MASSACHUSETTS INSTITUTE OF TECHNOLOGY ARTIFICIAL INTELLIGENCE LABORATORY Working Paper No. 192 February 1979. Logical Control Theory Applied to Mechanical Arms Ronald Joseph Pankiewicz A.I. Laboratory Working Papers are produced for internal circulation, and may contain information that is, for example, too preliminary or too detailed for formal publication. Although some will be given a limited external dis- tribution, it is not intended that they should be considered papers to which reference can be made in the literature. This report describes research done at the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for the Laboratory's artificial intelligence research is provided in part by the Advanced Research Projects Agency of the Department of Defense under Office of Naval Research contract N00014-75-C-0643. SMASSACHUSETTS INSTITUTE OF TECHNOLOGY 197 PAGE 2 Logical Control Theory Applied to Mechanical Arms by Ronald Joseph Pankiewicz Submitted to the Department of Electrical Engineering and Computer Science on January 19, 1979 in partial fulfillment of the requirements for the Degrees of Master of Science and Electrical Engineer. Abstract A new control algorithm based upon Logical Control Theory is developed for mechanical manipulators. The controller uses discrete tesselations of state space and a finite set of fixed torques to regulate non-rehearsed movements in real time. Varying effective inertia, coupling between degrees of freedom, and frictional, gravitional and Coriolis forces are readily handled. A logical controller was implemented on a mini-computer for the M ITzScheinman Vicarm. -The controller's--performance "compares- favorably- with- -that of- controllers designed according to existing methodologies as used, for example, in the control of present day industrial manipulators. Thesis supervisor: Berthold K. P. Horn, Associate Professor of Electrical Engineering and Computer Science PAGE 3 Acknowledgements I thank Berthold Horn for his patient and valuable guidance, his welcome insights and his critical readings of this document. Thanks also to Jon Doyle, Jan Galkowski and Marc Raibert for their encouragement, critical readings and contributions to this work, and to Meyer Billmers for his instruction on the use of the Mini-Robot System. I would also like to thank the Al Laboratory for providing me with the resources necessary to undertake this work. PAGE 4 Table of Contents Abstract . .. ... .... ..... .. 2 Acknowledgements ..... ..... ..... ... 3 Table of Contents ..... ..... ..... .. 4 List of Illustrations ........ 6 Overview ................. 7 Chapter I Introduction ........... ... 9 I.A General Background .............. 9 I.B Manipulators . ............... 10 I.C Control Techniques ............... 12 I.C.1 Movement Definition .......... 12 I.C.2 The Classical Approach .......... 13 I.C.3 The Modern Approach ........... 16 I.C.4 The State-Space Approach ......... 19 I.C.5 The Logical Approach ....... ... 20 Chapter II A Logical Controller .... ......... 24 II.A Introduction .............. 24 II.B Initial Approach . ............. 24 II.C Friction and Loading ... ......... 27 11. -:DOvercoming Osci lations ........... ... ; -...... .-43 II.E Constraining Intermediate Velocities ....... 46 II.F Introducting Gravity ............ 48 II.G Summary .. ............. 53 Chapter III A Multi-Link Arm ........... 55 III.A Introduction . 55 III.B Interacting Joints .... ......... 55 III.C Gravity and Oscillations ........... 64 III.0 Application ... .. .. ... .. .. 67 III.E Summary .... ..... ..... ... G69 Chapter IV What We've Done .............. 70 IV.A Introduction . .. ... .. .. .. 70 IV.B Review of the Logical Controller . ....... 70 PAGE 5 IV.C The Classical Controller ........... 73 IV.D The Modern Controller .... .... 73 IV.E The State-Space Controller ........... 75 IV.F The Variable-Structure-System Controller ..... 77 IV.G Summary .................. 79 Chapter V Where Do We Go From Here ........... 88 V.A Introduction . ... 88 V.B Dynamics Knowledge .. 80 V.C Comparison Testing . 85 V.0 Other Things in State-Space . 87 V.D.1 Collisions . .. 87 V.D.2 Dropped Objects .. 87 V.D.3 Failing Bearings and Actuators . .. 88 V.E Summary . ..... .... ..... 88 Bibliography . ... ........ 89 Appendix: A Model of the Arm . .... .. .. .. 93 PAGE 6 SList of Illustrations Chapter 1 Fig. 1.1 A logical controller 21 Chapter 2 Fig. II.1 Traditional bang-bang control . .. 11.2 Three control zones . ..... 11.3 Three control zones with delayed switching 11.4 Three control zones simplified . .. 11.5 System state-space trajectory . ... 11.6 Positioning the negative-torque line 11.7 Lower bound on joint velocity . 11.8 Lower bound on deceleration into origin 11.9 Positioning the positive-torque line II.10 Three-state logical controller 11.11 Complete set of three switching lines 11.12 Intermediate velocity regulation . 11.13 Gravitational force on limb . ... 11.14 Partitioning gravity-compensation Chapter 3 Fig. 111.1 Interacting degrees of freedom 111.2 Positioning the negative-torque lines 111.3 Positioning the positive-torque lines 111.4 Five control zones . ..... 111.5 Five-state logical controller . .. III.6 Complete set of five switching lines 111.7 Sample movement with Vicarm . .. Chapter 4 Fig. IV.1 Hysteresis in Controller's Behavior ...... 76 Appendix Fig. A.1 Photograph of the MIT Vicarm . ... ... 94 Fig. A.2 Drawing of the MIT Vicarm .. ....... 95 PAGE 7 Overview As highly nonlinear systems, mechanical manipulators are inherently difficult to control. The nonlinear characteristics are due to cross-coupling between degrees of freedom, changes in loading, and the effects of frictional, gravitational and Coriolis forces. Controllers designed with existing methodologies typically utilize complex compensation techniques to handle such- systems. These techniques have proven inadequate for the real-time regulation of manipulators executing nonrehearsed movements at a variety of speeds. This thesis explores the applicability of Logical Control Theory to the task of controlling a manipulator. Our research developed a logical controller that is essentially a finite state machine. It uses discrete tesselations of the system state-space and a set of fixed torques to regulate a mechanical arm. State-space trajectories acquired through modelling and observation are used to develop the tesselations. The number of different control torque levels and their magnitudes are determined on the basis of desired joint dynamics. The controller executes simple calculations to make logical decisions in regulating each_joint, making it ideally suited for implementation on a digital computer. Since calculations to service one degree of freedom typically require less than one half of a millisecond on a mini-computer, original movements can be executed in real time. The controller can regulate a mechanical arm with accuracy that is comparable to that achieved with existing approaches. Positioning accuracy to within *0.015 radians, and intermediate velocity regulation to within #0.15 radians per second was consistently realized in operating the three major joints of the MIT-Scheinman Vicarm in nonrehearsed PAGE 8 movements. Issues requiring further research were also identified. PAGE 9 I. Introduction I.A General Background , The modern history of manipulation began in 1947 at Argonne National Laboratory, where the first mechanical arms were developed for handling radioactive materials. These were master-slave devices, in which the remote hand replicated motions of a similar hand controlled by the operator. Force feedback was eventually added to enable the operator to tell what forces the hand was exerting [Goertz, 631 In 1961 Ernst developed a computer-controlled mechanical hand that had touch feedback [Ernst, 611 The hand could be used to explore regions, indentifying and moving objects. In 1968 Pieper analyzed the kinematics of manipulators and was able to plan collision-free movements through cluttered spaces [Pieper, 681. This work was followed by Kahn who analyzed arm dynamics and developed a "bang-bang" sub-optimal controller [Kahn & Roth, 71]. Since then, numerous efforts directed towards computer control have produced a variety of philosophies, of varying sophistication, on the control of mechanical manipulators [Finkel, 761 Reference [IITRI, 75] provides a sampling of the current state of the art. Today, applications of mechanical manipulators in industry include their performing boring and dangerous tasks which traditionally have required human participation. The benefits of such applications are increased productivity, production flexibility and improvement of individuals' working conditions. Potential applications includes the performing of routine tasks in hostile environments, such as explosive plants, PAGE 10 radioactive chambers or underground mines, or the performing of research in inaccessible environments, such as deep space or underseas. The successful utilization of mechanical manipulators for a large portion of these tasks depends on their having the dexterity and reprogrammibility characteristic of their human counterparts. At present, manipulators cannot match the combined speed, strength and agility of the human arm and hand which they are supposed to replace; in fact, the inability to efficiently and effectively control manipulators with such characteristics is the primary limitation in their widespread application [Bejczy, 761]. There exists no currently perfected scheme for the