Title Page
 Table of Contents
 Background and motivation for this...
 Optimization theory
 Philosophy of mechanism optimi...
 Precision position synthesis of...
 Optimization of the RCCC mecha...
 Optimization of the RSSR-SC and...
 Conclusions and recommendations...
 Biographical sketch

Title: Optimization of spatial mechanisms
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00099498/00001
 Material Information
Title: Optimization of spatial mechanisms
Physical Description: ix, 213 leaves : ill. ; 28 cm.
Language: English
Creator: Reinholtz, Charles Frederick, 1954-
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 1983
Copyright Date: 1983
Subject: Mathematical optimization   ( lcsh )
Mechanical movements   ( lcsh )
Kinematics   ( lcsh )
Mechanical Engineering thesis Ph. D
Dissertations, Academic -- Mechanical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis (Ph. D.)--University of Florida, 1983.
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Charles Frederick Reinholtz.
 Record Information
Bibliographic ID: UF00099498
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000473804
oclc - 11665790
notis - ACN9013


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Table of Contents
    Title Page
        Page i
        Page ii
        Page iii
        Page iv
    Table of Contents
        Page v
        Page vi
        Page vii
        Page viii
        Page ix
    Background and motivation for this research
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
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    Optimization theory
        Page 26
        Page 27
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    Philosophy of mechanism optimization
        Page 58
        Page 59
        Page 60
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        Page 72
        Page 73
    Precision position synthesis of spatial mechanisms
        Page 74
        Page 75
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        Page 109
        Page 110
    Optimization of the RCCC mechanism
        Page 111
        Page 112
        Page 113
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    Optimization of the RSSR-SC and RSSR-SS mechanisms
        Page 139
        Page 140
        Page 141
        Page 142
        Page 143
        Page 144
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    Conclusions and recommendations for future research
        Page 173
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    Biographical sketch
        Page 213
        Page 214
        Page 215
        Page 216
Full Text







With love to Jeri and Nicholas


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. Chen-Chi 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


The financial support of the Army Research Office

under grant DAAG29-K-0125 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.


ACKNOWLEDGEMENTS. . . . . . . . . . .iii

ABSTRACT. . . . . . . . . . . viii


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


3.1 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

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 Revolute-Spheric
(RS) Dyad. . . . . . . . 82
4.5 Synthesis of the Cylindric-Spheric
(CS) Dyad. . . . . . . .. 89
4.6 Synthesis of the Cylindric-Cylindric
(CC) Dyad. . . . . . . .. 96
4.7 Synthesis of the Revolute-Cylindric
(RC) Dyad. . . . . . . . .108
4.8 Conclusions of Precision Position
Synthesis. . . . . . . . .108


5.1 Problem Definition . . . .
5.2 Satisfying Additional Motion
Requirements . . . . .
5.3 The Grashof Condition. . . .
5.4 The Branch-Avoidance Condition .
5.5 The Order Condition. . . . .
5.6 Fixed-Pivot and Link-Length Ratio
Conditions . . . . . .
5.7 The Objective Function . . .
5.8 Numerical Example. . . . .

MECHANISMS. . . . . . . .

6.1 Problem Formulation. . . . .
6.2 Method of Design . . . . .
6.3 The Branch-Avoidance Condition .
6.4 The Grashof Condition. . . .
6.5 The Transmission Characteristic
Condition . . . . . .
6.6 Fixed-Pivot and Link-Length 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
. . .142
. . .145
. . .150

. . .157

. . .161

. . .162
. . .163
. . .164
. . 168

RESEARCH. . . . . . . . . .173

7.1 Conclusions. . . . . . . . .173
7.2 Recommendations for Future Research. . .175




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



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 dyad-based 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 dyad-based spatial mechanisms, is covered in Chapters


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, closed-form 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 four-link

RCCC and five-link RSSR-SC and RSSR-SS mechanisms. Finally,

Chapter Seven outlines procedures for the optimization of

other dyad-based spatial mechanisms, and offers suggestions

for further 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


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 age-old 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


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


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, cam-modulated 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 Four-Bar 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 path-generating

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 four-bar

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 five-bar 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 six-link function-generating 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 lower-pair 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 rigid-body guidance. He also showed

the function generation problem to be convertible to a

rigid-body guidance problem by inversion about the

input or output links.

The advancing computer technology of the late 1950's

gave rise to the first numerically-based 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 well-known 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 four-bar 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 Chi-Yeh (22) was published in

1966, synthesis methods invariably required the exact

satisfaction of some number (usually 3,4 or 5) of

precision conditions. Chi-Yeh's novel approach was to

minimize the structural error in the least-squares sense

at a much larger set of prescribed positions. Chi-Yeh'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 four-bar

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 four-bar 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 coplanar-motion generating four-bar

linkage. Second, they were the first to recognize the

simplification which resulted from using precision

position equality constraints to reduce the number of

free-choice variables.

Least-squares error synthesis of planar four-link

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 crank-and-

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 four-link


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 six-link and other

multi-loop 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 cam-modulated 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 four-bar linkages

where some of the prescribed positions are satisfied

exactly and the remaining positions are approximated in

the least-squares 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 four-link path-generating mechanism. Bagci

and Parekh (66) demonstrated the optimal design of

spherical four-bar and six-bar linkages. Bagci (67)

designed spherical four-link mechanisms to have optimal

transmission properties. Rao and Ambekar (68) considered

the design of spherical four-link function-generating

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 path-generating spatial

mechanisms with minimum structural error in the least-

squares sense. In a unique work, Bagci (73) optimized

screw-generating spatial mechanisms. Alizade, Rao and

Sandor (74,75) demonstrated the synthesis of spatial

function-generating 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

closed-form 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 labor-intensive 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 function-generating

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 four-bar 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 four-link 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 four-bar linkages. Strong

and Waldron (88) used joint displacements to determine

the mobility regions of four-bar 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 crank-type planar four-bar 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 lower-pair spatial mechanisms. This elegant and

powerful tool provided a concise method for formulating

and solving spatial mechanism problems. Often the screw

method provided closed-form 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 stretch-rotation tensors to find the

loop-closure 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

open-loop 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 closed-form method for the

synthesis of a spatial motion-generating 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


(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


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.



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)


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


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 function-generating

planar four-bar 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)

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 function-generating planar four-bar

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
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 four-bar mechanism is defined
as the acute angle between the coupler link and the output

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)

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 slider-crank 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


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)


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 four-bar linkage optimization problem

at hand,


t LD )

X = {A,B,C,y0,0 }

which minimizes

Z{F(6.) G(Oi)}2,

subject to



A B C2 + 1 + 2A

2 2 2
A B C + 1 2A


A A < 0






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


- 30 > 0

- 300 > 0





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 crank-rocker 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 so-called 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


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


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
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


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


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


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 right-hand 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 four-bar 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 well-known Freudenstein equation (1), which gives

the input/output angular relationship of the four-bar

function generator in terms of the linkage dimensions

K1cos(9i + 0) K2cos(4i + 0) + K3 =

cos(6i + 0 i 0) (2.24)


K -- K2 = KC = 1 (2.25)

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

right-hand 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


As in the previously-described 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 so-called 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)


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

U = U(X,r) = f(X) + r Z G.{g.(X)}
S j=l ~

+ 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

U(X,r) = f(X) + r Z < gj(X)) q

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.


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) ,


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'

5) Method of rotating coor-
dinates (Rosenbrock's

6) Simplex method

Descent Methods

1) Steepest descent

2) Conjugate gradient
method (Fletcher-

3) Newton's method

4) Variable metric
method (Davidon-

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


(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 cross-checking 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 two-dimensional

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 Ax-s 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


-70W c",art of the pattern search methodof
Hocoke and jeev'es

figure 2.4



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 four-link 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

real-world 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 four-bar function-generating

mechanism discussed in section 2.2, whereas twenty-six

such design variables may be needed to completely

specify a spatial four-link RCCC motion-generating

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 twenty-six design variables.

Since each variable may generally take on an infinite

number of values, a twenty-six 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 rigid-body 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-


(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 four-link

mechanisms, this can be expressed in terms

of a transmission angle.

(6) Link-length 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) Fixed-pivot 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


(9) Dynamic and elasto-dynamic property restrictions.

Velocities, accelerations, etc. and link

deflections must remain within prescribed

limits. This category also includes balancing


(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


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 rigid-body 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 free-choice 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 precision-position 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 fixed-pivot 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

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


(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 large-scale 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 smaller-scale problems. The trend,

therefore, should be toward more compact

iteratively-based 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

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


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 free-choice 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


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 four-bar 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 above-described method


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

closed-form 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.



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, closed-form solutions are developed

for rigid-body guidance problems using the RS, CS, CC and

RC dyads. Since the function generation problem can

generally be converted to a rigid-body guidance problem

by a process known as inversion (19), and since the path

generation problem can generally be treated as an

incompletely specified rigid-body 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 closed-form 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

free-choice parameters. All of the synthesis procedures

given here are for the maximum number of precision

positions which result in closed-form 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 two-link 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)


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 rigid-body

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 four-bar 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 single-input 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 single-input

figure 4.2 Two dyads with identical floating link
rotations joined to produce a four-bar

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 rigid-body 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 three-component 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 right-hand set of Cartesian axes;



Some of the dyad-based spatial
which can be synthesized using
of this chapter

the methods



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 (so-called 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 nine-component three-by-

three matrix when describing spatial angular displace-

ments; this is the so-called 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 Revolute-Spheric (RS) Dyad

The revolute-spheric 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 rigid-body guidance using linear, closed-form 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


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


o. locates the center of the moving coordinate
system relative to the fixed coordinate system

r. locates the spheric joint center in the jth

position relative to the moving coordinate


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



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 A2-AA 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
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 2-A

and A -A2 intersect at the tip of position vector a0

(as shown in figure 4.5). Denoting the perpendicular


^3 A2


figure 4.5 The plane of the RS dyad

distances from A 2-A1 and A 3-A2 to the tip of a0 by A2

and 13, the vector a can be expressed in the alternate


I0 = 2 2 + Al +

a0 3 + A2 + 3~ 2



Equating the right sides of equations 4.8 and 4.9 and

forming the vector product with p2 eliminates 12 and


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


A3 A2



explicitly for 13 by

(2 x 3 ) which gives

( 2 x ) (Cl + C2 3 C4
(32 x ) (* 2 x 3)


Ci = 2 x A

92 = x -1




C = 2 xA 2 (4.14)

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 Cylindric-Spheric (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
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
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


o., [R.], j = 1,2,3 (4.17)

Assuming the vector r arbitrarily again leads to the


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


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

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