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Dynamic modeling and advanced control of ground-based and space-based deployable manipulator systems Cao, Yang 2004

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DYNAMIC MODELING AND ADVANCED CONTROL OF GROUND-BASED AND SPACE-BASED DEPLOY ABLE MANIPULATOR SYSTEMS Yang Cao B.A. Sc., Shandong University, M.A.Sc., University of British Columbia A THESIS SUBMITTED IN PARTIAL F U L F I L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES Department of Mechanical Engineering We accept this thesis as conforrriing to the required standard The University of British Columbia May 2004 © Yang Cao, 2004 ABSTRACT The thesis focuses on several key issues associated with modeling, analysis, and advanced control of a class of multi-module manipulator systems, which can vary their geometric configuration to quickly adapt to the environment and the requirements of a task. The studied manipulator system has a combination of revolute and prismatic joints, which lends itself to distinct advantages; for example, reduced dynamic coupling effects and singularities, simpler kinematics, and ease of obstacle avoidance , for a specific number of degrees of freedom. In rigorous and hazardous environments, autonomous operation is preferred. A reliable and effective control approach is essential to properly carry out a task (e.g., trajectory tracking in parts joining, inspection and material application; satellite capturing for retrieval and repair) in an autonomous manner. In the present work, several controllers have been formulated, analyzed, designed, simulated, and tested. Specifically, linear quadratic regulator (LQR), feedback linearization technique (FLT), computed torque adaptive control, and neural network adaptive control, have been studied. Most of them are model-based controllers. In view of this and for use of computer simulations related to illustration and evaluation of the techniques developed in the thesis, complete dynamic formulations of ground-based and space-based models of the robotic system are presented in the thesis. The resulting coupled and nonlinear models may be incomplete and may have uncertainties in parameters. The neural-network adaptive controller developed here is able to accommodate these shortcomings. Details of this controller are presented and the performance is evaluated using several case studies. ii The various control methods studied in the present research have distinct advantages and disadvantages, which make them perform satisfactorily under some conditions while deteriorating under some other conditions. As a result it is desirable to have a supervisory system to monitor the performance of the robotic system and then switch on an appropriate control scheme depending on the system behavior. When the system operates in different modes of operation or carries out several tasks, then, it would be possible to optimize its performance through controller switching. A new control scheme termed Supervisory Control Switching System (SCSS) has been developed and evaluated during the concluding stage of the present investigation. It is capable of selecting the most suitable controller for a particular task or a situation, from several separately designed controllers. The present study lays a sound foundation for further exploration of this class of novel and useful manipulators. i n TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iv LIST OF SYMBOLS viii LIST OF FIGURES xii LIST OF TABLES xviii ACKNOWLEDGEMENT xix 1. INTRODUCION 1 1.1 Preliminary Remarks . . . . . . . 1 1.2 Literature Review on Control of Robotic Manipulators . . 11 1.3 Literature Review on Supervised Control Switching . . 14 1.4 Multi-module Deployable Manipulator System . . . 16 1.5 Scope of the Investigation . . . . . . 21 2. FORMULATION 25 2.1 Preliminary Remarks . . . . . . . 25 2.2 Modeling of Link and Joint Flexibilities . . . . 26 2.3 Space-Based Model . . . . . . . 31 2.3.1 Reference frames . . . . . . 31 2.3.2 Position vectors . . . . . . 34 2.3.3 Velocity vectors . . . . . . 36 2.3.4 Generalized coordinates . . . . . 37 2.3.5 Kinetic energy of the system . . . . . 39 iv 2.3.6 Gravitation potential energy . . . . . 45 2.3.7 Strain energy . . . . . . . 46 2.3.8 Equations of motion . . . . . . 47 2.4 Ground-Based Model . . . . . . 48 2.4.1 Position vectors . . . . . . 50 2.4.2 Velocity vectors . . . . . . 50 2.4.3 Generalized coordinates . . . . . 51 2.4.4 Kinetic energy of the system . . . . . 52 2.4.5 Gravitational potential energy and strain energy . . 56 2.4.6 Equations of motion . . . . . . 56 2.5 Verification of the Computer Code . . . . . 57 NEURAL NETWORK ADAPTIVE CONTROL OF D E P L O Y A B L E MANIPULATOR SYSTEM 61 3.1 Neural Network Control . . . . . . 61 3.1.1 Neural network approximations . . . . 61 3.1.2 Neural network dynamic modeling of robot manipulators . 64 3.1.3 Neural network based adaptive control . . . 69 3.2 Neural Network Adaptive Control of a Rigid Deployable Manipulator 72 3.2.1 Simulation study . . . . . . 74 3.3 Neural Network Adaptive Control of a Deployable Manipulator with Rigid Link and Flexible Joint . . . . . . 79 3.3.1 Singular perturbation theory . . . . . 81 3.3.2 Application to the single module deployable manipulator . 86 3.3.3 Controller design . . . . . . 89 3.3.4 Simulation study . . . . . . 90 3.4 Neural Network Adaptive Control of a Deployable Manipulator with Flexible Link and Flexible Joint . . . . . . 94 3.4.1 Controller design and simulation study . . . 96 3.5 Neural Network Adaptive Control of a Space Platform Based Deployable Manipulator System . . . . . . . 105 4. OBSERVER DESIGN FOR MODAL RATE SENSING . I l l 4.1 Description of the System . . . . . . I l l 4.2 Observer Design . . . . . . . 114 4.2.1 Full-order observer . . . . . . 115 4.2.2 Reduced-order observer . . . . . 117 4.3 Simulation Study . . . . . . . 117 4.3.1 Simulation results with full-order observer . . . 119 4.3.2 Simulation results of reduced-order observer. . . 119 5. SUPERVISORY CONTROL SWITCHING SYSTEM . . . 125 5.1 Feedback Linearization Technique (FLT) . . . . 125 5.2 Linear Quadratic Gaussian (LQG) Control . . . . 130 5.3 Linear Quadratic Regulator (LQR) Control . . . . 131 5.4 Adaptive Control . . . . . . . 133 5.5 A Comparison of Control Approaches . . . . 136 5.6 Supervisory Control Switching System (SCSS) . . . 138 5.6.1 Supervisory logic system . . . . . 140 vi 5.6.2 Switching compensator . . . . . 144 5.6.3 A sample decision making process . . . . 145 5.7 Simulation Study of Supervisory Control Switching System. . 146 5.7.1 Switching between FLT and LQR for the Space-based Flexible Model 146 5.7.2 Switching between N N A C and LQR for the Ground-based Flexible Model 156 6. CONLUDING REMARKS 165 6.1 Contributions . . . . . . . . 165 6.2 Conclusions . . . . . . . . 167 6.3 Recommendation for Further Works . . . . . 169 REFERENCES 171 vn LIST OF SYMBOLS A, B state-space representation of the flexible subsystem, Eq. (5.13) C M . center of mass d position of the manipulator base from the center of the platform ex tip vibration of manipulator link ep platform tip vibration EIP, Eli Flexural rigidity of the platform and the z'th link, respectively F or V vector containing the terms associated with the centrifugal, Coriolis, gravitational, elastic, and internal dissipative forces F],F2 deployment/retrieval forces at the prismatic joints one and two, respectively Jai rotary inertia at joint z J quadratic cost function which considers tracking errors and energy expenditure, Eq. (5.16) Kj Stiffness of the revolute j oint -^LQR optimal gain matrix, Eq. (5.17) Kp, Kv diagonal control matrices containing the proportional and derivative gains, respectively Id, ls length of deployable and slewing links, respectively length of the z'th manipulator link lp length of the platform L.H., L.V. local horizontal and local vertical, respectively ntd, ms mass of deployable and slewing links, respectively viii mai mass of the actuator rotor at joint i moase mass of the mobile base mp mass of the platform nipayioad mass of the payload M, V, G system mass matrix, nonlinear terms associated with the centrifugal, Coriolis, elastic, and internal dissipative forces, and gravitational vector, respectively M, V, G neural network estimations of the mass matrix (M) ,nonlinear terms (F), and gravitational terms (G) respectively; Eq. (3.15) to Eq.(3.17) M, K mass and stiffness matrices, respectively, corresponding to the linearized subsystem; Eq. (5.11) Mrr, M, rigid and flexible contributions to the system mass matrix M, respectively Mrj , M^r coupled contributions to the system mass matrix M N number of bodies (i.e. platform and manipulator units) in the system 0(N) order-JV P L Q R solution to the matrix Ricatti equation, Eq. (5.18) q set of generalized coordinates leading to the coupled mass matrix M q0 operation point used to linearize the governing nonlinear equation qr, rigid and flexible generalized coordinates, respectively qs specified or constrained coordinates Aqs desired variation of the specified or constrained coordinates qd desired value of q Q generalized forces, including the control inputs ix (2LQR, -ft LQ R symmetric weighting matrices which assign relative penalties to state errors and control effort, respectively; Eq. (5.16) t time T total kinetic energy of the system Tpr, Tpf torques provided by control momentum gyros for attitude control and vibration suppression, respectively Tx, T2 torques provided by actuators located at revolute joints one and two, respectively u{, v- longitudinal deformation and transverse elastic deformation u vector containing the FLT control inputs, Eq. (5.5) uL input determined from the Linear Quadratic Regulator, Eq. (5.17) W neural network output weights W estimations of the neural network output weights 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 JCL state vector for the flexible subsystem Greek Symbols ax, a2 rigid body rotations during slew maneuvers of modules one and two, respectively jBx, /32 contributions due to flexibility of revolute joints at modules one and two, respectively; x yt rotation of the frame Fj , attached to the module i , with respect to the frame A T time required for maneuver 6 true anomaly of the system T vector of the control torques or forces coi desired natural frequency associated with the error equation for rigid degrees of freedom, Eq. (5.9) r symmetric positive-definite constant matrices A a positive definite and diagonal matrix § elastic deformation of the z'th -1 body in the transverse direction y/p platform's pitch angle 0p , 0f spatially varying shape functions of the platform and the manipulator link, respectively 8, 8t time dependent generalized coordinates of the platform and the manipulator link, respectively Xj the fh eigenvalue (pi activation function of the z'th node of the neural network jUi center vector of the z'th node of the neural network a; variance in the Gaussian Radial Basis Function (RBF) 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. xi LIST OF FIGURES Figure 1-1 Canadarm being deployed from the space shuttle cargo bay. 4 Figure 1-2 Canadarm I assembling the first two units of International Space Station. 5 Figure 1-3 Artistic rendering of the International Space Station with its Mobile Servicing System (MSS) as prepared by the Canadian Space Agency. 6 Figure 1-4 All the space-based manipulators have used, so far, revolute joints thus permitting only slewing motion of links. 8 Figure 1-5 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. 9 Figure 1-6 Variable geometry manipulator showing obstacle avoidance character. 10 Figure 1-7 Schematic diagrams of space structure models: (a) Lips [20], rigid spacecraft with deployable beam-type members; (b) Ibrahim [25], rigid spacecraft with deployable beam- and plate-type members; (c) Shen [26], rigid spacecraft with slewing-deployable appendages; (d) Marom [27], flexible platform supporting one rigid slewing-deployable manipulator module and a rigid payload at the deployable link end (the revolute joint was flexible). 17 Figure 1-8 Two prototype deployable manipulator systems developed in the spacecraft Dynamics and Control Laboratory. 19 Figure 1-9 Challenges faced by studies aimed at dynamics and control of large, space-based systems. 22 xn Figure 2-1 Coordinates describing flexibility of revolute joints. 31 Figure 2-2 Description of the position vectors used in the formulation. 31 Figure 2-3 The Structure of a ground-based flexible manipulator with deployable links. 49 Figure 2-4 Verification of the conservation of energy. 60 Figure 3-1 Three-layer neural network. 62 Figure 3-2 Neural network implementation of Mkj(q). 66 Figure 3-3 Neural network implementation of Gdcj). 66 Figure 3-4 Neural network implementation of q, q ). 67 Figure 3-5 Structure of the neural network adaptive controller. 71 Figure 3-6 Model of the one-module deployable manipulator. 73 Figure 3-7 PD control of the single-module deployable manipulator: system performance and control inputs. 76 Figure 3-8 (a) Neural network adaptive control of the single-module deployable manipulator with adaptation gains rMk =diag[5.0], rvk =diag[5.0] and rGk =diag[10.0j. 78 xiii Figure 3-8 (b) Neural network approximation of elements in inertia matrix M and gravitational vector G(TMk - diag[5.0], rvk = diag[5.0] and rGt=diag[10.0]). 79 Figure 3-9 (a) Neural network adaptive control of the single-module deployable manipulator with adaptation gains ruk =diag[5.0], rvk =diag[5.0] and rck =diag[10.0]. 80 Figure 3-9 (b) Neural network approximation of elements in the inertia matrix M and gravitational vector G with adaptation gains rMk - diag[5.0], rvk = diag[5.0] and rGk =diag[10.0]. 81 Figure 3-10 A single-module deployable manipulator with joint flexibility. 82 Figure 3-11 Interaction of actuator dynamics with manipulator dynamics through flexible joint. 84 Figure 3-12 Neural network adaptive control of the deployable manipulator. 92 Figure 3-13 Comparison of the determinants of the actual system matrices ( |M|, |K|, \G\, dash-line) and the neural-network estimated ones ( |M|, |GJ, solid-line). 93 Figure 3-14 Structure of a deployable manipulator with flexible revolute joint and flexible arm. 94 Figure 3-15 Dynamic interactions of the deployable manipulator with flexible joint and flexible link. 96 xiv Figure 3-16 System response and the control input with Kv = diag[10, 50], Kd = 2, and Kp=5. 99 Figure 3-17 Tracking errors and joint vibration with Kv= diag[10, 50], Kd = 2, and Kp=5. 100 Figure 3-18 System response and the control input with Kv = diag[10, 50], Kd = 20, and Kp=5. 101 Figure 3-19 Tracking errors and joint vibration with Kv = diag[10, 50], Kd — 20, and Kp=5. 102 Figure 3-20 System response and the control input with Kv = diag[10, 50], Kd = 20, and Kp=50. 103 Figure 3-21 Tracking errors and joint vibration with Kv= diag[10, 50], Kd = 20, and Kp=50. 104 Figure 3-22 Space platform-based one-module rigid manipulator system. 106 Figure 3-23 Response of the system under neural network adaptive control during a simultaneous 20° slew and 7.5 m deployment maneuvers of the rigid one-unit manipulator system. 109 Figure 3-24 Parameter estimation results from the neural network (solid line) compared to the actual ones (dashed line). 110 xv Figure 4-1 A flexible multi-link deployable manipulator. 112 Figure 4-2 The structure of the system with observer. 114 Figure 4-3 Single module deployable manipulator with a flexible link. 118 Figure 4-4 Results from the full-order observer with gains L\ = 50 and Lj = 100. 120 Figure 4-5 Results from the full-order observer with gains L\ = 50 and L2 = 300. 121 Figure 4-6 Results from the reduced-order observer with gain G = -5. 122 Figure 4-7 Results from the reduced-order observer with gain G = -20. 123 Figure 5-1 Schematic diagram of a single-module space platform-based deployable manipulator system with flexible joint and flexible link 126 Figure 5-2 Schematic diagram showing of FLT control. 129 Figure 5-3 LQG control of the manipulator system. 130 Figure 5-4 Adaptive inertia-related control of a manipulator system. 135 Figure 5-5 Schematic diagram of the supervisory control switching system (SCSS). 139 Figure 5-6 The supervisory logic system. 141 Figure 5-7 A sample decision making process for controller switching. 145 xvi Figure 5-8 FLT controlled response of the space manipulator during a simultaneous 90° slew and a 7.5 m deployment maneuver: (a) rigid degrees of freedom and control inputs for module 1. 148 (b) flexible degrees of freedom. 149 Figure 5-9 FLT/LQR controlled response of the space manipulator during a simultaneous 90° slew and a 7.5 m deployment maneuver: (a) rigid degrees of freedom and control inputs for module 1. 151 (b) flexible degrees of freedom. 152 Figure 5-10 FLT/LQR controlled response of the space manipulator during a simultaneous 90° slew and a 7.5 m deployment maneuver: (a) rigid degrees of freedom and control inputs for module 1. 154 (b) flexible degrees of freedom. 155 Figure 5-11 Instantaneous control switching from NNAC to LQR: (a) System responses and the control inputs. 158 (b) Tracking errors (ee, ej) and joint vibration (6 - a). 159 Figure 5-12 Control switching from NNAC to LQR with a sine-on-ramp transition function (AT= 0.5s): (a) System responses and the control inputs. 161 (b) Tracking errors (ee, ei) and joint vibration (0 - a). 162 Figure 5-13 Control switching from NNAC to LQR with a sine-on-ramp transition function (AT= 2s): (a) System responses and the control inputs. 163 (b) Response of the tracking errors (ee, ei) and joint vibration (6 - a). 164 xvii Table 5-1 LIST OF TABLES A comparison of control approaches. 137 xviii ACKNOWLEDGEMENT I wish to thank my supervisor Prof. Clarence W. de Silva 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. At many stages in the course of this research project I benefited from his advice, particularly so when exploring new ideas. His positive outlook and confidence in my research inspired me and gave me confidence. His careful editing contributed enormously to the production of this thesis. I wish to thank my former supervisor Prof. Vinold J. Modi who sadly passed away during this study. His patience and great supervision from the very beginning is what ensured the existence of this thesis. Thanks also go to my colleagues and friends in our laboratory: Mr. Kenneth Wong, Mr. Jian Zhang, Mr. Poiloon Tang, Mr. Rick McCourt, and Mr. Duminda Wijewardene. 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 grants to Dr. CW. de Silva and Dr. V J . Modi from the Natural Sciences and Engineering Research Council (NSERC) of Canada. Additional funding from NSERC Postgraduate Scholarships, the University Graduate Fellowship and the Killam Predoctoral Fellowship is gratefully acknowledged. xix 1. INTRODUCTION 1.1 Preliminary Remarks The research presented here concerns dynamics and control of space-based robotic manipulators. While the conditions experienced by a space-based robot are not the same as those experienced by a ground-based robot, an investigation of the latter can benefit the former. In particular, since it is not convenient to carry out physical experiments on space-based manipulators, some validations may be done on ground-based manipulators in order to extend that knowledge to space-based manipulators. In view of this readily, the present research investigates both space-based and ground-based manipulators. In the 1960s, the two superpowers competed to be the first to send a man to the moon. Today, a similar race is underway among some nations, driven by technological advances. The objective in this case is to gain the control of the international robotics market including space applications. In the last ten years^  robotics systems have matured from laboratory demonstrations to become high speed and precision tools in almost all industrial fields. They have reached a point where commercially viable applications are not just futuristic concepts, but hard realities based on available capabilities [1]. Key features that make robotic systems attractive in an industrial environment are: decreased labor costs; increased precision and productivity; and production flexibility. The early robots were created using the combined technologies of teleoperators and numerical controlled milling machines. With the rapid growth of the robotics industry, standards for design, control and planning were created to facilitate compatibility, information exchange and technology transfer. There is a large number of standard robots used in the field today. Representative among them are: 1 • The Unimation Robot, installed for the first time in 1961, in a Trenton, New Jersey Plant of General Motors; • The Stanford Arm, developed at Stanford University in 1971; • The Cincinnati Milacron T 3 with computer control, introduced in 1974; • The PUMA Robot, developed by Unimation in 1978; • The SCARA Robot, developed at Yamanashi University in Japan in 1979. Robotic systems have been quite active in space applications as early as the 1960s [2]. In the late 60s, the unmanned U.S. 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 a 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 Viking landers, in 1976, used robotic manipulators to collect and process Martian soil samples. Several other countries around the world have endeavoured in space robotics development, notably Japan, Europe, and Canada. Canada in particular has chosen space robotics as one of its strategic development areas and continues to make significant technology investments on related activities. Perhaps the best known robotic system that has ever flown is the Shuttle Remote Manipulator System ("SRMS"), also known as the "Canadarm", used with the space shuttle. Since its debut in 1981, the Canadarm has successfully flown more than 50 missions on the U.S. fleet of five Space Shuttles (including Challenger). The Canadarm is a six (6) degree-of-freedom remote manipulator comprising an upper and lower arm boom, an end effector, 2 and a control workstation at the aft flight deck of the shuttle where the translational and rotational hand controllers direct the movement of the arm (see Figure 1-1). Although less utilized, the Canadarm can also be operated automatically using pre-programmed trajectories to complete specific maneuvers for the arm. Over the past two decades, the Canadarm has enabled space servicing missions such as payload/satellite deployment, maneuvering, servicing and retrieval, Extra-Vehicular-Activity (EVA) astronaut assisting, shuttle inspection and servicing, Orbital Replacement Unit (ORU) manipulation, as well as on-orbit construction and assembly. Perhaps its most dramatic success came in the 1993's Hubble servicing missions, 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. 'Unity' module with the Russian control module called 'Zarya' (Sunrise) (Figure 1-2), which had been launched a few weeks earlier, thus initiating construction of the International Space Station. For the Space Station, the Canadian contribution is through Canadarm I and an extension of the Canadarm in the form of Mobile Servicing System (MSS, Figure 1-3). It consists of the Space Station Remote Manipulator System (SSRMS) and Special Purpose Dexterous Manipulator (SPDM). The SSRMS was successfully launched, installed and tested in April 2001. Also known as "Canadarm2", the SSRMS is 17.6 meters long arm having 7 degrees of freedom and capable of handling large payloads of mass up to 100,000 kg, including the Shuttle itself. The MSS will play an important role in the construction, operation, and maintenance of the space station [3-5]. It will also assist in the Space Shuttle docking maneuvers; cargo handling; as well as assembly, release, and retrieval of satellites. 3 Figure 1-1 Canadarm being deployed from the space shuttle cargo bay. 4 Figure 1-2 Canadarm I assembling the first two units of International Space Station. 5 A number of other space robots have been proposed and some are under development. The European Robotics Arm (ERA), the Japanese Experiment Module Remote Manipulator System (JEMRMS), the U.S. Ranger Telerobotic Shuttle Experiment and the Canadian Special Purpose Dexterous Manipulator System (SPDM) are sophisticated manipulators, which serve as useful tools in the space exploration. All indications suggest a trend to accentuate future missions with robotic systems. 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 1-5), as in the case of the Canadarm and MSS 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 joints (Figure l-5a), 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-5b) with several advantages [6]. They include reduced coupling effects thereby resulting in relatively simpler equations of motion and inverse kinematics, fewer number of singularities, and more convenient obstacle avoidance compared to a purely revolute manipulator with the same number of degrees of freedom (Figure 1-6). Dynamics and control of such Multi-module Deployable Manipulator (MDM) system, free to traverse an orbiting elastic platform and carrying a payload, represent a challenging task. 7 8 Payload Module of Slewing-Deployable Links Orbit" Trajectory Platform Payload (a) Orbit Revolute Joint Platform Figure 1-5 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. 9 Figure 1 -6 Variable geometry manipulator showing obstacle avoidance character. 10 1.2 Literature Review on Control of Robotic Manipulators As can be expected, the amount of literature available on the subject of robotics is enormous. The objective here is to mention the main contributions that are directly relevant to the study in hand. Robotic manipulators are highly nonlinear, dynamically coupled, multi-axis, electro-mechanical systems that are commonly used in mechanical manipulation of objects. Tasks such as spray painting, welding and material handling in manufacturing, and accurate positioning as in anti-aircraft guns, missile platforms, and subsea operation machines may be carried out using robotic manipulators. In these tasks, the end-effectors are required to move from one place to another, or to follow some desired trajectories as close as possible at sufficiently fast operational speeds in a free workspace. Trajectory tracking control of robots is, thus, of practical significance, and is a fundamental need in robot control. Driven by the industrial desire for a high degree of automation, fast operation, and high precision, extensive research work has been carried out in the controller design for robot manipulators. Increasingly sophisticated tools based on nonlinear control theory and multivariable and optimal control have been developed for improved tracking performance. Concurrent advances in microprocessor technology have made the real-time implementation of complicated nonlinear control algorithms feasible and cost effective. Various control methods have been introduced in the literature such as Linear Quadratic Regulator (LQR), independent-joint proportional plus derivative (PD) control [7], feedforward compensation controller [8], computed torque control [9], adaptive model-based control [10], and neural network control [11]. 11 Despite the fact that in many systems a linear controller performs satisfactorily, a key assumption made is that the system operates over a small, closely controlled range in which the dynamics of the system are linear. However, the normal operating range of a robot is large, and its payload also varies. The linear model obtained at one operating point is not valid when the operating point changes over a wide region, where kinematic, dynamic and physical nonlinearities cannot be neglected. A linear controller performs poorly under these conditions because nonlinearities in the system are not properly compensated. The classical independent joint control of a robot is designed based on a theoretical linear model neglecting the nonlinear coupling forces associated with the mechanical motions of the robot. Currently, most of the industrial robot controllers in use are designed by assuming that each joint is decoupled from others, and applying either proportional plus derivative (PD) or proportional plus integral plus derivative (PID) control to each joint disregarding the dynamic interactions with the other joints. Then, the control performance degrades quickly as the operating speed of the robot increases. In order to achieve a pre-specified accuracy, the speed of motion and accordingly, the overall productivity, has to be severely restricted. As a result, a robotic manipulator controlled by linear techniques is only appropriate for tasks of relatively slow motion and limited precision. Consequently, various nonlinear control techniques have been investigated. Concurrent advances in microprocessor technology have also made the practical implementation of complicated nonlinear control algorithms feasible. Computed Torque Control using the Feedback Linearization Technique (FLT) is rather intuitive controller, which relies on the exact cancellation of all the nonlinear dynamics of the system, through appropriate feedback. The disadvantages of such a controller include the following: (i) exact 12 knowledge of the dynamics is required; (ii) robust closed-loop stability is difficult to guarantee i f accelerations are not measurable. These disadvantages have made the scheme unattractive in many industrial applications. It has been recognized that the performance of conventional approaches in high-speed robotic applications is greatly affected by parametric uncertainties. Adaptive control of robot manipulator has proven successful in dealing with modeling uncertainties by on-line tuning of parameters. A model-referenced adaptive controller developed by Dubowsky and De Forges in 1979 [12] has the disadvantage that the control action has to be generated much faster than the speed at which the nonlinear terms of the robot change. Another adaptive control strategy was proposed by Craig in 1985 based on the "Linear in the Parameters (LIP)" assumption [13]. This control strategy is called Adaptive Computed Torque Control. Unfortunately this method requires the measurement of accelerations. But most industrial robots have only position and velocity sensors, and since differentiation of velocity is not desirable in general, one must add additional costly sensors for measuring acceleration. In 1987, Slotine and L i [14, 15] used a judicious choice of inertia-related Lyapunov function in the stability analysis which utilizes physical properties inherent to a mechanical manipulator. With this Lyapunov function, they proposed Adaptive Inertia-Related Control which eliminated the requirement for measuring the acceleration. Most adaptive control strategies are model based control methods. They depend either on an exact model or on an exact model structure, such as the known regressor functions of the linear-in-the-parameter dynamics of the robot. Despite the existence of symbolic computational software, it is still very tedious and time-consuming to obtain the dynamic equations of a robot, not to mention the explicit regressor functions, particularly in 13 the presence of many degrees of freedom and complex kinematics and nonlinearities. Expressing of the dynamics as a product of the regressor and the parameter vector is prone to errors. Even under minor mechanical modifications, the same modeling process may have to be repeated for controller design. The situation is much more serious for robots having different structures and different numbers of degrees of freedom (DOF). Recently, considerable research has been carried out to use neural networks to over-come the need for linear-parametrization models. These applications utilize learning ability and the universal approximation property of neural networks [16] to emulate some nonlinear function J\x) up to a small error tolerance, and then apply an algorithm to update the parameters based on the output errors generated by a training set. Theoretically, neural networks can be used to approximate any dynamic function to any desired accuracy as long as the size of the network is sufficiently large. It has been shown that, by fully exploiting the structural properties of a system, the full dynamics of the system can be approximated by the neural networks of pre-determined structures [17]. The problems associated with neural network controllers include the difficulty of choosing the number of nodes, the number of layers, and the choice of basis functions. No matter how large the neural network is, a generalization error always exists. 1.3 Literature Review on Supervised Control Switching Introduction of adaptation in the feedback loop is a well-established approach of controlling highly nonlinear robotic systems. However, conventional continuous adaptation does not always perform satisfactorily. This is particularly true when the manipulator switches between different modes of operation and/or i f the closed-loop variables are not i 14 sufficiently excited. In such circumstances undesirable transients may typically arise because of slow adaptation. The classic gain-scheduling control is one option of tackling this type of problem. Even though it works well for linear plants, it is not suitable in general for robotic systems, which can be quite nonlinear and time varying. In recent years, the concept of control switching has emerged as an alternative approach. Two representative contributions are given now. In 1996, Zhou [18] presented some results on an adaptive switching control method for tracking control of a planar robot with two revolute joints. He used different inverse dynamics (ID) models for the robot, and a supervisor determined which one should be used in the closed-loop. Hocherman-Frommer [19] in 1998 proposed a controller switching method based on output prediction error. The method assumes that the system is linear and time-invariant. Several other investigations considered the case of linear system, which is not applicable in the present investigation. Furthermore, the switching may not involve different control strategies, unlike in the present work. In this thesis, we develop a new type of control called Supervisory Control Switching System (SCSS). the basic idea is to have a finite family of candidate controllers Q, so that the plant performs "satisfactorily" under control of one of the CkS, in the presence of possible operating conditions. Then, a suitably designed supervisory unit takes care of orchestrating the switching among the candidate controllers so as to ensure closed loop stability in the face of uncertainty. There are several advantages of SCSS, normally, • A priori knowledge on the system (e.g., the range of values of the uncertain physical parameters, the type of disturbances, etc.) can be more easily embodied in SCSS than in classic adaptive control. 15 • Control design is accomplished off-line, and, hence, some shortcomings of adaptive control (e.g., loss of stabilizability of the identified model, control complexity) are avoided. • The candidate controllers can be synthesized according to any criterion, irrespective of its complexity. Moreover, as far as one could determine, no research has been done on the control of this type of deployable manipulator system using SCSS. Therefore, there is a significant practical motivation for studying SCSS systems. 1.4 Multi-module Deployable Manipulator System The idea of deployable manipulator system was first initiated by Modi et. al. [20] for a space manipulator project. Lips and Modi [21] first studied at length the dynamics of spacecraft with a rigid central body connected to deployable beam-type members. Modi and Ibrahim [22] presented a relatively general formulation for this class of problems involving a rigid body supporting deployable beam-and plate-type members. Subsequently Modi and Shen [23] extended the study to account for deployment as well as slewing of the appendages. Lips [24], Ibrahim [25], and Shen [26] have reviewed this aspect of the literature in some detail. In these studies [20-26], although slewing and/or deployment were involved, each appendage was directly connected to the central body; i.e., a chain-type manipulator geometry of the links (appendages) was not involved. Figure 1-7 schematically shows these geometric structures. 16 gure 1-7 Schematic diagrams of space structure models: (a) Lips [20], rigid spacecraft with deployable beam-type members; (b) Ibrahim [25], rigid spacecraft with deployable beam- and plate-type members; (c) Shen [26], rigid spacecraft with slewing-deployable appendages; (d) Marom [27], flexible platform supporting one rigid slewing-deployable manipulator module and a rigid payload at the deployable link end (the revolute joint was flexible). 17 The new manipulator system, schematically shown in Figures 1-5 (a) and l-7(d), was first proposed for space application by Marom and Modi [27]. Planar dynamics and control of the one module mobile manipulator with a flexible revo lute joint, located on an orbiting flexible platform, were investigated. Results showed significant coupling effects between the platform and the manipulator dynamics. Modi et al. [28] as well as Hokamoto et al. [29] extended the study to the multimodule configuration, referred to as the Mobile Deployable Manipulator (MDM) system. The model, with an arbitrary number of modules, included flexibilities of both joints and links. Caron [30] presented an 0(N) formulation for studying planar dynamics and control of such complex dynamic systems. The dynamical parametric study [31] clearly shows the interactions that are present between the orbital motion, flexibility, librational dynamics, and manipulator maneuvers. Recently, Chen [32] extended Caron's study and presented an ordex-N formulation for three-dimensional motion of a mobile manipulator traversing an orbiting flexible platform. Two prototype deployable manipulator systems were built by Chu [33] and Wong [34] in 1997 and 1999 (Figure 1-8). Chu's model (the top unit in Figure 1-8) has two modules, while Wong's prototype has four modules (bottom unit in Figure 1-8). Each module has two links, one for slewing motion and the other for deploying motion. Experiments with these ground-based prototypes are valuable in the testing of control methodologies along with computer simulations; thus, providing an indirect means of verifying space-based modules as well. This thesis studies both ground-based and space-based models. It is important to point out some significant differences between manipulators supported on orbiting space platforms and their ground-based counterparts: 18 (a) Due to zero-gravity 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 more significant than the gravity effects in the study of space manipulators [35]. The large temperature variations encountered in space may significantly affect the system dynamics and control due to thermal deformations [36, 37]. (b) As the manipulator rests on a flexible orbiting platform, the dynamics of these two units are coupled [38, 39]. The manipulator maneuvers can affect the attitude of the platform and also can excite it to vibrate [40]. Conversely, the librational motion of the platform will affect performance of the manipulator. Fortunately, manipulator maneuvers in space tend to be relatively slow, permitting the end-effector dynamics to be less severe to approach equilibrium [41]. (c) Space manipulators tend to be large in size, lighter and highly flexible. Obviously, this will make the study of system dynamics and control, somewhat more complex. (d) The ratio of the payload mass to manipulator mass for a typical space-based system can be several orders of magnitude higher than that of a ground-based system [42]. For example, in the case of Canadarm I the ratio is 61.5. A corresponding ground-based manipulator used in nuclear industry (supplied by the same manufacturer) has the payload mass to manipulator mass ratio of 0.167 ! (e) Clearly, 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 [43]. Accordingly, more degrees of freedom are typically involved than required for a given task. 20 (f) Remote operation of a space-based manipulator will involve long time delays, an important factor in real-time control of the system. For the ROTEX teleoperation experiment it has reached seven seconds [44] ! These important differences emphasize the fact that the control of a large orbiting flexible platform (like the International Space Station) that supports a mobile elastic manipulator carrying a compliant payload, represents a rather challenging task. Major challenges presented by such large-scale systems are summarized in Figure 1-9. 1.5 Scope of the Investigation Based on the literature review carried out in this investigation, the following general observations can be made: (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 exclusively having revolute joints. (ii) Manipulators with both revolute and prismatic joints have received relatively little attention, and that too only recently. While the concept of space-based manipulators with slewing and deployable links was developed at the University of British Columbia (UBC), the main contributions in the field have also come from the same group. Here too, the initial focus has been on the dynamical response of the system. (iii) Control of a deployable manipulator using neural networks (NN) is still a new area. The use of NN is particularly appropriate for complex manipulator systems with significant nonlinearities and uncertainties. 21 cc U Q -J H U Pi H CC cu u a CU t a 0LH S3 2* <« •fl a> .S *C E j P j- z w CC WD o « CS S-£ a h * 3 -3 CU © » S3 E cs a > >> a V Q W .A IS cu • HH w fl cu s t-. -S s * © I , cu C * © CU < M o I -U M TS s S3 « ! ! -O Vi fl CC fl o S3 cy o -TS fl 93 fl _© V< CJ S £ fl S3 ••s 3 Cu u o <« fl o u s -a rt TS O cu S-cy S S3 S-S3 a, o cu fl •o cw u cu "o LH u a ts'« rt J5 Z S3 -a A < s o E © fl © s S3 fl O Z i . S3 = B 9 -O Vi z ^ ^ CC -a 2 & .BP o >>> fl « S © -cu s-to CU s--*-< Vi fl fl S3 o -J w 2 S 2 CU to cc 0) S-fl cu fl u TS fl 3 O u. o cc cc fl © > w © LH u fl o • HH S3 s-9 OX) fl o u WD a -o O TS CU S3 S S3 fl fl © CU s I < fl cu fl •—: 0"g Vi © S3 Z « fl CU a A o u '> fl CU CU fl CU Vi CU u P H CU -a CU cn S w 2 « C H W rt & CO I— o LU < I— LU O & _© "•C .2 -3 S3 rt "© CC -a A S3 2 •HH CU fl b£ S3 IH S3 o to 1-S3 — *^  S fl CJ CU • I s S 2 « i to to fl CU E fl o i -• HH > fl to Vi S O •HH S3 • •H u S3 E s-CU H 6X) S o 1-1 1 3 fl S3 1 t: o -C cc Vi B Vi ' f l S3 cu CU S © • HH CJ S3 HH CU fl CU s fl o s-fl a "« 1 .a a P H P H CC C/3 E CD -*-» >. Vi Xt CD Vi a CD a, U OO C+H o CD 3 C/3 o o Vi o 1 Cl >, O v> CD 00 a -d U ON <D to (N With this backdrop, the thesis investigates efficient control of deployable manipulator systems for both ground- and space-based cases. First, in Chapter 2, the governing equations of motion are derived for both space-based and ground-based systems. The formulation accounts for flexible joints and flexible links. The system kinematic equations are obtained. Next, expressions for the kinetic, gravitational forces and strain energies, and also the generalized forces are derived, and Lagrange's principle is applied to obtain the equations of motion. Validity of the formulation and the computer code is checked through the method of the conservation of energy. Rationalizing the appropriateness of neural networks in the control of a class of complex manipulator systems, Chapter 3 develops and implements a neural network adaptive controller for the deployable manipulator system. The controller is tested on different system structures, including a purely rigid system, a system with flexible joint and rigid link, and a system with flexible joint and flexible link. A comprehensive numerical study is carried out to examine the performance of the controller. For effective control of flexible modes of motion, it is important to measure modal speeds. Since it is not practical to make direct measurement, an observer is designed in Chapter 4 to estimate the rates of changes of flexible modes. Two types of observers are designed. One is a full-order observer and the other is a reduced-order observer. The observer algorithm is implemented and tested on a single-module deployable manipulator with flexible link. Chapter 5 first describes and compares several control techniques implemented on the deployable manipulator system. Based on the advantages and disadvantages established through this investigation, a new type of control - supervisory control switching system 23 (SCSS) - is developed. The SCSS is implemented and its performance is assessed using both space-based and ground-based systems. Chapter 6 concludes the thesis. It summarizes the important results of this investigation,- outlines the main contributions, and discusses avenues for further investigation. 24 2. FORMULATION OF THE P R O B L E M 2.1 Preliminary Remarks The present chapter develops the equations of motion for a space-based robotic system and a ground-based robotic system using the general model outlined in Chapter 1. Essentially, the space-based model consists of an orbiting beam-type platform with a translating base, which supports a manipulator consisting of an arbitrary number of flexible modules (units). The ground-based model does not have a supporting platform, and its based is fixed. Each manipulator module comprises two joints (degrees of freedom): One free to slew, through a revolute joint, while the other permitted to deploy (and retrieve), through a prismatic joint. The deploying segment (link) of one module is connected to the revolute joint of the next module. The links and the joints are considered flexible. The dynamic models developed in the present chapter are useful in the investigation of the control strategies as presented. As anticipated, the governing equations of motion are extremely lengthy, highly nonlinear, nonautonomous, and coupled. Interactions between orbital mechanics, librational motion, structural flexibility, and manipulator maneuvers for a space-based manipulator system are indeed quite complex. To get a better appreciation of the coupling effects, the attention is first focused on the planar dynamics of the system. In this case, all the motions are confined to the orbital plane. In the following development, the kinematic equations of the system are determined first. Issues pertaining to the modeling of body flexibility are examined. Next, the kinetic energy of the system is derived, which is followed by the derivation of expressions for the gravitational and strain potential energies. Energy dissipation mechanisms are presented, and 25 external generalized forces are examined. Subsequently, the Lagrangian principle is used to derive the equations of motion. The final section of the present chapter examines the use of Lagrange multipliers when some of the degrees of freedom are constrained. The energy approach to system modeling as used here is also useful in model validation by the method of energy conservation. 2.2 Modeling of Link and Joint Flexibilities Both space-based and ground-based models involve flexibilities. The dynamics of a robotic system with rigid components and localized (lumped) flexibilities can be described by a set of ordinary differential equations. However, the space platform and manipulator units form a highly flexible "distributed" system. The time dependent elastic deformations of such a system, with distributed parameters, require partial differential equations (PDE) for their description. Such continuous systems possess infinite degrees of freedom. For analysis and simulation purposes, it is necessary to approximate the distributed elastic deformations and inertial properties of the structure using ordinary differential equations (ODE) with a finite number of generalized coordinates. The finite element methods ( F E M ) or the system modes approach are commonly used for the discretization of the continuous elastic deformation and inertial properties of a spacecraft with fixed or variable geometry [45]. In the F E M , the structure is divided into a grid or mesh of finite elements. The continuous variables are thus represented as a number of discrete ones at the grid level. The accuracy of the approximation is improved by refining the mesh. This method offers the advantage of considering the integrated nature of the system. But for a complex system, the computational burden can be extensive, particularly in real-time applications. In the system modes approach, the system motion is approximated 26 by a finite number of "modes", as a modal summation, thereby converting the PDE model into a series of ODEs, one for each mode. Modeling a structure involving deployment joints often proves to be difficult with this approach, as the system modes need to be updated. The approach adopted in this thesis consists of dividing the system into components and modeling them separately, subject to the interface constraints. The platform as well as each of the manipulator units are taken to be separate components. The vibration of each component is expressed in the form of a product of spatially varying shape functions (0 t) and time dependent generalized coordinates (St). This approach is somewhat similar to the modal expansion approach except that the shape functions are not necessarily the mode shapes. The longitudinal deformation (ut) of the ith body is described by IX] (2.1) where j refers to the number of the shape function (or, generalized coordinate). The expression for the transverse elastic displacement (vt) is In the matrix form, the elastic deformation of the i'h body can be written as '0 IX 0 " 5ix - v l - _ 0 0 8iy_ 05 ii (2.2) (2.3) where: <P l x=[^x l, (/>ixl, (f)^]; 0iy =\j>iyl, (j>iyl, (j)iym\; Six=[Six\> Six2> •••» Sixs] ; a n d ^ > = L ^ i y l ' Siy2> 5iym\ 27 Here, Oi e 9? 2 x ( i + m ) and Si e ft(s+m), where s and m are the number of modes considered in the longitudinal direction and the transverse direction, respectively. The selection of accurate shape functions often reduces the number of modes required for a given level of accuracy. The shape functions used are a set of admissible functions, which typically correspond to the mode shapes of a similar, though simpler, problem. This is known as the assumed mode method [46]. For the sake of generality, the present formulation accounts for the longitudinal deformation of the bodies. However, this study focuses on the transverse displacements. Therefore, shape functions will not be specified for the longitudinal elastic displacements. However, it should be emphasized that they can be readily included in the formulation. In the present case, the platform and the manipulator units are assumed to behave as Euler-Bernoulli beams. This representation, which is valid for long slender beams, implies that the elastic deformation of the beam remains small and that the effects of rotary inertia and shear deformation are negligible. The equations of motion for an Euler-Bernoulli beam are obtained by applying Newton's second law to a differential beam element. The result is [47] dxz 2 \ EI dx2 d2v dr (2.4) where EI is the flexural rigidity of the beam and p is the mass per unit length. If EI is taken to be constant and a solution of the form v = ^ d0J(x)Sj(t) is assumed, the mode shapes j <f>j{x) can be written as [47] <t>j(x) = A\j s i n + A2j cos + A3j sinh + A4j- cosh (2.5) 28 where / is the length of the beam; X,- is the jth eigenvalue which, together with the coefficients AXj, A2j, A3J, and AAj, depends on the boundary conditions. The platform is assumed to behave as a free-free Euler-Bernoulli beam. Therefore, the boundary conditions at the two ends are zero moment and zero shear: EI, d2v dxi 0; EI = 0; X p 2 EI d2v pZxp 0; EI d3v P a 3 dxp = 0 (2.6) " 2 " 2 The mass of the manipulator is assumed not to affect the boundary conditions of the platform. The jth mode shape in the transverse direction (y) of the platform is then written as <Ppyj(x) = cos A 'PJ  x2 + cosh X PJ 2 / v cosh Xpj - cos Xpj sinh Xpj - sin Xpj sin X PJ P J p + sinh Xpj v 2 lPJ (2.7) where the eigenvalue Xpj can be obtained from the transcendental equation [47] cos Xpj cosh Xpj = 1. (2.8) The units of the manipulator (bodies 2 to AO are modeled as cantilever beams with tip masses. Although each unit is made of two links, a single shape function is used for the unit, since its actual vibration profile is expected to be continuous in reality. The tip masses model the inertia of the payload, as well as that of the manipulator units that are supported by the considered unit. In other words, the body supports a point mass, located at its tip, which is equal to the total mass of (N -j) units, plus that of the payload. Note that, as the cantilever 29 beam is used to model vibration, there is zero displacement and zero slope at its base, and zero moment at its tip. However, the shear force applied at the tip is nonzero, and is equal to the inertia force applied by the tip mass. The boundary conditions for this case can be summarized as: v , = o = ° ; dxi = 0 x,=0 EI, 'dxf = 0; EI, x,=l. • m, x,=l. 3\ (2.9) where mti is the tip mass. The jth mode shape in the transverse direction (y) for the platform is then written as ^yj(xi,h) = smAil - sinh A V sin Ajj + sinh Atj •—— < cos Au cos An + cosh A: 1 y v + cosh Ay (2.10) for i = 1, N. The eigenvalue A^ can be obtained from the following transcendental equation: cos AXj cosh AXj +1 = —— Ajj < sin Atj cosh Atj - cos Atj sinh Ajj J . Pi (2.11) where pt is the linear mass density of the beam. It should be pointed out that, because the length (/,) of a manipulator unit can vary with time (in the case of deployable links), the shape functions described by Eq. (2.10) will also be time-dependent. Although the concept of discretization through the application of admissible functions is approximate, it can be applied in many practical situations, while recognizing its limitations. Strictly speaking, modes are undefined for time-varying systems, as is the case during deployment and retrieval. 30 However, over the years, researchers have successfully used mode shapes as shape functions for discretization in a variety of situations including deployment of structures [48-50]. A comment concerning the representation of the flexibility of a revolute joint would be appropriate. The rotation yi of the frame Ft, attached to the module i, with respect to the frame F M has three contributions (Figure 2-1): elastic deformation of the z'th -1 body in the transverse direction ( ) ; rotation of the actuator rotor (a t), 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, ft + A - ( 2 - 1 2 ) 2.3 Space-Based Model 2.3.1 Reference frames A serial manipulator mounted on a space platform is shown in Figure 2-2. The inertial reference frame F0 is located at the center of mass of the earth. The x0, y0 plane defines the orbital plane and the z0 axis corresponds to the orbit normal. The position and the attitude of each body are described by assigning body-fixed frames to each element of the chain. Thus, the frame Fp is located at the center of mass of the platform. For each of the remaining manipulator units, the associated body-fixed frame is attached to the joint at the base of the unit. In other words, the reference frame F , is attached to the base of the ith manipulator unit. Since the platform and the manipulator units are beam-type bodies, the x, axis is along the length of the beam; the yt axis is perpendicular to the Xi direction, in the orbital plane; while z, is parallel to the orbit normal. Note that, for both the platform and the manipulator units, the stiffness, damping, and the inertial properties can vary along the x,-direction. 31 Coordinates describing the flexibility of revolute joints. 32 Figure 2-2 Description of the position vectors used in the formulation. 33 No orbital reference frame is defined at the center of mass of the complete system. For convenience, the orbital motion of the center of mass of the platform is described instead of the orbital motion of the center of mass of the entire system. The platform normally constitutes much of the mass of the system. Hence, the center of mass of the entire system will lie close to the center of mass of the platform. 2.3.2 Position vectors The position of an infinitesimal mass element dmp located on the platform, can be described with respect to the inertial frame as R*np =* + TpoDp =R + Tpo{rp+fp}, (2.12) where R is the vector from the inertia frame to the coordinate frame Fp and R = [R cos (9 i?sinc9] ; also, Tpo corresponds to the rotation matrix, which maps onto inertial coordinates the components of rp and / expressed in terms of the body-fixed coordinates. The position of an infinitesimal mass element drrii located on the ith body of the manipulator systems, can be described with respect to the inertial frame as i-l Rdm. =R + T„„D po -p xp=d X,!+T'°D' = R + Tpo{?p + fp} r+£TJO{rJ+fj} + Ti0 {n+fi} (2.13) R. dm,, = R dm. 7=1 . +Tl0{ri+fi] 34 where Tio is the rotation matrix for the components of ri and f . Particularly, the position of an infinitesimal mass element dni\ located on the first body of the manipulator system is Rdm, = R + Tpo Dp\r , + Tio D \ xp~a = R + T p o { r p + f p } +T]0{rl+f} xp-a R, dm„ + xp=d (2.14) Note that in Eq. (2.13) and (2.14), Tpo and Tio are defined as T = PO cos\//p -smyp sin y/p cos y/p T = *io cos - sm Yi sin Yi c o s Wi (2.15) where \f/p describes the orientation of the frame Fp and corresponds to the angle between the xp and x0 axes; \f/i describes the orientation of the frame F{ and corresponds to the angle between the xt and XQ 3.XCS. Note that rp is the position vector of the elemental mass dmp with respect to Fp, and rt is the position vector of the elemental mass dnii with respect to Ft. The coordinate system is chosen so that the x axis is along the platform and the manipulator. Hence, 7 P = \ _ X P °] ' 7 i = \ . X i °1 • (2.16) Finally, fp represents the elastic displacement of the mass element located at rp due to the flexibility of the platform, and f represents the elastic displacement of the mass element located at rt due to the flexibility of the manipulator link, thus fP=[0 v p ] T =[0 0 p S p J , f=[0 v , f =[0 0iSi]T. (2.17) 35 2.3.3 Velocity vectors The time derivative of Eq. (2.12) gives Rdmp = R = R cos 0 "-^sinc?" + sin 6 Rcos0 COS0 ~-Rsin0~ + sin 0 Rcos0 0 + PTpo [rp+fp}yp+ Tpo[pp+f} 0 + PT, po XP 0 0 8 _pup _ Vp+Tpo 0 8 p p _ (2.18) Velocity vector for dm\ on the first body of the manipulator system is d Rdm - dt f _ \ Rdmp \ XP=d; + PTl0Dl^+Ti0Dl 1 = R. ~dm„ + T, xp=d po 80r ^ S p dd p d+PT^^+f^^+rJfi+f = R dm„ + T, xp=d po 80r dd p d + PT, lo (2.19) lo 0X8X + h '80, d0^ 1 + 1 dx, dh A general form of the velocity vector for dnij (i = 2, 3, n) on the ith body of the manipulator system is d R dm. dt f _ = R dm. xi-\~h-\ + PTi0Diyi+Ti0Di (2.20) R dm. + PT„ xi-\ ~h~\ 0t8t + 'd<2>, d&> • + -v dxi dh J 36 where P = 0 -1 1 0 It is found that — dt f _ \ ~ Rdm,_, xi-\=li-\ ) This is because Xi-\~h-\ &\ , = 0. Note that, rt and Ot are both nonzero when the length of a body changes, for instance, during deployment. Specifically, 7 "1" / . • -i'—h ~ _0_ r; -(2.21) and 0t = — L + — -v dh J (2.22) Actuators which regulate the slewing of the manipulator units are also modeled. Since they are located at the manipulator joints, their inertial velocity, defined as Rai, is described by Rai ~ Rdm, xt-i~h-i (2.23) 2.3.4 Generalized coordinates Recall that the number of modes considered in the transverse direction is m. Then the kinematics problem of the space platform and the mobile base is described by np = 3+w+l generalized coordinates: R, 6, y/p, 8P and d. The coordinates R and 0 characterize the orbital motion of the center of mass of the platform, R being the orbital radius and 6 the true anomaly. Also d is the distance between the mobile base and the center of the platform. If the position and the orientation of the body-fixed frames are known relatively to the inertial frame, in addition to the length (/,•) and the elastic deformation (<$) of each body, then the kinematic description of the system is complete. In Eq. (2.13), the position of Ft is 37 specified directly relatively to the inertial frame by F0, and similarly, its orientation by the inertial angle y/j (Figure 2-1). Hence, y/i is defined as Vi=y'i-\+Zi-\+ai+Pi = wP +ZP +«i + A +6 + « 2 + Pi + & + • • •+« , • + Pt for i = 2, 3, . . . ,«. Particularly, for i = 1 one has ¥\ =VP+£p+ai+P\-Furthermore, 80p x=d and OX; for / = 1, 2, n. Hence, the time derivatives of Eq. (2.26) and (2.27) give d2<Z> • 80D 8„d + p dxr x=d 6 = d20, dxf 8, I; + ' 'I -I xp=d xi=lt (2.24) (2.25) (2.26) (2.27) (2.28) (2.29) Consequently, the kinematic problem of the manipulator system is described by nu = 3+m generalized coordinates: or,, /,-, and 8i. Therefore, the set of generalized coordinates required for complete description of the system kinematics can be written as q = [R 6 ¥ p 5p d ax px /, 8X an Pn ln 8nJ. (2.30) 38 2.3.5 Kinetic energy of the system The total kinetic energy (7) of the system contains contributions from the platform (Tp), the mobile base (Tbase), the actuator rotors at the joints ( r o ; ) , the manipulator modules (Tmi), and the payload (Tpayioad). Specifically, these contributions are TP=\\m Rdmp-Rdmpdmp; 2 J«» T = — R base 2 dmp * a *P=d'Rdmr m xp=d base ' (2.31a) (2.31b) \ ma\ Rdmn ' Rdmn + p V x=d x=d ) 1 " zi=2 ( mai ^ m , _ ! RdmiM V xi-i=li- -l (2.31c) xi-\~h-\ + JaiVf 1 " - -9 " Jm, T = — R payload ^ dmn •R X =1 dm,, m xn hi payload ' (2.3 Id) (2.3 le) where mp, mi,ase, mai, Jai, rrij, and mpayioad denote the mass of the platform, the mass of the mobile base, the mass of the actuator rotor at joint /, rotary inertia at joint i, the mass of the ith manipulator unit, and the mass of the payload, respectively. Furthermore, 77, = + + a i for j = 2, 3, ..., n and T]x = y/p + %p + ax. Hence, the total energy is T = Tp+ Tbase +Ta+Tm+Toayioad . (2.32) The velocity vectors can be written in the vector-matrix form. Specifically, the velocity vector for dmp on the platform can be rewritten as, 39 Rdm=o?R + o!>e + or¥p+oP8p of o{ oP ol o ... o (2.33) = opq where ,P -,P -cos 6 sin 6 -Rcos0 R sin 6 o!=PTpoDp; °4 = Tpo Or (->*) (->*) (->*„)• Velocity vector for dm\ on the first link of the manipulator system can be rewritten as d R-dm -dt f _ \ Rdmp X P = d y V + PTXoDx(¥p+ip+dx+/3x) lo 0X5X h f 80x 80^ 1 + 1 V J R dm„ + T, *P=d po 80r dd 8xx 8lx d + PTXoDx(¥p+ip+dx+j3x) , (2.34) + T, lo 0XSX + '' 80x | 80^ y 8xx 8l{ j 8xlx Substitute Eq. (2.28) into Eq. (2.32) and group terms corresponding to the derivatives of each of the generalized coordinates. One obtains 40 Rdm, = °lR + °2® + °\Wp + °\dp + °\d + °6^\ + °7 A + o\l\ + o\S{ + 0 + • • • + 0 o\ o\ o\ o\ o\ o\ Oj o\ ol o 0 (2.35) = oq where o\ = o(. o2=o%; o\=oB + PTXoDx; o\=o$+PTloDl—£-dx„ xp=d °5 = Tpo 80r dd X"=d 5, + PT]oD}—^ xp=d o\=o\=PTXoDx; (-^•dy and px) lo 1 r d0x +d0^ dx, dl i j ( - > ' l ) o{9=Tx lo 0 0X The velocity vector for dnij on the i'h (i = 1,2, n) link of the manipulator system can be rewritten as 41 Rdm, ~ Rdm, + PTioDi(yi_x+ii_x+dl+j3i) OA + h rd&i 80^ — L + — -v & i dh J R dm. +PTio A ( ^ P + i P + « i + A +6 + - + a / - i + A - i + « « • + A ) + 0 8 h f 80t d&> L + £ (2.36) Substitute Eq. (2.28) into the Eq. (2.36), and group the terms corresponding to the derivatives of each of the generalized coordinates. One obtains Rdm, = °\R + °\e + °lWP + o\8p + o\d + o'6dx + ol1px + o'slx + o'9Sx + ° i o « 2 + °\ l A + °\2h + °n52 +••• + °6+4(i-l)^i + °l+A(i-\)Pi + °8+4( / - l / ; + °9+4(M)< 5/ + 0 + • • • + 0 o\ o\ o\ o\ o\ o\ o'7 Og ol9 oj0 o\x o'X2 of 3 ••• °6+4(/-l) °7+4(M) °8+4(M) °9+4(i-l) " ' 0 = o'q (2.37) where o[=of: °2 = °2' (r>R) 7=1 o'4=op + PTXoDx p 8xr „=d J=2 8xj_x 42 po dd Xp=d sr - d20r 5P \ \x=d (-><ij and jd{) lo 1 1 + 1 dxx 3/j ^ d20 j=2 ebc. °9 = T\o 0 j=2 dx. y=2 (->dr2 and ^ 2 ) ° 1 2 = T2o 1 rd02 +d02^ V & 2 5/. 2 y v - i - d 0-, + y PT D • -—2- <?2; (->/2) °13 = J ; 2o 0o 4-^ - d0n y=3 0X2 °6+4(/-l) - °7+4(/-l) - PTIODT (->«,• and A ) °8+4(/-l) - 7)o 1 v 3x,. 5/,- j 8, °9+4(/-l) - 7/o 0, Consequently, the kinetic energy contributed from the manipulator system is 43 J J J_ ^ 7 m = - Z | R d m i - R d m i d m i 1 i=l 1 2* V f (o')T -o'dni: J (2.38) 9 If we define M/^as the mass matrix associated with the manipulator system, Eq. (2.38) can be rewritten in the more compact form Tm = X-qTMlinkq (2.39) Similarly, the kinetic energies of the space platform, the mobile base, the joint rotor, and the payload can also be written in the form of Eq. (2.39) as, 1 . T 1 . J" Tplatform ~ ^platform Q ' ^base ~ ^base 9 > 1 . T 1 • T Ta=~<l M a V ; Tpayload = ~^<1 Mpayload 9 > where Mpiatform, Mbase, Ma, and Mpayioad represent the mass matrices associated with the space platform, the mobile base, the joint actuator rotors, and the payload, respectively. Therefore, the total kinetic energy of the system can be expressed as platform base a xm payload = ^ qT (M'piatf0rm + Mbase + Ma + Mlink + Mpayload ) q (2.40) = -qTMq 2 where M is the overall mass matrix of the system. 44 2.3.6 Gravitation potential energy The gravitational potential energy of the platform is __(• tidmp _ , /udml Vg platform- }m _ || - || " }m n» R dm,, (2.41) P'R -R 12 ^dmp "~dmp where /u = Gme, G being the universal gravitational constant and me, the mass of the earth. Note that R dm. denotes the Euclidean norm value of vector Rdm . Using Eq. (2.12) to substitute for Rdm , Eq. (2.40) becomes, after some algebra, V =-V f " g platform ^ ^ DTD 2DTT D ^ 9 1 | PUP , zlJp1poLJP drrip . (2.42) Since the orbital radius is much greater than the geometrical dimensions of the orbiting system, we have R^>Dp. Expanding binomially the integrand and ignoring terms of order l/R4 and higher gives = _M^p_ + Jif r fjTbpdm p+2RTTpo\ Dpdm, Splatform ^ \ J m p P P P P J/Mp P f 2R: ¥ T p ° I 5 P 3 T P d m p TloR Similarly, the gravitational potential energy of the manipulator modules is Slink judrrij i=l ' \Kdm, Jm, N Rdm, IRJm, i 3 i 3 i d m i + 2 F m i T i o i 3 i d m i -^xL,Tu>iw!'dmi TlRdmi 2R. dm. (2.43) (2.44) 45 Note that Rdm = R^ix^l;) If the masses of the mobile base, the joint actuator rotors, and the payload are not negligible, their gravitational potential energy must be taken into account as well. Treating them as point masses, the gravitational potential energy of the mobile base is abase Rdmp(Xp=d) (2.45) The gravitational potential energy of the joint actuator rotors is N VSa ~ S II 5 /=! RdmXxi (2.46) The gravitational potential energy of the platform is V. Pm payload & payload (2.47) Hence, the total gravitational potential energy is the sum of Eq. (2.42) through Eq. (2.46); i.e., V„ = V„ +V- +Ve +V2 +Ve o & platform obase &a olink ° payload (2.48) 2.3.7 Strain energy Potential energy is also stored in the form of elastic deformations of the system. The total elastic potential energy (or, strain energy) is given by N 1 1 (rp-v ^ -I i ydXp j 2.. ^ N 1 J ^ ^ EI, 2 JO cbc dxt, (2.49) ' J where the first term represents the contribution from the deformation of the joints; the second and the third terms correspond to the elastic deformations of the platform and the manipulator links, respectively, in the transverse direction; AT, is the joint stiffness; and EI is 46 the structural stiffness of the bodies (platform and manipulator links) in the transverse direction. Note that the platform and the manipulator links are considered uniform beams in the present formulation; hence, EI is a constant. Also, as stated in section 2.2, deformations in the longitudinal direction are not considered here. 2.3.8 Equations of motion The equations of motion can be obtained using Lagrange's equations d_ dt T+ T - ° - ( 2 - 5 0 ) oq oq v 5 ? y where T represents the total kinetic energy and V represents the total potential energy including gravitational and elastic potential energy. The vector Q represents the nonconservative generalized forces. Substitute Eq. (2.39), (2.47) and (2.48) into Eq. (2.49). The equations of motion can be expressed as ! d(qTMq) dVe dV Mq + Mq~ V >+-^- + ?f- = Q, (2.51) 2 oq oq oq where Vg and Ve are defined in equations (2.47) and (2.48), respectively, and M is the mass matrix, as given by Eq. (2.39). Define •i d(qTMq] dVB dV F = Mq-—— L + —i- + ^ ± (2.52) 2 dq dq dq Eq. (2.50) can be rewritten in the following general form: M(q, i)q + F(q, q, t) = Q(q, q, t). (2.53) The vector F(q,q,t) contains terms associated with the centrifugal, Coriolis, gravitational, and elastic forces. 47 As can be seen from the foregoing derivation, the equations governing the dynamics of the manipulator system are highly nonlinear, nonautonomous and coupled. Eq. (2.53) describes the inverse dynamics of the system. For simulations, the forward dynamics of the system is required, and Eq. (2.53) must be solved for ij; thus q = M~\Q-F). (2.54) 2.4 Ground-Based Model The ground-based system is similar to the space manipulator system, except that it does not have the supporting platform and the mobile base. Figure 2-3 shows the structure of the ground-based model. The formulation considers a general rc-link deployable manipulator with flexible links and flexible joints. Modeling of the flexible links and joints is identical to what is described in sections 2.2 and 2.3. Now the inertial reference frame F0 is located at the base of the manipulator system. The reference frame F,- associated with each of the manipulator units is attached to the revolute joint at the base of the ih manipulator unit. Note that describes the orientation of the frame F,- and corresponds to the angle between the x, and x0 axes. Hence, for / = 2, 3, y/t is given by Vi=V'i-i+Zi-i+ai+Pi, (2-55) where , , and j3t are as defined in Figure 2-1 and y/\ ~ a \ + P\ • 48 Figure 2-3 The Structure of a ground-based flexible manipulator with deployable links. 49 2.4.1 Position vectors The position of an infinitesimal mass element drrij located on the i'h body of the manipulator system, can be described with respect to the inertial frame as i - i i-l + Tio Dt = HTJo{r-j + fj) = R dm. 1 \x -I + Tio{n+fi\> Tu, {?i+fi} (2.56) where Tio represents the rotation matrix, which maps the components of rt and f expressed in terms of the body-fixed coordinates, onto inertial coordinates. Also, Ti0 is given in Eq. (2.15), and rt and ft are given in Eq. (2.16) and (2.17). Particularly, the position of an infinitesimal mass element dm\ located on the first body of the manipulator system is Rdmx=Tl0 Dx=TXo {r\+f} (2.57) 2.4.2 Velocity vectors Velocity vector for dni\ on the first body of the manipulator module is = PT\0{7\+fx)n+Ti0{r-x + % (2.58) PT, lo X, X, A+T1O ' 0' 1 ai+PTLO 1 ,0A_ Ox5x A . 8,+Tu 1 dx{ 9/j where P -0 -1 1 0 A general form of the velocity vector for dmi (i = 2,3, . . . ,«) on the / body of the manipulator system is :th 50 - d f _ Rdm,=Jt Rdm, , x i - i = h - i J = R dm. xi-\~h-\ + PTioDiVsi+TioDi + PTto{rl+fi}vri+Ti0\Pi+}i (2.59) = R dm. + PT:r xi-\~h-\ 08 11 Vi+Tic 0t8{ + 30: 80; V dxi dli J Actuators regulating the slewing of the manipulator units are modeled in the same manner as in the space-based model. Here the velocity vector (R a i) is rewritten as Rai ~ Rdm^x Xi-X=h-2.4.3 Generalized coordinates The inertial angle y/i (i = 2,3, n) can be rewritten as Vi=Vi-\+Zi-\+ai+pi = <*! +#+£,+•• • + a M + pt_x + £M + at + Pi where 30 ' 3X: 8, (2.60) (2.61) Hence, the system dynamics can be explicitly expressed using the following variables: a{ rotation of the actuator rotor which corresponds to the controlled rotation of the revo lute joint; Pi elastic deformation of the joint i, which is due to flexible coupling; /, length of the z'th manipulator unit; 8i elastic deformation of the z'th manipulator unit. 51 Recall that the number of modes considered in the transverse direction is m. Hence the kinematics of each manipulator unit are described by nu = 3+m generalized coordinates: at, /?,, /,, and 5t. Therefore, the set of generalized coordinates required for complete description of the system kinematics can be written as q = [ax /J, /, d\ ••• an fin ln S„f. (2.62) 2.4.4 Kinetic energy of the system The total kinetic energy (T) of the system contains contributions from the actuator rotors at the joints (Tm), the manipulator modules (Tmi), and the payload (Tpayioaa). Specifically, they are Ta=\Ja\V\ 1<£ f ai dm. V R xi-\ ~h-\ Xi-\=li-+ Jaitf (2.63a) \ n - -Tm=-Ya\mRdmi-Rdmidmi ? " Jm, L 1=1 T = — R payload ^ dmn •R. X =/ dm,. m X =l payload • (2.63b) (2.63c) where mai, Jai, mi, and mpayioad denote the mass of the actuator rotor at joint /, rotary inertia at joint i, the mass of the ith manipulator unit, and the mass of the payload, respectively. Note that in Eq. (2.62a), 77,- =^ ,-_i +«,• for i = 2, 3, n and rjx =ax. Hence, the total kinetic energy is T = T + T +T a m payload' (2.64) Similarly, one can write the velocity vector in the matrix form. For instance, 52 where Rdmx = + °\ A + °\k + °\A + 0 + • • • + 0 o\ o\ o\ o\ o = oq 9 (2.65) o{=ol2=PTloDl; (—>CC\ and /?,) 03=Tl0 0 o\=Tx lo 1 dxx dlx (-+5xlx) The Velocity vector for dnij on the i'h (r = 1, 2, n) link of the manipulator system can be rewntten as Rdm, ~ Rdm, dm. xi-\~K-\ xi-i=li-i + PTiobi(Yi_x+ii_x+di+/3i) + Tic 0,5, + h rd0t d0^ — ' - + — -V dli J 5i h +pTio Di(vP+ip+d\+A+i\+---+a,--i+A-i + + <*i+A)+ rd0t 30^ — + — -v dh J (2.66) Substitute Eq. (2.29) into Eq. (2.66), and group the terms corresponding to the derivatives of each of the generalized coordinates. One obtains 53 Rdmt = + °2 A + + °\5\ + °5d2 + ^ 6 A> + °ll2 +08S2+--- + °4(i-\)+\di + °4(i-l)+2A- + °4(i-l)+Ji + °4(,-l)+44 + 0 + • • • + 0 „ ' „' „ ' „ ' „< °1 °2 °3 °4 °5 °6 °1 °8 °4(;-l)+l °4(/-l)+l °4(M)+1 °4(M)+1 " ' 0 (2.67) where °3 = Tio 1 y dx\ dl\ J ^ P T j o b s 9 2 0 y=2 7 dx} lo 0 0 + ±FTJODM i—t JO ] rs j=2 0 X \ j=2 o7=T: 2o 1 rd@2 | d<Z>2^  V 5 *2 2 y ^ <320, y=3 J dx 2 x2-l2 ols=T: 2o + y PTJ0D: —2 -Z—i J° J a„ L ^ 2 J y = 3 ^0 ^ a c 2 *2=/2 °4(/-l)+l - °4(M)+2 ~ PTioDi °4(M)+3 - ^ o 1 v 3*/ 9 / i y (->/i) 54 °4(/-l)+4 - TiQ Consequently, the kinetic energy contributed by the manipulator modules is 1 " - -Tm =-1(4,. -RdrndWi i=l V 2 (2.68) If we define Mu„k as the mass matrix associated with the manipulator modules, Eq. (2.68) can be rewritten in the more compact form 1 T Tm=-<1 MTlink <1 (2.69) Similarly, the kinetic energies of the space platform, the mobile base, the joint rotor, and the payload can be written in the form of Eq. (2.69); thus Ta=\qTMaq; _ 1 T Tpayload ~ ^ ^payload Q • where Ma and Mpayioa(i represent the mass matrices associated with the joint actuator rotors, and the payload, respectively. Therefore, the total kinetic energy of the system can be expressed as T = T + T +T 1 Aa m 1 payload 1 . qT [Ma + Mlink + Mpayload ) q (2.70) X-qJMq where M is the overall mass matrix of the system. 55 2.4.5 Gravitational potential energy and strain energy Define Ry^m as the projection of the position vector onto the y axis of the inertial frame F0. Then the gravitational potential energy contributed by the manipulator links is ^jink=tlsRy/midmi. (2.71) Similarly, the gravitational potential energy terms associated with the joint actuator rotors and the payload are Vs rotor-t,malgR^_\ , (2.72) <'=2 * M = ' M Vg _ payload = m payload £ R-dm„ _. • (2.73) Hence, the total gravitational potential energy is v =V +V +V (2 74) g g_Unk g_rotor g_ payload ^ * ' Strain energy has the same form as in the formulation of the space model, and it is rewritten here as 2.. \ 2 dxt. (2.75) 2.4.6 Equations of motion The equations of motion are obtained using the Lagrangian procedure, as described in section 2.3.8, and are not repeated here. 56 2.5 Verification of the Computer Code MAPLE and C languages are used to program the dynamic model developed in this chapter. The validity of the computer code is established through several checks. The size and complexity of the governing equations of motion, and the number of operations required to derive them, can lead to errors in formulation and programming. To some extent, these errors can be avoided through careful and systematic procedures for derivation and programming. Furthermore, errors can reveal themselves when the computer code is compiled and its constituent parts are linked, thereby resulting in compiling and linking errors. However, some errors may be quite elusive and require more precise checks. Ideally, the results obtained with the code should be compared with data collected from an actual spacecraft supporting a flexible manipulator. Unfortunately, few dynamics and control experiments of the type have been carried out in orbit. Furthermore, frequently, such information is not made available in open literature, and none for the manipulator system considered in the present work. The lack of relevant data does not permit a comparison of the simulation results with those for an actual space-based manipulator. A more convenient avenue is to check the conservation of energy for conservative systems. Similarly, the conservation of angular momentum may be verified as well. In this thesis, a check for conservation of energy is performed to establish the validity of the computer code and the formulation. In the absence of damping and external, nonconservative, generalized forces, the total energy of the system must remain constant. Thus, a variation of the total energy for a conservative system would indicate an error in the program or in the derivation of the 57 equations of motion. A check is considered successful if and only if the variation of the energy is found to be negligible and corresponds to the numerical noise. A single module manipulator system supported on a space the platform is used for model validation. The following numerical values are used in the simulation: Orbit: Circular orbit at an altitude of 400 km; period = 92.5 min. Platform: Geometry: a cylindrical unit with diameter = 3 m; axial to transverse inertia ratio of 0.005; Mass (mp) = 120,000 kg; Length (lp)= 120 m; Flexural Rigidity (EIP) = 5.5 x 108 Nm 2. Manipulator Position (d): d=0m. Manipulator Module (/j): Initial length of the manipulator module (i.e., /s+ deployed length, 7.5 + Id) is taken as 7.5 m, i.e. the deployable link is initially not extended. Here /s and k represent the lengths of the slewing link and the deployable link, respectively. Manipulator Links (Slewing and Deployable): Geometry: A cylindrical unit with axial to transverse inertia ratio of 0.005 Mass (m) = 200 kg; Length (/s, / d > m a x) = 7.5 m; Flexural Rigidity (El) = 5.5 x 105 Nm 2. 58 Revolute Joint: Mass (ma) = 20 kg; Moment of Inertia (Ja) =10 kgm ; Stiffness (K) = 104 Nm/rad . Note that the prismatic joint is treated as a part of the slewing link. Payload: None. Modes: For the modules: the fundamental mode for a cantilever beam with tip mass; for platform: free-free beam mode. The system is given an initial velocity (0) of 0.006428 rad/s and an initial tip deflection of 0.1 for the manipulator link. Figure 2-4 shows the variation of the total energy over 25 seconds. This variation is quite small and represents numerical noise with no discernible features. Note that its amplitude is just 10~12 %. Clearly, the total energy is considered to be conserved. With this substantiation of the formulation and the numerical code, one is able proceed with confidence to explore the avenues of control of the manipulator system. 59 _8 l 1 1 1 1 1 .0 5 10 15 20 25 Time (s) Figure 2-4 Verification of the conservation of energy. 3. NEURAL NETWORK ADAPTIVE CONTROL OF DEPLOYABLE MANIPULATOR SYSTEMS 3.1 Neural Network Control A serious limitation of conventional adaptive control, as applied to robotics, is the assumption of linearity in the unknown system parameters; specifically, /(*) = * ( * ) £ (3.1) where x is the vector of generalized coordinates, J[x) is a nonlinear robot function vector, R(x) is a regression matrix of known functions, and £ is a vector of unknown parameters of the robot (e.g., masses, friction coefficients). This assumption restricts the types of systems that are amenable to control. Some forms of friction for instance, are not linear in the parameters (LIP). Moreover, the LIP assumption requires the determination of the regression matrix for the system, which can involve tedious computations, and a new regression matrix must be computed for each different robot manipulator. In dynamic modeling, it is desirable to have generic nonlinear model structures. Neural networks are legitimate candidates for nonlinear modeling and control, substantiated by the fact that certain classes of feedforward neural networks possess the so-called universal approximation property [51, 52]. 3.1.1 Neural network approximations In control engineering, a neural network (NN) is often used to generate input/output maps using the property that a neural network of at least two layers can approximate an arbitrary function, under mild assumptions, to any desired accuracy (Figure 3-1). The approximation problem can be stated formally as given below 61 Definition 3.1 (Function Approximation): letj\x): 91" -» 91m be a smooth function. Then, given a compact set Se 91" and a positive number , there exists a two-layer NN such that f(x) = WT#(yTx) + £ (3.2) with \\s\\ < sN for all xeS , for some (sufficiently large) number L of hidden-layer neurons. The value s (generally a function of x) is the error in NN function approximation, and it decreases as the hidden layer size L increases. The associated NN structure is shown in Figure 3-1, where W= neural network output weights, V= neural network input weights, and (/>{= activation functions. Hidden layer Figure 3-1 Three-layer neural network. Note that, in this result, the activation functions are not needed on the output layer of the NN(i.e., the output-layer activation functions are linear as in Figure 3-1). Typical choices for <f> (•) include, the radial basis functions (RBF) and the sigmoid functions. The RBF network can be represented in a three-layer structure. The first layer is 62 the input layer which consists of the source nodes. The second layer is a hidden layer, composed of computation nodes, which are connected directly to all the nodes in the input layer. Each hidden node contains a parameter vector called a center. The node calculates the Euclidean distance between the center and the network input vector, and passes the results through a nonlinear activation function. The third layer is the output layer, which performs the summation of the weighted outputs from the hidden layers. The RBF networks have some useful properties which render them suitable for on-line nonlinear adaptive modeling and control. First, they belong to a class of linearly parameterized networks where the network output is related to the adjustable network weights in a linear manner, assuming that the basis function centers and variances are fixed a priori. Thus on-line learning rules can be used to update the weights, and the convergence results can be derived. Second, the activation functions of the RBF networks are localized, thus these networks store information locally in a transparent fashion. The adaptation in one part of the input space does not affect the knowledge stored in a different area; i.e., they have spatially localized learning capacity. Third, they exhibit a fast initial rate of learning convergence. This is because the network output is linearly dependent on the adjustable vector. When V = I, the three-layer feedforward neural network in Figure 3-1 becomes an RBF network. Then the output can be expressed as y = WT(j>(x). A variety of functions can be used as radial basis functions, such as the Gaussian function and Hardy's multiquadric function. In the present work we use Gaussian RBFs given by $(*) = exp -(x-MiYix-Mi) o-f (3.3) 63 where //, e91" is the center vector, and cn is the variance. Gaussian radial basis functions have some attractive properties: (i) they are bounded, strictly positive and absolutely integrable on 91" , (ii) their Fourier transforms are their own, and (iii) they are time-frequency localized. Because of these useful properties, they have been widely used in many applications including the control of nonlinear systems. 3.1.2 Neural network dynamic modeling of robot manipulators As in Eq. (2.3), robot dynamics may be described by M(q)q + V(q,q)q + G(q) = T, (3.4) as functions of position, velocity and acceleration. System identification for this model may be carried out by measuring applied forces/torques and motions. Suppose that a neural network is used to approximate the complete nonlinear dynamics of a robot manipulator, N(q,q, q) = M(q)q + V(q,q)q + G(q), (3.5) It can be seen that the neural network is a function of q, q, and q . Identification of the complete model (3.5) by a neural networks will need simultaneous measurement of all three motion variables. Because of the difficulty in measuring acceleration for system identification and control system design, the format in Eq. (3.5) can be modified to a model where neural networks are used to approximate different functions separately. It is well known that the elements of M(q) and G(q); i.e., Mkj(q) and Gk(q) respectively, are functions of q only and infinitely differentiable. Therefore, the neural network in Figure 3-1 whose inputs are q only, are sufficient to model these elements. Specifically, M/g(q) and Gk(q)can be modeled as Mkj=Wrk/Mk.(q) + eMkj(q) (3.6) 64 Gk=W^Gk(q) + SGk(q) (3.7) where WM and WGk are the vectors of neural network weights; <f>Mk. (q) and <f)Gk (q) are the vectors of activation functions; and eM . (q) and eG (q) are the approximation errors. The kj k elements of <j>Mk. (q) and <f>Gk (q) are the Gaussian radial basis functions as given in Eq. (3.3), with q as the input; thus, (j)Gk (x) = exp <<V2 -(q-MGk)T(<i-MGk) K) 2 (3.8) (3.9) Next, note that V(q,q) is a function of both q and^, and infinitely differentiable. Let - qT 9T T • The dynamic neural network approximator of Vkj(q,q) may be given by (3.10) where Wv is the vector of neural network weights; <bv (z) is the vector of activation kj rkj functions; and sv. (z) is the approximation error. Then the i-th element of <j)v (z) is given kj kj by # ty00 = exp - ( Z - M < / ( Z - ^ ) (3.11) The neural networks for My, Gk and Vkj are schematically shown in Figures 3-2, 3-3 and 3-4, respectively. 65 Gaussian RBFs Weights Figure 3-2 Neural network implementation of M/g(q). 1 Gaussian RBFs Weights Figure 3-3 Neural network implementation of Gk(q). 66 q q Gaussian RBFs Weights Figure 3-4 Neural network implementation of Vkj. To summarize, we have M(q)-. M2l M22 Mn\ Mn2 M, ln M. In M\ ] rM\ i T M\2tM\2 MXnVMXn T WM2J>M2n + SM1X 8Mn • M • M r ln (3.12) •M2n •M„ 67 V(q,q) = V2\ v22 In In Vn\ Vn2 ... v nn _ K (j>v "\\' "w Wy <f>y • KK KK KK - KK KK KK ¥ nn r nn + "2\ "\2 Yn2 (3.13) Wy(/)v+Ev\ G(q) KK KK K<f>G + SG " -- K (3-14) where = WG(f>G + EG; WT = w w w, Kkj = Kj Mkj wl K = I K <t>v "vkj rvkj KW T _ " wl wl Gk Gk W? Gk K K Gk. 68 and /, m, p denote the number of nodes of the neural network used for the elements of the system matrices Af, V, G. 3.1.3 Neural network based adaptive control The adaptive neural network control method uses the neural networks discussed in Section 3.1.2 to emulate the inertia matrix M(q), the Coriolis and centrifugal matrix V(q,q) and the gravitational forces G(q). The controller is based on a direct adaptive technique where the weights of the neural networks are updated on-line by a parameter adaptation algorithm requiring no matrix inversion. The estimates of M(q), V(q,q), and G (q) are obtained by replacing the true weight matrices WM, WV, and WQ by their estimates W M , WV and W G , respectively, i.e. M(q) = WTMcf>M{q), (3.15) V(q,q) = W?<f>M(z), (3.16) G(q) = W^M(q). (3.17) It should be noted that each element of M(q), G(q) and V(q,q) is a neural network. Therefore, parallel processing can be easily applied and the controller is suitable for real-time implementation. We use the control law: r = M(q) (qd + Ae) + V(q, q)(qd + Ae) + G(q) + Kve + KvAe, (3.18) where the tracking error e is defined as e-qd-q with qd denoting the desired trajectory, and A is a positive definite and diagonal matrix. With this control law, the stability property of the closed loop system is given by the following theorem. 69 Theorem 3.1 If Kv > 0, then the closed-loop system is asymptotically stable; i.e., s = Ae + e—>0ast—>oo under the following parameter adaptation laws WMkrrMk.<l>Mk.q.Sk, (3.19a) K=ry*\*4*» ( 3 - 1 9 b ) WGk=rGk<f>Gksk, (3.19c) where rMk., rVk,, rGk are symmetric positive-definite constant matrices, and WMk., WVk., WGk are elements of WM, Wv and WG , respectively; qs =qd + Ae and qs =qd + Ae ; qsj and qsj are the j-th elements of qs and qs, respectively; sk is the k-th element of s; e and e —>0 as t —> co; and all signals in the closed-loop system are bounded. The proof of Theorem 3.1 can be found in [53]. It is seen that the first three terms of Eq. (3.18) give the model-based linearizing controller TM =M(q)(qd+Ae) + V(q,q)(qd+Ae) + G(q) (3.20) and the next two terms give a PD-type controller. The proposed neural network adaptive control structure is shown in Figure 3-5. 70 o l-l fl o o CU _> o -t—» tu fl 13 t-l fl tu fl tu .fl o tu fl o CO m m tu i-< fl bp P-l 3.2 Neural Network Adaptive Control of a Rigid Deployable Manipulator The manipulator system under study is shown in Figure 3-6. It consists of two joints: one free to slew while the other permitted to deploy. Pertinent nonmenclature is defined below. C M . - center of mass; ni\ = mass of the arm segment connected to the slewing joint (termed the slewing link); m.2 = mass of the deploying segment of the arm (termed the deploying link); mp = payload attached at the end of the manipulator; I\,h = moment of inertia of the slewing and the deploying links (1= ml2/\2), respectively; l\, h = lengths of the slewing and the deploying links, respectively. The vector of generalized coordinates, which describes the complete motion of the manipulator model, is given by q = [l of, where: / = length between the origin and center of mass o f the deploying link. 6 = slewing angle associated with the revolute joint. The following assumptions are made in deriving the equations of motion: (i) A l l links are considered rigid. Rigidity is assumed due to the complexity and the amount of computation involved in modeling flexible links. Also , since the link lengths are relatively small, the flexibility effects are l ikely to be negligible. Modeling of flexibility is straightforward, however, as discussed elsewhere in the thesis. (ii) A l l links are considered uniform so that the center of mass is at the middle of each link. (iii) Friction is negligible 72 Figure 3-6 Model of the one-module deployable manipulator. In the Lagrangian formulation, the system matrices are given by: m2+mp M(q) = 0 ^ mtf + i m2l22 +m2r2+ mp (1 + lj-)2 (3.21a) V(q,q) = -mil6-mp{l + l±)e U [m2l + mp(l + ^ -)]0 [m2r + m(l + ^ -)]l (3.21b) m2g s i n 0 + mpg s i n 0 G(q) = \ I L (mlg± + m2gl + mpg(l + - J ) ) c o s 0 (3.21c) As seen, the governing equations of motion are nonlinear, nonautonomous and coupled. 73 3.2.1 Simulation study A. Trajectory Planning In the present study, a sinusoidal (sine-on-ramp) acceleration profile is adopted for prescribed maneuvers. It assures zero velocity and zero acceleration both in the beginning and at the end of the maneuver, thereby smoothing the structural response of the system. The trajectory of the specified maneuver is given by, td I In 7,71 —t \fd J • + q(0), (3.22) where qd is the desired trajectory; t is time; and td is the time required for the maneuver. In the simulation example, the following parameter values have been chosen for the desired trajectories of the two generalized coordinates: td =2 sec, qd(0) = [0.5m O.Oradf, qd(td) = [l.5m 2.0radf. B. Simulation Settings For the approximators M{q) and G(q), a static neural network with 117 nodes is chosen for each element, while a dynamic neural network with 468 nodes is chosen for each element of V(q,q), with the variances aM =av -o~G =2.0. The centers of the localized Gaussian RBFs are evenly distributed to span the input space. The weighting matrices WM, Wv and WG of the neural networks are initialized to zero. The following true parameters of the robot were used for the simulation, and are m\ = m2 = 0.8 kg; and l\ = l2 =1.0 m. Suppose that the robot initially rests at q0 = [0.5m 0.0rad]r , with 74 q0 = [O.Om/s O.Orad/s]r , and without a payload; thus, mp = 0.0 kg. The gains of the controller are chosen as Kv = diagflO.O 10.0], and A = diag[5.0 5.0]. In order to test the load disturbance rejection capabilities of the controller, a payload with mp = 1.0 kg is picked up by the robot at time t = 4.0 sec. In the figures showing the simulation results, dashed lines are used to represent the desired values and solid lines for the actual values. C. Non-Adaptive Case The control performance of the non-adaptive case is first investigated as a basis for comparison. In this case, the weight adaptation law (Eq. 3.19) of the neural network controller is not activated. The corresponding neural network model-based control signal rm (Eq. 3.20) is equal to zero since the weights of the neural networks are all initialized to zero. As a result, the controller is reduced to a simple PD controller. The position tracking performances of the robot and the corresponding control signal are shown in Figure 3-7. It can be seen that the PD controller results in a large tracking error and cannot effectively handle parameter uncertainties and changes in the system, such as load disturbances. 75 Figure 3-7 PD control of the single-module deployable manipulator: system performance and control inputs. 76 D. Adaptive Case In the case of adaptive control, the adaptation algorithm (Eq. (3.19)) is activated with rM =diag[1.0], rv =diag[1.0] and rG = diag[10.0]. The position tracking performances of the robot and the corresponding control inputs are shown in Figure 3-8(a). The results from the neural network approximation of the nonlinear terms of the model are also of interest to us. For brevity, only the estimations of elements in the inertia matrix M and the gravitational vector G are shown in Figure 3-8(b). Results clearly show that the tracking error is much smaller than in the non-adaptive case. This is due to the "learning" mechanism that is present in the adaptive control. The simulation results demonstrate that the adaptive neural network controller can effectively control an incompletely known nonlinear robotic system, in the presence of load disturbances. Notice from Figure 3-8(b) that there are estimation errors for the inertia matrix elements. But the estimates are bounded, which guarantees the tracking error to be asymptotically stable. Note further that the objective of control is not necessarily to estimate the model parameters accurately, but to perform the tasks accurately, which the robot seems to be able to do. Different tracking performances can be achieved by adjusting the parameter adaptation gains and other parameters such as the size of the neural network, and the centers and variances of the Gaussian RBFs. Only the effect of adjusting the parameter adaptation gains is checked here. In this case, the adaptation algorithm (Eq. (3.19)) is first activated with rM = diag[5.0], rv - diag[5.0] and rc = diag[10.0]. The position tracking performance of the robot and the corresponding control inputs are shown in Figure 3-9(a). The estimations of the elements of the inertia matrix M and the gravitational vector G are shown in Figure 3-9(b). As seen from Figure 3-9(a), the tracking errors decrease with the increase of the 77 adaptation gains, and they converge to zero in a short time, with a low level of oscillation. These effects are particularly obvious after the payload is added (r > 4). Slewing Motion Deployment Motion 2 4 6 Slewing Motion Error 0 2 4 6 8 Control Torque: Revolute Joint [m] 1 2 4 6 8 Deployment Motion Error 0.02 [m] 0 2 4 6 8 Control Force: Prismatic Joint Figure 3-8 (a) Neural network adaptive control of the single-module deployable manipulator with adaptation gains rMk = diag[5.0] , rvk - diag[5.0] and rGi=diag[10.0]. 78 Desired Actual M 2 2 2 Figure 3-8 (b) Neural network approximation of elements in inertia matrix M and gravitational vector G(TMk - diag[5.0], rvk = diag[5.0] and rGk = diag[10.0]). 79 Slewing Motion Deployment Motion 0.02 [rad] -0.02 2 4 6 Slewing Motion Error 0 2 4 6 8 Control Torque: Re volute Joint [m] 1 0 2 4 6 8 Deployment Motion Error 0.02 6/ [m] 0 2 4 6 8 Control Force: Prismatic Joint Figure 3-9 (a) Neural network adaptive control of the single-module deployable manipulator with adaptation gains ruk - diag[5.0] , rvk = diag[5.0] and rGt=diag[10.0]. 80 Desired Actual Figure 3-9 (b) Neural network approximation of elements in the inertia matrix M and gravitational vector G with adaptation gains ruk = diag[5.0], rvk = diag[5.0] and rGk =diag[10.0]. 81 3.3 Neural Network Adaptive Control of a Deployable Manipulator with Rigid Link and Flexible Joint Now consider the manipulator system shown in Figure 3-10. It has the same structure as in Figure 3-6 except that the revolute joint is flexible here. Symbols are as defined before, and those that have not defined are listed below: J = motor inertia; K = joint stiffness. 7777777 X Figure 3-10 A single-module deployable manipulator with joint flexibility. The vector of generalized coordinates, which describes the complete motion of the manipulator model, is given by q = [l 0 a f, where / represents the distance between the origin of the reference frame located at revolute joint and the center of mass of the deployable link; 0 denotes the slewing angle associated with the robot arm; and a denotes the motor rotor angular position. The equations of motion can be expressed in the vector-matrix form as 82 M(qr)qr + V(qr,qr)qr+G(qr) = K(a-G) (3.22a) Jd + K(a-0) = ra (3.22b) where and ra are the force and torque inputs for the deployable link and the revolute joint, respectively; qr = [/ 0]T; M(qr), V{qr,qr) and G(qr) are system matrices corresponding to the rigid model as described in Figure 3-6 and they are give by: M(qr) = m2 +mp 0 0 J M\L\ + T\ m2l2 + mil2 +™P(l + lp2 (3.23a) 0 -m2r0-m {l + —)9 [m2l + mp(l + lp]0 [m2i + mp(i + k)]i (3.23b) G(qr) = m2g sin 0 + mpg sin 0 (mlg | + m2gr + mpg(I + ^ -)) cos 0 (3.23c) Note from Eq. (3.22) that it is the joint elastic torque K(a-0) that drives the slewing of the rigid link. In fact, Eq. (3.22b) denotes the dynamics of the revolute joint motor with no damping. Figure 3-11 illustrates the control structure of this system. With the derived equations of motions, singular perturbation method is used to transform the original flexible joint robot model into a two-time-scale model; specifically, the well known steady-state-model and the boundary-layer-model. The singular perturbation theory and its application to the dynamic model are described. Here, only the basic concepts of the singular perturbation theory are given as far as it concerns the controller design. A more comprehensive treatment of this theory can be found in [54]. 83 Rigid Deployable Manipulator System (Eq. 3.22a) Joint Flexibility Motor Dynamics (Eq. 3.22b) > a Figure 3-11 Interaction of actuator dynamics with manipulator dynamics through flexible joint. 3.3.1 Singular perturbation theory The singular perturbation theory can be applied to analyze a model that depends on a scalar parameter e which is considered smaller than the other parameters in the model. The associated dynamic problem may be expressed as follows: For small values of s, the system has the so called two-time-scale-property. This means that the system can be virtually split up into two coupled subsystems, which describe a faster and slower part of the system. The singular perturbation theory then allows to make stability conclusions out of the properties of the separated subsystems which can be obtained when the parameter e tends to zero. x = fi(x,z,e,t), ez = f2(x,z,s,t). (3.24) 84 Letting the parameter s approach zero, which means that (3.24) is considered in steady state1 z\ = h(x,t), one obtains the so called steady-state-system on the one hand: x = fi(x,h(x,t),e,t). (3.25) On the other hand, by performing a change of coordinates y = z-h(x,t) (3.26) as well as a change of the time scale and time reference by introducing a new time variable v = (t-tQ)/e one gets the so called boundary-layer-model. Here the slow varying variables x and t are treated as constants with respect to the fast time variable v. ^ = f2(x,y + h(x,t),0,t). (3.27) dv Here the new variable y is the deviation of the fast variable z from its quasi steady state h, as give by Eq. (3.26) Equation (3.25) and (3.27) can serve as a starting point for stability analysis. In the following treatment the controller design for the flexible joint model will be based on Tychonov's theorem [52]. For brevity, we give here only the essential part of Tychonov's theorem without the mathematical intrications. The precise statement of Tychonov's theorem and its proof can be found in Theorem 9.1 of [54]. Basic statement of Tychonov's theorem: Assume that the steady-state-system (3.25) has a unique solution Xs(f) defined on a time interval [to, t\]. Assume further that the boundary-layer-model (3.27) is exponentially stable and its solution is denoted by jf(t). Then there exists a positive constant e such that for s<e the system (3.24) has a unique solution x(f), z(t) and the deviation of x(t) from xs(t) is of order son the time interval [to, t{\, i.e. 1 The situation, when e is set to zero will be called quasi steady state in the sequel. 85 x(t,s)-xs(t,s) = 0(s) Additionally, z(t,£)-h(x(t),t)-ys{t) = 0(s) holds on the time interval [to, t\]. It is important to note here that stronger statements may be concluded based on singular perturbation theory. This would then require exponential stability of the steady-state-system as well. In the adaptive control problem of a robotic manipulator, this would require a considerably more complex adaptation law. 3.3.2 Application to the single module deployable manipulator The approach proposed by Spong [55] is used to apply the singular perturbation theory on the single module deployable manipulator system with flexible joint. Note that Spong's approach is applied to a manipulator system with purely revolute joints, while our system has a deploying joint as well. For the present purpose, it is necessary to perform some modifications to the original equations of motion. First define V(qr,qr)qr = , G(qr)= Go where Vi, Ve, Gi, and Ge are terms associated with the generalized coordinates / and 0, as denoted by the corresponding subscripts. Furthermore, notice that the mass matrix M is diagonal; thus, MN = M 2 1 =0. Then Eq. (3.22a) can be written as MUI+V,+GI=T,. (3.28) M 2 2 0 + Vg + Ge = K(a - 0). (3.29) 86 The singular perturbation theory is introduced into the present problem in view of the fact that the stiffness K of the joint is usually quite high relative to the other parameters of the small value in fthen corresponds to a big value in the joint stiffness. It should be pointed out that the rationale for the particular choice of the "small" parameter in the two-time scale problem stems from physical reasoning. In particular, the preliminary simulations using the manipulator models have indicated the presence of vibrations of frequency that is much higher than the bandwidth of operation (or, rigid body dynamics) of the robot. In this context, one could consider the choice of joint stiffness AT in defining eio be related to the association of high-frequency vibrations to high joint stiffness. In another robotic system, the small parameter s might be associated with a system parameter other than the joint stiffness. Now define a new coordinate z according to system, and can be written as K = —j, where K\ is a constant and s is a small parameter. A z= K(a-6)=-t(a-9). (3.30) From Eq. (3.30), we have •• 6 •• n a=—z+9. (3.31) Substitute Eq. (3.30) and (3.31) into Eq. (3.22b). We have (3.32) From Eq. (3.29), 9 can be express as 0 = M-2\z-Ve-Ge). Then Eq. (3.32) can be written as 87 e2z + Kx {M22 + J~l)z = KxJ~xta + KXM22 (V0 +GS). (3.33) With Eq. (3.29) and Eq. (3.33), we obtain a model in the singular perturbation form M220 + Vo+Gg=z, (3.34a) e2z + Kx{M^+rx)z = KxrXTa+KxM22\Ve + Ge). (3.34b) 6 G\ and z = [z z] . The quasi steady state value h is given by h{6,6) = z £->0 (l + JM22Y T a |£_>o + JM22\Ve+Gd)_' (3.35) The steady-state-model and the boundary-layer-model can then be derived from Eq. (3.34). The Steady-State-Model: When the parameter s is set to zero, the steady-state-model for 6 is obtained from Eq. (3.34): ( M 2 2 + J )0 + Ve+Ge=ra £^0 = raslow, (3.36) where ra slow denotes the part of control input, which remains when s tends to zero. By combining Eq. (3.36) and (3.28), one obtains the steady-state-model: (M{qr) + J)qr + V(qr,qr)qr+G(qr) = ^a, slow slow (3.37) The Boundary-Layer-Model: Define a new coordinatey as it is in Eq. (3.26): y = z-h(0,d). (3.38) 88 Since z = K(a-6) is the joint elastic torque, which drives the slewing of the rigid link, it is clear that y corresponds to the deviation of the actual torque from its quasi steady state value h. Substitute (3.38) into (3.34b), introduce the fast time scale v-t/s and neglect all derivatives of h{6,6) with respect to v. This leads to p^ + Kx{J-x +M-22x){y + h) = KxJ-XTa + KXM22\V0+G9). (3.39) dv From Eq. (3.35), we have Or1 +M 2 -> = J-\tSl0W + M22l(V0 + G0) . (3.40) Substitute (3.40) into (3.39). We get the boundary-layer-model: ^ + K1(J~1 +M22l)y = KxJ-l(Ta-rBtdaw) = KXJ rajast As can be seen, by applying the singular perturbation approach to the original flexible jointed manipulator system, one gets two subsystems which may be controlled separately. Therefore, the motor torque to the revolute joint was slit up into a slow and a fast part xa slow and tajast, corresponding to the control inputs of the two subsystems. 3.3.3 Controller design In the context of Tychonov's theorem, on one hand our goal for the controller design is to get an exponentially stable boundary-layer system. On the other hand, convergence of the tracking error is desired for the steady-state-system. Note that the boundary-layer-system (Eq. 3.41) is linear. The fast control input Ta,fast is chosen as 89 ^jast=Kd(0-d) +Kp(0-a), (3.42) where Kd and Kp are control gains. It provides damping and additional stiffness. The steady-state model (Eq. 3.37) has the form of an equivalent rigid body model. Therefore, the neural network based adaptive controller is applied to control the steady-state model. Hence the design of the slow control rslow = [r, r a > i / o w ] uses the neural network based adaptive approach, and it is rewritten here as: Tslow=M(qr)(qrd+Ae)+ V(qr,qr)(qrd + Ae) + G(qr) + Kve + KvAe (3.43) where qra- represents the desired values for coordinates qr = [/ #]r. 3.3.4 Simulation study A sine-on-ramp acceleration profile is used for the prescribed maneuvers as in Eq. (3.22). The system generalized coordinates are initialized as *7(0) = [r(0) 0(0) a(0)] = [\m Odeg Odeg], and the desired end values are qd (td ) = [2m 90 deg 90 degf where td = 3 s. The adaptation algorithm is activated with rM = diag[2.0] , rv = diag[2.0] and rG =diag[5.0]. The controller gains are chosen as Kv= diag[10, 50], Kd = 0.1 and Kp = 0.5. In order to test the capability of load disturbance rejection of the controller, a payload of mass mp = 0.8 kg is picked up by the robot at time t = 5.0 sec. In figures 3-12 and 3-13 showing the simulation results, the solid lines are used to represent the actual values and the dashed lines for the desired values. 90 The position tracking performance of the robot is shown in Figure 3-12. Note that ei and ee denote the tracking errors of the prismatic and revolute joints, respectively, and 0-a represents the joint elastic motion (vibration). The results from the neural network approximation of the nonlinear terms of the model are also of interest to us. For brevity, only the determinants of the estimated and actual system matrices M, V, and G are shown in Figure 3-13. The simulation results demonstrate that the adaptive neural network controller can effectively control an unknown nonlinear robotic system even in the presence of sudden load disturbances. Notice from Figure 3-13 that there are estimation errors for the inertia matrix elements at steady state. The estimates are bounded, however, which guarantees the tracking error to be asymptotically stable. Furthermore, as before, the goal of control is achieving trajectory accuracy rather than the parameter estimation accuracy. 91 Deployment Motion Slewing Motion Figure 3-12 Neural network adaptive control of the deployable manipulator. 92 Ail and Af \V\ and \V\ G and \G\ 8 10 Time [s] Figure 3-13 Comparison of the determinants of the actual system matrices ( Af , V , G , dash-line) and the neural-network estimated ones ( Af V , solid-line). 93 3.4 Neural Network Adaptive Control of a Deployable Manipulator with Flexible Link and Flexible Joint Now consider the manipulator system shown in Figure 3-14. Both the robot arm and the revolute joint are considered flexible here. The vector of generalized coordinates, which describes the complete motion of the manipulator model, is given by q = [a 6 5 I ]r, where a denotes the motor rotor position; 6 denotes the slewing angle associated with the robot arm position; Sis the flexible generalized coordinate; and / represents the length of the manipulator. Figure 3-14 Structure of a deployable manipulator with flexible revolute joint and flexible arm. 94 The assumptions made in the present simulation are: (i) It is a planar system. Gravity effects are not relevant. (ii) Both deploying and slewing links are uniform and have the same flexural rigidity (ET). (iii) Only one flexible mode shape is used in modeling. (iv) Friction of the system is negligible. Using the Lagrangian procedure, the equations of motion of the system are described as ~Mrr Mrf~ 4_ V rr t + "o o " 9r Mfr Mff. 3f. i Vff. .if. 0 Kff_ .If. K(a-6) 0 (3.44a) Jd + K(a-9) = za (3.44b) where zl and ra are the force and the torque inputs for the deployable link and the revolute joint, respectively; qr = [/ #] r; and =5 ; the system matrices (M, V) are partitioned according to the rigid and flexible generalized coordinates. Also, K/f is the stiffness matrix. Since we only use one flexible mode, Kffis a constant in the present case. Dynamic interaction of the deployable manipulator with flexible joint and flexible link is schematically represented in Figure 3-15. From the figure it is evident that control of such a system would be more complicated than that of a manipulator with flexible joint only. In particular, one is faced with manipulating the intermediate variable a, which has subsequent influence on the variable of interest 6 (because of the flexible joint). Further more, one must control the revolute joint angle 9 of the manipulator in the presence of additional dynamics caused by the link flexibility. 95 Flexible Mode Link Flexibility Deployable Manipulator System (Eq. 3.44a) q r ( i ) • Joint Flexibility Motor Dynamics (Eq. 3.44b) a Figure 3-15 Dynamic interactions of the deployable manipulator with flexible joint and flexible link. 3.4.1 Controller design and simulation study Because of the presence of flexibilities in the revolute joint and the arm, singular perturbation theory is applied to decompose the manipulator into a slow subsystem, and a fast subsystem as in section 3.3. The fast control input ra jast is chosen as (3.45) where Kd and Kp are control gains. The controller provides damping (active) and additional stiffness (gain). As before, a neural network controller is designed for the slow subsystem, which corresponds to the rigid body dynamics of the manipulator: 96 rslow=M{qr){qrd+Ae)+ V{qr,qr){qrd+Ae) + G{qr) + Kve + KvAe (3.46) where qrd is the vector of desired values for coordinates, qr = [/ 0]T , and Tsiow = \_Ti Ta,shwJ • Note that the motor torque input for the revolute joint is ra = T ~\~ T a,slow a, fast ' The following numerical values are used in the simulation: ms = md = 2 kg ls = h = lm; J= 1 kgm ; ^=500Nm/rad; EI= 5.5x101 kg m3/s2; The system generalized coordinates are initialized as q(0) = [a(0) 0(0) S(0) r(0)] = [0deg Odeg 0.01 lm], and the desired values are qd (td) = [90 deg 90 deg 0 2mf where td = 4 s. Again, a sine-on-ramp profile is used for the maneuvers. The adaptation algorithm is activated with rM = diag[2.0], rv = diag[2.0], and rG = diag[5.0]. The gains for the slow controller are chosen as Kv = diag[10, 50]. For the fast controller Kd = 2 and Kp=5. Figure 3-16 presents the response of the system variables and the controller inputs. Figure 3-17 gives the tracking errors of the revolute joint {ee) and the deploying link {ei). The joint flexibility angle {0 - a) is also shown in Figure 3-17. From the result, it is seen that the system is able to follow the trajectories accurately. With this control structure, the maximum tracking errors for the revolute joint and the deployable link are around 1° and 0.03 m, respectively. The amount of control input is permissible. Furthermore, during the maneuver, the manipulator generates a maximum tip deflection {e=&S) of 0.06m and a joint 97 vibration (9- a) of amplitude less than 1°. Notice that the oscillations of the joint and the link converge to zero. Furthermore, the vibrations of the joint and the link are coupled, as expected. Initial vibrations are quickly damped out by the controller. Next, the fast controller gain Kd is increased by a factor of 10, which gives Kd = 20 and K is kept the same as before, i.e., Kp = 5. The objective of this simulation is to examine the effect of gain increase on the joint vibrations. Figure 3-18 shows the response of the system generalized coordinates and also the control inputs. Figure 3-19 presents the tracking errors and the joint vibrations. As a result of increasing the fast controller gain Kd , the high frequency vibrations in the system response are found to be significantly reduced, as can be seen from the joint vibrations and the tracking errors given in Figure 3-19. To examine the effect of fast controller gain Kp on the system response, Kp is increased to 50 while Kd is kept at 20. The response of the system variables is shown in Figure 3-20 and tracking errors of the revolute joint and deployable link are shown in Figure 3-21. Comparing with result shown in Figure 3-19, increasing Kp has affected the response of the slew motion (0). It reduces the number of limit cycle oscillations in the region as circled in Figure 3-21. However, the maximum tracking error of the slew motion (ee) increases because of higher Kp. 98 Deployable link motion Slewing motion 0 2 4 6 8 10 0 2 4 6 8 10 Force input for deployable link Torque input for acturator rotor 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 Time [s] Time [s] Figure 3-16 System response and the control input with KV= diag[10, 50], Kd = 2, and Kp=5. 99 Tracking error of the slew motion ee o -1 0 2 4 6 8 10 Tracking error of the deployable motion 0 2 4 6 8 10 Joint vibration e-a [deg] 0 2 4 6 8 10 Time [s] Figure 3-17 Tracking errors and joint vibration with Kv = diag[10, 50], Kd = 2, and K =5. 100 Deployable link motion Slewing motion 0 2 4 6 8 10 Force input for deployable link 0 2 4 6 8 10 Torque input for acturator rotor 0 2 4 6 8 10 Actuator rotor motion 0 2 4 6 8 10 Link tip deflection 0.05 [deg] -0 .05 Figure 3-18 System response and the control input with Kv = diag[10, 50], Kd = 20, and Kp=5. 101 Tracking error of the slew motion ei 0.03 0.02 0.01 Tracking error of the deployable motion -0.01 Joint vibration 9 - a [deg] -0.5 Figure 3-19 Tracking errors and joint vibration with Kv= diag[10, 50], /vrf = 20,and 102 Deployable link motion Slewing motion 0 2 4 6 8 10 0 2 4 6 8 10 Force input for deployable link Torque input for acturator rotor 0 2 4 6 8 10 0 2 4 6 8 10 Time [s] Time [s] Figure 3-20 System response and the control input with Kv = diag[10, 50], Kd = 20, and V s 50. 103 Tracking error of the slew motion ee o 0.03 Tracking error of the deployable motion Joint vibration 0-a [deg] 8 10 Time [s] Figure 3-21 Tracking errors and joint vibration with Kv = diag[10, 50], Kd = 20, and Kp=50. 104 3.5 Neural Network Adaptive Control of a Space Platform Based Deployable Manipulator System The neural network adaptive control scheme is applied now to a space platform-based manipulator system. Dynamics studies clearly establish a need for control, under critical combinations of system parameters, initial conditions and maneuvers, to maintain desirable performance of the manipulator [56]. The present section focuses on the control of the system with a platform-based one-unit (two links) rigid manipulator. To recapitulate, more frequently used symbols are summarized below and indicated in Figure 3-22: R altitude of the orbit; 9 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. y/p platform pitch; / length of module one; a rigid component of slew maneuvers associated with module one Using Lagrangian formulation method, the equations of motion can be expressed in the general form M(q,t)q + F(q,q,t) = T, (3.47) where M(q,t) is the system mass matrix; q is the vector of the generalized coordinates; and F(q,q,t) contains terms associated with centrifugal, Coriolis, and gravitational forces. 105 106 The vector r represents generalized forces, including the control inputs. The vector of generalized coordinates, which describes the complete motion of the manipulator model, is Orbit: Circular orbit at an altitude of 400 km; period = 92.5 min. Platform: Mass (mp) = 120,000 kg; Length (lp)= 120 m; Manipulator Position (d): d=60m. Manipulator Links (Slewing; and Deployable'): Mass (m) = 200 kg; Length (/s, / d , m a x ) = 7.5 m; Payload: Zero. The initial conditions for the controlled variables are y/p(0) = 0 deg, a(0) = 0 deg, and 1(0) = 7.5 m. The desired values for the controlled variables are \j/p(td) = 0 deg, a( td) = 20 deg, and l(td) = 15 m, where td = 6 sec. As before, a sine-on-ramp profile is used for the prescribed maneuver. To accentuate the dynamic response, the manipulator is located at the tip of the platform, i.e., distance between the mobile base and the center of mass of the platform is 60 m. The following numerical values are used in the simulation: 107 The control law is as given in Eq. (3.18), which is rewritten here: r = M(q) (qd + Ae) + V(q, q)(qd + Ae) + G(q) + Kve + KvAe, where M, V, and G are the system matrices estimated by the neural network. Since the orbital parameters R and 0 are not controlled variables, the control vector Kv is chosen as Kv= diag[0 , 0, 5000, 100, 50]. Figure 3-23 gives the plots of the response of the controlled variables (y/p, a and /)• Corresponding tracking errors {ev , ea, and et) and control inputs (ry , xa, and rt) are also shown in Figure 3-23. The maneuver sets the platform to librate through a negligible angle (permissible limit can vary from 0.1° to 1° depending on the mission). Results clearly show that the tracking errors for the revolute joint and the deployable line are quite small (e.g., maximum slew error is less than 0.5°). The neural network approximations of the nonlinear terms of the model are illustrated in Figure 3-24. For brevity, only the estimations of some elements in the inertia matrix M and gravitational vector G are shown. Estimation results show some discrepancy with the actual values. However, the error is bounded and the controller is still able to regulate the manipulator to follow the desired trajectory. This situation is acceptable in view of the fact that the purpose of the neural network controller here is not parameter estimation but rather accurate trajectory following. Furthermore, the results show the robustness of the neural network adaptive controller. 108 T v Initial Conditions: y/p = 0; a=0; . . . Desired Cj / \ \ d\ I = 7.5m; d= 60 m; \ no payload. k Desired Values: — Actual \ Y \ -7p = 0, a = 2 0 ° , / = 15m. \ \ sine-on-ramp; 6 s. Libration Motion Slewing Motion Deployable Link Motion 0 5 10 15 0 5 10 15 0 5 10 15 Libration Error Slewing Error Deploying Error 0 5 10 15 0 5 10 15 0 5 10 15 Libration Torque Slewing Torque Deploying Force 0 5 10 15 0 5 10 15 0 5 10 15 Time [s] Time [s] Time [s] ;ure 3-23 Response of the system under neural network adaptive control during a simultaneous 20° slew and 7.5 m deployment maneuvers of the rigid one-unit manipulator system. 109 M, 10.1 400 Vu 30 200 r -1000 Figure 3-24 Parameter estimation results from the neural network (solid line) compared the actual ones (dashed line). 110 4. OBSERVER DESIGN FOR MODAL RATE SENSING From the implementation point of view, full knowledge of all state variables is desirable for effective control of flexible multi-link manipulators. In fact, some multivariable control schemes such as the linear quadratic regulator (LQR) require the complete knowledge of the state vector. In practical systems, however, some of the needed state variables may not be measurable. In a manipulator with flexible links, the state variables include not only the modal coordinates but also the rates of changes of these coordinates. It is possible to measure the joint positions and velocities, and the shapes of the flexible modes of motion of a manipulator using encoders, tachometers, and strain gauges [57]. However, the measurement or direct computation of the rates of changes of flexible mode shapes can introduce unacceptable errors. One method of determining the modal velocity is to integrate the outputs of accelerometers installed along a link. Alternatively, analog or numerical differentiation may be carried out on the deflection variables. The former approach may be restricted due to economic considerations and bandwidth reduction, and the latter is often not practical because of noise problems. A nonlinear state observer may be used to overcome these limitations. In this chapter, an observer-based strategy is developed to estimate the rates of changes of flexible modes. 4.1 Description of the System Figure 4-1 shows the configuration of the system under study. The manipulator has an arbitrary number of modules, each carrying a slewing degree of freedom and a deployable degree of freedom, corresponding to a revolute joint and a prismatic joint. The following assumptions are made: 111 ss SV Figure 4-1 A flexible multi-link deployable manipulator. 112 1) The links are flexible and the joints are rigid. 2) The structural stiffness (El) is constant along each manipulator link. 3) The links move horizontally; hence, only the planar dynamics would be applicable. 4) The friction at the joints is negligible. Using Lagrangian formulation, the dynamics of the flexible multi-link manipulator shown in Figure 4-1 is expressed as: M{qr,qf) + "0 0" Qr + — 0 K 3f. 0 (4.1) where qr is the 2nth order column vector of rigid generalized coodinates, 9r= [h ®\ h &2 "" K @n] '•> 9f l s m e m t n order column vector of flexible generalized coordinates, q^=\5x 82 ••• Sm] ; VrX, Vr2, Vfl, and Vj-2 are terms due to gravity, Coriolis forces, and centrifugal forces; TT is a positive definite stiffness matrix, and Qr is an 2«xl vector of input torques for the rigid degrees of freedom. Since in the present system there are deployable links, stiffness matrix K is not constant, in general. Define H as the inverse of the mass matrix; i.e., H = M] = Hu Hx2 H2X H22 (4.2) which is appropriately partitioned. Also, define the state vector as x = \ qr qf qr qA. Then, Eq. (4.1) can be written in the state-space form x = f(x) + g(x)u, (4.3) 113 where u = Qr and /(*) = qf -Hu(Vr] +Vr2)-H]2(Vfl + VJ2+Kqf) #21 (Vrx +K2)~ H22 (Vfl +Vf2 + Kqf) g(x) = O H 21. (4.4) 4.2 Observer Design In this section, two types of observer are designed. One is a full-order observer, the other is a reduced-order observer. The variables that are measurable are the vectors qr, qr and qf, and the variables that are needed to be observed are the vectors qf. In the following, qf represents the observed value of qf and qf is the estimation error, i.e., qf=qf-qf. A block diagram showing the system with the observer is given in Figure 4-2. Reference Input Controller u Observer Manipulator System with Flexible Links Qr, <lr> qf Figure 4-2 The structure of the system with observer. 114 4.2.1 Full-order observer Define zx = qf and z2=qf • The dynamics of the flexible generalized coordinates can be written as zx =z2 z2 = -H2X (VrX + Vr2 (x)) - H22 (Vrl + Vr2 (x) + Kzx) + H2Xu (4.5) where xT = qTr qTj- qTr q^ as before. Now choose the following structure for the observer dynamics zx=z2+Lx(zx-zx) i2 = -H2l(Vrl + Vr2(xc)) - H22(Vfx + Vf2(xc) + Kzx) + H2Xu + (-H22K + L2)(zx-zx) (4.6) where xTc = qTr qTr z[ z( is the available state vector which may be used for control purposes; and Lx, and L2 are observer gain matrices to be selected. Next, define the estimation error variables as zx =zx - zx and z2 =z2 - z2 (4.7) Then, the observer error dynamics can be obtained by subtracting Eq. (4.6) from (4.5), which yields z = 0-zz + r-z(x,z2) (4.8) wherezT-^z[ z j ] and 0.= -Lx I -L2 0 (4.9a) H 21 dV. dz7 z2+Or(z22) + H 22 dV, dz7 z2+Of(z22) (4.9b) 115 The terms in Eq. (4.9b) have been obtained by using the Taylor series expansions of Vr2 and Vf2 • Specifically, since zf Z 2 - Z 2 , we may write Vr2 (Xc ) = Vr2 (ir A »*1 >*2 ) ~ V/2 (Xc ) = V/2 i9r >4r>Z\>Z2)-dK r2 dz2 dz2 z2+Or(z22) z2+Of(z22) (4.10) It should be noted that since the components of z2 in Vr2 and Vy2 are of second order (centrifugal terms), the Taylor series expansions given in (4.10) can be terminated after the quadratic terms. Moreover, considering a finite region O0 around the desired operating point (x, z2), it follows from Eq. (4.9) that || < k\z\. \iL2 is of the same order of magnitude as //21 and H22, the coefficient k is typically small. This follows by noting that dVr. dz0 l2 and dv Z2 are 0{qfqf). 0Z7 L Moreover, the matrix Oi can be made Hurwitz* by a proper choice of the gains L\, and Li, namely, by selecting L\ and L2 to be any positive definite matrices. Thus, we choose a Lyapunov function candidate V0 = z P^z , where Pi is the solution of the Lyapunov equation ®\Pi+Pi®i=-Qi- (4.11) ' Hurwitz matrices: a matrix A, Ae Rnxn, is said to be Hurwitz if its eigenvalues lie in the open left-half of the complex plane (C~), i.e., ^(A) e C~. 116 It follows that V0 <-(4lin(ez-)-2Aniax(Pj)^)H2 . Hence, provided that Amin(Qi)>2kAmax(Pi) it can be concluded that the error dynamics are locally asymptotically stable. 4.2.2 Reduced-order observer Consider the observation law 4 = - ^ 2 l ( ^ + ^ 2 ( ^ ) ) - ^ 2 2 ( ^ / l + ^ / 2 ( ^ ) + ^ l ) + ^ 2 l " + ^ 2 - 2 2 ) J (4-12) where G is a Hurwitz gain matrix and xc is as defined previously. Equation (4.12) reflects the fact that z2 = Of is not measurable. However, by taking Gz2 to the left-hand side of the equation and defining the auxiliary state variable y = z2 + Gz\, it can be shown that y = Gy-H2X(VrX+Vr2(xc))-H22(Vfx+Vf2(xc) + Kzx) (4.13) Similarly, defining an auxiliary state variable for the system dynamics as y - z2 + Gzx, and subtracting the second equation in the resulting dynamics from Eq. (4.12), it follows that if G is a Hurwitz matrix, the error dynamics are stabilized locally. 4.3 Simulation Study A planar single-module deployable manipulator with a flexible link is taken as an example to study the observer algorithms presented in the previous section. Figure 4-3 shows the structure of the manipulator. In the present study, only the first flexible mode is used and the damping is neglected. The system parameters for the simulation are given below. Masses of the slewing and the deployable link segments: m\ = 1 kg, m2 - 1 kg; 117 Link flexural rigidity: £ 7 = 5 . 5 x 1 0 Nm 2; Lengths of the slewing and the deployable link segment: l\ = h = 1 m. Figure 4-3 Single module deployable manipulator with a flexible link. The initial values of the generalized coordinates are q(P) = [0(0) 1(0) S(0)] = [0 2 0 . 0 2 ] . (4.14) Specifically, we give the tip of the manipulator an initial deflection. Both full- and reduced-order observer strategies are implemented to estimate the rate of change of the flexible coordinate (S). 118 4.3.1 Simulation results with full-order observer Since only one flexible mode is assumed, the observer gains (L\ and L2) in Eq. (4.9) are constant values. Different sets of observer gains are examined. Figure 4-4 shows the observation results with the observer gains L\ - 50 and L2 = 100. The top plot of Figure 4-4 shows the actual response 8 (solid line) and the estimated response 8 (dotted line). The bottom plot of Figure 4-4 gives the corresponding estimation error. As seen from this result, the observer has produced a maximum estimation error magnitude of approximately 0.04. The estimated 8 converges to the actual value in about 2 seconds. To examine the effect of the observer gains, L2 is increased to 300 and L\ is kept at 50. Figure 4-5 shows the simulation results. As seen, the increased gain has produced larger estimation error at the initial stage (about 0.1). But at the same, the time that is taken to converge to the actual value has decreased significantly (s 0.5 second only). Accordingly, a trade off may be struck between the estimation error and the estimation time. In particular, one needs to consider these two conflicting factors, in real-time implementation. 4.3.2 Simulation results of reduced-order observer The reduced-order observer is designed based on Eq. (4.13). As before, the observer gain G is a constant since there is only one flexible generalized coordinate here. Figures 4-6 and 4-7 show the results from the reduced-order observer with the gain values G = -5 and -20. As expected, when the magnitude of the observer gain is increased, the estimation error increases from approximately 0.1 to 0.3. The estimation time decreased from about 1.5 seconds to less than 0.5 second. 119 Observed Desired 8 and 8 0 Observation Error -0.04 Figure 4-4 Results from the full-order observer with gains L\ = 50 and Z-2 = 100. 120 Observed Desired Observed Desired From the simulation results, one can see that the reduced-order observer yields relatively larger estimation error than the full-order observer. The observer algorithm clearly shows that the full-order observer involves higher computation, especially when the manipulator system has more than one module. Hence, for real-time application both types of observers are applicable, but one needs to consider computation effort and estimation error at the same time. In the present study, the observer was implemented on a simple model, for the purpose of illustration and method validation. In a straight forward manner, the technique may be extended to a multi-module manipulator system, and more than one mode shapes may be incorporated for each module. 124 5. SUPERVISORY CONTROL SWITCHING SYSTEM Besides the neural network adaptive controller as presented previously in this thesis, several other controllers have been implemented on the manipulator system. In particular, Feedback Linearization Technique (FLT) based control, Linear Quadratic Gaussian (LQG) control, and Computed-torque Adaptive control have been implemented. The development and wide acceptance of these various control techniques provide testimony to the fact that no single controller can uniformly provide high performance in all applications and throughout the operating range of a given application. In this chapter, first we briefly describe several control techniques implemented by us in robotic systems, and compare them with regard to their strengths and weaknesses under various operating conditions. Based on the advantages and disadvantages, as experienced by us and through available knowledge, a new type of control system termed supervisory control switching system (SCSS) is developed. This control system has a high-level supervisory module, which monitors the performance of the robot and as required, switches on an appropriate controller for the particular operating condition. A knowledge-based system is used for performance monitoring and controller switching. 5.1 Feedback Linearization Technique (FLT) The FLT is an approach that is suitable for controlling a class of nonlinear systems. The procedure was pioneered by Bejczy [58]. It has been further developed and applied by many investigators resulting in a considerable body of literature [59-65]. The basic idea is to use a mathematical model of the nonlinear system to be controlled and establish a transformation that will decouple and linearize the dynamics of the system. The main 125 advantage of the feedback linearization technique over point-wise linearization is that once such a transformation (a nonlinear feedback signal) is.determined, a global linearization is achieved independent of the operating point. For use in the present chapter, a controller based on the FLT is developed as applied to a space platform-based single-module deployable manipulator with flexible joint and flexible link (Figure 5-1). FLT is used to regulate the rigid degrees of freedom of the manipulator; i.e., rotation of the revolute joint (a,), deployment of the link and attitude motion of the system (y/p). Flexible degrees of freedom of the manipulator (fix, flexibility of the revolute joint; ep = ®pSp, platform tip deflection; ex =0^ , tip deflection of module one) are left uncontrolled or controlled through coupling. However, the effectiveness of the controller is assessed using the original fully flexible system. Equations governing the dynamics of a flexible space-based manipulator can be written as M(q,t)q + F(q,q,t) = Q(q,q,t), (5.1) which may be partitioned into the form M„ \ K kr Qr 1 ' * * > = • . . . J. Mfr \ Mff. Hf. Ff. Qf v *—Y—' M k F Q where: M(q,t) is the system mass matrix composed of rigid (Mrr), flexible (M^) and coupling (M rj, Mjr) contributions; qr, q^ 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 126 Local Vertical Figure 5-1 Schematic diagram of a single-module space platform-based deployable manipulator system with flexible joint and flexible link. 127 control inputs. Subscripts r and / refer to contributions associated with rigid and flexible degrees of freedom, respectively. If only the rigid degrees of freedom are controlled, we have Mrrq\ + M^q f+Fr=Qr; Mfrqr + M ffqf +Ff = 0. A suitable choice for Qr would be Qr=M[(qr)d-u] + F, (5.3) with: M = Mrr - MrfM~ffM fr; F = Fr-MrfMffxFf. Here subscript 'a" refers to the desired value of a parameter. One way to form the control signal u is through proportional-derivative (PD) feedback; i.e., u = -Kv[{qr)d-qr]-Kp[(qr)d-qr], (5.5) where Kp and Kv are position gain and velocity gain, respectively. Let e = (qr )d -qr. Then the controlled equations of motion become: 0 = e + Kve + K e; qf=-M/XMfr[(qr)d-u)-Mff XFf. Now Qr is written as Qr = M(qr)d +F + M(Kve + Kpe), (5.7) which can be visualized as a combination of two controllers: the primary controller (Qr,P), and the secondary controller (Qr,s), as given by a, =M(qr)d +F; Q s = M(Kve + K e). (5.8) 128 The function of the primary controller is to offset the 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 5-2. It is not desirable for a robot to exhibit an overshoot, since this could cause impact and possible harmful consequences 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, suitable candidates for the PD gains Kp and Kv are the diagonal matrices 'oo2 0 1 2oox 0 0 cof_ 0 loo. so that the error dynamics (see Eq. (5.6)) would be critically damped, where co, is the undamped natural frequency associated with the error of the ith joint or link. Nonlinear Feedback Compensator Figure 5-2 Schematic diagram showing of FLT control. 129 5.2 Linear Quadratic Gaussian (LQG) Control Linear Quadratic Gaussian (LQG) control is a well established method of controlling a linear system [66-68]. A schematic diagram of LQG control is shown in Figure 5-3. Specifically, it uses a linear quadratic regulator (LQR) as described in the next section for full state feedback control. But since all states may not be measurable in general and since the measured response may contain random noise, it uses a Kalman filter to estimate the unrrieasurable states, and the optimal LQR gain is used to generate the control input. The optimal gain vector is obtained by solving the matrix Riccati equation, as indicated in the next section. It is important to note that the state feedback gain matrix K and the Kalman filter may be designed separately to yield a desired behavior in the closed-loop plant and the observer. This is referred to as the separation principle [66]. Since the manipulator system is nonlinear, the Kalman filter and the optimal LQR gain are designed using a model of the manipulator system linearized about an operating point. Reference Input r(t) Control Input "(') Process Noise w(t) Sensor Noise v(t) <8> + qif) <8> Manipulator Sensor Output Kalman Filter q(0 Estimated „ state C Estimated Output Optimal Gain K Figure 5-3 LQG control of the manipulator system. 130 5.3 Linear Quadratic Regulator (LQR) Control Linear Quadratic Regulator (LQR) is a linear optimal controller, and is designed based on a linearized model of the system. At the end of a large maneuver, the configuration of the robotic system remains nearly constant and then nonlinear effects can be neglected. This makes LQR control appropriate for fine-manipulation conditions of this type. First, the governing equations are linearized about an operating point q0. To that end, the following substitutions are made to the equations of motion of the system: q = q0+Aq; q = q0+Aq; q = qQ+Aq, (5.10) where A(*) denotes a variation around the operating point. Trigonometric functions are expanded in the Taylor series, and the terms of second and higher order in q, q and q are neglected. After some algebra, this leads to M(q0)Aq + K(q0)Aq = uL, (5.11) where M and K are the mass matrix and the stiffness matrix of the linearized system, respectively; and uL is the input as determined by the Linear Quadratic Regulator. Note that both M and K are evaluated at the operating point q0 and thus made locally time-invariant. For the single-module space platform-based deployable manipulator system (Figure 5-1), Aq-[Aep, Aax, Afix, Aex, Alx] , and the operating point is qL0=[0, axo, 0, 0, /10] . Solving for Aq, Aq = -M~lKAq + M~x uL, (5.12) which can be rewritten in the state-space form as 131 'Aq' Af --0 I - M X K 0 'Aq + 0 A T 1 _ A ^~B~^ u L ' (5.13) where xL e 5R10xl; A e 5R 1 0 x l°; and B e 5R10x5. 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,---,A9B]} = 1 0 . (5.14) The control input uL can be written as UL=-KU}RXL> (5-15) where KLqR is the optimal feedback gain matrix. It minimizes the following quadratic cost function (/) , which considers tracking errors and the energy expenditure: J = fQ(xlQLQRxL + ulRLQRuL)dt. (5.16) Here 2LQR a n d ^LQR a r e symmetric weighting matrices which assign relative penalties to state errors and control effort, respectively. The matrix RLQR is required to be positive definite while QLQR must be at least positive semi-definite. It can be shown that The optimal control input uL is given by UL = ~KLQRXL = ~RLQRB PLQRXL, (5-17) where P L Q R is the positive definite solution to the steady-state matrix Ricatti equation which, for infinite time, becomes P L Q R ^ + ^ 7 ' P L Q R - i , L Q R ^ i i R ^ L Q R + e L Q R = 0 - ( 5- 1 8) 132 5.4 Adaptive Control Even in a well-structured industrial facility, a robotic system may face uncertainty regarding the parameters describing its dynamical properties (e.g. mass, moments of inertia, payload, flexibility, etc.). It has been recognized that the performance of conventional approaches of control in high-speed applications is greatly affected by parametric uncertainties. Adaptive control has proven successful in dealing with modeling uncertainties of robot manipulators through on-line tuning of parameters. Conventional applications of adaptive control in robotics rely on the "linear in the parameters (LIP)" assumption first proposed by Craig in 1985 [13]. Thus f(x) = Y(x)<p, (5.19) where f[x) is a nonlinear function of robotic system; Y(x) is a regression matrix of known robot functions; and cp is a vector of unknown parameters (e.g., masses and friction coefficients). This separation of unknown parameters and known time functions is used in the formulation of the adaptive update rule and also in the stability analysis of the tracking error system. In 1987, Slotine and Li [14, 15] used a judicious choice of inertia-related Lyapunov function in the stability analysis, which utilizes physical properties inherent in a mechanical manipulator. The design procedure is briefly described now. Select the inertia-related Lyapunov-like function as V = ^ (sTM(q)s + yTT-xy), (5.20) where cp= <p-<p, <p is a vector used to represent a time-varying estimate of the unknown constant parameters; T is a diagonal positive definite matrix; s-Ae + e, and A is a positive 133 definite diagonal matrix such that A = diag (A\, Xn). The auxiliary signal s(t) may be considered as a filtered tracking error. With this choice of Lyapunov function, it has been proved that the tracking error e and its time derivative e are asymptotically stable and the parameter estimate q> is bounded [14, 15]. The adaptive inertia-related controller can be summarized as follows: Torque Controller r = Y(-)(p +Kve +KvAe , with Kv as a positive gain, (5.21) Update Rule fi = rYT(-)(Ae + e); (5.22) where Y(-)<p = M(q)(qd + Ae) + V(q, q)(qd +Ae) + G(q). (5.23) Note that the symbol 'A'denotes the estimated value of a parameter. A block diagram of the adaptive inertia related controller is shown in Figure 5-4. 134 Desired Values Figure 5-4 Adaptive inertia-related control of a manipulator system. 135 5.5 A Comparison of Control Approaches It is appropriate to compare the advantages and disadvantages of the various control strategies presented in this chapter. The comparison is summarized in Table 5-1. A s expected, no control strategy is perfect under all operating conditions. Each control methodology works best under certain conditions. For example, i f the system is linear, theoretically, one may use any one of the considered control approaches. But considering other issues such as computational effort and the ease of implementation, L Q G emerges as the most suitable candidate. For a nonlinear system, F L T based control yields the best performance i f exact knowledge of the system is known. Otherwise, a more robust controller, such as neural network adaptive control, would be necessary to overcome the uncertainties of the system. It is desirable to investigate whether these different controllers can be implemented together, and a suitable one among them is activated depending on its advantages for generating the best performance under the existing conditions. In this backdrop, we propose a new type of control system, the Supervisory Control Switching System (SCSS), as described next. 136 Table 5-1 A comparison of control approaches. Advantages Disadvantages LQG Control • Linear approach • Easy to implement • Guaranteed stability • Linear model needed - Weighting Matrices need to be tuned • A stability Margin cannot be specified FLT • Global linearization is achieved • Perfect control may be achieved over the entire operating region if model is accurate • High computation effort • Stability not guaranteed • Exact knowledge (model) of the system is necessary Adaptive Control • Applicable in time-variant system • Can successfully deal with model uncertainties • Linear-in-the-parameters assumption • On-line computational effort can be high « Model Based NN Adaptive Control « Linear in the tunable NN weights • Knowledge (explicit model) of the system is not required • Applicable to complex, nonlinear systems • Approximation error exists • Difficult to design • Model is implicit • Explicit analytical procedures are not available 137 5.6 Supervisory Control Switching System (SCSS) A schematic diagram for the supervisory control switching system developed in the present work is shown in Figure 5-5. There are four major modules in the system, namely, the sub-controllers module, the supervisory logic system, the switching compensator, and the manipulator system module. Their functions are outlined below. 1. The "sub-controllers" module contains controllers listed in Table 5-1. It is a bank of well designed and tested controllers. Before integrating them into this module, it is assumed that they individually perform well under normal operating conditions. 2. The supervisory logic system is the decision making module for control switching. It is placed in the loop that monitors the performance of both the system and the controller. A switching decision is sent out after evaluating the chosen states, based on some performance criteria. The decision could be done using heuristic methods, analysis, or even previous experience. In the present approach, the supervisory logic system module uses linguistic rules for decision-making. 3. Switching compensator is activated only when a controller is being switched. For a single controller, compensation is not needed because it is assumed that the controller performs well individually. But when control is switched from, say C\ to Ci, unexpected problems (related to stability, speed, accuracy, etc.) could arise. The switching compensator ensures a smooth transition of system operation from one controller to the next. This compensator is also monitored by the supervisory module. The sub-controllers (Q, C2, ..., Cn) have been developed previously in our work. Details of the supervisory logic module and the switching compensator are given now. 138 <u vi © Pi s s-o c « E u a W) O CO - J CC CC O cc » CO bO C o -4—» '% CO O o co CD co CD o i -bfj o "3 CD O CC IT) i n CD t H bp to ON 5.6.1 Supervisory logic system The decision making process for control switching is schematically represented in Figure 5-6. It consists of stages of system monitoring, information preprocessing, and decision making. This structure can be operationally expressed as: C = P®[(sl®Fl)®(s2®F2)®-®(sH®F„)] (5.24) where Fi - z'th sensory information prefilter; si = z'th sensor; P = supervisory logic system; C = control switching decision; © = a combinational operator; <8> = a transitional operator. A. Information preprocessing Sensory information (s,) may not be needed in its entirety and furthermore, it may be incompatible with the supervisory logic system. An information filter, (or preprocessor, or interpreter), denoted by F„ is necessary to convert a monitored signal into the required form. For example, the encoder signals from the joint motors of a manipulator are essential for servo control, but for switching purposes, the complete signal would not be needed. To ensure smooth transition, it must be preprocessed to obtain such features as offset, level of oscillations, current velocity, and the transient level. In addition to purely quantitative (analytical) preprocessing, some qualitative preprocessing (for example, using heuristics and past experience) might be useful. 140 Controller Switching Action C Information Preprocessing Supervisory Logic System P Sn-l System Monitoring Signals Figure 5-6 The supervisory logic system. 141 Let s be the filtered sensory information. Then the transitional operation of a feedback filter may be expressed as: s=F®s, (5.25) where S = [S\' S2> "' Sn ] > S = [S\> S2> " ' Sm] and m<n, which means that some unnecessary or redundant sensory information can be discarded to make the decision making process more efficient. B. Functional decision making module In Figure 5-6, the supervisory logic system is the decision making module for controller selection, and it is denoted by P. In the present system, module P functions like a linguistic knowledge base. It makes controller switching decision (actions) C using the context as determined by the preprocessed sensory information. Equation (5.24) may be rewritten as C = P®[s]®s2®---®sn]. (5.26) Here, the combinational operator aggregates the preprocessed information, and will take appropriate interpretations depending on the context information that is needed by the decision making module and the inference that is provided by the filter modules. The supervisory logic system may possess learning and self-organizing capabilities as well. For example, if one switching action apparently produces a worsening performance, the corresponding linguistic rule corresponding to that switching action should be deleted or modified. Similarly, the inferences that tend to improve the system performance may be 142 tagged as well and subsequently used to enhance the corresponding rules. Furthermore, new rules may be added to the supervisory logic system on the basis of experience and new knowledge. In this sense, the supervisory logic system need not be complete from the beginning. It is reconfigurable and may improve with time. C. Rule base Let vector E = {E{, E2, En) denote the set of "observable" state variables provided as the input for the rule base. Suitable candidates are tracking error (joint or end-effector), velocity, size of the maneuver, degree of the system nonlinearity, and performance of the controller. An example of typical linguistic rules is given below: If size of maneuver is small, then choose LQR. If size of maneuver is large, then choose non-linear controller. If system knowledge is complete, then choose FLT. If system knowledge is incomplete, then choose neural network adaptive controller. If velocity is too high, do not switch and continue monitoring. In a general form, the rule base is given by IF El THEN Q ELSE IF E2 THEN C 2 ELSE IF ••• E„ THEN C„ Here C, (i = 1, 2, ..., n) denote the available controllers. For implementation purposes, one needs to quantify the "observable" state variables for the rule base. For example, if we define 10° as the threshold for the size of maneuver, then, if the size of the maneuver is larger than 10°, it is treated as a large maneuver, and otherwise, it is a small maneuver. Alternatively, fuzzy membership functions maybe used to define the states of "small" and "large" in a fuzzy-logic decision making framework. 143 5.6.2 Switching compensator As mentioned before, unexpected problems (related to stability, speed, accuracy, and so on) could arise during controller switching. The purpose of the switching compensator is to ensure a smooth transition of operation as a new controller is switched on. Switching compensator takes inferred results from the supervisory logic module in activating an appropriate compensating action (see Figure 5-5). In particular, based on the given decisions of the logic module and the current performance of the system, it decides whether the system needs compensation. The switching compensator may possess its own intelligence. There are different ways of ensuring a smooth transition of operation during controller switching; for example, by using a smooth switching function and a proper switching duration. Also, the switching compensator can make use of electronic hardware such as filters or mechanical devices (e.g., dampers) to eliminate unwanted spikes, ripples, transients, and oscillations. In a rather elaborate system, the switching compensator may perform as a controller itself. This way, it is able to superpose its own control output onto the system, during switching. For example, when a new controller is switched on, the resulting closed-loop system will yield a new frequency response with a corresponding gain margin and a phase margin. In this manner, during switching, the stability margins of the system may vary. Compensation can ensure stability under such conditions. The switching compensator may continue to function in the normal control mode after controller switching, in order to ensure system stability. Design of such a compensation module can be a rather complex task. For simplicity, in this thesis only the switching function and duration are studied. 144 5.6.3 A sample decision making process A sample decision making process is indicated in Figure 5-7. In this switching scheme, only two state variables are used, i.e., size of the maneuver and knowledge of the system. This switching scheme is tested on both the ground- and the space-based models, as presented in the next section. Maneuver Starts Neural Network Adaptive Control Y Nonlinear Controller 1 FLT ^ ^Maneuver Maneuver Ends Figure 5-7 A sample decision making process for controller switching. 145 5.7 Simulation Study of Supervisory Control Switching System To examine the effectiveness. of the SCSS developed in this chapter, the Neural Network Adaptive Control (NNAC), FLT control, and L Q R control are used as the candidate controllers available for switching in the manipulator system. Both the space-based and ground-based deployable manipulator systems are used as test-beds. Note that the systems used in the present study here are fully flexible. Specifically, both manipulator joint and link are considered flexible. 5.7.1 Switching between FLT and LQR for the Space-based Flexible Model 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 108 N.m 2. Manipulator cylindrical with axial to transverse inertia ratio of 0.005; mass = 400 kg; Module: length = 7.5m each at start; flexural rigidity = 5.5 x 105 N.m 2. Revolute mass = 20kg; moment of inertia = 10 kg. m 2; stiffness = 1 .Ox 104 N.m/rad. Joints: Payload: no payload unless specified. Damping: assumed zero for all components. Mode: fundamental only. Damping is purposely not included to obtain conservative results and test the controller under severe conditions. Furthermore, the nature and precise quantification of damping in a space environment is still a subject of considerable debate. To assess the performance of the control system under a rather demanding situation, the platform is initially taken to be along the local horizontal (y/p = -90°) - an unstable 146 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 profile, completed in 0.01 of an orbit (55.50s), so that ax change from zero to 90° and /, from 7.5 m to 15 m. For the purpose of comparison, first the results of FLT control alone are presented in Figure 5-8. Note that FLT controller is designed as described in section 5.1 and it is used to regulate the rigid degrees of freedom only. The maneuver sets the platform librating with a peak amplitude of around 0.04°. After the maneuver is completed, the platform still has limit cycle type oscillations (of amplitude « 0.05°) which appear to persist due to vibration of the flexible joint (j3\), at steady state. The amplitude of joint vibration (J3X) amplitude is approximately 4° and the link tip deflection of the manipulator is approximately 0.1m. Clearly, the unmodeled dynamics of the flexible generalized coordinates affect the performance of the controller. Hence, active vibration control at the end of the large maneuver is necessary to ensure satisfactory tracking performance. Next, the switching scheme from FLT to LQR is demonstrated. The switching rule is as follows: When the manipulator is in a fixed configuration (no deployment, slew, or translation), the platform attitude (y/p) and the manipulator length (lx) are maintained using the FLT strategy. However, the joint rotation (ax), the platform tip vibration (ep), and the link tip deflection (e,) are all controlled using the LQR approach. During large slew and deployment maneuvers, variables y/p, ax and lx are regulated by the FLT controller until the 147 I .C.'s (Flexible d.o.f.): Controller Gains: ep = e\ = 0; y/p: Kp = 0.02; Kv=0.3. L.V. A=o. lx: Kp = 8; Kv=5.66. h \ I .C.'s (Rigid d.o.f.): ^ = - 9 0 ° , a,=0; a1:Kp = 25; Kv= 10. ,-vJv \eP /j = 7.5 m. Desired Values: ^ = - 9 0 ° , ax =90°; /j =15m. sine-on-ramp; 0.01 orbit. Figure 5-8 FLT controlled response of the space manipulator during a simultaneous 90° slew and a 7.5 m deployment maneuver: (a) rigid degrees of freedom and control inputs for module 1. 148 LC.'s (Flexible d.o.fY): Controller Gains: ep=ex = 0; y/p:Kp = 0m; £ v =0 .3 . L.V. A=o. lx: Kp = 8; Xv=5.66. h \ LC.'s (Rieid d.o.f.): y/p =-90°, ax =0; ax:Kp = 25; Kv = 10. D-i 3 \eP /j = 7.5 m. Desired Values: ^ = - 9 0 ° , =90°; lx =15m. sine-on-ramp; 0.01 orbit. 0 0.02 0.04 0 0.02 0.04 Tip Deflection of Manipulator Link 0.02 0.04 Figure 5-8 FLT controlled response of the space manipulator during a simultaneous 90° slew and a 7.5 m deployment maneuver: (b) flexible degrees of freedom. 149 manipulator reaches the vicinity of its target position. Structural vibrations are left uncontrolled. At the end of a large maneuver, the system configuration remains nearly constant and nonlinear effects can be neglected. This allows the Linear Quadratic Regulator to take over the control of ax to actively damp the flexible generalized coordinates which have been disturbed due to the maneuver. Essentially then, the FLT and LQR controllers handle different parts of the maneuver to ensure satisfactory performance in both gross and fine manipulations. Two types of switching are used here. One is instantaneous switching, i.e., LQR is activated immediately after the gross maneuver is completed. In the other scenario, a switching function is used to ensure a transitional period for switching between FLT and LQR. A. Instantaneous switching In this particular case, the LQR controller is only activated after 0.01 of an orbit, i.e., at the end of the maneuver. This means, in the first 0.01 of an orbit, the FLT is used to regulate the gross maneuver where it satisfactorily controls the rigid degrees of freedom. After the 0.01 orbit, the system enters the steady state phase and vibrates around the reference point (see Figure 5-9). At this stage, the LQR is activated, which begins to control vibrations. Note that the FLT controller is still active, which regulates the attitude of the platform and the link length. The LQR is quite effective in suppressing the vibrations of the joints and the platform which in turn helps eliminate the librational limit cycle response. The joint angles and link lengths attain and remain at their commanded values. Furthermore, the required control torques, in the steady state, are virtually negligible. Note the spikes in the control input as switching takes place. 150 I.C.'s (Flexible d.o.f.): Controller Gains: eP = e\ = 0; ^ : X P = 0.02; £ v =0 .3 . L.V. A 1 A=o. lx: Kp = $; Kv= 5.66. h \\ I.C.'s (Rieid d.o.f.): ysp=-90°, a,=0; a,: ¥PA bUj> /, =7.5m. r < 0.01 orbit, Kp = 25; ATV= 10. Desired Values: y/p=-90°, a, =90°; lx = 15m. r>0.01 orbit, LQR. Platform Libration -89.96 y/° -90.00 Momentum Gyro Torque 0.04 0.02 0.04 Deployment of Manipulator Link 0.04 Orbit 1000 0.02 Joint Actuator Torque 0.04 [Nm t = 0.01 orbit L Q R takes over for a, 0.02 Deployment Actuator Force 0.04 0.04 Orbit Figure 5-9 FLT/LQR controlled response of the space manipulator during a simultaneous 90° slew and a 7.5 m deployment maneuver: (a) rigid degrees of freedom and control inputs for module 1. 151 I.C.'s (Flexible d.o.f.): Controller Gains: y/p\Kp = 0m; Kv=0.3. L.V. A=o. lx: Xp = 8; Kv=5.66. h / I.C.'s (Rieid d.o.f.): y/p=-90°, a,=0; /,=7.5m. t < 0.01 orbit, £ p = 25; £ v = 10. Desired Values: •7 p =-90°, a, =90°; /, =15m. r>0.01 orbit, LQR. Platform Vibration P 0.0 0.00 0.02 0.04 0.06 0.00 Joint Vibration 0.02 0.04 0.06 Tip Deflection of Manipulator Link 0.00 0.02 0.04 Orbit 0.06 Figure 5-9 FLT/LQR controlled response of the space manipulator during a simultaneous 90° slew and a 7.5 m deployment maneuver : (b) flexible degrees of freedom. 152 B. Switching using a smooth transition function As seen in Figure 5-9, spikes exist in the control input at the instant of switching. This is because of the sudden change of the control action due to controller switching. In real-time applications, this may cause undesirable stresses on the drive hardware such as amplifiers. For example, the spike will cause a sudden change of current drawn from the amplifier, which may exceed its capacity. To eliminate this potential problem, a smooth transition function is used during switching. Let the toque output of the FLT and LQR controllers be 7FLT and TLQR, respectively. A transitional period is used to change the toque input into the system at the 0.01 orbit value when the large maneuver is completed and the control action is switched from FLT to LQR. Mathematically, the actual toque applied a the joint may be expressed as T=TFLTf(t) + TLQRf2(t), (5-27) where f(t) and/2(0 are the transition functions determining the level of torque from FLT and LQR controllers that are contributed to the overall drive torque. The transition functions, f(t) and f2(t), should satisfy the following end conditions /i(0) = l,/,Od7) = 0; f2(0) = 0,f2(AT) = l, where z i r i s the transition period. Here a sine-on-ramp profile is used for both f(t) and f2(t), while AT is chosen as 30 s ( « 0.005 orbit) to ensure a good performance. The results are shown in figures 5-10 (a) and 5-10 (b). It is noted that with the introduction of the switching function, the spikes caused by instantaneous switching have been suppressed. A smooth response is achieved both for the system variables and control inputs. At the same time, the vibrations of the joint and the tip converge to zero. This ensures good tracking performance. 153 I.C.'s (Flexible d.o.fV): Controller Gains: ep =^ = 0; y/p:Kp = 0.02; Kv=0.3. L.V. *7p I.C.'s (Rigid d.o.fV): /,: 7^ = 8; Kv=5.66. ^ = - 9 0 ° , a i = 0 ; a{: /j = 7.5 m. r < 0.01 orbit, ^ = 25; ^ v = 10. Desired Values: ^ = - 9 0 ° , a, =90°; /, = 15 m. r > 0.01 orbit, LQR. Platform Libration 0.04 -0.04 0.01 0.02 Joint Rotor Motion 0.03 Transition Region (30s) 0 0.01 0.02 0.03 Deployment of Manipulator Link 0.01 0.02 Orbit 0.03 1000 -1000 F< 0 Momentum Gyro Torque 0.01 0.02 Joint Actuator Torque 0.01 0.02 Deployment Actuator Force 0.01 0.02 Orbit 0.03 0.03 0.03 Figure 5-10 FLT/LQR controlled response of the space manipulator during a simultaneous 90° slew and a 7.5 m deployment maneuver: (a) rigid degrees of freedom and control inputs for module 1. 154 I.C.'s (Flexible d.o.f.): Controller Gains: y/p :KP = 0.02; £ v =0 .3 . L.V. ft 1 /]: = 8; Kv=5.66. h \\ I.C.'s (Rigid d.o.f.): ^ = - 9 0 ° , a,=0; ax: VPA /[ = 7.5 m. r<0.01 orbit, Kp = 25; Kv= 10. Desired Values: ^ = - 9 0 ° , a, =90°; /, =15m. r>0.01 orbit, LQR. Platform Vibration ,„ Joint Vibration 0.02 0.03 0 0.01 0.02 0.03 Orbit Figure 5-10 FLT/LQR controlled response of the space manipulator system during a simultaneous 90° slew and a 7.5 m deployment maneuver: (b) flexible degrees of freedom. 155 5.7.2 Switching between NNAC and LQR for the Ground-based Flexible Model The switching scheme developed in this chapter is now applied to a single-module flexible deployable manipulator system which is described in Figure 3-13 of Chapter 3. The neural network adaptive controller is as designed in Chapter 3, section 3.4. For comparison, the same system parameters, initial conditions, desired values and controller gains as in section 3.4 are used in the present simulation. Specifically, we use the following. System parameters: ms = m(i =2 kg 4 = /</ = lm; J= 1 kg. m ; K= 500 N.m/rad; EI= 5.5X101 kg. m3/s2; Initial conditions: q(0) = [a(0) 0(0) S(0) 1(0) ] = [Odeg Odeg 0.01 lm], Desired values: qd (td ) = [90 deg 90 deg 0 2mf where td = 4 s. Maneuver profile: sine-on-ramp profile. NNAC controller gain settings: The adaptation algorithms are activated with rM = diag[2.0], rv = diag[2.0] and rG = diag[5.0]. The gains for the slow controller are chosen as Kv= diag[10, 50]. For the fast controller Kd = 2 and Kp = 0.5. According to the switching scheme, NNAC controls the flexible system in the first 4 seconds (the maneuver time) and activates the LQR control starting from t = 4 s. Hence the operating point used to linearize the system is at the desired values, and the LQR gains are computed using the linearized model. As before, two types of switching are used here. One is instantaneous switching, where LQR is activated immediately after the gross maneuver is 156 completed. In the other scenario, a switching function is used to ensure a transitional period for the torque change between NNAC and LQR. A. Instantaneous switching In this case, the LQR controller completely takes over at the end of the maneuver (t = 4s) and NNAC is turned off. Figure 5-11(a) presents the time history of the system generalized coordinates and the control inputs. Note that, in the plot of the control inputs, the first four seconds of the curve is contributed by the NNAC controller and the rest is from the LQR controller. In Chapter 3, Figure 3-16 shows the results with NNAC control alone. It is seen that the vibration of the link tip continues even after 10 seconds of simulation although it is gradually approaching to zero. After switching to active LQR control, vibrations are suppressed within 1 second. Hence the LQR controller is quite effective in suppressing the vibrations of the joint and the link. The joint angles and link lengths attain and remain at their commanded values. A spike occurs in the plot of control input due to instantaneous switching from NNAC to LQR. This is especially obvious from the force input for the deployable link. To eliminate the spike, a smooth transition function is introduced next. B. Switching using a smooth transition function Let the drive torque due to NNAC and LQR control be denoted by TNNAC and TLQR, respectively. Using a sine-on-ramp type of switching function, the overall drive torque into the system during switching is given by 1 T ~ r L Q R A T f AT f *-* ^ t-tA sin In 2. " / 1 _ AT1 V + T N N A C 2TT t-t, V AT j) f i - - L AT ( t-t > In (5.28) d V AT J where AT is the transition period and tj is the maneuver time. 157 Deployable link motion Slewing motion 0 2 4 6 8 10 0 2 4 6 8 10 Drive force for deployable link Drive torque for acturator rotor 0 2 4 6 8 10 0 2 4 6 8 10 Time [s] Time [s] Figure 5-11 Instantaneous control switching from NNAC to LQR: (a) System responses and the control inputs. 158 Tracking error of the slew motion ei 0.03 0.02 0.01 -0.01 Tracking error of the deployable motion Joint vibration 9 - a [deg] 8 10 Time [s] Figure 5-11 Instantaneous control switching from N N A C to LQR: (b) Tracking errors (ee, ei) and joint vibration (9 - cc). 159 In the simulation study, we first use AT - 0.5s. Results are shown in figures 5-12(a) and 5-12(b). The control inputs show that the spike no longer exists, under smooth switching. Also, because the switching time is small, no significant deviation from the ideal case is observed in the response of the system generalized coordinates. Next we increase the switching time to AT= 2s. Simulation results are given in figures 5-13(a) and (b). Although the transition from NNAC to LQR becomes smoother, the time it takes to suppress the vibration has increased. For example, when AT = 0.5s, the joint vibration (6 - a) goes to zero in approximately 6s. When AT = 2s, however the joint vibration (0 - a) goes to zero in approximately 7s. 160 Deployable link motion Slewing motion i i i i , i , i i 1 , 1 1 . 1 1 0 2 4 6 8 10 0 2 4 6 8 10 Drive force for deployable link Drive torque for acturator rotor Time [s] Time [s] Figure 5-12 Control switching from NNAC to LQR with a sine-on-ramp transition function (AT= 0.5s): (a) System responses and the control inputs. 161 Tracking error of the slew motion 0.5 9-a [deg] -0.5 • V -0 2 4 6 8 10 Time [s] Figure 5-12 Control switching from NNAC to LQR with a sine-on-ramp transition function (AT = 0.5s): (b) Tracking errors (e$, ei) and joint vibration (9 - a). 162 Deployable link motion Slewing motion 0 2 4 6 8 10 0 2 4 6 8 10 Drive force for deployable link Drive torque for acturator rotor Time [s] Time [s] Figure 5-13 Control switching from NNAC to LQR with a sine-on-ramp transition function (AT= 2s): (a) System responses and the control inputs. 163 Tracking error of the slew motion eg 0 2 4 6 8 10 Tracking error of the deployable motion 0 2 4 6 8 10 Joint vibration 1 9-a [deg] 0 2 4 6 8 10 Time [s] Figure 5-13 Control switching from NNAC to LQR with a sine-on-ramp transition function (AT = 2s): (b) Tracking errors (eg, ei) and joint vibration (9 - a). 164 6. CONCLUDING REM ARKS The thesis presented comprehensive research in dynamic modeling and control of robotic manipulators. The class of manipulators treated in the thesis is termed variable geometry deployable manipulators. The main feature of the manipulators is the modular chain-like topology with each module possessing a deployable link (i.e., a prismatic joint) and a slewing (i.e., revolute) joint. This class of manipulators was found to have important advantages over those with revolute joints only. The dynamic models developed in the thesis include common nonlinearities and coupling in the robots. Actuator dynamics are included as well. Two application types of manipulator systems were useful in studying earth-based applications and has a fixed based. The space-based manipulator system is mounted on a space platform and incorporates the dynamic interactions in space. The dynamic models have been coded to generate computer models. They were used in the development of appropriate controllers and control systems for this class of manipulators. In the course of this research several important contributions have been made. This concluding chapter summarizes the significant contributions of the research. The main conclusions drawn are indicated. Possible avenues of future research, particularly in the area of supervisory control switching system (SCSS) as applied to multi-module robots, are mentioned. 6.1 Contributions The research presented in this thesis covered dynamic modeling and control of a novel class of manipulators. Several important contributions have been made in the course of the investigation. The main contributions are summarized below: 165 Complete dynamic models of a novel design of flexible space-based and ground-based deployable manipulators were developed. Each manipulator consists of multiple modules consisting of slewing (revolute) and deployable (prismatic) joints. The nonlinear model also accounts for joint flexibility and link flexibility. The space platform-based manipulator system involves orbital, librational and vibrational motions. The analytical models provide a basis for designing, controlling, and evaluation of such multi-module deployable manipulator systems, which represents a contribution of importance in the field of space robotics. A computer code was developed based on the analytical models, to simulate the dynamics of a deployable multi-module manipulator with flexible joints and flexible links. Simulation studies using the computer model are important in the design, development, and control of this class of manipulators. A neural network based adaptive control was developed. It was tested on manipulator systems with various structures, joint and link flexibilities, nonlinearities, and model uncertainties. The singular perturbation approach was used for control in the presence of flexibilities. The neural-network adaptive control promises to be efficient for the present class of manipulator systems with model uncertainties and system variabilities. A nonlinear state observer was developed to estimate the rates of changes of flexible modes. The observer is effective in the control of flexible manipulators when the state measurements are incomplete. 166 (v) Several other controllers have been developed and implemented on the manipulator system. They are a controller base on the Feedback Linearization Technique (FLT), Linear Quadratic Regulator (LQR), and Computed-torque Adaptive control. A single controller of this type is not uniformly effective for all tasks and under all conditions of operation. A comparative evaluation was carried out to establish conditions under which each controller is most effective. (vi) Based on the evaluation of the behavior and performance of the individual control techniques, a new type of control system termed the Supervisory Control Switching System (SCSS), was developed. The SCSS monitors the system performance and determines the best choice of controller under particular conditions of operation. It then switches on the selected controller. The switching method provides smooth transition between controllers in addition to instantaneous switching. The effectiveness of the SCSS and switching schemes was investigated. The application of SCSS on complex multi-module manipulators system represents an important practical contribution. 6.2 Conclusions The work reported in this thesis was aimed at the development of effective control methodologies for a useful class of manipulators in both ground-based and space-based applications. Development of suitable dynamic models and corresponding computer models was necessary for the investigation. From the studies carried out using these models several key conclusions can be made. 167 (a) Deployable multi-module manipulators, each module of which consisting of a deployable link and a revolute joint, have clear advantages over manipulators with purely revolute joints. They include simpler kinematics and dynamics, fewer singular configurations, and easier obstacle avoidance. (b) Conventional control techniques may not be uniformly effective in the control complex manipulator systems with nonlinearities, coupling, and model uncertainties. Novel approaches that integrate neural networks with adaptive control may find significant applicability. (c) With the learning ability and the universal approximation property, the neural network is able to emulate the nonlinear elements of the system matrices. (d) With neural network adaptive control, a robotic system can yield good tracking performance, particularly in the absence of flexibilities. For systems with flexible joints and links, some oscillations may not decay as the maneuver is completed. Active linear techniques such as LQR are effective in suppressing these oscillations. (e) An observer can provide accurate estimates of the rates of changes of flexible modes. Consequently, it can result in accurate control under conditions of incomplete state measurement. (f) For the space-based flexible manipulator system, significant coupling exists between the oscillations in the platform, links and joints, and system libration. In general, slewing and deployment maneuvers have a significant effect on the response of the flexible degrees of freedom. (g) The control strategy based on the FLT is quite effective in regulating the rigid-body motions of manipulator links and the attitude motion of the space platform. 168 (h) The supervisory control switching system (SCSS) provides accurate control with multiple control capabilities and smooth transition between controllers, in carrying out complex tasks and maneuvers with complex robotic systems. 6.3 Recommendations for Further Work Considering the diversity of research areas associated with the field of robotics, the present thesis should be viewed as just one step in the analysis and control of a particular class of space-based and ground-based deployable manipulators. There are several avenues which remain unexplored or demand further attention. In particular, the SCSS should be investigated further. Some possibilities are listed below: (i) Switching logic based on human experience and expertise may be implemented using a fuzzy logic paradigm. This itself is a good area of research. (ii) Stability analysis of SCSS. So far, the associated literature has only considered the stability of control switching for linear systems. That too is not a trivial task. In the present case of nonlinear control, new approaches will be necessary for stability analysis. For example, the development of a suitable Lyapunov function may be necessary. (iii) Trajectory tracking with and without SCSS. Execution of more complex maneuvers like obstacle avoidance and satellite capturing will be of practical importance. (iv) Comparative study of various optimal, adaptive, intelligent and hierarchical control strategies for the rigid and flexible dynamics of multi-module robotic systems will be appropriate switching logic may be developed accordingly. 169 Implementation of the SCSS in real-time experimentation using a laboratory manipulator system will demonstrate the effectiveness of the developed techniques. 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