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Kinematics of parallel manipulators with ground-mounted actuators

Material Information

Title:
Kinematics of parallel manipulators with ground-mounted actuators
Creator:
Weng, Tzu-Chen, 1956- ( Dissertant )
Sandor, George N. ( Thesis advisor )
Duffy, Joseph ( Reviewer )
Seirig, Ali ( Reviewer )
Matthew, Gary K. ( Reviewer )
Selfridge, Ralph ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1988
Language:
English
Physical Description:
vi, 190 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Axes of rotation ( jstor )
Coordinate systems ( jstor )
Cylinders ( jstor )
Degrees of freedom ( jstor )
End effectors ( jstor )
Infinitesimals ( jstor )
Inverse kinematics ( jstor )
Kinematics ( jstor )
Mechanism design ( jstor )
Robots ( jstor )
Actuators ( lcsh )
Dissertations, Academic -- Mechanical Engineering -- UF
Kinematics ( lcsh )
Manipulators (Mechanism) ( lcsh )
Mechanical Engineering thesis Ph. D
Robotics ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )
theses ( marcgt )

Notes

Abstract:
A great deal of research work has been focussed on the theoretical and experimental studies of mechanical manipulators in recent years. Almost all of these works are related to open-loop serial-link mechanisms, but only a few have dealt with multi-degree-of-f reedom parallel manipulators such as the Stewart platform or similar mechanisms . A new type of two-degree-of -freedom Rotary-Linear (R-L) actuator was adapted in this work. Several possible configurations of parallel six-degree-of-f reedom manipulators with ground-mounted actuators have been synthesized. With parallel configuration of the manipulators, the computations can be performed simultaneously. Therefore, the computation time will be significantly reduced. Workspace analysis of a six-degree-of -freedom parallel manipulator has been presented by determining the shapes and boundaries of the subworkspace and root regions in the subworkspace. The workspace of the manipulator is obtained as the common reachable region of the subworkspaces determined by the corresponding subchains. The orientation and the rotatability of the platform are also investigated. Finally, mechanical error analysis of the manipulator, due to the minor inaccuracies in displacements of the actuators, is studied by using the theory of screws.
Thesis:
Thesis (Ph. D.)--University of Florida, 1988.
Bibliography:
Includes bibliographical references.
Additional Physical Form:
Also Available on World Wide Web
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Tzu-Chen Weng.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
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20233905 ( OCLC )
AFM8966 ( NOTIS )

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KINEMATICS OF PARALLEL MANIPULATORS
WITH GROUND-MOUNTED ACTUATORS











By

TZU-CHEN WENG





















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

1988

:U OF F LIBRARIES











ACKNOWLEDGMENTS


The author wishes to express his sincere appreciation

to his committee chairman, Professor George N. Sandor, for

years of invaluable guidance, support and encouragement

during his graduate studies. The author also gratefully

acknowledges the advice and support given by the members of

his supervisory committee, Dr. Joseph Duffy, Dr. Ali Seireg,

Dr. Gary K. Matthew and Dr. Ralph G. Selfridge.

The author extends his gratitude to Dr. Dilip Kohli of

the University of Wisconsin--Milwaukee, for all his help in

the development of this work and Dr. Kenneth H. Hunt for his

advice during his visit at the University of Florida.

Special thanks are also extended to his colleagues and

fellow students, especially Mr. Yongxian Xu of the Dalian

Railway Institute, for their valuable suggestions.

The financial support of the National Science

Foundation under grant DMC-8508029 is gratefully

acknowledged.

Most of all, the author wishes to express his sincere

appreciation to his parents for their support and

encouragement which helped him throughout his graduate

studies. Finally, the author extends his deepest

appreciation to his wife, Han-Min, for her inspiration and

moral support, and for years of patience and encouragement.














TABLE OF CONTENTS


ACKNOWLEDGMENTS .................. ....................... ii

ABSTRACT ................................................ v

CHAPTERS

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

1.1 Literature Overview...................... 1
1.2 Serial and Parallel Manipulators......... 7
1.3 Summary. .................................. 10

2 TYPE SYNTHESIS AND INVERSE KINEMATICS OF THE
MANIPULATORS ................................. 12

2.1 Introduction.............................. 12
2.2 R-L (Rotary-Linear) Actuator............. 14
2.3 Type Synthesis............................ 18
2.4 Technical Discussion..................... 22
2.5 Inverse Kinematics....................... 30
2.5.1 Subchain (R-L)-R-S................ 32
2.5.2 Subchain (R-L)-P-S................ 37
2.5.3 Subchain (R-L)-S-R................. 39
2.5.4 Subchain (R-L)-S-P................ 45
2.6 Summary.................................. 50

3 WORKSPACE ANALYSIS OF THE MANIPULATOR .......... 52

3.1 Introduction............................. 52
3.2 Configuration of a Paralle Manipulator
with R-L Actuators........................ 55
3.3 The Subworkspace Analysis of the
Manipulator ............................ 57
3.3.1 Shapes of the subworkspace......... 59
3.3.2 Boundaries of the subworkspace
and root regions in the
subworkspace (infinitesimal
platform) ....................... 80
3.4 Conditions for No-Hole Workspace......... 106
3.5 Workspace of the Manipulator............. 110
3.6 Summary .................................. 118


iii






4 THE WORKSPACE OF THE MANIPULATOR WITH FINITE
SIZE PLATFORM.................................. 120

4.1 Introduction.............................. 120
4.2 Workspace of the Manipulator with
Infinitesimal Platform.................. 121
4.3 The Complete Rotatability Workspace
(CRW) and the Partial-Rotatability
Workspace (PRW)............. ........... 123
4.4 The Workspace of the Platform with
Given Orientation................. .......... 132
4.5 Summary..................... .............. 146

5 MECHANICAL ERROR ANALYSIS OF THE MANIPULATOR... 147

5.1 Introduction.............................. 147
5.2 Position Analysis........................ 148
5.3 Reciprocal Screws......................... 151
5.4 Screws of the Relative Motion of the
Joints.................................. 156
5.5 Jacobian Matrix........................... 162
5.6 Mechanical Error Analysis of the
Platform................................ 163
5.7 Summary................... ................ 168

6 CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER
RESEARCH.............. ....................... 171

6.1 Conclusions.............................. 171
6.2 Recommendations for Further Research..... 177

APPENDICES ................... ............ .............. 178

A ALTERNATIVE METHOD OF FINDING THE COORDINATES
OF JOINT C.................. .. ............. 178

A.1 Subchain (R-L)-R-S....................... 178
A.2 Subchain (R-L)-P-S....................... 180

B EQUATION OF A GENERAL FORM OF TORUS............. 183

REFERENCES................ .............. .............. 185

BIOGRAPHICAL SKETCH........................ ............. 190













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


KINEMATICS OF PARALLEL MANIPULATORS
WITH GROUND-MOUNTED ACTUATORS

By

TZU-CHEN WENG

December 1988


Chairman: Dr. George N. Sandor
Major Department: Mechanical Engineering

A great deal of research work has been focused on the

theoretical and experimental studies of mechanical

manipulators in recent years. Almost all of these works are

related to open-loop serial-link mechanisms, but only a few

have dealt with multi-degree-of-freedom parallel

manipulators such as the Stewart platform or similar

mechanisms.

A new type of two-degree-of-freedom Rotary-Linear (R-L)

actuator was adapted in this work. Several possible

configurations of parallel six-degree-of-freedom

manipulators with ground-mounted actuators have been

synthesized. With parallel configuration of the

manipulators, the computations can be performed

simultaneously. Therefore, the computation time will be

significantly reduced.






Workspace analysis of a six-degree-of-freedom parallel

manipulator has been presented by determining the shapes and

boundaries of the subworkspace and root regions in the

subworkspace. The workspace of the manipulator is obtained

as the common reachable region of the subworkspaces

determined by the corresponding subchains. The orientation

and the rotatability of the platform are also investigated.

Finally, mechanical error analysis of the manipulator,

due to the minor inaccuracies in displacements of the

actuators, is studied by using the theory of screws.













CHAPTER 1
INTRODUCTION



1.1 Literature Overview

Robotics has been a very popular subject to study in

the last few years. Researchers have developed many

advanced concepts and theories in kinematics, dynamics,

controls, actuators and sensors for the design of robots.

Recent areas of study also involve workspace, obstacle

avoidance, full rotational dexterity of the end-effector

and the control of flexible manipulator systems recently.

With the development of microprocessors, which has played

an important role in the rapid growth of industrial robots,

multi-degree-of-freedom mechanical systems are now becoming

a practical choice for use in automatic machinery. It has

been well recognized that, by using multi-degree-of-freedom

robotic manipulators with multiple actuators and automatic

control systems, we can achieve the goal of improving

efficiency, accuracy, reliability and reducing energy

consumption and cost of production in flexible

manufacturing systems.

Robotic manipulators currently in use in industry and

studied for research purposes are almost all traditional

open-loop serial-link manipulators in which the number of

degrees of freedom of the end-effector is equal to the sum

1






of the relative degrees-of-freedom of the joints in the

chain. There are only a small number of multi-degree-of-

freedom designs which involve multi-loop manipulator

linkages, with totally parallel or partially parallel

configurations. Since the design techniques for multi-loop

robotic manipulators are still in the infancy of their

development, investigating and developing the theoretical

background for multi-degree-of-freedom multi-loop robotic

manipulators may have a significant impact in the near

future in manufacturing industry.

For a given set of manipulator linkage dimensions, it

is necessary to determine all admissible positions of the

end effector. The collection of all such possible positions

is called the workspace of the manipulator. Recently

several methods have been proposed to determine the

workspace of a manipulator by showing possible extreme

positions of the end effector. These methods let us

calculate directly the boundaries of the workspace of a

given point or line on the hand.

Workspace analysis, generally, refers to determining

the boundaries of the workspace. Workspace synthesis, on

the other hand, consists of determining dimensions of the

manipulator linkage and ranges of joint motions for a

specified workspace.

One of the primary functions of the manipulator is to

have its end-effector reach a set of points in space with

prescribed positions and orientations. The manipulators







investigated early were almost all serial kinematic chains,

since these manipulators usually have larger workspaces and

more dexterous maneuverability than those of parallel

kinematic chains. However, serial chains have poor

stiffness and undesirable dynamic characteristics in high-

speed operation. Also, it is usually difficult to solve

their inverse kinematics problem. Therefore, mechanisms

based on parallel kinematic chains may have certain

advantages when dynamic loading is present and only limited

workspace is required.

Serial multi-degree-of-freedom manipulators have been

extensively investigated. Roth [1) studied the relationship

between the kinematic parameters of a manipulator and its

workspace. Shimano and Roth [2] presented the analytical

and geometrical conditions for a line on the hand to be at

the farthest distance from the base revolute pair. Sugimoto

and Duffy [3, 4] developed an algorithm to determine the

extreme distances of a robot hand. Kumar and Waldron [5]

developed the theory and algorithm for tracing the bounding

surfaces of a manipulator workspace. Sugimoto and Duffy [6]

and Sugimoto, Duffy and Hunt [7] investigated the

singularities in the workspace. Kumar and Waldron [8]

presented the algorithm for tracing the bounding surfaces of

manipulator workspaces. Tsai and Soni [9] presented the

study of determining the accessible region for two and

three-link robotic arms with pin-joints. Gupta and Roth

[10] presented some basic concepts regarding the workspace






shapes and structures of manipulators. Selfridge [11]

presented an algorithm for finding the boundary of reachable

volume of an arbitrary revolute-joint, serial-link

manipulator. Tsai and Soni [12] developed an algorithm for

the workspace of a general n R robot based on a linear

optimization technique and on small incremental

displacements applied to coordinate transformation equations

relating the kinematic parameters on the n R robot. Yang

and Lee [13] derived the equations representing the

boundaries of the workspace. Existence of holes and voids

in the workspace were also investigated. Lee and Yang [14]

have made a study of outlining the boundary of the

workspace, the quantitative evaluation of the volume, and

introduced a manipulator performance index. Hansen, Gupta

and Kazerounian [15] used a stable iterative algorithm for

inverse kinematic analysis to determine the approach angles

and lengths for reaching points in the workspace.

Freudenstein and Primrose [16] described the workspace of a

three-axis, turning-pair-connected robot arm of general

proportions in terms of the volume swept out by the surface

of a skew torus rotating about an offset axis in space.

Kohli and Spanos [17] studied the workspace analysis of

mechanical manipulators by using polynomial displacement

equations and their discriminants. Spanos and Kohli [18]

performed the study of workspace analysis of a class of

manipulators having the last three revolute joint axes

intersect orthogonally at a point. Cwiakala and Lee [19]







used an optimization technique to outline the boundary

profile of a manipulator workspace and perform quantitative

evaluation of the workspace volume. Tsai and Soni [20]

illustrated the general procedures to synthesize the

workspace of 3R, 4R, 5R and 6R robots. Tsai and Soni [21]

also considered the effects of kinematic parameters on the

workspace of general 3R robots. Oblak and Kohli [22] used

an analytical method, based on displacement equations, to

identify the Jacobian surface or a D-shaped surface which

the workspace of a regional structure is bounded by.

Davidson and Hunt [23] had a study of the rigid body

location and robot workspaces by using an enumeration

procedure. Davidson and Hunt [24] described plane

workspaces for robots by using a sweeping process and the

necessary equations for computer generation of plane-

workspace envelops and boundaries. Davidson and Pingail

[25] continued to generate envelope-surfaces for plane-

workspace of generally proportioned manipulators. Chen [26,

27] presented an analytical method for workspace analysis of

robot arms by using differential geometry. Kohli and Hsu

[28] studied the Jacobian analysis of workspaces of

mechanical manipulators by determining the maximum reach of

the manipulator within the intersection of the boundary with

a specified plane. Hsu and Kohli [29] dealt with closed-

form workspace analysis and used the Jacobian surfaces to

separate inaccessible regions, two- and four-way accessible

regions in both manipulator coordinates and Cartesian




6

coordinates. Palmquist [30] studied the reachable workspace

common to two planar RRR robots, dexterous relationship

between them and the kinematic motion capabilities of them.

There are few works that have dealt with parallel

multi-degree-of-freedom manipulators. The Stewart platform

[31] is a kind of parallel manipulator which has two plates

connected by six adjustable legs and is a six-degree-of-

freedom mechanism. It was originally used for flight

simulators and was suggested for applications on machine

tools and on space vehicle simulators. Asada and Ro [32]

applied a direct-drive arm to a closed-loop five-link

mechanism to overcome the problems encountered in open-loop

arrangements. Trevelyan, Kovesi and Ong [33] applied such a

mechanism to a sheep shearing robot. Bajpai and Roth [34]

studied the workspace and mobility of such a closed-loop

planar five-link mechanism. Yang and Lee [35] presented a

feasibility study of the Stewart platform as a robot

manipulator. The extreme ranges of motion, rotatability

and workspace were investigated and the workspace and the

maneuverability were found to be relatively restricted.

Fichter [36] also studied Stewart platform-based

manipulators, theoretical aspects of the generalized Stewart

platform, and practical considerations for building a

working machine. Again, Cwiakala [37] used the optimum

path search technique to find the section of the workspace

of the Stewart platform mechanism. Recently, Kohli, Lee,

Tsai and Sander [38] investigated manipulator configurations







with ground-mounted rotary-linear actuators; their direct

and inverse kinematics were also derived.

The majority of current industrial robots are used for

body guidance. One of the criteria in the control steps is

to reduce the positioning error to a limited range. The

techniques needed to solve such problems have been developed

in the study of closed-loop spatial linkages. Hartenberg

and Denavit [39] used a deterministic method to analyze the

mechanical error. Garrett and Hall [40] used a statistical

approach for mechanical error analysis. Dhande and

Chakraborty [41] presented a stochastic model of the planar

four-bar function generating linkage mechanism for error

analysis and synthesis for specified maximum of mechanical

error. Chakraborty [42) presented a probabilistic model of

linkage mechanisms considering tolerances on the link

lengths and clearances in the hinges, which may cause

mechanical error. Dhande and Chakraborty [43] studied the

effect of random error in the joint of spatial linkages and

developed a synthesis procedure to allocate tolerances and

clearances on different members of linkages to restrict the

output error within specified limits.



1.2 Serial and Parallel Manioulators

Open-loop serial-link manipulators have been the

subject of numerous investigations and have found

considerable applications in industry. In recent years,

there has been considerable increase in research in the






area of robotics and multi-degree-of-freedom programmable

automation devices. For being competitive in international

markets, the use of flexible manufacturing systems in

industries is becoming more and more important.

The heart of the flexible manufacturing system consists

of computer controlled multi-degree-of-freedom devices such

as robots and N.C. machines. The configurations which have

been widely used for these machines are serial-link (open-

chain type) arrangements, where one link is connected to

adjacent links by single-degree-of-freedom joints, each with

its separate actuator. Similarly, although N.C. machines

are also serial-link devices, the several degrees of freedom

are distributed between the work piece and the tool, which

decouples the motions associated with various groups of

several degrees of freedom. This simplifies kinematic,

dynamic, and control computations. The manipulator

mechanism generally consists of six links serially connected

via six single-degree-of-freedom separately actuated

revolute or prismatic joints. The end effector, which is

attached to the most distal link, imparts motion of six

degrees of freedom to the work piece. The motion and/or

force associated with these six degrees of freedom may be

controlled.

Although the mechanism may appear simple, the motion of

the end effector is related to joint motions by mathematical

transformations which are generally not easy to visualize.







Duffy [44] developed the theory which is applied to

the analysis of single-loop mechanisms, which are movable

polygons or open chains with one end fixed to the ground,

closed by an imaginary link between ground and a free end

where there is a mechanical hand or gripping device.

Hunt [45] has discussed all possible single- and multi-

degree-of-freedom kinematic pairs and used screw theory,

kinematic geometry and the techniques of linear algebra to

systematize in-parallel-actuated robot-arms. Mohamed and

Duffy [46] also applied screw theory to study the

instantaneous kinematics of the end-effector platform of

fully parallel robot-type device. Sugimoto [47] derived the

kinematic and dynamic model for a parallel manipulator by

using motor algebra and Newton-Euler formulation.

A comparison between the serial and parallel devices

in terms of some necessary and desirable performance and

control characteristics was presented by Cox [48]. There

are eight performance characteristics chosen as follows:


i. Range of motion

ii. Rigidity or stiffness and strength

iii. Complexity of end-effector positioning

formulation (computability)

iv. Complexity of system dynamics (computability)

v. Precision positioning

vi. Load carrying distribution through system

vii. Fabrication (economics)

viii. Compactness







Hunt [45, 49] also showed some possible alternative designs

for manipulators using parallel kinematic chains, and

pointed out that there are many intermediate possibilities

between purely serial and purely parallel kinematic

structures.

It is well recognized that more investigations in the

study of parallel manipulators are needed and they may have

potential usefulness in the manufacturing industry.



1.3 Summary

Conventional serial-linkage manipulators have each of

their up-to 6 actuators mounted on the joint they actuate.

This means that the mass of these actuators is added to the

link masses, which greatly increases the inertia seen by

actuators and links closer to the ground.

On the other hand, multi-loop manipulators with ground-

mounted actuators need to consider only the masses of the

links themselves. Also, the links can be lighter for the

same payload.

In Table 1.1, a comparison between serial and parallel

kinematic chains is shown, where X means more favorable

performance.

The successful completion of the study of parallel

manipulators would open up a new direction in the design of

robotic manipulators with advantages over present practice,

such as improved payload capacities, increased positioning

accuracy, greater economy in energy consumption, better







dynamic performance, increased speed with improved

precision, and reduced first cost.



Table 1.1 Performance characteristic between serial


and parallel


kinematic chains


Performance Serial Parallel
characteristic structure structure

Compactness X

Computation time X

Dexterous maneuverability X

Direct kinematics X

Inertia X

Inverse kinematics X

Payload capacity X

Power/weight X

Precision positioning X

Stiffness X

Workspace X











CHAPTER 2
TYPE SYNTHESIS AND INVERSE KINEMATICS
OF THE MANIPULATORS



2.1 Introduction

Industrial robots are available in a wide variety of

shapes, sizes and physical configurations. Generally, the

first three degrees of freedom (links) of the majority of

today's available robots are primarily used to achieve a

desired position for the origin of the wrist. These differ

considerably from one another and can be classified as

cartesian, cylindrical, spherical and revolute which are

shown in Fig. 2.1. The remaining degrees of freedom are

subsequently employed to achieve desired tool frame

orientations. For this purpose, almost all arrangements use

revolute pairs with their axes intersecting at a point. For

such geometries, the position of the common point of

intersection (wrist center) depends only on the first three

joint variables of the structure. Once these are computed,

the orientation of the hand can be attained by rotating the

last three joints only. However, the number of links can

be reduced by using joints with larger degrees of freedom

such as cylindric and spherical pairs.

A novel geometry of a ground-mounted two-degree-of-

freedom self-actuated joint connecting a manipulator link to


















.I







CYLINDRICAL
















REVOLUTE


CARTESIAN















SPHERICA
SPHERICAL


Figure 2.1 Four basic manipulator configurations




14

the ground is presented by Kohli, Lee, Tsai and Sandor [38].

It combines a rotary actuator and a linear actuator in such

a way that it imparts cylindrical (two-degree-of-freedom,

combined rotational and translational) relative motion to

the manipulator link with respect to ground. The rotary and

the linear actuators are independent of each other and do

not "see" each other's inertias. Based on this new

arrangement, several possible manipulator linkage

configurations with six degrees of freedom are described in

this chapter.



2.2 R-L (Rotary-Linear) Actuator

Two different configurations of R-L actuators are

shown in Figs. 2.2 and 2.3. In Fig. 2.2, C is a splined

shaft, and link E is mounted on the shaft and contains

internal splines in its hub. Therefore, link E can be

translated on shaft C and be rotated when shaft C rotates.

A is a linear actuator and is connected to bracket B which

is not splined and can freely slide on shaft C and makes the

link E slide on shaft C. The rotary actuator D rotates the

shaft C. The rotary and linear actuators thus rotate and

translate link E on shaft C without seeing each others'

inertia. The motion of link E is the same as that provided

by a cylinder pair with an axis which is the same as that of

shaft C. In such configuration, link E cannot rotate by 360

degrees due to interference between link E and bracket B.

In Fig. 2.3, the linear actuator is connected to link E

























Rotary Actuator


Splined Shaft


Linear Actuator


Figure 2.2 (R-L) actuator



























ROTARY ACTUATOR


LINEAR ACTUATOR


BEARINGS PULL AND PUSH
ARRANGEMENT


Figure 2.3 (R-L) actuator with 360 rotatability







through grooved hub B by means of pins or split ring P. In

this configuration link E can rotate a full 360 degrees.

The R-L actuator controls a rotation around and a

translation along the axis of a cylinder pair and is used

in type synthesis of parallel manipulators. The principal

advantages of using this type of actuator in the structure

of parallel manipulators are that, first, R-L actuators can

all be mounted on the ground. This reduces the necessary

load capacities of the joints which need support only the

mechanism links and the payload, whereas serial open-loop

robot manipulators must have joints that carry not only the

links and the payload, but also the actuators, their

controls and power conduits of all subsequent joints.

Secondly, with all three R-L actuators mounted on the

ground, the computations required for inverse kinematics and

thus the Jacobian matrix are significantly simplified.

In a manipulator configuration where all the actuators

could be mounted on the ground, the rotary and linear

actuators that form the R-L actuator could be off-the-shelf

items, since the power to weight ratio is not a major

concern in this situation. Thus the cost of the R-L

actuators can be considerably less than the actuators

currently being designed especially for and used in serial

link manipulators.







2.3 Type Synthesis

The earliest study of parallel manipulators is that of

Stewart's platform, as shown in Fig. 2.4, which has six

degrees of freedom. The actuators are mounted on the

floating links. Hunt [45] shows a three-degree-of-freedom

and a six-degree-of-freedom parallel manipulator, as shown

in Figs. 2.5 and 2.6 respectively, whose actuators are

mounted on the floating links and on the ground,

respectively. However, Hunt's six-degree-of-freedom

manipulator, as shown in Fig. 2.6, has additional six

redundant degrees of freedom: the axial rotation of the six

S-S links which causes uncontrolled wear in the S joints.

By using ground-mounted R-L actuators, we can reduce the

number of links of the mechanisms and still have six degrees

of freedom of the end effector, without any redundant

degrees of freedom in the mechanism.

With the R-L actuator ground-mounted, we can have

several possible configurations for each subchain of the

parallel manipulators with six degrees of freedom. These

configurations are as follows:



Dyads (R-L)-S-R (R-L)-R-S (R-L)-P-S

(R-L)-S-P (R-L)-C-C


Triads (R-L)-R-R-C (R-L)-C-R-R (R-L)-R-C-R

(R-L)-R-P-C (R-L)-R-C-P (R-L)-P-R-C

(R-L)-P-C-R (R-L)-C-R-P (R-L)-C-P-R

(R-L)-P-P-C (R-L)-P-C-P (R-L)-C-P-P





























































Figure 2.4 Stewart platform mechanism




























































Figure 2.5 Parallel platform-type manipulator with three
degrees of freedom




























































Figure 2.6 Parallel platform-type manipulator with six
degrees of freedom





22

These chains are shown in Figs. 2.7 and 2.8, where the order

of (R-L) can be reversed as (L-R).



2.4 Technical Discussion

Actuator. One of the major concerns in the design of

serial open-loop manipulators is to maximize the actuator

power/weight ratio, since some of the actuators must be

mounted on the moving links, as shown in Fig. 2.9, which

adds to the inertia of the actuators to the links' inertia

and decreases payload capacity. Therefore, the actuator

size increases from the distal joint to the proximal joint.

The manipulator becomes a massive linkage requiring bigger

actuator sizes and resulting in smaller payloads. If one-

degree-of-freedom actuators are to be used, this will

result in a five-loop linkage for a six-degree-of-freedom

robot manipulator. Further, if only one-degree-of-freedom

joints are used, the number of links in the linkage becomes

quite large. The number of links can, however, be reduced

by using joints with more than one degree of freedom, such

as cylinder and spherical pairs. The number of loops can

also be reduced, thereby reducing the number of links

further by devising and using two- or more-degree-of-

freedom self-actuated joints. A six-degree-of-freedom

parallel manipulator, where all actuators are ground-

mounted, is shown in Fig. 2.10.

Computation. The computation of inverse kinematics and

dynamics requires considerable time for serial-link

















01II


(R-L)-S-R


(R-L)-R-S


(R-L)-P-S


(R-L)-S-P


(R-L)-C-C


Figure 2.7 Possible configurations of dyads with six
degrees of freedom










I6
____ _









E^-^^r


(R-L)-R-R-C

(R-L)-C-R-R

(R-L)-R-C-R


(R-L)-R-P-C


(R-L)-R-C-P


(R-L)-P-R-C


Figure 2.8 Possible configurations of triads with six
degrees of freedom




25
(continued)


~c~Fi


(R-L)-P-C-R


(R-L)-C-R-P


(R-L)-C-P-R


(R-L)-P-P-C


(R-L)-P-C-P















Wrist-Swing
Actuator Hand-Twist
Actuator

Wrist-Bend
Actuator
Hand

Elbow
Actuator
0
Shoulder-
Bend
Actuator 0

Fixed
Base
Bae Shoulder-
ATwist
Actuator


Figure 2.9 Actuators in a serial manipulator





























Dyad 2






nk 4



Fixed
Base
Link 1


Figure 2.10 Actuators in a parallel manipulator






manipulators. Generally, computations of one link depend

upon other links. These computations must be done

serially, thus making parallel processing difficult and

ineffective in reducing computation time. With parallel-

type configuration in the manipulators, the computation can

be performed in parallel. Therefore, the computation time

will be significantly reduced. In general, the computations

required for inverse kinematics and Jacobian matrices will

be less complicated than those of serial open-loop

manipulators, but the computations for direct kinematics are

much involved.

Based on this new possibility, we describe possible

manipulator configuration linkages with six degrees of

freedom. Then we identify possible configurations in which

all actuators for actuating the manipulator linkage are

ground-mounted. The distinct advantage of being able to

put many actuators on the ground makes these manipulator

topologies appealing.

Degrees of freedom. In general, the mobility of a

kinematic chain can be obtained from the Kutzbach criterion.

The six-dimensional form of the criterion is given as



m = 6(n 1) 5j1 4j2 3j3 2j4 J5 (2.1)



where m = mobility of mechanism,

n = number of links,

ji = number of joints having i degrees of freedom.







Freudenstein and Maki [50] also show that a general

form of the degree-of-freedom equation for both planar and

spatial mechanisms can be written as


J
F = d(n j 1) + Efi Id (2.2)
i


where F = the effective degree of freedom of the assembly

or mechanism,

d = the degree of freedom of the space in which the

mechanism operates (for spatial motion d = 6, and

for planar motion and motion on a surface d = 3),

n = number of links,

j = number of joints,

fi = degree of freedom of i-th joint,

Id = idle or passive degrees of freedom.


The number of degrees of freedom that a manipulator

possesses is the number of independent position variables

which would have to be specified in order to locate all

parts of the mechanism. In the case of serial manipulators,

each joint displacement is usually defined with a single

variable; the number of joints equals the number of degrees

of freedom.

The number of degrees of freedom of multi-loop

manipulator linkages containing multi-degree-of-freedom

self-actuated joints can be determined simply by the

following equation:






n
Fc = E Fi 6(n 1) (2.3)
i=l


where Fc = the number of degrees of freedom of the multi-

loop mechanism,

Fi = the number of degrees of freedom of the i-th

subchain (leg),

n = the number of subchains (legs)



As shown in Fig. 2.10, there are three identical

subchains and each subchain has six degrees of freedom.

Therefore, the number of degrees of freedom of this type of

parallel manipulator can be calculated from Eq. (2.3) as


Fc = (6 + 6 + 6) 6(3 1) = 6





2.5 Inverse Kinematics

When the position of one link, generally the hand, is

specified and it is required to determine the position of

all other links, including the joint variables of actuated

joints which will move the hand to the specified position,

the method is called inverse kinematics. The determination

of the actuated joint variables for a specified position of

the hand is conducted by obtaining a set of equations

relating the actuated joint variables and constant

parameters of the manipulator linkages to the hand position

variables. In general, this set of equations is also





31

reduced to one equation of polynomial form in only one joint

variable. For a specified hand position, one proceeds to

find the roots of this displacement polynomial to determine

the joint variable. The degree of this polynomial also

determines the number of possible ways the desired hand

position can be reached.

Generally, the methods employed in solving the inverse

kinematics in robotics are either analytical or numerical.

An analytical solution is one that produces a particular

mathematical equation or formula for each joint variable

(rotation or translation) in terms of known configuration

values (length of the link, twist angle and offset), whereas

a numerical solution generally pertains to the determination

of appropriate joint displacements as the result of an

iterative computational procedure. It is noted that the

equations associated with the inverse kinematic problem are

nonlinear and coupled, and this nonlinear dependence is

basically trigonometric.

As shown in Figs. 2.7 and 2.8, there are five possible

dyads with six degrees of freedom and twelve possible triads

with six degrees of freedom. In order to reduce the number

of links in forming the mechanism and avoid the number of

translational joints greater than three in a loop, we only

consider subchains (R-L)-R-S, (R-L)-P-S, (R-L)-S-R and (R-

L)-S-P as shown in Fig. 2.7 in the following sections.






2.5.1 Subchain (R-L)-R-S

A schematic diagram of subchain (R-L)-R-S is shown in

Fig. 2.11. Since the position and orientation of the hand,

which is embedded in the platform, is known, we can find the

position and orientation of the coordinate system Cx3Y3z3

embedded in the sphere at point C with respect to the local

fixed coordinate system ox0y0z0 through systems Hx4y4z4 and

OXYZ by coordinate transformations. Also, we can write the

following equation to describe the position of point C with

respect to the local fixed coordinate system oxOy0z0 through

systems Bx2Y2z2 and Axlylzl as follows:

Co = AlA2C3 (2.4)

or

C, Cea -SOa 0 0 1 0 0 a

Cy SE, C8a 0 0 0 Cab -Sab 0

Cz 0 0 1 da 0 Sab Cab 0
1 0 0 0 1 0 0 0 1


C8b -Sb 0 bCEb 0

S6b Ceb 0 bSeb 0

0 0 1 sb 0

0 0 0 1 1
(continued)































1 x






0
0


X0



da


Figure 2.11 Subchain (R-L)-R-S








Cea(bC8b + a) sea(bsebCab sbSab)

sea(bCBb + a) + C8a(bSbbCab sbSab)
bSObSab + sbCab + da

1
(2.5)*

where the vector CO or its components CR (k= x, y and z)

denote the coordinates of point C with respect to the local

fixed coordinate system ox0Y0z0; the vector C3 denotes the

location of point C with respect to the coordinate system

Cx3Y3Z3; ea and eb are the rotation angles from xo to xl and

from x2 to x3, respectively; da is the translation of

cylindric joint A along the fixed axis z1 form the origin of

the local fixed coordinate system ox0YOz0; a and b are the

perpendicular distance between successive joint axes zl, z2

and z3, respectively; sb is the offset along the z2 axis; ab

is the twist angle between the axes zI and z2, and Ai, i = 1

and 2, represent the Hartenberg and Denavit [39] 4 x 4

homogeneous transformation matrices which relate the

kinematic properties of link i to link i-i and can be

derived as


COi -SeiCai SeiSai aiC8i

SEi CeiCai -ceiSai aiSEi
Ai = (2.6)
0 Sai Cai di

0 0 0 1



C can also be obtained by using the method in [44],
which is presented in Appendix A.1.







where CS and SG are shorthand for cos(8) and sin(8),

respectively. Similarly, Sc denotes the relative rotation

angle of joint C, db and dc denote the translations of

joints B and C along the moving axes z2 and z3,

respectively, and c denotes the length of link c. It is

noted that db, d, and c are zero for this subchain, but all

these notations are used throughout the following sections.

Premultiplying both sides of Eq. (2.4) by A2-1A1-1

yields


bCOb = CxC8, + CyS8a a (2.7)

bS8b = -CxSSaCab + CyCSaCab + CzSab daSab (2.8)

sb = CxS9SaSb CyCGaSab + CzCab daCab (2.9)


Squaring and adding Eqs. (2.7), (2.8) and (2.9) yields


b2 Sb2 = Cx2 + Cy2 + C 2 + da2 2Czda -

2aCxC8a 2aCySa + a2 (2.10)


From Eq. (2.9), we have


1
da = -- (CxSeSab CyCeaSab + CzCab sb)
Cab

Substituting da into Eq. (2.10) yields


E1X4 + E2X3 + E3X2 + E4X + E5 = 0 (2.11)

where

X = tan(9a/2)

D1 = CzCab sb

D2 = C2 y C 2 + C 2 + a2 b2 Sb2





36

Ei = D2C2b + 2aCxC2cb 2CyCzSabCab 2DiCzCab +

Cy2S2,b + D12 + 2D1CySab

E2 = -4(aCyC2ab + CxCzSabCab D1CxSab CxCyS2,b)

E3 = 2(D2C2ab 2D1CzCab + 2Cx2S2ab Cy2S2cb + D12)

E4 = -4(aCyC2ab + CxCzSabCab D1CxSab + CxCyS2ab)

E5 = D2C2ab 2aCxC2ab + 2CyCzSabCab 2DlCzCab +

Cy2S2ab + D12 2DiCySab


Since Eq. (2.11) is a fourth-degree polynomial

equation, there are up to four possible solutions for

variable X (or 8a). Back substituting 8a into Eqs. (2.7)

and (2.9), we can obtain up to four possible sets of

solutions of 9b and da, respectively. Therefore, the

subchain (R-L)-R-S has a fourth-degree polynomial equation

in inverse kinematics.

Numerical example. The given parameters are as

follows:

a = 2", b = 12", sb = 8", ab = 720 and C = [-4.86,

-11.60, 3.97]T


The four possible solutions are computed as


Solutions 8a da 9b
(deg.) (in.) (deg.)

1 29.932 1.557 180.299

2 -50.609 12.297 -71.132

3 17.534 7.618 212.427

4 -87.785 -5.592 38.407







2.5.2 Subchain (R-L)-P-S

Figure 2.12 shows a subchain (R-L)-P-S. The inverse

kinematics is similar to that of the subchain (R-L)-R-S.

can write the equation to express the location vector of

point C with respect to the local fixed coordinate system

ox0Y0Z0 as follows:


C = A1A2C3


Cx

Cy

C

1


0 E

CLb
:ab
0
o


i r
0

o

da

1






i t


COa(a + b)

SBa(a + b)
da +


+ seaSabdb

- CGaSabdb

Sabdb


(2.12)


(2.13)**


Premultiplying both sides of Eq. (2.12) by A2-1Al-1

yields

b = CxCea + Cysea a (2.14)

** CO can also be obtained by using the method in [44],
which is presented in Appendix A.2.





























zA x





0


YoY
I- 0 1Y


Figure 2.12 Subchain (R-L)-P-S







0 = -CxSeaCab + CyCGaCab + CzSab daSab (2.15)

db = CxSeSaSb CyCeaSab + CCab daCab (2.16)


Let tan(8g/2) = X, and then substituting CGa = (1 X2)/(1 +

X2) and Se, = 2X/(1 + X2) into Eq. (2.14), we obtain


(a + b + Cx)X2 2CyX + a + b Cx = 0 (2.17)


There are up to two possible solutions of X in Eq. (2.17),

or up to two possible solutions for 8a. Back substituting

these two possible solutions of 8a into Eqs. (2.15) and

(2.16), we will have up to two possible solutions of da and

db from each equation. Thus it is seen that this subchain

has a second-degree polynomial equation in inverse

kinematics.

Numerical example. The given parameters are as

follows:

a = 3", b = 2", ab = 600 and CO = [5.85, -0.13, 4.25]T


The two possible solutions are computed as


Solutions Sa da db
(deg.) (in.) (in.)

1 30.024 2.495 3.510

2 -32.570 6.005 -3.510




2.5.3 Subchain (R-L)-S-R

In Fig. 2.13, a schematic diagram of the subchain

(R-L)-S-R is shown. The spherical pair is kinematically






























X





0


Figure 2.13 Subchain (R-L)-S-R







equivalent to three revolute joints with three mutually

perpendicular concurrent axes. Since the orientation and

position of the hand, H, is given, we thus can obtain the

following equations with the assumption that z3 is parallel

to z4:


nx Sx ax PX

H n s ay Py
nz Sz az Pz

0 0 0 1


C8a -SOa 0 aC@a Cebl 0 S8bl 0

Sea C8a 0 aSea Sebl 0 -C8bl 0

0 0 1 da 0 1 0 0

0 0 0 1 0 0 0 1


C8b2 0 S8b2 0 Ceb3 -SOb3 0 bceb3

S8b2 0 -C8b2 0 Sb3 -C8b3 0 bSeb3
0 1 0 0 0 0 1 0

0 0 0 1 0 0 0 1


Cac -Sec 0 cC8c

SBe cec 0 cSec
(2.18)
0 0 1 sc

0 0 0 1
J

where components of the position (Px, Py and pz) and

orientation (nx, ny, nz, sx, Sy, Sz, ax, ay and az) of

system Hx4Y4z4 with respect to the local fixed coordinate

system oxOy0z0 are all specified; and 8bl, Ob2 and 8b3 are







the three rotational variables of the spherical joint B,

which play no part in the manipulation of the platform.

Equation (2.18) can be rewritten as


HO = A1A2A3

where

C8a


A1 = Se
0
SO



S8bl

A2 = 0
0




C(b3

Seb3
A3 =
0

b3
S8b3
0

0


(2.19)


S 0

a 0

1

0


SObl

-CObl

0

0


-Seb3

-Cab3

0

0


aCOa
ac(a

aS8a

da

1

-Ir
0 (

0

0

1


bCeb3

bSSb3

0

1


SEb2
-CGb2

0

0


-SOc

C8c

0

0


cC9c

cSOc

sc

1


Postmultiplying both sides of Eq. (2.19) by A3-1A2-1

yields Eq. (2.20). Premultiplying both sides of Eq. (2.19)

by A1-I and then postmultiplying both sides by A3-1 yields

Eq. (2.21).


H0 -1^
HOA3-A2-1

A1-1HOA3-1


(2.20)

(2.21)






Since ([A1](1,4))2 + ([A1](2,4))2 = ([HOA3-1A2-1](1,4 )2 +

([HOA3-1A2-1](2,4))2 is true, we can obtain the following

equation:


a2 = [-nx(c + bCSc) + bSxSec axSc + px]2 +

[-ny(c + bCSc) + bSyS8c aySc + py]2 (2.22)


Equation for ea is obtained since [A1](2,4) / [All(1,4)

[HOA3-1A2-1](2,4) / [HOA3-1A2-11(1,4) holds, and can be

expressed as


a= tan-l( -ny(c + bCec) + sybS9c aySc + py (2.23)
8, = tan- --- ) (2.23)
-nx(c + bcec) + sxbSSc axsc + Px


It is observed that [Al](3,4) = [HOA3-1A2-1](3,4) directly

implies the translation of joint A as


da = -nz(c + bCec) + szbSec azsc + Pz (2.24)


Let tan(e)/2) = X; then substituting C8c = (1 X2)/(1 + X2)

and SEc = 2X/(1 + X2) into Eqs. (2.22) yields


E1X4 + E2X3 + E3X2 + E4X + E5 = 0 (2.25)
where

D1 = (nx2 + ny2)b2

D2 = (sx2 + Sy2)b2
D3 = 2b(nx2c + ny2c + nxaxsc + nyaysc Pxnx Pyny)

D4 = -2b(nxSxC + nysyc + sxaxsc + yaysc sxpx -sypy)







D5 = -2b2(nx s + nysy)

D6 = nx22 + ny2c2 + aS2 ay2c2 + px2 + py2 +

2nxaxscc + 2nyay sc 2pxnxc 2pynyc 2axscpx -

2ayScPy a2

E1 = D1 D3 + D6

E2 = 2(D4 D5)

E3 = -2(D1 -2D2 Dg)

E4 = 2(D4 + D5)

E5 = D1 + D3 + Dg


It is seen that there is a maximum of four solutions of

X (or Bc) in Eq. (2.25). Back substituting the values of Sc

into Eq. (2.23) and (2.24) yields up to four sets of

solutions for ea and da-

From Eq. (2.21), we can find certain relationships by

equating corresponding elements of the two matrices on

either side of the equation. Thus we obtain the following

equations:


-axS8a + ayCea
8bI = tan-l( ----a-Sa + (2.26)
axC8a + aySea


eb2 = tan-l( aC + ) (2.27)
-azCbbl


nzSOc + SzC~c
9b3 = tan-1( n--Sc + ) (2.28)
szS9c + n+zCc







Since the subchain of (R-L)-S-R has a fourth-degree

polynomial equation in inverse kinematics, there are up to

four possible solutions in Eq. (2.25). However, the

possible solutions may be reduced due to special dimensions

of the subchain as shown in the following example.

Numerical example. The given parameters are as

follows:

a = 5", b = 3", ac = 0, c = 0.75", sc = 1.50" and


0.9300 -0.3323 -0.1574 8.3382

-0.3466 -0.9352 -0.0734 0.2201
HO =
-0.1228 0.1228 -0.9848 -1.5205

0 0 0 1
L


Since two of the solutions of Eq. (2.25) are complex

numbers, the remaining two possible real solutions are

computed as




Solutions c ea da 8bl 6b2 8b3
(deg.) (deg.) (in.) (deg.) (deg.) (deg.)

1 33.435 -1.008 0.559 26.009 -10.002 11.565

2 14.988 10.009 0.500 14.992 -10.002 30.012




2.5.4 Subchain (R-L)-S-P

The procedure of inverse kinematics of the subchain (R-

L)-S-P, as shown in Fig. 2.14, is similar to that of the






























z

0
0


Figure 2.14 Subchain (R-L)-S-P







subchain (R-L)-S-R in

following equation:


HO = A1A2A3

where


n"

"ny
HO =
nz

0


Sea

S9a
Al= :
0




C8bl

SEbl
A2 =
0

0


CEb3

S8b3
A3
0

0


2.5.3. Therefore, we can obtain the




(2.29)



ax Px

ay Py
az Pz
a py


0 1


0 aCQa

0 aSEa

1 da

0 1


Sbl 0 C8b2 0 58b2 0

CEbl 0 S8b2 0 -COb2 0

0 0 0 1 0 0

0 1 0 0 0 1
J


-SOb3

-COb3

0

0


bCOb3

bS9b3

0

1


where components of the position (Px, Py and pz) and

orientation (nx, ny, nz, sx, Sy, s,, ax, ay and a,) of

system Hx3Y3z3 with respect to the local fixed coordinate

system ox0YOz0 are all specified; and ebl, 9b2 and 9b3 are


-






the three rotational variables of the spherical joint B,

which play no part in the manipulation of the platform.

Postmultiplying both sides of Eq. (2.29) by A3-1A2-1

yields Eq. (2.30). Premultiplying both sides of Eq. (2.29)

by Al-1 and then postmultiplying both sides by A3-1 yields

Eq. (2.31).

Al = H0A3-1A2-1 (2.30)

A2 = A1-1HA3-1 (2.31)

Since ([Al](1,4))2 + ([A1](2,4) 2 = ([HOA3-1A2-11(1,4) 2 +

([HOA3-1A2-1](2,4))2 is true, we can obtain the following

equation:

(ax2 + ay2)dc2 + 2[(c +b)nxax + (c + b)nyay axpx

-aypy]dc + (c + b)2(nx2 + ny2) + px2 + py2 -
2(c + b)(nxPx + nypy) a2 = 0 (2.32)


Equation for 8, is obtained since [A1](2,4) / [A](1,4) =

[HOA3-1A2-1](2,4) / [HOA3-1A2-1]1,4) holds, and can be

expressed as


Sta -([ + b)ny ayd + y (2.33)
9a = tan-[ ] (2.33)
-(c + b)nx axdc + px


It is observed that [Al](3,4) = [HOA3-1A2-1](3,4) directly

implies the translation of joint A as


da = -(c + b)nz azdc + pz


(2.34)







From Eq. (2.31), we can find certain relationships by

equating corresponding elements of the two matrices on

either side of the equation. Thus we obtain the following

equations:


-axS8a + ayCOa
9bl = tan-1( -axSa + ayCS) (2.35)
axCea + aySqB

axC8a + aySea

Eb2 = tan-1( t C + ) (2.36)
-azCbl


sz
8b3 = tan-(--- ) (2.37)
-n"



There are two possible solutions for dc in Eq. (2.32),

since it is a quadratic polynomial equation. Back

substituting these solutions of dc into Eqs. (2.33) -

(2.37), respectively, we can obtain two possible sets of

solutions for 6a, da, 9bl, Bb2 and 8b3 from each equation.

Therefore, the subchain (R-L)-S-P has a second-degree

polynomial equation in inverse kinematics.

Numerical example. The given parameters are as

follows:

a = 3", b = 2", ac = 0, c = 0.25" and


0.7259 0.5900 0.3536 5.1151

0.4803 -0.8027 0.3536 3.4645
HO =
0.4924 -0.0868 -0.8660 0.4428

0 0 0 1







The two possible solutions are computed as


Solutions dc 8a da ebl 8b2 eb3
(in.) (deg.) (in.) (deg.) (deg.) (deg.)

1 2.499 30.001 1.499 14.999 30.004 9.997

2 14.089 239.999 11.536 -14.999 -30.004 189.997




2.6 Summary

The equations of inverse kinematics have been derived

for the dyads (R-L)-R-S, (R-L)-P-S, (R-L)-S-R and (R-L)-S-P

as shown in Fig. 2.7, each of which has six degrees of

freedom. Numerical examples have been shown to illustrate

the possible solutions for each subchain. The degrees of

polynomial equations in inverse kinematics for the dyads

with six degrees of freedom are summarized in Table 2.1.



Table 2.1 Degree of polynomial equation in inverse


Type of dyads Degree of polynomial equation

(R-L)-R-S 4

(R-L)-P-S 2

(R-L)-S-R 4

(R-L)-S-P 2





The detailed analysis and polynomial equation occurring

in inverse kinematics can be obtained using several methods.

The reader is advised to see the literature on kinematic


kinematics







analysis of spatial mechanisms, particularly the work by

Duffy [44].

Hunt [45] pointed out that a serially-actuated arm

accumulates errors from the shoulder out to the end-

effector; also, such arms often suffer from lack of rigidity

and, in the absence of sophisticated techniques of computer-

control compensation, are subject to load-dependent error.

They also suffer relatively low-frequency oscillations.

With in-parallel-actuation by ground-mounted actuators,

there are the advantages of both greater rigidity and

lightness of the linkage, but at the expense of more limited

workspace and dexterity. Since actuator-error is not

cumulative, greater precision is likely to be attainable

without excessive control complications. There should

surely be a future for in-parallel-actuation by ground-

mounted actuators in robotic devices.













CHAPTER 3
WORKSPACE ANALYSIS OF THE MANIPULATORS



3.1 Introduction

This chapter deals with the workspace of a parallel

manipulator having three rotary-linear (R-L) actuators on

grounded cylindric joints, three revolute and three

spherical pairs as shown in Fig. 3.1. The workspace is

defined as the reachable region of the origin of the moving

coordinate system embedded in the six-degree-of-freedom

platform of the manipulator. Since the mechanism consists

of three subchains, the workspace is the common reachable

region of three subworkspaces determined by the

corresponding subchains. The subworkspace described in this

chapter is defined as the workspace of the center of the

platform determined by a subchain regardless of the

constraints imposed by the other subchains. The dimensions

of the platform are considered to be infinitesimal and

therefore the workspace is determined without considering

the orientation of the platform in this chapter1. When the

R joint rotates about the C joint without translation and

the spherical joint rotates about R joint, the locus of the

spherical joint at the end of a dyadic subchain is the


1Workspaces with finite-size platform are derived in
Chapter 4.















































R-L Actuator


Figure 3.1 Six-degree-of-freedom closed-loop manipulator.
C: Cylindric joint, R: Revolute joint, S:
Spherical joint.







surface of a torus. The subworkspace of each open subchain

is the volume swept by this torus translated along the axis

of each ground-mounted (R-L) joint. In this chapter, the

shapes of the above-described torus of the subchain are

studied for different dimensions of the links. The

conditions on the dimensions of the links, for which the

subworkspace has no hole, are presented. Of course, an

infinitesimally small platform is not practical, because the

three spherical pairs supporting the platform coincide.

Therefore the platform has no controllability of its

orientation. To have controlled orientation, the platform

requires three controllable rotational degrees of freedom

with concurrent non-coplanar axes. This is attained by

placing the three spherical joints at finite distances from

one another. Nevertheless, the workspace study with

infinitesimally small platform is a useful step toward more

practical workspace studies with finite-size platforms

having controllable orientation, which will be covered in

Chapter 4.

One basic need in the design of mechanical manipulators

is to determine the shape of the workspace. Workspace

analysis of mechanical manipulators has been investigated by

many authors. Almost all the studies are related to

open-loop multi-degree-of-freedom serial-link mechanical

manipulators. Little work has been done in the area of

mechanical manipulators with parallel kinematic chains.







Therefore, theories for the workspace of such parallel

mechanical manipulators are needed.



3.2 Configuration of a Parallel Manipulator
with R-L Actuators

A six-degree-of-freedom parallel manipulator, where

all actuators are ground-mounted, is considered in this

chapter. It has three six-degree-of-freedom subchains, each

of which has a two-degree-of-freedom R-L actuator, which

controls both the rotation and the translation of a ground-

supported cylindrical joint. For reducing the number of

links in the subchains, spherical joints, which are three-

degree-of-freedom kinematic pairs, are used in the

subchains. A Cylindric-Revolute-Spherical ((R-L)-R-S) triad

may then be used as the subchain with the C joint (R-L

actuator) connected to the ground, and the S joint connected

to the end-effector platform of the manipulator. The axes

of the C joints (R-L joints) on the frame may be arranged in

several different configurations, such as star form,

triangle form, or parallel to each other, as shown in Fig.

3.2. Furthermore, they need not be coplanar, even if they

are not parallel to one other.

The lengths from the center of the platform, H, to the

centers of the spherical joints, C1, C2 and C3, affect the

volume of the workspace, and more significantly, the

rotatability of the platform about the center H. Thus, to

get a workspace of the manipulator with rotatability of the

end effector as large as possible, the lengths between H and




























































Figure 3.2 Three possible configurations of the
arrangements of the R-L actuators on the base.
Also, the R-L actuators need not be either
parallel or coplanar.







Ci (i = 1, 2 and 3) should be as small as possible, which is

consistent with controllability of end-effector orientation.

When the lengths are infinitesimal or, in the limit, zero,

the largest possible workspace with complete, but

uncontrollable rotatability of the platform results. In

this chapter, the equations of the workspace of the

manipulator with infinitesimal dimensions of the platform

are derived, i.e., with joints Cl, C2 and C3 infinitesimally

close to each other. In chapter 4, the workspace of a

similar parallel manipulator with a finite size platform

will be determined. Of course, controllability of the

orientation of the platform is reduced sharply as the

spherical joints approach one another. Therefore it must be

realized that there must be a practical trade-off between

the distances of the spherical joints from one another and

the controllability of platform orientation.



3.3 The Subworkspace Analysis of the Manipulator

The toroidal surface (torus) is the locus of a point

attached to a body that is jointed back to the reference

system through a dyad of two serially connected revolute

pairs. A general R-R dyad with a point C which traces the

surface of a general form of torus is shown in Fig. 3.3. It

is similar to the subchain (R-L)-R-S when the platform is

assumed to be infinitesimal and without the consideration of

translation along the axis of the cylindric (R-L) joint.

The shapes of the torus are illustrated first, then the



























































Figure 3.3 An R-R (Revolute Revolute) dyad with a point C
tracing a general form of torus. Note that a is
the common perpendicular of the axes of the
revolutes A and B.







equations of the boundaries of the subworkspace, which is

the volume swept by this torus as the R-L joint translates,

will be derived.



3.3.1 Shapes of the subworksoace

Fichter and Hunt [51] have geometrically described and

analyzed four forms of the torus, which are common,

flattened, symmetrical-offset and general forms as shown in

Fig. 3.4. They also introduced two types of bitangent-

plane2, A-type, whose quartic curve intersection with the

torus always encircles the OZ axis and B-type, whose points

of tangency are both on one side of the OZ axis. Any

bitangent-plane to any form of torus cuts the torus in two

circles of the same radius which intersect one another at

the two points of tangency. The curve of intersection of

the bitangent-plane and the torus can be obtained by the

simultaneous solution of the equations of the bitangent-

plane and the torus. The curve of intersection of A-type

bitangent-plane and a common torus (a > b) is shown in Fig.

3.5.

The equation of the surface of a completely general

form of torus can be expressed as follow:









2 A bitangent-plane has two points of tangency with a
toroidal surface.














AZ Edge view of A-type
bitangent-plane





I-.-




----~---^-D-0


Sa) Common torus


(b) Flattened torus


Figure 3.4 Diametral sections through tori: (a) common,
(b) flattened, (c) symmetrical-offset and (d)
general types.





61

(continued)


Edge view of B-type
bitangent-plane













) Symmetrical-offset torus

(c) Symmetrical-offset torus


Edge view of B-type
bitangent-plane


/ Edge view of A-type
bitangent-plane


Edge view of A-type
bitangent-plane


(d) General torus



















,urve of .ntersection
(common torus cna A- ype p:ane)
6









2-













-4
-6








6 -1 -2

X-axis, cnes


Figure 3.5 Intersection of the A-type bitangent-plane and
the right circular torus (a = 3", b= 2")







{(x2 + y2 + z2) (a2 + b2 + s2)}2

z scosa
= 4a2{b2 ()2} (3.1)3
sina



Common form. The common form of torus (right

circular), sometimes called the anchor-ring, is shown in

diametral section in Fig. 3.4(a). The axes of the two

revolute pairs are at right angle (a = 900). Their common

perpendicular is a and the offset between them is zero (s =

0) (see Fig. 3.3). The equation of the torus can be

expressed as


{(x2 + y2 + z2) (a2 + b2)}2 = 4a2(b2 z2) (3.2)



The difference between the lengths of the links affects the

shape of the torus, which is illustrated in Fig. 3.6. As a

> b shown in Fig. 3.6(a), the two circles in the diametral

section are separated by a distance of 2(a b). This kind

of torus is also shown in Fig. 3.7. When the two circles in

the diametral section are tangent at a point, the origin 0,

then a = b as shown in Fig. 3.6(b). The torus will

intersect itself when a < b as shown in Fig. 3.6(c). There

is a void in this kind of torus when a < b exists.

Flattened form. The flattened form of torus has no

offset (s = 0) either, but the axes of the two revolute




3 The equation of the general form of torus is derived
in Appendix B.






































Z








0


(C)
a -b



Figure 3.6 Diametral section of the common form of torus:
(a) a > b, (b) a = b and (c) a < b


I









-F


Figure 3.7 A right circular torus, a > b.







pairs are not at right angle (a 0 90). The equation of

this form of torus can be expressed as

z2
{(x2 y2 + z2) (a2 + b2))2 = 4a2(b2 __-
sin2a
(3.3)


The diametral plane cuts the torus in egg-shaped curves as

shown in Fig. 3.4(b). For different twist angles (300, 45

and 750) and dimensions of the links (a > b, a = b, and a <

b) with each specified twist angle, the shapes of the torus

are shown in Figs. 3.8 3.10, respectively. It is noticed

that the diametral sections of these tori are similar to

those of the common form when a > b and a = b, but flattened

and egg-shaped. But the tori in Figs. 3.8 3.10 do not

intersect themselves when a < b, which differs from the

schematic shown in Fig. 3.6(c).

Symmetrical-offset form. The symmetrical-offset form

has the axes of the two revolute pairs at right angle (a =

900) and with offset (s # 0). The equation of this form of

torus can be expressed as



((x2 + y2 + z2) (a2 + b2 + s2)}2 = 4a2(b2 z2)
(3.4)


The diametral section of this kind of torus is shown in Fig.

3.4(c). For a > b and small s, the shape of the torus as

shown in Fig. 3.11(a) is slightly different from that in

Fig. 3.6(a). As shown in Fig. 3.11(b), the inner walls of

the anchor-ring become flatter when a = b. The two closed















0


(a)







Z





(b)

















(e)


Figure 3.8 Diametral section of the flattened form of torus
(a = 300): (a) a > b, (b) a = b and (c) a < b








z

00


(C)


/1


Figure 3.9 Diametral section of the flattened form of torus
(a = 450): (a) a > b, (b) a = b and (c) a < b


r )


l l l l


i`:

















(a)














(b)










0








(c)



Figure 3.10 Diametral section of the flattened form of
torus (a = 750): (a) a > b, (b) a = b and
(c) a < b
















z


z


Figure 3.11 Diametral section of the symmetrical-offset
form of torus: (a) a > b, (b) a = b and
(c) a < b







curves in the diametral section become banana-shaped, as

shown in Fig. 3.11(c), when a < b.

General Form. The diametral section of the torus is

shown in Fig. 3.4(d). For different twist angles (30, 45

and 750) and dimensions of the links (a > b, a = b, and a <

b) with each specified twist angle, the shapes of the torus

are shown in Figs. 3.12 3.14, respectively. The closed

curves in the diametral section, while still more or less

banana-shaped, but now they are tilted over.

Equations (3.1) (3.4) are all of the fourth degree

and all forms of torus are thus quartic surfaces. The curve

of intersection between a torus and a general plane is a

quartic; also, in general, a straight line cuts any torus in

four points (real, imaginary, or coincident).

The volume and shape of the workspace are very

important for applications since they determine capabilities

of the robot. In order to obtain the optimum workspace, the

volume of the subworkspace of the corresponding subchain

generally should be as large as possible. Since most of

today's available industry robots have 0 or 900 twist

angles, we will discuss the following two cases with the

conditions of s = 0 and twist angle a = 0 or +900 applied,

respectively.



Case 1: s = 0 and a = 0 (or n)


Since the axes of the two revolute pairs are parallel

and there is no offset, the toroidal surface degenerates






















(a) "














(b)


Z


Figure 3.12 Diametral section of the general form of torus
(a = 300): (a) a > b, (b) a = b and (c) a < b


L









z


0


Figure 3.13 Diametral section of the general form of torus
(a = 45): (a) a > b, (b) a = b and (c) a < b


S:)-









Z






(a)


Z







b)



(b)


Figure 3.14 Diametral section of the general form of torus
(a = 75'): (a) a > b, (b) a = b and (c) a < b







into a plane. For different dimensions of the links a and

b, the shapes of the now planar toroidal surface are

illustrated in Fig. 3.15. The workspace resulting when this

surface is translated along the Z axis has no hole only when

a = b as shown in Fig. 3.15(b).

The workspace of this kind of subchain is generally the

volume between two coaxial cylinders when the translation

along the Z axis is in effect. Due to the limitation of the

rotation of the first revolute joint which is ground-

mounted, the workspace of this subchain is actually reduced

to the upper half (X 2 0) of the volume between the two

coaxial cylinders. Whenever the condition a = b exists, the

inner boundary disappears and there is no hole in the

workspace.



Case 2: s = 0 and a = +900


Since the two axes of the two revolute pairs are

perpendicular to each other and there is no offset, the

locus of point C is a torus, which is defined as the common

form of torus shown in Fig. 3.6. When the translation along

the Z axis is in effect, the workspace of the subchain can

be described as shown in Figs. 3.16 3.18, respectively.

In Fig. 3.16(a), the torus has a hole because of a > b.

When the translation d along the Z axis is in effect, we

obtain the workspace as the volume between the two coaxial

cylinders with radii of (a + b) and (a b), respectively

and height d, plus the volume of a half torus at each end.























a + b


Cross sections of the Workspace generated by
the planar R-R dyad: (a) a > b, (b) a = b and
(c) a < b (infinitesimal platform)


Figure 3.15


































VOID


(b) d 'VOID








Figure 3.16 Workspace with the common form of torus
(a > b); Along the Z axis: (a) d 2 2b and (b)
0 < d < 2b (infinitesimal platform)


































VOID



z





(b) -d VOID










Figure 3.17 Workspace with the common form of torus
(a = b); Along the Z axis: (a) d 2b and (b)
0 < d < 2b (infinitesimal platform)

































VOID




Z





VOID

(b)






Figure 3.18 Workspace with the common form of torus
(a < b); Along the Z axis: (a) d > 2b and (b)
0 < d < 2b (infinitesimal platform)





80

However, this is true only when the translation along the Z

axis, d, is greater than or equal to 2b. Otherwise, there

are additional voids that can be found as shaded areas

shown in Fig. 3.16(b). When the translation along the Z

axis is less than 2b, there is a void inside the workspace,

shown with a lentil-shaped cross section in Fig. 3.16(b).

When a = b as shown in Fig. 3.6(b), there is no hole as the

translation along the Z axis is in effect and d 2b. The

workspace is the volume of the cylinder with radius of

(a i b), height d, plus the volume of this kind of half

torus at each end, as shown in Fig. 3.17(a). If the

translation along the Z axis is less than 2b, voids can be

found even if the torus has no hole at all. The lentil-

shaped cross sections of the void can be shown as shaded

areas in Fig. 3.17(b). Finally, when the torus intersect

itself as shown in Fig. 3.6(c), there is a void inside in

this torus. As the translation d along the Z axis is in

effect and is greater than or equal to 2b, we obtain the

workspace as the volume of the cylinder with radius of (a +

b) and height d, plus the volume of this kind of half torus

at each end as shown in Fig. 3.18(a). Similarly, voids can

be found if the translation along the Z axis is less than

2b, which is shown in Fig. 3.18(b).



3.3.2 Boundaries of the subworkspace and root regions
in the subworkspace (infinitesimal platform)

In order to calculate the volume of the subworkspace,

we must find the boundaries (external and internal) of the






subworkspace. From the inverse kinematics of subchain (R-

L)-R-S solved in section 2.5.1, we know it has up to four

possible solutions for a given position of the S joint.

Once the orientation and position of the hand is given, we

may have up to 64 solutions for the manipulator. Therefore,

the study of root regions in the subworkspace is also

important.

A manipulator with R-L actuators in the subchains whose

axes are arranged in triangle form on the base can be

represented as shown in Fig. 3.19. The notation is as

follows:



OXYZ global fixed coordinate system.

i i-th subchain.

Ai, Bi R-L actuator joint and revolute joint in the

i-th subchain.

oixiYiZi local fixed coordinate systems with the zi

axis along the axis of the cylindric joint

Ai.

AixaiYaizai, BixbiYbizbi moving coordinate systems

embedded in joints Ai and Bi,

respectively.

8ai, 9bi relative rotation angle between successive

links.

di translation along the axis zi of cylindric

joint Ai from the origin of the local fixed

coordinate system oixiYizi.


















































z2, Za2


Figure 3.19 Notation of the manipulator with ground-mounted
R-L actuators arranged in triangle
configuration







ai, bi perpendicular distance between successive

joint axes zl, z2 and z3, respectively.

sbi, abi offset along axis zbi and twist angle between

zai and Zbi, respectively.
HCi approaching zero for infinitesimal platform.



As shown in Figs. 3.20 and 3.21 when sbi 7 0 or sbi = 0

respectively, if the revolute joint Bi makes a complete

rotation, the locus of the point Ci is a circle with respect

to the moving coordinate system BixbiYbizbi. It can also be

expressed with respect to the coordinate system AiXaiYaizai

from Eq. (2.5) as follows:



Xai = biCSbi + ai (3.5)

Yai = biSSbiCabi SbiSabi (3.6)

zai = biS9biSabi + sbiCabi (3.7)


The locus, generated by the point Ci turning around the

revolute joint Bi and the cylindric joint Ai, without

translating along the axis of the cylindric joint, can be

obtained by turning the circle, now represented by Eqs.

(3.5) and (3.6), about the zai axis. The shape of this

locus is the surface of a general form of torus as described

in section 3.3.1. The subworkspace of this subchain is the

volume swept by this toroidal surface as it translates along

the axis of the cylindric joint.


























































Figure 3.20 Subworkspace element of the i-th dyad of the
manipulator as the offset at joint Bi, sbi # 0
(infinitesimal platform)


























































Figure 3.21 Subworkspace element of the i-th dyad of the
manipulator as the offset at joint Bi, sbi = 0
(infinitesimal platform)







Since the manipulator's workspace with infinitesimal

platform is the common intersection of the three

subworkspaces of the three dyads that support the platform,

in general the ends of these subworkspaces need not be

considered. Therefore, only the projection of the

subworkspace onto the xy plane is of interest. For this

reason, henceforth r will designate only the xy projection

of the vector from the center Ai to the cylindric boundary

of the subworkspace. Accordingly:



r2 = xai + Yai2 = xi2 + i2


= (biCebi + ai)2 + (biS8biCabi sbiSabi)2 (3.8)



Subworkspace with offset sh = 04 (infinitesimal

platform). For the sake of simplicity and ease of

visualization, we will discuss the subworkspace with

infinitesimal platform and with offset sbi = 0 first. Eq.

(3.8) becomes



r2 = (biCebi + ai)2 + bi2S2ebiC2abi (3.9)



The projections on the oixiYi plane of the circles

described by Ci, which are ellipses, can be represented by

Eqs. (3.8) and (3.9) in terms of the parameter ebi. Figs.

3.20 and 3.21 show these ellipses when ai > bi. When


4See page 102 for subworkspace with offset sbi # 0.







cylindric joint Ai translates along its fixed axis zi, the

subworkspace with respect to the system AixiYizi is the

volume between two concentric cylinders without considering

the two ends of the subworkspace. The radii of the external

and the internal cylinders are respectively the maximum and

the minimum radii of the torus described by Ci without

translation. Taking the derivative of r expressed by Eq.

(3.9) with respect to 8bi and making the resulting

expression equal zero, we can obtain the maximum and minimum

radii of the torus from the following equation


dr
r ---- -biSebi(biCSbiS2abi + ai) = 0 (3.10)
d8bi

Then we obtain the following two equations



SBbi = 0 (3.11)
and
ai
Cebi = a(3.12)
biS2abi

When Sabi = 0 (abi = 0 or n) (the circles described by

Bi and Ci are coplanar), or ai > biS2obi, no real roots of

8bi can be found in Eq. (3.12). Hence we only have two

roots of 8bi from Eq. (3.11), i.e. 9 i = 0 and 8 i = n. In

this case, the values of r corresponding to 48i and 6ji are

respectively the maximum and minimum values of r.

Therefore, substituting the value of e8i and e8i into Eq.

(3.9) yields the maximum and minimum values of r as follows:









and

rmin2 = (ai bi)2 = r2i2, (3.14)


which are intuitively correct.

The relationship between r and ebi can be expressed in

a Cartesian coordinate system as shown in Fig. 3.22. It is

obvious that in the subworkspace where r2i < r (=

Sx2 + y2) < rji, for a given position of point Ci, which

implies a given value of r, there are two corresponding

solutions of 8bi. The subworkspace is a two-root region or

two-way accessible region. In other words, there are two

sets of Sai and ebi, or two kinematic branches of the i-th

dyad for reaching a given position of Ci. On the boundaries

where x + y2 = rli or r2i, there is only one solution

for 8bi. Thus the boundary surfaces are one-root regions.

(Recall that this is a planar R-R case).

In another case, when Sabi 0 (abi 0 or n) and ai

biS2abi, two additional roots of 9bi are found from Eq.

(3.12). Therefore in addition to rli and r2i there are

another two values of r, say r3i and r4i, which are also

limiting values of Eq. (3.9).

Substituting Eq. (3.12) into Eq. (3.9) yields


r3i2 = r4i2 = cot2abi(bi2S2obi ai2)


rmax2 (al + bi )2 = rli 2


(3.13)


(3.15)





















1-





0
9








3




0 Obi
0 90 180 270 360


Figure 3.22 Two solutions of 8bi corresponding to r2 < r <
rI (ai = 10, b = 5, bi = 0 and cbi = 0 or n)
(circles described by Bi and Ci without
translation of Ai are coplanar)







Thus, rli, r2i, r3i and r4i are the local minima or

local maxima of r expressed by Eq. (3.9). In order to find

the global minimum and maximum values, we take the second

derivative of r with respect to 9bi of Eq. (3.10), which

yields


d2r dr
r + (- )2 = -bi[biS2abi(2C2 bi 1) +
debi2 d8bi
aiCebi] (3.16)


At the position of local minima or maxima, the first

derivatives equal zero, and then


d2r
r
debi2 ebi



d2r
bi2
d8bi2 ebi


= -bi(biS2abi + ai) < 0




= -bi(bis2abi ai) < 0


(3.17)




(3.18)


At the position of r3 or r4,


d2r
r ----- -ai
drbi2 Gbi = arc cos ( --
biS2 bi


ai2 bi2S4abi
S2,bi


> 0 (3.19)


According to Eqs. (3.17) (3.19), rli and r2i are local

maxima, and r3i and r4i are equal local minima of r. Since







rli > r2i, the global maximum is rli, and there are two

equal global minima, namely r3i and r4i. The external and

internal cylinders of the subworkspace are then the

cylinders of radii of rli and r3i, respectively.

Figure 3.23 shows the relationship between r and 8bi in

a Cartesian coordinate system in this case. It is seen that

in the portion of the subworkspace where r2i < r < rli, for

a given position of point Ci, which implies for a given

value of r, there are two corresponding solutions of 9bi"

Thus this is the two-root region of the subworkspace. The

other portion of the subworkspace is the four-root region of

the subworkspace since four solutions of 9bi can be found in

that region for a given translation di (see Fig. 3.19) and a

given position of Ci. The cylinder of radius r = r2i inside

the subworkspace divides the subworkspace into a two-root

region and a four-root region, and the surface of the

cylinder of radius r2i itself is a three-root region. It

can also be seen that the external boundary (the surface of

the cylinder of radius r = r1i) is a one-root region, and

the internal boundary (the surface of the cylinder of radius

r = r3i) is a two-root region. Since the offset is zero and

the twist angle is not 90*, without the translation of Ai,

the locus of the positions of point Ci is the surface of a

flattened form of torus. The diametral section of this kind

of torus is shown in Fig. 3.24 and the root regions can also

be visualized easily from the figure.







































0o .. ... i 1 1 I I *I- 1 Obi
0 90 180 270 360













Figure 3.23 Two solutions of ebi corresponding to rf < r <
rI and four solutions of Obi corresponding to
r3 < r < r2 (ai = 2, bi = 10, sbi = 0 and
abi = n/3)




















Flattened Form


20



15 -




r2i


5- r3i(r4i) -----



0



-5



-io



-15



-20
-20 -10 0 10

zi-axis


Figure 3.24 Diametral section of flattened form of torus
(ai = 2, bi = 10, Sbi = 0 and abi = n/3)




Full Text
KINEMATICS OF PARALLEL MANIPULATORS
WITH GROUND-MOUNTED ACTUATORS
By
TZU-CHEN WENG
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
1988
JJ OF F LIBRARIES

UNIVERSITY OF FLORIDA
IIIIIIIIII ¡lili
3 1262 08552 4881

ACKNOWLEDGMENTS
The author wishes to express his sincere appreciation
to his committee chairman, Professor George N. Sandor, for
years of invaluable guidance, support and encouragement
during his graduate studies. The author also gratefully
acknowledges the advice and support given by the members of
his supervisory committee, Dr. Joseph Duffy, Dr. Ali Seireg,
Dr. Gary K. Matthew and Dr, Ralph G. Selfridge.
The author extends his gratitude to Dr. Dilip Kohli of
the University of Wisconsin--Milwaukee, for all his help in
the development of this work and Dr. Kenneth H. Hunt for his
advice during his visit at the University of Florida.
Special thanks are also extended to his colleagues and
fellow students, especially Mr. Yongxian Xu of the Dalian
Railway Institute, for their valuable suggestions.
The financial support of the National Science
Foundation under grant DMC-85G8029 is gratefully
acknowledged.
Most of all, the author wishes to express his sincere
appreciation to his parents for their support and
encouragement which helped him throughout his graduate
studies. Finally, the author extends his deepest
appreciation to his wife, Han-Min, for her inspiration and
moral support, and for years of patience and encouragement.
ii

TABLE OF CONTENTS
ACKNOWLEDGMENTS ii
ABSTRACT V
CHAPTERS
1 INTRODUCTION 1
1.1 Literature Overview 1
1.2 Serial and Parallel Manipulators 7
1.3 Summary 10
2 TYPE SYNTHESIS AND INVERSE KINEMATICS OF THE
MANIPULATORS 12
2.1 Introduction 12
2.2 R-L (Rotary-Linear) Actuator 14
2.3 Type Synthesis 18
2.4 Technical Discussion.. 22
2.5 Inverse Kinematics 30
2.5.1 Subchain (R-L)-R-S 32
2.5.2 Subchain (R-L)-P-S 37
2.5.3 Subchain (R-L)-S-R 39
2.5.4 Subchain (R-L)-S-P 45
2.6 Summary 50
3 WORKSPACE ANALYSIS OF THE MANIPULATOR 52
3.1 Introduction. 52
3.2 Configuration of a Paralle Manipulator
with R-L Actuators 55
3.3 The Subworkspace Analysis of the
Manipulator 57
3.3.1 Shapes of the subworkspace 59
3.3.2 Boundaries of the subworkspace
and root regions in the
subworkspace (infinitesimal
platform) 80
3.4 Conditions for No-Hole Workspace 106
3.5 Workspace of the Manipulator 110
3.6 Summary 118

4 THE WORKSPACE OF THE MANIPULATOR WITH FINITE
SIZE PLATFORM 120
4.1 Introduction 120
4.2 Workspace of the Manipulator with
Infinitesimal Platform 121
4.3 The Complete Rotatability Workspace
(CRW) and the Partial-Rotatability
Workspace (PRW) 123
4.4 The Workspace of the Platform with
Given Orientation 132
4.5 Summary 146
5 MECHANICAL ERROR ANALYSIS OF THE MANIPULATOR... 147
5.1 Introduction 147
5.2 Position Analysis 148
5.3 Reciprocal Screws 151
5.4 Screws of the Relative Motion of the
Joints 156
5.5 Jacobian Matrix 162
5.6 Mechanical Error Analysis of the
Platform 163
5.7 Summary 168
6 CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER
RESEARCH 171
6.1 Conclusions 171
6.2 Recommendations for Further Research 177
APPENDICES 178
A ALTERNATIVE METHOD OF FINDING THE COORDINATES
OF JOINT C 178
A. 1 Subchain (R-L)-R-S 178
A.2 Subchain (R-L)-P-S 180
B EQUATION OF A GENERAL FORM OF TORUS 183
REFERENCES 18 5
BIOGRAPHICAL SKETCH 190
iv

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
KINEMATICS OF PARALLEL MANIPULATORS
WITH GROUND-MOUNTED ACTUATORS
By
TZU-CHEN WENG
December 1988
Chairman: Dr. George N. Sandor
Major Department: Mechanical Engineering
A great deal of research work has been focussed on the
theoretical and experimental studies of mechanical
manipulators in recent years. Almost all of these works are
related to open-loop serial-link mechanisms, but only a few
have dealt with multi-degree-of-freedom parallel
manipulators such as the Stewart platform or similar
mechanisms.
A new type of two-degree-of-freedom Rotary-Linear (R-L)
actuator was adapted in this work. Several possible
configurations of parallel six-degree-of-freedom
manipulators with ground-mounted actuators have been
synthesized. With parallel configuration of the
manipulators, the computations can be performed
simultaneously. Therefore, the computation time will be
significantly reduced.
v

Workspace analysis of a six-degree-of-freedom parallel
manipulator has been presented by determining the shapes and
boundaries of the subworkspace and root regions in the
subworkspace- The workspace of the manipulator is obtained
as the common reachable region of the subworkspaces
determined by the corresponding subchains. The orientation
and the rotatability of the platform are also investigated.
Finally, mechanical error analysis of the manipulator,
due to the minor inaccuracies in displacements of the
actuators, is studied by using the theory of screws.
vi

CHAPTER 1
INTRODUCTION
1.1 Literature Overview
Robotics has been a very popular subject to study in
the last few years. Researchers have developed many
advanced concepts and theories in kinematics, dynamics,
controls, actuators and sensors for the design of robots.
Recent areas of study also involve workspace, obstacle
avoidance, full rotational dexterity of the end-effector
and the control cf flexible manipulator systems recently.
With the development of microprocessors, which has played
an important role in the rapid growth of industrial robots,
multi-degree-of-freedom mechanical systems are now becoming
a practical choice for use in automatic machinery. It has
been well recognized that, by using multi-degree-of-freedom
robotic manipulators with multiple actuators and automatic
control systems, we can achieve the goal of improving
efficiency, accuracy, reliability and reducing energy
consumptions and cost of production in flexible
manufacturing systems.
Robotic manipulators currently in use in industry and
studied for research purposes are almost all traditional
open-loop serial-link manipulators in which the number of
degrees of freedom of the end-effector is equal to the sum

of the relative degrees-of-freedom of the joints in the
chain. There are only a small number of multi-degree-of-
freedom designs which involve multi-loop manipulator
linkages, with totally parallel or partially parallel
configurations. Since the design techniques for multi-loop
robotic manipulators are still in the infancy of their
development, investigating and developing the theoretical
background for multi-degree-of-freedom multi-loop robotic
manipulators may have a significant impact in the near
future in manufacturing industry.
For a given set of manipulator linkage dimensions, it
is necessary to determine all admissible positions of the
end effector. The collection of all such possible positions
is called the workspace of the manipulator. Recently
several methods have been proposed to determine the
workspace of a manipulator by showing possible extreme
positions of the end effector. These methods let us
calculate directly the boundaries of the workspace of a
given point or line on the hand.
Workspace analysis, generally, refers to determining
the boundaries of the workspace. Workspace synthesis, on
the other hand, consists of determining dimensions of the
manipulator linkage and ranges of joint motions for a
specified workspace.
One of the primary functions of the manipulator is to
have its end-effector reach a set of points in space with
prescribed positions and orientations. The manipulators

3
investigated early were almost all serial kinematic chains,
since these manipulators usually have larger workspaces and
more dexterous maneuverability than those of parallel
kinematic chains. However, serial chains have poor
stiffness and undesirable dynamic characteristics in high¬
speed operation. Also, it is usually difficult to solve
their inverse kinematics problem. Therefore, mechanisms
based on parallel kinematic chains may have certain
advantages when dynamic loading is present and only limited
workspace is required.
Serial multi-degree-of-freedom manipulators have been
extensively investigated. Roth [1] studied the relationship
between the kinematic parameters of a manipulator and its
workspace. Shimano and Roth [2] presented the analytical
and geometrical conditions for a line on the hand to be at
the farthest distance from the base revolute pair. Sugimoto
and Duffy [3, 4] developed an algorithm to determine the
extreme distances of a robot hand. Kumar and Waldron [5]
developed the theory and algorithm for tracing the bounding
surfaces of a manipulator workspace. Sugimoto and Duffy [6]
and Sugimoto, Duffy and Hunt [7] investigated the
singularities in the workspace. Kumar and Waldron [8]
presented the algorithm for tracing the bounding surfaces of
manipulator workspaces. Tsai and Soni [9] presented the
study of determining the accessible region for two and
three-link robotic arms with pin-joints. Gupta and Roth
[10] presented some basic concepts regarding the workspace

4
shapes and structures of manipulators. Selfridge [11]
presented an algorithm for finding the boundary of reachable
volume of an arbitrary revolute-joint, serial-link
manipulator. Tsai and Soni [12] developed an algorithm for
the workspace of a general n R robot based on a linear
optimization technique and on small incremental
displacements applied to coordinate transformation equations
relating the kinematic parameters on the n R robot. Yang
and Lee [13] derived the equations representing the
boundaries of the workspace. Existence of holes and voids
in the workspace were also investigated. Lee and Yang [14]
have made a study of outlining the boundary of the
workspace, the quantitative evaluation of the volume, and
introduced a manipulator performance index. Hansen, Gupta
and Kazerounian [15] used a stable iterative algorithm for
inverse kinematic analysis to determine the approach angles
and lengths for reaching points in the workspace.
Freudenstein and Primrose [16] described the workspace of a
three-axis, turning-pair-connected robot arm of general
proportions in terms of the volume swept out by the surface
of a skew torus rotating about an offset axis in space.
Kohli and Spanos [17] studied the workspace analysis of
mechanical manipulators by using polynomial displacement
equations and their discriminants. Spanos and Kohli [18]
performed the study of workspace analysis of a class of
manipulators having the last three revolute joint axes
intersect orthogonally at a point. Cwiakala and Lee [19]

used an optimization technique to outline the boundary
profile of a manipulator workspace and perform quantitative
evaluation of the workspace volume. Tsai and Soni [20]
illustrated the general procedures to synthesize the
workspace of 3R, 4R, 5R and 6R robots. Tsai and Soni [21]
also considered the effects of kinematic parameters on the
workspace of general 3R robots. Oblak and Kohli [22] used
an analytical method, based on displacement equations, to
identify the Jacobian surface or a D-shaped surface which
the workspace of a regional structure is bounded by.
Davidson and Hunt [23] had a study of the rigid body
location and robot workspaces by using an enumeration
procedure. Davidson and Hunt [24] described plane
workspaces for robots by using a sweeping process and the
necessary equations for computer generation of plane-
workspace envelops and boundaries. Davidson and Pingail
[25] continued to generate envelope-surfaces for plane-
workspace of generally proportioned manipulators. Chen [26,
27] presented an analytical method for workspace analysis of
robot arms by using differential geometry. Kohli and Hsu
[28] studied the Jacobian analysis of workspaces of
mechanical manipulators by determining the maximum reach of
the manipulator within the intersection of the boundary with
a specified plane. Hsu and Kohli [29] dealt with closed-
form workspace analysis and used the Jacobian surfaces to
separate inaccessible regions, two- and four-way accessible
regions in both manipulator coordinates and Cartesian

6
coordinates. Palmquist [30] studied the reachable workspace
common to two planar RRR robots, dexterous relationship
between them and the kinematic motion capabilities of them.
There are few works that have dealt with parallel
multi-degree-of-freedom manipulators. The Stewart platform
[31] is a kind of parallel manipulator which has two plates
connected by six adjustable legs and is a six-degree-of-
freedom mechanism. It was originally used for flight
simulators and was suggested for applications on machine
tools and on space vehicle simulators. Asada and Ro [32]
applied a direct-drive arm to a closed-loop five-link
mechanism to overcome the problems encountered in open-loop
arrangements. Trevelyan, Kovesi and Ong [33] applied such a
mechanism to a sheep shearing robot. Bajpai and Roth [34]
studied the workspace and mobility of such a closed-loop
planar five-link mechanism. Yang and Lee [35] presented a
feasibility study of the Stewart platform as a robot
manipulator. The extreme ranges of motion, rotatability
and workspace were investigated and the workspace and the
maneuverability were found to be relatively restricted.
Fichter [36] also studied Stewart platform-based
manipulators, theoretical aspects of the generalized Stewart
platform, and practical considerations for building a
working machine. Again, Cwiakala [37] used the optimum
path search technique to find the section of the workspace
of the Stewart platform mechanism. Recently, Kohli, Lee,
Tsai and Sandor [38] investigated manipulator configurations

with ground-mounted rotary-linear actuators; their direct
and inverse kinematics were also derived.
The majority of current industrial robots are used for
body guidance. One of the criteria in the control steps is
to reduce the positioning error to a limited range. The
techniques needed to solve such problems have been developed
in the study of closed-loop spatial linkages. Hartenberg
and Denavit [39] used a deterministic method to analyze the
mechanical error. Garrett and Hall [40] used a statistical
approach for mechanical error analysis. Dhande and
Chakraborty [41] presented a stochastic model of the planar
four-bar function generating linkage mechanism for error
analysis and synthesis for specified maximum of mechanical
error. Chakraborty [42] presented a probabilistic model of
linkage mechanisms considering tolerances on the link
lengths and clearances in the hinges, which may cause
mechanical error. Dhande and Chakraborty [43] studied the
effect of random error in the joint of spatial linkages and
developed a synthesis procedure to allocate tolerances and
clearances on different members of linkages to restrict the
output error within specified limits.
1.2 Serial and Parallel Manipulators
Open-loop serial-link manipulators have been the
subject of numerous investigations and have found
considerable applications in industry. In recent years,
there has been considerable increase in research in the

8
area of robotics and multi-degree-of-freedom programmable
automation devices. For being competitive in international
markets, the use of flexible manufacturing systems in
industries is becoming more and more important.
The heart of the flexible manufacturing system consists
of computer controlled multi-degree-of-freedom devices such
as robots and N.C. machines. The configurations which have
been widely used for these machines are serial-link (open-
chain type) arrangements, where one link is connected to
adjacent links by single-degree-of-freedom joints, each with
its separate actuator. Similarly, although N.C. machines
are also serial-link devices, the several degrees of freedom
are distributed between the work piece and the tool, which
decouples the motions associated with various groups of
several degrees of freedom. This simplifies kinematic,
dynamic, and control computations. The manipulator
mechanism generally consists of six links serially connected
via six single-degree-of-freedom separately actuated
revolute or prismatic joints. The end effector, which is
attached to the most distal link, imparts motion of six
degrees of freedom to the work piece. The motion and/or
force associated with these six degrees of freedom may be
controlled.
Although the mechanism may appear simple, the motion of
the end effector is related to joint motions by mathematical
transformations which are generally not easy to visualize.

9
Duffy [44] developed the theory which is applied to
the analysis of single-loop mechanisms, which are movable
polygons or open chains with one end fixed to the ground,
closed by an imaginary link between ground and a free end
where there is a mechanical hand or gripping device.
Hunt [45] has discussed all possible single- and multi-
degree-of-freedom kinematic pairs and used screw theory,
kinematic geometry and the techniques of linear algebra to
systematize in-parallel-actuated robot-arms. Mohamed and
Duffy [46] also applied screw theory to study the
instantaneous kinematics of the end-effector platform of
fully parallel robot-type device. Sugimoto [47] derived the
kinematic and dynamic model for a parallel manipulator by
using motor algebra and Newton-Euler formulation.
A comparison between the serial and parallel devices
in terms of some necessary and desirable performance and
control characteristics was presented by Cox [48]. There
are eight performance characteristics chosen as follows:
i. Range of motion
ii. Rigidity or stiffness and strength
iii. Complexity of end-effector positioning
formulation (computability)
iv. Complexity of system dynamics (computability)
v. Precision positioning
vi. Load carrying distribution through system
vii. Fabrication (economics)
viii. Compactness

10
Hunt [45, 49] also showed some possible alternative designs
for manipulators using parallel kinematic chains, and
pointed out that there are many intermediate possibilities
between purely serial and purely parallel kinematic
structures.
It is well recognized that more investigations in the
study of parallel manipulators are needed and they may have
potential usefulness in the manufacturing industry.
1.3 Summary
Conventional serial-linkage manipulators have each of
their up-to 6 actuators mounted on the joint they actuate.
This means that the mass of these actuators is added to the
link masses, which greatly increases the inertia seen by
actuators and links closer to the ground.
On the other hand, multi-loop manipulators with ground-
mounted actuators need to consider only the masses of the
links themselves. Also, the links can be lighter for the
same payload.
In Table 1.1, a comparison between serial and parallel
kinematic chains is shown, where X means more favorable
performance.
The successful completion of the study of parallel
manipulators would open up a new direction in the design of
robotic manipulators with advantages over present practice,
such as improved payload capacities, increased positioning
accuracy, greater economy in energy consumption, better

il
dynamic performance, increased speed with improved
precision, and reduced first cost.
Table 1.1 Performance characteristic between serial
and parallel kinematic chains
Performance
Serial
Parallel
characteristic
structure
structure
Compactness
X
Computation time
X
Dexterous maneuverability
X
Direct kinematics
X
Inertia
X
Inverse kinematics
X
Payload capacity
X
Power/weight
X
Precision positioning
X
Stiffness
X
Workspace
X

CHAPTER 2
TYPE SYNTHESIS AND INVERSE KINEMATICS
OF THE MANIPULATORS
2.1 Introduction
Industrial robots are available in a wide variety of
shapes, sizes and physical configurations. Generally, the
first three degrees of freedom (links) of the majority of
today's available robots are primarily used to achieve a
desired position for the origin of the wrist. These differ
considerably from one another and can be classified as
cartesian, cylindrical, spherical and revolute which are
shown in Fig. 2,1. The remaining degrees of freedom are
subsequently employed to achieve desired tool frame
orientations. For this purpose, almost all arrangements use
revolute pairs with their axes intersecting at a point. For
such geometries, the position of the common point of
intersection (wrist center) depends only on the first three
joint variables of the structure. Once these are computed,
the orientation of the hand can be attained by rotating the
last three joints only. However, the number of links can
be reduced by using joints with larger degrees of freedom
such as cylindric and spherical pairs.
A novel geometry of a ground-mounted two-degree-of-
freedom self-actuated joint connecting a manipulator link to
12

13
r
:3w
V
4
\
REVOLUTE
Figure 2.1 Four basic manipulator configurations

14
the ground is presented by Kohli, Lee, Tsai and Sandor [38].
It combines a rotary actuator and a linear actuator in such
a way that it imparts cylindrical (two-degree-of-freedom,
combined rotational and translational) relative motion to
the manipulator link with respect to ground. The rotary and
the linear actuators are independent of each other and do
not "see" each other’s inertias. Based on this new
arrangement, several possible manipulator linkage
configurations with six degrees of freedom are described in
this chapter.
2.2 R-L (Rotary-Linear) Actuator
Two different configurations of R-L actuators are
shown in Figs. 2.2 and 2.3. In Fig. 2.2, C is a splined
shaft, and link E is mounted on the shaft and contains
internal splines in its hub. Therefore, link E can be
translated on shaft C and be rotated when shaft C rotates.
A is a linear actuator and is connected to bracket B which
is not splined and can freely slide on shaft C and makes the
link E slide on shaft C. The rotary actuator D rotates the
shaft C. The rotary and linear actuators thus rotate and
translate link E on shaft C without seeing each others'
inertia. The motion of link E is the same as that provided
by a cylinder pair with an axis which is the same as that of
shaft C. In such configuration, link E cannot rotate by 360
degrees due to interference between link E and bracket B.
In Fig. 2.3, the linear actuator is connected to link E

15
Figure 2.2 (R-L) actuator

16
ARRANGEMENT
Figure 2.3
(R-L) actuator with 360° rotatability

17
through grooved hub B by means of pins or split ring P. In
this configuration link E can rotate a full 360 degrees.
The R-L actuator controls a rotation around and a
translation along the axis of a cylinder pair and is used
in type synthesis of parallel manipulators. The principal
advantages of using this type of actuator in the structure
of parallel manipulators are that, first, R-L actuators can
all be mounted on the ground. This reduces the necessary
load capacities of the joints which need support only the
mechanism links and the payload, whereas serial open-loop
robot manipulators must have joints that carry not only the
links and the payload, but also the actuators, their
controls and power conduits of all subsequent joints.
Secondly, with all three R-L actuators mounted on the
ground, the computations required for inverse kinematics and
thus the Jacobian matrix are significantly simplified.
In a manipulator configuration where all the actuators
could be mounted on the ground, the rotary and linear
actuators that form the R-L actuator could be off-the-shelf
items, since the power to weight ratio is not a major
concern in this situation. Thus the cost of the R-L
actuators can be considerably less than the actuators
currently being designed especially for and used in serial
link manipulators.

19
2.3 Type Synthesis
The earliest study of parallel manipulators is that of
Stewart's platform, as shown in Fig. 2.4, which has six
degrees of freedom. The actuators are mounted on the
floating links. Hunt [45] shows a three-degree-of-freedom
and a six-degree-of-freedom parallel manipulator, as shown
in Figs. 2.5 and 2.6 respectively, whose actuators are
mounted on the floating links and on the ground,
respectively. However, Hunt's six-degree-of-freedom
manipulator, as shown in Fig. 2.6, has additional six
redundant degrees of freedom: the axial rotation of the six
S-S links which causes uncontrolled wear in the S joints.
By using ground-mounted R-L actuators, we can reduce the
number of links of the mechanisms and still have six degrees
of freedom of the end effector, without any redundant
degrees of freedom in the mechanism.
With the R-L actuator ground-mounted, we can have
several possible configurations for each subchain of the
parallel manipulators with six degrees of freedom. These
configurations are as follows:
Dyads
(R-D-S-R
(R-L)-S-P
(R-L)-R-S
(R-L)-C-C
(R-L)-P-S
Triads
(R-L)-R-R-C
(R-L)-R-P-C
(R-L)-P-C-R
(R-L)-P-P-C
(R-L)-C-R-R
(R-L)-R-C-P
(R-L)-C-R-P
(R-L)-P-C-P
(R-L)-R-C-R
(R-L)-P-R-C
(R-L)-C-P-R
(R-L)-C-P-P

19
Figure 2.4 Stewart platform mechanism

20
Figure 2.5 Parallel platform-type manipulator with three
degrees of freedom

Figure 2.6 Parallel platform-type manipulator with six
degrees of freedom

22
These chains are shown in Figs. 2.7 and 2.8, where the order
of (R-L) can be reversed as (L-R).
2.4 Technical Discussion
Actuator. One of the major concerns in the design of
serial open-loop manipulators is to maximize the actuator
power/weight ratio, since some of the actuators must be
mounted on the moving links, as shown in Fig. 2.9, which
adds to the inertia of the actuators to the links' inertia
and decreases payload capacity. Therefore, the actuator
size increases from the distal joint to the proximal joint.
The manipulator becomes a massive linkage requiring bigger
actuator sizes and resulting in smaller payloads. If one-
degree-of-freedom actuators are to be used, this will
result in a five-loop linkage for a six-degree-of-freedom
robot manipulator. Further, if only one-degree-of-freedom
joints are used, the number of links in the linkage becomes
quite large. The number of links can, however, be reduced
by using joints with more than one degree of freedom, such
as cylinder and spherical pairs. The number of loops can
also be reduced, thereby reducing the number of links
further by devising and using two- or more-degree-of-
freedom self-actuated joints. A six-degree-of-freedom
parallel manipulator, where all actuators are ground-
mounted, is shown in Fig. 2.10.
Computation. The computation of inverse kinematics and
dynamics requires considerable time for serial-link

53
(R-L)-S-R
(R-L)-R-S
(R-L)-P-S
(R-L)-S-P
(R-L)-C-C
Figure 2.7 Possible configurations of dyads with six
degrees of freedom

(R-L)-R-R-C
(R-L)-C-R-R
(R-L)-R-C-R
(R-L)-R-P-C
(R-L)-R-C-P
(R-L)-P-R-C
Figure 2.8 Possible configurations of triads with six
degrees of freedom

25
(continued)
(R-L)-P-C-R
(R-L)-C-R-p
(R-L)-C-P-R
(R-L)-P-P-C
(R-L)-P-C-P
(R-L)-C-P-P

26
Wrist-Swing
Actuator Hand-Twist
Figure 2.9 Actuators in a serial manipulator

3Í™
s*'
oí
-gtV*
,oH
0^
L-e*
Lt

28
manipulators. Generally, computations of one link depend
upon other links. These computations must be done
serially, thus making parallel processing difficult and
ineffective in reducing computation time. With parallel-
type configuration in the manipulators, the computation can
be performed in parallel. Therefore, the computation time
will be significantly reduced. In general, the computations
required for inverse kinematics and Jacobian matrices will
be less complicated than those of serial open-loop
manipulators, but the computations for direct kinematics are
much involved.
Based on this new possibility, we describe possible
manipulator configuration linkages with six degrees of
freedom. Then we identify possible configurations in which
all actuators for actuating the manipulator linkage are
ground-mounted. The distinct advantage of being able to
put many actuators on the ground makes these manipulator
topologies appealing.
Degrees of freedom. In general, the mobility of a
kinematic chain can be obtained from the Kutzbach criterion.
The six-dimensional form of the criterion is given as
m = 6(n - 1) - - 4 j 2 - 3 j 3 - 2i 4 - j5 (2.1)
where m = mobility of mechanism,
n = number of links,
jj_ = number of joints having i degrees of freedom.

29
Freudenstein and Maki [50] also show that a general
form of the degree-of-freedom equation for both planar and
spatial mechanisms can be written as
j
F = d(n - j - 1) + Efi - Td (2.2)
where F =
d =
n =
j =
fi =
!d =
the effective degree of freedom of the assembly
or mechanism,
the degree of freedom of the space in which the
mechanism operates (for spatial motion d = 6, and
for planar motion and motion on a surface d = 3),
number of links,
number of joints,
degree of freedom of i-th joint,
idle or passive degrees of freedom.
The number of degrees of freedom that a manipulator
possesses is the number of independent position variables
which would have to be specified in order to locate all
parts of the mechanism. In the case of serial manipulators,
each joint displacement is usually defined with a single
variable? the number of joints equals the number of degrees
of freedom.
The number of degrees of freedom of multi-loop
manipulator linkages containing multi-degree-of-freedom
self-actuated joints can be determined simply by the
following equation:

30
n
Fc = 2 Fi " 6ín ' D (2.3)
i=l
where Fc = the number of degrees of freedom of the multi¬
loop mechanism,
F; = the number of degrees of freedom of the i-th
subchain (leg),
n = the number of subchains (legs)
As shown in Fig. 2.10, there are three identical
subchains and each subchain has six degrees of freedom.
Therefore, the number of degrees of freedom of this type of
parallel manipulator can be calculated from Eq. (2.3) as
Fc =(6+6+6) - 6(3 - 1) = 6
2.5 Inverse Kinematics
when the position of one link, generally the hand, is
specified and it is required to determine the position of
all other links, including the joint variables of actuated
joints which will move the hand to the specified position,
the method is called inverse kinematics. The determination
of the actuated joint variables for a specified position of
the hand is conducted by obtaining a set of equations
relating the actuated joint variables and constant
parameters of the manipulator linkages to the hand position
variables. In general, this set of equations is also

31
reduced to one equation of polynomial form in only one joint
variable. For a specified hand position, one proceeds to
find the roots of this displacement polynomial to determine
the joint variable. The degree of this polynomial also
determines the number of possible ways the desired hand
position can be reached.
Generally, the methods employed in solving the inverse
kinematics in robotics are either analytical or numerical.
An analytical solution is one that produces a particular
mathematical equation or formula for each joint variable
(rotation or translation) in terms of known configuration
values (length of the link, twist angle and offset), whereas
a numerical solution generally pertains to the determination
of appropriate joint displacements as the result of an
iterative computational procedure. It is noted that the
equations associated with the inverse kinematic problem are
nonlinear and coupled, and this nonlinear dependence is
basically trigonometric.
As shown in Figs. 2.7 and 2.8, there are five possible
dyads with six degrees of freedom and twelve possible triads
with six degrees of freedom. In order to reduce the number
of links in forming the mechanism and avoid the number of
translational joints greater than three in a loop, we only
consider subchains (R-L)-R-S, (R-L)-P-S, (R-L)-S-R and (R-
L)-S-P as shown in Fig. 2.7 in the following sections.

32
2.5.1 Subchain IR-D-R-S
A schematic diagram of subchain (R-L)-R-S is shown in
Fig. 2.11. Since the position and orientation of the hand,
which is embedded in the platform, is known, we can find the
position and orientation of the coordinate system 0x37323
embedded in the sphere at point C with respect to the local
fixed coordinate system oxqyqZq through systems 11x47424 and
OXYZ by coordinate transformations. Also, we can write the
following equation to describe the position of point c with
respect to the local fixed coordinate system oxqYqZq through
systems BX2Y2Z2 anc^ AxlYlzl as follows:
C° = A^C3
(2.4)
or
cx
cea
-S0a
0
0
1
0
0
a
CY
sea
C0a
0
0
0
Cab
-Sdb
0
cz
0
0
1
da
0
Cab
0
1
0
0
0
1
0
0
0
1
C8b
-s©b
0
bceb
seb
C0b
0
bS0b
0
0
1
sb
0
0
0
1
(continued)

33
Fiqure
2.11
subch^11

34
C0a (bC0b + a) - S0a (bS©bCab - sbSab)
S0a(bC0b + a) + C0a(bS0bCab - sbSab)
bS0bSab + sbCctb + da
1
.
(2.5)*
where the vector C° or its components Cg (k= x, y and z)
denote the coordinates of point C with respect to the local
fixed coordinate system oxgygzg; the vector denotes the
location of point C with respect to the coordinate system
CX3Y3Z3; 0a and 0b are the rotation angles from xQ to x^ and
from X2 to X3, respectively; da is the translation of
cylindric joint A along the fixed axis z^ form the origin of
the local fixed coordinate system oxgygzg? a and b are the
perpendicular distance between successive joint axes z^, Z2
and Z3, respectively; sb is the offset along the Z2 axis; ab
is the twist angle between the axes z^ and Z2, and i = 1
and 2, represent the Hartenberg and Denavit [39] 4 x 4
homogeneous transformation matrices which relate the
kinematic properties of link i to link i-1 and can be
derived as
c©¿
-S0ÍCO.Í
S0^Sa-j_
aiCQi
SQi
C0iCai
-C®iSa±
0
Sa¿
Cai
di
0
0
0
1
* C° can also be obtained by using the method in [44],
which is presented in Appendix A.l.

35
where C© and S0 are shorthand for cos(0) and sini©),
respectively. Similarly, 0C denotes the relative rotation
angle of joint C, db and dc denote the translations of
joints B and C along the moving axes 22 and 23,
respectively, and c denotes the length of link c. It is
noted that db, dc and c are zero for this subchain, but all
these notations are used throughout the following sections.
Premultiplying both sides of Eq. (2.4) by A2”lA]_-l
yields
bC0b = CxC0a + CyS0a - a (2.7)
bS©b = -CxS0aCab + CyC©aCcxb + C2Sab - daSab (2.8)
sb = cxs0asab " cyc0aSab + czCab " daCab (2.9)
Squaring and adding Eqs, (2.7), (2.8) and (2.9) yields
b2 + sb2 = Cx2 + Cy2 + C22 + da2 - 2czda
2aCxC9a - 2aCyS0a + a2 (2.10)
From Eq. (2.9), we have
1
da = (CxS©aSab ~ CyC©aSab + C2Cab - sb)
Cab
Substituting da into Eq. (2.10) yields
EXX4 + E2X3 + E3X2 + E4X + E5 = 0 (2.11)
where
X = tan(0a/2>
D1 = czCab - sb
d2 = Cx2 + Cy2 + C,2 | a2 - b2 - sb2

36
Eb = D2C2ab + 2aCxC2ab - ZCyC^Sa^Caj-, - 2DbCzCab +
Cy2S2ctb + Db 2 + 2D1CySab
E2 - -*4 (aCyC2ab + CxC„SabCab - D]_CxSab - CxCyS2ab)
E3 = 2(D2C2ab - 2D1CzCab + 2Cx2S2ab - Cy2S2ab + D}2)
E4 = -4(aCyC2ctb + CxCzSabCab - DjCxSaj-, + CxCyS2ab)
E5 = D2C2ab - 2aCxC2ab + 2CvCzSabCab - 2DbCzCcLb +
Cy2S2ab + Db 2 - 2D1CySab
Since Eq. (2.11) is a fourth-degree polynomial
equation, there are up to four possible solutions for
variable X (or 0a). Back substituting 0a into Eqs. (2.7)
and (2.9), we can obtain up to four possible sets of
solutions of ©£, and da, respectively. Therefore, the
subchain (R-L)-R-S has a fourth-degree polynomial equation
in inverse kinematics.
Numerical example. The given parameters are as
follows:
a = 2", b = 12”, s^j = 8", = 72° and C° = [-4.86,
-11.60, 3.97]T
The four possible solutions are computed as
Solutions
0a
da
0b
(deg.)
( in. )
(deg.)
1
29.932
1.557
180.299
2
-50.609
12.297
-71.132
3
17.534
7.618
212.427
4
-87.785
-5.592
38.407

37
2.5.2 Subchain (R-L)-P-S
Figure 2.12 shows a subchain (R-L)-P-S. The inverse
kinematics is similar to that of the subchain (R-L)-R-S. We
can write the equation to express the location vector of
point C with respect to the local fixed coordinate system
ox0y020 as follows:
or
c° = a1a2c3
(2.12)
cx
C0a
-S0a
0
0
1
0
0
a
CY
S0a
C0a
0
0
0
Cab
-Sab
0
cz
0
0
1
da
0
Sab
Cab
0
1
.
0
0
0
1
0
0
0
1
10 0b 0
0 10 0 0
0 0 1 db 0
0 0 0 1 1
C©a (a + b) + S©aSabdb
S0a (a + b) - c©aSabdb
da + Sabdb
1
Premultiplying both sides of Eq. (2.12) by A2_^Ai~-1-
yields
b = CxC0a + CyS0a - a (2.14)
** C° can also be obtained by using the method in [44],
which is presented in Appendix A.2.

38
Figure 2.12 Subchain (R-L)-P-S

39
O - -CxS©aCa.b + CyC®aCcLb + ^z^ab “ ^a^ab (2.15)
dfr = CxS©aSaj2 - CyC0aSctj3 + CzCcij-¡ ~ daCCLj-, (2.16)
Let tan(0a/2) = X, and then substituting C0a = (1 - X2)/(l +
X2) and S0a = 2X/(1 + X2) into Eq. (2.14), we obtain
(a + b + Cx)X2 - 2CyX +a+b-Cx=0 (2.17)
There are up to two possible solutions of X in Eq. (2.17),
or up to two possible solutions for 0a. Back substituting
these two possible solutions of 0a into Eqs. (2.15) and.
(2.16), we will have up to two possible solutions of da and
d^ from each equation. Thus it is seen that this subchain
has a second-degree polynomial equation in inverse
kinematics.
Numerical example. The given parameters are as
follows:
a = 3", b =
2", ab = 60°
and C° =
[5.85, -0.13,
two possible
solutions are
computed
as
Solutions
0a
da
db
(in. )
(deg.)
(in. )
1
30.024
2.495
3.510
2
-32.570
6.005
-3.510
2.5.3 Subchain (R-L)-S-R
In Fig. 2.13, a schematic diagram of the subchain
(R-L)-S-R is shown. The spherical pair is kinematically

40
Figure 2.13 Subchain (R-L)-s-R

41
equivalent to three revolute joints with three mutually
perpendicular concurrent axes. Since the orientation and
position of the hand, H, is given, we thus can obtain the
following equations with the assumption that Z3 is parallel
to z4:
H° =
nx
SX
ax
Px
ny
SY
ay
Py
nz
sz
az
Pz
0
0
0
1
C0a
-s©a
0
aC©-
S0a
c©a
0
aS© -
0
0
1
0
0
0
1
c0b2
0
S0b2
s0b2
0
-C0b2
0
1
0
0
0
0
C0c
-S©c
0
cCQc
S0C
C0c
0
cS0c
0
0
1
sc
0
0
0
1
cebi
0
s0bl
0
sebi
0
-C0bi
0
0
1
0
0
0
0
0
1
c0b3
-S0b3
0
bceb3
s0b3
"c0b3
0
bS©b3
0
0
1
0
0
0
0
1
(2.18)
where components of the position (px, Py and pz) and
orientation (nx, ny, nz, sx, sy, sz, ax, ay and az) of
system Hx4y4z4 with respect to the local fixed coordinate
system oxqyqZq are all specified; and ©bl, ®b2 and ®b3 are

42
the three rotational variables of the spherical joint B,
which play no part in the manipulation of the platform.
Equation (2.18) can be rewritten as
AjA2A3
c0a
-S0a
0
aC®
S0a
C0a
0
aS©
0
0
1
¿a
0
0
0
1
c0bl
0 S0bl
0
ceb2
0
S©b2
0
S0bl
0 -C0bl
0
S0b2
0
-C0b2
0
0
1 0
0
0
1
0
0
0
0 0
1
0
0
0
1
c®b3
-S0b3
0
bceb3
C©c
-s©c
0
cC®c
s0b3
_ceb3
0
bs0b3
S0C
C®c
0
cS®c
0
0
1
0
0
0
1
sc
0
0
0
1
0
0
0
1
J
Postmultiplying both sides of Eq. (2.19) by A3_1A2_1
yields Eq. (2.20). Premultiplying both sides of Eq. (2.19)
by Aj^-1 and then postmultiplying both sides by A3'1 yields
Eq. (2.21).
(2.20)
AX = H°A3'1A2~1
a2 = A1'1H°A3-1
(2.21)

43
Since ([A]_](^4))2 + ( [ 1(2,4)^^ “ ( [H°A3~^A2~^ 1 (1,4 ) ) ^ +
([H°A3-^A2-1](2, 4j)2 is true, we can obtain the following
equation:
a2 = [-nx(c + bCQc) + bSxS0c - axSc + px]2 +
[-ny(c + bC0c) + bSyS0c - aySc + py]2 (2.22)
Equation for 0a is obtained since [A-^]^ 4) / [3 (i 4) =
[H°A3~^A2~'1- ] ( 2f4 ) / [H°A3_^A2-^ ] ( 4 t 4 ) holds, and can be
expressed as
-ny(c + bC©c) + svbS0c - aysc + py
0a = taiTM —) (2.23)
-nx(c + bC©c) + sxbS0c - axsc + px
It is observed that [A^]^ 4) = [H°A3-1A2-1 ] (3 4) directly
implies the translation of joint A as
da = -nz(c + bC0c) + szbS0c - a2sc + p2 (2.24)
Let tan(0c/2) = X; then substituting C0c = (1 - X2)/(l + X2)
and S0C = 2X/(1 + X2) into Eqs. (2.22) yields
EjX4 + E2X3 + E3X2 + E4X + E5 = 0 (2.25)
where
D1 = (nx2 + ny2)b2
D2 = < sx2 + sy2)t>2
D3 = 2b(nx2c + riy2c + nxaxsc + nyavsc - pxnx ~ Pyny)
D4 = -2b(nxsxc + riySyC + sxaxsc + syaysc - sxpx -sypy)

44
D5 = -2b^(nxsx + nySy)
E]_ = Dj_ - D3 + D6
E2 - 2(Ü4 - D5)
E3 = -2{DL -2D2 - DgJ
E4 = 2(D4 + D5)
Eg - D3 + D3 + Dg
It is seen that there is a maximum of four solutions of
X (or 0C) in Eq. (2.25). Back substituting the values of 9C
into Eq. (2.23) and (2.24) yields up to four sets of
solutions for 0a and da.
From Eq. (2.21), we can find certain relationships by
equating corresponding elements of the two matrices on
either side of the equation. Thus we obtain the following
equations:
0bl = tan 1(
-axS9a + ayC0a
(2.26)
ax0®a + ayS®a
0b2 = tan 1(
axc0a + ayS0a
(2.27)
azc0bl
0b3 = tan i(
sz00c + nz00c
nz0®c + sz^®c
(2.28)

45
Since the subchain of (R-L)-S-R has a fourth-degree
polynomial equation in inverse kinematics, there are up to
four possible solutions in Eq. (2.25). However, the
possible solutions may be reduced due to special dimensions
of the subchain as shown in the following example.
Numerical example. The given parameters are as
follows:
1.50" and
H° =
II
&
in
II
3", ac = 0°
, c = 0.75'
sc =
0.9300
-0.3323
-0.1574
8.3382
-0.3466
-0.9352
-0.0734
0.2201
-0.1228
0.1228
-0.9848
-1.5205
0
0
0
1
Since two of the solutions of Eq. (2.25) are complex
numbers, the remaining two possible real solutions are
computed as
Solutions
®c
®a
®bl
0b2
0b3
(deg.)
(deg.)
( in.)
(deg.)
(deg.)
(deg.)
1
33.435
-1.008
0.559
26.009
-10.002
11.565
2
14.988
10.009
0.500
14.992
-10.002
30.012
2.5.4 Subchain (R-L)-S-P
The procedure of inverse kinematics of the subchain (R-
L)-S-P, as shown in Fig. 2.14, is similar to that of the

46
Figure 2.14 Subchain (R-L)-S-P

47
subchain (R-L)-S-R in 2.5.3. Therefore, we can obtain the
following equation:
Ho “ a1a2a3
(2.29)
where
Ai
a2
nx
sx
ax
Px
nY
Sy
aY
Py
nz
SZ
az
Pz
0
0
0
1
C®a
-S0a
0
aC0-
S0a
cea
0
aS0j
0
0
1
0
0
0
1
C0bi
0
SQbi
0
s0bl
0
-C0bl
0
0
1
0
0
0
0
0
1
C0b2 0 S0b2 0
seb2 0 ~c0b2 0
0 10 0
0 0 0 1
=0b3
-S0b3
0
bC©b3
1
0
0
c
S0b3
-C0b3
0
bs©b3
o
1
0
0
0
0
1
0
0
0
1
dc
0
0
0
1
0
0
0
1
where components of the position (px, Py and pz) and
orientation (nx, ny, nz, sx, Sy, sz, ax, ay and az) of
system Hxjyjzj with respect to the local fixed coordinate
system oxqYqZq are all specified; and 0b3, 0b2 and 0b3 are

48
the three rotational variables of the spherical joint B,
which play no part in the manipulation of the platform.
Postmultiplying both sides of Eq. (2.29) by A3~^A2~^
yields Eq. (2.30). Premultiplying both sides of Eq. (2.29)
by Ai-^- and then postmultiplying both sides by A-a-^- yields
Eq. (2.31).
Ax = H°A3_1A2_1
A 2 = A1-1H'3A3-1
Since ( C Ax ] (i ^ 4 ) ) 2 + ( [ At_ j ( 2 , â– 
( [HaA3_1A2_:1‘ I ( 2, 4 ) i-s true,
equation:
(ax2 + ay2)dc2 + 2[(c +b
-aypy]dc + (c + b)2(nx2
2(c + b)(nxpx + nypy) -
(2.30)
(2.31)
l) ) 2 = ( Íh°a3"1aP5‘1 (1,4) ^ +
we can obtain the following
nxax t (c ■+■ b) ny-ay — axpx
+ ny:) + px2 + py2 -
a2 = 0 (2.32)
Equation for 0a is obtained since [A2](2 4) / [-%](]_ 4) =
[HaA3_-'-A2_l ] (2,4) / [H°A3-1a2-''- ] ( 2 14 j holds, and can be
expressed as
-(c + b)ny - aydc + py
ea = tan-1 [
-(c + b)nx - axdc + px
(2.33)
It is observed that t Ai](3,4) = [H°A3_1A2 1](3,4) directly
implies the translation of joint A as
d
-(c + b)nz - azdc + p
(2.34)

49
From Eq. (2.31), we can find certain relationships by
equating corresponding elements of the two matrices on
either side of the equation. Thus we obtain the following
equations:
®bl = tan_1
-axS©a + ayC0a
ax00a + ay^®a
0b2
axC0a + ayS0a
tan"1(
-a2C©bi
(2.35)
(2.36)
0b3 = tan"1(
(2.37)
There are two possible solutions for dG in Eq. (2.32),
since it is a quadratic polynomial equation. Back
substituting these solutions of dc into Eqs. (2.33) -
(2.37), respectively, we can obtain two possible sets of
solutions for ©a, da, 0bi, ©b2 ancl ®b3 from each equation.
Therefore, the subchain (R-L)-S-P has a second-degree
polynomial equation in inverse kinematics.
Numerical example. The given parameters are as
follows:
: 3", b =
2M, ac = 0 °,
, c = 0.25”
and
0.7259
0.5900
0.3536
5.1151
0.4803
-0.8027
0.3536
3.4645
0.4924
-0.0868
-0.8660
0.4428
0
0
0
1

50
The two possible solutions are computed as
Solutions
dc
0a
da
®bl
®b2
0b3
( in. )
(deg.)
! in. )
(deg.)
(deg.)
(deg.)
1
2.499
30.001
1.499
14.999
30.004
9.997
2
14.089
239.999
11.536
-14.999
-30.004
189.997
2.6 Summary
The equations of inverse kinematics have been derived
for the dyads (R-L)-R-S, (R-L)-P-S, (R-L)-S-R and (R-L)-S-P
as shown in Fig. 2.7, each of which has six degrees of
freedom. Numerical examples have been shown to illustrate
the possible solutions for each subchain. The degrees of
polynomial equations in inverse kinematics for the dyads
with six degrees of freedom are summarized in Table 2.1.
Table 2.1 Degree of polynomial equation in inverse
kinematics
Type of dyads
Degree of polynomial equation
(R-L)-R-S
4
(R-L)-P-S
2
(R-L)-S-R
4
(R-L)-S-P
2
The detailed analysis and polynomial equation occurring
in inverse kinematics can be obtained using several methods.
The reader is advised to see the literature on kinematic

51
analysis of spatial mechanisms, particularly the work by
Duffy [44].
Hunt [45] pointed out that a serially-actuated arm
accumulates errors from the shoulder out to the end-
effector; also, such arms often suffer from lack of rigidity
and, in the absence of sophisticated techniques of computer-
control compensation, are subject to load-dependent error.
They also suffer relatively low-frequency oscillations.
With in-parallel-actuation by ground-mounted actuators,
there are the advantages of both greater rigidity and
lightness of the linkage, but at the expense of more limited
workspace and dexterity. Since actuator-error is not
cumulative, greater precision is likely to be attainable
without excessive control complications. There should
surely be a future for in-parallel-actuation by ground-
mounted actuators in robotic devices.

CHAPTER 3
WORKSPACE ANALYSIS OF THE MANIPULATORS
3.1 Introduction
This chapter deals with the workspace of a parallel
manipulator having three rotary-linear (R-L) actuators on
grounded cylindric joints, three revolute and three
spherical pairs as shown in Fig. 3.1. The workspace is
defined as the reachable region of the origin of the moving
coordinate system embedded in the six-degree-of-freedom
platform of the manipulator. Since the mechanism consists
of three subchains, the workspace is the common reachable
region of three subworkspaces determined by the
corresponding subchains. The subworkspace described in this
chapter is defined as the workspace of the center of the
platform determined by a subchain regardless of the
constraints imposed by the other subchains. The dimensions
of the platform are considered to be infinitesimal and
therefore the workspace is determined without considering
the orientation of the platform in this chapter1. When the
R joint rotates about the C joint without translation and
the spherical joint rotates about R joint, the locus of the
spherical joint at the end of a dyadic subchain is the
^Workspaces with finite-size platform are derived in
Chapter 4.
52

53
Figure 3.1 Six-degree-of-freedom closed-loop manipulator.
C: Cylindric joint, R: Revolute joint, S:
Spherical joint.

surface of a torus. The subworkspace of each open subchain
is the volume swept by this torus translated along the axis
of each ground-mounted (R-L) joint. In this chapter, the
shapes of the above-described torus of the subchain are
studied for different dimensions of the links. The
conditions on the dimensions of the links, for which the
subworkspace has no hole, are presented. Of course, an
infinitesimally small platform is not practical, because the
three spherical pairs supporting the platform coincide.
Therefore the platform has no controllability of its
orientation. To have controlled orientation, the platform
requires three controllable rotational degrees of freedom
with concurrent non-coplanar axes. This is attained by
placing the three spherical joints at finite distances from
one another. Nevertheless, the workspace study with
infinitesimally small platform is a useful step toward more
practical workspace studies with finite-size platforms
having controllable orientation, which will be covered in
Chapter 4.
One basic need in the design of mechanical manipulators
is to determine the shape of the workspace. Workspace
analysis of mechanical manipulators has been investigated by
many authors. Almost all the studies are related to
open-loop multi-degree-of-freedom serial-link mechanical
manipulators. Little work has been done in the area of
mechanical manipulators with parallel kinematic chains.

55
Therefore, theories for the workspace of such parallel
mechanical manipulators are needed.
3.2 Configuration of a Parallel Manipulator
with R-L Actuators
A six-degree-of-freedom parallel manipulator, where
all actuators are ground-mounted, is considered in this
chapter. It has three six-degree-of-freedom subchains, each
of which has a two-degree-of-freedom R-L actuator, which
controls both the rotation and the translation of a ground-
supported cylindrical joint. For reducing the number of
links in the subchains, spherical joints, which are three-
degree-of-freedom kinematic pairs, are used in the
subchains. A Cylindric-Revolute-Spherical ((R-L)-R-S) triad
may then be used as the subchain with the C joint {R-L
actuator) connected to the ground, and the S joint connected
to the end-effector platform of the manipulator. The axes
of the C joints (R-L joints) on the frame may be arranged in
several different configurations, such as star form,
triangle form, or parallel to each other, as shown in Fig.
3.2. Furthermore, they need not be coplanar, even if they
are not parallel to one other.
The lengths from the center of the platform, H, to the
centers of the spherical joints, C]_, C2 and C3, affect the
volume of the workspace, and more significantly, the
rotatability of the platform about the center H. Thus, to
get a workspace of the manipulator with rotatability of the
end effector as large as possible, the lengths between H and

56
Figure 3.2 Three possible configurations of the
arrangements of the R-L actuators on the base.
Also, the R-L actuators need not be either
parallel or coplanar.

57
C-l (i = 1, 2 and 3) snouid be as small as possible, which is
consistent with controllability of end-effector orientation.
When the lengths are infinitesimal or, in the limit, tero,
the largest possible workspace with complete, but
uncontrollable rotatability of the platform results. In
this chapter, the equations of the workspace of the
manipulator with infinitesimal dimensions of the platform
are derived, i.e., with joints C^, Cj and C3 infinitesimally
close to each other. In chapter 4, the workspace of a
similar parallel manipulator with a finite site platform
will be determined. Of course, controllability of tne
orientation of the platform is reduced sharply as the
spherical joints approach one another. Therefore it must be
realized that there must be a practical trade-off between
the distances of the spherical joints from one another and
the controllability of platform orientation.
3.3 The Subworkspace Analysis of the Manipulator
The toroidal surface (torus) is the locus of a point
attached to a body that is jointed back to the reference
system through a dyad of two serially connected revolute
pairs. A general R-R dyad with a point C which traces the
surface of a general form of torus is shown in Fig. 3.3. It
is similar to the subchain (R-l)-R-S when the platform is
assumed to be infinitesimal and without the consideration of
translation along the axis of the cylindric (R-L) joint.
The shapes of the torus are illustrated first, then the

58
An R-R (Revolute - Revolute) dyad with a point C
tracing a general form of torus. Note that a is
the common perpendicular of the axes of the
revolutes A and B.
Figure 3.3

59
equations of the boundaries of the subworkspace, which is
the volume swept by this torus as the R-L joint translates,
will be derived.
3.3.1 Shapes of the subworkspace
Fichter and Hunt [51] have geometrically described and
analyzed four forms of the torus, which are common,
flattened, symmetrical-offset and general forms as shown in
Fig. 3.4. They also introduced two types of bitangent-
plane^, A-type, whose quartic curve intersection with the
torus always encircles the OZ axis and B-type, whose points
of tangency are both on one side of the OZ axis. Any
bitangent-plane to any form of torus cuts the torus in two
circles of the same radius which intersect one another at
the two points of tangency. The curve of intersection of
the bitangent-plane and the torus can be obtained by the
simultaneous solution of the equations of the bitangent-
plane and the torus. The curve of intersection of A-type
bitangent-plane and a common torus (a > b) is shown in Fig.
3.5.
The equation of the surface of a completely general
form of torus can be expressed as follow:
2 a bitangent-plane has two points of tangency with a
toroidal surface.

60
Figure 3.4 Diametral sections through tori: (a) common,
(b) flattened, (c) symmetrical-offset and (d)
general types.

(continued)
(d) General torus

62
f,urve of intersection
(common torus end A-type plane)
X-axis, inches
Figure 3.5
Intersection of the A-type bitangent-plane and
the right circular torus (a = 3”, b= 2")

{(x2 + y2 + z2) - (a2 + b2 + s2)}2
63
Z - SCCSCl
= 4a2{b2 - ( )2} ( 3 .1) 3
sina
Common form. The common form of torus (right
circular), sometimes called the anchor-ring, is shown in
diametral section in Fig. 3.4(a). The axes of the two
revolute pairs are at right angle (a = 90°). Their common
perpendicular is a and the offset between them is zero (s =
0) (see Fig. 3.3). The equation of the torus can be
expressed as
{(x2 + y2 + z2) - (a2 + b2)}2 = 4a2(b2 - z2) (3.2)
The difference between the lengths of the links affects the
shape of the torus, which is illustrated in Fig. 3.6. As a
> b shown in Fig. 3.6(a), the two circles in the diametral
section are separated by a distance of 2(a - b). This kind
of torus is also shown in Fig. 3.7. When the two circles in
the diametral section are tangent at a point, the origin 0,
then a = b as shown in Fig. 3.6(b). The torus will
intersect itself when a < b as shown in Fig. 3.6(c). There
is a void in this kind of torus when a < b exists.
Flattened form. The flattened form of torus has no
offset (s =0) either, but the axes of the two revolute
3 The equation of the general form of torus is derived
in Appendix B.

64
Figure 3.6 Diametral section of the common form of torus:
(a) a>b, (b) a = b and (c) a < b

65
Figure 3.7 A right circular torus, a > b.

66
pairs are not at right angle (a ?= 90°). The equation of
this form of torus can be expressed as
z2
{(x2 * y2 + z2) - (a2 + b2)}2 = 4a2(b2 - —)
sin2a
(3.3)
The diametral plane cuts the torus in egg-shaped curves as
shown in Fig. 3.4(b). For different twist angles (30°, 45°
and 75°) and dimensions of the links (a > b, a = b, and a <
b) with each specified twist angle, the shapes of the torus
are shown in Figs. 3.8 - 3.10, respectively. It is noticed
that the diametral sections of these tori are similar to
those of the common form when a > b and a = b, but flattened
and egg-shaped. But the tori in Figs. 3.8 - 3.10 do not
intersect themselves when a < b, which differs from the
schematic shown in Fig. 3.6(c).
Symmetrical-offset form. The symmetrical-offset form
has the axes of the two revolute pairs at right angle (a =
90°) and with offset (s i 0). The equation of this form of
torus can be expressed as
{(x2 + y2 + z2) - (a2 + b2 + s2)}2 = 4a2(b2 - z2)
(3.4)
The diametral section of this kind of torus is shown in Fig.
3.4(c). For a > b and small s, the shape of the torus as
shown in Fig. 3.11(a) is slightly different from that in
Fig. 3.6(a). As shown in Fig. 3.11(b), the inner walls of
the anchor-ring become flatter when a = b. The two closed

67
(a)
Figure 3.8 Diametral section of the flattened form of torus
(a = 300 ): (a) a > b, (b) a = b and (c) a < b

68
Figure 3.9 Diametral section of the flattened form of torus
(a = 45°): (a) a>b, (b) a=b and (c) a < b

69
Figure 3.10 Diametral section of the flattened form of
torus (a=75°): (a) a > b, (b) a = b and
(c) a < b

70
z
:
r\
U
0
i
rr
L
0
LJ
(b)
Figure 3.11 Diametral section of the symmetrical-offset
form of torus: (a) a > b, (b) a = b and
(c) a < b

71
curves in the diametral section become banana-shaped, as
shown in Fig. 3.11(c), when a < b.
General Form. The diametral section of the torus is
shown in Fig. 3.4(d). For different twist angles (30°, 45°
and 75°) and dimensions of the links (a > b, a = b, and a <
b) with each specified twist angle, the shapes of the torus
are shown in Figs. 3.12 - 3.14, respectively. The closed
curves in the diametral section, while still more or less
banana-shaped, but now they are tilted over.
Equations (3.1) - (3.4) are all of the fourth degree
and all forms of torus are thus quartic surfaces. The curve
of intersection between a torus and a general plane is a
quartic; also, in general, a straight line cuts any torus in
four points (real, imaginary, or coincident).
The volume and shape of the workspace are very
important for applications since they determine capabilities
of the robot. In order to obtain the optimum workspace, the
volume of the subworkspace of the corresponding subchain
generally should be as large as possible. Since most of
today’s available industry robots have 0° or ±90° twist
angles, we will discuss the following two cases with the
conditions of s = 0 and twist angle a = 0° or ±90° applied,
respectively.
Case 1: s = Q and a = 0° (or ± n)
Since the axes of the two revolute pairs are parallel
and there is no offset, the toroidal surface degenerates

72
z
c ^
0
Figure 3.12 Diametral section of the general form of torus
(a = 30°): (a) a > b, (b) a = b and (c) a < b

73
Figure 3.13 Diametral section of the general form of torus
(a = 45°): (a) a>b, (b) a = b and (c) a < b

74
(a)
Figure 3.14 Diametral section of the general form of torus
(a= 750 ): (a) a > b, (b) a = b and (c) a < b

75
into a plane. For different dimensions of the links a and
b, the shapes of the now planar toroidal surface are
illustrated in Fig. 3.15. The workspace resulting when this
surface is translated along the Z axis has no hole only when
a = b as shown in Fig. 3.15(b).
The workspace of this kind of subchain is generally the
volume between two coaxial cylinders when the translation
along the Z axis is in effect. Due to the limitation of the
rotation of the first revolute joint which is ground-
mounted, the workspace of this subchain is actually reduced
to the upper half (X > 0) of the volume between the two
coaxial cylinders. Whenever the condition a = b exists, the
inner boundary disappears and there is no hole in the
workspace.
Case 2: s = 0 and a - ±90°
Since the two axes of the two revolute pairs are
perpendicular to each other and there is no offset, the
locus of point C is a torus, which is defined as the common
form of torus shown in Fig. 3.6. When the translation along
the 2 axis is in effect, the workspace of the subchain can
be described as shown in Figs. 3.16 - 3.18, respectively.
In Fig. 3.16(a), the torus has a hole because of a > b.
When the translation d along the Z axis is in effect, we
obtain the workspace as the volume between the two coaxial
cylinders with radii of (a + b) and (a - b), respectively
and height d, plus the volume of a half torus at each end.

76
b - a
a + b
Figure 3.15 Cross sections of the Workspace generated by
the planar R-R dyad: (a) a>b; (b) a = b and
(c) a < b (infinitesimal platform)

(a)
Z
(b)
Figure 3
16 Workspace with the common form of torus
(a > b); Along the Z axis: (a) d > 2b and (b)
0 < d < 2b (infinitesimal platform)

Figure 3.17 Workspace with the common form of torus
(a = b) ; Along the Z axis*, (a) d > 2b and (b)
0 < d < 2b (infinitesimal platform)

79
Figure 3.18
Workspace with the common form of torus
(a < b); Along the Z axis: (a) d > 2b and (b)
0 < d < 2b (infinitesimal platform)

80
However, this is true only when the translation along the Z
axis, d, is greater than or equal to 2b. Otherwise, there
are additional voids that can be found as shaded areas
shown in Fig. 3.16(b). When the translation along the Z
axis is less than 2b, there is a void inside the workspace,
shown with a lentil-shaped cross section in Fig. 3.16(b).
When a = b as shown in Fig. 3.6(b), there is no hole as the
translation along the Z axis is in effect and d > 2b. The
workspace is the volume of the cylinder with radius of
(a + b), height d, plus the volume of this kind of half
torus at each end, as shown in Fig. 3.17(a). If the
translation along the Z axis is less than 2b, voids can be
found even if the torus has no hole at all. The lentil¬
shaped cross sections of the void can be shown as shaded
areas in Fig. 3.17(b). Finally, when the torus intersect
itself as shown in Fig. 3.6(c), there is a void inside in
this torus. As the translation d along the Z axis is in
effect and is greater than or equal to 2b, we obtain the
workspace as the volume of the cylinder with radius of (a +
b) and height d, plus the volume of this kind of half torus
at each end as shown in Fig. 3.18(a). Similarly, voids can
be found if the translation along the Z axis is less than
2b, which is shown in Fig. 3.18(b).
3.3.2 Boundaries of the subworkspace and root regions
in the subworkspace (infinitesimal platform)
In order to calculate the volume of the subworkspace,
we must find the boundaries (external and internal) of the

81
subworkspace. From the inverse kinematics of subchain (R-
U-R-S solved in section 2.5.1, we know it has up to four
possible solutions for a given position of the S joint.
Once the orientation and position of the hand is given, we
may have up to 64 solutions for the manipulator. Therefore,
the study of root regions in the subworkspace is also
important.
A manipulator with R-L actuators in the subchains whose
axes are arranged in triangle form on the base can be
represented as shown in Fig. 3.19. The notation is as
follows:
OXYZ
i
Af, Bj_
°ixiYizi
global fixed coordinate system,
i-th subchain.
R-L actuator joint and revolute joint in the
i-th subchain.
local fixed coordinate systems with the
axis along the axis of the cylindric joint
Ai.
Aixai^aizai• Bixbi^bizbi — moving coordinate systems
embedded in joints and B¿,
respectively.
0ai» ©bi — relative rotation angle between successive
links.
dj_ — translation along the axis z¿ of cylindric
joint from the origin of the local fixed
coordinate system OiX^yiZi.

82
Figure 3.19 Notation of the manipulator with ground-mounted
R-L actuators arranged in triangle
configuration

83
a¿, — perpendicular distance between successive
joint axes z-^, z2 and Z3, respectively.
sbi' abi — offset along axis zj^ and twist angle between
za¿ and , respectively.
HCj_ — approaching zero for infinitesimal platform.
As shown in Figs. 3.20 and 3.21 when s^i f 0 or s^ = 0
respectively, if the revolute joint Bj_ makes a complete
rotation, the locus of the point is a circle with respect
to the moving coordinate system Bi^biVbizbi* can also be
expressed with respect to the coordinate system AiXaiYaizai
from Eq. (2.5) as follows:
- bic0bi + ai
(3.
.5)
= - SbiSabi
(3.
.6)
= bis0biSabi + sbiCabi
(3.
.7)
The locus, generated by the point Cj_ turning around the
revolute joint Bj_ and the cylindric joint A^, without
translating along the axis of the cylindric joint, can be
obtained by turning the circle, now represented by Eqs.
(3.5) and (3.6), about the za-¡_ axis. The shape of this
locus is the surface of a general form of torus as described
in section 3.3.1. The subworkspace of this subchain is the
volume swept by this toroidal surface as it translates along
the axis of the cylindric joint.

84
Figure 3.20 Subworkspace element of the i-th dyad of the
manipulator as the offset at joint Bj_, s^ ^ 0
(infinitesimal platform)

85
Figure 3.21 Subworkspace element of the i-th dyad of the
manipulator as the offset at joint Bi; sbi =
(infinitesimal platform)
0

86
Since the manipulator’s workspace with infinitesimal
platform is the common intersection of the three
subworkspaces of the three dyads that support the platform,
in general the ends of these subworkspaces need not be
considered. Therefore, only the projection of the
subworkspace onto the xy plane is of interest. For this
reason, henceforth r will designate only the xy projection
of the vector from the center to the cylindric boundary
of the subworkspace. Accordingly:
r2 = xai2 + yai2 = Xi2 + Yi2
= (b-jCe^ + ai)2 + (b.jSebiCabi - sbiSabi)2 (3.8)
Subworkspace with offset = 0^ (infinitesimal
platform). For the sake of simplicity and ease of
visualization, we will discuss the subworkspace with
infinitesimal platform and with offset s^i = 0 first. £q.
(3.8) becomes
r2 = (b.jCebi + ai)2 * bi2S20biC2abi (3.9)
The projections on the o¿x¿y¿ plane of the circles
described by C¿, which are ellipses, can be represented by
Eqs. (3.8) and (3.9) in terms of the parameter 0^. Figs.
3.20 and 3.21 show these ellipses when a-j_ > b¿. When
^See page 102 for subworkspace with offset s^i t 0.

87
cylindric joint Aj_ translates along its fixed axis z¿, the
subworkspace with respect to the system Aj_x¿yj_z¿ is the
volume between two concentric cylinders without considering
the two ends of the subworkspace* The radii of the external
and the internal cylinders are respectively the maximum and
the minimum radii of the torus described by Cj_ without
translation. Taking the derivative of r expressed by Eq.
(3.9) with respect to 0bj_ and making the resulting
expression equal zero, we can obtain the maximum and minimum
radii of the torus from the following equation
dr
r = ) = 0 (3.10)
d8bi
Then we obtain the following two equations
S8bi = 0 (3.11)
and
cebi ^ (3.12)
bis abi
When Sa^i = 0 (a^ - 0 or ti) (the circles described by
and are coplanar) , or a-j_ > b^S^a]^, no real roots of
©bi can be found in Eq. (3.12). Hence we only have two
roots of from Eq. (3.11), i.e. ©¿j_ = 0 and 0£-¡_ = ti. In
this case, the values of r corresponding to ©¿j_ and ©§j_ are
respectively the maximum and minimum values of r.
Therefore, substituting the value of and 0¿j_ into Eq.
(3.9) yields the maximum and minimum values of r as follows:

max
(3.13)
2 = (at + bi)2 = rxi:
and
rmin^ = ^ai “ bj_)^ = ^23-2, (3.14)
which are intuitively correct.
The relationship between r and ©j^ can be expressed in
a Cartesian coordinate system as shown in Fig. 3.22. It is
obvious that in the subworkspace where r2i < r (=
) x^ + yj_^) < for a given position of point Cj_, which
implies a given value of r, there are two corresponding
solutions of ©kj_, The subworkspace is a two-root region or
two-way accessible region. In other words, there are two
sets of 9aj_ and ©^i' or two kinematic branches of the i-th
dyad for reaching a given position of Cj_. On the boundaries
where J x¿2 + = r^ or r2i/ there is only one solution
for Thus the boundary surfaces are one-root regions.
(Recall that this is a planar R-R case).
In another case, when Sa^j_ t 0 (a^ i 0 or n) and a¿ <
biS^d-bi, two additional roots of 0^ are found from Eq.
(3.12). Therefore in addition to r^ and r2i there are
another two values of r, say and r^, which are also
limiting values of Eq. (3.9).
Substituting Eq. (3.12) into Eq. (3.9) yields
r3i2 = b4i2 = cot2abi(bi2S2abi - aj_2)
(3.15)

89
r
Figure 3.22 Two solutions of 0b^ corresponding to r2 < r <
rl tai = 10, bj = 5, sbj, = C and abi = 0 or n)
(circles described by and without
translation of Aj_ are coplanar)

90
Thus, r^, r~2i, r3b and r4i are the local minima or
local maxima of r expressed by Eq. (3.9). In order to find
the global minimum and maximum values, we take the second
derivative of r with respect to 0bb of Eq. (3.10) which
yields
d2r dr
r — + (
d0bi2 d0bi
-bi[biS2c:bi(2C20bi - 1) +
aiCebi] (3.16)
At the position of local minima or maxima, the first
derivatives equal zero, and then
d2r
d0bi2
0bi = 0
= -bj_(bbS2abi + ab) < 0 (3.17)
d¿r
d0bi2
ebi = *
-b4 At the position of r3 or r,j,
d2r
d0bi
. 2
®bi = arc cos (
bis2abi
H2 - bi2s4abi
s2abi
> 0
(3.19)
According to Eqs. (3.17) - (3.19), and r2± are local
maxima, and r3j_ and r^ are equal local minima of r. Since

91
rli > r2i' t^ie global maximum is r^, and there are two
equal global minima, namely and r^. The external and
internal cylinders of the subworkspace are then the
cylinders of radii of r^i and r3¿, respectively.
Figure 3.23 shows the relationship between r and 0^ in
a Cartesian coordinate system in this case. It is seen that
in the portion of the subworkspace where r2j_ < r < r]_j_, for
a given position of point Cj_, which implies for a given
value of r, there are two corresponding solutions of 0^.
Thus this is the two-root region of the subworkspace. The
other portion of the subworkspace is the four-root region of
the subworkspace since four solutions of 0^^ can be found in
that region for a given translation dj_ (see Fig. 3.19) and a
given position of C¿. The cylinder of radius r = r2± inside
the subworkspace divides the subworkspace into a two-root
region and a four-root region, and the surface of the
cylinder of radius r2i itself is a three-root region. It
can also be seen that the external boundary (the surface of
the cylinder of radius r = r^) is a one-root region, and
the internal boundary (the surface of the cylinder of radius
r = r3is a two-root region. Since the offset is zero and
the twist angle is not 90°, without the translation of Aj_,
the locus of the positions of point is the surface of a
flattened form of torus. The diametral section of this kind
of torus is shown in Fig. 3.24 and the root regions can also
be visualized easily from the figure.

92
r
$b¡
Figure 3.23 Two solutions of 9^ corresponding to T2 < r <
r-L and four solutions of 6^ corresponding to
r3 < r < r2 (a^ = 2, b^ = 10, s^i = 0 and
abi = n/3)

93
Flattened Form
Figure 3.24 Diametral section of flattened form of torus
(a¿_ = 2, - 10, = 0 and a^i = n/3)

94
Furthermore, in the case when a¿ = b^S^ajj^, the roots
of ©^i of Eq. (3.12) are coincident and equal, i.e., 9§j_ =
®éi = ®Bi = TI* Thus, r3j_ = r^ = r2i' and their value is
the minimum of r, and is the radius of the internal cylinder
of the subworkspace.
The results derived are illustrated in Figs. 3.25 -
3.31, where the projections of the configurations of the
subchain are presented.
Figures 3.25 and 3.26 are shown for the case of = 0
or ti, i.e., Sa]-,^ = 0. The projection of the circle
generated by point on the A¿x¿y¿ plane is also a circle.
The radii of the external and internal cylinders of the
boundary are the maximum and minimum radii r of the torus.
It is also shown that the subchain can take two branches
for point to reach the same position in the
subworkspace, illustrating a two-root region.
Figures 3.27 and 3.28 are shown as the case of a^i ¿ 0
or ti and a¿ > biS2ab¿, The results are the same as in the
case shown in Figs. 3.25 and 3.26, except that the
projection of the circle of point Cj_ on the A^x^y^ plane is
an ellipse now.
Figures 3.29 and 3.30 are shown as the case of f 0
or rt and a¿ < bj_S2aj-¡j_. The internal boundary is the
cylinder of radius r3j_. The circle of radius r3^ on the
plane OiX^y^ is tangent to the ellipse at two points showing
the values of r3j_ and r^ (at the points of tangency) , which
are both the equal limiting values of the radius r of the

95
Figure 3.25 Two-root subworkspace as = 0 or n. and
ai > bi

96
Figure 3.26 Two-root subworkspace as ajji = 0 or n. and
ai <

97
Figure 3.27 Two-root subworkspace as > biS^a^ and
ai > bi

98
Figure 3.28
Two-root subworkspace as > biS^a^ and
ai < bi

99
Two-root subworkspace as a^ < bj_S^a^j_ and
r2 < r < r]_
Figure 3.29

100
Four-root subworkspace as a¿ < b¿S2ab^ and
r3 < r < r2
Figure 3.30

101
Figure 3.31 Two-root subworkspace as a^
t>iS2a.bi

102
torus. In Fig. 3.29, the subchain can take two branches for
point Cj_ to reach a certain position in the region of r2¿ <
r < r^i* In Fig. 3.30, the subchain can take four branches
for the point C¿ to reach a certain position in the region
of r3¿ (or r^) < r < r2j_- These figures illustrate that
these two regions are the two-root and four-root regions of
the subworkspace, respectively.
Figure 3.31 describes the case of ^ 0 or ti, and a¿
= b^S2ab¿. In this case, the circles of radii r2i and r3j_
(or r^j_) shown in Figs. 3.29 and 3.30 are coincident. Thus
the four-root region of the subworkspace degenerates to a
one-root cylindrical surface.
Subworkspace with Offset sy^ 0. Taking the
derivative of both sides of Eq. (3.8) with respect to ©j^
and letting the resulting expression equal zero, yields
“S0foi(biC0j-)i + a¿) + c0biCabi = 0 (3.20)
Expanding S©b:¡_ and C0bj_ in terms of tan-half -angle, X =
tan(9j3j_/2) , Eq. (3.20) can be rewritten in the form of a
quartic equation in X:
X4 T BX3 + CX2 +DX+E=0 (3.21)
where
+ b¿(1 - Cabi)
B = 4
3bis2abi

103
C = 0
ai “ - Ca^)
D = 4
Sbis2abi
E = 1
Solving the quartic equation Eq. (3.21), we can obtain
up to four real roots for X, and then of 9^. Back
substituting the roots of 0^ into Eq. (3.8), up to four
limiting values of the radius r of the torus are obtained.
In the case where only two limiting values of r exist,
one is the maximum rmax, and the other is the minimum rraan
in the range of 0 < 0^ < 2rt. Hence the subworkspace can
be expressed as
(3.22)
max
and
(3.23)
In the case where four limiting values of r exist, we
need to compare the four values and find the maximum and
the minimum which are respectively the radii of the
external and internal cylinders of the boundaries of the
subworkspace.
Furthermore, the relationship between r and 0^, in
the two cases, are shown schematically in Figs. 3.3 2 and
3.33 in Cartesian coordinate systems, respectively. In the
first case (Fig. 3.32) for a given value of r in the region

104
Figure 3.32 Schematics showing the relationship between r
and as s^i ^ 0. Two limit values of r
exist as aj_ = 2, bj_ = 12, = n/6 and s^i = 8

105
r
$bi
Figure 3.33 Schematics showing the relationship between r
and 0bi as sb^ # 0. Four limit values of r
exist as aj_ = 2, b^ = 12, ab^ = n./2.5 and
sbi = a-

106
rmin r < rmax^ two corresponding values of 0bj_ are found.
Hence, the subworkspace is a two-root region. In the other
case (Fig. 3.33), r reaches limiting values at four
positions in the range of 0 < <2n. We find that for
given values of r in the region of r2 < r < rmax, r2 < r <
r2, and rmj_n < r < r2, the numbers of corresponding are
two, four and two, respectively. Hence these regions in the
subworkspace are respectively two-, four- and two-root
regions. Since these involve the general form of torus,
the diametral sections of the torus are shown in Figs.
(3.34) and (3.35), respectively. The root regions in the
subworkspace can also be visualized easily from the
figures.
The procedure of finding the root regions in the
subworkspace can be summarized as shown in Fig. (3.36).
3.4 Conditions for No-Hole Workspace
Since the workspace of the manipulator is the common
region of the three subworkspaces, if none of the
subworkspaces has any holes, the workspace has no hole.
Subworkspaces in general, as described in the above
sections, have cylindrical internal boundaries. Under some
conditions of the dimensions of the manipulator, the radii
of the internal cylinders are zero. Thus the internal
boundaries disappear and the subworkspaces have no hole.
These conditions are listed in Table 3.1.

107
General Form
Zj-axis
igure 3.34 Diametral section of general form of torus
(a¿ = 2, bj_ = 12, = n/6 and s^i = 8)

108
General Form
Figure 3.35 Diametral section of general form of torus
(a¿ = 2, - 12, ct^i = n/2.5 and s^i = 8)

109
Figure 3.36 Procedure of finding the root regions in the
subworkspace as = 0

lie
Table 3.1 Conditions for No-Hole Workspace
| , 3'
Case
Radius of internal
cylinder
Condition for
no-hole
abi = ^ or 11
r2i2 " 2
bf = ai
°bi *
0 or
n
ai ^ biS2abi
ai < biS2abi
r3i2 =
cot2abi(bi2S2abi - ai2)
abi = ± Tt/2
3.5 Workspace of the Manipulator
The subworkspace analysis results in the equations of
the subworkspace i (i = 1, 2, 3) with respect to the local
fixed coordinate system oix^y^z^. Using coordinate
transformation matrix Agi from the global fixed system OXYZ
to the above-mentioned system, OiXj_yj_Z|, we transform these
equations to be expressed with respect to the global system
OXYZ.
-
•
xi
X
Vi
ii
>
It
H-
Y
zi
Z
1
1
-
-
In the case, for example, where the axes of the three
R-L actuator joints are arranged in equilateral triangle
form on the base as shown in Fig. 3.37, the matrices Agi (i
= 1, 2, 3) are as follows:

Ill
Figure 3.37 Coordinate systems of the ground-mounted R-L
actuators in the arrangement of equilateral
triangle form

112
A
gi
1000
0 1 0 h
0 0 10
0 0 0 1
A
32 ~
l
o
0
0
0 0
cos(-120) -sin(-120)
sin(-120) cos(-120)
0 0
0
h
0
1
10 0 0
0 -1/2 73/2 h
0 -73/2 -1/2 0
0 0 0 1
ftg3
10 0 0
0 cos(-240) -sln(-240) h
0 sin(-240) cos(-240) 0
0 0 0 1
10 0 0
0 -1/2 -73/2 h
0 73/2 -1/2 0
0 0 0 1
with the substitution of these matrices into Eg. (3.24),
we
obtain

113
X]_ = X
Yi = Y + h
z-j_ = Z
x2 = X
y2 = -(1/2)Y + (/T/2)Z + h
z2 = - ( 73"/2 ) Y - (1/2 ) Z
x3 = X
y3 = -(1/2)Y - (73/2)Z + h
z3 = (73/2IY - (1/2)Z
Thus the equations of the subworkspace can be expressed
as follows:
fei = *i2 + yi2 - Ui + bi»2 < o
and
FIi = xi2 + Yi2 - rIi2 s 0
where r^j_2 denotes r2i~ or r2j_2 (see Fig. 3.36), can be
respectively transformed to be expressed with respect to
the global OXYZ system as follows:
FE1 = X2 + (Y + h)2 - {a1 + b^2 < 0 (3.25)
fE2 = X2 + [-Y/2 + (VT/2)Z + h]2 - (a2 + b2)2 < 0
(3.26)
fE3 = X2 + [-Y/2 - (J2/2)Z + h]2 - (a3 + b3)2 < 0
(3.27)
FI1 = X2 + (Y + h)2 - rxl2 > 0 (3.28)

fi2 = X2 + I
:-y/2 +
(J2/2)Z
+ hi2 - rI22
> 0
114
(3.29)
Fi3 = X2 + (
'.-â– i/2 -
(73/21Z
+ h]2 - rI32
> 0
(3.30)
where subscripts
E and I
denote
the external
and
internal
boundary, respectively.
The workspace of the platform of the manipulator with
infinitesimal platform is the common reachable region of the
three subworkspaces. Therefore the workspace of the
manipulator is the region where Eqs. (3.25) - (3.30) are
satisfied simultaneously.
The workspace can be expressed graphically by plotting
the cross sections of the workspace on the planes normal to
the axes of the system OXYZ. For the cross sections, Eqs.
(3.25) - (3.27) are rewritten respectively as follows:
On plane OYZ, X = constant
Y = ± ,/u-l + b2)2 - X2 - h (3.31)
Y = (/3)Z + 2h ± 2 V(a2 + b2)2 - X2 (3.32)
Y = -(73)Z + 2h + 2 J{a3 + b3)2 - X2 (3.33)
On plane OZX, Y = constant
X = + 7(ai + b1)2 - (Y + h)2 (3.34)
X = ± J ( a2 + b2)2 - [-Y/2 + (V3/2)Z + h]2 (3.35)
X = ± J (a3 + b3)2 - [-Y/2 - (V3/2 ) Z + h]2 (3.36)
On plane OXY, Z = constant
X = ± Ju-i + bx)2 - (Y + h)2 (3.37)

115
X = + V(a2 + i¡212 - [-Y/2 + (V3/2ÍZ + h]2 (3.38)
X = + V(a3 + b3)2 - [-Y/2 - (V3/2IZ + h]2 (3.39)
Similarly, Eqs. (3.28) - (3.30) can be rewritten by way of
replacing the terms (a^ + b¡_)2 in Eqs. (3.31) - (3.39) by
rj-^2, and are not listed.
Numerical example. The given parameters are as
follows:
ai = a2 = a3 = 3", b3 = b2 = b3 = 3", h = 2" and L = 4"
The workspace of the manipulator is the common
reachable region of three subworkspaces determined by the
corresponding subchains. We thus can plot the cross section
of the workspace on the plane, say OYZ, X = 4, plane, which
is shown in Fig. 3.38. There are three pairs of parallel
lines which denote the intersections of the external
boundaries of subchains 1, 2 and 3, respectively, with the X
= 4 plane. The areas between those parallel lines are the
intersections of the workspaces of those subchains with the
X = 4 plane. It is noted that the lengths of link a and b
are equal, the radius of internal cylinder thus is zero, and
there is no hole in the subworkspace. In Fig. 3.39, the
cross section of the workspace on the plane OXY, Z = 0, is
shown. The contour of the external boundary of the first
subchain is a half circle and the contours of the second and
third subchains are coincident half ellipses. The shaded

axis (inches)
116
2
1
0
1
-2
-3
-A
-5
-6
WORKSPACE (SWIPi
on plane OYZ, X = 4
igure 3.38 Workspace of the manipulator on plane OYZ as
X = 4. *SubWorkspace, Infinitesimal Platform.
**See Eqs. (3.25) - (3.27).
-2 0 2-4 6 8 »
Z-axis (inches)

â– axis (inches)
117
WORKSPACE (SWIPf
on plane OXY, Z = 0
Y-axis (inches)
Figure 3.39 Workspace of the manipulator on plane OXY as
2=0. *SubWorkspacef Infinitesimal Platform

118
area is the intersection of the workspace of the manipulator
with infinitesimal platform with plane OXY, Z = 0.
3.9 Summary
A general approach for the workspace analysis of a
parallel manipulator (shown in Fig. 3.1), which is a new
area to be developed, has been presented in this chapter,
considering the platform to be infinitesimally small.
In order to get the workspace of the manipulator, the
shapes of the subworkspaces are illustrated first, then the
boundaries of the subworkspaces and the root regions in the
subworkspaces are determined.
Different combinations of joints in a chain create
different surfaces. Torfason and Crossley [52] showed the
surfaces generated by all closed spatial pairs. Jenkins,
Crossley and Hunt [53] used the generated surface method to
analyze the gross motion of the generating mechanism.
According to the equations derived in this chapter, the
workspace of the manipulator depends on the length of the
links, twist angles, offsets and the arrangements of the
R-L actuators on the base (Fig. 3.2). As to the design
requirements, the methods presented may help decide how to
choose those parameters to achieve a desired workspace. The
ground-mounted R-L actuators are initially assumed to have
no limits on translation along their axes; but the actually
needed maximum translations will be determined by the
required boundaries.

119
The dimensions of the end-effectcr-carrying platform,
on which the hand is mounted, are neglected in this chapter
in order that the workspace of this type of parallel
manipulator be more readily visualized at first. In Chapter
4, the workspace analysis in case of finite-size platform
will be discussed. One of the problems that will be
encountered in connection with the workspace is the
interference between the links, which will affect the
workspace. This is regarded as a subject for further study.

CHAPTER 4
THE WORKSPACE OF THE MANIPULATOR WITH FINITE SIZE PLATFORM
4.1 Introduction
This chapter deals with the workspace determined by the
geometric method which was developed in Chapter 3 as well as
the consideration of finite size platform of the
manipulator. The lengths from the center of the platform,
H, to the centers of the spherical joints, C^, C2 and C3,
affect the volume of the workspace, and more significantly,
the rotatability of the platform about the center H. Thus,
to get a workspace of the manipulator with rotatability of
the end effector as large as possible, the lengths between H
and (i = 1, 2, 3) should be as small as possible,
consistent with controllability of end-effector orientation.
When, in the limit, the lengths are infinitesimal or zero,
we get the largest theoretical workspace with complete, but
uncontrollable rotatability of the platform.
The workspace of the parallel manipulator is defined
here as the region within which every point can be reached
by the center of the platform, H, of the manipulator (Fig.
3.19). The volume of the workspace is correlated with the
orientation and the rotatability of the platform, because it
is often important for a designer to know the rotatability
120

121
of the platform about its center in various regions of the
workspace.
The surface where the platform can not rotate about H
in any direction, when the center H is on that surface, is
called the nonrotatability surface (NRS). The platform thus
cannot assume an arbitrary prescribed orientation when H is
on the NRS. The complete rotatability workspace (CRW)
refers to the workspace within which or on whose boundary
the platform can theoretically rotate in any direction. The
region between the nonrotatability surface and the external
boundary of the complete rotatability workspace can be
called the partial rotatability workspace (PRW). The
contours of the NRSS and CRSW of a subchain are shown
schematically in Fig. 4.1.
4.2 Workspace of the Manipulator with
Infinitesimal Platform
The workspace of the parallel manipulator is the common
region of the three subworkspaces which are, respectively,
the reachable regions of H, determined by subchain i (i = 1,
2 or 3), regardless of the constraints imposed by the other
two subchains. In the case of the manipulator with
hypothetically infinitesimal platform, point H is coincident
with the three centers of the spherical joints, at the
corners of a finite size platform. The subworkspace of
infinitesimal platform (SWIP) contains the reachable region
of point C¿. Symbolically, they can be expressed in the
form of the following inequalities:

122
The relationships between the external
boundaries of CRSW and NRSS of subchain i as
a¿ = (i = 1, 2 and 3)
Figure 4.1

123
(4.1)
(4.2)
Here the subscripts E and I denote respectively the external
and internal boundaries. For distinction of the coordinates
of the points on the boundaries of SWIP (subworkspace,
Infinitesimal Platform) from those of the points inside the
subworkspace of H, we use Xra, Ym and Zra to denote the
coordinates derived in Chapter 3 for infinitesimal platform,
and X, Y and Z are used in the following sections to denote
the coordinates when a finite size platform is considered.
The boundaries of the SWIP can then be respectively
expressed symbolically by the corresponding equalities as
(4.3)
(4.4)
4.3 The Complete Rotatabilitv Workspace (CRW) and
the Partial Rctatability Workspace (PRW)
The platform can rotate about point Cj_ describing a
whole sphere as the locus of point H, if we disregard link
interferences and the constraints imposed by the other two
subchains, even when point is already on the external
boundary of the i-th SWIP. This situation, with the finite
dimensions of the platform, is depicted in Fig. 4.2, where
are at the vertices of an equilateral triangle with H at

124
Figure 4.2
Rotatability of the finite size platform

125
its geometric center. Considering point H as the center of
the rotation of the platform, however, the platform can
usually rotate about it only in a portion of a sphere. The
platform can rotate about H only in such a direction that
the angle a between the velocity of point Cj_ moving with the
rotating platform and the gradient VF£j_, which points
towards the outside of the SWIP, is greater than n/2. When
the platform rotates so that HCj_ approaches the direction of
the gradient, then moves inside the SWIP. Therefore, it
does not prevent the platform from rotating. But in the
opposite direction, Cj_ would move outside the SWIP, and
therefore, it prevents the platform from rotating. This
means that at this position of H, the platform has partial
rotatability about the center H.
Referring to Fig. 4.2, let H¿c denote the point which
is on the internal extension of the line of the gradient
YFEi' and t^1€ distance from Hiq to the boundary equals the
length of Cj_H of the platform. The sense of C^Hj_c is
opposite to that of the gradient. If the point H takes the
place of H^c, the instantaneous velocity of C¿, when the
platform starts to rotate in any arbitrary direction, is
perpendicular to VFgj_. In this case point moves either
on the boundary or toward the inside of the SWIP.
Disregarding link interference, the subchain i does not
prevent the platform from rotating about point H, and
therefore, the platform is said to have complete
rotatability, so far as the i-th subchain is concerned. The

126
locus of all points H^c constitutes the external boundary of
the so called complete rotatability subworkspace (CRSW)
determined by the subchain i. The CRW (Complete
Rotatability Workspace) of the manipulator is the common
region of the three CRSWs. Moreover, let us consider the
other point on the line of the gradient VFEj_, and with
the same distance outside of the boundary as on the
inside. The sense of is the same as that of the
gradient. One can easily verify that the locus of these
Hj_N points constitutes the so-called nonrotatability
subsurface (NRSS). The region between the CRSW and NRSS is
called the partial rotatability subworkspace (PRSW). After
the three PRSWs of each of the three subchains have been
obtained, one can find the manipulator's PRW (Partial
Rotatability Workspace) as the common regions of the three
PRSWs.
It should be noticed that for the complete rotatability
subworkspace to exist, the short principal radius of
curvature of the surface of the external boundary should be
longer than L, the length of C^H of the platform. Otherwise
the arc drawn by point C-j_ centered at is on the outside
of the SWIP as shown in Fig. 4.3, and therefore, point Cj_
would prevent the platform from rotating about H. For the
manipulator with the R-L actuators with offset = 0, the
external boundary of the SWIP is the cylinder of radius
(ai + bi^ (see Fig. 3.21). Since the short principal radius
of curvature of a cylinder is the radius of the cylinder

127
Figure 4.3
Radius of curvature of the external boundary of
SWIP (SubWorkspace, Infinitesimal Platform)

128
itself, it is important to design the manipulator so that
the condition + bj_ > L be satisfied. Otherwise, the
complete rotatability subworkspace does not exist. In fact,
to make the CRSW as large as possible, the length of Cj_H,
and thus the size of the platform, ought to be as small as
possible. Mathematically, the coordinates of the point on
the CRSW and the NRSS can be expressed respectively as
X
xm
Y
=
Ym
2
(CRSW)
zm
and
X
xm
Y
=
Z
(NRSS)
zm
YFEi
L
IYFEi!
7FEi
L
lYFEi!
(4.5)
(4.6)
Generally the simultaneous solution of Eqs. (4.5) and
(4.3) represents the equation of the external boundary of
the CRSW, and the simultaneous solution of the Eqs. (4.6)
and (4.3) represents the equation of the NRSS, where both
are expressed in two of the three variables, Xm, Ym and Zm.
For obtaining the implicit equations of the boundaries of
the CRSW and NRSS correlating X, Y and Z, one can derive the
expressions of Xm, Ym and Zm in terms of X, Y and Z from
Eqs. (4.5) and (4.6), respectively, and then substitute the
resulting expressions into Eq. (4.3).

129
For the manipulator shown in Fig. 3.19 and with the
base arrangement in equilateral triangle form (see Fig.
3.37), Eqs. (4.3) for subchains 1, 2 and 3 have the
following forms (see Eq. (3.25) - (3.27))
fE1 = xra2 + (Yra + h)2 - (ax + b^2 = 0 (4.7)
fE2 = Xra2 + + V3Zm/2 + h)2 - (a2 + b2)2 = 0
(4.8)
fE3 = xra2 + (-Yra/2 ‘ ^W2 + h)2 - (a3 + b3)2 = 0
(4.9)
We now calculate the gradient of the first equation = 0,
Yfei as
3FEl/3Xm
2xm
3FEl/5Yra
=
2(Ym + h)
5FEl/3zm
0
|VFe1| = J (3FE1/3Xra)2 + (3FE1/3Yra)2 + (3FE1/3Zm)2
= 2 V Xm2 + (Yra +â–  h)2
= 2(ax + bx) (4.11)
Substituting expressions (4.10) and (4.11) into Eq. (4.5),
we obtain
X
Y
xrab
al + bl
(Yra + h)L
(4.12)
(4.13)
= Z
m
Z
al + bl
(4.14)

130
Solving Eq. (4.12) - (4.14) for Xm, Ym and Zm in terms of X,
Y and Z yields
X( )
Y( a-[_ + ) + hL
Ym = (4.16)
a^ + b]_ ~ L
z
m
Z
(4.17)
Finally, substituting expressions (4.15) - (4.17) into Eq.
(4.7) yields the equation of the external boundary of the
CRSW of the first subchain as
fEC1 = x2 + (Y + h)2 - (a2 + b3 - L)2 = 0 (4.18)
where the subscript C denotes complete rotatability.
Similarly, the equations of the external boundaries of
the CRSW of the second and third subchains are found to be
as follows:
fEC2 = X2 + (-Y/2 + 73Z/2 + h)2 - (a2 + b2 - L)2 = 0
(4.19)
fEC3 = X2 + (-Y/2 - V3Z/2 + h)2 - (a3 + b3 - L)2 = 0
(4.20)
In fact, the SWIP represented by Eq. (4.3) is a
cylinder of radius (a2 + b^), and the CRSW is a coaxial
cylinder of radius (a^ + b^ - L). Therefore one can easily

131
write Eqs. (4.21) - (4.23) for the NRSS (NonRotatability
Subsurface) directly by referring to Eqs. (4.18 - 4.20) as
follows:
fSN1 = X2 + (Y + h)2 - (¿i + bx + L)2 = 0 (4.21)
fSN2 = X2 + (-Y/2 + V3Z/2 + h)2 - (a2 + b2 + L)2 = 0
(4.22)
fSN3 = X2 + (-Y/2 - V3Z/2 + h)2 - (a3 + b3 + L)2 = 0
(4.23)
where the subscripts S and N denote surface and
nonrotatability, respectively.
Then we can obtain the external boundaries of the CRW
of the manipulator by solving Eqs. (4.18) - (4.20)
simultaneously. The PRW (Partial Rotatability Workspace)
can be obtained as the following inequalities are satisfied:
FECi 0 (4.24)
and
FSNi(X, Y, Z) < 0 (4.25)
where i = 1, 2 and 3.
The equations of the internal boundary of the CRSW of
subchains 1, 2, and 3, which can be derived in a similar
way, are expressed as follows:
(rn + L)2 > 0
FIC1 = x2 + (Y + h)2 -
(4.26)

= X2 + [-Y/2 + (73/21Z + h]2 -
132
fIC2
fIC3 = X2 + [-Y/2 - (V3/2)Z
0
(4.27)
h]2 - (rxi + L)2 > 0
(4.28)
where denotes r2i or r3¿ (see Fig. 3.36) and i = 1, 2
and 3. Once the internal boundaries of the three CRSWs have
been obtained, the internal boundary of the CRW can be found
as the common areas of the three CRSWs.
Numerical example. The given parameters are as
follows:
al = a2 = a3 = 3", b]_ = b2 = b3 = 3”, h = 2", L = 1"
and X = 4”
In Fig. 4.4, a cross section of the workspace of the
manipulator is plotted on the plane OYZ, with X = 4. It is
noticed that the complete rotatability workspace is very
limited for this kind of manipulator, especially when the
size of the platform is not small compared to the length of
the links.
4.4 The Workspace of the Platform with
Given Orientation
The workspace of the platform with given orientation on
a plane is defined here as the reachable region on a plane
of the center H of the platform, while the platform keeps
moving with a given orientation. The simplest case is that
the plane is perpendicular to one of the coordinate axes, X,

-axis (inches)
133
WORKSPACE (CRWI
on plane OYZ, X = 4
Z-axis (inches)
Figure 4.4 Complete rotatability workspace of the
manipulator on plane OYZ, with X = 4

134
Y or 2 of the fixed global system. All the points are in
that same plane.
When the point C-j_ of subchain i is on the intersection
of the boundary of the SWIP with the plane, which plane is,
for example, perpendicular to the X axis as shown in Fig.
4.5, the platform can rotate in the plane about the axis
through H and normal to the plane in a portion of a circle.
The platform can rotate only in such a direction that the
point Cmoves inside the SWIP.
In general, the intersection of the SWIP with the plane
are lines. They can be symbolically expressed as
FEiX and
F-m = *
where the superscript x denotes the normal of the plane, the
subscript E implies the external boundary, d is a constant
distance from the origin and i = 1, 2 and 3. Since the
procedures of the analysis for the external and internal
boundaries are similar to each other, we discuss only
positions on the external boundary in the following section.
The workspace inside the external boundary is expressed by
inequality (one for each subchain):
(4.29)
(4.30)
FEiX(xm' Ym'
Zra) <0, i = 1, 2, 3
(4.31)

135
Figure 4.5 Schematic drawing of the positions of points
and Hj_N when the platform keeps moving in a
plane perpendicular to the X axis of the global
coordinate system (no scale)

136
The gradient of F|?^x can be expressed as follow:
VFEix = [0, 3FEi*/3Ym, 3FEi;t/3Zm]T (4.32)
and points toward the outside of the boundary.
Let and be the points on the line of the
gradient such that the distance from these two points to
the boundary is L (the same as Cj_H), and the sense of C¿H¿C
is opposite to that of the gradient and the sense of Cj_Hj_N
is the same as that of gradient. It is seen that when the
center H takes the position of H¿c, the platform reaches a
limiting position in the direction of the gradient with
complete rotatability about H, provided we disregard link
interference and the constraints imposed by the other two
subchains. This means that the locus of points Hj_c
constitutes the external boundary of the CRSW with the
platform remaining in the given plane. Similarly, one can
see that the locus of points Hj_N constitutes the
intersection of the NRSS with the plane.
The coordinates of the points H¿c and when H is on
the plane can be expressed by Eq. (4.5) and (4.6) with X =
d. The simultaneous solution of Eqs. (4.5) and (4.29),
therefore, represents the intersection of the external
boundary of the CRSW with the plane X = d, and the
simultaneous solution of Eqs. (4.6) and (4.29) represents
the intersection of the NRSS with the plane X = d. Having
the equations of the intersection lines of the boundaries of

137
the three CRSWs and the three NRSSs of the three subchains
with the plane X = d, the intersections of the CRW and PRW
of the manipulator with the plane X = d are the common
regions of the former intersections.
For the manipulator shown in Fig. 3.19 and, with the
base arrangement in equilateral triangle form (Fig. 3.37),
for example, the intersection of the SWIP determined by the
first subchain with the Y2 plane (X = d) is described as
(see Eg. 3.31)
Yra = -h ± y(ai + bx)2 - Xm2 (4.33)
where Xra is a constant, and therefore this equation
represents two straight lines parallel to the Z axis. To
express the subworkspace, Eq. (4.33) ought to be written
individually for these two lines, two branches of the
intersections of the boundary with plane X = constant, as
fE1XCV Zm> = Ym + h - Jl+ bx)2 - xm2 < 0
(4.34)
and
fE1X!5W Yra, Zra) = -(Ym + h) - 7(ai + bj2 - Xm2
The gradients of the two half-planes defined by these
two inequalities are
VFe1x = (0, 1, 0)T (4.36)
and
VFEix = (0, -1, 0)T
(4.37)

138
Substituting the expressions (4.36) and (4.37) into Eq.
(4.5), the boundary of the CRW becomes these two lines:
X = xra
Y = Ym - L (4.38)
Z = zm
and
X = xm
Y = Ym + L (4.39)
Z = zm
Furthermore, substituting Eqs. (4.38) and (4.39) into
(4.34) and (4.35) respectively, we obtain the two
corresponding lines of the intersections of the CRSW with
the plane X = constant expressed in X, Y and Z as
Y + L + h - y(ax + bx)2 - X2 < 0
and
-(Y - L + h) - 7(ai + bx)2 - X2 < 0
where X is a constant.
The intersection with the plane can be expressed in one
equation which has a form similar to Eq. (4.33) as
Y = -h + 7(3! + bx)2 - X2 ± (-L) (4.42)
where X is a constant and therefore Eq. (4.42) also
represents two straight lines parallel to the Z axis.
(4.40)
(4.41)

139
Similarly, the equations of the external boundaries of
the CRSW of the second and third subchains are found to be
as follows:
fE2CX = X2 + (-Y/2 + 732/2 + h + L)2 - (a2 + b2)2 = 0
(4.43)
fE3CX = X2 + (-Y/2 - 73Z/2 + h + L)2 - (a3 + b3)2 = 0
(4.44)
Numerical example. The given parameters are as
follows:
al = a2 = a3 = 3", b-j_ = b2 = b3 = 3", h = 2", L = 1"
and X = 4"
In Fig. 4.6, a cross section of the workspace of the
manipulator is plotted on the plane X = 4. It is noticed
that the complete rotatability workspace is larger than the
cross section of the workspace as shown in Fig. 4.4.
However, the cross section of the workspace of the
manipulator is still very limited.
The equations for the NRSSs determined by the
corresponding subchains can be derived in a similar way and
are therefore not described here.
The foregoing discussion applies to the case where the
given plane is perpendicular to one of the axes of the
global coordinate system. In a general case, where the
plane is inclined to the axes of the system, the gradient of

140
WORKSPACE (CRW)
on plane X = 4
Z-axis (inches)
Figure 4.6 Complete rotatability workspace of the
manipulator on plane X = 4

141
the intersection of the SWIP with the given plane can be
obtained by coordinate transformation.
The given plane can be specified by its normal N and
the shortest distance to the plane from the origin of the
global system, d. We can set a new reference system Oxyz
with the x axis coincident with the normal of the plane, and
with the origin coincident with the origin of the old system
as shown in Fig. 4.7. The axis y is set to be perpendicular
to both the axis X of the old system and the normal. Let n,
r, and s now be unit vectors on the three new axes of x, y
and z, respectively. The unit vector n is given as
n = (nx, ny, nz)T
Then
i j k
r = i x n
10 0
(0, -n2, ny)T
nx ny n2
i j k
s = n x r
nx ny nz
0 -nz ny
The transformation matrix from the new system to the
old one can be written as

142
The positions of points Hj_c and Hj_N when the
platform keeps moving in a plane inclined to the
axes of the global coordinate system
Figure 4.7

143
1
3
X
rx
sx
O
3
X
0
ny2 + nz2
i
o
ny
rY
sy
0
ny
"nz
-nxny
0
nz
rz
sz
0
nz
ny
“nxnz
0
O
1
0
0
1
0
0
0
1
Using the transformation of
X
X
xnx
+ z(ny2
+ nz2)
Y
= A
y
_
xny
- ynz -
z nxny
Z
z
xnz
+ yny -
znxnz
1
1
1
the equation of the boundary of the SWIP originally
expressed with respect to the global system OXYZ can be
transformed to be expressed with respect to the new system
Oxyz. For example, Eq. (4.7) of the external boundary of
the SWIP of the first subchain can be transformed to be
expressed with respect to the new system as
FE1 “ txmnx + Zm(ny2 +• nz^)]2 + [X^ny - Ymnz
Zmnxnz + h]2-(a1 + b-|j2 = 0 (4.45)
where Xm is a constant.
Eq. (4.45) represents the intersection of the boundary
with the given plane with respect to the new system Oxyz.
In the above equations the subscript m denotes that the

144
corresponding coordinates are of the point on the SWIP. The
gradient of the intersection expressed with respect to the
new system can be obtained as
VFE1n = (0, 3FE1/3Ym, 3FE1/9Zm) (4.46)
where the superscript n denotes that the gradient is on the
plane whose normal is N, and from Eg. (4.45) one can obtain
3FE]_/3Ym — 2 (Xmny - Ymnz - Zmnxnz + h)(-nz) (4.47)
and
3FEl/5Zm = 2[Xmnx + Zm(ny2 + nz2)](ny2 + nz2) +
2(xmny ” Ymnz ” Zmnxnz + h)(-nxnz) (4.48)
Let Gy and Gz denote respectively ^FE^/3Ym and 9FE]_/ 3zm
expressed by Eqs. (4.47) and (4.48). Then we can obtain
|VFE1n| = J Gy2 + Gz2 (4.49)
Substituting Eqs. (4.46), (4.47), (4.48) and (4.49)
into Eq. (4.5), we obtain the coordinates of point the
limiting position of point H where the platform has complete
rotatability in the given plane, as follows:
x
y = Ym -
(4.50)
GyL/^Gy2 + G.
2
(4.51)

145
z = zm - GZL/Vay2 + Gz2 (4.52)
The simultaneous solution of Eqs. (4.50) - (4.52) and
(4.45) represents the equation of the external boundary of
the CRSW of the first subchain on the plane, correlating Ym
and Z^ expressed in the new system Oxyz.
Finally we can use the inverse transformation A-1
X
X
Y
= A'2
Y
z
Z
1
1
-
.
and transform the coordinates of the point on the boundary
defined by Eq. (4.50) - (4.52) combined with Eq. (4.45) to
be expressed with respect to the global system OXYZ.
That is the method of transforming the SWIP in the
plane to be expressed with respect to a new reference where
the given plane is perpendicular to one of the axes. Then
in the new reference the gradient of the boundary in the
given plane and the coordinates of the point on the CRSW or
NRSS are respectively calculated. Finally, the results are
transformed back to be expressed with respect to the global
system using inverse transformation. This procedure will be
taken for the three subworkspaces with complete rotatability
or nonrotatability of the platform, and the corresponding
workspace of the manipulator is obtained as the common
region of these three subworkspaces.

146
4.5 Summary
Two sorts of the workspace, the complete rotatability
workspace (CRW) and partial rotatability workspace (PRW), of
the parallel manipulator with finite size platform are
studied in this chapter. The nonrotatability surface (NRS)
and the boundaries of the complete rotatability workspace
are parallel to the boundary of the workspace of the
manipulator with infinitesimal platform (WIP). The complete
rotatability workspace is inside, and the nonrotatability
surface is outside of the WIP.
The cross sections of the CRW of the manipulator are
plotted on the plane 0Y2, with X = constant and on the
plane, X = constant, where the platform can only move
around, respectively. The workspace of this type of
manipulator is limited.
For a workspace with complete rotatability to be as
large as possible, the size of the platform should be as
small as possible. For a large partial rotatability
workspace, the size of the platform should be as large as
possible.

CHAPTER 5
MECHANICAL ERROR ANALYSIS OF THE MANIPULATOR
5.1 Introduction
In the control of robotic manipulators, the primary
task is to drive the links to their desired positions
corresponding to a given input. Actuators are the devices
that make the robot move. The mechanical error is defined
as the minor inaccuracies in displacements (translation and
rotation) of the actuators for a certain specified hand
position. In order to increase the accuracy of the desired
hand position, we must reduce all errors as far as possible.
In this chapter, a six-degree-of-freedom parallel
manipulator with ground-mounted actuators, as shown in Fig.
3.1, has been studied from the viewpoint of mechanical
error analysis. Ground-mounted actuators in parallel
manipulators avoid the dynamic errors caused in serial
manipulators by those actuators mounted on the movable
links, away from the ground. Therefore, the mechanical
errors of parallel manipulators with ground-mounted
actuators are due to the input angle and translation errors
of the actuators, if the tolerances of the link lengths are
not considered. In the study of position and orientation of
the hand (end-effector) embedded in the platform, we may use
inverse kinematics to transform the pose (position and
147

148
orientation) of the hand to a global coordinate system by
homogeneous transformation matrices. The determination of
joint velocities using screws, presented by Mohamed and
Duffy [46] and Sugimoto [47], is adapted in this chapter to
describe the velocities and small displacements.
With consideration of input errors as differential
motions, we will obtain the corresponding error movement of
the platform, in which the hand is embedded, due to the
inaccuracies of the actuators.
5.2 Position Analysis
A subchain of a six-degree-of-freedom parallel
manipulator, where all the actuators are ground-mounted, is
shown schematically in Fig. 5.1. Let H denote the position
vector of the hand which is embedded on the platform, and
1, m and n denote the unit vectors describing the hand
orientation of the moving system. By using homogeneous
transformation, we can describe the relative position and
orientation between any two coordinate frames. Therefore,
we can have the coordinate transformation matrix Tp as
follow:
lx mx nx
T lv my nY
p
ls m2 nz
0 0 0
(5.1)

149
The i-th subchain of the manipulator, with R-L
actuator at A¿
Figure 5.1

150
where lj_, and are the components of the basis unit
vectors of the hand coordinate system, are the
components of the hand position vector and i = x, y and z.
The position vector of point with respect to 0XY2 then
can be calculated from the following equation:
°Ci = Tp m(HCi)
(5.2)
where ra(HC^) denotes the vector HC^ with respect to the
moving coordinate system Hxyz. We can also find another
coordinate transformation matrix T¿, from c>iXj_yj_Zi to OXYZ,
which can be expressed as
xix
Yix
zix
°°ix
xiy
Yiy
2iy
0°iy
xiz
Yiz
ziz
Ooiz
0
0
0
1
According to the relationship between the coordinate
systems, we can have the following equations:
°Ci = TiCi
or
£i = (Ti)-loci
where
xix
xiy
xiz
-0Oj_
xi
l =
Yix
Yiy
Yiz
-Oo-j_
Yi
zix
ziy
ziz
•H
O
o
zi
(5.4)
(5.5)
(5.6)
0
0
0
1

151
and , Y.i and are unit vectors along the axes of
°ixiVizi-
The coordinates of point Cj_ with respect to the system
°ixi^izi can ke used to determine the necessary joint
displacements by inverse kinematics, which is described in
detail in Chapter 2.5.1. It is noted that there are four
possible sets of solutions for a given location and
orientation of the hand. Each set of solutions corresponds
to a specific configuration of branches of the subchains.
Calculation of Jacobian and error analyses should be
performed in the same branch.
In Fig. 5.1, the position vectors °Aj_, °B¿ and unit
vectors °za¿ and with respect to the global coordinate
system OXY2 can be calculated respectively by coordinate
transformation, and can be expressed as follows:
i = TiAi
(5.7)
°li = Titi
(5.8)
°£ai = Ti£ai
(5.9)
°Zbi = TÍ.ZM
(5.10)
5.3 Reciprocal Screws
A screw is defined by Ball [54] as a straight line with
which a definite linear magnitude termed the pitch is
associated. If (L, M, N; P, Q, R) are the Plücker
coordinates, as shown in Fig. 5.2, of the screw axis, (L, M,

152
Figure 5.2 Plücker line coordinates

153
N) are the direction ratios of the line and (P, Q, R) are
the x, y and z components of the moment of the line about
the origin, then we can express the screw coordinates as
(S ' SQ) where
= Li
+ Mi +
Nk
(5.
.11)
= Pi
+ Qi +
Rk
(5.
.12)
and
L = x2 - xx
M = y2 - yx
N = Z2 - Zi
P = y^N - z^M
Q = -X^N + ziL
R = x^M - y^L
Figure 5.3 shows a rigid body which is constrained to
move on a screw $2 = (S2; So2). A wrench (fj_; C^) of
intensity is applied to the body on a screw = (S^;
Sq^) and it produces a twist (u2; v2) of amplitude w2 on
the screw $2. Thus the moment of the force f_i about the
point A is a12a12 x f^, and the virtual work produced by the
wrench can be expressed as follow:
U — f^ • v2 + Ci ■ joJ2 ” al2—12 ^ ^1 *
(5.13)

154
¿12
Figure 5.3 A wrench on a screw

155
Substituting fj_ = = h^f^S^, u¿2 = w2^2 and
v_2 = b2^>2§.2 i-nto Eq. (5.13), the virtual work produced by
the wrench can thus be expressed as
U = fi^2 {(hi + h2)cosa]_2 - a^sina^} (5.14)
If the pitches h^ and h2, the twist angle a12, and mutual
perpendicular distance a-j^ between the axes S-j_ and §2 of the
two screws are chosen such that
(h]_ + h2)cosa12 “ al2s^na12 = 0 (5.15)
then the wrench cannot produce motion on the screw £2 no
matter how large is the intensity f. The screws $j and $2
are then said to be reciprocal. It is noted that Eq. (5.15)
is symmetrical, and a wrench on the screw cannot produce
motion on the screw .
Hunt [49] shows that, when the line of action of a
force passes through an axis of an revolute pair, the force
does not affect the rotation around the revolute pair
because it cannot exert any moment about it. Reciprocity
between a force F and an angular velocity co about an
revolute pair axis means that when a body is attached to a
base through a single revolute pair about which it can turn
at an angular velocity uj at any instant, a force F is
reciprocal to w when its contribution to the rate of working
is zero. For the prismatic pair, when the line of action of

156
a force is perpendicular to the direction of the prismatic
pair, there is no contribution by this force to the rate of
working. Also, an revolute pair perpendicular to the
direction of a couple also satisfies the condition of
reciprocity. All couples, no matter how they are oriented,
are reciprocal to a pure translation in a direction parallel
to a prismatic pair. Therefore, if a body is constrained to
move about an ISA (Instantaneous Screw Axis), c,, a wrench
acting on a screw V can contribute nothing to the rate at
which work is being done on the body. In such a
circumstance c and V are said to be reciprocal screws.
The necessary and sufficient condition for a pair of
screws, $_i = + £§ol and $.2 = §2 + e§o2' to be reciprocal
can thus be expressed as
$1 ° $2 = Si * So2 + S2 * Sol = 0 (5.16)
where the symbol O denotes the reciprocal product of a pair
of screws.
5.4 Screws of the Relative Motion of the Joints
In order to calculate the screws representing the
relative motion of the joints, we would like to have each
joint expressed with respect to the global system 0XY2.
Let ^jiíSji, S0j¿) denote the motion screws of the
joints with respect to the OXYZ system, where j = 1, 2,...,
6 denote the joints and i = 1, 2 and 3 denote the subchain.
Thus, representations of the screws are

157
Cylindric joint A¿ $3i = °Ai x S3i)
$2i = v2i(°' 5U)
Revolute joint Bj_ $3i = w3i Spherical joint Cj. £4i = “41(1, °Cj_ x i)
l5i = “5i(l. °Ci x D
¿6i = w6i<Ü- °Ci x k)
where S3i = °£ai and S34 = °zbi.
Since we deal with instantaneous kinematics, the
spherical joint may be considered as a combination of
three revolute joints whose concurrent axes are parallel to
the axes of the OXYZ system.
Mohamed and Duffy [46] used orthogonal screws to study
the instantaneous kinematics of the end-effector platform of
fully parallel robot type devices. They demonstrated that
for any parallel device the twist representing the
instantaneous motion of the end-effector platform is equal
to the sum of its partial twists. A partial twist is
defined as the twist representing the instantaneous motion
of the end-effector when all actuators other than the input
actuator are locked.
We can find the reciprocal screws, r$.3¿ and r$.2i>
reciprocal to all other screws in subchain i but and
$2i> respectively. As shown in Fig. 5.4, the reciprocal
screw r$.3¿ passes through the center of spherical joint C4,
intersects the axis of revolute joint and also lies on
the plane P which is perpendicular to the axis of cylindric
joint A4. In Fig. 5.5, the reciprocal screw r$.2i passes

158
Figure 5.4 Reciprocal screw reciprocal to all other
screws in subchain 1 but

159
Figure 5.5 Reciprocal screw r$^2i reciprocal to all other
screws in subchain i but i^i

160
through the center of spherical joint and intersects both
the axes of joints A^ and B¿. A numerical example is given
to illustrate how to obtain such reciprocal screws as
follow:
Numerical example. With the base arrangement of the
manipulator shown in Fig. 3.37 and the dimensions of the
first subchain given as follows:
al = 3, bi = 2, s^i = 1, h = 1, a-^i = -90° and °C^ =
[4.000, 0.000, 2.73 2]T,
one set of the four possible solutions can be obtained as
= 1, ©a2. = 0° and ©bi = -60° (see Section 2.5.1).
Therefore, the position vectors of joints and with
respect to the global coordinate system OXYZ can be
calculated as
0.000
3.000
°—1 =
-1.000
and °B^ =
-1.000
1.000
1.000
Therefore,
the
motion
screws
of the
joints with
the global
coordinate
system
OXYZ system can be
ill =
(0,
0,
1;
-1,
0,
0)
Í21 =
(0,
0,
0;
0,
0
1)
Í31 =
(0,
1,
0;
-1,
0
3)
\W
.t*.
H*
II
(1,
0,
0;
0,
2.732
0)
Í51 “
(0,
1,
0;
2.732,
0
4)
Í61 =
(0,
0/
1;
0,
-4
0)
respect to
expressed as

161
In order to find which is reciprocal to all other
screws in first subchain but $n, the following equations
must be satisfied:
L21prl + M2l2rl + N21Rrl + Lrlp21 + MrlQ21 + NrlR21 = 0
(5.17)
L31prl + M3lQrl + N31Rrl + Lrlp31 + MrlQ31 + NrlR31 = 0
(5.18)
L41prl + M4l2rl + N41Rrl + Lrlp41 + MrlQ41 »NrlR41 = 0
(5.19)
L51prl + M5l2rl + N51Rrl + Lrlp51 + Mrl®51 + NrlR51 = 0
(5.20)
L61prl + M6l2rl + N61Rrl + Lrlp61 + Mrl261 + NrlR61 = 0
(5.21)
Therefore, r$.n is obtained by selecting the ratio Rri/Mr^ =
4 and solving Eqs. (5.17) - (5.21) simultaneously, which
yields
r$H = (0, 1, 0? -2.732, 0, 4)
Similarly, r$.2i can be obtained by selecting the ratio
Rri/Mri = 4 and solving the Eqs. (5.18) -(5.22)
simultaneously, which yields
rÍ21 = (4, 1, 6.928; -2.732, -16.784, 4)
where
Lllprl + MllQrl + NllRrl + Lrlpll + Mrl^n + NriRll = 0
(5.22)

162
Similarly, the reciprocal screws r£ü and r$2i»
reciprocal to all other screws in subchain i (i = 2 and 3)
but and $.2i/ respectively, can be found as the positions
and orientations of joint Cj_ are given and therefore are not
listed here.
It is noted that a force dose not affect the rotation
around a revolute joint when its line of action is parallel
to or along the axis of revolute joint. This is also the
reciprocity condition.
5.5 Jacobian Matrix
Let oj$_ represent the motion screw of the platform with
respect to the global system OXYZ, which can be expressed as
6 3
4=2 E Ujiíji (5.23)
j=l i=l
Taking reciprocal products with reciprocal screws r£ü and
r*2i of
Eq. (5.
,23) respect
ively yields
to $
0 rÍli
= '"liili 0
rili
(5.
.24
and
to $
o r$2i
= lu2ii2i 0
ri2i
(5.
,25
where the symbol o denotes the reciprocal product of a pair
cf screws and i = 1, 2 and 3. Thus, we can obtain the
following equation:

163
‘rillT
$H°r$
11 0
0
0
0
0
“11
rÍ21T
0
Í21°rÍ21
0
0
0
0
“21
rÍ12T
“[S] =
0
0 $
12°rI
12 0
0
0
“12
rÍ22T
0
0
0
Í22°r
—22 0
0
"22
rÍ13T
0
0
0
0
Íl3°rS-13
0
“13
rÍ23T
0
0
0
0
0 $.23^—;23
“23
(5.26)
or
[A]u>[$] = [B] [u>a] (5.27)
where the components of [u>a] = u^i, “12' “22' “13'
U23)T are the intensities of the motion screws of the
actuators, and for the linear actuators of the joints,
“21 = v21> “22 = v22 and “23 = v23■
If [A] is nonsingular, we can write
i4$] = [A]”1[B] [ua] = [J][uia] (5.28)
where [J] = [A]"1[B] may be called a Jacobian matrix.
5.6 Mechanical Error Analysis of the Platform
The term of mechanical error defined here means the
minor movements of the platform due to the inaccuracies in
displacements (rotation and translation) of the actuators
for a certain specified position and orientation of the
platform. Let the errors of the displacements of the
actuators, [663], be expressed as

164
[fi©aJ = C u¡a ] fit #
(5.29)
where 6t denotes a very short period of time.
Then we have
wót[$J = [J][wa]6t
(5.30)
or
6©[$] = [J][60a]
(5.31)
where 60 denotes the minor angular displacement of the
platform about its axis due to the error input, 60a, of the
actuators.
The movement of the platform is generally specified by
the linear displacement of the center, H, and the angular
displacements of the hand coordinate system embedded in the
platform about the axes of the global coordinate system
OXYZ. In order to calculate the components of the minor
movements of the platform, the motion screws of the platform
may be expressed as
u>$ = w[s + SQ] = w[s + e (r x S + hS) ]
(5.32)
where r is the global position vector of an arbitrary point
on the axis of the motion screw of the platform, e denotes
the dual part of a vector and h is the pitch; this screw is
shown in Fig. 5.6.
Now, we have
h = S • SQ
(5.33)

165
COS
Figure 5.6 Screw representing the motion of the platform
with respect to the global system OXYZ

166
and
r x S = Sq - hS (5.34)
Since r = H + rH, we can have the following equations:
(H + rH) x S = SQ - hS (5.35)
or
rx S + hS = Sq - H x S (5.36)
The velocity of the center H of the platform can be
obtained as
vh = u(r^ x S + hS) (5.37)
or
X-H = ^t^o " M x (5.38)
Then the linear displacement of H can be obtained as
6dH = 6©(S0 - H x S) (5.39)
As we consider differential motions, let us define the
rotation matrix of a small rotation angle 6© about unit
vector S. For small angles, sinS© ~ 60, cos60 « 1 and VS©
= cosó© -1=0. Thus, we can have the rotation matrix
with small 6© as
VÓ0+C6©
SxSyV6©-SzS60
sxSzV60+SyS6©
0
yV6©+SzS60
Sy2V6©+C60
SySzV60-SxS6©
0
zV6©-SyS6©
SySzV60-fSxS6©
Sz2V60+C60
0
0
0
0
1
(continued)

167
1
Szó0
-Sy68
o
-SzS0 Sy66 O
1 -Sx60 O
sxse i o
O Q 1
(5.40)
Dividing the small rotation angles, 60, about S into three
components 60x, 60y and 60z about the axes of the global
system OXYZ, the rotation matrix can be expressed as
Rot(£0x, X) Rot(60y, Y) Rot(60z, Z)
0
1
66„
1
60z
-60y
0
0
-60x
1
0
-60z
1
60v
1—1
1
0
60y
0
1
-69z
0
r —
o
0
1
0
0
60z
1
0
0
>1
©
tO
1
0
1
0
0
0
1
0
0
0
0
1
0
0
0
1
68y
-60x
1
0
(5.41)
where 60x, 60y and 60z are all small angles. We therefore
simplify by neglecting the second and third order terms.
Comparing Eqs. (5.40) and (5.41), we obtain
60x = Sx60
6©y = Sy60 (5.42)
6SZ = Sz60

168
It is noted that a differential rotation 60 about an
arbitrary unit vector S is equivalent to three
differential rotations, 60x, 60y and 6©z, about the
X, Y and Z axes. Finally, the error movement of the
platform may be expressed as a vector as follow:
60S
S
. 6-H
So - H x S
60
(5.43)
The range of uncertainty of the position of any point
in the end-effector body, say, the point H, is shown in
Fig. 5.7. Such uncertainty in space can be approximated by
a small parallelepiped, whose three edges are 6h, P60 and
6p. Here, 6h is the axial position error, p is the distance
of point H from the error screw axis $_, 60 is the angular
error range and 6p is due to the uncertainty of position of
the screw axis.
5.7 Summary
The mechanical error analysis of a six degree-of-
freedom parallel manipulator has been studied in this
chapter. Matrix transformations, widely used in spatial
mechanisms, are applied. The screw theory, as applied to
determine the instantaneous kinematics of fully parallel
manipulators, is utilized. Numerical example is presented
to show how to determine the reciprocal screws in each
subchain. Velocities and small displacements of the joints

169
Figure 5.7 Range of uncertainty of the position of point H
on the end-effector

170
as well as the region of uncertainty of hand position due to
actuator errors are described.
The interferences between links, which affect the
working space and are related to the mechanical error of
the hand, are not considered in this chapter. The effect of
clearances in the joints and tolerances on link lengths and
the resulting uncertainty of the hand position in parallel
manipulators are not considered in this chapter either.
These are some of the subjects recommended for further study.

CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH
6.1 Conclusions
An industrial robot is defined by the U.S. Robot
Industries Association (RIA) as a "reprogrammable,
multifunctional manipulator designed to move material,
parts, tools, or specialized devices through variable
programmed motions for the performance of a variety of
tasks”. In the early development of industrial robots, they
operated in a fixed sequence and could repeat a sequence of
operations once they have been programmed. Later, the
robots were equipped with sensory devices that allowed them
to act in a not-completely defined environment. Now some
industrial robots have the intelligence to allow them to
make decisions.
It has been well recognized that, by using multi-
degree-of-freedom robot manipulators with multiple actuators
and automatic control systems, we can achieve the goal of
improving efficiency, accuracy, reliability and reducing
energy consumptions and cost of production in a flexible
manufacturing system.
Robot applications are continuously widening, their
functions are becoming more and more complicated and the
optimal designs are not yet established. Therefore, some
171

172
basic parameters of robots should be considered before the
applications to robotics, and they are listed as follows:
1. Mobility (degrees of freedom)
2. Workspace (volume, shape and degrees of
redundancy)
3. Agility (effective speeds of execution of
prescribed motions)
4. Positioning (accuracy and repeatability)
5. Dynamics behavior (structural stiffness, masses,
payload, damping coefficients and natural
frequencies)
6. Economics (cost, reliability, maintainability,
etc.)
In recent years, there has been considerable increase
in research in the area of robotics and multi-degree-of-
freedom programmable automation devices. Almost all the
industrial robots in use now are open-loop serial-link
manipulators with up to six degrees of freedom. Research in
the field of parallel manipulators is still in the infancy
of development. Therefore, the aim of this study is to
investigate and develop the theoretical background in the
kinematics of parallel robots. One of these kinds of
manipulators has been shown in Fig. 3.1.
The special feature of the proposed parallel
manipulator is that the novel geometry of a two-degree-of-
freedom self-actuated joint which combines a rotary actuator

173
and a linear actuator, is introduced. It imparts cylindric,
two-degree-of-freedom, combined rotational and translational
relative motion to the link with respect to ground. The
distinct advantage of using this kind of actuator, called R-
L (Rotary-Linear) actuator, is that the inertias of the
rotary and linear actuators are independent of each other.
In other words, the inertia of the rotary actuator is not
seen by the linear actuator and vice versa. Furthermore, in
case of parallel manipulators, all such R-L actuators can be
ground-mounted. Thus, the linkage itself can be made
lighter, and better dynamic performance can be achieved. It
is noted that one of the attractive utilizations of the R-L
actuator is that they can also be used in the military.
Since the actuators and their microprocessors are the most
sensitive parts of manipulators, they can be shielded from
external hazards as shown in Fig. 6.1. Therefore, only the
bare moving links with unpowered passive joints are exposed
and they can be quickly and easily replaced once they suffer
environmental or battle damage. Another attractive
utilization, as shown in Fig. 6.2, is that the manipulator
can be in an overhead, upside-down position for high-
precision micro-manipulation.
The use of ground-mounted R-L actuators in conjunction
with a parallel manipulator configuration promises to
provide some unique features which can be expected as
follows:

174
Figure 6.1 Shielded R-L actuators

175
Figure 6.2 Overhead Micro-manipulator

176
1. increased payload capacities with ground-mounted
actuators
2. increased positioning accuracy supported by the
parallel subchains (legs)
3. increased speed with lighter construction
4. better dynamic performance
5. greater economy in energy consumption and reduced
first cost
6. reduced computation time with parallel processing
7. restricted workspace
8. increased link interference
where items 1 to 6 are advantages, 7 and 8 are
disadvantages.
In this research, the type synthesis and inverse
kinematics of some possible subchains with six degrees of
freedom are covered. Workspace analysis of the parallel,
platform-type manipulator is presented, by determining the
shapes of the subworkspace and the boundaries of the
subworkspace. The root regions in the subworkspace is also
investigated. The workspace of the manipulator is then the
common reachable region of the three subworkspaces
determined by the corresponding subchain. In order to study
the positioning error of the hand (or end effector) of the
manipulator, the theory of screws is applied to determine
the instantaneous kinematics of the manipulator. The hand
position errors due to linear and angular inaccuracies in
the motions of the actuated joints are prescribed.

177
6.2 Recommendations for Further Research
One of the major difficulties in parallel manipulators
is the problem of link interference. Once the link
interference problem can be solved, the generalized
workspace can be found meaningfully.
It is recommended to simulate the motions of parallel
manipulators by computer graphics, since it will facilitate
understanding the geometry of the manipulators.
This study is concerned only with a kinematic
viewpoint. For the practical application of these types of
manipulators, studies of statics and dynamics should follow.
It is hoped that the results of this work will
contribute towards a basic understanding of the limitations
as well as the potential usefulness of parallel, platform-
type manipulators with ground-mounted actuators.

APPENDIX A
ALTERNATIVE METHOD OF FINDING THE COORDINATES OF JOINT C
A.1 Subchain (R-L)-R-S
By using the notations as shown in Fig. A.l and the
method presented in [443/ we can obtain the coordinates of
joint C with respect to the local coordinate system as
R - S-lSx + a12a.i2 + S22S2 + a23a23
(A.l)
where is the translation along the §1 axis, S22 is the
offset along the S2 axis, and a^2 and a2 3 are t^ie
perpendicular distances between successive joint axes.
We can rewrite Eq. (A.l) by using the Table A.l (set 1
of direction cosines - spatial heptagon) as follows:
Cl -Sl 0
0
1
O
1
S1 C1 0
{ sx
0
+ a12
0
+ S22
04
W
1
O
O
1
0
c12
*23
J21
J21
(A.2)
where IS 21 = S2C22 and U21
S2S12â– 
178

179
Figure A.l Subchain (R-L)-R-S

180
Table A.l Direction cosines - spatial heptagon (set 1)
Si(0
0
1 )
a12(1
. 0
0 )
S2 ( 0
-s12 -
c12 >
a23(c2
â–  U21
U21 >
S3(X2
Y2
Z2 )
a34 â–  ~u321
u321 )
S4(X32 ,
y32
z32 >
a45 > -u4321 '
u4321 )
§51X432 ,
y432 ’
z432 >
a56(w5432
• "u54321 -
u54321 >
—6^X5432 .
y5432 '
z5432 >
a67 - -u654321'
u654321>
§7 Y65432'
z65432>
a71 , -Si
0 )
Therefore, the coordinates of joint C with respect to the
local coordinate system can be expressed as follows:
cx = a12cl + S22s12s1 + a23
cy = a12s1 - S22s12Cx + a23 ( slc2 + C1S2C12)
cz = S1 + s22c12 + a23s2s12
The above expression are the same as those in Eq. (2.5)
except for different notation for elements of the subchain.
A.2 Subchain (R-L)-P-S
As shown in Fig. A.2, the coordinates of joint C with
respect to the local coordinate system can be obtained
similarly as
(A.3)
(A.4)
(A.5)

181
Figure A.2 Subchain (R-L)-P-s

182
5 - SiSi + a12a12 + S2Ü2 + a23a23
(A.6)
where S3 and S2 are the translation along the S3 and S2
axes, respectively, and a22 and a23 are the perpendicular
distances between successive joint axes.
We can rewrite Eq. (A.6) by using the Table A.l (set 1
of direction cosines - spatial heptagon) as follows:
o
w
1
t—1
u
1..
o
1
O
sx Cl 0
{ Si
0
+ a12
0
+ s2
-s12
0 0 1
1
o
i
c12
a23
J21
J21
(A.7)
where U22 — ®2^"12 U23 — a2a12*
Therefore, the coordinates of joint C with respect to the
local coordinate system can be expressed as follows:
Cx " a12cl + S2s12sl + a23cl
Cv - a12sx - S2s12c1 + a21s
Cz - si + S2c12
23sl
(A.8)
(A.9)
(A.10)
The above expression are the same as those in Eq. (2,13)
except for different notations for the elements of the
subchain.

APPENDIX B
EQUATION OF A GENERAL FORM OF TORUS
If the translation da along the z axis of the local
fixed coordinate system is disregarded in Eq. (2.5), (see
Fig. 2.11), we can obtain the following equations:
cx = C0a(bC©b + a) - S0a(bS©bCab - sbSab) (B.l)
Cy = S0a(bC0b + a) + C0a(bS©bCab - sbSab) (B.2)
Cz — bS0bSab + sbCab (B.3)
Squaring and adding Eqs. (B.l) - (B.3) yields
CX2 + cy2 + CZ2 = a2 + b2 + sb2 + 2abC0b (B.4)
Multiplying Eq. (B.4I by Sab yields
Sab{(Cx2 + Cy2 + Cz2) - (a2 + b2 + sb2)} = 2abC0bSab
(B.5)
Multiplying Eq. (B.3) by 2a yields
2a(Cz - sbCab) = 2abS0bSab IB.6)
Finally, squaring and adding Eqs. (B.5) and (B.6) and then
rearranging terms yields the following equation:
183

184
{(Cx2 + C2 + Cz2) - (a2 + b2 + sb2)}2
7 -) cz sbCab 0
= 4a2{b2 - ( )2}
(B.7)
Sab
If we replace Cx, Cy and C2 by x, y and z, respectively, sj,
by s and by a, we will get the same equation as shown in
Eq. (3.1), which represents the general form of torus [51].

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BIOGRAPHICAL SKETCH
Tzu-Chen Weng was born on September 29, 1956, in
Taipei, Taiwan, Republic of China. In 1979, he received a
Bachelor of Engineering degree in mechanical engineering
from Feng-Chia University in Taiwan. After two years of
military service in the Army, he came to the United States
and enrolled in graduate school at the University of Florida
in Fall 1982. In 1984, he received the degree of Master of
Science in mechanical engineering. He is currently pursuing
the Doctor of Philosophy in mechanical engineering at the
University of Florida.
190

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
George N. Sandor, Chairman
Professor of Mechanical Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Joseph Duffy
Professor of Mechanical Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Ali Seireg
Ebaugh Professor of Mechanical
Engineering

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Gary/fc. Matthew
Assgciate Professor of Mechanical
Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
mj/j
Ralph!Selfringe Y
Professor df Computer and
Information Science
This dissertation was submitted to the Graduate Faculty of
the College of Engineering and to the Graduate School and
was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
December 1988
¡UuM. $>mm.
Dean, College of Engineering
Dean, Graduate School

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