Motion control systems

MOTION CONTROL

SYSTEMS

Motion Control Systems, First Edition. Asif SŠabanovi´c and Kouhei Ohnishi.

© 2011 John Wiley & Sons (Asia) Pte Ltd. Published 2011 by John Wiley & Sons (Asia) Pte Ltd. ISBN: 978-0-470-82573-0

MOTION CONTROL

SYSTEMS

Asif Sˇabanovic´

Sabancı University, Turkey

Kouhei Ohnishi

Keio University, Japan

This edition first published 2011

2011 John Wiley & Sons (Asia) Pte Ltd

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Library of Congress Cataloging-in-Publication Data

Sˇabanovic´, Asif.

Motion control systems / Asif Sˇabanovic´.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-470-82573-0 (hardback)

1. Motion control devices. I. Title.

TJ214.5.S33 2011

621.4–dc22

2010041054

Print ISBN: 978-0-470-82573-0

ePDF ISBN: 978-0-470-82574-7

oBook ISBN: 978-0-470-82575-4

ePub ISBN: 978-0-470-82829-8

Set in 10/12pt Times by Thomson Digital, Noida, India.

Contents

Preface ix

About the Authors xi

PART ONE – BASICS OF DYNAMICS AND CONTROL

1 Dynamics of Electromechanical Systems 3

1.1 Basic Quantities 3

1.1.1 Elements and Basic Quantities in Mechanical Systems 3

1.1.2 Elements and Basic Quantities in Electric Systems 5

1.2 Fundamental Concepts of Mechanical Systems 7

1.2.1 The Principle of Least Action 7

1.2.2 Dynamics 8

1.2.3 Nonpotential and Dissipative Forces 9

1.2.4 Equations of Motion 10

1.2.5 Properties of Equations of Motion 14

1.2.6 Operational Space Dynamics 18

1.3 Electric and Electromechanical Systems 20

1.3.1 Electrical Systems 20

1.3.2 Electromechanical Systems 21

1.3.3 Electrical Machines 24

References 27

Further Reading 27

2 Control System Design 29

2.1 Basic Concepts 30

2.1.1 Basic Forms in Control Systems 31

2.1.2 Basic Relations 35

2.1.3 Stability 36

2.1.4 Sensitivity Function 37

2.1.5 External Inputs 38

2.2 State Space Representation 39

2.2.1 State Feedback 40

2.2.2 Stability 44

2.2.3 Observers 45

2.2.4 Systems with Observers 48

2.2.5 Disturbance Estimation 49

2.3 Dynamic Systems with Finite Time Convergence 51

2.3.1 Equivalent Control and Equations of Motion 52

2.3.2 Existence and Stability 53

2.3.3 Design 53

2.3.4 Control in Linear Systems 55

2.3.5 Sliding Mode Based Observers 56

References 59

Further Reading 59

PART TWO – ISSUES IN MOTION CONTROL

3 Acceleration Control 63

3.1 Plant 63

3.2 Acceleration Control 67

3.2.1 Formulation of Control Tasks 68

3.2.2 Equivalent Acceleration and Equivalent Force 74

3.3 Enforcing Convergence and Stability 85

3.3.1 Convergence for Bounded Control Input 90

3.3.2 Systems with Finite-Time Convergence 94

3.3.3 Equations of Motion 97

3.3.4 General Structure of Acceleration Control 105

3.4 Trajectory Tracking 107

References 114

Further Reading 114

4 Disturbance Observers 115

4.1 Disturbance Model Based Observers 118

4.1.1 Velocity Based Disturbance Observer 119

4.1.2 Position Based Disturbance Observer 121

4.2 Closed Loop Disturbance Observers 127

4.2.1 Internal and External Forces Observers 128

4.3 Observer for Plant with Actuator 132

4.3.1 Plant with Neglected Dynamics of Current Control Loop 133

4.3.2 Plant with Dynamics in Current Control Loop 136

4.4 Estimation of Equivalent Force and Equivalent Acceleration 140

4.5 Functional Observers 144

4.6 Dynamics of Plant with Disturbance Observer 149

4.6.1 Disturbance Estimation Error 150

4.6.2 Dynamics of Plant With Disturbance Observer 151

4.7 Properties of Measurement Noise Rejection 160

4.8 Control of Compensated Plant 164

4.8.1 Application of Estimated ^teq and q^€

eq 167

References 172

Further Reading 173

vi Contents

5 Interactions and Constraints 175

5.1 Interaction Force Control 176

5.1.1 Proportional Controller and Velocity Feedback 178

5.1.2 Environment with Losses 182

5.1.3 Lossless Environment 187

5.1.4 Control of Push Pull Force 191

5.2 Constrained Motion Control 193

5.2.1 Modification of Reference 195

5.2.2 Modification by Acting on Equivalent Acceleration 201

5.2.3 Motion Modification while Keeping Desired Force Profile 205

5.2.4 Impedance Control 209

5.2.5 Force Driven Systems 210

5.2.6 Position and Force Control in Acceleration Dimension 211

5.3 Interactions in Functionally Related Systems 215

5.3.1 Grasp Force Control 215

5.3.2 Functionally Related Systems 225

References 232

Further Reading 232

6 Bilateral Control Systems 233

6.1 Bilateral Control without Scaling 234

6.1.1 Bilateral Control Design 238

6.1.2 Control in Systems with Scaling in Position and Force 247

6.2 Bilateral Control Systems in Acceleration Dimension 251

6.3 Bilateral Systems with Communication Delay 256

6.3.1 Delay in Measurement Channel 257

6.3.2 Delay in Measurement and Control Channels 263

6.3.3 Closed Loop Behavior of System with Observer 267

6.3.4 Bilateral Control in Systems with Communication Delay 270

References 274

Further Reading 274

PART THREE – MULTIBODY SYSTEMS

7 Configuration Space Control 279

7.1 Independent Joint Control 280

7.2 Vector Control in Configuration Space 281

7.2.1 Selection of Desired Acceleration 282

7.3 Constraints in Configuration Space 290

7.3.1 Enforcement of Constraints by Part of Configuration Variables 303

7.4 Hard Constraints in Configuration Space 304

References 311

Further Reading 312

Contents vii

8 Operational Space Dynamics and Control 313

8.1 Operational Space Dynamics 314

8.1.1 Dynamics of Nonredundant Tasks 314

8.1.2 Dynamics of Redundant Tasks 315

8.2 Operational Space Control 318

8.2.1 Nonredundant Task Control 319

8.2.2 Redundant Task Control 328

References 336

Further Reading 336

9 Interactions in Operational Space 337

9.1 Task–Constraint Relationship 337

9.2 Force Control 341

9.3 Impedance Control 345

9.4 Hierarchy of Tasks 347

9.4.1 Constraints in Operational Space 347

9.4.2 Enforcing the Hierarchy of Tasks 352

9.4.3 Selection of Configuration Space Desired Acceleration 357

References 358

Further Reading 358

Index 361

viii Contents

Preface

This book is concerned with the development of an understanding of the design issues in

controlling motion within mechanical systems. There seems to be a never-ending discussion on

what motion control is – a new field or an extension or a combination of existing fields. Despite

this, both industry and academia have been involved in fulfilling real-world needs in

developing efficient design methods that will support never-ending requirements for faster

and accurate control of mechanical motion. High-precision manufacturing tools, product

miniaturization, the assembly of micro- and nanoparts, a need for high accuracy and fidelity of

motion in robot-assisted surgery – in one way or another all these employ motion control.

Looking back at its brief history, the concept of motion control was not well established in

the 1970s and 1980s. Many people still believed that controlling the torque needed for a load

should be achieved through velocity control. However, we found that torque and velocity could

be separately controlled. This was very effective for dexterous motion in robotics. We were

very excited and naturally wanted to announce this interesting finding and create a new field.

Meetings and discussions with other researchers and students encouraged us to create a new

workshop covering the problems of motion control. In March 1990, the first workshop

dedicated only to motion control (the IEEE International Workshop on Advanced Motion

Control – later known as AMC Workshops) was held at Keio University. To our surprise, there

were more than 100 papers presented at the workshop. Since then, many ideas, concepts and

results have come out. Subsequently, motion control gained visibility and attracted many

researchers. Time flows very fast and now it is time to summarize the results, particularly for the

new students coming into this field. We hope readers enjoy this book.

The intention of this book is to present material that is both elementary and fundamental,

but at the same time to discuss the solution of complex problems in motion control systems.

We recognize that the motion control system as an entity, separable from the rest of the

universe (the environment of the systems) by a conceptual or physical boundary, is composed

of interacting parts. This allows treatment of simple single degree of freedom systems as

well as complex multibody systems in a very similar if not identical way. By considering

complex motion control systems as physically or conceptually interconnected entities,

design ideas applied to single degree of freedom systems can also be applied with small

changes to complex multibody systems. Material in this book is treated in such a way that

the complexity of a system is gradually increased, starting from fundamentals shown in

the framework of single degree of freedom systems and ending with a treatment for the

control of complex multibody systems. Mathematical complexities are kept to a required

minimum so that practicing engineers as well as students with a limited background in

control may use the book.

This book has nine chapters, divided into three parts. The first part serves as an overview of

dynamics and control. It is intended for those who would like to refresh ideas related to

mathematical modeling of electromechanical systems and control. The first chapter is related

to the dynamics of mechanical and electromechanical systems. It presents basic ideas for

deriving equations of motion in mechanical and electromechanical systems. The second

chapter gives fundamental concepts in the analysis and design of control systems. Design is

discussed for systems with continuous and discontinuous control.

In the second part we discuss fundamentals of acceleration control framework for motion

control systems and give essential methods which are used in the third part of the book. Chapter 3

deals with single degree of freedom motion control system with asymptotic or finite time

convergence to the equilibrium. The design is based on the assumption that any disturbance due

to a change in parameters and interaction with environment should be rejected. In the fourth

chapter the design of a disturbance observer and the dynamics of a system with a disturbance

observer is discussed. Chapter 5 discusses the behavior of single degree of freedom motion

systems in interaction with the environment. While rejections of the interaction forces is a basic

requirement in Chapter 3, Chapter 5 considers modification of motion due to interaction. Such a

modification introduces a more natural behavior of the motion control system. The interaction

control is extended to controlling systems that need to maintain some functional relationship,

thus introducing a conceptual functional relationship between physically separated systems.

This serves as a background for a discussion of specific relationships – bilateral control –

discussed in chapter six for systems without and with a delay in the communication channels.

The third part extends the results obtained in part two to controlling fully actuated multibody

mechanical systems. Chapter 7 discusses the control of constrained systems in configuration

space and the enforcement of constraints by a selected group of degrees of freedom. In

Chapter 8 control design in operational space is carried out for nonredundant and redundant

tasks. The relationship between task and constraint is discussed and the similarities and

differences between the two are investigated. Chapter 9 discusses problems related to the

concurrent realization of multiple redundant tasks for constrained or unconstrained systems.

Problems in the hierarchy of the execution of multiple tasks are described.

The idea of writing this book stems from a long-term collaboration between the authors. It

began in early 1980s when we met at a conference in Italy and developed during Asif’s stay at

Keio. The book is the result of our discussions and common understanding of problems and

control methods applicable in the field of motion control. Obviously we do not pretend that it is

a final world; rather it is just a beginning, maybe a first step in establishing motion control as a

stand alone academic discipline. Results produced by many other authors are included in the

book in one way or another. Many authors, and especially our students, influenced our way of

treating certain material.

We would like to thank our numerous students, from whom we have learned a lot, and we

hope that they have learned something from us. We would like to express our sincerest thanks to

our families for their support during many years of research and especially during the

preparation of the manuscript.

Asif Sabanovi
c

Kouhei Ohnishi

x Preface

About the Authors

Asif Sabanovi
c is Professor of Mechatronics at Sabancı University, Istanbul, Turkey.

He received undergraduate and graduate education in Bosnia and Herzegovina, at the Faculty

of Electrical Engineering, University of Sarajevo. From 1970, for 20 years he was with

ENERGOINVEST-IRCA, Sarajevo, where he was head of research in sliding mode control

applications in power electronics and electric drives. He was Visiting Professor at Caltech,

USA, at Keio University, Japan, and at Yamaguchi University, Japan. He was Head of the

CAD/CAM and Robotics Department at Tubitak – MAM, Turkey. He has received Best Paper

Awards from the IEEE. His fields of interest include motion control, mechatronics, power

electronics and sliding mode control.

Kouhei Ohnishi is Professor of the Department of System Design Engineering at Keio

University, Yokohama, Japan. After receiving a PhD in electrical engineering from the

University of Tokyo in 1980, he joined Keio University and has been teaching, conducting

research and educating students for more than 30 years. His research interests include motion

control, haptics and power electronics. He received Best Paper Awards and a Distinguished

Achievement Award from the Institute of Electrical Engineers of Japan. He received the

Dr.-Ing. Eugene Mittelmann Achievement Award from the IEEE Industrial Electronics Society

(IES). He is an IEEE Fellow and served as President of the IEEE IES in 2008 and 2009.

He enjoys playing clarinet on holidays.

Part One

Basics of Dynamics

and Control

No mathematical representation can precisely model a real physical system. One cannot

predict exactly what the output of a real physical system will be even if the input is known, thus

one is uncertain about the system. Uncertainty arises from unknown or unpredictable inputs

(disturbance, noise, etc.), unpredictable dynamics and unknown or disregarded dynamics and

change of parameters. Yet, to design control systems one need a mathematical description of

the physical systems – plants – that will allow the application of mathematical tools to predict

the output response for a defined input, so that it can be used to design a controller. The models

should allow a design which leads to a control that will work on the real physical system. This

limits the details needed to describe the system and the scope of the details we will be including

in the mathematical models of physical systems.

Generally speaking the objective in a control system is to make some output behave in a

desired way by manipulating some inputs. The output of design is a mathematical model of a

controller that must be implemented. Motion control involves assisting in the choice and

configuration of the overall system or, in short, taking a system view of the overall

performance. For this reason it is important that an applied control framework not only leads

to good and consistent designs but also indicates when the performance objectives cannot be

met. In order to make sense of the issues involved in the design of a motion control system, a

short overview of the control methods for analysis and design are presented in Chapter 2. In

addition to classical frequency and state space methods, systems with finite-time convergence

are treated.

Motion Control Systems, First Edition. Asif SŠabanovi´ and Kouhei Ohnishi. c

© 2011 John Wiley & Sons (Asia) Pte Ltd. Published 2011 by John Wiley & Sons (Asia) Pte Ltd. ISBN: 978-0-470-82573-0

1

Dynamics of Electromechanical

Systems

In this chapter we will discuss methods of deriving equations of motion for mechanical and

electromechanical systems. We use the term equations of motion to understand the relation

between accelerations, velocities and the coordinates of mechanical systems [1]. For elec￾tromechanical systems the equations of motion, in addition to mechanical coordinates, also

establish the relationship between electrical system coordinates and their rate of change.

Traditionally, introductory mechanics begins with Newton’s laws of motion which relate

force, momentum and acceleration vectors. Analytical mechanics in the form of Lagrange

equations provides an alternative and very powerful tool for obtaining the equations of motion.

The Lagrange equations employ a single scalar function, and there are no annoying vector

components or associated trigonometric manipulations. Moreover, analytical approaches

using Lagrange equations provide other capabilities that allow the analysis of a wide range

of systems.

The advantage of using Lagrange equations is that they are applicable to an extensive field of

particle and rigid body problems, including electromechanical systems, by reducing derivation

to a single procedure while repeating the same basic steps. The procedure is based on scalar

quantities such as energy, work and power, rather than on vector quantities.

In this chapter only basic ideas will be discussed, without detailed and long derivations. Our

goal is to show ways of deriving the equations of motion as a first step for the later design of

control systems. The scope is to show basic procedures and their application to different plants

(mechanical, electrical, electromechanical) often used in motion control.

1.1 Basic Quantities

1.1.1 Elements and Basic Quantities in Mechanical Systems

Mechanics is based on the notion that the measure of mechanical interactions between systems

is force and/or torque (the turning effect of forces):

Motion Control Systems, First Edition. Asif SŠabanovi´c and Kouhei Ohnishi.

© 2011 John Wiley & Sons (Asia) Pte Ltd. Published 2011 by John Wiley & Sons (Asia) Pte Ltd. ISBN: 978-0-470-82573-0