
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00047158/00001
Material Information
 Title:
 Computeraided optimization in the dynamic analysis and parametric design of robotic manipulators
 Creator:
 YuanChou, HsinChien, 1950
 Publication Date:
 1985
 Language:
 English
 Physical Description:
 viii, 259 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Acceleration ( jstor )
Hands ( jstor ) Industrial robots ( jstor ) Inertia ( jstor ) Kinematics ( jstor ) Robotics ( jstor ) Stiffness ( jstor ) Structural deflection ( jstor ) Torque ( jstor ) Velocity ( jstor ) Dissertations, Academic  Mechanical Engineering  UF Manipulators (Mechanism)  Design and construction ( lcsh ) Mechanical Engineering thesis Ph. D Robots, Industrial  Design and construction ( lcsh )
 Genre:
 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1985.
 Bibliography:
 Bibliography: leaves 249258.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by HsinChien YuanChou.
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 University of Florida
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 University of Florida
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 The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. This item may be protected by copyright but is made available here under a claim of fair use (17 U.S.C. Â§107) for nonprofit research and educational purposes. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator (ufdissertations@uflib.ufl.edu) with any additional information they can provide.
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14393192 ( OCLC )

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COMPUTERAIDED OPTIMIZATION IN THE DYNAMIC ANALYSIS
AND PARAMETRIC DESIGN OF ROBOTIC MANIPULATORS
BY
HSINCHIEN YUANCHOU 1L !
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1985
With love to Ching Ying and WenYen
ACKNOWLEDGEMENTS
The author wishes to express his sincere appreciation to his supervisory committee chairman, Dr. Delbert Tesar, for his guidance, support and making this work possible. Much appreciation is also directed towards Mr. Mark Thomas for the encouragement, suggestions and sharing of knowledge.
He gratefully acknowledges the advice and support given by the members of his supervisory committee, Dr. Joseph Duffy, Dr. Ralph G. Selfridge, Dr. Gary K. Matthew and Dr. George N. Sandor.
The author would like to express his thanks to both past and present graduate students of the Center for Intelligent Machines and Robotics (CIMAR) for technical help and moral support. Particular thanks are extended to John Wonder, Behi Fariboz, and Gilbert Lovell.
Finally, the author extends his deepest appreciation to his wife and parents for the years of patience, encouragement and love they have provided.
TABLE OF CONTENTS
ACKNOWLEDGMENTS ........ ................... iii
ABSTRACT .......... ....................... vii
CHAPTER
INTRODUCTION .......................1
1.1 Subject Background and Dissertation
goals ......................... 1
1.2 Problems and Promise of Manipulators. . 3 1.3 Survey of Pertinent Literature ... ...... 8
1.3.1 Historical Development of
Optimization ...... ............ 9
1.3.2 Development of Manipulator
Dynamics ..... .............. 11
1.4 Scope of the Study .... ............ 13
2 OPTIMIZATION THEORY ..... ............. 15
2.1 Introduction to Optimization .......... 15
2.2 Definition of Optimization Problems . . . 16 2.3 Concept of Vector Norms ........... 18
2.4 Optimization Methods and Optimality
Conditions ................ 21
2.4.1 Classification of Optimization
Problems ..... .............. 21
2.4.2 OneDimensional Optimization
Methods ..... .............. 22
2.4.3 Unconstrained Optimization
Techniques .... .......... .. 30
2.4.4 Constrained Optimization
Techniques ..... ............. 39
2.5 Conclusions of Optimization Techniques. . 51
3 MODELING OF SERIAL ROBOTIC MANIPULATORS . . . 52
3.1 Introduction ..... ............... 52
3.2 Kinematic Arm Representation ......... .53
3.3 Kinematics of Serial Manipulators ... .. 53
3.3.1 Positioning of the Arm ......... .55
3.3.2 Kinematic Influence Coefficients
for Serial Manipulators ...... 58 3.3.3 Velocities for Serial Manipulators. 59
3.3.4 Accelerations for Serial
Manipulators .... ............ 64
3.4 Dynamics of Serial Manipulators ..... 67
3.4.1 Input Torques Due to Applied Loads. 68
3.4.2 Input Torques Due to the Inertial
Loads ..... ............... 71
3.4.3 The Generalized Actuator Stiffness
Matrix .................... 73
3.4.4 Vibrations of Serial Manipulators . 75
3.5 Conclusions of Modeling of Manipulators . 80
4 KINEMATIC AND DYNAMIC ANALYSIS OF ROBOTIC
MANIPULATORS ....... ................. 81
4.1 Introduction ..... ............... ...81
4.2 Arm Characteristics for the Industrial
Robot .................... 82
4.3 Workspace of the Industrial Manipulator . 84
4.4 Analysis of Actuator Torques Due to
Static Loads. .... ............... 95
4.5 Analysis of Actuator Torques Due to
Inertia .. ..................... 110
4.6 Analysis of the Manipulator Hand
Deflections ....................... 120
4.7 Analysis of the Natural Frequencies of
Manipulators ...... ............... 130
5 OPTIMIZING ACTUATOR PARAMETER DISTRIBUTIONS FOR
ROBOTIC MANIPULATORS BASED ON LOCAL DYNAMIC
CRITERIA ........ ................... 139
5.1 Introduction ..... ............... 139
5.2 Maximum Actuator Torques Due to Generic
External loads ..... .............. 140
5.3 Maximum Actuator Torques Due to Generic
Hand Velocity ..... .............. 147
5.4 Maximum Actuator Torques Due to Generic
Hand Acceleration .... ............ 156
5.5 Optimizing Actuator Compliance Distribution For a Known Load ... .......... 164
5.6 Optimizing Actuator Compliance Distribution For Generic Load ... .......... 176
5.7 Natural Frequencies Analysis Based on the
Local Optimal Actuator Stiffness
Distribution ..... ............... 180
6 OPTIMIZATION OF ACTUATOR PARAMETER DISTRIBUTIONS
FOR DESIGN OF ROBOTIC MANIPULATORS ........ .184
6.1 Introduction ..... ............... 184
6.2 Distribution of Actuator Sizing Due to
Static loads ...... ............... 186
6.3 Distribution of Actuator Sizing Due to
Inertial Loads ........ . ....... 201
6.4 Distribution of Actuator Compliance for
Generic Loads .. ............... ..215
7 CONCLUSIONS AND RECOMMENDATIONS ........ 220
7.1 Conclusions ...... ............... 220
7.2 Recommendations for Future Research . . . 222
APPENDICES
A DERIVATION OF THE KUHNTUCKER CONDITIONS. . . 228
B REVERSE DIJPLACMENT ANALYSIS OF CINCINNATIMILACRON T 776 ROBOT ..... .......... 235
B.I Introduction ...... ............... 235
B.2 Reverse Displacement Analysis  Part I. . 236 B.3 Reverse Displacement Analysis  Part II . 240
REFERENCES .......... ...................... 249
BIOGRAPHICAL SKETCH ....... ................. 259
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
COMPUTERAIDED OPTIMIZATION IN THE DYNAMIC ANALYSIS
AND PARAMETRIC DESIGN OF ROBOTIC MANIPULATORS By
HsinChien YuanChou
December, 1985
Chairman: Delbert Tesar
Major Department: Mechanical Engineering
Robotic manipulators provide general, programable motion paths and force functions to carry out highly dextrous and flexible processes. These systems are characterized by several degrees of freedom of controllable motion. As a consequence, the resulting mechanical structure contains a very large number of design values including geometric, mass, stiffness, strength and actuator parameters. The mathematical relations between the parameters and the manipulator's force and motion states are extraordinarily complex, nonlinear, and highly coupled. Therefore, the design of manipulators is an expensive, timeconsuming, and challenging task. Presently, the designers are lacking guidelines and well established design criteria.
This study brings together two fields: optimization theory and robotics. The goal is to develop analysis,
vii
design criteria and computational tools for optimizing the distribution of actuator parameters which satisfy load and precision requirements and increase the system's fundamental natural frequency for faster operations.
A rigidlink manipulator model is employed in this study. Computer generated plots are used to display extensive analysis information in a compact, understandable form. These plots show how system parameters vary as a function of hand position. Computeraided procedures for systems become an imperative in order to establish the dynamic design formulation, select rational design specifications, and evaluate the system's operating characteristics both locally and globally.
In order to develop a first level of computational and design methodology, local properties, i.e., properties at a specified configuration of the device, are considered first. A more complete methodology would result by considering a range of configurations in the workspace. This leads to the application of general nonlinear optimization techniques to develop design tools for manipulators. The design techniques developed here can greatly reduce the design cycle time for accurate, reliable manipulators, and also be used to improve or redesign existing manipulators to produce better dynamic performance. A detailed analysis for the industrial robot T3776 is presented.
viii
CHAPTER 1
INTRODUCTION
1.1 Subject Background and Dissertation Goals
The need to establish design criteria and design
processes for robotic manipulators has been recognized as a major subject at a Robotics Workshop [1]i held in 1978 at University of Florida and well documented [25]. This need is especially evident for next generation robotic systems which will be used for highspeed and precision operations, and will have more general geometric structures. Almost all manipulators in use today have very simple geometric structures, which can be analyzed by simple computational procedures but which do not provide the functional capacity of more general devices. The design of a general robotic manipulator is an expensive, timeconsuming and challenging task, because there is a large number of system design parameters. The magnitude of this task can be illustrated by noting that a general six degree of freedom serial manipulator can have as many as 18 geometric parameters, 60
1The numbers in brackets refer to references at the end of the document.
mass parameters and 42 stiffness parameters, along with 12 or more actuator parameters. The design and development of such a generic structure can cost millions of dollars. For
3
example, the Cincinnati Milacron T arm took 6 years to develop and an outlay of $6 million [4]. The NASA space shuttle manipulator is said to have cost about $100 million [6]. As requirements for precision operation, cyclic speed of operation, external loads and complexity of geometry increase, the ability to meet complex design objectives becomes more critical. The manipulator is proving to be one of the most difficult of our present day systems to design. One of the important means of breaking this technological barrier is the creation of a body of mathematical tools which is capable of making the design of new, more versatile systems feasible and the precision operation of these systems a reality.
The objective of this report is to develop analysis
procedures, design criteria and computational design tools for optimizing the distribution of actuator strength and stiffness parameters and increasing the system's fundamental natural frequency in a general robotic manipulator. The results should lead to improved system performance, economy and reliability, and should reduce the design cycle time which today is 6 to 8 years.
To analyze and design a manipulator requires a representative dynamic model of the system. A report [7]
establishes the rigidlink manipulator model and develops analysis methods, based on classical optimization, for evaluating the maximum accelerations, velocities and static load capacities for a given manipulator configuration. The model formulation explicitly shows how system strength and speed characteristics vary with the configuration of the system. Since this position dependence is nonlinear, general nonlinear optimization algorithms are required to isolate the "worstcase" hand positions. This work suggests some applications of nonlinear programming optimization in the analysis and design of manipulators, using position dependent kinematic and modeling coefficients to explicitly show the role of the significant parameters. Specific examples are employed to expose and demonstrate the procedures. Similar theoretical and computational tools are applicable to other related tasks, such as control of redundant manipulators [8101 or evaluation of a device's reachable workspace [11].
1.2 Problems and Promise of Manipulators
Robotic manipulators represent a technology which has rapidly become an important tool for increasing productivity and quality in industry. Together with the digital computer and computeraided manufacturing equipment they have caused a shift in the history of manufacturing similar to that which occurred when steam engines and electric
motors were substituted for human labor. In this new shift the main event is that the information flow is increasingly being automated.
The industrial robot industry emerged in the late
1960s and thus can be considered in its infancy (compare, for instance, with the machine tool industry). The following facts constitute a basis for the importance of this new industry:
1. it is an industry with a very high growth
potential;
2. it has become a symbol for flexible factory
automation in general and thus has large indirect
effects on the wide diffusion of CAD/CAM.
Industrial robots are a key component of flexible
manufacturing technologies because their reprogramability allows them to be quickly adapted to changes in the production process. There are many major benefits to be gained from using robots in manufacturing:
1. increased productivity;
2. increased product flexibility;
3. reduced labor cost;
4. elimination of dangerous and undesirable jobs;
5. improved product quality.
The introduction of robots led to high expectations for their use. However, during the past twenty years, robot applications have been limited primarily to the areas which are simple and most easily achieved industrial functions such as the following:
1. spot welding;
2. material handling;
3. machine loading;
4. inspection;
5. simple assembly.
Because the limitations of manipulator application, several problem areas have been identified in order to enhance robotics technology and applications. At the top of the list of problem areas is the issue of precision. Manufacturer's of robots rarely publish information regarding accuracy, preferring to describe repeatability instead. Repeatability is the ability of a manipulator to position the end effector at a taught position in space within a radius of the smallest sphere that can include all return points. Accuracy is the difference between the specified point and the center of the sphere. Figure 1.1 shows the concepts of accuracy and repeatability in the two dimensional case [121.
Center Target
, I ,
 R 04 R
A  Accuracy
R  Repeatability
Figure 1.1 Accuracy and repeatability
The absolute accuracy of most industrial robots is known to be not better than + 0.05 inches. Assembly and light machining operations require a positional accuracy of less than 0.005 inch; otherwise, offline programming cannot be used. The more difficult condition is to maintain accuracy while the manipulator experiences large load variations. Industrial robots are commonly available with repeatability of +0.008 inches. The same improvements are needed in repeatability as in positional accuracy.
A second problem area is the speed at which robots
move through a operating cycle. Typical robots of today do not work any faster than human workers. Highspeed operations for robots offer a serious challenge to the servo systems and control systems which must be.fast enough to accommodate the rapid changes in inertial characteristics of the robot as velocities and accelerations change during operations.
Third identified problem area is that current industrial robots are too large, too massive, and lacking in versatility. They can lift weights equal to only about 10% of their own weight. The prospects in this area for improved mass distribution in the system and the utilization of lightweight materials are very promising.
Another area is the need of great geometric dexterity. In material handling operations, assembly and nuclear maintenances, currently available industrial robots are not
adequately flexible to enable them to perform a variety of tasks. Most robots are constructed with six or less degrees of freedom. One of the best ways to increase dexterity is to add additional degrees of freedom to an arm. These redundant degrees of freedom make obstacle avoidance much more likely. Unfortunately, these redundant degrees of freedom make the control of such a manipulator very difficult.
The final important problem area is in the control
systems. Several subareas of improvements are required in robot control systems. Controllers need to be much more sophisticated in their ability to interact between manipulators and sensors in real time, because the flow of data from sensors in the system increases in order to increase manipulator's operational capacity and precision, and the complexity of this mathematical interface increases exponentially.
There are still many other problem areas, such as interfaces, sensors, programming languages, etc., which need improvement before a truly major impact of using robots for manufacturing will be felt on the economy.
1.3 Survey of Pertinent Literature
The level of interest in the application of optimization methods to mechanical design seems has increased dramatically during the past two decades. The main reason for
this is that the development of the digital computer has irrevocably changed the viewpoint and increased the computational ability of the engineering analyst. The ultimate goals are means to achieve mechanical designs which are globally optimum relative to all pertinent criteria.
A review of some historical developments and relevant literature in the areas of optimization theory and manipulator dynamics is given in the following two sections.
1.3.1 Historical Development of Optimization
Some early attempts to optimize were those of da Vinci (14521519) and Newton (16421727), who were both involved in specific mechanism design problems [13,14], and made modifications of previous designs to enhance their performance. Galileo (15641642) developed a rational optimization scheme [15] for selecting the form of a bent beam for uniform strength.
Optimization theory advanced significantly with the development of the differential calculus which introduced an elegant way for the evaluation of maxima and minima of differentiable functions. The foundations of the calculus of variations were laid by Bernouli [16], Euler [17] and Lagrange [17]. The method of optimization for constrained problems, which involves the addition of unknown multipliers became known by the name of its inventor, Lagrange. Cauchy (17891857)[19] proposed the steepest descent
algorithm to solve unconstrained minimization problems in 1847. In spite of these early contributions, very little progress was made until the middle of the twentieth century, when highspeed digital computers made the implementation of the optimization procedures possible and stimulated further research on new methods. Spectacular advances followed, producing massive literature on optimization techniques.
Since the simplex method of Dantzig [20] in 1949 for
linear programming problems and the work of Kuhn and Tucker [21] in 1951 on the necessary and sufficient conditions for the optimal solution of nonlinear programming problems laid the foundations for later development of the methods of constrained optimization. Bellman [22] developed the principle of optimality in 1957 for dynamic programming which is a mathematical technique well suited for the optimization of multistage decision problems. Significant developments have taken place in the area of gradient techniques with the introduction of the Variable Metric method suggested by Davidon [23] and modified by Fletcher and Powell [24]. This extremely successful method is an enormous improvement over simple steepest descent algorithms. The contributions of Zoutendijk [25] and Rosen [26, 27] to nonlinear programming during the early part of the 1960s have also been very significant. Although no single technique has been found to be universally applicable for
nonlinear programming problems, the penalty function algorithms developed by Fiacco and McCormick [28] made many difficult problems solvable by using the wellknown techniques of unconstrained optimization. Geometric programming was developed in the 1960s by Duffin, Zener and Peterson [29]. Gomory [30] did pioneering work in integer programming. Some papers that review the state of the art of optimization have been written by Fletcher [31], Dixon [32] and Powell [33,34,35].
1.3.2 Development of Manipulator Dynamics
The need for efficient mathematical formulations of manipulator dynamics has been recognized and the development of these formulations has rapidly grown in the past decade. The importance of dynamics of robotics manipulators stems from its use in simulation, analysis, control and design for the next generation robotic systems which will be used for highspeed and precision operations.
In 1965, Uicker [36] formulated a derivation of the equations of motion for spatial mechanisms using 4x4 transformation matrices and the Lagrange equations. These results were specifically written for open loop manipulators by Kahn [37], but these solutions require excessive computation and were recognized as impractical for real time control applications. Bejczy [38] considered neglecting the velocityrelated terms to obtain a simpler
approximation for actuator torques, but the results are inaccurate in many cases. In a parallel effort to reduce computation time, Whitney [39] and Raibert [40] considered the possibility of using lookup tables. A dramatic cut in the time required for calculations may be attained. However, the amount of storage necessary for control of a general spatial manipulator is extremely large. Furthermore, additional problems in storage and access of information may occur.
The inefficiency of the original Uicker/Kahn formulations, as well as other reasons, has led researchers to look for alternative formulations of manipulator dynamics. In the effort to make real time dynamic computation feasible, recursive algorithms have been developed using both NewtonEuler [41,42] and Lagrangian [43] dynamic formulations and Silver [44] has shown that with a proper choice the Lagrangian formulation is indeed equivalent to the NewtonEuler formulation. In these methods, the velocity and acceleration of each link are found sequentially starting with the fixed base link. Then, beginning at the free end and moving back to the fixed base, the force or torque on each successive joint is determined. These recursive methods are computationally the fastest for evaluating the actuator and link loads for a prescribed arm motion state with known external loads and manipulator properties. However, their use for design is limited to
trialanderror analysis, because the geometric properties of manipulators are hidden in their recurrence relations.
Recently, nonrecursive formulation which keeps the
dynamic terms and the important design parameters explicit in the equations of motion has been developed by Thomas and Tesar [5,7]. The method is especially wellsuited for manipulator systems because expressions for the kinematic influence coefficients, the basic components of the model, are extremely simple and compact for serial manipulators. The influence coefficients may also provide a highly efficient algorithm for evaluating the dynamic model in real time. Thus, the influence coefficient formulation may be advantageous for both the control and design of mechanisms and manipulators. This modeling approach has been extended to modeling and analysis of hybrid parallel/serial manipulators by Sklar and Tesar in 1984 [45]. More applications of influence coefficients for planar mechanisms can be found in [46,47,48].
1.4 Scope of the Study
This work brings together two fields, namely, optimization theory and robotics in order to develop design criteria and computational design tools for robotic manipulators. In Chapter 2 basic theorems of optimization are reviewed and some methods of nonlinear programming are discussed.
Chapter 3 of the report covers the kinematic aspects of the manipulator problem and the kinematic influence coefficients. Based on the kinematic influence coefficients, the dynamic model for dealing with applied loads, inertial loads, deflections, stiffness matrices, and natural frequencies are introduced.
Chapter 4 illustrates the use of the modeling formulation for the analysis of the characteristics of an industrial robot. Computer generated contour plots are employed to express the massive data in a compact form. In Chapter 5, some local dynamic criteria are established for distribution of certain design parameters which are modelled in Chapter 3.
The localized techniques in Chapter 5 are extended to develop global dynamic design criteria for general robotic manipulators. These design procedures make it possible to satisfy load and precision requirements and provide faster operation by increasing the frequencies of the fundamental vibrational modes. Chapter 7 offers conclusions and suggestions for future research work.
CHAPTER 2
OPTIMIZATION THEORY
2.1 Introduction to Optimization
Man's longing for perfection finds expression in the theory of optimization. It studies how to describe and attain what is best, once one knows how to measure and alter what is good or bad. Normally, one wishes the most, or maximum, good and the least, or minimum, bad. Optimum has become a technical term connoting quantitative measurement and mathematical analysis, whereas "best" remains a less precise word more suitable for everyday affairs. The technical verb optimize, a stronger word than "improve," means to achieve the optimum, and optimization refers to the act of optimizing. Thus optimization theory encompasses the quantitative study of optima and methods for finding them.
In order to have a solvable optimization problem, the desired benefit or the required effort must be expressible 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 the limitations on the behavior or
performance of a system. These limiting factors are called constraints, and they, too, must be expressible as functions of the design variables.
It can be seen from Figure 2.1 that the minimum value of a function, f(x), corresponds to the maximum value of the negative function, f(x). Thus, without loss generality, optimization can be taken to mean minimization since the maximum of a function can be found by seeking the minimum of the negative of the same function.
2.2 Definition of Optimization Problems
A common characteristic of all optimization problems is the existence of many feasible solutions. The seeking of the best possible solution depends on a clear definition of the interaction of all the design variables, an explicit statement of criterion or objective, and an effective search procedure for locating the optimum in accordance with the stated objective.
An optimization or a mathematical programming problem can be stated in the following general form [49].
Find X=[X ..., X nT which minimizes f(X)
subject to the equality constraints
g.(X) = 0, j=l, 2, ..., m (2.1)
and the inequality constraints
hi(X) < 0, j=m+l, m+2, ..., p (2.2)
Feasible region
f(X)
X =Minimum of f(X)
in a < X
X =Maximum of f(x)
in a < X < b
f(x)
Infeasible
region
Figure 2.1 Minimum of f(X) is same as the minimum of f(X)
f (X)
Infeasib
region
where X is called design vector, and f(X) is called the objective function. A necessary condition for existence of a design problem is n>m, in order to ensure the existence of free parameters. If there are no constraints (p=O), the problem is said to be unconstrained. An optimization problem is said to be linear if the objective function f(X) and all the constraint functions are linear in the n variables, X1, X2, ... X. Otherwise, the optimization problem is said to be nonlinear.
2.3 Concept of Vector Norms
Since manipulators and mechanisms operate in a multidimensional space, the objective and constraint functions usually contain vector quantities to describe important design criteria or dynamic phenomena. It is necessary to have some means of "measuring" vectors, in order to say that one vector is "larger" or "smaller" than another. The definition of a norm gives a rule for associating a nonnegative scalar with a vector [50], then the constraint and performance can be written in terms of scalar magnitudes associated with these vectors. The weighted pnorm of a vector u is defined as [6]
IIu 1Dp ( 1W u iP)I/p (2.3)
j J
where W. is the weighting factor for component j and the
s
summation is performed over all components of the vector.
While W. indicates the relative significance of component
J
j, the exponent p determines the importance of the largest components relative to the smaller ones. The larger the value of p, the more the vector norm depends on the largest scaled component and vice versa.
The most useful values of p are 1, 2 and o:
I jj1 = E 1w ujI, p=l (2.4)
II II 2 = E 1Wj u. 12)1/2, p=2 (2.5)
J
11 1K. = max(IW. uj1), p = (2.6)
J J J
In the two dimensional case , the curves in Figure 2.2 are effectively contour curves along which the norm is constant. These particular norms are the most popular because they are easily interpreted and their differentiation, with respect to the components u., produces tractable results. In general, the p=2 norm leads to higherorder equations in the optimization solutions. The other two norms often provide several individual cases which each must be solved.
1Nondiagonal weighting matrices [W] rotate these curves in the uplane.
20
1.5
1.0
.5
.5
1.0
1.5
2.0
Weighting values WI=I, Norm value K=1
W 2= 5
Figure 2.2 Curves of constant magnitude, Ij u p =K
p
2.4 Optimization Methods and Optimality Conditions
The mathematical programming problem stated in Section
2.2 has given rise to a great variety of solution methods. In order to select an efficient method for solving a particular problem, it is necessary to analyze and classify the problem.
2.4.1 Classification of Optimization Problems
When faced with any optimization problem, it is usually advantageous to determine the special characteristics that allow the problem to be solved more efficiently. Therefore, we consider how to classify optimization problems in order to enhance the efficiency of the resulting solution methods.
Optimization problems can be classified in several
ways. For example, it can be classified [49] on the basis of
1. the existence of constraints,
2. the nature of the design variables,
3. the physical structure of the problem, 4. the nature of the equations involved.
Although no set of classifications is ideal for every circumstance, the classification based on the nature of problem functions is extremely useful from the
computational point of view since there are many methods developed solely for the efficient solution of a particular class of problems. Table 2.1 gives a typical classification scheme based on the nature of the problem functions.
It is also highly desirable to organize the available nonlinear optimization approaches in order to get a feel for what is available, help classify past and present efforts, and perhaps enhance decisions in future efforts. Figure 2.3 shows such a representation.
In the next three sections, three types of methods will be described: onedimensional optimization, unconstrained optimization, and constrained optimization. The discussion has been restricted to certain methods which have proven useful in the present report.
2.4.2 OneDimensional Optimization Methods
We begin by considering the problem as follow:
Find X which minimizes f(X)
subject to XI < X < Xu
where f(X) is a real function of the real variable X, and Xu$ X, are given bounds on X. We consider this problem first because it is one of the simplest optimization problems and because many of the methods for solving more complex problems require the solution of this problem. The
Table 2.1 Classification of optimization problems based on the nature of the
problem functions
Properties of objective function
Function of a single variable Function of several variables Linear function Nonlinear function Quadratic function Sum of squares of linear function Sum of squares of nonlinear function Smooth nonlinear function Nonsmooth nonlinear function
Porperties of constraint functions
No constraints Simple bounds Linear functions Nonlinear functions Smooth nonlinear functions Nonsmooth nonlinear functions
1
OneDimensional Minimization
Unconstrained Optimization
Figure 2.3 Methods of Nonlinear Optimization
following two theorems provide the necessary and sufficient conditions for the relative or local minimum of a function of a single variable [49].
THEOREM 2.1 If a function f(X) is defined in the interval XI<_X
* X*
number at X=X , then f'(X )=O.
THEOREM 2.2 Let f'(X* )=f(X*)= ...=f(nl)(X*)=O, but f(n)(x*)40. Then f(X*) is I). a minimum value of f(X) if f(n)(x*)>0 and n is even, 2). a maximum value of f(X) if f(n)(x*)
Several numerical methods, as in Figure 2.3, are
available for solving the onedimensional minimization problem. These can be classified into two categories: elimination methods and interpolation methods. The elimination methods can be used for the minimization of discontinuous functions. The interpolation methods involve polynomial approximations to the given function.
Golden Section Method [49, 51, 52]
The basic idea behind the methods of Golden Section
and Fibonacci is to economize the number of function evaluations, trap a relative minimum in an interval and
successively decrease the size of the interval. The process thus gives successively better estimates for the location of the minimum point. The difference between these two methods is that in the Fibonacci method the total number of the experiments to be conducted has to be specified before beginning the calculation, whereas, this is not required in the Golden Section method.
The Golden Section search technique can be given in the form of a computational algorithm:
Step 1. First an upper bound qu must be found for q.
It is clear that 0 is a lower bound, ql. For a
chosen small step size 6 in q, let j be the
smallest integer such that Jk
f[X(i) + E 6(1.618) >
k=0 f[X(i) j k k=0
Then upper and lower bounds on the minimum,
q , are
J k j2 )k qu Z 6(1.618) q, E 6 (1.618 k=0
Step 2. Compute f(x(i)+q ) and f(X(i)+qb) where
q a = ql + .382 (q  ql) qb = q, + .618 (qu  ql)
Step 3. Compare f(x(i)+q a) and f(X(i)+qb) and go
to Step 4, 5 or 6.
Step 4. If f(X(')+qa)
By the choice of q a and qb' the new points
pl=ql and pu=qb. Compute now f(X(i)+p a where
pa=Pl+.382(puPl). Go to Step 7.
Step 5. If f(x(i)+qa)>f(x(i)+qb), then q aq*
P=q a and pu=qu so that pa=qbï¿½ Compute f(X(i)+
Pb) where pb=pl+.618(pupl). Go to Step 7.
Step 6. If f(x(i)+qa)=f(X(1)+qb), put ql=q and
qu=q b. Return to Step 2.
Step 7. If 1pu  pl1
accuracy, put q =.5(Pu+Pl) and stop.
Otherwise, put ql=pl, qa=Pa, qb= Pb and qu=Pu
and return to Step 3.
Quadratic Interpolation Methods [49, 51, 52, 53, 54]
The interpolation methods (see Figure 2.3) are generally more efficient than the Fibonaccitype methods; that is, fewer function evaluations are necessary for the minimum to be located to a specified accuracy. But these methods can fail or be inefficient if the objective function cannot be well approximated by a loworder polynomial.
The obvious idea of using a loworder polynomial for curve fitting is that we can replace a possibly complex function by a very simple function, the minimum of which is
known analytically. This will mean that finding the minimum will be a straightforward procedure. The use of quadratic or cubic polynomial means that we assume that the approximated function is relatively smooth and possibly unimodal in the interval being investigated. In many practical applications, a small enough interval assures validity of this assumption.
Box, Davies, and Swann [53] recommended that the algorithm of DSC be executed to bracket the minimum, X , by increasing the step size until the minimum is overshot and then a single quadratic interpolation is performed to estimate X ï¿½ Figure 2.4 illustrates the procedure and the algorithm is summarized as
Step 1. Evaluate f(X) at the initial point X(0).
If f(X(O)+ 6)
f(X(O)+ 6)>f(X(0)), let 6=6 and go to Step
2. Where 5 is a small step size.
Step 2. Compute x(i+l)=x(i) + 6.
Step 3. Compute f(X(i+l)).
Step 4. If f(X(i+l))
to Step 2 with i=i+l. If f(X(i+l))>f(X(i)),
denote X(i+l) by X(m), x(i) by X(ml)
etc., reduce 6 by onehalf, and return to
Steps 2 and 3 for one more (only) calculation.
x(m2) x(mi) x(m+l) X(m)
Figure 2.4 DSC unidimensional minimization
f (X)
Step 5. Of the four equally spaced values of X in the
set, X(m+l), X(m), X(ml) and X(m2), discard
either X(m) or X(m2), whichever is farthest
from the X corresponding to the smallest value
of f(X) in the set. Let the remaining three
values of X be denoted by Xa), x(b), and
X(c), where X(b) is the center point and
x(a)x( b) 6 and X(c)=x(b)+ 6.
Step 6. Carry out a quadratic interpolation to
,
estimate X
+ [f(X(a))  f(X(b))] 2[f(X(a))  2f(X(b)) + f(X(c)] Step 7. If If(x*)f(x(a))I< E and If(X*)f(X(b))I
terminate the process. Otherwise, reduce the
step size 5 and find the smallest value of
f(X(a)), f(X(b)), f(X(c)), and f(X*) and go
back to Step 1.
2.4.3 Unconstrained Optimization Techniques
We consider in this section a very important class of optimization problems, the class of problems of minimizing a real valued function of n real variables with no constraints. An unconstrained optimization problem can be stated in the standard form as
Find X such that the objective function
f(X) is minimum.
This problem can be considered as a particular case of the general nonlinear programming problem. The special characteristic of this problem is that the solution X need not satisfy any constraint. It is true that rarely a practical design problem would be unconstrained; still the study of this class of problems is important because most of the unconstrained optimization methods can be extended to handle constrained problems, either by directly considering the constraints or by transformation to an unconstrained problem.
The following two theorems provide the necessary and sufficient conditions for X to be a relative minimum or maximum of the unconstrained optimization problems [50].
THEOREM 2.3 A function f(X) has a relative minimum at
the point X if the first partial derivatives of f(X) at X is Vf(X ) = 0 and the matrix of second partial
derivatives (Hessian matrix) of f(X) at X
V 2f(X*) [2f/Xi 9X
is positive semidefinite.2
2A real symmetric matrix R is 1). positive definite if zTRz>o for all Z=O, 2). positive semidefinte if Z RZ>O for all Z.
THEOREM 2.4 A sufficient condition for a point X to be a relative minimum point is that the first partial
derivatives of f(X) at X are zero and the Hessian
matrix evaluated at X is positive definite.
Several methods are avilable for solving an unconstrained minimization problem. These methods can be classified into two categories: direct search methods and descent (or gradient) methods (see Figure 2.3). 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. In general, the direct search methods are less efficient than the descent methods.
Variable Metric Method [23, 24, 55]
Significant developments have taken place in the area of descent techniques with the introduction of the Variable Metric method by Davidon [23] and modification of the method by Fletcher and Powell [24]. This method is reported to be one of the most powerful known methods for general functions f(X) [55]. A major reason for the success of this method is its capability to accumulate information about the curvature of the objective surface during the
entire iterative process, even though only first order derivatives of the objective function need to be computed. The iterative procedure of this method can be stated as follows:
Step 1. Make an estimate X(0) of the minimum point
and choose a symmetric positive definite
matrix H(0). Usually H(0) is taken as the
identity matrix I.
Step 2. For i=0, 1, 2, ... compute
S(i) = H(i) vfT(x(i))
Step 3. Compute q=q(i) which minimizes f(X(i)+q si
Step 4. Compute
r(i) = q(i) S(i)
X(i+l) = X(i) + r(i)
H(i+l) = H(i) + Ai) + B(i)
where
y(i) VfT(X(i+l))  VfT(X(i)) A(i) = (r(i) r(i)T)/r(i)T y (i)
B(i) = _(H(i) y (i) y(i)T H(i)T)/y(i)T H(i) y (i)
Step 5. If IVf((X (i+l)) and 1X (i+l)  x(i)11 are
sufficiently small, terminate the process.
Otherwise, return to Step 2.
The Variable Metric method has many of the advantages of the second order gradient techniques without having some of their disadvantages. The advantages include fast convergence near a minimum and finiteness for a quadratic function. Due to the symmetry of H(i), the method is also comparable in its computer storage requirements. Comparison with other quadratically convergent methods shows that the Variable Metric method performs better for general nonquadratic functions.
Powell's Method
Occasionally in design and analysis applications, one is often faced with a problem in which computation of derivatives of the objective function is impossible or at least prohibitive from a computational point of view. There are many techniques for solving this sort of problem given in [49, 54, 56]. An efficient technique was developed by Powell [57] using conjugate directions.
The basic idea of the Powell's method can be quite
simply explained in twodimensional case and is illustrated in Figure 2.5. In descriptive terms the strategy is as follows:
I). Conduct a minimization along a line in each of the
coordinate directions in turn.
5
3
3
5
Figure 2.5 Strategy for Powell's method
3 1 1 3 5
2). Make a pattern move which is minimization along
the direction corresponding to the coordinate
moves in item I.
3). Repeat item I except that one of the coordinate
directions is replaced by the direction of the
last pattern move.
4). Continue in this fashion until the moves become
negligibly small.
The flow diagram for the Powell's method is given in Figure 2.6. A computational algorithm is presented as follows:
Step 1. Make an estimate of the minimum point x(O)
of f(X). Choose vectors SO, j1, 2,  .
in the coordinate directions of Rn
Step 2. Find q=q (k), k=l, 2, ..., n, which minimize
f(X(kl) + q S(k)), where
y (0) =X(i)
y(k) = (kl) + q(k)s(k) k = 1, 2, ..., n
and i is the number of iterations which have
been completed.
Starting point X, Max. No. of iteration IMAX
Set X  "+ q S s ( 0)/Ix ()
Yes
Figure 2.6 Flow chart for Powell's method
Step 3. Find the integer m, 1
A  f( (l))  f( (m))
is the largest.
Step 4. Define f1=f(y(0)), f2=f( (n) ), and
f3=f(2y(n)  y(0))
Step 5. If f3>fl or
(fl  2f2 + f3)(fl  f2  A)> .5 (fl  f3)2
put x(i+l)=Y(n). Terminate the process if
11x(+I ) x (i) II2 is sufficiently small.
Otherwise, return to Step 2 with the same set
of SO), j=l, 2, ..., n.
Step 6. If neither of the inequalities of Step 5 hold,
define s=y (n) (0) and find q=q which
minimizes f(y(n)+q s). Put x(i+l)=y(n)+q*s.
Terminate the process if Ix(i+l)x(i) 112 is
sufficiently small. Otherwise, return to Step
2 with the new set of vectors S(1), S(2)
_ , "'',
s5l (m+l) Sn
2.4.4 Constrained Optimization Techniques
The optimization problem defined in Section 2.2 can be rewritten in the following form:
Minimize f(X) subject to the constraints
C.(X) < 0, j = 1, 2, ..., p
and X e S, where S is some given set in Rn.
The set, S = {XI CJ(X)<0, j=l, 2, ..., p}, is called the feasible region. If the constraints are linear, the problem is considerably easier to solve than if they were nonlinear. If both the objective function and the constraints are linear, then we have a linear programming problem, the solution of which can be obtained by standard or revised simplex algorithm [20, 49]. We shall not discuss the simplex algorithm here.
The conditions to be satisfied at a constrained
minimum point, X , of the problem can be stated as follows:
3 f p 3Cj
+ E A = 0, i = 1, 2, ..., n (2.7)
DXi j=l aXi
x. C = 0, j = 1, 2, ..., p (2.8)
C. < 0, j = 1, 2, ..., p (2.9)
and
j = 1, 2, ..., p
where Xi, j=l, 2, ..., p are the Lagrange multipliers. These conditions are called KuhnTucker conditions [21] after the men who developed them. A detail derivation of these conditions are given in Appendix A. These conditions are, in general, not sufficient to ensure a relative minimum. Sufficient conditions for X to be a relative minimum of the problem are as follows:
af p C
+ E X axi jl 3 x A. C.ï¿½ = 0 ,
J 0J
C < 0, j
> , j=
i = 1, 2, ..., n
j = 1, 2, ..., p
1, 2, 1, 2,
"'', p
n n 2L
E E dXi dXk * i=l k=1 a Xi 9Xk X=X
is positive ,definite
where L is the lagrange function. Some good general discussions of the optimality conditions are given in [28, 58, 59]. In the following Sections, we shall discuss two solution techniques for solving constrained minimization problems.
(2.11)
and
(2.12)
(2.13) (2.14)
(2.15)
(2.10)
Xj > 0,
Gradient Projection Method
The gradient projection method of Rosen [26] is very efficient for solving problems with linear constraints. Although the method has been described by Rosen [27] for a general nonlinear programming problem, its effectiveness is mainly confined to problems in which the constraints are all linear. For the nonlinear constraints, very small steps must be taken, and a calculation to bring the new point back in the feasible region must be included in each cycle. So zigzagging will frequently occur in such situations.
The basic idea in the method for the linear constraints is try to move as close as possible to the direction of steepest descent, Vf(X), but still remain feasible. If moving in the direction of steepest descent would cause any of the constraints to be violated, X is changed along the projection of the negative gradient of the objective function onto the boundary of the feasible region.
Let us now summarize the gradient projection algorithm. A flow chart of this procedure is shown in Figure
2.7. It will be assumed that the initial point X(0) is admissible and lies in the intersection Q' of q linearly independent hyperplanes.
Read X (0) obj. function, con) straints and tolerances E1. S 2
Determine the hyperplanes HI..., Hq which form the r h q [T 1
intersection Q'; Form the matrices Nq1 [Nq Ngl
Stop and print XSX f i(X
No[
I [Set i=i+lj
inner product Z [af, /a
Set x (i+l)=x,(i+l) Add H to Q'.
m
s the inner pro
Ye duct negative I No
Figure 2.7 Flow chart of the gradient projection algorithm
Calculate the proj. matrix Pq,
3f(i)/3X, the gradient proj., Pq [f (i ax. Pq[_f (i)/fXI r, .E and r. = max. component of r'.
Set i=i+ll
Form the normalized gradient proj
Z (i) =eq [_af (i)/;X]/11 pq [_af (i) /aX][[ Determine the max. step size Tm; Calculate x'(i+I)=x(i) + T Z(i).
I Comniit~ ~fI(i+l)/~1~ pu~iiiatpI
Interpolate to fin e in the equation x (i+l)=xh+OT Z(i for which
IZ (i)T [_ f (i+l) /aX]
I
Step 1. Read in X(0), the objective function f(X),
the constraints and tolerances. Determine
the hyperplanes HI, H2, ..., Hq which
form the intersection Q'. Set i=1.
T
Step 2. Form the matrices N and [NT N ] q q q
Calculate the gradient vector at the point X( ) _f(i)/aX, the vector r=[NT N  NT q q q
[_qf(i)/3X], the projection matrix Pq, and the
gradient projection P [3f(i)/aX].
q
If IIPq [_@f(i)/X] 11 < E, and rq <0, where rq
is the maximum component of r, then x(i) is
the constrained global minimum and the
procedure is terminated; Otherwise, go to
Step 3.
Step 3. Determine whether or not a hyperplane should
drop from Q'. If eq[Df( )/x] JI0,
form the projection matrix Pq1' and go to Step 4. The other alternative is that the norm of the gradient projection is greater
than El. In this case, calculate a=maxoi},
i
where ai athe sum of the absolute values of the T 1I
ith row of the matrix [N N q]I If r > q q q
drop the hyperplane H from Q', form the
q
projection matrix Pq1. and go to Step 4. If
r q< , Q' remains unchanged.
Step 4. Compute the normalized gradient projection
z(i)
zM)  (Pq [ Df(i/;X])/ llPq [Df(i)/x]
and the maximum allowable step size Tm, where
T is the minimum positive value of the T.'s
m j
found by evaluating
T X(i) T (i)
T.=(v. n. )In. Z
J J  
for j corresponding to all hyperplanes not in the intersection Q'. The tentative next point
X'(i+l) is found from
x,(i+l) = x(i) + T z(i)
Step 5. Calculate the gradient at the point X'(i+1), if
z(i)T[af(x'(i+l))/axi > 0,
Set x(i+I)=xt(i+l); since X(i+l) lies in the
intersection of Q' and H (the hyperplane which
corresponds to the step size T determined in
m
Step 4), add H to Q', and return to Step 2.
m
On the othter hand, if
z (i)r[3f(x (i+l))/3X] < 0
find X(i+') by repeated linear interpolation
as in Figure 2.8. 0=0 corresponds to the point
(i) T fM
_ L  _ j
(i) T i) T 
Repeated linear interpolation
Figure 2.8
X), and e=1 corresponds to the point X,(i+l). eP the abscissa where the straight line from A to B has an ordinate of zero, is determined from the relationship
Z(0)T [f(i) /aX]
el z(i)T[_ f(i)/ X] _ z (i)T[_ f,(i+l) / X]
Next, we evaluate the gradient at the point
x.(i+l) . x(i) + 0 T z(i)
  1 m
and form the inner product
z ( i ) T [ _ f . i ) / X
If this inner product is positive, we use point C and B to interpolate again ( if the inner product had been negative, point C would have a negative ordinate and point A and C would be used for next interpolation). This procedure is repeated until a point X(i+l) is found where the magnitude of the inner product is less than a preassigned small positive number 62; that is,
1z(i)T [ f(i+1)/ X1I < E2
The intersection Q' remains unchanged, and the computational algorithm begins another iteration by returning to Step 2.
Exterior Penalty Function Methods
The basic idea of penalty function algorithms is to reduce the constrained problem to a sequence of unconstrained problems which can be solved by the methods of Section 2.4.3, the solutions of which converge to the solution of the original problem. Let the original optimization problem be the form:
Find X which minimizes f(X) subject to
g.(X) = 0, j = 1, 2, ..., m, and (2.16) h.(X) < 0, j = m+1, m+2, ..., p (2.17)
This is done by lumping the constraints with the objective function in such a fashion that minimizing the lumped function penalizes any violation of the constraints. The lumped function can be shown in the following form:
m
U(X,r) = f(X) + r E G [gM(X)] + j=l Jr H [h i(X)] (2.18)
j =m+l J
where G [gM(X)] and Hj [h (X)I are functions of the con straints gM(X) and hi(X), respectively and r is a positive constant known as the penalty parameter. This is the reason why these procedures are called the penality function methods and also known as "sequential unconstrained minimization techniques" or simply SUMT.
Two categories of penality function methods exist,
namely, interior methods and exterior methods. The interior methods must be supplied with a feasible starting point, X(0). As the penality parameter r is varied over successive minimizations, the solution of the constrained problem, always remaining within the feasible region, are called interior methods. The exterior methods do not require a feasible initial point, and generally converage to the constrained minimum from the infeasible region, hence the term exterior. The exterior methods have been judged to be generally superior to the interior methods [60].
In the exterior penality function methods, the lumped function is taken as
m2
U(Xr) = f(X) + r 2 [g+(X)] + j=l
r E q (2.19)
j=m+l
where the exponent q is a constant greater than one, and the singularity function is defined as
hi(XM, if hi(X) > 0
(constraint is violated)
= (2.15)
0, if h.(X) < 0
(constraint is satisfied)
It can be seen from Equation (2.19), that this
formulation is to assess an increasingly severe penality 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 (2.19) using a small value for the penalty parameter r for the first minimization. Subsequent minimizations, until the solution is forced to converge in the feasible region, is necessary. This is because, if r remained small, very small positive values of hi(X) and small errors of gj(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 also being forced toward the feasible region.
The flow chart of an exterior penalty function method is shown in Figure 2.9, and the algorithm is outlined in the following steps:
Step 1. Make an estimate X(0) of the solution of the
constrained optimization problem and suitable
value of the penalty parameter r. Set i=1.
Figure 2.9 Flow chart of the exterior penalty function method
Step 2. Find the point X(i) that minimizes the
function
m
U(X, r) = f(X) + r Z [g.(X)] 2 + p j=l
r Z q
j=m+l '
Step 3. Test whether the point X(i) satisfies all
the constraints. If X(i) is feasible, it is the desired optimum and hence terminate the
procedure. Otherwise, go to Step 4.
Step 4. Increase the penalty parameter r and set the
new value of i as original i plus one, and go
to Step 2.
2.5 Conclusions of Optimization Techniques
This chapter is not intended to be complete, but is
intended to offer some basic concepts and methods which are useful in a design environment. Many other methods have been proposed, and are worthy of careful consideration. There is no algorithm that is most suitable for all optimization problems. Many different circumstances can affect the choice of method. There are many papers [55, 60, 61, 62, 63] that give some guidance in choosing a suitable method for solving a particular problem alongwith some other computational details. There are also several geniunely useful textbooks [64, 65, 66] in the area.
CHAPTER 3
MODELING OF SERIAL ROBOTIC MANIPULATORS
3.1 Introduction
An important initial step in the analysis, design, or precision control of a complex mechanical function generator, such as a manipulator, is to obtain a representative model of the device. The model must be accurate enough to give results which satisfactorily describe the operation of the actual device, yet simple enough to be of practical use. To satisfy these requirements, the rigidlink manipulator model is employed in this dissertation. The model neglects deformations in the links and joint deformations which are not in the direction of the nominal joint motion. This is a good approximation for industrial manipulators in use today since most (90%) of the deformations seen in industrial robot structure are due to compliance in actuators and associated drive train components. Although some flexiblelink derivations of manipulator model are available [67, 68], these formulations are too complex to be of much practical use with near term computational capabilities.
3.2 Kinematic Arm Representation
The serial manipulator is represented kinematically as a sequence of rigid links jointed by one degree of freedom lowerpair connectors as shown in Figure 3.1. Associated with each joint are two geometric parameters: 1). S. (or Sj)  the offset distance along the joint axis, Sj, between the two links which the joint connects; and 2).e (or 0j )  the relative rotation about S between these two links. The link direction between the successive joints j and k is represented by the unit vector a jk which lies along the common perpendicular of S and Sk. The fixed link length, ajk, is the distance between two successive joint axes measured along the a jk vector. The fixed angular rotation of S. to Sk is jk and is the angle taken about a jk in a right hand sense. The fixed Cartesian coordinate system (X , Y , Z ) is located at the first joint with the Z axis along Sl Each link has a local coordinate system (X(J) Y(j) Z(j)) associated with it. The X(j) axis is along the a jk vector and Z(j) axis is along the S. vector.
3.3 Kinematics of Serial Manipulators
In the kinematic analysis of manipulators two types of problems may arise. The forward kinematic problem involves determining the position, velocity, and acceleration of all
54
ajk XQj)
S..
a23
22 .\2
2
z1 Z
Kinematic representation of manipulator
Figure 3.1
point in the manipulator when these quantities are known for the independent joint coordinates [41]. Many straight forward solutions to this type problem exist and it is only necessary to determine which formulation is the most efficient in a given situation. The reverse kinematic problem, that of finding the required joint coordinates in order to obtain a desired motion of the hand, is much more complicated and has received much research recently, e.g. [69, 70].
In general there are two approaches to the reverse
kinematic problem. One is an iterative numerical solution [71] in which the present location is used to determine the joint coordinates of the next desired endeffector location. The other approach, which is considerably more difficult, is to establish a closedform analytical procedure [72] in which the joint coordinates can be solved for any hand location in the system workspace. Appendix B employs this analytical procedure to derive a detailed position analysis for an example industrial manipulator.
3.3.1 Positioning of the Arm
As mentioned in Section 3.2, there are two coordinate systems, the absolute and bodyfixed references, as shown in Figure 3.1. One of the most popular methods for arm positioning involves the use of 3x3 rotation matrices. Let T. represent the rotational transformation matrix which
relates the jth bodyfixed coordinate system to a reference system with common origin to the jth system and with axes parallel to the fixed reference axes. If P(J) is a vector in terms of the local jth reference, its direction P in terms of the fixed global coordinates is
P = T. P(J) (3.1)
where the rotational transformation matrix can be expressed as
T = [ Sjxa. 3 .I (3.2)
 jk k ]
The location of the origin of the jth reference system can be defined by the vector R.. As shown in Figure 3.1, the
J
position vector R. can be found by sequentially summing the
J
link lengths and joint offsets starting at the fixed base:
j
R Sl1 + E {a(m )m(m)m + m S m (3.3)
m=2
Then, the transformation locating a point P, with local coordinates P(J), in terms of the fixed reference is
P = R. + T.PQ) (3.4)
 j
Equations (3.2), (3.3) and (3.4) show that the
position of any point in any link of the manipulator can be
readily evaluated as a linear combination of the unit vectors S. and a The S. and a. vectors can be obtained
 k " 3 k
recursively from the initial direction cosines
S l = ( X l ' Y 1 9 Z I)
= (0, 0, l)T (3.5)
a12  (X12, Y12' Z12)r
= (C1l SI, 0)r (3.6)
and the following expressions if j>1:
xjo
a Y Tj_1 S C_)j (3.7)
Zj C (j1)j
ajk = Y~J(j+I) = j I C(jI)j j( .8
Z(J+1) SS(j*)jS
where T. is defined by Equation (3.2) and C. = Cos j, S. = Sin 8j, etc. Equations (3.2  3.8) illustrate the arm position can be explicitly determined in terms of the fixed parameters of the manipulator geometry and the input reference parameters . which representing S or 0.
3.3.2 Kinematic Influence Coefficients for Serial
Manipulators
Kinematic influence coefficients have been used
extensively to create explicit dynamic model formulations for mechanisms [45, 46, 47, 48, 73]. Reference [5] for the first time develops the complete rigidlink dynamic model for all conceivable serial link manipulators based on an exceptionally accessible set of kinematic influence coefficients which are tabulated in Table 3.1. The philosophy there can be used to treat N degree of freedom manipulator to evaluate the effect of certain design parameters, such as mass content, actuator sizing, stiffness, and natural frequencies, ..., etc.
Suppose u f( ) represent any position function of a robot structure. The first and second order kinematic influence coefficients are the derivatives of the geometric position function u with respect to the input reference parameter(s) n' n~l, 2, ..., N.
[G In = auj (3.9)
jn
and
2
U.
[HIm. = (3.10)
[ m;n n
m n
The kinematic influence coefficients in Table 3.1 are provided by either the unit vector Sn of the joint axes or products of similar vectors. Because of the simplicity and compact notation of this result, real time modeling of present industrial robotic systems becomes a real possibility. This work is being pursued by J. Wander [80] in the University of Florida.
3.3.3 Velocities for Serial Manipulators
The positioning problem is by far the most demanding task in kinematic analysis and synthesis of manipulators. Having determined the joint inputs and resulting manipulator positions, the determination of velocities and accelerations is direct. The absolute velocity in a fixed reference of a link jk can be specified by the three component angular velocities of the link and the translational velocity of a certain point, P, which is fixed in the link. The absolute angular velocity of link jk of a serial manipulator is obtained from the angular velocity addition theorem as the sum of the relative angular velocities between preceding links in the serial chain
j ï¿½
W jk = e n n (3.11)
n=l
where 6n is the relative angular velocity between links ni and n. Those joints which are prismatic joints make no
contribution to the angular velocity (on=0). Equation (3.11) can be rewritten by using the rotational firstorder influence coefficients in Table 3.1 as
k = [Giki 4 (3.12)
The translational velocity, v, of a point p which is fixed in link jk can be found by differentiating Equations (3.3) and (3.4) with respect to time and regrouping terms, then the velocity becomes
J .
V = [Sn n + 6nin x (P  R n)] (3.13)
n=l
Since each joint n has only one degree of freedom, one of the two terms in and 6n is identically zero. Sn is zero for revolute joints and 8n is zero for prismatic joints. Therefore, the translational velocity of point P can also be written in terms of the translational fristorder influence coefficients as
v = [G p (3. 14)
p p
From Equations (3.12) and (3.14), the resultant
expressions for velocity will represent six equations which are linear in the N joint velocities. Therefore, the equations can be written as
P  R(3. 15) jk [G k]
{.Ijk} =[[
The velocity of the manipulator's endeffector can be expressed in a similar manner. Let VH be the hand velocity which consists, in general, of three components of its angular velocity H= WN(N+) and three components of the translational velocity vH of the reference point H in the endeffector. Then the six element vector VH and ; are related by
XH =Ll [GN+) LH] (3. 16)
H IN(N+ I L[G N(N+I)]1
or
H [J] (3.17)
Here, the 6xN matrix [J] is the Jacobian for the hand motion defined as
[J] = H I (3.18)
[GN(N+1)]j
The elements of the Jacobian [J] are complex functions of the joint positions and link geometries and are directly calculable once these joint values have been determined.
For six degree of freedom manipulators, if the hand velocity is known, the necessary joint velocities may be
found by inverting [J] in Equation (3.17)
;= [l1 1H (3.19)
However, even for six degree of freedom arms, there are numerous configurations of the arm in which [J] is singular. When [J] is singular, some rows or columns of the matrix become linearly dependent rows or columns and the rank of [J] is reduced. Realizing these facts, some singularities can be overcome by finding the rank of [J] and striking out the linearly dependent rows and corresponding columns in [J]. The resulting square matrix is nonsingular and may be inverted to find the independent joint motions.
The Jacobian for redundant system, however, is
nonsquare and hence is not directly invertable. Some consistent ways are needed to obtain the joint motions. A convenient criterion can be defined as [8]
Minimize = (;T [Wi] )1/2
Subject to H = [J]
in order to solve for the joint velocities where [W ] is a symmetric positive definite NxN weighting matrix. The Lagrangian for the constrained problem can be formulated as
63
L = l fj2 + (VT  ;T [j]) X (3.20)
H
where X is the Lagrangian multiplier vector. Differentiating L with respect to ' and X produces a set of linear equations.
= 0 = [W ] _ [j]T_ (3.21)
t.'T
 = 0 = VT [_ " (3.[2)
= [J(3.24) Substituting Equation (3.23) into Equation (3.24), we get
A = ([J [WI[jIT)I V (3.25)
 4H
From Equations (3.23) and (3.25), the joint velocities can be obtained as
[W ]I[j]T([J][W?]I[J]T)I V (3.26)
= 4) H (.6
3.3.4 Accelerations for Serial Manipulators
Having determined serial manipulator positions and
velocities, the determination of accelerations is straight forward. The angular acceleration of link jk can be found by differentiation of equation (3.12) to obtain
k = T [H_ + [Gik] ï¿½ (3.27)
3 [ j k k Note that [Hjk] is considered an NxN array, each component of which is a vector of length three. The joint velocities and accelerations are kept explicit by introducing the kinematic influence coefficient matrices. Analogously, the translational acceleration of a point in link jk is given by differentiating Equation (3,14) with respect to time and writing this acceleration as
a = dT[H I ; + [GpI (3.28)
p  p
Equations (3,27) and (3.28) show that the accelerations can be calculated as
{2~ T{ [+]JL~~ (3.29)
The acceleration of the endeffector is completely specified by its angular acceleration, CH= aN(N+I) and the
translational acceleration of the hand reference point, aH, as shown in Equation (3.29). The hand acceleration is related to the joint velocities and accelerations as
HI."
H [HH]
f~[H N(N+l) I e
AH = r[NNI]  :+ [J] _ (3.30)
Using the symbol Av to denote the velocityrelated terms of
H
the hand acceleration
AV = (3.31)
H[H N(N+l) J
and assuming the Jacobian is nonsingular, Equation (3.30) can be rewritten to get the joint accelerations corresponding to the velocityrelated acceleration and hand acceleration
[J]i {A  } (3.32)
H A
This equation again stresses the importance of the firstorder influence coefficients, or the Jacobian.
Since the Jacobian is not directly invertible for a redundant system, a criterion is defined here in order to get the joint acceleration by
Minimize 1 112 = .(T [W2]. )1/2
subject to AH = v + [J] ,
where [W] is a symmetric positive definite NxN weighting matrix. The Lagrangian can be expressed as
L 11211 + X T AH  A  [J]4} (3.33) Using the classical optimization techniques, the necessary conditions for an extreme of the constrained problem are
= [W l]I[j]TX (3.34) A  A [J 4 (3.35) Substituting Equation (3.34) into Equation (3.35), we get
([J] [W2Il[j]T)'{A  ) (3.36) From Equations (3.34) and (3.36), the joint acceleration can be obtained as
=[WfiI [ T ([J][W2Il[j]T)l {Al  AH1 (3.37) A similar equation is derived as Equation (3.26). From these equations the selection of coordinates for motion is arbitrary, and the kinematic influence coefficients are
proved a useful basis for changing reference systems [74]. In manipulator design, the designer may prefer to specify motion capabilities in terms of endeffector speeds and accelerations in the design processes.
3.4 Dynamics of Serial Manipulators
The relations between the manipulator joint variables and the time states of the links in a manipulator are needed for the dynamic analysis and response of the manipulator. The dynamic equations of motion must be formulated in a concise and efficient manner in order to be of use in the control and design of manipulators. A complete dynamic model is needed for manipulator control in order to determine the required input forces and torques, to evaluate the deformations due to loads and precision of the arm, to determine control properties, etc. In manipulator design it is essential to be able to recognize the dynamic effects of all parameters and the load imposed on the actuators for deciding on link and actuator sizing requirements. By evaluating the significance of the distribution of mass and stiffness parameters it may be possible to adjust these parameters to obtain higher lowest natural frequency and to enhance controllability, especially when precision control under external disturbance is required.
Based on the kinematic influence coefficients and the actual mass of the arm, inertial modeling matrices of the
serial manipulators were developed for the first time in the work [5] by Thomas and Tesar. These modeling matrices indicate the effect at the inputs of applied and inertial loads. The model leads to a simplified form of dynamic equations of motion for the rigidlink arm. The work must be considered as the foundation to this dissertation and will be briefly reviewed in the next two sections. Other dynamic phenomena, such as actuator stiffnesses and natural frequencies for the rigidlink manipulator model, will be derived by using the kinematic influence coefficients.
3.4.1 Input Torques Due to Applied Loads
The terminal device of a manipulator is usually subjected to some general loading conditions during execution of a task. For example, this might be a resisting force and a resisting torque resulting from a holedrilling operation. In the most general case, several forces and torques can be applied to different points and links in the manipulator. The effective load at each actuator due to these applied loads can be explicitly derived in terms of the firstorder influence coefficients for the points and links of load application.
First, consider the applied loads on link jk of the
manipulator as a force, f p, applied to point P in the link and a moment, m. k applied to the link. Let F(jk)
Jk n
represent the effective torque (or force, for prismatic
joints) on actuator n due to these applied loads and the method of virtual work can be used to determine an expression for this quantity as
F(jk) = [G IT f + [G ]T (3.38)
n Pn p jk n mjk (.8
Assuming that the force f and moment EN(N+l) at the hand
 H
are the only externally applied load on the arm, the effective loads at the joints can be written as
T
F [J] T F (3.39)
4 H
N(N+I) N(N+I) N(N+l)]T F = [fT where F =[FI F2 F ... and !H = ,I
4 1 2 ' *'N an H H'
MT ]T From Equations (3.17) and (3.39), the Jacobian
N(N+I)1 " provides a general transfer concept of velocities and loads between the generalized input coordinates and the fixed coordinates (see Figure 3.2).
The total effective load, F n, at actuator n due to
applied loads is found by summing the effective load from each applied force and torque:
L T N
Fn = [G ]n) + E (m k[H Jkn) (3.40)
p=l P j=l
where P is the total number of load points of interest in the manipulator.
Output velocity
Effective actuator load
[j]T
Figure 3.2 Linear velocity and load transformations
load
Input velocity
3.4.2 Input Torques due to the Inertial Loads
In this section the effective actuator torques due to the system inertia and a given dynamic state are presented. The NxN modeling matrices [I*] and [PI (n = 1, 2, ..., N) will be expressed in terms of the link mass distributions and the kinematic influence coefficients.
The kinetic energy of a rigidlink manipulator in
motion can be expressed as the sum of the kinetic energies of the individual link in the form N T
K.E. = (1/2) E (M j V + II Wk) (3.41)
j=l k cj cj jk jk jk
where Mjk is the mass of link jk, V is the velocity of
jk cj
the centroid of link jk, and [IIjk] is the 3x3 inertia tensor about the centroid for link jk. The kinetic energy of the system can be rewritten explicitly in terms of the joint velocities and the firstorder influence coefficients by using Equations (3.12) and (3.14), and is given as N *T T
K.E. = (1/2) Z (Mjk T[GcjI [G]cjE +
j=l
4T[Gjk][IlIjk] [Gjk] 4r)
= (1/2) ;r[I*] I (3.42)
where [I is the equivalent inertia matrix which represents the effective system mass seen at each input.
The scalar component in row m and column n of this matrix is defined as
N
Z (Mj [G ]T [
m;n j=l k Gcj m Gcj n
[Gjk]T[lk [Gkn) (3.43) jk m [Ijk j n)
The equivalent inertia matrix depends only on the mass properties of the links and the corressponding firstorder influence coefficients. Since the inertia matrix [IIjkI is symmetric, [1*1 is also a symmetric matrix.
The generalized inertia torques, i.e., the actuator torques required for driving the system inertia with a prescribed arm velocity and acceleration can be obtained by using Lagrange's equations given as
F d 3K.E. aK.E.
Fn=   (3.44)
n dt;n ) n
Differentiation of Equation (3.42) together with Equation (3.44) leads to
i T * T [P*(.
Fn  ;n  [ (3.45)
where [I ];n represents column n of the equivalent inertia
;n
matrix and [Pn] is the inertia power modeling matrix. The scalar component in row t and column m of [P can be expressed as
I* {M T [G .1+
[pn 1;m = jk [Hcj ];m cj n
[H IT [+ [
[Hk]l;m [ ljk] [G
[Gjk] T [ljk] ([Gjkin x [Gjk]m) (3.46)
The inertial modeling matrices defined by Equations
(3.43) and (3.46) indicate the physcial nature of the system in a given configuration and depend only on the position, geometry and mass distribution of the manipulator. The inertia torque in Equation (3.45) which each actuator must supply to drive the manipulator at the prescribed velocities and accelerations can now be readily evaluated using the inertial modeling coefficients. Note this inertia torque has two components: 1). an acceleration related term, and 2). a component which depends on the quadratic form of joint velocities. Massive manipulators and increased speeds of operation make the inertia torque even more important in the control and design of modern manipulators.
3.4.3 The Generalized Actuator Stiffness Matrix
Since the hand of a manipulator is subjected to some applied loads during operations, the compliance and deformation are distributed throughout its structure. Positiondependent influence coefficients again form a convenient basis for modeling actuator stiffness and compliance.
The deflection, ARH9 at the endeffector of a
manipulator due to an applied load, FH9 at the hand can be written as [75]
AH = [CH] F H (3.47)
where [CHI is a 6x6 positiondependent compliance matrix. In general, the vectors in Equation (3.47) contain three translational and three rotational components. Assuming that the flexibilities in the joint drives are the most significant compliance terms, the effective compliance matrix can be expressed in terms of the firstorder coefficients [J] that relate the motion at the hand to the joint motions.
The effective torques F at the joints due to a load FH applied at the endeffector reference point H are given in Equation (3.39). Using Hooke's law for elastic deformations, the deflections AO at the joints of a manipulator due to the effective torques F can be written as
A = [Ct] F (3.48)
where [Ct] is a NxN diagonal matrix of joint flexibilities. The deflection ARH at the hand due to the joint deflections can be expressed as
ARH = [J] A4 (3.49) substituting Equations (3.39) and (3.48) into Equation (3.49), the compliance matrix of equation (3.47) in the fixed coordinate system is
[C = [Jl[C ][J]T (3.50) [CH ] is generally symmetric but not diagonal, i.e., forces in one direction can produce hand deflections in several directions. The 6x6 stiffness matrix [KH] in the fixed reference of the equation
[KHI AR H = FH (3.51) can be obtained by inverting the matrix [CR]
[KH1] = [JIT [K [j (3.52) where the T superscript indicates the transpose of the inverse of the matrix and [K ] is a NxN diagonal matrix of the joint stiffness.
3.4.4 Vibrations of Serial Manipulators
An important issue in many manipulator tasks is the accuracy of the motion obtained. There are two distinct
types of structural deformations which limit this accuracy. These are referred to here as deformations in the large and vibrations in the small. Vibrations in the small are low magnitude, oscillatory deformations about the mean motion equilibrium as distinct from the deformation in the large which result from static loads. Vibrations in the small are caused by transient phenomena acting on the system such as from startup and stopping of the arm motion. The lowest natural frequencies of vibration are a good indicator of overall structural integrity of the manipulator and can lead to important design and control considerations.
For today's industrial manipulators, most of the
deformation energy of vibration appears in actuator and drive train components. Therefore, the rigidlink manipulator model which considers flexibility at joint in the direction of relative joint motion is a good approximation for industrial robots. The model in this case is identical to that formulated in Section 3.4.2, with spring terms K contributing to actuator torques. Assuming the vibrational deformations are small and the system damping is negligible. Let A4 n be a small magnitude of vibrational deformations about the gross motion equilibrium position at joint n, or
(3.53)
On ' n,nominal + AOn
Since vibrations about an equilibrium configuration are being considered, the gross motion torques balance in the equations of motion and Equation (3.45) leads to the vibrational equation of motion at joint n in the form:
T~ [II + A; [P1 A] K A 3.4
;n n n (354)
where the [I] and [Pn] are the inertial modeling matrices which are given in Equations (3.43) and (3.46), respectively. Ac is the vector of joint deflections and K is the stiffness at joint n.
n
The vibrational motion at each joint can be written as the superposition of N cyclic terms
N
n= E D Sin w.t, n = 1, 2, ..., N (3.55)
n j=l nj j
where 'nj is the magnitude frequency W . Considering only one mode of vibration at a time leads to the following time dependence for the vibrational motion
A n = On Sin wt (3.56) An = Dn W Cos Wt (3.57) A n =  n 2 Sin wt (3.58)
Performing an elementary order analysis by
substituting Equations (3.56), (3.57) and (3.58) into
Equation (3.54), the resulting equation indicates that the nonlinear term, A ;T[p*] A, can be neglected for small
 n
magnitudes of vibration. To show this, compare the relative magnitudes of the two terms on the left side of Equation (3.54). The components of [I *] are of same magnitude or slightly larger than the components of [P Similarly, the magnitudes of A4n and A n Apm are of the order
nu 0 2 (3.59)
n n
A 5 n A 2 (3.60)
A Am n m
Assuming that the vibrational deformations are less than +30 (a very large motion for a precision manipulator) gives
n m m n .05 A tn (3.61)
This implies that less than onetwentieth of the inertia torque due to vibration is in the velocityrelated term so this term can be neglected.
Equation (3.54) and this approximation lead to the reduced linear equations
L T I + Kn A = 0, n = 1, 2, ..., N (3.62)
Substituting for A~n and A&n from Equations (3.56) and
(3.58) and writing the equations for all N joints simultaneously gives the set of equations.
(w2[I*]  [K ) : 0 (3.63)
where [K ] is the NxN diagonal matrix of joint stiffness and 0=[(,O2,'',NIT is the vector vibration amplitudes. This is a standard eigenvalue problem. For (n are not to be all zero, then the determinant of the coefficient matrix must vanish; that is,
w2[I*]I;IKI ][*1;2 "'"Wll'i
*2[[1*K1 2 * ... W2[i*]2N 2 [1 ]2;1 W [1 ]2;2 K2 . . 2[1 ] 2;N = 0 (3.64)
W2[I*]N;I W2[I ]N;2 ... W2[I*]N;NKN
The evaluation of the determinant results in an Nthdegree algebraic equation in 2. The N roots of the characteristic equation are the square of a natural frequency (usually expressed in rad./sec.). It can be shown from matrix theory that the roots W 2 are all real and finite if the kinetic energy given in Equation (3.42) is positive definite and if both [I ] and [K] are real symmetric matrices. The eignvalues must, in general, be solved for
numerically. Several methods of solution are available for determination of a few or all eigenvalues and eigenvectors of the coefficient matrix, such as the Power method, the classical Jacobi's,..., etc.
3.5 Conclusions of Modeling of Robotic Manipulators
The underlying philosophy of this chapter has been the dependence of all kinematic relationships as functions of input parameters. The trend evident is the separation of the dynamic state from that which is solely kinematic. The kinematic properties are well known to be purely geometrical. The kinematic influence coefficients can be tabulated in an array as the dynamic modeling blocks. Once the geometry can be formalized in a logical manner, the superimposed dynamics problem becomes elementary.
The nature of the dynamic model derived in this chapter can be concluded as follows:
1). Kinematic influence coefficients are fundamental
to the dynamic model.
2). All influence coefficients can be calculated from
extremely elementary vector operations.
3). Model retains explicit role of all dynamic system
parameters.
4). Potential exists for real time modeling operation.
CHAPTER 4
KINEMATIC AND DYNAMIC ANALYSIS
OF ROBOTIC MANIPULATORS
4.1 Introduction
In the design of mechanical systems it is often necessary to investigate the performance of a system when one or more parameters of the system varies over a given range, because better design of manipulators requires improved understanding of the geometric and dynamic aspects of the important parameters. Since the model parameters presented in Chapter 3 play an important role in the dynamic behavior of robotic manipulators, an important problem in manipulator design is the investigation of the influence of the important parameters in the workspace of a manipulator.
Based on the dynamic model formulation given in the previous chapter, an analysis of the dynamic characteristics of a six degree of freedom industrial manipulator is illustrated in this chapter. The analysis on which to base the design methods involves the multivariable mathematical relations between the design parameters and the manipulator's force and motion states which are extraordinarily complex, nonlinear, and highly coupled. Computeraided
complex, nonlinear, and highly coupled. Computeraided procedures are imperative for the kinematic and dynamic analysis in order to understand the influence of the system. Computergenerated manipulator and parameter contour plots are employed to display the extensive analysis information in a compact format and to help the designer to develop an intuitive feel for his problem reenforced by the visual displays. The digitial computer has allowed the designer not only to quantitatively analyze the behavior but also to examine it qualitatively.
4.2 Arm Characteristics for the Industrial Robot
The manipulator investigated in this report is a large capacity six degree of freedom industrial type arm. Figure
4.1 contains a representative drawing of the manipulator showing its six rotational axes. Each of these revolute joints is driven by an electrical DC motor. The second and third axes are driven by two ballscrews which transfer the rotary motions from the drive DC motors to the linear motions along their axes. The wrist is one of its outstanding features. All three axes of the wrist are intersected at the same point and each axis is capable of a full 360 degree rotation. Due to the last two twist angles, it allows the end effector to be placed anywhere in a 238 degree cone with continuous rotation about the last axis.
Elbow axis
Three axis wrist
Shoulder axis
Base swivel
Figure 4.1 Six degree of freedom industrial robot
The kinematic representation of the industrial robot
using the notation established in Chapter 3 is displayed in Figure 4.2. Table 4.1 lists the corresponding kinematic parameters and the joint motion ranges. Because several link lengths and offsets are zero, the industrial manipulator is a special geometry of the general 6R robot and its position analysis simplifies considerably from the general case. Appendix B presents a detailed derivation of the reverse position analysis employed for the rest of this work.
The principal dynamic properties of interest are the
mass, rotational inertia, and the location of mass centroid for each link. Table 4.2 lists the dynamic properties for the example industrial robot. The hand reference point for this analysis is defined at the origin of the local coordinates for link 6. The hand location coordinates (XH, YH' ZH) refer to the position of this point.
4.3 Workspace of the Industrial Manipulator
Most of the performance characteristics of a robotic manipulator are functions of position and configuration in its workspace. Hence it is very important and useful to characterize the workspace of a manipulator and analyze the important parameters in the workspace.
Figure 4.3 shows the workspace of the example robot in the X*Z* plane (Y *=0). The working volume of the
34
s dW
2
a
;5 6
a2
y
Figure 4.2 Kinematic representation of the example manipulator
Kinematic parameters of the example manipulator
Joint ajk ajk S.. e . m
kjkjj mmn max No. (deg.) (in.) (in.) (deg.) (deg.)
1 90 0 32 135 135 2 0 44 0 30 117 3 90 0 0 45 60 4 61 0 55 180 180 5 61 0 0 180 180 6 0 6 180 180
Table 4.2 Inertia properties for the example manipulator
Link Centroid Mass Inertia
X Y Z I I I x y z
(i.)(im) (13 bmï¿½2
(in.) (lb) (10 3lb in. about centroid)
1 0 0 17 700 0 0 100 2 20 1 0 1500 20 180 150 3 4 7 0 1000 170 26 170
4 0 0 20 150 2 2 1.2 5 0 0 0 80 0.8 0.8 0.3 6 0 0 4 60 0.4 0.4 0.2
Table 4. 1
144
120
108
96
84
72 1 60 4 (in.)
48 36 24
12 7Z
12
24 I l l ,, L
36 24 12 0 12 24 36 48 60 72 84 96 108 X (in.) Figure 4.3 Workspace of the industrial robot in the X Z plane
industrial robot is limited by its geometry and the limits on the joint rotations as listed in Table 4.1. Although only the X Z plane of hand locations is shown, all other reachable hand positions can be achieved by rotating this plane about S axis. Such a rotation requires only a change in the value of first joint angle, so results for the entire working volume can be obtained by a straightforward transformation of the results presented in the workspace of the X Z plane.
In the following analysis of this chapter and next chapter, the orientation of the hand is held constant as S6= (1 0 0)T and a67'(0 1 0)T , and the hand reference point, or tool center point is moved throughtout the X Z plane. Figure 4.4 shows the effective workspace of the X Z plane for the hand orientation. Comparing these two workspaces, it is clear that the reachable workspace in Figure 4.4 has been significantly reduced in the X Z plane.
Figures 4.5  4.9 are computergenerated plots showing contours of constant joint angles. The outlines in the contour plots indicate the extent of the effective workspace in the X Z plane. The joint values from a rectangular data array are used for these plots, producing rough edges for the reachable area. If enough data points were employed, these edges would actually be piecewise smooth circular arcs. For this reason, the joint angle values at the edges of the reachable area in each figure
36 24 12 0 12 24 36 48 60 72 34 96 108 X (in.)
Figure 4.4
Reachable workspace of the industrial robot with the hand orienttion held as S6=(l 0 0)T and a=(0 1 0) in the X Z plane 6  (
are slightly unreliable, although the interior values and the important trends are accurately presented.
As mentioned before, the hand orientation is held
constant as S6=(1 0 0)T and a67=(0 1 0)T for these plots. The joint angles are generated by using the algorithms developed in Appendix B. For the orientation of the hand held in X direction, there are two feasible configurations for the hand locations in the X Z plane. For example, if we specified the manipulator's hand location as the vector (60 0 60)T inches and the end effector orientation as S6= (1 0 0)T and a67=(0 1 0)T, two sets of the feasible joint angles are given as
Joint Configuration A Configuration B
1 0.000 (Deg.) 0.000 (Deg.)
2 88.213 88.213 3 15.102 15.102 4 4.720 175.280 5 160.668 160.668 6 94.720 94.720
The joint angles of the first joint are zero degree for the end effector in this orientation and the X Z plane. Configuration A is arbitrarily chosen for the following analysis.
132
120 Units:
108 Degrees
96
84 z 72
(in) 60
48 36 24 12
0
12 i I I I. I I I I , I I
12 0 12 24 36 48 60 72 84 96 108 X (in)
132
120 Units:
108 Degrees
96 84 Z 72
(in) 60
48 36
24 12
0
12 ' 1 I I I I , I 1 , I I
12 0 12 24 36 48 60 72 84 96 108 X (in)
Figure 4.6 Joint angles of joint 3
Figure 4.5 Joint angles of joint 2
132 ,
20
72
(in) 60
48 36 24 12 0
12 L
12
160
Units: Degrees
(in) 60
48 36 24 12
12 r L.L
12 0
0 12 24 36 48 60 72 84 96 108
X (in)
a). Configuration A
Units: Degrees
I I I I J l I I I LI, I
12 24 36 48 60 72 84 96 108 X (in)
b). Configuration B
Figure 4.7 Joint angles of joint 4

Full Text 
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COMPUTERAIDED OPTIMIZATION IN THE DYNAMIC ANALYSIS AND PARAMETRIC DESIGN OF ROBOTIC MANIPULATORS BY HSINCHIEN ^ YU ANCHOU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1985
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With love to Ching Ying and WenYen
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ACKNOWLEDGEMENTS The author wishes to express his sincere appreciation to his supervisory committee chairman, Dr. Delbert Tesar, for his guidance, support and making this work possible. Much appreciation is also directed towards Mr. Mark Thomas for the encouragement, suggestions and sharing of knowledge . He gratefully acknowledges the advice and support given by the members of his supervisory committee, Dr. Joseph Duffy, Dr. Ralph G. Selfridge, Dr. Gary K. Matthew and Dr. George N. Sandor. The author would like to express his thanks to both past and present graduate students of the Center for Intelligent Machines and Robotics (CIMAR) for technical help and moral support. Particular thanks are extended to John Wonder, Behi Fariboz, and Gilbert Lovell. Finally, the author extends his deepest appreciation to his wife and parents for the years of patience, encouragement and love they have provided. iii
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TABLE OF CONTENTS ACKNOWLEDGMENTS iii ABSTRACT vii CHAPTER 1 INTRODUCTION 1 1.1 Subject Background and Dissertation goals 1 1.2 Problems and Promise of Manipulators. . . 3 1.3 Survey of Pertinent Literature 8 1.3.1 Historical Development of Optimization 9 1.3.2 Development of Manipulator Dynamics 11 1.4 Scope of the Study 13 2 OPTIMIZATION THEORY 15 2.1 Introduction to Optimization 15 2.2 Definition of Optimization Problems ... 16 2.3 Concept of Vector Norms 18 2.4 Optimization Methods and Optimality Conditions 2 1 2.4.1 Classification of Optimization Problems 2 1 2.4.2 OneDimensional Optimization Methods 22 2.4.3 Unconstrained Optimization T e chniques 3 0 2.4.4 Constrained Optimization Techniques 39 2.5 Conclusions of Optimization Techniques. . 51 3 MODELING OF SERIAL ROBOTIC MANIPULATORS ... 52 3.1 Introduction 52 3.2 Kinematic Arm Representation. 53 3.3 Kinematics of Serial Manipulators .... 53 3.3.1 Positioning of the Arm 55 3.3.2 Kinematic Influence Coefficients for Serial Manipulators 58 3.3.3 Velocities for Serial Manipulators. 59 iv
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3.3.4 Accelerations for Serial Manipulators 64 3.4 Dynamics of Serial Manipulators 67 3.4.1 Input Torques Due to Applied Loads. 68 3.4.2 Input Torques Due to the Inertial Loads 7 1 3.4.3 The Generalized Actuator Stiffness Matrix 7 3 3.4.4 Vibrations of Serial Manipulators . 75 3.5 Conclusions of Modeling of Manipulators . 80 4 KINEMATIC AND DYNAMIC ANALYSIS OF ROBOTIC MANIPULATORS 8 1 4.1 Introduction 81 4.2 Arm Characteristics for the Industrial Robot 82 4.3 Workspace of the Industrial Manipulator . 84 4.4 Analysis of Actuator Torques Due to Static Loads 9 5 4.5 Analysis of Actuator Torques Due to Inertia 110 4.6 Analysis of the Manipulator Hand Deflections 120 4.7 Analysis of the Natural Frequencies of Manipulators 130 5 OPTIMIZING ACTUATOR PARAMETER DISTRIBUTIONS FOR ROBOTIC MANIPULATORS BASED ON LOCAL DYNAMIC CRITERIA 139 5.1 Introduction 139 5.2 Maximum Actuator Torques Due to Generic External loads 140 5.3 Maximum Actuator Torques Due to Generic Hand Velocity 147 5.4 Maximum Actuator Torques Due to Generic Hand Acceleration 156 5.5 Optimizing Actuator Compliance Distribution For a Known Load 164 5.6 Optimizing Actuator Compliance Distribution For Generic Load 176 5.7 Natural Frequencies Analysis Based on the Local Optimal Actuator Stiffness Distribution 180 6 OPTIMIZATION OF ACTUATOR PARAMETER DISTRIBUTIONS FOR DESIGN OF ROBOTIC MANIPULATORS 184 6.1 Introduction 184 v
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6.2 Distribution of Actuator Sizing Due to Static loads 186 6.3 Distribution of Actuator Sizing Due to Inertial Loads 201 6.4 Distribution of Actuator Compliance for Generic Loads 215 7 CONCLUSIONS AND RECOMMENDATIONS 220 7.1 Conclusions 220 7.2 Recommendations for Future Research . . . 222 APPENDICES A DERIVATION OF THE KUHNTUCKER CONDITIONS. . . 228 B REVERSE DISPLACMENT ANALYSIS OF CINCINNATIMILACRON T776 ROBOT 235 B.l Introduction 235 B . 2 Reverse Displacement Analysis Â— Part I. 236 B . 3 Reverse Displacement Analysis Part II . 240 REFERENCES 249 BIOGRAPHICAL SKETCH 259 vi
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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 COMPUTERAIDED OPTIMIZATION IN THE DYNAMIC ANALYSIS AND PARAMETRIC DESIGN OF ROBOTIC MANIPULATORS By HsinChien YuanChou December , 1985 Chairman: Delbert Tesar Major Department: Mechanical Engineering Robotic manipulators provide general, programable motion paths and force functions to carry out highly dextrous and flexible processes. These systems are characterized by several degrees of freedom of controllable motion. As a consequence, the resulting mechanical structure contains a very large number of design values including geometric, mass, stiffness, strength and actuator parameters. The mathematical relations between the parameters and the manipulator's force and motion states are extraordinarily complex, nonlinear, and highly coupled. Therefore, the design of manipulators is an expensive, timeconsuming, and challenging task. Presently, the designers are lacking guidelines and well established design criteria. This study brings together two fields: optimization theory and robotics. The goal is to develop analysis, vii
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design criteria and computational tools for optimizing the distribution of actuator parameters which satisfy load and precision requirements and increase the system's fundamental natural frequency for faster operations. A rigidlink manipulator model is employed in this study. Computer generated plots are used to display extensive analysis information in a compact, understandable form. These plots show how system parameters vary as a function of hand position. Computeraided procedures for systems become an imperative in order to establish the dynamic design formulation, select rational design specifications, and evaluate the system's operating characteristics both locally and globally. In order to develop a first level of computational and design methodology, local properties, i.e., properties at a specified configuration of the device, are considered first. A more complete methodology would result by considering a range of configurations in the workspace. This leads to the application of general nonlinear optimization techniques to develop design tools for manipulators. The design techniques developed here can greatly reduce the design cycle time for accurate, reliable manipulators, and also be used to improve or redesign existing manipulators to produce better dynamic performance. A detailed analysis 3 for the industrial robot T 776 is presented. viii
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CHAPTER 1 INTRODUCTION 1 . 1 Subject Background and Dissertation Goals The need to establish design criteria and design processes for robotic manipulators has been recognized as a major subject at a Robotics Workshop [ 1 ] ^ held in 1978 at University of Florida and well documented [2Â—5] . This need is especially evident for next generation robotic systems which will be used for highspeed and precision operations, and will have more general geometric structures. Almost all manipulators in use today have very simple geometric structures, which can be analyzed by simple computational procedures but which do not provide the functional capacity of more general devices. The design of a general robotic manipulator is an expensive, timeconsuming and challenging task, because there is a large number of system design parameters. The magnitude of this task can be illustrated by noting that a general six degree of freedom serial manipulator can have as many as 18 geometric parameters, 60 ^The numbers in brackets refer to references at the end of the document. 1
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2 mass parameters and 42 stiffness parameters, along with 12 or more actuator parameters. The design and development of such a generic structure can cost millions of dollars. For 3 example, the Cincinnati Milacron T arm took 6 years to develop and an outlay of $6 million [4] . The NASA space shuttle manipulator is said to have cost about $100 million [6]. As requirements for precision operation, cyclic speed of operation, external loads and complexity of geometry increase, the ability to meet complex design objectives becomes more critical. The manipulator is proving to be one of the most difficult of our present day systems to design. One of the important means of breaking this technological barrier is the creation of a body of mathematical tools which is capable of making the design of new, more versatile systems feasible and the precision operation of these systems a reality. The objective of this report is to develop analysis procedures, design criteria and computational design tools for optimizing the distribution of actuator strength and stiffness parameters and increasing the system's fundamental natural frequency in a general robotic manipulator. The results should lead to improved system performance, economy and reliability, and should reduce the design cycle time which today is 6 to 8 years. To analyze and design a manipulator requires a representative dynamic model of the system. A report [7]
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3 establishes the rigidlink manipulator model and develops analysis methods, based on classical optimization, for evaluating the maximum accelerations, velocities and static load capacities for a given manipulator configuration. The model formulation explicitly shows how system strength and speed characteristics vary with the configuration of the system. Since this position dependence is nonlinear, general nonlinear optimization algorithms are required to isolate the "worstcase" hand positions. This work suggests some applications of nonlinear programming optimization in the analysis and design of manipulators, using position dependent kinematic and modeling coefficients to explicitly show the role of the significant parameters. Specific examples are employed to expose and demonstrate the procedures. Similar theoretical and computational tools are applicable to other related tasks, such as con trol of redundant manipulators [810] or evaluation of a device's reachable workspace [11]. 1 . 2 Problems and Promise of Manipulators Robotic manipulators represent a technology which has rapidly become an important tool for increasing productivity and quality in industry. Together with the digital computer and computeraided manufacturing equipment they have caused a shift in the history of manufacturing similar to that which occurred when steam engines and electric
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4 motors were substituted for human labor. In this new shift the main event is that the information flow is increasingly being automated. The industrial robot industry emerged in the late 1960s and thus can be considered in its infancy (compare, for instance, with the machine tool industry). The following facts constitute a basis for the importance of this new industry: 1. it is an industry with a very high growth potential; 2. it has become a symbol for flexible factory automation in general and thus has large indirect effects on the wide diffusion of CAD/CAM. Industrial robots are a key component of flexible manufacturing technologies because their reprogramability allows them to be quickly adapted to changes in the production process. There are many major benefits to be gained from using robots in manufacturing: 1. increased productivity; 2. increased product flexibility; 3. reduced labor cost; 4. elimination of dangerous and undesirable jobs; 5. improved product quality.
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5 The introduction of robots led to high expectations for their use. However, during the past twenty years, robot applications have been limited primarily to the areas which are simple and most easily achieved industrial functions such as the following: 1 . spot welding ; 2. material handling; 3. machine loading; 4 . inspection; 5. simple assembly. Because the limitations of manipulator application, several problem areas have been identified in order to enhance robotics technology and applications. At the top of the list of problem areas is the issue of precision. Manufacturer's of robots rarely publish information regarding accuracy, preferring to describe repeatability instead. Repeatability is the ability of a manipulator to position the end effector at a taught position in space within a radius of the smallest sphere that can include all return points. Accuracy is the difference between the specified point and the center of the sphere. Figure 1.1 shows the concepts of accuracy and repeatability in the two dimensional case [12].
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6 A Accuracy R Repeatability Figure 1.1 Accuracy and repeatability
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7 The absolute accuracy of most industrial robots is known to be not better than + 0.05 inches. Assembly and light machining operations require a positional accuracy of less than 0.005 inch; otherwise, offline programming cannot be used. The more difficult condition is to maintain accuracy while the manipulator experiences large load variations. Industrial robots are commonly available with repeatability of +0.008 inches. The same improvements are needed in repeatability as in positional accuracy. A second problem area is the speed at which robots move through a operating cycle. Typical robots of today do not work any faster than human workers. Highspeed operations for robots offer a serious challenge to the servo systems and control systems which must be. fast enough to accommodate the rapid changes in inertial characteristics of the robot as velocities and accelerations change during operations . Third identified problem area is that current industrial robots are too large, too massive, and lacking in versatility. They can lift weights equal to only about 10% of their own weight. The prospects in this area for improved mass distribution in the system and the utilization of lightweight materials are very promising. Another area is the need of great geometric dexterity. In material handling operations, assembly and nuclear maintenances, currently available industrial robots are not
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8 adequately flexible to enable them to perform a variety of tasks. Most robots are constructed with six or less degrees of freedom. One of the best ways to increase dexterity is to add additional degrees of freedom to an arm. These redundant degrees of freedom make obstacle avoidance much more likely. Unfortunately, these redundant degrees of freedom make the control of such a manipulator very dif f icult . The final important problem area is in the control systems. Several subareas of improvements are required in robot control systems. Controllers need to be much more sophisticated in their ability to interact between manipulators and sensors in real time, because the flow of data from sensors in the system increases in order to increase manipulatorÂ’s operational capacity and precision, and the complexity of this mathematical interface increases exponentially . There are still many other problem areas, such as interfaces, sensors, programming languages, etc., which need improvement before a truly major impact of using robots for manufacturing will be felt on the economy. 1 . 3 Survey of Pertinent Literature The level of interest in the application of optimization methods to mechanical design seems has increased dramatically during the past two decades. The main reason for
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9 this is that the development of the digital computer has irrevocably changed the viewpoint and increased the computational ability of the engineering analyst. The ultimate goals are means to achieve mechanical designs which are globally optimum relative to all pertinent criteria. A review of some historical developments and relevant literature in the areas of optimization theory and manipulator dynamics is given in the following two sections. 1.3.1 Historical Development of Optimization Some early attempts to optimize were those of da Vinci (14521519) and Newton (16421727), who were both involved in specific mechanism design problems [13,14], and made modifications of previous designs to enhance their performance. Galileo (15641642) developed a rational optimization scheme [15] for selecting the form of a bent beam for uniform strength. Optimization theory advanced significantly with the development of the differential calculus which introduced an elegant way for the evaluation of maxima and minima of differentiable functions. The foundations of the calculus of variations were laid by Bernouli [16] , Euler [17] and Lagrange [17]. The method of optimization for constrained problems, which involves the addition of unknown multipliers became known by the name of its inventor, Lagrange. Cauchy ( 1 7 8 91 8 5 7 ) [ 1 9 ] proposed the steepest descent
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10 algorithm to solve unconstrained minimization problems in 1847. In spite of these early contributions, very little progress was made until the middle of the twentieth century, when highspeed digital computers made the implementation of the optimization procedures possible and stimulated further research on new methods. Spectacular advances followed, producing massive literature on optimization techniques. Since the simplex method of Dantzig [20] in 1949 for linear programming problems and the work of Kuhn and Tucker [21] in 1951 on the necessary and sufficient conditions for the optimal solution of nonlinear programming problems laid the foundations for later development of the methods of constrained optimization. Bellman [22] developed the principle of optimality in 1957 for dynamic programming which is a mathematical technique well suited for the optimization of multistage decision problems. Significant developments have taken place in the area of gradient techniques with the introduction of the Variable Metric method suggested by Davidon [23] and modified by Fletcher and Powell [24]. This extremely successful method is an enormous improvement over simple steepest descent algorithms. The contributions of Zoutendijk [25] and Rosen [26, 27] to nonlinear programming during the early part of the 1960s have also been very significant. Although no single technique has been found to be universally applicable for
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11 nonlinear programming problems, the penalty function algorithms developed by Fiacco and McCormick [28] made many difficult problems solvable by using the wellknown techniques of unconstrained optimization. Geometric programming was developed in the 1960s by Duffin, Zener and Peterson [29]. Gomory [30] did pioneering work in integer programming. Some papers that review the state of the art of optimization have been written by Fletcher [31], Dixon [32] and Powell [33,34,35] . 1.3.2 Development of Manipulator Dynamics The need for efficient mathematical formulations of manipulator dynamics has been recognized and the development of these formulations has rapidly grown in the past decade. The importance of dynamics of robotics manipulators stems from its use in simulation, analysis, control and design for the next generation robotic systems which will be used for highspeed and precision operations. In 1965, Uicker [36] formulated a derivation of the equations of motion for spatial mechanisms using 4x4 transformation matrices and the Lagrange equations. These results were specifically written for open loop manipulators by Kahn [37], but these solutions require excessive computation and were recognized as impractical for real time control applications. Bejczy [38] considered neglecting the velocityrelated terras to obtain a simpler
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12 approximation for actuator torques, but the results are inaccurate in many cases. In a parallel effort to reduce computation time, Whitney [39] and Raibert [40] considered the possibility of using lookup tables. A dramatic cut in the time required for calculations may be attained. However, the amount of storage necessary for control of a general spatial manipulator is extremely large. Furthermore, additional problems in storage and access of information may occur. The inefficiency of the original Uicker/Kahn formulations, as well as other reasons, has led researchers to look for alternative formulations of manipulator dynamics. In the effort to make real time dynamic computation feasible, recursive algorithms have been developed using both NewtonEuler [41,42] and Lagrangian [43] dynamic formulations and Silver [44] has shown that with a proper choice the Lagrangian formulation is indeed equivalent to the NewtonEuler formulation. In these methods, the velocity and acceleration of each link are found sequentially starting with the fixed base link. Then, beginning at the free end and moving back to the fixed base, the force or torque on each successive joint is determined. These recursive methods are computationally the fastest for evaluating the actuator and link loads for a prescribed arm motion state with known external loads and manipulator properties. However, their use for design is limited to
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13 trialanderror analysis, because the geometric properties of manipulators are hidden in their recurrence relations. Recently, nonr ecur s i ve formulation which keeps the dynamic terms and the important design parameters explicit in the equations of motion has been developed by Thomas and Tesar [5,7]. The method is especially wellsuited for manipulator systems because expressions for the kinematic influence coefficients, the basic components of the model, are extremely simple and compact for serial manipulators. The influence coefficients may also provide a highly efficient algorithm for evaluating the dynamic model in real time. Thus, the influence coefficient formulation may be advantageous for both the control and design of mechanisms and manipulators. This modeling approach has been extended to modeling and analysis of hybrid parallel/ serial manipulators by Sklar and Tesar in 1984 [45]. More applications of influence coefficients for planar mechanisms can be found in [46,47,48]. 1 . 4 Scope of the Study This work brings together two fields, namely, optimization theory and robotics in order to develop design criteria and computational design tools for robotic manipulators. In Chapter 2 basic theorems of optimization are reviewed and some methods of nonlinear programming are discussed .
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14 Chapter 3 of the report covers the kinematic aspects of the manipulator problem and the kinematic influence coefficients. Based on the kinematic influence coefficients, the dynamic model for dealing with applied loads, inertial loads, deflections, stiffness matrices, and natural frequencies are introduced. Chapter 4 illustrates the use of the modeling formulation for the analysis of the characteristics of an industrial robot. Computer generated contour plots are employed to express the massive data in a compact form. I Chapter 5, some local dynamic criteria are established for distribution of certain design parameters which are modelled in Chapter 3. The localized techniques in Chapter 5 are extended to develop global dynamic design criteria for general robotic manipulators. These design procedures make it possible to satisfy load and precision requirements and provide faster operation by increasing the frequencies of the fundamental vibrational modes. Chapter 7 offers conclusions and suggestions for future research work.
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CHAPTER 2 OPTIMIZATION THEORY 2 . 1 Introduction to Optimization Man's longing for perfection finds expression in the theory of optimization. It studies how to describe and attain what is best, once one knows how to measure and alter what is good or bad. Normally, one wishes the most, or maximum, good and the least, or minimum, bad. Optimum has become a technical term connoting quantitative measurement and mathematical analysis, whereas "best" remains a less precise word more suitable for everyday affairs. The technical verb optimize, a stronger word than "improve," means to achieve the optimum, and optimization refers to the act of optimizing. Thus optimization theory encompasses the quantitative study of optima and methods for finding them. In order to have a solvable optimization problem, the desired benefit or the required effort must be expressible 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 the limitations on the behavior or 15
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16 performance of a system. These limiting factors are called constraints, and they, too, must be expressible as functions of the design variables. It can be seen from Figure 2.1 that the minimum value of a function, f(x), corresponds to the maximum value of the negative function, f(x). Thus, without loss generality, optimization can be taken to mean minimization since the maximum of a function can be found by seeking the minimum of the negative of the same function. 2 . 2 Definition of Optimization Problems A common characteristic of all optimization problems is the existence of many feasible solutions. The seeking of the best possible solution depends on a clear definition of the interaction of all the design variables, an explicit statement of criterion or objective, and an effective search procedure for locating the optimum in accordance with the stated objective. An optimization or a mathematical programming problem can be stated in the following general form [49]. Find X=[X 1 , X 2 , ..., X n ] T which minimizes f(X) subject to the equality constraints gj(X) =0, j=l, 2, . . . , m (2.1) and the inequality constraints h.(X) < 0, j=m+l , m+2 , . . . , p ( 2 . 2 )
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17 Figure 2.1 Minimum of f (X) is same as the minimum of f(X)
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18 where X Is called design vector, and f(X) is called the objective function. A necessary condition for existence of a design problem is n>m, in order to ensure the existence of free parameters. If there are no constraints (p=0), the problem is said to be unconstrained. An optimization problem is said to be linear if the objective function f(X) and all the constraint functions are linear in the n variables, X x , X 2 , ..., Otherwise, the optimization problem is said to be nonlinear. 2 . 3 Concept of Vector Norms Since manipulators and mechanisms operate in a multidimensional space, the objective and constraint functions usually contain vector quantities to describe important design criteria or dynamic phenomena. It is necessary to have some means of "measuringÂ” vectors, in order to say that one vector is "larger" or "smallerÂ” than another. The definition of a norm gives a rule for associating a nonnegative scalar with a vector [50] , then the constraint and performance can be written in terms of scalar magnitudes associated with these vectors. The weighted pnorm of a vector u is defined as [6] II H II p = ( 2 W u  p ) 1/p (2.3) j J where is the weighting factor for component j and the summation is performed over all components of the vector.
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19 While W indicates the relative significance of component j, the exponent p determines the importance of the largest components relative to the smaller ones . The larger the value of p, the more the vector norm depends on the largest scaled component and vice versa. The most useful values of p are 1, 2 and Â“ II i W. u . J J p=l (2.4) u  =( Z  W . u  2 ) 172 , p = 2 j (2.5) Â£ II 00 = max ( I W . u .  ) , j 2 2 ( 2 . 6 ) In the two dimensional case', the curves in Figure 2.2 are effectively contour curves along which the norm is constant. These particular norms are the most popular because they are easily interpreted and their differentiation, with respect to the components u^ , produces tractable results. In general, the p=2 norm leads to higherorder equations in the optimization solutions. The other two norms often provide several individual cases which each mus t be solved . Nondiagonal weighting matrices curves in the uplane. [ W ] rotate these
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20 Weighting values W =1, W2=1.5 Norm value K=1 Figure 2.2 Curves of constant magnitude. II =K
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21 2 . 4 Optimization Methods and Optimality Conditions The mathematical programming problem stated in Section 2.2 has given rise to a great variety of solution methods. In order to select an efficient method for solving a particular problem, it is necessary to analyze and classify the problem. 2.4.1 Classification of Optimization Problems When faced with any optimization problem, it is usually advantageous to determine the special characteristics that allow the problem to be solved more efficiently. Therefore, we consider how to classify optimization problems in order to enhance the efficiency of the resulting solution methods. Optimization problems can be classified in several ways. For example, it can be classified [49] on the basis o f 1. the existence of constraints, 2. the nature of the design variables, 3. the physical structure of the problem, 4. the nature of the equations involved. Although no set of classifications is ideal for every circumstance, the classification based on the nature of problem functions is extremely useful from the
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22 computational point of view since there are many methods developed solely for the efficient solution of a particular class of problems. Table 2.1 gives a typical classification scheme based on the nature of the problem functions. It is also highly desirable to organize the available nonlinear optimization approaches in order to get a feel for what is available, help classify past and present efforts, and perhaps enhance decisions in future efforts. Figure 2.3 shows such a representation. In the next three sections, three types of methods will be described: onedimensional optimization, unconstrained optimization, and constrained optimization. The discussion has been restricted to certain methods which have proven useful in the present report. 2.4.2 OneDimensional Optimization Methods We begin by considering the problem as follow: Find X which minimizes f(X) subject to X x < X < X u where f(X) is a real function of the real variable X, and X u , X are given bounds on X. We consider this problem first because it is one of the simplest optimization problems and because many of the methods for solving more complex problems require the solution of this problem. The
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problem functions 23
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Nonlinear Optimization 24 Figure 2.3 Methods of Nonlinear Optimization
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25 following two theorems provide the necessary and sufficient conditions for the relative or local minimum of a function of a single variable [49]. THEOREM 2.1 If a function f(X) is defined in the interval * * X. Q and n Â£ s even, 2). a maximum value of f(X) if f^ '(X ) <0 and n is even, 3). neither a maximum nor a minimum if n is odd. Several numerical methods, as in Figure 2.3, are available for solving the onedimensional minimization problem. These can be classified into two categories: elimination methods and interpolation methods. The elimination methods can be used for the minimization of discontinuous functions. The interpolation methods involve polynomial approximations to the given function. Golden Section Method [49, 51, 52] The basic idea behind the methods of Golden Section and Fibonacci is to economize the number of function evaluations, trap a relative minimum in an interval and
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26 successively decrease the size of the interval. The process thus gives successively better estimates for the location of the minimum point. The difference between these two methods is that in the Fibonacci method the total number of the experiments to be conducted has to be specified before beginning the calculation, whereas, this is not required in the Golden Section method. The Golden Section search technique can be given in the form of a computational algorithm: Step 1. First an upper bound q^ must be found for q. It is clear that 0 is a lower bound, q^. For a chosen small step size 6 in q, let j be the smallest integer such that f [X^ i) + Â£ k=0 . . j 1 , f + E 6(1 .618)*] k= 0 Then upper and lower bounds on the minimum, * q , are j k j T 2 k q = 2 6(1.618), q = Â£ 6 (1.618; U k= 0 1 k=0 Step 2. Compute f(X^*"^+q ) and f (X^ ^ ^+q^) , where q a = q i + * 382 ( q u ~ q i) q b = q i + * 618 ( q u Â“ q l ) Step 3. Compare f(X^ i ^+q ) and f(X^^+q, ) and go cL D to Step 4, 5 or 6.
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27 Step 4. If f ( 1 ^ +q a ) the new points p ^ = q ^ and P U = < 1^Compute now f(X^*^+P a ) where p a = p 1 +.382(p u ~p 1 ) Â• Go to Step 7. Step 5. If f(X (l) +q ) > f ( X ( 1 ) +q . ) , then q
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28 known analytically. This will mean that finding the minimum will be a straightforward procedure. The use of quadratic or cubic polynomial means that we assume that the approximated function is relatively smooth and possibly unimodal in the interval being investigated. In many practical applications, a small enough interval assures validity of this assumption. Box, Davies, and Swann [53] recommended that the al* gorithm of DSC be executed to bracket the minimum, X , by increasing the step size until the minimum is overshot and then a single quadratic interpolation is performed to estimate X . Figure 2.4 illustrates the procedure and the algorithm is summarized as Step 1. Evaluate f(X) at the initial point X^). If f(X (0) + 6)f (X (0) ) , let S~~6 and go to Step 2. Where 5 is a small step size. Step 2. Compute X^ + ^=X^^ + 6. Step 3. Compute f(X^ + ^). Step 4. If f ( X ^ ; Â’' + *" ^ ) f ( X ( 1 5 ) , denote x( 1+1 ) by X^Â™), by x( m1) , etc., reduce 5 by onehalf, and return to Steps 2 and 3 for one more (only) calculation.
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29 f (X) Figure 2.4 DSC unidimensional minimization
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30 Step 5. Of the four equally spaced values of X in the set, X (m+1) , X (m) , X (m_1) and X (m_2) , discard either X^ m ^ or X^ m 2 ^, whichever is farthest from the X corresponding to the smallest value of f(X) in the set. Let the remaining three values of X be denoted by X^ a \ X^ b ^, and X^ c \ where X^ b ^ i s the center point and X ( a ) = X ( b ) 5 and X^ c ^=X^ b ^+ 6. Step 6. Carry out a quadratic interpolation to Â•k estimate X X* = X (b) + [f (X (a) ) f (X (b) ) ] 2 [ f (X (a) ) 2f (X (b) ) + f (X (c) ] Step 7. If  f (X*)f (X (a) )  < Â£ and  f ( X* ) f ( X ( b ) )  < e , terminate the process. Otherwise, reduce the step size 6 and find the smallest value of f(X (a) ), f(X (b) ), f(X (c) ) , and f(X*) and go back to Step 1 . 2.4.3 Unconstrained Optimization Techniques We consider in this section a very important class of optimization problems, the class of problems of minimizing a real valued function of n real variables with no constraints. An unconstrained optimization problem can be stated in the standard form as
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31 Find X such that the objective function f(X) is minimum. This problem can be considered as a particular case of the general nonlinear programming problem. The special characteristic of this problem is that the solution X need not satisfy any constraint. It is true that rarely a practical design problem would be unconstrained; still the study of this class of problems is important because most of the unconstrained optimization methods can be extended to handle constrained problems, either by directly considering the constraints or by transformation to an unconstrained problem. The following two theorems provide the necessary and k sufficient conditions for X to be a relative minimum or maximum of the unconstrained optimization problems [50]. THEOREM 2.3 A function f (X) has a relative minimum at the point X* if the first partial derivatives of f(X) k k at X is Vf(X ) = 0 and the matrix of second partial * derivatives (Hessian matrix) of f(X) at X V 2 f(X*) = [ 3 2 f/3X. 3 X j ] 2 is positive semidefinite. 2 A real symmetric matrix R is 1). positive definite if Z^RZ>0 for all Z = 0, 2). positive semidefinte if Z^ T RZ^>0 for all _ Z .
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32 k THEOREM 2.4 A sufficient condition for a point X to be a relative minimum point is that the first partial derivatives of f(X) at X are zero and the Hessian * matrix evaluated at X is positive definite. Several methods are avilable for solving an unconstrained minimization problem. These methods can be classified into two categories: direct search methods and descent (or gradient) methods (see Figure 2.3). 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. In general, the direct search methods are less efficient than the descent methods. Variable Metric Method [23, 24, 55] Significant developments have taken place in the area of descent techniques with the introduction of the Variable Metric method by Davidon [23] and modification of the method by Fletcher and Powell [24]. This method is reported to be one of the most powerful known methods for general functions f(X) [55]. A major reason for the success of this method is its capability to accumulate information about the curvature of the objective surface during the
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33 entire iterative process, even though only first order derivatives of the objective function need to be computed. The iterative procedure of this method can be stated as follows : Step 1. Make an estimate X^) of the minimum point and choose a symmetric positive definite matrix Usually is taken as the identity matrix I. Step 2. For i=0, 1, 2, ... compute S^(i) = H (i) Vf T (X^ 1 ^) Step 3. Compute q = q^ li '^ which minimizes f(X^^+q S^ * ^ ) Step 4 . Compute _r X H where (i) = q ( i) S (i) (i+D = x (i) + Â£ (1) (i+1) = + A* 1 Â’ + B (i) z (i) = Vf T (X (i+1) ) Vf T (X (i) ) A* 1 Â’ = (r (i > r^ 1 )/r (1)T /i) B^ = (H^) y (i)T H ( i ) T )/z <1)T 5 . If Â« Vf(X (i + 1) )  and X (l+1) x (1) H' are sufficiently small, terminate the process. Otherwise, return to Step 2.
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34 The Variable Metric method has many of the advantages of the second order gradient techniques without having some of their disadvantages. The advantages include fast convergence near a minimum and finiteness for a quadratic function. Due to the symmetry of the method is also comparable in its computer storage requirements. Comparison with other quadrat ical ly convergent methods shows that the Variable Metric method performs better for general nonquadratic functions. Powell 1 s Method Occasionally in design and analysis applications, one is often faced with a problem in which computation of derivatives of the objective function is impossible or at least prohibitive from a computational point of view. There are many techniques for solving this sort of problem given in [49, 54, 56]. An efficient technique was developed by Powell [57] using conjugate directions. The basic idea of the Powell's method can be quite simply explained in twodimensional case and is illustrated in Figure 2.5. In descriptive terms the strategy is as follows: 1). Conduct a minimization along a line in each of the coordinate directions in turn.
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35 pH x Figure 2.5 Strategy for Powell's method
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36 2) . Make a pattern move which is minimization along the direction corresponding to the coordinate moves in item 1. 3) . Repeat item 1 except that one of the coordinate directions is replaced by the direction of the last pattern move. 4) . Continue in this fashion until the moves become negligibly small. The flow diagram for the Powell's method is given in Figure 2.6. A computational algorithm is presented as f o 1 lows : Step 1. Make an estimate of the minimum point X ^ ) of f(X). Choose vectors ^ , j = l, 2, Â•Â•Â•, n, in the coordinate directions of R n . Step 2. Find q = q^) > k=l, 2, Â•Â•Â•, n, which minimize f (2L< 1 > + q ^ k ^), where Z <Â°> X (1) y< k > yO1 ) + ,< k >S< k \ k 1. 2 n and i is the number of iterations which have been completed.
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37 Figure 2.6 Flow chart for Powell's method
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38 Step 3. Find the integer m, l 5 (f L f 3 ) 2 , put X ^ ^ = y^ n ^ . Terminate the process if  X^ + '*Â‘ ^ X^ ^ ^  2 is sufficiently small. Otherwise, return to Step 2 with the same set of S ( j\ j = l , 2 , . . . , n. Step 6. If neither of the inequalities of Step 5 hold, define s = y^ n ^y^^ and find q = q which minimizes f(y^ n ^+q s). Put X^ ^ =y^ n ^+q s. Terminate the process if  X^ + ^ ^ X^ ^  ^ is sufficiently small. Otherwise, return to Step 2 with the new set of vectors S^ , ^ ^ , ..., s (n) > Â• Â• Â• y Q y g ( m ~ 1 ) ^ s (m+l) s
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39 2.4.4 Constrained Optimization Techniques The optimization problem defined in Section 2.2 can be rewritten in the following form: Minimize f(X) subject to the constraints C j(X) < 0, j = 1, 2, . . . , p and X 6 S, where S is some given set in R n . The set, S = {X  Cj(X)<0, j=l, 2, ..., p}, is called the feasible region. If the constraints are linear, the problem is considerably easier to solve than if they were nonlinear. If both the objective function and the constraints are linear, then we have a linear programming problem, the solution of which can be obtained by standard or revised simplex algorithm [20, 49]. We shall not discuss the simplex algorithm here. The conditions to be satisfied at a constrained minimum point, X , of the problem can be stated as follows: 3f p 3C + E A, 1 = 0, i = 1, 2, . . . , n (2.7) 3X t j=l 3X. A. C. = 0, j = 1, 2, . . . , p (2.8) C. < o, j = 1, 2, . . . , p (2.9) and
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40 X j > 0, j = 1, 2, . . . , p (2.10) where Xj* j = l> 2, p are the Lagrange multipliers. These conditions are called KuhnTucker conditions [21] after the men who developed them. A detail derivation of these conditions are given in Appendix A. These conditions are, in general, not sufficient to ensure a relative minimum. Sufficient conditions for to be a relative minimum of the problem are as follows: 9 f p 9 C + E X . 1 = 9X Â± j=l J 9X i 0, i = 1 , 2 , ...Â» n (2.11) X j C. = 0, j = 1 , 2 , Â• Â• Â• , p (2.12) c, < 0 , j = 1 , 2 , ...Â» P (2.13) Aj > 0, j = 1, 2 , Â• Â• Â• , p (2.14) and n n 9 L E E dX i1 k= 1 9X i 8X k f is positive\ \definite (2.15) where L is the lagrange function. Some good general discussions of the optimality conditions are given in [28, 58, 59]. In the following Sections, we shall discuss two solution techniques for solving constrained minimization problems .
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41 Gradient Projection Method The gradient projection method of Rosen [26] is very efficient for solving problems with linear constraints. Although the method has been described by Rosen [27] for a general nonlinear programming problem, its effectiveness is mainly confined to problems in which the constraints are all linear. For the nonlinear constraints, very small steps must be taken, and a calculation to bring the new point back in the feasible region must be included in each cycle. So zigzagging will frequently occur in such situations . The basic idea in the method for the linear constraints is try to move as close as possible to the direction of steepest descent, Vf(X), but still remain feasible. If moving in the direction of steepest descent would cause any of the constraints to be violated, X is changed along the projection of the negative gradient of the objective function onto the boundary of the feasible region . Let us now summarize the gradient projection algorithm. A flow chart of this procedure is shown in Figure 2.7. It will be assumed that the initial point x/ ^ ) is admissible and lies in the intersection Q' of q linearly independent hyperplanes.
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42 Figure 2.7 Flow chart of the gradient projection algorithm
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43 Step 1. Read in _X' ^ , the objective function f ( X) , the constraints and tolerances. Determine the hyperplanes H ^ , H 2> which form the intersection Q'. Set i=l. T 1 Step 2. Form the matrices N and [N N 1 q q q Calculate the gradient vector at the point 3f^^/3X, the vector r=[N T N ] ^ N T Â— q q q [~9f^i)/^X]> projection matrix P , and the gradient projection [ 3 f ^ ^ ^ ] . If  P^ [ 3f ^ * V 3X]  _< and r <_0 , where r^ is the maximum component of _r, then _X^ ^ is the constrained global minimum and the procedure is terminated; Otherwise, go to Step 3 . Step 3. Determine whether or not a hyperplane should drop from Q'. If  P [ ~3f ^ ^ / 3_X ]  <_Â£ ^ , drop the hyperplane , which corresponds to r^XD, form the projection matrix P^_^, and go to Step 4. The other alternative is that the norm of the gradient projection is greater than e.. In this case, calculate $=max{ct}> 1 i 1 where a^sthe sum of the absolute values of the T 1 ith row of the matrix [N N ] . If r > 8 , q q q
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44 drop the hyperplane from Q', form the projection matrix P and go to Step 4. If r < 3 Â» Q' remains unchanged. 4 Â— Step 4. Compute the normalized gradient projection z ( i ) Z (1 ) = (P q [3f (i) /3X] )/ 1 P q [ 3 f (i) /8 X] II and the maximum allowable step size x , where r m T is the minimum positive value of the x. 's m J found by evaluating T . = ( v . n T X^ 1 ) )/n T z ( 1 ) J J J J for j corresponding to all hyperplanes not in the intersection Q*. The tentative next point X*(i + 1) i s found from x Â»(i+l) = + X m Â— Step 5. Calculate the gradient at the point if ^ (l)T [3f (X' (i+1 ^)/3X] > 0, Set X (i+1) =X' ( i+1 ) ; since X (l+1) lies in the intersection of Q' and (the hyperplane which corresponds to the step size determined in Step 4), add to Q', and return to Step 2. On the othter hand, if Z (i)T [3f (X' (i+1) )/3X] < 0 find X (1+1) by repeated linear interpolation as in Figure 2.8. 0=0 corresponds to the point
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45 Figure 2.8 Repeated linear interpolation
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46 and 0=1 corresponds to the point X'(i + 1 ). 0 ^, the abscissa where the straight line from A to B has an ordinate of zero, is determined from the relationship z/ i)T [af^/ax] 1 Z^ i)T [3f (i ^/3X] j^ i)T [ 3 f ' ( ' 1 + 1 ) /3X] Next, we evaluate the gradient at the point XÂ„ ( i+ 1 ) = x (i) + 0 T z (i) Â— Â— 1 m Â— and form the inner product Z (i)T [ 3 f " (i+ 1 ) /3X] If this inner product is positive, we use point C and B to interpolate again ( if the inner product had been negative, point C would have a negative ordinate and point A and C would be used for next interpolation). This procedure is repeated until a point X^ ^ is found where the magnitude of the inner product is less than a preassigned small positive number Â£ 2 > that is,  z (i)T [f (i+1) / X]  < e 2 The intersection Q' remains unchanged, and the computational algorithm begins another iteration by returning to Step 2.
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47 Exterior Penalty Function Methods The basic idea of penalty function algorithms is to reduce the constrained problem to a sequence of unconstrained problems which can be solved by the methods of Section 2.4.3, the solutions of which converge to the solution of the original problem. Let the original optimization problem be the form: Find X which minimizes f ( X) subject to g (X) = 0, j = 1, 2, ..., ra, and (2.16) h_. (X) Â£ 0, j = m+1, m+2 , ..., p (2.17) This is done by lumping the constraints with the objective function in such a fashion that minimizing the lumped function penalizes any violation of the constraints. The lumped function can be shown in the following form: m U ( X , r ) = f (X) + r Z G [g (X) ] + j=l 3 J j =m+l Hj lh. (X) ] (2.18) where G.[g.(X)] and H.[h.(X)] are functions of the con 3 3 Â— 3 3 straints g . ( X) and h.(X), respectively and r is a positive 3 J constant known as the penalty parameter. This is the reason why these procedures are called the penality function methods and also known as "sequential unconstrained minimization techniques" or simply SUMT .
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48 Two categories of penality function methods exist, namely, interior methods and exterior methods. The interior methods must be supplied with a feasible starting successive minimizations, the solution of the constrained problem, always remaining within the feasible region, are called interior methods. The exterior methods do not require a feasible initial point, and generally converage to the constrained minimum from the infeasible region, hence the term exterior. The exterior methods have been judged to be generally superior to the interior methods [60]. In the exterior penality function methods, the lumped function is taken as where the exponent q is a constant greater than one, and point, X^). As the penality parameter r is varied over m 2 U(X,r) = f(X) + r Z [g.(X)] Z + p J* 1 Â„ 3 r Â£ q jm+1 J (2.19) the singularity function is defined as if h j ( X) > 0 (constraint is violated) = < (2.15) 0 , if h j(X) < 0 (constraint is satisfied)
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49 It can be seen from Equation (2.19), that this formulation is to assess an increasingly severe penality 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 (2.19) using a small value for the penalty parameter r for the first minimization. Subsequent minimizations, until the solution is forced to converge in the feasible region, is necessary. This is because, if r remained small, very small positive values of h_.(X) and small errors of gj(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 also being forced toward the feasible region. The flow chart of an exterior penalty function method is shown in Figure 2.9, and the algorithm is outlined in the following steps: Step 1. Make an estimate ) of the solution of the constrained optimization problem and suitable value of the penalty parameter r. Set i = l .
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50 Figure 2.9 Flow chart of the exterior penalty function method
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51 Step 2. Find the point X^^ that minimizes the function m 2 U(X, r) = f(X) + r Z [g.(X)p + P j ' 1 c r E q j=m+l J Step 3. Test whether the point X ^ ^ ^ satisfies all the constraints. If X ^ 1 ^ is feasible, it is the desired optimum and hence terminate the procedure. Otherwise, go to Step 4. Step 4. Increase the penalty parameter r and set the new value of i as original i plus one, and go to Step 2 . 2 . 5 Conclusions of Optimization Techniques This chapter is not intended to be complete, but is intended to offer some basic concepts and methods which are useful in a design environment. Many other methods have been proposed, and are worthy of careful consideration. There is no algorithm that is most suitable for all optimization problems. Many different circumstances can affect the choice of method. There are many papers [55, 60, 61, 62, 63] that give some guidance in choosing a suitable method for solving a particular problem alongwith some other computational details. There are also several geniunely useful textbooks [64, 65, 66] in the area.
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CHAPTER 3 MODELING OF SERIAL ROBOTIC MANIPULATORS 3 . 1 Introduction An important initial step in the analysis, design, or precision control of a complex mechanical function generator, such as a manipulator, is to obtain a representative model of the device. The model must be accurate enough to give results which satisfactorily describe the operation of the actual device, yet simple enough to be of practical use. To satisfy these requirements , the rigidlink manipulator model is employed in this dissertation. The model neglects deformations in the links and joint deformations which are not in the direction of the nominal joint motion. This is a good approximation for industrial manipulators in use today since most (90%) of the deformations seen in industrial robot structure are due to compliance in actuators and associated drive train components. Although some flexiblelink derivations of manipulator model are available [67, 68], these formulations are too complex to be of much practical use with near term computational capabilities. 52
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53 3 . 2 Kinematic Arm Representation The serial manipulator is represented kinematically as a sequence of rigid links jointed by one degree of freedom lowerpair connectors as shown in Figure 3.1. Associated with each joint are two geometric parameters: 1). (or S^) the offset distance along the joint axis, S_^ , between the two links which the joint connects; and 2).0 (or Â®jj) Â“Â“ the relative rotation about between these two links. The link direction between the successive j joints j and k is represented by the unit vector a., which Â— J K. lies along the common perpendicular of S. and S. Â• The Â— j Â—K fixed link length, a jj c > is the distance between two successive joint axes measured along the a^^ vector. The fixed angular rotation of to is and is the angle taken about a^^ i n a right hand sense. The fixed Cartesian Jc jc jc coordinate system (X , Y , Z ) is located at the first joint with the Z axis along S_^ . Each link has a local coordinate system (X^^, Z^^) associated with it. The X ( axis is along the a^ vector and Z ^ axis is along the J3 . vector. 3 . 3 Kinematics of Serial Manipulators In the kinematic analysis of manipulators two types of problems may arise. The forward kinematic problem involves determining the position, velocity, and acceleration of all
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54 Figure 3.1 Kinematic representation of manipulator
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55 point in the manipulator when these quantities are known for the independent joint coordinates [41]. Many straight forward solutions to this type problem exist and it is only necessary to determine which formulation is the most efficient in a given situation. The reverse kinematic problem, that of finding the required joint coordinates in order to obtain a desired motion of the hand, is much more complicated and has received much research recently, e.g. [69, 70]. In general there are two approaches to the reverse kinematic problem. One is an iterative numerical solution [71] in which the present location is used to determine the joint coordinates of the next desired endeffector location. The other approach, which is considerably more difficult, is to establish a closedform analytical procedure [72] in which the joint coordinates can be solved for any hand location in the system workspace. Appendix B employs this analytical procedure to derive a detailed position analysis for an example industrial manipulator. 3.3.1 Positioning of the Arm As mentioned in Section 3.2, there are two coordinate systems, the absolute and bodyfixed references, as shown in Figure 3.1. One of the most popular methods for arm positioning involves the use of 3x3 rotation matrices. Let Tj represent the rotational transformation matrix which
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56 relates the j th bodyfixed coordinate system to a reference system with common origin to the j th system and with axes parallel to the fixed reference axes. If _P^ ^ Is a vector in terms of the local jth reference, its direction P in terms of the fixed global coordinates is where the rotational transformation matrix can be expressed as T j " [ jk j x ^jk Â— j 1 (3 ' 2) The location of the origin of the jth reference system can be defined by the vector . As shown in Figure 3.1, the position vector can be found by sequentially summing the link lengths and joint offsets starting at the fixed base: j R . = S . S , + Â£ {a/ , \ a , ... + SS} (3.3) Â— J 1Â—1 u 1 ( m1 )mÂ— (m1 )m mÂ— m J m= 2 Then, the transformation locating a point P, with local coordinates _P^ \ in terms of the fixed reference is P = R. + T.P (j ) (3.4) J jEquations (3.2), (3.3) and (3.4) show that the position of any point in any link of the manipulator can be
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57 readily evaluated as a linear combination of the unit vectors S. and a., . The S. and a., vectors can be obtained J J k j J k recursively from the initial direction cosines S l = (Xj, Yj, Zl ) T = (0, 0, 1) T (3.5) Â— 12 ^ X 1 2 Â’ Y 1 2 Â’ Z 12^ = (Cj, S lt 0) T and the following expressions if j>l: ( * > f \ X. 0 J * Y j ii H (_i. 1 i Â— * " s (jl)j * l Z 3 J > G (j1 )j J ' Y * 1 ( \ j(j + D c . j =< * Y j ( j + 1) Â’ Vi ( C (jDj s j , Z j(j + D, k s (ji)j s j, (3.6) (3.7) (3.8) where is defined by Equation (3.2) and = Cos 9^ , Sj = Sin , etc. Equations (3.2 3.8) illustrate the arm position can be explicitly determined in terms of the fixed parameters of the manipulator geometry and the input reference parameters which representing or 0^ .
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58 3.3.2 Kinematic Influence Coefficients for Serial Manipulators Kinematic influence coefficients have been used extensively to create explicit dynamic model formulations for mechanisms [45, 46, 47, 48, 73]. Reference [5] for the first time develops the complete rigidlink dynamic model for all conceivable serial link manipulators based on an exceptionally accessible set of kinematic influence coefficients which are tabulated in Table 3.1. The philosophy there can be used to treat N degree of freedom manipulator to evaluate the effect of certain design parameters, such as mass content, actuator sizing, stiffness, and natural frequencies, Â•Â•., etc. Suppose u. = f ( ) represent any position function of a robot structure. The first and second order kinematic influence coefficients are the derivatives of the geometric position function u^ with respect to the input reference parameter(s) (j)^, n=l, 2, ...Â» N. (3.9) and [H m n (3.10)
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59 The kinematic influence coefficients in Table 3.1 are provided by either the unit vector S n of the joint axes or products of similar vectors. Because of the simplicity and compact notation of this result, real time modeling of present industrial robotic systems becomes a real possibility. This work is being pursued by J. Wander [80] in the University of Florida. 3.3.3 Velocities for Serial Manipulators The positioning problem is by far the most demanding task in kinematic analysis and synthesis of manipulators. Having determined the joint inputs and resulting manipulator positions, the determination of velocities and accelerations is direct. The absolute velocity in a fixed reference of a link jk can be specified by the three component angular velocities of the link and the translational velocity of a certain point, P, which is fixed in the link. The absolute angular velocity of link jk of a serial manipulator is obtained from the angular velocity addition theorem as the sum of the relative angular velocities between preceding links in the serial chain 3 Â• U) = Z 0 S n= 1 n (3.11) where 6 n is the relative angular velocity between links n1 and n. Those joints which are prismatic joints make no
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60 contribution to the angular velocity (0^=0) . Equation (3.11) can be rewritten by using the rotational firstorder influence coefficients in Table 3.1 as w jk = [G jk ] 1 (3.12) The translational velocity, v of a point p which is fixed in link jk can be found by differentiating Equations (3.3) and (3.4) with respect to time and regrouping terms, then the velocity becomes ip S l^ n S n + e n S n x (P RÂ„)] (3.13) n= 1 Since each joint n has only one degree of freedom, one of Â• Â• Â• the two terms S and 6 is identically zero. S is zero for revolute joints and 0^ is zero for prismatic joints. Therefore, the translational velocity of point P can also be written in terms of the translational fristorder influence coefficients as Z p = [G p ] ! (3.14) From Equations (3.12) and (3.14), the resultant expressions for velocity will represent six equations which are linear in the N joint velocities. Therefore, the equations can be written as
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61 f V n 1 '[gj ^ = __E__ \ *r> _3J _ tG jk ] . ( 3 . 15 ) The velocity of the manipulator's endeffector can be expressed in a similar manner. Let be the hand velocity which consists, in general, of three components of its angular velocity u h =(J jj(n+ 1) and three components of the translational velocity _v^ of the reference point H in the endeffector. Then the six element vector _V^ and l l I I i i I as i O 1 1 lN(N+l)J _ [G N(N+1) L or v H = [J]Â± ( 3 . 16 ) ( 3 . 17 ) Here, the 6xN matrix [J] is the Jacobian for the hand motion defined as [J] Â£Â®h! _ [G N(N+1) 1 ( 3 . 18 ) The elements of the Jacobian [J] are complex functions of the joint positions and link geometries and are directly calculable once these joint values have been determined. For six degree of freedom manipulators, if the hand velocity is known, the necessary joint velocities may be
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62 found by inverting [J] in Equation (3.17) A = Ih (319) However, even for six degree of freedom arms, there are numerous configurations of the arm in which [J] is singular. When [J] is singular, some rows or columns of the matrix become linearly dependent rows or columns and the rank of [J] is reduced. Realizing these facts, some singularities can be overcome by finding the rank of [J] and striking out the linearly dependent rows and corresponding columns in [J]. The resulting square matrix is nonsingular and may be inverted to find the independent joint motions. The Jacobian for redundant system, however, is nonsquare and hence is not directly invertable. Some consistent ways are needed to obtain the joint motions. A convenient criterion can be defined as [8] Minimize fl j>J 2 = [w] cjj ) 1 ^ 2 Subject to V_ H = [J] (j) in order to solve for the joint velocities where [W ] is a symmetric positive definite NxN weighting matrix. The Lagrangian for the constrained problem can be formulated as
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63 L II Â± t + (Ih " i T [ J ] > A (3.20) where X is the Lagrangian multiplier vector. Differentiating L with respect to (p and X produces a set of linear equations . TTi ' 0 ' t"li l J l T A a (p 9L = 0 = (j) T [J] 3X Â“ H Â“ or i = [w ?] 1 [j] t x Ih = [J] i (3.21) (3.22) (3.23) (3.24) Substituting Equation (3.23) into Equation (3.24), we get A = ( [J] [W] 1 [J] T ) 1 V H (3.25) From Equations (3.23) and (3.25), the joint velocities can be obtained as i = [w] 1 [J] T ( [J] [w] _ 1 [ J] T ) 1 v H (3.26)
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64 3.3.4 Accelerations for Serial Manipulators Having determined serial manipulator positions and velocities, the determination of accelerations is straight forward. The angular acceleration of link jk can be found by differentiation of equation (3.12) to obtain V Â‘V i + !G jk' 1 (3.27) Note that [H., ] is considered an NxN array, each component Jk of which is a vector of length three. The joint velocities and accelerations are kept explicit by introducing the kinematic influence coefficient matrices. Analogously, the translational acceleration of a point in link jk is given by differentiating Equation (3,14) with respect to time and writing this acceleration as ip = i i + i G p i i (3.28) Equations (3,27) and (3.28) show that the accelerations can be calculated as fa p ii Â• T Â• ! tH p ] Â± . + [ V a . , ' Â— J k j i T [H jk u (3.29) The acceleration of the endeffector is completely specified by its angular acceleration, a H = a N (j^ + i) and the
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65 translational acceleration of the hand reference point, a.^ , as shown in Equation (3.29). The hand acceleration is related to the joint velocities and accelerations as f . T . i [n H ] i :t $ Â• f H N(N+l)^ i + m '
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66 Minimize f II2 = ($_ T [W^] ^ ^ V ** subject to A H = Ajj + [J] 4>. where [W^] is a symmetric positive definite NxN weighting matrix. The Lagrangian can be expressed as L = II i ll 2 + A T { A h Ah " (3.33) Using the classical optimization techniques, the necessary conditions for an extreme of the constrained problem are $ = [ WJ ] Â“ 1 [ J ] T A (3.34) Â—
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67 proved a useful basis for changing reference systems [74], In manipulator design, the designer may prefer to specify motion capabilities in terms of endeffector speeds and accelerations in the design processes. 3 . 4 Dynamics of Serial Manipulators The relations between the manipulator joint variables and the time states of the links in a manipulator are needed for the dynamic analysis and response of the manipulator. The dynamic equations of motion must be formulated in a concise and efficient manner in order to be of use in the control and design of manipulators. A complete dynamic model is needed for manipulator control in order to determine the required input forces and torques, to evaluate the deformations due to loads and precision of the arm, to determine control properties, etc. In manipulator design it is essential to be able to recognize the dynamic effects of all parameters and the load imposed on the actuators for deciding on link and actuator sizing requirements. By evaluating the significance of the distribution of mass and stiffness parameters it may be possible to adjust these parameters to obtain higher lowest natural frequency and to enhance controllability, especially when precision control under external disturbance is required. Based on the kinematic influence coefficients and the actual mass of the arm, inertial modeling matrices of the
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68 serial manipulators were developed for the first time in the work [5] by Thomas and Tesar. These modeling matrices indicate the effect at the inputs of applied and inertial loads. The model leads to a simplified form of dynamic equations of motion for the rigidlink arm. The work must be considered as the foundation to this dissertation and will be briefly reviewed in the next two sections. Other dynamic phenomena, such as actuator stiffnesses and natural frequencies for the rigidlink manipulator model, will be derived by using the kinematic influence coefficients. 3.4.1 Input Torques Due to Applied Loads The terminal device of a manipulator is usually subjected to some general loading conditions during execution of a task. For example, this might be a resisting force and a resisting torque resulting from a holedrilling operation. In the most general case, several forces and torques can be applied to different points and links in the manipulator. The effective load at each actuator due to these applied loads can be explicitly derived in terms of the firstorder influence coefficients for the points and links of load application. First, consider the applied loads on link jk of the manipulator as a force, f^, applied to point P in the link ( ik) and a moment, m., , applied to the link. Let F J Â— j K n represent the effective torque (or force, for prismatic
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69 joints) on actuator n due to these applied loads and the method of virtual work can be used to determine an expression for this quantity as F ( jk) n T [G ] L p J n f + P T n m Jk (3.38) Assuming that the force f_ H and moment Bn(N+ 1) at t * ie ^and are the only externally applied load on the arm, the effective loads at the joints can be written as *4 = Ih (3.39) where F> [fJ< N+ 1 >. FÂ»< N+1 >, .... F*< N+1 >] T and F fi = [fj. Â— N(N + 1)^ Â’ From Equations (3.17) and (3.39), the Jacobian provides a general transfer concept of velocities and loads between the generalized input coordinates and the fixed coordinates (see Figure 3.2). The total effective load, F L , at actuator n due to applied loads is found by summing the effective load from each applied force and torque: P N F*; = 2 ( f T [ G ] ) + I (m ., [ H ] ) n p = 1 1 p J n y J k L jk J n (3.40) where P is the total number of load points of interest in the manipulator.
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70 Figure 3.2 Linear velocity and load transformations
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71 3.4.2 Input Torques due to the Inertial Loads In this section the effective actuator torques due to the system inertia and a given dynamic state are presented. The NxN modeling matrices [I ] and [P*] (n = 1, 2, ..., N) will be expressed in terms of the link mass distributions and the kinematic influence coefficients. The kinetic energy of a rigidlink manipulator in motion can be expressed as the sum of the kinetic energies of the individual link in the form K.E. (1/2) Z (M. k V c . + [H jk l <Â£ Jk ) (3.41) where M ^ ^ is the mass of link jk, V. t * le ve lÂ° c i t y of the centroid of link jk, and [11^] is the 3x3 inertia tensor about the centroid for link jk. The kinetic energy of the system can be rewritten explicitly in terms of the joint velocities and the firstorder influence coefficients by using Equations (3.12) and (3.14), and is given as N Â• T T Â’ K.E. = (1/2) 2 (M * [G ] [G ]$ + 2=1 i T t G , k ] T [Hj k ] [G jk ] i T ) jk = (1/2) i T [ i* ] i (3.42) where [I ] is the equivalent inertia matrix which represents the effective system mass seen at each input.
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72 The scalar component in row m and column n of this matrix is defined as [I*] m ; n N = E 2=1 (M., [G . ] [6 . ] + J k cj m cj n [G.. ] T [II.. ] [ G . , ] ) j k J m jk J 1 jkn (3.43) The equivalent inertia matrix depends only on the mass properties of the links and the corressponding firstorder influence coefficients. Since the inertia matrix [II_.^] is * symmetric, [I ] is also a symmetric matrix. The generalized inertia torques, i.e., the actuator torques required for driving the system inertia with a / prescribed arm velocity and acceleration can be obtained by using Lagrange's equations given as 3 K . E . n dtV^^ 3K.E . 3<}) (3.44) Differentiation of Equation (3.42) together with Equation (3.44) leads to F n = I T U*] ;n + A T [P*l i (345) where [I ] > represents column n of the equivalent inertia 5 ft * matrix and [P ] is the inertia power modeling matrix. The scalar component in row t and column m of [ P n ] can be expressed as
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73 [P n ] Â£;m = S ^ M jk t H cj ^ Â£;m ^ G cj ^ n + [II jkl ^jkU + [G jk ] I^Jkl ([G jkJn x t G jkV> < 3 * 46 > The inertial modeling matrices defined by Equations (3.43) and (3.46) indicate the physcial nature of the system in a given configuration and depend only on the position, geometry and mass distribution of the manipulator. The inertia torque in Equation (3.45) which each actuator must supply to drive the manipulator at the prescribed velocities and accelerations can now be readily evaluated using the inertial modeling coefficients. Note this inertia torque has two components: 1). an acceleration related term, and 2). a component which depends on the quadratic form of joint velocities. Massive manipulators and increased speeds of operation make the inertia torque even more important in the control and design of modern manipulators . 3.4.3 The Generalized Actuator Stiffness Matrix Since the hand of a manipulator is subjected to some applied loads during operations, the compliance and deformation are distributed throughout its structure. Positiondependent influence coefficients again form a convenient basis for modeling actuator stiffness and compliance .
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The deflection, AR^ , at the end effector of a manipulator due to an applied load, , at the hand can be written as [ 75 ] Â“h I c hI h (347) where [C.,] is a 6x6 positiondependent compliance matrix. In general, the vectors in Equation (3.47) contain three translational and three rotational components. Assuming that the flexibilities in the joint drives are the most significant compliance terms, the effective compliance matrix can be expressed in terms of the firstorder coefficients [J] that relate the motion at the hand to the joint motions . The effective torques F^ at the joints due to a load F applied at the endeffector reference point H are given in Equation (3.39). Using Hooke's law for elastic deformations, the deflections Aj^ at the joints of a manipulator due to the effective torques _F^ can be written as A ! = (V IcJ, (3.48) where [ C ^ ] is a NxN diagonal matrix of joint flexibilities. The deflection Ar at the hand due to the joint deflections Â— H. can be expressed as
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75 AR H = [J] A_Â£ (3.49) substituting Equations (3.39) and (3.48) into Equation (3.49), the compliance matrix of equation (3.47) in the fixed coordinate system is [C H ] = [ J] [C^] [ J] T (3.50) [ C R ] is generally symmetric but not diagonal, i.e., forces in one direction can produce hand deflections in several directions. The 6x6 stiffness matrix [K^] in the fixed reference of the equation [K h ] AR h = I h (3.51) can be obtained by inverting the matrix [C^] [K h ] = [ J ] _T [ ] [J]' 1 (3.52) where the T superscript indicates the transpose of the inverse of the matrix and [ ] is a NxN diagonal matrix of the joint stiffness. 3.4.4 Vibrations of Serial Manipulators An important issue in many manipulator tasks is the accuracy of the motion obtained. There are two distinct
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76 types of structural deformations which limit this accuracy. These are referred to here as deformations in the large and vibrations in the small. Vibrations in the small are low magnitude, oscillatory deformations about the mean motion equilibrium as distinct from the deformation in the large which result from static loads. Vibrations in the small are caused by transient phenomena acting on the system such as from startup and stopping of the arm motion. The lowest natural frequencies of vibration are a good indicator of overall structural integrity of the manipulator and can lead to important design and control considerations. For today's industrial manipulators, most of the deformation energy of vibration appears in actuator and drive train components. Therefore, the rigidlink manipulator model which considers flexibility at joint in the direction of relative joint motion is a good approximation for industrial robots. The model in this case is identical to that formulated in Section 3.4.2, with spring terms K contributing to actuator torques. Assuming ~ the vibrational deformations are small and the system damping is negligible. Let A(j> n be a small magnitude of vibrational deformations about the gross motion equilibrium position at joint n, or Â“ n, nominal + ^n (3.53)
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77 Since vibrations about an equilibrium configuration are being considered, the gross motion torques balance in the equations of motion and Equation (3.45) leads to the vibrational equation of motion at joint n in the form: A$ T [I*] + A4) T [ P* ] AiK A <() Â— ; n Â— n Â— n n (3.54) where the [I ] and [P ] are the inertial modeling matrices which are given in Equations (3.43) and (3.46), respectively. AjÂ£ is the vector of joint deflections and is the stiffness at joint n. The vibrational motion at each joint can be written as the superposition of N cyclic terms N A = E $ . Sin 03. t, n = 1, 2, ..., N (3.55) n j=l n J 3 where is the magnitude frequency w one mode of vibration at a time leads dependence for the vibrational motion j* to Considering only the following time A *n = $ Sin ojt n (3.56) A^ n = $ 03 Cos 03 t n (3.57) 2 A0 n = 0 o) Sin cot n (3.58) Performing an elementary order analysis by substituting Equations (3.56), (3.57) and (3.58) into
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78 Equation (3.54), the resulting equation indicates that the magnitudes of vibration. To show this, compare the relative magnitudes of the two terms on the left side of Equation (3.54). The components of [I ] are of same Â•Jp magnitude or slightly larger than the components of [P ]. Assuming that the vibrational deformations are less than +3Â° (a very large motion for a precision manipulator) gives This implies that less than onetwentieth of the inertia torque due to vibration is in the velocityrelated term so this term can be neglected. Equation (3.54) and this approximation lead to the reduced linear equations Similarly, the magnitudes of A n and A n A m ate of the order (3.59) (3.60) (3.61) A$ T [I*] + K A = 0, t ; n n T n n = 1 , 2 , Â• Â• Â• Â» N (3.62) Substituting for A n and A$ n from Equations (3.56) and
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79 (3.58) and writing the equations for all N joints simultaneously gives the set of equations. (0) 2 [I*] [K ]) $ = 0 $ N ] is the vector vibration amplitudes. This is a standard eigenvalue problem. For $ n are not to be all zero, then the determinant of the coefficient matrix must vanish; that is, ,2 r * 2 * <*> [I 1 2 r r* w [I I , . ,K. 0) [I ] i ; 1 "l 2 ; 1 1 ; 2 Â“ 2 [ I * 1 2 ; 2K 2 1 ; N 2 r * Â• Â• . u [I 1 2 ; N = 0 (3.64) ? r * ] N ; 1 1 r * w 2 [I ] N; 2 ' ' * W ^ 1 ^ N ; N K N The evaluation of the determinant results in an Nthdegree 2 algebraic equation in 0) . The N roots of the characteristic equation are the square of a natural frequency (usually expressed in rad. /sec.). It can be shown from matrix theory that the roots w are all real and finite if the kinetic energy given in Equation (3.42) is positive definite and if both [I ] and [ ] are real symmetric matrices. The eignvalues must, in general, be solved for
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80 numerically. Several methods of solution are available for determination of a few or all eigenvalues and eigenvectors of the coefficient matrix, such as the Power method, the classical Jacobi's,..., etc. 3 . 5 Conclusions of Modeling of Robotic Manipulators The underlying philosophy of this chapter has been the dependence of all kinematic relationships as functions of input parameters. The trend evident is the separation of the dynamic state from that which is solely kinematic. The kinematic properties are well known to be purely geometrical. The kinematic influence coefficients can be tabulated in an array as the dynamic modeling blocks. Once the geometry can be formalized in a logical manner, the superimposed dynamics problem becomes elementary. The nature of the dynamic model derived in this chapter can be concluded as follows: 1) . Kinematic influence coefficients are fundamental to the dynamic model. 2) . All influence coefficients can be calculated from extremely elementary vector operations. 3) . Model retains explicit role of all dynamic system parameters . 4) . Potential exists for real time modeling operation.
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CHAPTER 4 KINEMATIC AND DYNAMIC ANALYSIS OF ROBOTIC MANIPULATORS 4 . 1 Introduction In the design of mechanical systems it is often necessary to investigate the performance of a system when one or more parameters of the system varies over a given range, because better design of manipulators requires improved understanding of the geometric and dynamic aspects of the important parameters. Since the model parameters presented in Chapter 3 play an important role in the dynamic behavior of robotic manipulators, an important problem in manipulator design is the investigation of the influence of the important parameters in the workspace of a manipulator. Based on the dynamic model formulation given in the previous chapter, an analysis of the dynamic characteristics of a six degree of freedom industrial manipulator is illustrated in this chapter. The analysis on which to base the design methods involves the multivariable mathematical relations between the design parameters and the manipulator's force and motion states which are extraordinarily complex, nonlinear, and highly coupled. Computeraided 81
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82 complex, nonlinear, and highly coupled. Computeraided procedures are imperative for the kinematic and dynamic analysis in order to understand the influence of the system. Computergenerated manipulator and parameter contour plots are employed to display the extensive analysis information in a compact format and to help the designer to develop an intuitive feel for his problem reenforced by the visual displays. The digitial computer has allowed the designer not only to quantitatively analyze the behavior but also to examine it qualitatively. 4.2 Arm Characteristics for the Industrial Robot The manipulator investigated in this report is a large capacity six degree of freedom industrial type arm. Figure 4.1 contains a representative drawing of the manipulator showing its six rotational axes. Each of these revolute joints is driven by an electrical DC motor. The second and third axes are driven by two ballscrews which transfer the rotary motions from the drive DC motors to the linear motions along their axes. The wrist is one of its outstanding features. All three axes of the wrist are intersected at the same point and each axis is capable of a full 360 degree rotation. Due to the last two twist angles, it allows the end effector to be placed anywhere in a 238 degree cone with continuous rotation about the last axis .
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83 Elbow axis Three axis wrist Figure 4.1 Six degree of freedom industrial robot
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84 The kinematic representation of the industrial robot using the notation established in Chapter 3 is displayed in Figure 4.2. Table 4.1 lists the corresponding kinematic parameters and the joint motion ranges. Because several link lengths and offsets are zero, the industrial manipulator is a special geometry of the general 6R robot and its position analysis simplifies considerably from the general case. Appendix B presents a detailed derivation of the reverse position analysis employed for the rest of this work . The principal dynamic properties of interest are the mass, rotational inertia, and the location of mass centroid for each link. Table 4.2 lists the dynamic properties for the example industrial robot. The hand reference point for this analysis is defined at the origin of the local coordinates for link 6. The hand location coordinates ( , Y^, Z^) refer to the position of this point. 4 . 3 Workspace of the Industrial Manipulator Most of the performance characteristics of a robotic manipulator are functions of position and configuration in its workspace. Hence it is very important and useful to characterize the workspace of a manipulator and analyze the important parameters in the workspace. Figure 4.3 shows the workspace of the example robot in the X Z plane (Y =0) . The working volume of the
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85 Â—34 Figure 4.2 Kinematic representation of the example manipulator
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86 Table 4.1 Kinematic parameters of the example manipulator Joint No. a., jk (deg.) a.. jk (in.) S. . 9 . 3 3 min (in.) (deg.) 0 max (deg.) 1 90 0 32 135 135 2 0 44 0 30 117 3 90 0 0 45 60 4 61 0 55 180 180 5 61 0 0 180 180 6 0 6 180 180 Table 4. 2 Inertia properties for the example man ipulator Link Centroid Mass Inertia X Y Z I I I x y z (in.) (1 V 3 2 (10 lb in. about centroid) m 1 0 0 17 700 0 0 100 2 20 1 0 1500 20 180 150 3 4 7 0 1000 170 26 170 4 0 0 20 150 2 2 1.2 5 0 0 0 80 0.8 0.8 0.3 6 0 0 4 60 0.4 0.4 0.2
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87 X ( in . ) * * Figure 4.3 Workspace of the industrial robot in the X Z plane
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88 industrial robot is limited by its geometry and the limits on the joint rotations as listed in Table 4.1. Although only the X Z plane of hand locations is shown, all other reachable hand positions can be achieved by rotating this plane about axis. Such a rotation requires only a change in the value of first joint angle, so results for the entire working volume can be obtained by a straightforward transformation of the results presented in k k the workspace of the X Z plane. In the following analysis of this chapter and next chapter, the orientation of the hand is held constant as S =( 1 0 0) T and a, 7 =(0 1 0) T , and the hand reference point, * k or tool center point is moved throUghtout the X Z plane. * * Figure 4.4 shows the effective workspace of the X Z plane for the hand orientation. Comparing these two workspaces, it is clear that the reachable workspace in Figure 4.4 has k k been significantly reduced in the X Z plane. Figures 4.5 4.9 are computergenerated plots showing contours of constant joint angles. The outlines in the contour plots indicate the extent of the effective k k workspace in the X Â— Z plane. The joint values from a rectangular data array are used for these plots, producing rough edges for the reachable area. If enough data points were employed, these edges would actually be piecewise smooth circular arcs. For this reason, the joint angle values at the edges of the reachable area in each figure
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89 Figure 4.4 Reachable workspace of the industrial robot with the hand orientation held as S^=(l 0 0) T and a_^y=(0 1 0) in the X Z plane
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90 are slightly unreliable, although the interior values and the important trends are accurately presented. As mentioned before, the hand orientation is held constant as S. Â»( 1 0 0) T and Â£ 67 =(0 1 0) T for these plots. The joint angles are generated by using the algorithms developed in Appendix B. For the orientation of the hand held in X* direction, there are two feasible configurations "ft ^ for the hand locations in the X Z plane. For example, if we specified the manipulator's hand location as the vector (60 0 6 0 ) T inches and the end effector orientation as S^ = (1 0 0) T and a 6? =(0 1 0) T , two sets of the feasible joint angles are given as Joint Configuration A Configuration B 1 0.000 (Deg.) 0.000 (Deg . 2 88.213 88.213 3 15.102 15. 102 4 4.720 175.280 5 160.668 160.668 6 94.720 94.720 The joint angles of the first joint are zero degree * * for the end effector in this orientation and the X Z plane. Configuration A is arbitrarily chosen for the following analysis.
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91 Figure 4.5 Joint angles of joint
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160 92 a) . Configuration
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93 Figure 4.8 Joint angles of joint
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94 a) . Configuration A b) Â• Configuration
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95 4 . 4 Analysis of Actuator Torques Due to Static Loads The first level dynamic analysis of a manipulator structure involves the determination of the actuator torques for each joint. The actuator torques are required to balance loads on the system and maintain the prescribed motion specifications. In general, there are four component loads to be considered at each joint n: 1) . the effective load due to static loads, F^; 2) . the actuator load due to inertial loads, F^; 3) . the equivalent load due to other system components s such as system springs and dampers, F^; A 4) . the driving torque supplied by the actuator, F^. The first two of these terms are covered in detail in Sections 3.4.1 and 3.4.2. The third term can be modeled by using similar formulations in Reference [48]. The actuator is a complex function of the actuator and control system properties. From the design point of view, these terms can be related as in the following equation F A = F L + F 1 + F S (4.1) n n n n The actuator inertial torque will be analyzed in the next section of this chapter. This section contains the analysis of the actuator torques resulting from several
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96 static loading conditions. The static loads investigated include gravity forces, the weight of the arm, and applied forces at the hand. In order to obtain a clear picture of the effects of general static loads, four basic loading cases are considered in this section: 1) . 150 lb applied force at the hand in the X direction. Â•k 2) . 150 lb applied force at the hand in the Y direction. * 3) . 150 lb applied force at the hand in the Z direction. 4) . Gravity loads on all links with no applied forces. Figure 4.10 4.20 show the contours of constant actuator static torque for each joint of the industrial robot. From the geometry of this robot, some equations of the actuator torque can be simplified and the results can be checked easily. The static actuator torques for each joint are discussed individually in the following pages. Since the industrial robot is mounted with the axis of Â•If first joint vertically upward in Z direction. Equation (3.38) can be reduced to: ^H Y H + + m H a H t Â—6 7 (4.2)
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97 where F^ is the resultant torque on joint one due to a torque applied to the last link and a force acting on the * * * hand reference point (X^, Y^, Z ^) . The x, y, and z superscripts refer to components of the applied loads. This equation shows that the actuator torque of the first joint is not affected by Z direction force and gravity loads. k k k k Also, Y =0 in X Z plane, X direction forces make no h. contribution to the actuator torque at the first joint. k k The actuator torque is proportional to the X^ for the Y direction force which is clearly shown in Figure 4.12. For the second joint, Equation (3.38) can also be simplified as F 2 ' Â£ iS X H f H "67 <4 Â’ 3) Equation (4.3) shows that there is no effect on the L * actuator torque FÂ£ due to the force in Y direction, the k k torque in the X direction and the Y direction. Figures 4.10 and 4.15 present the simple contours of constant L actuator torque F^. Figure 4.18 shows the large magnitude of the gravity loads. The magnitude of the actuator torque at joint 2 increases rapidly to a maximum of about 100,000 inlb as k the hand moves outward from the Z axis (see Figure 4.21). This is because the moment arms for the gravity loads increase as the centroids of the links move out from the axis .
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98 The actuator torque results for joint 3 are harder to visualize than the joint 2 results discussed above. This is partly due to the fact that the joint axis moves around the S 2 axis with a radius a 23 as the position of the hand k changes. In the X Z plane, the j3 2 axis always passes through the point (0 0 32) inches. There is no such fixed point for the axis. The direction cosines of are (0 T * 1 0) , hence, forces in the Y direction produce no actuator torques. Since the axis is parallel to and moves around S 2 on a circular arc about Z =72 inches ( s ^ + a 23 = 76 inches), the X* direction forces may intersect the joint 3 axis for hand positions in which is about Z =72 inches. The effects can be clearly seen from Figures 4.11 and 4.22. The magnitude of the joint torque is maximum in the top and bottom sections of the reachable space. These are the regions where the offset link is vertical, giving the largest moment arms about axis for X direction forces as shown in Figure 4.23. Offset link is nearly horizontal for hand positions about Z*=72 inches (see Figure 4.24). While this leads to the maximum magnitude of the joint torques due to Â•k gravity forces and applied forces in the Z direction. Figures 4.16 and 4.19 show the contours of joint torques at joint 2. The actuator torque results for the case of the applied force in the Z* direction are similar in nature to the gravity load results.
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99 It was mentioned previously that the geometry of the industrial robot is a special case of the general 6R manipulator geometry in that several joint axes intersect or are parallel, allowing a tractable position analysis. Since the axes of and are parallel and the axes of S, S and S, are intersected at a point, this produces Â—4 Â’ Â—5 Â—o some decoupling between the effects of the hand reference point location and the hand orientation. Because of the geometry of the industrial robot, the k k fourth joint axis, , always lies in the X Â— Z plane. For this reason, the joint axis intersects the line of k k action of the applied force in the X or Z direction. The actuator torque F^ is therefore zero in these cases. Figure 4.13 shows the torque at actuator 4. The actuator torques at joint 4 due to the applied force in the Y direction depend mainly on the angle between the axes of and S ^ . The joint torque function can be simply expressed as f H S 66 S1 " e <4 ' 4) for this industrial robot. The actuator torque F^ increase from zero when the 0 is zero degree, as shown in Figure 4.23, to a maximum magnitude that would reach 900 in. lb. if the magnitude of the 0 could be increased to 90 degrees (see Figure 4.23).
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100 Figure 4.10 Static torques at joint 2 due to Figure 4.11 Static torques at joint 3 due to a 150 lb force in the X direction a 150 lb force in the X direction
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132 101 Figure 4.12 Static torques at joint^.1 due to Figure 4.13 Static torques at joint ^4 due to a 150 lb force in the Y direction a 150 lb force in the Y direction
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132 102 CO o VO
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132 103 CO o rH cd o c u o Â•H QJ 4J U 2J Â„ v Cl ZJ <1) cr o M H a TO a) CM Â•H TO +J CJK Â•H CM O TJ (D 4J 4J cd c CO Â•H < 1 ) P 0 ) u 4 o H u O o 4 J uo rQ Â•H rH 4J cd o 4J in rH ui 0) H Z> 00 Â•H Pn
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104 Figure 4.17 Static torques at joint 5 due to 150 lb force in the Z direction
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105 Figure A. 18 Static torques at joint 2 due to Figure 4.19 Static torques at joint 3 due to gravity loads gravity loads
PAGE 114
106 Figure 4.20 Static torques at joint 5 due to gravity loads
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107 * a> X 6 3 .a 3 Â•H & CD X u u 3* . Â•H X 41 O ' 3 o Â•H 4J 03 3 00 Â•H 41 3 o CJ CO a) o 1 o 41 X 0) Â•H rH CL a l 03 CSl CM 03 U 3 00 Â•H Pn a) a) 3 ^ cr u H X a 03 CO X 03 O 3 O e ^ 3X0 e w cj o3 o3 XI 03 DM J3 03 4Â» .n 3 u Â•H 41 O 3i 10 O 03 3 41 O O *4 Â•H CM O U 41 U U 3 3 X 00 Â•H 0) Â•H O Â•H 41 Â•m rÂ— 1 3 3. o 44 0O O 3 CM 'd 03 H 3 00 Â•H P4
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* Â• 108 03 Figure 4.23 Configurations of the maximum joints torque of joint 3 for the applied forces in X directio
PAGE 117
(.9744 0 . 2247 ) 109 H II &0 03 "O H 3 03 o 3 Â•H cr 41 03 O 03 CO 0) 51 Â•H a Â•H 51 T3 3 .a Â•H rH 03 C 42 Â•r) o 44 m 44 r1 3 O 03 Â•H 03 rC XJ a u a cr 3 n 54 42 0 O 4J 41 03 42 5i U O rH u 1 O a 3 u 3 Â•H T3 a CN 03 03 O Â•H CN rÂ— 1 > 42 Â•H 4J 4J C Â•H Â•ro > 3 03 Â•H e 5i 2 00 03 e O *H 03 5i X 42 O to 4J 41 e 54 "3 a o 03 44 Â•H u rÂ— 1 CO Q41 3O 4J 3 3 c Â•H 03 0 o 42 C Â•H Â•rn u o 4J 03 Uh 'a 44 5i O 3 CJ 3 3 0) 00 03 54 Â•H 3 CO *H 44 CT T3 TD 3 U 3 O O CH< U U rH N3 CN
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110 Figures 4.14, 4.17 and 4.20 present the actuator torques on joint 5 for the loading cases. Due to the last two twist angles, the joint axis does not lie in the X Z plane except the joint angles of joint 4, < P^, are 0, +180. This effects can be clearly seen from these figures. For the joint angle cf> 5 =+ 180 as shown in Figure 4.8, the ic "k & joint axis, j>,. lies in the X Y plane of about Z =72 * inches and intersects the applied forces in the Y direction. The applied forces in the X direction intersect the fifth joint axis S^ 5 , so no actuator torque is produced. Since the point of force application H and the centroid of link 67 both lie on the final joint axis j> 6 Â» the lines of application of all applied loads and of the gravity load on the final link pass through this joint axis producing no resultant torque about the joint. The actuator torque of joint 6 is therefore zero for all of the static loading cases considered. 4 . 5 Analysis of Actuator Torques Due to Inertia The first step toward the analysis of the actuator torques for each joint is the determination of the actuator torques resulting from static loads in a range of hand locations which has been discussed in detail in Section 4.4. This section employs similar techniques to those used in the previous sections to analyze the relationships between the actuator torques and the arm motions of the
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Ill manipulators. A specified motion of the hand is using for investigating the actuator torques and joint motions of the industrial robot. It is generally more convenient and useful for designer to deal with the velocity and acceleration of the hand rather than of the joints themselves. Therefore, the model formulations of velocity and acceleration referenced to the hand motion, as developed in Sections 3.3.3 and 3.3.4, are preferred for the analysis in this section. From Equations (3.19), (3.31), (3.32) and (3.45), the inertia torque at joint n can be rewritten as llH n; + Â— H I p n ] (4 ' 5) where the handreferenced inertia modeling matrices [I ] H and [P ] are defined as n [I H ] = [I*] [J] _1 [p"i [Ji'hip*] i u H ] Knur 1 (4.7) m= 1 Â’ where T superscript indicates the transpose of the inverse of the matrix and [ ] , m= 1 , 2, ...,6, are K) * KlI h r] ^ H N(N+1) ^ lH 1 33 II [ H 5 ] H = t H N(N+l)^ (4.8) [**Â«] = [hsi. IÂ«h1 ^ h n(n+i)1
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112 Equation (4.5) shows that the inertia torques at the actuators depend on both the velocity and acceleration of the hand. The inertia torques F*, n=l, 2, ..., N, are usually largest during the acceleration and deceleration parts of the motion. These torques are also very significant during the high velocity operations. This can be seen from Equation (4.5). The velocityrelated inertia torques are proportional to the velocity squared, so doubling the speed multiplies this torque term by four. A simple hand motion is employed to see the dynamic properties of the industrial robot. Assuming zero hand acceleration, the inertia torques at actuator n due to a * hand velocity of magnitude 60 in/sec in the X direction * * are determined for a range hand locations in the X Z plane. Figures 4.26 4.29 show the resultant inertia torques for each joint of the example manipulator. The second joint has experienced large inertia torques for the prescribed hand motion. The corresponding joint velocities and accelerations at these joint is shown in Figures 4.30 4.37. Although only a simple motion considered at the hand, these plots show that the actuator inertia torque is the most difficult torque to conceptualize of those analyzed in this study. One reason the joint inertia torque is hard to visualize, the velocityrelated torque term is not a simple linear terra, and the formulation for [P^] is an extremely
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113 complex function. Some of the results are discussed in the f ollowing . The joint velocities about the axis of joint 2 remains nearly constant throughout this range of hand positions (Figure 4.30). The joint speed are high at the right and left edges of the working range, this is because that the joint 3 is at its joint rotational limits (see Figure 4.6). This is also why joint 3 has the large inertia torques at these regions. From Figures 4.26 and 4.34, the joint acceleration contours correspond very closely to the actuator torques. These figures also show that the inertia torque and joint acceleration are greatest along the outside boundary of the working volume and in the reachable points nearest to the axis of joint 2. Figures 4.31 and 4.35 indicate that the joint velocities and accelerations at joint 3 tend to increase as Â«p the hand extends outward from the Z axis. Conversely, the joint velocity is minimum in the upper left section of the working area because S^, is nearly vertical. The joint k acceleration is minimum in the lower left corner for Z about 35 inches. The joint inertia torques of joint 3 are very complex and hard to visualize the results, since this joint has the high joint speeds and the velocityrelated torque term is the dominant term of the joint inertia torque for the specified hand motion.
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in/sec 114 Figure 4.26 Inertia torques at joint 2 due to a Figure 4.27 Inertia torques at joint 3 due to a hand velocity of magnitude 60 in/sec hand velocity of magnitude 60 in/sec
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0] in/sec 115 Figure 4.28 Inertia torques at joint 4 due to a Figure 4.29 Inertia torques at joint 5 due to a hand velocity of magnitude 60 in/sec hand velocity of magnitude 60 in/sec
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0] in/ sec 116 G u g o w 4J \ G CD *H G G3 O vÂ£> CO G 41 'V g g Â•H 4J O *H Â•r) G DO U G G g CO 4t G O G Â•H O 4> PO *H O O rH 0) ;> CJ 0) o u Â•H * Â• 51 * o ca c >j ,c h ^d* a) V4 G DO Â•H Ph G O G O W G T3 o vD Csl G 4Â» TJ G G Â•H 4J O *H 4J G G g CO 44 G O G Â•H O 4J Po *H Â•HOt) O H O O O G iÂ— I O P4 G H *H > G G > 4J * G G X Â•H G O G G o ,Â£ h O co Â•
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0] in/sec 117 CO o U O *H O o o 0) iH O Vi CD T3 41 * fit) Â•H ffi o Â« c !) jfi iH T cO CJ CD O CO 4J Â£ CD Â•H Â£ T3 O vO 4J ^3 cn c 0 Â•rH u 1 Â— o Â•H Â•r) Â£ /N 00 o vÂ£> Â£ *H at CO a CO CO 44 u Â« CN Â£ X rÂ— 1 Â•H Â£ O Â£ Â£ & Â•H O CN CN m rH (D H 00 Â•H
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118 oo o sO ON co CM O vO 00 SÂ’ vO CO Q) U 03 O C G G u G G O O O O 4J vO uo CO a) n G 00 Â•H Pn 00 o X a o 0) 4J co Q) G G *H 03 O CM vO Â•u a) G T3 Â•H G O 4J Â•r) *H G 41 00 G cd G co G 4) O O 4J cd n o a cd Â•rH 4J O cd *H 03 G* G X Â•C C cd *H
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0] in/sec 119 Figure 4.36 Joint accelerations at joint 4 due to Figure 4.37 Joint accelerations at joint 5 due to a hand velocity of magnitude 60 in/sec a hand velocity of magnitude 60 in/sec
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120 For the industrial robot, the last few joints generally provide orientation of the hand. In this analysis the orientation of the hand is constant and the hand velocity is constant, the joint velocities of the last three joints remain in the low speeds which are clearly shown in Figures 4.32 4.33. The joint acceleration contours in Figures 4.36 and 4.37 correspond very closely to joint inertia torques in Figures 4.28 and 4.29, respectively. 4.6 Analysis of the Manipulator Hand Deflections Another important consideration for manipulator design and control is the relative magnitude of the system deformations under load. For precision control of a manipulator, a complete dynamic model can be used to predict and therefore compensate for the deformations. In manipulator design, the primary concern is to reduce the structural deformations and increase the rigidity of the arm. The objective of the section is to analyze how the hand deformation characteristics vary with the position in the effective workspace. A lumped mass model of deformatins for serial robotic manipulators has been developed in Section 3.4.3 where the compliance at each joint represents the actuator flexibility and the lumped masses include the actuator and link masses. This approximation greatly simplifies the computational procedure but should be used only when these joint
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121 deflections are the dominant deformation terms. The deflection, AR h Â» at the hand of a manipulator due to an applied load, F at the end effector can be rewritten as AR h = [J] [c ] [J] T F h (4.9) The basic assumption for this equation is that the deformations in manipulators are relatively small. The deformations due to the gravity loads in the arm are neglected. Knowing the loads acting and the joint stiffnesses (or compliances), three components of translational deflection and three components of rotational deflection at the hand can be determined. Because of the physical significance and convenience, the quadratic norm (P^=2) is useful for characterizing the hand deflection magnitude as I! RrII 2 = (lHt J ^ C (() ][J]TtW R ][Jl[C ( i , 1 [J]T H )1/2 (4>10) where [W D ] is a diagonal matrix of weighting factors. For simplicity, the weighting matrix is taken as an identity matrix in the analysis. A number of loading conditions are considered in this section in order to get a complete picture of the effects of the applied load.
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122 1) . 150 lb applied force at the hand in the X direction. k 2) . 150 lb applied force at the hand in the Y direction. k 3) . 150 lb applied force at the hand in the Z direction. 4) . 100 inlb moment applied to the last link in the * X direction. 5) . 100 inlb moment applied to the last link in the k Y direction. 6) . 100 inlb moment applied to the last link in the k Z direction . Figures 4.38 4.49 present the contours of the translational and orientational hand deflections due to each of the loading conditions described above with a set of joint stiffness as Joint Joint stiffness 1 lxlO 7 (inlb/rad) 2 7xl0 6 3 5x 1 0 6 4 lxlO 5 5 7xl0 4 6 7xl0 4
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Joint Stiffnesses K, Â— [1000 700 500 10 77] x 10 in lb/rad 123 e Â•H 00 o vO CTn 'dco CN o VO CO \Q x co X CM CN rH I c Â•H "d C OX Â•H d) X G rH 00 44 (U Cd 0) rH S ^ 41 co co Â•
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Joint Stiffnesses K, = [1000 700 500 10 7 7] x 10 inlb/rad 124 Figure 4.40 Magnitude of the translational hand Figure 4.41 Magnitude of the rotational hand dedeflection due to a 150 lb force in flection due to a 150 lb force in the >C the X direction direction
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Joint Stiffnesses K, = [1000 700 500 10 7 7] x 10 inlb/rad 125 Figure 4.42 Magnitudes of the translational hand Figure 4.43 Magnitudes of the rotational hand deflection due to a 150 lb force in deflection due to a 150 lb force in the Z* direction t ^ ie ^ direction
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Joint Stiffness K, = [1000 700 500 10 7 7] x 10 inlb/rad 126 Figure 4.44 Magnitudes of the translational hand Figure 4.45 Magnitudes of the rotational hand deflection due to a 100 inlb moment deflection due to a 100 inlb moment in the X* direction in the X direction
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Joint Stiffnesses K, = [1000 700 500 10 7 7 ] x 10 inlb/rad 127 Figure 4.46 Magnitudes of the translational hand Figure 4.47 Magnitudes of the rotational hand deflection due to a 100 inlb moment deflection due to a 100 inlb moment in the Y* direction in ^ direction
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Joint Stiffnesses K, = [1000 700 500 10 77] x 10 in lb/rad 128 Figure 4.48 Magnitudes of the translational hand Figure 4.49 Magnitudes of the rotational hand deflection due to a 100 inlb moment deflection due to a 100 inlb moment in the Z* direction in the Z direction
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129 Figure 4.50 Weak
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130 Figures 4.42 and 4.43 show that there are very large hand Â•k deflections due to the applied load in the Z direction. For the hand positions around the coordinates (92 0 60) inches, the hand deflections can be as much as .34 inch in magnitude of the translational hand deflection and .83 degree in magnitude of the orientational hand deflection which are about 60 times of the required accuracy, .005 inch, for assembly. Figure 4.50 shows these large deflections are caused by the large moment arms of the k joints, 2, 3 and 5, for the applied load in the Z direction, since the joint axes an d JI 3 are perpendicular to the X Z plane and the axis is lied in the X Y plane for Z about 60 inches. 4 . 7 Analysis of the Natural Frequencies of Manipulators As manipulator's operational speed increase, vibration considerations begin to play an important role in the design of the manipulator and its control systems. Vibrations in the small are due to transient phenomena acting on the system. This study in manipulator vibrations deals principally with determining the natural frequencies, ( n= 1 , 2,..., N) . The fundamental natural frequency gives one of the best overall indicators of the systemÂ’s structural integrity and the maximum speed for performing precise arm motions .
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131 A lumpedmass arm model has been developed in Section 3.4.4 for evaluating the natural frequencies of serial manipulators. Equation (3.63) shows those frequencies which can be determined from a standard eigenvalue problem, and can be rewritten in the following form. a) 2 [I*] [ ] =0 (4.11) Expanding the above characteristic determinant we 2 obtain an algebric polynominal of Nth order in U) which is known as the characteristic equation or frequency equation. Its roots are the desired eigenvalues. Since the joint stiffnesses in [ ] are large numbers, it is not a proper computational approach to obtain the eigenvalues by using a root solving procedure. If [I*] and [K^] are symmetric and positive definite, the eigenvalues are real and positive [77] . There are various numerical schemes for obtaining these eigenvalues of an eigenvalue problem involving the symmetric and positive definite matrices. The QR algorithm [78] is used here to determine the natural frequencies and the corresponding eigenvectors. Figures 4.51 4.54 show that the distributions of the first four natural frequencies of the industrial robot with a set of joint stiffnesses K =[1000 700 500 10 7 7] xlO Â“
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132 previous sections, the hand of the robot is in a single orientation defined by j5g = (l 0 0) > _5.67 = (Â® ^ ^) for the range of hand positions. From Figure 4.51, the fundamental natural frequency is the lowest along the outside boundary of the working volume because in this region where the arm is fully extended and S 2 has the greatest moment of inertia. The lowest fundamental frequency is about 6 Hz at the outside boundary. Since the stiffness of joint 2 is the dominant spring constant for the first mode (see the following eigenvectors), which implies the industrial robot to have structural weakness near the shoulder for the system. The operational frequencies of the arm, w, should be kept significantly lower than the lowest natural frequency (W<<6 Hz) in order to prevent dynamic coupling between the vibrations and gross motion deformations. Figure 4.52 show that the second natural frequency has low frequencies about 9 Hz along the lower right workspace boundary and high frequencies about 25 Hz in the upper section of the reachable area. From the analysis of the vibrational modes, the eigenvectors indicate the frequencies in the right and lower area are dominated by the first joint stiffness. When the end effector moves to the left and upper section, the dominant spring constant is switched to the joint 3. The reason is that the moment arm about the axis is large in the lower right section and small in the upper left section of the working area. This effect
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133 can be clearly seen from the following examples. The natural frequiencies and eigenvectors for hand locations at point A, B and C in Figure 4.52 are found as follows: 1). Hand location at point A, (48 0 60) inches Natural frequencies (Hz) 8.58 16.36 24.28 41.96 61.19 126.40 Corresponding natural modes (column vectors) 0.00 0.00 1.00 0.00 0.00 0.00 1.00 0.23 0.00 0.00 0.00 0.00 0.32 1.00 0.00 0.00 0.01 0.00 0.00 0.00 0.46 1 . 00 0.25 0.09 0.20 0.55 0.13 0.34 1 . 00 0. 12 0.00 0.01 0.02 0.17 0.09 1.00 Hand location at point B, (64 Natural frequencies (Hz) .8 0 60) i nches 7.77 17.81 17.82 41.90 6 1.17 126.32 Corresponding natural modes (column vectors) 0.00 1.00 0.27 0.00 0.00 0.00 1 . 00 0.13 0.27 0.00 0.00 0.00 0.38 0.49 1 . 00 0.00 0.01 0.00 0.00 0.27 0.07 1.00 0.25 0.09 0.21 0.36 0.55 0.34 1 . 00 0.12 0.00 0.01 0.00 0.17 0.09 1 . 00
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134 3). Hand location at point C, (72 0 60) inches Natural frequencies (Hz) 7.43 14.97 18.75 41.93 61.19 126.44 Corresponding natural modes (column ' vectors) 0.00 1.00 0.00 0.00 0.00 0.00 1 . 00 0.00 0.29 0.00 0.00 0.00 0.40 0.00 1 . 00 0.00 0.01 0.00 0.00 0.18 0.00 1 . 00 0.25 0.09 0.21 0.05 0.60 0.34 1 . 00 0.12 0.00 0.00 0.01 0.17 0.09 1 . 00 The results explicitly show that there are two equal natural frequencies near the location of point B and two vibrational modes are changed at that point. Figures 4.55 4.57 show the arm configurations at these points.
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CHAPTER 5 OPTIMIZING ACTUATOR PARAMETER DISTRIBUTIONS FOR ROBOTIC MANIPULATORS BASED ON LOCAL DYNAMIC CRITERIA 5 . 1 Introduction The analysis methods described in Chapter 4 give the actuator loads, hand deflections and natural frequencies for a given position with a prescribed applied load, velocity and acceleration state of the arm. These results of such analysis may be used to choose actuators of the appropriate size for any specified motion. The information may also be used to indicate the problems for redesign of mechanical structures. But this information is of only limited use to the designer because there is such a profusion of possible states to consider. In this chapter the optimization techniques are employed for condensing the information about actuator parameters by imposing motion magnitude constraints at the hand. The preceding chapter presented an extensive analysis of the industrial robot for a variety of applied loads and motions. Computer generated plots are used to display these analysis information in a compact, understandable form and explicitly show how system parameters vary as a 139
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140 function of hand positions, applied loads and motions. In order to establish a first level of understanding and computational methodology, the work in the rest of this chapter will consider local properties, i.e., properties at a specified position or configuration of the device. A more complete methodology would result by considering a range of configurations in the available workspace of the system. The global evaluation can only be established by including the functional dependence of the position vector as a variable in the optimization formulation. This leads to the application of nonlinear optimization techniques to develop general design tools for manipulators based on global dynamic criteria. This global optimization work will be presented in the next chapter. The industrial robot described in Chapter 4 is also employed for analysis of the results of local dynamic criteria in this chapter. The configurations investigated "k k consist of an X Â— Z plane of locations of the hand refer* ence point H with the hand pointing outward from the Z T axis in a single orientation defined by S^=(l 0 0) and a 67 =(0 1 0 ) T . 5 . 2 Maximum Actuator Torques Due to Generic External Loads Since manipulators are usually designed for a wide spectrum of applications, they must be capable of
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141 supporting a variety of applied loads. Hence, the fundamental question of this section can be stated as follows: Given a configuration of the manipulator, what is the worst joint load for each actuator with specified magnitude of a applied load on the end effector. The actuator torques F n are generally considered to be independent design parameters such that each should be considered separately. The static torque F^ at joint n due to the applied load F fl at point H on the end effector is given from Equation (3.39) as F n Ih [Â£Â„*<Â» Â£Â„> 1 Â• I H + Â£n Â• 2n(n+n (5 ' 1) From this equation, we know that F^ is a function of arm configuration and external loads. In this section the directions of the applied loads are left arbitrary, the maximum magnitude of load on joint n for a specific configuration is * 'in*'" + ( ">K N+l > > max < 5 ' 2 > where (f H ) max and (m N(N+ i)) max are the maximum magnitudes of the applied force and torque at the hand, respectively, and  S n x(HR )  is the perpendicular distance between the hand position and the joint axis S_ . Figure 5.1 shows the worst external loads for joint 2 of the industrial robot. The remainder of this section displays and discusses the
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142 Figure 5.1 The worst applied loads for joint
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143 results of the maximum actuator torques due to a force of magnitude 150 lb and a torque of magnitude 2000 inlb on * * the hand for a range of hand positions in the X Z plane. Since the axis of joint 1 is fixed and directed vertically, the worst applied force and torque are parallel to ^ A Y axis and Z axis, respectively. The perpendicular * 2 distance from the first joint axis to the point H is ((X^) +( Y* ) 2 ) 1 ^ 2 and, since Y* = 0 for the plane of hand positions H H * considered, the maximum joint torque is proportional to V This is readily seen in Figure 5.2. Figure 5.3 shows the maximum joint torques for joint 2 due to the generic loads. Because the joint torque is proportional to the perpendicular distance from the axis of joint 2 to the point of load application, the lines of constant maximum torque are circular arcs about the point * * (0 0 32) inches where joint axis intersects the X Z plane. The directions of the worst applied force are tangent to the contour curves. The maximum static torques on actuator 3 are plotted in Figure 5.4. Since the position of the third joint axis changes as the hand position varies, the maximum joint torques depend on the distance between the hand and the joint axis. Figure 5.4 shows the maximum contour values Â£ are around Z =72 inches where the offset links and Sgg are nearly parallel and the distance is maximum. This configuration is shown in Figure 4.24.
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144 As mentioned previously, the fourth joint axis lies in the X Z plane, the X and Z components of force have no effect on the actuator torque F^. The maximum joint torques on actuator 4 can be simply expressed as (F^) = (f u ) S,, Sin 9+ (m A7 ) (5.3) 4'max H max 66 67 max where 6 is the angle between the axes and as shown in Figure 4.25. It is interesting to note that the worst cases for joint 4 are in the top and bottom sections of the effective workspace as shown in Figure 5.5. Conversely, Figure 5.4 shows that these are the best configurations for supporting external loads at joint 3. Since the geometry of the industrial robot, the twist angle between and is 61Â° and the distance between and the hand reference point H is fixed, the maximum static torque at joint 5 is simplified as (F^) = ( f __ ) S,. sin a,, + (m A7 ) (5.4) v 5'max v H / max 66 56 67 max For (fÂ„) =150 lb and (ra,.,) =2000 inlb, the maximum H max 67 max static torque at joint 5 is a constant value, 2787.158 k k inlb, for all hand positions in X Z plane. In a similar manner, the maximum torque at actuator 6 is also a fixed value, 2000 inlb, for the reachable positions, since the applied forces pass through joint axis Â•
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145 * c N *H 00 o rH T3 o On o 04 o >d00 44 G 04 II X Â•H cd 04 O B O' Â•r) /O 4J vD O /*Â“S Cd e vO c 'w' *H CO oo v^ 0) nd 3 G rH cd II 41 X 04 CO cd rH & e rs 3 c o s HH 40 Â•rH w rH X 1 04 cd O G rH S +J H cn LO Q) U P 00 Â•H Eh 00 o vO < 00 04 o vO 00
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146 * c DM *H CO O vO On CO CM r^vO Â•* H X rH a o Â•h m cO 4J co e p B Â•H X cd X X cO B /~*N X 41 rO ^ rH I o p 44 H m LO
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147 5 . 3 Maximum Actuator Torques Due to Generic Hand Velocity The actuators of a manipulator must also be designed to provide sufficient driving torque to overcome the inertia forces induced by the prescribed magnitudes of the end effector velocity and acceleration . This section derives analysis methods to further condense the inertia torque results by prescribing the motion capabilities which are independent of direction. Instead of determining the actuator torques due to the hand motions in given directions as done in Section 4.5, the directions are left arbitrary and the maximum magnitudes of velocity and acceleration are chosen to find the maximum actuator inertia torque at each joint for a specific configuration. This assures that if the magnitudes of the arm velocity and acceleration are less than these specified values, none of the joint inertia torque are exceeded. The inertia loads at the actuators depend on both the velocity and acceleration. For simplicity, the first case to be considered in this section requires that the hand acceleration equal to zero (A =0). In order to obtain the maximum joint inertia torques (F^) for a prescribed hand J n max speed, regardless of the direction of the velocity vector V a , this problem can be stated as Â— H
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148 For a given configuration of the manipulator maximize the magnitude of actuator torque F^ for each joint subject to the constraint vÂ„ I Â» H  2 < (V H ) aax . The velocityrelated inertia torque F^ at actuator n is given from Equation (4.5) as K [pÂ“] V h (A h 0) (5.5) The hand velocity can be decomposed into magnitude and unit direction vector e as Â—v Â— H e Â— v and T Â— v [W 2 ] 1 v (5.6) (5.7) where [ W^. ] is a 6x6 diagonal matrix of weighting factors. Using Equations (5.6) and (5.7), the inertia torque F^ at actuator n becomes n [P H ] e n Â—v [ W 2 ] e 1 v Â— v o r [ W 2 ] ^P 11 ] e L v n Â—v (5.8) (5.9) The righthand side of Equation (5.9) is in the form of the Rayleigh quotient [50] which results in the solution
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149 n ; min n max (5.10) where X n are the eigenvalues of the symmetric matrix (l/2)[wj] ^[PÂ®] + [P^] T ] (5.11) Therefore, the inertia torque F 1 at actuator n for an end n effector velocity of magnitude is in the range n max (5.12) This result can be used to evaluate the maximum or mimimum value of actuator inertia torque for a specified magnitude of velocity, or to find the maximum velocity for prescribed joint torque limits. The latter case has been analyzed in detail in [6]. For the former case, the maximum magnitude of inertia torque at actuator n is given as Figures 5.6 5.10 show the maximum inertia torques for each joint due to the translational hand velocity of magnitude (v ) =59 in/sec of the industrial robot for a 11 max range of hand positions. The velocityrelated inertia torque is the most difficult torque term to conceptualize of those considered in this study. Notice that the n max
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150 to translational hand velocity to translational hand velocity ( v ) =60 in/sec = 60 in/sec
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zzi J 3ET 151 co o rH ^0 72 at id C0 03 09 /Â•s QJ ,C 3 *H cr rH 00 k to O 0 H o O Â•H CD vO K 03 HI CO X Â•H to Â— HI rH C H CO CD 0 Â•H J CM S 3) LD CD H 00 Â•H fH N 00 Â— I o Â»H co vO H> ON HÂ» Â•H _ G O Â•H o CD CM HI > 03 00 r"3 CO co o CD ,Â£ vO 3 c *H CT rH U G CO 'w' o 3 U a 03 CM rH I 00 m
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Units : 152 00 o MO (7\ 00 CM O vO 00
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153 e a e Â•H * X * X Figure 5.11 Configuration of the largest maximum Figure 5.12 Configuration of the smallest max inertia torque at the first three joints inertia torque of joint 1 for the for the translational hand velocity translational hand velocity
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154 K X! * X Figure 5.13 Configuration of the smallest maximum Figure 5.14 Configuration of the largest maximum inertia torque of joint 1 for the inertia torque of joint 1 for the translational hand velocity translational hand velocity
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155 c vÂ£> On W Â•K NJ e 3 & Â•H U o Â•41 c Â•H CN r*. * [S] Figure 5.15 Configuration of the smallest maximum Figure 5.16 Configuration of the largest max inertia torque of joints 2 and 3 for inertia torque of joints 2 and 3 the translational hand velocity the translational hand velocity
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156 smallest values of the maximum inertia torque occur near the middle of the effective workspace. This is good because this is where many of the robot's tasks are performed . Figure 5.6 shows the maximum inertia torques at the first joint due to the translational hand velocity. Positions A and D are the worst positions for the inertia load at this joint, since joint 1 sees large moments of inertia and has high joint velocity at these positions (see Figures 5.11 and 5.14). Positions B and C in Figure 5.6 are the positions where the inertia are well balanced for joint 1. Figures 5.12 and 5.13 plot the configurations of the industrial robot at these positions. Figures 5.7 and 5.8 show that joints 2 and 3 have quite similar contours of the inertia torque distribution in the reachable workspace. Figures 5.15 and 5.16 present the configurations of the industrial manipulator at the positions E and F in Figures 5.7 and 5.8. Note that joint 2 has the largest inertia throughout almost the entire working range. 5 . 4 Maximum Actuator Torques Due to Generic Hand Acceleration In a similar manner to that used in Section 5.3, assuming zero hand velocity (y_jj = 0), the maximum magnitude of inertia torque (F 1 ) at actuator n due to the maximum n max acceleration (A.,) in a given position can be found by Umax Â—
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157 Maximizing the actuator torque subject to 2 2 2 the condition A R = A H  2 < (A H ) max . The Lagrangian can be expressed as " F n + ,5 Â“ (A H ) max ) (5.14) where .5X is the Lagrangian multiplier. The necessary conditions for the maximum F^ give = [I H ] T + X[W 2 ] A = (i/X) [w 2 ] _ 1 [iV. d n , H [W a^ H " ^ A H ^ max ^ Â— H =
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158 max (Vmax U H i; ; ([ I H ] n; [ W 2]l 1/2 (5.18) Then, the maximum inertia load (F^) at actuator n due to 7 Ll 111 d A the acceleration of the hand can be determined as (F n 5 max [I 1 n; ^H^ax = (A ) ( [ I H ] [W^] _1 [I H ]^.) 1/2 (5.19) H max n; a n; Figures 5.17 5.21 present the maximum inertia torque results due to the translational hand acceleration of O magnitude 100 in/sec for the industrial robot. In order to have a better understanding of the results. Figure 5.22 displays the inertia matrix [I ] for several hand positions. The inertia torques necessary for accelerating the hand in the fixed coordinate system are given as F 1 = [I H ] A r (V r =0) (5.20) Figure 5.17 indicates that the worst areas of inertia load for joint 1 are in the top and bottom sections of the working space. The inertia matrices displayed in Figure 5.22 also confirm this effect. As an example of the position D, consider the value 169 in the first row (first joint), second column (Y* direction). This indicates that joint 1 must exert a torque of 169 inlb to accelerate the hand 1 in/sec in Y direction. In the region about the
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159 point B, the inertia torque of the arm at joint 1 attains its lowest values. As expected from the geometry of the * industrial robot, accelerating the hand in the Y direction requires more torque at joint 1 than accelerations in the other two directions. This can be clearly seen from the first three elements of first row of the inertia matrices [I H ] in Figure 5.22. The lowest values of the effective inertia torque about joint axis occur near the middle of the workspace. The inertia torques about joint 2 tend to increase inward and outward from this region, because the inertia torques increase as the hand moves in those directions. The U inertia matrices [I ] of positions E, F and G in Figure 5.22 show the second joint has minimum effective inertia at position F . The maximum inertia torque about the axis of joint 3 remains nearly constant throughout this range of hand positions. The largest values of the inertia are along the outside boundary of the working volume. For the industrial robot, the last three joints generally provide orienting of the hand. Therefore, the values of the maximum inertia torque for joints 4 and 5 due to the translational hand acceleration are low. The inertia torque at joint 6 has no effect due to the translational hand acceleration. This is because the mass centroid of the final link lies along the S, axis, such that
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160 Figure 5.17 Maximum inertia loads at joint 1 due Figure 5.18 Maximum inertia loads at joint 2 due to hand acceleration of the magnitude to hand acceleration of the magnitude (a ) =100 in/sec 2 (a ) =100 in/sec 2
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132 161 Figure 5.19 Maximum inertia loads at joint 3 due Figure 5.20 Maximum inertia loads at joint 4 due to hand acceleration of the magnitude to hand acceleration of the magnitude (a ) =100 in/sec^ (a ) =100 in/sec^
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Units 162 CM Figure 5.21 Maximum inertia loads at joint 5 due to hand acceleration of the magnitude (a ) =100 in/sec
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163 Figure 5.22 Effective inertia [I H ] for several hand positions in the reachable workspace
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164 accelerations of this centroid produce no moment about this axis. The first three components of sixth row of the [I ] matrices in Figure 5.22 are zero, also indicating that there is no effective inertia at joint 6 for translational motions. 5 . 5 Optimizing Actuator Compliance Distribution for A Known Load The compliance and deformation of a serial manipulator are distributed throughout its structure during operations. Increased compliance in one part of the chain can be compensated by increasing stiffness in another component. Consequently, optimization techniques are useful for rationally obtaining the best structural stiffness distribution. The modeling matrices of actuator stiffness and compliance in Section 3.4.3 form a convenient basis for understanding and computation for applying these design techniques . The deflection, ARÂ„ , at the end effector of a manipuH lator due to an applied load, F_^ , at the hand can be explicitly written as a linear combination of the individual joint compliances A i H = Â£ (5.21) T Â„ , Â„ r r.l is the vector of joint where C^ = [Ci> c 2Â» " ' > Â°N J compliances (diagonal elements of [C ,]). The component in
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165 row m and column n of the position and force dependent matrix [ JF ] is [JF1 = [J] ( [ J ] FÂ„) L J m;n 1 J m;n 1 J ;nÂ— ( 5 . 22 ) where fJl is the element of the Jacobian [J] in row m L m ; n and column n, and [J]. n represents column n of the Jacobian . Now it is possible to synthesize the individual joint compliances C^. The manipulator must satisfy a set of performance criteria which could be represented by limits on the magnitude of the hand deflection. However, there are real costs associated with decreasing the joint compliances, such as the need for larger actuators (more weight), increased gear reduction (more friction and backlash), more expensive matrials and components, etc. Therefore, a balanced design formulation can be of the form: For a given manipulator configuration maximize the compliance magnitude II C II subject to the Â— tp p hand deflection constraint  ARgf Â£ ^axÂ’ where AR is the magnitude of the maximum max allowable deflection at the end effector. Because of its physical significance and convenience, the quadratic norm (P =2) is useful for the hand deflection.
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Squaring this norm, the deflection constraint is represented as a quadratic function of the component compliances: Â„ II ? = cT [A] C, < AR 2 H "2 Â— Â— Â— m max (5.23) where [A($, F h , W r )] = [JF] T [W 2 ] [JF] (5.24) Here [ ] is a 6x6 diagonal matrix of weighting factors W_ r that represent the relative significance of the different components of the hand deflection to a given operation. The deflection is formulated in terms of compliances rather than stiffnesses because flexibilities in a serial structure are a linear combination (see Equation (5.21)). However, maximizing the pnorm of joint flexibilities with constraints on the total deflection produces an undesirable solution the resulting distribution often contains compliances which are zero. A simple two degrees of freedom manipulator in Figure 5.23 illustrates this effect. Table 5.1 lists the corresponding kinematic data, the applied force and the weighting matrices. Figure 5.24 shows that this result is physically impossible since no actuator can be infinitely stiff. This situation can be remedied by bounding the component stiffness, in which case the associated stiffness will be at their upper limit in the optimal distribution (see Figure 5.25).
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167 Figure 5.23 A two degree of freedom manipulator Table 5.2 The kinematic parameters, applied load and weighting matrices of a system with two flexible joints a). Kinematic parameters and joint angles Joint a.. a., S.. Joint angles 3 k jk 33 No. (deg) (in) (in) (deg) 1 0 20 0 0 2 15 0 90 b) . Maximum allowable deflection at hand AR = 0.001 in max c) . Applied load F u = [100 0 0 0 0 0] T lb d) . Weighting matrices [W A ] X tv 1 2 x 2 6x6
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1x10 168 00 CN
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169 An alternate approach is to minimize a pnorm of the flexibilities with p c <0 which corresponds to the minimization of the norm of the component stiffness. In this case, the optimal distribution will never contain components which are infinitely stiff. For example, minimizing a linear combination (p c =_ l) Â°f the stiffnesses subject to the quadratic constraint (p^=2) on the hand deflection leads to a nonlinear set of equations which must be satisfied by the optimal compliance distribution. A nonlinear optimization technique or graphical procedure could be used to solve these equations. Another useful formulation is the p = Â“ form which minimizes the maximum c stiffness (or maximize the smallest compliance). The problem can be formulated as For a given manipulator configuration maximize the smallest (weighted) compliance subject to the constraint Â“h ! A Bh >2 [ Â“r 2 ' fi H < Â“max for a known load F^^ . Often, the optimal solution for this problem will occur when the weighted values of all component flexibilities are equal (see Figure 5.26). The magnitude of the component stiffnesses are determined by the constraint on the deflection AR .
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170 This solution is optimal if 3 1 AR  0 / 3 C >0 for all Â— ri z n Â— components C n * However, if one or more of the derivatives 3 II 2 /3C m are negative, then the stiffness can be further decreased by replacing the constraint W C =W C with J r Â° m m n n 3 II Ar  _ / 3 C =0. This means that for all component compliances which are not equal to smallest weighted flexibility, the gradient of  Ar [ ^ (considered as a function of C^) must be normal to these component axes. This formulation for the optimal compliance distribution (p c = <Â», p r = 2) is equivalent to the general quadratic programming problem with linear inequality constraints. The solution procedure mentioned above involves a slight modification of the simplex algorithm for linear programming [20]. Figure 5.27 illustrates this alternate situation for a simple system composed of two compliant components. Other formulations for compliance distribution lead to the same type of problem and similar solution procedures (i.e., p c =l, p r =l). In addition for this implied comparative review of these various formulations, a major issue to be addressed is the selection of the values for weighting factors. These weighting factors must be determined by physical design considerations of the system. For example, W should be greater than W Â£ to reflect the fact that increasing the stiffness of the distal actuators (i.e., by using larger prime movers) is more costly in terms of
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1x10 ' 1x10 171 Figure 5.26 Optimal distribution has equal (weighted) Figure 5.27 Optimal distribution of joint compliance component compliances has 9  AR  2/3C^=0
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172 dynamic performance (larger inertia loads) than increasing the stiffness for the actuators at the base of the arm. The optimal actuator compliance distribution of the example industrial robot for a known load on the hand is determined in the following analysis. A 150 lb force in the X* direction is applied to the hand reference point and the maximum allowable magnitude of the hand deflection is 0.005 inches. The values p c = Â°Â° and p r = 2 a re selected here in terms of a quadratic programming algorithm for solving for the compliance distribution. Figures 5.28 5.33 show the optimal compliance distributions of all actuators for a plane of hand positions in the manipulator's reachable space. The contours of constant value show the minimum joint stiffnesses such that the prescribed precision is attained. Figures 5.28 5.30 indicate that large stiffness values are required for joints 2 and 3 when the hand is in the top or bottom sections of the effective workspace. This is because the moment arms for the applied load increase as the hand moves further away from the horizontal planes containing the axes and . Figure 5.29 shows that the magnitude of the Â•k stiffness is minimum in the area near the Z =32 inches, since the joint 2 is constrained to pass through the point with coordinates (0 0 32) inches. As mentioned above, link k S ^ is nearly horizontal for hand positions in which is
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173 igure 5.28 Stiffness of joint 1 with 150 lb force Figure 5.29 Stiffness of joint 2 with 150 lb force in X x direction and the maximum magnitude in X' v direction and the maximum magniof the hand deflection .005 in. tude of the hand deflection .005 in.
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xlO inlb/rad 174 Figure 5.30 Stiffnesg of joint 3 with 150 lb force Figure 5.31 Stiffnes^ of joint 4 with 150 lb force in the X* direction and the maximum in the X direction and the maximum magnitude of the hand deflection 0.005 magnitude of the hand deflection 0.005
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175 Figure 5.32 Stiffness of joint 5 with 150 lb force Figure 5.33 Stiffness of joint 6 with 150 lh force in the X* direction and the maximum in the X direction and the maximum magnitude of the hand deflection 0.005 magnitude of the hand deflection 0.005
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176 about 72 inches. This leads to minimum required stiffness * for joint 3 due to applied forces in the X direction. 5 . 6 Optimizing Actuator Compliance Distribution for Generic Loads Since manipulators are designed for flexibility of applications, they must be capable of supporting a variety of applied loads while maintaining positional accuracy. Hence, the stiffness design task might best be formulated as For a given manipulator configuration find the optimal compliance distribution such that the norm of the deflection AR^ produced by any applied load of magnitude less than ( F n^max will be less than AR max (F ) is the largest magnitude of external load that can H'max be applied to the end effector in all directions. Generally, the quadratic norm (p^=2) is desirable for the load magnitude since there is no directional preference associated with this norm. Using p f = 2 for the deflection magnitude also leads to a complex set of nonlinear equations for the optimal compliance distribution. These equations can again be solved by numerical techniques. Alternatively, the value 00 or 1 could be used for p r to reduce the
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177 solution complexity. Here, P r =l is selected since it produces more conservative results (see Figure 2.2). Optimizing the compliance with P c = Â°Â° as in the previous section, the stiffness distribution problem can be stated as For a given manipulator configuration minimize (maximize) the maximum stiffness (smallest compliance) component 1/W^C^ (or W^C^) subject to the deflection constraint I Ar h  1 . a 1 [W r ] Ar h < AR max for all applied loads F_^ satisfying S Hh* I Â‘ la tw] f h < (F H )^ ax . Here, [W r ] and [W f ] are diagonal matrices of weighting factors for components of ARÂ„ and FÂ„. Each component (0.=1 Â— rl Â— n J or aj=l) of a is chosen such that a^AR^M). Assuming o_ is known, the deflection magnitude is a bilinear function of FÂ„ and C, of the form Â— H Â— (p AR H" 1 = F h [JR] C
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178 The Lagrangian for the constrained optimization problem can be formulated as L II ii^ll _oo + K 1 II II 1 K 2 II Â— H I! 2 (5.27) where K 1 and are La S ratl g e multipliers. The optimality condition on _F^ is 3L 3 II A R 1 2 = 0 = Kj Â— + 2 k 2 [Wj] F r (5.28) Substituting Equation (5.25) into Equation (5.28) leads to the result that is a linear function of I H = Y [ ] ~ 1 [JR] (5.29) where Y = K ^/2i<2 is an unknown scalar multiplier. The value for Y can be found by substituting Equation (5.29) into the load constraint yielding (F H^max = H [W f^ H = Y 2 [ JR ] T [W^]1 [JR] (5.30) Similarly, the constraint on the deflection, Equation (5.25) , becomes Ar I = ycj [JR] T [ W J ] Â“ 1 [JR] C, < AR Â— H 11 1 Â— (p f J Â—
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179 Squaring this equation and dividing by Equation (5.30) leads to the quadratic compliance constraint equation as cl [B] C, < (AR 2 / ( FÂ„ ) 2 ) (5.32) Â— 1 J Â— Â— v max v H'max where [B] = [JR] T [W 2 ] 1 [JR] (5.33) As the Equations (5.23) and (5.24) in the previous section, this formulation leads to a quadratic programming problem. The same solution procedure can be employed, but with the additional constraint that 0_.AR^>J3 for all deflection components. The stiffness distribution problem can be rewritten as For a given manipulator configuration maximize the smallest compliance component subject to the quadratic compliance constraint d [B] C, < (AR 2 /(FÂ„) 2 ). Â— 1 J Â— 0. As mentioned before, the selection of the appropriate pnorms for the vector quantities leads to a quadratic programming problem as described in Section 5.5. The same
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180 solution procedure can be employed to obtain the joint compliances necessary to keep the hand deflection less than 0.005 inches for applied loads up to 150 lb in magnitude. Figure 5.34 shows the contours of the optimal compliance magnitude II C^] _
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Units : 181 CO o vO av co 04 o vO 00 OÂ’ * co X H 03 3Â£ Â§ * a u 1 I . 1 ,  ,  i 1 1 i 1 1 U_ i_L_ u_jl 04 Â•H X 04 o CO vO
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182 In this section the same analysis procedure is employed to obtain the natural frequencies of the optimal distribution of actuator stiffness which presented in the previous section for the industrial manipulator. The joint stiffnesses necessary to keep the end effector deflection less than 0.005 inches for applied forces up to 150 lb in magnitude are K^=[6.4 6.4 4.48 1.92 1.28 .64]xl0 inlb/ rad . Figures 5.35 and 5.36 show that the distributions of the first two natural frequencies of the industrial robot * * with the improved stiffness distribution for the X Z plane of hand locations. Applying the optimal stiffness distribution criteria to the industrial robot significantly improves the precision and increases the lowest natural frequency of the system. This improved distribution of joint stiffness increases the fundamental natural frequency by a factor about ten.
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183 * e M *H 00 O vO CTn vO CO * X < CM O CM iÂ— l I X *H 00 O vO ON < 1 * CO CM O vO 00
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CHAPTER 6 OPTIMIZATION OF ACTUATOR PARAMETER DISTRIBUTIONS FOR DESIGN OF ROBOTIC MANIPULATOR 6 . 1 Introduction The previous chapter has developed design criteria and computational design tools for optimizing the distribution of actuator parameters for a given manipulator configuration. The computational methods and the local optimization criteria based on the rigid link, flexible joint model prove useful for design because they condense the information about actuator parameters by imposing motion and load magnitude constraints at the end effector for a given configuration. Computer generated plots for compactly dis playing extensive dynamic analysis results and computer graphics for illustrating robot motion have been employed to help designers select rational design specifications. The optimization schemes also expand the designer's understanding of the geometric and dynamic aspects of systems. The model formulation directly indicates the position dependent nature of the results augumented by the contour plots which show how system strength and stiffness characteristics vary a$ a function of hand position and arm 184
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185 configuration. These local design criteria in Chapter 5 are only of limited use to the designer, since there are so many configurations of a minipulator in its workspace to consider. Therefore, the localized techniques are extended to develop global design criteria for general robotic manipulators in this chapter. These global evaluations can be established by including the position vector ^ as a variable in the optimization formulations. Since these formulations are position dependent and nonlinear, general nonlinear optimization algorithms are required to find the "worstcase" hand positions and arm configurations. These global design procedures make it possible to satisfy load, dynamic state and precision requirements in the workspace and provide faster operation by increasing the frequencies of the fundamental vibration modes. It is believed that the design criteria for actuator parameters represent a significant step forward in the stateoftheart of design of robotic manipulators. The pattern search method of Powell and the exterior penalty method presented in Chapter 2 are employed for the examples of this chapter. The Powell's method is a direct, sequential stepping technique. This method is quite effective in solving optimization problems. It should be cautioned, however, that none of the optimization methods can discriminate between a local and a global minimum. Therefore, it is necessary to start the computational
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186 process at several initial estimates to determine the global minimum. 6 . 2 Distribution of Actuator Sizing Due to Static Loads Actuators are the muscles of a manipulator. They are also the primary source of the system strength and flexibility constraints. An essential initial design step involves how to determine the actuator static load capacity of each joint for a given load handling capacity of the device. The load handling capacity is considered here to be the largest load which can be applied to the hand of a robotic manipulator in any direction. This load capacity is directly related to size distribution of actuators that drive the system. In order to assure that there is no actuator overload due to an applied load with magnitude less than the system load handling capacity during operation, the maximum load handling actuator capacity of each joint is the basic specification we should consider. The maximum actuator capacities are generally independent of each other and each of the actuators should be investigated separately. For simplicity, the first case to be considered requires that we neglect the gravity loads. The maximum magnitude of load on joint n due to the externally applied loads at the hand for a configuration is derived in
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187 Equation (5.2). A formal statement for obtaining the global maximum actuator capacity can be stated as For an N degree of freedom manipulator, find the maximum actuator load capacity for each joint, (F^) , subject to the load constraints II H II 2 1 ( f H^max and HÂ— N ( N+l ) H 2 ^ m N ( N+l ) ^ max and the joint angle constraints ^^min Â± ( ^ i } max Â’ 1 = ^ 2 Â» N * Figures 6.1 6.4 show the contours of the maximum static torques at joints due to the applied loads, =150 lb and (m^.,) =2000 inlb, at the hand in the work67'max space. The outermost contours indicate the workspace in the X Z plane. These plots are based on data from discrete points, which leads to the rough edges for the working area. The actual workspace of the industrial robot has been shown in Figure 4.3. Since the industrial robot is fixed with the axis of the first joint pointing vertically upward in the direction of the Z axis and the origin of the second joint at the point (0 0 32) inches, the contour plots of the maximum static torques at these joints displayed in Figures 6.1 and 6.2 are similar to Figures 5.2 and 5.3 where the last link is held as S^ = (l 0 0) T and Â£ 67 =(0 1 0) T . Figure 6.1 shows
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188 the first joint load is not affected by the Z direction loads. The contours of the maximum static torques are circular arcs about the origin of joint 2, because the axis If S 2 is parallel to the Y direction as shown in Figure 6.2. The shaded regions of Figures 6.3 and 6.4 represent the largest values of the maximum static loads which can be reached when the hand position is in these areas. The largest torque values are about 11150 inÂ— lb and 2900 inlb for joints 3 and 4, respectively. Figures 6.5 and 6.8 display the two worst configurations for joint 3 at the hand positions which are not in the shaded area of Figure 6.3. Figures 6.6 and 6.7 are also the worst configurations for joint 3 with the same magnitudes of loads on the hand which is in the shaded region. Some of the worst configurations for joint 4 are displayed in Figures 6.9 6.12. As in Section 5.2, the maximum static torques of joints 5 and 6 are constants for all the positions in the workspace. The maximum static torques due to the external loads, (f H ) max = 150 lb and ( m 67>max =2000 in_lb Â’ for a11 the joints of the example industrial robot are listed in Table 6 . 1 . The following investigation of the actuator static loads will include the effects of the gravity forces from the weight of the arm and the applied loads at the hand. The limits of the actuator loads for supporting the static loads depend, in general, on the position and configuration
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189 Â•o
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190 o rn <1) rH 3 I "3 3 Â•H o u o 3 o Â•H CN O II 03 CO 0) 3 cr U 03 X 03 2 vÂ£> 6 II a Â•H X 03 <41 2 w I M > u I o Â£ X 03 0) an N 4J 33 3 *H a) h 3 00 Â•H Pn o 4J rQ 0) rH 3 1 T3 3 Â•H cn O 4J O 3 o Â•H CN o II Â•rj X 3 U 2 03 /*\ CO vO a) 3 3 cr X) o 3 h 3 a) o o 03 Â•H Â« Â— i Pn CO 03 O M 4J lO P CO rH O II Â£ 2 X 3 o3 a) 2 ac *H Â“N H X 32 C3 Ml 3 s *H CO Â• a; i( 3 Â•H a
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191 robot for joint 3 due to (f ) =150 lb robot for joint 3 due to (f ) =150 lb and (m, n ) =2000 inlb witn â€¢He hand and (m,7 ) =2000 inlb witn Â’tfie hand at (lÂ§ 7 O m ?^6) in. at (96 0 60^ in.
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192 robot for joint 3 due to (f ) =150 lb robot for joint 3 due to (f ) max =150 lb and (n^^ax 312000 with tfie hand and ( m 67) max = 2000 inlb with the hand at (30 6 43) in. at (18 0 0) in.
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193 vO * X tH T) 3 o c Â•H m co u 1 Â— 1 rC u II X x aj 3 co,n Eu c , N Â•H K.C 4J U aj w H c S h o Â•U X3 4) 1 Â— 1 O 01 1 3 C 3 X) H o Â•H
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4z 194 robot for joint 4 due to ( f H) ma x =150 lb robot for joint 4 due to (f H ) max =150 lb and (m 67 ) ma =2000 inlb with the hand and (m 67 ) max =2000 inlb with the hand at (36 0 72) in. at (66 0 0) in.
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195 Table 6.1 Maximum actuator static torques due to (fÂ„) =150 lb and (m, 7 ) =2000 inlb v H ; max v 67'max Joint 1 2 3 4 5 6 Maximum static torque 16866 inlb 17250 11150 2900 2787 2000 (see Section 4.4) of the manipulator. The maximum actuator torque of each joint due to the static loads for a manipulator configuration can be obtained from Equations (3.40) and ( 5 . 2 ) as < F i;>maxS n X (HR n )  + (Â«Â„ ( N + 1 ) > max + (6 ' l) where f. is the gravity force of link jk and [G. is the j c J nth component of the firstorder influence coefficients for the centroid of link jk. The optimization problem of actuator static sizing can be formulated as follows:
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196 For an N degree of freedom manipulator, find the maximum actuator load capacity for each joint n subject to the gravity forces, the external load constraints II 1^2 ^H^max and H Â— N ( N+1 ) H 2 (m N(N+l) ) max H ' max and the joint angle constraints < *1 < <*i> i 'max * Consider the magnitudes of the applied loads at the hand as described in the previous example and the gravity loads at the centroids of all links. Table 6.2 lists the worst joint torque on each input for the industrial robot. The effective gravity loads are generally significant at all joints except joint 1 and 6. For the second joint, the gravity loads have a very significant effect on this joint. This is because joint 2 sees the largest effective arm massand must support the full weight of the arm. Therefore, a manipulator with a geometric configuration like that of the industrial robot, joint 2 should generally have the largest load capacity. Figures 6.13 ~ 6.17 display the worst configurations for each joint.
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197 N rH CO 03 O H3 Â•H m o3 Ji ri O 4Â» II tH CO X P 03 64J p /''"N Â«H Â•H 33 > 41 03 a) U X 00 4J O 4Â» T3 41 g O (1) 03 3 G 'O rO o rH Â•H CN 1 u G o3 4J *H U G P Â•H O 00 O O Â•H *r) O 41 CN G H II O O X u 44 03 e u U /~s CO O r. u o O 6 & Ji ^ 3 Â•H 35 > 41 CO O U 13 41 3 O
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198 * x rÂ£3 rH (0 c d O X) Â•H m cd rH O II rH X X 3 03 >> T3 Eh 3 /Â—s *H Â•H Â£C> 4i 03 o o a 13 U ^ VO rH VD 0) 3 00 Â•H pH K X #* tH tH CO 03 O HJ Â•H m cd H rH O 4J II rH CO X 3 cd T3 EH 3 /Â— s *H Â•H PC > 41 03 0) U rC 00 u O H T3 41 3 O CD 03 3 3 T3 rQ o rH Â•H cn I H 3 03 4J *H 3 Â•H O 00 o o Â•H Â•r> O 41 CM 3 U  o o X a 41 cd E H H CO o u rH vÂ£> o O E in rH Â• vD cd H 3 00 Â•H Ph
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199 Top view Side view Figure 6.17 Worst due to loads configuration of (f ) =150 lb, H max the industrial robot for joint 4 (m,.,) =2000 inlb and gravity 6/ max
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200 o X 03 s /*Â“N ps 41 O P Q) 3 T3 e Â•H O Â•rn P O 41 O o p 03 W Â•H oj P 03 P O CO tH 3 T3 >s C 4J Â•H *H > a; oj & p P CxO 41 OD O C 03 CO C rP O tH Â•H I P c 03 *rl P 3 O tuO O Â•H O P CN e ii o x U 03 & P XN co P VO o e ^ w 00 tH vÂ£>
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201 Table 6.2 Maximum static torques due to (f,,) =150 lb (m,,) =2000 inlb and gravity Ioats v 67 'max Joint Maximum static torque 1 16866 inlb 2 113299 3 31615 4 3020 5 2892 6 2000 6 . 3 Distribution of Actuator Sizing Due to Inertial Loads Sections 5.3 and 5.4 considered the maximum actuator torques due to system inertia content with the specified magnitudes of the hand velocity and acceleration A^ for a specific configuration. The actuator torques obtained in this manner assure that if the magnitudes of the arm velocity and acceleration are less than these prescribed values of the maximum hand velocity and acceleration, none of the joints will be overloaded, regardless of the path direction for the specific configuration. Although this represents a significant step to condense the dynamic information for a manipulator configuration, very many configurations must be investigated to give the designer comprehensive knowledge of the load and motion relationships for a specific arm. This section employs the nonlinear optimization methods to develop the global design criteria and further
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202 condense the dynamic information for general robotic manipulators. These design criteria will force the actuators to satisfy the system strength and dynamic respons requirements. They then will be useful throughout the workspace of the manipulator structure. These optimization computational methods are analogous to those employed for finding the actuator static load capacity in the previous section. The actuator inertia torque can be split into its velocityrelated and accelerationrelated components as shown in Equation (4.5). First consider the hand velocity constraints with the hand acceleration equal to zero. Section 5.3 discusses the nature of the velocityrelated inertia loads in detail. The velocityrelated torques are generally very sigificant at all joints. This is because the velocityrelated inertia torques are proportional to the hand velocity squared. Therefore the torque component rapidly becomes important as manipulator speeds increase. Recall that the maximum magnitude of velocityrelated inertia torque at joint n with a specified hand speed for a given configuration is (see Section 5.3) (r) n max max ( (X n max ) ( 6 . 2 ) where (VÂ„) H ' max is the maximum magnitude of hand velocity and X n are the eigenvalues of the symmetric matrix d/2) [W^ ] 1 [ [PH] + [PH] T ] (6.3)
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203 Therefore, the optimization criterion for obtaining the maximum required strength of inertia torque F X at each joint regardless of the direction of the hand velocity V_^ in the reachable workspace can be stated as Maximize the magnitude of the inertia torque F X for each joint subject to the hand n velocity constraint V H 1Ih1 2 Â„ax and the joint angle constraints "Mmin <*1< <*iW 1 * ' 2 Â’ Â•Â•Â•Â• N ' The velocityrelated inertia torque is the most difficult actuator torque term to conceptualize of those considered in this report. By applying the criterion to the example industrial robot, Table 6.3 lists the maximum inertia torque for each joint due to the translational hand velocity V =60 in/sec. Figure 6.19 6.24 show the worst configurations of the inertia torque due to the hand velocity V H It is interesting to note that the last link of the industrial robot is folded back in these configurations. This is because the last link has smallest mass and the inertia torque is a quadratic form of the hand velocity. Decreasing the distance between the hand and the axis of each joint, the velocityrelated inertia torque at each joint can be significantly increased.
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204 i co a cO G G Q) Â•H g 4H a) >> o G 4J "O H g a o CM o Â•H rH JJ h u Â•H G O X) GO *n G Â•H c0 41 H 4C G O O 4) rH O CO > 41 u Â•rt G o GO Â•rH G *H CO 4H H rG G o 0 41 rH CJ CO cu G Â•Â— * G O CO cr *h H H U o O CO 3 4J rH cr rH vÂ£> Q) U G GO Â•H
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205 i CO cd G Â•H cd 41 U 4J (D G CD Â•H 4J CD C o 4J u 44 CD O G X) G O Â•H 4J 4J cd G Â•H G O GO T) Â•H 44 J4 G O o 44 a CD u G CO o< o O 4J cs Â• vO
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Joint angles Joint angles f 0.000 'I a * ( 0.000 'j 206 00 OJ X vÂ£> O Â«H O O co o cn m o ^oo^oo rH O cn vo cn rH o O I "XÂ” el CN A* X CO a cd g a) Â•H cd CO +J u M H c aj Â•H c a) Â•H r 1 o zÂ» vÂ£5 a) II rC c PC 4J H > 41 0J X O G 4J X Â•H g O o vO O Â•H Â«H 4J H a) cd c > M Â•H O X) 00 Â•rj g Â•H cd 4H H rC G O O 41 rH a cd u P to Â•H pH I CO o cd G Â•H cd co U H \ u H G 0) Â•H c 0) Â•H G O U vÂ£5 OJ II fl O PC H H > 41 0) X O G H X *H G O O m o Â•H rH H H 0) Cd G > H Â•H G O X to Â•r) G Â•H Cd 41 M rG G O O 41 rH o cd QJ G H P o CO cr *h H H 4J o o cd & U rH co CN vD QJ U P to Â•H pH
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207 Table 6.3 Maximum inertia torques due to the translational hand velocity Vjj = 60 in/sec Joint 1 2 3 4 5 6 Maximum inertia torque 3.571xl0 6 inlb 3 . 109xl0 7 2.880xl0 6 7 . 189xl0 6 5 . 968x10"* 2.636xl0 3 Similarly, for a specified acceleration magnitude with the hand velocity equal to zero are considered. The maximum inertia torque at actuator n due to the specified magnitude of hand acceleration for a manipulator configuration is given by Equation (5.19). Therefore, the global optimization problem of obtaining the necessary inertia torque for accelerating the hand in its workspace can be formulated as Maximize the magnitude of inertia torque F ^ for each joint subject to the hand acceleration constraint T 9 1/2 AÂ„ II Ah 1 2 Â‘ * ( VÂ».x and the joint angle constraints (Â‘Mmin < ^i 1 (Â‘MmaxÂ’ i = 2 Â» Â•Â•Â•Â» N Â‘
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208 Table 6.4 presents the maximum torque for each joint 2 due to the translational hand acceleration A^=100 in/sec for the industrial robot. Figure 6.25 6.29 display the worst configurations of the inertia torque for each joint due to the hand acceleration and the corresponding inertia matrix [ I 11 ] . Table 6.4 Maximum inertia torques due to the translational hand acceleration A^=100 in/sec Joint Maximum inertia torque 1 9.818xl0 4 inlb 2 3.539xl0 4 3 5 . 07 4x 1 0 3 4 1 . 0 14x 10 3 5 8 . 8 OOx 1 0 1 6 0 Since the actuators of the manipulator must be designed to support applied loads and provide sufficient driving torque to overcome the inertia loads induced by the prescribed endeffector velocity and acceleration A^ during its operations. The remainder of this section will derive the criterion for the maximum inertia torque which include the velocityrelated term and the accelerationrelated term. This criteron for actuator sizing will be developed for all positions of the manipulator. The inertia modeling matrices are position dependent terms (see
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Hand position 209 V c 00 CO o VO 00 CO G5 rH CN m m CO CN CN o m CN
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Hand position ( 97.59 .0 210 c0 Â£ o Â•H rÂ— i U& CO M CO Â£ c0 U d) & Â£ o 4 J d) P H 3 U U d) Â£ Â•H 60 04 c Â•H U T 3 Â£ Â£ Â•H O O o< Â•rÂ“) CO
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Hand position (96.16 .0 211 Figure 6.27 Worst configuration of the inertia torque for joint 3 due to the translational hand acceleration A^=100 in/sec^ and the corresponding inertia matrix [I ]
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Hand position ( 86.55 .0 212 7 CO r>. CO ON CO 04 ' orH rH rH 04 04 ON i CO  04 1 nO 1 ON 00 NO rH in O 00 O NO uO 00 o 04 T3 uO ON rH o * M * X cd g o Â•H i Â— i WPS cd H rH iÂ— i CO C X cd Â•H u u 4J cd 0) 6 rC 41 cd Â•H o 44 4Â» H 04 04 G 3 Â•H TJ 00 G Â•H 4J c G Â•H O 0 Cu Â•ro CO (14 U O H MH O U 0) 3 (14 cr .e H 4> O 44 no G cd cd Â•H 4104 U O cu 04 G CO Â•H G
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Hand position ( 85.42 .7 213 c Â•H m cn co CO 00 NT rH CO c X CN 1 00 1 6 Â• 3
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214 (4.6) and (4.7)) which relate the motion states, V R and A R , of the device to the corresponding generalized inertia forces F 1 at the actuators. The global evaluation of the n maximum inertia torques can be established by including the position vector as a variable in the optimization formulation . The maximum inertia load (F^;) at actuator n due to n max the hand velocity and acceleration for a given configuration can be obtained from Equations (5.13) and (5.19) as i (F ) v n max The objective of this section is to determine the largest values of (F ) in order to assure that the Â° n max actuators of the device have the sufficient driving torques for all positions and configurations in the workspace. The optimization problem can be formulated as Maximize the magnitude of inertia torque for each joint subject to the hand acceleration constraint A H = ^ 2 = ^ Â— H ^ W a ^ Â— H ^ ^ A H ^ max Â’ the hand velocity constraint V H V H  2 v 2 1V h ) 1/2 < (V H ) max and the joint angle constraints ^i^min Â— Â— ^i^maxÂ’ = ( V H } max max((X n ) minÂ’ (X n ) max ) +
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215 6 . 4 Distribution of Actuator Compliance for Generic Loads Generally, it is desireable to increase the operating speed of the manipulator. Precision of operation will also become increasingly important as development proceeds. For precision control of a manipulator, a complete dynamic formulation is needed to predict and compensate for the system deformations under loads. A formulation of forceÂ— deformation relations to determine the joint compliance (or stiffness) has been established in Section 3.4.3. Based on this formulation, some local optimization criteria for distribution of actuator compliance are presented in Sections 5.5 and 5.6. The basic idea of these local design criteria is to determine a set of joint compliances which satisfy the precision requirements under a known magnitude of load at the hand with increasing magnitude of these compliances. Because there are generally costs associated decreasing the joint compliance. In order to design a system for supporting a variety of applied loads and maintaining positional accuracy at the hand for all configurations in the workspace, the stiffness design task in Section 5.6 can be restated as Find the optimal compliance distribution such that the norm of the deflection ARjj produced by any applied load magnitude less than (F H )max wil1 be leSS than AR maxÂ‘
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216 Apply the local compliance criterion presented in Section 5.6 to the example industrial robot for a range of hand positions. Figure 5.34 shows the maximum compliance magnitude II in the X Z plane with the hand orientation as 1 6 = (1 0 0) T and a. 67 =(0 1 0) T . The magnitude  .C^fl^ k decreases rapidly as the hand moves outward from the Z axis. This plot also indicates that the maximum compliance magnitude, C (4>)= lcj , is a position dependent function. In order to enhance the overall stiffness of the structure and to guarantee a specified level of precision under a known magnitude of load, the smallest value of the maximum compliance magnitude is the basic specification we should consider. From the global point of view, the design task can be formulated as: Minimize the maximum compliance magnitude function, C ( d> ) , subject to the joint max Â— angle constraints < h < 1 ' 2 "Â• Where C (
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217 Based on this global optimization criterion, the same industrial robot is used to investigate the optimal distribution actuator compliance to keep the endeffector deflection less than 0.005 inches for applied forces up to 150 lb in magnitude. Figure 6.30 shows the worst configuration of the required joint stiffness for the example robot. Table 6.6 Optimal distribution of joint compliance for hand deflection less than 0.005 in. and applied loads 150 lb in magnitude Joint 1 2 3 4 5 6 Joint compliance 1.453x10 ^ rad/inlb 1 .453xl0 _ 9 2.07 5xl0 9 4.842xl0 _9 7 . 263xl0 9 1 .453xl0' 8 Now it becomes useful to applythese joint stiffnesses inthe natural frequency analysis of the industrial robot. * * The configurations investigated consist of an X Z plane of the locations of the hand reference point H with the * hand pointing outward from the Z axis in a single T T orientation defined by S^=(l 0 0) and = 1 0) * Figure 6.31 displays the distribution of the fundamental natural frequency and that the lowest first natural frequencies are along the outer boundary of the workspace at about 64 Hz .
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Joint angles 218 0 0
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Units: Hz 219 H Figure 6.31 First natural frequency distribution of the industrial robot with the joint stiffnesses K^=[6.89 6.89 4.82 2.07 1.38 0.69] x 10Â® inlb/rad
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CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS t 7 . 1 Conclusions A systematic and unified approach for analyzing and distributing the actuator parameters of robotic manipulators has been presented. The design criteria and computational methods developed in this work can greatly expand the designer's understanding and reduce the design cycle time for accurate, reliable manipulators. The techniques developed in this effort can also lead to improved manipulator control algori thorns which consider how the effective stiffness, strength and speed characteristics vary through out the workspace, and make adjustment of the distribution of system parameters to enhance controllability, especially when precision control under external disturbance is required . Since the actuator /drive components are the primary source of system flexibility and strength constraints, the rigidlink manipulator model is employed in this study. The two principal objectives are: 220
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221 1. Develop computerassisted methods for characterizing the effective strength, stiffness, natural frequency and deflection of the overall system as functions of hand position and individual properties . 2. Develop design criteria and computational methods for optimizing actuator strength and stiffness distributions in general robotic manipulators in order to meet static load, inertia load and precision requirements and to allow faster operation by increasing the frequencies of the fundamental vibration modes. In order to provide enhanced system design and control over the design process, computer generated plots for compactly displaying extensive dynamic analysis information and computer graphics for illustrating robot motion have been used to help designers select rational design specifications. These would give the designer the powerful tools to compute and interactively display the effects of changes in the design parameters on these key criteria, and to obtain a more balanced distribution. Hopefully, the design and computational methodology introduced herein will provide the impetus for future efforts towards manipulator design and optimization.
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222 7 . 2 Recommendations for Future Research It is believed that the material in this study represents a significant step forward in the design of robotic manipulators. Yet it must be recognized that all the research to date in this area represents only a small portion of what can potentially be accomplished. There are several logical extensions of the presented work. This section should by no means be considered as a complete listing of the unsolved manipulator optimization problems. Rather, it is a brief summary of a few of the more obvious problems which would follow as an extension and an improvement of this work. As has been previously mentioned in Chapter 1, there is a large number of system design parameters associated with a general manipulator. These parameters need to be determined in order to meet given criteria. One of the most important and most difficult topics, which remains to be studied, is how to distribute the mass parameters in a appropriate manner, since massive manipulators and increased speeds of operation make the inertia terms even more important in control and design of modern manipulators. Distribution of mass parameters for a general six degree of freedom robot is exceedingly difficult; there are some sixty parameters for such a system. Synthesis processes are required to reduce the size of mass parameters before the optimization procedures can be implemented. In
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223 general, the effective inertia loads acting at the joints can be minimized by an improved distribution of mass parameters. For example, applying the cubic power law to the distribution of mass in a simple threelink planar arm increases the fundamental frequency by a factor of three over the value for an arm with the same overall mass and joint stiffness but a flat mass distribution. To date, very little consideration has been given to develop design tools for the geometry of robotic manipulators. There are three parameters associated with the position of a joint axis relative to a neighboring joint axis in a manipulator. Therefore, a general six degree of freedom manipulator will contain 18 geometric design parameters which control the reach, workspace, dexterity and obstacle avoidance capability. Consequently, a major optimization results. No one clearly understands how to rationally select these design parameters for better kinematic capacibilities of the system, but some of the general objectives for design would be to determine the best set of parameters in order to 1. enhance dexterity, 2. expand operating space, 3. reduce the possibility of unreachable zones, 4. reduce the singular configurations.
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224 Most existing robotic manipulators are serial in structure. One of the pressing problems facing the designer in robotics is the high level of deformation in serial devices. Parallel structure is generally stronger since loads are distributed throughout the system subchains while serial structure requires each link to support all loads from that link out to the free end of the arm. The Florida Shoulder, which is a typical parallel actuated device, is illustrated in Figure 7.1. Hybrid Structures, containing both parallel and serial components as shown in Figure 7.2, have a great promise for high precision manipulators. Unfortunately, these systems are much more difficult to model, design, and control. The design criteria developed in this work can be extended to properly select the actuator sizing and stiffness for these devices to have better dynamic performance. Further information about parallel and hybrid systems is given in [45, 73 and 79]. The author realizes that many of the problems that have been mentioned here will be extremely difficult to solve, and, of course, countless other important problems exist which have not been discussed here. It is evident that much more research is necessary before a comprehensive and systematic design procedure is available for such devices. The author hopes this study has made at least a small contribution toward the goal of developing design criteria and methods for robotic manipulators.
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225 Figure 7.1 Three degree of freedom Florida Shoulder module
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226 Hand (endeffector) Figure 7.2 A robotic manipulator of hybrid structure
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APPENDIX A DERIVATION OF THE KUHNTUCKER CONDITIONS
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In attempting to develop algorithms or computational tools for solving nonlinear problems, it is useful to have some information concerning the characteristics of an optimal solution. In Chapter 2 a set of the KuhnTucker conditions were stated which an optimal solution must satisfy for a constrained optimization problem. In this appendix the conditions will be derived in detail. Let us consider the minimization of a general function with respect to an ndimens ional vector X, subject to m inequality constraints. Minimize f(X) subject to C (X) < 0, j = 1, 2, . . . , m where X=[ Xl ,x 2 , ... ,x n ] T . All equality constraints are taken into account implicitly in here. The inequality constraints can be transformed into equality constraints by introducing nonnegative slack variables, S., defined such that J Cj (X) + sj = 0, j = 1, 2, ..., m ( A Â• 1 ) or S = tc j (X) 1 1/2 _> Â°. j = 1, 2, .... ra (A. 2) 228
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229 The problem is converted as follow: Minimize f(X) subject to C. (X) + Sj = 0, j = 1, 2, . . . , m The Lagrange function L can be constructed as L(X,S,X) = f(X) + Z X.[C.(X) + S?] (A. 3) j1 3 2 ~ J T where S= [ S , , SÂ„ , . . . , S ] is the vector of slack variables Â— 12 m T and X=[ Xj , X 2 , Â• Â• Â• , X ] is the vector of Lagrange multipliers . Now, a set of necessary conditions for existence of a minimum to the Lagrange function is that 9 L ( X , S , X ) 8x . 3 Â— 0, i 1 , 2 , Â• Â• Â• , n 9L ( X , S , A ) 9 A . 3 Â— 0, jÂ— 1, 2, ...,m 9L ( X , S , X ) 3s j Â— 0, jÂ— 1, 2, ...,m (A. 4) (A. 5) (A. 6) These three equations can be rewritten as 9 f ( X ) m 9 C . ( X ) + L A. Â— Â— = 0, i = 1, 2, n (A. 7) 3x i j = l J
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230 ( X) + s j = Â°. j = 1, 2, (A. 8) j= 1, 2, (A. 9) Equation (A. 8) ensures the constraints Cj (JS.) Â» j = 1 , 2 , . Â• Â• , m , are satisfied, while Equation (A. 9) implies that either X =0 or S.=0. If A.=0, it means that the constraint is j J J inactive. Those constraints which are satisfied with straints while those that are satisfied with equality sign, C =0, are termed as active constraints. On the other hand, 3 if S j =0 > ic means that the constraints is active at the optimum point . Let the set indicate the indices of those constraints which are active at the optimum point and let H 2 include the indices of all the inactive constraints. Thus for jÂ£H^, the constraints are active, and for The constraints are inactive, and Equation (A. 7) can be simplified as strict inequality sign, C^<0, are called as inactive con3f (X) 9C.. (X) + I X . Â— *3x i j6H 1 3x. = 0, i = 1, 2, Â• Â• Â• > n (A. 10) Similarly, Equation (A. 8) can be written as Cj(x) = 0, (jeHj) C .(X) + S 0, (j6H 2 ) (A. 11) (A. 12)
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231 Equations (A. 10), (A. 11) and (A. 12) represent n+g+(mg)= n+m equations in n+m unknowns x^ ( i= 1 , 2 n) , X. (j 6 Hj) and Sj ( 36 H 2 ) where g denotes the number of active constraints . Equation (A. 10) can also be expressed as 3f(X) 3C.(X) I X , i = 1, 2, n (A. 13) 3x Â± jSHj. j 3x. These equations can be written collectively as Vf (X) = Z X. VC . (X) (A. 14) jeHj. J 2 where Vf(X) and VC^.(X) are the gradients of the objective function and j th constraint, respectively. Thus the negative of the gradient of the objective function can be expressed as a linear combination of the gradients of the active constraints at the optimum point. Before showing that the X ' s (j 6 H^) have to be positive in the case of a minimization problem, let's define the term of feasible direction of a vector. The search direction satisfying that a small move in that direction from a point X violates no constraints is called a feasible direction. Thus for problems with sufficiently smooth constraint surfaces, any vector jS satisfying the relation
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232 C.(X + aj; ) = S 1 . VC (X) < 0 (A. 15) da 1 a=0 can be called a feasible direction. T By premultiplying both sides of Equation (A. 14) by which is a feasible direction, we obtain S T . Vf(x) = Z' A. S T . Vc.(X) (A. 16 ) j6H x J J Since S is a feasible direction, it should satisfy the relations S T . VC^X) < 0, ( j SH x ) (A. 17) Thus if Aj > 0 (jeH^, the quantity S 1 . Vf ( X) can be seen to be positive always. Where Vf(X^) indicates the gradient T direction and S^.Vf(X) represents the component of the increment of f(X) along the direction If ^ .Vf(3()>0, the function value increases as we move along the direction S. Hence, if A^'s (jeH^ are positive, we will not able to find any direction in the feasible domain along which the function value can be further decreased. Since the point at which Equation (A. 17) is assumed to be optimum, A ^ ' s ( j S H ^ ) have to be positive. Thus, the conditions to be satisfied at a constrained Â£ minimum point, X , of the problem can be expressed as
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233 3f(X) 3C (x) + Â£ X. Â— = 0, i = 1, 2, n (A. 18) J 6 H 1 3x i and Xj > 0, ( j 6 Hj) (A. 19) These are the KuhnTucker conditions. If the set of active constraints is not know, the alternative form of the KuhnTucker conditions can obtained from Equations (A. 2), (A. 7), (A. 8) and (A. 9) and stated as follows: 3f ( X) 3C . ( X) + Â£ X. 1 = 0, i = 1, 2, n (A. 20) 3x Â± j = l J 9x Â± Cj (X) < 0, j 1, 2, .... m (A. 21) Xj Cj (X ) = 0, j = 1, 2, .... m (A. 22) and Xj>Â°, j = 1 , 2 , . . . , m (A. 23) These conditions are, in general, not sufficient to ensure a relative minimum. However, the KuhnTucker conditions are the necessary and sufficient for a global minimum of convex programming problems
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APPENDIX B REVERSE DISPLACEMENT ANALYSIS Â°F CINCINNATI MIL AC RON T 776 ROBOT
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B.l Introduction A major problem in the operation of a manipulator controlled by a computer which is directing the endeffector to move from one position to another has been to obtain a set of joiirt displacments which will position the endeffector at a given point with a specified orientation within the workspace. A general theory [72, 76, 81] for the algebraic analysis of spatial mechanism geometry which has been developed by J. Duffy to solve this problem. This can be accomplished by joining the hand to the first or grounded joint by a hypothetical link. In this way the problem reduces to the analysis of a corresponding closed spatial mechanism with mobility one. In order to solve for joint displacements in this fashion it is necessary to derive a polynomial equation in the first joint displacement. The degree of the polynomial is the number of maximum real modes for an arbitrary hand location and orientation within the workspace. The central problem becomes the derivation of this polynomial which is the socalled inputoutput equation of a corresponding closedloop spatial mechanism. Once this polynomial is known it is a relatively simple task to develop an algorithm for computing the remaining joint displacements. It is thus possible to compute joint displacements for large finite hand 235
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236 displacements, and it is relatively simple to derive an algorithm to guide the hand along some path between positions . The purpose of this appendix is to derive a detailed position analysis procedure for the CincinnatiMilacron electric T^776 robot as shown in Figure 4.1. The kinematic representation of the industrial robot is displayed in Figure 4.2 and the corresponding kinematic parameters are listed in Table 4.1. There are two parts in the reverse solution procedure to generate the joint displacements for the specified hand location and orientation of the example manipulator. First, a hypothetical link is used to form a closedÂ— loop mechanism of the open manipulator chain in order to find 0y , a 71 , a 71 Â» (0 1 ()) 1 )Â» s 77 > S* Once the loop has been closed, the remaining joint displacements can be determined. B . 2 Reverse Displacement Analysis Part I The manipulator hand is considered to be a rigid body which moves in threedimensional space. Its position can ^ & be specified by three coordinates P^(X H , Y^, Z fl ) , and by the direction cosines Â£g 7 ( X 6 7 Â» Y^y, Z ^y) , S_y( x 7 Â» Yy , Zy) of the hand and gripper axes. It is assumed in the following analysis that these nine quantities are specified and that it is required to compute the corresponding joint displaceClearly , it requires only six independent ments .
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237 parameters to locate a rigid body in threedimensional space and the above parameters are therefore overabundant by three. This is accounted for since Â£67 Â• Â£67 1 (B.l) (B.2) (B.3) Figure B.l illustrates the hand joined to the first joint in the chain by a hypothetical link a ji and offset and S ^ thus forming a closed spatial mechanism. It is now necessary to determine the six constraint parameters S 77 , a 7iÂ’ Â®7Â’ a 71* ^11 anc * Â®1 ^1* * * * * * * . . * Â„* transformation, [ ( X R , Y^, Z^) , ( X 67Â’ X 67* ^67^Â’ o* Z*)]Z>[S 77 , a 71 , 0 7 ,O 71 , S* L , e i (j)^, is derived in the following steps: Step 1. Calculate the value of C 71 C 7 1 = Â—7 * 1 If C 71 =+l, go to Step 2. Otherwise, go to Step 6 . Step 2. C 71 =j1 and therefore S 7 ^ = 0. This is because the vectors S^ and S_^ are either parallel or antiparallel. Let S 77 =0. Step 3. From the vector loop equation (see Figure B.l) * V 4a 7 ^ Â£71 ^ii S 1 = 0 A the hypothetical link length a 71 and S^ can be expressed as
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238 Figure B.l Completion of the spatial loop
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239 Step 4 . Step 5 . Step 6 . Step 7 . * S 11 = S^l . V 1 * l a 7 1 = v + S u Sjl if a 71 ^o, compute ay ^ Â—11 ~ Â§.1^/ a 7 1 and go to Step 8. If a^ = 0 , let aj ^=Â£57 an d Â®7 = 0 and go to Step 9 . Find the unit vector a_y ^ and the length Sy^ Â£71 = (JL7 x Sj) / S 7 x SJ S 7 1 Â“ 71 * ( 7 x 1^ From the vector loop equation 1 + a 71 Â±71 + S 11 1 = Â° the hypothetical link length ay^ tical offset length Sy^ and S ^ expressed as , hypothecan be S 77 (V . aj ^ x _Sj)/ a 71 " (S 7 . V x Sj)/ S ?1 S 1 1 = (I7 Â• Â£71 x D / s 7 1 Step 8 . Find 0 y c 7 = Â£67 Â• 71 s 7 = 17 * 67 x 7 1 Step 9. Compute (0^ ^ ) Cos ( 0 ^ ^ ) ^71 * Â— Sin( 0 ^ = S^ . ajy x i
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240 B . 3 Reverse Displacement Analysis Part II Taking the information obtained from the first part of the reverse displacement analysis, it is possible to find the following joint displacements, 0^, Q^y Â® 3 > Â®4Â’Â®5Â’ and 0 . Write the vector loop equation for the closedloop 6 mechanism 0 S 11 1 + a 23 23 + S 44 44 + S 66 6 + S 7 7 Â—7 + a 7 1 71 (B.4) With this equation and information obtained in PartI, we proceed to the calculation of 9^. The steps for obtaining all joint displacements are shown in Figure B.2. Taking dot product of the vector loop equation, Equation (B.4), with 0 (S *11 Â— 1 + a 23 Â—2 3 + S 4 4 Â—4 + S 66 + s 77 I7 + a 7 1 Â—7 1 ^ Â’ Â—2 (B.5) Due to the actual physical arrangement of the example robot the following is true, (B.6) ( B . 7 ) then. Equation (B.5) becomes
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241 Figure B.2 Steps for obtaining all joint displacements
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242 66 Â—6 ' 2 + S 7 7 Â—7 * 12 + a 7 1 7 1 * 2 66 Z 7 1 + S 7 7 Z 1 + a 7 1 U 12 (B.8) where Z 7 1 = S 7 S 1 C 7 1 C 7 C 1 Z 1 = _S 7 1 C 1 U 12 S 1 (B.9) (B. 10) (B. 11) Substituting the Equations (B.9), (B.10) and (B.ll) into Equation (B.8), then Equation (B.8) becomes an inputoutput equation which 0^ is a function of 0y 0 = S 1 (S 66 S 7 + a 71 ) + C 1 (_S 66 C 7 C 71 S 77 S 71 } = S 1 B 7 + C 1 A 7 ( B Â• 12) where A 7 " S 6 6 C 7 C 7 1 S 77 S 71 B 7 = S 6 6 S 7 + a 7 1 (B. 13) (B. 14) 0j^ and ^ can be obtained as (B. 15) (B . 16) 6 = Tan 1 (S l /C l ) = T an" 1 ( A ? / B ? ) 4Â» = 9j (6 1 !< 135 Â°>
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243 The vector from the second joint to the wrist can be defined as 1 ~ (S 66 Â—6 + S 7 7 Â—7 + a 7 1 7 1 + S 11 1 ) (B. 17) Since V, a 23 and S^. form a triangle and the lengths of these three sides are known, can be solved by using law of cosines : S 3 ' 'll' 2 a 23 S 44>' 2a 23 S 44
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244 (V ^ ^44 ^ 3 ^ 'Â—12^ ^ a 2 3 ^44 ^ 3 ) 44 J 3 ( B . 2 3 ) + S ,. , + 2 a 23 T 4 4 23 s 44 a 3 (V Â« a ^ 2 ) ( ^2 3 S 44 s 3 )+(v . d 12 )(s 44 C 3 ) (B . 24) a 2 3 + S 44 + 2 a 2 3 S 44 S 3 then, angle 0 2 0 2 = Tan _1 (S 2 /C 2 ) , (30Â° < Â© 2 < 117Â°) (B.25) Joint angle 0^ is determined from a spherical cosine law in [76] Z 5 _ Z 3 2 1 7 (B .26) and Z 5 ' C 5 6 C 4 5 S 56 S 45 C 5 ( B . 2 7 ) where ^ 2,217 Can 0 ^ ta ^ ne ^ from Table B.l. From Equations (B.26) and (B.27), C,. can be solved as C 5 " (C 56 C 4 5 Z 3217 )/S 56 S 45 ( B . 2 8 ) and therefore, s 5 = +(1 c 2 )Â‘ /2 (B . 29)
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245 Joint angle 0^ is determined from a pair of spherical sine and sinecosine laws in [76], G 5 4 Â“ X 3 2 1 7 W 54 = Y 3217 (B . 30) ( B . 3 1 ) Using the expressions for G , W 54 > X 3217 and Y 3 2 1 7 in Table B.l yields, (S 56 C 5 C 4 5 + C 5 6 S 45 )S 4 + (S 56 S 5 )C 4 = X 3 2 1 7 (S 56 C 4 5 C 5 + C 5 6 S 45 )C 4 + (_S 56 S 5 )S 4 = v 3217 ( B . 3 2 ) (B.33) then, using Cramer's rule S 4 X 3 2 1 7 S 56 S 5 Y 3 2 1 7 S 5 6 C 4 5 C 5 + C 56 S 45 A ( B . 3 4 ) C 4 S 5 6 C 45 C 5 + C 5 6 S 45 X 3 2 1 7 ~ S 5 6 S 5 Y 3 2 1 7 A ( B . 3 5 ) Where A is the determinant of the coefficients of Equations ( B . 3 2 ) and (B.33) as
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246 A = (S 56 C 4 5 C 5 + C 5 6 S 45 )2 + (S 56 S 5 )2 (B Â‘ 36) The remaining joint angle 0 6 is determined from a pair of spherical sine and sinecosine laws in [76], G 6 ~ X 5 4 3 2 1 \ = Y 5432 1 ( B . 3 7 ) ( B . 3 8 ) Using the s implications in Table B.l yields the explicit solution, S 6 " X 5 4 3 2 1 C 6 = Y 54321 ( B . 39 ) ( B . 4 0 ) then, the joint angle 0 & 8 6 T.n'hSj/Cj) ( B 4 3 The reverse displacement analysis for T 776 robot has been accomplished by solving for the six joint angles of the corresponding heptagon spatial mechanism.
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247 Table B.l Definition of direction cosine notation and s impl if i cat i o ns X 3 2 1 7 C 2+3 X 17 " S 2+3 Y 17 Y 3 2 1 7 = c 34 (s 2+3 X 1 7 + C 2 + 3 Y 1 7 ) s 34 Z 1 7 " ~ Z 1 7 Z 3 2 1 7 = S 3 4 ( S 2+3 X 1 7 + C 2+3 Y 17 ) + c 34 Z 1 7 Â“ S 2 + 3 X 17 + C 2+3 Y 17 X 17 = C 1 X 7 Â‘ Â’ S 1 Y 7 Y 1 7 " C 12 (S 1 X 7 ' C 1 V S 1 2 X 7 = Z7 II r^H S 12 (S 1 X 7 ' * C 1 V + C 1 2 X 7 = S 1 X 7 + Cj y 7 11 IX! S 67 S 7 = " S 7 Y 7 " < S 71 C i 67 + C 71 S 6 7 C 7 ) = C 7 1 C 7 Z 7 = C 7 1 C 6 7 S 71 S 67 C 1 J = S 7 1 C 7 G 5 4 = S 56 (S 5 E 4 + C 5 V C 5 6 G 4 E 4 = C 4 E 4 = C 4 5 S 4 G 4 = s 45 s 4 W = S^(S, U, + c, V.) 56 Â“ 5 6 v Â“ 5 " 4 ' Â”5 C 56 W 4 U 4 = Â“ S 4 ^4 = C 4 5 C 4 W4 s 4 5
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248 Table 1 X 54 3 2 1 Y 5 4 3 2 1 X 4 3 2 1 Y 432 1 Z 4 3 2 1 X 321 = Y 3 2 1 Z 3 2 1 = . 1 , Continued " C 5 X 4 3 2 1 " S 5 Y 432 1 = c 56 (s 5 x 4321 + c 5 y 4321 ) s 56 z 4321 = c 4 x 321 s 4 y 321 C 45< S 4 X 3 2 1 + C 4 1 : 32p S 45 Z 3 2 1 s 45 (s 4 X 32 i + C 4 1 : 321 ) + c 45 Z 3 2 1 C 2+3 X 1 S 2 + 3 Y 1 C 34 ( S 2 + 3 Xi + C 2 + 3 V s 34 X 1 = S 3 4 ^ S 2 + 3 Xi + C 2 + 3 V + c 34 Z 1 S 2 + 3 X 1 + C 2+3 Y 1 12 S 1 = S 1 ( S 7 1 C 1 2 + C 7 1 S 1 2 C l } = _C 7 1 C 1 71 C 1 2 Â“ S 7 1 S 1 2 C 1 = _S 7 1 C 1 67 S 6 = S 6 67 C 6 = C 6
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REFERENCES Tesar, D., Ed., Proceeding of the National Science Foundation Workshop on the Impact of the Academic Community of Required Research Activities for Generalized Robotic Manipulators , University of Florida , Gainesville , FL, February, 1978 . Kugath , D.A., and Wilt, D.R., "Problems in Selection of Design Parameters Affecting Manipulator Performance," First CISMIFTOMM Symposium on Theory and Practice of Robots and Manipulators , Vol. 2, Udine, Italy, September, 1973, pp. 313332. Roth, B., Rasteger, J., and Scheinman, V., "On the Design of Computer Controlled Manipulators, First CISMIFTOMM Symposium on Theory and Practice of Robots and Manipulators, Udine, Italy, September, 1973, Vol. 1, pp. 93113. Tesar, D., and Thomas, M., "Assessement for the Mathematical Formulation for Design and Digital Control of Programmable Manipulator Systems," NSF Grant ENG7820112 and DOE Contract No. ER78S056102, Center for Intelligent Machines and Robotics, University of Florida, Gainesville, FL, 1979. Thomas, M., and Tesar, D., "Dynamic Modeling of Serial Manipulator Arms," ASME , Journal of Dynamic Systems, Measurement and Control , Vol. 104, No. 3, September, 1982, pp. 218228. Thomas, M., YuanChou, H.C., and Tesar, D., Optimal Actuator Sizing for Robotic Manipulators Based on Local Dynamic Criteria," ASME, Journal of Mechanisms, Transmissions, and Automation in Design , Vol. 107, June, 1985 , pp. 163169.
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250 [7] Thomas, M., and Tesar, D., "Dynamic Modeling and Analysis of RigidLink Serial Manipulators,Â” DOE Grant No. DEACO 57 9ER1 00 1 3 , Center for Intelligent Machines and Robotics, University of Florida, Gainesville, FL, November, 1982. [8] Whitney, D.E., "The Mathematics of Coordinated Control of Prosthetic Arms and Manipulators," ASME , Journal of Dynamic Systems, Measurement and Control , Vol. 94, December 1972, pp. 303309. [9] Ozgoren , M.K., "Optimization of Manipulator Motions, Proceeding of Symposium on Theory and Practice of Robots and Manipulators, Warsaw, Poland, September, 1976 . [10] Brooks, T.L., "Optimal Path Generation for Cooperating or Reduntant Manipulators," ASME, Computers in Engineering 1982, Vol. 2, Robo ts and Robotics , Proceedings of the Second International Computer Engineering Conference, San Diego, CA, August, 1982, pp. 119122. [11] Lin, Y.C., and Duffy, J., "The Mapping and Structure of the Workspace of Robot Manipulators with Revolute and Prismatic Pairs," ASME, Computers in Engineering 1982, Vol. 2, Robots a nd Robotics , Proceedings of the Second International Computer Engineering Conference, San Diego, CA, August, 1982, pp. 165172. [12] Albertson, P., "Verifying Robot Performance, Robotics Today, October, 1983, pp. 3336. [13] Ar t obo 1 e v sk i i , 1. 1., Mechanisms for the Generation of Plane Curves, Macmillan, New York, 1964. [14] Newton, I., "The Enumeration of Curves of the Third Degree," Russian translation in the book: Newton, I., Mathematical Works , ONTINKTP, USSR, 1937. [15] Galileo Galilei linceo, Discorsi e dimostrazioni mathematiche intorno, a due nuove scienze attenenti alia mecanica et i movimenti local!, Leida 1638.
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251 [16] Bernoulli, Johan, Letter to Leibnitz on the the beams of uniform strength, 1687. [17] Euler, Leonhard, "Methodus inveniendi Lineas curvas in maximi minimive proprietate gaudentes , " (1744), in A History of Mathematics, edited by Cajori, F., Macmillan, New York, NY, 1919, 232 pp. [18] Lagrange, J.L., Sur la figure des collonnes , Miscellanea Tnurinensia, 1770. [19] Cauchy, Auguste L., "Methode generale pour la resolution des systemes d'equations simultanees, Computes Rendus , Academic de Sciences, Paris, 25, 1847 , pp . 536 . [20] Dantzig, G.B., Linear Programming and Extensions , Princeton University Press, Princeton, NJ, 1963. [21] Kuhn, H.W., and Tucker, A.W., "Nonlinear Programming," in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability , edited by Neyman, JÂ«, University of California Press, Berkeley, CA, 1951, pp 487492. [22] Bellman, R.E., Dynamic Programming , Princeton University Press, Princeton, NJ, 1957. [23] Davidon, W.C., "Variable Metric Method of Minimization,Â” Argonne National Laboratory Report No. ANL5990, 1959. [24] Fletcher, R., and Powell, M.J.D., "A Rapidly Convergent Descent Method for Minimization, C omp u ter Journal, Vol. 6, No. 2, 1963, pp* 163Â—168. [25] Zoutendijk, G., Methods of Feasible Directions , Elsevier Publishing Company, Amsterdam, Netherlands, 1960.
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252 [26] Rosen, J.B., "The Gradient Projection Method for Nonlinear Programming: Part I, Linear Constraints, SIAM Journal , Vol. 8, 1960, pp. 181217. [27] Rosen, J.B., "The Gradient Projection Method for Nonlinear Programming: Part II, Nonlinear Constraints," SIAM Journal , Vol. 9, 1961, pp Â• 414432 . [28] Fiacco, A.V., and McCormick, G.P., Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley & Sons, New York, NY, 1968. [29] Duffin, R.J., Peterson, E., and Zener, C., Geometric Programming, John Wiley & Sons, New York, NY, 1967. [30] Gomory, R.E., "An AllInteger Integer Programming Algorithm," Chapter 13 in Industrial Scheduling , edited by Muth, J.F., and Thompson, G.L., PrenticeHall, Inc., Englewood Cliffs, NJ, 1963. [31] Fletcher, R., "A Review of Methods for Unconstrained Optimization," in Optimization , edited by Fletcher, R., Academic Press, London, 1969, pp. 112. [32] Dixon, L.C.W., "Nonlinear Programming: A view of the State of the Art," in Towards Global Optimization , edited by Dixon, L.C.W., and Szego, G.P., NorthHo Hand Publishing Co., New York, NY 1975 , pp. 320340 . [33] Powell, M.J.D., "A Survey of Numerical Methods for Unconstrained Optimization," in the Symposium on Optimization, Sponsored by the Air Force Office of Scientific Research, at the 1968 National Meeting of Society for Industrial and Applied Mathematics, Toronto, Canada, June 1114, 1968, pp.321. [34] Powell, M.J.D., "A view of Unconstrained Optimization," in Optimization in Action , edited by Dixon, L.C.W., Academic Press, New York, NY, 1976, pp. 117152 .
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253 [35] Powell, "Optimization Algorithms in 1979," in Optimization Techniques, Part 1, edited by Iracki, K., Malanowski, K., and Walukiewicz, S., Springer Verlag, New York, NY, 1980, pp. 8398. [36] U i eke r , J.J.,Jr., "On the Dynamic Analysis of Spatial Linkages Using 4x4 Matrices," Ph.D. Dissertation, Northwestern University, 1965. [37] Kahn, M.E., "The Near MinimumTime Control of OpenLoop Articulated Kinematic Chains," Ph.D. Dissertation, Stanford University, 1969. [38] Bejczy, A.K., "Robot Arm Dynamics and Control," NASA Technical Memorandum 33669, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, February, 1974. [39] Whitney, D.E., "StateSpace Models of Remote Manipulation Tasks," IEEE, Journal of Automatic Controls , Vol. 14, December, 1969 , pp. 617. [40] Raibert, M.H., "Mechanical Arm Control Using a StateSpace Memory," Society of Manufacturing Engineers Technical Paper MS77742, 1977. [41] Orin, D.E., McGhee, R.S., Vukobrato vie , M., and Hartoch, G., "Kinematic and Kinetic Analysis of OpenChain Linkages Utilizing NewtonEuler Methods, Case Western Reserve University Internal Report #EEAP1 7 8 , Cleveland, OH, 1978 . [42] Luh , J.Y.S., Walker, M.W., and Paul, R.P.C., "OnLine Computational Scheme for Mechanical Manipulators," ASME, Journal of Dynamic Systems, Measurement, and Control, Vol. 102, 1980, pp. 6976. [43] Hollerbach , J.M., "A Recursive Lagrangian Formulation of Manipulator Dynamics and a Comparative Study of Dynamics Formulation Complexity, IEEE, Journal of Systems, Man, Cybernetics , Vol. SMC10, No. 11, 1980, pp. 730Â—736.
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254 [44] Silver, W.M., "On the Equivalence of Lagrangian and NewtonEuler Dynamics for Manipulators," Proceeding o f the 1981 Joint Automatic Controls Conference , Charlottesville, VA, 1981, Paper TA2A. [45] Sklar, M., and Tesar, D., "The Analysis of the Hybrid Prarllel/ Serial Robotic Manipulators," Center for Intelligent Machines and Robotics, University of Florida, Gainesville, FL, April, 1984. [46] Benedict, C.E., and Tesar, D., "Dynamic Response Analysis of QuasiRigid Mechanical Systems Using Kinematic Influence Coefficients, Journal of Mechanisms , Vol. 6, 1971, pp. 383403. [47] Benedict, C.E., and Tesar, D., "Model Formulation of Complex Mechanisms with Multiple Inputs: Part IGeometry," ASME , Journal of Mechanical Design , Vol. 100, October, 1978, pp. 747754. [48] Benedict, C.E., and Tesar, D., "Model Formulation of Complex Mechanisms with Multiple Inputs: Part II The Dynamic Model," ASME, Journal of Mechanical Design , Vol. 100, October, 1978, pp. 755761 . [49] Rao, S.S., Optimization Theory and Applications , A Halsted Press Book, John Wiley & Sons, New York, NY, 1979. [50] Noble, B., and Daniel, J.W., Applied Linear Algebra , Second edition, PrenticeHall, Inc., Englewood Cliffs, NJ, 1977, pp. 130132. [51] Wilde, D.J., Optimum Seeking Methods , PrenticeHall, Englewood Cliffs, NJ , 1964. [52] Haug, E.J., and Arora, J.S., Applied Optimal Design: Mechanical and Structural Systems , John Wiley & Sons, New York, NY, 1979. [53] Box, M.J., Davies, D., and Swann, W.H., "Nonlinear Optimization Techniques,Â’ Chemical Industries Monograph 5, Oliver and Boyd, Edinburgh, 1970.
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255 [54] Jacoby, S.L.S., Kowalik, J Â• S . , and Pizzo, J.T., 1 1 e r a t i ve Methods for Nonlinear Optimization Problems , PrenticeHall, Englewood Cliffs, NJ , 1972. [55] Box, M.J., "A Comparsion of Several Current Optimization Methods and the Use of Transformations in Constrained Problems," Computer Journal , Vol. 9, 1966, pp. 6777. [56] Wilde, D.J., and Beighter, C.S., Foundations of Optimization, PrenticeHall, Englewood Cliffs, N J , 1967 . [57] Powell, M.J.D., "An Effective Method for Finding the Minimum of Functions of Several Variables Without use of Derivatives," Computer Journal , Vol. 7, 1966, pp. 155162. [58] Powell, M.J.D., "Introduction to Constrained Optimization," in Numerical Methods for Constrained Optimization, edited by Gill, P.E., and Murray, W., Academic Press, New York, NY, 1974, pp. 1Â—28. [59] Avriel, M., Nonlinear Programming: Analysis and Methods , PrenticeHall, Englewood Cliffs, NJ , 1976 . [60] Eason, E.D., and Tenton, R.G., A Comparsion of Numerical Optimization Methods for Engineering Design,Â” ASME , Journal of Engineering for Industry , Series B, Vol. 96, No. 1, 1974, pp. 196200. [61] Colville, A.R., "A Comparative Study of Nonlinear Programming Codes,Â” in Proceedings of the Princeton Symposium on Mathematical Programming , edited by Kuhn, H., 1970, pp. 487501. [62] Himmelbau, D.M., "A Uniform Evaluation of Unconstrained Optimization Techniques,Â” in Numerical Methods for Nonlinear Optimization , edited by Lootsma , F . A . , Academic Press, London , 1972, pp . 6997 .
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256 [63] Powell, M.J.D., "Variable Metric Methods for Constrained Optimization," in Computing Methods in Applied Sciences and Engineering, 1977, I , Third International S ymp o s i um , edited by Glowinski , RÂ«, and Lions, J.L., 1977, pp. 6272. [64] Fox, R.L., Optimization Methods for Engineering Design , AddisonWesley , Reading, MA , 1971. [65] Dixon, L.C.W., Nonlinear Optimization , English Universities Press, London, 1972. [66] Gill, P.E., Murray, W., and Wright, M.H., Practical Optimization , Academic Press, London, 1981. [67] Maatuk, J., "A Study of the Dynamics and Control of Flexible Spatial Manipulators," Ph.D. Dissertation, University of California, Los Angles, CA, 1976. [68] Book, W.J., "Modeling, Design and Control of Flexible Manipulator Arms,Â” NASA Contract NAS828055, MIT, Cambridge, MA , 1974. [69] Duffy, J, and Crane, C., "A Displacement Analysis of the General Spatial SevenLink, 7R Mechanism, Mechanisms and Machine Theory , Vol. 15, 1980, pp. 153169 . [70] Tesar, D., "Proposal for the Analytical Experimental and Design Studies of General Robotic Manipulators," College of Engineering, University of Florida, Gainesville, FL, 1978. [71] Pieper, D.L., and Roth, B., "The Kinematics of Manipulators under Computer Control, Proceed! ng s 1 1 International Congress on Theory of Machines and Mechanisms, Vol. 2, 1969, pp. 159168. [72] Duffy, J., Analysis of Mechanisms and Robot Manipulators , John Wiley & Sons, New York, NY, 1980.
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257 [73] [74] Tesar , D . , and Co*, D.J.. "The Â£" 4 1981 Â• . A Tovar D "The Generalized Coordinate Selection <,1 t.i 1 Â»Â«. pp. 206 Â— 217. Thomas M., YuanChou, H.C., and Tesar, D., Optima Actuator Stiffness Distribution for IEEE> Manipulators Based on Robotics and pp * 275281 . T ar ,A Tpqar D., "Kinematic l7M Control ol\ DU R ob^i = 'iaÂ„ipniatorÂ’Â»ith c aJnilateral SÂ“lu5Â«rSUi;.. i.iÂ«..iÂ«T of Florida, Gainesville, FL, May, [77] Hildebrand, F.B., Methods of Applied 77 . PrenticeHall, Englewood Cliffs, nj , [75] [78] Isaacson, E., and Mnnisric al Methods 1966 . Keller, H.B., Analysis of John Wiley & Sons, New York NY, Â„ . , M instantaneous Kinematics and Joint Dis [79] Mohamed, M., Ins tanca lv p aral lel Robot Devices, placement Analysis of Florld a, GainesPh.D. Dissertation, University ville, FL, December, 1983. ft , T p "RealTime Computation of Influence 1801 Coefficient* Base (^Dynamic Modeling Matrice^ fÂ°. . Unive rl i ty 3 Qf P plor ida^Â°Gaines vi lie , FL, August, 1985
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258 [81] Lipkin, H. and Duffy, J., "A Vector Analysis of Robot Manipulators,Â” in Recent Advances in Robotics , Edited by G. Beni and S. Hackwood, John Wiley & Sons, New York, NY, 1985, pp. 175241.
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259 BIOGRAPHICAL SKETCH HsinChien YuanChou was born May 2, 1950, in SanHsia, Taiwan. In 1951, his family moved to Hualien, Taiwan, where he attended primary and secondary schools. He received a high school diploma from SuiWei High School in 1968, and an associate degree in engineering mechanics with honors from Southern Asia Engineering College in 1974. In 1977, he received a Bachelor of Science in physics from Tamkang College of Arts & Sciences. In September, 1978, he enrolled in graduate school at Old Dominion University in Norfolk, Virginia, and received a Master of Engineering in engineering mechanics in 1980. He is currently pursuing the Doctor of Philosophy Degree in mechanical nngineering at the University of Florida. He was married to Ms. ChingYing Lin in June 1980. They are blessed with one child, WenYen, born October, 10, 1983, in Gainesville, Florida. *
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Delbert Tesar, Chairman Adjunct Professor of Mechanical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. L L Joseph Duffy 1 J Professor of Mechanical Eri gineer ing I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. George/ N . Professor Sandor of Mechanical Engineering
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. "Gary K. /Matthew A s s o c i at e Professor of Mechanical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. (I / J [A ' // ^ Ralph G. Sel^ridge Professor of Computer and OH SciSHC6 This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. D e cemb e r 19 8 5 iUJl&A Msb Dean, College of Engineering Dean, Graduate School

