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  • OBSERVER-BASED FEEDBACK CONTROL METHODS FOR AN UNDERACTUATED ROBOT SYSTEM

    by

    Guoyu Wang

    B.Eng., Tsinghua University, Beijing, P.R.China, 1995

    A THESIS SUBMITTED IN PARTIAL FULFILLMENT

    O F THE REQUIREMENTS FOR THE DEGREE O F

    MASTER OF APPLIED SCIENCE

    in the School

    of

    Engineering Science

    @ Guoyu Wang 2003

    SIMON FRASER UNIVERSITY

    November 2003

    All rights reserved. This work may not be

    reproduced in whole or in part, by photocopy

    or other means, without the permission of the author.

  • APPROVAL

    Name: Guoyu Wang

    Degree: blaster of Applied Science

    Title of Thesis: Observer-Bascd Feedback Control Mcthods for an Under-

    actuated Robot System

    Examining Committee: Dr. John Dill, Chair

    Professor, School of Engineering Science

    -on Fraser University

    Simon Fraser University

    V - Dr. William Gruver, Supervisor

    Professor, School of Engineering Science

    Simon Fraser Univefsity

    Dr. Shahram ~ a ~ a n d e j , t ~ x a 4 n e r

    Professor, School of Engineering Science

    Simon Fraser University

    Date Approved:

  • PARTIAL COPYRIGHT LICENCE

    I hereby grant to Simon Fraser University the right to lend my thesis, project or

    extended essay (the title of which is shown below) to users of the Simon Fraser

    University Library, and to make partial or single copies only for such users or in

    response to a request from the library of any other university, or other educational

    institution, on its own behalf or for one of its users. I further agree that permission for

    multiple copying of this work for scholarly purposes may be granted by me or the

    Dean of Graduate Studies. It is understood that copying or publication of this tvork

    for financial gain shall not be allowed without my written permission.

    Title of Thesis/Project/Extended Essay:

    OBSERVER-BASED FEEDBACK CONTROL METHODS FOR

    AN UNDERACTUATED ROBOT SYSTEM

    Author: (Signature)

  • Abstract

    This thesis is devoted to nonlinear control and state variable estimation of a typical

    underactuated mechanical system, called "Pendubot".

    In most cases, the controllers applied to mechanical systems require information

    of all states for calculating the feedback control signal. For the Pendubot, states

    are comprised of joint angular positions and velocities. The joint angular positions

    could be measured accurately. However,the joint angular velocities are obtained either

    by differentiating the measured joint angular positions or through tachometers in

    practical situations. In both cases, the velocity signal is contaminated by noise. This

    degrades the performance of closed-loop Pendubot system.

    The main focus of this thesis is on how to accurately estimate the joint angular

    velocities of the Pendubot. To develop advanced observers, model based approaches

    are considered. High-gain observer (HGO) and sliding mode observer (SMO) are

    designed and applied respectively. In order to fully investigate the effect of these

    two observers on the Pendubot, controllers based on each of them are applied to the

    Pendubot system in both simulation and real-time implementation. The controller

    employed here includes two parts. One is called swing up controller and swings up

    the links of the Pendubot from their downward positions to pre-chosen equilibrium

    positions. The other one is named balancing controller, which is used to balance the

    Pendubot at the equilibria.

    As a linear form observer, HGO is simple and easily implemented. The SMO

    covers all the nonlinearity of the system and therefore is more robust to noise in mea-

    surements. Based on both simulation and implementation of the combined system,

    it is concluded that the HGO and SMO are both more effective than the traditional

    difference method in supplying accurate estimation of joint angular velocities of the

  • Pendubot and more resistant to noise contamination.

    In summary, this work aims to study the issues on accurate estimation of the joint

    angular velocities of the Pendubot using available noisy measurements. The results of

    this thesis implies a better control of the dynamic states of this underactuated system

    with substantial modelling uncertainties and disturbances.

  • Acknowledgements

    I would like to thank my senior supervisor Dr. Mehrdad Saif, for his great support and

    guidance throughout the course of this thesis project. I also would like to thank Dr.

    William Gruver and Dr. Shahram Payandeh for being my Supervisory Committee

    members.

    I gratefully acknowledge Mr. Lucky One and Mr. Gary Houghton for their help

    on setting up the electrical and mechanical parts of the Pendubot.

    I also very appreciate my colleagues for their enthusiastic help on troubleshooting

    of using softwares and assembling of the hardware.

  • Dedication

    I dedicate this thesis to my wife Xuesong, my lovely son Richard and my parents.

  • Table of Contents

    . . Approval 11

    ... Abstract 111

    Acknowledgements v

    Dedication vi

    Table of Contents vii

    List of Figures x

    1 Introduction 1

    1.1 Motivations and Objectives . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The Pendubot System . . . . . . . . . . . . . . . . . . . . . . . . . . 4

    1.2.1 Description of the Hardware . . . . . . . . . . . . . . . . . . . 4

    1.2.2 Description of the Software . . . . . . . . . . . . . . . . . . . 6

    1.3 Layout of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

    1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    2 Mathematical Model of the Pendubot 9

    2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

    2.2 Mathematical Description of the System . . . . . . . . . . . . . . . . 11

    2.2.1 System Motion Equation . . . . . . . . . . . . . . . . . . . . . 11

    2.2.2 System Properties . . . . . . . . . . . . . . . . . . . . . . . . . 14

    vii

  • 2.3 System Identification . . . . . . . . 2.3.1 CAD Solid Model . . . . . .

    2.3.2 Energy Equation Method . 2.3.3 Optimization Method . . . .

    2.3.4 Final results . . . . . . . . .

    2.4 The State Space Based Description

    2.5 The Equilibrium Manifold . . . . . . . . . . . . . . . . . . . 2.6 Summary

    3 The Feedback Control 23

    3.1 Swing up Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

    3.1.1 Control of the First Link . . . . . . . . . . . . . . . . . . . . . 24

    3.1.2 Control of the Second Link . . . . . . . . . . . . . . . . . . . . 27

    3.1.3 Reference Trajectories Selection . . . . . . . . . . . . . . . . . 28

    3.2 Balancing Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    3.3 Combining and Implementing the Controllers . . . . . . . . . . . . . 32

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Summary 33

    4 The Observer Design 34

    4.1 Review of Observer Design Methods . . . . . . . . . . . . . . . . . . . 35

    4.2 High Gain Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

    4.2.1 Observer Structure . . . . . . . . . . . . . . . . . . . . . . . . 37

    4.2.2 Prerequisite on Parameters . . . . . . . . . . . . . . . . . . . . 39

    4.3 Sliding Mode Observer . . . . . . . . . . . . . . . . . . . . . . . . . . 43

    4.3.1 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    4.3.2 Definition of Coefficient Matrices . . . . . . . . . . . . . . . . 48

    4.3.3 Convergence Rate . . . . . . . . . . . . . . . . . . . . . . . . . 51

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Summary 53

    5 Combined System Performance 5 6

    5.1 Scheme of the Combined System . . . . . . . . . . . . . . . . . . . . 56

    5.2 Simulation of the Combined System . . . . . . . . . . . . . . . . . . . 57

    ... Vll l

  • 5.3 High-Gain Observer Based Combined System . . . . . . . . . . . . . 58

    5.3.1 Preliminary Calculation . . . . . . . . . . . . . . . . . . . . . 58

    5.3.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 60

    5.4 Sliding Mode Observer Based Combined System . . . . . . . . . . . . 66

    5.4.1 Preliminary Calculation: . . . . . . . . . . . . . . . . . . . . . 66

    5.4.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 67

    5.5 The Noisy Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

    5.5.1 High Gain Observer Based System . . . . . . . . . . . . . . . 70

    5.5.2 Sliding Mode Observer Based System . . . . . . . . . . . . . . 71

    5.5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

    6 Real-Time Implementation of the Combined System 75

    6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

    6.2 Sampled-Data Control of the Pendubot . . . . . . . . . . . . . . . . . 76

    6.3 Real-TimeAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

    6.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 81

    7 Conclusions 85

    Bibliography 8 9

  • List of Figures

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Pendubot 5

    . . . . . . . . . . . . . . . . . . 2.1 Coordinate Description of the Pendubot 12

    . . . . . . . . . . . . . . . . 2.2 The Optimization Method for Identification 18

    . . . . . . . . . . . . . . . . . . . 2.3 The Estimation of Inertial Parameters 19

    . . . . . . . 3.1 Block Diagram of the Partial Feedback Linearization Control 26

    . . . . . . . . . . . . . . . . . . 3.2 Feedback Control Algorithm Description 33

    . . . . . . . . . . . . . . . . . . . The Scheme of the Combined System

    Link 1's angular position and its estimation via high gain observer . . . .

    Link 2's angular position and its estimation via high gain observer . . . .

    Link 1's angular velocity and its estimation via high gain observer . . . .

    Link 2's angular velocity and its estimation via high gain observer . . . .

    The Errors between system states and estimations by applying the sliding

    mode observer. the initial condition being x(0) = (01010.51 0.5) . . . . . .

    The Errors between system states and estimations by applying the sliding

    mode observer. the initial condition being Z(O) = (010131 3) . . . . . . . .

    The errors between system states and estimations with white noise added

    . . . . . . . . . . . . . . . . . . . to the high gain observer based system

    The errors between system states and estimations with noise added to the

    sliding mode observer based system . . . . . . . . . . . . . . . . . . . . .

    . . . . . . 6.1 Configuration of the Sampled-Data Pendubot Hardware System 77 . . . . . . . . . . . . . . . . . . . . . . . 6.2 Structure of the real-time Code 80

  • 6.3 Measurements of Links' positions for the observer based system and differ-

    ence method based system. . . . . . . . . . . . . . . . . . . . . . . . . . 83

    6.4 Errors between the estimations and measurements in real-time implementation. 84

  • Chapter 1

    Introduction

    In the last three decades, the conception and use of robots has evolved from science fic-

    tion fantasies to computer-controlled electromechanical devices integrated into a wide

    variety of industrial environments. Recently, various kinds of robots have been de-

    signed for different purposes and the vast majority of robot manipulators in operation

    are being used for stuffing printed circuit boards with IC components, welding and

    painting car bodies on assembly lines, inspecting and repairing structures in nuclear,

    undersea and underground environments, and even picking oranges and harvesting

    grapes. The applications range from pick and place operations, to mobile cameras

    and other inspection equipment, to delicate assembly tasks involving matching parts.

    PlI. Robotics is a broad field that includes many areas such as physics, mechanical

    design, motion analysis and planning, actuators and drivers, control design, sensors

    and state estimation, signal and image processing, computer algorithms, study of the

    behavior of machines, animals, and even human beings [41].

    Recently, in the robotic field, much attention has been given to the control of

    mechanical systems where the number of control inputs is less than that of generalized

  • CHAPTER 1. INTRODUCTION 2

    coordinates. Such mechanical systems are said to be underactuated systems [16].

    Underactuated mechanical systems arise in several ways. The most obvious way

    is from intentional design as in the brachiation robot of Fukuda [29] or the Acrobot

    [2]. These systems are multi-link robots that have motors placed at only some of

    the joints. The underactuated systems also arise in mobile robot systems where, for

    example, a manipulator arm is attached to a mobile platform, a space platform, or

    an undersea vehicle [19]. In this case, additional controls are often used in order to

    maintain attitude control of the base. If these additional actuators fail or are turned

    off to conserve fuel, then the overall system is underactuated. Underactuated systems

    can also happen in the mathematical model used for control design as, for example,

    when the joint flexibility is included in the model [36]. In this case, all mechanical

    systems are underactuated if one wishes to control flexible modes that are not directly

    actuated or even to include actuator dynamics in the model description.

    Based on the above discussion, one can see that the underactuated systems appear

    in a broad range of applications including robotics, aerospace systems, marine systems,

    flexible systems, mobile systems, and locomotive systems. In addtion to give a more

    detailed consideration of system property (e.g. flexible modes), underactuated systems

    have certain advantages comparing to fully-actuated systems, such as enhancement

    of fault tolerant ability as well as reduction of cost, weight, and hardware complexity.

    Due to its wide applications and various advantages, the underactuated system has

    been under intense studies recently. In order to investigate the control methodologies

    in the underactuated system, the PENDUBOT (short for PENDUlum ROBOT),

    which is a typical underactuated robotic system, has been studied as a model in this

    work, and it will be discussed in detail in the following section.

  • CHAPTER 1. INTRODUCTION 3

    The Pendubot is a two-link revolute planar robot with one actuator a t the shoul-

    der. It possesses a wide range of applications for control, state estimation and fault

    diagnosis research in the field of robotics. Based on this complicated nonlinear sys-

    tem, advanced control strategies, accurate state estimation and even fault diagnosis

    methods that are suitable for applying into robotic manipulators can be designed and

    implemented.

    1.1 Motivations and Objectives

    The underactuated systems, for example, the Pendubot in this study, are not only

    nonlinear but also underactuated which make them very complicated. Therefore, ad-

    vanced controllers are required to keep these systems stable and function as expected.

    To design controllers, it is often assumed that all state variables are already known

    and could be used as the input to the controller algorithm in order to calculate...

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