) [Tj]m X n (([Tkj)k+l + [Tk)k+l)+(:) xE mk( (-[Tk])k + [Tkjk)+ () X flk (([TkJ)k+1 + [Tk]k+1);f x—!-dm1=[Tj x f [TJ x [T]dmi=1x ++ ([j+1)i=1 j=1x((frkJ)k+1 + [Tk1k+l).So finallyzJ (x)dm1,=i=1 mJ [T} x (-[Tj})dm + f [T] x [T]dmm2140+ (_1) [7]jrn x mk(([Tk]) bk + [Tk]k)+ E [7]m x E ((-[Tkj) k+1 + [Tk]k+l) (flk÷1 () + f)where: f = 0, (k > i — 1)f=1, (ki—1)+ ( j+1(j+1() +f)) x1=1 j=1where: f = 0, (j > j — 1)f=1, (ji—1)i—i i—i(j+1) (([Tkj) ak+1 + [Tk]k+l)fj=1 k=1where: f = n1 (i + n (f)), j kf=nk+l(1+n+l()), k>j= HSF = HQ + 77w + H1 = system angular momentumDetails of’q, w, fz, are listed in Appendices K, L, and M, respectively.—T —TLSF=f2 HSF.141Appendix K: 71 - SYSTEM ANGULAR MOMENTUMDUE TO TRANSLATION W.R.T. SYSTEMFRAME“Q x[TkJkmk(yj)x ‘([Tk]k+I (nk+i () +f))where:f=O, (k>i—1)f = 1, (k i—i)+ ( [Tj’ () + f)) x [T]m1=1 j=1where:f=O, (j > i—i)f = 1, (i i — 1)1—1 1—1(j÷ xj=1 k=1where: f = n1 (i + k+i ()), j kf=nk+l(1+n+l()), k>jThusHQ =[NbJ1 +where:[N61 [Jhb] [7:][NQ.]= [1ha1 [T_].The vectors and Nh1a3 were described in Appendix I.142Appendix L: Tw - SYSTEM ANGULAR MOMENTUMDUE TO ROTATION W.R.T. THE SYSTEMFRAME= f [T} xm+[7m xx (([Tk])k+l (flk+1 () +f))where: f 0, (k > i — 1)f = 1, (k i—i)([Tjij+i(nj+i() +f)) x ([T])m1 21where: f = 0, (j > i — 1)f=1, (ji—1)i—i i—i(j+1) ([Tkj)k+lfj=1 k=1where: f = n1 (i + ki ()), kf=nk+l(1+n+l()), k>jThus3(1—1)Hw= E [Iw:jwherei 1 3(k—i)[I] = [Ihj(ll [CJ) + [Tk][Imk}( II [CJ),j1 km 31+1143with: m = 2 for i=1,2,3; m = 3 for i=4,5,6; etc. The matrices [Ihim I and [Imk]were described in Appendix H.144Appendix M: - SYSTEM ANGULAR MOMENTUMINTEGRAL= J [T] x [T]dm1=1 Tfl1-where:[1(6,]=[2] [1(][Kg]=J[1([i] + [S][u])d.145Appendix N: TL - LIBRATIONAL ENERGY OF THESYSTEMTL= 2’M= 1.T[][I] = J [(F ) [ U] — dmm1= [1h, I + [Ti] [Im] [T]T.Here [I] represents the inertia diadic of the system (w.r.t. the system frame). Again,note the similarity between this formula and that for the parallel-axis theorem. Thematrices [Ih11] and [Im.] were given earlier in Appendix H.146Appendix 0: TROT - ENERGY DUE TO ROTARY INERTIA AND TORSIONContribution to the kinetic energy from rotational motions of the system elementsdm1 w.r.t. frame i (i.e. rotary inertia and torsion effects) can be written asTROT I [‘ezwhere: Teç = angular velocity of mass element drn1with respect to the body frame;[I I = inertia diadic per unit length ofmass element dm:.NowTROT = j f dlwhere:frat =. [ ‘ez I 5e1= [P1. ]Consider the ith body, foregoing the subscript i. For a 1-2-3 rotational sequence,C23 33 0[P1I= i C23 C3 0L.0 1Imposing the restriction that [Iei] be principal,147F J 0 01[Iezl=i 0 Jy 0[0 0 JFurthernore, J = J, = 0.5 * J , hence:F’ 0 01[I,j=J10 1 0LO 0 2and0 2521[p1}T[1][p= J 0 1. 0L 282 0 2But assuming small deflections, we get:F 1 0 2a][pjT[j][p]= J i 0 1 0L2a2 0 2and so now,frot = [1ei ] ZT= .T[p1[1 {Peilá= J(a + à + 2à +4c21&3).Conducting mathematical operations on T70, and rearranging, one obtains:d 8Trot —( b) = [MrOtb] ô + Vratb.For m °bl , &iMrotb (m, n) = Jxb b1n dzb148o < m (obl + 0b2), Obl < n (°bi + 0b2)Mratb (m, n) = J f b2n dzb;(Obi + 0b2) <m o, (obl + 0b2) <n 0bMrotb (m, n) = 2Jxb 17b3m 7b3n dzb;m 0b1 , (obl + °b2) <n OMrot& (m, n) = 2Jzb E bj f b1m b3n b2j dzb;j=c’61+ 1bMrotb (n, m) = M,.otb (m, n)The remaining elements of [Mrotb j are zero.For m(ob1+0b2) °bV0 (m) = bjbk J 7b1mb23k dzb;°b1 k=(obl+0b2)+10b1 <m (Obl + 062)= 0;(061 + °b2) <m 06°bl (061+062)Vrot6(m)=2J6 kbb f 7b3mllbli?7b2j dz&;$=1 2061+1 lb149Obl <m (o —f- 0b2)ãTr °bl(m) = 2J : bbk J 7b2m1i3k dzb.i=1 1bThe remaining elements of 8Tr0tb (rn) are zero for m 0bl and for (obi -f- 0b2) < m0b•150Appendix P: UL - GRAVITATIONAL POTENTIAL ENERGYThe gravitational potential energy for a single rigid body was obtained by Brereton56.The conventional approach for determining the gravitational potential energy for asystem of flexible bodies is the same as above — the system of bodies is treated as a(Csingle body” librating about the system center of mass. The resultant expressiondiffers only in that the inertia diadic in Brereton’s case is for a single rigid body,while that in the general version reflects the instantaneous inertia properties of thesystem of bodies w.r.t. the system frame. Brereton’s procedure is repeated here forcompleteness.To determine the gravitational potential energy, consider the vector ,which locates an arbitrary mass element dM w.r.t. the center of force. The vectorlocates the system center of mass w.r.t. the center of force, andI riI, r3 )locates mass element dM w.r.t. the system center of mass. The gravitational potentialenergy of the system is given byI dMULMI-PIt should be noted that the projection of a unit vector (directed along the local vertical151axis) onto the system frame axes is given by the direction cosine vectorIi)113 )withl+l+l=1.The distance between the mass element and the center of force can be written as= [(ri + rli)2 + (r2 + rl2) + (r3 + 7c13)2 jr [1 + --(rili + r21 + r31) + --(r + r + r) }Substituting for j, one obtains=[1 + !(rili + r21 + r31) + + r + r) j+ dMT MCarrying out the binomial expansion(1±z) — 1 + n(n-I-1)x2 n(n+1)(n+2)x3 +and truncating the series givesUL=[1 — ‘(r1 + r21 + r31) —+ + r21 +r31)2 ] dMBut because the integration is w.r.t. the system c.m.FdM =152and tJL reduces toUL = + J— 6 (l’j ‘2 1112 + T2 T3 1213 + f3T1 13 li)] dMA part of the integrand can be rearranged as follows:(r +r +r)—3(rl +rl +r1)- 3i) + r(1 - 3l) + r(1 - 3l)= r (—2 + 3(1—l)) + r (—2 + 3(1—l))+ r(—2-I- 3(1 — l))= r (—2 + 3(l + l)) + r (—2 + 3(l + 1))+ r(—2 + 3(l + l))= —[(r +r)+ (r-f-r) + (r +r)]+3[(r+r)l + (r+r)l + (r+r)l].With this, the potential energy expression becomesUL =___-2r31MRT + r) + (r + r) + (r H- r)] dM+ 4J [(r H- r)l + (r +r)l + (r + r)l— 2 fr’ r2 1112 + r2 T3 2’3 + r3r1 1311) } dMFrom Appendix N,-T]dM.153So finally, the gravitational potential energy can be written asUL = _. — 5j-tr[I] +4JrcT{I]ircRecall that the librational generalized coordinate vector, , describes the orientationof the system frame w.r.t. the orbital frame. Here pitch, roll, and yaw describerotations of the system frame about the orbit normal, local horizontal, and localvertical axes, respectively.‘I I pitch(about axis 2) ‘I= =‘ roll(about axis 1)( b3 ) i yaw(about axis 3) jFor a pitch-roll-yaw sequence of the libration rotations, the vector T has the form—I—s1c&3 +c1sJ2i,b3 11rc = .s&1s’çb3 + cJ’s’z/.’i,bI.. cb1’çb2 )154Appendix Q: [M] — MASS MATRIXElements of the mass matrix [M] are given by the following expressions:[m] =[P]T[I][ j;[m.] = [p]T[1][p][ms.] = [P ]T [[N ] ( [A] + [S ] [B])+ [Naj1 ]([D] + [S ][G]) + [Ks.]];[m.] = [p]T [1w3.2] [P3_2[()]+[I3,]T) [P3]; fori = j[m71j = [P3]T[J31[P1; fori <j[mj = -[F31] HQw3.[m.] = 1[p3]T[J31 2][P31_[1)]; for3i < 3j —2[] [j]T[I2 p][j(7j)]; for3i> 3j —2‘QW31 = [[N3b1]([A1 + [S ][B1)+ [Nw3aj1J([D1]+ [5i ][G1) + [K351]][m.s.] = 1([]T + [B]S)([Mb.b.] + [Mbb1]T)([A] + [S][B])+ ([D1]T + [GiIT[Si]T)([Mai+iai+ij+ JT)([D] + [S][G])+ ([A]T + [B]T[S]T)[Mbjaj÷l ]([D] + [s][c])+ ([DjT + [G]S)[Mb0÷1]T([] + [6][B])155+ [Iç.]; fori=j.[m6.6.] =([AjT +[BjS])[Mb.b.]([A] + [Sj[B])+ + [G]T[SV)[Mai+iaj÷i }([Dj + [s][c])+ + [Bi}T[Si]T)[Mb.a.1}([D] + [s][G])+ + [G]T[S]T)[Mbjaj+l iT([A1 + [SJ[B])for i <j.[m.]= . [P_2 II ‘jOj) I;[m.]= [(i)1T[p3. ]T([j.23 2 I +[13i-2,3i-2 1T)*[P3_2}[?7j()] + [Mot.j, fori =[mA] = [ i() ]T[p 1T [I3-2,3j-2]*[P3_2I[77(a)] , fori < j.Details of matrix [M70] can be found in Appendix 0.156Appendix R: VECTOR {Rq}The elements of the vector {Rq} are given by the following expressions:rem=[p]T((d[JJ)j[J] )dtrem+[pJT HSFremI IHSFrem z (-[K.]) +E (-[Nb])m1i=11+ [NJ[ s ][B] + ([N.j)i=2I I+ [NaJj_i + [Na[ &_i ][G11ii=23(1—1)+ (‘i)j= 13(1—1)+ [I,,]fj= 1where:f= (-,) , for j = 1,3,4,6,7,9,1O...etc.remf= -Zij , for j=2,5,8,11...etc.i = index for joint of concern, k = 3 * iy,rem = ([pkT1 jT(+ [pk]T([1wk1T)1J )wk+[pkjTrI lT(dj)L ““dt rem+ +[Pk}THQg)157+ [(d[p ]T)[I ]T_ + [Pk]T({If,k}T)f]3(1—1)j=k+ {PkjT([Ik,J)jt [P]T{I fwhere:f= (p,) ,for j= 1,3,4,6,7,9,1O...etc.remZij , for j=2,5,8,11...etc.3(1—1)+ 1 [P]TI,,jfwhere:f= (-j) , for j= 1,3,4,6,7,9,1O...etc.remf = for j = 2, 5,8, 11...etc.;HQWk = (([NWkb])tj + [Nwkb][j J[B])+ 2(([NWka 1) j + [Nwka 1[ si-i ][G1]i+ +i- index for body of concern, k = i + 1, n = j — 1.= (-_[i<6jT) rE+[KJT(_ri)t rem+ [BjT S.JT[NoJ T+ ([A]T ++ ([A]T +t rem(include the following for i < 1)158÷[G]T[.’rN ]TI a+ ([D]T + [Gj]T[aj]T)([Nok]T)i+ ([D1]T + [G]T5)[N lTId&akj‘dt rem( end of include)3(1—1)1 1( d [K5]T) zz ++ Jj=13(1—1)1+ [B]T[E]T[N&rj3=13(1—1)1+ E ([A]T +j=13(1—1)d ‘1+ ([A} +• j=1(include the following for i < 1)3(1—1)1+ E [Gt]TEiJNwjakj 13(1—1)1+ ([D}T + [GtjT[Si]T)({Nwak]T)j3=13(1—1)1+ ([D] + [GilT [S]T)[N jT()3=1( end of include)++ [j= i+ ([A]T + [Bj]T[6]T)([Mb.b.])mjj159+ ([A1]T + [B1]T[Sl]T)[Mb.b]([ E ][B])][ ([B1]T]T)[Mbb.] mb+ ([A1]T + [B1]T[8)( [M1jr)+ ([A1]T + [B1]T[Sl]T)[MbbJ([E2j=2([B]T[E1T)[Mb10a+ ([A1]T ++ ([A1]T + [B1]T[Sl]T)[Mba.](n + [En ][G])](include the following for i < 1)2k([Gl]T[El)[Makaj j+ ([D1]T + [G1]T[5)( [MakQ])+ ([D1]T + [G1]T[Sl]T)[Maka ](n + [En ][c])]k([G1]T[E]T)[Mj+ ([D1]T + [G1}T[6iJT)([Ma.ak]T)j+ ([D1]T +[Gi]5l)[Majak](n+ [E ][G])J+ [ ([Gl][El)[Mbak]mjj+ ([D1]T + [Gj]T[Sl]T)([Mbjak]T)rnjj+ ([D1]T + [G1]T[Sl]T)[Mbjak ]T([ E ][B])]( end of include)160i = index for body of concern, k = 3i — 2RA:,rem = 1”rot(if i = 1, stop here)+[()jT( ([pkjT)[Jwk]r+ [pklT(drl ‘Tdt .1 iw.i )+ [Pk]T{IwkjT(_!) )dt+ [ Th(h) iT ( ( d rpk}T) QWk + [Pk]T7f )i QWk5k+ {([PkjT) [I,k]TW + [Pk]T([I,k]T)i= IT” d+[Pk]T[I, j I1-em3(1—1)+ [)]T {([Pk]T)[Ik,j] + [Pk]T([Ik,])3=k+[Pk]T[Ik,j](4.j) IremDetails of vector can be found in Appendix 0. The vectors (j) andrem( d represent the terms remaining from vectors (-) and ( d ,i, respeci) remtively, after the second derivative terms have been removed.161

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