DYNAMICS AND CONTROL OF A NOVEL MANIPULATOR WITH DEPLOYABLE AND SLEWING LINKS Jian Zhang B. Eng., University of Electronic Science and Technology of China, China, 1982 M. Eng., Guizhou University of Technology, China, 1987 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENT OF THE DEGREE OF Master of Applied Science in The Faculty of Graduate Studies Department of Mechanical Engineering We accept this thesis as conforming to the required standard The University of British Columbia April 2002 © Jian Zhang, 2002 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that the permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is under stood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Jian Zhang The University of British Columbia Department of Mechanical Engineering 2324 Main Mall Vancouver, B.C, Canada V6T 1Z4 A B S T R A C T The thesis investigates the planar dynamics and control of a novel, flexible, multi-module manipulator with slewing and deployable links, which may be used in space - as well as ground - based operations. The system is composed of a flexible orbiting platform supporting two modules connected in a chain topology. Each module consists of two links: one free to slew while the other permitted to deploy and retrieve. There are three major aspects to the study. To begin with, a detailed dynamical response study is undertaken which assesses the influence of initial conditions, system parameters, and manipulator maneuvers on the system response. Results suggest that, under critical combinations of parameters, the response may not conform to the acceptable limit. This points to a need for active control. Next, the study focuses on the behavior of the system using two different control methodologies: (i) the nonlinear Feedback Linearization Technique (FLT) applied to rigid degrees of freedom with flexible generalized coordinates indirectly regulated through coupling; (ii) an integration of the FLT and Linear Quadratic Regulator (LQR) to achieve active control of both rigid and flexible degrees of freedom. Finally, the thesis presents the development of an intelligent hierarchical system for the vibration control of the manipulator. The emphasis is on the use of knowledge-based tuning of the low-level direct controller so as to improve the performance of the control system. For this purpose, first a fuzzy inference system (FIS) is developed. The FIS is then combined with a conventional modal controller to construct a hierarchical control system. Specifically, a knowledge-based fuzzy system is used to tune the parameters of the modal controller. The effectiveness of the hierarchical control system is assessed through simulation studies. Results show that the knowledge-based hierarchical control system is quite effective i i in suppressing vibrations due to a wide variety of disturbances, and performance of the modal controller can be significantly improved through knowledge-based tuning. Such a comprehensive investigation involving dynamics and controlled performance of a robotic system should prove useful in the design of this new class of manipulators. The study lays a sound foundation for further exploration of this class of novel manipulators. iii T A B L E OF CONTENTS A B S T R A C T i i T A B L E OF CONTENTS iv L I S T OF S Y M B O L S vi L I S T OF FIGURES xi L I S T OF T A B L E S xv A C K N O W L E D G E M E N T xvi 1. I N T R O D U C T I O N 1 1.1 Preliminary Remarks . . . . . . . 1 1.2 A Brief Review of the Relevant Literature . . . . 6 1.2.1 Characteristics of space-based manipulators . . . 7 1.2.2 Dynamics and control of space-based manipulators . . 8 1.3 Scope of the Investigation . . . . . . 12 2. D Y N A M I C A L I N V E S T I G A T I O N OF A SPACE-BASED M U L T I - M O D U L E M A N I P U L A T O R S Y S T E M 15 2.1 Background to the system model . . . . . 15 2.2 Simulation Methodology . . . . . . 17 2.3 Simulation Considerations . . . . . . 20 2.4 System Response . . . . . . . 27 2.4.1 Effect of manipulator location and orientation . . 27 2.4.2 System response to initial disturbances . . . 34 2.4.3 System response in presence of manipulator maneuvers . . 40 iv 3. SYSTEM CONTROL USING FLT/LQR 54 3.1 Control Methodologies . . . . . . 56 3.1.1 FLT control 58 3.1.2 FLT/LQR control 60 3.2 Simulation Results and Discussion: Commanded Maneuvers . 64 3.2.1 FLT control 64 3.2.2 FLT/LQR control 69 4. TUNED MODAL CONTROL 85 4.1 Control System Development . . . . . 85 4.1.1 Eigenvalue assignment. . . . . . 85 4.1.2 Hierarchical structure . . . . . . 87 4.1.3 Performance specification, evaluation, and classification. . 90 4.1.4 Fuzzy tuner layer . . . . . . 92 4.1.5 Construction of fuzzy inference system . . . 93 4.2 Ground-Based Simulation . . . . . . 96 4.2.1 Modeling of a ground-based manipulator system . . 96 4.2.2 Control system and simulation results . . . 98 4.3 Concluding Remarks . . . . . . . I l l 5. CONCLUDING REMARKS 114 5.1 Contributions . . . . . . . . 114 5.2 Concluding Remarks. . . . . . . . 115 5.3 Recommendations for Future Work . . . . . 117 REFERENCES 118 V L I S T O F S Y M B O L S A, B state-space representation of the flexible subsystem, Eq. (3.15) ACCP acceptable G.M. center of mass d position of the manipulator base from the center of the platform DDAMPt fuzzy variable, change in modal damping ratio DFREQi fuzzy variable, change in modal frequency e orbital eccentricity ex, e2 tip vibrations of modules one and two, respectively ep platform tip vibration EId, EIS flexural rigidity of deployable and slewing links, respectively EIP flexural rigidity of the platform ERR vectors of indices of deviation (Eqs. 4.9, 4.10, 4.11) F vector containing the terms associated with the centrifugal, Coriolis, gravitational, elastic, and internal dissipative forces F,, F 2 deployment/retrieval forces at the prismatic joints one and two, respectively HIUN highly unsatisfactory IjZ moment of inertia of the revolute joint J quadratic cost function which considers tracking errors and energy expenditure, Eq. (3.18) Kj stiffness of the revolute joint ^ L Q R optimal gain matrix, Eq. (3.19) vi Kp, Kv diagonal control matrices containing the proportional and derivative gains, respectively ld, ls length of deployable and slewing links, respectively lx, l2 lengths of the manipulator modules one and two, respectively lp length of the platform L.H. , L . V . local horizontal and local vertical, respectively INSP in specification ma, ms mass of deployable and slewing links, respectively mj mass of the revolute joint mp mass of the platform M system mass matrix M , K mass and stiffness matrices for the linearized system, respectively; Eq. (3.12) M , K mass and stiffness matrices, respectively, corresponding to the elastic subsystem; Eq. (3.13) Mrr, M ff, rigid and flexible contributions to the system mass matrix M, respectively Mrf , Mfr coupled contributions to the system mass matrix M N number of bodies (i.e. platform and manipulator units) in the system NDIM needs improvement NL negative large NM negative moderate NS negative small O(N) order-TV OFSd desired offset at steady-state (Eq. 4.11) vii OFSi index of deviation of offset at steady-state OFSr actual offset at steady-state OVSd desired overshoot (Eq. 4.10) OVSi index of deviation of overshoot OVSr actual overshoot OVSP over specification PL positive large P L Q R solution to the matrix Ricatti equation, Eq. (3.20) PM positive moderate PS positive small q set of generalized coordinates q0 operation point used to linearize the governing nonlinear equation qr, qj 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 GLQR ' ^LQR symmetric weighting matrices which assign relative penalties to state errors and control effort, respectively; Eq. (3.18) RSTd desired rise time (Eq. 4.9) RSTt index of deviation of rise time viii RSTr actual rise time t time T total kinetic energy of the system Tpn 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 vector containing the FLT control inputs, Eq. (3.5) uL input determined from the Linear Quadratic Regulator, Eq. (3.17) 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 X L state vector for the flexible subsystem ZR zero Greek Symbols a,, a2 rigid body rotations during slew maneuvers of modules one and two, respectively /?,, f32 contributions due to flexibility of revolute joints at modules one and two, respectively; Yi rotation of the frame F , , attached to the module i , with respect to the frame F M ' conjugate pair of eigenvalues of linear system Eq. (4.5) IX AT time required for maneuver, Eq. (2.4) r time from start of maneuver rp librational period of the platform Q)i modal frequency, Eq. (4.5) a>¥ frequency of platform librational motion cop platform's bending natural frequency coJl first revolute joint's torsional natural frequency comX first module's bending natural frequency C0j2 second revolute joint's torsional natural frequency 0)m2 second module's bending natural frequency B,i elastic deformation of the i t h -1 body in the transverse direction y/p platform's pitch angle modal damping ratio (Eq. 4.5) 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. X LIST OF F I G U R E S Figure 1-1 Artist view of the International Space Station with its Mobile 2 Servicing System (MSS) as prepared by the Canadian Space Agency. Figure 1-2 A l l the space-based manipulators have used, so far, revolute joints 4 thus permitting only slewing motion of links. Figure 1-3 Variable geometry manipulator showing: (a) single module with a 5 pair of slewing and deployable links; (b) Several modules connected to form a snake-like geometry. Figure 1-4 Variable geometry manipulator showing obstacle avoidance 6 character. Figure 1-5 Schematic diagrams of space structure models: (a) Lips [33], rigid 10 spacecraft with deployable beam-type members; (b) Ibrahim [34], rigid spacecraft with deployable beam- and plate-type members; (c) Shen [35], rigid spacecraft with slewing-deployable appendages; (d) Marom [36], flexible platform supporting one rigid slewing-deployable manipulator module and a rigid payload at the deployable link end. The revolute joint is flexible. Figure 2-1 Schematic diagram of the mobile flexible variable geometry 16 manipulator, based on an elastic space platform. Figure 2-2 Coordinates describing flexibility of revolute joints. 18 Figure 2-3 Normalized time histories of the sinusoidal maneuvering profile 21 showing displacement, velocity, and acceleration. Figure 2-4 Schematic diagram of a two-module, flexible, variable geometry 23 manipulator, based on an elastic space platform, considered for study. Figure 2-5 Schematic diagram showing important parameters appearing in 26 the response study. Figure 2-6 System equilibrium as affected by the manipulator's geometry and 28 location on the platform. The diagram shows a case where the links are locked at ax =90°, cc2 =0 and lx =l2 -1.5 m. The equilibrium configuration deviates from the local vertical by about 0.17°. Figure 2-7 Direction of rotation of the platform, initially in the local horizontal equilibrium position, due to a change in the manipulator orientation. 30 XI It is governed by the location of the center of mass in different quadrants. Figure 2-8 System response with manipulator located at: (a) center of the 31 platform (d = 0); (b) tip of the platform (d = 60 m). Figure 2-9 System response when the platform is initially around the local 33 horizontal. Figure 2-10 Response of the system to an initial disturbance of 0.5m tip 36 displacement of the platform: Z7 = l2 = 7.5 m. Figure 2-11 Response of the system to an initial disturbance of 0.5m tip 3g displacement of the platform: l} = l2= 15 m. Figure 2-12 System response to the manipulator's tip displacement of 0.2 m. 39 Figure 2-13 System response showing the effect of number of modes to represent 42 flexibility. Note, the fundamental mode is quite adequate to capture physics of the system dynamics. Figure 2-14 A schematic diagram showing two maneuvers for capture of payload 43 brought by a hovering space shuttle Figure 2-15 System response in presence of maneuver: effect of joint flexibility. 45 Figure 2-16 System response in presence of maneuver: effect of platform 4g flexibility. Figure 2-17 System response in presence of maneuver: effect of link flexibility. 4g Figure 2-18 System response in presence of maneuver: effect of maneuver 50 speed. Figure 2-19 Effect of maneuver speed: maximum stretch configuration. 51 Figure 2-20 System response, even under a set of conservative values for payload 53 (2000 kg), joint stiffness (5xl0 4 Nm/rad) and maneuver speed ( 0.05 orbit), is unacceptable. This suggests a need for control. Figure 3-1 A two-module manipulator system showing frequently used 55 symbols. Figure 3-2 Schematic diagram of the manipulator system showing the location 57 of the control actuators. The two torques Tp//2, opposing each other, control the platform's vibration by regulating the local slope. xii Figure 3-3 FLT-based control scheme showing inner and outer feedback loops. 61 Figure 3-4 Block diagram illustrating the combined FLT/LQR approach applied 65 to the manipulator system. Figure 3-5 Response of the system with rigid degrees of freedom controlled by the FLT: (a) rigid degrees of freedom and control inputs for platform and 66 module 1. (b) rigid degrees of freedom and control inputs for module 2. 67 (c) flexible degrees of freedom. 68 Figure 3-6 FLT/LQR controlled response of the system: (a) rigid degrees of freedom and control inputs for platform and 70 module 1. (b) rigid degrees of freedom and control inputs for module 2. 71 (c) flexible degrees of freedom. 72 Figure 3-7 FLT/LQR controlled response of the system showing the effect of payload: (a) rigid degrees of freedom and control inputs for platform and 74 module 1. (b) rigid degrees of freedom and control inputs for module 2. 75 (c) flexible degrees of freedom. 76 Figure 3-8 Effect of speed of maneuver on the FLT/LQR controlled response: (a) rigid degrees of freedom and control inputs for platform and 78 module 1. (b) rigid degrees of freedom and control inputs for module 2. 79 (c) flexible degrees of freedom. 80 Figure 3-9 FLT/LQR controlled response as affected by the revolute joint stiffness: (a) rigid degrees of freedom and control inputs for platform and 82 module 1. (b) rigid degrees of freedom and control inputs for module 2. 83 (c) flexible generalized coordinates. 84 Figure 4-1 Schematic representation of the three-level controller. 89 Figure 4-2 Rulebase for the control parameter tuning. 95 Figure 4-3 Membership functions for the fuzzy performance attributes. 97 Figure 4-4 Membership functions for the fuzzy tuning actions. 97 xiii Figure 4-5 Configuration of the single-module manipulator with revolute and 98 prismatic joints. Figure 4-6 The flexible revolute joint model. Here J represents the rotor's mass 99 moment of inertia. Figure 4-7 Parameters for the ground-based simulation. 101 Figure 4-8 System response to an initial displacement at the revolute joint: 102 (a) controlled by LQR; (b) controlled by hierarchical controller. Figure 4-9 System response at each tuning action. 104 Figure 4-10 Evolution of the system pole locations during tuning: 105 (a) pole locations for the first mode; (b) pole locations for the second mode. Figure 4-11 System response to an initial velocity j3 (2° per second) applied at the revolute joint: (a) controlled by LQR; (b) controlled by hierarchical controller. Figure 4-12 System response at each tuning action. Figure 4-13 Evolution of the system pole locations during tuning: (a) pole locations for the first mode; (b) pole locations for the second mode. Figure 4-14 System response while going through a maneuver: (a) controlled by the FLT/LQR; (b) controlled by the FLT/Tuned 1 1 0 Modal Control. Figure 4-15 System response at each tuning action when FLT/Tuned Modal Control is applied. Figure 4-16 Evolution of the system pole locations during tuning: (a) pole locations for the first mode; (b) pole locations for the second mode. 106 107 108 112 113 xiv LIST OF T A B L E S Table 2-1 Important factors affecting the system performance. 22 Table 2-2 Fundamental natural frequencies of the components in absence of 35 payload: ly =l2 = 7.5m . Table 2-3 Fundamental natural frequencies of the components in absence of 25 payload: /, = l2 = 15m . Table 4-1 Mapping from the index of deviation to a discrete performance index. g\ Table 4-2 Heuristics of modal control tuning. 94 Table 4-3 Fuzzy labels of performance indices 94 Table 4-4 Tuning fuzzy sets and representative numerical values 96 XV ACKNOWLEDGEMENT I wish to thank my supervisor Prof. Vinod J. Modi for his guidance and support throughout my graduate studies. His invaluable insight on a wide variety of subjects greatly enriched my knowledge and took me to a new level. I wish to thank my supervisor Prof. Clarence W. de Silva for his support and advice towards the completion of my research project. Thanks are also due to my past and present colleagues and friends from all around the world. They included: Mr. Yang Cao, Mr. Kenneth Wong, Dr. Yuan Chen, Mr. Jooyeol Choi, Mr. Mark Chu, Dr. Ayhan Akinturk, Mr. Jean-Francois Goulet, Mr. Vijay Deshpande, Mr. Mathieu Caron, Mr. Vincent Den Hertog, Dr. Shinji Hokamoto, Dr. Sandeep Munshi, Mr. Ben Triplett, and Dr. Seiya Ueno. 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 was provided through a Strategic Research Grant 5-82268 from the Natural Sciences and Engineering Research Council of Canada (NSERC). It was jointly held by Dr. de Silva and Dr. Modi at the University of British Columbia, and Dr. A . K . Misra at McGi l l University. The project was also supported by NSERC Operating grants held by Dr. Modi and Dr. de Silva. xvi 1. I N T R O D U C T I O N 1.1. Preliminary Remarks Since introduced in the 1960's, robotic systems have become one of the key technologies playing a significant role in the development of space projects [1]. In the late 60s, the unmanned Surveyor lunar mission used a rudimentary manipulator arm to dig and collect soil samples. The versatility of the space robots was demonstrated during the Surveyor 7 mission where the manipulator was employed to jab open an instrument that had failed to deploy automatically. In 1970, and again in 1973, the Soviet Lunakhod rovers surveyed large areas of the moon and used a deployable arm to lower an instrumentation package to the surface. The Viking landers, in 1976, used robotic manipulators to collect and process Martian soil samples. The Canadian contribution to space robotics has been through the now famous Canadarm, introduced in 1981. It has played diverse, significant roles in almost all NASA's Space Shuttle missions: platform to support astronauts; position experiment modules; satellite launch and retrieval; loosen a jammed solar panel; even knocked-off a block of ice from a clogged waste water vent [2]. Perhaps its most dramatic success came in 1993 when it successfully retrieved the malfunctioning Hubble Space Telescope, placed it in the cargo bay for repair and relaunched it. In December 1998, it assisted in the integration of the U.S. 'Unity' module with the Russian control module called 'Zarya' (Sunrise), launched a few weeks earlier, thus initiating construction of the International Space Station. For the Space Station, which is likely to be operational in near future (around 2006), the Canadian contribution is through an extension of the Canadarm in the form of Mobile Servicing System (MSS, Figure 1-1). It consists of the Space Station Remote Manipulator 1 Figure 1-1 Artist view of the International Space Station with its Mobile Servicing System (MSS) as prepared by the Canadian Space Agency. 2 System (SSRMS) and Special Purpose Dexterous Manipulator (SPDM). 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; handle cargo; as well as assemble, release, and retrieve satellites. This evolved version is referred to as C A N A D A R M 2. A number of other space robots have been proposed and some are under development. The American Extravehicular Activity Helper/Retriever (EVAHR) and Ranger Telerobotic Flight Experiment, as well as the Japanese ETS-VII, are examples of free-flying telerobotic systems which will be used for satellite inspection, servicing and retrieval [6,7]. Thus manipulators are serving as useful tools in the space exploration. A l l indications suggest the trend to accentuate with future missions becoming more dependent on robotic systems. As the Space Station will operate in the harsh environment at an altitude of about 400 - 500 km, it is desirable to minimize extravehicular activity by astronauts. Robotics is identified as one of the key technologies to reach that goal. It is important to point out that all the space-based robotic devices mentioned above use revolute joints, i.e. links are free to undergo slewing motion (Figure 1-2), as in the case of the Canadarm and 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 links (Figure l-3a), one free to slew (revolute joint) while the other is permitted to deploy and retrieve (prismatic joint). A combination of such modules can lead to a snake-like variable geometry manipulator (Figure l-3b) with several advantages [8]. It reduces coupling effects 3 •4 Orbit Station Figure 1-2 A l l the space-based manipulators have used, so far, revolute joints thus permitting only slewing motion of links. 4 Figure 1-3 Variable geometry manipulator showing: (a) single module with a pair of slewing and deployable links; (b) several modules connected to form a snake-like geometry. 5 resulting in relatively simpler equations of motion and inverse kinematics, decreases the number of singularities, and facilitates obstacle avoidance for comparable number of degrees of freedom (Figure 1-4). Dynamics and control of such Multi-module Deployable Manipulator ( M D M ) system, free to traverse an orbiting elastic platform and carrying a payload, represent a challenging task. 1.2. A Brief Review of the Relevant Literature As can be expected, the amount of literature available on the subject of robotics is literally enormous. The objective here is to touch upon contributions directly relevant to the study in hand. Obstacle Figure 1-4 Variable geometry manipulator showing obstacle avoidance character. 6 1.2.1. Characteristics of space-based manipulators There are several significant differences between the orbiting space platform supported manipulators and their ground-based counterparts: (a) Due to zero-weight condition at the system center of mass and microgravity field elsewhere, the environmental torques due to free molecular flow, Earth's magnetic field and solar radiation can become significant [9-11]. (b) As the manipulator rests on a flexible orbiting platform, their dynamics are coupled [12,13]. The manipulator maneuvers can affect attitude of the platform as well as excite it to vibrate [14]. Conversely, the librational motion of the platform would affect the manipulator's performance. Fortunately, manipulator maneuvers in space tend to be relatively slow permitting the end-effector to approach equilibrium [15]. (c) Space manipulators tend to be large in size, lighter and highly flexible. Obviously, this will make the study of system dynamics, and its control, a formidable task. (d) The ratio of the payload to manipulator mass for a typical space-based system can be several orders of magnitude higher [16]. For example, in case of the Canadarm the ratio is 61.5. The corresponding ground-based manipulator used in nuclear industry (supplied by the same manufacturer) has the payload to manipulator mass ratio of 0.167 ! (e) Obviously, space manipulators are not readily accessible for repair in case of, say, joint failure. This requires incorporation of a level of redundancy in their design [17]. Correspondingly, more degrees of freedom are involved than required for a given task. 7 (f) Remote operation of a space-based manipulator would involve time delays, an important factor in control of the system. For the R O T E X teleoperation experiment it reached seven seconds [18] ! These important differences emphasize the fact that one cannot entirely rely on the vast body of literature available for ground-based manipulators. We will have to explore and understand distinctive character of the space robotic systems. Dynamics and control of a large orbiting flexible platform (like the International Space Station), supporting a mobile elastic manipulator, carrying a compliant payload represent a class of problems never encountered before. It is only recently, some of the issues mentioned here have started to receive attention. Obviously, there is an enormous task facing space dynamicists and control engineers that will keep them occupied for years to come. The points which concern us are the nonlinear, nonautonomous and coupled character of the governing equations of motion, relatively low frequencies, and development of a controller, preferably robust. In the present study, focus is on the behavior of the basic system. Even here, the dynamics and control are rather involved. To help appreciate complex interactions between various degrees of freedom, influence of solar radiation, signal time-delay, and such complicating factors is purposely not addressed here. Of course, these issues are important, however, they are beyond the scope of the present thesis, in the available time. 1.2.2. Dynamics and control of space-based manipulators From the observations made earlier, it is apparent that space manipulators, as well as large flexible space structures in general, have unveiled a new and challenging field of space dynamics and control. Over the years, a large body of literature has evolved, which has been 8 reviewed quite effectively by a number of authors including Meirovitch and Kwak [19], Roberson [20], Likins [21], as well as Modi et al. [22 - 26]. In the majority of studies aimed at manipulators, only revolute joints were involved, i.e. links were permitted to undergo only slewing motions. On the other hand, several space structures feature deployment capabilities. For instance, a large solar array was deployed from the Space Shuttle cargo-bay during the Solar Array Flight Experiment (SAFE), in September 1984. Cherchas [27], as well as Sellappan and Bainum [28], studied the deployment dynamics of extensible booms from spinning spacecraft. Lips and Modi [29,30] have studied at length the dynamics of spacecraft with a rigid central body connected to deployable beam-type members. Modi and Ibrahim [31] presented a relatively general formulation for this class of problems involving a rigid body supporting deployable beam-and plate-type members. Subsequently Modi and Shen [32] extended the study to account for deployment as well as slewing of the appendages. Lips [33], Ibrahim [34], and Shen [35] have reviewed this aspect of the literature in some detail. In the above mentioned studies [29-35], although slewing and/or deployment were involved, each appendage was directly connected to the central body, i.e. a manipulator-type chain geometry of the links (appendages) was not involved. Figure 1-5 shows schematically the different models described above. The new manipulator system, schematically shown in Figures l-3(a) and l-5(d), was first proposed for space application by Marom and Modi [36]. Planar dynamics and control of the one module mobile manipulator with a flexible revolute joint, located on an orbiting flexible platform, were investigated. Results showed significant coupling effects between the platform and the manipulator dynamics. Control of the system during tracking of a specified 9 Deployable Deployable Figure 1-5 Schematic diagrams of space structure models: (a) Lips [33], rigid spacecraft with deployable beam-type members; (b) Ibrahim [34], rigid spacecraft with deployable beam- and plate-type members; (c) Shen [35], rigid spacecraft with slewing-deployable appendages; (d) Marom [36], flexible platform supporting one rigid slewing-deployable manipulator module and a rigid payload at the deployable link end. The revolute joint is flexible. 10 trajectory, using the computed torque technique, proved to be quite successful. Modi et al. [37] as well as Hokamoto et al. [38] extended the study to the multimodule configuration, referred to as the Mobile Deployable Manipulator (MDM) system. The model, with an arbitrary number of modules, accounted for the joint as well as link flexibility. A relatively general formulation for three-dimensional dynamics of the system in orbit was the focus of the study by Modi et al., while Hokamoto et al. explored a free-flying configuration. More recently, Hokamoto et al. [40] studied control of a single unit system and demonstrated successful tracking of a straight-line trajectory at right angle to the initial orientation of the manipulator. A comment concerning a rather comprehensive study by Caron [12] would be appropriate. He has presented an O(A0 formulation for studying planar dynamics and control of such formidable systems. The dynamical parametric study [42] clearly shows involved interactions between the orbital motion, flexibility, librational dynamics, and manipulator maneuvers. Furthermore, Caron [12] successfully demonstrated control of a single module (i.e. two links) manipulator, free to traverse a flexible platform, using the Feedback Linearization Technique (FLT) applied to rigid degrees of freedom, and suppression of flexible members' response through the Linear Quadratic Regulator (LQR). Recently, Chen [43] extended Caron's study and presented an order-Af formulation for three-dimensional motion of a mobile manipulator traversing an orbiting flexible platform. Control of a single module manipulator, i.e. with two flexible links and an elastic revolute joint, was investigated with the Feedback Linearization Technique applied to the rigid degrees of freedom. The controlled response of the system during commanded maneuvers of the manipulator was surprisingly good. Goulet [44] studied control of a single-unit rigid 11 manipulator with a knowledge-based hierarchical approach. The control strategy proved quite successful during pick-and-place operations as well as trajectory tracking. Cao [45] investigated the planar dynamics and control of a variable geometry manipulator with two deployable modules. A composite control system comprising FLT and LQR was used for both regulation of rigid degrees of freedom and suppression of vibration of flexible members. The FLT was also used to track several prescribed trajectories. Based on the literature review, 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 having revolute joints. (ii) Manipulators with revolute as well as prismatic joints have received relatively little attention, and then only recently. As the concept of space-based manipulators with slewing and deployable links was developed at the University of British Columbia, the few contributions in the field have also come from the same source. Here too, focus has been on the dynamical response of the system during a specified maneuver. (iii) A few reported control studies involve one-module manipulator, i.e. the system comprised of two links: One free to slew while the other is permitted to deploy. The control of multimodule manipulators has received little attention [45]. (iv) Tuned modal control has not received the needed attention. 1.3. Scope of the Investigation With this as background, the thesis investigates planar dynamics and control of a two-module (four links) flexible manipulator based on an elastic orbiting platform. To begin with, in Chapter 2, the model used is explained. A detailed dynamical response study 12 follows which assesses the influence of initial conditions, system parameters, and manipulator maneuvers. In a sense, this part of the study complements the results obtained by Cao [45] by providing response information for different, severe maneuvers. Results suggest that under critical combinations of system parameters and disturbances the response may not conform to the acceptable limits. This points to a need for active control. Three different control methodologies are used (Chapters 3,4): (a) The nonlinear Feedback Linearization Technique (FLT) is applied to rigid degrees of freedom with flexible generalized coordinates left uncontrolled actively but indirectly regulated through coupling. (b) A synthesis of the FLT and Linear Quadratic Regulator (LQR) to achieve control of both rigid (FLT) and flexible degrees of freedom (LQR). Developments in (a) and (b) are reported in Chapter 3. (c) A hierarchical control system, using a modal controller and a fuzzy tuning structure, for active suppression of system vibration (Chapter 4). Chapter 4 focuses on the development and implementation of an intelligent hierarchical controller for the vibration control of the deployable manipulator. The emphasis is on the use of knowledge-based tuning of the low-level controller so as to improve the performance of the control system. For this purpose, a fuzzy inference system (FIS) is developed first. The FIS is then combined with a conventional modal controller to construct a hierarchical control system. Specifically, a knowledge-based fuzzy system is used to tune the 13 parameters of the modal controller. The effectiveness of the hierarchical control system is investigated through simulation studies. The thesis ends with a brief review of important conclusions, significant original contributions and suggestions for future study (Chapter 5). 14 2. DYNAMICAL INVESTIGATION OF A SPACE-BASED MULTI-MODULE MANIPULATOR SYSTEM 2.1. Background to the System Model The order-JV Lagrangian formulation for the novel variable geometry manipulator, developed by Caron [12], is used for the dynamic investigation and controller design. The distinct features of the system model used may be summarized as follows: (a) The manipulator with an arbitrary number of modules, each carrying a slewing and a deployable link thus involving both revolute and prismatic joints, is supported by a mobile base free to traverse a platform. The platform is in an orbit around Earth (Figure 2-1). (b) The supporting platform, manipulator modules and revolute joints are treated as flexible. Prismatic joints are considered as integral parts of modules. (c) The module is permitted to have variable mass density, flexural rigidity and cross-sectional area along its length. (d) The system is permitted to undergo planar librational as well as vibrational motions. The slewing maneuver at any joint is confined to the plane of the orbit. (e) The damping is accounted for through Rayleigh's dissipation function. (f) The governing equations account for gravity gradient effects, shift in center of mass as well as change in inertia due to maneuvers and flexibility. Such a manipulator with a combination of revolute and prismatic joints is able to change its geometry, has a marked decrease in dynamical coupling, a reduction in the number of 15 Figure 2-1 Schematic diagram of the mobile flexible variable geometry manipulator, based on an elastic space platform. 16 singularity conditions, and can negotiate obstacles with ease [8]. Note, the model considered is relatively general and applicable to a large class of space - as well as ground - based manipulator systems. The Lagrangian approach adopted for derivation of the governing equations is particularly well suited to the flexible multibody system, with a large number of degrees of freedom, under consideration. It automatically satisfies holonomic constraints while the nonholonomic constraints can be accounted for, quite readily, using Lagrange multipliers. The form of the equations of motion conveys a clear physical meaning in terms of contributing forces. Equally important is the fact that the equations are well suited for a controller design. Furthermore, validity of the formulation and numerical integration code can be checked with ease through the conservation of energy for nondissipative systems. A comment concerning representation of the revolute joint's flexibility would be appropriate. The rotation yi of the frame F, , attached to the module i, with respect to the frame F,_i has three contributions (Figure 2-2): elastic deformation of the i t h -1 body in the transverse direction (£,-); rotation of the actuator rotor ( « r ; ) , 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 (2.1) 2.2. Simulation Methodology The equations governing the dynamics of the robotic systems mentioned above are highly nonlinear, nonautonomous, and coupled. They can be expressed in the general form M(q,t)q + F(q,q,t)=Q(q,q,t), (2.2) 17 Body i-1 Coordinates describing flexibility of revolute joints. 18 where M(q,t) is the system mass matrix; q is the vector of the generalized coordinates; F(q,q,t) contains terms associated with centrifugal, Coriolis, gravitational, elastic, and internal dissipative forces; and Q{q,q,t) represents generalized forces, including the control inputs. Equation (2.2) describes the inverse dynamics (i.e. forces corresponding to a specified motion) of the system. For simulations, forward dynamics is of interest, and Eq. (2.2) must be solved for q , q = Ml(Q-F). (2.3) The solution of these equations of motion generally requires 0(N ) arithmetic operations, where N represents the number of bodies (modules) considered in the study. In other words, the number of computations required by the 0(N3) algorithm will vary as the cube of the number of modules. It also depends on the number of generalized coordinates associated with each module. Clearly, the computation cost can become prohibitive for a system with a large number of modules or generalized coordinates. Hence, the development of the O(A0 formulation by Caron [12], where the number of arithmetic operations increases linearly with the number of bodies (or degrees of freedom) in the system, promises a significant saving in the computational cost. Equally important is a possibility of real-time implementation of a control strategy. It is often useful to specify some of the generalized coordinates. For example, cases where the length of the units is varied in a specified manner, or where joints are locked in place at given angles, require the use of specified coordinates. These coordinates are prescribed through constraint relations which are introduced in the equations of motion through Lagrange multipliers. 19 In the present study, a sinusoidal acceleration profile is adopted for prescribed maneuvers. It assures zero velocity and acceleration at the beginning and end of the maneuver, thereby reducing the structural response of the system. The maneuver time-history considered is as follows, where qs is the specified or constrained coordinate; Aqs is its desired variation; r is the time; and Ar i s the time required for the maneuver. The time histories for qs, qs, and qs are plotted in Figure 2-3. A F O R T R A N program for the dynamical simulation of the system, as developed by Caron [12], integrates the acceleration vector q numerically using Gear's method, which is well suited for stiff systems of ordinary differential equations. To reduce computational time during simulations, a symbolic manipulation routine ( M A P L E V) is used in order to obtain analytical expressions for the integrals of the shape functions. Furthermore, efficient matrix multiplication algorithms are developed to take advantage of the structure of various matrices involved. 2.3. Simulation Considerations The system performance is governed by a large number of parameters. Some of the important variables are listed in Table 2-1. Obviously, a systematic change of these variables would lead to a large volume of information. However, it would also demand considerable amount of time, effort and computational cost. Hence, one is forced to focus on cases that are likely to provide useful trends. These include the manipulator position; platform, link and joint flexibility; number of (2.4) 20 0 / V • • "L" •— ^ —• ^y \ \ • ./• \ \ \ \ \ \ \ N N \ / \ / q \ / * * \ / - q \ / q \ / • * \ / \ / \ / ' s . ,y 0.5 Figure 2-3 Normalized time histories of the sinusoidal maneuvering profile showing displacement, velocity, and acceleration. 21 modes; profile and speed of a maneuver; and mass of the payload. Even with these selected parameters, the task is formidable. Hence only a few typical results corresponding to a two-module manipulator, i.e. with four links, are presented for conciseness (Figure 2-4). Table 2-1 Important factors affecting the system performance. Parameters . orbit eccentricity mass of : platform; links; joints; payload stiffness of : platform; links; joints damping of : platform; links; joints . link length Initial Conditions . platform attitude . manipulator location and orientation . deformation of platform, manipulator links, joints Maneuvers . type : slewing; deployment; retrieval; base translation . amplitude . speed Discretization . shape of admissible functions (modes) number of admissible functions (modes) 22 Figure 2-4 Schematic diagram of a two-module, flexible, variable geometry manipulator, based on an elastic space platform, considered for study. 23 Numerical values used in the analysis, unless specified otherwise, are indicated below: Orbit: Circular orbit at an altitude of 400 km; period = 92.5 min. Platform: Geometry: circular cylindrical with diameter = 3 m; axial to transverse inertia ratio of 0.005; Mass (mp) = 120,000 kg; Length (lp) = 120m; Flexural Rigidity (EIP) = 5.5xlO 8 Nm 2 . Manipulator Position (d): d = 0 or 60 m. Manipulator Module (lx, Z2): Initial length of the manipulator module (i.e. Zs + deployed length, 7.5 + Id) is taken as 7.5 m, i.e. the deployable link is initially not extended. Here Zs, Id represent lengths of slewing and deployable links, respectively. Manipulator Links (Slewing and Deployable): Geometry: circular cylindrical with axial to transverse inertia ratio of 0.005 Mass (ms, md) = 200 kg; Length (Zs, Z d,m a x) = 7.5 m; Flexural Rigidity (£7„ EId) = 5.5xlO 5 Nm 2 . Revolute Joint: Mass (mj) = 20 kg; 24 Moment of Inertia (Ijz) = 10 kgm2; Stiffness (Kj) = 104 Nm/rad . Note, the prismatic joint is treated as a part of the slewing link. Payload: Nominally zero. Specified in figure legend when different. Modes: Fundamental mode for a cantilever beam with tip mass for modules, free-free beam mode for platform. Note, subscripts d, j, m, p, and s correspond to deployable link, revolute joint, manipulator, platform and slewing link, respectively. Initially the platform is in equilibrium either along the local vertical (stable) or aligned with the local horizontal (unstable) position. The manipulator links are aligned with the platform, i.e. they are also along the local vertical or local horizontal before the maneuver. The damping is purposely assumed to be zero in all components to obtain conservative estimate of the response, i.e. the damping coefficient for joints (Cj) as well as structural damping coefficients for manipulator links (£m) and the platform (£ ) are considered zero. More important specified and response parameters are summarized below and indicated in Figure 2-5: d position of the base from the center of the platform; ex, e2 tip vibrations of modules one and two, respectively; ep platform tip vibration; x, y body fixed coordinate system with x aligned with the undeformed axis of the platform and y normal to x in the orbital plane; 2 5 To Earth Center Figure 2-5 Schematic diagram showing important parameters appearing in the response study. 26 ax,a2 rigid body rotations, during slew maneuvers of modules one and two, respectively; /?!,/?2 contributions due to flexibility of revolute joints at modules one and two, respectively; yr platform pitch libration. 2.4. System Response 2.4.1. Effect of manipulator location and orientation At the outset it must be recognized that the platform itself has two equilibrium positions: one is along the local vertical, which is stable; the other is aligned with the local horizontal, which is unstable. The presence of manipulator, when aligned with the platform, has virtually no effect on the equilibrium because the geometry remains effectively unchanged, as well as relatively massive (120,000 kg) character of the platform compared to the manipulator (800 kg). However, with different orientations of the modules, the geometry changes, i.e. platform and the manipulator no longer remain in alignment with the local vertical or the local horizontal position. The system's new equilibrium orientation slightly deviates depending on the slew and deployment characters as well as location of the manipulator on the platform. For example, consider the equilibrium position corresponding to the case when the manipulator is located at the tip (d = 60 m), av -90°,a2 =0, and the deployable links of both modules remain unextended ( l x - l 2 - 7.5 m) as indicated in Figure 2-6. The deviation in the equilibrium position (y/' ) from the local vertical (or local horizontal) is around 0.17°. This acts as a small disturbance and sets the platform 27 T Figure 2-6 System equilibrium as affected by the manipulator's geometry and location on the platform. The diagram shows a case where the links are locked at a, =90°, a 2 =0 and /, = l2 =7.5m. The equilibrium configuration deviates from the local vertical by about 0.17°. 28 oscillating. The local vertical orientation being stable, the platform tends to move towards and oscillate about the new equilibrium position. On the other hand, the local horizontal position being unstable, the platform moves away from it; the direction of rotation being governed by the position of the center of mass (Figure 2-7). Figure 2-8 shows the effect of a two-module manipulator located at the center of the platform (d = 0) and at the platform tip (d - 60 m). The effects of variations in module length and payload are also considered. In all the cases, the modules are locked in position with ax =90°, a2 -0. In Figure 2-8(a), the base of the manipulator is located at the center of the platform, and four cases with different system configurations are considered: (1) with both modules undeployed and no payload attached to the tip of the manipulator; (2) with both modules undeployed and a payload of 2000kg attached to the tip of the manipulator; (3) with both modules fully deployed and no payload attached to the tip of the manipulator; (4) with both modules fully deployed and a payload of 2000kg attached to the tip of the manipulator. In Figure 2-8(b), the base of manipulator is located at the tip of the platform, and four similar cases are considered. As pointed out before, the presence of the manipulator results in a shift in the center of mass causing a pitch moment, which increases as the manipulator base moves towards the platform tip. It can be seen from Figure 2-8(a) that when the manipulator is located at the center of the platform, the pitch vibration amplitude is rather small (about 0.001°), and the change of length and payload has virtually insignificant effect on the pitch angle. On the other hand, with d = 60 m, the maximum platform deviation from the local vertical reaches a value as high as 3.41° when the manipulator carries a payload of 2000kg and modules are 29 Local Vertical Figure 2-7 Direction of rotation of the platform, initially in the local horizontal equilibrium position, due to a change in the manipulator orientation. It is governed by the location of the center of mass in different quadrants. J 30 a i L.V. C M . Parameters: EIs=EId=5.5xl05 Nm 2 £ 7 p = 5 . 5 x l 0 8 N m 2 ^=1.0xl0 4 Nm/rad Initial Conditions: ¥p = /3x=j32 = 0 ep = ei = e 2 = 0 Specified Coord.: ax = 90°, a2 = 0 . /j = l2 =7.5 m, Payload = 0 lx = l2 =7.5 m , Payload = 2000kg lx = l2 =15 m, Payload = 0 lx =l2 =15 m, Payload = 2000kg (a) d = 0 0.001 h vP o -0.001 ' I \ \ N \ _ \ \ _ -. A' -I 0.5 Orbit (b) d = 60m - 3 h T . -' s / / /' / / / / / / / /' / / / -I 0.5 Orbit Figure 2-8. System response with manipulator located at: (a) center of the platform (d = 0); (b) tip of the platform (d = 60 m). 31 fully deployed. With no payload and the modules undeployed, the deviation can still reach to 0.35°. This may appear small, however, depending on the mission, the permissible platform deviation may be as small as 0.1°. This suggests a need of active control for the system. Figure 2-9 considers the cases where the platform is initially located around the local horizontal. The manipulator is at the platform tip, and the modules are locked in position with a, =90°, a 2 =0. In Figure 2-9(a), three different initial positions around the local horizontal are considered with the modules undeployed and no payload at the tip. When the platform is aligned with the local horizontal, i.e. y/p(0) =-90°, in absence of the manipulator, the platform would have stayed there, as it should be an equilibrium position. However, due to presence of the manipulator, the new equilibrium orientation corresponds to -90.17°. So the initial orientation of the platform deviates by 0.17° from its unstable equilibrium position now. Consequently, this causes the platform to swing away from the position. When the platform reaches yi = 90°, there is similar phenomenon and the platform continues to rotate in the anticlockwise sense. Thus, in absence of dissipation, one has a perpetually rotating platform. Of course, the energy input corresponds to the initial deviation of the platform from the new equilibrium position. Note, when the platform is initially set at its equilibrium position, y/ =-90.17°, it remains there as it should. Thus a small change in initial conditions near the unstable configuration can lead to widely different system response. This is further emphasized in Figure 2-9 (a) when the platform is initially at y/p(0) =-91°. Note, now the system center of mass is in the fourth quadrant and experiences a clockwise moment. The platform continues to swing until its velocity becomes zero at yr =90.66°. The return journey in the 32 L.V. Parameters; EIs=EId=5.5xl05 Nm 2 £ 7 p = 5 . 5 x l 0 8 N m 2 ^=1.0xl0 4 Nrn/rad Payload = 0 C. M. d= 60m Initial Conditions: A=A=o Cp = ei = e 2 = 0 Specified Coord. 0^ = 90°, a 2 =0 (a) li=h= 7.5 m Orbit (b) l1=l2- 15 m Figure 2-9. System response when the platform is initially around the local horizontal. 33 anticlockwise sense starts and the platform continues to swing back and forth around the stable local vertical position. This is in sharp contrast to the results with other two initial conditions in Figure 2-9(a). Similar system response can be observed in Figure 2-9(b) where the manipulator modules are fully deployed and the new unstable equilibrium position is V„ =-90.35°. 2.4.2. System response to initial disturbances The effects of various external disturbances, such as spacecraft docking, satellite capture, or impacts with external bodies, can be incorporated into the model through suitable initial conditions. This section investigates the system response to such external disturbances. The external disturbances considered here are in the form of tip deflection of the platform and of the manipulator. It should be recognized that the system consists of elastic and rigid degrees of freedom with a wide variation in their natural frequencies. Obviously, depending on the disturbance, a number of them may be excited revealing complex interactions at different frequencies. Therefore, the knowledge of the natural frequencies of the flexible components of the system will be helpful to have a better understanding of the coupling effects. Tables 2-2 and 2-3 list the natural frequencies of various elastic members in the system. With the manipulator located at the tip of the platform and joints locked in the position as before (a x - 90°, a2 = 0, d = 60m ), the tip of the platform, aligned with the local vertical, was given an initial disturbance of 0.5m. Two cases are considered: with the modules undeployed and fully deployed. Figure 2-10 shows the system response with two modules undeployed. The platform vibrates at its natural frequency (0.18Hz ) with a period of around 0.001 orbit (~ 6s). Note, due to coupling effects, the librational motion of the 34 Table 2-2 Fundamental natural frequencies of the components in absence of payload: l x = l 2 - 7.5m Component Frequency, Hz Period Second Orbit Platform, Libration 3.12xl0"4 3204 0.6 Platform, Bending 0.18 5.56 l.OxlO"3 Module 1, Bending 5.85 0.17 3.1xl0"5 Module 2, Bending 10.50 0.10 1.8xl0"5 Joint 1, Torsion 0.06 16.67 3.0xl0"3 Joint 2, Torsion 0.39 2.56 4.6xl0"4 Table 2-3 Fundamental natural frequencies of the components in absence of payload: lx =l2 = 15m Component Frequency, Hz Period Second Orbit Platform, Libration 3.12X10"4 3204 0.6 Platform, Bending 0.18 5.56 l.OxlO3 Module 1, Bending 1.52 0.66 1.2xl0"4 Module 2, Bending 2.68 0.37 6.7X10"5 Joint 1, Torsion 0.03 33.33 6.0xl0"3 Joint 2, Torsion 0.19 5.26 9.5xl0"4 35 C. M. Parameters: EIs=EId= 5 . 5 x l 0 5 N m 2 EIP= 5 .5xl0 8 N m 2 A}=1.0xl0 4Nrn/rad Specified Coordinates : h = h~ 7-5m; ax = 90°, a2 = 0 Initial Conditions: = 0.5m, = e2= 0 Payload = 0 0.02 2 0 Platform Libration 0.01 0.02 First Joint Vibration 0 0.01 0.02 Second Joint Vibration 0.02 Platform Tip Vibration -0.5 0 0.01 0.02 Tip Deflection of Module 1 0.01 -0.01 0 0.01 0.02 Tip Deflection of Module 2 0.01 m -0.01 0.01 Orbit 0.02 Figure 2-10 Response of the system to an initial disturbance of 0.5m tip displacement of the platform: 1; = l2 = 7.5 m. 36 platform is induced. The pitch librational response is modulated at the platform frequency. The flexibility effects of the manipulator joints and modules (links) are also apparent. A l l the joint and module vibrations are modulated at the platform natural frequency, and the vibrations of modules and joint 2 are coupled with vibrational frequencies of the platform and joint 1. The platform tip motion and the corresponding tip moment (free-free beam) excite joints of the manipulator, resting on the platform, with peak amplitudes of /?i ~ 2° and L\ ~ 1°. The links are also excited and vibrate with peak amplitudes of ex = 0.007m and e2 = 0.002m. Figure 2-11 shows the system response with two modules fully deployed. Due to the change of the module length, the natural frequencies of links and joints also change. A beat phenomenon can be observed in the vibrations of joint 2 and tip of module 2. The beat character of the response is attributed to the proximity of the joint 2 and platform natural frequencies ( a>j2 =0.19 Fiz, Q)p = 0.18 Hz ). The beat period approximately corresponds to 222s (0.04 orbit). The platform vibration, serving as an exciting source, induces the beat vibration of joint 2, which, in turn, causes the beat vibration of module 2. Due to near resonance condition, a relatively large amplitude of 8° in joint 2 vibration results. The tip deflections of both modules 1 and 2 are much larger than the previous undeployed case (Figure 2-10) as can be expected. Once more, the modulation at platform's frequency can be observed in platform libration as well as joint and module vibrations. Figure 2-12 shows the system response when an initial deflection of 0.2 m is given at the tip of module 2, with two modules fully deployed and the same manipulator configuration as above (ax =90°, a2 = 0, d = 60m ). The complex coupling effects among the flexible degrees are quite apparent. 37 \ L.V. Parameters: EIs=EId= 5.5xl05Nm2 FJP= 5.5 xlO 8 Nm2 ^=1.0xl0 4 Nm/rad Initial Conditions: <?p = 0.5m, gj = e2= 0 Specified Coordinates : lx = l2 = 15 m; ax = 90°, a2 = 0 Payload = 0 e2 CM. Platform Libration Platform Tip Vibration 0 0.01 0.02 First Joint Vibration o 0 0.01 0.02 Second Joint Vibration 0 0.01 0.02 Tip Deflection of Module 1 « i 0 0 0.01 0.02 Tip Deflection of Module 2 0.02 Figure 2-11 Response of the system to an initial disturbance of 0.5m applied at the platform tip: l} = l2= 15 m. 0.02 38 Parameters: L.V. EIS= EId= 5.5xl05Nm2 Initial Conditions: £/p=5.5xl08 Nm2 Kj=1.0 xlO 4 Nm/rad e p = 0, <?, = 0, e2= 0.2 m Specified Coordinates : lx = l2 = 15 m; Payload = 0 ax = 90°, a2 = 0 e2 C. M. Platform Libration Platform Tip Vibration 0 0.01 0.02 ' 0 0.01 0.02 Second Joint Vibration Tip Deflection of Module 2 Orbit Orbit Figure 2-12 System response to the manipulator's tip displacement of 0.2 m. 39 In absence of damping, the tip continues to oscillate (e2) at a constant amplitude and with a frequency of 2.7 Hz (= 150 cycles in 0.01 orbit). This, in turn, excites joint 2 through coupling (/%), which oscillates at its natural frequency of 0.19 Hz (approximately 11 oscillations in 0.01 orbit) with high frequency module 2 oscillations superposed. Joint 1 displays a rather complex response showing coupling effects of L \ , ei and e2, while an apparent coupling effect of e2 can be observed in the tip vibration of module 1. The platform tip vibration is modulated at the fh. frequency, and the platform pitch response is modulated at f3\ frequency. As expected, the platform tip response exhibits the typical beat phenomenon, as its frequency is quite close to that of (coj2 =0.19 Hz, cop =0.18 Hz) as mentioned earlier. Thus, manipulator maneuvers at critical frequencies can lead to resonance and unacceptable system behavior. This is explored in the following section. 2.4.3. System response in presence of manipulator maneuvers In the study so far, the manipulator joints were locked in position, i.e. there were no maneuvers involved. The next logical step is to assess the effect of the manipulator executing slewing and deployment maneuvers. The attention was turned to assess the influence of several important parameters such as number of modes used in the flexibility discretization; platform, link and joint flexibility; and speed of maneuver. The manipulator is located at the platform tip (d = 60m) to impose more severe disturbance through its prescribed maneuver. Effect of Higher Order Modes The system flexibility, i.e. elastic character of the members such as platform, links, joints, etc., was discretized using assumed modes. One of the important problems faced by 40 dynamicists is the number of modes needed to predict the system response accurately. This is important as the number of modes directly affects the computational effort. To get insight into this aspect, the flexibility of the platform and each of the manipulator links was represented by up to the first three modes. The system response to the manipulator's simultaneous 90° slew and 7.5 m deployment maneuvers in 0.01 orbit was evaluated (Figure 2-13). It is apparent that the contribution from the second and third modes is negligible. The fundamental mode is able to capture the system dynamics quite well. In the subsequent studies, the member flexibility is represented only by the fundamental mode. Influence of the Revolute Joint Stiffness The manipulator is free to traverse the length of the platform. So it was thought appropriate to include its prescribed translation, with slewing and deployment, to make the maneuver more general. Such translational maneuver of the manipulator may occur during, say, capturing of a payload carried by a space shuttle (Figure 2-14). The manipulator is initially aligned with the platform. Two different configurations are indicated in Figure 2-14: (a) illustrates the case where the manipulator approaches the payload with ^ =a2 =90°; (b) modules vertical to provide the maximum stretch with ax = 90° ,a 2 - 0 . Such studies may help assess the effect of system parameters such as revolute joint stiffness, platform flexibility, link flexibility, maneuver speed, etc. on the capture performance. 41 L . V . Parameters: £7,=£7 r f =5.5xl0 5 Nm 2 EIp=5.5xlO* N m 2 Initial Conditions: »% = /?,= A = 0 A: ;=1.0xl0 4 Nm/rad ep = e\ = e 2 = 0 Specified Coordinates : Payload = 0 Zj =/ 2 =10m—> 15m; = ctr2 = 0 —> 90° in 0.01 orbit. a ^ t / * 2 C. M . Pi 1 mode 3 modes Platform Libra tion 0 0.01 0.02 First Joint Vibration 0 0.01 0.02 Second Joint Vibration A° o 0.02 0.0005 > 0 -0.0005 Platform Tip Vibration 0 0.01 0.02 Tip Deflection of Module 1 -0.005 0 0.01 0.02 Tip Deflection of Module 2 0.001 e2 0 -0.001 0.02 Figure 2-13 System response showing the effect of number of modes to represent flexibility. Note, the fundamental mode is quite adequate to capture physics of the system dynamics. 42 Hovering space shuttle (a) Figure 2-14 A schematic diagram showing two different maneuvers for capture of payload carried by a hovering space shuttle: (a) ax=a2= 90°; (b) . ax = 9 0 ° , a 2 =0. 43 The prismatic joint, being a part of the slewing link, has its flexibility accounted through the link elasticity. Here the focus is on the revolute joint. Three different values of the joint stiffness were considered. It should be emphasized that the value of Kj = l x l0 4 Nm/rad is quite low compared to the conventional value. This low value was purposely chosen to accentuate the effect of maneuver-induced disturbances. Thus Kj = 5xl0 3 Nm/rad represents an extremely flexible joint. Figure 2-15 shows the system response as affected by the variation of joint stiffness. As can be expected, the joint flexibility affects the manipulator dynamics significantly, however its influence on the platform response is relatively small. The platform pitch libration (y/p) as well as tip vibration (e ) are now modulated because of the joint flexibility, however ep continues to remain negligibly small (~2mm maximum amplitude) while the if/presponse remains essentially unchanged. On the other hand, amplitude of the first joint vibration can reach as high as 30°. Although e2 is measured with reference to the link-fixed coordinate system, it increases significantly with a decrease in the joint flexibility suggesting coupling between the two. Thus careful planning of the joint stiffness in the manipulator design is essential, and a need for active control to meet mission specifications may be necessary. Effect of Platform Stiffness Figure 2-16 shows the system response when the platform stiffness was varied over two orders of magnitude, from £ 7 p = 5 . 5 x l 0 7 t o 5 .5x l0 9 Nm 2 . It is of interest to note that the 44 C. M . Parameters: EIs=EId=5.5xl05Nm2 £ / p = 5 . 5 x l 0 8 N m 2 Specified Coordinates : lx =l2 =7.5m—> 15m; ax = a2 = 0 —> 90°; d = 60m -> 30m in 0.01 orbit. Initial Conditions: = = e 2 = 0 Payload - 0 •-• Kj = 5 .0xl0 3 Nm/rad — Aj= l .OxlO 4 Nm/rad •-- Kj= 5.0X10 4Nm/rad Platform Libration -0.2 -0.4 0.02 First Joint Vibration 0.04 0.02 0.04 Second Joint Vibration 0.02 Orbit 0.04 Platform Tip Vibration 0.002 -0.002 30 j\ A / / \ A / \ i \ ! * 0.3 0 \ / \ / Wi v\ /V |V i /' i i i «, 0 -30 \ / \ /' V V \ / \ / \ ; -0.3 0 0.02 0.04 Tip Deflection of Module 1 0 0.02 0.04 Tip Deflection of Module 2 0.05 e2 0 -0.05 0.04 Figure 2-15 System response in presence of maneuver: effect of joint flexibility. 45 L . V . i Parameters: £ / ,=£ / r f =5 .5x l0 5 Nm 2 ^=1.0xl0 4 Nm/rad Initial Conditions: V gP = g l = e 2 = 0 ax 1 V / ^ \ \ > Specified Coordinates : Payload = 0 lx =l2 =7.5m—> 15m; ax = a2 = 0 -» 90°; J = 60m -> 30m in 0.01 orbit. a2 C M . £ / p = 5 . 5 x l 0 7 N m 2 £ / p = 5 . 5 x l 0 8 N m 2 £ / p = 5 . 5 x l 0 9 N m 2 0 0.02 0.04 0 0.02 0.04 Orbit Orbit Figure 2-16 System response in presence of maneuver: effect of platform flexibility. 46 flexural rigidity of the platform has virtually no effect on the platform libration, joint vibration and the manipulator tip response. Of course, the platform tip deflection, being the parameter directly related to the platform flexibility, showed some effect. However, it was virtually negligible until the platform flexibility reached a value of 5 .5x l0 7 Nm 2 , the lowest considered in the study. This is an important piece of information and can be used to advantage in the system design. It suggests that the platform flexibility has insignificant influence on the manipulator dynamics. Thus treating the platform as rigid can give information of considerable importance with a large saving in the computational effort. Flexibility of Links The link stiffness was changed by an order of magnitude above and below the nominal values EIs=EId= 5.5x10 Nm . The results are presented in Figure 2-17. As anticipated, the link stiffness has virtually no effect on the system response in iff p , ep and (3{ degrees of freedom except for the phase shift, which is of little consequence here. However, as can be expected, the manipulator tip deflection increases significantly, in a nonlinear fashion, as the links' flexural rigidity diminishes. The corresponding increase in the period of tip oscillations is also apparent. Note, for a decrease in the link flexibility by an order of magnitude from its nominal value, the tip oscillation amplitude of the first module increases to =3m! Although in the real-life situation there will be some structural damping present, the system design is usually a compromise between several competing factors. For example, increased flexural rigidity to reduce tip deflection would demand an increase in weight. 47 C M . Parameters: £ / p = 5 . 5 x l 0 8 N m 2 « / = 1 . 0 x l 0 4 N m / r a d Specified Coordinates: Zj = / 2 =7.5m—>15m; ax - a2 = 0 —> 90°; d = 60m -^30m in 0.01 orbit. Initial Conditions: ep = e \ = e 2 = 0 Payload = 0 £ / , = £ ^ = 5 . 5 x l 0 4 N m 2 £ ; / ,=£ / d =5 .5x l0 5 Nm 2 £7 ,=£7 d =5 .5x l0 6 Nm 2 P: 0 PI 0 Platform Libration 0 0.02 0.04 First Joint Vibration V> " >\ i\ 0.02 Second Joint Vibration 0.04 0.02 Orbit 0.04 0.002 -0.002 Platform Tip Vibration 0 0.02 0.04 Tip Deflection of Module 1 e, 0 m s n , / ' M '» /« / I I I , ' , I I I I , , « » ' » ' » ' l ' \ , 0 0.02 0.04 Tip Deflection of Module 2 1 v. ' v , / v M ; ' in in I, - w ' I v v' 0.02 Orbit 0.04 Figure 2-17 System response in presence of maneuver: effect of link flexibility. 48 Thus the results suggest that there are situations where active control of the system may be necessary. Effect of Maneuver Speed Figure 2-18 shows the effect of maneuver speed on the resulting system dynamics. The objective now is to assess the influence of faster and slower maneuvers compared to the nominal one. Intuitively, one would expect the faster maneuver to accentuate the response. This is borne out by the simulation results. Note, the faster maneuver affects both the platform as well as the manipulator dynamics, although the effect on the former is relatively small. The vibration amplitude at the first joint can reach nearly 50°, while the tip oscillation amplitude of the first module exceeds lm. The similar trend can be observed in Figure 2-19 where the manipulator moves from its initial position to the maximum stretch configuration. However the amplitudes of vibration are slightly smaller compared to the case shown in Figure 2-18. The results suggest inherent danger associated with relatively faster maneuvers and thus provides useful information for the system operation. In the investigations so far, the maneuver was purposely considered rather severe to check performance under demanding situations. In a real-life operation, the maneuver will likely proceed differently, and take place in a longer time. To assess the system performance under a real-life operational situation, consider a rather conservative situation of the maneuver: a, = 180° —> 90°, <x, =-180° —> 0, /1=/2=7.5m —>15m completed in 0.05 orbit, for a manipulator with Kj = 5xl0 4 Nm/rad. A payload of 2,000 kg is also included which was not present before. Note, the maneuver rate is rather slow, the joint stiffness is quite high and the payload is somewhat moderate (a typical communications satellite may 49 L . V . Parameters: £/ ,=FJ r f =5.5xl0 5 Nm 2 £ / p = 5 . 5 x l 0 8 N m 2 Initial Conditions: *\d\ Kj=l.0xl04 Nm/rad g P = g l = e 2 = 0 Specified Coordinates: Payload = 0 lx-l2 =7.5m—> 15m; Duration of manuever C. M . = #2 = 0 —> 90° ; d= 60m —>30m. 0.005 orbit (27.75s) 0.01 orbit (55.5s) 0.02 orbit ( I l l s ) Figure 2-18 System response in presence of maneuver: effect of maneuver speed. 50 L . V . Parameters: EIs=EId=5.5xW5 N m 2 £ 7 p = 5 . 5 x l 0 8 N m 2 Initial Conditions: £ ; = 1 . 0 x l 0 4 N m / r a d e p = g l = e 2 = 0 Specified Coordinates: Payload = 0 l{ =l2 =7.5m—> 15m; Duration of manuever c1. M . e2 = 0 -> 90°; a2 = 0; J = 60m -> 30m. 0.005 orbit (27.75s) 0.01 orbit (55.5s) 0.02 orbit ( I l l s ) Platform Libration ¥P - ° - 2 P\ 0.02 0.04 First Joint Vibration 0 0 0.02 0.04 Second Joint Vibration PI 25 0 \ ^ \ / -A JKX f ^ -25 0.02 Orbit 0.04 Platform Tip Vibration 0 0.02 0.04 Tip Deflection of Module 1 0 0.02 0.04 Tip Deflection of Module 2 0.04 Figure 2-19 Effect of maneuver speed: maximum stretch configuration. 51 weigh 1 to 4 tons). Furthermore, the manipulator is folded in the stored orientation along the platform (or, = lS0°,a2 = -180°). Response results are presented in Figure 2-20. It is apparent that the peak platform libration far exceeds the permissible range of 0.1° -1.0°. The maximum joint vibrations of /?, = 0.25° and /?2 =0.3° would lead to the manipulator tip deviation from the desired position by ~ 20cm along the x axis (Figure 2-5). Thus a need for control is clearly indicated and this is the subject of the following chapter. 52 C. M . Parameters: FJ,=£7 r f =5.5xl0 5 Nm 2 ^ = 5 . 0 x l 0 4 Nm/rad Specified Coordinates : Zj =l2 =7.5m—> 15m; a,= 180° ^ 9 0 ° ; a2 = -180° -» 0 in 0.01 orbit; d=60m. Initial Conditions: Payload = 2000 kg Platform Libration Platform Tip Vibration PI 0.5 First Joint Vibration 0.05 Second Joint Vibration 0 0.05 0.1 Tip Deflection of Module 1 0 0.05 0.1 Tip Deflection of Module 2 Figure 2-20. System response, even under a set of conservative values for payload (2000 kg), joint stiffness (5xlO 4Nm/rad) and maneuver speed (0.05 orbit), is unacceptable. This suggests a need for control. 53 3. S Y S T E M C O N T R O L USING F L T / L Q R The previous chapter clearly established a need for control, under critical combinations of system parameters, initial conditions and maneuvers, to maintain desirable performance of the manipulator. In particular, joint stiffness, maneuver speed and payload constituted parameters having significant effect on the system dynamics. The present chapter focuses on the next logical step of control of the system with a platform-based two-unit (four links) manipulator. As before, the platform, modules and joints are treated as flexible members. To recapitulate, more frequently used symbols are summarized below and indicated in Figure 3-1: 0 true anomaly with reference at perigee; C M . center of mass; d position of the manipulator C M . ; L . V . local vertical; L . H . local horizontal. Rigid Degrees of Freedom \f/p platform pitch; lx,l2 lengths of modules one and two, respectively; ax,a2 rigid components of slew maneuvers associated with modules one and two, respectively. Flexible Degrees of Freedom /?j,/?2 flexible components of slew maneuvers associated with modules one and two, respectively; ep platform tip deflection; 54 55 ex,e2 tip deflections of modules one and two, respectively. 3.1. Control Methodologies The torques and forces responsible for slewing and deployment maneuvers are illustrated in Figure 3-2. The objective here is to control the operational behavior of the system. For example, the slewing maneuvers will arise with application of torques (TX,T2) provided by actuators located at revolute joints of the manipulator. Similarly, forces (FX,F2) for the deployment and retrieval of units are provided by actuators at the prismatic joints of the manipulator. In addition to joint actuators which regulate the manipulator link dynamics, there are Control Momentum Gyros (CMGs). They are used to regulate the platform orientation as well as its vibration. CMGs located near the center of the platform contribute the torque (Tpr) which controls the rigid body motion of the platform, i.e. its attitude or pitch response. On the other hand, a pair of CMGs, located symmetrically about the platform's center and providing equal torques (Tpf/2) in the opposite sense, control its elastic vibration by regulating the local slope. The present section is concerned with the selection of control inputs which lead to acceptable motion of the system. Two different control strategies are considered: (i) Nonlinear Feedback Linearization Technique (FLT) applied to the rigid degrees of freedom, with flexible generalized coordinates indirectly affected through coupling but not actively controlled. The FLT leads to uncoupled linearized equations of motion which are then subjected to the conventional PD control. 56 Controller Figure 3-2 Schematic diagram of the manipulator system showing location of the control actuators. The two torques Tp//2, opposing each other, control the platform's vibration by regulating the local slope. 57 (ii) The classical Linear Quadratic Regulator (LQR), based on a linear approximation of the flexible subsystem, is designed for active vibration suppression [46,47]. Both rigid as well as flexible degrees of freedom are now controlled through the FLT and LQR, respectively. 3.1.1. F L T control The FLT is an approach particularly suited to a class of nonlinear systems. The procedure was pioneered by Bejczy [48]. It has been further developed and applied by many investigators resulting in a considerable body of literature [49-55]. The basic idea is to use a mathematical model and find a transformation to decouple and linearize the dynamics of the controlled system. The main advantage of the feedback linearization over point-wise linearization is that once such a transformation is determined, a global linearization is achieved independent of the operating point. In the present study, a controller based on the FLT is designed to regulate the rigid degrees of freedom, i.e. rotations of the revolute joints (ax,a2), deployment of the links (l{,l2), and attitude motion of the system {y/p). However, the effectiveness of the controller is assessed using the original fully flexible system so that the potential effects of uncontrolled dynamics can be investigated. Equations governing dynamics of a flexible space-based manipulator can be written as M(q,t)q + F(q,q,t)=Q(q,q,t), (3.1) Mrr •• Mrf~ 1 " Qr ... \ Mfr Mff_ K J J M q F Q 58 where: M(q,t) is the system mass matrix composed of rigid (Mrr), flexible (Mff) and coupled (Mrf , Mfr) contributions; qr, qf are rigid and flexible generalized coordinates, respectively; F{q,q,t) contains terms associated with centrifugal, Coriolis, gravitational, and elastic forces; and Q(q,q,t) represents nonconservative generalized forces including the control inputs. Subscripts r and / refer to contributions associated with rigid and flexible degrees of freedom. If only the rigid degrees of freedom are controlled: Mrrqr+Mrfqf+Fr =Qr; M frqr + M sqf +Ff = 0. (3.2) A suitable choice for Qr would be Qr=M[(qr)d-u] + F, (3.3) with: M = Mrr-MrfMffMfr; F=Fr-MrfM-jFf. (3.4) Here subscript d refers to the desired value of a parameter. One way to select the control signal u is the Proportional-Derivative (PD) feedback, i.e. u = -Kv[(qr)d-qr]-Kp[(qr)d-qrl (3-5) where Kp and Kv are position and velocity gains, respectively. Let e = (qr )d -qr, then the controlled equations of motion become: 0 =e +Kve + Ke; (3.6) 9f=-M/lMA^r)d-U]-Mff~lFf Now Qr can be written as Qr = M(qr)d +F+ M(Kve + Kpe), (3.7) 59 which can be visualized as a combination of two controllers: the primary (Qr,P)'-Qr,p=M(qr)d+F; (3.8) and the secondary (Qr,s), QrtS=M(Kve + Kpe). (3.9) The function of the primary controller is to offset nonlinear effects inherent in the rigid degrees of freedom; whereas the secondary controller ensures a robust behavior. A block diagram of the control procedure is presented in Figure 3-3. It is undesirable for a robot to exhibit an overshoot, since this could cause impact if, for instance, a desired trajectory terminates at the surface of a workpiece. Therefore, to ensure asymptotic and critically damped behavior of the closed-loop system, a suitable candidates for the PD gains, Kp and KVt would be diagonal matrices such that ~<o\ 0 " '2cox 0 0 co] 0 2COi where coi is the desired natural frequency associated with the i joint or link error (Eq. 3.6). 3.1.2. F L T / L Q R control A combined FLT/LQR approach was also applied to the two-unit manipulator system. When the manipulator is in a fixed configuration (no deployment, slew, or translation), the platform attitude (y/p) and manipulator length (/,, l2) are maintained using the FLT strategy. However, the joint rotation (ax,a2), the platform's tip vibration (ep), as well as the links' tip deflections (ex, e2) are controlled using the LQR approach. During large slew and deployment maneuvers, variables y/p, ax,a2, lx, and l2 are regulated by the FLT 60 Kn Kv (qr)d Nonlinear Inner Loop M Qr Linear System Manipulator System qr Qr Figure 3-3 FLT-based control scheme showing inner and outer feedback loops. 61 controller until the manipulator reaches the vicinity of its target position. Structural vibrations are left uncontrolled. At the end of the large maneuver, the system's configuration remains nearly constant and nonlinear effects can be neglected. This allows the Linear Quadratic Regulator to takeover the control of ax and a2 to actively damp flexible generalized coordinates disturbed due to the maneuver. The optimal LQR controller is designed based on a linearized model of the system. To begin with, the governing equations are linearized about an operating point q0. To that end, the following substitutions are made in the left hand side of Eq. (3.1): q = q0+Aq; q = qQ + Aq ; q = q0+Aq. (3.11) Trigonometric functions are expanded in the Taylor series, and the second and higher order terms in q, q and q are neglected. After some algebra, this leads to M(q0)Aq + K(q0)Aq = Q = Q, (3.12) where M and K are the mass and stiffness matrices for the linearized system, respectively. Note, both M and K are evaluated at the operating point qQ and thus made time-invariant. Since the system's librational motion and the deployment length of the manipulator are controlled by the FLT, only the linearized equations governing the elastic degrees of freedom are needed. The decoupled vibrational subsystem is now described by M(q0)AqL+K(q0)AqL = uL, (3.13) where AqL = [ep,Aal,/3i,el,Aa2,/32,e2], with Aax and Aa2 as the deviations of the slew angles from their desired values; M and K are the mass and stiffness matrices, respectively, corresponding to the elastic subsystem; and uL is the input determined from the 62 Linear Quadratic Regulator to control e , Aax, ex, Aa 2 and e2. Note, the operational point is qLQ = [0, axo, 0,0, a20,0,0]r . Solving for AqL, AqL = -M~lKAqL+M~l uL, (3.14) which can be rewritten in the state-space form as ML. 0 -M AK 0 A + 0 M"1 x,. B u L ' (3.15) where xL e 9 t 1 4 x l ; A £ S^ 1 4 x 1 4; and B e 9 l 1 4 x 5 . For simplicity, it is assumed that all states are available, thus making the system observable. Controllability of the system is assured if and only if rank{[B, AB, A2B, AUB]} = 14. (3.16) The control input uL can be written as " L = - # L Q R * L > ( 3 - 1 7 ) where KLQR is the optimal feedback gain matrix. It minimizes a quadratic cost function J which considers tracking errors and energy expenditure, J = fQ(xTLQhQRxL+uTLRLQRuL)dt. (3.18) Here <2LQR a n d ^ L Q R 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 <2LQR c a n be positive semi-definite. The optimal control input uL is given by U L = - ^ L Q R - * - L = ~RLQ_RB F L Q R X L , (3.19) 63 where P L Q R is the positive definite solution to the steady-state matrix Ricatti equation which, for infinite time, becomes PLQRA + ATPLQR ~PLQRBR-LC,RBTPLQR + 6LQR =0. (3.20) The FLT/LQR control strategy is indicated in Figure 3-4. Note, qr represents generalized coordinates controlled by the FLT. 3.2. Simulation Results and Discussion: Commanded Maneuvers The same numerical values are used in the simulation as indicated in Chapter 2. Only the fundamental mode shape is considered for representation of the flexibility. Damping is purposely not included to obtain conservative results and test the controller under severe conditions. Furthermore, character and precise value of damping in the space environment is still somewhat uncertain. To assess the system control under a rather demanding situation, the platform is initially taken to be along the local horizontal (y/p = - 9 0 ° ) , an unstable equilibrium configuration. The manipulator is located at the platform tip to accentuate the maneuver effect, and is initially aligned with the platform (in the folded condition for storage). The system, without any payload, is commanded to undergo simultaneous slew and deployment maneuvers, in a sine-on-ramp profile (Figure 2-3), completed in 0.01 orbit (55.50s), so that ax changes from 180° to 90°; a2 from - 1 8 0 ° to zero; and lx,l2 from 7.5 m to 15 m. 3.2.1. FLT control Figure 3-5 shows the F L T controlled response of the platform libration (y/p), its tip vibration (ep ), the first and second modules' revolute joint rotations {ax,/3x,a2,P2)> a s w e ^ as the modules' tip deflections (ei,e2). The maneuver sets the platform librating with a peak 64 Qr' Qr (qr)dMr)dMr)d FLT Controller LQR Controller Space Manipulator System q,q &qL Figure 3-4 Block diagram illustrating the combined FLT/LQR approach applied to the manipulator system. 65 I.C.'s (Flexible d. off.): Controller Gains: eP = ei = e 2 = ° ; ¥ p : Kp = 0.02; K= 0.3. L.V. 2 Jft+Pi A = A = o. Z„Z2: K p = 8 ; £ v = 5 . 6 7 . ' e I.C.'s (Rigid d. off.): <Fp = - 9 0 ° ; Z, = l2 =7.5m; Of, : Kp = A9, K=U. a, = 180° ,a 2 = - 1 8 0 ° ; Desired Values: a2: Kp=25; K= 10. 2% = - 9 0 ° ; l x = l 2 = 15m; ax -90°,«2 = 0; Platform Libra tion -89.98 r ¥ -90 -90.02 Momentum Gyros Torque 0 0.02 0.04 1st Joint Rotor Motion 0.02 1st Joint Actuator Torque 0.04 500 Ti o -500 0 0.02 0.04 0 0.02 0.04 Deployment of Module 1 1st Deployment Actuator Force 0.02 Orbit 0.04 0.02 Orbit 0.04 Figure 3-5. Response of the system with rigid degrees of freedom controlled by the FLT: (a) rigid degrees of freedom and control inputs for platform and module 1. 66 I.C.'s (Flexible d. off.): Controller Gains: ep = ei = e2=0> ¥p: Kp = 0.02; K= 0.3. L.V. 2 Jg^A ft=02=<>- Z„Z2: K p = 8 ; £ v = 5 . 6 7 . I.C.'s (Rigid d. off.): 1 1 ^ f r ^ 1 Wp = - 9 0 ° ; Z, =l2=1.5m- or,: ^ p = 49, K = U. ax = 1 8 0 ° , a 2 = - 1 8 0 ° ; Desired Values: a2: Kp=25; KV=10. Wp =-90° ; Z[ = Z2 = 15m; a, =90°,tf 2 =0; 2nd Joint Rotor Motion 2nd Joint Actuator Torque Orbit Orbit Figure 3-5. Response of the system with rigid degrees of freedom controlled by the FLT: (b) rigid degrees of freedom and control inputs for module 2. 67 I.C.'s (Flexible d. off.): Controller Gains: eP = e \ = e 2 = ° ; L.V. 2 Jtfr*02 px=p2 = o. Z„Z2: Kp=8; K=5.61. I.C.'s (Rigid d. off.): | 1 U U A Wp = - 9 0 ° ; /2 =l2 = 7.5m; ax: Kp = 49, KV=U. a, = 180°,tf2 = - 1 8 0 ° ; Desired Values: a2: Kp=25; Kv=10. f p = -90° ; /, = l2 = 15m ; a{ = 9 0 ° , a 2 =0; Platform Vibraiton 1st Joint Vibration PI o 0.04 Tip Deflection of Module 1 0 0.02 0.04 2nd Joint Vibration PI 0 0.02 0 0.02 0.04 Tip Deflection of Module 2 -0.02 0.02 Orbit 0.04 0.04 Figure 3-5. Response of the system with rigid degrees of freedom controlled by the FLT: (c) flexible degrees of freedom. 68 amplitude of around 0.02° which is rather small (permissible limit can vary from 0.1° to 1° depending on the mission). The platform returns to the original local horizontal orientation in less than 100 s even in presence of such a severe maneuver ! The negligible (~ 0.01°) limit cycle type oscillations persist due to vibrations of the flexible joints (J3\,fii). The tip response of the massive platform, as expected, is also vanishingly small (= 0.5 - 1 mm). The steady state joint vibration (j3\, jii) amplitudes (3° and 1°, respectively) may be considered acceptable recognizing the fact that the flexible generalized coordinates are uncontrolled, the disturbance is unusually severe and the inherent structural damping is not accounted for. Clearly, the unmodeled dynamics of the flexible generalized coordinates affects the performance of the controller. However, the controller demands remain rather modest, considering the fact that the torque output of Control Momentum Gyros can be as high as 15,000 Nm, revolute joint actuator of the manipulator is able to provide a torque of 4,000 Nm, and the force produced by prismatic joint actuator can be as high as 500 N. The five controlled variables (y/p,ax,lx,a2,md l2) show satisfactory response even during such a large maneuver and display only small oscillations after the desired values are reached. Thus the FLT control of the rigid degrees of freedom does provide encouraging results. 3.2.2. F L T / L Q R control With the control of rigid as well as flexible degrees of freedom (FLT/LQR approach), the situation further improves remarkably (Figure 3-6), particularly in the steady state librational and vibrational responses. In this particular case, the LQR controller is only activated after 0.01 orbit, i.e. at the end of the maneuver. This means, in the first 0.01 orbit, the FLT is used to regulate the large maneuver where it satisfactorily controlled the rigid 69 I.C.'s (Flexible d. off.): Controller Gains: eP = ei = e 2 = ° ; Wp: Kp = 0m; K=0.3. L.V. 2 > r ^ A A= A = o. I.C.'s (Rieid d. off.): IJ2: Kp=8; Kv=5.61. ^ = - 9 0 ° ; Zj =Z2 = 7.5m; ax: t< 0.01 orbit, £ p = 49, Kv = 14; a, = 180° ,a 2 = - 1 8 0 ° ; Desired Values: r>0.01 orbit, LQR. Wp = -90° ; a2: t<0.01 orbit,Kp=25; Kv = 10; Z[ = Z2 = 15m ; = 90°, a2 = 0; t>0.0l orbit, LQR. Platform Libration -89.98 ¥ -90 -90.02 0.02 0.04 1st Joint Rotor Motion a, 0.02 0.04 Deployment of Module 1 15 -12 -9 -6 -0 0.02 0.04 Orbit Momentum Gyros Torque 0.06 0.02 0.04 0.06 1st Joint Actuator Torque 500 Ti o -500 0.06 0 0.02 0.04 0.06 1st Deployment Actuator Force 0.06 0.02 0.04 Orbit 0.06 Figure 3-6. FLT/LQR controlled response of the system: (a) rigid degrees of freedom and control inputs for platform and module 1. 70 C M I.C.'s (Flexible d. off.): ep = e \ = e 2 = 0 ' A = A = o. I.C.'s (Rigid d. off.): A ¥p = - 9 0 ° ; Z, = l2 =7.5m; eP ax =180°, a 2 = - 1 8 0 ° ; Desired Values: = -90° ; /, = l2 = 15m; a. = 90°, a2 = 0; Controller Gains: Wp: Kp = 0.02; Kv=03. \,l2: Kp=8; K=5.67. ax: t < 0.01 orbit, ATp = 49, ATV = 14; r>0.01 orbit, LQR. a2: t <0.01 orbit,K p= 25; ^ v = 10; r>0.01 orbit, LQR. o r . -120 2nd Joint Rotor Motion 2nd Joint Actuator Torque 0.06 Deployment of Module 2 0.02 0.04 Orbit 2nd Deployment Actuator Force F2 0.06 0.02 0.04 Orbit 0.06 Figure 3-6. FLT/LQR controlled response of the system: (b) rigid degrees of freedom and control inputs for module 2. 71 I.C.'s (Flexible d. off.): Controller Gains: eP = ei = e2=0> Wp: Kp = 0.02; K= 0.3. L.V. 2 Jtf^fa A= J32 = 0. I.C.'s (Rigid d. off.): Z„Z2: # „ = 8 ; K=5.61. Wp = - 9 0 ° ; Zj =l2=1.5m; ax: t< 0.01 orbit, Kp = 49, £ v = 14; ax = 1 8 0 ° , a 2 = - 1 8 0 ° ; Desired Values: f>0.01 orbit, LQR. Wp =-90° ; a 2 : r < 0.01 orbit, Kp=25; Kv = 10; l x - l 2 - 15m; c^ j = 90°,or 2 = 0; r>0.01 orbit, LQR. Platform Vibraiton 1st Joint Vibration A° o 0 0.02 0.04 0.06 Tip Deflection of Module 1 0 0.02 0.04 0.06 2nd Joint Vibration 0.02 0.04 0.06 Tip Deflection of Module 2 '2 0 -o.oi y 0.02 0.04 0.06 Orbit 0.02 0.04 0.06 Orbit Figure 3-6. FLT/LQR controlled response of the system: (c) flexible degrees of freedom. 72 degrees of freedom. After 0.01 orbit, the system approaches the steady state phase and vibrates around the reference point. At this stage, the LQR begins to control vibrations. Note, the FLT controller is still active to regulate the platform's attitude and length of the links. The LQR is quite effective in suppressing the joints and platform vibrations which, in turn, help eliminate the librational limit cycle. The joint angles and link lengths attain and remain at their commanded values. Furthermore, the control torques required, in the steady state, are virtually negligible. Effect of payload In the previous two cases, the payload was purposely taken to be zero to help isolate coupling effects. The next logical step was to assess the influence of a point mass payload at the manipulator's tip. Three values of the payload ratio (mass of the payload / mass of the manipulator) were considered: 1, 2 and 5; which correspond to the payloads of 400 kg, 800 kg and 2,000 kg, respectively. The initial configuration of the manipulator remains the same as described in the previous cases. The maneuver, as before, involves a simultaneous -90° slew at revolute joint 1, 180° at revolute joint 2, and 7.5m deployment of the links in a sine-on-ramp profile. It is desired that the maneuver be finished in 0.01 orbit. Figure 3-7 presents results as affected by the payload. The controller gains used by the FLT are indicated in the legend. The gains were purposely kept fixed to help assess robustness of the controller. As before, the LQR becomes effective at 0.01 orbit, i.e. when the maneuver is completed. At the outset, it is apparent that the manipulator is able to attain the commanded values of slew and deployment even in presence of payloads (Figures 3-7a, 7b). As can be anticipated, the peak control efforts increase with an increase in the payload, 73 C. M I.C.'s (Flexible d. off.): eP = ei = e 2 = ° ; / j , = / ? 2 = o. I.C.'s (Rigid d. off.): Wp = - 9 0 ° ; lx =/ 2 =7.5m; ax = 1 8 0 ° , a 2 = - 1 8 0 ° ; Desired Values: Wp = -90 ; l x = l 2 - 15m; ax = 90°,«r2 = 0; Controller Gains: V p : tf, = 0.02; K= 0.3. Kp=8; K=5.67. a,: Kp = 49, KV=14. a. : K-25; K=\0. Payload = 400 kg Payload = 800 kg Payload = 2000 kg Platform Libration -89.98 ¥ o -90 -90.02 h 0.02 0.04 0.06 0.08 1st Joint Rotor Motion 0.02 0.04 0.06 0.08 Deployment of Module 1 0 0.02 0.04 0.06 0.08 Orbit 0.1 Momentum Gyros Torque 0.02 0.04 0.06 0.08 1st Joint Actuator Torque 100 Tj 0 -100 h 20 F 0 0.02 0.04 0.06 0.08 0.1 1st Deployment Actuator Force 0 0.02 0.04 0.06 0.08 0.1 Orbit Figure 3-7. FLT/LQR controlled response of the system showing the effect of payload: (a) rigid degrees of freedom and control inputs for platform and module 1. 74 L.V CM. I.C.'s (Flexible d. off.): ep = ex = e 2 = ° ; A = A = °-I.C.'s (Rigid d. off.): W = - 9 0 ° ; /, =/ 2 =7.5m; a, 180° ,a 2 = - 1 8 0 ° ; Desired Values: ! r% = -90° ; / ,= / ,= 15m; ax = 90°, a 2 = 0; Controller Gains: Wp: Kp = 0.02; Kv=0.3. lx,l2:Kp=%; K=5.61. ax: K = 49, iSTv= 14. : ^ = 2 5 ; tfv= 10. Payload = 400 kg Payload = 800 kg Payload = 2000 kg 2nd Joint Rotor Motion 2nd Joint Actuator Torque 0 0.02 0.04 0.06 0.08 Deployment of Module 2 0.02 0.04 0.06 0.08 Orbit T2 F2 2nd Deployment Actuator Force 0.02 0.04 0.06 0.08 0.1 Orbit Figure 3-7. FLT/LQR controlled response of the system showing the effect of payload: (b) rigid degrees of freedom and control inputs for module 2. 75 L.V C M . I.C.'s (Flexible d. off.): Controller Gains: ep = ei = e 2 = ° ; fl=/?2=0. I.C.'s (Rigid d. off.): ¥ p = - 9 0 ° ; Z, = Z2 =7.5m; a, = 1 8 0 ° , a 2 = - 1 8 0 ° ; Desired Values: ¥ p =-90 ; Z, = Z2 = 15m; a, = 9 0 ° , a2 = 0; Wp: Kp = 0.02; K= 0.3. Z P Z 2 : AT, = 8; ATV= 5.67. or,: ^ = 49, ATV= 14. a2: Kp=25; Kv= 10. Payload = 400 kg Payload = 800 kg Payload = 2000 kg Platform Vibraiton 1st Joint Vibration r , 1 1 • — V J \) -1 1 1 1 1 1 -0.001 0 0.02 0.04 0.06 0.08 0.1 0 Tip Deflection of Module 1 PI 0.02 0.04 0.06 0.08 0.1 2nd Joint Vibration -0.001 0.02 0.04 0.06 0.08 0.1 Tip Deflection of Module 2 0.02 0.04 0.06 0.08 0.1 Orbit 0.02 0.04 0.06 0.08 0.1 Orbit Figure 3-7. FLT/LQR controlled response of the system showing the effect of payloads: (c) flexible degrees of freedom. 76 however the additional demands are rather modest and remain well within the permissible limits. Note, the control effort is required for a relatively longer period of time, however, after approximately 0.08 orbit, the system settles down to the new equilibrium position, very close to the original, and the control effort required is virtually negligible or very small. Flexible degrees of freedom are also controlled, in presence of the payload, quite effectively (Figure 3-7c). Effect of Maneuver Speed Another important system parameter is the speed of the maneuver. The same maneuver as before was considered in absence of a payload. Of course, one would like to complete the maneuver as quickly as possible without adversely affecting the performance. Three different values were considered: 0.005 orbit (fast), 0.01 orbit (nominal), and 0.03 orbit (slow). Results are presented in Figure 3-8. It is apparent that the manipulator attains the commanded values rather quickly after the specified period even in the case of a fast maneuver. The force and torque demands remain modest. Note, the peak platform deviation from the unstable equilibrium position is around 0.05° (Figure 3-8a) for the fast maneuver, and virtually negligible for the slow case of 0.03 orbit. The peak torques and forces encountered are well within the accepted limit (Figure 3-8a,b). Even the flexible degrees of freedom are controlled rather well with the equilibrium configuration regained in less that 0.03 orbit ( = 167 s). Effect of Revolute Joint Stiffness Stiffness of revolute joints also represents a significant variable. Its effect on the controlled performance while executing the same maneuver in 0.01 orbit, with no payload, 77 C. M I.C.'s (Flexible d. off.): eP = ei = e 2 = ° ; fl = A = o. I.C.'s (Rigid d. off.): Wp = - 9 0 ° ; /, = Z2 =7.5m; a, = 180°,a 2 = - 1 8 0 ° ; Desired Values: 2% = -90° ; /, = U= 15m; a, =90 , a 2 =0; Controller Gains: y/p: 7^ = 0.02; K v =0.3. Z,,/,: £ p = 8 ; £ v = 5 . 6 7 . or,: ^ p = 49, «TV= 14. a 2 : ^ = 2 5 ; JSTV= 10. Maneuver speed: 0.005 orbit 0.01 orbit 0.03 orbit -89.95 Vf -90 Platform Libration Momentum Gyros Torque -90.05 h o r , 0 0.02 0.04 0.06 1st Joint Rotor Motion 0.02 0.04 Deployment of Module 1 15 12 9 6 0.02 0.04 Orbit 0 0.02 0.04 0.06 1st Joint Actuator Torque T, 0.06 0 0.02 0.04 0.06 1st Deployment Actuator Force , 1 • 1 1 ; m \j /' -" • 1 t_ L_ -0.06 -200 0.02 0.04 Orbit 0.06 Figure 3-8. Effect of speed of maneuver on the FLT/LQR controlled response: (a) rigid degrees of freedom and control inputs for platform and module 1. 78 I.C.'s (Flexible d. off.): eP = ei = e 2 = ° ; I.C.'s (Rigid d. off.): Wp = - 9 0 ° ; /, = l2 =7.5m; = 1 8 0 ° , a 2 = - 1 8 0 ° ; Desired Values: ¥ p = - 9 0 ° ; h = h = ; ax = 9 0 ° , er2 = 0; Controller Gains: ¥ p : Kp = 0.02; K=03. lltl2: Kp=S; K=5.61. a,: Kp = 49, K=U. a2: Kp=25; K= 10. Maneuver speed: 0.005 orbit 0.01 orbit 0.03 orbit 2nd Joint Rotor Motion 2nd Joint Actuator Torque a. -180 K-0 0.02 0.04 Deployment of Module 2 0.06 2nd Deployment Actuator Force F2 0.02 0.04 Orbit 0.06 0.02 0.04 Orbit 0.06 Figure 3-8. Effect of speed of maneuver on the F L T / L Q R controlled response: (b) rigid degrees of freedom and control inputs for module 2. 79 C. M. I.C.'s (Flexible d. off.): eP = ei = e 2 = ° ; A= A = o. I.C.'s (Rigid d. off.): Wp = - 9 0 ° ; Z, = Z2 =7.5m; = 180°,a 2 = - 1 8 0 ° ; Desired Values: ^ p = -90° ; Zj = Z2 = 15m; a, = 9 0 ° , a 2 =0; Controller Gains: * „ = 0.02; * „ = 0.3. Z„Z 2: £ , = 8; Kv=5.61. a,: =49, tf = 14. : ^ p = 2 5 ; K = 10. Maneuver speed: 0.005 orbit 0.01 orbit 0.03 orbit Platform Vibraiton 1st Joint Vibration xl 0" 3 m i t ii ii nt I'I H i1 0.02 0.04 0.06 Tip Deflection of Module 1 0.02 0.04 0.06 Tip Deflection of Module 2 0.02 0.04 Orbit 0.06 PI PI 0.02 0.04 2nd Joint Vibration 0.02 0.04 Orbit 0.06 - 1 « 1 1 : ' ll - II, . i<l 1 y \ > i > | / ^ II ,i "« |! • | I 1 1 > 0.06 Figure 3-8. Effect of speed of maneuver on the FLT/LQR controlled response: (c) flexible degrees of freedom. 80 was also assessed. These results are presented in Figure 3-9. Two stiffness values, one below (soft) and the other above (hard) the nominal value of l x l O 4 Nm/rad were considered. Even in the demanding situation presented by the soft spring, the system settles down to the commanded values in around 0.04 orbit (= 167 s). As before, the demands on control forces and torques continue to remain modest. It is important to point out that gains during the studies aimed at assessing the influence of payload, maneuvering speed and stiffness variations are intentionally kept the same to demonstrate robust character of the FLT/LQR control. Based on the investigation reported in this chapter, it can be concluded that both the FLT by itself as well as a synthesis of the FLT and LQR, or other linear control procedure, appear quite promising. They should receive further attention in refining their implementation. 81 L.V C M I.C.'s (Flexible d. off .) : ep = e i = e2=0> A= 02 = o. I.C.'s (Rigid d. off .) : W p = - 9 0 ° ; lx = l2 =7.5m; ax = 180° ,a 2 = - 1 8 0 ° ; Desired Values: Wp = -90° / j —12 15m; ax = 90°, a2 = 0; Controller Gains: Wp: * p = 0 . 0 2 ; K= 0.3. lx,l2: Kp=&; K=5.61. ax: Kp = 49, K=U. a2: Kp=25; Kv=10. Joint stiffness: 5x l0 3 Nm/rad l x l O 4 Nm/rad 5xl0 4 Nm/rad -89.98 -90.02 a, 180 150 120 90 Platform Libration Momentum Gyros Torque 0.02 1st Joint Rotor Motion 0 0.02 1st Joint Actuator Torque 0.04 , ! , 500 Nm L v -• \ • Ti o /AV V -500 1 1 , —1 1 I_ 0 0.02 Deployment of Module 1 0.04 0.02 0.04 1st Deployment Actuator Force Fi 0.02 Orbit 0.04 0.02 0.04 Orbit 0.06 Figure 3-9. FLT/LQR controlled response as affected by the revolute joint stiffness: (a) rigid degrees of freedom and control inputs for platform and module 1. 82 I.C.'s (Flexible d. off .) : ep = ei = e2=0> I.C.'s (Rigid d. off .) : Wp = -90° ; Z, = Z2 = 7.5m; a, =180°,a2 =-180°; Desired Values: Vp = -90° ; Zj = l2 - 15m; ax = 90°, a2 = 0; Controller Gains: ¥ p : Kp = 0.02; K = 0.3. ZPZ2: 7^=8; Kv=5.61. a,: Kp = 49, Kv=14. a2: Kp=25; K = 10. Joint stiffness: 5xl0 3 Nm/rad lxlO 4 Nm/rad 5xl0 4 Nm/rad 2nd Joint Rotor Motion 0 0.02 Deployment of Module 2 0.02 Orbit 2nd Joint Actuator Torque "Nm A' /' V A I1 -I • 0.04 0.02 0.04 2nd Deployment Actuator Force F2 0.04 0.02 Orbit 0.04 Figure 3-9. FLT/LQR controlled response as affected by the revolute joint stiffness: (b) rigid degrees of freedom and control inputs for module 2. 83 I.C.'s (Flexible d. off .) : ep = ei = e2=°; I.C.'s (Rigid d. off .) : 2% =-90°; Z, =Z2 = 7.5m; a, = 180°,a2 =-180°; Desired Values: Wp = -90°; Z, = Z2 = 15m; ax = 90°, a 2 =0; Controller Gains: Wp: Kp = 0.02; K=Q3. lltl2: Kp = 8; K=5.61. ax: Kp = 49, K = U. a2: Kp=25; KV=10. Joint stiffness: 5xl0 3 Nm/rad lxlO 4 Nm/rad 5xl0 4 Nm/rad Platform Vibraiton 1st Joint Vibration 'p o -l "x10"3m 1 ' i 1 i PI 0 0.02 0.04 0.06 Tip Deflection of Module 1 0.02 2nd Joint Vibration 0.06 0.01 -o.oi \ 0 0.02 0.04 0.06 Tip Deflection of Module 2 0.02 0.04 Orbit 0.06 0.02 0.04 Orbit 0.06 Figure 3-9. FLT/LQR controlled response as affected by the revolute joint stiffness: (c) flexible degrees of freedom. 84 4. T U N E D M O D A L C O N T R O L Among the fundamental developments in the modern control theory are the two sets of analytical results that underlie the linear quadratic regulator (LQR) and ei gen structure assignment regulator (EAR). Design and implementation of practical control of flexible structures have been accomplished using both the design techniques. In the LQR approach, the central feature is the minimization of a quadratic performance index, subject to a linearized system model. However, a major drawback of the LQR is that it has no direct control of the system eigenstructure, which determines not only the level of stability but also the specific nature of the response to a control input (e.g., a step function). The LQR method does not involve the assignment of the system eigenstructure in a specified manner. Consequently, it is desirable to employ a control strategy that has the capability to modify the system eigenstructure appropriately to meet specified requirements. Such a control approach would prove more effective if the capability of the parameter tuning is available as well. To this end, a modal control strategy is introduced here. An intelligent control system, which combines a modal controller and a fuzzy tuning structure, is developed to 'intelligently' assign the system eigenstructure so as to obtain better performance of the controller in terms of response speed, overshoot, and steady state offset. Simulation studies have been carried out using this intelligent control system to suppress vibrations of a ground-based deployable manipulator. The approach may be conveniently applied to a space-based manipulator as well. 4.1. Control System Development 4.1.1. Eigenvalue assignment When the manipulator configuration is fixed or nearly fixed, as is the case near the end of maneuvers, a linear approximation may be applied. The linearized equation can be written as 85 ML ML. 0 / M K 0 + 0 M " 1 XL A 5 It can be expressed as xL = AxL + BuL, (4.2) where the time-dependent state vector xL contains generalized coordinates and their first time derivatives. The square matrix A is composed of the matrices of mass, damping and stiffness. The term BuL{t) represents the effect of control action, with uL{t) and B being the control force (torque) vector and actuator placement (i.e. where the actuator is located) matrix, respectively. As is normally the case in such studies, all states are assumed to be available thus making the system observable [46]. By introducing state feedback, the control input uL can be written as uL = -KxL. (4.3) Thus one obtains a closed loop system xL=(A-BK)xL. (4.4) In Eq. (4.4), matrix A - BK decides the modal parameters of the closed loop system, such as the modal frequencies, damping ratios and mode shapes [61]. A relation exists, between the modal parameters of the system and the eigen-parameters of matrix A-BK, as eigen-parameters decide the controlled behavior of the closed loop system, Eq.(4.4). To obtain the relation explicitly, it is useful to define some notations. Assuming A-BK to be a matrix of real-numbers, the eigenvalues and eigenvectors of A-BK appear as conjugate pairs. Let /L,._, and A\{ be the ith pair of eigenvalues, and z2i_l and z2,be the corresponding /th pair of eigenvectors. Also let 0Jt, £t and ny denote, respectively, the modal frequency, damping ratio and mode shape of the rth mode. Then we have: 86 4--i = -£^+M>/w?; for i = 1 to n , (4.5) and z2,-i = j ^ " 1 " ' ' | ; Z2,- = | ^ n ' } ; for / = 1 to n , (4.6) where j = , and n is the number of degrees of freedom of the system. Equations (4.5) and (4.6) give a one-to-one mapping between the system modal parameters and the ei gen-parameters of matrix A-BK. Therefore, if the modal parameters coi £ and « ( are specified in the domain, one can calculate the corresponding eigenvalues and eigenvectors for the closed loop system using Eq. (4.5) and (4.6). Moreover, according to Eq. (4.4), if one can modify and assign the eigenstructure at desired values by selecting proper feedback matrix K, the modal property of the system can be modified accordingly. This is the essence of the modal control. It is also the reason why modal control is also called eigenvalue assignment control. 4.1.2. Hierarchical structure The control system developed for the deployable manipulator system has a three-level structure. This hierarchical structure combines the advantages of a crisp controller, i.e. a modal controller, with those of a soft, knowledge-based, supervisory controller. The overall structure can be developed into three main layers. Bottom Layer The first layer deals with information coming from sensors attached to the system. This type of information is characterized by a large amount of high resolution data points produced and collected at high frequency. The crisp controller used is a state feedback 87 regulator with feedback gain matrix determined using the eigenstructure assignment approach. The control algorithm can be described as x = Ax + Bu ; (4.7) u=-Kx ; where u is the control action and K the feedback matrix. Second Layer The data processing for monitoring and evaluation of the system performance occurs in the intermediate layer. Here high-resolution, crisp data from sensors are filtered to allow representation of the current state of the manipulator. This servo-expert layer acts as an interface between the crisp controller, which regulates the servomotors at the bottom layer, and the knowledge-based controller at the top layer. The intermediate layer handles such tasks as performance specification, response processing, and computation of performance indices. This stage involves, for example, averaging or filtering of the data points, and computation of the rise time, overshoot, and steady state offset. Top Layer The knowledge base and the inference engine in the uppermost layer are used to make decisions that achieve the overall control objective, particularly by improving the performance of low-level direct control. This layer can serve such functions as monitoring the performance of the overall system, assessment of the quality of operation, tuning of the low-level controllers, and general supervisory control. In this layer, there is a high degree of information fuzziness and a relatively low control bandwidth. Figure 4-1 presents the hierarchical structure of the three-level control system. 88 1-1 c o o > o I o <D c o c <D CO a, )-< o '-4—» 03 fi CD -C o on ON OO 4.1.3. Performance specification, evaluation, and classification The desired performance of the system is specified in terms of the following time domain parameters: • Rise time (RSTd ); • Overshoot, if underdamped (OVSd ); • Offset at steady state (OFS d ). We use these three parameters to present the desired performance of the system. The rise-time is chosen as the time it takes for the response to reach 95% of the desired steady-state response. The overshoot is calculated at the first peak of the response, as a percentage of the desired steady-state response. The percentage steady-state offset is computed by taking the difference between the average of the last third of the response and the desired response. The corresponding time domain parameters are obtained from the response of the actual system, with the subscript r referring to the real system response as: RSTr, OVSr, andOFSr. Once evaluated, the parameters of the real system are compared with the desired ones to get the index of deviation. For each performance attribute, an index of deviation is calculated using the following equation, , , . . , .m •, i ithdesired attribute .. Index of deviation of i attribute = 1 — . (4.0) / actual attribute The index is defined in such a way that a value of 1 corresponds to the worst-case performance, while a value of 0 means the actual performance of the system, for that particular attribute, exactly meets the specification. The indices are calculated according to 90 RST, = 1 - = ERR(l), (4.9) RSTr OVSt = 1 - = ERR(2), (4.10) OVSr OFS OFSi = 1 d- = ERRQ). (4.11) OFSr These indices represent the performance of the system and hence should correspond to the context of the rulebase of system tuning. The index of deviation is therefore fuzzified into membership values according to the five selected primary fuzzy states: Highly Unsatisfactory (HIUN), Needs Improvement (NDIM), Acceptable (ACCP), In Specification (INSP) and Over Specification (OVSP). In order to obtain a discrete set of performance indices K(i), threshold values 77/(0 a r e defined for each index of deviation over the interval -°° to 1, as given in Table 4-1. Table 4-1 Mapping from the index of deviation to a discrete performance index. Discrete Performance Index K(i) Index of Deviation 5 ERR(i) < 0 4 0 < ERR(i) < TH(1) 3 TH(1) < ERR(i) <TH(2) 2 TH(2) < ERR(i) < TH(3) 1 TH(3) < ERR(i) < 1 The performance indices obtained in this manner are the input to a Fuzzy Inference System (FIS) which tunes the modal frequencies and damping ratios of the closed-loop system. The 91 output from FIS is the tuning action that is used to update modal frequencies and modal damping ratios of the closed-loop system. Therefore, closed-loop poles can be modified correspondingly. 4.1.4. Fuzzy tuner layer At the highest level of the hierarchical structure, there is a knowledge base for tuning a crisp controller. This knowledge may originate from human experts or some form of archives, and is expressed as linguistic rules containing fuzzy terms. For each status (context) of the system, a conceptual abstraction is computed, and the expert knowledge is transformed into a mathematical form by the use of the fuzzy set theory and fuzzy logic operations. A Fuzzy Inference System (FIS) has been built using the Matlab Toolbox to this end. To construct the system, one must first assign a membership function to each of the performance indices and tuning parameters. Then the knowledge base should be created. Taking performance indices as the input, Fuzzy Inference System carries out such tasks as fuzzification of the performance indices, operations of the fuzzy set, and defuzzifying of the tuning actions. The output of the FIS is a crisp tuning action corresponding to numerical context values of the system condition. The tuned parameters are chosen to be the modal frequencies and modal damping ratios. As mentioned before, eigenstructure of the closed loop system plays a key role in determining the system performance. Required performance can be achieved by properly assigning the system eigenstructure. There are relationships between system eigenstructure and system modal parameters (Eqs. 4.5, 4.6). They provide a way to modify the eigenstructure by tuning modal frequencies and modal damping ratios. These modal parameters are physically meaningful and hence chosen as the tuned parameters. 92 If coi and represent the /th modal frequency and damping ratio, respectively, the relationship between modal parameters and system eigenstructure is given by Eqs. (4.5) and (4.6). A t each tuning step, the values of coi and are updated according to the tuning actions obtained from the Fuzzy Inference System (FIS). Once updated, the new values of parameters coi and are used to determine the new desired eigrnstructure of the system. Relations used for updating coni and are: where the subscript 'new' denotes the updated value and 'o ld ' refers to the previous value. The incremental action taken by the fuzzy controller is denoted by L\coi and A£t. Parameters 0)isen and £ i s e n are introduced to adjust sensitivity of tuning, when needed. 4.1.5. Construction of fuzzy inference system The expert tuning knowledge for a modal controller may utilize heuristics such as those given in Table 4-2. One may define the primary fuzzy sets for the performance indices for each context variable (i.e., RST, OVS, and OFS) as given in Table 4-3. Fuzzy tuning variables are defined as follows: DFREQ,. = Change in the /th modal frequency; DDAMP,. = Change in the /th modal damping ratio. (4.12) 93 Table 4-2 Heuristics of modal control tuning Actions for Performance Improvement Context Modal Frequency coni Modal Damping Ratio £ Rise Time (RST) Increase Decrease Overshoot (OVS) Increase Offset (OFS) Increase Decrease Table 4-3 Fuzzy labels of performance indices Performance Index Context Fuzzy Set Notation Fuzzy Value 1 HIUN Highly Unsatisfactory 2 NDIM Needs Improvement 3 ACCP Acceptable 4 INSP In Specification 5 OVSP Over Specification Each tuning variable may be expressed with the fuzzy sets and representative numerical values that are listed in Table 4-4. The rulebase for control parameter tuning is given in Figure 4-2. 94 If RST is HIUN, then DFREQ, is PL, DDAMP, is NM, or If RST is NDIM, then DFREQ, is P M , DDAMP is NS, or If RST is ACCP, then DFREQ i is PM, DDAMP is ZR, or If RST is INSP, then DFREQ, is ZR, DDAMP, is ZR, or If RST is OVSP, then DFREQ, is MS, DDAMP. is ZR, or If OVS is HUIN, thenDFPLQ, is /VM, DDAMP is FL , or If OVS is NDIM, then DFREQ is/VS, DDAMP, is P M , or If OVS is ACCP, then DFREQ i sZP, DDAMP, i sP5, or If OVS is INSP, then DFREQ. is ZR, DDAMP. i sZF , or If OVS is OVSP, then DFREQ, is PS, DDAMP. is/VS, or If OFS is HIUN, then DFREQ, is PM, DDAMP ,. is /VS, or If OFS is NDIM, then DFREQ is PS, DDAMP, is TVS, or If OFS is ACCP, then DFREQ t isZfl , DDAMP. is MS, or If OFS is INSP, then DFREQ. is ZR, DDAMP t isZ/?, or If OFS is OVSP, then DFREQ, is NS, DDAMP. i sZP, Figure 4-2 Rulebase for the control parameter tuning. 95 Table 4-4 Tuning fuzzy sets and representative numerical values Tuning Fuzzy Set Integer Value Notation Fuzzy Value PL Positive Large 3 PM Positive Moderate 2 PS Positive Small 1 ZR Zero 0 NS Negative Small -1 NM Negative Moderate -2 NL Negative Large -3 Triangular membership functions for the performance attributes RST, OVS, OFS, and for the fuzzy tuning actions DFREQn DDAMP, are given in Figure 4-3 and Figure 4-4, respectively. Each fuzzy action or condition quantity has a representative value, which is assigned a membership grade equal to unity. The decreasing membership grade around that representative value introduces a degree of fuzziness. 4.2. Ground-Based Simulation 4.2.1. Modeling of a ground-based manipulator system The ground-based manipulator system considered for fuzzy tuning modal control is shown in Figure 4-5. The system consists of a single module manipulator carrying a point-mass payload held by the end effector. The module has two rigid links. The first l ink 96 Figure 4-3 Membership functions for the fuzzy performance attributes. Membership Grade . NL NM NS ZR PS PM PL i i 1 1 ^ • - 3 - 2 - 1 0 1 2 3 Tuning Action Figure 4-4 Membership functions for the fuzzy tuning actions. 97 Prismatic Figure 4-5 Configuration of the single-module manipulator with revolute and prismatic joints. undergoes slewing motion through a flexible revolute joint. The other link can be deployed and retrieved by the rigid prismatic joint. The motion of the manipulator is confined to the horizontal plane, i.e. the gravity effects are not present. The revolute joint is considered flexible. As before, it is modeled by a linear torsional spring, with stiffness K and damping C, that connects the rotor of the servomotor to the slewing link. Figure 4-6 shows a schematic diagram of the flexible joint with details. The angular motion of the rotor, with respect to stator, is denoted by a . The angular deformation of the torsional spring is represented by ft . Thus 0 = a + /? is the total angular displacement of the slewing link. 4.2.2. Control system and simulation results As mentioned before, the hierarchical structure used combines the advantages of a crisp modal controller with those of a soft, knowledge-based, supervisory controller. The three layers of the structure implement such tasks as collection of information coming from 98 Figure 4-6 The flexible revolute joint model. Here J represents the rotor's mass moment of inertia. 99 sensors, data processing and information abstraction, as well as general supervisory control. This hierarchical control system is used to suppress the vibrations of the manipulator system described in Figure 4-5. The effectiveness of the control system is assessed by investigating simulation cases of suppression of vibrations caused by different initial disturbances. The length of the module may be specified at fixed value through Lagrange multiplier. Therefore, the subsystem considered for control simulation has two degrees of freedom, with a, j3, a, ft as the system state variables. The parameters for the first simulation case are given in Figure 4-7. The initial feedback control gain is determined using the LQR method. Based on this, tuning action takes place. The tuning process consists of analyzing the response to an initial disturbance listed in Figure 4-7, with respect to the performance requirements of rise time, overshoot, and steady-state error. The feedback gain matrix is updated by the supervisory controller accordingly. Figure 4-8(a) shows the system response when controlled using the LQR strategy. The initial displacement ( = 2°) of the torsional spring at revolute joint results in vibration at a and /?, and the suppression of the vibrations can be observed due to the application of the LQR controller. As can be seen, the convergence speed is slow in this case, and significant vibration remains after 10 seconds. Figure 4-8(b) shows the results after fuzzy tuning is applied. It can be observed that the convergence speed is much faster compared to the LQR controlled case. Within 4 seconds, the vibrations at every degree of freedom have been eliminated. Further more, the peak amplitudes of the vibrations are evidently reduced. 100 (a) Time, s Time, s Figure 4-8 System response to an initial displacement at the revolute joint: (a) controlled by L Q R ; (b) controlled by hierarchical controller. 102 The system response after each tuning action is shown in Figure 4-9. At each step, the rise time, the overshoot, and the offset are computed and then fuzzified. For each of these attributes, a context value is thus obtained. The intelligent supervisor uses these context values to determine the appropriate inferences, i.e. the required tuning actions. The evolution of the system pole locations after each tuning action is shown in Figure 4-10. The second case studies the controlled behavior of the system when an initial velocity is given as a disturbance. The initial conditions for this case become: a(0) = 0; ,0(0) = 0; a(0) = 0;/?(0) = 2° / s . The rest of the parameters remains the same as in Figure 4-7. Figures 4-11 (a) and 11 (b) show the system response when controlled by the LQR and the hierarchical control system, respectively. It can be seen that, when controlled by the hierarchical approach, the joint vibration is eliminated much faster than that by the LQR. The hierarchical control system is also able to reduce the peak amplitudes of vibrations significantly. The system response after each tuning action is shown in Figure 4-12, and the evolution of the system pole locations after each tuning action is shown in Figure 4-13. In the above two cases, the response of a ground-based deployable manipulator, experiencing vibrations due to the initial disturbances at the revolute joint, was studied. To further evaluate the effectiveness of the hierarchical control system, a case of simultaneous 30° slew and 0.5 m deployment in 10 seconds was considered. Now the slew motion at the revolute joint and deployment at the prismatic joint are controlled by the FLT. The desired profiles are described as 103 Step 1 Figure 4-9 System response at each tuning action. 104 (a) 0.25 0.2 0.15 0.1 ca 0.05 ton o •0.05 -0.1 •0.15 -0.2 •0.25 f T \ 7 V w \ \ \ V 1 1 1 -0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Real (b) a ton 03 B 2 0 -2 -4 -6 -8 Si 7 A -0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Real Figure 4-10 Evolution of the system pole locations during tuning: (a) pole locations for the first mode; (b) pole locations for the second mode. 105 Figure 4-11 System response to an initial velocity ft (2° per second) applied at the revolute joint: (a) controlled by LQR; (b) controlled by hierarchical controller. 106 Step 1 Time, s Time, s Figure 4-12 System response at each tuning action. 107 Figure 4-13 Evolution of the system pole locations during tuning: (a) pole locations for the first mode; (b) pole locations for the second mode. 108 where qs is the specified set of coordinates (ex,I); Aqs is its desired variation (Aa,Al); r is the time; and Ar i s the time required for the maneuver. As can be expected, the large scale motions at both revolute and prismatic joints would result in vibration at the revolute joint, and this would persist if damping is not present in the system. This suggests a need for active control to suppress the maneuver-induced vibration. To that end, two control approaches are considered: the LQR and Tuned Modal Control. They are applied after the FLT-regulated maneuver is finished. Figure 4-14 (a) shows the system response when controlled by FLT/LQR. The rigid degrees of freedom are regulated very well within the first 10 seconds. After that, the LQR is applied to suppress the maneuver-induced vibration at the flexible revolute joint. One can see that the LQR is effective but its convergence speed is slow. Figure 4-14 (b) shows the system response when the synthesis of the FLT and Tuned Modal Control is employed. To obtain faster convergence speed, tuning action is carried out based on the LQR feedback gain matrix. It can be seen from Figure 4-14 (b) that, after a few tuning steps, much faster convergence speed is achieved. The vibration at the flexible revolute joint is quickly eliminated right after completion of the maneuver, without any oscillations. Therefore, the developed knowledge-based tuning system is quite effective in improving the controller performance in presence of maneuvers. It should be pointed out that, by changing weight matrix of the LQR, a faster response than that shown in Figure 4-14 (a) may be achievable. However it is still significant to evaluate the effectiveness of the 'intelligent' tuning system in improving the controller performance. 109 o 1-1 > o U T3 O -a c 3 03 CJ . . -C <D "O C OJ 03 3 B S -C O se j=i . -Pi 1 £ C O 73 P - i CD CD 7S s ° S § t o <-> 00 A 4J I i 3 DO • i-H PH The system response after each tuning action is shown in Figure 4-15, and the evolution of the system pole locations is shown in Figure 4-16. 4.3 Concluding Remarks A knowledge-based hierarchical control system is developed for the vibration control of a manipulator system. For this purpose, a fuzzy inference system (FIS) is established first. The FIS is then combined with a crisp modal controller to construct a hierarchical control system. The effectiveness of the hierarchical control system is investigated through three simulation cases. In the first two cases, the system is experiencing vibration due to the initial disturbance at the revolute joint. The third case considers a system going through a simultaneous slew and deployment maneuver. The knowledge-based hierarchical control system proves to be quite effective in all the three cases. The performance can be improved by further tuning. 111 Step 1 Time, s Time, s Time, s Step 4 Time, s Time, s Time, s Figure 4-15 System response at each tuning action when FLT/Tuned Modal Control is applied. 112 " f r - " - " - " - " J L J - • ' " - K A r -0.2 -0.15 -0.1 -0.05 0 Real - f - v - , - ^ 1 ! T 1 i n 1 J i L J i T 1 p-""" - A ' " 1 •0 " " " ; -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Real Figure 4-16 Evolution of the system pole locations during tuning: (a) pole locations for the first mode; (b) pole locations for the second mode. 113 5. C O N C L U D I N G R E M A R K S The thesis studies dynamics and control of a variable-geometry manipulator which may be used in space - as well as ground - based operations. To begin with, a detailed dynamical response study is undertaken which assesses the influence of initial conditions, system parameters, and manipulator maneuvers. This is followed by an investigation on control of the manipulator system. Three different control methodologies are used: Feedback Linearization Technique (FLT), its synthesis with the Linear Quadratic Regulator (LQR), and Modal Control with Intelligent Tuning. A three-layer hierarchical control system, consisting of a modal controller and a knowledge-based decision making system, is developed for monitoring and tuning of the controller. The simulation experiments are carried out on controlled response of space-as well as ground-based deployable manipulator using different control approaches. 5.1. Contributions The main contributions of the thesis can be summarized as follows: (a) A detailed dynamical study of a novel flexible space-based manipulator provides better understanding of such a complex system and would help towards its efficient design. (b) • Control of this novel manipulator has received relatively little attention. The study clearly establishes that even the FLT by itself can regulate the rigid degrees of freedom quite well. This would make control implementation using the FLT relatively simple. Although the flexible joint vibrations ( / 5 j , / ? 2 ) persist, their amplitudes remain rather small (1°- 3°) and would be acceptable in most situations. This is an important piece of information from control consideration. 114 • Synthetic control using the FLT as well as LQR for rigid and flexible degrees of freedom, respectively, improves the control performance significantly. Now all vibrations are damped within 0.04 orbit, representing considerable improvement in the control performance. It is important to emphasize that the synthetic control is quite robust. (c) The Modal Control with Intelligent Tuning has received virtually no attention. Here tuning knowledge of the system was gained through control expertise, and expressed mathematically as a set of fuzzy rules, using fuzzy logic operations. Design of a hierarchical control system, comprising a modal controller and a knowledge-based decision making system, applicable to a prototype variable-geometry robotic manipulator is an important development of a far-reaching consequence. Ground-based manipulator was controlled and tuned successfully in simulations. (d) Such a comprehensive numerical simulation study of modal control with intelligent tuning, and its application in the control of a manipulator system, has not been reported before, and represents an important step forward. 5.2. Concluding Remarks Based on the investigation, following general conclusions can be made: (a) • Significant coupling exists between the platform, link and joint vibrations, as well as system libration. Pronounced interactions were observed between the joint and link vibrations. In general, slewing and deployment maneuvers have a significant effect on the response of flexible degrees of freedom. • When the manipulator base is located near the platform's extremity, slewing and deployment maneuvers can result in significant rigid body motion of the platform. 115 Excitation of the system's flexible degrees of freedom can significantly deteriorate the accuracy of the manipulator. In general, payload, speed of maneuver and joint flexibility represent three important parameters governing the response of the system. As can be expected, heavier payload, faster speed of maneuver and reduced joint stiffness affect the manipulator tip dynamics adversely. The system exhibits unacceptable response under critical combinations of parameters. The control strategy based on the FLT is found to be quite effective in regulating the rigid-body motion of manipulator links as well as the attitude motion of the platform. Active control of flexible degrees of freedom using the LQR, together with the FLT for rigid generalized coordinates, significantly improves the situation. The controller is quite robust and continues to be effective even in presence of heavy payloads, fast maneuvers and reduced stiffness of the revolute joints. The results should prove useful in the design of this new class of promising manipulators. Fuzzy logic offers a mathematical framework to acquire and manipulate information available from human experts, particularly in linguistic form, and to make inferences for practical uses such as control and system tuning. Monitoring and controller tuning of a manipulator could be accomplished by the use of knowledge-based techniques in a fuzzy logic framework, and thereby reducing the degree of complexity of processing low-level data. The performance of modal controller for the vibration control of a manipulator can be significantly improved through knowledge-based tuning. • The three-layer control system, developed in the present work, promises to be quite effective where active vibration control is needed, (d) The methodology used in the present development is general enough to be useful in many other applications, particularly where a process plant is complex and difficult to model and expert knowledge is available. 5.3. Recommendations for Future Work Based on the results obtained, several avenues are suggested for possible consideration for future work which are likely to be beneficial: (i) Comparative study of various optimal, adaptive, intelligent and hierarchical control strategies to regulate the rigid and flexible dynamics of single and multi-module systems. Introduce an intelligent tuning system as well to the rigid degree regulator, e.g. the FLT, and synthesize with tuned modal controller to implement intelligent tuning control of both rigid and flexible degrees of freedom of a manipulator. (ii) Undertake ground-based experiments to help validate numerical simulation results. Incorporate a flexible joint and a flexible link in the present prototype experimental system for more realistic simulation studies. (iii) More detailed investigation on application of Eigenstructure Assignment Regulator (EAR) for the flexible deployable manipulator system, e.g. decoupling of vibrations of different modes, modification of dominating vibration modes, 'intelligent' assignments of both eigenvalues and eigenvectors, etc. (iv) Study the use of prismatic joint actuators for active suppression of vibrations. 117 REFERENCES [I] Evans, B., "Robots in Space," Spaceflight, Vol . 35, No. 12, 1993, pp. 407-409. [2] Canadian Science and Technology, O'Brien Publishing, Ottawa, Ontario, Fall 1995, p . l l . [3] Bassett, D.A., Wojik, Z.A., Hoefer, S., and Wittenborg, J., "Mobile Servicing System Assembly, Checkout and Operations," 46th International Astronautical Congress, Oslo, Norway, October 1995, Paper No. IAF-95-T.3.04. 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[30] Lips, K.W., and Modi, V.J . , "Three-Dimensional Response Characteristics for a Spacecraft with Deploying Flexible Appendages", Journal of Spacecraft and Rockets, Vol. 4, 1981, pp. 650-656. [31] Modi, V.J . , and Ibrahim, A . M . , "A General Formulation for Librational Dynamics of Spacecraft with Deploying Appendages", Journal of Guidance, Control, and Dynamics, Vol . 7, 1984, pp. 563-569. [32] Modi, V.J . , and Shen, Q., "On the Dynamics of Spacecraft with Flexible, Deployable and Slewing Appendages", Advances in the Astronautical Sciences, Editors: A . K . Misra et al., Univelt Incorporated Publisher for the American Astronautical Society, San Diego, U.S.A., Vol . 85, Part I, pp. 877-896. [33] Lips, K.W., Dynamics of a Large Class of Satellites with Deploying Flexible Appendages, Ph.D. Thesis, University of British Columbia, Vancouver, B.C., Canada, 1980. [34] Ibrahim, A . M . , Mathematical Modeling of Flexible Multibody Dynamics with Application to Orbiting Systems, Ph.D. Thesis, University of British Columbia, Vancouver, B.C., Canada, 1988. [35] Shen, Q., On the Dynamics of Spacecraft with Flexible, Deployable and Slewing Appendages, Ph.D. Thesis, Simon Fraser University, Burnaby, B.C., Canada, 1993. [36] Marom, I., and Modi, V.J . , "On the Dynamics and Control of the Manipulator with a Deployable Arm", AAS/AIAA Astrodynamics Specialist Conference, Victoria, B.C., Canada, August 1993, Paper No. AAS-93-670; also Advances in the Astronautical Sciences, Editors: A . K . Misra et al., Univelt Incorporated Publisher for the American Astronautical Society, San Diego, U.S.A., Vol . 85, Part III, pp. 2211-2229. [37] Modi, V.J . , Chen, Y . , Misra, A .K . , de Silva, C.W., and Marom, I., "Nonlinear Dynamics and Control of a Class of Variable Geometry Mobile Manipulators", 120 Proceedings of the Fifth International Conference on Adaptive Structures, Sendai, Japan, December 1994, Editor: J. Tani et al., Technomic Publishing Co. Inc., Lancaster, PA, U.S.A., pp. 140-149. [38] Hokamoto, S., Modi., V.J . , and Misra, A .K . , "Dynamics and Control of Mobile Flexible Manipulators with Slewing and Deployable Links", A A S / A I A A Astrodynamics Specialist Conference, Halifax, Nova Scotia, Canada, August 1995, Paper No. AAS-95-322; also Advances in the Astronautical Sciences, A A S Publications office, San Diego, California, U.S.A., Editors: K . Terry Alfriend et al., Vol . 90, pp. 339-357. 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Dynamics and control of a novel manipulator with deployable and slewing links Zhang, Jian 2002
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Title | Dynamics and control of a novel manipulator with deployable and slewing links |
Creator |
Zhang, Jian |
Date Issued | 2002 |
Description | The thesis investigates the planar dynamics and control of a novel, flexible, multimodule manipulator with slewing and deployable links, which may be used in space - as well as ground - based operations. The system is composed of a flexible orbiting platform supporting two modules connected in a chain topology. Each module consists of two links: one free to slew while the other permitted to deploy and retrieve. There are three major aspects to the study. To begin with, a detailed dynamical response study is undertaken which assesses the influence of initial conditions, system parameters, and manipulator maneuvers on the system response. Results suggest that, under critical combinations of parameters, the response may not conform to the acceptable limit. This points to a need for active control. Next, the study focuses on the behavior of the system using two different control methodologies: (i) the nonlinear Feedback Linearization Technique (FLT) applied to rigid degrees of freedom with flexible generalized coordinates indirectly regulated through coupling; (ii) an integration of the FLT and Linear Quadratic Regulator (LQR) to achieve active control of both rigid and flexible degrees of freedom. Finally, the thesis presents the development of an intelligent hierarchical system for the vibration control of the manipulator. The emphasis is on the use of knowledge-based tuning of the low-level direct controller so as to improve the performance of the control system. For this purpose, first a fuzzy inference system (FIS) is developed. The FIS is then combined with a conventional modal controller to construct a hierarchical control system. Specifically, a knowledge-based fuzzy system is used to tune the parameters of the modal controller. The effectiveness of the hierarchical control system is assessed through simulation studies. Results show that the knowledge-based hierarchical control system is quite effective in suppressing vibrations due to a wide variety of disturbances, and performance of the modal controller can be significantly improved through knowledge-based tuning. Such a comprehensive investigation involving dynamics and controlled performance of a robotic system should prove useful in the design of this new class of manipulators. The study lays a sound foundation for further exploration of this class of novel manipulators. |
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Thesis/Dissertation |
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FileFormat | application/pdf |
Language | eng |
Date Available | 2009-09-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0090507 |
URI | http://hdl.handle.net/2429/13038 |
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Master of Applied Science - MASc |
Program |
Mechanical Engineering |
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Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2002-05 |
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Scholarly Level | Graduate |
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