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

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DYNAMICS AND CONTROL OF ORBITING D E P L O Y A B L E M U L T I M O D U L E MANIPULATORS Yang Cao B.A.Sc, Shandong University, China, 1993 A THESIS SUBMITTED LN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE 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 this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada Date (POT, / f r , / * , ? ? DE-6 (2/88) ABSTRACT This thesis focuses on the planar dynamics and control of a variable geometry manipulator which may be used in space- as well as ground-based operations. The system is composed of a flexible orbiting platform supporting two modules connected in a chain topology. Each module consists of two links: one free to slew while the other permitted to deploy. The model used and the governing order-N equations of motion, as developed by Caron, are explained. A detailed dynamical response study is undertaken which assesses the influence of initial conditions, system parameters, and manipulator maneuvers on the system response. Results suggest that under critical combinations of system parameters and disturbances the response may not conform to the acceptable limit. This points to a need for active control. Two different control methodologies are used: (i) the nonlinear Feedback Linearization Technique (FLT) applied to rigid degrees of freedom with flexible generalized coordinates indirectly regulated through coupling; (ii) a synthesis of the F L T and Linear Quadratic Regulator (LQR) to achieve active control of both rigid and flexible degrees of freedom. Furthermore, the F L T is used to track several prescribed trajectories with considerable accuracy. 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 of this class of novel manipulators. ii T A B L E O F C O N T E N T S A B S T R A C T i i T A B L E O F C O N T E N T S i i i L I S T O F S Y M B O L S v i L I S T O F F I G U R E S x i L I S T O F T A B L E S xv i i A C K N O W L E D G E M E N T xv i i i 1. I N T R O D U C T I O N 1 1.1 Preliminary Remarks . . . . . . . 1 1.2 A Br ie f Review of the Relevant Literature . . . . 3 1.2.1 Characteristics of space-based manipulators . . . 3 1.2.2 Dynamics and control of space-based manipulators . . 8 1.3 Scope of the Investigation . . . . . . 13 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 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 of manipulator location and orientation . . 27 2.4.2 Response due to platform's tip excitation . . . 38 2.4.3 Response to manipulator tip displacement . . . 41 in 2.4.4 Response in presence of manipulator maneuvers . . 44 2.4.5 Effect of system parameters . . . . . 47 3. SYSTEM CONTROL 60 3.1 Control Methodologies . . . . . . 62 3.1.1 F L T control 64 3.1.2 F L T / L Q R control . . . . . . 66 3.2 Simulation Results and Discussion: Commanded Maneuvers . 70 3.2.1 F L T control 72 3.2.2 F L T / L Q R control . . . . . 76 3.3 Trajectory Tracking . . . . . . . 91 4. GROUND BASED EXPERIMENTS . 106 4.1 System Description . . . . . . . 106 4.1.1 Manipulator base . . . . . . 108 4.1.2 Manipulator module . . . . . . 108 4.1.3 Elbow joint . . . . . . . 112 4.2 Hardware and Software Control Interface . . . . 112 4.3 Digital Control of the Ground-Based Manipulator . . . 117 4.3.1 Discretization of inner nonlinear loop . . . 119 4.3.2 Joint velocity estimates from position measurements . 119 4.3.3 Discretization of outer PD/PID control loop . . . 120 4.4 Controller Implementation . . . . . . 122 4.4.1 Dynamical equations for the ground-based Manipulator . 122 iv 4.4.2 Control system parameters . . . . . 124 4.4.3 Controller design . . . . . . 126 4.5 Trajectory Tracking . . . . . . . 127 4.5.1 Straight line trajectory. . . . . . 128 4.5.2 Circular trajectory . . . . . . 135 5. CONCLUDING REMARKS 142 5.1 Contributions . . . . . . . . 142 5.2 Conclusions . . . . . . . . 143 5.3 Recommendations for Future Work . . . . . 144 REFERENCES 146 APPENDIX I: SPECTRAL DENSITY ANALYSIS OF DYNAMICAL RESPONSE 151 V LIST OF SYMBOLS A, B state-space representation of the flexible subsystem, Eq . (3.14) C M . center of mass d position of the manipulator base from the center of the platform e orbital eccentricity e,, e 2 tip vibrations of modules one and two, respectively ep platform tip vibration Eld, EI S Flexural rigidity of deployable and slewing links, respectively E I P Flexural rigidity of the platform F vector containing the terms associated with the centrifugal, Coriolis, gravitational, elastic, and internal dissipative forces Fx ,F2 deployment/retrieval forces at the prismatic joints one and two, respectively I J Z moment of inertia of the revolute joint J quadratic cost function which considers tracking errors and energy expenditure, Eq. (3.17) Ki spacecraft's inertia parameter, (I -Ix)l Iz, Eq . (2.5) Kj Stiffness of the revolute joint ^ L Q R optimal gain matrix, Eq . (3.18) Kp, Kv diagonal control matrices containing the proportional and derivative gains, respectively Id, ls length of deployable and slewing links, respectively lx, l2 lengths of the manipulator modules one and two, respectively vi lp length of the platform L.H. , L.V. local horizontal and local vertical, respectively ma, ms mass of deployable and slewing links, respectively nij mass of the revolute j oint mp mass of the platform M system mass matrix M, F estimates of the discretized mass matrix and nonlinear terms M and F, 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) Mrr, Mjy, rigid and flexible contributions to the system mass matrix M, respectively Mrf , Mfr 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 q0 operation point used to linearize the governing nonlinear equation qk discretized generalized coordinates q qr, qf rigid and flexible generalized coordinates, respectively qs specified or constrained coordinates Aqs desired variation of the specified or constrained coordinates v i i qa desired value of q Q generalized forces, including the control inputs ( ? L Q R , i ? L Q R symmetric weighting matrices which assign relative penalties to state errors and control effort, respectively; Eq. (3.17) t time T total kinetic energy of the system Tpr, Tpf torques provided by control momentum gyros for attitude control and vibration suppression, respectively T,, T2 torques provided by actuators located at revolute joints one and two, respectively H vector containing the F L T control inputs, Eq. (3.5) uL input determined from the Linear Quadratic Regulator, Eq . (3.16) x, y, z body coordinate system; in equilibrium x, y in the orbital plane with x along the local vertical, y along the local horizontal and z aligned with the orbit normal xd, yd desired trajectory for the manipulator tip xt state vector for the flexible subsystem Greek Symbols ax, a2 r igid body rotations during slew maneuvers of modules one and two, respectively J3X , / ? 2 contributions due to flexibility of revolute joints at modules one and two, respectively; viii / , rotation of the frame F i , attached to the module i , with respect to the frame F i . i AT time required for maneuver, Eq . (2.4) vk digital joint velocity estimate, Eq. (4.6) 6 true anomaly of the system T time from start of maneuver rk digital control input for the robot arm, Eq. (4.4) rp librational period of the platform coi desired natural frequency associated with the error equation for rigid degrees of freedom, Eq. (3.9) co¥ frequency of platform librational motion a)p platform's bending natural frequency cojX first revolute joint's torsional natural frequency comX first module's bending natural frequency G)j2 second revolute joint's torsional natural frequency a)m2 second module's bending natural frequency elastic deformation of the i t h -1 body in the transverse direction y/p platform's pitch angle A dot above a character refers to differentiation with respect to time. A boldface italic character denotes a vector quantity. 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 FIGURES Figure 1-1 Artist view of the International Space Station with its Mobi le 2 Servicing System (MSS) as prepared by the Canadian Space Agency. Figure 1-2 A l l the space-based manipulators have used, so far, revolute joints 4 thus permitting only slewing motion of links. Figure 1-3 Variable geometry manipulator showing: (a) single module with a 5 pair of slewing and deployable links; (b) Several modules connected to form a snake-like geometry. Figure 1-4 Variable geometry manipulator showing obstacle avoidance 6 character. Figure 1-5 Challenges faced by studies aimed at dynamics and control of large, 9 space-based systems. Figure 1-6 Schematic diagrams of space structure models: (a) Lips [33], rigid 11 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) Marom [36], flexible platform supporting one rigid slewing-deployable manipulator module and a rigid payload at the deployable link end. The revolute joint was flexible. Figure 2-1 Schematic diagram of the mobile flexible variable geometry 16 manipulator, based on an elastic space platform. Figure 2-2 Coordinates describing flexibility of revolute joints. 18 Figure 2-3 Normalized time histories of the sinusoidal maneuvering profile 21 showing displacement, velocity, and acceleration. Figure 2-4 Schematic diagram of a two-module, flexible, variable geometry 23 manipulator, based on an elastic space platform, considered for study. Figure 2-5 Schematic diagram showing important parameters appearing in 26 the response study. Figure 2-6 Equilibrium of the system as affected by the manipulator's geometry 29 and location on the platform. The diagram shows a case where the links are locked in position at ax = a2 = 90° and /, =l2 = 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 30 equilibrium position, due to a change in the manipulator orientation. It is governed by the location of the center of mass in different quadrants. Figure 2-8 System response with a two-module manipulator located at different 31 positions on the platform: (a) d = 60 m; (b) d = 0. Figure 2-9 System response as affected by the initial orientation of the platform: (a) ^ ( 0 ) = +10°; 33 (b) ^ ( 0 ) = -10°; 34 (c) ^ ( 0 ) = -80°. 35 Figure 2-10 System response with the manipulator when the platform is 37 initially at: (a) y/ (0) = -90°, i.e. aligned with the local vertical; (b) ^,(0) = ^  =-90.13°; (c) ^ ( 0 ) = - 9 1 ° . Figure 2-11 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; 39 (b) long duration evolution of the platform's pitch libration. 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; 45 (b) short duration behavior of rigid and flexible degrees of freedom. 46 Figure 2-14 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; 48 (b) short duration behavior of rigid and flexible degrees of freedom. 49 Figure 2-15 System response as affected by payload with the manipulator 50 completing simultaneous slew and deployment maneuvers in 0.03 orbit. 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 54 be a rather critical parameter. Figure 2-18 System response as affected by the number of modes used in 56 flexibility discretization. xii Figure 2-19 A likely procedure would involve slew and deployment separately instead of all four joints maneuvering at the same time. 57 Figure 2-20 System response, even under a set of conservative values for 58 payload (2,000 kg), joint stiffness ( 5 x l 0 4 Nm/rad) and maneuver speed (0.05 orbit), is unacceptable. This suggests a need for control. Figure 3-1 A two-module manipulator system, showing frequently used 61 symbols. Figure 3-2 Schematic diagram of the manipulator system showing the location 63 of the control actuators. The two torques Tpf/2, opposing each other, control the platform's vibration by regulating the local slope. Figure 3-3 FLT-based control scheme showing inner and outer feedback loops. 67 Figure 3-4 Block diagram illustrating the combined F L T / L Q R approach applied 71 to the manipulator system. Figure 3-5 Response of the system during a simultaneous 90° slew and 7.5m deployment maneuver of the two-unit manipulator with rigid degrees of freedom controlled by the F L T : (a) rigid degrees of freedom and control inputs for module 1; 73 (b) rigid degrees of freedom and control inputs for module 2; 74 (c) flexible degrees of freedom. 75 Figure 3-6 F L T / L Q R controlled response of the two-unit system during a simultaneous 90° slew and a 7.5 m deployment maneuver of the manipulator units: (a) rigid degrees of freedom and control inputs for module 1; 77 (c) rigid controlled degrees of freedom and control inputs for 78 module 2. (c) flexible degrees of freedom. jg Figure 3-7 F L T / L Q R controlled response of the system during a simultaneous 90° slew and 7.5 m deployment maneuver of the two-unit manipulator with different payload ratios: (a) rigid degrees of freedom and control inputs for module 1; 81 (b) rigid degrees of freedom and control inputs for module 2; 82 (c) flexible degrees of freedom. 83 Figure 3-8 Effect of the speed of maneuver on the F L T / L Q R controlled response: (a) platform and module 1; 85 (b) module 2; 86 (c) flexible degrees of freedom. 87 Xlll 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 (b) module 2; 89 (c) flexible generalized coordinates. 90 Figure 3-10 Tracking of a straight line using the second module and the F L T : (a) manipulator tip trajectory and errors; 93 (b) response of the platform and rigid degrees of freedom with 94 control inputs; (c) response of flexible degrees of freedom. 95 Figure 3-11 Tracking of a horizontal straight line with two revolute joints using the F L T : (a) initial configuration and the trajectory error; 97 (b) response of rigid degrees of freedom and control inputs; 98 (c) response of flexible degrees of freedom. 99 Figure 3-12 Tracking of a circle using the two revolute joints and the F L T for control: (a) manipulator tip trajectory and error during the period of 0-200 s; 101 (b) manipulator tip trajectory and error during the period of 200-400s; 102 (c) response of rigid degrees of freedom and control inputs; 103 (d) response of flexible degrees of freedom. 104 Figure 4-1 The prototype manipulator system. 107 Figure 4-2 M a i n components of the two-module manipulator system. 109 Figure 4-3 M a i n components of the manipulator base assembly. 110 Figure 4-4 Prismatic joint mechanism which provides the deployment and 112 retrieval capability. Figure 4-5 M a i n components of the elbow joint assembly. 113 Figure 4-6 Open architecture of the manipulator control system for a single 114 joint. Figure 4-7 Digital robot control scheme. 118 Figure 4-8 Two-module ground based manipulator system. 123 Figure 4-9 Schematic diagram of the swing test to determine the moment of 125 inertia of a manipulator link with different lengths. xiv Figure 4-10 Schematic diagrams for straight line tracking using: (a) one revolute 129 joint and one prismatic joint; (b) two revolute joints. Figure 4-11 Straight line tracking using one revolute and one prismatic joint 130 under the PID control: (a) tip trajectory; (b) joint motion and the corresponding control signals. Figure 4-12 Straight line tracking using two revolute joints under the PID 132 control: (a) tip trajectories; (b) joint motion and corresponding control signals. Figure 4-13 Straight line tracking using one revolute and one prismatic joint with 133 the F L T : (a) tip trajectory; (b) joint motion and the corresponding control signals. Figure 4-14 Straight line tracking using one revolute and one prismatic joint: 134 comparison of the F L T and PID procedures. Figure 4-15 Schematic diagram showing tracking of a circular trajectory using 136 module 2. Figure 4-16 Tracking of a circle, at a speed of 0.314 rad/s, using the PID control: 137 (a) tip trajectories; (b) joint dynamics and control signals. Figure 4-17 Tracking of a circle, at a speed of 0.628 rad/s, using the PID control: 138 (a) tip trajectories; (b) joint motion and control signals. Figure 4-18 Circle tracking behavior under the PID control at a speed of 0.209 139 rad/s: (a) tip trajectories; (b) joint dynamics and control effort. Figure 4-19 Tracking of a circle, at a speed of 0.314 rad/s, with the F L T : (a) tip 140 trajectories; (b) joint dynamics and control effort. Figure 1-1 Initial deflection of 5° applied to the revolute joint of module 1: (a) system response; 154 (b) power spectral density distribution. 155 Figure 1-2 Module 1 tip deflection of 0.2m: (a) system response; 156 (b) power spectral density distribution. 157 Figure 1-3 Module 2 tip deflection of 0.2m: (a) system response; 158 (b) power spectral density distribution. 159 XV Figure 1-4 Module two joint deflection of 5°: (a) system response; 160 (b) power spectral density distribution. 161 xvi LIST OF TABLES Table 2-1 Important factors affecting the system performance. 22 Table 2-2 Fundamental natural frequencies of the components, forming the 43 platform-based manipulator system, in absence of payload. Table 4-1 Swing-test results for the ground-based robot. 125 Table 4-2 Controller gains for the prismatic joint of module 2. 127 Table 4-3 Controller gains for the revolute joint of module 2. 127 Table 4-4 Controller gains for the revolute joint of module 1. 128 XVll ACKNOWLEDGEMENT I wish to thank my supervisor Prof. Vinod J. M o d i for his guidance, teaching and friendship throughout my graduate studies. His constant encouragement took me on an incredible journey of experience and learning throughout my graduate program. I wish to thank my supervisor Prof. C. W. de Silva for his support and advice towards the completion of my 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 . Mark Chu, 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 my horizons. They have made my stay in the University of 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 of British Columbia, and Dr. A . K . Misra at M c G i l l University. xviii 1. I N T R O D U C T I O N 1.1 Pre l iminary 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 of 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 of the moon and used a deployable arm to lower an instrumentation package to the surface. The V i k i n g landers, in 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 in 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 of 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 in the cargo bay for repair and relaunched it. In December 1998, it assisted in the integration of the U .S . 'Uni ty ' module with the Russian control module called 'Zarya' (Sunrise), launched a few weeks earlier, thus initiating construction of the International Space Station. For the Space Station, which is scheduled to be operational in year 2004, the Canadian contribution is through an extension of the Canadarm in the form of Mobile Servicing System (MSS, Figure 1-1). It consists of the Space Station Remote Manipulator System (SSRMS) and Special Purpose Dexterous Manipulator (SPDM) . The M S S w i l l play 1 an important role in the construction, operation, and maintenance of 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 Artist view of the International Space Station with its Mobi le Servicing System (MSS) as prepared by the Canadian Space Agency. A number of other space robots have been proposed and some are under development. The American Extravehicular Activi ty Helper/Retriever ( E V A H R ) and Ranger Telerobotic Flight Experiment, as well as the Japanese ETS-VI I , are examples of 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 of 400 km, 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 in the case of the Canadarm and M S S abode the International Space Station. With this as background, the thesis undertakes a study aimed at a novel flexible multimodule manipulator capable of varying its geometry. Each module consists of two links (Figure l-3a), one free to slew (revolute joint) while the other is permitted to deploy and retrieve (prismatic joint). A combination of 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 of motion and inverse kinematics, decreases the number of singularities, and facilitates obstacle avoidance for comparable degrees of freedom (Figure 1-4). Dynamics and control of 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 of literature available on the subject of 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 (b) Payload Module of Slewing-Deployable Links 4 Trajectory Figure 1-3 Variable geometry manipulator showing: (a) single module with a pair of slewing and deployable links; (b) several modules connected to form a snake-like geometry. Figure 1-4 Variable geometry manipulator showing obstacle avoidance character. 6 Due to zero-weight condition at the system center of mass and microgravity field elsewhere, the environmental torques due to free molecular flow, Earth's magnetic field and solar radiation can become significant in the study of space manipulators [9]. The large temperature variations encountered in 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 of the platform as well as excite it to vibrate [14]. Conversely, the librational motion of 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 in size, lighter and highly flexible. Obviously, this w i l l make the study of system dynamics , and its control, a formidable task. The ratio of the payload to manipulator mass for a typical space-based system can be several orders of magnitude higher [16]. For example, in case of 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 of 0.167 ! Obviously, space manipulators are not readily accessible for repair in case of, say, joint failure. This requires incorporation of a level of redundancy in their design [17]. Correspondingly, more degrees of freedom are involved than required for a given task. (f) Remote operation of a space-based manipulator would involve time delays, an important factor in control of 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 of literature available for ground-based manipulators. We w i l l have to explore and understand distinctive character of the space robotic systems. Dynamics and control of a large orbiting flexible platform (like the International Space Station), supporting a mobile elastic manipulator, carrying a compliant payload represent a class of problems never encountered before. Major challenges presented by such large-scale systems are summarized in Figure 1-5. It is only recently, some of 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 of the governing equations of motion, relatively low frequencies, and development of 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 of space dynamics and control. Over the years, a large body of literature has evolved, which has been reviewed quite effectively by a number of authors including Meirovitch and Kwak [19], Roberson [20], Likins [21], as well as M o d i et al. [22 - 26]. In the majority of studies aimed at manipulators, only revolute joints were involved, i.e. links were permitted to undergo slewing motion. On the other hand, several space 8 6 era' CD U l n 3* £L CT* 3 tra CT cn P ^ o CT cr CT' CO p CT P. & P o co P-O o 3 o o p 1-1 era CT CO T3 P O CT cr P CO CT P-CO '< CO ° s s H O Q 03 03 5 05 m O > m ~n n m o — i c/> o 25 w H ts O 50 o w 03 C/3 hd o 25 OS H 03 o o *1 > o o 25 © 25 o H ^ s > HH C/3 Cd C/3 O td r w H o H O H d c/i C/3 O a w r "I d o HH HH O H 25 K n o HH o d H HH o 25 O 3 d o r HH 2! 0 O H w w 2! § * d 03 **5 *d W h H n r d C/3 H O n o d d d O 03 CZl M r 25 o 25 d HH 2! H O 2! O o d C/3 O o w d o d n H o 2 O d C/3 w 25 03 O 03 25 O n H d H o 03 o hd H hH d 3 03 W tr 1 w n H HH o 25 25 © o 25 d 2 o o n d O 03 d o H 25 H o r C/3 •< C/3 H M „ H> 25 d d *5 *d Q d^ d » 0 3 g CZl d 25 n o 25 H O r o H H w Cd > 25 O HH o H K 03 O 25 O ts 25 HH o 25 s w 25 H d W X HH Cd d w C/3 H d n H d 03 •H O 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 (SAFE) , in September 1984. Cherchas [27], as well as Sellappan and Bainum [28], studied the deployment dynamics of extensible booms from spinning spacecraft. Lips and M o d i [29,30] have studied at length the dynamics of 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 of problems involving a rigid body supporting deployable beam-and plate-type members. Subsequently M o d i and Shen [32] extended the study to account for deployment as well as slewing of the appendages. Lips [33], Ibrahim [34], and Shen [35] have reviewed this aspect of 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 chain geometry of the links (appendages) was not involved. Figure 1-6 shows schematically the different models described above. The new manipulator system, schematically shown in Figures l-3(a) and l-6(d), was first proposed for space application by Marom and M o d i [36]. Planar dynamics and control of 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 of the system during tracking of a specified trajectory, using the computed torque technique, proved to be quite successful. M o d i et al. [37] as wel l as Hokamoto et al. [38] extended the study to the multimodule configuration, referred to as the Mobi le Deployable Manipulator ( M D M ) system. The model, with an arbitrary number of modules, accounted for the joint as well as link flexibility. A relatively 10 Figure 1-6 Schematic diagrams of 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) Marom [36], flexible platform supporting one rigid slewing-deployable manipulator module and a rigid payload at the deployable link end. The revolute joint was flexible. l l general formulation for three-dimensional dynamics of the system in orbit was the focus of 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 of a single unit system and demonstrated successful tracking of a straight-line trajectory at right angle to the initial orientation of 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 of a single module (i.e. two links) manipulator, free to traverse a flexible platform, using the Feedback Linearization Technique (FLT) applied to rigid degrees of freedom, and suppression of flexible members' response through the Linear Quadratic Regulator (LQR) . Recently, Chen [43] extended Caron's study and presented an order-N formulation for three-dimensional motion of a mobile manipulator traversing an orbiting flexible platform. Control of 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 of freedom. The controlled response of the system during commanded maneuvers of the manipulator was surprisingly good. Goulet [44] studied control of 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 of literature dealing with modeling, dynamical performance and control of space-based manipulators, most of 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 of the system. (iii) A few reported control studies involve one-module manipulator, i.e. the system comprised of two links: One free to slew while the other is permitted to deploy. The control of multimodule manipulator does not seem to have received attention. 1.3 Scope of the Investigation With this as background, the thesis investigates planar dynamics and control of a two-module (four links) flexible manipulator based on an elastic orbiting platform. To begin with, in Chapter 2, model used and the governing order-A^ equations of motion, as developed by Caron [12], are explained. A detailed dynamical response study is undertaken which assesses the influence of initial conditions, system parameters, and manipulator maneuvers. Results suggest that under critical combinations of system parameters and disturbances the response may not confirm to the acceptable limit. This points to a need for active control. Control of 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 of freedom with flexible generalized coordinates indirectly regulated through coupling, and (b) a synthesis of the F L T and L Q R to achieve control of both rigid and flexible degrees of freedom. Also, several prescribed trajectories are tracked with the F L T applied to rigid degrees of 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 Chu [8], is used to assess effectiveness of Proportional-Integral-Derivative (PID) and F L T control procedures in performing several trajectory tracking maneuvers. This brings to light, quite vividly, problems of friction and backlash often present in real-life manipulator systems. The thesis ends with a brief review of important conclusions, significant original contributions and suggestions for future study (Chapter 5). 14 2. DYNAMICS 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 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 of the system model used may be summarized as follows: (a) The manipulator with an arbitrary number of 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 of 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 of the orbit. (e) The damping is accounted for through Rayleigh's dissipation function. (f) The governing equations account for gravity gradient effects, shift in center of mass as well as change in inertia due to maneuvers and flexibility. A s pointed out in Chapter 1, such a manipulator with a combination of revolute and prismatic joints is able to change its geometry, has a marked decrease in dynamical coupling, a reduction in the number of singularity conditions, and can negotiate obstacles with ease. Note, the model considered is relatively general and applicable to a large class of space- as well as ground-based manipulator systems. 15 Prismatic Figure 2-1 Schematic diagram of the mobile flexible variable geometry manipulator, based on an elastic space platform. 16 The Lagrangian approach adopted for derivation of the governing equations is particularly wel l suited to the flexible multibody system, with a large number of degrees of freedom, under consideration. It automatically satisfies holonomic constraints while the nonholonomic constraints can be accounted for, quite readily, using Lagrange multipliers. The form of the equations of motion conveys a clear physical meaning in terms of contributing forces. Equally important is the fact that the equations are readily amenable to stability study and well suited for controller design. Furthermore, validity of the formulation and numerical integration code can be checked with ease through the conservation of energy for nondissipative systems. A comment concerning representation of the revolute joint's flexibility would be appropriate. The rotation of the frame F, , attached to the module i, with respect to the frame F,-_i has three contributions (Figure 2 - 2 ) : elastic deformation of the ith -1 body in the transverse direction rotation of the actuator rotor (at), which corresponds to the controlled rotation of the revolute joint; and elastic deformation of the joint i (/?,) which could be due to, for instance, flexible coupling. Hence, 2.2 S i m u l a t i o n M e t h o d o l o g y The equations governing the dynamics of 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 1 7 Body M Figure 2-2 Coordinates describing flexibility of 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) of the system. For simulations, forward dynamics is of interest, and Eq . (2.2) must be solved for q, q = M\Q-F). (2.3) The solution of these equations of motion generally requires 0(N ) arithmetic operations, where N represents the number of bodies (modules) considered in the study. In other words, the number of computations required by the 0 ( N3) algorithm w i l l vary as the cube of the number of modules. It also depends on the number of generalized coordinates associated with each module. Clearly, the computation cost can become prohibitive for a system with a large number of modules or generalized coordinates. Hence, the development of the 0(7V) formulation by Caron [12], where the number of arithmetic operations increases linearly with the number of bodies (or degrees of freedom) in the system, promises a significant saving in the computational cost. Equally important is a possibility of real-time implementation of a control strategy. It is often useful to specify some of the generalized coordinates. For example, cases where the length of the units is varied in a specified manner, or where joints are locked in place at a specified angle, require the use of specified coordinates. These coordinates are prescribed through constraint relations which are introduced in the equations of 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 of the maneuver, thereby reducing the structural response of the system. The maneuver time-19 history considered is as follows, qs(r) = -^J-<r sin AT 2K '2K ^ T (2.4) where qs is the specified or constrained coordinate; Aqs is its desired variation; r is the time; and A r i s the time required for the maneuver. The time history for qs, qs, and qs are plotted, for the case of Aqs =1 and Ar=\, in Figure 2-3. A F O R T R A N program for the dynamical simulation of 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 in order to obtain analytical expressions for the integrals of the shape functions. Furthermore, efficient matrix multiplication algorithms are developed to take advantage of the structure of various matrices involved. 2.3 Simulat ion Considerations The system performance is governed by a large number of parameters. Some of the important variables are listed in Table 2-1. Obviously, a systematic change of these variables would lead to a large volume of information. However, it would also demand considerable amount of 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 of modes; profile and speed of maneuver; and mass of the payload. Even with these selected parameters, the task is formidable. Hence only a few typical results corresponding to a two-module manipulator, i.e. with four links, are presented for conciseness (Figure 2-4). 20 Figure 2-3 Normalized time histories of the sinusoidal maneuvering profile showing displacement, velocity, and acceleration. 21 Table 2-1 Important factors affecting the system performance. Parameters orbit eccentricity mass o f : platform; links; joints; payload stiffness o f : platform; links; joints . damping o f : platform; links; joints . link length Initial Conditions . platform attitude manipulator location and orientation deformation of platform, manipulator links, joints Maneuvers type : slewing; deployment; retrieval; base translation , amplitude speed Discretization . shape of admissible functions (modes) o number of admissible functions (modes) 22 Figure 2-4 Schematic diagram of a two-module, flexible, variable geometry manipulator, based on an elastic space platform, considered for study. 23 Numerical values used in the analysis, unless specified otherwise, are indicated below: Orbit: Circular orbit at an altitude of 400 km; period = 92.5 min. Platform: Geometry: circular cylindrical with diameter = 3 m; axial to transverse inertia ratio of 0.005; Mass (mp)= 120,000 kg; Length (Ip)= 120 m; Flexural Rigidity (EI P) = 5.5 x 10 8 N m 2 . 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 / s , U represent lengths of slewing and deployable links, respectively. Manipulator Links (Slewing and Deployable): Geometry: circular cylindrical with axial to transverse inertia ratio of 0.005 Mass (ms, md) = 200 kg; Length (/s, / d > m a x ) = 7.5 m; Flexural Rigidity (EI S , EI d ) = 5.5 x 10 5 N m 2 . Revolute Joint: Mass (mj) = 20 kg; 24 Moment of Inertia (L z) = 10 kgm 2 ; Stiffness (Kj) = 10 4 Nm/rad . Note, the prismatic joint is treated as a part of 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 of 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 of the response, i.e. the damping coefficient for joints (Cj) as well as structural damping coefficients for manipulator links ( £ m ) and the platform (£" ) are considered zero. More important specified and response parameters are summarized below and indicated in Figure 2-5: d position of the base from the center of the platform; ex ,e2 tip vibrations of modules one and two, respectively; ep platform tip vibration; x, y body fixed coordinate system with x aligned with the undeformed axis of the platform and y normal to x in the orbital plane; 25 Figure 2-5 Schematic diagram showing important parameters appearing in the response study. 2 6 ax,a2 rigid body rotations, during slew maneuvers of modules one and two, respectively; fti,j02 contributions due to flexibility of revolute joints at modules one and two, respectively; y/p platform pitch libration. 2.4 System Response This section studies effect of system parameters and disturbances, in the form of initial conditions as well as manipulator maneuvers, on the resulting response. It clearly shows that under certain combinations of 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 At 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 of manipulator, when aligned with the platform, has virtually no effect on the equilibrium as the geometry remains effectively unchanged, as wel l as relatively massive (120,000 kg) character of the platform compared to the manipulator (800 kg). However, with different orientations of the modules, the geometry changes, i.e. they no longer remain in alignment with the local vertical or the local horizontal position. The system's new equilibrium orientation slightly deviates depending on the slew and deployment characters as well as location of 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), ax = a2 = 90°, and the deployable links of both modules remain unextended (l{ = l2 =7.5 m) as indicated in Figure 2-6. The deviation in the equilibrium position (y/pe) 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. On the other hand, the local horizontal position being unstable, the platform moves away from it; the direction of rotation being governed by the position of the center of 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 of 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 (/, = / 2 =7.5 m). Both modules are locked in position with a \ - a2 - 90° • This results in a shift in the center mass causing a pitch moment, which 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 of y/pfi~ 0.13°. The librational period of 0.6 orbit matches precisely with the established value for a long spacecraft in a circular orbit. On the other hand, with the manipulator at the center of the platform, the center of mass is slightly below the local horizontal (i.e. -xcm). This leads to the pitch angle for equilibrium, y/p e , that is positive. The amplitude of oscillation, as can be expected, is rather small (« 0.01°) due to an insignificant pitching moment caused by the shift in the center of mass because of the presence of the manipulator. 28 4 7 . 5 m 7 . 5 m Figure 2-6 Equilibrium of the system as affected by the manipulator's geometry and location on the platform. The diagram shows a case where the links are locked in position at ear, = a2 = 90° and /, =l2 = 7.5m. The equilibrium configuration deviates from the local vertical by « 0.13°. 2 9 L o c a l Ver t ica l 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. 3 0 L . V . J = 60m C M . Parameters: E I p = 5 . 5 x l 0 8 N m 2 ; E L = 5 . 5 x l 0 5 N m 2 ; E I d = 5 . 5 x l 0 5 N m 2 ; Kj = 1 .0x l0 4 Nm/rad . Specified Coordinates: or, = a2 = 90°; /, = l2 = 7.5m. Initial Conditions: Vp=Q> eP = 0> A = 0, ex= 0; & = 0, e2=0. d = 60m d = 0 Platform Libra t ion 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 = / 2 = 7.5m and ax = a2 = 90°. Three different initial positions of the platform with respect to the local vertical are considered: y/p(0) = +10°, - 1 0 ° and - 8 0 ° . In all the three cases, the platform oscillates about the stable equilibrium position close to the local vertical (y/ = - 0 . 1 3 ° ) . With 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 in Case (a) is slightly larger than 10° (» 10.13°) as the oscillations take place about the equilibrium position, which is now at y/ e = - 0 . 1 3 ° . On the other hand, the amplitude in Case (b) is slightly lower than 10° (« 9.87°). Similarly in Case (c), the starting position is 79.87° away from the equilibrium location leading to the resulting amplitude of 79.87°. A word about the pitch librational period would be appropriate. It can be obtained from the governing equation [9] (l + ecos0)y/p -2esin0(y/'p +\) + 3Kl siny/ pcosy/ p = 0 , (2.5) where: y/ = pitch librational angle; e = orbital eccentricity; 6 = true anomaly; Kj = spacecraft's inertia parameter, (I -Ix)/Iz; x, y, z - body coordinate system; in equilibrium x, y in the orbital plane with x along the local vertical, y along the local horizontal and z aligned with the orbit normal. I L . V . Parameters: EIp = 5 . 5 x l 0 8 N m 2 ; E L = 5 . 5 x l 0 5 N m 2 ; E I d = 5 . 5 x l 0 5 N m 2 ; Kj = l . O x l 0 4 Nm/rad. Specified Coordinates: or, = or2 = 90°; lx=l2= 7.5m. Initial Conditions: ^,= +10°, ep = 0; A = 0, e, = 0; A2 = 0, e 2=0. Platform Libration 1 i 1 1 1 1 1 r J 1 1 i i I i i I 0.5 1 1.5 Orbit Figure 2-9 System response as affected by the initial orientation of the platform: (a) {/,(()) = +10° 33 L . V . Parameters: E I p = 5 . 5 x l 0 8 N m 2 ; E L = 5 . 5 x l 0 5 N m 2 ; E I d = 5 . 5 x l 0 5 N m 2 ; Kj = l . O x l 0 4 Nm/rad. Specified Coordinates: ax = a2 = 90°; lx=l2= 7.5m. Initial Conditions: yp=-\0°, ep = 0; A = 0, ex= 0; A2 = 0, e2=0. Platform Libration T 1 1 1 1 r 10 -10 n r 9 . 7 3 7 ° J i I i i I i i _ 0.5 1 Orbit 1.5 Figure 2-9 System response as affected by the initial orientation of the platform: (b) ^ ( 0 ) = - 1 0 ° . 34 L.V. Parameters: Initial Conditions: E I p = 5 . 5 x l 0 8 N m 2 ; ^ = - 8 0 ° , ep = 0; E I s = 5 . 5 x l 0 5 N m 2 ; E I d = 5 . 5 x l 0 5 N m 2 ; A = 0, ex =0; /? 2 = 0, e2=0. - i V / Kj = l . O x l 0 4 Nm/rad. A Specified Coordinates: ax=a2= 90°; /, = l2 = 7.5m. Platform Libra t ion Orbi t Figure 2-9 System response as affected by the initial orientation of the platform: ( c ) ^ ( 0 ) = - 8 0 ° . 35 Primes denote differentiation with respect to 6. For a circular orbit, e = 0, i.e. y/"p + lKism\i/pcasy/p = 0 . In the case of a long spacecraft like a 120m long uniform platform (120,000 kg) with a small two-module manipulator (800 kg), Thus the librational period rp is given by TP = ^= radian per pitch cycle, V3 = - J = orbit per pitch cycle, V3 = 0.577 orbit per cycle. Note, in Figures 2-9(a) and 2-9(b), the pitch amplitude is relatively small and the linear approximation is valid giving the pitch period of around 0.58 orbit. However, with large amplitude oscillations in 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/p (0) = - 9 0 ° . In absence of the manipulator, the platform would have stayed there, being an equilibrium position. However, due to presence of the manipulator ( / 1 = / 2 = 7 . 5 m , ax = a2 = 90°), the new equilibrium orientation corresponds to y/pe = - 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 of mass is now in quadrant III causing the platform to continue rotation in the anticlockwise sense. Thus, in absence of dissipation, one has a perpetually rotating platform ! O f course, the energy input corresponds to the initial deviation of the platform from the new equilibrium position. 36 i L . V . h a, 1 i 1 a. ft-: Parameters: E I p = 5 . 5 x l 0 8 N m 2 ; E L = 5 . 5 x l 0 5 N m 2 ; E I d = 5 . 5 x l 0 5 N m 2 ; K j = 1 .0x l0 4 Nm/rad . Specified Coordinates: or, = or2 = 90°; l\ =l2= 7.5m. Initial Conditions: e p = 0; A = 0, e, = 0; A = 0, e 2 =0. ^ = - 9 0 ° ; -90 .13° ; ^ = - 9 1 ° . Platform Libra t ion 270 180 90 P 0 -90 -180 -270 i 1 r \ r (a) / ¥ °(0) = - 9 0 / ° / / V°(0) = -90.133° (b) X ¥ ° ( 0 ) = - 9 1 ° \ \ j • i 0.5 Jc)_ .. i 1 Orbit 1.5 Figure 2-10 System response with the manipulator when the platform is initially at: (a) (0) = - 9 0 ° , i.e. aligned with the local vertical; (b) WP<P) = VP.e = - 9 0 . 1 3 ° ; (c) ^ ( 0 ) = - 9 1 ° . 37 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 in 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/p = - 9 1 ° . Note, now the system center of mass is in the fourth quadrant and experiences a clockwise moment. The platform continues to swing until its velocity becomes zero at y/p = +91°. The return journey 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 in Figure 2-10(a) and (b). 2.4.2 Response due to platform's tip excitation With the manipulator located at the tip and joints locked in position as before (/j = / 2 =7.5m, (X\ = a 2 = 9 0 ° , d = 60m), the end of the platform, aligned with the local vertical, was given an initial disturbance of 0.5m (Figure 2-11). The platform vibrates at its natural frequency with a period of around 0.001 orbit (« 6s). Note, due to coupling effects, the librational motion of the platform is induced. The pitch librational response is modulated at the platform frequency. The flexibility effects of the manipulator joints and modules (links) are also apparent. The platform tip motion and the corresponding tip moment (free-free beam) excite joints of the manipulator, resting on the platform, with peak amplitudes of f?i ~ 6° and {h « 25°. The beat character of the response is attributed to the proximity of the joint 2 and platform frequencies ( < » - 2 « 0 . 2 1 Hz , & ^ « 0 . 1 8 Hz) . The beat period approximately corresponds to 0.012 orbit (« 0.015 Hz) . One can discern presence of the low 38 Parameters: EIp = 5 . 5 x l 0 8 N m 2 ; E L = 5 . 5 x l 0 5 N m 2 ; E I d = 5 . 5 x l 0 5 N m 2 ; Kj = l . O x l 0 4 Nm/rad . Specified Coordinates: ax = a2 = 90°, lx=l2 = 7.5m. Initial Conditions: y,p=0,ep = 0.5m; A = o. e i = ° ; /? 2 = 0, e2=0. 0.03 o 0 l 25 r -25 Platform Libration (a) 0.01 0.02 First Joint Vibration 0.01 0.02 Second Joint Vibration 0.01 Orbit 0.02 Platform Tip Vibration 0.5 ev 0 -0.5 m 0 0.01 0.02 Tip Deflection of Module 1 0.02 e, 0 -0.02 m 0.01 0.02 Tip Deflection of Module 2 0.07 V m e2 0 -0.07 0.01 Orbit 0.02 Figure 2-11 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. 39 Parameters: E I p = 5 . 5 x l 0 8 N m 2 ; E L 5 . 5 x l 0 5 N m 2 : E I d = 5 . 5 x l 0 5 N m 2 ; Kj = 1 . 0 x l 0 4 N m / r a d . Specified Coordinates: ax=a2= 90°, lx=l2= 7-5m. Initial Conditions: i//p=0, ep = 0.5m; A = o. e i = ° ; A2 = 0, e2=0. Platform Libra t ion -0.3 h Orbi t Figure 2-11 Response of the system to an initial disturbance of 0.5m tip displacement of the platform: (b) long duration evolution of the platform's pitch libration. 40 frequency beat induced modulations in the platform's librational (y/p) as well as tip (ep) responses. The large amplitude vibration of the second revolute joint is due to near resonance condition, its frequency being close to the platform's bending frequency (co ~ 0.18 H z , « 0.21 Hz) which here acts as the forcing frequency. Long duration evolution of the librational response is presented in Figure 2-11(b). it clearly shows, as expected, a period of around 0.6 orbit. The high frequency modulations are at the platform's bending frequency (0.18 Hz). Small ridges at the periphery are due to the beat phenomenon as observed in Figure 2-11(a). 2.4.3 Response to manipulator tip displacement It should be recognized that the system consists of elastic and rigid degrees of freedom with wide variations in their natural frequencies. Obviously, depending on the disturbance, a number of them with significant response would be excited revealing complex interactions at different frequencies. This is precisely the case with the manipulator displacement of 0.2m at the tip of module 2 (Figure 2-12). To help appreciate involved coupling effects, it was desirable to establish natural frequencies of 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 of the response were obtained. The results are given in Appendix I. Natural frequencies of various system components thus obtained are summarized in Table 2-2. With the manipulator tip initially deflected through 0.2m, the system is set vibrating. In absence of damping, the tip continues to oscillate (e2) at a constant amplitude and with a 41 Parameters: EIp = 5 . 5 x l 0 8 N m 2 ; E I s = 5 . 5 x l 0 5 N m 2 ; E I d = 5 . 5 x l 0 5 N m 2 ; Kj = l . O x l 0 4 Nm/rad . Specified Coordinates: ax=a2= 90°, lx=l2= 7.5m. Ini t ial Condit ions: y/p = 0 , ep = 0; A = 0> ex = 0; A2 = 0, e2=0.2m. Platform Libration Platform Tip Vibration 0.01 0.02 First Joint Vibration 0.01 0.02 Second Joint Vibration 0.01 Orbit 0.02 'P o -6h x l 0 " J m 0 0.01 0.02 Tip Deflection of Module 1 0 0.01 0.02 Tip Deflection of Module 2 0.2 m llMllliiBlBIBll liiililiil •n n -0.2 IIHI l l i l l i l l i l l l 0 0.01 0.( Orbit Figure 2-12 System response to the manipulator's tip displacement of 0.2 m. 42 Table 2-2 Fundamental natural frequencies of the components, forming the platform-based manipulator system, in absence of payload. Component Frequency, H z Period Second Orbit Platform, Libration 3.12xl0" 4 3204 0.6 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" 3 Joint 2, Torsion 0.21 4.762 8.6xl0" 3 43 frequency of 5.2 H z (« 290 cycles in 0.01 orbit). This, in turn, excites joint 2 through coupling (ft) which oscillates at its natural frequency of 0.21 H z (approximately 12 oscillations in 0.01 orbit) with high frequency module 2 oscillations superposed. Joint 1 displays a rather complex response showing coupling effects of p\ and e 2 (Figure I-2b). The platform tip shows a typical beat response, as its frequency is quite close to that of fii (a)J2 « 0 . 2 1 Hz , ap « 0 . 1 8 Hz) as mentioned earlier. The beat frequency of 0.015 H z corresponds to a period of « 66s. The platform pitch response is modulated at the h\ frequency with high frequency contribution from e 2 appearing at the peaks. 2.4.4 Response in presence of manipulator maneuvers In the study so far, the manipulator joints were locked in position, i.e. there were no maneuvers involved. The next logical step is to assess the effect of the manipulator executing slewing and deployment maneuvers. The manipulator is taken to be at the tip of the platform, i.e. d = 60m. It undergoes simultaneous slewing (ax, a2 from 0 to 90°) and deployment (/,, l2 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 of the response is quite similar to that observed in Figure 2-8(a). However, because of the initial disturbance, the amplitude is higher. Effect of the maneuver on the flexible degrees of freedom can be appreciated from Figure 2-13(b). Note, both the joints are set into vibration with a characteristic frequency having a period of « 28s (6 oscillations in 0.03 orbit). The module tip oscillations display the same frequency. Their amplitude due to flexibility of the links alone (e,, e2) are rather significant, around 10cm and 44 Parameters: E I p = 5 . 5 x l 0 8 N m 2 ; E I s = 5 . 5 x l 0 5 N m 2 ; E I d = 5 . 5 x l 0 5 N m 2 ; Initial Conditions: yp=0, ep = 0; A = 0, ex= 0; A 2 = 0, e2=0. Kj = 1.0x10 4 Nm/rad. Specified Coordinates: or, = a2 = 0->90°, /, = l2 = 7.5m->15m in 0.01 orbit. Platform Libration Orbit 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. 45 Parameters: E I p = 5 . 5 x l 0 8 N m 2 ; E I s = 5 . 5 x l 0 5 N m 2 ; E I d = 5 . 5 x l 0 5 N m 2 ; Kj = l . O x l 0 4 Nm/rad. Specified Coordinates: ax=a2= 0->90°, /, = l2 Initial Conditions: ^ p = 0 , e p = 0; A = 0, e, = 0; A 2 = 0, e 2 =0. = 7 . 5 m ^ l 5 m in 0.01 orbit. Platform Libration 0.01 0.02 0.03 0.04 First Joint Vibration 0.01 0.02 0.03 0.04 Second Joint Vibration 0.02 Orbit 0.03 0.04 Platform Tip Vibration 0.01 0.02 0.03 0.04 Tip Deflection of Module 1 '/ 0 0 0.01 0.02 0.03 0.04 Tip Deflection of Module 2 e7 0 0.04 Figure 2-13 System response in presence of a manipulator maneuver with the platform initially aligned with the local vertical: (b) short duration behavior of rigid and flexible degrees of freedom. 46 2cm, respectively, in the steady state. Note, the flexible joint vibrations with steady state amplitudes of around 1.3° (/%) and 4.5° (J3\) would accentuate the effect leading to a significant deviation of 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 in Figure 2-14(a). A s anticipated, the system becomes unstable and the anticlockwise rotational motion, similar to that observed in Figure 2-10(a), sets in. It is interesting to observe (Figure 2-14b) that response of the flexible generalized coordinates remains essentially the same as that for the platform along the local vertical (Figure 2-13b). Hence in 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 of several important parameters such as payload, speed of maneuvers, joint flexibility, and number of modes used in the flexibility discretization. In all the cases, the initial orientation of the platform (aligned with the local vertical) and the manipulator maneuver are purposely kept the same as before to assist in the comparison of data and isolate parameter effects. Effect of Payload Figure 2-15 presents the influence of payload carried by the manipulator during the execution of a prescribed maneuver of slew and deployment ( a x , a2 from 0 to 90° and / , , l2 from 7.5m to 15m; maneuvers completed in 0.03 orbit). Note, the maneuver time is 47 L . V . Al Parameters: ex E I p = 5 . 5 x l 0 8 N m 2 ; fi or, E L 5 . 5 x l 0 5 N m 2 E I d = 5 . 5 x l 0 5 N m 2 ; Initial Conditions: y/p = - 9 0 ° , e p = 0; A = = o; A = 0, e 2 =0. Kj = 1.0x10 4 Nm/rad. Specified Coordinates: ax=a2 = 0->90°, lx=l2 = 7.5m^>15m in 0.01 orbit. Platform Libration 270 h 180 h P 90 h Orbit Figure 2-14 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. 48 I l . v . JX_ a2 \ Parameters: EIp = 5 . 5 x l 0 8 N m 2 ; E I s = 5 . 5 x l 0 5 N m 2 ; E I d = 5 . 5 x l 0 5 N m 2 ; Kj = l . O x l 0 4 Nm/rad. Specified Coordinates: ax=a2= 0->90°, /, = l2 Initial Conditions: ¥ p = - 9 0 ° , ep = 0; A = 0, e, = 0; A 2 = 0, e 2 =0. = 7 . 5 m ^ l 5 m in 0.01 orbit. Platform Libration Platform Tip Vibration -90.3 0.01 0.02 0.03 First Joint Vibration 0.01 0.02 0.03 Second Joint Vibration 0.04 0.04 0.04 -5 0 e4 0 -0.02 0.01 0.02 0.03 0.04 Tip Deflection of Module 1 0 0.01 0.02 0.03 0.04 Tip Deflection of Module 2 0.02 0.03 Orbit 0.04 Figure 2-14 System response in presence of a manipulator maneuver with the platform initially aligned with the local horizontal: (b) short duration behavior of rigid and flexible degrees of freedom. 49 Parameters: EIp = 5 . 5 x l 0 8 N m 2 ; EI s = 5 . 5 x l 0 5 N m 2 ; E I d = 5 . 5 x l 0 5 N m 2 ; Kj = l . O x l 0 4 Nm/rad. Specified Coordinates: ax=a2= 0->90°; /, = l2 =7.5m->15m; in 0.03 orbit. Initial Conditions: y/p = 0 , ep = 0; A = 0, ex =0 ; A = 0, e 2 =0 . Payload: 0; 400 kg; 2,000 kg Platform Libration Platform Tip Vibration 0.01 0.02 0.03 0.04 First Joint Vibration 0.01 0.02 0.03 0.04 Second Joint Vibration 0.05 0.05 0 0.01 0.02 0.03 0.04 0.05 Orbit xl0~ 3m i 1 1 1 1 i \ / \ i \ i \ / \ — i p 1 1 • \ • " \ \ \ jS i \ / • \ 7 \ — 7 — \.> i • ~ \ " V " ~" ' 0.03 0.02 0.01 h 0 -0.01 0.01 0.02 0.03 0.04 0.05 Tip Deflection of Module 1 m i i i i i i -/ \ / / / \ \ / V / V. A / 7 - \ " A V / / 1 I 0.01 0.02 0.03 0.04 0.05 Tip Deflection of Module 2 0.05 e2 0 -0.05 0 0.01 0.02 0.03 0.04 0.05 Orbit Figure 2-15 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 of 0.01 orbit) due to the presence of a payload. A s can be expected, with an increase in the payload ratio from 0 to 5 (i.e. from 0 to 4,000 kg), the joint oscillations reach a peak value of « 6° for the first joint and around 3° for the second joint. Although the manipulator tip deflection (e 2 ) , which is measured relative to fii, appears 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 in the speed of maneuver; (iii) introduction of an active control. A s maneuver in the present case is only one of the numerous ones the manipulator w i l l be called upon to execute, and the speed of maneuver is rather modest, introduction of control is the logical solution. Influence of Speed of Maneuver Figure 2-16 shows the effect of speed of maneuver (in absence of 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 of 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. Again, such large amplitude joint 51 Parameters: EIp = 5 . 5 x l 0 8 N m 2 ; E I s = 5 . 5 x l 0 5 N m 2 ; E I d = 5 . 5 x l 0 5 N m 2 ; Kj = l . O x l 0 4 Nm/rad . Specified Coordinates: a{ = a2= 0->90°; h  =h =7.5m—>15m. Initial Conditions: ^ = 0 , ep = 0; A = °> e\ = o; A2 = 0, e2=0. Speed of Maneuver: 0.005 orbit; 0.01 orbit; 0.03 orbit. Platform Libration 0 0.01 0.02 0.03 0.04 First Joint Vibration 0 0.01 0.02 0.03 0.04 Second Joint Vibration 0.01 0.02 0.03 0.04 Orbit 0.06 0.03 ep 0 -0.06 Platform Tip Vibration m i i i i i i i i i i i i i n ; [' i1 •' "I'I ' 1 1 n i i ! ' t" i i " 1 1 'I''II'!I''"II''!"!. \ ' « 11 '. H ii " I'IIII!" I 1 " " ! ! ii t ii . i -'i ii |i • 111 ' i i n , 11 -' i | j l l " | ii | i i i 1 if ii ' I ' l u i i ' i i ! ! ' , ] ! 1 : ? : : I| u l[ n ll ) \'< li i 1 ,i II i1 i N i' i . ' i ' ! ' i i 1 lis -—i . ' K i 0.01 0.02 0.03 0.04 Tip Deflection of Module 1 0.01 0.02 0.03 0.04 Tip Deflection of Module 2 0.02 0.03 Orbit 0.04 Figure 2-16 Influence of the speed of maneuver on the system response. 52 oscillations ( /? 1 > m a x « 3 5 ° , A 2 , m a x - 1 2 ° ) as well as link flexibility effects (ex,e2) 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, in 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 of the platform from the local vertical of around 0.3°. The peak value of the platform's tip vibrations also remains essentially unaffected. A s anticipated, it is the manipulator's response that is affected the most and it follows the expected trend: amplitudes of joints and module tip vibrations increase as the joints' torsional rigidity diminishes. To limit the tip deflection of a fully deployed manipulator of 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 4 Nm/rad. Again this points to the need of an active control. Effect of Number of Modes Analysis of large scale flexible systems has often raised a question concerning representation of elastic deformations through admissible functions. Complexity of the 53 Parameters: E I p = 5 . 5 x l 0 8 N m 2 ; E L = 5 . 5 x l 0 5 N m 2 E I d = 5 . 5 x l 0 5 N m 2 ; Specified Coordinates: ax=a2= 0->90°; l2 = 7.5m^-15m. Initial Conditions: yp=0,ep = 0; A = °> ex= 0; A2 = 0, e2=0. Joint Flexibility (Kj): 5 x l 0 3 Nm/rad; l x l O 4 Nm/rad; 5 x l 0 4 Nm/rad. Platform Libration Platform Tip Vibration 0.01 0.02 0.03 First Joint Vibration 0.01 0.02 0.03 Second Joint Vibration 0.01 0.02 Orbit 0.03 2 . x l 0 " 3 m I ' I - I 1 -if \Vi !j x 1 • % J ,", - '« ^ r 1 1 1 I 1 1 \ 1 / -2 \ 1 — 1 1 1 1 • 0.01 0.02 0.03 Tip Deflection of Module 1 0.2 m A'' A ; ' \ I \ / / V \ ' / \ / ' \ e, o / M A / > \ ; / \ \i 1 * \J \"\ fi V-\ j i \ -0.2 \ i —i 1 \ i \ i \ / 0.01 0.02 0.03 Tip Deflection of Module 2 0.04 - m —i 1 1 1 — A'' 1 1 1 \ ; A n r \ 1 / A /A ' V\ ' / '\- AM 2 0 .//; \\ /k /\\/]\ J; XJ \\J\ A  ''' \ 'V \ / -i V -- \ •J7 V-A v 7 ''-\ \'J L \ / ' V \l -0.04 -1 0.01 0.02 Orbit 0.03 Figure 2-17 System response as affected by the joint stiffness which appears to be a rather critical parameter. 54 formulation as wel l as the computational effort involved significantly increase with the number of modes used for flexibility discretization. So the question is: how many modes are necessary to capture physics of the problem ? To help answer the question, the system with a 800 kg payload performing the same maneuver as before, i.e. ax - a 2 =0 - » 90°, lx =l2 =7.5m - » 15m in 0.01 orbit, was considered. Response results were obtained using one-mode as wel l as three-mode flexibility representation for the platform and the modules (Figure 2-18). It is apparent that, even in this demanding situation leading to large amplitude of oscillations in virtually all degrees of freedom ( A , A2 > 3 0 ° ! el,e2~ l m , y/p » 1°), the fundamental mode is able to predict the 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 l ikely proceed differently (e.g. Figure 2-19), and take place in a longer time. The illustration shows execution of 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 of module 2 (Phase II) and 90° slew o f module 1 (Phase III) represents the procedure expected to be used in practice. To assess the system performance under a realistic situation, consider a rather conservative situation of the same maneuver (a x - a2 = 0 —» 90°, lx =l2 =7.5m —»15m) completed in 0.05 orbit, for a manipulator with Kj = 5 x l 0 4 Nm/rad, and carrying a payload of 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 in Figure 2-20. It is apparent that the peak platform libration 55 Payload Parameters: E I p = 5 . 5 x l 0 8 N m 2 ; E I s = 5 . 5 x l 0 5 N m 2 ; E I d = 5 . 5 x l 0 5 N m 2 ; Kj = 1 .0x l0 4 Nm/rad . Specified Coordinates: ax=a2 = 0->90°; /, = / 2 = 7.5m->15m; in 0.01 orbit. Initial Conditions: Vp=0, ep = 0; A = o, «i = o; A = 0, e2=0. Number of Mode: 1 mode; 3 modes. Pavload: 800 kg Platform Libration 0.01 0.02 0.03 First Joint Vibration 0.01 0.02 0.03 Second Joint Vibration 0.01 0.02 Orbit 0.03 Platform Tip Vibration 0.01 0.02 0.03 Tip Deflection of Module 1 0.01 0.02 0.03 Tip Deflection of Module 2 0.01 0.02 Orbit 0.03 Figure 2-18 System response as affected by the number of modes used in flexibility discretization. 56 (a) I fi 7.5m 7 — • -7.5m Initial Configuration (b) I (c) £ fi 15m 7 • a 15m Phase I: Deployment and Payload Pick-up Phase II: Slew, a2 = 0 -^90° 90° B (d) I D Phase III: Slew, ar3= 0 ->90c 90 c Figure 2-19 A likely procedure would involve slew and deployment separately instead of all four joints maneuvering at the same time. 57 Parameters: EIp = 5 . 5 x l 0 8 N m 2 ; E I s = 5 . 5 x l 0 5 N m 2 ; E I d = 5 . 5 x l 0 5 N m 2 ; Initial Conditions: y/p = 0 , ep = 0; A = °> e i = A = 0, e2=0. Payload Kj = 5.0 x lO 4 Nm/rad . Specified Coordinates: ax=a2= 0->90°; /, = l2 = 7.5m—> 15m; in 0.05 orbit. Pavload: 2,000 kg -1.6h P, o 0.15 -0.15 Platform Libration 0.05 0.1 First Joint Vibration 0.05 0.1 Second Joint Vibration Platform Tip Vibration p 0 0.05 0.1 Tip Deflection of Module 1 0.05 0.1 Tip Deflection of Module 2 0.01 h -0.01 Figure 2-20 System response, even under a set of conservative values for payload (2,000 kg), joint stiffness ( 5 x l 0 4 Nm/rad) and maneuver speed (0.05 orbit), is unacceptable. This suggests a need for control. 58 far exceeds the permissible range of 0.1° - 1.0°. The maximum joint vibrations of /?i ~ 0.23° and fi2 « 0.09° would lead to the manipulator tip deviation from the desired position 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 of 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 of system parameters, initial conditions and maneuvers, to maintain desirable performance of 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 of control of the system with a platform-based two-unit (four links) manipulator. A s before, the platform, modules and joints are treated as flexible members. To recapitulate, more frequently used symbols are summarized below and indicated in Figure 3-1: 6 true anomaly with reference at perigee; C M . center of mass; d position of the manipulator form C M . ; L . V . local vertical; L . H . local horizontal. Rig id Degrees of Freedom y/ platform pitch; /, , / 2 lengths of modules one and two, respectively; ax, a2 rigid components of slew maneuvers associated with modules one and two, respectively. Flexible Degrees of Freedom flexible components of slew maneuvers associated with modules one and two, respectively; ep platform tip deflection; 60 Payload Figure 3-1 A two-module manipulator system showing frequently used symbols. 61 ex,e2 tip deflections of modules one and two, respectively. 3.1 Con t ro l Methodologies The torques and forces responsible for slewing and deployment maneuvers are illustrated in Figure 3-2. The objective here is to control operational behavior of the system. For example, the slewing maneuvers w i l l arise with application of torques (TX,T2) provided by actuators located at revolute joints of the manipulator. Similarly, forces (FX,F2) for the deployment and retrieval of units are provided by actuators at the prismatic joints of the manipulator. In addition to joint actuators which regulate the manipulator link dynamics, there are Control Momentum Gyros (CMGs) . They are used to regulate the platform orientation as well as its vibration. C M G s located near the center of the platform contribute the torque (Tpr) which controls the rigid body motion of the platform, i.e. its attitude or pitch response. On the other hand, a pair of C M G s , located symmetrically about the platform's center and providing equal torques (Tpj-/2) in the opposite sense, control its elastic vibration by regulating the local slope. The present section is concerned with the selection of control inputs which w i l l result in the desired motion of the system. Two different control strategies are considered: (i) Nonlinear Feedback Linearization Technique (FLT) applied to the rigid degrees of freedom, with flexible generalized coordinates indirectly affected through coupling but not actively controlled. The F L T leads to uncoupled linearized equations of motion which are then subjected to the conventional P D control. 62 Control ler Figure 3-2 Schematic diagram of the manipulator system showing the location of the control actuators. The two torques Tpf/2, opposing each other, control the platform's vibration by regulating the local slope. 63 (ii) The classical Linear Quadratic Regulator (LQR) , based on a linear approximation of the flexible subsystem, is designed for active vibration suppression [45,46]. Both rigid as well as flexible degrees of 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 in a considerable body of literature [48-54]. The basic idea is to use a mathematical model and find a transformation to decouple and linearize the dynamics of the controlled system. The main advantage of the feedback linearization over point-wise linearization is that once such a transformation is determined, a global linearization is achieved independent of the operating point. In the present study, a controller based on the F L T is designed to regulate the rigid degrees of freedom, i.e. rotations of the revolute joints (ax,a2), deployment of the links (lx,l2), attitude motion of the system (i//p). However, the effectiveness of the controller is assessed using the original fully flexible system so that the potential effects of uncontrolled dynamics can be investigated. Equations governing dynamics of a flexible space-based manipulator can be written as M(q,t)q + F{q,q,t) = Q{q,q,t), (3.1) Mrr Mrf~ 4r Qr 1 ' ' ' ' ' ' f = ' 1 ' " f Mj, Mff_ Ff. Qf ~M 64 where: M(q,t) is the system mass matrix composed of rigid (Mrr), flexible ( A T ^ ) and coupled (Mrf, Mfr) contributions; qr, qf are rigid and flexible generalized coordinates, 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. If only the rigid degrees of freedom are controlled: Mrrqr+Mrfqf+Fr=Qr: Mfrq\ +Mffqf+Ff=0. (3.2) A suitable choice for Qr would be Qr=M[(qr)d-u]+F, (3.3) with: M = Mrr-MlfMjflMfr; F = Fr-MrfMj}Ff. (3.4) Here subscript 'd' refers to the desired value of a parameter. One way to select the control signal « is the Proportional-Derivative (PD) feedback, i.e. U = -KV[(qX -9r]-Kpl(9r)d ~<lrl (3-5) where Kp and Kv are position and velocity gains, respectively. Let e = (qr )d -qr, then the controlled equations of motion become: 0 = e + Kve + Kpe\ qf=-M/XMfr[(qr)d-u]-MflFf. (3.6) N o w Qr can be written as Qr = M(qr)d +F + M(Kve + K.e) , (3.7) 65 which can be visualized as a combination of two controllers: the primary (Qr,p); and the secondary (Qr,s); Qr,P=M{qr)d +F; Qrs = M(Kve + Kpe). (3.8) The function of the primary controller is to offset nonlinear effects inherent in the rigid degrees of freedom; whereas the secondary controller ensures a robust behavior. A block diagram of the control procedure is presented in 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 of a workpiece. Therefore, to ensure asymptotic and critically damped behavior of the closed-loop system, a suitable candidate for the P D gains, Kp and KVt would be diagonal matrices such that '' a>\ 0 If) 0 0 a>? 0 2coi where is the desired natural frequency associated with the i joint or link error (Eq. 3.6). 3.1.2 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 in a fixed configuration (no deployment, slew, or translation), the platform attitude (y/p) and manipulator length (Ix,l2) are maintained using the F L T strategy. However, the joint rotation (ax,a2), the platform's tip vibration (e ), as well as the links' tip deflections (ex,e2) are controlled using the L Q R approach. During large slew and deployment maneuvers, variables y/p, ax,a2, lx, and l2 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 of the large maneuver, the system's configuration 66 Kv Nonlinear Inner Loop ^ M Qr Linea r System Manipu la to r System 9r Figure 3-3 FLT-based control scheme showing inner and outer feedback loops. 67 remains nearly constant and nonlinear effects can be neglected. This allows the Linear Quadratic Regulator to take over the control of ax and a2 to actively damp flexible generalized coordinates disturbed due to the maneuver. The optimal L Q R controller is designed based on a linearized model of the system. To begin with, the governing equations are linearized about an operation point q0. To that end, the following substitutions are made in the left hand side of Eq . (3.1): q = q0+Aq; q = q0+Aq; q = q0+Aq. (3-10) 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(q0)Aq + K(q0)Aq = Q = Q, (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 q0 and thus made time-invariant. Since the system's librational motion and the deployment length of the manipulator are controlled by the F L T , only the linearized equations governing the elastic degrees of freedom are needed. The decoupled vibrational subsystem is now described by M(q0)AqL +K(q0)AqL=uL, (3-12) where AqL = [ep, Aax,/3x,ex,Aa2, f?2,e2] > w r t n Aax and Aa2 as the deviations of the slew angle from their desired values; M and K are the mass and stiffness matrices, respectively, corresponding to the elastic subsystem; and uL is the input determined from the Linear Quadratic Regulator to control ep, Aax, ex, Aar 2 and e2. Note, the operational point is qL0 = [0, axo, 0,0, a20,0,0]r . Solving for AqL, 68 Aq L = —M~{ KAqL + M~l uL, (3.13) which can be rewritten in the state-space form as AqL ML M]K 0 ML. + M u L i (3.14) B where xL e 9 T 4 x l ; A e 5 R 1 4 x 1 4 ; and B e SR 1 4 " . For simplicity, it is assumed that all states are available, thus making the system observable. Controllability of the system is assured i f and only i f rank{[B, AB, A2B, AnB]} = 14. (3.15) The control input uL can be written as U L ~ ^LQRXL i (3.16) where ^ L Q R is the optimal feedback gain matrix. It minimizes a quadratic cost function / which considers tracking errors and energy expenditure, J = \T(xlQLQRxL + ulRLQRuL)dt. (3.17) Here QLQR and RLqR are symmetric weighting matrices which assign relative penalties to state errors and control effort, respectively. The matrix i ? L Q R is required to be positive definite while QLqR can be positive semi-definite. The optimal control input uL is given by UL - KLQRXL - RLQRB PLQRXL> (3.18) where PLQR is the positive definite solution to the steady-state matrix Ricatti equation which, for infinite time, becomes - ^ L Q R ^ + A PL0R PL0RBRLQRB -PLQR+2 Q R A Q R - " - " - L Q R - " 1 L Q R ^ J i L Q R (3.19) 69 The F L T / L Q R control strategy is indicated in Figure 3-4. Note, qr 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 of 400 km; period = 92.5 min. Platform: cylindrical with axial to transverse inertia ratio of 0.005; mass =120,000 kg; length = 120m; flexural rigidity = 5.5 x 10 8 N m 2 . Manipulator cylindrical with axial to transverse inertia ratio of 0.005; mass = 400 kg Module: each; length = 7.5m each at start; flexural rigidity = 5.5 x 10 5 N m 2 . Revolute mass = 20kg; moment of inertia = 10 kg m 2 ; stiffness = l . O x l O 4 Joints: 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 of damping in the space environment is still a subject of 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, in a sine-on-ramp 70 F L T Control ler QLO> 1LO qr>qr Space Manipu la to r System q,q &qL, AqL Figure 3-4 Block 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 ax,a2 change from zero to 90° and lx,l2 from 7.5 m to 15 m. 3.2.1 F L T control Figure 3-5 shows the F L T controlled response of the platform libration ( y/p ), its tip vibration (ep), the first and second modules' revolute joint rotations (<2r1,/?i,^ 2'A)' a s we^ as the modules' tip deflections (ex, e2). The maneuver sets the platform librating with a peak amplitude of 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 in presence of such a severe maneuver ! The negligible (» 0.05°) limit cycle type oscillations persist due to vibrations of the flexible joints (j3\,/%). The tip response of 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 of the flexible generalized coordinates affects the performance of the controller. However, the controller demands remain rather modest. The five controlled variables (y/p,ax,lx,a2,and l2) show satisfactory response even during such a large maneuver and display small oscillations after the desired values are reached. Thus the F L T control of the rigid degrees of freedom does provide encouraging results. 72 I.C.'s (Flexible d.o.f.): Controller Gains: eP = e\ = ei = 0; y/p:Kp = 0.02; A = A = 0 - /^ v =0.29. / / ^ \ I.C.'s (Rigid d.o.f.): orl5 a2 : Kp = 25; L . V . y/p = - 9 0 ° , a x = a 2 - 0; /, = l2 = 7.5 m. Kv=\0. lx:Kp = %; Kv=5.67. Desired Values: l2:Kp = 4; Kv=4. ¥ p = - 9 0 ° ; ax, a2 = 0 -> 90° and / , , l2 = 7.5m ^ 15m; sine-on-ramp; 0.01 orbit. Platform Libration Momentum Gyros Torque -89.96 -90.04 0.00 a, 15 12 0.02 0.04 First Joint Rotor Motion 0.06 0.00 0.02 0.04 0.06 Deployment of Module 1 0.00 0.02 0.04 0.06 First Joint Actuator Torque i N m 1000 TX 0 -1000 0.00 0.02 0.04 0.06 First Deployment Actuator Force m i . i 30 : / F, 0 - / 5.01 : / 5.00 AAAAAAA -30 - / 4.99 I V U U U V V I / 0.02 0.03 0.04 0.05 -60 - , , , 1 , , I — . c 0.00 0.02 0.04 Orbit 0.06 0.00 0.02 0.04 0.06 Orbit Figure 3-5 Response of the system during a simultaneous 90° slew and 7.5m deployment maneuver of the two-unit manipulator with rigid degrees of freedom controlled by the F L T : (a) rigid degrees of freedom and control inputs for module 1. 73 I .C. 's (Flexible d.o.f.I: eP = e \ = ei = °; I .C. 's (Rigid d .o .D: if/p = - 9 0 ° , ax=a2 =0; l\=l2= 7.5 m. Desired Values: Control ler Gains : ^ : Kp = 0.02; ^ v = 0 . 2 9 . ccx, a2:Kp = 25; l2: Kp = 4; Kv Kv= 10. 8; Kv=5.61. 4. ^ = - 9 0 ° ; a l 5 a 2 =0 -> 90° and / 2 = 7.5m -> 15m; sine-on-ramp; 0.01 orbit. Figure 3-5 Response of the system during a simultaneous 90° slew and 7.5m deployment maneuver of the two-unit manipulator with rigid degrees of freedom controlled by the F L T : (b) rigid degrees of freedom and control inputs for module 2. 74 I .C. 's (Flexible d.o.f.): ep=e\=e2= °; A=A=o. xei I .C. 's (Rigid d.o.f.): y/p = - 9 0 ° , ax = a2 =0; Ii=l2= 7.5 m. Desired Values: Control ler Gains : y/p\Kp = 0.02; ^ v = 0 . 2 9 . cXy, cc2 '• Kp = 25; Kv= 10. l]:Kp = S; Kv=5.67. l2:Kp = 4- Kv=4. ¥ p = - 9 0 ° ; ^ , ^ = 0 ^ 90° and / , , l2 = 7.5m -> 15m; sine-on-ramp; 0.01 orbit. First Joint Vibration Platform Vibration P° 0 0.04 0.02 0.02 0.04 Second Joint Vibration 0.06 0.00 0.02 0.04 0.06 Tip Deflection of Module 2 0.02 0.04 Orbit 0.06 0.00 0.02 0.04 0.06 Tip Deflection of Module 1 0.00 0.02 0.04 Orbit 0.06 Figure 3-5 Response of the system during a simultaneous 90° slew and 7.5m deployment maneuver of the two-unit manipulator with rigid degrees of freedom controlled by the F L T : (c) flexible degrees of freedom. 75 3.2.2 F L T / L Q R control With the control of rigid as well as flexible degrees of freedom ( F L T / L Q R approach), the situation further improves remarkably (Figure 3-6), particularly in 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 of the maneuver. This means, in the first 0.01 orbit, the F L T is used to regulate the large maneuver where it satisfactorily controlled the rigid degrees of 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 of 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, in 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 of the payload ratio (mass of the payload / mass of the manipulator) were considered: 1, 2 and 5; which correspond to the payloads of 400 kg, 800 kg and 2,000 kg, respectively. The initial configuration of the manipulator remains the same as described in the previous cases. The maneuver, as before, involves a simultaneous 90° slew of the revolute joints and 7.5m deployment of the links in a sine-on-ramp profile. It is 76 I .C. 's (Flexible d.o.f.): Control ler Gains : ^ = e i = e 2 =0; i//p:Kp = 0.02; Kv=0.3. L . V . i A = A = o . I .C. 's (Rigid d.o.f.): / , : Xp = 8; 7^v=5.66. /2: Kp = 4; Kv=4. y/p = - 9 0 ° , ar, =ar2 =0; ax, a2: VPA /j = l2 = 7.5 m. t < 0.01 orbit, KP = 25; KV= 10. Desired Values: ^ = - 9 0 ° , or, =ar2 = 9 0 ° ; l\=l2= 15 m. t > 0.01 orbit, L Q R . Platform Libration -89.96 -90.00 -90.04 0.00 0.02 0.04 0.06 First Joint Rotor Motion a, 0.00 0.02 0.04 0.06 Deployment of Module 1 15 12 m 0.00 0.02 0.04 Orbit 0.06 Momentum Gyros Torque 1000 Tx 0 -1000 0.00 0.02 0.04 0.06 First Joint Actuator Torque Nm J = 0.01 orbit LQR takes over for a. 0.00 0.02 0.04 0.06 First Deployment Actuator Force 0.00 0.02 0.04 Orbit 0.06 Figure 3-6 F L T / L Q R controlled response of the two-unit system during a simultaneous 90° slew and a 7.5 m deployment maneuver of the manipulator units: (a) rigid degrees of freedom and control inputs for module 1. 77 I.C.'s (Flexible d.o.f.I: A=A=o. = 0; I.C.'s (Rigid d.o.f.): WP = - 9 0 ° , ax = a2 = 0; /, = / 2 =7.5m. Desired Values: y/p = - 9 0 ° , or, = a2 = 9 0 ° ; /, =l2 = 15 m. Controller Gains: y/p :KP = 0.02; iC v =0.3. /,: KP = S; Kv=5.66. l2: Kp ax, a2 4; Kv=4. t < 0.01 orbit, Kp = 25; Kv= 10. t > 0.01 orbit, L Q R . Second Joint Rotor Motion 0L 0.00 0.02 0.04 0.06 Deployment of Module 2 0.00 0.02 0.04 Orbit 0.06 300 T2 0 -300 Second Joint Actuator Torque Nm t = 0.01 orbit LQR takes over for OL 20 F 0 1 2 0.00 0.02 0.04 0.06 Second Deployment Actuator Force -20 -40 0.00 0.02 0.04 Orbit 0.06 Figure 3-6 F L T / L Q R controlled response of the two-unit system during a simultaneous 90° slew and a 7.5 m deployment maneuver of the manipulator units: (b) rigid degrees of freedom and control inputs for module 2. 78 I .C. 's (Flexible d .o .D: e„ =ei =e 2 =0; ^ X 1 I .C. 's (Rig id d.o.f.I: = - 9 0 ° , ax - a 2 = 0; li=l2= 7.5 m. Desired Values: ^ p = - 9 0 ° , ^ =a2 =90 c Control ler Gains : ^ :KP = 0.02; Kv=0.3. / , : A> = 8; Kv=5.66. l2: Kp = 4; Kv=4. ax, a2: t < 0.01 orbit, ATP = 25; £ v = 10. t > 0.01 orbit, L Q R . lx=l2= 15 m. First Joint Vibration Platform Vibration K o 0.00 0.04 0.02 0.00 -0.02 0.02 0.04 Second Joint Vibration 0.06 0.02 0.04 0.06 Tip Deflection of Module 2 0.00 0.02 0.04 Orbit 0.06 > 0.0 0.00 0.02 0.04 0.06 Tip Deflection of Module 1 0.02 0.04 Orbit 0.06 Figure 3-6 F L T / L Q R controlled response of the two-unit system during a simultaneous 90° slew and a 7.5 m deployment maneuver of the manipulator units: (c) flexible degrees of freedom. 79 desired that the maneuver be finished in 0.03 orbit. A s in the case of the uncontrolled dynamical study (Figure 2-15), a longer maneuver time is used here because of 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 in the legend. The gains were purposely kept fixed to help assess robustness of the controller. 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 of slew and deployment even in presence of payloads (Figures 3-7a, 7b). A s can be anticipated, the peak control efforts increase with an increase in the payload, however the additional demands are rather modest and remain wel l within the permissible limits. For example, with the largest payload of 2,000 kg, the absolute peak value of the C M G demand changes from around 3000 N m (no payload) to 4400 N m while the force (F2) 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 of 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 of freedom are also controlled, in presence of payload, quite effectively (Figure 3-7c). Effect of Maneuver Speed Another important system parameter is the speed of the maneuver. The same maneuver as before was considered in absence of payload. 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 Payload I.C.'s (Flexible d.o.f.): e = e, = e2 = 0; /, a2 p A i.c.'s (Controlled d.o.f.): ^ ^ = - 9 0 ° ; a 1 = f l r 2 = 0 ; X i • > / 1 = / 2 = 7 . 5 m . Controller Gains: y/p:Kp = 0.02; Kv=0.29. ax, a2:Kp = 25; Kv= 10. /,, l2:Kp = S; Kv=5.67. Payload Ratio: 1:1; 2:1; Desired Values: 5:1. y/p = - 9 0 ° , ax,a2 =0 - » 90° and / l 5 / 2 = 7.5m -> 15m; sine-on-ramp; 0.03 orbit. Platform Libration -89.94 -90.00 -90.06 0.00 a, 0.02 0.04 0.06 First Joint Rotor Motion 0.02 0.04 0.06 Deployment of Module 1 0.04 Orbit 0.08 0.08 0.08 4000 Momentum Gyros Torque 1000 500 T, 0 -500 -1000 0.02 0.04 0.06 0.08 First Joint Actuator Torque Nm /,.--. \ / . \ \ 0.00 0.02 0.04 0.06 0.08 First Deployment Actuator Force 0.08 Figure 3-7 F L T / L Q R controlled response of the system during a simultaneous 90° slew and 7.5 m deployment maneuver of the two-unit manipulator with different payload ratios: (a) rigid degrees of freedom and control inputs for module 1. 81 I.C.'s (Flexible d.o.f/1: eP = e\ = ei = 0; A=A=o. ^ I.C.'s (Controlled d.o.f.): ^ /, = / 2 =7.5m . Desired Values: Controller Gains: y p : K p = 0.02; Kv=0.29. a{, a2:Kp = 25; Kv= 10. /], l2: Kp = 8; Kv = 5.67. Payload Ratio: 1:1; 2:1; 5:1. y/p = - 9 0 ° , ax,a2=0^> 90° and / , , l2 = 7.5m -> 15m; sine-on-ramp; 0.03 orbit. Second Joint Rotor Motion « 2 0.02 0.04 0.06 Deployment of Module 2 0.04 Orb i t 0.08 0.08 Second Joint Actuator Torque 0.02 0.04 0.06 0.08 Second Deployment Actuator Force 0.00 0.02 0.04 Orb i t 0.06 0.08 Figure 3-7 F L T / L Q R controlled response of the system during a simultaneous 90° slew and 7.5 m deployment maneuver of the two-unit manipulator with different payload ratios: (b) rigid degrees of freedom and control inputs for module 2. 82 Controller Gains: yp:Kp = 0.02; Kv=0.29. ax, a2:Kp = 25; Kv= 10. /[, l2 '• Kp = 8; Kv = 5.67'. Payload Ratio: 1:1; 2:1; Desired Values: 5:1. y/p = - 9 0 ° , o r , , ^ =0 -> 90° and lx, l2 = 7.5m -> 15m; sine-on-ramp; 0.03 orbit. I.C.'s (Flexible d.o.L): eP = e \ = e2 = 0; J^1 I.C.'s (Controlled d.o.f.): = - 9 0 ° ; a-j = or2 = 0; l\=l2= 7.5 m . First Joint Vibration Platform Vibration P" 0.0 4.0 2.0 Pa 0.0 -2.0 0.00 P 0 0.02 0.04 0.06 Second Joint Vibration 0.08 0.02 0.04 0.06 0.08 Tip Deflection of Module 1 1.0 • Xl0"2 m I—, i 0.5 0.0 -0.5 \ i ^ Y 0.02 0.04 0.06 Tip Deflection of Module 2 0.08 0.00 0.02 0.04 Orbit 0.06 0.08 0.00 0.02 0.04 0.06 0.08 Orbit Figure 3-7 F L T / L Q R controlled response of the system during a simultaneous 90° slew and 7.5 m deployment maneuver of the two-unit manipulator with different payload ratios: (c) flexible degrees of 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 of 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 of 0.03 orbit. The peak torques and forces encountered are wel l within the accepted limit (Figure 3-8a,b). Even the flexible degrees of freedom are controlled rather well with the equilibrium configuration regained in less that 0.03 orbit ( « 167 s). Effect of Revolute Joint Stiffness Stiffness of revolute joint also represents a significant variable. Its effect on the controlled performance while executing the same maneuver in 0.01 orbit, with no payload, was also assessed. These results are presented in Figure 3-9. Two stiffness values, one below (soft) and the other above (hard) the nominal value of l x l O 4 Nm/rad were considered. Even in the demanding situation presented by the soft spring, the system settles down to the commanded values in 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 of payload, maneuvering speed and stiffness variations are intentionally kept the same to demonstrate robust character of 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 I.C.'s (Flexible d.o.f.): eP = e \ = e2 = o; A =A=o. ^ I.C.'s (Controlled d.o.f.): y/p = - 9 0 ° ; ax=a2 =0; /, = / 2 =7 .5m . Desired Values: ¥ p = - 9 0 ° , alta2=0^> 90°; - — / 2 = 7.5m —> 15m; sine-on-ramp. Controller Gains: yp:Kp = 0.02; Kv=0.29. or,, tf2:i^ = 25; 10. , l2: Kp = 8; ^ v = 5.67. Maneuver Speed: 0.005 orbit; 0.01 orbit; 0.03 orbit. Platform Libration -89.9 -90.1 Momentum Gyros Torque 10000 0.06 -10000 H 0.04 0.06 First Joint Rotor Motion 0 0.02 0.04 0.06 Deployment of Module One 0.02 0.04 Orbit 0.06 4000 First Joint Actuator Torque -4000 0 0.02 0.04 0.06 First Deployment Actuator Force 0.02 0.04 Orbit 0.06 Figure 3-8 Effect o f the speed of maneuver on the F L T / L Q R controlled response: (a) platform and module 1. 85 fr I.C.'s (Flexible d.o.L): Controller Gains: eP = e\ = e2 = o; ^ :KP = 0.02; Kv=0.29. ax, a2: Kp = 25; Kv= 10. L . V . j ly 4&\ ... I.C.'s (Controlled d.o.f/l: y/p = - 9 0 ° ; or, = ar2 =0; / , , / 2 : = 8; Kv = 5.67. Maneuver Speed: 3 ^ = / 2 = 7.5 m . Desired Values: 0.005 orbit; 0.01 orbit; y/p = - 9 0 ° , ^ , ^ 2 = 0 - > 9 0 ° , 0.03 orbit. /!, l2 = 7.5m - » 15m; sine-on-ramp. Second Joint Rotor Motion 0.02 0.04 0.06 Deployment of Module Two 0.02 0.04 0.06 Orbit Second Joint Actuator Torque 1600 -1600 h 0 0.02 0.04 0.06 Second Deployment Actuator Force -150 0.04 0.06 Orbit Figure 3-8 Effect o f the speed of maneuver on the F L T / L Q R controlled response: (b) module 2. 86 I.C.'s (Flexible d.o.f.I: eP = e\ =  ei = °; A=A=o. J^ i I.C.'s (Controlled d.o.f.I: y/p = - 9 0 ° ; a, =a2 = 0; l\=l2= 7.5 m . Desired Values: y/p = - 9 0 ° , ar,,<ar2 =0 -> 90°; Controller Gains: yp:Kp = 0.02; i : v =0.29 . ax, a2:Kp = 25; ^ v = 10. , l2: Kp = 8; iT v = 5.67. Maneuver Speed: 0.005 orbit; 0.01 orbit; 0.03 orbit. lx, l2 = 7.5m -> 15m; sine-on-ramp. First Joint Vibration Platform Vibration 0.06 0.02 0.04 0.06 Second Joint Vibration 0 0.02 0.04 0.06 Tip Deflection of Module 2 0.02 0.04 Orbit 0.06 0 Tip Deflection of Module 1 0.06 Figure 3-8 Effect o f the speed of maneuver on the F L T / L Q R controlled response: (c) flexible degrees of freedom. 87 h a*, I.C.'s (Flexible d.o.f.): eP = e \ = ei = °; A=A=o. ^ I.C.'s (Controlled d.o.f.): y/p = - 9 0 ° ; ax = a2 =0; l\=l2- 7.5 m . Desired Values: y/p = - 9 0 ° , ax,a2 = 0 - > 9 0 ° ; Controller Gains: y/p :KP = 0.02; Kv=0.29. a{, a2:Kp = 25; Kv= 10. /], l2: Kp — 8; Kv = 5.67. Joint Stiffness: 5 x l 0 3 Nm/rad; l x l O 4 N m / r a d ; 5 x l 0 4 Nm/rad. lx, l2 = 7.5m —t 15m; sine-on-ramp. Platform Libration -89.96 h ¥p "90 -90.04 h 90 a, Momentum Gyros Torque 5000 v 0 -5000 \ 0.01 0.02 First Joint Rotor Motion 0.03 0.02 First Joint Actuator Torque 0 0.01 0.02 0.03 Deployment of Module One 0.01 0.02 0.03 1 1 1 1 1 1 l \ 1000 - Nm i 1c-Ts 0 JP \\\ / • 1 1 1 1 1 -1000 \J ' 1 0.03 First Deployment Actuator Force i 0 0.01 0.02 Orbit 0.03 -50 0.01 0.02 Orbit 0.03 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: eP = e\ = e i = °; A=A=o. I.C.'s (Controlled d.o.f/1: y/p = - 9 0 ° ; ax = a2 = 0; lx = l2 = 7.5 m . Desired Values: y/p = - 9 0 ° , ax,a2=0^ 90°; 5 x l 0 4 Nm/rad. lx, l2 = 7.5m - » 15m; sine-on-ramp. Controller Gains: y/p:Kp = 0.02; Kv=0.29. ax, a2:Kp = 25; Kv= 10. /], l2: Kp = 8; Kv = 5.67. JoinlStiffness: 5 x l 0 3 Nm/rad; l x l O 4 Nm/rad; 90 « 2 15 7.5 Second Joint Rotor Motion Second Joint Actuator Torque "•• ' — 1 1 1 1 400 . Nm 1 1 1 1 ^ 2 0 / V 7 ' J -400 — i 1 1 i 0 0.01 0.02 0.03 Deployment of ModuleTwo 0 0.01 0.02 0.03 Second Deployment Actuator Force 1 1 1 1 1 m N 0 ^ 2 \ //' J -40 • . I . 0.01 0.02 Orbit 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: (b) module 2. 89 I.C.'s (Flexible d.o.f.): eP = e \ = ei = °; ^ I.C.'s (Controlled d.o.f.): y/p = - 9 0 ° ; a x = a 2 - 0; /, = l2 = 7.5 m . Desired Values: ¥ p = - 9 0 ° , alta2=0-> 90°; Controller Gains: y/p :KP = 0.02; Kv=0.29. ax, a2: Kp = 25; Kv= 10. /,, l2\ Kp = 8; Kv = 5.67. Joint Stiffness: 5 x l 0 3 Nm/rad; l x l O 4 Nm/rad; 5 x l 0 4 Nm/rad. /,, l2 = 7.5m —> 15m; sine-on-ramp. First Joint Vibration Platform Vibration PS 0 -4 "p 0 0.01 0.02 Second Joint Vibration 0.03 0.01 0.02 0.03 Tip Deflection of Module 1 0.1 0.05 1 1 1 m -0 ' / / V • N \ / v -0.05 0 0.01 0.02 0.03 Tip Deflection of Module 2 0.01 0.02 Orbit 0.03 0.03 e 2 0 -0.03 0.01 0.02 Orbit 0.03 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 Track ing Tracking of prescribed trajectories represents one of the important tasks a manipulator is called upon to perform. This section considers trajectories in the form of straight line and circle in the vertical x, y - plane using the F L T applied to the rigid degrees of freedom. Note, the system requires two coordinates to specify the manipulator's tip position. However, there are four actuators {T\, F\, T2, F2) in the form of two revolute (ax,a2) and two prismatic (l\, h) joints. Thus there are two redundant coordinates. A trajectory involves both a path and the time evolution of the path. In the present study, trajectories follow sine-on-ramp profile as mentioned earlier (Figure 2-3). Three cases are considered: (i) Tracking of a 10 m long straight line perpendicular to the platform using revolute and prismatic joints of module two. Joints of module one are held fixed. The time allocated to complete the task is 120 s. (ii) Tracking of a 10 m long straight line along the platform, in 120 s, using revolute joints of modules one and two. Prismatic joints are locked in position. (iii) Tracking of a circle with 3 m radius in 200s using two revolute joints, i.e. prismatic joints are locked as in Case (ii) keeping the module lengths fixed. Case (i): Straight Line Perpendicular to the Platform The manipulator is located at the tip of the platform with modules initially aligned with the platform. The initial lengths of 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 of a2 and l2 can be readily obtained from: 91 (3.20a) h = -Ji-52+y2d m; (3.20b) with 10 (t-120 sin( t) m. 120 yd = 120 The tracking performance and system response are presented in Figure 3-10. Initial configuration of the manipulator, its trajectory and tip errors are shown in Figure 3-10(a). Note, the peak deviation of the tip from the desired position is « 4.5 mm in the x direction with a steady state oscillation amplitude, about the terminal point, of « 1 mm. Thus, the maximum error in the x direction is less than 0.12% of the trajectory's length. The corresponding value in 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 of the flexible degrees of freedom (Figure 3-10c) clearly shows that the effect of the module elasticity on the tip error (e2) is negligible (« 0.26 mm maximum). The major contribution to the error arises from the joint flexibility (/?, m a x « 0.06°, J32 m a x ~ 0.03°). Case (ii): Straight Line along the Platform Here tracking of 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 lx =l2 =7.5 m (Figure 3-1 la). The task-time of 120 s corresponds to « 0.02 orbit. The desired time history for the trajectory is given by: 92 4.5 Platform First Module^ Second Module \ d = 60m 10m 75m 12 r y, m X, m Tracking Error Along x Direction -0.01 y, m Second Module's Tip Motion 0.02 0.04 0.06 Tracking Error Along y Direction 12 t = 120 s 10 -8 Desired 6 4 2 -0 t = 0 74.99 75 0.02 0.04 Orbit 0.06 75.01 x, m Figure 3-10 Tracking of 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: Controller Gains: ep = e, = e2 = 0; y/p:Kp = 0.02; Kv=0.29. ll:Kp = &3; KV=5J6. L.V. j h /7\ fA^i I.C.'s (Controlled d.o.f.): l2: Kp = 1; Kv — 2. ' fr y/p = - 9 0 ° ; ^ = ^ = 0 ; ax:Kp = 8.3; KV=5J6. /, = l2 = 7.5 m. a2:Kp = 6.67; ^ v = 5.16. Maneuver: 10m straight line tracking using the second module. Platform Libration Momentum Gyros Torque -89.9 V0 -90.0 -90.1 0.00 60 « 2 0.00 14 0.00 0.02 0.04 0.06 Second Joint Rotor Motion 0.02 0.04 0.06 Deployment of Module 2 0.02 0.04 Orbit 0.06 100 Pr 0 -100 0.00 0.02 0.04 0.06 Second Joint Actuator Torque 0.00 0.02 0.04 0.06 Second Deployment Actuator Force F, 0.0 0.00 0.02 0.04 0.06 Orbit Figure 3-10 Tracking of a straight line using the second module and the F L T : (b) response of the platform and rigid degrees of freedom with control inputs. 94 I.C.'s (Flexible d.o.f.): Controller Gains: eP = e \ = ei = 0; yp:Kp = 0.02; Kv=0.29. 1}:KP = S3; KV=5J6. L . V . | JAa' A ^ I.C.'s (Controlled d.o.f.): l2 '• Kp = 1; Kv=2. ¥P 1 y/p = - 9 0 ° ; ar, = ar2 =0; al:Kp = &3; Kv=5.16. /, = l2 = 7.5 m. a2:Kp = 6.67; ^ = 5 . 1 6 . Maneuver: 10m straight line, tracking using the second module. First Joint Vibration Platform Vibration 0.03 0.00 -0.03 h 0.02 0.04 0.06 Second Joint Vibration 0.00 0.02 0.04 0.06 Tip Deflection of Module 2 e2 10 0.00 0.02 0.04 Orb i t 0.06 0.00 0.02 0.04 0.06 Tip Deflection of Module 1 0.00 0.02 0.04 Orb i t 0.06 Figure 3-10 Tracking of a straight line using the second module and the F L T : (c) response of flexible degrees of freedom. 95 yd = 1 . 5 m ; xd =(10/120)(r-(120/2^)sin[(2^/120)r] m; with the joint angles obtained using simple inverse kinematics: a X r l =tan + —cos" rl?+l22-(x2d+y2^ 2/ , / 2 Tt 2 ' a l d = K - cos 1 2/ , / 2 (3.21a) (3.21b) Note, the initial manipulator configuration represents a kinematic singularity requiring axd , d2d to be oo. To overcome this situation, ax was given a small positive value (5°) at t=0. This also resulted in a small negative value (-10°) for a2 • A s the use of revolute joints leads to increased coupling effects, the error is anticipated to be higher. Figure 3-9(a) shows the maximum error of 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 lb) . With 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\t m a x « 40 N m , T2> m a x ~ 10 Nm) . Flexible degrees of 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 of the joints. Note, the peak values of both J3X and P2 are now higher than before (J3xmax * 0 . 2 ° , /32max * 0.05° as against j3Xmax * 0 . 0 6 ° , J3ljmm « 0.03° for Case (i)). This reflects in a higher peak error. Contribution of modules' elasticity continues to remain small 96 y A 4 .5m Platform First Module Second Module r* ' 10m J = 60m 75m y, m 10 First Module Locus of the First Module 's Tip Second Module Locus of the Second Module's Tip Desired Trajectory 60 65 70 75 x, m Figure 3-11 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.): ep = e\ = e = 0: 2 A =A=o. I.C.'s (Controlled d.o.f.): y/p = - 9 0 ° ; ax = 5°; a2 = - 1 0 ° ; lx = / 2 =7.5m. Maneuver: 10m straight line tracking using the two revolute joints. Controller Gains: yp:Kp = 0.02; Kv=0.29. = 8.3; KV=5J6. l2:Kp = 4; Kv= 4. ar,: A"„ = 32; ^ v = 11.3. flr2:Xp = 25; ^ v = 10. Platform Libration 0.02 0.04 Second Joint Rotor Motion 0.02 0.04 Orb i t 400 200 V 0 Momentum Gyros Torque -200 0.00 0.02 0.04 First Joint Actuator Torque 0.00 0.02 0.04 Second Joint Actuator Torque 0.00 0.02 0.04 Orb i t Figure 3-11 Tracking of a horizontal straight line with two revolute joints using the F L T : (b) response of rigid degrees of freedom and control inputs. 98 I.C.'s (Flexible d.o.Q: Controller Gains: L . V . ; eP = e \ = ei = °; ^ I.C.'s (Controlled d.o.f.): y/p: Kp = 0.02; Kv=0.29. li:Kp = S3; KV=5J6. l2:Kp = A; Kv= 4. ¥P 1 A ^ = - 9 0 ° ; ax=5°; ax:Kp = 32; Kv= 11.3. a2 = - 1 0 ° ; lx = / 2 =7.5m. a2 :KP = 25; Kv= 10. Maneuver: 10m straight line tracking using the two revolute joints. First Joint Vibration Platform Vibration P° 0.0 r 0.00 0.02 0.04 Second Joint Vibration 0.06 0.00 -0.06 0.00 0.02 0.04 Tip Deflection of Module 2 0.00 0.02 0.04 Orb i t 0.00 0.02 0.04 Tip Deflection of Module 1 et 0 0.00 0.02 0.04 Orb i t Figure 3-11 Tracking of a horizontal straight line with two revolute joints using the F L T : (c) response of flexible degrees of freedom. 99 (< 0.8 mm). Case (iii): Circular Trajectory Tracking of a circular trajectory, of 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 /, =l2 = 7.5 m and the base is located at x = 6 m, y = 1.5 m. The tracking period is 200 s. The trajectory can be represented as (JC - 64) 2 + (y - 8.5) 2 = 3 2 , (3.22) or in terms of time of 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 of 3 m and a period of 200s. The point, and hence the tip of the manipulator, moves at a uniform speed of V = ^ jx2+y2 =0.03^- m/s. (3.24) Figure 3-12(a) shows tracking of the circle (with reference to x', 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, in terms of the orbital period of 92.5 minutes, 0.01 orbit corresponds to 55.5 s. The error is rather large with a peak value of around 7 cm. However, the error diminishes significantly during the second period of 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 of straight lines, the circle represents a large disturbance to the platform as well as the manipulator. Though within the 100 8.5m *1.5m Platform 60m Starting Position •Second Module First Module Figure 3-12 Tracking of a circle using the two revolute joints and the F L T for control: (a) manipulator tip trajectory and error during the period of 0-200 s. 101 Platform 60m . Starting Position Second Module First Module y, m Desired Trajectory Actual Trajectory Starting Point, t = 200 s Equations of the Desired Circle: x ' = 4 + 3cos(0.0lTtt) m; / = 7 + 3sin(O.Ol7Ct) m. Time History of Tracking Error Second Module 200 300 400 (s) 0 10 15 x, m Figure 3-12 Tracking of a circle using the two revolute joints and the F L T for control: (b) manipulator tip trajectory and error during the period of 200-400 s. 102 I.C.'s (Flexible d.o.L): Controller Gains: e p = e \ = e 2 = 0; iyp\Kp = 0M; Kv=0.29. A=A=o. / , : i ^ = 8.3; Kv=5.16. L . V . j I.C.'s (Controlled d.o.f.): l2:Kp = 4; Kv=4. ^ = - 9 0 ° ; ax = - 3 . 7 ° , ax:Kp = 64; Kv= 16. a2 =97 .4° ; ^ = / 2 =7.5m. Maneuver: a2:Kp = 50; 14.14. Tracking a circle using the two revolute joints. Platform Libration Momentum Gyros Torque 0.00 160 0.02 0.04 0.06 First Joint Rotor Motion 0.02 0.04 0.06 Second Joint Rotor Motion 0.00 0.02 0.04 0.06 First Joint Actuator Torque 0.00 0.02 0.04 0.06 Second Joint Actuator Torque Figure 3-12 Tracking of a circle using the two revolute joints and the F L T for control: (c) response of rigid degrees of freedom and control inputs. 103 I.C.'s (Flexible d.o.D: Controller Gains: eP = e\ = ei = °; y/p :KP = 0.02; Kv=0.29. A=A=o. lx:Kp = 8.3; KV=5J6. L . V . j I.C.'s (Controlled d.o.f.): l2:Kp = A; Kv= 4. ^ = - 9 0 ° ; ax = - 3 . 7 ° , ' ^ : i : p = 64; # v = 16. a2 =97 .4° ; lx =l2 =7.5m. Maneuver: a2:Kp = SQ; ^ v = 14.14. Tracking a circle using the two revolute joints. First Joint Vibration Platform Vibration 0.00 0.02 0.04 0.06 Orb i t Figure 3-12 Tracking of a circle using the two revolute joints and the F L T for control (d) response of flexible degrees of freedom. 104 permissible limit, the peak platform libration is « 0.007° with the C M G output of « 1,100 N m . The revolute joints also demand higher torques with peak values reaching « 200 N m and « 15 N m for a{ and a2, respectively (Figure 3-12c). The tip tracking errors are primarily contributed by joint and module flexibility effects. Both of them are relatively high during tracking of the first circle (i.e. time < 0.04 orbit) with peak values reaching: e i . » « « * 2 m m ; e 2 i m f l X * 3 m m ; A.max ~ Pl.max X 0.85°. These lead to the high value of error observed during tracking of the first circle. However, during the period of 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 of time and should become negligible for t > 0.06 orbit. Note, the flexible degrees of freedom are not actively controlled by the F L T . They are regulated only indirectly through coupling. In practice, the presence of structural damping would help reduce the error. O f course, one can improve the tracking performance by actively controlling the flexibility degrees of freedom at a cost of 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 of the main reasons for the necessity of 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 of the space-age in 1957, around 20,000 spacecraft have been launched. Every one of 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 of the variable geometry manipulator using the ground-based prototype facility designed by Chu [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 of the Department of Mechanical Engineering, University of British Columbia. The prototype manipulator, employed in the experimental study, consists of a fixed base that supports two modules of 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 of 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 of revolute joints actuated by D C servo-motors. The deployment and retraction are carried out with prismatic joints consisting of lead-screw and roller-nut assemblies, each of which transforms the rotational motion of 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 Man ipu la to 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 mm thrust bearing located between the pivot plate and the top plate of the base carries the weight of the manipulator. This bearing also provides the slewing freedom about the rotational axis. A second bearing is located under the top plate of the base and is held in 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 of 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 of the first module of the manipulator system (Figure 4-3). 4.1.2 Man ipu la to r modules Both modules of 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 Slewing Flexible Coupling • Deployable L ink 1 A x i a l Bearing Flexible Coupling i iJ D C Servo-Motor 1= Lock Nut •Pivot Shaft Gear ^ Head (1:20) •Pivot Plate - Thrust Bearing \ ^ Top Plate of the Base Base Housing Optical Encoder D C Servo-Motor Figure 4-3 Ma in components of the manipulator base assembly. 110 second one is at the end of the first module, i.e. at the elbow joint. The deployment is realized with the transformation of the rotational motion of the motor that drives the lead screw into translational motion of a roller nut that is fixed to the deployable link (Figure 4-4). The pitch of the lead screw of the first module is 2.5 mm (i.e. the deployable link moves 2.5 mm per revolution) while it is 1 mm for the second module. 4.1.3 Elbow joint The joint connects the deployable end of module 1 to the slewing link of module 2. The structural connection consists of two pivot plates bolted onto the deployable end of 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 of one closed control loop of 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 of 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 of a digital control system depends on an operating system to handle multiple events within specified time i l l L i n e a r A x i a l B e a r i n g R o l l e r N u t F i g u r e 4-4 P r i s m a t i c j o i n t m e c h a n i s m w h i c h p r o v i d e s the d e p l o y m e n t a n d r e t r i e v a l c a p a b i l i t y . 1 1 2 Figure 4-5 Main components of the elbow joint assembly. 113 User Interface (QNX) Desired Position / p s s g £ 5 | g S l S \ , , Feedback Signal Control Signal MFIO -3A Interface Power Ampli f ier D C Servo-Motor Position Feedback Optical Encoder Manipulator Joint (one degree of freedom) Figure 4-6 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 PCs 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 of 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 of 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 of 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 in C or through the use of 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 of 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 of 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 of ratio 1:20 is used. The gear head reduces the speed while increasing the output torque of the motor by a factor of 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 of each motor is sensed through the use of the optical encoder attached to the motor shaft. The encoders have the offset track configuration (two tracks with their windows having an offset of 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 of 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 of 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 of the ground-based manipulator model. He has also reported a PID control o f the system. Objective here is the real-time implementation of the control algorithms, developed earlier, on the ground-based system. From the governing equations of motion, it follows that the manipulator, as a control plant, is nonlinear, non-autonomous and coupled. That makes the Feedback Linearization Technique (FLT) , 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 of comparison with the F L T . Robot control schemes often involve a great deal o f computation for the evaluation of nonlinear terms. Therefore, they are implemented as digital control laws on digital signal processors (DSPs). A s obtained before for the F L T in the continuous time domain, the control input can be written as (Eq. 3.3) Qr=M[(qr)d-u]+F, or in the general form r(t) = M(q)[qd-u] + 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 qk = q(kT) > qk=q(kT), (4.2) 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 rk . The F L T digital control law amounts to selecting M = M(q(kT)), F = F(q(kT),q(kT)), (4.3) and a digital P D outer loop control signal uk where uk=-Kvek-Kpek. The robot arm digital control input can now be written as rk = M(qk )(qkd + Kvek + Kpek ) + F(qk, qk ) , (4.4) where the tracking error is e(t) = qd if) - q{t) with subscript 'd' representing, as before, the desired trajectory. 9k 9kd ak qk ! Inner Loop F(Mk) 9kd i Kv i *k L M(qk) k5 Z O H Robot arm 9k / 9k Figure 4-7 Digital robot control scheme. Depending on the manipulator configuration, one may use different sampling rates for qk and qk as against that for M and F, e.g. the inner nonlinear loop can be sampled more slowly than the outer linear feedback loop. 118 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 in 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 Eq. (4.3), which appear in 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 vk. Then a filtered derivative can be used to compute vk from qk using 119 vk = <Jvk-\ + (ak -<lk-\)lT> (4.6) 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 in the outer linear loop for computing ek, but also to evaluate the inner nonlinear term F(qk,qk) in Eq. (4.4). 4.3.3 Discretization of outer PD/PID control loop A PID controller represents the outer feedback loop. From this continuous-time PID controller, a digital PID controller for the outer loop may be designed as explained by Lewis A continuous PID controller that only uses joint position measurements q(t) is given 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 in the s-plane at s = -N/TD . The value for N is often in the range 3 to 10; it is usually a fixed number. O f course, the P D controller is a special case of the PID procedure. A common approximate discretization technique for converting the continuous-time controller Kc(s) to the digital controller K(z) is the Bi-Linear Transform ( B L T ) where: [56]. by: u = -Kc(s)e; Kc(s) = k 1 + + (4.7) 120 K(z) = Kc(s'); 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 Z~ l-sT/2 (4.8) 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 Eq . (4.7) yields K(z) = k 1 + T z + l T, T u z -• + - Dd •1 •1 T z-cr (4.9) with the discrete integral and derivative time constants: Tld=2T,; T N T D d \ + NTj2TD ' (4.10) (4.11) and the derivative-filtering pole at cr -1 - NT/2TD l + NT/2TD It is easy to implement this digital outer-loop filter in terms of difference equations on a DSP. First, one writes K(z) in terms of z _ 1 , which is the unit delay in the time-domain (i.e. a delay of T seconds), as K(z~x) = k u T 1 + z-' | TDd 1 + z-TId l-z- T \-oz~x (4.12) 121 N o w let the control input uk be related to the tracking error as uk=K(z-l)ek. (4.13) Then uk may be computed from past and present values of ek using auxiliary variables as follows: v'k=v'k_l+{TlTId){ek+ek_x); (4.14) vDk=vlx+(TDdlT){ek-ek_x); (4.15) uk=-k(ek+v'k+v?). (4.16) The variables v[ and vk represent the integral and derivative portions of the digital PID 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 of the ground-based prototype manipulator can be obtained quite readily by eliminating the orbital motion, gravity gradient, and the flexible degrees of 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 V 2 ' (4-17) where: m2 Is2, Id2 h ls2, ld2 F2, T2 a2 mass of module 2, m^i + md2; mass of slewing and deployable links, respectively, of module 2; mass moments of inertia of slewing and deployable links, respectively, of module 2; length of module 2; lengths of slewing and deployable links, respectively, module 2; force and moment, respectivley, at joint 2; slew angles at joints 2. nent of inertia of module 2 about the swing axis. y 4 Module 2 nid2, I<n mS2, Is2 Module 1 > i 111 r Figure 4-8 Two-module ground based manipulator system. 123 4.4.2 Con t ro 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 of the manipulator system are: Slewing arm length 0.3 m; Maximum extension of the deployable arm 0.2 m; Slewing arm sweep range -135 to 135 deg; Maximum rotational speed 60 deg/s; Maximum deployment speed 0.04 m/s. Values of the moments of inertia Isi, and Idi were obtained experimentally through swing (pendulum) tests as shown in Figure 4-9. A typical test involved application of a small displacement from the vertical and counting the number of cycles over a known period of time. The moment of inertia about the swing axis is given by J = ! ! ^ = n ^ _ t ( 4 1 8 ) co 4 / r where T = period of oscillation; m = mass; and / = distance from the swing axis to the center of mass. The parameter values determined in this fashion are shown in Table 4-1. The moments of inertia were measured for the whole module with three different positions of 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 in the experiments. The parameters for the manipulator are: Module 1 • slewing link mass w s i = 4.3 kg; • slewing link length / s i = 0.3 m; 124 Center of Mass VA \ Figure 4-9 Schematic diagram of the swing test to determine the moment of inertia of a manipulator link with different lengths. Table 4-1 Swing-test results for the ground-based robot. Mass (kg) /(m) T(s) / ( k g m 2 ) Position Module 1 4.5 0.276 1.343 0.557 fully retracted 0.322 1.444 0.750 middle 0.373 1.560 1.015 fully deployed Module 2 2.4 0.106 1.021 0.066 fully retracted 0.119 1.063 0.080 middle 0.129 1.109 0.094 fully deployed 125 • deployable link mass - 0.2 kg; • 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 = 0.2 kg; • deployable link length = 0.3048 m; Based on system parameters, the time constant varied in the range of 0.23 s to 0.28 s. 4.4.3 Control ler design Design of the PID 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 of values for the PID gains. Due to the nonlinear and non-autonomous nature of the system, these parameters need to be refined and tuned for improved performance of the system. With linear representation of the results from the swing test (Table 4-1), one has I = 4.206 x l 0 " 8 « + 0.066, (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(qd-u) + F, (4.20) 126 where: M = md2 0 0 I F = -mdl{l2-l-f){a2)2 2m2(l2-^-)i2d2 V ; r = -a2_ Ji. and u = -Kpe - Kve with e(t) = qd (t) - q(t) . The F L T involves compensation for change in the 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 PID controller. 4.5 Trajectory Track ing Once the controller is designed, a series of trajectories tracking tests were performed using both the F L T and PID algorithms. These tests fall into two main categories: a) straight line trajectories; b) circle tracking. The PID and F L T gains used are shown in Tables 4-2 through 4-4. Table 4-2 Controller gains for the prismatic joint of module 2. Module 2 (Prismatic) Kp Kd Ki PID 0.5 0.001 0.0001 F L T 0.5 0.001 Table 4-3 Controller gains for the revolute joint of module 2. Module 2 (Revolute) Ka Kt PID 3 0.001 0.001 F L T 0.5 0.001 127 Table 4-4 Controller gains for the revolute joint of module 1. Module 1 (Revolute) Kd Kt PID 4 0.001 0.001 These gains were used throughout the set of experiments. With the F L T , a P D controller is used in the outer loop, for the error and the derivative of 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: el , ea^ , ea^ tracking errors at prismatic, revolute joint 1 and revolute joint 2, respectively; II , Ia^, Ia^ driving currents at prismatic and revolute joints, respectively; a\, (%2 rotation angles at revolute joints 1 and 2, respectively; A l2 change in length of the deployable link of module 2. 4.5.1 Straight line trajectory The first tracking test involves making the tip of the manipulator to follow a straight line by using one-revolute joint and one prismatic joint of the outer module (module 2) while the inner module is kept locked. The PID controller is used to perform the test. The line is located 41.5 cm from the base of module 2 and its length is 20 cm along the y direction. Figure 4-10(a) shows the location of the tracked line and the configuration of the manipulator. The specified tracking time is 4 seconds, and the tip trajectory has a prescribed sine-on-ramp profile as given in Eq . (2.4). The results of the tracking experiment are shown in Figure 4-11 (prescribed profile in dotted). It is seen that the manipulator tip follows 1 2 8 <2l 20 cm (a) 41.5 cm 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; Ja,>A 0 10 Time, s Time, s Figure 4-11 Straight line tracking using one revolute and one prismatic joint under the PID control: (a) tip trajectory; (b) joint motion and the corresponding control signals. 130 the straight line with reasonable accuracy. The saturation in Ia 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. Now a distance of 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 of the same module, since the reaction force passes through the center of the joint. When two revolute joints are used, the reaction torque due to the rotational dynamics of the second module w i l l try to rotate the first module in the counter-clockwise direction resulting in over-rotation of the first module. Thus the tracking error is biased in the negative direction of the jy-axis, as is evident from the experimental results shown in Figure 4-12 (a). From the plot of ea^ in Figure 4-12 (b), it is clear that error is in the negative side, as a result of 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 4-11 for the PID control. Again, the execution time is set at 4 seconds and the length of the line is 20 cm. The results are shown in Figures 4-13. B y comparing the behaviors of the PID and F L T controllers, following observations can be made: (i) The F L T has an adaptive capability with respect to the variation of the mass matrix. It also has a compensation capability for dynamical coupling. Thus, the F L T produces more active control signals (high frequency Ia and i / 2 ) . (ii) The PID controller is simpler, requires less computational power, and does 131 y, cm 10 8 6 4 2 0 -2 -4 -6 -8 -10 (a) -i 1 1 1 1 1 1 r Actual Trajectory _i i_ _i i_ 74 76 78 80 82 84 86 88 90 92 x, cm (b) - ~ — 10 Figure 4-12 Straight line tracking using two revolute joints under the PID 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 of the robot (the experimental, Ziegler-Nichols method is used here to tune the PID 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 of the prototype robotic system. The tip trajectories of the robot under the control of PID and F L T separately, are plotted in Figure 4-14. It is clear that at the expense of the computation cost, the F L T gives better tracking accuracy than the PID. Note that towards the end of 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 38 39 40 41 42 43 44 45 46 x, cm Figure 4-14 Straight line tracking using one revolute and one prismatic joint: comparison of the F L T and PID procedures. 1 3 4 4.5.2 C i r c u l a r trajectory In order to further investigate the system, tracking of a circular trajectory was undertaken as a typical test-case. To begin with, tracking was carried out under the PID control. Different trajectory speed profiles were employed to assess the effectiveness of the controller. The circular trajectory is defined as: where: Px , Py are tip positions in the x and y directions, respectively; r is the radius of the circle; and co is the angular velocity of 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 of tracking. The first experiment of tracking a circular trajectory uses <v= OAn; i.e. 20 seconds a circle. The results are shown in Figure 4-16. The maximum errors occur at locations where joints change their directions of 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 in causing the motion error. This can be seen when the speed of 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 (4.21) 135 Figure 4-15 Schematic diagram showing tracking of a circular trajectory using module 2. corresponding control effort also has increased. The repetitiveness of 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 of 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 in the case of 20 seconds a circle (Figures 4-19). A s before, the performance of the F L T controller is better than that of the PID controller. It compensates for the dynamical coupling effect, which significantly reduces the error of 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, cm 40 42 44 46 48 50 52 x, cm (b) 0 10 20 30 40 0 10 20 30 40 Time, s Time, s Figure 4-16 Tracking of a circle, at a speed of 0.314 rad/s, using the PID control: (a) tip trajectories; (b) joint dynamics and control signals. 137 A / 2 , cm 10 15 20 10 15 20 -1 5 10 15 20 0 5 10 15 20 Time, s Time, s Figure 4-17 Tracking of a circle, at a speed of 0.628 rad/s, using the PID 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 60 0 20 40 60 Time, s Time, s gure 4-18 Circle tracking behavior under the PID control at a speed of 0.209 rad/s: (a) tip trajectories; (b) joint dynamics and control effort. 139 v, cm 100 , ° o 'a0 (a) (b) 0 10 20 30 40 -0.5 0 ^\r>r/lA^^^ 0 10 20 30 40 la^A 0 -1 1 0 10 20 30 40 -1 0 10 20 30 40 Figure 4-19 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 of the variable-geometry manipulator system and its controllers: • Prismatic joints have lower dynamical coupling, and are preferable for executing high precision tasks. • Kinematic redundancy of a robot is useful in 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 PID 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 of the F L T algorithm to account for friction should provide improved control performance. 141 5. CONCLUDING REMARKS 5.1 Contributions The main contributions of the thesis can be summarized as follows: (i) A detailed dynamical study of a novel flexible space-based manipulator, involving two modules and accounting for orbital, librational as wel l as vibrational interactions, has not been reported before. Providing understanding of such complex interactions is indeed a contribution of importance. (ii) Control of 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 (LQR) is designed to suppress vibrations arising in the manipulator links, joints, as well as in the platform. Such synthesis of the F L T and L Q R to control both rigid and flexible degrees of freedom represents a significant contribution. (iv) A n open architecture experiment is set up for a ground-based prototype manipulator. It makes implementation of different control strategies possible. (v) It was indeed fortunate to have a two-module ground-based prototype manipulator designed and constructed by Chu [8]. This has made it possible to assess performance of control strategies not only through numerical simulations but also with ground-based experiments. Such correlation study for the novel robotic system is indeed rare. (vi) Such a comprehensive study involving numerical simulations as wel 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 of a novel manipulator studied here. The objective has been to understand the system behavior at the fundamental level, through the study of 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 of 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 of the platform. (c) Excitation of the system's flexible degrees of freedom can significantly deteriorate the accuracy of the manipulator. (d) In general, payload, speed of maneuver and joint flexibility represent three important parameters governing the response of the system. A s can be expected, heavier payload, faster speed of maneuver and reduced joint stiffness affect the manipulator tip dynamics adversely. (e) The system exhibits unacceptable response under critical combinations of parameters. The control strategy based on the F L T is found to be quite effective in regulating the rigid-body motion of manipulator links as well as the attitude motion of the platform. The unmodeled flexibility of the platform, joints, and manipulator links has virtually no effect on the performance of 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 of flexible degrees of 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 in presence of heavy payloads, fast maneuvers and reduced stiffness of the revolute joints. The results should prove useful in the design of this new class of promising manipulators. (g) The prototype manipulator with an open architecture is an effective way of evaluating performance of 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 of backlash, friction and, at times, less than smooth operation. 5.3 Recommendations for Future W o r k Considering the diversity of research areas associated with the field of space robotics, the present thesis should be viewed as an initial step in the analysis and development of this particular class of space manipulators. There are several avenues which remain unexplored or demand more attention. 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K . , de Silva, C .W. , and Marom, I., "Nonlinear Dynamics and Control of a Class of Variable Geometry Mobi le Manipulators", 148 Proceedings of the Fifth International Conference on Adaptive Structures, Sendai, Japan, December 1994, Editor: J. Tani et al., Technomic Publishing Co. Inc., Lancaster, P A , U . S . A . , pp. 140-149. [38] Hokamoto, S., Mod i . , V . J . , and Misra, A . K . , "Dynamics and Control o f Mobile Flexible Manipulators with Slewing and Deployable Links", A A S / A I A A Astrodynamics Specialist Conference, Halifax, Nova Scotia, Canada, August 1995, Paper No . AAS-95-322; also Advances in 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 Space-Based 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 of Flexible Space Robots with Deployable Links", Transactions of the Japan Society for Mechanical Engineers, V o l . 62, No. 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 No. AIAA-96-3626-CP, pp. 506-513. [42] Caron, M . , M o d i , V . J . , Pradhan, S., de Silva, C .W. , and Misra, A . K . , "Planar Dynamics of 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 No. AIAA-96-3625-CP, pp. 491-505. [43] Chen, Y . , On the Dynamics and Control of Manipulators with Slewing and Deployable Links, Ph.D. Thesis, University of 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 of 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] De Silva, C .W. , Vibration Fundamentals and Practice, C R C Press, Boca 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 . , March 1985, pp. 954-959. Spong, M . W . , and Vidyasagar, M . , "Robust Nonlinear Control of 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 , " On the Nonlinear Slewing Dynamics and Control o f the Space Station Based Mobi le 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 , No. 2, March/Apri 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, Boca Raton, Florida, U . S . A . , pp. 15-19. 150 APPENDIX I: SPECTRAL DENSITY ANALYSIS OF DYNAMICAL RESPONSE To help establish the natural frequencies of various system components and coupling effects, Power Spectral Density (PSD) distribution of the system dynamical response was carried out. O f interest are the platform's rigid body librational frequency as wel l as natural frequencies of the elastic degrees of freedom. To that end, an initial disturbance was given to the system, in the desired degree of freedom, and the corresponding response plots were obtained (y/ , ep, J3X, el, J32, e2). The individual response plot was subjected to the power spectral density analysis to arrive at the characteristic frequency as well as coupling contributions from other degrees of freedom (Figures 1-1 to 1-4). The symbols used to designate various characteristic frequencies are as follows: co¥ frequency of platform librational motion; co platform's bending natural frequency; cojX first joint's natural frequency; a>mX first module's bending natural frequency; coj2 second joint's natural frequency; com2 second module's bending natural frequency. Note, the librational frequency of the platform pitch motion as wel l as its fundamental natural frequency in bending were obtained in Chapter 2 (Figure 2-11) as co¥ « 3x IO"4 H z and 0.18 Hz . Information in 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 of the first module was given an initial disturbance of 5° (Jh(0) = 5°) and the system response was recorded (Figure I-la). The P S D analyses of the responses associated with various flexible components are presented in Figure I-1(b). The revolute joint's natural frequency was found to be cojX « 0.08 Hz . Major peaks representing contributions from other flexible components are also indicated to help assess coupling effects. First Module Tip Deflection Tip of the first module was given an initial deflection of 0.2 m (e;(0)= 0.2m). The system response and P S D plots are shown in Figures I-2(a) and I-2(b), respectively. The natural frequency of the first module's tip oscillations is found to be comX « 5.85 Hz . 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 of the first module. Note, dynamics of the first module excites revolute joint of the second module (/? 2 ) at its characteristic frequency ( a)j2 = 0.21 Hz) . O f course, tip of the module two should also vibrate at its natural frequency (co m l ) , however its P S D measure, 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 of 0.2 m (Figure 1-3). A s apparent from Figure I-3(b), a)m2 « 8.5 Hz . Coupling contributions can also be discerned, 152 particularly from its own joint (co j 2) as well as from module one (comX, co^). Second Module Joint Deflection With 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 of the second joint {coj2) is about 0.21 Hz . The bending of module two is also excited at com2 ~ 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 com] and a>jX. 153 Parameters: E I p = 5 .5x l0 8 N m 2 ; E I s = 5 . 5 x l 0 5 N m 2 ; E I d = 5 . 5 x l 0 5 N m 2 ; Kj = l . O x l 0 4 Nm/rad. Specified Coordinates: ax=a2 = 90°, lx=l2 = 7.5m. Initial Conditions: Vp=0,ep = 0; A = 5°, ex = 0; A 2 = 0, e2=0. Platform Libration Platform Tip Vibration 0.01 0.02 First Joint Vibration 0.01 0.02 Second Joint Vibration 0.02 0 0.01 0.02 Tip Deflection of Module 1 ei 0 0.01 0.02 Tip Deflection of Module 2 0.02 Figure 1-1 Initial deflection of 5° applied to the revolute joint of module 1: (a) system response. 154 10 5 0 4 2 0 i 5000 VP L I - I I i 0 20 10 0 2000 1000 0 4 2 0 0 0 J2 0 0), ml 4 6 Frequency, Hz 10 10 I I i i A A.. i i 10 ei 10 A 10 e2 10 Figure 1-1 Initial deflection of 5° applied to the revolute joint of module 1: (b) power spectral density distribution. 155 L . V . Parameters: Initial Conditions: E I p = 5 . 5 x l 0 8 N m 2 ; p , = G , ep = 0; E I s = 5 . 5 x l 0 5 N m 2 ; Px = 0, ex = 0.2m; V , E I d = 5 . 5 x l 05 N m 2 ; Kj = 1.0 x l O 4 N m / r a d . A = 0 , e 2=0. Specified Coordinates: ax=a2 = 90°, lx=l2 = 7.5m. Platform Libration Platform Tip Vibration Figure 1-2 Module 1 tip deflection of 0.2m: (a) system response. 156 1 1 1 1 vl. i i 2000 200 1000 0 20 10 0 4 6 Frequency, H z I I I ^2 1 J- 1 _ J 1 10 Figure 1-2 Module 1 tip deflection of 0.2m: (b) power spectral density distribution. 157 Parameters: E I p = 5 . 5 x l 0 8 N m 2 ; E I s = 5 . 5 x l 0 5 N m 2 ; E I d = 5 . 5 x l 0 5 N m 2 ; Kj = l . O x l 0 4 Nm/rad. Specified Coordinates: ax=a2 = 90°, lx=l2 = 7.5m. Initial Conditions: y/p = 0 , ep = 0; A = 0, ex =0; 02=0, e2=0.2m. Platform Libration Platform Tip Vibration 0.01 0 . 0 2 First Joint Vibration 0.01 0 . 0 2 Second Joint Vibration 0.01 Orbit 0 . 0 2 SP 0 x l ( T m ft 0.01 0 . 0 2 Tip Deflection of Module 1 0.01 0 . 0 2 Tip Deflection of Module 2 -0.2 F 0 . 0 2 Figure 1-3 Module 2 tip deflection of 0.2m: (a) system response. 158 4 2 0 4 2 0 400 1 1 1 1 VP t l 2000 200 4 6 Frequency, H z 10 I I I i ^2 i i i 10 Figure 1-3 Module 2 tip deflection of 0.2m: (b) power spectral density distribution. 159 Parameters: E I p = 5 . 5 x l 0 8 N m 2 ; E I s = 5 . 5 x l 0 5 N m 2 ; E I d = 5 . 5 x l 0 5 N m 2 ; Kj = l . O x l 0 4 Nm/rad. Specified Coordinates: ax=a2= 90°, lx=l2= 7.5m. Init ial Conditions: y/p = 0 , ep = 0; A = 0, ex = 0; A 2 = 5°, e 2 =0. Platform Libration Platform Tip Vibration 0.01 0.02 First Joint Vibration 0.01 0.02 Second Joint Vibration 0.02 0.01 0.02 Tip Deflection of Module 1 0.01 0.02 Tip Deflection of Module 2 0.02 Figure 1-4 Module two joint deflection of 5°: (a) system response. 160 10 5 0 20 10 0 2000 1000 0 10 5 0 5000 I I I I 0 4 6 Frequency, H z 10 I I ei I -10 Figure 1-4 Module two joint deflection of 5°: (b) power spectral density distribution with frequency. 161 

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