Dynamics and Robust Control of Robot-environment Interaction (New Frontiers in Robotics) [1 ed.] 9789812834751, 9812834753

Citation preview

NEW FRONTIERS IN ROBOTICS Series Editors: Miomir Vukobratovic (“Mihajlo Pupin” Institute, Serbia) Ming Xie (Nanyang Technological University, Singapore)

Published Vol. 1

Haptics for Teleoperated Surgical Robotic Systems by M. Tavakoli, R. V. Patel, M. Moallem & A. Aziminejad

Vol. 2

Miomir Vukobratovic “Mihajlo Pupin” Institute, Belgrade, Serbia

Dragoljub Surdilovic Fraunhofer Institute for Production Systems and Design Technology IPK, Berlin, Germany

Yury Ekalo Research and Engineering Center of St Petersburg Electrotechnical University, Russia

Dusko Katic “Mihajlo Pupin” Institute, Belgrade, Serbia

World Scientiic NEW JERSEY

•

LONDON

•

SINGAPORE

•

BEIJING

•

SHANGHAI

•

HONG KONG

•

TA I P E I

•

CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

DYNAMICS AND ROBUST CONTROL OF ROBOT-ENVIRONMENT INTERACTION New Frontiers in Robotics — Vol. 2 Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-283-475-1 ISBN-10 981-283-475-3

Printed in Singapore.

Preface

When writing a book that is concerned with the robot’s physical contact and interaction with an environment, first and foremost we ought to consider the following contradictory situation. During the last two decades, the control of robot contact motion (also referred to as compliant motion) has emerged as one of the most attractive and fruitful research areas in robotics. The initial investigations in the field were motivated by the practical needs for automating complex tasks mainly performed by humans, such as assembly, deburring, etc. The control of physical robot interaction is still a challenging research issue, recently addressing the emerging fields of human-robot interaction systems, human augmentations and enhancements, haptic rendering, rehabilitation robotics, etc. However, in spite of considerable research efforts and results achieved, the applications of compliance control in the industry and service fields are still insignificant in comparison with widespread free-space robot applications, such as pick-and-place or seam-tracking tasks. The majority of industrial robot assembly applications utilize passive compliance devices (Remote Center of Compliance - RCC) compensating for misalignments of parts with specific geometry. Other applications employ additional passive or active axes with simple compliance control algorithms. More sophisticated robotic systems that would involve the programming and control of the interaction with a complex, dynamic and variable environment are still missing in practice. There are many different reasons for such a situation. Based on almost twenty years of research and experience with implementing the compliantmotion control algorithms in industrial and other robotic systems we would try to identify the most critical causes which in our opinion mainly inhibited a more widespread application of interactive robotic systems. First, the development of a controller for contact tasks has proven to be quite difficult, largely due to the stability problems that arise in the dynamic interaction during robot’s physical contact with an environment. The interaction control problems are still insufficiently specified, and their structural relationships to classical servo control design problems and methodologies are not completely clear. One

v

vi

Dynamics and Robust Control of Robot-Environment Interaction

further limitation is the absence of a widely accepted framework for the synthesis of the interaction control parameters that would ensure the stability of both the contact transition and interaction processes, and guarantee the desired contact performance. The existing design procedures based on robot passivity appear to be exceedingly conservative in the applications in which the interaction between an industrial robot and a stiff environment should be controlled. The established interaction control algorithms are mostly concerned with particular problems, usually at the lowest servo-control layer, and their integration into a complete control system, which appears to be a very tedious task, is still missing. Further, many of the proposed control algorithms are based on the computedtorque method and are closely related to direct-drive robotic systems. The directdrive technology is, however, still seldom used in robotic practice, due to several serious problems related to the large mass, overheating, and high costs of directdrive actuators. On the other hand, direct-drive robots appear to be quite suitable for advanced experimental investigations of robot control in research laboratories. The popular computed-torque control technique requires real-time computation of complete dynamic models of the robot and environment, which makes its realization rather complex. This approach works reasonably well in direct-drive robots when their dynamic parameters have been correctly identified. In industrial robots, however, the performance improvements which can be achieved with these algorithms are not in proportion with the implementation efforts. Due to quite different performance, the results obtained for direct-drive robots, although experimentally verified, cannot be applied onto industrial robots. The investigations of compliant motion control are usually concerned with the nonlinear effects in robot and environment dynamics, rather than with the problems encountered in conventional robotic systems, such as Coulomb friction, control time delay, practical limitations of computer and sensory systems, etc. Still, there is another problem concerning the knowledge of the environment model. The majority of proposed algorithms exhibit good performance only under the assumption that an accurate model of the environment and its parameters (e.g. contact location, stiffness) are available. In real stiff environments, however, this condition is quite restrictive and non-realistic, since it is difficult to identify the parameters using available sensory and computer control systems. All these problems make the existing results of compliant motion control difficult to implement from a practical point of view. This requires further

Preface

vii

investigation efforts on reliable and simple, but nevertheless robust, control solutions. Despite the progress that has been made during the last decade, there are still research issues insufficiently investigated, such as: • the stability of contact transition is not clearly addressed in the literature. The reliable necessary and sufficient conditions that would ensure the maintenance of a stable contact during transition are still missing. • the problem of handling an inadvertent loss of contact has not been solved yet. Furthermore, of special practical importance are the following topics, deserving further computational/experimental study: Design of robust compliance motion control to improve disturbance rejection capabilities; • Definition of measures and criteria to evaluate the compliant motion capabilities of industrial robots in relation to performing a task, taking into account the distortion of friction and other similar disturbances in the arm; • Comparison of the available algorithms and definition of benchmark tests; • Development of reliable control schemes based on a unified approach to force, position and impedance control, which can be applied in conventional industrial robotic systems. Therefore, this book is aimed at considering the interaction control problems from a broad and comprehensive point of view. The problem of robotenvironment interaction control is tackled taking into account different issues, such as: mathematical modeling of contact and interaction kinematics and dynamics, stability of coupled systems and contact transition process, various interaction control algorithms and techniques, robustness against disturbances and model uncertainties, programming and planning of simple contact tasks, sampled systems control effects, as well as practical control synthesis and design for industrial implementations. Last but not least, practical knowledge and experience gained in the developing and implementing various interactive robotic systems is reflected in this book. The contents of the book are organized as follows. Chapter 1 provides a comprehensive review of various compliant motion control methods proposed in the literature. It covers some early ideas and their later improvements, as well as new control concepts and recent trends in this field. Before reviewing many of the results, a categorization of compliant motion tasks and proposed control concepts is established based on various

viii

Dynamics and Robust Control of Robot-Environment Interaction

classifying criteria. In this survey, particular attention is paid not only to traditional indices of control performance, but also to the reliability and applicability of algorithms and control schemes in industrial robotic systems. These systems are widely employed in practice, providing a reasonable background for compliance motion control implementation. However, compliance control is a very attractive control approach in the new emerging technologies, such as service robotics (e.g. surgical and rehabilitation robots), virtual reality and haptics, telemanipulation, human augmentation and assistance. These fields apply new and quite different robotic structures in comparison to conventional industrial robotic systems (e.g. direct-drive robots, parallel and wire manipulators, etc.). This chapter provides a historical perspective, summarizes contributions of the most relevant or representative investigations and methods, and identifies the interaction control problems that are still open, requiring additional research efforts. It provides useful information, especially for younger roboticians having no previous work experience. Chapter 2 is devoted to the unified approach to dynamic control of the robot interacting with a dynamic environment. The unified position-force control differs essentially from the conventional hybrid position/force control schemes. A dynamic approach to controlling simultaneously both the position and force in an environment with completely dynamic reactions has been established. The approach of dynamic interaction control defines two control subtasks responsible for the stabilization of robot position and interaction force. The both control subtasks utilize dynamic models of the robot and environment in order to ensure tracking of both the nominal motion and force. Special attention is given not only to the synthesis of control laws ensuring stability of robot’s desired motions and desired interaction forces of the robot and environment, but also to the definition of possible motions of the robot and its possible interaction forces in contact tasks. The concept of the family of transient responses with respect to the robot’s motion and its force of interaction with environment is formulated. It allows one to set and then solve the problem of the synthesis of control laws that not simply stabilize the motion and force of interaction of the robot with its environment, but also solves the problem of stabilization with the preset quality of transient processes. Significant attention is given to the analysis of the influence of the constraints imposed on the state, control, and interaction force on transient responses, taking into account the inadequacy of dynamics models of the robot and environment and/or external perturbations. The adaptive control scheme proposed in this chapter enables one to solve contact tasks for robots with both stationary and nonstationary dynamics. The elaborated stability test

Preface

ix

may be used either to check the stability of the specified control laws, or to establish procedures for the synthesis of parameters of different control laws. Hence, the control synthesis becomes much more accurate and effective, i.e. higher robustness of the control to the uncertainties in the robot and environment models can be ensured, which is one of the most relevant aspects in the potential industrial applications of robots in numerous technological tasks where the robot is interacting with the environment (e.g. in cutting, deburring, etc.). The developed control algorithms also appear very promising for interaction with a virtual environment in high dynamic haptic systems, as well as for controlling novel dual-arm robots and bimanual contact tasks. Chapter 3 is concerned with the design of compliance control algorithms that are reliable and robust for implementing in industrial robots and advanced interaction systems (e.g. haptic interfaces, surgical and rehabilitation robots, human enhancers, collaborative robots, etc.). The problems and research issues that are associated with the design of robust impedance control algorithms for stable interaction with a passive environment are in the main focus of this chapter. The basic control development problems, such as stability, performance and robustness of impedance control algorithms are addressed. These problems are considered at the lowest servo control layer. For the sake of simplicity, the impedance control design problem is split into two subproblems concerning the realization of the target impedance and selection of target impedance parameters which ensure specific desired task performance, as well as common control design requirements, such as stability, fast reaction and robustness. The stability of the interaction between the robot and environment in contact, which is essential for the impedance control synthesis, is defined by means of the coupled stability. For the examination of stability we have applied a common approach utilizing the properties of the system at equilibrium, and various modern control techniques (e.g. positivity and H ∞ control concepts, etc.). We have further considered the stability of the contact transition process. Although recognized to be most fundamental in contact tasks control synthesis, this problem has not been addressed appropriately until now. Especially, the contact stability in industrial robotic systems has not been explored adequately. Several practical contact stability definitions are proposed in order to clearly make distinction between the contact stability and coupled stability, often mistaken in the literature. Based on these definitions, new stability conditions are proposed. The concept of dynamic systems passivity and robust stability analysis are applied to obtain the reliable conditions for ensuring contact stability. The established stability criteria provide the basis for examining the effects of impedance control

x

Dynamics and Robust Control of Robot-Environment Interaction

parameters on the transition process stability. The analysis/synthesis oriented stability examinations allow the tuning of target impedance parameters in order to meet both interaction performance and stability. Within the robust stability analysis framework, the generalized contact stability condition is also derived, ensuring both contact and coupled system stability. The stability analysis is pursued in discrete-time and sampled-data interaction systems. In order to evaluate the derived contact stability criteria a set of several hundred contact transition experiments was designed and realized using an industrial robot. The aim was to capture a practical parametric contact stability limit in a typical real robot-environment interaction system. On the basis of tests performed it is demonstrated that the most accurate contact stability results (the parametric limits closest to the experimental bounds) provides the passivity-based criterion for the sampled-data system. Robust contact stability always ensures a safe transition and appears to be very practical for the control synthesis in an uncertain robot-environment interaction system. Chapter 4 addresses the synthesis of the adopted second-order target impedance model for a generic contact task. The contact task consists of the realization and maintenance of a stable contact with the environment. The interaction force should be kept within the prescribed limits, dependent on the position tolerances and environmental stiffness. This assignment is intrinsically involved in almost all robot interaction tasks. The chapter considers the algorithms for the practical impedance control design in industrial robotic systems. The developed algorithms integrate the theoretical and practical stability results dealt with in the previous chapter. The considered impedance control synthesis addresses basic control design problems at the servo-control layer. The impedance control design has been established for a reliable decoupled compliance geometric model that allows a relatively simple parameterization of the target impedance behavior. For the fundamental and common interaction tasks, the compliance parameters can be chosen independently of the interaction system configuration. More complex robotenvironment interactions were also considered based on the spatial compliance model. The control synthesis consists of the straightforward steps of computing the target impedance parameters and impedance compensator gains. All input parameters to the design algorithm have been explicitly specified. The feasibility of the developed algorithms was demonstrated using experiments with two industrial robot systems. Finally, a reliable geometric and control framework for the implementation of compliance control in industrial and other advanced robotic systems has been developed and presented. Several practical and robust

Preface

xi

control algorithms at higher planning and programming control layers were designed and tested. The essential algorithms support setting of the compliance parameters, such as the C-frame location and impedance gains, as well as continuous switching of compliance control and variation of parameters. These features are proven to be essential for a stable and robust execution of the compliance control tasks. Powerful sets of control functions, also presented in this chapter, integrate the basic compliance control algorithms in the forward robot control. These functions perform all of the computations and management of the parameters between the convenient robot position control system and impedance control kernel. Finally, some new commands, providing a flexible user-interface, are designed and implemented in a high-level robot programming environment. The new programming language commands are illustrated by means of several examples. An essential design requirement was to combine the user’s experience with robot motion programming and simple understandable physical behavior of the impedance control which mimics a variable spatial mass-damper-spring system. The experimental testing within the space control system SPARCO has clearly proven the reliability and robustness of the presented high-level compliance control algorithms. Certainly, a basic precondition for the implementation of compliant motion control is the design of a robust servo impedance controller, which ensures stable transition and coupling with the environment. However, the control integration and programming issues, which are often underestimated in the literature, are essential for a customary and efficient application of impedance control in practical contact tasks. A proper selection of the C-frame location and target impedance gains is crucial for a successful execution of the impedance control tasks. This selection should be compatible with the very nature of the motion constraints, i.e. contact task geometry and physical task characteristics (e.g. force-motion relationships). The experience gained in performing the compliance tasks presented here is essential for compliance control design and implementation in a wide range of tasks in industrial and service robotics. The robust control framework and the new contact stability theory established for the control synthesis of the interaction between an impedancecontrolled robot and a passive environment are expanded in Chapter 5 to the control and synthesis of haptic interfaces interacting with a virtual environment. This rapidly emerging technology imposes high requirements on the interaction stability and robustness of the control system in spite of considerable control computation efforts and time lags. Recently, the new interactive systems concerning the interaction between a human and a robotic device, as well as with robot’s physical or virtual dynamic environments, have aroused a strong research

xii

Dynamics and Robust Control of Robot-Environment Interaction

interest. To the novel interactive systems belong kinesthetic displays and haptic interfaces, teleoperation systems, human enhancers and augmentation devices, rehabilitation robots, robot assistants and collaborative robots, etc. These systems are designed to produce/receive kinesthetic stimuli for/from human movements, as well as to provide the user with a realistic feeling of the contact and dynamic interaction with the close, remote or virtual environments. The advanced interaction systems have recently found very attractive applications in surgical and rehabilitation robotics, power assist-devices, training simulation systems, etc. The most critical issue in these systems is how to ensure stable and safe interaction with a high fidelity of reproduction of a virtual environment. This is a challenging task when taking into account serious problems such as unknown and variable human dynamics, commonly non-linear environmental characteristics, as well as various disturbances in computer-controlled systems. This chapter considers the stability of the interaction of a human-robotenvironment (real or virtual) system based on the robust control design approach. The proposed new interaction stability paradigm ensures contact stability during all phases of the interaction. Moreover, the new design framework realizes low-impedance performance allowing considerable reduction of high apparent industrial robot inertia and stiffness. The defined stability indices take into account the relevant effects in the robot control systems, such as time lags and sampling data effects, as well as the uncertainties in the environment and realized target admittance models. The synthesis of robust control laws is confirmed to be very efficient for the stabilization of the interaction between a robot and a stiff and force-delayed environment taking into account the desired interaction performance. The testing of this approach in various robotic systems demonstrates the feasibility and reliability of the interaction control approach even for relatively high control rates and lags. The advantage of robust stability is particularly demonstrated in the interaction control of novel intelligent power-assist handling systems with significant perturbations in the force and position measurements. This shows the practical applicability of the novel stability criteria for haptic systems. Chapter 6 covers some advanced control techniques. As robotic systems make their way into standard practice, they have opened the door to a wide spectrum of complex applications. Such applications usually demand highly intelligent robots. Future robots are likely to have greater sensory capabilities, more intelligence, higher levels of manual dexterity, and the mobility, compared to humans. In order to ensure high-quality control and performance in robotics, new intelligent control techniques must be developed that will be capable of

Preface

xiii

coping with task complexity, multi-objective decision making, large volumes of perception data and substantial amounts of heuristic information. Soft Computing paradigms consisting of complementary elements of Fuzzy Logic, Neural Computing and Evolutionary Computation are viewed as the most promising methods towards intelligent robotic systems. The specific emphasis in research is given on the development of efficient learning rules for robotic connectionist training and synthesis of neural learning algorithms for low-level control in the domain of robotic compliance tasks. The synthesis of new advanced learning algorithms for robotic contact tasks by nonrecurrent and recurrent connectionist structures is presented in this chapter as the main research contribution. The main concern of this chapter, which provides a survey of connectionist algorithms for robotic contact tasks, is the development of learning control algorithms as an upgrade of conventional non-learning control laws for robotic compliance tasks (algorithms for stabilization of robot motion, stabilization of robot interaction force and impedance algorithms). In view of the important influence of the robot environment, a new comprehensive learning approach, based on simultaneous classification of robot environment and learning of robot uncertainties, is also presented. The book is addressed to a wide audience of scientists, practitioners and scholars dealing with interactive robotic systems. It is our hope that the material presented in this book will be useful to a wide range of readers, ranging from undergraduate and graduate students, new and advanced academic researchers, to the technical specialists (mechanical, electrical, computer or systems engineers). It can also be adopted as a textbook for a graduate course on advanced robotic systems.

The authors Belgrade, December 2007

Contents

Preface .......................................................................................................................................

v

1. Control of Robots in Contact Tasks: A Survey ................................................................... 1.1 Introduction ............................................................................................................... 1.2 Contact Tasks ............................................................................................................ 1.3 Classification of Constrained Motion Control Concepts ........................................... 1.4 Model of the Robot Performing Contact Tasks ......................................................... 1.5 Passive Compliance Methods .................................................................................... 1.5.1 Non-adaptable compliance methods ............................................................. 1.5.2 Adaptable compliance methods .................................................................... 1.6 Active Compliant Motion Control Methods.............................................................. 1.6.1 Impedance control......................................................................................... 1.6.1.1 Force-based impedance control ...................................................... 1.6.1.2 Position-based impedance control .................................................. 1.6.1.3 Other impedance-control approaches ............................................. 1.6.2 Hybrid position/force control........................................................................ 1.6.2.1 Explicit force control...................................................................... 1.6.2.2 Position-based (implicit) force control ........................................... 1.6.2.3 Other force control approaches....................................................... 1.6.3 Force-impedance control............................................................................... 1.6.4 Unified position-force control....................................................................... 1.7 Contact Stability and Transition ................................................................................ 1.8 Compliance Planning................................................................................................. 1.9 Haptic Systems Control ............................................................................................. 1.10 New Robot Application ............................................................................................. 1.11 Conclusion................................................................................................................. Bibliography........................................................................................................................

1 1 1 2 10 14 14 15 18 18 22 24 27 28 30 37 39 40 43 46 56 60 64 65 67

2. A Unified Approach to Dynamic Control of Robots........................................................... 2.1 Introduction ............................................................................................................... 2.2 Dynamic Environments ............................................................................................. 2.2.1 Model of a dynamic environment ................................................................. 2.2.1.1 Kinematic-dynamic constraints ...................................................... 2.2.1.2 Pure dynamic environment............................................................. 2.2.1.3 Linear impedance model ................................................................ 2.3 Synthesis of Control Laws for the Robot Interacting with Dynamic Environment ... 2.3.1 Stabilization of motion with the preset quality of transients......................... 2.3.2 Stabilization of interaction force with the preset quality of transients .......... 2.3.3 Concluding discussion ..................................................................................

77 77 80 81 82 82 83 84 86 91 100

xv

xvi

Dynamics and Robust Control of Robot-Environment Interaction 2.4

Analysis of Transient Processes................................................................................. 2.4.1 Task setting ................................................................................................... 2.4.2 Motion transient processes............................................................................ 2.4.3 Force transient processes............................................................................... 2.4.4 Numerical example ....................................................................................... 2.4.5 Effect of sensor errors on the transient processes ......................................... 2.5 Adaptive Stabilization of Motion and Forces ............................................................ 2.5.1 Introduction................................................................................................... 2.5.2 Task setting ................................................................................................... 2.5.3 General scheme of robot adaptive control in contact tasks ........................... 2.5.4 Adaptive stabilization of programmed motions and forces........................... 2.6 Position-Force Control – A Generalization ............................................................... 2.6.1 Models of robot and environment dynamics. Task setting ........................... 2.6.2 Control laws stabilizing the interaction force................................................ 2.6.3 Example ........................................................................................................ 2.6.4 Conclusion .................................................................................................... 2.7 Position-Force Control in Cartesian Space ................................................................ 2.7.1 Introduction................................................................................................... 2.7.2 Task setting ................................................................................................... 2.7.3 Relation to previous results........................................................................... 2.7.4 Control laws for specified force dynamics.................................................... 2.7.5 Example ........................................................................................................ 2.7.6 Conclusion .................................................................................................... 2.8 New Realization of Hybrid Control........................................................................... 2.8.1 Introduction................................................................................................... 2.8.2 Revised hybrid control procedure ................................................................. 2.8.3 Case study ..................................................................................................... 2.9 Impedance Control – A Special Case of the Unified Approach ................................ 2.9.1 Introduction................................................................................................... 2.9.2 Improved impedance control......................................................................... 2.9.3 Case study ..................................................................................................... 2.9.4 Concluding remarks ...................................................................................... 2.10 Stability of Robots Interacting with Dynamic Environments .................................... 2.10.1 Introduction................................................................................................... 2.10.2 Practical stability of robots interacting with dynamic environment.............. 2.10.3 Mathematical model...................................................................................... 2.10.4 Formulation of the control task ..................................................................... 2.10.5 Control law ................................................................................................... 2.10.6 Practical stability analysis ............................................................................. 2.10.7 Example ........................................................................................................ 2.10.8 Conclusion .................................................................................................... 2.10.9 Practical stability - A remark ........................................................................ Appendix A Proof of Theorem 1 ....................................................................................... Appendix B Proof of Theorem 2 ....................................................................................... Appendix C Proof of Theorem 3 ....................................................................................... Appendix D Proof of Theorem 4 ....................................................................................... Appendix E Proof of Theorem 5 ....................................................................................... Appendix F Proof of Theorem 6 ....................................................................................... Appendix G Proof of Lemma ............................................................................................

101 102 104 110 119 121 128 128 129 134 139 146 146 149 155 157 158 158 159 160 161 166 169 170 170 173 177 182 182 182 186 190 191 191 194 195 196 198 199 203 207 208 211 215 217 219 221 223 225

Contents

xvii

Appendix H Proof of Theorem 7 ....................................................................................... Appendix I Some Basic Relations.................................................................................... Appendix J Proof of Theorem 8 ....................................................................................... Appendix K Prior Assertions and Proof of Theorem 10.................................................... Bibliography........................................................................................................................

229 232 237 242 250

3. Impedance Control .............................................................................................................. 3.1 Introduction ............................................................................................................... 3.2 Control Objectives..................................................................................................... 3.3 Impedance Control Scheme ....................................................................................... 3.4 Impedance Control Synthesis .................................................................................... 3.4.1 Effects of impedance control ........................................................................ 3.4.2 Common impedance control law................................................................... 3.4.3 Interactive system behavior - coupled stability ............................................. 3.4.3.1 Linearized interaction models......................................................... 3.4.3.2 Coupled system stability................................................................. 3.4.3.3 Coupling of passive systems........................................................... 3.4.3.4 Robust coupled system stability ..................................................... 3.4.3.5 Coupled system performance ......................................................... 3.4.3.6 Performance of the controller Gf = Gt-1 .......................................... 3.5 Improved Impedance Control .................................................................................... 3.5.1 Improved control law .................................................................................... 3.5.2 Coupled system performance ........................................................................ 3.5.3 Target impedance model realization ............................................................. 3.6 Typical Impedance Contact Behavior........................................................................ 3.7 Contact Transition Stability....................................................................................... 3.7.1 Definition of the contact stability.................................................................. 3.8 Contact Stability Conditions...................................................................................... 3.8.1 Time domain analysis ................................................................................... 3.8.1.1 Constant velocity phase contact ..................................................... 3.8.1.2 Constant acceleration/deceleration phase contact........................... 3.8.2 Passivity-based contact transition stability analysis...................................... 3.8.3 Robust transition stability - generalized contact stability.............................. 3.8.4 Equivalence of robust- and passivity-based contact stability........................ 3.9 Influence of Non-Linear Effects on Contact-Stability ............................................... 3.9.1 Coulomb’s friction ........................................................................................ 3.9.2 Control lags and sampling effects ................................................................. 3.9.2.1 Ideal target system with force delay ............................................... 3.9.2.2 Robust and passivity-based contact stability of discrete-time system............................................................................................. 3.9.2.3 Contact stability of sampled-data system ....................................... 3.10 Evaluation of Contact Transition Stability Conditions.............................................. 3.10.1 Contact transition performance indices......................................................... 3.10.2 Contact transition assessment........................................................................ 3.10.3 Upper limits on target impedance frequency ................................................ 3.11 Conclusion................................................................................................................. Bibliography........................................................................................................................

255 255 256 261 266 267 268 269 270 272 277 280 284 287 288 290 294 296 301 319 320 332 333 335 339 340 344 347 351 351 352 354 365 369 375 379 381 394 398 401

xviii

Dynamics and Robust Control of Robot-Environment Interaction

4. Practical Synthesis of Impedance Control ........................................................................... 4.1 Introduction ............................................................................................................... 4.2 Influence of the Target Parameters on the Impedance Control Performance............. 4.2.1 Influence of target frequency and mass on contact transition ....................... 4.2.2 Reduction of force overshoots – Hogan’s target impedance model (generalized stiffness control) ....................................................................... 4.3 Selection of Target Impedance Parameters – Impedance Control Design at Lower Control Layer ................................................................................................. 4.3.1 Specification of impedance control geometry............................................... 4.3.2 Specification of input design parameters - user interface.............................. 4.3.3 Design algorithm........................................................................................... 4.4 Synthesis of Impedance Control at Higher Layers .................................................... 4.4.1 World model for the impedance control ....................................................... 4.4.2 Impedance control integration into forward industrial robot position control 4.4.3 Impedance control functions and algorithms ................................................ 4.4.3.1 End-effector position modification................................................. 4.4.3.2 Selection and initialization of impedance gain parameters............. 4.4.3.3 Computation of path correction...................................................... 4.4.3.4 Delta x to delta T transformation (deltax_2_deltaT) ...................... 4.4.3.5 Compensation for payload and inertial effects ............................... 4.4.3.6 Sensor frame to compliance frame transformation (Sf2cf)............. 4.4.3.7 Force setpoint check ....................................................................... 4.4.3.8 Force sensor data preprocessing ..................................................... 4.4.3.9 Contact check ................................................................................. 4.4.4 Higher control layers algorithms................................................................... 4.4.4.1 Impedance control operating modes............................................... 4.4.4.2 Change of impedance gains (relax) ................................................ 4.4.5 Actions and tasks control algorithms ............................................................ 4.4.5.1 Grasping/detach action ................................................................... 4.4.5.2 Insertion/extraction......................................................................... 4.5 Conclusion................................................................................................................. Bibliography........................................................................................................................

405 405 406 406 409 414 416 426 431 451 452 457 461 461 463 463 464 465 466 467 467 468 468 468 469 471 473 476 485 487

5. Robust Control of Human-Robot Interaction in Haptic Systems ........................................ 5.1 Introduction ............................................................................................................... 5.2 Haptic System Structures........................................................................................... 5.3 Haptic Rendering ....................................................................................................... 5.4 Robust Control of Haptic Systems Interaction .......................................................... 5.4.1 Admittance display control ........................................................................... 5.4.2 Impedance display control ............................................................................ 5.5 Conclusion................................................................................................................. Bibliography........................................................................................................................

491 491 492 496 498 499 510 516 517

6. Intelligent Control Techniques for Robotic Contact Tasks ................................................. 6.1 Introduction ............................................................................................................... 6.2 The Role of Learning in Intelligent Control Algorithms for Compliant Tasks ......... 6.3 A Survey of Intelligent Control Techniques for Robotic Contact Tasks ...................

519 519 523 532

Contents The Synthesis of New Connectionist Learning Control Algorithms for Robotic Contact Tasks............................................................................................... 6.4.1 The background of the new connectionist control synthesis......................... 6.4.2 Model of robot interacting with dynamic environment – task setting........... 6.4.3 Factors affecting task performance and stability in robotic compliance control........................................................................................ 6.4.4 The comprehensive connectionist control algorithm based on learning and classification for compliance robotic tasks............................... 6.4.5 The genetic-connectionist algorithm for compliant robotic tasks ................. 6.4.6 GA tuning of PI force feedback gains........................................................... 6.4.7 Case studies................................................................................................... 6.5 Connectionist Reactive Control for Robotic Assembly Tasks by Soft Sensored Grippers ..................................................................................................................... 6.5.1 Analysis of the assembly process with soft fingers....................................... 6.5.2 Assembly process.......................................................................................... 6.5.3 Learning compliance methodology by neural networks ............................... 6.5.4 Experimental results...................................................................................... 6.6 Intelligent Control of Contact Tasks in Humanoid Robotics..................................... 6.6.1 Introduction................................................................................................... 6.6.2 Definition of control problem and advanced control methods for humanoid robots ........................................................................................... 6.6.3 The model of the system ............................................................................... 6.6.3.1 Model of the robot’s mechanism .................................................... 6.6.3.2 Definition of control criteria........................................................... 6.6.3.3 Gait phases and indicator of dynamic balance................................ 6.6.4 Hybrid integrated dynamic control algorithm with reinforcement structure. 6.6.4.1 Dynamic controller of trajectory tracking ...................................... 6.6.4.2 Compensator of dynamic reactions based on reinforcement learning structure ............................................................................ 6.6.4.3 Impact-force controller ................................................................... 6.6.4.4 Conflict between controllers........................................................... 6.6.5 Simulation studies......................................................................................... 6.7 Conclusion................................................................................................................. Bibliography........................................................................................................................

xix

6.4

552 552 554 557 559 567 569 570 575 577 578 579 580 584 584 587 593 593 594 596 598 600 601 605 606 608 616 618

Instead of Conclusion ................................................................................................................

625

Index ..........................................................................................................................................

629

Chapter 1

Control of Robots in Contact Tasks: A Survey

1.1 Introduction This chapter provides the state of the art in compliant motion control. It covers some early ideas and their later improvements, as well as some new control concepts and recent trends in this field. A comprehensive review of various compliant motion control methods proposed in the literature would certainly be very voluminous, since the research in this area has rapidly been growing in the recent years. Therefore, for practical reasons, a limited number of the most relevant or representative investigations and methods are discussed. Before reviewing many of the results, a categorization of compliant motion tasks and proposed control concepts will be made based on various criteria. In this survey particular attention is paid not only to traditional indices of control performance, but to the reliability and applicability of algorithms and control schemes in industrial robotic systems. These systems are widely employed in practice, and they provide a reasonable background for compliance motion control implementation. However, compliance control is a very interesting control approach in the new emerging technologies such as service robotics (e.g. surgical and rehabilitation robots), virtual reality and haptics, telemanipulation, human augmentation and assistance. These fields apply new and quite different robotic structures in comparison to conventional industrial robotic systems (e.g. direct-drive robots, parallel and wire manipulators, etc.). The development of interaction control algorithms for new robotic applications will also be addressed in this chapter. 1.2 Contact Tasks Regarding the nature of the interaction between a robot and its environment robotic applications can be categorized in two classes. The first one is related to non-contact tasks, i.e. unconstrained motion in a free space, without any relevant environmental influence exerted on the robot. In the non-contact tasks the robot’s own dynamics has a crucial influence upon its performance. A limited number of 1

2

Dynamics and Robust Control of Robot-Environment Interaction

most frequently performed simple robotic tasks in practice such as pick-andplace, spray painting, gluing or welding, belong to this group. In contrast to these tasks, many complex advanced robotic applications such as assembly and machining require the manipulator to be mechanically coupled to the other objects. In principle, two basic contact task subclasses can be distinguished. The first one covers essential force tasks whose very nature requires the end-effector to establish the physical contact with the environment and exert a process-specific force. In general, these tasks require a simultaneous control of both the end-effector position and interaction force. Typical examples of such tasks are machining processes such as grinding, deburring, polishing, bending, etc. In these tasks, the force is an inherent part of the process and plays a decisive role for its fulfillment (e.g. metal cutting or plastic deformation). In order to prevent overloading or damage to the tool during operation, this force must be controlled in accordance with some definite task requirements. The prime emphasis of the tasks within the second subclass lies on the endeffector motion, which has to be realized close to the constrained surfaces (compliant motion). A typical representative of such tasks is the part-mating process. The problem of controlling the robot during these tasks is, in principle, the problem of accurate positioning. However, due to imperfections inherent in the process, sensing and control system, these tasks are inevitably accompanied by the occurrence of contact with constrained surfaces, which results in the appearance of reaction forces. The measurement of interaction force provides useful information for error detection and an appropriate modification of the prescribed robot motion. Compliance, i.e. accommodation [1], can be considered as a measure of the ability of a manipulator to react to interaction forces. This term refers to a variety of different control methods in which the end-effector motion is modified by contact forces. The future will certainly hold more tasks for which the interaction with the environment is fundamental. Recent medical robot applications in surgery (e.g. spine surgery, neurosurgical and microsurgical operations, knee and hip joints replacement) may also be considered as being essentially contact tasks. Comprehensive research programs in automated construction, agriculture and food industry focus on the robotization of several representative contact tasks such as underground excavation, meat deboning, etc. 1.3 Classification of Constrained Motion Control Concepts The previous classification of elementary robotic tasks provides a framework for further systematization of the concepts concerning the robotized compliant

Control of Robots in Contact Tasks: A Survey

3

motion control. The problems encountered in controlling compliant motion have been extensively investigated and several control strategies and schemes have been proposed and elaborated. These methods can be systematized according to the different criteria. The primary systematization can be made in relation to the sort of compliance. According to this criterion, two basic groups of control concepts for compliant motion can be distinguished (Fig. 1.1): i)

ii)

Passive compliance, in which the end-effector position is accommodated by the contact forces themselves, due to the compliance inherent in the manipulator structure, servos or special compliant devices; Active compliance, in which the compliance is provided by constructing a force feedback in order to achieve a programmable robot reaction, either by controlling interaction forcea or by generating task-specific compliance at the robot end-point.

Regarding the possibility of adjusting the system compliance to specific process requirements, passive compliance methods can be classified as adaptable and non-adaptable. Based on the dominant sources of compliance, the following subgroups of methods within these groups can be distinguished (Fig. 1.2): i) • •

ii) • •

Fixed passive compliance: Methods based on the compliance inherent in the robot’s mechanical structure, such as elasticity of the arm, joints and end-effectors [2]; Methods that use specially constructed passive deformable structures attached close to the robot end-effectors and specially designed for particular problems. The best known is the “Remote Center Compliance”-RCC element [3]. Adaptable passive compliance: Methods based on the devices with tunable compliance [4], Compliance achieved by the adjustment of joint servo-gains [5].

The basic classification of active compliance control methods can be done based on the previous underlying contact task classification into essential and potential. Using the terminology of bond-graph formalisms, the robot behavior

a By force we mean force and torque and accordingly position should be interpreted as position

and orientation.

4

Dynamics and Robust Control of Robot-Environment Interaction

in the essential contact tasks can be generalized as a source of the effort (force) that should raise a flow (motion) reaction of the environment objects. The robot behavior associated with the second task’s subclass corresponds to the impedance, since this is characterized by the robot’s motion reaction to external forces exerted by the environment.

Task-Oriented Compliance of Robots

passive

active

Solution: controller or structural inherent compliance (system response in accordance to arising contact force)

Solution: force feedback controlled compliance (system response in accordance to task specific criteria)

Fig. 1.1 Basic classification of robot compliance

In view of the above, the active control force methods can be classified into the following two groups (Fig. 1.3): i)

ii)

Force, i.e. position/force control in general, or admittance control, where both desired interaction force and robot position are controlled. In force control, a desired force trajectory is commanded and force should be measured to realize the feedback control; Impedance control [6], which uses the different relationships between the acting forces and manipulator position to adjust the mechanical impedance of the end-effector to the external forces. The impedance control problem can be defined as a requirement for designing a controller so that the interaction forces govern the difference between nominal and actual positions of the end-effector according to the target impedance law. Impedance control is essentially based on position control and requires position commands and position measurements in order to close the feedback loop. In addition, force measurements are needed to realize the target impedance behavior.

Position/force control methods can be divided into:

Control of Robots in Contact Tasks: A Survey

i)

ii)

5

Hybrid position/force control, where position and force are controlled in a non-conflicting way in two orthogonal subspaces defined in a task-specific frame (compliance or constraint frame). In the force controlled end-effector degrees of freedom (DOFs) the contact force is essential for performing the task, while in the position DOFs, the motion is most important. Force is imposed and controlled along directions that are constrained by the environment, while position is controlled in those directions in which the manipulator is free to move (unconstrained). Note that the term hybrid control is usually referred to the method of Raibert and Craig [7]. However, in accordance with the view of Mason [1], the definition of this term is used here in a more general sense and it refers to any controller that is based on the division into force- and position-controlled directions. Unified position-force control, which differs essentially from the above conventional hybrid control schemes. Vukobratovic and Ekalo [8] have established a dynamic approach, to control simultaneously both the position and force in an environment with completely dynamic reactions. The approach of dynamic interaction control [8] defines two control subtasks responsible for the stabilization of robot position and interaction force. Both control subtasks utilize dynamic models of the robot and environment in order to ensure tracking of both the nominal motion and force. Parallel position/force control [9] is based on the appropriate tuning of position and force controllers. The force loop is designed to dominate the position control loop along constrained task directions where a force interaction is expected. From this viewpoint, the parallel control can be considered as a combination of impedance and force control.

Taking into account the way in which the force information is included in the forward control path, the following position/force control schemes can be distinguished: i)

Explicit or force-based [7, 10, 11] algorithms, where force control signals (i.e. the difference between the desired and the actual force) are used to generate the torque inputs for the actuators at the robot’s joints.

6

Dynamics and Robust Control of Robot-Environment Interaction

ii)

Implicit or position-based algorithms [12, 13], where the force control error is converted to an appropriate robot motion adjustment in force-controlled directions and then added to the positional control loop.

Similar to the above classification, impedance control methods can also be divided in accordance with the way in which the robotic mechanism is treated, either as an actuator (i.e. source) of position or a force actuator. However, as already noted, the aim in impedance control is rather to provide specific relationships between the effort and motion than to follow a prescribed force trajectory as in case of force control. Considering the arrangement of position and force sensor and control signals within control loops (inner or outer), the following two common approaches to the issue of providing task-specific impedance via feedback control can be distinguished [14]: i)

ii)

Position-mode or outer-loop control, where a target impedance control block relating the force exerted on the end-effector and its relative position is added in an additional control loop around the position-controlled manipulator. Here, an inner loop is closed on the basis of the position sensor with an outer loop closed around it, based on the force sensor [15, 16]; Force-mode or inner-loop control, where position is measured and force command is computed to satisfy target impedance objectives [14].

Regarding the force-motion relationship, i.e. the impedance order, impedance control schemes can be further categorized into: stiffness control [17], damping control [18], and general impedance control [19, 20], using the zero-, first- and second-order impedance model respectively. There are also additional criteria for a further detailed classification of active compliant motion control concepts. For example, we can categorize methods with respect to the source of force information (with or without direct interaction force sensing), allocation of force sensor (wrist, torque sensor at joints, force sensing pedestal, force sensor placed at the contact surface, sensors at robot links, fingers, etc.). In order to avoid the problems associated with noncollocation between measurement of contact forces and actuation at robot joints, which can cause instability [21], it was also proposed to use redundant force information, combining the joint force sensing with one of the above force sensing principles.

Control of Robots in Contact Tasks: A Survey

7

Regarding the space in which the active force control is performed, one can distinguish: i)

ii)

Operational space control techniques, where the robot control is taking place in the same frame in which robot actions are specified [22, 23]. This approach requires the construction of a model describing the system dynamic behavior as perceived at the point of an end-effector, where the task is specified (operational point, i.e. coordinate frame). Traditionally, compliant motion is specified using the task or compliance frame approach [1]. This geometrical approach introduces a Cartesian compliant frame with orthogonal force and position (velocity) controlled directions. In order to overcome the limitations of this approach, new methods were proposed [24, 25]. These approaches, referred to as explicit task specification of compliant motion, are based on the model of the constraint topology for every contact configuration and utilize projective geometry metrics in order to define a hybrid contact task. Joint space control, where control objectives and actions are mapped into joint space [26]. Associated with this control approach are transformations of action attributes, compliance and contact forces from the task to the joint space.

Further, considering control issues, such as variations of control parameters (gains) during execution, one can distinguish: i)

ii) iii)

Non-adaptive active compliance control algorithms which use fixed gains, assuming small variations in the robot and environment parameters, Adaptive control, which can adapt to the process variations [27, 28], and Robust control approaches that can handle model imprecision and parametric uncertainties within specified bounds [29, 30].

Depending on the extent the system dynamics is involved in the applied control laws, it is further possible to distinguish: i)

Non-dynamic, i.e. kinematic model-based algorithms, such as hybrid control [7], stiffness control [17], etc., which approximate the contact problem considering its static aspects only;

8

Dynamics and Robust Control of Robot-Environment Interaction

ii)

Dynamic model-based control schemes, such as resolved acceleration control [31], dynamic hybrid control [11], constrained robot control [32], dynamic position/force control in contact with dynamic environment [8, 33], based on complete dynamic models of the robot and environment that take into account all dynamic interactions between position- and force- controlled directions.

Although relatively low velocities are involved in contact motion, high dynamic interaction (i.e. exchange of energy) between the robot and its environment affects the control system significantly and can jeopardize the stability of the control system [34]. Consequently, the roles of the robot dynamics [35] and environment dynamics [8, 33] in the control of compliant motion are of essential importance. Kinematic algorithms are mostly based on Jacobian matrix computation, while the complexity of dynamic methods is much higher [36]. The seminal hybrid control method proposed by Raibert and Craig [7] provides essentially a quasi-static approach to compliance control based on an idealized simple geometric model of a constrained motion task (Mason’s constraint-frame formalism). In the hybrid control, the dynamics of both the robot and environment (i.e. dynamic interaction) is neglected. The dynamic hybrid control [11] and constrained motion control [32] consider constraints on the robot motion described in the form of algebraic equations defining a hyper surface. These methods take both the robot dynamic models and the model of environment into account in order to synthesize the dynamic control laws that ensure an admissible robot motion under the constraint and realize desired interaction forces. A further generalization of the constrained motion problem leads to introducing active dynamic contact forces (dynamic environment), also described by differential equations. In a dynamic environment, the interaction forces are not compensated for by constraint reactions, they rather produce active work on the environment. Obviously, the contact with a dynamic environment requires considering the overall system dynamics, involving the robot and interaction models, in order to obtain an admissible robot motion and interaction forces. In the papers dealing with the dynamic control of robots interacting with dynamic environment [8, 33] Vukobratovic and Ekalo considered “purely dynamic” interaction without passive reactions. De Luca and Manes [37] have proposed a convenient model structure that handles a more general case in which purely kinematic constraints on the robot effector exist together with dynamic interactions.

9

Control of Robots in Contact Tasks: A Survey

Passive Compliance

fixed

inherent compliance of robot structure

adaptable

additional compliance devices (RCC)

additional adjustable compliance devices

servo gain adjustment

Fig. 1.2 Passive compliance classification

Active Compliance

Force Control Solution: direct control of interaction force

Impedance Control Solution: control of dynamic robot reaction to contact forces according to target force/motion relationship

hybrid position/force control

unified postion/force control

stiffness

position and force control in two orthogonal subspaces

control of both position and force along each task space direction

position proportional

F=k ⋅∆ x

damping general (accomodation) impedance ⋅ ⋅ ⋅∆ x F=k ⋅∆ x F=M ⋅ ∆⋅⋅ x+D ⋅∆ x+k velocity proportional

position, velocity and acceleration proportional

Fig. 1.3 Active compliance control methods

Although being very inclusive, the above classification cannot encompass all of the concepts that have been proposed up to now. Namely, some of the elaborated approaches combine two or more different methods categorized in distinct groups, and attempt to use benefits of each to compensate specific disadvantages of single solution strategies. To such methods belong compliant motion control approaches that combine force and impedance control [12, 38]. Some methods integrate control and mechanical system design [39]. This approach is based on micro-macro manipulator structures, providing an inherently stable and well-suited subsystem for high-bandwidth active force

10

Dynamics and Robust Control of Robot-Environment Interaction

control. Note that the above terminology represents, in some measure, a trade-off among different nomenclature used in the literature. For instance, Mason [1] denominates the control concepts specifying a linear relation between the effector force and position as explicit feedback, while Whitney [6] refers the explicit control to techniques having a desired force input rather than position or velocity input. The introduced classification and terminology reflect, in our opinion, in a most suitable way the essential aspects of appropriate control strategies. The above classification is summarized in (Figs. 1.1, 1.2, 1.3). 1.4 Model of the Robot Performing Contact Tasks Here we consider in brief the simplified models of robot constrained motion to be used for the analysis of contact motion control concepts. In order to form the mathematical model which describes the dynamics of a closed-configuration manipulator, let us consider an open robotic structure whose terminal link (endeffector) is subjected to a generalized external force (Fig. 1.4). The dynamics model of a rigid manipulation robot interacting with the environment is described by the vector differential equation in the form:

H (q )qɺɺ + h(q, qɺ ) = τ a + J T (q ) F

(1.1)

where q = q(t ) is an n-dimensional vector of robot generalized coordinates; H ( q ) is an n × n positive definite matrix of inertia moments of the manipulator mechanism; h( q, qɺ ) is an n-dimensional nonlinear vector function involving centrifugal, Coriolis and gravitational moments, i.e. h(q, qɺ ) = h (q, qɺ ) + g ( q ) , where g ( q ) is the vector of gravitational moments and h ( q, qɺ ) is the Coriolis and centrifugal vector component; τ a = τ a (t ) is an n-dimensional vector of generalized joint-axes driving torques; J T ( q ) is an n × m Jacobian matrix relating joint space velocity to task space velocity; F = F (t ) is an mdimensional vector of external forces and moments acting on the end-effector. The dynamic model of the actuators (we confine ourselves to robot manipulators driven by DC motors) that drive the robot joints has to be added to the above equations. It is convenient to adopt this model in a linear form. Taking into account that electric time constants of actual DC motors driving almost all commercial robotic systems are very low, we shall adopt a second-order model of actuators

Control of Robots in Contact Tasks: A Survey

n 2j I mj qɺɺmj + n2j bmj qɺmj + τ aj = n jτ mj

11

(1.2)

where qmj is the output angle of motor shaft after reducer; n j is the gear ratio; I mj is the inertia of the motor actuator; bmj is the motor viscous friction coefficient; τ mj is the control input to the actuator j (i.e. motor torque), while j denotes the j -th local subsystem (m stands for “motor”). The torque produced by the motor is proportional to the armature current, that is

τ mj = kmj imj

(1.3)

where kmj is the torque constant. If we assume the stiffness at the joints (gears) to be infinite, the relations between the mechanism coordinate q j coincide with those of the actuator coordinate qmj . The dynamic models of the actuators and mechanical part of the robot are related by joint torques (loads). If we introduce τ aj from (1.2) into (1.1) we get the overall model of the robotic mechanism in the joint coordinate space

H (q) qɺɺ + Bm qɺ + h (q, qɺ ) + g (q) = τ q + J T (q) F

(1.4)

where

H (q ) = H (q ) + I m = H (q ) + diag (n 2j I mj ) (1.5)

Bm = diag (n 2j bmj )

and τ q is a nx1 vector of input torques at joint shaft (after reducer), which for n = 6 has the form T

τ q = [ n1τ m1.... n6τ m 6 ] . The above dynamic model can be transformed into an equivalent form which is more convenient for the analysis and synthesis of a robot controller for contact tasks. When the manipulator interacts with the environment it is very convenient to describe its dynamics in the space where the manipulation task is described, rather than in joint coordinate space (also termed configuration space). The endeffector position and orientation with respect to a reference coordinate system can be described by a 6-dimensional vector x. Using the robot Jacobian matrix we can transform the robot dynamic model (1.4) from the joint into the endeffector coordinate system

Λ ( x ) ɺɺ x + B ( x ) xɺ + µ ( x, xɺ ) + p ( x ) = τ + F

(1.6)

12

Dynamics and Robust Control of Robot-Environment Interaction

Fig. 1.4 Open kinematic chain exposed to the action of an external force

where the relationships between the corresponding matrices and vectors from equations (1.4) and (1.6) are given by the following equations

Λ ( x)

=

J − T ( q ) H ( q ) J −1 ( q )

B( x)

=

J −T (q ) Bm J −1 (q )

µ ( x, xɺ ) =

J −T (q ) h (q, qɺ ) − Λ ( x) Jɺ (q ) qɺ −T

p( x)

=

J

τ

=

J −T ( q ) τ q

(1.7)

(q) g (q )

Description, analysis and control of manipulator systems with respect to the dynamic characteristics of their end-effectors are referred to as the operational space formulation [22]. Analogous to the joint space quantities, Λ ( x ) is the operational space inertia matrix, µ ( x, xɺ ) is the vector of Coriolis and centrifugal forces, p ( x ) is the vector of gravity terms, and τ is the applied input control force in the operational space. The above Cartesian dynamic model covers a large class of different robotic structures such as industrial robots, parallel manipulator, wire robots, etc.

Control of Robots in Contact Tasks: A Survey

13

Since the mechanical interaction process is generally very complex and difficult to describe mathematically in an exact way, we are compelled to introduce certain simplifications and thus partly idealize the problem. In practice the interaction force F is commonly modeled as a function of the robot dynamics, i.e. end-effector’s motion (position, velocity and acceleration) and control input

F = F ( x, xɺ , ɺɺ x ,τ , d , d e )

(1.8)

where d and d e denote the sets of robot and environment model parameters, respectively. The following general work environment models have been mostly applied in the literature for describing robot constrained motion: rigid hypersurface [11, 32], dynamic environment [8, 37], and compliant environment [40]. In the case when the environment does not possess the displacements (DOFs) that are independent from the robot motion, the mathematical model of the environment dynamics in the frame of robot coordinates can be described by nonlinear differential equations [8]

M ( q )qɺɺ + L( q, qɺ ) = S T ( q ) F

(1.9)

where M ( q) is a non-singular n × n matrix; L( q, qɺ ) is a nonlinear ndimensional vector function; S T ( q) is an n × n matrix with rank equal to n. Then, the system (1.4)-(1.9) describes the dynamics of robot interacting with dynamic environment. We assume that for the contact cases all the mentioned matrices and vectors are continuous functions of the arguments. In the operational space the model of a purely dynamic environment has the form [40]

( x) ɺɺ x + ( x, xɺ ) = − F

( x ) = − S − T ( q ) M ( q ) J −1 ( q ) ( x, xɺ ) = − S −T (q) L(q, qɺ ) +

( x) Jɺ (q)qɺ

In effect, a general environment model involves geometrical (kinematic) constraints plus dynamic constraints [37]. An example of such dynamic environment is when the robot is turning a crank or sliding a drawer whose dynamics is relevant for the robot motion and cannot be neglected. For the control design purposes it is customary to utilize the linearized model of manipulator and environment. The applicability of linearized model in constrained motion control design, especially in industrial robotic systems, was

14

Dynamics and Robust Control of Robot-Environment Interaction

demonstrated in [41, 42]. Neglecting the nonlinear Coriolis and centrifugal effects due to relatively low operating velocities (“rate linearization”) during contact and assuming the gravitational effect to be ideally compensated for, we obtain the linearized model around a nominal trajectory in the Cartesian space x0 in the form

Λ( x0 ) ɺɺ x + B ( x0 ) xɺ = τ ( x0 ) + F

(1.10)

In passive linear environments, it is convenient to adopt the relationship between forces and motion around the contact point in the form (linear elastic environment)

− F = M e ɺɺ p + Be pɺ + K e p

(1.11)

where p denotes the end-effector penetration through the surface defined by p = x − xe , xe is the contact point location, M e , Be and K e are the inertial, damping and stiffness matrices, respectively. 1.5 Passive Compliance Methods In accordance with the above classification, we shall firstly review the compliant control methods based on passive accommodation (with no actuator involved). Passive compliance in general is a concept often used in practice to overcome the problems arising from positional and angular misalignments between the manipulator and its working environment. 1.5.1 Non-adaptable compliance methods The passive compliance method, which is based on the inherent robots structural elasticity, is more interesting as a theoretical solution than as a feasible approach. This method supposes that the compliance of the mechanical structure has a determining role in the compliance of the entire system. However, this assumption is opposite to the real performance of commercial robotic systems, which are designed to achieve high positioning accuracy, whereas elastic properties of the arms are not significant. This method does not offer any possibility to adapt system compliance to the various task requirements. The idea to utilize flexible manipulator arms as an instrumented compliant system [2] is coupled with additional problems due to complex modeling and controlling of elastic robots.

Control of Robots in Contact Tasks: A Survey

15

The method based on mechanical compliance devices, in principle, also utilizes structural compliance, but most influential source of multi-axes compliance in this case is a specially constructed device, whose behavior is known and sufficiently repeatable. By this means relatively good performances, especially in the robotic assembly, have been achieved. Different types of such devices have been developed, the best known being the RCC, “Remote Center Compliance” [3], developed in the Charles Stark Draper Laboratory. RCC is designed to achieve the workpiece rotation around a defined center of compliance. The compliance center is referred to as a point such that the force applied at this point causes only translation, while the torque applied around an axis through this point will cause pure rotation of the workpiece (Fig. 1.5). A crucial feature of the RCC is that it consists of translational and rotational parts, which allows lateral and angular errors to be accommodated independently. An improvement of RCC represents IRCC, “Instrumented Remote Center Compliance” [43], which provides the fast error absorption characteristic of an RCC and measurement characteristic of a multi-DOF sensor. The information about contact forces and deformations can be used for task monitoring, calibration, contour following, or for positioning feedback.

Fig. 1.5 Remote Center Compliance (RCC)

1.5.2 Adaptable compliance methods Further development of the RCC has led to adaptable compliant devices [4] which enable the location of the center of compliance to be automatically controlled to a prescribed extent in accordance with involved parts of different

16

Dynamics and Robust Control of Robot-Environment Interaction

length and weight. These devices are usually instrumented so to provide information about end-point deflections for robot control.

Fig. 1.6 Passive gain adjustment scheme

The controller gain adjustment method is based on the compliance of the robotic controller and attempts to provide a universally programmable passive compliance at the end-point, by a relatively simple adjustment of the servo gains. The basic principle is to tune the positional servo gains in order to make the robot behave as a linear six-dimensional spring in Cartesian space with programmable stiffness. Therefore, taking into account the relationship between the forces exerted upon the robot and its reaction (stiffness-like behavior), the gain adjustment method is often considered as being equivalent to the impedance (i.e. stiffness) control. The basic gain adjustment control scheme is sketched in Fig. 1.6, where x0 and q0 are the nominal Cartesian and joint position vectors respectively; f −1 ( x ) denotes the inverse kinematic transformation; q is the actual joint position; gˆ ( q ) is the computed gravitational torque, while the control torque τ q is defined as

τ q = k p ( q0 - q ) + gˆ ( q )

(1.12)

where k p represents the joint stiffness matrix which should be tuned in order to achieve the arm to behave with the desired stiffness K S . The relationship between the joint and Cartesian stiffness matrices is given by

k p = J T KS J K S = J − T k p J −1

(1.13)

Control of Robots in Contact Tasks: A Survey

17

where J (q ) is a Jacobian matrix relating velocities (i.e. forces) between a Cartesian frame fixed at the compliance center and joint coordinate space. At the center of compliance the Cartesian stiffness matrix is diagonal, but, according to (1.13), the corresponding joint stiffness k p is a fully symmetric matrix. This means that the joint stiffness matrix is highly coupled and a position error at one joint will affect the commanded torque at all other joints. This equation represents the central formulation of active gain adjustment methods. However, although this stiffness-like behavior could be theoretically adjusted on-line during task running, we have classified this method as being passive compliance, because the compliant motion is performed in a purely passive way by the action of external forces, rather than by the force feedback as in the active stiffness control. In reality, this concept is coupled with several problems. The desired springlike behavior cannot be accurately realized by the major of contemporary robotic systems. The nonlinearities such as friction and backlash in mechanical transmission, or process frictional phenomena like jamming, can largely destroy the force-position causality based on the stiffness law. Furthermore, by setting the control gains in some directions to be very low, we make the entire system more sensitive to perturbations. All these facts make the performance of this control approach uncertain, imposing the necessity to introduce additional sensor information in order to monitor the task execution. The principle of adaptable control gains is more suitable for the direct drive, multifingered or wrist hands, but in this case the method appears to be very close to the above ones that use special adaptable compliant-devices. Concerning the use of passive gain adjustment concept in industrial practice, it should be taken into account that conventional robotic systems are nonbackdriveable, because of high gear ratios and Coulomb friction/stiction effects at joints (the order of equivalent friction force in the Cartesian space is about 102 N). Hence, although a compliant control is applied, a force exerted at the end-effector will not cause a corresponding detectable displacement at joints. Therefore, the method may be applied only in the manipulation tasks permitting large interaction forces. Due to relatively high costs and low robustness of force sensors, however, there has been evident recently an increasing interest of industrial robot manufacturers to apply this method in specific tasks such as

18

Dynamics and Robust Control of Robot-Environment Interaction

handling of castings (e.g. Soft-servo or Soft-float industrial robot control functions). 1.6 Active Compliant Motion Control Methods The active compliance control methods utilize in a best way the reprogrammability of manipulation robots, representing their major characteristic (i.e. the ability to switch from one production task to another). 1.6.1 Impedance control First implementations of force feedback control of a manipulator belong to the impedance control [6]. The impedance control problem can be defined as designing a controller so that interaction forces govern the error between desired and actual positions of the end-effector. The control input describing a desired target impedance relation may, in principle, have an arbitrary functional form, but it is commonly adopted in the form of a linear second-order differential equation describing the simple six-dimensional decoupled mass-spring-damper mechanical system. The reason for this lies in the fact that the dynamics of a second-order system is well understood and familiar. Lee and Lee [44] have developed a control algorithm, referred to as generalized impedance control, by introducing a higher order impedance relation between position and force errors, which includes force derivatives. In other words, impedance control is a general approach to contact task control in which the robot behaves as a mass-spring-dashpot system whose parameters can be specified arbitrarily. This can be achieved by feedback control using position and force sensing. The following control objective should be attained

F = M t ( ɺɺ x − ɺɺ x0 ) + Bt ( xɺ − xɺ0 ) + K t ( x − x0 ) = M t ɺɺ e + Bt eɺ + K t e

(1.14)

or in the s-domain

F ( s ) = Z t ( s ) ( x − x0 ) = Z t ( s ) e = ( M t s 2 + Bt s + K t ) ( x − x0 )

(1.15)

2 where Z t ( s ) = M t s + Bt s + K t is the target robot impedance in Cartesian space; x0 describes the desired position trajectory, x is the actual position

Control of Robots in Contact Tasks: A Survey

19

vector; e is the position control error; F is the external force exerted upon the robot, and M t , Bt , and K t are positive definite matrices which define the target impedance, where K t is the stiffness matrix, Bt is the damping matrix and M t is the inertia matrix. The diagonal elements of these target model matrices describe the desired robot mechanical behavior during the contact. One of the most common approaches for representing robot’s and object’s positions in robot programs is based on coordinate frames. Therefore it is convenient to describe the robot impedance reaction to the external forces also with respect to a frame, referred to as compliance or C-frame. Along each of Cframe directions, the target model describes the mechanical system with the programmable impedance (mechanical parameters), presented in Fig. 1.7, where, for simplicity sake, only spring elements are depicted. The model describes a virtual spatial system consisting of mutually independent spatial mass-damperspring subsystems in six Cartesian directions.

Fig. 1.7 Target stiffness model in the C-frame

The target impedance matrices can be selected to correspond to various objectives of the given manipulation task [14]. Obviously, high stiffness is selected in the directions where the environment is compliant and positioning accuracy is important. Low stiffness is selected in the directions where small interaction forces have to be maintained. Large Bt values are specified when

20

Dynamics and Robust Control of Robot-Environment Interaction

energy is to dissipate, and M t is used to provide smooth transient behavior in the system response during the contact. In order to assess how well a designed impedance controller meets the above control objective, it is customary to specify various performance criteria. A reasonable measure capable to express the performance of the impedance control represents the difference between the target model and the real system behavior described by actual robot motion and interaction forces [45]. Depending on which of these physical quantities is used to characterize the system behavior (force or position), the impedance control error can be expressed by means of force measure (force mode error)

e f = M t ( ɺɺ x − ɺɺ x0 ) + Bt ( xɺ − xɺ0 ) + K t ( x − x0 ) − F

(1.16)

or by position measure (position model error)

e p = x − x0 − δ x f

(1.17)

where the target model-based position deviation δ x f is obtained as the solution of the target model differential equation

F = M tδ ɺɺ x f + Btδ xɺ f + K tδ x f

(1.18)

for the initial conditions: F ( t0 ) = 0; δ x f ( t0 ) = δ x0 .

Fig. 1.8 Damping control

The control goal defined above can be achieved using various control strategies. Impedance control represents a strategy for constrained motion rather

Control of Robots in Contact Tasks: A Survey

21

than a concrete control scheme. Various control concepts and schemes have been established and proposed for controlling the relation between robot motion and interaction force. One of the first approaches to impedance control was proposed by Whitney [18] (Fig. 1.8). In this approach, referred to as damping or accommodation control, the force feedback is closed around the velocity control loop. The interaction force is converted into velocity modification command by a constant damping coefficient K f . On a simplified example of discrete time force control, Whitney defined the condition for system stability during the contact

0 < T K f Ke < 1

(1.19)

where T is the sampling period, K f is the force control gain (damping coefficient), and K e is the stiffness of the environment. This condition implies that if K e is high, the product TK f must be small. To avoid large contact forces, a very high sampling rate, i.e. small T , is required. Alternatively, for the contact with a very stiff object Whitney proposed to introduce a passive compliance in order to make the equivalent environmental stiffness (includes the stiffness of robot structure, environment, sensor, etc.) K e smaller. Salisbury [17] proposed to modify the end-effector position in accordance with the interaction force (Fig. 1.9). This concept is based on a generalized stiffness formulation F = Kδ x , where δ x is a generalized displacement from the nominal commanded end-effector position, and K is a six-dimensional stiffness matrix. Based on the difference between desired and actual end position, a nominal force is computed and converted into the joint torque using the transpose of the Jacobian matrix. This force is then used to determine the torque error at each joint, which is further used to correct the applied torque, so that a desired force (i.e. stiffness) is maintained at the robot hand. The requirements on the stiffness matrix elements and their design for specific tasks are considered in [6]. The above impedance control schemes are simple and relatively easy to implement. However, in these approaches the achieved closed-loop impedance behavior in the Cartesian space depends on robot configuration. Obviously, to replace a highly nonlinear robot dynamic model with a linear time-invariant target system (e.g. mass-damper-spring system), requires in general a control law that will compensate for the relevant system nonlinearities (model-based dynamic control). The most common and general impedance control concept was

22

Dynamics and Robust Control of Robot-Environment Interaction

proposed by Hogan [19]. The author has defined a unified theoretical framework for understanding the mechanical interactions between physical systems.

Fig. 1.9 Stiffness control

1.6.1.1 Force-based impedance control Most of the impedance control algorithms utilize the computed torque method to cancel the nonlinearity in robot dynamics in order to achieve linear target impedance behavior. This popular approach requires computation of a complete dynamic model of the robot’s constrained motion, which makes its realization rather complex. An important drawback of this approach is also the sensitivity to model uncertainties and parameter variations. Performance improvements that can be achieved with the algorithms in industrial robotics are not in proportion to the implementation efforts. Hogan [46] has proposed several techniques with and without force feedback for modulating the end-point impedance of a general nonlinear manipulator. Supposing that the Cartesian dynamic model perfectly matches the real system, Hogan proposed the following nonlinear control law

τ = Λˆ M t −1 [ K t ( x0 − x ) − Bt xɺ + F ] + µˆ ( x, xɺ ) + pˆ ( x ) − F

(1.20)

to be applied in order to attain a reasonable target impedance behavior in the ideal case in the form

F = M t ɺɺ x + Bt xɺ + K t ( x − x0 )

(1.21)

Control of Robots in Contact Tasks: A Survey

23

The control scheme corresponding to the above control law is sketched in Fig. 1.10. In this figure a distinction is made between the active force exerted by the robot ( F ) and the reactive external force ( F ), which can be computed assuming a simple spring-like environmental model

F = K e ( x − xe ) = − F

(1.22)

where K e is the stiffness of the environment. This control law represents essentially a nonlinear control algorithm which combines the inverse control technique [47] (terms also used are computed torque method, nonlinear decoupling) and force-based (inner loop) impedance control. In the force-based impedance control algorithms (Fig. 1.10), in general, an expected reference force is computed to satisfy the desired impedance specification based on the position error and target impedance FC ( s ) = Gt ( s )( x0 − x)(Gt ( s ) = Z t ( s )). The expected contact force FC is compared with the actual force sensed by the force sensor and a force error is computed. This error is further multiplied

ˆ M −1 . Finally, the product is summed with dynamic with the inertia matrices Λ t compensation terms (Coriolis and gravitation vectors) and feed-forward force F to obtain Cartesian control force, which is further transferred to the robot joint T

via the transposed Jacobian J , to get the actuator torque control inputs. It is relatively easy to prove that the control law

τ = Λˆ {ɺɺ x0 + M t −1 [ K t ( x0 − x ) + Bt ( xɺ0 − xɺ ) + F ]} + µˆ ( x, xɺ ) + pˆ ( x ) − F

(1.23)

realizes the impedance control behavior specified in (1.15). The reason why impedance control methods based on force control input cannot be suitably applied in commercial robotic system lies in the fact that commercial robots are designed as “positioning devices”. In the above methods the driving torque vector ensuring the desired target impedance behavior has been computed and then multiplied by the transpose of the Jacobian matrix, to be realized around the actuated robot joints. However, the realization of computed torque in commercial robotic systems is not accurate because the local servos are position-controlled and there is no force feedback with respect to the torques around the robot’s joints. Consequently, the realization of desired torques is poor, since high friction and other nonlinearities in the transmission mechanisms contribute significantly to the inaccuracy of current-torque causality. The implementation of force-based impedance control under above-mentioned

24

Dynamics and Robust Control of Robot-Environment Interaction

difficulties can be only successful in a new generation of direct drive robotic systems [48] with accurate joint torque control. It may also be remarked that force-based impedance control requires a completely new control system to be implemented.

Fig. 1.10 Force-based dynamic impedance control

1.6.1.2 Position-based impedance control In commercial robotic systems it is feasible to implement only the position-mode impedance control by closing a force-sensing loop around position controller. Position- based impedance control is most reliable and suitable for implementation in industrial robot control systems since it does not require any modification of conventional positional controller. Practically, two basic impedance control schemes with an internal position control can be distinguished [49]. The first scheme is sketched in Fig. 1.11. In this control system, an inner position control loop is closed based on the position sensing, with an outer loop closed around it, based on the force sensing. The force loop is naturally closed when the end-effector encounters the environment. The outer loop includes a force feedback compensator GF , basically representing admittance since its role is to shape the relation between the contact force and corresponding nominal position modifications ∆x f . This block is imposed on the system to regulate the force response to the commanded and −1 actual motion according to the target admittance Z t . Other control blocks in Fig. 1.11 represent a common industrial robot position control system involving

Control of Robots in Contact Tasks: A Survey

25

the following transfer function matrices: Gr -position control regulator, Gs robot plant and Ge -environment. The position correction ∆x F is subtracted from the nominal position x0 and the command input vector for the positional controller, referred to as reference position xr , is computed. A good tracking of the reference position has to be realized by the internal position controller. −1 Practically, assuming GF = Z t , the position error input to the position controller ∆xr becomes

∆xr = xr − x = x0 − ∆xF − x = x0 − x − Z t −1F = e p

(1.24)

This means that the control system depicted in Fig. 1.11 utilizes the positionrelated impedance model error e p (1.17) to realize the target impedance behavior. Practically, the impedance model error e p is fed forward to the position controller Gr in order to be nullified within the internal position control loop. Since the purpose of the control system in Fig. 1.11 is to control position, it will be referred to as position impedance model error control.

Fig. 1.11 Position model error impedance control

Another position-based impedance control structure is depicted in Fig. 1.12. This scheme provides a generalization of the original scheme proposed by Maples and Becker [15] and is referred to as outer/inner loop stiffness control. The control scheme consists of two parallel feedback loops superimposed to the internal position control and closed using measurements from both the wrist force sensor and position sensors. By analyzing the control scheme one can see that the position error e = ∆x0 = x0 − x is multiplied by the task-specific target impedance G t ( s ) = Z t ( s ) to provide a nominal (reference) force F0 which corresponds to the target impedance behavior at the output. The tracking of this force is realized by the next feedback loop closed on the sensed force F . In the

26

Dynamics and Robust Control of Robot-Environment Interaction

ideal case we have F0 = F , describing the target behavior. Thus, Fig. 1.12 shows basically a force control system with target impedance added to regulate the motion response to the interaction force. Following the control flow we see that the force-error in this control scheme corresponds to the previously defined force impedance model error (1.17)

∆F = F0 − F = Gt ( x0 − x ) − F = e f

(1.25)

Therefore, the control system shown in Fig. 1.12 we will refer to as force model error impedance control. Similarly to the previous system (Fig. 1.11), the model error (1.25) is further relayed to the internal control part, to reduce this error to zero. However, in contrast to the position model error control shown in Fig. 1.11, where the position model error is eliminated by the internal position control, in the control system depicted in Fig. 1.12 the regulation of the model error is realized by means of the compensator GF . In order to retain the internal position control loop, the implicit force control structure is implemented by passing the force error ∆F through the admittance filter GF , providing the nominal path modification ∆x F . The position correction is further added to the Cartesian nominal position x0 , and via reference position xr fed forward to the position servo. Obviously, to achieve ∆F = e f → 0 as t → ∞ which ensures a steady-state position deviation ( x0 − x )∞ = e∞ corresponding to the target impedance (stiffness) model, the regulator GF has to include an integral control term. Both control approaches (schemes) utilize basically similar concepts to realize the target impedance model by reducing the impedance model errors e p and e f to zero. Each of them has specific advantages and disadvantages [49]. The e p -based scheme (Fig. 1.11) is essentially simpler and easier for implementation. This scheme, under some circumstances, allows the realization of different target impedances. However, their realization is done by setting the compensator GF to the target admittance, while the feedback control is to be undertaken by the position controller. This is similar to an “open-loop” target impedance control. Contrarily to this, in the force model-error control scheme (Fig. 1.12), the target impedance is specified in the outer-loop using the Gt block, while the role of internal loop compensator GF is to ensure tracking of

Control of Robots in Contact Tasks: A Survey

27

the selected model using the force feedback. The main problem with the e f based scheme lies in the transition from and to the contact (constrained motion). The external impedance loop in this scheme is closed even in the free space, when the actual contact force is zero, and thus affects the position control performance. The compensator GF has to be tuned to achieve the required control goal in the contact with a stiff environment, e.g. a large amount of damping, in order to ensure a stable transition. However, this is usually contrary to the position control performance needed in the free space. In the e p -based scheme, however, the force feedback loop is closed naturally by the physical contact. The position-based impedance approach, in general, suffers from the inability to provide very soft impedance because of the limits in the accuracy of the position control system and sensor resolution. This approach is mainly suitable for the applications where high position accuracy is required in some Cartesian directions, which is realized by a stiff and robust joint control. The force- (i.e. torque)-based approach is better suited to providing small impedance (stiffness and damping) while reducing the contact force. From a computational viewpoint, this approach is reasonable for the applications where manipulator gravity is small, and slow motion is required. In other cases, manipulator modeling details (i.e. complete dynamic models) are needed. Contrary to the position-based impedance control, the force-based control is mainly intended to be applied in robotic systems with relatively good causality between joint torques and endeffector forces such as in direct-drive manipulators. A detailed consideration of the synthesis of position-based impedance control for industrial robots is presented in Chapter 3. 1.6.1.3 Other impedance-control approaches Considerable research efforts have been made to develop adaptive impedance control algorithms. Daneshmend et al. [27] have proposed a model-reference adaptive control scheme with Whitney’s damping control loop. Several authors have pursued Craig’s adaptive inverse dynamic control algorithms [50] and expanded it to suit the application for contact motion. Lu and Goldenberg [45] proposed a sliding-mode-based control law for impedance control. The proposed

28

Dynamics and Robust Control of Robot-Environment Interaction

controller consists of two parts: nominal dynamic model to compensate for nonlinearities in the robot dynamics, and a compensator ensuring the impedance error (i.e. the difference between the nominal target model and the actual impedance) converges asymptotically to zero on the sliding surface. In order to cope with the chattering effects in the variable structure sliding-mode control, a continuous switching algorithm in a small region around sliding surface is proposed. Al-Jarah and Zheng [51] proposed an interesting adaptive impedance control algorithm intended to minimize the interaction force between the manipulator and the environment.

Fig. 1.12 Force-model-error-based impedance control

Dawson et al. [30] have developed a robust position/force control algorithm based on the impedance approach. The control scheme consists of two blocks: a “desired trajectory generator”, which computes the modified command position (based on the target impedance model, the nominal position and force measurements), and a controller involving a PD regulator and a robust controller. The purpose of the robust controller is to ensure that the control tracking error (i.e. difference between target and actual robot impedance) converges asymptotically to zero in spite of model uncertainties within specified bounds. 1.6.2 Hybrid position/force control This approach is based on a theory of compliant force and position control formalized by Mason [1] and it concerns with a large class of tasks involving partially constrained motion of the robot. Depending on the specific mechanical and geometrical characteristics of the given contact problem, this approach makes a distinction between two sets of constraints upon the robot’s motion and

Control of Robots in Contact Tasks: A Survey

29

contact forces. A set of constraints that occur as a natural consequence of the task configuration, i.e. of the nature of the desired contact between an endeffector held by robot and constrained surface, is called natural constraints. Physical objects impose natural constraints. As already mentioned, a suitable frame in which the task to be performed is easily described, i.e. in which constraints are specified, is referred to as the constraint-frame (terms also used are task- or compliance-frame) [52]. For example, in the “surface sliding” contact task it is customary to adopt the Cartesian constraint-frame as sketched in Fig. 1.13. Assuming an ideally rigid and frictionless contact between the endeffector and the constraint surface, it is obvious that the natural constraints restrict the end-effector motion in the -z-direction, as well as rotations about the x and y axes. The frictionless contact prevents the forces in these directions and the torque around z-axis to be applied. In order to specify the task to be realized by the robot with respect to a compliant frame it is necessary to introduce the so-called artificial constraints, and they have to be imposed by the control system. These constraints essentially partition the possible DOFs of motion in those that must be position-controlled and those which should be force-controlled, in order to perform the given task. It is obvious to define an artificial constraint with respect to force when there is a natural constraint on the end-effector motion in this direction (i.e. DOF), and vice versa (Fig. 1.13) (the index “-z” denotes constraints in the negative z direction).

Fig. 1.13 Specification of “surface sliding” hybrid position/force control task

30

Dynamics and Robust Control of Robot-Environment Interaction

To implement hybrid position/force control, a diagonal Boolean matrix S, called the compliance selection matrix [7], has been introduced in the feedback loops in order to filter out sensed end-effector forces and displacements which are inconsistent with the contact task model. In accordance with the specified artificial constraints the i-th diagonal element of this matrix has the value 1 if the i-th DOF with respect to the task-frame is to be force-controlled, and the value 0, if it is position-controlled. According to Mason [1], to specify a hybrid contact task, the following information sets have to be defined i) ii) iii)

Position and orientation of the task frame. Denotation of position- and force-controlled directions with respect to the task frame (selection matrix). Desired position and force setpoints expressed in the task frame.

Once the contact task is specified, the next step is to select the appropriate control algorithms. The relevant methods are discussed below. 1.6.2.1 Explicit force control The most important method within this group is certainly the algorithm proposed by Raibert and Craig [7]. Figure 1.14 presents the control scheme that illustrates the main idea of this method. The control consists of two parallel feedback loops, the upper one for the position and the lower one for the force. Each of these loops utilizes separate sensor systems. The positional loop utilizes the information obtained from the positional sensors at the robot joints, and the force loop is based on force sensing data. Separate control laws are adopted for each loop. The central idea of this hybrid control method is to apply two outwardly independent control loops assigned to each DOF in the task frame. Both control loops cooperate synchronously to control each of the manipulator joints.

Fig. 1.14 Explicit hybrid position/force control

Control of Robots in Contact Tasks: A Survey

31

At first glance, this concept appears to be ideally suited to solve the hybrid position/force control problems. However, a deeper insight into the method reveals some essential difficulties related to this concept. The first problem is related to the opposite requirements of the hybrid control concept concerning position and force control subtasks. Precisely, it is required that the position control is highly stiff in order to keep the positioning errors in the selected directions as small as possible, while the force control requires a relatively low stiffness of the robot (corresponding to the desired force) in the force-controlled direction with respect to the task-frame, in order to ensure that the end-effector behaves compliantly with the environment. As already explained above, the explicit hybrid control attempts to solve this problem by control decoupling in two independent parts that are position- and forcecontrolled (Fig. 1.14). In the force-controlled directions the position errors are put to zero by multiplication with the selection matrix orthogonal complement (position selection matrix) defined as S = I − S b. This would mean that the position control part does not interfere with the force control loop. However, that is not the actual case. The joint space nature of robot control realization results in the coupling between the position- and force-control loops which have been previously mathematically decoupled in the task-frame. Assuming a proportional plus differential (PD) position control law, and that the force control consists of a proportional plus integral controller (PI) with the respective gains K Fp and K Fi , as well as a force feedforward part, the control law according to the scheme in Fig. 1.14 can be written in the Cartesian space as t

τ = K p S ∆x + K v S ∆xɺ + K Fp S∆F + K Fi S ∫ ∆Fdt + F0

(1.26)

0

Based on the relationships between the Cartesian and joint space gains, Zhang and Paul [26] have proposed an equivalent hybrid control law in the joint space t   T  ɺ τ q = J τ = k p J S J∆q + kv J S J∆q + J  K Fp S∆F + K Fi S ∫ ∆Fdt + F0  0   T

−1

−1

(1.27)

b Note that for sake of simplicity it is assumed that the task-frame coincides with the Cartesian

frame. Generally, the selection matrix S is not diagonal in Cartesian space [35].

32

Dynamics and Robust Control of Robot-Environment Interaction

Since each robot joint contributes to the control of both position and force, couplings in the manipulators mechanical structure (implied in the Jacobian matrix) cause that a control input to the actuator, corresponding to the force loop (e.g. force-controlled directions), produces additional forces in positioncontrolled directions in the task frame, and vice versa. It is obvious from (1.26) that by setting the position errors in the force-controlled directions to zero (i.e. by filtering the position error through S ), the position feedback gains in all directions are actually changed in comparison with the position control in free space because the entire system becomes more sensitive to perturbations. As a consequence, the performance of a robot is not unique with this scheme for all robotic configurations and for all position/force commanded directions. Moreover, one can find certain robot configurations for which, depending on selected force and position directions, the robot becomes unstable with the control law (1.26). This can be easily demonstrated on a simplified linearized robot model (1.10)

Λ ( x ) ɺɺ x =τ + F

(1.28)

Let us analyze the case when the manipulator is in free space and a noncontacting environment (e.g. in the transition phase when the force-controlled robot is approaching a contact surface after being switched from the positioncontrol mode). Let us assume that some directions (e.g. those orthogonal to the contact surface) have been selected for force control and the remaining ones for position control. Taking into account that the force is zero, substituting (1.26) in (1.28) yields

Λ ( x ) ɺɺ x + K v S xɺ + K p S x = K v S xɺ0 + K p S x0

(1.29)

with a robot closed-loop system matrix

 0 A =  −1 Λ K pS

I  Λ K v S  −1

(1.30)

To analyze the stability of this system we should determine the eigenvalues of A. As was shown in [53] there are a number of configurations in which the closed-loop matrix becomes unstable. Even if we introduce the feedback loops with respect to the integral of position errors in directions which are position-

Control of Robots in Contact Tasks: A Survey

33

controlled, it is always possible that an unstable configuration may appear. It must be noted that these unstable configurations build working subspaces quite far away from singular positions where the system matrix A is intrinsically unstable due to the degeneration of the Jacobian matrix. Moreover, it was found that in a robot position only alterations of the selection matrix can cause switching of robot’s behavior from stable to unstable, and vice versa. The instability was experimentally tested and proven using the industrial robot control systems [53]. Although the above stability analysis was based on a linearized model and, therefore, has some limitations, it provides a simple explanation in revealing the nature of stability problems in hybrid position/force control. Since the instability is influenced only by the robot’s position and selection matrix, this phenomenon is referred to as kinematic instability [54]. This phenomenon does not depend on whether the robot is in contact with the constraint surface or not. However, if it is in contact, the analysis of this problem is complicated by the force/position relationship and experimental tests become highly dangerous. It may be concluded that the kinematic instability problem encountered in the considered explicit hybrid position/force control represents a serious deficiency of this method and reduces significantly its applicability. In order to overcome the difficulties related to the kinematic instability, Zhang [55] has proposed to introduce an additional selection of input forces. In other words, the input torques from the position and force control parts (Fig. 1.14) are decoupled in the task-frame before they are applied to the joints. In the above case, when the robot is in free space, the joint torque from the position control part (1.27) is initially transferred into the Cartesian compliant frame, then multiplied with the selection matrix and again transferred back using the static force transformation (i.e. Jacobian matrix), which provides the following control law for the position loop

τ q p = J T S J −T k p J −1S J ∆q + J T S J −T kv J −1 S J ∆qɺ

(1.31)

It is relatively easy to prove that the linearized model (1.29) becomes kinematically stable with this control law. However, similarly to the original control scheme, the eigenvalues of the system change not only with variation of the robot’s configuration, but already with the given task, i.e. selection matrix.

34

Dynamics and Robust Control of Robot-Environment Interaction

This causes the robot performance to be strongly dependent on the configuration and selection of controlled directions. Fisher and Mujtaba [56] have shown that the kinematic instability is not inherent to the explicit hybrid position/force control scheme, but it is rather a result of an inappropriate mathematical formulation of position/force decomposition via the selection matrix S. It was demonstrated that in the original hybrid control formulation (1.26)-(1.27), the position control loop is responsible −1 for inducing the instability, precisely the term J S J in (1.27). The crucial error in the position control loop is, in the authors’ opinion, made by the decomposition of the robot coordinate (DOF) into the position- and forcecontrolled. Instead to compute the selected position-controlled DOF and the corresponding selected joint errors, respectively, based on

x p = Sx

(1.32)

∆q p = J −1∆x p = J −1S ∆x = J −1SJ ∆q

(1.33)

and the authors proposed to use the “correct” relationship between the selected Cartesian errors and the joint errors

∆x p = ( SJ ) ∆q

(1.34)

Taking into account the selection matrix structure, it is obvious that ( SJ ) is a singular matrix (with zero rows corresponding to the force DOFs). Hence, the selected joint errors equivalent to the selected Cartesian position error are obtained as the minimal 2-norm solution +

+

+

+

∆q p = ( SJ ) ∆x p = ( SJ ) S ∆x = ( SJ ) ∆x = ( SJ ) J ∆q

(1.35)

or, in general, when the robot is in a singular position, or it has a redundant number of joints, with an additional term from the null-space of the Jacobian J +

∆q p = ( SJ ) ∆x +  I − J + J  zq

(1.36)

where zq is an arbitrary vector in the joint space, the sign “+” denotes the MoorPenrose pseudo-inverse matrix. Thus, for the case (1.35) the control law of the position hybrid control part becomes

Control of Robots in Contact Tasks: A Survey +

35

+

τ q p = k p ( S J ) J ∆q + kv ( S J ) J ∆qɺ

(1.37)

To determine how the above kinematic transformations can induce instability of the hybrid control, the authors have defined a sufficient condition for kinematic stability. From the viewpoint of control, this criterion prevents the second-order system gain matrices (1.26) become negative definite, which is a necessary and sufficient condition for the system instability [55]. Testing the kinematic stability conditions for both original and “correct” selection and position error transformation solutions, the authors have proven that in the first case the instability can occur. The new hybrid control scheme, however, always satisfies the kinematic stability condition (it is always possible to find a vector zq to ensure the kinematic stability). The second problem is related to some dynamic stability issues in force control [57]. These effects concern: high-gain effect of force sensor feedback (caused by high environment stiffness), unmodeled high frequency dynamic effects (due to the arm and sensor elasticity), contact with a stiff environment, non-collocated sensing and control, etc. To overcome the dynamics problems of hybrid position/force control several researchers have pursued the idea of including the robot dynamics model into the control law. The resolved acceleration control originally formulated for the position control [58] belongs to the group of dynamic position control algorithms. Shin and Lee [31] have extended this approach to the hybrid position/force control. The joint space implementation of the proposed control scheme is sketched in Fig. 1.15.

Fig. 1.15 Resolved acceleration-motion force control

36

Dynamics and Robust Control of Robot-Environment Interaction

In this scheme, the driving torque compensates for the gravitational, centrifugal and Coriolis effects, and the feedback gains are adjusted according to the changes in the inertial matrix. An acceleration feed-forward term is also included to compensate for the changes along the nominal motion in position directions. Finally, the control inputs are computed by

τ = Λˆ ɺɺ x ∗ + µˆ ( x, xɺ ) + pˆ ( x ) + Sf ∗

(1.38)

where ɺɺ x ∗ is the commanded equivalent acceleration

ɺɺ x ∗ = ɺɺ x0 + K v ( xɺ0 − xɺ ) + Κ p ( x0 − x )

(1.39)

and f ∗ is the command vector from the force control parts, whose form depends on the applied control law. To minimize the force error it is convenient to introduce the PI force regulator of the form t

f ∗ = K fp ( F0 − F ) + Κ fi ∫ ( F0 − F ) dt

(1.40)

0

Khatib [22] has introduced an “active damping” term into the force control part to avoid bouncing and minimize force overshoots during transition (impact effects)

τ f = Sf ∗ − Λˆ SK vf xɺ

(1.41)

where K vf is a diagonal Cartesian damping matrix. Bona and Indri [59] have proposed further modifications of the control scheme. To compensate for the coupling between force and position control loops, as well as for the disturbance on the position controller due to reaction force, the authors modified the position control law to

τ p = Λˆ S  ɺɺ x ∗ − Λˆ −1 ( Sf ∗ − F ) + µˆ ( x, xɺ ) + pˆ ( x )

(1.42)

If the dynamic modeling used for computation of the control law is exact, the above control law provides the complete decoupling between position and force control in the task frame, i.e. the following closed-loop behavior

ˆ −1Sf ∗ − S Λ ˆ −1 F − SK xɺ ɺɺ x = Sxɺɺ∗ + S Λ vf

(1.43)

A comprehensive experimental evaluation and comparison of explicit force control strategies has been presented in [60].

Control of Robots in Contact Tasks: A Survey

37

1.6.2.2 Position-based (implicit) force control The practical reason why the methods based on explicit force control can not be suitably applied in commercial robotic system is the same as in the force-based impedance control and lies in the fact that commercial robots are designed as “positioning devices”. In addition, since there is no position feedback loop in the force-controlled direction, the robot will move due to various disturbances acting upon it, such as the controller and sensor drifts, etc. The implementation of explicit force control under above difficulties, as already mentioned, could be only successfully performed by a new generation of direct-drive robots. In commercial robotic system it is most promising to implement implicit or position-based force control by closing a force-sensing loop around the position controller as in the scheme shown in Fig. 1.16. This scheme involves an additional explicit force block, proposed in [61], whose role will be explained in the text to follow. In the implicit force control the input to the force controller is the difference between desired and actual contact force in the task frame. The output is an equivalent position in force-controlled directions that is used as the reference input to the position controller. According to the hybrid position/force control concept the equivalent position in force direction x0F is superimposed onto the orthogonal vector x0P in the compliance frame, which defines the nominal position in the orthogonal position-controlled directions. The robot behavior in the force direction is practically affected only by the acting force. The position controller remains unchanged, except for the additional transformations between the Cartesian and task frames, which have to be introduced since in general case these two frames do not coincide. Since the position controller provides a basis for realization of force control, this concept is referred to as implicit or positionbased force control [15], or external force control [13]. The role of the force control block in this scheme is twofold: firstly, to compensate for the effects of the environment (contact process), and secondly, to realize tracking of the desired force. Another important feature of the forcecontrolled manipulator is the ability to respond to positional variations of the contact surfaces. Commonly, a PI force controller is applied. A more complex force controller, including the compensation of the internal position control effects, has been proposed in [61], where an explicit force control block is included in the control scheme (Fig. 1.16). This scheme combines the implicit and explicit controls with the aim of using the benefits of the both (robustness and reliability of implicit force control, and fast reaction of the explicit one) and compensating for specific disadvantages of single-force control approaches.

38

Dynamics and Robust Control of Robot-Environment Interaction

Fig. 1.16 Implicit hybrid position/force control (an explicit control block is included according to [61])

The main features of implicit force control scheme are its reliability and robustness. Implemented in commercial robotic systems, this scheme is neither configuration dependent, nor sensitive to parameter variation. However, this scheme also exhibits some drawbacks. The accuracy of contact forces is mainly limited by the precision of robot positioning (sensor resolution). It can be especially disturbed when a contact with a very stiff environment is involved. Fortunately, an inherent compliance of the robot structure or force sensor is always present, and it reduces the equivalent system stiffness. The performance of implicit force control is significantly limited by the bandwidth of the position controller. A slightly higher bandwidth can be achieved using a compensator of a higher order. However, due to coupling between the position- and forcecontrolled DOFs, it remains questionable whether force control may become significantly faster or not.

Control of Robots in Contact Tasks: A Survey

39

1.6.2.3 Other force control approaches The next group of algorithms refers to more complex constraints on the robot motion which are described by a set of rigid hypersurfaces in the space of the end-effector Cartesian coordinates [11], or in the joint coordinate space [32]. The system model is described by a typical set of linearly implicit second-order differential-algebraic equations (mechanical differential-algebraic equations). This model is used to compute the control law whose functions are to linearize and decouple the system dynamics, as well as to decompose the control problem into position-and force-controlled directions. To improve the concept reliability, the dynamic hybrid control has been extended to the case of unknown environments, consisting of hypersurfaces [62]. The improved control schemes involve on-line identification algorithms based on force and position measurements, and adaptive control mechanisms. However, as already mentioned, the adaptive constrained motion control is theoretically attractive, but still impractical in reality. The hybrid position-force task specification has also been a subject of considerable investigations. Lipkin and Duffy [24] showed that the Mason’s position-force decomposition approach based on “geometrical orthogonality” is in fact erroneous. The resulting planning for hybrid control is invariant neither with respect to the origin translation nor the change of unit length. The authors have proposed a more general and mathematically consistent invariant hybrid task formulation based on screws algebra. In this approach, the complementarities between motion (modeled by a twist) and force (represented by a wrench) is expressed via the reciprocity relationship that is independent of the coordinate frame, scaling or units. In certain simple tasks and reference frames, both conventional and reciprocity-based decomposition, show the same results. However, the reciprocity-based approach provides a more general decomposition, applicable to the cases when the freedom and constraint subspaces do not span a six-dimensional space or have nonzero intersections, but also to the case when a manipulator has less than six DOFs [63]. If the specified twist and wrench are consistent with the environment (i.e. the freedom and constraint equations are satisfied), the specified task is feasible for hybrid control. In the opposite case, it is necessary to filter the specified twist and wrench to obtain a kinestatically realizable control action (so-called kinestatic filtering). A procedure to apply the reciprocity-based task decomposition to manipulator dynamics in order to obtain equations of motion relevant for hybrid control has been presented in [64]. Several model-based tools for tasks

40

Dynamics and Robust Control of Robot-Environment Interaction

specification using this approach are presented in [22]. It has been shown that the reciprocity concept is not only well suited for nominal specification of arbitrary motion constraints, but also serves definition of possible uncertainties and on-line identification and observation of real motion constraints. An overall hybrid position-force control scheme based on the general decomposition formalism including identification of geometrical uncertainties has been proposed in [25]. 1.6.3 Force-impedance control There have been several attempts to combine impedance and force control with the aim of using benefits and overcoming specific disadvantages of single control approaches. Although it is possible, under some circumstances, to demonstrate a correspondence between force and impedance control laws [65], there are essential differences between these main constrained motion control concepts. The main advantage of the impedance control over the force control is the easier task specification and programming. A contact task is practically specified in terms of motion sequences, hence the impedance control does not require any modifications of conventional free-space planning control concepts and algorithms (the programmer can take advantage of his own experience and existing off-line programming tools). Moreover, the impedance control can be activated in free space during the approach motion. Thus, it can be applied for the transition to and from the constrained motion, without specific control switching algorithms. As mentioned above, the impedance control realizes the closed loop position control in free space, while in case of the contact with rigid environments it offers force open-loop capabilities. In contrast to this, the force (admittance) control approach allows closed-loop force control capabilities in contact, but exhibits open-loop position control characteristics in free space. Therefore, the activation of force control in free space is only possible under specific circumstances. In general, however, a discontinuous control strategy is required for the transition from non-contact to contact motion phase, or vice versa. The control structure changes in the most critical phase when the manipulator is in contact with the environment, and this represents a major drawback of force control. To cope with unexpected collisions, additional sensors (e.g. distance)

Control of Robots in Contact Tasks: A Survey

41

have to be integrated in the control system. The fundamental superiority of the force control implies, however, from the fact that the interaction force is a result of the control action, rather than of the actual deviation of environment position and chosen target impedance. In Goldenberg’s algorithm [38], force control is closed around an internal impedance control loop. Desired force and force error are used to compute an equivalent desired relative motion of the end-effector. In addition, the impedance control is included with the aim to achieve a suitable relationship between force and relative motion during contact. This is realized in the internal velocity loop by adjusting compensator gains to obtain the target impedance. A similar reliable position-based force-impedance control scheme suitable for implementation in industrial robots has been proposed in [66]. In this scheme an external implicit force controller loop is closed around an internal position-based impedance controller (Fig. 1.17). The main goal of the internal loop is to achieve target impedance while the external loop takes care of desired force realization. In this scheme, the selection between position- (i.e. impedance) and force-controlled directions is not needed. Indeed, both impedance and force control affect all directions. The disproportion between motion and force planning is not critical in the control scheme (Fig. 1.17), since the internal control loop behaves as a low-stiffness target impedance system allowing relatively large differences between the input position command and real robot position at the output. In contrast to this, the internal loop in the implicit force control (Fig. 1.16) is a very stiff position control, so that the selection is inevitable.

Fig. 1.17 Position-based force-impedance control scheme

42

Dynamics and Robust Control of Robot-Environment Interaction

Anderson and Spong [12] have proposed an approach to controlling contact forces, referred to as hybrid-impedance control algorithm. The kernel part of the algorithm is Raibert-Craig’s hybrid position/force control scheme, with the selection matrix applied to decompose position- and force-controlled subspaces. Both control parts use the feedback of contact force to realize desired system impedance (position-based and force-based impedance control) along each DOF. A controller that combines an internal position control, a position-based impedance compensator, and a desired force feedforward has been described in [67]. The authors proposed integral control actions to be applied for both impedance (damping control) and force filters to ensure the compliance and desired steady-state force. A conceptually different approach to position/force control, referred to as parallel control (Fig. 1.18), has been proposed in [9]. Contrary to the hybrid control, the key feature of the parallel approach is to have both force and position control along the same task space direction without any selection mechanism. Generally, both position and force cannot be effectively controlled in an uncertain environment. Therefore, the logical conflict between the position and force actions is managed by imposing the dominance of the force control action over the position one along the constrained task direction where force interaction is expected. The force control is designed to prevail over the position control in constrained motion directions. This means that the force tracking is dominant in the directions where interaction with the environment is expected, while the position control loop allows the compliance, i.e. deviation from the nominal position in order to attain the desired forces. For this reason the parallel control method can be considered as a force-impedance control approach.

Fig. 1.18 Parallel position-force control

For a parallel controller case, consisting of a PD action in the position loop, and a PI control in the force loop, together with the gravity compensation and

Control of Robots in Contact Tasks: A Survey

43

desired force feedforward, a set of sufficient local asymptotic stability conditions has been derived in [68]. Stability analysis and simulation results on an industrial robot also are included. These conditions imply a relatively high damping (i.e. velocity gains) to ensure the system’s stability. 1.6.4 Unified position-force control Vukobratovic and Ekalo [8] have established a unified approach to control simultaneously position and force in an environment with completely dynamic reactions. This fully dynamic approach to the control of robot’s interacting with dynamic environment will be presented in a very condensed way. In the case when the environment does not possess displacements (DOFs) that are independent of robot motion, the model (1.9) provides a mathematical description of the environment dynamics in terms of the robot coordinates (motion). Then the system (1.1)-(1.9) describes the dynamics of the robot interacting with dynamic environment. It is assumed that in the case of contact all mentioned matrices and vectors are continuous functions of the arguments. It is also assumed that the robot is constantly in a unilateral contact with the environment. Further, it is taken that n=m, where n is the number of robot DOFs and m is the number of contact force components. The general case, when n>m, has been considered in [69]. In the case of contact with the environment, the robot control task can be described as the robot motion along a programmed trajectory q p (t ) representing a twice-continuously differentiable function, when a desired force of interaction Fp (t ) acts between the robot and the environment. Based on the nonlinear model, the programmed motion q p (t ) and desired force Fp (t ) must satisfy the relations:

Fp (t ) ≡ f ( q p (t ), qɺ p (t ), qɺɺp (t )) f ( q, qɺ , qɺɺ) = ( S T ( q)) −1[ M ( q)qɺɺ + L( q, qɺ )]

(1.44)

The control goal of a robot interacting with dynamic environment can be formulated in the following way. Let us define the control τ (t ) for t ≥ t0 that is to satisfy the target conditions:

q(t ) → q p (t ), F (t ) → Fp (t ), as

t→∞

(1.45)

Two alternative questions can be phrased concerning the control design problem. Can we choose such a control law which, by satisfying the preset robot

44

Dynamics and Robust Control of Robot-Environment Interaction

motion quality, would enable the attainment of the control goals that satisfy the relation (1.45)? Is it possible to choose the control law in such a way as to ensure the preset quality of the robot interaction force, and also the attainment of the control goals? The answer to the first question is quite simple [8, 33]. The inverse dynamics methods ensure a desired motion quality and at the same time guarantee a stable interaction force. The answer to the second question depends on the environmental dynamics. The task of stabilizing the programmed interaction force ( PFI ) Fp (t ) can be tackled by considering a family of transient processes with respect to force, in the form µɺ = Q ( µ ), µ = F (t ) − Fp (t ) and choosing a continuous vector function Q (Q (0) = 0) of dimension n, such that the asymptotic stability as a whole is ensured for the trivial solution of µ (t ) ≡ 0 . Let us consider the “pure force control” according to the assumption that m = n , i.e. when the number of the contact force components is equal to the number of the powered DOFs of the robot. For convenience, when describing the quality of transient processes with reference to perturbation force dynamics, µɺ = Q ( µ ) , we shall use an equivalent relation of the form: t

µ (t ) = µ0 + ∫ Q ( µ (ω ))d ω

(1.46)

t0

Without loss of generality, we can adopt µ0 ≡ 0 , because the stabilization of µ in the sense of preset quality (1.46) directs stabilization according to the preset quality µɺ = Q ( µ ) , independently of the value of µ0 . Let us consider only one of the possible control laws with the feedback loops with respect to q, qɺ and F of the form [8, 33] 



t







t0

 

  τ = H (q ) M −1 ( q) − L(q, qɺ ) + S T ( q)  Fp + ∫ Q( µ (ω )) dω   + h(q, qɺ ) − J T ( q) F (1.47)

By applying this control law to the robot dynamics model (1.1) we obtain the following law for the robot operating in contact with the environment: t   M ( q ) qɺɺ + L ( q, qɺ ) = S T ( q )  Fp + ∫ Q ( µ (ω ))d ω    t0  

45

Control of Robots in Contact Tasks: A Survey

Taking into account the environment dynamics model (1.9), we obtain the following closed-form control system: t   S ( q )  µ (t ) − ∫ Q ( µ (ω ))d ω  = 0   t0   T

(1.48) t

∫

and, because rank(S) = n , (1.48) is equivalent to: µ (t ) = Q ( µ (ω ))d ω , from t0

where, µɺ (t ) = Q ( µ (ω )) follows directly. In this way, the control law (1.47) ensures the desired quality of stabilization of ( PFI ) Fp (t ) . The stability of the real motion (position) when asymptotic stability of the contact force is fulfilled has been considered in [8, 33, 69]. Sufficient conditions for constrained motion stability based on generalized Lyapunov’s theorem on stability in the first approximation of the system with perturbation have been derived. The theorem defines conditionally the properties of “internal stability” of the environment because the fulfillment of stability conditions depends in general not only on the environment dynamics but also on the nature of the programmed motion. It should be emphasized that without knowing a sufficiently accurate environment model it is not possible to determine the nominal contact force. Besides, the insufficiently accurate environment dynamics model can significantly influence the contact task performance. Inaccuracies of the robot and environment dynamic models, as well as dynamic control robustness will be considered in Chapter 2. The problems arising from the uncertainty of parameters may also be resolved by applying the knowledge-based techniques. Taking into account external perturbations, which do not expire with time, as well as model and parameter uncertainties, it may be difficult to achieve asymptotic (exponential) stability of the system. Therefore, it may be of practical interest to demand a more relaxed stability condition, i.e. to consider the socalled practical stability of the robot around the desired position and force trajectories by specifying the finite regions around them within which the robot actual position and force have to be during the task execution. More details on the synthesis and stability of dynamic control of the robot’s interacting with dynamic environment, called unified approach, will be presented in Chapter 2.

46

Dynamics and Robust Control of Robot-Environment Interaction

1.7 Contact Stability and Transition The type of contact tasks may vary substantially in dependence of specific requirements, but in all cases the robot realizing a contact task has to perform three kinds of motion: · · ·

gross motion, related to robot’s movement in free space (freemotion mode), compliant or fine motion, related to robot’s movement constrained by an environment, and transient motion, representing all transition phases between the freespace and compliant motions.

From a practical point of view, the contact transition can be considered as stable if the contact is not lost after the manipulator hits the environment. A stable contact transition can be characterized by non-zero force (after the contact is detected), positive penetration of the manipulator end-point into the environment, nonappearance of bouncing, etc. A most critical issue in transition control is the initial impact against a stiff environment. Obviously, a stable controller should ensure the passage through the transition phase while maintaining the contact until all impact energy has been absorbed. Many researchers have shown that in most of the proposed control algorithms instability occurs when the contact between the end-effector and environment is stiff. However, the investigations have primarily been concerned with the question of coupled stability (i.e. will the robot remain stable when it is interconnected with the environment?) of robots and the environment under various control algorithms, while assuming the manipulator being initially and constantly remaining in contact with the environment. Surprisingly, relatively little research has addressed the problem of contact transition stability (i.e. will the robot during transition from free- to contact-motion establish a continuous contact with the environment without multiple impacts?), which is most fundamental for performing contact tasks. The contact transition stability problem is important for both unilateral (i.e. force) and bilateral (geometric) constraints. Namely, a bilateral constraint is usually achieved by closing the gripper, which is due to position misalignment usually realized through a unilateral contact between the gripper jaws and grasping object. In the impedance control, contact stability issues have mainly been considered on the basis of simplified models of the interaction between a target impedance system and the environment. Colgate and Hogan [70] have defined

Control of Robots in Contact Tasks: A Survey

47

necessary and sufficient conditions to ensure the stability of a linear robotic system coupled to a linear environment. The authors have applied the network theory to describe the manipulator and environment interactive behavior at the equilibrium point. In a passive stationary environment, two time-invariant networks coupled along interaction ports (Fig. 1.19) can represent the interactive model around the equilibrium p0 ( ∞ ) = p0∗ , where p0 denotes the penetration into the environment. The coupling imposes the velocities of robot and environment at contact point to be equal, while the forces acting upon the robot and environment have opposite signs (action and reaction). If the environmental transfer matrix Ge ( s ) s is positive real, representing any passive Hamiltonian environment, then a necessary and sufficient condition to ensure stability of linearized robotic control system is that the realized admittance sGt −1 ( s ) be positive real [70]. In other words, it should represent the driving point impedance of a passive network. Considering a SISO system, the coupled stability has been proven using the Nyquist criterion and the property of positive real transfer function having a limited phase by ± 90 deg. [70]. Then, it is relatively easy to prove that the mapping of the Nyquist contour of a positive real environmental impedance

Ge ( s ) s through an, also positive real, admittance sGˆ t −1 ( s ) , altering the phase by ± 90 deg. and changing the magnitude by a factor 0 to ∞ , provides a stable system, i.e. a stable Nyquist plot of the open-loop coupled system transfer function.

Fig. 1.19 Robot-environment interaction model

System passivity concept provides a relatively simple test for the assessment of coupled system stability. In this test, only passivity of the environment should be proven, without an accurate knowledge of parameters. Assuming again that an ideal target impedance response (1.15) is being realized, the passivity of target

48

Dynamics and Robust Control of Robot-Environment Interaction

admittance sGt ( s ) implies positive definite matrices M t , Bt and K t , and consequently, the closed-loop system should be stable in contact with any passive environment to which it is directly coupled. The explicit design of a positive-real robot control system, however, may in practice become cumbersome [71]. Moreover, various practical control implementation effects, including computational time delay and sampling effects, as well as unmodeled dynamics (e.g. high-order actuator and arm dynamic effects), may result in a non-passive real impedance control response. The above stability results can be practically extended to nearly-passive control systems. However, in this case a passive environment can be found which destabilizes the coupled system. In order to simplify the coupled stability analysis, Colgate and Hogan [70] have introduced the term worst or the most destabilizing environment, denoting the most critical environmental for the coupled system stability. Such −1 environmental impedance Ge ( s ) s shapes the Nyquist contour of sGˆ t ( s ) by minimizing the distance from the critical point -1 to the nearest point on the −1 Nyquist plot of the loop transfer function Ge ( s ) Gt ( s ) . Taking into account that the driving point impedance of simple passive environmental models, such as mass or spring ( M e s and sK e ), perform the maximum rotation in the Nyquist plane, the authors have found that the worst passive environment for the coupled stability consists of a set of pure masses and springs. The beauty of the passivity theory is that it guarantees the phase lead or lag of any passive block to be no more than 90 deg. Thus, when two passive blocks are connected together as shown in Fig. 1.19, the maximum phase lead or lag of the overall open-loop system will be limited to no more than 180 deg. Therefore, it guarantees stability regardless of the overall gain. This result can be obtained from either the Nyquist or Bode analysis. The problem with any practical application such as a robot performing impedance control is that additional phase lags caused by cut-off (analogue and digital) filters and sampling delays make the overall phase lag to be more than 180 deg. at high frequencies. However, the realization of a passive system in a real digital robot controller imposes a fundamental limit on the reduction of apparent robot inertia to a maximum of 50% [72]. In industrial robots with apparent Cartesian end-point masses of several hundred kilograms, that is an exceedingly conservative condition. Newman [73] proposes a natural admittance control (NAC) approach that provides considerable compensation for friction, however, without −1

Control of Robots in Contact Tasks: A Survey

49

significant improvement of achievable target admittance reduction which does not violate the passivity constraints. To solve this critical limitation in the implementation, the author proposed in [74] the insertion of a mechanical filter between the manipulator and the environment in order to achieve low-impedance performance. However, this is a specific solution, rather than a general approach. The reduction of apparent inertia appears also to be essential in human-robot systems. In order to achieve valuable performance of the human-robot interaction, Buerger and Hogan have suggested [75] relaxing the restrictive passivity condition and designing the interaction system either by taking into account the limited knowledge of the particular environment or by lowering the target inertia beyond the passivity threshold. If both the environment and realized admittance are stable, the coupled stability of the interactive system in Fig. 1.19 can also be assessed by means of small gain theorem. This theorem states that a feedback loop composed of stable operators will certainly remain stable if the product of all operator gains is smaller than unity

Ge ( jω ) Gˆ t −1 ( jω )

∞

the above condition also requires the target stiffness to be higher than the environmental one. Moreover, no target model, i.e. the compliance feedback compensator G f (Fig. 1.11), can be found to allow interaction with an infinitely rigid environment ( → ∞ ). Therefore, one of the main conclusions in [16] is related to the practical need for an intrinsic compliance, either in the robot or in the environment for achieving interactive stability. The problems with control design taking into account environmental models and parameters are the uncertainties and nonlinearities in the real systems. A promising approach to cope with these problems is provided by robust-control theory. The robust control framework considers simplified linearized models while taking into account specific nonlinear effects by means of weighting functions in the frequency domain. In [42, 76], this approach is applied to derive new stability criteria for the design of low-impedance interaction in industrial robot systems. The robust interaction stability paradigm ensures contact stability during all phases of interaction. Moreover, the new design framework realizes low-impedance performance, allowing considerable reduction of high apparent industrial robot inertia and stiffness. The novel stability criteria are established based on robust control theory and taking into account the estimates of environmental stiffness, tolerating thereby large uncertainties and variations in the industrial environments. These criteria are proved by extensive tests involving industrial and space robots, and have been extended to control synthesis of human robot interaction systems (haptic admittance displays and rehabilitation robots) [77]. The contact transition stability conditions require interaction force, i.e. actual penetration to be positive p(t ) = x − xe > 0, t ≥ t0 , or the position deviation to be smaller than the nominal penetration e (t ) ≤ p0 (t ) i.e.

51

Control of Robots in Contact Tasks: A Survey

e(t ) p (t ) − p (t ) x0 (t ) − x(t ) = 0 = ≤1 p0 (t ) p0 (t ) x0 (t ) − xe

(1.51)

In other words, this relation implies the actual end-effector position during a stable contact transition to be always located between the position of environment and the nominal position. Since this contact stability condition is based on a simple geometric consideration it is referred to as geometric criterion. The advantage of the geometric criterion is that it compares two time signals. The norm comparison offers the possibility to apply relatively simple and efficient system theory formalisms for the contact stability analysis. In [76, 77], this criterion has been utilized to derive robust contact stability condition ensuring both coupled and contact transition stability based on

sup

e (t ) 2 p0 ( t )

< W ( s ) ( I + Ge −1 ( s ) Gt ( s ) )

2

−1

≤1

(1.52)

∞

where stable weighting transfer function matrix W ( s ) describes the uncertainties of environmental model and target impedance realization. In a SISO system, this condition implies

ξt ≥

1 2

(

)

1 + 2κ − 1

(1.53)

where ξ t = Bt 2 M t K t and κ = K e K t are target damping and stiffness ratios, respectively. However, in spite of an effective and simple formulation, this criterion only ensures sufficient contact stability conditions, but not the necessary ones. Consequently, the obtained contact stability indices may be conservative. It should be mentioned that the damping ratio bound (1.53) is still smaller than in the most commonly applied in practice dominant real-pole solution [78], imposing ξ t ≥ 1 + κ . A very important advantage of the input/output criterion (1.52) is that it can be applied for the continuous as well as discrete systems, including the time lags. The control time lags have been identified in [76, 77] as the critical destabilizing contact transition effect. In general, the retarded system requires a significantly higher amount of damping in order to stabilize the transition process with delayed force signals. Robust contact stability analysis and control synthesis are dealt with in detail in Ch. 3.

52

Dynamics and Robust Control of Robot-Environment Interaction

Typical transition results obtained experimentally in position-based impedance and force control during contact with a stationary environment are presented in (Fig. 1.20) [42]. In order to compare force and impedance control, a nominal force corresponding to the steady-state impedance interaction force F∞ = p0∞ K e K t ( K e + K t ) , where p0∞ denotes the stationary value of the nominal penetration, has been selected, and an integral force controller with the same bandwidth as the impedance control has been synthesized. As expected, it can be remarked that the force transition in hybrid control is characterized by lower overshoots. The reason for this is that the force control represents an explicit goal in the hybrid control that is realized by an appropriate control structure and design. In the impedance control, however, the aim is to passively modify a preplanned motion in accordance to the interaction forces. Therefore, the force transition in the impedance control is to a great extent influenced by the selected target impedance parameters and nominal motion [42]. In contrast to impedance control algorithms, which provide the same control structure for the three motion phases, in force control schemes the transition to and from contact motion is usually based on discontinuous control. There are two basic concepts for the change of the control strategy from the position control to the force control, operating (a) in free space, where the transition is realized in the force control mode, and (b) after the contact has been established. Most of force control algorithms execute the transition control in the force mode. The reason for this is that the impact force can be very large, especially due to the relatively high approach velocities and delay in the stiff position controller. One of methods to reduce impact is to use soft force sensors [79], i.e. passive compliance, but this reduces the position accuracy during position control. The underlying idea of the majority of methods concerned with the impact problem is to increase damping in the collision direction [22]. Assuming a simplified stiffness model of the environment, the damping effect can be achieved by utilizing either the force derivatives or the approach velocity feedbacks. However, both methods have practical limits. The force signals are usually noisy and the derivation is inaccurate. Qian and De Schutter [80] have proposed lowpass sensor filtering and nonlinear damping to cope with the transition problem. However, in the commonly slow approach motion before contact the velocity sensing is not reliable. Moreover, in a stiff environment, relatively fast oscillations in force and robot velocity can cause the instability due to time (phase) lags between the sensing and control actions. These difficulties have been addressed in several works, proposing design of a stable force controller without velocity

Control of Robots in Contact Tasks: A Survey

53

measurement [81], and even without sensing the end-effector contact force [82], for a system with well-known dynamics. However, these innovative schemes are relatively complex to implement and require further experimental tests.

Fig. 1.20 Comparison of contact transition performance: impedance vs. implicit hybrid control (experiments with Manutec R3 robot; target impedance parameters: Mt = 10 kg, ξt = 8, Kt = 1500 N/m; environment: Ke = 60000 N/m; bandwidth of the integral force control 2 Hz)

In the other transition control concept, the approach phase is realized in the position control mode, and when contact is established, the control is switched to force control mode. Numerous discontinuous transition control algorithms have

54

Dynamics and Robust Control of Robot-Environment Interaction

been elaborated and experimentally tested using simple test-systems. Considering discontinuous controller as an entire generalized system, Mills and Lokhorst [83] have proposed a discontinuous control scheme which guarantees the following performance: global asymptotic stability of the closed-loop system, asymptotic trajectory tracking of position and force inputs; and reestablishment of contact after an inadvertent loss. Wu et al. [84] have proposed to add a positive acceleration feedback to the force control in the impact direction. In addition, a switching control strategy is introduced to eliminate unexpected bouncing. A similar control strategy for the transition problem in both force and impedance control has been developed by Volpe and Khosla [85]. The authors recommended a positive force feedback to be used during transition, and integral force control after a stable contact is established. A force-regulated switch triggers the transition from position control to impact control, while several options are proposed for the further switch to the integral force control. Based on the equivalency between the force and impedance control, the authors have established the transition stability condition for the impedance control that imposes the mass ratio (robot inertia over target inertia) to be less than one. Based on the experiments with this control approach, realized on a direct-drive robot at very high impact velocity (0.7 m/s) with a relatively stiff environment (104 N/m), the authors have demonstrated the reliability of the established criteria. However, these results on direct-drive robots are not applicable to industrial robots. The performance of industrial robot systems, such as very high Cartesian inertia (> 500 kg), extremely stiff position controller (several million N/m equivalent stiffness in Cartesian space) and considerable time lags (from 2 to 20 ms), causes switching algorithms to be quite difficult to implement. Gorinevski et al. [86] have examined the transition problem in both impedance and general force control during the contact with stationary and dynamic environments. The authors tested two control approaches: linear control and sliding-mode control. The influence of several effects, such as the time delay and elasticity of robot end-effector, transmissions and mechanical structure, on the contact stability has been examined. The contact stability criteria for singleand two-DOF systems are derived in the explicit closed form in terms of control gains and limits on the robot and environment velocities. Several authors have considered transition control as a short-impulse dynamic problem. This model, however, is valid for very fast systems (e.g. micro-macro manipulators), but they are still rare in practice. In industrial robotic systems, the transition problem can be accurately analyzed in a finite time period. The

Control of Robots in Contact Tasks: A Survey

55

features of almost all industrial control systems still do not provide mechanisms to control short-impulse impact effects. McClamorch and Wang [32] emphasized the important role of constraints in contact dynamics. They presented global conditions for tracking based on a modified computed torque and local conditions for feedback stabilization using a linear controller. The closed- loop properties in the case of force disturbances, dynamics in the force feedback loops, or uncertainties in constrained functions have also been investigated. Eppinger and Seering [87] studied the influence of unmodeled dynamics on contact task stability, introducing additional (elastic) DOFs of both the robot and environment. A treatment of the contact stability that considers the environment as a nonlinear dynamic system is given in [34]. It is shown that if the impedance control is applied, enabling the robot to be asymptotically stable in free space, the robot interacting with the environment is a passive system and is stable in isolation. However, the conclusion is valid only if the robot in contact is at rest, and for this reason the result cannot be considered complete. The stability issue, i.e. the establishment of the conditions under which a particular control law guarantees the stability of the robot in contact with the environment, is of essential importance. In [8, 33, 88], the control laws stabilizing simultaneously the motion of the robot and the forces of its interaction with a dynamic environment are synthesized, ensuring the exponential stability of the closed-loop systems (based on the analysis of a complete dynamic model of the robot and dynamic environment). The papers formulated conditions ensuring asymptotically stable position of the system in the first approximation (local stability). It has been emphasized that the character of the mentioned position stability depends particularly on the nature of programmed (desired) motion. In spite of sufficient conditions of the linearized system asymptotic stability being conservative, the fact is that the dynamic character of interaction of the environment with the robot can lead to positional instability. This problem deserves full attention of the researchers, as well as of the designer of robotic controllers dedicated to the diverse contact tasks. Therefore, it should be emphasized that linear analysis gives a very important criterion that must be fulfilled by any force-based law. However, the model uncertainties representing a crucial problem in control of the robot’s interacting with a dynamic environment have not been appropriately addressed yet. Therefore, it can be difficult to achieve the asymptotic (exponential) stability of the system (unless robust control laws including factors for compensating these perturbations and uncertainties are used). Inaccuracies of the robot and environment dynamic models, as well as dynamic control robustness, have been considered in [89, 90, 91]. The problem

56

Dynamics and Robust Control of Robot-Environment Interaction

arising from the uncertainties of parameters may also be resolved by applying the knowledge-based techniques [92]. Taking into account external perturbations and model uncertainties, it may be difficult to achieve asymptotic (exponential) stability. Therefore it is of practical interest to require less restrictive stability conditions, i.e. to consider the so-called practical stability of the system. An approach to the analysis of the practical stability of manipulation robots interacting with a dynamic environment based on a centralized model of the system is presented in [93]. The conditions for testing practical stability of a robot interacting with its dynamic environment were given in [93, 94] and the test were performed using two very representative control laws. The first one is the pure position dynamic control (based on the so-called inverse dynamic technique, or computed-torque method), where the desired position trajectories are calculated based on the desired position and force trajectories using the dynamic model of the environment. The other control law considered belongs to the hybrid position/force control schemes, where the complete dynamic model of the interactive system is taken into account: in the directions in which the desired position trajectories are specified the control law attempts to stabilize position, while in the directions in which force trajectories are specified, the control law focuses on the force. The elaborated stability test may be used either to check the stability of the specified control laws, or to establish procedures for the synthesis of parameters of different control laws. The derived stability may be too conservative due to a number of linearizations (approximations) made. More refined approximations, e.g. by taking into account possible dependence of the model elements on the parameters, may lead to a less conservative test. 1.8 Compliance Planning In spite of the existence of numerous sophisticated compliance and interaction control strategies and schemes, these advanced control capabilities have not been implemented yet in commercial robotic control systems. There are several reasons for this. The majority of compliance control concepts are concerned with particular problems, mainly at lower control levels. Combining various algorithms and control concepts and their integration in conventional robot position control systems is complex and tedious. Most of the studies on the impedance control are primarily related servo control. Except the seminal works on the compliance control [1, 18] the contact task planning and programming

Control of Robots in Contact Tasks: A Survey

57

issues have usually been neglected in the research studies. However, these subjects are essential for practical application of compliance control. The planning of complex impedance control tasks in terms of spatial geometrical and compliance parameters is still an insufficiently treated problem. In complex geometrical tasks, the relationship between forces and displacements becomes rather complex. Practically, in the common linear second-order impedance target system (1.14)-(1.15) the mass and damping elements specify system dynamics (i.e. transition processes), while the stiffness provides apparently simple relationship between displacements and forces (steady-state behavior). However, due to the configuration dependence of the spatial stiffness it is difficult to specify and select compliance parameters as well to easily understand the spatial compliance relationships. The reasoning about admissible or realizable interaction forces during design is not a simple task, and the practical goal is to explore means of facilitating it. As proposed by Fasse and Broenink [95], the specification would be simplified if the compliance parameters could be chosen independently of the interaction system configuration (e.g. reference frame, contact point, etc.). In the compliance control design it is quite desirable to specify target stiffness matrix independently of the robot/environment configuration. The authors proposed the application of principal (eigen) characteristic, i.e. (singular) values, as quite suitable for specification of stiffness (compliance) submatrices. Since the stiffness submatrices are positive semi-definite, the eigen values are identical to the singular ones. The characteristic values are independent of the rigid-body rotation occurring in the transformations. Hence, the appropriate parameterization of stiffness and compliance matrices involves two sets of parameter: the set of non-spatial parameters consisting of principal values of compliance submatrices and the set of spatially-dependent parameters describing rigid body transformation (rotation matrix and displacement vector). This parameterization provides the basis for computation of the centers of stiffness and compliance [96]. Dynamic impedance control algorithms based on resolved acceleration control approach are proposed in [97, 98, 99] to control six-dimensional impedance interaction in industrial robot control systems. The orientation

58

Dynamics and Robust Control of Robot-Environment Interaction

representation based on quaternion was identified as a most reliable and robust approach for specification of impedance for rotational DOFs [97]. Considerable research efforts have been devoted to the synthesis of admittance for a reliable and robust assembly [100, 101]. The main goal is to replace spatial intuitive reasoning with more systematic synthesis approaches. A reliable admittance design, which defines an appropriate mapping of interaction forces into robot velocities for a particular assembly task, requires the contact forces always reduce the existing position misalignments. This approach is referred to as force-assembly and has been proven [100] to be quite efficient for insertion of workpieces at infinitesimal errors. A practical compliance control planning and programming approach was realized in the ESA Space robot control system (SPARCO) [102, 103]. The SPARCO control system provides an integrated compliance control framework based on the industrial robots standards. In order to avoid the nominal motion perturbation it is recommended to keep the nominal position constant during force realization using this control approach. Such strategy was realized in the SPARCO control system by means of the “apply-force” control function. Hence, in order to retain the simple and obvious impedance control structure, it is more reliable to activate the force control in the contact realized by the impedance control. This strategy is integrated in the SPARCO control system, and was proven in numerous experiments to be quite reliable and robust [102]. The SPARCO control system involves impedance- and integral-force control algorithms at the servo control level, as well as compliance control monitoring, planning and programming control functions at higher control levels. This control system utilizes extended PDL2 language commands to specify and program a compliance task, such as to define compliance frame (relative to the tool frame), to select pre-designed impedance control gains (e.g. highimpedance, medium-damping, etc.) or to explicitly specify target impedance parameters, to activate and deactivate impedance control, to control interaction force, as well as to monitor the contact and force, etc. A specific control algorithm referred to as relax supports a continuous change of impedance control gains (i.e. target impedance models). The SPARCO system utilizes simple and robust position controllers of industrial robotic systems that exhibit quite

Control of Robots in Contact Tasks: A Survey

59

desirable control performance, such as diagonal dominance, spatial roundness (Fig. 1.4) and normality, in order to realize simple and efficient position-based impedance and force control. Safe and robust executions of SPARCO interaction tasks in numerous tests demonstrated the feasibility and reliability of the applied interaction control concepts in an uncertain and stiff environment ( K e ≈ 100000 N/m, position misalignments to 10 mm) (Fig. 1.21). The SPARCO is implemented in a space control platform that is used in the European space programs. A more comprehensive presentation of the impedance control synthesis at a higher control level based on spatial intuitive reasoning, as well as integration of impedance control in forward industrial robot controllers, are presented in Chapter 3.

Fig. 1.21 SPARCO testbed

The SPARCO system considers a six-dimensional decoupled compliance model with respect to a compliance C-frame of reference (Fig. 1.7). The origin of this frame represents the compliance (impedance) center. Interaction forces and moments move this frame relative to the initial unloaded position according to the selected compliance model. The angle-axis representation has been applied to compose elemental rotation displacements. A specific planning

60

Dynamics and Robust Control of Robot-Environment Interaction

problem is to specify the location of the C-frame so that equilibrium can be achieved [102]. In the space automation tasks with relatively simple geometry, such as grasping and mating of cylindrical parts, drawer open/close tasks, etc., this approach was demonstrated to be reliable and robust. 1.9 Haptic Systems Control Considerable research interest is related to new interactive systems designed for the interaction between human and a robotic device, or with remote or virtual dynamic environments. To the new interactive systems belong kinesthetic displays and haptic interfaces, teleoperation systems, human enhancers and augmentation devices, etc. These systems are designed to produce/receive kinesthetic stimuli for/from human movements, as well as to render a realistic feeling of contact and dynamic interaction with nearby, remote or virtual environments. The advanced interaction systems have found very attractive applications in surgical and rehabilitation robotics, power assist-devices, training simulation systems, etc. The most critical issue in these systems is to ensure stable and safe interaction with a high rendering performance. This is a challenging task, when taking into account serious problems, such as unknown and variable human dynamics, common nonlinear environmental characteristics, as well as various disturbances in computer-controlled systems. Essentially, the basic interaction chain in a haptic display consists of three principal elements (Fig. 1.22): human operator (H), haptic device (D) and virtual environment (VE). Common model of a haptic interface is presented in (Fig. 1.23). The middle element in this model is a haptic device, which is, based on an analogy with electrical network circuits, represented as a so-called two-port network. A haptic device interconnects the human with the virtual environments (both linked as one-port network) via force and velocity I/O signal pairs, describing the exchange of energy between the blocks. This representation has been demonstrated to be very useful in the analysis of teleoperation and haptic systems [12, 104]. Since the haptic device is computer-controlled, critical SD effects on the interaction system stability (control delay and sample-and-hold effects) must be also introduced in the interaction model. The main role of SD control system is to measure and render I/O signals via the haptic interface, and thus provide the operator with an enforced sense of haptic (or kinesthetic) presence in a virtual environment. Depending on the signals measured in the haptic interface, two system classes may be distinguished: impedance and admittance displays. In the

Control of Robots in Contact Tasks: A Survey

61

impedance display the velocity (position) of the haptic mechanism is measured and a command force is rendered. The admittance display ensures tracking of a command position using interaction force/moment measurements in the handle. Impedance displays are commonly lightweight back-drivable active mechanisms (e.g. Phantom), while position-controlled high-inertia manipulators have often been used as admittance displays. An impedance-controlled industrial robot could be considered as an admittance display rendering the target admittance. Although study and modeling of human motor control and spatial limbs dynamics are fundamental challenges in biomechanics and neuroscience, the understanding of human interaction with a dynamic environment is still insufficient. The key quantity describing human arm dynamic interaction is the end-point impedance [105]. Numerous studies have demonstrated surprising human capabilities to adapt the arm impedance to variable interaction conditions and perturbations, even so to perform mechanically unstable tasks [106]. The Cartesian end-point arm impedance is nonlinear and non-symmetric spatial impedance, combining passive and active components [105]. However, in the control analysis human impedance is commonly, for the sake of simplicity, considered as linear variable impedance, often with one or two DOFs. Likewise, the haptic display dynamics can be considered as a linear admittance, while the environment is generally represented by nonlinear impedance. However, the performance obtained with the basic interaction system shown in Fig. 1.22 is commonly poor and, therefore, such an interaction structure is not feasible. The essential interaction problems, such as unpredictable and unknown human behavior and nonlinear and highly variable environment, cannot be successfully resolved with this interaction system architecture. Such a haptic system could interact well with a particular environment and for a specific human behavior, but in other cases performance might be bad, moreover the interaction might be unstable (especially in the contact with a stiff virtual wall). Generally, it is not possible to guarantee the stability of interaction with the simple haptic interface control system presented in Fig. 1.22. Obviously, a more sophisticated controller is required to take into account the human and environmental dynamics. The synthesis of such a controller is, however, an extremely complex task. In order to simplify design and improve the stability of the haptic interaction system, Colgate et al. [107] have proposed to couple an additional block, referred to as virtual compliance or virtual coupling (Fig. 1.23), between the haptic device and the virtual environment. The virtual coupling is commonly selected as impedance, i.e. admittance. The virtual coupling provides a simple but nevertheless stable and robust haptic controller. For a particular haptic

62

Dynamics and Robust Control of Robot-Environment Interaction

device a corresponding virtual coupling might be designed regardless of the simulated virtual worlds and real human behavior. The main design goal is to ensure passive behavior of the coupled subsystem consisting of the virtual coupling and haptic display, thereby also taking into account critical SD effects (sampling and control delay). By these means, when taking into account that the human performs almost passive and stable interaction with a passive system, the stability of the entire haptic system may be ensured under all operating conditions if the virtual environment is passive. A haptic interface operates as a connection between human and virtual environments. The transparency and the Z-width are the main measures of performance of a haptic display. Transparency represents the degree to which velocities and forces (at the human and environment sides) match each other. The Z-width [108] of a haptic interface can be defined as the achievable range of impedance that can stably be presented to the operator. An ideal haptic interface can simulate free motion without inertia or friction, as well as render infinitely rigid and massive virtual objects. The primary concern in haptic systems is to achieve stable interaction under all operating conditions and for all simulated virtual environments, without undesired oscillations that would degrade virtual surface rendering. However, that is a challenging goal, because several destabilizing effects tend to jeopardize interaction stability. In order to ensure stability, almost all modern haptic systems utilize the advantages of the virtual coupling concept and implement the control architectures similar to the on sketched in Fig. 1.23.

Fig. 1.22 Elemental network model of a haptic system

Fig. 1.23 Haptic interface with virtual coupling (admittance)

Control of Robots in Contact Tasks: A Survey

63

The stability of a haptic interaction system is commonly considered on the basis of the passivity theory. Explicit conditions for the passivity of a haptic system including a linear haptic device, a virtual coupling and a virtual environment, have been derived in [109] taking into account sampling and computation delay effects. The authors argued the essential relevance of physical damping parameters for the enhancement of system passivity and interaction stability. For a simple SISO coupling system consisting of the haptic-device, i.e. admittance Z D = 1 ( ms + b ) , and the virtual-coupling impedance ZV = Bs + K , the stability (passivity) condition imposes

b>

KT +B 2

(1.54)

where T is the control sampling time. In this elemental case of haptic interface, the virtual coupling represents a virtual wall, which consists of the parallel connection of the virtual stiffness K and the virtual damping B, to be rendered to the human. Hence, the condition (1.54) means that physical damping must be involved in the system in order to ensure a stable interaction with the virtual wall. Higher sampling rates (i.e. smaller T) facilitate the implementation of stiffer walls. Brown and Colgate [110] derived similar expressions for the minimum mass of the virtual wall that can be simulated passively. However, the stability conditions that were obtained appear to be quite conservative. Moreover, these criteria imply physics-based system design that is not always reliable. For example, higher additional damping (1.54) allows higher virtual impedance to be realized, but thereby the impedance of the haptic device must also be increased. Adams [111] has proposed an approach for a virtual coupling design based on the network stability that appears to be less conservative than the passivity-based synthesis. The stability of two-port network consisting of the haptic device and the virtual coupling guarantees the stability of a haptic interface when coupled with any passive virtual environment and human operator. Miller et al. [112] have extended the passivity-based approach to haptic systems involving nonlinear and time-delayed virtual environments. Hannaford and Ryu [113] have applied time-domain passivity analysis in order to improve the system performance in contact with a very stiff and delayed environment. However, numerous experiments clearly showed that the contact stabilization with stiff, delayed and nonlinear environments still represents the crucial problem in haptic interfaces. A specific problem in haptic interfaces is the lack of objective stability testing. Human exhibits good capability to stabilize (damp)

64

Dynamics and Robust Control of Robot-Environment Interaction

the interaction with a slightly oscillating environment. Therefore, the loss of contact and bouncing in the haptic interfaces appear to be less critical [113], comparing with contact stability problems in industrial robots (see Chapter 3 and Chapter 5). However, these oscillations can jeopardize the interaction fidelity. In the majority of experiments the increasing of sampling rates and reducing of force magnitudes have been recognized as promising measures to reduce bouncing. Chapter 5 presents an extension of the new robust control design framework established for the control synthesis of interaction between an impedancecontrolled robot and a passive environment [76, 77] to other interactive systems with physical or virtual interfaces (admittance and impedance displays). 1.10 New Robot Applications Several authors have considered the development of interaction control algorithms based on soft-computing techniques [114, 115]. Fuzzy and neuralnetwork-based control algorithms provide promising approaches to cope with nonlinear environment effects. However, further investigations are needed to improve robustness, design and reliability of soft-computing methods based compliance control. The development of specific interaction control algorithms for new robotic structures, such as parallel robots [116], dual and cooperative arms [117], wire robots [118], etc., has also attracted considerable research attention. The design of light-weight robotic arms with integrated torque sensors at robot joints has been demonstrated to be a very promising concept for future safe robotic systems suitable for dynamic interaction control [119]. Passivity-based control design techniques have demonstrated good potential for interaction stabilization of rigid [120] and flexible-joints robots [121, 122] with a stiff environment. Specific interaction control algorithms suitable for the application in surgical and rehabilitation robotics have also attracted research interest [123]. The impedance control has also appeared as a very promising approach to dynamic interaction control in humanoid gait [124, 125]. The compliance control represents the basic control approach in new robot applications in rapidly emerging fields requiring physical interaction between the robot and operator such as human extenders and power-assists devices, rehabilitation robots, etc (see only a few of selected papers [126, 127, 128]). The control algorithms applied in these fields are essentially based on the above control approaches. The integral force control is basically applied in hand-driven

Control of Robots in Contact Tasks: A Survey

65

intelligent assistant devices [128] to realize the movement of large payloads by a human operator handling a force sensor interface. The synthesis of passive and robust control laws, discussed above, is crucial for ensuring a stable interaction between human, robot and a passive environment in intelligent assistance devices. It is worth mentioning an experimental comparison of robot interaction control schemes [129]. Position based force and impedance control has been implemented and tested in an industrial experimental robotic system. The authors have pointed out superior performance of the dynamic-model-based control schemes in comparison to so-called “static-model” algorithms designed to achieve desired steady-state performance. Good performance of parallel position-force control has also been reported. However, further investigations are still required concerning practical needs and benefits of the complete robot dynamic model computations in the interaction control algorithms intended for industrial robot control systems interacting with a passive environment and in relatively slow contact tasks, which are most frequently performed in practice. Definitions of benchmark tests and objective criteria for comparison of various compliance control schemes are also required. Significant research efforts have also been made in the direction of practical design of interaction control [130]. Computer-aided procedures for design of hybrid position/force and impedance control were developed in [131] and [132], respectively. Publication of the first monographs focusing on interaction control [133, 134] has also contributed to the further investigations in this field. Finally, the integration of compliance and visual control [135, 136] should also be mentioned as a very attractive research topic that is quite promising for further improvement of interaction control capabilities and performance. 1.11 Conclusion During the past two decades, compliant motion control has emerged as one of the most attractive and fruitful research areas in robotics. The control of constrained motion of robots is still a challenging research area, whose successful solution will considerably influence further applications of robots in the industry and emerging service fields. Widespread applications of impedance control in industrial robotic systems are still a challenging problem. One of limitations is the absence of a widely-accepted framework for the synthesis of impedance control parameters that would ensure the stability of both contact transition and interaction processes and guarantee desired contact performance.

66

Dynamics and Robust Control of Robot-Environment Interaction

In conclusion, it is perhaps of interest to indicate some of possible future research subjects. A clear formulation and specification of hybrid control and impedance control are further required in order to integrate these control approaches in commercial industrial robotic systems. Especially, the synthesis of compliance control for specific contact tasks at higher motion planning and programming control levels is essential for the realization of first industrial robot systems with compliance control capabilities. Further simulation and experimental tests of the proposed compliant motion control algorithms, such as parallel position-force control, adaptive and variable structure algorithms, and, particularly, dynamic control of a robot interacting with a dynamic environment, are also of interest. Particular attention should be paid to solving contact task control problems caused by uncertainties and nonlinear effects in the environment and robotic system, such as friction, multipoint contact, elasticity, time lags in computer controlled systems, etc. In the impedance control, further advances are to be expected in the adaptation of target impedance to complex task requirements. Design and optimization of robot admittance for specific tasks represents a challenging practical task. The compliant motion capability analysis of industrial robots and requirements on the next robot generation from the viewpoint of contact-task applications are of interest to designers. Robust control provides a practical approach for control design of compliance motion. Research on human-robot interaction, understanding of human motoric functions and diseases, modeling and estimation of human impedance, as well as the development of stable and safe interaction algorithms, represent further very attractive topics in new robotic applications supporting physical interactions with a human. Nonlinear control concepts might be useful to bridge the gap between geometric and dynamic constraints. Sophistication of the robustness concept is another worthwhile theoretical challenge. From a more practical point of view, future research should produce systems with improved intermediate and high level performance and user friendliness and safety. Comparison of the available algorithms, definition of benchmark tests, investigation of compliant control in uncertain and dynamic environment, examination of nonlinear effects in the robot and environment dynamics, specification and design of force and impedance control tasks, solving control problems at higher control levels, etc., are certainly some areas deserving further computational/experimental studies.

Control of Robots in Contact Tasks: A Survey

67

Bibliography [1]

Mason, M.T., Compliance and Force Control for Computer Controlled Manipulators, IEEE Transactions on Systems, Man and Cybernetics, Vol. SMC-11, No. 6, 418-432, 1981.

[2]

Pfeifer, F., Richter, K., Wapenhans H., Elastic Robot Trajectory Planning with Force Control, Proceedings, IFIP International Symposium on Theory of Robots, Rome, 201-212, 1990.

[3]

Whitney, D.E., Nevins J.L., What is Remote Centre Compliance (RCC) and What Can It Do, in Robot Sensors Volume 2 - Tactile and Non-Vision, IFS Publication, Springer-Verlag, Berlin, 3-17, 1986.

[4]

Cutkosky M.R., Wright P.K, Active Control of a Compliant Wrist in Manufacturing Tasks, in Robot Sensors Vol. 2 - Tactile and Non-Vision, IFS Publication, SpringerVerlag, Berlin, 17-33, 1986.

[5]

Simons J., Van Brussel H., Force Control Schemes for Robot Assembly, International Trends in Manufacturing Technology: Robotic Assembly, IFS Publication, SpringerVerlag, Berlin, 1985.

[6]

Whitney D.E., Historical Perspective and State of the Art in Robot Force Control, The Inter. Journal of Robotic Research, Vol. 6, No.1, 3-14, 1987.

[7]

Raibert M.H., Craig J.J., Hybrid Position/Force Control of Manipulators, ASME Journal of Dynamic Systems, Measurement and Control, Vol. 102,126-133,1981.

[8]

Vukobratovic M., Ekalo, Y., Unified Approach to Control Laws Synthesis for Robotic Manipulators in Contact with Dynamic Environment, Tutorial S5: Force and Contact Control in Robotic Systems, Proc. IEEE, Int. Conference on Robotics and Automation, 213-229, 1993.

[9]

Chiaverini S. and Sciavicco L., Force/Position Control of Manipulators in Task Space with Dominance in Force, Proceedings of 2nd IFAC Symposium on Robot Control (SYROCO), Karlsruhe, 137-143, 1988.

[10]

Wedel D.L., Saridis G.N., An Experiment in Hybrid Position/Force Control of a Six DOF Revolute Manipulator, Proceedings of IEEE International Conference on Robotics and Automation, 1638-1642, 1988.

68

Dynamics and Robust Control of Robot-Environment Interaction

[11]

Yoshikawa T., Sugie T., Tanaka M., Dynamic Hybrid Position/Force Control of Robot Manipulators-Controller Design and Experiment, IEEE Journal of Robotics and Automation, Vol.4, No. 6, 693-705, 1988.

[12]

Anderson R., Spong M., Hybrid Impedance Control of Robotic Manipulators, IEEE Journal of Robotics and Automation, Vol. 4, No. 5, 549-556, 1988.

[13]

De Schutter J., Van Brussel H., Compliant Robot Motion II: A Control Approach Based on External Control Loops, The International Journal of Robotic Research, Vol. 7, No. 4, 18-33, 1988.

[14]

Lawrence D.A., Impedance Control Stability Properties in Common Implementations, Proc. IEEE International Conference on Robotics and Automation, Vol. 2, 1185-1190, 1988.

[15]

Maples J.A., Becker J.J., Experiments in Force Control of Robotic Manipulators, Proc. of IEEE Int. Conf. on Robotics and Automation, Vol. 2, 695-703, 1986.

[16]

Kazerooni H., Waibel B.J., Kim S., On the Stability of Robot Compliant Motion Control: Theory and Experiments, Journal of Dynamics, Systems, Measurement and Control, Vol. 112, No. 3, 417-426, 1990.

[17]

Salisbury J.K., Active Stiffness Control of a Manipulator in Cartesian Coordinates, Proceedings 19th IEEE Conference on Decision and Control, 95-100, 1980.

[18]

Whitney D.E., Force Feedback Control of Manipulator Fine Motions, ASME J. of Dynamics, Systems, Measurement and Control, Vol. 99, No. 2, 91-97, 1977.

[19]

Hogan N., Impedance Control: An Approach to Manipulation, Part I- Theory, Part IIImplementation, Part III- Application, Journal of Dynamics Systems, Measurement and Control, Vol. 107, No. 1, 1-24, 1985.

[20]

Kazerooni K.H., Sheridan T., Houpt P., Robust Compliant Motion for Manipulators, Part I: The Fundamental Concepts of Compliant Motion; IEEE Journal of Robotics and Automation, Vol. RA-2, No. 2, 83-92, 1986.

[21]

Eppinger S.D., Seering W.P., Understanding Bandwidths Limitation in Robot Force Control, Proceedings of IEEE International Conference on Robotics and Automation, Raleigh, NC, 904-909, 1987.

[22]

Khatib O., A Unified Approach for Motion and Force Control of Robot Manipulators: The Operational Space Formulation, IEEE Journal of Robotics and Automation, Vol. 3, No.1, 43-53, 1987.

[23]

De Luca A., Manes C., Nicolo F., A Task Space Decoupled Approach to Hybrid Control of Manipulators, Proc. IFAC SYROCO ´88, Karlsruhe, 157-162, 1988.

[24]

Lipkin H, Duffy J., Hybrid Twist and Wrench Control for a Robotic Manipulator, Trans. ASME Journal of Mech., Trans. and Aut. in Design, Vol. 110, No. 2, 138-144, 1988.

Control of Robots in Contact Tasks: A Survey

69

[25]

De Shutter J., Bruyninckx.H., Model-based Specification and Execution of Compliant Motion, Tutorial M6, IEEE Conference on Robotics and Automation, ICRA’92 Nice, 74-84, 1992.

[26]

Zhang H., Paul R., Hybrid Control of Robot Manipulator, Proc. of IEEE Int. Conf. on Robotics and Automation, 602-607, 1985.

[27]

Daneshmend L., Hayward V., Pelletier M., Adaptation to Environment Stiffness in the Control of Manipulators, Experimental Robotics I, (Hayward V. and Khatib O. Eds.), Springer-Verlag, Berlin, 150-156, 1990.

[28]

Stokic D., Vukobratovic M., Surdilovic D., An Adaptive Hybrid Control Scheme for Manipulation Robots with Implicit Force Control, Proc. V ICAR, Pisa, 1505-1508, 1991.

[29]

Kazerooni K.H., Sheridan T., Houpt P., Robust Compliant Motion for Manipulators, Part II Design Method, IEEE Journal of Robotics and Automation, Vol. RA-2, No. 2, 93-106, 1986.

[30]

Dawson D.M., Lewis F.L., Dorsey J.F., Robust Force Control of a Robot Manipulator, The Int. Journal of Robotic Research, Vol. 11, No.4, 312-319, 1992.

[31]

Shin K.G., Lee C.P., Compliant Control of Robotic Manipulators with Resolved Acceleration, Proceedings of 24th IEEE Conference on Decision and Control, Vol. 1, 350-357, 1985.

[32]

Mc Clamroch H., Wang D., Feedback Stabilization and Tracking of Constrained Robots, IEEE Transaction on Automatic Control, Vol. 33, No. 5, 419-426, 1988.

[33]

Vukobratovic, M., Ekalo, Y., New Approach to Control of Robotic Manipulators Interacting with Dynamic Environment, Robotica, Vol. 14, 31-39, 1996.

[34]

Hogan N., On the Stability of Manipulator Performing Contact Tasks, IEEE Journal of Robotics and Automation, Vol. 4., No. 6, 677-686, 1990.

[35]

An C.H., Hollerbach J.M., The Role of Dynamic Models in Cartesian Force Control, The Int. Jour. on Rob. Research, Vol. 8., No. 4, 51-72, 1989.

[36]

Kircanski N., Computational Aspects of Force/Impedance Control Algorithms, Tutorial S5: Force and Contact Control in Robotic Systems (Ed. Vukobratovic M. and Nakamura Y., IEEE International Conference on Robotics and Automation, 33-48, 1993.

[37]

De Luca A., Manes C., Modeling of Robots in Contact with a Dynamic Environment, IEEE Transactions on Robotics and Automation, Vol. 10, No. 4, 542-548, 1994.

[38]

Goldenberg A.A., Force and Impedance Control of Robot Manipulators, Proceedings Winter Annual Meeting of ASME, 161-168, 1987.

[39]

Sharon A., Hogan N., Hardt D., More Analysis and Experimentation of a Micro/Macro Manipulator System, Proc. ASME Winter Annual Meeting, Boston, 417-422, 1987.

70

Dynamics and Robust Control of Robot-Environment Interaction

[40]

Vukobratovic M., Stojic R., On Position-Force Control of Robot Interacting with Dynamic Environment in Cartesian Space, Journal of Dynamic Systems, Measurement and Control, Vol. 118, 187-192, 1996.

[41]

Goldenberg A.A., Analysis of Force Control Based on Linear Models, Proceedings of IEEE International Conference on Robotics and Automation, Nice, 1348-1353, 1992.

[42]

Surdilovic D., Contact Stability Issues in Position-based Impednace Control: Theory and Experiments, Proceedings of IEEE International Conference on Robotics and Automation, Minneapolis, 1675-1680, 1996.

[43]

DeFazio T.L., Seltzer D.S., Whitney D.E., The IRCC Instrumented Remote Center Compliance, in International Trends in Manufacturing Technology: Robotic Assembly, IFS Publications, Springer-Verlag, 33-45, Berlin, 1985.

[44]

Lee S., Lee H.S., Generalized Impedance of Manipulators: Its Application to Force and Position Control, Proc. Fifth International Conference of Advanced Robotics ICAR’91, 1477-1480, 1991.

[45]

Lu Z., Goldenberg A.A., Robust Impedance Control and Force Regulation: Theory and Experiments, The International Journal of Robotics Research, Vol. 14. No. 3, 225-237, 1995.

[46]

Hogan N., Stable Execution of Contact Tasks Using Impedance Control, Proceedings of IEEE International Conference on Robotics and Automation,, Raleigh, North Carolina, 1047-1054, 1987.

[47]

Vukobratovic M., Stokic D., Kircanski N., Non-Adaptive and Adaptive Control of Manipulation Robots, Monograph, Scientific Fundametals of Robotics 5, SpringerVerlag, Berlin, 1985.

[48]

Visher D., Khatib O., Design and Development of Torque-controlled Joints, Proceedings of The First International Symposium on Experimental Robotics, 271-276, 1989.

[49]

Vukobratovic M., Surdilovic D., Control of Robotic Systems in Contact Tasks, Chapter 21 in The Mechanical Systems Design Handbook: Modeling, Measurement and Control (Ed. Nwokah O.I. and Hurmuzlu Y.), CRC Press, Boca Raton, 2002.

[50]

Craig J.J., P. Hsu, S. S. Sastry, Adaptive Control of Mechanical Manipulators, The International Journal of Robotic Research, Vol. 6, No.2, 16-28, 1987.

[51]

Al-Jarrah O.M., Zheng Y.F., Intelligent Compliant Motion Control, Proceedings of IEEE International Conference on Robotics and Automation, Minneapolis, Vol. 3, 2610-2615, 1996.

[52]

Paul and R.P., Shimano B, Compliance and Control, Proceedings of. Joint Automatic Control Conference, San Francisco, 694-699, 1976.

[53]

Stokic D.M., Surdilovic D.T., Position/Force Control of Industrial Robots: Implementation Problems, Report IPK-Berlin, 1990.

Control of Robots in Contact Tasks: A Survey

71

[54]

An C.H., Hollerbach J., Kinematic Stability Issues in Force Control of Manipulators, Proceedings of IEEE International Conference on Robotics and Automation, Raleigh, 897-903, 1987.

[55]

Zhang H., A Kinematically Stable Hybrid Control Formulation, Proc. of the Third International Symposium ISRAM ´90, British Columbia, 287-292, 1990.

[56]

Fisher W.D., Mujtaba M.S., Hybrid Position/Force Control: A Correct Formulation, The International. Journal of Robotics Research, Vol. 11, No. 4, 299-311, 1992.

[57]

An C.H., Hollerbach J., Dynamic Stability Issues in Force Control of Manipulators, Proceedings. of IEEE International Conference on Robotics and Automation, Raleigh, 890-896, 1987.

[58]

Luh J.Y.S., Fisher W.D., Paul R.P.C., Joint Torque Control by a Direct Feedback for Industrial Robots, IEEE Trans. on Automatic Control, Vol. AC-28, No. 2, 153-161, 1983.

[59]

Bona B., Indri M., Exact Decoupling of the Force-Position Control Using the Operational Space Formulation, Proceedings of IEEE International Conference on Robotics and Automation, 1394-1398, Nice, 1992.

[60]

Volpe R., Khosla P., An Experimental Evaluation and Comparison of Explicit Control Strategies for Robotic Manipulators, Proc. of IEEE Int. Conference on Robotics and Automation, Nice, 1387-1393, 1992.

[61]

Stokic D.M., Surdilovic D.T., Simulation and Control of Robotic Deburring, International Journal of Robotics and Automation, Vol. 5, No. 3, 107-115, 1990.

[62]

Yoshikawa T., Sudou A., Dynamic Hybrid Position/Force Control of Robot Manipulators: Estimation of Unknown Constraint, Proceedings of IEEE International Conf. on Robotics and Automation, Cincinnati, 1231-1236, 1990.

[63]

Bruyninckx H., De Schutter J., Allota B., Model-Based Constrained Motion: Introducing On-Line Identification and Observations of Motion Constraints, Proceedings of. Fifth International Conference of Advanced Robotics, ICAR’91, Pisa, 976-981, 1991.

[64]

Sinha P., Goldenberg A.A., A Unified Theory for Hybrid Control of Manipulators, Proceedings of IEEE International Conference on Robotics and Automation, Atlanta, Vol. 1, 343-348, 1993.

[65]

Volpe R., Khosla P., The Equivalence of Second-Order Impedance Control and Proportional Gain Explicit Force Control, The International Journal of Robotics Research, Vol. 14, No. 6, 574-589., 1995.

[66]

Surdilovic D., Kirchhof J., A New Position Based Force/Impedance Control for Industrial Robots, Proceedings of IEEE International Conference on Robotics and Automation, Minneapolis, Minnesota, 629-634, 1996.

[67]

Mayeda H., Ikeda N., Miyaji K., Position/Force/Impedance Control for Robot Tasks, Proceedings of the Second ISMCR´92, Tsukuba, Japan, 551-558, 1992.

72

Dynamics and Robust Control of Robot-Environment Interaction

[68]

Chiaverini S., Sicilliano B., Villani L., Force/Position Regulation of Compliant Robot Manipulators, IEEE Transactions on Automatic Control, Vol. 39, No. 3, 647-652, 1994.

[69]

Vukobratovic M., Stojic R., Ekalo Y., Contribution to the Position-Force Control of Manipulation Robots Interacting with Dynamic Environment – A Generalization, IFAC Automatica, Vol. 34, No. 10, 1219-1226, 1998.

[70]

Colgate J. E., Hogan M., Robust Control of Dynamically Interacting Systems, International Journal of Control, Vol. 48, No. 1, 65-88, 1988.

[71]

Chapel J.D., Su R., Coupled Stability Characteristics of Nearly Passive Robots, Proceedings of IEEE International Conference on Robotics and Automation, Nice, 1342-1346, 1992.

[72]

Colgate J. E., Hogan N., An Analysis of Contact Instability in Terms of Passive Physical Equivalents, Proceedings of IEEE Int. Conference on Robotics and Automation, 404-409, 1989.

[73]

Newman W. S., Stability and Performance Limit of Interaction Controllers, Trans. of the AMSE Journal of Dynamic Systems, 114, No. 4., 65-88, 1992.

[74]

Dohring M. and Newman W., “Admittance Enhancement in Force Feedback of Dynamic Systems”, Proc. of ICRA, Washington DC, 638-643, 2002.

[75]

Buerger S. P. and Hogan N., Relaxing Passivity for Human-Robot Interaction, Proceedings of the Int. Conference on Intelligent Robot and Systems, Beijing, 45704575, 2006.

[76]

Surdilovic D., Robust Control Design of Impedance Control for Industrial Robots, Proceedings of IROS’07, San-Diego, 3572-3579, 2007.

[77]

Surdilovic D., Radojicic J., 2007, Robust Control of Interaction with Haptic Interfaces, Proceedings of ICRA’07, Rome, 3237-3244, 2007.

[78]

Benady M., Flash T., Gershon D., Robot Learning of Contact Tasks, in Robotic Systems and AMT, (Editors Halevi G., Yudilevich I., and Weill I.), Elsevier Science Publishers, Amsterdam 147-159, 1990.

[79]

Xu Y., Paul R.P., Corke P. I., Hybrid Position Force Control of Robot Manipulator with an Instrumented Compliant Wrist, Proceedings of First International Symposium on Experimental Robotics, Springer-Verlag, Berlin, 244-271, 1990.

[80]

Quian H.P., De Schutter J., Introducing Active Linear and Nonlinear Damping to Enable Stable High Gain Force Control in Case of Stiff Contact, Proc. of IEEE International Conf. on Robotics and Automation, Nice, 1374-1379, 1992.

[81]

Siciliano B., Villani L., A Force/Position Regulator for Robot Manipulators Without Velocity Measurements, Proceedings of IEEE International Conference on Robotics and Automation, Minnesota, 2567-2572, 1996.

Control of Robots in Contact Tasks: A Survey

73

[82]

De Quieroz M., Dawson D., Burg T., Position/Force Control of Robot Manipulators without Velocity/Force Measurements, Proceedings of IEEE International Conf. on Robotics and Automation, Minnesota, 2561-2566, 1996.

[83]

Mills J., Lokhorst D., Control of Robotic Manipulators During General Task Execution: A Discontinuous Control Approach, The International Journal of Robotic Research, Vol. 12, No. 2, 146-162, 1993.

[84]

Wu Y., Tarn T.J., Xi N., Isidori A., On Robust Impact Control via Positive Acceleration Feedback for Robot Manipulators, Proceedings of the IEEE International Conf. on Robotics and Automation, Minneapolis, 1891-1896, 1996.

[85]

Volpe R., Khosla P., A Theoretical and Experimental Investigation of Impact Control for Manipulators, The International Journal of Robotics Research, Vol. 12, No. 4, 351-365, 1993.

[86]

D.M. Gorinevsky, A.M. Formalsky, A.Y. Shneider, Force Control of Robotic Systems, CRC Press, 1997.

[87]

Eppinger S.D., Seering W.P., Introduction to Dynamic Robot Models for Robot Force Control, IEEE Control System Magazine, Vol. 7, No. 2., 48-52, 1987.

[88]

Vukobratovic M., The Role of Environment Dynamics in Contact Force Control of Manipulation Robots, ASME Journal Dynamic Systems, Measurement, and Control, Vol. 119, 86-89, 1997.

[89]

Ekalo Y., Vukobratovic M., Robust and Adaptive Position-Force Stabilization of Robotic Manipulators in Contact Tasks, Robotica, Vol. 11, 373-386, 1993.

[90]

Ekalo Y., Vukobratovic M., Adaptive Stabilization of Motion and Forces in Contact Tasks for Robotic Manipulators with Nonstationary Dynamics, International Journal Robotics and Automation, Vol. 9., No. 3, 91-98, 1994.

[91]

Ekalo Y., Vukobratovic M., Quality of Stabilization of Robot Interacting with Dynamic Environment, Journal of Intelligent and Robotic Systems, Vol. 14, 155-179, 1995.

[92]

Vukobratovic M., Katic D., Robust Stabilizing Position-Force Control of Robots Interacting with Dynamic Environment by Learning Connectionist Structure, IFAC Automatica, Vol. 32, No. 12, 1733-1739, 1996.

[93]

Stokic D., Vukobratovic M., Practical Stabilization of Robots Interacting with Dynamic Environment by Decentralize Control, Int. J. on Robotics and Automation, Vol. 13, Issue 4, 137-146, 1998.

[94]

Stokic D., Vukobratovic M., An Improved Method for Analysis of Practical Stability of Robots Interacting with Dynamic Environment, Int. J. Izvestya of Russian Academy of Science: Theory and Control Systems, No.4, 186-192, 2000.

[95]

Fasse E., Broenink F.J., A Spatial Impedance Controller for Robotic Manipulation, IEEE Trans. on Robotics and Automation, Vol. 13, 546-556, 1997.

[96]

Selig J.M., The Spatial Stiffness Matrix from Simple Stretched Springs, Proceedings of the 2000 ICRA, San Francisco, 3314-3319, 2000.

74

Dynamics and Robust Control of Robot-Environment Interaction

[97]

Siciliano B., Villani L., Six-Degree-of-Freedom Impedance Robot Control, Proceedings of the 8th ICAR, Monterey, 387-392, 1997.

[98]

Caccavale F., Siciliano B., Villani L., Robot Impedance Control with Nondiagonal Stiffness, IEEE Transactions on Automatic Control, Vol. 44, 1943-1946, 1999.

[99]

Caccavale F., Natale C., Siciliano B., Villani L., Six-dof Impedance Control Based on Angle/Axis Representations, IEEE Transactions on Robotics and Automation, Vol. 15, No. 2, 289-300, 1997.

[100]

Schimmels J.M. and Peshkin M.A., “Force-assembly with Friction”, IEEE Transaction on Robotics and Automation, Vol. 10, No. 4, 465-479, 1994.

[101]

Huang S. and Schimmels J., “Admittance Selection for Force-Guided Assembly of Polyhedral Parts in Single-Point Contact”, Proccedings of the 20004 Int. Conference on Robotics and Automation, New Orleans, Louisiana, 3577-3582, 2004.

[102]

Surdilovic D., Synthesis of Impedance Control Laws at Higher Control Levels: Algorithms and Experiments, Proceedings of IEEE International Conference on Robotics and Automation, Leuven, Belgium, 306-311, 1998.

[103]

Surdilović D., Bernhardt R., Colombina G., Grassini F., A Space Robot Control with Advanced Sensor-Based Capabilities: Terrestrial Spin-off, 27th ISIR, Milan, pp. 243248, 1996.

[104]

Adams R., Hannaford B, Stable Haptic Interaction with Virtual Environments, IEEE Trans. on Robotics and Automation, Vol. 15, No. 3, 465-474, 1999.

[105]

Hogan N., Controlling Impedance at the Man/Machine Interface, Proceedings of IEEE International Conference on Robotics and Automation, Scottsdale, Arizona, 1626-1631, 1989.

[106]

Burdet E., Osu R., Franklin D., Millner T., Kawato M., The central nervous system stabilizes unstable dynamics by learning optimal impedance, Nature, Vol. 414, 446-449, 2001.

[107]

Colgate J. E., Stanley M.C., Brown J.M., Issues in the Haptic Display of Tool Use, Proceedings IEEE/RSJ International Conference on Intelligent Robots and Systems, Pitsburgh, PA, 140-145, 1995.

[108]

Colgate J. E., Brown J.M., Factors Affecting the Z-Width of a Haptic Display, Proceedings of IEEE International Conference on Robotics and Automation, San Diego, 3205-3210, 1994.

[109]

Colgate J. E., Schenkel G.G., Passivity of a Class of Sampled-Data Systems: Application to Haptic Interfaces, Proceedings American Control Conference, Baltimore, 3236-3240, 1994.

[110]

Brown M., Colgate E., Minimum Mass for Haptic Display Simulations, DSC-Vol. 64, Proceedings of the ASMEDynamic Systems and Control Division, ASME, 249-256, 1998.

Control of Robots in Contact Tasks: A Survey

75

[111]

Adams R., Stable Haptic Interaction With Virtual Environment, Ph. D. Thesis, University of Washington, 1999.

[112]

Miller B., Colgate E., Freeman R., Guaranteed Stability of Haptic Systems with Nonlinear Virtual Environments, IEEE Transaction on Robotics and Automation, Vol. 16, Nr. 3, 712-719, 2000.

[113]

Hannaford B., Ryu J-H., Time Domain Passivity Control of Haptic Interfaces, Proceedings of IEEE International Conference on Robotics and Automation, Seoul, Korea, 1863-1869, 2001.

[114]

Baptista L.F., Sousa J.M., de Costa J.S., Force Control of Robotic Manipulators Using a Fuzzy Predictive Approach, Journal of Intelligent and Robotic Systems, Vol. 20, 359-376, 2001.

[115]

Lin S.T., Tzeng S.J., Neural Network Force Control for Industrial Robots, Journal of Intelligent and Robotic Systems, Vol. 20, 253-268, 1999.

[116]

Fasse E., Gosselin C., Spatial-Geometric Impedance Control of Gough-Stewart Platforms, IEEE Trans. on Robotics and Automation, Vol. 15, 281-288, 1999.

[117]

Surdilovic D., De Bartolomeo M., Grassini F., Synthesis of Impedance Control for Complex Co-operating Robot Assembly Tasks, Proceedings AIM’01 Conference, Como, Italy, 2001, 1181-1187, 2001.

[118]

Morizono T., Kurahashi K., Kawamura S., Realization of Virtual Sports Training System with Parallel Wire Mechanism, Proceedings ICRA 97’, Alberquerque, New Mexico, 3025-3030, 1997.

[119]

Hirzinger G., Albu-Schäffer A., Hähnle M., Schaefer I., Sporer N., On a New Generation of Torque Controlled Light-Weight Robots, Proceedings of 2001 ICRA, Seoul, Korea, 3356-3361, 2001.

[120]

Stramigioli S., Modeling and IPC Control of Interactive Mechanical Systems: A Coordinate-free Approach, Springer-Verlag, Berlin, 2001.

[121]

Ott C., Albu-Schäffer A., Kugi A., Stramigioli S. and Hirzinger G., A Passivity Based Cartesian Impedance Controller for Flexible Joint Robots – Part I: Torque Feedback and Gravity Compensation, Proccedings of the 20004 Int. Conference on Robotics and Automation, New Orleans, Louisiana, 2659-2665, 2004.

[122]

Albu-Schäffer A., Ott C. and Hirzinger G., A Passivity Based Cartesian Impedance Controller for Flexible Joint Robots – Part II: Full State Feedback, Impedance Design and Experiments, Proccedings of the 20004 Int. Conference on Robotics and Automation, New Orleans, Louisiana, 2666-2672, 2004.

[123]

Rembold U., Burghart C., Surgical Robotics: An Introduction, Journal of Intelligent and Robotic Systems, Vol. 30, 1-28, 2001.

[124]

Chung P.H., Impedance Control and Modulation for Stable Footing in Locomotion of Biped Robots, Proc. ICRA 99, Detroit-Michigan, 1786-1791, 1999.

76

Dynamics and Robust Control of Robot-Environment Interaction

[125]

Silva F.M., Machado T.J.A., Goal-Oriented Biped Walking Based on Force Interaction Control, Proceedings of 2001 ICRA, Seoul, Korea, 4122-4127, 2001.

[126]

Hannaford B., Feeling is Believing: History of Telerobotics Technology, in The Robot in the Garden: Telerobotics and Telepistemology in the Age of the Internet (Ed. K. Goldberg), MIT Press, 1999.

[127]

Reinkensmeyer D.J., Rehabilitators, in Biomedical Engineer’s Handbook (Ed. M. Kutz), McGraw Hill, 2002.

[128]

Kazerooni H., A New Dimension in Robotics for Automobile Assembly: The Return of the Worker, Proceedings 6th Annual Symposium on Haptic Interfaces for Virtual Environment and Telemanipulator Systems, Dallas, Texas, 214-222, 1997.

[129]

Chiaverrini, Siciliano B., Villani L., A Survey of Robot Interaction Control Schemes with Experimental Comparison, IEEE/ASME Transactions on Mechatronics, Vol. 4, No. 3, 273-285, 1999.

[130]

De Schutter J., Bruyninckx H., Zu W-H., Spong M.W., Force Control: A bird’s eye view, in Control Problems in Robotics and Automation, Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin, Vol. 230, 1-17, 1998.

[131]

Natale C., Koeppe R, Hirzinger G., An Automatic Procedure for Force Controller Design, IEEE Transaction on Mechatronics, Vol. 05, No. 02, 122-132, 2000.

[132]

Surdilovic D., Impedance Control Design Toolbox – User’s Guide, IPK-Berlin, 2000.

[133]

Siciliano B., Villani L., Robot Force Control, Kluwer Academic Publishers, Boston, MA, 2000.

[134]

Natale C., Interaction Control of Robot Manipulators: Six-degrees-of-Freedom Tasks, Series: Springer Tracts in Advanced Robotics, Vol. 3, Springer-Verlag, Berlin, 2003.

[135]

Yoshikawa T., Force Control of Robot Manipulators, Proceedings of the 2000 IEEE International Conference on Robotics & Automation, San Francisco

[136]

Nakabo Y. and Ishikawa M., Visual Impedance Using 1ms Visual Feedback systems, Proceedings of the 1998 Int. Conference on Robotics & Automation, Leuven, Belgium, 2333-2338, 1998.

Chapter 2

A Unified Approach to Dynamic Control of Robots

2.1 Introduction The difficulties encountered in solving the problem of simultaneous stabilization of programmed motion and desired contact force of the robot interacting with its environment, have probably been the reason for introducing the simplifying idea of splitting the task into the motion control and interaction force control. This idea enabled Raibert and Craig [1] and Mason [2] to formulate the approach to manipulator control called the hybrid position/force control. The basic idea of this approach is to divide the control task into two independent subtasks in a certain coordinate space {Z } . One subtask is the robot’s motion control along a predetermined part of the coordinates (directions), and the other is the control of the interaction force of the robot and environment along the rest of the coordinates (directions). From the mathematical point of view, the essence of the hybrid control concept is in the following. It is assumed that there exists a coordinate space {Z } and its partition into a direct sum of the subspaces {X } and {Y } : {Z } = { X } ⊕ {Y } , so that the dynamic model of manipulation robot written in the coordinate space {Z } is split into two independent equations:

U1 ( x, xɺ, ɺɺ x) = τ x U 2 ( Fy ) = τ y where τ x , τ y are the corresponding control inputs. However, there are no real examples of a robot interacting with its dynamic environment in which such partition would be possible. Even for a simplest manipulation mechanism interacting with the environment, the following partition is obtained: 77

78

Dynamics and Robust Control of Robot-Environment Interaction

U1 ( x, xɺ, ɺɺ x, Fx ) = τ x U 2 ( y, yɺ , ɺɺ y, Fy ) =τ y In Section 1.6.2 of Chapter 1, the reader can find a detailed discussion of the position/force hybrid control. The orthogonality between the constraint force and the direction of unconstrained motion is assumed and has been used in the majority of the works. The weakness of this approach related to the notion of “orthogonality” lies not only in the fact that it is not correct to use the term “orthogonality” itself, but also in the fact that, in finding directions along which motion and force are “orthogonal”, the followers of hybrid control commit a mistake. Namely, for the task of stabilization in these directions they use feedback loops with respect to motion or force only. The criticism refers to the basic idea of the position/force stabilization based on traditional hybrid control concept, and not to the realization of this possibility itself by means of a certain procedure. However, there have been no attempts to realize the hybrid control concept by some other means, save the “orthogonal complements”. The idea of splitting the task of the robot interacting with the environment into the task of position control in certain directions and force control in the other, represents by itself a more profound idea than the idea of hybrid control based on the “orthogonal complements”. This Chapter is devoted to the unified approach to dynamic control of the robots interacting with a dynamic environment, which differs essentially from the above conventional hybrid control schemes. A dynamic approach to control simultaneously both the position and force in an environment with completely dynamic reactions has been developed. The approach of dynamic interaction control defines two control subtasks responsible for the stabilization of robot position and interaction force. Both control subtasks utilize dynamic model of the robot and the environment in order to ensure the tracking of both the nominal motion and force. Special attention is paid not only to the synthesis of the control laws that ensure stability of robot’s desired motions and desired interaction forces of the robot and environment, but also to the definition of possible motions of the robot and its possible interaction forces in contact tasks. These motions and interaction forces are called programmed. The concept of the family of transient processes with respect to the robot’s motion and its force of interaction with environment

A Unified Approach to Dynamic Control of Robots

79

is formulated. It allows one to set and solve the problem of synthesis of control laws that not simply stabilize the motion and force of the robot’s interacting with its environment, but also enables solving of the stabilization problem with the preset quality of transient processes. Thus, two types of control laws can be synthesized. One type of control laws solve the problem of exponential stabilization of programmed motions of the robot with the predetermined quality of transient processes with respect to position, and simultaneously stabilize exponentially the programmed interaction forces of the robot and environment. The other type of control laws solve the problem of exponential stabilization of programmed interaction forces of the robot and environment with the predetermined quality of transient processes with respect to force, and simultaneously stabilize exponentially the programmed motions of the robot. It is necessary to note that the laws of former type solve the problem of robot’s control both in free space and at its contact with the environment. In the general case, for the control laws of second type it was possible to obtain only a sufficient condition of exponential stability of motion when the exponential stability of the contact force is achieved. The special case of the environment dynamics model for which this type of control laws gives both necessary and sufficient conditions of exponential stability of motion is also considered. For the control laws of both types significant attention is paid to the analysis of the influence on transient processes of the constraints imposed on the state, control and interaction force, taking into account the inadequacy of dynamics models of the robot and environment and/or external perturbations. In view of the fact that in the course of operation, the robot parameters, especially the viscous friction coefficients at the manipulator’s drives, may vary with time, the unknown drifts of parameters are of interest. The adaptive control scheme proposed in this chapter enables one to solve contact tasks for robots with both stationary and nonstationary dynamics. One of the parameters of the scheme, such as the algorithm adaptation processing speed, determines the “speed” of the parameters’ drift to which the adaptive control system has time to adapt without violating the a priori constraints of the control, motion, and interaction forces. Classes of stabilized motions and forces and their stabilization accuracy are considered in reference to the level of initial and external perturbations of dynamics of the robot and its environment, as well as of the sensors errors, processing speed of the adaptation algorithm, and of other parameters of the adaptive control scheme.

80

Dynamics and Robust Control of Robot-Environment Interaction

As was mentioned above, the control laws that stabilize simultaneously both the robot motions and interaction forces are presented. These control laws possess exponential stability of closed-loop systems and ensure the preset quality of transient processes of motions and interaction forces. However, for the control laws stabilizing the desired interaction force with preset quality of transient process only sufficient conditions for the exponential stability of motion are given. In the cases when the environment dynamics can be approximated sufficiently well by a linear time-invariant model in Cartesian space, necessary and sufficient conditions for the exponential stability of both motion and force will be derived and the corresponding control laws will be defined. In addition to the schemes that use adaptive control laws, the problem arising from the uncertainties of parameters may also be resolved by applying the knowledge-based techniques. Taking into account external perturbations that do not expire with time and model uncertainties, it may be difficult to achieve asymptotic (exponential) stability. Therefore, it is of practical interest to demand less restrictive stability conditions, i.e. to consider the so-called practical stability of the system. The practical stability of the robot around the desired position and force trajectories is defined by specifying the finite regions around the desired position and force trajectories within which the robot position and interaction force have to be during the task execution. Practical stability tests will be demonstrated in this chapter using two very representative control laws. 2.2 Dynamic Environments Control of commercial robotic systems in free space does not usually require so accurate knowledge of the system’s dynamics since simple servo controllers are often capable of ensuring satisfactory positioning of industrial robots. As already mentioned, however, to control the robots involving constrained motion, sophisticated and accurate models of the entire system, encompassing the robotic mechanism and its environment, need to be studied carefully. During the execution of a contact task, the kinematic structure of the robot changes from an open to closed kinematic chain. The contact with the environment imposes some kinematic and dynamic constraints on the motion of the robot end-effector. One of the most difficult aspects of dynamic modeling is related to the interactions with the bodies in contact.

A Unified Approach to Dynamic Control of Robots

81

2.2.1 Model of a dynamic environment The reaction force is influenced not only by the robot’s motion, but also by the very nature of the environment. Since the mechanical interaction process is generally very complex and difficult to describe mathematically in an exact way, we are compelled to introduce certain simplifications, and thus partly idealize the problem. An ideal frictionless environment, imposing pure kinematic constraints on the end-effector motion is characterized by a static force balance at the contact. On the other hand, the interaction forces in contact with a dynamic environment are not completely compensated for by the constraint reactions. These forces produce active work on the environment, causing also its motion. The relationship between the interaction force and environment dynamics is usually described by nonlinear second-order ordinary differential equations:

M ( s )ɺɺ s + L( s, sɺ) = ϕ sT ( s ) F

(2.1)

and by algebraic equations connecting the end-effector and environment coordinates

x = ϕ ( s ) ; s = ϕ −1 ( x) xɺ =

(2.2)

∂ϕ sɺ = ϕ s ( s ) sɺ ∂ sT

where x ∈ R 6

– vector of the end-effector coordinates (position and orientation),

s∈ Rd

ϕ ( s) ∈ R

– vector of the environment coordinates (displacements), 6

– nonlinear kinematic function,

M (s)

– d × d nonsingular mass matrix,

L( s, sɺ) ∈ R d

– nonlinear dynamic function,

F =− F

– 6-dimensional vector of generalized forces acting on the environment,

ϕs (s)

– 6 × d Jacobian matrix,

d

– dimension of the coordinates’ vector s .

82

Dynamics and Robust Control of Robot-Environment Interaction

The general multibody environment topology involves rigid and/or flexible bodies with passive elements, such as inertia, springs and dampers, but without actuators. Furthermore, the friction and contact forces can be included. The case when the environment DOFs are powered by additional actuators leads to a specific redundant control problem, referred to as cooperative manipulation, which has been studied in a number of works (see, for example, [3, 4]). An example may be two cooperating robots handling an object. Depending on the dimension d of the coordinates’ vector s , as well as on the rank of the Jacobian matrix ϕ s , several passive dynamic environment cases can be distinguished. 2.2.1.1 Kinematic-dynamic constraints In effect, a general model of the environment involves geometrical (kinematic) constraints plus dynamic constraints [5]. An example of such dynamic environment would be a robot turning a crank or sliding a drawer (Fig. 2.1), the dynamics of which is relevant to the robot’s motion and cannot be neglected.

Fig. 2.1 Dynamic environment interactions

2.2.1.2 Pure dynamic environment Pure dynamic environment would correspond to an environment consisting of a mechanical system rigidly coupled to the end-effector. Thus, no kinematic

A Unified Approach to Dynamic Control of Robots

83

constraints on the robot’s end-effector motion are imposed, which results in a relatively simple mapping between the robot and environment coordinates. Such environment was considered in [6, 7]. This model may represent an elastodynamic environment, which is in a linearized form (linear impedance model) considered below. Another representation of this general environment model combines a passive mechanical system and real deformable constraints (joints) (Fig. 2.1) along the different DOFs. 2.2.1.3 Linear impedance model The linear impedance model provides a significant simplification of the environment dynamics modeling. In principle, this concept is based on the assumption of small movement in the neighborhood of the equilibrium contact point xe ∈ R 6 describing a contact frame (position and orientation). Thus, the dynamic interaction can be described by the convenient and well-studied linear multi-DOF system model (impedance causality).

M e ( ɺɺ x − ɺɺ xe ) + Be ( xɺ − xɺe ) + K e ( x − xe ) = − F

(2.3)

where M e is a positive-definite environment mass or inertia matrix; Be and K e are semi-definite environment damping and stiffness matrices, respectively. The term

p = x − xe

(2.4)

refers to the end-effector penetration into the environment. Accordingly

p0 = x0 − xe

(2.5)

will be termed as the nominal position, and could be taken to mean a position planning failure due to tolerances, or a desired ingoing into the environment. It should be mentioned that the computation of the penetration vector components associated with rotational DOFs is dependent on the form used to describe the end-effector orientation. If the orientation vector has been applied, the subtraction of orientation subvectors in equations (2.4), (2.5) is symbolic and denotes special algebraic formalisms for subtracting orientation vectors [8].

84

Dynamics and Robust Control of Robot-Environment Interaction

2.3 Synthesis of Control Laws for the Robot Interacting with Dynamic Environment The model of the dynamics of a robot interacting with its environment is described by vector differential equation of the form:

H (q )qɺɺ + h(q, qɺ ) = τ + J T (q ) F

(2.6)

where, q = q (t ) is the n -dimensional vector of the robot generalized coordinates; H ( q ) is the n × n positive definite matrix of inertia moments of the manipulation mechanism; h( q, qɺ ) is the n -dimensional nonlinear function of centrifugal, Coriolis’ and gravitational moments; τ = τ (t ) is the n dimensional vector of the input control; J T (q ) is the n × m Jacobian matrix connecting the velocities of the robot end-effector and velocities of the robot generalized coordinates; F = F (t ) is the m -dimensional vector of generalized forces or generalized forces and moments acting on the end-effector from the environment. The dimension of the vector F can be adopted in the Cartesian coordinates to be 3 or 6. Presently, for simplicity sake and without loss of generality it will be assumed that n = m (in general n ≥ m ). Because the inverse of the matrix H ( q ) exists, let us write the equation of the robot dynamics model (2.6) in the form solved with respect to the higher derivative:

qɺɺ = − H −1 (q ) h(q, qɺ ) + H −1 (q )τ + H −1 (q ) J T (q ) F

(2.7)

and denote the right-hand side of (2.7) by Φ ( q, qɺ , τ , F ) . Then, this equation can be written in the form:

qɺɺ = Φ ( q, qɺ ,τ , F ) , q(t0 ) = q0 , qɺ (t0 ) = qɺ0

(2.8)

which also includes the initial conditions q (t0 ) = q0 , qɺ (t0 ) = qɺ0 . It is assumed that the function Φ satisfies all the conditions needed for the existence and uniqueness of the solution of the system of differential equations (2.8) on [t0 , +∞) . The robot dynamics model (2.6) can also be solved with respect to the control variable τ :

τ = U ( q, qɺ , qɺɺ, F )

(2.9)

85

A Unified Approach to Dynamic Control of Robots

ɺɺ, F ) = H (q)qɺɺ + h(q, qɺ ) − J (q) F . where U ( q, qɺ , q Let us consider now the mathematical model of environment. It represents one of the most complex and delicate problems in the contact tasks control. As we have seen, to describe the environment we should generally use a different set of generalized coordinates, say q e , when the environment has autonomous DOFs. In the case when the environment does not possess displacements that are independent of the robot’s motion (the environment is firmly attached to the robot’s end-effector), the environment dynamics can be described by the nonlinear differential equationa: T

M 1 ( s )ɺɺ s + L1 ( s, sɺ) = − F

(2.10)

s = ϕ (q) or in the frame of generalized coordinates of the robot (joint variables):

M (q )qɺɺ + L(q, qɺ ) = S T (q ) F

(2.11)

where M (q ) is a nonsingular n × n matrix; L( q, qɺ ) is a nonlinear n dimensional vector function; S T ( q ) is the n × n matrix of rank n , i.e. rank ( S ) = n . We also assume that all the mentioned matrices and vectors are continuous functions of their arguments. By using the inverse matrix M, equation (2.11) can be written in the form:

qɺɺ =ψ ( q, qɺ , F )

(2.12)

where ψ ( q, qɺ , F ) = − M (q) L(q, qɺ ) + M (q) S (q) F . We assume that the vector function ψ satisfies all the conditions for the existence and uniqueness of the solution in an initial value problem for the system of differential equation (2.12) on [t0 + ∞) . Let us solve (2.11) with respect to the force F. Due to the fact that rank ( S ) = n , we have: −1

F = ( S T (q) )

−1

−1

T

[ M (q)qɺɺ + L(q, qɺ )]

a For difference’s sake, when the environment possesses autonomous DOFs, we introduce the

coordinates of passive environment instead of the coordinates

qe .

s

86

Dynamics and Robust Control of Robot-Environment Interaction

In the general case ( n ≥ m ), it is supposed that rank ( S ) = m and, instead of the −1 inverse to the matrix S T ( q ) its left inverse matrix S ( q ) S T ( q ) S ( q ) is used. Evidently, S ( q ) S T ( q ) is the Gramm matrix. ɺɺ) , it can be By denoting the right-hand side of this equation with f ( q, qɺ , q written in the form:

(

)

F = f ( q, qɺ , qɺɺ)

(2.13)

where f is a continuous function of its arguments. Any specific force F (t ) by which robot acts on the environment causes a unique motion of the robot q (t ) in this environment for the prescribed initial conditions q (t0 ) = q0 , qɺ (t0 ) = qɺ0 . The solvability property of the model equation (2.11) with respect to the force action presumes that if the robot moves in the environment in conformity with the function q (t ) , then the force:

F (t ) ≡ f ( q(t ), qɺ (t ), qɺɺ(t ) ) is the only force that causes this motion. In the case of contact with the environment the robot control task can be described as the robot’s motion along a programmed (desired) trajectory q p (t ) when a programmed (desired) force of interaction Fp (t ) is acting between the robot and the environment. The desired programmed motion ( PM ) q p (t ) and desired programmed force of interaction ( PFI ) Fp (t ) cannot be arbitrary. The two functions must satisfy the following relation:

Fp (t ) ≡ f ( q p (t ), qɺ p (t ), qɺɺp (t ) ) ,

∀ t ≥ t0

(2.14)

The control goal of the robot interacting with dynamic environment can be formulated in the following way: Synthesize the control τ (t ) for t ≥ t0 so that to satisfy the target conditions:

q (t ) → q p (t ) as t → ∞, F (t ) → F p (t ) as t → ∞

(2.15)

2.3.1 Stabilization of motion with the preset quality of transients Let us describe the deviation of the real motion q (t ) from the programmed one by the transient process:

A Unified Approach to Dynamic Control of Robots

87

η (t ) = q (t ) − q p (t ) It is natural to determine in advance the requirements which the transient process should obey. These requirements allow certain quality of the transient processes to be set in advance. As certain initial perturbations are always present, i.e. η0 =η (t0 ) = q (t0 ) − q p (t0 ) ≠ 0 , these requirements should not refer to a particular function η (t ) , but to the whole family of functions {η (t )} , each of which possesses its concrete value at the initial moment. Therefore, the family of desired transient processes can be given by the vector differential equation [9, 10]:

ηɺɺ = P(η ,ηɺ )

(2.16)

where P is an n -dimensional vector function continuous over the whole set of arguments, such that equation (2.16) has a unique trivial solution η (t ) ≡ 0. Then, the fulfillment of the requirements that have been imposed on the transient process is ensured by the choice of the function P , and the robot control should be designed in such a way that in the absence of any perturbations except for the initial ones, the robot dynamics equation coincides with equation (2.16). We shall a priori adopt that the choice of the function P in (2.16) ensures asymptotic stability in the whole of trivial solution of the system (2.16). The function P can be adopted, for example, in the form: P(η ,ηɺ ) = Γ1ηɺ + Γ2η . Then, (2.16) acquires the form:

ηɺɺ = Γ1ηɺ + Γ2η

(2.17)

where Γ1 and Γ2 are the constant n × n matrices. Let us turn from equation (2.17) to a system of 2n -order differential equations of the form:

xɺ = Γ x

(2.18)

where

0 η  x=  , Γ= n ηɺ  Γ 2

In  . Γ1 

0 n and I n are the respective zero and unit matrices of dimension n × n . Then, the matrices Γ1 and Γ2 can be chosen in such a way that the eigenvalues

88

Dynamics and Robust Control of Robot-Environment Interaction

of the matrix Γ possess negative real parts. In this way asymptotic stability as a whole of the solution of equation (2.18), and thus of equation (2.17), is achieved. The task of stabilizing the PFI Fp (t ) can be tackled in an analogous way, by considering the family of transient processes with respect to force in the form [10]:

µɺ (t ) = Q( µ ) , µ (t ) = F (t ) − F p (t )

(2.19)

and by choosing a continuous vector function Q (Q (0) = 0) of dimension n , such that the asymptotic stability of the trivial solution of µ (t ) ≡ 0 as a whole, is ensured. At this point, a crucial question is whether one can generally achieve that both the perturbed robot’s motion and perturbed robot’s force of interaction with the environment satisfy simultaneously equations (2.16) and (2.19). The answer is negative. Let us explain this. Let τ = τ (t ), t ≥ t0 be a control law, such that the robot’s dynamics equation (2.6) with the control law is equivalent to the reference equation (2.16). Then, it follows from (2.16):

qɺɺ = qɺɺp + P (η ,ηɺ )

(2.20)

On the other hand, by comparing (2.13) and (2.14), we obtain the following relation between the real and programmed force of interaction, irrespective of the control law:

F − Fp = f ( q, qɺ , qɺɺ) − f ( q p , qɺ p , qɺɺp )

(2.21)

or, by taking into account (2.20):

µ (t ) = f (η + q p , ηɺ + qɺ p , qɺɺp + P(η ,ηɺ ) ) − f ( q p , qɺ p , qɺɺp )

(2.22)

This shows that the character of change of µ (t ) is fully defined by the righthand side of (2.22) and depends on the function f , determining the model of the external environment dynamics, on the function P that determines the transient process character of the robot’s motion and on the control τ , via q and qɺ . In this way, µ (t ) is actually a function which cannot be arbitrary and which, being the solution of equation (2.19), is dependent of the function Q that has been chosen in advance, and which is relatively arbitrary. This means that the quality

A Unified Approach to Dynamic Control of Robots

89

of transient processes is controlled only in an implicit way, via the function P , and is strongly dependent on the model of the environment. Let us formulate two alternative questions: Is it possible to choose such control law that would attain the control goals, defined by the relations (2.15), and at the same time ensure the preset quality of robot’s motion, specified by (2.16)? Is it possible to choose a control law that would ensure the preset quality of mutual force interaction between the robot and environment, defined by (2.19), and also attaining of the control goals? If the answer to the first question is positive (which will be discussed in the text to follow), the answer to the second question depends on the environment dynamics model, and this is going to be considered in the next section. Let us synthesize the control law τ (t ) in such a way to ensure the desired quality of the robot’s motion (2.16). Then the relation (2.20) has to be satisfied in the closed loop. This can be achieved by adopting the control law that has feedback loops with respect to q, qɺ and F [6]:

τ = H (q )  qɺɺp + P(η ,ηɺ )  + h ( q, qɺ ) − J T (q ) F

(2.23)

By substituting (2.23) into the robot’s dynamics model (2.6), we obtain the equation of a closed-form system:

H (q )  qɺɺ − qɺɺp − P (η ,ηɺ )  = 0

(2.24)

Because H ( q ) is a positive definite matrix, det H ≠ 0 , and consequently, (2.24) is equivalent to (2.16). Due to the property of the robot’s function P , which ensures asymptotic stability of the system (2.16) as a whole with respect to (η ,ηɺ ) , we will have:

η (t )  → 0 and ηɺ (t )  →0 t →∞ t →∞

(2.25)

and, since P is a continuous function and P (0, 0) = 0 , it follows that

ηɺɺ (t )  →0 t →∞

(2.26)

By taking into account the previously obtained relations (2.21) or (2.22), and also the continuity of the function f , we obtain:

µ (t )  →0 t →∞

(2.27)

90

Dynamics and Robust Control of Robot-Environment Interaction

Thus, the control goal is achieved, i.e. the real robot’s motion is close to the programmed motion and simultaneously the real force F (t ) is approaching the programmed force of interaction F p (t ) between the robot and environment. Let us consider one more control law of the robot motion stabilization without utilizing force feedback when the interaction force between the robot and environment is exists. For this purpose, let us modify the control law (2.23) using the relations (2.13) and (2.20). We obtain:

τ = H (q )  qɺɺp + P(η ,ηɺ )  + h(q, qɺ ) − J T (q ) f ( q, qɺ , qɺɺp + P(η ,ηɺ ) ) (2.28) Let us show that the control law (2.28), with the feedback loops only with respect to q and qɺ , solves the task of stabilizing PM (2.25), (2.26) and, consequently, in accordance with (2.27), the control task itself under the condition of the robot’s contact with the environment and under some supplementary assumption. By substituting the control (2.28) into the robot’s dynamics model (2.6) we obtain:

(

H (q ) [ηɺɺ − P (η ,ηɺ ) ] = J T (q ) F − f ( q, qɺ , qɺɺp + P (η ,ηɺ ) )

)

(2.29)

Taking into account the equality (2.13) we have:

(

H (q ) [ηɺɺ − P (η ,ηɺ ) ] = J T (q ) f ( q, qɺ , qɺɺ) − f ( q, qɺ , qɺɺp + P (η ,ηɺ ) )

)

(2.30)

or, by taking into account the form of the function f : −1

H (q ) [ηɺɺ − P (η ,ηɺ ) ] = J T (q ) ( S T (q ) ) M (q ) [ηɺɺ − P (η ,ηɺ )]

(2.31)

Therefore

( H (q) − J (q) ( S (q) ) T

T

−1

)

M (q ) [ηɺɺ − P (η ,ηɺ ) ] = 0

(2.32)

Let us suppose that for all q , i.e. for an arbitrary robot configuration, the supplementary condition −1 det  H (q ) − J T (q ) ( S T (q ) ) M (q )  ≠ 0  

(2.33)

is fulfilled. Then, the equation of the closed-loop system (2.32) is equivalent to the reference equation (2.16). Although the control law (2.28) does not use

91

A Unified Approach to Dynamic Control of Robots

force feedback, it is explicitly dependent on the environment dynamics:

(

τ = U q, qɺ , qɺɺp + P(η ,ηɺ ), f ( q, qɺ , qɺɺp + P(η ,ηɺ ) )

)

(2.34)

through the function f , and cannot be used to control the manipulator in free space. 2.3.2 Stabilization of interaction force with the preset quality of transients As shown in the preceding section, the answer to the first question was quite simple: the inverse dynamics method ensures achievement of a desired motion quality and at the same time guarantees stability of the interaction force. Let us consider now the issue of stabilizing the force of interaction of the robot with the environment. Let us discuss first several control laws ensuring the solution of this task. For convenience, when describing the quality of transient processes with respect to force (2.19), we shall use an equivalent relation of the form [6, 10, 11]: t

µ (t ) = µ0 + ∫ Q ( µ (ω ) ) d ω

(2.35)

t0

Without loss of generality we can adopt that µ0 = 0 , since the stabilization of µ in the sense of preset quality (2.35) implies the stabilization according to the preset quality (2.19). Let us consider the control law with the feedback with respect to q, qɺ and F , of the form: t    T τ = H ( q ) M (q )  − L (q, qɺ ) + S ( q )  Fp + ∫ Q ( µ (ω ) ) d ω      t0    + h(q, qɺ ) − J T (q ) F −1

(2.36)

By substituting this control law into the robot’s dynamics model (2.6) we obtain the following law of robot’s functioning in the contact with the environment: t   M (q )qɺɺ + L(q, qɺ ) = S (q )  Fp + ∫ Q ( µ (ω ) ) d ω    t0   T

(2.37)

92

Dynamics and Robust Control of Robot-Environment Interaction

Taking into account the environment dynamics model (2.11) we obtain the following closed-loop control system: t  S T (q )  µ (t ) − ∫ Q ( µ (ω ) ) d ω  t0 

  =0  

and, because rank ( S ) = n , this equation is equivalent to the equation: t

µ (t ) = ∫ Q ( µ (ω ) ) dω

(2.38)

t0

wherefrom, equation (2.19) follows right away. In this way the control law (2.36) ensures a desired quality of stabilization of PFI Fp (t ) . Let us note that the control law (2.36) can be rewritten in a more compact form as: t     τ = U  q, qɺ ,ψ  q, qɺ , Fp + ∫ Q ( µ (ω ) ) dω  , F      t0    

(2.39)

Let us consider another control law of the form t   τ = U  q, qɺ ,ψ (q, qɺ , F ), Fp + ∫ Q ( µ (ω ) ) dω  (2.40)   t0   involving feedback loops with respect to q, qɺ and F , and let us demonstrate that it also ensures a desired quality of stabilizing PFI Fp (t ) . In a developed

form, this control law can be written as

τ = H (q) M −1 (q)  − L(q, qɺ ) + S T (q) F  + h(q, qɺ ) − t   − J ( q)  Fp + ∫ Q ( µ (ω ) ) d ω    t0   T

(2.41)

Substituting τ from (2.41) into (2.6) we obtain the closed-form equation:

H (q )qɺɺ = H (q ) M −1 (q )  − L(q, qɺ ) + S T (q ) F  t   + J T (q )  µ − ∫ Q ( µ (ω ) ) d ω    t0  

(2.42)

93

A Unified Approach to Dynamic Control of Robots

Taking into account that for the contact of the robot with environment the equation (2.11) holds, i.e.

− L(q, qɺ ) + S T (q ) F ≡ M (q )qɺɺ the relation (2.42) is equivalent to the relation: t  J (q )  µ − ∫ Q ( µ (ω ) ) d ω  t0  T

  =0  

Assuming that rank J ( q ) = n for all robot configurations {q} , we obtain a closed-form equation of the system in the form (2.35). The requirement rank J ( q ) = n is quite natural if it is borne in mind that from the point of view of physics this means that the magnitude of the force of interaction of the robot (2.6) with the environment (2.11) should be uniquely determined by the function of control and by the robot’s motion. Moreover, it can be stated that in the case of robot’s contact with environment no configurations {q} for which rank J ( q ) < n can arise, as this would mean that the interaction force satisfying the equation of environment’s dynamics (2.13) would not be uniquely determined by the function of motion q (t ). Finally, let us consider one more control law stabilizing PFI Fp (t ) with a preset quality of transient processes, using no explicit force feedback, but the implicit one, via the integral of the difference F (t ) − F p (t ). Let the control law be given in the form:



  

t

  

t



τ = U  q, qɺ ,ψ  q, qɺ , Fp + ∫ Q ( µ (ω ) ) dω  , Fp + ∫ Q ( µ (ω ) ) dω   

t0

t0

 

(2.43)

or, in a more developed form, as t t      T T τ = HM  − L + S  Fp + ∫ Q( µ )d ω   + h − J  Fp + ∫ Q( µ )dω  .     t0 t0    

−1

The closed form of the control system is then t t      Hqɺɺ = HM −1  − L + S T  Fp + ∫ Q( µ )dω   + J T  µ − ∫ Q( µ )dω       t0 t0     

(2.44)

94

Dynamics and Robust Control of Robot-Environment Interaction

ɺɺ from equations (2.44) and (2.12) we obtain By eliminating q t t     − L + S T F = − L + S T  Fp + ∫ Q( µ )d ω  + MH −1 J T  µ − ∫ Q( µ )d ω      t0 t0    

or

t  S T  µ − ∫ Q( µ )dω  t0 

t    −1 T = MH J µ − Q( µ ) d ω  .   ∫    t0   

Wherefrom, t  −1 T T S − MH J µ − ( )  ∫ Q(µ )dω t0 

  =0  

(2.45)

Let us set the supplementary requirement as

rank  S T (q ) − M (q ) H −1 (q ) J T (q )  = n

(2.46)

for all possible robot configurations {q} . Then, the relation (2.45) will be identical to (2.19) and, consequently, under the supplementary assumption (2.46), the control law (2.43) also ensures stabilization of the PFI Fp (t ) with a preset quality of transient processes. Let us notice that because of the continuity of the robot’s motion, the violation of the conditions (2.46) over a finite or countable set of points does not entail violation of the closed-loop control functioning of the form (2.38). The violation of this condition may generally arise only if the inequality

rank  S T (q ) − M (q ) H −1 (q ) J T (q )  < n has over some time interval twice continuously differentiable solution q (t ), t ∈ (ta , tb ), ta < tb . This issue requires additional investigations of dynamics models of the robot and environment. However, attention has to be paid to the possibility of appearance of the regimes analogous to sliding regimes in the tasks of dynamic objects control [12]. So, we have synthesized three control laws, (2.39), (2.40), and (2.43), each of which ensures a desired quality of stabilizing the PFI Fp (t ) . Now let us consider the question of whether there is a sufficient condition (and what it should be) for the convergence of the real motion q (t ) to the

A Unified Approach to Dynamic Control of Robots

95

programmed one q p (t ) , when asymptotic stability of the contact force takes place. To discuss this issue, let us return to the environment dynamics model (2.11) and, by taking into consideration the relation between PM q p (t ) and PFI

FP (t ) in the form (2.14), we obtain: M ( q) qɺɺ − M (q p ) qɺɺp + L(q, qɺ ) − L( q p , qɺ p ) −  S T ( q) − S T (q p )  Fp = S T ( q) ( F − Fp ) or in the deviation form

M (η + q p )ηɺɺ +  M (η + q p ) − M (q p )  qɺɺp −  S T (η + q p ) − S T ( q p )  Fp

+ L(η + q p ,ηɺ + qɺ p ) − L(q p , qɺ p ) = S T (η + q p ) ( F − Fp )

(2.47)

Let us introduce the notation:

K (η ,ηɺ, t ) = M −1 (η + q p ) {L(η + q p ,ηɺ + qɺ p ) − L(q p , qɺ p ) +  M (η + q p ) − M (q p )  qɺɺp −  S T (η + q p ) − S T (q p )  Fp } Then, the relation (2.47) can be written in a more compact form

ηɺɺ + K (η ,ηɺ, t ) = M −1 (η + q p ) S T (η + q p ) ( F − Fp )

(2.48)

Let us note that the continuous vector function K satisfies the property K (0, 0, t ) ≡ 0, ∀t ≥ t0 . Consider the system of differential equations:

ηɺɺ + K (η ,ηɺ , t ) = 0

(2.49)

Evidently, η (t ) ≡ 0 is the trivial solution of equation (2.49). In fact, this equation represents the environment dynamics equation (2.12) written in the form of deviations for F (t ) = F p (t ). It is clear that the environment dynamics should satisfy the property of asymptotic stability (desirable in the whole) of the trivial solution of the system (2.49) which, along with the stability of η (t ) , should ensure fulfillment of the limiting condition:

η (t ) → 0 as t → ∞

(2.50)

This means that in the most favorable case of robot control, when the real interaction force of the robot with environment, F (t ) , coincides all the time

96

Dynamics and Robust Control of Robot-Environment Interaction

with PFI FP (t ) , the real motion q (t ) should be close to q p (t ) , and converge to it when t → ∞ . In the opposite case, F (t ) ≠ F p (t ) , the stabilization of the force of interaction of the robot with environment does not solve the contact task as a whole in a satisfactory way. Let us consider sufficient conditions (in general form) for the stabilization of PM q p (t ) under which asymptotic stability of the solutions of (2.49) entails the asymptotic stability of the perturbed motion η (t ) in (2.48) and the fulfillment of the limiting condition (2.50). For this purpose, assuming the vector function K is continuously differentiable, let us consider the first approximation of this function in the neighborhood of the point (η ,ηɺ ) = (0, 0) :

K (η ,ηɺ , t ) =

∂K (t )ηɺ ∂ηɺ

+

∂K (t )η ∂η

(η ,ηɺ ) = (0,0)

where

2

2

α 0 (η ,ηɺ , t ) = o ( η + ηɺ ) as

+ α 0 (η ,ηɺ , t )

(η ,ηɺ ) = (0,0)

η → 0, ηɺ → 0 . Because of

K (0, 0, t ) ≡ 0, ∀t ≥ t0 it is evident that α 0 (0, 0, t ) ≡ 0, ∀t ≥ t0 . Assuming that the vector function α 0 (η ,ηɺ, t ) is smooth, let us introduce the notation:

x  x1 =η , x2 =ηɺ , x =  1  ,  x2  In  On      ∂K ∂K A(t ) = −  − (t ) (t ) ∂ηɺ  ∂η  (η ,ηɺ ) = ( 0 , 0 ) (η ,ηɺ ) = ( 0 , 0 )    0   0   α ( x, t ) =  , β ( x , t ) =  ,  −1 T α 0 ( x1 , x2 , t )   M ( x1 + q p (t ) ) S ( x1 + q p (t ) )  where A is a continuous 2n × 2n matrix function, α is a smooth 2n vector function, and β is a 2n × n matrix function. The system of differential equations (2.48) can be rewritten in the form:

97

A Unified Approach to Dynamic Control of Robots

xɺ = A(t ) x + α ( x, t ) + β ( x, t ) µ (t )

(2.51)

where α ( x, t ) = o ( x ) as x → 0 , α (0, t ) ≡ 0, ∀t ≥ t0 and sup A(t ) < ∞ t

because of q p (t ) ∈ Vq , qɺ p (t ) ∈ Vqɺ , Fp (t ) ∈VF . Here, Vq , Vqɺ , VF are some known bounded sets. This system is the one with the perturbation β ( x, t ) µ (t ) for the system of differential equations (2.49) which, with the adopted notation, has the form:

xɺ = A(t ) x + α ( x, t )

(2.52)

Here, x(t ) ≡ 0 is the trivial solution of (2.52). Sufficient conditions for asymptotic stability of the system of differential equation (2.51), together with the fulfillment of the condition x(t ) → 0 as t → ∞ , are given by the following theorem. Theorem 1. Let the environment dynamics satisfy the following conditions: 1. The first approximation system:

xɺ = A(t ) x

(2.53)

is regular [13], i.e. there exists the limit t

2n 1 Sp ( ) A ω d ω = σ and σ = αk , ∑ 0 0 t →∞ t ∫ k =1 t0

lim

where α k ( k = 1, 2,… , 2n) are characteristic indicesb of the solutions of the system (2.53), and Sp A is the trace of matrix A ; 2. All characteristic indices α k ( k = 1, 2,… , 2n) are negative.

t ∈ [t0 , + ∞) then the number 1 (or the symbols +∞ or −∞ ) defined by the formula χ [ f ] = lim ln f (t ) is called x →∞ t Lyapunov’s characteristic index (shortly the characteristic index) of the function f .

b If

f

is the complex-valued function of the real variable

98

Dynamics and Robust Control of Robot-Environment Interaction

Let equation (2.19) that define the quality of transient processes with respect to interaction force be such that the following estimate holds for an arbitrary solution µ (t ) :

µ (t ) ≤ C2e − λ (t −t ) µ (t0 ) 0

(2.54)

with the positive constant C2 and index λ > 0 . Let the number γ satisfy the inequalities:

max α k < −γ < 0 k

−λ < −γ

(2.55)

sufficiently small initial perturbations x(t0 ) and µ (t0 ) = F (t0 ) − Fp (t0 ) , the transient processes defined by the system of differential equations (2.51) with respect to motion and its derivative behave according to the inequality Then,

with

 b µ (t0 )  − γ (t −t0 ) , ∀t ≥ t0 x(t ) ≤  a x(t0 ) + e λ −γ   with the positive constants a and b which consequently fulfill the limiting condition:

x(t ) → 0 as t → ∞

(2.56)

which represents the exponential stability of the system (2.51) under the condition of small initial perturbations. The proof of Theorem 1 is given in Appendix A. In a sense, this theorem represents the Lyapunov’s generalized theorem on stability in the first approximation for a system with perturbations [13, 14]. In this way, with the conditions of the theorem being fulfilled, the control laws (2.39), (2.40), (2.43) ensure a desired quality of stabilization of PFI FP (t ) and also of PM q p (t ) , due to the fulfillment of the following conditions, comes out from (2.56):

η (t ) → 0, ηɺ (t ) → 0, as t → ∞ . Let us notice that conditions 1 and 2 of Theorem 1 define the property of the environment which might be called “internal stability”. Besides, the stabilization

A Unified Approach to Dynamic Control of Robots

99

of PM in the case of using force stabilization control laws takes place only if the rate of stabilization of force interaction λ is higher than the guaranteed rate γ (inequality (2.55)) of stabilization of motion. Here, sufficient conditions for the exponential stability of the first approximation of the system (2.51) under condition of small perturbations are given by conditions 1 and 2. If these conditions are fulfilled, the system will be stable. It has to be emphasized that the fulfillment of the conditions of asymptotic stability depends not only on the environment dynamics but also on the programmed displacement (motion). For instance, let us adopt a nonlinear model of environment dynamics with the nonlinearity in the form of a cubic spring:

F (t ) = me ɺɺ x(t ) + be xɺ (t ) + k1 x(t ) + k2 x3 (t )

(2.57)

where me > 0 is a parameter representing the equivalent environment mass;

be > 0 is the corresponding environment damping; k1 ≥ 0 and k2 > 0 are the environment stiffness coefficients. According to equation (2.51), the deviation form of (2.57) becomes: 2

k + 3k2 x p (t ) b ηɺɺ(t ) = − e ηɺ (t ) − 1 η (t ) − me me

(2.58)

1 k − 2 (η 3 (t ) + 3η 2 (t ) x p (t ) ) + µ (t ) me me It is obvious that

k2 3 (η (t ) + 3η 2 (t ) x p (t ) ) = o( η me

1+ε

) as η → 0 .

Here 0 < ε < 1 . For simplicity let us suppose that desired displacement x p (t ) = x p = const . According to equation (2.57), the desired force of interaction must be equal to Fp (t ) = Fp = k1 x p + k2 x3p . The characteristic polynomial of the linear part of the system (2.58) is given by 2

k1 + 3k2 x p b det( sI − A) = s + e s + me me 2

(2.59)

100

Dynamics and Robust Control of Robot-Environment Interaction

where

0   A =  k1 + 3k2 x 2p −  me

1   be  . − me 

The eigenvalues of the characteristic equation corresponding to polynomial (2.59) are

s1,2 =

−be ± be2 − 4me (k1 + 3k2 x 2p ) 2me

.

Let the parameters be , me , k1 of the cubic spring meet the condition b < 4me k1 . Then, independently of desired displacement x p , characteristic indices of the linear part of the system (2.58) : α1 = Re s1 , α 2 = Re s2 will be negative. Therefore the second condition of Theorem 1 is fulfilled. Let us consider the first condition of Theorem 1. It is also fulfilled because: 2 e

t

t

∫

∫

0

0

b b 1 1  be  lim Sp( A) dω = lim −  dω = − e and α1 + α 2 = − e  t→∞ t t→∞ t  m  me me e t t

(2.60)

Based on Theorem 1 we can conclude that any control law asymptotically stabilizing the desired force of interaction Fp for the cubic spring considered guarantees asymptotic stabilization of the corresponding desired displacement x p as well. 2.3.3 Concluding discussion We have focused our attention on the problem of position stabilization when asymptotic stability of the contact force was ensured. This task is a basic issue in the control of a robot interacting with its dynamic environment. The theorem ensuring asymptotically stable position of the system in the first approximation (local stability) formulates sufficient conditions under which such stability is achieved. It should be emphasized that the character of this position (displacement) stability depends particularly on the nature of programmed motion. Nevertheless, it should be pointed out that the presented linear analysis gives a very important criterion that must be verified for any force-based control

A Unified Approach to Dynamic Control of Robots

101

law. However, if the environment model (2.11) is not known to a sufficiently accurate degree, it is not possible to determine the nominal contact force F p (t ) . Besides, an insufficiently accurate environment dynamics model can significantly affect the contact task execution. The inaccuracies of the robot and environment dynamics models, as well as dynamic control robustness, are considered in [6, 15-17]. The problems arising from the parametric uncertainties may also be resolved by applying the knowledge-based techniques (fuzzy logic and neural networks) [18]. Taking into account external perturbations, which do not expire with time, as well as the model and parameter uncertainties, it may be difficult to achieve asymptotic (exponential) stability of the system unless robust and adaptive control laws that include a factor for compensating these perturbations and uncertainties are used. Therefore, it may be of practical interest to require more relaxed stability conditions, i.e. to consider the so-called practical stability of the robot around the desired position and force trajectories by specifying the finite regions around them within which the robot’s actual position and force have to be during the task execution, and by assuming that the inaccuracies of the model parameters (of both the robot and environment) are bounded. The conditions for practical stability of the robot interacting with dynamic environment enable study of the model uncertainty issue in control of robots in this class of tasks without any approximation, i.e. to correctly examine the influence of these uncertainties upon different control laws. The test conditions for practical stability of the robot interacting with dynamic environment have been derived in [19, 20]. A wider discussion of stability of the robot interacting with dynamic environment is given in Section 2.10. 2.4 Analysis of Transient Processes In this section we give the estimates of transient processes with respect to both the robot’s motion and its force of interaction with the environment. Here we shall formulate sufficient conditions for the fulfillment of the constraints imposed on the state, control, and interaction force, taking into account the inadequacy of the dynamics models of the robot and environment and/or external perturbations.

102

Dynamics and Robust Control of Robot-Environment Interaction

2.4.1 Task setting Dynamics model of the robot interacting with the environment is described by the vector differential equation (2.6). This equation can be written in the form (2.8). Let the real robot dynamics be written in the form of the vector differential equation:

qɺɺ = Φ ( q, qɺ ,τ , F ) + r (t )

(2.61)

where the n -dimensional continuous vector function r (t ) represents the inaccuracy of robot’s dynamics description and/or the uncontrollable external perturbations. We shall assume that the vector function Φ satisfies conditions sufficient for the existence and uniqueness of the solution for equations (2.8) and (2.61) with the initial conditions q (t0 ) = q0 , qɺ (t0 ) = qɺ0 . Let the dynamics model of the environment with which robot is interacting be described by the nonlinear differential equation (2.11). Let the real environment dynamics be described by the vector differential equation:

M (q)qɺɺ + L(q, qɺ ) = S T (q) [ F − ρ (t ) ]

(2.62)

where the n -dimensional vector function ρ (t ) represents the inaccuracy of describing environment dynamics by the mathematical model (2.11) and/or the uncontrollable external perturbations of the environment. The real environment can be represented by the equation:

F = f ( q, qɺ , qɺɺ) + ρ (t )

(2.63)

The goal of the control of the robot in contact with its environment is to realize the programmed motion ( PM ) q p (t ) in the presence of the programmed force acting on the environment. Namely,

Fp (t ) = f ( q p (t ), qɺ p (t ), qɺɺp (t ) ) Since

some

initial

perturbations

are always present, i.e. q (t0 ) ≠ q p (t0 ), qɺ (t0 ) ≠ qɺ p (t0 ), F (t0 ) ≠ Fp (t0 ), the control goal is transformed into the task of stabilizing PM q p (t ) :

A Unified Approach to Dynamic Control of Robots

103

q (t ) → q p (t ), qɺ (t ) → qɺ p (t ), as t → ∞ and the task of stabilizing PFI Fp (t ) :

F (t ) → F p (t ), as t → ∞ . In Section 2.3, these two tasks are solved for the case of the absence of the model’s inaccuracy and external perturbations r (t ) and ρ (t ) . Because of the uncontrollable behavior of the introduced functions r (t ) and ρ (t ) , the control goal will be formulated in the form of the following two goal conditions: 2

2

q (t ) − q p (t ) + qɺ (t ) − qɺ p (t ) < ε 2

(2.64)

F (t ) − Fp (t ) < δ

(2.65)

where ε , δ > 0 are the given numbers. These goal conditions should be fulfilled for any t ≥ t p , where t p is a time instant, t p − t0 is the time interval of the transient process. Let us note that the goal condition (2.64) is equivalent to the inequality:

x(t ) − x p (t ) < ε ,

(

T

)

(

T

)

where x(t ) = qT (t ), qɺ T (t ) , x p (t ) = qTp (t ), qɺ Tp (t ) . Here, and in the text to follow, the norm ⋅ denotes the Euclidean norm. The robot’s motion q (t ), qɺ (t ), the control action τ (t ) , and the force of interaction of the robot with environment F (t ) , are usually constrained by the conditions of the technological task to be performed. Let us preset these constraints in the form of the following relations:

q(t ) ∈ Vq ⊂ R n , ∀t ≥ t0 qɺ (t ) ∈ Vqɺ ⊂ R n , ∀t ≥ t0 n

τ (t ) ∈ Vτ ⊂ R , ∀t ≥ t0 n

F (t ) ∈ VF ⊂ R , ∀t ≥ t0

(2.66) (2.67) (2.68) (2.69)

where Vq , Vqɺ , Vτ , V F are the prescribed, open, bounded, and simply connected sets in the corresponding spaces, and Vτ is the closure of the set Vτ in R n . We will assume that the levels of inaccuracy of the robot and environment dynamics models and/or external perturbations are bounded, i.e.

104

Dynamics and Robust Control of Robot-Environment Interaction

r (t ) ≤ Cr , ∀t ≥ t0 ,

(2.70)

ρ (t ) ≤ Cρ , ∀t ≥ t0

(2.71)

Under these conditions, the control task can be formulated in the following way. For the complete equations of the robot model (2.61) and the external environment (2.62), including the disturbances r (t ) and ρ (t ) satisfying the constraints (2.70) and (2.71), the admissible control action τ (t ), t ≥ t0 , i.e. the action satisfying the constraint (2.68), should be synthesized in such a manner that the real motion q (t ), qɺ (t ) satisfies the constraints (2.66) and (2.67), the generated interaction force satisfies the constraint (2.69), and the goal conditions (2.64), (2.65) are fulfilled starting from the time instant t p . 2.4.2 Motion transient processes In Section 2.3.1, two control laws were synthesized that stabilize robot’s motion in accordance with the given quality of transient processes (2.16), (2.17). The first control law (2.23) has a more general character, ensuring the solution of the motion stabilization task q p (t ) not only when the robot is in contact with the environment but also in a free space. The second control law, (2.28), is applied only when the robot is in contact with the environment. The control law (2.28), in contrast to (2.23), does not use force feedback. There is a constraint on the applicability of this control law, i.e. the condition (2.33) has to be satisfied for all possible configurations {q} of the robot in contact with the environment. Both control laws considered ensure also a simultaneous stabilization of F p (t ) , but the quality of this stabilization cannot be set a priori in an arbitrary way. In this section the classes of the programmed motions q p (t ) and

{

programmed interaction forces

{F (t )} p

}

for which the control laws (2.23) and

(2.28) solve the control tasks (2.64) and (2.65) in the presence of the constraints (2.66) – (2.69), will be described. Stability of these control laws under external perturbations of the robot’s and environment dynamics, or subject to their

105

A Unified Approach to Dynamic Control of Robots

inaccuracies, will be demonstrated, and the estimate of transient process time in the case of motion and force stabilization will be presented. Let us consider the task of robot control subject to the constraints (2.66) – (2.69). The PM q p (t ) and PFI Fp (t ) have to satisfy the constraints (2.66), (2.67) and (2.69), i.e. the conditions:

q p (t ) ∈ Vq , ∀t ≥ t0

(2.72)

qɺ p (t )∈ Vqɺ , ∀t ≥ t0

(2.73)

Fp (t )∈ VF , ∀t ≥ t0

(2.74)

must be fulfilled. For the existence of the control task solution it is necessary to demand the control:

τ p (t ) = U ( q p (t ), qɺ p (t ), qɺɺp (t ), Fp (t ) )

(2.75)

be admissible. Let us suppose that there exists such a bounded open set Vqɺɺ in R n , that:

U : Vq × Vqɺ × Vqɺɺ × V F → Vτ

(2.76)

Then, if PM q p (t ) satisfies the constraints (2.72), (2.73) and the condition:

qɺɺp (t ) ∈ Vqɺɺ ⊂ R n , ∀t ≥ t0 and PFI

(2.77)

Fp (t ) satisfies (2.74), then, in accordance with (2.76), the control

τ p (t ) = U ( q p , qɺ p , qɺɺp , Fp ) will be admissible for any t ≥ t0 . Consequently, the existence of the set Vqɺɺ that ensures fulfillment of the relation (2.76) is a natural sufficient condition for the existence of the solution of the task of robot control in contact with environment in the presence of the constraints (2.66) – (2.69). In view of the fact that some initial perturbations are always present, the exact realization of PM q p (t ) and PFI Fp (t ) is not possible. Hence, we shall require PM q p (t )

together with its first two derivatives qɺ p (t ) and

qɺɺp (t ), to be distant from the bounds of the corresponding sets Vq , Vqɺ , Vqɺɺ for some “reserves” δ1 , δ 2 , δ 3 > 0 ; i.e.

106

Dynamics and Robust Control of Robot-Environment Interaction

q p (t )∈ Vqδ1 ⊂ Vq , ∀t ≥ t0

(2.78)

qɺ p (t ) ∈ Vqɺδ 2 ⊂ Vqɺ , ∀t ≥ t0

(2.79)

qɺɺp (t ) ∈ Vqɺɺδ3 ⊂ Vqɺɺ ,

(2.80)

∀t ≥ t0

and, PFI Fp (t ) be distant from the bound of the set V F for a “reserve” σ > 0 , i.e.:

Fp (t ) ∈ VFσ ⊂ VF , ∀t ≥ t0

(2.81)

Here, Vqδ1 ,Vqɺδ 2 , Vqɺɺδ 3 ,VFσ respectively, are the δ1 , δ 2 , δ 3 and σ narrowings of the regions Vq ,Vqɺ , Vqɺɺ , VF . The ε -narrowing of the open set A in R n is defined as a such non-empty subset Aε that:

A = ∪ Bε ( x) x∈ Aε

where Bε (x) is an open solid sphere in R n with the center at the point x and the radius ε . For instance, if A = Bc ( x0 ) is an open solid sphere in R n , with the center at the point x0 and the radius c > ε , then: Aε = Bc −ε ( x0 ). Let the deviations of the real robot motion q (t ) and its derivatives qɺ (t ) and

qɺɺ(t ) from q p (t ), qɺ p (t ), qɺɺp (t ) , in the sense of Euclidean norm, be not greater than δ1 , δ 2 , δ 3 respectively, and let the deviation of the real interaction force

F (t ) of the robot in contact with the environment from Fp (t ) be not greater than σ . Then, the real robot motion and the real interaction force will satisfy the constraints (2.66), (2.67), (2.77) and (2.69). Let us consider the control laws (2.23) and (2.28) that stabilize PM q p (t ). We will suppose that the function P , determining the family of transient processes (2.16), is of the form (2.17):

P(η ,ηɺ ) = Γ1ηɺ + Γ2η As the real robot’s dynamics is described by (2.61) and the real environment’s dynamics by (2.62), the closed-loop system functioning with the control laws (2.23) and (2.28) applied, in contrast to (2.17), will have the form:

ηɺɺ = Γ1ηɺ + Γ2η +

(t )

(2.82)

107

A Unified Approach to Dynamic Control of Robots

where (t ) = r (t ) in the case when the control law (2.23) is applied, and

(t ) = D1 ( q(t ) ) r (t ) + D2 ( q(t ) ) ρ (t ) in the case of applying the control law (2.28). Here −1

D1 (q ) =  H (q ) − J T (q ) ( S T (q )) −1 M (q )  H (q ) −1

D2 (q ) =  H (q ) − J T (q ) ( S T (q )) −1 M (q )  J T (q ) Let all eigenvalues λ1 , λ2 ,..., λ2 n of the matrix

0 Γ = n Γ 2

In  Γ1 

be real, different and negative. Let us estimate the solution of the system (2.82). For that purpose let us rewrite it in the form of a 2n -dimensional system of differential equations:

xɺ = Γx + ɶ ( t ) η 

(2.83)

 0   . Then, the solution (2.83) can be written in the  (t ) 

where x =   , ɶ (t ) =  ɺ

η 

following form: t

x(t ) = eΓ (t −t0 ) x(t0 ) + ∫ eΓ (t −ω ) ɶ (ω )d ω

(2.84)

t0

Let T be a non-singular transformation that transforms the matrix Γ into its diagonal form:

T −1Γ T = diag(λ1 , λ2 ,..., λ2 n ) Then

(

)

eΓ (t −ω ) = T diag eλ1 ( t −ω ) , eλ2 (t −ω ) ,..., eλ2 n (t −ω ) T −1 and consequently: 1/ 2

e

Γ ( t −ω )

≤ T T

−1

 2 n 2 λi ( t −ω )  ∑e   i =1 

≤ T T −1

2n max eλi ( t −ω ) = CΓ e − λ (t −ω ) i

108

Dynamics and Robust Control of Robot-Environment Interaction

where

C Γ = 2n T T −1 , λ = − max λi . i

By using the obtained inequality, let us estimate by norm the right-hand side , and of equation (2.84), taking into account that ɶ =

(t ) ≤ C ,

∀t ≥ t0 ,

where C = Cr , if the control law (2.23) is used, and:

C = d1Cr + d 2Cρ if the control law (2.28) is applied. Here d1 = sup D1 (q) , d 2 = sup D2 (q ) where the suprema are taken over all q ∈ Vq . Let us note that Cr and C ρ are the estimates of perturbation levels given by (2.70) and (2.71). We shall, then, have: t

x(t ) ≤ e

Γ ( t − t0 )

x(t0 ) + ∫ eΓ (t −ω )

(ω ) d ω ≤ CΓ e− λ (t −t0 ) x(t0 ) +

t0

(2.85)

t

+ CΓ C

∫e

− λ ( t −ω )

d ω ≤ CΓ e

− λ ( t −t0 )

x(t0 ) +

CΓ C

λ

t0

This gives: 2

2

2

2

η (t ) ≤ CΓ η (t0 ) + ηɺ (t0 ) e− λ (t −t ) + CΓC λ −1 0

ηɺ (t ) ≤ CΓ η (t0 ) + ηɺ (t0 ) e− λ (t −t ) + CΓC λ −1 0

(2.86) (2.87)

Let us estimate now the transient processes in the case of stabilizing the

PFI Fp (t ). As the real environment dynamics is described by (2.62) and the functioning of the closed-loop control system proceeds according to (2.82), the difference between the real and programmed interaction force

µ (t ) = F (t ) − F p (t ) satisfies the relation

109

A Unified Approach to Dynamic Control of Robots

µ (t ) = f ( q, qɺ , qɺɺp + Γ1ηɺ + Γ 2η + (t ) ) + ρ (t ) − f ( q p , qɺ p , qɺɺp )

(2.88)

We will assume that the function f , describing the environment dynamics, satisfies the Lipschitz conditions with the constants Li (i = 1, 2,3) for any i -th variable on the set of Vq × Vqɺ × Vqɺɺ . Then, from (2.88) we will have

µ (t ) ≤ L1 η + L2 ηɺ + L3 Γ1ηɺ + Γ 2η + (t ) + ρ (t ) ≤ ( L1 + L3 Γ 2

) η +(L

2

+ L3 Γ1

) ηɺ

+ L3C + Cρ

By using the estimates (2.86) and (2.87), from the last inequality we obtain: 2

2

µ (t ) ≤ L CΓ η (t0 ) + ηɺ (t0 ) e− λ (t −t ) + L CΓC λ −1 + L3C + Cρ where

0

(2.89)

L = L1 + L2 + L3 ( Γ1 + Γ2 ).

The obtained estimates of transient processes (2.86), (2.87), (2.89) show that sufficiently small inaccuracy levels of the robot and environment models, and/or of the external perturbations, C r and C ρ , after a certain time of transient processes, ensure fulfillment of the goal conditions (2.64) and (2.65). If, in addition, the initial perturbations are sufficiently small, the a priori constraints (2.66) – (2.69) can be fulfilled, too. More precisely, the following theorem holds.

Theorem 2. Let PM q p (t ), together with its derivatives qɺ p (t ) and qɺɺ p (t ) , satisfy the inclusions (2.78) – (2.80), and let PFI FP (t ) satisfy (2.81). If the initial perturbations η (t0 ), ηɺ (t0 ) and the inaccuracy levels of the models and/or external perturbations C r and C ρ satisfy the inequalities: 2

2

CΓ η (t0 ) + ηɺ (t0 ) + CΓ C λ −1 < min {δ1 , δ 2 }

(Γ

1

+ Γ2

)

(C 2

2

Γ

2

)

η (t0 ) + ηɺ (t0 ) + CΓC λ −1 < δ 3 , 2

(2.90) (2.91)

LCΓ η (t0 ) + ηɺ (t0 ) + LCΓ C λ −1 + L3C + Cρ < σ ,

(2.92)

CΓ C λ −1 < ε ,

(2.93)

−1

C ( LCΓ λ + L3 ) + Cρ < δ ,

(2.94)

110

Dynamics and Robust Control of Robot-Environment Interaction

then, for any t ≥ t0 and any control law (2.23) or (2.28): 1. the real robot’s motion q (t ), qɺ (t ) will satisfy the constraints (2.66) and (2.67); 2. the control law will be admissible, i.e. it will satisfy the constraint (2.68); 3. the real interaction force F (t ) will satisfy the constraint (2.69). Besides, the goal condition (2.64) will be realized not later than: 2 2  C η (t0 ) + ηɺ (t0 ) t + 1 ln Γ , if CΓ tp1 =  0 λ ε − CΓC λ −1  t0 , if not and the goal condition (2.65) not later than:

2

η (t0 ) + ηɺ (t0 )

2

+ CΓC λ −1 ≥ ε

2 2  LCΓ η (t0 ) + ηɺ (t0 ) 1 t + ln ,  0 λ δ − C ( LCΓ λ −1 + L3 ) − Cρ  2 2 tp2 =  if LCΓ η (t0 ) + ηɺ (t0 ) + C ( LCΓ λ −1 + L3 ) + Cρ ≥ δ    t0 , if not

(2.95)

(2.96)

Consequently, the both conditions (2.64) and (2.65) will be fulfilled not later

{

than: tp = max tp1 , tp2

}.

The proof of Theorem 2 is given in Appendix B. Let us note that in the conditions of Theorem 2, the initial force perturbation µ (t0 ) = F (t0 ) − Fp (t0 ) is not present. This is not unexpected because, in view of the relation (2.88), which is fulfilled when using the control laws (2.23) and (2.28), the initial perturbation µ (t0 ) is uniquely determined by the initial perturbations η (t0 ) and ηɺ (t0 ) .

2.4.3 Force transient processes Let us consider the control laws stabilizing the robot-environment interaction force with the preset quality of transient processes defined by the relation [10, 11, 16]:

111

A Unified Approach to Dynamic Control of Robots t

µ (t ) = ∫ Q ( µ (ω ) ) d ω

(2.97)

t0

where µ (t ) = F (t ) − F p (t ) , Q is a vector function characterizing quality of the transient processes. Due to equation (2.97), a differential equation of the form (2.19) will hold:

µɺ (t ) = Q ( µ (t ) ) Usually, the choice of the function Q , which defines the character of the transient processes, relies upon the relation (2.19), and (2.97) is considered as the reference equation in the system of control laws. Of the three control laws, (2.36), (2.41) and (2.43), considered in [10, 16], the first two: t      τ = U q, qɺ ,ψ  q, qɺ , Fp + ∫ Q ( µ (ω ) ) dω  , F      t0     t   τ = U  q, qɺ ,ψ ( q, qɺ , F ) , Fp + ∫ Q ( µ (ω ) ) d ω    t0   utilize the feedback with respect to q, qɺ and force F , and they are not limited

by the supplementary conditions of their applicability. At the same time, the third control law, (2.43):



  

t

  

t



t0

 

τ = U  q, qɺ ,ψ  q, qɺ , Fp + ∫ Q ( µ (ω ) ) dω  , Fp + ∫ Q ( µ (ω ) ) dω   

t0

has a constraint on its applicability in the form of the supplementary condition (2.46):

rank  S T (q ) − M (q ) H −1 (q ) J T (q )  = n which should be fulfilled for all possible configurations {q} that appear in its contact with the environment. The control laws (2.36), (2.41), (2.43) ensure a desired quality of realization of the robot-environment interaction force F p (t ) . At the same time, stabilization of the robot’s motion q p (t ) is also achieved, but the character of that motion cannot be prescribed in advance in an arbitrary way.

112

Dynamics and Robust Control of Robot-Environment Interaction

{

}

In this section we will describe the class of programmed motions q p (t ) , the class of programmed interaction forces F p (t ) and the class of environment’s dynamics equations for which the control laws (2.36), (2.41), (2.43) solve the control task (2.64), (2.65) under the constraints (2.66) – (2.69). We will demonstrate the stability of these control laws in the presence of external perturbations of the robot and environment dynamics, as well as in respect to the models inaccuracies in any finite time interval, giving also an estimation of the time of transient processes in the stabilization of the motion and force. Let us consider the task of robot control under the constraints (2.66) – (2.69). Let PM q p (t ) and PFI Fp (t ) satisfy the conditions (2.78) – (2.81), and let the set Vqɺɺ ⊂ R n be defined by the relation (2.76). It is easy to prove (and this will be done in Theorem 4) that, if the control laws (2.36), (2.41), (2.43) ensure the necessary closeness of q (t ) to q p (t ) and F (t ) to F p (t ) for any t ≥ t0 , they are admissible and will ensure fulfillment of the constraints (2.66), (2.67) and (2.69). Hence, the basic issue in using these control laws is the influence of the robot and environment models inaccuracies and/or external perturbations r (t ) and ρ (t ) on the transient processes. For simplicity, we will assume that the function Q , determining the family of transient processes in (2.19), has the form:

{

}

Q ( µ ) = Rµ where R is a constant n × n matrix. Since the real robot’s dynamics is described by (2.61) and the real environment dynamics by (2.62), the functioning of the control laws (2.36), (2.41), (2.43), in contrast to (2.97), will correspond to: t

µ (t ) = R ∫ µ (ω )dω + D1 ( q(t ) ) r (t ) + D2 ( q(t ) ) ρ (t )

(2.98)

t0

where the n × n matrix D1 and the n × n matrix D2 are of the form: −1

D1 (q ) = ( S T (q ) ) M (q ) D2 (q ) = I n in the case of the control law (2.36),

(2.99)

113

A Unified Approach to Dynamic Control of Robots −1

D1 (q ) = − ( J T (q ) ) H (q)

(2.100)

D2 (q) = D1 (q) M −1 (q ) S T (q) in the case of using the control law (2.41), and −1

D1 (q ) = ( D(q) ) M (q)

(2.101)

−1

D2 (q) = ( D(q) ) S T (q) in the case of using the control law (2.43), where:

D ( q ) = S T ( q ) − M ( q ) H −1 ( q ) J T ( q ) . −1

Let us note that, due to the condition (2.46), the inverse matrix ( D ( q ) ) exists. Assume the vector functions r and ρ are differentiable for t ≥ t0 . If we introduce the vector function:

g (t ) = D1 ( q(t ) ) r (t ) + D2 ( q (t ) ) ρ (t ) then, equation (2.98) can be written in the form: t

µ (t ) = R ∫ µ (ω ) dω + g (t ) t0

By differentiating this expression with respect to t , we obtain the equation:

µɺ = R µ (t ) + gɺ (t ) Its solution can be written in the form t

µ (t ) = e R (t −t ) µ (t0 ) + ∫ e R (t −ω ) gɺ (ω )d ω 0

t0

Integrating the expression e R ( t −ω ) gɺ (ω ) by parts, we obtain t

µ (t ) = e

R ( t − t0 )

( µ (t0 ) − g (t0 ) ) + g (t ) + R ∫ e R (t −ω ) g (ω )dω

(2.102)

t0

Let on the set Vq the vector functions D1 (q ) and D2 (q ) be bounded:

Di (q) ≤ di , i = 1,2

(2.103)

The inequalities (2.103) give the estimate of the vector function norm g (t ) :

114

Dynamics and Robust Control of Robot-Environment Interaction

g (t ) ≤ d1Cr + d 2Cρ

(2.104)

Let all eigenvalues λ1 , λ 2 ,..., λ n of the matrix R be real, different and negative, and let λ = − max λi . Based on the inequality (2.104), analogously to i the estimation of x(t ) for equation (2.84), we obtain the estimate of the transient processes with respect to force:

µ (t ) ≤ CR e− λ (t −t ) ( µ (t0 ) + g (t0 ) ) + g (t ) + 0

(2.105)

+ R CR λ −1 ( d1Cr + d 2Cρ ) ≤ CR e − λ (t −t0 ) µ (t0 ) + Cµ where

CR = 2n R R −1 , Cµ = (1 + CR + R CR λ −1 ) ( d1Cr + d 2Cρ ) .

Let us estimate now the transient process η (t ) = q (t ) − q p (t ) when using the control laws (2.36), (2.41), (2.43), which stabilize PFI Fp (t ) . Since the real environment dynamics is described by (2.62), and PM q p (t ) and PFI Fp (t ) satisfy the constraint equation (2.14), or, which is the same, the equation:

M (q p )qɺɺp + L(q p , qɺ p ) = S T (q p ) F p then, by subtracting it from (2.62), we obtain:

ηɺɺ + K (η ,ηɺ , t ) = M −1 (η + q p ) S T (η + q p )  F − Fp − ρ (t )  where:

K (η ,ηɺ, t ) = M −1 (η + q p ) {L(η + q p ,ηɺ + qɺ p ) − L(q p , qɺ p ) + +  M (η + q p ) − M ( q p )  qɺɺp −  S T (η + q p ) − S T ( q p )  Fp

}

We are not going to repeat here the procedure of obtaining the system in its first approximation for the vector function K in order to examine asymptotic stability of the perturbed motion η (t ) in (2.48) for the case of an ideal model of the environment with which the robot is in dynamic interaction. In contrast to that case, in this case we deal with the real environment, so that the perturbation equation in its first approximation differs from the previous case (2.51) only by one perturbation term β ( x, t ) ρ (t ) , and it has the form:

xɺ = A(t ) x + α ( x, t ) + β ( x, t ) µ (t ) − β ( x, t ) ρ (t )

(2.106)

To estimate the transient process x (t ), let us consider the behavior of the solution of equation (2.106) in an arbitrary finite time interval [t0 , T ] on the set:

115

A Unified Approach to Dynamic Control of Robots

Ω = {( x, t ) ∈ R 2 n × R / x < h, t0 ≤ t ≤ T } where

h = min{δ 1 , δ 2 }

(2.107)

and the values δ 1 and δ 2 determine the measure of the closeness of the real motion q (t ), qɺ (t ) to the programmed one q p (t ), qɺ p (t ) in (2.78) and (2.79). Because α ( x, t ) = o ( x ) at x → 0 , α (0, t ) ≡ 0, ∀t ≥ t0 and due to smoothness of the vector function α ( x, t ) , a positive constant C1 and constant p > 1 can be found, such that: p

α ( x, t ) ≤ C1 x , ∀( x, t ) ∈ Ω

(2.108)

Besides, due to the continuity of the function β ( x, t ) , a positive constant M can be found, such that:

β ( x, t ) ≤ M , ∀( x, t ) ∈ Ω

(2.109)

Let us assume that the system of first approximation of equation (2.106):

xɺ = A(t ) x

(2.110)

is regular, and its characteristic indices α 1 ,..., α 2 n are negative. Let us carry out transformation of the system (2.106):

x = y e −γ (t −t0 )

(2.111)

where the value γ satisfies the inequality:

max α k < −γ < 0 k

Then, we will have:

yɺ = B (t ) y + αɶ ( y, t ) + βɶ ( y, t ) µ (t ) − βɶ ( y, t ) ρ (t ) where:

B (t ) = A(t ) + γ I 2 n ,

αɶ ( y, t ) = eγ (t −t ) α ( y e −γ (t −t ) , t ) , 0

0

βɶ ( y, t ) = eγ ( t −t ) β ( y e −γ (t −t ) , t ) . 0

Evidently, x (t0 ) = y (t0 ).

0

(2.112)

116

Dynamics and Robust Control of Robot-Environment Interaction

In Appendix A (proof of Theorem 1) we show that the first approximation system of (2.112):

yɺ = B (t ) y

(2.113)

is also regular, and its characteristic indices are negative. Let us replace (2.112) by the equivalent integral equation: t

y (t ) = H (t ) y (t0 ) + ∫ K (t , ω ) αɶ ( y (ω ), ω ) + βɶ ( y (ω ), ω ) µ (ω ) − t0

(2.114)

− βɶ ( y (ω ), ω ) ρ (ω )] d ω where H (t ) is the normed (i.e. H (t0 ) = I 2 n ) fundamental matrix of the system (2.113), and K (t , ω ) = H (t ) H −1 (ω ) is the Cauchy matrix. Since all characteristic indices of (2.113) are negative, the following estimate of its fundamental matrix H (t ) at any t ≥ t0 is valid:

H (t ) ≤ C 2 (C 2 ≥ 1)

(2.115)

By taking into account the regularity of the system (2.113) it can be stated that a constant C 3 can be found, such that:

K (t , ω ) ≤ C 3

(2.116)

t0 ≤ ω < t ≤ T . Then, the following theorem on estimation of the transient processes norm in (2.106) is justified.

at

Theorem 3. Let the environment dynamics be such that (a) the first approximation system (2.110) is regular; (b) all characteristic indices α k ( k = 1, 2,..., 2n) in (2.110) are negative. Let the index λ of stabilizing PFI Fp (t ) in the estimate (2.105) satisfy the inequality:

−λ < −γ Then, if the initial perturbation levels

x(t0 ) , µ (t0 )

and the levels of

inaccuracy of the models and/or the external perturbations C r and C ρ satisfy the inequality:

117

A Unified Approach to Dynamic Control of Robots

C2 x(t0 ) +

≤2

−

1 p −1

MCRC3 MC3 µ (t0 ) + Cµ + Cρ ) eγ (T −t0 ) − 1 ≤ ( λ −γ γ

(

)

1    γ  p −1  min   , h  C1C3    

(2.117)

the following estimation of transient processes in the time interval [t0 , T ] is valid:

 b µ (t0 )  − γ (t −t0 ) + C eγ (T −t0 ) − 1 x(t ) ≤  a x(t0 ) + e λ −γ  

(

1

where:

1

)

(2.118)

1

a = 2 p −1 C2 , b = 2 p −1 MCR C3 , C = 2 p −1 MC3γ −1 ( Cµ + Cρ ) .

From the obtained estimation of transient processes (2.118) on a finite time interval [t0 , T ] , it follows that for any t ∈ [t0 , T ] holds:

max { η (t ) , ηɺ (t )

}≤

 b µ (t0 )  − γ (t −t0 ) 2 2 ≤  a η (t0 ) + ηɺ (t0 ) + + C eγ (T −t0 ) − 1 e λ −γ  

(

)

(2.119)

The proof of Theorem 3 is given in Appendix C. Let us show that in the case when the conditions of Theorem 3 are fulfilled one can choose sufficiently small levels of inaccuracy of the robot and environment dynamics models and/or external perturbations C r , C ρ , as well as of the initial perturbations with respect to position and force so as the fulfillment of the goals (2.64) and (2.65) after some time instant smaller than T − t0 will be guaranteed, as well as the fulfillment of the a priori constraints (2.66) – (2.69) for all t ∈ [t0 , T ] .

Theorem 4. Let (a) the function ψ , describing the environment dynamics, satisfy the Lipschitz conditions with respect to any of the i -th variables with the constants Li (i = 1, 2,3) ;

ɺɺ p (t ) , satisfy the (b) PM q p (t ), together with its derivatives qɺ p (t ) and q inclusions (2.78), (2.79), (2.80), and PFI Fp (t ) the inclusion (2.81);

118

Dynamics and Robust Control of Robot-Environment Interaction

(c) the conditions of Theorem 3 be fulfilled. If the levels of initial perturbation η (t0 ) , ηɺ (t0 ) , µ (t0 ) and the levels of inaccuracy of the models and/or external perturbations C r and C ρ , in addition to the inequality (2.117), satisfy also the inequalities:

C R µ (t 0 ) + C µ < d1Cr + d 2Cρ
0 and of programmed (nominal) interaction force along the y -axis

(2.133)

121

A Unified Approach to Dynamic Control of Robots

Fy p (t ) = Fy0 = F 0 = const The complement force and motion components are obtained from (2.131), (2.132)

Fx p (t ) = Fx0 (t ) = hx V0 +ν F 0 ,

y p (t ) = y 0 (t ) =

1 0 F Ky

(2.134)

In order to stabilize nominal motion and force, the control law (2.23) is applied with closed-loop dynamics equation (2.24), and therefore (2.16) is expressed in the form:

ηɺɺx + 2ς xcω xcηɺ x + ω xc2 η x = 0

(2.135)

ηɺɺy + 2ς ycω ycηɺ y + ω yc2 η y = 0

(2.136)

It is obvious that unless the models of robot and environment have some uncertainties, (2.135), (2.136) represent the closed-loop system behavior. In order to study the influence of the inaccuracy of the robot and environment dynamics the following functions

 rx 0 sin (Ω x t )  r (t ) =    ry 0 sin (Ω y t ) 

 ρ x 0 cos (Ω x t )    ρ y 0 cos (Ω y t ) 

ρ (t ) = 

are introduced into the robot and environment equations (see (2.61), (2.62)). The bounds on the motion and force deviations from their nominal values in the presence of the inaccuracies r (t ) and ρ (t ) are given by (2.85), (2.89). For some parameter values the tracking of the nominal motion and force is shown in Fig. 2.3, while Fig. 2.4 shows the bounds along with the motion and force deviations from their nominal values.

2.4.5 Effect of sensor errors on the transient processes In this section we deal with the stability of the considered control laws in respect of the errors of the sensors of position, velocity and force, as well as with the effect of these errors on the accuracy of transient processes in the stabilization of PM q p (t ) and PFI Fp (t ) under the conditions of the constraints (2.66) – (2.69).

122

Dynamics and Robust Control of Robot-Environment Interaction

Fig. 2.3 Tracking the nominal force and motion

Fig. 2.4 Norm and norm bounds for the motion and force

123

A Unified Approach to Dynamic Control of Robots

Let the functions of sensor errors for any t ≥ t0 be described by the equalities

∆ q (t ) = qˆ (t ) − q (t ) , ∆ qɺ (t ) = qˆɺ (t ) − qɺ (t ) ,

(2.137)

∆ F (t ) = Fˆ (t ) − F (t ) , where qˆ, qɺˆ , Fˆ are the sensors’ readings of position, velocity and force. Let the accuracy of these sensors be determined by the values δ q , δ qɺ , δ F , so that ∆ q (t ) ≤ δ q ,

∀t ≥ t0 ,

∆ qɺ (t ) ≤ δ qɺ ,

∀t ≥ t0 ,

∆ F (t ) ≤ δ F ,

∀t ≥ t0 .

(2.138)

Let us consider the control laws (2.23) and (2.28), stabilizing PM q p (t ) with the given quality of transient processes, which, after taking into account the sensors’ errors assume the form:

( ) ( ) τ = U ( qˆ , qɺˆ , qɺɺ + Γ ( qɺˆ − qɺ ) + Γ ( qˆ − q ) , f ( qˆ , qˆɺ , qɺɺ + Γ ( qˆɺ − qɺ ) + Γ ( qˆ − q ) ) ) τ = U qˆ , qˆɺ , qɺɺp + Γ1 qˆɺ − qɺ p + Γ 2 ( qˆ p − q p ) , Fˆ 1

p

1

p

p

p

2

(2.139)

p

2

(2.140)

p

Obviously, the control τ (2.139) satisfies the equation

(

)

(

qɺɺp + Γ1 qɺˆ − qɺ p + Γ 2 ( qˆ − q p ) = Φ qˆ , qɺˆ ,τ , Fˆ

)

Subtracting this equation from the equation of robot’s dynamics (2.61) we obtain the following closed-loop system

ηɺɺ = Γ1ηɺ + Γ 2η + (t )

(2.141)

where

(

(t ) = r (t ) + Γ1∆ qɺ (t ) + Γ 2 ∆ q (t ) + Φ ( q, qɺ ,τ , F ) − Φ qˆ , qˆɺ ,τ , Fˆ

)

(2.142)

The same form as (2.141) has the equation of closed-loop control system when using the control law (2.140). Thereby, the function (t ) is of the form

124

Dynamics and Robust Control of Robot-Environment Interaction

(t ) = D1 ( q (t ) ) r (t ) + D2 ( q (t ) ) ρ (t ) + Γ1∆ qɺ (t )

(

+ Γ 2 ∆ q (t ) + ϕ ( q, qɺ ,τ ) − ϕ qˆ , qˆɺ ,τ

(2.143)

)

where the vector function ϕ is determined by the expression −1

ϕ ( q, qɺ ,τ ) = [ H (q) − N (q ) M (q) ]

( −h(q, qɺ ) + τ + N (q) L(q, qɺ ) )

and the vector functions D1 ( q ), D2 ( q ) are determined in the section 2.4.2,

(

and N ( q ) = J T ( q ) S T ( q )

)

−1

.

Indeed, by eliminating the force F from the equations of the robots dynamics (2.61) and environment dynamics (2.62), we obtain:

qɺɺ = D1 ( q(t ) ) r (t ) + D2 ( q(t ) ) ρ (t ) + ϕ ( q, qɺ ,τ )

(2.144)

On the other hand, the control law (2.140) satisfies obviously the equation

(

)

(

qɺɺp + Γ1 qɺˆ − qɺ p + Γ 2 ( qˆ − q p ) = ϕ qˆ , qɺˆ ,τ

)

(2.145)

By subtracting (2.145) from (2.144) we obtain the desired representation (2.141), (2.143). As the dependence of the estimates of transient processes (2.85), (2.86), (2.87), (2.89) on the applied control laws (2.23), (2.28) is fully determined by the estimate C of the perturbation function (t ) in equation (2.82), then, on the basis of (2.142), (2.143) it can be concluded that the analogous relations hold also in respect of the control laws (2.139), (2.140). Hence, the following theorem, similar to Theorem 2, is justified

Theorem 5. Let PM q p (t ) , together with its derivatives qɺ p (t ), qɺɺp (t ) , satisfy the inclusions: δ +δ q

q p (t ) ∈ Vq 1

δ 2 +δ qɺ

qɺ p (t )∈ Vqɺ

,

∀t ≥ t0 ,

(2.146)

,

∀t ≥ t0 ,

(2.147)

δ + Γ1 δ qɺ + Γ 2 δ q

qɺɺp (t ) ∈ Vqɺɺ 3

,

∀t ≥ t0 ,

(2.148)

and let PFI Fp (t ) satisfy the inclusion σ + max {δ F ,( L1 + L3 Γ 2 )δ q + ( L2 + L3 Γ1 ) δ qɺ }

Fp (t ) ∈ VF

,

∀t ≥ t0

(2.149)

125

A Unified Approach to Dynamic Control of Robots

Let on the sets Vq , Vqɺ , Vτ , VF of the values of arguments the vector functions Φ and ϕ satisfy the Lipschitz conditions with respect to each variable with the constants LΦq , LΦqɺ , LΦτ , LΦF for the function Φ and Lϕq , Lϕqɺ , Lϕτ for the function ϕ . Then, for the control laws (2.139), (2.140) Theorem 2 holds, with the constant C being:

C = Cr + Γ1 δ qɺ + Γ 2 δ q + LΦq δ q + LΦqɺ δ qɺ + LΦF δ F for the control law (2.139) and

C = d1Cr + d 2Cρ + Γ1 δ qɺ + Γ 2 δ q + Lϕq δ q + Lϕqɺ δ qɺ for the control law (2.140). The proof of Theorem 5 is given in Appendix E. Let us consider the control laws (2.39), (2.40), (2.43) ( Q ( µ ) = R µ ), stabilizing PFI Fp (t ) with the preset quality of transient processes, which, after taking into account sensors’ errors assume the form: t     ˆ ˆ τ = U  qˆ , qɺ , ψ  qˆ , qɺ , Fp + R ∫ Fˆ − Fp (ω )dω  , Fˆ      t0    

(



(

)

t

)

(

)



τ = U  qˆ , qˆɺ ,ψ qˆ , qɺˆ , Fˆ , Fp + R ∫ Fˆ − Fp (ω ) d ω   

t0

t    ˆ ˆ τ = U  qˆ , qɺ ,ψ  qˆ , qɺ , Fp + R ∫ Fˆ − Fp (ω ) d ω  ,    t0    t  Fp + R ∫ Fˆ − Fp (ω )d ω   t0 

(

(

(2.150)

 

(2.151)

)

(2.152)

)

ɺɺ from the equation of robot’s After eliminating the second derivative q dynamics (2.61) and environment’s dynamics (2.62), we obtain

F = D1 (q) r (t ) + D2 (q) ρ (t ) + G ( q, qɺ ,τ , F ) where

D1 (q ), D2 (q ) are given by (2.99) and

(2.153)

126

Dynamics and Robust Control of Robot-Environment Interaction

G ( q, qɺ ,τ , F ) = (2.154)

−1

= ( S T (q ) )  M (q ) H −1 (q ) (τ − h(q, qɺ ) + J T (q ) F ) + L(q, qɺ )  for the control law (2.150); D1 ( q ), D2 ( q ) are determined by (2.100) and

G ( q, qɺ ,τ , F ) = −1

= ( J T (q ) )  H (q ) M −1 (q ) ( S T (q ) F − L(q, qɺ ) ) + h(q, qɺ ) − τ 

(2.155)

for the control law (2.151); D1 ( q ), D2 (q ) are determined by (2.101) and −1

G ( q, qɺ ,τ , F ) = ( D(q) )  M (q) H −1 (q) (τ − h(q, qɺ ) ) + L(q, qɺ ) 

(2.156)

for the control law (2.152). On the other hand, the control laws (2.150) – (2.152) satisfy the equation t

(

)

(

Fp + ∫ Fˆ − Fp (ω ) dω = G qˆ , qˆɺ ,τ , Fˆ

)

(2.157)

t0

with the function G determined by the respective formulas (2.154), (2.155), (2.156). By subtracting equation (2.157) from (2.153) we obtain that for each of the control laws (2.150), (2.151), (2.152) the closed-loop control system obeys the equation t

t

t0

t0

µ (t ) = R ∫ µ (ω )dω + R ∫ ∆ F (ω )dω + D1 ( q(t ) ) r (t ) + D2 ( q(t ) ) ρ (t ) +

(

+ G ( q(t ), qɺ (t ),τ (t ), F (t ) ) − G qˆ (t ), qɺˆ (t ),τ (t ), Fˆ (t )

(2.158)

)

which differs from equation (2.98) because of the presence of the additional terms on the right-hand side of (2.158). Let us consider the behavior of the system (2.158) over a finite time interval [t0 , T ] on the set

Ω = Vq × Vqɺ × Vτ × VF . Equation (2.158) can be written in the form: t

t

µ (t ) = R ∫ µ (ω )dω + R ∫ ∆ F (ω )dω + g* (t ) t0

t0

(2.159)

127

A Unified Approach to Dynamic Control of Robots

g* (t ) = D1 ( q(t ) ) r (t ) + D2 ( q(t ) ) ρ (t ) + where

(

)

+ G ( q(t ), qɺ (t ),τ (t ), F (t ) ) − G qˆ (t ), qɺˆ (t ),τ (t ), Fˆ (t ) .

By differentiating (2.159) with respect to t , we obtain the equation:

µɺ (t ) = Rµ (t ) + R∆ F (t ) + gɺ* (t ) Its solution can be written in the form t

µ (t ) = e R (t −t ) µ (t0 ) + ∫ e R (t −ω ) ( R∆ F (ω ) + gɺ* (ω ) ) dω 0

t0

Integrating the last expression by parts, we obtain

µ (t ) = e R (t −t ) ( µ (t0 ) − g* (t0 ) ) + g* (t ) + 0

t

+ R∫ e

t

R ( t −ω )

∆ F (ω )d ω + R ∫ e R ( t −ω ) g* (ω )d ω

t0

(2.160)

t0

Let on the set Ω the vector-function G satisfy the Lipschitz conditions with respect to each variable with the constants LGq , LGqɺ , LτG , LGF . Then, we will have

g* (t ) ≤ C g* , where

∀t ∈ [t0 , T ]

C g* = d1Cr + d 2Cρ + LGq δ q + LGqɺ δ qɺ + LGF δ F .

From the equality (2.160) we obtain the following estimation of the transient processes with respect to force:

µ (t ) ≤ CR e− λ (t −t ) ( µ (t0 ) + g* (t0 ) ) + g* (t ) + 0

+ R CR λ −1δ F + R CR λ −1Cg* ≤ CR e− λ (t −t0 ) µ (t0 ) + Cµ where

(2.161)

Cµ = C g* (1 + CR + R CR λ −1 ) + R CR λ −1δ F .

Because the estimates (2.161), (2.105) differ only by the value of the constant Cµ which depends on the control law applied, and Cµ → 0 when Cr , Cρ , δ q , δ qɺ , δ F → 0 the following theorem, analogous to Theorem 4, is justified.

Theorem 6. Let the vector-function G satisfy the Lipschitz conditions on the set Ω with respect to the each variable q, qɺ , F with the constants LGq , LGqɺ , LGF correspondingly.

128

Dynamics and Robust Control of Robot-Environment Interaction

ɺɺp (t ) , satisfy the Let PM q p (t ) , together with its derivative qɺ p (t ), q inclusions δ +δ q

q p (t ) ∈ Vq 1

δ +δ qɺ

qɺ p (t ) ∈ Vqɺ 2

δ3 +( qɺɺp (t ) ∈ Vqɺɺ

,

∀t ≥ t0

,

∀t ≥ t0

L1 + L3 LGq

)δ +( q

L2 + L3 LGqɺ

(2.162) (2.163)

)δ

qɺ + L3δ F

max

{

LGF

} , ∀t ≥ t 0

,1

(2.164)

and let PFI Fp (t ) satisfy the inclusion

{

σ + max δ F , LGq δ q + LGqɺ δ qɺ + LGF δ F

Fp (t ) ∈ VF

} , ∀t ≥ t 0

(2.165)

Then, for the control laws (2.150), (2.151), (2.152) Theorem 4 is justified. The proof of Theorem 6 is given in Appendix F. As a supplement to the above discussion let us point out that the application of the control laws (2.139), (2.140), (2.150), (2.151), (2.152), which take into account the errors of the sensors of position, velocity, and force under the conditions of a priori constraints (2.66) – (2.69) and in the presence of external perturbations narrows the classes of stabilized PM q p (t ) and PFI Fp (t ) and worsens the accuracy of stabilization in comparison with the ideal control laws (2.23), (2.28), (2.39), (2.40), (2.43). This is related to the fact that sensors’ inaccuracies introduce additional components into the estimates used in the conditions of Theorems 2 and 4. The preservation of these estimates may be ensured by means of an increase of the quantities δ 1 , δ 2 , δ 3 , σ that narrow the classes of stabilized PM and PFI in addition to their narrowing caused by the conditions of Theorems 5 and 6, and also by lowering the accuracy of stabilization of ε and δ in the goal conditions (2.64), (2.65).

2.5 Adaptive Stabilization of Motion and Forces 2.5.1 Introduction In adaptive cases, the goal of stabilization of motion and contact forces of the manipulators interacting with dynamic environment especially for the robots with nonstationary dynamics is attained by designing a special adaptive control scheme and using finite-convergence adaptation algorithms [21].

A Unified Approach to Dynamic Control of Robots

129

In view of the fact that in the course of technological operation the robot parameters, especially the viscous friction coefficients at the manipulator’s drives, may vary with time, the unknown drifts of parameters are of interest. There is a sufficiently broad array of adaptive control laws [6, 22-24] for solving contact tasks by manipulators with stationary dynamics. However, there are practically no results for the robots with nonstationary dynamics. The adaptive control scheme proposed here enables solving contact tasks for robots with both stationary and nonstationary dynamics. One of the parameters of the scheme, such as the algorithm adaptation processing speed, determines the “speed” of the parameters’ drift to which the adaptive control system has time to adapt without violating the a priori constraints that are usually imposed on the control, motion, and interaction forces. Let us notice that an additive component of the external perturbation of the robot dynamics in the proposed scheme can be considered as a time-dependent robot parameter. Therefore this control scheme can be used for adaptation to the essentially inadequate description of the robot’s dynamics by its mathematical model. The extent of parameters’ drift and the degree of inadequacy of the robot model are determined by the class of functions of uniformly bounded variation. Here we consider the classes of stabilized motions and forces and their stabilization accuracies subject to the level of initial and external perturbations of dynamics of the robot and its environment, as well as of the sensors errors, processing speed of the adaptation algorithm, and of other parameters of the adaptive control scheme.

2.5.2 Task setting Let us consider a manipulation robot whose mechanism model is described by the vector differential equation of the form

H (q, ξ1 )qɺɺ + h(q, qɺ , ξ1 ) = τ + J T (q, ξ1 ) F

(2.166)

where, q = q (t ) is the n -vector of the generalized coordinates; H ( q, ξ1 ) is the n × n positive-definite matrix of inertia moments of the manipulation mechanism; h( q, qɺ , ξ1 ) is the n -vector of the centrifugal, Coriolis’ and gravitational moments; τ is the n -vector of the torques at the robot joints; J T (q, ξ1 ) is the n × m matrix connecting the robot end-effector velocities with the velocities of the generalized coordinates; F is the m -vector of the generalized forces, or of the generalized forces and moments acting on the robot

130

Dynamics and Robust Control of Robot-Environment Interaction

end-effector, and ξ1 is the k1 -vector of the manipulation mechanism parameters. We will assume that n ≥ m (general case). The robot actuators considered are the electromechanical drives with reducers, whose dynamics is described with a high degree of accuracy by the following system of differential equations [4]:

LR

diR + RiR + Eqɺ = u dt

J R qɺɺ + K v qɺ + τ = DiR

(2.167) (2.168)

where LR , R, K v , E and D are the n × n diagonal matrices representing the respective parameters of the actuators such as the electric motor armature inductivity, armature resistances, coefficients of viscous friction at the joints, and electromechanical and mechanoelectrical constants, including the reduction ratio of the reducers; J R is the diagonal n × n matrix of inertia moments of the actuators, reduced to the output shafts; u is the n -vector of the electric motor controlling voltage, and iR is the n -vector of the electromotor rotor current. To obtain a generalized model of the robot dynamics, let us eliminate the vector of moments τ from (2.166) and (2.168). Then, the vector of currents iR will be described via the generalized coordinates q :

iR = D −1 ( J R + H (q, ξ1 ) ) qɺɺ + K v qɺ + h ( q, qɺ , ξ1 ) − J T ( q, ξ1 ) F  . Substituting this expression into (2.167) we obtain an equation that connects the vector of the controlling voltages u to the vector of generalized coordinates q in the form:

u = U ( q, qɺ , qɺɺ, ɺɺɺ q , F , Fɺ , ξ ) where:

U ( q, qɺ , qɺɺ, ɺɺɺ q , F , Fɺ , ξ ) = A ( q, ξ ) ɺɺɺ q + B ( q, qɺ , ξ ) qɺɺ + C ( q, qɺ , ξ ) + + a ( q, ξ ) Fɺɺ + b ( q, qɺ , ξ ) F , A ( q, ξ ) = LR D −1 ( J R + H ( q, ξ1 ) ) ,  n ∂H ∂h −1 B ( q, qɺ , ξ ) = LR D −1  ∑ qɺi + K v +  + RD ( J R + H ( q, ξ1 ) ) , ∂ qɺ   i =1 ∂ qi

(2.169)

131

A Unified Approach to Dynamic Control of Robots

C ( q, qɺ , ξ ) = RD −1 ( K v qɺ + h ( q, qɺ , ξ1 ) ) + Eqɺ + LR D −1

∂h qɺ , ∂q

a ( q, ξ ) = − LR D −1 J T ( q, ξ1 ) , n   ∂ JT b ( q, qɺ , ξ ) = − D −1  LR ∑ qɺi + R J T ( q, ξ1 )  .  i =1 ∂ qi 

(

Here, the k -vector of robot parameters ξ = ξ1T , ξ 2T

T

)

consists of the k1 -

vector ξ1 of the manipulation mechanism parameters and the k 2 -vector ξ 2 of the actuator parameters ( k = k1 + k 2 ) . Let us note that the presented form of the matrices A, B, a, b and of the vector C , correspond to a constant value of the vector ξ 2 of the parameters of the actuators. If this vector is a time function, these matrices will have a more unwieldy form, but the form of robot dynamics equation (2.169) will be preserved. In that case, the vector of parameters ξ should be supplemented with the components corresponding to the derivatives of the vector ξ 2 (t ) . On the basis of the positive-definiteness of the matrix A( q, ξ ), let us write (2.169) in the form solved with respect to the highest derivative:

ɺɺɺ q = Φ ( q, qɺ , qɺɺ, F , Fɺ , u , ξ )

(2.170)

where

Φ ( q, qɺ , qɺɺ, F , Fɺ , u , ξ ) = = A−1 ( q, ξ ) u − B ( q, qɺ , ξ ) qɺɺ − C ( q, qɺ , ξ ) − a ( q, ξ ) Fɺ − b ( q, qɺ , ξ ) F  . Let the real robot dynamics satisfy the differential equation

ɺɺɺ q = Φ ( q, qɺ , qɺɺ, F , Fɺ , u , ξ ) + r (t )

(2.171)

where the n -vector r (t ) represents the uncontrollable external perturbation, which can also be treated as the inadequacy of description of the robot dynamics by the mathematical model (2.170). We will assume that the vector functions Φ and r satisfy conditions sufficient for the existence and uniqueness of the solutions of (2.170) and ɺɺ(t0 ) = qɺɺ0 . (2.171) for the initial conditions q (t0 ) = q0 , qɺ (t0 ) = qɺ0 , q

132

Dynamics and Robust Control of Robot-Environment Interaction

Let us describe the dynamics model of the environment interacting with the robot by a nonlinear vector differential equation:

M (q )qɺɺ + L(q, qɺ ) = S T (q ) F

(2.172)

with the nonsingular and continuous n × n matrix M (q ), continuous n -vector function L( q, qɺ ) , and continuous n × m matrix S T (q ) of the environment, reduced to the robot coordinates frame

{q} .

The environment dynamics

parameters have been described in [25]. In the Cauchy form, the equation of the environment dynamics model (2.172) can be written as

qɺɺ = ψ ( q, qɺ , F ) where ψ ( q, qɺ , F ) = M −1 ( q )  S T ( q ) F − L( q, qɺ )  . We will assume that (2.172) has the following property:

rank S (q ) = m,

∀q ∈ R n

which means that the equation is uniquely solvable with respect to force:

F = f ( q, qɺ , qɺɺ) where

−1

f ( q, qɺ , qɺɺ) = ( S (q ) S T (q ) ) S (q ) [ M (q )qɺɺ + L(q, qɺ )] .

Let the real environment dynamics satisfy the vector differential equation

F = f ( q, qɺ , qɺɺ) + ρ (t )

(2.173)

where the vector function ρ (t ) of the dimension m represents the uncontrollable perturbations of the environment. The goal of the control of the robot in contact with its environment is the realization of the programmed (desired) motion q p (t ) and the programmed (desired) force of robot’s interaction with environment Fp (t ) . The programmed robot motion ( PM ) q p (t ) and the programmed force of interaction ( PFI ) Fp (t ) must satisfy the constraint equation

Fp (t ) ≡ f ( q p (t ), qɺ p (t ), qɺɺp (t ) ) ,

∀t ≥ t 0

(2.174)

Because of the existence of initial perturbations and uncontrollable external perturbations r (t ) and ρ (t ) , exact realization of PM q p (t ) and PFI Fp (t )

133

A Unified Approach to Dynamic Control of Robots

is not possible. Hence, we will formulate the control goal in the form of the following two conditions on the analogy of (2.64), (2.65)

x (t ) − x p (t ) < ε F (t ) − Fp (t ) < δ

(2.175)

where ε , δ > 0 are the given numbers, T

T

x (t ) = ( qT (t ), qɺ T (t ), qɺɺT (t ) ) , x p (t ) = ( q p T (t ), qɺ pT (t ), qɺɺpT (t ) ) . Here, and below, the norm of a vector signifies the Euclidean norm. The goal conditions must be fulfilled starting from a time instant t p ≥ t0 .

ɺɺ(t ), the control action u (t ) , and the force of The robot motion q (t ), qɺ (t ), q interaction of the robot and environment F (t ) , are usually constrained by the conditions of the technological task to be performed. Let us set up these constraints in the form of the relations:

q(t ) ∈ Vq ⊂ R n , qɺ (t ) ∈ Vqɺ ⊂ R n , qɺɺ(t ) ∈ Vqɺɺ ⊂ R n , ∀t ≥ t0 u (t ) ∈ Vu ⊂ R n ,

∀t ≥ t0

F (t ) ∈ VF ⊂ R m , ∀t ≥ t0

(2.176) (2.177) (2.178)

where Vq , Vqɺ , Vqɺɺ , Vu , and VF are the preset open, bounded and simply connected sets of the corresponding spaces, and Vu is the closure of the set Vu in

Rn . We will assume that the robot parameters are unknown and can change as time functions ξ = ξ (t ) in an unpredictable manner. We will also assume that the closed convex set Vξ ∈ R k , constraining the parameter variations is known:

ξ (t ) ∈ Vξ ,

∀t ≥ t0

(2.179)

Also, we assume that the levels of external perturbation of the robot and environment are bounded:

r (t ) ≤ Cr ,

ρ (t ) ≤ Cρ , ∀t ≥ t0

(2.180)

and that the function ρ (t ) is differentiable and the norm of its derivative is also bounded:

134

Dynamics and Robust Control of Robot-Environment Interaction

ρɺ (t ) ≤ Cρɺ ,

∀t ≥ t0

(2.181)

Under these conditions, the task of controlling robot in contact with its environment can be formulated in the following way. For the complete equations of dynamics of the robot (2.171) and of the environment (2.173), it is necessary to synthesize the admissible (i.e. satisfying the constraint (2.177)) control action u (t ), t ≥ t0 , which, in solving the contact task, will ensure that the real robot

ɺɺ(t ) satisfies the constraints (2.176), that the real force of motion q (t ), qɺ (t ), q interaction of the robot with environment F (t ) satisfies the constraint (2.178), and that the goal conditions (2.175) are fulfilled starting from a time instant t p ≥ t0 . 2.5.3 General scheme of robot adaptive control in contact tasks We will assume that the function f , describing the environment dynamics, possesses continuous partial derivatives with respect to each variable. After differentiating, the environment dynamics equation (2.173) becomes

Fɺ = f ( q, qɺ , qɺɺ, ɺɺɺ q ) + ρɺ (t )

(2.182)

where

f ( q, qɺ , qɺɺ, ɺɺɺ q ) = N (q) M (q )ɺɺɺ q + m(q, qɺ )qɺɺ + l (q, qɺ ) .

Here

N (q ) =  S (q ) S T (q )  S (q ),

−1

n

m(q, qɺ ) = ∑ i =1 n

l (q, qɺ ) = ∑ i =1

∂ [ N (q ) M (q) ] ∂ qi

qɺi + N (q )

∂ L(q, qɺ ) , ∂ qɺ

∂N (q) ∂ L(q, qɺ ) qɺi L(q, qɺ ) + N (q) qɺ . ∂ qi ∂q

Substituting F and Fɺ from (2.173) and (2.182) into the robot dynamics equation (2.171), and solving it with respect to the highest derivative, we obtain the following equation of interaction of the robot and environment:

ɺɺɺ q = ϕ ( q, qɺ , qɺɺ, u , ξ ) +

+ A −1 ( q, ξ )  A ( q, ξ ) r (t ) − a ( q, ξ ) ρɺ (t ) − b ( q, qɺ , ξ ) ρ (t ) 

(2.183)

135

A Unified Approach to Dynamic Control of Robots

where

A ( q , ξ ) = A ( q, ξ ) + a ( q, ξ ) N ( q ) M ( q ) ,

ϕ ( q, qɺ , qɺɺ, u, ξ ) = A −1 ( q, ξ ) u − ( B ( q, qɺ , ξ ) + a ( q, ξ ) m(q, qɺ ) ) qɺɺ − −C ( q, qɺ , ξ ) − a ( q, ξ ) l (q, qɺ ) − b ( q, qɺ , ξ ) f ( q, qɺ , qɺɺ)  . We will assume that ∀(q, ξ ) ∈ Vq × Vξ , the matrix A( q, ξ ) is invertible, and the matrix and vector functions continuous. Then the quantities:

ϕ (⋅), A −1 (⋅), A(⋅), a (⋅), and b (⋅) are

Cϕ = sup ϕ ( q, qɺ , qɺɺ, u , ξ ) , C A = sup A −1 ( q, ξ ) , C A = sup A ( q, ξ ) , Ca = sup a ( q, ξ ) , Cb = sup b ( q, qɺ , ξ ) ,

ɺɺ ∈Vqɺɺ , u ∈Vu , and ξ ∈Vξ , where the suprema are taken over all q ∈ Vq , qɺ ∈Vqɺ , q are bounded. Hence, for an arbitrary admissible control u (t ), t ≥ t0 , ensuring fulfillment of the inclusion

( q(t ), qɺ (t ), qɺɺ(t ) )∈Vq × Vqɺ × Vqɺɺ ,

∀t ≥ t0 ,

(2.184)

because of (2.183) and inequalities (2.180), (2.181), the inequality

ɺɺɺ q (t ) ≤ Cɺɺɺq ,

∀t ≥ t0

(

(2.185)

)

is satisfied, where Cɺɺɺq = Cϕ + C A C ACr + Ca Cρɺ + Cb Cρ . Furthermore, we will assume that on the set Vq × Vqɺ × Vqɺɺ × R n × R m × R m × R k the vector function U is linear with respect to the parameters:

U ( q, qɺ , qɺɺ, ɺɺɺ q , F , Fɺ , ξ ) = G ( q, qɺ , qɺɺ, ɺɺɺ q , F , Fɺ ) ξ

(2.186)

Here, G (⋅) is a known continuous matrix function of the dimension n × k . The property (2.186) of the robot dynamics model (2.169) is well known, and is frequently used in the synthesis of adaptive control laws [9, 22-24]. We will also assume that there exist the open, simply connected and bounded sets Vɺɺɺq ⊂ R n and VFɺ ⊂ R m such that

U : Vq ×Vqɺ ×Vqɺɺ ×Vɺɺɺq ×VF ×VFɺ ×Vξ → Vu

(2.187)

136

Dynamics and Robust Control of Robot-Environment Interaction

This property is a sufficient condition for the existence of the admissible control in the realization of PM q p (t ) and PFI Fp (t ) by the robot under constraint conditions (2.176) – (2.178) and in the absence of initial, external, and parametric perturbations. Let us define a scheme of the function of robot’s control system in the following way. Let ξ 0 be an arbitrary vector in Vξ . Let the set of time constants

t0 , t1 ,… , tk and the set of vectors ξ0 , ξ1 ,… , ξ k in Vξ be determined. For t ≥ tk let us consider the control law of the form

(

) )

(

uk (t ) = U qˆ , qɺˆ , qɺɺˆ , q∗ , Fˆ , f qˆ , qɺˆ , qɺɺˆ , q∗ , ξ k

(

)

(

)

(

(2.188)

)

q p + Γ1 qɺɺˆ − qɺɺp + Γ 2 qɺˆ − qɺ p + Γ 3 qˆ − q p , Γ1 , Γ 2 , Γ 3 are the where q ∗ = ɺɺɺ

n × n matrices, such that the eigenvalues of the matrix  0n Γ =  0n Γ3

In 0n Γ2

0n  I n  Γ1 

ɺɺˆ , Fˆ are negative, real and different quantities; qˆ, qˆɺ , q

are the respective

indications of the position, velocity, acceleration and force sensors. Here, 0 n and I n are the respective zero and unit matrices of dimension n × n . It is easy to see, that the matrix Γ will have eigenvalues λ1 , λ 2 ,..., λ3 n if its submatrices Γ1 , Γ 2 , Γ 3 have the following representation

Γ1 = diag ( λ1 λ2 λ3 , … , λ3n − 2 λ3n −1 λ3n ) , Γ 2 = diag ( −λ1λ2 − λ1λ3 − λ2 λ3 , … , − λ3n − 2 λ3n −1 − λ3n − 2 λ3n − λ3n −1λ3n ) , Γ3 = diag ( λ1 + λ2 + λ3 , … , λ3n − 2 + λ3n −1 + λ3n ) . Let us for t ≥ t k determine the auxiliary function

(

(

) )

uɶk (t ) = U qˆ , qˆɺ , qɺɺˆ , ɺɺɺ qˆ , Fˆ , f qˆ , qɺˆ , qɺɺˆ , ɺɺɺ qˆ , ξ k

where ɺqɺˆɺ is the measured or calculated estimate of ɺqɺɺ . Consider the inequality of the form

(2.189)

137

A Unified Approach to Dynamic Control of Robots

uk (t ) − uɶk (t ) < h ,

t ≥ tk

(2.190)

where h > 0 is a control scheme parameter. Let tk ′ ≥ tk be the first time instant in which the inequality (2.190) is violated, that is, at that time instant

uk (tk ′ ) − uɶk (tk ′ ) ≥ h

(2.191)

Let

(

ξ k +1 = Α ξ k , tk ′

)

(2.192)

be a correction algorithm defining the estimate ξ k +1 of the current value of the unknown parameters vector ξ (t ) at the time instant tk ′ , such that ξ k +1 ∈ Vξ . The time instant t k +1 , following t k in the sequence t0 , t1 , t2 ,… , tk , is determined by the equality

tk +1 = tk ′ + θ where θ is the time needed to calculate a new estimate of the parameters vector ξ k +1 according to the algorithm (2.192). In this way, over the time interval [tk , tk +1 ) , the control low is defined by (2.188), while the duration of this time interval is determined by the duration of fulfillment of the inequality (2.190), and by the time interval θ . The number θ plays the role of the control scheme parameter characterizing the processing speed of the adaptation algorithm under consideration. Note that a similar scheme has been utilized for adaptive robot control in free space [9, 22-24]. To solve the inequalities (2.190) one can use any algorithm out of the class of finite-convergence algorithms. As an example of the correction algorithm of estimates (2.192), we will consider an algorithm of the form [6]

 G T (tk ′ ) uk (tk ′ ) − uɶk (tk ′ )  ξ k +1 = PrVξ ξ k + 2  G (tk ′ ) 

(

)    

(2.193)

where

(

(

G (tk ′ ) = G qˆ (tk ′ ), qɺˆ (tk ′ ), qɺɺˆ (tk ′ ), ɺɺɺ qˆ (tk ′ ), Fˆ (tk ′ ), f qˆ (tk ′ ), qɺˆ (tk ′ ), qɺɺˆ (tk ′ ), ɺɺɺ qˆ (tk ′ ) and PrV is the orthogonal projector onto the set Vξ . ξ

))

138

Dynamics and Robust Control of Robot-Environment Interaction

Let us denote by

∆ q (t ) = qˆ (t ) − q(t ), ∆ qɺ (t ) = qˆɺ (t ) − qɺ (t ), ∆ qɺɺ (t ) = qɺɺˆ − qɺɺ(t ), ∆ɺɺɺq (t ) = ɺɺɺ qˆ (t ) − ɺɺɺ q (t ), ∆ F (t ) = Fˆ (t ) − F (t ) the deviation of the corresponding estimates of motion and force from their real values. Let the accuracy of these estimates be bounded by the quantities δ q , δ qɺ , δ qɺɺ , δɺɺɺq and δ F , such that:

∆ q (t ) ≤ δ q , ∆ qɺ (t ) ≤ δ qɺ , ∆ qɺɺ (t ) < δ qɺɺ ,

(2.194)

∆ɺɺɺq (t ) ≤ δɺɺɺq , ∆ F (t ) ≤ δ F , ∀t ≥ t0 Let us denote by

CG = sup G ( q, qɺ , qɺɺ, ɺɺɺ q , F , f ( q, qɺ , qɺɺ, ɺɺɺ q ))

(2.195)

where the supremum is taken over all

q , F ) ∈ Vq × Vqɺ × Vqɺɺ × Y ×VF , ( q, qɺ, qɺɺ, ɺɺɺ

Y = { y ∈ R n / y ≤ Cɺɺɺq + δ ɺɺɺq } .

Because the matrix and vector functions G and f are continuous, the number

C G is finite. We

assume n

that m

the

vector

m

k

function

U

satisfies

on

the

set

Vq ×Vqɺ ×Vqɺɺ × R × R × R × R the Lipschitz conditions for each variable, with the Lipschitz constants Luq , Luqɺ , Luqɺɺ , Luɺɺɺq , LuF , LuFɺ and Luξ , respectively. Due to the linearity of U with respect to ɺɺɺ q , F , Fɺ and ξ this function unconditionally satisfies the Lipschitz conditions with respect to these variables if ( q, qɺ , qɺɺ) ∈Vq × Vqɺ ×Vqɺɺ . In view of the assumption on continuous differentiability of the function f we can state that this function satisfies for the set Vq × Vqɺ × Vqɺɺ the Lipschitz conditions for each variable, with some Lipschitz constants Lqf , Lqfɺ , Lqɺfɺ . We will assume that the vector function f also satisfies the Lipschitz conditions for each variable with the constants equaling the respective Lqf , Lqfɺ , Lqɺɺf and Lɺɺɺqf on the set Vq ×Vqɺ ×Vqɺɺ × R n .

A Unified Approach to Dynamic Control of Robots

139

2.5.4 Adaptive stabilization of programmed motions and forces Let us give the following accepted definition. The variation (or the total variation) of the real-valued function g (t ) , defined on the interval [a, b ] is the quantity N

Var ( g ) = sup ∑ g (tk ) − g (tk −1 ) [a , b]

k =1

where the supremum is taken over all partitions a = t0 < t1 < … < tn = b of the interval [a, b ] . If there exists a constant M such that Var ( g ) ≤ M , the function [a , b]

g is called the function of bounded variation over the interval [a, b ] . The function g :[t0 , ∞ ) → R will be called the function of uniformly bounded variation of order p if on an arbitrary interval [ a, b ] ⊂ [t0 , ∞ ) of the length p the function g is a function of bounded variation with one and the same constant M ( p ). Consider the scheme of adaptive control presented in the preceding section. Let us denote by r( a, b) the number of corrections (the number of time instants

tk ′ ) of the algorithm (2.193) on the interval [a, b] . We will prove first an auxiliary statement asserting that under certain conditions the number of corrections of the algorithm (2.193) is bounded on any finite time interval [a, b ] by the majorant independent of the adaptation algorithm processing speed θ , if the components ξ j (t ), j = 1,2,..., k of the vector function ξ (t ) are the functions of bounded variation on [a, b ] .

Lemma: Let [a, b ] be an arbitrary interval contained in [t0 , ∞) , and for every time instant t ∈ [a, b] during the functioning of the closed-loop control system defined by the control law (2.188), the following conditions are fulfilled: 1. 2.

(qˆ (t ), qɺˆ (t ), qɺɺˆ (t ), ɺɺɺ q p (t ) + Γ1 (qɺɺˆ (t ) − qɺɺp (t )) + + Γ 2 (qɺˆ (t ) − qɺ p (t )) + Γ3 (qˆ (t ) − q p (t ))) ∈Vq × Vqɺ ×Vqɺɺ ×Vɺɺɺq

( q(t ), qɺ (t ), qɺɺ(t ) )∈Vq × Vqɺ × Vqɺɺ

140

Dynamics and Robust Control of Robot-Environment Interaction

3. 4.

Fˆ (t ) ∈ V F f (qˆ (t ), qˆɺ (t ), qɺɺˆ (t ), ɺɺɺ q p (t ) + Γ1 (qɺɺˆ (t ) − qɺɺp (t )) + Γ 2 (qˆɺ (t ) − qɺ p (t )) + Γ 3 (qˆ (t ) − q p (t ))) ∈ VFɺ

Let the adaptation scheme parameter h be chosen in such a way that the real number h 0 = h − 2 h1 > 0 , where

h1 = Luq δ q + Luqɺ δ qɺ + Luqɺɺ δ qɺɺ + Luɺɺɺq (δɺɺɺq + Cr ) + LuF δ F

(

+ LuFɺ Lqf δ q + Lqfɺ δ qɺ + Lqɺɺf δ qɺɺ + Lɺɺɺqf δɺɺɺq + Cρɺ

)

If the components ξ j (t ), j =1,..., k of the vector function ξ (t ) are the functions of bounded variations on [a, b ] , the number of corrections r ( a, b) of the algorithm (2.193) is bounded from above by the quantity

 k  Κ  ∑ Var (ξ j )  =  j =1 [ a , b] 

( diam V ) ξ

2

k

+ 4 Cξ

(ξ ) ∑ Var [ ] j

a,b

j =1 −2 G

(2.196)

hh 0 C

which is not dependent on the parameter θ of the control scheme. Here Cξ = max ξ , diamVξ is the diameter of the set Vξ . ξ ∈Vξ

The proof of Lemma is given in Appendix G. Let us consider the transient processes occurring during the realization of the adaptive control low (2.188). Evidently, the control u k (t ) satisfies the equation

(

)

(

)

(

ɺɺɺ q p + Γ1 qɺɺˆ − qɺɺp + Γ 2 qɺˆ − qɺ p + Γ 3 ( qˆ − q p ) = ϕ qˆ , qˆɺ , qɺɺˆ , Fˆ , uk , ξ k

)

(2.197)

where



( (

(

)

)

ϕ qˆ , qɺˆ , qɺɺˆ , Fˆ , uk , ξ k = A −1 ( qˆ , ξ k ) uk − B qˆ , qɺˆ , ξ k + a ( qˆ , ξ k ) m (qˆ , qɺˆ ) qɺɺˆ −

(

)

(

)

)

−C qˆ , qˆɺ , ξ k − a ( qˆ , ξ k ) l (qˆ , qˆɺ ) − b qˆ , qɺˆ , ξ k Fˆ  and the auxiliary function uɶk (t ) corresponding to this control (relation (2.189)) satisfies the equation

(

ɺɺɺ qˆ = ϕ qˆ , qˆɺ , qɺɺˆ , Fˆ , uɶk , ξ k

)

(2.198)

141

A Unified Approach to Dynamic Control of Robots

By extracting (2.197) from (2.198) we obtain the relation

(

)

(

)

ɺɺɺ qˆ − ɺɺɺ q p − Γ1 qɺɺˆ − qɺɺp − Γ 2 qɺˆ − qɺ p − Γ 3 ( qˆ − q p ) =

(

)

(

= ϕ qˆ , qɺˆ , qɺɺˆ , Fˆ , uɶk , ξ k − ϕ qˆ , qɺˆ , qɺɺˆ , Fˆ , uk , ξ k

)

which is equivalent to the equation

ηɺɺɺ = Γ1ηɺɺ + Γ 2ηɺ + Γ3η + (t )

(2.199)

where η (t ) = q (t ) − q p (t ) ,

(t ) = −∆ɺɺɺq (t ) + Γ1∆ qɺɺ (t ) + Γ 2 ∆ qɺ (t ) + Γ3 ∆ q (t ) +

(

) (

)

+ ϕ qˆ , qˆɺ , qɺɺˆ , Fˆ , uɶk , ξ k − ϕ qˆ , qɺˆ , qɺɺˆ , Fˆ , uk , ξ k . Let the vector function ϕ satisfy the Lipschitz condition for the variable u on the set Vq ×Vqɺ ×Vqɺɺ ×V F × R n × R k with the Lipschitz constant Lu . Then,

(t ) ≤ δɺɺɺq + Γ1 δ qɺɺ + Γ 2 δ qɺ + Γ3 δ q + Lu uk − uɶk

(2.200)

If the lemma conditions are fulfilled ∀t ≥ t0 , the inequality ɺɺɺ qˆ (t ) ≤ Cɺɺɺq + δɺɺɺq is true, and inasmuch as q ∗ (t ) ∈ Vɺɺɺq , then q ∗ (t ) ≤ Cɺɺɺq . Hence, in addition to the estimate (2.200) for the norm of the function

(t ) , the following estimate holds:

(t ) ≤ 2 Cɺɺɺq + 2 δɺɺɺq + Γ1 δ qɺɺ + Γ 2 δ qɺ + Γ3 δ q , ∀t ≥ t0

(2.201)

Let us split the interval of integration [t0 , t ] into two sets:

At = { t ′∈[t0 , t ] / uk (t ′) − uɶk (t ′) < h } Bt = { t ′∈[t0 , t ] / uk (t ′) − uɶk (t ′) ≥ h } Then, according to the inequalities (2.200), (2.201), we will have

δɺɺɺq + Γ1 δ qɺɺ + Γ 2 δ qɺ + Γ 3 δ q + Lu h , (t ′) ≤   2 Cɺɺɺq + 2 δɺɺɺq + Γ1 δ qɺɺ + Γ 2 δ qɺ + Γ 3 δ q ,

(

Let us introduce the notation x = η T ,ηɺ T ,ηɺɺT

if t ′ ∈ At if t ′ ∈ Bt T

)

, ɶ (t ) = ( 0, 0,

(2.202) T

T

(t ) ) .

Then, (2.199) can be rewritten in the form

xɺ (t ) = Γx(t ) + ɶ (t )

(2.203)

142

Dynamics and Robust Control of Robot-Environment Interaction

Its solution has the form t

x(t ) = eΓ (t −t0 ) x(t0 ) + ∫ eΓ (t −ω ) ɶ (ω ) dω

(2.204)

t0

If λ = − max λi , where λi are the different, real and negative eigenvalues of the i

matrix Γ , then

eΓ (t −t0 ) x (t0 ) ≤ CΓ x (t0 ) e− λ ( t −t0 ) , CΓ ≥ 1 Hence, the following estimate of the solution of equation (2.203), x(t ) = x (t ) − x p (t ) , holds:

x(t ) ≤ CΓ e− λ (t −t0 ) x(t0 ) + CΓ ∫ e − λ (t −ω )

(ω ) d ω + CΓ ∫ e− λ (t −ω )

At

< CΓ x(t0 ) e

− λ ( t −t0 )

(ω ) d ω
0 , with the constants M j ( p ) . Then, when the lemma conditions are fulfilled the number of corrections of the adaptation algorithm on the interval [t0 , t0 + p ] is bounded:

 k  r(t0 , t0 + p) < Κ  ∑ M j ( p)   j =1 

143

A Unified Approach to Dynamic Control of Robots

and, as the processing speed of the adaptation algorithm equals θ , it is evident that



k





j =1



µ es ( Bt ) < Κ  ∑ M j ( p)  θ . Consequently, in accordance with the inequality (2.205) on the interval [t0 , t0 + p ] the following estimate holds:

 x(t ) < CΓ x(t0 ) e − λ (t −t0 ) + δ + Lu CΓ λ −1h + C Κ  

k

∑M j =1

j

 ( p) θ 

(2.206)

In accordance with the lemma, the function Κ (⋅) is not dependent of θ . Then the parameters h and θ of the control scheme can be chosen in such a way that the sum Lu CΓ λ −1h + C Κ (⋅) θ will be as small as desired. By subtracting (2.174) from (2.173), we obtain the following expression for the deviation µ (t ) = F (t ) − F p (t ) :

µ (t ) = f ( q, qɺ , qɺɺ) + ρ (t ) − f ( q p , qɺ p , qɺɺp )

(2.207)

Taking into account the Lipschitz conditions for the vector function f , and also the inequality (2.206), we obtain the following estimate for transient process with respect to force for all t ∈ [ t0 , t0 + p ] :

µ (t ) ≤ Lqf η + Lqfɺ ηɺ + Lqɺɺf ηɺɺ + Cρ ≤ L x(t ) + Cρ <  k < LCΓ x(t0 ) e − λ (t −t0 ) + Lδ + LLu CΓ λ −1h + LC Κ  ∑ M j ( p )  j =1 f f f where L = Lq + Lqɺ + Lqɺɺ .

(2.208)  + C θ  ρ 

An α -narrowing of the open set A ⊂ R n is defined as the non-empty subset Aα such that A = ∪ Bα ( x) , where the uniting is taken over all x ∈ Aα . Here,

Bα ( x) is an open solid sphere in R n of the radius α > 0 , with the center at the point x . Theorem 7. Let the components ξ j (t ), j = 1,..., k of the robot parameters ξ (t ) be functions of the uniformly bounded variations of order p > 0 with the constants M j ( p ) .

144

Dynamics and Robust Control of Robot-Environment Interaction

Let the accuracies of stabilization ε and δ satisfy the inequalities

a CΓε + a and

R = Γ1 + Γ 2 + Γ3 , r = Γ1 δ qɺɺ + Γ 2 δ qɺ + Γ3 δ q . Let us determine the class PFI

{F (t )} by means of the inclusions: p

L δ 0 + Cρ + δ F F

Fp (t ) ∈ V , Fɺ (t ) ∈ V ɺC1δ 0 + C2 , p

where

∀t ≥ t0

F

(2.213)

∀t ≥ t0

(2.214)

C1 = Lqf + Lqfɺ + Lqɺɺf + Lɺɺɺqf R , C2 = Lqf δ q + Lqfɺ δ qɺ + Lqɺɺf δ qɺɺ + Lɺɺɺqf r .

Let the levels of the external perturbation C r , C ρ , C ρɺ , the sensors’ errors

δ q , δ qɺ , δ qɺɺ , δ ɺqɺɺ , δ F , and the parameters h and θ of the adaptation scheme be such that these classes of the h 0 = h − 2h1 > 0 .

PM

and

PFI

are nonempty and

If the level of initial perturbation x (t0 ) satisfies the inequality

x (t0 ) < ε and the order p satisfies the equality

  C LCΓ 2 2 p = max  p1 = ln Γ , p2 = ln  λ ε −a λ δ − La − Cρ  

145

A Unified Approach to Dynamic Control of Robots

then, at every t ≥ t0 , for the transient processes of motion and force determined by action of the adaptive control law (2.188), the following estimates hold: − λ ( t −t0 ) + a, if t ∈ [t0 , t0 + p ] C x(t0 ) e x(t ) = x (t ) − x p (t ) <  Γ ε , if t ≥ t0 + p

(2.215)

 LCΓ x(t0 ) e− λ (t −t0 ) + La + Cρ , if t ∈ [t0 , t0 + p ] µ (t ) = F (t ) − Fp (t ) <  (2.216) δ , if t ≥ t0 + p Here, for every t ≥ t0 , the following statements hold: ɺɺ(t ) satisfies the constraints (2.176); (a) the real robot motion q (t ), qɺ (t ), q (b) the control u (t ) = u k (t ) is admissible, that is, the constraint (2.177) is fulfilled; (c) the real force of interaction of the robot with the environment, F (t ) , satisfies the constraint (2.178). Besides, the goal conditions (2.175) will be fulfilled for robot’s motion not later than the time instant

 1 CΓ x(t0 ) , if CΓ x(t0 ) + a ≥ ε t0 + ln tp1 =  λ ε −a t , if not 0

(2.217)

and for the force of interaction of the robot with environment not later than

 1 LCΓ x(t0 ) , if LCΓ x(t0 ) + La + Cρ ≥ δ t0 + ln λ δ − La − Cρ tp2 =  t , if not 0

(2.218)

The proof of Theorem 7 is given in Appendix H. The adaptive control scheme proposed here enables one to solve the contact task for robot with both stationary and nonstationary dynamics. We consider the classes of stabilized motions and forces and their stabilization accuracies depending on the level of initial and external perturbations of dynamics of the robot and its environment, as well as of the sensors’ errors, processing speed of the adaptation algorithm, and other parameters of the adaptive control scheme. Because the results presented are based on the realistic assumptions about the type and character of perturbations acting on the environment-robot system to

146

Dynamics and Robust Control of Robot-Environment Interaction

which the adaptive control scheme is to be applied, the obtained solution gives a justification of the possibility of its practical implementation in the controllers with adaptive features. It should also be emphasized that the proposed adaptive control laws provide stability of both interaction force and motion in a theoretically correct way, so that simulation experiments are not needed. The issues concerning practical implementation of the results presented in this section and the possible implications of these results for modification and generalization of the standard procedure of impedance control deserve further systematic research.

2.6 Position-Force Control – A Generalization 2.6.1 Models of robot and environment dynamics. Task setting Consider a robot interacting with its dynamic environment, assuming that the robot coordinates q and velocities qɺ are sufficient to represent the state of the environment, and that the number of interacting forces is smaller than the number of DOFs. The robot interacting with environment (see Appendix I and also [26]) can be modeled as:

H (q )qɺɺ + h(q, qɺ ) = τ + J T (q ) F

(2.219)

Mɶ (q )qɺɺ + Lɶ (q, qɺ ) = − J T (q ) F

(2.220)

where q is an n -dimensional vector of generalized robot coordinates; H ( q ) is an n × n matrix of inertia of the manipulator mechanism; Mɶ (q ) is n × n matrix representing environment inertia, h( q, qɺ ), Lɶ ( q, qɺ ) are n -dimensional nonlinear functions of centrifugal, Coriolis’ and gravitational terms; τ is an n dimensional vector of inputs; F is an m -dimensional vector of generalized forces acting on the end-effector from the environment side; J T ( q ) is an n × m full-rank matrix connecting the vector F with the generalized forces associated to the generalized robot coordinates q . Note that, by assumption, the interaction force F is uniquely determined by the robot motion q (t ) . However, the converse is not true. Namely, the matrix

Mɶ (q ) is positive semidefinite and rank Mɶ (q ) = m , so equation (2.220) contains only m independent differential equations. Hence, for a given interaction force F (t ) equation (2.220) does not have a unique solution with

A Unified Approach to Dynamic Control of Robots

147

ɺɺ , and the robot motion q (t ) cannot be determined by integrating respect to q equation (2.220). However, the interaction force F (t ) determines in full the motion of the environment (see Remark 1 in Appendix I). For these reasons it is more convenient to use an equivalent representation of the environment dynamics in the form: M (q )qɺɺ + L(q, qɺ ) = S T (q ) F (2.221) ɺ where M (q ) is a continuous m × n full rank matrix, L( q, q ) is a continuous m -vector function, and S (q ) is a nonsingular continuous m × m matrix. Note: For simplicity and without loss of generality in Sections 2.1 – 2.4 the case when the number n of generalized coordinates is equal to the number m of generalized forces is considered. A general case ( n ≥ m ) dealt with in Section 2.5 differs from the case n = m only by the representation of environment equation in the form solved with respect to force: −1

F = ( S (q ) S T (q ) ) S (q ) [ M (q )qɺɺ + L(q, qɺ )] as against

F = ( S T (q) )

−1

[ M (q)qɺɺ + L(q, qɺ )] .

In this section we make generalization of the environment vector equation for the case when the number of its one-dimensional linearly independent equations is smaller than n and equal to the number m which simultaneously coincide with the number of generalized forces. The matrix M (q ) can be partitioned into the submatrices M 1 , M 2 such that: M ( q ) =  M 1 ( q ) M 2 ( q )  . Here M 1 ( q ) is m × ( n − m) matrix and

M 2 ( q ) is m × m matrix with rank M 2 (q ) = m, ∀q ∈ R n . Then for the  q (1)  , where q (1) ∈ R n − m , q (2) ∈ R m , the expression M (q )qɺɺ can (2)  q  

vector q = 

be written as follows

M (q )qɺɺ = M 1 (q )qɺɺ(1) + M 2 (q )qɺɺ(2) It is assumed that the conditions for the existence and uniqueness of the solutions of (2.219), (2.221), with the initial conditions q (t0 ) = q0 , and

qɺ (t0 ) = qɺ0 are satisfied. As we have seen in the previous sections, the goal of the control of a robot in contact with its environment is to realize the programmed motion ( PM ) q p (t )

148

Dynamics and Robust Control of Robot-Environment Interaction

in the presence of the programmed force interaction ( PFI ) Fp (t ) with the environment, where q p (t ) and Fp (t ) must satisfy (2.219), (2.221). It is possible to choose q p (t ) and then determine Fp (t ) from (2.221), or alternatively, to (2) specify Fp (t ) and programmed motion q (1) p (t ) , whereby the subvector q p (t )

can be obtained by integrating (2.221). The control task is to define the control τ , which for all of the initial conditions q (t0 ) ≠ q p (t0 ), qɺ (t0 ) ≠ qɺ p (t0 ), F (t0 ) ≠ Fp (t0 ) , satisfies the following goal conditions:

q (t ) → q p (t ), qɺ (t ) → qɺ p (t ), t → ∞

(2.222)

F (t ) → Fp (t ), t → ∞

(2.223)

with the quality of transient processes specified in advance. However, the transient processes for both motion and force cannot be arbitrarily specified. Namely, because of equation (2.221), the motion transient process fully determines the force transient process. On the other hand, the force transient process determines the motion transient process only partially. For a given force transient process, part of motion coordinates q (1) (t ) has to be specified in order to determine the rest of motion coordinates. For these reasons, two subtasks can be identified. The first one is the attaining of the control goal (2.222), (2.223) with specified motion transient processes, and the second one is the attaining of the control goal (2.222), (2.223) with the specified processes for the force F (t ) and motion coordinates q (1) (t ) . The task of Fp (t ) stabilization can be defined as follows: Ensure the fulfillment of the condition (2.223) with the preset quality of transient processes specified by the integral equation: t

µ (t ) = ∫ Q ( µ (ω ) ) d ω

(2.224)

t0

where µ (t ) = F (t ) − Fp (t ) ; Q is a continuously differentiable vector function characterizing the quality of transient processes. Equation (2.224) can be presented in an equivalent form as: µɺ (t ) = Q ( µ (t ) ) (2.225)

A Unified Approach to Dynamic Control of Robots

149

However, the achievement of Fp (t ) stabilization can ensure the stabilization of m robot DOFs only, represented by the vector q (2) . So, for the force stabilization task to be well defined, it is also required to set the condition:

ɺ (1) ɺ (1) q (1) (t ) → q (1) p (t ), q (t ) → q p (t ), t → ∞

(2.226)

which should be realized in such a way to ensure the desired quality of transient processes defined by the equation:

ηɺɺ1 = P1 (η1 ,ηɺ1 )

(2.227)

where η1 (t ) = q (1) (t ) − q (1) p (t ) ; P1 is ( n − m ) -dimensional vector function continuously differentiable with respect to all arguments, such that equation (2.227) has the trivial solution η1 = 0 . The control should be synthesized in such a way that the perturbed robot motion and interaction force in closed-loops satisfy (2.225), (2.227). It is a priori adopted that the choice of the functions P1 and Q ensures exponential stability (thus implying asymptotic stability) in the whole of the trivial solution of the systems (2.225), (2.227). The control laws stabilizing the motion with the required quality of transient processes were presented in [10, 11]. The answer to the question how to ensure the contact force stability is quite simple: computed torque method [27, 28] ensures that the desired motion quality is achieved and, at the same time, guarantees that interaction force is asymptotically stable. In the text to follow we will focus our attention only on the control laws that stabilize the interaction force and on the way how to satisfy the stable position requirements. For the case when the number of contact force components is greater than the number of robot DOFs, this delicate task needs new analyses.

2.6.2 Control laws stabilizing the interaction force The control laws proposed in [10, 11] ensure force stabilization not only in case when m = n but also in case when m < n . These control laws can be easily modified to suit the case considered in Section 2.6.1. To define control laws, we introduce the notation: t

∆F = ∫ Q ( µ (ω ) ) dω t0

(2.228)

150

Dynamics and Robust Control of Robot-Environment Interaction

ɺ  Ψ (q, qɺ , F ) = M 2−1 (q )  S T (q ) F − L(q, qɺ ) − M 1 (q ) ( qɺɺ(1) p + P1 (η1 ,η1 ) )   qɺɺ(1) + P1 (η1 ,ηɺ1 )  q( F ) =  p   Ψ ( q, qɺ , F ) 

(2.229) (2.230)

and formulate the following assertion.

Assertion: The robot and environment dynamics equations (2.219), (2.221) for each of the following control laws:

ɺ   qɺɺ(1) p + P1 (η1 ,η1 ) τ I = H (q )   + h(q, qɺ ) − J T (q) F  Ψ ( q, qɺ , Fp + ∆F )  ɺ   qɺɺ(1) p + P1 (η1 ,η1 ) T  + h(q, qɺ ) − J (q ) ( Fp + ∆F ) ɺ q , q , F Ψ ( )  

τ II = H (q ) 

τ III

ɺ   qɺɺ(1) p + P1 (η1 ,η1 ) = H (q)   + h(q, qɺ ) − J T (q) ( Fp + ∆F ) ɺ  Ψ ( q, q, Fp + ∆F ) 

(2.231)

(2.232)

(2.233)

in a closed loop are equivalent to the reference equations (2.225), (2.227). Due to notation (2.230) the control laws (2.231) – (2.233) can be rewritten in the following way

τ I = H (q)q( Fp + ∆F ) + h(q, qɺ ) − J T (q) F

(2.234)

τ II = H (q) q( F ) + h(q, qɺ ) − J T (q) ( Fp + ∆F )

(2.235)

τ III = H (q)q( Fp + ∆F ) + h(q, qɺ ) − J T (q ) ( Fp + ∆F )

(2.236)

The three control laws defined by the above assertion stabilize the desired interaction force with a specified quality of transient processes. These control laws utilize the feedback with respect to q, qɺ and force F . Let us show that these control laws ensure force stabilization. To illustrate this, let us present the proof of Assertion for the control law (2.231) separately. Substituting the control law (2.231) into the robot dynamics equation (2.219) we obtain:

H (q ) ( qɺɺ − q ( Fp + ∆F ) ) = 0

A Unified Approach to Dynamic Control of Robots

151

which, in view of nonsingularity of the matrix H ( q ) , directly implies:

ɺ qɺɺ(1) − qɺɺ(1) p − P1 (η1 , η1 ) = 0

(2.237)

qɺɺ(2) = Ψ ( q, qɺ , Fp + ∆F )

(2.238)

Since rank M 2 ( q ) = m the environment dynamics (2.221) can be written in the form:

qɺɺ(2) = M 2−1 (q )  S T (q ) F − L(q, qɺ ) − M 1qɺɺ(1) 

(2.239)

Let us eliminate qɺɺ(2) from (2.238) and (2.239). Then, in accordance with the notation (2.229), we will have

ɺ S T (q ) ( F − Fp − ∆F ) − M 1 ( qɺɺ(1) − qɺɺ(1) p − P1 (η1 , η1 ) ) = 0 Taking into account the identity (2.237) we obtain:

S T (q ) ( F − Fp − ∆F ) = 0 Since S T ( q ) is by assumption nonsingular, it follows that the closed-loop robot dynamics is described by the equations:

ηɺɺ1 = P1 (η1 ,ηɺ1 ) µɺ (t ) = Q ( µ (t ) ) and the control goal is fulfilled. Below we give the integral proof of Assertion. Let us add to each of the control laws (2.234) – (2.236) one by one three evident identities. Then we obtain three pairs of the following expressions:

 H (q )q ( Fp + ∆F ) + h(q, qɺ ) = τ I + J T (q ) F  T M (q )q ( Fp + ∆F ) + L(q, qɺ ) = S (q ) ( Fp + ∆F ) 

(2.240)

H (q) q( F ) + h(q, qɺ ) = τ II + J T (q) ( Fp + ∆F )   M (q) q( F ) + L(q, qɺ ) = S T (q) F 

(2.241)

H (q )q ( Fp + ∆F ) + h(q, qɺ ) = τ III + J T (q ) ( Fp + ∆F )   M (q )q ( Fp + ∆F ) + L(q, qɺ ) = S T (q ) ( Fp + ∆F ) 

(2.242)

152

Dynamics and Robust Control of Robot-Environment Interaction

Subtracting the equations of the expressions (2.240), (2.241), (2.242) from (2.219) and (2.221), respectively, we obtain the relations:

0   qɺɺ − q ( Fp + ∆F )   H (q)  M (q) − S T (q)    =0 µ − ∆F    

(2.243)

 H (q ) − J T (q)   qɺɺ − q( F )      =0 0   µ − ∆F   M (q)

(2.244)

 H (q) − J T (q)   qɺɺ − q( Fp + ∆F )     =0 T µ − ∆F   M (q) − S (q)  

(2.245)

Since all the system matrices in the above equations are nonsingular (see Remarks 4-6 in Appendix I), it follows that for all control laws τ I ,τ II ,τ III one of the following conditions is fulfilled:

qɺɺ − q ( Fp + ∆F ) = 0  qɺɺ − q ( F ) = 0   or  µ − ∆F = 0  µ − ∆F = 0 

(2.246)

Using (2.228) – (2.230) we obtain:

ɺ qɺɺ(1) − qɺɺ(1) p − P1 (η1 , η1 ) = 0

(2.247)

t

µ = ∫ Q ( µ (ω ) ) dω

(2.248)

t0

which finally proves that the robot environment with each of the control laws (2.231) – (2.233) behaves according to the reference equations:

ηɺɺ1 = P1 (η1 ,ηɺ1 )

(2.249)

µɺ (t ) = Q ( µ (t ) )

(2.250)

Consider now the question: is it sufficient to stabilize Fp (t ) and q (1) p (t ) in

(

order to have the real motion converging to the programmed one q (t ) → q p (t ), when t → ∞ ) ? Note: The above assertion ensures that the conditions (2.223), (2.226) are satisfied. To have stability of the closed-loop system ensured (i.e. the condition (2.222) fulfilled) it is necessary (and sufficient too) to ensure that

A Unified Approach to Dynamic Control of Robots

ɺ (2) (t ) → qɺ (2) q (2) (t ) → q (2) p (t ), q p (t ), t → ∞

153

(2.251)

Let any of the control laws (2.231) – (2.233) be applied onto the system (2.219), (2.221). Using the environment dynamics model (2.221) we obtain:

M (q)qɺɺ − M (q p )qɺɺp + L(q, qɺ ) − L(q p , qɺ p ) = S T (q) F − S T (q p ) Fp Since, according to Assertion, the system (2.219), (2.221) in a closed loop with the control laws (2.231) – (2.233) is equivalent to the system in deviation form (2.225), (2.227), the preceding relation reduces to the equation:

ηɺɺ2 + K (η ,ηɺ , t ) = M 2−1 (η + q p ) S T (η + q p ) ( F − Fp )

(2.252)

where

η2 (t ) = q (2) (t ) − q (2) p (t ), η (t ) = q (t ) − q p (t ) and:

K (η ,ηɺ , t ) = M 2−1 (η + q p ) {L (η + q p ,ηɺ + qɺ p ) − L (q p , qɺ p ) T T +  M (η + q p ) − M (q p )  qɺɺp −  S (η + q p ) − S (q p )  Fp

+ M 1 (q p +η ) P1 (η1 ,ηɺ1 )} . If F (t ) = Fp (t ) , this equation reduces to:

ηɺɺ2 + K (η ,ηɺ , t ) = 0

(2.253)

Evidently, the trivial solution η 2 (t ) = 0 is stable only if the environment has such properties that ensure fulfillment of the condition q (2) (t ) → q (2) p (t ) , as

t → ∞ Hence, if F (t ) ≠ Fp (t ), ∀t ≥ t0 , the stabilization of Fp (t ) does not necessarily ensure stability of the motion in contact with the environment. It is shown that the motion of the robot and environment in a closed loop is described by equations (2.249), (2.252):

ηɺɺ1 = P1 (η1 ,ηɺ1 ) ηɺɺ2 + K (η ,ηɺ , t ) = M 2−1 (η + q p ) S T (η + q p ) ( F − Fp ) Let us formulate sufficient conditions for stabilization of q (2) p (t ) , i.e. the conditions for which exponential stability of the trivial solution of the system (2.253) induces exponential stability of the trivial solution of the system (2.252).

154

Dynamics and Robust Control of Robot-Environment Interaction

Introduce the notation:

η  η  x   ηɺ1  x1 =  1  , x 2 =  2  , x =  1  , ( x1 ) =    P(η1 ,ηɺ1 )  ηɺ1  ηɺ 2   x2  0   ηɺ2   ( x2 , x1 , t ) =  , β ( x2 , x1 , t ) =  −1   T  − K (η ,ηɺ , t )   M 2 (η + q p (t ) ) S (η + q p (t ) )  Then, the system (2.249), (2.252) can be rewritten as:

xɺ1 =

( x1 )

(2.254)

xɺ2 = ( x2 , x1 , t ) + β ( x2 , x1 , t ) µ

(2.255)

On linearizing the equation (2.255) in the neighborhood of x = 0 , we get:

xɺ 2 = A(t ) x 2 + B(t ) x1 + α ( x 2 , x1 , t ) + β ( x 2 , x1 , t ) µ

(2.256)

where A(t ) and B (t ) are 2m × 2m and 2m × 2( n − m) matrices, respectively,

α ( x2 , x1 , t ) = o ( x ) when x → 0 and sup A(t ) < ∞ since q p (t ), qɺ p (t ), Fp (t ) t

belong to the bounded regions. We assume here that all vector functions possess the same smoothness as in Section 2.3.2. Let the vector functions P1 and Q in (2.225), (2.227) ensure that for their arbitrary solutions µ (t ), η1 (t ) the following estimates hold:

µ (t ) ≤ D1e − λ (t −t ) µ (t0 )

(2.257)

x1 (t ) ≤ D2 e − λ (t −t0 ) x1 (t0 )

(2.258)

0

with the constants D1 , D2 , λ > 0 . Sufficient conditions for the exponential stability of the solution of the system of differential equations (2.254), (2.255) are given by the following theorem analogous to the Theorem 1:

Theorem 8. Let the following conditions be satisfied [10, 11]: 1. The linear system

xɺ 2 = A(t ) x 2 is regular, i.e. there exists the limit

(2.259)

A Unified Approach to Dynamic Control of Robots

155

t

2m 1 lim ∫ Sp A(ω ) dω = σ 0 and σ 0 = ∑ α k , t →∞ t i =1 t0

where α k ( k = 1,2,..., 2m) are characteristic indices of the solution of the system (2.259), Sp A is the trace of matrix A . 2. All characteristic indices α k ( k = 1, 2,… , 2m) are negative. 3. The index λ in (2.257), (2.258) and the number γ > 0 satisfy the inequalities:

max α k < −γ < 0 , k

−λ < −γ

then, for sufficiently small initial perturbations x (t0 ) and µ (t0 ) , the transient process of the system (2.255) satisfies the inequality:

 b x1 (t0 ) + c µ (t0 )  − λ (t −t0 ) , x2 (t ) ≤  a x2 (t0 ) + e λ −γ  

∀t ≥ t0

(2.260)

for some positive constants a, b, c , and consequently, the system (2.254), (2.255) is exponentially stable. The proof of Theorem 8 is given in Appendix J. The control laws (2.231) – (2.233) ensure a desired quality of stabilization of the robot interaction force Fp (t ) with the environment. At the same time, the stabilization of the robot motion q p (t ) can also be achieved for sufficiently small perturbations.

2.6.3 Example To illustrate the obtained theoretical results, let us consider the following hypothetical contact task: the 2-DOF sliding manipulator has to move a tool over the support which behaves as a system with distributed parameters (Fig. 2.5). The control goal is to realize the nominal motion along the x -axis x p (t ) = x 0 (t ) = V0 t , V0 = const and nominal force along the y -axis

Fy p (t ) = Fy0 = F 0 = const. The component of contact force along the x -axis is the sum of the friction terms:

156

Dynamics and Robust Control of Robot-Environment Interaction

Fx = hx xɺ +ν Fy sgn xɺ

(2.261)

while the contact force component Fy is the sum of inertial, frictional and elastic terms:

Fy = m y ɺyɺ + h y yɺ + k y y

(2.262)

Fig. 2.5 Robot in contact with dynamic environment

It is adopted in (2.261), (2.262) that m y = me , where me is the equivalent mass representing the contribution of the environment inertia. Further, h y , hx denotes viscous friction, k y is the environment stiffness and ν is the static friction coefficient. Using the notation given in Fig. 2.5 the robot dynamics model is:

(m1 + m 2 ) ɺxɺ = τ x − Fx

(2.263)

m 2 ɺyɺ = τ y − Fy

(2.264)

After introducing the notation

τ x  (m + m2 ) 0   x , q =   , τ =   , F =  Fy  , H (q ) =  1 0 m2   y  τ y   h xɺ   −ν sgn xɺ  h(q, qɺ ) =  x  , J (q ) =  ,  0   −1 

M (q ) = 0 m y  , L(q, qɺ ) = hy yɺ + k y y

S T (q ) = [1]

157

A Unified Approach to Dynamic Control of Robots

equations (2.261) – (2.264) reduce to (2.219), (2.221). Determine now the remaining nominal motion component from (2.262):

y p (t ) = y 0 (t ) =

1 0 F ky

(2.265)

and choose equations specifying the desired motion and force transient processes in the form:

where

ηɺɺx + 2 ς xcω xcηɺ x + ω xc2 η x = 0

(2.266)

µɺ + T yc µ = 0

(2.267)

η x = x − x p = x − x 0 , µ = F − Fy = F − F 0 p

and

ς xc , ω xc , T yc

are

constants. Let us apply the control law (2.233), which in this case is:

 0 t  0 2 ɺ ɺɺ ɺ τ x = (m1 + m2 ) ( x − 2ς xcω xcη x − ωxcη x ) + hx x +ν  F + ∫ µ d ω  sgn xɺ (2.268)   t0    m  m2   0 t τ y =  + 1   F + ∫ µ d ω  − 2 (hy yɺ + k y y ) m   t0  y   my

(2.269)

It is obvious that with this control law equations (2.266), (2.267) represent the closed-loop system behavior. In order to examine the nominal motion and force stability, the environment equation (2.262) has to be considered in the deviation form:

m yηɺɺy + h yηɺ y + k yη y = µ

(2.270)

It is easy to check that if the conditions of Theorem are satisfied, i.e. equation (2.270) has characteristic values α 1 , α 2 < − γ < 0 , for some γ , and

ς xc , ω xc , T yc are chosen such that the characteristic values of (2.266), (2.267) are smaller than α 1 , α 2 , then the closed-loop system is stable. 2.6.4 Conclusion The control methodology presented in this section is applicable to a wide class of robotic tasks. It exhibits the same features as in the case when the dimension of the control input vector and contact force vector are equal. Namely, the synthesis of control laws is performed in order to ensure local asymptotic

158

Dynamics and Robust Control of Robot-Environment Interaction

stability of the desired robot behavior. Stabilization of both desired robot motion and interaction force is realized simultaneously, for the cases in which environment dynamics is described by second-order differential equations. All necessary modifications of control laws stabilizing the interaction force of the robot and environment have been performed and the corresponding stability theorem has been proved. The presented theorem formulates sufficient conditions for asymptotic stability of the system in the first approximation (local stability). It should be emphasized that the character of this position (displacement) stability depends substantially on the nature of programmed motion and programmed force of interaction. Nevertheless, the presented linear analysis gives a very important criterion that must be verified for any force-based control law. It should be pointed out that the inadequate accuracy of the environment dynamics model can significantly influence the contact task performance. Inaccuracies of the robot and environment dynamics models, as well as dynamic control robustness are considered in Sections 2.4 and 2.5.

2.7 Position-Force Control in Cartesian Space 2.7.1 Introduction All the results obtained previously on the basis of dynamic models of the robot and environment in the system of generalized coordinates can be, without loss of generality, directly transposed into the Cartesian coordinate frame of the robot’s end-effector. In this operation, the representations of the robot and environment dynamics models as the systems of differential equations (2.6) and (2.11) retain their form. In the previous sections, the control laws stabilizing simultaneously the robot motion and interaction force have been synthesized. These control laws possess exponential stability of closed-loop systems and ensure the preset quality of transient processes of motion and interaction force. However, for the control laws stabilizing the desired interaction force with a preset quality of transient processes, only sufficient conditions for the exponential stability of motion were given. This section presents a more detailed description of the environment dynamics. In the cases when the environment dynamics can be approximated sufficiently well by a linear time-invariant model in the Cartesian space, necessary and sufficient conditions for the exponential stability of both motion and force are derived, and the corresponding control laws are defined.

159

A Unified Approach to Dynamic Control of Robots

2.7.2 Task setting Consider the model of the robot interacting with its environment (2.6):

H (q )qɺɺ + h(q, qɺ ) =τ + J T (q ) F The model of environment dynamics can be described by equation (2.11):

M (q )qɺɺ + L(q, qɺ ) = S T (q ) F In this section, as in all previous sections except for Sections 2.5, 2.6, to simplify the analysis, we adopted that m = n , which means that the number of components of interactive force is equal to the number of powered DOFs. Using the operational space approach [29], the end-effector equations of motion can be written in the task frame. Let z be an n -dimensional vector of external coordinates specifying position and orientation in the chosen task frame, and let Τ be a vector of the control force coordinates. If the relation between z and q is defined by the bijective function χ :

z = χ (q) then zɺ =

(2.271)

∂χ (q )qɺ = J (q )qɺ , ɺɺ z = J (q )qɺɺ + Jɺ (q )qɺ and equations (2.6), (2.11) ∂q

can be transformed so to acquire the forms:

Λ ( z ) ɺɺ z + v ( z , zɺ) = Τ + F

(2.272)

( z ) ɺɺ z + ( z , zɺ ) = − F where:

(2.273)

Λ ( z ) = J −T ( χ −1 ( z ) ) H ( χ −1 ( z ) ) J −1 ( χ −1 ( z ) ) ,

(

v ( z , zɺ ) = J −T ( χ −1 ( z ) ) h χ −1 ( z ), J −1 ( χ −1 ( z ) ) zɺ − Λ ( z ) Jɺ ( χ −1 ( z ) ) J −1 ( χ −1 ( z ) ) zɺ,

)

Τ = J − T ( χ −1 ( z ) ) τ ,

( z ) = − S −T ( χ −1 ( z ) ) M ( χ −1 ( z ) ) J −1 ( χ −1 ( z ) ) ,

(

( z , zɺ) = − S −T ( χ −1 ( z ) ) L χ −1 ( z ), J −1 ( χ −1 ( z ) ) zɺ +

( )

Here J −T (⋅) = J −1

T

(⋅) .

( z ) Jɺ ( χ −1 ( z ) ) J −1 ( χ −1 ( z ) ) zɺ.

)

160

Dynamics and Robust Control of Robot-Environment Interaction

It is assumed that all the functions in (2.6), (2.11), i.e. (2.272), (2.273) are continuously differentiable with respect to all variables, ensuring thus the existence and uniqueness of the solutions for (2.272), (2.273), with the initial conditions z (t0 ) = z0 , zɺ (t0 ) = zɺ0 . Let z p (t ) be the desired motion and let Fp (t ) be the desired interaction force, satisfying (2.273). The control objective is to realize the asymptotically stable z p (t ), Fp (t ) , i.e. to achieve:

( z (t ), zɺ(t ), F (t ) ) → ( z p (t ), zɺ p (t ), Fp (t ) ) ,

t →∞

It is additionally required that the closed-loop system possesses contact force transient processes (force dynamics) specified by the equation:

µɺɺ = Q( µ , µɺ )

(2.274)

where µ (t ) = F (t ) − Fp (t ) . Alternatively, it may be required that the closedloop system possesses motion transient processes (motion dynamics) specified by the equation:

ηɺɺ = P(η ,ηɺ )

(2.275)

where η (t ) = z (t ) − z p (t ) . The vector functions P and Q are assumed to be continuously differentiable with respect to all arguments, so that equations (2.274) and (2.275) have the asymptotically stable trivial solutions η = 0 and µ = 0 , respectively. Note that because of (2.273) Q ( µ , µɺ ) and P (η ,ηɺ ) can not be arbitrarily specified at the same time. Thus the force stabilization task and motion stabilization task should be distinguished. 2.7.3 Relation to previous results The computed torque-based control law has been proposed to stabilize the desired motion z p (t ) with the quality of transient processes specified by (2.275), and also to stabilize the contact force Fp (t ) :

(

Τ = U z , zɺ, ɺɺ z p + P (η ,ηɺ ), f ( z , zɺ, ɺɺ z p + P (η ,ηɺ ) )

z , F ) = Λ ( z ) ɺɺ z + v ( z , zɺ ) − F , f ( z , zɺ, ɺɺ z) = − where U ( z , zɺ, ɺɺ

)

(2.276)

( z ) ɺɺ z − ( z , zɺ ) .

A Unified Approach to Dynamic Control of Robots

161

By assumption, the vector function P (η ,ηɺ ) has the property of assuring an asymptotic stability of the entire trivial solution η = 0 of the system (2.275), i.e. η (t ) → 0, ηɺ (t ) → 0, t → ∞ . Because of the continuity of the function f we have: F (t ) → Fp (t ), t → ∞ . We give here the control laws: t t      T = U z , zɺ, Ψ  z , zɺ, Fp + ∫ Q ( µ (ω ) ) dω  , Fp + ∫ Q ( µ (ω ) ) dω      t0 t0    

where Ψ ( z , zɺ, F ) = −

−1

(2.277)

( z ) [ F + ( z , zɺ) ] , to stabilize a desired interaction

force with the quality of transient processes specified by the equation:

µɺ = Q ( µ )

(2.278)

The control laws are applicable only if the matrix ( p ) is invertible. However, these control laws allow stable motion of the robot in contact with environment if the environment possesses the properties that ensure fulfillment of the limiting condition z (t ) → z p (t ), t → ∞ , i.e. the environment stabilizes motion of the robot in contact. 2.7.4 Control laws for specified force dynamics Let us observe first the relation between the closed-loop motion and force dynamics. Assume that the robot is in closed loop and that the corresponding motion and force dynamics are described by (2.275), (2.274), i.e. by the vector functions P (η ,ηɺ ) and Q ( µ , µɺ ) , respectively. If the motion dynamics is specified (vector function P (⋅, ⋅) given), then, because of (2.273), the vector function Q ( µ , µɺ ) is uniquely determined. Furthermore, if η = 0, ηɺ = 0 , then µ = 0 . However, in the converse case, having Q( µ , µɺ ) specified, the vector function P (η ,ηɺ ) is not generally uniquely determined. Moreover, µ = 0, µɺ = 0 does not imply η = 0 , therefore the force stabilization does not necessarily imply the motion stabilization. To simplify derivation, let us focus on a particular class of control laws (2.276) (obtained after subtracting equation (2.273) from (2.272) and compensating for all nonlinearities)

162

Dynamics and Robust Control of Robot-Environment Interaction

(

τ = J T ( χ −1 ( z ) ) ( Λ ( z ) +

)

( z ) ) ( ɺɺ z p + Γ1ηɺ + Γ 2η ) + v ( z , zɺ) + ( z , zɺ) (2.279)

where z = χ ( q ) , which enable linear closed-loop dynamics

ηɺɺ = Γ1ηɺ + Γ2η

(2.280)

where Γ1 and Γ2 are n × n constant matrices. If Γ1 , Γ 2 are chosen so that

det ( − I λ 2 + Γ1λ + Γ 2 ) is Hurwitz, equation (2.280) represents a class of

admissible motions, and the convergence to zero of the errors is ensured. However, the force dynamics is determined by Γ1 , Γ2 . Our aim is to attain the given force transient processes (2.274), still ensuring stability of the motion defined by (2.280) with some matrices Γ1 , Γ 2 . However, the problem is to determine if any of the corresponding matrices exist, and how to find them. To obtain the correspondence between the motion and force dynamics, assume that in the neighborhood of the nominal trajectory the environment dynamics equation (2.273) can be approximated sufficiently well by the equation

µ = Mηɺɺ + Lhηɺ + Lkη

(2.281)

where M , Lh , Lk are n × n constant matrices of inertia, viscous friction and stiffness, respectively. This approximation is often used in Cartesian space. Note also that (2.281) does not impose additional restrictive conditions such as real positiveness of the corresponding transfer function [30].

η 

µ 

 

 

Defining x =   , Γ = Γ 2  Γ1  , L =  Lk  Lh  , y =   , equations     ηɺ µɺ (2.280), (2.281) can be written as:

0 xɺ =  n Γ 2

In  x ≡ Γη x Γ1 

µ = ( MΓ + L) x ≡ Cx

(2.282) (2.283)

From (2.283) we have

 C  η  y =   ɺ ≡ α x Γ C η   η  In a more expanded form, the matrix α can be written as follows:

(2.284)

163

A Unified Approach to Dynamic Control of Robots

 MΓ + L

MΓ + L



2 k 1 h α = ( M Γ + L ) Γ ( M Γ + L ) Γ + M Γ 2 + Lk  1 h 2 1 h 1 

If the matrix α is nonsingular, then (2.282), (2.284) yield

yɺ = Qµ y , where Qµ = α Γηα −1

(2.285)

Introducing the notation:

α 0 = Lk + M Γ 2 , α i +1 = βi Γ 2 β0 = Lh + M Γ1 , βi +1 = α i + βi Γ1 , i = 0,1

(2.286)

µɺɺ = Q1 µɺ + Q2 µ

(2.287)

we obtain:

where

[Q2 |

α β  Q1 ] = [α 2 | β 2 ]α −1 , α =  0 0  .  α1 β1 

Therefore, the linear closed-loop motion dynamics (2.280) induces the linear force dynamics given by (2.287). Furthermore, the motion and force transient processes are the same. It follows from the fact that because of (2.285) the matrices Γη , Qµ have equal eigenvalues. Note: When the matrix α is singular, the force dynamics in closed loop is still linear, but it can not be represented in the form of (2.287). In fact, the following cases are possible: (i) rank α < n . Force variables µ are not independent (see (2.283), (2.284)). However, this contradicts the assumption that there are n independent forces. (ii) rank α = n . It is easily seen from (2.283), (2.284) that in this case the variable µɺ is dependent on µ and the induced force dynamics equation reduces to the degenerative form:

Q1 µɺ + Q2 µ = 0

(2.288)

Namely, let Yi , i = 1, 2,..., 2n be the rows of α . Then n of them (say first

n ) are linearly independent, and the matrix Q2 =  qij  containing no zero n× n row, such that Yi + n =

n

∑q Y j =1

ij

j

, exists. Therefore

164

Dynamics and Robust Control of Robot-Environment Interaction n

n

j =1

j =1

µɺ i = Yi + n x = ∑ qijY j x = ∑ qij µ j . When any n rows of α are linearly independent, (2.288) is obtained. (iii) n < rank α < 2n . Induced force dynamics is represented by the linear system consisting of k equations of the type (2.287) and n − k equations of the type (2.288), where k = rank α − n . Singular matrix α usually appears when the force vector µ , still being linearly independent, effectively depends on some (but not on all) DOFs. For example, if η i is the i -th generalized coordinate, then µ j = k jη i , µ i = hiηɺ i are linearly independent, but they are only influenced by the i -th DOF. Although in this case there are n independent forces to be controlled, it can be seen that the environment configuration space is a subspace of the robot configuration space, which contradicts the assumption that the environment possesses m = n DOFs. The relation between stability characteristics of the motion and force dynamics equations (2.280), (2.287) is given by the following theorem based on the well-known linear system theorem [31]. Theorem 9. A nonsingular coordinate transformation (preserving eigenvalues) T

T

between the motion η T , ηɺ T  and force  µ T , µɺ T  variables will exist if and only if the system (2.282), (2.283) is completely observable and the observability index is 2. A corollary of the theorem is that if a desired force dynamics is specified in the form of equation (2.287) and all characteristic roots are stable, then the corresponding motion dynamics equation will have a stable solution with the same characteristic roots, provided that the matrix α is nonsingular. Finally, let us answer the question on how to find the corresponding matrices Γ1 , Γ2 , having the matrices Q1 , Q2 specified, ensuring stable force dynamics. Let us introduce the matrices:

c0 =  Lk

| Lh − I n  , c1 = [ I n | M ]

then the matrix C can be written as C = c0 + c1Γη . It follows from (2.287) and (2.284)

A Unified Approach to Dynamic Control of Robots

165

 C  η  µ  µɺɺ = [Q2 | Q1 ]   = [Q2 | Q1 ]     µɺ  C Γη  ηɺ  ɺɺ yields the relation: Differentiating (2.283) twice and substituting µ

 C  C Γη2 = [ Q2 | Q1 ]   C Γη 

(2.289)

which can be reduced to:

(c0 + c1Γη ) Γη2 = Q1 (c0 + c1Γη )Γη + Q2 (c0 + c1Γη )

(2.290)

c1Γη3 + (c0 − Q1c1 )Γη2 − (Q1c0 + Q2 c1 )Γη − Q2 c0 = 0

(2.291)

i.e.

Given the environment parameters Lk , Lh , M (i.e. c0 , c1 ) and desired (stable) force dynamics specified by Q1 , Q2 , the solution (if it exists) of the obtained equation yields the matrices Γ1 , Γ2 . If there exists a matrix Γη , satisfying equation (2.291), and if the matrix α is nonsingular, then the desired force dynamics specified by (2.287) can be realized by the control law (2.279). At the same time, the closed-loop robot motion will be stable, with the motion dynamics satisfying (2.280) and force dynamics satisfying (2.287). The proposed control law is applicable under sufficient condition that the matrix α is nonsingular. However, this condition is also necessary for the existence of the solution of the problem posed in Section 4.2. Namely, according to Note, the matrix α is singular if either m < n (not considered) or closedloop force dynamics equation has a degenerative form. Since the matrix α depends on Γ1 , Γ2 and thus on Q1 , Q2 , the latter case arises from an inappropriate specification of desired force dynamics. A question should be posed: Is it possible to have the matrices Q1 , Q2 always chosen so as to ensure nonsingularity of the matrix α ? The answer gives the following theorem. Theorem 10. Given the n × n matrices M , Lh , Lk , with rank [ Lk | Lh | M ] = n , the matrices Q1 , Q2 can always be chosen so that the corresponding matrices Γ1 , Γ2 exist and the matrix α is nonsingular.

166

Dynamics and Robust Control of Robot-Environment Interaction

Proof of the theorem is given in Appendix K. Note that due to the correspondence between the closed-loop motion and force dynamics, the control law stabilizes the overall system with either motion or force of transient processes being preset. 2.7.5 Example To illustrate the obtained theoretical results let us consider the following hypothetical contact task: The planar two-link revolute manipulator, whose workspace is the horizontal plane, has to move the workpiece over a flat support which may rotate about the horizontal axis (Fig. 2.6).

p

x p

y

q

m

2

y R

m2 ϕ

m1

l2 l1

q

1

Fg

O h R ϕ mg

y

0

Fig. 2.6 Robot in contact with dynamic environment

The task is to realize the nominal motion along the x -axis:

x p (t ) = x 0 (t ) = V0 (t ), V0 = const > 0

(2.292)

and nominal contact force between the end-effector and the workpiece along the y -axis:

A Unified Approach to Dynamic Control of Robots

Fy p (t ) = Fy0 = F 0 = const > 0

167

(2.293)

Adopting that q = [ q1 q2 ] , neglecting the joints friction, and using the T

notation given in Fig. 2.6, the matrices in the robot dynamics model (2.6) can be written as:

(m1 + m2 )l12 + m2l2 (l2 + 2l1 cos q2 ) H (q) =  m2l2 (l2 + 2l1 cos q2 ) 

m2l2 (l2 + 2l1 cos q2 )  m2l22 

(2.294)

− qɺ (qɺ + 2qɺ1 ) h(q, qɺ ) = m2 l1l 2 sin q 2  2 22  qɺ 2  

(2.295)

− (l sin q1 + l 2 sin (q1 + q 2 )) − l 2 sin (q1 + q 2 ) J (q) =  1   l1 cos q1 + l 2 cos (q1 + q 2 ) l 2 cos (q1 + q 2 ) 

(2.296)

Introduce the Cartesian position vector of the manipulator end-effector

z =[x

y ] and contact force vector F =  Fx Fy  . The relation between z T

T

and q is:

 l1 cos ( q1 ) + l2 cos ( q1 + q2 )  z = χ (q) =   l1 sin ( q1 ) + l2 sin ( q1 + q2 ) − y0  The contact force component Fy (assuming y small, y < 0 ) may be approximated by a sum of the inertial Fi = ( m + me ) ɺɺ y , the frictional Fh = hy yɺ , and the gravitational Fg = − k y y terms:

Fy = (m + me ) ɺyɺ + h y yɺ − k y y

(2.297)

while the component of contact force along the x -axis, Fx , is a sum of the inertial and frictional terms:

Fx = mxɺɺ + hx xɺ + ν x ( Fy − me ɺɺ y )sgn ( xɺ )

(2.298)

It has been adopted that m in (2.297), (2.298) is the workpiece mass, me is the support equivalent mass, whereas hy , hx and ν x denote the viscous friction coefficients and static friction coefficient respectively; k y = hme g / R 2 , where

168

Dynamics and Robust Control of Robot-Environment Interaction

R and h are the distances shown in Fig. 2.6, and g is the gravity acceleration. Let us determine the remaining force and motion components from (2.297), (2.298):

Fx p (t ) = Fx0 (t ) = hxV0 + ν x F 0

y p (t ) = y 0 (t ) = −

(2.299)

1 0 F ky

(2.300)

and set the control goal be exponentially stable nominal system behavior, with the force transient processes specified by the equation:

µɺɺ = Q1 µɺ + Q2 µ

(2.301)

with µ being exponentially stable. The linearization of (2.297), (2.298) for

V0 + ηɺx > 0 gives (2.281) with

 hx ν x hy  0 −ν x k y  m ν x m  M = L L , , = =   0 − k  h k  hy  y   0 m + me  0  Choose the matrices Q1 , Q2 such that the matrix Qµ has the eigenvalues

λ1 , λ 2 , ..., λ 2 n and the conditions of Lemmas 5 and 6, given in Appendix K, are satisfied. Then the matrix α is nonsingular. Solving (2.291) yields the matrices Γ1 , Γ2 . As was shown in Section 2.7.4, the control law

z = χ (q)

(

τ = J T (q) ( Λ ( z ) +

(2.302)

( z ) ) ( ɺɺ z p + Γ1ηɺ + Γ 2η ) + v ( z , zɺ ) + ( z , zɺ )

)

(2.303)

ensures that in the closed-loop, force and motion deviations µ behave according to (2.301). In fact, the environment dynamics equations in deviation form (2.297), (2.298) become:

µ y = (m + me )ηɺɺy + h yηɺ y − k yη y

(2.304)

µ x = mηɺɺx + hxηɺx +ν x ( µ y − meηɺɺy )

(2.305)

Since the control law (2.303) ensures (ηɺɺ,ηɺ ,η ) → 0 as t → ∞ , it is clear that (2.304), (2.305) implies ( µɺ , µ ) → 0 .

A Unified Approach to Dynamic Control of Robots

169

Now we consider the case when the control law (2.277) is applied. Let us specify the desired force transient processes by the equations:

µɺ x = − Q11µ x , Q11 > 0

(2.306)

µɺ y = − Q22 µ y , Q22 > 0

(2.307)

As shown in [10], the control law (2.277) will certainly ensure that in a closed loop the force deviations µ x , µ y satisfy the above equations, i.e. the desired force dynamics will be achieved. However, the motion will be unstable. Namely, the behavior of the closed-loop system will be described by equations (2.303) – (2.307). Since equations (2.304), (2.307) are independent of the variables η x , µ x , the characteristic polynomial associated to the overall system will have the factor:

Py (λ ) = ( (m + me )λ 2 + hy λ − k y ) ( λ + Q22 )

and unstable roots because k y > 0 . Therefore, in spite of achieving ( µɺ , µ ) → 0 as t → ∞ , the motion of the robot in contact is unstable. Note that if the environment dynamics equations had stable solutions (the case when − k y > 0 ) the conditions of Theorem 1 in Section 2.3 would be satisfied, and the motion would also be asymptotically stable. 2.7.6 Conclusion The task of stabilizing both the robot motion and interaction force simultaneously in Cartesian space, within the scope of the unified approach to control laws synthesis for robot manipulator in contact with dynamic environment, has been considered. The task is solved under less restrictive conditions imposed on environment dynamics than in Section 2.3, where some particular environment properties are required to ensure the overall system stability. The one-to-one correspondence between the closed-loop motion and force dynamics equations is obtained and a unique control law that ensures system’s stability and preset either motion or force transient processes, is proposed. The perturbed environment dynamics equation was assumed to be linear in Cartesian space, so that the obtained results are applicable to nonlinear environment dynamics in the range of validity of the linear approximation.

170

Dynamics and Robust Control of Robot-Environment Interaction

2.8 New Realization of Hybrid Control 2.8.1 Introduction In Chapter 1 we presented sufficiently clear hybrid control. In the introduction to Chapter 2 we also gave a brief account of the basic idea of such control. As we have already pointed out, the basic idea of this control is that the control task can be divided into two independent subtasks in a certain coordinate space. One is the robot motion control along a predetermined part of the coordinates (directions) and the other is the control of the interaction force of the robot and environment along the rest of the coordinates (directions):

U1 ( x, xɺ, ɺɺ x ) =τ x

(2.308)

U 2 ( Fy ) = τ y

(2.309)

where τ x , τ y are the corresponding control inputs. However, there are no instances of the robot’s interaction with the environment in which the possibility of such partition would exist. Even for the simplest manipulation mechanism interacting with environment, the following partition is obtained:

U1 ( x, xɺ, ɺɺ x, Fx ) = τ x

(2.310)

U 2 ( y, yɺ , ɺɺ y, Fy ) = τ y

(2.311)

where equations (2.310) and (2.311) represent the robot’s dynamics model in two subspaces; τ x and τ y are the driving forces or torques acting in the corresponding subspaces. The holonomic constraints in the traditional hybrid control are not the only limitations that are important. It should be mentioned that for the Cartesian coordinate robots the controllers with first and higher order integrators can compensate to a certain extent for the perturbations due to friction forces. For the non-Cartesian robots the compensation for friction forces is practically impossible. All these problems become evident if we consider the simplest example of a two-link planar manipulator with the joints of sliding type. In the coordinate frame illustrated in Fig. 2.7, the equations of robot dynamics can be written in the form: (m1 + m2 ) ɺɺ x = τ 1 − kFsgn ( xɺ ) (2.312)

A Unified Approach to Dynamic Control of Robots

m2 ɺyɺ = m2 g + τ 2 − F

171

(2.313)

Fig. 2.7 Simplest planar manipulator

where mi is the mass of the i -th ( i = 1, 2 ) link of the manipulator, F is the reaction force by which the environment is acting on the manipulator’s endeffector, k is the friction coefficient and g is the gravitational constant. Let us suppose that the manipulator has to perform the following simple technological operation. A pressure force F p (t ) should be generated onto the environment along the coordinate y , and the preset motion x p (t ) = vt realized with a constant velocity v along the coordinate x . Evidently, equations (2.312), (2.313) represent the partitioned forms of the robot dynamics equations (2.310) and (2.311), and there is no any other form of partitioning. It is easy to see that for k ≠ 0 , the task of stabilizing the motion x p (t ) using the hybrid control concept does not have a solution. Namely, in this case there is no the control action τ 1 that is independent of the force that realizes PM x p (t ) , even in case of the absence of initial perturbations. In fact, the control action should then satisfy the equation:

(m1 + m 2 ) ɺxɺp = τ 1 − kF (t ) and, because of ɺɺ x p (t ) ≡ 0 , this equation becomes:

τ 1 = kF (t ) Evidently, no equivalent replacement of force increments with position increments is possible because these increments occur in the mutually independent orthogonal directions.

172

Dynamics and Robust Control of Robot-Environment Interaction

As for the stabilization of interaction force along the coordinate y , even in the simplest case, the task is unsolvable. Let e.g. the environment dynamics in the coordinate frame considered be described by the relation:

F = k1 y

(2.314)

where k1 is the environment stiffness coefficient, and PFI is constant, i.e.

F p (t ) = F p . Then, the coefficients α 1 and α 2 of the control law with the force

feedback only: t

τ 2 = − m2 g + α1 ( F − Fp ) + α 2 ∫ ( F − Fp ) (ω ) dω + Fp

(2.315)

0

cannot be chosen in such a way that F (t ) → F p as t → ∞ . In fact, by substituting (2.315) into (2.313) we obtain the following equation of the closedloop control system: t

m2 ɺɺ y = (α1 − 1) ( F − Fp ) + α 2 ∫ ( F − Fp ) (ω )d ω

(2.316)

0

ɺɺ = k ɺyɺ . Since Fɺɺ (t ) = 0 , by differentiating From (2.314) it follows that F 1 p (2.316) we obtain

m2 ɺµɺɺ = (α 1 − 1) µɺ + α 2 µ k1

(2.317)

where µ (t ) = F (t ) − F p . The characteristic equation is of the form:

m2 3 λ − (α 1 − 1)λ − α 2 = 0 k1 It is obvious that since the coefficient at λ 2 is equal to zero, the equation roots λ1 , λ 2 , λ3 satisfy the identity λ1 + λ2 + λ3 ≡ 0 . Therefore, the equation (2.317) has no stable solutions (not to mention asymptotically stable solutions). Thus we can not determine any of the preset roots λ1 , λ 2 , λ3 ensuring µ (t ) → 0 when t → ∞ independently of initial perturbations of the force. Let us notice that in the case of a sufficiently small friction coefficient k (e.g. the environment consists of homogeneous ice), the task of motion

A Unified Approach to Dynamic Control of Robots

173

stabilization can be solved with satisfactory accuracy if, instead of the dynamics equation (2.312), we solve the same equation but without perturbations:

(m1 + m2 ) ɺxɺ = τ 1 Let us note that the presence of friction during the contact with the environment does not represent only problem of hybrid control. Let the environment equation have a more general form than equation (2.314):

F = k1 ɺɺ y + k2 ( y, yɺ ) Then, the task of synthesis of the control law stabilizing force of the robot interacting with the environment in the y -axis direction using force feedback only becomes a task of the same level of complexity as the task of contact control itself. It is evident that for the motion stabilization along the x -axis and force stabilization along the y -axis it is necessary to use both the position and force feedback loops. In this case the control laws:

τ 1 = (m1 + m2 )  ɺɺ x p + γ 1 ( xɺ − xɺ p ) + γ 2 ( x − x p )  + kFsgn ( xɺ ) t  m  m2   τ 2 =  + 1  Fp + r ∫ ( F − Fp ) (ω )d ω  − 2 k2 ( y, yɺ ) − m2 g  k1  k1   t0

can easily solve the stabilization task of PM x p (t ) along the x -axis and

PFI Fp (t ) along the y -axis, respectively, if γ 1 , γ 2 and r are negative constants. 2.8.2 Revised hybrid control procedure The critical points emphasized above concern the very idea of position/force stabilization based on the conventional hybrid control concept rather than the special algorithm implementing it. One such algorithm is based on the concept of “orthogonal complements”. Its fallacy has been rightly pointed out by Duffy [32]. No attempts have been made to realize hybrid control on the basis of a concept other than the orthogonal complements. The idea of splitting the control task for a robot interacting with environment into the task of position control in certain directions and the task of force control in the others seems to be more profound than the idea of the hybrid control based on orthogonal complements.

174

Dynamics and Robust Control of Robot-Environment Interaction

Here we assume again that the environment dynamics model is described in the robot’s coordinate frame by a system of nonlinear differential equations of the form (2.11), i.e.

M (q )qɺɺ + L(q, qɺ ) = S T (q ) F where M (q ) is a nonlinear n × n matrix; L(q, qɺ ) is a nonlinear n -vector function; S T (q ) is an n × n matrix with rank ( S ) = n . We can represent the equation describing the environment dynamics as follows: qɺɺ = Ψ ( q, qɺ , F ) (2.318)

Ψ ( q, qɺ , F ) = − M −1 (q) L(q, qɺ ) + M −1 (q) S T (q ) F . Here we assume that m = n , i.e. the number of interaction force components

where

is equal to the number of powered DOFs. Let us suppose that there exists a function of coordinate transformation w :{ z} → {q} which transforms the robot dynamic model (2.6) into two equations (2.310) and (2.311). The same function of coordinate transformation ensures partition of the environment dynamics model (2.318) into two independent equations:

ɺɺ x = Ψ1 ( x, xɺ , Fx ) , ɺɺ y = Ψ 2 ( y, yɺ , Fy )

(2.319)

which admit a unique representation in the form:

Fx = f1 ( x, xɺ , ɺɺ x) ,

Fy = f 2 ( y, yɺ , ɺɺ y)

(2.320)

Let us consider two systems of equations taking into account (2.310) and (2.311), and let the first system

U1 ( x, xɺ, ɺɺ x, Fx ) = τ x ,

ɺɺ x = Ψ1 ( x, xɺ , Fx )

(2.321)

describe the robot’s interaction with the environment in the coordinate space {X }. Let the second system

U 2 ( y, yɺ , ɺɺ y, Fy ) = τ y ,

ɺɺ y = Ψ 2 ( y, yɺ , Fy )

(2.322)

describe the robot’s interaction with the environment in the coordinate space {Y } . It is obvious that both systems (2.321) and (2.322) are completely independent of each other. The former system stabilizes the robot motion

175

A Unified Approach to Dynamic Control of Robots

(

(

x p (t ) ɺɺ x p (t ) ≡ Ψ1 x p (t ), xɺ p (t ), Fx p (t ) force

))

independently of the magnitude of

Fx (t ), t ≥ t0 , whereas the latter stabilizes the interaction force

(

(

Fy p (t ) ɺɺ y p (t ) ≡ Ψ 2 y p (t ), yɺ p (t ), Fy p (t )

))

independently of the position

magnitude y (t ), t ≥ t0 . The contact task can then be solved by stabilizing the position (motion) in the coordinate space { X } and the interaction force in the coordinate space {Y } . In contrast to the conventional hybrid control concept, the control laws ensuring stabilization of the motion and interaction force utilize both the position and force feedback loops. Let us write equations (2.310) and (2.311) solved with respect to second derivatives:

ɺɺ x = Φ1 ( x, xɺ ,τ x , Fx ) ɺɺ y = Φ 2 ( y, yɺ ,τ y , Fy ) Then, using the control laws (2.23), (2.28) or (2.34) from Sections 2.3.1 we solve the task of stabilizing x p (t ) , whereas the laws (2.39), (2.40), (2.43) from Section 2.3.2 solve the task of stabilizing the interaction force Fy p (t ) . In addition, the stabilization of x p (t ) is carried out with the preset quality of transients determined by (2.16) and stabilization of Fy p (t ) with the preset quality of transients determined by the relation (2.19). The mentioned control laws that stabilize PM x p (t ) are:

(

τ x = U1 x, xɺ, ɺɺx p + P ( x − x p , xɺ − xɺ p ) , Fx

(

)

(

τ x = U1 x, xɺ, ɺɺ x p + P ( x − x p , xɺ − xɺ p ) , f1 x, xɺ , ɺɺ x p + P ( x − x p , xɺ − xɺ p )

(2.323)

))

(2.324)

The control laws (2.39), (2.40), (2.43) that stabilize PFI Fy p (t ) are of the form: t     τ y = U 2  y, yɺ , Ψ 2  y, yɺ , Fy + ∫ Q Fy − Fy dω  , Fy      t    

(

p

0

p

)

(2.325)

176

Dynamics and Robust Control of Robot-Environment Interaction

t   τ y = U 2  y, yɺ , Ψ 2 ( y, yɺ , Fy ) , Fy p + ∫ Q Fy − Fy p dω    t0   t    τ y = U 2  y, yɺ , Ψ 2  y, yɺ , Fy p + ∫ Q Fy − Fy p d ω  , Fy p    t0    t  + ∫ Q Fy − Fy p d ω   t0 

(

)

(

(

(2.326)

)

(2.327)

)

It is easy to see that analogous control laws can also be applied in a more general case, when 1. representations (2.310), (2.311) are of the form

U1 ( x, xɺ, ɺɺ x, y, yɺ , Fx , Fy ) = τ x

(2.328)

U 2 ( y, yɺ , ɺɺ y, x, xɺ, Fx , Fy ) = τ y

(2.329)

under the condition of their unique solvability via the functions Φ1 and

Φ 2 with respect to second derivatives ɺxɺ and ɺyɺ . 2. the equation of the environment dynamics model (2.11) or (2.12) can be presented in the form of the two equations

ɺɺ x = Ψ1 ( x, xɺ, y, yɺ , Fx , Fy )

(2.330)

ɺɺ y = Ψ 2 ( y, yɺ , x, xɺ, Fx , Fy )

(2.331)

under the condition of their unique solvability with respect to Fx and Fy :

Fx = f1 ( x, xɺ , ɺɺ x, y, yɺ , Fy )

(2.332)

Fy = f 2 ( y, yɺ , ɺɺ y, x, xɺ, Fx )

(2.333)

In this case PM x p (t ), y p (t ) and PFI Fx p (t ), Fy p (t ) should satisfy the system of equations

( ) (t ) = f ( y (t ), yɺ (t ), ɺɺ y (t ), x (t ), xɺ (t ), F (t ) )

Fx p (t ) = f1 x p (t ), xɺ p (t ), ɺɺ x p (t ), y p (t ), yɺ p (t ), Fy p (t )

(2.334)

Fy p

(2.335)

2

p

p

p

p

p

xp

177

A Unified Approach to Dynamic Control of Robots

Then, to stabilize PM x p (t ) in the space { X } we can apply, for instance, the control law (an analogue of the control law (2.323))

(

τ x = U1 x, xɺ, ɺɺ x p + P ( x − x p , xɺ − xɺ p ) , y, yɺ , Fx , Fy

)

(2.336)

and, to stabilize PFI Fy p (t ) in the space {Y } we can apply, for instance, the control law (an analogue of the control law (2.325)) t     τ y = U 2  y, yɺ , Ψ 2  y, yɺ , x, xɺ, Fx , Fy + ∫ Q Fy − Fy dω  , x, xɺ, Fx , Fy  (2.337)     t    

(

p

p

)

0

One of these control laws is used to solve the task of stabilizing PM and PFI in the considered example of the two-link manipulator given in Fig. 2.7. 2.8.3 Case study In order to get a better insight into the efficiency of the proposed hybrid dynamic position-force control laws in this section we present simulation results of the manipulation robot’s model with two DOFs (Fig. 2.8).

Fig. 2.8 Model of a 2-DOF planar manipulator

We consider a typical robot task in which the robot end-effector realizes the desired motion x p (t ) along the x -axis and the required contact force Fy p in the y -direction. The environment model defined in the direction perpendicular to the sliding surface (the y -direction in Fig. 2.8) is assumed to be in the form of linear impedance, while the model of force in the horizontal plane ( x -

178

Dynamics and Robust Control of Robot-Environment Interaction

direction in Fig. 2.8) is defined by the friction force model of the end-effector. The models of active forces acting upon the tip of the robot’s end-effector are represented by the following relations:

− Fy = M ɺɺ y + B yɺ + K y

(2.338)

Fx = −ν Fy sgn ( xɺ (t ))

(2.339)

where M , B and K are the environment equivalent mass, damping and stiffness parameters, respectively. For a relatively rigid environment we assume the following hypothetical values of its parameters: M = 0.5 [ kg ] ,

K = 10 5 [N / m ] , whereas Coulomb’s friction coefficient of the robot end-effector, ν = 0.3 . The returned motion of the robot end-effector is imposed from X 0 to X 1 and back, with the periodically varying

B = 268 [ N /(m / s) ]

and

velocity v(t ) = v max sin (ωt ) . The maximal length of the end-effector path along the y -axis: lmax = 0.5 [ m] . Maximal feedrate of the manipulator’s end-effector:

v max = 0.2 [ m / s ] , while the frequency of the imposed end-effector feedrate xɺ p (t ) = v(t ) is assumed to be ω = 0.8 [rad / s ] . Maximal output torque of the

actuator shaft at both mechanism’s joints is τ max = ± 50 [Nm] . The robot contact task is to synthesize such control law which would ensure the desired motion precision x p (t ) and the interaction force Fy p (t ) = 11[ N ] . The robot links are l1 = l 2 = 1[m] , and their masses m1 = 1.3 [kg ] and

m2 = 0.9 [kg ] . The initial joint position deviations are: ∆q1 (0) = 0.0306 [rad ] and ∆q 2 = − 0.0611[rad ] . This causes the end-effector position displacement in the x -direction ∆x(0) = − 0.045 [m] . As a consequence, the contact force error in the initial time instant is ∆Fy (0) = −6.98 [N ] . In this planar example, the entire manipulator workspace {Z } (Fig. 2.8) can

be decoupled in two “orthogonal subspaces”,

{X}

and {Y } . Thus z = [ x, y ]T ,

F =  Fx , Fy  . It should be noticed that the robot’s motions in the planes { X } and {Y } are mutually interconnected (see the models (2.338) and (2.339)). This T

fact is of crucial importance for the understanding of the differences between the

179

A Unified Approach to Dynamic Control of Robots

conventional hybrid position/force control and the proposed dynamic positionforce control. The former is based on the concept of the so-called “orthogonal complements”, while the proposed control laws are based on the simultaneous control of robot position and interaction contact force. Below we present simulation results that emphasize the necessity of applying the “corrected” position-force control laws. Using the environment model (2.338) and the model of manipulation robot in Cartesian space ((2.272) in Section 2.7)), the conventional hybrid control law in the robot task space is expressed by the general relation (1.41) – (1.43) in Chapter 1. For comparison sake, along with the conventional hybrid control, we apply dynamic position-force control laws (2.336), (2.337), which control simultaneously the end-effector position and contact interaction force during the robot’s operation. More concretely, the conventional hybrid control law in robot task space is described by the expression [29]:

τ = J T (q ) {( I 2 − S ) [ Λ( z ) ɺɺzc + v ( z , zɺ) ] − SFc }

(2.340)

where

ɺzɺc = ɺzɺp − K v ( zɺ − zɺ p ) − K p ( z − z p ) , t

Fc = Fp − K fp ( F − Fp ) − K fi ∫ ( F − Fp )d ω + Λ ( z ) K fv zɺ 0

The subscript “p” denotes programmed values of the particular variables. The unit matrix I 2 and the selectivity matrix S are defined in the form:

0 0  1 0 I2 =  ; S =   0 1  0 1  The matrices K (⋅) in the control law (2.340) represent the corresponding diagonal matrices of the regulator of control gains k(⋅) . Instead of the implementation of the conventional hybrid control (2.340) we propose implementation of the above dynamic position-force control laws (2.336), (2.337). In this case the contact force Fy (t ) is controlled explicitly. Then, the laws (2.336), (2.337) can be written in the compact vector form:

180

Dynamics and Robust Control of Robot-Environment Interaction

τ x  τ =   = J T (q ) {Λ ( z ) ɺɺzc + v ( z, zɺ ) − Fc } τ y 

(2.341)

where

x p − kv ( xɺ − xɺ p ) − k p ( x − x p )   ɺɺ  , ɺɺ zc =  − 1 ( B yɺ + K y + Fy )  M  

Fx   ω t  Fc =   Fy p + ∫  −κ fp ( Fy − Fy p ) − κ fi ∫ ( Fy − Fy p ) dω1 t0  t0 

   .  dω    

Simulation results obtained by implementing the control laws (2.340) and (2.341) under the imposed motion conditions are presented in Fig. 2.9. The following values of parameters were used: k p = 631 [ s −2 ] ,

k v = 40 [ s −1 ] , κ fp = 38 [ s −1 ] , κ fi = 355 [ s −2 ] , k fp = 2.76 , k fi = 35.53 and k fv = 0.1 [ N s / kg m] . A comparative analysis shows that in this case the conventional hybrid position/force control (2.340) can ensure neither the desired stable motion of the robot’s end-effector nor the required contact force. On the contrary, the proposed dynamic control law (2.341) with respect to position and contact force ensures a stable programmed motion x p (t ) of the end-effector in the horizontal direction and a desired contact force Fy p (t ) in the vertical direction. Besides, the control law (2.341) ensures both the stability of the desired quality of dynamic behavior in all directions of motion. The dynamic position control law (2.336), combined with explicit control (2.337)c in the y -direction, ensures simultaneous position-force control of the end-effector. Which control law will be implemented with the force control law (2.336) in a particular robot contact task it will depend on the concrete requirements concerning the quality of robot task performance. We have to

c In addition to the explicit force control law (2.337), there are two more variants of this control

such as implicit and explicit/implicit force control, analogous to those represented by (2.326) and (2.327) for a somewhat simpler case of coupling in respect of force and position.

A Unified Approach to Dynamic Control of Robots

181

underline again that the stability of robot motion (positions), as well as of contact forces, is guaranteed by the application of any of the proposed control laws pairs: (2.336) with (2.337), (2.326) or (2.327), respectively [33].

Fig. 2.9 Accuracy indices of the nominal trajectory tracking and desired contact force during motion of the end-effector along the x -axis and y -axis using control laws: a) (2.340) and b) (2.341)

182

Dynamics and Robust Control of Robot-Environment Interaction

2.9 Impedance Control – A Special Case of the Unified Approach 2.9.1 Introduction Impedance control represents a strategy for constrained motion rather than a concrete control scheme. The objective of this control concept is to achieve specific mechanical impedance at the manipulator end-effector. This objective imposes a desired relationship between the position error and force acting on the end-effector. Various control concepts and schemes have been established and proposed for controlling the relation between the robot’s motion and interaction force. It should be emphasized that despite of the existence of a large number of impedance control algorithms only a few of them can be applied in the existing industrial robot control systems. In Section 1.6.1 of Chapter 1 we gave a brief survey of the state-of-the-art in the domain of impedance control, whereas impedance control as active compliant motion control will be presented in a much wider form in Chapter 3. 2.9.2 Improved impedance control Let us start with the general impedance control concept presented in [34]. The basic idea of this approach consists in generating a control law for the robot, in which the closed-loop control system functions in accordance with the differential equation:

F = M ′( ɺɺ x − ɺɺ x0 ) + B′( xɺ − xɺ0 ) + K ′( x − x0 )

(2.342)

where the constant m × n matrices M ′, B′, K ′ represent the matrices of inertia, damping and stiffness of the overall interactive system. These matrices can be chosen by the control system designer in dependence of the goals to be achieved by the robot in the technical task to be performed. Equation (2.342) can be written in the robot’s joint coordinates in the form:

F = M (q ) (qɺɺ − qɺɺp ) + B (q ) (qɺ − qɺ p ) + K (q ) (q − q p )

(2.343)

where the joint-coordinates dependent m × n matrices M ( q ), B ( q ), K ( q ) represent the matrices of inertia, damping and stiffness of the whole interactive system, too. Equation (2.343) is derived from (2.342) by transpose Jacobian matrix multiplication, and using the known kinematic relations between the Cartesian and joint coordinates.

A Unified Approach to Dynamic Control of Robots

183

For the sake of simplicity and without loss of generality, the impedance control will be considered in the coordinate frame q by putting m = n . The goal of impedance control is then to ensure the preset impedance of the closed-loop system defined by (2.342). Hereby, it is assumed that the environment dynamics model is also given in the impedance form:

F = Mɶ ′ ɺɺ x + Bɶ ′ xɺ + Kɶ ′ x

(2.344)

or in its alternative form in the joint coordinate space:

F = Mɶ (q )qɺɺ + Bɶ (q )qɺ + Kɶ (q )q (2.345) ~ ~ ~ where M ( q ), B ( q ), K ( q ) and Mɶ ′, Bɶ ′, Kɶ ′ are the corresponding sets of matrices of inertia, damping and stiffness of the environment. The same notation for F , used in (2.342) and (2.343), has evidently a different physical meaning, i.e. force and moment, respectively. In accordance with the definitions of PM (programmed motion) and PFI (programmed force of interaction), these components of the desired dynamic behavior have to satisfy the equation of environment dynamics, i.e.:

Fp (t ) = Mɶ (q p ) qɺɺp (t ) + Bɶ (q p ) qɺ p (t ) + Kɶ (q p ) q p (t ), ∀t ≥ t0

(2.346)

Let us assume that the task of impedance control has been solved, i.e. that such control τ has been synthesized that ensures the closed-loop control system functioning in accordance with (2.342). Assuming the differences in the environment dynamics matrices for the actual and nominal motion to be negligible, it follows from (2.344) and (2.345) that the differences between the actual and programmed robot force interaction with the environment are determined by the equation:

F − Fp = Mɶ (qɺɺ − qɺɺp ) + Bɶ (qɺ − qɺ p ) + Kɶ (q − q p )

(2.347)

By subtracting this equation from (2.343) we find that the functioning of the closed-loop system is governed by the equation:

Fp = ( M − Mɶ )ηɺɺ + ( B − Bɶ )ηɺ + ( K − Kɶ )η

(2.348)

where η (t ) = q (t ) − q p (t ). Evidently, the control system designer can choose the matrices M , B, K in such a way that the trivial solution of the unperturbed motion:

( M − Mɶ )ηɺɺ + ( B − Bɶ )ηɺ + ( K − Kɶ )η = 0

(2.349)

184

Dynamics and Robust Control of Robot-Environment Interaction

is, say, exponentially stable, whereby all its solutions comply with the inequality:

max { η (t ) , ηɺ (t ) } ≤ C e − λ (t −t0 )

η (t0 ) + ηɺ (t0 ) 2

2

(2.350)

where C is a constant and the index λ > 0. In that case, an arbitrary solution of the differential equation (2.348) with the perturbation F p (t ) will be stable, whereby the estimation of stabilizing accuracy implies the following:

max { η (t ) , ηɺ (t ) } ≤ Ce− λ (t −t0 )

η (t0 ) + ηɺ (t0 ) + CCF / λ 2

2

where C F is a constant which bounds the norm of PFI Fp (t ) for every t ≥ t0 , i.e. F p (t ) ≤ C F . It is clear that for a large λ , or, which is the same, for a small

C F , the impedance control solves satisfactorily the task of stabilizing motion of the robot in contact with its environment (2.345). A question arises as to whether this condition implies also that the interaction force F (t ) is close to F p (t ) , i.e. whether the impedance control solves fully the control task of the robot in contact with the dynamic environment. Evidently, if

F p (t ) is changing slowly with time, and its norm F p (t ) is small, the answer is positive. In fact, because η and ηɺ are small, it follows from (2.347) that for a small F p (t ) , ηɺɺ will be small too. And, because the connection between the deviation of the force from its programmed value F (t ) − F p (t ) and deviation of the motion from the programmed one η (t ) = q (t ) − q p (t ) is determined by (2.347), the small values η , ηɺ , ηɺɺ cause µ (t ) = F (t ) − F p (t ) to be small as well. It is easy to see that all these supplementary requirements related to the small value of C F and to the slowly changing F p (t ) are explained by the presence of the perturbation F p (t ) on the left-hand side of equation (2.348), describing the behavior of the closed-loop control system. It is therefore fully justified to introduce a correction into the impedance equation (2.343), by replacing it with the following equation:

F − F p = M (qɺɺ − qɺɺp ) + B (qɺ − qɺ p ) + K (q − q p )

(2.351)

185

A Unified Approach to Dynamic Control of Robots

Thus the closed-loop system will evidently operate according to (2.349) without the perturbation F p (t ) . In that case the estimation (2.350) will hold, i.e.

η (t ), ηɺ (t ) → 0 when t → ∞ and, because of equation (2.349), the relation ηɺɺ(t ) → 0 when t → ∞ will also hold. Hence, bearing in mind (2.347), it immediately follows that F (t ) → F p (t ) at t → ∞ . In this way, the correction carried out by the impedance yields an ideal solution of the contact task. Let us now focus our attention on the fact that in the case when impedance equation is of the form (2.351), any impedance control solving the task of attaining the preset impedance of the closed-loop system is actually a special case of the control law (2.23), with a suitably chosen vector function P in the case when environment dynamics is described by the differential equation (2.345). This relates to the fact that the realization of the preset impedance goal in the form (2.351) leads to the closed-system equation in the form of (2.349), which is a special case of the equation ηɺɺ = P (η ,ηɺ ) (see (2.16)), specifying the preset quality of stabilization, where:

P (η ,ηɺ ) = ( M − Mɶ )−1 ( Bɶ − B )ηɺ + ( Kɶ − K )η 

(2.352)

The impedance equation (2.351), solvable with respect to second derivative, is of the form:

ηɺɺ = M −1 (− Bηɺ − Kη + F − Fp ) Therefore,

( M − Mɶ )−1 ( Bɶ − B)ηɺ + ( Kɶ − K )η  = P(η ,ηɺ ) = ηɺɺ = M −1 (− Bηɺ − Kη + F − Fp ). In that case, the robot control law (2.23), based on the impedance equation (2.351), can be written as

τ = H (q)  qɺɺp + M −1 (− Bηɺ − Kη + F − Fp )  + h(q, qɺ ) − J T (q) F

(2.353)

This relation provides an ‘ideal’ solution of the task of motion and force stabilization. The control law synthesized on the basis of impedance control (2.343), i.e. on the basis of the equation without correction, has a form analogous to (2.353), if the term − F p is omitted:

τ = H (q)  qɺɺp + M −1 (− Bηɺ − Kη + F )  + h(q, qɺ ) − J T (q) F Indeed, this control law has been used for the synthesis of adaptive impedance control of robots and manipulators [35].

186

Dynamics and Robust Control of Robot-Environment Interaction

2.9.3 Case study Simulation studies were conducted on the MANUTEC-r3 industrial robot (Fig. 2.10) to verify the described impedance control scheme. The parameters of the mechanical structure and actuators were taken from [36]. Tables 2.1–2.3 show the values of kinematic and dynamic parameters of the robot. The actuators’ parameters can be found in [36].

Fig. 2.10 MANUTEC-r3 robot: (a) View, (b) Kinematic scheme

Table 2.1 Vector from the joint center to the center of mass of the augmented link i (link i and electric motor rotor i + 1), resolved in the body fixed frame of link i

Rxi Ryi Rzi

i

1

2

3

4

5

6

[m] [m] [m]

0.000 -0.172 0.000

-0.295 0.000 0.172

0.000 -0.064 -0.034

0.000 -0.410 0.000

0.000 0.000 0.023

0.000 0.000 -0.020

Table 2.2 Mass of the link (mi) (i = 1, n) i mi [kg]

1

2

3

4

5

6

20.000

56.500

26.400

28.700

5.200

1500

187

A Unified Approach to Dynamic Control of Robots

Table 2.3 Inertia tensor of the augmented link i (link i and electric motor rotor i + 1), with respect to the center of mass of the augmented link, resolved in the body fixed frame of link i i Ixxi [kg m2] Iyyi [kg m2] Izzi [kg m2]

1

2

3

4

5

6

0.0000 1.1600 0.0000

2.5800 2.7300 0.6400

0.2790 0.4130 0.2400

1.6700 1.6700 0.0810

1.2500 1.5300 0.8100

0.0000 0.0000 0.0002

Fig. 2.11 Robot machining process under consideration

The example of robotic deburring has been selected for simulation experiments using the conventional and improved impedance control schemes. The rotational-milling work tool of the robot performs surface processing in the π-plane (Fig. 2.11). The robot end-effector (tool) moves along a prescribed, nominal trajectory 1-2 (Fig. 2.11). The total trajectory is 200 [mm] long. The robot’s task is to carry out machining of the work-surface along the prescribed trajectory with a prescribed contact force FN0 = 5 [ N ] and the prescribed

188

Dynamics and Robust Control of Robot-Environment Interaction

velocity v f = 25  mms −1  . The adopted general model of impedance was the environment model (2.344). The corresponding matrices in the environment model were adopted in the form: Inertia matrix: Mɶ ′ = diag {mɶ i′}[ kg ], mɶ 1′ = 28.173, mɶ ′2 = 101.424, mɶ 3′ = 28.173, mɶ ′4 = 10.142, mɶ 5′ = 10.142, mɶ 6′ = 10.142.

(2.354)

Damping matrix:

Bɶ ′ = diag {bɶi′}[ N /(ms −1 )] bɶ1′ = 1061.571, bɶ2′ = 4458.599, bɶ3′ = 1061.571, bɶ4′ = 445.859, bɶ′ = 445.859, bɶ′ = 445.859. 5

(2.355)

6

Stiffness matrix: Kɶ ′ = diag {kɶi′}[ Nm −1 ] kɶ1′ = 10 4 , kɶ2′ = 10 5 , kɶ3′ = 10 4 , kɶ4′ = 10 4 , kɶ5′ = 10 4 , kɶ6′ = 10 4 ,

(2.356)

The matrices of the overall interactive closed-loop control system (2.342) were cast in the following forms:

M ′ = diag {mi′}[kg ], m1′ = 100, m′2 = 100, m3′ = 100, m4′ = 10,

m5′ = 10,

m6′ = 10,

B ′ = diag {bi′}[ N / ms −1 ], b1′ = 6000, b2′ = 6000, b3′ = 6000, b4′ = 600, K ′ = diag {k i′}[ Nm −1 ], k1′ = 10 4 ,

b5′ = 600,

b6′ = 600,

k 2′ = 10 5 , k 3′ = 10 4 ,

k 4′ = 10 4 , k 5′ = 10 4 , k 6′ = 10 4.

(2.357)

(2.358)

(2.359)

A Unified Approach to Dynamic Control of Robots

189

Fig. 2.12 Simulation results: (a) for conventional impedance control; (b) for improved impedance control

The following initial conditions were used in the simulation: initial deviation of the end-effector tip ∆y = + 30 [µ m] and ∆z = + 50 [µ m ] ; the robot gripper started with a zero initial velocity; the position deviations resulted in an initial deviation of the contact force of ∆FN = + 4.375 [N ] , which means that the robot actuator at the instant t = 0 was overloaded because the nominal force value was FN (t ) (t = 0) = 0 [N ] . The setting time of the desired contact force was

t = 0.5 [sec] . The simulation results are presented in Fig. 2.12. The relevant

190

Dynamics and Robust Control of Robot-Environment Interaction

indices of control quality are the shapes of transient processes of force components FN , FT and F f and the deviation of the positions ∆q, ∆x, ∆y and

∆z of the internal and external robot coordinates. The results presented in Fig. 2.12 show that both control laws ((2.343) and (2.351)) ensured a satisfactory quality of dynamic behavior. The control law realizing conventional impedance (2.343) gave worse dynamic feature than the control law realizing improved impedance (2.351); in the latter case the overshoot of the contact force in the setting time interval was reduced. The control law corresponding to the improved impedance equation (2.351) exhibited a decrease in the mean force error squares of about 8 %. It should be noticed that the parameters of the systems in contact were selected in a way that yielded no phenomena that would strongly favor the improved impedance control.

2.9.4 Concluding remarks In concluding the discussion of impedance approach to control tasks let us note that even without the mentioned correction of impedance equation, in contrast to hybrid control, the control laws synthesized on the basis of target impedance (2.343), solve satisfactorily the task of control of robot in contact with its environment. Indeed, because of the equivalence of the impedance equations (2.343) and (2.348) for the environment (equation (2.345)), the matrices M , B and K can be selected on the basis of equation (2.348), to ensure the convergence η (t ) and ηɺ (t ) to zero for t → ∞ . In principle, this can always be achieved because the left-hand side of equation (2.348) represents the known time function F p (t ) . But in this case, for the different PFI Fp (t ) it is necessary to consider each time different matrices M , B and K . However, even so, it is not possible to ensure that the second derivative ηɺɺ(t ) tends to zero for t → ∞ , and this can in some cases worsen significantly the quality of stabilization of the contact force. For this reason, as well as from the point of view of obtaining “ideal” control laws, the use of the target impedance with correction (2.351) seems to be unavoidable. From a theoretical point of view it was shown that the impedance control is only a special case of active compliant control scheme. This leads to the disappearance of the principal difference between the active compliance tasks in which impedance control is much different from the position-force control. A

A Unified Approach to Dynamic Control of Robots

191

direct aim is to improve the transient dynamic process immediately after the contact. In this way, the impedance control performance is improved in the very instant when its quality is of the decisive practical importance, and thus the impedance control can also become competitive in other contact tasks. Finally, it should be added that it is not possible to determine the nominal contact force Fp without having a sufficiently accurate environment model. Moreover, an insufficiently accurate environment dynamics model can significantly affect the contact task execution. Inaccuracies of the robot and environment dynamics models have been considered in [6, 15]. The problems arising from parameter uncertainties may also be resolved by applying knowledge-based techniques [37]. Let us also note that the adaptive control laws given in Section 2.5 can cope with the task of motion and force stabilization even in case of an essential inadequacy of the models of dynamics of the robot and environment, and thus with an inaccurate calculation of the force Fp .

2.10 Stability of Robots Interacting with Dynamic Environments 2.10.1 Introduction The synthesis of control of robots in the so-called constrained motion control tasks, i.e. the tasks in which manipulation robots are coming into contact with the environment, has attracted much attention in the last two decades or so. One of the most delicate problems in position-force control of robots interacting with dynamic environment is the stability of both desired motions and interaction forces. A multitude of various control approaches such as hybrid control, stiffness control, impedance control, etc. points to the control stability tasks as a problem which is not yet satisfactorily solved, both from the theoretical and practical standpoint. In almost all approaches when considering specific contact tasks, simplifications are introduced in the modeling of robot and environment dynamics in order to obtain simpler control algorithms. A very popular approach is to describe the environment by a set of algebraic equations, assuming that the motion of the robot in contact is kinematically constrained [1, 2]. Colgate and Hogan consider the environment to be a linear time-invariant dynamic system [30]. In both cases experimental verification led to the discovery of instability caused by the environment dynamics [38]. Ann and Holerbach have identified a new form of instability in the force control of manipulators [39]. That form of instability occurs only in multi-joint

192

Dynamics and Robust Control of Robot-Environment Interaction

manipulators and is caused by an inverse kinematic transformation in the feedback path. It has been shown that kinematic instability occurs not only at the points of kinematic singularities, where the Jacobian inverses are not defined, but in a wide range of the manipulator work space, where the Jacobian inverses are well defined. Mc Clamroch and Wang emphasized the important role of constraints in the constrained dynamics [40]. They presented global conditions for the tracking based on a modified computed torque controller, and local conditions for feedback stabilization using a linear controller. The closed-loop properties in the case of force disturbances, dynamics in the force feedback loops, or uncertainty in the constraint functions, were also investigated. Eppinger and Seering have studied the influence of unmodeled dynamics on contact task stability, introducing additional (elastic) DOFs of both robot and environment [41]. Nevertheless, they concluded that the environment dynamics cannot cause instability. A treatment of the contact task stability, considering the environment as a nonlinear dynamic system is given in [42]. It is shown that if impedance control is applied, enabling the robot to be asymptotically stable in free space, then the robot interacting with environment is a passive system and is stable in the isolation. However, the conclusion is valid only if the robot in contact is at rest and for this reason the result can not be considered complete. The stability of the robot and environment taken as a whole, using unstructured models for the dynamic behavior of the robot and the environment, has been investigated in [43]. The objective was to define sufficient conditions for stability of the closed-loop system. The basic idea has been to start from the very general (“unstructured”) model of the robot in contact with environment, where both the robot dynamics and environment dynamics are presented by general functions, for which only input/output characteristics are taken into account (structure is totally ignored). This approach is then applied to the “structured” model of the robot and environment, where infinitely rigid environment model is assumed. Due to the fact that the structure of the model is totally ignored, the obtained results are extremely conservative. In the case of structured model the same “conservativism” is preserved, since structural features of the model are nowhere specifically used to derive the stability conditions. Both robot and environment dynamics are “approximated” by the “gain” of mapping of the input trajectory into the force vector. This means that the stability conditions set upon the “allowed” control law “size” take into account robot dynamics, environment dynamics and the position control law –

A Unified Approach to Dynamic Control of Robots

193

all together – through one “gain”. No attempt has been made to show how the results of the stability analysis can actually be used for control synthesis when the desired force trajectory has to be achieved, and when the model of environment dynamics is to be assumed. In order to extend the hybrid strategy to the cases when the environment also exhibits a dynamic behavior De Luca and Manes [25, 44] introduced a suitable modeling of the robot-environment interaction. The paper [44] describes how to use this modeling technique to design hybrid control laws in the presence of environment dynamics. The fundamentals of the dynamic control of robotic manipulators interacting with dynamic environment [6, 10, 11, 15, 16, 45], are presented in Sections 2.3.2 and 2.3.3. Attention was focused just on the stabilization of position when asymptotic stability of the contact force was ensured. This task is the basic problem of controlling a robot interacting with dynamic environment. The theorem ensuring the asymptotically stable position of the system in the first approximation (local stability) formulates the sufficient conditions under which it is achieved. However, without knowing sufficiently accurate environment model it is not possible to determine the nominal contact force Fp (t ) . Besides, an insufficiently accurate environment dynamics model can significantly affect the contact task performance. Inaccuracies of the robot and environment dynamics models, as well as dynamic control robustness are considered in [6, 15, 16]. The problems arising from parametric uncertainties may also be resolved by applying the knowledge-based techniques (fuzzy logic and neural networks) [46]. Taking into account external perturbations, which do not expire with time, and model and parameter uncertainties, it may be difficult to achieve asymptotic (exponential) stability of the system unless use is made of robust and adaptive control laws that include a factor to compensate for these perturbations and uncertainties. Therefore, it may be of practical interest to demand more relaxed stability condition, i.e. to consider the so-called practical stability of the robot around the desired position and force trajectories by specifying the finite regions around them within which the robot actual position and force have to be during the task execution, and by assuming that the inaccuracies of model parameters (of the robot and environment) are bounded. The conditions for the practical stability of the robot interacting with dynamic environment enable one to study the model uncertainty issue in control of robots in this class of tasks without any approximations, i.e. to correctly examine the influence of these uncertainties upon different control laws. The test conditions for practical stability of the robot interacting with dynamic environment will be presented in the text to follow.

194

Dynamics and Robust Control of Robot-Environment Interaction

2.10.2 Practical stability of robots interacting with dynamic environment Numerous applications of robots in the industry, agriculture, mining, health care, etc. require effective solutions for the control of robots in the so-called constrained motion control tasks, i.e. the tasks in which manipulation robots are coming into contact with the environment. The synthesis of control of robots for such tasks has attracted significant attention in the last 25 years (see e.g. [1, 2, 5, 29, 47, 48]). A special problem represents the stability issue, i.e. the establishment of the conditions under which a particular control law guarantees stability of the robot in contact with its environment. A number of papers have considered the stability aspects assuming different approximate models of the robot and the environment [6, 10, 38, 43]. In [6, 10], the control laws stabilizing simultaneously the robot motion and its interaction force with dynamic environment have been synthesized to ensure exponential stability of the closedloop systems (based on the analysis of the complete dynamics models of the robot and dynamic environment). However, the model uncertainties, representing the crucial problem in control of robots interacting with the dynamic environment, have not been appropriately addressed yet. The uncertainties of the environment dynamics model in certain technological tasks may have especially strong influence because of the difficulties in the identification/prediction of the environment parameters and behavior of the environment. Therefore, it may be difficult to achieve asymptotic (exponential) stability of the system (unless robust and adaptive control laws including factors for compensating these perturbations and uncertainties are used). It is of practical interest to require a more relaxed stability condition, i.e. to consider the so-called practical stability of the system. An approach to the analysis of practical stability of manipulation robots interacting with the dynamic environment based on centralized model of the system has been presented in [19, 49]. In [50], an approach to stability analysis of the robots in such control tasks based on the decomposition of the system into two subsystems (one associated to the ‘position’- controlled and the other to the ‘force’-controlled part) has been presented. In this section, a new approach is presented following the basic idea of the decomposition-aggregation method for the stability analysis of large-scale systems in which the system is decomposed into ‘subsystems’, each associated to one DOF in Cartesian (or, task) space. The dynamic interactions among these ‘subsystems’ are taken into account within the ‘aggregation’ phase of the method. The objective is to establish for the first time less conservative conditions for

A Unified Approach to Dynamic Control of Robots

195

robot’s practical stability, suitable for the analysis and synthesis of decentralized control laws. The conditions are derived using the method of Michael [51], which has been already effectively applied for the practical stability analysis of the robot moving in a free space [52]. The inaccuracy in modeling, constraints upon motion, control, and interaction forces are considered in a general form. The control law considered belongs to the so-called hybrid position-force control schemes, where the overall dynamics model of the system is taken into account.

2.10.3 Mathematical model We will consider a robot in contact with dynamic environment, assuming that the contact is constantly maintained. The issue of keeping of contact and impact between the robot and the environment (i.e. the discontinuity of the model, since the robot can only push and not pull the environment) is out of the scope of this section. The overall dynamics model of the robot with n ≤ 6 DOFs and the dynamics model of the environment are considered in Cartesian space. The model of dynamics of the mechanical part of the robot can be written in the form: ɺ d ) = ( J -1 )T ( z, d )τ + F Λ ( z, d ) ɺɺ z + ρ ( z, z, (2.360) where z = χ ( q ) is the n -vector of the robot Cartesian coordinates (see (2.271)); q is the n -vector of the robot internal coordinates; Λ ( z, d ) is the

ɺ d ) is the n -dimensional nonlinear vector function n × n inertia matrix; ρ ( z, z, of Coriolis’, centrifugal and gravity moments; d is the l -vector of parameters which belongs to the constrained set D ; J ( z, d ) is the n × n Jacobian matrix; τ is the n -vector of driving torques (inputs); F is the n -vector of Cartesian forces, generalized interaction forces (forces and moments) acting upon the endeffector of the robot. For the sake of simplicity we shall: (1) consider the second order models of actuators, which are assumed to be included in the robot dynamics model, and (2) write the functions without arguments. We shall assume that the environment does not introduce any extra DOFs in the overall system. The model of dynamic environment can be written as:

ɺ d)= − S F M ( z, d ) ɺɺ z + L ( z, z,

(2.361)

where M is the n × n matrix; L is the n -dimensional nonlinear vector function, and S is an n × n matrix function of Cartesian coordinates, representing transformation between forces acting upon the robot end-effector

196

Dynamics and Robust Control of Robot-Environment Interaction

and forces (moments) of the dynamic environment, with a rank equal n. (This is the property of the environment model (2.361), whose solution is unique with respect to the force F .) We shall assume that S = I . It is assumed that all mentioned matrices and vectors are continuous functions of their arguments. The model of the robot in the state space can be defined as:

xɺ = f ( x, d ) + B ( x, d )τ + G ( x, d ) F

(2.362)

where x = ( z T , zɺT )T is a 2n -dimensional state vector; f ( x, d ) is a 2n -vector; B ( x, d ) and G ( x, d ) are 2n × n matrices.

2.10.4 Formulation of the control task Let us assume that in m1 directions ( m1 < n ) the desired force trajectories

Fp(1) (t ) are specified, where Fp(1) (t ) is m1 -vector, while in n1 ( n1 < n ) (1) directions, the desired trajectory z (1) p (t ) is specified, where z p (t ) is n1 -vector

and n1 + m1 = n . This means that in some directions are specified only desired forces and in some directions only desired position trajectories. Let us introduce the following notation: z p (t ) =

(

)

( z (1)p ) (t ), ( z (2)p ) (t ) , where z (2)p (t ) is the T

T

T

m1 -vector of the nominal trajectories of the Cartesian coordinates in the directions in which the force trajectories are specified. Note that the trajectories z (2) p (t ) are not specified in advance, but have to be determined on the basis of the environment model. Similarly, the vector of desired force trajectories can be denoted as Fp (t ) =

(( F )

(1) T p

)

(t ), ( Fp(2) ) (t ) , where Fp(2) (t ) is the n1 -vector T

T

of nominal force trajectories in the directions in which forces are acting upon the robot, but the nominal trajectories of the Cartesian space coordinates are specified. The force trajectories Fp(2) (t ) are not specified in advance, but have to be calculated based on the dynamics model of the environment. The nominal trajectories of the forces and Cartesian coordinates must satisfy the environment model, i.e.:

M ( z p (t ), d ) ɺɺ z p (t ) + L ( z p (t ) , zɺ p (t ), d ) = − Fp (t )

(2.363)

A Unified Approach to Dynamic Control of Robots

197

Let us introduce the following notations:

M M =  11  M 21

M 12  T and L =  L1T ,L2T   M 22 

(2.364)

where the dimensions of the matrix M 11 are m1 × n1 , of M 12 m1 × m1 , of M 21

n1 × n1 , and of M 22 n1 × m1 , while L1 and L2 are the vectors of dimensions m1 and n1 respectively. The desired trajectory in the state space is denoted by

x p (t ) = ( z pT (t ), zɺ pT (t ) ) . Now, the control task is specified as a task of T

practical stability of the robot around the nominal trajectory x p (t ) in the following form: Let t1 be the predefined time period and Τ = [0, t1 ] . The robot

ɶ control has to ensure that ∀x0 ∈ R 2 n and ∀d ∈ D x(0) = x0 ∈ X 0

imply

ɶ t ), ∀ t ∈Τ , where X( ɶ t ) are the finite regions in the state space around x(t )∈ X( ɶ is the prescribed nominal trajectory x (t ) defined for every point of time t , X p

0

the finite region in the state space around the prescribed nominal trajectory ɶ , x (t ) ∈ X( ɶ t) , x p (0) in the initial instant. It is assumed that x p (0) ∈ X 0 p

ɶ ⊂ X(0) ɶ ∀ t ∈Τ , X . This formulation of the control task can be interpreted in 0 the following way. Given the desired trajectory x p (t ) , an initial error around the nominal trajectory is allowed such that the initial state x(0) = x0 must belong to

ɶ around x (0) ; the control has to ensure the robot’s the predefined region X 0 p state follows the desired trajectory x p (t ) with an allowed error which is constrained by the requirement that the state of the robot x(t ) in each point of

ɶ t ) around the nominal trajectory time must belong to the predefined region X(

x p (t ) in the predefined time period Τ . This must be provided for all admissible parameters d . Because of (2.363), and, as was shown in [10], the fulfillment of the specified control task also guarantees tracking of the desired force trajectories Fp (t ) , i.e. it guarantees that ∀ F0 ∈ R n and ∀ d ∈ D

F (0) = F0 ∈ Fɶ0

ɶ (t ) , ∀ t ∈ Τ , where Fɶ (t ) are the finite imply F (t ) ∈ F

regions in the n -dimensional space around the nominal trajectory

Fp (t )

198

Dynamics and Robust Control of Robot-Environment Interaction

ɶ is a region in the n -dimensional space defined for every point of time t , and F 0 around the nominal trajectory Fp (0) in the initial time. Note that the regions

ɶ and X( ɶ t ) . In order Fɶ0 and Fɶ (t ) must correspond to the respective regions X 0 to simplify the stability analysis let us consider specific forms of the finite ɶ and X( ɶ t) : regions X 0

{

}

{

}

ɶ t ) = x ∈ R 2 n / x − x (t ) < X e −α t , X( p t

ɶ = x ∈ R 2 n / x − x (0) < X , X p 0 0

∀ t ∈ Τ , where X t > X 0 > 0 , α > 0 . Here X t , X 0 , α denote real-valued positive numbers, ⋅ denotes Euclidean norm of the corresponding vector. Let

(

us denote by ∆x(t ) = x(t ) − x p (t ) = ∆z T (t ), ∆zɺT (t )

)

T

a 2n -dementional

vector of the state deviation of the real trajectory x(t ) from the desired nominal

ɶ and x(t )∈ X( ɶ t ), ∀ t ∈Τ trajectory x p (t ) . Then the inclusions x(0) = x0 ∈ X 0 will be fulfilled if ∆x(0) < X 0 and ∆x(t ) < X t e −α t , ∀ t ∈Τ .

2.10.5 Control law The dynamic position-force control law is considered (see Fig. 2.13) in the form:

τ = U * ( z , zɺ, ɺɺzc , F ) = ( J * ) (Λ*ɺɺzc + ρ * − F ) T

(2.365)

where J * , Λ* , ρ * denote the matrices and vector corresponding to J , Λ, ρ from the model (2.360) but with the assumed parameter values d = d 0 ∈ D . This means that we assume that the parameters values are not accurately known. The variable zc in (2.365) is governed by the equations: (1)   ɺɺ ɺ (1) ) z (1) p + P1 ( ∆z , ∆z ɺɺ zc =  * , (1) ɺ (1) ), Fp(1) + ∆F (1) )  z (1) p + P1 ( ∆z , ∆z W ( z , zɺ, ɺɺ (1) * (1) * (1) * ɺɺ W * ( z , zɺ, ɺɺ z (1) p , Fp ) = ( M 12 ) ( Fp − M 11 z p − L1 ) ,

−1

∆F (1) = K 1F ∫ ( F (1) (t ) − Fp(1) (t ))dt , P1 (∆z (1) , ∆zɺ (1) ) = K11∆z (1) + K 21 ∆zɺ (1) ,

199

A Unified Approach to Dynamic Control of Robots

where

M 11* , M 12* , L*1 denote the matrices and vector corresponding to

M 11 , M 12 , L1 from the model (2.364) but with the assumed parameter values d 0 ; F (1) is an m1 -subvector of the force vector F for which the nominal trajectories

are

specified

in

advance;

F=

(( F ) , ( F ) (1) T

),

T (2) T

1F ∆z (1) = z (1) (t ) − z (1) is an m1 × m1 matrix of force p (t ) is an n1 -vector; K

feedback gains; K11 and K 21 denote the n1 × n1 matrices of the position and velocity feedback gains, respectively. It is assumed that the quadratic matrix M 12* is nonsingular. For the sake of simplicity we shall assume that both matrices are diagonal, K11 = diag ( K1ii ) and K 21 = diag ( K 2ii ) . Note that, contrary to the so-called classical hybrid control schemes [1, 2], the control law (2.365) takes into account complete dynamic models of the robot and environment, as well as the interaction among directions in which position is controlled and directions in which force is controlled. This means that both position and force feedback loops are used in all directions. The control law may be considered in a more general form, i.e. P1 and ∆F (1) can be defined in a more general form. Note also that the control law (2.365) implies that the contact forces are directly measured by appropriate force sensors. The force sensors are normally based on measuring the ‘deformation’ and, therefore, their ability to measure highly dynamic changes of forces is limited. Therefore, certain limits on the allowable frequency of change of forces should be taken into account.

2.10.6 Practical stability analysis Here we will establish a procedure for the analysis of practical stability of the robot’s interacting with dynamic environment, based on the decompositionaggregation approach. The closed-loop model of the robotic system (model of deviation around the desired nominal trajectory x p (t ) ) in the state space is obtained by combining the robot model in the state space (2.362) and the corresponding control law (2.365):

∆xɺ = ∆f (∆x, x*p , d ) + ∆G (∆x, d ) F

(2.366)

where ∆f (∆x, x*p , d ) is a 2n -vector and ∆G ( ∆x, d ) is a 2n × n matrix. In [6, 10], it has been shown that the application of the control law (2.365) ensures

200

Dynamics and Robust Control of Robot-Environment Interaction

desired tracking of the prescribed nominal force trajectory (i.e. desired transient behavior of the force F (t ) ) when an ideal model of the system used in the control law is assumed. Since the model of the system (robot + environment) in (2.365) is not ideal (because of the parameters’ uncertainties) the desired force transient process cannot be perfectly achieved. However, it can be shown that by an appropriate selection of force feedback gains, and under assumption of limited deviations of the assumed model parameters from the actual values, the control law (2.365) can guarantee the force transient process satisfies the conditions of practical stability, i.e.

ɶ t ), ∀ t ∈ Τ F (t ) ∈ F(

(2.367)

Fig. 2.13 Dynamic position-force control law

{

}

where Fɶ (t ) = F ∈ R n / F − Fp (t ) < F t e− β t , ∀ t ∈ Τ . Here F t and β are positive numbers. If we denote by ∆F (t ) = F (t ) − Fp (t ) an n -dimensional vector-function of the deviation of the force vector F (t ) around the desired nominal trajectory Fp (t ) , then the inequality ∆F (t ) < F t e − β t will be fulfilled for each t ∈ Τ . The values F t and β  correspond to the values X t and α , since

ɶ = ∪ Fɶ (t ) corresponds to the region X ɶ = ∪ X( ɶ t) . the region F t ∈Τ

t ∈Τ

Starting from the assumption that the force transient process satisfies (2.367), it has to be examined whether the control law (2.365) can ensure the practical stability of the overall system. This means that the conditions under which the

A Unified Approach to Dynamic Control of Robots

201

proposed control law fulfills the specified control task, i.e. the conditions of practical stability of the robotic system have to be derived. On the other hand, if the system satisfies the conditions of practical stability with respect to the ɶ and X( ɶ t ) , then the force satisfies also the condition (2.367). For regions X 0 this reason, in what follows we will check the practical stability of the system ɶ , X( ɶ t ) , Τ ), which implies that the force satisfies (2.367). with respect to ( X 0 It can be shown [51, 52] that the system is practically stable with respect to ɶ ɶ t ) , Τ ) as defined above if there exists a real-valued continuously ( X 0 , X( differentiable function v(t , x) and a real-valued function of time Ψ (t ) which is integrable over the time interval Τ such that

vɺ(t , x) ≤ Ψ (t ), t

∫ Ψ(t ′)dt ′ < v

∀x ∈ X(t ), ∀t ∈T

ɶ t) ∂X( m

ɶ

− vM∂X0 ,

∀t ∈T

(2.368) (2.369)

0

ɶ denotes the boundary of the corresponding region and ∂X ɶ t ) \ X e−α t . In (2.368) vɺ denotes the time derivative of the function X(t ) = X( t

where

v(t , x) along the solution of the closed-loop system. The terms vm and vM in (2.369) denote the respective minimum and maximum values of v at the corresponding boundaries of the regions. Note that (2.368) and (2.369) are sufficient but not necessary conditions for the practical stability of the system. For justification of this method for testing the practical stability, see [51, 52]. Following the decomposition-aggregation approach to system stability analysis one can decompose the system (2.366) into a set of n subsystems: the position-controlled subsystems (i = 1,...,n1) and the force-controlled subsystems (i = n1 +1,…,n):

∆xɺ i = Aii ∆x i + ∆f i (∆x, x*p , ∆F , Fp , d ) + ∆Gi (∆x, d ) F , i = 1,…, n

(2.370)

where ∆xi (t ) = x i (t ) − xip (t ) = (∆z i (t )T , ∆ɺz i (t )T )T , i = 1,… , n and ∆f i and

∆Gi are the vectors and matrices of the appropriate dimensions, while the 2 × 2 matrices Aii are given by:

 0 Aii =  ii ii Q11K1

 , i = 1,… , n1 ;  Q K  1

ii 11

ii 2

202

Dynamics and Robust Control of Robot-Environment Interaction

 0 1  Aii =  ii Pi , i = n1 + 1,… , n ii Vi  Q22 L1 Q22 L1  and Q22ii are the diagonal (constant) elements of the n1 × n1

where Q11ii

and n2 × n2 matrices representing estimates of Q11 = Λ−11 J −T J * Λ*1 , and

Q22 = Λ−21 J −T J *Λ*2 ( Λ−11 , Λ−21 , Λ*1 , Λ*2 are the corresponding submatrices of the P matrices Λ −1 and Λ* ); L1Pi and LVi 1 represent elements of the vectors L1 and

LV1 estimating stability factors in the dynamics model of the environment and integral force feedback, i.e.

(M )

* −1 12

L*1 − ( M 120 ) L10 + ( M 12* ) K 1F ∫ ( F (1) (t ) − Fp(1) (t ))dt −1

−1

= L1P ∆z (2) + LV1 ∆zɺ (2) + L1R Let us consider the practical stability of the decoupled subsystems

∆xɺ i = Aii ∆xi . We may assume that the regions of practical stability can be presented in the form:

ɶ =X ɶ (1) × ..... × X ɶ (n) X 0 0 0 where

{

and

ɶ t) = X ɶ (1) (t ) × ..... × X ɶ ( n ) (t ) X(

}

ɶ (i ) = x (i ) ∈ R / x (i ) − x (i ) (0) < X (i ) , X 0 p 0

{

}

ɶ (i ) (t ) = x (i ) ∈ R / x (i ) − x ( i ) (t ) < X (i ) e −αi t , ∀ t ∈ Τ , and X p t

X t(i ) > X (0i ) > 0, α i > 0, ∀ t ∈Τ, ∀ i = 1,… , n . Let

us

select

the

vi (t , x) = ( ∆x i T H i ∆x matrices

of

the

)

i 1/ 2

functions

vi (t , x) , i = 1,… , n

in

the

form:

, i = 1,… , n , where H i are the positive definite

appropriate

dimensions

and

∆xi (t ) = x (i ) (t ) − x (pi ) (t ) ,

t ∈Τ, i = 1,… , n . The derivative of the functions vi along the solutions of the decoupled subsystems can be written as:

vɺi (t , x) = ( grad vi ) Aii ∆xi ≤ − γ i vi ≤ − γ i′ ∆xi T

(2.371)

where γ i = min λ ( Aii ) , λ ( Aii ) denotes the eigenvalues of the corresponding matrix and γ i′ = γ i λmin ( H i ) . The conditions (2.371) are valid provided

203

A Unified Approach to Dynamic Control of Robots

that the matrices H i are selected so as to satisfy H i Aii + Aii T H i ≤ − 2γ i H i . The coupling members in the subsystems (2.370) can be estimated in the following form (summation indices being omitted): (1) 1 (1) 2 (1) ( grad vi )T ∆fi (∆x, x (1) p , ∆F , Fp , d ) < ∑ ξ ij ( x p ) (t ) + ∑ ξ ij ( Fp ) (t ) + j

j

4 5 + ∑ ξij3 ( xɺ (1) p ) (t ) + ∑ ξ ij ∆x j + ∑ ξ ij ∆F j

(2.372) (2.372)

j

( grad vi )T ∆Gi (∆x, d ) F < < ∑ ξij6 ∆x j + ∑ ξij7

( (F ) p

j

)

(2.373)

+ ∆Fj , i = 1,… , n

where ξ ijk , k = 1, 2,… , nk ( nk denotes an appropriate number corresponding to

k ) are the real numbers (note that these numbers may also be negative). The ɶ t ) , ∀ t ∈Τ and inequalities (2.372) and (2.373) must be valid for ∀ x ∈ X( ∀ d ∈ D . The practical stability conditions of the overall system can be established by considering the derivative of the function vi along the solutions of the coupled subsystems (2.370) (aggregation principle). Based on (2.371) – (2.373), the candidates for the functions Ψ i (t ) for each subsystem i can be obtained in the form:

( )

Ψ i (t ) = −γ i Xt( i ) e −αi t + ∑ (ξij4 + ξij6 )Xt e−α t + ∑ ξij1 x (1) p

(

+ ∑ ξij2 Fp(1)

)

j

( )

(t ) + ∑ ξij5 F t

j

j

( )

(t ) + ∑ ξij3 xɺ (1) p

( )

e − β t + ∑ ξij7 ( Fp ) (t ) + F t j

j

e− β t ,

j

(t ) +

i = 1, 2

(2.374) By substituting (2.374) into (2.369) we obtain the practical stability test for the robot interacting with dynamic environment when the control law (2.365) is applied.

2.10.7 Example Let us consider the following simple example: n = 2 DOFs, acting upon the dynamic environment (see Fig. 2.14). The model of the environment in the x direction is assumed to be of the form:

204

Dynamics and Robust Control of Robot-Environment Interaction

me ɺxɺ + k D xɺ + ke ( x − xe ) = Fx

(2.375)

while in the y -direction the model is given by:

µe Fx = Fy where me is the equivalent mass of the environment in the x -direction; k D is the equivalent damping (not shown in the figure); ke is the equivalent stiffness,

xe is the equivalent position, and µe represents the equivalent friction. The system parameters include those of the robot (masses of the links m1 and m2 , moments of inertia of the links J1 and J 2 , lengths of the links l1 and l2 , and the distances between the joints and the mass centers of the links l01 and l02 ) and the environment parameters ( me , k D , ke , xe and µe ).

Fig. 2.14 Robot interacting with the environment

For the sake of simplicity let us assume that the parameters of the robot are precisely known, and only parameters of the environment are not sufficiently well known, but the lower and upper bounds of their allowable values are being known. This means that the set D of the allowable values of the parameters is defined as:

{

D = d : m1 = m10 , m2 = m20 , J1 = J10 , J 2 = J 20 , l1 = l10 , l2 = l20 , l01 = l010 , l02 = l020 ,

mel < me < meu , k Dl < k D < k Du , kel < ke < keu , xel < xe < xeu , µel < µe < µeu } (2.376) where superscript "0" denotes the ‘nominal’ values of the parameters and the indices l and u stand for the lower and upper bounds of the allowable

205

A Unified Approach to Dynamic Control of Robots

parameters values. Let us consider the following simple task: let in the x direction be defined the desired force Fp(1) (t ) = Fx0 (t ) = F 0 c = const , while in 0 0c the y -direction the desired position trajectory is imposed z (1) p (t ) = y (t ) = v t ,

where v 0 c = const . Let us consider how the dynamic position-force control law (2.365) can fulfill this control task. In the y -direction, the position control law is applied. Let us assume that the position, K111 , and the velocity, K 211 , feedback gains are being selected in such a way that the solutions of the characteristic equation of the closed-loop system K111 + K 211s + s 2 = 0 are −σ 1 and −σ 2 . In this case

K 211 = σ 1 + σ 2 and K111 = σ 1 σ 2 . In the x -direction, the force control law is applied. According to (2.365) this control law has the form:

τ = U * ( z, zɺ, ɺɺzc , F ) = ( J * ) (Λ* ɺɺzc + ρ * − F ) T

(2.377)

  K111 ( y − v 0 c t ) + K 211 ( yɺ − v 0 c )   ɺɺ zc = −1 ( me* ) [ F 0 c + K 1F ∫ ( Fx − F 0 c )dt − k D* xɺ − ke* ( x − xe* )]   where me* , k D* , ke* and xe* represent the prescribed ‘nominal’ values of the environment model (which must be within the allowed region of values of these parameters). From the assumption of the precise knowledge of robotic parameters it follows that J * = J , Λ* = Λ, ρ * = ρ . Therefore, one may assume that the control law perfectly compensates for the robot dynamics and for the forces in both directions. However, the control of the force Fx is realized using unknown parameters of the environment model. To apply the method proposed for practical stability analysis, the system can be ‘decomposed’ in two subsystems. The first subsystem is associated to the y direction in which the position of the robot tip is controlled, while the second subsystem is associated to the force-controlled x -direction. In this particular case the subsystem matrices Aii get the simple forms:

 0 A11 =  11  K1

1  K 211 

(2.378)

206

Dynamics and Robust Control of Robot-Environment Interaction

0  *  k A22 = − e + K 1F k Dl  m* me* e  ii

K 1F

1  mel k D*  − me* me* 

(2.379)

ii

since the elements Q11 and Q 22 are equal to unity. To define the matrices H i , which enable one to estimate the stability degree of the matrices Aii , it is convenient to introduce the transformation of the state coordinates ∆xi = T1i ∆xˆ i in such a way that the stability test of the decoupled subsystems in the transformed state space leads to (2.371). Assuming that the matrix H i in the function vi (t , x) is equal to I 2 , it is obtained H i Aˆ iiT + Aˆ iiT H i ≤ −2γ i H i , where

γ 1 = min { σ 1 , σ 2 } and γ 2 = min λ ( A22 ) . The stability test in this case leads to the following relations: (1)

ˆ −γ 1 X 0

1 − e −α1t

α1

1− e ˆ (2) −γ 2 X 0

−α 2 t

α2

(2)

(1)

(1)

ˆ e−α1t − X ˆ , ∀ t ∈Τ 1, ∀( x, t ) ∈ Ω p

(A.1)

Due to the continuity of the function β ( x, t ), which has a special form, such constant M can be determined that

β ( x, t ) ≤ M , ∀( x, t ) ∈ Ω

(A.2)

If in the system (2.51) we carry out the transformation

x = y e−γ (t −t0 )

(A.3)

yɺ = B(t ) y + αɶ ( y, t ) + βɶ ( y, t ) µ (t )

(A.4)

we will get where

B(t ) = A(t ) + γ I 2 n

αɶ ( y, t ) = eγ (t −t ) α ( ye −γ (t −t ) , t ) 0

0

(A.5)

βɶ ( y, t ) = eγ (t −t ) β ( ye −γ (t −t ) , t ) 0

0

whereby x(t0 ) = y (t0 ) . Then the system

yɺ = B(t ) y

(A.6)

212

Dynamics and Robust Control of Robot-Environment Interaction

is regular. Indeed, if β k (k = 1, 2,… , 2n) are characteristic indices of the linear system (A.6), it is obvious that

β k = α k + γ < 0 (k = 1, 2,… , 2n) Taking into account the regularity of the system (2.52) and formula (A.5) we get t

t

2n 2n 1 1 Sp ( ) lim Sp ( ) 2 2 B ω d ω = A ω + n γ d ω = α + n γ = βk [ ] ∑ ∑ k t →∞ t ∫ t →∞ t ∫ k = k = 1 1 t0 t0

lim

Therefore, the system (A.6) is regular. Let H (t ) ( H (t0 ) = I 2 n ) be the normed fundamental matrix of the system (A.6). The system of equations (A.4) can be replaced by the integral equation t

y (t ) = H (t ) y (t0 ) + ∫ K (t , ω ) αɶ ( y (ω ), ω ) + βɶ ( y (ω ), ω ) µ (ω )  dω

(A.7)

t0

where the matrix K (t , ω ) = H (t ) H −1 (ω ) is a Cauchy matrix [13, 53]. As all the characteristic indices β k (k = 1,2,...,2n) of the linear system (A.6) are negative, there is a positive constant C3 such that

H (t ) < C3 for t ≥ t0

(C3 ≥ 1)

Besides, we can introduce the following estimate [13] of the Cauchy matrix K (t , ω ) for the regular system with negative characteristic indices

K (t , ω ) < C4 eε 0 (ω −t0 ) for t0 ≤ ω < t < ∞ and for an arbitrarily small constant ε 0 > 0 . On the basis of (A.1), (A.2), (2.54) we have the following estimate

αɶ ( y, t ) + βɶ ( y, t ) µ (t ) ≤ eγ (t −t ) α ( ye −γ (t −t ) , t ) + β ( ye−γ (t −t ) , t ) µ (t ) 0

0

0

≤ eγ (t −t0 )C1e −γ p (t −t0 ) y + eγ ( t −t0 ) MC2e − λ ( t −t0 ) µ (t0 ) p

= C1e − ( p −1)γ (t −t0 ) y + MC2 e − ( λ − γ ) (t −t0 ) µ (t0 ) p

Taking into account the relation (2.55) let us select the positive number ε 0 so small to satisfy the inequalities

213

A Unified Approach to Dynamic Control of Robots

δ 0 = ( p − 1)γ − ε 0 > 0 and λ − γ − ε 0 > 0 . Hence, estimating with respect to the norm the right and left side of the integral equation (A.7) we have t

y (t0 ) + ∫ K (t , ω ) αɶ ( y (ω ), ω ) + βɶ ( y (ω ), ω ) µ (ω ) d ω

y (t ) ≤ H (t )

t0 t

≤ C3 y (t0 ) + ∫ C4C1e[ 0 

ε − ( p −1) γ ] (ω −t0 )

p y (ω ) + MC4C2 e − ( λ −γ −ε 0 )(ω −t0 ) µ (t0 )  dω 

t0

t

= C3 y (t0 ) + ∫ C5e[ 0

ε − ( p −1) γ ] (ω − t0 )

y (ω ) d ω + p

t0

(

)

C6 1 − e − ( λ −γ −ε 0 ) (t −t0 ) µ (t0 ) ≤ λ −γ − ε0 C6 µ (t0 ) t p ε − ( p −1) γ ] (ω −t0 ) ≤ C3 y (t0 ) + + ∫ C5e[ 0 y (ω ) dω , λ − γ − ε 0 t0 +

where

(A.8)

C 5 = C 4 C1 , C 6 = MC 4 C 2 .

Then the estimate (A.8) implies t C6 µ (t0 ) p y (t ) ≤ C3 y (t0 ) + + C5 ∫ e −δ 0 (ω −t0 ) y (ω ) d ω λ −γ −ε0 t0

(A.9)

From the inequality (A.9), according to the Bihari lemma [13, 54], we have

C3 y (t0 ) + y (t ) ≤

C6 µ (t0 ) λ −γ −ε0

  C µ (t0 )  1 − ( p − 1) C3 y (t0 ) + 6 λ − γ − ε 0   

p −1

C5   δ0  

1 p −1

(A.10)

if only

 C µ (t0 )  ( p − 1) C3 y (t0 ) + 6  λ −γ −ε0  

p −1

C5

δ0

t0 , such that

x(t ′) = CΓ x(t0 ) e− λ (t ′−t0 ) + a

(H.1)

Consequently, taking into account the theorem conditions CΓε + a < δ 0 ,

x (t0 ) < ε , for ∀t ∈ [t0 , t ′] holds the inequality: x(t ) ≤ CΓ x(t0 ) e − λ ( t −t0 ) + a < CΓε + a < δ 0 Hence, according to (2.211), for ∀t ∈ [t0 , t ′] holds the inclusions:

( qˆ (t ), qˆɺ (t ), qɺɺˆ (t ) ) ∈V ×V ×V

( q (t ), qɺ (t ), qɺɺ(t ) ) ∈Vq × Vqɺ × Vqɺɺ , Because of the inequality

q* (t ) − ɺɺɺ q p (t ) ≤ Γ1

( ηɺɺ + δ ) + Γ ( ηɺ qɺɺ

2

+ δ qɺ ) + Γ3

q

qɺ

(η

+ δq )

≤ R x(t ) + r < Rδ 0 + r

qɺɺ

(H.2)

(H.3)

and according to (2.212), for ∀t ∈ [t0 , t ′] , the following inclusion is fulfilled:

(

)

(

)

q* (t ) ≡ ɺɺɺ q p (t ) + Γ1 qɺɺˆ − qɺɺp + Γ 2 qɺˆ − qɺ p + Γ3 ( qˆ − q p ) ∈ Vɺɺɺq

(H.4)

From the inequality (H.4) follows that for ∀t ∈ [t0 , t ′] the inequality analogues to (2.208) is fulfilled:

µ (t ) < L ( CΓ x(t0 ) e − λ (t −t ) + a ) + Cρ < Lδ 0 + Cρ . 0

According to (2.213) we will have:

F (t ) ∈ VF , Fˆ (t ) ∈ VF

(H.5)

230

Dynamics and Robust Control of Robot-Environment Interaction

On the basis of (H.3), (H.4) we obtain:

(

)

(

) (

)

(

)

f qˆ , qˆɺ , qɺɺˆ , q* − Fɺp = f qˆ , qɺˆ , qɺɺˆ , q* − f q p , qɺ p , qɺɺp , ɺɺɺ q p ≤ Lqf η + δ q +

(

)

(

)

ɺɺ + δ qɺɺ + Lɺɺɺfq q* − ɺɺɺ + Lqfɺ ηɺ + δ qɺ + Lqɺɺf η q p ≤ C1 x(t ) + C2 < C1δ 0 + C2 Hence, in accordance with (2.214), for ∀t ∈ [t0 , t ′] :

(

)

f qˆ (t ), qˆɺ (t ), qɺɺˆ (t ), q* (t ) ∈VFɺ

(H.6)

On the basis of the mapping (2.187), relations (H.2), (H.4) – (H.6) and due to the fact that ∀k ξ k ∈ Vξ , we conclude that for ∀t ∈ [t0 , t ′] the following inclusion is fulfilled:

uk (t ) ∈ Vu

(H.7)

The relations (H.2), (H.4) – (H.6) guarantee the fulfillment of the Lemma conditions on the interval [t0 , t ′] and, consequently, of the estimate (2.206). In particular, it holds for t = t ′ :

x(t ′) < CΓ x(t0 ) e − λ (t ′−t0 ) + a which contradicts the equality (A.1). In this way, the estimate (2.215) has been proven for ∀t ∈ [t0 , t0 + p ] . It necessitates the fulfillment on this closed interval of the inequality (2.216) and of the relations (H.2), (H.4) – (H.7). As x(t0 ) < ε < 1 , then from the estimate (2.215) we have:

x(t0 + p / 2) < CΓ x(t0 ) e

−λ

p 2

+ a ≤ CΓ x(t0 ) e

−λ

p1 2

+ a = CΓ x(t0 ) e

ln

ε −a CΓ

+ a = x(t0 ) (ε − a ) + a < ε

x(t ) < ε , ∀t ∈ [t0 + p / 2, t0 + p]

(H.8)

In particular, the inequality (H.8) is fulfilled for t = t0 + p . Hence, the time instant t0 + p can be regarded as the initial one in the analysis of the robot’s behavior in contact with the environment. Since functions ξ j (t ) are the functions of uniformly bounded variations of order p , the relations (H.2), (H.4) – (H.7) and the Lemma’s conditions are fulfilled on the interval [t0 + p, t0 + 2 p ] again. Then ( ε < 1 ):

231

A Unified Approach to Dynamic Control of Robots

x(t0 + 3 p / 2) < CΓ x(t0 + p ) e

−λ

p 2

+ a < x ( t0 + p ) (ε − a ) + a

< ε (ε − a ) + a < ε , x(t ) < ε , ∀t ∈ [t0 + 3 p / 2, t0 + 2 p]

(H.9)

In particular, the inequality (H.9) is fulfilled for t = t0 + 2 p . By induction, we obtain that the relations (H.2), (H.4) – (H.7) are fulfilled for ∀t ≥ t0 . Hence, the conclusions (a), (b), (c) of Theorem have been proven. Let us prove the second inequality in the formula (2.215). On every closed interval of the length p the following estimate holds: ɶ x(t ) < CΓ x(tɶ ) e− λ (t −t ) + a, t ∈ [tɶ, tɶ + p], tɶ ≥ t0 .

Then, taking into account (2.209), from inequality (H.8) we obtain for

∀tɶ ∈ [t0 + p / 2, t0 + p ] :

x(tɶ + p / 2) < CΓ x(tɶ ) e This means that

−λ

p 2

+ a < x ( tɶ ) (ε − a) + a < ε (ε − a) + a < ε .

x(t ) < ε , ∀t ∈ [t0 + p, t0 + 3 p / 2] . Together with the

inequality (H.9) we have x(t ) < ε , ∀t ∈ [t0 + p, t0 + 2 p ] . By repeating this procedure we can conclude that x(t ) < ε , ∀t ≥ t0 + p . Hence, the estimate (2.215) is completely proven. As the estimate (2.216) holds for ∀t ∈ [t0 , t0 + p ] then taking into account the formula (2.209), we obtain:

µ (t0 + p / 2) ≤ LCΓ x(t0 ) e = LCΓ x(t0 ) e

ln

−λ

p 2

+ La + Cρ ≤ LCΓ x(t0 ) e

−λ

p2 2

+ La + Cρ

δ − La − Cρ LCΓ

+ La + Cρ = x(t0 ) (δ − La − Cρ ) + La + Cρ
0 is a fixed number. Because of the smooth vector function α ( x 2 , x1 , t ) = o( x ) at x → 0 , the positive constants C1 , C 2 and the constant

p > 1 can be found, such that:

α ( x2 , x1 , t ) ≤ C1 x1 + C2 x2 p

p

∀( x, t ) ∈Ω

(J.2)

Because of the continuity of the functions B (t ), β ( x2 , x1 , t ) in (2.256) it is possible to find the positive constants M 1 , M 2 such that:

B(t ) ≤ M 1 ,

β ( x2 , x1 , t ) ≤ M 2 , ∀( x, t )∈Ω

(J.3)

Because of theorem conditions the system of the first approximation of equation (2.256):

xɺ 2 = A(t ) x 2

(J.4)

is regular and characteristic indices α1 ,… , α 2 m of (J.4) are negative. Let us carry out the transformation of (2.256):

x2 = ye −γ ( t −t0 )

(J.5)

where the number γ satisfies the inequality:

max α k < −γ < 0 k

Then, we will have:

yɺ = Aɶ (t ) y + Bɶ (t ) x1 + αɶ ( y, x1 , t ) + βɶ ( y, x1 , t ) µ where:

Aɶ (t ) = A(t ) + γ I 2 m ,

(J.6)

238

Dynamics and Robust Control of Robot-Environment Interaction

Bɶ (t ) = eγ (t −t0 ) B (t ) ,

(J.7)

αɶ ( y, x1 , t ) = eγ (t −t )α ( ye −γ (t −t ) , x1 , t ) , 0

0

βɶ ( y, x1 , t ) = eγ (t −t ) β ( ye −γ ( t −t ) , x1 , t ) . 0

0

Evidently, x2 (t0 ) = y (t0 ) . The linear system:

yɺ = Aɶ (t ) y

(J.8)

is also regular, and all its characteristic indices are negative. In fact, if β k (k = 1, 2,… , 2m) are the characteristic indices of the linear system (J.8), then we have:

βk = α k + γ < 0

(k = 1, 2,… , 2m)

Taking into account the regularity of the system (2.259) and relation (J.7) we have: t

t

2m 2m 1 1 lim ∫ SpAɶ (ω )dω = lim ∫ (SpA(ω ) + 2mγ )dω = ∑ α k + 2mγ = ∑ β k t →∞ t t →∞ t k =1 k =1 t0 t0

Hence, the system (J.8) is regular. Let H (t ) be the normed ( H (t0 ) = I 2 m ) fundamental matrix of the system (J.8). Then, equation (J.6) is equivalent to the integral equation: t

y (t ) = H (t ) y (t0 ) + ∫ K (t , ω )[ Bɶ (ω ) x1 (ω ) + αɶ ( y (ω ), x1 (ω ), ω ) t0

(J.9)

+ βɶ ( y (ω ), x1 (ω ), ω ) µ (ω )] d ω where K (t , ω ) = H (t ) H −1 (ω ) is the Cauchy matrix. Since all characteristic indices of (J.8) are negative, the following estimate of its fundamental matrix H (t ) for all t ≥ t0 is valid:

H (t ) ≤ C3 , (C3 ≥ 1)

∀t > t0

(J.10)

The Cauchy matrix K (t , ω ) for a regular system with negative characteristic indices can be estimated in the following way [13]:

239

A Unified Approach to Dynamic Control of Robots

K (t , ω ) ≤ C4 eε 0 (ω −t0 ) , t0 ≤ ω < t < ∞

(J.11)

where C 4 > 0 is a suitably chosen constant, and ε 0 > 0 is an arbitrarily small constant. On the basis of (J.2), (J.3) and (2.257), (2.258) we obtain the following estimates:

Bɶ (t ) x1 = eγ (t −t0 ) B (t ) x1 ≤ eγ (t −t0 ) M 1 D2 e − λ (t −t0 ) x1 (t0 ) ,

αɶ ( y, x1 , t ) = eγ ( t −t ) α ( x2 , x1 , t ) ≤ eγ (t −t ) (C1 x1 + C2 ye −γ (t −t 0

p

0

0)

p

)≤

≤ eγ (t −t0 ) (C1 D2p e − λ p ( t −t0 ) x1 (t0 ) + C2 e −γ p (t −t0 ) y ), p

p

βɶ ( y, x1 , t ) µ (t ) = eγ (t −t ) β ( ye−γ (t −t ) , x1 , t ) µ (t ) ≤ 0

0

≤ eγ (t −t0 ) M 2 D1e− λ (t −t0 ) µ (t0 ) , Bɶ (t ) x1 + αɶ ( y, x1 , t ) + βɶ ( y, x1 , t ) µ (t ) ≤ Bɶ (t ) x1 + αɶ ( y, x1 , t ) +

(

p + βɶ ( y, x1 , t ) µ (t ) ≤ eγ (t −t0 ) M 1 D2 e− λ (t −t0 ) x1 (t0 ) + C1 D2p e − λ p (t −t0 ) x1 (t0 ) +

+ C2 e − γ p ( t − t 0 ) y Since e

p

+ M 2 D1e − λ (t −to ) µ (t0 )

− λ ( p −1)( t −t0 )

).

≤ 1 , choosing the constant M 3 such that:

M 1 D2 + C1 D2p x1 (t0 )

p −1

≤ M3

it is finally obtained:

Bɶ (t ) x1 + αɶ ( y, x1 , t ) + βɶ ( y, x1 , t ) µ (t ) ≤ ≤ e − ( λ −γ ) (t −t0 ) ( M 3 x1 (t0 ) + M 2 D1 µ (t0 ) ) + C2 e− ( p −1) γ (t −t0 ) y . p

Let us choose a positive number ε 0 , small enough to satisfy the inequalities:

δ 0 = ( p − 1) γ − ε 0 > 0 and λ − γ − ε 0 > 0 Therefore, by estimating the norms of both sides of the integral equation (J.9) we will have:

240

Dynamics and Robust Control of Robot-Environment Interaction t

y (t ) ≤ H (t ) y (t0 ) + ∫ K (t , ω ) Bɶ (ω ) x1 (ω ) + αɶ ( y (ω ), x1 (ω ), ω ) + t0 t

p ε − ( p −1) γ ] (ω − t0 ) + βɶ ( y (ω ), x1 (ω ), ω ) µ (ω ) dω ≤ C3 y (t0 ) + ∫ C4C2 e[ 0 y (ω ) + 

+C4 ( M 3 x1 (t0 ) + M 2 D1 µ (t0 ) ) e

t0

− ( λ −γ −ε 0 )( ω −t0 )

 d ω = C3 y (t0 ) + (J.12)

t

+ ∫ C5 e[ 0

ε − ( p −1) γ ] (ω −t0 )

y (ω ) dω + p

t0

C6 x1 (t0 ) + C7 µ (t0 ) 1 − e − ( λ −γ −ε 0 ) (t −t0 ) ≤ λ − γ − ε0

(

)

C6 x1 (t0 ) + C7 µ (t0 ) t p ε − ( p −1) γ ] (ω −t0 ) ≤ C3 y (t0 ) + + ∫ C5 e[ 0 y (ω ) dω λ − γ − ε0 t0 where C 5 = C 4 C 2 , C 6 = C 4 M 3 , C 7 = C 4 M 2 D1 . Then, the estimate (J.12) can be written as follows:

y (t ) ≤ C3 y (t0 ) +

t C6 x1 (t0 ) + C7 y (t0 ) p + C5 ∫ e −δ 0 (ω −t0 ) y (ω ) dω (J.13) λ − γ − ε0 t0

From the following inequality, in accordance with the Bihari lemma [13, 54], it follows that:

C3 y (t0 ) + y (t ) ≤

C6 x1 (t0 ) + C7 µ (t0 ) λ − γ − ε0

  C x (t ) + C7 µ (t0 )  1 − ( p − 1) C3 y (t0 ) + 6 1 0  λ − γ − ε0   

p −1

C5   δ0  

1 p −1

(J.14)

only if

 C x (t ) + C7 µ (t0 )  ( p − 1) C3 y (t0 ) + 6 1 0  λ − γ − ε0  

p −1

C5

δ0

> 1 . Now we can simply formulate the notion of coupled stability. Definition 3.1 (Coupled system stability) The interaction between a robotic manipulator under impedance control and a passive environment is said to be stable if the equilibriums p* and F* are stable (in the sense of Liapunov). Examining the interaction system models (3.19)-(3.29), it is obvious that for a stable position control system G p ( s ), consisting of a stable and proper regulator and a stable robotic plant, the coupled stability will be ensured if the transfer matrices

[I + Gˆ

t

−1

(s )Ge (s )]

−1

[

and/or I + Ge (s )Gˆ t

−1

(s )]

−1

(3.34)

a

are stable, assuming thereby both environment and target impedance to be also stable. Feedback system configurations suitable for the analysis of the coupled stability problem (3.34) based on the above interactive models are presented in (Fig. 3.7). To attain more realistic models, the force disturbance df is introduced to represent unmodeled effects, sensory noise etc. Practically, the feedback system in (Fig. 3.7a), corresponding to the coupled impedance model (3.28), can a A rational transfer-function matrix is stable if it is proper and has no poles in the closed right

half-plane (Maciejowski, 1989). The properness is required in order to ensure bounded outputs with bounded inputs.

274

Dynamics and Robust Control of Robot-Environment Interaction

be obtained directly from the adopted impedance control scheme (Fig. 3.4) by assuming an ideal position servo system (Fig. 3.8). The remaining configurations corresponding respectively to the penetration and deviation models are not immediately correlated to the physical control scheme on (Fig. 3.4). Therefore, we will first concern the configuration on (Fig. 3.7a). Definition 3.2 (Internal stability) The feedback system shown in (Fig. 3.7) is internally stable if and only if the transfer function matrix

(

)

−1  I + Gˆ t −1 ( s ) Ge ( s )  p    ˆ =  −1  F   I + Ge ( s ) Gˆ t −1 ( s ) Ge ( s ) 

(

)

( I + Gˆ (s ) G (s )) Gˆ ( I + G (s ) Gˆ (s )) −1

−1

t

e

−1

e

−1

t

t

−1

( s )  p0 

 d    f  (3.35)

is exponentially stable. The general purpose of this definition is to exclude unstable pole-zero cancellation between blocks in (Fig. 3.7) (hidden unstable modes), which cannot be detected by Nyquist-like tests. In general case, it is necessary to check all four of these transfer matrices to ensure internal stability. However, if the environment and realized target admittance blocks are both stable, or at least Gt−1 ( s ) is stable, it is enough to check the exponential stability of (3.34). Generally this is fulfilled if and only if i)

det[ I + Ge ( s )Gˆ t−1 ( s )] has no zeros in the closed right half-plane (CRHP);

ii)

[ I + Ge ( s )Gˆ t−1 ( s )]−1 Ge ( s ) is analytic (i.e. has non zeros) at every CRHP pole of Ge ( s ) including infinity.

In an ideal environment (3.24) it is sufficient only to check the first condition. In the ideal case when both Gt ( s ) and Ge ( s ) are diagonal matrices, it is quite easy to check the stability of (3.34). In multivariable systems, however, det [ I + Ge(s) Gt-1(s) ] becomes much more complicated and difficult to use for the synthesis of a coupled stabilizing impedance controller (i.e. target impedance). The problem is also the uncertainty of the environment, and the

275

Impedance Control

question remains whether the adopted model will provide acceptable performance upon real contact (robust control problem).

Fig. 3.7 Feedback configurations for investigation of coupled stability

If both the environment and the realized admittance are stable, the coupled stability of the interactive system in (Fig. 3.7) can also be assessed by means of the small gain theorem [8]. The small gain theorem states that a feedback loop composed of stable operators will certainly remain stable if the product of all operator gains is smaller than unity −1 Ge ( jω)Gˆ t ( jω) < 1

(3.36)

∞

It is relatively easy to prove that the stable loop transfer matrix

(

)

−1 −1 −1 Ge ( jω) Gˆ t ( jω) in conjunction with (3.36) implies I + Ge (s ) Gˆ t (s ) also

to be stable [2]. The small gain theorem provides a quite general law, valid for

276

Dynamics and Robust Control of Robot-Environment Interaction

continuous- or discrete-time, SISO and MIMO, linear and non-linear systems, and it is also the convergence criterion used in many iterative processes. Furthermore, the norm inequality criterion (3.36) can easily be extended to maintain the uncertainties in the target system and environment models.

Fig. 3.8 Impedance control in an ideal position servo system

However, the small gain theorem only gives the sufficient stability conditions, which in many cases are too conservative to be of much use in practical contact tasks. For example, assuming that ideal second-order target impedance has been achieved Gˆ t = Gt , the condition (3.36) implies the admissible target stiffness to be K t ≥ K e in order to ensure stable interaction. In real stiff environments this is without any practical relevance. This result is similar to the stability analysis performed by Kazerooni et al. [7]. The established interaction stability criterion practically implies that the gain of feedback compensator (i.e., the target admittance for the control law 3.15) should be limited by the magnitude of the sum of environmental admittance and robot position control sensitivity. For a SISO system this imposes the condition

K t ≥ min (K p ,K e ) in the steady state. In a direct drive robotic system having significantly lower position control stiffness (due to elimination of the transmission) in comparison to industrial robots, this condition might provide reliable target models for practical tasks. However, in industrial robots, with quite stiff servos gains ( K p > K e and S p ≈ 0 ), the above condition also requires the target stiffness to be higher than the environmental one. Moreover, no target model, i.e. the compliance feedback compensator G f can be found to enable interaction with

Impedance Control

277

an infinitely rigid environment ( K e → ∞ ). Therefore, one of the main conclusions in [7] has pointed out the practical needs for intrinsic compliance either in the robot or in the environment for the interactive stability. The conservativeness of this result was observed in experiments performed by McCormick and Schwartz [9], and will also be confirmed later in this work based on both stability analysis and experiments. 3.4.3.3 Coupling of passive systems A power tool for the analysis of the coupled stability problem is provided by passivity theory [10], i.e. the theory of positive systems, which plays a key role in network theory (passive networks).

Definition 3.3 (Real Positivity of a Matrix) A matrix H (s ) of real rational functions is positive real if i) All elements of H (s ) are analytic in the open RHP ( Re[s ] > 0 ) (i.e. they have no poles in RHP); ii) The eventual poles on the axis [ ] = are distinct, and the associated residue matrix of H (s ) is a positive semidefinite Hermitian; iii) The matrix H ( jω ) + H T (− jω ) is a positive semidefinite Hermitian for all real values of ω which are not a pole of any element of H (s ) . If positive real H (s ) is the closed loop transfer matrix of a linear timeinvariant multivariable system

xɺ = Ax + Bu y = Cx + Du

(3.37)

i.e.

H (s ) = D + C(sI − A)−1 B

(3.38)

then the system (3.37) is a positive system. For x (0) = 0 and for any input vector function u (t ) and corresponding solutions of the system (3.37), the following inequality represents an equivalent time-space definition of positive systems [10]

278

Dynamics and Robust Control of Robot-Environment Interaction t1

∫y

T

(t )u (t )dt ≥ 0

(3.39)

0

Assume p0 to be constantb and consider a static environment. Then the interactive model configuration (Fig. 3.7a) may be modified to present two timeinvariant networks coupled along interaction ports (Fig. 3.9). The coupling requires the velocities of robot and environment at contact point to be equal, while the forces acting upon robot and environment have opposite signs (action and reaction). If the environmental transfer matrix Ge ( s ) / s is positive real, representing any passive Hamiltonian environment, then the following theorem holds.

Theorem 3.2 (Passivity and coupled stability): A necessary and sufficient condition to ensure stability of a linearized robotic control system under impedance control, when coupled with any passive Hamiltonian environment (linear or non-linear), is that the realized admittance sGˆ t−1 ( s ) be positive real.

Proof : It is based on Popov’s lemma [11] stating that any block obtained from the feedback combination of two positive real blocks also is positive real, i.e. hyperstable. In other words, the feedback interconnection of passive systems is again passive. For the considered interaction system in (Fig. 3.9) consisting of a t1

passive environment, satisfying

∫F

T

(t ) pɺ (t )dt ≥ 0 , the sufficient stability

0

condition implies t1

∫F

T

(t )eɺ(t )dt ≥ 0

0

Considering the input/output relationship between the interaction force F(t) and the motion deviation eɺ(t ) , expressed by the admittance sGˆ t−1 ( s ) , it follows

t  ∫0 F (t ) ∫0 K (τ − t ) F (τ )dτ  dt ≥ 0 t1

T

b That corresponds to the analysis of coupled system stability around the equilibrium point.

(3.40)

Impedance Control

279

where K (τ − t ) is integral kernel. A necessary and sufficient condition for the input/output inner product (3.40) to be positive or null is that K (τ − t ) be a positive definite kernel [10], i.e. that the corresponding Laplace transform ∞

sGˆ t −1 ( s ) = K(t)e− st dt

∫ 0

need be a positive real matrix of rational functions of the complex variable s = jω. Initially, the passivity was utilized in works of Colgate and Hogan [12-13] to examine the robot contact instability phenomena associated with force feedback. However, the passivity referred to the entire robot control system instead of the realized admittance. Considering a SISO system, the coupled stability has been proven using the Nyquist criterion and the property of positive real transfer function having a limited phase by ± 90 degrees [10]. Then, it is relatively easy to prove that the mapping of the Nyquist contour of a positive real environmental impedance Ge ( s ) / s through an also positive real admittance sGˆ t−1 ( s ) , altering the phase by ± 90o and changing the magnitude by a factor 0 to ∞ , provides a stable system, i.e. a stable Nyquist plot of the open loop coupled system transfer function [12].

Fig. 3.9 Passivity model of manipulator/environment interaction

The system passivity concept provides a relatively simple test for the assessment of coupled system stability. In this test only passivity of the environment should be proven, without an accurate knowledge of parameters. Assuming again that ideal target impedance response (3.1)-(3.2) is realized, the

280

Dynamics and Robust Control of Robot-Environment Interaction

passivity of the target admittance sGt−1 ( s ) implies positive definite matrices M t , Bt and K t , and consequently, the closed loop system should be stable in contact with any passive environment to which it is directly coupled. The explicit design of a positive real robot control system, however, practically may become cumbersome [12]. Moreover, various practical control implementation effects, including computational time delay and sampling effects, as well as unmodeled dynamics (e.g. high order actuator and arm dynamic effects), may result in a non-passive real impedance control response [14]. The above stability results practically can be extended to nearly-passive control systems. However, in this case a passive environment can be found which destabilizes the coupled system. In order to simplify the coupled stability analysis, Colgate and Hogan [12] have introduced the term worst or most destabilizing environment denoting the most critical environmental for the coupled system stability. Such environmental impedance Ge ( s ) / s shapes the Nyquist contour of sGˆ t−1 ( s ) by minimizing the distancec from the critical point -1 to the nearest point on the Nyquist plot of the loop transfer function Ge ( s )Gt−1 ( s ) . Taking into account that driving point impedances of simple passive environmental models, such as mass or spring ( M e / s and sKe), perform the maximum rotation in the Nyquist plane, the authors have found that the worst passive environment for the coupled stability consists of a set of pure masses or springs. 3.4.3.4 Robust coupled system stability An efficient and useful framework to cope with the uncertainties in control system analysis and synthesis is provided by the robust control concept [2]. The uncertainties in the coupled system in (Fig. 3.7) occur both in the environment (e.g. uncertain model, parameters) and in the realized admittance (e.g. due to sampling effects, unmodeled dynamics, etc.).

c The distance from the critical point -1 to the nearest point on the Nyquist plot of the loop transfer

function is inversely proportional to the infinity norm of sensitivity transfer function

1/ s ∞ .

Impedance Control

281

Fig. 3.10 Interaction model with target system uncertainty

Generally, for the majority of practical contact problems, the assumption of a passive interactive environment yields a reliable model. In a passive environment, the simplest approach to handling environmental model and parameter uncertainties provides the above mentioned worst environment concept, identifying certain types of environment as being most destabilizing for a specific target system. The effects of uncertainties in the realized admittance on the coupled system stability can be examined by means of the basic feedback configuration in (Fig. 3.7c). This scheme is represented by the deviation-coupled model in which the resulting admittance can be considered as a “plant”. Basically, there are two approaches to dealing with the problem of plant uncertainty in a closed loop system: i)

ii)

unstructured [15], which usually assumes additive or multiplicative frequency-dependent perturbations of bounded magnitudes, but without considering physical origins and parametric models of the perturbations; structured [16-17] which accounts for the structure of the perturbations by assuming the parametric variations in the plant physical model to be within some range.

The first method we are going to apply, provides a simple input-output oriented approaches, which is similar to classical feedback-system design and analysis methods and can easily can be extended to practical sampled control problems. However, the price that may be paid is conservativeness, since in

282

Dynamics and Robust Control of Robot-Environment Interaction

general the unstructured concepts are more conservative than the structured methods, which consider the physical plant model. Consider the interaction model on (Fig. 3.9) and assume that the achieved admittance Gˆ t−1 ( s ) can be represented as the target one Gt−1 ( s ) perturbed by a multiplicative perturbation (Fig. 3.10) −1 Gˆ t (s ) = (I +

t

(s ))Gt −1 (s )

(3.41)

Conveniently, the perturbation can be presented in the form [2] t

(s ) = ∆(s )Wt (s )

(3.42)

where Wt ( s ) is a fixed stable transfer function matrix, and ∆(s ) is a variable stable transfer function matrix satisfying

∆ (s ) ∞ ≤ 1

(3.43)

Commonly Wt ( s ) is chosen as a diagonal stable, proper and minimum-phase transfer function matrix. For a SISO system (3.41) provides

Gt ( jω ) −1 ≤ Wt ( jω ) Gˆ t ( jω )

(3.44)

This inequality describes a disk in the complex plane with center at 1 and radius Wt ( jω ) (disk uncertainty). The purpose of ∆ is to account for the phase uncertainties and to scale the magnitude of perturbation. For some uncertainties models it is possible to allow ∆(s ) to be unstable, but then it is important to assume that the nominal and perturbed plant (target admittances) have the same number of unstable poles. Perturbation, which does not cancel unstable poles, is said to be allowable. Now, the following robust stability test can be introduced Theorem 3.3 (Robust coupled stability): A sufficient condition to guarantee that instability cannot occur for any possible allowable multiplicative perturbations of the target admittance satisfying (3.43), in contact with a passive stable environment is

(

Wt (s ) I + Ge

−1

(s )Gt (s ))

−1

≤1 ∞

(3.45)

283

Impedance Control

Proof : Since for an allowable perturbation the generalized Nyquist theorem implies that no locus of Gt−1 ( s )Ge ( s ) passes through -1 as the elements of

Gt−1 ( s ) vary, it follows

(

det I + Gˆ t

−1

( jω )Ge ( jω )) ≠ 0

which is the same as

(

)

σ I + Gˆ t −1 (s )Ge (s ) > 0 or

σ ( I + Gt −1 ( jω) Ge ( jω) +

{

−1 = σ ( t Gt −1Ge ) + 

−1 t

t

+ I  

t

which will hold if

{

Gt −1 ( jω) Ge ( jω))

( jω)

}

Gt −1 Ge > 0

}

−1 σ (Gt −1Ge ) + I  ∆ t −1 + I > 0



or

σ

(∆ ) −1

t

−1 σ (Gt −1Ge ) + I  > 1





where σ and σ denote smallest and largest principal gains (singular values) of transfer matrices, respectively. Hence the largest singular value is bounded by

σ

( ∆t ) σ

( Ι + G −1G )−1  < 1 e t  

and concerning (3.42)-(3.43) −1 −1 σ ( ∆Wt )σ ( Ι + Ge −1Gt )  < σ ( ∆ ) σ (Wt )σ ( Ι + Ge −1Gt ) 









 < σ W σ  Ι + G −1G −1  < 1 ( t ) ( e t)   from which taking into account the definition and basic properties of the ⋅ operator norm, one obtains the desired result (3.45). Condition (3.45) implies the bounds on the coupled system norm

(I + G

e

−1

( jω )Gt ( jω ))

−1 ∞

≤ Wt

−1

( jω )

∞

(3.46)

284

Dynamics and Robust Control of Robot-Environment Interaction

or on the perturbation norm

(

Wt ( jω ) ∞ ≤ I + Ge

−1

( jω )Gt ( jω )) ∞

(3.47)

in order to ensure robust coupled stability. It should be pointed out that the transfer function matrix

(I + G

e

−1

( jω )Gt ( jω ))

−1

(3.45)-(3.47) represents the

deviation model (3.29), i.e. the relationship between nominal penetration p0 and position deviation e. There is a practical control engineering graphical interpretation of the norm bounds (3.45)-(3.46) [2]. For a SISO system

Wt ( jω )

1 W ( jω )Gt ( jω) Ge ( jω) = t 1 + Gt ( jω ) Ge ( jω ) ∞ 1 + Gt ( jω) Ge ( jω)

≤1 ∞

imposes

Wt ( jω )Gt ( jω) Ge ( jω) < 1 + Gt ( jω ) Ge ( jω ) which says that at every frequency ω, the critical point (-1,0) lies outside a disk with center at the open-loop transfer function L( jω) = Ge ( jω ) Gt ( jω ) of the coupled system, with radius Wt ( jω )L( jω ) . Practically, Theorem 3.3 represents an application of the basic robust stability theorem [15] to the coupled system. As will be demonstrated later, this theorem provides essential and quite usable results for both coupled system control analysis and synthesis. Moreover, we will extend this theorem to an examination of the stability of contact transition process to and from constrained motion, based on a simple physical interpretation of the condition (3.46). 3.4.3.5 Coupled system performance The basic coupled system configurations in (Fig. 3.7-3.10) may also be used to analyze the performance of the interactive system, taking into account the uncertainties of the achieved impedance, i.e. admittance. Consider again the deviation coupled model sketched in (Fig 3.7c) and (Fig, 3.10). Using this model we can specify the performance of the coupled system in a similar way as in common feedback control systems by assessing the tracking of a reference signal. However, the specific goal is thereby not to achieve the model output

285

Impedance Control

(position deviation e) to perfectly track the input (nominal penetration p0), but rather to bound their normed difference

e − p0 p = ≤ε p0 p0

(3.48)

This inequality practically restrains the penetration i.e. the interaction force during contact, which is an essential practical control design constraint. Taking into account the input-output Theorem 3.1 and that the control error is shaped by the sensitivity functions, the performance specification can be given as

(

S ( jω ) ∞ = I + Gt

−1

( jω)Ge ( jω))

−1

≤ε

(3.49)

∞

or in general

(

W ( jω)S ( jω) ∞ = W ( jω ) I + Gt

−1

( jω)Ge ( jω))

−1

≤1

(3.50)

∞

where W (s ) is a frequency dependent weighting function. The above inequality specifies so-called nominal coupled system performance. Based on (3.50) we can define the performance of the perturbed system, focusing again on multiplicative perturbations (Fig. 3.11). Analogously to the robust stability, we introduce the robust performance test specified by the following condition.

Theorem 3.4 (Robust coupled performance): A necessary and sufficient condition for the robust performance of the interactive system is

(

W1 (s ) I + Gt

−1

(s )Ge (s ))

−1

(

+ W2 (s ) I + Ge

−1

(s )Gt (s ))

−1

≤1

(3.51)

∞

The proof is given in [2] for a general SISO feedback system. The significance of the Theorems 3.3 and 3.4 is that they define robust stability and performance in terms of acceptable forms (shapes) of a nominal target system taking into account possible destabilizing environments. Based on modern H ∞ synthesis techniques [3, 18], these theorems allow us to optimize the weighted performance objectives (3.45) and (3.50-51) over the set of admissible target models. This general control shaping technique is referred to as singular value shaping. However, as demonstrated by Stein and Doyle [15], the singular value loop shaping techniques are effective for control design of spatially round systems, but they can be excessively conservative in the so-called skewed systems,

286

Dynamics and Robust Control of Robot-Environment Interaction

characterized by large condition numbers of system matrices. Therefore, an alternate design paradigm has been proposed to handle skewed problems [18, 19], which utilizes the equivalent performance criteria but tighter matrix measure robustness norms. This approach is based on the new matrix functions

Fig. 3.11

M −∆

block diagram

Km called diagonally perturbed multivariable stability margin (MSM) introduced by Safonov and Athans [20], the reciprocal of which is known as µ, i.e. the structured singular value (SSV) [21]. For a general nominal feedback system with a block-structured transfer matrices M (s) and the block-diagonal perturbations ∆ (so called robust M - ∆ diagram, Fig. 3.11), these functions are defined as

K m (M ) =

1 = inf σ (∆) det ( I − M∆) = 0 µ (M ) ∆

{

}

(3.52)

or in other words, K m (M ) is the smallest σ (∆ ) which can destabilize the system ( I − M∆ ) −1 . If no ∆ exists such that det ( I − M∆ ) = 0 , then

K m (M ) = ∞ i.e. µ (M ) = 0 [18]. The measure can also be expressed as M

µ

= sup µ (M ( jω )) ω

Since the function µ (M ) accounts for the structure of the perturbations and systems in order to assess the instability and has similar properties as singular values, the terms structured singular value is suggestive but rather misleading since ⋅

µ

is not a norm. Doyle [21] has proved a number of properties of µ(⋅) .

The most important of these for stability analysis are as follows

287

Impedance Control

µ ( AB ) ≤ σ ( A)µ (B ) ρ (M ) ≤ µ (M )

(3.53)

Using the structured singular values, the robust coupled stability theorem can be expressed as Theorem 3.5 (Structured robustness of coupled stability): A sufficient condition to guarantee that instability cannot occur for any possible allowable multiplicative perturbations of the target admittance satisfying (3.43), in contact with a passive stable environment is

(

Wt (s ) I + Ge

−1

(s )Gt (s ))

−1

≤1

(3.54)

µ

(

)

Proof : Denote M (s ) = Wt (s ) I + Ge −1 (s )Gt (s ) write based on (3.53)

−1

. Since σ (∆ ) ≤ , we can

sup[ρ (M∆ )] ≤ sup[µ (M∆ )] ≤ sup[µ (M )σ (∆ )] ≤ M ω

Thus, if M

ω

µ

ω

≤ 1 then ρ (M∆ ) ≤ 1 and I − M

µ

cannot be singular (on the

imaginary axis). Hence the coupled system remains stable. In order to provide practical tests for stability robustness using structured singular values, several numerical procedures for computing µ(⋅) have been proposed [19]. The advantage of µ control analysis and synthesis procedures is that they combine both structured and unstructured uncertainties. However, they are coupled with very demanding computations. 3.4.3.6 Performance of the controller Gf = Gt-1 Now let’s analyze the performance of the convenient impedance control law (3.19) by assessing the achievable interactive system behavior based on the above introduced evaluation criteria and measures. As already mentioned, in the considered linearized robot/environment control model, the control laws G f ( s ) = Gt−1 ( s ) provides the following admittance/impedance behavior

288

Dynamics and Robust Control of Robot-Environment Interaction −1 −1 Gˆ t (s ) = G p (s ) Gt (s )

(3.55)

−1 Gˆ t (s ) = Gt (s ) G p (s )

For the sake of simplicity consider a SISO system. The first observation is that the realized admittance has a higher order than the desired second-order model (3.2). Generally, such a system is much more difficult to stabilize when is coupled to a stiff environment. Concerning the passivity of the realized admittance, if we select passive target admittance sGt−1 ( s ) , the passivity of (3.55) is not ensured, since a passive position control behavior G p ( s ) is difficult to achieve in practice. Commonly implemented manipulator position control laws (e.g. integral feedback) violate the passivity condition. That means that there are some passive environments (pure stiffness and masses) that can destabilize the coupled system. Considering the robust stability, we can write −1 −1 Gˆ t (s ) = (I − S p (s ))Gt (s )

and taking the position control sensitivity S p ( s ) as the model perturbation weight function, the robust coupled stability Theorem 3.3 implies

(

S p (s ) I + Ge Since

S p (s)

∞

−1

(s )Gt (s ))

−1

≤1

(3.56)

∞

≤ 1 , the coupled stability with the controller G f = Gt−1

is

guaranteed if

(I + G

e

−1

(s )Gt (s ))

−1

≤1

(3.57)

∞

is fulfilled. This is a quite conservative condition, which is difficult to realize. In effect, it allows a multiplicative uncertainty of the target admittance to be large almost 100% before instability occurs.

3.5 Improved Impedance Control The above analysis and example have clearly demonstrated that in spite of simple and straightforward control law providing relatively small model error, the most frequently applied impedance compensator (3.19) exhibits poor coupled

Impedance Control

289

performance. Although, following the robust control concept, we can select the target admittance model, i.e. the compensator (3.19), which will guarantee the coupled stability in contact with a stiff environment based on (3.56)-(3.57), this could be a very conservative design providing a sluggishd system behavior and large contact forces. Therefore, in the impedance control approach pursued in this Chapter a more accurate realization of the target model becomes quite important for improving the coupled system performance. As already mentioned, by decomposing the impedance control synthesis problem into the selection of target model and its realization, we have consequentially simplified the design. A propriety target model can be selected to meet robust coupled performance in contact with an environment. Then the impedance compensator should be designed to realize this target model as accurately as possible since, as demonstrated above, relatively small perturbances caused by the position controller still can jeopardize the desired contact behavior. An outstanding reason to follow this approach is that in an industrial robot control system we can improve the accuracy of the achieved impedance control in a rather simple way. As discussed in the Chapter 1, in order to realize a given target impedance system it is necessary to cancel robot dynamic and position control effects, for example using the inverse dynamics control law. However, the implementation of the non-linear dynamic control law in industrial robots is still very complex. Even more, the benefits of this control are not clear, since at usually low velocities in contact tasks, the friction effects in high-gear joint transmission systems play a dominant role in the dynamic behavior, rather than non-linear inertial effects. The joint friction is quite difficult to be compensated for due to the highly complex non-linear and variable nature [22]. As already mentioned, the industrial position controller is sufficiently reliable and robustly designed to cope with friction and other non-linear control effects. This was the main reason to retain the internal position controller in the considered compliance control systems. Utilizing performance of the industrial robot position control system in the Cartesian space (i.e. compliance frame), expressed by diagonal dominance and spatial roundness of position control transfer matrix, we can relatively simple implement a dynamic impedance control approach based on linear compensation technique.

d As will be demonstrated later, the robust stability conditions (3.56-57) imply relatively large

target damping.

290

Dynamics and Robust Control of Robot-Environment Interaction

3.5.1 Improved control law Substituting

x − x0 = Gt

−1

(s )F

(3.58)

in (3.11) we derive the expression for the position modification ∆x f which ensures the realization of the target model in the form

∆x f = G p

−1

(s )[(Gt −1 (s ) − S p (s )Gs (s ))F − S p (s )x0 ]

(3.59)

The corresponding impedance control scheme (Fig. 3.12) includes the compensator

G f (s ) = G p

−1

(s )(Gt −1 (s ) − S p (s )Gs (s )) = G p −1 (s )Gt −1 (s ) − Gr −1 (s )

(3.60)

and an additional nominal position feed-forward term

Gp

−1

(s )S p (s )x0 = Gr −1 (s )Gs −1 (s )x0

(3.61)

The position control output corresponding to (3.59)-(3.61) is

(

)

τ = Gr ( s ) x0 − x f − x = Gr ( s ) ( x0 − x )

(

)

+ Gs −1 ( s ) + Gr ( s ) Gt −1 ( s ) F + Gs −1 ( s ) x0 + F

(3.62)

It is relatively easy to show that in the linearized robot control system, this control law provides an equivalent effect to that of the computed-torque based impedance control (3.23). Essentially, the main issue in (3.60) is to compensate for dynamic effects in the forward position control in order to achieve the given target model, the effect of which is similar to the non-linear control goal. The difference is that the control law defined in (3.59)-(3.61) is based on linearized compensation techniques, which are less complex than the computation of nonlinear robot dynamics. However, the impedance compensator (3.60) includes the inverse position control system G p−1 ( s ) , which generally is not well suited for use in a compensator, since the inverse systems principally produce large control signals, amplify high-frequency noise and may introduce unstable pole-zero cancellations. In spite of this, we can require that G p−1 ( s ) shows inverse characteristics over some finite frequency range only. In order to obtain a realizable (proper) compensator we can employ a low-pass filter (by inserting some more poles), or even utilize low-pass performance of the target admittance

291

Impedance Control

Gt−1 ( s ) . Moreover, taking the advantages of the diagonally dominant industrial robot position control system we can easily determine G p−1 ( s ) by inverting the dominant diagonal part of the position control transfer matrix −1 G p −1 ( s ) = diag ( Gɶ pii ( s ) )   

for (i=1,...6).

Fig. 3.12 Target model realizing scheme

Furthermore, G p (s ) is spatially round in industrial robots, assuring that the behavior of the position control system is independent of either the Cartesian direction or compliance frame selection in a large work area far from singularities. These characteristics, quite important in position control in order to ensure uniform performance in Cartesian space, allow the impedance control to be implemented in a simple way, in the same spirit as the position controller, using a constant compensator (3.60) for a given target admittance. For the considered position controller (3.4)-(3.9) and adopted target impedance (3.1), in the case of a SISO system the compensator (3.60) has the form

G f (s ) =

( Λ − M t )s 3 + (B − Bt )s 2 + (K p − K t )s + K I

(K

p

(

s + K I ) M t s 2 + Bt s + K t

)

(3.63)

292

Dynamics and Robust Control of Robot-Environment Interaction

For a stable target admittance and position regulator the above transfer function also is stable. However, the occurrence of zeros in (3.63) can produce interesting effects upon position-correction/contact-force relationship. Most frequently in practice the aim is to reduce the large industrial robot impedance and to realize a compliant behavior by selecting a target system with M t , Bt , K t Λ, B, K p the

(

) (

)

feedback system in (Fig. 3.12) becomes positive. That means, if it is required to achieve an impedance higher than that of the robot control system, the positive force feedback should be applied. It is a quite interesting result, which can be generalized to the overall impedance control. The problem with the compensator law (3.60), however, is that its second term depends on the actual robot configuration and compliance frame directions. Practically, we have demonstrated that the performance of industrial robot position transfer matrix G p ( s ) is quite robust with respect to the change of the robot configuration and compliance frame selection, but that is not valid for separated regulator Gr (s ) and robot plant matrices Gs ( s ) . However, considering that the purpose of this term is to compensate for the disturbance force, the effect of which on impedance control model error is due to stiff position control system insignificant, we can simply neglect this term and get

G f (s ) = G p

−1

(s )Gt −1 (s )

(3.64)

or in the SISO system case

( )=

Λ

(

+ +

+

)(

+ +

+

)

(3.65)

Impedance Control

293

The role of the last feedforward term G p−1 ( s ) S p ( s ) x0 in the control law (3.59) is to compensate for the convenient position control error during acceleration and deceleration motion phases, which also influences the impedance model error (3.20). Considering the expressions for the closed loop position control and sensitivity transfer matrices (3.10)-(3.12), one obtains the feedforward prefilter in (Fig. 3.12) in the form

Gp

−1

(s )S p (s ) = Gr −1 (s )Gs −1 (s )

(3.66)

which gives the SISO system

Λs 3 + Bs 2 Λs + B ɺxɺ0 x0 = K ps + KI K ps + KI

(3.67)

This transfer function can be realized as a phase-lag filter of nominal acceleration obtained from the interpolator. However, the problem is again the dependence of the transfer function (3.66) on the robot position. This requires changing the impedance control parameters dependent not only on the current task, which prescribes corresponding target impedance behavior and C-frame location, but also in the function of robot configuration. Practically, the control law (3.59) ensures good model tracking close to a working position for which the gains (3.64) have been computed. However, if this position is changing during a task, it is necessary to compute on-line Cartesian robot dynamic and position control parameters in order to compensate for variable dynamic effects. This causes the impedance control law implementation to become quite complex, similar to the case the dynamic robot control (e.g. feedforward compensation or computed torque method). Computing the entire non-linear robot dynamic model in the real time, we lose the advantages of the simple position control structure and performance. However, assuming that the nominal motion exhibits slow acceleration/ deceleration in the vicinity of constraints and during contact, which is a reliable premise due to in general unknown constraints, we can also neglect the feedforward term (3.66)-(3.67), and thus substantially simplify the control law

~ −1 −1 G f (s ) = G p (s )Gt (s )

(3.68)

~

−1 where G p−1 = diag (G pii ) is the inverse of the estimated position control

transfer matrix, i.e. of its dominant diagonal part. The controller (3.68) practically consists of a diagonal and for a given task constant compensator. In order to obtain a realizable compensator G f ( s ) , the convolution

294

Dynamics and Robust Control of Robot-Environment Interaction

~ −1 −1 G pii (s )Gtii (s ) should provide a proper and stable transfer function. In other ~ words G pii (s ) has to be a stable minimum-phase function fulfilling the following condition

(

( (

)

))

(

) + size ( num (Gɶ ( s )))

size den (Gtii ( s )) + size den Gɶ pii ( s ) ≤ size num (Gtii ( s ))

(3.69)

pii

where num(⋅) and den (⋅) denote the numerator and denumerator of transfer functions respectively. 3.5.2 Coupled system performance The benefits of the new control law are obvious. By keeping the simple diagonal control structure and constant gains values (for a selected target model), the control law provides more accurate target model realization and better coupledperformance. In accordance with equations (3.23)-(3.27), the target impedance/ admittance achieved by the control law (3.68) has the form

~ −1 −1 −1 Gˆ t (s ) = G p (s )G p (s ) Gt (s ) (3.70) ~ −1 Gˆ t (s ) = Gt (s )G p (s ) G p (s ) ~ which in the ideal case G p (s ) ≡ G p (s ) , provides Gˆ t (s ) = Gt (s ) . As already demonstrated in practice, we can relatively easily and accurately estimate the robot position control function by step response or dynamic parameter estimation experiments [25]. In any method, due to identification errors and variation of the position dependent inertia matrix, the diagonal constant gain

~

estimation matrix G p ( s ) differs from the actual closed loop control transfer matrix. However, taking into account the performance of the commercial industrial robot controllers, especially the diagonal-dominance of constant rotor inertia terms, these variations could be considered to be small, i.e.

~ −1 −1 G p (s ) = G p ( I +

)

with ∆ > 1 in order to achieve a well damped interaction, i.e. the above coupled system, taking into account that K t 0, t ≥ t 0

(3.91)

2. The interaction force is non-negative

F (t,D0 ) ≥ 0, t > t0

(3.92)

3. The position deviation is less than the nominal penetration

e(t,D0 ) < p 0 (t ), t > t 0

(3.93)

i.e.

e(t,D0 ) p0 (t )

=

p 0 (t ) − p(t,D0 ) p0 (t )

=

x0 (t ) − x(t,D0 ) x0 (t ) − xe

p0, which provides a negative position error e. However, in a contact between a reliable industrial robotic system with realistic stiff environment, the actual motion is, as already mentioned, nearly stopped by the resistant force and impedance control effects. Hence this case does not have practical relevance. The advantage of the geometric criterion (3.93)-(3.94) over the previous ones (3.91)-(3.92) is that it compares two time signals. For this purposes various norms can be applied

e ≤ p0

(3.95)

such as the 1-norm (l1) ∞

e 1 = ∫ e(t ) dt t0

2-norm (square root of the signal energy) 1

∞ 2 2  e 2 = ∫ e(t ) dt  t  0  or ∞-norm (

∞

)

e

∞

:= sup e(t ) t

The norm comparison offers possibilities to apply relatively simple and efficient control techniques for the contact stability analysis. In that case, however, the criterion (3.95) only ensures sufficient contact stability conditions, but not the necessary ones. Obviously, even when e ≤ p0 is filled there may exist time intervals in which e(t ) > p0 (t ) (see the example in Fig. 3.31). Consequently, the obtained contact stability indices might be conservative. Definition 3.4 supposes that the robot/environment system still meets the coupled stability conditions. From this viewpoint, the contact transition stability implies stronger conditions than the coupled stability. As will be demonstrated that is a quite realistic postulate.

331

Impedance Control

Fig. 3.34 Geometric contact stability

The stability of the contact transition can be reasonably defined using the practical stability concept [32]. From a practical view the contact transition can be considered stable if the deviations of the actual position remain within certain bounds determined by the environmental surface. Practical stability generally does not require a stable equilibrium in the sense of Lyapunov. For the test of contact stability one defines a region X0 (t) around the nominal trajectory x0 (t), i.e. relative to the contact point a pre-defined region E (t) around p0 (t ) (Fig. 3.35). To guarantee the contact this region should be selected to ensure

{

[

E (t ) ≤ p0 (t ) . A finite time period T = t , t ∈ t0 , t1

]} and admissible initial

values region DI should also be selected.

Definition 3.5 (Practical Contact Transition Stability) The contact transition is said to be stable if ∀p (0 ) ∈ D0 , D0 ⊂ DI and ∀t ∈ T the deviation between the nominal penetration p0 (t ) and the actual penetration p (t ) , e(t ) = p0 (t ) − p (t ) , is constrained by E (t ) around p0 (t ) , i.e. e(t ) ∈ E (t ) . Definition 3.6 (Domains of contact stability) The sets of all initial values D0 and parameters Dt for which the contact transition is stable define the domains of contact stability.

332

Dynamics and Robust Control of Robot-Environment Interaction

Fig. 3.35 Practical contact stability region

For the developed position based impedance control law the domain of stability in the parametric space is determined by the target impedance and

{

}

environment parameters Dt = M t , Bt , K t , K e ,⋯ . It is customary to use a normalized set of parameters such as

Dt = {M t , ξ t , ωt , κ,⋯}

(3.96)

3.8 Contact Stability Conditions In this section reliable explicit conditions and domains D0 and Dt under which the proposed impedance control law fulfils the specified impedance control tasks and ensures a stable contact transition will be determined. The main goal is to examine the parametric domains of the contact stability, and to benefit from the simple impedance control law and design (practically based on the selection of target impedance parameters and approaching velocity), rather than to develop complex transition control algorithms. It is desirable to keep the domains D0 and Dt as large as possible in order to meet the requirements of a broad class of contact tasks and to ensure robustness against environmental and robot uncertainties. Contact stability tests and conditions may be used to either check the stability of the specified control laws or to define procedures for the synthesis of the parameters. As will be demonstrated, the established contact stability conditions can easily be extended to other compliance control laws as well as non-linear interaction models.

333

Impedance Control

3.8.1 Time domain analysis A good place to begin the analysis is the ideal deviation impedance model (3.29) realized by the improved control law (3.68) in each constrained direction (for the sake of simplicity the environment is presented as a stiffness)

(

)

ɺeɺ + 2ξtωt eɺ + ωt 2 + ωe 2 e = ωe 2 p0

(3.97)

where analogous to the target frequency, ω represents the equivalent frequency of the environment

ωe =

Ke Mt

(3.98)

The relationship between the frequencies is determined by the stiffness ratio

κ = ωe ωt >> 1

( ) is a continuous, monotone It is assumed that the nominal trajectory increasing and bounded function, with initial conditions D0 (3.88) and the ∗ constant maximum value (maximum penetration) p0 (t , t ≥ T ) = p0 ∗ and corresponding to the stationary nominal goal position ɺ ( ≥ ) = ɺɺ ( ≥ ) = . The target and environmental parameters are assumed to be constant. Under these circumstances, the position deviation e(t ) and the actual penetration p (t ) reach equilibrium as → ∞ ∗

∗

=

+κ κ = +κ

∗

(3.99) ∗

The equilibrium of the second-order perturbed system (3.97) is asymptotically stable for ξt > 0 , which ensures negative real parts for characteristic equation eigenvalues

(

)

λ2 + 2ξtωt λ + ωt 2 + ωe 2 = 0 λ1, 2 = −ξtωt ± jωt 1 + κ − ξt 2 = −ξtωt ± jωd where

(3.100)

334

Dynamics and Robust Control of Robot-Environment Interaction

ωd = ωt 1 + κ − ξt 2 Thereby it is assumed that ξt < 1 + κ which corresponds to the undercritical damping case. Increasing target damping is a common practical approach in the impedance control to stabilize contact between the robot and the environment. Basically the strategy to maintain a stable contact is to enlarge ξ t to achieve a critical or overcritical coupled system (3.97) [29], i.e. a dominant real pole of the coupled target-model/environment transfer function. That implies ξ t ≥ 1 + κ . However, the high damping gives the robot sluggish behavior. That is not desirable during contact establishment, since it generally produces large forces and long transition times. Therefore in our consideration we are looking for the minimum damping needed to ensure both stable contact and satisfactory robot performance. The stability of the coupled system (3.97) can be easily proven using the Liapunov method. For this purpose we utilize an equivalent model obtained by shifting the origins to the equilibrium (3.97) yielding

(

)

(

ɺeɺ + 2ξtωt eɺ + ωt 2 + ωe 2 e = ωe 2 p0 − p0∗

)

(3.101)

Choose a Liapunov function such as

V=

1 ɺ2 2 2 e + ωt + ωe e 2 ≥ 0 2

[ (

) ]

(3.102)

which is a positive definite function (except at the origin) representing total position deviation energy. Then

(

)

∗ 2 Vɺ = −2ξtωt eɺ 2 − ωe p0 − p0 eɺ ≤ 0

(3.103)

Considering typical contact transition behavior sketched in (Fig. 3.29), we can see that the deviation velocity ɺ = ɺ remains positive as the nominal motion ∗ → ), and takes relatively small negative values after the goal increases ( nominal position is reached. Hence the above condition requires a minimum amount of damping ξt > 0 to ensure coupled system stability. However, as demonstrated in experiments, that is not valid for contact stability. To obtain conditions ensuring contact stability by the Liapunov method, it is necessary to find a more reliable Liapunov function, which is generally a difficult problem. Estimation of contact transition stability measures (3.91)-(3.94) based on the Liapunov function (3.102) might be useless for the transition stability analysis. For the considered simple linear system (3.97), more accurate bounds can be obtained by solving the differential equation taking into account the initial

335

Impedance Control

conditions (3.88). For the considered under-critical damping case ( ξt < 1 + κ ) the solution has the form [33]

e(t ) =

ωe 2 t p0 (ζ ) e −ξ ω (t −ζ ) sin ωd (t − ζ ) dζ ∫ ωd t t

(3.104)

t

0

( ) represents a common As mentioned, the nominal penetration function ( ) is an at least C2nominal motion profile. In industrial robotic systems continuous function, having continuous second (acceleration profiles) or higher derivatives. Thus, we can apply partial integration upon (3.104), taking into account the initial conditions (3.88), which yields e(t ) = +

v κ  2ξ t pɺ 0 (t ) + 0 e −ξ ω (t −t ) sin[ωd (t − t0 ) − 2θ ] p 0 (t ) − 1+ κ  ωd (1 + κ )ωt t t

0

 −ξtωt (t −ζ ) ɺ ɺ ( ) ( ) p ζ e sin [ ω t − ζ − 2 θ ] d ζ  d ωd t∫0 0  1

t

(3.105) where

sin θ =

1 + κ − ξt 1+ κ

2

; cos θ = −

θ = atan2 (sin θ , cos θ );

π 2

ξt 1+ κ

(3.106)

≤θ ≤π

The above equations provide a starting point to analyze nominal system transition behavior dependent on a nominal motion phase in which the robots impacts the environment (see Fig. 3.30-3.32). 3.8.1.1 Constant velocity phase contact Consider firstly the constant velocity phase impact (Fig. 3.31). Since the nominal acceleration is zero, i.e.

ɺpɺ0 (t ) = 0; pɺ 0 (t ) = v 0 = const; p (t ) = v0 (t − t0 ) it follows from (3.105)

(3.107)

336

e(t ) =

Dynamics and Robust Control of Robot-Environment Interaction

 2v0ξ t v κ  + 0 e −ξ ω (t −t ) sin[ωd (t − t0 ) − 2θ ] v0 (t − t0 ) − (1 + κ )ωt ωd 1+ κ   0

t t

(3.108)

Time-differentiation yields the position-error velocity

eɺ(t ) =

  1+ κ v0 1 + e −ξtωt (t −t0 ) sin[ωd (t − t0 ) − θ ] 2 1+ κ  1 + κ − ξt  

κ

(3.109)

The most critical point for the lost of contact C on the phase diagrams (Fig. 3.29) corresponds to the minimum penetration. This point can be obtained from the condition

pɺ (tC ) = 0

(3.110)

which is equivalent to

eɺ(tC ) = pɺ 0 (tC )

(3.111)

Substituting (3.107) and (3.109) in (3.111) and assuming of simplicity yields the transcendental equation

e−ξ t ω t t sin (ωd t − θ ) = The solution of this equation

1 + κ − ξt

κ 1+ κ =

2

=

sin θ

κ

=

for the sake

(3.112)

depends on the target impedance

parameters ξ t , ωt and environment/target system stiffness ratio κ . Generally we can apply some known numerical methods to compute this solution. However, taking into account that in real applications κ is large (usually κ ≈ 50 …100 ), i.e. the term on the right side is small, it is possible relatively simply to obtain an approximate solution. For this purpose we can utilize the diagram in (Fig. 3.36) illustrating the graphical solution of the transcendental equation (3.112) for a typical transition case. Apparently for small sin θ κ the time period

needed to reach the minimum penetration after initial impact is

quite close to the second zerog of sin (ωd t − θ ) corresponding to

g The first zero

t = θ ωd

correspond to maximum penetration after impact (see Fig. 3.31).

Impedance Control

tC ≈

π +θ ωd

337

(3.113)

i.e. tC ≈ [(2k + 1)π + θ ] ωd (k = 1,2,…) for following minima along phase trajectory. In the example in (Fig. 3.36), the numerical solution of (3.112) is = , while the approximate solution according to (3.113) provides ≈ (s).

Fig. 3.36 Graphical solution of minimum penetration transcendental equation for constant velocity phase impact ( ξ t = 2, ωt = 8.7rad s , κ = 40 )

The contact stability conditions imply the following equivalent inequalities to be fulfilled

( )≥

( ) ≤ ( )

(3.114)

Substituting (3.107)-(3.108) and (3.113) in (3.114) yields π +θ

e tan θ ≤

π +θ − 2 cosθ κ sin θ

(3.115)

338

Dynamics and Robust Control of Robot-Environment Interaction

For the known stiffness ratio κ we can numerically solve the above transcendental equation in terms of the angle θ in order to obtain the parametric stability domain. The contact transition is stable for

θ ≥ θC min

(3.116)

where θ C min is a solution of the inequality (3.115). The minimum amount of target damping needed to guarantee a stable contact transition is then obtained from (3.106)

ξt ≥

1+ κ 1 + tan 2 θC min

(3.117)

The following observations are indicative for constant velocity contact transition stability analysis. •

In the considered ideal transition case the contact stability does not depend nor the target frequency ωt . on the initial impact velocity

•

The relevant parameters are only the target damping ξ t and stiffness ratios

κ

Fig. 3.37 Contact transition stability bounds in the parametric space Dt (ξ t , κ ) (constant velocity case)

339

Impedance Control

Fig. 3.37 presents the bound of contact stability in the parameter space (the stability domain Dt) for the considered constant velocity phase transition. Obviously, as the stiffness ratio increases, i.e. as the environment becomes stiffer and the target system more elastic, a larger amount of damping is needed to stabilize the contact transition. 3.8.1.2 Constant acceleration/deceleration phase contact We will omit the mathematical considerations for this transition case (for more details see [34]). It is worth mentioning that the obtained contact stability model is significantly more complex than the previous model for the constant velocity transition (3.107)-(3.117). In the considered case the contact stability depends not only on the impedance and environmental parameters, but also on the initial contact conditions (impact velocity and acceleration). In a case where nominal acceleration/deceleration is very small, the contact transition behavior is similar to the constant velocity case. However, dynamical nominal motion changes during contact cause transition behavior to be highly shaped by the impact velocity and acceleration ( and ), as well as by the nominal motion profile.

Fig. 3.38 Influence of impact acceleration sign on contact transition stability ( M t = 20 kg, ξt = 1.05 , K t = 1500 N/m, κ = 40 , v0 = 0.005 m/s, m/ =±

)

340

Dynamics and Robust Control of Robot-Environment Interaction

Thereby, as it was verified in [34], in the considered trapezoidal nominal > ), i.e. increasing nominal profile the acceleration in the contact direction ( velocity, improves the contact stability, while the nominal motion deceleration after impact destabilizes the contact. That is illustrated in the example in (Fig. 3.38), where a transition at the stability limit ( p0 = e , i.e. p = 0 ) with constant impact velocity (

= ), grows stable ( p0 > e ) when

> , i.e.

< . instable ( p0 < e ) for Solving the contact transition model in the considered acceleration/deceleration contact transition phase in terms of the system parameters is considerably more complex and involves significant numerical stability and solution separation problems in transcendental equations. Therefore, this model is reliable to use as an analysis tool, for testing the contact stability for a set of selected impedance and initial impact parameters, however not for synthesis purposes. This example shows that the time domain contact stability analysis and synthesis are, even for the simplified ideal second-order target impedance model, extremely complex and only efficient in specific cases, such as a transition with constant nominal velocity. Therefore, it is practical to apply concepts that simplify the contact transition stability analysis, providing a usable frame for the development of impedance control design procedures. As a rule, the price which is paid is conservativeness. 3.8.2 Passivity-based contact transition stability analysis Passivity theory provides a common technique for the stability analysis of both linear and non-linear control systems. As demonstrated in [35], passivity theory is closely related to fundamental stability analysis techniques, such as the smallgain theorem, Liapunov stability theory, L2-gains and H ∞ techniques. In the prior analysis passivity methods have been used to establish conditions for the coupled stability of a robot contacting an arbitrary passive environment −1 (Theorem 3.2). These conditions imply the realized target admittance s Gˆ t (s ) , describing the input/output relationship between the interaction force F and motion deviation eɺ to be a positive real matrix. Unfortunately, this result is obtained considering the coupled stability around the equilibrium point ( pɺ 0 (t ) = 0 ), and cannot be directly applied for the analysis of contact transition stability.

341

Impedance Control

Likewise, it is possible to apply the passivity theorem, i.e. the concept of positive dynamic systems, to obtain the conditions ensuring contact stability. To solve the contact stability using the passivity approach, first one must relate this problem to a feedback system and input/output relationships. Fig. 3.7 represents basic linearized feedback systems describing the interaction of a robot under impedance control and a passive environment. The input in these systems, the nominal penetration p0 (t ) , is a monotone-increasing positive function

p0 (t,D0 ) = x0 − xe ≥ 0,

t ≥ t0

The contact stability criteria indicate the output interaction force F(t), i.e. the penetration p (t ) to be non-negative during transition (3.91)-(3.92)

p(t, D0 ) = x − xe ≥ 0,

t ≥ t0

Theorem 3.6 (Passivity and contact stability): A sufficient condition to ensure contact stability of a linearized robotic control system under impedance control, during transition from the free space to a unilateral contact with any passive environment, is that the feedback system with the input-output pair satisfy the passivity condition

{p , p} 0

t

∫ p(τ )

T

p0 (τ )dτ ≥ 0

(3.118)

o

Proof : Substituting p(t ) = p0 (t ) − e(t ) in the above inequality yields t

t

t

o

0

0

2 T T ∫ p(τ ) p0 (τ )dτ = ∫ p0 (τ ) dτ − ∫ e(τ ) p0 (τ )dτ ≥ 0

(3.119)

This inequality implies that the “nominal penetration power” pushing the robot in the direction of the environment should be greater than the “position deviation power”, which draws the robot actual position away from the nominal one to the contact surface (see Fig. 3.34). Obviously, if during transition

e(t ) ≤ p0 (t ) for ≥

(3.120)

then the contact is stable and (3.118) is satisfied. This theorem, however, defines sufficient conditions in a stable contact, rather than necessary prerequisites ensuring stable contact transition. In practice this means that though the integral inequalities (3.118)-(3.119) are satisfied, the

342

Dynamics and Robust Control of Robot-Environment Interaction

contact could be lost for a short instant. Despite this limitation, the above theorem provides a quite practical result. Theorem 3.7 (Passivity-based contact stability criterion): A necessary and sufficient condition for the input-output pair {p0 ,p} to satisfy passivity (3.118)

[

]

is that the transfer function matrix I + Gˆ −1 (s )G (s ) t e

−1

be positive real.

Proof : For the considered linear interaction system and initial conditions (3.88), the actual penetration is t

p(t ) = ∫ K (τ , t ) p0 (τ )dτ

(3.121)

o

A necessary and sufficient condition for K (τ, t) = K (τ - t) to be a positive definite kernel is that its Laplace transform

[

]

∞

−1 −1 I + Gˆ t (s )Ge (s ) = ∫ K(t) e −st dt 0

be a positive real matrix of rational functions of the complex variable Definition 3.3 provides conditions for the transfer matrix

[

]

−1 −1 G ( s ) = I + Gˆ t (s )Ge (s )

= ω. (3.122)

to be positive real. For a SISO system with

G (s ) =

Gt(s) Ge(s) + Gt(s)

these conditions imply

  Gt ( jω) Re ≥0  Ge ( jω) + Gt ( jω ) Substituting: Ge ( jω ) = K e , and ωt =

(3.123)

Gt ( jω) = K t − M t ω2 + jωωt ,

Bt = 2ξt M t K t

K t M t in the above inequality provides

(K(

)(

)

− M t ωt 2 K )( e +e Kt −t M t ωt 2 +) Bt tω2

t t

((

2

))

= M t 2 ω 4 + M t 4ξ t 2 K t − 2 K t − K e ω 2 + K t ( K e + K t ) ≥ 0

343

Impedance Control

A necessary and sufficient condition ensuring that the obtained biquadratic (quartic) trinomial is non-negative for any real ω is that the discriminant satisfies 2

(

)

2

2

2

D = M t K e + 2 K t − 4ξ t K t − 4M t K t (K e + K t ) 4

(

= ωt κ + 2 − 4ξ t

) − 4ω

2 2

t

4

(1 + κ ) ≤ 0

which yields the contact stability criterion

ξt ≥

1 1+ κ −1 2

(

)

(3.124)

That is a simple, nevertheless very usable outcome for the synthesis of impedance control law, i.e. selection of target impedance parameters ensuring stable contact transition. As in the “exact-solution” (3.115)-(3.117) derived for the constant velocity phase transition, in the condition (3.124), obtained using passivity-based contact stability, the relevant parameters for the contact maintenance are only the target damping ξ and stiffness ratio κ . Moreover, the amount of damping needed to stabilize the contact transition is again increasing with rising environmental-stiffness/target-stiffness ratios. The obtained minimum damping values (Fig. 3.39) indeed are higher than in the “exact solution”, however, more than two times smaller than in the so-called “dominant real-pole” criterion proposed in [29]. This condition imposes a critically damped + ξ ω + ω ( + κ ) , i.e. total impedance

ξ ≥

+κ

(3.125)

The effectiveness of the obtained contact stability criteria can easily be proven by a simulation using the ideal interaction model (3.97) with variable initial conditions ( v0 , ) and environmental stiffness ( κ ). It can be remarked that in almost all transition tests with realistic impact velocities and accelerations

< (e.g. v0 < 0.1 m/s, ) even the “exact solution” (3.115)-(3.117) ensures a stable transition. However, in critical deceleration-phase transitions with a very small nominal penetration (i.e. goal position close to the environmental surface), a slightly larger amount of damping than provided by the passivity-based criterion (3.124) is still needed to prevent the lost of contact. Therefore this criterion can be used as “absolutely safe” for the ideal system. In the ideal case, assuming a diagonal nominal target model and a decoupled environment, the criterion (3.124) can be applied for the target impedance

344

Dynamics and Robust Control of Robot-Environment Interaction

synthesis in the real six-dimensional task space. In other words, the target impedance parameters can be tuned along each C-frame direction to meet (3.124). In a more general case, the impedance control synthesis should be based

[

]

on ensuring that the transfer matrix I + Gˆ −1 (s )G (s ) t e defined “worst case” environment.

−1

is positive real for a

Fig. 3.39 Contact transition stability bounds

3.8.3 Robust transition stability - generalized contact stability As already pointed out, the established geometric contact transition stability criteria (3.91)-(3.94) offer the possibility to apply different norms for contact signals: p (t ) , p0 (t ) , and e(t ) , in order to test their relationships, i.e. the performance of the transition process.

Theorem 3.8 (Robust contact transition stability criterion): A sufficient condition for a stable contact transition of a linearized robotic control system under impedance control from the free space to a unilateral contact with any passive environment, is that the 2-norm/2-norm system gain of the feedback

345

Impedance Control

{p0 ,e}, i.e. ∞ − norm of the corresponding −1 transfer function matrix [I + Ge −1 (s )Gt (s )] , is less than 1. system with the input-output pair

Proof : It is based on Theorem 3.1 (see Eq. 3.8) defining bounds for the 2-norm input/output gain. In the considered case the relationship between the nominal penetration and position deviation signals “energy” is limited by

e p0

[

2

≤ I + Ge

−1

(s )Gt (s )]

−1

2

(3.126)

∞

A stable contact transition is characterized by (3.94)

e(t ) < p 0 (t ) imposing ∞

∫ e (t )dt 2

e p0

2

= 2

0 ∞

[

≤ I + Ge

−1

(s )Gt (s )]

−1

2

∫ p (t )dt

0

ω

within a frequency interval of interest. The above relation represents a generalization of the maximum modulus theorem for matrix functions. In a SISO system, or in the case of an ideal diagonally decoupled interaction model, the principal gain is equal to the magnitude of the { } transfer function

σ {G ( jω)} = G ( jω) =

Ge ( jω) Ge ( jω) + Gt ( jω)

(3.129)

2 Substituting Ge ( jω ) = K e and Gt ( jω) = K t − M t ω + jωωt yields

G ( jω) = =

(K

Ke K e + Kt − M t ω2 + jωBt Ke

e + Kt − M t ω

)

2 2

2

+ ω Bt

2

(

)

 K e + K t − M t ω2 − jωBt   

Thus the robust contact stability criterion becomes

G ( jω ) =

Ke

(K

e

+ Kt − M t ω

Substituting Bt = 2ξt M t K t and ωt =

ω − ω

( +κ −

)

2 2

2

+ ω Bt

2

≤1

(3.130)

K t M t yields

)

(

ξ ω +ω

κ+

)≥

(3.131)

The obtained biquadratic (quartic) trinomial will be non-negative for all real frequencies of interest ω ∈ ( ∞] if the discriminant is non-positive

= ω which implies

( +κ −

ξ

)

− ω

(

κ+

)≤

Impedance Control

+κ − ξ

≤

347

+ κ

from which the robust contact stability condition

ξt ≥

1 2

( 1 + 2κ − 1)

(3.132)

follows. As in the previously derived contact stability conditions (time-domain “exact solution”, passivity-based positive real requirement (3.124) and dominant real-pole limit (3.125)), in the robust-stability infinity-norm criterion (3.132) the contact stability bound is only determined by the target damping and stiffness ratios. The minimum target damping values required by the bound (3.132) for various target stiffness ratios κ are compared in (Fig. 3.40) with the stability limits prescribed by prior contact stability criteria.

3.8.4 Equivalence of robust- and passivity-based contact stability As may be remarked from (Fig. 3.40), the infinity-norm boundary line is placed between the positive-real and dominant real-pole bounds. A further important observation concerns similar forms of the expressions for the contact stability conditions (3.124) and (3.132) obtained in the passivity-based and robuststability methods respectively. Practically, the positive-real stability limit provides the same stability bound as the robust-stability condition when the stiffness ratio, i.e. the stiffness of the environment, is taken to be twice the real stiffness.

Fig. 3.40 Comparison of contact transition stability bounds

348

Dynamics and Robust Control of Robot-Environment Interaction

This relationship is the result of the equivalence between positive-realness, the small-gain theorem and the ∞ norm. The principal basis for this correlation is the transformation referred to as the bilinear sector transform [37], or in a more general form as Cayley transformation [38]. This transformation can convert a positive real problem concerning a system within the sector [ ∞ ) into small-gain i.e. vice versa.

∞

problem relating systems inside the sector [−

], or

Definition 3.7 (Sector transformation) Let T (s) and G (s) denote the systems lying within sectors [− ] and [ ∞ ) respectively. Then the bilinear sector transformation is defined by −1

T(s) = sec tf (G(s)): = (I − G(s))(I + G(s))

(3.133)

We define further

herm ( A(s)) =

1 A(s) + A∗ (s) ; 2

(

)

Im ( A(s)) =

1 A(s) − A∗ (s) ; 2

(

)

where the asterisk means conjugate. When A( s ) ∈ [− 1,1], then A( s ) when A(s ) ∈ sector

[

(3.134) ∞

≤ 1 , and

∞ ) , then herm ( A ( s )) ≥ 0 for s = jω, ∀ω. Thus,

using the above definitions and notation, we have the following theorem Theorem 3.10 (Passivity and H ∞ -norm) For the sytems related by (3.134)

G(s ) ∞ ≤ 1 if and only if herm (T ( s )) ≥ 0 . Proof : See [37]. Hence, the following theorem expresses the equivalence between passivitybased contact stability and the infinity norm condition used in the robust contact stability. Theorem 3.11 (Passivity-based and robust contact stability) The input-output pair {p0 ,p} will satisfy passivity, i.e. the corresponding transfer function matrix will satisfy −1 herm   I + Gˆ t −1 ( s )Ge ( s )  ≥ 0  

349

Impedance Control

which ensures the stable contact transition, if and only if the equivalent system {p0 ,e } describing the interaction between the same target impedance and a “halved” environment satisfies the robust contact stability −1

 I + 2Ge −1 ( s ) Gˆ t ( s ) ≤ 1   Proof : It is relatively easy to confirm that −1

 I + Gˆ t −1 ( s ) Ge ( s ) = sec tf  

  I + 2G −1 s Gˆ s  −1  e ( ) t ( )  

In the SISO contact stable transition case the ensures

{p0 , e}

(3.135)

input/output system

Ge ( jω) ≤1 Ge ( jω) + Gt ( jω)

(3.136)

for the target damping values

ξt ≥

1 1 + 2κ − 1 2

(

) ∈ [ ∞ ) satisfies

while the associated sector-transformed system

  Gt ( jω )  herm (T ( jω)) = herm   Gt ( jω) + 2Ge ( jω) 

(3.137)

The passivity (i.e. positivity) based contact stability Theorems 3.6 and 3.7 imply the transfer function relating {p0 , p} input/output system to fulfil

  

( ω) ( ω) +

(

  ω) 

⇔

( ω) ( ω)

+

( ω)

≤

(3.138)

In the parametric space this yields the condition (3.103)

ξt ≥

1 1+ κ −1 2

(

)

Basically Theorem 3.11 defines the equivalence between robust and passivity-based contact transition stability. The interaction system considered in the robust contact stability relates a sector-transformed passive system that has the same form as the coupled system considered in the passivity-based contact stability theory. However, this sector-transformed system involves the double

350

Dynamics and Robust Control of Robot-Environment Interaction

size of the environmental parameters (3.138). That means that the robust contact stability and the passivity based stability are essentially correspondent and have the same system-theoretical background. However, the robust contact stability permits 100% of environmental uncertainties and therefore is more conservative than the passivity-based stability (see Fig. 3.40). Finally, it should be pointed out that derived robust and passivity-based contact stability conditions are based on the sufficient stability criteria, but not the necessary ones. That means that both stability conditions are satisfied in a stable contact transition; however, the accomplishment of these conditions does not guarantee a stable contact. According to (3.91)-(3.94) the contact stability is defined in terms of the relationships between instantaneous time signals characterizing the contact transition (p0, p and e). However, the established stability criteria in Theorems 3.6-9 are based on the relationships between the 2norms of these signals. The squares of these norms define the signals energy obtained by integrating the instantaneous signal power over time (see 3.95). Consequently, though the contact stability norm relationships have been fulfilled over the specified transition time, the contact could be lost in instantaneous interval when the stability relationship between direct signals is momentarily violated as demonstrated in (Fig. 3.33). Therefore, a practical way to test the applicability and reliability of sufficient contact stability criteria is to test them experimentally.

Experiment 3.4:

Preliminary tests of contact stability bounds

For the testing of the established contact transition stability conditions, initial impedance control trials with the improved impedance law, presented in the Experiment 3.2 (Fig. 3.25-3.26), have been used. In each experiment the transition has been characterized by analyzing the force component in the contact direction. Dependent on the minimum force peak, which otherwise occurs in the first phase immediately after impact (Fig. 3.25), the transition process has been evaluated as: instable (contact is lost, e.g. Fig. 3.25-a), at stability limit (e.g. Fig. 3.25-d), or stable (e.g. Fig. 3.25-b, c). Each experiment is presented as a point in (κ , ξt ) diagram specifying the contact transition of a SISO system. The results are presented in (Fig. 3.40), which also depicts theoretical stability bounds for different criteria. Thereby, the following symbols have been used to characterize the transition experiments in (Fig. 3.40): x

-unstable transition;

Impedance Control

o +

351

-at (closely to) stability limit; -stable transition;

Considering the fitting of the experimental results by theoretical stability criteria bound lines, it may be concluded that the most reliable contact stability limit provides an infinity-norm based criterion, i.e. robust contact stability (3.132).

3.9 Influence of Non-Linear Effects on Contact-Stability Indeed the result of the last experiment is surprising, since according to the above discussion robust contact stability limit is expected to be more conservative than passivity-based and exact solutions contact stability bounds. Obviously the experiment differs from the simulation tests based on the ideal target impedance model (Fig. 3.30-3.33), which have proved the theoretical transition stability results. Hence, it may be concluded that real systems involve additional effects that were not considered in the previous analysis. Both theoretical analysis and simple simulation tests of non-linear effects in the coupled impedance model indicate the relevance of Coulomb’s friction, quantization and roundoff, as well as control time lag effects.

3.9.1 Coulomb’s friction The influence of the Coulomb friction on the contact transition process can be analyzed by introducing the equivalent Cartesian friction in the considered impedance control models. For example, substituting the impedance control law (3.64) into the error-model (3.13) and introducing the friction disturbance force yields

e = Gt

−1

(s )F + S p (s )x0 + S p (s )Gs (s )(F + Fc )

(3.139)

where Fc is Cartesian Coulomb friction force. From the above equation we can see that the Coulomb friction has the equivalent effect on the transition process as the perturbation contact force. In industrial robotic systems with very stiff position controller, i.e. very small sensitivity transfer function matrix Sp(s) within the frequency interval of interest, the perturbation friction effect is significantly diminished. Moreover, considering the motion resistant effect of the Coulomb friction, it may be stated that Fc supports the end-effector to remain

352

Dynamics and Robust Control of Robot-Environment Interaction

in contact, rather than to destabilize the contact establishment. Thus, the stability results obtained with the ideal model are on the safe side. The perturbation effects of non-linear Coulomb friction are crucial in the steady-state behavior causing limit cycle oscillations, which can be observed in the experiments (Fig. 3.25). The quantization and round-off phenomena have similar effects on the feedback system performance. These effects are caused by signal conversion in analog-to-digital and digital-to-analog converters and floating-point computations. Generally in modern control system these effects give raise to relative small errors [39]. However, in contact with a stiff environment, these effects may become significant and could jeopardize steady-state behavior. Therefore these effects will be considered separately below.

3.9.2 Control lags and sampling effects A control delay in real sampling-data control system exhibits quite the opposite effect on the transition process. In the considered interaction process and impedance control system the presence of a delay time mainly is caused by the retardation between input control signals and control reactions. Generally real digital control systems operate in a continuous framework, but some signals are sampled in certain time intervals in order to be processed by digital computers operating at fixed rates. The most critical delay from the view of the stability of contact transition process concerns the tardiness of the contact force information. An extensive force sensor processing computation is needed to obtain instrumental virtual contact force information in a selected compliance, i.e. task-frame, to be used in the impedance control. This processing includes: filtering, compensation for offset, coupling and payload, as well as transformation in different frames. The other major cause of the retarded contact force information is the structure of the selected implicit position control system (Fig. 3.4). Basically, in this scheme the sampling period of the outer impedance control is limited by the sampling rate of the internal position controller. The sampled-data (SD) model of the considered impedance controlled interaction system in (Fig. 3.41) involves the plants (the robot mechanical part and the environment, i.e. contact process) operating in continuous time and the controllers (position and impedance) operating in discrete time. The control model consists of samplers (A/D converters), discrete-time controllers implemented in a digital computer and holding elements, which support D/A

Impedance Control

353

conversion of control signals by holding them constant over the sampling intervals. The unilateral delay is presented with a nonlinear unilateral-limiter block. The symbol ⊗ denotes the sampled representation of the signal at the sampling instant. The blocks in this figure represent various control tasks and sampled-data computation effects, such as the reading and processing of force information, computation of position and impedance control laws, process and control monitoring, communication with the operator, etc. These functions are usually realized in several control tasks running at various sampling time intervals, including different control delays. Such complex multirate sampling systems, however, are time-varying even for a time-invariant controlled plant, and quite difficult to analyze. Therefore, usually various simplifications are needed to consider dominant sampling-time effects. Generally, the presence of a controldelay in feedback system may cause the self-exciting oscillations and even the instability of the system [40, 33, 41]. As already mentioned, in high dynamic processes, such as the contact transition process, the control delay appears to be especially critical.

Fig. 3.41 Sampled-time impedance control system

There are essentially three approaches to design a digital controller [42]: i) analog design and digital implementation,

354

Dynamics and Robust Control of Robot-Environment Interaction

ii) iii)

plant discretization and design in discrete time, direct sampling-data control design.

Each of these approaches has specific advantages and limitations. The advantage of the first (most common) approach is that the design is performed in continuous time, where the performance specification is most obvious. However, this method provides good performance only for very small sampling periods. The problems with the second approach is related to discrete-time performance specifications, as well as control redesign when the sampling time has to be changed. The third approach considers accurately the performance of SD systems; however, the design is more complicated because SD systems are time varying. In the following sections the effects of a force delay and sampling-data in the impedance control contact transition will be analyzed. Thereby, the simple and reliable contact transition analysis/synthesis approach based on the passivity and robust control design will be pursued and extended to delayed and SD systems. 3.9.2.1 Ideal target system with force delay The simplest model of a time-delayed control system assumes that all control signals are continuous (i.e. very fast sampled signals, with a rate T < 1 ms) except for the delayed force information. The approximation of the force aftereffect in a pure stiff environment is based on the interaction force model that depends on penetration at some preceding time instance. Introduce the delayed contact force

F (t − τ ) = K e p (t − τ ) = K e [ p0 (t − τ ) − e(t − τ )]

(3.139a)

where τ is the time delay. Assuming p0 (t ) to be a monotone, relatively slowly increasing function during transition, we can write p0 (t − τ ) ≈ p0 (t ) without loss of generality and

F (t − τ ) ≈ K e [ p0 (t ) − e(t − τ )] Moreover, it is relatively easy to show that this assumption does not affect the passivity and infinity norm of the corresponding interaction transfer function, thus providing a suitable model for the comparison of the time-domain stability conditions with those obtained by passivity-based and robust control approaches. Substituting the above expression into the ideal target model system (3.17)

Impedance Control

355

yields the linear difference-differential equation (i.e. differential equation with retarded, or deviating argument)

ɺeɺ(t ) + 2ξ t ωt eɺ(t ) + ωt 2e(t ) + ωe 2e(t − τ ) = ωe 2 p0 (t )

(3.140)

with constant coefficients and constant deviating argument. Consider first the coupled stability problem around the equilibrium point. Similarly to the linear non-retarded systems, the coupled stability analysis is based on an investigation of the location of the roots of the corresponding characteristic quasipolynomial [33, 41]

φ ( z ) = z 2 + 2ξtωt z + ωt 2 + ωe 2e −τz = 0

(3.141)

A necessary and sufficient condition for the asymptotic stability of the solution of the linear stationary equation with deviating argument is negativity of the real parts of all the roots of the characteristic quasipolynomial (3.141). However, φ ( z ) has an infinite set of roots, the computation of which is relatively complex. Therefore, to check the stability several practical tests of negativity of the real part of all roots of the quasipolynomial have been developed (see [41]. The method of D-partitions determines the regions of asymptotic stability in the parametric space by separating the quasipolynomial coefficients values into intervals providing at least one zero on the imaginary = does not provide a reliable parametric solution axis. Since the case 2 2 ( ωt + ωe ≠ 0 ), we should look for pure imaginary roots (at the stability limit) by substituting = ( ∈ ℜ > ) into (3.141), which yields 2

2

− y 2 + 2ξtωt iy + ωt + ωe (cos τy − i sin τy ) = 0 or the system of transcendental equations

a = 2ξtωt = 2

2

sin τy 2 ωe y

(3.142)

2

b = ωt = y − ωe cos τy The above equations define D-partitions boundaries in the parametric plane 2 2ξtωt , ωt . The typical form of this D-partition is sketched in (Fig. 3.42). A

(

)

reliable part of this boundary corresponding to the target frequency parameter ωt ≤ 100 (rad/s) is sketched by the bold line. It is relatively easy to prove that the first transcendental equation

356

Dynamics and Robust Control of Robot-Environment Interaction 2

2ξtωt y = ωe sin τy does not provide real solutions for y representing the intersection between the 2

straight line 2ξtωt y crossing the origin and the sine functions ωe sin τy for

ξt >

τωe 2 2ωt

Fig. 3.42 D-partition boundaries ( τ = 0.008 sec,

(3.143)

=

N/m,

= 2

kg)

As y → 0 the D-boundary line approaches the point (τωe ,0) . In the example in (Fig. 3.42) the above inequality defines the region of the parametric 2 plane on the right side from the line 2ξtωt = τωe = 48 As can been seen from (Fig. 3.42), within this region the quasipolynomial φ ( z ) has no pure imaginary = + roots, i.e. above this region only conjugate complex roots are possible. In order to check if this region is the region of asymptotic stability

357

Impedance Control

( < ) we should check the sign of the real part of the root x in the neighborhood of the D-boundary. Based on (3.141)-(3.142) it can be written

∂φ 2 2 dz = 2 z + a − ω e τe −τz dz = 2 x + 2iy + a − ω e τ (cosτy − i sin τy ) ( dx + iy ) ∂z ∂φ ∂φ ∂φ dz = − da − db = − zda − db = − ( x + iy ) da − db ∂z ∂a ∂b

(

)

[

]

from which the differential of the real part of the root is

  (x + iy ) da + db dx = − Re   2  2 x + 2iy + a − ω e τ (cos τy − i sin τy )  Usually it is sufficient to test the differential upon considering the variation of only one parameter whose change guarantees passage across the examined D-partition. Assuming across the D-boundary: x = 0 and = , yields

=−

+ω τ

(

−ω τ

τ

) +(

τ +ω τ

τ

)

> +ω τ τ > , it may be and also Considering that for is opposite the sign of . Hence, the directions concluded that the sign of = ξ ω relative to the D-partition of increasing value of the parameter boundary determine the regions of asymptotic stability in the parametric plane (Fig. 3.42). Based on the above analysis it can also be concluded that the condition (3.143) (see Fig. 3.42) is necessary and sufficient to ensure the coupled stability of the system (3.140) describing the interaction between an ideal targetimpedance and a pure stiffness environment including delayed force feedback information. Unlike the interaction without retardation (3.103) in which ξ > was sufficient to ensure the coupled stability, the control lags require a much more complex stability condition (3.143) to be guaranteed. In this case the coupled stability depends on the time delay, environmental stiffness and target impedance frequency parameters. As with the time-domain contact transition stability analysis of an ideal coupled system without retardation (3.97), the goal is to establish the parametric

358

Dynamics and Robust Control of Robot-Environment Interaction

{

}

domain of stability Dt = M t , ξ t , ωt , κ ,⋯ of the considered system (3.140) which ensures the geometric contact stability criteria (3.91)-(3.95). The stability analysis of differential equations with delayed argument generally is very complex (see [33, 40, 41]). However, in some specific second-order delayed systems [43] it is possible to establish the equivalence with the systems without time lags. That makes it possible to utilize previous contact stability results. Consider the initial value problem (IVP)

ɺeɺ(t ) + 2ξ t ωt eɺ(t ) + ωt 2e(t ) + ωe 2e(t − τ ) = ωe 2 p0 (t )

( )

( )

e 0 − = 0; eɺ 0 − = 0

(3.144)

In the general case for arbitrary initial functions e(t − τ ) ≡ φ (t − τ ) when t − τ < 0 , the solution space of the above type of equations is, according to the conditions for the existence and uniqueness of solutions [43], infinitedimensional. However, for a fixed initial function (3.144) the space of solutions is two-dimensional, which is also the case for equations without delay. Then the following theorems define the equivalence between differential equations of the second order with and without delayed argument [43]. Theorem 3.12 (Common fundamental systems of solutions of homogenous system) Let the functions u1 (t ) and u2 (t ) belong to the two-dimensional space

L2φ of linearly independent solutions of the homogenous equation with delayed argument (fundamental system of solutions) 2 2 eɺɺ(t ) + 2ξ t ωt eɺ(t ) + ωt e(t ) + ωe e(t − τ ) = 0

( )

( )

e 0 − = 0; eɺ 0 − = 0

(3.145)

Then there exists a unique differential equation without delay of the form

eɺɺ(t ) + peɺ(t ) + qe(t ) = 0 for which u1 (t ) and u2 (t ) also form a fundamental system of solution. Theorem 3.13 (Common fundamental systems of solutions of non-homogenous 2 system) Let u1 (t ), u 2 (t ) ∈ Lφ be a fundamental system of solutions of the IVP

(3.145), and let v(t ) be a particular solution of (3.144). Then there exists a function

359

Impedance Control

ψ (t ) = vɺɺ(t ) + pvɺ(t ) + qv(t ) such that the solution of non-homogenous equation without delay

ɺeɺ(t ) + peɺ(t ) + qe(t ) = ψ (t )

( )

(3.146)

( )

e 0 − = 0; eɺ 0 − = 0

coincides with the general solution of the equivalent delayed system (3.144).

Proof : It is based on basic properties of second-order differential equations with delayed arguments (see [43]) and well-known theorems of fundamental systems for linear differential equations without delay. The equivalence between second-order systems with and without delay based on the above theorems provides useful tools to enlarge the result of the contact stability analysis to the case involving time lags. Using well-known formulas for representing the coefficient of a linear homogenous differential equation without z1,2 t

x ±iy

= e , see for delay in term of the fundamental system of solutions ( u1, 2 = e example [44]), we can obtain the relationship between the coefficients of the equivalent systems p = −(z1 + z 2 ) = 2ξ t ωt + ωe

2

D(u1,u 2 ,τ ) W (u1,u2 )

(3.147)

W (u1,u 2 ,τ ) q = z1 z2 = ωt + ωe W (u1,u 2 ) 2

2

where the Wronskian W and the determinant D have the forms

D(u1,u2 ,τ ) = W (u1,u 2 ) =

u1 (t ) u 2 (t ) u1 (t − τ ) u2 (t − τ )

u1 (t ) u2 (t ) u (t − τ ) u2 (t − τ ) ; W (u1,u2 ,τ ) = 1 ɺu1 (t ) uɺ2 (t ) uɺ1 (t ) uɺ2 (t )

Without loss of generality we can assume coefficients of the form 2

2

2

p = 2ξtωt ; q = ωt + ωe = ωt (1 + κ )

(3.148)

where ξ t and κ represent the damping and stiffness ratios of the equivalent system without delay. Substituting in the above equations:

u1,2 = e

z 1,2 t

 − ξ ω ± iω 1+ κ − ξ 2  t t t t t 

= e( x ± iy )t = e

= e −ξ t ω t t ± iω d t

(3.149)

360

Dynamics and Robust Control of Robot-Environment Interaction

yielding

= ξω − =ω

ω ω

ξωτ

+κ

+ω

(ω τ )

(ω τ + θ )

ξωτ

(3.150)

+κ−ξ where

cosθ = −

ξt

;sin θ =

1+ κ

1 + κ − ξt

2

1+ κ

;ωd = ωt 1 + κ − ξt

2

From (3.148)-(3.150) it follows for the relationship between the target coefficients of the equivalent systems

ξt = ξ t − κ=

κ 2 1 + κ − ξt

κ 1+ κ 1 + κ − ξt

2

2

eξ t ωtτ sin (ωdτ )

eξ t ωtτ sin (ωdτ + θ )

or explicitly

ξt = ξt + κ =κ

sin (ω d τ ) 2 1 + κ sin (ω d τ + θ )

κ

sin θ

(3.151)

e −ξ tωtτ

sin (ω dτ + θ )

Assuming that the nonhomogenous equivalent systems with and without delay (3.145) and (3.146) respectively have the same particular solution, it is relatively easy to derive the relation between the nonhomogenous parts applying the method of variation of the constants

u 1 (ζ ) 2

ψ (t ) = ω e p 0 (t ) + ω e

2

t

∫τ

t−

t −τ ≤ ζ ≤ t

u 2 (ζ )

u 1 (t − τ ) u 2 (t − τ ) W (u 1 (ζ ), u 2 (ζ ))

ψ (ζ ) dζ

(3.152)

361

Impedance Control

Substituting (3.149) in the above equation provides an integral equation of Volterra type of the second kind

ψ (t ) = ω e 2 p0 (t ) +

ωe 2 ωd

t

∫τe

−ξ t ωt (t −τ −ζ )

sin ω d (t − τ − ζ )ψ (ζ ) dζ

(3.153)

t−

For the continuous and bounded kernel (3.143) this equation has a unique and continuous solution that may be obtained by the method of iteration (successive approximations). Furthermore, it can be written

ψ (t ) = ωe 2 p0 (t ) where the equivalent nominal motion function p0 (t ) has the form

p0 (t ) = p0 (t ) +

1

ωd

t

∫τe

−ξtωt (t −τ −ζ

)

sin ωd (t − τ − ζ ) p0 (ζ )dζ

(3.154)

t−

The equivalence with the system without retardation is now completely defined. The established analogy allows us to obtain solutions of the considered IVP’s with retardation (3.144) and (3.146) using the previous results on the contact transition in the system without time lags (3.97) and (3.104)-(3.106) respectively. The solution of the equivalent IVP without retardation has the known form

ωe 2 t −ξ ω (t −ζ ) e(t ) = e sin ωd (t − ζ ) p0 (ζ )dζ ωd ∫0 t t

or

e(t ) = +

v κ  2ξt pɺ 0 (t ) + 0 e −ξ ω (t −t ) sin [ωd (t − t0 ) − 2θ ]  p0 (t ) − ωd (1 + κ )ωt 1+ κ  t t

0

 ɺpɺ0 (ζ )e −ξtωt (t −ζ ) sin [ωd (t − ζ ) − 2θ ] dζ  ωd t∫0  1

t

(3.155) Together with the geometric contact transition stability condition

e(t ) ≤1 p0 (t )

(3.156)

362

Dynamics and Robust Control of Robot-Environment Interaction

the derived model (3.150)-(3.155) defines a mathematical frame that can be applied for the impedance control synthesis in a system with retardation, i.e. for

{

}

computing the domain Dt = M t , ξ t , ωt , κ ,⋯ of target impedance parameters that ensures a stable contact transition. However, the control design algorithms based on iterative numerical solving of the systems of integral and transcendental equations appear to be extremely complex and time consuming. A practical solution can be obtained by neglecting the difference between nominal motion functions p0 (t ) and p0 (t ) , corresponding to the systems with and without retardation respectively. Usually this difference is for a considerable delay very small. In general, the difference between non-homogenous functions of the equivalent systems with and without retardation (3.144) and (3.146) is bounded by [43]

sup ψ (t ) − f (t ) ≤ f 0 K 0 τe K0τ [0,t )

where

(3.157)

f (t ) denotes the non-homogenous part of the retarded system 2

( f (t ) = ωe p0 (t ) ), f 0 = sup f (t ) , K (t , ζ ) is the kernel of the integral equation (3.152) and K 0 = sup K (t,ζ ) , Substituting (3.152)-(3.153) in (3.157) yields

K0 = ω e2

sin ω d τ

ωd

eξtωtτ

and

sup p0 (t ) − p0 (t ) ≤ [0,t )

1 ∗ p0 K 0 τe K0 τ 2 ωe

In a “worst-case” transition with: κ = 100 , τ = 0.01 (s), M t = 10 (kg) and ∗ K t = 1500 (N/m) the above bound is about . This means that the nominal motion function can be applied in the system without time lag without loss of accuracy. Then, the algorithm runs as follows. Algorithm 3.1 Computation of the contact stability domain Dt (ξt , κ ) for a SISO delayed system Step 1: For known κ , ωt and τ , and assuming the most common constant velocity phase impact, one computes the equivalent stiffness ratio κ and damping ratio ξ t at the stability limit for the equivalent system without delay.

Impedance Control

363

These parameters are obtained by solving the system of transcendental equations describing the equivalence between delayed and non-delayed interaction systems (3.151) and contact stability bound for the constant velocity transition (3.115), respectively

κ

sin θ

e − ξ t ω tτ − κ = 0

sin (ω τ + θ ) d

e(π +θ ) tan θ + 2 cosθ −

π +θ =0 κ sin θ

Step 2: The damping ratio ξ t of the original system with delay, which ensures the same stable contact transition (i.e the same basis solution of IVP) as in the equivalent system without delay is then computed from (3.151)

ξt = ξt +

κ

sin (ω dτ )

2 1 + κ sin (ω dτ + θ )

The contact transition stability bounds obtained by the above algorithm for various values of the computation time lags τ are presented in (Fig. 3.43). The stability limit defined by the robust contact stability, which in the previous experiment was identified as the most reliable one (see Fig. 3.40), has also been sketched. Obviously, compared to the ideal case τ = 0 , also shown in the figure, the control delay requires significantly higher damping ratios in order to ensure a stable transition. These curve diagrams also give the explanation why the most reliable stability results in the experiments have been provided by the robust stability criterion that is more conservative than the other criteria. Namely, in the performed experiments the time delay in the implemented control system was approximately τ ≈ 0.008 (s). For the stiffness ratios κ applied in the Experiment 3.4 ( 40 ≤ κ ≤ 100 , see Fig. 3.40), the robust stability bound, computed for the system without delay, quite well matches the “exact” stability limit obtained for the ideal time-domain model with retarded force signal and corresponding to the time lag τ = 0.008 (s).

364

Dynamics and Robust Control of Robot-Environment Interaction

Fig. 3.43 Contact stability bounds of the retarded systems: time-domain analysis (“exact solution”) ( M t = 10 kg, K t = 1500 N/m, ωt = 12.25 rad/ )

Fig. 3.44 Dependence of minimum damping on target frequency – time domain delayed system analysis ( M t = 152 − 0.6 kg, K t = 1500 N/m, ωt = 3.14 − 50.26 rad/ )

365

Impedance Control

Practically this experiment also proves the stability analysis of the retarded system based on the linear impedance model (3.144) and its corresponding system without control delay (3.146). It is important to remark that unlike the contact stability of the ideal coupled system, the stability bounds of the delayed system depends on the target frequency ω t . An increasing target frequency for a constant time lag and stiffness ratio requires even higher values of the damping ratio to guarantee stable transition. This relationship using the considered transition experiment example is illustrated on (Fig. 3.44). Numerical determination of the stability limits based on the Algorithm 3.1, however, becomes very sensitive to the selection of the initial values for higher target frequencies. 3.9.2.2 Robust and passivity-based contact stability of discrete-time system In spite of the relative simple analytical approach and resulting reliable contact stability bounds, target impedance system synthesis based on the Algorithm 3.1 has two main limitations. The first one concerns the numerical problems in solving the system of transcendental equations describing the relationship between equivalent systems with and without delay. Second, it is difficult to generalize this closed-form model approach to MIMO or sampled-data impedance digital control systems (Fig. 3.41), as well as to include modeling uncertainties. As still demonstrated above, the H ∞ induced norm, describing the maximum energy gain measure, is quite useful in analyzing the performance and synthesis of stable interacting impedance control systems. In linear systems the result of H ∞ -norm based synthesis can be directly applied in both continuous and discrete time control. A common proximal method of converting a continuous (analog) system to a digital system with the same properties is based on the bilinear transform, a special case of which is the Tustin transform

s=

2 z −1 T z +1

(3.158)

where T denotes the sampling period and = is the delay operator for fixed sampling period. The bilinear transform has the advantage of mapping the left half s-plane into the unit disk in the z-plane, ensuring a stable continuous system to remain stable after discretization. For the contact stability analysis, a key property of the Tustin transform is relevant: it preserves the H ∞ norm.

366

Dynamics and Robust Control of Robot-Environment Interaction

Considering the correspondence between the H ∞ norm and the passivity, Tustin’s method is also a passivity preserving discretization technique. Hence, the results of stable interacting system synthesis in the continuous-domain can be applied in the discrete-time and vice versa. If the Tustin transform is applied to a stable system considered in passivity-based or infinity-norm based contact transition stability theory, (3.122) and (3.128) respectively, then the resulting discrete system also is contact stable. Translating a linear continuous system, represented by the transfer function G (s), into the corresponding discrete-time transfer function H (z) using the Tustin transform is done by substituting (3.158)

G (s ) → H ( z ) = G (2 T z − 1 z + 1) For a stable discrete time interaction system matrix considered in the robust contact stability Theorem 3.8 (3.128), the induced infinity norm for testing the contact stability conditions is defined in terms of the frequency-dependent singular values −1 jωT   2 e jωT − 1   − 1   2 e jωT − 1   −1  2 e   ≤ 1 (3.159)   I + Ge   Gt  sup σ W  jωT jωT jωT ω  T e + 1   T e + 1     T e + 1  

or for the SISO case, assuming unity weighting matrix and a pure stiffness environment,

Ge ( z ) Gt ( z ) + Ge ( z )

2

∞

ωe T 2(z + 1 )2 = 2 2 4( z − 1) + 4ξ t ωtT z 2 − 1 + (1 + κ ) ωt T

(

)

≤1

(3.160)

∞

ω

= Substituting in (3.160), for which according to the maximum modulus principle the supreme norm (magnitude) is obtained, it is relatively easy to derive the contact stability condition in the parametric space. As expected, the obtained stability condition is the same as in the continuous system (3.132)

ξt ≥

1 2

( 1 + 2κ − 1)

Likewise, the passivity based contact stability analysis (see the Theorem 3.7) for the discretized system generated using the passivity preserving Tustin transform will provide the identical condition as in the continuous case (3.144). The robust and passivity-based contact transition stability theorems allow the force control delay to be considered in a relatively simple manner. Let us

367

Impedance Control

consider for this purpose the interaction model in (Fig. 3.45) with transport force delay modeled by the transfer function e-τs , where τ is the time delay. The infinity norm for testing the robust contact stability according to Theorem 3.8 for the interaction system with force retardation becomes

[G (s )e e

-τs

]

−1

+ Gt (s ) Ge (s ) e -τs

∞

[

]

−1

= Ge (s )e -τs + Gt (s ) Ge (s )

≤1

(3.161)

∞

Furthermore, adding the model uncertainties in the form (3.42) yields

[

]

−1

W (s ) Ge (s )e -τs + Gt (s ) Ge (s )

≤1

(3.162)

∞

where W (s ) is a stable transfer matrix describing the perturbations effects in the realized target model (i.e. robot) or in the environment. Based on the relationships between passivity-based and robust contact stability (3.138), the infinity norm for testing the passivity-based contact stability for delayed system has the form

[G (s )e e

-τs

]

−1

2 + Gt (s ) Ge (s )

≤2

(3.163)

∞

For a SISO interaction system without delay the passivity-based and robust contact stability provide the stability criteria in the simple explicit parametric forms (3.124) and (3.132), respectively. In the case with time delay, however, the determination of the target impedance parameters ensuring a stable transition based on criteria (3.161) and (3.163) becomes much more complex. For example, the robust stability implies

σ {G ( jω)} = G ( jω) = Substituting Ge ( jω ) = K e transformations

(

Ge ( jω) ≤1 Ge ( jω) e -jωτ + Gt ( jω)

and Gt ( jω ) = K t − M t ω 2 + jωωt

(3.164) yields after

)

ω4 − 2ωt 2 1 + κ − 2ξt 2 ω2 + ωt 4 (1 + 2κ ) ≥ −4ωt 2 κ sin 2 (

ωτ  2 ωτ  )  ω + 2ξt ωt ctg ( )ω + ωt 2  2  2 

(3.165)

The biquadratic parabola on the left side is identical to the non-delayed case (3.131). In the delayed system, however, the transcendental quadratic function on the right side of (3.165) complicates computation of ξ which satisfies the

368

Dynamics and Robust Control of Robot-Environment Interaction

robust stability inequality. The target system synthesis is based on the numerical solution of transcendental inequalities. An alternative general design approach is to select some of the target parameters based on the given task and environment knowledge. The remaining parameters (usually target damping ratio) should be varied (increased) and the corresponding infinity norm (i.e. principal gains of matrix transfer functions) should be computed using well-known numerical methods and functions (e.g. MATLAB’s sigma function) until contact stability inequalities are not fulfilled. Taking into account the preservation of the infinity norm by the bilinear transformation, the equivalent robust and passivity-based contact stability conditions in the time-discrete domain respectively have the form −1

 2 z − 1    2 z − 1  -nT  2 z −1   2 z − 1  W  Ge   Ge   z + Gt    T z + 1   T z +1   T z +1  T z +1

≤1 ∞

(3.166) −1

  2 z − 1  -nT  2 z − 1   2 z −1  Ge  T z + 1  z 2 + Gt  T z + 1  Ge  T z + 1        

≤2 ∞

where the delay time is assumed to be a multiple of the sample time τ = nT (commonly n = 1). The discrete time model is more suitable to describe transport delay effects, including also different sampling and delay times.

Fig. 3.45 Interaction system with force delay

Impedance Control

369

For a practical SISO interaction case, the obtained stability limits in the parametric domain are presented in (Fig. 3.46). It can be remarked that as with the results of the prior time domain analysis, the increasing delay time requires a larger amount of damping to stabilize the interaction. As expected, the robust stability criterion requires higher damping ratios at the stability limit (robust stability assumes a twice as stiff environment than in reality, i.e. in the passivitybased one). As pointed out, an important difference of stability analysis of a time delayed system compared to the ideal case without delay is that the minimum target-damping ratio stabilizing the contact does not depend only on the stiffness ratio and time delay, rather also on the target frequency ω t . The effects of the target frequency on the parametric contact stability limits are illustrated in (Fig. 3.47). Apparently, for a constant delay the increasing target frequency requires still higher damping to stabilize contact transition. 3.9.2.3 Contact stability of sampled-data system The previous analysis, however, does not cover the most common class of SD systems in which the plant operates in the continuous time while the controller operates in the discrete time. In the considered interaction systems (Fig. 3.4) the robot Gs (s ) and the environment Ge (s ) are continuous time components while the position Gr (z ) and impedance controllers G f (z ) , as well as the nominal motion interpolator, are discrete time components. As mentioned, the continuous and discrete parts are interfaced with each other using A/D and D/A converters, i.e. sampler S and hold ZOH operators (Fig. 3.41). Real A/D and D/A converters are electronic devices possessing some dynamic characteristics. However, for the sake of simplicity, we will assume ideal system elements: as samplers S that instantaneously retrieves input, and the hold operators H immediately providing signals at the sampling instant. Furthermore, assume that all digital control components are synchronized at a unique sample rate T (usually real controllers are realized as multitasking systems running at different sample frequencies). As the most critical control delay, the force signal retardation will be taken into account. We will assume this time delay in the form τ = nT , with the transfer function e-nTs , or z − n in the discrete-time domain, where n denotes the number of sampling intervals (usually n = 1 , though, in some implementations impedance control sample rate can be significantly larger than the position control rate, causing an even longer force delay).

370

Dynamics and Robust Control of Robot-Environment Interaction

Fig. 3.46 Contact stability bounds of the delayed systems (s-domain analysis): passivity-based (above) and robust (below) contact stability ( M t = 10 kg, K t = 1500 N/m, ωt = 12.25 rad/ )

Impedance Control

Fig. 3.47 Dependence of minimum damping on the target frequency (s-domain analysis): passivity-based (above) and robust (below) contact stability ( M t = 152 − 0.6 kg, K t = 1500 N/m, ωt = 3.14 − 50.26 rad/ )

371

372

Dynamics and Robust Control of Robot-Environment Interaction

The above idealization of the real SD interaction system is needed to simplify the system modeling facing the most relevant effects, and thus to make contact stability analysis and impedance control synthesis much easier. Consider the model of the SD interaction system in (Fig. 3.41). Assuming again that the internal position controller is compensated for using the outer loop impedance compensator (3.72), implemented in discrete form as

G f (z) = G p

−1

(z )Gt −1 (z )

the obtained SD transfer function of the interaction system {p0 , e} considered in the robust (i.e. generalized) contact stability theory and its limited stability norm condition have the form

[I + (G G (z)) s

e

−1

]

−1 G s ( z )Gˆ t ( z ) G p ( z )

[

]

−1 ≤ I + (G s Ge(z)) −1G s ( z )Gˆ t ( z ) ∞