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Dynamics and control of orbiting deployable multimodule manipulators Cao, Yang 1999

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DYNAMICS AND C O N T R O L O F ORBITING D E P L O Y A B L E MULTIMODULE MANIPULATORS Yang Cao B.A.Sc, Shandong University, China, 1993 A THESIS SUBMITTED LN PARTIAL F U L F I L M E N T OF THE REQUIREMENT FOR THE D E G R E E OF Master of Applied Science in  The Faculty of Graduate Studies Department of Mechanical Engineering We accept this thesis as conforming to the required standard  The University of British Columbia October 1999 ©Yang Cao, 1999  In  presenting  degree freely  at  this  the  thesis  in  partial  fulfilment  University  of  British  Columbia,  available for reference  copying  of  department publication  this or  thesis by  of this  for  his thesis  and  study.  for  her  permission.  Department The University of British Vancouver, Canada  Date  DE-6 (2/88)  (POT,  /fr  Columbia  , / * , ? ?  requirements  agree  may  be  It  is  representatives.  financial  the  I agree that the  I further  scholarly purposes  or  of  gain shall  not  that  an  advanced  Library shall  permission for  granted  by  understood be  for  the that  allowed without  head  make  it  extensive of  copying  my or  my written  ABSTRACT  This thesis focuses on the planar dynamics and control o f a variable geometry manipulator which may be used in space- as well as ground-based operations. The system is composed o f a flexible orbiting platform supporting two modules connected i n a chain topology. Each module consists o f two links: one free to slew while the other permitted to deploy. The model used and the governing order-N equations o f motion, as developed by Caron, are explained. A detailed dynamical response study is undertaken which assesses the influence o f initial conditions, system parameters, and manipulator maneuvers on the system response.  Results suggest that under critical combinations o f system parameters and  disturbances the response may not conform to the acceptable limit. This points to a need for active control. T w o different control methodologies are used: (i) the nonlinear Feedback Linearization Technique ( F L T ) applied to rigid degrees o f freedom with flexible generalized coordinates indirectly regulated through coupling; (ii) a synthesis o f the F L T and Linear Quadratic Regulator ( L Q R ) to achieve active control o f both rigid and flexible degrees o f freedom. Furthermore, the F L T is used to track several prescribed trajectories considerable accuracy.  with  Finally, a two unit ground-based prototype manipulator, designed  and constructed by Chu, is used to assess effectiveness  o f the Proportional-Integral-  Derivative (PID) and F L T control procedures in performing several trajectory tracking maneuvers. The study lays a sound foundation for further exploration o f this class o f novel manipulators.  ii  T A B L E O F  CONTENTS  A B S T R A C T  ii  T A B L E O F CONTENTS  iii  LIST O F S Y M B O L S  vi  LIST O F FIGURES  xi  xvii  LIST O F T A B L E S  xviii  A C K N O W L E D G E M E N T  1.  INTRODUCTION  1.1  Preliminary Remarks .  1.2  A B r i e f Review o f the Relevant Literature  1.3 2.  1  .  .  .  .  .  .  .  . .  1 .  3  1.2.1  Characteristics o f space-based manipulators .  .  .  3  1.2.2  Dynamics and control o f space-based manipulators .  .  8  Scope o f the Investigation  .  .  .  .  .  .  13  D Y N A M I C S O F SPACE BASED M U L T I - M O D U L E M A N I P U L A T O R S  15  2.1  Background to Formulation  .  15  2.2  Simulation Methodology  .  .  .  .  .  .  17  2.3  Simulation Considerations  .  .  .  .  .  .  20  2.4  System Response  .  .  .  27  .  .  .  .  .  .  .  .  .  2.4.1  Effect o f manipulator location and orientation  .  .  27  2.4.2  Response due to platform's tip excitation  .  .  .  38  2.4.3  Response to manipulator tip displacement  .  in  .  .  41  3.  Response in presence o f manipulator maneuvers  2.4.5  Effect o f system parameters  .  .  .  . .  . .  3.2  3.3  60  Control Methodologies 3.1.1  F L T control  3.1.2  F L T / L Q R control  .  .  .  .  .  3.2.1  F L T control  3.2.2  F L T / L Q R control  System Description  62 64  .  .  .  .  .  .  66 .  70 72  .  .  .  .  .  .  .  .  .  GROUND BASED EXPERIMENTS 4.1  .  Simulation Results and Discussion: Commanded Maneuvers  Trajectory Tracking  44 47  SYSTEM CONTROL 3.1  4.  2.4.4  .  .  .  76  .  91  .  106  .  .  .  .  .  .  106  4.1.1  Manipulator base  .  .  .  .  .  .  108  4.1.2  Manipulator module  .  .  .  .  .  .  108  4.1.3  E l b o w joint  .  .  .  .  .  .  112  .  .  .  .  112  .  4.2  Hardware and Software Control Interface  4.3  Digital Control of the Ground-Based Manipulator  .  .  .  117  4.3.1  Discretization o f inner nonlinear loop  .  .  .  119  4.3.2  Joint velocity estimates from position measurements  .  119  4.3.3  Discretization o f outer P D / P I D control loop .  .  120  4.4  Controller Implementation 4.4.1  .  .  .  .  . .  Dynamical equations for the ground-based Manipulator  iv  .  122 .  122  4.5  5.  4.4.2  Control system parameters  4.4.3  Controller design  Trajectory Tracking  .  4.5.1  Straight line trajectory.  4.5.2  Circular trajectory  .  . .  .  . .  . .  .  .  .  .  . .  . .  124  .  . .  .  .  126  . .  .  . .  .  127 128  .  CONCLUDING REMARKS  135 142  5.1  Contributions .  .  .  .  .  .  .  .  142  5.2  Conclusions  .  .  .  .  .  .  .  143  5.3  Recommendations for Future W o r k .  .  .  .  .  144  .  REFERENCES  146  APPENDIX I: SPECTRAL DENSITY ANALYSIS OF DYNAMICAL RESPONSE  151  V  LIST OF SYMBOLS  A, B  state-space representation o f the flexible subsystem, E q . (3.14)  CM.  center o f mass  d  position o f the manipulator base from the center o f the platform  e  orbital eccentricity  e,, e  tip vibrations o f modules one and two, respectively  2  e  platform tip vibration  p  Eld, E I EI  S  Flexural rigidity o f deployable and slewing links, respectively Flexural rigidity o f the platform  P  F  vector containing the terms associated with the centrifugal, Coriolis, gravitational, elastic, and internal dissipative forces  F ,F  deployment/retrieval forces at the prismatic joints one and two, respectively  I  moment o f inertia o f the revolute joint  x  2  JZ  J  quadratic  cost  function  which  considers  tracking  errors  and  energy  expenditure, E q . (3.17) Ki  spacecraft's inertia parameter, (I  Kj  Stiffness o f the revolute joint  ^LQR  optimal gain matrix, E q . (3.18)  K, K  diagonal control matrices containing the proportional and derivative gains,  p  v  -I )l x  I , E q . (2.5) z  respectively Id, l  length o f deployable and slewing links, respectively  s  l, l x  2  lengths o f the manipulator modules one and two, respectively  vi  l  length of the platform  L . H . , L.V.  local horizontal and local vertical, respectively  p  ma, m  mass of deployable and slewing links, respectively  s  nij  mass of the revolute j oint  m  mass of the platform  M  system mass matrix  M, F  estimates of the discretized mass matrix and nonlinear terms M and F,  p  respectively; Eq. (4.3) M, K  mass and stiffness matrices for the linearized system, respectively; Eq. (3.11)  M, K  mass and stiffness matrices, respectively, corresponding to the elastic subsystem; Eq. (3.12)  M,  Mjy,  rr  M  , M  rf  fr  rigid and flexible contributions to the system mass matrix M, respectively coupled contributions to the system mass matrix M  N  number of bodies (i.e. platform and manipulator units) in the system  O(N)  order-TV solution to the matrix Ricatti equation, Eq. (3.19)  q  set of generalized coordinates leading to the coupled mass matrix M  q  operation point used to linearize the governing nonlinear equation  q  discretized generalized coordinates q  0  k  q,  q  r  f  q  rigid and flexible generalized coordinates, respectively specified or constrained coordinates  s  Aq  s  desired variation of the specified or constrained coordinates  vii  qa  desired value o f q  Q  generalized forces, including the control inputs , i?  ( ? L Q R  symmetric weighting matrices which assign relative penalties to state errors  L Q R  and control effort, respectively; E q . (3.17) t  time  T  total kinetic energy of the system torques provided by control momentum gyros for attitude control and  T , Tf pr  p  vibration suppression, respectively torques provided by actuators located at revolute joints one and two,  T,, T  2  respectively vector containing the F L T control inputs, E q . (3.5)  H u  input determined from the Linear Quadratic Regulator, E q . (3.16)  x, y, z  body coordinate system; in equilibrium x, y in the orbital plane with x along  L  the local vertical, y along the local horizontal and z aligned with the orbit normal x, y  desired trajectory for the manipulator tip  xt  state vector for the flexible subsystem  d  d  Greek Symbols rigid body rotations during slew maneuvers o f modules one and two,  a, a x  2  respectively J3 , / ? X  2  contributions due to flexibility o f revolute joints at modules one and two, respectively;  viii  /,  rotation o f the frame F i , attached to the module i , with respect to the frame Fi.i  AT  time required for maneuver, E q . (2.4)  v  digital joint velocity estimate, E q . (4.6)  6  true anomaly o f the system  T  time from start o f maneuver  r  digital control input for the robot arm, E q . (4.4)  r  librational period o f the platform  k  k  p  co  desired natural frequency associated with the error equation for rigid degrees  i  o f freedom, E q . (3.9) co  ¥  frequency o f platform librational motion  a)  p  platform's bending natural frequency  co  first revolute joint's torsional natural frequency  co  first module's bending natural frequency  jX  mX  G)j  second revolute joint's torsional natural frequency  a)  second module's bending natural frequency  2  m2  elastic deformation o f the i y/  p  t h  -1 body in the transverse direction  platform's pitch angle  A dot above a character refers to differentiation with respect to time. character denotes a vector quantity.  A boldface italic  A boldface character denotes a matrix quantity.  Subscripts ' p ' a n d ' d ' correspond to the platform and deployable link, respectively. Subscript 's' refers to the slewing link or a specified coordinate.  LIST O F F I G U R E S Figure 1-1  Artist view o f the International Space Station with its M o b i l e Servicing System ( M S S ) as prepared by the Canadian Space Agency.  2  Figure 1-2  A l l the space-based manipulators have used, so far, revolute joints thus permitting only slewing motion o f links.  4  Figure 1-3  Variable geometry manipulator showing: (a) single module with a pair o f slewing and deployable links; (b) Several modules connected to form a snake-like geometry.  5  Figure 1-4  Variable geometry character.  avoidance  6  Figure 1-5  Challenges faced by studies aimed at dynamics and control o f large, space-based systems.  9  Figure 1-6  Schematic diagrams o f space structure models: (a) Lips [33], rigid spacecraft with deployable beam-type members; (b) Ibrahim [34], rigid spacecraft with deployable beam- and plate-type members; (c) Shen [35], rigid spacecraft with slewing-deployable appendages; (d) M a r o m [36], flexible platform supporting one rigid slewingdeployable manipulator module and a rigid payload at the deployable link end. The revolute joint was flexible.  11  Figure 2-1  Schematic diagram o f the mobile flexible variable geometry manipulator, based on an elastic space platform.  16  Figure 2-2  Coordinates describing flexibility o f revolute joints.  18  Figure 2-3  Normalized time histories o f the sinusoidal maneuvering profile showing displacement, velocity, and acceleration.  21  Figure 2-4  Schematic diagram o f a two-module, flexible, variable geometry manipulator, based on an elastic space platform, considered for study.  23  Figure 2-5  Schematic diagram showing important parameters appearing in the response study.  26  Figure 2-6  Equilibrium o f the system as affected by the manipulator's geometry  29  manipulator  showing  obstacle  and location on the platform. The diagram shows a case where the links are locked in position at a = a x  2  = 90° and /, =l  2  = 7.5m. The  equilibrium configuration deviates from the local vertical by « 0.13°.  XI  Figure 2-7  Direction of rotation of the platform, initially in the local horizontal equilibrium position, due to a change in the manipulator orientation. It is governed by the location of the center of mass in different quadrants.  30  Figure 2-8  System response with a two-module manipulator located at different positions on the platform: (a) d = 60 m; (b) d = 0.  31  Figure 2-9  System response as affected by the initial orientation of the platform:  Figure 2-10  (a) ^ ( 0 )  = +10°;  33  (b) ^ ( 0 )  = -10°;  34  (c) ^ ( 0 ) = -80°.  35  System response with the manipulator when the platform is initially at: (a) y/ (0) = -90°, i.e. aligned with the local vertical;  37  (b) ^,(0) = ^ Figure 2-11  =-90.13°; (c) ^ ( 0 ) = - 9 1 ° .  Response of the system to an initial disturbance of 0.5m tip displacement of the platform: (a) short duration behavior of rigid and flexible degrees of freedom; (b) long duration evolution of the platform's pitch libration.  39 40  Figure 2-12  System response to the manipulator's tip displacement of 0.2 m.  42  Figure 2-13  System response in presence of a manipulator maneuver with the platform initially aligned with the local vertical: (a) long duration evolution of the platform pitch libration; (b) short duration behavior of rigid and flexible degrees of freedom.  45 46  System response in presence of a manipulator maneuver with the platform initially aligned with the local horizontal: (a) long duration evolution of the platform pitch libration; (b) short duration behavior of rigid and flexible degrees of freedom.  48 49  Figure 2-14  Figure 2-15  System response as affected by payload with the manipulator completing simultaneous slew and deployment maneuvers in 0.03 orbit.  50  Figure 2-16  Influence of the speed of maneuver on the system response.  52  Figure 2-17  System response as affected by the joint stiffness which appears to be a rather critical parameter.  54  Figure 2-18  System response as affected by the number of modes used in flexibility discretization.  56  xii  Figure 2-19  A likely procedure would involve slew and deployment separately instead o f all four joints maneuvering at the same time.  57  Figure 2-20  System response, even under a set o f conservative values for payload (2,000 kg), joint stiffness ( 5 x l 0 Nm/rad) and maneuver speed (0.05 orbit), is unacceptable. This suggests a need for control.  58  Figure 3-1  A two-module symbols.  used  61  Figure 3-2  Schematic diagram o f the manipulator system showing the location of the control actuators. The two torques T f/2, opposing each other, control the platform's vibration by regulating the local slope.  63  Figure 3-3  FLT-based control scheme showing inner and outer feedback loops.  67  Figure 3-4  B l o c k diagram illustrating the combined F L T / L Q R approach applied to the manipulator system.  71  Figure 3-5  Response o f the system during a simultaneous 90° slew and 7.5m deployment maneuver o f the two-unit manipulator with rigid degrees of freedom controlled by the F L T : (a) rigid degrees o f freedom and control inputs for module 1; (b) rigid degrees o f freedom and control inputs for module 2; (c) flexible degrees o f freedom.  4  manipulator  system,  showing  frequently  p  Figure 3-6  Figure 3-7  Figure 3-8  F L T / L Q R controlled response o f the two-unit system during a simultaneous 90° slew and a 7.5 m deployment maneuver o f the manipulator units: (a) rigid degrees o f freedom and control inputs for module 1; (c) rigid controlled degrees o f freedom and control inputs for module 2. (c) flexible degrees o f freedom.  73 74 75  77 78 jg  F L T / L Q R controlled response o f the system during a simultaneous 90° slew and 7.5 m deployment maneuver o f the two-unit manipulator with different payload ratios: (a) rigid degrees o f freedom and control inputs for module 1; (b) rigid degrees o f freedom and control inputs for module 2; (c) flexible degrees o f freedom.  81 82 83  Effect o f the speed o f maneuver on the F L T / L Q R controlled response: (a) platform and module 1; (b) module 2; (c) flexible degrees o f freedom.  85 86 87  Xlll  Figure 3-9  Figure 3-10  Figure 3-11  Figure 3-12  F L T / L Q R controlled response as affected by the revolute joint stiffness during a manipulator maneuver: (a) platform and module 1; (b) module 2; (c) flexible generalized coordinates.  88 89 90  Tracking o f a straight line using the second module and the F L T : (a) manipulator tip trajectory and errors; (b) response o f the platform and rigid degrees o f freedom with control inputs; (c) response o f flexible degrees o f freedom.  95  Tracking o f a horizontal straight line with two revolute joints using the F L T : (a) initial configuration and the trajectory error; (b) response o f rigid degrees o f freedom and control inputs; (c) response o f flexible degrees o f freedom.  97 98 99  93 94  Tracking o f a circle using the two revolute joints and the F L T for control: (a) manipulator tip trajectory and error during the period o f 0-200 s; (b) manipulator tip trajectory and error during the period o f 200400s; (c) response o f rigid degrees o f freedom and control inputs; (d) response o f flexible degrees o f freedom.  102 103 104  Figure 4-1  The prototype manipulator system.  107  Figure 4-2  M a i n components o f the two-module manipulator system.  109  Figure 4-3  M a i n components o f the manipulator base assembly.  110  Figure 4-4  Prismatic joint mechanism which provides the deployment and retrieval capability.  112  Figure 4-5  M a i n components o f the elbow joint assembly.  113  Figure 4-6  Open architecture o f the manipulator control system for a single joint.  114  Figure 4-7  Digital robot control scheme.  118  Figure 4-8  Two-module ground based manipulator system.  123  Figure 4-9  Schematic diagram o f the swing test to determine the moment o f inertia o f a manipulator link with different lengths.  125  xiv  101  Figure 4-10  Schematic diagrams for straight line tracking using: (a) one revolute joint and one prismatic joint; (b) two revolute joints.  129  Figure 4-11  Straight line tracking using one revolute and one prismatic joint under the P I D control: (a) tip trajectory; (b) joint motion and the corresponding control signals.  130  Figure 4-12  Straight line tracking using two revolute joints under the P I D control: (a) tip trajectories; (b) joint motion and corresponding control signals.  132  Figure 4-13  Straight line tracking using one revolute and one prismatic joint with the F L T : (a) tip trajectory; (b) joint motion and the corresponding control signals.  133  Figure 4-14  Straight line tracking using one revolute and one prismatic joint: comparison o f the F L T and P I D procedures.  134  Figure 4-15  Schematic diagram showing tracking o f a circular trajectory using module 2.  136  Figure 4-16  Tracking o f a circle, at a speed o f 0.314 rad/s, using the P I D control: (a) tip trajectories; (b) joint dynamics and control signals.  137  Figure 4-17  Tracking o f a circle, at a speed o f 0.628 rad/s, using the P I D control: (a) tip trajectories; (b) joint motion and control signals.  138  Figure 4-18  Circle tracking behavior under the P I D control at a speed o f 0.209 rad/s: (a) tip trajectories; (b) joint dynamics and control effort.  139  Figure 4-19  Tracking o f a circle, at a speed o f 0.314 rad/s, with the F L T : (a) tip trajectories; (b) joint dynamics and control effort.  140  Figure 1-1  Initial deflection o f 5° applied to the revolute joint o f module 1: (a) system response; (b) power spectral density distribution.  154 155  Module 1 tip deflection o f 0.2m: (a) system response; (b) power spectral density distribution.  156 157  Figure 1-2  Figure 1-3  Module 2 tip deflection o f 0.2m: (a) system response; (b) power spectral density distribution.  XV  158 159  Figure 1-4  Module two joint deflection o f 5°: (a) system response; (b) power spectral density distribution.  xvi  160 161  LIST OF TABLES Table 2-1  Important factors affecting the system performance.  22  Table 2-2  Fundamental natural frequencies o f the components, forming the platform-based manipulator system, i n absence o f payload.  43  Table 4-1  Swing-test results for the ground-based robot.  125  Table 4-2  Controller gains for the prismatic joint o f module 2.  127  Table 4-3  Controller gains for the revolute joint o f module 2.  127  Table 4-4  Controller gains for the revolute joint o f module 1.  128  XVll  ACKNOWLEDGEMENT  I wish to thank m y supervisor Prof. V i n o d J. M o d i for his guidance, teaching and friendship throughout m y graduate studies. H i s constant encouragement took me on an incredible journey o f experience and learning throughout m y graduate program.  I wish to thank m y supervisor Prof. C . W . de Silva for his support and advice towards the completion o f m y research project.  Thanks are also due to my colleagues and friends. In the alphabetical order I wish to thank: M r . Mathieu Caron, M r . Jooyeol Choi, M r . M a r k C h u , M r . Vijay Deshpande, M r . Jean-Francois Goulet, M r . Vincent Den Hertog, Dr. Sandeep Munshi, M r . Behara Patnaik, M r . Kenneth Wong, and M r . Jian Zhang. They have shared their knowledge, experience, and culture with me and have thus broadened m y horizons. They have made my stay i n the University o f British Columbia very enjoyable.  Funding for this research project has been provided through a Strategic Research Grant 5-82268 from the Natural Sciences and Engineering Research Council o f Canada. It is jointly held by Dr. de Silva and Dr. M o d i at the University o f British Columbia, and Dr. A . K . M i s r a at M c G i l l University.  xviii  1. I N T R O D U C T I O N 1.1  Preliminary Remarks Robotic systems have been used in space as early as the 1960s [1]. In the late 60s, the  unmanned Surveyor lunar mission used a rudimentary manipulator arm to dig and collect soil samples. The versatility o f the space robots was demonstrated during the Surveyor 7 mission where the manipulator was employed to jab open an instrument that had failed to deploy automatically. In 1970, and again in 1973, the Soviet Lunakhod rovers surveyed large areas o f the moon and used a deployable arm to lower an instrumentation package to the surface. The V i k i n g landers, i n 1976, used robotic manipulators to collect and process Martian soil samples. The Canadian contribution to space robotics has been through the now famous Canadarm, introduced in 1981. It has played diverse, significant roles i n almost all N A S A ' s Space Shuttle missions: platform to support astronauts; position experiment modules; satellite launch and retrieval; loosen a jammed solar panel; even knocked-off a block o f ice from a clogged waste water vent [2]. Perhaps its most dramatic success came in 1993 when it successfully retrieved the malfunctioning Hubble Space Telescope, placed it i n the cargo bay for repair and relaunched it. In December 1998, it assisted i n the integration o f the U . S . ' U n i t y ' module with the Russian control module called 'Zarya' (Sunrise), launched a few weeks earlier, thus initiating construction o f the International Space Station. For the Space Station, which is scheduled to be operational i n year 2004, the Canadian contribution is through an extension o f the Canadarm i n the form o f Mobile Servicing System ( M S S , Figure 1-1). It consists o f the Space Station Remote Manipulator System ( S S R M S ) and Special Purpose Dexterous Manipulator ( S P D M ) . The M S S w i l l play  1  an important role in the construction, operation, and maintenance o f the space station [3-5]. It w i l l also assist in the Space Shuttle docking maneuvers; handle cargo; as well as assemble, release, and retrieve satellites.  Figure 1-1  A  Artist view o f the International Space Station with its M o b i l e Servicing System (MSS) as prepared by the Canadian Space Agency.  number o f other space robots have been proposed and some are  under  development. The American Extravehicular Activity Helper/Retriever ( E V A H R ) and Ranger Telerobotic Flight Experiment, as well as the Japanese E T S - V I I , are examples o f free-flying telerobotic systems which w i l l be used for satellite inspection, servicing and retrieval [6,7]. Thus manipulators are serving as useful tools in the space exploration. A l l indications suggest the trend to accentuate with future missions becoming more dependent on robotic systems. A s the Space Station w i l l operate in the harsh environment at an altitude o f 400 k m , it is desirable to minimize extravehicular activity by astronauts. Robotics is identified as one  2  of the key technologies to reach that goal. It is important to point out that all the space-based robotic devices mentioned above use revolute joints, i.e. links are free to undergo slewing motion (Figure 1-2), as i n the case o f the Canadarm and M S S abode the International Space Station. W i t h this as background, the thesis undertakes a study aimed at a novel flexible multimodule manipulator capable o f varying its geometry. Each module consists o f two links (Figure l-3a), one free to slew (revolute joint) while the other is permitted to deploy and retrieve (prismatic joint). A combination o f such modules can lead to a snake-like variable geometry manipulator (Figure l-3b) with several advantages [8]. It reduces coupling effects resulting in relatively simpler equations o f motion and inverse kinematics, decreases the number o f singularities, and facilitates obstacle avoidance for comparable degrees o f freedom (Figure 1-4). Dynamics and control o f such Multi-module Deployable Manipulator ( M D M ) system, free to traverse an orbiting elastic platform and carrying a payload, represent a challenging task.  1.2  A Brief Review of the Relevant Literature A s can be expected, the amount o f literature available on the subject o f robotics is  literally enormous. The objective here is to touch upon contributions directly relevant to the study in hand.  1.2.1  Characteristics of space-based manipulators There are several significant differences between the orbiting space platform  supported manipulators and their ground-based counterparts:  3  4  Payload  (b)  Module of SlewingDeployable Links  Figure 1-3  4 Trajectory  Variable geometry manipulator showing: (a) single module with a pair of slewing and deployable links; (b) several modules connected to form a snakelike geometry.  Figure 1-4  Variable geometry manipulator showing obstacle avoidance character.  6  Due to zero-weight condition at the system center o f mass and microgravity field elsewhere, the environmental torques due to free molecular flow, Earth's magnetic field and solar radiation can become significant i n the study o f space manipulators [9]. The large temperature variations encountered i n space may significantly affect the system dynamics and control due to thermal deformations [10,11]. A s the manipulator rests on a flexible orbiting platform, their dynamics are coupled [12,13]. The manipulator maneuvers can affect attitude o f the platform as well as excite it to vibrate [14]. Conversely, the librational motion o f the platform would affect the manipulator's performance. Fortunately, manipulator maneuvers in space tend to be relatively slow permitting the end-effector to approach equilibrium [15]. Space manipulators tend to be large i n size, lighter and highly flexible. Obviously, this w i l l make the study o f system dynamics , and its control, a formidable task. The ratio o f the payload to manipulator mass for a typical space-based system can be several orders o f magnitude higher [16]. For example, i n case o f the Canadarm the ratio is 61.5. The corresponding ground-based manipulator used in nuclear industry (supplied by the same manufacturer) has the payload to manipulator mass ratio o f 0.167 ! Obviously, space manipulators are not readily accessible for repair in case of, say, joint failure. This requires incorporation o f a level o f redundancy i n their design [17]. Correspondingly, more degrees o f freedom are involved than required for a given task.  (f)  Remote operation o f a space-based manipulator would involve time delays, an important factor in control o f the system. For the R O T E X teleoperation experiment it reached seven seconds [18] ! These important differences emphasize the fact that one cannot entirely rely on the  vast body o f literature available for ground-based manipulators. W e w i l l have to explore and understand distinctive character o f the space robotic systems. Dynamics and control o f a large orbiting flexible platform (like the International Space Station), supporting a mobile elastic manipulator, carrying a compliant payload represent a class o f problems never encountered before. Major challenges presented by such large-scale systems are summarized in Figure 1-5. It is only recently, some o f the issues mentioned here have started to receive attention. Obviously, there is an enormous task facing space dynamicists and control engineers that w i l l keep them occupied for years to come. The points which concern us are the nonlinear, nonautonomous and coupled character o f the governing equations o f motion, relatively low frequencies, and development o f a controller, preferably robust.  1.2.2  Dynamics and control of space-based manipulators From the observations made earlier, it is apparent that space manipulators, as well as  large flexible space structures in general, have unveiled a new and challenging field o f space dynamics and control. Over the years, a large body o f literature has evolved, which has been reviewed quite effectively by a number o f authors including Meirovitch and K w a k [19], Roberson [20], Likins [21], as well as M o d i et al. [22 - 26]. In the majority o f studies aimed at manipulators, only revolute joints were involved, i.e. links were permitted to undergo slewing motion. O n the other hand, several space  8  6  era' C/3  n  25  >o o *1  CD  H o  oH  hd  O H  ^ s > O HH w w 25 oo H o d 2! § d 25 © 25 * d C/3 OS 03 **5 w H td c/i *d HH r W H Cd w o  Ul  HH  n 3*  hH  £L  ts H O  CT*  3  tra CT cn P^  o  CT  ° s s H O Q  cr  C/3  o w  CT' CO  p  O  CT P.  a  3  "I  d o  w C/3  o  d o  o  n r d  H O 2!  C/3  O  H  o d  HH  2!  O H 25 K  0  C/3  &  >  P  o co  05 m  P-  ~n n  O  o  m o —i c/>  3  o o p  1-1  CT  o  CO  T3  03  cr  03  P CO  CT PCO  '<  o d n H  era  P O CT  n O o d hd d 2 d o 03 o C/3 H w w hH o O d O  d  2  25  d  03 3  O  03  03 W  25  tr  w 1  O n H  d  25  H  *5 *d  o r  C/3  •<  C/3 H n HH H H o M „ 03 d 25 25 H O 25 H 25 w o© Cd H>  03  > 25  CO  O  o 25  HH  o H K  O  03 HH  2!  r  HH HH  •H  CZl  r  03  nM or d 25 do d 25 O d  O  O  03  m  25  O o  50  5  H  O  n dHo 25 o Q ^d H d » 03  gd Or CZl  o d O W X ts HH Cd 25 d HH w C/3 o H 25 d sn 25  w 25  H  H  d 03  O -<  O C/)  structures feature deployment capabilities. For instance, a large solar array was deployed from the Space Shuttle cargo-bay during the Solar Array Flight Experiment ( S A F E ) , in September  1984.  Cherchas [27], as well as Sellappan and B a i n u m [28], studied the  deployment dynamics o f extensible booms from spinning spacecraft.  Lips and M o d i [29,30]  have studied at length the dynamics o f spacecraft with a rigid central body connected to deployable beam-type members.  M o d i and Ibrahim [31] presented a relatively general  formulation for this class o f problems involving a rigid body supporting deployable beamand plate-type members.  Subsequently M o d i and Shen [32] extended the study to account  for deployment as well as slewing o f the appendages. Lips [33], Ibrahim [34], and Shen [35] have reviewed this aspect o f the literature in some detail. In the above mentioned studies [29-35], although slewing and/or deployment were involved, each appendage was directly connected  to the central body, i.e. a manipulator-type  (appendages) was not involved.  chain geometry  o f the links  Figure 1-6 shows schematically the different  models  described above. The new manipulator system, schematically shown i n Figures l-3(a) and l-6(d), was first proposed for space application by M a r o m and M o d i [36]. Planar dynamics and control o f the one module mobile manipulator with a flexible revolute joint, located on an orbiting flexible platform, were investigated. Results showed significant coupling effects between the platform and the manipulator dynamics. Control o f the system during tracking o f a specified trajectory, using the computed torque technique, proved to be quite successful. M o d i et al. [37] as w e l l as Hokamoto et al. [38] extended the study to the multimodule configuration, referred to as the M o b i l e Deployable Manipulator ( M D M ) system.  The model, with an  arbitrary number o f modules, accounted for the joint as well as link flexibility. A relatively  10  Figure 1-6  Schematic diagrams o f space structure models: (a) Lips [33], rigid spacecraft with deployable beam-type members; (b) Ibrahim [34], rigid spacecraft with deployable beam- and plate-type members; (c) Shen [35], rigid spacecraft with slewing-deployable appendages; (d) M a r o m [36], flexible platform supporting one rigid slewing-deployable manipulator module and a rigid payload at the deployable link end. The revolute joint was flexible.  ll  general formulation for three-dimensional dynamics o f the system i n orbit was the focus o f the study by M o d i et al., while Hokamoto et al. explored a free-flying configuration. More recently, Hokamoto et al. [40] studied control o f a single unit system and demonstrated successful tracking o f a straight-line trajectory at right angle to the initial orientation o f the manipulator. A comment concerning a rather comprehensive study by Caron [12] would be appropriate. He has presented an 0(N) formulation for studying planar dynamics and control of such formidable systems.  The dynamical parametric study [42] clearly shows involved  interactions between the orbital motion, flexibility, librational dynamics, and manipulator maneuvers.  Furthermore, Caron [12] successfully demonstrated control o f a single module  (i.e. two links) manipulator, free to traverse a flexible platform, using the Feedback Linearization Technique ( F L T ) applied to rigid degrees o f freedom, and suppression o f flexible members' response through the Linear Quadratic Regulator ( L Q R ) . Recently, Chen [43] extended Caron's study and presented an order-N formulation for three-dimensional motion o f a mobile manipulator traversing an orbiting flexible platform. Control o f a single module manipulator, i.e. with two flexible links and an elastic revolute joint, was investigated with the Feedback Linearization Technique applied to the rigid degrees o f freedom.  The controlled response o f the system during commanded maneuvers o f the  manipulator was surprisingly good.  Goulet [44] studied control o f a single-unit rigid  manipulator with a knowledge-based hierarchical approach.  The control strategy proved  quite successful during pick-and-place operations as well as trajectory tracking. Based on the literature review, following general observations can be made:  12  (i)  Although there is a vast body o f literature dealing with modeling, dynamical performance and control o f space-based manipulators, most o f it is concerned with systems having revolute joints.  (ii)  Manipulators with revolute as well as prismatic joints have received relatively little attention, and that too only recently.  A s the concept o f space-based manipulators  with slewing and deployable links was developed at U . B . C , the few contributions in the field have also come from the same source.  Here too, focus has been on the  dynamical response o f the system. (iii)  A few reported control studies involve one-module manipulator, i.e. the system comprised o f two links: One free to slew while the other is permitted to deploy. The control o f multimodule manipulator does not seem to have received attention.  1.3  Scope of the Investigation W i t h this as background, the thesis investigates planar dynamics and control o f a two-  module (four links) flexible manipulator based on an elastic orbiting platform.  To begin  with, i n Chapter 2, model used and the governing order-A^ equations o f motion, as developed by Caron [12], are explained.  A detailed dynamical response study is undertaken which  assesses the influence o f initial conditions, system parameters, and manipulator maneuvers. Results suggest that under critical combinations o f system parameters and disturbances the response may not confirm to the acceptable limit. This points to a need for active control.  Control o f multimodule manipulators with slewing and deployable links logically follows the dynamical study. Two different control methodologies are used (Chapter 3):  13  (a)  the nonlinear Feedback Linearization Technique applied to rigid degrees o f freedom with flexible generalized coordinates indirectly regulated through coupling, and  (b)  a synthesis o f the F L T and L Q R to achieve control o f both rigid and flexible degrees o f freedom.  Also, several prescribed trajectories are tracked with the F L T applied to rigid degrees o f freedom. So far the control studies were conducted using numerical simulations. In Chapter 4, a two-unit ground-based prototype manipulator, designed and constructed by C h u [8], is used to assess effectiveness o f Proportional-Integral-Derivative (PID) and F L T control procedures in performing several trajectory tracking maneuvers.  This brings to light, quite vividly,  problems o f friction and backlash often present i n real-life manipulator systems. The thesis ends with a brief review o f important conclusions, significant original contributions and suggestions for future study (Chapter 5).  14  2. D Y N A M I C S O F S P A C E - B A S E D M U L T I - M O D U L E M A N I P U L A T O R 2.1  Background to Formulation It was mentioned earlier that the order-Af Lagrangian formulation for the novel  variable geometry manipulator was developed by Caron [12].  The distinct features o f the  system model used may be summarized as follows: (a)  The manipulator with an arbitrary number o f modules, each carrying a slewing and a deployable link thus involving both revolute and prismatic joints, is supported by a mobile base free to traverse a platform.  The platform is in an  orbit around Earth (Figure 2-1). (b) The supporting platform, manipulator modules and revolute joints are treated as flexible. Prismatic joints are considered as integral parts o f modules. (c)  The module is permitted to have variable mass density, flexural rigidity and cross-sectional area along its length.  (d) The system is permitted to undergo planar librational as well as vibrational motions. The slewing maneuver at any joint is confined to the plane o f the orbit. (e)  The damping is accounted for through Rayleigh's dissipation function.  (f)  The governing equations account for gravity gradient effects, shift i n center o f mass as well as change in inertia due to maneuvers and flexibility.  A s pointed out i n Chapter 1, such a manipulator with a combination o f revolute and prismatic joints is able to change its geometry, has a marked decrease in dynamical coupling, a reduction i n the number o f singularity conditions, and can negotiate obstacles with ease. Note, the model considered is relatively general and applicable to a large class o f space- as well as ground-based manipulator systems.  15  Prismatic  Figure 2-1  Schematic diagram o f the mobile flexible manipulator, based on an elastic space platform.  16  variable  geometry  The Lagrangian approach adopted for derivation o f the governing equations is particularly well suited to the flexible multibody system, with a large number o f degrees o f freedom, under consideration.  It automatically satisfies holonomic constraints while the  nonholonomic constraints can be accounted for, quite readily, using Lagrange multipliers. The form o f the equations o f motion conveys a clear physical meaning i n terms o f contributing forces. Equally important is the fact that the equations are readily amenable to stability study and well suited for controller design. Furthermore, validity o f the formulation and numerical integration code can be checked with ease through the conservation o f energy for nondissipative systems. A comment concerning representation o f the revolute joint's flexibility would be appropriate. The rotation  o f the frame F, , attached to the module i, with respect to the  frame F,-_i has three contributions (Figure 2 - 2 ) : elastic deformation o f the i -1 body in the th  transverse direction  rotation o f the actuator rotor (a ), t  which corresponds to the  controlled rotation o f the revolute joint; and elastic deformation o f the joint i (/?,) which could be due to, for instance, flexible coupling. Hence,  2.2  Simulation  M e t h o d o l o g y  The equations governing the dynamics o f the robotic systems mentioned above are highly nonlinear, nonautonomous, and coupled. They can be expressed in the general form M(q,t)q + F(q,q,t) = Q(q,q,t),  (2.2)  where M(q,t) is the system mass matrix; q is the vector o f the generalized coordinates; F(q,q,t) contains terms associated with centrifugal, Coriolis, gravitational, elastic, and  17  Body M  Figure 2-2  Coordinates describing flexibility o f revolute joints.  18  internal dissipative forces; and Q(q,q,t) represents generalized forces, including the control inputs. Equation (2.2) describes the inverse dynamics (i.e. forces corresponding to a specified motion) o f the system. For simulations, forward dynamics is o f interest, and E q . (2.2) must be solved for q, q = M\Q-F).  (2.3)  The solution o f these equations o f motion generally requires 0(N  ) arithmetic  operations, where N represents the number o f bodies (modules) considered i n the study. In other words, the number o f computations required by the 0 ( N ) 3  algorithm w i l l vary as the  cube o f the number o f modules. It also depends on the number o f generalized coordinates associated with each module. Clearly, the computation cost can become prohibitive for a system with a large number o f modules or generalized coordinates. Hence, the development o f the 0(7V) formulation by Caron [12], where the number o f arithmetic operations increases linearly with the number o f bodies (or degrees o f freedom) in the system, promises a significant saving i n the computational cost. Equally important is a possibility o f real-time implementation o f a control strategy. It is often useful to specify some o f the generalized coordinates. For example, cases where the length o f the units is varied in a specified manner, or where joints are locked in place at a specified angle, require the use o f specified coordinates.  These coordinates are  prescribed through constraint relations which are introduced in the equations o f motion through Lagrange multipliers. In the present study, a sinusoidal acceleration profile is adopted for prescribed maneuvers.  It assures zero velocity and acceleration at the beginning and end o f the  maneuver, thereby reducing the structural response o f the system. The maneuver time-  19  history considered is as follows,  q (r) = -^ -<r J  s  2K  AT  where q  s  sin  '2K  ^  (2.4)  is the specified or constrained coordinate; Aq  T  s  is its desired variation; r is the  time; and A r i s the time required for the maneuver. The time history for q , s  plotted, for the case o f Aq =1 and Ar=\, s  q , and q s  s  are  in Figure 2-3.  A F O R T R A N program for the dynamical simulation o f the system integrates the acceleration vector q numerically using Gear's method, which is well suited for stiff systems of ordinary differential equations.  To reduce computational time during simulations, a  symbolic manipulation routine ( M A P L E V ) is used i n order to obtain analytical expressions for the integrals o f the shape functions.  Furthermore, efficient matrix multiplication  algorithms are developed to take advantage o f the structure o f various matrices involved.  2.3  S i m u l a t i o n Considerations The system performance is governed by a large number o f parameters. Some o f the  important variables are listed in Table 2-1. Obviously, a systematic change o f these variables would lead to a large volume o f information. However, it would also demand considerable amount o f time, effort and computational cost. Hence, one is forced to focus on cases that are likely to provide useful trends. These include the manipulator position; platform, link and joint flexibility; number o f modes; profile and speed o f maneuver; and mass o f the payload. Even with these selected parameters, the task is formidable. Hence only a few typical results corresponding to a twomodule manipulator, i.e. with four links, are presented for conciseness (Figure 2-4).  20  Figure 2-3  Normalized time histories o f the sinusoidal maneuvering showing displacement, velocity, and acceleration.  21  profile  Table 2-1 Important factors affecting the system performance.  orbit eccentricity  Parameters  mass o f : platform; links; joints; payload stiffness o f : platform; links; joints  Initial Conditions  .  damping o f : platform; links; joints  .  link length  .  platform attitude manipulator location and orientation deformation o f platform, manipulator links, joints  type : slewing; deployment; retrieval; base translation  Maneuvers ,  amplitude speed  Discretization  .  shape o f admissible functions (modes)  o  number o f admissible functions (modes)  22  Figure 2-4  Schematic diagram o f a two-module, flexible, variable geometry manipulator, based on an elastic space platform, considered for study.  23  Numerical values used i n the analysis, unless specified otherwise, are indicated below: Orbit: Circular orbit at an altitude of 400 k m ; period = 92.5 min. Platform: Geometry: circular cylindrical with diameter = 3 m ; axial to transverse inertia ratio of 0.005; Mass (m )=  120,000 kg;  Length (I )=  120 m;  p  p  Flexural Rigidity (EI ) = 5.5 x 10 N m . 8  2  P  Manipulator Position (d): d = 0 or 60 m. Manipulator Module (/,. / ) : 2  Initial length of the manipulator module (i.e. 4 + deployed length, 7.5 + k) is taken as 7.5 m, i.e. the deployable link is initially not extended. Here / , U s  represent lengths of slewing and deployable links, respectively. Manipulator L i n k s (Slewing and Deployable): Geometry: circular cylindrical with axial to transverse inertia ratio o f 0.005 Mass (m , m ) = 200 kg; s  d  Length (/ , / s  d>max  ) = 7.5 m;  Flexural Rigidity (EI , E I ) = 5.5 x 10 N m . 5  S  d  Revolute Joint: Mass (mj) = 20 kg;  24  2  Moment o f Inertia (L ) = 10 k g m ; 2  z  Stiffness (Kj) = 10 Nm/rad . 4  Note, the prismatic joint is treated as a part o f the slewing link. Payload: Nominally zero. Specified in figure legend when different. Modes: Fundamental mode for a cantilever beam with tip mass for modules, free-free beam mode for platform. Note, subscripts d, j , m, p, and s correspond to deployable link, revolute joint, manipulator, platform and slewing link, respectively. Initially the platform is in equilibrium either along the local vertical (stable) or aligned with the local horizontal (unstable) position. The manipulator links are aligned with the platform, i.e. they are also along the local vertical or local horizontal before the maneuver.  The system is subjected to a maneuver o f 90° in  slew and 7.5 m deployment. Maneuver time is variable. The damping is purposely assumed to be zero in all components to obtain conservative estimate o f the response, i.e. the damping coefficient for joints (Cj) as well as structural damping coefficients for manipulator links ( £ ) and the platform (£" ) are considered zero. m  More important specified and response  parameters are summarized below and indicated i n Figure 2-5: d  position o f the base from the center o f the platform;  e ,e  tip vibrations o f modules one and two, respectively;  e  platform tip vibration;  x, y  body fixed coordinate system with x aligned with the undeformed axis o f the  x  p  2  platform and y normal to x in the orbital plane;  25  Figure 2-5  Schematic diagram showing important parameters appearing in the response study.  26  a ,a x  rigid body rotations, during slew maneuvers o f modules one and two,  2  respectively; fti,j0  2  contributions due to flexibility o f revolute joints at modules one and two, respectively;  y/  platform pitch libration.  p  2.4  System Response This section studies effect o f system parameters and disturbances, i n the form  of initial conditions as well as manipulator maneuvers, on the resulting response.  It  clearly shows that under certain combinations o f system's physical properties and disturbances, the response can become unacceptable suggesting a need for control.  2.4.1  Effect of manipulator location and orientation A t the outset it must be recognized that the platform, a long flexible beam, has two  equilibrium positions: along the local vertical (stable) and aligned with the local horizontal (unstable).  The presence o f manipulator, when aligned with the platform, has virtually no  effect on the equilibrium as the geometry remains effectively unchanged, as w e l l as relatively massive (120,000 kg) character o f the platform compared to the manipulator (800 kg). However, with different orientations o f the modules, the geometry changes, i.e. they no longer remain i n alignment with the local vertical or the local horizontal position.  The  system's  and  new  equilibrium orientation  slightly deviates  depending  on the  slew  deployment characters as well as location o f the manipulator on the platform. For example, consider the equilibrium position corresponding to the case when the manipulator is located  27  at the tip (d = 60 m), a = a = 90°, and the deployable links o f both modules remain x  2  unextended (l = l =7.5 m) as indicated i n Figure 2-6. {  position (y/ ) pe  2  The deviation i n the equilibrium  from the local vertical (or local horizontal) is around 0.13°. This acts as a  small disturbance and sets the platform oscillating.  The local vertical orientation being  stable, the platform tends to move towards and oscillate about the new equilibrium position. O n the other hand, the local horizontal position being unstable, the platform moves away from it; the direction o f rotation being governed by the position o f the center o f mass with respect to the local horizontal (Figure 2-7). Figure 2-8 shows the effect o f a two-module manipulator located at the center o f the platform (d = 0) and at the platform tip (d = 60 m). Each module is taken to be 7.5 m long, i.e. the links are not deployed (/, = / =7.5 m). B o t h modules are locked i n position with 2  a  \ - 2 - 90° • This results i n a shift i n the center mass causing a pitch moment, which a  increases as the base supporting the manipulator moves towards the platform tip. Note, with d - 60 m , the peak platform deviation from the local vertical is « 0.27°. This may appear small, however, depending on the mission, the permissible deviation may be as small as 0.1°. The system oscillates about the new equilibrium position o f y/ ~ pfi  0.13°.  The librational  period o f 0.6 orbit matches precisely with the established value for a long spacecraft i n a circular orbit.  O n the other hand, with the manipulator at the center o f the platform, the  center o f mass is slightly below the local horizontal (i.e. -x ). cm  This leads to the pitch angle  for equilibrium, y/ , that is positive. The amplitude o f oscillation, as can be expected, is p e  rather small ( « 0.01°) due to an insignificant pitching moment caused b y the shift i n the center o f mass because o f the presence o f the manipulator.  28  4 7.5m  7.5m  Figure 2-6  Equilibrium o f the system as affected by the manipulator's geometry and location on the platform. The diagram shows a case where the links are locked i n position at ear, = a  2  = 90° and /, =l  2  = 7.5m. The equilibrium  configuration deviates from the local vertical by « 0.13°.  29  Local Vertical  Figure 2-7  Direction o f rotation o f the platform, initially i n the local horizontal equilibrium position, due to a change in the manipulator orientation. It is governed by the location o f the center o f mass i n different quadrants.  30  L.V.  Parameters: EI = 5.5xl0 Nm ;  Initial Conditions:  EL = 5.5xl0 Nm ;  A = 0, e = 0;  EI = 5 . 5 x l 0 N m ;  & = 0,  8  2  Vp > =Q  p  5  J = 60m  5  2  P  e  >  =  0  x  2  d  Kj = 1 . 0 x l 0 N m / r a d .  e =0. 2  4  C M .  Specified Coordinates: or, = a = 90°; /, = l = 7.5m. 2  d = 60m d=0  2  Platform L i b r a t i o n  Figure 2-8  System response with a two-module manipulator located at different positions on the platform: (a) d = 60 m ; (b) d = 0.  31  Figure 2-9 shows the effect o f initial conditions on the resulting response. The manipulator is located at the platform tip (d — 60m) with the same orientation as before, i.e. /j = / = 7.5m and a = a = 90°. 2  x  2  Three different initial positions o f the platform  with respect to the local vertical are considered: y/ (0) = + 1 0 ° , - 1 0 ° and - 8 0 ° . In all the p  three cases, the platform oscillates about the stable equilibrium position close to the local vertical (y/  = - 0 . 1 3 ° ) . W i t h a small damping, the system would have returned to the  equilibrium position, i.e. essentially to the local vertical orientation. Note, the amplitude of oscillations i n Case (a) is slightly larger than 10° ( » 10.13°) as the oscillations take place about the equilibrium position, which is now at y/  = - 0 . 1 3 ° . O n the other hand,  e  the amplitude i n Case (b) is slightly lower than 10° ( « 9.87°). Similarly i n Case (c), the starting position is 79.87° away from the equilibrium location leading to the resulting amplitude o f 79.87°. A word about the pitch librational period would be appropriate.  It can be obtained  from the governing equation [9] (l + ecos0)y/ -2esin0(y/' p  where:  p  +\) + 3K siny/ cosy/ l  p  p  =0,  (2.5)  y/ = pitch librational angle; e = orbital eccentricity; 6 = true anomaly; Kj = spacecraft's inertia parameter, (I  -I )/I ; x  z  x, y, z - body coordinate system; i n equilibrium x, y i n the orbital plane with x along the local vertical, y along the local horizontal and z aligned with the orbit normal.  IL.V.  Parameters: EIp = 5 . 5 x l 0 N m ;  Initial Conditions:  EL = 5.5xl0 Nm ;  A = 0,  EI = 5 . 5 x l 0 N m ;  A = 0, e =0.  8  ^,= +10°, e = 0;  2  5  p  2  5  2  2  d  e, = 0; 2  Kj = l . O x l 0 N m / r a d . 4  Specified Coordinates: or, = or = 90°; l =l = 2  x  2  7.5m.  Platform Libration  J  1  1  i  1  1  i  i  0.5  1  1  1  I  i  i  1  1  r  I  1.5  Orbit  Figure 2-9  System response as affected by the initial orientation of the platform: (a) {/,(()) = +10°  33  L.V.  Parameters: EI = 5.5xl0 Nm ;  Initial Conditions:  EL = 5.5xl0 Nm ;  A = 0,  e = 0;  EI = 5 . 5 x l 0 N m ;  A = 0,  e =0.  8  y =-\0°,  2  p  5  2  5  e = 0;  p  2  p  x  2  d  2  Kj = l . O x l 0 N m / r a d . 4  Specified Coordinates: a = a = 90°; l =l = 2  x  x  2  7.5m.  Platform Libration T  1  1  1  1  r  i  i  n  r  9.737°  10  -10  J  0.5  i  I  1  I  i  i_  1.5  Orbit  Figure 2-9  System response as affected by the initial orientation o f the platform: (b) ^ ( 0 ) = - 1 0 ° .  34  L.V.  Parameters: EI = 5.5xl0 Nm ;  Initial Conditions:  EI = 5 . 5 x l 0 N m ;  A = 0, e = 0 ;  EI = 5 . 5 x l 0 N m ;  /? = 0,  8  ^=-80°,  2  p  5  2  s  5  A  p  x  2  d  -iV/  e = 0;  2  e =0. 2  Kj = l . O x l 0 N m / r a d . 4  Specified Coordinates: a =a = 90°; /, = l = 7.5m. x  2  2  Platform L i b r a t i o n  Orbit  Figure 2-9  System response as affected by the initial orientation o f the platform: ( c ) ^ ( 0 ) = -80°.  35  Primes denote differentiation with respect to 6. For a circular orbit, e = 0, i.e. y/" + lK sm\i/ casy/ p  i  p  =0.  p  In the case o f a long spacecraft like a 120m long uniform platform (120,000 kg) with a small two-module manipulator (800 kg),  T = ^= V3 P  Thus the librational period r is given by p  radian per pitch cycle,  = -J= V3  orbit per pitch cycle,  = 0.577  orbit per cycle.  Note, i n Figures 2-9(a) and 2-9(b), the pitch amplitude is relatively small and the linear approximation is valid giving the pitch period o f around 0.58 orbit.  However, with large  amplitude oscillations i n Case (c), the nonlinearities significantly affect the period, which is now « 1.17 orbits. Figure 2-10(a) considers the case where the platform is initially aligned with the local horizontal, i.e. y/ (0) = - 9 0 ° . In absence o f the manipulator, the platform would have stayed p  there, being an equilibrium position. (/ =/ =7.5m, 1  y/  pe  2  a =a x  2  = 90°),  the  However, due to presence o f the new  equilibrium  orientation  manipulator  corresponds  to  = - 9 0 . 1 3 ° . This, being the unstable equilibrium position, causes the platform to swing  away from it. When the platform reaches y/ = p  90°, the same thing happens. A s seen in  Figure 2-7, the center o f mass is now i n quadrant III causing the platform to continue rotation in the anticlockwise sense. Thus, in absence o f dissipation, one has a perpetually rotating platform ! O f course, the energy input corresponds to the initial deviation o f the platform from the new equilibrium position.  36  i  Parameters: EI = 5.5xl0 Nm ;  Initial Conditions:  EL = 5.5xl0 Nm ;  A = 0, e, = 0;  EI = 5.5xl0 Nm ;  A = 0, e = 0 .  8  L.V.  e = 0;  2  p  a, 1i  5  1  h  p  2  5  2  d  2  Kj = 1.0xl0 Nm/rad. 4  a.  ft-:  ^=-90°;  Specified Coordinates:  -90.13°;  or, = or = 90°; 2  l\ =l =  ^=-91°.  7.5m.  2  Platform L i b r a t i o n i  270  1  r  \  180  / (a)  90 ¥  P  r  °(0) = - 9 0 ° / /  /  0  (b)  V°(0) = -90.133° -90 X -180 ¥  °(0)=-91°\  \  Jc)_  -270 j  •  i  ..  i  1  0.5  1.5  Orbit  Figure 2-10  System response with the manipulator when the platform is initially at: (a) W <P) = P  (0) = - 9 0 ° ,  i.e.  aligned  with  V .e = - 9 0 . 1 3 ° ; (c) ^ ( 0 ) = - 9 1 ° . P  37  the  local  vertical;  (b)  Note, when the platform is initially set at its equilibrium position, y/  = - 9 0 . 1 3 ° , it  remains there as it should (Figure 2-10b). Thus a small change i n initial conditions near the unstable configuration can lead to widely different system responses.  This is further  emphasized in Figure 2-10(c) where the platform is initially at y/ = - 9 1 ° . Note, now the p  system center o f mass is i n the fourth quadrant and experiences a clockwise moment.  The  platform continues to swing until its velocity becomes zero at y/ = +91°. The return journey p  in the clockwise sense starts and the platform continues to swing back and forth around the stable local vertical position. This is in sharp contrast to the results i n Figure 2-10(a) and (b).  2.4.2  Response due to platform's tip excitation W i t h the manipulator located at the tip and joints locked i n position as before  (/j = /  2  =7.5m, (X\ = a  2  = 9 0 ° , d = 60m), the end o f the platform, aligned with the local  vertical, was given an initial disturbance o f 0.5m (Figure 2-11). The platform vibrates at its natural frequency with a period o f around 0.001 orbit (« 6s). Note, due to coupling effects, the librational motion o f the platform is induced. The pitch librational response is modulated at the platform frequency.  The flexibility effects o f the manipulator joints and modules  (links) are also apparent. The platform tip motion and the corresponding tip moment (freefree beam) excite joints o f the manipulator, resting on the platform, with peak amplitudes o f f?i ~ 6° and {h « 25°. The beat character o f the response is attributed to the proximity o f the joint 2 and platform frequencies  (<»- «0.21 2  Hz, &^«0.18  Hz).  The beat period  approximately corresponds to 0.012 orbit (« 0.015 H z ) . One can discern presence o f the low  38  Initial Conditions:  Parameters: EIp = 5 . 5 x l 0 N m ;  y, =0,e  EL = 5.5xl0 Nm ;  A = o. i = ° ; /? = 0, e =0.  8  2  5  p  2  e  EI = 5 . 5 x l 0 N m ; 5  = 0.5m;  p  2  d  2  2  Kj = l . O x l 0 N m / r a d . 4  Specified Coordinates: a = a = 90°, l =l x  2  x  2  (a)  Platform Libration  = 7.5m. Platform Tip Vibration  m  0.5  0.03  v  e  0  -0.5 0.01  0.02  0  0.01 Tip Deflection of Module 1  First Joint Vibration 0.02  o l  0.02  m  e, 0  0  -0.02 0.01  0.01  0.02  Tip Deflection of Module 2  Second Joint Vibration 25  0.02  0.07 V m  r  e  2  0  -0.07  -25 0.01  0.01  0.02  Orbit  Orbit Figure 2-11  Response o f the system to an initial disturbance o f 0.5m tip displacement o f the platform: (a) short duration behavior o f rigid and flexible degrees o f freedom.  39  0.02  Initial Conditions:  Parameters: EI = 5.5xl0 Nm ;  i// =0,  EL  A = o. A = 0,  8  2  p  2  EI = 5.5xl0 Nm ; 5  e = 0.5m;  p  5.5xl0 Nm : 5  2  d  2  p  e  i  =  ° ;  e =0. 2  Kj = 1 . 0 x l 0 N m / r a d . 4  Specified Coordinates: a =a = x  2  90°, l =l = x  2  7-5m.  Platform L i b r a t i o n  -0.3 h  Orbit  Figure 2-11  Response o f the system to an initial disturbance o f 0.5m tip displacement o f the platform: (b) long duration evolution o f the platform's pitch libration.  40  frequency beat induced modulations in the platform's librational (y/ ) as well as tip (e ) p  responses. resonance  p  The large amplitude vibration o f the second revolute joint is due to near condition, its frequency  (co ~ 0.18 H z ,  being close to the platform's bending  frequency  « 0.21 H z ) which here acts as the forcing frequency.  L o n g duration evolution o f the librational response is presented in Figure 2-11(b). it clearly shows, as expected, a period o f around 0.6 orbit. The high frequency modulations are at the platform's bending frequency (0.18 H z ) . Small ridges at the periphery are due to the beat phenomenon as observed i n Figure 2-11(a).  2.4.3  Response to manipulator tip displacement It should be recognized that the system consists o f elastic and rigid degrees o f  freedom with wide variations i n their natural  frequencies.  Obviously, depending on the  disturbance, a number o f them with significant response would be excited revealing complex interactions at different  frequencies.  This is precisely the case with the manipulator  displacement o f 0.2m at the tip o f module 2 (Figure 2-12). To help appreciate involved coupling effects, it was desirable to establish natural frequencies o f the elastic members and dominant frequencies affecting a given response.  To that end the platform, two revolute  joints, and two modules were individually subjected to an initial disturbance and power spectral density functions o f the response were obtained. The results are given i n Appendix I. Natural frequencies o f various system components thus obtained are summarized in Table 2-2. W i t h the manipulator tip initially deflected through 0.2m, the system is set vibrating. In absence o f damping, the tip continues to oscillate (e ) at a constant amplitude and with a 2  41  Initial Conditions:  Parameters: EIp = 5 . 5 x l 0 N m ;  y/ = 0 , e = 0;  EI = 5 . 5 x l 0 N m ;  A = 0>  EI = 5 . 5 x l 0 N m ;  A = 0, e =0.2m.  8  2  5  p  2  s  5  2  p  2  d  e = 0; x  2  Kj = l . O x l 0 N m / r a d . 4  Specified Coordinates:  a =a = x  90°, l =l =  2  x  7.5m.  2  Platform Libration  Platform Tip Vibration xl0" m J  'P  o  -6h  0.01  0.02  0.01  0  Tip Deflection of Module 1  First Joint Vibration  0.01  0.02  Tip Deflection of Module 2 m 0.2  llMllliiBlBIBll liiililiil  IIHIl l i l l i l l i l l l  -0.2 0  0.02  0.01  Orbit  Orbit  Figure 2-12  •nn 0.01  0  0.02  Second Joint Vibration  0.01  0.02  System response to the manipulator's tip displacement o f 0.2 m.  42  0.(  Table 2-2  Fundamental natural frequencies o f the components, forming the platform-based manipulator system, i n absence o f payload.  Period Component  Frequency, H z Second  Orbit  3204  0.6  Platform, Libration  3.12xl0"  Platform, Bending  0.18  5.555  l.OxlO"  3  Module 1, Bending  5.85  0.171  3.1xl0"  5  Module 2, Bending  8.50  0.117  2.1xl0"  5  Joint 1, Torsion  0.08  12.500  2.3 xlO"  Joint 2, Torsion  0.21  4.762  8.6xl0"  4  43  3  3  frequency o f 5.2 H z (« 290 cycles in 0.01 orbit). coupling (ft)  This, in turn, excites joint 2 through  which oscillates at its natural frequency o f 0.21 H z (approximately 12  oscillations in 0.01 orbit) with high frequency module 2 oscillations superposed. displays a rather complex response showing coupling effects o f p\ and e  2  Joint 1  (Figure I-2b). The  platform tip shows a typical beat response, as its frequency is quite close to that o f fii (a)  J2  «0.21 Hz, a  « 0 . 1 8 H z ) as mentioned earlier.  p  corresponds to a period o f « 66s.  The beat frequency o f 0.015 H z  The platform pitch response is modulated at the h\  frequency with high frequency contribution from e appearing at the peaks. 2  2.4.4  Response in presence of manipulator maneuvers In the study so far, the manipulator joints were locked i n position, i.e. there were no  maneuvers involved.  The next logical step is to assess the effect o f the manipulator  executing slewing and deployment maneuvers.  The manipulator is taken to be at the tip o f  the platform, i.e. d = 60m. It undergoes simultaneous slewing (a , x  deployment (/,, l  2  a  2  from 0 to 90°) and  from 7.5m to 15m) in 0.01 orbit. Figure 2-13(a) shows the librational  response when the platform is initially aligned with the local vertical, a stable equilibrium position.  The character o f the response is quite similar to that observed i n Figure 2-8(a).  However, because o f the initial disturbance, the amplitude is higher. Effect o f the maneuver on the flexible degrees o f freedom can be appreciated from Figure 2-13(b). Note, both the joints are set into vibration with a characteristic frequency having a period o f « 28s (6 oscillations in 0.03 orbit). The module tip oscillations display the same frequency. Their amplitude due to flexibility o f the links alone (e,, e ) are rather significant, around 10cm and 2  44  Initial Conditions:  Parameters: EI = 5.5xl0 Nm ;  y =0,  EI = 5 . 5 x l 0 N m ;  A = 0, e = 0;  EI = 5.5xl0 Nm ;  A = 0,  8  2  p  5  p  2  s  5  e = 0; p  x  2  2  d  e =0. 2  Kj = 1.0x10 Nm/rad. 4  Specified Coordinates: or, = a = 0 - > 9 0 ° , /, = l = 7.5m->15m i n 0.01 orbit. 2  2  Platform Libration  Orbit  Figure 2-13  System response i n presence o f a manipulator maneuver with the platform initially aligned with the local vertical: (a) long duration evolution of the platform pitch libration.  45  Initial Conditions:  Parameters: EI = 5.5xl0 Nm ;  ^ =0,e  EI = 5 . 5 x l 0 N m ;  A = 0, e, = 0;  EI = 5.5xl0 Nm ;  A = 0, e = 0 .  8  2  p  p  5  2  s  5  2  d  2  p  = 0;  2  Kj = l . O x l 0 N m / r a d . 4  Specified Coordinates: a =a = x  2  0 - > 9 0 ° , /, = l = 7 . 5 m ^ l 5 m in 0.01 orbit. 2  Platform Libration  0.01  0.02  0.03  Platform Tip Vibration  0.04  0.01  '/  0.02  0.03  0  0.04  0.01  0.02  0.03  7  0  0.04  0.04  Orbit Figure 2-13  0.04  Tip Deflection of Module 2  e  0.03  0.04  0  Second Joint Vibration  0.02  0.03  Tip Deflection of Module 1  First Joint Vibration  0.01  0.02  System response i n presence o f a manipulator maneuver with the platform initially aligned with the local vertical: (b) short duration behavior o f rigid and flexible degrees o f freedom.  46  2cm, respectively, in the steady state. Note, the flexible joint vibrations with steady state amplitudes o f around 1.3° (/%) and 4.5° (J3\) would accentuate the effect leading to a significant deviation o f the manipulator tip from its desired position. This suggests a need for control. The librational response with the platform initially aligned with the local horizontal and the manipulator performing the same maneuver is presented i n Figure 2-14(a).  As  anticipated, the system becomes unstable and the anticlockwise rotational motion, similar to that observed i n Figure 2-10(a), sets in.  It is interesting to observe (Figure 2-14b) that  response o f the flexible generalized coordinates remains essentially the same as that for the platform along the local vertical (Figure 2-13b). Hence i n the subsequent study, focus is on the platform along the stable equilibrium position.  2.4.5  Effect of system parameters Next, the attention was turned to assess the influence o f several important parameters  such as payload, speed o f maneuvers, joint flexibility, and number o f modes used in the flexibility discretization. In all the cases, the initial orientation o f the platform (aligned with the local vertical) and the manipulator maneuver are purposely kept the same as before to assist in the comparison o f data and isolate parameter effects.  Effect of Payload Figure 2-15 presents the influence o f payload carried by the manipulator during the execution o f a prescribed maneuver o f slew and deployment ( a , a x  l  2  from 7.5m to 15m; maneuvers completed in 0.03 orbit).  47  2  from 0 to 90° and / , ,  Note, the maneuver time is  Initial Conditions:  Parameters:  L.V.  e  EI = 5.5xl0 Nm ;  y/ = - 9 0 ° , e = 0; p  p  EL  A=  = o;  8  x  Al  2  p  5.5xl0 Nm 5  2  A = 0, e = 0 .  EI = 5 . 5 x l 0 N m ; 5  2  d  or, fi  2  Kj = 1.0x10 Nm/rad. 4  Specified Coordinates: a =a x  2  = 0 - > 9 0 ° , l =l x  2  = 7.5m^>15m i n 0.01 orbit.  Platform Libration 270 h  180 h  P  90 h  Orbit  Figure 2-14  System response i n presence o f a manipulator maneuver with the platform initially aligned with the local horizontal: (a) long duration evolution o f the platform pitch libration.  48  I  a  JX_ 2  l.v.  Initial Conditions:  Parameters:  \  EIp = 5 . 5 x l 0 N m ; 8  = - 9 0 ° , e = 0;  2  ¥  p  p  EI = 5 . 5 x l 0 N m ;  A = 0, e, = 0;  EI = 5 . 5 x l 0 N m ;  A = 0, e = 0 .  5  2  s  5  2  d  2  2  Kj = l . O x l 0 N m / r a d . 4  Specified Coordinates: 0->90°, /, = l = 7 . 5 m ^ l 5 m in 0.01 orbit.  a =a = x  2  2  Platform Libration  Platform Tip Vibration  -90.3 0.01  0.02  0.03  0.04  0.01  0.03  0.04  0  -5  0.02  0.03  Tip Deflection of Module 1  First Joint Vibration  0.01  0.02  0  0.04  0.01  0.02  0.03  0.04  Tip Deflection of Module 2  Second Joint Vibration  e  4  0  -0.02 0.02  0.04  0.03  Orbit Figure 2-14  System response i n presence o f a manipulator maneuver with the platform initially aligned with the local horizontal: (b) short duration behavior o f rigid and flexible degrees o f freedom.  49  0.04  Parameters: EIp = 5 . 5 x l 0 N m ;  Initial Conditions:  EI = 5 . 5 x l 0 N m ;  A = 0,  e =0;  EI = 5 . 5 x l 0 N m ;  A = 0,  e =0.  Kj = l . O x l 0 N m / r a d .  Payload:  8  y/ = 0 , e = 0;  2  5  p  2  s  5  2  d  4  p  x  2  2  0; 400 k g ;  Specified Coordinates: a =a =  x  0->90°;  2,000 k g  /, = l =7.5m->15m; 2  in 0.03 orbit. Platform Libration  Platform Tip Vibration i  xl0~ m 3  /  1  i  1  1  p  1  — i  1  1  i i  \ \  /  \ ~ \  \  \  i  jS  \  0.02  0.03  0.04  \.>  0.01  0.05  First Joint Vibration  •"\  " V " ~" '  \ — 7 —  \7  /  i  •  0.02  0.03  •  0.01  •  \ \ \  0.04  0.05  Tip Deflection of Module 1  0.03  i  i  m  i  i  i  i  -  0.02 0.01  /  V /  0  h  0.02  0.03  0.04  \\ V  .  / A  /  7 -  -0.01 0.01  \  ///  1  0.01  0.05  0.02  \  " A  V //  I  0.03  0.04  0.05  Tip Deflection of Module 2  Second Joint Vibration 0.05  e  2  0  -0.05 0  0.01  Figure 2-15  0.02 0.03 Orbit  0.04  0  0.05  0.01  0.02 0.03 Orbit  0.04  0.05  System response as affected by payload with the manipulator completing simultaneous slew and deployment maneuvers in 0.03 orbit.  50  purposely taken longer (0.03 orbit instead o f 0.01 orbit) due to the presence o f a payload. A s can be expected, with an increase i n the payload ratio from 0 to 5 (i.e. from 0 to 4,000 kg), the joint oscillations reach a peak value o f « 6° for the first joint and around 3° for the second joint. Although the manipulator tip deflection ( e ) , which is measured relative to fii, appears 2  rather modest (« 7.5cm), that is somewhat misleading as pointed out before. Accounting for the two revolute joints' elastic deformations, the peak tip deviation from the desired orientation would be more than 2m along as well as normal to the platform. Obviously this would be unacceptable. Thus, one would be faced with: (i)  limiting the load carried to a lower value;  (ii)  reduction i n the speed o f maneuver;  (iii)  introduction o f an active control.  A s maneuver i n the present case is only one o f the numerous ones the manipulator w i l l be called upon to execute, and the speed o f maneuver is rather modest, introduction o f control is the logical solution.  Influence of Speed of Maneuver Figure 2-16 shows the effect o f speed o f maneuver (in absence o f payload).  Three  different speeds are considered: simultaneous slew and deployment maneuvers completed in 0.03 orbit (slow), 0.01 orbit (nominal rate), 0.005 orbit (fast). In general, the response results show trends as anticipated.  The platform being massive (120,000 kg), the effect o f  maneuvering speed on the peak librational response is the same, around 0.3°. The speed affects only the local response character. Note the high frequency modulations, for the fast maneuver case, are due to flexible joint oscillations.  51  Again, such large amplitude joint  Initial Conditions:  Parameters: EIp = 5 . 5 x l 0 N m ;  ^ = 0 ,  EI = 5 . 5 x l 0 N m ;  A  EI = 5 . 5 x l 0 N m ;  A = 0,  Kj = l . O x l 0 N m / r a d .  Speed of Maneuver:  8  2  5  2  s  5  2  d  e = 0; p  = °>  e =0.  2  4  2  0.005 orbit; 0.01 orbit;  Specified Coordinates: a{ = a = 0->90°; 2  0.03 orbit.  h h =7.5m—>15m. =  Platform Libration  = o;  \  e  Platform Tip Vibration  0.06  m  i  i  i  i  i  i  i i  i  i  11  t" i  i  1  p  '  0  i |•' j"I'I" |' ii | l  l  i if  i ii  i  : I|  0.02  0.03  0.04  0.01  0.01  0.02  0.03  0.01  0.04  0.02  0.03  .  '  0.02  0.02  0.02  0.04  ?::  1  ,i II i i N i' i . ' 1  1  n  ll K  i  0.03  0.03  0.03  Orbit  Orbit Figure 2-16  i n , 11  1  0.04  0.04  Tip Deflection of Module 2  Second Joint Vibration  0.01  I " " ! ! 111' i ) \'< li i i ' ! ' i i lis u •  1  Tip Deflection of Module 1  First Joint Vibration  0  'i ii |i  I'IIII!"  1  —i  0.01  1  'I'luii'ii!!',]!  l[  0  nii!'  i "  'I''II'!I''"II''!"!.  1  -  -0.06  n ; [' ii \ ' « '. H ii " t ii . i 11  0.03 e  i i  i  Influence o f the speed o f maneuver on the system response.  52  0.04  oscillations ( / ?  1>max  « 3 5 ° , A 2 , x - 1 2 ° ) as well as link flexibility effects (e ,e ) m a  x  2  would be  unacceptable. Design and operation criteria for the manipulator w i l l have to be modified to limit the maneuvering speed, increase joints' torsional rigidity and, most importantly, include an active control strategy.  Effect of Joint Stiffness The earlier results clearly showed the important role played by the joint response. To help arrive at an acceptable value for the joint flexibility, three different cases were considered: Kj = 5 x l 0  3  (low), l x l O  4  (considered nominal) and 5 x l 0  4  (high) Nm/rad. The  manipulator executes the same maneuver as before, i n 0.01 orbit and without a payload. Response results are presented in Figure 2-17.  The librational response remains virtually  unaffected by the joint stiffness, with peak deviation o f the platform from the local vertical o f around 0.3°. unaffected.  The peak value o f the platform's tip vibrations also remains essentially  A s anticipated, it is the manipulator's response that is affected the most and it  follows the expected trend: amplitudes o f joints and module tip vibrations increase as the joints' torsional rigidity diminishes.  To limit the tip deflection o f a fully deployed  manipulator o f 30m length to, say, 3cm would require the joint deflection to be less than 0.1° ! Thus the joint rigidity should be at least 5 x l 0 Nm/rad. A g a i n this points to the need 4  of an active control.  Effect of Number of Modes Analysis o f large scale flexible systems has often raised a question concerning representation o f elastic deformations through admissible functions.  53  Complexity o f the  Parameters: EI = 5.5xl0 Nm ;  Initial Conditions: y =0,e  p  = 0;  EL = 5.5xl0 Nm  A = °>  e = 0;  EI = 5 . 5 x l 0 N m ;  A = 0,  e =0.  Specified Coordinates:  Joint Flexibility (Kj):  8  2  p  5  p  2  5  2  d  a =a = x  2  x  2  2  5 x l 0 Nm/rad;  0->90°;  3  l x l O Nm/rad;  l = 7.5m^-15m.  4  2  5 x l 0 Nm/rad. 4  Platform Libration  Platform Tip Vibration '  I  2 .xl0" m  I  I  -  1  3  if  0.01  0.02  \Vi  !j x1 \ 1 J • % -2  0.03  ,",  -  '«  ^  r 1  1  —1  1  1 I  1  1  1  0.01  •  0.02  0.03  Tip Deflection of Module 1  First Joint Vibration 0.2  m  A'' A ' ;  o  e,  /  M A  \ I \ V \ / >  \ ;  \  1  / / ' / \  \"\ fi  /  \i \J \ i  / '\ \\ j i i \ i \/  *  -0.2 —i  0.01  0.02  \  1 /  1  \  1  0.01  0.03  V-  0.02  0.03  Tip Deflection of Module 2  Second Joint Vibration 0.04 - m  2  0  —i  1  1  1—  A''  1  1  1  A n \ / A /A ' V\ ' / '\A V ' \ /M -i V .//; \\ /k /\\/]\- A\ \ • '''/\ 7 -A 7 ''-\ \  r  ;  1  J  J; XJ \\J\ \'J  V  v  L  \/ ' V  \l  -0.04 0.01  0.02  0.01  0.03  0.02 Orbit  Orbit Figure 2-17  1  System response as affected by the joint stiffness which appears to be a rather critical parameter.  54  0.03  formulation as w e l l as the computational effort involved significantly increase with the number o f modes used for flexibility discretization. So the question is: how many modes are necessary to capture physics o f the problem ? To help answer the question, the system with a 800 k g payload performing the same maneuver as before, i.e. a - a x  2  =0 - » 90°,  l =l x  2  =7.5m - » 15m i n 0.01 orbit, was  considered. Response results were obtained using one-mode as w e l l as three-mode flexibility representation for the platform and the modules (Figure 2-18). It is apparent that, even i n this demanding situation leading to large amplitude o f oscillations i n virtually all degrees o f freedom ( A , A2 > 3 0 ° ! e ,e ~ l  2  l m , y/ » 1°), the fundamental mode is able to predict the p  system dynamics quite well. In the investigations so far, the maneuver was purposely considered rather severe to check performance under demanding situations. In a real-life operation, the maneuver w i l l likely proceed differently (e.g. Figure 2-19), and take place i n a longer time. The illustration shows execution o f a task requiring the manipulator to pick-up a payload from position ' A ' (e.g. from the Space Shuttle) to a desired position ' B ' . The sequence o f maneuvers involving deployment (Phase I), 90° slew o f module 2 (Phase II) and 90° slew o f module 1 (Phase III) represents the procedure expected to be used i n practice. To assess the system performance under a realistic situation, consider a rather conservative situation o f the same maneuver (a  x  - a  2  = 0 —» 90°,  l =l x  2  =7.5m —»15m)  completed i n 0.05 orbit, for a manipulator with Kj = 5 x l 0 Nm/rad, and carrying a payload o f 4  2,000 kg. Note, the maneuver rate is rather slow, the joint stiffness is quite high and the payload is somewhat moderate (a typical communication satellite weighs 3 to 5 tons). Response results are presented i n Figure 2-20. It is apparent that the peak platform libration  55  Parameters: EI = 5.5xl0 Nm ;  Initial Conditions:  EI = 5 . 5 x l 0 N m ;  A  EI =5 . 5 x l 0 N m ;  A = 0,  Kj = 1 . 0 x l 0 N m / r a d .  Number of Mode:  8  2  p  5  2  s  5  2  d  4  V =0, p  = o,  a =a x  2  p  «i = o; e =0. 2  1 mode; 3 modes.  Specified Coordinates: Payload  e = 0;  = 0->90°;  /, = / = 7.5m->15m; 2  in 0.01 orbit. Platform Libration  0.01  0.02  Platform Tip Vibration  0.03  0.01  0.02  0.01  0.03  0.02  0.02  0.01  0.03  0.02  Orbit  Orbit Figure 2-18  0.03  0.03  Tip Deflection of Module 2  Second Joint Vibration  0.01  0.02  Tip Deflection of Module 1  First Joint Vibration  0.01  Pavload: 800 k g  System response as affected by the number o f modes used i n flexibility discretization.  56  0.03  7.5m  7.5m fi  (a)  I7 — • -  Initial Configuration  15m  (b)  fi  I7  15m  •  Phase I: Deployment and Payload Pick-up  Phase II: Slew, a = 0 -^90° 2  90°  a (c)  £  D B  Phase III: Slew, ar = 0 ->90  c  3  90  (d)  Figure 2-19  c  I  A likely procedure would involve slew and deployment separately instead o f all four joints maneuvering at the same time.  57  Parameters: EIp = 5 . 5 x l 0 N m ;  Initial Conditions:  EI = 5 . 5 x l 0 N m ;  A = °> A = 0,  8  5  2  2  s  EI = 5 . 5 x l 0 N m ; 5  2  d  y/ = 0 , e = 0; p  p  e  i =  e =0. 2  Kj = 5.0 x l O N m / r a d . 4  Specified Coordinates: a =a = 0->90°;  Payload  x  Pavload: 2,000 k g  2  /, = l = 7.5m—> 15m; i n 0.05 orbit. 2  Platform Libration  Platform Tip Vibration  p  0  -1.6h 0.05  0.05  0.1  Tip Deflection of Module 1  First Joint Vibration  P,  0.1  o  0.05  0.05  0.1  0.1  Tip Deflection of Module 2  Second Joint Vibration 0.15 0.01 h  -0.01  -0.15  Figure 2-20  System response, even under a set o f conservative values for payload (2,000 kg), joint stiffness ( 5 x l 0 Nm/rad) and maneuver speed (0.05 orbit), is unacceptable. This suggests a need for control. 4  58  far exceeds the permissible range o f 0.1° - 1.0°.  The maximum joint vibrations o f /?i ~  0.23° and fi « 0.09° would lead to the manipulator tip deviation from the desired position 2  by « 5cm along the x axis and « 9cm in the y direction. Thus a need for control is clearly indicated and this is the subject o f the following chapter.  59  3. S Y S T E M C O N T R O L The previous chapter  clearly established  a need  for  control, under  critical  combinations o f system parameters, initial conditions and maneuvers, to maintain desirable performance o f the manipulator. In particular, joint stiffness, maneuver speed and payload constituted parameters having significant effect on the system dynamics. The present chapter focuses on the next logical step o f control o f the system with a platform-based two-unit (four links) manipulator. members.  A s before, the platform, modules and joints are treated as flexible  To recapitulate, more frequently used symbols are summarized below and  indicated in Figure 3-1: 6  true anomaly with reference at perigee;  CM.  center o f mass;  d  position o f the manipulator form C M . ;  L.V.  local vertical;  L.H.  local horizontal.  R i g i d Degrees o f Freedom y/ /, , /  platform pitch; lengths o f modules one and two, respectively;  2  a, a x  2  rigid components o f slew maneuvers associated with modules one and two, respectively.  Flexible Degrees o f Freedom flexible components o f slew maneuvers associated with modules one and two, respectively; e  p  platform tip deflection;  60  Payload  Figure 3-1  A two-module manipulator system showing frequently used symbols.  61  e ,e x  3.1  2  tip deflections o f modules one and two, respectively.  C o n t r o l Methodologies The torques and forces responsible for slewing and deployment maneuvers  are  illustrated i n Figure 3-2. The objective here is to control operational behavior o f the system. For example, the slewing maneuvers w i l l arise with application o f torques (T ,T ) X  provided  2  by actuators located at revolute joints o f the manipulator. Similarly, forces (F ,F ) X  2  for the  deployment and retrieval o f units are provided by actuators at the prismatic joints o f the manipulator. In addition to joint actuators which regulate the manipulator link dynamics, there are Control Momentum Gyros ( C M G s ) . They are used to regulate the platform orientation as well as its vibration. C M G s located near the center o f the platform contribute the torque (T ) which controls the rigid body motion o f the platform, i.e. its attitude or pitch pr  response. O n the other hand, a pair o f C M G s , located symmetrically about the platform's center and providing equal torques (T j-/2) in the opposite sense, control its elastic vibration p  by regulating the local slope. The present section is concerned with the selection o f control inputs which w i l l result in the desired motion o f the system. Two different control strategies are considered: (i)  Nonlinear Feedback Linearization Technique ( F L T ) applied to the rigid degrees o f freedom, with flexible generalized coordinates indirectly affected through coupling but not actively controlled. The F L T leads to uncoupled linearized equations o f motion which are then subjected to the conventional P D control.  62  Controller  Figure 3-2  Schematic diagram o f the manipulator system showing the location o f the control actuators. The two torques T f/2, opposing each other, control the platform's vibration by regulating the local slope. p  63  (ii)  The classical Linear Quadratic Regulator ( L Q R ) , based on a linear approximation o f the flexible subsystem, is designed for active vibration suppression [45,46]. Both rigid as well as flexible degrees o f freedom are now controlled through the F L T and L Q R , respectively.  3.1.1  F L T control The F L T is an approach particularly suited to a class o f nonlinear systems. The  procedure was pioneered by Bejczy [47]. It has been further developed and applied by many investigators resulting i n a considerable body o f literature [48-54]. The basic idea is to use a mathematical model and find a transformation to decouple and linearize the dynamics o f the controlled system. The main advantage o f the feedback linearization over point-wise linearization is that once such a transformation is determined, a global linearization is achieved independent o f the operating point. In the present study, a controller based on the F L T is designed to regulate the rigid degrees o f freedom, i.e. rotations o f the revolute joints (a ,a ), x  2  deployment o f the links (l ,l ), x  2  attitude motion o f the system  (i// ). p  However,  the effectiveness o f the controller is assessed using the original fully flexible system so that the potential effects o f uncontrolled dynamics can be investigated. Equations governing dynamics o f a flexible space-based manipulator can be written as M(q,t)q  M  + F{q,q,t)  M~  rr  rf  4r 1 ' ' '  M_  Mj,  ff  (3.1)  = Q{q,q,t),  Qr '''  f.  F  ~M  64  f  =  '  1 ' "  Qf  f  where: M(q,t) is the system mass matrix composed o f rigid (M ), rr  coupled (M , rf  M ) contributions; q , q fr  r  flexible ( A T ^ ) and  are rigid and flexible generalized coordinates,  f  respectively; F(q,q,t) contains terms associated with centrifugal, Coriolis, gravitational, and elastic forces; and Q(q, q, t) represents nonconservative generalized forces including the control inputs.  Subscripts r and /  refer to contributions associated with rigid and flexible  degrees o f freedom. I f only the rigid degrees o f freedom are controlled: M q +M q +F =Q : rr  r  rf  f  r  M q\  r  (3.2)  +M q +F =0.  fr  ff  f  f  A suitable choice for Q would be r  Q =M[(q ) -u]+F, r  r  (3.3)  d  with: M =  M -M Mj M ; l  rr  F =  lf  f  fr  (3.4)  F -M Mj}F . r  rf  f  Here subscript 'd' refers to the desired value o f a parameter. One way to select the control signal « is the Proportional-Derivative (PD) feedback, i.e. U  = - [(qX -9r]- pl(9r) ~<lrl K  (-) 3  K  V  d  where K and K are position and velocity gains, respectively. L e t e = (q ) -q , p  v  r  d  r  then the  controlled equations o f motion become: 0 = e + K e + K e\ v  p  q =-M/ M [(q ) -u]-Mf F . X  f  (3.6)  l  fr  r  d  f  N o w Q can be written as r  Q = M(q ) r  r  d  +F + M(K e + K.e) , v  65  (3.7)  5  which can be visualized as a combination o f two controllers: the primary (Q , ); and the r p  secondary (Q , ); r s  Q , =M{q ) r P  r  +F;  d  Q  rs  = M(K e + K e). v  p  (3.8)  The function o f the primary controller is to offset nonlinear effects inherent i n the rigid degrees o f freedom; whereas the secondary controller ensures a robust behavior. A block diagram o f the control procedure is presented i n Figure 3-3. It is undesirable for a robot to exhibit an overshoot, since this could cause impact if, for instance, a desired trajectory terminates at the surface o f a workpiece. Therefore, to ensure asymptotic and critically damped behavior o f the closed-loop system, a suitable candidate for the P D gains, K and K would be diagonal matrices such that p  Vt  If)  '' a>\ 0  0  0 a>? where 3.1.2  0 2co  i  is the desired natural frequency associated with the i joint or link error (Eq. 3.6). F L T / L Q R control A combined F L T / L Q R approach was also applied to the two-unit manipulator system  (Figure 3-2). When the manipulator is i n a fixed configuration (no deployment, slew, or translation), the platform attitude (y/ ) and manipulator length (I ,l ) are maintained using p  x  2  the F L T strategy. However, the joint rotation (a ,a ), the platform's tip vibration (e ), as x  2  well as the links' tip deflections (e ,e ) are controlled using the L Q R approach. During large x  2  slew and deployment maneuvers, variables y/ , a ,a , p  x  2  l , and l x  2  are regulated by the F L T  controller until the manipulator reaches the vicinity o f its target position. Structural vibrations are left uncontrolled. A t the end o f the large maneuver, the system's configuration  66  Nonlinear Inner L o o p  Linear System  ^ Qr  M  Manipulator System  K  v  Figure 3-3 FLT-based control scheme showing inner and outer feedback loops.  67  9r  remains nearly constant and nonlinear effects can be neglected. This allows the Linear Quadratic Regulator to take over the control o f a  x  and a  2  to actively damp flexible  generalized coordinates disturbed due to the maneuver. The optimal L Q R controller is designed based on a linearized model o f the system. To begin with, the governing equations are linearized about an operation point q . To that 0  end, the following substitutions are made in the left hand side o f E q . (3.1): q = q +Aq;  q = q +Aq;  0  q = q +Aq.  0  (3-10)  0  Trigonometric functions are expanded in the Taylor series, and the second and higher order terms in q, q and q are neglected. After some algebra, this leads to M(q )Aq  + K(q )Aq  0  = Q = Q,  0  (3-11)  where M and K are the mass and stiffness matrices for the linearized system, respectively. Note, both M and K are evaluated at the operating point q  and thus made time-invariant.  0  Since the system's librational motion and the deployment length o f the manipulator are controlled by the F L T , only the linearized equations governing the elastic degrees o f freedom are needed. The decoupled vibrational subsystem is now described by M(q )Aq 0  where Aq  L  = [e , Aa ,/3 ,e ,Aa , p  x  x  x  +K(q )Aq =u ,  L  f? ,e ] >  2  2  w  r  0  L  n  Aa  t  2  (3-12)  L  and Aa  x  2  as the deviations o f the slew  angle from their desired values; M and K are the mass and stiffness matrices, respectively, corresponding to the elastic subsystem; and u  L  Quadratic Regulator to control e , p  q  = [0, a , 0,0, a ,0,0]  r  L0  xo  20  Aa , x  . Solving for  e,  is the input determined from the Linear  Aar and e .  x  2  Aq , L  68  2  Note, the operational point is  Aq = —M~ KAq  + M~  {  L  u,  l  L  (3.13)  L  which can be rewritten i n the state-space form as  Aq  L  MK  0  ]  ML  +  ML.  uL  M  (3.14)  i  B  where x  e 9T  L  4 x l  ; A e 5R  14x14  ; and B e S R " . For simplicity, it is assumed that all states are 14  available, thus making the system observable. Controllability o f the system is assured i f and only i f rank{[B, AB, A B,  A B]}  2  = 14.  n  (3.15)  The control input u can be written as L  U  where ^  Q  L  R  L  ~  ^LQR L  (3.16)  i  X  is the optimal feedback gain matrix. It minimizes a quadratic cost function /  which considers tracking errors and energy expenditure, J = \ (xlQ x  +  T  LQR  Here  and R q  QQ L  L  R  L  L  R  L  R  i? Q L  is required to be positive  R  L  L  PQ  L  can be positive semi-definite. The optimal control input u is given by  U  where  LQR  are symmetric weighting matrices which assign relative penalties to  R  state errors and control effort, respectively. The matrix definite while Q q  (3.17)  ulR u )dt.  -  KLQR L X  -  RLQRB  PLQR L> X  (3.18)  is the positive definite solution to the steady-state matrix Ricatti equation which,  for infinite time, becomes -^LQR^ +  A PL Q R L0R  PL QBR R - " -B "-LQR-"  A  L0R  LQR  69  1  -PLQR+2 LQR ^ J i L Q R  (3.19)  The F L T / L Q R control strategy is indicated i n Figure 3-4. Note, q  r  represents generalized  coordinates controlled by the F L T .  3.2  Simulation Results and Discussion: Commanded Maneuvers The numerical values used in the simulation are summarized below:  Orbit:  circular at an altitude o f 400 k m ; period = 92.5 min.  Platform:  cylindrical with axial to transverse inertia ratio o f 0.005; mass =120,000 kg; length = 120m; flexural rigidity = 5.5 x 10 N m . 8  Manipulator Module:  2  cylindrical with axial to transverse inertia ratio o f 0.005; mass = 400 kg each; length = 7.5m each at start; flexural rigidity = 5.5 x 10 N m . 5  Revolute Joints:  2  mass = 20kg; moment o f inertia = 10 k g m ; stiffness = l . O x l O 2  4  Nm/rad. Payload:  no payload unless specified.  Damping:  assumed zero for all components.  Mode:  fundamental.  Damping is purposely not included to obtain conservative results and test the controller under severe conditions. Furthermore, character and precise value o f damping i n the  space  environment is still a subject o f considerable debate. To assess the system control under a rather demanding situation, the platform is initially taken to be along the local horizontal (y/  p  = - 9 0 ° ) , an unstable equilibrium  configuration. The manipulator is located at the platform tip to accentuate the maneuver effect, and is initially aligned with the platform. The system, without any payload, is commanded to undergo simultaneous slew and deployment maneuvers, i n a sine-on-ramp  70  q >q r  r  FLT Controller Space Manipulator System  QLO> 1LO  q,q  &q , Aq L  L  Figure 3-4 B l o c k diagram illustrating the combined F L T / L Q R approach applied to the manipulator system.  71  profile (Figure 2-3), completed in 0.01 orbit (55.50s), so that a ,a x  and l ,l x  3.2.1  2  2  change from zero to 90°  from 7.5 m to 15 m.  F L T control Figure 3-5 shows the F L T controlled response o f the platform libration ( y/ ), its tip p  vibration (e ), p  the first and second modules' revolute joint rotations  a sw e  (<2r ,/?i,^2'A)' 1  ^  as the modules' tip deflections (e , e ). The maneuver sets the platform librating with a peak x  2  amplitude o f around 0.04° which is rather small (permissible limit can vary from 0.1° to 1° depending on the mission). The platform returns to the original local horizontal orientation in less than 100 s even i n presence o f such a severe maneuver ! The negligible (» 0.05°) limit cycle type oscillations persist due to vibrations o f the flexible joints (j3\,/%). The tip response o f the massive platform, as expected, is also vanishingly small (« 0.2 - 2 mm). The steady state joint vibration {J3\, J3i) amplitudes (4° and 2°, respectively) may be considered acceptable recognizing the fact that the flexible generalized coordinates are uncontrolled, the disturbance is unusually severe and the inherent structural damping is not accounted for. Clearly, the unmodeled dynamics o f the flexible generalized coordinates affects  the  performance o f the controller. However, the controller demands remain rather modest. The five controlled variables (y/ ,a ,l ,a ,and p  x  x  2  l ) show satisfactory response even during such 2  a large maneuver and display small oscillations after the desired values are reached. Thus the F L T control o f the rigid degrees o f freedom does provide encouraging results.  72  I.C.'s (Flexible d.o.f.): e P  e  p  = 0.02;  p  /^ =0.29.  A=A=0I.C.'s (Rigid d.o.f.): L.V.  Controller Gains: y/ :K  = \ = i = 0; e  v  or a : K = 25; l5  2  p  K =\0.  y/p = - 9 0 ° , a = a - 0; x  / / ^ \  v  2  /, = l = 7.5 m.  l :K  = %; K =5.67.  Desired Values:  l :K  = 4;  2  ¥  p  x  2  p  p  v  K =4. v  = - 9 0 ° ; a , a = 0 -> 90° and / , , l = 7.5m ^ x  2  2  15m;  sine-on-ramp; 0.01 orbit. Platform Libration  Momentum Gyros Torque  -89.96  -90.04 0.00  0.02  0.04  0.06  0.00  First Joint Rotor Motion  0.02  0.04  0.06  First Joint Actuator Torque iNm  1000  a, T  X  0.00  0.02  0.04  0.06  -1000  Deployment of Module 1  m 15  /  :  /  5.01  :  /  5.00  /  4.99  Figure 3-5  AAAAAAA  I VU  0.04  0.06  First Deployment Actuator Force  -30  U U V VI  0.02 0.03 0.04 0.05  -  0.00  0.02  F, 0  12 - /  0.00 30  i  i.  0  ,  0.02  ,  Orbit  ,  1  0.04  ,  ,  I—  -60 . c  0.00  0.06  0.02  Orbit  0.04  0.06  Response o f the system during a simultaneous 90° slew and 7.5m deployment maneuver o f the two-unit manipulator with rigid degrees o f freedom controlled by the F L T : (a) rigid degrees o f freedom and control inputs for module 1.  73  I.C.'s (Flexible d.o.f.I: = \ =i  e  e  P  e  = °;  Controller Gains: ^ : K = 0.02; p  ^ =0.29. v  cc , a :K  I.C.'s ( R i g i d d . o . D : if/ = - 9 0 ° , a =a p  x  l\=l = 2  x  K = 10. v  8;  7.5 m.  =-90°; a  l 5  = 25;  p  =0;  2  a  K =5.61. v  l : K = 4; K  Desired V a l u e s : ^  2  2  2  =0 -> 90° and  p  v  /  2  4.  = 7.5m -> 15m;  sine-on-ramp; 0.01 orbit.  Figure 3-5  Response o f the system during a simultaneous 90° slew and 7.5m deployment maneuver o f the two-unit manipulator with rigid degrees o f freedom controlled by the F L T : (b) rigid degrees o f freedom and control inputs for module 2.  74  I.C.'s (Flexible d.o.f.):  p= \= 2=  e  e  y/ \K  °;  e  Controller Gains: p  A=A=o. xei  = 0.02;  p  ^ =0.29. v  I.C.'s ( R i g i d d.o.f.):  cXy, cc '• Kp = 25; 2  y/ = - 9 0 ° , a = a =0; p  x  Ii=l = 2  p  v  l :K  7.5 m.  ]  Desired V a l u e s : ¥  K = 10.  2  p  l :K 2  p  = S; = 4-  K =5.67. v  K =4. v  = - 9 0 ° ; ^ , ^ = 0 ^ 90° and / , , l = 7.5m -> 15m; 2  sine-on-ramp; 0.01 orbit. Platform Vibration  First Joint Vibration  0.02  0.04  0.06  0.00  Second Joint Vibration  P°  0.02  0.04  0.06  Tip Deflection of Module 1  0  0.00 0.04  0.02  0.04  0.06  0.00  0.02  0.04  Orbit  Tip Deflection of Module 2  0.06  0.02  0.02  0.04  0.06  Orbit Figure 3-5  Response o f the system during a simultaneous 9 0 ° slew and 7.5m deployment maneuver o f the two-unit manipulator with rigid degrees o f freedom controlled by the F L T : (c) flexible degrees o f freedom.  75  3.2.2  F L T / L Q R control W i t h the control o f rigid as well as flexible degrees o f freedom ( F L T / L Q R approach),  the situation further improves remarkably (Figure 3-6), particularly i n the steady state librational and vibrational responses. In this particular case, the L Q R controller is only activated after 0.01 orbit, i.e. at the end o f the maneuver. This means, i n the first 0.01 orbit, the F L T is used to regulate the large maneuver where it satisfactorily controlled the rigid degrees o f freedom.  After 0.01 orbit, the system enters the steady state phase and vibrates  around the reference point. A t this stage, the L Q R begins to control vibrations. Note, the F L T controller is still active to regulate the platform's attitude and length o f the links. The L Q R is quite effective in suppressing the joints and platform vibrations which, in turn, help eliminate the librational limit cycle. The joint angles and link lengths attain and remain at their commanded values. Furthermore, the control torques required, i n the steady state, are virtually negligible.  Effect  ofpayload  In the previous two cases, the payload was purposely taken to be zero to help isolate coupling effects. The next logical step was to assess the influence o f a point mass payload at the manipulator's tip. Three values o f the payload ratio (mass o f the payload / mass o f the manipulator) were considered: 1, 2 and 5; which correspond to the payloads o f 400 kg, 800 kg and 2,000 kg, respectively. The initial configuration o f the manipulator remains the same as described i n the previous cases.  The maneuver, as before, involves a simultaneous 90°  slew o f the revolute joints and 7.5m deployment o f the links in a sine-on-ramp profile. It is  76  L.V.i  I.C.'s (Flexible d.o.f.):  Controller Gains:  ^  i// :K  =  e  i = 2 =0; e  = 0.02;  p  K =0.3. v  A=A=o.  / , : Xp = 8; 7^ =5.66.  I . C . ' s ( R i g i d d.o.f.):  /:  y/ = - 9 0 ° , ar, =ar =0;  a,  /j = l = 7.5 m.  t < 0.01 orbit, K = 25; K = 10.  Desired V a l u e s : ^ = - 9 0 ° , or, =ar = 9 0 ° ;  t > 0.01 orbit, L Q R .  v  p  VPA  p  K = 4;  2  x  2  p  K =4. v  a: 2  P  2  V  2  l\=l = 2  15 m .  Platform Libration  Momentum Gyros Torque  -89.96  -90.00  -90.04 0.00  0.02  0.04  0.06  0.00  First Joint Rotor Motion  0.02  1000 Nm  0.04  0.06  First Joint Actuator Torque J = 0.01 orbit LQR takes over for a.  T 0 x  a, -1000 0.00  0.02  0.04  0.06  0.00  Deployment of Module 1 15  0.02  0.04  0.06  First Deployment Actuator Force  m  12  0.00  0.02  0.04  0.06  0.00  Orbit Figure 3-6  0.02  0.04  0.06  Orbit  F L T / L Q R controlled response o f the two-unit system during a simultaneous 90° slew and a 7.5 m deployment maneuver o f the manipulator units: (a) rigid degrees o f freedom and control inputs for module 1.  77  Controller Gains: y/p :K = 0.02; i C = 0 . 3 .  I.C.'s (Flexible d.o.f.I: = 0;  P  v  A=A=o.  /,: K = S;  I.C.'s (Rigid d.o.f.):  l:  K  W = - 9 0 ° , a = a = 0;  a,  a  /, = /  t < 0.01 orbit, Kp = 25; K = 10.  P  P  x  2  2  x  2  =7.5m.  p  4;  K =5.66. v  K =4. v  2  v  t > 0.01 orbit, L Q R .  Desired Values: y/ = - 9 0 ° , or, = a p  =90°;  2  /, =l = 15 m. 2  Second Joint Rotor Motion  Second Joint Actuator Torque 300  Nm t = 0.01 orbit LQR takes over for OL  T  0L  2  0  -300 0.00  0.02  0.04  0.06  0.00  Deployment of Module 2  20  F 1  0.02  0.04  0.06  Second Deployment Actuator Force  0  2  -20  0.00  0.02  0.04  -40 0.00  0.06  0.04  0.06  Orbit  Orbit  Figure 3-6  0.02  F L T / L Q R controlled response o f the two-unit system during a simultaneous 90° slew and a 7.5 m deployment maneuver o f the manipulator units: (b) rigid degrees o f freedom and control inputs for module 2.  78  Controller Gains: ^ :K = 0.02; K =0.3.  I.C.'s (Flexible d . o . D : e„ =ei = e ^X  1  2  =0;  P  I . C . ' s ( R i g i d d.o.f.I: = -90°, a - a x  li=l = 2  2  K =4.  K = 4; p  v  a:  x  2  P  v  t > 0.01 orbit, L Q R . 2  2  l:  v  t < 0.01 orbit, AT = 25; £ = 10.  p  x  K =5.66.  a,  7.5 m.  Desired V a l u e s : ^ = - 9 0 ° , ^ =a l =l =  / , : A> = 8; 2  = 0;  v  =90  c  15 m.  Platform Vibration  First Joint Vibration  > 0.0  0.02  0.04  0.06  0.00  0.02  0.04  0.06  Tip Deflection of Module 1  Second Joint Vibration  K o  0.00 0.04  0.02  0.04  0.06  0.02  Tip Deflection of Module 2  Orbit  0.04  0.06  0.02 0.00 -0.02 0.00  Figure 3-6  0.02  Orbit  0.04  0.06  F L T / L Q R controlled response o f the two-unit system during a simultaneous 90° slew and a 7.5 m deployment maneuver o f the manipulator units: (c) flexible degrees o f freedom.  79  desired that the maneuver be finished i n 0.03 orbit.  A s i n the case o f the uncontrolled  dynamical study (Figure 2-15), a longer maneuver time is used here because o f the presence of the payload. Figure 3-7 presents results as affected by the payload. The controller gains used by the F L T are indicated i n the legend. robustness o f the controller.  The gains were purposely kept fixed to help assess  The L Q R becomes effective at 0.03 orbit, i.e. when the  maneuver is completed. A t the outset, it is apparent that the manipulator is able to attain the commanded values o f slew and deployment even i n presence o f payloads (Figures 3-7a, 7b). A s can be anticipated, the peak control efforts increase with an increase i n the payload, however the additional demands are rather modest and remain w e l l within the permissible limits. For example, with the largest payload o f 2,000 k g , the absolute peak value o f the C M G demand changes from around 3000 N m (no payload) to 4400 N m while the force (F ) 2  at the prismatic joint o f module two has corresponding variations from 32 N to 56 N . Note, the control effort is required for a relatively longer period o f time, however,  after  approximately 0.08 orbit, the system settles down to the new equilibrium position, very close to the original, and the control effort required is virtually negligible. Flexible degrees o f freedom are also controlled, in presence o f payload, quite effectively (Figure 3-7c).  Effect of Maneuver Speed Another important system parameter is the speed o f the maneuver. maneuver as before was considered i n absence o f payload.  The same  O f course, one would like to  complete the maneuver as quickly as possible without adversely affecting the performance. Three different values were considered: 0.05 orbit (fast), 0.01 orbit (nominal), and 0.03 orbit  80  Controller Gains: y/ :K = 0.02; K =0.29.  I.C.'s (Flexible d.o.f.):  Payload  /,  ep = e, = e = 0;  p  2  2  a  p  v  a , a :K x  A  2  = /  2  x  K =5.67. v  1:1;  = 7 . 5 m . Desired Values: y/ = - 9 0 ° , a ,a 2  p  v  = S;  p  X  1  = 25; K = 10.  Payload Ratio:  ^=-90°;a =flr =0; 1  p  /,, l :K  i.c.'s (Controlled d.o.f.): ^ i • > /  2  =0 - » 90° and /  2  l 5  2:1;  5:1. / = 7.5m -> 15m; 2  sine-on-ramp; 0.03 orbit. Platform Libration  Momentum Gyros Torque  4000  -89.94  -90.00  -90.06 0.00  0.02  0.04  0.06  0.02  0.08  First Joint Rotor Motion  1000  0.06  0.08  First Joint Actuator Torque Nm  500  /,.--. \  /. \  T, 0 a,  0.04  -500 \  -1000 0.02  0.04  0.06  0.08  0.00  Deployment of Module 1  0.04  0.02  0.04  0.06  0.08  First Deployment Actuator Force  0.08  0.08  Orbit  Figure 3-7  F L T / L Q R controlled response o f the system during a simultaneous 90° slew and 7.5 m deployment maneuver o f the two-unit manipulator with different payload ratios: (a) rigid degrees o f freedom and control inputs for module 1.  81  I.C.'s (Flexible d.o.f/1:  Controller Gains:  = \ = i = 0; A=A=o.  y :K  e  e  e  p  P  ^  p  0.02;  =  a , a :K {  2  K =0.29. v  = 25; K = 10.  p  v  /], l : K = 8; K = 5.67.  I.C.'s (Controlled d.o.f.):  2  p  v  Payload Ratio: ^  /, = / = 7 . 5 m .  1:1;  2  Desired Values: y/ = - 9 0 ° , a ,a =0^> p  x  2:1;  5:1. 90° and / , , l = 7.5m -> 15m;  2  2  sine-on-ramp; 0.03 orbit.  Second Joint Rotor Motion  Second Joint Actuator Torque  «2  0.02  0.04  0.06  0.08  0.02  Deployment of Module 2  0.04  0.04  0.06  0.08  Second Deployment Actuator Force  0.00  0.08  Orbit  0.02  0.04  0.06  0.08  Orbit  Figure 3-7 F L T / L Q R controlled response o f the system during a simultaneous 90° slew and 7.5 m deployment maneuver o f the two-unit manipulator with different payload ratios: (b) rigid degrees o f freedom and control inputs for module 2.  82  I.C.'s (Flexible d.o.L): =  e P  e  \ = 2 e  =  Controller Gains: y :K  0;  p  = 0.02;  p  a , a :K x  J^  1  = 25; K = 10.  p  2  v  p  v  Payload Ratio:  = - 9 0 ° ; a-j = or = 0; 2  2  v  /[, l '• K = 8; K = 5.67'.  I.C.'s (Controlled d.o.f.):  l\=l =  2  K =0.29.  1:1;  7.5 m .  2:1;  Desired Values: 5:1. y/ = - 9 0 ° , o r , , ^ =0 -> 90° and l , l = 7.5m -> 15m; p  x  2  sine-on-ramp; 0.03 orbit. Platform Vibration  First Joint Vibration  P" 0.0  P  0.02  0.04  0.06  0  0.08  0.02  Second Joint Vibration  4.0  0.04  0.06  0.08  Tip Deflection of Module 1 1.0 • Xl0" m 2  2.0  Pa  0.5  I—, i  0.0  0.0  \  i  ^  -2.0 0.00  0.02  0.04  0.06  -0.5 0.00  0.08  0.02  0.04  0.06  0.02  0.04  0.06  0.08  Orbit  Tip Deflection of Module 2  0.00  Y  0.08  Orbit Figure 3-7  F L T / L Q R controlled response o f the system during a simultaneous 90° slew and 7.5 m deployment maneuver o f the two-unit manipulator with different payload ratios: (c) flexible degrees o f freedom.  83  (slow). Results are presented in Figure 3-8. It is apparent that the manipulator attains the commanded values rather quickly after the specified period even in the case o f a fast maneuver.  The force and torque demands  remain modest. Note, the peak platform deviation from the unstable equilibrium position is around 0.1° (Figure 3-8a) for the fast maneuver and virtually negligible for the slow case o f 0.03 orbit.  The peak torques and forces encountered are w e l l within the accepted limit  (Figure 3-8a,b).  Even the flexible degrees o f freedom are controlled rather well with the  equilibrium configuration regained in less that 0.03 orbit ( « 167 s).  Effect of Revolute Joint Stiffness Stiffness o f revolute joint also represents a significant variable.  Its effect on the  controlled performance while executing the same maneuver i n 0.01 orbit, with no payload, was also assessed.  These results are presented in Figure 3-9.  T w o stiffness values, one  below (soft) and the other above (hard) the nominal value o f l x l O Nm/rad were considered. 4  Even i n the demanding situation presented by the soft spring, the system settles down to the commanded values i n around 0.03 orbit (« 167 s). A s before, the demands on control forces and torques continue to remain modest. It is important to point out that gains during the studies aimed at assessing the influence o f payload, maneuvering speed and stiffness variations are intentionally kept the same to demonstrate robust character o f the F L T / L Q R control. Based on the investigation reported in this chapter, it can be concluded that both the F L T by itself as well as a synthesis of the F L T and L Q R , or other linear control procedure, appear quite promising. They should receive further attention in refining their implementation.  84  Controller Gains: y :K = 0.02; K =0.29.  I.C.'s (Flexible d.o.f.): =  e P  e  \ = 2 e  =  o;  p  A =A=o. ^  v  or,,  y/ = - 9 0 ° ; a =a x  2  =0;  tf :i^  = 25;  2  , l : K = 8; ^  I.C.'s (Controlled d.o.f.): p  p  2  p  Desired Values: = - 9 0 ° , a a =0^> lt  /  2  2  90°;  0.01 orbit; 0.03 orbit.  -—  = 7.5m —> 15m; sine-on-ramp.  Platform Libration  -89.9  5.67.  0.005 orbit;  2  p  v  Maneuver Speed:  /, = / = 7 . 5 m .  ¥  10. =  Momentum Gyros Torque 10000  -10000 H  -90.1  0.04  0.06  First Joint Rotor Motion  0.06  First Joint Actuator Torque  4000  -4000 0  0.02  0.04  0  0.06  0.04  0.06  0.02  Orbit  Figure 3-8  0.04  0.06  First Deployment Actuator Force  Deployment of Module One  0.02  0.02  Orbit  0.04  0.06  Effect o f the speed o f maneuver on the F L T / L Q R controlled response: (a) platform and module 1.  85  Controller Gains:  I.C.'s (Flexible d.o.L): fr  = \ = 2 = o;  e  e  P  ^  e  :K = 0.02;  K =0.29.  P  v  a , a : K = 25; K = 10. x  L.V.j  /,, / :  I.C.'s (Controlled d.o.f/l: ly  4&\ ... y/  p  3  2  2  p  =  v  8; K  = v  5.67.  Maneuver Speed:  = - 9 0 ° ; or, = ar = 0 ; 2  0.005 orbit;  ^ = / = 7.5 m . 2  0.01 orbit; 0.03 orbit.  Desired Values: y/ = - 9 0 ° , ^ , ^ 2 = 0 - > 9 0 ° , p  /!, l = 7.5m - » 15m; sine-on-ramp. 2  Second Joint Rotor Motion  Second Joint Actuator Torque 1600  -1600 h 0.02  0.04  0  0.06  Deployment of Module Two  0.02  0.04  0.06  Second Deployment Actuator Force  -150 0.02  0.04  0.06  0.04  Orbit  Orbit  Figure 3-8  0.06  Effect o f the speed o f maneuver on the F L T / L Q R controlled response: (b) module 2.  86  I.C.'s (Flexible d.o.f.I: e  P  e  p  A=A=o. J^i  Controller Gains: y :K  = \ = i = °;  e  = 0.02; i : = 0 . 2 9 .  p  v  a , a :K x  2  = 25; ^ = 10.  p  v  , l : K = 8; i T = 5.67. I.C.'s (Controlled d.o.f.I: Maneuver Speed: y/ = - 9 0 ° ; a, =a = 0; 0.005 orbit; l\=l = 7.5 m . 0.01 orbit; Desired Values: 0.03 orbit. y/ = - 9 0 ° , ar,,<ar =0 -> 90°; 2  p  p  v  2  2  p  2  l , l = 7.5m -> 15m; sine-on-ramp. x  2  Platform Vibration  First Joint Vibration  0.02  0.06  0.04  0.06  Tip Deflection of Module 1  Second Joint Vibration  0  0  0.02  0.04  0.06  0.06  Tip Deflection of Module 2  0.02  0.04  0.06  Orbit  Figure 3-8  Effect o f the speed o f maneuver on the F L T / L Q R controlled response: (c) flexible degrees of freedom.  87  I.C.'s (Flexible d.o.f.): = \ = i = °;  e  h  e  P  *,  a  ^  Controller Gains: y/ :K = 0.02; K =0.29.  e  p  P  v  A=A=o.  a , a :K  I.C.'s (Controlled d.o.f.):  /], l : K — 8; K = 5.67.  {  2  x  l\=l 2  p  2  v  5 x l 0 Nm/rad; 3  Desired Values: y/ = - 9 0 ° , a ,a x  v  p  7.5 m .  p  = 25; K = 10.  Joint Stiffness:  y/ = - 9 0 ° ; a = a = 0 ; p  2  lxlO Nm/rad; 5 x l 0 Nm/rad. 4  = 0->90°;  2  4  l , l = 7.5m —t 15m; sine-on-ramp. x  2  Platform Libration  Momentum Gyros Torque 5000  -89.96 h ¥p "90  0  v  -90.04 h 0.01  1  0.02  First Joint Rotor Motion 1  -5000 \  1  1  0.02  0.03  First Joint Actuator Torque 1000 - N m  1  i 1c-  1 l\  90  T 0  /  a,  s  -1000 •  0  1  1  0.01  1  1  0.02  0.03  1  JP  \\\  \J ' 1  0.01  0.03  0.02  0.03  First Deployment Actuator Force  Deployment of Module One  i 0 -50 0.01  0.02  0.03  0.01  Orbit  0.02  0.03  Orbit  Figure 3-9 F L T / L Q R controlled response as affected by the revolute joint stiffness during a manipulator maneuver: (a) platform and module 1.  88  I.C.'s (Flexible d.o.f.I: e  Controller Gains: y/ :K = 0.02; K =0.29.  = \ = i = °; e  P  e  p  p  v  A=A=o.  a , a :K  I.C.'s (Controlled d.o.f/1:  /], l : K = 8; K = 5.67.  x  x  2  = 25; K = 10.  p  v  p  v  JoinlStiffness:  y/ = - 9 0 ° ; a = a = 0; p  2  2  l = l = 7.5 m .  5 x l 0 Nm/rad;  Desired Values: y/ = - 9 0 ° , a ,a =0^  l x l O Nm/rad;  x  3  2  p  x  4  90°;  2  5 x l 0 Nm/rad. 4  l , l = 7.5m - » 15m; sine-on-ramp. x  2  Second Joint Actuator Torque 1 1 400 . N m  Second Joint Rotor Motion "••  '—  1  1  1  1  1  90  /  «2  0  ^2  V 7' -400  J  0  0.01  0.02  15  1  1  1  0  1  1  i  0.02  0.03  \  //'  -40 0.01  0.02  •  0.01  0.03  Orbit  Figure 3-9  0.01  N  m  J  i  Second Deployment Actuator Force  1  ^ 2  7.5  —  0  0.03  Deployment of ModuleTwo  1  1  .  0.02  Orbit  I  .  0.03  F L T / L Q R controlled response as affected by the revolute joint stiffness during a manipulator maneuver: (b) module 2.  89  Controller Gains:  I.C.'s (Flexible d.o.f.): = \ =i  e  e  P  e  y/ :K = 0.02;  = °;  p  K =0.29.  P  v  a , a : K = 25; K = 10. x  ^  x  p  =  2  v  8; K  = v  5.67.  Joint Stiffness:  y/ = - 9 0 ° ; a = a - 0; p  2  /,, l \ Kp  I.C.'s (Controlled d.o.f.): 2  /, = l = 7.5 m .  5 x l 0 Nm/rad;  Desired Values: = - 9 0 ° , a a =0->  l x l O Nm/rad; 5 x l 0 Nm/rad.  3  2  ¥  lt  p  4  90°;  2  4  /,, l = 7.5m —> 15m; sine-on-ramp. 2  First Joint Vibration  Platform Vibration  "p  0.01  0.02  0  0.01  0.03  Second Joint Vibration  0.1  0.02  Tip Deflection of Module 1 1  1  1  m  0.05  PS  0  • \ / N  0  v  -4  0  0.01  0.02  -0.05  0.03  0.03  -  ' //  V  0.01  0.02  0.03  Orbit  Tip Deflection of Module 2 0.03  e  2  0  -0.03 0.01  0.02  0.03  Orbit  Figure 3-9  F L T / L Q R controlled response as affected by the revolute joint stiffness during a manipulator maneuver: (c) flexible generalized coordinates.  90  3.3  Trajectory T r a c k i n g Tracking  o f prescribed trajectories  manipulator is called upon to perform.  represents one  o f the  important  tasks a  This section considers trajectories in the form o f  straight line and circle in the vertical x, y - plane using the F L T applied to the rigid degrees o f freedom. position. a ,a )  (  x  2  Note, the system requires two coordinates to specify the manipulator's tip  However, there are four actuators {T\, F\, T , F ) i n the form o f two revolute 2  and two prismatic (l\, h) joints.  2  Thus there are two redundant coordinates.  trajectory involves both a path and the time evolution o f the path.  A  In the present study,  trajectories follow sine-on-ramp profile as mentioned earlier (Figure 2-3). Three cases are considered: (i)  Tracking o f a 10 m long straight line perpendicular to the platform using revolute and prismatic joints o f module two. Joints o f module one are held fixed. The time allocated to complete the task is 120 s.  (ii)  Tracking o f a 10 m long straight line along the platform, i n 120 s, using revolute joints o f modules one and two.  Prismatic joints are locked in  position. (iii)  Tracking o f a circle with 3 m radius in 200s using two revolute joints, i.e. prismatic joints are locked as i n Case (ii) keeping the module lengths fixed.  Case (i): Straight Line Perpendicular to the Platform The manipulator is located at the tip o f the platform with modules initially aligned with the platform.  The initial lengths o f module one and module two are 7.5 m.  The  manipulator tip is commanded to move from the x, y coordinates (75m, 1.5m) to (75m, 11.5m) in 120 s. Desired time histories o f a  2  and l  2  91  can be readily obtained from:  (3.20a)  h = -Ji-5 +y 2  10  with  yd  The tracking performance  =  120  (t-  (3.20b)  ;  2  m d  120  sin(  120  t) m.  and system response are presented  in Figure 3-10.  Initial  configuration o f the manipulator, its trajectory and tip errors are shown i n Figure 3-10(a). Note, the peak deviation o f the tip from the desired position is « 4.5 m m in the x direction with a steady state oscillation amplitude, about the terminal point, o f « 1 mm. Thus, the maximum error i n the x direction is less than 0.12% o f the trajectory's  length.  The  corresponding value i n the y direction is « 0.4 %. A s can be expected, the maneuver has virtually no effect on the platform orientation and the peak torque required to maintain the local horizontal (unstable) position is only « 80 N m (Figure 3-10b).  Demands on the  revolute and prismatic joints are also rather small; only 5 N m and 0.7 N , respectively. Response o f the flexible degrees o f freedom (Figure 3-10c) clearly shows that the effect o f the module elasticity on the tip error (e ) is negligible (« 0.26 m m maximum). The major 2  contribution to the error arises from the joint flexibility (/?,  m a x  « 0.06°, J3  2m a x  ~ 0.03°).  Case (ii): Straight Line along the Platform Here tracking o f a horizontal straight line near the x-axis using the two revolute joints is considered.  A s before, the manipulator is initially aligned with the platform and the  module lengths are held fixed at l =l x  2  =7.5 m (Figure 3-1 la).  The task-time o f 120 s  corresponds to « 0.02 orbit. The desired time history for the trajectory is given by:  92  First Module^  Second Module \ 10m  4.5  d = 60m  Platform  75m 12 r y, m  X, m  Tracking Error Along x Direction  y, m  Second Module's Tip Motion t = 120 s  12 10 -0.01 0.02  0.04  0.06  -  8  Desired  6  Tracking Error Along y Direction  4 -  2  t= 0  0 74.99 0.02  Figure 3-10  Orbit  0.04  75  75.01 x,  0.06  m  Tracking o f a straight line using the second module and the F L T : (a) manipulator tip trajectory and errors.  93  I.C.'s (Flexible d.o.f.I: e = e, = e = 0; p  L.V.j  fA^i  p  2  p  l :K  /7\  h  Controller Gains: y/ :K = 0.02; K =0.29. l  '  fr  v  = &3;  p  K =5J6. V  I.C.'s (Controlled d.o.f.):  l : K = 1; K — 2.  y/ = - 9 0 ° ; ^ = ^ = 0 ;  a :K  = 8.3;  /, = l = 7.5 m.  a :K  = 6.67; ^ = 5.16.  p  2  p  x  2  2  p  p  v  K =5J6. V  v  Maneuver: 10m straight line tracking using the second module.  Momentum Gyros Torque  Platform Libration 100  -89.9  V  0  -90.0  P  r  -90.1 0.00  60  0.02  0.04  0.06  0  -100 0.00  Second Joint Rotor Motion  0.02  0.04  0.06  Second Joint Actuator Torque  «2  0.00  14  0.02  0.04  0.06  0.00  Deployment of Module 2  0.02  0.04  0.06  Second Deployment Actuator Force  F, 0.0  0.00  0.02  0.04  0.00  0.06  Orbit Figure 3-10  0.02  0.04  0.06  Orbit  Tracking o f a straight line using the second module and the F L T : (b) response o f the platform and rigid degrees o f freedom with control inputs.  94  Controller Gains:  I.C.'s (Flexible d.o.f.):  y :K  = \ = i = 0;  e  e  P  e  p  = 0.02;  p  1 :K }  L.V.|  ^  ¥P 1 JA '  A  a  I.C.'s (Controlled d.o.f.):  l '• K = 1;  y/ = - 9 0 ° ; ar, = ar = 0 ;  a :K  /, = l  a :K  p  2  = 7.5 m.  K =5J6. V  K =2.  p  l  2  2  v  = S3;  P  2  K =0.29.  p  v  = &3;  p  K =5.16. v  = 6.67; ^ = 5 . 1 6 .  Maneuver: 10m straight line, tracking using the second module.  Platform Vibration  First Joint Vibration  0.02  0.04  0.00  0.06  0.02  0.04  0.06  Tip Deflection of Module 1  Second Joint Vibration 0.03  0.00  -0.03 h 0.00  0.02  0.04  0.00  0.06  2  0.04  0.06  Orbit  Tip Deflection of Module 2  e  0.02  10  0.00  0.02  0.04  0.06  Orbit Figure 3-10  Tracking o f a straight line using the second module and the F L T : (c) response o f flexible degrees o f freedom.  95  y  =1.5m;  d  x =(10/120)(r-(120/2^)sin[(2^/120)r] m ; d  with the joint angles obtained using simple inverse kinematics:  l?+l -(x y ^  r  a  a  X r l  l  d  =tan  = K -  + —cos"  cos  2  2  2  2  Tt  d+  2/,/  2 '  2  (3.21b)  1  2/,/  (3.21a)  2  Note, the initial manipulator configuration represents a kinematic singularity requiring a  xd  d  ,  to be oo. T o overcome this situation, a was given a small positive value (5°) at t=0.  2d  x  This also resulted i n a small negative value ( - 1 0 ° ) for a • A s the use o f revolute joints leads 2  to increased coupling effects, the error is anticipated to be higher. Figure 3-9(a) shows the maximum error o f about 2.8 % . Effect o f the straight line tracking maneuver on the platform, though relatively higher than that in the previous case, still remains rather small (Figure 3-1 l b ) . W i t h a peak demand of « 330 N m , the platform is able to maintain its equilibrium position along the local horizontal. Control demands at revolute joints also remain modest (T\  tm a x  « 40 N m , T  2>m a x  ~  10 N m ) . Flexible degrees o f freedom exhibit trends (Figure 3-1 l c ) which are similar to those observed before (Figure 3-10c), i.e. trajectory tracking error is primarily contributed by the flexibility o f the joints. Note, the peak values o f both J3 and P are now higher than before X  (J3  xmax  * 0 . 2 ° , /3  2max  * 0.05° as against j3  Xmax  * 0 . 0 6 ° , J3  2  ljmm  « 0.03° for Case (i)). This  reflects i n a higher peak error. Contribution o f modules' elasticity continues to remain small  96  y  A  First Module  Second Module  4.5m r*  ' 10m Platform  J = 60m 75m  y, m 10 First Module Locus of the First Module's T i p  Second Module  Locus of the Second Module's Tip  Desired Trajectory 60  Figure 3-11  65  70  75  x, m  Tracking of a horizontal straight line with two revolute joints using the F L T : (a) initial configuration and the trajectory error.  97  I.C.'s (Flexible d.o.f.):  Controller Gains:  e = \ = 2 = 0:  y :K  e  e  p  p  p  A =A=o.  = 0.02;  K =0.29. v  = 8.3;  K =5J6. V  I.C.'s (Controlled d.o.f.):  l :K  y/ = - 9 0 ° ; a = 5°;  ar,: A"„ = 32; ^ = 11.3.  a  flr :Xp  p  2  x  2  p  = 4; K = 4. v  v  = - 1 0 ° ; l = / =7.5m. x  2  2  = 25; ^ = 10. v  Maneuver: 10m straight line tracking using the two revolute joints.  Platform Libration 400  Momentum Gyros Torque  200 V  0 -200 0.00  0.02  0.04  First Joint Actuator Torque  0.02  0.00  0.04  Second Joint Rotor Motion  0.02  0.04  Second Joint Actuator Torque  0.00  0.04  Orbit  Figure 3-11  0.02  0.02  0.04  Orbit  Tracking o f a horizontal straight line with two revolute joints using the F L T : (b) response o f rigid degrees o f freedom and control inputs.  98  Controller Gains: y/ : K = 0.02; K =0.29.  I.C.'s (Flexible d.o.Q: e = \ = i = °; e  e  p  P  p  l :K i  L.V.;  ^  A  ¥P 1  = S3;  p  I.C.'s (Controlled d.o.f.):  l :K  ^=-90°;  a :K  a =5°;  2  x  2  =7.5m.  K =5J6. V  = A; K = 4.  p  x  x  a =-10°; l =/  2  v  v  = 32; K = 11.3.  p  v  a :K = 25; K = 10. 2  P  v  Maneuver: 10m straight line tracking using the two revolute joints.  Platform Vibration  First Joint Vibration  P°  0.0  r  0.00  0.02  0.04  0.00  0.02  0.04  Tip Deflection of Module 1  Second Joint Vibration  0.06 e  t  0  0.00 -0.06 0.00  0.02  0.04  0.00  0.02  0.04  Orbit  Tip Deflection of Module 2  0.00  0.02  0.04  Orbit Figure 3-11  Tracking o f a horizontal straight line with two revolute joints using the F L T : (c) response o f flexible degrees o f freedom.  99  (< 0.8 mm).  Case (iii): Circular Trajectory Tracking o f a circular trajectory, o f radius 3 m and center located at x = 64m, y = 8.5m, using two revolute joints represents a relatively challenging task (Figure 3-12a). The module lengths are fixed at /, =l = 7.5 m and the base is located at x = 6 m , y = 1.5 m. The 2  tracking period is 200 s. The trajectory can be represented as (JC - 6 4 ) + (y - 8.5) = 3 , 2  2  (3.22)  2  or in terms o f time o f 200 s permitted to complete the trajectory: j c - 6 4 = 3cos(0.0brt) m ;  (3.23a)  y - 8 . 5 = 3sin(0.0Lrt) m.  (3.23b)  It can be visualized as a point moving on a circle with a radius o f 3 m and a period o f 200s. The point, and hence the tip o f the manipulator, moves at a uniform speed o f V = ^jx +y 2  2  =0.03^- m/s.  Figure 3-12(a) shows tracking o f the circle (with reference to x',  (3.24) y' coordinates) and the  associated error. The period covered is the first 200 s, i.e. the time taken to complete the first circular trajectory.  Note, i n terms o f the orbital period o f 92.5 minutes, 0.01 orbit  corresponds to 55.5 s. The error is rather large with a peak value o f around 7 cm. However, the error diminishes significantly during the second period o f 200 s to 400 s as shown in Figure 3-12(b).  The maximum error is reduced to « 7mm ! This can be explained quite  readily by referring to the response plots in Figures 3-12(c) and 3-12(d). A t the outset it is apparent that compared to tracking o f straight lines, the circle represents a large disturbance to the platform as well as the manipulator. Though within the  100  Starting Position 8.5m •Second Module  *1.5m  Platform  Figure 3-12  60m  First Module  Tracking o f a circle using the two revolute joints and the F L T for control: (a) manipulator tip trajectory and error during the period o f 0-200 s.  101  . Starting Position  Second Module  Platform  y,  60m  First Module  m  Desired Trajectory Actual Trajectory  Equations of the Desired Circle: Starting Point, t = 200 s  x ' = 4 + 3cos(0.0lTtt) m; / = 7 + 3sin(O.Ol7Ct) m.  Time History of Tracking Error  Second Module  0  200  10 Figure 3-12  400 (s)  300  15  x,  m  Tracking o f a circle using the two revolute joints and the F L T for control: (b) manipulator tip trajectory and error during the period o f 200-400 s.  102  Controller Gains:  I.C.'s (Flexible d.o.L): e  L.V.j  p =  e  \ =  e  2 =  iy \K  0;  p  p  = 0M;  A=A=o.  / , : i ^ = 8.3;  I.C.'s (Controlled d.o.f.):  l :K  ^ = -90°; a =-3.7°,  a :K  a =97.4°; ^ = /  a :K  2  p  x  x  2  2  =7.5m.  = 4;  2  p  p  K =0.29. v  K =5.16. v  K =4. v  = 64; K = 16. v  = 50;  14.14.  Maneuver: Tracking a circle using the two revolute joints.  Platform Libration  0.02  0.04  Momentum Gyros Torque  0.00  0.06  First Joint Rotor Motion  0.00  0.02  0.04  0.02  0.04  0.06  First Joint Actuator Torque  0.06  0.00  Second Joint Rotor Motion  0.02  0.04  0.06  Second Joint Actuator Torque  160  Figure 3-12  Tracking o f a circle using the two revolute joints and the F L T for control: (c) response o f rigid degrees o f freedom and control inputs.  103  I.C.'s (Flexible d.o.D:  Controller Gains:  e  y/ :K = 0.02;  L.V.j  = \ = i = °; e  P  e  p  K =0.29.  P  v  A=A=o.  l :K  = 8.3;  I.C.'s (Controlled d.o.f.):  l :K  = A; K = 4.  x  ^ =-90°; a =-3.7°, x  a  2  = 9 7 . 4 ° ; l =l x  2  =7.5m.  p  2  p  ' ^ : i : a :Kp 2  K =5J6. V  v  p  = 64; # = 16. v  = SQ; ^ = 14.14. v  Maneuver: Tracking a circle using the two revolute joints.  First Joint Vibration  0.00  0.02  0.04  Platform Vibration  0.06  Orbit  Figure 3-12  Tracking o f a circle using the two revolute joints and the F L T for control (d) response o f flexible degrees o f freedom.  104  permissible limit, the peak platform libration is « 0.007° with the C M G output o f « 1,100 Nm.  The revolute joints also demand higher torques with peak values reaching « 200 N m  and « 15 N m for a  {  and a , 2  respectively (Figure 3-12c).  The tip tracking errors are  primarily contributed by joint and module flexibility effects. Both o f them are relatively high during tracking o f the first circle (i.e. time < 0.04 orbit) with peak values reaching: e  i.»««*  2 m m  ;  e  A.max ~  2 i m f l X  *3mm;  Pl.max  X  0.85°.  These lead to the high value o f error observed during tracking o f the first circle. However, during the period o f 200-400 s ( « 0.04 to 0.08 orbit) they reduce significantly resulting in better tracking performance.  Results suggest that the tracking error would progressively  reduce with the passage o f time and should become negligible for t > 0.06 orbit. Note, the flexible degrees o f freedom are not actively controlled by the F L T . They are regulated only indirectly through coupling. reduce the error.  In practice, the presence o f structural damping would help  O f course, one can improve the tracking performance by actively  controlling the flexibility degrees o f freedom at a cost o f increased demand on power and the controller complexity.  105  4. G R O U N D B A S E D E X P E R I M E N T S Experiments in space are very costly and time consuming. They can become prohibitive and infeasible in many cases. That is one o f the main reasons for the necessity o f lengthy mathematical modeling and investigation through computer simulation. A s an alternative to space-based experimentation, one often turns to prototypes located on Earth. O f course, for practical reasons, no ground-based setup can simulate the space environment exactly. However, a carefully designed ground-based facility can be used to advantage in assessing the performance trends. Furthermore, once the ground-based computer simulations are verified through prototype experiments, it is possible to justify, by induction, their validity in space where the forces are significantly small. In fact, since the beginning o f the space-age in 1957, around 20,000 spacecraft have been launched. Every one o f them was primarily designed through extensive numerical simulations, complemented by a few simplified ground-based experiments. The objective here is to evaluate real-time controlled performance o f the variable geometry manipulator using the ground-based prototype facility designed by C h u [8].  4.1  System Description Figure 4-1 shows the manipulator system that has been developed and located in the  Space Dynamics and Control Laboratory o f the Department o f Mechanical Engineering, University o f British Columbia. The prototype manipulator, employed i n the experimental study, consists o f a fixed base that supports two modules o f the robot connected in series. Each module has two links: one able to slew, and the other free to deploy and retract. The manipulator workspace has the shape o f a human heart, extending 2 m from top to bottom  106  107  and 2.5 m across. Rotational motion is made possible through the use o f revolute joints actuated by D C servo-motors. The deployment and retraction are carried out with prismatic joints consisting o f lead-screw and roller-nut assemblies, each o f which transforms the rotational motion o f a servo-motor into the translational motion o f a deployable link (Figure 4-2). Actuator motors integrated with optical-encoder motion sensors are interfaced with a Pentium 200 M H z M M X P C through a three-axis multi-function input/output motion control card. The manipulator is essentially rigid.  4.1.1  M a n i p u l a t o r base The fixed base supports the manipulator system. The first module is attached to the  pivot plate, which is threaded to the pivot shaft. A n 80 m m thrust bearing located between the pivot plate and the top plate o f the base carries the weight o f the manipulator. This bearing also provides the slewing freedom about the rotational axis. A second bearing is located under the top plate o f the base and is held i n place with a lock nut. A flexible coupling connects the pivot shaft to a gear head with a speed reduction ratio 20:1, which amplifies the torque that is delivered by the D C servo-motor. The rotational motion o f the base motor is transmitted in series, through the gear box, the flexible coupling, the pivot shaft, and finally through the pivot plate holding the slew end o f the first module o f the manipulator system (Figure 4-3).  4.1.2  M a n i p u l a t o r modules Both modules o f the prototype manipulator system are identical, each having one  revolute joint and one prismatic joint. The first revolute joint is located at the base, while the  108  Module 1  Flexible Coupling  Slewing  D C Servo-Motor J  • Deployable Link 1  •Pivot Plate  1= A x iiai al Bearing  L o c k Nut  - Thrust Bearing \ ^ Top Plate o f the Base  •Pivot Shaft Flexible Coupling  Gear ^ Head (1:20)  Optical Encoder  DC Servo-Motor  Figure 4-3  Base Housing  M a i n components o f the manipulator base assembly.  110  second one is at the end o f the first module, i.e. at the elbow joint. The deployment is realized with the transformation o f the rotational motion o f the motor that drives the lead screw into translational motion o f a roller nut that is fixed to the deployable link (Figure 4-4). The pitch o f the lead screw o f the first module is 2.5 m m (i.e. the deployable link moves 2.5 m m per revolution) while it is 1 m m for the second module.  4.1.3  Elbow joint The joint connects the deployable end o f module 1 to the slewing link o f module 2.  The structural connection consists o f two pivot plates bolted onto the deployable end o f module 1. These plates support the slewing motor and the gear head. The elbow joint is supported on a flat structure within the workspace, through a spherical joint. The mechanism that provides the rotational motion, at the elbow joint, is identical to the one located at the base (Figure 4-5).  4.2  Hardware and Software Control Interface The hardware o f one closed control loop o f the prototype manipulator mainly consists  of an I B M compatible host computer, an M F I O 3 A motion control input/output interface, a power amplifier, and D C servo-motors with built-in optical encoders (Figure 4-6). The control structure for each degree o f freedom is identical. Computer System The computer used for control purposes is a Pentium 200 M H z M M X I B M compatible, with Q N X as the operating system. Real-time application o f a digital control system depends on an operating system to handle multiple events within specified time  ill  Linear Axial Bearing  F i g u r e 4-4  Prismatic  joint  Roller Nut  mechanism  retrieval capability.  112  which  provides  the  deployment  and  Figure 4-5  Main components of the elbow joint assembly.  113  User Interface (QNX) Desired Position  /pssg£5|gSlS\,,  Feedback Signal  Control Signal  MFIO-3A Interface  Power Amplifier  Position Feedback  Figure 4-6  D C ServoMotor  Manipulator Joint (one degree o f freedom)  Optical Encoder  Open architecture of the manipulator control system for a single joint.  114  constraints. The more responsive the operating system, the more 'room' a real-time application has for maneuvering to meet its deadlines. The Q N X operating system provides multitasking, priority-driven preemptive scheduling, and fast context switching.  Motion control interface card A n M F I O - 3 A high-speed interface card for P C s is used for multi-axis, coordinated motion control. It is a multifunction input/output (I/O) card for motion control applications using a P C . It has three-channel 16-bit digital-to-analog converter ( D / A ) ; three quadrature encoder inputs; 24 bits o f programmable digital I/O, synchronized data reading and writing; a programmable interval timer; and a watchdog timer. The card has a S Y N C signal, which allows for the synchronisation o f data acquisition and analog output. The data from the D / A converters are latched into registers through the S Y N C signal. The D / A channels have 16-bit o f resolution. The encoder signals are digitally filtered for noise suppression. The programmable interval timer can generate timed intervals from 0.25 p.s to 515 seconds. The feedback controller runs as a task in the Q N X operating system. The card may be programmed either by accessing the hardware at the register level i n C or through the use o f Precision MicroDynamics' C subroutine library. The source code is compiled with the Watcom-C compiler for Q N X .  The C-library provides access to the M F I O - 3 A hardware  with routines to initialize the board; set up the Programmable Interval Timer (PIT), watchdog timer, and S Y N C signal; start the PIT and watchdog timer; set up the interrupts; read and write the digital I/O; read and write the encoders; and write the digital-to-analog converters.  115  Linear power amplifier Linear amplifiers are used with the joint motors o f the prototype manipulator. Their function is to transform the pulse train, i.e. the +/- 10 V signal from the controller, into current to drive the joint motors. The amplifier gain is set so that a 10V command generates the maximum drive current.  DC Servo-motors Slewing motors o f the manipulator are Pittman 14202 (109 oz-in peak torque) and 9413 (16 oz-in peak torque) for modules 1 and 2, respectively. A N E M A 23-20 reduction gear head o f ratio 1:20 is used. The gear head reduces the speed while increasing the output torque o f the motor by a factor o f twenty. The deployment motors are Pittman 9414 (24 oz-in peak torque). The motors operate through the D C current supplied by the power amplifier, in response to a controller signal.  Optical encoders The position o f each motor is sensed through the use o f the optical encoder attached to the motor shaft. The encoders have the offset track configuration (two tracks with their windows having an offset o f 1/4 pitch with respect to each other). A n encoder disk has two identical tracks, each having 1000 windows. A third track with a lone window generates a reference pulse for every revolution [53]. The physical resolution o f the encoders is 0.09°. The signal from the encoder is monitored at every sampling interval by the controller whose objective is to correct any deviation o f the actual joint position from the desired one.  116  4.3  Digital Control of the Ground-Based Manipulator Chu [8] has carried out extensive numerical simulation study aimed at dynamics o f  the ground-based manipulator model.  H e has also reported a P I D control o f the system.  Objective here is the real-time implementation o f the control algorithms, developed earlier, on the ground-based system. From the governing equations o f motion, it follows that the manipulator, as a control plant, is nonlinear, non-autonomous and coupled. That makes the Feedback Linearization Technique ( F L T ) , sometime called the computed torque method [53], a reasonable choice as a control procedure for this manipulator. A typical Proportional-Integral-Derivative (PID) controller is also implemented for the purpose o f comparison with the F L T . Robot control schemes often involve a great deal o f computation for the evaluation o f nonlinear terms. Therefore, they are implemented as digital control laws on digital signal processors (DSPs).  A s obtained before for the F L T i n the continuous time domain, the  control input can be written as (Eq. 3.3) Q =M[(q ) -u]+F, r  r  d  or in the general form r(t) = M(q)[q -u] d  + F(q,q).  (4.1)  The objective here is to discretize the above relation for the digital implementation. One approach to this end is shown in Figure 4-7. q(t) and q(t) are sampled to define q = q( ) > kT  k  q =q(kT),  (4.2)  k  with T as the sample period. Typically, a sample period in robotic applications can vary from about 1 to 20 ms. A zero-order hold is used to reconstruct the continuous time control input  117  r(t), needed for the actuators, from the samples r . The F L T digital control law amounts to k  selecting M = M(q(kT)),  F = F(q(kT),q(kT)),  (4.3)  and a digital P D outer loop control signal u where k  u =-K e -K e . k  v  k  p  k  The robot arm digital control input can now be written as r = M(q )(q k  k  kd  +Ke v  k  +Ke p  k  ) + F(q , q ) , k  (4.4)  k  where the tracking error is e(t) = q if) - q{t) with subscript 'd' representing, as before, the d  desired trajectory.  9k 9kd  k  ! Inner Loop  (Mk)  F  k  a  M(q ) k  i  q  k5  ZOH  Robot arm  K  v  /  i  L  *k  9k  9kd  Figure 4-7  Digital robot control scheme.  Depending on the manipulator configuration, one may use different sampling rates for q and q as against that for M and F, e.g. the inner nonlinear loop can be sampled more k  k  slowly than the outer linear feedback loop.  118  9k  4.3.1  Discretization of the inner nonlinear loop There is no exact way to discretize the nonlinear dynamics. Given a nonlinear state-  space system x = f(x,u), Euler's approximation yields  One relies on disretizing the robot arm dynamics in such a way that energy and momentum are conserved at each sampling instant [55]. Unfortunately, this results i n extremely complicated discrete dynamical equations, even for simple robot arms. Thus it is difficult to derive guaranteed digital control laws. Here, only approximations given by E q . (4.3), which appear i n the inner nonlinear loop, are considered.  4.3.2  Joint velocity estimate from position measurements For a continuous-time robot controller, it is assumed that both the joint position and  velocity are available exactly. In fact, it is usual to measure joint position using an optical encoder, and then estimate joint velocity from these position measurements. Simply computing the joint velocity using the Euler approximation  °k=("k-<lk-x)lT,  (4.5)  is virtually doomed to failure, since this high-pass filter amplifies the encoder measurement noise. Let the joint velocity estimates be v . k  compute v from q k  k  using  119  Then a filtered derivative can be used to  k  v  =  k-\ + ( k  <Jv  a  (4.6)  -<lk-\)l > T  where cr is the design parameter, ideally equal to zero. Equation (4.6) corresponds to a pole at z = 1 with faster response, but with some filtering to reject unwanted sensor noise. It should be noted that velocity estimates are not only used i n the outer linear loop for computing e , but also to evaluate the inner nonlinear term F(q ,q ) k  4.3.3  k  in E q . (4.4).  k  Discretization of outer PD/PID control loop A P I D controller represents the outer feedback loop. From this continuous-time P I D  controller, a digital P I D controller for the outer loop may be designed as explained by Lewis [56]. A continuous P I D controller that only uses joint position measurements q(t) is given by: u=  -K (s)e; c  K (s) = k 1 +  (4.7)  +  c  where k is the proportional gain, Tj is the integration time-constant or 'reset' time, and To is the derivative time-constant. Rather than employing pure differentiation (Eq. 4.5), a 'filtered derivative' is used which has a pole far left i n the s-plane at s = -N/T  D  . The value for N is  often i n the range 3 to 10; it is usually a fixed number. O f course, the P D controller is a special case o f the P I D procedure. A common approximate discretization technique for converting the continuous-time controller K (s) to the digital controller K(z) is the Bi-Linear Transform ( B L T ) where: c  120  K(z) =  K (s'); c  s =•  2 z-1 T z +l  This corresponds to approximating integration by the trapezoidal rule. Under this mapping, stable continuous systems with poles as s are mapped into stable discrete systems with poles at •_ l + sT/2 ~  (4.8)  l-sT/2  Z  The finite zeros also map according to this transformation. However, the zeros at infinity in the s-plane map into zeros at z = - 1 . Using the B L T to discretize E q . (4.7) yields  K(z) = k 1 + T  T  z +l  T, Dd  •1  u  z  - •1  T  z-cr  •+ -  (4.9)  with the discrete integral and derivative time constants: (4.10)  T =2T,; ld  T  N  (4.11)  T  \ + NTj2T  D d  '  D  and the derivative-filtering pole at  cr -  1-  NT/2T  D  l + NT/2T  D  It is easy to implement this digital outer-loop filter in terms o f difference equations on a D S P . First, one writes K(z) i n terms o f z , which is the unit delay in the time-domain (i.e. _ 1  a delay o f T seconds), as  K(z~ ) = k x  u  T 1 + z-' T  Id  121  l-z-  |  T  Dd  T  1 + z\-oz~  x  (4.12)  N o w let the control input u be related to the tracking error as k  u =K(z- )e .  (4.13)  l  k  Then u  k  k  may be computed from past and present values o f e using auxiliary variables as k  follows: ' = ' _ TlT ){e +e _ );  (4.14)  v =vl +(T lT){e -e _ );  (4.15)  u =-k(e +v' +v?).  (4.16)  v k  v k  l+{  Id  k  k  x  D  k  x  k  The variables v[ and v  k  Dd  k  k  k  x  k  represent the integral and derivative portions o f the digital P I D  controller, respectively. These difference equations are easily implemented through an appropriate software.  4.4  Controller Implementation  4.4.1  Dynamical equations for the ground-based Manipulator Equation governing dynamics o f the ground-based prototype manipulator can be  obtained quite readily by eliminating the orbital motion, gravity gradient, and the flexible degrees o f freedom from the space-based formulation for the system [12]. A s the F L T is a model-based controller, it is rather computationally intensive. During experiments with the prototype , the inner module (module 1) was held fixed (Figure 4-8). The corresponding governing equations are:  122  + m S2  2'  (4-17)  V  where:  m  mass o f module 2, m^i + md ;  2  2  mass o f slewing and deployable links, respectively, o f module 2; Is2, Id2  mass moments o f inertia o f slewing and deployable links, respectively, o f module 2;  h  length o f module 2;  ls2, ld2  lengths o f slewing and deployable links, respectively, module 2;  F,  force and moment, respectivley, at joint 2;  2  a  2  T  2  slew angles at joints 2. nent o f inertia o f module 2 about the swing axis.  y4 Module 2 nid2, I<n  m 2, Is2 S  Module 1  > i 111  r  Figure 4-8  Two-module ground based manipulator system.  123  4.4.2  C o n t r o l system parameters The F L T has several desirable properties as pointed out before. However, in order to  implement the F L T with accuracy, the system model and associated parameters must be known precisely. Design specifications o f the manipulator system are: Slewing arm length  0.3 m ;  M a x i m u m extension o f the deployable arm  0.2 m ;  Slewing arm sweep range  - 1 3 5 to 135 deg;  M a x i m u m rotational speed  60 deg/s;  M a x i m u m deployment speed  0.04 m/s.  Values o f the moments o f inertia I i, and I i were obtained experimentally through s  d  swing (pendulum) tests as shown in Figure 4-9. A typical test involved application o f a small displacement from the vertical and counting the number o f cycles over a known period o f time. The moment o f inertia about the swing axis is given by  J  =  !  !  ^  =  co  n ^ _  t  (  4  1  8  )  4/r  where T = period o f oscillation; m = mass; and / = distance from the swing axis to the center of mass. The parameter values determined in this fashion are shown i n Table 4-1.  The  moments o f inertia were measured for the whole module with three different positions o f the deployable link.  For the F L T control, a single-module manipulator (module 2) was used,  hence only the parameters for this module are relevant i n the experiments. The parameters for the manipulator are: Module 1 •  slewing link mass  wsi  •  slewing link length  / s i = 0.3 m ;  124  =  4.3 kg;  Center o f Mass  VA \  Figure 4-9  Schematic diagram o f the swing test to determine the moment o f inertia o f a manipulator link with different lengths.  Table 4-1 Mass (kg)  Module 1  Module 2  4.5  2.4  Swing-test results for the ground-based robot. /(m)  T(s)  /(kgm )  Position  0.276  1.343  0.557  fully retracted  0.322  1.444  0.750  middle  0.373  1.560  1.015  fully deployed  0.106  1.021  0.066  fully retracted  0.119  1.063  0.080  middle  0.129  1.109  0.094  fully deployed  125  2  - 0.2 kg;  •  deployable link mass  •  deployable link length  /di = 0.3048 m ;  Module 2 •  slewing link mass  ms2 = 2.2 kg;  •  slewing link length  /s2 = 0.3 m ;  •  deployable link mass  •  deployable link length  = 0.2 kg; = 0.3048 m ;  Based on system parameters, the time constant varied in the range o f 0.23 s to 0.28 s.  4.4.3  C o n t r o l l e r design Design o f the P I D controller is based on the experimental approach for the selection  of gains as proposed by Ziegler and Nichols [57]. In this context a linear and time-invariant system is assumed. The parameters obtained from the Ziegler-Nichols method give an initial set o f values for the P I D gains. Due to the nonlinear and non-autonomous nature o f the system, these parameters need to be refined and tuned for improved performance o f the system. W i t h linear representation o f the results from the swing test (Table 4-1), one has I = 4.206 x l 0 " « + 0.066, 8  (4.19)  where n (4000 counts/mm) is the encoder reading from the actuator o f the prismatic joint. The F L T algorithm can be expressed as r = M(q -u) d  + F,  126  (4.20)  where:  0  0  I  d2  M=  -m {l - -f){a ) l  m  dl  2  2  V  2  F =  - 2_  2m (l -^-)i d 2  2  2  ;  2  and  r=  a  Ji.  u = -K e - K e with e(t) = q (t) - q(t) . The F L T involves compensation for change i n the p  v  d  mass matrix M as the system moves and also for the dynamical coupling term F. A s a result, it should provide better performance compared to the fixed gain P I D controller.  4.5  Trajectory T r a c k i n g Once the controller is designed, a series o f trajectories tracking tests were performed  using both the F L T and P I D algorithms. These tests fall into two main categories: a) straight line trajectories; b) circle tracking. The P I D and F L T gains used are shown i n Tables 4-2 through 4-4.  Table 4-2  Controller gains for the prismatic joint o f module 2.  Module 2 (Prismatic)  K  K  Ki  PID  0.5  0.001  0.0001  FLT  0.5  0.001  Table 4-3  p  d  Controller gains for the revolute joint o f module 2.  Module 2 (Revolute)  Ka  Kt 0.001  PID  3  0.001  FLT  0.5  0.001  127  Table 4-4  Controller gains for the revolute joint o f module 1.  Module 1 (Revolute) PID  K  Kt  0.001  0.001  d  4  These gains were used throughout the set o f experiments. W i t h the F L T , a P D controller is used in the outer loop, for the error and the derivative o f the error, as mentioned before. A n y other suitable controller may be used for the outer-loop after linearization, for example the Linear Quadratic Gaussian ( L Q G ) or Hoc procedure [58]. Important symbols involved in the trajectory tracking are defined below: e  l  ,e^ ,e^  tracking errors at prismatic, revolute joint 1 and revolute joint 2, respectively;  I  I  , I ^, I ^  driving currents at prismatic and revolute joints, respectively;  a\, (%2  rotation angles at revolute joints 1 and 2, respectively;  a  a  a  a  A l  2  4.5.1  change in length o f the deployable link o f module 2.  Straight line trajectory The first tracking test involves making the tip o f the manipulator to follow a straight  line by using one-revolute joint and one prismatic joint o f the outer module (module 2) while the inner module is kept locked. The P I D controller is used to perform the test. The line is located 41.5 cm from the base o f module 2 and its length is 20 cm along the y direction. Figure 4-10(a) shows the location o f the tracked line and the configuration o f the manipulator. The specified tracking time is 4 seconds, and the tip trajectory has a prescribed sine-on-ramp profile as given i n E q . (2.4). The results o f the tracking experiment are shown in Figure 4-11 (prescribed profile i n dotted). It is seen that the manipulator tip follows  128  20 cm  <2l 41.5 cm  (a)  41.5 cm  41.5 cm  /V/>  \  (b)  gure 4-10  Schematic diagrams for straight line tracking using: (a) one revolute joint and one prismatic joint; (b) two revolute joints.  129  (a)  y, cm  (b)  a;  a,>A  J  0  10  Time, s  Time, s Figure 4-11  Straight line tracking using one revolute and one prismatic joint under the P I D control: (a) tip trajectory; (b) joint motion and the corresponding control signals.  130  the straight line with reasonable accuracy. The saturation i n I  a  is caused by the safety limit  of the current that is transmitted to the motor, which is constrained to 1 A . Figure 4-10 (b) shows another manipulator configuration for straight-line tracking. N o w a distance o f 20 cm is tracked along the x direction using two revolute joints, in 8 seconds. Note, the prismatic joints are locked with each module length fixed at 41.5 cm. A s pointed out in Chapter 1, a prismatic joint has the advantage that it does not possess dynamic coupling with the revolute joint o f the same module, since the reaction force passes through the center o f the joint. When two revolute joints are used, the reaction torque due to the rotational dynamics o f the second module w i l l try to rotate the first module in the counterclockwise direction resulting in over-rotation o f the first module. Thus the tracking error is biased i n the negative direction o f the jy-axis, as is evident from the experimental results shown in Figure 4-12 (a). From the plot o f e ^ a  in Figure 4-12 (b), it is clear that error is in  the negative side, as a result o f the dynamical coupling.  In a robotic system that has  kinematic redundancy, it w i l l be possible even to maneuver the first module to an appropriate position and then use only the prismatic joint for the line tracking. This w i l l lead to virtually no error in the tracking. Next, the F L T controller is used to carry out the same task as that shown in Figures 411 for the P I D control. Again, the execution time is set at 4 seconds and the length o f the line is 20 cm. The results are shown in Figures 4-13. B y comparing the behaviors o f the P I D and F L T controllers, following observations can be made: (i) The F L T has an adaptive capability with respect to the variation o f the mass matrix. It also has a compensation capability for dynamical coupling. Thus, the F L T produces more active control signals (high frequency I  a  and i / ) . (ii) The P I D controller is simpler, requires less computational power, and does 2  131  (a) 10  -i  1  1  1  1  1  1  r  8 6 4 y, c m  2 0 -2 Actual Trajectory  -4 -6 -8 -10  74  76  78  _i  80  i_  82  84 x, cm  _i  86  i_  88  90  92  (b)  -~—  10  Figure 4-12  Straight line tracking using two revolute joints under the P I D control: (a) tip trajectories; (b) joint motion and corresponding control signals.  132  (a)  y, cm  (b)  5  10  Time, s gure 4-13  Straight line tracking using one revolute and one prismatic joint with the F L T : (a) tip trajectory; (b) joint motion and the corresponding control signals.  133  not depend on a model o f the robot (the experimental, Ziegler-Nichols method is used here to tune the P I D gains). One is able to tune the F L T controller by incorporating a multiplicative confidence factor into the dynamical compensation term F, and gradually increasing it as more experience is gained through experimentation. This approach is used for tuning the F L T controller o f the prototype robotic system. The tip trajectories o f the robot under the control o f P I D and F L T separately, are plotted in Figure 4-14. It is clear that at the expense o f the computation cost, the F L T gives better tracking accuracy than the P I D . Note that towards the end o f the tracking, both controllers produce larger errors. This is due to increased dynamical coupling and the greater effort that is required for synchronizing the two joints at the end o f the trajectory. Another major source o f error is the unmodeled friction. It causes a steady-state error and stick-slip motion. These nonlinear effects cause vibrations and reduce the tip position accuracy. 25  20  15  10  y, cm 5  0  -5 37  Figure 4-14  38  39  40  41  42  43  44  45  46  x, cm Straight line tracking using one revolute and one prismatic joint: comparison o f the F L T and P I D procedures.  134  4.5.2  C i r c u l a r trajectory In order to further investigate the system, tracking o f a circular trajectory was  undertaken as a typical test-case. To begin with, tracking was carried out under the P I D control. Different trajectory speed profiles were employed to assess the effectiveness o f the controller. The circular trajectory is defined as:  (4.21)  where: P , P x  y  are tip positions i n the x and y directions, respectively; r is the radius o f the  circle; and co is the angular velocity o f the circular motion. For instance, co = O.ln corresponds to a tip motion around the circle in 20 seconds. Figure 4-15 schematically shows the tracked circle and the corresponding manipulator configurations at three different instants during tracking. The radius r is taken to be 10 cm for all the cases. Each circle is tracked twice to check the repetitiveness o f tracking. The first experiment o f tracking a circular trajectory uses <v= OAn; i.e. 20 seconds a circle. The results are shown i n Figure 4-16. The maximum errors occur at locations where joints change their directions o f motion. Due to the Coulomb friction, the joints have nonlinear dead zones. The signal from the controller is generated based on the motion error. Once a joint stops, the error must be large enough to overcome the static friction. This, in turn, causes a larger error. When the joint starts to move, the smaller dynamic friction results in a lower error. O f course, dynamical coupling also plays a role i n causing the motion error. This can be seen when the speed o f the profile increases from 20 seconds a circle to 10 seconds a circle (Figures 4-17). It is clear that with the increased speed the tip trajectory is not as smooth as the one in Figure 4-16. This is mainly due to the dynamical coupling. The  135  Figure 4-15  Schematic diagram showing tracking o f a circular trajectory using module 2.  corresponding control effort also has increased. The repetitiveness o f the trajectory worsens as well due to the same reason. When the tracking speed decreases from 20 seconds a circle to 30 seconds a circle (Figure 4-18), the repetitiveness improves but the error is slightly larger than that for the 20 seconds a circle. Here, dynamical coupling is lower, which makes the trajectory smoother and the level o f repetition better. However, when the desired trajectory moves slower, the error increases gradually and the control effort needed to overcome the steady-state friction takes a longer time to accumulate. This causes the nonlinear dead-zone effect to worsen. The F L T controller was also used i n the case o f 20 seconds a circle (Figures 4-19). A s before, the performance o f the F L T controller is better than that o f the P I D controller. It compensates for the dynamical coupling effect, which significantly reduces the error o f the revolute joint. It is seen that, in area C , the tip error is low. In other areas ( A , B , D ) , the  136  (a)  y, c m  40  42  44  46 x,  48  50  52  cm  (b)  0  10  20  30  40  0  Time, s Figure 4-16  10  20  30  40  Time, s  Tracking of a circle, at a speed o f 0.314 rad/s, using the P I D control: (a) tip trajectories; (b) joint dynamics and control signals.  137  A / , cm 2  5  10  15  20  10  15  20  -1  Time, s Figure 4-17  0  5  10  15  20  10  15  20  Time, s  Tracking of a circle, at a speed o f 0.628 rad/s, using the P I D control: (a) tip trajectories; (b) joint motion and control signals.  138  (a)  y, cm  0  L  42  44  46  48  50  52  x, cm (b)  0  20  40  0  60  Time, s gure 4-18  20  40  60  Time, s  Circle tracking behavior under the P I D control at a speed o f 0.209 rad/s: (a) tip trajectories; (b) joint dynamics and control effort.  139  (a)  v, cm  (b) 100  ,°  o  'a  ^\r>r/ A^^^  0  0  l  0  10  20  30  -0.5  40  0  10  20  30  40  10  20  30  40  1  la^A  0  -1 0  Figure 4-19  10  20  30  -1 0  40  Tracking of a circle, at a speed of 0.314 rad/s, with the FLT: (a) tip trajectories; (b) joint dynamics and control effort.  140  dynamical compensation effort is evident. The F L T controller tries to compensate for the nonlinearity  and  dynamics, and  consequently  reduces  the  error  and  improves  the  repetitiveness. Again, it may be pointed out that the F L T needs a higher level o f active control, and also has a higher bandwidth as expected. The ground-based experiments verify several distinguishing characteristics o f the variable-geometry manipulator system and its controllers: •  Prismatic joints have lower dynamical coupling, and are preferable for executing high precision tasks.  •  Kinematic redundancy o f a robot is useful i n task planning to improve the tracking accuracy.  •  The F L T controller is efficient, robust, and stable. It gives satisfactory performance, but at a computational cost. It is suitable for the variable-geometry manipulator system.  •  The P I D controller is simple and fast. It works well with most trajectory following cases. Its parameters need to be fine tuned once they are assigned by a technique such as the Ziegler-Nicholes approach. Friction plays a significant role in causing tracking errors. It should be carefully  modeled and compensated for. This is a difficult job, however, due to the highly nonlinear and time varying nature o f friction. Even in presence o f friction, the F L T is able to handle the control task in a robust manner. Modification o f the F L T algorithm to account for friction should provide improved control performance.  141  5. CONCLUDING REMARKS  5.1  Contributions  The main contributions o f the thesis can be summarized as follows: (i)  A detailed dynamical study o f a novel flexible space-based manipulator, involving two modules and accounting for orbital, librational as w e l l as vibrational interactions, has not been reported before. Providing understanding o f such complex interactions is indeed a contribution o f importance.  (ii)  Control o f this novel manipulator has received virtually no attention. The study lays a sound foundation to build on with the nonlinear Feedback Linearization Technique (FLT).  (iii)  A two-module manipulator system is linearized and a Linear Quadratic Regulator ( L Q R ) is designed to suppress vibrations arising in the manipulator links, joints, as well as in the platform. Such synthesis o f the F L T and L Q R to control both rigid and flexible degrees o f freedom represents a significant contribution.  (iv)  A n open architecture experiment is set up for a ground-based prototype manipulator. It makes implementation o f different control strategies possible.  (v)  It was indeed fortunate to have a two-module ground-based prototype manipulator designed and constructed by C h u [8]. This has made it possible to assess performance of control strategies not only through numerical simulations but also with groundbased experiments. Such correlation study for the novel robotic system is indeed rare.  (vi)  Such a comprehensive study involving numerical simulations as w e l l as ground-based experiments represents an important step forward.  142  5.2  Conclusions The project was so formulated as to emphasize the dynamics and control o f a novel  manipulator studied here. The objective has been to understand the system behavior at the fundamental level, through the study o f representative cases, and establish trends. Based on the investigation, the following general conclusions can be made: (a)  Significant coupling exists between the platform, link and joint vibrations, as well as system libration. The most pronounced coupling was observed between the joint and link vibrations.  In general, slewing and deployment maneuvers have a significant  effect on the flexible degrees o f freedom response. (b)  When the manipulator base is located near the platform's extremity, slewing and deployment maneuvers can also result in significant rigid body motion o f the platform.  (c)  Excitation o f the system's flexible degrees o f freedom can significantly deteriorate the accuracy o f the manipulator.  (d)  In general, payload, speed o f maneuver and joint flexibility represent three important parameters governing the response o f the system. A s can be expected, heavier payload, faster speed o f maneuver and reduced joint stiffness  affect the manipulator tip  dynamics adversely. (e)  The system exhibits unacceptable response under critical combinations o f parameters. The control strategy based on the F L T is found to be quite effective in regulating the rigid-body motion o f manipulator links as well as the attitude motion o f the platform. The unmodeled flexibility o f the platform, joints, and manipulator links has virtually no effect on the performance o f the F L T controller. It is able to regulate the elastic degrees o f freedom rather well through coupling. The controller is quite robust.  143  (f)  Active control o f flexible degrees o f freedom using the L Q R , together with the F L T for rigid generalized coordinates, significantly improves the situation.  The controller is  quite robust and continues to be effective even i n presence o f heavy payloads, fast maneuvers and reduced stiffness o f the revolute joints. The results should prove useful in the design o f this new class o f promising manipulators. (g)  The prototype manipulator with an open architecture is an effective way o f evaluating performance o f various control strategies.  (h)  The ground-based experiments generally validated trends indicated by the numerical simulation results. This is encouraging as the prototype system has limitations in terms o f backlash, friction and, at times, less than smooth operation.  5.3  Recommendations for F u t u r e W o r k Considering the diversity o f research areas associated with the field o f space robotics,  the present thesis should be viewed as an initial step i n the analysis and development o f this particular class o f space manipulators. There are several avenues which remain unexplored or demand more attention. Some o f the more interesting and useful aspects include: (i)  path planning and inverse kinematics with emphasis on obstacle avoidance, as well as minimization o f structural vibrations and base reaction; effect o f redundancy on system performance;  completion o f a given mission with one or more joints  inoperational; (ii)  dynamics and control o f satellite capture;  (iii)  comparative study o f various optimal, adaptive, intelligent and hierarchical control strategies to regulate the rigid and flexible dynamics o f single and multi-module systems;  144  (iv)  more  two-dimensional ground-based  experiments  to  help  validate  numerical  simulation results; (v)  animation o f simulation results for visual appreciation o f the physics o f the problem;  (vi)  incorporate another data acquisition board or upgrade the current one so that all four axis o f the ground based manipulator can be controlled;  (vii)  introduce an 'eye-in-the-sky' camera to determine the actual location o f the endeffector, and strain gauges to sense the flexibility o f the links  (viii)  develop a graphical user interface for the robot system and its controller.  145  REFERENCES [I]  Evans, B . , "Robots i n Space," Spaceflight, V o l . 35, N o . 12, 1993, pp. 407-409.  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A A S - 9 3 - 6 7 0 ; also Advances in the Astronautical Sciences, Editors: A . K . M i s r a et al., Univelt Incorporated Publisher for the American Astronautical Society, San Diego, U . S . A . , V o l . 85, Part III, pp. 2211-2229.  [37]  M o d i , V J . , Chen, Y . , Misra, A . K . , de Silva, C . W . , and M a r o m , I., "Nonlinear Dynamics and Control o f a Class o f Variable Geometry M o b i l e Manipulators",  148  Proceedings of the Fifth International Conference on Adaptive Structures, Sendai, Japan, December 1994, Editor: J. Tani et al., Technomic Publishing C o . Inc., Lancaster, P A , U . S . A . , pp. 140-149. [38]  Hokamoto, S., M o d i . , V . J . , and Misra, A . K . , "Dynamics and Control o f Mobile Flexible Manipulators with Slewing and Deployable Links", AAS/AIAA Astrodynamics Specialist Conference, Halifax, N o v a Scotia, Canada, August 1995, Paper N o . A A S - 9 5 - 3 2 2 ; also Advances i n the Astronautical Sciences, A A S Publications office, San Diego, California, U . S . A . , Editors: K . Terry Alfriend et al., V o l . 90, pp. 339-357.  [39]  Hokamoto, S., and M o d i , V . J . , "Nonlinear Dynamics and Control o f a Flexible SpaceBased Robot with Slewing-Deployable Links", Proceedings of the International Symposium on Microsystems, Intelligent Materials and Robots, Sendai, Japan, September 1995, Editors: J. Tani and M . Esashi, pp. 536-539.  [40]  Hokamoto, S., and M o d i , V . J . , "Formulation and Dynamics o f Flexible Space Robots with Deployable Links", Transactions of the Japan Society for Mechanical Engineers, V o l . 62, N o . 596, 1996, pp. 1495-1502.  [41]  Hokamoto, S., Kuwahara, M . , M o d i , V . J . , and Misra, A . K . , "Control o f a Flexible Space-Based Robot with Slewing-Deployable Links", Proceedings of the AIAA/AAS Astrodynamics Conference, San Diego, California, U . S . A . , July 1996, A I A A Publisher, Paper N o . A I A A - 9 6 - 3 6 2 6 - C P , pp. 506-513.  [42]  Caron, M . , M o d i , V . J . , Pradhan, S., de Silva, C . W . , and Misra, A . K . , "Planar Dynamics o f Flexible Manipulators with N Slewing Deployable Links", Proceedings of the AIAA/AAS Astrodynamics Conference, San Diego, California, U . S . A . , July 1996, A I A A Publisher, Paper N o . A I A A - 9 6 - 3 6 2 5 - C P , pp. 491-505.  [43]  Chen, Y . , On the Dynamics and Control of Manipulators with Slewing and Deployable Links, P h . D . Thesis, University o f British Columbia, Vancouver, B . C . , Canada, July 1999.  [44]  Goulet, J.F., Intelligent Hierarchical Control of a Deployable Manipulator, M . A . S c . Thesis, University o f British Columbia, Vancouver, B . C . , Canada, June 1999.  [45]  Ogata, K . , Modern Control Engineering, Prentice-Hall, Upper Saddle River, N . J . , U . S . A . , 1997, pp. 915-935.  [46]  D e Silva, C . W . , Vibration Fundamentals and Practice, C R C Press, B o c a Raton, Florida, U . S . A . , 1999, pp. 798-816.  [47]  Bejczy, A . K . , Robot Arm Dynamics and Control, J P L T M 33-669, California Institute of Technology, Pasadena, California, U . S . A . , 1974.  149  Singh, S.N., and Schy, A . A . , "Invertibility and Robust Nonlinear Control o f Robotic Systems", Proceedings of the 23rd Conference on Decision and Control, Las Vegas, Nevada, U . S . A . , December 1984, pp. 1058-1063. Spong, M . W . , and Vidyasagar, M . , "Robust Linear Compensator Design for Nonlinear Robotic Control", Proceedings of the IEEE Conference on Robotics and Automation, St. Louis, Missouri, U . S . A . , M a r c h 1985, pp. 954-959. Spong, M . W . , and Vidyasagar, M . , "Robust Nonlinear Control o f Robotic Manipulator", Proceedings of the 24th Conference on Decision and Control, Fort Lauderdale, Florida, U . S . A . , December 1985, pp. 1767-1772. Spong, M . W . , "Modeling and Control o f Elastic Joint Robots", Journal of Dynamics Systems, Measurement and Control, V o l . 109, 1987, pp. 310-319. M o d i , V . J . , Karray, F . , and N g , A . C , " O n the Nonlinear Slewing Dynamics and Control o f the Space Station Based M o b i l e Servicing System", Nonlinear Dynamics, V o l . 7, 1995, pp. 105-125. De Silva, C . W . , Control Sensors and Actuators, Prentice-Hall, Englewood Cliffs, N . J . , U . S . A . , 1989, pp. 196-199, pp. 218-244. De Silva, C . W . , and Mcfarlane, A . G . J . , Knowledge-Based Control with Application to Robots, Springer-Verlag, Berlin, Germany, 1989, pp. 87-98. Neuman, C P . , and Tourassis, V . D . , "Discrete Dynamic Robot Models", IEEE Trans. System Man and Cybernetics, V o l . S M C - 1 5 , N o . 2, M a r c h / A p r i l , 1985, pp. 193-204. Lewis, F . L . , Applied Optimal Control and Estimation, Prentice-Hall, Englewood Cliffs, N . J . , U . S . A . , 1992. Kranklin, G . F . , Powell, J.D., and Emami-Naeini, A . , Feedback control of Dynamics Systems, Addison-Wesley Publishing Company, Inc., Massachusetts, U . S . A . , 1986, pp.103-106. De Silva, C . W . , Intelligent Control: Fuzzy Logic Applications, C R C Press, B o c a Raton, Florida, U . S . A . , pp. 15-19.  150  APPENDIX I : SPECTRAL DENSITY ANALYSIS OF DYNAMICAL RESPONSE  To help establish the natural frequencies o f various system components and coupling effects, Power Spectral Density (PSD) distribution o f the system dynamical response was carried out. O f interest are the platform's rigid body librational frequency as w e l l as natural frequencies o f the elastic degrees o f freedom. To that end, an initial disturbance was given to the system, i n the desired degree o f freedom, and the corresponding response plots were obtained (y/ , e , J3 , e , J3 , e ). The individual response plot was subjected to the power p  X  l  2  2  spectral density analysis to arrive at the characteristic frequency as well as coupling contributions from other degrees o f freedom (Figures 1-1 to 1-4). The symbols used to designate various characteristic frequencies are as follows:  co  ¥  co  frequency  o f platform librational motion;  platform's bending natural frequency;  co  jX  first joint's natural frequency;  a>  first module's bending natural frequency;  co  second joint's natural frequency;  co  second module's bending natural frequency.  mX  j2  m2  Note, the librational frequency o f the platform pitch motion as w e l l as its fundamental natural frequency in bending were obtained i n Chapter 2 (Figure 2-11) as co « 3x IO" H z and 4  ¥  0.18 H z . Information i n Figures 1-1 to 1-4 was used to prepare Table 2-2 presented on page 42.  151  First Module Revolute Joint The revolute joint o f the first module was given an initial disturbance o f 5° (Jh(0) = 5°) and the system response was recorded (Figure I-la). The P S D analyses o f the responses associated with various flexible components are presented i n Figure I-1(b). joint's natural frequency was found to be co  jX  « 0.08 H z .  Major peaks  The revolute representing  contributions from other flexible components are also indicated to help assess coupling effects.  First Module Tip Deflection Tip o f the first module was given an initial deflection o f 0.2 m (e;(0)= 0.2m). The system response and P S D plots are shown i n Figures I-2(a) and I-2(b), respectively. The natural frequency o f the first module's tip oscillations is found to be co  mX  « 5.85 H z . It is  apparent from Figures T l ( b ) and I-2(b) that there is a strong coupling between the tip and the joint vibrations o f the first module. Note, dynamics o f the first module excites revolute joint of the second module (/? ) at its characteristic frequency ( a)j = 0.21 H z ) . O f course, tip o f 2  2  the module two should also vibrate at its natural frequency ( c o ) , however its P S D measure, ml  being relatively small, is not apparent. Therefore, it was determined through an independent module two tip excitation.  Second Module Tip Deflection Here the tip was subjected to an initial displacement o f 0.2 m (Figure 1-3). A s apparent from Figure I-3(b), a)  m2  « 8.5 H z . Coupling contributions can also be discerned,  152  particularly from its own joint (co ) as well as from module one (co , co^). j2  mX  Second Module Joint Deflection W i t h a 5° initial deflection for the second joint (/%(0) = 5°), Figure 1-4 gives the system response and corresponding spectral plots. It is found that the natural frequency o f the second joint {co ) is about 0.21 H z . The bending o f module two is also excited at co ~ j2  m2  8.5 H z (Figure I-4a, ei) modulating the p\ low frequency contributions. Note also the energy transfer to module one as suggested by the spectral peaks corresponding to co and a> . m]  153  jX  Initial Conditions:  Parameters: EI = 5.5xl0 N m ;  V =0,e  EI = 5 . 5 x l 0 N m ;  A = 5°, e = 0;  EI = 5.5xl0 Nm ;  A = 0, e =0.  8  2  p  5  p  2  s  5  = 0;  p  x  2  d  2  2  Kj = l . O x l 0 N m / r a d . 4  Specified Coordinates: a =a x  2  = 90°, l =l x  Platform Libration  0.01  0  0.02  0.01  0.02  Tip Deflection of Module 1  i  e  0  0.01  0.02  0.02  Tip Deflection of Module 2  Second Joint Vibration  0.02  0.02  Figure 1-1  = 7.5m.  Platform Tip Vibration  First Joint Vibration  0.01  2  Initial deflection o f 5° applied to the revolute joint o f module 1: (a) system response.  154  10  VP  5 0L  I  -  i  I  I  10  4 2 0  10  i  5000  I  i  I  i  A  0 A.. 0 20  i  i  10 ei  0),m l  10 0 0  10  2000 1000  A J2  0  10  4  e2  2 0 0  4  6  Frequency, Hz  Figure 1-1  Initial deflection o f 5° applied to the revolute joint o f module 1: (b) power spectral density distribution.  155  10  L.V.  Initial Conditions:  Parameters: EI = 5.5xl0 Nm ;  p , = G , e = 0;  EI = 5 . 5 x l 0 N m ;  P = 0, e = 0.2m;  EI = 5 . 5 x l 0 N m ;  A=0,  8  2  p  5  p  2  x  s  5  V ,  2  d  x  e =0. 2  Kj = 1.0 x l O N m / r a d . 4  Specified Coordinates: a =a x  2  = 90°, l =l x  Platform Libration  Figure 1-2  2  = 7.5m.  Platform Tip Vibration  Module 1 tip deflection of 0.2m: (a) system response.  156  1  1  vl.  1  1  i  i  2000  200  1000  0 20  I  I  I  ^2  10 0  J-  1  4  1  6  1  _J  10  Frequency, H z  Figure 1-2  Module 1 tip deflection of 0.2m: (b) power spectral density distribution.  157  Initial Conditions:  Parameters: EI = 5.5xl0 Nm ;  y/  EI = 5 . 5 x l 0 N m ;  A =  EI = 5 . 5 x l 0 N m ;  02=0, e =0.2m.  8  2  p  5  p  2  s  5  2  d  = 0 , e = 0; p  0, e = 0 ; x  2  Kj = l . O x l 0 N m / r a d . 4  Specified Coordinates: a =a x  2  = 90°, l =l x  2  = 7.5m.  Platform Libration  Platform Tip Vibration xl(T m ft P  S  0.01  0  0.02  0.01  Tip Deflection of Module 1  First Joint Vibration  0.01  0.01  0.02  -0.2  F 0.02  0.02  Orbit Figure 1-3  0.02  Tip Deflection of Module 2  Second Joint Vibration  0.01  0.02  Module 2 tip deflection o f 0.2m: (a) system response.  158  4  1  1  1  1  VP  2 0  t  l  10  4 2 0 400  2000  200  I  I  I  i  i  i  4  6  i  ^2  10  Frequency, H z  Figure 1-3  Module 2 tip deflection o f 0.2m: (b) power spectral density distribution.  159  Initial Conditions:  Parameters:  = 0 , e = 0;  EI = 5.5xl0 Nm ;  y/  EI = 5 . 5 x l 0 N m ;  A = 0, e = 0;  EI = 5 . 5 x l 0 N m ;  A = 5°, e = 0 .  8  2  p  5  p  2  s  5  p  x  2  d  2  2  Kj = l . O x l 0 N m / r a d . 4  Specified Coordinates:  a =a = x  2  Platform Libration  0.01  x  2  7.5m.  Platform Tip Vibration  0.02  0.01  0.01  0.02  0.02  Tip Deflection of Module 2  Second Joint Vibration  0.02  0.02  Figure 1-4  0.02  Tip Deflection of Module 1  First Joint Vibration  0.01  90°, l =l =  Module two joint deflection of 5°: (a) system response.  160  10  I  I  I  I  5 0  10  20 10 0 2000 1000 0 10 5 0  I  I  ei  -  I  10  0  5000  4  6  Frequency, H z  Figure 1-4  Module two joint deflection o f 5°: (b) power spectral density distribution with frequency.  161  

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