KINEMATICS OF PARALLEL MANIPULATORS
WITH GROUNDMOUNTED ACTUATORS
By
TZUCHEN 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 WisconsinMilwaukee, 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 DMC8508029 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, HanMin, 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 RL (RotaryLinear) Actuator............. 14
2.3 Type Synthesis............................ 18
2.4 Technical Discussion..................... 22
2.5 Inverse Kinematics....................... 30
2.5.1 Subchain (RL)RS................ 32
2.5.2 Subchain (RL)PS................ 37
2.5.3 Subchain (RL)SR................. 39
2.5.4 Subchain (RL)SP................ 45
2.6 Summary.................................. 50
3 WORKSPACE ANALYSIS OF THE MANIPULATOR .......... 52
3.1 Introduction............................. 52
3.2 Configuration of a Paralle Manipulator
with RL 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 NoHole 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 PartialRotatability
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 (RL)RS....................... 178
A.2 Subchain (RL)PS....................... 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 GROUNDMOUNTED ACTUATORS
By
TZUCHEN 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 openloop seriallink mechanisms, but only a few
have dealt with multidegreeoffreedom parallel
manipulators such as the Stewart platform or similar
mechanisms.
A new type of twodegreeoffreedom RotaryLinear (RL)
actuator was adapted in this work. Several possible
configurations of parallel sixdegreeoffreedom
manipulators with groundmounted 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 sixdegreeoffreedom 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 endeffector
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,
multidegreeoffreedom mechanical systems are now becoming
a practical choice for use in automatic machinery. It has
been well recognized that, by using multidegreeoffreedom
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
openloop seriallink manipulators in which the number of
degrees of freedom of the endeffector is equal to the sum
1
of the relative degreesoffreedom of the joints in the
chain. There are only a small number of multidegreeof
freedom designs which involve multiloop manipulator
linkages, with totally parallel or partially parallel
configurations. Since the design techniques for multiloop
robotic manipulators are still in the infancy of their
development, investigating and developing the theoretical
background for multidegreeoffreedom multiloop 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 endeffector 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 multidegreeoffreedom 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
threelink robotic arms with pinjoints. 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 revolutejoint, seriallink
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
threeaxis, turningpairconnected 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 Dshaped 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 envelopesurfaces 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 fourway 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
multidegreeoffreedom manipulators. The Stewart platform
[31] is a kind of parallel manipulator which has two plates
connected by six adjustable legs and is a sixdegreeof
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 directdrive arm to a closedloop fivelink
mechanism to overcome the problems encountered in openloop
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 closedloop
planar fivelink 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 platformbased
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 groundmounted rotarylinear 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 closedloop 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
fourbar 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
Openloop seriallink 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 multidegreeoffreedom 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 multidegreeoffreedom devices such
as robots and N.C. machines. The configurations which have
been widely used for these machines are seriallink (open
chain type) arrangements, where one link is connected to
adjacent links by singledegreeoffreedom joints, each with
its separate actuator. Similarly, although N.C. machines
are also seriallink 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 singledegreeoffreedom 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 singleloop 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
degreeoffreedom kinematic pairs and used screw theory,
kinematic geometry and the techniques of linear algebra to
systematize inparallelactuated robotarms. Mohamed and
Duffy [46] also applied screw theory to study the
instantaneous kinematics of the endeffector platform of
fully parallel robottype device. Sugimoto [47] derived the
kinematic and dynamic model for a parallel manipulator by
using motor algebra and NewtonEuler 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 endeffector 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 seriallinkage manipulators have each of
their upto 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, multiloop 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 groundmounted twodegreeof
freedom selfactuated 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 (twodegreeoffreedom,
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 RL (RotaryLinear) Actuator
Two different configurations of RL 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 (RL) actuator
ROTARY ACTUATOR
LINEAR ACTUATOR
BEARINGS PULL AND PUSH
ARRANGEMENT
Figure 2.3 (RL) 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 RL 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, RL 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 openloop
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 RL 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 RL actuator could be offtheshelf
items, since the power to weight ratio is not a major
concern in this situation. Thus the cost of the RL
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 threedegreeoffreedom
and a sixdegreeoffreedom 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 sixdegreeoffreedom
manipulator, as shown in Fig. 2.6, has additional six
redundant degrees of freedom: the axial rotation of the six
SS links which causes uncontrolled wear in the S joints.
By using groundmounted RL 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 RL actuator groundmounted, we can have
several possible configurations for each subchain of the
parallel manipulators with six degrees of freedom. These
configurations are as follows:
Dyads (RL)SR (RL)RS (RL)PS
(RL)SP (RL)CC
Triads (RL)RRC (RL)CRR (RL)RCR
(RL)RPC (RL)RCP (RL)PRC
(RL)PCR (RL)CRP (RL)CPR
(RL)PPC (RL)PCP (RL)CPP
Figure 2.4 Stewart platform mechanism
Figure 2.5 Parallel platformtype manipulator with three
degrees of freedom
Figure 2.6 Parallel platformtype manipulator with six
degrees of freedom
22
These chains are shown in Figs. 2.7 and 2.8, where the order
of (RL) can be reversed as (LR).
2.4 Technical Discussion
Actuator. One of the major concerns in the design of
serial openloop 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
degreeoffreedom actuators are to be used, this will
result in a fiveloop linkage for a sixdegreeoffreedom
robot manipulator. Further, if only onedegreeoffreedom
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 moredegreeof
freedom selfactuated joints. A sixdegreeoffreedom
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 seriallink
01II
(RL)SR
(RL)RS
(RL)PS
(RL)SP
(RL)CC
Figure 2.7 Possible configurations of dyads with six
degrees of freedom
I6
____ _
E^^^r
(RL)RRC
(RL)CRR
(RL)RCR
(RL)RPC
(RL)RCP
(RL)PRC
Figure 2.8 Possible configurations of triads with six
degrees of freedom
25
(continued)
~c~Fi
(RL)PCR
(RL)CRP
(RL)CPR
(RL)PPC
(RL)PCP
WristSwing
Actuator HandTwist
Actuator
WristBend
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 openloop
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
groundmounted. 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 sixdimensional 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 degreeoffreedom 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 ith 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 multiloop
manipulator linkages containing multidegreeoffreedom
selfactuated 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 ith
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 (RL)RS, (RL)PS, (RL)SR and (R
L)SP as shown in Fig. 2.7 in the following sections.
2.5.1 Subchain (RL)RS
A schematic diagram of subchain (RL)RS 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 (RL)RS
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 ii 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 A21A11
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 fourthdegree 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 (RL)RS has a fourthdegree 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 (RL)PS
Figure 2.12 shows a subchain (RL)PS. The inverse
kinematics is similar to that of the subchain (RL)RS.
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 A21Al1
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 (RL)PS
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 seconddegree 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 (RL)SR
In Fig. 2.13, a schematic diagram of the subchain
(RL)SR is shown. The spherical pair is kinematically
X
0
Figure 2.13 Subchain (RL)SR
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 A31A21
yields Eq. (2.20). Premultiplying both sides of Eq. (2.19)
by A1I and then postmultiplying both sides by A31 yields
Eq. (2.21).
H0 1^
HOA3A21
A11HOA31
(2.20)
(2.21)
Since ([A1](1,4))2 + ([A1](2,4))2 = ([HOA31A21](1,4 )2 +
([HOA31A21](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)
[HOA31A21](2,4) / [HOA31A211(1,4) holds, and can be
expressed as
a= tanl( 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) = [HOA31A21](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 = tanl( aSa + (2.26)
axC8a + aySea
eb2 = tanl( aC + ) (2.27)
azCbbl
nzSOc + SzC~c
9b3 = tan1( nSc + ) (2.28)
szS9c + n+zCc
Since the subchain of (RL)SR has a fourthdegree
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 (RL)SP
The procedure of inverse kinematics of the subchain (R
L)SP, as shown in Fig. 2.14, is similar to that of the
z
0
0
Figure 2.14 Subchain (RL)SP
subchain (RL)SR 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 A31A21
yields Eq. (2.30). Premultiplying both sides of Eq. (2.29)
by Al1 and then postmultiplying both sides by A31 yields
Eq. (2.31).
Al = H0A31A21 (2.30)
A2 = A11HA31 (2.31)
Since ([Al](1,4))2 + ([A1](2,4) 2 = ([HOA31A211(1,4) 2 +
([HOA31A21](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) =
[HOA31A21](2,4) / [HOA31A21]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) = [HOA31A21](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 = tan1( axSa + ayCS) (2.35)
axCea + aySqB
axC8a + aySea
Eb2 = tan1( 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 (RL)SP has a seconddegree
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 (RL)RS, (RL)PS, (RL)SR and (RL)SP
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
(RL)RS 4
(RL)PS 2
(RL)SR 4
(RL)SP 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 seriallyactuated 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 loaddependent error.
They also suffer relatively lowfrequency oscillations.
With inparallelactuation by groundmounted 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 actuatorerror is not
cumulative, greater precision is likely to be attainable
without excessive control complications. There should
surely be a future for inparallelactuation 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 rotarylinear (RL) 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 sixdegreeoffreedom
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 finitesize platform are derived in
Chapter 4.
RL Actuator
Figure 3.1 Sixdegreeoffreedom closedloop 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 groundmounted (RL) joint. In this chapter, the
shapes of the abovedescribed 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 noncoplanar 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 finitesize 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
openloop multidegreeoffreedom seriallink 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 RL Actuators
A sixdegreeoffreedom parallel manipulator, where
all actuators are groundmounted, is considered in this
chapter. It has three sixdegreeoffreedom subchains, each
of which has a twodegreeoffreedom RL 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
degreeoffreedom kinematic pairs, are used in the
subchains. A CylindricRevoluteSpherical ((RL)RS) triad
may then be used as the subchain with the C joint (RL
actuator) connected to the ground, and the S joint connected
to the endeffector platform of the manipulator. The axes
of the C joints (RL 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 RL actuators on the base.
Also, the RL 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 endeffector 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 tradeoff 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 RR 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 (RL)RS when the platform is
assumed to be infinitesimal and without the consideration of
translation along the axis of the cylindric (RL) joint.
The shapes of the torus are illustrated first, then the
Figure 3.3 An RR (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 RL 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, symmetricaloffset and general forms as shown in
Fig. 3.4. They also introduced two types of bitangent
plane2, Atype, whose quartic curve intersection with the
torus always encircles the OZ axis and Btype, whose points
of tangency are both on one side of the OZ axis. Any
bitangentplane 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 bitangentplane 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 Atype
bitangentplane 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 bitangentplane has two points of tangency with a
toroidal surface.
AZ Edge view of Atype
bitangentplane
I.
~^D0
Sa) Common torus
(b) Flattened torus
Figure 3.4 Diametral sections through tori: (a) common,
(b) flattened, (c) symmetricaloffset and (d)
general types.
61
(continued)
Edge view of Btype
bitangentplane
) Symmetricaloffset torus
(c) Symmetricaloffset torus
Edge view of Btype
bitangentplane
/ Edge view of Atype
bitangentplane
Edge view of Atype
bitangentplane
(d) General torus
,urve of .ntersection
(common torus cna A ype p:ane)
6
2
4
6
6 1 2
Xaxis, cnes
Figure 3.5 Intersection of the Atype bitangentplane 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 anchorring, 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 eggshaped 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 eggshaped. 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).
Symmetricaloffset form. The symmetricaloffset 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 anchorring 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 symmetricaloffset
form of torus: (a) a > b, (b) a = b and
(c) a < b
curves in the diametral section become bananashaped, 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
bananashaped, 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 RR 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 lentilshaped 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)RS 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 RL 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 ith subchain.
Ai, Bi RL actuator joint and revolute joint in the
ith 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 groundmounted
RL 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 ith dyad of the
manipulator as the offset at joint Bi, sbi # 0
(infinitesimal platform)
Figure 3.21 Subworkspace element of the ith 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 tworoot region or
twoway accessible region. In other words, there are two
sets of Sai and ebi, or two kinematic branches of the ith
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 oneroot regions.
(Recall that this is a planar RR 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 tworoot region of the subworkspace. The
other portion of the subworkspace is the fourroot 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 tworoot
region and a fourroot region, and the surface of the
cylinder of radius r2i itself is a threeroot region. It
can also be seen that the external boundary (the surface of
the cylinder of radius r = r1i) is a oneroot region, and
the internal boundary (the surface of the cylinder of radius
r = r3i) is a tworoot 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
ziaxis
Figure 3.24 Diametral section of flattened form of torus
(ai = 2, bi = 10, Sbi = 0 and abi = n/3)
