JIEGAO WANG

KINEMATIC ANALYSIS, DYNAMIC ANALYSIS AND STATIC BALANCING OF PLANAR AND SPATIAL

PARALLEL MECHANISMS OR MANIPULATORS WITH REVOLUTE ACTUATORS

Thèse

présentée

à la Faculté des études supérieures

de 1'Cniversité Laval

pour l'obtention

du grade de Philosophiae Doctor (PhB.)

Département de génie mécanique

FACULTÉ DES SCIENCES ET DE GESIE

LXIVERSITÉ LAVAL

QCEBEC

@ Jiegao Wang, 1997

National Librat-y 1*1 of ,nada Bibliothèque nationale du Canada

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Abstract

This thesis deals with the kinematic analysis, dynamic analysis and static balancing of

planar and spatial parallel mechanisms or manipulators with revolute actuators.

The inverse kinematics of each mechanism is first computed and a nen- general al-

gorithm is used to locate the boundaries of the workspace of the mechanism. Two

approaches. narnely. the algebraic formulation and the vector formulation are used to

derive the velocity equations of the mechanisms. The singularity loci is then determined

using the velocity equations. The approach of t-ector formulation is a new approach

and leads to simpler expressions for the determination of the singularity loci . Kine-

matic optirnization of mechanisms 11-ith reduced degrees of freedom is also discussed in

the thesis. The generalized reduced gradient method of optimization is used to find

the optimal sohtions of the link parameters which enable the dependent Cartesian

coordinates to follow an ideal trajectory as closely as possible when the independent

Cartesian coordinates pass through some prescribed points.

A new approach for the dynamic analysis of parallel mechanisms or manipulators

is proposed in the thesis. This approach is based on the principle of virtual n-ork. -4s

compared to the conventional approach of Newton-Euler? the new approach wilI lead

to a faster algorithm for derivation of the generalized forces: which is useful for the

control of a mechanism or manipulator. The Xen~on-Euler approach is also used for

dynamic anaIysis of parallel mechanisms or manipulators. Since the constraint forces

Résumé

Cette thèse porte sur l'analyse cinématique et dynamique ainsi que sur l'équilibrage

statique des mécanismes et manipulateurs parallèles plans et spatiaux avec actionneurs

rotoïdes.

Le problème géométrique inverse de chaque mécanisme est d'abord résolu et un

nouvel aigorithme général est proposé pour la détermination des frontières de l'espace

atteignable. Deus approches, soit la formulation algébrique et la formulation vecto-

rielle. sont utilisées pour obtenir les équations de vitesse des mécanismes. Le lieu des

configurations singulières est ensuite déterminé en utilisant les équations de vitesse. La

formulation vectorielle est une nouvelle approche qui conduit à des expressions plus sim-

ples pour les lieux de singularité. L'optimisation cinématique de mécanismes à degré de

liberté réduit est aussi traitée dans cette thèse. La méthode du gradient généralisé est

utilisée afin de trouver des valeurs optimales des paramètres géométriques permettant

aux coordonnées cartésiennes dépendantes de suivre une trajectoire prédéterminée en

fonction des coordonnées cartésiennes dépendantes en certains points prescrits.

Une nouvelle approche pour l'analyse dynamique des mécanismes et manipulateurs

parallèles est aussi proposée dans cette thèse. Cette approche est basée sur le principe

du travail virtuel. En comparaison avec la formulation traditionnelle de Newton-Euler.

la méthode proposée ici conduit à un algorithme plus rapide pour le calcul des efforts

articulaires. ce qui est intéressant pour la commande. Les équations de Xewton-Euler

Foreword

This thesis describes the most important aspects of my work in the Robotics Laboratory

of the Department of Mechanical Engineering of Laval university? Québec. Canada.

Firstly. 1 ivould like to thank rny supervisor Professor Clément 51. Gosselin for

his invaluabie guidance. stimulating discussions and continuous encouragements in the

whole research as weil as the critical review of the manuscript and the translation of

the abstract of the thesis into French.

GratefuI thanks go to Professor Marc J. Richard for his prerevieir- of nly thesis aiid

to Professor Li Cheng and Professor J. J. 3IcPhee of the University of FVaterloo for

having accepted to examine m?; thesis and for their precious remarks.

FinalIy, 1 would like to express my gratitude to the colleagues in the Roboeic Lab-

oratory, namely Xartin Jean, Boris Mayer St-Onge, Rémi Ricard and others, for their

help in rnany aspects.

Contents

Short Abstract

Abstract

Résumé

Foreword

Contents

List of Tables

List of Figures

1 Introduction

2 Kinematic analysis

. . . . . . . . . . . . . . . . . . . . . 2.1 Inverse kinematics and workspace

2.1.1 Planar paralle1 manipulators with revolute actuators . . . . . . .

2.1.1.1 Planar two-degree-of-freedom manipulator . . . . . . . . . . 2.1.1.2 Planar three-degree-of-freedom manipulator . . . . . . . . .

2.1.1 -3 Branches of paraIIel manipulators . . . . . . . . . . . . . . 2.1.2 Spatial parallel manipulators with revolute actuators . . . . . . .

2.1.2.1 Spatial four-degree-of-freedom parallel manipulator with rev-

olute actuators . . . . . . . . . . . . . . . . . . . . . . . . .

2.1.2.2 SpatiaI five-degree-of-freedom parallel manipulator with rev-

olute actuators . . . . . . . . . . . . . . . . . . . . . . . . .

vii

vii

2.1.2.3 Spatial six-degree-of-freedom parallel manipulator wit h revo-

. . . . . . . . . . . . . . . . . . . . . . . . . . lute actuators

. . . . . . . . . . . . . . . . . . 2.2 Velocity equations and singularity loci

. . . . . . . . . . . . . . . . . . . . . 2.2.1 Planar parallel manipulators

. . . . . . . . . . . . . . 2.2.1.1 Tmdegree-of-freedom manipulator

. . . . . . . . . . . . . 2.2.1.2 Three-degree-of-freedom manipulator

. . . . . . . . . . . . . . . . . . . . . 2.2.2 Spatial parallel manipulators

2.2.2.1 Spatial four-degree-of-freedom parallel manipulators with rev-

olute actuators . . . . . . . . . . . . . . . . . . . . . . . . .

2.2.2.2 Spatial five-degree-of-freedom parallel manipulators wit h rev-

. . . . . . . . . . . . . . . . . . . . . . . . . o h t e actuators

2.2.2.3 Spatial six-degree-of-freedom parallel manipulators ivith rev-

o h t e actuators . . . . . . . . . . . . . . . . . . . . . . . . .

2.3 Eiinematic optimization of mechanisms with reduced degrees of freedom

. . . . . . . . . . . . . 2.3.1 Planar two-degree-of-freedom manipulator

. . . . . . . . . . . . . 3 - 3 2 Spatial four-degree-of-freedorn manipulator

. . . . . . . . . . . . . 2.3.3 Spatial five-degree-of-freedom manipulators

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Conclusion

Dynarnic analysis

. . . . . . . . 3.1 -4pproach using the principle of virtual work

3.1.1 Planar parallel manipuIators nith revolute actuators

3.1.1 -1 Two-degree-of-freedorn rnanipulator . . . . . . . . . . . . . 3.1.1.2 Three-degree-of-freedom manipulator

3.1.2 Spatial parallel manipulators with revolute actuators

3.1.2.1 Four-degree-of-freedom manipulator . . . . . . .

3.1.2.2 Five-degree-of-freedom manipulator . . . . . . .

3.1 . 2.3 Sis-degree-of-freedom manipulator . . . . . . .

. . . . . . . . . . . . . . . . . . 3.2 Yewton-Euler formulation

3.2.1 Planar parallel manipulators with revolute actuators

3.2.1.1 Two-degree-of-freedom manipulator . . . . . . .

3.2.1.2 Three-degree-of-freedom manipulator . . . . . .

3.2.2 Spatial parallel manipulators with revo1ute actuators

3.2.2.1 Four-degree-of-freedom manipulator . . . . . . .

viii

3.2.2.2 Five-degree-of-freedom rnanipulator . . . . . . . . . . . . . . 118

. . . . . . . . . . . . . . 3.2.2.3 Six-degree-of-fkeedom manipulator 123

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Conclusion 125

4 Static bdancing 128

4.1 Balancing with countenveights . . . . . . . . . . . . . . . . . . . . . . . 129

4.1.1 Planar parallel manipulators with revolute actuators . . . . . . . 129

. . . . . . . . . . . . . . 4.1.1.1 Two-degree-of-freedom manipulator 129

4.1.1.2 Three-degree-of-freedom manipulator . . . . . . . . . . . . . 133

4.1.2 Spatial parallel manipulators with revohte actuators . . . . . . . 136

. . . . . . . . . . . . . . 4.1 2 . 1 Four-degree-of-freedom manipulator 137

. . . . . . . . . . . . . . 4.1.2.2 Five-degree-of-freedom manipulator 143

. . . . . . . . . . . . . . 1.1.2.3 Six-degree-of-freedom manipulator 147

. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Balancing with springs 151

4.2.1 Planar parallel manipulators with revolute actuators . . . . . . . 151

. . . . . . . . . . . . . . 1.2.1.1 Two-degree-of-freedom manipulator la1

4.2.1.2 Three-degree-of-freedom manipuiator . . . . . . . . . . . . . 154

4.2.2 Spatial parallel manipulators with revolute actuators . . . . . . . 156

. . . . . . . . . . . . . . 4.2.2.1 Four-degree-of-freedorn manipulator 156

4.3.2.2 Five-degree-of-freedom manipulator . . . . . . . . . . . . . . 159

. . . . . . . . . . . . . . 4.2.2.3 Six-degree-of-freedorn manipulator 163

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -1.3 Conclusion 170

Conclusion 172

Bibliography

A Polynomial formulation of the singularity loci of planar parallel ma- nipulator 182

. . . . . . . . . . . . . . . . . . . . .4.1 Two-deg~ee-of-freedom manipulator 183

. . . . . . . . . . . . . . . . . . . .4.2 Three-degree-of-freedom manipulator 183

B Generai expressions of det(A*) for planar parallel manipulators 186

C Simplification of Jacobian matrix 188

D Expressions associated with the elements of the matrix CI and vector d f 192

List of Tables

2.1 Data set for the two-dof parallel rnanipulator, lengths are in meters and

angles in degrees, K = -1. . . . . . . . . . . . . . . . . . . . . . . . . 69

2.2 Data set for the 4-dof parallel manipulator, lengths are in meters and

angles in degrees. K I = -1 and Ir; = 1. . . . . . . . . . . . . . . . . . 74

2.3 Data set for the 5-dof parallel manipulator, lengths are in meters and

angles in degrees, KL = -1. . . . . . . . . . . . . . . . . . . . . . . . . 78

List of Figures

1.1 Two different types of robotic manipulators . . . . . . . . . . . . . . . .

1.2 ParaIIel mechanisms used for robot manipulators . . . . . . . . . . . . .

2.1 Planar two-degree-of-freedom parallel manipulator . . . . . . . . . . . .

2.2 Planar three-degree-of-freedom parallel manipulator . . . . . . . . . . .

2.3 IYorkspace of the tn-O-dof manipulator . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . 2.4 Rorkspace of the three-dof manipulator

2 . 5 Spatiai four-degree-of-freedom parallel manipulator \vit h revolute act u-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ators

2.6 Spatial five-dof parallel manipulator mith revolute actuators . . . . . . .

2.7 Spatial six-dof parallel manipulator with revolute actuators . . . . . . .

2.8 Geometry of the fifth leg of the four-dof mechanism . . . . . . . . . . .

2.9 Configuration of the ith actuated joint of the four-dof manipulator with

revolute actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.10 Example of the workspace of the four-dof mechanism . . . . . . . . . .

3.11 ExampIe of the workspace of the five-dof mechanism . . . . . . . . . . .

2.12 Example of the workspace of the six-dof mechanism . . . . . . . . . . . .

2.13 Loop vectors associated with the twc-degree-of-freedorn manipulator . . 37

2.14 Singularity locus and workspace of the two-dof manipuIator+ . . . . . . 39

2.15 Singularity locus and workspace of the two-dof manipulator . . . . . . . 40

2.16 Loop vectors associated with the three-degree-of-freedom manipulator . 41

2.17 Three-degree-of-freedom manipulator in a configuration corresponding

. . . . . . . . . . . . . . . . . . . . . . to the second type of singularity 42

2.18 Singularity locus and workspace of the three-dof manipulator with branches

1-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.19 Singularity locus and workspace of the three-dof manipdator with branches

5.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.20 Position and ~elocity vectors associated with the spatial four-degree-oi-

freedom parallel manipulator with re~oIute actuators . . . . . . . . . . .

2.21 léctor li2 represented in spherical coordinates . . . . . . . . . . . . . . .

2.32 Singularity locus and workspace of the four-dof manipdator . . . . . . .

2-23 Position and velocity vectors associated with the spatial five-degree-of-

freedoni parallel mechanism tvith remlute actuators . . . . . . . . . . . .

2.24 Singiilarity Iocus and workspace of the fi~e-dof rnanipulator . . . . . . .

2.25 Position and velocity vectors associated wit h the spatiai sis-degree-of-

freedom paralle1 mechanisrn with revolute actuators . . . . . . . . . . . .

2.26 Singularity locus and workspace of the sis-dof manipulator . . . . . . . .

2.27 First leg of the planar two-degree-of-freedom parallel manipulator . . . .

2.28 Esample of optirnization synthesis of the two-dof mechanism . . . . . . .

2.29 Esample of optimal synthesis of the spatial four-dof manipulator . . . .

2.30 Esample of optimal synthesis of spatial five-dof rnanipulator . . . . . . .

3.1 Vectors Ii, and Ill represented in spherical coordinates . . . . . . . . . .

xii

3.2 Body forces acting on the links of the manipulator with revolute actuators . 97

3.3 Body forces acting on the links of the manipulator with revolute actuators . 104

. . . . . . 3.4 Generdized actuator force in the planar two-dof mechanism 110

3.5 Generalized force in the planar three-dof rnechanism at the actuated

j o i n t s l a n d 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

3.6 Generalized force in the planar three-dof mechanism at the actuated joint

3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I l 2

3.7 The constraint forces between the fifth leg and the platform . . . . . . . 113

3.8 The constraint forces on the platform . . . . . . . . . . . . . . . . . . . 114

3.9 The forces acting on the two links of the ith leg for the manipulator n-ith

revolute actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

3.10 Generalized force for the spatial four-dof mechanism a t the actuated

joints 1 to 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.11 The action forces of 6th leg . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.12 The action forces of the platform . . . . . . . . . . . . . . . . . . . . . . 121

3.13 Generalized force for the spatial five-dof mechanism at the actuated

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joints 1 t o 4 124

3.14 Generalized force for the spatial five-dof mechanism a t the actuated joint

5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

3.15 Generalized force for the spatial six-dof mechanism at the actuated joints

1 to 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

3.16 Generalized force for the spatial sLu-dof mechanism at the actuated joints

5 and 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.1 Planar two-degree-of-freedom mechanism . . . . . . . . . . . . . . . . . . 130

4.2 Tm-degree-of-freedom balanced mechanism with countertveights . . . . 132

. . . . . . . . . . . . . . . . . 4.3 Planar three-degree-of-freedom mechanism 133

... Xl l l

Three-degree-of-freedom balanced mechanism with countemeights . . . . 136

Geometric representation of the spatial four-degree-of-freedom system

with revolute actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . 13'1

Geometry of the fifth leg . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Geometry of the i th leg . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Four-degree-of-freedom balanced rnechanism with revolute actuators us-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ing countemeights 143

Geometric representation of the spatial five-degree-of-freedom mecha-

. . . . . . . . . . . . . . . . . . . . . nism n-ith revolute actuated joints 143

. . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Geometry of the s k t h ieg 144

4.1 1 Five-degree-of-freedom balanced mechanism with revolute actuators us-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ing counterweights 147

4.12 Spatial six-degree-of-freedom parallel mechanism u-ith revolute actuators . 145

. . . . . . . . . . . . . . . . . 4.13 Complete balancing using countenveights 150

. . . . . . . . . . . . 4.14 Geometry and kinematic architecture of the i t h leg 15'2

4.1.5 Planar two-degree-of-freedam balanced mechanism with springs . . . . . 1.54

4.16 Planar three-degree-of-freedorn balanced mechanism with springs . . . . 1.57

. . . . . . . . . . . . . . . . . . 4.17 Architecture and geometry of the ith leg 137

4.18 Four-degree-of-freedom balanced mechanism with revolute actuators us-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ing springs 159

4.19 Five-degree-of-freedom balanced mechanism with revolute actuators us-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ing springs 162

. . . . . . . . . . . . 4.20 Geometry and kinematic architecture of the ith leg 162

. . . . . . . . . . . . . . . . . . . . . . 4.21 Balanced mechanism with springs 166

4.22 Alternative architecture of Ieg for the 6-dof parallel mechanism with

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . revolute actuators 167

xiv

4.23 Balanced mechanism with springs. . . . . . . . . . . . . . . . . . . . . . 170

Chapter 1

Introduction

.\ccotding to their geometric architecture. robot manipulators are classified as serial

robot manipulators and parallel robot manipulators. Each type of robotic manipulator

has its advantages and drawbacks and is suitable for different applications. Figure 1.1

represents simple planar tn-degree-of-freedom serial and parallel manipulators. The

serial manipulator (Figure l . l (a)) consists of two moving links and two revolute joints.

Two revolute actuators are respectively mounted at joint O and O1 with joint variables

O1 and &. Point P is the position of the end-effector of the manipulator. The paraIleI

manipulator represented in Figure l . l (b ) which will be studied in this thesis consists of

two kinematic chains connecting the fixed base to the end-effector of the manipulator:

the two kinematic chains are 002P and 0i0304P respectively. Two revolute actuators

are mounted at joints O and O1 with joint variables O1 and 02, while joints 02. o3 and

O4 are passive revolute joints. Similady, point P is the position of the end-effector of

the manipulator.

(a) Planar two-dof serial manipulator.

. .

(b) Planar two-dof paralIel manipuIator.

Figure 1.1: Two different types of robotic manipulators.

In general. an n-degree-of-freedom parallel manipulator is one in which the end-

effector is connected to the base with n distinct kinematic chains and in which one

joint of each of these chains is actuated.

Comparing the two types of robot manipulators, one can realize that serial robot

manipulators have a simpler structure, wider reachable area and relatively simpler

kinematics. These advantages led to an estensive use of this type of manipulator in

the industn, for instance, for assembling, welding, painting, etc. However, since the

serial structure leads to low rigidity, this type of manipulator has smaller load capacity.

Iacks stiffness and cannot reach high dynamic performances. Therefore. the' are not

suitable for some applications where large load or high speed and accuracy are needed.

-4s opposed to serial robot manipulators, parallel robot manipulators consist of sev-

eral kinematic chains connecting the fked base to the end-effector, which leads to sev-

eral advantages. For esample, the weight of the manipulator is reduced and the rigidity

of the manipulator is increased. Therefore, the manipulators have large load ability

good stiffness and high dynarnic performances and are suitable for applications requir-

ing large load ability or high speed and accuracy. However, comples parallel structures

a1sa lead to some drawbacks such as a smaller workspace and a complex mathemat-

ical mode1 as compared to the serial robot manipulators. Two typical examples of

(a) Stewart platform. (b) Agile-eye.

Figure 1.2: Parallel mechanisms used for robot manipulators.

the application of parallel manipulators are the spatial SLX-degree-of-freedom parallel

mechanism n-ith prismatic actuators [48] (Gough-Stewart platform Figure 1.2(a)) and

the spherical three-degree-of-freedom parallel mechanism with revolute actuators [lï]

(agile-eue Figure 1.2(b)). The former is used for flight simulators and the latter is used

for the orientation of a camera at high speed. The? respectic-ely illustrate the cases of

large load and high speed and accuracy.

-\lthough parallel manipulators have received more and more attention over the

iast tn-O decades jsee for instance, j3j. j4j: [loi, i l l j , j13j, jl4j. j15j and [ l ï ] , and many

more), their applications have been limited to only a few areas so far. The comples

mathematical models of parallel robot manipulators is the main factor to hinder their

applications. Therefore, the development of efficient approaches for the kinematic and

dynamic analysis is estremely important for the study and the application of the mech-

anisms.

It can be noticed in the literature that parallel mechanisms or manipulators with

prismatic actuators have received more attention than those with revolute actuators.

Indeed, parallel mechanisms or manipulators with prismatic actuators have a simpler

architecture and kinematic mode1 than those with revolute actuators. For example. the

singularity loci of planar three-degree-of-freedom parallel manipulators with prismatic

actuators is quadratic (471. However, for the planar three-degree-of-freedom parallel

manipulator with revolute actuators, the cun7es representing the singularity loci in

the Cartesian space are of a very high degree ([19] and [20]). However, as will be

seen in the thesis, paralle1 mechanisms or manipulators with revolute actuators can be

statically balanced. -4 statically balanced parallel rnanipulator could be easier to control

and leads to higher energy efficiency. Additionally, since the architecture is sirnilar.

the approaches used in the kinematic and dynamic analysis of parallel mechanisms or

manipulators with revolute actuators can be directly applied to paralle1 mechanisms

or manipulators with prismatic actuators [58]. Therefore, in this thesis, only parallel

mechanisms or manipulators n-ith revolute actuators are discussed, which does not limit

the generality of the study of paralle1 mechanisms or manipulators.

Depending on the applications, the paraIlel devices studied here ma? be calIed

.'mechanisrns" or "manipulators"; both terms can be found in the literature. In general.

the term "manipulator" is used in robotic applications while the more general term

"mechanisrn" is used in a11 other applications. In this thesis. both terms will be used

and wi11 be considered equivalent since the present work is generic and is not application

dependent.

In Chapter 2: the kinematic analysis of parallel mechanisms or manipulators is

presented. Several important issues are addressed. narnely. inverse kinematics and

workspace analysis. the velocity equations and singularity loci and the kinematic opti-

mization of mechanisms with reduced degrees of freedom.

The determination of the workspace of parallel manipulators has been studied by

many authors. see for instance, [l], [2], [4]. [ll], [13], [37], [.38], [JO] and [65]. '\Iost

of the approaches proposed by these authors for the determination of the workspace

mainly use analytical methods to find the esplicit expressions of the boundap of the

workspace. For some simple planar mechanisms or for the spatial sis-dof (where "dof*

stands for "degree of freedom") parailel mechanism with prisrnatic actuators 1131, the

boundaries of the workspace consist of low degree curves or surfaces. It is possible to

find an explicit expression which will Iead to a faster algorithm for Iocating the limits

of the workspace. However, for some cornples rnulti-degree-of-freedom spatial parallel

mechanisms, especially. paralle1 rnechanisms or manipulators with revolute actuators.

it is very difficult or impossible to find esplicit expressions of the boundary of the

workspace since the curves or surfaces of the boundary are of a very high degree.

In Chapter 2, a new numerical algorithm for the determination of the workspace is

proposed. The compact expressions corresponding to the workspace boundaries of

paralle1 mechanisms or manipulators are first obtained from the analysis of the solution

of the inverse kinematics. Then, a numerical procedure is used to locate the curve of

the boundary of the workspace. -4lthough this algorithm is much slower than those

developed by the authors mentioned above, it is a general algorithm and it can be

applied to any type of planar or spatial parallel manipulator.

Kinematic singularity is an important issue. When a mechanism or manipulator is

in a singular configuration, the output link or the end-effector gains one or more degrees

of freedom and the input links or the actuators lose their independence which means

that the manipulator is uncontrollable. Such singular configurations must be avoided.

Singularity loci consist of the sets of reachable end-effector poses corresponding to the

singular configurations of the mechanism or manipulator. The singularity problem of

parallel manipulators has been studied by some authors. The singularity of single-

loop kinematic chains is analyzed in [24] and [SOI. In [3]. [30]. 1391 and [%] ïarious

criteria and chssifications for the singularities of paralle1 rnechanisms tvere analyzed and

presented. However, these studies do not provide a systematic approach to determine

the singularity loci.

Gosselin and Angeles in [16] used the Jacobian matrices of the input-output ve-

locity equations of parallel manipulators to classify the different types of singularities.

Three generaI types of singularities which can occiir in parallel manipulators have been

identiEed. This classification has been further refined in [66] where several sub-classes

of singularities are defined. The approach presented in [16] alIows the determination

of singularity loci using velocity equations of manipulators and will therefore be used

in the thesis. Xccording to this singularity classification. the first type of singularity

corresponds to the boundary of the workspace, which has been obtained using the new

algorithm rnentioned above and the third type of singularity can be avoided by proper

arrangement of the architecture of the manipulator. Therefore, the second type of sin-

gularity loci is the major singularity tvhose loci must be determined using the Jacobian

matrix of the velocity equations. In other words: the velocity equations are critical

in the determination of the singularities of type two. Usually. the velocity equations

are obtained through differentiation of the kinematic equations of manipulators. This

approach leads to a direct relationship between the joint and Cartesian velocities and

will be referred to here as the algebraic formulation. It will be used for the derivation of

the velocity equations. However, a t the same time, a new approach for the derivation

of the velocity equations of complex manipulators is introduced and will be referred to

here as the vector formulation.

The new approach of derivation of the velocity equations consists in writing the ve-

locity equations using components of angular velocities of the links to form the Carte-

sian velocity vector and relative velocities of some chosen points of the links to form

the joint coordinate vector ([.58], [60] and [61]). This leads to a redundant formulation.

i.e.. one in which the number of Cartesian or joint velocity components is larger than

the degree of freedom of the mechanism. Homever, the Jacobian matrices obtained are

rather simple and sparse and can be written in terms of the components of vectors asso-

ciated n-ith the geometry of the mechanism. Although the dimensions of the Jacobian

matrices obtained with this method are larger than with the algebraic approach. the

espression for the determinant - which is used for the determination of the singularity

loci - is in general simpler. which leads to faster algorithms for the computation of the

loci. On the other hand, for purposes of control of a mechanism. the algebraic approach

leads to a more direct relationship between the joint and Cartesian velocities.

Finally. it is pointed out that the vector formulation characterizes the local behavior

of the mechanism esâctly as the algebraic formulation does and hence. singularity loci

obtained with both approaches mil1 lead to identical results.

For mechanisms with reduced degrees of freedom, such as planar two-dof as well

as spatial four- and five-dof parallel mechanisms, since some Cartesian coordinates

cannot be prescribed-they are dependent on the prescribed independent Cartesian

coordinates-the kinematic optimization of the mechanisms is introduced in Chapter

2. The optimal synthesis of planar linkages has been addressed by many authors (for in-

stance: [12], [32] and man? others). The synthesis of multi-loop spatial mechanisms has

also been studied in [44] using the Generalized Reduced Gradient method of optimiza-

tion. However, the optimal synthesis of the trajectories of the dependent Cartesian

coordinates of spatial parallel mechanisms or manipulators with reduced degrees of

freedom has received less or no attention. The optimum design of these mechanisms

is important for their practical applications. In Chapter 2, the optimization of planar

two-dof as well as spatial four- and five-dof parallel mechanisms is presented. The

Generalized Reduced Gradient method of optimization is used.

In a context of design and control, the dynamic analysis of paralIe1 manipuIators is

an important issue (51, Cg], (431, [14], [17] and [34]. Indeed, it provides information on

the internd constraint forces and moments on the links as well as on input forces or

torques at the actuâtors for a specified trajectory.

The Xewton-Euler equations of motion have been used by several authors for the

dynamic analysis of spatial parallel manipulators, for instance, in [10]: [Id] and [49].

In this approach the links constituting the manipulator are isolated and the Yen-ton-

Euler equations are written for each link. Then, al1 interaction forces and moments

between the links are obtained. In [36], the dynamic analysis of a three-degree-of-

freedom parallel manipulator using a Lagrangian approach is presented. Hoivever.

because of the complesity of the kinematic mode1 of the spatial parallei rnanipulator.

some assumptions have to be made to simplify the espressions of the kinetic energ'- and

potential energ-: Therefore. this approach is not general and efficient for the dynamic

analysis of parallel mechanisms or manipulators.

In Chapter 3. the dynarnic analysis of planar and spatial paralle1 mechanisms or

manipuIators with revolute actuators is performed. Two different approaches are used.

The first approach is based on the principle of virtual work. This is a new approach

which is applied to paralle1 manipulators for the first time [59!. In this approach the

inertial forces and moments are determined using the linear and angular accelerations.

Then, the equiIibrium of the whole manipulator is considered and the principle of

virtual work is applied to derive the input forces or torques. Since constraint forces

and moments do not need to be computed, this approach Ieads to faster cornputational

algorithms, which is an important advantage for the purposes of control of a rnanipula-

tor. The other approach is based on the Newton-Euler formulation mentioned above. It

is useful for the purposes of design. However, the computational algorithms are slower

than with the nem approach. Therefore, the tnTo approaches have their own âdvantages

and drawbacks and each of them is suitable for different applications.

The balancing of mechanisms has been an important research topic for several

decades (see for instance [3Z] for a literature review). -4 balanced mechanism Ieads

to better dynamic characteristics and less vibrations caused by motion. Static and

dy-narnic balancing of planar linkages have been studied extensively in the literature

(see for instance [6], [23], [51], [52] and [64]). Mechanisrns are said to be force-bdanced

when the total force applied by the mechanism on the fixed base is constant for any

motion of the mechanisrn. In other words, a mechanism is force-balanced when its

global center of mass remains fi'red, for any arbitrary motion of the mechanism. This

condition is very important in machinery since unbalanced forces on the base will lead

to vibrations. Wear and other undesirable side effects. For robotic manipulators or

motion simulation devices, however, the forces on the base are usually not critical and

designers are mostly concerned with the torques (or forces) which are required at the

actuators to maintain the manipulator or mechanism in static equilibrium. Hence. in

this context. manipulators or mechanisrns are said to be staticalit. balanced. when the

weight of the links does not produce any torque (or force) at the actuators under static

conditions. for any configuration of the manipulator or mechanism. This condition

is aIso referred to as grarity compensation. Gravity-compensated seria1 manipuiators

have been designed in [26], [42]. [53], [S6] and [Di] using countenveights. springs and

- sometimes cams andior pulleys. -1 hybrid direct-drive gravity-compensated manipula-

tor has also been developed in 1311. hloreover. a general approach for the equiIibrium

of planar linkages using springs has been presented in [33]. The balancing of spatial

rnechanisms has also been studied. for instance in [7] and [S Ï ] .

However. the static balancing of spatial multi-ciegree-of-freeciom paraIlel manipu-

lators or mechanisms h a received very iittle attention. Since spatial parallel rnani-

pulators find more and more applications in robotics and flight simulation. the static

balancing of spatial parallel manipulators becomes an important issue. -4s mentioned

above, a statically balanced parallel manipulator is one in which the actuators do not

contribute to supporting the weight of the moving links, for al1 configurations. Hence.

the actuators are used only to impart accelerations to the moving links, n-hich leads to

a reduction of the size and power of the actuators and results in the improvement of

the accuracy of the control. In flight simulation, for instance, since the payload is Yery

large (usually in the order of tons) and the motion of the platform of the mechanism is

rather slow. the forces or torques exerted at the actuated joints are mainly due to the

wight of the platform. Hence, if the mechanism is statically balanced, the actuating

forces or torques d l be greatly reduced, which wiIl result in significant improvements

of the control and energy efficiency.

In Chapter 4, the static balancing of planar and spatial parailel mechanisms or

manipulators with revolute joints is addressed. Two approaches of static balancing are

presented, namely, i) static baiancing using countemeights and ii) using springs ([21]

and [62]). When the mechanism is balanced using countemeights, a mechanism with a

fixed global center of mass is obtained. In other words, the static balancing is achieved

in any direction of the Cartesian space of the mechanism. This property is useful for

applications in which the mechanism is needed to be statically balanced in al1 directions

(e.g. if the mechanism is to be installed in an a r b i t r q direction with respect to the

gravity acceleration vector). However. for sume paraIleI mechanisms. static balancing

with countenveights is difficult to realize. For esarnple. in flight simulators. since the

mass of the platform is yen; large. the countenveights required would in general be too

large to be practical. However, springs can be used in such instances. li-hen springs - or other elastic elements - are used. the total potential energy of the manipulator -

gravitational and elastic - can be kept constant and the weight of the whole manipula-

tor can be balanced with a much smaller total mass than when using counterweights. as

pointed out in [53]. However. a mechanism which is statically balanced using springs

will be statically balanced for only one direction of the gravitu vector. which may

be unsuitable for some applications. Therefore, both rnethodoiogies are discussed in

Chapter 4 since they each have their own merit.

Chapter 2

Kinemat ic analysis

Kinematic modeling is essential for the analysis and design of a paraIlel mechanism.

C s u a l l ~ the kinernatic analpis consists of forward kinematics, inverse kinematics. de-

termination of the workspace and singularity analysis. Ii'nlike serial robotic manipu-

lators, the fomard kinematic analysis of parallel manipulators is more cornples than

its inverse kinernatic analysis. For instance, for the spatial six-degree-of-freedom par-

allel manipulator R-ith prismatic actuators (the Gough-Stewart ptatform) the inverse

kinematic analysis is very simple while the fomard kinematic analysis is estremel? dif-

ficult and only numericaI solutions can be obtained. Fortunately, the solution of the

forward kinematics of a parallel rnanipulator is not necessary for its kinematic design.

hIoreover: the fonvard kinernatic analysis of different types of paraIIel manipulators is

often similar. Therefore, the fomrard kinematic analysis of paralle1 mechanisms is not

discussed in this thesis.

The workspace of a manipulator is defined as the set of Cartesian poses (position

and orientation) that the end-effector of the manipulator can reach. The workspace

determines the volume or hyper-volume in which the manipulator can perform tasks.

Therefore, the determination of the workspace is an important first step for the design

of a manipulator. Moreover, inside the workspace of a manipulator there exist some

positions and orientations in which the end-effector gains or loses one or more degrees

of freedom and therefore the manipulator becomes uncontrollabie. These positions

and orientations are cailed singular configurations and they must be considered in the

design of the manipulator. The singular configurations of parallel manipulators can be

determined using the velocity equations [16]. The determination of the workspace and

the singularity loci ni11 be discussed here.

For rnechanisms with reduced degrees of freedorn' for instance. the planar two-

degree-of-freedom and spatial four- and five-degree-of-freedom paraIlel mechanisms.

only a subset of the three Cartesian coordinates x, y and 4 (for planar parallel mech-

anisms) or of the sis Cartesian coordinates x. y, z. d. 6' and v (for spatial parallel

mechanism) are independent (where @. 0 and .iI! are 3 Euler angles). If the manipula-

tors' dependent coordinates are required to follow sorne desired trajectories as closely

as possible when its independent coordinates pass exactly through the specified trajec-

tories: there exists a kinematic optimization problem which can be solved to obtain a

set of optimal linkage parameters. In this chapter, the optimization problem of planar

rnrn-degree-of-freedom and spatial four- and five-degree-of-freedorn parallel niechariisms

is presented in a separate section.

Therefore, the kinematic analysis presented here consists of the inverse kinematics

and workspace analysis. the velocitv equations and singularity loci as tvell as the kine-

matic optimization of mechanisrns with reduced degrees of freedorn. These topics will

be addressed in the following sections.

2.1 Inverse kinematics and workspace

If x is the Cartesian coordinate vector and 6 is the actuated joint coordinate vector.

the inverse kinematic problem can be stated as: given the position and orientation of

the pIatform x, find the corresponding actuated joint variables 0. Since the trajecto-

ries are usually given in Cartesian space while the actuators are mounted at joints: the

solution of the inverse kinematic problem is necessaq for controliing a manipulator.

In this section, the inverse kinematics of the planar two- and three-degree-of-freedorn

as weli as spatial four-, five- and six-degree-of-freedom parallel mechanisms with rev-

olute actuators is solved. It is the foundation for further analysis and design of the

mechanisms.

Moreover, as mentioned above, the determination of the workspace of a manipulator

is a basic requirement for its kinematic design. Based on the fact that the workspace

of manipulators is related to the solution of the inverse kinematics, a nem general

numerical aigorithm for the determination of the workspace is proposed. This algorithm

is used to solve high degree nonlinear equations and to plot the resulting cun-es in

a two-dimensional plane. In the algorithm, the bisection method is combined with

Yewton-Raphson's method to search for the roots of the equations. If it is assumed

that a cross-section of the workspace in the x - y plane is to be found, for given values

of other independent Cartesian coordinates, the procedure c m be described as folloms:

Step 1: In the x - y plane, divide the x axis and the y axis in m and n sections

respectiveIy. T-lt the same time. compute the values of 6 a t points (x,, yJ) (i = 1.2. .... m

and j = 1.2, .... n). where d is an espression arising from the solution of the inverse

kinematic problem and which maj- be reaI (when the prescribed Cartesian coordinates

are inside the workspace) or complex (when the prescribed Cartesian coordinates are

outside the workspace). A typica1 esample is the square root of an espression. There-

fore: the boundary of the workspace can be obtained as the locus of points for which

b = O.

Step 2: Check the values of 6 at points (xi, Yi) and (xi, y,+l) ( j = 1,2, ..., n) and

if one of them is real while the other is not then it means that there is a root in this

interval. Use Newton-Raphson's method to search for the root of equation b = O in the

int erval.

Step 3: Repeat step 2 m times, that is, let i = 1,2, ..., m. Finally, al1 the points of

the boundaries of the workspace of the manipulator are obtained.

Step 4: Plot these points in the x - y plane.

This algorîthm is much slower thaa the algorithm presented in [13]. However, it is

more general and can be applied to any type of manipulator. This algorithm is used

here to plot the boundaries of the workspace of the studied planar and spatial parallel

mechanism.

2.1.1 Planar parallel manipulators with revolute actuators

Planar two- and three-degree-of-freedom parallel manipulators are represented in Fig-

ures 2.1 and 2.2 respectively. -411 their joints are of the revolute type. The two-degree-

of-freedom manipuiator can be used to position a point on the plane and the Cartesian

coordinates associated ivith this manipurator are the position coordinates of one point

of the platform, noted (x: y) (Figure 2.1). The three-degree-of-freedom manipulator

can be used to position and orient a body on the pIane and hence. in this case the

Cartesian coordinates are the position of one point of the pIatform. noted (x. y) . and

its orientation, given by angle 4 (Figure 2.2). Vector 8 represents the actuated joint

coordinat es of the planar parallel manipulator and is defined as 0 = [ Bi ... 8, lT where n is the number of degrees of freedom of the manipulator studied. The only ac-

tuated joints are those which are directly connected t o the fised link (see for instance

[16]. f46] and [19]).

Figure 2.1: Planar tw-degree-of-freedom parallel manipulator.

Figure 2 2: Planar t hree-degree-of-freedom parallel manipulator.

2.1.1.1 Planar two-degree-of-freedom manipulator

1) Inverse kinematics

This manipulator. illustrated in Figure 2.1, has four movable links and five revolute

joints (identified as O1 to Us). The tmo links whose length are I L and 1' are the input

links and the joints OI and O2 are the only actuated joints. The lengths of the other

two links are noted i3 and l4 respectively. Moreover, l5 is the distance from joint

O5 to point P ( x , y). Point P(r , y ) is the point to be positioned by the manipulator.

The origin of the fised Cartesian coordinate system is located on joint O1. 1,Ioreover.

( x o l : go*) and ( x O 2 , yo2) are the coordinates of points U1 and 0 2 respectively. and one

has x , ~ = y , ~ = g,2 = 0-

From the geornetry of the linkage one can write

- COS a,)' + (Y21 - l Z f 2 sin = l?, i = 1: 2

where the intermediate variables ~2~ and ~2~ (i = 1 ,2 ) are defined as

and where angles al and ûr2 represent respectively the orientation of the links of length

l3 and I4 with respect to the b e d link.

From eq.(2.1), one obtains

sin ai

where

where h;: is the ith branch index, which can be used to distinguish the four branches of

the inverse kinematic problem. Finally. the solution of the inverse kinematic problem

for this manipulator can be obtained as

8, = atan2[(yîZ - li+* sin a,) , (x2, - li+? COS a,)]: i = 1.2 (3.12)

where Oi(i = 1, 2) are defined as the angles between the link of length 1, and the x a i s

of the fised Cartesian frame and atan2 is the inverse tangent function which uses 2

arguments and returns a unique value.

II) Determination of the workspace

Considering eqs.(2.6), (2.7): it is clear that whether or not angIe ai is real depends

on Si. If 6, < 0, û.i is not real; if Si > 0, a, can be real; and if &=O, a, has a unique

reaI solution. Because û i is related to the solution of the inverse kinematics of the ma-

nipulator, in fact it means that when Si = O the inverse kinematic problem has f ewr

solutions. Recalling the properties of the first type of singularity of parallel manipula-

tors [16], it folloms that the workspace boundaries of the planar parallel manipulators

consist of the set of configurations which satisfy the following equation in the Cartesian

space of the manipulator.

The problem of determining the boundaries of the workspace of paraIlel rnanipula-

tors is equivalent to solving equation (2.13). Since the latter equation is highly non-

linear, it is usually impossible to obtain analytical solutions. Therefore. the numencal

method mentioned in a preceding section can be used to solve the equation and plot

the workspace limits in a two-dimensiond plane.

-4 numericd example is now given to illustrate the application of the algorithm for

pIotting the boundaries of the workspace.

For this manipulator, let = 0.8, l2 = 1.0, l3 = 1.5, l4 = O.& l5 = 0.6. xol = 0.0.

yol = O.O? xO2 = 2.0 and y02 = 0.0-

Figure 2.3 show ttwo different workspaces with two different values of the branch

indes h',, namel- the workspace of the manipulator is different with KI = 1 and

K I = -1. This is because angle a2 is a function of angle al. In other words, there are

two different workspaces for this manipulator. depending on the value of K I .

(a) Itorkspace 1 of the two-dof manip (b) Workspace 2 of the two-ciof ma-

ulator R-ith Kr = f l . nipulator a-ith Ki = -1.

Figure 2.3: Workspace of the two-dof manipulator

2.1.1.2 Planar t hree-degree-of-freedom manipulator

1) Inverse kiaematics

A planar three-degree-of-freedorn rnanipdator consists of seven movabIe links and

nine revolute joints (identified as Oi to U9f2 as indicated in Figure 2.2. Such an architec-

ture has been studied in [15]. The moting link with three revolute joints is considered

as the ptatform of the manipdator. The three rinks connected to the base are the

input links and the three joints Cl1, 0 2 and 0 3 are the actuated joints. Joints O1 and

are respectively the origin of the fked Cartesian coordinate frarne and the point to

be positioned by the end effector. Tt is assumed here that the three input links have

the same Iength 1, and that the other three links which connect the input links to the

platform have a length 1 2 .

From the geornetry of the linkage, one can m i t e

where a, is defined aç the angle between the ith link of Iength l2 and the L asis of

the fixed Cartesian frame: 13 and l4 are as indicated on Figure 2.2 and where (x,,. y,).

i=1.2,3 are the coordinates of point 0,: with x , ~ = yol = y,2 = y,g = 0: and

Frurii eq. (2.14) one obtains

where

u-here Ki is defined as the ith branch index [14], which can be used to distinguish the

eight branches of the inverse kinematic problem.

Finallx the solution of the inverse kinematic problem for this manipulator is ob-

tained as

= atan2[(y2i - l2 sin C Y ~ ) ( x ~ ~ - i2 COS ai)], i = 1 2 , 3

where Bi(i = 1: 2,3) is defined as the angle between the ith link of length 1 , and the x

axis of the fised Cartesian frame.

II) Determination of the workspace

The workspace boundaries of the planar three-degree-of-freedom parallel manipu-

lator consist of the set of configurations which satisfS. the follon-ing equation in the

Cartesian space of the manipulator.

Similarly to the previous case, in order to obtain the boundaries of the workspace of

this type of parallel manipulator. the algorithm presented in the first section of this

chapter is used ta solve eq.(2.28).

.A numerical example is now presented in order ta illustrate the determination of

the workspace.

The resulting workspace is represented in Figure 2.4.

2.1.1.3 Branches of parailel manipulators

In planar or spatial mechanisms with multiple chahs: branches usually esist. Different

branches may have different kinematic characteristics. Sometimes? it is impossible for a

given mechanism to assume several branches without being disassembled. Hence. i t is

important to be able to distinguish the different branches and to know their kinematic

characteristics in order to choose a configuration which is best suitable for the mani-

pulator to be designed.

Figure '1.4: Workspace of the three-dof manipulator.

From eqs.(2.6) and (3.7) and h m eqs.(2.31) and (2.22), i t is clear that angles

a, (i = 1.2 for the trw-degree-of-freedorn manipulator and i = 1.2 .3 for the three-

degree-of-freedorn manipulator) have two different solutions which depend on the value

of h',. Therefore the inverse kinematics of the manipulators have different sets of

soIutions and h; is the ith branch indes of the manipulator. I t represents the two

different configurations of the ith leg of the manipulator. Hence. the three-degree-of-

freedom manipulator has eight different branches caused bu the different configurations

of its three legs and. similarly. the two-degree-of-freedorn manipulator has four difkrent

branches.

2.1.2 Spatial parallel manipulators wit h revolute actuators

Spatial four-, five- and six-degree-of-freedom parallel mechanisms are respectively rep-

resented in Figures 2.& 2.6 and 2.7.

A fised reference frame O - xyz is attached to the base of the manipulator and a

moving coordinate frame 0'-x'y'z' is attached to the platform. Moreover, the points of

attachrnent of the actuated legs to the base are noted Oi and the points of attachment

of ail legs to the platform are noted Pi, with i = 1:. . . ! n where n is equal to 5: 6 and

6 respectively, for each of the manipulators studied here. Point O = O, is located at

(a) CAD mode1 (b) Schematic representation

Figure 2.5: Spatial four-degree-of-freedom parallel manipulator with revolute actuators.

the center of the joint connecting the nth leg to the base.

The Cartesian coordinates of the platform are given by the position of point O' with

respect to the fised frame. noted p = [x y zjT and the orientation of the platform

(orientation of frame O' - xfy'z' with respect to the fived frame). represented bu three

Euler angles dt 6 and c or by the matri\; Q.

If the coordinates of point P, in the moving reference frame are noted (a,. b,. c , ) and

if the coordinates of point O, in t h e fixed frarne are noted (x,,: yi,, z,,), then one has

where pi is the position vector of point Pi espressed in the fised coordinate frame - and rvhose coordinates are defined as (xi, Yi, 2,) -: p: is the position vector of point

Pt expressed in the rnoving coordinate frame, and p is the position vector of point O'

expressed in the fised frame a. defined above. One can then m i t e

where Q is the rotation matrix corresponding to the orientation of the platform of the

manipulator with respect to the base coordinate frame. This rotation mat ri^ can be

(a) C.4D mode1

L . .

(b) Schematic representation

Figure 2.6: Spatial five-dof parallel manipulator with revolute actuators.

n-ritten as a function of the three Euler angles defined above. Uï th the Euler angle

convention used in the present n-ork. this rnatriu is written as

-soc,

ivhere s, denotes the sine of angle x R-hile c, denotes the cosine of angle x.

2.1.2.1 Spatial four-degree-of-freedom parallel manipulator wit h revolute

actuat ors

1) Inverse kinematics

-4s represented in figure 2.5, this type of manipulator consists of five kinematic

chains. numbered from 1 to 3. connecting the fised base to a moving platforrn. Four of

these kinematic chains have the same topology. The kinematic chains associated mith

these four legs consist - from base to platform - of a fixed actuated revoIute joint,

a moving link, a Hooke joint, a second moving link and a spherical joint attached to

the platform. The fifth chain connecting the base to the platform is not actuated and

has an architecture which differs from the other chains. I t consists of a revolute joint

attached to the base, a moving link and a spherical joint attached to the platform. This

- - -

(a) CAD mode1 (b) Schematic representation

Figure 2.7: Spatial six-dof parallel manipulator with revolute actuators.

last leg is used to constrain the motion of the platform to only four degrees of freedom.

It is pointed out. however. that the manipulator could also be built using only four

legs. i.e.. by removing one of the four identical legs and actuating the first joint of the

special leg. Both arrangements lead to similar kinematic equations.

Since the platform of the manipulator has four degrees of freedom. only four out of

the sis Cartesian coordinates of the platform are independent. In the present study.

the independent Cdrtesian cuordi~iaces have been chosen as [x. y. z. O ) since it is as-

sumed that the manipulator will be used to position a point in space while specifying

a rotation about one axis. The two remaining coordinates. i.e.. Euler angles û and @

can be determined using the constraints associated with the special 5th leg. -4lthough

this choice of coordinates is arbitrary, it can be easily justified by the applications.

Moreover, the analysis reported here can easily be repeated with a different choice of

coordinates. which would lead to very similar results.

Hence, the four independent coordinates (x, y, z , d ) are first specified and the re-

maining Cartesian coordinates describing the pose of the platform are then determined

using the kinematic constraints associated with the special leg. This constitutes the

first step of the solution of the inverse kinematic problem. Hence, eq.(2.30) is first

written for chah 5, Le.,

PS - P = QP;

Taking the square of the n o m of both sides of eq.(2.32), one then obtains

(z5 - x ) ~ + (y5 - y)2 + (z5 - z ) ~ =a: + bg +cg (2.33)

Since the unactuated leg is mounted on a passive h e d revolute joint (Figure 2.5). point

P5 is constrained to move on a circle and hence: the coordinates of this point can be

written as

xg =Essina, y 5 = 0 . Z ~ = & C O S Q (2.34)

tvhere a is the angle defined by link OP5 with respect to the Z avis of the fised

coordinate frarne, as illustrated in Figure 2.8. and l5 is the length of Iink OP5 of the

fifth kinematic chain. It is noted that the fixed coordinate frarne is defined such that

the axis of the fised revolute joint is along the 1' avis of the fixed frame and point O

is located such that point Pi is constrained to move in the S Z plane.

Figure 2.8: Geornetry of the fifth leg of the four-dof mechanism.

Cpon substitution of eq.(2.34) into eq.(2.33) and further simplification. one then

obtains

il41 COS a + B4t sin a = C41 (2.35)

where

The soIution of eq(2.35) then leads to

where K I = k1 is the branch indes associated with the fifth kinematic chain of the

manipulator and where

= .4i1 + ~ 4 2 ~ - Ci1 (3.41)

Once angle a is obtained. eq.(2.32) can be used to compute the dependent Euler angles

v and B. Indeed. multiplying the second component of the latter equation bu cos O and

subtracting from the first component times sin Q. one obtains

.q4? COS li' + BJ2 sin w = CJ2 (2.42)

w h ere

which leads to

tvhere K2 = f 1 is another branch index associated with the orientation of the platform

Moreover, the last two component of eq.(2.32) can be rewritten as

where

cg sin # 1 cos 6 -y - a5 cosdsin (u - b 5 c o s d c o s ~

15c0sa - z 1 Hence. once angles 4 and @ are known, angle 0 can be determined by solving eq.(2.49)

for ts.

Having obtained the Euler angles. the pose of the platform (its position and ori-

entation) is completely known. The rest of the inverse kinematic procedure therefore

consists in computing the actuated coordinates for a given Cartesian pose of the plat-

form. This problem is rather straightforward.

Figure 3.9: Configuration of the i th actuated joint of the four-dof manipulator with

revolute actuators.

Figure 2.9 represents the configuration of the i th actuated joint of the manipulator

with revolute actuators. Point 0: is defined a,s the center of the Hooke joint connecting

the ttvo moving links of the ith actuated leg. Moreover, the Cartesian coordinates of

point 0: expressed in the fixed coordinate frame are noted ( x i l , yil, zil). Since the a ~ i s

of the &xed revolute joint of the ith actuated leg is assumed to be parallel to the xy

plane of the h e d coordinate frame (Figure 2.9). one can mite

where y, is the angle between the positive direction of the x ais of the base coordinate

frame and the auis of the ith actuated joint while pi is the joint variable - rotation angle

around the fked revolute joint - associated with the ith actuated Ieg. Moreover, l t l is

the length of the first link of the ith actuated leg. From the geometry of the mechanism.

one can write

where x , . y,. 2, have been previously defined as the coordinates of point P, and l i2 is the

length of the second link of the ith actuated leg.

Substituting eqs.(2.50). (2.51) and (2 .52) into equation (2.23): one obtains

mhere

R, = (y, - yf0 ) COS ., - (x t - x f 0 ) sin :lt

which leads directly to

& T , - K i 3 S f f l . COS pi = * 2 =1, ...: 4 e2 + Sf

where Kt3 = f 1 is the branch index of the manipulator associated with the configura-

tion of the ith leg and

The solution of the inverse kinematic problem is then completed by performing

pi = atan2[sin pi, cos pi] , i = 1.. , . $ 4 (2.08)

Since two solutions are obtained for each of the pi's, it is clear that the inverse

kinematic problern of this manipulator leads to 64 solutions.

II) Determination of the workspace

Simiiarlÿ to what was done for pIanar parailel mechanisms, the first type of sin-

gularity will be obtained by finding the boundaries of the Cartesian workspace of the

manipulator. Hence. the locus of the limit of the workspace is given by the follolving

equation

xhere LJ1. &2 and I ; are respectively defined in eqs.('i.42). (2.48) and (2.57) and the

same algorithm is used here to determine the curves corresponding to this locus.

An esample is non: given to illustrate the determination of the workspace of this

type of mechanism. The parameters used in this example are given as

and the Cartesian coordinates being imposed are respectivelely

- * j' X

O = - y = 0.0 and d = - x = -1.25 10 ? 10'

Figure 2.10 shows two sections of the workspace for this example.

(a) X section of the workspace (over z (b) A section of the worbpace (over y

and 2). and 2).

Figure 2.10: Esample of the workspace of the four-dof mechanism

2.1.2.2 Spatial five-degree-of-freedom parallel manipulator wit h revolut e

actuators

1) Inverse kinematics

-4s represented in Figure 2.6. the spatial five-degree-of-freedom paralle1 manipulator

mechanisrn consists of sis kinematic chains, numbered from 1 to 6. connecting the fixed

base to a moring platforrn. Five of these kinematic chains have the same topolog'..

The kinematic chains associated with these five legs consist - from base to platform

- of a fked actuated revohte joint, a moving link. a Hooke joint, a second moving link

and a spherical joint attached to the platform (Figure 2.6). The sisth chain connecting

the base to the platform is not actuated and has an architecture which differs from

the other chains. It consists of a Hooke joint attached to the base, a moving link

and a spherical joint attached to the platform. This 1 s t leg is used to constrain the

motion of the platform to only five degrees of freedom. Similarly to the case of the

four-dof mechanism, this mechanism could also be built using only five legs. Le., b~

removing one of the five identical legs and actuating the first joint of the special leg.

Both arrangements lead to similar kinematic equations.

Since the platform of the mechanism has five degrees of freedom, only five of the

six Cartesian coordinates of the platform are independent. In the present study, the

independent Cartesian coordinates have been chosen as (x: y, z, 4: 8) since it is assumed

that the mechanism will be used to position a point in space while specifying two

independent rotations. The remaining coordinate, Le., Euler angIe I,!J can be determined

using the constraints associated with the special 6th leg. Sirnilarly to the four-degree-

of-freedom parallel mechanism, although this choice of coordinates is arbitrary. it can

be justified by the applications to be considered? i.e., the position and orientation of

avisymmetric bodies, and the analysis reported here can easily be repeated with a

different choice of coordinates, which would lead to very similar results.

Hence. the five independent coordinates ( x . y, z, 4: 8 ) are first specified and the

remaining Cartesian coordinate describing the pose of the platform is then determined

using the kinematic constraints associated with the speciaI leg. Hence. eq.(2.30) is first

written for chain 6. i.e..

p6 = P + QP; (2.60)

or. in terms of the components.

+CG COS O sin O (2.61)

gs = y + a 6 ( s i n o c o s 8 c o s ~ t c o s 0 s i n d ~ ) + b 6 ( - s i n o c o s ~ s i n ~ ~ + c o s ~ c o s z ~ )

+cg sin d~ sin 8 (2.62)

2~ = zf a 6 ( - s i n 8 c o s ~ ~ ) + b ~ s i n ~ s i n e . + c ~ c o s ~ ( 2 . 6 3 )

l~loreover. from the geometry of the special leg, one can wite

where is the length of the special leg.

Squaring both sides of equations (2.61). (2.62) and (2.63) and adding, one then

ob tains

-4 cos u: + B sin u: = C (2 .65)

where

A = 2b6y cos 4 + aex cos (4 - 8 ) + a ~ x cos (4 + 8) - 2b6x sin @ + as y sin (4 - 8 )

-Sa6% sin 9 + a6y sin (4 + 8 ) (2.66)

B = îa6y cos 4 - b6x cos (4 - 8) - bsx cos ( 4 + 8 ) - 2a6x sin Q, - b6y sin (4 - O)

+2b6z sin 0 - b6ysin (4 + 0) (2.67)

C = 1 ~ - ( a ~ + b ~ + c ~ + ~ ~ + ~ ~ + Z * + C ~ ~ C O S ( @ - ~ ) + ~ C ~ Z C O S O - C ~ ~ C O S ( @ + O )

-c6x sin (6 - 8) + ccx sin (4 + O ) ) (2-68)

The solution of eq.(2.65) then leads to

sin zO = BC + &.-ta

*A2 + B2

rs-here h6 = 5 1 is the branch index associated wîth the sivth kinematic chain of the

mechanism and where

1 = -42 + B~ - C- (2.71)

Having obtained the last Euler angle. the pose of the platform (its position and orien-

tation) is completely known. The rest of the inverse kinematic procedure therefore con-

sists in computing the actuated coordinates for a given Cartesian pose of the platform.

It is identical to what is encountered in the four-degree-of-freedom parallel rnechanism

and the procedure is not repeated here.

Since two solutions are obtained for each of the pi's, it is rlear that the in!-erse

kinematic problem of this mechanism leads to 64 solutions, which can be distinguished

using branch indices to K6.

II) Determination of the workspace

Similarly, for this mechanism the locus of the limit of the workspace. namelu. the

first type of singularity locus, are given by the foIlowing equation

where is defined in eq.(2.57) and 1 is defined in eq.(2.71).

An esample is now given to illustrate the determination of the workspace of this

type of mechanism. The parameters used in this example are given as

and the three fked Cartesian coordinates are respectively given by

Figure 2.11 shows two sections of the workspace for the above esample.

(a) -4 section of the workspace (over x (b) A section ~f the workspace (over y

and 2). and 2).

Figure 2.11: Esample of the workspace of the five-dof mechanism.

2.1.2.3 Spatial six-degree-of-freedom parallel manipuiat or wit h revolute

actuators

AS represented in Figure 2.7, spatial six-degree-of-freedom parallel manipulator consists

of six identical legs connecting the base to the platform. Each of these legs consists of

an actuated revolute joint attached to the base, a first moving link, a passive Hooke

joint, a second moving link and a passive spherical joint attached to the platform. .A

parallel manipulator of this type was described in [8].

The procedure for the cornputation of the inverse kinematics therefore consists in

computing the actuated joint coordinates for a given Cartesian pose of the platform.

It is esactly identical to the procedure of the last part of the computation of the

inverse kinematics of the four- and five-degree-of-freedom parallet mechanisms and is

not repeated here.

Similarly, for this mechanism the locus of the limit of the workspace is given by the

following equation

where 1; is defined in e q . ( X Ï )

However: it is pointed out here that for this rnechanism there esists an algorithm

to find the analytical description of the boundary of the workspace [13].

.An esample is now given to illustrate the determination of the workspace of t h

type of mechanism. The parameters used in this esample are given as

and the four fixed Cartesian coordinates are respectively given by

Figure 2-12 short's two sections of the workspace for this exampIe.

(a) -4 section of the n-orkspace (over J (b) -4 section of the workspace (over y

and z ) . and 2).

Figure 3.13: Esample of the workspace of the sis-dof mechanism.

Velocity equations and singularity loci

The velocity equations of manipulators are important for their kinematic analysis.

Indeed, since the velocity equations represent the linear mapping between the actuated

joint velocities and the Cartesian velocities: theÿ characterize the kinematic accuracy of

a manipulator and they allow the determination of the singularities. Here, two different

methods will be used to derive the velocity equations of the manipulators studied. Each

of these methods has its own advantages and drawbacks, as will be shown belon?.

Moreover, the determination of the singular configurations of mechanisms is an issue

of the utmost importance. since when the singularities occur, the end-effector of the

manipulator gains or loses one or more degree(s) of freedom and therefore becomes

uncontrollable. Hence, such configurations must be avoided.

In [16]. three general types of singularities which can occur in parallel manipulators

have been identified. '1Ioreover. in [66], this classification has been further refined and

several types of singularities have been defined according to their physical interpre-

tation. However. the mathematical description of these types of singularities is more

difficult to obtain and hence. the classification proposed in [16] will be used here. In-

deed. the mathematical espressions allowing the identification of these singularities is

readily available.

In x-hat follows. ~ e c t o r 8 is used to denote the actuated joint coordinates uf the nia-

nipulator. representing the vector of kinematic input. 'vforeover. vector x denotes the

Cartesian coordinates of the manipulator gripper, representing the kinematic output.

The velocity equations of the manipulator can be written as

where

and where A and B are square matrices of dimension n, calIed Jacobian matrices. with

n representing the number of degrees of freedom of the manipulator.

Referring to eq.(2.74), Gosselin and Angeles [16] have defined three types of singu-

larities which can occur in paraiiel manipulators.

1) The first type of singularity occurs when det(l3) = O. In such a situation, the

gripper of the manipulator loses one or more degrees of freedom and lies in a deadpoint

position. In other words, the gripper can resist one or more forces or moments nithout

exerting any torque or force at the actuated joints. These configurations correspond

to a set of points defining the outer and internal boundaries of the workspace of the

manipulator. They have been studied in the previous section of this chapter.

II) The second type of singularity occurs when det(A) = O. As opposed to the first

one. the gripper of the manipulator gains one or more degrees of freedom, namely, it

cannot resist the forces or moments from one or more directions even when al1 actuated

joints are locked. The actuated joints are at a deadpoint. This kind of singularity

corresponds to a set of points within the xorkspace of the manipulator.

III) The third kind of singularity occurs when the positioning equations degenerate.

This type of singularity is also referred to as an architecture singularity [61. Only

when the parameters of a manipulator satis- certain special conditions can this kind

of singularity occur. It corresponds to a set of configurations where a finite motion of

the gripper of the manipulator is possible even if the actuated joints are Iocked (referred

to as self-motion in [27Il [28] and [29]) or where a finite motion of the actuated joints

produces no motion of the gripper.

The singularity classification rnethod presented abox-e is applicable to any parallel

manipulator. In this Chapter. it is used to analyze the second type of singularity

of the manipulators introduced earlier in order to find Cartesian loci associated with

this type of singularity. For the first type of singularity, the loci can be obtained

by computing the boundary of the workspace of the manipulator. which has been

completed in the previous section and will not be discussed any further here. The second

type of singularity loci will be found from the espressions of the determinant of the

Jacobian matrices of the mechanisms. as will be shown in the following. Furthermore.

it is assumed that the third type of singularity is avoided by a proper choice of the

kinematic parameters.

In [47], the singularity loci of pIanar three-degree-of-freedom parallel manipula-

tors with prismatic actuators have been obtained. Because of the prismatic actuators,

the mathematical expressions obtained were very simple and the loci were shown to

correspond to quadratic forms. However, for manipulators with revolute actuators.

expressions are much more complex as d l be shown next and the simple approach

used in the latter reference cannot be applied directly Similady, for spatial six-degree-

of-freedom parallel manipulators with prismatic joints, an expression for the singularïty

loci has been obtained in [di]. It was shown. in the latter reference. that for a given

orientation of the platform, the loci were surfaces of degree 3. However: this approach

cannot be directly applied to manipulators with revolute actuators, since the joint

coordinates and the Cartesian coordinates both appear in the Jacobian rnatis.

2.2.1 Planar parallel manipulators

The velocity equations of planar two- and three-degree-of-freedom parallel mechanisms

or manipulators will first be derived. and then used to determine the singularity loci of

type II.

Let a, be the two-dimensional position vector connecting the ith joint O, to one

of its neighboring joints. as indicated in Figures 2.13 and 2.16 (escept a3 and as on

Figure 2.13. which connect O3 and O5 to P and as in Figure 2.16. which connects O;

to 0 9 ) . and let dl be the angular velocity of the ith link. From these definitions. it is

possible to obtain the velocity equations. as will now be illustrated.

2.2.1.1 Two-degree-of-freedom rnanipulator

Using the closed loop of the two-degree-of-freedom manipulator 0103050402r one can

write

where

is a 2 x 2 operator rotating an arbitrary two-dimensiond vector counterclock~vise

Figure 2.13: Loop vectors associated n-ith the two-degree-of-freedom manipulator.

through an angle of 90" [45]. C'sing eq.(2.73) as the velocity equation of the manipula-

tor is equivalent to using the angular velocity of links of length l3 and l4 to represent

its Cartesian velocity instead of using the velocity of one of the points on the platform.

Then. frorn eq.(2.75): one obtains

where

Equation (2.76) can be used as the velocity equation of the two-degree-of-freedom

manipulator since for this manipulator, we can use the angular velocities of links 3 and

4 as the output vector. From eq.(2.76), A* is then written as

( 1 3 - 15) COS QI -LJ COS a2 A * = [

( 1 3 - 1 5 ) sin ai -14 sin a2 I

From eq.(2.77), one then obtains

From eq.(2.78), it is clear that when 01 = a:!+nn, n = O, f 1: f 2, ...: then det(A*) =

O. In other words, if the two vectors a3 and a4 are aligned, the manipulator is in a

configuration which corresponds to the second type of singularity The polynomial

formulation of the singularity loci for this type of mechanism is given in ilppendix A.

It leads to a curve of degree 6.

The determination of the singularity loci of the manipulators consists in finding

the roots of equation det(A8) = O. This can be accomplished using the expressions of

eqs. (2.78) direct lu. The detailed expressions associated wit h the equation are g i ~ e n in

-4ppendLx B.

The above equations can be solved using a numerical procedure in order to determine

the singularity loci for each of the branches and to represent them graphically. This

results in the singularity locus according to each branch of the manipulator within its

workspace. The singularity loci of al1 the branches of the manipdator can aIso be

superimposed on the workspace.

IYith the same parameters used in the esample on the determination of the workspace

of the manipulator. one can obtain the singularity loci over the workspace in each

branch. as shown in Figures 2.14 and 2.15.

The manipulators in Figures 2.14(b) and 2.14(d) are showri in a singular configu-

ration while they are represented in a nonsingular configuration in Figures 2.14(a) and

2.14(c). The singularity loci of al1 branches in the two workspaces of the manipulator

are shown in Figures 2.15(a) and 2.15(b), respectively.

2.2.1.2 Three-degree-of-freedom manipulator

From Figure 2.16, it is clear that this manipulator contains two independent closed

loops: for instance 010407080502 and 01040708090603. Therefore, by using the

loop constraints. one can write the following velocity equations:

(a) Branch 1: KI = - 1. K2 = -1.

(c) Branch 3: KI = + 1. & =

-i.

(b) Branch 2: K, =

-1, h-2 = +l.

(d) Braxh 4: hPl =

+l. K3 = +l.

Figure 2.14: Singularity locus and workspace of the two-dof manipulator.

dlEal + d4E* + d7E* = w3Ea3 + u16Ea6 (2.80)

From eqs. (2.79) and (2.80), one obtains

AWwo + B*W[ = O

(a) Singuiarity Iocus of workçpace 2. (b) Singularity locus of worlrspace 2.

Figure 2.15: Singularity Iocus and workspace of the two-dof manipulator.

Equation (2.81) can be used as the velocity equation of the three-degree-of-freedom

rnanipulator. where the angular velocities of the input links are used as the joint ve-

locities and the angular velocities of the other moving links are used as the Cartesian

velocities.

Comparing eqs.('l.51) and (2.74). it is possible to use the angular velocities of links

4, 5 . 6 and 7 as output velocities. and hence, to write

Therefore, considering eqs. (2.82); (2.83) and (2.81) together, it can be concluded

that if det(A8) = O. then the rnanipulator is in a singular configuration corresponding

to the second type of s ingu la r i - In other words, matrices A* and A are equivalent

for the purpose of determining the second type of singularity of the manipulator. In

a singular configuration. since the nullspace of A* is not empty, there esist nonzero

vectors wo which will be mapped into the origin by A*.

Figure 2.16: Loop vectors associated with the three-degree-of-freedom manipulator.

It can be easily shown that when the three vectors cq, as and are parallel to one

another or intersect a t a comrnon point Q(x,: y,) . as represented in Figure 2.17. then

det (Ae) = O. namely, the second type of singularity of the rnanipulator occurs. This

result has been obtained in [23].

Matris A' can be written as

1 l2 sin ai -12 sina, O l3 sin d A' =

O -i2 cas a3 l4 COS d

where the notation for the Iink lengths refers to Figure 2.2.

From eq.(2.84), one has

det(AB) = 1; sin ci(& cos al sin a;! cos a3

-lq sin al cos (Y* COS (YS

-13 COS û1 COS a2 sin a3

+13 sin al cos a2 COS 0 3 )

+z: COS 4(i4 sin al cos a2 sin 03

-Iq COS a, sin a2 sin a3

-13 sin al sin a;! cos u3

+13 COS QI sin a:! sin a3)

and two cases may mise.

a) If %( as and a~ are paraIlel to one another, then

Figure 2-17: Three-degree-of-freedom manipulator in a configuration corresponding to

the second type of singularity.

The substitution of eqs(3.86) and (2.87) into eq.(2.85). leads to

b) If the three vecton 4, ai and intersect a t a cornmon point Q ( x q . yq) (as

represented in Figure 2.17). then one can mi te

tan ÛLX, - y, + (y21 - x21 tan a l ) = O

tan aax, - y, + (y22 - 5 2 2 tan a*) = O

tan û 3 x q - y, + (y23 - 5 2 3 tan cr3) = O

where al, a:! and a3 # n/2 + mn, m = O, f 1, f 2, ....

Eqs.(2.88)-(2.90) constitute a linear system of 3 equations in two unknowns xq, y,.

Hence, the following condition is required to obtain a real solution for x, and y,:

where H is a matrk defined as

[ tan al -1 (yz1 - xZ1 tan a l )

H= tanu;? -1 (y2* -x22tanû2)

tan a3 - 1 - xZ3 tan as)

The substitution of eqs.(2.15)-(2.20) into eq.(2.91) leads to

det(H) = det (A') cos 01 cos a2 cos as = O

Since cos a1 COS a2 COS a3 # O: from (2.88)-(MO), one finally obtains

Therefore. in the formulation derived above, matrices A' can be used to investigate the

singularities of both manipulators.

Similarly to the previous case. the polynomial formulation of the singularity loci

for this type of mechanism is also found in Appendis -1 and the detailed expressions

associated with the equation are given in -4ppendis B. -1 polynomial of degree 64 is

obtained in this case.

For this manipulator. let I I = 1.2, l2 = 1.5, l , = 0.7: 1, = 1.6. x,, = 0.0: y,, = 0.0.

xO2 = 1.0, y02 = 0.0. x , ~ = 2-4. yos = 0.0 and Q = ~ / 1 2 .

The singularity loci of the eight branches of the manipulator are shotvn on Fig-

ures 2.18 and 2.19, where the outer and interna1 solid curves are the boundaries of the

workspace of the manipulator-they are also the locus of the first type of singularity

of the manipulator-and where the cunres within the workspace are the locus of the

second type of singularity of the manipulator.

The branch configurations of the three-degree-of-freedom manipulator are shown on

each graph. Figures 2.18(b), 2.18(c) and 2.19(d) show the rnanipulator in a configura-

tion corresponding to a singularity of type II while on the other graphs. the manipulator

is represented in a nonsingular configuration.

(a) Branch 1: h'l =

-lqK2 = -1. hj = -1.

(b) Branch 2: h'l = -1, K2 = -1. K3 = +l.

(c) Branch 3: K I = (d) Branch 4: Ki =

-1.- = +1.h-3 = -1. - l . K 2 = +l.h3 = i-1.

Figure 2.18: Singularity locus and workspace of the three-dof manipulator

branches 1-4.

2.2.2 Spatial parallel manipulators

In this subsection, the velocity equations of the spatial four-, five- and six-degree-of-

freedom parallel mechanisms or manipulators are first derived using two approaches.

namely, the algebraic and vector formulation.

The first approach presented for the derivation of the velocity equations consists in

a direct differentiation of the kinematic equations derived above. I t is the most popular

approach used for the derivation of the velocity equations of parallel manipulators [16].

The second approach, as mentioned in the introduction, is a new approach. In fact,

(a) Branch 5: KI =

+1,K2 = -1,K3 = -1.

(c) Branch 5: KI =

-il.K2 = +l.K3 = -1.

(b) Branch 6: Kt =

+1. K2 = -1, K3 = +l.

(d) Branch 6: KI =

+I ,& = +l.& = +l.

Figure 2.19: Singularity locus and workspace of the three-dof manipulator wi th

branches 5-8.

it is the extension of the approach used above for the planar parallel mechanisms to

comples spatial parallel mechanisms.

2.2.2.1 Spatial four-degree-of-freedorn parallel manipulators with revolute

actuators

1) Algehraic formulation

Differentiating eq. (2.54) with respect to timel one obtains

lii (Si COS pi - & sin & ) P i

where x,, yi and i, can be obtained by the differentiation of eq.(2.30) with respect to

time, namely

where one has. for i = 1 to 4:

C:, = a,(-sin#cosûcos(1:-cosQsinv) +bi(sindcos6sin~-cos~cos~~r:)

+ci ( - sin Q sin 8)

C:, = a i ( - c o s d s i n û c o s ~ ) + b i ( c o s ~ s i n 6 s i n ~ ) + c , c o s ~ c o s û

c:, = a, (- cos o cos 6 sin w - sin @cos G) + bi(- cos 4 cos 6 cos zr + sin d sin v)

Cio = a, (COS 4 COS O COS y - sin O sin ~ 2 ) + bi (- cos d~ cos 6 sin v - sin o cos u )

+c, cos 6 sin 19

C e = a i ( - s i n ~ s i n û c o s ~ ) + b i ( s i n o s i n 9 s i n ~ ) + c i s i n ~ c o s ~

CÇ, = a, (- sin Q cos t9 sin c: + cos d cos .c) + b, ( - sin O cos 9 cos u! - cos o sin L!)

C o = O

Ci, = a i ( - cos6cos~) i -6 , cosOs in~-c5s in6

c:* = ai (sin I9 sin zt) + b, sin 8 cos

Substituting eqs.(2.94)-(2.96) into eq.(2.93) then leads to

where vectors t and i, are defined as

Moreover, matrices 8, and K, can be written as Br = diag[b; b; 6; 641 and

with. for i = 1.. . . : 4,

br = lil(Sicospi -&sinpi) , i = 1 . . . . : 4

k> = xi - xio + lil sin yi cos pi

k:, = y, - y,, - lZ1 cos sin p,

k:. = 2, - :,, - lil sin pl

= C:,(x, - x,, + lIl sin -/, cos p l )

+C;*(Y~ - Y10 - 111 COS 7, sin P,)

+Cro(=, - =,, - lil sin pl)

1"15 = C:e(~, - xl0 + 111 sin r, cos p,)

+Cie(~i - Y ~ O - 111 cos 11 sin PI)

+Cfe(=l - zto - lI1 sin pl)

= C:e (xl - xIO + Ill sin Î, cos pl )

+Cfiv(yl - Y10 - 111 cos 31 sin PI 1 +Cf,,(=, - zI0 - lIl sin pl)

Then, from the geometry of the 5th leg of the manipulator. Le.. from eq.(2.32), one can

w i t e 2 2 x 5 + g =Es, and y5 = O (2.99)

Differentiating eqs. (2.99), one then obtains

xjx5 +z& = 0. and j15 = O (2.100)

where x5: y5. Zj can be obtained by the differentiation of eqs. (2.32). i.e..

Substituting eqs.(2.101)-(2.103) into eqs.(2.100) then leads to an expression of the

dependent Cartesian velocities as a function of the independent velocities which can be

w i t t en as

where x = [ 3 ~ i dlT and

where

with C = (qC& + I ~ C ~ ~ ) C ; , -CEe (xjC&+r&~,). Eq.(?.lO-l) represents the kinernatic

velocity constraints associated with the special unactuated kinematic chain - the fifth

ieg - connecting the base to the platform.

FinaHy, substituting eq.(2.104) into eq.(2.97), one obtains the veiocity equations of

the four-degree-of-freedom manipulator 116th revolute actuators. Le..

where A, is a 4 bÿ 4 matrix which is defined a s A, = K,J,.

II) Vector formulation

Considering the quantities defined in Figure 2.20, one can write the velocity equa-

tions associated with the closed loop O ~ O : E P ~ ~ ! for i = 1, . . . , 4 , as

where w,, ( 2 = 1,. . . , 4 and j = 1,2) is the vector of the angular veIocity of the j t h

link of the ith leg. while l i l (i = 1,. . . ,4 ) is the vector connecting point Oi to point 0:

and I i2 ( 2 = 1.. - . , 4 ) is the vector connecting point 0: to point P,. Eq.(2.107) can be

Figure '2.20: Position and velocity vectors associated with the spatial four-degree-of-

freedorn paraIlel manipulator with revoiute actuators.

rewrit ten as

-Lilei& = Li2ut2 + V ~ W i- Cuc, i = 1, . . . , 4 (2.108)

ivhere the cross products appearing in eq.(2.101) have been written in matris form and

where is the onIy nonvanishing component of vector us: ive.. the latter vector cari

be written as

ws = [ O uc O I T (2.109)

and where vector wtl has been written as

W ~ I = piei, i = 1,. . . , 4 (2.110)

with ei the unit vector associated with the direction of the asis of the actuator. The

matrices used above are defined as

Vi = [Il x v;, i = 1, . . .?- l

C = C p 5 Z 0 p5zIT

Li1 = [l] x lii, i = 1: .. ..4

Li2 = [1] x li2, i = 1. . . . , 4

where 1 stands for the 3 x 3 identity matrix. Vectors li2(i = 1,. . . , 4 ) are easily wï t ten

p , (x,. y,. z)

O,? :

Figure 2.21: Léctor lI2 represented in sphericai coordinates.

using sphericai coordinates. as shown in Figure 2.21. Hence. one has

where

Substituting eqs(2.112) into eq.(?.108) and using eq.(2.110), one obtains the velociry

equations of the manipulator as

B r u @ = J r v w

where Br, is a 12 x 4 matris defined as

where O denotes a zero 3-dimensional column vector, with p the joint velocity vector

defined as /> = [ P I h h D 4 1 T , with vector w a 12-dimensional vector defined as

and with J,, a 12 x 12 matriv defined as

where O denotes a 3 x 2 zero m a t e and where

J,, =

III) Determination of the singularity loci

O SP O 0 v 2 c O O S3 O v3 C

The determination of the second type of singularity loci consists in finding the roots

of the following equations: Le..

det (A,) = O or det (J , , ) = O (2.119)

It can be noticed that the procedure to obtain matriv A, using the algebraic approach

is much more complicated than the one followed to obtain matriv J,. using the T-ector

rnethod. which means that the espressions of the elements of matris A, will be miich

more comples than the espressions of the elements of matris J,,. However. the former

matrix is of dimension 4 x 4 while the latter is of dimension 12 x 12 and the expression

of the determinant of a larger matriu will be much more complicated than for a smaller

matriu. On the other hand, matriv J,,, is rather sparse and it is therefore easy to reduce

the expression of its determinant to the expression of the determinant of a 4 x 4 matris.

The procedure for this simplification is given in Appendix C. Hence, globally, the use

of the matrices obtained with the vector method will provide simpler expressions which

will therefore lead to faster algorithms.

The loci of the singularities of the second type are determined using

From the above equations. it is clear, by inspection, that D, # O. Therefore.

eq.(2.120) reduces to

det (J) = O (2.121)

where J is a 4 x 1 matrix and hence. the determination of the singularity loci is greatly

simplified.

Al1 the esamples presented below have been produced using the two sets of velocity

equations described above (O btained with the algebraic method and with the vector

nietliiid). Bùth approadies have given identical results and the vector method has ied

to faster cornputation times.

Since the manipuIator has four degrees of freedom, its workspace is four-dimensional.

In order to be able to visualize the results: a computer program has been written

in which it is possible to fix two of the Cartesian coordinates and to obtain a two-

dimensional section of the workspace associated with the two other coordinates and on

which the limits of the workspace and the singularity loci can be plotted.

Csing the parameters used in the example of determination of the workspace of the

mechanism, one can obtain the singularity locus over the workspace in two sections, as

represented in Figures 2.22.

(a) X section of the singularity loci (b) -2 section of the singularity loci

(over x and :) for the manipuiator and (over y and u) for the manipulator and

rc-ith Kdi = 1.& = 1.1; = l(i = with = l,hF4? = 1.1; = l(i = 1.. ... 4 ) . 1,. -. .4).

Figure 2.22: Singuiarity locus and workspace of the four-dof manipulator.

2.2.2.2 Spatial five-degree-of-freedom parallel manipulators with revolute

actuators

1) Algebraic formulation

Sirice the identical actuated legs of the four-. five- and six-degree-of-freedom mech-

anisms have the same architecture. one can directly use eq.(2.5-4) for the derivation of

the velocity equations of the mechanism.

Differentiating eq.(2.54) with respect to tirne: one obtains eqs.(2.93)-(2.96) with the

same coefficients, except that i ranges nouT from 1 to a.

One then obtains an equation identical to eq.(2.91) but with

8 = [ P i A 6 3 8 ,&lT B, = diag[b; b; 63 b; b;J

and

where the kil are defined after eq.(2.98) but should now be taken for i = 1:. . . 5 .

Nowl in order to include the constraint associated with the special sixth leg eq.(2.64)

is differentiated nith respect to time, which leads to

Setting i = 6 in eq~~(2.94)-(2.96)-which is justified since the sixth leg is non- considered-

and subçtituting the latter equations in eq.(2.123). one obtains

with

and

and

Eq.(2.124) represents the kinematic velocity constraints associated with the special

unactuated kinematic c h i n - the sixth leg - connecting the base to the platform.

Finally, substituting eq.(2.124) into eq.(2.97) written for the five-degree-of-freedom

mechanism one obtains the velocity equations of this mechanism, i.e.:

BrP = A,x (2.126)

where A, is a 5 by 5 matrix which is defined as

A, = K,Jr

II) Vector formulation

Considering the quantities defined in Figure 2.23. one can write the veIocitx equa-

tions associated with the closed loop O ~ O ~ P ~ P ~ O . for i = 1, . . . , S ! as

W ~ I x lil + x Ii2 +O x ~i = W C x pst i = 1: - - - . 3 (2.127)

where oij (i = l t . . . . 5 and j = I f 2) is the angular velocity vector of the j th link of

the ith leg. while lil (i = 1.. . . , 5 ) is the position vector from point Oi to point 0: and

li2 (i = 1 . . . . , 5 ) is the position vector from point 0: to point Pz.

Eq.(2.127) can be rewritten as

-lil x ~~1 = l i2 x w,? f vi x w - p6 x W C ! i = 1:. . . . S (2.138)

Moreover. the cross products appearing in eq.(3.128) can be written in matris form.

which leads to

-Llletei = LZ2wi2 + V,W - P6w6. i = 1.. . . . 3 (2.129)

where vector wt l has been written as

W ~ I = piei, i = 1 , . . . , 5 (2.130)

where ei is the unit vector associated with the direction of the avis of the ith actuator.

Moreover, the matrices used above are defined as

Figure 2.23: Position and velocity vectors associated with the spatial fiz-e-degree-of-

freedom parallel mechanism with ret-olute actuators.

where lilzI lily and lilz are the components of vector 1,1 and Il2=. 112y and 1,2z are the

components of vector li2-

One can represent vectors li2(i = 1.. . . . 5 ) and vector PB using spherical coordinates.

as shosvn in Figure2.21. Hence, one has

and

W6 = G 6 ~ 6 s

mhere G, is defined in eq.(2.113) (non- for i=l , . . . ,6) and

Substituting eqs.(2.131) and eq.(2.132) into eq. (2.129): one obtains

11- here

Tz=Li2GiI i = 1: ...

and Vi and Si are as previously defined.

Finally, the velocity equations of the five-degree-of-freedom mechanism with revo-

lute actuators is obtained Gom eq.(2.134) as

SIoreover. vector w is a 15-dimensional vector defined as

where Br, is a 15 x 5 rnatrix defined as

and matri. J,, a 15 x 15 matri. defined as

B,, =

where O denotes a 3 x 2 zero matris.

III) Determination of the singularity loci

The determination of the second type of singularity loci consists in finding the roots

of the following equations, Le.,

det(A,) = O (2.140)

where O denotes a 3-dimensional zero column vector. and p is the joint veiocity vector

previously defined.

- -Luel O O O O - O -L2& 0 O O

O O 4 3 1 e 3 O O

0 0 0 -L41e4 0

- O O O O -Lw% -

Since matris J,, is rather sparse, it is therefore easy to reduce the expression of its

determinant to the expression of the determinant of a 5 x 5 matrix. Hence. the use of

the matrix obtained with the vector rnethod wiIl lead to faster algorithms.

(2.13'7)

where

i= 1

with vector d, defined as

where

wit h

From the above equations. it is clear. by inspection. that Do # O. Therefore.

eqs.(?.lclO) and (2.141) can be reduced to

det (J) = O (2.148)

where J is a 5 x 3 matrix and hence. the determination of the singularity loci is greatly

simplified.

IV) Example

Two examples are given to illustrate the results. Since the mechanisnis have five

degrees of freedom. their workspace is five-dimensional. In order to be able to visualize

the results, three of the Cartesian coordinates are fixed and one then obtains a two-

dimensional section of the workspace associated with the other two coordinates on

which the limits of the workspace and the singularity loci can be plotted.

C'sing the same parameters as the ones used in the example on the determination of

the workspace of the mechanism, one can obtain the singularity locus over the workspace

in two sections, a s represented in Figures 2.24.

(a) a section of the aorkspace (over z (b) a section of the workspace (over y

and 2) with K, = 1. (i = 1 , . . . -6). and z ) with K, = 1Ji = 1.. . . .6).

Figure 2.24: Singularity locus and workspace of the five-dof manipulator

2.2.2.3 Spatial six-degree-of-freedom parailel manipulators with revolute

actuators

1) AIgebraic formulation

Similarly to the previous cases. differentiating eq.(2.54) with respect to time. one

obtains

(SL COS p, - R, sin p, )p , = (1, - x,, + 4 1 sin -f, cos p,)x, -+ (y, - y,, - l , , cos -., sin p, )y ,

+(zl - Zia - Ill sin p, ) i , , i = 1, . . . . 6 (2.149)

Substituting eqs.(2.94)-(2.96) into eq.(2.149), one obtains the velocity equations of the

sis-degree-of-freedom mechanism with revolute actuators. i.e..

ahere vector t has been defined above and where @ = [ P I fiÎ fi3 b5 & l T and

matrices Br and Kr can be written as

Br = diag [bf b; bj b; bg bk ] (2.151)

and

where the ki j are defined after eq.(2.98) but should now be taken for i = 1: . . . , 6.

Since no constraining kinematic chahs are introduced in this manipulator. the ve-

locity equations are rather simple.

II) Vector formulation

Similady, considering the quantities defined in Figure 2.25: one can mi te the veloc-

ity equations associated with the closed loop o ~ o : P ~ P ~ o ~ , for i = 1.. . . .5 , as

where w,, ( i = 1, . . . . 6 and j = 1.2) is the angular velocity vector of the j th link of

the i th leg. while I l l ( i = 1.. . . .6) is the position vector from point O, t o point 0: and

l l2 ( i = 1. . . - - 6 ) is the position vector from point 0: to point PL.

Eq(2.149) c m be rewritten as

The cross products appearing in eq.(2.154) can be written in matris form

where vector wil has been written as

where ei is the unit vector associated with the direction of the axis of the ith actuator.

Moreover, the matrices used above are defined after eq.(2.112), but now for i = 1 , . . . .6.

where E , I , , E i l g and l t l = are the cornponents of vector l i l and li2zl l i2y and lizz are the

components of vector liz: and where matrices Vi has also been defined after eq.(2.111).

Figure 2.25: Position and velocity vectors associated with the spatiai sis-degree-of-

freedorn parallel mechanism n-ith revolute actuators.

One can represent vectors li2(i = 1,. . . ! 6) and vector p6 using spherical coordinates.

as represented in Figure2.21. Hence, one has

where Gi r a s defined in eq.(2.113) and w,, = [c i , 3,lT? i = 1.. . . 6-

Substituting eqs.(2.l;6j and eq.(S.i57j into eq.(2.155). one obtains

where

T,=Li2Gi, i = 1 , ..., 6

and V i and S, are as previously defined.

Finally, the velocity equations of the mechanism are obtained frorn eq.(2.158) as

where Bru is a 15 x 6 matrix defined as

where O denotes a 3-dimensional zero column vector, and p is the joint velocity vector

previousIÿ defined.

Moreover, vector w is a 15-dimensional vector defined as

.4gain, the determination of the second type of singularity loci consists in finding

the roots of the following equations. i.e.,

and rnatris J, , a 15 x 13 matrix defined as

Vatrix J,, is rather sparse and it is therefore easy to reduce the expression of its

determinant to the expression of the determinant of a 5 x 5 matriu. Hence, globally the

use of the matrices obtained with the vector method wilI provide simpler expressions

which will therefore lead to faster algorithrns.

J, , =

-Tl O O O O VI -TG

O T2 O O O V2 -T6 0 0 Tg 0 0 V3 -Tg

0 0 0 Tq 0 V4 -Tg - O O O O T5 Vj -TG

where O denotes a 3 x 2 zero matris.

III) Determination of the singdarity loci

where

with vector h2 defined above, with vector d, defined in eq.(2.144) and where matriv J

can be written as

where h and u are defined respectivelÿ in eq~(2.146) and (2.147).

From the above equations, it is clear, by inspection, that Do # O. Therefore.

eqs.(2.161) and (2.163) can be reduced to

where J is a 5 x 3 matris and hence, the determination of the singularity loci is greatIy

simplified.

N) Example

Similarly to the other cases, in order to illustrate the results. two esamples are

presented below. Since the mechanism has s is degrees of freedom. its workspace is

sis-dimensional. In order to be able to visualize the results. one has to fis four of

the Cartesian coordinates and then obtain a two-dimensional section of the workspace

associated with the other two coordinates on which the limits of the workspace and the

singularity loci can be plotted.

Using the same parameters as the ones used in the esample on the determination of

the workspace of the mechanism, one can obtain the singularity locus over the workspace

in two sections, as represented in Figures 2.26.

(a) A section of the workspace (over x (b) -4 section of the workspace (over y

and z ) with K, = 1. (i = 1.. . . .6). and z ) with h', = 1. (i = 1.. . . .6).

Figure 2.26: Singularity locus and workspace of the SLY-dof manipulator.

2.3 Kinematic opt imization of mechanisms wit h re-

duced degrees of freedom

In this section. the optimization of planar two-degree-of-freedom as well as spatial four-

and five-degree-of-freedom parallel manipulators is addressed. The objective is to syn-

thcsizc manipulators in rvhich the uncontrollable dependent Cartesiân coordinates i d I

foHosv certain trajectories which are functions of the independent Cartesian coordinates

and which are prescribed by design. This may be suitable in certain applications where

the dependent Cartesian coordinates would be required to follow prescribed - and fixed

- trajectories. The dependent Cartesian coordinates are first expressed as functions

of the relative Iinkage parameter of the manipulator using the kinematic equations.

The objective functions are formed using the least square method. The limits of the

workspace of the manipulators are used as the constraints in the optimization in order

to ensure that ail specified trajectory points are in the Cartesian space and located

inside the workspace of the optirnized manipulators.

2.3.1 Planar two-degree-of-freedom manipulator

Referring to Figure 2.1, it is clear that the orientation of the end-effector of the two-

degree-of-fieedom pIanar mechanism is only related to the two links of the k s t leg.

Kence, in order to simplify the study, it is assumed that the mechanism studied consists

of two moving links and two revolute joints, as illustrated in Figure 2.27, and the lengths

of the two links of the second leg can be arbitrarily chosen to meet the kinematic

requirements of the resulting mechanism.

Two reference coordinate frames are defined. Coordinate fiame O -x-y is attached

to the base and its origin O lies at the center of the joint connecting the first moving

link and the base. Coordinate frame O' - x' - y' is 6 ~ e d to the second moving link

and its origin O' is coincident with point P where the end-effector of the manipulator

is assumed to be positioned. Point Pl denotes the center point of the joint connecting

the two moving links.

Let I I denote the length of the first moving link. (at 6) the coordinates of point Pl

in the moving reference frame O' - x' - y' and (x: y) the coordinates of point P in the

fixed reference frame O - x - y. From the geometry of the manipulator, one can \\-rite

Figure 2.27: First leg of the planar two-degree-of-freedom parallel manipulator.

the following vector equation

PI = P + Q ~

where

r = [ a blT

Q is the orientation rnatrix of the moving frame with respect to the fked frame, which

cos 4 -sin qi Q = [

sin 4 cos 4 1 where 4 is the angle between coordinate axes x' and x.

If one chooses any two coordinates of the three Cartesian coordinates xl y and 4. for instance x and y, as the independent coordinates of the rnanipulator, the dependent

angle ai can be computed from equation (2.170).

From equation (2.170) one can m i t e

Squaring both sides of equations (2.173) and (2.174) and then adding Ieads to

-4 cos q5 t i l sin ei = C (2.175)

where

B = 2(ay - hx)

From equation (2.175) one obtains

sin Q = BC + KA&

A* + B2

COS dl = AC + K B ~

A* + B2

where

Variables 4 and K are two important factors in the optimization of the manipula-

tor. 4 is the discriminant of equation (2.175). A value of A larger than zero means

that point P(s? y) is located inside the workspace of the manipulator. h' is the branch

index of the manipulator, its two values determine the tu70 possible configurations of

the manipulator.

From eq.(2.175), it is clear that when z and y are given, angle 6 is only reIated to

linkage parameters I I , a and 6. Therefore, these three linkage parameters are chosen a s

optimization variables.

-4ssuming that the two-degree-of-freedom manipulator is used to position n points

of the plane from P ( x r 2 gr) to P(z,, y,) while the orientation of the end-effector of

the manipulator corresponding to these points is required to be as close as possible to

prescribed values noted & to 6,: one can formulate the optimization problem as follows

Subject to:

where

gi(x) = A i = l , . . . ! n

where W is a 2n by 2n weighting m a t r ~ v and e is a 2n dimensional error vector, written

where sin oi? cos @i and Ai (i = 1,2! ...: n) are computed according to eqs.(2.179),

(2.180) and (2.182) respectively.

e =

- sin G1 - sin 3, cos dl - cos 3,

...

... sin 4, - sin 4,

- - COS 4n - COS 4,

The generalized reduced gradient method is used here [63]. This method is an

extension of the reduced gradient method which is used to soIve equality-constrained

optimization problems, where one adds one slack variable to each inequality constraint

thereby transforming the inequdity-constrained optimization problern into an equality-

constrained one.

This method is efficient for both objective and coristraint functions which are highIy

nonlinear. In the method. a search direction is found such that any active constraints

remain precisely active for some srnaIl move in this direction. If a move is made.

because of nonlinearity, some currently active constraint does not remain precisely

satisfied? Newton's method is used to return to the constraint boundary.

An algorithm n-ritten in FORTRAN language by Wang and S ie [63] is directly

applied for the optimization. The espressions of the partial derivatives of the objective

and constraint functions are needed. They can be obtained as follows:

Assuming W = 111. Il] denoting a 271 by 2n identity matris. differentiating equation

(2.186) with respect to x one then has

From equation (2.185) one can obtain

cos @,, ( $)+

- sin 4 n ( s ) T

where $$ ean be obtained from equation (2.175)

where 5, 2 and 2 can be easily obtained from equations (2.116)-(2.178) as

Table 2.1: Data set for the tn-o-dof parallel manipulator, lengths are in meters and

angles in degrees, K = - 1.

The partial derivatives of the constraint functions can be obtained by differentiating

equation (2.181) with respect to x, Le.,

Prescribing 10 points of the -Y-Y plane, Pl? P2: ...: Plo, as well as a set of orientation - - -

angles corresponding to each of these points and noted dl. 4, ...,Q,,. it is desired to

find the most suitable three linkage parameters Il: a and b so that when the end-effector

of the manipulator passes through points Pi (i = 1: 2. ..., 10) its orientation angles Oi

(i = 1.2. ...! 10) are as close as possible to the specified values & (i = 1.2: .... 10). An

esample data set is given in Table 2.1.

Csing the program GRGSI (Generalized Reduced Gradient Slethod) [63] one obtains

the following results for the optimization

and

The square of the sum of the errors gives

The corresponding error curve is s h o w in Figure 2.28. This exarnple illustrates that

when the end-effector of the two-dof manipulator passes through a set of prescribed

points. it is possible that its orientation angles can be close to the desired values by an

optimal choice of the linkage parameters.

Figure 2.28: Esample of optimization synthesis of the two-dof mechanism.

2.3.2 Spatial four-degree-of-freedom manipulator

Similady. the optimization problem for the four-degree-of-freedom manipulator can be

stated as:

Specifying a set of points P,(x,, Yi, Zir ui)i = 1. . . . . n in the four dimensional space

and the corresponding sets of Euler angIes and Fi: determine a set of optimal pa-

rameters ls, as, b5 and cg of the manipulator, for which. when the manipulator passes

through points P,, the two dependent Euler angles 8, and u:, are as close as possible to

their prescribed value ë, and Ti.

I t can be formulated as

subject to:

g i ( x ) 3 0, z = 1,. ..,

wtiere F ( x ) is the objective function and gi is the constraint function.

where W1 and W2 are two weighting matrices, and

sin el - sin ë1 1 I sin QI - sin 3, cos el - cos ë1 COS el - cos ii,

-.. ...

... . . . s i n e n - s i n ë n 1 7 "= 1 sin SIn - sin G,, cos 0 - cos a,, cos @,.' - cos Gn

The constraint function is n-ritten as

where

where g(jt2}i(x) are the constraints associated with the j t h actuated leg while ~ l . 4 ~ . AJ2

and I.; have been defined in eqs.(2.41), (2.48) and (2.57) respectively

The generalized reduced gradient method is used for the optimization problem.

The espressions of the partial derivatives of the objective and constraint functions are

written as follows

Assuming W1 = W2 = [l], a i t h [l] the identity matrix,

Differentiating equations (2.198) with respect to the optimization variables x! one

where 2 and $ (i = 1,2. ...' n) can be obtained by differentiating equations (2.42)

and (2.49) respectively. Le.,

and 2. 2 and 2 ,an be easily obtained by differentiating equations (2.43)-(2.45)

as

Differentiating equation ('1.33) with respect to x, one has

0 '

O

dx 1

(2.208)

and similarly? b-J differentiating eqs.(2.36)-(2.38), one then obtains

aiia, -- dB;, &

-0: -- ax

- O 7 -- ax (2.209)

O- -0 - -u , sin mi cos ai$ -

8 4 2 -- - dx

The partial derivatives of the constraint functions can then be obtained as follows

'O - -sin&(sinûi + ul cosai%)-

1

O

-ul sin d, cos ai % ac;, ax -ul sin @, COS ai%

(2.207)

and

where

where xi, yi and zi are the three components of vector pi, narnely

The same optimization program GRG51 used for the optimization of the two-dof

mechanism is used here. -1 numerical esample illustrating the application of the above

procedure to the four-degree-of-freedom manipulator is now presented.

Given 10 points. Pi, P2? .... Pro, for n-hich the four independent Cartesian coordi-

nates are prescribed as weI1 as two sets of dependent orientation angles associated to - - - -

t hese points. ël? &, .... eI0 and vl . v2. .... L : ~ ~ , find the most suitable four linkage param-

eters 1 3 . a5: b5 and CS so that when the end-effector of the manipulator passes through

points Pt ( i = 1: 3. .... 10) its two sets of dependent orientation angles el.&..... OIo

and wl : q ) ~ , ...? ZYLO are as close as possible to the specified values $1, $2, ..-. Bi,-, and

The parameters used in this exampie are given as

Table 2.2: Data set for the ddof parallel manipulatort lengths are in meters and angles

in degrees, Kl = -1 and h; = 1.

The specified Cartesian poses are represented in Table 2.2.

The results of the optimization are

which leads to

B = [7.676 11.312 13.150 19.221 23.563 28.258 33.284 38.831 45.043 32.2111T

The l e a s square sum of the error gives

The error curws are shown in Figure 3.29.

It can be seen from the esampie that by an optimal choice of the parameters of the

special leg, the two dependent Cartesian coordinates assume the desired values when

the independent Cartesian coordinates pass through the prescribed trajectories.

2.3.3 Spatial five-degree-of-freedom manipulators

The optimization problem of the five-degree-of-freedom manipulator can be stated as:

Specifying a set of points Pi(si, pi, zi? Qi, Bi) i = 1,. . . n in the five-dimensional

space and the corresponding set of Euler angles qi: find the four optimal parameters

p, P,

(a) First dependent Euler angle 0 (b) Second dependent Euler angle c

Figure 2-29: EsampIe of optimal synthesis of the spatial four-dof manipulator

of the manipulator Ig: act b6 and CG: for which, when the manipulator passes through

points P,. the dependent EuIer angle tOi is as close as possible to its prescribed value

This can be espressed mathematically as

subject to:

g,(x) 2 0, z = 1:2, ...? n

11-here n is the number of prescribed Cartesian points.

where W is a n by n weighting matrix and

r sin - sin Tl cos - cos F,

The constraint function is

where

*.. -

sin & - sin $,,

. cos $ - cos zo,

and wi-here g(,,,,,(x) are the constraints associated with the j t h actuated leg. moreover.

A and 1 ; have been defined in eqs.(2.57) and ('2.66) respectiveiy.

Similarly to the previous case, we assume W = [l]. with [II an identity matris.

One can \\-rite

Differentiating equations (2.215) with respect to x' one then obtains ad T cos Zb(=)

a~ T - sin c ! ( ~ )

.*.

... au T cos d(=) aiU T - s i n d ( = )

where 2 (i = 1: 2: .... n) ,an be obtained by differentiating equation (2.66)

aA1 aea where =, and can be easily obtained from equations (2.66)-(2.68) as

O

dAz 2(xi COS di COS Bi + Yi sin $i COS Bi - zi sin Bi) - = I dx 2 (y i COS c$i - xi sin di) (2 .222)

O O

6'B' 2(yi COS 4i - Xi sin 4i) _ [ dx 2(x i COS 4i COS Bi + Yi sin di COS Bi - sin di)

1 O

2~ 1

d C i -2u2 âx

! (2.224)

-2U3

-2(u4 - Xi COS @i sin 8, + y, sin di sin di + ri COS O,

The partial derivat ives of the constraint functions can be respectively obtained as fol-

1on.s

and

tvhere

similarly, where xi! yi and 2, are the three components of vector pi , i.e..

Again: the same numerical algorithm is used for the optimization.

Given 10 points Pl P2, ..., Plo, for which the five independent Cartesian coordinates

are prescribed as well as the dependent angle associated to these points, find the most

Table 2.3: Data set for the 5-dof paralIel rnanipulator, lengths are in meters and angles

in degrees, KI = -1.

suitable four linkage parameters 15: as, b5 and c5 so that $1-hen the end-effector of the

nianipulator passes through points P, (i = 1.2, 10) its dependent orientation angles - - -

v2, .... v10 are as close as possible to the prescribed vaIues Z!J,? u,. .... a IO -

The parameters used in this esample are given as

The specified Cartesian poses are represented in Table 2.3.

The results of the optimization are

which leads to

with = -1. The least square sum of the error gives

The error curve is shown in Figure 2.30.

Figure 2.30: Esample of optimal sythesis of spatial five-dof manipulator.

Sirnilarly. one can realize from the example that by an optimal choice of the param-

eters of the special kg. the dependent Cartesian coordinates can be close t.o the desired

values when the five independent Cartesian coordinates pass through the prescribed

trajectories.

SIoreover. it can be noticed from the examples that the results obtained for the

five-degree-of-freedom manipulator are better than those obtained for the four-degree-

of-freedom manipulator. This is because the five-degree-of-freedom manipulator has

more degrees of freedom and it is therefore easier to adapt to the prescribed poses.

Conclusion

The kinematic analysis of planar and spatial parailel mechanisms has been addressed

in this chapter. The inverse kinematics of these mechanisms has been solved and their

workspace has been determined using a simple novel algorithm. This general numerical

algorithm can be applied to the determination of the workspace of any type of planar

and spatial parallel mechanism.

The velocity equations of the mechanisms have then been derived using two a p

proaches, namely, the algebraic and the vector formulation. The latter approach is a

new approach which can lead to simple expressions of the determinants of the Jaco-

bian matrices of the mechanisms. The singularity loci of the mechanisrns have been

deterrnined by the two approaches. both approaches Ieading to identical results. The

algorithm for the determination of the singularity loci using the latter approach is

however. much faster than the former one.

The kinematic optimization of mechanisms with reduced degrees of freedom has

also been discussed. The generalized reduced gradient method has been used for the

optimization and led to a fast converge. The dependent Cartesian coordinates of the

mechanisms can follow desired trajectories as closely as possible ben the independent

Cartesian coordinates pass through the prescribed points using the optimization proce-

dure presented in this chapter. This property is important for the practical applications

of mechanisms v.-ith reduced degrees of freedom.

Chapter 3

D ynamic analysis

The dynamic analysis of planar and spatial parallel manipulators is presented in this

chapter. .-\ new approach based on the principle of virtual work is first used to derive

the generalized input forces of the manipulators. This approach is efficient and suit-

able for the control of manipulators. Shen, the conventiona1 approach used for dynamic

analysis of parallel mechanisms or manipulators, namely, the Yewton-Euler equations,

is also applied to derive the generalized input forces of the manipulators. Since the

constraint forces between the links are computed, this approach leads t o a slower al-

gorithm compared to the approach based on the principle of virtual work: however. it

is useful for the simulation and design of the manipulators. F i n a l l ~ the corresponding

algorithms are compared and numerical examples are given in order t o illustrate the

results.

3.1 Approach using the principle of virtual work

The approach based on the principle of virtud work consists in expressing the inertial

force and moment acting on the links of the manipulator and then considering the ma-

nipuIator to be in "static" equilibrium. The virtuai displacement of each Iink caused

by the virtual displacement of each actuated joint is computed. Finallc., the principle

of virtual work is applied to obtain the actuator forces or torques.

3.1.1 Planar parallel manipulators with revolute actuators

In chapter 2. the inwrse kinematics of the planar two- and three-degree-of-freedom

mechanisms is cornputed and the orientations of the al1 moving links of the two mecha-

nisms have been determined. Therefore, the determination of the position and orienta-

tion of the moving links is not mentioned here for reason of simplicity. The procedure

for deriving the generalized forces or torques using this approach. therefore. will consist

of the following steps: velocity analysis, acceleration analysis, computation of the iner-

- tial forces and moments, determination of the virtual displacements and determination

of generalized input forces or torques.

1 j télocicy analysis

From the two kinematic chains of the mechanism (see Figure 2-13)? one can write

where w , (i = 1,. . . ? 4) and ai (i = 1: . . . , 5 ) are respectively the vector of the angular

velocity and position vector of the ith moving link, vector p is the given Cartesian

velocity vector of the end-effector, Le.,

Eqs. (3.1) and (3.2) can be rewritten in matrix f o m as

where

-il sin O1 -13 sin cri

I I cos BI l3 cos al

-Z2 sin O2 -lq sin 0 2

2 - l2 COS e2 l4 COS a2 1

x + w315 sin al b2 = p - w 3 x a 5 =

i - w31j COS al 1 mhere al1 quantities in the above equations are as defined in Chapter 2.

The solution of eqs. (3.4) and (3.5) respectively leads to

II) Accelerat ion analysis

Differentiating eqs(3.1) and (3.2). one then obtains

where p is the given acceleration vector of the end-effector, and

Similarly, eqs.(3.8) and (3.9) can be rewritten as

where

% = [*?] . i2 = [:?] From eqs.(3.11) and (3.12), one c m obtain

III) Computation of the forces and torques acting on the center of mass of each link

The force acting on the center of rnass of each link consists of two parts: the inertial

force and the gravity force. The moment acting on each link is the inertial moment.

In order to compute inertial forces. one must first determine the Iinear accelcration

of the center of mass of each Iink.

One can write

where Q, and 1, are respectively the orientation matris of the i th Iink and the position

rector of the length of the link while r, is the position vector from the center of mass of

the ith link to the center of the rel-olute joint connecting the current link to the former

link, and

QI = [ (3.18)

Then, the force and moment acting on the center of mass of each link can be directly

computed as follows

where f, and mi denote the inertial force and moment acting on the ith link. lloreover.

1, and wi denote the moment of inertia and gravity vector of the i th link.

VI) Computation of the vîrtual displacements of the links

The computation of the virtual displacements of the links is the most important

step for the determination of the generalized input forces. For this mechanisrnt one

directly uses the velocity equations obtained in Chapter 2. From eq.(2.76) one has

where A*-' is the inverse matrix of matriv A'.

Now, let 68, be the vinual angular displacement of the ith Ieg corresponding to

angles 8; (i = 1.2) and dw,,;? be the virtual angdar displacement corresponding to

angles n, (j = 1,2) .

From eq. (3.21) one can obtain

where

Ha~ ing obtained the virtual angular displacements of each of the links of the ma-

nipulator. the virtuaI linear displacements of the center of mass of each link can be

computed as follows

where S,(i = 1: 2) and 6,(j = 3,4) are respectively the virtual Iinear displacement of

the center of m a s of the 1.2nd and 3,4th links.

V) Computation of the actuator force/torque

FinaIly, the principle of virtual work can be applied to compute the actuating

torques

where T~ is the actuating force or torque at the i th actuated joint.

1) velocity analysis

From the three kinematic closed loops of the mechanism (see Figure 2.16): one can

write

where w, : i = 1,. . . .6 is the vector of the angdar velocity of the ith link while p and

w7 are respectively the vectors of the linear and angular velocity of the end-effector.

L Y l Rewriting eqs.(3.26)-(3.28) in rnatrix form, one has

where

-l1sinO1 -13sinal

- [j:] A, = [ 771 - 1 . b, =f i = [il 1 1 ~ ~ ~ 8 1 /3COSQ~

-12 sin O2 -15 sin a? - 1 . A2= [ 1 x + d717 sin O

7 2 - . b 2 = p - w 7 x a 7 = l2 COS O2 l5 COS Q:! y - ~ * ; l ; COS O

-13 sin B3 -16 sin cvg - [ i 3 ] , & = [ 1 x + u718 sin o - , b 3 = p - w ; x a 8 =

b4'6 l3 COS O3 l6 COS a3 y - dglg cos 0

where al1 quantities in the above equations are as defined in Chapter 2.

The solution of eqs.(3.30)-(3.32) respectively leads to

II) Acceleration analysis

Differentiating eqs.(3.26) to (3.28), one then obtains

where p is the given acceleration vector of the end-effector, and

Eqs(3.36) to (3.38) can be rewritten as

From ~qr.(3.39) and (3.41). one can finally obtain

III) Computation of the forces and torques acting on the center of mass of each Iink

One c m mi t e

where Q is the orientation mat+ of the 7th link, Qi and 1, are respectively the orien-

tation matrix of the i th link and the position vector of the Iength of the link while r,

is the position vector from the center of m a s of ith link to the center of the revolute

joint connecting the current link to the former link, and

[cos@ - s i n 4 cos Bi - sin Bi

Q = [sin 4 cos4 1 , ~ i = [ I i = l , 2 . 3

sin Bi cos Bi cos cri -sin CY,

Q i = 1 , i = 4 , 5 , 6 i i = [ l ] ; i = I 7 s inai C O S û i

The force and moment acting on the center of mass of each link can be directly computed

as follori-s

n-here f, and rn, denote the inertial force and moment acting on the i th link and 1, is

the inertial matris of the Iink.

1-1) Computation of the virtual displacements of the links

For this mechanism. one directly uses the velocity equations obtained in Chapter 2.

Form eq.(?.81) one has

where A*-' is the inverse matris of matrix A'.

Similarly to the previous case, let dei be the virtual angular displacement of the i th

leg corresponding to angles O, (i = 1: 2: 3) and 6w,4 be the virtual angular displace-

ments of the j th Ieg corresponding to angles a, ( j = 1: 2 ,3 ) , 6w; be the virtual angular

displacement of the end-effector and 6r and 6y be the virtual linear displacements of

the end-effector along axes x and y.

Frorn eq.(3.51) one can obtain

where

Having obtained the virtual angular displacements of each of the links of the manipula-

tor, the virtual linear displacements of the center of mass of each link can be computed

as follows

where 6,(1 = 1: 2 . 3 ) and 6,(j = 4: 5,6) are respectively the virtual linear displacement

of the center of mass of the 1.2.3rd and 4,S16t h link. 67 is the virtual linear displacement

of the center of mass of the 7th link. bp is the vector of virtual displacement of the

VI) Computation of the actfiator force/torque

Finally, by application of the principle of virtual work one can obtain the actuating

torques

where T, is the actuating force or torque at the ith actuated joint.

3.1.2 Spatial parallel manipulators with revolute actuators

In this subsection, the computation of the kinematics, velocity and acceleration anal-

yses are first performed. Shen, the principle of virtual work is applied to derive the

generalized forces of three types of spatial paraIlel manipulators. For the spatial four-

Figure 3.1 : \Tectors Ii, and represented in spherical coordinates.

and five-degree-of-freedom paralIel manipulators, the algorithms for the computation

of the dependent Cartesian coordinates from the independent Cartesian coordinates

presented in Chapter 2 will be used directly and are therefore not repeated here.

3.1.2.1 Four-degree-of-freedom manipulat or

1) Inverse kinernatics

For the four-dof mechanism, once angles 0 and v have been determined. the s is

Cartesian coordinates of the platform are available and one can directly compute the

position of point Pt.

where Q,, and Q,, are respectively the rotation matrices describing the orientation of

the upper and lower links of the i th leg with respect to the base coordinate frame.

Moreover, li, and liI are the vectors from 0: to P, and from O, to 0: expressed in their

local frarnes, respectively, as represented in Figure 3.1. One has,

m-here angles a,: 3,: -i, and pi are as defined in Figure 3.1. Equation (3.57) consists of

three scalar equations with three unknowns pi, cri and gi: and can be rewritten as

Zi2 COS al sin ,Bi = Xi - l i l COS 7, sin pi - Xi,

1i2 sin a , sin ,Ji = Yi - l t l sin Yi sin p, - y,,

l i2 COS ,3, = 4, - l i l COS pi - Go

From eqs.(3.58)-(3.60), one can obtain

which leads directly to

sin pi = Sir, + ~ ~ ~ & f l . i = 1, ..., 4

&? + Si! R,r - K , ~ S ~ &

COS pi = i = 1 , . . . , 4 &2 + s;

with Ki3 = il the branch index of the manipulator associated with the configuration

of the ith leg. Finally, one has

sin f i = \/l - cos2 A, (O < A 5 li)

COS a, = (xi - lil COS 7, sin pi - xi,) / ( l I 2 sin pi)

sin ai = (y* - l i l sin ~i sin pi - y io) / ( l i2 sin Oi)

and the three variables p,, a, and allow one to completely determine the position

and orientation of the two links in the ith leg.

II) Gélocity andysis

The linear and angular velocities of al1 moving links will be computed from the

given independent Cartesian velocities of the platform x: y. t and L*,. where the anguIar

velocity of the platform is defined as w = [ar3 uY: LI=]^.

First. the tno dependent cornponents of the angular velocity. d, and J:: need to be

determined using a special kinematic chain,

One can mite the linear velocity of point P5 as

1,h COS a

-lja sin a

Equation (3.68) can be rewritten in matrix form as

A q = b

where

t5 COS Q: Cs i

*=[ O Y 21: q = b=[!+6;dz! -&sina -6; O W: - b5u,

and [a;. bk, 4jT = ap; and eq.(3.70) is easily solved for vector 7 .

Hallng obtained the six Cartesian velocity components x, y, 2 , u,, w, and w,, the

linear velocity of the center of joint Pi connecting each leg to the platform can be

computed as follows 1 .

pi = p + w x Qpi, z = 1, ... , 4 (3.72)

Sloreover, pi can be espressed using the angular velocities of the i th leg, i.e..

where w,, and wIl are the angular velocities of upper and lower links of the ith kg, Le..

-pi sin y,

Equation (3.73) can be rewritten in matri.; form as

where

I l l COS COS p. -ll7 sin a, sin ,g1 lI2 cos a , cos 3 ,

Cl. = sin 3 cos p. -lZ2 cos a , sin û, l t o sin a. cos 4 - l I 1 sin pl O - l I2 sin 3,

A., = ( b , & 3,1T

Solving eq.(J.ij). one readily obtains Pl . ci, and j,. Once these three quantiries are

known: the velocities of the links are easily determined.

III) -1cceleration analysis

The linear and angular accelerations of each of the moving bodies will now be

determined from the given Cartesian accelerations of the platform, Le.: x, y, 2 and 2,.

First, the tnlo dependent acceleration components cj, and ij, must be determined.This

can be achieved by using the kinematic geometx-y of the fifth leg.

Differentiating eq.(3.68) with respect to time, one obtains the espression of the

linear acceleration of point P5 a s follows:

where L5ii COS CY - l5ci2 sin a x &

ps = O ] p=[i], & = [ - ] (3-78)

- lS& sin a - l 5 r i 2 COS û W=

Since the veIocity anaiysis has already been performed, eq.(3.77) consists of three scalar

equations ahich contain only three unhowns: &, W, and 2,. Therefore, it is easy to

solve for these unknowns.

Equation (3.77) is rewritten in matrix form as

u-here matris A has been previouslj- defined and where [a, a, a,IT = w x (w x QP;).

Equation (3.79) is readily solved for f-/' which leads to the desired acceleration

. components. -411 the Cartesian accelerations of the platform are then knoivn. and the

linear and angular accelerations of each of the leg bodies can then be determined.

The linear and angdar accelerations of the two links of each of the legs can be

obtained from the linear accelerations of points P, which have been computed from the

six Cartesian accelerat ion component S.

Differentiating eq.(3.73) tvith respect to time. one obtains

mhere

Equation (3.81) can then be rewritten in matrix form as

C i t X i r = ~ i i i = 1 , . . . , 4

where Ci, is given in eq.(3.76) and where

A,, = [Pi ai AlT

si = [sZ sY s , ] T

where

The solution of eq.(3.83) for h,, will then allow the determination of the linear and

angular accelerations of the moving bodies of each of the legs.

12) Computation of the inertial forces and torques acting on the center of mass of

each link

In order to compute inertial forces. one must first determine the Iinear acceleration

of the center of mass of each link.

One can write

a,, = ~ i + ~ l u ~ Q 1 u ~ i u + W i u ~ ( ~ l u ~ Q l u ~ l u ) . i = l ..... 4 (3.86)

1 = wil x Qilïil + wil x (wil x Q i l ~ l l ) 7 i = 1:. . . 4 (3.87)

where a,, and a,[ are the linear accelerations of the center of mass of the upper and

lower links of the ith leg, vectors r,, and r,l are the position vectors of the center of

mass of the upper and lower link of the i th leg and are espressed in the leg's reference

frame! while w,, and wil are the weight vectors, and

mhere g is the gravitational acceleration.

The linear acceleration of the center of mass of the 5th leg and the platform can be

computed as follows

as = W5 x 4 5 1 5 + ~5 x (US x Q5r5)

a,., = p + w x Q r P + w x ( ~ x Q r p )

where is the linear acceleration of the center of mass of the 5th leg, Q5 is the

orientation matrix of coordinate frame O - X ~ Y J Z ~ with respect to the k e d coordinate

frame O - xyz, w s is the angular velocity of the 5th kg, W 5 is its ang-ular acceleration,

r g is the vector connecting point O to the center of mass of the 5th leg noted Cs: i.e..

cos a O s i n a O O O

QS = [ 0 1 O ] . -.=[;] i , = [ n ] p g i i

- s ina O COSLL TS

and a, is the linear acceleration of the platform, and where r p is the vector connecting T point O' to the center of mass of the platform, nainel- rp = [ x p y, 7 ] .

Then, the force and moment acting on the center of mass of each link can be directly

computed as folloms

ftu = - r n , , a , , f ~ , , ~ i = Z : . . . : 4

fil = -milail+wil, i = 1 ..... 4 d

mi. = - x ( ~ r u ~ ~ u ~ ~ ~ i u ) i = 1, . . . .4

where fiui miu, fil and m , ~ denote the force acting on the upper link, the moment acting

on the upper link, the force acting on the Iower link and the moment acting on the

lower link of the ith leg, Iiu and Iiu are respectively the inertia tensor of the upper and

loiver links u-ith respect to their center of mass.

Finally, one has

Figure 3.2: Body forces acting on the links of the manipulator u-ith revolute actuators.

where f5 and m5 denote the force and moment acting on 5th k g vhile fp and mp denote

the force and moment acting on the platform, Ig and 1, are the inertia tensor of the

5th leg and the platform with respect to their center of rnass.

The forces and moments acting on the center of mass of each link of this type of

manipiiIator are represented schematically in Figure 3.2.

I,-) Computation of the virtual displacements of the links

The virtual linear displacements of the center of mass and the virtual angular dis-

placements ofeach link d l be obtained from the given joint virtual displacements of the

manipulator. This is the most important step for the determination of the generalized

input forces by this approach.

From the kinematic geometry of the 5th leg, one can m i t e

xi++ = l 2 5

Y5 = 0

Differentiating eqs.(3.100) and (XlOl), one obtains

y5 = O (3.103)

Letting& = [ I S y5 iS5]Tandsubstitutingeq.(3.68) intoeqs.(3.102) and (3.103), one

obtains

t = J,x, or 6t = JJx (3.104)

where bt = [bz 6y 6z du. bw, bu, jT, and

Differentiating eq.(3.61). one obtains

l i l (St COS pz - 8 sin p,)dp, = (xi - x,, - l I1 sin y, cos p,)dx,

+(y i - Y,, + LI cos 7, sin p,)dy,

+(z,-zlo+I,lsinp,)dz,. 2 = 1 ..... 4 (3.106)

Substituting eq.(3.73) into eq.(3.106). one obtains

where

Substituting eq.(3.104) into eq.(3.107) one finaily obtains

where

A, = K,J,

From eq.(3.108), one cm obtain

bx' = A~'B,~~', i = 1, ... $ 4

From eqs.(3.70) and (3.75), one can obtain

and

SA;, = c$dPit i, j = 1 .... ,4

where C$ is the inverse of matrix C,, which was preriously obtained. and

and finaily. the linear virtual displacements are given as

where al1 quantities are as previously defined.

IV) Computation of the actuator force/torque

The principle of virtual work can then be applied and Ieads to

where T, is the actuating force or torque a t the ith actuated joint.

1) Inverse kinematics

Since the computation of the dependent Cartesian coordinate @ from the specified

independent Cartesian coordinates x, y: t, 0 has been completed in Chapter 2, the

only task related to the inverse kinematics which must still be reformed is to obtain

the position and orientation of the links of the manipulator from the known pose of

the platform. However, this procedure is exactly identical to what has been done in

the computation of the inverse kinematics of the spatial four-dof parallel manipulator

since the architecture of the actuated Iegs is the same for the two types of mechanisms.

Hence, it is not repeated here.

II) i'elocity analysis

In this section, the linear and angular velocities of al1 moving links will be computed

from the given independent Cartesian velocities of the platform x1 y! 2 : zz and *,,. n-here

the angular velocity of the platform is defined as w = [sl,: d,. jrZlT.

First. the dependent component of the angular vdocity, d2, needs t o be determined

using the special kinematic Chain.

One can m i t e the linear velocities of point P6 as

Equation (3.114) can be rewritten in matrix form

where

and [a i , b i , i6] = ~ p ; j and eq.(3.115) is easiiy solved for vector q.

Having obtained the six Cartesian velocity components x, y, 2: d,: sr,, and (+;=! the

linear velocities of the center of joints Pi connecting each leg to the platform can be

computed as f o l l o~s

pz = p + w x i = 1: ..., 5 (3.116)

The rest of the procedure is to determine the velocities of the other bodies from

the linear velocity of points Pl. This is identical to what was presented in the previous

subsection on the velocity analysis of the spatial four-dof parallel mechanism.

III) Acceleration anaiysis

The linear and angular accelerations of each of the moving bodies wiI1 now be

determined from the given Cartesian accelerations of the platform. i.e.. x. ÿ. 5 . 2, and

L',, .

First. the dependent acceleration component W= must be determined. This can be

achieved by using the kinernatic geometry of the special leg.

Differentiating eq.(3.114) with respect to time and letting i = 6' one obtains the

expression of the linear acceleration of point P6 as follows:

where

where

it6 = 4( j6 cos a COS (l6 - fi6 sin b6 sin a6 - sin f 6 COS a6

Since the velocity analysis has already been performed, eq. (3.11 7) consists of t hree

scalar equations which contain onIy three unknowns: &,Gy and ;jz. Therefore, it is

easy to solve for these unknowns.

Equation (3.117) is rewritten in matrix form as

T where = [G6 2 , ) and e = [ e . e, e,lT, and

and matris A mas previously defined and where [a, a, azIT = w x (W x Q P ~ ) .

Equation (3.119) is readily solved for j l . which leads to the desired acceleration

components. -411 the Cartesian accelerations of the platform are then knom. and the

linear and angular accelerations of each of the leg bodies of the two types of manipu-

lators can then be determined.

The rest of the procedure is to obtain the linear and angular accelerations of the

two links of each of the legs from the linear accelerations of points P, and can be found

in the acceleration analysis of the spatial four-dof parallel mechanism.

VI) Computation of the inertial forces and torques acting on the center of mass of

each link

Because of the similarity of the architectures of the five-dof and four-dof mecha-

nisms, only the formulas for the computation of the inertial forces and torques of the

special6th leg are different and the others can be found in the previous subsection on

the four-dof mechanism.

The linear acceleration of the center of rnass of the 6th leg and the platform can be

computed as follows

where % and a, are respectively the linear accelerations of the center of mass of the

6th leg and the platform, and where we is the angular velocity of the 6th leg, w6 is

its angular acceleration and where r6 is the vector connecting point O to the center of

mass of the 6th leg noted Cc, and where

The force and moment acting on the center of rnass of the link of the 6th leg can

be directly cornputed as follows

where f6 and m6 denote the force and moment acting on the 6th leg while f, and m,

denote the force and moment acting on the platform, I6 and 1, are the inertia tensors

of the 6th leg and the platforrn with respect to their center of mass.

The forces and moments acting on the center of rnass of each link of this type of

manipulator are represented in Figure 3.3.

Figure 3.3: Body forces acting on the links of the manipulator ~vi th revolute actuators.

1') Computation of the virtual displacements of the links

In this section. the virtual linear displacement of the center of mass and the virtual

angular displacement of each link are obtained from the given joint virtual displacement

of the manipulator. It is the crucial step for determining the generalized input force by

this approach.

From the kinematic geornetry of the 6th leg, one can write

9 2 r; + y, + ri = rg

Differentiating eq.(3.128)? one has

Letting fi6 = [f6 ?j6 &]* and substituting q (3 .114) into eqs.(3.129), one obtains

where bt is as previous defined, and

Differentiating eq. (3.61): one obtains

Substituting eq. (3.72) into eq.(3. lX?), one obtains

where

Br = diag[bl b2 b3 b4 6 5 1

Substituting eq.(3.130) into eq.(3.133) one finally obtains

B,bp = AJx

where

A, = KrJr

Sirnilarly, one can obtain

bxi = &-'B,&~J'

where bx' and bpi are as previously defined.

Then, one can obtain

6 4 = ~ - l & '

and

S A : ~ = C ; ~ ~ ~ ~ j = l , ...: 5

where

SA:, = [ S p i do; d,Oj ] , j = l2 - . . - 5

and finally. the linear virtrral displacements are given as

where al1 quantities are as as previously defined.

II') Computation of the actuator force/torque

The principle of virtual work can then be applied and leads to

where T, is the actuating force or torque at the i th actuated joint.

3.1.2.3 Six-degree-of-freedom manipulator

Since the architectures of the siu- and five-degree-of-freedom manipulators are very

similar: the procedure for the computation of the inverse kinematics, the velocity anal-

ysis: the acceleration analysis and the computation of the inertial forces or moments

for the six-degree-of-fieedom manipulator c m be directIy obtained from the analysis

of the five-degree-of-freedom manipdator. Therefore, for the 6-dof manipulator it is

only needed to describe the final two steps, namely, the computation of the virtual

displacements of the links and the determination of the generalized forces/torques.

V) Computation of the virtual displacements of the Iinks

The virtual linear displacement of the center of mass and the virtual angular dis-

placement of each link are computed from the given joint virtual displacement of the

manipulator.

Differentiating eq. (3.61), one obtains

Substituting eq.(3.72) into eq.(3.140), one obtains

Br6p = A,bx

where

d'x = [6x by 6 2

Br = diag [ b l b2

where

Then, one can obtain

6xi = A Y ' B , ~ ~ '

where 6x1 and 6pi are as previously defined.

Then, one c m also obtain 6,,i = A-L&.,i

where

SA;, = [6p; bn; 63;] , j = 1,. . . y 6

and finalIl the linear virtual displacements are given as

where al1 the quantities are as previously defined.

I l ' ) Computation of the actuator force/torque

The principle of virtual work can then be applied and leads to

where r, is the actuating force or torque at the ith actuated joint.

3.2 Newton-Euler formulation

In order to verify the resuits obtained using the approach presented in the previous

section. a second approach based on the Newton-Euler formulation is now deveIoped.

Each link of the rnanipulator is considered individualiy and constraint forces between

the links are computed. -4lthough this approach leads to a slower computational aigo-

rithm than the approach based on the principle of virtual work, the interna1 constraint

forces between the links are computed here, which may be useful for design and simula-

tion of a mechanism or manipulator. Therefore, this approach is suitable for the design

of the mechanism. This approach has been used by other authors for the dynamic

analysis of 6-dof parailel mechanisms (see for instance [14] and [22]).

3.2.1 Planar parallel r;=cmipulators with revolute actuators

Since the application of the Newton-Euler formulas to solve the dynamic problem of

planar mechanisms is rather straightforward, the detailed procedure for the dynarnic

analysis of planar two- and three-degree-of-freedom manipulators is not given here.

However, in order to verify the results obtained with the approach based on the principle

of virtual work, two computation prograrns corresponding to the two approaches have

been written for the two types of planar parallel mechanisms. Final15 an esample

is given for each manipulator and it can be verified from the esamples that the tnro

approaches lead to identical results.

3.2.1.1 Two-degree-of-freedom manipulator

An esample for the planar two-degree-of-freedom manipulator Kith revolute actuators is

now given to illustrate the resdts. It is assumed that the end-effector of the manipulator

follows a simple trajectory involving vertical motion dong the direction of the y a~is.

The trajectory is described in cletail belon-. The force and torque needed to produce

the specified motion are obtained by the approach based on the principle of virtual

work and the Xewton-Euler formulation. The two approaches lead to identical results

and the approach based on the principle of virtual work leads to a faster algorithm.

The parameters used in this example are

O O O O O

11 = I2 = L4 =

The specified trajectory is

with KI = -1: K2 = 1, the masses are given in tons: the lengths in meters and the

angles in radians.

The generalized input forces obtained for the two actuated joints are represented in

Figure 3.4.

(a) Torque at the actuated joint 1. (b) Torque at the actuated joint 2.

Figure 3.4: Generalized actuator force in the planar two-dof mechanism

3.2.1.2 Three-degree-of-freedom manipulator

Similarly, an example for the planar three-degree-of-freedom manipulator 116th revolute

actuators is now given to illustrate the results. It is assumed that the end-effector of the

manipulator follows a simple trajectory involving vertical motion aIong the direction

of the y avis with the orientation angle d h e d . The trajectory is described in detail

below. The force and torque needed to produce the specified motion are obtained by

the two approaches.

The parameters used in this example are

The specified trajectory is

n-ith Kl = -1. h2 = 1: K3 = l1 the masses are given in tons. the lengths in meters

and the angles in radians.

The generalized input forces obtained for the three actuated joints are represented

in Figures 3.5 and 3.6.

(a) Torque at the actuated joint 1. (b) Torque at the actuated joint 2.

Figure 3.3: Generalized force in the planar three-dof mechanism at the actuated joints

1 and 2.

Figure 3.6: Generalized force in the planar three-dof mechanism at the actuated joint

3.

3.2.2 Spatial parallel manipulators with revolute actuators

As mentioned in the introduction and in the previous subsection. the dynamic analysis

of spatial parallel manipulators using the Sewton-Euler formulation has been studied

by many authors. for example. the dynamic analysis of spatial sis-degree-of-freedom

parallel manipulators using the latter approach [IO] and [l-l]. Therefore. similarly

to the planar paraIlel manipulators. the procedure for the dynamic analysis of the

spatial sis-degree-of-freedom parallel manipulator using the Newton-Euler formulation

is not ciiscussed here and only an esampie is given to verify the results obtained with

the approach based on the principle of virtual work. Kowever. for the spatial parallel

manipulators with reduced degree of freedom. namely, the spatial four- and five-degree-

of-freedom parallel manipuIators. since their architectures differ from the spatial sis-

degree-of-freedorn parallel manipulator, the procedures for the dpamic analysis using

the Xewton-Euler formulation is different and will be described belon--

3.2.2.1 Four-degree-of-fkeedom manipulator

The constraint forces between the 5th leg and the platforrn and the associated coordi-

nate frame are represented in Figure 3.7. Here it is assumed that gj and f5 are normal

Figure 3.7: The constraint forces between the fifth leg and the platform.

to and along the direction of the 5th leg OP5, namely

g5 = g5y f5- [O] f j

!doreover. the linear acceleration of the center of mass of the 5th leg can be obtained

as follo~vs

a5 = w 5 x Q 5 r 5 + u 5 x (wj x Qjrj) (3.149)

where Q5: w 5 : cbj and r j are as previously defined.

Considering axis y, one obtains one element gsZ of the force gj by the application

of the Euler equation around this auis. This Euler equation can be u-ritten as

d -Qs[(gs + f5) x kJ + 7 ~ 5 x Q5r5 = m5as x Q 5 ~ 5 + z ( ~ ~ ~ 5 Q : w j ) (3.150)

where l5 = [O, O> 15JT is the vector connecting point O to P5 and w~ = [O: O. -mjglT is

the rveight of the 5th leg, where g is the gravitational acceleration and I5 is the inertia

tensor of the 5th Ieg with respect to its center of m a s .

Substituting eq.(3.91) into eq.(3.150) and extracting the second cornponent. one

then obtains

gst = [m5g~5 sin û. - ( I ~ ~ +- ~ n ~ r ~ ) a J / / ~ (3.131)

Figure 3.8: The constraint forces on the platform.

tvhere IS9 is the moment of the inertia of the 5th leg about the y axis with respect to

its center of mas .

It is then assumed that forces gi(i = 1,2,3.1)-their determination will be dis-

cussed below-and gsl are known. and therefore forces fi(i = I l . . . : 5) and gj, can be

determined by the application of the Kewton-Euler equations to the platforrn of the

rnanipulator in the coordinate system of the platform. Figure 3.8 shonrs the platforrn

nf the manipiilatm and the forces acting on it. Forces giz, giy ( i = 1:. . . . 4 ) and g,, are

obtained from the ith ( i = 1, . . . . 4 ) and 5th leg respectively as will be shown belou-.

The remaining unknown action forces are only siu elements, i.e., fi ( i = 1:. . . , 5) and

g5,. Hence, these unknown forces can be obtained by the application of Xewton-Euler

equations to the platform of the manipulator, i.e.,

where a,. is the acceleration of point 0' and where it is assumed that the center of mass

of the platform lies in 0' and 1, is the inertia tensor of the platform body expressed in

the local frame, and

with i = 1,. . . ? 5.

Equations (3.152) and (3.153) consist of siu scalar equations and six unknowns,

which can be mitten as

d/ = Cff (3.135)

where

where the elements of vector df and matris CI are obtained from eqs(3.152) and

(3.133). Equation (3.155) is readily solved for vector f . The determination of the forces

g,(i = 1,. . . ! 4) mentioned above is nom- discussed. Moreover, the detailed expressions

associated with the elements of the matris Cf and wctor di are given in -4ppendis D.

Figure 3.9 represents the forces acting on the two links of each leg. First. considering

the upper link, by the application of the Euler equation with respect to point 0:. one

can obtain force g,. One has

where g, is a force vector normal to the avis of the second moving link and ai, is the

linear acceleration of the center of m a s of the upper link of the i th leg. They are

expressed in the fked coordinate system. Vectors riu and wiu have been previously

defined. and

and

where mi, and mil are the masses of the upper and lower links of the i th leg. ivhile Iiu and III are their inertia tensors.

Taking the first two components of eq.(3.156) and solving for the two unknon-ns gi,

and gi,. one obtains

Substituting eqs(3.158) and (3.159) into equation (3.155): one obtains the forces f

In order to compute the generalized input forces (here the ith generalized input

force is the torque r, exerted around the revolute joint connecting the leg to the base of

the manipulator). one must first find the constraint forces in the joint connecting the

two links of the leg. This can be achieved by applying Newton's equation to the upper

link, Le.,

Q i u ( d i - & - f i ) + wiU =miu+u, i = 1, ..., 4 (3.160)

where d, is the constraint force between the two links of the i th leg.

Figure 9: The forces acting on the two links of the i th leg for the rnanipulator with

revolute actuators.

From eq(3.160). one obtains

g,, - F,,, cos a, cos 3, + Fiuy sin a, - (FI,, - m,,g) cos 0, sin 3,

dl = [ gLy - FLUY COS aI - F,,, COS 3L sin a, - (FI,, - m,,gj sin a, sin 3, (3.161)

fi - (F,,, - m,,g) cos 3: i- F,,= sin 5'z

where F,,,. F,,,. FI,,. JI,,,. JU,,, and -If,,, are previously defined.

Considering asis y,i. thc application of EuIer3 equation relative to this a i s on tilt.

lower link. one finally obtains the input torque as

mhere j, is the unit vector associated with the direction of the axis of the i th actuated

revolute joint.

.4n esample for the four-degree-of-freedom manipulator with revohte actuators is

now given t o ilIustrate the results. I t is assumed that the end-effector of the manipu-

lator folIows a simple trajectory involving vertical motion along the direction of the z

asis with the other three Cartesian coordinates fked. The trajectory is described in

detail below. The force and torque needed to produce the specified motion are obtained

using the procedures presented above. The tnro approaches lead to identical results and

the approach based on the principle of virtual mork leads to a faster algorithm.

The parameters used in this example are

miu =mil = 0.1, ri, = 1.2, r , ~ = 0.7, (i = 1,2,3.4): rp = O

. The specified trajectory is

11-ith KI = 1. = 1 and KI3 = - 1. ( i = 1.2.3.4). the masses are given in tons. the

lengths in meters and the angles in radians.

The generalized input forces obtained for the four actuated joints are represented

in Figure 3.10.

3.2.2.2 Five-degree-of-freedom manipulator

Similarly, the constraint forces between the 6th leg of this manipulator and the platform

as well as the associated coordinate frames are represented in Figure 3.11. Here it is

assumed that gs and f6 are normal to and dong the direction of the 6th leg OP6, namely

(a) Torque at the actuated

joint 1.

(b) Torque at the actuated

joint 2.

(c) Torque at the actuated

joint 3.

(d) Torque at the actuated

joint 4.

Figure 3.10: Generalized force for the spatial four-dof rnechanism at the actuated joints

1 to 4.

3~Ioreover, the linear acceleration of the center of mass of the 6th leg can be obtained

as follows

a6 = ~6 X Q6r6 + W6 X ( ~ 6 X Qsrsf (3.165)

where al1 the matrices and vectors are as previously defined.

Considering point 0, one obtains two elements g6, and g,, of the action force g bu

the application of the Euler equation relative to this point. This Euler equation can be

wntten as

l

Figure 3.11: The action forces of 6th leg.

where l6 = [O. 0 . 1 ~ 1 ~ is the vector connecting point O t o P6 and w6 = [O. O. r n G g l T is

the weight of the 6th Ieg, n-here g is the gra~l ta t ional acceleration and I6 is the inertia

tensor of the 6th leg with respect to its center of mass.

Substituting eqs.(3.122) and (3.123) into eq(3.166) and solving the equations. one

then obtains 9~~ and gsy.

Similarly, it is then assumed that forces g i ( i = 1- . . . : 5)-their determination w-il1

be discussed belon--are known, and then forces f,(i = 1,. . . . 6 ) can be deterniined by

the application of the Xewton-Euler equations to the platform of the manipulator in

the coordinate system of the platform.

Figure 3.12 shows the platforrn of the manipulator and the forces acting on it.

Forces gi, and gi, ( i = 1, . . . ,5) are obtained from the ith (i = 1,. . . , 5 ) leg as will

be shown below. The remaining unknown action forces are only six elements. i.e..

f, (i = 1, . . . : 6). Hence, these unknown forces can be obtained by the application of

Newton-Euler equations to the platform of the manipulator, i.e.,

5 d 4 6 4 X QP'~ + [Qi C(fi + pi ) ] X QP; = d ? ( ~ & ~ T ~ )

i=l

Figure 3.12: The action forces of the platform.

where ao1 is the acceleration of point 0' and where it is assumed that the mass center

of the platform lies in 0' and 1, is the inertia tensor of the platform body espressed in

the local frame. and

f, =

with i = 1.. . . ,6.

Equations (3.167) and (3.168) consist of six scaiar equations and sis unknowns.

which can be written as

dj = Cif (3.110)

where

where the elements of vector df and matLu CI are obtained from eqs.(3.167) and

(3.168). Equation (3.170) is readily solved for vector f. The determination of the

forces g,(i = 1, . . ., 5) mentioned above is completed using the same procedure as in

the case of the four-dof manipulator. This procedure is therefore not repeated here.

The rest of the procedure is to compute the generalized actuating forces. It is

identical to the procedure in the spatial four-dof paralle1 mechanism.

Therefore. the expression for the computation of the generalized forces of the mech-

anism can be written as

-Ln esample for the five-degree-of-freedom manipulator with revolute actuators is

given to illustrate the results. It is assumed that the end-effector of the manipulator

folloti~s a simple trajectory involving a vertical motion along the direction of the z &\ris

with the other four Cartesian iïsed coordinates. The trajectory is described in detail

below. The force and torque needed to produces the specified motion are obtained by

the two approaches.

The parameters used in this example are given as

The specified trajectory is

The resulting generalized input forces for five actuated joints are represented in

Figures 3.13 and 3.14.

3 -2.2.3 Six-degree-of-freedom manipulat or

Since the Xewton-Euler formulation has been apptied to the sis-dof manipuIator by

other authors, no derivation is presented here. An esample is simply given to iIlustrate

the results. In this esample , it is assumed that the end-effector of the manipuIa-

tor folIows a simple trajectory involving a vertical motion along the direction of the

z asis with the other five Cartesian h e d coordinates. The trajectory is described in

detail belon.. The discussed abow torqucs needed to produce the specified niotim are

obtained by the two approaches.

The parameters used in this example are given as

(a) Torque at the actuated

joint 1.

(b) Torque at the actuated

joint 2.

(c) Torque at the actuated

joint 3.

(d) Torque at the actuated

joint 4.

Figure 3.13: Generalized force for the spatial five-dof rnechanism a t the actuated joints

1 to 4.

m,, =mil =0.1, r,, = 2 . 2 , r,r= 1.7. ( i = I o . . . , 5 ) , r, = O

The specified trajectory is

Figure 3.14: Generaiized force for the spatial five-dof mechanism a t the actuated joint - 3.

The resulting generalized input forces for the sis actuated joints are represented in

Figures 3.15 and 3.16.

Conclusion

The dynamic anaIysis of planar and spatial paraHel manipulators has been addressed

in this chapter. The analysis of the position, velocity and acceleration of these manipu-

Iators has been performed. Two different methods for the derivation of the generalized

input forces have been presented and each method h a its own advantages. Finally. es-

amples have been given for each manipulator in order to illustrate the results. Paralle1

niauipulators are of interest for nianu applications in robotics and in flight simuiation.

The dynamic analysis is an important issue for the design and control of the rnanipu-

lators and can be efficiently handled with the procedures described in this chapter.

. .

w t ( r a d )

(a) Torque at the actuated

joint 1.

(b) Torque at the actuated

joint 2.

b ut ( rad )

( c ) Torque at the actuated

joint 3. (d) Torque at the actuated

joint 4.

Figure 3.13: Generalized force for the spatial six-dof mechanism at the actuated joints

1 to 4.

(a) Torque at the actuated

joint 5.

(b) Torque at the actuated

joint 6.

Figure 3.16: Generalized force for the spatial six-dof mechanism at the actuated joints

5 and 6 .

Chapter 4

S t at ic balancing

The static balancing of planar and spatial paraIlel mechanisms or manipulators with

revolute actuators is studied in this chapter. Static balancing is an important issue in

the design of parallel manipulators and mechanisms. Indeed, if a mechanism is stati-

cally balanced, its actuators will not contribute to supporting the weight of the links

in any configuration. Since parallel mechanisms are often used in applications in tvhich

large loads are involved, static balancing can lead to significant improvements in the

efficiency of the mechanisms. Two static balancing methods, namely. using counter-

weights and using springs, are used here. The first method leads to mechanisms tvith a

stationary global center of mass while the second approach Ieads to mechanisms whose

total potential energy (including the elastic potential energy stored in the springs as

well as the gravitational potential energy) is constant. The position vector of the global

center of mass and the total potential energy of the manipulator are first expressed as

functions of the position and orientation of the platform as well as the joint coordinates.

Then, the kinematic constra.int equations of the mechanism are introduced in order to

eliminate some of the dependent variables from the expressions. Finally, conditions for

static balancing are derived from the resulting expressions and examples are given in

order to illustrate the design methodologies.

4.1 Balancing wit h counterweight s

In this section, the conditions for the static balancing of the mechanisrns with coun-

tenveights are derived. This is accomplished by specikng that the global center of

rnass of the mechanism be fised. This property is useful for applications in n-hich the

rnanipulator or mechanism is required to be statically balanced for al1 directions. In

other wordst the resulting manipulator mould be staticaly balanced for an? direction

of the gravit'. vector, which is a desirable property for portable systems n-hich rnw be

mounted in different orientations.

4.1.1 Planar parallel manipulators with revolute actuators

4.1.1.1 Two-degree-of-freedom manipulator

This mechanism is represented in Figure 4.1. The black dots represent the center of

mass of the links while m,. 1,. r, and c, ( i = 1. . . . . 4) are respective1~- the n ia s . the

length. the distance from the joint to the center of mass and the angIe between the Iink

and the line connecting the joint and the center of mass of the i th link. Moreover. it is

assumed that the direction of the gravity is along the negative direction of the y axis.

The espression of the global center of mass of the mechanism can be written as

ivhere !Li and r are the total mass and the vector of the global center of rnass, ivhile r,

is the vector of the center of mass of the ith link.

l'ectors ri can be espressed as functions of the orientation angles of the links. Le..

T , C O S ( ~ I + ~ I ) ] , [ II cos + r2 COS (al + @2) r2 =

TI sin (61 + $1) II sin el + rp sin (aI + &) I

Figure 4.1 : Planar tn-O-degree-of-freedom mechanism.

I4 COS 82 + 7-3 COS (a2 f &3) f IO r4 COS ( 8 2 + 'li.'4) + xo r3 = [

l4 sin 192 + r3 sin (a2 + tb3) + y0 rl sin (02 + $4 j + y0

where xo and y0 are the position coordinates of point O1.

The substitution of eqs(4.2) and (4.3) into eq.(l . l) leads to

r, = (mlr l cos cl + mzll) cos 61 - mlrl sin ~I sin BI

jm4r4 cos c14 + m314 j cos O2 - m4r4 sin lu4 sin B2

m2r2 cos zb2 cos al - mprp sin 1~12 sin 0 1

m3r3 cos q3 COS a2 - m3r3 sin $73 sin a2 + (m3 + m4}xo

r , = (mlr l cos + m211) sin Or - mlrl sin QL COS Bi

(m4r4 cos w4 + m314) sin B2 - m4r4 sin d4 cos û2

m2r2 cos Si2 sin al - m2r2 sin G2 COS al

m3r3 cos q3 sin a2 - m3r3 sin Q3 COS a2 + (m3 + m4) y0

From the closed loop of the mechanism 002030401i one can obtain the following

constraint equations

1 1 COS el + l 2 COS û.1 = xo + i3 COS 02 + 1.4 COS O2 (4.7)

I I sin O1 + l2 sin al = y. + l3 sin û2 + 4 sin B2 (4.8)

Eliminating cosaz and sina2 in eq.(4.4) by substitution of eqs.(4,7) and (4.8) into

eqs(4.5) and (4.6): one has

r, = AI cos + BI sin 65 + A2 cos 82 + B2 sin Oz

+A3 COS QI + B3 sin QI + C, T, = AI sin Br t BI cos O1 + -4z sin 82 + B2 COS O2

+.43 sin QI + B3 COS Q I + C,,

i2 B3 = m2r2 sin 7j2 + -m3r3 sin Q3 13

and C, and C, are constants which are independent from the joint variables. Le..

In order for the position vector of the global center of mass of the mechanism to be

constant: the coefficients of the variables 01,02 and û1 in the above expressions must

vanish. Therefore, the conditions for the static balancing of the two-degree-of-freedom

manipulator should be

;l,=O: Bi=O. i = 1 ? 2 , 3 (1.18)

+An example is now given in order to illustrate the application of the balancing

conditions derived above to this type of mechanism.

For the two-dof mechanisrn, let

Figure 1.2: Tm-degree-of-freedom balanced mechanism with çounterweights.

X , = 1.5. YI = O

where the masses are given in kilograms and the lengths in meters.

From equations (4.18) one obtains

ml = 1 (kg), m4 = 3 (kg). rl = rz = 0.5. $L = d:., = 7i

The balanced mechanism is represented schematically in Figure 4.2 where the size

of the black dots is roughly proportional to the mass of the links. The center of mass

of this mechanism wiil remain fixed for any configuration and hence, the actuators wi11

never contribute to supporting the weight of the links. This mechanism is graklty-

compensated for an\: orientation and magaitude of the gravity vector.

Figure 4.3: P lanar t hree-degree-of-freedom mechanism.

4.1.1.2 Three-degree-of-freedom manipulator

This rnechanism is represented in Figure 4.3. Similarlyl the black dots represent the

center of mass of the links while m i j , l,,. T i j and u,, (i = 1: 2,3 j = ll 2) are respec-

tively the m a s , the length, the distance from the joint to the center of mass and the

angIe between the link and the line connecting the joint and the center of mass of the

j t h link of the ith Ieg.

The espression of the global center of mass of the mechanism can be writ ten as

rrhcrc m , ~ . ml? : r , ~ and rt2 arc the mass and the position vector of the center of mass

of the two linlis of the i th leg. m3 and r3 are the mass and the position vector of the

center of mass of the end-effector and JI and r are the total mass and the position

vector of the global center of mass.

Vectors r, can be espressed as functions of the orientation angles of the links. Le..

L i l COS 8, + T i 2 COS (ai f Q i ) f XO, ri2 =

l i l sin 8, + r i 2 sin (ai + ?,hi) + yoi

u-here xoi and y,,, with i = 2 , 3 are the position coordinates of joint O,.

Substituting eq~(4.20)-(4.22) into eq.(4.19) leads to

where

+(m22~22 COS &2) COS a2 + (m22r22 sin ~ 5 2 2 ) sin a:!

+(m32~32 COS ~ 3 ~ ) COS a3 + (m32r32 sin dii32) sin a 3 + C, (1.24)

r, = (ml l r I l sin oI1) sin O1 + [mll r l l cos + (ml2 + m3)lll] cos 01

+(m21r21 sin d21) sin 02 + (rn21r21 COS Ib21 + m12121) COS 02

+(m31r31 sin 1b31) sin 63 + (~7231~31 COS Ib31 + m32131) cos03

i-(m12r12 sin d12) sin ÛI + (~212~12 cos $12 + m3112) cos al

+ (m22 T~~ sin 7.!922) sin û 2 + (731227-22 COS ,$22) COS a:!

+(rn32r32 sin c&) sin û3 + (m32~32 COS d~32) COS û3 + C, (4.25)

where C, and C, are constant terms.

From the geometric relations of the mechanism (from the closed loops). one can

obtain the following kinematic constraint equations

Eliminating cos a2: sin 02, COS a3 and sin a 3 in eq. (4.23) by substitution of eqs. (4.26)-

(4.29) into eqs.(4.24) and (4.25)1 one has

T, = .-II cos BI + BI sin 81 + -42 cos O2 + B2 sin O2 + A3 cos O3 + B3 sin O3

+.il4 cos a 1 + B4 sin <YI + Ag COS 4 + B5 sin 4 + C: (4.30)

r, = Al sin 81 + 3 1 cos el + A2 sin + B2 cos O2 + A3 sin O3 + B3 cos 83

+.4c sin al + B4 cos al + .15 sin 4 + Bs cos q!~ + C; (4.31)

where

112 112 .-ls = r n l z r l 2 cos d~~~ + rn3l12 + -rn32r32 COS d 3 2 + - r n 2 2 ~ 2 2 COS d772 (4.38) h 2 122

and where C: and C; are constant terms which can be rr i t ten as

Similarly to the previous case, in order for the position vector of the global center of

mass of the mechanism be constant, the coefficients of variables el , 02, 03: C#I and al in

the above expressions must vanish. Therefore, the conditions for the static balancing

of the three-degree-of-freedom mechanism should be

.ln example is now given to illustrate the results derived above.

For the three-dof mechanism, let

Figure 4.4: Three-degree-of-freedom balanced mechanism with countenveights.

ml* = 5 (kg). ml? = 3 (kg). rll = rl2 = 0.5

The balanced mechanism is represented schematically in Figure 4.4.

4.1.2 Spatial parallel manipulators with revolute actuators

The conditions for the static balancing of spatial parallel mechanisms will non- be

derived using countenveights.

Static balancing using countenveights consists in ensuring that the global center

of mass of the mechanism remains fked for any configuration of the mechanism. In

other words, the resulting manipulator would be statically balanced for anÿ direction of

the gravity vector - and hence the weight of the mechanism does not have any effect

on the actuators -, which is a desirable property for portable systems which may be

mounted in different orientations.

Figure 4.5: Geometric representation of the spatial four-degree-of-freedom system with

revolute actuators.

4.1.2.1 Four-degree-of-fkeedom manipulator

The spatial four-dof parallel mechanism is represented in Figure 4.5. .A reference coor-

dinate frame attached to each link must first be defined.

The coordinate frame of the base. designated as the O - xl y , t frame is fised to

the base with its Z-axis pointing vertically upward. Similarlÿ. the moving coordinate , , l

frame O' - x , y . z is attached to the pIatform, as represented in Figure 4.5.

The Cartesian coordinates of the platform are given by the position of point O' with

respect to the fixed frame. noted p = [x: y: clT and the orientation of the platform (the

orientation of frame O' - x'y'z' with respect to the fixed frame). represented by matris

Q, which can be written as q l l q12 413

(4.43)

q31 432 933

where the entries can be espressed as functions of Euler angles, quadratic invariants.

linear invariants or any other representation.

Finally, the coordinates of point Pi (Figure 4.5) relative to the moving coordinate

frame of the platform are noted (ai, bit c,) with i = 1,. . . 5.

Figure 4.6: Ceurnetri. of the fifth leg.

where p, ( i = 1:. . . . 4 ) is the position vector of point Pz espressed in the fked coordinate

frame, is the position vector of point P, expressed in the nioving coordinate frame.

and

iéc tor p5 is the position vector of point P5 expressed in the fked coordinate frame. as

represented in Figure 4.6 and can be espressed as

If it is assumed that the center of mass of the 5th leg is located on the line connecting

point O and Psi one can compute the position vector of the center of mass of the 5th

leg? as represented in Figure 4.6! Le.:

where r5 is the position vector of the center of mass of the 5th leg and l5 is the length

of the leg and lBc is the distance from O to C5.

The two links of the ith leg of the mechanism are represented schematically in

Figure 4.7. -4 reference frarne noted Oii - xi, zi is attached to the first Link of the

leg. Point Oil is located a t the center of the first revolute joint. The coordinates of

point Oil expressed in the base coordinate fiame are (xio7 yioi zio), where i = 1,. . . ,4.

Moreover, the unit vectors defined in the direction of axes z,, y* and zi are noted x , ~ , yil

and z,l, respectively.

Vector zil is defined along the axis directed from point Oil toward point 0i2 while

vector xl l is defined along the direction of the first revolute joint axis. Finallc vector

y, , is defined as

Also, points Cl[ and Cl, denote respectively the center of mass of the Iower and upper

link of each leg.

Let Bi be the joint variable associated with the first revolute joint of the ith leg

and 3, be the angle between the positive direction of the x avis of the base coordinate

frame and the coordinate axis x , ~ . where it is assumed that vector xi1 is contained in

the xy pIane of the fised reference frame (Figure 4.7). One can write the rotation

matris giving the orientation of frame Oil - x i . Y i , q with respect to the reference frame

attached to the base as

cos yi - sin ;ii cos Bi sin -ji sin O,

QrI = sin 7, COS ri COS Bi - COS yi sin 8, , i = 1. . . . ,c l [ 0 sin Bi cos ei 1 (4.49)

SIoreover, it is assumed that the center of mass of the second link of the ith leg lies on

line Ol2PZI as represented in Figure 4.7- One can then \-rite

where pil and r,, are respectively the position vectors of points 0i2 and Cll1 espressed in

the base coordinate frame, as represented in Figure 4.7, while I i I is the vector pointing

from Oil to Oiz and espressed in the local coordinate frame: and

where l I l is the distance from Oil to 0i2-

Eq.(4.50) can be written in component form as

Figure 4.7: Geometry of the ith leg.

Then. one c m cornpute the position vector of the center of mass of the second link of

the ith Ieg from the position vectors of points 0i2 and Pt as

1 1 , - ( p i ) i = 1 ..... 4 (1.53)

Il,

rr-hcrc r,, is the position vcctor of the center of m a s of the upper link of the itli k g

and ~vr-here l,, and i,, are respectively the distance from Oiz to P, and from 0t2 to

C,,. Sloreover. position vector p, can be expressed as a function of the position and

orientation of the platform. i.e..

where

p = [il, [il i = ~ . 4

!Vith al1 the above definition, the global center of mass of the manipuiator can be

where hl is the total mass of al1 moving links of the mechanism, m,, mg, mi, and mil

are respectively the masses of the platform, of the special5th leg, of the upper link and

lower link of the ith leg, and

mhile r, and ril are respectively the position vectors of the center of mass of the platform

and of the lower links of the ith leg, namely

where c, and c,l are the position vectors of the center of mass of the platform and the

lower link of the ith leg espressed in the local reference frame, and whose components

are given as

Substituting eqs.(4.47), (4.55): (4.60) and (4.61) into eq.(4.58), one then obtains

where

where D,,, Dyo and D,, are constant quantities, and

IE is clear that when the coefficients of the joint and Cartesian variables in the expres-

sions of r,: r,, and r= vanish, the global center of mass of the manipulator wiII be fised.

Hence, one obtains the conditions for static balancing as follo~vs

An esample is now given in order to illustrate the application of the baiancing

conditionsl derived above: to this type of mechanism.

For the 4-dof manipulator with revolute actuators studied here. let

where the masses are given in kilograms and the lengths in meters.

From equations (4.67) one obtains

Figure 4.8: Four-degree-of-freedom balanced mechanism with revolute actuators using

countenveights.

Figure 1.9: Geornetric representation of the spatial five-degree-of-freedom rnechanism

with revolute actuated joints.

2, = 0: y, = O: + = 12 (cm): Ijc = -$O (cm)? m j = 24 (kg)

The baIanced mechanism is represented schematically in Figure 4.8.

4.1.2.2 Five-degree-of-freedom manipulator

The spatial five-dof parallel mechanism is represented schematically in Figure 4.9.

For this mechanism, the coordinate frames attached to the tn-O links of the identical

actuated legs and to the platform are the same as the ones defined in the case of spatial

four-dof parallel mechanism. Hence, one can write

Figure 4.10: Geometry of the sixth leg.

where p, and (i = 1,. . . - 6 ) are the position vector of point P, espressed in the h e d

coordinate frame and the rnoving coordinate frarne respectively. and

x 5 2

p = [!] pi= [ - i l . & = [II. i - l . . 6 (4.69)

- &2

One can compute the position vector of the center of mass of the 6th kg. narnely.

the position vector of point Cs. as represented in Figure 4.10.

where r6 is the position vector of the center of mass of the 6th leg.

Similarly to the previous case, the position vector of the center of mass of the second

link of the i th leg can be determined from the position vectors of points 0t2 and PI as

where riu7 Liu and fi, are as previously defined.

Then, the global center of mass of the manipulator c m be espressed as

where llf is the total mass of dl rnoving links of the mechanism, m,, m6, mi, and mn

are respectively the masses of the platform, of the special sixth leg, of the upper link

and lower link of the i th leg, and

while r, and ril are respectively the position vectors of the center of mass of the platfom

and of the lower links of the ith leg, namely

where c, and cil are the position vectors of the center of m a s of the platform and the

first link of the i th leg expressed in the local reference frame. and whose cornponents

are given as

Substituting eqs.(4.70) (4.71): (1.74) and (1.73) into eq.(4.72). one then obtains

where Dzo, D,, and DzO are constant quantities, and

Similarly. when the coefficients of the variables such as D,(i = 1' . . . 12) vanish.

the global center of m a s of the manipulator will be fiued. Thereby. one obtains the

conditions for static baiancing as follows

.in esample is non* given in order to illustrate the application of the balancing

conditions derived above to this type of mechanism.

For the 3-dof manipuiator n-ith revolute actuators: let

a l = -0.5. . bl = -0.5. cl = -0.3. a2 = 0.5. .b2 = -0.5. C? = -0.3

a3 = 0.5. b3 = 0.3, CJ = -0.3, a4 = -0.5. b4 = 0.5, c4 = -0.3

Uj =O.z, b5 = o . Cg =-0.3. a6 = O ? b6 = 0, CG = -0.3

xlo = -1.5: ylo = -1.5, zlo = O, ~ 2 , = 1.5, = -1.5. t20 = O

xgo = 1.5, ~3~ = 1.8, = 0, ~4~ = -1.3, ~4~ = 1.5, ~4~ = O

xzo = 1.3: Yso = 0, zjo = 0: x6, = 0, 960 = 0: 260 = 0

7r 7r 3Ïr 'ii 7

"(1 = --. 72 = -: y3 = - 4 = --

-fs = - 4 4 4 ' 4

Yic = 0: zic = -0.5. xp = O

where the masses are given in kilograms and the lengths in meters.

From equations (4.81) one obtains

Figure 1.11: Five-degree-of-freedom balanced mechanism with revolute actuators using

countenveights.

m,, = 4 (kg), mi, = 13 (kg) (i = 1, . . . -6)

y, = 0, +, = 30 (cm)

The balanced mechanism is represented schematically in Figure 4.11.

4.1.2.3 Six-degree-of-freedom manipulator

The spatial sis-dof parallel mechanism is represented schematically in Figure 4.13.

The coordinate frarne attached to the platform and the two links of the actuated

leg are the same as the ones defined for the spatial four-dof parallel mechanism.

b ï t h al1 the definitions of the vectors given for the case of the spatial four-dof

parallel mechanisrn, one has

where ri, is the position vector of the center of mass of the upper Iink of the i th kg.

and where Li,, Li, are as previously defined.

Figure 4.12: Spatial six-degree-of-freedom paraIlel mechanism with revoIute actuators.

Shen. the global center of mass of the mechanism, noted r can be u-ritten as

where 3.l is the total mass of al1 moving links of the mechanisrn, m,. miu and m , ~ are

respectively the masses of the platform: the upper link and lower Iink of the ith Ieg,

and

while r, and ril are respectively the position vectors of the center of mass of the platform

of the mechanism and of the center of m a s of the lower link of the ith Leg, namely

where c, and cil are the position vectors of the center of mass of the platform and of

the lower links expressed in the local reference frame, and whose components are given

Substituting eqs.(4.82), (4.85) and (4.86) into eq.(4.83), one then obtains

where

where D,,. Dy, and D,, are constant coefficients: and where

If the coefficients of the joint and Cartesian variables in eqs.(4.89)-(4.91) vanish. the

global center of mass of the manipulator will be fised. Therefore. one obtains the

conditions for static balancing as follows

An example is nonr given in order to illustrate the application of the balancing

conditions to the type of mechanism described above. For this mechanism. let

-- - - - - -

Figure 4.13: Complete baiancing using counterweights.

yic = O, z,, = -0.5

where the masses are given in kilograms and the lengths in meters.

From eqs.(4.92), one obtains

2 6 ~ = -0-5. miu = 4 (kg), mi[ = 13 (kg) (i = 1,. . . .6)

q, = O (m), y, = 0, z, = 0.3 (m)

The balanced manipulator is represented schematicaily in Figure 4.13.

As can be realized from the figure. large countenveights are necessal to balance

the mechanism.

4.2 Balancing with springs

In this section, the p l ana and spatial paraIlel mechanisms or manipulators will be

statically balanced using springs. Static bdancing using springs consists in ensuring

that the total potential energy of the rnechanisrn is kept constant, which means that

the weight of the mechanism does not have any effect on the actuators. Moreover, using

this approach, the weight of the whoIe mechanisrn can be balanced with a much smaller

total mass than when using countenveights. This approach may be more appropriate

for applications in which large counterweights would be impractical. Houlever, it should

be noted that the mechanisms obtained with this approach will be balanced for only

ori_e direction and magnitude of the gravity vector.

4.2.1 Planar parallel manipulators wit h revolute actuators

In order to use springs to balance the manipulator. a special architecture (similar to

what was used in [54/) is proposed for the legs. -4s represented in Figure 4.14. a

parallelogram four-bar linkage is used instead of the first link of the ith leg. This

enables the attachment of a spring t o the upper link of the leg and to a support

which is maintained vertically. -1 spring is also attached to the parallelogram. The

upper link of the leg is then mounted on a revolute joint with a horizontal axis. The

new architecture is kinernatically equivalent to the prm-ious one. Hon-ever, the nen-

architecture now allows the use of springs for the static balancing of the mechanism.

Moreover, it is pointed out that the global center of mass of the parallelogram and the

center of mass of the replaced first link of the ith leg can be handled similarly.

4.2.1.1 Two-degree-of-freedom manipulator

The expression of the total potential energy of the mechanism can then be written as

where V, and V, are respectively the the gravitational potential energy of the rnech-

anism and the elastic potentia1 energy stored in the springs. Theses quantities can be

Figure 4.14: Geometry and kinernatic architecture of the ith leg.

wi t ten: for ~ h i s mechanism as

where r, is defined in eq.(4.6), g is the gravitationai acceIeration. kif is the stiffness of

the lower spring of the ith leg, eir is the length of the lower spring of the i t h leg, k,, is

the stiffness of the upper spring of the i th leg and ei, is its length. I t is assumed here

that the undeformed length of the spring is equal to zero in order to obtain complete

balancing [54]. -4s shon?i in [54], this condition can easily be met in a practical design

using, for instance: cables and pulleys.

Using the Iaw of cosines, the effective Iength of the springs can be written as

eil = Jh:, + el - 2 hiidil sin O,, i = 1.2 (1.96) - -

ei, = Jh:; i- eu - 2ht,diu sin ai, i = 1 ,2

where hii and dil are the distances fiom the revolute joint located at Oil to the attach-

ment points of the lower spring (Figure 4.14) while hi, and diu are the sarne distances

for the upper spring.

Moreover, from eq.(4.8) one can obtain

1 sin a:, = - ( I I sin B1 + l2 sin a l - lq sin û2 - y,-,) (4.98)

13

Substituting eqs.(4.96)-(4.98) into eqs(4.94) and (4.95) and then substituting the

latter equations into eq.(4.93), one obtains

V = (.41g - 2kllhlldll - 2k2uh2ud2u) sin el + Blg cos 61

(--I2g - 2k11 h21d21 + 3k2,h2,d2,,) sin e2 + B2g COS 02

(A3g - 2kluhludlu - 2k2uh2ud2u) sin al + B3g cos û1 + D, (4.99)

From eq(4.99) one can finally obtain the conditions for the static balancing of the

manipulaior with springs as follows

where -4, and Bi (i = 1,2,3) have been defined in eqs.(l.ll)-(1.16).

..An exarnple is now given in order to illustrate the application of the balancing

conditions derived above.

For the two-dof mechanism, let

Figure 4-15: PIanar tm-degree-of-freedorn balanced mechanism with springs.

X I = 1.5. y1 = 0

where the masses are given in kilograms and the lengths in meters.

From eqs.(4100)-(4.105). one obtains

The balanced mechanisrn is represented schernatically in Figure 4.15.

4.2.1.2 Three-degr ee-of-freedom manipulator

SimiIarlyt the expression of the total potentiat energy of the planar three-degree-of-

freedom manipulator can also be written as

where V, and V, are respectively the gravitational potential energy of the mechanism

and the elastic potential energy stored in the springs. These quantities can be written.

for this mechanism as

where r, is defined in eq.(4.25).

Using the law of cosines, the length of the springs can be written as

where hi[ and dil are the distances from the revolute joint located at Oil to the attach-

ment points of the lower spring (Figure 4.14) while hi, and di, are the same distances

for the upper spring.

Xloreover, from eqs.(1.27) and (4.29) one has

1 sin a3 = -(il1 sin el + lL2 sin al $- 1, sin Q - /31 sin 83 - go2) (4.112)

132

Substituting eqs.(4.109)-(4.112) into eqs.(4.107) and (1.108) and then substituting

the latter equations into eq.(4.106). one can obtain

112 112 ( - 4 - 2kluhludlu - 2-hnud2u - 2-h3ud3u) sin01 + B4 cos al 122 132

13 4 (-45 - -h2ud2u - 2-h3ud3u) sin 6 + B5 cos d + Dy 122 132

From eq.(4.113) one can finally obtain the conditions for the static balancing of the

manipulator with springs as folloms

Similarly to the previous case, an example is non: given to illustrate the results

derived above.

For the three-dof mechanism. let

m3 =mil = mi2 = 1 (i = 1,2,3)

hil = h,, = 0.5 ( i = 1' 2. 3),dZu = d3u = 0.5

l3 = 0.6. l4 = 1.2. Ili = 1,2 = 1 ( i = 1.2:3)

ril = ri2 = 0.5, r3 = 0.6, îi.li = ei2 = O (i = 1: 3'3)

= 1.5, X O ~ = 3, y01 = ~2 = O

n-here the masses are given in kilograms and the lengths in meters.

From eqs.(4.113)-(4.123) one obtains

k,, = 10I\;/m. kZl = JOX/m (i = 1, '2.3)

dil = 0.5 (i = 1,2. 3)dlu = 0.5

The baIanced mechanism is represented schematically in Figure 4.16.

4.2.2 Spatial parallel manipulators wit h revolute actuators

4.2.2.1 Four-degree-of-freedom manipulat or

SimiIarly to what --as obtained for the planar manipulators, the expression of the total

potential energy of this spatial manipulator can be written as

Figure 4.16: Planar three-degree-of-freedom balanced mechanism a i t h springs.

Figure 4.17: Architecture and geometq- of the ith leg.

where \-, and V, are respectively the the gravitational potential energ?. of the mani-

pulator and the elastic potential energy stored in the springs. These quantities can be

espressed as

where rz is defined in eq.(4.66) I , is the stiffness of the j th spring. el is its lengfh and

other quantities are as previously defined.

For this type of manipulator, a spring can be used in each revolute joint connecting

the i th leg to the base of the manipulator, where i = 1,. . . ,S. This is represented

in Figure 4.17 where the geometnc parameters are defined. Therefore, the potential

energy stored in the spring attached to the i th 1eg can be written as

where. from the law of cosines

and

e5 = {hg + d;: - 2hJdj cos ol

Substituting eqs(4.125) and (4.126) into eq.(4.124) leads to

1 l 4 +(Dl? - ?k5h5d5) cos a + -k(h2 + d2) + - li,(h: + d:) + D,. (4.130)

2 2 t = l

From eq.(4.130) one can finalIy obtain the conditions for the static balancing of the

manipulator when springs are used as follows

where coefficients Dl , i = 1:. . . .12 are the ones used in eq.(4.67).

-An esample is now given in order to illustrate the application of the balancing

conditions derived above to this type of mechanism with springs.

For the 4-dof manipulator with revoIute actuators, let

Figure 4.18: Four-degree-of-freedom balanced mechanisrn with revolute actuators using

springs.

11-here the masses are given in kilograms and the Iengths in meters.

From equations (-I.l31)-(4.133) one obtains

The balanced mechanism is represented schematicaIly in Figure 4.18.

4.2.2.2 Five-degree-of-freedom manipulator

Similarly to the previous case, the expression of the total potential energy of the ma-

nipulator can be written as

V = V , + V , (1.134)

where V, and V, are respectively the gravitational potential energy of the manipulator

and the elastic potential energy stored in the springs, which can be expressed as

where r- is defined in eq.(4.80) and other quantities are a s previously defined. As in

the previous case, a spring is attached to each of the links mounted on the base, as

illustrated in Figure 4.17.

From the law of cosines, one still has

and

e6 = Jha + 4 - 2h6d6 cos a

Substituting eqs.(4.135) and (4.136) into eq. (4.134) leads to

Frorn eq.(4.139) one can finally obtain the conditions for the static balancing of the

manipulator when springs are used as folIows

where coeEcients Di: i = 1, . . . , 1 4 are the ones used in eq. (4.81).

An example is now given in order to illustrate the application of the balancing

conditions derived above to this type of mechanism with springs.

For the 5-dof manipulator with revolute actuators, let

'y, = -- - 4 ? / 2 = - 7 7'3 = - 4 4 - = - 4

where the masses are given in kilograms and the lengths in meters.

From equations (4.140)-(4.145) one obtains

y,, = O. k, = 2 N/cm(i = 1,. . . , 5 )

The balanced mechanism is represented schematically in Figure 4.19.

Figure 4.19: Five-degree-of-freedom balanced mechanism ai th revolute actuators using

springs.

Figure 4.20: Geometry and kinematic architecture of the ith kg.

In order to use springs to balance the manipulator, a special architecture of the leg

similar to what was used in the case of static balancing of planar parallel mechanism

using springs (Figure 4.14) is proposed for the legs. As represented in Figure 4.20, a

parallelogram four-bar linkage is used instead of the first link of the ith leg. The upper

link of the leg is then mounted on a revolute joint with a horizontal axis which is in

turn rnounted on a revolute joint n i th a vertical LUS. The latter two joint form a Hooke

joint. This new architecture is equivalent to the original one.

Using the new architecture of the leg, the expression of the total potential energv

of the mechanism can then be written as

where V, and Y, are respectively the gravitational potential energy and the elastic

potential energy stored in the springs. Theses quantities can be written. for this mech-

anism as

V, = r-g + Dc (4.147)

V, = 1 C (kzle; f kiue:u) (4.148)

j=l - where r z is defined in eq.(1.91) and Dc is a constant which arises from the distance

d,. which increases the gravitational potential energ- This constant can be written

as D, = rn& + ~ ; , ( m ~ ~ d ~ ) . Again, g is the gravitational acceleration, kil is the

stiffness of the lower spring of the ith leg, eii is the length of the lower spring of the

ith leg, kiu is the stifhess of the upper spring of the ith leg and ei, is its length. It is

also açsumed here that the undeformed length of the

to obt.ain complete balancing [54].

Csing the law of cosines: the length of the springs

springs is equal to zero in order

can be written as

iwhere hi( and dii are the distances from the revolute joint located a t Oil to the attach-

ment points of the lower spring (Figure 4.20) while hiu and di, are the same distances

for the upper spring. cosBi caa be expressed as a function of angle ai as me11 as the

position and orientation of the platform, Le.,

and where t i and t i l are respectively the third components of the position vectors pi

and pi,.

Substituting eqs.(4.147), (4.148) and (4.119)-(4.151) into eq.(4.146): one then ob-

tains 6

lil V = C [ ( D , ~ - 2ki[hil& - 2ki,hiudiuT) COS Oi + Di+6g sin Bi]

From eq.(4.152) one can finallu obtain the conditions for the static balancing of the

manipulator with springs as follows

where coefficients Di are the ones used in eq.(4.81).

An esample is now given in order to ilhstrate the application of the balancing

conditions to this type of mechanism.

For the 6-dof manipulator with revolute actuators presented above, let

y,, = O ( i = 1 ..... 5)z,, = l,, = O.5m[i = 1 ,.... 6)

n-here the masses are given in kilograms and the lengths in meters.

From eqs.(4.153)-(4.158) one obtains

z, = O (m), y, = 0. zp = 0.45 (m)

The balanced mechanism is represented schematically in Figure 4.21. Since each

leg of the rnechanism has an identical architecture, onl? one Ieg is represented in the

figure.

As can be clearIÿ seen from the figure, the use of spnngs has ailowed one to eliminate

the counterweights. Hoivever, the resulting mechanism will be statically balanced if and

only if the gravity vector is aligned with the negative direction of the z mis of the fised

reference frame and if its magnitude is maintained.

.An alternative achitecture is now introduced for this type of manipulator. In the

new architecture, each of the legs is mounted on a passive revolute joint having a vertical

axis of rotation, as shown in Figure 4.22. The leg itself is a pianar mechanism with a

parallelogram ABCD, a distal link CP, and a spherical joint at point Pt. Additionally.

Figure 1.21: Balanced mechanism with sprint

a second parallelogram mechanism BEFC is introduced in the leg. as represented in

Figure 1.22. The second parailelogram mechanism is used to actuate the link CP

thereby improving the mechanical advantage. Link BE is the actuated Iink. aithough

this achitecture is more cornplex chan the previous one. it may have design advantages.

The potential energv of the springs used in the manipulator can be written as

where

e , ~ = Jh:u + eu - 2hIudiu sin di

where d, and Bi are respectiveiy the angles betmeen links BC and BE and the coordinate

axis x,.

Figure 4.22: Alternative architecture of leg for the 6-dof parallel mechanism trith rev-

olute actuators.

The gravitational potential energ-. of the manipulator can be espressed as

where wil and zci2 are respectively the lengths of links BE and EF, mtl and m,? are

their corresponding masses, and ri1 and ri2 are the distances of the centers of mass Cil and Ctp of the two links t o points B and E.

Since the manipulator consists of five independent kinematic closed loops. one can

write

From eq. (4.163) one has

1 sin di = - (q, + 111 sin & + I l , sin d l - t;., - liu sin Oi

lil

Substituting eq.(4.164) into eqs.(4.159) and (4.162), one then obtains the total potential

energy of this type of manipulator as

where

Dl =

Similarly to the previous cases, if the coefficients of the configuration variables in

eq.(4.165). Le.: D,(i = 1, . . . ,101 vanish, the total potentia1 energy will be constant.

Therefore, the conditions for the static balancing for this type of manipulator are

-4n example is now given in order to iIlustrate the application of the balancing

conditions to this type of mechanism.

For the alternative 6-dof manipulator presented above, let

al = -0.5. . bl = -0.3, cl = -0.3, a;! = 0.5, , b2 = -0.5, c:! = -0.3

a3 = 0.5. b3 = 0.5. c3 = -0.3, a4 = -0.5, b4 = 0.5, c4 = -0.3

as = 1.0. b5 = 0: cg = -0.3, as = -1.0. b6 = 0. CG = -0.3

hZl = h,, = 0.5. dil = d[, = O.5(i = 1.. . . :6)

where the masses are @en in kilograms and the lengths in meters.

From eqs.(4.166) one obtains

The balanced mechanism is represented scheinatically in Figure 4.23. Similarily to

the previous case, since each leg of the mechanism has an identical architecture, only

one leg is represented in the figure.

Figure 4.23: Balanced rnechanism with springs.

4.3 Conclusion

The static balancing of planar and spatial parallel mechanisms or manipulators has

been addressed in this chapter. Two static balancing approaches. namely. with coun-

terweights and with springs, have been introduced. The espressions of the position

vector of the global center of mass and of the potential energi- of the mechanisrn have

been derived. The kinematic constraint equations of the rnechanism have then been

used to eliminate some dependent variables from these expressions. The sets of equa-

tions of static balancing have finally been obtained from the resdting expressions.

Examples have also been given in order to illustrate the results.

It has been shown that the planar and spatial parallel mechanisrns presented in

the chapter can be statically balanced by the two approaches discussed above- Each

approach is suitable to different applications. The static balancing of planar and spatial

parallel mechanisms is of great interest, especially in applications where the mo~ing

masses are large since it leads to a substantial reduction of the actuator torques. The

conditions derived here can be used directly to design balanced systerns.

Conclusion

The kinematic analysis. dynamic analysis and static balancing of planar two- and three-

degree-of-freedom as w l l as spatial four-, five- and six-degree-of-freedom parallel mech-

anisms or manipulators have been addressed in the thesis.

-ifter having described the architecture of each of the mechanisms. the inverse kine-

matics has been computed for each of them and a new general algorithm for the de-

termination of the b o u n d q of the workspace of parallel mechanisms or manipulators

has been proposed. This algorithm has then been used to obtain the workspace of

the planar and spatial rnechanisms studied in the thesis. It has been shown that this

algorithm is general and can be applied to any type of parallel rnechanism or manip-

ulator. The velocity equations of the mechanisms have then been derived using two

approaches, nameIy, the algebraic formulation and the vec tor formulation. The latter is

a new approach which provides an equation of the velocity relations between the joint

velocities and the angular velocities of the moving links of the mechanism. The velocity

equations obtained using the two approaches have been used for the determination of

the singularity loci and the velocity equation obtained using the new approach leads to

a faster computational aIgorithm for the determination of the singularity loci,

The kinematic optimization of pianar and spatial paralle1 mechanisms or rnanip-

ulators with reduced degrees of freedom has also been discussed in this thesis. The

Generalized Reduced Gradient method of optimization has been used and led t o a fast

converging algorithm. The optimum design of the parallel mechanisms is useful for

their applications where the dependent Cartesian coordinates are required to follow

some prescribed trajectories as closely as possible.

The dynamic analysis of planar and spatial parallel mechanisms or manipulators

has been addressed in this thesis. To this end, the analysis of the position, velocity

and acceleration of these manipulators has been performed. Two different methods for

the derivation of the generalized input forces have been presented. The first approach

is a nem approach based on the principle of virtual work. It has been verified that

this approach is efficient and leads to a faster algorithm for the determination of the

generalized input forces, which is useful for the control of a manipulator. The second

approach has been used for the dynarnic analysis of parallel mechanisms or manipulators

by several researchers. It is suitable for the purpose of the design and simulation of

a manipulator. It has been used here mainly to verify the results obtained with the

new approach. Finallu, esamples have been given in order to illustrate the results. The

dynamic analysis is an important issue for the design and control of the manipulators

and can be efficiently handled with the procedures described in this thesis.

The static balancing of planar and spatial parallel mechanisms has also been ad-

dressed in this thesis. Two static balancing approaches, namely. with countenveights

and with springs have been used. To this end, the expressions of the position vectors

of the global center of mass and the potential energy of the mechanisms h a w been

derived. The sets of equations of static balancing have finally been obtained frorn these

expressions. Esamples have been given in order to illustrate the results. The esampies

are provided for ilhstrative purposes only. Indeed, it is clear. from the equations. that

infinitelu many staticaily balanced mechanisms exist, for each of the architectures stud-

ied here. Moreover, it is also found, by inspection of the equations, that balancing is

always possible for any given value of the geometric parameters. This is an interesting

result since it allows the kinematic design of a mechanism to be completed using any

criterion and the balancing to be performed a posteriori.

It has been clearly shown that the types of planar and spatial parallel mechanisms

studied here can be statically balanced using either one of the two approaches pre-

sented in this paper. Each approach has its own advantages and is suitable to different

applications. In al1 cases, the mechanisms obtained are perfectly balanced, i.e., no

toque is required a t the actuators to maintain the mechanisrn in static equilibrium for

any configuration. Static balancing of parallel mechanisrns is of g e a t interest and c m

be used in the design of mechanisms for robotics, flight simulators and several other

applications involving large Ioads or the simulation of free-floating conditions.

In the kinematic and dynamic anaiysis of mechanisms, although several types of

spatial parallel mechanisms with specified architectures have been studied here, the

structure for these mechanisms is not unique. There exisaçt other structure arrangements

for these mechanisms, for instance, exchanging the Hooke joint connecting the moving

links and the revolute joint attached to the base of the i th leg. The algorithms presented

in the thesis foi the determination of the workspace and singularity loci as well as

for the derivation of the generalized actuator forces can be applied to other spatial

parallel mechanisms ui th different structures. Moreover, if the deformations of the

links are considered, considering the links of the mechanisrns as flexible bodies. the

results obtained from the kinematic and dynamic analysis should be more accurate.

The static balancing of the parallel mechanisms or manipulators using counter-

weights and springs have been presented in this thesis. Hontever, how to build the prac-

tical balanced system. especially, the arrangement of the counterweights and springs.

will be north further studying. It would also be useful for some applications to consider

the dynamic balancing of parallel mechanisms.

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Appendix A

Polynomial formulation of the

singularity loci of planar parallel

manipulat or

-4s explained in Chapter 2, eqs.(2.78) and (2.85) are the expressions describing the sin-

gularity loci of the two- and three-degree-of-freedom manipulators, respectively. Since

these expressions contain some square roots-when they are expressed as functions of

the Cartesian coordinates- it is difficult to extract from them the information about

the characteristics of the singularity locus. In this appendiu, the polynomial expres-

sions of the singularity loci of the manipulators wi1l be found. Polynomial expressions

are useful since they can provide some insight into the locus and they may lead to

al ternative numerical solutions.

A. 1 Two-degree-of-freedom manipulator

Solving eq.(2.1) (with i = 1) and (2.78) for cosa l and sin al, one obtains

Cl COS (Y2 COSQl =

al cos a2 + bl sin a 2

cl sin a 2 s ina l =

al cos a2 + bl sin a 2

Substituting eqs.(-4.1) and (-4.2) into sin2 ÛL + cos2 al = 1

relation

al cos a 2 + bi sin a2 = Scl

(A. 1)

(-4-3)

then leads to the following

where S = f 1 denotes the type of singularity. When S = -1, links l3 and L4 are aligned

whereas when S = +I. the? are folded.

Then. letting i = 3 in eq.(S.l), the substitution of eqs. (-4.1) and (-4.2) into eq.(2.1)

leads to

a2 cos a:! + b2 sin a2 = c2 (-4.4)

From eqs.(.4.3) and (A.4). one obtains

Sclb2 - cZbr COS Qq =

a1b2 - a261

By squaring both sides of eqs.(.l.5) and (-4.6) and adding, one then obtains the poly-

nomial espression of the singularity loci of this manipulator. rvhich can be writtcn

The latter expression, eq.(.4.7) is a polynomial of degree 6 in x and y v-here mised

terms are present. The curves representing the singularity loci in the Cartesian space

are therefore of degree 6. The detailed expression is rather comples and is not given

here because of space limitation but it can be obtained from the author upon request.

A. 2 Three-degree-of- freedom manipulator

Similarly, for the three-degree-of-freedm manipulator, one can rewrite eq.(2.85) as

a0 cos al + bo sin al = O (-4.8)

where

a. = l4 sin 4 sin cr2 cos C Y ~ - l4 cos q5 sin a 2 sin a 3

+13 COS 4 sin a2 sin a3 - l3 sin c$ cos û 2 sin a 3

bo = Z4 cos 4 cos a 2 sin a3 - l4 sin 4 cos a2 cos a 3

+13 sin 4 cos a2 COS a3 - l3 COS 4 sin a2 cos 0 3

Letting i = 1: eq.(2.14) c m be rewritten as

Solving eqs.(A.8) and (-4.9) for cos crl and sin al and then substituting the results into 2 equation sin al + cos2 a1 = 1, one obtains an equation involving only 0 2 and û 3 . i.e..

Eq.(.\.lO) can be rewritten as

c,~ cos2 a 2 + ces cos a2 sin a2 + cs2 sin2 a:! = O (-1.1 1)

where coefficients Ccar Ces and Cs2 are functions of cos QJ and sin as.

Letting i = 2 in eq.(2.14), one obtains the second equation involving a 2 . i.e..

a2 cos a 2 + b2 sin a2 = c:! (-4.12)

Eliminating sin CQ from eqs.(X.11) and (-4.12): one then obtains

~2 cos2 û3 + D~ COS a3 + DO = O

E2 cos2 û3 -i- El cos a3 + EO = O

where

Using the resultant [33] on eqs. (-4- 13) and (-4.14) to eliminate a*, one obtains

Cc4 cos4 a3 i- Cdsl cos3 as sin a3 + Cb2 COS* C Y ~ sin2 a 3

+Cclr3 COS 0 3 sin3 a3 + Csg sinq a3 = O (-4.15)

Similarily, Ietting i = 3 in eq.(2.14) one obtains the equation involving a3, Le.,

a3 cos 013 + b3 sin a3 = c3 (A.16)

EIiminating sin a3 from eqs.(-4.15) and (A.16) then leads to

Csing the resultant once again on eqs.(A.l/) and (-4.18): one obtains the espression

describing the singularity locus of this manipulator. The expression is a poIynomia1

in x and y in which the highest degree of y is 64 while the highest degree of x is 48.

Therefore, the curves representing the singularity loci in the Cartesian space are of a

very high degree. This result is in contrast with the results obtained in [47] for manipu-

lators with prismatic actuators, which lead to quadratic singularity loci. Moreover: the

reason for the difference in degree between x and y is the assumption on the geometry

( k e d pivots are assumed to be aligned on the x axis).

Appendix B

General expressions of det (A*) for

planar parallel manipulators

For the two-degree-of-freedom manipulator, one has

For the three-degree-of-freedorn rnanipulator: one has

where

L

(a: + bf)(ag + b;)(a; + g ) ( ~ 1 ~ 2 ~ 3 ( ( ( a l b 3 - a3bl)b213 $- ( a 2 h - alb2)b314) COS 4

+ ( ( ~ 3 b l - alb3)a213 + (a162 - a2bl)a314) sin&)

( ~ 2 ~ 3 ( ( ( b l b 3 A ala3)6213 + (blb2 + ala2)b314) cos

+((a1a3 - b1b3)a2& - 16162 + ala2)ad4) sin 4 j G (clc3(((alb3 - a3bl)a213 - (ala2 f blb2)b3/4) COS 9

+((alb3 - a361)6213 + (a1a2 - b 1 6 ~ ) ~ 3 1 4 ) sin #)K2

( ~ 1 ~ 2 ( ( ( a l a 3 + blb3)k?/3 + Ia2b1 - alb2)~314) cos

+(-(ala3 + b1b3)a213 + (a2b1 - aIb2)b3E4) sin 4)hj

( c d ( - ( a ~ a s + hb3)~213 + ( a h - alk!)b3/4) COS 0

+(-(ala3 + bl b3)b2l3 + (a l b2 - a2bl)a314) sin @)KI&

(ca( ( (a lb3 - a3bl)b213 + (a la l + blb2)a314) COS d

Appendix C

Simplification of Jacobian matrix

Thc original Jacobian matri.; of the four-degree-of-freedom rnanipulator cari be n-ritten

10 O o s , and its determinant can actually be written as

where

Csing similar operations in the other r o m and columns and then exchanging some

rows of the matris: one can obtain

where Li= # 0: l iy # O and l iz # O ( i = 1: 2,3 ,4) .

mhich can be written in another form

where E is the 8 by 8 identity matrix, O the 4 by 8 zero matrix, D the 8 by 4 matrix

and J the reduced 4 by 4 matrix.

and

FinaIl- one obtains

Appendix D

Expressions associated with the

elements of the matrix Cf and

vector df

The detailed expressions of the elements of Cf and d, can be written as:

TEST TARGET (QA-3)

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