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MODELING AND CONTROL OF A ROBOTIC JOINT

WITH IN-PARALLEL REDUNDANT ACTUATORS

Benoit Boulet

B.Sc.A. Universite Laval 1990

Department of Electrical Engineering

McGill University, Montreal

A thesis submitted to the Faculty of Graduate Studies and Research

in partial fulllment of the requirements for the degree of

Master of Engineering

August 1992

c Benoit Boulet

Abstract

The rst part of this thesis includes a brief comparison between electricmotors and

hydraulic actuators for high performance robotics applications. Hydraulic actuators

with fast valves are shown to be superior because of their large torque to mass ratios

and their extended bandwidth. One such hydraulic actuator is characterized and its

highly nonlinear dynamics are modeled and identied. A simulator implementing

this dynamic model is shown to predict the system's behavior satisfyingly. A lead-lag

force controller that yields a large bandwidth and good accuracy is also designed.

The second part is devoted to the modeling and control of an in-parallel actuated,

redundant, revolute joint mechanism. An autonomous kinematic calibration method

is presented, and tested on a prototype of the joint. The actuator forces are optimized

to reduce internal force, and to minimize their maximum magnitude. A method to

generate a pre-load force in the joint to eliminate backlash is also presented. Finally,

a PD controller, a robust PID controller, and a robust H1-optimal controller are

designed to control the joint angle. Results are presented for position and impedance

control experiments, and the PD and H1-optimal controllers are shown to be superior

to the PID controller in terms of trajectory tracking and robustness to variations in the

joint's inertia. A variable bandwidth, nonlinear position controller is also developed

and tested experimentally.

Resume

La premiere partie de cette these inclut une breve comparaison entre les moteurs

electriques et les actionneurs hydrauliques pour la robotique de haute performance.

On montre que les actionneurs hydrauliques equipes de valves rapides representent un

meilleur choix a cause de leur rapport couple/masse favorable et leur grande largeur de

bande. Un actionneur de cette categorie est caracterise et sa dynamique non-lineaire

est modelisee et identiee. Il est montre qu'un simulateur de ce modele predit le

comportement du systeme de maniere satisfaisante. Un reseau correcteur de force

du type integral avec avance de phase qui donne une bonne precision et une grande

largeur de bande est egalement concu.

La deuxieme partie est consacree a la modelisation et a la commande d'une ar-

ticulation redondante parallele. Une methode de calibration cinematique autonome

est presentee et testee sur un prototype de l'articulation. Les forces des actionneurs

sont optimisees pour reduire la force interne et pour minimiser leur amplitude max-

imale. Une methode pour generer une force de pre-tension dans l'articulation est

egalement presentee. Finalement, un compensateur PD, un compensateur PID ro-

buste et un compensateur optimal H1 robuste sont concus pour commander l'angle

de l'articulation. Des resultats experimentaux sont presentes pour la commande de

position et d'impedance, et il est demontre que les compensateurs PD et H1 orent

une meilleure performance que le compensateur PID en termes d'asservissement a

une trajectoire et de robustesse face a des variations dans l'inertie de l'articulation.

Un compensateur non-lineaire a largeur de bande variable est egalement developpe

et teste experimentalement.

i

Acknowledgements

First, I wish to thank my supervisor, professor Vincent Hayward, because this

thesis would simply not have been possible without his great help and encouragement.

His amazing curiosity and imagination have inspired me throughout the course of this

work and led me to believe that doing research can be actually a lot of fun.

I also wish to express my gratitude to the research engineer Chafye Nemri who

spent countless hours writing code and setting up the systems so that the experiments

could be carried out successfully. Professor Laeeque Daneshmend was also of great

help to model properly the hydraulic actuators. John Foldvari skillfully machined all

the parts needed to build the prototype of the redundant parallel joint.

I would like to express my appreciation to my colleagues and friends at the McGill

Research Centre for Intelligent Machines who were always very helpful whenever I

needed assistance. In particular, I would like to thank Marc Bolduc and Robert

Lucyshyn who accepted the ungrateful job of reviewing some of the chapters of this

thesis.

I am very grateful for the constant encouragement and support of Isabelle Lemay

and of my parents.

Finally, I would like to acknowledge the nancial support of the Natural Sciences

and Engineering Research Council of Canada and of La Fondation Desjardins.

To Isabelle.

ii

Claims of Originality

The author of this thesis claims the originality of:

(1) The nonlinear dynamic model of the asi high performance hydraulic actuator.

(2) The extension of an autonomous kinematic calibration method developed in

[Bennett and Hollerbach, 1991] for serial manipulators to a redundant, parallel joint

mechanism.

(3) The optimization of the joint's actuator forces as a minimum-norm problem for-

mulated in a dual Banach space.

(4) The theoretical and experimental comparison of two robust position control schemes

for the redundant, parallel joint mechanism.

(5) The development of a variable-bandwidth, nonlinear position controller for the

redundant, parallel joint mechanism.

(6) The experimental results from various position, impedance and force control ex-

periments performed on the hydraulic actuators and on the redundant, parallel rev-

olute joint.

iii

Contents

1 Introduction 1

2 Modeling of a Hydraulic Actuator for Robotics 8

2.1 A Comparison Between Electric and Hydraulic Actuators for Robotics

Applications : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8

2.1.1 Power to Mass and Torque or Force to Mass Ratios : : : : : : 9

2.1.2 Force or Torque Bandwidth : : : : : : : : : : : : : : : : : : : 11

2.1.3 Linearity of the Force or Torque Characteristic : : : : : : : : : 13

2.1.4 Additional Comments : : : : : : : : : : : : : : : : : : : : : : 14

2.2 A Brief Discussion on Hydraulic Valves : : : : : : : : : : : : : : : : : 16

2.3 Modeling of the ASI High Performance Hydraulic Actuator : : : : : : 18

2.3.1 Actuator Overall Properties : : : : : : : : : : : : : : : : : : : 18

2.3.2 Physical Modeling : : : : : : : : : : : : : : : : : : : : : : : : 22

2.3.3 Experimentation : : : : : : : : : : : : : : : : : : : : : : : : : 30

2.3.4 Simulation Results : : : : : : : : : : : : : : : : : : : : : : : : 39

2.3.5 Discussion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 44

3 Modeling of a Redundant, Parallel Revolute Joint 45

3.1 Redundancy and Antagonism : : : : : : : : : : : : : : : : : : : : : : 45

3.2 Kinematics of the Parallel, Redundant, Revolute Joint : : : : : : : : 49

iv

3.2.1 Inverse Kinematics : : : : : : : : : : : : : : : : : : : : : : : : 50

3.2.2 Direct Kinematics : : : : : : : : : : : : : : : : : : : : : : : : : 50

3.2.3 Velocity Mapping: The Jacobian Matrix : : : : : : : : : : : : 50

3.3 Autonomous Kinematic Calibration of the Revolute Joint : : : : : : : 52

3.3.1 The Levenberg-Marquardt Algorithm : : : : : : : : : : : : : : 55

3.3.2 Experimental Results : : : : : : : : : : : : : : : : : : : : : : : 56

4 Optimization of the Joint's Actuator Forces 59

4.1 Actuator Forces to Joint Torque Mapping: The Transposed Jacobian 59

4.2 Optimization of Actuator Forces Seen as a Minimum-Norm Problem : 60

4.2.1 Minimum 2-norm Optimal Vector of Forces : : : : : : : : : : : 61

4.2.2 Minimum1-norm Optimal Vector of Forces : : : : : : : : : 61

4.2.3 Addition of a Pre-Load Force on the Joint : : : : : : : : : : : 67

5 Control of the Redundant Parallel Joint 68

5.1 Dynamics of the Revolute Joint : : : : : : : : : : : : : : : : : : : : : 69

5.2 Position Control : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 69

5.2.1 A Simple PD Controller : : : : : : : : : : : : : : : : : : : : : 69

5.2.2 A Robust State Feedback Controller Based on the Internal

Model Principle : : : : : : : : : : : : : : : : : : : : : : : : : 73

5.2.3 An H1-Optimal Robust Controller : : : : : : : : : : : : : : : 80

5.3 A Variable Bandwidth, Nonlinear Controller : : : : : : : : : : : : : : 89

5.3.1 Implementation : : : : : : : : : : : : : : : : : : : : : : : : : : 92

5.4 Impedance Control of the Joint : : : : : : : : : : : : : : : : : : : : : 93

5.5 Experimental Results : : : : : : : : : : : : : : : : : : : : : : : : : : : 96

5.5.1 Position Control Experiments : : : : : : : : : : : : : : : : : : 97

5.5.2 Impedance and Force Control Experiments : : : : : : : : : : : 117

v

6 Conclusion 123

6.1 Suggestions for Further Research : : : : : : : : : : : : : : : : : : : : 126

6.1.1 Modeling of the ASI Hydraulic Actuator : : : : : : : : : : : : 126

6.1.2 Parallel, Redundant Revolute Joint : : : : : : : : : : : : : : : 127

A Partial Derivatives of FB 135

B Calibration Algorithm Implemented on MatlabTM 137

vi

List of Tables

2.1 Measured Force Sensor Parameters : : : : : : : : : : : : : : : : : : : 31

2.2 Friction Measurements : : : : : : : : : : : : : : : : : : : : : : : : : : 33

3.1 Calibrated Kinematic Parameters of the Parallel Revolute Joint : : : 58

vii

List of Figures

1.1 A Six-DOF Parallel Manipulator : : : : : : : : : : : : : : : : : : : : 3

1.2 A Six-DOF Serial Manipulator : : : : : : : : : : : : : : : : : : : : : : 3

2.1 Colocated Sensor and Actuator : : : : : : : : : : : : : : : : : : : : : 12

2.2 (a) Spool-Type and (b) Suspension-Type Valves : : : : : : : : : : : : 16

2.3 The asi Hydraulic Actuator : : : : : : : : : : : : : : : : : : : : : : : 20

2.4 Block Diagram of the Closed-Loop Model : : : : : : : : : : : : : : : : 23

2.5 Valve Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 25

2.6 Valve Static Force Characteristic : : : : : : : : : : : : : : : : : : : : 26

2.7 Actuator Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 30

2.8 (a) Open-Loop Static Force Characteristic, (b) Valve Hysteresis : : : 32

2.9 Experimental Hydraulic Damping Eect : : : : : : : : : : : : : : : : 34

2.10 (a) prbs Input, (b) System and arx Model Outputs : : : : : : : : : 36

2.11 Poles and Zeros of Identied Transfer Function : : : : : : : : : : : : : 37

2.12 Closed-Loop Frequency Responses: (a) Magnitude, (b) Phase (Kf =

2:44) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 38

2.13 Closed-Loop Force Responses: (a) f = 63 Hz, (b) f = 20 Hz : : : : : 39

2.14 Closed-Loop Force Response to a 120 N Square-Wave Input (Kf = 2:44) 40



2.17 Analog Circuit Implementing the Lead-Lag Controller : : : : : : : : : 43

viii

2.18 Closed-Loop Force Response to a 200 N Step Input, Lead-Lag Controller 43

3.1 Parallel Revolute Joint with Actuator Redundancy : : : : : : : : : : 46

3.2 Prototype of the Parallel Revolute Joint with Actuator Redundancy : 46

3.3 Geometry of the Parallel Revolute Joint : : : : : : : : : : : : : : : : 49

5.1 Block Diagram of the PD Controller : : : : : : : : : : : : : : : : : : 71

5.2 Bode Plot of the Cascaded PD Controller and Plant (Open-Loop) : : 72

5.3 Block Diagram of the Closed-Loop Robust Control System : : : : : : 72

5.4 Block Diagram of the Decoupling Part of the PID Robust Controller : 78

5.5 State Space Diagram of the PID Robust Control System : : : : : : : 79

5.6 Bode Plot of the Cascaded PID Controller and Plant (Open-Loop) : : 80

5.7 Nyquist Plot of the Cascaded PID Controller and Plant (Open-Loop) 81

5.8 Block Diagram of the H1-Optimal Control System : : : : : : : : : : 82

5.9 Magnitude of the Weighting Function W (s) on the j!-axis : : : : : : 87

5.10 Bode Plot of the Closed-Loop Transfer Function H(s) : : : : : : : : : 88

5.11 Bode Plot of the Open-Loop Transfer Function C(s)G(s) : : : : : : : 90

5.12 Nyquist Plot of the Open-Loop Transfer Function C(s)G(s) : : : : : 90

5.13 Sensitivity Magnitudes for the PD, PID and H1 Control Systems : : 91

5.14 Block Diagram of the PD Control System Seen as an Impedance Con-

trol System : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 94

5.15 PD control of the angle : I = I = 0:71, 2-norm optimal vector of forces 98

5.16 Actuator lengths 1 and 2: PD controller, I = I = 0:71, 2-norm

optimal vector of forces : : : : : : : : : : : : : : : : : : : : : : : : : : 99

5.17 Joint torque : PD controller, I = I = 0:71, 2-norm optimal vector of

forces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 99

5.18 Actuator forces: PD controller, I = I = 0:71, 2-norm optimal vector

of forces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 100

ix

5.19 PD control of the angle : I = I = 0:71, 1-norm optimal vector of

forces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 101

5.20 Joint torque : PD controller, I = I = 0:71, 1-norm optimal vector

of forces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 102

5.21 Actuator forces: PD controller, I = I = 0:71, 1-norm optimal vector

of forces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 102

5.22 PD control: I = I = 0:71, 50 N pre-load force : : : : : : : : : : : : : 103

5.23 Actuator forces: PD controller, I = I = 0:71, 50 N pre-load force : : : 104

5.24 Pre-load force: PD controller, I = I = 0:71, 50 N pre-load force : : : 104

5.25 PD control of the angle : I = 0:71, I = 0:36, 2-norm optimal vector

of forces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 105

5.26 Joint torque : PD controller, I = 0:71, I = 0:36, 2-norm optimal

vector of forces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 105

5.27 Actuator forces: PD controller, I = 0:71, I = 0:36, 2-norm optimal

vector of forces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 106

5.28 PID control of the angle : I = I = 0:71, 2-norm optimal vector of

forces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 107

5.29 Joint torque : PID controller, I = I = 0:71, 2-norm optimal vector

of forces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 107

5.30 Actuator forces: PID controller, I = I = 0:71, 2-norm optimal vector

of forces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 108

5.31 H1 control of the angle : I = I = 0:71, 2-norm optimal vector of forces109

5.32 Joint torque : H1-optimal controller, I = I = 0:71, 2-norm optimal

vector of forces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 109

5.33 Actuator forces: H1-optimal controller, I = I = 0:71, 2-norm optimal

vector of forces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 110

x

5.34 PID control of the angle : I = I = 0:71, 2-norm opt. vector of forces,

no feedforward : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 110

5.35 H1 control of the angle : I = I = 0:71, 2-norm opt. vector of forces,

no feedforward : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 111

5.36 PID control of the angle : I = 0:2, I = 0:71, 2-norm opt. vector of

forces, no feedforward : : : : : : : : : : : : : : : : : : : : : : : : : : : 112

5.37 PID control of the angle : I = 0:71, I = 0:36, 2-norm opt. vector of

forces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 112

5.38 H1 control of the angle : I = 0:71, I = 0:36, 2-norm opt. vector of

forces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 113

5.39 Nonlinear Controller + PID: I = I = 0:71, Fa = 200 N, 2-norm opt.

vector of forces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 114

5.40 Actuator forces: nonlinear controller + PID, I = I = 0:71, Fa = 200 N,

2-norm optimal vector of forces : : : : : : : : : : : : : : : : : : : : : 114

5.41 Actuator forces: PID controller only, I = I = 0:71, 2-norm optimal

vector of forces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 115

5.42 Nonlinear Controller + H1: I = I = 0:71, Fa = 200 N, 2-norm opt.

vector of forces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 116

5.43 Actuator forces: nonlinear controller + H1, I = I = 0:71, Fa = 200 N,


5.44 Actuator forces: H1-optimal controller only, I = I = 0:71, 2-norm


5.45 Contact experiment, PD controller: I = I = 0:71, 2-norm optimal

vector of forces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 118

5.46 Actuator forces: contact experiment with PD controller, I = I = 0:71,


xi

5.47 Contact experiment, H1-optimal controller: I = I = 0:71, 2-norm


5.48 Actuator forces: contact experiment with H1-optimal controller, I =

I = 0:71, 2-norm optimal vector of forces : : : : : : : : : : : : : : : : 120

5.49 Regulation of force applied on the environment with H1-optimal con-

troller, 2-norm optimal vector of forces : : : : : : : : : : : : : : : : : 121

5.50 Regulation of force applied on the environment with H1-optimal and

nonlinear controller, 2-norm optimal vector of forces : : : : : : : : : : 122

xii

Chapter 1

Introduction

For the last two decades, robotics research has increasingly focused on sensor-based

robotics as the tasks performed by robots have become more complex and delicate.

The robot controllers have reached a high degree of complexity as they are required

to process vast amounts of sensory data for feedback of position, velocity, force, visual

information, etc.

Such controllers were developed not only because of task complexity, but also in

an attempt to overcome many problems that stemmed from current industrial robot

designs inadequate for many sensor-based applications, such as contact tasks. Many

interesting control algorithms which compute joint torque or force commands, of-

ten taking the full nonlinear robot's dynamic model into account, were developed

(see, for example [Luh, Walker and Paul, 1980], [Khatib, 1987] and [Slotine, 1985]),

but few of them have yet been tested and compared experimentally. This is because

many robotics research laboratories lack a suitable robot manipulator system with

good joint torque control and sensing capabilities [An, Atkeson, Hollerbach, 1988].

Paradoxically, even though some control strategies were developed to address exper-

imental problems observed on industrial manipulators, they can not be implemented

on these robots because they lack good joint torque control, accurate sensors, high

1

CHAPTER 1. INTRODUCTION 2

bandwidth data transmission channels, and exible electronic architectures. But this

just re ects the fact that time has come to focus on technology in robotics in order to

keep up with the theory that promises much better performance. As a result, many

researchers are now concentrating on more basic issues such as actuator, sensor, and

manipulator designs. This research will most likely result in a new breed of high per-

formance robots over the next decade which could nd new applications in industry

and in society. Some interesting new robot manipulators are already available, such as

the Sarcos Dextrous Arm [Jacobsen et al., 1991] which is a hydraulic, seven-degree-of-

freedom (7-DOF), redundant, serial manipulator including full torque sensing at the

joints. In [Hayward, 1991], the author proposes a promising anthropomorphic 7-DOF

manipulator consisting of two spherical joints for the shoulder and the wrist, and a

revolute joint for the elbow. This design also features actuator redundancy in the

three joints, i.e. four linear actuators are used to control three DOFs in orientation

for the spherical joints, while the elbow joint is driven by two linear actuators.

Parallel manipulators (see Figure 1.1) possess properties complementary to those

of their serial counterparts (see Figure 1.2) [Hunt, 1983]. They can handle heavy

loads and their dynamics are dominated by the dynamics of the actuators and the

load. From this standpoint, it seems easier to improve their dynamics by using

feedback control since they suer less from dynamic coupling eects such as those

produced by the limbs of serial robots, or from friction, backlash, or joint compliance

imputed to transmission devices such as gears or harmonic drives. However, the

forces applied by the linear actuators act on the same rigid endplate and hence they

must be coordinated to prevent large internal forces from damaging the manipulator.

Moreover, since the load's inertial parameters aect directly the overall manipulator

dynamics, they must be estimated or identied. These considerations somewhat

complicate controller design.

Parallel manipulators suer unfortunately from a small workspace which limits


Endplate

Linear Actuators

Moving

Fixed Base

Figure 1.1: A Six-DOF Parallel Manipulator

Figure 1.2: A Six-DOF Serial Manipulator


their use. Nevertheless, in some cases, it is possible to increase signicantly their

workspace by adding redundant actuators, as proposed in [Hayward, 1988, 1991] for

a parallel spherical joint. In fact, actuator redundancy in parallel manipulators of-

fers many other advantages such as: elimination of Hunt-type singularities, improve-

ment of dexterity, possibility of controlling internal forces, smoother maximumoutput

torque and force characteristics over the workspace, etc. [Hayward, 1988]. Further-

more, colocated sensors and actuators add the advantage of redundant sensing which

allows better-conditioned measurements of endpoint position and orientation, and

also the possibility of implementing an autonomous kinematic calibration scheme as

discussed in Chapter 3.

This thesis is divided into two parts. The rst part is devoted to the modeling and

control of a high performance hydraulic actuator for robotics while the second part

deals with the modeling and control of an in-parallel actuated, redundant, revolute

joint mechanism which is actuated by the hydraulic actuators analyzed in the rst

part. This division follows what is believed by the author to be the right sequence

of steps in analysis and design for model-based control of a robot, in the spirit of the

current trend in robotics research discussed earlier.

It is argued that in most cases, the rigid-body dynamics of the robot, if they

are known to a good accuracy, do not represent the main restriction on closed-loop

performance. Indeed, any bulky, conventional robot equipped with perfect sensors and

actuators at the joints could reach almost any desired performance level, provided

the kinematic and dynamic models would be faithful. Thus, actuator and sensor

technology is the main limitation, and hence a detailed characterization and modeling

of the actuators that are to be used to actuate a robot should be performed rst and

foremost. This rst step should then be followed by the design of good force or torque

controllers for these actuators, since most advanced robot control laws calculate force

or torque commands to be applied at the joints. It seems worth putting some eort


on choosing the right actuator and on its modeling and control until acceptable force

bandwidth and accuracy are attained. Then, advanced position or impedance control

laws can be tested on the robot with the assumption that the actuators can be modeled

as pure force or torque sources, provided the estimates of the kinematic parameters

are reasonably good.

Chapter 2 is devoted to the characterization, modeling, and identication of a high

performance, piston-type hydraulic actuator. It also includes a brief comparison be-

tween electric and hydraulic actuators for high performance robotics. In Chapter 3, a

very simple, one-degree-of-freedom, redundant, parallel robot is analyzed. It consists

of two linear actuators linked to a revolute joint through a lever arm of xed length.

Its antagonistic conguration is shown to yield a large workspace free of singularities,

well-conditioned angle and torque sensing, a smooth maximum torque characteristic,

and the possibility of controlling internal forces. The kinematics of this parallel joint

are derived and an autonomous kinematic calibration scheme based on a new method

presented in [Bennett and Hollerbach, 1991] is developed and experimentally tested

on a prototype of the mechanism actuated by the ASI hydraulic actuators discussed

in Chapter 2.

Chapter 4 addresses the problem of optimization of actuator forces for the parallel

joint mechanism. This problem bears some resemblance with the problem of optimiz-

ing forces when two or more manipulators share a common load [Nahon, 1990], and

also with the problem of grasping an object with a robotic hand [Kerr and Roth, 1986].

Two solution vectors corresponding to two dierent objectives are calculated in the

form of minimum-norm vectors of actuator forces. The rst optimal solution mini-

mizes the internal force in the joint while the second solution minimizes the maximum

magnitude of the actuator forces.

Finally, Chapter 5 explores position and impedance control of the redundantly-

actuated parallel joint. In robotics, it is always necessary for a position or impedance


controller to perform well under large variations in the inertial parameters as the

manipulator moves, picks up an unknown load, makes contact with its environment,

etc. To deal with this uncertainty in the plant's parameters, two classes of controllers

can be used. The rst class includes those controllers which adapt themselves to

these changes by identifying them in real time. They are referred to as adaptive

controllers (see [Astrom, 1983] for a survey of adaptive control theory and its appli-

cations). The second class includes those controllers which are xed but guarantee

performance measure bounds for given uncertainty bounds. They are referred to as

robust controllers. Robust control of uncertain systems is currently a very active

area of research of control theory and it has been approached in a variety of ways,

leading to important new theories such as the H1 control theory [Zames, 1981],

[Zames and Francis, 1983], [Francis, Helton and Zames, 1984], and the robust ser-

vomechanismtheory [Davison and Ferguson, 1981], [Francis and Wonham, 1976]. The

approaches dier in the plant models, the uncertainty models, and the performance

measures used.

The uncertain dynamics of robot manipulators lend themselves naturally to ro-

bust controller designs. One particularly interesting approach is to linearize ap-

proximately the robot's dynamics by feedback and then to use a linear robust con-

troller [Slotine, 1985], [Spong and Vidyasagar, 1987], [Spong and Vidyasagar, 1989],

[Kuo and Wang, 1991]. In Chapter 5, two robust linear position controllers are

designed for the parallel joint, one being based on the Internal Model Principle,

[Francis and Wonham, 1976] and the other being based on the siso H1 sensitivity

minimization method [Zames and Francis, 1983]. It is shown that the H1 controller

is more robust to variations in joint's inertia. A simple PD controller is also proposed.

Dierent position, impedance, and force control experiments were conducted and re-

sults are presented and compared. A nonlinear, variable-bandwidth, controller which

is designed based on a property of the actuator's hydraulic damping characteristic is


also tested experimentally in Chapter 5.

Chapter 2

Modeling of a Hydraulic Actuator

for Robotics

2.1 A Comparison Between Electric and Hydraulic

Actuators for Robotics Applications

Hydraulic actuation used to be, and in many cases remains, the technique of choice for

high performance robotics applications. However, this type of actuation is presently

not receiving a great deal of attention from the robotics research community despite

its often ignored advantages. This may be due, in part, to unjustied prejudice against

hydraulic systems on the part of robot designers in the research community. Hydraulic

actuation is often believed to be dirty, noisy, inaccurate, inadequate for force control,

complicated to use, dangerous, expensive, and hard to package. These descriptions

do indeed apply to certain, general purpose, hydraulic actuators. However, hydraulic

actuators specically designed for robotics and other demanding applications, such

as the one discussed in this chapter, overcome many of these alleged shortcomings

and oer a unique set of performance characteristics.

8

CHAPTER 2. MODELING OF A HYDRAULIC ACTUATOR FOR ROBOTICS 9

In this section, we will establish a brief comparison between electricDC motors and

hydraulic actuators in terms of their merits for high performance robotics applications.

An extensive comparative analysis of actuator technologies for robotics can be found

in [Hollerbach, Hunter and Ballantyne, 1991]. Some of the most desirable features of

actuators for use in serial or parallel manipulators may be argued to be:

(i) High power to mass ratio

(ii) High force to mass or torque to mass ratio

(iii) High force or torque bandwidth

(iv) Linear input-output force or torque characteristic

(v) Low volume

(vi) Low heat loss

This list is, of course, not exhaustive. For example, ease of maintenance, reliability

and robustness are other peripheric requirements.

2.1.1 Power to Mass and Torque or Force to Mass Ratios

A high power to mass ratio is obviously a general requirement in robotics. A high

torque to mass ratio is particularly important in serial-type manipulators where the

proximal joints have to compensate high torques or forces due to gravity and inertial

forces acting on the outer links and on the load. In that case, unless the designer

is willing to use high transmission ratios between the actuators and the joints, the

motors actuating the proximal joints will have to be more powerful, and hence heavier,

which leads to a pyramidal design with poor eciency (see [Hayward, 1991]). This

eect gets even worse as the actuators' torque to mass ratios become lower, because

then the motors used at each joint will increase the weight of the limbs. One way

to get around this problem would be to locate the actuators driving the distal joints

closer to the base of the robot by using some kind of transmission mechanism, but


this is always at the cost of an increase in complexity and with a possible loss of

bandwidth and accuracy.

Electric DC motors cannot be made to have a sucient torque to mass ratio

to be integrated in a truly useful 6-DOF direct-drive arm because of the magnetic

nature of the driving force itself: unless the armature windings are made of su-

perconducting material, one just cannot apply enough current to these motors to

generate the torque required for the manipulator to support itself against gravity

without overheating them. Power dissipation is indeed a serious problem in direct-

drive DC and AC motors; a new design being currently developed at McGill in-

cludes an integrated water-cooling system to address this problem, as described in

[Hollerbach, Hunter and Ballantyne, 1991]. According to these authors, the torque

to mass ratio of electric DC motors is approximately limited to 6 Nm/kg, while the

power to mass ratio can be made reasonably high (e.g. 100 W/kg) with an adequate

power amplier. However, the peak mechanical power from which the power to mass

ratio is derived is obtained at high angular velocities and thus gears must be used

to scale down the angular velocity in a range suitable for robot joints. On the other

hand, linear or rotary hydraulic actuators oer much higher torque to mass or force

to mass ratios|at least one order of magnitude higher than what can be expected

from good DC motors| that allow for compact actuator designs. The force or torque

delivered depends essentially on the supply pressure which thus provides a design pa-

rameter to tradeo maximum output thrust with precision of the machining and seal

design to prevent leakage. Power to mass ratios of hydraulic actuators are generally

comparable to, or higher than, the ratios obtained with DC motors. Overheating of

hydraulic actuators or valves is seldom a problem because the circulating uid carries

the heat away to a heat-sink. Thus, ecient cooling of the important parts is intrinsic

to the system.


2.1.2 Force or Torque Bandwidth

Force bandwidth is also important for at least two reasons. Firstly, most control

algorithms used in robotics generate control torques or forces to be applied at the

joints. This implies that the actuators have to be able to track the desired forces or

torques as accurately as possible in the frequency bands of interest. Slowly-responding

actuators will invalidate most of the controller design made under this assumption

and might also cause the entire closed-loop system to display unpredictable behavior.

Secondly, as the robot makes contact with the environment, it is desirable to adjust

its mechanical impedance for a smooth, stable interaction. Modulation of the end-

eector's mechanical impedance will be possible only if actuators with good force-

tracking capabilities are used (see [An, Atkeson, Hollerbach, 1988]).

DC motors have, in principle, a virtually innite open-loop torque bandwidth since

the magnetic force acts almost instantly on the rotor when the armature current is ap-

plied. But if a closed-loop torque control must be implemented to reduce the eect of

disturbances such as friction, then the loop bandwidth will be constrained by the dy-

namics acting between the motor shaft and the sensor [Eppinger and Seering, 1987],

and also by the sensor stiness itself. The reason behind this appears to be the laws

of nature which are the most conveniently exploited for force or torque transduction.

In particular, forces or torques are related to displacement, velocity, and acceleration

signals, which can then be measured by many dierent means. Hence, a closed-loop

force or torque control system is generally nothing but a high gain position, velocity

or acceleration control system which strives to track very small sensor movements.

From that point of view, it is easy to explain most instability problems that have

been observed by investigators in many robot force control experiments: the corre-

sponding servos become unstable because of the high equivalent feedback gains (see

[Eppinger and Seering, 1987], [An, Atkeson, Hollerbach, 1988]).

The closed-loop bandwidth of a DC motor-based joint torque control system is


Ks

bs

Sensor

DC Motor

Figure 2.1: Colocated Sensor and Actuator

limited mostly by the system's inertia, the torque sensor dynamics, and also possibly

by the friction, backlash and compliance of the gearbox when torque amplication

is needed [Vischer and Khatib, 1990]. It should be noted that, according to linear

theory, the closed-loop bandwidth of a system with colocated DC motor and torque

sensor as shown at Figure 2.1 could be extended as far as required since the open-

loop transfer function from current to sensed torque is only of second order. A PD

controller could achieve any second-order closed-loop transfer function, regardless of

the sensor stiness. So the constraints have to come mostly from practical limits

on the armature current usually set by the heat transfer capacity of the motor. If

a good current amplier is used, the torque should follow the supplied current very

closely over a large frequency band. Thus, in order to increase the closed-loop torque

bandwidth of a system with a large inertia, a lot of gain must be provided at high

frequency by the controller and therefore current saturation may occur. Certain

current ampliers also exhibit low slew rates which also reduce performance. These

nonlinear eects often limit the eective attainable bandwidth in most systems.


The open-loop force bandwidth of hydraulic actuators is not almost innite as for

DC motors; it generally depends on the valve and uid dynamics. Moreover, under

feedback control with a colocated sensor, the force bandwidth is not really limited

by the system's inertia. That can be explained by the fact that the available output

force is dominant when compared to inertial eects, i.e. saturation in valve position

(and hence in force) occurs only for very large discrepancies between the desired and

the sensed forces. Consequently, careful design of the valve and the hydraulic lines

can lead to a fast-responding force-controlled actuator as will be shown in the next

chapter on modeling of a hydraulic actuator commercialized by Animate Systems

Incorporated (asi), Salt Lake City, Utah, USA.

2.1.3 Linearity of the Force or Torque Characteristic

Linearity of the force characteristic is important when the actuator is used in a higher-

level control system: it simplies analysis and controller design. DC motors are based

on a physical principle which is essentially linear, i.e. the interaction between a mag-

netic eld and an electric current, but several nonlinearities can result in actual

designs. These nonlinearities can be attributed to: magnetic material causing sat-

uration and hysteresis, geometry of the windings or the parts producing the eld,

power electronics, and so on. On the other hand, hydraulic actuators are intrinsi-

cally nonlinear devices. Thus, a good, robust force control law should be used for

hydraulic actuators to linearize their force characteristic and to prevent any unstable

behavior (such as limit cycles) from occurring. Fortunately, the high saturation levels

of hydraulic force allow the use of a large feedback gain in some high performance

hydraulic actuators such as the asi actuator. Hence, reasonable performance appears

to be attainable with a simple linear lead-lag controller, provided a thorough stability

analysis can be conducted by simulation, and provided a good model of the actuator

including all the nonlinearities is available.


2.1.4 Additional Comments

The advantages of hydraulic actuation are obtained at the price of an increase in power

equipment complexity because a pump, hydraulic lines and possibly an accumulator

must be purchased. Also, oil leakage is almost unavoidable in any hydraulic system,

but it can be reduced to an acceptable level by proper design. Finally, even though

large accelerations can be reached, the maximum shaft or piston velocity quickly

saturates for a given valve opening . This saturation corresponds to the maximum ow

of oil through the valve orices, and higher velocities can be achieved by increasing

the supply pressure. Direct-drive electric motors do not have that speed limitation

and therefore are able to track much faster trajectories than can hydraulic actuators.

Hence, a trajectory generator providing setpoints to a controlled hydraulic robot

should check whether the actuators' velocity limits will be attained and reject any

trajectory that does not respect these constraints. Otherwise, large tracking errors

will result.

In this section, we have seen that hydraulic actuators possess many of the desirable

features an actuator should have for high performance robotics applications. For

example, the asi linear hydraulic actuator analyzed in the next chapter oers a force

to mass ratio of approximately 1.8 kN/kg and a closed-loop force bandwidth of 100 Hz.

Also, the parallel revolute joint discussed in Chapters 3, 4 and 5 which is actuated by

two of these actuators features a torque to mass ratio of about 20 Nm/kg in the middle

of its workspace for a supply pressure of 345 N/cm2, but this ratio could be raised as

high as 40 Nm/kg if the joint and lever arm would be made of lightweight material.

Furthermore, hydraulic piston-type actuators remain the only practical choice for

actuation of parallel manipulators with fair or high payload handling capabilities.

Another interesting feature of hydraulic motors is their great mechanical stiness

when the valve is closed. This is due to the very low compressibility of oil. This high

stiness oers two advantages: rstly, far less feedback gain is required to hold the


load in a xed position; secondly, a high mechanical resonant frequency results from

the load acting on the equivalent sti spring [Blackburn, Reethof and Shearer, 1960].

Dierent valve designs are brie y described in the next section.


To cylinder

Supply

ReturnReturn

(a)

To cylinder

Supply

Return Return

xv

xv

(b)

Spool Valve Suspension Valve

Figure 2.2: (a) Spool-Type and (b) Suspension-Type Valves

2.2 A Brief Discussion on Hydraulic Valves

Hydraulic valves exist in many dierent varieties, but it is possible to conveniently

classify them by their operating member which acts to control the oil ow. That way,

at least four large classes can be dened: the spool-type, the suspension-type, the

poppet-type and the apper-nozzle-type valves. Moreover, these single-stage valves

can be combined to form two-stage devices, where one valve uses hydraulic power

to position the main one. Figure 2.2 shows the two types of single-stage devices

considered here: the spool valve and the suspension valve.

The spool valve is simple to fabricate and actuate, and that makes it very popular

for common applications. However, its dynamics are dominated by the spool's inertia

and the friction between the spool and the valve body, both of these factors severely


limiting the valve's bandwidth and accuracy. Hence it would be of limited use in high

performance robotics applications.

Suspension valves, on the other hand, do not suer from friction eects because

the operating member is not in contact with the receptor part. Also, the inertia of

the moving part can be made very small as in jet-pipe valves which belong to the

suspension valve family. Even though jet-pipe valves seem to have the requirements

for demanding applications, they also are very dicult to analyze because of the

intricate ow phenomena occuring around the stem that create disturbing forces on

it. Nevertheless, careful design based on empirical knowledge can lead to a very

useful and reliable valve, as the one mounted on the ASI actuator analyzed in the

next section (see [McLain et al., 1989] and [Blackburn, Reethof and Shearer, 1960]).


2.3 Modeling of the ASI High Performance Hy-

draulic Actuator

As the objectives in advanced manipulator research become increasingly demand-

ing, the interaction among various components of the system, and the impact of this

interaction on overall manipulator performance, becomes progressively more impor-

tant. This necessitates an integrated approach to manipulator design: encompassing

the kinematic, structural, actuation, sensing, and control aspects of the manipulator

within a unied design process. Hence, detailed knowledge of actuator properties, and

the nature of the limits on actuator performance, are a prerequisite for the integrated

design of advanced manipulators. Actuator characteristics are of special relevance to

control law design.

This chapter focuses on the modeling and system identication of one particular

high performance hydraulic actuator built by asi. A physical model is derived for this

actuator, and the parameters of the various components of this model are identied

experimentally. The overall force loop performance of the actuator is investigated

with a simple linear proportional controller, and it is compared to the predictions of

a software simulator which implements the physical model. A linear lead-lag controller

is also designed to achieve better force accuracy.

2.3.1 Actuator Overall Properties

Modern quick release exible supply lines make connecting and disconnecting a hy-

draulic unit almost as easy as connecting or disconnecting an electrical component.

With proper design, leakage has been reduced to a minimum and can be easily con-

trolled. We used an acoustically isolated conventional hydraulic supply which is not

noisier than say, a ventilated backplane chassis, and not more expensive than a bank


of good quality dc motor ampliers. The actuator itself is completely noise-free even

at maximumthrust, that is approximately 900 N for 345 N/cm2 (500 psi) supply. The

turbulent ow is conned inside a solid metal manifold from which no audible (at least

in our lab) acoustical noise can escape. This contrasts with some electro-mechanical

equipment driven by switching power supplies. Also, the produced mechanical signal

(force or velocity) is almost perfectly free of noise. This is typied by the sensation

of smoothness when the controlled hydraulic actuator is made to interact with the

experimenter's hand.

The device discussed here is a linear piston-type actuator driven by an integrated

high-bandwidth jet pipe suspension valve, and tted with a force sensor. It is very

compact, mechanically robust, and its mass is about 0.5 kg . A view of the actuator

without the lvdt position sensor is shown in Figure 2.3. For a 73 mm stroke, the

overall dimensions are 25 X 55 X 139 mm. Since it is a force-controlled device,

it must include some elasticity which is almost entirely lumped in the force sensor

mounted directly on the cylinder. It thus may be considered as an active instrumented

structural member easily integrated in a larger assembly.

The standard servo system available for the actuator includes a controller card

which can be accessed by a host computer. The card features on-board analog linear

controllers whose gains can be programmed from a host computer, allowing gain

scheduling. Digital control is also possible since the valve current can be specied as

desired. The system state variables can be accessed either digitally via an on-board

analog to digital converter, or directly by measuring the analog signals.

These actuators must be essentially seen as force producers due to the four-way

jet pipe design of its single-stage electromagnetic valve. The force output primarily

results from the dierential pressure across the lines leading to the chambers on

each side of the piston. The pressure imbalance due to the suspension deviation is

the fundamental operational mechanism. Because the valve is piggy-backed on the


Manifold

Valve

Load Cell

Cylinder

Figure 2.3: The asi Hydraulic Actuator

piston, a very direct connection between suspension deviation and force output is

established. In addition, no solid friction force intervenes in the valve's operation as

the valve's tip does not contact the receiving plate where the valve orices are located.

Force control resolution is limited by the residual solid friction forces as seen at

the piston rod in closed-loop operation. Thus, resolution depends on the ability of the

internal driving force to overcome these forces, and by the resolution of the sensor

itself. The closed-loop force feedback gain can be fairly high, hence the eects of

residual friction can be made quite small. Consequently, sensor stiness determines

the basic tradeo between force control bandwidth and resolution. All other things

being equal, a more compliant sensor will cause a lower resonant frequency, hence

a lower force control bandwidth, but a higher force resolution, since larger elastic

displacements are easier to measure accurately.

Among the several major nonlinear characteristics of this actuator, hydraulic


damping has a notable eect on performance. Hydraulic damping is a force which

opposes the piston motions due to the circulation of oil through the valve orices.

For a xed valve current that species a certain valve position, the eect is very

small at low velocities, which make it dicult to assess, but increases faster than

linearly for a certain velocity range, past which the characteristic curve tapers o.

We conjecture that this eect is attributed to ow forces which become signicant

enough to force the opening of the valve. This phenomenon happens only when an

external force applied on the piston adds up to the uid pressure to produce higher

velocities (and thus higher ow rates) than usually obtained. Thus, the force response

bandwidth, kept at a maximum for small amplitude motions such as constrained or

contact motions, is drastically decreased for fast motions of an inertial load in free

space because the resultant velocities are in the range where the damping is expo-

nential, enhancing stability. Hence the actuator has the intrinsic property to adapt

its natural impedance characteristic to the type of tasks required in robotics. At the

limit, when the fully opened valve forces maximum ow in and out the chambers,

velocity saturates and is maintained constant for large variations of the disturbing

load forces, as the thrust force would augment rapidly should the velocity drop. At

the other end of the spectrum, when the velocity is small, the suspension deviation

has a direct impact on the force output, resulting in high bandwidth force control.

High reliability is facilitated by a very small number of parts of which only two are

moving parts: the bending jet pipe and the piston, not counting the lvdt position

sensor. Solid friction only occurs between the piston, the rod and the cylinder in

the entire assembly. The force sensor has inherent mechanical overload protection

which enhances further reliability. Furthermore, elastic displacements are sensed by

a noncontact Hall-eect transducer. Finally the actuator can reach its mechanical

travel limit at full valve opening without incurring any damage as the oil, forced

out of the vanishing chamber volume, smoothly damps the motion to a stop. In


these conditions, no external mechanical stops are required since they are built-in the

actuator and can be adjusted to any requirement.

In summary, this actuator may be characterized as a direct-drive device since

the power derived from the input uid pressure is almost directly applied to the

load without any need for a motion transmission mechanism, with the valve acting

as a variable gain amplifying element. It can thus be conceptually compared to an

operational amplier producing the best of its performance when linearized by high

feedback gains.

In the coming sections, we shall dwell in some detail into the modeling of this

device with a view to its use for force control.

2.3.2 Physical Modeling

A \gray-box" model approach was adopted since a number of the system parameters

were not known and in most cases were unavailable information. Some \reverse

engineering" was performed to develop an understanding of how the system elements

were designed. The model includes linear dynamics in conjunction with nonlinear

elements. These are hysteresis, static valve force characteristic, hydraulic damping

and friction. These nonlinearities play an important role in the actual system and

must be included if the model's predictions are to be a good approximation of the

actuator's behavior. A block diagram of the closed-loop model is presented in Figure

2.4. The linear blocks represent the valve, uid, and force sensor dynamics, which

are respectively denoted as G(z), D(z) and S(s). Zero-order holds are used at the

outputs of the discrete-time blocks but are not shown in the gure.

The supply pressure (345 N/cm2) was the only available a priori information before

asi kindly agreed to provide us with proprietary information regarding the geometry

of the valve. This information was needed to calculate the valve force versus the valve

pipe tip position static characteristic ~F (xv) i.e. the static hydraulic force applied on


K

K

a

a

KfKs

+

−

−+

−

i x

x F F

F

F

v

v

x FF ~

~v v

v

d

f

p

s

p

sG(z) D(z) S(s)

F (v ,x )d p v

s

s

K a = .0013

des

Valve Hysteresis

Valve Dyn. Fluid Dyn.Static Force Char.

Friction

Hydraulic Damping

Sensor Dyn. SensorStiffness

Figure 2.4: Block Diagram of the Closed-Loop Model

the piston when it is constrained to a null velocity. All the other system parameters

were unknown and had to be measured or identied.

The unknown, but measurable, model parameters were the sensor calibration, the

force sensor dynamics and stiness, the valve hysteresis characteristic, the friction

characteristic and the hydraulic damping eect. The unknown, but identiable, model

parameters were the valve and uid dynamics.

Valve Static Force Characteristic

A mathematical model of the valve static force characteristic ~F (xv) was worked out.

Four assumptions were made. Firstly, the ow through the valve orices was assumed

to vary with the square root of the pressure dierence across the orices. If q denotes

the ow through an orice, P is the pressure drop across the orice, a is the orice

area, Cd is the discharge coecient (Cd < 1) and is the uid density, the relationship


can be written as

q = aCd

s2

pP : (2:1)

Orice discharge coecients could not be measured and were estimated, based

on values given in [Blackburn, Reethof and Shearer, 1960, pp. 181183]. The second

assumption was that direct leakage from the valve pipe tip to the return chamber is

negligible. This is justied considering that even if there is some leakage, its eect

should be mostly independent of xv and should roughly be equivalent to a drop in

pressure at the end of the supply line, thereby aecting only the saturation force

values but not the general shape of the function. Hence we dene ~Ps to be the

stall force measured at full valve opening divided by the eective piston area and

use it as a new \supply" pressure in the equations. The valve pipe tip, the uid

lines, the cylinder and the piston of a general asymmetric hydraulic actuator with a

suspension-type valve are shown in Figure 2.5. Referring to this gure, the third and

fourth assumptions made were the following: the return chamber pressure is null, i.e.

Pr = 0; the supply pressure Ps, and hence ~Ps, remain constant. These assumptions

usually hold in many hydraulic systems. The expression needed to calculate the

steady-state force with respect to the valve pipe tip position is now derived.

The ows q1 and q2 are calculated using equation 2.1:

q1 = Cd

s2

a2qP1 a1

q~Ps P1

; (2:2)

q2 = Cd

s2

a3

q~Ps P2 a4

qP2

: (2:3)

But we also have

q1 = A1vp; (2:4)

q2 = A2vp: (2:5)

We seek an expression for the static force ~F applied on the piston when it is

constrained to a xed position, i.e. vp = 0. Since the ows q1 and q2 must be null by


Supply

Return Return

xv

PP

q

q

AA1

1 2 2

2

1

a a aa 1 3 42

P Pr r

xp

Ps~

Figure 2.5: Valve Model


equations 2.4 and 2.5, We can solve equations 2.2 and 2.3 for P1 and P2. We get

P1 =a21

~Psa21 + a22

; (2:6)

P2 =a23

~Psa23 + a24

: (2:7)

The force ~F is then essentially the dierence between the forces applied on each

side of the piston. The orice areas a1, a2, a3 and a4 vary with the valve position xv

according to their geometry.

~F (xv) =a21A1

~Psa21 + a22

a23A2~Ps

a23 + a24: (2:8)

-800

-600

-400

-200

0

200

400

600

800

1000

-4 -3 -2 -1 0 1 2 3 4 5

x10-3

Valve Static Force Characteristic

For

ce (

N)

Valve Position xv (cm)

Figure 2.6: Valve Static Force Characteristic

The characteristic is shown in Figure 2.6. Each saturation force corresponds to

the area on each side of the piston multiplied by the pressure ~Ps. It can be observed

in Figure 2.6 that the output force has a positive bias when the valve pipe tip rests

at the center. This obviously results from the asymmetry in the eective areas on

each side of the piston. One diculty is to determine the valve's tip position when

it is at rest, but there is a way around this problem. An oset in valve current can


be set to approximately compensate any mechanical bias such that the output force

is about zero. If this new valve resting position is taken to be the origin, then the

model can be adjusted correspondingly by adding a negative oset to xv before the

function simulating the force characteristic is invoked.

Sensor Calibration and Dynamics

Calibration of the position sensor is performed by adjusting an oset and a gain and

by measuring the piston stroke. The force sensor is calibrated similarly but its stiness

has to be measured. The force sensor essentially consists of a U-shaped piece of steel

with no solid contact, thus a mass-spring-damper second-order dynamic model was

chosen. We assumed that the xture used for experimentation was perfectly rigid,

although actual results showed signicant bending.

Valve Hysteresis

Valve hysteresis is signicant and a model was built to account for it. The model

is based on a technique described in [Frame, Mohan and Liu, 1982]. It is capable

of generating minor loops from the knowledge of the major hysteresis loop. In the

model, the input to the hysteresis block is the valve current iv and the output is the

dc valve pipe tip position ~xv. Hysteresis output is usually chosen to be the valve's

motor torque but we could not measure it. Hence, although the relationship between

iv and xv would normally include the valve dynamics, we had to separate the dc

hysteresis characteristic from the dynamics which relate the static and actual valve

positions, Xv(z)= ~Xv(z) = G(z).

The valve hysteresis is included in the system static force characteristic which can

be measured easily. Friction is also included in the static force characteristic and a

constant Coulomb friction force was added to cancel approximately its eect in the

measurements. We then used the inverse of the calculated nonlinear valve static force


function to obtain the lower and higher parts of the ~xv(iv) hysteresis major loop from

the dc characteristic data.

Friction Model

The friction model includes kinetic friction only. Numerical oscillation problems were

avoided in the simulator by using a modied Dahl model (see [Threlfall, 1978]). The

expression of the time-derivative of the friction force is:

@Ff@t

= (Ff Fc sgn(vp))2vp; (2.9)

where = 100;

Fc = 7 N (Coulomb Friction).

The parameter in equation (2.9) is set to a suitable value for a fast transient in Ff

towards the Coulomb friction Fc or Fc when vp changes sign. It should be noted

that the use of this friction model which, in steady-state, is equivalent to a simple

Coulomb friction model, was only intended for improving the numerical integration

and not for modeling the actual Dahl eect.

Hydraulic Damping Eect

The hydraulic damping force depends on the valve pipe tip position and on the piston

velocity. The family of curves used to model this eect is based on experimental

data and thus it includes the ow forces acting on the valve pipe tip. Although

the valve position can not be measured, we used the knowledge of the desired input

currents and found the corresponding valve positions by applying these current values

to the hysteresis model. The ow forces on the valve pipe and the uncertainty in the

hysteresis model limit our ability to predict the valve position accurately.


Identication of Valve and Fluid Dynamics

The valve and uid dynamics had to be identied for parametrization of the linear

blocks in the model. All the linear dynamics were identied as a whole and several

assumptions were made in order to be able to select the right poles and zeros for each

transfer function.

It was assumed that the valve was the most restrictive limit to the open-loop band-

width and this was based on the gures used for a similar valve in [McLain et al., 1989].

A second-order model with two distinct real poles was expected to give good results

because of severe damping applied on the valve pipe tip by the uid in the return

chamber.

For the uid dynamics, the supply and return lines were assumed to be lumped-

parameter linear second-order systems. The parameters are the uid inertia, the

uid and line compliance and the orice resistance. The chambers on each side

of the piston were assumed to be lumped-parameter rst-order linear systems, the

parameters being the uid compliance and the orice resistance. The overall uid

dynamic model order is six.

Two poles should be related to the force sensor dynamics in the identied linear

transfer function which should be of the tenth order. These poles were expected to

be complex and located below the force sensor's natural frequency because of the

hydraulic damping eect, which is assumed to be small since the prbs input used for

identication had a low amplitude.

Actuator Model

A diagram of the physical actuator model is shown in Figure 2.7. The dynamic and

output equations relating the hydraulic force F to the sensed force Fs are:

F (xv) Fd(xv; vp) Ff (vp) = mxs + bs _xs + ksxs (2.10)


Fs = ksxs: (2.11)

where:

Fd(xv; vp) hydraulic damping force xs force sensor de ection

Ff(vp) friction force m actuator mass minus piston mass

vp piston velocity bs, ks force sensor parameters

m

Ff(vp)Fd(vp,xv)

Force Sensor

ks F(xv)

xs

bs

xp

Figure 2.7: Actuator Model

2.3.3 Experimentation

Measurement of Force Sensor Characteristics

As a rst experiment, we had to measure the force sensor characteristics. We di-

rectly measured the force sensor stiness by locking the piston to the mount and by

measuring the total sensor de ection as full output force was applied in both direc-

tions. Then, by using the known saturation force values, we were able to calculate


the sensor stiness. One disadvantage of this method is that full sensor de ection

probably covers a nonlinear domain of the sensed force Fs versus sensor position xs

relationship.

The force sensor impulse response was also measured by gently knocking the

actuator with a piece of metal while it was held vertically. The damping factor ,

natural frequency !n, damping coecient bs and sensor stiness ks were then looked

up in [Gille, Decaulne and Pelegrin, 1985] or calculated and are listed in Table 2.1. It

was noted that the impulse response gave much better results than the step response

because in the latter case, lateral modes were excited and masked the eect of the

desired axial mode. The sensor stiness value used in the model for simulation is the

one derived from the impulse response experiment. Equation (2.12) shows the force

sensor transfer function S(s) used in the model.

S(s) =178:6

s2 + 758:4s + 6712857(2:12)

actuator mass ma 0.612 kgactuator mass minus piston mass m 0.560 kgsensor stiness ks (direct) 43659 N/cm

(impulse) 37592 N/cm

natural frequency !n =qks=ma 2478 rad/s (394 Hz)

damping factor 0.14viscous damping coecient bs 4.25 N/cm/s

Table 2.1: Measured Force Sensor Parameters

Measurement of Open-Loop Static Force Characteristic

With the piston locked to the mount, the open-loop static force characteristic was

recorded while the valve current was slowly varied step by step following a triangular

input. The current driver sensitivity allowed .488 mA increments in valve current.


The static force characteristic and the calculated hysteresis major loop are shown in

Figure 2.8.

-800

-600

-400

-200

0

200

400

600

800

1000

-1 -0.5 0 0.5 1

Open-Loop DC Force Characteristic

Valve Current iv (A)

Outpu

t Forc

e Fs (

N)

(a)

-3

-2

-1

0

1

2

3

4x10-3

-1 -0.5 0 0.5 1

Valve Current iv (A)

Valve

Posit

ion xv

(cm)

Valve Hysteresis

(b)

Figure 2.8: (a) Open-Loop Static Force Characteristic, (b) Valve Hysteresis

Measurement of Friction

Kinetic friction and stiction were measured with the oil drained from the actuator

(some oil was left, providing lubrication). The main disadvantage of this method is

that friction is likely to change when the pressure across the piston varies as the seal

gets squeezed. In situ dierential pressure measurements would give more accurate

assessment of the phenomenon. Stiction was measured as the force at the breaking

point where the piston starts moving. Coulomb friction was measured by pulling

on the piston by hand and recording the force and the piston velocity. Results are

collected in Table 2.2.


stiction (pushing on piston) 27 N(pulling on piston) -15 N

Coulomb friction 7 N

Table 2.2: Friction Measurements

Measurement of Damping Eect

The hydraulic damping experiment was carried out with the actuator mounted ver-

tically such that weights could be hung to the piston (which was free to move). The

procedure was as follows: we used a certain valve current as input to the open-loop

system and measured the piston steady-state velocity without any load. The cor-

responding steady-state force applied by the uid pressure on the piston could not

be measured directly but was found later by locking the piston and measuring the

output force for the same input current. The sequence of applied currents was similar

to follow approximately the same hysteresis curve. Then, for the same input current,

dierent masses were hung to the piston and the corresponding steady-state velocities

were measured. For each of these masses, the total force applied on the piston could

be calculated as the sum of the measured hydraulic force, the gravitational force

acting on the mass, and the kinetic friction force opposing the movement. It was

assumed that the hydraulic reaction force was equal to that sum. This procedure,

which provided experimental data for one value of valve current, was repeated for

dierent valve currents in order to be able to t a family of curves to the data.

A family of hyperbolic tangents whose magnitudes, scalings and positions with

respect to the origin depend on the valve position xv has been tted to the exper-

imental data (see equation 2.13 below). A linear damping term was added. Cubic

splines were used for interpolation between the experimental values of A(xv), s(xv)

and d(xv). We also assumed that the curves were odd, i.e. for a negative piston veloc-

ity vp, the damping force would be Fd(jvpj). The curves tted to the experimental


data are shown in Figure 2.9.

It is interesting to note how the incremental hydraulic damping force decreases

as vp increases past a certain value depending on the valve position, whereas the

damping force was expected to follow the usual small orice quadratic relationship

between the ow and the pressure. As stated earlier on, this is probably due to the

ow forces acting on the valve which would tend to open it as the piston velocity

increases, thus causing the incremental force to get smaller.

0

200

400

600

800

1000

1200

0 0.5 1 1.5 2 2.5 3 3.5*

*

**

*

*

*

*

**

*

*

*

*

**

*

*

*

*

**

*

*

*

*

**

*

*

*

*

**

*

*

*

*

**

*

*

*

*

**

*

*

* *

**

*

*

0

.021

.28

.56

.93

1.261.48

1.691.89

valve pos. x10 cm-3

Damping Force, Hyperbolic Tangent Model

Piston Velocity vp (cm/s)

Dam

ping

For

ce F

d (N

)

Figure 2.9: Experimental Hydraulic Damping Eect

Fd = A [tanh(s (jvpj+ d)) tanh(sd)] sgn(vp) + 40vp; (2.13)

where A = A(xv), s = s(xv), d = d(xv):

The model uses bandlimited dierentiation (a pole added at 40 Hz) to reduce

numerical noise problems arising in the nonlinear damping loop and to improve the


closed-loop response.

Identication of the Linear Part

Separate identication of the valve dynamics and the uid dynamics was not possible;

we had to identify the linear part as a whole. With the piston xed at midstroke

position, a low-amplitude pseudo-random binary sequence (prbs) input was applied

to the open-loop system so that we could assume that the system was operating in

a linear region. It should be noted that the uid compliance in the cylinder depends

on the piston position and reaches a maximum when the piston is at a point where

both chamber volumes are equal. Therefore, the case studied here was nearly the

most adverse condition to stable control when considering only the uid dynamics

(see [Walters, 1967], pp. 5051). The sampling frequency was 5000 Hz.

An arx model was estimated using a least-squares method on matlabTM (arx

command) and the best t was given by a tenth-order model with two delays as

predicted:

(1+a1z1+a2z

2+ +a10z10)Y (z) = (b3z3+ b4z

4+ + b10z10)U(z); (2:14)

where a1 = 0:4223; a2 = 0:3765; a3 = 0:2802; a4 = 0:1959; a5 = 0:1930;

a6 = 0:2234; a7 = 0:0532; a8 = 0:0051; a9 = 0:0903; a10 = 0:0394;

b3 = 0:0997; b4 = 0:1360; b5 = 0:0258; b6 = 0:1094; b7 = 0:4047;b8 = 0:1323; b9 = 0:4253; b10 = 0:7851:

The prbs input and the system and arxmodel outputs are shown respectively in

Figure 2.10 (a) and (b).The pole-zero plot of the identied model is shown in Figure

2.11: as can be seen, the zeros of the identied model lie outside the unit circle. This

indicates that the system identication technique has yielded a non-minimum phase

model. The physical system has several components which are actually distributed

parameter systems, e.g. hydraulic uid and lines, valve stem exure, etc., and there


also exist possibilities of multiple transmission paths due to the mechanics of the

test set-up. Hence the non-minimum phase nature of the model appears justied.

Fortunately, these non-minimumphase zeros are clustered at high frequencies. Hence

controller design can be based upon frequency separation, by using an additional

compensator block which lters out the high frequency behavior.

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 0.1 0.2 0.3

PRBS Input

Valve

Curr

ent iv

(A)

time (s)(a)

-20

-15

-10

-5

0

5

10

15

0 0.1 0.2 0.3

System and ARX Model Outputs

time (s)

Outpu

t Forc

e Fs (

N)

(b)

--- : ARX Model

Figure 2.10: (a) prbs Input, (b) System and arx Model Outputs

Dynamics

As it was pointed out earlier on, the valve dynamics should have the lowest bandwidth

and therefore the only two identied real poles plus a zero at z = 0 were selected for

G(z):

G(z) =0:01203z

(z 0:9762)(z 0:4947); jzj > 0:9762 (2.15)

For the uid dynamics, we picked the three pairs of complex poles at high fre-

quencies and the three pairs of complex non-minimum phase zeros. We also chose the

only real zero at z = 2:0282 and placed two poles at z = 0 to make D(z) causal (see


-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

o

o

o

o

o

o

o

x

x

x

x

x

x

x

x

x

x

Poles and Zeros of Identified Model

Rez

Imz

Figure 2.11: Poles and Zeros of Identied Transfer Function

equation (2.16)). These two poles get cancelled with the zero of G(z) and a zero at

z = 0 attributed to the force sensor dynamics in the identied transfer function. For

the sensor dynamics, a pair of complex poles 1 around 300 Hz and one zero at z = 0

were disregarded.

D(z) =:7(z 2:0)(z2 1:7z + 2:2)(z2 + :14z + 1:4)(z2 + 2:2z + 1:3)

z2(z2 + 1:55z + :67)(z2 + :04z + :66)(z2 + :63z + :47);(2.16)

jzj > 0:819

Open-Loop and Closed-Loop Force Bandwidth

The open-loop force bandwidth has been measured with the piston locked to the

mount, and was found to be in the vicinity of 20 Hz. It was noted that open-loop

1these poles are at z = 0:5797+ 0:2408j and z = 0:5797 0:2408j


control was impractical because of the presence of hysteresis and friction.

Assuming that the force closed-loop system is linear for a given amplitude of the

sinusoidal input, frequency responses were experimentally obtained and are shown in

Figure 2.12 (a) and (b) for dierent amplitudes of the input and for a force feedback

gain of 2.44. The roll-o on the magnitude Bode plot (Figure 2.12 (a)) indicates

that the system is at least of the ninth order. The closed-loop bandwidth is around

100 Hz and decreases for higher input amplitudes. This is due in part to saturating

nonlinear elements in the system but also to the nonlinear hydraulic damping. The

6 dB bandwidth goes as high as 196 Hz for low-amplitude inputs.

-50

-40

-30

-20

-10

0

10

10-1 100 101 102 103

(dB)

freq. (Hz)

Closed-Loop Magnitude

-450

-400

-350

-300

-250

-200

-150

-100

-50

0

10-1 100 101 102 103

Closed-Loop Phase

freq. (Hz)

(degre

es)

20 N _ (highest) 20 N _ (highest)

30 N - - 30 N - -120 N ... 120 N ...

220 N ._._ 220 N ._._

510 N _ 510 N _

(a) (b)

Figure 2.12: Closed-Loop Frequency Responses: (a) Magnitude, (b) Phase (Kf =2:44)

A limit cycle has been observed for feedback gain values of 3.66 and higher (sus-

tained oscillations at frequencies around 95 Hz). It was also observed that the closed-

loop responses to sinusoid inputs (see Figure 2.13 (a) and (b)) present little distortion

given the degree of nonlinearity of the system. The slight distortions seen for high-

amplitude, low-frequency responses (e.g. Figure 2.13 (b)) are probably due to the

piece of aluminum on which the actuator was mounted: the assembly was such that


this part of the xture bent signicantly for high output forces. The xture also had

an asymmetric, nonlinear, stiness characteristic, so that it absorbed some elastic

energy from the system and then suddenly released it as it moved back and forth.

This could be observed for open-loop responses as well.

-250

-200

-150

-100

-50

0

50

100

150

200

250

0 0.01 0.02 0.03 0.04

Closed-Loop Resp. to Sin. Input

time (s)

Force

Outp

ut Fs

(N)

-600

-400

-200

0

200

400

600

800

0 0.05 0.1 0.15

Closed-Loop Resp. to Sin. Input

time (s)

Force

Outp

ut Fs

(N)

(a) (b)

Figure 2.13: Closed-Loop Force Responses: (a) f = 63 Hz, (b) f = 20 Hz

2.3.4 Simulation Results

A software simulator has been built using simulabTM. The simulator includes

discrete-time and continuous-time linear transfer functions. Zero-order holds are used

at the output of the discrete-time blocks. The fth-order Runge-Kutta integration

algorithm was chosen for the simulations because it is well adapted for systems fea-

turing many nonlinearities.

The experimental and model closed-loop force responses (with feedback gain

Kf = 2:44) to a square-wave input agree fairly well for amplitudes of 120 N and

520 N (see Figures 2.14 and 2.15). The 20 N model response is overdamped when

compared to the experimental response (see Figure 2.16). This is not surprising since


the level of uncertainty in the combined eects of hydraulic damping and friction lies

in the range of the 20 N response. The simulations show that the hydraulic damping

and friction models are not really satisfactory at low velocities and low pressures.

The kinetic friction might be less than expected for low pressures across the piston as

it could explain why the simulated 20 N response is overdamped. Another potential

source of error comes from the fact that the hysteresis model can not easily reproduce

the small minor loops. The lower and higher parts used to construct the major loop

were experimentally obtained and although some ltering was done on them, they

are not locally perfectly smooth. Moreover, using the inverse of the valve static force

function amplied these imperfections.

-200

-150

-100

-50

0

50

100

150

200

250

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Closed-Loop Force Response to a Square-Wave Input

time (s)

Force

Outp

ut Fs

(N)

_ : experimental

---: simulation

Figure 2.14: Closed-Loop Force Response to a 120 N Square-Wave Input (Kf = 2:44)

A Lead-Lag Force Controller

In Section 2.1, it was mentioned that actuators for high performance robotics should

be able to track force trajectories accurately, i.e. they should have a high force

bandwidth and good accuracy at low frequencies. The proportional force controller


-600

-400

-200

0

200

400

600

800

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2


Force

Outp

ut Fs

(N)

time (s)

_ : experimental

---: simulation


-50

-40

-30

-20

-10

0

10

20

30

40

50

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2


time (s)

Outpu

t Forc

e Fs (

N)

---: simulation

_ : experimental



implemented on the asi actuator yielded satisfactory dynamic performance for our

purpose, i.e. for actuating the parallel, redundant revolute joint which is the main

topic of the next chapters. However, poor force tracking capability was observed at

very low frequencies, with force errors as high as 10 N to 20 N being common. Hence a

linear analog lead-lag controller was designed to get more gain at low frequencies while

maintaining sucient damping. Since no systematic design method exists to calculate

the parameters of a linear lead-lag controller achieving some pre-specied performance

when the plant is nonlinear, we rst designed the lag part to get the desired accuracy

at low frequencies without changing the open-loop gain at high frequencies. This

lag controller alone yielded a very undamped force response, thus a lead part was

added to provide about 55 of phase lead around 60 Hz. The analog active circuit

implementing the lead-lag controller is shown in Figure 2.17. The transfer function

of the resulting compensator is

C(s) = Kf

R2 +R3

R0 +R1

(R0C1s+ 1)(R2R3C2

R2+R3

s+ 1)

(R0R1C1

R0+R1

s+ 1)(R3C2s+ 1): (2:17)

Numerically, with the component values given in Figure 2.17 and with Kf =

6:84, we getC(s) = 7:5

(:008s + 1)(:027s + 1)

(:0008s + 1)(:3s+ 1): (2:18)

The closed-loop system response with the lead-lag compensator to a step of 200 N

is shown in Figure 2.18. The steady-state error is approximately 4 N and the risetime

from 10 % to 90 % of the nal value is 0.01 s. The overshoot is 16 % of the nal

value and the 5 % settling time is approximately 0.1 s. When the desired force is

set to 0, the residual error is less than 1 N, which is close to the sensitivity of 0.37 N

provided by the 12-bit analog to digital converter.


V

V

R

C

R

R

R

C

−

+

i

0

1

1

2

3

2

o

R = 90 k

R = 10 k

R = 10 kR = 100 kC = .09 FC = 3 F

0

1

2

3

1

2

ΩΩΩΩ

µµ

Figure 2.17: Analog Circuit Implementing the Lead-Lag Controller

0

50

100

150

200

250

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Force Step Response with Lead-Lag Controller

time (s)

Outpu

t For

ce F

s (N)

Figure 2.18: Closed-Loop Force Response to a 200 N Step Input, Lead-Lag Controller


2.3.5 Discussion

A complete nonlinear model of the high performance asi hydraulic servosystem has

been obtained, simulated and validated. The model's ability to reproduce experi-

mental closed-loop force responses for dierent amplitudes indicates that it could be

a valuable tool for the design of better digital nonlinear force control laws. It could

also be useful to explain the sytem's behavior. The model can be easily extended for

simulation of position and impedance control.

The implications of the frequency response of the hydraulic actuator for control

law design are encouraging: it is predominantly low-pass and the non-minimumphase

zeros are clustered at high frequencies. Based on the dynamics, together with the

large gains possible due to the high saturation level of the actuator, it appears ideally

suited to feedback modulation of impedance over a wide range.

The next chapter is devoted to the analysis and design of a simple one-degree-of-

freedom parallel mechanism actuated by two ASI hydraulic actuators characterized

in this chapter.

Chapter 3

Modeling of a Redundant, Parallel

Revolute Joint

In this chapter, we turn our attention to the design and control of a simple one-

degree-of-freedom parallel manipulator with a redundant actuator as the one shown

at Figure 3.1. The mechanism is composed of two linear actuators which are linked

to a revolute joint through a moment arm of xed length. Figure 3.2 shows a picture

of the joint prototype.

The actuators apply forces which are mapped into torques at the joint by a time-

varying linear transformation which depends on the joint angle. A torque applied at

the joint produces a movement governed by the joint's dynamic equations, and one

of our goals will be to control accurately the joint movement, or trajectory, i.e. the

angle (t) in Figure 3.3, by acting on the actuator forces.

3.1 Redundancy and Antagonism

At this point, a reasonable question arises: why adding a second actuator when

the joint can be controlled by only one? There are several reasons for this. In

45

CHAPTER 3. MODELING OF AREDUNDANT, PARALLEL REVOLUTE JOINT 46

Figure 3.1: Parallel Revolute Joint with Actuator Redundancy

Figure 3.2: Prototype of the Parallel Revolute Joint with Actuator Redundancy


[Hayward, 1988], the author presents an extensive survey of the properties of re-

dundancy in robot manipulators. First, using two actuators eliminates singularities

inside the workspace. For a single actuator mechanism, this type of singularity hap-

pens when the line of action of the actuator is collinear with the moment arm. In

such a case, a nite joint angular velocity maps to a null actuator velocity while a

nite actuator force maps to a null joint torque. This situation is clearly undesirable

and must be avoided by design, for example by restricting the range of the actua-

tor so that the workspace does not include the two possible singular congurations.

Furthermore, when the mechanism is nearly singular, there is a signicant loss of

accuracy when the force and position sensors are colocated, and they can hardly be

relied upon to calculate the joint angle, velocity, and torque due to uncertainty in

measurement and in the kinematic parameters. For example, if the joint angle un-

dergoes a small change while the position sensor is close to being singular, only a

very small sensor displacement will result, and if this displacement is too small to

be measured accurately, then the change in joint angle will remain almost entirely

unknown. Also, the kinematic parameters describing the geometry of the mechanism

map nonlinearly into the jacobian matrix which relates innitesimal linear sensor

displacements to innitesimal joint angles. This adds even more uncertainty in the

value of the joint angle calculated using estimates of the kinematic parameters. The

same kind of argument could be given on velocity or torque measurements. Adding

a second actuator eliminates this type of singularity: when one actuator aligns with

the moment arm, the other can be used to measure the state of the system and to

control it [Hayward, 1991]. Thus, a larger worskspace free of singularities is obtained.

Of course, the antagonistic nature of this mechanismhas some implications regard-

ing movement coordination and resulting internal forces. Uncoordinated application

of forces can result in excessive stress on the passive joints and may damage the

mechanism or cause rapid wear. But antagonism also oers advantages which can


be noticed by observing animal joints. For instance, the human shoulder, which can

be approximated by a three-degree-of-freedom ball-and-socket joint, uses muscle an-

tagonism to generate a pre-load force which keeps the humeral head (ball) and the

glenoid (socket) rmly together (see [Hayward, 1988]).

Control of internal forces in a redundant mechanism is made possible by the re-

dundancy in the number of actuators with respect to the number of task space degrees

of freedom. In this case, the mapping from actuator forces and torques to task space

forces and torques has a non-empty nullspace of dimension (nactDOF ntaskDOF ).

Thus, for a desired wrench in task space, there exists an innity of solutions from

which we can pick one satisfying a given constraint or optimizing a given objec-

tive function. In Chapter 4, we will present the solutions to two such optimization

problems, one of which minimizes the internal forces while the other minimizes the

maximum of the two actuator force magnitudes. we will also describe a method

to calculate actuator forces that will keep the joint preloaded, which thus prevents

backlash from aecting performance.

Another important feature oered by antagonism is that the maximum torque

available at each joint angle has a smoother characteristic than for the single actuator

case. This is helpful when the controller calculates forces based on a torque value

required to reduce the trajectory error, considering that the actuators have a given

saturation level. Only very small torques could be produced by a single-actuator

mechanism operating near a singularity, even for maximum actuator output force.

The parallel, redundant, revolute joint is a simple mechanism which entails most

of the features found in more complex redundant parallel mechanisms, such as the

3-DOF parallel wrist mechanism described in [Hayward, 1991] and in [Kurtz, 1990].

Hence, it appears to be a good starting point to analyze this class of manipulators

and to design and test force optimization techniques and control laws.


ρ ρ

λ

λ

k k

r rγ

ψ

1

1 2

2

1

2

Joint Axis

P P

A B

1 2

= k + r

= k + r

1

2

actuator 1 actuator 2

Figure 3.3: Geometry of the Parallel Revolute Joint

3.2 Kinematics of the Parallel, Redundant, Rev-

olute Joint

In this section, the kinematics of the planar parallel revolute joint with actuator

redundancy are derived. The notation used is illustrated at Figure 3.3.


3.2.1 Inverse Kinematics

The law of cosines applied to triangles A and B yields two expressions for the actuator

lengths 1 and 2:

1 =q21 + r2 21r cos ; (3.1)

2 =q22 + r2 22r cos( ) : (3.2)

3.2.2 Direct Kinematics

Equations 3.1 and 3.2 can be inverted; they yield equations 3.3 and 3.4 for the joint

angle .

= arccos

21 + r2 21

21r

!; (3.3)

= arccos

22 + r2 22

22r

!: (3.4)

Since the joint angle may become negative or greater than the angle , the

redundancy in sensing allows us to determine its sign. Assuming that the arccos

function always gives an answer in the rst or the second quadrant (which is the case

in the c language), the idea is to check whether 1 is greater than 2, and if it is,

equation 3.3 is used to calculate . In the opposite case, equation 3.4 must be used.

This method also has the advantage of always using the best conditioned sensor to

calculate the joint angle.

3.2.3 Velocity Mapping: The Jacobian Matrix

We are now interested in calculating the time-varying linear transformation mapping

the joint angular velocity to the actuator linear velocities. We thus dene the 2x1

jacobian matrix J (or more simply the jacobian) of the mechanism by the following


relationship. 264 _1

_2

375 = J _ : (3:5)

It can be calculated by dierentiating equations 3.1 and 3.2:

_1 =1r sin

1_ ; (3.6)

_2 =2r sin( )

2_ : (3.7)

Hence,

J =

264 1r sin

1

2r sin( )2

375 : (3:8)

By using the principle of virtual work, it can be easily shown that the joint output

torque is related to the actuator forces by the tranpose of the jacobian:

= JTf ; (3:9)

where f = [f1 f2]T is the vector of forces applied by the actuators.


3.3 Autonomous Kinematic Calibration of the Rev-

olute Joint

In order to be able to calibrate the kinematic parameters of the parallel revolute joint,

most kinematic calibration methods (see for example [Whitney, Lozinski, Rourke, 1986])

would require that an external device be used to measure the joint angle with sucient

accuracy. Then a nonlinear search of the kinematic parameters could be implemented

to nd the best set of parameters satisfying the consistency equations which relate the

measured joint angle to the kinematic parameters through the geometry of the mech-

anism. Instead of using external measurements, Bennett and Hollerbach observed

in [Bennett and Hollerbach, 1991] that single-loop closed kinematic chains formed by

serial-type manipulators can be calibrated when the manipulator is redundant with

respect to the task, i.e. when it has more degrees of freedom than required to perform

the task. The key idea is to avoid the need of having external measurements of the

endpoint position or orientation by constraining some of the cartesian space degrees

of freedom to a known position or orientation and then to use redundancy in joint

sensing to eliminate the unsensed passive degrees of freedoms of the task kinematics.

These results can be directly used to calibrate the parallel revolute joint even

though it belongs to the class of parallel manipulators. The following observations

can be made on the structure of this mechanism (see Figure 3.3). First, it is composed

of two closed kinematic chains (triangles A and B), each formed by the actuator and

the moment arm, and both of which are position constrained in a plane at the joint's

rotation axis. If we put a coordinate system at that point, we then readily know that

the endpoint position of the two kinematic chains is the origin, and two loop closure

equations could be derived in the x and y coordinates for each closed kinematic chain.

The passive task degree of freedom is the joint angle , which must be eliminated

from the loop closure equations.


In fact, it is simpler not to use a coordinate system but to use only one loop

closure equation derived from triangle geometry for each closed kinematic chain. For

example, the law of cosines is applied to the two kinematic chains shown at Figure

3.3, and we get equation 3.10 for chain A and equation 3.11 for chain B.

21r cos = 21 + r2 21; (3:10)

22r cos( ) = 22 + r2 22 (3:11)

We can rewrite equation 3.11 by collecting all the terms on the left-hand side, and

we dene FB = FB(1m; 2m; 10; 20; r; k1; k2; ) as

FB = 22r cos cos + 22r sin sin 22 r2 + 22 = 0; (3:12)

where i = im + i0; i = ki + r; i = 1; 2:

The angle can be eliminated in equation 3.12 by using equation 3.10. We restrict

the analysis to the subworkspace > 0. Equation 3.15 is the loop closure equation

needed for the nonlinear iterative search. Such a method is required because the

kinematic parameters do not map linearly into FB. We see that equation 3.15 can

be scaled arbitrarily, that is, any scaled version of the mechanism will satisfy it, and

hence one of the kinematic parameters must be specied at the outset for the solution

to be unique in a small neighborhood around it.

cos =21 + r2 21

21r; (3:13)

sin =

0@1

21 + r2 21

21r

!21A1=2

; (3:14)

FB =21

cos (21 + r2 21) + 22r sin

0@1

21 + r2 21

21r

!21A1=2

22 r2 + 22:

(3:15)


The Levenberg-Marquardt algorithm is used to identify the kinematic parameters

(k1, k2, 10, 20, ) of the mechanism because it is numerically robust. This algo-

rithm is based on a linearization of the nonlinear equation and thus needs the partial

derivatives of FB with respect to each of the kinematic parameters except the moment

arm's length r which is used to scale the mechanism and is supposed to be known with

good accuracy. These partial derivatives are listed in appendix A. A large numberM

of actuator position measurements (1m; 2m) must be taken inside the subworkspace

decribed by > 0. Now dene the 5x1 jacobian matrix Ci as follows, where the

superscript i denotes the ith conguration in which the mechanism was placed to take

the ith measurement (i1m; i2m).

Ci =

"@F i

B

@k1

@F iB

@k2

@F iB

@10

@F iB

@20

@F iB

@

#: (3:16)

Let

C =

2666664C1

...

CM

3777775 ; F =

2666664F 1B

...

FMB

3777775 ; =

26666666666664

k1

k2

10

20

37777777777775; =

266666411m 12m...

...

M1m M2m

3777775 :

The explicit dependency of F on and is stressed when we write

F = F(; ): (3:17)

The rst order dierential of F with respect to is

F = C; (3:18)

In the case where C has full column rank (i.e. is one to one since M 5),

can be calculated as follows

= (CTC)1CTF: (3:19)


For a given estimate of the vector of kinematic parameters which diers from

the true vector by approximately , the dierential F can be calculated easily:

F = F F c = Fc; (3:20)

where F c = F(; ); F = F(; ) = 0 assuming is noise-free.

Given these relationships, it is possible to use a Newton-like method to search

for the optimal set of kinematic parameters, such as the aforementioned Levenberg-

Marquardt algorithm.

3.3.1 The Levenberg-Marquardt Algorithm

This iterative scheme consists of linearizing the nonlinear function F about the cur-

rent estimate of the parameters k, then calculating a correction vector

k+1of

the parameters by using a modied pseudo-inverse of the jacobian matrix C, and

then adding this correction vector to the current estimate to form the new estimate.

The algorithm iterates until, for example, the 1-norm of F gets smaller than a

pre-specied threshold. It should be noted that C and F must be evaluated at each

iteration using the current estimate k. The algorithm implemented to identify the

kinematic parameters of the parallel mechanism is outlined here.

(0) Set k to 0, and to small values. Use a reasonable rst estimate 0.

(1) Calculate Fkc (

k; ) and C(

k; ) ,

(2) Calculate the correction vector k+1

and the new estimate k+1

:

k+1

= (CkTCk + I)1CkTFkc ,

k+1

= k+

k+1.

(3) Let k = k + 1, goto (1) and iterate until kFkck1 < .

The diagonal matrix I is needed to avoid numerical singularities when the jaco-

bian matrix C loses rank, and it helps nd the desired solution even if a bad initial


estimate of the parameters is used. In a practical implementation, has to be grad-

ually reduced as the estimate approaches the solution, otherwise it will reduce the

rate of convergence of the iteration. Introduction of this diagonal matrix also corre-

sponds to minimizing the criterion kCFk2+T which penalizes large

variations in the parameters during the search.

3.3.2 Experimental Results

In simulations where the actuator position measurements were assumed to be perfect,

the kinematic parameters were quickly recovered, even for bad initial estimates. How-

ever, our rst try to estimate the kinematic parameters of the real mechanism proved

unsuccessful, even though the moment arm's length had been measured with an ac-

curacy of about 0:05 mm (r = :06374 m). We used a 12-bit A/D converter which

provided a sensitivity of about 1:78x105m, but the actuator position signals coming

from the LVDT sensors were corrupted by noise and for each static conguration we

had to average the position measurements over 100 samples. Further simulations in

which zero-mean uniformly-distributed white noise of magnitude 106 m (less than

the A/D converter sensitivity) was added to the measurements failed to nd a con-

sistent set of parameters. Hence we had to modify our scheme to deal with this level

of uncertainty in measurement.

Apart from the noisy measurements, the failure of the search was also believed

to partly result from three other possible causes. The rst one is the geometric

model which might not be consistent with the real mechanism. A second possible

explanation is that we did not use enough measurements (we usedM = 50), although

this hypothesis was quickly abandoned after the simulations had shown that using

100 measurements corrupted by the noise described above would not improve the

search. The third possible reason is that a priori xing of r might not constrain the

search to a small enough region in parameter space. Intuitively, if the sensitivity of


FB with respect to r is small, many dierent sets of parameters could reach a given

bound on the 1-norm of F c. One solution would be to accurately measure a second

parameter to which FB would be more sensitive, and then to estimate the remaining

four parameters. Even if FB is not really more sensitive to variations in the second

xed parameter, the search should be better constrained in the neighborhood of the

optimal solution because of the fact that there are now two xed parameters which

reduce uncertainty in the estimate of the others.

Not much could be done to improve the kinematic model so it was decided to

investigate the third possible cause. We xed k2 because 2 could be easily measured

and it was believed that FB was sensitive enough to it. This does not restrict the

usefulness of the autonomous kinematic calibration method because the hardest pa-

rameters to measure are the actuator dead lengths 10 and 20 and the angle . The

jacobian matrix Ci and the vector of unknown kinematic parameters have to be

redened as follows.

Ci =

"@F i

B

@k1

@F iB

@10

@F iB

@20

@F iB

@

#; (3:21)

=

2666666664

k1

10

20

3777777775: (3:22)

With these new denitions and the same denitions of C, F and as before, the

Levenberg-Marquardt algorithm outlined above can be used directly to estimate the

kinematic parameters. The actual algorithm implemented on matlabTM is presented

in appendix B. Simulations were run with the same noisy measurement matrix as

described above, i.e. with a zero-mean uniformly-distributed white noise between

106 m, and the parameters were recovered to within about .25 mm and 105 rad.

Simulations with noise of maximum amplitude 105 m produced estimates which had

one length parameter o by as much as 1 mm, but the angle was recovered with an


accuracy of 104 rad. Since an A/D converter with more than 12 bits of resolution

was not available at that time, we tried to estimate the kinematic parameters of the

real mechanism using the 12-bit converter.

Before we could use the autonomous calibration scheme for the real mechanism,

we had to accurately measure 2. It was measured to be 28:66 :05 cm, thus the

parameter k2 = 2 r is k2 = 22:29 :05 cm. The kinematic parameters of the

redundant, parallel revolute joint were successfully identied starting from two very

dierent initial estimates. The lengths appeared to be exact to within about 1 mm,

and the angle was identied to within about :001 rad. These results are somewhat

satisfying considering the simulation results with a noise level comparable to the

quantization eect. The kinematic parameters are shown at Table 3.1.

k1 22.30 cmk2 (measured) 22.29 cmr (measured) 6.374 cm10 22.09 cm20 22.04 cm .633 rad

Table 3.1: Calibrated Kinematic Parameters of the Parallel Revolute Joint

Chapter 4

Optimization of the Joint's

Actuator Forces

4.1 Actuator Forces to Joint Torque Mapping:

The Transposed Jacobian

The joint output torque is related to the actuator forces by the tranpose of the

jacobian:

= JTf ; (4:1)

where 2 R, JT = [j1 j2] 2 R2, f = [f1 f2]T 2 R2:

Equation 4.1 can be explicitly written as:

=

"1r sin

1

2r sin( )

2

#f : (4:2)

59

CHAPTER 4. OPTIMIZATION OF THE JOINT'S ACTUATOR FORCES 60

4.2 Optimization of Actuator Forces Seen as a

Minimum-Norm Problem

An innity of solutions exists for the vector of actuator forces in equation 4.1, and we

shall present two methods for nding the minimum-norm optimal solution according

to two dierent denitions of the norm, one in Hilbert space, the other in a Banach

space.

The rst optimization problem may be stated as follows: given the relationship

between and f , nd the optimal minimum-norm vector ~f that satises it when the

norm is chosen to be the usual Hilbert space norm:

kfk = (f21 + f22 )1=2: (4:3)

The second minimumnorm optimization problem is formulated in a Banach space

with a norm dened as

kfk1 = max(jf1j; jf2j): (4:4)

Both methods lead to very simple solutions which can be calculated in real time.

The 2-norm optimal vector produces a resultant force which is orthogonal to the

moment arm, the latter being aligned with the nullspace of the transposed jacobian, so

that this resultant force delivered by the actuators contributes entirely to the torque

applied at the joint. It also comes from a linear transformation applied to the desired

joint torque and is thus easy to calculate even for systems having more degrees of

freedom.

The 1-norm optimal solution minimizes the maximum absolute value of the ac-

tuator forces at any time and can therefore be regarded as a way to reduce stress

on the force sensors. It also enables the system to reach higher output torques given

the force limits of the actuators because as will be seen below, the 1-norm solution

vector has forces that are equal in magnitude, and that magnitude is minimal. On


the other hand, a high output force may be applied on one of the actuators that is, or

close to being, singular, which is not the case for the 2-norm optimal solution which

always makes an actuator close to, or at, a singularity very compliant.

4.2.1 Minimum 2-norm Optimal Vector of Forces

The minimum 2-norm optimal vector of actuator forces corresponds to the solution

of equation 4.1 obtained by using the pseudo-inverse of the transposed jacobian JT

which is an onto transformation. The optimal vector is simply

~f = J(JTJ)1; (4:5)

which turns out to be

~f =

j21 + j22

264 j1

j2

375 : (4:6)

This expression would be the most ecient one in an actual implementation pro-

vided the jacobian has already been computed. A more explicit expression of ~f is

now given in terms of the kinematic parameters:

~f =12

r(2122 sin

2 + 2221 sin

2( ))

264 21 sin

12 sin( )

375 : (4:7)

It is interesting to note that when an actuator aligns with the moment arm, for

example actuator 1 when = 0, the force it applies is 0 since sin(0) = 0. Therefore

actuator 1 becomes very compliant around = 0.

4.2.2 Minimum1-norm Optimal Vector of Forces

The optimization problem is stated as follows: given a torque , is there a minimum-

norm vector ~f satisfying equation 4.1 when the norm is dened as in equation 4.4

above, and if so, can we nd its expression? We use the theory of dual Banach


spaces as presented in [Luenberger, 1969] to solve this problem. Roughly speaking,

the dual space of linear functionals of a normed space plays a role analogous to the

inner product in Hilbert space and with careful interpretation, the projection theorem

which gives the solution to minimum-norm problems in Hilbert space can be extended

to arbitrary Banach spaces.

First, recall that lp spaces are spaces of innite sequences of real numbers with

norm dened as

kxkp = +1Xi=1

jxijp!1=p

; (4:8)

where x 2 lp, x = fx1; x2; x3; : : :g, xi; p 2 R, 1 p < +1:

For p = +1, the l1 space is the space of bounded sequences of real numbers, and

the norm is dened as

kxk1 = supijxij: (4:9)

Our purpose will be to use a subspace of l1 in which vectors have only their rst

two elements not identically zero, and also a subspace of l1 dened similarly. Also

needed is the following denition of a normed dual space.

Denition 1 Let X be a normed linear vector space. The space of all bounded linear

functionals on X is called the normed dual of X and is denoted as X . The norm of

an element x 2 X is dened as

kxk = supkxk1

jx(x)j; (4:10)

where x(x) denotes the value of the linear functional x at x 2 X .

The value of x at x 2 X will also be denoted by hx; xi or by hx; xi to suggestthe analogy with the inner product in Hilbert space. It can be shown that the

space X is a Banach space, i.e. a complete normed linear space. For the special

case where X is the lp space, it can also be shown that the normed dual of X is lq


where1=q + 1=p = 1. In that case, all bounded linear functionals x 2 X lq can

be expressed as

hx; xi =+1Xi=1

ii; (4:11)

where x = f1; 2; 3; : : :g 2 lp, f1; 2; 3; : : :g = y 2 lq, i; i 2 R:

We say that X is equivalent to lq in the sense that there is a one to one corre-

spondence between the sequence y = f1; 2; 3; : : :g 2 lq and the linear functional

in equation 4.11, and every element of lq denes a member of X . From now on,

vectors in the dual space X of an lp space will be referred to as x 2 X or as y 2 lq,depending on the context.

We are now ready to show that the normed dual space X of the space X of pairs

of real numbers x = (1; 2) with norm dened as kxk1 =P2i=1 jij is the space of

pairs of real numbers y = (1; 2) with norm dened as in equation 4.4.

Proposition 1 The normed dual space X of the space X of all pairs of real numbers

with norm dened as

kxk1 =2Xi=1

jij; (4:12)

where x = (1; 2) 2 X ;

is the space of all pairs of real numbers y = (1; 2) with norm dened as

kxk = kyk1 = maxi=1;2

jij: (4:13)

Moreover, the value of x at x can be expressed as

hx; xi =2Xi=1

ii: (4:14)

Proof: First, we consider a subspace V1 of l1 consisting of all sequences of the

form v = f1; 2; 0; 0; 0; : : :g where 1; 2 2 R, and a subspace W1 of l1 consisting of

all sequences of the form w = f1; 2; 0; 0; 0; : : :g where 1; 2 2 R. We see that


V1 and W1 are only extensions of X and X obtained by padding zeros to get

innite sequences. We know that the dual space of l1 is l1 from [Luenberger, 1969].

Therefore, it is clear that the dual space of W1 is V1, and that all bounded linear

functionals of V1 can be written as

hw; wi = hv; wi =+1Xi=1

ii =2Xi=1

ii: (4:15)

The norm of w is

kwk = kvk1 = maxi=1;2;:::

jij = maxi=1;2

jij: (4:16)

Since the summation in equation 4.15 runs from i = 1 to 2, and the norms of W1

and V1 use only the rst two elements of the sequences, it is readily seen that the

zero padding is not necessary for equations 4.15 and 4.16 to hold. Hence X is the

normed dual space of X with norm dened as in equation 4.13.

Q.E.D.

The 1x2 transposed jacobian JT in equation 4.1 will be regarded as an element of

X while f will be considered as belonging to X . Equation 4.1 can be rewritten as

hf; JT i = : (4:17)

Now the problem is: nd an f 2 X such that equation 4.17 is satised and kfk1is minimum, or, in other words, nd an element f of the coset K dened by equation

4.17 such that its norm is minimum. Since we have formulated the optimization

problem in a dual space, the minimum-norm vector is guaranteed to exist by virtue

of theorem 2, Section 5.8 of [Luenberger, 1969], which we have adapted to the problem

at hand and present here as Theorem 1. But before, we need the following denitions

of alignment and orthogonality properties in dual spaces.

Denition 2 A vector x 2 X is said to be aligned with a vector x 2 X if

hx; xi = kxkkxk.


Denition 3 The vectors x 2 X and x 2 X are said to be orthogonal if hx; xi = 0.

Denition 4 Let U be a subset of a normed linear space X . The orthogonal comple-

ment of U , denoted U?, consists of all elements x 2 X orthogonal to every vector

in U .

Theorem 1 Let M denote the one-dimensional subspace generated by JT 2 X , and

let f be a vector in X . Then,

d = minhf;JT i=

kfk = minm2M?

kf mk = supx2M;kxk1

hx; fi; (4:18)

where f satises hf ;JT i = , and the minimum is achieved for some m0 2 M?. If

the supremum on the right is achieved for some x0 2 M, then f m0 is aligned with

x0.

The proof can be found in [Luenberger, 1969]. Corollary 1 of Theorem 2 in Section

5.8 of the same reference is also useful and we state it here as Corollary 1 adapted to

our Theorem 1.

Corollary 1

d = minhf;JT i=

kfk = maxkaJT k11

hf ; aJT i = maxkaJT k11

a; (4:19)

where f satises hf ;JT i = and a 2 R:

Furthermore, the optimal vector ~f is aligned with the optimal aJT , i.e.

h~f ; aJT i = k~fk1kaJTk1: (4:20)

This immediately follows once we notice that the subspace M can be described

by aJT , a 2 R. If we explicitly write the norms in equation 4.20, we get

h~f ; aJT i = jaj2X

k=1

jjkjmaxi=1;2

j~fij; (4:21)


but

h~f ; aJT i = a2Xi=1

ji ~fi = a: (4:22)

Hence,

maxi=1;2

j~fij2X

k=1

jajjjkj =2Xi=1

aji ~fi: (4:23)

This occurs only when the following conditions are met:

(i) j ~f1j = j ~f2j = k~fk1(ii) sign( ~fi) = sign(aji); i = 1; 2

Also, since

d = minhf;JT i=

kfk = k~fk1 = maxi=1;2

j~fij; (4:24)

it follows that

j~fij = maxkaJT k11

a , i = 1; 2: (4:25)

For a complete characterization of the optimal vector ~f , we need to nd maxkaJT k1 a .

We maximize a under the constraint

2Xi=1

jajjjij 1; (4:26)

jaj 1P2i=1 jjij

: (4:27)

Hence,

a =sign( )P2i=1 jjij

(4:28)

maximizes a . Then, from condition (i) above,

k~fk1 = j ~f1j = j ~f2j = a =j jP2i=1 jjij

: (4:29)

Finally, using condition (ii), the optimal vector of forces ~f is

~f =

jj1j+ jj2j

264 sign(j1)

sign(j2)

375 ; (4:30)


where sign() =

8><>:1 , < 0

+1 , 0; 2 R:

We see that its expression is very simple and can be easily computed in real time.

4.2.3 Addition of a Pre-Load Force on the Joint

It may be desirable to apply a small pre-load force in a joint in order to reduce the

eect of a loosened bushing or bearing. It can be accomplished by using a force vector

lying in the nullspace of JT . Such a vector would be aligned with the moment arm

and thus would not contribute to the movement of the joint. Let n be a unit vector

lying in N (JT ) and pointing outward from the joint axis. It can be expressed as

n =

8><>:

[j2(j21 + j22)1=2 (1 + j22=j

21 )1=2]T ; > 0

[(1 + j21=j22)1=2 (1 + j22=j

21)1=2]T ; 0

: (4:31)

A pre-load force of Fpl newtons will be applied on the joint if a force vector Fpln is

added to the 2-norm optimal vector ~f corresponding to a desired torque . It should

be noted that the 1-norm optimal vector of forces cannot be used for that purpose.

Chapter 5

Control of the Redundant Parallel

Joint

In robotics, it is necessary for a controller to perform reasonably well under many

circumstances because the inertial parameters and friction eects may vary in large

proportions when a manipulator moves, picks up a load, makes contact with the

environment, etc. This is especially true for hydraulic robots since they can handle

heavy payloads, in which case their inertial parameters undergo large variations.

Hence a good controller should be able to either adapt itself to these changes or at

least guarantee that the system will remain stable at all times while maintaining a

certain desired performance level. This latter property is referred to as the robustness

of a control system as discussed in Chapter 1. The aim of this chapter is to compare

three linear, time-invariant position controllers for the parallel joint, and not to nd

the best controller possible. Thus the class of adaptive controllers is deliberately

excluded at the outset because it is believed that uncertainty in the system should

be taken care of in the design stage, not while the system is operating. Furthermore,

linear time-invariant control systems are much simpler and easier to analyze than are

time-varying control systems.

68

CHAPTER 5. CONTROL OF THE REDUNDANT PARALLEL JOINT 69

Two approaches are investigated for the robust position control of the revolute

joint. These approaches consist of applying linear control schemes to the system.

Linearization of the plant's dynamics is often performed by feeding back the nonlinear

dynamic terms in complex robotic systems, although here we deal with a plant that

is essentially linear if we neglect friction. Unmodeled nonlinearities, disturbances and

variations in system parameters form the uncertainty, and the robustness properties of

the two control laws should reduce its eect. A simple PD controller is rst designed

for purpose of comparison.

5.1 Dynamics of the Revolute Joint

We consider the case where the redundant revolute joint is carrying an inertial load

with unmodeled friction which is seen as a disturbance:

= I + f ; (5:1)

where I is the joint inertia and f is the friction torque.

5.2 Position Control

5.2.1 A Simple PD Controller

The problem of designing a simple position controller is straightforward for the pure-

inertia case under study. We propose the following linear proportional-derivative

(PD) control law:

= Kp( des ) +Kd( _ des _ ): (5:2)

The joint angle can not be measured directly|the direct kinematics must be

solved rst. Taking the Laplace transforms of both sides of (5.1) and (5.2) and


equating them, we get

I (s)s2 +Kd (s)s+Kp (s) = Kd des(s)s+Kp des(s) f (s); (5:3)

(s)

des(s)=

Kp +Kds

Is2 +Kds+Kp: (5:4)

Equation 5.4 is the closed-loop transfer function from the desired angle des to

the actual angle . If the desired trajectory is specied not only by the angle but

also by its rst two derivatives, the desired angular acceleration can then be used as

a feedforward term. Also, multiplying the control by an estimate of the inertia I will

make the resulting closed-loop second-order system behave as if it had a unit inertia:

= I( des +Kp( des ) +Kd( _ des _ )): (5:5)

This control law corresponds to the computed torque method [Paul, 1972].

Using (5.1) and (5.5), and letting I = I,

( des ) +Kd( _ des _ ) +Kp( des ) = f : (5:6)

With e = des , equation 5.6 can be written as:

e+Kd _e+Kpe = f : (5:7)

This is the error equation, and if f is identically zero or an impulse, e will tend to

zero at a rate governed by the values of Kp and Kd. However, in a real system, the

friction torque may be a persisting disturbance (e.g. stiction), and in that case the

error will not be null in steady-state. In the next section, we develop a robust PID

controller which can drive the error to zero even in the presence of a constant torque

disturbance.

A block diagram of the PD controller is shown in Figure 5.1. The controller

calculates a control torque but the inputs to the system are desired actuator forces as


γ

γ

γ

τ

ρ

ρ+

+ +

−+

−+

KK

IJoint and Actuator

DynamicsForce

Optimizer

f

Kinematics

γ

γ

(J J) JT −1 T

d

d

d

dp

^ −

−

−

es

es

es K( )

Figure 5.1: Block Diagram of the PD Controller

discussed in Chapter 4. Thus, if we use the 2-norm optimal solution for the actuator

forces, the control law can be expressed as:

~f =( des +Kp( des ) +Kd( _ des _ ))I12

r(2122 sin

2 + 2221 sin

2( ))

264 21 sin

12 sin( )

375 : (5:8)

The feedback gains Kp and Kd can be chosen based on desired natural undamped

frequency and damping factor. A reasonable choice would be an undamped natural

frequency !n = 25 rad/s (corresponding to 4 Hz) and a damping factor = :71. Thus

Kp = !2n = 632; (5:9)

Kd = 2!n = 35:7: (5:10)

The Bode plot of the cascaded plant and controller is shown in Figure 5.2.


-50

0

50

100

100 101 102 103 104

Frequency (rad/sec)

Gai

n dB

-180

-160

-140

-120

-100

-80

100 101 102 103 104

Frequency (rad/sec)

Phas

e de

g

Figure 5.2: Bode Plot of the Cascaded PD Controller and Plant (Open-Loop)

τ γ γ

γ

w

n

γ1__

I s2

m

d

(disturbance)

(noise)

C(s)−es + +

+

+

Figure 5.3: Block Diagram of the Closed-Loop Robust Control System


5.2.2 A Robust State Feedback Controller Based on the

Internal Model Principle

The closed-loop system is shown in Figure 5.3. We will assume that we have only an

estimate of the joint's inertia I denoted as I. The goal is that the system be able

to track the reference input des(t) and to reject a possibly persistent disturbance w

in the face of uncertainty in the knowledge of the system's inertia and measurement

noise. This is known as the robust servomechanism problem.

Theory

The rst method we use is based on the Internal Model Principle of control theory

[Francis and Wonham, 1976]. This principle states that a regulator synthesis (i.e.

the whole closed-loop system) is structurally stable only if the controller utilizes

feedback of the regulated variable, and incorporates in the feedback path a suitably

reduplicated model of the dynamic structure of the exogeneous signals which the

regulator is required to process. A system is said to be structurally stable if it remains

stable for slight variations in its state-space model parameters.

Although this principle is important in theory, it has a major drawback in practice:

it is impossible to have a priori knowledge of the eect of uncertainty and disturbance.

In fact, these may well be stochastic signals. Hence, attempting to build an inter-

nal model of a disturbance may have limited success in that context. Nevertheless,

[Kuo and Wang, 1991] demonstrated the usefulness of the method when the uncer-

tainty, which combines the eect of uncertainty in the robot's dynamic model and a

disturbance at the output, is assumed to be constant ( _e = 0) but of unknown magni-

tude. In this case, the resulting controller is just a classical PID controller, that is, an

integral term is added to the usual state feedback controller. This fact was pointed

out in [Desa and Roth, 1985].


The approach taken by Kuo and Wang is as follows. First decouple and linearize

the system, making it second-order in each decoupled direction in task space by a

proper choice of feedback gains Kp and Kd. Then apply the robust linear control

on the linearized system, which amounts to adding an integrator on the error in the

feedback when a constant disturbance model is used.

The internal model approach is attractive because it is possible to set the rate at

which the error will be driven to zero by selecting the poles of a linear subsystem.

We lose this advantage if we only merely try to put integrators with gains in the

compensator. Here, we follow the approach taken by Kuo and Wang to design a

PID controller for our simple system, i.e. we will assume that the uncertainty signal

combining the uncertainty in the plant's parameters and disturbance is a constant.

Referring to Figure 5.3 for the notation, we rst apply a state feedback to equation

5.1,

I = I(Kd _ m Kp m +Kpu); (5:11)

where u is a linear input.

The reference trajectory is known and we dene ud with the following equation:

Kpud = des +Kd _ des +Kp des: (5:12)

Now by subtracting equation 5.12 from equation 5.11, we get

Kp(u ud) =I

I des +Kd( _ m _ des) +Kp( m des): (5:13)

Dene y = des as the tracking error and v = uud as the input from the robust

part of the controller. Also, let I = I +I. We have

Kpv = y I

Iw +

I

I +Kd( _ + _n _ des) +Kp( + n des); (5:14)

Kpv = y +Kd _y +KpyIIw +

I

I +Kd _n+Kpn| z e

; (5:15)


y +Kd _y +Kpy = Kpv e: (5:16)

The uncertainty is

e = Kd _n+Kpn I

Iw +

I

I : (5:17)

Equation 5.17 tells us that for a small I and low-power noise, a slowly-varying e

implies that the disturbance w is also slowly-varying, since w must have its energy at

low frequency. The assumption that e be constant may be justied for slowly-varying

disturbances as well, which for example may be due to friction. In other words,

integral control works also well for slowly-varying uncertainty. Equation 5.16 can be

rewritten in state-space form.

Let x =

264 y

_y

375 : Then,

_x =

264 0 1

Kp Kd

375

| z A

x+

264 0

Kp

375

| z B1

v +

264 0

1

375

| z B2

e; (5:18)

y = [1 0]| z C

x: (5:19)

We dene the following quantities:

z = _x; (5:20)

s = _v: (5:21)

Take the vector z to be a new state vector, and s to be a new input. Dierentiating

(5.20) and using (5.19) and (5.21) yields

_z = A _x+B1 _v +B2 _e| z 0

; (5:22)

_z = Az + B1s; (5:23)


_y = Cz: (5:24)

Now, let be a new state vector formed as follows:

=

2666664y

_y

y

3777775 =

264 y

z

375 : (5:25)

The following state and output equations can be obtained:

_ =

26666640 1 0

0 0 1

0 Kp Kd

3777775

| z G

+

2666664

0

0

Kp

3777775

| z H

s; (5:26)

y = [1 0 0]| z F

: (5:27)

The output of this system, y, is what we want to drive to zero at a desired rate.

We now show that this system is controllable. We form the controllability matrix C

and verify that it has full rank.

C = [H GH G2H]; (5:28)

C =

2666664

0 0 Kp

0 Kp KdKp

Kp KdKp K2dKp K2

p

3777775 : (5:29)

Since C is lower triangular with nonzero entries in the diagonal, it has full rank, and

thus the system is controllable. Hence we can apply a state feedback to select the

closed-loop poles at will, thereby specifying the rate at which y will be reduced to 0.

Let the gain matrix be [K1 K2 K3]. Then

s = K1y K2 _y K3y: (5:30)


But

v =Z t

0s()d = K1

Z t

0y()d K2y K3 _y: (5:31)

We see that v represents a PID control input. The overall control is

u = K1

Z t

0y()d K2y K3 _y +

1

Kp des +

Kd

Kp_ des + des; (5:32)

= I[Kd _ m Kp m + des +Kd _ des +Kp des KpK2( des)

KpK3( _ _ des)KpK1

Z t

0( () des())d]: (5.33)

Since and _ are unknown, we use their measurement m and _ m in the control

signal . Collecting the gains together, we nd the familiar PID control law plus a

feedforward term

= I[ des + ~Kp( des m) + ~Kd( _ des _ m) + ~Ki

Z t

0( des() m())d]; (5:34)

where ~Kp = Kp +KpK2;

~Kd = Kd +KpK3;

~Ki = KpK1:

Controller Design

We now procede to compute the controller parameters for the real system. The

principal guideline is to keep the poles of the robust part to the left of the ones

introduced for the mass-spring-damper part. The system dynamics are described by

equation 5.1, and we rst apply a PD controller to this system as shown in Figure

5.4.

We pick Kp=25.3 and Kd=7.14 such that the undamped natural frequency is .8

Hz and the damping factor is = :71, the latter so chosen as to obtain a maximally


K s

K

1__

u

I I s

p

d

2

γτ

+

− +

−

^ m

w + n

++

Figure 5.4: Block Diagram of the Decoupling Part of the PID Robust Controller

at magnitude of the frequency response. Now using equation 5.26,

_ =

26666640 1 0

0 0 1

0 25:3 7:14

3777775

| z G

+

2666664

0

0

25:3

3777775

| z H

s; (5:35)

y = [1 0 0]| z F

: (5:36)

The state vector contains the trajectory error y which we want to drive to

zero. We now select feedback gains [K1 K2 K3] to get real closed-loop poles at 1 Hz,

p1 = p2 = p3 = 6:28. The desired characteristic polynomial is then calculated as

q(s) = (s+ 6:28)3 = s3 + 18:85s2 + 118:4s + 248:1: (5:37)

The characteristic polynomial of the inner PD control loop is

p(s) = s3 + 7:14s2 + 25:3s: (5:38)


yyyyvK

K

K

K

K

K

p

d

p

1

2

3

−

−−

+

−−

ud +

s s s1 1 1’’’

Figure 5.5: State Space Diagram of the PID Robust Control System

By referring to Figure 5.5, we see that the feedback gains must be picked such

that

K1Kp = 248:1;

K2Kp = 118:4 25:27 = 93:13;

K3Kp = 18:85 7:14 = 11:71:

And nally, the PID gains are computed

~Kp = Kp +KpK2 = 25:27 + 93:13 = 118:4;

~Kd = Kd +KpK3 = 7:14 + 11:71 = 18:85;

~Ki = KpK1 = 248:1:

Note that the resulting PID gains are equal to the coecients of the desired

characteristic polynomial in this case. The control is given by

= I[ des+118:4( des m)+18:85( _ des _ m)+248:1Z t

0( des() m())d]: (5:39)

The Bode plot of the cascaded plant and controller is shown in Figure 5.6 while the


-50

0

50

100

150

10-1 100 101 102

Gai

n dB

Frequency (rad/sec)

Bode Plot of C(s)G(s), Kp=118.4 Kd=18.85 Ki=248.1

-300

-250

-200

-150

-100

-50

10-1 100 101 102

Frequency (rad/sec)

Phas

e (d

eg)

Figure 5.6: Bode Plot of the Cascaded PID Controller and Plant (Open-Loop)

Nyquist plot is shown in Figure 5.7. From the latter gure, it is not possible to con-

clude that the system is stable because it does not show the loops at innity, but the lo-

cus actually crosses the real axis twice clockwise to the left of the critical point (at -1)

and hence the system is stable (see [Vidyasagar, Bertschmann and Sallaberger, 1988]).

The gain margin is about -18 dB while the phase margin is close to 70. Therefore

the system should show robustness against unmodeled phase lag but not in the face

of uncertainty in the joint's inertia, especially if it is underestimated.

5.2.3 An H1-Optimal Robust Controller

Theory

The second method investigated for the robust position control of the revolute joint

is the H1 sensitivity minimization method which originated in the work of Zames in


-25

-20

-15

-10

-5

0

5

10

15

20

25

-30 -25 -20 -15 -10 -5 0

Real Axis

Imag

Axi

s

Nyquist Plot of Cascaded PID Controller and Plant

Figure 5.7: Nyquist Plot of the Cascaded PID Controller and Plant (Open-Loop)

[Zames, 1981]. The problem is the following: given a single-input, single-output linear

time-invariant plant aected by external disturbances and of which we have only an

approximate model, can we design a controller such that the maximum value of the

weighted sensitivity of the closed-loop system on the j!-axis is minimized? This

corresponds to a minimax problem of the sensitivity function which is the transfer

function from the eect of perturbations in plant's parameters and disturbance to the

output. A block diagram of the closed-loop system where the disturbance is taken to

be the main input is shown in Figure 5.8. The transfer function from w to is easily

calculated to be:

Hw(s) =1

1 + C(s)G(s): (5:40)

The transfer function from the input des to the output is simply

H(s) =C(s)G(s)

1 + C(s)G(s): (5:41)


G(s) C(s)

γ

γ

w

n

dτ

+

++ +

−

+

γ−

es

Figure 5.8: Block Diagram of the H1-Optimal Control System

Now the sensitivity S(s) is dened as follows:

S(s) =@H(s)=H(s)

@G(s)=G(s): (5:42)

and carrying out the calculations we nd

S(s) =1

1 + C(s)G(s); (5:43)

which is identical to Hw(s). Thus we see that minimizing the weighted sensitivity has

two advantages: it desensitizes the system from variations in the plant's parameters

and it attenuates the eect of the disturbance (see [Zames and Francis, 1983]).

The H1 method often leads to a conservative design in terms of tracking perfor-

mance since it is the solution to a worst-case problem. Nevertheless, the controller

will guarantee great robustness as long as the disturbance aecting the system lies in

the L2 space, i.e. if w(t) is a nite-energy signal.

In our previous design, we assumed that the eect of uncertainty and disturbance

was a constant signal although we recognized the fact that the resulting PID controller

works also well for slowly-varying disturbances. Here we have to assume that the


disturbance is varying but eventually dies out. This is not so bad an assumption if

we consider that the revolute joint is programmed to follow a trajectory and then

stops, which is often the case in robotics. Then the disturbance would come mainly

from dynamic eects like viscous friction in the bearings or other unmodeled dynamics

while the joint follows its trajectory.

It is important to note that an H1-optimal compensator is often not practical if

the weighting function is not chosen carefully, because it might not take into account

the limitations imposed by the actuator dynamics and the sensor noise. For example,

anH1 design sometimes results in a closed-loop magnitude that does not go to zero as

! tends to innity, thereby demanding an enormous amount of power to be delivered

at high frequencies. Nevertheless, an H1-optimal design can be a good starting point

if the weighting function is chosen carefully and if the high frequency behavior can

be xed without disturbing too much the sensitivity function in the passband.

First of all, let us recall the so-called interpolation conditions that are required for

internal stability of the closed-loop system, assuming that the compensator C(s) and

the plantG(s) are represented by minimal realizations. These interpolation conditions

are directly derived from the internal stability conditions stating that S(s), G(s)S(s)

and H(s)=G(s) must all be stable, and they must be satised when G(s) has poles

or zeros in the closed right-half plane.


condition 1:

If G(s) has a closed right-half plane pole of multiplicity m at p0, S(s) must have a

zero of same multiplicity at the same location.

S(p0) =dS(s)

ds

s=p0

=d2S(s)

ds2

s=p0

= : : : =dm1S(s)

dsm1

s=p0

= 0: (5:44)

condition 2:

If G(s) has a closed right-half plane zero of multiplicitym at z0, then S(s) must equal

one at that point, and the (m 1) rst derivatives of S(s) evaluated at z0 must be

zero.

S(z0) = 1; (5:45)

diS(s)

dsi

s=z0

= 0; i = 1; : : : ;m 1: (5:46)

For a plant with no closed right-half plane poles and zeros, the design of the

compensator is as follows: pick a desirable sensitivity function (or, equivalently, a

closed-loop transfer function) and calculate the controller from

C(s) =H(s)

S(s)G(s): (5:47)

If we assume that we have perfect actuators and perfect sensors, we see that we

have total freedom over the choice of S(s) (or H(s)). For example, we could use an

H1-optimal sensitivity function corresponding to a certain weighting function W (s)

which will be described later. For the case where G(s) has closed right-half plane

poles or zeros, the interpolation conditions pose constraints on the choice of S(s).

Now assume we pick a minimum-phase, proper, rational weighting function W (s)

and we seek to minimize the H1 norm of the weighted sensitivity S(s)W (s):

minimize = kSWk1 = sup!jS(j!)W (j!)j: (5:48)

We have to use a weighting function because for a strictly proper plant G(s), the H1

norm of the sensitivity is always greater or equal to 1 (see [Zames and Francis, 1983])


as we now show.

lim!!1

S(j!) = 1 lim!!1

G(j!)

"C(j!)

1 + C(j!)G(j!)

#= 1; (5:49)

since G(s) is strictly proper and C(s)=(1 + C(s)G(s)) is proper. Therefore

kSk1 1: (5:50)

By using a weighting function, we can make jS(j!)j small at the frequency bands

of interest by choosing a large jW (j!)j at those frequencies. This fact comes from

the nature of the optimal weighted sensitivity which is a Blaschke product:

S(s)W (s) = kBG(s) B(s): (5:51)

This result is derived from functional analysis. The Blaschke product BG(s) is formed

with the open RHP poles of G(s). Suppose these poles are denoted as pi, i = 1; : : : ; r,

then BG(s) is

BG(s) =rYi=1

s+ pis+ pi

: (5:52)

Note that BG(s), like any Blaschke product, is stable and all-pass, i.e. jBG(j!)j = 1.

The Blaschke product B(s) is needed only if the plant has RHP zeros which is not

the case for the revolute joint. Therefore, B(s) = 1 for our purposes and the constant

k can be made as small as desired.

It should be noted that the product S(s)W (s) is in H1 even though W (s) does

not necessarily belong to this space. The restriction is, if G(s) has poles on the

j!-axis, W (s) must have these poles with at least the same multiplicity; this is just

a restatement of the rst interpolation condition when equation 5.51 holds: S(s)

is required to have cancelling zeros for these poles but since the magnitude of the

Blaschke product in the right-hand side of (5.51) is 1 on the j!-axis, W (s) must be

innite at those points. This guarantees that the product S(s)W (s) belongs to H1

as long as the weighting function W (s) is not allowed to have poles or zeros in the

open right-half plane.


Controller Design

Now we are ready to design a desirable optimal sensitivity function by picking a

suitable weightingW (s). First we notice that jW (j!)j inverts jS(j!)j and that W (s)

must have two poles at s = 0 since the plant model is G(s) = 1=Is2. We also observe

that BG(s) = 1 because G(s) has no open RHP poles. Hence S(s) = k=W (s), so

that the sensitivity is just the inverse of the weighting function. Knowing this and

recalling that the closed-loop transfer function possess the same poles as the sensitivity

function, reasonable zeros are selected for W (s) which become the closed-loop poles.

Finally, after a few trials, we chose W (s) to be:

W (s) =(s=!a + 1)3

s2(s=b+ 1): (5:53)

The magnitude of this weighting function is plotted in Figure 5.9. The corre-

sponding optimal sensitivity is

S(s) =ks2(s=b+ 1)

(s=!a + 1)3: (5:54)

The gain k is now adjusted such that S(j!) ! 1 as ! ! 1 to make sure that

jH(j!)j rolls o at high frequencies as fast as the second order plant. Thus the

actuators will not be delivering too much power at high frequencies.

S(j!) =k!2(j!=b + 1)

(j!=!a + 1)3; (5:55)

lim!!1

S(j!) =k!3

a

b= 1; (5:56)

k =b

!3a

: (5:57)

We now calculate the closed-loop transfer function H(s). Its Bode plot is shown

in Figure 5.10.

H(s) = 1 S(s) =(s=!a + 1)3 b

!3a

s2(s=b+ 1)

(s=!a + 1)3: (5:58)


-40

-30

-20

-10

0

10

20

30

40

10-1 100 101 102 103

freq. (rad/s)

Magnitude of W(s)

(dB

)

w = 15 rad/sa

b = 45 rad/s

Figure 5.9: Magnitude of the Weighting Function W (s) on the j!-axis

We pick b = 3!a, hence k = 3=!2a, and

H(s) =3s=!a + 1

(s=!a + 1)3: (5:59)

A low sensitivity at frequencies below !a and a well-behaved closed-loop frequency

response are obtained. The sensitivity is

S(s) =3s2(s=3!a + 1)

!2a(s=!a + 1)3

: (5:60)

We now proceed to obtain the controller C(s):

C(s) =H(s)

G(s)S(s); (5:61)

C(s) =I!2

a(3s=!a + 1)

3(s=3!a + 1): (5:62)

C(s) turns out to be a classical lead controller. This may seem surprising at rst, but

the H1 procedure, being the solution to a worst-case problem, bears some analogy


-80

-60

-40

-20

0

20

10-1 100 101 102 103

Frequency (rad/sec)

Gai

n dB

-200

-150

-100

-50

0

10-1 100 101 102 103

Frequency (rad/sec)

Phas

e de

g

Bode Plot of H(s)

Figure 5.10: Bode Plot of the Closed-Loop Transfer Function H(s)

with the classical approach using the gain and phase margins, which is also a worst-

case approach (see [Zames and Francis, 1983]). But, of course, it is clear that the

structure of the controller always depends on the choice of the weighting function

W (s).

The controller must be transformed to the z-domain because it will be imple-

mented on a digital computer. For that purpose, we use the bilinear transformation

because it is simple and yields a controller whose frequency response on the unit disc

closely matches the frequency response of the analog controller.

s 2(1 z1)

Ts(1 + z1); (5:63)

where Ts is the sampling period. The expression for the discrete-time controller is

C(z) =I!2

a(6 + !aTs)(1 +!aTs6!aTs+6

z1)

(2 + 3!aTs)(1 +3!aTs23!aTs+2

z1)

: (5:64)


The Bode plot of the cascaded plant and (continuous-time) controller is shown in

Figure 5.11 with !a = 15rad/s while the Nyquist plot is shown in Figure 5.12. The

Nyquist locus does not cross the real axis to the left of the critical point and the

system is thus stable. It can also be observed that the phase margin is a little more

than 50 while the gain margin is innite, which shows good robustness properties.

The sensitivity magnitudes of the three controllers developed so far are plotted

in Figure 5.13, and it can be seen that, paradoxically, the H1-optimal controller

yields the worst sensitivity magnitude of the three designs. This is due to its low

bandwidth set by !a = 15 rad/s. The lead controller could not be applied at higher

frequencies on the real system without causing the actuator force loops to break into

limit cycles. More interestingly, it can be noticed that, even though the H1-optimal

design displays the poorest sensitivity magnitude, it is, on the other hand, more

robust to variations in the joint's inertia than the PID design because it has an innite

gain margin. This is an important remark since the joint's inertia is likely to vary

signicantly in many applications. Furthermore, it reveals what may be a weakness

of the sensitivity magnitude minimization approach which does not seem to capture

all the information contained in the classical gain and phase margins approach.

5.3 A Variable Bandwidth, Nonlinear Controller

In Chapter 2, it was shown that the actuator dynamics exhibit four types of nonlin-

earities, namely valve hysteresis, valve static force characteristic, hydraulic damping,

and friction. Nonlinearities in a system are often seen as major limitations for con-

trol system performance, but here we have adopted a more positive approach and

tried to see what benets could be obtained from the nonlinear hydraulic damping

characteristic.


-100

-50

0

50

100

10-1 100 101 102 103

Frequency (rad/sec)

Gai

n dB

-180

-160

-140

-120

10-1 100 101 102 103

Frequency (rad/sec)

Phas

e de

g

Bode Plot of C(s)G(s)

Figure 5.11: Bode Plot of the Open-Loop Transfer Function C(s)G(s)

-8

-6

-4

-2

0

2

4

6

8

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

Real Axis

Imag

Axi

s

Nyquist Plot of C(s)G(s)

Figure 5.12: Nyquist Plot of the Open-Loop Transfer Function C(s)G(s)


-120

-100

-80

-60

-40

-20

0

20

10-1 100 101 102 103

(dB

)

frequency (rad/s)

Sensitivity Function Magnitudes for the Three Controller Designs

PD : _

PID : - -

H-infinity : ...

Figure 5.13: Sensitivity Magnitudes for the PD, PID and H1 Control Systems

Looking back at Figure 2.9, we can see that one of the interesting features of the

family of curves is that the characteristic is shifted to the right as the valve opens up.

This has the eect of reducing the hydraulic damping force at constant velocity and

thus it increases the open-loop force bandwidth, possibly allowing the use of higher

feedback gains.

In robotics, it is desirable to maintain a high force bandwidth as contact with the

environment occurs, but this bandwidth is often limited by the actuators themselves

since the inertial eects of the structure get negligible after contact. With the ASI

hydraulic actuators, it appears possible to modulate the bandwidth by using the

valve position, but altering xv of course changes the output force. This is where the

antagonistic nature of the redundantly-actuated joint comes into play: given a desired

torque, an innite number of solutions is available for the actuator forces. If a solution

vector with high force magnitudes is picked, the valves will open up, thereby shifting


the damping characteristics to the left and extending the open-loop bandwidth.

It is tempting to try to establish an analogy with a pair of antagonistic muscles ac-

tuating a human joint: for fast movements, the antagonistic muscles generally operate

at high forces which, for the most part, cancel each other (see [Gottlieb et al., 1990]),

i.e. a signicant component of the vector of forces is in the nullspace of the transposed

jacobian which maps forces to output torque.

However, the Hill dynamic model of the muscle, used by a large number of re-

searchers in biomechanics (see for example [Winters, 1990]), assumes that the non-

linear damping term representing the viscosity of a contracting muscle varies with a

fractional power of the velocity, whereas in our model it grows exponentially up to

a certain velocity. Furthermore, the elasticity of muscles is their dominant property,

not the damping. Hence, the analogy can not be carried any further since hydraulic

actuators and muscles possess quite dierent macroscopic properties.

5.3.1 Implementation

The variable bandwidth controller can be implemented very easily and is in fact just

an enhancement of any of the three linear controllers already developed for the revo-

lute joint. The only additional computation to be done is to calculate the nullspace

of the 1x2 transposed jacobian JT . An expression for a unit vector n lying in N (JT )

was given in the previous chapter (equation 4.31).

After calculations of the control torque and either the optimal 2-norm or 1-

norm control forces have been completed by the linear controller, the nonlinear part

calculates a vector of forces lying in the nullspace of the transposed jacobian. These

two force vectors are then added together to produce the actual vector of desired

actuator forces.

The choice of the norm Fn of the vector of forces lying in N (JT ) which is one-

dimensional can be based on the following heuristic: pick a magnitude Fn such that


the absolute values of the resulting control forces are equal to, or greater than, some

force Fa, i.e.

pick Fn 2 R such that min(jfc1 + Fnn1j; jfc2 + Fnn2j) = Fa; (5:65)

where fc1 and fc2 are the control forces calculated by the linear controller.

With this, we make sure that the valves operate at or above the desired opening

specied by the lower force limit Fa since we know the static relationship between F

and xv.

To implement the above scheme on a digital computer, suppose n has already

been calculated and the lower bound on the absolute values of the actuator forces is

Fa. We seek a magnitude Fn such that the following inequalities are both satised

with one of the two actually becoming an equality.

jfc1 + Fnn1j Fa; (5:66)

jfc2 + Fnn2j Fa: (5:67)

First, we check if jfc1j and jfc2j are both greater than Fa. If it is the case, no

change to the vector of control forces is required as it already satises equations 5.66

and 5.67 with Fn = 0. Otherwise the following Fni's and Fn are computed.

Fn1 =Fa fc1n1

; Fn2 =Fa fc1

n1; Fn3 =

Fa fc2n2

; Fn4 =Fa fc2

n2;

Fn = maxi=1;:::;4

(Fni): (5:68)

5.4 Impedance Control of the Joint

In this section, we will consider the impedance control problem of the parallel revolute

joint, and more specically, we will analyze the compliance provided by the three

linear position controllers discussed in Section 5.2. Let us rst consider the PD


γ

τ

ρ

ρ+ +

−+

−+

KK

IJoint and Actuator

DynamicsForce

Optimizer

f

Kinematics

γ

γ

(J J) JT −1^ ^ ^T

dp

^ −

−

−

K( )^

0

τd

des

+

+

^

^

Figure 5.14: Block Diagram of the PD Control System Seen as an Impedance ControlSystem

control law. The impedance of a mechanical system is dened as the transfer function

between an external force or torque input and a linear or angular velocity output. If

we look at the PD controller as an impedance controller, the block diagram of this

new control system with an external torque d as input and the angular velocity _ as

output can be drawn as shown in Figure 5.14.

Assuming that the models are exact, that is

K() K(); (5.69)

J J ; (5.70)

I I; (5.71)

and neglecting the actuator dynamics, we can calculate the transfer function between


s (s) and d(s), which turns out to be the second-order impedance of a mass-spring-

damper system.s (s)

d(s)=

1Is

s2 +Kvs+Kp: (5:72)

This impedance has the most useful form and its damping factor and undamped

natural frequency can be adjusted at will with Kp and Kd.

Now suppose we add an integral term ~Ki to get a PID compensator. We show

that the resulting impedance gets undesirably sti .

s (s)

d(s)=

1Is2

s3 + ~Kvs2 + ~Kps+ ~Ki

: (5:73)

If d(s) is a constant torque disturbance (d(s) = d=s), the nal value of is:

limt!+1

(t) =

"dIs

s3 + ~Kvs2 + ~Kps+ ~Ki

#s=0

= 0: (5:74)

This represents an innitely sti system at low frequencies. Hence, in the passage

from free space motion to contact, the integral term should be disabled to regain

the compliance oered by the PD control law for a smooth interaction with the

environment.

Finally, we calculate the impedance oered by a lead controller assuming the

kinematic and dynamic models are exact. Suppose the lead compensator is written

as

C(s) =IKc(s=a+ 1)

s=b+ 1; b > a: (5:75)

Then the corresponding impedance is

s (s)

d(s)=

1Is(s=b+ 1)

1bs3 + s2 + Kc

as+Kc

: (5:76)

If d(s) is a constant torque disturbance (d(s) = d=s), the nal value of is:

limt!+1

(t) =

"dI(s=b+ 1)

1bs3 + s2 + Kc

as+Kc

#s=0

=d

IKc: (5:77)


Thus the static compliance is 1IKc

. We see that equation 5.76 is similar to the

one obtained with the PD controller when b is large enough, hence the system should

behave nicely as contact with the environment occurs. This is not surprising since at

the limit when b tends to +1, the lead controller degenerates into a PD controller.

For the previously developed H1 lead compensator, the impedance is

s (s)

d(s)=

1Is( 1

3!as+ 1)

13!a

s3 + s2 + !as+!2a3

: (5:78)

Neglecting the term in s3 in the denominator, the damping factor of the resulting

second-order system isp3=2 = 0:866, so that the response to an external torque

disturbance is nicely damped, without showing any overshoot. Moreover this damping

factor is independent of the choice of the frequency !a. However the static compliance

does depend on !a since its value is3I!2a

.

5.5 Experimental Results

In this section, experimental results are shown and compared for dierent position,

impedance and force control experiments using the controllers developed so far.

In all the position control experiments, the trajectories are composed of linear

segments with parabolic blends, except in the step response experiments. The main

property of these trajectories is that the angular acceleration is discontinuous at

the junctions between the linear and parabolic segments. In most manipulators,

these trajectories would induce undesirable structural vibrations since the control

torques would also be discontinuous if the desired angular acceleration is used as a

feedforward term. But the parallel joint has almost all of its exibility lumped in

the force sensors, thus the vibrations can be measured and controlled. Hence the use

of linear trajectories with parabolic blends seems more interesting for performance

assessment. Furthermore, the actuators force bandwidth is sucient to allow fast


changes in the control torque signal.

5.5.1 Position Control Experiments

The rst set of trajectory tracking experiments involves the PD controller developed

in Section 5.2.1. The angular velocity signal is calculated from the linear actuator

velocities by using the jacobian. These actuator velocities are calculated as the sum

of analog derivatives of the signals coming from the lvdts and digital derivatives of

the force sensor de ections obtained by dividing the measured forces by the corre-

sponding sensor stiness. The digital dierentiators are implemented as sixth-order

FIR lters. The actuator velocities had to be ltered at 10 Hz, otherwise it would

excite a resonance around 60 Hz to 80 Hz in the actuator closed-loop force responses.

A second-order digital all-pole low-pass lter with a double pole at z = :9391 elim-

inated that instability problem. The two dierent optimal vector of actuator forces

are used and the inertia I is set to two dierent values in order to test the closed-loop

robustness. Figure 5.15 shows the angle calculated from the actuator displacement

measurements using the calibrated direct kinematics. The force sensor de ections

xfs1 and xfs2 are also taken into account in 1m and 2m, i.e. im = LV DT xfsi,

where xfsi = Fsi=Ksi; i = 1; 2. The 2-norm optimal vector of forces is calculated in

real time, and a sampling frequency of 1000 Hz is used, for this and all the other

experiments. Also, the actuator force controllers are the analog lead-lag compensator

presented in Chapter 2.

The tracking performance shown in Figure 5.15 is quite satisfying, although there

are small jumps in the angle measurement at the passage from a linear to a parabolic

segment. This is due to the sudden application of high forces (see Figure 5.18) coming

from the feedforward of the angular acceleration. These forces create a torque on the

shaft linking the actuators to the moment arm, and since it was a bit loose, the

actuators could travel almost freely for a very short time, hence creating the illusion


-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5

PD controller

join

t ang

le g

amm

a (r

ad)

time (s)

--- : desired trajectory

Figure 5.15: PD control of the angle : I = I = 0:71, 2-norm optimal vector of forces

that the joint angle undergoes a fast change on the plots. Figure 5.16 shows the

actuator lengths 1 and 2 where we can also see these abrupt changes. This eect is

analogous to backlash, however it depends on the joint angle and on the magnitude

of the forces applied by the actuators.

The torque applied at the joint and shown in Figure 5.17 is found using equation

4.1 where the jacobian is calculated with the calibrated kinematic parameters. The

dashed curve represents the computed control torque, and it is also the noisiest of

the two plotted signals. We see that the actual torque follows the desired torque

reasonably well although it does not quite reach the peaks, and drifts a bit past

t = 1 s. It is easier to see where the error comes from by looking at the actuator

forces in Figure 5.18.

We see that the actuators do not respond fast enough to the sharp peaks, and we

also observe that the drifting is due to actuator 2, but it is unclear what causes it. It

was shown in Chapter 2 that when the hydraulic actuators are position constrained,


0.222

0.224

0.226

0.228

0.23

0.232

0.234

0.236

0 1 2 3

time (s)

actua

tor le

ngth

rho1 (

m)

PD controller

0.22

0.225

0.23

0.235

0.24

0.245

0 1 2 3

PD controller

time (s)

actua

tor le

ngth

rho2 (

m)

Figure 5.16: Actuator lengths 1 and 2: PD controller, I = I = 0:71, 2-norm optimalvector of forces

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 0.5 1 1.5 2 2.5

time (s)

torq

ue ta

u (N

m)

PD controller

--- : desired control torque

Figure 5.17: Joint torque : PD controller, I = I = 0:71, 2-norm optimal vector offorces


-250

-200

-150

-100

-50

0

50

0 1 2 3

time (s)

actua

tor 1

force

(N)

PD controller

--- : desired force

-200

-150

-100

-50

0

50

100

0 1 2 3

PD controller

actua

tor 2

force

(N)

time (s)

--- : desired force

Figure 5.18: Actuator forces: PD controller, I = I = 0:71, 2-norm optimal vector offorces

they can track fast desired force trajectories accurately, but when they move, hy-

draulic eects take place and might become signicant. Furthermore, when the force

control experiments were conducted in Chapter 2, only one actuator was actually

loading the pump. But in the position control experiments, both actuators move,

thereby demanding more ow from the pump, which might cause pressure drops in

the lines. One possible solution to this would be to put an accumulator close to each

valve.

In the next experiment the same PD controller is used, the only change is the real-

time optimization of forces: the 1-norm optimal vector of forces is tested. Figure

5.19 shows the tracking performance which is pretty close to the performance obtained

using the 2-norm optimal vector of forces.

Figures 5.20 and 5.21 show the control torque and forces respectively. The com-

puted and actual torques look like the ones measured in the previous experiment,

thus showing that (1) the computed jacobian is close to the true one, and (2) force

optimization is feasible and useful. Looking at the control forces, one may notice


-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5

join

t ang

le g

amm

a (r

ad)


PD controller

time (s)

Figure 5.19: PD control of the angle : I = I = 0:71, 1-norm optimal vector offorces

that their magnitudes are always equal, a property of the minimum1-norm vector

of forces that was pointed out in Section 4.2.2. It is also interesting to see how force

tracking gets better when an actuator gets close to being singular. For example, ac-

tuator 1 closely follows its desired force input at the beginning of the trajectory (rst

peak) where it is almost singular, but the tracking error gets larger around t = 1 s

(negative peak). Exactly the opposite happens for actuator 2. This phenomenon can

again be attributed to hydraulic eects (e.g. the nonlinear hydraulic damping eect

discussed in Chapter 2) and we can make the general statement that force tracking

performance degrades as the magnitude of the piston velocity grows larger.

In Section 4.2.3, a method to generate a pre-load force on the joint was developed

and experimental results are now presented for a desired pre-load force of 50 N pushing

along the moment arm towards the joint axis. It can be observed in Figure 5.22 that

there is a degradation in the angle tracking performance, although a large part of it

may be attributed to the previously discussed torque eect on the shaft linking the

actuators to the moment arm.


-10

-8

-6

-4

-2

0

2

4

6

8

10

0 0.5 1 1.5 2 2.5

time (s)

PD controller

torq

ue ta

u (N

m)


Figure 5.20: Joint torque : PD controller, I = I = 0:71, 1-norm optimal vector offorces

-200

-150

-100

-50

0

50

100

150

200

250

0 1 2 3

actua

tor 1

force

(N)

time (s)

PD controller

--- : desired force -250

-200

-150

-100

-50

0

50

100

150

200

0 1 2 3

time (s)

actua

tor 2

force

(N)

PD controller

--- : desired force

Figure 5.21: Actuator forces: PD controller, I = I = 0:71, 1-norm optimal vectorof forces


-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5

PD controller

time (s)

join

t ang

le g

amm

a (r

ad)


Figure 5.22: PD control: I = I = 0:71, 50 N pre-load force

Figures 5.23 and 5.24 show the actuator forces and the pre-load force on the joint.

We rst notice that actuator 2 started oscillating at high frequency around t = :8 s

for a short while which suggests that the linear lead-lag force controller might not be

very robust. It can also be noticed that the internal force is not accurate, but this

is not the main objective. Rather, the objective is to keep the force bias along the

moment arm of the same sign at all times to avoid backlash at the joint.

The closed-loop robustness of the PD control system is tested by changing the

joint's inertia to I = :36. The minimum 2-norm vector of control forces is used. The

angle is plotted in Figure 5.25, showing only a slight dierence from the response

obtained when I = I = 0:71. This is not surprising since the PD control system is

quite similar to the H1-optimal control system, thus it should possess a comparable

sensitivity, and hence be robust to perturbations in the inertial parameters.

The control torque and forces are plotted in Figures 5.26 and 5.27 respectively.

We now turn our attention to the robust position control problem as we will

compare experimental results obtained from the two designs developed in Sections


-250

-200

-150

-100

-50

0

50

100

150

200

0 1 2 3

time (s)

PD controller

--- : desired force

actua

tor 1

force

(N)

-400

-300

-200

-100

0

100

200

0 1 2 3ac

tuator

2 for

ce (N

)

time (s)

PD controller

--- : desired force

Figure 5.23: Actuator forces: PD controller, I = I = 0:71, 50 N pre-load force

0

20

40

60

80

100

120

0 0.5 1 1.5 2 2.5

pre-

load

forc

e (N

)

PD controller

time (s)

--- : desired pre-load force

Figure 5.24: Pre-load force: PD controller, I = I = 0:71, 50 N pre-load force


-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5

time (s)

PD controller

join

t ang

le g

amm

a (r

ad)


Figure 5.25: PD control of the angle : I = 0:71, I = 0:36, 2-norm optimal vector offorces

-6

-4

-2

0

2

4

6

0 0.5 1 1.5 2 2.5

PD controller

time (s)

torq

ue ta

u (N

m)


Figure 5.26: Joint torque : PD controller, I = 0:71, I = 0:36, 2-norm optimal vectorof forces


-140

-120

-100

-80

-60

-40

-20

0

20

40

0 1 2 3

PD Controller

time (s)

actua

tor 1

force

(N)

--- : desired force-120

-100

-80

-60

-40

-20

0

20

40

60

0 1 2 3

actua

tor 2

force

(N)

PD Controller

time (s)

--- : desired force

Figure 5.27: Actuator forces: PD controller, I = 0:71, I = 0:36, 2-norm optimalvector of forces

5.2.2 and 5.2.3, namely the robust PID controller and the H1-optimal controller.

The rst controller to be examined is the PID, which has the same digital lter on

the actuator velocities as the PD controller. The desired and actual trajectories d

and are shown in Figure 5.28. The 2-norm optimal vector of forces is implemented

for this experiment.

The response looks a bit sluggish but that is to be expected since the open-

loop crossover frequency of the PID control system is less than that of the PD, but

comparable to that of the H1 design. The control torque and forces are shown in

Figures 5.29 and 5.30 respectively.

The same experiment is repeated with the H1-optimal controller and the trajec-

tories are plotted in Figure 5.31. We notice that the tracking performance is perhaps

slightly better in this case, but it is not as good as the performance attained with

the PD controller. This is due to the unstable behavior of the actuators when the

frequency !a in the H1-optimal controller is raised beyond 15 rad/s: the bandwidth

of the H1 control system is limited to this frequency whereas the bandwidth of the


-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5

Robust PID controller

time (s)

join

t ang

le g

amm

a (r

ad)


Figure 5.28: PID control of the angle : I = I = 0:71, 2-norm optimal vector offorces

-8

-6

-4

-2

0

2

4

6

8

0 0.5 1 1.5 2 2.5


time (s)

torq

ue ta

u (N

m)


Figure 5.29: Joint torque : PID controller, I = I = 0:71, 2-norm optimal vector offorces


-200

-150

-100

-50

0

50

0 1 2 3

Robust PID Controller

time (s)

actua

tor 1

force

(N)


-150

-100

-50

0

50

0 1 2 3

actua

tor 2

force

(N)

Robust PID Controller

time (s)

--- : desired force

Figure 5.30: Actuator forces: PID controller, I = I = 0:71, 2-norm optimal vector offorces

PD control system is approximately 25 rad/s (see Figure 5.2). The control torque

and the control forces are plotted in Figures 5.32 and 5.33. It can be noticed that

the force control signals sent to the actuators by the computer are much more noisy

in this experiment than in the previous one, but their magnitudes are smaller.

In the next trajectory tracking experiments with the two robust controllers, the

feedforward of the desired angular acceleration is disabled for a better comparison.

The PID-controlled angle is plotted in Figure 5.34 while the H1-controlled angle

is shown in Figure 5.35. The 2-norm optimal vector of forces is used.

Under these conditions, the PID controller performs better, and it is clear that

in this case the feedforward of the desired angular acceleration helps a lot for good

tracking of a trajectory, independently of the structure of the controller used.

In the discussion on the robustness of the PID control system in Section 5.2.2,

we pointed out that the system would not be robust in the face of uncertainty in the

inertia. Looking at the Bode plot of Figure 5.6, we see that if the inertia I is larger

than its estimated value I, then the open-loop gain is less than expected and the


-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5

Robust H-infinity controller

time (s)


join

t ang

le g

amm

a (r

ad)

Figure 5.31: H1 control of the angle : I = I = 0:71, 2-norm optimal vector of forces

-6

-4

-2

0

2

4

6

0 0.5 1 1.5 2 2.5

time (s)


torq

ue ta

u (N

m)


Figure 5.32: Joint torque : H1-optimal controller, I = I = 0:71, 2-norm optimalvector of forces


-200

-150

-100

-50

0

50

0 1 2 3

time (s)


--- : desired force

actua

tor 1

force

(N)

-140

-120

-100

-80

-60

-40

-20

0

20

40

0 1 2 3ac

tuator

2 for

ce (N

)


time (s)

--- : desired force

Figure 5.33: Actuator forces: H1-optimal controller, I = I = 0:71, 2-norm optimalvector of forces

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5


time (s)

join

t ang

le g

amm

a (r

ad)


Figure 5.34: PID control of the angle : I = I = 0:71, 2-norm opt. vector of forces,no feedforward


-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5


time (s)

join

t ang

le g

amm

a (r

ad)


Figure 5.35: H1 control of the angle : I = I = 0:71, 2-norm opt. vector of forces,no feedforward

crossover frequency is lower, which decreases the phase margin and could eventually

destabilize the system. We can observe this phenomenon in Figure 5.36, where the

estimated value of the inertia I = 0:71 is set to I = 0:2. No feedforward is used and

the 2-norm optimal vector of forces is chosen.

The next two experiments are designed to evaluate and compare the robustness of

the PID and the H1-optimal controllers for an overestimated inertia, that is I = 0:36

while I = 0:71. The trajectories are plotted in Figures 5.37 and 5.38. TheH1-optimal

controller clearly oers more robustness and better tracking performance under these

conditions.

In Section 5.3, a variable-bandwidth, nonlinear controller exploiting the hydraulic

damping nonlinear characteristic of the actuator was developed, and a set of step

response experiments are performed to test its eectiveness. The desired step d is

at the middle of the workspace and goes from 0.2 rad to 0.4 rad at t = 1 s. We rst

test the nonlinear controller combined to the PID, and the step responses with and

without the nonlinear controller are shown in Figure 5.39. It should be noted that


-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.5 1 1.5 2 2.5


time (s)

join

t ang

le g

amm

a (r

ad)


Figure 5.36: PID control of the angle : I = 0:2, I = 0:71, 2-norm opt. vector offorces, no feedforward

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5

time (s)


join

t ang

le g

amm

a (r

ad)


Figure 5.37: PID control of the angle : I = 0:71, I = 0:36, 2-norm opt. vector offorces


-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5


time (s)

join

t ang

le g

amm

a (r

ad)


Figure 5.38: H1 control of the angle : I = 0:71, I = 0:36, 2-norm opt. vector offorces

there is no feedforward of the desired angular velocity _ d (an impulse). The control

forces for the nonlinear controller combined to the PID and for the PID only are

plotted in Figures 5.40 and 5.41.

Looking at Figure 5.39, it is seen that the step response obtained with the nonlin-

ear controller rises slightly faster than the one obtained with the linear PID controller,

however the settling time of the former appears to be longer. It can also be observed

that the high forces produced by the nonlinear controller tend to create high stress on

the mechanism which results in undesirable discontinuities in the angle measurement.

Figure 5.40 clearly shows that the forces are always less than -200 N and that the

force tracking is quite good when the joint moves in the middle of its workspace.

The nonlinear controller is now combined to the H1-optimal controller and the

procedure is repeated. The step responses are plotted in Figure 5.42 and the control

forces are shown in Figures 5.43 and 5.44. Here it is not clear whether the rise

time is actually less for the nonlinear controller considering the distortion in the


0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.5 1 1.5 2 2.5

PID + Nonlinear Controller

time (s)

join

t ang

le g

amm

a (r

ad)

... : desired trajectory

--- : PID controller only

Figure 5.39: Nonlinear Controller + PID: I = I = 0:71, Fa = 200 N, 2-norm opt.vector of forces

-600

-500

-400

-300

-200

-100

0

100

0 1 2 3


time (s)

actua

tor 1

force

(N)

--- : desired force

-600

-500

-400

-300

-200

-100

0

0 1 2 3

actua

tor 2

force

(N)


time (s)

--- : desired force

Figure 5.40: Actuator forces: nonlinear controller + PID, I = I = 0:71, Fa = 200 N,2-norm optimal vector of forces


-150

-100

-50

0

50

100

150

200

250

0 1 2 3

PID controller

time (s)

actua

tor 1

force

(N)


-400

-300

-200

-100

0

100

0 1 2 3

actua

tor 2

force

(N)

PID controller

time (s)

--- : desired force

Figure 5.41: Actuator forces: PID controller only, I = I = 0:71, 2-norm optimalvector of forces

angle measurement. But again, the settling time is considerably shorter for the H1-

optimal controller alone and the step response it produces is better. In both cases the

computed control forces reach high magnitudes well beyond the saturation forces of

the actuators, but that is to be expected since the lead controller is essentially a digital

bandlimited dierentiator which acts on the error between d and . Consequently,

this error being discontinuous at t = 1 s, the computed control torque contains a

pulse that must be produced by the control forces.

Although it is demonstrated that the variable-bandwidth, nonlinear controller

could produce slightly shorter rise times in step responses, the benets obtained

are outweighted by the problems occuring because of the high forces applied on the

mechanism. A stronger, more rigid prototype of the mechanism for which higher

values of the lower bound on the force magnitudes Fa could be used would have to

be built in order to test further the properties of this controller.


0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.5 1 1.5 2 2.5

H-infinity + Nonlinear Controller

time (s)

joint

angle

gamm

a (rad

)

... : desired trajectory

--- : H-infinity only

Figure 5.42: Nonlinear Controller + H1: I = I = 0:71, Fa = 200 N, 2-norm opt.vector of forces

-600

-500

-400

-300

-200

-100

0

100

0 1 2 3


time (s)

actua

tor 1

force

(N)

--- : desired force

-3000

-2500

-2000

-1500

-1000

-500

0

0 1 2 3

actua

tor 2

force

(N)


time (s)

--- : desired force

Figure 5.43: Actuator forces: nonlinear controller + H1, I = I = 0:71, Fa = 200 N,2-norm optimal vector of forces


-200

0

200

400

600

800

1000

1200

0 1 2 3

H-infinity Controller

time (s)

actua

tor 1

force

(N)


-2000

-1500

-1000

-500

0

500

0 1 2 3

actua

tor 2

force

(N)


time (s)

--- : desired force

Figure 5.44: Actuator forces: H1-optimal controller only, I = I = 0:71, 2-normoptimal vector of forces

5.5.2 Impedance and Force Control Experiments

Unstable interactions between a robot and its environment have been a major source

of concern for many researchers in robotics. One way to deal with this class of

problems is to actively control the impedance of the manipulator [Whitney, 1977].

The impedance oered by the PD and the H1-optimal controllers was analyzed in

Section 5.4, and two experiments were designed to verify the theoretical results. The

two experiments are essentially the same as the position control experiments except

that they involve a desired trajectory which goes past a rigid object introduced in

the workspace. No special contact detection algorithm is implemented, thus, after

contact, the actuators just apply increasingly higher forces as the trajectory error

rises. The point of contact is at the end of the rod, about 0.49 m away from the joint

axis. The trajectory is plotted in Figure 5.45 for the PD control system. It can be

noticed that the environment (a long beam made of aluminum) bends signicantly

after contact is made. Nevertheless, no bouncing is apparent on this gure which is


quite satisfying. The forces are shown in Figure 5.46 and we can see that actuator 2

saturates.

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0 0.5 1 1.5 2 2.5

PD Controller

time (s)

joint

angle

gamm

a (rad

)


Figure 5.45: Contact experiment, PD controller: I = I = 0:71, 2-norm optimal vectorof forces

Figure 5.47 shows the trajectory when the H1-optimal controller is used in the

contact experiment. Again there is no apparent bouncing at the instant where con-

tact occurs, and the environment bends less than it did in the previous experiments

because the open-loop gain of the H1 control system is lower than that of the PD

control system. The control forces produced by the H1-optimal controller are shown

in Figure 5.48, and we can see that they are about ve times smaller than the control

forces produced by the PD controller.

After contact with the environment has been made and detected, it is usually

desirable to regulate the force the robot applies or to make its end-eector track

desired force trajectories. The next experimental results are obtained from basically

the same experiment as before, except that after contact the position controller is

disabled and the force applied on the environment by the end of the rod attached to


-50

0

50

100

150

200

250

300

350

400

450

0 1 2 3

PD Controller

time (s)

actua

tor 1

force

(N)


0

200

400

600

800

1000

1200

1400

1600

1800

0 1 2 3

actua

tor 2

force

(N)

PD Controller

time (s)

--- : desired force

Figure 5.46: Actuator forces: contact experiment with PD controller, I = I = 0:71,2-norm optimal vector of forces

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0 0.5 1 1.5 2 2.5


time (s)

joint

angle

gamm

a (rad

)


Figure 5.47: Contact experiment, H1-optimal controller: I = I = 0:71, 2-normoptimal vector of forces


-20

-10

0

10

20

30

40

50

60

70

0 1 2 3


time (s)

actua

tor 1

force

(N)


0

50

100

150

200

250

300

0 1 2 3ac

tuator

2 fo

rce (N

)


time (s)

--- : desired force

Figure 5.48: Actuator forces: contact experiment with H1-optimal controller, I =I = 0:71, 2-norm optimal vector of forces

the joint (see picture in Figure 3.2) is regulated by the analog lead-lag force controllers.

Contact detection is implemented as a simple check on the dierence between two

samples of the force applied by actuator 2. When this dierence exceeds a threshold

of 10 N, the digital position controller is disabled and the minimun 2-norm vector

of forces corresponding to a torque of -4.91 Nm is used as input to the analog force

controllers, such that the force applied on the environment is approximately 10 N. This

detection scheme is not meant to be general but works well in this case. The rst plot

(Figure 5.49) shows the force applied on the aluminum beam based on measurements

of forces applied by the actuators. Of course these measurements do not indicate

what is really happening at the interface between the beam and the robot's endpoint

because the mechanism is not perfectly rigid. Hence, a ner regulation requires that

a force sensor be mounted at the endpoint.


-5

0

5

10

15

20

0 0.5 1 1.5 2 2.5


time (s)

force

(N)

--- : desired force on environment

Figure 5.49: Regulation of force applied on the environment with H1-optimal con-troller, 2-norm optimal vector of forces

The nonlinear controller is also tried in combination with the H1-optimal con-

troller to see if a faster regulation occurs, but this is not the case. The force applied

on the beam is plotted in Figure 5.50.


-5

0

5

10

15

20

25

30

0 0.5 1 1.5 2 2.5


time (s)

force

(N)

--- : desired force on environment

Figure 5.50: Regulation of force applied on the environment with H1-optimal andnonlinear controller, 2-norm optimal vector of forces

Chapter 6

Conclusion

In Chapter 2, a brief comparative analysis of two dierent actuator technologies for

high performance robotics was presented. Electric motors were compared to hydraulic

actuators in terms of their merits on important characteristics such as torque to mass

ratio, torque bandwidth and linearity. Hydraulic actuators with fast suspension-type

valves were found to be superior mainly because of their high torque to mass ratios,

but also because of their extended force bandwidth. Moreover, their force or torque

characteristic can be linearized by feedback.

A high performance hydraulic actuator commercialized by asi was then analyzed

and its highly nonlinear dynamics were modeled. A software simulator implementing

the model in a force control loop demonstrated its usefulness to predict the force

response of the actual system. A simple analog lead-lag force controller was shown

to yield good force bandwidth (100 Hz) and good tracking accuracy.

In Chapter 3, a robotic revolute joint mechanism actuated in-parallel by two

antagonistic hydraulic actuators was designed, built and tested. This mechanism

was shown to oer many advantages over a non-redundant design, such as a large

workspace free of singularities, well-conditioned torque and angle sensing, and the

123

CHAPTER 6. CONCLUSION 124

possibility of controlling internal forces. Although the parallel joint is not truly direct-

drive since the forces are transmitted through a moment arm, it compares nonetheless

favorably to any direct-drive electric motor currently available. For example, its

torque to mass ratio is almost ten times larger than it is for most DC motors.

The kinematics of the parallel joint were derived and an autonomous kinematic

calibration scheme that requires no external measurement of the joint angle was

developed and tested. The kinematic parameters of the prototype were successfully

identied even though the measurement matrix was signicantly noisy. This was

achieved by constraining the nonlinear search to a smaller region in parameter space

by xing (i.e. measuring) two length parameters.

The mapping from actuator forces to joint torque is onto, and hence, an innity

of solutions for the vector of actuator forces exist for a given output torque. This

allowed the calculation of two dierent optimal solutions in the form of minimum-

norm vectors of forces in Chapter 4. The rst minimum-norm vector was a least-

squares solution minimizing the internal force in the joint. The second minimum-norm

vector minimized the maximummagnitude of the actuator forces. It was characterized

and calculated using the theory of dual Banach spaces. Both optimal solutions were

shown to have simple expressions suitable for real-time control. In Chapter 5, it was

demonstrated that these optimal force commands could be easily calculated in less

than 1 ms on a sky Challenger C30 TM computer.

Finally, three position controllers were designed in Chapter 5, of which one was

a simple PD controller while the other two were robust controllers|one based on

the Internal Model Principle and the other based on single-input single-output H1

sensitivity minimization theory. Although the design based on the Internal Model

Principle was carried out in state space, the result was shown to be a simple PID con-

troller for a constant uncertainty model. This controller is robust to slight variations

in the state space model but this has no physical meaning in terms of input-output


behavior. Hence, the H1 optimal controller appears to be better from the robustness

standpoint because it is robust to variations in the input-output map, i.e. the transfer

function. This explains partly why H1 theory is so popular at the moment.

A series of position control experiments were conducted to assess performance of

the three controllers. The PD controller displayed the best tracking performance,

mainly because the resulting closed-loop system had a larger bandwidth (4 Hz) than

with the other two robust controllers (2.5 Hz). The PID controller showed reasonable

tracking performance, but it was not as robust as the H1 lead controller when bad

estimates of the inertia were used. The angle response of the H1-optimal control

system followed the desired trajectory satisfyingly. Feedforward of the desired angu-

lar acceleration was shown to help greatly trajectory tracking independently of the

structure of the controller, even though the linear trajectory with parabolic blends

displayed discontinuities in its angular acceleration.

A nonlinear controller was designed to modulate the system's bandwidth by im-

posing a lower limit on the valve position magnitudes. This has the eect of shifting

the nonlinear damping characteristic to the right, and hence, it reduces the hydraulic

reaction force for a given piston velocity. Antagonistic actuation allowed the imple-

mentation of this idea on the joint since the vector of actuator forces computed by

the nonlinear controller lies in the nullspace of the transposed jacobian. Only slightly

faster rise times were recorded in the angle step responses with this nonlinear con-

troller, but some problems occured due to the high forces applied by the actuators.

These problems outweighted the small increase in bandwidth obtained.

When a robot makes contact with a sti environment, its impedance determines

how the end-eector will react to the sudden impact. Thus the impedance of the

three joint position control systems were calculated in Chapter 5. The PD controller

and the lead controller yielded impedances similar to that of a nicely damped mass-

spring-damper system, which is what we want. On the other hand, the impedance


obtained from the PID controller was shown to be undesirably sti at low frequency

because of the integral term|a well-known fact.

Contact experiments were designed and conducted to see if the system would re-

main stable at all, and satisfying angle responses were observed, showing no apparent

bouncing of the end-eector on the hard surface. Force control was also implemented

by detecting the collision and specifying a desired constant torque command, which

was then translated to actuator force commands to the analog force controllers. These

experiments also proved to be successful as the force applied at the endpoint would

rapidly settle to the desired value without showing excessive oscillations.

6.1 Suggestions for Further Research

6.1.1 Modeling of the ASI Hydraulic Actuator

Further investigation into the identication of the linear dynamics of the actuator is

required to ascertain whether the non-minimumphase zeros in the model are artefacts

of the system identication technique, or whether they can be related to specic

distributed-parameter components of the overall system. Since non-minimum phase

zeros place absolute limits on the achievable sensitivity minimization using feedback

control, establishing the physical meaning of these zeros would be of relevance in

further rening the actuator design to achieve even higher performance.

Better experiments will have to be designed for more satisfying models of the

hydraulic damping and friction characteristics. The hydraulic actuator model could

be used to assess the attainable range of mechanical impedance. It would also be

desirable to see the eect of reducing the model order by neglecting some of the uid

high frequency dynamics. The system's ability to act as a force servo or regulator

while the piston is moving will also have to be better assessed and improved.


Given the fact that the model developed in this research for the asi hydraulic ac-

tuator seems reasonably faithful, it would be interesting to build more sophisticated

linear or possibly nonlinear digital force controllers for it. For example, a nonlinear

observer-based controller could be developed for the model and tested on the real

actuator in order to see to what extent the model can be used as an observer of

the system states, and to nd out if the performance attained would surpass that

obtained by using a simple analog lead-lag compensator. Of course, such a controller

would require a very fast computer and therefore might not be practical. Yet, prelim-

inary experiments have shown that two single-input single-output digital lead force

controllers could be implemented easily on a sky Challenger C30TM computer at a

sampling frequency of 1000 Hz.

The simple linear lead-lag controller used in this research did not show good

robustness properties as it would sometimes make the system break into a limit

cycle for a short time without apparent reason. Hence a more robust force control

law should be developed before a higher level control scheme can be condently

implemented on a multi-actuator mechanism.

6.1.2 Parallel, Redundant Revolute Joint

The mechanism we dealt with in this research has essentially linear dynamics if we

neglect the actuators, but multi-DOF parallel manipulators generally possess highly

nonlinear dynamics. The two linear robust control methods investigated in Chapter 5

could be extended to these mechanisms after implementation of a linearization scheme

performed by feeding back the nonlinear terms of the dynamic equations, as done in

the computed torque method. A convenient characterization of the uncertainty and

disturbance vectors aecting the output of the linearized system should then lead to

an ecient closed-loop control system with nice robustness properties.

A simulator including the nonlinear actuator dynamicmodel could be developed to


test dierent position and impedance control laws for hydraulically actuated parallel

manipulators.

The autonomous kinematic calibration procedure tested on the parallel revolute

joint could be extended to more complex parallel mechanisms with closed kinematic

chains and partially constrained endpoints. An interesting problem is when there is

no closed-form solution to the direct kinematics of a parallel mechanism: how can one

prove convergence when two iterative schemes are used at the same time to calculate

the endpoint position and to estimate the kinematic parameters?

Finally, new criterions could be used in the force optimization problem, and an-

tagonistic actuation should be investigated further as it has already proven to be a

promising idea in robotics.

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Appendix A

Partial Derivatives of FB

These expressions can be used only when > 0.

@FB@k1

= 2 cos

1 r2 21

21

! 2 sin

0@ 21r

21 + r2 21

!2

1

1A1=2 1 r2 21

21

!

@FB@k2

= 22 + cos

1(21 + r2 21) + 2r sin

0@1

21 + r2 21

21r

!21A1=2

@FB@r

= 2k2 4r + cos

"(k1 k2)

1 +

r2 2121

!+ 22

1 +

r

1

#

+ 2 sin

264(k2 + 2r)

0@1

21 + r2 21

21r

!21A1=2

2

0@ 21r

21 + r2 21

!2

1

1A1=2 (k1 + 2r)(21 k21)

2r21

!375

135

APPENDIX A. PARTIAL DERIVATIVES OF FB 136

@FB@10

= 221

cos 1 +22 sin 1

1

0@ 21r

21 + r2 21

!2

1

1A1=2

@FB@20

= 22

@FB@

= 21

sin (21 + r2 21) + 22r cos

0@1

21 + r2 21

21r

!21A1=2

Appendix B

Calibration Algorithm

Implemented on MatlabTM

% This command file implements the Levenberg-Marquardt algorithm to

% calibrate the kinematic parameters of the parallel revolute joint given

% a reasonable estimate. The matrix rhomeas must be in the workspace.

% The file kincalibjoint.mex4 calculates the partial derivatives of FB

% and also FB itself.

% The parameters r (parmest(3)) and lambda2 (parmest(2))

% must be measured accurately because they scale the mechanism.

sigma = 1e-8;

alpha = .001; % This constant controls the speed of convergence

% and gets us through the singularities.

epsilon=.0000001;

FBmin = 1;

i=0;

137

APPENDIXB. CALIBRATIONALGORITHM IMPLEMENTEDONMATLABTM138

normFB = 2*sigma;

% Make sure that the results of the squareroot operations in

% kincalibjoint.mex4 are real

if (parmest(1)-parmest(3)) >= (parmest(4)-max(-rhomeas(:,1)))

parmest(1)=parmest(4)-max(-rhomeas(:,1))-epsilon+parmest(3);

end

while ( (normFB > sigma) & (i < 10000) )

% Calculate the jacobian matrix C and FB

for i=1:50

C(i,:)=kincalibjoint(parmest',rhomeas(i,:));

end

% Calculate deltaparm and new estimate

FB=C(:,7);

C=[C(:,1) C(:,4:6)];

D=inv(C'*C+alpha*eye(4));

deltaparm=-D*C'*FB;

parmest=[(parmest(1)+deltaparm(1));parmest(2);parmest(3); ..

(parmest(4:6)+deltaparm(2:4))];

if max(abs(1000*FB)) < alpha

alpha = alpha/1000;

end

APPENDIXB. CALIBRATIONALGORITHM IMPLEMENTEDONMATLABTM139

normFB=max(abs(FB))

if normFB < FBmin

FBmin = normFB;

bestparmest = parmest;

end

% Make sure that the results of the squareroot operations in

% kincalibjoint.mex4 are real

if (parmest(1)-parmest(3)) >= (parmest(4)-max(-rhomeas(:,1)))

parmest(1)=parmest(4)-max(-rhomeas(:,1))-epsilon+parmest(3);

end

clear C

i=i+1;

end

parmest

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