Group Title: Kinematics of parallel manipulators with ground-mounted actuators
Title: Kinematics of parallel manipulators with ground-mounted actuators /
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Title: Kinematics of parallel manipulators with ground-mounted actuators /
Physical Description: vi, 190 leaves : ill. ; 28 cm.
Language: English
Creator: Weng, Tzu-Chen, 1956-
Publication Date: 1988
Copyright Date: 1988
 Subjects
Subject: Manipulators (Mechanism)   ( lcsh )
Actuators   ( lcsh )
Kinematics   ( lcsh )
Robotics   ( lcsh )
Mechanical Engineering thesis Ph. D
Dissertations, Academic -- Mechanical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
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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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Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: alephbibnum - 001131657
oclc - 20233905
notis - AFM8966

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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)




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