Adaptive Robust Control of Nonlinear Systems with Application to Control of

Mechanical Systems

by

Bin Yao

B.Eng. (Beijing University of Aeronautics and Astronautics, P.R.China ) 1987M.Eng. (Nanyang Technological University, Singapore) 1992

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Mechanical Engineering

in the

GRADUATE DIVISION

of the

UNIVERSITY of CALIFORNIA at BERKELEY

Committee in charge:

Professor Masayoshi Tomizuka , ChairProfessor Karl J. HedrickProfessor S. Shankar Sastry

1996

1

Abstract

Adaptive Robust Control of Nonlinear Systems with Application to Control of Mechanical

Systems

by

Bin Yao

Doctor of Philosophy in Mechanical Engineering

University of California at Berkeley

Professor Masayoshi Tomizuka , Chair

This dissertation focuses on the high performance robust control of nonlinear systems in the presence

of parametric uncertainties and uncertain nonlinearities (e.g., disturbances) and its application to the

control of mechanical systems. A new approach, adaptive robust control (ARC), is proposed. The

approach e�ectively combines the design techniques of adaptive control (AC) and deterministic robust

control (DRC) and improves performance by preserving the advantages of both AC and DRC. Speci�-

cally, the approach guarantees a superior performance in terms of both transient error and �nal tracking

accuracy in the presence of parametric uncertainties and uncertain nonlinearities. This result overcomes

the drawbacks of AC and makes the approach attractive to real applications. Through parameter adap-

tation, the approach achieves asymptotic tracking in the presence of parametric uncertainties without

using a high-gain in the feedback loop, which implies that the control input is smooth. In this sense,

ARC has a better tracking performance than DRC. The design is conceptually simple and amenable to

implementation.

A general framework of the proposed ARC is formulated in terms of adaptive robust control

(ARC) Lyapunov functions. Through backstepping design, ARC Lyapunov functions can be successfully

constructed for a large class of multi-input multi-output (MIMO) nonlinear systems transformable to a

semi-strict feedback form.

The method is applied to the control of robot manipulators in several applications. For

trajectory tracking control, two ARC algorithms are developed: adaptive sliding mode control (ASMC)

and desired compensation ARC (DCARC). ASMC is based on the sliding mode control (SMC) and

the conventional adaptation law that uses the actual state variables in the regressor. DCARC uses

the desired trajectory information in the regressor. Three di�erent adaptive or robust control schemes

are also derived for comparison: a simple nonlinear PID type robust control, a gain-based nonlinear

PID type adaptive control, which requires no model information, and a combined parameter and gain-

based adaptive robust control. All the algorithms are implemented and compared on a two-link direct

drive robot. Comparative experimental results show the importance of the controller structure and the

parameter adaptation. The proposed DCARC is found to provide the best tracking performance without

increasing the control bandwidth and the control e�ort.

For a constrained robot manipulator, the end-e�ector of which is in contact with rigid surfaces,

a new constrained dynamic model is obtained to account for the e�ect of contact surface friction. The

ARC scheme utilizes a PI type force feedback control structure with a low proportional gain to avoid the

2

acausality problem. Possible impact problems caused by losing contact are alleviated by the guaranteed

transient performance. An adaptation law driven by both motion and force tracking errors guarantees

asymptotic motion and force tracking without any persistent excitation conditions. Simulation results

verify the e�ectiveness of the method.

For the coordinated control of multiple robot manipulators handling a constrained object, a set

of transformed dynamic equations are obtained in the joint space. In the transformed domain, internal

force and external contact force have the same form and can be dealt with in the same way as in the

constrained motion problem. A coordinated motion and force ARC controller is developed. It possesses

the same nice properties as the ARC constrained motion controller mentioned above.

Motion and force tracking control of robot manipulators in contact with unknown sti�ness

environments is formulated. An ARC motion and force controller is developed to deal with unknown

robot parameters and surface parameters, such as sti�ness and friction coe�cients, as well as uncertain

nonlinearities caused by modeling errors.

Professor Masayoshi TomizukaDissertation Committee Chair

iii

Contents

List of Figures vi

List of Tables viii

1 Introduction 11.1 Control of Uncertain Nonlinear Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Adaptive Control (AC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Deterministic Robust Control (DRC) . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Motivations and Contributions of the Dissertation . . . . . . . . . . . . . . . . . . . . . 61.3.1 General Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.2 General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Outline of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

I Adaptive Robust Control - Theory 14

2 Control of a First-order Uncertain System 152.1 Deterministic Robust Control (DRC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Adaptive Control (AC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Adaptive Robust Control (ARC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Adaptive Robust Control of SISO Nonlinear Systems in a Semi-Strict Feedback form 243.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 Smooth Projection and Positive De�nite Function V� . . . . . . . . . . . . . . . . . . . 253.3 Backstepping Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.1 Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3.2 Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.3 Step i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.4 Step n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 Guaranteed Transient Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 General Framework of Adaptive Robust Control 384.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2 ARC Lyapunov Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

iv

4.3 Adaptive Robust Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Backstepping Design via ARC Lyapunov Functions 445.1 Initial MIMO Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.2 Augmented MIMO Nonlinear Systems I . . . . . . . . . . . . . . . . . . . . . . . . . . 455.3 Backstepping Design I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.4 Augmented MIMO Nonlinear Systems II . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6 Adaptive Robust Control of MIMO Nonlinear Systems 546.1 MIMO Semi-Strict Feedback Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.2 Backstepping Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.2.1 Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.2.2 Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.2.3 Step i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.2.4 Step r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606.2.5 Guaranteed Transient Performance . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

II Adaptive Robust Control - Applications 68

7 Trajectory Tracking Control of Robot Manipulators 697.1 Dynamic Model of Robot Manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.2 Control Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7.2.1 Adaptive Sliding Mode Control (ASMC) . . . . . . . . . . . . . . . . . . . . . . 707.2.2 Desired Compensation Adaptive Robust Control (DCARC) . . . . . . . . . . . . 747.2.3 Nonlinear PID Robust Control (NPID) . . . . . . . . . . . . . . . . . . . . . . . 777.2.4 Nonlinear PID Adaptive Control (PIDAC) . . . . . . . . . . . . . . . . . . . . . 787.2.5 Desired Compensation Adaptive Robust Control with Adjustable Gains (ARCAG) 80

7.3 Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7.4.1 Performance Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847.4.2 Controller Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847.4.3 Comparative Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 86

7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

8 Other Applications 938.1 Constrained Motion and Force Control of Robot Manipulators . . . . . . . . . . . . . . 93

8.1.1 Dynamic Model of Constrained Robots . . . . . . . . . . . . . . . . . . . . . . . 938.1.2 Adaptive Robust Control of Constrained Manipulators . . . . . . . . . . . . . . . 968.1.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028.1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

8.2 Coordinated Control of Multiple Robot Manipulators . . . . . . . . . . . . . . . . . . . 1048.2.1 Dynamic Model of Robotic Systems . . . . . . . . . . . . . . . . . . . . . . . . 1088.2.2 Adaptive Robust Control of Coordinated Manipulators . . . . . . . . . . . . . . 1118.2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8.3 Motion and Force Tracking Control of Robot Manipulators in Contact With UnknownSti�ness Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

v

8.3.1 Dynamic Model of a Manipulator in Contact with a Sti� Environment . . . . . . 1148.3.2 ARC Motion and Force Tracking Control . . . . . . . . . . . . . . . . . . . . . . 1168.3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

9 Conclusion 1239.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239.2 Suggested Ideas for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Bibliography 126

vi

List of Figures

2.1 Nondecreasing n-th smooth projection map . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1 Tracking errors in the presence of parametric uncertainties . . . . . . . . . . . . . . . . 353.2 Estimated parameters in the presence of parametric uncertainties . . . . . . . . . . . . 353.3 Control input in the presence of parametric uncertainties . . . . . . . . . . . . . . . . . 353.4 Tracking errors in the presence of parametric uncertainties and small disturbances (d1 =

d2 = 0:02) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.5 Estimated parameters in the presence of parametric uncertainties and small disturbances

(d1 = d2 = 0:02) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.6 Control input in the presence of parametric uncertainties and small disturbances (d1 =

d2 = 0:02) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.7 Tracking errors in the presence of parametric uncertainties and large disturbances (d1 =

d2 = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.8 Estimated parameters in the presence of parametric uncertainties and large disturbances

(d1 = d2 = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.9 Control input in the presence of parametric uncertainties and large disturbances (d1 =

d2 = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.1 Tracking errors in the presence of parametric uncertainties . . . . . . . . . . . . . . . . 636.2 Tracking errors in the presence of parametric uncertainties . . . . . . . . . . . . . . . . 646.3 Estimated parameters in the presence of parametric uncertainties . . . . . . . . . . . . 646.4 Control inputs in the presence of parametric uncertainties . . . . . . . . . . . . . . . . 646.5 Control inputs in the presence of parametric uncertainties . . . . . . . . . . . . . . . . 656.6 Tracking errors in the presence of parametric uncertainties and disturbances (d1=d2=2) 656.7 Tracking errors in the presence of parametric uncertainties and disturbances (d1=d2=2) 656.8 Estimated parameters in the presence of parametric uncertainties and disturbances

(d1=d2=2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.9 Control inputs in the presence of parametric uncertainties and disturbances (d1=d2=2) 666.10 Control inputs in the presence of parametric uncertainties and disturbances (d1=d2=2) 666.11 Tracking errors in the presence of parametric uncertainties . . . . . . . . . . . . . . . . 676.12 Tracking errors in the presence of parametric uncertainties . . . . . . . . . . . . . . . . 676.13 Estimated parameters in the presence of parametric uncertainties . . . . . . . . . . . . 67

7.1 Berkeley/NSK Two-Link Direct-Drive Manipulator . . . . . . . . . . . . . . . . . . . . 827.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837.3 Transient Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.4 Final Tracking Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.5 Average Tracking Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

vii

7.6 Control E�ort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887.7 Control Chattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.8 Estimated payloads approach their true values . . . . . . . . . . . . . . . . . . . . . . 897.9 Estimated Feedback Gains K� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917.10 Joint Tracking Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927.11 Joint Control Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

8.1 Con�guration of the Robot Moving on a Semi-circle Surface . . . . . . . . . . . . . . . 1038.2 Position Tracking Error ep in the Presence of Parametric Uncertainties . . . . . . . . . 1058.3 Force Tracking Error ef in the Presence of Parametric Uncertainties . . . . . . . . . . . 1058.4 Interaction Force in the Presence of Parametric Uncertainties . . . . . . . . . . . . . . 1068.5 Estimated Parameters in the Presence of Parametric Uncertainties . . . . . . . . . . . . 1068.6 Position Tracking Error ep in the Presence of Parametric Uncertainties and Disturbances 1068.7 Force Tracking Error ef in the Presence of Parametric Uncertainties and Disturbances . 1078.8 Estimated Parameters in the Presence of Parametric Uncertainties and Disturbances . . 1078.9 Joint Torque in the Presence of Parametric Uncertainties and Disturbances . . . . . . . 1078.10 Con�guration of a Robotic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088.11 A Manipulator in Contact With a Sti� Environment . . . . . . . . . . . . . . . . . . . 114

viii

List of Tables

7.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

ix

Acknowledgements

I would like to express my deepest gratitude to Professor Masayoshi Tomizuka for in uencing my way

of thinking and for helping and supporting my research.

I would also like to thank Professor Karl Hedrick and Professor Shankar Sastry for their

invaluable comments as members of my dissertation committee.

I would like to thank all my friends in Berkeley and all members of Professor Tomizuka's

research group, with whom I shared many good and bad moments during my study here. I would

especially like to thank Professor Hui Peng, Dr. George T. C. Chiu, Dr. Yean-Ren Hwang, Dr. Liang-

Jong Huang, Dr. Satyajit Patwardhan, Professor Addisu Tesfaye, Dr. Wei-Hsin Yao, Dr. Eugene David

Tung, Dr. Thomas M. Hessburg, Dr. Perry Li, Rob Bickel, Mohammed Al-Majed, Prabhakar Pagilla,

Chieh Chen, and Lin Guo for their inspiring discussions and warm friendship, and Victor Chu and Carlos

Osorio for their help with computer software in the laboratory.

Finally, I would like to thank all my family members for their encouragement and my dearest

Dorothy for her love and support.

1

Chapter 1

Introduction

1.1 Control of Uncertain Nonlinear Dynamics

Although linear control theory has evolved a variety of powerful methods and has had a long

history of successful industrial applications, it has found to be inadequate in many applications, for

many reasons such as increasingly stringent performance requirements and large operating range, which

invalidate the use of linearized models. Many physical systems have so-called "hard nonlinearities",

such as Coulomb friction, saturation, dead zones, backlash, and hysteresis. These nonlinearities are non-

smooth or discontinuous, and do not allow linear approximations. They often cause undesirable behavior

in the control system, such as instability and limit cycles if not properly handled. It may be necessary

to apply nonlinear control to obtain acceptable performance. The design of nonlinear controllers is not

necessarily complex. For example, in robot control, it is easier to design a stabilizing nonlinear controller

than a stabilizing linear controller. Also, with the advances of low-cost microprocessors, it is neither

di�cult nor costly to implement nonlinear controllers. All these factors have made nonlinear control

increasingly more popular, and the �eld has grown quickly during the past twenty years.

Earlier results in nonlinear control [47] required exact knowledge of the system dynamics. In

reality, though we may apply physical laws to model the system and �nd the shapes of the nonlinear

functions, parameters of the system (e.g., the inertia parameters of a new object grasped by a robot)

may depend on operational conditions and may not be precisely known in advance. Because of factors

such as aging e�ect, the parameters may also be slowly time-varying. These types of uncertainties

are called parametric uncertainties and may cause the control law designed based on the nominal

model unstable or degrade its performance. In mechanical systems, nonlinearities such as nonlinear

friction force and backlash as well as external disturbances cannot be modeled exactly. These types of

uncertain nonlinearities can be classi�ed as unknown nonlinear functions. On the whole, the system may

be subjected to both parametric uncertainties and unknown nonlinear functions. Control of uncertain

nonlinear dynamics is, thus, essential for successful applications. In fact, during the past twenty years,

the control of uncertain dynamics has been very popular. Numerous algorithms have been proposed,

which can basically be classi�ed into two classes: adaptive control and deterministic robust control.

2

1.2 Previous Work

1.2.1 Adaptive Control (AC)

Biological systems cope easily and e�ciently with changes in their environments. As interests

in control theory have shifted over the years to the control of systems with large uncertainty, e�orts

are naturally made to incorporate in them characteristics similar to those in living systems; numerous

words, such as adaptation, learning, pattern recognition and self-organization, were introduced into

the control literature. Among those words, adaptive control, which was born in the late 1950s to

deal with parametric uncertainties, was the �rst introduced. Since then, it has remained to be a

mainstream research activity, with hundreds papers published on it every year, and has become a well-

formed discipline, especially for linear systems.

Earlier results in adaptive control were developed for linear time-invariant (LTI) systems [68,

4, 85, 104] described by

_x = A(�)x+B(�)u

y = C(�)x+D(�)u(1.1)

where � represents the vector of parameters that are unknown but constant. For LTI systems with relative

degree one (the relative degree r of a LTI system is equal to the number of poles minus the number

of zeros of its transfer function), a stable adaptive controller was proposed in [88] with the concept of

positive realness playing an important role. With the concept of the augmented error introduced by

Monopoli [77], the general problem with r � 2 was �nally solved around 1980 [87, 78, 34, 68]. These

breakthrough results made researchers feel that the era of practical adaptive control had �nally arrived.

However, it was soon realized that the above adaptive control derived for the ideal case would result in the

parameter error growing without bound and destabilizing the system when bounded disturbances were

present [27]. It was also shown, primarily by simulations, that other perturbations, such as time-varying

parameters and un-modeled dynamics [101], could result in instability. All of these clearly indicated that

new approaches were needed to assure the boundedness of all the signals in the system and led to a

body of work referred to as the robust adaptive control theory. Two distinct approaches were taken to

achieve robustness. One is to use the appropriate reference input. The other is to modify the adaptation

law. It was realized even in the 1960s [3] that for parameter convergence the reference input should

satisfy certain conditions, generally referred to as persistent excitation (PE) conditions. Narendra and

Annaswamy [83] demonstrated that the degree of persistent excitation would determine whether or not

the system would be robust in the presence of speci�ed disturbances, i.e., it was shown that in the absence

of disturbances but with a persistently exciting input, an adaptive system is uniformly asymptotically

stable. In view of the importance of PE in adaptive systems, Boyd and Sastry [6, 104] used frequency

domain methods to show that if the spectral measure of the input was concentrated at at least n points,

the state of an n-th order dynamical system would be persistently exciting. For most applications such

as trajectory tracking control, the reference input (or desired output trajectory) is speci�ed by the task

and normally does not satisfy the PE condition. Thus, the �rst approach has limitations in practical

problems. As to the modi�cation of adaptation laws to achieve robustness, several approaches were

proposed. One was the use of a dead zone [27, 93] in the adaptation law. In this approach, it has

to be assumed that the magnitude of the disturbance was known and asymptotic stability was lost.

By introducing an additional term in the adaptation law (referred to as �-modi�cation), Ioannou and

Kokotovic [46] achieved the uniform stability at large. However, when the disturbance was not present,

3

the error would no longer tend to zero and asymptotic stability was lost. To overcome this drawback,

�-modi�cation was proposed in [84]. More recently, by assuming that parameters lie in a known compact

set, projection methods presented by Sastry in [104] and by Goodwin and Mayne in [33] have become

popular to achieve robustness. Other recent developments in adaptive control of linear systems include

relaxing the assumptions under which stable adaptive control is possible.

Another drawback of adaptive control is that the transient performance is not clear. It was

shown in [171] that poor initial parameter estimates may result in unacceptable poor transient behavior.

The design of adaptive controllers with improved transient performance is a current research topic. Fu

[30] introduced a variable structure control (VSC) design for a relative degree two plants and Narendra

and Boskovic [86] proposed a combined direct, indirect, variable structure method. However, transient

performance under these methods is still not guaranteed and the resulting controllers are discontinuous,

which leads to control chattering. With a known high frequency gain, an L1 formulation was used in

[24] to improve transient performance of continuous model reference adaptive control (MRAC). The

assumption of known high frequency gain was relaxed in [92] and a di�erent interpretation using modi�ed

high order tuning was given in [134]. However, in all of these controllers, only parametric uncertainties

were considered and robustness was not discussed.

In trying to extend the above adaptive schemes from linear systems to nonlinear systems,

one was faced with considerable obstacles. One important factor was the lack of a systematic design

methodology for nonlinear feedback. As such, adaptive nonlinear control started with speci�c problems,

e.g., trajectory tracking control of rigid robot manipulators. A robot arm is constructed to simulate a

human being's arm to accomplish a variety of tasks and has been widely used in industry to increase

exibility and productivity. Thus, high performance control of robots is of practical signi�cance. Since

robot dynamics are described by a set of highly coupled nonlinear di�erential equations, control of such

a system is challenging, and has been extensively studied during the past decade. Earlier results, such as

the computed torque method [119, 81], which utilized the feedback linearization method [47], required

exact knowledge of the robot dynamics. It was soon found that such methods could not perform well

in practical application because of parametric uncertainties such as the payload. A nonlinear adaptive

method that guarantees asymptotic stability without any approximation of nonlinear dynamics was �rst

developed by Craig, Hsu, and Sastry [23] around 1986. The requirement of acceleration measurements

and invertibility of the estimate of the inertia matrix was later removed by Slotine and Li [110, 111], Wen

and Bayard [135], Sadegh and Horowitz [103], and Middleton and Goodwin [75]. Sadegh and Horowitz

presented an adaptive scheme [103] which used reference trajectory information rather than actual state

information, and a locally exponentially stable adaptive algorithm [102] under the assumption of (semi)

persistent excitation. Recently, Whitcomb, et al. [138] presented comparative experiments for di�erent

adaptive control algorithms.

Motivated by the initial success of the adaptive control of robot manipulators, the adaptive

control of general nonlinear systems has also undergone rapid developments during the past ten years

[65, 105, 95], leading to global stability and tracking results for reasonably large classes of nonlinear

systems [59, 62, 52]. Earlier results [82, 105, 123, 95, 96, 121, 9, 54, 26] were based on the feedback

linearization method. Because of the parameter-dependent forms of feedback linearization conditions

and the "certainty-equivalence" implementation, restrictions had to be imposed either on the location of

unknown parameters or on the type of nonlinearities. Accordingly, the earlier results could be classi�ed

into two categories: the nonlinearity-constrained schemes [82, 105, 123, 95, 96], which do not restrict

the location of unknown parameters but impose restrictions on the nonlinearities of the original system

4

as well as on those appearing in the transformed error system, and the uncertainty-constrained schemes

[121, 9, 54], which impose restrictions on the location of unknown parameters but can handle all

types of nonlinearities. Speci�cally, in the �rst category, as long as the norm of perturbing nonlinear

terms was dominated by an a�ne function, for all initial estimates lying in some open neighborhood

of the true values in the parameter space, global convergence results were obtained in [82] for pure-

feedback systems by updating estimates of both the feedback terms and the coordinate transformations

that were required to linearize the system. Sastry and Isidori [105] solved the problem of adaptive

asymptotic tracking of feedback linearizable minimum phase nonlinear systems (including pure-feedback

systems). Overparametrization was required and some restrictive assumptions on nonlinearities, such as

the change of coordinates being globally Lipschitz in terms of states, were made. An indirect scheme

(indirect adaptive control di�ers from direct adaptive control in that it relies on an observation error

to update the parameters rather than relying on the output error) was proposed in [123] to overcome

the overparametrization problem. The restrictive PE condition, an additional assumption required by

the indirect scheme, was then eliminated by a "semi-indirect" scheme [123], which combined parameter

estimation elements from both the direct and the indirect approaches. Global stabilization was achieved

in [95] for feedback stabilizable nonlinear systems, a larger class of nonlinear systems than feedback

linearizable nonlinear systems. For the uncertainty-constrained schemes, assuming that the matching

condition (loosely speaking, the matching condition implies that control and uncertainty enter the

system dynamics via the same channel) was satis�ed, a feedback control scheme was developed in [121]

for stable regulation of a class of nonlinear plants with parametric and dynamic uncertainties, and the

estimate of stability region was given. The matching condition was relaxed to the extended-matching

condition in [9, 54]. Praly, et al. [96] uni�ed and generalized most of the earlier results by introducing

a novel Lyapunov function for the design of direct schemes and by generalizing equation error �ltering

and regressor �ltering for the design of indirect schemes. The key assumption in this approach was that

a Lyapunov-like function existed and depended on unknown parameters in a particular way. Depending

on the properties of this function, various designs were possible, including feedback linearization designs

when this function was quadratic in the transformed coordinates. Output-feedback designs were studied

in [55, 71, 72].

It soon became clear that the "certainty-equivalence" adaptive controllers based on the feed-

back linearization technique were unable to achieve stability without restrictions on nonlinearities. New

thinking was needed for the systematic design of adaptive nonlinear controllers, resulting in the exciting

era of adaptive nonlinear control [65]. The new thinking employed a recursive design methodology

| backstepping. With this methodology, the construction of feedback control laws and associated

Lyapunov functions became systematic. Strong properties, such as global or regional stability and

tracking, were built into the nonlinear system in multiple steps, never higher than the system order.

In contrast to feedback linearization methods that required cancelation of all nonlinearities, the back-

stepping design avoided wasteful cancelations and retained useful nonlinearities. Backstepping designs

were exible and allowed a choice of design tools for dominating, or adapting to, uncertain nonlin-

earities. Speci�cally, Kanellakopoulos, et al. [56, 60] presented a systematic design of globally stable

and asymptotically tracking adaptive controllers for a class of nonlinear systems transformable to a

parametric strict-feedback canonical form (local results for parametric pure-feedback systems). The

number of overparametrization was reduced in half in [50], and the overparametrization problem was

soon eliminated by Krstic, et al. [62] by elegantly introducing the concept of tuning function. Recently,

the nonlinear damping was introduced by Kanellakopoulos [52, 53] to improve transient performance.

5

Generalization to output-feedback design was presented in [66, 67]. The nonlinear design method was

also applied to linear systems in [63, 64]. Compared to the previous traditional adaptive control schemes

for linear systems, which could not resolve the con ict between their linear form and their nonlinear

nature, the new nonlinear design achieved stronger stability and convergence properties with a much

more transparent and straightforward design procedure. These improvements o�ered new insights into

the �eld of adaptive control.

1.2.2 Deterministic Robust Control (DRC)

One of the earliest approaches to the control of uncertain systems was sliding mode control

(SMC) or variable structure control (VSC) [48, 127, 128, 129, 114, 174, 165, 166, 28, 90, 31, 143],

which was �rst studied in the Soviet Union in the 1960's [48] and was introduced to western researchers

by Utkin [127, 128, 129]. The central feature of SMC is sliding mode, in which the dynamic motion of

the system is e�ectively constrained to lie within a certain subspace of the full state space. The sliding

mode is achieved by altering the system dynamics along some sliding surfaces in the state space so

that the system state is �rst brought to these surfaces or their intersection surface and is made to stay

on them thereafter. During the sliding mode, the system is equivalent to an unforced system of lower

order, termed the equivalent system, which is insensitive to both parametric uncertainties and unknown

nonlinear functions when the matching condition is satis�ed.

The design of a SMC system consists of two stages. In the �rst stage, sliding surfaces are

selected so that the equivalent system is asymptotically stable and has a desired dynamic response.

This stage may be completed without any assumptions about the form of the control functions. The

static design of sliding surfaces was presented in [25] and dynamic sliding mode design was studied in

[143, 12, 167, 154]. In the second stage, a control law is determined depending on the speci�c plant and

the chosen sliding surfaces to ensure that the chosen sliding mode is attained. Among SMC schemes for

robot manipulators, there have been proposals to make each sliding surface attractive. This approach

makes the problem complicated, resulting in a control law de�ned implicitly by a set of fairly complicated

algebraic inequalities [165, 166, 106]. By exploiting the passivity of robot dynamics, other researchers

obtained simple control laws, which made the system state attracted to the intersection of the surfaces

without necessarily reaching each individual one [90, 161, 141, 117]. Recently, a dynamic sliding mode

controller, in which a dynamic compensator is introduced in forming the sliding surfaces, was employed

in [143] to ensure that the system achieved a desired second-order model to realize several control

purposes, such as impedance control, hybrid motion/force control, and constrained motion control.

Reaching transients were also eliminated so that the system was maintained in the sliding mode all the

time. Robust sliding mode control in the form of MIMO input-output (I/O) linearization was considered

by Fernandez and Hedrick in [28]. Hedrick, et al., applied SMC to the control of automotive engines

[79, 16, 37], aircraft ight control [40], electronics suspension control [1] and "platoon control" in

automated highway systems [41]. Observers based on SMC were discussed in [113].

One of the drawbacks of the SMC is that, in general, it only applies to the uncertain systems

which satisfy the matching condition. The most severe drawback of the SMC is that the control

law is discontinuous across sliding surfaces. Such control laws lead in practice to control chattering,

which involves high frequency control activity and may excite neglected high-frequency dynamics. To

remove control chattering, smoothing techniques, such as a boundary layer [106, 112], have to be

employed. However, such a modi�cation can guarantee the tracking error only within a prescribed

6

precision. Although transient performance is still preserved at large, asymptotic stability is lost and a

trade-o� exists between control bandwidth and tracking precision.

Another general deterministic robust control (DRC) technique has been developed based on

Lyapunov's second method originally by Leitmann, et al. [39, 69, 21]. For uncertain systems satisfying

the matching condition, a stabilizing discontinuous min-max control law was developed in [39, 38].

Like smoothed SMC, a continuous approximation of the min-max control law that guaranteed globally,

uniformly, ultimately bounded (GUUB) stability instead of asymptotic stability was presented in [22].

Although the matching condition is met in many important applications, such as mechanical systems,

it is still very restrictive. Subsequently, much e�ort has been devoted to loosening the restrictions

imposed by the matching condition. Two main approaches have been used to tackle this issue. The

�rst one studies the robustness of the controlled system against the mismatched uncertainty. In this

approach, the uncertainty is �rst decomposed into two categories, the matched and the mismatched.

The controller is designed assuming no mismatched uncertainty. A passive stability analysis is then made

for mismatched uncertainty. The framework was �rst introduced by Barmish and Leitmann [5] for linear

systems. Subsequent results were presented in [15, 13, 99]. Since this approach is based on the stability

margin of the stabilized nominal system, certain restrictions on the mismatched uncertainty have to be

made and the design procedure is not systematic. The second approach looks for a structural condition

under which a systematic robust control design may be applied. This approach imposes restrictions

on the location of uncertainty as in uncertainty-constrained adaptive nonlinear schemes. Along this

line, Thorp and Barmish [124] presented a robust control design for linear uncertain systems satisfying

a generalized matching condition. In extending the results to uncertain nonlinear systems, once, the

backstepping procedure played an important role. Marino and Tomei [73] solved the robust stabilization

problem of nonlinear systems with vanishing uncertainties and satisfying the strict feedback condition

( similar to the parametric-strict feedback condition). The case of nonvanishing uncertainties, which

allows bounded disturbances and tracking, was solved by Freeman and Kokotovic in [29] by extending

the results of [73]. A di�erent approach, multiple surface sliding mode control, was presented by Won

and Hedrick in [140]. The approach used a series of simple Lyapunov functions instead of the whole

Lyapunov function in the backstepping design and made each sliding surface attractive outside a user-

de�ned boundary layer thickness. Based on backstepping, Qu [97] presented the generalized matching

condition for nonlinear systems in a pure-feedback form.

1.3 Motivations and Contributions of the Dissertation

1.3.1 General Methodology

In spite of the recent rapid advances in adaptive nonlinear control, one problem remains un-

solved, i.e., unknown nonlinear functions have not been considered. All the adaptive nonlinear controllers

mentioned in Section 1.2.1 dealt with the ideal case of parametric uncertainties only. Nonlinearities of

the system were assumed known and unknown parameters were assumed to appear linearly with respect

to these known nonlinear functions. The integral adaptation laws developed for linear systems may

lose stability when even a small disturbance is present. Considering that every real system has some

sorts of disturbances, we wonder if we can safely implement such adaptive controllers. This is more

serious for nonlinear systems, as shown in [100] for the adaptive control of robot manipulators. As

in the adaptive control of linear systems, one may apply similar remedies to nonlinear systems. For

7

example, the adaptation law may be modi�ed to achieve stability for bounded disturbances [100]. How-

ever, such modi�cations do not guarantee tracking accuracy since the steady state tracking error can

only be shown to stay within an unknown ball, whose size depends on the disturbances Furthermore,

transient performance is unknown. In [94], by using a variant of the �-modi�cation and backstepping

procedure, Polycarpou and Ioannou presented a robust adaptive control design for a class of single input

single output (SISO) nonlinear systems in a "semi-strict" feedback form, which allowed both parametric

uncertainties and unknown nonlinear functions. However, transient performance was not guaranteed

and asymptotic stability was lost even in the presence of parametric uncertainties only.

Despite the above drawbacks of adaptive control, one should realize that the main advantage

of adaptive control lies in the fact that, through on-line parameter adaptation, parametric uncertainties

can be eliminated and, thus, asymptotic stability or zero �nal tracking error can be achieved in the

presence of parametric uncertainties without using high-gain feedback. New thinking should be adopted

to utilize this advantage judiciously.

On the other hand, the deterministic robust control (DRC) mentioned in Section 1.2.2 em-

ploys proper controller structures to attenuate the e�ect of the model uncertainties coming from both

parametric uncertainties and unknown nonlinear functions. In general, it can guarantee transient per-

formance and certain �nal tracking accuracy. However, DRC does not discriminate between parametric

uncertainties and unknown nonlinear functions and the control law uses �xed parameters. Model un-

certainties coming from parametric uncertainties cannot be reduced. In order to reduce tracking errors,

the feedback gains must be increased, resulting in high-gain feedback and increased bandwidths of

closed-loop systems. Theoretically, SMC can use discontinuous control laws and some of the so-called

continuous DRC schemes [98, 97] can use in�nite gain feedback control to achieve asymptotic track-

ing. However, those are impractical and unachievable solutions because of �nite bandwidths of physical

systems.

In view of the above drawbacks and advantages of both adaptive control (AC) and DRC,

this dissertation will propose a new approach, adaptive robust control (ARC). which uses both means

| proper controller structure and parameter adaptation | to reduce tracking errors. The DRC tech-

nique will be used to design a baseline control law (proper controller structure) to guarantee transient

performance and certain �nal tracking accuracy. On top of it, parameter adaptation will be used to

reduce the model uncertainties coming from parametric uncertainties (as in AC) and to improve track-

ing performance. In other words, the robust control problem is formulated under the general setting

of DRC, but the di�erence between parametric uncertainties and unknown nonlinear functions is rec-

ognized and parameter adaptation is used to reduce the parametric uncertainties. In general, DRC

design needs the modeling uncertainties to be bounded by some functions with known shapes but the

estimated parameters by AC design may be unbounded in the presence of unknown nonlinear functions.

By formulating the robust control problem under the general setting of DRC, when one designs either

the baseline robust control law or the parameter adaptation law, one always keeps in mind the above

con icts between the DRC design and the AC design and solves the con icts at the beginning. In such

a way, stronger stability results can be obtained. Such a formulation has several advantages: it can

naturally eliminate the transient problem and robustness problem of adaptive control, while at the same

time, improve the tracking performance of DRC by reducing model uncertainties. The qualitative results

obtained [154, 153, 152, 159, 157, 158, 156, 160] well re ect this philosophy. In general, in the presence

of both parametric uncertainties and unknown nonlinear functions, the same qualitative results as DRC

are achieved. Furthermore, if the model is accurate | i.e., in the presence of parametric uncertainties

8

only | asymptotic tracking is achieved without using high-gain feedback as in AC.

The above idea is simple and natural. In fact, during the past several years, some researchers

in the two �elds have been trying to achieve that goal. However, they all failed in one way or another.

Researchers in the robust adaptive control �eld [100, 94] tended to formulate the problem for parametric

uncertainties �rst and then to robustify the schemes. This approach inevitably complicated the problem

because it lost the whole picture and leaded to conservative results | only stability was achieved

and nothing could be obtained about performance. On the other hand, researchers in the DRC �eld

realized that parameter adaptation could reduce the control e�ort [107, 110, 35] but did not consider

its destabilizing e�ect and the main advantages of the AC and DRC methods. Thus, when parameter

adaptation was introduced in DRC design, as in the adaptive sliding mode control in [107], transient

performance was lost and a discontinuous control law had to be used, since the traditional proof in AC

was used. Furthermore, unlike the original SMC schemes for which smoothing techniques have been

developed for the discontinuous control law, the scheme in [107] cannot directly employ the smoothing

techniques since it is not robust to any approximation errors. This robustness problem was corrected

in [107] by stopping adaptation inside the boundary layer. However, transient performance was not

guaranteed and asymptotic stability could not be achieved in the presence of parametric uncertainties

only.

Finally, we would like to di�erentiate our algorithms from other adaptive robust control algo-

rithms that have appeared in the literature [14]. Instead of true parameter adaptation, those algorithms

in [14] used adaptation to adjust some of the feedback gains to achieve stability when the bounds of

modeling uncertainties were unknown. So, their main purpose was to relax the conditions under which

stabilization was possible. In general, those schemes do not provide better performance than their DRC

counterparts when the bounds of modeling uncertainties are known. By contrary, our algorithms use

true parameter adaptation to improve performance instead of relaxing the stabilizing conditions. These

claims are veri�ed by the experimental results shown in chapter 7.

1.3.2 General Form

The proposed ARC is formulated for general MIMO nonlinear systems in terms of the concept of

adaptive robust control (ARC) Lyapunov functions. The formulation reduces the ARC of a system to the

problem of �nding an ARC Lyapunov function for the system. By using backstepping design procedure,

we may successfully construct ARC Lyapunov functions for a class of MIMO nonlinear systems in a

semi-strict feedback form. The form is very general and includes mechanical systems, such as robot

manipulators. In the absence of unknown nonlinear functions, the form reduces to a parametric-strict

feedback form, which extends the parametric-strict feedback form used in general adaptive nonlinear

control [62, 65] in several ways. First, it is a MIMO version. A MIMO parametric-strict feedback form

was also presented in [65] but it allowed coupling among di�erent input channels of each layer at the

last step only. Second, the form allows parametric uncertainties at each layer's input channels also,

which increases the di�culty in the design of a pure adaptive control law considerably. Third, the last

layer's state equations do not have to be completely linearly parametrized (linear parametrization is a

requirement in [62, 65]). This extension is vital for applications, such as control of robot manipulators,

where the dynamics cannot be linearly parametrized in the state equations.

9

1.3.3 Applications

The proposed ARC is applied to the control of robot manipulators in several ways as explained

below.

Trajectory Tracking Control of Robot Manipulators

Industrial manipulators are commonly used in tasks such as painting, welding and material

handing. In these tasks, their end-e�ectors are required to move from one place to another in a free

workspace or to follow desired trajectories. In order to meet increased productivity requirement as well as

tight tolerance requirements, it is essential for the manipulator to follow a desired trajectory as close as

possible at fast speed. Thus, trajectory tracking control of robot manipulators is of practical signi�cance.

It is also the simplest but most fundamental task in robot control [81]. Because of these factors, during

the past decade, numerous adaptive algorithms and DRC algorithms have been proposed. In addition

to those schemes, there are also some adaptive schemes [17, 116, 115] termed as performance-based

(or direct) adaptive control in [18], in which adaptation laws are used to adjust controller gains instead

of true parameters. Thus, these gain-based schemes share the same properties as the adaptive robust

scheme in [14]. They are claimed to be simple, computationally e�cient and require very little model

information. Robustness to bounded disturbances is also guaranteed. However, they can only guarantee

tracking errors within certain bounds even when the system is subject to parameter uncertainties only.

Some comparative experiments were carried out in [138] to test some of the model-based

(or parameter-based) adaptive algorithms. However, the tested algorithms belonged to the same class.

Facing so many algorithms and so many qualitatively di�erent approaches, one has di�culty choosing a

suitable one for a particular application since each algorithm has its own claim. Thus, it is of practical

signi�cance to test qualitatively di�erent approaches on the same machine to understand their funda-

mental advantages and drawbacks. To work toward that direction, in addition to the proposed ARC,

several typical robust and adaptive control algorithms are also developed for comparison. Speci�cally,

two ARC schemes, one based on the conventional adaptation law structure [110] and one using the

idea of desired compensation adaptation law [103], are �rst developed by applying the proposed ARC.

Then, a very simple nonlinear PID scheme is proposed, which can guarantee the stability and requires

little model information. By adjusting the feedback gains on-line, a simple gain-based adaptive control

is also suggested to remove the requirements in choosing feedback gains in the nonlinear PID scheme.

By combining the design techniques of the gain-based adaptive control with the proposed ARC, a new

adaptive robust scheme is also proposed to remove the conditions on the selection of the controller

gains. Finally, all schemes, as well as two benchmark adaptive control schemes [110, 103], are imple-

mented and compared. Experimental results are presented to show the advantages and the drawbacks

of each method. Comparative experimental results show that importance of using both proper controller

structure and parameter adaptation in designing high-performance robust controllers. It is observed that

the proposed ARC achieves the best tracking performance in the experiments. Detailed conclusions are

given in chapter 7.

Constrained Motion and Force Control

Another important class of tasks requires the robot end-e�ector to make contact with its

environment. Typical examples of such tasks are contour following, grinding, scrubbing, deburring, as

10

well as those related to assembly and with multi-arm robot systems. In these applications, the contact

force between the end-e�ector and the environment is generated, which modi�es the dynamics of robot

manipulator and creates some problems that do not exist in the free motion of robot systems. Research

in this area has focused on simultaneous control of motion and force. Depending on the contact

environment, di�erent approaches [139, 143] have been proposed.

The �rst type of motion and force control considers the robot whose end-e�ector is in contact

with rigid surfaces [81, 132, 133, 74, 76, 163, 44, 162, 20, 146, 148, 147, 142, 168, 118, 45, 49, 108,

8, 70, 10]. In many cases the contact surface sti�ness is so large that the surfaces must, in practical

terms, be viewed as rigid. Such a view may be appropriate to prevent damage of either the workpiece

or the end-e�ector.

Typical example of constrained motion is contour following, in which the robot end-e�ector

is required to move along a very sti� or rigid contact surface. In the normal direction of the surface,

the end-e�ector's motion is restricted by the surface, and the robot can only move along the tangent

direction of the surface. Correspondingly, contact force exists in the normal direction of the surface and

no force but that of friction occurs along the tangent direction. This unique duality will be used in the

subsequent formulation of constrained motion.

When the robot moves on rigid surfaces, holonomic kinematic constraints are imposed on

the robot motion that correspond to some algebraic constraints among the manipulator state variables.

Dynamics of such a robot system is described by a set of nonlinear di�erential-algebraic equations, which

is called singular system [74] or descriptor system [76]. The objective is to control both the motion on

the constraint surfaces and the generalized constrained force.

A general theoretical framework of constrained motion control was rigorously developed by

McClamroch and Wang [74]. The proposed controller was based on a modi�cation of the computed

torque method. A Lyapunov's direct method was utilized by Wang and McClamroch [133, 132] to

develop a class of decentralized position and force controllers. Mill and Goldenberg [76] applied descriptor

theory to constrained motion control. The controller was derived based on a linearized dynamic model

of the manipulator. State feedback control and dynamic state feedback control were utilized to linearize

the robot dynamics with respect to motion and contact force in [163], and [168], respectively.

The above methods are based on the exact model of constrained robot dynamics. As in the

case of free motion, robust control methods are needed. There are many papers applying the two robust

control methods to constrained motion of robot manipulators: adaptive constrained motion control [10,

118, 49, 149, 2] for parametric uncertainties only, and SMCmotion and force control [148, 143, 142, 145].

Basically, adaptive constrained motion control methods proposed in [108, 10, 118, 49] are all based on

the reduced dynamic model proposed in [74], which enable motion and force controllers to be designed

separately. It should be noted that this model is only valid for frictionless contact surfaces, while most

real contact surfaces have friction. Furthermore, the previous parameter adaptation laws proposed are

only driven by motion tracking error. Thus, the force tracking error can be guaranteed to be only

bounded unless some persistent excitation conditions are satis�ed | these are di�cult to verify and

depend on speci�c desired motion trajectories. Although, theoretically, the force tracking error can be

made small by using a large proportional force feedback gain [10, 49], the gain for the proportional force

feedback is severely limited in applications because of the acausality that arises from the rigid body

dynamics assumed in the modeling of the robot [91]. In fact, recent one-dimensional force experimental

results presented by [130] and [91] suggest that the best force tracking performance is achieved by

integral (I) force feedback or PI force feedback control. Considering these factors, we propose a new

11

transformed constrained dynamic model that is suitable for controller design and is also valid for friction

surfaces with unknown friction coe�cients in [149, 155]. The resulting adaptive controller guarantees

asymptotic motion and force tracking without persistent excitation, and has the expected PI type force

feedback control structure with a low proportional force feedback gain.

It should be noted that all the above force controllers are synthesized based on the assumption

that the robot keeps contact with the surface when the controller is applied. This assumption is valid

only if the controller has good transient performance since, otherwise, drastic transient response may

cause the robot to lose contact with the surface, thus voiding the obtained result. Therefore, it is

important to design a motion and force controller with a guaranteed transient performance. This goal

is achieved by applying the proposed ARC and using our previous general formulation of constrained

motion in [149, 155]. Dynamic motion sliding mode and �ltered force tracking error are used to enhance

the dynamic response of the system. The suggested control law can achieve asymptotic motion and

force tracking without persistent excitation condition in the presence of parametric uncertainties, and

has a guaranteed transient performance with a prescribed �nal tracking accuracy in the presence of both

parametric uncertainties and external disturbances or modeling errors. Simulation results illustrate the

proposed motion and force controller.

Coordinated Control of Multiple Robot Manipulators

For assembly-related tasks, such as heavy material handling, several manipulators are required

to grasp a common object. In these applications, a set of homogeneous constraints are imposed on the

positions of the manipulators. As a result, degrees of freedom (DOF) of the whole system decrease, and

internal forces exerted on the object by the manipulators are generated. These internal forces do not

a�ect the motion of the object. To control the robots cooperatively, a number of control methods have

been proposed. In closed kinematic chain methods [120, 80, 42, 172, 173, 32, 146], only the position

of the whole system is controlled. Hence, the joint torque for a particular load of the object cannot

be uniquely determined and load distribution is required. In hybrid position/force control methods

[61, 125, 170, 137, 136, 126, 131], both position and internal force of the whole system are controlled.

The DOF lost in the arm con�guration from the imposed kinematic constraints is introduced as the

DOF needed to control the internal forces of the system [61]. This scheme is important especially when

the object is fragile or needs operations such as compression, tension, and torsion.

The problem of a constrained object grasped by multiple manipulators has been considered

in [169, 164, 43, 148]. The methods in [169, 164] were based on the exact model of the system, and

the adaptive law derived from Popov hyperstability theory [43] needs the measurements of acceleration

and force derivative. A set of transformed dynamic equations of the robotic system were obtained in

the joint space in [148], in which internal force and constrained force had the same form and could

be controlled in the same method. The VSC method was used to deal with the problem of parameter

uncertainties as well as external disturbances. However, the e�ect of friction force on the object was

not considered, and the transformation was basically formed by a position relationship but may not be

easily obtained.

In this dissertation, we apply the ARC to the robust control of motion, internal force, and

external contact force control of multiple manipulators handling a constrained object in the presence of

both parametric uncertainties and disturbances. Parametric uncertainties may exist in the manipulators

and in the object and in the friction coe�cients of contact surfaces. A set of transformed dynamic

12

equations are obtained in the joint space, in which internal force and external contact force have the

same form [150]. Thus, internal force control and external contact force control can be dealt with in

the same way as in constrained motion and force control. The resulting controller possesses those nice

properties mentioned in the above subsubsection.

Motion and Force Tracking Control of Robot Manipulators In Unknown Sti�ness

Environments

In addition to constrained motion, another important class of contact tasks is when the robot

end-e�ector comes in contact with surfaces that are not so rigid and can be modeled as sti�ness

environments. Typical examples include the deburring process. The objective in these applications is

the same as that in constrained motion | i.e., control of motion along the tangent direction of the

surface and control of force along the normal direction of the surface.

There are only a few published papers addressing motion and force tracking control in the

presence of unknown environmental sti�ness. Carelli, et. al. proposed an adaptive force control method

to estimate unknown parameters of the robot and the environmental sti�ness in [11]. The inertia matrix

of the robot is assumed to remain constant. Because of the highly nonlinear and coupled nature of

the robot dynamics and the wide working range of the robot, this assumption is usually not satis�ed.

Recently, a variable structure adaptive (VSA) method was developed by Yao, et. al. in [144] to solve this

problem. This method resulted in a two-loop control system. VSC method was used in the inner-loop

that forced the system to reach and be maintained on a dynamic sliding mode provided by the outer-loop

design. In the outer loop, the adaptive control method was used to estimate environmental sti�ness

and provide the system with good force tracking property. However, the resulting VSC control law

was inherently discontinuous and the associated chattering problem had not been analyzed. In [151],

we developed an adaptive motion and force control algorithm to eliminate the chattering problem.

However, transient performance was not guaranteed when disturbances appeared. The e�ect of time-

varying equilibrium position was not considered.

In this dissertation, we show that motion and force tracking control of such a system falls

nicely into the proposed semi-strict feedback form with a relative degree two. The proposed ARC is

applied and the resulting controller needs measurements of position, velocity and interaction force only.

Transient performance is also guaranteed when disturbances appear.

1.4 Outline of the Dissertation

The dissertation is organized into two parts. Part one deals with general theory and Part two

talks about applications. Part one consists of the following chapters:

� Chapter 2 uses a simple �rst-order system to illustrate the general idea of the proposed ARC. An

adaptive controller and a DRC controller are also constructed for comparison.

� Chapter 3 generalizes the proposed ARC to a class of SISO nonlinear systems with arbitrary known

relative degrees and transformable to a semi-strict feedback form.

� Chapter 4 introduces the concept of ARC Lyapunov functions and presents a general framework

of ARC for general nonlinear systems via ARC Lyapunov functions.

13

� Chapter 5 talks about the systematic construction of ARC Lyapunov functions via the backstepping

design procedure.

� Chapter 6 solves the ARC of a class of MIMO nonlinear systems with arbitrary known relative

degrees and transformable to a semi-strict feedback form.

Part two consists of the following chapters:

� Chapter 7 applies the proposed ARC to trajectory tracking control of robot manipulators. Several

conceptually di�erent adaptive and robust control algorithms are also developed for comparison.

Comparative experimental results on a SCARA robot are presented.

� Chapter 8 applies the proposed ARC to constrained motion and force control, coordinated control

of multiple manipulators, and motion and force control of robot manipulators in unknown sti�ness

environments.

Finally, Chapter 9 concludes the dissertation and discusses possible future research directions.

14

Part I

Adaptive Robust Control - Theory

15

Chapter 2

Control of a First-order Uncertain

System

In this chapter, we consider the tracking control of a �rst-order nonlinear system to illustrate

the basic ideas of the proposed adaptive robust control (ARC) scheme. The results can also be used

later in the backstepping design for general nonlinear systems.

The �rst-order nonlinear system under consideration is described by

_x = f(x; t) + u x; u 2 R (2.1)

where u is the control input. Normally, it is very hard to determine the exact form of the nonlinear

function f(x; t). In this chapter, we describe it in two parts. The �rst part represents all the terms

that can be modeled and linearly parametrized: i.e., this part normally represents the terms derived by

physical laws or certain kinds of approximation and its form or base shape is usually available but its

magnitude may not be known in advance. For the �rst order system (2.1), it is assumed to be described

by ��(x; t), where �(x; t) is a known shape function and � is an unknown magnitude parameter. The

second part is used to represent terms that cannot be modeled or linearly parametrized as well as those

which may be due to external disturbances and modeling simpli�cations, such as neglecting Columb

friction. This part is denoted by �(x; t). Therefore,

f(x; t) = �'(x; t) + �(x; t) (2.2)

For controller design, it is necessary to make some reasonable assumptions about the prior knowledge of

the plant. The more we know about the plant | i.e., the more strict the assumptions are | the better

the nominal performance of the resulting controller will likely be. However, if the assumptions are too

strict, the actual plant may not satisfy them and, thus, the obtained nominal performance may likely be

useless. Although the exact value of the parameter � and the modeling error�(x; t) may not be known,

the extent of the parametric uncertainty and modeled errors can often be predicted in advance. For

example, when a robot picks up an object, although we may not know the exact mass property of the

object, we know the maximum payload the robot is going to pick up. Thus, we can make the following

reasonable and practical assumptions that � and �(x; t) are bounded by some known parameters or

known functions, i.e.,

� 2 ��= (�min ; �max)

j�(x; t)j � �(x; t)(2.3)

16

where �min, �max, and �(x; t) are the known scalars and the known function respectively. In this

dissertation, all functions involved in the design are assumed to be bounded with respect to (w.r.t.)

time t (e.g., for �(x; t), there exists a function �p(x) such that 8t; j�(x; t)j � �p(x)), and have �nite

value when all their variables except t are �nite (e.g., x 2 L1 =) �(x; t) 2 L1 ).

Let xd(t) be the desired output, which is assumed to be bounded with bounded derivatives

up to a su�cient order. The control problem can be formulated as that of designing a control law for u

such that, under assumption (2.3), the system is either globally, ultimately, uniformly bounded (GUUB)

stable or asymptotically stable, and the output x tracks xd(t).

To illustrate what we want to do, two popularly used nonlinear synthesis methods, deterministic

robust control (e.g. sliding mode control) and adaptive control, are �rst applied. After that, the proposed

new adaptive robust control is naturally introduced by e�ectively combining the two methods.

2.1 Deterministic Robust Control (DRC)

Since the system (2.1) has relative degree one, sliding mode control (SMC) can be applied. A

dynamic sliding mode is employed to enhance the dynamic response of the system in sliding mode and

eliminate the unpleasant reaching transient [154].

Let a dynamic compensator be

_xc = Acxc +Bce xc 2 Rnc Ac 2 Rnc�nc Bc 2 Rnc

yc = Ccxc yc 2 R Cc 2 R1�nc (2.4)

where e = x � xd(t) is the tracking error and constant matrices (Ac; Bc; Cc) are chosen to ensure

that the resulting dynamic sliding mode exhibits the desired dynamics. (Ac; Bc; Cc) is controllable and

observable. The sliding mode controller is designed to make the following quantity remain zero.

z = e+ yc

= x� xr xr�= xd(t)� yc

(2.5)

Transfer function from z to e is

e = G�1z (s)z (2.6)

where

Gz(s) = 1 +Gc(s) Gc(s) = Cc(sInc � Ac)�1Bc (2.7)

and In represents an n � n identity matrix. From (2.7), G�1z (s) can be arbitrarily assigned by suitably

choosing dynamic compensator transfer function Gc(s) as long as G�1z (s) has relative degree zero.

During the sliding mode, z = 0 and the system response is governed by the free response of transfer

function G�1z (s). Therefore, as long as G�1z (s) is stable, the resulting dynamic sliding mode will be

stable and is invariant to various modeling errors. Furthermore, the sliding mode can be arbitrarily

shaped to possess any exponentially fast converging rate since poles of G�1z (s) can be freely assigned.

In addition to these results, G�1z (s) can be chosen to minimize the e�ect of z on e when ideal sliding

mode fz = 0g cannot be exactly achieved in practice.

The rest of the design is to construct a control law such that the sliding mode is reached. The

control law is suggested asu = uf + usuf = _xr(t)� �'(x; t)

us = �kz � h(x; t)sgn(z)(2.8)

17

where � 2 � is the estimate of �, sgn(:) denotes the discontinuous sign function de�ned as sgn(z) = 1

if z > 0 and sgn(z) = �1 if z < 0, and h(x; t) is any known bounding function satisfying

h(x; t) � j � ~�'(x; t) + �(x; t)j 8� 2 � (2.9)

where ~��= � � � is the estimation error. The required known function h(x; t) exists since the extent of

uncertainties is known. For example, let

h(x; t) = (�max � �min)j'(x; t)j+ �(x; t) (2.10)

h(x; t) can also be chosen in other ways to simplify the on-line computation time.

Theorem 1 The control law (2.8) guarantees that the system (2.1) is exponentially stable and its

output tracks the desired trajectory asymptotically. 4Proof. Choose a positive de�nite (p.d.) function as

Vs =12z

2 (2.11)

From (2.1), (2.5), (2.8) and (2.9), its time derivative is

_Vs = z _z = z[�~�'(x; t) + �(x; t) + us]

� jzjj � ~�'(x; t) + �(x; t)j � kz2 � h(x; t)jzj� �kz2 = �2kVs

(2.12)

Therefore,

Vs(t) � exp(�2kt)Vs(0) (2.13)

which implies that z exponentially decays to zero. This result leads to the theorem 1 since the sliding

mode is exponentially stable. 4Corollary 1 If the initial value xc(0) of the dynamic compensator (2.4) can be chosen to satisfy

Ccxc(0) = �e(0) (2.14)

then the system is maintained in the sliding mode all the time and reaching transient is eliminated, i.e.,

z(t) = 0; 8t. 4

Proof. If (2.14) is satis�ed, z(0) = 0 and Vs(0) = 0. From (2.13), z(t) = 0; 8t. 4Usually, the control law (2.8) is discontinuous across the sliding surface since it contains

sgn(z). Such a control law leads to control chattering in practice. To overcome this problem, the

above ideal SMC law can be smoothed by replacing discontinuous robust control term h sgn(z) by a

continuous function �h (h sgn(z)). �h (h sgn(z)) is required to satisfy the following two conditions:

i: z �h (h sgn(z)) � 0

ii: hjzj � z�h (h sgn(z)) � "(t)(2.15)

where "(t) is any bounded time-varying positive scalar, i.e., 0 < "(t) � "max. "(t) is used to measure

the approximation error. The SMC law is thus smoothed to

u = uf + us us = �kz � �h (h sgn(z)) (2.16)

where uf is the same as before.

18

Theorem 2 If the smoothed SMC control law (2.16) is applied, the system is exponentially stable at

large with a guaranteed transient performance and �nal tracking accuracy. 4Proof. From (2.1) and (2.16), error dynamics is given by

_z + kz + �h (h sgn(z)) = �~�'(x; t) + �(x; t) (2.17)

Following the same steps as in (2.12) and noting (2.15), the time derivative of Vs is now given by

_Vs � jzjj � ~�'(x; t) + �(x; t)j � kz2 � z�h (h sgn(z))

� �kz2 + hjzj � z�h (h sgn(z)) � �2kVs + "(t)(2.18)

soVs(t) � exp(�2kt)Vs(0) +

R t0 exp(�2k(t� �)"(�)d�

� exp(�2kt)Vs(0) + "max

2k [1� exp(�2kt)] (2.19)

This implies that the system is exponentially stable at large with a guaranteed transient performance

and �nal tracking accuracy since, theoretically, both the exponentially decaying rate 2k and the bound

of the �nal tracking error z(1),q

"max

k , can be freely adjusted by the controller parameters k and ".

4Remark 1 Two examples of the required continuous function �h (h sgn(z)) are as follows.

� Continuous Modi�cation (CM).

First, as in most smoothed SMC schemes [112, 154], we use the continuous saturation function

sat�

z�z

�to replace sgn(z). In order to take into account the time-varying nature of h, the

strength of the discontinuity, we use a time-varying boundary layer thickness given by �z =4"

h+"h,

where "h is any small positive number to avoid the possible singularity in case that h = 0. Thus,

�h (h sgn(z)) = h(x; t) sat�(h+"h)z

4"

�(2.20)

Obviously, (2.20) satis�es condition i of (2.15). When jzj � �z , we have, hjzj � zhsat�

z�z

�= 0.

When jzj � �z , we have

hjzj � zhsat�

z�z

�� hjzj � h2z2

4" = 1" [�(12hjzj � ")2 + "2] � " (2.21)

Thus, condition ii of (2.15) is satis�ed.

� Smooth Modi�cation (SM) .

Later, when we extend the methodology from relative degree one systems to general relative degree

n systems, we will use the backstepping design procedure, which recursively requires the derivatives

of the control components at each step. In such a case, a smooth modi�cation is preferred. For

this purpose, similar to [94], a smooth approximation of sgn(:) by tanh(:) function is used by

considering the following nice properties of tanh(:):

tanh(0) = 0 tanh(1) = 1 tanh(�1) = �10 � juj � u tanh( u"z ) � �"z 8u 2 R and "z > 0

(2.22)

where � = 0:2785. Letting "z ="�h , we have

�h (h sgn(z)) = h(x; t) tanh��h(x;t) z

"(t)

�(2.23)

Noting (2.22), (2.23) satis�es the conditions i and ii of (2.15). 4

19

2.2 Adaptive Control (AC)

In this subsection, the conventional AC [112, 85] is applied. The adaptive control is formulated

for parametric uncertainties only, i.e., for the case where �(x; t) = 0.

Let the control law be

u = ufa + usaufa = _xd(t)� �'(x; t); usa = �ke (2.24)

with � updated on-line by_� = '(x; t)e (2.25)

Theorem 3 In presence of parametric uncertainties only (� = 0), if the adaptive control law (2.24)

with the update law (2.25) is applied, the system output follows the desired output asymptotically, i.e.,

e �! 0 when t �! 1.

Additionally, if the desired trajectory satis�es the following persistent excitation (PE) condition

R t+Tt j'(xd(�); �)j2d� � "p 8t � t0 (2.26)

where T; t0 and "p are some positive scalars, then, the estimated parameter � converges to its true value

(i.e., ~� �! 0 when t �! 1). 4

Proof: Substituting (2.24) into (2.1), the error dynamics is

_e+ ke = �~�'(x; t) (2.27)

The time derivative of the positive de�nite (p.d.) function

Va =12e

2 + 12 ~�2 (2.28)

is_Va = e[�~�'(x; t)� ke] + 1

~�_� = �ke2 (2.29)

which implies that e 2 L2 \L1 and ~� 2 L1. From (2.27), _e 2 L1 and thus e is uniformly continuous.

By Barbalat's lemma [112], e �! 0 1 and asymptotic tracking is achieved. Furthermore, from (2.27),

since all terms except _e are uniformly continuous, _e is uniformly continuous. Applying Barbalat's lemma

again, _e �! 0. From (2.27), ~�'(x; t) �! 0. Thus, the PE condition (2.26) will guarantee that ~� �! 0.

4

2.3 Adaptive Robust Control (ARC)

The advantage of the adaptive control in section 2.2 is that, through on-line parameter adap-

tation, it can reduce the model uncertainty ~�� ( in fact, ~�� �! 0). Thus, we can obtain asymptotic

stability or a zero steady state tracking error without using high gain feedback (asymptotic stability

is achieved for any gain k). However, there are two main drawbacks. First, transient performance of

the system is not clear. Second, unknown nonlinear functions, such as external disturbance, are not

1For a vector �, which is a function of time, � �! 0 denotes the asymptotic convergence of �

20

considered, and it is well known that the integral type adaptation law (2.25) may su�er from param-

eter drifting and destabilize the system in the presence of even a small disturbance and measurement

noise[100] when certain PE conditions are not satis�ed. Considering that every real system is always

subjected to some sorts of disturbances, it is natural to wonder if the above adaptive controller can be

safely implemented. As contrast to adaptive control, transient performance and �nal tracking accuracy

are guaranteed in the smooth SMC design in section 2.1 for both parametric uncertainties and external

disturbances. This result makes the SMC design attractive for applications. From (2.17), we can see

that the SMC reduces the tracking error by attenuating the e�ect of modeling uncertainties (the left side

of (2.17) can be considered as a nonlinear �lter and the right side represents modeling uncertainties). In

order to reduce the tracking error, we have to use large feedback gains, i.e., large k or small ". However,

since the bandwidth of any real system is limited, there will be a practical upper bound on the feedback

gains that we can use. This fact limits the tracking accuracy that DRC can achieve in practice although

theoretically it can achieve arbitrarily small �nal tracking errors. For any chosen feedback gains, from

(2.17), the real tracking error is proportional to the modeling uncertainty, �~�'+ �. Therefore, if we

can introduce parameter adaptation in the DRC design to reduce the modeling uncertainty coming from

the parametric uncertainty, ~�', as in adaptive control, we may further improve the tracking accuracy.

This is the rationale of the proposed adaptive robust control (ARC).

The proposed ARC is to combine the design methodologies of DRC and AC to keep the

advantages of the two methods while overcoming the previously mentioned drawbacks. In other words,

we will try to use both means | proper controller structure and parameter adaptation | to reduce

the tracking error. The way to do so is to use the DRC technique to design a baseline controller

(proper controller structure) to guarantee transient performance and prescribed �nal tracking accuracy

for both parametric uncertainties and disturbances. On top of it, we will also use the adaptive control

technique to update the parameters to obtain asymptotic output tracking in the presence of parametric

uncertainties. To do so, we have to solve the con icts between the two design methodologies. DRC

requires knowledge of the bounds of modeling uncertainties, but the estimated parameters by AC may not

be bounded in the presence of unknown nonlinear functions. Thus, we have to modify the conventional

adaptation law in such a way that it can guarantee that the estimated parameters stay in a prescribed

uncertainty range all the time even in the presence of unknown nonlinear functions. Such a modi�cation

should not damage the correct estimation process for parametric uncertainties. In this dissertation, this

modi�cation is achieved by generalizing the smooth projection used in [122].

Let "� be an arbitrarily small positive real number. There exists a real-valued, su�ciently

smooth nondecreasing function � (Fig.2.1) de�ned by

�(�) = � 8� 2 �

�(�) 2 ��= [�min � "� ; �max + "�] 8� 2 R (2.30)

with bounded derivatives up to order n � 1: i.e., there exist constants, c�i > 0; i = 1; : : : ; n� 1, such

that

j did�i�(�)j � c�i 8� 2 R; i = 1; : : : ; n� 1 (2.31)

Let

V�(~�; �) =1

R ~�0 (�(� + �)� �)d� > 0 (2.32)

21

θ

)

θimin imax

imaxθ

θimin

εθ

θε

ν

π (ν

Figure 2.1: Nondecreasing n-th smooth projection map

From assumption (2.3) and (2.30), �(� + �) � � is a nondecreasing function of � that passes through

the origin (�(0 + �)� � = 0). Thus, V�(~�; �) is positive de�nite w.r.t. ~�. Furthermore,

@@~�V�(~�; �) =

1 (�(�)� �) (2.33)

which will later be used in the stability analysis.

The suggested adaptive robust control law has the same structure as the smoothed SMC

control (2.16) but with a projected parameter, ���= �(�), instead of a �xed estimate. It is given by

u = uf + usuf = _xr(t)� ��'(x; t)us = �kz � �h (h(x; t)sgn(z))

(2.34)

where, similar to (2.9), h(x; t) is any function satisfying

h(x; t) � j � (�� � �)'(x; t) + �(x; t)j 8�� 2 �

(2.35)

For example, let h(x; t) be

h(x; t) = (�max � �min + "�)j'(x; t)j+ �(x; t) (2.36)

� is updated on-line by the following adaptation law

_� = '(x; t)z (2.37)

Theorem 4 If the control law (2.34) with (2.35) and (2.37) is applied to the system described by (2.1),

the following results hold:

22

A. In the presence of both parametric uncertainties and unknown nonlinear functions, the control input

is bounded and Vs is bounded above by (2.19), in which the exponential converging rate and the

bound of �nal tracking error can be freely adjusted by the controller parameters in a known form.

B. In the presence of parametric uncertainties only, (i.e., �(x; t) = 0), in addition to the result in A,

the system output tracks the desired output asymptotically. Furthermore, if the PE condition

(2.26) is satis�ed, the estimated parameter converges to its true value. 4

Proof. From (2.1) and (2.34), the error dynamics is given by

_z + kz + �h (hsgn(z)) = �~��'(x; t) + �(x; t) (2.38)

where ~���= �� � � is bounded for any �, which is an important property used later in the proof.

In view of the similarity between the error dynamics (2.17) and (2.38) and the choice of the

function h(x; t) by (2.35), the time derivative of Vs can be described by (2.18) with ~� replaced by~�� . Eq. (2.19) is still valid and the control input (2.34) is bounded. This fact proves the result A in

Theorem 4.

When �(x; t) = 0, noting (2.33), (2.38), and condition i of (2.15), the time derivative of the

p.d. function Vt = Vs + V� is

_Vt = z _z + 1 ~��_�

= z[�~��'(x; t)� kz � �h (hsgn(z))] + ~��'(x; t)z

= �kz2 � z�h (hsgn(z)) � �kz2(2.39)

which implies that z 2 L2 \ L1. From (2.38), _z 2 L1, which leads to z �! 0 by Barbalat's lemma.

Similar to the proof for the adaptive controller, the PE condition (2.26) will guarantee the convergence

of the estimated parameter, which leads to the result B in Theorem 4. 4

Remark 2 The above theorem shows that the proposed ARC retains the results of both DRC and AC.

This fact naturally eliminates the drawbacks of each of the two methods. The main drawbacks of AC

| the transient problem and the non-robustness to the unknown nonlinear functions | are overcome

by A in the above theorem. The drawback of DRC | large �nal tracking errors | is overcome by

the improved performance in B. Therefore, the control law e�ectively combines the DRC design with

the AC design and achieves the expected goal. The analysis is qualitatively di�erent from the adaptive

robust control [100] for bounded disturbances in that not only is robustness obtained for a more general

class of disturbances but performance robustness is also guaranteed by the suggested controller | i.e.,

arbitrarily fast exponential convergence can be provided and the �nal tracking error can be adjusted by

the controller parameters independent of the magnitude of disturbances. }

Remark 3 One of the good features of the propose ARC is that its underline control law is a DRC type

robust control law. The adaptation loop can be switched o� at any time and the resulting control law

is a DRC law. The result in A of the above theorem is still valid in such a case. }

Remark 4 In general, if we choose "(t) as a time-varying positive scalar converging to zero, i.e.,

"(1) = 0 (or exponentially converging to zero, i.e., "(t) � "maxexp(��"t) for some �" > 0 and

"max > 0), from the �rst inequality of (2.19), Vs converges to zero asymptotically (or exponentially).

23

Thus, asymptotic output tracking (or exponential output tracking) can be obtained even in the presence

of unknown nonlinear functions. The same is true for the smoothed SMC law (2.16) and the following

analysis is also applied to it. Notice that although the control laws (2.34) are continuous for any �nite

time t, they tend to the ideal SMC law (2.8) as t �! 1 (in�nite gain feedback). Therefore, control

chattering will appear when t �! 1 and it is not surprising to see that the ideal performance of SMC

law is obtained. This result corresponds to some of the continuous robust control techniques (e.g., in

[98]). However, to truly remove control chattering, a large " and a small k have to be chosen to avoid

very high gain caused by the limited bandwidth of the system in practice. Within the allowable limit

in which control chattering is not excited, however, the larger the e�ective gain, the smaller the �nal

tracking error. Since the system under consideration is nonlinear, it is not obvious how to choose k

and "(t). Here, we roughly analyze the system in the following way. Normally, the dynamics around

the sliding mode | i.e., dynamics about z given by (2.38) | is the fastest, and the system tracks the

desired trajectory closely around the sliding mode, fz = 0g. This is true especially in the case when the

initial tracking error is zero. Therefore, we can assume x � xd(t) and treat the right hand side of (2.38)

as slowly varying disturbances on the fast �rst-order dynamics about z with the e�ective proportional

gain @@z [kz+�h (h sgn(z))] jz=0;x=xd . For the SM (2.23), the e�ective gain is k+ �h(xd(t);t)

2

"(t) . Suppose

that the allowable limit is ka. Then we can choose a time varying " as

"(t) = �h(xd(t);t)2

ka�k (2.40)

so that the e�ective gain is at the allowable limit all the time to minimize the tracking error. A similar

idea is used later in the experiments [154] to reduce the output tracking error. }

24

Chapter 3

Adaptive Robust Control of SISO

Nonlinear Systems in a Semi-Strict

Feedback form

In chapter 2, we presented the adaptive robust control (ARC) technique for a simple �rst-order

nonlinear system (relative degree one). In this chapter, the ARC technique will be generalized to a class

of SISO nonlinear systems with arbitrary known relative degrees and transformable to a semi-strict-

feedback form. This generalization is achieved by combining the general deterministic robust control

design technique with the well-known adaptive algorithms in [62, 59, 52] that were originally developed

for SISO nonlinear systems in a parametric strict-feedback form.

3.1 Problem Formulation

We consider the SISO nonlinear system transformable to the following semi-strict-feedback

form_xi = xi+1 + �T'i(x1; : : : ; xi; t) + �i(x; t) 1 � i � n � 1

_xn = �(x)u+ �T'n(x; t) + �n(x; t)

y = x1

(3.1)

where x = [x1; : : : ; xn]T . 'i(x1; : : : ; xi; t) 2 Rp; i = 1; : : : ; n; are the known shape functions, which

are assumed to be su�ciently smooth and, similar to (2.3), � = [�1; : : : ; �p]T 2 Rp and �i(x; t) are the

vector of unknown constant parameters and unknown nonlinear functions, respectively. For simplicity,

denote �xi = [x1; : : : ; xi]T (in general, �i;j denotes the j-th element of �i, ��i denotes [�T1 ; : : : ; �Ti ]T ).

� and �i's are assumed to satisfy

� 2 ��= f� : �min < � < �max; g

j�i(x; t)j � �i(�xi; t) i = 1; : : : ; n(3.2)

where �min = [�1min; : : : ; �pmin]T 2 Rp, �max = [�1max; : : : ; �pmax]T 2 Rp, and �i(�xi; t)'s are known1. The operation < for two vectors is performed in terms of the corresponding elements of the vectors

(e.g., �min < � means that �jmin < �j ; 8j).1If � is a vector or matrix with elements being functions, � is said to be known if its elements are known functions

of their variables

25

When �i(x; t) = 0; 8i, i.e., in the absence of unknown nonlinear functions, if we treat xi+1as the control input of the _xi dynamics, the _xi dynamics depends only on the states of its previous

dynamics, i.e., x1; : : : ; xi. In other words, only the feedback signals determine the dynamics. Such a

form is called strict-feedback form and is studied in [62, 59, 52]. Eq. (3.1) is called a semi-strict-

feedback form in that the bounding function �i(�xi; t) is required to be the function of xj ; j � i and t

only, but �i(x; t) may contain some bounded functions of xj ; j > i, thus violating the strict-feedback

property. Some examples of (3.1) can be found in [94].

Let yd(t) be the desired output trajectory, which is assumed to be bounded with bounded

derivatives up to n-th order. The control problem is stated as that of designing a bounded control law

for the input u such that, under the assumption of (3.2), the system is stable and the output y tracks

yd(t) asymptotically or in a GUUB fashion.

3.2 Smooth Projection and Positive De�nite Function V�

To begin the controller design, we would like to de�ne the multi-variable counterpart of the

smooth projection (2.30) �rst. De�ne a smooth projection � : Rp ! Rp by

�(�) = [�1(�1); : : : ; �p(�p)]T (3.3)

for each vector � 2 Rp with � = [�1; : : : ; �p]T , in which each �i : R ! R is de�ned by (2.30) with

�i = [�imin; �imax] and �i= [�imin � "�i; �imax + "�i]. Its j-th derivative is de�ned by �(j)(�) =

[ dj

d�j1�1(�1); : : : ;

dj

d�jp�p(�p)]

T . Thus,

�(�) = � 8� 2 �

�(�) 2 ��= f� : �i 2 �i 8ig 8� 2 Rp

�(j)(�) 2 �j�= f� : j�ij � c�ijg 8� 2 Rp j � n

(3.4)

where � and �j are compact sets and � is known. Let �� be the smooth projection of �, the estimate

of �, de�ned by �� = �(�), and ��(i)� = [�(�); : : : ; �(i)(�)]T . De�ne ~� = � � �, ~�� = �� � �, and

V�(~�; �) =Pp

i=11 i

R ~�i0 (�i(�i + �i)� �i)d�i i > 0 (3.5)

From (3.2) and (3.4), V�(~�; �) is positive de�nite w.r.t ~� for each � 2 �. Furthermore,

@@~�V�(~�; �) = [ 1 1 (�1(�1)� �1); : : : ; 1

p(�p(�p)� �p)] = ~�� T��1 (3.6)

where � = diagf 1; : : : ; pg

3.3 Backstepping Design Procedure

The design follows the recursive backstepping procedure in [62, 59, 52], which proceeds in the

following steps.

26

3.3.1 Step 1

Let ~�1(x; t) = �1(x; t). The �rst equation of (3.1) can be rewritten as

_x1 = x2 + �T'1(x1; t) + ~�1(x; t) (3.7)

In (3.7), by viewing x2 as a virtual control, we can design for it a control law �1 such that x1 tracks

its desired trajectory x1d(t). This design can be done by the ARC method presented in Chapter 2 by

noticing that j ~�1(x; t)j � ~�(x1; t)�= �(x1; t). From (2.34), the control law �1 is given by 2.

�1(z1; ��; t) = �1f + �1s�1f(z1; ��; t) = _x1d(t)� �� T'1(z1 + x1d(t); t)

�1s(z1; ��; t) = �k1z1 � h1(z1; ��; t) tanh

��h1(z1;��;t)z1

"1(t)

� (3.8)

where x1 = z1 + x1d(t) has been used and h1 is any function with continuous partial derivatives up to

(n� 1)-order. h1 satis�es

h1(z1; ��; t) � sup�2� j � ~��T'1(z1 + x1d; t) + ~�(x; t)j (3.9)

which is possible since � is a known compact set and the bounding function of ~� is known. For

example, let h1 be any su�ciently smooth function satisfying

h1(z1; ��; t) �Pp

j=1 �jM j'1j(z1 + x1d; t)j+ ~�1(z1 + x1d; t) (3.10)

where �jM = �jmax � �jmin + "�j .

Remark 5 An easy way to obtain a smooth h1 is to use (3.10). Since the right hand side of (3.10) is

continuous but may not be su�ciently smooth because of the non-di�erentiability of absolute operator

j � j at the origin, h1 can be chosen as equal to the right hand side of (3.10) by replacing operator j � jby any su�ciently smooth operation As(�) satisfying As(�) � j � j ; 8� 2 R. For example, a simple

smooth operator As is given by

As(�) =p�2 + r 8� 2 R (3.11)

where r is any positive scalar. }

The same as in [62], if x2 were the real control input, then the adaptation law would be given

by (2.37) and the design would be �nished. Since it is not the case, we postpone the choice of the

adaptation law and use the �rst tuning function [62]

�1(z1; t) = �1(z1; t)z1 �1(z1; t)�= '1(z1 + x1d(t); t) �1 2 Rp (3.12)

to denote the essential part of the adaptation law (2.37). De�ne the di�erence between the actual value

of x2 and its desired value �1 in (3.8) to be the second error variable

z2 = x2 � �1(z1; ��; t) (3.13)

2For simplicity, dynamic compensator (2.4) is dropped o� | i.e., sliding surface z1 reduces to z1 = x1 � x1d(t).The following design procedure will still be valid if a dynamic compensator is added. Furthermore, the smoothmodi�cation (2.23) is used.

27

Substituting (3.8) and (3.13) into (3.7), the �rst error subsystem S1 becomes

_z1 + k1z1 + h1 tanh��h1 z1"1(t)

�= z2 � ~�T� �1(z1; t) + ~�1(x; t) (3.14)

Choose Vs1 =12z

21 . From (3.14), its time derivative is

_Vs1 = z1z2 � [k1z21 + h1z1 tanh��h1 z1"1(t)

�� ~�1(x; t)z1]� ~�T� �1(z1; t) (3.15)

3.3.2 Step 2

From (3.1) and (3.13), noticing (3.8) and (3.14), we have

_z2 = x3 + �T'2(�x2; t) + �2(x; t)� [@�1@z1_z1 +

@�1@�

_� + @�1

@t ]

= x3 + (�� � ~��)T'2(�x2; t) + �2(x; t)

�@�1@z1

[�1s + z2 � ~�T�'1(z1 + x1d; t) + ~�1(x; t)]� @�1@�

_� � @�1

@t

= x3 � �2f � �02s � ~��T�2 + ~�2(x; t)� p2( _� � ��2)

(3.16)

in which by treating x1 and x2 as functions of z1; z2; � and t, i.e, x1 = z1 + x1d(t) and x2 = z2 +

�1(z1; �(�); t), and noticing that � appears in �1 only in the form of �(�), we have de�ned the functions

�2f , �02s, �2,

~�2(x; t) and p2 as

�2f (�z2; �(�); t) = ��(�)'2(�x2; t) + @�1f@t

�02s(�z2; ��(1)� ; t) = @�1

@z1[�1s + z2] +

@�1s@t + @�1

@���2

�2(�z2; �(�); t) = '2(x1; x2; t)� @�1@z1

�1(z1; t)~�2(x; t) = �2(x; t)� @�1

@z1~�1(x; t)

p2(z1; ��(1)� ; t) = @�1

@�(z1; �(�); �

(1)(�); t)

(3.17)

where �(1)(�) has appeared in the above expressions because of the term @�1@�

. �2(�z2; �(�); t) is the

second tuning function, which will be de�ned later. From (3.2) and (3.17),

j ~�2(x; t)j � ~�2(�z2; ��; t)�= �2(�x2; t) + j@�1@z1

j�1(x1; t) (3.18)

Similar to (3.9) and (3.10), there exists a known function h2(�z2; ��; t) with continuous partial

derivatives up to (n� 2) -order such that

h2(�z2; ��; t) � �TM j�2(�z2; ��; t)j+ ~�2(�z2; ��; t) � j � ~�� T�2 + ~�2(x; t))j (3.19)

where �M = [�1M ; : : : ; �pM ]T , �iM is de�ned in (3.10), and j � j is in the sense of element operation

if � is a vector or matrix. Since x3 is the virtual control for (3.16), let z3 = x3 � �2, where �2 is the

desired control law for x3, which will be speci�ed later. Consider the augmented p.d. function

Vs2 = Vs1 +12w2z

22 (3.20)

where w2 > 0 is any weighting. From (3.15) and (3.16), its derivative is given by

_Vs2 = z1z2 � [k1z21 + h1z1 tanh��h z1"1(t)

�� ~�1(x; t)z1]� ~�� T �1

+w2z2[z3 + �2 � �2f � �02s � ~��T�2 + ~�2(x; t)� p2( _� � ��2)]

= w2z2z3 � [k1z21 + h1z1 tanh��h1 z1"1(t)

�� ~�1(x; t)z1]� ~�� T [�1 + w2z2�2]

+w2z2[1w2z1 + �2 � �2f � �02s + ~�2(x; t)]� w2z2p2(

_� � ��2)

(3.21)

28

where the term z1z2 has been grouped together with �2 since it is going to be dealt with via �2 in this

design step, and the term w2z2z3 has been separated from the rest since it is going to be dealt with at

the next step. De�ne the second tuning function �2 and the function ~p2 as

�2(�z2; �(�); t) = �1(z1; t) + w2z2�2(�z2; �(�); t)

~p2(�z2; ��(1)� ; t) = w2z2p2(z1; ��

(1)� ; t)

(3.22)

With the choice of

�2(�z2; �(�); t) = �2f (�z2; �(�); t) + �2s(�z2; �(�); t)

�2s�= �02s + �002s �002s

�= � 1

w2z1 � k2z2 � h2 tanh

��h2 z2"2(t)

� (3.23)

(3.21) becomes

_Vs2 = w2z2z3 �P2j=1 wjzj [kjzj + hj tanh

��hj zj"j

�� ~�j(x; t)]� ~��

T �2 � ~p2(_� � ��2) (3.24)

From (3.16) and (3.23), the second error subsystem becomes

_z2 = z3 + �002s � ~��T�2 + ~�2(x; t)� p2( _� � ��2) (3.25)

3.3.3 Step i

We will use mathematical induction to explain the remaining intermediate design steps. In the

following, we treat xj ; j � i as the function of z1; : : : ; zj; � and t, i.e., xj = zj + �j�1(�zj�1; ��(j�2)� ; t)

(for simplicity, denote �0(t) = x1d(t)). Thus, we can recursively de�ne the following functions for step

j from those in the previous steps

�j(�zj ; ��(j�2)� ; t) = 'j(z1 + x1d; : : : ; zj + �j�1; t)�Pj�1

k=1@�j�1

@zk�k

~�j(x; t) = �j(x; t)�Pj�1

k=1@�j�1

@zk~�k(x; t)

pj(�zj�1; ��(j�1)� ; t) =

@�j�1

@�+Pj�1

k=1@�j�1

@zkpk p1 = 0

~pj(�zj ; ��(j�1)� ; t) = ~pj�1 + wjzjpj ~p1 = 0

�j(�zj ; ��(j�2)� ; t) = �j�1 + wjzj�j

(3.26)

Let zj+1 = xj+1 � �j and choose the desired control function �j(�zj ; ��(j�1)� ; t) as

�j = �jf (�zj ; ��(j�2)� ; t) + �js(�zj ; ��

(j�1)� ; t) (3.27)

where

�jf�= ��(�)'j(�xj ; t) + @�(j�1)f

@t

�js�= �0js + �00js

�0js�=

Pj�1k=1

@�j�1

@zk[�00ks + zk+1] +

@�(j�1)s

@t +Pj�1

k=1@�j�1

@zkpk�(�j � �k) +

@�j�1

@���j

�00js�= �wj�1

wjzj�1 � kjzj � hj tanh

��hj zj"j(t)

�� ~pj�1��j

(3.28)

and hj(�zj ; ��(j�2)� ; t) is any positive function that is speci�ed in each step. Then, the j-th error subsystem

may be assumed to be

_zj = zj+1 + �00js � ~�� T�j + ~�j(x; t)� pj( _� � ��j) (3.29)

29

The augmented p.d. function is

Vsj = Vs(j�1) + 12wjz

2j wj > 0 (3.30)

and its derivative is given by

_Vsj = wjzjzj+1 �Pj

k=1 wkzk[kkzk + hk tanh��hk zk"k

�� ~�k(x; t)]

�~�� T �j � ~pj(_� � ��j)

(3.31)

It is easy to check that the �rst two steps satisfy the above general forms. So we assume that they

are valid for step j, 8j � i � 1, and show that they are also true for step i to complete the induction

process. From (3.2) and (3.26)

j ~�i(x; t)j � ~�i(�zi; ��(i�2)� ; t)

�= �i(�xi; t) +

Pi�1j=1 j@�i�1

@zjj~�j(�zj ; ��(j�2)� ; t) (3.32)

There exists a known function hi(�zi; ��(i�2)� ; t) with continuous partial derivatives up to (n � i)-order

such that

hi(�zi; ��(i�2)� ; t) � �TM j�i(�zi; ��(i�2)� ; t)j+ ~�i(�zi; ��

(i�2)� ; t) (3.33)

From (3.1) and (3.26), noting (3.27) and (3.29) for step j < i, we have

_zi = xi+1 + �T'i(�xi; t) + �j(x; t)

��Pi�1

j=1@�i�1

@zj[zj+1 + �00js � ~�� T�j + ~�j(x; t)� pj(

_� � ��j)] + @�i�1

@�

_� + @�i�1

@t

�= zi+1 + �i � �if � �0is � ~�� T�i + ~�i(x; t)� pi( _� � ��i)

(3.34)

where �if and �0is satisfy the de�nition (3.28), and �i and pi are given by (3.26). If �i is chosen to be

in the form of (3.27), (3.34) reduces to the form (3.29). Furthermore, the derivative of Vsi is

_Vsi = wi�1zi�1zi �Pi�1

k=1 wkzk [kkzk + hk tanh��hk zk"k(t)

�� ~�k(x; t)]� ~��

T �i�1

�~pi�1( _� � ��i�1) + wizi[zi+1 + �00is � ~�� T�i + ~�i(x; t)� pi( _� � ��i)

= wizizi+1 �Pi

1 wkzk [kkzk + hk tanh��hk zk"k(t)

�� ~�k]� ~��

T�i � ~pi(_� � ��i)

(3.35)

which agrees with the general form (3.31). This completes the induction process.

3.3.4 Step n

This is the �nal design step. By letting xn+1 = �(x)u, the last equation of (3.1) has the same

form as the intermediate step i � n� 1. Therefore, the general form ((3.26) to (3.31)) applies to Step

n. Since u is the real control, we can choose it as

u = 1�(x)�n(�zn;

��(n�2)� ; t) (3.36)

where �n is given by (3.27), in which hn is determined from (3.33) with j = n. By doing so, zn+1 = 0.

Specify the adaptation law as_� = ��n (3.37)

Then, the n-th error subsystem (3.29) becomes

_zn = �00ns � ~�� T�n + ~�n(x; t) (3.38)

30

and the derivative of the augmented p.d. function Vsn is given by

_Vsn = �Pnk=1 wkzk [kkzk + hk tanh

��hk zk"k

�� ~�k(x; t)]� ~��

T �n (3.39)

Theorem 5 With the control law (3.36) and the adaptation law (3.37), the following results hold for

the system (3.1) if the assumption (3.2) is satis�ed:

A . In the presence of both parametric uncertainties and unknown nonlinear functions, the control

input is bounded. The system is stable and the output tracking error exponentially converges to a

ball whose size can be freely adjusted by controller parameters in a known form. Vsn is bounded

above byVsn(t) � exp(�2kvt)Vsn(0) +

R t0 exp(�2kv(t� �)"v(�)d�

� exp(�2kvt)Vsn(0) + "vmax

2kv[1� exp(�2kvt)] (3.40)

where kv=minfk1; : : : ; kng, "v(t)=Pnk=1 "k(t), and "vmax = maxt "v(t).

B . In the presence of parametric uncertainties only (i.e., �i(x; t) = 0, 8i), in addition to the result

in A, the system output tracks the desired output asymptotically. 4

Proof. From (3.26), �n =Pn

k=1 wkzk�k . Noticing (3.33), (3.32) and the condition ii of

(2.15), (3.39) becomes

_Vsn = �Pnk=1 wkzk[kkzk + hk tanh

��hk zk"k

�� ~�k(x; t) + ~�� T�k]

� �Pnk=1fwk[kkz2k + hkzk tanh

��hk zk"k(t)

�� hkjzkj]

� �Pnk=1 wkkkz

2k +

Pnk=1 "k � �2kvVsn + "v(t)

(3.41)

which leads to (3.40). Since �(j)(�); 0 � j � n � 1, are bounded for any � and all the terms involved

are bounded functions w.r.t. t, it is easy to check that (3.40) also guarantees that all the variables in

(3.26) to (3.28) are bounded for j, which implies that the state x is bounded and so is the control input

(3.36). Since keyk2 = kz1k2 � 2Vsn, A of the Theorem is thus proved.

When �i(x; t) = 0; 8i, from (3.26), ~�i(x; t) = 0; 8i. Choose a p.d. function Van as

Van = Vsn + V�(~�; �). Noticing (3.6), (3.39), and (3.37), we have

_Van = _Vsn +@V�(~�;�)

@~�

_�

= �Pnk=1 wkzk [kkzk + hk tanh

�hk zk�"k(t)

�]� ~�� T �n + ~�� T��1 _�

= �Pnk=1 wkzk [kkzk + hk tanh

�hk zk�"k(t)

�] � �Pn

k=1 wkkkz2k

(3.42)

Therefore, z = [z1; : : : ; zn]T 2 Ln2 2 Ln1. It is also easy to check that _z is bounded. So, z �! 0 by

Barbalat's lemma, and B of the Theorem is proved. 4

3.4 Guaranteed Transient Performance

In Theorem 5, the exponentially converging rate, 2kv, can be any value by adjusting the

controller gains k = [k1; : : : ; kn]T and the �nal tracking accuracy, kz(1)k �q

�"vmax

kv, can be made

arbitrarily small by increasing k = [k1; : : : ; kn]T and decreasing " = ["1; : : : ; "n]T . However, Vsn(0)

also depends on k and ", and thus the transient behavior of the error system may not be improved

31

by increasing k and reducing " if Vsn(0) increases. To deal with this problem, the idea of trajectory

initialization in [53] will be used to render z(0) = 0 independent of the choice of k and ".

Let x1d(t) be the trajectory created by the following n-th order stable system

x(n)1d + �1x

(n�1)1d + : : :+ �nx1d = y

(n)d + �1y

(n�1)d + : : :+ �nyd (3.43)

where yd(t) is the desired output. Recursively de�ne the following functions

h1(x1; �; t) = �T'1(x1; t)

hi(�xi; �; t) = �T'i(�xi; t) +Pi�1

k=1@hi�1(�xi�1;�;t)

@xk[xk+1 + �T'k(�xk; t)]

+@hi�1(�xi�1;�;t)@t i = 2; : : : ; n� 1

(3.44)

Lemma 1 Each part of the control functions �j = �jf + �js can be written in the following forms

A: �jf = x(j)1d � hj(�xj ; ��; t) + fbj(�zj ; ��

(j�2)� ; t) where every term in fbj contains

either zk or tanh��hk zk"k

�as a factor for some k � j:

B: every term in �js contains either zk or tanh��hk zk"k

�as a factor for some k � j

(3.45)

Proof. We prove the lemma by induction. First, from (3.8), �1f satis�es A for fb1 = 0

and B is obviously satis�ed. Thus, we assume that A and B are valid for 8j � i � 1, and we prove

that they are also valid for i. From B, 8j � i � 1 and 8l, all the terms in@l�js@tl

contain either zk or

tanh��hk(�zk;t); zk

"k(t)

�as a factor for some k � j. Thus, from (3.26) and (3.28), �is satis�es B. So (B)

is true for i. From (3.28) and A for 8j � i� 1 and noticing @xk@t =

@�k�1(�zk�1 ;��k�2� ;t)

@t , we have

�if = ���'i(�xi; t) + x(i)1d �

Pi�1k=1

@hi�1(�xi�1;�� ;t)@xk

@�k�1@t � @hi�1(�xi�1;�� ;t)

@t +@fb(i�1)(�zj ;��

(i�3)� ;t)

@t

= x(i)1d � hi(�xi; ��; t) + fbi(�zi; ��

(i�2)� ; t)

(3.46)

where

fbi = �Pi�1k=1

@hi�1(�xi�1;��;t)@xk

[xk+1 + �T�'k � @�k�1

@t ] +@fb(i�1)(�zj;��

(i�3)� ;t)

@t

= �Pi�1k=1

@hi�1

@xk[xk+1 + �T� 'k � @�k�1f

@t � @�k�1s

@t ] +@fb(i�1)(�zj ;��

(i�3)� ;t)

@t

= �Pi�1k=1

@hi�1

@xk[xk+1 � �kf � @�k�1s

@t ] +@fb(i�1)(�zj;��

(i�3)� ;t)

@t

= �Pi�1k=1

@hi�1

@xk[zk+1 + �ks � @�k�1s

@t ] +@fb(i�1)(�zj ;��

(i�3)� ;t)

@t

(3.47)

It is thus obvious that A is satis�ed for j = i since every term in �ks and @�k�1s

@t contains either zl or

tanh��hl zl"l

�as a factor for some l � k. 4

Lemma 2 If the initial conditions x1d(0); : : : ; x(n�1)1d (0) of the �ltered trajectory x1d(t) are chosen as

x1d(0) = x1(0)

x(i�1)1d (0) = xi(0) + hi�1(�xi�1(0); ��(0); 0); i = 2; : : : ; n

(3.48)

then z(0) = 0. 4

32

Proof. We use induction to prove the above lemma. It is obvious that z1(0) = 0. So we

assume that x1d(0); : : : ; and x(i�2)1d (0) have been chosen to render zj(0) = 0; 8j � i� 1. Then, from

(3.45) and noticing that zj(0) = 0; 8j � i� 1, we have

�i�1f(0) = x(i�1)1d (0)� hi�1(�xi�1(0); ��(0); 0)

�i�1s(0) = 0(3.49)

By choosing x(i�1)1d (0) according to (3.48), we have zi(0) = xi(0)� �i�1(0) = 0, which completes the

proof. 4Remark 6 From (3.1), in the absence of unknown nonlinear functions, the i-th derivative of the output

is

y(i)j�j=0 = xi + hi(�xi; �; t) (3.50)

Thus the above trajectory initialization (3.48) can be considered as placing the initial condition x(i)1d (0)

at the best initial estimate of y(i)(0) by substituting ��(0) for �. A similar implication is �rst observed

in [65] for the adaptive control of parametric strict-feedback systems. Also, from this implication,

trajectory initialization can be performed independently from the choice of controller parameters such

as k and ". 4Theorem 6 Given the desired trajectory x1d(t) generated by (3.43) with the initial conditions (3.48),

the following results hold for the system (3.1) if the control law (3.36) and the adaptation law (3.37)

are applied and the assumption (3.2) is satis�ed:

A . In the presence of both parametric uncertainties and unknown nonlinear functions, the control

input is bounded and Vsn is bounded above by

Vsn(t) � R t0 exp(�2kv(t� �)"v(�)d� � �"vmax

2kv[1� exp(�2kvt)] (3.51)

Transient performance and �nal tracking accuracy of the output tracking can be freely adjusted

by controller parameters in a known form.

B . In the presence of parametric uncertainties only, in addition to the result in A, the system output

tracks the desired output asymptotically. 4Proof. Noting Theorem 5, we only have to show that the output tracking error has a

guaranteed transient performance. Since the initial conditions x1d(0); : : : ; x(n�1)1d (0) chosen by (3.48)

are independent of the choice of the controller parameters k and ", the trajectory planning error,

ed(t) = x1d(t)� yd(t), can be guaranteed to possess any good transient behavior by suitably choosing

the Hurwitz polynomial Gd(s) = sn+�1sn�1+ : : :+�n without being a�ected by k and ". On the other

hand, such a desired trajectory initialization renders z(0) = 0 by Lemma 2. From (3.40), (3.51) is true,

which indicates that z1 can be made arbitrarily small by increasing k and decreasing ". Therefore, any

good transient performance of the output tracking error e = y � yd = z1(t) + ed(t) can be guaranteed

by the choice of the controller gains k and " in a known form. 4Remark 7 Similar comments as in remarks 2 to 4 in Chapter 2 can be made for the above ARC

law. The above design method can be extended to the case of bounded time-varying parameters with

bounded derivatives up to a su�ciently high order. Results similar to the one in B of Theorem 6 can

be obtained. Robustness to the neglected high frequency dynamics may also be obtained since our

controller guarantees exponential stability at large. }

33

Remark 8 If wi = 1; 8i; and the parameter projection is not used, the adaptation law (3.37) reduces

to the one used in [62, 53]. The reason of introducing the weighting wi is to gain more freedom in

shaping the transient performance since only the tracking error z1 is of our concern. }

3.5 Simulation Results

Consider the following relative degree 2 nonlinear system:

_x1 = x2 + �1x21 + �1(x2) �1(x2) = d1 sin(r1x2)

_x2 = u + �2(x21 + x22) sin3(t) + �2(x) �2(x) = d2 cos(r2x1x2)

y = x1

(3.52)

where �1 and �2 are unknown parameters satisfying (3.2) in which �1min = �3; �1max = 0; �2min = �4and �2max = 0, r1 and r2 are assumed to be unknown, d1 and d2 are also unknown but are bounded

by d1 � d1M = 2 and d2 � d2M = 2 respectively. It is observed that (3.52) is not in a strict feedback

form but satis�es the semi-strict feedback form (3.1) since

j�1(x2)j � �1 = d1Mj�2(x)j � �2 = d2M

(3.53)

From (3.10), we choose h1(z1; t) = �1M(z1 + x1d)2 + �1 where �� = [1; 1]T . �1 is then determined by

(3.8), where k1 = 5 and "1 = 0:5�, and the projection �i in de�ning �� is speci�ed by

�i(�i) =

8><>:�imax + ��i[1� exp(� 1

��i(�i � �imax))] if �i > �imax

�i �i 2 [�imin; �imax]

�imin � ��i[1� exp( 1��i(�i � �imin))] �i < �min

(3.54)

which is monotone increasing and has a continuous �rst derivative. From (3.17), �2f ; �02s; �2 and p2

can be obtained. �2 is formed from (3.22) where w2 = 0:5. h2 is determined by (3.19) and is given by

h2 = �TM j�2j+ �2 + �1j@�1@z1j (3.55)

Therefore, �2 can be determined by (3.23), where k2 = 5 and "2 = �, and the control law is given by

(3.36). Estimated parameters are updated by (3.37) where � = diagf500000; 150000g.The desired output is yd = 0:1sin(0:5�t) and xd and _xd are calculated by (3.43) with initial

values given by (3.48) where �1 = 80 and �2 = 1600. Actual plant parameters are �1 = �2; �2 =

�3; r1 = 2 and r2 = 3 with initial estimates �1(0) = 0 and �2(0) = 0. Sampling time is 1ms. Three

controllers are run for comparison:

ARC : The proposed adaptive robust controller as described in the above.

DRC : The same control law as in ARC but without using the parameter adaptation law (3.37). In

such a case, the proposed control law is equivalent to the conventional DRC law.

AC : By setting hi = 0 and without using parameter projection, i.e., letting �(�) = �, the suggested

control law is equivalent to the nonlinear adaptive control law in [62].

34

To test the nominal performance, simulations were run for parametric uncertainties only, i.e.,

d1 = d2 = 0. The tracking error z1 is shown in Fig. 3.1, from which we can see that all three controllers

have very good tracking ability. ARC has a much better �nal tracking accuracy than DRC since the

estimated parameters approach their desired values as shown in Fig. 3.2, and has a better transient

response than AC. This result substantiates the necessity of using parameter adaptation to improve

tracking performance. As shown in Fig.3.3, control inputs do not exhibit chattering.3

To test the performance robustness, small disturbances were �rst added to the system, i.e.,

d1 = d2 = 0:02 in (3.52). Fig. 3.4 shows the tracking error z1. Roughly speaking, the tracking

performance of DRC remains unchanged but the AC's performance degrades a lot because of the wrong

fast-changing parameter estimates shown in Fig. 3.5. Although the ARC's performance also degrades, it

still achieves the best tracking performance. Control inputs shown in Fig. 3.6 do not exhibit chattering.

Large disturbances were also added to the system, i.e., d1 = d2 = 2 in (3.52). As shown in

Fig. 3.7, ARC achieves the same tracking performance as DRC although its estimated parameters vary

quite wildly ( Fig. 3.8). AC has the worst tracking ability and a large control e�ort since its estimated

parameters are unbearably large. Fig. 3.9 presents the control inputs for ARC and DRC, which do not

exhibit control chattering. All these results illustrate the e�ectiveness of the proposed ARC.

3.6 Conclusions

In this chapter, we have presented a systematic design of adaptive robust controllers for a

class of SISO nonlinear systems transformable to a semi-strict feedback form in the presence of both

parametric uncertainties and unknown nonlinear functions. By utilizing prior knowledge of the bounds

of parametric uncertainties and the bounding function of unknown nonlinear functions, we use a simple

smooth projection of the estimated parameters updated by an adaptation law similar to that in [62, 52] in

designing the robust control law to combine the backstepping adaptive control [62, 52] with conventional

DRC techniques. This approach preserves the advantages of the two methods while eliminating their

drawbacks. Simulation results validate the analysis.

3Only the input during the �rst 0.8 second is presented in order to show the large transient of the initial inputclearly

35

0 0.5 1 1.5 2 2.5 3 3.5 4−2

−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3

Time(s)

Tra

ckin

g er

ror

z1Solid: ARC Dashed: DRC Dashdot: AC

Figure 3.1: Tracking errors in the presence of parametric uncertainties

0 0.5 1 1.5 2 2.5 3 3.5 4−6

−5

−4

−3

−2

−1

0

1

Time(s)

Est

imat

ed p

aram

eter

s

Solid: theta1 (ARC) Dashed: theta2 (ARC)Dashdot: theta1 (AC) Dotted: theta2 (AC)

Figure 3.2: Estimated parameters in the presence of parametric uncertainties

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−2

0

2

4

6

8

10

12

14

Time(s)

Con

trol

inpu

t

Solid: ARC Dashed: DRC Dotted: AC

Figure 3.3: Control input in the presence of parametric uncertainties

36

0 0.5 1 1.5 2 2.5 3 3.5 4−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−3

Time(s)

Tra

ckin

g er

ror

z1

Solid: ARC Dashed: DRC Dashdot: AC

Figure 3.4: Tracking errors in the presence of parametric uncertainties and small disturbances(d1 = d2 = 0:02)

0 0.5 1 1.5 2 2.5 3 3.5 4−6

−5

−4

−3

−2

−1

0

1

2

3

Time(s)

Est

imat

ed p

aram

eter

s

Solid: theta1 (ARC) Dashed: theta2 (ARC)

Dashdot: theta1 (AC) Dotted: theta2 (AC)

Figure 3.5: Estimated parameters in the presence of parametric uncertainties and small distur-bances (d1 = d2 = 0:02)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−2

0

2

4

6

8

10

12

14

Time(s)

Con

trol

inpu

t

Solid: ARC Dashed: DRC Dotted: AC

Figure 3.6: Control input in the presence of parametric uncertainties and small disturbances(d1 = d2 = 0:02)

37

0 0.5 1 1.5 2 2.5 3 3.5 4−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Time(s)

Tra

ckin

g er

ror

z1

Solid: ARC Dashed: DRC Dashdot: AC

Figure 3.7: Tracking errors in the presence of parametric uncertainties and large disturbances(d1 = d2 = 2)

0 0.5 1 1.5 2 2.5 3 3.5 4−50

0

50

100

150

200

250

300

350

Time(s)

Est

imat

ed p

aram

eter

s

Solid: theta1 (ARC) Dashed: theta2 (ARC)

Dashdot: theta1 (AC) Dotted: theta2 (AC)

Figure 3.8: Estimated parameters in the presence of parametric uncertainties and large distur-bances (d1 = d2 = 2)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−30

−25

−20

−15

−10

−5

0

5

10

15

Time(s)

Con

trol

inpu

t

Solid: ARC Dashed: DRC Dotted: AC

Figure 3.9: Control input in the presence of parametric uncertainties and large disturbances(d1 = d2 = 2)

38

Chapter 4

General Framework of Adaptive Robust

Control

In this chapter, we will present a general framework of the adaptive robust control of nonlinear

systems using the ARC Lyapunov functions.

4.1 Problem Formulation

Considering the following general MIMO nonlinear system:

_x = f(x; �; t) + B(x; �; t)u+D(x; t)�(x; �; u; t)

y = h(x; t)(4.1)

where y 2 Rm and u 2 Rm are the output and input vectors respectively, x 2 Rn is the state vector,

� 2 Rp is the vector of unknown parameters, h(x; t); f(x; �; t);B(x; �; t), and D(x; t) 2 Rn�ld are

known, and �(x; �; u; t) 2 Rld represents the vector of unknown nonlinear functions such as disturbances

and modeling errors. Similar to (2.3), we make the following reasonable and practical assumptions:

Assumption 1 Parametric uncertainties and the unknown nonlinear functions satisfy

� 2 ��= f� : �min < � < �max g

�(x; �; u; t) 2 ��= f� : k�(x; �; u; t)k � �(x; t) g

(4.2)

where �min; �max and �(x; t) are known. �Let yd(t) 2 Rm be the desired outputs at t, and let the output tracking errors be denoted as

ey = y � yd(t). The adaptive robust control problem can now be formulated as that of designing a

control law for the inputs u such that, under the assumption of (4.2), the system is globally stable and

the output tracking has a prescribed transient performance and �nal tracking accuracy. Furthermore,

in the presence of parametric uncertainties only, asymptotic output tracking should be achieved.

4.2 ARC Lyapunov Functions

Since almost all adaptive nonlinear controllers and deterministic robust controllers were syn-

thesized by Lyapunov functions, it is natural that Lyapunov functions will be utilized here to formulate

39

the general problem. In addition, to solve the con ict between DRC and AC, the smooth projection

presented in Chapter 3 will be utilized. Namely, we only use the projected parameter estimates �� and

the derivatives of the projection, �(j)(�), as they belong to those compact sets in (3.4) for any �.

Let V (x; �; ��(lV )� ; t) be a positive semi-de�nite (p.s.d.) function with continuous partial deriva-

tives (lV is any index), which satis�es the following assumptions:

Assumption 2 Bounded V means bounded x, and guaranteed transient performance of V (t) is equiv-

alent to the guaranteed transient performance of output tracking error ey (e.g., guaranteed exponential

convergence of V (t) �! 0 means guaranteed exponential convergence of ey �! 0). �

Basically, Assumption 2 says that the stability and performance of the nonlinear system (4.1)

can be converted to the study of the stability and performance of the scalar function V , which is much

easier to deal with. The control law we seek consists of two parts given by

u(x; ��(lu)� ; t) = ua(x; ��

(lu)� ; t) + us(x; ��

(lu)� ; t) (4.3)

where lu is an index, ua functions as an adaptive control law and us a robust control law to be designed

within an allowable set u.

Assumption 3 There exists a continuous control law ua(x; ���lu ; t) such that 8us 2 u:

@V@x [f(x; �; t) +B(x; �; t)(ua + us)] +

@V@t � �W (x; �; ��

(lr)� ; t) + ~�T� �(x; ��

(lr)� ; u; t) + @V

@��� (4.4)

or, equivalently, _V j�=0� �W + ~�T� � +@V@�(_� + ��) (4.5)

where lr = maxflV + 1; lug, �(x; ��(lr)� ; u; t) is a known function, _V j�=0 represents the derivative of V

under the condition that � = 0, and W (x; �; ��(lr)� ; t) is any continuously di�erentiable p.s.d. function

which satis�es the condition that asymptotic convergence of W means asymptotic output tracking, i.e.,

W �! 0 =) ey �! 0. �

Assumption 3 guarantees that there exists an adaptive control law to achieve asymptotic

output tracking in the presence of parametric uncertainties only as shown later.

Remark 9 In [65], the adaptive control Lyapunov function (aclf) is introduced for the following single

input plant

_x = f(x) + F (x)� + b(x)u (4.6)

A smooth function V (x; �) , positive de�nite (p.d.) and radially unbounded in x for each �, is called

an aclf for (4.6) if there exists a s.p.d. � such that

infu2Rn@V@x

hf(x) + F (x)

�� + �(@V@� )

T�+ b(x)u

io< 0 (4.7)

It can be proved that there exists an aclf V for (4.6) i� there exists a control u = �(x; �) such that

@V (x;�)@x

hf(x) + F (x)� + B(x)�(x; �)

i� �W (x; �) + ~�T �(x; �) + @V (x;�)

@���(x; �) (4.8)

for some W � 0 and �(x; �). Comparing (4.8) with Assumption 3 in (4.4) for the system (4.6), we can

see that they have a similar structure except that we use the projected estimated parameters and add

a robust term. 4

40

In the presence of unknown nonlinear functions, principally, there is no way that we can

estimate the unknown parameters accurately since the model is not accurate. The best we can do is to

use DRC to synthesize a robust control law.

Assumption 4 . There exists a us(x; ��(lu)� ; t) 2 u such that 8� 2 � and 8� 2 �:

@V@x [f(x; �; t) + B(x; �; t)(ua + us) +D�(x; �; u; t)]+ @V

@t

�@V@���(x; ��

(lr)� ; u; t) � ��V V + cV (t)

(4.9)

or, equivalently,_V � ��V V + cV (t) +

@V@�(_� + ��) (4.10)

where �V > 0 and cV (t) is a bounded positive scalar, i.e., 0 � cV (t) � cVmax. Both �V and cV (t)

are supposed to be freely adjusted by some controller parameters in a known form without a�ecting the

initial value of V, V (0). �

Normally, Assumption 4 can be satis�ed since all the unknown terms involved belong to some

known compact sets or a bounded range.

De�nition 1 A continuously di�erentiable p.s.d. function V (x; �; ��(lV )� ; t) is called an adaptive robust

control (ARC) Lyapunov function for (4.1) if it satis�es Assumptions 2-4 for some continuous control

functions ua(x; ��(lu)� ; t) and us(x; ��

(lu)� ; t) and the adaptation function �(x; ��

(lr)� ; u; t). �

4.3 Adaptive Robust Control

In the above section, we introduced the concept of ARC Lyapunov functions. In this section,

we will utilize this idea to solve the ARC of (4.1).

Theorem 7 If there exists an ARC Lyapunov function V for (4.1), then, by using the control law (4.3)

and the following adaptation law

_� = ���(x; ��(lr)� ; u(x; ��

(lu)� ; t); t) (4.11)

the following results hold if Assumption 1 is satis�ed:

A. In general, the control input and the system state are bounded with V bounded above by

V (t) � exp(��V t)V (0) +R t0 exp(��V (t� �))cV (�)d�

� exp(��V t)V (0) + cV max

�V[1� exp(��V t)] (4.12)

Output tracking is guaranteed to have arbitrary good transient performance and �nal tracking

accuracy.

B. If, after a �nite time, there are no unknown nonlinear functions, i.e., �(x; �; u; t) = 0; 8t � t0;

for some �nite t0, in addition to the result in A, the system outputs track the desired outputs

asymptotically. 4

41

Proof. Noting (4.11), from Assumption 4, we have

_V � ��V V + cV (t) (4.13)

which leads to (4.12). From Assumption 2, the system state x is bounded and thus the control input

is bounded. Since the exponentially converging rate, �V , and the bound of the �nal tracking errors,cV max

�V, can be freely adjusted by controller parameters in a known form without a�ecting V (0), any

prescribed good transient performance of V can be guaranteed. Therefore, from Assumption 2, output

tracking can be guaranteed to have any good transient performance and �nal tracking accuracy. This

proves A of the Theorem.

Now consider the situation that �(x; �; u; t) = 0; 8t � t0; for some �nite t0. Since x is

bounded as shown in A, from (4.11),_�(t) 2 Lp1; 8t. Thus, �(t0) is bounded. Choose a p.s.d. function

as

Va(x; �; �; t) = V (x; �; ��(lV )� ; t) + V�(~�; �) (4.14)

Then, Va(t0) is bounded. 8t � t0, noticing (3.6), (4.5), and (4.11), the derivative of Va along (4.1) is

_Va = _V j�=0 +@@~�V�(~�; �)

_� � �W; 8t � t0 (4.15)

Therefore, W 2 L1 and Va 2 L1. Since u 2 Lm1, _x 2 Ln1 and_� 2 Lp1. These imply that W is

uniformly continuous. By Barbalat's lemma [112], W �! 0 and thus, from Assumption 3, asymptotic

output tracking is achieved. 4

Remark 10 In the absence of adaptation (i.e., � = 0), the proposed ARC law reduces to a DRC law

and Result A of Theorem 7 still holds. Therefore, the adaptation loop can be switched o� at any time

without a�ecting the stability. 4

Remark 11 Although � is not guaranteed to be bounded in the presence of unknown nonlinear functions,

the stability and the performance of the controller is not a�ected since only the bounded projection and

its bounded derivatives are used in the design. Furthermore, since_� is bounded, for any �nite time, �(t)

is bounded. In application, the execution time is always �nite, so the issue of boundedness of � is not

essential here. In addition, in view of Remark 10, either reinitialization or switching o� the adaptation

can be used in case that unbearable wrong adaptation is observed. }

Corollary 2 For the system (4.1) under Assumption 1, if there exists an ARC Lyapunov function

V (x; �; t), which is not a function of �, then, by using the control law (4.3) and the following modi�ed

adaptation law_� = ��[l�(�) + �(x; ��

(lr)� ; u(x; ��

(lu)� ; t); t)] (4.16)

where l�(�) is any vector of functions that satis�es the following conditions

i. l�(�) = 0 if � 2 �

ii. ~�T� l�(�) � 0 if � 62 �(4.17)

we have the results in Theorem 7. 4

42

The reason for using (4.16) is that by suitably choosing l�(�), we can make the parameter

estimation process more robust and guarantee that � is bounded since l�(�) acts like a nonlinear damping

term.

Proof. Since V is not a function of �, @V@�

= 0. Thus, (4.13) and (4.12) are not a�ected,

and the results in A of Theorem 7 remain valid.

When 8t � t0;� = 0, from Assumption 3 and (4.17), following the same proof as in (4.15),

we have_Va � �W � ~�T� l�(�) � �W (4.18)

Thus, the results in B of Theorem 7 remain valid. 4

Remark 12 Sometimes, the right hand side of adaptation law (4.16) can be discontinuous since it

only causes_� to be discontinuous, and � is still continuous which is normally used in the control law.

Therefore, the discontinuous modi�cation law l�(�) may be allowed. In such a case, the widely used

projection method in adaptive systems [104, 33] can be employed, which is de�ned for any bounded

open convex set � as described in the following.

De�ne a set, 0� = ��12 (�), which is a bounded open convex set. Let @� denote the

boundary of �, �Pr(�) the projection of the vector � onto the hyperplane tangent to @0� at �� 1

2� �,

and �perp the unit vector perpendicular to the tangent hyperplane of � at � 2 @�, pointing outward.

Then, l�(�) is given by

l�(�) =

8>><>>:

0 if

(� 2 �

� 2 @� and �Tperp�� � 0

�� � ��12 �Pr

��� 1

2 ��

� 2 @� and �Tperp�� < 0

(4.19)

Let �� be the closure of �. The above choice of l�(�) guarantees that � 2 �� no matter

what the control law is and what the error dynamics are as long as the initial estimate is in � . This is

because the resulting_� in (4.16) always points inside or along the tangent plane of �� when � 2 @�.

Clearly, condition i of (4.17) is satis�ed. When � 2 @�, by the de�nition of �perp,

(�12 �perp)

T��12 ~� = �Tperp

~� � 0 8� 2 � (4.20)

Thus, �12 �perp is along the outward normal direction of @0� at �

� 12 �, and �Pr

��� 1

2 ��is given by

�Pr��� 1

2 ��

= �� 12 � � cn� 1

2 �perp (4.21)

where scalar cn is

cn =(�

12 �perp)T (��

12��)

k� 12 �perpk2

=��Tperp��k� 1

2 �perpk2(4.22)

Thus, when � 2 @� and �Tperp�� < 0, from (4.22), cn > 0, and, from (4.19) and (4.21),

~�l�(�) = ~�h�� � ��

12 �Pr

��� 1

2 ��i

= cn~�T �perp � 0(4.23)

Noting (4.19), (4.23), and the fact that 8� 2 ��; �(�) = �, we have that 8� 2 @�; ~��l�(�) � 0.

Since 8t; �(t) 2 �� and � is open, � 62 � is equivalent to � 2 @�. Thus, condition ii of (4.17) is

satis�ed and (4.19) satis�es all the required conditions.

43

For the open convex set � given by (4.2), with the modi�cation (4.19) in which � is a

diagonal matrix, the adaptation law (4.16) becomes

_�i =

8>>>>>><>>>>>>:

0 if �i = �imax and (��)i < 0

�(��)i

8><>:�imin < �i < �imax

�i = �imax and (��)i � 0

�i = �imin and (��)i � 0

0 �i = �imin and (��)i > 0 }

(4.24)

Remark 13 Some continuous modi�cations are given in [154] by generalizing �-modi�cation. �

44

Chapter 5

Backstepping Design via ARC Lyapunov

Functions

In Chapter 4, we presented a general framework of the adaptive robust control via ARC

Lyapunov functions and converted the ARC of a nonlinear system into the problem of �nding an ARC

Lyapunov function. In this chapter, we will address the problem of systematic construction of ARC

Lyapunov functions. We use the same strategy as in the systematic design of adaptive controllers

[65] { using backstepping procedure to recursively enlarge the applicable nonlinear systems. Namely,

we assume that an ARC Lyapunov function is known for an initial system and construct a new ARC

Lyapunov function for the augmented system by adding another nonlinear system to the back of the

initial system.

5.1 Initial MIMO Nonlinear Systems

Consider the following initial MIMO system

_xI = f0I (xI ; t) + FI(xI ; t)� +BI (xI ; �; t)uI +DI(xI ; t)�I(xI ; �; uI ; t)

yI = hI(xI ; t) uI ; yI 2 RmI xI 2 RnI FI 2 RnI�p BI 2 RnI�mI(5.1)

which satis�es the general setting in section 4.1 with the norm of the vector of unknown nonlinear

functions �I bounded by a known function �I(xI ; t), i.e., k�Ik � �I(xI ; t). In addition, we make the

following assumptions:

Assumption 5 B can be linearly parametrized by �, i.e.,

BI(xI ; �; t) = B0I (xI ; t) + B1

I (xI ; �; t) (5.2)

where B1I (xI ; �; t) is linear w.r.t. �.

1 }Assumption 6 There exists an ARC Lyapunov function VI(xI ; ��

(lI)� ; t) for the system (5.1) with the

associated control

uI = �I(xI ; ��(kI)� ; t) = �Ia(xI ; ��

(kI)� ; t) + �Is(xI ; ��

(kI)� ; t) (5.3)

and the adaptation function �I(xI ; ��(kI)� ; �I ; t). }

1For a matrix �, � is said to be linear w.r.t. � if all its elements are linear functions of �

45

Since B1I is linear w.r.t. �, there exist known matrices Gr

I(xI ; �; t) and GlI(xI ; �; t), which are

linear w.r.t. �, such that

B1I (xI ; �; t)vr = Gr

I(xI ; vr; t)� 8vr 2 RmI

vTl B1I (xI ; �; t) = �TGl

I(xI ; vl; t) 8vl 2 RnI(5.4)

We call GrI and G

lI the right and the left substitution matrices of B1

I w.r.t. �, respectively. By de�nition,

VI satis�es the assumptions 2- 4, in which Assumptions 3 and 4 are rewritten as

B2. @VI@xI

�f0I + FI� +BI (xI ; �; t)�I

�+ @VI

@t � �WI(xI ; ��kI� ; t) + ~�T� �I +

@VI@�

��I

B3. @VI@xI

�f0I + FI� +BI (xI ; �; t)�I +DI�I

�+ @VI

@t � ��VIVI + cVI(t) +@VI@�

��I(5.5)

where �VI and cVI can be freely adjusted by some controller parameters, say CI . For convenience,

denote

DVI =@VI@xI

�f0I + FI� +BI (xI ; �; t)�I +DI�I

�+ @VI

@t � @VI@�

��I (5.6)

In the following, for a system matrix �, � is obtained by substituting the projected estimated parameters

for the unknown parameters in � (e.g., BI = BI (xI ; ��; t)). ~� refers to the estimation error of �, i.e.,~� = � � �.

5.2 Augmented MIMO Nonlinear Systems I

Consider the following MIMO nonlinear system with the state vector �e = [xTe ; �T ]T , the

input vector ue 2 Rm, and the output vector ye 2 Rm

_xe = f0e (x; t) + Fe(x; t)� +Be(x; �; t)ue +De(x; t)�e(x; �; ue; t)

_� = �0�(x; t) + �1

�(x; t)� � 2 Rn�

ye = xe

(5.7)

where x = [xTI ; xTe ; �

T ]T 2 Rn and n = nI +m+ n�. In (5.7), the dynamics are allowed to depend

on the states of the initial system: i.e., the matrices f0e ; Fe; Be; De; �0�, and �1

� can be functions

of xI as well as �e. We make the following assumptions:

Assumption 7 Unknown nonlinear functions �e are bounded by k�ek � �e(x; t) where �e(x; t) is

known. }

Assumption 8 Be(x; �; t) is nonsingular for any � 2 � and Be can be linearly parametrized by �, i.e.,

Be = B0e (x; t) + B1

e (x; �; t) where B1e (x; �; t) is linear w.r.t. �. }

Assumption 9 The �- subsystem is bounded-input bounded-state (BIBS) stable w.r.t. the input

(xI ; xe). }

From Assumption 8, the system (5.7) has relative degree one from ue to ye. � functions as the

vector of its internal states and Assumption 9 assures that the internal dynamics of the system (5.7) is

stable. Since � can be chosen arbitrarily close to �, without loss of generality, we assume that Be (=

B(x; ��; t)) is nonsingular also. For convenience, Ne and Ue denote Ne = [0 Im�mI]T 2 Rm�(m�mI )

and Ue = [ImI0] 2 RmI�m in the following.

46

Now, augment the system by connecting the �rst mI outputs of the system (5.7) to the

inputs of the initial system (5.1): i.e., uI = �xemI= Uexe. The remaining outputs of the system (5.7),

xe(mI+1); : : : ; xem, are combined with the outputs of the initial system to form the new outputs of the

augmented system and the inputs of (5.7) become the inputs of the augmented system. The augmented

system thus has the dimension n and is described by

_xI = f0I (xI ; t) + FI (xI ; t)� + BI(xI ; �; t)�xemI+DI(xI ; t)�I(xI ; �; �xemI

; t)

_xe = f0e (x; t) + Fe(x; t)� +Be(x; �; t)u+De(x; t)�e(x; �; u; t)

_� = �0�(x; t) + �1

�(x; t)� � 2 Rn�

y = [yTI ; (NTe xe)

T ]T

(5.8)

where u 2 Rm and y 2 Rm. We make the following compatibility assumption about the connection:

Assumption 10 The initial output of the added system is compatible with the required initial input of

the original system, i.e.,

�xemI(0) = �I(0) (5.9)

Similar to (5.4), we use Gre and Gl

e to denote the right and the left substitution matrices of

B1e . (5.8) can be rewritten in the standard form (4.1) where

f =

264f0I + FI� +BI �xemI

f0e + Fe�

�0� + �1

��

375 B =

264

0

Be

0

375 D =

264DI 0

0 De

0 0

375 � =

"�I

�e

#(5.10)

5.3 Backstepping Design I

In this section, we will construct an ARC Lyapunov function for the augmented system (5.8)

based on the ARC Lyapunov function for the initial system (5.1). The key point is that, since the control

�I can achieve the ARC for the initial system (5.1) and the inputs of (5.1) are �xemI, we can design a

control law for the system (5.7) such that �xemItracks �I and other outputs track their desired values.

To this end, de�ne the tracking error z 2 Rm and V as

z(x; ��(kI)� ; t) = xe � ��(xI ; ��

(kI)� ; t) ��

�= [�TI (xI ;

��(kI)� ; t); (NT

e yd(t))T ]T

V (x; ��(lV )� ; t) = VI(xI ; ��

(lI)� ; t) + 1

2zTEz

(5.11)

where E is any symmetric positive de�nite (s.p.d.) matrix and lV = maxflI ; kIg.

Lemma 3 V in (5.11) satis�es Assumption 2 for the system (5.8). }

Proof. From (5.11), V being bounded means that VI and z are bounded. By the Assumption

2 for VI , xI 2 LnI1 and thus �I 2 LmI1 , which leads to xe 2 Lm1. From Assumption 9, � 2 Ln�1 . Thus,

x is bounded. Furthermore, since 0 � VI � V , the guaranteed transient of V means the guaranteed

transient of VI . Thus, Assumption 2 is satis�ed by V for (5.8). 4

Lemma 4 If L(x; ��(l�)� ; t) is nonsingular, by letting

u = fus : zTELus � 0g (5.12)

47

and choosing ua as

ua(x; ��(l�)� ; t) = L�1[�E�1Qz + �1e(x; ��

(l�)� ; t)] (5.13)

where Q is any s.p.d. matrix, Assumption 3 is satis�ed by V for the system (5.8) with the following

adaptation function

�(x; ��(lV )� ; u; t) = �I(x; ��

(kI)� ; �I ; t)� �T (x; ��

(lV )� ; u; t)Ez (5.14)

L, �1e , and � in the above equations are de�ned by

�1e(x;��(l�)� ; t) = �0e(x;

��(lV )� ; t)� �0�� � �0�

�@VI@�

�T � UTe@�I@�

�(�I � �0TEz)

L(x; ��(l�)� ; t) = Be +B1

e (x;��@VI@�

�T; t)� UT

e

266664

zTEB1e(x;�

�@�I1@�

�T; t)

...

zTEB1e (x;�

�@�ImI

@�

�T; t)

377775

�(x; ��(lV )� ; u; t) = �0(x; ��

(lV )� ; t) +Gr

e(x; u; t)

(5.15)

where

�0e(x; ��(lV )� ; t) = �E�1

�@VI@xI

(xI ; ��(lI)� ; t)B0

I (xI ; t)Ue�T

+UTe@�I@xI

(xI ; ��(kI)� ; t)[f0I (xI ; t) +B0

I �xemI]� f0e (x; t) + @��

@t (xI ;��(kI)� ; t)

�0(x; ��(lV )� ; t) = E�1UT

e GlTI (xI ;

�@VI@xI

�T; t)

�UTe@�I@xI

(xI ; ��(kI)� ; t)[FI(xI ; t) +Gr

I(xI ; �xemI; t)] + Fe(x; t)

(5.16)

}

Proof. From (5.11),

@V@x =

h@VI@xI

; 0; 0i+ zTE

h�UT

e@�I@xI

; Im; 0i

@V@t = @VI

@t � zTE @��@t

@V@�

= @VI@�� zTEUT

e@�I@�

(5.17)

Noticing (5.2), (5.4), and (5.6), from (5.10) and (5.17), we have

@V@x [f +Bu +D�]+ @V

@t

= @VI@xI

�f0I + FI� +BI�I +BIUez +DI�I

�+ zTE

h�UT

e@�I@xI

(f0I + FI�

+BI �xemI+DI�I) + f0e + Fe� +Beu+De�e

�+ @VI

@t � zTE @��@t

= DVI +@VI@�

��I + zTE

�E�1

�@VI@xI

(B0I +B1

I (x; �; t))Ue�T � UT

e@�I@xI

[f0I + FI�

+(B0I +B1

I (xI ; �; t))�xemI+DI�I ] + f0e + Fe� + Beu� ~Beu+De�e � @��

@t

o= DVI +

@VI@�

��I + zTEf��0e + �0� + Beu�Gre(x; u; t)~�� + ~�g

(5.18)

where~�(x; ��

(kI)� ; u; t) = �UT

e@�I@xI

DI�I +De�e (5.19)

48

Noticing (5.17), (5.14), and (5.16), (5.18) can be rearranged as

@V@x [ f +Bu +D�]+ @V

@t

= DVI +@VI@�

�(� + �TEz)� zTE�~�� + zTEf��0e + �0�� + Beu+ ~�g= DVI +

@V@��� � ~�T� �

TEz + zTEf���@VI@�

�T+ UT

e@�I@�

�� � �0e + �0�� + Beu+ ~�g= DVI +

@V@��� � ~�T� �

TEz + zTEf�0��@VI@�

�T+ B1

e (x;��@VI@�

�T; t)u

+UTe@�I@�

�(�I � �0TEz)� UTe@�I@�

�GrTe (x; u; t)Ez� �0e + �0�� + Beu+ ~�g

= DVI +@V@��� � ~�T� �

TEz + zTEf��1e + Lu + ~�g(5.20)

in which

@�I@�

�GrTe Ez =

26664

@�I1@�

�GrTe Ez

...@�ImI

@��GrT

e Ez

37775 =

266664

uTB1Te (x;�

�@�I1@�

�T; t)Ez

...

uTB1Te (x;�

�@�ImI

@�

�T; t)Ez

377775

=

266664

zTEB1e(x;�

�@�I1@�

�T; t)

...

zTEB1e (x;�

�@�ImI

@�

�T; t)

377775u

(5.21)

is used and l� = maxflV ; kI + 1g. When � = 0, from (5.19), ~� = 0. Thus, by letting � = 0 in

(5.20), from B2 of (5.5) and (5.13), we have that 8us 2 u

LS of (4.4) � �WI + ~�T� �I +@V@��� � ~�T� �

TEz + zTEf�E�1Qz + Lusg� �W + ~�T� � +

@V@���

(5.22)

where W =WI + zTQz. Thus, Assumption 3 is satis�ed. 4

Lemma 5 If we can choose us 2 u such that

zT [E(L� ~B1e )us + �3e ] � "e(t) (5.23)

where "e is a design parameter and

�3e(x; ��(l�)� ; u; t) = Ef��(x; ��(lV )� ; ua(x; ��

(l�)� ; t); t)~�� + ~�(x; ��

(kI)� ; u; t)g (5.24)

then, Assumption 4 is satis�ed by V for the system (5.8) with �V = minf�VI ; 2�min(Q)�max(E)

g and cV =

cVI + "e. }

Proof. From (5.20), (5.13), B3 of (5.5), and (5.23),

LS of (4.9) � ��VIVI + cVI � ~�T� �TEz + zTEf�E�1Qz + Lus + ~�g

� ��V V + cVI + zTEf��0~�� � ~Beu + Lus + ~�g= ��V V + cVI + zT

nE[L� B1

e (x; ~��; t)]us + �3e

o� ��V V + cV

(5.25)

which completes the proof. 4

49

Remark 14 One solution to (5.23) can be found in the following way. Noticing (5.19), similar to (3.9),

there exists a function h(x; ��(l�)� ; t) satisfying

h(x; ��(l�)� ; t) � sup�2�; �2�

k�3e(x; ��(l�)� ; u; t)k (5.26)

For example, let

h � �MkE�e(x; ��(lV )� ; ua; t)k+ kEUTe@�I@xI

DIk�I(xI ; t) + kEDek�e(x; t) (5.27)

where �M = k�max � �min + "�k. Choose �u such that

�u(x; ��(l�)� ; t) � sup�2� kEB1

e(x; ~��; t)L�1E�1k (5.28)

which is not di�cult to calculate since B1e is linear w.r.t. ~�� . We assume that �u < 1. In the absence

of input channel parametric uncertainties (i.e., B1e = 0), we can set �u = 0. So as long as the input

channel uncertainties are not big, �u < 1 can be satis�ed. By choosing

us(x; ��(l�)� ; t) = � 1

4(1��u)"eh2L�1E�1z (5.29)

from (5.27) and (5.28), we have

LS of (5.23) � 14(1��u)"eh

2(�kzk2 + kzkkEB1e(x;

~��; t)L�1E�1kkzk) + kzkk�3ek

� �( 12p"ehkzk � p"e)2 + "e � "e

(5.30)

Thus, (5.23) is satis�ed. }In viewing of Lemmas 3 to 5, we have the following theorem:

Theorem 8 Under the Assumptions 5-10, if L is nonsingular and (5.23) is satis�ed, V de�ned by (5.11)

is an ARC Lyapunov function for the augmented system (5.8) with the control functions ua given by

(5.13) and us determined from (5.23). The adaptation function � is given by (5.14). 4Remark 15 Noticing that the last two terms of L in (5.15) are linear w.r.t. � and Be is nonsingular,

nonsingularity of L can be guaranteed by using a small adaptation rate �. Also, since our controller

guarantees transient performance, the states can be restricted with a known compact region. Thus, an

allowable range of � without making L singular may be calculated o�-line. }

5.4 Augmented MIMO Nonlinear Systems II

In the above section, we constructed an ARC Lyapunov function for the augmented system

(5.8). The state equations of the added system (5.7) are required to be linearly parametrized by the

unknown parameter vector � when �e = 0. For most mechanical systems, such as robot manipulators,

their state equations cannot be linearly parametrized in terms of a set of unknown parameters. To include

those applications, let us consider the following nonlinear system with the state vector �e = [xTe ; �T ]T ,

the input vector ue 2 Rm, and the output vector ye = xe 2 Rm

_xe = M�1(xI ; �; t)�f0e (x; t) + Fe(x; t)� + F�(x; t)�

+Be(x; �; �; t)ue +De(x; t)�e(x; �; �; ue; t)]

_� = ��(x; �; �; t) � 2 Rn�

(5.31)

in which � 2 Rl� is a vector of some unknown parameters, �e satis�es Assumption 7, and the �-

subsystem satis�es Assumption 9. We make the following assumptions:

50

Assumption 11 � 2 ��= f� : �min < � < �max g where � is a known set. }

Assumption 12 Be is nonsingular and

Be(x; �; �; t) = B0e (x; t) +Be�(x; �; t) +Be�(x; �; t) (5.32)

where Be�(x; �; t) and Be�(x; �; t) are linear w.r.t. � and � respectively. }

Assumption 13 .

P1 . M(xI ; �; t) is a s.p.d. matrix and there exist positive scalars km and kM such that kmIm �M(xI ; �; t) � kMIm.

P2 . M(xI ; �; t) =M0(xI ; t) +M�(xI ; �; t) in which M� is linear in terms of �. }

Basically, the above system has the same meaning as that in (5.8) except that its state

equations cannot be linearly parametrized because of the appearance ofM�1(xI ; �; t) in the right hand

side of (5.31) (although M is assumed to be linearly parametrized in terms of �, in general M�1

cannot be linearly parametrized). ��(x; �; �; t) is not required to be linearly parametrized. Introducing

M greatly expands the applicability of the method since most mechanical systems, including robot

manipulators, satisfy (5.31) but not (5.8) as will be shown later.

Now connecting the system (5.31) to the initial system (5.1) in the same way as in (5.8), we

obtain the following augmented system

_xI = f0I (xI ; t) + FI(xI ; t)� +BI (xI ; �; t)�xemI+DI(xI ; t)�I(xI ; �; �xemI

; t)

_xe = M�1(xI ; �; t)�f0e + Fe(x; t)� + F�(x; t)� +Be(x; �; �; t)u+De(x; t)�e

�_� = ��(x; �; �; t) � 2 Rn�

y = [yTI ; (NTe xe)

T ]T

(5.33)

We assume that Assumption 10 is satis�ed and proceed to construct an ARC Lyapunov function for

(5.33). De�ne z as in (5.11) and V as

V (x; ��(lV )� ; �; t) = VI(xI ; ��

(lI)� ; t) + 1

2zTM(xI ; �; t)z (5.34)

Similar to (5.4), we let GrM(x; �; t) and Gl

M(x; �; t) denote the right and left substitution matrices of the

matrixM�(x; �; t) in terms of �. Let Gre�(x; �; t) and Gl

e�(x; �; t) denote the right and left substitution

matrices of the matrix Be�(x; �; t) (in terms of �) respectively, and Gre�(x; �; t) and Gl

e�(x; �; t) forBe�(x; �; t) (in terms of �).

Lemma 6 V in (5.34) satis�es Assumption 2 for the system (5.33). }

Proof. In viewing P1 of Assumption 13, Lemma 6 can be proved in the same way as in

Lemma 3. 4

Lemma 7 If L(x; ��(l�)� ; ��; t) is nonsingular, by letting u = fus : zTLus � 0g and and choosing ua

as

ua(x; ��(l�)� ; ��; #�; t) = L�1(x; ��(l�)� ; ��; t)[�Qz + �1e(x;

��(l�)� ; ��; #�; t)] (5.35)

V of (5.34) satis�es Assumption 3 for the system (5.33). The detailed expressions of L and �1e will be

given in the following proof. }

51

Proof. From (5.11) and (5.34),

@V@�

= @VI@�

� zTM(xI ; �; t)UTe@�I@�

(5.36)

From (5.33), each component of _M is

_Mij =@Mij

@xI[f0I + FI� + Gr

I(xI ; �xemI; t)�] +

@Mij

@xIDI�I +

@Mij

@t(5.37)

Since M(xI ; �; t) can be linearly parametrized in terms of � (P2 of Assumption 13), so do @Mij

@xIand

@Mij

@t . Therefore, when �I = 0, from (5.37), _M(xI ; �; t) can be linearly parametrized in terms of �; �

and #, where # = [�1�T ; �2�

T ; : : : ; �l��T ]T 2 Rl�p. For simplicity, let �e denote �e = [�T ; �T ; #T ]T in

the following. Thus, there exist known vectors or matrices dM (xI ; �; t); DM�(xI ; �; t); DM�(xI ; �; t)and DM#(xI ; �; t) such that

12v

T _M(xI ; �; t)v = vT [dM(xI ; v; t) +DM�(xI ; v; t)�+DM�(xI ; v; t)�

+DM#(xI ; v; t)#+ ~�M(x; v; �e; t) 8v (5.38)

where ~�M linearly depends on �I and can be bounded by a known function �M , i.e.,

k ~�M(x; v; �e; t)k � �M (x; v; t) (5.39)

Viewing (3.2) and Assumption 11, # 2 #, where # is a known bounded set given by

# = f# : #min < # < #max g. So we can de�ne �� = ��(�), the projection of �, and #� = �#(#),

the projection of #, in the same way as in (3.4).

From (5.33), noting (5.6), the derivative of V is

_V = @VI@xI

_xI +@VI@�

_� + @VI

@t + zT (M _xe �M _��+ 12_Mz)

= DVI +@VI@�

(_� + ��I) + zT

�@VI@xI

BI(x; �; t)Ue�T

+zT�f0e + Fe� + F�� + Be(x; �; �; t)u+De�e

�MUTe

�@�I@xI

(f0I + FI� +BI �xemI+DI�I) +

@�I@�

_�

��M @��

@t +12_Mz

� (5.40)

Since M� is linear w.r.t. �, there exists a matrix Dp#(x; t), whose elements are known functions, such

that

M�(xI ; �; t)UTe [FI� + B1

I (xI ; �; t)�xemI] = Dp#(x; t)# (5.41)

Noting (5.36), using similar techniques as in (5.18) and substituting (5.38) and (5.41) into (5.40), we

have_V = DVI +

@V@�

_� + @VI

@���I + zT

n��0e + �0�� + �0�� + �0##+ Beu

�Gre�(x; u; t)

~�� �Gre�(x; u; t)

~�� + ~�o (5.42)

where

�0e(x; ��(lV )� ; t) = �

�@VI@xI

(xI ; ��(lI)� ; t)B0

I (xI ; t)Ue�T � f0e (x; t) +M0

@��@t (xI ;

��(kI)� ; t)

+M0(xI ; t)UTe@�I@xI

(xI ; ��(kI)� ; t)[f0I (xI ; t) + B0

I �xemI]� dM(xI ; z; t)

�0�(x;��(lV )� ; t) = UT

e GlTI (xI ;

�@VI@xI

�T; t) + Fe(x; t) +DM�(xI ; z; t)

�M0(xI ; t)UTe@�I@xI

(xI ; ��(kI)� ; t)[FI(xI ; t) + Gr

I(xI ; �xemI; t)]

�0�(x;��(kI)� ; t) = F�(x; t)� Gr

M(xI ;�UTe@�I@xI

[f0I (xI ; t) +B0I (xI ; t)�xemI

] + @��@t

�; t)

+DM�(xI ; z; t)

�0#(x;��(kI)� ; t) = DM#(xI ; z; t)�Dp#(x; t)

~� = De�e �MUTe@�I@xI

DI�I + ~�M

(5.43)

52

De�ne��(x; ��

(lV )� ; u; t) = �0�(x;

��(lV )� ; t) + Gr

e�(x; u; t)

�0� (x;��(lV )� ; t) = �I(x; ��

(kI)� ; �I ; t)� �0T� (x; ��

(lV )� ; t)z

��(x; ��(lV )� ; u; t) = �0� (x;

��(lV )� ; t)� GrT

e� (x; u; t)z

(5.44)

Then, from (5.36),

@VI@�

��I = @V@���� + zT

�MUT

e@�I@�

��� + ����@VI@�

�T�

= @V@���� + zT

�M0U

Te@�I@�

��� + GrM(xI ; UT

e@�I@�

��� ; t)� + ����@VI@�

�T� (5.45)

Further, de�ne

��(x; ��(lV )� ; u; t) = �0�(x;

��(lV )� ; t) + Gr

M(xI ;�UTe@�I@�

���(x; ��(lV )� ; u; t)

�; t) + Gr

e�(x; u; t)

��(x; ��(lV )� ; u; t) = ��T� (x; ��(lV )� ; u; t)z

�#(x; ��(kI)� ; t) = ��0T# (x; ��

(kI)� ; t)z

�e(x; ��(lV )� ; u; t) = [�T� ; �

T� ; �

T# ]

T

~�e� = [~�T� ;~�T� ;

~#T� ]T

�0e(x;��(lV )� ; t) = [�0T� ; �0T� ; �0T# ]T

�e(x; ��(lV )� ; u; t) = [�T� ; �

T� ; �

0T# ]T

(5.46)

Then, substituting (5.45) into (5.42), we have

_V = DVI +@V@�(_� + ���) + zT

�M0U

Te@�I@�

��� +GrM (xI ; U

Te@�I@�

��� ; t)�� + ����@VI@�

�T��0e + �0��� + �0� �� + �0##� + Beu + ~�

o� zT�� ~�� + �T�

~�� + �T#~#�

= DVI +@V@�(_� + ���)� �TI ~�� + �Te

~�e� + zTn��1e + Lu+ ~�

o (5.47)

where

�1e(x;��(l�)� ; ��; #�; t) = �M(xI ; ��; t)UT

e@�I@�

��0� � �0���@VI@�

�T+ �0e(x;

��(lV )� ; t)� �0e �e�

L(x; ��(l�)� ; ��; t) = Be + B1

e (x;��@VI@�

�T; t)�M(xI ; ��; t)U

Te ZB(x;

��(l�)� ; t)

ZB(x; ��(l�)� ; t)

�=

266664

zTBe�(x;��@�I1@�

�T; t)

...

zTBe�(x;��@�ImI

@�

�T; t)

377775

(5.48)

in which transformations similar to the one in (5.21) have been used. When � = 0, from (5.43), ~� = 0.

Then, from (5.47), B2 of (5.5), and (5.35), by noting that V does not depend on � and #, we have

that 8us 2 u

LS of (4.4) � �W + ~�Te��e +@V@���� = �W + ~�Te��e +

@V@�e

�e�e (5.49)

where W = WI + zTQz and �e = diagf�; �� ; �#g, in which �� and �# are any s.p.d. matrices.

Thus, Assumption 3 is satis�ed by V in terms of the augmented unknown parameter vector �e. 4

53

Lemma 8 If we can choose a us 2 u such that

zT [(L� ~Lu)us + �3e ] � "e(t) (5.50)

where

~Lu(x; ��(l�)� ; ��; t) = Be�(x; ~��; t) + Be�(x; ~��; t)�M�(x; ~��; t)UeZB(x; ��

(l�)� ; t)

�3e(x; ��(l�)� ; ��; #�; u; t) = ��e(x; ��(lV )� ; ua; t)~�e� + ~�

(5.51)

then, Assumption 4 is satis�ed by V for the system (5.33) with �V = minf�VI ; 2�min(Q)kM

g and

cV = cVI + "e. }

Proof. Noting (5.44), (5.46), (5.35), and (5.48), from (5.47), B3 of (5.5), and (5.50),

LS of (4.9) � ��VIVI + cVI � ~�T� �T� z +

~�T� �� +~#T� �# + zT f�Qz + Lus + ~�g

= ��V V + cVI + zTn��� ~�� � �� ~�� � �# ~#� + Lus + ~�

o= ��V V + cVI + zT [(L� ~Lu)us + �3e ] � ��V V + cV

(5.52)

which completes the proof. 4

Remark 16 One solution to (5.50) can be found in the same way as in Remark 14 except that h and

�u are given by

h � �eMk�e(x; ��(lV )� ; ua; t)k+ kMkUTe@�I@xI

DIk�I + kDek�e + �M (x; z; t)

�u(x; ��(l�)� ; t) � sup�2� k~LuL�1(x; ��

(l�)� ; t)k (5.53)

where �eM = k�emax � �emin + "�ek. }

In viewing of Lemmas 6 to 8, we have the following theorem:

Theorem 9 Under the Assumptions 5-7, 9, 10, and 11-13, if L is nonsingular and (5.50) is satis�ed, V

de�ned by (5.34) is an ARC Lyapunov function for the augmented system (5.33). The control function

ua is given by (5.35), us is determined from (5.50), and the adaptation function �e for the augmented

parameter �e is given by (5.46). 4

Remark 17 Although the state equations of the system (5.33) cannot be linearly parametrized, by

introducing M as a weighting matrix in forming V in (5.34), _V can be linearly re-parametrized, which

makes the utilization of adaptive control possible. }

54

Chapter 6

Adaptive Robust Control of MIMO

Nonlinear Systems

In Chapter 3, we presented the ARC for a class of SISO nonlinear systems transformable to a

semi-strict-feedback form. In this chapter, by recursively applying the backstepping design procedure in

chapter 5, we solve the ARC for a class of MIMO nonlinear systems with arbitrary known relative degrees

and transformable to a semi-strict-feedback form. The form extends the SISO semi-strict feedback form

and allows parametric uncertainties in the input channels also. This class of systems include mechanical

systems, such as robot manipulators, as will be shown in Part 2 of this dissertation.

6.1 MIMO Semi-Strict Feedback Form

The proposed MIMO semi-strict feedback form is an inter-connection of r subsystems. The

i-th subsystem is a MIMO nonlinear system with the state vector �i = [xTi ; �Ti ]

T ; xi 2 Rni ; �i 2 Rmi ,

the input vector vi 2 Rmi and the output vector xi, which is described

_xi = f0i (��i; t) + Fi(��i; t)� + Bi(��i; �; t)vi +Di(��i; t)�i(�; �; t)

_�i = �0i (��i; t) + �1

i (��i; t)� 1 � i � r � 1(6.1)

where ��i = [�T1 ; : : : ; �Ti ]

T , � = ��r . The vectors or matrices, f0i ; Fi; Bi; Di; �0i ; and �1

i ; are

known functions of their variables, which include ��i�1, the states of all its previous subsystems. �i is

the vector of unknown nonlinear functions. The r-th subsystem has the same meaning as (6.1) except

that its dynamics are in the form of (5.31), which are described by

_xr = M�1(��r�1; �; t)[f0r (�; t) + Fr(�; t)� + F�(�; t)�

+Br(�; �; �; t)vr +Dr(�; t)�r(�; �; t)]

_�r = �r(�; �; �; t)

(6.2)

We will use the following notation: �xi = [xT1 ; : : : ; xTi ]

T , ��i = [�T1 ; : : : ; �Ti ]

T , x = �xr, � = ��r,

Ni = [0 Imi�mi�1 ]T 2 Rmi�(mi�mi�1); and Ui = [Imi�1 0] 2 Rmi�1�mi ; 8i.

We assume that 0 = m0 = m1 � m2 � : : : � mr = m. Now, connect the �rst mi outputs

of the i + 1-th subsystem to the inputs of the i-th subsystem, i.e., vi = �xi+1;mi= Ui+1xi+1. The

remaining outputs of the i + 1-th subsystem, yi+1b = NTi+1xi+1, become the i + 1-th output block of

55

the system. The inputs of the r-th subsystem are the real inputs u. The entire system is thus described

by_xi = f0i (��i; t) + Fi(��i; t)� + Bi(��i; �; t)�xi+1;mi

+Di(��i; t)�i(�; �; t)

_�i = �0i (��i; t) + �1

i (��i; t)� 1 � i � r � 1

_xr = M�1(��r�1; �; t)[f0r + Fr� + F�(�; t)� +Br(�; �; �; t)u+Dr(�; t)�r]

_�r = �r(�; �; �; t)

y = [yT1b; : : : ; yTrb]

T 2 Rm

(6.3)

We make the following assumptions about the system (6.3):

Assumption 14 . 8i � r�1; Bi(��i; �; t) is nonsingular and Bi(��i; �; t) = B0i (��i; t)+B

1i (��i; �; t)where

B1i (��i; �; t) is linear w.r.t. �. Similarly, Br(�; �; �; t) is nonsingular and Br = B0

r (�; t) +Br�(�; �; t) +

Br�(�; �; t), where Br� and Br� are linear w.r.t. � and � respectively. }

Assumption 15 . M(��r�1; �; t) is a s.p.d. matrix and there exist positive scalars km and kM such

that kmIm � M(��r�1; �; t) � kMIm. Furthermore, M(��r�1; �; t) = M0(��r�1; t) +M�(��r�1; �; t) inwhich M� is linear w.r.t. �. }

Assumption 16 The �i- subsystem is bounded-input bounded-state (BIBS) stable w.r.t. the input

(��i�1; xi). }

Assumption 17 There exist known functions �i(��i; t) such that

k�i(�; �; t)k � �i(��i; t) i = 1; : : : ; r (6.4)

In Eq. (6.3), the output vector y is partitioned into r blocks, and the outputs of the i-th

block, yib (empty if mi = mi�1), have a relative degree r � i + 1. In this way, we can have relative

degrees ranging from 1 to r and solve the problem that di�erent outputs of a MIMO system can have

di�erent relative degrees.

Similar to the SISO system (3.1), we call (6.3) a semi-strict feedback form in that only the

bounding functions of the unknown nonlinear functions �i are required to be the function of ��i and t

only, and �i(�; �; t) can contain bounded functions of �j ; j > i and thus damage the strict-feedback

property. For SISO systems (i.e., mr = 1), when �i(�; t) = 0, B1i = 0; 8i; (i.e., in the absence of

unknown nonlinear functions and parametric uncertainties in the input channels) and � does not present

in the r-th subsystem, (6.3) reduces to the parametric strict-feedback form in [62] where an adaptive

controller was developed. For the SISO parametric-strict feedback form, the case of Bi(��i; �; t) being

an unknown positive scalar bi is also studied in [65], where over-parametrization about bi is used, i.e.,

two parameter estimates are needed for one bi.

6.2 Backstepping Design Procedure

In this section, the backstepping design procedure in Chapter 5 will be applied to solve the ARC

of the system (6.3). In the following recursive design, we need to use trajectory initialization to satisfy the

compatibility conditions of the connections like (5.9) to obtain the guaranteed transient performance as

in Chapter 3. For this purpose, instead of tracking the desired outputs yd(t) = [yTd1b; : : : ; yTdrb]

T directly,

56

we will deign the controller to track the �ltered outputs yt(t) = [yTt1b; : : : ; yTtrb]

T , in which the i-th block

of outputs, ytib 2 Rmi�mi�1 , are created by the following (r � i+ 1)-th order stable system

(y(r�i+1)tib � y(r�i+1)dib ) + �i1(y

(r�i)tib � y

(r�i)dib ) + : : :+ �i(r�i+1)(ytib � ydib) = 0 (6.5)

Such a procedure enables us to choose the initial conditions ytib(0); : : : ; y(r�i)tib (0) freely. Also, we would

like to know the explicit dependence of the control law on yt and its derivatives so that we can perform

trajectory initialization. In the general development in Chapter 5, yt and its derivatives are considered

as known functions of t and thus only t appears in the expression of the control law. In the following,

they will appear as variables instead of functions of t in any function.

Let Gri (��i; �; t) and Gl

i(��i; �; t) denote the right and the left substitution matrices of the matrix

B1i (��i; �; t), G

rr�(�; �; t) and Gl

r�(�; �; t) for the matrix Br�(�; �; t), Grr�(�; �; t) and Gl

r�(�; �; t) for thematrix Br�(�; �; t), and G

rM(��r�1; �; t) and Gl

M(��r�1; �; t) for the matrixM�(��r�1; �; t). Also, denote�y(k)tib = [yTtib; : : : ; (y

(k)tib )

T ]T ; 8k. Recursively de�ne yju by y1u = yt1b; : : : ; yju = [y(1)Tj�1u; yTtjb]

T =

[y(j�1)Tt1b ; : : : ; yTtjb]

T 2 Rmj , and Lju by L1u = L1; : : : ; Lju = [UTj Lj�1u Nj ]Lj. The design proceeds

in the following steps:

6.2.1 Step 1

The �rst system is de�ned as the �rst subsystem of (6.3), which is described by

_x1 = f01 (�1; t) + F1(�1; t)� + B1(�1; �; t)�x2;m1 +D1(�1; t)�1

_�1 = �01(�1; t) + �1

1(�1; t)�

�y1 = y1b = x1

(6.6)

By treating �x2;m1 as the control for (6.6), comparing (6.6) with the system (5.8), and noting Assumptions

14, 16 and 17, we can see that the �rst system can be considered as a special case of (5.8) with mI = 0.

x1; �1; �x2;m1 ; f01 ; F1; B1; D1, and �1 in (6.6) correspond to xe; �; u; f

0e ; Fe; Be; De; and �e in

(5.8), respectively. Therefore, we can apply the backstepping design procedure in section 5.3 to �nd an

ARC Lyapunov function V1 for the �rst system. V1 is given by (5.11), i.e.,

V1(x1; yt1b(t)) =12z

T1 E1z1

z1(x1; yt1b) = x1 � �0(yt1b) �0(yt1b) = yt1b(6.7)

with the associated control functions given by (5.13) and (5.23). For simplicity, in the following, usis chosen according to (5.29). Noting (5.19), (5.14), and (5.15), after some tedious substitutions and

calculations, we obtain the �nal form of the control as

�1(�1; ��; �y(1)t1b(t); t) = �1a + �1s

�1a(�1; ��; �y(1)t1b(t); t) = L�11 [y

(1)t1b � f01 � �01�� � E�1

1 Q1z1]

�1s(�1; ��; �y(1)t1b(t); t) = � 1

4(1��u1)"e1h21L

�11 E�1

1 z1

(6.8)

where�01(�1; t) = F1L1(�1; ��; t) = B1 = B1(�1; ��; t)

h1(�1; ��; �y(1)t1b(t); t) � �MkE1[�

01 +Gr

1(�1; �1a; t)k+ kE1D1k�1(�1; t)�u1(�1; ��; t) � sup�2� kE1B

11(�1;

~��; t)L�11 E�1

1 k(6.9)

57

The adaptation function is given by (5.14) where

�1(�1; ��; �y(1)t1b; t) = � �

�01 + Gr1(�1; �1; t)

�TE1z1 (6.10)

By choosing

yt1b(0) = x1(0) (6.11)

from (6.7), we have

z1(0) = 0 (6.12)

From (6.8), the control law has the following structure

�1 = L�11u (�1; �� ; t)y(1)t1b + �1y(�1; ��; �y

(1)t1b; t)� �1p(�1; ��; yt1b; t)

�1y = �1s � L�11uE�11 Q1z1

�1p = L�11u [f01 + �01�� ]

(6.13)

with the property that every element of �1y contains z1 as a factor.

6.2.2 Step 2

Now, augment the �rst system by the second subsystem in the same way as in section 5.2 to

obtain the second system. Then, the second system has the state vector ��2, the input vector �x3;m2 ,

and the output vector �y2 = [yT1b; yT2b]

T . From (6.12) and (6.13), �1y(0) = 0. By choosing

y(1)t1b(0) = L1u(�1(0); ��(0); 0)[�x2;m1(0) + �1p(�1(0); ��(0); yt1b(0); 0)]

= B(�1(0); ��(0); 0)[�x2;m1(0) + f01 (�1(0); 0)+ �01(0)��(0)](6.14)

we have,

�x2;m1(0) = �1(0) (6.15)

Thus, with �1; x2; �2; �x3;m2 , and �y2 corresponding to xI ; xe; �; u, and y in (5.8) respectively,

the second system satis�es all the assumptions in section 5.2, and we can apply the backstepping

design results again to obtain an ARC Lyapunov function V2 with the associated control law �2 and

the adaptation function �2. Detailed expressions will be obtained from the general expressions in the

following.

6.2.3 Step i

In general, 8i � r � 1, the i-th system of (6.3) is the ��i-system with the input vector

�xi+1;mi2 Rmi and the output vector �yi 2 Rmi , i.e.,

_xj = f0j (��j ; t) + Fj(��j ; t)� +Bj(��j ; �; t)�xj+1;mj+Dj(��j ; t)�j(�; �; t)

_�j = �0j(��j ; t) + �1

j(��j ; t)� 1 � j � i

�yi = [�yTi�1; yTib]T = [yT1b; : : : ; y

Tib]

T

(6.16)

58

The (i� 1)-th system can be rearranged as

_��i�1 =

266666664

f01 + B01(�1; t)�x2;m1

�01(�1; t)...

f0i�1(��i�1; t)�0i�1(��i�1; t)

377777775+

266666664

F1(�1; t) +Gr1(�1; �x2;m1 ; t)

�11(�1; t)...

Fi�1(��i�1; t)�1i�1(��i�1; t)

377777775�

+

266666664

0

0...

Bi�1(��i�1; �; t)0

377777775�xi;mi�1 +

266666664

D1

0. . .

Di�10

377777775

2664

�1...

�i�1

3775

�yi�1 = [yT1b; : : : ; yT(i�1)b]

T

(6.17)

which has the form of the xI -subsystem in (5.1) with ��i�1 corresponding to xI in (5.1). Thus, the

i-th system can be considered as the system obtained by augmenting the i � 1-th system by the i-th

subsystem in the same way as in section 5.2. xi; �i; �xi+1;mi; f0i ; Fi; Bi; Di, and �i in (6.16)

correspond to xe; �; u; f0e ; Fe; Be; De; and �e in (5.8) respectively.

Lemma 9 8i < r, the backstepping design results in section 5.3 can be recursively applied to �nd an

ARC Lyapunov function, Vj(��j ; ��(j�1)� ; �y

(j�1)t1b ; : : : ; �y

(0)tjb ; t), an associated control law �j(��j ;

��(j)� ; �y

(j)t1b; : : : ; �y

(1)tjb ; t),

and an adaptation function �j(��j ; ��(j)� ; �y

(j)t1b; : : : ; �y

(1)tjb ; t) for each j-th system where j � i. At each step

j, by de�ning

zj = xj � ��j�1; ��j�1 = [�Tj�1; yTtib]

T (6.18)

zj(0) = 0 is achieved by choosing the initial values of the �ltered reference trajectories, yju(0).

Furthermore, �j has the following structure

�j = L�1ju (��j ; ��(j)� ; �y

(j�1)t1b ; : : : ; �y

(0)tjb ; t)y

(1)ju + �jy(��j ; ��

(j)� ; �y

(j�1)t1b ; : : : ; �y

(0)tjb ; y

(1)ju ; t)

��jp(��j ; ��(j)� ; �y(j�1)t1b ; : : : ; �y

(0)tjb ; t)

(6.19)

with the property that every term in �jy contains a zk as a factor for some k � j. }

Proof. We proceed to prove the lemma by induction. We assume that Lemma 9 is true for

i� 1 and are going to show that it is true for i-th system to complete the proof.

Since zk(0) = 0; 8k � i � 1; and every term in �i�1y contains a zl as a factor for some

l � i� 1, we have �i�1y(0) = 0. Thus by choosing

y(1)i�1u(0) = Li�1u(��i�1(0); ��

(i�1)� (0); �y

(i�2)t1b (0); : : : ; �y

(0)i�1t(0); 0)[�xi;mi�1(0)

+�jp(��i�1(0); ��(i�1)� (0); �y

(i�2)t1b (0); : : : ; �y

(0)i�1t; t)]

(6.20)

from (6.19), we have

�xi;mi�1(0) = �i�1(0) (6.21)

Thus, all Assumptions in section 5.2 are satis�ed by the i-th system and the backstepping design results

in section 5.3 can be applied. An ARC Lyapunov function Vi can be found by (5.11), which is rewritten

59

asVi(��i; ��

(i�1)� ; �y

(i�1)t1b ; : : : ; ytib; t) = Vi�1 + 1

2zTi Eizi =

Pij=1

12z

Tj Ejzj

zi(��i; ��(i�1)� ; �y

(i�1)t1b ; : : : ; ytib; t) = xi � ��i�1

��i�1(��i; ��(i�1)� ; �y

(i�1)t1b ; : : : ; ytib; t) = [�Ti�1(��i; ��

(i�1)� ; �y

(i�1)t1b ; : : : ; �y

(1)ti�1b; t); y

Ttib]

T :

(6.22)

The associated control is given by (5.13) and (5.29). By noting (5.19), (5.14), and (5.15), and after

some tedious substitutions and calculations, we obtain the �nal form of the control as

�i(��i; ��(i)� ; �y

(i)t1b; : : : ; �y

(1)tib ; t) = �ia + �is

�ia = L�1in�E�1

i Qizi � E�1i UT

i B0Ti�1Ei�1zi�1 + UT

i

Pi�1j=1

h@�i�1

@xj(f0j + B0

j �xj+1;mj)

+@�i�1

@�j�0j

i+ UT

i

"Pi�1j=1

Pi�jk=0

@�i�1

@y(k)tjb

y(k+1)tjb +

@�i�1

@t

#+Niy

(1)tib � f0i

��0ih�� � �

Pi�2j=1(

@�j

@�)TUjEj+1zj+1

i� UT

i@�i�1

@��(�i�1 � �0Ti Eizi)

o�is = � 1

4(1��ui)"eih2iL

�1i E�1

i zi

(6.23)

where

�0i = E�1i UT

i GlTi�1(��i�1; Ei�1zi�1; t)� UT

i

Pi�1j=1

n@�i�1

@xj[Fj

+Grj(��j ; �xj+1;mj

; t)] + @�i�1

@�j�1j

o+ Fi

Li = Bi +B1i (��i;��

Pi�2j=1(

@�j

@�)TUjEj+1zj+1; t)� UT

i ZBi

ZBi =

26664

zTi EiB1i (��i;�(

@�i�1;1

@�)T ; t)

...

zTi EiB1i (��i;�(

@�i�1;mi�1

@�)T ; t)

37775

hi � �MkEi[�0i + Gr

i (��i; �ia; t)k+Pi�1

j=1 kEiUTi@�i�1

@xjDjk�j + kEiDik�i

�ui � sup�2� kEiB1i (��i;

~�� ; t)L�1i E�1

i k

(6.24)

The adaptation function is given by (5.14)

�i = �i�1 ���0i + Gr

i (��i; �i; t)�TEizi (6.25)

Noting (6.19) for �i�1, the terms in �i which contain y(1)iu are

L�1i [UTi

Pi�1j=1

@�i�1

@y(i�j)tjb

y(i�j+1)tjb +Niy

(1)tib ] + �is(��i; ��

(i)� ; �y

(i)t1b; : : : ; �y

(1)tib ; t)

= L�1i [UTi

@�i�1

@y(1)i�1u

y(2)i�1u +Niy

(1)tib ] + �is

= L�1i [UTi L

�1i�1uy

(2)i�1u +Niy

(1)it ] + L�1i

@�i�1y

@y(1)j�1u

y(2)j�1u + �is

= L�1iu y(1)iu + �iy

(6.26)

where�iy = L�1i [UT

i@�i�1y

@y(1)i�1u

y(2)i�1u � kzi

1��uih2iE

�1i zi] (6.27)

Since zk ; 8k � i� 1, does not depend on y(1)i�1u and every term of �i�1y has a zk as a factor, so does

@�i�1y

@y(1)i�1u

. It is thus clear from (6.27) that every term of �iy has a zk as a factor for k � i. Thus, �i has

the form (6.19). Obviously, by choosing yiu(0) = [y(1)Ti�1u(0); y

Ttib(0)]

T in terms of (6.20) and

ytib(0) = NTi xi(0) = [xi;mi�1+1(0); : : : ; xi;mi

(0)]T (6.28)

we have zi(0) = 0 by (6.22). (6.22), (6.23), (6.25), (6.20), (6.28), and (6.26) agree with the general

conjectures about the k-th system for k � i� 1 and, thus, are true for every system by induction. 4

60

6.2.4 Step r

By using the general formula in step i, we can recursively �nd the ARC Lyapunov function for

each system until the r� 1-th system. Then, by augmenting the r� 1th system by the r-th subsystem

in the same way as in the augmented system (5.33), we obtain the r-th system, which is the entire

system (6.3). Similar to Step i, if we choose y(1)r�1u(0) in the same way as in (6.20), the compatibility

condition (5.9) will be satis�ed and it is easy to check that all Assumptions for the augmented system

(5.33) are satis�ed. Thus, we can apply the backstepping design results in section 5.4 to obtain an ARC

Lyapunov function for the system (6.3). The �nal form of the ARC Lyapunov function Vr is given by

Vr(�; ��(r�1)� ; �; �y

(r�1)t1b ; : : : ; ytrb; t) =

Pr�1j=1

12z

Tj Ejzj +

12z

Tr M

�1(��r�1; �; t)zrzr(�; ��

(r�1)� ; �y

(r�1)t1b ; : : : ; ytrb; t) = xr � ��r�1 ��r�1 = [�Tr�1; yTtrb]

T :(6.29)

The associated control law is

�r(�; ��(r)� ; �y

(r)t1b; : : : ; �y

(1)trb; t) = �ra + �rs

�ra = L�1rn�Qrzr � UT

r B0Tr�1Er�1zr�1 � �0�

h�� � �

Pr�2j=1(

@�j

@�)TUjEj+1zj+1

i��0� �� � �0##� +M0fB � f0r � MUT

r@�r�1

@���0� � dM(��r�1; zr; t)

o�rs = � 1

4(1��ur)"er h2rL

�1r zr

(6.30)

where�0� = UT

r GlTr�1(��r�1; Er�1zr�1; t)�M0U

Tr

Pr�1j=1

n@�r�1

@xj[Fj

+Grj(��j ; �xj+1;mj

; t)] + @�r�1

@�j�1j

o+ Fr +DM�(��r�1; zr; t)

�0� = �r�1 � �0T� zr

fB = UTr

Pr�1j=1

h@�r�1

@xj(f0j + B0

j �xj+1;mj) + @�r�1

@�j�0j

i+UT

r

"Pr�1j=1

Pr�jk=0

@�r�1

@y(k)tjb

y(k+1)tjb + @�r�1

@t

#+Nry

(1)trb

�0� = F� �GrM (��r�1; fB; t) +DM�(��r�1; zr; t)

�0# = DM#(��r�1; zr; t)�Dp#(�; t)

Lr = Br +B1r (�;��

Pr�2j=1(

@�j@�

)TUjEj+1zj+1; t)� MUTr ZBr

ZBr =

26664

zTr B1r�(�;�(

@�r�1;1

@�)T ; t)

...

zTr B1r�(�;�(

@�r�1;mr�1

@�)T ; t)

37775

(6.31)

and the bounding functions hr and �ur satisfy the following conditions

hr � �eMk�e(�; ��(r)� ; �ra; t)k+ kMPr�1

j=1 kUTr@�r�1

@xjDjk�j + kDrk�r + �M (�; zr; t)

�ur � sup�2� k( ~Br� + ~Br� � ~M�UrZBr)L�1k(6.32)

in which the form of the function �e is de�ned by

��(�; ��(r)� ; �; t) = �0�(�;

��(r)� ; t) +Gr

r�(�; �; t)��(�; ��

(r)� ; �; t) = �0� (�;

��(r)� ; t)� GrT

r� (�; �; t)zr��(�; ��

(r)� ; �; t) = �0� + Gr

M(��r�1;�UTr@�r�1

@����(�; ��

(r)� ; �; t)

�; t) + Gr

r�(�; �; t)�e(�; ��

(r)� ; �; t) = [�T� ; �

T� ; �

0T# ]T

(6.33)

61

The forms of the vectors or matrices dM ; DM�; DM�; DM#; �M and Dp# used in the above are obtained

from (5.38), (5.39) and (5.41). The adaptation function is

�e =

2664

��(�; ��(r)� ; �r; t)

��T� (�; ��(r)� ; �r; t)zr

��0T# (�; ��(r)� ; t)zr

3775 (6.34)

Theorem 10 If the actual control law

u = �r(�; ��(r)� ; ��; #�; �y

(r)t1b; : : : ; �y

(1)rt ; t) (6.35)

with the adaptation law ,

_�e = ��e�e(�; ��r�; ��; #�; �y(r)t1b; : : : ; �y

(1)rt ; �r; t) (6.36)

is applied to the system (6.3) under the Assumptions 14-17, the following results hold

A. In general, the control input and the state are bounded with Vr bounded above by

Vr(t) �R t0 exp(��Vr(t� �))cVr(�)d� � cVrmax

�Vr[1� exp(��Vrt)] (6.37)

where �Vr = minfmini�r�1f2�min(Qi)�max(Ei)

g; 2�min(Qi)kM

g and cVr =Pr

i=1 "ei. Output tracking is

guaranteed to have arbitrary good transient performance and �nal tracking accuracy.

B. If, after a �nite time, there are no unknown nonlinear functions, i.e.,�i = 0; 8t � t0; for some �nite

t0, in addition to the result in A, the system outputs track the desired outputs asymptotically. 4Proof. By using the trajectory initialization (6.20) and (6.28), zi(0) = 0 for all i's. Thus,

Vr(0) = 0. The above theorem is then a Direct application of Theorem 7. 4

6.2.5 Guaranteed Transient Performance

From (6.37, any good transient performance about the Vr can be guaranteed by adjusting

controller parameters Qi and "ei. This result in turn guarantees any good transient performance about

tracking error vector et(t) = y � yt since Vr is a p.s.d. quadratic function of zi, and etib = yib � ytib,

the i-th block of et, is the vector of last mi�mi�1 elements of zi. If the above trajectory initialization

is independent of Qi and "ei, from (6.5), the trajectory planning error, ed(t) = yt(t)� yd(t); convergesto zero exponentially and can be guaranteed to possess any good transient behavior when one suitably

chooses the Hurwitz polynomials Gid(s) = s(r�i+1) + �i1sr�i + : : :+ �i(r�i+1) without being a�ected

by Qj and "ei. Thus, the actual output tracking error, e = et + ed, can be guaranteed to have any

good transient performance. So, in the following, the same as in the SISO case in section 3.4, we

illustrate that the above trajectory initialization actually places y(k)tib (0) at the estimated value of y

(k)ib (0)

by neglecting all unknown nonlinear functions and using ��(0) for � and thus the initialization process

is independent of the controller gains.

From (6.11), yt1b(0) = y1b(0). From (6.3),

y(1)1b = h1(��1; x2; �; t) +D1(��1; t)�1

h1 = f01 (��1; t) + F1(��1; t)� +B1(��1; �; t)x2(6.38)

From (6.14) and (6.38),

y(1)t1b(0) = h1(��1(0); x2(0); ��(0); 0) = y

(1)1b (0) j�=0; �=�� (0)

(6.39)

which supports the above claim. The general proof is very tedious and is omitted.

62

6.3 Simulation Results

Consider the following time-varying system with two inputs u = [u1; u2] and two outputs

_x1 = �1x21 + �2x2;1 + �1(x2)

_x2 =

"(x21 + x22;1) sin

3(t)

x31 + 5x2;2

#�1 + u+

"x1x2;2

#�2(x)

y = [x1; x2;2]T

(6.40)

where�1(x2) = d1 sin(r1x2)

�2(x) = d2 cos(r2x1x2)(6.41)

�1 2 [�1; 1], and �2 2 [10; 15]. r1, r2, d1 2 [0; d1M], and d2 2 [0; d2M] are assumed to be unknown.

It can be observed that (6.40) has input channel uncertainty for x1 dynamics (term �2x2;1) and are

not in a strict feedback form, but it does satisfy the MIMO semi-strict feedback form (6.3) since

j�1(x2)j � �1 = d1M and j�2(x)j � �2 = d2M . Therefore, we can apply the results in section 6.2 to

obtain an ARC law.

In simulation, actual plant parameters are �1 = 1, �2 = 14, r1 = 2, and r2 = 3. Sam-

pling time is 1ms. The same projections as in (3.54) are used. The desired outputs are yd =

[sin(0:5�t); 0:5sin(1:2�t)]T , which have two frequency components. The �lter output trajectories

are created by (6.5) where �11 = 80, �12 = 1600, and �21 = 40. Controller parameters are E1 = 1,

E2 = I2 , Q1 = 10; Q2 = diagf100; 100g, � = diagf100; 500g, kz1 = 0:1, and kz2 = 0:1. Three

controllers are run for comparison:

ARC : the proposed adaptive robust control as described above;

DRC : Deterministic Robust Control | the same control law as in ARC but without the using parameter

adaptation law;

AC : Adaptive Control, which is obtained by setting hi = 0 and without using parameter projection

| i.e., letting �(�) = � in ARC.

To test the nominal performance, a simulation is run for the parametric uncertainties only

(i.e., d1 = d2 = 0). Tracking errors z1 = y1 � yt1 and z2;2 = y2 � yt2 are shown in Fig. 6.1 and

Fig. 6.2 respectively. We can see that all the three controllers have very good tracking ability. The

estimated parameters in ARC and AC approach their desired values as shown in Fig. 6.3. The proposed

ARC has a better transient and a much better �nal tracking accuracy than RC. AC also has a good

�nal tracking accuracy but has the worst transient response. These results substantiate the necessity

of using parameter adaptation to improve �nal tracking accuracy and using robust control to improve

transient performance. Control inputs of all three controllers are smooth and more or less the same as

shown in Fig. 6.4 and Fig. 6.5.

To test the performance robustness, large disturbances are added to the system (i.e., let

d1 = d2 = 2 in (3.52)). The tracking errors z1 and z2;2 are shown in Fig. 6.6 and Fig. 6.7, and

the estimated parameters are shown in Fig. 6.8. The ARC still achieves the best tracking performance

although its estimated parameters run quite wildly. AC has a very large tracking error (actually diverging)

since its estimated parameters run wildly and have a diverging trend. Control inputs are smooth as shown

in Fig. 6.9 and Fig. 6.10. All these results illustrate the e�ectiveness of the proposed ARC.

63

The simulation is also run to track a less rich output trajectory, yd = [(1 � exp(�4t));0:5(1 � exp(�4t))], which exponentially decays to a constant value and thus does not have much

frequency content after a short time. In the presence of parametric uncertainties, the tracking errors

z1 and z2;2 are shown in Fig. 6.11 and Fig. 6.12 . It can be seen that the ARC still achieves perfect

tracking and has a pretty good parameter estimation as shown in Fig. 6.13. It is interesting to observe

that AC diverges quickly during the initial transient because of its wrong parameter adaptation while it

is supposed to achieve asymptotic stability in theory. This result veri�es the non-robustness nature of

pure adaptive control since the closed loop system may be unstable under the e�ect of sampling only

(in this simulation, the only approximation comes from the discrete implementation of the continuous

control laws). This observation further substantiates the necessity of using robust control in the design

of a baseline control law.

6.4 Conclusions

In this chapter, by using the backstepping design procedures in Chapter 5, we have solved

the ARC of a class of MIMO nonlinear systems in a semi-strict feedback form. The form allows both

parametric uncertainties and unknown nonlinear functions. Parametric uncertainties are also allowed

in the input channels of each layer and the linear parametrization requirement is relaxed to include

mechanical systems. Simulation results show that the proposed ARC has a better tracking performance

than its DRC counterpart and a better performance robustness than AC.

0 1 2 3 4 5 6 7 8−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

Time (sec)

Tra

ckin

g er

ror

of z

1

Solid: ARC Dotted: DRC Dashdot: AC

Figure 6.1: Tracking errors in the presence of parametric uncertainties

64

0 1 2 3 4 5 6 7 8−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

Time (sec)

Tra

ckin

g er

ror

of z

2Solid: ARC Dotted: DRC Dashdot: AC

Figure 6.2: Tracking errors in the presence of parametric uncertainties

0 1 2 3 4 5 6 7 80

5

10

15

Time (sec)

Est

imat

ed p

aram

eter

s

Solid: theta1 (ARC) Dashdot: theta2 (ARC)

Dashed: theta1 (AC) Dotted: theta2 (AC)

Figure 6.3: Estimated parameters in the presence of parametric uncertainties

0 1 2 3 4 5 6 7 8−2

0

2

4

6

8

10

12

14

Time (sec)

Con

trol

inpu

t u1

Solid: ARC Dotted: DRC Dotted: AC

Figure 6.4: Control inputs in the presence of parametric uncertainties

65

0 1 2 3 4 5 6 7 8−4

−3

−2

−1

0

1

2

3

4

5

Time (sec)

Con

trol

inpu

t u2

Solid: ARC Dotted: DRC Dotted: AC

Figure 6.5: Control inputs in the presence of parametric uncertainties

0 1 2 3 4 5 6 7 8−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Time

Tra

ckin

g er

ror

of z

1

Solid: ARC Dashed: DRC Dashdot: AC

Figure 6.6: Tracking errors in the presence of parametric uncertainties and disturbances (d1=d2=2)

0 1 2 3 4 5 6 7 8−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Time

Tra

ckin

g er

ror

of z

2

Solid: ARC Dashed: DRC Dashdot: AC

Figure 6.7: Tracking errors in the presence of parametric uncertainties and disturbances (d1=d2=2)

66

0 1 2 3 4 5 6 7 8−10

0

10

20

30

40

50

60

70

Time

Est

imat

ed p

aram

eter

s

Solid: theta1 (ARC) Dashdot: theta2 (ARC)

Dashed: theta1 (AC) Dotted: theta2 (AC)

Figure 6.8: Estimated parameters in the presence of parametric uncertainties and disturbances(d1=d2=2)

0 1 2 3 4 5 6 7 8−4

−2

0

2

4

6

8

10

12

14

Time

Con

trol

inpu

t u1 Solid: ARC Dashed: RC Dotted: AC

Figure 6.9: Control inputs in the presence of parametric uncertainties and disturbances (d1=d2=2)

0 1 2 3 4 5 6 7 8−6

−4

−2

0

2

4

6

Time

Con

trol

inpu

t u2

Solid: ARC Dashed: RC Dotted: AC

Figure 6.10: Control inputs in the presence of parametric uncertainties and disturbances(d1=d2=2)

67

0 1 2 3 4 5 6 7 8−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Time

Tra

ckin

g er

ror

of z

1 Solid: ARC Dotted: DRC Dashdot: AC

Figure 6.11: Tracking errors in the presence of parametric uncertainties

0 1 2 3 4 5 6 7 8−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

Time

Tra

ckin

g er

ror

of z

2 Solid: ARC Dotted: DRC Dashdot: AC

Figure 6.12: Tracking errors in the presence of parametric uncertainties

0 1 2 3 4 5 6 7 8−2

0

2

4

6

8

10

12

14

16

Time

Est

imat

ed p

aram

eter

s

Solid: theta1 (ARC) Dashdot: theta2 (ARC)

Dashed: theta1 (AC) Dotted: theta2 (AC)

Figure 6.13: Estimated parameters in the presence of parametric uncertainties

68

Part II

Adaptive Robust Control - Applications

69

Chapter 7

Trajectory Tracking Control of Robot

Manipulators

In this chapter, the proposed ARC is applied to the trajectory tracking control of robot ma-

nipulators. Two schemes are developed: adaptive sliding mode control (ASMC) is based on SMC and

the conventional adaptation law structure in which the regressor uses the actual state feedback infor-

mation; desired compensation adaptive robust control (DCARC) is based on the desired compensation

adaptation law structure, in which the regressor uses the desired trajectory information only. In addition,

three conceptually di�erent adaptive and robust control schemes | a simple nonlinear PID type robust

control, a gain-based nonlinear PID type adaptive control, and a combined parameter and gain based

adaptive robust control | are derived for comparison. All algorithms, as well as two other established

adaptive schemes, are implemented and compared on the UCB/NSK SCARA direct drive robot.

7.1 Dynamic Model of Robot Manipulators

A dynamic equation of a general rigid link manipulator having n degrees of freedom in free

space can be written as [81]

M(q; �)�q + C(q; _q; �) _q +G(q; �) + ~f(q; _q; t) = u (7.1)

where q 2 Rn is the joint displacement vector, � 2 Rl� is the vector of a suitably selected set of the robot

parameters, u 2 Rn is the applied joint torque, M(q; �) 2 Rn�n is the inertia matrix, C(q; _q; �) _q 2 Rn

is the Coriolis and centrifugal force, G(q; �) 2 Rn is the gravitational force, and ~f(q; _q; t) 2 Rn is the

vector of unknown nonlinear functions such as external disturbances and joint friction.

Equation (7.1) has the following properties that will facilitate the controller design [81, 109,

89, 103, 98].

Property 1 . M(q; �) is a symmetric positive de�nite (s.p.d.) matrix, and there exists km > 0 such

that kmIn�n � M(q; �). Furthermore, for the robot with all joints revolute or prisma, there exists

kM > 0 so that M(q; �) � kMIn�n. For a general robot, M(q; �) � kMIn�n is valid for any �nite

workspace q = fq : kq � q0k � qmaxg where q0 and qmax are some constants.

Property 2 . The matrix N(q; _q; �) = _M(q; �)� 2C(q; _q; �) is a skew-symmetric matrix.

70

Property 3 . M(q; �); C(q; _q; �), and G(q; �) can be linearly parametrized in terms of �. Therefore,

we can write

M(q; �)�qr + C(q; _q; �) _qr + G(q; �) = f0(q; _q; _qr; �qr) + Y (q; _q; _qr; �qr)� (7.2)

where Y 2 Rn�l� , _qr and �qr are any reference vectors.

We assume that Assumption 11 in section 5.4 is satis�ed and the disturbance ~f(q; _q; �; t) can

be bounded by

k ~f(q; _q; t)k � hf(q; _q; t) (7.3)

where hf (q; _q; t) is a known scalar function. We can now formulate the trajectory tracking control of

robot manipulators as follows:

Suppose qd(t) 2 Rn is given as the desired joint motion trajectory. Let e = q(t)� qd(t) 2 Rn

be the motion tracking error. For the robot manipulator described by (7.1), design a control law u so

that the system is stable and q tracks qd(t) as close as possible.

7.2 Control Algorithms

7.2.1 Adaptive Sliding Mode Control (ASMC)

Let x1 = q and x2 = _q. In state space, (7.1) can be rewritten as

_x1 = x2_x2 =M�1(x1; �)[�C(x1; x2; �)x2 � G(x1; �) + u� ~f(x1; x2; t)]

y = x1

(7.4)

Noting (7.3) and Properties 1 and 3, we can see that (7.4) is in the semi-strict feedback form (6.3)

with a relative degree r = 2 and satis�es all assumptions in section 6.1. Thus, we can apply the general

results in section 6.2 to obtain an ARC controller. However, since the �rst equation of (7.4) does not

have any modeling uncertainties, (7.4) satis�es the matching condition. The controller design can,

thus, be simpli�ed and treated as a relative degree one design, which is much easier to deal with, and

powerful SMC techniques can be used in designing the baseline robust control law. The detailed design

procedure follows.

Similar to the DRC controller in section 2.1, a dynamic sliding mode is employed to eliminate

the unpleasant reaching transient and to enhance the dynamic response of the system in sliding mode.

Let a dynamic compensator be

_z = Azz +Bze z 2 Rnc Az 2 Rnc�nc Bz 2 Rnc�n

yz = Czz +Dze yz 2 Rn Cz 2 Rn�nc Dz 2 Rn�n (7.5)

where (Az ; Bz; Cz; Dz) is controllable and observable. The sliding mode controller is designed to make

the following quantity remain zero.

� = _e + yz � 2 Rn

= _q � _qr _qr�= _qd(t)� yz

(7.6)

Transfer function from � to e is

e = G�1� (s)� (7.7)

71

where

G�(s) = sIn +Gc(s) Gc(s) = Cz(sInc � Az)�1Bz +Dz (7.8)

The state space realization of (7.7) is

_x� = A�x� + B�� y� = C�x� (7.9)

where x� = [zT ; eT ]T 2 Rnc+n and

A� =

"Az Bz

�Cz �Dz

#B� =

"0

In

#C� = [0 ; In] (7.10)

From (7.7), G�1� (s) can be arbitrarily assigned by suitably choosing a dynamic compensator transfer

function Gc(s) as long as G�1� (s) has relative degree one. Since during sliding mode, � = 0, the system

response is governed by the free response of transfer function G�1� (s). Therefore, as long as G�1� (s)

is stable, the resulting dynamic sliding mode will be stable and is invariant to various modeling errors.

Furthermore, the sliding mode can be arbitrarily shaped to possess any exponentially fast converging

rate, since poles of G�1� (s) can be freely assigned. In addition, G�1� (s) can be chosen to minimize

the e�ect of � on e when the ideal sliding mode f� = 0g cannot be exactly achieved in practice. The

equivalent results in state space can be stated as follows: there exists an s.p.d. solution P� for any

s.p.d. matrix Q� for the following Lyapunov equation,

AT� P� + P�A� = �Q� (7.11)

Furthermore,�min(Q�)�max(P�)

can be arbitrarily shaped by assigning the poles of A� to the far left plane to

obtain any exponentially fast converging rate. In addition, when Cz is of full column rank, the initial

value z(0) of the dynamic compensator (7.5) can be chosen to satisfy

Czz(0) = � _e(0)�Dze(0) (7.12)

then �(0) = 0. It is shown in [154] that choosing the initial value z(0) in such a way guarantees that

the system is maintained in the sliding mode all the time and the reaching transient is eliminated when

ideal sliding mode control is applied. Therefore, in the following, such a choice is made and �(0) = 0

is used whenever dynamic sliding mode is used.

Noting (7.6) and Property 3, (7.1) can be rewritten as

M(q; �) _�+ C(q; _q; �)� + f0(q; _q; _qr; �qr) + Y (q; _q; _qr; �qr)� + ~f(q; _q; t) = u (7.13)

Let h�(q; _q; _qr; �qr) be a bounding function satisfying

kY (q; _q; _qr; �qr)~��k = kY �� � Y �k � h�(q; _q; _qr; �qr) 8�� 2 � (7.14)

For example, choose

h�(q; _q; _qr; �qr) = kY (q; _q; _qr; �qr)k �M (7.15)

where �M = k�max � �min + "�k. De�ne

hs(q; _q; _qr; �qr; t) = hf (q; _q; t) + h�(q; _q; _qr; �qr; t) (7.16)

72

The following continuous control law is suggested

u = ua + �h(�hs �k�k)

ua = f0(q; _q; _qr; �qr) + Y (q; _q; _qr; �qr)�� �K��(7.17)

where K� is any s.p.d. matrix and �h(�hs �k�k) is a continuous approximation of the ideal SMC control,

�hs �k�k with an approximation error "(t).

De�nition 2 For any discontinuous vector like �h �k�k where h is a positive scalar function and � is a

vector of functions, its continuous approximation, �h(�h �k�k), with an approximation error "(t) is de�ned

to be a vector of functions that satis�es the following two conditions:

i. �T �h(�h �k�k) � 0

ii. hk � k+ �T�h(�h �k�k) � "(t)

(7.18)

}

Remark 18 A natural generalization of the concept of boundary layer [106] to multiple input/output

cases is given by

�h(�h �k�k) = �(1 + �1h)h

�k�k+�(t) (7.19)

where �1 > 0 is any positive scalar, and �(t) is any bounded time-varying positive scalar, i.e., 0 ��(t) � �max, which has the role of boundary layer thickness. It is easy to show [154] that (7.18) is

satis�ed for " = �(t)4�1

. }

Remark 19 A smooth �h(�h �k�k) = [�h1; : : : ; �hn]T is given by

�hi = �htanh�

h�i�i(t)

�(7.20)

From (2.22), condition i of (7.18) is satis�ed and

�T�h = �Pni=1 h �i tanh

�h�i�i

��Pn

i=1(��i � hj �i j) � �Pn

i=1 �i � hk � k (7.21)

Thus, condition ii of (7.18) is satis�ed for " = �Pn

i=1 �i. }

Remark 20 The same as in Remark 4 in Chapter 2, in order to achieve a good tracking accuracy, a

time varying �(t) similar to (2.40) has to be employed, which is quite complicated and is not easily

implemented. To overcome this problem, the following modi�cation is suggested: 1

�h(�hs �k�k) =

8>><>>:�Ks� if k�k � �h �h

�= �(t)

hs(q; _q; _qr;�qr;t)+"1

�(1� c1)Ks� � c1hs �k�k �h � k�k � (1 + "2)�h

�hs �k�k k�k � (1 + "2)�h

(7.22)

where Ks is any s.p.d. matrix, c1 =k�k��h"2�h

, and "1 and "2 are any positive scalars. It can be shown

[154] that (7.18) is satis�ed for " = (1 + "2)�(t).

The above modi�cation is quite simple and yet it provides the desired properties { namely,

around sliding mode fk�k = 0g, a �xed feedback gain matrix is employed all the time and thus can be

1� is replaced by � here to make the presentation clear.

73

chosen near its allowable limit without inducing control chattering. We can also tune the gain around

each joint separately since it is a gain matrix instead of a nonlinear scalar gain. When the system is

away from sliding surfaces, the original nonlinear feedback control law is employed to guarantee the

stability at large. It is shown in [154] by both simulation results and experimental results that the above

modi�cation can achieve a better tracking performance than (7.19). }

Lemma 10 The following p.s.d. function V

V = 12�

TM(q; �)� (7.23)

is an ARC Lyapunov function for (7.1) with the control (7.17) and the adaptation function given by

� = Y T (q; _q; _qr; �qr)� (7.24)

}

Proof. From Property 1, Assumption 2 in section 4.2 is satis�ed by V for (7.1). From

Property 2, 12�

T _M� = �TC�. Noting (7.13) and (7.17), the derivative of V is

_V = �TM _� + 12�

T _M� = �T [M(q; �) _�+ C(q; _q; �)�]

= �T [Y (q; _q; _qr; �qr)~�� � ~f (q; _q; t)�K�� + �h(�hs �k�k)]

(7.25)

When ~f = 0, noting condition i of (7.18), we have

_V j ~f=0� ��TK�� + ~�T� YT (q; _q; _qr; �qr)� (7.26)

Thus, Assumption 3 (4.4) is satis�ed for W = �TK��.

In general, when ~f 6= 0, from (7.3), (7.14), (7.16), (7.25), and condition ii of (7.18),

_V � k�k[kY (q; _q; _qr; �qr)~��k+ k ~f(q; _q; t)k]� �TK�� + �T �h(�hs �k�k)

� k�khs � �TK�� + �T�h(�hs �k�k) � ��TK�� + " � ��V V + "

(7.27)

where �V =2�min(K�)

kM. Thus, Assumption 4 (4.9) is satis�ed, which completes the proof. 4

Noting that V is not a function of �, we can use the adaptation law (4.16) (replacing � with

�) with discontinuous modi�cation (4.19) | i.e., (4.24), which is rewritten here as

_�i =

8>>>>>><>>>>>>:

0 if �i = �imax and (��)i < 0

�(��)i

8><>:�imin < �i < �imax

�i = �imax and (��)i � 0

�i = �imin and (��)i � 0

0 �i = �imin and (��)i > 0

(7.28)

The above results are summarized in the following theorem.

Theorem 11 If the control law (7.17) with the adaptation law (7.28) is applied to the manipulator

described by (7.1), the following results hold:

74

A. In general, all the signals in the system remain bounded and tracking errors, e and _e, exponentially

converge to some balls with size proportional to ". Furthermore, the tracking error � is bounded

by

k�(t)k2 � 2

km[exp(��V t)V (0) +

Z t

0exp(��V (t� �))"(�)d�] (7.29)

In addition, if (7.12) is satis�ed, then V (0) = 0 in (7.29).

B. If after a �nite time, ~f = 0, then the following are true:

a) � �! 0; e �! 0 _e �! 0 when t �! 1 i.e., the robot follows the desired motion

trajectories asymptotically.

Additionally, if the desired motion trajectory satis�es the following persistent excitation con-

dition Z t+T

tY T (qd; _qd; _qd; �qd)Y (qd; _qd; _qd; �qd)d� � "dIkE 8t � t0 (7.30)

where T; t0 and "d are some positive scalars, and q(3)d (t) is bounded, then

b) ~� �! 0 when t �! 1: i.e., estimated parameters converge to their true values. 4

Remark 21 The extra freedom in choosing dynamic sliding mode G�1� (s) can be utilized to minimize

the e�ect of a non-zero � on the tracking error e. For example, if the system is mainly subject to

some constant disturbances, a constant steady state � may appear. By including a di�erentiator in

the numerator of G�1� (s), e.g., G�1� (s) = ss2+kps+ki

In, which can be realized by choosing the dynamic

compensator parameter as Cz = In ; Az = 0 ; Bz = kiIn ; Dz = kpIn, a zero steady state tracking

error e(1) can be obtained. }

Remark 22 By setting us = 0 in (7.17), without using parameter projection and any modi�cation to the

adaptation law, and taking o� the dynamic compensator (i.e., letting Cz = 0; Az = 0; Bz = 0; Dz > 0

in (7.5)), the control law (7.17) reduces to Slotine and Li's well-known adaptive algorithm (SLAC),

which is also tested later for comparison. }

7.2.2 Desired Compensation Adaptive Robust Control (DCARC)

The regressor Y in adaptation function (7.24) depends on the actual state. In [103], Sadegh

and Horowitz proposed a desired compensation adaptation law (DCAL), in which the regressor is calcu-

lated by reference trajectory information only. By doing so, one obtains a resulting adaptation law that

is less sensitive to noisy velocity signals and has a better robustness as well as a signi�cantly reduced

amount of on-line computation. Comparative experiments in [138] demonstrated the superior tracking

performance of the DCAL. Inspired by these results, a desired compensation adaptive robust control

(DCARC) is proposed in this subsection.

It is shown in Appendix 1 that there are known non-negative bounded scalars 1(t), 2(t),

3(t), and 4(t), which depend on the reference trajectory and A� only, such that the following inequality

is satis�ed

kf0(q; _q; _qr; �qr) + Y (q; _q; _qr; �qr)� � f0(qd; _qd; _qd; �qd)� Y (qd; _qd; _qd; �qd)�k� 1kx�k+ 2k�k+ 3k�kkx�k+ 4kx�k2

(7.31)

75

1; 2; 3, and 4 can be determined o�-line. Similar to (7.14), there exists a known scalar function

h�(qd; _qd; �qd) such that

kY (qd; _qd; _qd; �qd)~��k = kY �� � Y �k � h�(qd; _qd; �qd) 8�� 2 � (7.32)

Since h�(qd; _qd; �qd) depends on reference trajectory only, it can be determined o�-line, one of the

advantages of this scheme. Similar to (7.16), de�ne

hs(q; _q; t) = hf (q; _q; t) + h�(qd; _qd; �qd) (7.33)

The following continuous robust control law is suggested

u = ua + �h(�hs �k�k)

ua = f0(qd; _qd; _qd; �qd) + Y (qd; _qd; _qd; �qd)�� �K�� �Kxx� � 5kx�k2�(7.34)

where K� > 0 is an s.p.d. matrix, 5 is a positive scalar, Kx = BT� P�, in which P� is determined by

(7.11), and �h is a continuous approximation of �hs �k�k with an approximation error ".

Lemma 11 The following p.s.d. function

V = 12�

TM(q; �)� + 12x

T� P�x� (7.35)

is an ARC Lyapunov for (7.1) with the control law (7.34) and the adaptation law

� = Y T (qd; _qd; _qd; �qd)� (7.36)

Proof. Noting Property 1,

1

2kmk�k2 + 1

2�min(P�)kx�k2 � V � 1

2kMk�k2 + 1

2�max(P�)kx�k2 (7.37)

Thus, Assumption 2 is satis�ed by V . Noting (7.13), (7.9) and (7.11), di�erentiating V with respect

to time yields

_V = �T [M(q; �) _�+ C(q; _q; �)�] + 12x

T� (A

T� P� + P�A�)x� + xT� P�B��

= �T [u� f0(q; _q; _qr; �qr)� Y (q; _q; _qr; �qr)� � ~f + BT� P�x�]� 1

2xT� Q�x�

(7.38)

Substituting the control law (7.34) into (7.38) and noting (7.31), we can obtain

_V = �T [Y (qd; _qd; _qd; �qd)~�� + f0(qd; _qd; _qd; �qd)� f0(q; _q; _qr; �qr) + Y (qd; _qd; _qd; �qd)�

�Y (q; _q; _qr; �qr)� � ~f �K�� � 5kx�k2� + �h]� 12x

T� Q�x�

� ��TK�� � 5kx�k2k�k2 � 12x

T� Q�x� + �TY (qd; _qd; _qd; �qd)~�� � �T ~f + �T �h

+ 1k�kkx�k+ 2k�k2 + 3k�k2kx�k+ 4kx�k2k�k(7.39)

Applying the inequality

w1jy1jjy2j � w2y21 + w3y

22 8y1; y2 2 R w1; w2; w3 � 0 (7.40)

where 4w2w3 = w21 to (7.39), we have,

_V � ��TK�� � 5kx�k2k�k2 � 12x

T� Q�x� + �TY (qd; _qd; _qd; �qd)~�� � �T ~f + �T�h

+ 6k�k2 + 7kx�k2 + 2k�k2 + 8k�k2 + 9k�k2kx�k2 + 10kx�k2 + 11kx�k2k�k2= ��T [K� � ( 2 + 6 + 8)In]� � xT� [

12Q� � ( 7 + 10)In+nc ]x�

�[ 5 � ( 9 + 11)]kx�k2k�k2 + �TY (qd; _qd; _qd; �qd)~�� � �T ~f + �T �h

(7.41)

76

where 6 7 =

14

21

8 9 =14

23

10 11 =14

24

(7.42)

By choosing controller parameters K�; Q�, and 5 as

�min(K�) � "3 + 2 + 6 + 8�min(Q�) � 2("3 + 7 + 10)

5 � 9 + 11

(7.43)

where "3 is any positive scalar, (7.41) becomes

_V � �"3(k�k2 + kx�k2) + �TY (qd; _qd; _qd; �qd)~�� � �T ~f + �T �h (7.44)

When ~f = 0, noting condition i of (7.18), (7.44) becomes

_V j ~f=0 � �"3(k�k2 + kx�k2) + ~�T� YT (qd; _qd; _qd; �qd)� (7.45)

Thus, Assumption 3 (4.4) is satis�ed for W = "3(k�k2 + kx�k2).In general, when ~f 6= 0, from (7.3), (7.32), (7.33), (7.37), and condition ii of (7.18), (7.44)

becomes_V � �"3(k�k2 + kx�k2) + hsk�k+ �T �h � ��V V + " (7.46)

where �V is a positive scalar satisfying

�V � 2"3maxfkM ; �max(P�)g

(7.47)

Thus, Assumption 4 (4.9) is satis�ed if �V can be freely adjusted, which is shown in the following

remark. This completes the proof. 4

Remark 23 In (7.46), the exponential convergence rate �V can be any large value by choosing the

controller parameters as follows. Noting �V is bounded below by (7.47) and kM is a �xed constant,

�V can be any large value as long as we can arbitrarily choose "3 and "3�max(P�)

. Therefore, �rst set

"3 to its desired value and let Q� satisfy (7.43), in which 7 and 10 can be any �xed values. Then,

choose the dynamic compensator parameter A� such that the solution P� of (7.11) makes "3�max(P�)

big

enough. 1; 2; 3, and 4 in (7.31) can then be determined, and 6; 8; 9 and 11 can be calculated

to satisfy (7.42). Finally, choose K� and 5 such that (7.43) is satis�ed. In this way, theoretically, any

fast exponential convergence rate can be achieved. }

Noting that V is not a function of �, we can use the adaptation law (7.28) to achieve ARC

of the robot manipulator (7.1), which is summarized in the following theorem.

Theorem 12 If the control law (7.34) and the adaptation law (7.28) with � given by (7.36) is applied

to the manipulator described by (7.1), the same results as Theorem 11 can be obtained. 4

77

Remark 24 In the control law (7.34) and the adaptation function (7.36), the regressor

Y (qd; _qd; _qd; �qd) is a function of the reference trajectory only and, thus, can be calculated o�-line.

In addition to the reduction of on-line computation time, this result also removes the problem of

noise correlation between the estimation error and the adaptation signals, especially when the velocity

measurement is noisy in implementation [103], and, thus, enhances the performance robustness of the

resulting adaptive control law. }Remark 25 By setting us = 0 in (7.34), without using parameter projection and any modi�cation to the

adaptation law, and taking o� the dynamic compensator (i.e., letting Cz = 0; Az = 0; Bz = 0; Dz > 0

in (7.5)), the control law (7.34) reduces to the well-known desired compensation adaptation law (DCAL)

by Sadegh and Horowitz [103], which is also implemented for comparison. }

7.2.3 Nonlinear PID Robust Control (NPID)

In this subsection, a simple robust control with nonlinear PID feedback structure is designed.

We assume that only bounded disturbances appear | i.e., hf in (7.3) is a constant instead of a function

of states.

The following simple control structure is suggested

u = fc � (K� + 5kx�k2)� �Kxx� (7.48)

where fc is any constant vector that is used to cancel the low frequency component, K� > 0 is a s.p.d.

matrix, 5 is a positive scalar, and Kx = BT� P�, in which P� is determined by (7.11).

For the p.s.d. function V given by (7.35), its derivative is given by (7.38). Substituting control

law (7.48) into (7.38) and following similar steps as in (7.39) and (7.41), we have

_V = �T [fc �K�� � 5kx�k2� � f0(q; _q; _qr; �qr)� Y (q; _q; _qr; �qr)� � ~f ]� 12x

T� Q�x�

� �T [fc � f0(qd; _qd; _qd; �qd)� Y (qd; _qd; _qd; �qd)� �K�� � 5kx�k2� � ~f ]

�12x

T� Q�x� + 1k�kkx�k+ 2k�k2 + 3k�k2kx�k+ 4kx�k2k�k

� k�k [kfc � f0(qd; _qd; _qd; �qd)� Y (qd; _qd; _qd; �qd)�k+ hf ]� �TK��

� 5kx�k2k�k2 � 12x

T� Q�x�

+ 6k�k2 + 7kx�k2 + 2k�k2+ 8k�k2 + 9k�k2kx�k2 + 10kx�k2 + 11kx�k2k�k2

(7.49)

where 6; 7; 8; 9; 10, and 11 satisfy (7.42). De�ne

c0(t) = kfc � f0(qd; _qd; _qd; �qd)� Y (qd; _qd; _qd; �qd)�k+ hf (7.50)

Noting that c0 is bounded, we can choose the controller parameters K�; Q�, and 5 as

�min(K�) � "3 + 2 + 6 + 8 +c20

4"(t)

�min(Q�) � 2("3 + 7 + 10)

5 � 9 + 11

(7.51)

Then, (7.49) becomes

_V � k�kc0 � c204"k�k2 � "3(k�k2 + kx�k2)

� "� ( c02p"k�k � p"� "3(k�k2 + kx�k2) � ��V V + "

(7.52)

where �V satis�es (7.47). This leads to the following theorem.

78

Theorem 13 If the simple control law (7.48) with controller parameters satisfying (7.51) is applied to

the robot manipulator described by Eq. (7.1) with bounded modeling error (7.3), then all signals in the

system remain bounded and tracking errors, e(t) and _e(t), exponentially converge to some balls, the

sizes of which are proportional to ". }

Remark 26 By choosing the dynamic compensator as an integrator, x� consists of e andR t0 e; thus,

control law (7.48) may be considered as a nonlinear PID feedback control, which is quite easy to

implement since it does not require any model information, except some bounds in choosing controller

parameters. }

7.2.4 Nonlinear PID Adaptive Control (PIDAC)

Feedback gains in the nonlinear PID robust controller are required to satisfy the condition

(7.51), in which the lower bounds are not quite straightforward to calculate. Although analytic formulas

exist to calculate them, as given in the above development, often the calculated lower bounds are so

conservative and so large that they actually may not be used in implementation because of the limited

bandwidth of physical systems. Also, the constant feedforward control term fc may not quite match

the low frequency component of the feedforward term because of parametric uncertainties. In this

subsection, a gain-based nonlinear PID adaptive controller is proposed to solve these di�culties.

First, choose any Q� > 2"3I and thus determine Kx = BT� P� by (7.11). There exist 7 and

10 satisfying (7.51), and 6 and 11 satisfying (7.42). This means that there exist constant �K� and � 5such that (7.51) is satis�ed. In the following, we do not need to calculate �K� and � 5, but only need to

know their existence. The following control law is suggested:

u = fc � (K� + 5kx�k2)� �Kxx� (7.53)

Let �K be the independent components of K�. For example, if we want a diagonal K�, �K consists of

the n diagonal elements only. �K represents its estimate. Then we can write

�K�� = YK(�)��K K�� = YK(�)�K~K�� = (K� � �K�)� = YK(�)~�K ~�K = �K � ��K

(7.54)

where YK(�) is a known function. The gain adaptation law is chosen as

_f c = �0f [��00f (fc � fc0)� �]_�K = �0�K [��00�K(�K � �K0) + YK(�)

T�]_ 5 = �0 [��00 ( 5 � 50) + kx�k2k�k2]

(7.55)

where �0f ;�00f ;�

0�K ;�

00�K;�

0 , and �

00 are any constant s.p.d. matrix or scalars; fc0; �K0, and 50 are the

corresponding initial estimates. Choose a p.d. function as

Va = V +1

2~fTc �

0f�1 ~fc +

1

2~�TK�

0�K

�1 ~�K +1

2~ T5 �

0 �1~ 5 (7.56)

where ~fc = fc � fc; ~ 5 = 5 � � 5, and V is as de�ned by (7.35). Rewrite (7.53) as

u = �u+ ~fc � YK(�)~�K � ~ 5kx�k2��u = fc � ( �K� + � 5kx�k2)� �Kxx�

(7.57)

79

Noting that _~� = _� with adaption law (7.55), and following similar derivations as in (7.49) and (7.52),

we can obtain

_Va = �T [ ~fc � YK(�)~�K � ~ 5kx�k2�] + �T [�u� Y (q; _q; _qr; �qr)� � ~f + BT� P�x�]

�12x

T� Q�x� + ~fTc [��00f (fc � fc0)� �] + ~�TK [��00�K(�K � �K0) + YK(�)

T�]

+~ T5 [��00 ( 5 � 50) + kx�k2k�k2]� "� �T [ �K� � ( 2 + 6 + 8 +

c204")In]� � xT� [12Q� � ( 7 + 10)In+nc ]x�

�[� 5 � ( 9 + 11)]kx�k2k�k2 � ~fTc �00f~fc � ~fTc �

00f (fc � fc0)� ~�TK�

00�K

~�K�~�TK�

00�K(

��K � �K0)� ~ T5 �00 ~ 5 � ~ T5 �

00 (� 5 � 50)

� "� "3(k�k2 + kx�k2)� ~fTc �00f~fc � ~fTc �

00f (fc � fc0)� ~�TK�

00�K

~�K�~�TK�

00�K(

��K � �K0)� ~ T5 �00 ~ 5 � ~ T5 �

00 (� 5 � 50)

(7.58)

Case I : First, consider the case that the initial estimates �K0 and 50 satisfy the condition (7.51) for

fc = fc0. Since the only condition in choosing fc; ��K; and � 5 is that they should satisfy the condition

(7.51), we can choose fc = fc0; ��K = �K0 and � 5 = 50. In such a case, (7.58) becomes

_Va � "� "3(k�k2 + kx�k2)� ~fTc �00f~fc � ~�TK�

00�K

~�K � ~ T5 �00 ~ 5

� ��0V Va + "(7.59)

where

�0V � 2minf"3; �min(�

00f); �min(�

00�K); �

00 g

maxfkM ; �max(P�); �max(�0�1f ); �max(�

0�1�K); 1=�

0 g

(7.60)

So,

Va ��exp(��0V t)Va(0) +

Z t

0exp(��0V (t � �))�(�)d�

�(7.61)

Case II : Now, consider the general case that the initial estimates �K0 and 50 may not

satisfy the condition (7.51). From (7.58):

_Va � � ~fTc (�00f � 14In) ~fc � ~�TK(�

00�K � 15I)~�K � ~ T5 (�

00 � 16)~ 5 + 17

� ��00V Va + 17(7.62)

where 14; 15, and 16 are any positive scalars such that 14 < �min(�00f ); 15 < �min(�00�K); 16 < �00 , and

17 = " +k�00

f(fc�fc0)k24 14

+k�00

�K( ��K��K0)k24 15

+k�00 (� 5� 50)k2

4 16

�00V � 2minf"3; �min(�00f )� 14; �min(�00�K)� 15; �00 � 16g

maxfkM ; �max(P�); �max(�0�1f

); �max(�0�1�K

); 1=�0 g(7.63)

So,

Va ��exp(��00V t)Va(0) +

Z t

0exp(��00V (t � �)) 17(�)d�

�(7.64)

Cases I and II lead to the following theorem by taking (7.56) into consideration.

Theorem 14 If the control law (7.53) with the adaptation law (7.55) is applied to the robot manipulator

described by Eq. (7.1) with bounded modeling error (7.3), all signals in the system remain bounded.

Furthermore,

A . If the initial estimates �K0 and 50 satisfy the condition (7.51) for fc = fc0, the tracking errors

are bounded by (7.61), i.e., tracking errors exponentially converges to some balls whose sizes are

proportional to controller parameter ".

80

B . In general, the tracking errors are bounded by (7.64). 4

Remark 27 The above adaptive controller does not require any model information and has a simple

nonlinear PID feedback structure. Thus, it can be easily implemented and costs little computation

time, however, bounded disturbances are assumed in the development, and asymptotic stability is not

guaranteed even in the presence of parameter uncertainties only. Also, when the initial estimates do not

satisfy the condition (7.51), the error bound 17 in (7.64) is not guaranteed to be reduced by suitably

choosing controller gains and theoretical performance may not be guaranteed. }

7.2.5 Desired Compensation Adaptive Robust Control with Adjustable Gains

(ARCAG)

The DCARC scheme in subsection 7.2.2 requires that feedback gains satisfy condition (7.43),

which has the same drawback as the nonlinear PID robust control (NPID) scheme, as pointed out in the

above subsection. In this subsection, by incorporating a gain-based adaptive control synthesis technique

into the design of the DCARC scheme, a new adaptive robust controller is proposed to overcome this

di�culty.

As in the above subsection, choosing any Q� > 2"3I and obtaining Kx = BT� P� by (7.11),

there exist constant �K� and � 5 such that (7.43) is satis�ed. Since �K� and � 5 are unknown, instead

of using constant feedback gains K� and 5 in (7.34), we will adjust them as in the above gain-based

adaptive control. The resulting control law is given by

u = ua + �h(�hs �k�k)

ua = f0(qd; _qd; _qd; �qd) + Y (qd; _qd; _qd; �qd)�� � K�� �Kxx� � 5kx�k2�(7.65)

in which the parameter adaptation law for � is the same as in DCARC, and the gain adaptation laws

are suggested as_�K = �0�K [��00�K(�K � �K0) + YK(�)

T�]_ 5 = �0 [��00 ( 5 � 50) + kx�k2k�k2]

(7.66)

Choose a positive de�nite (p.d.) function as

Vp = V +1

2~�TK�

0�K

�1 ~�K +1

2~ T5 �

0 �1~ 5 (7.67)

where V is de�ned by (7.35). Rewrite (7.65) as

u = �u� YK(�)~�K � ~ 5kx�k2��u = f0(qd; _qd; _qd; �qd) + Y (qd; _qd; _qd; �qd)�� � ( �K� + � 5kx�k2)� �Kxx� + �h

(7.68)

and de�ne _V j�u as (actually the derivative of V under the control �u as shown later)

_V j�u = �T [�u� f0(q; _q; _qr; �qr)� Y (q; _q; _qr; �qr)� � ~f + BT� P�x�]� 1

2xT� Q�x� (7.69)

Noting (7.38) and (7.66), we have

_Vp = _V j�u +�T [�YK(�)~�K � ~ 5kx�k2�] + ~�TK�0�K

�1 _�K + ~ T5 �0 �1 _ 5

= _V j�u �~�TK�00�K

~�K � ~�TK�00�K(

��K � �K0)� ~ T5 �00 ~ 5 � ~ T5 �

00 (� 5 � 50)

(7.70)

81

Noting that _V j�u has the same form as the _V in (7.38) with u replaced by �u and that �u is the same as

the control (7.34) used in DCARC with gains satisfying (7.43), all the derivations from (7.38) to (7.46)

remain valid if we replace _V by _V j�u. Thus, in general, from (7.46),

_V j�u � �"3(k�k2 + kx�k2) + " (7.71)

and when ~f = 0, from (7.45),

_V j�u � �"3(k�k2 + kx�k2) + ~�T� YT (qd; _qd; _qd; �qd)� (7.72)

From (7.71) and (7.70),

_Vp � �"3(k�k2 + kx�k2) + "� ~�TK�00�K

~�K�~�TK�

00�K(

��K � �K0)� ~ T5 �00 ~ 5 � ~ T5 �

00 (� 5 � 50)

(7.73)

From (7.73), following similar arguments as in case I and case II of subsection 7.2.4, we have the

following theorem.

Theorem 15 If the control law (7.65) with the adaptation law (7.28) and (7.66) is applied to the robot

manipulator described by Eq. (7.1), all signals in the system remain bounded. Furthermore,

A. If the initial estimates �K0 and 50 satisfy the condition (7.43), then,

Vp ��exp(��0Vpt)Vp(0) +

Z t

0exp(��0Vp(t � �))"(�)d�

�(7.74)

where �0Vp is a scalar satisfying

�0Vp � 2minf"3; �min(�

00�K); �

00 g

maxfkM ; �max(P�); �max(�0�1�K); 1=�

0 g

(7.75)

and, thus, tracking errors exponentially converge to some balls whose sizes are proportional to the

controller parameter ".

B. In general, the tracking errors are bounded by

Vp ��exp(��00Vpt)Vp(0) +

Z t

0exp(��Vp(t � �)) 18(�)d�

�(7.76)

where

18 = "+k�00

�K( ��K��K0)k24 15

+k�00 (� 5� 50)k2

4 16

�00Vp � 2minf"3; �min(�00�K)� 15; �00 � 16g

maxfk00; �max(P�); �max(�0�1�K

); 1=�0 g(7.77)

4

In the following, we will show that this controller can actually do more than what stated in

the above theorem, a reasonable assertion in view of the great performance o�ered by its counterpart

DCARC. We consider the nominal case of no disturbances | i.e., ~f = 0. From (7.72) and (7.70),

_Vp � �"3(k�k2 + kx�k2) + ~�T� YT (qd; _qd; _qd; �qd)� � ~�TK�

00�K

~�K�~�TK�

00�K(

��K � �K0)� ~ T5 �00 ~ 5 � ~ T5 �

00 (� 5 � 50)

(7.78)

82

If the initial gain estimates satisfy the condition (7.43), we can set ��K = �K0 and � 5 = 50. Thus, Vpsatis�es Assumption 2 (i.e., (4.4)) with � given by (7.36) and

W = �"3(k�k2 + kx�k2)� ~�TK�00�K

~�K � ~ T5 �00 ~ 5 (7.79)

Thus, the adaptation law (7.28) guarantees that W �! 0 and asymptotic tracking is achieved. So we

have the following theorem.

Theorem 16 In the absence of disturbances (i.e., ~f = 0), if the initial gain estimates �K0 and 50satisfy condition (7.43), asymptotic tracking is achieved. 4

7.3 Experimental Set-up

Experiments are conducted on the planar UCB/NSK two axis SCARA direct drive manipulator

system. The robot (Fig. 7.1) consists of four major mechanical parts, two NSK direct drive motors

(Model 1410 for the �rst axis with maximum torque 245 Nm and Model 608 for the second axis with

maximum torque 39.2 Nm), and two aluminum links. The actual link lengths between the centers of

joints are 0.36m and 0.24m respectively.

MASS 4.85 kg

LENGTH 380mm2

MASS 10.6kgINERTIA 0.565kgmLENGTH 610mm

2

MAX TORQUE 245NmMAX SPEED 1.1rps

ROTOR INERTIA 0.267kgm

NSK RS 1410

ENCORDER RES. 153,600cpr2

NSK RS 608MAX TORQUE 39.2NmMAX SPEED 1.1rps

INERTIA 0.099kgm

ENCODER RES. 153,600cpr

Motor 1

ROTOR INERTIA 0.0077kgm2

Motor 2

Link 2

Link 1

Figure 7.1: Berkeley/NSK Two-Link Direct-Drive Manipulator

Fig. 7.2 shows the experimental set-up. A 486 PC equipped with IBM Data Acquisition and

Control Adapters (DACA) board is used to control the entire setup. Each DACA board contains two 12

bit D/A and four 12 bit A/D converters. The three-phase sensor feedback signal is fed through a 10-bit

Resolver to Digital Converter (RDC), which provides a motor position resolution of 153,600 pulses per

revolution (or 4:09�10�5 rad). The velocity signal is then obtained by the di�erence of two consecutive

position measurements with a �rst-order �lter 2. At each sampling time, based on the digital feedback

signals of the position and velocity, the torque control input for each joint is calculated in the 486 PC

and sent to the DACA board. The real-time code is written in C language. The analog torque inputs

from the DACA board are used to drive each motor through two NSK ampli�ers. The NSK motors

2The robot is equipped with tachometers to measure the joint rotation velocities, which are fed to the 486 PCthrough the A/D channels of the IBM DACA board, but the signals are too noisy and not used.

83

NSK TWO LINK

POSITIONDECODER

IBM DACA BOARD

REAL-TIME CONTROL

486 PC

NSK SERIES 1.0AMPLIFIER

AMPLIFIERSERIES 1.5

NSK

A/D0 VELOCITY 2

D/A0 TORQUE CMD

A/D1 VELOCITY 1

D/A1 TORQUE CMD

POWER LINE

FEEDBACK SIGNALS

FEEDBACK SIGNALS

POWER LINE

MANIPULATORDIRECT DRIVE

Figure 7.2: Experimental Setup

are variable reluctance motors. The ampli�ers contain digital communication circuits that convert the

torque commands into the necessary three-phase communication current to drive the motors in a torque

mode. Multiple poles are used within the motor to produce high torque output. To make such direct

drive motors behave like conventional DC motors, internal nonlinear feedback is used. Details of the

experimental setup and modeling can be found in [57].

The matrices in dynamic equation (7.1) are given by [57]

M(q) =

"p1 + 2p3Cq2 p2 + p3Cq2

p2 + p3Cq2 p2

#

C(q; _q) =

"�p3 _q2Sq2 �p3( _q1 + _q2)Sq2p3 _q1Sq2 0

#

G(q) = 0

(7.80)

where Cq2 = cos(q2); Sq2 = sin(q2); p1, p2, and p3, the combined robot and payload parameters, are

given by p1 = pa1 + 0:194mp; p2 = pa2 + 0:0644mp; and p3 = pa3 + 0:0864mp; respectively, mp is

the payload mass, and pa1 = 3:1623; pa2 = 0:1062; and pa3 = 0:17285 are the robot parameters. The

friction term Ff (q; _q) is lumped into ~f(q; _q; t) and is bounded by (7.3), where hf = 9. In the experiment,

only payload mass mp is unknown with the maximum payload, mpmax = 10kg. Thus, letting � = mp

and � = ( �0:00001; mpmax+ 0:00001), (7.2) can be formed. Since all the controllers are supposed

to deal with model uncertainties, the initial estimate of the payload is set to 9kg, with an actual value

in experiments being around 1kg. All experiments are conducted at a sampling time �T = 1ms.

84

7.4 Experimental Results

All schemes presented before were implemented and compared. In addition, Slotine and Li's

adaptive algorithm [110] and Sadegh and Horowitz's DCAL [103], which achieves the best tracking

performance in the experiments reported by Whitcomb, et al [138], are also implemented for comparison.

7.4.1 Performance Indexes

Since we are interested in tracking performance, sinusoidal trajectories with a smoothed initial

starting phase are adopted for each joint. In this experiment, the desired joint trajectories are qd =

[1:5(1:181�0:3343exp(�5t)�cos(�t�0:561)) ; 1:3045�0:538exp(�5t)�cos(43�t�0:697))]T (rad),

which are reasonably fast. Zero initial tracking errors are used and each experiment is run for ten

seconds, i.e, Tf = 10s.

Commonly used performance measures, such as the rising time, damping and steady state

error, are not adequate for nonlinear systems like robots. In [138], the scalar valued L2 norm given

by L2[e(t)] = ( 1Tf

R Tf0 ke(t)k2dt)1=2 is used as an objective numerical measure of tracking performance

for an entire error curve e(t). However, it is an average measure, and large errors during the initial

transient stage cannot be predicted. Thus, the sum of the maximal absolute value of tracking error

of each joint, eM = e1M + e2M , is used as an index of measure of transient performance, in which

eiM = maxt2[0;Tf]fjei(t)jg. The maximal absolute value and the average tracking error of each joint

during the last three seconds are de�ned by eiF = maxt2[Tf�3;Tf ]fjei(t)jg and L[eif ] =13

R TfTf�3 jeijdt

respectively. Then, eF = e1F + e2F and L[ef ] = L[e1f ] + L[e2f ] are used as indexes to measure the

steady state tracking error, The average control input of each joint, L[ui] =1Tf

R Tf0 juijdt, is used to

evaluate the amount of control e�ort. The average of control input increments of each joint is de�ned

by L[�ui] =1

10000

P10000k=1 jui(k�T ) � ui((k � 1)�T )j. The sum of the normalized control variations

of each joint, cu =P2

i=1L[�ui]L[ui]

, is used to measure the degree of control chattering.

7.4.2 Controller Gains

The choice of feedback gains is crucial to achieve a good tracking performance for all con-

trollers. A discussion of the gain tuning processes for each controller follows in detail. In general, the

larger the feedback gains (especially, the gain K�), the smaller the tracking errors. However, if the

gains are too big, the robot will be subject to severe control chattering and a large noisy sound can

be heard. After the gains exceed certain limits, the structural resonance is excited because of severe

control chattering and the system goes unstable. Thus, in order to achieve a fair comparison, we will

try to tune gains of each controller such that the tracking errors of each controller are minimized while

maintaining the same degree of control chattering for all controllers.

ASMC: Adaptive Sliding Mode Control .

As explained in Remark 21 in Section 7.2.1, a dynamic compensator (nc = 2) is formed by (7.5),

in which Az = 0I2; Bz = 400I2; Cz = I2; Dz = 40I2 with initial values calculated on-line by

(7.12). Such a choice of gains guarantees that the resulting sliding mode is critically damped with

corner frequency w� = 20.

85

The adaptation law is given by (7.28) where � = 10. Thus, � = �� and h�(q; _q; _qr; �qr) can be

determined by (7.15). The control law is then formed by (7.17), in which K� = diagf40; 5g andus is determined by (7.22) where Ks = diagf60; 4g; "1 = 1, "2 = 0:5, and � = 300.

SLAC: Slotine and Li's Adaptive Algorithm .

The control law is formed as explained in Remark 22 in Section 7.2.1, in which Dz = 20I2is used to provide the same corner frequency w� for the sliding mode as in ASMC. A large

K� = diagf180; 15g is used to produce roughly the same degree of control chattering as ASMC.

This gain is slightly larger than the combined feedback gain for �, K� +Ks in ASMC.

DCARC: Desired Compensation Adaptive Robust Control .

The same dynamic compensator as ASMC is used. Letting Q� = diagf105; 104g, P� is calculatedfrom (7.11) and the resulting gain matrix Kx is

[120; 0; 1250; 0; 0; 12; 0; 125]. The control law is given by (7.34), in which K� = diagf100; 8gand 5 = 1000. us is given by (7.22), in which Ks = diagf60; 4g; "1 = 1; "2 = 0:5, � = 200,

and hs is calculated by (7.33). The adaptation law is given by (7.28) with � given by (7.36) and

� = 10.

DCAL: Sadegh and Horowitz's Desired Compensation Adaptation Law .

The control law is formed as explained in Remark 25 in Section 7.2.2, in which Dz = 20I2 as in

SLAC. By using the same Q� as in DCARC, the resulting Kx is [0; 0; 2500; 0; 0; 0; 0; 250]. A large

K� = diagf170; 14g is used to produce roughly the same degree of control chattering as DCARC

and the rest of controller parameters are the same as in DCARC.

DCRC: Desired Compensation Robust Control .

The control law is the same as in DCARC except not to use the adaptation law. In such a case,

the proposed DCARC reduces to a robust control (termed as DCRC(I) in the following).

To verify the e�ect of using a dynamic compensator, the same control law is applied, but without

using the dynamic compensator, i.e., without the integrator, which is obtained by setting Cz =

0; Az = 0; Bz = 0; Dz = 20I . Correspondingly, Kx = [0; 0; 2500; 0; 0; 0; 0; 250] by using the same

Q� (termed DCRC(NI) in the following).

NPID: Nonlinear PID Robust Control .

The control law is given by (7.48) with the same 5 and Kx as in DCRC. fc = 0. A large

K� = diagf160; 12g is used. NPID(I) stands for integrator case and NPID(NI) for no integrator

case as in DCRC.

PIDAC: Nonlinear PID Adaptive Control .

The control law is given by (7.53) with the sameKx as DCRC and a diagonalK� = diagf�K1; �K2g.The gain adaptation law is given by (7.55), where �K0 = [10; 1]T , 50 = 1000, fc0 = 0;�f

0 =diagf10; 2g; �f 00 = diagf0:1; 0:1g; �0�K = diagf1000; 10g; �00�K = diagf0:0002; 0:02g; �0 =

104, and �00 = 2� 10�5.

ARCAG: Desired Compensation Adaptive Robust Control with Adjustable Gains

86

The control law is given by (7.65) with the same Kx, us, and the parameter adaptation law as

DCARC and a diagonal K� = diagf�K1; �K2g. The gain adaptation law is given by (7.66) where

�K0 = [20; 3]T , 50 = 500. �0�K = diagf1000; 10g, �00�K = diagf0:00003; 0:003g, �0 = 104,

and �00 = 10�4.

7.4.3 Comparative Experimental Results

As in [138], we �rst test the reliability of the results by running the same controller several

times. It is found that the standard deviation of the error from di�erent runs is negligible.

The experimental results are shown in the following table (unit is rad for tracking errors and

Nm for control input torques).

Table 7.1: Experimental Results

Controller eM eF L[ef ] L2[e] L[u1] L[u2] cuASMC 0.0301 0.0167 0.0058 0.0058 32.1 6.2 0.54

SLAC 0.0520 0.0325 0.0160 0.0133 32.8 6.2 0.55

DCARC 0.0201 0.0134 0.0039 0.0039 30.6 6.4 0.41

DCAL 0.0353 0.0199 0.0092 0.0081 30.3 6.3 0.43

DCRC(I) 0.0256 0.0227 0.0077 0.0081 30.3 6.3 0.42

DCRC(NI) 0.0690 0.0486 0.0175 0.0209 29.6 6.1 0.40

NPID(I) 0.0202 0.0195 0.0066 0.0061 30.5 6.4 0.41

NPID(NI) 0.0386 0.0346 0.0151 0.0145 29.8 6.3 0.40

PIDAC 0.0705 0.0158 0.0057 0.0070 30.4 6.3 0.44

ARCAG 0.0364 0.0119 0.0035 0.0045 30.2 6.3 0.42

The above results are also displayed in Fig. 7.3 to Fig. 7.4.

Based on the above experimental data, the following general results can be concluded:

a . Parameter Adaptation Improves Tracking Accuracy

If we compare the parameter-based adaptive controllers with their robust counterparts, i.e.,

DCARC versus DCRC(I), DCAL versus DCRC(NI), then we can see that, in terms of both �-

nal tracking accuracy (Fig. 7.4) and average tracking errors (Fig. 7.5), parameter adaptation

reduces the tracking errors around a factor of 2. The parameter-based adaptive controllers also

have better transient performance (Fig. 7.3). The improvement comes from the fact that the

estimated payloads approach their true values, which is shown in Fig. 7.8. This result veri�es the

importance of introducing parameter adaptation. All controllers use almost the same amount of

control e�ort and have the same degree of control chattering, as shown in Fig. 7.6 and Fig. 7.7,

and thus the comparison is fair.

b . Dynamic Compensator Improves Tracking Accuracy

Comparing the controllers having dynamic compensators with their counterparts not employing

dynamic compensators, i.e., DCRC(I) versus DCRC(NI) and NPID(I) versus NPID(NI), we can

see that introducing dynamic compensators reduces the tracking errors by more than a factor of

2 in terms of all the performance indexes, as shown in Fig. 7.3 to Fig. 7.5. The comparison is

87

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

AS

MC

SLA

C

DC

AR

C

DC

AL

DC

RC

(I)

DC

RC

(NI)

NP

ID(I

)

NP

ID(N

I)

PID

AC

AR

CA

G

Max

imum

Tra

ckin

g E

rror

(ra

d) eM

Figure 7.3: Transient Performance

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

AS

MC

SLA

C

DC

AR

C

DC

AL

DC

RC

(I)

DC

RC

(NI)

NP

ID(I

)

NP

ID(N

I)

PID

AC

AR

CA

G

Fin

al T

rack

ing

Err

or (

rad)

eFL[ef]

Figure 7.4: Final Tracking Accuracy

88

0

0.005

0.01

0.015

0.02

0.025

AS

MC

SLA

C

DC

AR

C

DC

AL

DC

RC

(I)

DC

RC

(NI)

NP

ID(I

)

NP

ID(N

I)

PID

AC

AR

CA

G

Tra

ckin

g E

rror

(ra

d)L2[e]

Figure 7.5: Average Tracking Errors

0

5

10

15

20

25

30

35

AS

MC

SLA

C

DC

AR

C

DC

AL

DC

RC

(I)

DC

RC

(NI)

NP

ID(I

)

NP

ID(N

I)

PID

AC

AR

CA

G

Co

ntr

ol I

np

ut

(Nm

)

L[u1]L[u2]

Figure 7.6: Control E�ort

89

0

0.1

0.2

0.3

0.4

0.5

0.6

AS

MC

SLA

C

DC

AR

C

DC

AL

DC

RC

(I)

DC

RC

(NI)

NP

ID(I

)

NP

ID(N

I)

PID

AC

AR

CA

G

Cu

Figure 7.7: Control Chattering

0 1 2 3 4 5 6 7 8 9 10−2

0

2

4

6

8

10

Time

Est

imat

ed P

aylo

ad

Solid: DCARC Dashdot: ASMC

Dashed: DCAL Dotted: SLAC

Figure 7.8: Estimated payloads approach their true values

90

fair, as shown by the control e�ort in Fig. 7.6, and the degree of control chattering in Fig. 7.7.

This result supports the importance of employing proper controller structure.

c . Desired Compensation Improves Tracking Accuracy

Comparing the controllers having desired compensation with their counterparts using actual state

in model compensation design, i.e., DCARC versus ASMC and DCAL versus SLAC, we can see

that, in terms of all performance indexes (Fig. 7.3 to Fig. 7.5), the controllers with desired com-

pensation have a better tracking performance. They also have a less degree of control chattering,

as shown in Fig. 7.7.

d . Gain-based Adaptive Controllers via Robust Controllers

If we compare the gain-based adaptive controllers with their robust counterparts, i.e., PIDAC

versus NPID(I) and ARCAG versus DCARC, we can see that gain-based adaptive controllers can

have a large stability margin for the choice of feedback gains since they can use small initial gain

estimates. Because of the small initial estimates, they have larger initial tracking errors or poorer

transient response, as seen from Fig. 7.3. The estimated feedback gains (e.g., K� shown in

Fig. 7.9) increase quickly to some values that are slightly larger than the �xed feedback gains

used in their robust counterparts (e.g., when t = 10s, K�(t) = diagf180; 12:6g for PIDAC but

K� = diagf160; 12g for NPID(I)). This is the reason that they achieve a slightly better �nal

tracking accuracy, as shown in Fig. 7.4. We should keep in mind, however, that this advantage

comes from the slightly increased degree of control chattering, as shown in Fig. 7.7. Therefore,

in practice, gain-based adaptive controllers do not o�er any advantage in improving tracking

performance. They may be used in the initial gain-tuning process to obtain the lower bound of

the stabilizing feedback gains instead of using a troublesome and conservative theoretical formula

like (7.51). However, caution should be taken. Large dampings (e.g., �00�K and �00 in (7.55))

should be used; otherwise, the resulting �nal estimates may be too big that they may exceed the

practical limits and destabilize the system because of their gain adaptation nature.

Since the proposed DCARC possesses all the desirable good qualities | parameter adaptation,

dynamic compensator, and desired compensation | it is natural that it achieves the best tracking

performance, as seen from Fig. 7.3 to Fig. 7.5, by using the same amount of control e�ort (Fig. 7.6)

and control chattering (Fig. 7.7). These facts show again the importance of using the both means,

parameter adaptation and proper controller structure, in designing high performance controllers, which

is the main theme of the proposed ARC. Using either one of them alone is not enough | in fact, in

these experiments, probably because the e�ect of link dynamics is not so severe and the disturbances

and measurement noise are not so small, the simple NPID robust controller out-performs DCAL, the

adaptive controller that achieves the best tracking performance among existing adaptive controllers.

The tracking errors of DCARC are plotted in Fig. 7.10 and the control inputs are shown in

Fig. 7.11. Those spikes of the tracking errors after the initial transient occur at the time when the joint

velocities change their directions. Thus, they are mainly caused by the discontinuous Columb friction.

7.5 Conclusions

In this chapter, the proposed ARC is applied to the trajectory tracking control of robot ma-

nipulators. Two schemes are developed: ASMC is based on the conventional adaptation structure and

91

0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

160

180

200

Time

Est

imat

ed F

eedb

ack

Gai

ns

Solid: Kxi1 (PIDAC) Dashdot: Kxi1 (ARCAG)

Dashed: Kxi2 (PIDAC) Dotted: Kxi2 (ARCAG)

Figure 7.9: Estimated Feedback Gains K�

DCARC is based on the desired compensation adaptation structure. A dynamic sliding mode is used to

enhance the system response. In addition, several conceptually di�erent robust and adaptive controllers

are also constructed for comparison | a simple nonlinear PID type robust control, and a simple gain-

based adaptive control, which requires almost no model information, and a combined parameter and

gain-based adaptive robust control. All algorithms, as well as two existing adaptive control algorithms,

SLAC and DCAL, are implemented on a two-link SCARA type robot manipulator. Comparative experi-

mental results show the importance of using the both means, proper controller structure and parameter

adaptation, in designing high performance controllers. It is observed that in these experiments, the

proposed DCARC achieves the best tracking performance without increasing control e�ort.

92

0 1 2 3 4 5 6 7 8 9 10−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

Time

Tra

ckin

g E

rror

s (r

ad)

Solid: Joint 1 Dashed: Jpint 2

Figure 7.10: Joint Tracking Errors

0 1 2 3 4 5 6 7 8 9 10−80

−60

−40

−20

0

20

40

60

80

100

Time

Join

t Tor

que

(Nm

)

Solid: Joint 1 Dashed: Jpint 2

Figure 7.11: Joint Control Torque

93

Chapter 8

Other Applications

8.1 Constrained Motion and Force Control of Robot Manipulators

In this section, the proposed ARC is applied to solve the motion and force control of constrained

robot manipulators.

8.1.1 Dynamic Model of Constrained Robots

When the robot end-e�ector comes in contact with its environment, interaction forces/moments

develop between the end-e�ector and the environment. In a Cartesian coordinate system, let x 2 Rn0

denote the vector of the position/orientation of the robot end-e�ector and F 2 Rn0 the vector of

interaction forces/moments on the environment exerted by the robot at the end-e�ector. Forces are

decomposed along the Cartesian axes and moments are decomposed along the rotation axes de�ning the

angles of the orientation, which may not be orthogonal. For example, the three axes de�ning the three

Euler angles are not orthogonal. To account for the e�ect of the interaction forces, dynamic equation

(7.1) is modi�ed to [81]

M(q; �)�q + C(q; _q; �) _q +G(q; �) + ~f(q; _q; t) + JT (q)F = u (8.1)

where J(q) = @x(q)=@q 2 Rn0�n is the Jacobian matrix.

In this section, it is assumed that the robot is nonredundant 1 (i.e., n0 = n) and the position,

velocity, and constrained force measurements are all available. J is assumed to be nonsingular in a

�nite work space q. The robot end-e�ector in contact with rigid constraint surfaces is considered.

It is assumed that the end-e�ector is initially in contact with the constraint surfaces, and the control

exercised over the constrained force is such that the force will always hold the end-e�ector on the

constraint surfaces. Later, we will show how to choose controller parameters to satisfy this assumption.

Suppose that the environment is described by a set of m rigid hypersurfaces [81, 74, 163]

�(x) = 0 �(x) = [�1(x); : : : ; �m(x)]T m � n (8.2)

which are mutually independent, and �i(x) is assumed to be twice di�erentiable with respect to x. The

interaction force F can be written as

F = Fn + Ft = DT (x)�+ Atft(�; vend; �); D(x) = @�(x)@x

(8.3)

1The assumption of the robot being nonredundant can be easily removed as shown in the next section

94

where � 2 Rm is a vector of Lagrange multipliers associated with the constraints which usually represent

normal contact force components, Fn = DT (x)� represents the constraint force (i.e., the normal contact

force in the Cartesian space), and Ft = Atft(�; vend; �) is the vector of friction forces, the directions

of which are speci�ed by At, the unit tangent directions of the surfaces, with opposite sign to the

end-e�ector velocity vend. The magnitude ft(�; vend; �) is linearly proportional to the normal contact

force Fn or �. Thus, we can write

Ft = [LT (�; x; _x) + ~LTf (x; _x)]�; L ; ~Lf 2 Rm�n (8.4)

in which LT (�; x; _x)� is used to describe the modeling part of the friction{L is linear with respect to

the unknown friction coe�cients � 2 Rk� with known shape function{ and ~LTf (x; _x)� represents the

modeling error. In general, L and ~Lf are di�erentiable except at points where vend changes direction on

the surfaces, i.e., vend = 0. Those points are not considered. In the assumption of frictionless contact

surfaces, i.e., Ft = 0, (8.3) reduces to the form given by [74, 76].

When motion of the robot is constrained to be on the surfaces (8.2), only (n�m) coordinates

of the position vector can be speci�ed independently [81, 142]. Control of all position coordinates of

the robot is unnecessary, and only (n�m) position coordinates need to be controlled in the constrained

motion of the robot. Therefore, motion control is in the (n � m) mutually independent curvilinear

coordinates, (x) = [ 1(x); : : : ; n�m(x)]T . (x) are assumed to be twice continuously di�erentiable

and independent of �(x) in the �nite workspace q. Thus, once (x) is regulated to the desired

value d(t), combining with the constraints (8.2), the con�guration of robot is uniquely determined.

The generality of choosing (x) gives us great exibility in implementation. It can be selected as

some joint angles qi, some end-e�ector coordinates xi, or some task space coordinates, in which the

resulting controller will be implemented in the joint space, Cartesian space, or task space respectively.

For example, since D(x) is of full rank m, without the loss of generality, we can assume that the �rst

m columns of D(x) are independent. In this case, we can choose (x) = [xm+1; : : : ; xn]T .

De�ne a set of curvilinear coordinates as [143, 163]

r = [rTf ; rTp ]T rf = [�1(x); : : : ; �m(x)]

T

rp = [ 1(x); : : : ; n�m(x)]T(8.5)

Di�erentiating (8.5), we obtain

_r = Jx _x = Jq _q (8.6)

where

Jx =@r(x)@x Jx = [D(x)T JTxp]

T Jxp =@(x)@x 2 R(n�m)�n

Jq =@r(x(q))

@q Jq = Jx(x(q))J(q) Jq ; Jx 2 Rn�n (8.7)

Using transformations (8.5) and (8.6) in (8.1) and multiplying both sides by J�Tq , the dynamic equation

(8.1) with the constraints (8.2) and the interaction force (8.3) can be expressed in terms of r as

M(r; �)�r+ C(r; _r; �) _r+G(r; �) +B0(�; r; _r)�+ ~fr(r; _r; �; t) = ur

r =

"0

rp

#B0 =

"Im0

#+ B(�; r; _r)

(8.8)

95

or

M12(r; �)�rp+ C12(r; _r; �) _rp+ G1(r; �) + (Im +B1)�+ ~fr1(r; _r; �; t) = ur1M22(r; �)�rp+ C22(r; _r; �) _rp+ G2(r; �) + B2�+ ~fr2(r; _r; �; t) = ur2

(8.9)

where

M(r; �) =

"M11(r; �) M12(r; �)

M21(r; �) M22(r; �)

#= J�Tq (q)M(q; �)J�1q (q)

C(r; _r; �) =

"C11 C12

C21 C22

#= J�Tq C(q; _q; �)J�1q � J�Tq M(q; �)J�1q

_JqJ�1q

G(r; �) =

"G1(r; �)

G2(r; �)

#= J�Tq (q)G(q; �)

B(�; r; _r) =

"B1

B2

#= J�Tx LT (�; x; _x)

~fr(r; _r; �; t) =

"fr1fr2

#= ~frp(r; _r; t) + ~Fr�(r; _r; t)�

~frp�= J�Tq (q) ~fq(q; _q; t)

~Fr��= J�Tx

~LTf (x; _x)

ur =

"ur1ur2

#= J�Tq (q)u

(8.10)

In (8.8), the constraints are simply described by rf = 0. The robot motion is thus uniquely determined

by the coordinates rp. Also, the constraint force Fn has a simple structure in the new coordinate

system, i.e., J�Tq Fn = [Im 0]T�. In the absence of the surface friction forces and the unknown

nonlinear functions, B1 = 0; B2 = 0, and ~fr = 0. The constraint force � does not appear in the second

equation of (8.9). Therefore, motion control can be designed based on the reduced order equation

without considering force control. This is the basic strategy adopted by most previous researchers in

this area [74, 133, 76, 58, 163, 168, 7, 19, 36]. Clearly, in the presence of the surface friction forces,

motion and force equations are coupled and a new strategy should be adopted.

Let Kf = diagf kf1; : : : ; kfm g and Gf = diagf gf1; : : : ; gfm g be constant diagonal matrices

with kfi > 0 and gfi � 0; i = 1; : : : ; m. By adding Gf� to both sides of the �rst equation of (8.9),

adding and subtracting M21(r; �)Kf� to the right hand of the second equation of (8.9), and noting

� = K�1f Kf�, Eq. (8.9) can be rewritten in a concise form as

H(rp; �)v + Ch(rp; _rp; �) _r+G(rp; �) +Bm(�; �; rp; _rp)�+ ~fr(rp; _rp; �; t) = ur + �Gf� (8.11)

where

96

v =

"Kf�

�rp

#

H(rp; �) =

"(Im + Gf)K

�1f M12(r; �)

M21(r; �) M22(r; �)

#

Ch(rp; _rp; �) =

"0 C12

C21 C22

#

Bm(�; �; rp; _rp) = B(�; r; _r) +B0m(rp; �) B0m(rp; �) = �"

0

M21(r; �)Kf

#

�Gf =

"Gf

0

#

(8.12)

Equation (8.11), which possesses some nice properties introduced in the following, is the basic equation

for our controller design. The physical meaning of introducing Kf and Gf in (8.11) will become apparent

later in the controller design.

The following properties are obtained for Eq. (8.11) in Appendix 2.

Property 4 . For the �nite work space q in which Jq is nonsingular, H(rp; �) is a s.p.d. matrix

for su�ciently small �max(Kf) = maxifkfig. Furthermore, for �max(Kf) � 1k00r, we have k0rIn �

H(rp; �) � k00hIn where k00h = k00r � k0r + 1+�max(Gf )�min(Kf )

. 2

Property 5 . The matrix Nh(rp; _rp; �) = _H(rp; �)� 2Ch(rp; _rp; �) is a skew-symmetric matrix.

Property 6 H(rp; �); Ch(rp; _rp; �); G(rp; �); and Bm(�; �; rp; _rp) are linear w.r.t. the combined robot

parameters and surface friction coe�cients, �c = [�T ; �T ]T 2 Rk� where k� = l� + k�, i.e.,

H(rp; �)zv + Ch(rp; _rp; �)zr +G(rp; �) + Bm(�; �; rp; _rp)�

= fc(rp; _rp; zr; zv; �) + Yc(rp; _rp; zr; zv; �)�c(8.13)

where zr and zv are any reference values, and fc and Yc are known.

Noting that ~fr(rp; _rp; �; t) is linear w.r.t �, we make the following assumption:

j ~fr(rp; _rp; �; t)j � �(rp; _rp; �; t)�= �p(rp; _rp; t) + ��(rp; _rp; t)k�k (8.14)

where �p(rp; _rp; t) and ��(rp; _rp; t) are known functions.

Let rpd(t) = (x(qd(t))) 2 Rn�m be the desired robot motion trajectory and �d(t) 2 Rm be

the desired constrained force trajectory, which are su�ciently smooth. Let ep(t) = rp(t) � rpd(t) and

ef (t) = �(t)��d(t) be tracking errors of the motion and constrained force respectively. The constrained

motion and force control problem can now be stated as follows, under parametric uncertainties and

the modeling error (8.14), design a control law for the actuator torque u or ur such that the robot

manipulator described by (8.11) is stable and the motion and the constrained force of the robot track

their desired values as close as possible.

8.1.2 Adaptive Robust Control of Constrained Manipulators

In this subsection, by using a dynamic sliding mode, ARC of constrained manipulators is

presented.

2k0r and k00r are some positive constants de�ned in Appendix 2

97

Dynamic Motion Sliding Mode

The same strategy as in Chapter 7 is used to design a dynamic motion sliding mode controller.

The dynamic motion sliding mode controller is designed to make the following quantity remain zero.

�p = _ep + yp; �p 2 R(n�m) (8.15)

where yp is the output of a np-th order dynamic compensator given by

_zp = Apzp +Bpep; zp 2 Rnp

yp = Cpzp +Dpep; yp 2 R(n�m) (8.16)

Constant matrices (Ap; Bp; Cp; Dp) are chosen in the same way as in Chapter 7 to guarantee that the

resulting motion sliding mode has prescribed good qualities. The initial value zp(0) of the dynamic

compensator (8.16) is chosen as

Cpzp(0) = � _ep(0)�Dpep(0) (8.17)

when Cp has rank n�m.

Dynamic Force Sliding Mode

In (8.11), the relationship between the constraint force � and the control input ur is static

instead of dynamic. This static relationship poses some di�culties in the dynamic control of the

constraint force since the force tracking error ef cannot be used to form force switching functions. In

[149], the integral of force tracking error, If =R t0 ef(�)d�, was used to form force switching functions.

Instead of controlling the constraint force directly, we stabilize If to control the constraint force indirectly.

Here, in order to broaden the generality of the force controller and possibly to speed up the force response

in the sliding mode, we also use the following �ltered force tracking error in forming force switching

functions:

_zf = �Af zf + Afef ; zf 2 Rm; Af 2 Rm�m (8.18)

where Af = diagf�f1; : : : ; �fmg is any s.p.d. diagonal matrix. The force switching functions are design

as

�f = Cfzf +DfIf �f 2 Rm; Cf ; Df 2 Rm�m (8.19)

where Cf andDf are any p.s.d. diagonal matrices satisfying CfAf+Df = Kf . Let Cf = diagfcf1; : : : ; cfmgwith cfi � 0 and Df = diagfdf1; : : : ; dfmg with dfi � 0. Transfer function from �f to If and zf are

If = G�f(s)�f G�f (s) = diagf1

�fis+1

kfi�fi

s+dfi; i = 1; : : : ; mg

zf = Gzf (s)�f Gzf (s) = diagf skfi

�fis+dfi

; i = 1; : : : ; mg(8.20)

From (8.20), both G�f(s) and Gzf (s) are stable. Thus during the force sliding mode, the �ltered force

tracking error zf and the integral force tracking error If will converge to zero. When Cf = 0, Df = Kf .

The force switching functions (8.19) reduce to those used in [149], in which only the integral of force

tracking errors is used in forming switching functions. In such a case, G�f(s) = diagf 1kfi; i = 1; : : : ; mg,

and Gzf (s) = diagf skfi(

1�fi

s+1); i = 1; : : : ; mg, which are stable.

98

Adaptive Robust Control Law

By using the dynamic force sliding mode, the constrained force is regulated indirectly by

controlling If and the �ltered force tracking error zf . The state of the entire system thus includes the

state variables of the original dynamics (8.11), the state variables of the dynamic compensators (8.16)

and (8.18) and If , i.e.,

x = [rTp ; _rTp ; zTp ; z

Tf ; I

Tf ]

T (8.21)

From (8.15) and (8.19), the switching functions and their derivatives are

� =

"�f�p

#=

"0

_rp

#� zr

_� = v � zv(8.22)

where v is de�ned in (8.12) and the reference velocity zr and the acceleration zv are given by

zr(x) =

"zfrzpr

#=

"��f

_rpd � yp

#

zv(x) =

"zvfzvp

#=

"Kf�d + CfAfzf

�rpd � _yp

# (8.23)

Note that zv 6= _zr as opposite to the case in adaptive motion control [109, 103, 51, 89, 154]. Both zrand zv are calculable feedback signals.

Let hs be a bounding function satisfying

hs(x; �c�; �; t) � sup �2�fkYc(rp; _rp; zr; zv; �)~�c� � ~fr(rp; _rp; �; t)kg (8.24)

For example, let

hs = kY (rp; _rp; zr; zv; �)k�cM + �p(rp; _rp; t) + ��(rp; _rp; t)k�k (8.25)

where �cM = k(�cmax��cmin+"c�)k. By the de�nition (8.13), Yc is linear w.r.t. �. Thus, from (8.24)

or (8.25), hs can be a linear function of k�k, i.e.,

hs = hp(x; �c�; t) + h�(x; �c�; t)k�k (8.26)

for some positive functions hp and h�.

The control torque is suggested to be

ur = Hzv + Chzr + G+ (Bm � �Gf)��K�� + �h(�hs �k�k)

= fc(rp; _rp; zr; zv; �) + Yc(rp; _rp; zr; zv; �)�c� � �Gf��K�� + �h(�hs �k�k)

(8.27)

where K� is a s.p.d. matrix, �, zr; and zv are de�ned by (8.22) and (8.23), respectively, and hbar is a

continuous approximation of the ideal SMC control �hs �k�k with an approximation error ".

Substituting the control law (8.27) into (8.11) and noting (8.22), the error dynamics are

obtained as

H _� + Ch� = ~Hzv + ~Chzr + ~G+ ~Bm�� ~fr(rp; _rp; �; t)�K�� + �h

= Yc(rp; _rp; zr; zv; �)~�c� � ~fr(rp; _rp; �; t)�K�� + �h(8.28)

99

Noting Property 4, a p.s.d. function is chosen as

V =1

2�TH(rp; �)� (8.29)

with1

2k0rk�k2 � V � 1

2k00hk�k2 (8.30)

From Property 5, 12�

T _H(rp; �)� = �TCh(rp; _rp; �)�. Noting (8.28), di�erentiating V with respect to

time yields_V = �TH _� + 1

2�T _H� = �TH _� + �TCh�

= �T [Yc(rp; _rp; zr; zv; �)~�c� � ~fr]� �TK�� + �T�h(8.31)

Noting (8.14), (8.24), and condition ii of (7.18),

_V � k�khs � �TK�� + �T�h � ��V V + " (8.32)

where �V is a positive scalar satisfying

�V � �min(K�)

k00h(8.33)

Noting (8.30), k�k exponentially converges to a known setn� : k�(1)k �

q2"max

k0r�V

o, and the expo-

nentially converging rate �V and the bound of the �nal tracking errors,q

2"max

k0r�V, can be freely adjusted

by the controller parameters " and K� in a known form. Since the sliding mode designed by (8.15) and

(8.20) are exponentially stable with predetermined transient performances, ep; _ep; zp; zf , and If expo-

nentially converge to some known sets whose size can be freely adjusted by the controller parameters

in a known form. Therefore, the system is exponentially stable at large with a guaranteed transient

performance.

In the above, we have shown the exponential stability of the state x. However, we have not

shown that the force tracking error ef is bounded and thus the control torque may be in�nite or ill-

de�ned as it contains �. In the following, this condition will be examined as it reveals some consequences

of the causality problem important in the force control of constrained motion. The relationship between

the constraint force and the control input is static and a small integral force tracking error or the �ltered

force tracking error does not necessarily mean a small force tracking error. This point has been neglected

by most previous researchers.

From (8.19) and (8.18), _�f = Kfef � CfAfzf . Noting the de�nition of H in (8.12) and

lumping the terms containing � or ef together, Eq. (8.28) can be rewritten as

"Im + Gf M12

M21Kf M22

# "ef_�p

#= w(x; t; �c�; �c�) + ~BF (rp; _rp; ~�c�)ef + �h (8.34)

where~BF (x; t; ~�c�) = ~Bm(rp; _rp; �c�)� ~Fr�(rp; _rp; t)

w(x; t; �c�; �c�)=

"(Im +Gf)K

�1f

M21

#CfAfzf � Ch� + ~Hzv + ~Chzr + ~G

� ~frp �K�� + ~BF�d

(8.35)

100

Multiplying both sides of (8.34) by MI�= [Im ; �M12M

�122 ] , we can eliminate _�p to obtain ef :

[Im + Gf �M12M�122 M21Kf ]ef =MI [w(x; t; �c�; �c�) + ~BF ef + �h] (8.36)

If the above equation has a �nite solution ef all the time, the control torque (8.27) is �nite since all

terms except ef are bounded as shown before. Noting (8.26), in general, �h is a function of � also since

it is an approximation of �hs �k�k . Thus, it is not so easy to gain insight into how to choose controller

parameters to guarantee Eq. (8.36) to have a �nite solution ef . However, noting that Im + Gf , part

of the coe�cient matrix of ef at the left hand side of Eq. (8.36), is an s.p.d. matrix, we proceed in the

following way.

Since �h is an approximation of �hs �k�k , we can assume that

k�hk � p(�) + hs (8.37)

for some non-negative function p(�) � 0. For example, for the smooth approximation like (7.20),

p(�) = 0, and for the continuous approximation like (7.22), p(�) = kKs�k. Noting (8.36 ), (8.26), and

(8.37), we have

�min(Im +Gf )kefk2� eTf (Im + Gf) ef = eTfM12M

�122 M21Kfef + eTfMI [w + ~BF ef + �h]

� kM12M�122 M21Kfkkefk2 + kefk

hkMIwk+ kMI

~BF kkefk+kMIk(p(�) + hp + h�k (kefk+ k�dk)]

(8.38)

De�ne

��=�min(Im + Gf)� kM12M�122 M21Kfk � kMI

~BF k � kMIkh� (8.39)

If

�� � "� (8.40)

for some "� > 0, then, from (8.38),

kefk � 1��fkMIwk+ kMIk (p(�) + hp + h�k�dk)g (8.41)

Thus kefk is bounded since the right hand side of (8.41) depends on the bounded state x only. How

to choose controller gains to guarantee (8.40) will be given later.

In the above, we have shown that the suggested control law guarantees a certain transient

performance and �nal tracking accuracy with a bounded control. In the following, we will show that

if the dynamic model is accurate, i.e., in the presence of parametric uncertainties only ( ~fr = 0 in Eq.

(8.8)), asymptotic motion and force tracking can be obtained through parameter adaptation without

using a high-gain in the feedback loop. From (8.31) and condition ii of (7.18),

_V = �TYc(rp; _rp; zr; zv; �)~�c� � �TK�� (8.42)

Thus, Assumption 2 (4.4) is satis�ed for W = �TK�� and

� = Y Tc (rp; _rp; zr; zv; �)� (8.43)

Since V is not a function of �c, we can use an adaptation law like (4.16), i.e.,

_�c = ��[l�(�c) + � ] (8.44)

101

where the adaptation function � is given by (8.43) and l� satis�es the conditions like (4.17). Then,

asymptotic output tracking can be obtained, i.e., � �! 0. Thus, x �! 0. Since _x is bounded, x is

uniformly continuous. So, all terms in the right hand side of (8.41) are uniformly continuous, and thus

ef is uniformly continuous. Since If �! 0, by applying Barbalat's lemma, ef �! 0. The above results

can be summarized in the following theorem.

Theorem 17 For the constrained robot manipulator described by Eq. (8.11) with the modeling error

(8.14), in the �nite workspace q, with a su�ciently small Kf and (8.40) being satis�ed, the following

results hold if the control law (8.27) with the adaptation law (8.44) is applied:

a). In general, the system is exponentially stable at large with a guaranteed transient performance and

�nal tracking accuracy, i.e., ep; _ep; zp; zf and If exponentially converge to some balls whose sizes

can be freely adjusted by controller parameters in a known form. The control is bounded and efis bounded by (8.41).

b). In the presence of parametric uncertainties only, i.e, ~fr = 0, the system is asymptotically stable in

the sense that ep; _ep; zp; zf ; If �! 0. Furthermore, ef �! 0, i.e., asymptotic force tracking is

achieved. 4

Remark 28 There are several ways to guarantee �� > 0 as required by (8.40). We classify them in the

following three cases:

Case 1 . Consider the case that the friction coe�cient vector � is known and there is no contact

friction modeling uncertainty (i.e., ~Lf = 0). From (8.10), ~Fr� = 0 and ~fr = ~frp(rp; _rp; t). Thus

we can set �� = 0 in (8.14). By setting � = �, we have, ~B = 0. From (8.12), ~Bm = ~B0m(rp; ~��).Noting that the only term containing � in Yc ~�c� is ~Bm�, from (8.24) and (8.26), we can choose

h� = sup�2�

k ~M21kkKfk � sup�2�

k ~B0mk = sup�2�

k ~Bmk (8.45)

From (8.39),

����min (Im + Gf)� kM12M�122 M21kkKfk � 2kMIk sup�2�fk ~M21kgkKfk (8.46)

By choosing a small weighing matrix Kf such that

�max(Kf) <1

sup��2�

nkM12M

�122 M21k+ 2kMIk sup�2� k ~M21k

o+ "�

(8.47)

(8.40) is satis�ed.

Case 2 . In the above development, only the normal contact force Fn or � is assumed to be measurable.

If the total interaction force F between the end-e�ector and the contact surface can also be

measured by the force sensor, Ft can be calculated from (8.3). Thus, B�+ ~Fr� = J�Tx Ft can be

directly obtained. The control law (8.27) can be simpli�ed to

ur = Hzv + Chzr + G+ (B0m(rp; ��)� �Gf )�+ J�Tx Ft �K�� + �h (8.48)

102

Noting Property 6 (8.13), we can write

H(rp; �)zv+ Ch(rp; _rp; �)zr + G(rp; �) +B0m(rp; �)�=f�(rp; _rp; zr; zv; �) + Y�(rp; _rp; zr; zv; �)�

(8.49)

where f� and Y� are the vector and matrix of known functions respectively. The control law

(8.48) can be rewritten as

ur = f�(rp; _rp; zr; zv; �) + Y�(rp; _rp; zr; zv; �)�� + J�Tx Ft � �Gf��K�� + �h (8.50)

and thus we can eliminate the need for estimating � in �c. In such a case, we only need to

estimate �, and an adaptation law similar to (8.44) can be used to estimate �. The adaptation

function becomes

� = Y T� (rp; _rp; zr; zv; �)� (8.51)

By doing so, we can also eliminate B� and ~Fr� in the subsequent development and, thus, we can

choose h� in the same way as in Case 1. This case is then identical to the Case 1 and (8.40) can

be guaranteed by choosing a su�cient small weighing matrix Kf .

Case 3 . Now consider the general case that only � can be measured and �c has to be estimated as

before. From (8.24), we can choose

h� = sup �2� k ~BF (rp; _rp; ~��)k (8.52)

By choosing Gf such that

�min(Gf) > �1 + "� + sup�2�f�maxkM12M�122 M21Kfkg

+2kMIkmax �2� k ~BF k(8.53)

(8.40) can be satis�ed. Thus, theoretically, (8.40) can be guaranteed by choosing a relatively large

proportional (P) force feedback gain Gf . However, as shown in [149], because of the causality

problem, the allowable force feedback gain Gf is severely limited in implementation. Roughly

speaking, in the absence of modeling uncertainties and parametric uncertainties, �max(Gf) < 1 is

required to guarantee stability for discrete implementation of the continuous control law no matter

how fast the sampling rate is going to be. Fortunately, in general, � and friction modeling error

is small, which implies that k ~Bk and k ~Fr�k are small. By choosing a small kKfk, ~BF � ~B� ~Fr�and k ~BF k is small. Thus, the right hand side of (8.53) can be made less than zero so that it is

not necessary to impose a lower bound on Gf and a small or zero force feedback gain control is

possible. }

8.1.3 Simulation Results

Consider a two-DOF SCARA robot moving on a rigid semi-circle surface S as shown in Fig.

8.1. The matrices M;C, and G in (8.1) are the same as in (7.80). The forward kinematics 3 and the

Jacobian matrix are

x =

"x

y

#=

"l1Cq1 + l2Cq12 � d

l1Sq1 + l2Sq12

#

J(q) =

"�l1Sq1 � l2Sq12 �l2Sq12l1Cq1 + l2Cq12 l2Cq12

# (8.54)

3In this subsection, we use x to denote the x-coordinate and use x to represent the position vector.

103

where Cq1 = cos(q1) ; Cq2 = cos(q2) ; Cq12 = cos(q1 + q2) , Sq1 = sin(q1) ; Sq2 = sin(q2), Sq12 =

sin(q1 + q2), l1 = 0:36m, l2 = 0:24m, and d = 0:5m. In this simulation, all parameters of the robot

are assumed unknown. Thus, � = [p1; p2; p3]T . The actual value of � is � = [3:694; ; 0:353; 0:363]T

and its initial estimate is assumed to be � = [0:18; 0:18; 1:8]T.

2

X

Y

R

dq

q

ffn t

θΦ(x)=0

1

Figure 8.1: Con�guration of the Robot Moving on a Semi-circle Surface

The surface S, with an unknown dry friction coe�cient � = 0:2, is described by

�(x) =qx2 + y2 �R = 0 R = 0:2 m (8.55)

The task space is de�ned asr = [rf ; rp]T

rf =px2 + y2 �R

rp = R� � = tan�1(x=y)(8.56)

Notice that rp is orthogonal to the curvilinear coordinate of rf . The interaction force on the surface is

given by (8.3) where

Fn =

"sin(�)

cos(�)

#fn

Ft = �"

cos(�)

�sin(�)

#�sgn( _rp)fn

(8.57)

where � = fn 2 R represents the normal contact force component. Task space equation (8.8) can be

obtained and the transformed dynamic equation (8.11) is thus derived. The forms of fc and Yc are

obtained from (8.13). Let �cmin = [0; 0; 0; 0]T and �cmax = [0:6; 0:6; 6; 0:4]T in de�ning �c

np = 1, Ap = 0; Bp = 1; Cp = 400, and Dp = 40 are chosen for (8.16) and �p is de�ned

by (8.15). Af = 10 is chosen in (8.18) and �f is formed by (8.19) where Cf = 0:001; Df = 0:01

and Kf = 0:02. Smooth projections like (3.54) are used, in which the bounding parameters are

�cmin = [0; 0; 0; 0]T ; �cmax = [0:6; 0:6; 6; 0:4]T , and "� = [0:01; 0:01; 0:01; 0:01]T . hs is chosen

by (8.25) where �p = 6 and �� = 0. A continuous approximation of �hs �k�k similar to (7.22) is used to

104

obtain �h, in which �1 = 1; �2 = 0:5; � = 20; and Ks = diagf100; 100g. The control torque can then

be calculated from (8.27) where Gf = 0 and K� = diagf1000; 1000g. The adaptation law is given by

(8.44), where � = diagf150; 150; 150; 5g and l� = 0. The desired motion and force trajectories are

rpd = �R�6 (1 + cos(0:5�t)) and fnd = �15 + 5cos(�t). Sampling time is 0.005s.

For comparison, two control laws are simulated: one is the proposed ARC and the other is the

deterministic robust control (DRC) law obtained by switching o� the adaptation law in the ARC. The

�rst set of simulations is run under ~fr = 0 to test the nominal performance. Position and force tracking

errors are shown in Fig. 8.2 and Fig. 8.3, respectively. The interaction force of ARC is shown in Fig 8.4.

It is seen that the performance of either of the two is satisfactory. However, ARC has a much better

motion and force tracking performance than DRC although its estimated parameters do not converge to

their true values as shown in Fig. 8.5. This shows an advantage of introducing parameter adaptation.

The sudden changes occurring at about t = 2s; 4s; 6s in these �gures are caused by surface friction

force because of the changes of motion direction of the robot end-e�ector as shown in Fig. 8.4.

To test the performance robustness, another set of simulations is run under a large external

disturbance ~fr = [5sin(0:25t); 2:5sin(0:4t)]T . Position and force tracking errors are shown in Fig.

8.6 and Fig. 8.7, respectively. It is seen that both control laws still have a satisfactory tracking

performance. Comparing to DRC, parameter adaptation errors of ARC in Fig. 8.8 do not a�ect the

tracking performance of ARC much. This veri�es the robustness of the proposed ARC to unknown

nonlinear functions. Joint torque of ARC shown in Fig. 8.9 does not exhibit control chattering.

8.1.4 Conclusions

The proposed ARC is applied to the constrained motion control of robot manipulators in the

presence of both parametric uncertainties in the robot and contact surface as well as external distur-

bances. Instead of the reduced constrained dynamic model obtained for frictionless contact surfaces,

a new transformed constrained dynamic model, which is suitable for the controller design, has been

proposed to deal with contact surfaces with or without friction. Dynamic motion sliding mode and

�ltered force tracking error are used to enhance the dynamic response of the system. An ARC law with

unknown parameters updated by both motion and force tracking errors has been suggested to achieve

asymptotic motion and force tracking without persistent excitation condition in the presence of para-

metric uncertainties, and a guaranteed transient performance with prescribed �nal tracking accuracy

in the presence of both parametric uncertainties and external disturbances or modeling errors. The

suggested control law has the expected PI type force feedback control structure with a low P-gain to

avoid the acausality problem. Simulation results demonstrated the advantage of the proposed method.

8.2 Coordinated Control of Multiple Robot Manipulators

In this section, the proposed ARC is applied to solve the coordinated motion and force control of

multiple manipulators handling a common constrained object in the presence of parametric uncertainties

in the manipulators, the object, and the contact surfaces as well as unknown nonlinearities resulting

from modeling errors.

105

0 1 2 3 4 5 6 7 8−8

−6

−4

−2

0

2

4

6

8x 10

−5

Time (sec)

Tra

ckin

g er

ror

(m)

Solid: ARC Dashdot: DRC

Figure 8.2: Position Tracking Error ep in the Presence of Parametric Uncertainties

0 1 2 3 4 5 6 7 8−2

−1

0

1

2

3

4

Time (sec)

Tra

ckin

g er

ror

(N)

Solid: ARC Dashdot: DRC

Figure 8.3: Force Tracking Error ef in the Presence of Parametric Uncertainties

106

0 1 2 3 4 5 6 7 8−25

−20

−15

−10

−5

0

5

Time (sec)

For

ce (

N)

Solid: desired fn Dotted: actual fn

Dashdot: Friction force

Figure 8.4: Interaction Force in the Presence of Parametric Uncertainties

0 1 2 3 4 5 6 7 8−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Time (sec)

Solid: beta 1 Dot: beta 2 Dashdot: beta 3 Dash: mu

Figure 8.5: Estimated Parameters in the Presence of Parametric Uncertainties

0 1 2 3 4 5 6 7 8−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1x 10

−4

Time (sec)

Tra

ckin

g er

ror

(m)

Solid: ARC Dashdot: DRC

Figure 8.6: Position Tracking Error ep in the Presence of Parametric Uncertainties andDisturbances

107

0 1 2 3 4 5 6 7 8−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Time (sec)

Tra

ckin

g er

ror

(N)

Solid: ARC Dashdot: DRC

Figure 8.7: Force Tracking Error ef in the Presence of Parametric Uncertainties and Disturbances

0 1 2 3 4 5 6 7 8−4

−2

0

2

4

6

8

Time (sec)

Solid: beta 1 Dot: beta 2 Dashdot: beta 3 Dash: mu

Figure 8.8: Estimated Parameters in the Presence of Parametric Uncertainties and Disturbances

0 1 2 3 4 5 6 7 8−10

−5

0

5

Time (sec)

Join

t tor

que

(Nm

)

Solid: joint 1 Dotted: joint 2

Figure 8.9: Joint Torque in the Presence of Parametric Uncertainties and Disturbances

108

8.2.1 Dynamic Model of Robotic Systems

Consider k ni-joint serial link manipulators handling a rigid object in a n0 dimensional workspace

where n0 � ni. It is assumed that all robot end-e�ectors grasp the object �rmly at k speci�ed points

without any relative motion among the end-e�ectors and the object. However, this assumption could be

relaxed and the analysis could be extended to the robust control of multi�ngered hands [81] where rela-

tive motion among the �ngers and the object exists and non-holonomic constraints result. Let OXY Z

be the Cartesian reference frame, oxyz be the object frame �xed relative to the object with the origin

at its mass center, and oeixeiyeizei be the end-e�ector frame of the i-th robot located at the grasp

point as shown in Fig. 8.10.

Z

Robot 1

Robot i

Robot k

Object

Rigid surfaces

O X

Y

Zei

X

Y

O

ZYei

Xei

Oei

XekOek

Yek

Zek ...

...

Figure 8.10: Con�guration of a Robotic System

Let xei 2 Rn0 represent the position vector of the i-th manipulator's end-e�ector in OXY Z

and x 2 Rn0 represent the position vector of the object represented in OXYZ. Use vei 2 Rn0 to denote

the velocity of the i-th manipulator's end-e�ector expressed in the moving coordinate frame oeixeiyeizeiand v 2 Rn0 to denote the object's velocity expressed in the moving coordinate frame oxyz. vei and v

are related to _xei and _x by some nonsingular transformation matrices respectively:

vei = Jci(xei) _xeiv = Jo(x) _x

(8.58)

where Jci and Jo depend on xei and x respectively. In case that xei and x are the R3 translation

position vectors only, Jci is the R3�3 rotation matrix from the frame OXY Z to the coordinate frame

109

oeixeiyeizei, and Jo is the R3�3 rotation matrix from the frame OXY Zto the coordinate frame oxyz.

The dynamic equation of each manipulator can be described by an equation like (8.1). In (8.1), the

force vector is represented in the reference frame OXYZ. Here, we project the force vector in the

moving end-e�ector frame oeixeiyeizei. Thus, the dynamic equation of i-th manipulator can be written

as

Mi(qi; �i)�qi + Ci(qi; _qi; �i) _qi + Gi(qi; �i) + ~f(qi; _qi; t) + JTei(qi)fei = ui (8.59)

where, as in (8.1), qi 2 Rni is the joint displacement vector, ui 2 Rni is the applied joint torque vector,

Mi 2 Rni�ni ; Ci _qi 2 Rni , Gi 2 Rni and ~fi are the inertia matrix, the vector of Coriolis and centrifugal

force, the vector of gravity, and the vector of modeling error including external disturbances respectively,

fei 2 Rn0 is the vector of the interaction force on the object exerted by the i-th manipulator at the

end-e�ector represented in oeixeiyeizei, and Jei(qi) is the corresponding manipulator Jacobian matrix

given by

Jei(qi) = Jci(xei(qi))Ji(qi); Ji =@xei(qi)@qi

(8.60)

Jei(qi) is assumed to be of full rank in a �nite work space qi .

Stacking all the dynamic equations of the manipulators together, we can write them in a

concise form

Mm(q; �m)�q + Cm(q; _q; �m) _q +Gm(q; �m) + ~fm(q; _q; t) + JTe (q)fe = u (8.61)

where

Mm(q; �m) = diagfM1(q1; �1) : : : Mk(qk; �k)g 2 Rns�ns ns =Pk

i=1 niCm(q; _q; �m) = diagfC1(q1; _q1; �1) : : : Ck(qk; _qk; �k)gGm(q; �m) = [GT

1 (q1; �1) : : : GTk (qk; �k)]

T

~fm(q; _q; t) = [fT1 (q1; _q1; t) : : : ~fTk (qk; _qk; t)]T

Je(q) = diagfJe1(q1) : : : Jek(qk)g 2 Rn0k�ns

q = [qT1 qT2 : : : qTk ]T 2 Rns

�m = [�T1 : : : �Tk ]T

fe = [fTe1 fTe2 : : : fTek ]T 2 Rn0k

u = [uT1 uT2 : : : uTk ]T 2 Rns

(8.62)

The dynamic equation of the grasped object expressed in OXY Z can be written as

M0(x; �0)�x+ C0(x; _x; �0) _x+ G0(x; �0) + ~f0(x; _x; t) = F � Fc (8.63)

where M0; C0; G0; and ~f0 are the inertia matrix, the vector of Coriolis and centrifugal force, the vector

of gravity, and the vector of modeling error including external disturbances respectively, Fc is the vector

of contact force on the environment exerted by the object, and F is the resultant force on the object

exerted by the end-e�ectors. F is given by

F =Pk

i=1 JTo (x)L

Ti fei = ATfe

AT �= JTo (x)[L

T1 LT2 : : : LTk ]

(8.64)

where Li 2 Rn0�n0 is the constant nonsingular transformation matrix from oxyz to oeixeiyeizei, i.e.,

vei = Liv (8.65)

110

Remark 29 . By the assumption of rigid grasps, the object position is uniquely determined by the

position of each manipulator, i.e.,

x = �i(qi) i = 1; : : : ; k (8.66)

From (8.58), (8.60), and (8.65),

@�i(qi)

@qi= J�1o (x)L�1i Jei(qi) (8.67)

Although we will not use the analytical expression of (8.66), we do need its existence to formulate the

problem. In other words, all constraints are holonomic constraints. }

We now consider the situation when the object comes in contact with rigid environment as in

section 8.1. Similar to (8.2), we assume that the environmental constraints are described by nc � n0mutually independent smooth hypersurfaces [74, 163]:

�0(x) = 0 �0(x) = [�01(x); : : : ; �0nc(x)]T (8.68)

The contact force Fc has a form similar to (8.3) and (8.4):

Fc = Fcn + Fct

Fcn = JTc (x)� Jc(x) =@�0(x)@x

Fct = [LTc (�; x; _x) + ~LTcf (x; _x)]�

(8.69)

where � 2 Rnc is a vector of Lagrange multipliers associated with the constraints. It is necessary to

control both the motion of the object on the constraint surfaces (8.68) and the generalized constrained

force �.

Consider AT : Rn0k ! Rn0 . AT is of full row rank and, therefore, the pseudo-inverse (AT )+

is given by A(ATA)�1. Also A+ = (ATA)�1AT . Thus, A+T �= (AT )+ = (A+)T [61]. From (8.64)

A+ = J�1o (x)(Pk

i=1LTi Li)

�1[LT1 LT2 : : : LTk ] 2 Rn0�n0k (8.70)

In calculating A+, only J�1o (x) is needed to be calculated on-line since Li is a constant matrix that

only depends on the position of grasp point of the i-th end-e�ector and is independent of the motion

of the object. Furthermore, analytic expression of J�1o (x) can be obtained.

Given the resultant force F , the end-e�ector force fe that satis�es (8.64) can be represented

by

fe = A+TF +Wfint W 2 Rn0k�(n0k�n0) fint 2 Rn0(k�1) (8.71)

where W is the matrix of orthonormals that are generated from the linearly independent vectors of the

null space of AT , i.e., ATW = 0. fint represents the vector of internal force components [164]. From

(8.64), ATW = 0 means

[LT1 LT2 : : : LTk ]W = 0 (8.72)

Since Li is a constant matrix, W can also be a constant matrix and can be calculated o�-line. This

is another advantage of using the moving end-e�ector frame and the object frame instead of using the

reference frame OXY Z in studying the relative motion among manipulators and the object. To ensure

111

some necessary coordinations among the manipulators, it is also needed to control the internal force

fint in addition to the motion and contact force control of the grasped object [169, 164, 148].

In the above constrained robotic system, the imposed n0k internal holonomic constraints

(8.66) and the nc external constraints (8.68) reduce the DOF of the system fromPk

i=0 ni to np =Pki=0 ni � n0k � nc = ns � m where m

�= n0k � n0 + nc. Therefore, np independent generalized

coordinates are su�cient to characterize the con�guration of the constrained robotic system, which are

assumed as rp = (q) = [ 1(q) ; : : : ; np(q)]T . rp are assumed to be twice continuously di�erentiable.

Note that np =Pk

i=1(ni � n0) + (n0 � nc). In them, (n0 � nc) generalized coordinates are used to

parametrize the motion of the object on the constraint surfaces. (ni � n0) generalized coordinates

are used to characterize self-movement of the i-th manipulator when it is redundant. The choice of

generalized coordinates rp depends on speci�c task requirements and is exible. For example, (n0�nc)generalized coordinates used to characterize the motion of the object on the constraint surfaces can

be chosen as the independent parameters that describe the surfaces. The others can be chosen as the

redundant joint angle of each redundant manipulator. Another easy way is to select the independent

joint angle qij as rp, which may be convenient for implementation. These independent joint angles can

be identi�ed in the same way as in the constrained motion control.

Let f 2 Rm be the vector of forces to be controlled, i.e., f = [fTint; �T ]T . Suppose

that rpd(t) = (qd(t)) is given as the desired motion trajectory of the robotic system and fd =

[fTintd(t) �Td (t)]T is the vector of desired internal force and external contact force trajectories. These

trajectories may be calculated o�-line to optimize some performance indexes. Let ep = rp(t) � rpd(t)

and ef = f(t)�fd(t) be the motion and force tracking errors respectively. The objective can be stated

as that of designing a control law for the actuator torque u so that ep �! 0 and ef �! 0 as t �! 1.

8.2.2 Adaptive Robust Control of Coordinated Manipulators

The equations and constraints in the above subsection are derived directly from the physical

laws. The internal force fint and the constraint force � look di�erent, and it is hard to get a clue about

how to control them. In the following, we will reformulate these equations and constraints to obtain an

equivalent set in the joint space in terms of q and f in a form in which descriptions of internal force

and constraint force are uni�ed to facilitate the design of control algorithms.

Let ve�= [ve1; : : : ; vek]

T . From (8.65), (8.58), and (8.64),

ve = A _x (8.73)

Multiplying it by A+,

_x = A+ve = A+Je(q) _q = J(q) _q J(q)�= A+Je(q) 2 Rn0�ns

�x = J(q)�q + _J _q(8.74)

Substituting (8.63) and (8.69) into (8.71)

fe = A+T fM0�x+ C0 _x +G0 + ~f0 + [JTc + LTc (�; x; _x) + ~LTcf ]�g+Wfint (8.75)

Let � = [�Tm; �T0 ]

T . Substituting (8.75) into (8.61) and noting (8.74), the resulting equation can be

written as

M(q; �)�q + C(q; _q; �) _q + G(q; �) + ~f (q; _q; t) + [DT (q) + TTc (�; x; _x) + ~TT

c ]f = u (8.76)

112

whereM(q; �) =Mm(q; �m) + JT (q)M0(x; �0)J(q)

C(q; _q; �) = Cm(q; _q; �m) + JT (q)C0(x; _x; �0)J(q) + JT (q)M0(x; �0) _J(q)

G(q; �) = Gm(q; �m) + JT (q)G0(x; �0)~f(q; _q; t) = ~fm(q; _q; t) + JT ~f0(x; _x; t)

D(q) =

"WTJe(q)

Jc(x)J(q)

#

Tc(�; q; _q) =

"0

Lc(�; x; _x)J(q)

#

~Tc =

"0

~Lcf (x; _x)J(q)

#

(8.77)

Since the constraint manifold is f�0(x) = 0; Jc(x) _x = 0g, noting (8.74), we have

Jc(x)J(q) _q = Jc(x) _x = 0 (8.78)

Multiplying ve = Je(q) _q by WT and noting that WTA = (ATW )T = 0 and (8.73),

WTJe(q) _q = WTve = WTA _x = 0 (8.79)

Combining (8.78) and (8.79), the system satis�es the constraints

D(q) _q = 0 (8.80)

where D(q), a full row rank matrix, is de�ned in (8.77).

It can be seen that the reformulated equation (8.76) under the velocity constraints (8.80)

resembles the constrained motion equation (8.1) of a single manipulator if we write the constraint (8.2)

in terms of velocity. In (8.76), the descriptions of internal force fint and external constrained force �

are uni�ed so that they can be controlled in a similar fashion. To make this clear, let us do the following

transformation, which is similar to the coordinate transformation (8.6) in the constrained motion control

except that no explicit expression of the coordinates rf is de�ned here to simplify the derivation and

computation.

Since rp is the vector of the generalized coordinates of the system, q = q(rp). Explicit

expression of q(rp) is not necessary for the following development and it is only used symbolically.

Di�erentiating rp,

_rp = Jrp _q; Jrp =@(q)

@q(8.81)

Combining (8.81) with the constraints (8.80), _q can be expressed in terms of the generalized velocity

_rp by

_q = J�1q (q)

"0

_rp

#; Jq(q)

�=

"D(q)

Jrp(q)

#2 Rns�ns (8.82)

where Jq(q) is nonsingular because of the fact that D(q) is of full row rank and rp is the vector of the

complete generalized coordinates of the system. Multiplying both sides of (8.76) by J�Tq and noting

(8.82) and its derivative, we have

M(rp; �)

"0

�rp

#+ C(rp; _rp; �)

"0

_rp

#+G(rp; �)

+(

"Im0

#+B(�; rp; _rp))f + ~fr(rp; _rp; f; t) = ur

(8.83)

113

orM12(rp; �)�rp+ C12(rp; _rp; �) _rp +G1(rp; �) + (Im + B1)f + ~fr1 = ur1M22(rp; �)�rp+ C22(rp; _rp; �) _rp +G2(rp; �) +B2f + ~fr2 = ur2

(8.84)

where M(rp; �), C(rp; _rp; �) , G(rp; �), and ur are calculated in the same expressions as in (8.10) 4

but with M(q; �), C(q; _q; �) , G(q; �), and u given by (8.76), and

B(�; rp; _rp) = J�Tq TTc (�; q; _q)

~fr(rp; _rp; f; t) = J�Tq [ ~f(q; _q; t) + ~TTc f ]

(8.85)

It can now be seen that (8.83) and (8.84) have the same form as (8.8) and (8.9) in the constrained

motion control with � in (8.8) and (8.9) corresponding to f in (8.83) and (8.84). Furthermore, since the

dynamic equations of each robot manipulator and the object all have the properties similar to Properties

1 to 3 in section 7.1, it is easy to check that the reformulated equation (8.77) also has those properties.

Thus, we can proceed the design of the coordinated motion and force controller exactly in the same way

as in the constrained motion control in section 8.1 by replacing � by f at each step. The detailed design

procedure is thus trivial but tedious, and it is omitted. Roughly, from (8.83) or (8.84), an equation

similar to (8.11) can be formed with exactly the same expressions for H(rp; �); Ch(rp; _rp; �); G(rp; �),

Bm(�; �; rp; _rp) and �Gf . Similar properties as Properties 4 to 6 in section 8.1.1 can then be obtained.

By designing motion and force sliding modes in the same way as in section 8.1.2 with exactly the same

expressions for all formula involved, we can obtain the control torque

ur = Hzv + Chzr + G+ (Bm � �Gf )f �K�� + �h

= fc(rp; _rp; zr; zv; f) + Yc(rp; _rp; zr; zv; f)�c� � �Gff �K�� + �h(8.86)

where

zv(x) =

"zvfzvp

#=

"Kffd + CfAfzf

�rpd � _yp

#(8.87)

and other symbols have the same expressions as those in (8.27) with � replaced by f . The adaptation

law has the same expression as (8.44) and the adaptation function � has the same expression as (8.43)

with � replaced by f .

Finally, the same qualitative results as in Theorem 11 can be obtained and the resulting

coordinated motion and force ARC law possesses the same good qualities as those mentioned in section

8.1.

8.2.3 Conclusions

The coordinated motion and force control of robot manipulators handling a constrained object

in the presence of parametric uncertainties and modeling errors in the manipulator and the object has

been studied. E�ect of contact surface friction was considered. A set of transformed dynamic equations

was obtained in the joint space to unify the descriptions of internal force and external contact force.

Thus, in principle, the coordinated control of multiple manipulators is equivalent to the constrained

motion control of a single robot. The same design technique as in the constrained motion control is

employed to derive an ARC coordinated motion and force controller.

4replace r in (8.10) by rp

114

8.3 Motion and Force Tracking Control of Robot Manipulators in

Contact With Unknown Sti�ness Environments

In this section, the proposed ARC is applied to solve the motion and force control of a robot

manipulator in contact with a sti� environment in the presence of parametric uncertainties and unknown

nonlinearities in both the manipulator and the contact surface.

8.3.1 Dynamic Model of a Manipulator in Contact with a Sti� Environment

Now consider the robot manipulator described by (8.1) in contact with a sti� environment.

The undeformed environment is described by a set of m time-varying hypersurfaces as shown in Fig.

8.11

�(x; t) = �e(t) �(x; t) = [�1(x; t); : : : ; �m(x; t)]T m � n (8.88)

which are mutually independent for any t. �e(t) = [�e1; : : : ; �em]T represents the equilibrium position

of the undeformed environment and is unknown. We assume that the derivative of �e(t) is bounded,

i.e.,

k _�e(t)k � �e (8.89)

Usually, �e is unknown but constant. Then _�e = 0 and (8.89) is trivially satis�ed.

Figure 8.11: A Manipulator in Contact With a Sti� Environment

Suppose that there exists a set of (n�m) scalar functions f 1(x; t); : : : ; n�m(x; t)g suchthat f�i(x; t); i = 1; : : : ; m; j(x; t); j = 1; : : : ; n�mg are mutually independent for any t. Similar to

(8.5), the task space is de�ned as

r = [rTf ; rTp ]T rf = [�1(x; t); : : : ; �m(x; t)]T 2 Rm

rp = [ 1(x; t); : : : ; n�m(x; t)]T 2 Rn�m (8.90)

Di�erentiating (8.90), we have

_r = Jr _x+ vt = Jq _q + vt (8.91)

115

whereJr =

@r(x;t)@x ; Jr = [JTrf JTrp]

T ; Jrf 2 Rm�n; Jrp 2 R(n�m)�n

Jq =@r(x(q);t)

@q ; Jq = JrJ; J = @x(q)@q ; Jq; Jr 2 Rn�n

vt =@r(x;t)@t vt 2 Rn

(8.92)

Multiplying both sides of (8.1) by J�Tq and noting (8.90) and (8.91), we obtain

M(r; t; �)�r+ C(r; _r; t; �) _r+ G(r; t; �)+Dt(r; _r; t; �) + ~f(r; _r; t) + Fr = ur (8.93)

whereM(r; t; �) = J�Tq (q; t)M(q; �)J�1q (q; t)

C(r; _r; t; �) = J�Tq C(q; _q; �)J�1q � J�Tq M(q; �)J�1q_JqJ

�1q

G(r; t; �) = J�Tq G(q; �)

Dt(r; _r; t; �) = �M(r; t; �) _vt� C(r; _r; t; �)vt~f(r; _r; t) = J�Tq (q; t) ~f(q; _q; t)

Fr = J�Tr F

ur = J�Tq (q; t)u

(8.94)

In de�ning the task space (8.90), the directions of curvilinear coordinates rf are aligned with

the normal directions of the undeformed environment (Fig. 8.11 ). Without loss of generality, we assume

that rf are aligned with the outer normal directions of the contact surfaces. Therefore, the subspace

rf 2 Rm in fact represents the constrained subspace in which force tracking control is required and

the subspace rp 2 Rn�m can be considered as the unconstrained subspace in which motion control is

needed. Such a de�nition has a clear physical meaning.

Along the normal directions of contact surfaces, the environment is assumed to be represented

by an elastic model with an unknown constant symmetric positive de�nite (s.p.d.) sti�ness matrix Ke

( either from the force sensor or from the contact surfaces), i.e.,

fn = Ke(rf � rfe(t)) or rf = Kffn + rfe fn � 0 (8.95)

where fn 2 Rm is the vector of normal contact force components, rfe(t) = �e(t) represents the

unknown equilibrium position, and Kf = K�1e 2 Rm�m is a unknown constant s.p.d. compliance

matrix. Since the contact surfaces are unilateral, fn � 0. As in the constrained motion control, it is

assumed that the end-e�ector is initially in contact with the surfaces, and that fn � 0 is never violated

after the control torque is applied. If the exact force tracking control can be achieved and force transient

response is not wild, which will be the case of the proposed controller, the assumption that fn � 0

during the operation can be justi�ed since the desired force trajectory must satisfy fnd < 0.

As in (8.3) and (8.4), the total interaction force F in (8.1) can be modeled by (Fig. 8.11)

F = Fn + FtFn = N(x; t)fnFt = [LT (�; x; _x; t) + ~LTf (x; _x; t)]fn

(8.96)

where N 2 Rn�m is a matrix with i-th column being the unit outer normal vector of the i-th contact

surface, i.e., Ni =(JTrf)i

k(JTrfi

)ik . Thus, Fr in (8.93) can be written as

Fr = Lr(�; r; _r; t)fn + ~Lr(r; _r; t)fnLr = J�Tr [N(x; t) + LT (�; x; _x; t)]~Lr = J�Tr

~LTf (x; _x; t)

(8.97)

116

where Lr represents the modeling part of the contact force and ~Lr represents the modeling error of the

contact force in task space. Lr can be linearly parametrized in terms of �. Thus, we can write

Lr(�; r; _r; t)fn = f�(r; _r; fn; t) + Y�(r; _r; fn; t)� (8.98)

where f� and Y� are known.

From Properties 1 to 3 in section 7.1, the following properties can be obtained for (8.93) by

using the same techniques as in Appendix 2 [146].

Property 7 For the �nite workspace q in which Jq is nonsingular, M(r; t; �) is a s.p.d. matrix with

k0rIn�n � M(r; t; �) � k00r In�n; 8q 2 q; t 2 R, where k0r = kmc21; k00r = kM

c22; c1 =

supq2q ;t2R[�max(Jq(q; t))]; and c2 = inf q2q;t2R[�min(Jq(q; t))].

Property 8 The matrix _M(r; t; �)� 2C(r; _r; t; �) is a skew-symmetric matrix.

Property 9 M(r; t; �); C(r; _r; t; �); G(r; t; �), and Dt(r; _r; t; �) can be linearly parametrized in terms

of �. Thus, we can write

M(r; t; �)zv + C(r; _r; t; �)zr +G(r; t; �) +Dt(r; _r; t; �)

= f�(r; _r; zr; zv; t) + Y�(r; _r; zr; zv; t)�(8.99)

where zr and zv are any reference values, and f� and Y� are related to f0 and Y in (7.2) by

f�(r; _r; zr; zv; t) = J�Tq (q; t)f0(q; _q; _qr; �qr)

Y�(r; _r; zr; zv; t) = J�Tq (q; t)Y (q; _q; _qr; �qr)(8.100)

in which_qr = J�1q (q; t)zr�qr = J�1q (q; t)[zv � _Jq _qr]

(8.101)

We assume that the modeling errors are bounded by some known functions, i.e.,

k ~f(r; _r; t) + ~Lr(r; _r; t)fnk � �r(r; _r; fn; t) (8.102)

where �r is known. We can now formulate the robust motion and force control problem as follows:

Suppose that rpd(t) 2 Rn�m is given as the desired motion trajectory in the unconstrained

subspace and fnd(t) 2 Rm is the desired force trajectory in the constrained subspace. Let ep =

rp(t) � rpd(t) 2 Rn�m and ef = fn(t) � fnd(t) 2 Rm be the tracking errors of motion and force

respectively. Consider the robot manipulator described by (8.93), whose end-e�ector is in contact with

the sti� surfaces (8.88) with the interaction force given by (8.97). Under the modeling errors (8.102),

with the robot parameters �, the contact surface parameters Ke;�e(t), and � being unknown, design

a control law and some parameter adaptation laws so that ep(t) and ef (t) are as small as possible.

8.3.2 ARC Motion and Force Tracking Control

In this subsection, the ARC is applied to solve the above robust motion and force tracking

control. Unlike in the previous applications where the designs are essentially for systems with relative

117

degree one, here, we have to deal with a relative degree two system because the derivative of the contact

force, _fn, is not available. To make this clear, let

x1 = [xT1;1; xT1;2]T x1;1 = fn x1;2 = rp

x2 = _r(8.103)

Noting (8.95), (8.93) and (8.97), the system can be represented by

_x1 = B1x2 +D1�1

_x2 =M�1(r; t; �)[�C(r; x2; t; �)x2 �G(r; t; �)�Dt(r; x2; t; �)

�Lr(�; r; x2; t)x1;1 + ur +�2]

y = x1

(8.104)

where

B1 =

"Ke 0

0 In�m

#

D1 = [Im 0]T

�1 = �Ke _rfe(t)

�2 = � ~f(r; x2; t)� ~Lr(r; x2; t)x1;1

(8.105)

Unlike (7.4) where the �rst equation does not have any modeling uncertainties, the �rst equation of

(8.104) has parametric uncertainties in B1 and the unknown nonlinearities �1 caused by the unknown

sti�ness and the unknown time-varying equilibrium position rfe(t). Also, if we treat r in the second

equation of (8.104) as a known quantity since r and _r are measurable5, noting (8.98), (8.102), (8.89),

and Properties 7 and 9, it can be checked out that (8.104) is in the semi-strict feedback form (6.3)

and satis�es all the Assumptions in section 6.1. Thus, in principle, we can apply the general results in

section 6.2 to obtain an ARC controller. However, in order to take into account of the special structure

of the robot dynamics, we proceed the design in the following way. The design parallels the recursive

backstepping design procedure in section 6.2. An ARC Lyapunov function is �rst constructed for the

�rst equation of (8.104). Then, using the backstepping results in section 5.4, an ARC Lyapunov function

is found for the whole system.

The �rst equation of (8.104) is actually made of two decoupled equations, i.e., the force

equation_fn = Kex2;1 + �1 (8.106)

and the motion equation

_rp = x2;2 (8.107)

Thus, in the following, ARC laws will be constructed for the force and motion equations separately. For

the force equation, since recent one-dimensional force experimental results [130, 91] have showed that

integral force feedback control has some advantages such as stronger robustness to the measurement

time delay and the removal of steady state force tracking error, we introduce the integral of force

tracking error, If = If(0)+R t0 ef (�)d�, in the design. Also, since Ke is a s.p.d. matrix, it will be easier

to design a control law based on the estimate of Kf = K�1e instead of the estimate of Ke. Considering

5Otherwise, we have to write r as a function of x1. The relationship r(x1) is unknown because of the unknownsti�ness and the unknown equilibrium. Then, terms like M(r(x1); t; �) cannot be linearly parametrized.

118

these factors, from c83p1w), equations for If and the force are

_If = ef = fn � fnd(t)Kf

_fn = x2;1 + ~�1~�1 = � _rfe

(8.108)

De�ne a switching-function-like vector �f as

�f = ef +D1If (8.109)

where D1 is a s.p.d. matrix. By choosing the initial value of If as If (0) = D�11 ef(0), we have

�f (0) = ef(0) +D1If (0) = 0 (8.110)

From (8.109), we note_�f = _fn � � �

�= _fnd �D1ef (8.111)

which will be utilized later. Denote the set of independent unknown parameters of Kf as � 2 Rk� where

k� � 12m(m+ 1) because of the symmetry of Kf . Then, we can write

Kf � = f�(�) + Y�(�)�

Kf � = f�(�) + Y�(�)��(8.112)

where f� and Y� are known. Choose a p.s.d. function Vf as

Vf =12wf�

Tf Kf�f (8.113)

where wf > 0 is any weighting factor. Since that �f �! 0 means that If �! 0 and ef �! 0, it can be

checked out that Assumption 1 in section 4.2 is satis�ed by Vf for the system (8.108). Noting (8.108)

and (8.112), the derivative of Vf is

_Vf = wf�Tf Kf

_�f = wf�Tf (Kf

_fn �Kf�)

= wf�Tf [x2;1 +

~�1 � f�(�)� Y�(�)�] (8.114)

Noting (8.89), as before, we can �nd a bounding function hf (�; ��; t) such that

k�Tf ( ~�1 � Y�(�)~��)k � hf(�; �� ; t) (8.115)

Let the control law for x2;1 be

uf (�; �f ; ��; t) = ufa(�; �f ; ��) + ufs(�; �f ; ��; t)

ufa(�; �f ; ��) = Kf� �D2�f= f�(�) + Y�(�)�� �D2�f

(8.116)

where D2 2 Rm�m and �1 2 Rkf�kf are any constant s.p.d. matrices, and ufs is a di�erentiable

continuous approximation of the discontinuous control �hf �fk�fk . ufs satis�es the same two conditions

as in (7.18) with an approximation error "f . Let _Vf juf denote _Vf under the condition that x2;1 = uf .

From (8.114), when ~�1 = 0, with the control x2;1 = uf ,

_Vf juf = �wf �Tf D2�f + wf�

Tf Y�(�)

~�� + wf�Tf ufs

� �wf �Tf D2�f + wf�

Tf Y�(�)

~��(8.117)

119

In general case that ~�1 6= 0,

_Vf juf = �wf�Tf D2�f + wf�

Tf [~�1 + Y�(�)~�� ] + wf�

Tf ufs

� �wf�Tf D2�f + wfk�fkhf + wf�

Tf ufs

� �wf�Tf D2�f + wf"f � ��VfVf + wf"f

(8.118)

where �Vf =2�min(D2)�max(Kf )

. From (8.117) and (8.118), Assumption 2 and 3 in section 4.2 are satis�ed by

Vf for the system (8.108) with the following adaptation function

�f = wfYT� (�)�f (8.119)

Thus, Vf is a valid ARC Lyapunov function for (8.108).

Since the position equation (8.107) has no modeling uncertainties, we can use the technique in

designing dynamic sliding mode in section 7.2.1 to obtain a stabilizing control for it. Namely, a dynamic

compensator that has the same form as (8.16) is introduced and the constant matrices (Ap; Bp; Cp; Dp)

are determined in the same way as in Chapter 7 to guarantee that the resulting sliding mode, �p = 0,

has prescribed good qualities. �p is given by (8.15). Choose the target control law for x2;2 as

up = _rpd(t)� yp (8.120)

Then, if x2;2 = up, from (8.15), we have

�p = x2;2 � ( _rpd + yp) = 0 (8.121)

Thus, the motion sliding mode is achieved and the resulting system is stable. In state space, the motion

subsystem has the state x�p = [zTp ; eTp ]

T and a representation like (7.9)

_x�p = A�px�p +B�p�p y�p = C�px�p

A�p =

"Ap Bp

�Cp �Dp

#B�p =

"0

In�m

#C�p = [0 In�m]

(8.122)

Similar to (7.11), for any s.p.d. matrixQ�p, there exists an s.p.d. solution P�p for the following Lyapunov

equation

AT�pP�p + P�pA�p = �Q�p (8.123)

Then, it can be shown that

Vp =12x

T�pP�px�p (8.124)

is a Lyapunov function or an ARC Lyapunov function for the motion subsystem under the control

x2;2 = up.

Now, an ARC law is designed for the second equation of (8.104) so that its output x2 tracks

its desired value u1d = [uTf ; uTp ]T given by (8.116) and (8.120). The same as in section 5.4, the ARC

law is determined to make a p.s.d. function like (5.34) an ARC Lyapunov function. The p.s.d. function

is given by

V = Vf + Vp +12z

T2M(r; t; �)z2 (8.125)

where z2 = x2 � u1d = _r � u1d. Before we move on to the design of the control law, the same as in

section 6.2, we have to use the trajectory initialization to achieve that z2(0) = 0 so that Assumption 1

120

can be satis�ed by V . From (8.110) and (8.116), ufs(0) = 0 and ufa(0) = Kf �(0). Thus, z2(0) = 0

can be guaranteed if we choose the initial values of the desired motion and force trajectories as

_fnd(0) = K�1f (0)rf(0) +D1ef (0)

_rpd(0) = _rp(0) + yp(0)(8.126)

Noting that _� = �fnd+D1_fnd�D1

_fn and _�f = _fn� �, by di�erentiating (8.116), we can write

_uf = Y1(�; �f ; ��(1)� ; t) + Y2(�; �f ; ��; t) _fn +

@uf@�

(_� � ���f ) (8.127)

where Y1 and Y2 are calculable and given by

Y1 =@uf@� (

�fnd +D1_fnd)� @uf

@�f�+

@uf

@����f +

@uf@t

Y2 = �@uf@� D1 +

@uf@�f

(8.128)

Noting (8.95), _u1d can be decomposed into the following terms

_u1d = zv +

"Y20

#(Ke _rf �Ke _rfe) +

"@uf@�

0

#( _� � ���f) (8.129)

where

zv =

"Y1(�; �f ; ��

(1)� ; t)

_up

#(8.130)

zv is calculable based on the measurements of position, velocity, and force only. Similar to (5.41), there

exists known Y3(r; �; �f ; ��; _rf ; t) and Y#(r; �; �f; �� ; _rf ; t) such that

M(r; t; �)

"Y2(�; �f ; ��; t)Ke _rf

0

#= Y3 + Y## (8.131)

where # represents a set of suitably selected unknown constants whose elements are the products of the

elements of � and Ke. Noting (8.114), (8.123), and Property 8, we have

_V = _Vf juf +wf �Tf (x2;1� uf )� 1

2xT�pQ�px�p + xT�pP�pB�p(x2;2� up) + zT2 (M _z2 + Cz2)

= _Vf juf �12x

T�pQ�px�p + zT2 [M( _x2 � _u1d) + C(x2 � u1d) + u0r]

(8.132)

where

u0r =

"wf�f

BT�pP�px�p

#(8.133)

Substituting the second equation of (8.104) into (8.132) and noting (8.129) and (8.131), we have

_V = _Vf juf �12x

T�pQ�px�p + zT2 [�G(r; t; �)�Dt(r; _r; t; �)� Lr(�; r; _r; t)fn

+ur +�2 �M _u1d � Cu1d + u0r]= _Vf juf �1

2xT�pQ�px�p + zT2 fur �Mzv � Cu1d � G�Dt � Lrfn

�Y3 � Y##+ u0r �M

"@uf@�

0

#(_� � ���f ) + ~�2g

(8.134)

where

~�2 =M

"Y20

#Ke _rfe + �2 (8.135)

121

Noting that the only term in V that contains � is uf , we have,

@V@�

= �zT2M"

@uf@�

0

#(8.136)

Let the control law be

ur = ura + ursura = f�(r; _r; u1d; zv; t) + Y�(r; _r; u1d; zv; t)�� + f�(r; _r; fn; t)

+Y�(r; _r; fn; t)�� + Y3 + Y##� � u0r �Kzz2

(8.137)

where Kz > 0 and urs is a continuous approximation of the discontinuous control �hz z2kz2k that satis�es

the same two conditions as in (7.18) with an approximation error "z . As before, the bounding function

hz is chosen so that the following inequality is satis�ed

hz � kY� ~�� + Y�~�� + Y# ~#� + ~�2k (8.138)

Noting (8.99), (8.98), and (8.136), and substituting (8.137) into (8.134), we have

_V = _Vf juf �12x

T�pQ�px�p � z2Kzz2

+zT [urs + Y� ~�� + Y�~�� + Y# ~#� + ~�2] +@V@�(_� � ���f)

(8.139)

In the absence of uncertain nonlinearities, i.e., �i = 0, from (8.108) and (8.135), ~�i = 0. From (8.117)

and noting that V does not contain �, �, and #, we have

_V � �wf �Tf D2�f � 1

2xT�pQ�px�p � z2Kzz2 + �Tf

~�� + zT2 urs + zT2 Y�~��

+zT2 Y� ~�� + zT2 Y#~#� +

@V@�(_� � ���f )

� ��V V + �Te~�e� +

@V@�e

(_�e � �e�e)

(8.140)

where

�V = minf2�min(D2)�max(Kf )

;�min(Q�p)

�max(P�p); 2�min(Kz)

k00rg

�Te = [�Tf ; zT2 Y� ; zT2 Y�; zT2 Y#]

�e = [�T ; �T ; �T ; #T ]T

�e = diagf��; �� ; ��; �#g

(8.141)

Thus Assumption 2 in section 4.2 is satis�ed by V for the system (8.104) with W = �V V and the

adaptation function �e for the extended parameter set �e. Furthermore, since V does not contain �, �,

and #, we can use the following adaptation law

_� = ����f_� = ��� [l�(�) + Y T

� z2]_� = ���[l�(�) + Y T

� z2]_# = ��#[l#(#) + Y T

# z2]

(8.142)

where l�, l�, and l# are modi�cation functions de�ned in section 4.3.

In general case that �i 6= 0, from (8.138) and (8.118),

_V � �wf�Tf D2�f + wf"f � 1

2xT�pQ�px�p � z2Kzz2 + zT2 urs + kz2khz + @V

@�(_� � ���f )

� ��V V + " + @V@�(_� � ���f)

(8.143)

where " = wf"f + "z . Thus, Assumption 3 in section 4.2 is satis�ed and V is a valid ARC Lyapunov

function. The above results can now be summarized in the following theorem.

122

Theorem 18 When the robot manipulator described by (8.93) comes in contact with the sti� surfaces

(8.88) and the interaction force is given by (8.97), the following results hold if the control law (8.137)

with the adaptation law (8.142) is applied:

a). In general, ep; zp; ef , and If exponentially converge to some balls whose size can be freely adjusted

by controller parameters in a known form. The control input is bounded.

b). When the system does not have uncertain nonlinearities, i.e., the equilibrium position �e is un-

known but constant and �2 = 0 in (8.104), asymptotic motion and force tracking control is

achieved, i.e., ep �! 0 and ef �! 0. 4

8.3.3 Conclusions

In this section, the proposed ARC is applied to solve the problem of robust motion and force

tracking control of robot manipulators in contact with sti� environments. Parametric uncertainties can

exist in the robot dynamics and the contact surfaces' friction coe�cients and sti�ness matrix. Uncertain

nonlinearities can also be allowed in the modeling of the robot dynamics and the contact surfaces.

123

Chapter 9

Conclusion

9.1 Conclusions

In this dissertation, an adaptive robust control (ARC) method is proposed in order to design a

high performance robust controller in the presence of parametric uncertainties and uncertain nonlinear-

ities. The approach improves performance by preserving the advantages of both adaptive control (AC)

and deterministic robust control (DRC) while removing their drawbacks. This is achieved by selecting

the controller structure and parameter adaptation properly. Proper controller structure attenuates the

e�ect of model uncertainties from parametric uncertainties and uncertain nonlinearities. Thus, transient

performance and �nal tracking accuracy can be guaranteed in general. Proper parameter adaptation

reduces the model uncertainties to further improve the tracking performance. Thus, asymptotic track-

ing (or zero �nal tracking error) can be achieved without using high-gain feedback in the presence of

parametric uncertainties only. The design is conceptually simple and is attractive in applications because

of its high performance and strong robustness (exponential stability at large).

The concept of adaptive robust control (ARC) Lyapunov functions is introduced to formulate a

general framework of the ARC of a multi-input multi-output (MIMO) nonlinear system. The formulation

reduces the adaptive robust control of a system into the problem of �nding an ARC Lyapunov function for

the system. Such a formulation enables us to use the backstepping design procedure to systematically

enlarge the applicable nonlinear systems. Two backstepping design procedures are developed. By

applying the backstepping design procedure recursively, we have successfully constructed ARC Lyapunov

functions for a class of MIMO nonlinear systems transformable to a semi-strict feedback form. The form

allows coupling and parametric uncertainties in the input channels of each layer and includes mechanical

systems, such as robot manipulators.

The method is applied to the control of robot manipulators for di�erent tasks. Trajectory

tracking control of robot manipulators is comprehensively studied. Two ARC algorithms are developed:

adaptive sliding mode control (ASMC) is based on SMC and the conventional adaptation law structure,

in which the regressor uses actual state feedback information; desired compensation ARC (DCARC)

employs a regressor depending on the desired trajectory information only. In addition, three di�erent

adaptive or robust control schemes are derived for comparison: a simple nonlinear PID type robust

control, a gain-based nonlinear PID type adaptive control, which requires no model information, and a

combined parameter and gain-based adaptive robust control. All algorithms, as well as two benchmark

algorithms, Slotine and Li's SLAC and Sadegh and Horowitz's DCAL, are implemented and compared

124

on a two-link direct drive robot. Comparative experimental results show the importance of using both

means, proper controller structure and parameter adaptation, in designing high performance controllers.

It is observed that in these experiments, the proposed DCARC achieves the best tracking performance

without increasing the control bandwidth and the control e�ort.

The motion and force control of a constrained robot manipulator in contact with rigid surfaces

is studied with a consideration of realistic aspects, such as contact surface friction and acausality

problems. A new constrained dynamic model is obtained. Dynamic motion sliding mode and �ltered force

tracking error are used to enhance the dynamic response of the closed-loop system. The resulting ARC

algorithm has the expected PI type force feedback control structure with a low proportional (P) gain.

The guaranteed transient performance of the controller alleviates the impact problems caused by losing

contact. By using both motion and force tracking errors in updating estimated parameters, asymptotic

motion and force tracking is achieved without any persistent excitation conditions. Simulation results

are given and verify the e�ectiveness of the method.

The coordinated control of multiple robot manipulators handling a constrained object is solved.

A set of transformed dynamic equations are obtained in the joint space to unify the expression of internal

force and external contact force. Such a formulation facilitates the controller design and provides a

di�erent insight to the coordinated control of multiple robot manipulators. The same design technique

as in constrained motion control is used to design the coordinated motion and force ARC controller.

The hybrid motion and force control of robot manipulators in contact with sti�ness environ-

ments is considered. The formulation is very general and allows both robot parameters and surface

parameters, such as sti�ness and friction coe�cients, to be unknown. The formulation also allows

uncertain nonlinearities, such as modeling errors.

9.2 Suggested Ideas for Future Research

Future research topics on theoretical and experimental developments of high performance

robust controllers include the following:

� For the proposed ARC, it is only shown that, if uncertain nonlinearities disappear after a �nite

time, model uncertainties resulting from parametric uncertainties can be eliminated by parameter

adaptation. When uncertain nonlinearities exist all the time, with the degree of uncertain non-

linearities relatively small and model uncertainties mainly coming from parametric uncertainties,

then it seems intuitively that model uncertainties should still be reduced by parameter adaptation

and an improved tracking performance should still be obtained. This fact has been veri�ed by

the experimental results in Chapter 7 where disturbances are small but nonzero all the time and

by the simulation results in other chapters. At the same time, some quantitative analysis is still

helpful in understanding when an improved tracking performance can be guaranteed even with

the presence of uncertain nonlinearities.

� The proposed semi-strict feedback form only allows M�1 at the last layer. For control of exible

joint robot manipulators, the term like M�1 may appear in the intermediate layer. Thus, it is of

practical signi�cance to generalize the proposed ARC for a more general form.

� In this dissertation, we only deal with full-state feedback. In some applications, only output is

measurable. Thus, ARC by output feedback is one of the promising directions to go.

125

� For the motion control of robot manipulators in Chapter 7, we only use a simple integrator in

designing the dynamic siding mode. It is worthwhile to investigate using other more involved

techniques, such as frequency shaping optimal control technique, in designing the dynamic siding

mode.

126

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Appendix 1

For revolute joint robot, q appears in M(q; �); C(q; _q; �), and G(q; �) in the form of cos(qi) or

sin(qi) only. Therefore, M(q; �); C(q; _q; �), and G(q; �) and their partial derivatives are uniformly

bounded with respective to q. Applying the mean value theorem, there exist non-negative scalars,

�1(�qd); �3( _qd); �4( _qd), and �7 such that

kM(qd; �)�qd �M(q; �)�qdk � �1(�qd)kekkC(q; _qd; �)k � �4( _qd)

kC(qd; _qd; �) _qd � C(q; _qd; �) _qdk � �3( _qd)kekkG(qd; �)�G(q; �)k � �7kek

(.1)

Since C(q; _q; �) is linear w.r.t. _q, there exist non-negative scalars, �5( _qd) and �6 such that

kC(q; _q; �)k = kC(q; _qd + _e; �)k � �5( _qd) + �6k _ek (.2)

Let �z = kCz ; Dzk and noticing that _e = � � [Cz; Dz]x�, we have,

k _ek � k�k+ �zkx�k (.3)

Noting C(q; _q; �) _qd = C(q; _qd; �) _q, from (.1) to (.3),

kM(qd; �)�qd �M(q; �)�qrk � kM(qd; �)�qd �M(q; �)�qdk+ kM(q; �)(�qd� �qr)k� �1(�qd)kek+ k00k[Cz; Dz]A�kkx�k+ k00k[Cz; Dz]B�kkx�k

kC(qd; _qd; �) _qd � C(q; _q; �) _qrk � kC(qd; _qd; �) _qd � C(q; _qd; �) _qdk+ kC(q; _qd; �) _qd�C(q; _q; �) _qdk+ kC(q; _q; �) _qd � C(q; _q; �) _qrk � �3( _qd)kek+�4( _qd)k _ek+ kC(q; _q; �)kk[Cz; Dz]x�k � [�3( _qd) + �4( _qd)�z+�5( _qd)�z]kx�k+ �4k�k+ �6�zk�kkx�k+ �6�

2zkx�k2

(.4)

From (.1) and (.4), it is clear that there are known constants, 1; 2; 3, and 4, such that (7.31) is

satis�ed.

Appendix 2

From Properties 1 and 2, the following properties can be obtained for Eq. (8.8) [146, 148].

Property 10 . For the �nite work space q in which Jq is nonsingular, M(r; �) is a s.p.d. matrix with

k0rIn �M(r; �) � k00rIn 8q 2 q (.5)

where k0r = k0

c21k00r = k00

c22c1 = supq2q [�max(Jq(q))] c2 = infq2q [�min(Jq(q))]. �(�) denotes

singular values of �, and �max (or �min ) is the maximum ( or minimum) value of �.

Property 11 . The matrix N(r; _r; �) = _M (r; �)� 2C(r; _r; �) is a skew-symmetric matrix.

136

Now, rewriting H as

H =

"(I +Gf )K

�1f �M11(r; �) 0

0 0

#+M(r; �) (.6)

If �max(Kf) � 1k00r, then, �minf(I + Gf)K

�1f g � �minfK�1

f g = 1�max(Kf )

� k00r . From Property 10,

M11(r; �) � k00r Im. Thus, (I + Gf )K�1f �M11 is a symmetric positive semide�nite (s.p.sd.) matrix.

From (.6), Property 4 is established.

From (8.12) and (.6)

Nh = _M(r; �)� 2C(r; _r; �)�"

_M11(r; �)� 2C11(r; _r; �) 0

0 0

#(.7)

From Property 11, _M11 � 2C11 is a skew- symmetric matrix. Thus, Property 5 is true.

In view of Property 3 and that B(�; rp; _rp) is linear w.r.t. �, Property 6 is obviously true.