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Quality index and kinematic analysis of spatial redundant in-parallel manipulators

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 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 1. Introduction
 2. Spatial geometry and static...
 3. The optimum quality index for...
 4. The kinematic analysis of the...
 5. The optimum quality index for...
 6. The forward kinematic analysis...
 7. The optimum quality index for...
 8. The forward kinematic analysis...
 Conclusions
 Appendix A. Constants for the forward...
 Appendix B. Constants for the forward...
 List of references
 Biographical sketch
 
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Title:
Quality index and kinematic analysis of spatial redundant in-parallel manipulators
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xii, 152 leaves : ill. ; 29 cm.
Language:
English
Creator:
Zhang, Yu, 1969-
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Mechanical Engineering thesis, Ph.D   ( lcsh )
Dissertations, Academic -- Mechanical Engineering -- UF   ( lcsh )
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theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph.D.)--University of Florida, 2000.
Bibliography:
Includes bibliographical references (leaves 148-151).
Statement of Responsibility:
by Yu Zhang.
General Note:
Printout.
General Note:
Vita.

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University of Florida
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MISSING IMAGE

Material Information

Title:
Quality index and kinematic analysis of spatial redundant in-parallel manipulators
Physical Description:
xii, 152 leaves : ill. ; 29 cm.
Language:
English
Creator:
Zhang, Yu, 1969-
Publication Date:

Subjects

Subjects / Keywords:
Mechanical Engineering thesis, Ph.D   ( lcsh )
Dissertations, Academic -- Mechanical Engineering -- UF   ( lcsh )
Genre:
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph.D.)--University of Florida, 2000.
Bibliography:
Includes bibliographical references (leaves 148-151).
Statement of Responsibility:
by Yu Zhang.
General Note:
Printout.
General Note:
Vita.

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 025877150
oclc - 47122855
System ID:
AA00018876:00001

Table of Contents
    Title Page
        Page i
        Page ii
    Dedication
        Page iii
    Acknowledgement
        Page iv
    Table of Contents
        Page v
        Page vi
    List of Tables
        Page vii
    List of Figures
        Page viii
        Page ix
        Page x
    Abstract
        Page xi
        Page xii
    1. Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
    2. Spatial geometry and statics
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
    3. The optimum quality index for a spatial redundant 4-4 in-parallel manipulator
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
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        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
    4. The kinematic analysis of the spatial redundant 4-4 in-parallel manipulator
        Page 43
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        Page 51
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        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
    5. The optimum quality index for a spatial redundant 4-8 in-parallel manipulator
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
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        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
    6. The forward kinematic analysis of the spatial redundant 4-8 in-parallel manipulator
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
    7. The optimum quality index for a spatial redundant 8-8 in-parallel manipulator
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
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        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
    8. The forward kinematic analysis of the spatial redundant 8-8 in-parallel manipulator
        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
        Page 118
        Page 119
        Page 120
    Conclusions
        Page 121
        Page 122
    Appendix A. Constants for the forward kinematic analysis of the redundant 4-4 in-parallel manipulator
        Page 123
        Page 124
        Page 125
        Page 126
        Page 127
        Page 128
        Page 129
    Appendix B. Constants for the forward kinematic analysis of the redundant 8-8 in-parallel manipulator
        Page 130
        Page 131
        Page 132
        Page 133
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        Page 141
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        Page 143
        Page 144
        Page 145
        Page 146
        Page 147
    List of references
        Page 148
        Page 149
        Page 150
        Page 151
    Biographical sketch
        Page 152
        Page 153
        Page 154
        Page 155
Full Text










QUALITY INDEX AND KINEMATIC ANALYSIS OF
SPATIAL REDUNDANT IN-PARALLEL MANIPULATORS

















By

YU ZHANG


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2000



























Copyright 2000

by

Yu Zhang


























To my wife, Ying, and our parents.















ACKNOWLEDGMENTS

I want to express my deep and sincere gratitude to Dr. Joseph Duffy, my

supervisor during my Ph.D. study, for providing me with the opportunity to complete my

study under his exceptional guidance. Without his untiring patience, constant

encouragement, guidance and knowledge this work would not have been possible. I

would also like to thank my supervisory committee members, Dr. Carl D. Crane, Dr.

Gloria J. Wiens, Dr. Ali A. Seirig, and Dr. Ralph Selfridge. I am grateful for their

willingness to serve on my committee, providing me help whenever needed and for

reviewing this dissertation. I especially thank Professor Chonggao Liang of Beijing

University of Posts and Telecommunications for educating me on the various aspects of

mechanism analysis and design. Also, I would like to thank all my colleagues in the

Center for Intelligent Machines and Robotics for their help and support.

Finally, I would like to thank my lovely wife, Ying Zhu. Her love, support and

encouragement has had made my life rich and complete. I am grateful to my parents and

parents-in-law for their constant support and encouragement throughout my educational

endeavors.















TABLE OF CONTENTS

page

ACKNOWLEDGMENTS ............................................................................................. iv

LIST O F TAB LES ........................................................................ .................................... vii

LIST O F FIG U RES....................................................................................................viii

A B ST R A C T ....................................................................................................................... xi

1. INTRODUCTION..................................................................................................... 1

1.1 Redundant Parallel Manipulators ..................................... ............................ 1
1.2 Q quality Index................................................................................................... 6
1.3 Outline of Dissertation .................................................................................. 9

2. SPATIAL GEOMETRY AND STATICS ........................................ .......................... 12

2.1 Plicker Line Coordinates............................................................................ 12
2.2 Statics of a Rigid Body................................................................................ 15
2.3 The Statics of a Parallel Manipulator.......................................................... 20

3. THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT
4-4 IN-PARALLEL MANIPULATOR .............................................................. 24

3.1 Determination of /detJmJ. .............................................................................25
3.2 Im plem entation.............................................................................................. 30

4. THE KINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT
4-4 IN-PARALLEL MANIPULATOR .............................................................. 43

4.1 Inverse Kinematic Analysis..............................................................................44
4.2 Forward Kinematic Analysis....................................................................... 47
4.2.1 Introduction ............................................................................................ 47
4.2.2 Coordinate Transformations.................................................................48
4.2.3 Constraint Equations ............................................................................ 51
4.2.4 The Solution ........................................................................................... 52
4.2.5 Numerical Verification......................................................................... 55









5. THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT
4-8 IN-PARALLEL MANIPULATOR ................................................................... 58

5.1 Determination of /detJ.J .....................................................................59
5.2 Implementation.............................................................................................. 64

6. THE FORWARD KINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT
4-8 IN-PARALLEL MANIPULATOR................................................ ................ 79

6.1 Forward Kinematic Analysis............................... .............................................. 79
6.2 Numerical Verification.................................................................................. 82

7. THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT
8-8 IN-PARALLEL MANIPULATOR ............................. ............................ 85

7.1 Determination of 4detJ.J. ...................................................................... 85
7.2 Im plem entation.............................................................................................. 94

8. THE FORWARD CINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT
8-8 IN-PARALLEL MANIPULATOR .................................................................... 110

8.1 C oordinate System s........................................................................................... 110
8.2 Constraint Equations ................. ......................................................................... 113
8.3 Equation Solution .............................................................................................. 114
8.4 N um erical V erification...................................................................................... 118

9. CO N CLU SIO N S ................................................................................................... 121

APPENDIX A: CONSTANTS FOR THE FORWARD KINEMATIC ANALYSIS OF
THE REDUNDANT 4-4 IN-PARALLEL MANIPULATOR ................................. 123

APPENDIX B: CONSTANTS FOR THE FORWARD KINEMATIC ANALYSIS OF
THE REDUNDANT 8-8 IN-PARALLEL MANIPULATOR ................................. 130

LIST OF REFEREN CES ................................................................................................ 148

BIOGRAPHICAL SKETCH..................................................................................... 152















LIST OF TABLES


Table Page

Table 4.1: Numerical results of the redundant 4-4 in-parallel manipulator .................. 56

Table 4.2: A numerical example for the special case of the redundant 4-4 in-parallel
m anipulator..................................................... ................................... 57

Table 6.1: Numerical results of the redundant 4-8 in-parallel manipulator .................. 83

Table 6.2: A numerical example for the special case of the redundant 4-8 in-parallel
m anipulator..................................................... ................................... 84

Table 8.1: Numerical results of the redundant 8-8 in-parallel manipulator .................... 119

Table 8.2: A numerical example for the special case of the redundant 8-8 in-parallel
m anipulator.............................................................................................. 119















LIST OF FIGURES


Figure Page

Figure 1.1: A planar parallel x-y manipulator with one redundant actuator.................... 2

Figure 1.2: A 2-DoF planar parallel manipulator....................................................... 3

Figure 1.3: A redundant 2-DoF planar parallel manipulator......................... .............. 3

Figure 1.4: Planar view of spatial nonredundant 4-4 in-parallel manipulators.................. 4

Figure 1.5: Self-deployable space structure ............................................. ................ 11

Figure 2.1: Determination of a line ........................................................................... 13

Figure 2.2: Pliicker line coordinates............................................................................... 15

Figure 2.3: Representation of a force on a rigid body................................... .......... .. 16

Figure 2.4: Dyname and wrench ................................................. .......................... 19

Figure 2.5: A 6-6 in-parallel manipulator ................................................................. 21

Figure 3.1: A redundant 4-4 in-parallel manipulator .................................... ............ 24

Figure 3.2: Plan view of the redundant 4-4 in-parallel manipulator ............................. 24

Figure 3.3: Plan view of the optimal configuration of the redundant 4-4 in-parallel
manipulator with the maximum quality index ................................... 30

Figure 3.4: Quality index for platform vertical movement .............................................31

Figure 3.5: Quality index for platform horizontal translation ........................................... 34

Figure 3.6: Platform rotations about the y'-axis........................................ ............ .... 35

Figure 3.7: Quality index for platform rotations about the x'- and y'-axes..................... 37

Figure 3.8: Platform rotations about the z-axis ........................................... ........... ... 38








Figure 3.9: Quality index for platform rotation about the z-axis ....................................40

Figure 3.10: Plan view of the singularity position of the redundant 4-4 in-parallel
m anipulator when z = 900................................................................ 41

Figure 4.1: Coordinate systems of a redundant 4-4 in-parallel manipulator................... 45

Figure 4.2: Coordinate transformations..................................................................... 49

Figure 5.1: A redundant 4-8 in-parallel manipulator ................................... ............. 58

Figure 5.2: Plan view of the redundant 4-8 in-parallel manipulator ............................. 58

Figure 5.3: Plan view of the optimal configuration of the redundant 4-8 in-parallel
manipulator with the maximum quality index ................................... 63

Figure 5.4: Compatibility between the redundant 4-4 and the 4-8 parallel manipulators. 64

Figure 5.5: Quality index for platform vertical movement ............................................. 65

Figure 5.6: Quality index for platform horizontal translation with different values of f. 70

Figure 5.7: Platform rotation about the y'-axis ........................................... .............. 71

Figure 5.8: Quality index for platform rotations about the x'- and y'-axes..................... 73

Figure 5.9: Platform rotation about the z-axis............................................ ............... 74

Figure 5.10: Quality index for platform rotation about the z-axis ..................................76

Figure 5.11: Plan view of the singularity position of the redundant 4-8 in-parallel
manipulator when Oz = 90 ........................................ .................... ... 77

Figure 6.1: Coordinate systems of a redundant 4-8 in-parallel manipulator................. 80

Figure 6.2: Leg relations ............................................................................................. 81

Figure 7.1: A redundant 8-8 in-parallel manipulator ................................... ............ 86

Figure 7.2: Plan view of the redundant 8-8 in-parallel manipulator ............................. 86

Figure 7.3: Plot of f(a, fl) = 2a 2a 2 +1 = 0 ........................................................ 90

Figure 7.4: Plot of h vs. raand f with a = 1 ...................................................................... 92

Figure 7.5: Plot of /det J.J vs. a with a = 1 .................................... ............... .. 93

Figure 7.6: An example of redundant 8-8 manipulator in optimal configuration............. 93









Figure 7.7: Quality index for platform vertical movement ............................................... 96

Figure 7.8: Reduction of the size of the redundant 8-8 in-parallel manipulator ............... 96

Figure 7.9: Quality index for platform horizontal translation with different values of al00

Figure 7.10: Platform rotations about the y'-axis....................................................... 102

Figure 7.11: Quality index for platform rotations about the x'- and y'-axes................... 105

Figure 7.12: Platform rotations about the z-axis ........................................................... 106

Figure 7.13: Quality index for platform rotation about the z-axis ................................ 108

Figure 7.14: Plan view of the singularity position of redundant 8-8 in-parallel
manipulator when 8z = 900................................................................. 109

Figure 8.1: Coordinate systems of a redundant 8-8 in-parallel manipulator.................. 11















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

QUALITY INDEX AND KINEMATIC ANALYSIS OF
SPATIAL REDUNDANT IN-PARALLEL MANIPULATORS

By

Yu Zhang

December 2000


Chairman: Dr. Joseph Duffy
Major Department: Mechanical Engineering

Parallel manipulators have been the subject of much investigation over the last

decade because of their inherent advantages of load carrying capacity and spatial rigidity

compared to serial manipulators. Usually they have the same number of actuators as their

degree of freedom, but in some cases, it may be interesting to have more actuators than

needed and to consider redundant parallel manipulators. Redundancy in actuation can be

used to increase dexterity, to reduce or even eliminate singularities, to increase reliability,

to simplify the forward kinematics, and to improve load distribution in actuators. The

purpose of this work is to design and analyze several spatial redundant parallel

manipulators.

The proposed quality index will assist a designer to choose the relative

dimensions of the fixed and moving platforms, locate joint centers in the fixed and

moving platforms, determine an optimum position which would be an 'ideal' location of








the workspace center, and determine acceptable ranges of pure translations and pure

rotations for which the platform is stable.

The quality index for redundant parallel manipulators is defined as a dimensionless

ratio that takes a maximum value of 1 at a central symmetrical configuration that is

shown to correspond to the maximum value of the square root of the determinant of the

product of the manipulator Jacobian by its transpose. The Jacobian matrix is none other

than the normalized coordinates of the leg lines. When the manipulator is actuated so that

the moving platform departs from its central configuration, the determinant always

diminishes, and, as is well known, it becomes zero when a special configuration is

reached (the platform then gains one or more uncontrollable freedoms). It is shown that

the quality index 2, for which 012< 1, can be used as a constructive measure of not only

acceptable and optimum design proportions but also an acceptable operating workspace

(in the static stability sense).

We also studied the forward kinematic analysis of the redundant in-parallel

manipulators to determine the position and orientation of the platform, given the leg

lengths.













CHAPTER 1
INTRODUCTION

Parallel manipulators have been studied extensively over the last decade with

their high structural stiffness, position accuracy and good dynamic performance. Usually

they have the same number of actuators as their degree of freedom, but in some cases, it

may be interesting to have more actuators than needed to overcome disadvantages of the

nonredundant parallel manipulators shown by Merlet [25].


1.1 Redundant Parallel Manipulators

A number of redundant parallel manipulators have been studied in literature, for

example, the development of a direct-drive redundant parallel manipulator for haptic

displays by Buttolo and Hannaford [2, 3], the design of a 2-DoF parallel manipulator

(Figure 1.1) with actuation redundancy for high speed and stiffness-controlled operation

by Kock and Schumacher [17], and the addition of a redundant (fourth) branch to three-

branch manipulators for the purpose of uncertainty elimination and assembly mode

reduction by Notash and Podhorodeski [28].

Maeda et al. [24] also designed a redundant wire-driven parallel manipulator that

is suitable especially for high speed assembling of lightweight objects such as

semiconductors. By studying a parallel machining center, O'Brien and Wen [29]

examined the effectiveness of singularity modification through redundant actuation and

suggested that augmenting the actuation of a mechanism provides a mechanically feasible

means of increasing kinematic manipulability.
























Figure 1.1: A planar parallel x-y manipulator with one redundant actuator
(Adapted from: Kock and Schumacher, A Parallel X-y Manipulator with Actuation
Redundancy for High Speed and Active Stiffness Applications (1998) [17])



Leguay-Durand and Reboulet [23] studied a redundant spherical parallel

manipulator and showed that actuator redundancy removes singularities and improves

dexterity in an enlarged workspace. Using a conditioning measure, they compared the

redundant spherical parallel manipulator with an equivalent nonredundant structure and

found notably improved uniformity of dexterity for the redundant structure. Similar

results were also found by Kurtz and Hayward [19].

Kokkinis and Millies [18] found that actuation redundancy allows the selection of

optimal joint torque for a given load. Nakamura and Ghodoussi [27] also showed that the

redundant actuation could increase the payload and improve the dynamic response of

manipulators.

Dasgupta and Mruthyunjaya [6, 7] saw the redundancy of parallel manipulators as

the series-parallel dual part of redundancy in serial manipulators. They proposed the








concept of force (static) redundancy for redundancy in parallel manipulators in contrast to

kinematic redundancy (widely studied in literature) in serial manipulators.

In summary, redundant parallel manipulators have the following advantages:

1. Redundancy in actuation can be used to increase dexterity and reduce or even

eliminate singularities of parallel manipulators (Perng and Hsiao [30]). Usually,

parallel manipulators have a high stiffness, except in some special positions or

postures where the platform has self-motion and may even collapse. These singular

configurations may cause serious damage to the manipulator and/or objects in its

environment. Redundant legs can be used to pull out the platform from singularity

positions. For example, for the 2-DoF planar parallel manipulator shown in Figure

1.2, its singularity positions can be found on the line joining the two fixed pivot

positions, and this singularity can be eliminated by adding another leg as shown in

Figure 1.3. The new redundant parallel manipulator is entirely free from singularity as

long as its three fixed pivots are noncollinear.












Figure 1.2: A 2-DoF planar parallel Figure 1.3: A redundant 2-DoF planar
manipulator parallel manipulator


Here is another example, considering the two cases of spatial nonredundant 4-4

in-parallel manipulators shown in Figure 1.4. They are in singularity positions when








their platforms are parallel to the base. Such singularity is dangerous because it is not

immediately obvious from its configuration and if we build such manipulators, they

may collapse immediately when their platforms are parallel to the base. One possible

solution is to add another two legs to form a redundant 4-4 in-parallel manipulator

(Chapter 3) (Figure 3.1).









SI "G






b_ b


(a) (b)

Figure 1.4: Planar view of spatial nonredundant 4-4 in-parallel manipulators



2. Redundancy in actuation can be used to increase the reliability of in-parallel

manipulators (Shin and Lee [33]). That is, even if some of the actuators fail, a

manipulator can still operate normally as long as the number of operating actuators is

not less than the mobility of the manipulator. Thus such a redundant system has a

failure tolerance, which is increasingly important in robotics, especially when robots

and manipulators are used in remote or harsh environments such as space, deep sea,

nuclear plants and for bomb disposal. Because these environments do not allow








immediate human intervention for repair or recovery, the ability of a robot or a

manipulator to cope with the failures becomes desirable.

3. The information from the length of the redundant legs can be used to simplify the

forward kinematics. When controlling a parallel manipulator, we need to do the

forward kinematic analysis, i.e., to determine the configuration of the moving

platform given all the leg lengths. This analysis is usually difficult as it involves a set

of nonlinear equations and, generally, there is more than one solution. For example,

the forward analysis for the general 6-6 parallel manipulator requires the solution of a

40th degree polynomial (Raghavan [31]) the solution of which is clearly impractical

for real-time implementation. The additional information from the redundant legs

reduces many uncertainty positions and even can obtain a unique solution to the

forward analysis.

4. The actuator forces and joint torques in the redundant parallel manipulators are not

uniquely determined. This characteristic can be used to optimize some criteria. For

example, the joint torque required for a given motion can be minimized. Accordingly,

it is possible to increase the payload of a closed-link mechanism by adding redundant

actuators.

Some other advantages of using redundant actuators are increasing workspace

while improving dexterity, having autonomous calibration, and building variable

geometry trusses. Possibilities of redundancy in parallel manipulators and their effective

use have not been studied extensively until now. The purpose of this work is to design

and analyze several spatial redundant parallel manipulators.








1.2 Quality Index

Parallel manipulators have better load carrying capacity and spatial rigidity than

serial manipulators. However, the complexity of the kinematics of parallel manipulators

makes it more difficult for a designer to determine a set of kinematic and geometry

parameters that will efficiently produce prescribed performances. Indeed, the behavior of

parallel manipulators is far less intuitive than that of serial manipulators. The geometric

properties associated with singularities, for example, may be much more difficult to

identify directly (Fichter [10] and Merlet [26]). Therefore, more systematic analysis and

optimization tools are needed to make parallel manipulators more accessible to designers.

At this time little information is available to assist designers in thefollowing task:

(a) Choose the relative sizes of the fixed and moving platforms.

(b) Locate the positions of the centers of the spherical joints in the base and the

centers in the moving platform.

(c) Determine an optimum position that would be an ideal 'center' location of the

workspace.

(d) Determine acceptable ranges of pure translations of the platform for which the

platform is stable (i.e., not too close to a singularity). However, the question

"How close is too close?" is often hard to answer.

(e) Determine acceptable ranges of pure rotations of the platform for which the

platform is stable.

(f) Determine the ranges of leg displacements.

These considerations are the reasons that the quality index was proposed.

The quality index was defined initially for a planar 3-3 in-parallel device by the

dimensionless ratio (Lee, Duffy, and Keler [22])








Idet J
S= det(1.1)
IdetJ.I

where J is the three-by-three Jacobian matrix of the normalized coordinates of three leg

lines. Then it was defined for an octahedral in-parallel manipulator by Lee, Duffy, and

Hunt [21] and 3-6, 6-6 in-parallel devices by Lee and Duffy [20]. For these cases J is the

six-by-six matrix of the normalized coordinates of the six leg lines. For these fully

symmetrical nonredundant parallel manipulators the quality index takes a maximum

value of A = 1 at a central symmetrical configuration that corresponds to the maximum

value of the determinant of the six-by-six Jacobian matrix (i.e., det J = det Jm) of the

manipulator. When the manipulator is actuated so that the moving platform departs from

its central configuration, the determinant always diminishes, and, as is well known, it

becomes zero when a special configuration is reached (the platform then gains one or

more uncontrollable freedoms).

In this dissertation, the quality index is extended for redundant manipulators by

the dimensionless ratio


detJJT (1.2)

This makes complete sense because the Cauchy-Binet theorem detJJT =LA +A, +* .+A ,


has geometrical meaning. Here, each A (1 i m = () is simply the determinant of the


6x6 submatrices of J which is a 6xn matrix. This is clear when n = 6, (1.2) reduces to

(1.1). It has been shown by Lee et al. [21] that by using the Grassmann-Cayley algebra

(White and Whiteley [35]), for a general octahedron, when the leg lengths are not

normalized, detJ has dimension of (volume)3 and it is directly related to the products of









volumes of tetrahedra that form the octahedron. In this way detJ and VdetJJT have

geometrical meaning.

We mention in passing the work of Cox [4] and Duffy [8], both of which cover

special configurations of planar motion platforms. Hunt and McAree [14] go into

considerable detail regarding the general octahedral manipulator. Its special

configurations are described in the context of other geometrical properties. A few papers

were published on the optimal design of nonredundant parallel manipulations (see for

example Gosselin and Angeles [11, 12], Zanganeh and Angeles [36]).

Zanganeh and Angeles [36] point out problems with quantities such as condition

number due to the inherent inhomogeneity of the columns of the Jacobian, J. This is

precisely why equations (1.1) and (1.2) are adopted as an index of quality rather than

other well-established methods (found in books on theory of matrices and linear algebra)

that lead (via norms, diagonalization and singular values, etc.) to properties that relate to

'conditioning'. All such methods are based implicitly on the presumption that a column-

vector (say, of a six-by-six matrix) can be treated as a vector in 96. However, the six

elements in the column of a typical robot Jacobian are the normalized coordinates of a

screw (almost always of zero pitch; i.e., a line); in a metrical coordinate frame three of

them are dimensionless and three have dimension [length], such a length being the

measure of the moment about a reference point of a unit force. The column generally

comprises two distinct vectors (each of them in 9%3). For the legs of the nonredundant and

redundant manipulators it is not possible to remove all the length dimensions from their

coordinates. Even the adoption of some artificial length unit fails, simply because a

moment can never be converted to a pure force. Moreover, any index of quality derived








from such textbook techniques is likely to vary according to the coordinate frame in

which the Jacobian is formulated. Our method works for two reasons: first, the

determinant of a (square) Jacobian of line coordinates depends solely on the

configuration in 9t3 of the actuated axes and not on the coordinate frame in which the line

coordinates are determined. The second reason is that equations (1.1) and (1.2) are

dimensionless ratios, and our quality indices are always independent of the choice of

units of length measurement.

Unlike the case of a mechanism designed for a specific task, the tasks to be

performed by a manipulator are varied. Hence, there should not be any preferred general

orientation for which the manipulator would have better properties. It suggests that the

manipulator should be symmetrical. Such symmetrical configurations may not always

exist, of course. However, except for unusual applications (and there will undoubtedly be

some where for example unusual loads must be sustained) we are safe in seeking

centrally symmetrical designs to which we can assign the highest quality index A= 1, or

close to it. For these cases, contours of quality index help to determine a realistic

workspace volume that is free from singularities. Therefore, we are concerned primarily

with symmetrical redundant parallel manipulators in this dissertation.


1.3 Outline of Dissertation

A simple introduction to the screw theory is presented in Chapter 2 to provide

insight into how a screw-based Jacobian matrix of a parallel manipulator is determined.

In Chapters 3 and 4 a spatial redundant 4-4 in-parallel manipulator is studied first.

The device consists of a square platform and a square base connected by eight actuated

legs. As in Chapter 3, the quality index for the redundant 4-4 parallel manipulator is








determined. To achieve the maximum quality index for a redundant 4-4 in-parallel

manipulator with platform side a, the base has side Za and the perpendicular distance

between the platform and the base is a The kinematic analysis of the redundant 4-4


in-parallel manipulator is studied in Chapter 4. The derivation of forward kinematic

equations for position and orientation of the platform is described.

Chapters 5 and 6 extend the study to a redundant 4-8 in-parallel manipulator with

a square platform and an octagonal base. The octagonal base is formed by separating

from each vertex of a square by a small distance. The quality index for this manipulator is

determined in Chapter 5. The compatibility between the redundant 4-4 and the 4-8

parallel manipulators also is discussed in this Chapter. Chapter 6 solves the forward

kinematics of the redundant 4-8 parallel manipulator by transferring the problem to the

corresponding redundant 4-4 case.

Finally, in Chapters 7 and 8, a redundant 8-8 in-parallel manipulator is studied.

The device has an octagonal platform and a similar octagonal base connected by eight

legs. Such arrangement avoids using double-spherical joints because they can produce

serious mechanical interference. However, by using the quality index determined in

Chapter 7, the best design can be obtained when the pair of separated joints in the base

and top platform are as close as possible. In Chapter 8, the kinematic analysis of the

redundant 8-8 parallel manipulator is performed. The forward analysis gives a much

simpler solution than that of the nonredundant case.

Using quality index, variable motions are investigated for which a moving

platform rotates about a central axis or moves parallel to the base. The quality index can

be used as a constructive measure not only of acceptable and optimum design proportions








but also of an acceptable operating workspace (in the static stability sense). Moreover,

analysis of these redundant in-parallel manipulators can be used to model and design a

self-deployable space structure that has a pair of flexible antenna platforms in the base

and top platform as shown in Figure 1.5 (Duffy et al. [9] and Knight et al. [16]).


I>


Figure 1.5: Self-deployable space structure













CHAPTER 2
SPATIAL GEOMETRY AND STATICS

Chapter 1 showed that the quality index of parallel manipulators is based on the

Jacobian matrix. This chapter, which is mostly a general background in screw theory

(Ball [1]), provides insight into how the Jacobian matrix of parallel manipulators is

determined. Firstly, we review some basic concepts of spatial geometry and screw theory.


2.1 Plucker Line Coordinates

Two distinct points ri(xl, Yi, zi) and r2(x2, Y2, Z2) can be connected by a line in

space. The vector S whose direction is along the line can be written in the form

S = r2 r. (2.1)

Alternatively this may be expressed as

S = Li + Mj + Nk (2.2)

where

L=x2-xI, M=y2-yi, N=z2-Z1 (2.3)

are defined as the direction ratios of the line and they are related to the distance ISI

between the two points by

L2 +M2 + N2 = IS12 (2.4)

where the notation 1I denotes absolute magnitude.

Often L, M, and N are expressed in the form

X2 -X1 Y2 yl Z2 --Z1
L = 2N M=Y2., N=, (2.5)
ISI ISI ISI

which consists of unit direction ratios of the line, and (2.4) reduces to








L2 +M2 +N2 = 1. (2.6)

If r represents a vector from the origin to any general point on the line (Figure 2.1), then

the vector r-r1 is parallel to S and therefore the equation of the line can be written as

(r ri) x S = 0 (2.7)

and in the form

r x S = So (2.8)

where

So = ri x S (2.9)

is the moment of the line about origin O and is clearly origin dependent. Further, because

So=rlXS, the vectors S and So are perpendicular and as such satisfy the orthogonality

condition

SSo = 0. (2.10)


Figure 2.1: Determination of a line








The coordinates of a line are written as [S; So]' and are referred to as the Pliicker

coordinates of the line [13]. The coordinates [S; So] are homogeneous since from (2.8)

the coordinates [kS; kSo] (k is a non-zero scalar) determine the same line.

Expanding (2.9) yields

i jk

So = xi y, zl (2.11)
LM N

which can be expressed in the form

So = Pi + Qj + Rk (2.12)

where

P= yN zM,
Q= zL xN, (2.13)
R= xM y1L.

From (2.2) and (2.12) the orthogonality condition S-So-0 can be expressed in the form

LP + MQ + NR= 0. (2.14)

The Pliicker coordinates of the line [S; So] now can be written in terms of their

components as [L, M, N; P, Q, R], which are known as the ray coordinates for a line

(Figure 2.2). Unitized coordinates for a line can be obtained by imposing the constraint

that ISI=1. The Pliicker coordinates thus must satisfy equations (2.6) and (2.14) and hence

only four of the six scalars L, M, N, P, Q, and R are independent. It follows that there are

4 2
04 lines in space2.




SThe semi-colon is introduced to signify that the dimension of ISI is different from ISol.
2 Systems of lines and their properties, -o' (line series), 02 (congruence), _3 (complex),
are described by Hunt [13] which contains an extensive bibliography on the subject.












R
R


S

YKN
^'L
Zl





y


Figure 2.2: PlUcker line coordinates



A straightforward method to obtain the Plucker coordinates was given by

Grassmann (Hunt [13]) by expressing the coordinates of the points rl(xi, yi, zl) and r2(x2,

Y2, Z2) in the array


X1 Yl Zl
1 x, y, z,
x1 2 Y2 Z2

and by expanding the sequence of 2x2 determinants

1 x, 1= z


y2 2 x,
Y2 Z2 Z2 X2
P= ,z Q z X,


1 z2
N= ,
1 z2

R= x1 Yi
x2 Y2


2.2 Statics of a Rigid Body

The concepts developed in the previous section now can be applied directly to the

statics of a rigid body. A line $ with ray coordinates [S; So] (where ISI = 1) can be used to


(2.15)


(2.16)








express the action of a force upon a body (Figure 2.3). Because the body is rigid, the

point of application can be moved anywhere along the line.


Z






f=f


r





Figure 2.3: Representation of a force on a rigid body


As illustrated by Figure 2.3, a force f can be expressed as a scalar multiple JS of

the unit vector S that is bound to the line $. The moment of the force f about a reference

point O is mo which can be written as mo = rxf where r is a vector to any point on the

line $. This moment can also be expressed as a scalar multiple fSo where So is the

moment vector of the line $ (i.e., So = rxS). The action of the force upon the body thus

can be expressed elegantly as a scalar multiple f$ of the unit line vector, and the

coordinates for the force are given by

f$ =f[S; So = [f; mo] (2.17)


where S-S=l and S-So=0.








Clearly, when the reference point O is coincident with A, then mo=0 and the

coordinates of the force are [f; 0]. Therefore, f is a line bound vector that is invariant with

a change of coordinate systems while mo is origin dependent.

An important special case is [0; mo] which can be considered as the resultant of a

pair of equal and opposite forces with coordinates [f; mo1] =f[S; Sol] and [-f; m02] =f[-S;

S02], where ISI=1. The coordinates of the resultant [0; ma]= [0; mol+m2] =f[0; Sol+S02]

are not a line bound vector, but a pure couple. The couple can be considered as equivalent

to a force of of infinitesimal magnitude (16Sf-+0) acting along a line that is parallel to the

lines of action of the pair of parallel forces. The line of action of of is infinitely distant

with coordinates [0; mo], such that Ipl= o where p is the vector from the origin

perpendicular to the line of action of sf, and the moment of the force 8f about the origin

is pxSf=mo. A pure couple thus can be represented as a scalar multiple of a line at

infinity.

The problem of determining the resultant of an arbitrary system of forces with

coordinates [fl; moI], [f2; mo2], ..., [fn; mon] acting on a rigid body is essentially the

determination of the quantity

^v = [f ; mo], (2.18)

where

n n
f=If, and mo= m,. (2.19)
i=1 i=i

It is assumed at the outset that a reference point 0 was chosen so that the forces

acting on the rigid body were translated to point 0 and so that moments mol, mo2, ..., mon

were introduced to yield an equivalent system of forces and torques that act on the rigid








body. Therefore, the line of action of the resultant force f passes through point O and the

resultant moment mo is a couple [0; me]. In general f and mo are not perpendicular (i.e.,

f-moe0). The new quantity with coordinates ^v' = [f; mo] therefore is not a force and was

defined as a dyname by Plicker.

Because in general f mo # 0, it is not possible to translate the line of action of

force f through some point other than point O and to have the translated force produce the

same net effect on the rigid body as the original dyname. The moment mo, however, can

be resolved into two components, ma and mt, which are respectively parallel and

perpendicular to f (Figure 2.4a) and

mo = ma + mt. (2.20)

The moment ma can be determined as

m. = (mo S) S (2.21)

where S is a unit vector in the direction of the resultant force f. The moment mt is then

determined as

mt = mo ma. (2.22)

The line of action of force f now can be translated so that the force with coordinates

[f; mt] plus the moment [0; a] (Figure 2.4b) is equivalent to the dyname [f;mo].

Therefore, the dyname is represented uniquely by a force f acting on the line [S; Sot]

mi
(where Sot= --) and a parallel couple ma. This parallel force-couple combination was
f

called a wrench by Ball [1].

From (2.18) and (2.20), the wrench which is equivalent to the dyname [f; mo]

can be expressed in the form





19

^v = [f; mo] = [f; mt + ma] = [f; mnt + [0 ; ma].

Clearly, [f; mt] is a pure force because f-mt=0.




9 1A


ma

r

rxf=mt y
x


(a) Dyname, [f; mo]


(b) Wrench, [f; mt] + [0; na]


Figure 2.4: Dyname and wrench


Further, because ma is parallel to f, then

ma = hf

where h is a non-zero scalar which is called the pitch of the wrench. From (2.20)


f mo = f ma


(2.24)


(2.25)


and from (2.24) and (2.25), the pitch h is given by


h= f m= f m
f f f f


(2.26)


Substituting (2.24) into (2.23), together with (2.22) allows the wrench 1v to be

written as


wV = [f ; mo hf] + [0; hfJ.


(2.23)


(2.27)








Thus, the coordinates for the line of action of the wrench are [f; mo-hf] and from (2.27)

the equation for the line is

r x f = mo hf. (2.28)

In the same way as the action of a force can be expressed as a scalar multiple of a

unit line vector, a wrench can be expressed elegantly as a scalar multiple of a unit screw $

where

$ = [S ; Sol (2.29)

and where S.S=1. From (2.26), the pitch of the screw is given by

h = S So. (2.30)

Further, from (2.2) and (2.12),

h =LP + MQ + NR. (2.31)

Therefore, Ball [1] defined a screw as "a line with an associated pitch". Following (2.28),

the Plicker coordinates for the screw axis are [S; So-hS] and the equation of the axis is

r x S = So hS. (2.32)


2.3 The Statics of a Parallel Manipulator

Figure 2.5 illustrates a nonredundant 6-63 in-parallel manipulator. The device has

a moving platform and a fixed base connecting by six legs each of which is the same

kinematic chain. The prismatic joint in each leg is actuated and the moving platform has

six degrees of freedom.






3 These numbers indicate the number of connecting points in the top and base platform
respectively.











$5 moving
$1 6 $ platform











base


Figure 2.5: A 6-6 in-parallel manipulator


Consider that the six leg forces with magnitudes fi,f2, ...,f6 are generated in each

of the lines $1, $2, ..., $6. The resultant wrench ^ =[f; mo] acting upon the moving

platform due to these six leg forces is given by

*v = [fi; moi] + [f2; mo2] + ... + [f6; mos], (2.33)

or in the alternative form,

wv =f [S1; Soil +f2[S2; S02] + ... +f6[S6; Soi] (2.34)

where [Si; Soil (ISil=l, i=1...6) are the Piicker line coordinates of the six legs. Further,

(2.34) can be expressed in the matrix form

i = JF (2.35)

where = [f, m,] and F = [fi, /f, f4, f5, f6, f are 6x1 column vector. J is a 6x6

matrix of line coordinates given as








S, S, S3 S4 Ss S6
J = (2.36)
.S01 So2 S03 S04 S05 S06

and is called the Jacobian matrix, or simply Jacobian, which enables us to determine the

resultant wrench *^ = [f; mo] produced by six actuator forces generated in the legs. It

should be noted that for redundant parallel manipulators the Jacobian matrix is not

square.

The transpose of the Jacobian matrix relates the infinitesimal displacements 81i in

each leg to the infinitesimal displacement twist4 of the platform and

61 = JTSD (2.37)

where 1 =[ [81, 312,..., S16]T and 8D = [x, Sy, 8z; &P, 8(Py, ypz ]T. Here, 8x, Sy,

and 8z are the infinitesimal displacement of a point in the moving platform coincident

with a reference point O which is chosen to be the origin of a fixed coordinate system on

the fixed base. The quantities 8x, Spy, and 8iz are infinitesimal rotations of the

moving platform about the axes of the fixed reference coordinate system.

In summary, the Jacobian matrix of parallel manipulators serves two distinct

purposes. In its ordinary form the columns of which are the coordinates of the actuator

lines (normalized), it enables us to obtain from actuated force inputs the wrench at the

end effector platform. In its transposed form the Jacobian can give the relative speeds

required at each actuator that corresponds to a given twist to be executed by the platform.

The first of these gives the instantaneous solution to a problem of static equilibrium; the

second, the solution of first order kinematic compatibility. When the Jacobian matrix is


4 An infinitesimal twist is also a scalar multiple of a unit screw, as the scalar is an
infinitesimal rotation with unit of radian.






23


singular (i.e., its rank is less than six) the actuators (i) cannot equilibrate a general wrench

applied to the platform and (ii) cannot on their own prevent a transitory uncontrollable

movement of the platform. This latter phenomenon is associated with the platform's

gaining one or more freedoms when all the actuators are locked. The platform is then in a

singularity position.













CHAPTER 3
THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT
4-4 IN-PARALLEL MANIPULATOR

A spatial redundant 4-4 in-parallel manipulator is shown in Figure 3.1. The device

has a square platform and a square base connected by eight legs. Figure 3.2 shows a plan

view of this manipulator, where the moving platform is symbolically represented by four

pairs of concentric spherical joints located at points A, B, C, and D, and the fixed base is

represented by another four pairs of concentric spherical joints located at points E, F, G,

and H. This manipulator is said to be redundant since the platform and the base are

connected by eight actuated legs.


Figure 3.1: A redundant 4-4 in-parallel
manipulator


Figure 3.2: Plan view of the redundant 4-4
in-parallel manipulator








3.1 Determination of detJ.JJ.

The moving platform of the redundant 4-4 parallel manipulator shown in Figure

3.2 is located at its central symmetrical configuration and is parallel to the base with a

distance h. At this configuration, the manipulator is fully symmetric and each leg has the

same length. Clearly, at such position the platform is most stable from the geometric

static point of view. When the platform departs from this central symmetric position, the

platform will lose its geometric symmetry and the eight leg lengths will be different.

Therefore, it is reasonable to assume that at the central symmetric configuration shown in

Figure 3.2, it is possible to determine the values of square base side b and height h based

on square platform side a so that a maximum value of the square root of the determinant

of the product of the manipulator Jacobian by its transpose, i.e., detJ.J. may be

obtained.

Firstly, the coordinates of the points A, B, C, and D on the platform and E, F, G,

and H on the base are determined with the origin of a fixed coordinate system placed at

the center of the square base as shown in Figure 3.2, and

A a h B 0 h} CO0 O a h), D (- a 0 h
2 2 2 2
(3.1)
E b 0 o, F(b -b 0, G 0 H o b- o0.
(2 2 2 2 2 2 2 2

Then, using the Grassmann method described in Chapter 2 to calculate the

Pliicker line coordinates of the eight leg lines, i.e., counting the 2x2 determinants of the

various arrays of the joins of the pairs of points EA, FA, FB, GB, GC, HC, HD, and ED.








For example, the coordinates of the line $1 are obtained using the coordinates of points E

and A in (2.15) to form the array


-b
2

1 0


b
2
2a
2


and using (2.16) to yield


V2a -b bh
h;
2 2


i

h




bh 42ab
2 4


Similarly from points F and A,


Sr = b V2a-b
2 2


bh bh F2ab
h; 2 4
2 2 4


From points F and B,


S3 b
2


b bh
2 h;
2~ 2


F2ab
4


From points G and B,


S4 Va-
2


b
2'


bh bh
h; 2
2 2


From points G and C,


S5 = -2
2


Va --b
2 h;
2


bh bh
2 2


From points H and C,


s, b
2


V2a-b
2


bh bh 2ab
2 2' 4


From points H and D,


(3.2)


(3.3)


(3.4)


bh
2


(3.5)


4Vab
4


(3.6)


2ab]
4


(3.7)


(3.8)


S- b
2









-[ -a-b b
7=[2 2'


hbh bh ijab
2 2 4


(3.9)


From points E and D,


Sbh
h;
2


bh FJab]
2 4


(3.10)


It should be noted that the above Plicker line coordinates are not normalized and each

line must be divided by 1, = S,I (i = 1, 2, ..., 8). Hence, the normalized Jacobian matrix

of the eight leg lines (now all reduced to unit length) can be expressed as


1 3T T T T T 17 8
1= 12 13 14 15 16 17 18


(3.11)


Since the device is in a symmetrical position, the normalization divisor is the

same for each leg, namely li = (i = 1, 2, ..., 8), and for every leg


= L2 +M2+N2 = a2 -2ab+b2+2h2).


From (3.3) to (3.10), the Jacobian matrix in (3.11) becomes


b
d, di
2


-d, -di


b
d, di 2
2


h h h h h h h h


(3.12)


bh
2


bh
2


bh bh
2 2
rab 2jab
4 4


bh
2
42ab


bh
2
v2ab


4 4


bh bh
2 2
j2ab /jab
4 4


J-
I
1


bh
2
bh
2
,2ab
4


ii


(3.13)


bh
2
bh
2
42ab
4


Ss = ,-a-b b
2 2





28

where

d 2a -b
d, =-
2

Using equation (3.13), the determinant of the product J J T can be expressed in the

form

d2 0 0 0 -d3 0
0 d2 0 d3 0 0

1 0 0 8h2 0 0 0
detJJ 2(3.14)

-d3 0 0 0 2b2h2 0

0 0 0 0 0 a2b2

where

d2 =2(a 2 2ab + b2),

d3 = (a2a-2b)bh.

Expanding (3.14) and using (3.12), then extracting the square root yields

/ v 32ia'b'hi
Vdet JJ = 322a. (3.15)
(a2 ab +b + 2h2 Y

Assuming the top platform size a is given, now taking the partial derivative of (3.15) with

respect to h and b respectively and equating to zero yield

96F2 a3b3h2(a2 aab+b2 -2h2) (3.16)
(a2 ab+b2+ 2h2)4

and

962a3b2h3(a2 -b2 +2h2) (3.17)
(a2- -Fab+b2 +2h)4 7)








When a, b, and h are not equal to zero, equations (3.16) and (3.17) give

a2 4ab+b2-2h2 =0, (3.18)

a2 -b2 + 2h2 =0. (3.19)

Adding (3.18) and (3.19) yields

2a2 2ab=0. (3.20)

Solving the above equation, we obtain

b = 2a (3.21)

Further, substituting (3.21) into equation (3.19) yields

a2 -2h2 =0. (3.22)

There are two solutions for h in the above equation, here we only take the positive

solution (the negative solution is simply a reflection through the base)

h= (3.23)


Finally, substituting (3.21) and (3.23) into (3.15) we get

vdetJ,Jt = (det J ), = 4Va3 (3.24)

where Jm denotes the Jacobian matrix for the configuration at which the 4-4 redundant

parallel manipulator has a maximum quality index. This optimum configuration is shown

in Figure 3.3.

























Figure 3.3: Plan view of the optimal configuration of the redundant 4-4 in-parallel
manipulator with the maximum quality index



3.2 Implementation

From the definition of quality index (see (1.2)) and (3.24), the quality index for

the redundant 4-4 parallel manipulator shown in Figure 3.1 becomes

-dettjj
AVa= (3.25)

The variation of the quality index now is investigated for a number of simple motions of

the top platform. Here, an optimal redundant 4-4 parallel manipulator with platform side

a = 1, and thus base side b = 2 is taken as an example.

First, consider a pure vertical translation of the platform from the central

symmetric position shown in Figure 3.2 along the z-axis while remaining parallel to the

base. For such movement, from (3.15) and (3.25), the quality index is given by





31


8b3h3
8b 3 h (3.26)
(a2 2.Iab 2 + 2h2

With a = 1 and b = /2, this reduces to

16/2h3
S= 2h (3.27)
(1+ 2h 2)3

a
and is plotted in Figure 3.4 as a function of h. It shows that at height h = = the
T2 2

quality index of the redundant 4-4 parallel manipulator has a maximum value, A = 1.



1.0..........



0


0.0-z----n----i---i---i--
0.4


0.2

0.0-
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Height h

Figure 3.4: Quality index for platform vertical movement



We now derive an expression for detJJT when the platform of the redundant

4-4 parallel manipulator is translated away from its central location while remaining

parallel to the base at height h. Assume the center of the moving platform to move to

point (x, y, h), then the coordinates of the points A, B, C, and D on the platform become









Ax y--- h,
2


C x y+ a h\
2


B +DiF y h ,


D x-1 y h.
2


The coordinates of points E, F, G, and H on the base can be found from (3.1). Thus, the

Plicker line coordinates for each of the eight leg lines can be determined as


Sb .V2a-b
,= x+-, y- 2 ,
2 2


b J2a-b
2 = x- y a b
2 2

S3=[x,2a- a-b +b




S54= x+- -, y- b


S [ b i2a-b
S =x-, y+ 2
2 2'


S b 42a-b
S= x+a-, y+ ,

7 = 2a-b b
= x +-, y + ,
2 2'


$ = a-b b
=x-2 y +2'


bh bh b(2x 2y+ V2a)
2' 2' 4 J'


h; bh bh b(2x + 2y -2a)
2' 2' 4


bh bh b(2x+2y+J2a)
2' 2' 4


bh bh b(2x- 2y+ ia)
2' 2' 4


Sbh bh b(2x- 2y -f2a)
2' 2' 4


Sbh bh b(2x+2y + 2a)]
2 2' 4


Sbh bh b(2x + 2y a)1
2' 2 4

bh bh b(2x 2y ]2a)
Sbh bh b(2x-2y-4 2a)
2 2 4


(3.28)


(3.29)



(3.30)



(3.31)



(3.32)



(3.33)



(3.34)



(3.35)



(3.36)







The above coordinates are not normalized and each row must be divided by the

corresponding leg length. The Jacobian matrix J then can be constructed by using (3.11).

Further, substituting and expanding VdetJJr yields


det J =a3b3h32(1l +12 +I2 +l 2)(l +142 +12 +12)
S1112131415161718


(3.37)


where the leg lengths are


I b 2 -b 2 2,
11 = x+ + y- +h 2,





, = x- + y a b +h 2,



7 a-b 2 b 2
= x + y- +h
2 2


1If ,b2F2a b I 2,
2 2

S2a-b2
14 = X+ + y +h2,


18= x- b+ y+ +h 2


l2 f
8=x- + y+ +h'
2 4 +2.


vJ
With a = 1, b = -2, and h = -, from (3.25) and (3.37), the quality index A becomes
2

S detJJT x2 +y2 +1
det jT (x4+2xy2+y4 + 22+1) (4+2x2y24 +2y2+1) (3.38)

which is plotted in Figure 3.5(a) as a function of x and y. Figure 3.5(b) shows the

contours of the quality index for this platform horizontal movement. The contours are

labeled with values of constant quality index and they are close to being concentric

circles of various radii. When x ory is infinite, A=0, and when x=y=0, A= 1.
























1 -1 -1.0 -0.5 0.0 0.5 1.0

(a) (b)

Figure 3.5: Quality index for platform horizontal translation


To illustrate the variation of the quality index during some simple rotations of the

platform, a new coordinate system x'y'z' is attached to the square platform. The origin of

the new coordinate system is located at the center of the top platform and the coordinate

system is oriented such that its x'-axis is passing through vertex B, its y'-axis then is

passing vertex C, and its z'-axis is normal to the square platform ABCD. Thus, when the

platform locates at its initial central position shown in Figure 3.2, the x'- and y'-axes are

parallel to the x- and y-axes on the base respectively.

Figure 3.6 illustrates a side view of the redundant 4-4 parallel manipulator when

the square platform ABCD is rotated by an angle 0y about the y'-axis from its initial

position. For such platform rotation, the coordinates of the vertices A, B, C, and D

become








Af0 -2a h BF cosO 0 h+- sinO ,
2 2 I 2

C 0 V2a h D -a-cosoy 0 h- sin m
2 2 2
\i /~ \1 /(~cs, o _~ i


(3.39)


Figure 3.6: Platform rotations about the y'-axis


It should be noted that the positions of line $1, $2, $5, and $6 do not change during

this movement and their corresponding Plicker line coordinates can be obtained from

(3.3), (3.4), (3.7), and (3.8) respectively. The Plicker coordinates for the line $3, $4, $7,

and $s are now given by

S, _2acosb, b b asiny +2h
2 2' 2


(3.40)


b(j2a sin O + 2h) b(12a sin 9O + 2h) F2abcos OY
4 4 4









[2acos0, b
S4 =
2

b(Jfa sin O, + 2h)
4


7 4/acosO b
2

b(r2a sin 9, 2h)
4


8 2acos9 -b
2

b([2a sin 09 2h)
4


b 2asin O +2h
2 2

b(r2a sin Y + 2h)
4


b
2


(3.41)


V2abcos ,
4


J2a sin y 2h
Y l


b(2a sin 0Y 2h)
4


b
2'


(3.42)


F2ab cos 0,
4


V/2asin 0 -2h
2


b(f2a sin 9, 2h)
4


V2ab cos O0
4


Since the configuration of the manipulator keeps symmetric about the x-axis

during the platform rotation about the y'-axis, from Figure 3.6, we have

1 = l2 = 15 = 16 = = 4 and 1, = 1g. The corresponding Jacobian matrix then can be

determined by (3.11), and further detJJ" becomes


det JJ = N2ab (((4h2 +(2a2 -22ab+b2)sin20 )(12 +12)
41 12 3

4h(-2a b)(l2 -12) sin ,O + 2(4h2 cos2 O, + b2 sin2 6 )12)


(h2(4h2 cos2 y +b2 sin2 0y)(l2 +l) -4bh3(l2 -12)cosO, sinOy +

((4a2h2 cos2 0y + a2b2sin 20 + 2b2h2 -8N2abh2 cosOr)sin2 O +


8h4 cosO ,)12))


(3.44)


where


(3.43)





37


13= -]b(bacos -b +b'+(2 asin, + 2h)2


1, = -ia cos9y -b) + b2 + (,a sin 2h)2,

and I can be found from (3.12).


With a =1, b = 2, and h= from (3.25) and (3.44), the quality index
2

becomes

/detJJT J (3-cosO,)(6cos30 cos20, -7cosy +4)
V(3A =(3.45)
/det J. J 2(2cos2 O, -4cosOy +3)

Since the redundant 4-4 in-parallel manipulator is fully symmetric at its central

configuration shown in Figure 3.2, the same result can be obtained when the platform is

rotated about the x'-axis.

From (3.45), the variation of the quality index for rotations about the x'- and y'-

axes is drawn in Figure 3.7.



1.0 ..- .....I.......... . -. ...... ................



I 0.6 . ........ .......... ........ ........... .



0.2

0,0 .i0i i
-90 -60 -30 0 30 60 90
Rotation Angle 0 (degree)

Figure 3.7: Quality index for platform rotations about the x'- and y'-axes









Figure 3.8 illustrates a plan view of the redundant 4-4 parallel manipulator with

the moving platform ABCD rotated 6a about the z-axis. The x and y coordinates of the

vertices A, B, C, and D then become

xA = rsin0., YA =-rcosOZ,

Xg =rcos90, yB = rsin 0,
(3.46)
xc =-rsin0z, Yc = rcosO,

XD = -rcos z, Yo = -r sin 0z

a
where r= From (3.46)


XA + XB + xC + XD = 0,
(3.47)
YA + Y + Yc + YD =0.

The complete set of coordinates of points A, B, C, and D are therefore

A(xA YA h), B(x, YB h), C(xc yc h), D(xD Y, h) (3.48)

where h is the height of the moving square ABCD above the base square EFGH.

Yi





I


Figure 3.8: Platform rotations about the z-axis








Then the coordinates for the corresponding lines $1, $2, ..., $8 are given by


S= [X + b,












s,6 -. +
[ b 2 b
92 r -A



b


2






9= xc +-,

S b 2
7 = xD +- ,

S 2'
9, = x, +b,
=[ b 2


b
YA + 2,
2

b
YA +


b
Ys +-
2

b
YB 2'

b


b
Yc -2


b



Yo +
2


Sbh
h; 2


bh
h;
2

bh
h; 2
2


bh b(xA Y)]
2 2


bh
2'

bh
2 '


bh bh
h;
2" 2


bh
h;
2


bh
2


b(x, + y )
2 J

b(x, + y )
2 J


b(x y)]


b(xc yc)
2


Sbh bh b(xc + Yc)
2 2 2


bh
h; ~
2


bh b(x, + y,)
2 2 J


bh bh b(x, y,)
2 2 2


It is apparent from Figure 3.8 that =13 = 15 = 1, and 12 =4 =16 = The

corresponding Jacobian matrix can be determined by (3.11). Furthermore, calculating

detJJT yields


42a3'b3h'Icos90,
detJJ'= a3b1
1 2


(3.57)


where


(3.49)


(3.50)


(3.51)


(3.52)


(3.53)


(3.54)


(3.55)


(3.56)





40


= 2asin0 +b + fV2acos -b 2 ,2
I- =--- +h,2
2 2


a I sin0, -b r2 2acos9 -b 2
12 ..2 2 + h .


J2
With a = 1, b = N2, and h = from (3.25) and (3.57), the quality index becomes
2

SJdet JT Icos0 (3
Vdet J mJ (2cos2 0o 4cosO, +3)

This is plotted in Figure 3.9 and it shows how the quality index varies as the platform is

rotated about the vertical z-axis through its center. The eight legs are adjusted in length to

keep the platform parallel to the base at a distance h. It is shown in the figure that the

manipulator has the highest quality index A = 1 when 0~ = 0', and A = 0 (singularity)

when 0= 90.


1.0

0.8 ..

0.6 -.---

| 0.4 ........ ......... ............. .... .....

0.2 -

0.0 iI
-90 -60 -30 0 30 60 90
Rotation Angle 0 (degree)

Figure 3.9: Quality index for platform rotation about the z-axis








It is interesting to note that the redundant 4-4 parallel manipulator shown in

Figure 3.1 always becomes singular when its platform rotates z=900 about the z-axis

from its central symmetric position. This can be seen from (3.57), Jdet JJ = 0 when 60

= 900. Figure 3.10 illustrates the singularity position of the redundant 4-4 parallel

manipulator when a6 = 900. It is not immediately obvious from the figure why the eight

connecting legs are in a singularity position. This kind of singularity has been discussed

in detail by Hunt and McAree [14]. They explain that at such position, even when all

eight leg actuators are locked, the connectivity between the base and moving platform is

one. The moving platform can move instantaneously on a screw reciprocal5 to the eight

leg forces on the z-axis with pitch hz, i.e., a screw with coordinates

[0, 0, 1; 0, 0, he]. (3.59)


y
H G















Figure 3.10: Plan view of the singularity position of the redundant 4-4 in-parallel
manipulator when Oz = 900


s When a wrench acts on a rigid body in such a way that it produces no work while the
body is undergoing an infinitesimal twist, the two screws are said to be reciprocal.








Now when az=90, from (3.49) through (3.56), the component of moments

about the z-axis for each of the eight legs are all the same

Viab
N, =h and R, 2 = (i = 1, 2,...,8). (3.60)
4

The coordinates for the eight legs become


= L, M,, h; Pi, Q,, (3.61)


Hence, from (3.59) and (3.61),

Nliab Jiab
hhh =0 or hz =T (3.62)
4 4h

It follows that all eight legs lie on a linear complex, which is a three-parameter system of

linearly dependent lines (Hunt [13]).













CHAPTER 4
THE KINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT
4-4 IN-PARALLEL MANIPULATOR

The kinematic analysis of in-parallel manipulators deals with the study of the

platform motion determined by the leg displacements. Two problems can be

distinguished for the kinematic aspects: inverse kinematics and forward kinematics. The

inverse kinematics problem, i.e., finding the leg lengths for a given location (position and

orientation) of the mobile platform (a difficult problem for serial manipulators), is

straightforward for parallel manipulators. On the other hand, the forward kinematics

problem, i.e., finding the platform location from a given set of leg lengths, is much more

difficult. In general, this problem has more than one solution for nonredundant cases. As

an example, the forward analysis for the general 6-6 platform requires the solution of a

40th degree polynomial (Raghavan [31]), the solution of which is clearly impractical for

real-time implementation. A lot of methods have been presented to solve various types of

nonredundant parallel manipulators as summarized by Dasgupta and Mruthyunjaya [7].

However, few works have been done on the kinematic analysis of redundant

parallel manipulators, particularly the forward kinematics. A similar problem for

determining a unique position and orientation of the platform of a general geometry

parallel manipulator by using seven linear transducers has been solved by Innocenti [15].

He constructed a 146x147 constant matrix for solving the problem, which makes the

computation time still larger than real time. Also, Innocenti's method produces only one

solution for a general geometry parallel manipulator, but when the fixed base of a








manipulator is in a plane, there should be two solutions instead of one since the

manipulator can have two reflection configurations through the base according to the

same set of readings from linear transducers no matter how many transducers are used.

Therefore, his method may not be correct to solve the problem when the base of a parallel

manipulator is in a plane.

Zhang, Crane and Duffy [37] have performed the forward kinematic analysis on a

special redundant parallel mechanism whose platform and base are connected by a ball-

and-socket joint with four legs to determine the orientation of the platform.

In this chapter the kinematic analysis of the redundant 4-4 in-parallel manipulator

shown in Figure 3.1 is performed. From here on, the notations from Crane and Duffy [5]

are used to describe the coordinates of point and the transformation matrix. For example,

the notation 'PA is used to indicate the coordinates of a point A as measured in a

coordinate system I and 2R is defined as the orientation of the coordinate system 2

relative to the coordinate system 1.


4.1 Inverse Kinematic Analysis

The objective of the inverse kinematic analysis of the redundant 4-4 parallel

manipulator is to find the eight leg lengths for a given position and orientation of the

square moving platform.

In Figure 4.1, coordinate systems 1 and 2 have been attached to the fixed base and

the moving platform respectively. The origins of the coordinate systems 1 and 2 are

located at points E and A receptively. The coordinates of points E, F, G, and H on the

base are known in terms of the coordinate system 1 and








0 b b 0

PE = 0 'P = 0, P = b, 'PH = b, (4.1)



while the coordinates of points A, B, C, and D on the platform are known in terms of the

coordinate system 2 and

0 a a 0 = (4.2)

2 A = 0 2 PPB = 0 2C = a 2 D = a (4.2)


Figure 4.1: Coordinate systems of a redundant 4-4 in-parallel manipulator








It is well known that the location of a rigid body in space can be described by the

position and orientation of a coordinate system attached to the body with respect to a

fixed reference frame. Thus, for the inverse analysis, the position and orientation of the

coordinate system 2 is considered known, and can be given by the position vector 'PA,

which defines the position of the origin A of the system 2 relative to the origin E of the

fixed frame 1, and a rotation matrix 2R, which is a 3x3 matrix whose columns are the

unit vectors along the coordinate axes of the system 2 as measured in the system 1. Then

the coordinates of points B, C, and D in terms of the coordinate system 1 become

'PB = 'PA + 2R2pB,

'Pc =PA + A R2Pc, (4.3)

'PD= 'PA+ R2 PD.

Finally, the eight leg lengths can be calculated by

11 =('PA-'PE)('PA-'PE)

12 = PA-'PF'PA-'PF)

13 =('PB-'PF 'PB-' PF)

14 = ('PB-PG ) PB-'PG)
(4.4)
15 =( Pc-' P) 'Pc-'P).

16 =('Pc-'PH 'Pc-'PH),

17 =('PD-'PH (PD-'PH)

e e (' PD-uct of two vects.

where represents the dot product of two vectors.








Hence, for a given location of the moving platform, there is only one possible

solution for each leg length.


4.2 Forward Kinematic Analysis

The objective of the forward kinematic analysis of the redundant 4-4 parallel

manipulator is to find the location of the moving platform given the actuator

displacements of all the eight legs. Thus, the coordinates of points A, B, C, and D

measured in the coordinate system 1 shown in Figure 4.1 need to be determined for a

given platform side a, base side b, and eight leg lengths 1i (i= 1, 2,..., 8).

4.2.1 Introduction

The forward analysis is performed in detail in this section and thus provides a

unique solution for the location of the moving platform above the base platform together

with a reflected solution through the base for an arbitrarily specified set of eight leg

lengths. However, extreme care must be taken in applying this analysis since what

appears to be an arbitrary set of leg lengths may well be special and the solution will fail.

For such cases, the constraint equations employed in the analysis presented here become

linearly dependent in one way or another.

A class of special cases has been reported by Selfridge [32] where he obtained a

pair of assembly configurations (as opposed to a unique solution) above the base and a

corresponding pair of reflected solutions through the base. It is interesting to note that

one class of solutions reported by Selfridge [32] occurs when the platforms are parallel,

the odd leg lengths are all equal and 1, = = 1 = 1 = 1 = Further, the even leg lengths are

all equal and 12 =4 1 = 1 = 1'. This class of solution embraces the workspace

generated by a rotation of the top platform about the z-axis (Figure 3.8). While this does









not raise a problem with the quality index analysis it is important to recognize that these

are in fact a pair of assembly configurations above the base platform. A numerical

example is present in section 4.2.5. All this of course raises the issue of other classes of

special cases that are worth further investigation.

4.2.2 Coordinate Transformations

First, the coordinates of any point in coordinate system 2 need to be transferred to

coordinate system 1. To do so, coordinate system 2 may be obtained by initially aligning

it with coordinate system 1 and then introducing the following transformations:

1. Rotate the coordinate system 2 by an angle 01 about the x-axis until the y-axis is in the

plane defined by points A, E, and F, and the scalar product of the y-axis with the

vector S1 is positive as shown in Figure 4.2.

2. Translate the origin from point E to F along the positive x-axis.

3. Rotate by an angle i about its current z-axis, which causes the x-axis to point along

the vector S2.

4. Translate the origin from F to A along the negative x-axis.

5. Rotate the coordinate system about its current x-axis by an angle 0z until the y-axis is

in the plane defined by points A, F, and B, and the scalar product of the y-axis with

the vector S3 is positive.

6. Rotate by an angle 2 about its current z-axis, which causes the x-axis to point along

the vector S3.

7. Rotate by an angle 03 about its current x-axis until the y-axis points along the vector

S4.









S4A
Y2
DL_


Figure 4.2: Coordinate transformations


The coordinates of points A, B, C, and D may now be expressed in the coordinate

system 1 as

PA= R, [T, + R3(T4+RRRR 2A,
IPB = R, [T2 +R,(T4 +R,RR 2PB),

'Pc = R, T + R3(T4 +R5R6R7 2)], (4.5)
SPD= R, [T2 +R3(T4 +RRR,72P)],


where









1 0 0 b cos -sin 1 0 -1,

R,= 0 cos -sin8, T2= 0, R3= sin cos 0 T4= 0 ,

0 sin cos8, 0 0 0 1 0

1 0 0 cosO2 -sin 0 1 0 0

R,= 0 cosO2 -sin2n R6 = sin2 cosQ2 0 R7= 0 cos03 -sinO3 ,

0 sin 2 cos2 J 0 0 1 0 sin 0 cos03

and the coordinates of points A, B, C, and D are known in terms of the coordinate system

2 and are written as 2P^, 2pB' 2Pc, and 2P.

The angles A and A are shown in Figure 4.2 as the inner angles of the triangles

AEF and AFB, respectively. Therefore, the angles 0i and 0 are constrained to lie in the

range of 0 to n. The cosines of 01 and 0 may be determined from a planar cosine law as

b2 +122 l2
Cos 0 =
2bl2
(4.6)
a2 +12 -12
2 _
cos, 22 3
2al2

and the values of 0 and 0 are determined as the inverse cosine value in the range of 0 to

7E.

The coordinates of points A, B, C, and D as measured in the coordinate system 1

have been written as a function of the parameters 01, 6O, and 03. The objective now is to

determine these parameters that will locate points A, B, C, and D such that they satisfy

the distance constraints with points E, F, G, and H.








4.2.3 Constraint Equations

Since the eight leg lengths, ii (i= 1, 2, ..., 8), have been given for the forward

analysis, the distance between points A, B, C, and D on the platform and points E, F, G,

and H on the base must satisfy these leg lengths as shown in Figure 4.2. The coordinates

of points A, B, C, D, E, F, G, and H all have been expressed in terms of the coordinate

system 1 (see (4.5) and (4.1)), and the distance between these points may be expressed in

the coordinate system 1 as

('PB-'P,)(IPB- PG)= 1 (4.7)

(' Pc- PG).(PC'-'G ) = 5 (4.8)

('PC1 P)'(lP-'P,)=12, (4.9)

('PDo-PH)(' PD-'PH)= 1, (4.10)

('PD PE) IPD-- PE) = (4.11)

Note that three constraint equations are not written for the distance between points A and

E, A and F, and B and F. The distance between these points will be equal to 11, 12, and 13,

respectively. These three leg lengths have been used in the transformation of coordinate

systems, which relates the coordinate systems 1 and 2 included rotation angles 1 and 2

and origin translation distance 12.

Equations (4.7) through (4.11) may be expanded and factored into the form

2abs s s 2bc, (as cc2 + s (ach -12))+a2 +b2 -2ac 1 +12 12 =0, (4.12)

2abs, (s2 s2 + c0 s2c3 + c2s3) + 2bc (ac. s2s3 acA s c2 a(cA c, 2 S s )c3 -
s (ac -12))+2a2 +b2-2ac 12+1 -lI =0, (4.13)








2absi (s s2 + ch s2c3 + cs3) + 2bc, (ac, (s2s3 s c2) + a(c, cc,2 + s5A ss )c3 -
s. (ac, -12))+ 2abs (s2s3 -s c2)- 2a(bs cO c2 + (bc, 1)s )c, + (4.14)
2a2 + 2b2 + 2abc ch 2bc 12 2ac 12 +12 -l =0,

2abs, (c s2c3 + c2s3)+ 2bc, (acA s2s3 a(c c c2 s s0 )c3 + l2so )+
(4.15)
2abs s2s3 -2a(bs c c +(bc -l2)s )c +a2 +2b2 2bcA 1 +12 -l2 =0,

2abs s2s3 -2a(bs c c2 +bc sh -12sh)c +a2 +b2 -2bcl2 +l1 -12 = 0 (4.16)

after recognizing that s2 + c2 =1 and s2 + c2 =1, and where si, ci, (i= 1, 2, 3) represent

the sine and cosine of 9i and s c j, (j= 1, 2) represent the sine and cosine of O. The

objective now is to determine values for tO, 02, and 03 which will simultaneously satisfy

the five equations represented by (4.12) through (4.16).

4.2.4 The Solution

To solve equations (4.12) (4.16), we consider sl, cl, s2, C2, 53, and c3 as

independent variables. It gives us three more equations since

sin2 O + sin2 i9 = s2 + c = (i= 1, 2,3). (4.17)

Now equations (4.12) (4.16) will be manipulated to eliminate si, ci, s3, and c3 first. The

algebra to achieve this is what follows.

Adding equations (4.12) and (4.15), and then subtracting (4.14) yields

kc +k2c2 + k3 =0, (4.18)
where k1, k2, and k3 are known constants and are defined in Appendix A. Similarly, an

equation that is linear in c2 and c3 is generated by subtracting (4.14) from the sum of

(4.13) and (4.16):


k2c2+k4c3 + k =0.


(4.19)









Solving equations (4.18) and (4.19) for ci and c3 respectively yields

k2 k3
c, =- C2 (4.20)
k, kI


c3 = 2 (4.21)
k4 k4

Now substituting the expressions for cl and c3 into equations (4.12) and (4.16)

produces

k6sIs2 + k7c + kc2 + k = 0, (4.22)

k6s2s3 + klos + kc2 + k12 =0. (4.23)

Solving the above two equations for sl and s3 respectively yields

k7c 2 + kSc2 + k9
s, = (4.24)
k6s2


s3 = 1 (4.25)
k6 s2

Finally, substituting the expressions for si, s3, cl, and c3 into equations (4.15) and

s, + c2 = 1 for i = 1 and 3 produces three equations in two variables, s2 and c2. Further,

replacing s2 with 1- c2, it is interesting to note that s2 cancels from these equations. This

leaves following three equations in only one unknown, c2

4
Eq,(c2)= IM,c =0 (i = 1,2,3) (4.26)
j=0

where the constants Mi are defined in Appendix A. The objective now is to determine

value for c2 that simultaneously satisfy the three equations represented by (4.26).









Multiplying the three equations in (4.26) by c2, we obtain three additional

equations. Thus, a total of six equations in the unknown c2 are obtained. These equations

can be written in matrix form as

My = 0 (4.27)

where

M,4 M13 M12 M11 M10 0 c

M24 M23 M22 M21 M20 0 C2

M34 M33 M32 M31 M30 0 C
M= y=
0 M14 M,3 M12 M,, M,1o

0 M24 M23 M22 M21 M20 2

0 M34 M33 M32 M31 M30


Here, we treat c2, c2, c2, c2, c2, and 1 as unknowns and thus equation (4.27) can be

regarded as a homogeneous linear system in six unknowns. The trivial solution of y=0

is not feasible, since the last element of y must equal 1. Solutions other than the trivial

solution exist only if the homogeneous equations are linearly dependent, and as such the

determinant of the matrix M must equal zero. Evaluating this determinant and seeing

how close it is to zero will provide an indication of the quality of the measured data (i.e.,

the platform side a, base side b, and the joint positions) and the sensed data (i.e., the eight

measured displacements 11, 12, ..., Is). The issue of how close to zero is satisfactory is not

addressed in this dissertation.

The six equations represented by (4.27) may now be rearranged into the form

Ux = v (4.28)


where









M14 M13 M12 M11 M1 0
5
c2
AM24 M23 22 M21 M20 0
4
C2
MM M33 M32 M31 M30 0
U= x= c v=
0 MI4 M13 M12 M11 -M10
2
C2
0 M24 M23 M22 M21 -M20
c2J
0 M34 M33 M32 M31. -M30

Equation (4.28) represents six linear equations in five unknowns. The vector x may be

solved for by selecting any five of those equations. The term c2 is the fifth component of

the vector x and unique value for this term is thereby determined. However, it should be

noted that since 92 is in the range of 0 to 2x, there are actually two solutions of 92 for a

value of C2. Thus, the manipulator has two configurations for a given set of leg lengths.

These two configurations are due to a reflection through the base plane.

For each value of 02, corresponding values for cl, C3, si, and s3 can be calculated

from (4.20), (4.21), (4.24), and (4.25) respectively. Then, values for 01 and 03 can be

determined. Finally, the coordinates of points A, B, C, and D in terms of the coordinate

system 1 can be obtained by substituting A9, 02 and 93 into (4.5).


4.2.5 Numerical Verification

In this section, a numerical example is presented for a redundant 4-4 parallel

manipulator to verify the analysis. The dimensions of the manipulator are measured in an

arbitrary length unit and given as follows:

platform side a = 10, base side b = 15.

A set of leg lengths are given as








1, =13.62421, 12 =10.40411, 13=14.47201, 14=11.16409,

15 = 16.34095, 16= 17.59696, 17 =16.22984, 18 = 15.92500.

The numerical results are presented in Table 4.1 and two configurations are shown to be

reflecting through the base plane. Thus, a unique configuration may be easily determined

by checking the sign of z coordinate of one of the platform joints. In order to verify these

results, an inverse kinematic analysis was performed. All solutions reproduced the correct

leg lengths.


Table 4.1: Numerical results of the redundant 4-4 in-parallel manipulator

No. O (deg.) 62 (deg.) 03 (deg.) P PB 'PC PD

10.079 16.119 8.921 2.881
1 -105.534 133.523 -27.872 2.455 10.327 15.045 7.173
8.832 10.077 15.168 13.923


10.079 16.119 8.921 2.881
2 105.534 -133.523 27.872 2.455 10.327 15.045 7.173
-8.832 -10.077 -15.168 -13.923


In the following example the above solution failed because equation (4.7) through

(4.11) become linearly dependent while equation (4.9) through (4.11) are redundant for

the system. Using the same dimensions as the above example, the leg lengths now

become:


S=13 1=17 ==18 and 12 =14 =16 = 1'= 16.








The numerical results are presented in Table 4.2. It is apparent that there are now

two configurations above the base plane and further two solutions reflected through the

base.




Table 4.2: A numerical example for the special case of
the redundant 4-4 in-parallel manipulator

No. h I PA 1 PB 'PC IPD

2.267 -6.698 -2.267 6.698
1 5.199 6.698 2.267 -6.698 -2.267
5.199 5.199 5.199 5.199


2.267 6.698 2.267 -6.698
2 15.099 -6.698 2.267 6.698 -2.267
15.099 15.099 15.099 15.099


2.267 -6.698 -2.267 6.698
3 -5.199 6.698 2.267 -6.698 -2.267
-5.199 -5.199 -5.199 -5.199


2.267 6.698 2.267 -6.698
4 -15.099 -6.698 2.267 6.698 -2.267
15.099 -15.099 -15.099 15.099














CHAPTER 5
THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT
4-8 IN-PARALLEL MANIPULATOR

A redundant 4-8 parallel manipulator is shown in Figure 5.1, which is derived

simply by separating the double ball-and-socket joints in the base of the redundant 4-4

manipulator shown in Figure 3.1. A plan view of the redundant 4-8 parallel manipulator

is shown in Figure 5.2. The device has a square platform of side a and an octagonal base

formed by 4 pairs of joints ED and EA, FA and FB, GB and Gc, and He and HD. Each pair

of joints is separated from a vertex of a square of side b by a distance fb for which

1 1
Ol< -. Clearly the platform is degenerate when f = -
2 2












$ /
GF


E EA b B


Figure 5.1: A redundant 4-8 in-parallel Figure 5.2: Plan view of the redundant 4-8
manipulator in-parallel manipulator








5.1 Determination of det JJT

The moving platform of the redundant 4-8 parallel manipulator shown in Figure

5.2 is located at its central symmetrical configuration and is parallel to the base with a

distance h. By analogy with the redundant 4-4 parallel manipulator, a maximum value of

the square root of the determinant of the product of the manipulator Jacobian by its

transpose, i.e., /detJ.JT may be obtained for this configuration.

A fixed coordinate system is placed at the center of the octagonal base as shown

in Figure 5.2. Then, the coordinates of the points A, B, C, and D on the platform are


2 2 2 2
Af0 ]a h, BL- 0 hJ], C0 Vi h}, D( /a 0 hJ) (5.1)

The coordinates of the points EA, FA, FB, GB, Gc, Hc, Ho, and ED on the base are

EA-d, -_ o10 L,F d o1 F -d4 o0, GB( d4 0),
2 2 2 2 ) ^ )
(5.2)
Gc d4 b o Hc -d4 b 0 H -_ d4 0o, E -d4 0

where

(1- 2/)b
d4- =
2
Counting the 2x2 determinants of the various arrays of the joins of the pairs of points

EAA, FAA, ..., EoD yields the Plicker line coordinates of the eight leg lines. That is, from

points EA and A,

J =d4. J2a-b bh; d4h, (5.3)
2 2 2


From points FA and A,









$2 = I-d4,


bh iad4
h; -, d4h, -
2 2


From points FB and B,


3 = 2a -b, d4
2


bh
h; -d4h, ,
2


Viad4
2


From points GB and B,


4 a-b
S,2a b
2


bh
d4, h; d4h, ,
2


V2ad4
2


From points Gc and C,


s= -d4,


From points Hc and C,


$6 = d-4,


bh _-2ad4
h; h, d4h,
2 2


From points HD and D,


7 =[a b
I 2a-b
2


bh
-d4, h; d4h, ,
2'


From points ED and D,


S-2a -a-b
2


d4, h;


bh i2ad4
-d4h, -
2 2


The normalized Jacobian matrix of the eight leg lines can be expressed in the


form


i1 1 3 ^T ^5T T 6T 7T 18
1 12 13 14 15 16 17 I1


(5.11)


5a -b
2


(5.4)


(5.5)


(5.6)


bh
h; -,
2


-d4h ad
2


VZa-b
2




-5a-b
2


(5.7)


(5.8)


V2ad4
2


(5.9)


(5.10)






61


where 11, 12, ..., 1 are the leg lengths and since the device is in a symmetrical position, 11

= 12 = = 1 = I, and


l= L2 +M +N2 = [a2 -l ab+(2f2 -2 p+1)b2 +2h2.


From (5.3) to (5.10), (5.11) becomes


h h


(5.12)


J
I


bh
2


bh
2


-d4h d4h


bh
d4h -d4h
2
if2ad4, 2ad4 l2ad4
S2 2 2


where same as (3.13), di


bh
2
i2ad4
2


-d4h d4h


,2ad4
2


2lad4
2


bh
2
I2ad4
2


_2a-b
2


From equation (5.13), the determinant of the product JJ T becomes

4(d 2+d2) 0 0 0 2(2d2-bd,)h


0 4(d2 +d'2)


0 2(bd,-2d2)h


0 8h2


detJJT= 1
12


0 2(bd,-2d2)h 0 (b2+4d2 )h2


2(2d2-bd )h


0 0 0 (b2+4d2)h2


0 0 0


4a2d2


Expanding (5.14) and using (5.12), then extracting the square root, we obtain


d4h -d4h


(5.13)


bh
2
iFad4
2


. (5.14)


d4 -d,








-j-jT = 322(1 2#)3 a 3b3h3
vdet JJ = -,--- ----- f--------. (5.15)
[a2- 2ab+(22-2 p +)b2+2h 2'

Assuming the top platform size a is given and taking the partial derivative of (5.15) with

respect to h and b respectively and equating to zero yield

96,f2(1- 2f)3'a3b3h2[a 2-_-ab+(2'2 -2p + )b2 -2h2]= (5.16)
----- -- &------------- = U (5. 16)
[a2 2ab+(2/f2 -2l + l)b2 +2h24

and

9642(-1+2)3 a3b 2h [-a2 +(2f 2 -2f +1)b2 2h2]
[a2 2ab+(2f2 -2,f+l)b2 +2h2 (517)

1
Note that we already assumed f -, then when a, b, and h are not equal to zero, from

equations (5.16) and (5.17), we get

a2 2ab+(2fi2 -2f + l)b2 -2h2 =0, (5.18)

-a2 +(2fl2 2 + 1)b2 -2h2 =0. (5.19)

Subtracting (5.19) from (5.18) yields

2a 2 ab = 0, (5.20)

and thus

b = 42a. (5.21)

Substituting (5.21) into (5.19) gives

(4 82 -4f +1)a 2 2h2 =0. (5.22)

The above equation yields two solutions for h, here we only take the positive solution

h = (1 2f)a. (5.23)
V-2









Therefore, when b = -2a and h= (1 2f)a, the redundant 4-8 parallel manipulator


is at the maximum quality index as shown in Figure 5.3, and from (5.15)

qdetJ.J =( detJJ') = 44a' (5.24)

where Jm denotes the Jacobian matrix of this configuration. It is interesting to note that

this maximum value of det JJ T is independent to the value of f.


Figure 5.3: Plan view of the optimal configuration of the redundant 4-8 in-parallel
manipulator with the maximum quality index


From (3.24) and (5.24), it shows that both the redundant 4-4 and the 4-8 parallel

manipulators have the same maximum value of [det JJ. Figure 5.4 illustrates the

compatibility of these two results. It can be observed that as the distance between the

pairs of separation points of the double ball-and-socket joints E, F, G and H of the

original 4-4 manipulator increases, the height h at which the manipulator has the


ED___
EEA d a" FA








maximum quality index decreases (see (5.23)) from h 7= ( = 0, concentric ball-and-



2
socket joints) to h = 0 (f/ = -, platform is degenerate).






$ $6 $ 4
H G B-


$8 $1 $2 $3



EA b FA

E F

Figure 5.4: Compatibility between the redundant 4-4 and the 4-8 parallel manipulators
(1h. 2f)a
(h, =)



5.2 Implementation

From (1.2) and (5.24), the quality index for the redundant 4-8 parallel manipulator

shown in Figure 5.1 can be expressed as


(5.25)


Sdet JJ
4,r= .


In this section, a redundant 4-8 parallel manipulator with a = 1 and b = 22 is taken as an

example for the investigation of the variation of the quality index A during a number of

platform movements.









The first platform movement we studied is a pure vertical translation of the

platform along the z-axis that passes through the center of the platform. From (5.15) and

(5.25), the quality index for this movement becomes

8(1- 2f)3b3h3'
[a2 Nab+(2p2 -2f +l)b2 +2h2]3

With a = 1 and b =2 this reduces to

S16r2(1- 23)3 h3
Al= (- ,)h 3 (5.27)
[(1-2f)2 +2h2]3

and is plotted in Figure 5.5(a) as a function of h and ft. Figure 5.5(b) plots the variation

curves of the quality index for several different values of fl. From these figures, we can

see the height (hm) at which the manipulator has the maximum quality index is reduced as

p increases. Each value of f designates the distance between the separation points in the

base and is a first design parameter. Clearly, f= 0 is the best overall design.


h
1.0 ..

=0
0.8 .
0. x= 1/8
1 0.6 ...-...o.. ............. ....................

0

0. 0. .. ........... ....................
~-




11.5 0.3
h" 2 .2/5
2.5 0.1 0.0
30 0 1 2 3
Height h
(a) (b)

Figure 5.5: Quality index for platform vertical movement






66


The second platform movement is a pure horizontal translation of the platform

away from its initial location at height h. To derive an expression for detJJr we

assume the center of the platform move to point (x, y, h), then the coordinates of the

points A, B, C, and D on the platform become


Ax



C x


2ia
2


ya h*}
y+ h h,
2


B x+ -
2


D x--
2


y h


y hJ.


(5.28)


The coordinates of points EA, FA, FB, GB, Gc, He, Ho, and ED on the octagonal base can

be found from (5.2). Calculating the Pliicker line coordinates for each of the eight leg

lines yields


S2= x d,

S

S3 = x+-



S4 = X+ -V



S = x-d,



S, = x+d4,


Via -b
y- .
2


Ja-b
y---
2


a -b
2


a-b
2


bh
h; -,
2'


bh
h; -
2


y+d4, h; -d4h,



y-d4, h; d4h, -


Via-b
y+-
2


Via -b
2


bh
, h; ,
2


bh
h; -,
2


dh, bx+d,4(Va-2y)
d~h, -
2


bx-d4(I2a-2y)
-dh,


bh d4 (-2a+2x)+by
2' 2


bh d4(,2a+2x)-by
2' 2


d4h, bx-d4(Via+2y)
-d4h, 2,
2


bx+d4(2a + 2y)]
d4h, ,
2


(5.29)



(5.30)



(5.31)



(5.32)


(5.33)


(5.34)








7 X Fa b
7 =x a- b
2


y-d4, h; d4h,


bh
2


d4 (2a 2x)- by
2


S8 = I


42a-b
2


d4, h;-d4h, bh d,(F2a-2x)+by
y+d,, h; -dh, ,
2' 2


(5.36)


(1- 2f)b
where same as (5.2), d4 -
2

The Jacobian matrix can then be constructed by using (5.11). Further, det JJ

becomes


detJJT =


a25 + + 2 2 +12 +12)
lll2131415161718


(5.37)


where the leg lengths are


11= (x+d42 + Y a-b +h2,
2


=, j .22 2a -b +h2
12= (x-d4)2+ y-- +h ,
2,e f


+(y+ d4)2 +h2,


13= (x+- 2a-b I


15= (x-d4)+ 2 a-b +h2

S-b 2 ,2
17= x- +(y-d4 )+h2 ,
2


14 = (x+ 2a b
2


+(y-d4)2 +h2,


16 = x+d4)2++ i2a-b +h2,
2


Via-bx2
18 x -2a b
2


+(y+d4)2 +h2.


(1-2f8)a = 1-2,8
With a = 1, b = V2, and h = hm = 2(1 from (5.25) and (5.37), the quality
2 Vf2


index becomes


(5.35)








VdetJJ _(1-2f1)6(l +1+2+13+1 )
t = 2222 (5.38)
detJmJ 411 12134

where

11 = 1, = x2 +2(1-2)x+ y +(1-2f)2,

12 =l5 =Vx2 -v(l-2f)x+ y2 +(1-2i)2 ,

13 =18 = x2 +f(1-2fi)y+y2 +(1-2f)2,

14 =17 = Vx2 2(1-2f8)y+ y2 +(1-2f#)2.

In Figure 5.6, the quality index and its contours as the platform is translated away

from the central location while remaining parallel to the base at hm, are drawn for various

values of jf. It should be noted that when fl=0, the 4-8 manipulator becomes the 4-4

manipulator and its corresponding quality index is drawn in Figure 3.5. Comparing

Figure 5.6(a)-(d) and Figure 3.5, it is clear that the smaller f, the larger workspace area of

the platform is with high quality index.

Now we attach a new coordinate system x'y'z' to the square platform. This

coordinate system may be obtained from the platform configuration shown in Figure 5.2

by initially aligning it with the xyz coordinate system on the base and then raising it by a

distance h along the z-axis to the top platform. Thus, the x'- and y'-axes are parallel to the

x- and y-axes respectively when the platform locates at its initial central position shown

in Figure 5.2.

We are interested in deriving detJJr when the platform rotates about the x'-

and y'-axes from its central position. Here, we only derive the platform rotation about the






69


y'-axis. But the result to be derived is the same for the platform rotation about the x'-axis

since the redundant 4-8 parallel manipulator is fully symmetric.


1.0


0.5


A 0.0


-0.5


-1.0


-1.0 -0.5 0.0
x


0.5 1.0


2 N1
(a) =- at h,= = 0.14
5 10


1 -1


-0.5 -


-1.0 -


-1.0 -0.5 0.0 0.5 1.0


1 -5
(b) =- at h, = 6 0.24
3 6


S............... i . ......... ............. i .......... . .




- .. . . ."............. 4 ........... i .............. -






. ........... . . . . . .. . . . . . . .


----.......... --............... ..............--.................--
.."............... ;................ .. ............. ............... .. .


.. ............. ........... ........................ i












1.0 -


0.5 -


~ 0.0 -


-0.5 -


-1.0 -


1 -1


1
(c) =- at hm
4


-1.0 -0.5 0.0 0.5 1.0


= = 0.35
4


1.0 -


0.5 -


p0.0-


-0.5 -


-1.0 -


I -1


-1.0 -0.5 0.0 0.5 1.0


1 a
(d) #=- at h, =- = 0.53
8 8


Figure 5.6: Quality index for platform horizontal translation with different values of 6


............... ..............
--b








Figure 5.7 illustrates a side view of the moving platform ABCD rotated 0y about

the y'-axis. The coordinates of the vertices A, B, C, and D become


A f- l2a h, B cos9 0 h+ sinej,
2 2 2
(5.39)
C 0 a h D - ca os9, 0 h---a sin
2 2 ( 2

and the coordinates of vertices EA, FA, FB, GB, Gc, Hc, Ho, and ED on the base can be

found from (5.2).


Figure 5.7: Platform rotation about the y'-axis


Note that the positions of line $1, $2, $5, and $6 do not change during this platform

rotation and their corresponding Plicker line coordinates can be obtained from (5.3),

(5.4), (5.7), and (5.8) respectively. The Plucker coordinates for the line $3, $4, $7, and $8

are now given by









[V acos0 -b
S3 = -2ac 2Oy-b d4,
2

d4 (V2a sin ,y + 2h)
2


2a sin y + 2h
2


(5.40)


b(V2a sin Oy + 2h)
4


F2ad4 cos O
2


^IFacosOy -b V2asin OY +2h
24 2 -d4 2 $


(5.41)


d4 (V2a sin Oy + 2h)
2


7 2acos Oy-b
2

d4,(2asin Oy 2h)
2


b(-2a sin 0y + 2h)
4


i4 a sin 0 -2h
2


b( 2a sin O, 2h)
4


2ad4 cos ,0
2


(5.42)


V2ad4 cos Oy
2


Viacos6/-b
8 /2a cos =Y b
S2

d4 ( a sin 0y -2h)
2


F2a sin 0y 2h
2

b(VF2a sin 0, 2h)
4


(5.43)


i2ad4 cos Oy
2


(1-2fl)b
with d4 =-
2

From Figure 5.7, we have 1, = 12 = l5 = 6 = 1, 13 14, and 17 = 1. The

corresponding Jacobian matrix can now be obtained by (5.11), and further detJJT can


be determined. With a =1, b = 2, and h = h= -2 from (5.25), the quality index
= rm 52) teqait ne


then becomes






73


detJJ (1-2i)4 212 + 2+12
= = 2(1-2) 4213 +17 (2(5-(42 -4f8 +2)(cosO, +
VdetJ.Jr 41413

cos 30,) + (16 84 -32l3 +24/82 8 2)(cos20, +1)) + (l2 + 72)(14/2 -

6f +1+ 8f3' (f 2)(cos20 + 1) + 2P8(5f# 1)cos20y) + (l3 -l 7)(8f8' -

12w2 + 6 -l)sin 202

where


(5.44)


= 1-2fi,

13 = 4f2 -4f, + 2- cos0Y -(2f -1)sin 0,,

17 = 4fi2 -4/f+2-cos9 +(2/i-1)sin9y.


Figure 5.8(a) plots the quality index as a function of y4 and ft. Figure 5.10(b)

presents the change of the quality index for several different values of f.


...I......... I ..................







-90 -60 -30 0 30 60 9
Rotation Angle 6 (degree)


Figure 5.8: Quality index for platform rotations about the x'- and y'-axes


1.

0.

0.2.


0400.1
-60040 0.
460 0.1
80 0









A plan view of the redundant 4-8 parallel manipulator with the moving platform

ABCD rotated 06 about the z-axis is shown in Figure 5.9. The x and y coordinates of the

vertices A, B, C, and D become

A = r sin O, YA = -r cos 0,

x, =r cos0 9, y = rsin 0,
(5.45)
xc =-rsin O, Yc = rcos0~,

XD = -r cos 0z, Yo = -r sin Oz

a
where r = -


The complete set of coordinates of points A, B, C, and D are therefore

A(xA YA h), B(xB y, h), C(xc Yc h), D(x, yD h) (5.46)

where h is the height of the moving square platform above the octagonal base.


Figure 5.9: Platform rotation about the z-axis





75


The coordinates for the corresponding lines $1, $2, ... $8 are then given by


A-
S, = xA +d4,


[
2 =XA -d4,


b
2












b
94 = xB

[ 2

5 = xc -d4,






9 = [XD +-

2
b
98 = xD +- ^


b
y, +- ,
2

b
YA +-,
2


bh
h; --,
2


bh
h; --,
2


bh
y, +d4, h; -d4h, --,
2

bh
yB -d4, h; d4h,
2


b
Yc --
2

b
Yc -,
2


bh
h; -,
2

bh
h; ,
2


bh
YD -d4, h; d4h, -,
2


Y +d4, h;


b(x, y + 2py,)]
d4h, 2



2
b(x, + Y, 2fAYA ) ]
-dh, 2,


b(x, 2pxB +y)
2b(x

b(xB 2xB -yB)
2J


b(xc Yc + 2fpyc)
- d4h, -


b(xc + yc- 2pfyc)
d4h, 2


b(x, 2fpx + y,)
2


b(x, 2fpx y,)
2


bh
-d4h, 2'
2


From Figure 5.9, we have I1 = l1 = l = 17 and 12 = 14 = 16 = I. The corresponding

Jacobian matrix can now be determined by (5.11), and further ddetJJT becomes

de--- 4[2a3b3h3(1- 28)3'cos 0
=detJJ = ,- 33 (5.55)


where


I, =l- (iasinO, +b-2b))2 +(Jacosz -b +4h2


12 =V(a sin -b+ 2+ 2) +(,iacosO, b +4h2


(5.47)


(5.48)


(5.49)


(5.50)


(5.51)


(5.52)


(5.53)


(5.54)






76



With a =1, b = and h = = -2, from (5.25) and (5.55), the quality index


becomes


A, = (1- 2)'6 ICS (56)

(2(2f2 -2fi+1)(cosO, -2)cosOz +16f84 -328' +28fl2 -12,+3)

This is plotted in Figure 5.10(a) as a function of f and O. It shows how the quality index

varies as the platform is rotated about the vertical z-axis through its center. The eight legs

are adjusted in length to keep the platform parallel to the base at a distance hm. Figure

5.10(b) illustrates the variation of the quality index for several different values of f. It is

shown in these figures that the manipulator has the highest quality index A = 1 when

Oz=0, and A = 0 (singularity) when 0= 900. As can be seen in Figure 5.10(b), a slight

change of 6z under a large f has a much greater impact on the quality index than that of

the same change under small 8.


(a)


u.u 0 I i I
-90 -60 -30 0 30 60 90
Rotation Angle 0 (degree)

(b)


Figure 5.10: Quality index for platform rotation about the z-axis


................... ..... . ......... 7 .........--


=1/8
1/4......


I II


LI









Again, from Figures 5.8 and 5.10 we can see clearly that better designs are

obtained as f/ reduces to zero. Hence the best 4-8 parallel manipulator design is obtained

when the pair of base joints are as close as possible.

Since from (5.55), detJJ =0 when z=900, a redundant 4-8 parallel

manipulator always becomes singular when its platform rotates a=900 about z-axis

from its central symmetric position. Figure 5.11 illustrates the singularity position when

z=90.


y
He Gc G

HD $ GB

$6 B-$4



C x




ED, FB
EA FA F

Figure 5.11: Plan view of the singularity position of the redundant 4-8 in-parallel
manipulator when 6O = 900



In complete analogy with the redundant 4-4 parallel manipulator presented in

Chapter 3, when 09=900, the moving platform of the redundant 4-8 parallel manipulator

can move instantaneously on a screw reciprocal to the eight leg forces on the z-axis with

J2ab
pitch h = T- This is because for OL=900, from (5.47) through (5.54), the
4h






78


component of moments about the z-axis for each of the eight legs all are equal to

4ab
4













CHAPTER 6
THE FORWARD CINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT
4-8 IN-PARALLEL MANIPULATOR

The kinematic analysis of the redundant 4-4 parallel manipulator has been

performed in Chapter 4. It is shown that the inverse kinematics is straightforward for

parallel manipulators while the forward kinematics is difficult. In this chapter the forward

kinematics of the redundant 4-8 parallel manipulator shown in Figure 5.1 is studied. It

will be shown how this problem can be easily solved by transferring it to the

corresponding redundant 4-4 case which then can be solved by using the method

presented in Chapter 4.


6.1 Forward Kinematic Analysis

In Figure 6.1, coordinate systems 1 and 2 have been attached to the fixed base and

the moving platform of a redundant 4-8 parallel manipulator, respectively. The origins of

the coordinate systems 1 and 2 are located at points E and A receptively. The objective of

the forward kinematic analysis of the redundant 4-8 parallel manipulator is to find the

position and orientation of the moving platform given all the eight leg lengths.

Eight dash lines connecting the moving platform and the base are drawn in Figure

6.1. If we connect the platform with the base by legs along these dash lines to replace the

original legs, we obtain a redundant 4-4 parallel manipulator with the platform location

same as the original 4-8 manipulator. Thus, if we can determine the distances, loi (i= 1, 2,

..., 8), between the platform and the base along the dash lines, the forward kinematic

problem will have been solved by using the method presented in Chapter 4 for the









redundant 4-4 parallel manipulator. The objective now is to determine values for 10o, 102,

.... s08 from the original leg lengths i, 12, ..., Is.


XI

Figure 6.1: Coordinate systems of a redundant 4-8 in-parallel manipulator


Observing Figure 6.1, we find 11, 12, 101, and 102 are in the same plane defined by

points A, EA, and FA while 13, 4, 103, and l04 are in the plane defined by points B, FB, and

GB, 15, 16, los, and 1o6 are in the plane defined by points C, Gc, and Hc, and 17, 18, 107, and los

are in the plane defined by points D, HD, and ED. Thus, loi can be determined from li in

the same plane. For example, in the plane defined by points A, EA, and FA as shown in

Figure 6.2, we have






81


11o = J2b2 + 1 2fb cos( ,,

o2 = V/(1- )2 b2 + -2(1 f)bl cos4p,,


where


(1- 2)2b2 +12 -12
cos = 2b( -
2b(1- 2,)1,


Figure 6.2: Leg relations


Similarly, the other leg lengths can be obtained

l03 = 4f2b2 +1 3 2f0bl3 cos( p2),

104= V(1- _f)2 b2 +12 -2(1-f)bl3cosqp,,

lo = 2b +12 -2fbl, cos( r- ),


(6.1)

(6.2)


(6.3)

(6.4)

(6.5)









lI = 4(1- f)2b2 +12 2(1-/ )bl, cos(p3, (6.6)


S= J2b2 +12 2fbl7 cos('r (4), (6.7)


ls = 4(l- f)2b2 +12 2(1- f)bl7 cos 4 (6.8)

where

(1-2p)2b2 +2 122
cos =-3
2b(l- 2f)l3

(1 2,)2b2 +l1-l2
cos P = --
2b(1 -2 f)1l

(1-2)b2 +12- _12
COS V4
2b(1- 2f)17

Using the values of lot, 102, ..., lo8 as input leg lengths to the forward kinematic

analysis presented in Chapter 4, the position and orientation of the moving platform will

be determined.


6.2 Numerical Verification

A numerical example is presented for a redundant 4-8 parallel manipulator to

verify the analysis. The dimensions of the manipulator are measured in an arbitrary

length unit and given as follows:

1
platform side a = 10, base side b = 15, = -
8

A set of leg lengths are given as

I, = 12.21787, 12 = 9.15596, 13 =12.83105, 14 = 7.52035,

1, =13.47917, 16 =13.13367, 17 =13.88865, 1 = 14.04687.








Thus, the input leg lengths for the forward analysis of the corresponding redundant 4-4

parallel manipulator are obtained from (6.1) through (6.8) and

10, = 13.59387, 102= 9.87590, 03 = 9.87590, 14 = 7.94680,

15 = 14.41631, 1 = 13.98464, 107 = 14.72302, 10 = 14.92181.

The numerical results are presented in Table 6.1. It has been verified by an inverse

kinematic analysis that all solutions reproduced the correct leg lengths.




Table 6.1: Numerical results of the redundant 4-8 in-parallel manipulator

No. 91 (deg.) 9 (deg.) 0 (deg.) PA 'PB 'PC 'PD

10.409 14.940 7.091 2.560
1 -112.939 122.570 -28.524 3.408 12.304 16.592 7.696
8.052 7.475 11.948 12.525


10.409 14.940 7.091 2.560
2 112.939 -122.570 28.524 3.408 12.304 16.592 7.696
-8.052 -7.475 -11.948 -12.525




Similar to the redundant 4-4 parallel manipulator, there is a special solution when

the platform rotates about the z-axis (Figure 5.9). For example, when the leg lengths now

become

1, =13 =15 =1=1=18 and 12 =14 =6 = = '=16

for the redundant 4-8 parallel manipulator in the first example, two configurations above

the base plane with another two reflected through the base are obtained as shown in Table

6.2.








Table 6.2: A numerical example for the special case of
the redundant 4-8 in-parallel manipulator

No. h IPA I PB P IPD

3.022 -6.393 -3.022 6.393
1 7.498 6.393 3.022 -6.393 -3.022
7.498 7.498 7.498 7.498


3.022 6.393 -3.022 -6.393
2 15.748 -6.393 3.022 6.393 -3.022
15.748 15.748 15.748 15.748


3.022 -6.393 -3.022 6.393
3 -7.498 6.393 3.022 -6.393 -3.022
-7.498 -7.498 -7.498 -7.498


3.022 6.393 -3.022 -6.393
4 -15.748 -6.393 3.022 6.393 -3.022
-15.748 -15.748 -15.748 -15.748













CHAPTER 7
THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT
8-8 IN-PARALLEL MANIPULATOR

A redundant 8-8 in-parallel manipulator is shown in Figure 7.1, which is derived

by separating the double ball-and-socket joints in the base and the top platform of a 4-4

manipulator shown in Figure 3.1. A plan view of the redundant 8-8 in-parallel

manipulator is shown in Figure 7.2. The device has eight legs connecting an octagonal

platform and a similar octagonal base. The octagonal top platform is formed by 4 pairs of

joints Al and A2, BI and B2, C1 and C2, and D1 and D2. Each pair of joints is separated

I
from a vertex of a square of side a by a distance ca for which O 2

octagonal base is formed by 4 pairs of joints ED and EA, FA and Fa, GB and Gc, He and

HD, and each of them is separated from a vertex of a square of side b by a distance fib for

I
which 0:<5 <-. This design has the distinct advantage that it completely avoids the
2

mechanical interference problem associated with the design of double spherical joints.


7.1 Determination of .det J., .

The moving platform of the redundant 8-8 parallel manipulator shown in Figure

7.2 is located at its central symmetrical configuration and is parallel to the base with a

distance h. It will be shown in this chapter that a maximum value of the square root of the

determinant of the product of the manipulator Jacobian by its transpose, i.e., Jdet J J ,

may be obtained from this symmetric configuration.




























Figure 7.1: A redundant 8-8 in-parallel
manipulator


Figure 7.2: Plan view of the redundant 8-8
in-parallel manipulator


First, a fixed coordinate system is placed at the center of the octagonal base as

shown in Figure 7.2. Then the coordinates of the points A,, A2, B1, B2, C1, C2, Di, and D2

on the platform are


V2a
d5
2


-d5 h,


A2(d5


i2a
d,5-
2


B2NF-d5 d5 h
2


(7.1)


Va d5
2

- 2a
-2
-- d
2


2ia
-d,
2


-d5 h)


where


d5 2aa
ds5=-


A, d,


fB, f-d
2


C(ds

DC ds
D, d,


C2 d5


,V2a
d5 i
2







The coordinates of the points EA, FA, FB, GB, Gc, Hc, Ho, and ED on the base are

EA -d4 0 o1, d4 0 FB -d4 o0 GB d4 0,
2 2 b2 ) ,)
(7.2)
G d 02 H -d4 b H b- d4 0, ED(- -d4 0 .
(1- 2 2)b
Same as (5.2), d4 = (
2
Now the Pliicker line coordinates of the eight leg lines of the redundant 8-8

parallel manipulator can be obtained by counting the 2x2 determinants of the various

arrays of the joins of the pairs of points EAAI, FAA2, ..., EoD2. From points EA and A1, we

get

= d4-d5, d5 -d,, h; --, d4h, d6 (7.3)
[ 2bh

where

d6 = (2afl 2a 2fl +)ab and d, -
4 2
Similarly, from points FA and A2,

2= [d5-d4, d5-d1, h; --bh, -d4h, -d (7.4)
Sb2

From points FB and B1,

S3= d,-d5, d4-d,. h; -d4hd, d6] (7.5)

From points GB and B2,

4 =d-ds, d5-d4, h; d4h, ---, d6 (7.6)


From points Gc and C1,








bh 1
S= d,-d, d, -d5, h; -, -d4h, d6. (7.7)

From points Hc and C2,

Si =d4 -d, d,-d5, h; -,b d4h, -d (7.8)
Sbh2

From points HD and DI,

S = [d5-d, d5-d4, h; d4h,2 d6]. (7.9)

From points Eo and D2,

bh 1
S8=[ d-dd, d4-ds, h; -d4h, -, -d6 (7.10)

The above coordinates are not normalized and each leg line needs to be reduced to unit

length. Then, the normalized Jacobian matrix of the eight leg lines can be expressed in

the form

"or Sl T T g T T gr 1
1 j_4 5 7 8(7.11)
1 12 13 14 15 16 17 18

where i, 12, ..., 18 are leg lengths.

Here, the device is in a symmetrical position so that the normalization divisor is

the same for each leg, namely 11 = 12 = ... = I = 1, and for every leg

1= L +M2 +N2

= 2[(2a2-2a+l)a2 + i(2a-I)ab+(2'l2-2i+l)b2 +2h2]. (7.12)

From (7.3) to (7.10), (7.11) becomes