OPTIMIZATION OF
SPATIAL MECHANISMS
By
CHARLES FREDERICK REINHOLTZ
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
1983
With love to Jeri and Nicholas
ACKNOWLEDGMENTS
I wish to express my sincere appreciation to the
chairman of my supervisory committee, Dr. George N. Sandor,
for his guidance, encouragement and support throughout my
undergraduate and graduate studies. He has probably ful
filled more of Rudyard Kipling's "ifs" than any other
person I have known. I hope I have become a little more
like him.
I gratefully acknowledge the advice and support
given by the members of my supervisory committee, Dr.
Joseph Duffy, Dr. Ralph Selfridge, Dr. ChenChi Hsu and
Dr. Gary Matthew.
Special thanks are extended to Dr. Robert Gaither,
Chairman of the Department of Mechanical Engineering, for
encouraging my graduate studies and for his strong contri
bution toward my professional development.
Thanks are also due to the many fellow students and
associates who provided technical help and moral support.
In particular I thank Mr. Keith Soldner, Mr. Mark Thomas,
Mr. Xirong Zhuang, Mr. Manuel Hernandez and Mr. Ashitava
Ghosal.
The financial support of the Army Research Office
under grant DAAG29K0125 is gratefully acknowledged.
I am indebted to my father and mother for twenty
nine years of unfailing love and support.
Finally, I extend my deepest appreciation to my wife,
Jeri, for the unselfish time she devoted to the typing of
this manuscript, and for the years of patience and
encouragement she has provided.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS. . . . . . . . . . .iii
ABSTRACT. . . . . . . . . . . viii
CHAPTER
BACKGROUND AND MOTIVATION FOR THIS
RESEARCH. . . . . . . . . . 1
1.1 Introduction . . . . . . . . 1
1.2 The Elements of Mechanism
Optimization . . . . . . . 4
1.3 History and Literature Review. . . .7
1.4 Conclusions of the Literature Review . 23
2 OPTIMIZATION THEORY . . . . . . . 26
2.1 Introduction to Optimization . . . 26
2.2 Formal Definition of the
Optimization Problem . . . . . 28
2.3 The Mechanism Optimization Problem . . 30
2.4 Solving the Mechanism Optimization
Problem . . . . . . . .. 37
2.4.1 Constrained Nonlinear
Optimization. . . . . .. 38
2.4.2 Indirect Constrained Nonlinear
Programming Techniques. . .... 40
2.4.3 Unconstrained Nonlinear
Optimization. . . . . .. 50
2.4.4 Hooke and Jeeves' Nonlinear
Programming Method. . . ... 53
3 PHILOSOPHY OF MECHANISM OPTIMIZATION. .. .. . 58
3.1 The Need for a General Philosophy. .. . 58
3.2 Objectives and Constraints of
Mechanism Optimization . . . . 60
3.3 Observations and Trends Affecting
Mechanism Optimization . . . . 67
3.4 Development of a General Mechanism
Optimization Philosophy. . . . .. 69
4 PRECISION POSITION SYNTHESIS OF SPATIAL
MECHANISMS. . . . . . . . . 74
4.1 Introduction to Precision Position
Synthesis. . . . . . . .. 74
4.2 Dyadic Synthesis of Mechanisms . . . 75
4.3 Position and Orientation of a Body
in Space . . . . . . . . 79
4.4 Synthesis of the RevoluteSpheric
(RS) Dyad. . . . . . . . 82
4.5 Synthesis of the CylindricSpheric
(CS) Dyad. . . . . . . .. 89
4.6 Synthesis of the CylindricCylindric
(CC) Dyad. . . . . . . .. 96
4.7 Synthesis of the RevoluteCylindric
(RC) Dyad. . . . . . . . .108
4.8 Conclusions of Precision Position
Synthesis. . . . . . . . .108
5 OPTIMIZATION OF THE RCCC MECHANISM. . . .111
5.1 Problem Definition . . . .
5.2 Satisfying Additional Motion
Requirements . . . . .
5.3 The Grashof Condition. . . .
5.4 The BranchAvoidance Condition .
5.5 The Order Condition. . . . .
5.6 FixedPivot and LinkLength Ratio
Conditions . . . . . .
5.7 The Objective Function . . .
5.8 Numerical Example. . . . .
6 OPTIMIZATION OF THE RSSRSC AND RSSRSS
MECHANISMS. . . . . . . .
6.1 Problem Formulation. . . . .
6.2 Method of Design . . . . .
6.3 The BranchAvoidance Condition .
6.4 The Grashof Condition. . . .
6.5 The Transmission Characteristic
Condition . . . . . .
6.6 FixedPivot and LinkLength Ratio
Conditions . . . . . .
6.7 Satisfying Additional Motion
Requirements . . . . .
6.8 The Order Condition. . . . .
6.9 The Objective Function . ..
6.10 Numerical Example. . . . .
. . .112
. . .117
. . .121
. . .126
. . .128
. . .129
. . .134
.139
. . .139
. . .142
. . .145
. . .150
. . .157
. . .161
. . .162
. . .163
. . .164
. . 168
7 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE
RESEARCH. . . . . . . . . .173
7.1 Conclusions. . . . . . . . .173
7.2 Recommendations for Future Research. . .175
APPENDICES
1 RCCC MECHANISM OPTIMIZATION PROGRAMS. . .179
2 RSSRSR MECHANISM OPTIMIZATION PROGRAMS . .189
REFERENCES. . . . . . . . . . . .201
BIOGRAPHICAL SKETCH . . . . . . . . .213
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
OPTIMIZATION OF SPATIAL MECHANISMS
By
Charles Frederick Reinholtz
August, 1983
Chairman: George N. Sandor
Cochairman: Joseph Duffy
Major Department: Mechanical Engineering
The material in this dissertation can be effectively
divided into two subtopics: philosophy of optimal mechanism
design, and optimization of dyadbased spatial mechanisms.
The first subtopic, philosophy of optimal mechanism
design, is intended to be general in nature, applying to all
types of mechanisms, both higher and lower pair, and both
planar and spatial. This is covered in Chapters One through
Three. Chapter One examines past approaches to mechanism
optimization. Chapter Two is a brief review of optimization
theory, particularly as it applies to mechanism optimization.
Chapter Three draws upon the insights gained in the first
two chapters to formulate a general approach to the mechanism
optimization problem.
The second subtopic of this dissertation, optimization
of dyadbased spatial mechanisms, is covered in Chapters
viii
Four through Seven. This is actually a rather limited
example of applying the philosophy developed in the first
three chapters. Nevertheless, the mechanisms treated in
this section are believed to represent some of the most
useful motion generating spatial mechanisms, and, therefore,
those for which improved design theories are most urgently
needed. In Chapter Four, closedform synthesis equations
are derived for dyads containing revolute (R), spheric (S)
and cylindric (C) pairs. Chapters Five and Six present
detailed examples of the optimization of the fourlink
RCCC and fivelink RSSRSC and RSSRSS mechanisms. Finally,
Chapter Seven outlines procedures for the optimization of
other dyadbased spatial mechanisms, and offers suggestions
for further research.
CHAPTER 1
BACKGROUND AND MOTIVATION FOR THIS RESEARCH
1.1 Introduction
The design of mechanisms, like almost all design
problems, is an iterative process. Generally, a
mechanism can be synthesized to meet some of the design
requirements, but then it must be analyzed or tested to
determine whether or not it satisfies the remaining
requirements. Most often it does not, and the designer
must "go back to the drawing board" and synthesize a
new mechanism which again must be analyzed or tested.
This second design may also not satisfy all the design
requirements, but it is likely to be somewhat of an
improvement over the first design, since the designer
had the "experience" of the first design on which to
base this second synthesis. The third design attempt
will likely be better still, since now two points of
reference are available. This iterative process is
continued until a satisfactory solution is obtained.
It is probably the oldest and most often used form of
optimization.
For the purpose of this dissertation, optimization
may be considered to be the process of seeking the best
result under a given set of circumstances. The questions
of what optimization processes are available, what is
meant by the "best" result and how these "circumstances"
are incorporated into the solution will be topics of
considerable importance. The following paragraphs
briefly introduce these concepts.
As has already been stated, optimization may be any
process which seeks to find the best result to a given
problem. Available optimization methods include the
intuitive successive improvement approach described
previously; classical optimization methods, such as the
Lagrange multiplier method, based on the principles of
variational calculus; and a large and rapidly expanding
body of knowledge known as mathematical programming.
The last of these, mathematical programming, is most
important with respect to this work. In a sense,
mathematical programming methods are an extension and
formalization of the ageold successive improvement
process. These newer methods, though, have become quite
sophisticated, employing a variety of numerical techniques
and almost always requiring programming on a digital
computer. Also, by formulating some measurement of the
quality of a given design, the burden of decision
making after each iteration can be removed from the
designer. The most general form of mathematical
programming aims at solving nonlinear types of problems
and, not surprisingly, is called nonlinear programming.
It will soon become evident that mechanism optimization
generally involves nonlinear programming. A detailed
discussion of optimization methods is given in Chapter
Two.
The second question to be addressed is what is meant
by the "best" result. Suppose, for example, that an
airplane strut is to be designed for a given design load
with maximum stiffness, but with minimum weight. It is
apparent that these represent competing design require
ments, and that some compromise between the two must be
reached. Here, the designer's judgment must enter the
process and he must decide what importance or "weight"
to assign to each requirement. Of course, mechanism
design problems are rarely this simple, and will usually
involve a much larger number of design requirements. As
a result, finding the best design usually involves
some amount of subjective opinion. Even if the importance
of the various design requirements were exactly known,
the best mathematical solution would probably not
correspond to the best physical solution. This is because
there are inherent errors and approximations present in
the modeling and analysis of any real system. Thus,
although mathematicians may speak of a "globally optimum"
or best solution, the best that can be hoped for in an
engineering setting is to find some approximation to the
best solution.
The final topic of this section addresses the
question of what role the "circumstances" play in
determining a best design. Will the machine designed
to run at 60 revolutions per minute (rpm) perform
acceptably at 600 rpm? Perhaps not, since the circum
stances or "constraints" under which the design was
developed have been altered. Constraints may be cast in
the form of equality or inequality conditions, and they
may be explicit or implicit. In any case, it is evident
that the circumstances surrounding a design will also
have an effect on which solution is best.
1.2 The Elements of Mechanism Optimization
Generally speaking, the optimization of any system
or component comprises four principal elements. These are
(1) the choice of a conceptual design;
(2) the development of an analyzable model
of the physical system from which the
design variables and the governing
equations may be extracted, both for
synthesis and analysis;
(3) the setting up of a scalar function
(called the objective function) of the
design variables which serves as a
measure of the effectiveness of the
system, accompanied by specifying
all design constraints; and finally,
(4) finding the set of values of the design
variables which produce the best value
of the objective function of part (3)
and are consistent with all design
constraints.
The first three of these elements are primarily
dependent on the specific problem at hand; they will
collectively be called the problem formulation. The
fourth element is purely an application of optimization
theory. One of the primary objectives of this dissertation
is to develop a method of mechanism problem formulation
in such a way as to accurately reflect the problem
requirements, while maintaining a simple enough structure
to allow efficient and sufficiently accurate solutions
by one of the available optimization methods.
The first element of the problem formulation to be
addressed is choosing a conceptual design. The designer
must decide what types of mechanisms, such as linkages,
cams, belts, gears, cammodulated linkages, etc., are
best suited to solve a particular problem. Following
this, he must decide the particular configuration of the
mechanism chosen. If a linkage has been selected, should
it have four links or six links? Should it be planar or
spatial? The above decisions are known as type and
number synthesis, respectively. Although these are
extremely important decisions in the optimization process,
a detailed discussion of them is beyond the scope of this
work. Throughout the remainder of this dissertation, it
will be assumed that a specific mechanism type has been
selected, and optimization must proceed from this point.
The second element of problem formulation consists
of synthesis, modeling and analysis of the system concep
tualized above. In the present work, this stage will
draw heavily upon the classical theories of mechanism
synthesis and analysis. Here the term analysis includes,
among other things, application of Grashof's condition
(or a similar spatial condition), branch and order deter
mination and, of course, position, velocity, and
acceleration analysis. Typically, the mechanism to be
optimized will be modeled as having rigid links and zero
clearance in the joints, although other assumptions such
as elastic links and joint clearances and/or compliances
may be applied where appropriate.
The third and final element of the problem formulation
is the setting up of the objective function and the design
constraints. The objective function is a scalar function
of the design variables whose numerical value reflects
the quality of the system being designed. In designing
a mechanism for a specified performance, it may be
desired to minimize the sum of the squares of the
structural errors at a large number of positions. Or
perhaps it is desired to minimize the maximum deviation
of the transmission angle from the ideal value of 900
throughout the motion cycle. Maybe some weighted
combination of these two requirements in the objective
function is best. It is evident that the designer must
possess a thorough understanding of the requirements
of the system being designed if he is to choose an
effective objective function.
1.3 History and Literature Review
This dissertation brings together two fields, namely,
optimization theory and mechanism science,which histor
ically have a related and remarkably similar development.
Both disciplines have their roots in early civilization,
although neither was formally recognized until quite
recently. Both disciplines were topics of interest and
were significantly advanced by the great mathematicians
of the seventeenth and eighteenth centuries. Newton,
Euler and Lagrange are but a few of the men whose names
1Inherent deviation from the prescribed performance,
expressed as a scalar or vector quantity.
are common to both subjects. Even with the contributions
of these and many other great men, development of both
subjects remained slow and sporadic well into the
twentieth century.
By about 1945, both mechanism science and optimization
theory had reached somewhat of a plateau. Graphical
methods of planar mechanism design and analysis were well
established, although their usefulness was limited by the
sometimes inaccurate and tedious nature of graphical
constructions. Classical optimization methods, based on
calculus, were also well established, but were useful for
solving only a limited class of problems. Both fields,
mechanism science and optimization theory, were advancing
at a relatively slow pace. The possibility of solving
new types of problems or more difficult problems seemed
distant indeed.
Then came the dawning of the computer age. Suddenly,
the potential arose for handling large amounts of informa
tion, and for performing tedious calculations quickly and
accurately. The growth rate of both fields virtually
exploded. Kinematicians adopted algebraic methods which
were soon translated into computer code. It became
possible to generate hundreds or even thousands of
mechanisms as possible candidates to solve a given problem.
It soon became apparent that the computer could generate
more solutions than the designer could reasonably assess.
The next logical step was to let the computer examine
the relative merits of each solution, presenting to
the designer only those mechanisms which met some
preestablished criteria. It was quickly recognized
that the rapidly advancing field of optimization theory
was exactly the tool needed for this purpose. The
premise was a simple one: to allow the computer to
perform a logical search for the mechanism or mecha
nisms which best satisfied the designer's requirements
under the given circumstances. This is the foundation
of practically all modern mechanism optimization.
The previous section gave a very brief summary of
the early history of kinematics and optimization.
This section outlines the developments in kinematics
since about 1950, particularly as they relate to
synthesis and optimization of spatial mechanisms.
Many of the references cited here deal exclusively
with planar mechanisms. This is to be expected since
often, although not always, the development of planar
methods of synthesis and optimization have been, or
can be, extended directly to spatial mechanisms.
Also, the inclusion of these planar references will
help to explain how the trends toward many of the
currently used methods were established.
Optimization theory, as a subject unto itself, is
not directly treated in this literature review. Rather,
in the later section on optimization methods, a number
of standard references are cited.
The pioneer of modern kinematics in the United States
is generally considered to be Freudenstein. His 1955
paper, "Approximate Synthesis of FourBar Linkages," (1)
probably marks the beginning of the shift in emphasis
from graphical to analytical methods. The expression,
"Approximate Synthesis," was widely used during this era
to denote precision position synthesis to approximate a
given function. It should not be confused with its more
recent use to denote synthesis of a mechanism which
approximately satisfies a large number of precision
positions. Freudenstein's later work with Sandor (2) in
1959 employed complex number theory and a programmed IBM
650 digital computer in the synthesis of pathgenerating
mechanisms. This work, perhaps more than any other, marks
the marriage of the kinematician to the digital computer.
Other important contributions from this time period
dealing with planar mechanisms include the fourbar
linkage atlas of Hrones and Nelson (3), the additional
works of Freudenstein (4,5) and the combined work of Roth
and Freudenstein (6), to name a few. In the latter work,
path generating geared fivebar linkages are synthesized
using numerical methods and a digital computer. The work
of A.S. Hall (7), along with his organizational involve
ment in the early "Conferences on Mechanisms" from 1953
to 1962, gave inspiration and insight to many of the
researchers who followed him. McLarnan (8) extended
the earlier work of Freudenstein to include the synthesis
of planar sixlink functiongenerating mechanisms.
Analytical studies of spatial mechanism synthesis
in the United States also received attention during the
1950's. Denavit and Hartenberg (9) provided what has
become standard symbolic notation for the description
of the kinematic properties of lowerpair mechanisms.
In a later paper (10), Denavit and Hartenberg extended
the precision position approach of Freudenstein (1) to
the synthesis of spatial RSSR and RCCC mechanisms, and
showed the synthesis equations to be linear up to a
limited number of precision positions.
It should be noted that a number of German and
Russian scholars were also contributing to the advancement
of knowledge on spatial mechanisms at this time. Notable
among these are Beyer (11), Dimentberg (12), Novodvorskii
(13), Stepanov (14), and Levitskii and Shakvazian (15).
For a more detailed review of these works, the reader is
referred to the survey articles of Beyer (16), Harrisberger
(17) and Yang (18).
Until about 1965, the only spatial mechanism
synthesis problems covered in the literature involved
coordinating motions of input and output links, or as
it is commonly known, the function generation problem.
Wilson (19) changed this trend by introducing the
problem of spatial rigidbody guidance. He also showed
the function generation problem to be convertible to a
rigidbody guidance problem by inversion about the
input or output links.
The advancing computer technology of the late 1950's
gave rise to the first numericallybased attempts to
optimize planar mechanisms. Freudenstein (20) devised
an iterative scheme for respacing the precision points
in order to find the linkage which best approximates a
given function. Starting with the wellknown Chebychev
spacing of the precision points, a mechanism was
synthesized. An analysis routine then evaluated the
resulting structural error over the range of the desired
function. Following this, a new "modified Chebychev"
spacing of the precision points was created, and the
analysis step repeated. The resulting errors of the two
trials were then compared, and, based on the result, a
new spacing could be projected.
Roth, Freudenstein and Sandor (21) used a digital
computer to synthesize planar fourbar path generating
linkages with optimum transmission characteristics. This
was done by synthesizing linkages to satisfy four precision
conditions in addition to requiring the maximum and minimum
values of the transmission angle to deviate from 90 by
the same numerical amount. Synthesized mechanisms were
then compared by means of a "quality index" formula and
iterations were performed until the mechanism with the
best value of the index could be found.
Until the work of ChiYeh (22) was published in
1966, synthesis methods invariably required the exact
satisfaction of some number (usually 3,4 or 5) of
precision conditions. ChiYeh's novel approach was to
minimize the structural error in the leastsquares sense
at a much larger set of prescribed positions. ChiYeh's
method involved taking the partial derivatives of the
objective function with respect to the design variables
(the linkage dimensions), and setting these equal to
zero. This resulted in a system of nonlinear equations
which were then solved numerically.
Nonlinear programming was first introduced to
mechanism optimization by Fox and Willmert (23) in 1967.
Their design objective was to synthesize a fourbar
linkage whose coupler point would generate, as closely
as possible, a given curve, and whose crank rotations
would be as close as possible to the desired values.
Constraints were imposed which limited forces and torques
within the linkage, restricted fixed pivot location,
limited link length ratios, assured the desired Grashof
type and required the output positions to be in a
specified order. Also, the possibility of limiting
velocities and accelerations below a certain value was
discussed. This paper is remarkable in that it introduced
nonlinear programming to kinematic synthesis in addition
to developing a number of the constraint conditions which
are still in common use today. In fact, many of the more
recent works in this area have been in an effort to find
more efficient optimization algorithms; however, the
objective function and constraint equations have been
virtually the same as those suggested by Fox and Willmert.
Freudenstein, in his discussion of the paper (23),
described the nonlinear programming method as a "natural
tool" in the area of kinematic synthesis.
Closely following, but apparently independent from,
the work of Fox and Willmert, was a paper by Tomas (24)
which also treated linkage synthesis as a nonlinear
optimization problem. Noteworthy about the work of Tomas
was the first treatment of the function generation problem
by nonlinear programming, and the first published example
of optimization with respect to dynamic properties of
the mechanism.
A different tack to the planar linkage optimization
problem was presented by Garrett and Hall (25). They
chose to generate, by means of a digital computer, a set
of random fourbar linkages. Each of these linkages was
then analyzed to find the one which best suited the
design requirements. Following this, an expanded set
of random linkages was generated in the neighborhood
of this "best" design, and repeating the analysis step,
a new best linkage was selected. This refinement process
was repeated until the desired accuracy was obtained, or
until the method converged. Although it is usually
inefficient, this method is quite simple and is almost
guaranteed to converge. Furthermore, it may have a
better chance of finding the global optimum (26),
although no proof of this is known to exist.
Eschenbach and Tesar (27) contributed to the theory
of planar linkage optimization on two important fronts.
First, they treated the formerly unsolved problem of
optimum design of a coplanarmotion generating fourbar
linkage. Second, they were the first to recognize the
simplification which resulted from using precision
position equality constraints to reduce the number of
freechoice variables.
Leastsquares error synthesis of planar fourlink
mechanisms was a popular subject during the 1960's, and
attracted the interest of such authors as Lewis and Gyory
(28), Levitskii and Sarkisian (29) and McLarnan (30)
among others. Although none of these papers directly
employ nonlinear programming methods, they do point out
the importance given to the problem of minimizing
structural error. Among these works, only the paper of
Lewis and Gyory (28) considers the need to satisfy
additional constraints. Here, a test is performed to
be sure the synthesized mechanism is of the crankand
rocker type. Structural error minimization was also
the objective of Sandor and Wilt (31) in their novel
paper on the optimal synthesis of a geared fourlink
mechanism.
Tomas (32) discounts the importance of minimizing
structural error in many practical problems, and again
presents the idea of optimization with respect to the
dynamic properties of the mechanism. Benedict and Tesar
(33) reinforce this concept by demonstrating the design
of a complex stamping and indexing machine with optimal
torque balance. Improved dynamic properties and
balancing of planar mechanisms were also the objective
of a number of other papers, including those by Berkof
and Lowen (34,35), Huang, Sebesta and Soni (36), Sadler
and Mayne (37) and Elliot and Tesar (38). Kaufman and
Sandor (39) developed a complete force balancing method
for spatial mechanisms.
Variations of the planar mechanism optimization
problems already mentioned have been the subject of
numerous other papers. Most of these papers attempt to
improve the efficiency of the optimization process,
either by using slightly different problem formulations,
or by using one of the recently improved optimization
methods. These include the works of Golinski (40),
Alizade, Novruzbekov and Sandor (41), Rose and Sandor
(42), Savage and Suchora (43), Nolle and Hunt (44),
Bagci and Brosfield (45) and Bagci and Lee (46).
As work in the area of mechanism optimization
progressed, the importance of good problem formulation
combined with an efficient numerical optimization method
became apparent. Kinematicians began to modify existing
optimization methods, or, in some cases, create new
methods, which would be tailored to the peculiarities
of mechanism design. Huang (47) developed a mechanism
optimization scheme based on sensitivity coefficients.
Lee and Freudenstein (48) introduced hueristic combina
torial optimization in the kinematic design of mechanisms.
A hueristic method for sorting mechanism variables into
independent groups was proposed by Datseris and
Freudenstein (49). Sutherland and Siddall (50) presented
a method for optimization using what they called "inverse
utilities" as a basis for comparing the undesireable
characteristics of a mechanism. Spitznagel (51) and
Spitznagel and Tesar (52) developed a very effective
technique for optimizing planar mechanisms based on
Burmester synthesis followed by sequential filtering.
Other interesting and important problems concerning
planar mechanisms were also being addressed in the
literature. Optimal synthesis of sixlink and other
multiloop mechanisms was studied by Chen and Dalsania
(53), Prasad and Bagci (54), Sallam and Lindholm (55),
Mariante and Willmert (56) and Spitznagel (57). Dhande
and Chakraborty (58) were the first to use stochastic
methods to optimize mechanisms considering the effects
of tolerances and clearances. Sevak and McLarnan (59)
considered the problem of optimizing mechanisms with
flexible links. Huey and Dixon (60) studied the
optimization of cammodulated linkages for path and
function generation. Kramer and Sandor (61) and Kramer
(62) developed an optimization technique based on what
they termed "selective precision," wherein structural
errors of the mechanism are held within a prescribed
tolerance at each of the prescribed positions. Sutherland
(63) presented a method for designing fourbar linkages
where some of the prescribed positions are satisfied
exactly and the remaining positions are approximated in
the leastsquares error sense. Chouby and Rao (64)
suggest a direct method for minimizing the structural
error together with the mechanical error due to manu
facturing tolerances on the link dimensions.
Application of numerically based optimization methods
to the design of spherical and spatial mechanisms became
a topic of interest in the early 1970's. Stridher and
Torfason (65) minimized the structural error in a
spherical fourlink pathgenerating mechanism. Bagci
and Parekh (66) demonstrated the optimal design of
spherical fourbar and sixbar linkages. Bagci (67)
designed spherical fourlink mechanisms to have optimal
transmission properties. Rao and Ambekar (68) considered
the design of spherical fourlink functiongenerating
mechanisms with minimum structural error subject to
link length and transmission angle constraints.
Early efforts to optimize spatial mechanisms were
concerned mainly with either minimizing structural error
or maximizing force or load transmission, or some combi
nation of the two. Research in this category includes
the work of Sutherland and Siddall (50), Hamid and Soni
(69) and Shoup, Steffen and Weatherford (70). Gupta
(71) demonstrated the synthesis of RSSR and RSRC spatial
linkages with minimum structural error, subject to
branching, mobility and transmission constraints. Suh
and Mechlenburg (72) synthesized pathgenerating spatial
mechanisms with minimum structural error in the least
squares sense. In a unique work, Bagci (73) optimized
screwgenerating spatial mechanisms. Alizade, Rao and
Sandor (74,75) demonstrated the synthesis of spatial
functiongenerating mechanisms with minimum error subject
to transmission angle constraints. Suh and Radcliffe
(76) used nonlinear programming methods for the precision
position synthesis of spatial mechanisms. Although
closedform solutions were possible for some of the
mechanisms synthesized by Suh and Radcliffe, their work
demonstrates that it is possible to circumvent much of
the laborintensive manipulation of algebraic equations
if one is willing to pay for more computer time.
Bagci and Falconer (77) optimized the transmission
characteristics of RSSR and RSSP functiongenerating
spatial mechanisms. Soylemez and Freudenstein (78)
found the optimum dimensions of an RSSR function
generating mechanism when the extreme positions of the
input and output links are given. Karelin (79) obtained
formulas for determining the optimum link sizes and
slider offset for the RSSP mechanism given the required
stroke. His design was subject to constraints on the
pressure angle.
A number of other papers which do not fall directly
in the category of optimization are, nevertheless,
important to the present study. These can be classified
into two groups: papers dealing with determining the
quality of an existing mechanism and papers dealing
with precision position synthesis of spatial mechanisms.
The first group is important to this study because
all mechanism optimization techniques are based on
quality comparisons. Perhaps the simplest and best
known quality criterion is the Grashof mobility test
(80) for planar fourbar linkages. Filemon (81) extended
the usefulness of this test by determining the regions
of mobility along Burmester's centerpoint curve.
Jenkins, Crossley and Hunt (82) studied the gross
motion attributes of certain spatial mechanisms based
on the intersections of surfaces generated by points
within the mechanism. Duffy and Gilmartin (83,34,85)
generalized Grashof's work by determining the limit
positions and the mobility of fourlink spatial and
spherical mechanisms using the laws of spatial and
spherical triangles. Gupta and Radcliffe (86) used
geometric methods and design charts to determine the
mobility of planar and spatial mechanisms. Waldron
and Stevensen (87) worked on the development and
application of conditions on branching, mobility and
order of positions in planar fourbar linkages. Strong
and Waldron (88) used joint displacements to determine
the mobility regions of fourbar linkages. Sutherland
(89) developed an index for determining the quality of
force and motion transmission of planar and spatial
mechanisms. Gupta (90,91) developed theories for
synthesizing cranktype planar fourbar linkages with
transmission angle control. Gupta and Tinubu (92)
proposed a theory for synthesizing planar and spatial
bimodal function generating mechanisms which were free
from branching problems. Zhuang and Sandor (93,94)
determined the branching condition for a variety of
spatial mechanisms containing spheric joints.
Several early works dealing with the precision
position synthesis of spherical and spatial mechanisms
have already been discussed. Although the subject is
too broad to fully cover here, a few additional references
are cited to give the reader an idea of the methods
currently available.
Dimentberg (95) introduced the screw method to the
study of lowerpair spatial mechanisms. This elegant and
powerful tool provided a concise method for formulating
and solving spatial mechanism problems. Often the screw
method provided closedform solutions to problems which
otherwise require numerical solutions when formulated
using vector or matrix methods. Sandor (96) and Sandor
and Bisshopp (97) introduced methods of dual number
quaternions and stretchrotation tensors to find the
loopclosure equations of spatial mechanisms. Beran (98)
used dual complex numbers to synthesize the RCCC
mechanism for multiply separated positions. Tsai and
Roth (99) used screw triangle geometry to synthesize
openloop kinematic chains. Kohli and Soni (100)
developed spatial mechanism synthesis procedures based
on pair geometry constraints and successive screw
displacements. Suh (101) employed 4x4 matrices in the
synthesis of spatial mechanisms.
Recently, Sandor, Kohli, Reinholtz and Ghosal (102)
developed an analytical closedform method for the
synthesis of a spatial motiongenerating mechanism
for three prescribed positions based on vector geometry.
Sandor, Kohli, Zhuang and Reinholtz (103) extended this
work to four prescribed precision positions; their
solution involves finding the simultaneous solutions
of two cubic equations in three unknown variables. The
value of one of the variables is assumed arbitrarily in
solving the equations.
While fairly extensive, this literature review is
by no means complete, and the reader is encouraged to
peruse the excellent review articles by Fox and Gupta
(104) and Root and Ragsdell (105).
1.4 Conclusions of the Literature Review
The foregoing literature review reveals a number of
important points:
(1) In recent years, optimization theory
has been widely recognized as an
important and natural tool to be
used in the design of mechanisms.
(2) No single mechanism optimization
procedure has received widespread
acceptance. This is partially due
to the newness of the optimization
methods themselves, and partially
due to the wide variety of problems
which occur in mechanism design.
(3) In a majority of the papers reviewed
dealing with optimization, the
objective function was in some way
a measure of the structural error,
although no reason for this choice
was generally given. It is believed
that this approach arose as a
"natural extension" of the methods
of precision position synthesis.
(4) The authors who chose to optimize with
respect to properties other than
structural error most often incorporated
either transmission quality or some
dynamic property of the mechanism in
the objective function. Quite often
in these works equality constraints
were used to force the solution
mechanism to pass through a small
number (usually 3 or 4) of precision
positions.
(5) Very few designers have attempted to
devise a general philosophy or strategy
for mechanism optimization. As a result,
many of the methods developed have
limited application or are relatively
25
inefficient or both.
(6) The literature contains a number of
very fine examples of planar mechanism
optimization. Also, several useful
techniques have been developed for the
synthesis and analysis of spatial
mechanisms. However, the extension of
these techniques to spatial mechanism
optimization has been quite limited.
CHAPTER 2
OPTIMIZATION THEORY
2.1 Introduction to Optimization
Optimization has previously been defined as the act
of seeking the best result under a given set of circum
stances. Finding the best result ultimately means
minimizing something (such as the required effort), or
maximizing something (such as the desired benefit). In
order to have a solvable optimization problem, the
desired benefit or the required effort must be express
ible as a function of a set of variables over which the
designer has control. These variables are called the
design variables. Limits on the values of design
variables may result from such things as limited
material supplies or limited production capabilities.
In either case, these limiting factors are called
constraints, and they, too, must be expressible as
functions of the design variables.
From figure 2.1 it can be seen that the maximum
value of a function, f(x), corresponds to the minimum
value of the negative of the function, f(x). Therefore,
f (x)
f (x)
I f(x*) = minimum of f(x)
x* x
if(x*) = maximum of f(x)
/f(x)
figure 2.1
Example showing the maximum of f(x) to be the
same as the minimum of f(x)
no generality is lost by assuming the optimization problem
to always be one of function minimization.
2.2 Formal Definition of the Optimization Problem
An optimization problem or a mathematical programming
problem can be stated in the following general form
(104,106):
x2
Find X = (2.1)
which minimizes f(X,O) (where 0 is an independent set
of input motion variables having a predetermined range),
subject to inequality constraints
g.(X,O) < 0, j = 1,2,...,m (2.2)
and equality constraints
(X,G) = 0, j = m+1l, m+2, ..., p (2.3)
Here X is called the design vector, and f(X,0) is called
the objective function. The design vector, X, is composed
of the design variables x1, x2, ..., Xn, which may be
written in transposed ixn matrix form as {xl,x2,...,xn} ,
where the superscript T denotes the transpose. The vector
e = {61, 2',"...'6 qT of a set of independent parameters
may typically comprise quantities such as time or
position. Most types of optimization problems do not
contain this independent parameter vector, and, hence,
most textbooks do not include it in defining the
standard optimization problem. However, in mechanism
design, the independent parameters) representing the
mechanism input(s) often appear in the design equations.
When the vector 6 is present in the objective function
or in any of the constraint equations, the problem will
be called a parametric programming problem. This type
of problem is generally much more difficult to solve than
nonparametric problems, because for each new design vector,
X, that is tried, the objective function and the constraints
must be evaluated at every possible value of the independent
input parameterss. In this dissertation, a great deal
of emphasis will be placed on avoiding the need to involve
these independent parameters, while assuring optimal
performance throughout their range.
Finally, it should be pointed out that the components
of the design vector, X, may also be functions of inde
,pendent parameters. This occurs, for example, when link
shapes are considered variables in the design of a
mechanism which is to have maximum stiffness. This type
of optimization problem is known as a dynamic programming
problem. No dynamic programming problems have been
included among the examples of this dissertation.
2.3 The Mechanism Optimization Problem
To put the preceding definition of the optimization
problem in a clearer light, it will now be cast in the
framework of application to the design synthesis of a
simple mechanism.
Suppose it is desired to design a functiongenerating
planar fourbar linkage for a number of arbitrarily
prescribed positions greater than five. Since, in this
case, five is the maximum number of positions that can
be satisfied exactly (unless input and output scale
factors are made components of the design vector), the
best that can be hoped for is to minimize the structural
error at the prescribed positions. A logical and popular
approach to this problem has been to minimize the sum of
the squares of the resulting structural errors at the
prescribed positions. Thus, if the output angle, i, is
some known function, F, of the input angle, 6., at each
of the n prescribed positions, then
,i = F(6i), i = 1,2,...n, (2.5)
and the objective function (O.F.) to be minimized is
n 2
O.F. = Z {F(6 ) G(6 )} (2.6)
i=l
where G(6.), i = 1,2,...,n are the output positions
actually generated by the mechanism. The mechanism to
be optimized is shown in figure 2.2.
figure 2.2 The functiongenerating planar fourbar
linkage
The design vector consists of the design variables
which are the link lengths A,B,C,D and the starting
angles 60 and 40" For a given value of 6=. the values
of these variables determine the values of F(8.) and
G(e6). The values of the variables A,B,C,D,60 and 0
are to be found which minimize the function 2.6.
The reader familiar with kinematics will know that
the relative, rather than the absolute, proportions of
this mechanism determine the input/output functional
relationship. Therefore, D, the length of the grounded
link, can be set equal to unity without loss of generality.
The design vector now becomes
X1 A
x2
T
X = x3 = C = {A,B,C,0 0 (2.7)
x4 0
While finding the value of X which minimizes the
function 2.6 may be quite an interesting problem in
itself, in practice there are usually a number of other
requirements the linkage must satisfy. For example, the
transmission angle labeled i in figure 2.2, must be
IThe transmission angle in a fourbar mechanism is defined
as the acute angle between the coupler link and the output
link.
held reasonably close to 900 to ensure effective conver
sion of the force in the coupler to torque about the
output crank pin. In practice, a value of p less than
about 30 will generally be unacceptable. This require
ment can be expressed in the form of an inequality
constraint equation
p > 300 or p 30 > 0 (2.8)
However, this is not an acceptable form of the equation
because the transmission angle, i, is not a design
variable or an independent parameter. However, p can
be expressed in terms of the design variables and the
independent parameter 6 as follows (107,pp319):
A2 B2 C2 + 1 2Acos(8 + (2.9)
p = arccos 2BC (2.9)
Upon substituting this result into the transmission angle
constraint, equation 2.8 becomes
ar A2 B2 C2 + 1 2Acos( + 0 30 > 0 (2.10))
arccos 30 > 0 (2.10)
2BC
Unfortunately, this is a parametric constraint since
it contains the independent parameter 0. It therefore
seems that equation 2.10 must be evaluated at every possible
value of 3, from 6=00 to 6=3600, for every set of design
variables tried, to be sure the constraint is satisfied.
Fortunately, this is not usually necessary because, as
shown by Roth, Freudenstein and Sandor (21), the trans
mission angle, p, is a maximum when e + 60 = 180 and a
minimum when 9 + e 0 = 00, provided these positions are
real, i.e., if the linkage closes in these positions of
the input link. This demonstrates an extremely important
concept: the need for parametric constraints can
sometimes be eliminated by determining the critical
values of the independent parameter.
Nonparametric constraints can result from a number
of requirements. For example, it is usually desirable to
limit the ratio of link lengths within a mechanism;
otherwise solutions will result with extremely long
coupler links and short cranks (for practical purposes,
these become slidercrank mechanisms). To eliminate these
unwanted solutions, the following constraints are specified:
A < A < A
B < B < B
C < C < C (2.11)
or, expanding these into the standard form of separate
inequalities,
A A < 0 A A < 0
B B < 0 B B < 0
S C 0 C 0 (2.12)
C C< 0 C C < 0 (2.12)
where, again, the length of the fixed link, D, is arbi
trarily set equal to unity, and the subscripts i and u
refer to the lower and upper limits of the link length.
Equality constraints in linkage synthesis problems
most often result from precision position requirements.
Referring to the notation used in equation 2.6, suppose
that, at the three positions q,r and s, the actual
mechanismoutput is required to be the same as the
specified output, then
F(ei) = G(6 ), i = q,r,s (2.13)
or
F(8e ) G(8e ) = 0, i = q,r,s (2.14)
are the needed equality constraints. Positions q,r and s
may now be removed from consideration in equation 2.6,
although this is not mandatory since they do not contribute
to the value of the objective function. An important
point which will be studied in detail in section 2.3.2
is that equality constraints can very often be used to
eliminate variables from consideration, without any loss
of generality in the problem formulation.
Summarizing the fourbar linkage optimization problem
at hand,
find
T
t LD )
X = {A,B,C,y0,0 }
which minimizes
Z{F(6.) G(Oi)}2,
subject to
arccos
arccos
A B C2 + 1 + 2A
2BC
2 2 2
A B C + 1 2A
2BC
A A < 0
A A
u
B B
B B
u
CL C
C C
u
F(6.) G(6.) = 0,
1 1
i = q,r,s
Notice that the critical values (9 + 60) = 0' and
(a + 60) = 1800 have been substituted into constraint
equation 2.10 to yield equation 2.17 and 2.18. Also,
the positions at which the function must be exactly
i = 1,2,...,q,...,r,...,s,...,n
(2.16)
 30 > 0
 300 > 0
(2.17)
(2.18)
(2.19)
(2.20)
satisfied, ei, i = q,r,s, have not been removed from
consideration in the objective function.
A number of other constraints could be added to
this problem. For example, it is often desirable to
have a crankrocker type linkage, which can be driven
from a continuously rotating prime mover. The inequality
constraint which ensures this type of mechanism is
derived from the socalled Grashof criterion. Other
constraint conditions will be discussed in Chapter
Three. The present problem is sufficiently general for
use in discussing the various methods available for
solving the general nonlinear optimization problem which
typically results from mechanism optimization; these
solution methods are presented in the following section.
2.4 Solving the Mechanism Optimization Problem
Optimization problems may be either linear or non
linear. Nonlinear problems occur when the objective
function or any of the constraint equations are nonlinear
in any of the design variables. It will not surprise the
experienced kinematician to know that almost all
mechanism optimization problems fall into this category.
Even judging from the simple example of the previous
section, it becomes evident that these types of
problems are generally nonlinear, and usually involve
a large number of constraints. For this reason, the
remainder of this section will deal exclusively with
methods for solving constrained, nonlinear optimization
problems.
2.4.1 Constrained Nonlinear Optimization
A number of methods are currently available for
solving constrained nonlinear optimization problems.
While some of these methods are more widely used than
others, no single method is best suited to solve every
type of problem. In fact, even small changes in the
way a problem is formulated can grossly alter the effec
tiveness of the optimization procedure being used. This
explains, in part, the large number of past approaches
which have been taken in mechanism optimization, and
the claims, often conflicting or confusing, about the
efficiency and the effectiveness of these approaches
(104,105).
Although it is possible to solve some constrained
nonlinear optimization problems using the classical
techniques of variational calculus, the complexity of
most mechanism optimization problems renders this approach
impractical. Therefore, the remainder of this section
will be devoted to the iterative solution methods
known as constrained nonlinear mathematical programming,
or simply constrained nonlinear programming.
Constrained nonlinear programming techniques can
be divided into two distinct groups: direct methods and
indirect methods. Table 2.1 shows the various techniques
that come under these headings (106).
Table 2.1
Constrained Nonlinear Programming Techniques
Direct Methods Indirect Methods
1) Heuristic search 1) Transformation of
variables
2) Constraint approximation
2) Penalty functions
3) Feasible directions
Classification is based on the manner in which the
constraints are handled. The direct methods deal with
the constraints explicitly, whereas the indirect methods
first transform the constrained problem into an uncon
strained problem, and then solve this new easier problem
using one of the unconstrained nonlinear programming
methods.
All of the methods listed in Table 2.1 are poten
tially useful in mechanism optimization. However, it
will not be practical to discuss each of these methods
in detail. Since the indirect methods are used in
working the examples of this dissertation, these will
be discussed in greater detail. Complete details on
all of the methods listed are available in the
references (106,108,109).
2.4.2 Indirect Constrained Nonlinear Programming Techniques
As can be seen from Table 2.1, two methods come under
this heading: transformation of variables and penalty
functions.
In the transformation of variables technique, the
design variables are changed in such a way as to auto
matically ensure constraint satisfaction. There are
two cases where this is possible: (1) when the constraints
are simple, explicit functions of the decision variables,
and (2) when equality constraints can be used to eliminate
variables.
As an example of the first case, recall the first
of the link length constraint equations 2.11 of section
2.2, repeated here as equation 2.21.
A < A < A (2.21)
1 u
where A was the length of the input crank of the four
bar linkage of figure 2.1, and AL and Au were the upper
and lower limits placed on this length. These constraints
can be automatically satisfied by transforming the
variable A to the form
A = A + (A A )sin A*, (2.22)
where A* is the new variable which can take on any
numerical value. This technique is called change of
variables. Notice the A will always be between A, and
Au for any value of A* in equation 2.22.
While this approach seems quite promising at first,
experience has shown that, unless all the constraints
can be transformed in this way, it is probably better
not to use the transform at all (102). This is because
substitution of the righthand side of equation 2.22
into the objective function may distort it to the point
where it is more difficult to minimize than the
original function, when other constraints are present.
As a result, this approach appears to be impractical
for most mechanism optimization problems, although the
author believes further study in this area is warranted.
The second case for which transformation of variables
is sometimes possible occurs when equality constraints
are present. For example, in section 2.2, equation 2.13
expresses a set of three precision position requirements
for the fourbar linkage of figure 2.1
F(6.) = G(6.), i = q,r,s (2.23)
where the functions F(6.) and G(O.) express the desired
and the generated output angular positions of the mechanism,
respectively, at a given input position 8.. Recall from
equation 2.5 that F(9 ) = i. is a known, prescribed
function. Equation 2.23 may be rewritten in the form of
the wellknown Freudenstein equation (1), which gives
the input/output angular relationship of the fourbar
function generator in terms of the linkage dimensions
K1cos(9i + 0) K2cos(4i + 0) + K3 =
cos(6i + 0 i 0) (2.24)
where
K  K2 = KC = 1 (2.25)
C A 2AC
and 80 and 0 are the starting angles (see figure 2.2).
Equation 2.24 is linear in the coefficients Kl,K2 and K3,
and can be written three times corresponding to the three
values of i (i.e. i = q,r and s). It is therefore a
relatively easy matter to solve for the values of KI, K2
and K3 in terms of the angles 6., pi i = q,r,s, and
6eo' 0 (101). With K K2 and K3 known, the link lengths
may be determined from the relations
A = ; C = 1 ; B = {A2 + C2 + 1 2ACK }3 (2.26)
K2 K1
Notice that the only design variables remaining in the
righthand sides of equations 2.26 are the starting angles
60 and 0. Anywhere A,B and C appear in the remaining
constraint equations or in the objective function they
may be replaced by the equivalent expressions given in
equation 2.26. This technique is called elimination of
variables.
As in the previouslydescribed change of variables
technique, the objective function and the remaining
constraints will undoubtedly be distorted when the non
linear expressions for A,B and C of equation 2.26 are
substituted into them. However, experience has shown
that it is generally beneficial, in this case, to make
the substitution (26,51,52,63,106). Obviously, the
greater the degree of nonlinearity of the expressions
used to eliminate variables, the less beneficial this
type of substitution becomes.
It has already been noted that the role of the
indirect methods of constrained nonlinear programming is
to transform the constrained problem into an equivalent
unconstrained problem. Occasionally, the transformation
of variables method can accomplish this goal by itself.
More often, however, some of the constraint equations
are too complex for this, and the socalled penalty
function approach must be employed.
To demonstrate the penalty function approach,
consider the following simple problem, where the parametric
vector, 0, has been omitted for clarity.
Find X which minimizes f(X)
subject to
g. (X) < 0, j = 1,2,...,m (2.27)
and
Z.(X) = 0, j = m+l,m+2,...,p
This constrained problem is converted into an unconstrained
problem by constructing a new function to be minimized of
the form
m
U = U(X,r) = f(X) + r Z G.{g.(X)}
S j=l ~
p
+ r Z L.{Z.(X)} (2.28)
j=m+l ~
where G.{g.(X)} and L.{i.(X)} are functions of the constraint
functions gj(X) and Zj(X), respectively and r is a positive
constant called the penalty parameter. The solution of
the unconstrained problem of equation 2.28 can be made to
converge to the solution of the original problem of
equation 2.27 by repeating the minimization process for
a progressively larger series of values of the penalty
parameter, r. For this reason, the penalty function
methods are often referred to as "sequential unconstrained
minimization techniques" or simply SUMT.
Two categories of penalty function methods exist,
namely, interior methods and exterior methods. The
interior methods must be supplied with a feasible
starting vector, Xi (i.e. g (X1) < 0 for all j). As
the parameter r is varied over successive minimizations,
the solution of the unconstrained problem converges to
the solution of the constrained problem, always remaining
within the feasible region. Since the search is conducted
within the feasible region, these are called interior
methods. The exterior methods do not require a feasible
starting vector, and generally converge to the constrained
minimum from outside the feasible region, hence the
term exterior. The exterior methods have been judged
to be generally superior to the interior methods (109);
therefore, these will be reviewed in greater detail.
A typical exterior penalty function form of equation
2.28 is
m
U(X,r) = f(X) + r Z < gj(X)) q
j=l
p 2
+ r Z {Z. (X)} (2.29)
j=m+l ~
where, again, r is a positive penalty parameter, q is a
constant greater than one, and the singularity function
gj (X) if g (X) > 0
S0 if g (X) < 0
It can be seen that the effect of this formulation
is to assess an increasingly severe penalty on the value
of U(X,r) as the constraints become violated by larger
amounts. The most successful way to find the true
constrained minimum of the original function has generally
been to minimize equation 2.23 using a small value for
the parameter r for the first minimization. Subsequent
minimizations use successively larger values of this
parameter, until the solution is essentially forced to
converge in the feasible region. This is necessary
because, if r remained small, very small positive values
of g (X), even though infeasible, would not contribute
much of a penalty to U(X,r) and the solution might remain
infeasible. On the other hand, if r were made initially
very large, the contribution of f(X) would be negligible,
and the solution may not converge to the minimum f(X)
within the constrained region. In other words, the
effects should be balanced, so that the solution is
urged toward the minimum of f(X) at the same time it is
being forced toward the feasible region. The simple
example which follows will help to demonstrate some
of these concepts.
Find X = {x } which minimizes f(X) = x1
subject to the constraint
3 x1 < 0 (2.31)
The objective function f(X) and the constraint
boundary are plotted in figure 2.3. The constrained
minimum is clearly at xI = 3. Now construct the exterior
penalty function
U(X,r) = x1 + r 3 x 12 (2.32)
Table 2.2 gives various values of U(X,r) versus x1 for
several values of r.
The resulting curves are plotted in figure 2.3. It
is clear from this figure that, as r tends toward infinity,
the solution to the unconstrained objective function of
equation 2.32 will approach the solution to the constrained
problem of equation 2.31. It may be noted that, in this
case, the solution will reach the feasible region only in
the limit as r approaches infinity. This usually is not
troublesome for practical problems because the constraints
are rarely known exactly, and some allowance must be made
for errors when formulating them.
The preceding discussion focused on the use of
penalty function methods and change of variable techniques
to transform constrained optimization problems into
Table 2.2
Example showing the effects of the penalty parameter, r.
U(X,r)
xl r=0.25 r=0.5 r=l
0 2.25 4.50 9.00
0.25 2.02 3.90 7.69
0.50 1.81 3.38 6.50
0.75 1.64 2.91 5.44
1.00 1.50 2.50 4.50
1.25 1.39 2.16 3.69
1.50 1.31 1.86 2.75
1.75 1.27 1.66 2.44
2.00 1.25 1.50 2.00
2.25 1.27 1.40 1.69
2.50 1.31 1.38 1.50
2.75 1.39 1.41 1.43
3.00 1.50 1.50 1.50
Bold squares indicate the tabulated minimum of U(X,r).
xl = 3
infeasible region
r= 1
feasible region
f(X) = Ix
1" J
figure 2.3 Penalty function example
f (X) ,
U(X,r)
unconstrained ones. The question still remains of how
these unconstrained problems may be solved; this is the
topic of the next section.
2.4.3 Unconstrained Nonlinear Optimization
Even though the vast majority of nonlinear optimiza
tion problems involve constraints, most of the available
nonlinear optimization techniques have been developed
for solving unconstrained problems. This does not
represent a serious limitation, however, because most of
these methods can be extended to handle constrained
problems, either by directly considering the constraints
or by transformation to an unconstrained problem as
discussed in the previous section.
Unconstrained minimization methods may be divided
into two groups: direct search methods and descent (or
gradient) methods. The gradient methods require either
an analytical or a numerical derivative of the objective
function with respect to the design variables, whereas
the direct search methods do not. Some of the commonly
available techniques in both groups are listed in Table
2.3 (106).
Of the methods listed in Table 2.3, the random
search and the grid search are known to be quite ineffi
cient. However, these methods tend to be reliable when
minimizing discontinuous, sharply varying or
nondifferentiable functions. They may also be useful
for finding feasible solutions to initiate some of the
more efficient methods.
Table 2.3
Unconstrained Minimization Techniques
Direct Search Methods
1) Random search
2) Grid search
3) Univariate search
4) Pattern search (Powell's
method, Hooke and Jeeves'
method)
5) Method of rotating coor
dinates (Rosenbrock's
method)
6) Simplex method
Descent Methods
1) Steepest descent
method
2) Conjugate gradient
method (Fletcher
Reeves)
3) Newton's method
4) Variable metric
method (Davidon
FletcherPowell)
An excellent comparison of many of the numerical
optimization methods commonly used to solve mechanical
design problems was made by Eason and Fenton (109). They
point out that the ideal computer code for design optimi
zation should solve any problem conveniently and at
moderate cost. No code tested by them fulfilled this
requirement, but one method did stand out above the
others, namely, the pattern search method of Hooke and
Jeeves. A number of other important conclusions were
reached in this study; some of these are summarized
below:
(1) Derivatives of the objective function are
often difficult or impossible to calculate
analytically for many mechanical design
type problems, and must be approximated
by numerical methods, if needed.
(2) If derivatives must be calculated numeri
cally, the direct methods (which do not
require derivatives) are generally superior
to the gradient methods.
(3) Automatic scaling of the design variables
within the computer program generally
increases the efficiency of an algorithm.
(4) The most general methods (those which
could solve the most types of problems)
were not necessarily slow, nor did they
require the greatest amount of computer
code to program.
(5) The cost of preparing a problem for
computer solution may be greater than the
execution cost. The algorithm should,
therefore, be convenient to use.
(6) A computer package containing an assortment
of optimization methods would generally be
preferred to any single method. This
would allow crosschecking of results,
and obviously would allow more types of
problems to be solved than would any of
the methods individually.
Development of a computer optimization package as
described in conclusion number (6) above would be a
major undertaking. Also, such a package would require
a relatively large amount of computer storage. Still,
this would be the preferred approach in terms of
efficiency and generality. Since no such computer
package is readily available, and since computer storage
will often be an important consideration, it was decided
to use the pattern search method of Hooke and Jeeves in
working the examples of this dissertation. Later results
will show this method to be quite effective in solving
mechanism optimization problems. Good results using
this method for optimizing planar mechanisms were also
reported by Kramer and Sandor (61) and Kramer (62).
2.4.4 Hooke and Jeeves' Nonlinear Programming Method
Hooke and Jeeves' search method is a direct, sequen
tial stepping technique consisting of alternating
exploratory and pattern moves. The exploratory move
seeks to determine the local behavior of the objective
function, and the pattern move uses this information in
an attempt to "leapfrog" to an improved position. Figure
2.3 demonstrates this procedure for a twodimensional
problem, and the algorithm is outlined in detail below.
(1) Starting from an arbitrarily selected point
0 0 }T
X located at X = {x try an explor
atory move by changing x by a predetermined
positive step Axl. Evaluate the objective
function at this new point. If its value is
improved, this step is retained, and becomes
the new base point. If the value of the
objective function is not improved, a negative
step, Ax1, is taken and the objective function
is reevaluated. If the move is successful,
the point is retained as the new base point.
If both steps fail, no move is made. At
this point it is often beneficial to adjust
the value of Ax1, and save this information
for future exploratory searches in the xl
direction. A successful step move would
suggest an increase in the value of Axl. If
neither step were successful, the value of
Axl should be decreased. Starting from the
best point found in the Xl search (labeled
X in figure 2.3) a similar search is made
in the x2 direction. The best point found
in this search is labeled X2
figure 2.3 The pattern search method of Hooke and Jeeves
(2) A pattern move is now attempted by repeating
all the successful moves of the exploratory
search from point X After the initial
search, this move may also include the
previous exploratory moves and the previous
pattern search.
(3) If the pattern move of part (2) is successful,
it is retained as point X and the exploratory
search begins at this point. If the pattern
move fails, the next exploratory search begins
at point X2.
(4) This process is repeated until the values of
the Axs are below a certain preset limit.
At this point the minimum is presumed to
have been reached.
A flow chart of this procedure is shown in figure 2.4.
While simple in concept, the Hooke and Jeeves'
method is extremely powerful. It should be cautioned,
however, that none of the optimization methods discussed
can discriminate between a local and a global minimum.
Geven ten different starting points, the Hooke and Jeeves'
method may converge to ten different local minima. As
discussed at length in the next chapter, this will
seldom represent a serious problem in mechanism
optimization.
70W c",art of the pattern search methodof
Hocoke and jeev'es
figure 2.4
CHAPTER 3
PHILOSOPHY OF MECHANISM OPTIMIZATION
3.1 The Need for a General Philosophy
Although theories for the kinematic synthesis and
analysis of many types of spatial mechanisms are well
developed and readily available, complete design theories
for even the simplest fourlink spatial mechanisms are not
yet developed to the point where they are practical for
use by the industrial machine designer. This "gap"
exists because of the difficulty involved in selecting
a single mechanism which satisfies all of the various
realworld design requirements. This dissertation
advocates optimization theory as the proper tool for
bridging this gap. But while the application of
optimization theory to planar mechanism design has risen
at a seemingly exponential rate, there has been little
application to spatial mechanism design. The author
believes the reason is the relatively greater complexity
and correspondingly greater number of parameters
typically encountered in spatial mechanism design.
For example, only six design variables are needed to
specify the planar fourbar functiongenerating
mechanism discussed in section 2.2, whereas twentysix
such design variables may be needed to completely
specify a spatial fourlink RCCC motiongenerating
mechanism.1 Even when a problem is rather poorly
formulated, optimization can often be performed in a
reasonable amount of time when dealing with six or ten
design variables. However, the situation becomes decidedly
different when dealing with twentysix design variables.
Since each variable may generally take on an infinite
number of values, a twentysix parameter problem
represents on the order of 26 possible solutions!
However, this does not mean that the application of
optimization theory to problems involving a large number
of design variables should be avoided. Quite the contrary,
optimization theory offers the only real hope of solving
such difficult problems. The point is that, as the
number of design variables increases, so does the need
for an efficient problem formulation coupled with an
efficient optimization method and a means for reducing
the number of parameters. Accordingly, this chapter
attempts to define the goals of mechanism optimization,
and to develop guidelines for obtaining these goals in
See the derivation of the RC and CC dyad synthesis
equations in Chapter Four.
the most efficient manner; collectively, this will be
called the philosophy of mechanism optimization.
3.2 Objectives and Constraints of Mechanism Optimization
There are typically a number of requirements which
must be taken into account when designing a mechanism.
Some of these are listed and described below.
(1) Motion specification. Most often a mechanism
is required to generate (a) a functional
relationship between input and output members;
(b) a point path, sometimes coordinated with
input motion, or (c) a rigidbody motion.
Other motion requirements are possible, but
these three have proven to be adequate for
the vast majority of problems.
(2) Branch avoidance. Most mechanisms can be
assembled in two or more distinct configura
tions while keeping the links connected in
the same order. Each distinct configuration
is called a branch. The mechanism will
generally operate in one branch. Dissassembly
is required to move it into a different branch.
Branch avoidance implies the design of
mechanisms such that the entire motion
specification lies on one branch only.
(3) Order of positions. Precision positions
must occur in the prescribed sequence and
sense. For example, if the prescribed
positions were given in the order 1,2,3,4,
a mechanism generating these same positions,
but in the order 1,3,2,4, would be unaccept
able.1
(4) Grashof's condition. This refers to the
relative rotatability of links within a
mechanism. Often it is desirable to drive
the input link of the mechanism from a
continuously rotating source. Such an input
is called a crank.
(5) Transmission characteristics. This refers
to the effectiveness of the mechanism in
transforming work at the input to work at
the output. For some three and fourlink
mechanisms, this can be expressed in terms
of a transmission angle.
(6) Linklength ratio restrictions. Quite
often, when trying to optimize the trans
mission characteristics of a mechanism, some
1See Strong and Waldron (88) for a complete discussion
of the order problem.
of the links within the mechanism become
relatively much longer than others. This
may cause manufacturing difficulties as well
as other problems. To avoid this effect,
limits may be placed on the allowed ratio
of link lengths.
(7) Fixedpivot location restrictions. Fixed
pivots must be located so they do not inter
fere with other components. They must also
be placed where they can be connected to the
machine frame.
(8) Workspace restrictions. The operating space,
or workspace, of the mechanism must often be
restricted to avoid interference with other
components.
(9) Dynamic and elastodynamic property restrictions.
Velocities, accelerations, etc. and link
deflections must remain within prescribed
limits. This category also includes balancing
requirements.
(10) Tolerance and clearance effect restrictions.
The errors in mechanism output due to tolerances
and clearances must be held within prescribed
limits. Since these are not usually deter
ministic quantities, stochastic methods must
often be employed.
Of course, all of these requirements will not apply to
every problem. Quite often the designer can use his
experience and judgment to eliminate several of these
from consideration. For example, dynamic properties
would probably not be a restrictive factor in a linkage
designed for the purpose of guiding open an automobile
hood. The reader can undoubtedly think of many other
examples.
Classical mechanism design procedures generally
begin by considering the motion specification. Typically,
this results in a set of precision position requirements
for function, path or rigidbody motion generation.
These requirements are expressed as equality constraints
which are then solved to yield the dimensions of one
or more mechanisms which, at least mathematically,
satisfy the precision positions. This procedure,
commonly known as mechanism synthesis, is fascinating
because of its unusual combination of geometric and
mathematical complexities coupled with a concrete
physical phenomenon. Unfortunately, this fascination
has sometimes led to the feeling in academics that
precision position synthesis is tantamount to mechanism
design. This is not true, since each of the previously
discussed requirements, and perhaps more, must be
considered when designing a mechanism.
Precision position synthesis is so routinely used
as the first step in mechanism design that most designers
do not stop to question why. Are the precision conditions
of greater importance than the other requirements listed?
Is a mechanism with branching problems more acceptable
than a mechanism that does not satisfy the prescribed
motion? The answer to both of these questions, of course,
is "no," since a mechanism must satisfy all of the
design requirements if it is to be a workable solution.
Why then should mechanism design always begin with
precision position synthesis? Why not begin designing
a mechanism by first considering, for instance, the
Grashof condition? From the set of all possible
mechanisms of a given type, the subset consisting of
only those of the desired Grashof type could be selected.
From this subset, mechanisms meeting the specified
precision conditions could then be selected, yielding
an even smaller subset. Following this, the link
length ratio requirements could be applied as a third
requirement, and so forth. This approach seems practical
enough, and yet, to the author's knowledge, it has
never been applied.
The reason for using precision position synthesis
as the first step in mechanism design is not that it
is the most important consideration. Rather, it is
used as a first step because,among all the requirements
listed at the beginning of this section, only the
precision position specifications are generally in the
form of equality constraints. Thus, it is often possible
to use the elimination of variables technique discussed
in section 2.4.2. In addition, it may be possible
for the designer to choose the number of equality
constraints by choosing the number of specified precision
positions. This gives the designer control over the
number of design variables he will have to work with.
Although used intuitively for years, the concept of
precision position specifications to reduce the number
of freechoice variables was formally advanced by Tesar
(26), Eschenbach and Tesar (27), Tesar and Spitznagel
(51,52) and Sutherland (63), among others. Of these
works, only Sutherland's addresses the possibility of
approximately satisfying additional motion requirements.
Two principal drawbacks may exist to using precision
condition equality constraints to eliminate variables.
First, as discussed in section 2.4.2, this technique
introduces nonlinearities which tend to distort the
objective function and the remaining constraints. Thus,
for example, using a three precisionposition solution
in which the unknown variables are nonlinear and must be
solved for numerically, even though one more variable is
eliminated in the latter case. The second drawback is
that the designer may not have a need to satisfy any
precision conditions at all, but rather, may require
some approximate motion specification to be met. In this
case, a great many possible solutions are lost by
arbitrarily setting up precision position requirements.
If no acceptable design can be found from this reduced
solution set, the designer may be forced to eliminate
some, or all, of the precision conditions.
Although not so obvious or as well developed in the
literature, it may sometimes be possible to use other
design requirements as equality constraints to eliminate
variables. For example, one variable could be eliminated
by specifying the coupler link to be twice as long as the
input crank. Or perhaps the minimum and maximum trans
mission angles could be specified to have a certain
numerical value. Again, the designer must be careful
not to introduce extreme nonlinearities in the remaining
design equations. Although not explored in detail here,
this appears to be a novel and potentially useful area
of research.
Eschenbach and Tesar (27) have suggested dividing
the design requirements into two groups: necessary
requirements and desirable requirements. In their work,
the necessary requirements are the branching, Grashof
and order conditions, plus the requirement of satisfying
four precision conditions. The remaining requirements
are treated as desirable conditions. The necessary
requirements are considered to be go no go conditions,
because they are either acceptable or they are unaccept
able. In other words, these are the constraint conditions.
The desirable conditions collectively correspond to the
objective function, the idea being to find the most
desirable value for a given combination of these conditions.
It is doubtful that the division of necessary and
desirable conditions suggested by Eschenbach and Tesar
(27) will apply to every mechanism design problem. For
example, satisfying four precision positions may well
be a desirable condition, while fixedpivot location may
be a necessary condition. However, their work is
indicative of the importance placed on satisfying the
first four conditions listed at the beginning of this
section, namely, motion specification, branch avoidance,
order of positions and Grashof's condition.
3.3 Observations and Trends Affecting Mechanism
Optimization
The development of a general philosophy for mechanism
optimization undoubtedly must involve some amount of sub
jective opinion and speculation. Some of the concepts
presented in this section break away from established
trends in mechanism optimization. This is partially
due to a reevaluation of mechanism.optimization procedures
based on past literature, and partially due to the rapidly
changing role of the computer in engineering design.
Although it is difficult to make broad generalizations
about any subject, the following observations seem to
apply:
(1) The need to find a globally optimum solution
has, at times, been overplayed in the
literature on mechanism optimization. Most,
if not all, practical mechanism design
problems are of such a complex and multi
faceted nature that it is impossible to
precisely define what is meant by optimum.
Furthermore, finding the global optimum is
not really necessary for many practical
mechanism design problems. Therefore, the
objective of this dissertation is to use
optimization theory to find designs which
are workable solutions, rather than emphasizing
finding the optimum solution. For this reason,
care has been used to call the present work,
"Optimization of Spatial Mechanisms,"
rather than, for example, "Optimum Design of
Spatial Mechanisms."
(2) Centralized largescale digital computers
are being replaced in many applications by
smaller and more local ones. While the
capabilities of these smaller machines
have increased enormously in recent years,
storage capacity often limits their useful
ness to smallerscale problems. The trend,
therefore, should be toward more compact
iterativelybased programs which require
less storage.
(3) Since designers will more often be using
local, and perhaps personal, computers,
the algorithms developed should be easy to
program and readily adaptable to a variety
of problems. This means that the design
theories should, so far as possible, be
based on more easily understood concepts.
(4) Based on points (2) and (3) above, the
optimization method employed should be
simple in concept, require little computer
storage, and should be capable of solving a
wide variety of problems. The Hooke and
Jeeves' method described in Chapter Two
possesses these qualities.
3.3 Development of a General Mechanism Optimization
Philosophy
In the past, many authors have developed optimization
methods which do not include a parameter reduction step,
for example (25,54). A simple flow chart for this
method is shown in figure 3.1. These methods might
be called design by analysis, because there is no
synthesis step involved. The advantage of this approach
is that no potential solutions are lost through parameter
reduction. Given enough time and computer resources,
this would be the preferred method. However, this
approach will generally be too inefficient for use on
complex mechanism design problems where a large number
of variables are present, such as spatial mechanism
design.
Undoubtedly, the most popular approach to mechanism
optimization has been the "standard" precision position
approach shown in figure 3.2. This method is generally
more efficient than the design by analysis technique,
because the number of freechoice parameters has been
reduced. However, this method is still lacking in the
respect that no attempt has been made to distinguish
nonparametric constraints from parametric ones. In
addition, this method does not recognize the possibility
of eliminating variables other than by precision position
synthesis. Also, no provision is generally included to
allow additional approximate motion specifications.
A flowchart depicting the method of sequential
filters, as presented by Spitznagel and Tesar (51,52),
is shown in figure 3.3. It is an improvement over the
71
problem parametric and non
formulation parametric optimization
figure 3.1 Design by analysis
problem precision parametric and
formulation  position  nonparametric 
ormuatio synthesis optimization
figure 3.2 Standard approach to mechanism optimization
figure 3.3 The method of sequential filters
Problem parameter nonparametric parametric
formulation reduction optimization optimization
figure 3.4 The mechanism optimization method
used in this dissertation
NOTE: Feedback may occur to any previous block in all figures.
standard approach because it recognizes many of the
special properties which result when dealing with a
planar fourbar linkage. However, no distinction is
made between parametric and nonparametric constraints.
Also, only precision position synthesis is considered as
a parameter reduction tool and no provision is made for
satisfying additional approximate motion specifications.
Sutherland (63) was the first to recognize the
potential need for and the usefulness of satisfying both
precision and approximate motion specifications simul
taneously. Although not a complete optimization scheme
in itself, this feature is both a desirable and a
necessary component of any general optimization scheme.
The most general and efficient mechanism optimization
scheme should utilize the best features from all of the
above methods. It should incorporate parameter reduction,
either using precision position synthesis or perhaps
using other equality constraints. It should also incor
porate approximate motion specification in addition to
precision conditions. Finally, it should make a
distinction between parametric constraints and nonpara
metric ones, and should use special properties of the
mechanism to eliminate parametric constraints, when
possible. One such method is illustrated by the flow
chart, figure 3.4. While simple (and perhaps obvious)
in concept, application of the abovedescribed method
73
will often require a deep understanding of the specific
problem at hand. Spatial mechanism design, the topic
of the next four chapters, demonstrates this point.
Chapter Four discusses parameter reduction by means of
closedform precision position solutions for various
spatial dyads. Chapters Five and Six demonstrate
methods for formulating some of the constraints for the
RCCC and the RSSR mechanisms in nonparametric form.
These are then used, along with the precision position
synthesis methods of Chapter Four, to demonstrate the
optimization of these mechanisms.
CHAPTER 4
PRECISION POSITION SYNTHESIS OF SPATIAL MECHANISMS
4.1 Introduction to Precision Position Synthesis
The preceding chapters emphasized the need for
parameter reduction in mechanism optimization. Clearly,
precision position synthesis is often of importance, and
offers a readily available means for reducing the number
of parameters.
In this chapter, closedform solutions are developed
for rigidbody guidance problems using the RS, CS, CC and
RC dyads. Since the function generation problem can
generally be converted to a rigidbody guidance problem
by a process known as inversion (19), and since the path
generation problem can generally be treated as an
incompletely specified rigidbody guidance problem, the
solutions developed in this chapter have a broad spectrum
of applications. Furthermore, as shown in the following
section, the dyads treated here can be assembled into a
variety of mechanisms, giving the procedures an even
wider applicability.
Synthesis procedures for some of the dyads treated
in this chapter have been discussed elsewhere (76,102,
110, 111, 112). However, these works generally do not
emphasize finding closedform linear solutions, a topic
of considerable importance when the synthesis process is
to be used in an optimization loop. A detailed discussion
of these synthesis procedures was also felt to be needed
in order to provide a uniform notation for later reference
and to provide a better understanding of the available
freechoice parameters. All of the synthesis procedures
given here are for the maximum number of precision
positions which result in closedform linear solutions.
4.2 Dyadic Synthesis of Mechanisms
For the purposes of this work, dyads may be thought
of as building blocks for mechanism synthesis. A dyad
is a twolink kinematic chain composed of a grounded
link and a floating link, and having two degrees of
freedom. The grounded link is joined to ground through
one kinematic pair and joined to the floating link
through another. For simplicity and clarity, the
concept of dyadic mechanism synthesis will be explained
using the planar dyad shown in figure 4.1. Extension of
these concepts to spatial dyads will readily follow.
Begin by considering a set of discrete planar
precision positions, defined by f(x yj), j=1,2,3, as
shown in figure 4.1. Any dyad whose tracer point can
physically reach all of these positions can generate
tracer point
/
/ f (x2y2)
Sf (xl,Y)
'I//I
figure 4.1 A planar dyad
these points. In fact, it is possible to generate an
infinite number of such positions. If, however, rotations
of the floating link or of the grounded link are specified
as 6j, j=1,2,3 at each position, the problem becomes much
more difficult so much so, in fact, that it is now
possible to generate a maximum of only five such positions.
When the rotations of the floating link are specified,
the problem becomes identical to the planar rigidbody
guidance problem. Classical Burmester theory is a well
known method for solving this problem. If two different
dyads can be found which satisfy the given motion
requirements, their floating links may be rigidly
connected to form a constrained planar fourbar linkage,
as shown in figure 4.2.
It should be apparent that spatial mechanisms can
be constructed from spatial dyads in a similar fashion.
Of course, the problems become more complex since the
motions are now in three dimensions, and because a
greater number of joint types, or pairs, must be consi
dered. For a discussion of the various types of joints
used in spatial mechanisms see Harrisberger (17).
When designing singleinput mechanisms, dyads
usually must be connected in combinations which result
in a single degree of freedom, although a number of
exceptions to this rule exist (see, for example, Shigley
and Uicker (113)). In any case, several singleinput
figure 4.2 Two dyads with identical floating link
rotations joined to produce a fourbar
linkage
spatial mechanisms which can be synthesized from the
dyads treated in this chapter are shown in figure 4.3.
Thus, for example, if the RC and CC dyads can be inde
pendently synthesized for three rigidbody positions,
they can be joined together to form an RCCC mechanism
also capable of generating these three positions.
4.3 Position and Orientation of a Body in Space
One of the fundamental tools necessary for designing
spatial mechanisms is the ability to describe the motion
of rigid bodies in space. Since the examples presented
in this work deal only with finitely separated positions,
the present discussion is concerned with describing
finite displacements; extension of this concept to include
higher order properties can be found in many standard
texts, for example (76).
The total displacement of a rigid body in space can
always be considered to be the sum of an angular rotation
and a linear displacement of a reference point fixed in
the moving body. The displacement of a point in space
is easily described by a single threecomponent vector.
However, describing angular displacements in space is
not so easy nor is the method so obvious. Some of the
most popular methods are (1) Euler angles; (2) angular
rotation about an axis in space; (3) prescribed order
of rotations about a righthand set of Cartesian axes;
RCCC
RSCR
Some of the dyadbased spatial
which can be synthesized using
of this chapter
mechanisms
the methods
RSSR
RSSRSC
figure 4.3
and (4) the direction cosines of a pair of independent
unit vectors fixed in the body. Simultaneous translation
and rotation can be specified in a single quantity in
several ways, such as, for example, (1) quaternions (96);
(2) 3x3 matrices and tensors with dual numbers (98);
(3) 4x4 matrices using homogeneous coordinates (101);
and (4) dual quaternions (97), etc. Coaxial translations
and rotations (socalled screw displacements) can also
be expressed by way of any of these methods.
Regardless of the specific method being used, it
is important to realize that only three independent
parameters are needed to describe spatial angular
displacements. Nevertheless, it is almost always
most convenient to work with a ninecomponent threeby
three matrix when describing spatial angular displace
ments; this is the socalled rotation matrix. For a
given angular displacement of a body, all of the methods
listed above will lead to rotation matrices. For this
reason, the rotation matrix will be the basic tool used
in this dissertation to describe spatial angular displace
ments. For detailed discussions on finding the rotation
matrix from the angular displacement description methods
listed above, see (76,114).
The rotation matrix is extremely convenient for
specifying finite rotations in space. For example, if
v, and v2 represent two arbitrarily oriented spatial
positions of the vector v, then the rotation from
position 1 to position 2 can be expressed in the form
2 = [R] v (4.1)
where [R] is a rotation matrix (76). Equation 4.1 can
be expanded in terms of components to give
V2 r11 r12 r13 lx
2y = r21r22 r23 vly (4.2)
2z 31 r32 r33 Vlz_
4.4 Synthesis of the RevoluteSpheric (RS) Dyad
The revolutespheric dyad (RS dyad for short) is
shown schematically in the jth position with associated
vectors in figure 4.4. It is one of the simplest dyads
to synthesize and is perhaps the most useful. The RS
dyad can be synthesized for up to three precision positions
of rigidbody guidance using linear, closedform solution
procedures (103).
The following vectors are used in figure 4.4 to
describe the RS dyad:
a locates the revolute joint center relative to
the fixed coordinate system
A. locates the spheric joint center in the jth
Y
the moving body
in the jth position
a 0a
figure 4.4 The RS dyad and associated vectors in the
jth position
position relative to the fixed coordinate
system
o. locates the center of the moving coordinate
3
system relative to the fixed coordinate system
r. locates the spheric joint center in the jth
position relative to the moving coordinate
system
s unit vector (denoted by the hat) along the
fixed revolute axis
The synthesis procedure can be viewed intuitively
as follows. Select an arbitrary point fixed in the
moving rigid body as the location of the spheric joint.
The location of this point at the three prescribed
positions defines a plane (three points define a plane),
and a circle lying in this plane (three points also define
a circle). Since the revolute joint physically constrains
points to lie on a circle, its axis must pass through the
center of this circle, and must also be along the normal
to the plane containing the three points. The mathemat
ical formulation of this easily visualized procedure is
developed below.
The positions of a body in space are specified by
giving the location and orientation of the moving coordi
nate system embedded in the body at each position, as
described in section 4.3. Thus, for three prescribed
positions, the given quantities are
?. and [Rj] j = 1,2,3 (4.3)
where o. is a three component vector, and [R.] is the
rotation matrix rotating the moving body from a reference
position to position j. Displaced, rather than absolute,
positions of the moving body are of importance here.
Therefore, the starting position, defined by ol and [R ],
may be selected arbitrarily, and the other two positions
measured relative to the first. Choosing [R1] to be a
3x3 identity matrix has the effect of making the fixed
and moving coordinate systems parallel in the initial
position. Since this assumption generally simplifies
the design equation, it will be employed throughout
the remainder of this chapter. The initial position
vector, Ol, may be taken to be any convenient point in
the moving body.
Referring to figure 4.4, begin the RS dyad synthesis
by assuming a value of r the vector locating the
spheric joint relative to the origin of the moving
coordinate system in the initial position. Since the
locations of the moving origin, o., and rotation matrices,
[R.], j = 1,2,3 have been specified, the vector A.,
locating the spheric joint center, can be expressed as
A. = o. + r.,
3 3
j = 1,2,3
(4.4a)
or
A. = o. + [R. ]r j = 1,2,3 (4.4b)
It has already been noted that the three positions
of the spheric joint define a plane whose normal is
parallel to the fixed revolute axis s Since the
~
vectors A2AA and A A2 lie in this plane, the unit
vector s, along the revolute axis is defined by
(A2 A ) x (A3 A2)
S= ~ ~ ~ ~ (4.5)
S (A2 A ) x (A A2)i
The plane containing the vectors A A and A A2
and normal to s is shown in figure 4.5. It is necessary
1
to determine the vector a locating the intersection of
this plane with the fixed revolute axis. This is
accomplished by first determining the unit vectors p
and p which are respectively perpendicular to the vectors
A A and A A2
p = s x (A l)/J A A (4.6)
P = s x (A3 A2)/ A3 A21 (4.7)
The perpendicular bisectors of the vectors A 2A
and A A2 intersect at the tip of position vector a0
(as shown in figure 4.5). Denoting the perpendicular
87
^3 A2
A
y
figure 4.5 The plane of the RS dyad
distances from A 2A1 and A 3A2 to the tip of a0 by A2
and 13, the vector a can be expressed in the alternate
forms
A A
I0 = 2 2 + Al +
A A
a0 3 + A2 + 3~ 2
(4.8)
(4.9)
Equating the right sides of equations 4.8 and 4.9 and
forming the vector product with p2 eliminates 12 and
yields
0
2^ 2 + 2 x A1 2 x
A3p2 x p3 + A2 x 2 x
Equation 4.10 can now be solved
forming the scalar product with
A2 A
2
A3 A2
2
(4.10)
explicitly for 13 by
(2 x 3 ) which gives
( 2 x ) (Cl + C2 3 C4
(32 x ) (* 2 x 3)
where
Ci = 2 x A
A A
92 = x 1
(4.11)
(4.12)
(4.13)
C = 2 xA 2 (4.14)
A A
C4 x A2 (4.15)
? 2
The value of 13 thus obtained can be back substi
tuted into equation 4.9 to determine a The starting
position of the grounded link, represented by the vector
a can be obtained from the expression
a1 = A a0 (4.16)
This completely determines the dimensions and the
starting position of the RS dyad.
4.5 Synthesis of the CylindricSpheric (CS) Dyad
The procedure for synthesizing the CS dyad is
similar to the RS dyad synthesis procedure. Additionally,
the motion of the grounded cylindric joint along its axis
must be considered. This scalar variable is denoted by
S. in figure 4.6, which shows the 1st and jth position of
3
the CS dyad.
The CS dyad synthesis procedure can be visualized
as follows. Referring to figure 4.6, once again assume
the vector r locating the spheric joint relative to
the origin of the moving xyz coordinate system in its
initial position. Also, assume the orientation of the
jth position
ist position
figure 4.6 CS dyad in the ist and the jth position
cylindric joint axis, s in the fixed XYZ coordinate
system. Since the spheric joint is physically constrained
to lie on a cylinder about s1, projections of its
location onto a plane normal to s must lie on a circle.
The procedure is thus to project the three locations of
the spheric joint onto a plane normal to s These three
V
points define a circle within the plane, and the remainder
of the procedure becomes the same as for the RS dyad.
This procedure is described mathematically below.
As before, three positions of the moving body are
specified
o., [R.], j = 1,2,3 (4.17)
Assuming the vector r arbitrarily again leads to the
result
A. = o. + [R.]r j = 1,2,3 (4.18)
~J ~J J ~1
The orientation of the cylindric joint axis is now
assumed arbitrarily by specifying two of its components.
The third component can be found from the unit vector
identity
2 2 2
x + sly + sz = 1 (4.19)
Figure 4.7 shows the vector s1 in true length. The
relationship between the vector s and the scalar
