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SYNCHRONIZATION OF MECHANICAL SYSTEMS

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Synchronization_of_Mechanical_Systems/981238605X/files/00005___f17d6498073887b6efb62b0c5a57230f.pdfPreface

Synchronization is everywhere! This is the feeling one may get once alerted for it. Everyone is familiar with all kinds of biological rhythms ('biological clocks') that create some kind of conformity in time and in nature. This includes for instance neural activity and brain activity, but also the cardiac vascular system. Clearly, there are numerous other examples to be men-tioned, sometimes much more controversial like the claimed synchronicity of the monthly period of nuns in a cloister, and so on.

Synchronous motion was probably first reported by Huygens (1673), where he describes an experiment of two (marine) pendulum clocks hang-ing on a light weighted beam, and which exhibit (anti-)frequency synchro-nization after a short period of time. Synchronized sound in nearby organ tubes was reported by Rayleigh in 1877 [Rayleigh (1945)], who observed similar effects for two electrically or mechanically connected tuning forks. In the last century synchronization received a lot of attention in the Rus-sian scientific community since it was observed in balanced and rotors and vibro-exciters[Blekhman (1988)]. Perhaps an enlightening potential new ap-plication for coordinated motion is in the use of hundreds of piezo-actuators in order to obtain a desired motion of a large/heavy mechanical set-up like for instance an airplane or MRI-scanner, [Konishi et al. (1998)], [Konishi et al. (2000)]; or the coordination of microactuators for manipulation at very small scales, [Sitti et al. (2001)].

In astronomy synchronization theory is used to explain the motion of ce-lestial bodies, such as orbits and planetary resonances, [Blekhman (1988)]. In biology, biochemistry and medicine many systems can be modelled as os-cillatory or vibratory systems and those systems show a tendency towards synchronous behavior. Among evidences of synchronous behavior in the natural world, one can consider the chorusing of crickets, synchronous flash

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light in a group of fire-flies, and the metabolic synchronicity in yeast cell suspension, see [Winfree (1980)].

The subject of synchronization has received huge attention in the last decades, in particular by biologists and physicists. This attention probably centers around one of the fundamental issues in science, namely curiosity: how come we find synchronous motion in a large ensemble of identical systems? Also, new avenues of potential use of synchronicity are now being explored.

Synchronization has much in common and is in sense equivalent to-coordination and cooperation. In ancient times it was already understood that joint activity may enable to carry out tasks that are undoable for an individual.

Our interest in the subject of synchronization is strongly influenced by a desire to understand what the basic ingredients are when coordinated mo-tion is required in an engineering system. We therefore have concentrated in this book on synchronization or coordination of mechanical systems, like in robotic systems. This allows to delve, on the one hand, in the theoretic foundations of synchronous motion, but, on the other hand, made it pos-sible to combine the theoretical findings with experimental verification in our research laboratorium.

This book concentrates therefore on controlled synchronization of me-chanical systems that are used in industry. In particular the book deals with robotic systems, which nowadays are common and important systems in production processes. However, the general ideas developed here can be extended to more general mechanical systems, such as mobile robots, ships, motors, microactuators, balanced and unbalanced rotors, vibro-exciters.

The book is organized as follows: Chapter 1 gives a general introduction about synchronization, its definition and the different types of synchronization.

Chapter 2 presents some basic material and results on which the book is based. In Section 2.1 some mathematical tools and stability concepts used throughout the book are presented. The dynamic models of rigid and flexible joint robots are introduced in Section 2.2, including their most important properties. The experimental set-up that will be used in later chapters is introduced in Section 2.3, where a brief description of the robots and their dynamic models is presented.

Chapter 3 addresses the problem of external synchronization of rigid joint robots. The synchronization scheme formed by a feedback controller and model based observers is presented and a stability proof is developed.

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Simulation and experimental results on one degree of freedom systems are included to show the applicability and performance of the proposed controller. The main contribution of this chapter is a gain tuning proce-dure that ensures synchronization of the interconnected robot systems.

The case of external synchronization for flexible joint robots is addressed in Chapter 4. The chapter starts by explaining the differences between rigid and flexible joint robots and the effects on the design of the synchronization scheme. The synchronization scheme for flexible joint robots and stability analysis is presented. The chapter includes a gain tuning procedure that guarantees synchronization of the interconnected robot systems. Simulation results on one degree of freedom systems are included to show the viability of the controller.

The problem of internal (mutual) synchronization of rigid robots is treated in Chapter 5. This chapter presents a general synchronization scheme for the case of mutual synchronization of rigid robots. The chapter includes a general procedure to choose the interconnections between the robots to guarantee synchronization of the multi-composed robot system. Simulation and experimental results on one degree of freedom systems are included to show the properties of the controller.

Chapter 6 presents a simulation and experimental study using two rigid robot manipulators and shows the applicability and performance of the syn-chronization schemes for rigid joint robots. Particular attention is given to practical problems that can be encountered at the moment of implemen-ting the proposed synchronization schemes. The robots in the experimental setup have four degrees of freedom, such that the complexity in the imple-mentation is higher than in the simulations and experiments included in Chapters 3 and 5.

Further extensions of the synchronization schemes designed here are discussed in Chapter 7. Some conclusions related to synchronization in general and robot synchronization in particular are presented in Chapter 8.

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Synchronization_of_Mechanical_Systems/981238605X/files/00009___f813bc4b26f3d0e8677cc63a5f88ae2f.pdfContents

Preface v

1. Introduction 1

1.1 General introduction 1 1.2 Synchronization 3 1.3 Synchronization in robotic systems 5

1.3.1 Velocity and acceleration measurements 6 1.3.2 Joint flexibility 7 1.3.3 Friction phenomena 7

1.4 Problem formulation 8 1.4.1 External synchronization of rigid joint robots . . . . 8 1.4.2 External synchronization of flexible joint robots . . . 9 1.4.3 Mutual (internal) synchronization of rigid joint robots 10

1.5 Scope of the book 12 1.6 Outline of the book 13

2. Preliminaries 15

2.1 Mathematical preliminaries and stability concepts 15 2.1.1 Basic definitions 15 2.1.2 Lyapunov stability 18 2.1.3 Stability of perturbed systems 21

2.2 Dynamic models of robot manipulators 23 2.2.1 Rigid joint robots 23 2.2.2 Flexible joint robots 24 2.2.3 Properties of the dynamic model of the robots . . . . 26 2.2.4 Friction phenomena 27

ix

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2.2.4.1 Friction compensation 29 2.2.4.2 Friction phenomena in rigid joint robots . . 30 2.2.4.3 Friction phenomena in flexible joint robots . 31

2.3 Experimental setup 31

3. External synchronization of rigid joint robots 35

3.1 Introduction 35 3.2 Synchronization controller based on state feedback 37 3.3 Synchronization controller based on estimated variables . . 38

3.3.1 Feedback control law 38 3.3.2 An observer for the synchronization errors 39 3.3.3 An observer for the slave joint variables 39 3.3.4 Synchronization closed loop error dynamics 41 3.3.5 Stability analysis 43

3.3.5.1 Lyapunov function 45 3.3.5.2 Time derivative of the Lyapunov function . 45

3.4 Gain tuning procedure 49 3.5 Friction compensation 50 3.6 Simulation and experimental study 52

3.6.1 Simulation and experimental results 53 3.6.2 Comparative results for different controllers 55 3.6.3 Sensitivity to desired trajectories 56 3.6.4 Disturbance rejection 57

3.7 Concluding remarks and discussion 58

4. External synchronization of flexible joint robots 61

4.1 Introduction 61 4.2 Synchronization controller based on state feedback 64 4.3 Synchronization controller based on estimated variables . . 65

4.3.1 An observer for the synchronization errors 66 4.3.2 An observer for the slave variables 66 4.3.3 Synchronization closed loop error dynamics 67 4.3.4 Stability analysis 69

4.3.4.1 Lyapunov function 71 4.3.4.2 Time derivative of the Lyapunov function . 72

4.4 Gain tuning procedure 77 4.5 Simulation study 78 4.6 Concluding remarks and discussion 80

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5. Mutual synchronization of rigid joint robots 83

5.1 Introduction 83 5.2 Synchronization controller based on state feedback 85

5.2.1 Synchronization closed loop error dynamics 87 5.2.2 Stability analysis 87 5.2.3 Algebraic loop 92

5.3 Synchronization controller based on estimated variables . . 92 5.3.1 An observer for the joint variables 94 5.3.2 Synchronization closed loop error dynamics 95 5.3.3 Stability analysis 97

5.3.3.1 Lyapunov function 98 5.3.3.2 Time derivative of the Lyapunov function . 100

5.4 Gain tuning procedure 102 5.5 Friction compensation 103 5.6 Simulation and experimental study 104

5.6.1 Simulation and experimental results 105 5.6.2 Comparison between synchronization and tracking

controllers 107 5.6.3 Sensitivity to desired trajectory 107 5.6.4 Disturbance rejection 109

5.7 Concluding remarks and discussion I l l

6. An experimental case study 113

6.1 Introduction 113 6.2 The CFT transposer robot 113

6.2.1 Joint space dynamics 115 6.3 External synchronization of a complex multi-robot system 115

6.3.1 Performance evaluation 119 6.4 Mutual synchronization of a complex multi-robot system . 123 6.5 Conclusions and discussion 127

7. Synchronization in other mechanical systems 131

7.1 Leader-follower synchronization of mobile robots 131 7.1.1 Kinematic model of the mobile robot 132 7.1.2 Leader-follower synchronization controller 133 7.1.3 Simulation study 134

7.2 Control of differential mobile robots via synchronization . . 135 7.2.1 Model of the differential mobile robot 137

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7.2.2 Synchronization control strategy and controller . . . 138 7.2.3 Simulation study 141

7.3 Attitude formation of multi-satellite systems 143 7.3.1 Dynamics of the satellite system 143 7.3.2 Synchronization strategy and controller 145 7.3.3 Simulation study 147

7.4 Discussion 149

8. Conclusions 151

Appendix A Proof of Lemma 3.2 155

Appendix B Proof of Theorem 3.2 159

Appendix C Proof of Lemma 4.1 163

Appendix D Proof of Lemma 4.3 167

Appendix E Proof of Proposition 4.1 171

Appendix F Proof of Lemma 5.3 173

Appendix G Proof of Theorem 5.3 177

Appendix H Dynamic model of the CFT robot 181

Bibliography 197

Index 203

Synchronization_of_Mechanical_Systems/981238605X/files/00013___4d49a139ac3bbe0d622b76bc5e710119.pdfChapter 1

Introduction

1.1 General introduction

Nowadays the developments in technology and the requirements on effi-ciency and quality in production processes have resulted in complex and integrated production systems. In actual production processes such as ma-nufacturing, automotive applications, and teleoperation systems there is a high requirement on flexibility and manoeuvrability of the involved sys-tems. In most of these processes the use of integrated and multi-composed systems is widely spread, and their variety in uses is practically endless; assembling, transporting, painting, welding, just to mention a few. All these tasks require large manoeuvrability and manipulability of the exe-cuting systems, often even some of the tasks can not be carried out by a single system. In those cases the use of multi-composed systems has been considered as an option. A multi-composed system is a group of individual systems, either identical or different, that work together to execute a task. In practice many multi-composed systems work either under cooperative or under coordinated schemes. In cooperative schemes there are intercon-nections between all the systems, such that all systems have influence on the combined dynamics, while in coordinated schemes there are only in-terconnections from the leader or dominant system to the non-dominant ones. Therefore in coordinated schemes the leader system determines the synchronized behavior of all the non-dominant systems. Note that coor-dinated and cooperative systems are nothing else that a requirement of synchronous behavior of the multi-composed system. Synchronization, co-ordination, and cooperation are intimately linked subjects and very often, mainly in mechanical systems, they are used as synonymous to describe the same kind of behavior.

1

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The synchronization phenomenon was perhaps first reported by Huygens (1673), who observed that a pair of pendulum clocks hanging from a light weight beam oscillated with the same frequency. Synchronized sound in nearby organ tubes was reported by Rayleigh in 1877 [Rayleigh (1945)], who observed similar effects for two electrically or mechanically connected tuning forks. In the last century synchronization received a lot of attention in the Russian scientific community since it was observed in balanced and unbalanced rotors and vibro-exciters [Blekhman (1988)]. Nowadays, there are several papers related with synchronization of rotating bodies and elec-tromechanical systems [Blekhman et al. (1995)], [Huijberts et al. (2000)]. On the one hand, rotating mechanical structures form a very important and special class of systems that, with or without the interaction through some coupling, exhibit synchronized motion, for example the case of vibro-machinery in production plants, electrical generators, unbalanced rotors in milling machines [Blekhman (1988)]. On the other hand, for mechanical systems synchronization is of great importance as soon as two machines have to cooperate. The cooperative behavior gives flexibility and manoeu-vrability that cannot be achieved by an individual system, e.g. multi fin-ger robot-hands, multi robot systems and multi-actuated platforms [Brunt (1998)], [Liu et al. (1999)], teleoperated master-slave systems [Dubey et al. (1997)], [Lee and Chung (1998)]. In medicine master-slave teleoperated sys-tems are used in surgery giving rise to more precise and less invasive surgery procedures [Hills and Jensen (1998)], [Guthart and Salisbury (2000)]. In aerospace applications coordination schemes are used to minimize the er-ror of the relative attitude in formations of satellites [Wang et al. (1996)], [Kang and Yeh (2002)]. The case of group formation of multiple robotic vehicles is addressed in [Yamaguchi et al. (2001)] while ship replenishment is treated on [Kyrkjeb and Pettersen (2003)]. Perhaps an enlightening potential new application for coordinated motion is in the use of hundreds of piezo-actuators in order to obtain a desired motion of a large/heavy me-chanical set-up like for instance an airplane or mri-scanner, [Konishi et al. (1998)], [Konishi et al. (2000)]; or the coordination of microactuators for manipulation at very small scales, [Sitti et al. (2001)]. Nevertheless mechanical systems are not the only application in which syn-chronization plays an important role. In communication systems synchro-nization is used to improve the efficiency of the transmitter-receiver sys-tems, also synchronization and chaos have been used to encrypt information (potentially) improving security in the transmissions [Pecora and Carroll (1990)], [Cuomo et al. (1993)], [Kocarev et al. (1992)].

Synchronization_of_Mechanical_Systems/981238605X/files/00015___9f1c094b4b1a172a3be140d425f2d9f5.pdfIntroduction 3

The importance of synchronization does not only lie in the practical ap-plications that can be obtained, but also in the many phenomena that can be explained by synchronization theory. In astronomy synchroniza-tion theory is used to explain the motion of celestial bodies, such as orbits and planetary resonances, [Blekhman (1988)]. In biology, biochemistry and medicine many systems can be modelled as oscillatory or vibratory systems and those systems show a tendency towards synchronous behavior. Syn-chronous activity has been observed in many regions of the human brain, relative to behavior and cognition, where neurons can synchronously dis-charge in some frequency ranges [Gray (1994)]. Synchronous firing of car-diac pacemaker cells in human heart has been reported in [Torre (1976)]. Meanwhile evidence of synchronicity among pulse-coupled biological oscil-lators has been presented in [Mirollo and Strogatz (1990)]. Among evi-dences of synchronous behavior in the natural world, one can consider the chorusing of crickets, synchronous flash light in group of fire-flies, and the metabolic synchronicity in yeast cell suspension, see [Winfree (1980)]. Notice that synchronization in the above mentioned physical phenomena, such as biology and astronomy, appears in a natural way and is due only to the proper couplings of the systems, which is called self-synchronization. This is the main difference with respect to the practical applications of syn-chronization theory, where the synchronous behavior is induced by means of artificial couplings and inputs, such as feedback and feedforward con-trollers. This is the so-called controlled synchronization. This book focuses on controlled synchronization of robot systems, which nowadays are common and important systems in production processes. However, the general ideas developed here can be extended to more general mechanical systems, such as mobile robots, ships, motors, balanced and un-balanced rotors, vibro-exciters. For better understanding of the controlled synchronization problem first a general definition of synchronization is in-troduced, second the particular problems in the case of controlled robot synchronization are briefly listed.

1.2 Synchronization

According to [Blekhman et al. (1995)] synchronization may be denned as the mutual time conformity of two or more processes. This conformity can be characterized by the appearance of certain relations between some functionals or functions depending on the processes. In this book we deal

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in particular with synchronization with respect to a function depending on the state of the coupled system. Furthermore, based on the type of interconnections (interactions) in the system, different kinds of synchronization can be defined [Blekhman et al. (1997)].

In case of disconnected systems that present synchronous behavior this is referred to as natural synchronization, e.g. all precise clocks are synchronized in the frequency domain.

When synchronization is achieved by proper interconnections in the systems, i.e. without any artificially introduced external ac-tion, then the system is referred to as self-synchronized. A classi-cal example of self-synchronization is the pair of pendulum clocks hanging from a light weight beam that was reported by Huygens (1673). He observed that both pendulums oscillated with the same frequency. Another example is the synchronization of celestial bo-dies, such as rotation of satellites around planets.

When there exist external actions (input controls) and/or artificial interconnections then the system is called controlled-synchronized. Examples of this case are most of the practical applications of syn-chronization theory such as transmitter-receiver systems.

Depending on the formulation of the controlled synchronization problem distinction should be made between internal (mutual) synchronization and external synchronization.

In the first and most general case, all synchronized objects occur on equal terms in the unified multi-composed system. Therefore the synchronous motion occurs as the result of interaction of all elements of the system, e.g. coupled synchronized oscillators, co-operative systems.

In the second case, it is supposed that one object in the multi-composed system is more powerful than the others and its motion can be considered as independent of the motion of the other objects. Therefore the resulting synchronous motion is predetermined by this dominant independent system, e.g. master-slave systems.

From the control point of view controlled synchronization problem is the most interesting, i.e. how to design a controller and/or interconnections that guarantee synchronization of the multi-composed system with respect

Synchronization_of_Mechanical_Systems/981238605X/files/00017___d8071134a4aa35d5e2dfac0bfd33e4aa.pdfIntroduction 5

to a certain desired functional. The design of the controller and/or inter-connections is mainly based on the feedback of the variables or signals that define the desired synchronous behavior.

1.3 Synchronization in robotic systems

Robot manipulators are widely used in production processes where high flexibility, manipulability and manoeuvrability are required. In tasks that cannot be carried out by a single robot, either because of the complexity of the task or limitations of the robot, the use of multi-robot systems working in external synchronization, e.g. master-slave and coordinated schemes, or mutual synchronization, e.g. cooperative schemes, has proved to be a good alternative. Coordinated and cooperative schemes are important illustra-tions of the same goal, where it is desired that two or more robot systems, either identical or different, work in synchrony [Brunt (1998)], [Liu et al. (1997)], [Liu et al. (1999)]. This can be formulated as a control problem that implies the design of suitable controllers to achieve the required syn-chronous motion.

In synchronization of robotic systems there exist several fundamental pro-blems. First, a functional with respect to which the desired synchronization goal is described, has to be formulated. For this the type of robots and the variables of interest have to be taken into account. For robot synchroniza-tion the functionals can be defined as the norm of the difference between the variables of interest e.g. positions, velocities. Second, the couplings or interconnections and the feedback controllers to ensure the synchronous behavior have to be designed. The interconnections between the robots can be the feedback of the difference between the variables of interest. Finally conditions to guarantee the synchronization goal have to be determined. The problem of synchronization of robotic systems seems to be a straight-forward extension of classical tracking controllers, however it implies chal-lenges that are not considered in the design of tracking controllers. The interconnections (interactions) between the robots imply control problems that are not considered in classical tracking controllers. However the inter-connections cannot be neglected since they are precisely what determine the synchronized behavior and therefore the synchronization functional. Most of the tracking controllers are only based on the signals of the controlled system, i.e. the desired and controlled position, velocity and acceleration. Therefore any external signal, like the signals due to couplings, is consi-

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dered as a disturbance and its effects are supposed to be minimized or even canceled by the controller. Besides the fundamental problems of robot synchronization there exist other problems to be taken into account. Problems can arise because of the particular structure of the robots, such as type of joints (rigid or elastic), kinematic pairs (prismatic, rotational, etc.), transmission elements (gears, belts). Furthermore available equipment, for instance position, velocity and acceleration measuring capabilities, noise in the measurements and time delays, might cause other problems in robot synchronization. Some of the frequently encountered problems in robot synchronization are briefly discussed in the following sections. Note that there are many other types of coordination problems, such as coordination of underactuated or redundant manipulator systems. How-ever these problems are beyond the scope of this book. When redundant robot manipulators are considered the excess of actuated joints with res-pect to the degrees of freedom can be used to optimize certain functional, avoid singularities, pay load distribution, etc.. For this purpose synchro-nization schemes between the redundant joints can be used, however those synchronization schemes are different from the joint coordinate schemes considered in this book. On the other hand, for underactuated manipula-tors there are less actuated joints than degrees of freedom. In this situa-tion the movement of the non-actuated joints is subject to holonomic or non-holonomic constraints. These constraints establish relations between the actuated and non-actuated joints, such that they can be considered as being synchronized, with the constraints as synchronization function-a l . However, because there is no actuation the synchronization behavior of the non-actuated joints is achieved by the design of the robot and its own dynamics, therefore it is in fact self-synchronization. We now briefly discuss some of the frequently encountered problems in robot synchronization.

1.3.1 Velocity and acceleration measurements

The controlled synchronization problem is further complicated by the fact that frequently only position measurements of the robots are available or reliable, due to a lack of velocity and acceleration measuring equipment or noise in the measurements. In practice, robot manipulators are equipped with high precision position sensors, such as optical encoders. Meanwhile new technologies have been

Synchronization_of_Mechanical_Systems/981238605X/files/00019___d19d95066e3e4827ef4befee4fd639f3.pdfIntroduction 7

designed for measuring angular velocities and accelerations, e.g. brush-less AC motors with digital servo-drivers, microcontroller based measure-ments [Laopoulos and Papageorgiou (1996)], digital processing [Kadhim et al. (1992)], [Lygouras et al. (1998)], linear accelerometers [Ovaska and Valiviita (1998)], [Han et al. (2000)]. However, such technologies are not very common in applications yet. Therefore, very often the velocity mea-surements are obtained by means of tachometers, which are contaminated by noise. Moreover, velocity measuring equipment is frequently omitted due to the savings in cost, volume and weight that can be obtained. On the other hand acceleration measurements are indirectly obtained by pseudo-differentiation and filtering of the position and/or velocity direct measure-ments, such that the measurement noise is amplified and corrupts the ac-celeration measurements even farther.

1.3.2 Joint flexibility

Joint flexibility (also called joint elasticity) is caused by transmission ele-ments such as harmonic drives, belts or long shafts and it can be modelled by considering the position and velocity of the motor rotor and the posi-tion and velocity of the link [De Luca and Tomei (1996)], [Book (1984)], [Spong (1987)]. Therefore the model of a joint flexible robot has twice the dimension of an equivalent rigid robot, and thus the controllers for flexible joint robots are more complex than those for rigid joint robots. It has been shown that joint flexibility considerably affects the performance of robot manipulators since it is major source of oscillatory behavior [Good et al. (1985)]. This means that, to improve the performance of robot manipu-lators, joint flexibility has to be taken into account in the modelling and control of such systems.

1.3.3 Friction phenomena

Friction phenomena play an important role in control of robot manipula-tors. In high performance robotic systems, friction can severely deteriorate the performance. Bad compensation of the friction phenomena generates oscillatory behavior like limit cycles or stick-slip oscillations, introduces tracking errors, and in some cases can generate instability of the system [Armstrong-Helouvry (1993)], [Olsson and Astrom (2001)]. There is a great variety of friction models proposed in literature [Armstrong-Helouvry et al. (1994)], [Olsson et al. (1998)], and each can be classified

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with respect to their detail in describing surface contact properties occur-ring on a microscopic and macroscopic level. Which model is more suitable for modelling and control purposes depends on the physical friction phe-nomena observed in the system such as stiction, Stribeck curve effects, viscous friction, etc. and on the velocity regime in which the system is supposed to work, i.e. slow, medium, or high speed. One major limitation in modelling friction phenomena is the complexity of the models and the drawback for parameter identification and control that it implies.

1.4 Problem formulation

The problem of synchronization of robot systems is very wide depending on the kind of robots and their structural and measuring limitations, not to mention the possible synchronization goals. Therefore, this book is re-stricted to rigid and flexible rotational joint robots and internal or mutual synchronization (cooperative scheme) and external synchronization (coor-dinated and master-slave schemes). The robots considered are fully actuated, i.e. the number of actuators is equal to the number of joints. It is also assumed that all the robots in the synchronization system have the same number of joints and equivalent joint work spaces, i.e. any possible configuration of a given robot in the system can be achieved by any other robot in the system. It does not imply that the robots are identical in their physical parameters, such as masses, inertias, etc.

Based on the robot manipulator structure described above and the possible synchronization schemes, the synchronizing problems addressed here can be formulated as follows.

1.4.1 External synchronization of rigid joint robots

Consider a multi-robot system formed by two or more rigid joint robots, such that the motion of one of the robots is independent of the other ones. This dominant robot will be referred to as the master robot. The mas-ter robot is driven by a controller already designed and not relevant for the synchronization goal. In the ideal case the controller ensures conver-gence of the master robot angular positions and velocities to a given desired trajectory. Then the goal is to design interconnections and feedback con-trollers for the non-dominant robots, hereafter referred to as slaves, such

Synchronization_of_Mechanical_Systems/981238605X/files/00021___f777e73d6098a20429cc0ebfb35a763f.pdfIntroduction 9

that their positions and velocities synchronize to those of the master robot. For the design of the slave interconnections and controllers it is assumed that only the master and slave angular positions for all joints are available for measurement. Furthermore only the dynamic model of the slave robot is assumed to be known, which complicates the reconstruction of the master angular velocity and acceleration since the master robot dynamic model is unknown. Notice that the goal is to follow the trajectory of the master robot and not the desired trajectory of it, since this might not be achieved because of noise, parametric uncertainty or unmodelled dynamics of the master robot, like friction, unknown loads, etc.

Slave robot 1

Desired trajectory

qd % *

Trn Control

Master robot

f) )

// 0. An equilibrium point x* has the property that for any t > 0 if the state of the system starts at x* it will remain at x* for all future time.

Definition 2.3 A ball (sphere) of radius r around the origin is denoted by Br i.e.,

Br = {xeR"| | | :E| | < r }

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The ball with center xc, and radius r is denned by

Br(xc) = {x R| \\x - xc\\ < r}

when considering also the equality < rather that Rm is said to be continuous at a point x if given any e > 0 a constant 6 > 0 exists such that

\\x-y\\ 0 a constant 6 > 0 exists (depending only on e) such that (2.4) holds.

Often uniform continuity of a function / :'. of the following lemma.

Lemma 2.2 Consider a differentiable function f : M G R exist such that

can be verified by means

-> R. If a constant

sup xeM. dx

(x) Rn be a continuously differentiable function, such that

V(0) = 0 and V(x) > 0 in D - {0}

V(x) < 0 in D

Then x = 0 is stable. Moreover, if

V(x) < 0 in D - {0}

then x = 0 is asymptotically stable.

Notice that for asymptotic stability it is required that V(x) < 0 in D - {0}. However, there are some auxiliary theorems that allow to conclude asymp-totic stability when V(x) < 0. For autonomous systems it is possible to prove asymptotic stability when V(x) < 0 by considering LaSalle's The-orem, while for non-autonomous systems Barbalat's Lemma is useful in proving asymptotic stability.

Theorem 2.2 LaSalle's Theorem: [Khalil (1996)] Given the system (2.10) suppose that a Lypaunov function candidate V is found such that, along the solution trajectories

V