Analysis of the Relationship

between the Physical and the Mathematical Kinematic Parameters

in Robotic Manipulator Parameter Estimation Algorithms

A Thesis

Submitted to the Faculty

of

Drexel University

by

Ambarish Goswami

in partial fulfillment of the

requirements for the degree

of

Master of Science

in

Mechanical Engineering

September 1988
 ii

ACKNOWLEDGEMENTS

It is my great privilege to express my deep gratitude to Professor James R. Bosnik, my

Thesis Advisor, for his inspiring guidance, encouragement, and suggestions

throughout the course of this research. He spent a considerable amount of his valuable

time to help me at every stage of this work.

Special thanks are due to Professor Sorin Siegler and Professor Ajmal Yousuff for their

ever-willingness to help me. Professor Leon Bahar and Professor Yousuff took pain in

reading the manuscript and added many valuable comments. My heartfelt thanks go to

my fellow graduate students at Hess Laboratory for their warm friendship.

Finally, I would like to express my most sincere appreciation to Dr. R. Raychowdhury

and Dr. M. K. Raychowdhury, my sister and my brother-in-law, for providing me with

continuous affection and motivation throughout the research program.

 iii

TABLE OF CONTENTS

Page
LIST OF TABLES ................................................................................. v

LIST OF ILLUSTRATIONS .....................................................................vi

LIST OF SYMBOLS ............................................................................ viii

ABSTRACT ........................................................................................ x

Chapter

1 INTRODUCTION........................................................................... 1

1.1 Problem Background ........................................................... 1

1.2 Literature Survey ................................................................ 3
1.3 Problem Statement .............................................................. 6
1.4 Chapter Summary ............................................................... 7

2 KINEMATIC MODELING ................................................................ 8

2.1 Introduction ...................................................................... 8

2.2 Kinematic Models for Parameter Estimation ................................. 8
2.3 Description of a Suitable Model .............................................. 13
2.4 Chapter Summary .............................................................. 16

3 LEAST SQUARES PARAMETER ESTIMATION TECHNIQUES ................ 18

3.1 Introduction ..................................................................... 18

3.2 Quadratic Model of The Objective Function................................. 20

3.3 Optimization Techniques ...................................................... 24

3.3.1 Steepest Descent Method.......................................... 24
3.3.2 Gauss-Newton Algorithm......................................... 25
3.3.3 Disadvantages of the Simpler Methods.......................... 26
3.3.4 Levenberg-Marquardt Algorithm................................. 27

3.4 Singular Value Decomposition................................................ 28

3.5 Chapter Summary .............................................................. 32

4 SIMULATION, RESULTS, AND DISCUSSION .................................... 33

4.1 Introduction ..................................................................... 33

4.2 Kinematic Description of the Manipulator................................... 33
4.3 Synthesis of Data............................................................... 36
 iv

Chapter Page

4.4 Results and Discussion ........................................................ 37

4.4.1 Effect of Parameters on EECF Pose ............................. 37
4.4.2 Ill-Conditioning of the System ................................... 45
4.4.3 Parameter Redundancy ............................................ 50
4.4.4 Effect of Parameters on the Objective Function ................ 55
4.4.5 Analysis of Solution of Levenberg-Marquardt Algorithm..... 63
4.4.6 Analysis of Solution of Singular Value Decomposition ....... 66
4.4.7 Extent of Redundancy ............................................. 66
4.4.8 Effect of Data Collection at Multiple Sites....................... 70

4.5 Chapter Summary .............................................................. 75

5 SUMMARY, CONCLUSIONS, AND FUTURE WORK............................ 77

5.1 Summary ........................................................................ 77

5.2 Conclusions..................................................................... 78
5.3 Future Work .................................................................... 80

LIST OF REFERENCES ........................................................................ 82

 v

LIST OF TABLES

Table Page
4.1 Nominal Values of Parameters for Synthesized Data ............................. 35

4.2 Joint Voltages and EECF Pose Components for 5 Postures..................... 42

4.3 Jacobian Matrix (Radians, Inches) from 3 Postures

without Parameter Redundancy ..................................................... 46

4.4 Hessian Matrix (Radians, Inches) from 3 Postures

without Parameter Redundancy ..................................................... 47

4.5 Hessian Matrix (Degrees, Inches) from 3 Postures

without Parameter Redundancy ..................................................... 49

4.6 Hessian Matrix from 34 Postures with One Redundancy ........................ 51

4.7 Hessian Matrix from 34 Postures with One Identical Redundancy ............. 52

4.8 Typical Convergence Cases for Parameters with an

Identical Redundant Set .............................................................. 65

4.9 Typical Jacobian Matrix from one posture in a

Multiple Sites per Posture Data Collection Scheme ............................... 71
 vi

LIST OF ILLUSTRATIONS

Figure Page

2.1 Two Typical Coordinate Frame Attachments ......................................... 12

2.2 Summary of Kinematic Notation ....................................................... 14

2.3 General Six Degree-of-Freedom Robotic Manipulator............................... 17

3.1 Least Squares Estimation Model (3-D Case).......................................... 30

4.1 Coordinate Frame Attachments for GE Model A4

Robot Manipulator ....................................................................... 34

4.2 EECF Pose Characteristic Curves for Linear Parameters............................ 39

4.3 EECF Pose Characteristic Curves for Angular Parameters ......................... 40

4.4 EECF Pose Characteristic Curves for Joint Transducer Parameters ............... 41

4.5 EECF Pose Characteristic Curves

for Joint Transducer Parameters (360° Variation) .................................... 44

4.6 Sum-of-Squares Characteristic Curves for Link Lengths

(p 1, p3, p8, p26) .......................................................................... 56

4.7 Sum-of-Squares Characteristic Curves for Link Euler

Angles (p 4 , p1 6 , p2 8 , p3 0 )............................................................... 57

4.8 Sum-of-Squares Characteristic Curves for Link Euler

Angles (p 5 , p1 2 , p1 7 )..................................................................... 58

Figure Page

4.9 Sum-of-Squares Characteristic Curves for Joint

Zero Point Offsets (p31, p32, p33, p34)................................................ 60
 vii

4.10 Sum-of-Squares Characteristic Curves for Joint

Transducer Output Slopes (p35, p36, p37, p38)....................................... 61

4.11 Detailed Sum-of-Squares Characteristic Curves for p38 ............................. 62

4.12 Location of Sites on the Manipulator Body ........................................... 72

 vii

LIST OF SYMBOLS

di joint variable, prismatic, joint i

d 0i parameter, joint transducer zero-point offset, prismatic, joint i

D transformation matrix, used in gradient methods of optimization

f residual vector

g gradient vector

H Hessian matrix

I identity matrix

J Jacobian matrix
ki parameter, joint transducer output slope, joint i

n total number of independent kinematic parameters

np number of prismatic joints in a manipulator

nr number of revolute joints in a manipulator

ns number of calibration sites per posture

N total number of independent kinematic and electrical parameters

p parameter vector

∆p parameter update vector

t scalar, determines step length in gradient method of optimization

T transformation matrix, aggregate

T i,J transformation matrix, joint i

T i,L transformation matrix, link i

Vi joint transducer output voltage, joint i

xG global x-axis

xi parameter, translation along local x-axis, link i; also, the axis itself
 ix

yG global y-axis

yi parameter, translation along local y-axis, link i; also, the axis itself

zG global z-axis

zi parameter, translation along local z-axis, link i; also, the axis itself

αi parameter, rotation about local z-axis, joint i

βi parameter, rotation about local y-axis, joint i

γi parameter, rotation about local x-axis, joint i

θi joint variable, revolute, joint i

θ0i parameter, joint transducer zero-point offset, revolute, joint i

φ sum-of-squares of residuals (the objective function)

 x

ABSTRACT

Analysis of the Relationship

between the Physical and the Mathematical Kinematic Parameters

in Robotic Manipulator Parameter Estimation Algorithms

Author: Ambarish Goswami

Advisor: Dr. James R. Bosnik

Although kinematic parameter estimation is well-established as a technique for

improving end effector positioning accuracy of a robotic manipulator, little attention has

been given to the relationship between the optimal mathematical parameters and the

corresponding physical parameters of the manipulator. Such a relationship is very

much desirable from the standpoint of preventive maintenance and isolation of sources

of damage in the manipulator body. In this work, choice of kinematic model is shown

to have a strong effect on this relationship, as well as on the calibration process in

general. It is observed that the most desirable kinematic model for a given manipulator

includes redundant parameters, which interact numerically among themselves during the

solution process and complicate interpretation of results. A kinematic model with no

redundant parameter, on the other hand, contains too few parameters to get a

comprehensible relation between the physical dimensions of the manipulator and the

mathematical parameters. Multiple point data collection in a single posture is shown to

be of no significant help in preserving the mathematical/physical parameter relationship.

 1

Chapter 1

INTRODUCTION

1.1 Problem Background

Kinematic positioning accuracy is one of the foremost concerns in the design of robotic

manipulators, and consequently has received wide attention from the research

community since the emergence of the robots with off-line programming capability.

Accuracy, in this context, can be described as the maximum difference between the

pose of the end of a manipulator's arm, as calculated using the controller's kinematic

model, and the true pose of the end of the arm. One approach to improving accuracy is

to specify tighter manufacturing tolerances on a manipulator's components. This

approach, however, does not provide a long lasting solution and, moreover, leads to

considerable increases in manufacturing costs. The end user, therefore, is often forced

to compromise between cost and accuracy. Calibration, a second approach to

improving accuracy, is in many ways preferable to the first, as described below.

The kinematic accuracy of a robotic manipulator depends on the accurate knowledge of

the parameters which describe the manipulator's mechanical links and joint transducers.

For various practical reasons, an accurate estimate of some of the parameters may not

be available, and, in this case, numerical re-evaluation of these parameters in a

calibration scheme typically produces greatly improved end effector positioning

accuracy. The basic aim of robot calibration is to replace the initially estimated set of

kinematic parameters with a new set of "optimal" parameters which reduces the

aggregate end effector positioning error over the calibration space. To achieve this

goal, the optimized parameters do not need to have any particular relationship to the

actual physical parameters, and often, in fact, do not.

 2

In the current work, the possibility of establishing a relationship between the set of

optimal (mathematical) parameters and the actual (physical) dimensions of the

manipulator's kinematic components is explored. Knowledge of this relationship will

be important for establishing approaches to the analytical tracking of component wear or

damage in predictive maintenance programs for robotic manipulators. In a predictive

maintenance scheme, the optimal parameters could be included in a time-history

database and the behavior of each of them observed over time for the purpose of

identifying those parameters whose values are changing more than an expected or

acceptable amount.

Choice of kinematic model is shown to have an important effect on both the robustness

of the calibration algorithm and the mathematical/physical parameter relationship.

Numerical simulations, utilizing both actual experimental data and synthesized data,

have been run to compare several kinematic model/optimization algorithm

combinations. These simulations have brought to light a number of properties of the

kinematic models and the optimization algorithms which have important effects on the

mathematical/physical parameter relationship. These properties will be useful in current

efforts to refine this relationship for applications in predictive maintenance.

The following definitions will be used throughout this work:

• position: the translational relationship of one coordinate frame to another

• attitude: the rotational relationship of one coordinate frame to another

• pose: the position and attitude of one coordinate frame with respect to another

• posture: the unique sequential combination of a set of manipulator joint variables

 3

• parameter: a quantity which describes some physical or mathematical aspect of a

manipulator's kinematic model, assumed to be constant in magnitude over some

useful period of time

• global coordinate frame: coordinate frame fixed to the ground with respect to

which the pose of the end effector coordinate frame (EECF) is measured

• redundancy: the situation in which a kinematic model of a manipulator contains

more parameters than are needed to describe the change in the pose of its EECF

• site: a point on a manipulator body where pose measurements are taken externally

1.2 Literature Survey

The technique of calibration has been adopted by many researchers as a primary

approach to dealing with the problem of end effector coordinate frame (EECF) pose

error. A calibration scheme may be formulated as a dimensional synthesis of

manipulator parameters based on data taken from external measurements (Sommer and

Miller, 1981). The functional inputs (manipulator joint transducer signals) must be

recorded along with the desired functional outputs (actual poses of the EECF). An

iterative numerical optimization routine may then be used to adjust manipulator

parameters so as to minimize the aggregate error of the EECF pose over the entire

workspace of the manipulator.

From a kinematic standpoint, parameter estimation may be expressed as a multiple-point

synthesis of a spatial open-loop mechanism for rigid-body guidance. Similar syntheses

of closed loop spatial linkages have been performed by Tull and Lewis (1968) and
 4

Chen and Chan (1974). One of the restricted-step nonlinear least-squares optimization

schemes popularly known as the Levenberg-Marquardt algorithm has been shown to

work well for closed-loop mechanisms. Based on this knowledge, an efficient

algorithm for simultaneous optimization (re-evaluation of the entire set of parameters)

was developed by Sommer and Miller (1981) for the calibration of instrumented spatial

linkages.

Several varying approaches to kinematic model selection and static calibration of robotic

manipulators have been proposed. Whitney et al. (1984) were the first to publish an

investigation of robot calibration which included actual measurement results. Their

approach was unique in that it included the effects of non-geometric errors in the

calibration process. More recently, Stone et al. (1986) presented a calibration method

which utilized a different kinematic model from previous works. In these results, the

authors found "dramatic" differences between optimized parameters and the

corresponding physical manipulator parameters. Kirchner et al. (1987) proposed a

method of calibration based on the Denavit-Hartenberg notation (Denavit and

Hartenberg, 1955) which utilizes a parameter perturbation approach to avoid the

problem of redundancy in the parameter set. These authors outlined an approach to deal

with parameter redundancy through careful selection of the calibration postures.

According to Roth et al. (1987), manipulator calibration techniques may contain up to

three different sublevels. At the first sublevel, the joint sublevel, calibration deals with

the determination of the parameters describing joint transducers. At the second

sublevel, the kinematic model sublevel, calibration deals with the determination of the

geometric parameters describing the links and certain non-geometric parameters

describing phenomena such as gear backlash and joint compliance. The third sublevel

of calibration deals with the determination of the inertial properties of the various links

and is known as the dynamic sublevel. The calibration techniques dealing with the first
 5

two sublevels are sometimes called static calibration or kinematic parameter

estimation.

The scope of this work includes the first two sublevels, although in the present form it

does not take into account the difficult-to-model non-geometric errors of the second

sublevel. The non-geometric errors are caused by link compliance, gear train

compliance, motor-bearing wobble, gear backlash etc. and in the present kinematic

model they are assumed to be negligible. Judd and Knasinski (1987) observed that

about 95% of the RMS value of the EECF positioning error is caused by errors in the

geometric parameters. Therefore the geometric parameters of the links along with the

electrical parameters of the joint transducers serve to model a manipulator almost

completely. Manipulators are made up of a wide variety of driving mechanisms and it

is extremely difficult to develop a model or a set of models which adequately describes

the effects of the non-geometric parameters for an arbitrary manipulator. The cost and

complexity involved in analyzing these effects on each individual manipulator after it is

manufactured do not appear to be practical (Stone, 1987). Also, it has been assumed

here that the resolution of the encoders providing joint position feedback is infinite.

In all of the references described above, the process of calibration was aimed primarily

at reducing the end effector positioning error. The relationship between a physical

parameter and its optimized value was first considered by Bosnik (1986). Aside from

extending and applying the algorithm developed by Sommer and Miller (1981), this

work described the potential application of the mathematical/physical parameter

relationship in identifying and isolating points of damage or wear before such problems

might become discernible by other means. If a static calibration is performed on a

robotic manipulator at regular time intervals, then the values of the various parameters

can be compared to their corresponding values during previous tests, for the purpose of

identifying those parameters whose values are changing by more than an acceptable or

expected amount. After additional experience with the procedure, it may be possible to
 6

make specific maintenance or repair recommendations for a manipulator on the basis of

its calibration history. The mathematical/physical parameter relationship is obtained as

an outcome of the calibration process itself, and no extra experimental setup or

calculation is required. All that is needed is to modify the calibration algorithm in such

a way that the results can be conveniently interpreted. It is an objective of the current

work to develop a calibration technique which more closely relates the mathematical

(theoretical) to the physical (actual) parameters.

1.3 Problem Statement

The primary objective of this study is to investigate the relationship between the

physical features of the links and the joints of a manipulator and the parameters included

in the kinematic model for that manipulator. A secondary objective is to be able to track

the changes in these physical features from time to time by comparing the earlier values

and the most recent optimal values which are obtained through parameter estimation

algorithm in a robot calibration technique.

Specifically, meeting these objectives involves :

1. modification of an existing parameter estimation algorithm (Bosnik, 1986) to enable

it to calculate the singular values and/or eigenvalues of the Jacobian and of the

Hessian matrices of the manipulator system,

2. testing of the performance of the modified algorithm using synthesized data with

known performance criteria,

3. interpretation of the results,

 7

4. investigation of the characteristics of the objective function with respect to various

kinematic and electrical parameters,

5. development of an algorithm accepting pose error data from multiple sites in a single

posture, and

6. final interpretation of the results with reference to the extent of redundancy in a

kinematic model.

1.4 Chapter Summary

This chapter describes the motivation for the present work and includes a literature

survey which outlines the work of previous researchers in the area of the present work.

The scope of this work is explained and the specific sub-problems to be tackled are

listed.
 8

Chapter 2

KINEMATIC MODELING

2.1 Introduction

A kinematic model can be defined as a systematic mathematical relationship between

manipulator joint positions and the pose of the EECF. Coordinate frames are attached

to different suitable places in the manipulator body and a relationship between two

successive frames is obtained from manipulator geometry. The local relationships

between the frames are concatenated, typically by matrix transformation methods, to get

a relationship between the global coordinate frame and the EECF of the manipulator.

Variables are included in the relationship so that any change in the EECF pose due to

joint movements can be properly described. The inputs to a kinematic model are

therefore the joint variables and the output is the EECF pose with respect to the global

coordinate frame. The choice of a kinematic model is very important in calibration,

often determining the success or failure of a particular calibration scheme.

2.2 Kinematic Models for Parameter Estimation

The choice of a kinematic model also impacts the desired mathematical/physical

parameter relationship. Everett et al. (1987) attempted to categorize the desirable

properties of a kinematic model to be used for calibration of manipulators having

revolute and prismatic joints only. Three main properties were defined. First, the

model should contain a sufficient number of parameters to express any possible

variation in the kinematic structure of the robot; models possessing this property are

complete. Second, the model should have a clear functional relationship to other
 9

acceptable models; models possessing this property are equivalent. Third, a small

variation in the geometry of the robot should effect small only changes in the model

parameters; models possessing this property are proportional. It was mentioned in the

same paper that the properties of completeness and proportionality in a kinematic model

are necessary for the model to be used in a general calibration scheme. These

definitions of completeness, equivalence and proportionality are also useful in

describing the ability of a kinematic model to preserve the physical/mathematical

parameter relationship.

Although manipulator links are considered to be perfectly rigid in all current modeling

methods, the process of parameter estimation may be visualized as allowing any

possible variation to the mathematically flexible links and/or joint transducers so that

the aggregate end effector positioning error is reduced. "Any possible variation"

implies any change in the dimensions or characteristics of a manipulator component

which render the features of that component different from its ideal or initially estimated

features. The optimized parameter values obtained as a result of the calibration

(optimization) process are again attributed to rigid manipulator components, and the

pose of the EECF is subsequently calculated in terms of those constant parameters. The

quality of completeness is related to the capability of a kinematic model to describe "any

possible variation" in the physical kinematic structure.

In the present work, the objective is to relate the optimized value of the parameters to

the physical features of the links and the joints of the manipulator. It is expected that a

change in a dimension of a component will be reflected as a change in its corresponding

mathematical parameter. Thus it has to be ensured that the kinematic model actually

describes the physical dimensions and that each possible independent variation of the

robot components are modeled separately. This is achieved by considering a link in

three-dimensional space with a coordinate frame attached to each of its two ends.

Keeping one frame fixed, the pose of the other frame can be changed in six independent
 10

ways — three corresponding to translation (along the fixed frame x, y, and z axes) and

three corresponding to rotation (about the fixed frame x, y, and z axes). In other

words, a combination of six primitive transformations is needed to compose the total

transformation between the two frames (Stone, 1987). These are the kinematic features

of the link under consideration.

Each of the six kinematic features described above is independent in the sense that each

has an effect of its own and cannot be duplicated by any combination of the other five.

Mathematically speaking, the complete relationship between two unconstrained

coordinate frames can be represented by a vector in a six-dimensional space. The

translational and rotational directions are represented by the six basis vectors of that

six-dimensional space. For any model to be able to reflect the change in any of the six

kinematic features of a link, it has to have at least six parameters (a necessary condition)

spanning the six-dimensional space (a sufficient condition) described above. Zeigert

and Datseris (1988) arrived at a similar conclusion .

It is also to be understood that the modeling must be completely unconstrained.

Constrained models may describe the pose of the end effector with a smaller number of

parameters, but they fail to describe each of the kinematic features of the links

separately. This prevents the gaining of a clear picture of the mathematical/physical

parameter relationship.

The quality of proportionality in a kinematic model affects both the stability of the

numerical optimization algorithm and the interpretation of the optimized parameters. It

has been noted (Everett, 1987; Hayati, 1983) that the Denavit-Hartenberg notation lacks

this property in case where the model includes two parallel or nearly parallel joints.

Proportionality of a kinematic model is an outgrowth of the physical nature of a

manipulator, and if the parameters of a model depict, at least to some extent, the
 11

physical dimensions of the links, then proportionality of the kinematic model will be

inherently achieved.

It is interesting to note that most of today's popular kinematic models do not satisfy the

fundamental requirements above. The ubiquitous Denavit-Hartenberg model uses only

three parameters per link (namely, link length, link twist, and joint offset), and

therefore is not generally a useful model for calibration purposes. The more closely the

mathematical parameters relate to the physical shape of a link, the more closely will the

optimized parameters tend to relate to the true shape of the link. A better description of

the mathematical/physical parameter relationship will be obtained if the line joining the

two coordinate frames travels through the body of the link instead of traveling through

space. Although achieving this goal may be difficult for any arbitrarily shaped link, for

most common industrial robots this condition is ensured by merely establishing the

coordinate frame origins inside the body of the robot. Figure 2.1 illustrates two typical

coordinate frame choices, one of which reflects the kinematic features of the link (the

unprimed frames) and the other which does not (the primed frames). When such

information has been lost due to choice of coordinate frame locations, it cannot be

regained through calibration.

 12

Figure 2.1: Two Typical Coordinate Frame Attachments

 13

2.3 Description of a Suitable Model

The kinematic notations proposed by Sheth and Uicker (1971) are well-suited for

calibration. In the current work a similar model used by Bosnik (1986) is adopted. It

is closely related to that used by Sheth and Uicker, is easy to understand in terms of

mathematical/physical relationships, and is simple to assign to the kinematic chains.

Figure 2.2 summarizes the notation used in this work. Under this system, the link

transformation matrix is

TL = \B\BC\[(\A\AC\VS9\HS16\CO4(cαcβ,cαsβsγ-

sαcγ,cαsβcγ+sαsγ,x,sαcβ,sαsβsγ+cαcγ,sαsβcγ-cαsγ,y,-

sβ,cβsγ,cβcγ,z,0,0,0,1)) , (2.1)

where x, y, and z represent the position of the distal coordinate frame origin with

respect to the proximal coordinate frame; α, β, and γ represent a set of ZYX-ordered

Euler angles relating the distal coordinate frame to the proximal frame; and cα =

cosine(α), sα = sine(α), etc.

The manipulator joint variables must be measured accurately during the course of an

experimental session. For the revolute and prismatic joints commonly found in robotic

manipulators, joint transducers in common use include potentiometers, synchros,

resolvers, contact encoders, and non-contact magnetic and optical encoders. The

number of parameters chosen for each joint depends upon the nature of the

characteristic curve of the transducer at that particular joint. Assuming all transducers to

be linear in nature, we must add two parameters per joint, expressing the joint variable

for joint "i" in terms of its parameters as

 14

Figure 2.2: Summary of Kinematic Notation

 15

di = d0i + kiVi (2.2)

for prismatic joints, or

θi = θ0i + kiV i (2.3)

for revolute joints, where d0 and θ0 are the joint transducer zero point offsets, k is the

joint transducer output slope, and V is the joint transducer output voltage. The joint

transducer model need not be linear, of course, and additional terms may be added as

necessary to accommodate nonlinear transducer characteristics. The joint

transformation matrices, then, are

T p,J = [ 1,0,0,0,0,1,0,0,0,0,1,d,0,0,0,1 ] (2.4)

for prismatic joints, and

T r,J = [ cθ,-sθ,0,0,sθ,cθ,0,0,0,0,1,0,0,0,0,1 ] (2.5)

for revolute joints, where again cθ = cosine(θ) and sθ = sine(θ).

The most general robotic manipulator is an open-loop six degree-of-freedom (DOF)

kinematic chain composed of a base and an end effector connected by five intermediate

links and six intermediate joints, as shown in Figure 2.3. Allowing six parameters per

link and two parameters per joint, a total of 54 parameters is required to completely

describe the manipulator for calibration purposes via this model. This, therefore, is the

minimum set of parameters required to maintain the physical/mathematical parameter

relationship. The aggregate transformation matrix relating the EECF to the global
 16

coordinate frame may then be expressed as a sequential multiplication of the 13 local

homogeneous transformation matrices, as follows:

T = T 1,L T 1,J T 2,L T 2,J T 3,L ... T 6,L T 6,J T 7,L . (2.6)

The global position and attitude of the EECF can be extracted from the matrix T above

by any standard method found in textbooks (Fu et al., 1987).

2.4 Chapter Summary

In this chapter it was emphasized that the choice of a particular kinematic model is

important in calibration. Often, the choice becomes crucial, determining the success

and failure of the calibration scheme. To model all the kinematic features of a

manipulator, six parameters are needed per link and two parameters per joint

(considering linear joint transducer characteristics). A suitable model was described

which possesses all the necessary qualities of a model to be used for a parameter

estimation procedure.
 17

Figure 2.3: General Six Degree-of-Freedom Robotic Manipulator

 18

Chapter 3

LEAST SQUARES PARAMETER ESTIMATION TECHNIQUES

3.1 Introduction

The various methods of calibration which have been proposed differ mainly in the

choice of their respective kinematic models. Regardless of the kinematic model chosen,

some differences between the true pose of the EECF (as measured externally) and the

pose predicted by the internal kinematic model will be observed for any real robot.

These differences, known as residuals, will be observed at each calibration posture.

The aggregate sum-of-squares of the residuals φ may be calculated as

e e e
φ = \I\SU(i=1,n, [(xG\O(i,e)-xGi)2+(yG\O(i, )-yGi)2+(zG\O(i, )-zGi)2+(αG\O(i, )-
α Gi)2+(β Gi,e-β Gi)2+(γGi,e-γGi)2]) , (3.1)

where n is the number of calibration postures, the superscripted terms are the externally

measured global position and attitude values, and the unsuperscripted terms are the

global position and attitude values of the EECF as predicted by the internal kinematic

model. The parameter values used by the kinematic controller of the manipulator may

not be the parameters' actual physical values. The sum-of-squares of the residuals φ is

the objective function, and it is the goal of the optimization to minimize this to ensure

best possible performance of the manipulator. The set of parameters for which φ is

minimum is the desired or optimal set.

In the procedure described above, data is collected at only one site on the manipulator

body in a single posture, namely at the EECF. It is also possible to collect data at

several different points on the manipulator in a single posture. In this latter case, the
 19

objective function will be the aggregate sum of the squares of the positional and

orientational errors at all the sites and summed over all the postures.

In the above discussions it has implicitly been assumed that the external measurements

are perfect. This assumption might lead to erroneous calibration results if the

instruments used for external pose measurements contain errors. Considering the fact

that the magnitude of accuracy in the measuring devices (sonic digitizer, as an example)

are much more than those of the typical manipulators, it can be safely said that the

minimization of the objective function (Equation 3.1) would contribute significantly to

the kinematic accuracy of the manipulator.

Regardless of what external measuring method is chosen, three basic considerations

must be satisfied as listed by Bosnik (1986): (1) positioning precision and knowledge

of the relative position and attitude at each calibration posture must be significantly

better than the desired manipulator accuracy, (2) measurement of the transducer output

signals must be at least as accurate as desired manipulator accuracy, (3) a sufficient

number of calibration postures should be measured so as to representatively populate

and span the desired manipulator workspace. Also, the number of calibration postures

should be greater than one-sixth the number of estimated parameters (if using all six

pose quantities at each calibration posture, that is three radial vector components and

three Euler angles).

 20

3.2 Quadratic Model of The Objective Function

The multivariable function φ depends on all the 54 parameters describing the

manipulator and it can be expressed as the second-order expansion of a Taylor series as

φ(p+∆p) ≅ φ(p) + gT∆p + \F(1,2!) ∆pTH∆p , (3.2)

where

pT = {p1 p2 p3 ... p54} (3.3)

is the transpose of the vector of parameter initial guesses,

∆pT = {∆p1 ∆p2 ∆p3 ... ∆p54} (3.4)

is the transpose of the vector of parameter updates at each iteration,

gT = ∇φ = { \F(∂φ,∂p1) \F(∂φ,∂p2) \F(∂φ,∂p3) ... \F(∂φ,∂p54) } (3.5)

is the transpose of the gradient vector, and

H = ∇ 2φ =
\B\BC\[(\A\AC\VS9\HS16\CO5(\F(∂2φ,∂p12),\F(∂2φ,∂p1∂p2),\F(∂2φ,∂p1∂p3),...,\F(
∂ 2 φ,∂p 1 ∂p 54 ),\F(∂ 2 φ,∂p 2 ∂p 1 ),\F(∂ 2 φ,∂p 2 2 ),\F(∂ 2 φ,∂p 2 ∂p 3 ),...,\F(∂2 φ,∂p 2 ∂p 54 ),\F(∂
2 φ,∂p
3 ∂p 1 ),\F(∂
2 φ,∂p
3 ∂p 2 ),\F(∂
2 φ,∂p 2 ),...,\F(∂ 2 φ,∂p ∂p ),\o(., , . ),\o(., , . ),\o(.,
3 3 54 . . .
,.) ,
,\o(.,.,.),\F(∂ 2 φ,∂p 54 ∂p 1 ),\F(∂ 2 φ,∂p 54 ∂p 2 ),\F(∂ 2 φ,∂p 54 ∂p 3 ),...,\F(∂ 2 φ,∂p 54 2 ))) (3.6)
 21

is the Hessian matrix, which is always square, symmetric, and quadratic. The gradient

vector ideally has zero magnitude at the minimum of the objective function. The

right-hand side of Equation 3.2 is the quadratic model of the actual function on the

left-hand side. This model will be reasonably accurate if its application is restrained to a

sufficiently small neighborhood of the current objective function value.

The Hessian matrix must be either positive definite or positive semi-definite.

Geometrically, an nth-order positive definite Hessian matrix represents a paraboloid in

(n+1)-dimensional space and its contour represents an n-dimensional ellipsoid. In the

special case of three-dimensional space, the surface will be a paraboloid opening toward

the positive z-axis. For equal eigenvalues, the horizontal cross section (contour) of this

paraboloid will be a circle, and for unequal eigenvalues, the cross section will be an

ellipse. The shape of the ellipse will depend on the relative magnitudes of the

eigenvalues, the major axis being in the direction of the eigenvector associated with the

smaller eigenvalue. In any case, if the Hessian is positive definite, the associated

surface as well as the associated function will have a definite minimum corresponding

to a unique set of variables. The position of the minimum is at the vertex of the

paraboloid.

If one of the system eigenvalues becomes zero, the major axis of the n-dimensional

surface becomes infinitely long, and the Hessian is positive semi-definite. In three

dimensions the surface will be an infinitely long parabolic trough. Thus, one of the

vertical sections will be a straight line and another will be a parabola opening

indefinitely towards the positive z-axis. The surface associated with a positive

semi-definite matrix does have a definite minimum, but different combinations of

dependent variables may yield the same minimum value. Once the bottom (minimum)

of the trough is reached, no change in the magnitude of the function will be observed

along any of the directions associated with zero eigenvalues. A positive semi-definite
 22

Hessian is, of course, singular, and represents a system with an infinite number of

valid solutions.

Another important matrix associated with a robotic system is the Jacobian matrix J. For

a robotic system, a typical element of J expresses the change in the objective function at

a particular posture with respect to a change in one of the parameters. It is also possible

to express the elements of this matrix as a change in one of the global pose components
(xG, yG, zG, αG, βG, γG) of the EECF (as extracted from the aggregate transformation

matrix) at a particular posture with respect to a change in one of the kinematic

parameters. Recalling that there are 54 parameters in the proposed kinematic model for

a 6-DOF manipulator, the Jacobian matrix for a single posture takes on the expanded

form

Ji =

\B\BC\[(\A\AC\VS9\HS16\CO5(\F(∂xGi,∂p1),\F(∂xGi,∂p2),\F(∂xGi,∂p3),...,\F(∂xGi,∂p

54),\F(∂yGi,∂p1),\F(∂yGi,∂p2),\F(∂yGi,∂p3),...,\F(∂yGi,∂p54),\F(∂zGi,∂p1),\F(∂zGi,∂p2

),\F(∂zGi,∂p3),...,\F(∂zGi,∂p54),\F(∂α Gi,∂p1),\F(∂α Gi,∂p2),\F(∂α Gi,∂p3),...,\F(∂α Gi,

∂p 54),\F(∂β Gi,∂p1),\F(∂β Gi,∂p2),\F(∂β Gi,∂p3),...,\F(∂β Gi,∂p54),\F(∂γGi,∂p1),\F(∂γGi

,∂p 2 ),\F(∂γ Gi ,∂p 3 ),...,\F(∂γ Gi ,∂p 54 ))) , (3.7)

where "i" is the posture number. Each calibration posture adds 6 similar rows to the

total system Jacobian matrix J. In almost all the cases, the Jacobian matrix is a

rectangular matrix with number of rows exceeding the number of columns.

The definitions expressed in Equations 3.5 and 3.6 can now be rewritten in terms of the

Jacobian as

g = -2JTf (3.8)
 23

where f is the vector of residuals, and

H = 2JTJ . (3.9)

The above results can be found in most of the textbooks dealing with the minimization

of least squares error (Fletcher, 1980; Nash, 1979). Subsequently in this work, the

name Hessian is loosely used to represent the JTJ matrix, although they are not exactly

the same in the stricter sense.

3.3 Optimization Techniques

3.3.1 Steepest Descent Method

The steepest descent method is one of the simplest forms of gradient search methods.

The gradient search methods, in general, are based on the fact that the gradient of a

function points in the direction of its maximum rate of increase. Therefore, if the search

direction is chosen to be opposite to that of the gradient, it will be pointed in the

direction of the maximum rate of decrease of the function. This search technique

continues iteratively, until the minimum of the function is reached.

As described by Nash (1979), the basic iteration step taken by most of the descent

methods is

∆p = -tDg , (3.10)

where D is a matrix defining a transformation of the gradient and t is a scalar describing

the length of the step taken in the direction opposite to that of the gradient. For the

steepest descent method, D is chosen to be an identity matrix and t is chosen in such a

 24

way that the objective function is reduced in that particular step. In particular, if t=1,

the steepest descent step is determined as

∆p = -g . (3.11)

The elements of the update vector are added to the corresponding elements of the

parameter vector to give the optimal set of parameters for that particular iteration. This

process is continued until one or more pre-defined convergence criteria (corresponding

to a minimum objective function value) are met. For a steepest descent method, it is not

necessary to use the quadratic model of the objective function.

3.3.2 Gauss-Newton Algorithm

The Gauss-Newton algorithm is one of the most basic second-order optimization

techniques for non-singular systems. It utilizes the fact that the magnitude of the

gradient of a function at a minimum is zero. In this technique, the transformation

matrix D in Equation 3.10 is replaced by H-1, the inverse of the Hessian matrix.

Commonly, the value of t remains unity. The optimization scheme proceeds in an

iterative way. In each iteration an update vector ∆p is obtained via the equation

∆p = -H-1g , (3.12)

which minimizes the objective function in a small neighborhood of its current value.

Using the relations in Equations 3.8 and 3.9, the above equation can be rewritten as

∆p = (JTJ)-1(JTf) . (3.13)
 25

This is the normal equation, a well-known part of least squares estimation. Based on

the kinematic model chosen for this work, Equation 3.13 represents the solution of 54

parameter updates in a single iteration. Since the Jacobian matrix is not square,

Equation 3.13 cannot be written in any simpler form.

To ensure that the optimal parameter set will increase end effector positioning accuracy

throughout the desired workspace, calibration data should be collected at a number of

calibration postures sufficient to populate and span the workspace. This typically

requires many more calibration postures than the minimum that would be needed for the

evaluation of Equation 3.13 (Sommer and Miller, 1981).

3.3.3 Disadvantages of the Simpler Methods

Although the steepest descent method is guaranteed to reach a local minimum or a

saddle point, the greatest disadvantage with this method is that it is often very

inefficient. There are common cases in which the successive directions chosen by this

method are nearly opposites, and which are both nearly perpendicular to the direction in

which the minimum is to be found . This progression, sometimes called the

hemstitching pattern, is due to the search directions which are not linearly independent

(Nash, 1979). In computational terms, the steepest descent method requires an

excessively large number of iterations for convergence and, therefore, is not appealing

from a practical standpoint.

The Gauss-Newton method fails in case where the Hessian matrix is not positive

definite. It was observed by Fletcher (1980) that, even in case of positive definite

Hessians, convergence may not occur. For an nth-order system, the Gauss-Newton

method requires evaluation of n2 second derivatives, n first derivatives, and also the

inversion of a matrix. For a complicated function, these may be difficult to perform.

 26

For a function with a positive semi-definite Hessian, Equation 3.13 will represent an

infinite number of valid solutions.

3.3.4 Levenberg-Marquardt Algorithm

The difficulties encountered in the steepest descent method and in the Gauss-Newton

methods are rectified in an algorithm developed by Levenberg (1944) and Marquardt

(1963) which is popularly known as the Levenberg-Marquardt algorithm. This

method has been applied successfully in various optimization schemes. According to

this algorithm, Equation 3.13 should be modified as

∆p = (JTJ+λI)-1(JTf) , (3.14)

where λ is an adjustable scalar and I is an identity matrix of the same order as that of

the Hessian. This approach essentially adds a positive quantity to each of the diagonal

elements of the Hessian. A sufficiently large λ will always be available which will

force the Hessian to be non-singular.

This method is a mixture of first order gradient search techniques and second order

Gauss-Newton techniques. For parameter estimates far from optimum, the quantity λ

typically increases and the method resembles a steepest descent technique. As the

search comes closer to the minimum, λ typically becomes very small, making the

search direction almost coincident with the Gauss-Newton search direction. This

technique successfully minimizes the objective function.

On some occasions matrix I in Equation 3.14 is replaced by the diagonal matrix B2,

where the elements of B are defined as

 27

Bii = √
Hii for Hii ≠ 0
(3.15)
Bii = 1 for Hii = 0 .

This replacement provides a better scaling of the parameters in that it helps to improve

the numerical conditioning of the system.

Geometrically, the Levenberg-Marquardt algorithm transforms the original surface in

such a way that the unique minimum of the transformed surface is equal to the

minimum (minima) of the original surface. Thus, good updates are obtained and

relatively few steps are required to reach the minimum of the objective function.

3.4 Singular Value Decomposition

Singular value decomposition is one of the most robust mathematical approaches to

solve a least squares problem. Although it can be directly applied to evaluate the

solutions as represented in Equation 3.13, it is possible to avoid the calculations in

Equations 3.8 and 3.9 and to solve a simpler system, as follows. Equation 3.13 can be

rewritten in terms of the Jacobian matrix as

J∆p = f . (3.16)

The above system of equations has a solution if f lies in the column space of J. For a

typical manipulator, the vector representing the residuals does not lie in the column

space of J. This means that, in general, there is no parameter update vector for which

each of the residuals is identically equal to zero. Thus, a solution for Equation 3.16 is

not obtained. This is due to the fact that all the kinematic features of the manipulator are

not modeled and a re-evaluation of the modeled parameters might not always
 28

compensate for the unaccounted and unknown errors of the unmodeled ones. In such a

case, the algorithm actually attempts to identify a parameter set such that the quantity

||J∆p-f|| is minimum (Strang, 1976). In terms of the three dimensional case, illustrated

in Figure 3.1, the condition to be satisfied is that the vector joining the point P (where

OP represents the optimal parameter set) and F (where OF represents the residual

vector) must be perpendicular to the (planar) column space.

The uniqueness of a solution in a least squares scheme is ensured if the columns of the

Jacobian are linearly independent. If they are not, the system has an infinite number of

valid solutions. This is because the null space of the Jacobian matrix is non-zero. In

this case the point P is still uniquely determined by being closest to F, but it can now be

expressed as a non-unique combination of the columns of the Jacobian, and again an

infinite number of solutions results.

One of the ways to deal with such a problem is to make a choice according to the rule

that, among all solutions, the optimal solution (a vector) is the one that has the

minimum magnitude. This can be achieved by the singular value decomposition

technique. In this method, the m×n Jacobian matrix (where m = 6 times the number of

postures, n = number of parameters, and m > n) can be factored into

J = Q1 E Q 2 , T , (3.17)
 29

Figure 3.1: Least Squares Estimation Model (3-D Case)

 30

where Q1 is an m×m orthogonal matrix, Q2 is an n×n orthogonal matrix and E is

expressed as

E = [ Σ,0,0,0 ] , (3.18)

where Σ = diagonal(σ1, σ2, σ3, … σr), σi being the ith singular value of the

Jacobian matrix and r being the rank of the Jacobian. The inverse of the Jacobian is

obtained by

J + = Q2 E + Q 1 , T , (3.19)

where J+ is called the pseudoinverse or the generalized inverse of the Jacobian and E+

is the pseudoinverse of E. E+ is obtained by merely replacing the singular values of E

with their respective reciprocals. Equation 3.16 becomes

∆p = J+f . (3.20)

In a similar way the pseudoinverse of JTJ can also be calculated to evaluate Equation
3.13. For a symmetric matrix such as the Hessian, the columns of Q2 will be a set of

orthonormal eigenvectors. It is general practice to set very small singular values to zero

rather than to invert them; otherwise, the very large reciprocals of the small singular

values overwhelm the pseudoinverse.

 31

3.5 Chapter Summary

The least squares parameter estimation scheme was detailed in this chapter. The

objective function was expanded as a second-order Taylor series and optimization

techniques were employed to minimize this function. Two simple techniques, namely,

the steepest descent method and the Gauss-Newton algorithm were described. These

simpler methods, though easy to implement, suffer from some inherent difficulties.

There are more efficient algorithms which are based on the simpler algorithms but

which rectify most of their computational difficulties. Two such algorithms are the

Levenberg-Marquardt algorithm and the singular value decomposition algorithm.

 32

Chapter 4

SIMULATION, RESULTS, AND DISCUSSION

4.1 Introduction

Several kinematic model/calibration algorithm combinations have been investigated,

using both actual external position data collected from the manipulator and synthesized

position data for the same manipulator. The optimization algorithms were implemented

in VAX FORTRAN (version 4.8) and were run on a VAX 11/750 minicomputer.

Some subroutines were taken from the IMSL mathematical and statistical subroutine

library.

4.2 Kinematic Description of the Manipulator

Most of the investigations were based on a General Electric Model A4 industrial

manipulator. As shown in the Figure 4.1, the A4 is a 4 DOF manipulator of the RRPR

type, having five links and four joints. Coordinate frames are attached to both ends of

each of the links. The base coordinate frame can be suitably fixed at a point during

calibration which is convenient for data collection. According to the kinematic model

used in this work (discussed in Section 2.3) 38 parameters (30 geometric and 8

electrical) are needed to describe the A4 completely. The nominal values of these

parameters are listed in Table 4.1, where the lengths are measured in inches and the

angles are measured in degrees. The parameters are numbered in a sequence for easy

reference (parameter numbers appear in parentheses following each parameter). Table

4.1 should be read as follows: the x-coordinate length of the first link is 15.080 inches
and it is parameter p1. In the ideal
 33

Figure 4.1: Coordinate Frame Attachments for GE Model A4 Robot Manipulator

 34

Table 4.1: Optimal Values of Parameters for Synthesized Data (Notations in

Parentheses are Parameter Identifiers):

Link 1 Link 2 Link 3 Link 4 Link 5

x (in) 15.080 (p1) 0.000 (p7) 0.000 (p13) 0.000 (p19) 0.875
(p25)

y (in) -8.500 (p2) 15.750 (p8) 11.810 (p14) 0.000 (p20) -0.125
(p26)

z (in) 10.540 (p3) 0.000 (p9) 0.000 (p15) 1.500 (p21) 3.000
(p27)

α (deg) 0.000 (p4) 0.000 (p10) 0.000 (p16) 0.000 (p22) 171.000
(p28)

β (deg) 180.000 (p5) 0.000 (p11) 0.000 (p17) 0.000 (p23) 0.000
(p29)

γ (deg) 0.000 (p6) 0.000 (p12) 0.000 (p18) 0.000 (p24) 90.000
(p30)

Joint 1 Joint 2 Joint 3 Joint 4

θ oi 90.000 (p31) -117.000 (p32) 0.250 (p33) 200.000 (p34)

(degrees) (degrees) (inches) (degrees)

ki 7.200 (p35) 7.200 (p36) 1.400 (p37) 9.000 (p38)

(degrees/volt) (degrees/volt) (inches/volt) (degrees/volt)

TYPE revolute revolute prismatic revolute

 35

model the four joint axes are parallel, regardless of posture, and this makes the A4 a

rather interesting case for study, as will be described subsequently.

4.3 Synthesis of Data

The parameter estimation algorithms were tested on synthesized data. This approach

provided a theoretical case for which the minimum value of the objective function φ

was known to be zero and the corresponding optimal parameter set was known in

advance. The optimal parameters were arbitrarily fixed to be the design parameters,

since that set of parameters was readily available.

The forward kinematic program generated the position and orientation of the EECF for

the optimal set of parameters and for an arbitrary set of joint voltages. This was

repeated for about 30 different sets of voltages for which the end effector populated the

workspace of the manipulator reasonably well. The next step was to perturb the values

of some of the parameters to simulate the condition in which those parameters had

undergone some distortion in course of the operation of the manipulator. The

calibration algorithm was then run and it was observed whether the perturbed values

returned to their original values when the objective function converged to zero. This

allowed for a reasonable comparison of the performance of various kinematic models

and optimization schemes.

In case of multiple sites per posture calibration scheme, positional and orientational data

were collected not only at the EECF but also at some other convenient sites on the

manipulator body. As the position and orientation of a site with respect to its nearest

proximal coordinate frame are not known exactly, for each of the sites, six parameters

are added to the original list of parameters. These site parameters are not very important

in a calibration scheme because the knowledge of the exact position of a site with
 36

respect to the previous coordinate frame does not directly help in reducing the EECF

positioning error. Nevertheless it is required to include those parameters in the

optimization scheme to gain some indirect advantages which will be described later in

this chapter.

4.4 Results and Discussion

4.4.1 Effect of Parameters on EECF Pose

If the repetitive multiplication of the matrices of the right hand side of Equation 2.6 is

performed, the translational and rotational quantities can be extracted from the final

transformation matrix on the left hand side. Each of the six pose components will be

functions of the kinematic parameters. Although the functions are complex in nature, it

is possible to get a good idea about the effect of the kinematic parameters on the pose of

EECF. The functions describing these relations should in general change from one site

to another in the manipulator workspace. A similar analysis was reported by Kirchner,

et al. (1987).

Another way to view this situation is to draw the characteristic curves of each of the

EECF pose components as they change with respect to the kinematic parameters.

Although, at first sight, it appears to take a long time to draw all the 228 possible curves

(for 38 parameters and 6 pose components), a pattern of behavior of the parameters

begins to emerge after only a few curves are drawn. Therefore, to demonstrate the

effect of variation of the kinematic parameters on the EECF pose component, only a

small subset of all of the possible curves are required. These curves give instant visual

representation of the nature of the characteristics of the parameters and of their effects

on EECF pose.
 37

A representative group of these curves are shown in the Figures 4.2, 4.3, and 4.4. The

EECF pose components are plotted along the vertical axes and the parameters along the

horizontal axes. When a certain EECF pose component does not depend on a certain

parameter, its characteristic graph with respect to that parameter is a horizontal straight

line. For each of the curves, the parameters are varied for 100 units, 50 units on each

side of their nominal values. In the particular posture considered for drawing the

above-mentioned figures, the joint voltages are -6.883V, 13.301V, 4.122V, and

-28.645V for joints 1, 2, 3, and 4, respectively, and the corresponding joint variables

are 40.443°, -21.233°, 6.021 inches, and -57.809°. With the help of forward

kinematics, the coordinates of the origin of the EECF were found to be at 28.576,

13.995, and 0.020 (all in inches) with respect to the global origin. The attitude of the

EECF, represented by ZYX-ordered Euler angles, was found to be 47.600°, 0°, and

-90° respectively. Table 4.2 lists the joint variables and EECF pose components for 5

typical postures. Data from posture #4 of this table were selected for drawing the

curves.

In each of Figures 4.2, 4.3, and 4.4, a vertical dotted line is drawn on the parameter

axis at the nominal parameter value. Where it meets the curve, another line is drawn

horizontally to meet the vertical axis at the corresponding EECF pose component value.

The nominal value of a parameter and the corresponding pose component can also be

obtained from forward kinematics. Thus we can compare the known analytical solution

and the solution exhibited by a curve as a means of confirming the validity of the curve.

Figure 4.2 shows the characteristic curves for linear kinematic parameters. All the

curves are seen to be linear. Also, the EECF attitudes are found to be independent of

the linear parameters, as shown in Figures 4.2c and d.

 38

Figure 4.2: EECF Pose Characteristic Curves for Linear Parameters

 39

Figure 4.3: EECF Pose Characteristic Curves for Angular Parameters

 40

Figure 4.4: EECF Pose Characteristic Curves for Joint Transducer Parameters
 41

Table 4.2: Joint Voltages and EECF Pose Components for 5 Postures:

Posture Joint Voltages (volt)

Joint 1 Joint 2 Joint 3 Joint 4

1 -11.053 12.369 4.009 -11.583

2 -12.696 9.037 1.493 5.156

3 -13.197 9.964 3.714 -1.898

4 -6.883 13.301 4.122 -28.645

5 -8.793 12.922 3.391 -20.333

6 -9.353 6.637 1.760 11.523

7 -8.917 11.267 1.496 5.202

8 -6.367 12.479 1.369 -4.141

9 -6.034 12.081 0.159 -11.348

10 -6.420 6.843 0.314 -14.182

Posture EECF Pose Components

x (in) y (in) z (in) α (deg) β (deg) γ (deg)

1 14.071 19.083 0.177 -69.233 0.000 -90.000

2 6.098 14.220 3.669 175.940 0.000 -90.000

3 5.120 15.465 0.590 -123.644 0.000 -90.000

4 28.576 13.995 0.020 47.600 0.000 -90.000

5 21.851 17.546 1.042 -10.730 0.000 -90.000

6 12.889 13.330 3.326 111.850 0.000 -90.000

7 20.452 16.645 3.695 132.262 0.000 -90.000

8 30.381 14.222 3.873 -170.732 0.000 -90.000

9 30.125 14.500 5.567 -105.410 0.000 -90.000

10 20.507 14.236 5.350 -39.404 0.000 -90.000

 42

In Figure 4.3, some of the characteristic curves for the angular parameters are shown.

They are mostly nonlinear relationships as a result of the sine and cosine terms in the

coordinate transformation matrices of the kinematic model. Typical characteristic

curves for joint transducer parameters are shown in Figure 4.4. All of the EECF pose

components are either independent with respect to the variation of the parameters

representing the transducer output slopes or repeat themselves after definite intervals.
Considering Equation 2.3, it is seen that with θ0i being kept constant, the characteristic

curve for ki repeats itself after every (360/Vi) degrees, where Vi represents the voltage

of the joint in consideration at the particular posture. This is observed in Figure 4.4c,

where the EECF global position repeats itself after every 52.303° interval.

It can also be verified that all the EECF pose components repeat themselves after every

360° interval with respect to the variation of the link Euler angles and the zero point

offsets for the revolute joints. This is demonstrated in Figure 4.5, where each of the

angular parameters has been varied for 370°, starting at 5° less than the nominal values.

The sudden jump of the characteristic curve in Figure 4.5c is attributed to the fact that

an angle less than -180° has been converted to a positive angle. For instance, -181° is

converted to 179°, and so on. In any case, we have a curve which repeats itself after

every 360° variation of the parameter.

 43

Figure 4.5: EECF Pose Characteristic Curves for Joint Transducer Parameters
(360° Variation)
 44

4.4.2 Ill-Conditioning of the System

During the computational runs, the Jacobian matrix was typically found to be

moderately ill-conditioned owing to the wide variation of magnitude of its elements.

This is shown in Table 4.3, which represents a typical Jacobian matrix (see Equation

3.7). For this particular run, 8 parameters were chosen for optimization, and EECF

positioning error data from the first 3 postures of Table 4.2 were considered. In Table

4.3, the parameters are shown at the top of each column. Depending on the posture,

the changes in positions of the EECF with respect to the changes in the Euler angles of

the most proximal link give rise to elements of the largest magnitudes. The elements

representing, for instance, the changes in positions of the EECF with respect to the

changes in length parameters are of considerably smaller magnitudes. The

ill-conditioning was found to get worse when the Hessian matrix was formed from the

Jacobian matrix using Equation 3.9. Table 4.4 shows the Hessian matrix calculated

from the Jacobian in Table 4.3. By comparing terms in Tables 4.3 and 4.4 the

worsening condition of the Hessian may be observed.

The numerical ill-conditioning of the Hessian is an important issue, as the Hessian must

be inverted to get updates, and ill-conditioning introduces numerical error during

inversion. Severe ill-conditioning often produces numerical behavior similar to

singularity, which makes it impossible to invert a matrix or gives a meaningless

numerical solution.

It was found that the ill-conditioning of the Hessian could be avoided to a great extent

by judicious choice of angular and linear units. Although the inputs and outputs of

angular data in the optimization algorithm were in degrees, all the numerical operations

in FORTRAN were done in radians. The Hessian matrix of Table 4.4 also employs

radians as its angular unit. An improvement in the conditioning of the Hessian was

observed by using degrees as the angular units (as permitted by VAX FORTRAN)
 45

instead of using
 46

Table 4.3: Jacobian Matrix (Radians, Inches) from 3 Postures without Parameter

Redundancy:

p1 p2 p3 p4 p5 p6 p14 p29

x 1.00 0.00 0.00 -22.53 -10.55 0.00 -0.33 0.00

y 0.00 1.00 0.00 13.44 0.00 -10.58 0.94 0.00

z 0.00 0.00 1.00 0.00 -13.47 -22.47 0.00 0.00

α 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00

β 0.00 0.00 0.00 0.00 0.67 0.74 0.00 -1.00

γ 0.00 0.00 0.00 0.00 0.74 -0.67 0.00 0.00

...........................................................................................
x 1.00 0.00 0.00 -26.06 -9.51 0.00 -0.05 0.00

y 0.00 1.00 0.00 6.70 0.00 -9.56 1.00 0.00

z 0.00 0.00 1.00 0.00 -6.75 -26.02 0.00 0.00

α 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00

β 0.00 0.00 0.00 0.00 0.98 -0.19 0.00 -1.00

γ 0.00 0.00 0.00 0.00 -0.19 -0.98 0.00 0.00

...........................................................................................
x 1.00 0.00 0.00 -27.58 -10.36 0.00 -0.30 0.00

y 0.00 1.00 0.00 -1.08 0.00 -10.43 0.95 0.00

z 0.00 0.00 1.00 0.00 1.03 -27.56 0.00 0.00

α 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00

β 0.00 0.00 0.00 0.00 0.35 -0.93 0.00 -1.00

γ 0.00 0.00 0.00 0.00 -0.93 -0.35 0.00 0.00

 47

Table 4.4: Hessian Matrix (Radians, Inches) from 3 Postures without Parameter
Redundancy:

p1 p2 p3 p4 p5 p6 p14 p29

p1 3.00 0.00 0.00 -76.16 -30.42 0.00 0.07 0.00

p2 0.00 3.00 0.00 19.07 0.00 -30.57 2.89 0.00

p3 0.00 0.00 3.00 0.00 -19.18 -76.05 0.00 0.00

p4 -76.16 19.07 0.00 2177.32 771.53 -195.04 18.00 0.00

p5 -30.42 0.00 -19.18 771.53 540.33 449.73 -0.81 -2.01

p6 0.00 -30.57 -76.05 -195.04 449.73 2256.54 -29.49 0.38

p 14 0.07 2.89 0.00 18.00 -0.81 -29.49 3.00 0.00

p 29 0.00 0.00 0.00 0.00 -2.01 0.38 0.00 3.00

Note: The Singular Values for this Matrix are 2488.575, 2370.407, 121.603, 5.398,
2.969, 0.149, 0.100, and 0.005.
 48

radians. The resulting Hessian is shown in Table 4.5. The quotient of the largest and

the smallest eigenvalues of a symmetric matrix is known as its condition number, and is

an indication of the severity of the matrix's ill-conditioning. The larger the condition

number, the poorer the conditioning of the matrix. Notice that the condition number

decreased from about 446781 (in Table 4.4) to a much smaller value of 61 (in Table

4.5). Even after removing the ill-conditioning caused by unwise choice of physical

units, solution of Equation 3.13 was found impossible to obtain due to singularity of

the Hessian matrix as discussed further below.

During the computational runs, the theoretically singular Hessians, however, were

often found to be numerically nonsingular, and thus invertible. It was interesting to

note that, in many of the cases where the Hessian was nearly singular but invertible,

one of the major goals of calibration was still attained, namely, the objective function

was reduced considerably. The parameter updates in those cases were strange,

however, and the parameters wandered numerically even at solution convergence. The

subsequent "optimal" parameters, therefore, did not accurately reflect the physical

kinematic characteristics of the manipulator. Even if the sole objective of optimization

is that of reducing the end effector positioning error, this is not a reliable approach as it

is not possible to predict when the numerical error will be sufficient to make a

theoretically singular matrix invertible.

 49

4.5: Hessian Matrix (Degrees, Inches) from 3 Postures without Parameter

Redundancy:

p1 p2 p3 p4 p5 p6 p14 p29
p1 3.00 0.00 0.00 -1.33 -0.53 0.00 0.07 0.00

p2 0.00 3.00 0.00 0.34 0.00 -0.53 2.89 0.00

p3 0.00 0.00 3.00 0.00 -0.34 -1.33 0.00 0.00

p4 -1.33 0.34 0.00 3.67 0.23 -0.06 0.32 0.00

p5 -0.53 0.00 -0.34 0.23 3.17 0.14 -0.01 -2.01

p6 0.00 -0.53 -1.33 -0.06 0.14 3.69 -0.51 0.38

p 14 0.07 2.89 0.00 0.32 -0.01 -0.51 3.00 0.00

p 29 0.00 0.00 0.00 0.00 -2.01 0.38 0.00 3.00

Note: The Singular Values for this Matrix are 6.274, 5.338, 4.604, 4.402, 1.963,
1.891, 0.945, and 0.102.
 50

4.4.3 Parameter Redundancy

By calculating the singular values of the full Hessian matrix (which is of order 38 in the

present model), it was found to be severely rank deficient. Mathematically, the

presence of zero singular values in the Hessian means that there are linear dependencies

among its columns. From a physical standpoint it means that there are more parameters

in this kinematic model than are needed to completely describe a change in the EECF

pose of the manipulator being observed. These extra parameters will, henceforth, be

termed redundant parameters and the phenomenon will be referred to as redundancy.

In the general case, it is difficult to identify the columns in the Hessian which are

linearly dependent. The singular value decomposition only yields the number of

dependency relations among the Hessian columns. Table 4.6 shows a Hessian matrix

which was formed from the data of 34 different postures of the A4 manipulator. Eight

parameters were chosen to be re-evaluated (optimized). By singular value

decomposition, the matrix is shown to have one zero singular value, which implies that

there is one linear dependency among its columns. For a general manipulator, the

parameters tend to be highly interactive in behavior, and in a typical case one parameter

can depend on many other parameters, making the physical interpretation of the

dependency relation extremely complicated.

There is, however, a special case of redundancy which is easy to detect and interpret.

For the purposes of this work, that particular type of redundancy will be termed

identical redundancy . In this case, the respective derivatives of all the EECF pose

components with respect to the identically redundant parameters will be of equal

magnitude regardless of the posture. This results in two or more identical columns in

the Jacobian, and consequently in the Hessian. Table 4.7 shows this special case of
redundancy. Parameters p3 and p9 are identically redundant, and they give rise to two

columns whose corresponding elements are

 51

Table 4.6: Hessian Matrix from 34 Postures with One Redundancy:

p3 p4 p5 p6 p7 p8 p10 p13

p3 34.00 0.00 -1.37 -10.66 0.00 0.00 0.00 0.00

p4 0.00 38.53 1.42 -0.18 10.25 -5.68 -35.69 8.18

p5 -1.37 1.42 35.59 0.38 3.08 -2.47 -0.57 2.73

p6 -10.66 -0.18 0.38 38.27 -2.46 -3.07 -0.49 2.39

p7 0.00 10.25 3.08 -2.46 34.00 0.00 -0.91 4.28

p8 0.00 -5.68 -2.47 -3.07 0.00 34.00 5.68 -27.51

p 10 0.00 -35.69 -0.57 -0.49 -0.91 5.68 35.43 -7.00

p 13 0.00 8.18 2.73 2.39 4.28 -27.51 -7.00 34.00

Note: The Singular Values for this Matrix are 83.406, 54.315, 46.259, 37.513,
31.466, 24.735, 6.142, and 0.000.
 52

Table 4.7: Hessian Matrix from 34 Postures with One Identical Redundancy:

p1 p2 p3 p4 p5 p9 p11 p12

p1 34.00 0.00 0.00 -10.66 -4.55 0.00 -3.07 -2.47

p2 0.00 34.00 0.00 1.37 0.00 0.00 2.46 -3.07

p3 0.00 0.00 34.00 0.00 -1.37 -34.00 5.68 -0.98

p4 -10.66 1.37 0.00 38.53 1.42 0.00 1.36 0.75

p5 -4.55 0.00 -1.37 1.42 35.59 1.37 23.28 18.93

p9 0.00 0.00 -34.00 0.00 1.37 34.00 -5.68 0.91

p 11 -3.07 2.46 5.68 1.36 23.28 -5.68 35.64 -0.06

p 12 -2.47 -3.07 -0.91 0.75 18.93 0.91 -0.06 35.08

Note: The Singular Values for this Matrix are 71.005, 65.746, 45.86, 37.938,
30.283, 24.997, 5.032, and 0.000.
 53

of equal magnitude (although opposite in sign in this particular case). This is a case of

direct linear dependency and consequently there is one zero singular value for this

matrix, as is shown in the table. Identical redundancy is easy to detect — a visual

inspection of the Hessian is sufficient to tell which of its columns are equal and thus

which parameters are redundant.

Most commonly, identical redundancy occurs in a robotic manipulator (and in its

corresponding kinematic model) if there are two (or more) joints whose axes are parallel

in all postures. In the manipulator studied here, the observed conditions for identical

redundancy can be classified as follows:

1. For any two or more parallel joints, the parameters of the related link dimensions

along the joint axes (Z-component) are redundant. This condition gives rise to 4
identical redundancies among the 5 parameters p3, p9, p15, p21, and p27.

2. For two consecutive parallel joints, the Z-Euler angle α for the link between the

parallel joints is redundant with the zero-point offset of the distal joint, if the distal
joint is revolute. Thus the parameter pairs of p4-p31, p10-p32, and p22-p34 are

identically redundant. If the distal joint is prismatic, its zero-point offset has units

of length and it does not match up with the angle α of the preceding frame.

3. For two parallel revolute joints with a prismatic joint in between, another

redundancy is obtained if the axis of the prismatic joint coincides with the distal

revolute joint. In this case, the Z-Euler angle α for the proximal revolute joint is

found to be identically redundant with the zero-point offset of the distal revolute
joint. Parameter p16 is found to be identically redundant with parameter p34 for of

this reason.
 54

4. For one or more parallel prismatic joints, additional redundancies result. First of

all, the length parameter along the joint axis becomes redundant with the zero-point
offset of the prismatic joint. Parameters p21 and p33 are therefore identically

redundant. If the distal joint is prismatic, the parameters related to the length

components along the X and Y directions also become identically redundant.

Parameter pairs p13-p19 and p14-p20 are found to be identically redundant for this

reason. This does not happen if the distal joint is revolute and the frontal joint is

prismatic.

Thus it is found that there are in total 11 identical redundancies in the GE model A4

manipulator. If the same manipulator is modeled with a different kinematic model, the

number of identical redundancies may change. The first condition stated above will,

however, be satisfied by any kinematic notation (complete or incomplete) which assigns

its coordinate axes along the axes of the joints. In the Denavit-Hartenberg model, for

instance, the joint offsets for any two consecutive parallel axes will be redundant.
 55

4.4.4 Effect of Parameters on the Objective Function

Considerable insight into the nature of the objective function can be gained by

observing its changes with respect to the change in only one kinematic parameter at a

time. The curve thus obtained will represent the intersection of the multivariable

objective function surface with a plane along the axis representing that particular

parameter. Following the reasoning described in section 3.2, a curve obtained in such a

way is expected to be a parabola, and the magnitude of the objective function calculated

at the optimal values of the parameters is located at the vertex of the parabola. In the

present case, with synthesized data, this minimum is zero.

Figures 4.6 to 4.10 are all drawn considering the aggregate sum-of-squares errors from

ten postures, which are shown in Table 4.2. Figure 4.6 shows the characteristic curves

for link length parameters. The parameter value is varied for 10 units on both sides of

the optimal value. The curves are smooth and are of perfectly parabolic shape. The

minima for these curves are at the optimal values for these parameters, as seen in the

figures. As far as these parameters are concerned, the optimization algorithm will have

no difficulty in reaching the minimum value in a very few iterations.

Figures 4.7 shows typical characteristic curves for link Euler angles. The parameters

are varied for 370°, starting at 5° less than their nominal values. The curves are of a

parabolic nature and repeat themselves after every 360° interval of variation. Also, the

curves are symmetrical in nature about the nominal parameter values. The repetition of

these characteristic curves confirms the fact that after every 360° rotation of these

parameters, the manipulator configuration is repeated. Figure 4.8 shows additional

characteristic curves for link Euler angles. They are also smooth, symmetrical about

nominal values, and self-

 56

Figure 4.6: Sum-of-Squares Characteristic Curves for Link Lengths

(p 1, p3, p8, p26)
 57

Figure 4.7: Sum-of-Squares Characteristic Curves for Link Euler Angles

(p 4 , p16, p28 , p30 )
 58

Figure 4.8: Sum-of-Squares Characteristic Curves for Link Euler Angles

(p 5 , p12 , p17 )
 59

repeating after every 360°. However, these curves are clearly not parabolic. The

reason for their deviation from parabolic shape is not yet clearly known.

As shown in Figure 4.9, the characteristic curves for the parameters representing the

joint transducer zero point offsets are essentially of the same nature as those of the link
length and link Euler angle parameters. The curve corresponding to p33, which is

related to a prismatic joint, is similar to that of a link length parameter and does not

repeat itself, as seen in Figure 4.9c.

As explained below, the characteristic curves for the transducer output slopes (revolute

joints) are the most complex and interesting in the sense that they do not have a smooth

parabolic shape at wide variations from the optimal value. These curves are shown in

Figures 4.10a, b, c, and d. The characteristics of the transducer output slope of the 4th
joint (p38) is studied in a greater detail in Figure 4.11. For two different postures of the

4th joint (posture #1 and posture #2 in Table 4.2), the variation of the objective function

with respect to parameter 38 is shown in Figures 4.11a and b. The joint voltages are

-11.583 V and 5.156 V, and the curves repeats themselves after every 31.08° and

69.819°, respectively. These two curves have only one minimum in common, which

corresponds to the optimal value. If data from two different postures are taken

simultaneously and plotted, Figure 4.11c is obtained. In this case the unique minimum

is clearly seen. Finally, Figure 4.11d shows the characteristic curve for 5 simultaneous

postures, and it is seen to be highly non-smooth in nature.

 60

Figure 4.9: Sum-of-Squares Characteristic Curves for Joint Zero-Point Offsets

(p 31 , p32 , p33 , p34 )
 61

Figure 4.10: Sum-of-Squares Characteristic Curves for Joint Transducer Output

Slopes
(p 35 , p36 , p37 , p38 )
 62

Figure 4.11: Detailed Sum-of-Squares Characteristic Curves for p38

 63

The lack in smoothness in the φ characteristics for transducer output slopes is

important, as it creates a number of valleys at some distance from from the optimal

value. Mathematically, this function cannot be called a "convex function." This is

dangerous, because if the initial estimate of such a parameter is far away from the

optimal value, or if one of the valleys happens to be near to the optimal value, there is a

possibility that the optimization algorithm will get stuck in one of those local minima

and will fail to converge to the global minimum. Some such cases have been

encountered where the magnitude of the gradient converges to a small value, but the

objective function remains relatively large. Most of the algorithms available will fail to

deal effectively with such a situation. Fortunately, in most cases, the initial guesses of

the parameter values (which, in the present work, are the same as the design values) are

sufficiently close to the corresponding optimal values and the possibility of the

termination of the optimization algorithm before reaching the actual minimum is

relatively rare.

4.4.5 Analysis of Solution of Levenberg-Marquardt Algorithm

It has been experimentally verified that the Levenberg-Marquardt algorithm does not

preserve the mathematical/physical parameter relationships if the kinematic model

contains redundant parameters. While the updated parameters serve to increase end

effector positioning accuracy, they do not, in general, correspond to actual dimensions

in the case of the redundant parameters. If the minimum of the objective function is

zero, the algebraic sums of different groups of redundant parameters in the optimal set

are found to be the same as they are in the physical set. Otherwise, the algebraic sums,

rather than the individual parameters, are found to converge towards a certain value

corresponding to the minimum of the objective function. During optimization, these

parameters merely readjust themselves so as to produce the unique algebraic sum.

 64

As an example, it has been found that the parameters p3, p9, p15, p21, p27 and p33 of

the GE A4 form an identically redundant set. It has also been found that during

optimization, the relationship that is preserved is

p3 - [p9 + p15 + p21 + p27 + p33] = 5.79 . (4.1)

The optimal values for these parameters are found from Table 4.1, and they also satisfy

the above equation.

In Table 4.8 results from 5 separate runs are tabulated. Attention is focussed on 8

parameters which are listed at the top of the table. For a particular run, the initial

guesses are listed in the first row and corresponding optimal values reached are listed in

the second row. The optimal values correspond to the objective function when it

reaches its minimum (zero in this case), and they are obtained as the output of the

Levenberg-Marquardt algorithm. It is observed that, in each of these cases, the

parameters which do not belong to the identically redundant set converge back to their

optimal values. This is the proof of correct performance of the optimization scheme.

The parameters belonging to the redundant set do not, however, converge to their

individual optimal values, but rather, they always satisfy Equation 4.1. The data of

Table 4.8 also demonstrates the dependence of optimal values on the initial guesses of

identically redundant parameters. Thus, if a certain parameter in a group of redundant

parameters undergoes some physical change, in the optimal set that change might not be

attributed to that particular parameter, but possibly to all the parameters in that group.
 65

Table 4.8: Typical Convergence Cases for Parameters with an Identical Redundant Set:

p1 p3 p9 p 10 p 15 p 21 p 27 p 33

Run 1
14.08 10.00 -0.40 2.00 0.80 1.00 3.00 0.35
15.08 10.09 -0.49 0.00 0.71 0.91 2.91 0.26

Run 2
14.08 12.00 -0.60 2.00 0.40 1.20 4.00 0.45
15.08 11.87 -0.47 0.00 0.53 1.33 4.12 0.57

Run 3
17.08 9.00 0.10 2.00 -0.20 0.30 2.80 0.50
15.08 8.97 0.13 0.00 -0.17 0.33 2.83 0.08

Run 4
17.08 13.00 0.60 2.00 -0.40 1.00 2.50 0.15
15.08 12.44 1.16 0.00 0.16 1.56 3.06 0.71

Run 5
14.08 12.00 -0.60 3.00 0.40 1.20 4.00 0.45
15.08 11.86 -0.47 0.00 0.52 1.32 4.13 0.57
 66

4.4.6 Analysis of Solution of Singular Value Decomposition

The kinematic model that has been used herein contains more parameters than are

actually needed to describe the kinematic relationship between EECF and global frame.

Therefore, some of the columns of the system Jacobian matrix are linearly dependent.

This results in an infinite number of valid solutions to Equation 3.16. The solution

vector with minimum magnitude can be chosen as the unique solution. The imposed

condition of minimum magnitude is a purely mathematical one and does not necessarily

have any relationship with the physical dimensions of the system concerned. In other

words, the method of singular value decomposition is not useful in preserving

mathematical/physical parameter relationships.

This technique of solving a singular system is, however, extremely robust. Even if it

fails to satisfy the particular need of this work, the method is strongly recommended for

those parameter estimation processes where the reduction of the objective function is

more important than obtaining a physically meaningful optimal set.

4.4.7 Extent of Redundancy

It was previously mentioned that the full Hessian matrix was found to have a number of

zero singular values. This number is equal to the number of extra parameters which are

included in the kinematic model but which are not needed to describe the pose of the

EECF. In other words, this number indicates the extent to which the chosen model is

redundant.

In order get a systematic feeling for redundancy in kinematic models, the mathematical

model of a two link 1-DOF manipulator was analyzed. According to the model used,

the 1-DOF manipulator needed (6 x 2 + 2)=14 parameters for its description. The
 67

parameter values were not kept fixed to the respective values of the original model but

were changed arbitrarily in different runs. In each run, the Hessian was constructed

and the corresponding singular values were calculated.

Regarding the simulation runs, it has been observed that, in a numerical calculation of

singular values, it often becomes very difficult to decide which of the singular values

are zero and which are not. A common way is to look for distinct gap in the

magnitudes of the singular values. In many cases, it is up to the user to decide a

threshold magnitude. A good knowledge of the numerical features of the system and

experience with similar problems are very important and useful in making this decision.

The 1-DOF manipulator model was extended up to a 5-DOF manipulator by increasing

1-DOF in each step. Each of the models was tried with widely varying link

dimensions, link Euler angles, and joint electrical parameters. The sequence and types

of joints were also varied arbitrarily. Eventually it became clear that the number of zero

singular values was following a definite pattern.

If the number of zero singular values is subtracted from the total number of parameters

in the Hessian matrix, we get the number of linearly independent columns in it. This is

the maximum number of independent parameters that is needed to completely describe

the kinematic features of the manipulator. It was found that this number is a function of

only the number and types of joints the manipulator has. If N is the number of

independent kinematic parameters, the function can be expressed as

N = 5nr + 3np + 6 , (4.2)

where nr and np are the number of revolute joints and the number of prismatic joints,

respectively. It is interesting to note that N does not depend on the shape or size of the

links.
 68

It was also found that each of the joint transducers contribute one independent

parameter regardless of the joint type. Therefore, if the electrical parameters for the

joint transducers are not optimized, the maximum number of independent parameters

will be given by the expression

n = 4nr + 2np + 6 . (4.3)

The above equation was first formulated from an entirely different approach by Everett,

et al. (1987) as a condition of "completeness" of a kinematic model. It was shown that

any model, in which the number of kinematic parameters is smaller than that shown in

Equation 4.3 could not describe the manipulator completely. This result is also

supported by Everett and Suryohadiprojo (1988) in yet another analytical approach.

Another way of verifying Equation 4.3 is by considering the kinematic features of the

joints. Denavit and Hartenberg (1955) discussed the minimum number of independent

parameters that is required to uniquely specify the joint axes. A prismatic joint is

characterized by a vector direction in space with no location specified. Only two

parameters are needed to describe this. A revolute joint, on the other hand, is

characterized by the location and direction of a vector in space, and consequently needs

four parameters to describe it. The global coordinate frame is not constrained in any

way with regard to the most proximal local frame, and, therefore, needs a complete set

of six parameters. To reduce or remove redundancy, then the number of parameters in

the kinematic model of a manipulator must satisfy the condition

n ≤ 4nr + 2np + 6 . (4.4)

It is to be understood that the above condition is not a sufficient condition regarding the

absence of redundancy. In other words, the condition does not guarantee that there will
 69

be no redundancy in the kinematic model. For instance, if a manipulator has parallel

joints, the length components along the joint axes will be identically redundant

regardless of whether the total number of parameters satisfies the Equation 4.4. As an
example, at the design values of the parameters (as shown in Table 4.1), p3 and p9 are
identically redundant. If, the parameter p5 is perturbed from its optimal value of 180°,

the global coordinate frame and the frame at the first joint no longer remain parallel.
Consequently, p3 and p5 are no longer identically redundant and the number of

non-zero singular values increases by one.

In Section 2.3, it was concluded that six parameters per link and two or more

parameters per joint (depending on linear or nonlinear encoder characteristics) are the

minimum number of independent parameters which can account for all the geometric

errors in the manipulator and thus preserve an understandable mathematical/physical

parameter relationship during optimization. However, we see that in actuality, a lesser

number of parameters is sufficient to describe the pose of the EECF in space. Thus a

contradiction is reached. On one hand, a proper description of the links for a

comprehensive calibration results in a singular system which is impossible to solve

while preserving the separate identities of all the parameters. On the other hand, if a

non-singular system with just the necessary number of parameters (as defined by

Equation 4.3) is obtained, it will be very difficult, if not impossible, to relate a change

in one of the optimized parameters to a change in the physical characteristic of a link or

in the electrical characteristic of a joint encoder.

4.4.8 Effect of Data Collection at Multiple Sites

It is known that, to improve positional and orientational accuracy of a manipulator,

calibration has to be done on the basis of data collected at the EECF. It is implicitly

understood at this point that only the final accuracy of the EECF is desirable rather than
 70

the accuracy of each and every part of the manipulator body. In other words, from the

accuracy standpoint, it is not very important for the ith joint, for instance, to be at the

point in space where it is expected to be, as long as the EECF is at the expected position

and attitude. In order to improve accuracy of a certain point on the manipulator body, it

is necessary that data be collected from that particular point for the estimation of the

necessary parameters.

It was proposed by Bosnik (1986) that data collection at several different points on the

manipulator body might help in improving the conditioning of the system as far as the

evaluation of Equation 3.13 is concerned. This proposition was based on the

observation that by collecting data at several well-chosen points on the manipulator, it is

possible to make the Jacobian matrix appear to be better conditioned in that it will have

no identically equal columns. Also, it intuitively appears that collection of more data is

equivalent to the addition of more information to the system model, thereby tending to

reduce redundancy. However, it was found that this effort was not successful in

producing a non-singular system matrix. The procedure and results are explained

below.

As shown in Figure 4.12, three different sites were chosen on the manipulator body, in

addition to the site at the tip of the end effector, which became the fourth site. In each

posture, positional and orientational data of each site was measured externally. In order

to calculate the residuals at a particular site, these externally measured pose components

are to
 71

Table 4.9: Typical Jacobian Matrix from One Posture in a Multiple Sites per Posture

Calibration Scheme:
 72

Figure 4.12: Location of Sites on the Manipulator Body

 73

be compared with the respective pose components predicted by the internal control

algorithm of the manipulator. Due to the fact that exact pose of none of the coordinate

frames attached to the sites are available, six unknown parameters are to be added for

each new site. These parameters typically relate the position and orientation of a site

with respect to the joint frame immediately preceding it. As described at the end of

Section 4.3, it is not required to optimize these parameters from a standpoint of

improving EECF positioning accuracy. They are included in the optimization scheme to

test their usefulness in reducing the redundancy of the system.

Table 4.9 shows a typical Jacobian matrix for the above-mentioned arrangement. In a

single posture, data were collected from all four sites on the manipulator, thereby

producing 24 rows. Each site except the last produced six additional parameters, thus

increasing the number of columns to 56 from of 38. Parameters of link #5 are the same

as the parameters of site #4. In the figure, each of the blocks represents a 6×6

submatrix. Some of the submatrices contain only zero elements, and these are shown

in the figure by a single zero inside the respective block. This pattern is repeated in

each posture.

The important observation regarding Table 4.9 is that there are no identically equal
columns in the Jacobian. For instance, in Table 4.7 it was found that the parameters p3
and p9 gave rise to equal columns. Table 4.9 shows that multiple site data collection

can reduce identically equal columns in the Jacobian.

From Table 4.9, it is possible to see that there can be very few identical columns
corresponding to the link parameters. For instance, it is known that the parameters p3,

p9, p15, and p21 give rise to identical columns in the Jacobian matrix. Table 4.9

confirms that multiple point data collection removes these identical columns.

Consequently, it can be expected that there will be a reduction in the number of

redundancies. This, however, does not happen in practice.

 74

During computational runs it was observed that the number of redundancies in a

manipulator remains unchanged even after utilizing positional and orientational data

from additional sites. This failure can be viewed most simply by realizing that each site

adds six equations to the system and as well as six unknowns to the system. The rank

of the system therefore remains unaltered. By calculating the singular values for

different Hessians in a similar way, as described in the previous section, it was found
that the number of independent parameters Ns in the case of a multiple sites per posture

calibration can be calculated as

Ns = 5nr + 3np + 6ns , (4.5)

where ns is the number of sites on the manipulator body including the site at the end

effector tip. The above expression shows clearly that the implementation of multiple

sites per posture data collection in the calibration algorithm does not help in reducing the

number of redundancies.
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4.5 Chapter Summary

This chapter describes the computational analysis done under the scope of the present

work, including the following :

1. A graphical approach was proposed for obtaining the relationships between the

EECF pose components and the kinematic parameters .

2. It was shown that the Jacobian matrix and especially the Hessian matrix are typically

ill-conditioned. An effective way of reducing the ill-conditioning was suggested.

3. It was observed that there are more parameters in the kinematic model than are

actually needed to completely describe the EECF pose. This phenomenon was

termed redundancy and was shown to be responsible for creating singularities in the

system matrices.

4. A graphical approach was shown to be very useful for depicting the relationship

between the objective function and the kinematic parameters of the model. This

approach also helps in visualizing the multi-dimensional surface which is a

characteristic of the objective function.

5. The solution techniques of the Levenberg-Marquardt algorithm and the singular

value decomposition were shown not to be very effective in preserving the

physical/mathematical parameter relationship.

6. The number of independent parameters needed to completely describe the EECF

pose was found to be a function of the number and the types of joints a manipulator

has. This set of parameters was found to be inadequate to describe the kinematic
 76

features of a manipulator and, therefore, is not suitable for a simultaneous

optimization scheme.

In summary, therefore, it was observed that the inclusion of all the parameters for the

preservation of the physical/mathematical parameter relationship and obtaining a

non-singular system matrix are contradictory demands and cannot be satisfied

simultaneously. Moreover, for certain manipulators it may be impossible to get a

non-singular matrix even with the minimal number of parameters, as defined by

Equation 4.3. The collection of data at multiple sites per posture was found not to be

useful in reducing redundancy.

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Chapter 5

SUMMARY, CONCLUSIONS, AND FUTURE WORK

5.1 Summary

Calibration has been found by many investigators to be a practical solution to the

problem of increasing the accuracy of the EECF pose of a robotic manipulator. By an

appropriate choice of kinematic model and calibration algorithm, the optimized

kinematic parameters can be made to represent closely the actual physical dimensions of

the manipulator in many cases. Such information is helpful in identifying sources of

error, implementing predictive maintenance schemes, and hence, in predicting

component damage or wear before it might be discernible by other means. The

preservation of these mathematical/physical parameter relationships has been a primary

goal of the current work.

An appropriate choice of kinematic model and coordinate frame locations on the

manipulator structure has been found to affect both success of the calibration algorithm

and preservation of the mathematical/physical parameter relationships. A minimum of

six parameters per link and two parameters per joint (considering linear joint transducer

characteristics) is necessary for full description of the manipulator for calibration

purposes. The model proposed by Sheth and Uicker (1971) is useful in calibration,

although a slightly different model (Bosnik, 1986) was used in the current work. The

various kinematic model/calibration algorithm combinations were evaluated via a

numerical implementation of the Levenberg-Marquardt nonlinear least squares

optimization algorithm, utilizing both experimentally collected and synthesized data.

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The existence of redundant kinematic parameters in the manipulator kinematic model

complicates the problem of preserving the mathematical/physical parameter relationships

considerably. The redundant parameters interact among themselves during optimization

in such a way that the final result depends on the initial guesses and also in such a way

so as to satisfy a condition which does not have any direct correlation with the physical

features of the manipulator.

The straightforward method of collecting data at different sites on the robot body was

attempted with the intention of reducing redundancy. This attempt was not successful,

as the extra information provided by additional data was nullified by the extra number

of parameters incorporated into the model. The number of redundant parameters

remained unchanged.

5.2 Conclusions

The primary goal of this work was to explore the possibility of establishing a

relationship between the physical parameters of a manipulator and the corresponding

optimal mathematical parameters obtained as a result of calibration. This knowledge

has potential application in tracking the wear and damage of various parts of the

manipulator body.

The comparison of the initial parameters and the changed parameters is physically

meaningful only if the manipulator possesses a clearly discernible mathematical/physical

parameter relationship. It was shown that, in a useful kinematic model, there should be

six parameters per link and two parameters per joint for purposes of calibration.

During the computational runs it became evident that the EECF pose changes can be

completely described with fewer independent parameters than the number required by
 79

the "useful" model described above. Thus the original set of parameters describing the

kinematic features include "useful" redundant parameters, making the system singular.

It therefore becomes impossible to get a meaningful solution as far as the preservation

of physical/mathematical parameter relationship is important. In this work it also

became apparent that it is impossible to get rid of redundancy in a complete calibration

procedure which requires simultaneous optimization of all the parameters and uses a

kinematic model that includes a full description of each individual link and joint.

One way to reduce the severity of rank deficiency of a system is to optimize fewer

parameters. The exclusion of some of the parameters from the optimization scheme will

yield useful results only if their values are known to a high degree of accuracy. The

number of redundant parameters in the system may be reduced or eliminated by using

such a technique, although the opportunity to obtain meaningful mathematical/physical

parameter relationships is concomitantly reduced. Some guidelines are available; it is

generally known, for example, that a major portion of the total kinematic error in a

manipulator is typically due to inaccurate knowledge of the joint transducer zero-point

offset values (Bosnik, 1986). Judd and Knasinski (1987) observed that inaccurate

knowledge of these parameters might contribute up to 95% of the RMS error in the

EECF pose. Therefore, zero-point offsets should generally not be removed from the

optimized parameter set.

The parameter identification method suggested by Stone (1987) is interesting from the

standpoint of its effects on parameter redundancy. Stone's technique does not rely on

the link dimensions supplied by the manufacturer for initial parameter guesses; rather,

parameters are deduced from performance data of the manipulator. The technique

employs a variation of the multiple site data collection scheme described above and

performs separate local optimization at each joint. The set of parameters suggested by

Stone exceeds the number of parameters that can be had in a system without

redundancy.
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5.3 Future Work

The research work presented here can be extended towards improving the calibration

procedure of a manipulator. Some salient points are:

1. As mentioned in Chapter 2, our present knowledge about the behavior of the non-

geometric errors is far from satisfactory at present. These errors are caused by

factors such as link compliance, gear backlash, gear train compliance, motor-

bearing wobble, etc., and may contribute significantly to the end effector

positioning error. A systematic way of parameterize these errors will represent an

important step towards a more complete static calibration.

2. The elastic effects which occur during the operation of manipulators are another

class of factors which contribute to the kinematic error. There is already a trend in

the industry towards manufacturing manipulators which are lighter in weight and/or

faster in operation. Elastic effects will play a significant role in the accuracy of this

new generation robots and an effort towards accurately modeling these effects will

be valuable.

3. The dynamic and vibrational factors also contribute to the kinematic errors of a

robot during its operation. Consequently, dynamic calibration will considerably

improve the accuracy of the robot from this standpoint. The dynamic factors cannot

be accounted for in a static calibration scheme and therefore new methods need to be

developed and/or refined.

In all likelihood, an integrated approach towards calibration will eventually emerge.

This approach will attempt to minimize the observed errors in a manipulator by suitably
 81

adjusting all the static, elastic, and dynamic parameters simultaneously. A final scheme

might be devised in which a robot will automatically "sense" the need for calibration

(when its observed error exceeds some threshold value) and will run the calibration

algorithm without the input of a human operator. Thus the internal control parameters

will be continuously updated and will maintain the manipulator performance at an

optimum level. Such an approach would provide the greatest gain in performance

which appear to be attainable through robot calibration.

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