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Representations of the Euclidean group and its applications to the kinematics of spatial chains

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Representations of the Euclidean group and its applications to the kinematics of spatial chains
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Rico Martinez, Jose Maria, 1954-
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ix, 159 leaves : ill. ; 28 cm.

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Algebra ( jstor )
Axes of rotation ( jstor )
Euclidean space ( jstor )
Geometric translations ( jstor )
Kinematics ( jstor )
Lie groups ( jstor )
Mathematical vectors ( jstor )
Mathematics ( jstor )
Quaternions ( jstor )
Rigid structures ( jstor )
Clifford algebras ( lcsh )
Dissertations, Academic -- Mechanical Engineering -- UF
Kinematics ( lcsh )
Mechanical Engineering thesis Ph. D
Vector spaces ( lcsh )
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bibliography ( marcgt )
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Thesis (Ph. D.)--University of Florida, 1988.
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Includes bibliographical references.
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Jose Maria Rico Martinez.

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REPRESENTATIONS OF THE EUCLIDEAN GROUP AND ITS APPLICATIONS
TO THE KINEMATICS OF SPATIAL CHAINS


















By

JOSE MARIA RICO MARTINEZ


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1988


pp p RJTY, OF FLORIDA LIBRARIES


































Copyright 1988

by

Jose Maria Rico Martinez














ACKNOWLEDGMENTS

The author wishes to thank firstly his advisor

Dr. Joseph Duffy for his guidance during the selection and development of the contents of this dissertation. Without his encouragement, in times when nothing fruitful seemed to evolve from the approaches followed, without his geometrical insight, and his quest for clarity, this dissertation would have plenty of errors. Nonetheless, the author is to blame for the remaining ones. Secondly, deep felt thanks go to the members of the supervisory committee for their invaluable criticism and for their teachings in the classroom. The faculty of the Mechanical Engineering Department, and in particular the faculty of the Center for Intelligent Machines and Robotics, must be thanked for the development of an atmosphere conductive to research. The contributions of the faculty of this center, including the visiting professors, can be found in many parts of this work. Special gratitude is owed to visiting professors Eric Primrose and Kenneth H. Hunt for their insightful comments. My thanks go also to my fellow students for their friendship and kindness. The economic support from the Mexican Ministry of Public Education and the Consejo Nacional de Ciencia y Tecnologia (CONACYT) is dutifully acknowledged. Finally, the author thanks his wife


iii








and children for bearing together four years of uncertainties, new experiences, and, hopefully, familiar growth.















TABLE OF CONTENTS


ACKNOWLEDGMENTS ......................................... iii

LIST OF FIGURES ......................................... vii

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

CHAPTERS

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

1.1 The Role of the Euclidean Group ............... 3
1.2 The Euclidean Group as a Lie Group ............ 5
1.3 Grassmann and Clifford Algebra Versus Standard
Vector Calculus .................................. 9
1.4 Objectives and Organization of the Work ....... 11


2 EUCLIDEAN SPACE AND EUCLIDEAN TRANSFORMATIONS ..... 15

2.1 Physical Space as a Euclidean Space ........... 16
2.2 Free Vector Algebraic Structure in ExE ........ 17 2.3 Bound Vector Algebraic Structure in ExE ....... 23 2.4 Euclidean Mappings .............................. 26
2.5 Properties of the Euclidean Mappings .......... 31
2.6 Translations ..................................... 35
2.7 Rotations ........................................ 37
2.8 Decomposition of Euclidean Mappings ........... 40
2.9 Composition of Euclidean Mappings .............. 46

3 REPRESENTATION OF THE EUCLIDEAN GROUP .............. 48

3.1 Fundamentals of Representation Theory ......... 50
3.2 The Affine "Representation" of the Euclidean
Group ......................................... 53
3.3 The 4x4 Matrix Representation of the Euclidean
Group ......................................... 56
3.4 Spin Representation of the Euclidean Group .... 61
3.5 The Restriction of the Group Action of the Spin
Representation on [el,e2,e3] .................... 71
3.6 Invariants of the Representation of the
Euclidean Group .................................. 73









page

3.7 The Biquaternion Representation of the Euclidean
Group ......................................... 74
3.8 The Representation of the Induced Line
Transformation ................................... 78
3.9 Screw Representation of the Euclidean Group ... 81

4 THE EUCLIDEAN GROUP AS A LIE GROUP ................. 87

4.1 Lie Groups ....................................... 88
4.2 The Lie Algebra of the Euclidean Group ........ 90
4.3 The Adjoint Representation of the Euclidean
Group ......................................... 95
4.4 The Euclidean Group as a Semi-Riemannian
Manifold ......................................... 97
4.5 Analysis of the Structure of the Set of Second
Derivatives of the Euclidean Group at the
Identity ........................................ 102

5 APPLICATIONS ........................................ 105

5.1 The Principle of Transference ................. 105
5.2 Analysis of the Dualization Process ........... 108
5.3 Statement and Proof of the Principle of
Transference .................................... 114

CONCLUSIONS ............................................. 126

APPENDICES

A CLIFFORD ALGEBRAS ................................... 128

B A PHYSICAL INTERPRETATION OF THE ADJOINT
TRANSFORMATION ...................................... 144

REFERENCES .............................................. 148

BIOGRAPHICAL SKETCH ..... .................................. 158















LIST OF FIGURES


Figure 5.1 Figure 5.2


page

Skeletal Spatial Kinematic Chain .......... 115 Skeletal Spherical Chain Associated with the Spatial Chain of Figure 5.1 ........... 116


vii














Abstract of the Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

REPRESENTATIONS OF THE EUCLIDEAN GROUP AND ITS APPLICATIONS TO THE KINEMATICS OF SPATIAL CHAINS By

JOSE MARIA RICO MARTINEZ

August 1988

Chairman: Dr. Joseph Duffy
Major Department: Mechanical Engineering

A coordinate free analysis of the Euclidean group is performed by employing the theory of affine and orthogonal spaces. A unified treatment of several of the most used representations of the Euclidean group, including homogeneous transformations, spin and biquaternion representations, is obtained by applying the theory of group representations. Several results related to the Clifford algebra of the positive definite three-dimensional orthogonal space are proved. These results are employed in the deduction of a generalized spin representation of the Euclidean group. The action of the spin representation over the points of a three-dimensional space yields the particular form of the spin representation that is commonly used in kinematics. The biquaternion representation is obtained from the spin representation, and the two distinct roles played by the


viii








dual unit in these last two representations are explained. The Euclidean group is analyzed as a Lie group. A representation of the Lie algebra of the Euclidean group is obtained from the spin representation of the Euclidean group. The Lie algebra operations are compactly expressed in terms of Clifford algebras. The Euclidean group is characterized as a six-dimensional semi-Riemannian manifold with a hyperbolic metric. The structure of the second derivatives of the Euclidean group at the identity is analyzed. The mathematical apparatus developed is applied to the principle of transference. A thorough analysis and proof of the principle of transference are given.














CHAPTER 1
INTRODUCTION

A striking feature that any new student of kinematics and robotics has to endure, during his or her study of the subject, is the great variety of mathematical techniques that it is necessary to master, or at least be acquainted with, in order to obtain a working knowledge of the recent advances in kinematics and robotics. These mathematical systems range from very elementary ones, such as complex numbers (Blaschke and Miller [1956], Rooney [1978b], Sandor and Erdman [1984]), standard vector algebra (Brand [1947], Chace [1963], Lipkin and Duffy [1985]) to more sophisticated structures such as matrix algebra (Suh and Radcliffe [1978], Bottema and Roth [1979], Angeles [1982]), quaternions (Blaschke [1960]), dual numbers (Rooney [1978b], Duffy [1980]), biquaternions (Yang [1963], Keler [1970a]), screw calculus (Yuan et al. [1971a], Yuan et al. [1971b], Hunt [1978], Phillips [1982]), line geometry (Weiss [1935], Woo and Freudenstein [1969]), and Grassmann algebra (Pengilley and Browne [1987]). All these techniques have been employed in an effort to disentangle the secrets of kinematics.

Yang [1963], in his dissertation, provides a list of references dealing with early attempts to provide mathematical structures suitable for dealing with kinematic

1










analysis and synthesis. Some of these structures disappeared because they rendered an incomplete description of the kinematic phenomena, or proved to be too cumbersome to deal with. Others, like matrix analysis and biquaternions, have remained and become common tools for kinematicians. However, the newer attempts to provide better mathematical tools (Ho [1966], Hiller and Woernle [1984], Pennock and Yang [1984], Casey and Lam [1986], McCarthy [1986b]), and the rediscovery of old mathematical structures in a disguised, or slightly different form (Waldron [1973], Sugimoto and Matsumoto [1984], Nikravesh [1984], Wehage [1984], Nikravesh et al. [1985], Sugimoto [1986], Lin [1987], Agrawal [1987]) suggest a lack of agreement, or even a lack of discussion, about the criteria that a mathematical system suitable for solving kinematics problems must satisfy.

For a newcomer, the problem is aggravated because some of the mathematical structures have several distinct names, for example, the biquaternions (Clifford [1882]) are also called dual complex numbers (Keler [1970a], Keler [1970b], Beran [1977]), dual quaternions (Yang [1963], Wittenburg [1977], Wittenburg [1984]), and dual number quaternions (Agrawal [1987]). Furthermore, the description of the mathematical system employed by a kinematician is, sometimes, incomplete or obscure due to a lack of mathematical training, or an emphasis on the application of the mathematical system rather than on logical subtleties.










1.1 The Role of the Euclidean Group

A common characteristic of most of the mathematical structures discussed in the previous section is that they are equipped with means to represent general rigid body motions, or special cases such as planar or spherical motions. In fact, it is well known that complex numbers represent rotations around a fixed axis, and hence the planar displacements which are of interest in planar kinematics. Quaternions and matrices represent the rotations around a fixed point, that are the transformations analyzed in spherical kinematics. Finally, dual quaternions, screws, and matrices represent general spatial movements of a rigid body. Moreover, the mathematical structures mentioned above allow the representation of the geometrical entities--such as points, lines, and planes which belong to a rigid body--by using elements of the structure. Furthermore, the structures allow the determination of the image, under a rigid body displacement, of these entities by manipulating their corresponding elements. In the language of modern mathematics, the set of points and lines of rigid bodies, together with the set of rigid body displacements, forms a model of Euclidean geometry (Giering [1982]).

From the previous paragraph, it follows that the set of rigid body displacements is of paramount importance in the study of kinematics. Actually, this set forms a group under the composition operation, and it is called the Euclidean










group. This fact is latent even in the writings of the nineteenth century kinematicians (Study [1903]), and it was known to physicists and mathematicians of the early part of this century (Weiss [1935]). However, the group property appears to have been overlooked by the kinematicians of the fifties and early sixties. Blaschke and Muller [1956] did mentioned the group property for the set of planar displacements. Later Blaschke [1960] and Muller [1962] did mention the group property of spherical displacements and Suh and Radcliffe [1967] proved the group property for the set of planar motions. Suh and Radcliffe [1968] further suggested that the results were also true for the spatial case. Bottema and Roth [1979] provided a synthetic proof of the general case.1

The fact that the set of rigid body displacements forms a group under composition has far reaching implications, for it permits the use of the existing framework of group theory and group representations (Herstein [1975], Rose [1978], Fraleigh [1982], Naimark and Shtern [1982]).2 In particular, it allows the introduction of the important concept of isomorphism. Two mathematical structures are isomorphic if there is a bijective mapping (one-to-one and onto) that


iFrom now on, only general spatial displacements will be considered.
2Recently, the theory of groups has been applied to robotics, but in the field of artificial intelligence (Popplestone [1984]).








5

preserves all the operations defined in the structure. If two structures are isomorphic, then, loosely speaking, they are identical; just the names of the elements and the appearance of the operations have changed. Furthermore, any calculations performed in one structure can be equally carried out in the other.

Hence, after two structures have been proven to be

isomorphic, the criteria for selecting the most appropriate one for purposes of representing physical phenomena are reduced to

1. The compactness of the resulting expressions,

2. The number of additions or multiplications of real

(or complex) numbers that the operations require, and

3. The previous knowledge of the structure.

The work most closely related to the objectives

addressed here is that of Rooney (Rooney [1977], Rooney [1978a]), who discusses several possible representations of rotations around a fixed point, and general spatial displacements. However, no use of the key concept of isomorphism is made, and the spin representation of the Euclidean group is not mentioned.



1.2 The Euclidean Group as a Lie Group

In the previous part of this introduction, the Euclidean group has been regarded just as an algebraic group whose elements are transformations of the Euclidean space. This










characterization suffices when the kinematic problems to be solved are related to the action, over the points and lines of a rigid body, of a finite number of elements of the Euclidean group (For example, synthesis of mechanisms with a finite number of precision points). The solutions of these problems were precisely the main objectives of most of the kinematical research done in the mid 50s and 60s when kinematics was re-discovered in American universities. Only some authors of the German school (Blaschke and Miller [1956], Blaschke [1960], Maller [1962], Maller [1970]) pursued the analysis of the structure of the Euclidean group itself. In this way, they adhered to the tradition of the kinematicians of the late nineteenth century and of early this century (Stdphanos [1883], Study [1903], Weiss [1935]). In the late 70s and early this decade, Roth and his coworkers (Bottema and Roth [1979], De Sa [1979], Ravani [1982], Ravani and Roth [1984], McCarthy [1986a]) re-discovered and generalized the kinematic mappings. These kinematic mappings are representations of the Euclidean group, although in the initial writings of Roth's school the group structure, a key property, was overlooked.

The main objective of these new researches was to

provide a sense of "metric" in the Euclidean group itself and to perform synthesis of mechanisms, where the synthesis criterion is the "closeness" of the synthesized motion of a rigid body with respect to some specified motion. Thus, the








7

number of precision points is infinite. It turns out that the solution of this problem requires to consider the Euclidean group as a Lie group. This fact was not however recognized in their studies.

The origin of the theory of Lie groups goes back to

1880, when Sophus Lie, attempting to emulate Felix Klein's employment of group theoretical concepts in the study of geometry, introduced the idea of continuous groups of transformations (Lie [1975]), later known as Lie groups. The theory of Lie groups proceeded to develop throughout the first part of this century. Concurrently, significant advances were made in differential geometry, including Riemannian and non-Riemannian geometry, and differentiable manifolds, primarily as a result of the newly discovered theory of relativity. The interchange of ideas between these two fields led to their blossoming in the 50s. Today the references about Lie groups and differential geometry are numerous. Furthermore, the depth of the references is variable, ranging from elementary introductions (Spivak [1965], O'Neill [1966], Boothby [1975], DoCarmo [1976], Sattinger and Weaver [1986]), to research publications (Chevalley [1946], Hermann [1966], Belinfante and Kolman [1972], Gilmore [1974], Helgason [1978], Spivak [1979], Warner [1983], O'Neill [1983]).

The theory of Lie groups and Lie algebras has been applied, in an unconscious way, to kinematics for a long










time. Indeed, Karger and NovAk [1985] show that the algebra of infinitesimal screws is isomorphic to the Lie algebra of the Euclidean group. Hence, Ball's contributions (Ball [1900]), and all the research along this line (Yuan et al. [1971a], Yuan et al. [1971b], Yang [1974], Hunt [1978]) can be regarded as an application of the Lie group theory to spatial kinematics.

The conscious application of Lie group theory to spatial kinematics is more recent. This delay can be explained by a two-part argument; first, with the discovery of relativity theory, most of the research done by physicists and applied mathematicians was directed toward new orthogonal spaces, such as Lorentz space and Minkowsky space and their transformations group. Only recently there has been an interest in regarding the Euclidean group as the transformation group of the Euclidean space (Patera et al. [1975a), Patera et al. [1975b], Beckers et al. [1977]). On the other hand, the contribution of kinematicians was hampered, probably, by a lack of mathematical education. As far as the author is aware, the first application of Lie groups to kinematics is due to Herve [1978]. However, attempts to provide a sound mathematical foundation of kinematics (Karger and Novak [1985], Selig [1986]), and the recent studies on the control of manipulators (Loncaric [1985], Brockett [1983]) have sparked the interest in the relationship between Lie groups and spatial kinematics.











1.3 Grassmann and Clifford Algebra Versus Standard Vector Calculus

The problem of searching for a suitable mathematical structure for the study of kinematics can be placed in the more general framework of looking for appropriate mathematical structures for dealing with physical phenomena at large.

In an exciting narrative, Crowe [1985] describes how vector algebra came into being as a hybrid creature of the two leading mathematical systems of the last half of the nineteenth century, quaternions due to Hamilton, and extensor algebra, also known as Grassmann algebra after its developer. The use of this new vector algebra drew heated rebuttals from the quaternion followers. However, after some years, the vector algebra was so widely accepted that the only mementos of the quaternions were some notational conventions, and the curiosity of a mathematical structure that is unique in many ways. On the other hand, Grassmann had such an obscure life (Crowe [1985]) that, except for Germany, his native country, he lacked followers that could provide support for his algebra.

Recently, however, there has been a renewed interest in the design of mathematical systems for the description of physical phenomena. On one side, there has been an increasing interest in Grassmann work (Dieudonne [1979], Barnabei et al. [1985], Stewart [1986], Pengilley and Browne [1987]); on the other hand, Hestenes (Hestenes [1966], Hestenes [1971],











Hestenes and Sobczyk [1984], Hestenes [1985], Hestenes [1986]) has claimed that Clifford algebra provides more comprehensive tools, than vector calculus, for dealing with physical problems, including those from classical mechanics.

Clifford algebra (Crumeyrolle [1974], Brooke [1980], Porteous [1981], Brackx et al. [1982]) is in a sense a misnomer since it really comprises a family of algebras, each of these algebras is closely related to the symmetric bilinear form of the corresponding orthogonal space (Dieudonn& [1969], Kaplansky [1969], Porteous [1981], Scharlau [1985]). The first Clifford algebras were formulated by Clifford [1876], under the name of geometric algebras. During several decades, their study lay dormant, or was made without connecting it to Clifford's original ideas. However, in the 50s (Riesz [1958]), the interest in Clifford algebras was renewed, and it continues up to the present (Chisholm and Common [1985]). Clifford algebras have been used in kinematics research under several disguised names. It can be shown (Lam [1973], Porteous [1981]) that dual numbers, complex numbers, quaternions, dual quaternions, and Grassmann algebras are all special cases of Clifford algebras.

It is still too early to decide if these structures will replace vector calculus as the most used mathematical physics tool, but it is evident that they warrant a close examination.










1.4 Objectives and Organization of the Work

An outline of the objectives and organization of the work is now given.

Chapter 2 deals with the analysis of the Euclidean space and the Euclidean group. Although the results obtained are not new, the methods employed--the theory of affine and orthogonal spaces as developed in Porteous [1981]--yield a coordinate free analysis that, as far as the author is aware, is new. After proving several classical results, the chapter ends with the characterization of the Euclidean group as the semi-direct product of the normal subgroup of the translations times the subgroup of the rotations around an arbitrarily fixed point.

Following the suggestions of section 1.1, Chapter 3 uses the group theoretical tools of isomorphism and group representation to provide a unified treatment of the mathematical structures mostly employed in spatial kinematics. In fact, it is shown that the representation of the Euclidean groups by means of affine orthogonal mappings of a three-dimensional vector space, homogeneous 4x4 matrices, spin groups, and biquaternion groups, all produce isomorphic algebras, and therefore, alternative models of Euclidean geometry. The development of the spin representation, as shown in sections 3.4 and 3.5, together with the connection between the spin and biquaternion representations, in section 3.7, is thought to be original.










Since the development of the spin representation of the Euclidean group relies heavily on the Clifford algebras of some orthogonal spaces, Appendix A was written to cover some important concepts and results on this topic. There, four key results, which are required in the course of sections 3.4 up to sections 3.8, are proven for the first time.3 Furthermore, the representation of the induced line transformation, by means of spin groups, as shown in section 3.8, is a novel and interesting by-product of the main development of this chapter. Finally, the screw representation of the Euclidean group allows the unification of the ideas of screws and of the screw triangle with the remaining representations of the Euclidean group.

In Chapter 4, the Euclidean group is analyzed again, but this time, its topological and metric characteristics, inherited by every Lie group, are taken into consideration. After one introductory section, an explicit representation of the Lie algebra of the Euclidean group is obtained (section

4.2) from the spin representation of the Euclidean group. The development employs Clifford algebras. An advantage of using Clifford algebras is that the structure is robust enough to handle both the spin representation of the Euclidean group and its Lie algebra in a very compact way. Furthermore following the ideas of Karger and Novak [1985], it is shown

3The origin of this work was an initial attempt to look into Clifford algebras and what they could offer to the field of spatial kinematics.










that the Lie algebra of the Euclidean group and the algebra of infinitesimal screws (Dimentberg [1965], Hunt [1978], Duffy [1985]) are isomorphic.

Section 4.3 analyzes the last of the representation of the Euclidean group namely, the adjoint representation. This representation maps elements of the Euclidean group into automorphisms of its Lie algebra. Since the elements of the Lie algebra can be interpreted as the infinitesimal elements of the group, it is clear that the derivation of this representation requires the topological properties of a Lie group, which are totally disregarded in Chapter 3.

Section 4.4 establishes the semi-Riemannian manifold

structure of the Euclidean group. The development makes use of bi-invariant metrics defined on the Lie algebra of the Euclidean group. This subject has already been studied by Loncaric [1985] and Lipkin [1985], in the latter case by resorting to the algebra of infinitesimal screws. During the process, it is necessary to employ the concept of invariance under the adjoint mapping. This concept is explained in Appendix B. The characterization of the Euclidean group as a semi-Riemannian manifold with a hyperbolic metric is thought to be new.

The space of second derivatives of the Euclidean group at the identity is studied in section 4.5. This space is, similar to the Lie algebra of the Euclidean group, a










six-dimensional orthogonal space with a non-degenerate hyperbolic metric.

In chapter 5, the theory developed in the previous part of this study is applied to spatial kinematics. There, an analysis and proof of the principle of transference (Rooney [1975] and Selig [1986]) are carried out. The principle has a history of failed attempts and controversial results. In that chapter, a complete proof of the principle of transference is given; and the relation between the principle of transference and the Hartenberg and Denavit notation (Denavit and Hartenberg [1955]), which was previously overlooked, is explicitly indicated.

Finally, the more relevant conclusions of this study,

together with suggestions for further studies, are presented.














CHAPTER 2
EUCLIDEAN SPACE AND EUCLIDEAN TRANSFORMATIONS

In this chapter a coordinate free analysis of the

Euclidean group is undertaken. In section 2.1, the physical space is modelled as a three dimensional Euclidean space. Sections 2.2 and 2.3 explain how to induce a free vector and a bound vector space structure into the Euclidean space; the fundamentals of such constructions are the theory of affine spaces and orthogonal spaces as developed in Porteous [1981] and Kaplansky [1969]. Section 2.4 defines Euclidean mappings as bijective mappings, of the set of points, which preserve both the affine and orthogonal structure of a Euclidean space. The lack of structure in the physical space prevents a direct probe into the properties of Euclidean mappings, and the remainder of the section is directed toward the analysis of the orthogonal mappings induced, by the Euclidean mappings, onto the free vector space structure introduced in section 2.2. Using the results of section 2.4, the properties of Euclidean mappings are investigated in section 2.5, and their structure as an algebraic group is proved. Sections 2.6 and 2.7 are dedicated to the analysis of two important classes of subgroups of the Euclidean group, namely translations, and rotations leaving an arbitrary point of the Euclidean space fixed. The decomposition of Euclidean

15










mappings, in terms of a translation and a rotation around a fixed point, is analyzed in section 2.8 together with some invariant characteristics of the mappings. Finally in section

2.9, the composition rule for Euclidean mappings is re-examined and expressed in terms of the components, and the structure of the Euclidean group, as the semi-direct product of the normal subgroup of the translations by the subgroup of the rotations that leave an arbitrary point fixed, is disclosed.



2._1 Physical Space as a Euclidean Space

For purposes of study of classical mechanics, physical space can be modelled by a three-dimensional Euclidean space (Porteous [1981]); equivalently, the space can be regarded as a set of points E coupled with a mapping 6 from the cartesian product ExE into R3, a three-dimensional vector space endowed with a positive definite quadratic form, defined by

6 : ExE _ R3 6(M,N) = M - N (1) with the additional properties

1. For every NcE, the restriction mapping
6N : E , R.3 6(M,N) = M - N (2)

is a bijection and

2. For all L,M,NEE, the triangle axiom holds

(L - M) + (M - N) = L - N (3)










This property can be expressed in terms of the mapping 8 by

6(L,M) + 6(M,N) = 6(L,N) (4) The elements of R3 are called vectors; in particular the vector M g N is called the difference vector of the pair of points (M,N) in ExE. This notation is specially useful since it emphasizes that the elements of R3 are dependent upon the points of E.

In fact, the existence of a mapping 6 : ExE _ R3 fulfilling the above axioms ensures the existence of a family of similar mappings

SI : ExE _ R3 611(M,N) = AS(M,N) for each AeR, A j 0

all of which also satisfy the axioms. The specific value of A depends on the arbitrary election of a unit of length which cannot intrinsically be preselected.

It is important to recognize that the set E is at the outset completely void of algebraic structure; hence, there is no meaningful way for adding, subtracting or multiplying the points of E. However the rich algebraic structure already existing in the quadratic vector space R3 can be employed to induce some algebraic structures in E.



2.2 Free Vector AlQebraic Structure in ExE

The first of the algebraic structures, induced by R3

into ExE, to be studied is that of the free vectors. Before










proceeding, it is necessary to note that ExE supports an equivalence relationship

(M,N) z (P,Q) * 6(M,N) = 6(P,Q) (1) where M,N,P,QeE.

This relationship has the following properties:

1. It is reflexive. Trivially

6(M,N) = 6(M,N) thus (M,N) z (M,N)

2. It is symmetric. Let (M,N) z (P,Q); then

6(M,N) = 6(P,Q) and thus 6(P,Q) = 6(M,N). Therefore
(P,Q) (M,N).

3. It is transitive. Let (M,N) = (P,Q) and

(P,Q) z (R,S); then 6(M,N) = 6(P,Q) and

6(P,Q) = 6(R,S). Thus 6(M,N) = 6(R,S) and therefore
(M,N) = (R,S).

Further since 6 : ExE _ R3 is surjective, it is possible to associate with every element v of R3 an equivalence class

v = {(M,N)e ExE I 6(M,N) = v) (2) In the field of classical mechanics v is usually referred as a free vector. The set of all free vectors will be denoted by
IR3 = U v (3) VER3

The equivalence relationship thus has induced a partition of ExE.










It will now be demonstrated that R3 has the structure of a real orthogonal vector space (Porteous [1981]); hence, it can be considered as a real orthogonal space of free vectors.

The first step in this demonstration is to recognize that there exists an induced mapping
6: _ R3 6(y)= v (4) which is well defined and bijective.

1. 6 is well defined. Let (M,N),(P,Q)ev; therefore

6(M,N) = v = 6(P,Q) and 6(M,N) = v = 6(P,Q).

2. 6 is injective. Let y,El3 such that

j(y) = v = w = 6(H). It follows, from the partition properties (Fraleigh [1982]), that v C w and w C y,

and therefore v = w.

3. 6 is surjective. Since 6 : ExE _ R3 is surjective,
V vER3 there is a (M,N)eExE such that 6(M,N) v.

Further, let vER3 such that (M,N)ev,then 6(y) = v

and 6 is surjective.

This bijective mapping provides a way for translating

the algebraic structure of R3 into an algebraic structure of



The set R3 together with the operations of addition and scalar multiplication defined by

v + w = (v + w) V vwER3 (5a) and


X(v) = (Xv) V XeR and V vER3


(5b)










form a real vector space isomorphic to R3

i. R3 is closed under addition.

V vjCR3 3 v,weR3 such that, for M,N,P,QcE,

(M,N)ev 8 S(M,N) = v, and (P,Q)ew * 6(P,Q) = w.

Since R3 is a vector space (v + w)IR3 , and since 6 : R3 _ R3 is surjective, there is a (v + w)ER3.

2. Addition is associative.
V v, w,xE_3

(Y + w) + x = (v + w) + x = [(v + w) + x] = [v + (w + x)] = v + (w + x) =v + (w + x)

3. Addition is commutative.
V v weR3

v + w = (v + w) = (w + v) = w + v

4. Existence of an additive identity.

V vER 3 OER3 such that

v + 0 = (v + 0) = v It is interesting to note that the equivalence

class of ExE, corresponding to 0, is given by

0 = ((M,N)eExE I M = N} This can be demonstrated by firstly considering

(M,M)eExE where M is arbitrary. From the triangle

axiom

6(M,M) + 6(M,M) = 6(M,M) Therefore 6(M,M) = 0 and (M,M)eO. Secondly consider










(M,N)eExE with M : N. Therefore (M,N)O.Q, for if

(M,N)eO, then both

6N(M) = 6(M,N) = 0 and 6N(N) = 6(N,N) = 0
a contradiction to the injectivity of 6N.

5. Existence of an additive inverse.
VvER3 3 (-v) ER3 such that

v + (-v) = [v + (-v)] = 0

It will now be demonstrated that if

v = {(M,N)EExE I 6(M,N) = v),

then

(-v) = {(N,M)eExE I (M,N)ev)

Employing the triangle axiom

6(M,N) + 6(N,M) = 6(M,M)

Since (M,M)cO and assuming (M,N)Ev, then

6(N,M) = -S(M,N) = -v

and thus (N,M)e(-v).

6. R3 is closed under scalar multiplication.

VvER3 3vER3 such that for M,NcE, (M,N)ev

6(M,N) = v; further, since R3 is a vector space,
V NER XvER3. Moreover, since6 : R3 _ R3 is surjective, 3 a (Lv)ER3 such that for P,QeE,

(P,Q)E(_v) 4 6(P,Q) = Xv.

7. Scalar multiplication satisfies V X,gER and vER3

X(Av) = X(1a) = [X(Uv) ] = [vLOV] = (A)v










8. Scalar multiplication and addition satisfy V X,geR

and v,wER3

(X + ) [(X + u)v] = (Xv + Uv) = (x) + (Uv)

and
X(Y + W) = [X(v + w)) = (Xv + Xw) = (Lv) + (Xw)



Furthermore, the mapping 8 : R3 _ R3 is also linear since VX,AeR and Vv,wER3

A(XV + Aw) = A(Xv + Uw) = Xv + Aw = NS(y) + A(Y)

Thus & is a linear isomorphism; then R3 and R3 are isomorphic as vector spaces.

Finally, it is also possible to introduce an orthogonal structure on R3 by defining the symmetric bilinear form

I3xR3 , - (vW) = (v,w) Vx,wER3 (6) It is straightforward to show the form is well defined and indeed symmetric. Further, the mapping 6 : _ - R3 preserves the form, for trivially

(A(Y),6(w)) = (vw) (7) and therefore R3 and R3 are isomorphic as real orthogonal vector spaces.

It is important to recognize that in the process of

providing R3 with an orthogonal vector space structure it is unnecessary to make any arbitrary choice of origin. The next section introduces a structure that, unlike the structure considered here, is origin dependent.










2.3 Bound Vector Algebraic Structure in ExE

The Euclidean space E is endowed with a second algebraic structure induced by means of the mapping

So : E - R3 60(M) = 6(M,O) = M 0 O (1) By property 1 of the mapping 6, the mapping 60 is bijective; thus it can be used to induce a vector space and ultimately an orthogonal space structure in E. Here the point OcE is an arbitrary reference point or origin which, under the mapping 60, is set equal to the zero vector.

In particular it is possible to define the operations of addition and scalar multiplication of points of E by

M + N = 60-1(60(M) + 60(N)) V M,NEE (2) and

XM = 60-1(x60(M)) V MeE, V XeR (3)

It will now be demonstrated that the points of E together with these operations have the structure of a real vector space.

1. E is closed under addition.

V M,NcE 60(M),60(N)ER3. Since R3 is a vector space,

60(M) + 60(N)eIR3 and thus 60-1(60(M) + 60(N))eE.

2. Addition is commutative.

V M,NeE

M + N = 60-1(60(M) +60(N)) = 60-I(60(N) + 60(M)) = N + M










3. Addition is associative.

V L,M,NcE

(L + M) + N = 60-1[60(L + M) + 60(N)]

= 60-I(60[60-1(60(L) + 60(M))] + 60(N))

= 60-1[(S0(L) + S0(M)) + 60(N)] = 60-1[60(L) + (60(M) + 60(N))]

= 60-1(60(L) + 6o1-I(60(M) + S0(N))])

= 60-1160(L) + 6O(M + N)]

=L + (M + N)

4. Existence of an additive identity.

From section 2.2

60(0) = 6(0,0) = 0

Therefore, V MEE

M + 0 = 60-1(60(M) + 60(0)) = 60-1(60(M)) = M

It has been shown that the point 0 acts, in this

structure, as the additive identity.

5. Existence of an additive inverse.

Let MEE be such that 60(M) = vER3; then 3 -MeE such

that -M = 60-1(-v), hence

M + (-M) = 60-1160(M) + 60(-M)] = 60-1[v + (-v)] = 0

6. E is closed under scalar multiplication.

V MeE, 60(M)ER3. Since R3 is a real vector space,

V NI, X60(M)ER3, and thus 60-I[?60(M)]eE.










7. The scalar multiplication satisfies

V X,geR and MeE

x(AM) = 60-1(60(M)) = 60-1(x60[60-1(A60(M))1}

60-1[(XA)60(M)] = (XA)M.

8. The scalar multiplication and addition satisfy

V X,AeR and M,NEE

(X + 14)M = 60-1[(N + A)60(M)

= 60-I[X60(M) + A60(M)]

= 60-1[60 (XM) + 60(AM)] = XM + AM

and

X(M + N) = 60-1[X60(M + N)] = 60-1{X[60(M) + 60(N)])

= 60-1[X60(M) + X60(N)] = XM + XN

where the last equality, in each of the two previous

sequences of equalities, holds due to the

bijectivity of 60.

Rewriting equations 2 and 3 in the form

60(M + N) = 60(M) + 60(N) V M,NEE (2a) and

60(XM) = X60(M) V MeE, V XeR (3a) it is apparent that 60 : E _ R3 is a linear isomorphism. Additionally, by providing E with an orthogonal structure, induced from R3, according to

ExE - R3 (M,N) = (60(M),60(N)) V M,NcE (4) the linear isomorphism 60 becomes an orthogonal isomorphism, for it trivially preserves the form.










A prominent consideration is that some of the results obtained within the realm of this algebraic structure are only valid for the specific selection of the origin 0; hence, they are not Euclidean geometry properties.



2.4 Euclidean Mappings

In this section an analysis of mappings of the Euclidean space which preserve the orthogonal structure of R3 is presented. Such mappings P : E - E must preserve the quadratic form of the associated orthogonal space,R3, given by

(8(M,N),6(M,N)) = (6(*M,pN),8(,PM,,PN)) V M,NcE (1) It can be shown (Porteous [1981]) this condition is equivalent to the preservation of the related symmetric bilinear form or inner product

(6(M,N),6(P,Q)) = (6(,PM,,PN),6(,PP,,iQ)) V M,N,P,QEE (2)

Mappings with this property are called Euclidean mappings, and it is straightforward to show that this definition is independent of the particular value of A chosen in the mapping 6. : ExE _ R3. Furthermore, any Euclidean mapping induces a mapping of R3 into itself according to the rule

:O -3 O R3(v) = 6(%PM.,\N) where (M,N)ev (3) Since the definition of p involves a choice, it will now be shown that b is indeed well defined.










Let (M,N),(P,Q)ev, and assume 6(,M,bN) = vI and

6('P,'PO) = v2, where Y2 can be orthogonally decomposed (Kaplansky, [1969]) as Y2 = XvI + u with ue[vl] --the orthogonal complement of [Vl]--, it suffices to show v, = Y2.

1. V, = 0; then

0 = (0,.Q) = (Vl,Vl) = (6(_M0_N),6 _MoN
=(6(M.N) ,6.))

Thus 6(M.N) = 0, and (M,N)eO. If (P,Q)EO then

6(P,Q)= 0 and

0 = (6(P,Q),6(P,Q)) = (6(PP,4Q),6(PP,4Q)) = (v2,v2)

Thus v2 = 0 = vI.

2. v, 6 0; then

(Vl,Vl) = (6(,kM.PN),6(6MPN)) = (6(M.N),6(MN))

- (6(MN),6(P.Q)) = (6(,M.,PN),6('P.,PO))

- (Vl,V2) = (Vl1 + u) = X(Vl,Vl) + (VlU)
X (VlVYl)

Therefore X = 1 and thus Y2 = vI + u. Consider again

the relationship above

(Vl,l)= (6(,PM.,N),6(JM.'PN)) = (6(M.N,6(M.N))

- (6(PO),6(P,O)) = (6(,P.j Q),6('4P.'PO))

- (v2,v2) = (vI + u,v1 + u)

= (VlVl) + (uu)

Therefore (uu) = 0. Since R3 is positive definite,

u = 0 and Y2 = vI.

Furthermore, it can be proved that the induced mapping










is independent of the scale of length singled out by the mapping 6 : ExE _ R3.

It is also evident that if p(y) = 3, then the image of

any vey, under the induced mapping on R3, will be given by w. Since R3 and R3 are orthogonally isomorphic, the induced mappings on R3 and n3 will be both denoted by !p, the distinction will be clear from the context.

The following proposition provides more insight into the nature of the induced linear transformation.

Proposition 1. The mapping ! preserves the inner product of IR3.

Proof: Let v,weCR3 with v = 6(M,N), w = 6(P,Q) for some M,N,P,QEE. Then

(!Pv,w) = (6(,PM,,PN),6('P,,PQ)) = (6(M,N),6(P,Q)) = (v,w)

However, by definition of the induced orthogonal structure of R3, (v,w) = (x,w), and therefore U = (x'W) (5)
This result is, in fact, a logical consequence of the particular way in which the induced orthogonal structure of R3 as well as the induced mapping P have been defined.

It will now be demonstrated that the induced mapping
R3 _ R3 is orthogonal. Since it has already been shown

that A preserves the inner product, it suffices to show that A is a linear mapping. The proof is accomplished by extending and slightly modifying a result reported in Angeles [1982].








29

Proposition 2. Every mapping of a positive definite (or negative definite) real orthogonal space into itself which preserves the inner product is linear.

Proof: Let X be a positive definite (or negative

definite) real orthogonal space, * a mapping which preserves the inner product, and v,weX, define

e = P(v + w) - 4(v) - 4(w) Then

(e,e) = (q(v + w) - '(v) - '(w),4,(v + w) - '(v) - P(w))

= ('(v + w), (v + w)) + ( (v), (v) ) + ( (w), (w) )

- 2(q,(v + w) ,p(V)) - 2(,(v + w),P(w))

+ 2 (p (v) ,p (w) )

= (v + w, v + w) + (v,v) + (w,w) - 2(v + w,v)

-2(v + w,w) +2(v,w)

= (v,v) + (w,w) + 2(v,w) + (v,v) + (w,w) - 2(v,v)

- 2(w,v) - 2(v,w) - 2(w,w) + 2(v,w) = 0 Since X is a definite orthogonal space e = 0, and 'p(v + w) = #(v) + 4M(w), then 4 is additive. Further, let veX, v : 0 ,and XeR; then

*(Xv) can be expressed in the form

(Xv) = aP (v) + w

where aER, and we['(v)]-L. Consider now the inner product

X(vv) = (vXv) = ((v),#(Xv)) = ((v),a"(v) + w)

= a((v),(v)) + ((v),w) = a(O(v),4(v))

= a(v,v)


Thus X = a, and










X2(V,V) = (Xv,Xv) = (Xp(V) + w,X4(v) + w)

= X2(' (v),p(v)) + 2X(,P(v),w) + (w,w)

= X2(v,v) + 0 + (w,w)

Therefore (w,w) = 0. Since X is a definite orthogonal space, then w = 0; thus * is homogeneous and linear.

Corollary 3. Every mapping of a positive definite (or negative definite) real orthogonal space into itself which preserves the inner product is orthogonal.

In particular, these two last results apply to mappings which preserve the inner product in R3.

After having shown that the induced Euclidean mappings are linear transformations, it appears natural to inquiry whether the mappings are injective and/or surjective. The answer to these questions can be obtained from a result reported by Porteous [1981].

Proposition 4. Every orthogonal mapping of a

non-degenerate orthogonal space into itself is injective.

Proof: Let X be a non-degenerate orthogonal space,

v,weX, and %P an orthogonal map satisfying *(v) = *(w); then

*(v - w) = q,(v) - P(w) = 0, and for every ueX

0 = (0,4P(u))= 0 (v - w),qi(u)) = (v - w,u)

Since X is non-degenerate,

v - W= 0 or v = w

Corollary 5:. *Every orthogonal mapping of a

non-degenerate, finite-dimensional orthogonal space into itself is a bijection.










Once more, these results apply in particular to the orthogonal mappings induced in R3 by Euclidean mappings.

Summarizing the results, it has been shown that the

induced mappings of Euclidean transformations are orthogonal automorphisms of the real orthogonal space R3. It is well known that the orthogonal automorphisms of R3 form a group, called the orthogonal group and denoted by 0(3). However there are two classes of orthogonal mappings. Those which preserve the intrinsic orientation of R3 form a subgroup of the orthogonal group, denoted by SO(3), and called the special orthogonal group. Those which do not preserve the orientation of R3 do not form a subgroup. Bottema and Roth [1979] show that the induced orthogonal mappings must preserve the orientation of R3; hence they must be elements of SO(3). Even though the definition of Euclidean mapping is broad enough to include mappings which do not preserve the orientation of R3, those will not be considered anymore.



2.5 Properties of the Euclidean Mappings

In the previous section Euclidean mappings and their induced transformations were defined, but the lack of structure in the Euclidean space E hampered any effort to directly find their characteristics. After searching the properties of the induced mappings, it is possible to pursue the investigation of the Euclidean ones.








Proposition 1. Let * : E - E be a Euclidean map, and : R3 R3 the induced (orthogonal) map; then for any NeE
- = (6@ (N))-Ik6N (1)

Proof: Let MeE be arbitrary; then
((6 (N))-Ik6N) (M) = ((8@(N))-Ik)6N(M)

- ((6, (N))-ip) 6 (M,N)
=(6* (N))- (1 (6 (M, N))

= (6p (N)) (6 (PM,4N))

- (56 (N) )6, (N)(#(S)) = ('P) Corollary 2. Every Euclidean mapping is bijective.
Proof: By definition 6N and (6*(N))-I are bijections, and by corollary (4.5) !p is also a bijection; since * is a composition of bijective mappings, * is also bijective.
Proposition 3. The composition of Euclidean mappings is also a Euclidean mapping. Furthermore, the induced mapping is the composition of the corresponding induced mappings.
Proof: Let *1,*2 be Euclidean mappings, 1,4k2 be their associated mappings, and M,NcE be arbitrary; then
(6(2'P@IM,@2'PIN),6(2IM,p2'IN)) = (6(qPIM,qIN),6(*IM,'PIN)) = (6(M,N),6(M,N))
Thus *2q1 is also Euclidean. Assume now _ : R3 - R3 is the induced mapping of 'P2*1, and S(M,N) = v; then
_1Wv)= 6(P2'P1M,@2'PIN) = 6(P2(i1M),*2(,IN)) =1-2(6(OIM,PIN) = !2P41(6(M,N)) = 421-l(V)










Therefore

t= A2Ai (2) Proposition 4. The inverse of a Euclidean mapping is

also Euclidean, and its induced mapping is the inverse of the induced mapping of the original Euclidean mapping.

Proof: Let * be a Euclidean mapping, and P its induced mapping; then

(6(4'M,qN),6(4'M,4N)) = (6(M,N),S(M,N))

Renaming M' = *M and N' = *N; then M = *-1M' = *-14M, and N = *-1N' = *-1*N are, respectively, the unique images of M' and N' under *-1. Therefore
(c(M',N'),6(M',N')) =((-I'-1),(IM,1N) V M',N'EE; hence *-1 is also Euclidean. Assume now - : R3 R 13 is the induced mapping of *-1 and 6(M',N') = v; then

= I3(y) = 6(M',N') = S(+1I'41N')

= !ki(6('Pf1M',*1-1N') = I (6(M',N')) = kI (M) Thus 13 = jl, and

I = (-1)-iI3 = (i)-1 (3) Proposition 5. The set of Euclidean mappings together with the composition operation form a group, called the Euclidean group, and denoted by E(3).

Proof: By proposition 3, the composition of two

Euclidean mappings is also a Euclidean mapping; thus the set is closed under the operation.










The composition of Euclidean mappings being a special case of the composition of arbitrary mappings, which is associative, is also associative.

Consider the mapping i : E - E such that i(M) = M V MEE. It is straightforward to prove that t is a Euclidean mapping, and it behaves as the identity element of the group.

By corollary 2, every Euclidean mapping is invertible, and, by proposition 4, its inverse is also a Euclidean mapping.

Now that it has been established that Euclidean mappings constitute a group, proposition 3 provides a proof for the following statement:

Corollary 6. The mapping ' : E(3) - SO(3), which assigns to every Euclidean mapping its induced orthogonal mapping is a group homomorphism.

It is a well known fact that the composition of

Euclidean mappings is not commutative (Bottema and Roth [1979]). Therefore the Euclidean group is not abelian. It now appears natural to inquire about the possible existence and properties of subgroups of the Euclidean group. It will be shown, in the next two sections, that there are two important classes of subgroups of the Euclidean group namely translations, and rotations about a fixed point.










2.6 Translations

A translation is defined as a mapping T : E - E with the property that there exists a veR3 such that 6(TM,M) = v V MeE (I) More precisely, r is called a translation of E by the vector

V.

Proposition 1. A translation is a Euclidean map.

Proof: Let M,NcE be arbitrary; then by the triangle axiom

6(rM,rN) = S(rM,M) + 6(M,N) + 6(N,rN) = 6(rM,M) + 6(M,N) - 6(rN,N) by definition S(rM,M) = S(rN,N); therefore 6(rM,rN) = 6(M,N) (2) Hence the preservation of the quadratic form is immediate, for

(6(rM,rN),6(rM,TN)) = (6(M,N),6(M,N)) V M,NeE

The following result provides a useful characterization of a translation.

Proposition 2. A Euclidean mapping * is a translation if and only if 1- = 13, where 13 is the identity mapping in R_3.

Proof: Assume * is a translation; then by equation (2)

P(5(M,N)) = 6(,P(M),P (N)) = 6(M,N)

Therefore P = 13. Assume there are two points M,NeE with

6(,PM,M) A 6(4N,N) ; then

IP(6(M,N)) = 6(qM,4N) = 6(qM,M) + 6(M,N) + 8(N, *N)

= 6(M,N) + [6(,PM,M) - S(,PN,N)]










Since 6(4'M,M) : 6('PN,N), then

(6(M,N)) 8(M,N) and JA 13

Proposition 3. The subset of all translations is closed under the composition operation; furthermore, composition of translations is commutative.

Proof: Let 71,T2 be two translations, with vl,v2ER3

their associated vectors, and MeE be arbitrary; then using the triangle axiom

6(72'IM,M) = 6(T271M,TlM) + 8('1M,M) = v2 + V1

Thus 7271 is a translation, and the associated vector is the sum of the associated vectors. Moreover

6(r172M,M) = 6(T172M,T2M) + 6(r2M,M) = v1 + v2

Since addition in R3 is commutative, v1 + v2 = v2 + vl, and

6C'r2rlM,rlT2M) = 6(r2r1M,M) + 6(M,Trlr2M) = 6(72rIM,M) - 6(7r12M,M) = (v2 + v1) - (vI + v2) = 0 Thus

T2T1M = r1T2M V MeE Proposition 4. Let r : E - E be the translation of E by a vector v; then r-1 : E - E is the translation of E by the vector -v.

Proof: Let MeE be arbitrary, with TM = N, and

r-1N = r-lrM = P, for some PeE. Applying the triangle equality

6(P,M) = 6(P,N) + 6(N,M) = 6(r-IN,N) + 6(rM,M) = -v + v = 0










Thus P = M, or r-lrM = M, hence r-lr = t, and this result coupled with the commutativity of the composition proves the assertion. Here t denotes the identity Euclidean mapping.

Proposition 5. The set of all translations forms a normal abelian subgroup, denoted by T, of the Euclidean group.

Proof: By proposition 3, the set is closed under composition, and the composition is commutative. By proposition 4, the set is closed under the operation of taking inverses. Finally, let r and 'p be an arbitrary translation and Euclidean mapping respectively; consider the mapping * = r'-l; then
= _ -i = ! P -i = 1 13 (4)-i = i3 Thus by proposition 2, is a translation.

Further, with the usual definition of a scalar multiple of a mappings, this group can be made isomorphic to the vector space R3.



2.7 Rotations

A Euclidean mapping p : E - E is called a rotation if

there exists a point PEE such that pP = P. P is then called a fixed point of the rotation.

Proposition 1. The set of rotations having a common fixed point P, denoted by np, forms a subgroup of the Euclidean group.

Proof: Let pl,P2enp; then










p2P1P = p2(PlP) = p2P = P and p2PlEf2p. Trivially, the identity mapping, t : E - E, belongs to np, for tP = P. Let penp; then

P = tP = (p-p)p = p-1(pP) = p-1p

Thus, p-lefp and the set is closed under the operation of taking inverses.

It is noteworthy to recognize that for any point PeE there is a subgroup np of the rotations leaving P fixed. Sometimes the fixed point will be used as a suffix of a rotation as a way of specifically stating the invariance of that point.

The following result characterizes the equivalence of two rotations.

Proposition 2. Two rotations P, and P2 are equal if, and only if, they have a common fixed point and the same induced orthogonal mapping.

Proof: Let P be the common fixed point, and consider an arbitrary MeE; then

6(PlM,P2M) = S(PlM,plP) + 6(p1P,p2P) + 6(P2P,P2M) = P16(M,P) + 6(P,P) + Q26(P,M) = P1[6(MP) + 6(P,M)] + 0 = I[6(M,M)] = P-1(0) = 0
Thus, pl(M) = P2(M) V MEE, and P1 = P2Assume P1 and P2 do not have a common fixed point; then if P is any fixed point of Pl, plP = P; however by assumption p2P Y P; hence plP 2 p2P and P, 6 P2-








39

Finally, assume P1 and P2 have P as a common fixed point but pi 30 P-2; then 3vIR3 such that pl(y) 7 P2(y). Let MeE be such that pp(M) = v; then applying the triangle equality

6(P1M,P2M) = 6(PlM,P) + 6(P,P2M)

= 6(PlM,PlP) - 6(P2M,p2P)

= P16(M,P) - P26(M,P) = Rlv - P2v 7 0 Thus plM 3 p2M and P, ; P2If a rotation p : E - E is also a translation, then the vector associated with the translation is given--employing the fixed point P of p--by

6(pP,P) = 6(P,P) = 0

It follows that for every MEE

6(pM,M) = 0; thus pM = M

Therefore p is the identity mapping. Conversely, the identity mapping is the unique Euclidean mapping which is both a rotation and a translation; hence npnT = (t).

Given a non-identity rotation p : E - E, then it can be shown that the induced mapping p is a non-identity proper orthogonal transformation of R3. Further, from the theory of proper orthogonal mappings (Herstein [1975]), 1 is the unique real eigenvalue of any non-identity proper orthogonal mapping of R3; Euler's theorem will now be proved using these results.

Proposition 3 (Euler's Theorem). Every rotation

p : E - E has associated a pointwise fixed affine line, which is called the rotation axis of p.










Proof: Let PEE be any fixed point of p, and vER3 be an eigenvector associated with the eigenvalue 1; consider QeE given by

6(Q,P) = Xv for some XER Then applying the triangle equality

6(pQ,Q) = 6(pQ,pp) + 6(pP,P) + 6(P,Q)

= [6(Q,P)] + 6(P,P) + 4(P,Q)

= P(Xv) + 0 - 6(Q,P) = Xv - Xv = 0

Thus pQ = Q. Furthermore, the pointwise fixed affine line is given by

{QeE I 6(Q,P) = Xv for some XeR) (i)



2.8 Decomposition of Euclidean Mappings

After establishing that translations and rotations

around a fixed point are subgroups of the Euclidean group, it appears natural to ask whether it is possible to decompose an arbitrary Euclidean mapping in terms of a translation and a rotation. The answer is in the affirmative.

Proposition 1. Every Euclidean mapping can be written as the composition of a translation and a rotation whose fixed point is arbitrarily chosen. Moreover, the decomposition is unique up to the selected fixed point of the rotation.

Proof: Let * : E - E be an arbitrary Euclidean map, and PeE be an arbitrary point; denote v = 6(*P,P) and define

pp : E - E pp = (Tv)-' * = T.v *










Then

6(ppP,P) = 6(ppP,PP) + &(*P,P) = 6(r.v'P,41P) + v = -v + V = 0 Thus ppP = P, and pp is indeed a rotation leaving the point P fixed; hence

= TvpP (1) There are two special cases of this proof. If 6(qPP,P) = 0, then *P = P, and * is a rotation; therefore trivially 'P = t 1P

If there is a vER3 such that 6(*M,M) = v V MeE, then ' is a translation and trivially 'P = 1P t

Now assume that 'p has two decompositions with the same fixed point P

,P = TvPP and 'P = w(pp)* Since v = 6(*P,P) = w, then rv = rw; therefore vpp rv 'P (pp), and (7v) (rvpp) = (7v)-1(rv(pp)*); thus pp = (Pp)*

A similar proof shows that any Euclidean mapping can also be decomposed as 'P = P('Pp)Tw (2) where w = 6(\PP,P) for an arbitrary PEE. Moreover by proposition 5.3

P = 13P = _vap = = -(PP) 1w = P(*P) 13 = P-(\PP) Thus


p = P ('PP)










Similarly, it is a routine task to verify that if a Euclidean mapping * is decomposed as

%P = Tvpp or P = p(,PP) TW (4a) Then, the inverse mapping can be decomposed as

*-i = TW(P(PP))-I or 'P-i = PP-1.v (4b)

Although the selection of different points, as fixed

points of the rotation, leads to distinct decompositions, it will be proved, analogously to the proof of equation 3, that the induced orthogonal mapping remains invariant.

Proposition 2. The induced orthogonal mapping of an

arbitrary Euclidean mapping is independent of the fixed point chosen to accomplish the decomposition.

Proof: Let 'P : E - E be an arbitrary Euclidean mapping with two distinct decompositions

* = rvPp and ' = rwPQ Then, by proposition 5.3,

RP = 13RP = IvRP = wPQ I3RQ = RQ
This result coupled with the invariance of the induced orthogonal mapping under changes of the scale of length ensures that the induced orthogonal mapping is a true Euclidean property of any Euclidean mapping.

On the other hand, the selection of different points, as fixed points of the rotation, also leads to distinct values of the translation vector; however, those vectors still have interesting invariant characteristics whose investigation leads up to some of the results credited to Rodrigues (Gray








43

[1980]),and Chasles (Bottema and Roth [1979]), and ultimately to the screw representation of Euclidean mappings.

Proposition 3. Let + be a Euclidean mapping decomposed as 4 = Tvpp and * = rwpQ, where P,Q are two arbitrary distinct points and let u be an eigenvector related to the eigenvalue 1 of the common induced orthogonal mapping. The components of the translation vectors along the eigenvector are equal.

Proof: It is obvious that (v,u) = (w,u) * (v - w,u) = 0; then it suffices to show that (v - w)e[u]�.

By definition v = 6(*P,P) and w = 6(qQ,Q); hence by invoking the triangle equality

v - w = 6(,'P,P) - 6(* Q,Q)

- 6(4P,4Q) + 6(4'Q,Q) + 6(QP) - 6(4'Q,Q)

= [6(P,Q)] - 6(P,Q)

Expressing 6(P,Q) = Xu + y with ye[u]L, then

v - w = 40(Xu + y) -(Xu + y) = Xju + jy - Xu - y Since by assumption !pu = u, then

v - w =Xu + y-u -y = y - y Finally, since Sb is orthogonal, then

(v - w,u) = (y - y,u) = ( y,u) - (y,u)

= (Uy, u) - (y,u) = (y,u) - (y,u) = 0

Thus (v - w)e[u]-, which establishes the result.

The component of the translation vector, along the

eigenvector u, which by the previous result is independent of the fixed point chosen during the decomposition. It does










however depend on the scale of length chosen in the mapping

6 : ExE _ R3, and thus, it is not a Euclidean property of the mapping. In view of this result, the question arises whether there is a point S such that * = rrPS with r = Nu, for some NeD, and if it exists then where is it located.

Proposition 4. If a point S, satisfying the conditions shown above exist, then all the points on the affine line

{PI 6(P,S) = gu for some AeU} (5) satisfy the same conditions. Additionally, all the points belonging to this line are displaced equally by the Euclidean mapping *.

Proof: Consider an arbitrary point P belonging to the affine line given by equation 5; then

6(,PP,P) = 6('PP,4,S) + 6(*S,S) + 6(S,P)

= P[6(P,S)] + Xu - S(P,S)

= 4P(Au) + Xu - Au = Au + Nu - Au = Xu

Proposition 5. Let * be a Euclidean mapping; then for any decomposition 4 = TwPQ the restriction mapping 13 - RQ : [u]-- - [u]-- is a bijective linear mapping.

Proof: Let ye[u]-�; then

((13 - PQ)Y, =) = - P, = (Y,1) - (Q,)

= (x, U) - (aQX, Q1) = 0 - (y, ) = 0 Thus the transformation indeed maps [u]' into [u]1.

Assume that yeker(I3-PQ); then

0 = (13 - P-Q)Y = Y - Q . 'QY = Y










If y = 0, then the result follows for the function is injective and consequently bijective. Assume v j 0; then y is an eigenvector of pQ associated with the eigenvalue 1; hence y = Au for some AeM a contradiction to ye[u]-.

Proposition 6. Let * : E - E be an arbitrary Euclidean

mapping; then a unique affine line can be found such that for all the points S that belong to the line

P= TrPS with r = Nu

where u is an eigenvector associated with the eigenvalue 1 of the induced orthogonal mapping.

Proof: By proposition 4, it suffices to find a point in the affine line. Let Q be an arbitrary point; then

* = TwPQ

Let u be an eigenvector associated with the unit eigenvalue of PQ; then w = giu + y with AjeR, and ye[u]-.

Let S be another arbitrary point; then 6(S,Q) = A2u + x with A2ER and xe[u]. Applying the triangle equality

S(4IS,S) = 6(TwPQS,S) = 6(TwPQS,PQS) + 6(PQ,S)

= w + 6(PQSPQQ) + 6(Q,S) = w + PQ(6(SQ)) - 6(S,Q)

= w + PQ(A2u + x) - (A2u + x)

= Aiu + y + A2u + 2Q(X) - A2u - x

= Aiu + y - (13 - PQ)X
By proposition 5, the mapping (13 - PlQ) : [u]- - [u]- is bijective; hence a solution of (13 - P.Q)X = y (6)










always exists, and it is unique. Hence, a point S can be found such that

r = 6(*S,S) = giu

This affine line is commonly referred as the screw axis of the Euclidean mapping. It is certainly a Euclidean invariant of the Euclidean mapping.



2.9 Composition of Euclidean Mappings

Now that the decomposition and properties of an

arbitrary Euclidean mapping have been examined, it seems natural to inquire how the composition of two arbitrary Euclidean mappings can be related to this decomposition. This provides the necessary foundation for a further result associated with the structure of the Euclidean group.

Proposition 1. Let #1 = rtPo and P2 = rv(Po)* be two

arbitrary Euclidean mappings decomposed using a common point 0; then

2'Pi = TvT (Po)t(P)*PO () Proof: Consider 'P21i = (7v(PO)*)(TtPo) = Tv((Po)*Tt)Po; then (po)*rt is a Euclidean mapping transforming point 0 into

(po)*M where 6(M,O) = t. According to proposition 8.1

(PO)*rt = Tw(PO)'

where

w = 6((po)*M,O) = ((po)*M, (po)*O) = (pO)*6(M,O) = (po)*t and

(2)* = 0)*3 = (0)* rt = 1w(PO)' = I3(PO) ' = (PO)










Then, by proposition 7.2
(Po)* = (Po) ' and (PO)*rt = (P0)*t (PO)* (2) Thus finally

'2"1 = TvT(Po)*t(Po)*(Po)

Proposition 2. The quotient group E(3)/T is isomorphic to np, where P is an arbitrary point of E.

Proof: Consider the mapping

�p : E(3) - np *p = pp, where P = rtPp It is easy to notice that �p assigns to an arbitrary Euclidean mapping its rotation part in its decomposition with respect to the point P. It will now be shown that �p is a group homomorphism onto np with kernel T, then resorting to the so-called isomorphisms theorems (Herstein [1975] pp. 59) the result will follow.

Let *Iq42eE(3), where *i = rtPP and \P2 = rv(PP)*; then

�P(P2*l) = PP(rvr(p)*tPP*PP) = (PP)*pP = �P*2�P@I Let ppenp be arbitrary; then trivially ppEE(3) and �ppp = �p(tpp) = pp
Thus the mapping is surjective. Finally

+cKer� 0 @' = t* *' = rti for some tER3 P = rt for some teR3 o 'PeT. This result gives a complete characterization of the Euclidean group. In algebraic terms, it is defined as the semidirect product of T by 11p with action pp (Rose [1978] pp. 208-210).














CHAPTER 3
REPRESENTATION OF THE EUCLIDEAN GROUP

In the previous chapter it was emphasized that the

Euclidean group exists even if there is no prior selection of a reference system in the Euclidean space. In this chapter several homomorphic representations of the Euclidean group will be reviewed.

It is important to know that in all the usual

representations of the Euclidean group it is required to choose a fixed frame of reference. This frame of reference consists of a point 0, called the origin, a scale of length, and three directions given by an orthonormal basis of the positive definite orthogonal space R3.

This fact has deep implications since (see section 2.3) the selection of a point and a scale of length sets up a bijective relationship between the points of the Euclidean space and the elements of R3. Further, choosing a second orthonormal reference system in a moving rigid body such that in the initial position of this body both reference systems coincide, there is a bijective relationship between the elements of the Euclidean group and the possible positions of the moving rigid body, whose set is usually called configuration space.










A significant difficulty, however, is that it is

necessary to test any result obtained for a particular choice of a reference system for invariance with respect to the action of the Euclidean group and a change of the scale of length before the result is declared a Euclidean geometry property.

No attempt here is made to exhaust all possible

representations of the Euclidean group (Miller [1964], Rooney [1978a]), and only those which are amply used in the kinematics field or shed additional insight are discussed. Further, the analysis of the representations of the Euclidean group is, in this chapter, restricted to algebraic requirements leaving aside metric or topological requirements which will be treated in chapter 4.

The main achievements of this chapter are firstly the development of a unified treatment of the algebraic representations of the Euclidean group used in kinematics, and the identification of the so-called kinematic mappings as representations of the Euclidean group. Secondly, the deduction of the spin representation of the Euclidean group, developed in sections 3.4 to 3.6, as well as the connection of the spin representation with the biquaternion representation, proved in section 3.7, is also new. Finally, the application of the spin representation to the induced line transformation, given in section 3.8, is an interesting










and novel byproduct of the approach developed in this chapter.



3.1 Fundamentals of Representation Theory

Let G be a group and X a real vector space; a

representation T of the group G into L(X), the algebra of linear operators of X, is a mapping T: G - L(X) satisfying

1. T(g2g1) = T(g2) T(gl) for all g1,g2eG (1) 2. T(e) = IX where e is the identity of G (2)

The first property shows that the representation is a group homomorphism; furthermore, since for any geG

IX = T(e) = T(gg-1) = T(g) T(g-1) and

IX = T(e) = T(g-1g) = T(g-1) T(g), then

T(g-I) = (T(g)]-i (3) Hence all the operators T(g) are non-singular. Consequently, T(G) forms a subgroup of the group of units of the algebra L(X).

If X is finite-dimensional, T is called a

finite-dimensional representation of G. Otherwise, T will be an infinite-dimensional representation. This work considers only finite-dimensional representations of the Euclidean group.

Given a group G and a non-empty set S, if for any gEG and seS there is a gseS such that








51

g2g1s = g2(gls) for all g1,g2eG and seS (4a) and

es = s for eeG, the group identity, and V seS, (4b) then G is said to act upon the set S. Rose ([1978] pp. 68-70) shows that the mapping

*g : S - S 'Pg(S) = gs (5) is a bijective mapping of S; i.e. OgeES, where ES is the symmetric group of the set S. Furthermore the mapping

* : G - ES g = % g (6) is a group homomorphism. Since G is isomorphic to a group of bijective mappings or transformations of the set S, then G is called a transformations group. It follows that the Euclidean group acts upon the points of the Euclidean space. Further, since for any specified points M and N of the Euclidean space there is an element (in fact, an infinite number of elements) of the Euclidean group which transforms M into N, the action of the Euclidean group over the points of the Euclidean space is called transitive. Moreover, the set of points of the Euclidean space is said to be a homogeneous set of the Euclidean group (Naimark and Shtern [1982]).

From a strictly theoretical point of view, the previous definition of a group representation is sufficient for any class of groups, including groups of transformations. However, it is highly desirable to use the representation of the group for obtaining the image of an element of the set








52

under any of the transformations. This can be accomplished as follows:

Let G be a group of transformations of a set S; then an additional injective mapping, from the set S to a vector space, 4 : S - X is required such that

4* : S - O(S) _ X �*(s) = �(s) V seS

is bijective, and the following diagram commutes 0*
S I4(S) C X g 1J1T(g) S O4(S) g X

or alternatively for all sES I*gs = T(g)4*s (7) Since 4* is bijective, it is possible to write g = �*-iT(g)�* (8) or

T(g) = *gI,-1 (9) This procedure is accomplished only if an injective mapping I and subsequently 4* which satisfies equation (7) can be found. Then T(g) is linear.

The technique illustrated above can be reversed; viz. given an injective mapping 4 : S - X, the mapping

4* : s - �(s) 4*(s) = I(s) V seS

is bijective, and it is feasible to define its inverse mapping as

�*- 4 : (S) - S *-lv = s (10) where seS is the unique element satisfying 4*s = v.










By completing the following diagram


S '(S) X
g 1 1. T(g)



the transformation T(g) : X - X can be defined as

T(g)v = �*g�*-lv Vvel(S) and T(g)v = v Vv &(S) (11)

The mapping T which assigns T(g) to g satisfies

T(e)v = I*et*-lv = I*es = I*s = v VvEI(S) Thus T(e)=Ix, and

T(g2g1) = 1*g2g11*-I = *g2�.-.*$g1*l = T(g2)T(gl) Hence, T is a group homomorphism from G to the group of bijective mappings of X; however, in this case, there is no assurance that the mappings T(g) : X - X are linear.



3.2 The Affine "Representation" of the Euclidean Group1

It follows from the decomposition theorem (2.8.1) that any element * of the Euclidean group E(3) is uniquely determined by

1. If one chooses a point 0 of the Euclidean space and a

scale of length, then the decomposition of P = Ttpo yields the two other required pieces of information.

2. There is a vector t associated with the translation

Tt.


1Strictly the term representation as used in the
previous section is confined to linear mappings. Here the term is used to include affine transformations that in general are not linear.










3. There is a proper orthogonal mapping pO associated

with the rotation.

The first condition (see section 2.2) sets up a

bijection between the points of the Euclidean space and the vectors of p3, with

So : E _ R3 0(M) = 8(M,O) (1) as the bijective mapping.

It will now be shown that the Euclidean mapping

induces a mapping 'P, in general nonlinear, of R3 into R3 in the following way:

Consider two distinct Euclidean spaces; one of them,

denoted by E, moves freely and the other, denoted by Z', is fixed. Let 0 be a reference point of E, and 0' be the point of Z' initially coincident with 0. Let MEZ be an arbitrary point with v = 80,(M); then define the induced mapping O' : R3 _. R3
as

P o,(V)= 6((M),O') = ( (M), (O)) + 6(P(0),0')

= sk(6(M,O) + 6(rtPo(O),O')

= Po(V) + 6(rtO,O') = Po(V) + t (2)
This class of non-linear affine transformations, with a proper orthogonal linear part, is a subgroup of the three-dimensional affine transformations group. Here it is denoted by ASO(3).

It is simple to show that the following diagram commutes










60 , R
E :flR3

1:01
E R

Thus, except for the non-linearity of *0,, the mapping

A : E(3) - ASO(3) A4 = *0, (3) seems a good choice for a "representation" of the Euclidean group. It remains to show that A is a group homomorphism.

Let '1 = ruPO, and *2 = rt(Po)* be two arbitrary Euclidean mappings. Proposition 2.9.1 shows that

*2*1 = TtT(po)*u(PO)*(po); Hence for any vER3,

A('2'1)v = (po)*(po)v + (po)*u + w = [A(*2)]l[(Po)v + u]
= [A (P2) ][A (P1)]v

Therefore

A('P2'P1) = A(P2)A(4P1) (4) In fact, it is possible to show that A is an isomorphism.

This "representation" of the Euclidean group given by equation (3) is far too abstract for calculations. This problem is overcome by selecting an orthonormal basis of R3. It is well known (Herstein [1975]) that this selection sets up an algebra isomorphism between the proper orthogonal transformations of R3 and the 3x3 proper orthogonal matrices.2 The image of pO under this mapping will be from here on represented by R.




2An orthogonal matrix is said to be proper if its determinant is positive.








56
Consider now the set of ordered pairs (R,t), where R is a 3x3 orthogonal matrix, and t is an element of R3, with the operation

(R2,t2)(Rl,tI) = (R2R1, R2tl + t2) (5) It is straightforward to show this set is a group formed by the semi-direct product of the subgroup (R,O) acting on the normal subgroup (0,t). In particular the identity is given by (13,0), where 13 stands for the 3x3 identity matrix, and the inverse of (R,t) is given by (R,t)-I = (R-1, -R-1t) (6) Furthermore, the action of this group upon an arbitrary vER3, where v = 60,(M), is

v' = Rv + t (7) This group is sometimes referred to as the group of symmetries of R3. This result has influenced many to identify the Euclidean group with the group of isometries of R3. However, it is necessary to realize that the identification requires an arbitrary selection of the scale of length and an origin. Hence the identification is not natural (Loncaric (1985]).


3.3 The 4x4 Matrix Representation of the Euclidean Group

In the previous section it has been demonstrated that after selecting a point 0, scale of length, and an orthonormal basis of R3, any element of the Euclidean group can be represented by the ordered pair (R,t). Further, the








57
image, M', of an arbitrary point M of the Euclidean space is obtained through

x' = R x + t, (1) where x' and x are respectively the images of M' and M under the mapping

S0, : E R33 60, (P) = 6(P,O') (2)

Nonetheless, the map given by equation (1) is nonlinear. Thus, it is not a representation of the Euclidean group in the sense of section 3.1. Here, following the ideas developed in section 3.1, a novel approach is presented which leads to a group theoretical vindication of the 4x4 matrix representation of the Euclidean group. This representation, although widely used in computer graphics and robotics, has been recently the subject of some criticism (Angeles [1982]).

Consider the mapping

tk : R3 . M4 tk(vl,v2,v3) = (vl,v2,v3,k) (3) where keR and k :P 0. This map is obviously injective. Thus t*k : R3 tk(R3) t*k(vl,v2,v3)=(vl,v2,v3,k) V(vl,v2,v3)e6R3 is a bijective mapping.

For an arbitrary mapping (R,t)eASO(3) consider the transformation defined according to equation (1.11) t*k(R,t)t*k-l : R4 , R4, (4) The image of an arbitrary (vl,v2,v3,k)Etk(R3) will be given by











(t*k(R,t)t*k-l)(vl,v2,v3,k) = (t*k(R,t)) (vl,v2,v3)

= t*k(rllvl + r12v2 + r13v3 + t1, r21vI + r22v2 + r23v3 + t2, r31vI + r32v2 + r33v3 + t3) = (rllv1 + r12v2 + r13v3 + ti, r21vI + r22v2 + r23v3 + t2, r31vI + r32v2 + r33v3 + t3,k) The restriction of the mapping t*k(R,t)t*k-l to

ik(R3) C R' can be functionally represented by the matrix expression


(t*k(R,t)t*k-1) (Vl,V2,v3,k)=


rll r21 r31

0


or alternatively by


(i*k(R,t)t*k-i) (v,k) =
j0


r12 r13 tl/k r22 r23 t2/k r32 r33 t3/k

0 0 1


t/k

1iik


Let the set of all matrices of the form given by (6b) be denoted by Ek(4), where k 4 0 is arbitrarily fixed. According to section 3.1, the mapping


a : ASO(3) - Ek(4) a(R,t) = (7) is a group homomorphism. Furthermore, it is easy to prove that the mapping is injective and surjective; hence, a is a group isomorphism.


V1

V2 V3

k


(6a)


(6b)










The group Ek(4) is a subgroup of M4x4, the group of non-singular 4x4 real matrices; thus a is a matrix representation of the Euclidean group. Moreover, since the image of an arbitrary point vER3, under a Euclidean mapping, can be found via equation (5), a is a representation in the extended sense of section 3.1.

For an arbitrary k : 0 any of the matrices belonging to Ek(4) represents a linear automorphism of R4. However the restriction of this linear mapping to ik(R3) is itself not a linear mapping. In fact, tk(R3) is not even a linear subspace of R4. Nevertheless, in no part of the theory of representation of groups of transformations, is there any requirement that the action of the representation be linear.

Equally and perhaps more important, it will now be

proved that Ek(4) can be regarded as an orthogonal group. This is to say, all the transformations of Ek(4) preserve the symmetric bilinear form associated with an orthogonal vector space (Brooke [1980]).

Consider the orthogonal space R10,3. This is a vector space endowed with the following symmetric bilinear form
(x,y) : RI03XRI,0,3 _. R

((xl,x2,x3,xo),(Yl,Y2,Y3,Yo))= x1Y1 + x2Y2 + x3Y3 + 0xOY0 (8)

It is well established (Herstein [1975]) that a linear

mapping, M, of R1,0,3 will preserve the bilinear form if, and only if,


MTMt = T










where r is the matrix representative of the bilinear form with respect to the canonical basis of R1,0,3. Therefore, r is given by


r = (10)



Expressing the matrix M in four blocks which correspond to those of T, one obtains


MMl (11)



Further, using the condition expressed by equation (9) yields MIIMI1t = 13 (12a) M21 = 0 (12b)

Moreover, if one requires firstly that the matrix M belong to the special orthogonal group SO(1,0,3), and secondly preserve the semi-orientations of the space--i.e. McSO+(I,0,3)--, then the following additional constraints are imposed

M22 > 0 (12c) IM1111M221 = 1 (12d) IM111 = 1 (12c) Therefore M11 must be a proper orthogonal matrix of

dimension 3; i.e. it is a three-dimensional rotation matrix R. Since {M111 = 1 from equation (12d) M22 = 1, and













R M1 R M I 2/k
M = = for some k 4 0 (13)



This is the same matrix given by equation (7).



3.4 Spin Representation of the Euclidean Group

The problem of representing a Euclidean motion has thus far been translated into one of representing orthogonal automorphisms of R11013. However, R1,0,3 is a degenerate orthogonal space, viz. there are non-zero elements x of 1,0,3 which are orthogonal to the entire space, or equivalently (x,y) = 0 V yeR1,0,3. This constitutes a major disadvantage. Not only are degenerate orthogonal spaces not as well understood as their non-degenerate counterparts, but a multitude of important results for non-degenerate spaces do not apply unrestrictedly to degenerate spaces. For instance, the decomposition of arbitrary orthogonal automorphisms in terms of hyperplane reflections, an important result which is attributed to Dieudonne [1955], does not apply to degenerate orthogonal spaces.

Brooke [1980] overcame this problem of degeneracy by regarding R1,0,3 as an orthogonal subspace of a non-degenerate orthogonal space of higher dimension. It can easily be proved that R1,4 (short for R0,1,4), a five-dimensional orthogonal space with the symmetric bilinear form










1'4xR1,4 _+ IR (xy) = ((Xl,x2,x3,X4,X5),(yl,y2,y3,y4,y5)) = -xlyl + x2Y2 + x3Y3 + x4Y4 + x5y5, (1) is the non-degenerate orthogonal space of the lowest dimension which contains an orthogonal subspace isomorphic to R1,0,3. Brooke proved that

Spin+(l,0,3) = Spin+(l,4)nStab(e0 + e4)

using the theory of matrix representation of spin groups.

Here, a novel, simpler and more direct proof using

solely the theory of spin groups is given. Spin groups are subgroups of the Clifford algebra (see Appendix A) of the corresponding orthogonal space, which represent the orientation preserving orthogonal automorphisms of the space. They were devised by Cartan, and initially applied to relativistic physics. Initially interest is focussed upon the subgroup of orthogonal automorphisms of Ri,4, which leaves invariant an orthogonal subspace isomorphic to R1,0,3 embedded in i1,4. Later it will be required that the restriction of these orthogonal automorphisms to the orthogonal subspace isomorphic to R1,0,3 be a Euclidean mapping.

The dimensions of its Clifford algebra R1,4 and its even Clifford algebra R1,4 0 are respectively

dim R1,4 = 25 and dim R1,40 = 24 (2)

Let {e0,ele2,e3,e4} be an orthonormal basis of R1,4 with


and (ei,ei) = +1


(e0,e0) = -1


Vi = 1,2,3,4 (3)









Then, a basis for the even Clifford algebra R1,40 will be

{e$,e2e3,e3el,ele2; e0e4,e0e4e2e3,e0e4e3ele0e4ele2;

e0e1e2e3,e0e1,e0e2,e0e3; e1e2e3e4,e1e4,e2e4,e3e4}

This seemingly odd arrangement of the basis elements suggests the possibility of expressing an arbitrary element of R1,40 as

g = g, + e0e4g2 + e0g3 + g4e4, (4) where

g, = a0e + a1e2e3 + a2e3el + a3ele2 (5a) 92 = b0e + ble2e3 + b2e3el + b3ele2 (5b) 93 = c0e1e2e3 + clel + c2e2 + c3e3 (5c) 94 = d0e1e2e3 + d1el + d2e2 + d3e3 (5d) Obviously g1,g2ER30, and g3,g4e1R31, the even Clifford algebra of M0,3 and its complement respectively. Furthermore, P30 is isomorphic to the quaternion skewfield M.

A straightforward calculation shows that the conjugate of g is given by

g- = gl- + g2-e4e0 - g3-e0 - e4g4- (6) Thus, it follows from proposition A.3.11, appendix A, that a necessary and sufficient condition for g1,40 to belong to Spin(l,4) is that

g-g=� (7) This condition can be expressed, in terms of g1,g2,93 and g4, as follows








64

glg1 - g2g2 - g3-g3 + g4 = � (8a) gl-g3 - g2-g4 + g3-g1 - g4-g2 = 0 (8b) gl-g4 + g3-g2 + g4-g1 + g2-g3 = 0 (8c) g1-g2 - g2-g1 + g3-g4 - g4-g3 = 0 (8d)
The interest will be focussed on the orthogonal

automorphisms of R1,4 that leave invariant an orthogonal subspace, of R114, orthogonally isomorphic to R1,013. The existence of an orthogonal subspace of R1,4 isomorphic to R1,0,3 is clarified by firstly noticing that

((-e0 + e4),(-e0 + e4)) = 0 (9a) and

((e0 + e4),(e0 + e4)) = 0 (9b) and secondly by recognizing that (-e0 + e4),(e0 + e4)e[ele2,e3]1. Therefore, either [-e0 + e4]O[ej,e2,e33 or [e0 + e4)e[ele2,e3] form an orthogonal subspace of R1,4 isomorphic to R1,0,3. The orthogonal subspace [e0 + e4]*[ele2,e3] will be used from here on. A completely analogous development could be performed using [-e0 + e4]O[ele2,e3].

An element geSpin(1,4) is said to belong to the normalizer of [e0 + e4] (Rose [1978]), a subgroup of Spin(l,4), if

gxg-le[e0 + e4] Vxc[e0 + e4]
Since the action of Spin(l,-4) on R1,4 is linear, it suffices to test (e0 + e4); viz.

g(e0 + e4)g-1 = A(e0 + e4) for some AR (10)









However geSpin(1,4); thus g-g = �1, and it is easy to show that gg- = �1; hence g-i = �g-, (11) and the condition given by equation (10) becomes
g(e0 + e4)g- = A(e0 + e4) for some peR (12) In terms of the gi's, this condition translates into

gg1- - g1g2- - g2gl- + g2g2- ER (13a)

1g3--g2g3--g3g1-+g3g2-+g1g4--g2g4 --g4g1-+g4g2- = 0 (13b)
glg3-g2g3-+g3gl--g3g2-+glg14-g2g4-+g4gl--g4g2- = 0 (13c)
g3g3- + g4g4- + g3g4- + g4g3- = 0 (13d)
which can be simplified and expressed in the form

gg1- - g1g2- - g2g1- + g292- ER (14a) (g1 - g2)(g3 + g4)- = 0 (14b) (g3 + g4)(g1 - g2)- = 0 (14c) (g3 + g4)(g3 + g4)- = 0 (14d) Proposition A.4.1, in appendix A, shows that equation (14d) is satisfied if, and only if, 94 =c -93 (15)
This solution also satisfies equations (14b) and (14c). Further, since g1,g2ER30, then glgl-,g2g2-R, and the equation (14a) reduces to g2gl- + g1g2-IR (16) Substituting equation (15) into equations (4) and (8) yields
g = g, + e0e4g2 + e0g3 + (-g3)e4

= g, + e0e4g2 + (e0 + e4)g3 (17)










and

g1g1 - g2g2 = �i (18a) (g1 + g2)-g3 + g3-(g1 + g2) = 0 (18b) (g1 - g2)-g3 + g3-(g1 - g2) = 0 (18c) g1-g2 - g2-g= 0 (18d) Summarizing, the necessary and sufficient conditions for geSpin(l,4)nNormalizer[e0 + e4] are given by equations

(16),(17) and (18). Further, proposition A.4.2, in appendix A, shows that provided g, j' 03,

gl-g2 - g2-g1 = 0 g2 = Xg1 for some XER (19) This result satisfies the condition expressed by (16), viz.

g2g1- + g1g2- = Xggl- + gl(Xgl-) = 2Xglgl-eR and simplifies (17) and (18)

g = gl(l + Xe0e4) + (e0 + e4)g3 (20) (1 - X2)gl-gI = �1 (21a) g1-g3 + g3-g1 0 (21b) It is interesting to note that condition (21a) rules out the possibility of X = �1, and this result leads to the reduction of conditions (18c) and (18d) to (21b).

Finally, proposition A.4.3, in the appendix A, proves that provided g, 7 0,

glg3 + g3-g1 = 0 g3 = dg, for some dER3 This result is perhaps one of the most significant contributions of this development, and as far as the author

3If it is assumed that g, = 0, a subset of Spin(l,4) is obtained which is not closed under the Clifford product. Hence the subset cannot be homomorphic to the Euclidean group.









is aware the proof is new. From this result, equations (20) and (21) can be further reduced to

g = gl(l + Xe0e4) + (e0 + e4)dgI (22) (1 - X2)gl-gI = �1 (23) Although equation (21b) is eliminated, the representation given by equations (20) and (21), and the representation given by equations (22) and (23) are to be regarded as equally desirable.

Equations (22) and (23) are the simplest representations of orthogonal automorphisms of R1,4 which leave invariant the subspace [e0 + e4]. The additional conditions that they must satisfy to be orthogonal automorphisms of a subspace of R114 isomorphic to R110,3 are now examined in detail.

Consider an arbitrary element v of RI,4; then

v = a0(e0 + e4) + a4(-e0 + e4) + y, (24a) where

y = ale, + a2e2 + a3e3, (24b) and the action of geSpin(l,4)nNormalizer[e0 + e4] upon v will be

v' = gvg- = [gl(l + Xe0e4) + (e0 + e4)dgl]v [(1 + Xe4e0)gI- + gl-d(e0 + e4)] A lengthy computation yields v'= [a0(l - X)2glgI- - 2(1 - X)(d,glygl-) - 4a4glgl-(d,d)]

((e0 + e4)] + a4(l + X)2glgl-(-e0 + e4) + (1 - X2)glygI- 4a4(1 + X)glgl-d (25)








68
This transformation will represent a Euclidean motion of the orthogonal subspace [(-e0 + e4), el, e2, e3] if, and only if,

(1 + X)2glgI- = 1 (26a) and

(1 - X2)glygI- - 4a4(1 + X)glgl-d=hyh- + t Vye[ele2,e3] (26b) where hER30 with hh- = 1, and te(el, e2, e3] = R3.

These equations coupled with equation (23) lead to the following system of equations (1 - X2)glg- = �1 (27a) (1 + X2)glg- = 1 (27b) (1 - X2)glyg- = hyh- with hh- = 1 V ye[el,e2,e3] (27c) 4a4(1 + X)glgl-d = -t (27d) It is easy to see for there to be a solution to this system of equations that X = 0 (28a) Therefore

gg1- = 1 (28b) d = -t/(4a4) (28c) The final expression for the elements that satisfy the conditions indicated above is

g = g, - (e0 + e4)tgl/(4a4) (29) A simple calculation reveals that, for an arbitrarily fixed a4 5 0, the set of these elements form a subgroup of Spin(l,4)nNormalizer[e0 + e4], which will be denoted by










Spin+(1,0,3). It is claimed here that this is the most general spin representation of the Euclidean group E(3).

After this important result it is now necessary to

determine the action of the group upon the elements of the orthogonal space. Substituting X = 0 into equation (25), the group action becomes

v' = gvg- = [a0 + (t,glygl-)/(2a4) - (t,t)/(4a4)](e0 + e4) + a4(-e0 + e4) + glYg1 + t (30)

Choosing a0 = 0, one obtains an orthogonal subspace of R1,4 isomorphic to R1103', this selection reduces the group action to

V' = -[(t,glygl-) + (t,t)/2]/(2a4)(e0 + e4)

+ a4(-e0 + e4) + g1Yg1- + t (31)

In particular, the image of

w = O(e0 + e4) + a4(-e0 + e4) + 0 will be

w= -(t,t)/(4a4)(e0 + e4) + a4(-e0 + e4) + t

At the outset this result is surprising since the original vector has only one component along (-e0 + e4) whilst its image has in general components along all the elements of the basis {(e0 + e4), (-e0 + e4), el, e2, e3) of Rl,4. However a simple calculation shows that (w,w) = 0 = (w',w')

This phenomenon results from the fact that both (e0 + e4) and (-e0 + e4) are isotropic directions of the orthogonal space IRI,4 (eqn.(9)).








70

Finally, the following two properties of (e0 + e4) given by

(e0 + e4)2 = -((e0 + e4),(e0 + e4)) = 0 (eqn.(9b)) and

ei(eO + e4) = -(e0 + e4)ei V i = 1,2,3 (32) make it possible to introduce the dual unit c (Clifford [1876]), in the form

e = (e0 + e4) (33) with the properties

62 = 0 (34a) a= ae V aeIR30 (34b) ea = - ae V aCR31 (34c) Thus the general spin representation of the Euclidean group can now be transformed into g = g, - ctgl/(4a4) (35) with the usual restriction, gg1- = 1 (eqn.(28b)) It is easy to recognize that g, represents the rotational part of the mapping, while t is associated with the translational part. In particular, if the Euclidean motion represents a rotation of e degrees around an axis given by the unit vector 0 = u23eI + u31e2 + u12e3, then g, will be

g, = C(8/2) + S(e/2)(u23e2e3 + u31e3eI + ul2ele2), (36) where C(8/2) = Cos(8/2) and S(8/2) = Sin(e/2).

Although equations (35) and (28b) provide, indeed, a representation of the Euclidean group, the identification










given by (33) does not preserve the group action (30). Therefore a new group action will be required, this problem is addressed in the following section. Further, equation (33) is the first introduction of the dual unit c in this work; a second role of the dual unit e will be employed in the biquaternion-representation of the Euclidean group (see section 3.7).


3.5 The Restriction of the Group Action of the Spin

Representation on [el-e2se3l

Section 3.4 showed that the restriction on [ele2,e3] of the action of the group Spin+(l,0,3) on R1,4 constitutes a Euclidean motion. Although equation (4.30) of the group action includes the isotropic vectors (e0 + e4) and (-e0 + e4), it is clear that their computational importance is non-existent. Furthermore, the introduction of the dual unit e (eqn.(4.33)) did not preserve the natural group action of the spin groups over the orthogonal space. Thus, it appears natural to search for another group action of Spin+(l,0,3) on [el,e2,e3] exclusively; the new action is required to produce the same effect on [el,e2,e3] as the original one.

A guideline is evident, the new action cannot be of the form

y' = gyg- with ye[el,e2,e3]

because this action consists only of successive application










of the Clifford product. Since the Clifford product is linear, for y = 0

y' = gOg- = 0

Therefore this action cannot represent a translation.

Consider the set {f, el, e2, e3}; due to the properties of e (eqn.(4.34)), this set can be regarded as an orthonormal basis of the degenerate space R1,0,3. Then the mapping
. : R3 - R1,0,3 t(y) = 1 + ey (la) is evidently injective; hence
t* : IR 3 -. I(R3) t*(y) = t(y) VyeIR3 (1b) is bijective. Consider the following action, which resembles that proposed by Porteous [19811 and used by Selig [1986]
-g a: Spin+(1,0,3)xR1,0,3 - R1,0,3

ag(y) = [gl - etgl/(4a4)][l + Ey1][g1- - egl-t/(4a4)]

= 1 + E[glygl- - t/(2a4)] (2) Therefore

(L*)-lagt*(y) : R3 _ R3 y' = glygl- - t/(2a4) (3) will be a Euclidean motion if, and only if,

-t/(2a4) = t (4) this condition requires a4 = -1/2. Thus, for this specific group action, the representation of the Euclidean group is g = g, + ctgl/2 (5) Using this result, the group action transforms into

(t*)-lagt*(y) : R3 _ R3 Y' = glygl- + t (6) This is precisely the action proposed by Porteous and used by Selig. However, the particular value of a4 depends upon the










mapping t : R3 - R1,0,3, and other mappings are certainly possible.



3.6 Invariants of the Representation of the Euclidean Group The invariants associated with the representation of the Euclidean group will be now indicated.

If the representation provided by equation (4.29), or its equivalent equation (4.35), is used, the unique representation invariant is g1g1- = 1 (eqn.(4.28b)) However, if we use the representation provided by the equations (4.20), and (4.21), the substitution of (4.28a) leads to

g = g, + (e0 + e4)g3, (la) and the substitution of (4.33) finishes the transformation into

g= g 1+ 6g3 (1b) with the conditions

gg1- = 1 (eqn.(4.28b)) and

gl-g3 + g3-g1 = 0 (eqn.(4.21b)) Hence, the invariants of this representation are precisely those given by (4.28b), and (4.21b).







74

3.7 The Biquaternion Representation of the Euclidean Group Section 3.4 proved that the most general representation of the Euclidean group is of the form

g = g, - (e0 + e4)tgl/(4a4) (eqn. (4.29)) where g1IR30 with ggl- = 1, tep3, and a4ER with a4 J 0.

In the process of proving proposition A.4.3, in appendix A, it is shown that if t = tle, + t2e2 + t3e3, (1) and

g, = aoe + aje2e3 + a2e3e, + a3ele2, (2) then

tgI = c0ele2e3 + cleI + c2e2 + c3e3, (3) where

cI = a0tI + a3t2 - a2t3 (4a) c2 = -a3tI + a0t2 + a1t3 (4b) c3 = a2tI - a1t2 + a0t3 (4c) co = alt, + a2t2 + a3t3 (4d) However, it is possible to rewrite (3) in the form

tgl = ele2e3(c0 - cle2e3 - c2e3eI - c3ele2) (5) Further, a simple calculation shows that co - cle2e3 - c2e3eI - c3ele2 = -(tle2e3 + t2e3eI + t3ele2) (aoe + aje2e3 + a2e3eI + a3ele2) (6) If t* is the image of t under the mapping S: g3 P fl 30 P(t) = j(tleI + t2e2 + t3e3) = tle2e3 + t2e3eI + t3ele2 = t*, (7) then the representation of the Euclidean group becomes








75

g = g, + (eo + e4)ele2e3t*gl/(4a4) (8)

Since

[(e0 + e4)ele2e3]2 = 0 (9a) and

[(e0 + e4)ele2e3]ei = -ei[(eo + e4)ele2e3] V i = 1,2,3, (9b) it is possible to introduce the substitution

6 = [(e0 + e4)ele2e3] (10) with the usual properties 62 = 0 (eqn.(4.34a)) ca= ae V aER30 (eqn.(4.34b)) a= -ae V aER31 (eqn.(4.34c)) Then the representation of the Euclidean group will be g = gl + ct*gl/(4a4) (11) with

g -= 1 (eqn.(4.28b)) This expression can be regarded-as a generalized

biquaternion representation, and it exhibits the second role played, in this work, by the dual unit e. It can be easily proved that the set of elements of the form given by equation

(8) together with the Clifford product forms a group. Certainly, the same is true for the set of elements of the form given by equation (11), but because of the introduction of the dual unit, the Clifford product becomes the usual biquaternion product (also called the dual quaternion product, Yang [1963]). The set of all elements of the form

g = hI + ch2 where hl,h2 6 R30 = 1H (12)









is correctly called the ring of biquaternions or dual quaternions and is denoted here by 21H.

Certainly the biquaternion representation, given by

equation (11), can be used to obtain the image of any point, of the Euclidean space, under a Euclidean mapping. Consequently, the biquaternion representation provides a representation of the Euclidean group in the extended sense of section 3.1.

Consider the mapping
t : R3 , 2H f(y) = i(y1el + Y2e2 + Y3e3)

= 1 + e(yle2e3 + Y2e3el + y3ele2) = 1 + Ey* (13)

Evidently the map is injective; hence
t* : [R3 -. I(R3) i*(y) = t(y) V yER3 is bijective. Consider the action ag : 2M , 2E

ag(y) = [g, + et*gl/(4a4)](I + ey*)[gl- + egl-t*/(4a4)]

= 1 + E[gly*gl- + t*/(2a4)]; (14) then the mapping
(i*)-lgt*(y) : R3 . R3 y' = glygl- + t/(2a4) (15) is a Euclidean mapping if, and only if, t/(2a4) = t (16) This condition requires that a4 = 1/2. Thus, for this specific group action, the representation of the Euclidean group is


g = gl + 7(t*gl)/2


(17)









Again, it is necessary to realize that this particular

value of a4 depends upon the injection t : R3 _ 2M, and other injections are certainly possible.

Of course, if the starting point of this analysis is the representation given by g = g, + (e0 + e4)g3 (eqn.(6.1a)) with

gg1l =1 (eqn.(4.28b)) g1-g3 + g3-g1 = 0, (eqn.(4.21b)) then, following the same procedure, and introducing the dual unit (eqn.(10)), the representation can be written as

g = g, + [(e0 + e4)e1e2e3](e1e2e3g3)
= g, + e(ele2e3g3) (18) Since g3, ele2e3cR31, then ele2e3g3cR30 M. Calling g3* = ele2e3g3 (19) the representation becomes g = g1 + 6g3* (20) with

g1g1- = 1 (eqn.(4.28b)) and

gl-g3* + g3*g1 = 0 (21) These equations constitute the foundations of the

kinematic mapping proposed by Ravani and Roth [1984], which associates a Euclidean mapping with a point of the dual four-dimensional sphere.









3.8 The Representation of the Induced Line Transformation

Throughout this study, the Euclidean space has been considered as an aggregate of points satisfying certain properties; accordingly, the Euclidean group was considered as a group of transformations acting upon this aggregate of points. This point of view is not unique, and no claim is made that it is the most appropriate for the study of spatial kinematics.

Since the middle of last century (Klein [1939], Giering [1982]), it was well known that the Euclidean space can be deemed as a collection of straight lines (or even a collection of planes, a possibility which will not be considered here). It is useful to let the Euclidean group act upon this set of lines, and to investigate its effects on the set.

Following the fundamentals of chapter 2, a straight line passing through two points, M and N, of the Euclidean space is defined as

MN = (PeE I 6(P,M) = X6(N,M) for some XeR) (1) This definition exhibits the two characteristics of a line in the Euclidean space, a point M lying on the line, and a vector 6(M,N) directed along the line.

Choosing a length scale and a reference system, with origin 0, one obtains

y = 6(M,O) and w = 6(N,M) where y,weR3 (2) Then it is possible to define the six-dimensional vector







79

p = (w, wxy) (3) where x stands for the usual vector product. Since any other vector w' = Aw, for some non-zero MeR, also lies along the same line, it is possible to use Ap = (Aw, Awxy) (4) It can be proved that there is a bijective correspondence between the collection of sets of six-dimensional vectors given by equation (4), and the lines of the Euclidean space. Therefore, for a given p = (w,wxy), the set Ap = {jp Ifor some nonzero AeR) is called the homogeneous Plucker coordinates of the line represented by p.

McCarthy [1986b] shows that when a Euclidean mapping given by

y' = Ry + t (5) acts on the Euclidean space, the Plucker coordinates of the image of p = (w,wxy) are


, = (6)
(xy) T R x


where T stands for the skew-symmetric matrix representing the usual three-dimensional vector product; viz.

0 -t3 t2

T= t3 0 -tI (7)

-t2 tI 01

The transformation given by equation 6 is usually referred as the induced--by a Euclidean motion--line









transformation (Klein [1939]). Using these fundamental results, McCarthy reformulates the equation into a three-dimensional dual-matrix equation, and uses this dual-matrix form to provide another illustration of the closure equation for spatial mechanisms.

In what follows, it will be shown that these rather

cumbersome representations of the induced line transformation are unnecessary. In fact, the same representation, of the Euclidean group, can be employed regardless of whether the action of the group is upon points or lines. This result is overlooked in the literature on kinematics.

Let y = S(M,O), and z = 6(N,O); then the Placker

coordinates of the line passing through M and N are, in terms of the Clifford algebra R3, given by p = (y - z) + (zy - yz)/2, (8) where juxtaposition symbolizes the Clifford product. Let L3 represent the set of all lines of the Euclidean space, and consider the mapping

t : L3 - R1,0,3

t(p) = t[(y - z) + (zy - yz)/2] = (y - z) + e(zy - yz)/2 (9) Ostensibly, the mapping is injective; hence

t* : L3 - t(L3) with t*(p) = t(p) VpcL3 is a bijection. Then consider the action











ag(t


Fin


ag : E(3)xt*(L3) - i*(L3)
*(P))= [g1 + (tgl/2)6]{(y - z) + [(zy -yz)/2]e)

[gl - (gl-t/2)6]

= g1(y - z)g1- + 6{gl(zy - yz)gl+ t[gl(y - z)gl-] - [gl(y - z)gl-]tl/2

ally, the induced line transformation is given by

t*-la gt* : L3 - L3

i*-lagt*(P) = gl(Y - z)gl- + E{gl(zy - yz)gl+ t[gl(y - z)gl-] - [gl(y - z)gl-]t)/2


3.9 Screw Representation of the Euclidean Group

It has been known, since the last century (Ball [1900]), that a Euclidean motion can be represented, after selecting a reference system, by a twist applied to a screw. The screw is defined as an arbitrary line endowed with a pitch. Section

2.8, demonstrated that the component of the translation vector along the rotation axis is not a Euclidean invariant; therefore the pitch of the screw--the ratio of the magnitude of this component to the rotation angle--is not a Euclidean invariant. In order to underscore this point, a screw will be represented here by


$ = (O, ii, r, d)


(1)


where


e = Rotation angle, or twist, with 0 e 5 r.










1 = Unit vector associated to the rotation axis, such

that the rotation is perceived, from the tip of 0,

as counterclockwise. If e = 0, according to this

rule, 0 is undefined; in this case, 0 must lie along

the translation vector.

r = Perpendicular vector from the origin of the

reference system to the line associated to the screw

(r.11 = 0).

d = Magnitude of the component of the translation vector

along the axis of rotation. If e = 0, dq is the

total translation vector.

This ordering separates those screw parameters which are indeed Euclidean invariants (e,O), from those which depend upon the selection of the reference system (r,d). The restrictions on 9 and Q aim to avoid any multiplicity of the representation. This idea was outlined, for the pure rotation case, by Altmann [1986].

The set of all tetrads (9, i, r, d) satisfying the condition given in the definition above is denoted by S. From what has been pointed out, it follows that the mapping

Z : E(3) - S Z(P) = (9, C1, r, d) = $ (2) is a bijection. This bijection permits to furnish the set S with a group structure, according to the rule

$21 = $2 $1 = E[Z-1($2) Z-($1)] (3) This rule assigns to $2 $1 the screw of the element of








83
Euclidean group obtained by composing E-($2) and E-1($1), in this order.

It is desired to obtain the four parameters of $21 in

terms of the parameters of $1 and $2. This problem was solved by Halphen (Beggs [1983)), using descriptive geometry methods, a century ago. However, an analytical solution did not come out easily, even though Rodrigues (Gray [1980]) provided, in 1840, the correct equation for the composition of general spatial rotations. In fact, as late as 1963, Paul [1963] published a didactic paper about composition of finite rotations. With regard to the more general problem of analytically composing general spatial motions, Beatty [1966] acknowledged-many geometrical proofs but none analytical. Then Beatty provides a vector and matrix algebra solution of the composition of spatial motions, and the determination of some characteristics of the resultant motion.

The problem was partially solved by Roth [1967], who

introduced the screw triangle as an spatial generalization of the planar pole triangle. This screw triangle permits finding the composite screw in terms of the two composing screws. However, the formuli exhibit there assume the origin of the coordinate system lying on one of the composing screws. Recently, Phillips and Zhang [1987] considered the relationship-between the screw triangle and its infinitesimal counterpart the cylindroid.










Here, the problem will be posed as follows: Given $1 = (81, Ul, rl, dl), and $2 = (82, Q2. r2, d2), find equations that provide $21 = (821, Q21, r21, d21) in terms of $1 and $2- Moreover, the problem will be solved within the realm of the Clifford algebra R3The results obtained in section 2.9, concerning the semi-direct structure of the Euclidean group with the rotation as the direct part, indicate that the parameters related to the rotation (821, A21) are independent of the parameters related to the translation. Therefore, they must be calculated first.

Let a, = XleI + X2e2 + X3e3, and a2 = . lel + A2e2 + A3e3 be the unit vectors associated to the rotation axes; then the elements of R30 2 M representing the rotations are

1= Cl + Sl(Xle2e3 + X2e3eI + X3ele2) (4a) 92 C2 + S2(gle2e3 + A2e3el + A3ele2), (4b) where Ci = Cos(9i/2), and Si = Sin(9i/2); therefore, the element representing the composite rotation is

921 = g2g1 = [C2CI - S2Sl(Al + + X3/3))
+ e2e3[S2CA1 + C2SlX1 + S2S1(t2X3 - A3X2)] + e3el[S2ClM2 + C2SIX2 + S2SI(3X1 - A3)]

+ ele2[S2Cl3 + C2SIX3 + S2Sl(A1X2 - A2Xl)] (5)

From equation (5), it follows that the angle of the composite rotation satisfies

C21 = C2CI - S2SI(XA1 + X2M2 + X313) (6)








85

Since by restriction 0 5 921 : r, the sine of the angle must be positive; hence

S21 = [1 - (C21)2]k (7) and the unit vector associated to the composite rotation is

U21 = {eI[S2CIAI + C2SIAI + S2SI(A2X3 - A3X2)]

+ e2(S2CIA2 + C2SlX2 + S2SI(A3X1 - AIX3))

+ e3[S2CA3 + C2SIX3 + S2S1(AIX2 - A2XI)])/$21 (8)

The calculation of the parameters related to the

translation basically involves the reconstruction, from the screw parameters, of the translation vector associated with the origin of the reference system, followed by the calculation of the corresponding vector of the composite motion, and a final decomposition.

A simple calculation shows that the translation of the origin of the reference system due to the motion represented by $1 and $2 are

t= r, - glrlgj- + dj1~ (9a) t2= r2 - g2r292- + d202 (9b) Thus the translation vector of the origin of the reference system under the composite motion is

t21 = g2(rl - glrlgl- + d1]g2- + r2 - g2r2g2- + d2Q2 (10)

From equation (10), the components of the translation vector along the rotation axis and perpendicular to it are

d2=121 (t21 + 021t21021)/2 (la)

t21 = (t21 - 021t21021)/2 (llb)










Finally, the-perpendicular vector from the origin of the reference system to the screw axis is obtained from t2 = r21 - g21r21g21- (12a) Premultiplying by g21-, and postmultiplying by g21, equation (12a) becomes

g21t212= g21-r21g21 - r21 (12b) Since r21 is perpendicular to the rotation axis associated with g21, one obtains

g21r21g21- = r21Cose21 + w (13a) and

g21-r21g21 = r21Cos821 - w (13b) with wE[r21, U21]'.

The substitution of equations (13a) and (13b) into (12a) and (12b) yields

t21 = r21(l - Cose21) + w (14a) and

g21t211g21 = -r21(l - Cos821) + w (14b) Subtracting equation (14b) from (14a), and solving for r21, one obtains

r2= (t21 _L - g21-t21-Lg21)/[2(l - CosE21)] (15)

These equations provide the composition laws for the group of tetrads of quantities representing the screws. Of course, if the reference system is selected such that the origin is located on one of the axes of the composing screws, these equations collapse to those provided by the screw triangle.














CHAPTER 4
THE EUCLIDEAN GROUP AS A LIE GROUP

Thus far, this thesis has dealt exclusively with the

algebraic aspects of the Euclidean group and its action upon the Euclidean space. In this chapter the topological and metrical characteristics of the Euclidean group will evolve when the group is regarded as a Lie Group.

It is important to recognize that there is an algebra

associated with every Lie group. In the case of the Euclidean group, this algebra will be identified with the algebra of infinitesimal screws or screws for short. Furthermore, this algebra is endowed with symmetric bilinear forms which remain invariant under the action of the group into itself. This property provides the Euclidean group with the structure of a semi-Riemmanian manifold with a hyperbolic metric tensor.

Even though the theory of Lie groups, differential

manifolds and Riemannian geometry is relatively recent, the wealth of information it yields is extraordinary. It is therefore not possible here to give details of all the background material. Reference is however made to the standard works in the relevant subjects.










4.1 Lie Groups

Definition 1. A Lie group G (Boothby (1975)) is an algebraic group which simultaneously is a differentiable manifold, and such that the mappings

GxG - G (g2,g1) -* g2g1 (1) and

G - G g -9g1 (2) are differentiable mappings.

A differentiable manifold is, loosely speaking, a set with sufficient topological structure that permits the development of a differential calculus on the set. A formal definition is provided below (Boothby [1975]).

Definition 2. A differentiable manifold M is a Hausdorff space with a countable basis of open sets, such that each point of M has a neighborhood homeomorphic to an open set of an. Each pair (U,I), where U is an open set of M and 0 is a homeomorphism of U to an open set of an, is called a coordinate neighborhood. In addition, the differentiable manifold must have a family I = {(Ua,))) of coordinate neighborhoods such that

1. The U. form a covering of M,

2. For any a,fl the coordinate neighborhoods are

compatible; viz. either UanUp = 4, or the mappings

-()-I : an - an and a(n)- : nn n

are diffeomorphisms,








89

3. If a coordinate neighborhood (V,*) is compatible with

every (U.,a)', then (V,*)el.

In order to prove that the Euclidean group E(3) is a Lie group, perhaps, the best approach (Belinfante and Kolman [1972]) is to consider firstly the general linear group GL(n), which is the group of all non-singular nxn real matrices. This group is a manifold since the n2 entries of an arbitrary matrix can be used as the image of an element of GL(n) under a homeomorphism from GL(n) to Rnxn. Furthermore, since the determinant function is continuous, the elements of the group GL(n) form an open subset. Moreover, the entries of a product of matrices are polynomials in the entries of the factors; whilst the entries of the inverse of a matrix are rational functions of the entries of the original matrix. Thus, the functions are analytical and therefore differentiable. Hence, GL(n) is a Lie group.

The final step is to make use of a result (Warner

[1983)) which states that closed algebraic subgroups of a Lie group are Lie subgroups. It has already been shown (see section 3.3) that E(3) = SO+(I,0,3) is a subgroup of GL(4), and that the conditions imposed on E(3) produce a closed subset of GL(4). Hence, the result proves that E(3) is a Lie group.










4.2 The Lie Algebra of the Euclidean Group

An important property of Lie groups is the existence of an algebra of vector fields defined on the group, which is isomorphic to an algebra of tangent vectors belonging to the tangent space at the identity element (O'Neill [1983]). Since both algebras are isomorphic, any one of them is referred as the Lie algebra of the corresponding Lie group. If G is a Lie group, it is customary to represent its Lie algebra by 9. In particular, the Lie algebra of the euclidean group E(3) is represented by e(3).

Here an explicit representation of e(3) is provided. Of course this representation can be obtained using anyone of the representations of the Euclidean group given in chapter 3 (Karger and Novdk [1985]). Nonetheless, the spin representation is used here. An advantage of using this Clifford algebra based representation is that Clifford algebra is robust enough to handle concisely algebraic manipulations of the new algebra e(3) without having to resort to a new algebraic structure.

Let an arbitrary motion of a rigid body be given by

g = g, + etgl/2 (eqn. (3.5.5)) where g, g1, and t are functions of time; it was previously shown that g, can be expressed in the form gi=C(8/2)+S(0/2)(u23e2e3 + u31e3eI + u12ele2) (eqn. (3.4.36)) with


U232 + U312 + u122 = 3,


(1a)










and t is given by

t = tie, + t2e2 + t3e3 (ib) If g represents the identity mapping t in the Euclidean group, then 6 = 00; thus g, = 1, and t = 0, with

0 = u23e2e3 + u31e3eI + u12ele2 unitary but otherwise unrestricted.

The derivative of the motion, represented by g, at the identity t of the Euclidean group is given by

g(t) = g1(t) - e[t(t)gl(t) + t(t)gl(t)]/2 (2) where indicates the derivative with respect to time. Since

= (S(8/2))/2 + O(C(e/2))(u23e2e3 + u31e3el + u12ele2)/2

+ S(e/2)(u23e2e3 + u31e3el + u12ele2) (3a) and

= tlel + t2e2 + t3e3, (3b) then

g1(t) = 6(t)(u23e2e3 + u31e3el + u12ele2)/2 (4a) and

t(t) = t1(t)el + t2(t)e2 + t3(t)e3 (4b) Hence, finally

g(t) = O(t)(u23e2e3 + u31e3eI + u12ele2)/2

- e(tl(t)el + t2(t)e2 + t3(t)e3)/2 (5) Incorporating the scalar terms into the components of the vector, the tangent space at the identity--i.e. the collection of the derivatives evaluated at the identity of all possible motions--will be




Full Text

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REPRESENTATIONS OF THE EUCLIDEAN GROUP AND ITS APPLICATIONS TO THE KINEMATICS OF SPATIAL CHAINS By JOSE MARIA RICO MARTINEZ A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1988 v ||miyBRSlTY_OF FLORIDA LIBRARIES

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Copyright 1988 by Jose Maria Rico Martinez

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ACKNOWLEDGMENTS The author wishes to thank firstly his advisor Dr. Joseph Duffy for his guidance during the selection and development of the contents of this dissertation. Without his encouragement, in times when nothing fruitful seemed to evolve from the approaches followed, without his geometrical insight, and his quest for clarity, this dissertation would have plenty of errors. Nonetheless, the author is to blame for the remaining ones. Secondly, deep felt thanks go to the members of the supervisory committee for their invaluable criticism and for their teachings in the classroom. The faculty of the Mechanical Engineering Department, and in particular the faculty of the Center for Intelligent Machines and Robotics, must be thanked for the development of an atmosphere conductive to research. The contributions of the faculty of this center, including the visiting professors, can be found in many parts of this work. Special gratitude is owed to visiting professors Eric Primrose and Kenneth H. Hunt for their insightful comments. My thanks go also to my fellow students for their friendship and kindness. The economic support from the Mexican Ministry of Public Education and the Consejo Nacional de Ciencia y Tecnologia (CONACYT) is dutifully acknowledged. Finally, the author thanks his wife iii

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and children for bearing together four years of uncertainties, new experiences, and, hopefully, familiar growth . iv

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TABLE OF CONTENTS Rag e ACKNOWLEDGMENTS iii LIST OF FIGURES vii ABSTRACT viii CHAPTERS 1 INTRODUCTION 1 1.1 The Role of the Euclidean Group 3 1.2 The Euclidean Group as a Lie Group 5 1.3 Grassmann and Clifford Algebra Versus Standard Vector Calculus 9 1.4 Objectives and Organization of the Work 11 2 EUCLIDEAN SPACE AND EUCLIDEAN TRANSFORMATIONS 15 2.1 Physical Space as a Euclidean Space 16 2.2 Free Vector Algebraic Structure in ExE 17 2.3 Bound Vector Algebraic Structure in ExE 23 2.4 Euclidean Mappings 26 2.5 Properties of the Euclidean Mappings 31 2.6 Translations 35 2.7 Rotations 37 2.8 Decomposition of Euclidean Mappings 40 2.9 Composition of Euclidean Mappings 46 3 REPRESENTATION OF THE EUCLIDEAN GROUP 48 3.1 Fundamentals of Representation Theory 50 3.2 The Affine "Representation" of the Euclidean Group 53 3.3 The 4x4 Matrix Representation of the Euclidean Group 56 3.4 Spin Representation of the Euclidean Group .... 61 3.5 The Restriction of the Group Action of the Spin Representation on [ 62 , 62 , 63 ] 71 3.6 Invariants of the Representation of the Euclidean Group 73 v

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p age 3.7 The Biquaternion Representation of the Euclidean Group 7 4 3.8 The Representation of the Induced Line Transformation 78 3.9 Screw Representation of the Euclidean Group ... 81 4 THE EUCLIDEAN GROUP AS A LIE GROUP 87 4.1 Lie Groups 88 4.2 The Lie Algebra of the Euclidean Group 90 4.3 The Adjoint Representation of the Euclidean Group 95 4.4 The Euclidean Group as a Semi-Riemannian Manifold 97 4.5 Analysis of the Structure of the Set of Second Derivatives of the Euclidean Group at the Identity 102 5 APPLICATIONS 105 5.1 The Principle of Transference 105 5.2 Analysis of the Dualization Process 108 5.3 Statement and Proof of the Principle of Transference 114 CONCLUSIONS 12 6 APPENDICES A CLIFFORD ALGEBRAS 128 B A PHYSICAL INTERPRETATION OF THE ADJOINT TRANSFORMATION 144 REFERENCES 148 BIOGRAPHICAL SKETCH 158 vi

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LIST OF FIGURES pa ge Figure 5.1 Skeletal Spatial Kinematic Chain 115 Figure 5.2 Skeletal Spherical Chain Associated with the Spatial Chain of Figure 5.1 116 vii

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Abstract of the Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy REPRESENTATIONS OF THE EUCLIDEAN GROUP AND ITS APPLICATIONS TO THE KINEMATICS OF SPATIAL CHAINS By JOSE MARIA RICO MARTINEZ August 1988 Chairman: Dr. Joseph Duffy Major Department: Mechanical Engineering A coordinate free analysis of the Euclidean group is performed by employing the theory of affine and orthogonal spaces. A unified treatment of several of the most used representations of the Euclidean group, including homogeneous transformations, spin and biquaternion representations, is obtained by applying the theory of group representations. Several results related to the Clifford algebra of the positive definite three-dimensional orthogonal space are proved. These results are employed in the deduction of a generalized spin representation of the Euclidean group. The action of the spin representation over the points of a three-dimensional space yields the particular form of the spin representation that is commonly used in kinematics. The biquaternion representation is obtained from the spin representation, and the two distinct roles played by the viii

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dual unit in these last two representations are explained. The Euclidean group is analyzed as a Lie group. A representation of the Lie algebra of the Euclidean group is obtained from the spin representation of the Euclidean group. The Lie algebra operations are compactly expressed in terms of Clifford algebras. The Euclidean group is characterized as a six-dimensional semi-Riemannian manifold with a hyperbolic metric. The structure of the second derivatives of the Euclidean group at the identity is analyzed. The mathematical apparatus developed is applied to the principle of transference. A thorough analysis and proof of the principle of transference are given. ix

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CHAPTER 1 INTRODUCTION A striking feature that any new student of kinematics and robotics has to endure, during his or her study of the subject, is the great variety of mathematical techniques that it is necessary to master, or at least be acquainted with, in order to obtain a working knowledge of the recent advances in kinematics and robotics. These mathematical systems range from very elementary ones, such as complex numbers (Blaschke and Muller [1956], Rooney [1978b], Sandor and Erdman [1984]), standard vector algebra (Brand [1947], Chace [1963], Lipkin and Duffy [1985]) to more sophisticated structures such as matrix algebra (Suh and Radcliffe [1978], Bottema and Roth [1979], Angeles [1982]), quaternions (Blaschke [I960]), dual numbers (Rooney [1978b], Duffy [1980]), biquaternions (Yang [1963], Keler [1970a]), screw calculus (Yuan et al. [1971a], Yuan et al. [1971b], Hunt [1978], Phillips [1982]), line geometry (Weiss [1935], Woo and Freudenstein [1969]), and Grassmann algebra (Pengilley and Browne [1987]). All these techniques have been employed in an effort to disentangle the secrets of kinematics. Yang [1963], in his dissertation, provides a list of references dealing with early attempts to provide mathematical structures suitable for dealing with kinematic 1

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2 analysis and synthesis. Some of these structures disappeared because they rendered an incomplete description of the kinematic phenomena, or proved to be too cumbersome to deal with. Others, like matrix analysis and biquaternions, have remained and become common tools for kinematicians . However, the newer attempts to provide better mathematical tools (Ho [1966], Hiller and Woernle [1984], Pennock and Yang [1984], Casey and Lam [1986], McCarthy [1986b]), and the rediscovery of old mathematical structures in a disguised, or slightly different form (Waldron [1973], Sugimoto and Matsumoto [1984], Nikravesh [1984], Wehage [1984], Nikravesh et al. [1985], Sugimoto [1986], Lin [1987], Agrawal [1987]) suggest a lack of agreement, or even a lack of discussion, about the criteria that a mathematical system suitable for solving kinematics problems must satisfy. For a newcomer, the problem is aggravated because some of the mathematical structures have several distinct names, for example, the biquaternions (Clifford [1882]) are also called dual complex numbers (Keler [1970a], Keler [1970b], Beran [1977]), dual quaternions (Yang [1963], Wittenburg [1977], Wittenburg [1984]), and dual number quaternions (Agrawal [1987]). Furthermore, the description of the mathematical system employed by a kinematician is, sometimes, incomplete or obscure due to a lack of mathematical training, or an emphasis on the application of the mathematical system rather than on logical subtleties.

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1 . 1 The Role of the Euclidean Group A common characteristic of most of the mathematical 3 structures discussed in the previous section is that they are equipped with means to represent general rigid body motions, or special cases such as planar or spherical motions. In fact, it is well known that complex numbers represent rotations around a fixed axis, and hence the planar displacements which are of interest in planar kinematics. Quaternions and matrices represent the rotations around a fixed point, that are the transformations analyzed in spherical kinematics. Finally, dual quaternions, screws, and matrices represent general spatial movements of a rigid body. Moreover, the mathematical structures mentioned above allow the representation of the geometrical entities — such as points, lines, and planes which belong to a rigid body — by using elements of the structure. Furthermore, the structures allow the determination of the image, under a rigid body displacement, of these entities by manipulating their corresponding elements. In the language of modern mathematics, the set of points and lines of rigid bodies, together with the set of rigid body displacements, forms a model of Euclidean geometry (Giering [1982]). From the previous paragraph, it follows that the set of rigid body displacements is of paramount importance in the study of kinematics. Actually, this set forms a group under the composition operation, and it is called the Euclidean

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4 group. This fact is latent even in the writings of the nineteenth century kinematicians (Study [1903]), and it was known to physicists and mathematicians of the early part of this century (Weiss [1935]). However, the group property appears to have been overlooked by the kinematicians of the fifties and early sixties. Blaschke and Muller [1956] did mentioned the group property for the set of planar displacements. Later Blaschke [1960] and Muller [1962] did mention the group property of spherical displacements and Suh and Radcliffe [1967] proved the group property for the set of planar motions. Suh and Radcliffe [1968] further suggested that the results were also true for the spatial case. Bottema and Roth [1979] provided a synthetic proof of the general case. 1 The fact that the set of rigid body displacements forms a group under composition has far reaching implications, for it permits the use of the existing framework of group theory and group representations (Herstein [1975], Rose [1978], Fraleigh [1982], Naimark and Shtern [1982]). 2 In particular, it allows the introduction of the important concept of isomorphism. Two mathematical structures are isomorphic if there is a bijective mapping (one-to-one and onto) that -'-From now on, only general spatial displacements will be considered. 2 Recently, the theory of groups has been applied to robotics, but in the field of artificial intelligence (Popplestone [1984]).

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5 preserves all the operations defined in the structure. If two structures are isomorphic, then, loosely speaking, they are identical; just the names of the elements and the appearance of the operations have changed. Furthermore, any calculations performed in one structure can be equally carried out in the other. Hence, after two structures have been proven to be isomorphic, the criteria for selecting the most appropriate one for purposes of representing physical phenomena are reduced to 1. The compactness of the resulting expressions, 2. The number of additions or multiplications of real (or complex) numbers that the operations require, and 3. The previous knowledge of the structure. The work most closely related to the objectives addressed here is that of Rooney (Rooney [1977], Rooney [1978a]), who discusses several possible representations of rotations around a fixed point, and general spatial displacements. However, no use of the key concept of isomorphism is made, and the spin representation of the Euclidean group is not mentioned. 1 . 2 The Euclidean Group as a Lie Group In the previous part of this introduction, the Euclidean group has been regarded just as an algebraic group whose elements are transformations of the Euclidean space. This

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6 characterization suffices when the kinematic problems to be solved are related to the action, over the points and lines of a rigid body, of a finite number of elements of the Euclidean group (For example, synthesis of mechanisms with a finite number of precision points) . The solutions of these problems were precisely the main objectives of most of the kinematical research done in the mid 50s and 60s when kinematics was re-discovered in American universities. Only some authors of the German school (Blaschke and Muller [1956], Blaschke [1960], Muller [1962], Muller [1970]) pursued the analysis of the structure of the Euclidean group itself. In this way, they adhered to the tradition of the kinematicians of the late nineteenth century and of early this century (Stephanos [1883], Study [1903], Weiss [1935]). In the late 70s and early this decade, Roth and his coworkers (Bottema and Roth [1979], De Sa [1979], Ravani [1982], Ravani and Roth [1984], McCarthy [1986a]) re-discovered and generalized the kinematic mappings. These kinematic mappings are representations of the Euclidean group, although in the initial writings of Roth's school the group structure, a key property, was overlooked. The main objective of these new researches was to provide a sense of "metric" in the Euclidean group itself and to perform synthesis of mechanisms, where the synthesis criterion is the "closeness" of the synthesized motion of a rigid body with respect to some specified motion. Thus, the

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7 number of precision points is infinite. It turns out that the solution of this problem requires to consider the Euclidean group as a Lie group. This fact was not however recognized in their studies. The origin of the theory of Lie groups goes back to 1880, when Sophus Lie, attempting to emulate Felix Klein's employment of group theoretical concepts in the study of geometry, introduced the idea of continuous groups of transformations (Lie [1975]), later known as Lie groups. The theory of Lie groups proceeded to develop throughout the first part of this century. Concurrently, significant advances were made in differential geometry, including Riemannian and non-Riemannian geometry, and differentiable manifolds, primarily as a result of the newly discovered theory of relativity. The interchange of ideas between these two fields led to their blossoming in the 50s. Today the references about Lie groups and differential geometry are numerous. Furthermore, the depth of the references is variable, ranging from elementary introductions (Spivak [1965], O'Neill [1966], Boothby [1975], DoCarmo [1976], Sattinger and Weaver [1986]), to research publications (Chevalley [1946], Hermann [1966], Belinfante and Kolman [1972], Gilmore [1974], Helgason [1978], Spivak [1979], Warner [1983], O'Neill [1983]). The theory of Lie groups and Lie algebras has been applied, in an unconscious way, to kinematics for a long

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8 time. Indeed, Karger and Novak [1985] show that the algebra of infinitesimal screws is isomorphic to the Lie algebra of the Euclidean group. Hence, Ball's contributions (Ball [1900]), and all the research along this line (Yuan et al. [1971a], Yuan et al. [1971b], Yang [1974], Hunt [1978]) can be regarded as an application of the Lie group theory to spatial kinematics. The conscious application of Lie group theory to spatial kinematics is more recent. This delay can be explained by a two-part argument; first, with the discovery of relativity theory, most of the research done by physicists and applied mathematicians was directed toward new orthogonal spaces, such as Lorentz space and Minkowsky space and their transformations group. Only recently there has been an interest in regarding the Euclidean group as the transformation group of the Euclidean space (Patera et al. [1975a], Patera et al . [1975b], Beckers et al. [1977]). On the other hand, the contribution of kinematicians was hampered, probably, by a lack of mathematical education. As far as the author is aware, the first application of Lie groups to kinematics is due to Herve [1978]. However, attempts to provide a sound mathematical foundation of kinematics (Karger and Novak [1985], Selig [1986]), and the recent studies on the control of manipulators (Loncaric [1985], Brockett [1983]) have sparked the interest in the relationship between Lie groups and spatial kinematics.

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9 1 . 3 Grassmann and Clifford Algebra Versus Standard Vector Calculus The problem of searching for a suitable mathematical structure for the study of kinematics can be placed in the more general framework of looking for appropriate mathematical structures for dealing with physical phenomena at large. In an exciting narrative, Crowe [1985] describes how vector algebra came into being as a hybrid creature of the two leading mathematical systems of the last half of the nineteenth century, quaternions due to Hamilton, and extensor algebra, also known as Grassmann algebra after its developer. The use of this new vector algebra drew heated rebuttals from the quaternion followers. However, after some years, the vector algebra was so widely accepted that the only mementos of the quaternions were some notational conventions, and the curiosity of a mathematical structure that is unique in many ways. On the other hand, Grassmann had such an obscure life (Crowe [1985]) that, except for Germany, his native country, he lacked followers that could provide support for his algebra. Recently, however, there has been a renewed interest in the design of mathematical systems for the description of physical phenomena. On one side, there has been an increasing interest in Grassmann work (Dieudonne [1979], Barnabei et al. [1985], Stewart [1986], Pengilley and Browne [1987]); on the other hand, Hestenes (Hestenes [1966], Hestenes [1971],

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10 Hestenes and Sobczyk [1984], Hestenes [1985], Hestenes [1986]) has claimed that Clifford algebra provides more comprehensive tools, than vector calculus, for dealing with physical problems, including those from classical mechanics. Clifford algebra (Crumeyrolle [1974], Brooke [1980], Porteous [1981], Brackx et al. [1982]) is in a sense a misnomer since it really comprises a family of algebras, each of these algebras is closely related to the symmetric bilinear form of the corresponding orthogonal space (Dieudonne [1969], Kaplansky [1969], Porteous [1981], Scharlau [1985]). The first Clifford algebras were formulated by Clifford [1876], under the name of geometric algebras. During several decades, their study lay dormant, or was made without connecting it to Clifford's original ideas. However, in the 50s (Riesz [1958]), the interest in Clifford algebras was renewed, and it continues up to the present (Chisholm and Common [1985]). Clifford algebras have been used in kinematics research under several disguised names. It can be shown (Lam [1973], Porteous [1981]) that dual numbers, complex numbers, quaternions, dual quaternions, and Grassmann algebras are all special cases of Clifford algebras. It is still too early to decide if these structures will replace vector calculus as the most used mathematical physics tool, but it is evident that they warrant a close examination.

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11 1.4 Objectives and Organization of the Work An outline of the objectives and organization of the work is now given. Chapter 2 deals with the analysis of the Euclidean space and the Euclidean group. Although the results obtained are not new, the methods employed — the theory of affine and orthogonal spaces as developed in Porteous [1981] — yield a coordinate free analysis that, as far as the author is aware, is new. After proving several classical results, the chapter ends with the characterization of the Euclidean group as the semi-direct product of the normal subgroup of the translations times the subgroup of the rotations around an arbitrarily fixed point. Following the suggestions of section 1.1, Chapter 3 uses the group theoretical tools of isomorphism and group representation to provide a unified treatment of the mathematical structures mostly employed in spatial kinematics. In fact, it is shown that the representation of the Euclidean groups by means of affine orthogonal mappings of a three-dimensional vector space, homogeneous 4x4 matrices, spin groups, and biquaternion groups, all produce isomorphic algebras, and therefore, alternative models of Euclidean geometry. The development of the spin representation, as shown in sections 3.4 and 3.5, together with the connection between the spin and biquaternion representations, in section 3.7, is thought to be original.

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12 Since the development of the spin representation of the Euclidean group relies heavily on the Clifford algebras of some orthogonal spaces, Appendix A was written to cover some important concepts and results on this topic. There, four key results, which are required in the course of sections 3.4 up to sections 3.8, are proven for the first time. 3 Furthermore, the representation of the induced line transformation, by means of spin groups, as shown in section 3.8, is a novel and interesting by-product of the main development of this chapter. Finally, the screw representation of the Euclidean group allows the unification of the ideas of screws and of the screw triangle with the remaining representations of the Euclidean group. In Chapter 4, the Euclidean group is analyzed again, but this time, its topological and metric characteristics, inherited by every Lie group, are taken into consideration. After one introductory section, an explicit representation of the Lie algebra of the Euclidean group is obtained (section 4.2) from the spin representation of the Euclidean group. The development employs Clifford algebras. An advantage of using Clifford algebras is that the structure is robust enough to handle both the spin representation of the Euclidean group and its Lie algebra in a very compact way. Furthermore following the ideas of Karger and Novak [1985], it is shown 3 The origin of this work was an initial attempt to look into Clifford algebras and what they could offer to the field of spatial kinematics.

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13 that the Lie algebra of the Euclidean group and the algebra of infinitesimal screws (Dimentberg [1965], Hunt [1978], Duffy [1985]) are isomorphic. Section 4.3 analyzes the last of the representation of the Euclidean group namely, the adjoint representation. This representation maps elements of the Euclidean group into automorphisms of its Lie algebra. Since the elements of the Lie algebra can be interpreted as the infinitesimal elements of the group, it is clear that the derivation of this representation requires the topological properties of a Lie group, which are totally disregarded in Chapter 3. Section 4.4 establishes the semi-Riemannian manifold structure of the Euclidean group. The development makes use of bi-invariant metrics defined on the Lie algebra of the Euclidean group. This subject has already been studied by Loncaric [1985] and Lipkin [1985], in the latter case by resorting to the algebra of infinitesimal screws. During the process, it is necessary to employ the concept of invariance under the adjoint mapping. This concept is explained in Appendix B. The characterization of the Euclidean group as a semi-Riemannian manifold with a hyperbolic metric is thought to be new. The space of second derivatives of the Euclidean group at the identity is studied in section 4.5. This space is, similar to the Lie algebra of the Euclidean group, a

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14 six-dimensional orthogonal space with a non-degenerate hyperbolic metric. In chapter 5, the theory developed in the previous part of this study is applied to spatial kinematics. There, an analysis and proof of the principle of transference (Rooney [1975] and Selig [1986]) are carried out. The principle has a history of failed attempts and controversial results. In that chapter, a complete proof of the principle of transference is given; and the relation between the principle of transference and the Hartenberg and Denavit notation (Denavit and Hartenberg [1955]), which was previously overlooked, is explicitly indicated. Finally, the more relevant conclusions of this study, together with suggestions for further studies, are presented.

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CHAPTER 2 EUCLIDEAN SPACE AND EUCLIDEAN TRANSFORMATIONS In this chapter a coordinate free analysis of the Euclidean group is undertaken. In section 2.1, the physical space is modelled as a three dimensional Euclidean space. Sections 2 . 2 and 2 . 3 explain how to induce a free vector and a bound vector space structure into the Euclidean space; the fundamentals of such constructions are the theory of affine spaces and orthogonal spaces as developed in Porteous [1981] and Kaplansky [1969]. Section 2.4 defines Euclidean mappings as bijective mappings, of the set of points, which preserve both the affine and orthogonal structure of a Euclidean space. The lack of structure in the physical space prevents a direct probe into the properties of Euclidean mappings, and the remainder of the section is directed toward the analysis of the orthogonal mappings induced, by the Euclidean mappings, onto the free vector space structure introduced in section 2.2. Using the results of section 2.4, the properties of Euclidean mappings are investigated in section 2.5, and their structure as an algebraic group is proved. Sections 2.6 and 2.7 are dedicated to the analysis of two important classes of subgroups of the Euclidean group, namely translations, and rotations leaving an arbitrary point of the Euclidean space fixed. The decomposition of Euclidean 15

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16 mappings, in terms of a translation and a rotation around a fixed point, is analyzed in section 2.8 together with some invariant characteristics of the mappings. Finally in section 2.9, the composition rule for Euclidean mappings is re-examined and expressed in terms of the components, and the structure of the Euclidean group, as the semi-direct product of the normal subgroup of the translations by the subgroup of the rotations that leave an arbitrary point fixed, is disclosed. 2 . 1 Physical Space as a Euclidean Space For purposes of study of classical mechanics, physical space can be modelled by a three-dimensional Euclidean space (Porteous [1981]); equivalently, the space can be regarded as a set of points E coupled with a mapping
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17 This property can be expressed in terms of the mapping <5 by S (L,M) + 6 (M, N) = S (L, N) (4) The elements of IR 3 are called vectors; in particular the vector M 5 N is called the difference vector of the pair of points (M,N) in ExE. This notation is specially useful since it emphasizes that the elements of IR 3 are dependent upon the points of E. In fact, the existence of a mapping <5 : ExE -*• IR 3 fulfilling the above axioms ensures the existence of a family of similar mappings 6 H : ExE -» IR 3 5 ^ (M, N) = /x<5(M,N) for each /xelR, /i f 0 all of which also satisfy the axioms. The specific value of /i depends on the arbitrary election of a unit of length which cannot intrinsically be preselected. It is important to recognize that the set E is at the outset completely void of algebraic structure; hence, there is no meaningful way for adding, subtracting or multiplying the points of E. However the rich algebraic structure already existing in the quadratic vector space IR 3 can be employed to induce some algebraic structures in E. 2 . 2 Free Vector Algebraic Structure in ExE The first of the algebraic structures, induced by IR 3 into ExE, to be studied is that of the free vectors. Before

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18 proceeding, it is necessary to note that ExE supports an equivalence relationship (M,N) * (P/Q) o <5 (M, N) = 5 (P, Q) (1) where M,N,P, QeE. This relationship has the following properties: 1. It is reflexive. Trivially S (M, N) = S (M, N) thus (M, N) « (M,N) 2. It is symmetric. Let (M, N) a (P,Q) ; then <5(M,N) = S (P, Q) and thus 5(P,Q) = <5(M,N). Therefore (P,Q) a (M,N) . 3. It is transitive. Let (M, N) a (P,Q) and (P,Q) a (R, S) ; then 6(M,N) = 5(P,Q) and 6 (P/Q) = 5 (R, S) . Thus <5(M,N) = <5(R,S) and therefore (M,N) a (R, S ) . Further since S : ExE -* IR 3 is surjective, it is possible to associate with every element v of IR 3 an equivalence class v = { (M, N) eExE | 5 (M, N) = v} (2) In the field of classical mechanics v is usually referred as a free vector. The set of all free vectors will be denoted by l 3 = U v (3) veIR 3 The equivalence relationship thus has induced a partition of ExE.

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19 It will now be demonstrated that R 3 has the structure of a real orthogonal vector space (Porteous [1981]); hence, it can be considered as a real orthogonal space of free vectors. The first step in this demonstration is to recognize that there exists an induced mapping 6. : R 3 -*• IR 3 i.(Y) = v (4) which is well defined and bijective. 1. S_ is well defined. Let (M, N) , (P, Q) ev; therefore 6 (M, N) = v = S (P, Q) and S(M,N) = v = 6(P,Q) . 2. S_ is injective. Let v,weR 3 such that i.(v) = v = w = <£(w) . It follows, from the partition properties (Fraleigh [1982]), that v C w and w c v, and therefore v = w. 3 . S_ is surjective. Since 6 : ExE -* IR 3 is surjective, V veIR 3 there is a (M, N) eExE such that <5(M,N) = v. Further, let veR 3 such that (M,N)ev,then i.(v) = v and 5_ is surjective. This bijective mapping provides a way for translating the algebraic structure of R 3 into an algebraic structure of R 3 . The set R 3 together with the operations of addition and scalar multiplication defined by v + w = (v_+_w) V v,weR 3 (5a) and V AeR and V veR 3 A(v) = (Av) (5b)

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form a real vector space isomorphic to IR 3 1. K 3 is closed under addition. V v,weK 3 3 v,weIR 3 such that, for M,N,P,QeE, (M,N)ev o 6 (M, N) = v, and (P,Q)ew o
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21 (M, N) eExE with M f N. Therefore (M,N)^0, for if (M,N) eO, then both 6 n (M) = S (M, N) = 0 and 5 N (N) = S(N,N) = 0 a contradiction to the injectivity of
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22 8. Scalar multiplication and addition satisfy V A,/zeIR and v, we[R 3 (A + n)v = f (A + u)v ] = ( Av + zzv ) = (Av) + (uv) and A (v + w) = f A fv + w) l = ( Av + Aw ) = (Av) + (Aw) Furthermore, the mapping 5. : IR 3 IR 3 is also linear since VA,^zeIR and Vv,weK 3 A (Av + /zw) = 6 f Av + uw ) = Av + /zw = A5.(v) + /z5.(w) Thus i is a linear isomorphism; then IR 3 and K 3 are isomorphic as vector spaces. Finally, it is also possible to introduce an orthogonal structure on K 3 by defining the symmetric bilinear form H 3 xIR 3 -*• [R (v, w) = (v, w) Vv,weK 3 (6) It is straightforward to show the form is well defined and indeed symmetric. Further, the mapping 6_ : K 3 -* IR 3 preserves the form, for trivially (!(Y) ,!(w) ) = (v,w) (7) and therefore IR 3 and K 3 are isomorphic as real orthogonal vector spaces. It is important to recognize that in the process of providing JR 3 with an orthogonal vector space structure it is unnecessary to make any arbitrary choice of origin. The next section introduces a structure that, unlike the structure considered here, is origin dependent.

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33N'W A W + N = ( (w) °9 + (N) 0 S , ) I _ 0 S> = ( (N) °?+ (w) 0 S>) I _ 0 S > = N + H •SAT^B^numiOO ST UOTqxppY 'Z *a3((N)°ff + (W)°S , ) I _ 0 ? snqq pub e HP(N)°? + (w)°P 'aoBds aoqoaA b sx c h aouxs * C HP (N) °p ' (w) °p 33N'W A •uoxqxppB aapun pasojo st a ’I •aoBds aoqoaA IBaa b jo aanqonaqs aqq aABq suoxqBaado asaqq qqxw aaqqa&oq 3 jo squxod aqq qBqq paqBjiqsuouiap aq aou njw qx (O HP Y A '33W A ( (W)°?y) I _ 0 S> = WY PUB (2) 33N'W A ((N)°S> + (W) 0 ?)-,-0 ? = N + N Aq a jo squxod jo uoxqBOXidxqxnux xiBxeos pub uoxqxppB jo suoxqBaado aqq auxjap oq axqxssod sx qx JBxnoxqjBd ui •aoqoaA oaaz aqq oq x^nba qas sx 'Oy BuxddBux aqq xapun 'qoxqw uxfixao xo quxod aouaxajax AaBaqxqaB ub sx 330 quxod aqq aaaH *3 ux aanqonaqs aoads x^uoboqqao ub A xojBuixqxn pub aoBds aoqoaA b aonpux oq pasn aq ubo qx snqq .'aAxqoaCxq sx Oy fiuxddBui aqq 'p BuxddBui aqq jo x Aqaadoad Ag (I) 0 B W = (o'W)S = (W)°? e ifl 3 : °P fiuxddBui aqq jo susaui Aq paonpux aanqonjqs oxBuqafix^ puooas b qqxw pawopua sx a aoBds UBapxxona aqx, axa ux aanqonaqs oxBaqaoxv uoqoaA punog z ’ Z ZZ

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*33 C (w) 0 9y] T _ 0 9 snqq. PUB ' e a3(w) 0 9y 'ypy a 'eoeds jo^osa TBaJC b si c a aouxs * c y]3(w) 0 9 'a3 W A •uoxq.BOTidTq.inui .ibtbos aapun pasojo si a *9 0 = [ ( A— ) + a ] t _0 9 = [(w -)°9 + (w) °9 ] x _°9 = (W-) + H aouaq y (A-) T _Os> = hqeqq. qons sawE uaqq. .' e ifl3A = (w)°s> q.Bqq. qons aq a 3 W q.aa •asxaAUi aAxqxppB ub jo aouaqsxxa ’S •Aqxquapx aAxqxppB aqq sb ' a;tnq.onjq.s sxqq. ux 'sq,OB o qxixod aqq q.Bqq. u^oqs uaaq sBq qi W = ((w) 0 9) x _ 0 9 = ((o)°9 + (W) °S> ) x0s> = O + H 33W A 'eaojajiaqj 0 = (0'0)S> = (0)°9 Z ’ Z uoxqoas uioaa •Aq.xq.uapx aAxqxppB ub jo aouaqsxxa (N + W) + 3 = [ (N + W)°9 + (l) °9 ] -[_°9 = { [ ( (N) °9 + (W) °P ) x0s> ] 0s> + (3E) °S > > x0s> = [ ( (N) 0 9 + (w)°9) + (T) 0 ?]-!;. 0 ? = C(N)°S> + ((w)°9 + (3)°9)] T _°9 = { (N) °9 + [((W)°9 + (3) °9 ) x0 ^ 1 °? ) x0 ? = [ (N) °9 + (W + 3) °9 ] x-°9 = N + (W + 3) 33N'H'3 A \Z •aAxqBToossB sx uoxqxppv *G

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25 7. The scalar multiplication satisfies V X,/xeR and MeE MMM) =
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26 A prominent consideration is that some of the results obtained within the realm of this algebraic structure are only valid for the specific selection of the origin 0; hence, they are not Euclidean geometry properties. 2 . 4 Euclidean Mappings In this section an analysis of mappings of the Euclidean space which preserve the orthogonal structure of R 3 is presented. Such mappings ^ : E -+ E must preserve the quadratic form of the associated orthogonal space, R 3 , given by ( <5 (M, N) , 5 (M, N) ) = (6(^M,^N) ,6(^M,^N) ) V M, NeE (1) It can be shown (Porteous [1981]) this condition is equivalent to the preservation of the related symmetric bilinear form or inner product ( 6 (M, N) , <5 (P, Q) ) = (tf (*M,*N) ,«(*P,*Q)) VM,N,P,QeE (2) Mappings with this property are called Euclidean mappings, and it is straightforward to show that this definition is independent of the particular value of n chosen in the mapping 6 ^ : ExE -* R 3 . Furthermore, any Euclidean mapping induces a mapping of R 3 into itself according to the rule ± : R 3 ->• R 3 ^(v) = S where (M,N)ev (3) Since the definition of ± involves a choice, it will now be shown that ± is indeed well defined.

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27 Let (M,N) , (P,Q) ev, and assume 6 (4M.4N) = V;l and 5 (4P . 40) = v 2 , where v 2 can be orthogonally decomposed (Kaplansky , [1969]) as v 2 = Av-l + u with ue[v 1 ] ± — the orthogonal complement of [v^] — , it suffices to show = v 2 . 1 . Yi = 0 ; then 0 = (0,0) = (v lf v x ) = ( 6(4 M,4N) . 6(4 M.4N)) = ( 6IM.N) , 6(M,N) ) Thus 6 (M.N) = 0, and (M,N)eO. If (P,Q)eO then <5(P,Q) = 0 and 0 = ( <5 (P, Q) ,6(P,Q) ) = (6(4P,4Q) ,6(4P,4Q) ) = (Y 2 ,v 2 ) Thus v 2 = 0 = Vi|_ . 2 . V;l f 0; then (Si# Xi) = (
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28 is independent of the scale of length singled out by the mapping S : ExE -*• IR 3 . It is also evident that if ±(v) = w, then the image of any vev, under the induced mapping on IR 3 , will be given by w. Since IR 3 and K 3 are orthogonally isomorphic, the induced mappings on IR 3 and K 3 will be both denoted by ± f the distinction will be clear from the context. The following proposition provides more insight into the nature of the induced linear transformation. Proposition 1. The mapping ± preserves the inner product of K 3 . Proof: Let v,weK 3 with v =
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29 Proposition 2. Every mapping of a positive definite (or negative definite) real orthogonal space into itself which preserves the inner product is linear. Proof: Let X be a positive definite (or negative definite) real orthogonal space, -p a mapping which preserves the inner product, and v,weX, define e = v// (v + w) p (v) p (w) Then (e, e) = (4>(v + w) ^ (v) P(w),p(v + w) ^ (v) ^ (w) ) = (p (V + W),p(V + W) ) + (p (V) ,p (V) ) + (P (w) ,p (w) ) 2 (p (v + W) f'P(V) ) 2 (p (v + w) ,p (w) ) + 2 (+(V) ,* (W) ) = (V + W, V + w) + ( v , v) + (w , w) 2 (v + w, v) -2 (v + w,w) +2(v,w) = (V,v) + (w,w) + 2 (v, w) + (v,v) + (w,w) 2 (v, v) 2(w,v) 2(v,w) 2(w,w) + 2(v,w) = 0 Since X is a definite orthogonal space e = 0, and 'Mv + w) = P(v) + p( w), then p is additive. Further, let veX, v ^ 0 ,and XeIR; then p (Xv) can be expressed in the form ^(Xv) = ap (v) + w where aelR, and we[^(v)]L . Consider now the inner product Mv,v) = (v , Xv ) = (p (v) ,p (Xv) ) = (p (v) ,ap (v) + w) = a (p (v) ,xp (v) ) + (xp(v), w) = a(P(v) ,P(v) ) = a(v,v) Thus X = a, and

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30 X 2 (v,v) = (Av,Av) = (A^ (v) + w,A^/(v) + w) = A 2 ('P (v) (v) ) + 2A ('p (v) , w) + (w, w) = A 2 (v, v) + 0 + (w,w) Therefore (w,w) = 0. Since X is a definite orthogonal space, then w = 0; thus ^ is homogeneous and linear. Corollary 3. Every mapping of a positive definite (or negative definite) real orthogonal space into itself which preserves the inner product is orthogonal. In particular, these two last results apply to mappings which preserve the inner product in R 3 . After having shown that the induced Euclidean mappings are linear transformations, it appears natural to inquiry whether the mappings are injective and/or surjective. The answer to these questions can be obtained from a result reported by Porteous [1981]. Proposition 4. Every orthogonal mapping of a non-degenerate orthogonal space into itself is injective. Proof: Let X be a non-degenerate orthogonal space, v,weX, and ^ an orthogonal map satisfying 'P (v) = '/'(w); then 'P (v w) = 'P (v) v// ( w) = 0, and for every ueX 0 = (0,^(U)) = ('p (v w),*(u)) = (v w, u) Since X is non-degenerate, v w = 0 or v = w Corollary 5:. Every orthogonal mapping of a non-degenerate, finite-dimensional orthogonal space into itself is a bijection.

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31 Once more, these results apply in particular to the orthogonal mappings induced in R 3 by Euclidean mappings. Summarizing the results, it has been shown that the induced mappings of Euclidean transformations are orthogonal automorphisms of the real orthogonal space R 3 . It is well known that the orthogonal automorphisms of R 3 form a group, called the orthogonal group and denoted by 0(3) . However there are two classes of orthogonal mappings. Those which preserve the intrinsic orientation of R 3 form a subgroup of the orthogonal group, denoted by SO (3) , and called the special orthogonal group. Those which do not preserve the orientation of R 3 do not form a subgroup. Bottema and Roth [1979] show that the induced orthogonal mappings must preserve the orientation of R 3 ; hence they must be elements of SO (3) . Even though the definition of Euclidean mapping is broad enough to include mappings which do not preserve the orientation of R 3 , those will not be considered anymore. 2 . 5 Properties of the Euclidean Mappings In the previous section Euclidean mappings and their induced transformations were defined, but the lack of structure in the Euclidean space E hampered any effort to directly find their characteristics. After searching the properties of the induced mappings, it is possible to pursue the investigation of the Euclidean ones.

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32 Proposition 1. Let ^ : E -* E be a Euclidean map, and * : K 3 -+ IR 3 the induced (orthogonal) map; then for any NeE + = (<5^ (N) ) "^N (!) Proof: Let MeE be arbitrary; then ( ( (N) ) ~ 1 ^ <5 n) ( M ) = ( ( S + (N) ) -1 ^) 5 n( m ) = ((^(N)) _1 ^)5(M,N) = (5^(N))" 1 (^(^(M,N))) = (^(N)) 1 («(^W) = (<^(N))"%(N) (+W) = + (M) Corollary 2. Every Euclidean mapping is bijective. Proof: By definition <5 N and ( IR 3 is the induced mapping of + 2 + 1 , and 5(M,N) = v; then 1(Y) = 6 (+ 2 +l*,+2+l*) = S(+ 2 (+l*) , + 2(+l*)) = ± 2 ( S ( + 1 M ' + 1 N ) = +.2*1 ( ^ (M, N) ) = ±2±i(v)

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33 Therefore # = ± 2 ±1 ( 2 ) Proposition 4. The inverse of a Euclidean mapping is also Euclidean, and its induced mapping is the inverse of the induced mapping of the original Euclidean mapping. Proof: Let ^ be a Euclidean mapping, and ± its induced mapping; then ($(*M,*N) ) = ( <5 (M, N) , 6 (M, N) ) Renaming M' = ^ M and N' = ^N; then M = ^ _ 1 M' = and N = = 'p-l'pN are, respectively, the unique images of M' and N' under 'P~ 1 . Therefore ( <5 (M ' , N ' ) , <5 (M ' , N ' ) ) = (5(v// _ 1 M' ,^ _ 1 N') , 6 (^ 1 M' ,^ _ 1 N') ) V M',N'eE; hence \// -1 is also Euclidean. Assume now $ : R 3 -* R 3 is the induced mapping of and <5 (M' ,N' ) = v; then V = I 3 (v) = 5 (M ' , N ' ) = = ,^/ 1 " 1 N') = £ii( 6 (M',N')) ± 1 $(Y) Thus I 3 = and £ = (£i)" 1 I 3 = (±i)~ 1 (3) Proposition 5. The set of Euclidean mappings together with the composition operation form a group, called the Euclidean group, and denoted by E(3). Proof: By proposition 3, the composition of two Euclidean mappings is also a Euclidean mapping; thus the set is closed under the operation.

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34 The composition of Euclidean mappings being a special case of the composition of arbitrary mappings, which is associative, is also associative. Consider the mapping i : E -* E such that 1 (M) = M V MeE. It is straightforward to prove that i is a Euclidean mapping, and it behaves as the identity element of the group. By corollary 2, every Euclidean mapping is invertible, and, by proposition 4, its inverse is also a Euclidean mapping. Now that it has been established that Euclidean mappings constitute a group, proposition 3 provides a proof for the following statement: Corollary 6. The mapping ^ : E(3) -* SO (3), which assigns to every Euclidean mapping its induced orthogonal mapping is a group homomorphism. It is a well known fact that the composition of Euclidean mappings is not commutative (Bottema and Roth [1979]). Therefore the Euclidean group is not abelian. It now appears natural to inquire about the possible existence and properties of subgroups of the Euclidean group. It will be shown, in the next two sections, that there are two important classes of subgroups of the Euclidean group namely translations, and rotations about a fixed point.

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35 2 . 6 Translations A translation is defined as a mapping r : E -* E with the property that there exists a veIR 3 such that 5 ( rM, M) = v V MeE (1) More precisely, r is called a translation of E by the vector v. Proposition 1. A translation is a Euclidean map. Proof: Let M, NeE be arbitrary; then by the triangle axiom 6(tM,tN) = 6(rM,M) + 5(M,N) + 6(N,rN) = <5 ( rM, M) +
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36 Since $(v//M,M) f
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37 Thus P = M, or t -1 t M = M, hence r -1 r = i , and this result coupled with the commutativity of the composition proves the assertion. Here i denotes the identity Euclidean mapping. Proposition 5. The set of all translations forms a normal abelian subgroup, denoted by T, of the Euclidean group . Proof: By proposition 3, the set is closed under composition, and the composition is commutative. By proposition 4, the set is closed under the operation of taking inverses. Finally, let r and ^ be an arbitrary translation and Euclidean mapping respectively; consider the mapping § = then $ = ± t ±~ 1 = ± t £ _1 = ± I 3 ( ±)~ 1 = I 3 Thus by proposition 2, $ is a translation. Further, with the usual definition of a scalar multiple of a mappings, this group can be made isomorphic to the vector space R 3 . 2 . 7 Rotations A Euclidean mapping p : E -* E is called a rotation if there exists a point PeE such that pP = P. P is then called a fixed point of the rotation. Proposition 1. The set of rotations having a common fixed point P, denoted by n p , forms a subgroup of the Euclidean group. Proof: Let p p ,p 2 en P ; then

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38 p 2 PiP P2(Pl p ) ~ P 2 P “ p and p 2 P ien p . Trivially, the identity mapping, i : E -* E, belongs to n p , for IP = P. Let pen p ; then P = iP = (p -1 p) P = p -1 (pP) = p _1 P Thus, p -1 efi p and the set is closed under the operation of taking inverses. It is noteworthy to recognize that for any point PeE there is a subgroup n p of the rotations leaving P fixed. Sometimes the fixed point will be used as a suffix of a rotation as a way of specifically stating the invariance of that point. The following result characterizes the equivalence of two rotations. Proposition 2. Two rotations p^ and p 2 are equal if, and only if, they have a common fixed point and the same induced orthogonal mapping. Proof: Let P be the common fixed point, and consider an arbitrary MeE; then 6( Pl M,p 2 M) = 6( Pl M, Pl P) +
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39 Finally, assume and p 2 have P as a common fixed point but q_i f q_2 ' then 3veK 3 such that £.i(v) f 2.2 (•) • Let MeE be such that pp (M) = v; then applying the triangle eguality 5( Pl M,p 2 M) = 5( Pl M,P) + 5(P,P 2 M) = S^M^P) S (p 2 M, p 2 P) = fi.i$(M,P) p. 2 6(M,P) = fi. 2 v f 0 Thus p^M fp 2 M and p^ fp 2 . If a rotation p : E -* E is also a translation, then the vector associated with the translation is given — employing the fixed point P of p — by S(pP,P) = <5 (P, P) = 0 It follows that for every MeE
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40 Proof: Let PeE be any fixed point of p, and veR 3 be an eigenvector associated with the eigenvalue 1; consider QeE given by
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41 Then S (p p P, P) = 6(p P P,^P) + 5 ('PP, P) =
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42 Similarly, it is a routine task to verify that if a Euclidean mapping ^ is decomposed as 'P = T v p p or yft = P(^,p)T w (4a) Then, the inverse mapping can be decomposed as 'P ~ 1 = T -w(P (*P) ) _1 or ^ _1 = Pp 1 r_ v ( 4 b) Although the selection of different points, as fixed points of the rotation, leads to distinct decompositions, it will be proved, analogously to the proof of equation 3, that the induced orthogonal mapping remains invariant. Proposition 2. The induced orthogonal mapping of an arbitrary Euclidean mapping is independent of the fixed point chosen to accomplish the decomposition. Proof: Let ^ : E -*• E be an arbitrary Euclidean mapping with two distinct decompositions 'P = VP and 'P = t wPq Then, by proposition 5.3, E-P = I 3 P.P = LyP-P = ± = L-wB-Q = I 3 P.Q = £Lq T his result coupled with the invariance of the induced orthogonal mapping under changes of the scale of length ensures that the induced orthogonal mapping is a true Euclidean property of any Euclidean mapping. On the other hand, the selection of different points, as fixed points of the rotation, also leads to distinct values of the translation vector; however, those vectors still have interesting invariant characteristics whose investigation leads up to some of the results credited to Rodrigues (Gray

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43 [1980]), and Chasles (Bottema and Roth [1979]), and ultimately to the screw representation of Euclidean mappings. Proposition 3. Let ^ be a Euclidean mapping decomposed as 4* = t v p p and 4* = t w Pq, where P,Q are two arbitrary distinct points and let u be an eigenvector related to the eigenvalue 1 of the common induced orthogonal mapping. The components of the translation vectors along the eigenvector are equal. Proof: It is obvious that (v,u) = (w,u) o (v w,u) = 0; then it suffices to show that (v wjefu]1 -. By definition v = 6(4*P,P) and w = 6(4*Q,Q); hence by invoking the triangle equality v w = $(*P,P) 6 (4>Q , Q) = 6(4* P,4*Q) + S(4> Q,Q) + 6(Q,P) 6(4>Q,Q) = 2fc[«(P/Q) ] 5(P,Q) Expressing 5(P,Q) = Xu + y with yefu]1 -, then v w = ±(\u + y) -(Xu + y) = X^.u + ^y Xu y Since by assumption ±u = u, then v w = Xu + ±y Xu y = 4lY ~ y Finally, since ± is orthogonal, then (v w,u) = (±y y,u) = (±y,u) (y,u) = (±y,±u) (y,u) = (y,u) (y,u) = 0 Thus (v wjefu]1 -, which establishes the result. The component of the translation vector, along the eigenvector u, which by the previous result is independent of the fixed point chosen during the decomposition. It does

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44 however depend on the scale of length chosen in the mapping 5 : ExE -* IR 3 , and thus, it is not a Euclidean property of the mapping. In view of this result, the question arises whether there is a point S such that ^ = r r p g with r = Xu, for some XeIR, and if it exists then where is it located. Proposition 4. If a point S, satisfying the conditions shown above exist, then all the points on the affine line { P |
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45 If Y = 0, then the result follows for the function is injective and consequently bijective. Assume y f 0 ; then y is an eigenvector of p.g associated with the eigenvalue 1 ; hence y = mu for some ne\R a contradiction to yefu]1 -. Proposition 6 . Let ^ : E -*• E be an arbitrary Euclidean mapping; then a unique affine line can be found such that for all the points S that belong to the line 'P = T rPS with r = Xu where u is an eigenvector associated with the eigenvalue 1 of the induced orthogonal mapping. Proof: By proposition 4, it suffices to find a point in the affine line. Let Q be an arbitrary point; then 'P = t w pq Let u be an eigenvector associated with the unit eigenvalue of q_ q; then w = Hju + y with /x^IR, and ye[u]-*-. Let S be another arbitrary point; then 6 (S,Q) = p 2 u + x with jU2 elR and xefu] 1 . Applying the triangle equality S('pSfS) — 6 ( T w p qS , S) = 6 (t w PqS , PqS ) +
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46 always exists, and it is unique. Hence, a point S can be found such that r = S ('//S , S) = /i^u This affine line is commonly referred as the screw axis of the Euclidean mapping. It is certainly a Euclidean invariant of the Euclidean mapping. 2 . 9 Composition of Euclidean Mappings Now that the decomposition and properties of an arbitrary Euclidean mapping have been examined, it seems natural to inquire how the composition of two arbitrary Euclidean mappings can be related to this decomposition. This provides the necessary foundation for a further result associated with the structure of the Euclidean group. Proposition 1. Let ^ = t^Pq and = r v (p 0 )* be two arbitrary Euclidean mappings decomposed using a common point 0 ; then *2*1 = T v T ( £o )*t (p O>* p O (1) Proof: Consider ^ 2*1 = ( r v (p 0 ) *) ( T t Po) = r v ( ( Po) * r t) Po • then (Po)*' r t a Euclidean mapping transforming point 0 into (Pq) *M where
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47 Then, by proposition 7.2 (po)* = (p o) ' and (Po)* r t = t (£. 0 ) *t (p o) * ( 2 ) Thus finally *2*1 = r v 7 ( ^ o) * t (Po)*(Po) Proposition 2. The quotient group E(3)/T is isomorphic to ftp, where P is an arbitrary point of E. Proof: Consider the mapping $p : E ( 3 ) -* n p §px// = p p, where ^ = r-^Pp It is easy to notice that $p assigns to an arbitrary Euclidean mapping its rotation part in its decomposition with respect to the point P. It will now be shown that $p is a group homomorphism onto ftp with kernel T, then resorting to the so-called isomorphisms theorems (Herstein [1975] pp. 59) the result will follow. Let *i,* 2 eE(3) , where = r^pp and ^ 2 = r v (pp)*; then $p(*2*l) = $ p( 7 ’v r ( £p ) *t Pp*Pp) = (Pp)*Pp = *p*2 $ P*l Let ppeHp be arbitrary; then trivially p p eE(3) and 4>pPp = fp(lpp) = pp Thus the mapping is surjective. Finally 'pe Kerf «> = i <* ^ = r -j1 for some teIR 3 ^ for some teIR 3 <* ^eT. This result gives a complete characterization of the Euclidean group. In algebraic terms, it is defined as the semidirect product of T by n p with action p_p (Rose [1978] pp. 208-210) .

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CHAPTER 3 REPRESENTATION OF THE EUCLIDEAN GROUP In the previous chapter it was emphasized that the Euclidean group exists even if there is no prior selection of a reference system in the Euclidean space. In this chapter several homomorphic representations of the Euclidean group will be reviewed. It is important to know that in all the usual representations of the Euclidean group it is required to choose a fixed frame of reference. This frame of reference consists of a point 0, called the origin, a scale of length, and three directions given by an orthonormal basis of the positive definite orthogonal space IR 3 . This fact has deep implications since (see section 2.3) the selection of a point and a scale of length sets up a bijective relationship between the points of the Euclidean space and the elements of IR 3 . Further, choosing a second orthonormal reference system in a moving rigid body such that in the initial position of this body both reference systems coincide, there is a bijective relationship between the elements of the Euclidean group and the possible positions of the moving rigid body, whose set is usually called configuration space. 48

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49 A significant difficulty, however, is that it is necessary to test any result obtained for a particular choice of a reference system for invariance with respect to the action of the Euclidean group and a change of the scale of length before the result is declared a Euclidean geometry property. No attempt here is made to exhaust all possible representations of the Euclidean group (Miller [1964], Rooney [1978a]), and only those which are amply used in the kinematics field or shed additional insight are discussed. Further, the analysis of the representations of the Euclidean group is, in this chapter, restricted to algebraic requirements leaving aside metric or topological requirements which will be treated in chapter 4. The main achievements of this chapter are firstly the development of a unified treatment of the algebraic representations of the Euclidean group used in kinematics, and the identification of the so-called kinematic mappings as representations of the Euclidean group. Secondly, the deduction of the spin representation of the Euclidean group, developed in sections 3.4 to 3.6, as well as the connection of the spin representation with the biquaternion representation, proved in section 3.7, is also new. Finally, the application of the spin representation to the induced line transformation, given in section 3.8, is an interesting

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and novel byproduct of the approach developed in this chapter. 50 3 . 1 Fundamentals of Representation Theory Let G be a group and X a real vector space; a representation T of the group G into L(X) , the algebra of linear operators of X, is a mapping T: G -* L(X) satisfying 1. T(g 2 gi) = T (g 2 ) T(g x ) for all g 1 ,g 2 eG (1) 2. T(e) = I x where e is the identity of G (2) The first property shows that the representation is a group homomorphism; furthermore, since for any geG Ix = 1( e ) = T(gg -1 ) = T (g) T (g -1 ) and Ix = T ( e ) = T(g _1 g) = T(g -1 ) T(g), then T (g -1 ) = [T (g) ] -1 (3) Hence all the operators T(g) are non-singular. Consequently, T(G) forms a subgroup of the group of units of the algebra L(X) . If X is finite-dimensional, T is called a finite-dimensional representation of G. Otherwise, T will be an infinite-dimensional representation. This work considers only finite-dimensional representations of the Euclidean group . Given a group G and a non-empty set S, if for any geG and seS there is a gseS such that

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51 92 < ?i s = 92(9l s ) for a11 9l^2 eG and seS (4a) and es = s for eeG, the group identity, and V seS, (4b) then G is said to act upon the set S. Rose ([1978] pp. 68-70) shows that the mapping 'Pg : S S ^g(s) = gs (5) is a bijective mapping of S; i.e. ^g€Z s , where 2 S is the symmetric group of the set S. Furthermore the mapping * : G -* 2 S *g = (6) is a group homomorphism. Since G is isomorphic to a group of bijective mappings or transformations of the set S, then G is called a transformations group. It follows that the Euclidean group acts upon the points of the Euclidean space. Further, since for any specified points M and N of the Euclidean space there is an element (in fact, an infinite number of elements) of the Euclidean group which transforms M into N, the action of the Euclidean group over the points of the Euclidean space is called transitive. Moreover, the set of points of the Euclidean space is said to be a homogeneous set of the Euclidean group (Naimark and Shtern [1982]). From a strictly theoretical point of view, the previous definition of a group representation is sufficient for any class of groups, including groups of transformations. However, it is highly desirable to use the representation of the group for obtaining the image of an element of the set

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52 under any of the transformations. This can be accomplished as follows : Let G be a group of transformations of a set S; then an additional injective mapping, from the set S to a vector space, $ : S -*• X is required such that $* : S ^ $(S) C X §*(s) = §(s) V seS is bijective, and the following diagram commutes #* s * $(S) c X g T(g) $ s — * $(S) c X or alternatively for all seS $*gs = T(g)$ *s (V) Since $* is bijective, it is possible to write g = $* -1 T(g) $* (8) or T(g) = $*g$* -1 (9) This procedure is accomplished only if an injective mapping $ and subsequently $* which satisfies equation (7) can be found. Then T(g) is linear. The technique illustrated above can be reversed; viz. given an injective mapping $ ; S -*• X, the mapping $* : S -*• $(S)
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53 By completing the following diagram (S)cx t (g) (S)CX the transformation T(g) : X -*• X can be defined as T(g)v = $*g$* -1 v Vve§(S) and T(g)v = v Vv^§(S) (11) The mapping T which assigns T(g) to g satisfies T(e)v = $*e$* _1 v = §*es = $*s = v Vve$(S) Thus T(e)=I x , and T(g 2 gi) = §*g 2 gi^* -1 = $*g 2 $* -1 $*gi$* -1 = T(g 2 )T(g 1 ) Hence, T is a group homomorphism from G to the group of bijective mappings of X; however, in this case, there is no assurance that the mappings T(g) : X -*• X are linear. 3 . 2 The Affine "Representation" of the Euclidean Group 1 It follows from the decomposition theorem (2.8.1) that any element ^ of the Euclidean group E(3) is uniquely determined by 1. If one chooses a point 0 of the Euclidean space and a scale of length, then the decomposition of ^ = r-^-p o yields the two other required pieces of information. 2. There is a vector t associated with the translation r t^Strictly the term representation as used in the previous section is confined to linear mappings. Here the term is used to include affine transformations that in general are not linear.

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54 3 . There is a proper orthogonal mapping g_ 0 associated with the rotation. The first condition (see section 2.2) sets up a bijection between the points of the Euclidean space and the vectors of R 3 , with S 0 : E -+ R 3 Sq (M) = S (M, 0) (1) as the bijective mapping. It will now be shown that the Euclidean mapping \p induces a mapping 'Pqi , in general nonlinear, of R 3 into R 3 in the following way: Consider two distinct Euclidean spaces; one of them, denoted by 2, moves freely and the other, denoted by S', is fixed. Let 0 be a reference point of 2, and O' be the point of 2' initially coincident with 0. Let Me2 be an arbitrary point with v = 5 0 ' (M) ; then define the induced mapping 'P o' ' R 3 -* R 3 as ^O'(v) = S ('P (M) , 0 ' ) =
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55 3 '/'O' 3 Thus, except for the non-linearity of ^ 0 »/ the mapping A : E ( 3 ) ASO ( 3 ) = ^ 0 . (3) seems a good choice for a "representation” of the Euclidean group. It remains to show that A is a group homomorphism. Let ^ 2 . = T uPO' anc * ^2 = T t (Po) * two arbitrary Euclidean mappings. Proposition 2.9.1 shows that ^2*1 = T t T ( £q) * u (Po) *(Po) i Hence for any veIR 3 , A(v// 2 ^i)v = (e o)*(£ 0 ) v + (P-o) * u + w = C A (^2) ] [ (P-o) v + u 3 = [A(^ 2 )][A(^ 1 )]v Therefore A(^2'Pi) = A (v// 2 ) A ('//^) (4) In fact, it is possible to show that A is an isomorphism. This "representation" of the Euclidean group given by equation (3) is far too abstract for calculations. This problem is overcome by selecting an orthonormal basis of IR 3 . It is well known (Herstein [1975]) that this selection sets up an algebra isomorphism between the proper orthogonal transformations of IR 3 and the 3x3 proper orthogonal matrices. 2 The image of £_ 0 under this mapping will be from here on represented by R. 2 An orthogonal matrix is said to be proper if its determinant is positive.

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56 Consider now the set of ordered pairs (R,t) , where R is a 3x3 orthogonal matrix, and t is an element of IR 3 , with the operation (R 2 • t 2 ) (Ri • t].) = (R 2 R 1/ R 2tl + t 2 ) (5) It is straightforward to show this set is a group formed by the semi-direct product of the subgroup (R,0) acting on the normal subgroup (0,t). In particular the identity is given by ( I 3 , 0 ) , where I 3 stands for the 3x3 identity matrix, and the inverse of (R,t) is given by (R, t) -1 = (R _1 , -R -1 t) (6) Furthermore, the action of this group upon an arbitrary veIR 3 , where v = 6 0 « (M) , is v' = Rv + t (7) This group is sometimes referred to as the group of symmetries of IR 3 . This result has influenced many to identify the Euclidean group with the group of isometries of K 3 . However, it is necessary to realize that the identification requires an arbitrary selection of the scale of length and an origin. Hence the identification is not natural (Loncaric [1985] ) . 3 . 3 The 4x4 Matrix Representation of the Euclidean Group In the previous section it has been demonstrated that after selecting a point 0, scale of length, and an orthonormal basis of IR 3 , any element of the Euclidean group can be represented by the ordered pair (R,t). Further, the

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57 image, M', of an arbitrary point M of the Euclidean space is obtained through x' = R x + t, (1) where x' and x are respectively the images of M' and M under the mapping
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58 (i*l c (R f t)l* ]c “ 1 ) (v lf v 2 ,v 3 ,k) = (i* k (R,t)) (v lf v 2f v 3 ) = l *k( r ll v l + r 12 v 2 + r 13 v 3 + tl/ r 21 v l + r 22 v 2 + r 23 v 3 + t 2 / r 31 v l + r 32 v 2 + r 33 v 3 + t 3 ) = ( r ll v l + r 12 v 2 + r 13 v 3 + t-L, r 21 v l + r 22 v 2 + r 23 v 3 + t 2 < r 31 v l + r 32 v 2 + r 33 v 3 + ( 5 ) The restriction of the mapping i * k (R, t) i * k -1 to ^-k^ 3 ) c k4 can be functionally represented by the matrix expression (i*k(R,t)i* k _1 ) (v 1 ,v 2 ,v 3/ k)= or alternatively by (i* k (R,t)l*k1 ) (v f k) = r ll r 12 r 13 ti/k V 1 r 21 r 22 r 23 t 2/ k v 2 r 3 1 r 32 r 3 3 t3/ k v 3 0 0 0 1 k R t/k V 0 1 k (6a) (6b) Let the set of all matrices of the form given by (6b) be denoted by E k (4) , where k f 0 is arbitrarily fixed. According to section 3.1, the mapping a : ASO ( 3 ) -*• E k (4) CT(R,t) R t/k 0 1 (7) is a group homomorphism. Furthermore, it is easy to prove that the mapping is injective and surjective; hence, a is a group isomorphism.

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59 The group E k (4) is a subgroup of M 4x4 , the group of non-singular 4x4 real matrices; thus a is a matrix representation of the Euclidean group. Moreover, since the image of an arbitrary point veIR 3 , under a Euclidean mapping, can be found via equation (5) , a is a representation in the extended sense of section 3.1. For an arbitrary k f0 any of the matrices belonging to Efc ( 4 ) represents a linear automorphism of IR 4 . However the restriction of this linear mapping to t^IR 3 ) is itself not a linear mapping. In fact, i] c (IR 3 ) is not even a linear subspace of IR 4 . Nevertheless, in no part of the theory of representation of groups of transformations, is there any requirement that the action of the representation be linear. Equally and perhaps more important, it will now be proved that E}.(4) can be regarded as an orthogonal group. This is to say, all the transformations of £^(4) preserve the symmetric bilinear form associated with an orthogonal vector space (Brooke [1980]). Consider the orthogonal space IR 1 ' 0 / 3 . This is a vector space endowed with the following symmetric bilinear form (x,y) : IR 1 / 0 / -^xIR 1 / 0 / 3 -* IR ( (x 1 ,x 2 ,x 3 ,x 0 ) , (y 1 ,y 2 ,y 3 ,y 0 ) )= xiYi + x 2 y 2 + X 3 y 3 + ox 0 y 0 (8) It is well established (Herstein [1975]) that a linear mapping, M, of IR 1 / 0 / 3 will preserve the bilinear form if, and only if, Mt-M 1 = r (9)

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where r is the matrix representative of the bilinear form with respect to the canonical basis of R 1 ' 0 ' 3 . Therefore, r is given by 60 (10) Expressing the matrix M in four blocks which correspond to those of t , one obtains M 11 M12 21 M 22 ( 11 ) Further, using the condition expressed by equation (9) yields MllMll 1 " = I3 ( 12a ) M 2 i = 0 (12b) Moreover, if one requires firstly that the matrix M belong to the special orthogonal group SO (1,0, 3), and secondly preserve the semi-orientations of the space — i.e. MeSO + (l,0,3) — , then the following additional constraints are imposed m 22 > 0 (12c) Mill |M 22 I = 1 ( 12d) 1 Mu 1 = 1 (12c) Therefore M-^ must be a proper orthogonal matrix of dimension 3; i.e. it is a three-dimensional rotation matrix R. Since | | = 1 from equation (12d) M 22 = 1/ and

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M for some k f 0 (13) R m 12 R M' 12 /k 0 1 0 1 This is the same matrix given by equation (7) . 3 . 4 Spin Representation of the Euclidean Group The problem of representing a Euclidean motion has thus far been translated into one of representing orthogonal automorphisms of IR 1 / ^ , 3 ^ However, IR 3 -'* 3 ' 3 i s a degenerate orthogonal space, viz. there are non-zero elements x of IR 1 ' 0 ' 3 W Hi c h are orthogonal to the entire space, or equivalently (x,y) = 0 V yelR 1 ' 0 ' 3 . This constitutes a major disadvantage. Not only are degenerate orthogonal spaces not as well understood as their non-degenerate counterparts, but a multitude of important results for non-degenerate spaces do not apply unrestrictedly to degenerate spaces. For instance, the decomposition of arbitrary orthogonal automorphisms in terms of hyperplane reflections, an important result which is attributed to Dieudonne [1955], does not apply to degenerate orthogonal spaces. Brooke [1980] overcame this problem of degeneracy by regarding IR 1 ' 0 ' 3 as an orthogonal subspace of a non-degenerate orthogonal space of higher dimension. It can easily be proved that IR 1 ' 4 (short for R 0 ' 1 ' 4 ), a five-dimensional orthogonal space with the symmetric bilinear form

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62 IRl^xIR 1 ' 4 IR (x,y) = ( (x 1 / x 2 ,X3,x 4 / x 5 ) , (y 1 ,y 2 ,Y 3 /Y 4 /Y 5 ) ) = ~ x lYl + x 2 Y 2 + x 3Y3 + x 4Y4 + X 5Y5 / (!) is the non-degenerate orthogonal space of the lowest dimension which contains an orthogonal subspace isomorphic to D* 1 ' 0 ' 3 . Brooke proved that Spin + (1, 0, 3) = Spin + (1,4) nStab (e 0 + e 4 ) using the theory of matrix representation of spin groups. Here, a novel, simpler and more direct proof using solely the theory of spin groups is given. Spin groups are subgroups of the Clifford algebra (see Appendix A) of the corresponding orthogonal space, which represent the orientation preserving orthogonal automorphisms of the space. They were devised by Cartan, and initially applied to relativistic physics. Initially interest is focussed upon the subgroup of orthogonal automorphisms of D* 1 ' 4 , which leaves invariant an orthogonal subspace isomorphic to IR 1 ' 0 ' 3 embedded in IR 1 ' 4 . Later it will be required that the restriction of these orthogonal automorphisms to the orthogonal subspace isomorphic to IR 1 ' 0 ' 3 be a Euclidean mapping. The dimensions of its Clifford algebra 0 * 1,4 an<1 its even Clifford algebra 0*i, 4 ° are respectively dim 0 *i , 4 = 2 5 and dim fl*i, 4 ° = 2 4 ( 2 ) Let {eo' e l' e 2' e 3' e 4) be an orthonormal basis of D * 1 ' 4 with ( e O' e o) ~ -1 and ( e i , e i) = +1 Vi = 1,2, 3, 4 (3)

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63 Then, a basis for the even Clifford algebra IR-j ^ 0 will be e 2 e 3 ' e 3 e l • e l e 2 ' e 0 e 4 ' e 0 e 4 e 2 e 3 / e 0 e 4 e 3 e l > e 0 e 4 e l e 2 ' e 0 e l e 2 e 3 / e 0 e l ' e 0 e 2 / e 0 e 3 ' e l e 2 e 3 e 4 ' e l e 4 ' e 2 e 4 < e 3 e 4 ) This seemingly odd arrangement of the basis elements suggests the possibility of expressing an arbitrary element of R 1/4 ° as g = gi + e o e 4g2 + e og3 + 94 e 4' ( 4 ) where gi = a 0 e ({) + a l e 2 e 3 + a 2 e 3 e l + a 3 e l e 2 ( 5a ) g2 = b 0 e (|) + b l e 2 e 3 + b 2 e 3 e l + b 3 e l e 2 ( 5b ) 93 = c 0 e l e 2 e 3 + c l e l + c 2 e 2 + c 3 e 3 ( 5c ) 94 = d 0 e l e 2 e 3 + d l e l + d 2 e 2 + d 3 e 3 ( 5d ) Obviously g 1 ,g 2 eIR 3 0 , and g 3 ,g 4 elR 3 1 / the even Clifford algebra of IR 0 ' 3 and its complement respectively. Furthermore, IR 3 0 is isomorphic to the quaternion skewfield IH . A straightforward calculation shows that the conjugate of g is given by g = gi “ + g 2 ”e 4 e 0 g 3 ~e 0 e 4 g 4 " ( 6 ) Thus, it follows from proposition A. 3. 11, appendix A, that a necessary and sufficient condition for gelRj ^ 0 to belong to Spin (1,4) is that g-g = ±1 (7) This condition can be expressed, in terms of gi,g2'93 and g 4 , as follows

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64 ) and secondly by recognizing that ( -e 0 + e 4 ) f ( e o + e 4) e [ e l/ e 2 ' e 3 ]"*• Therefore, either [ _e o + e 4 ] ® [ e i / e 2 ' e 3 ] or [ e o + e 4 1 ® [ e l / e 2 > e 3 ] form an orthogonal subspace of R 1 ' 4 isomorphic to R 1 ' 0 ' 3 . The orthogonal subspace [e 0 + 64 ]© [e^, > e 3 ] will be used from here on. A completely analogous development could be performed using [-e 0 + e 4 ]® [e x , e 2 , e 3 ] . An element geSpin(l,4) is said to belong to the normalizer of [e 0 + e 4 ] (Rose [1978]), a subgroup of Spin(l, 4) , if gxg -1 e[e 0 + e 4 ] Vxe[e 0 + e 4 ] Since the action of Spin(l,-4) on R 1 ' 4 is linear, to test (e 0 + e 4 ) ; viz. g(e 0 + e 4 )g _1 = /x(e 0 + e 4 ) it suffices for some /xeR ( 10 )

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65 However geSpin(l,4); thus g”g = ±1, and it is easy to show that gg“ = ±1; hence g " 1 = ±g _ , (ii) and the condition given by equation (10) becomes g(e 0 + e 4 )g“ = n(e 0 + e 4 ) for some /xelR (12) In terms of the g^'s, this condition translates into gigi" 9 i 92~ 9291" + 9292" eK ( 13a ) <3l ( 33~( 32 < 33~-<33 < 3l~ +( 33 ( 32~ +< 3l c 34~( 32 < 34--
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9l gi " g2"92 = ±! (18a) (gi + g2)"93 + g3 _ (gi + g 2 ) = 0 ( 18b ) (gi g2)"g3 + g3~(gi g 2 ) = 0 use) gi"g2 g2"gi = ° (i8d) Summarizing, the necessary and sufficient conditions for geSpin (1, 4) nNormalizer [e 0 + e 4 ] are given by equations (16), (17) and (18). Further, proposition A. 4. 2, in appendix A, shows that provided g 3 f 0 3 , gi“g 2 “ g 2 "gi = o » g 2 = Agi for some XeIR (19) This result satisfies the condition expressed by (16), viz. g2gi" + gig2 _ = x gigi“ + gi( x gi“) = 2Xg 1 g 1 _ eiR and simplifies (17) and (18) g = g x (l + Xe 0 e 4 ) + (e 0 + e 4 )g 3 (20) (1 X 2 )g 1 "g 1 = ±1 (21a) gi"g3 + g3 _ gi = ° ( 21 b) It is interesting to note that condition (21a) rules out the possibility of X = ±1, and this result leads to the reduction of conditions (18c) and (18d) to (21b) . Finally, proposition A. 4. 3, in the appendix A, proves that provided g 3 f 0, gi'g 3 + g 3 "gi = 0 «g 3 = dg! for some deK 3 This result is perhaps one of the most significant contributions of this development, and as far as the author 3 If it is assumed that g 1 = 0, a subset of Spin (1,4) is obtained which is not closed under the Clifford product. Hence the subset cannot be homomorphic to the Euclidean group.

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67 is aware the proof is new. From this result, equations (20) and (21) can be further reduced to g = g x (l + Xe 0 e 4 ) + (e 0 + e 4 )dg 1 (22) (1 X 2 )g 1 "g 1 = ±1 (23) Although equation (21b) is eliminated, the representation given by equations (20) and (21) , and the representation given by equations (22) and (23) are to be regarded as equally desirable. Equations (22) and (23) are the simplest representations of orthogonal automorphisms of IR 1 ' 4 which leave invariant the subspace [eg + e 4 ]. The additional conditions that they must satisfy to be orthogonal automorphisms of a subspace of IR 1 ' 4 isomorphic to IR 1 ' 0 ' 3 are now examined in detail. Consider an arbitrary element v of IR 1 ' 4 ; then v = a 0 (e 0 + e 4 ) + a 4 (-e 0 + e 4 ) + y, (24a) where y = a-^eiL + a 2 e 2 + a 3 e 3 , (24b) and the action of geSpin ( 1 , 4 ) nNormalizer [eg + e 4 ] upon v will be V = gvg" = [ gi (l + Xe 0 e 4 ) + (e 0 + e^dg^Jv [(1 + Xe 4 e 0 )g 1 “ + g 1 "d(e 0 + e 4 )] A lengthy computation yields v'= [a 0 (l “ x ) 2 gigi~ 2(1 X) (d^yg-L ) 4a 4 g 1 g 1 “ (d,d) ] [(e 0 + e 4 ) ] + a 4 (l + X) 2 g 1 g 1 “ (-e 0 + e 4 ) + (1 X 2 )g 1 yg 1 “ 4a 4 ( 1 + XJg^^d (25)

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68 This transformation will represent a Euclidean motion of the orthogonal subspace [(-eg + e 4 ) , e^, e 2 , e 3 ] if, and only if r (1 + X) 2 g 1 g 1 " = 1 (26a) and (1 X 2 )g 1 yg 1 " 4a 4 (l + X) g 1 g 1 "d=hyh“ + t Vye [e lf e 2 , e 3 ] (26b) where hdR 3 ° with hh“ = 1, and teO^ e 2 , e 3 ] = IR 3 . These equations coupled with equation (23) lead to the following system of equations (1 X 2 )g 1 g 1 = ±1 (27a) (1 + X 2 )g igi = l (27b) (1 X 2 )g 1 yg 1 " = hyh" with hh“ = 1 V ye[e 1 ,e 2 ,e 3 ] (27c) 4a 4 ( 1 + XJg-Lg-^d = -t (27d) It is easy to see for there to be a solution to this system of equations that X = 0 (28a) Therefore gigi= 1 (28b) d = -t/(4a 4 ) (28c) The final expression for the elements that satisfy the conditions indicated above is g = gi (e 0 + e 4 )tg 1 /(4a 4 ) (29) A simple calculation reveals that, for an arbitrarily fixed a 4 ^ 0, the set of these elements form a subgroup of Spin(l,4)nNormalizer[e 0 + e 4 ], which will be denoted by

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69 Spin + (1 , 0 , 3 ) . It is claimed here that this is the most general spin representation of the Euclidean group E(3). After this important result it is now necessary to determine the action of the group upon the elements of the orthogonal space. Substituting X = 0 into equation (25) , the group action becomes v' = gvg" = [a 0 + (t,g 1 yg 1 “)/(2a 4 ) (t,t)/(4a 4 ) ] (e 0 + e 4 ) + a 4 (-e 0 + e 4 ) + + t (30) Choosing ag = 0, one obtains an orthogonal subspace of R 1 ' 4 isomorphic to R 1 ' 0 ' 3 , this selection reduces the group action to v' = -[ (t^yg! ) + (t,t)/2]/(2a 4 ) (e 0 + e 4 ) + a 4 (-e 0 + e 4 ) + g^gx" + t (31) In particular, the image of w = 0 (e 0 + e 4 ) + a 4 (-e 0 + e 4 ) + 0 will be w' = ~(t,t)/ (4a 4 ) (e 0 + e 4 ) + a 4 (-e 0 + e 4 ) + t At the outset this result is surprising since the original vector has only one component along (-eg + e 4 ) whilst its image has in general components along all the elements of the basis {(e 0 + e 4 ) , (-e 0 + e 4 ) , e^/ e 2 , e 3 } of R 1 ' 4 . However a simple calculation shows that (w,w) = 0 = (w 1 ,w’ ) This phenomenon results from the fact that both (e 0 + e 4 ) and (-e 0 + e 4 ) are isotropic directions of the orthogonal space R 1 ' 4 (eqn. (9) ) .

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70 Finally, the following two properties of (e 0 + e 4 ) given by (e 0 + e 4 ) 2 = ( (e 0 + e 4 ) , (e 0 + e 4 )) =0 (eqn.(9b)) and e i (e 0 + e 4 ) = (e 0 + e 4 )e i V i = 1,2,3 (32) make it possible to introduce the dual unit e (Clifford [1876] ) , in the form e = (e 0 + e 4 ) (33) with the properties 6 2 = 0 (34a) ea = ae V aeK 3 ° (34b) ea = ae V aeH^ 1 (34c) Thus the general spin representation of the Euclidean group can now be transformed into g = gi etg 1 /(4a 4 ) (35) with the usual restriction, g^r = 1 (eqn. (28b) ) It is easy to recognize that g-^ represents the rotational part of the mapping, while t is associated with the translational part. In particular, if the Euclidean motion represents a rotation of 9 degrees around an axis given by the unit vector u = U 2 3 e 3 + u 31 e 2 + u 12 e 3 ' then g 3 will be gi = c ( 0 /2) + S (9/2) (U 2 3 e 2 e 3 + u 31 e 3 e l + u 12 e l e 2)' ( 36 ) where C(9/2) = Cos(9/2) and S(9/2) = Sin(9/2) . Although equations (35) and (28b) provide, indeed, a representation of the Euclidean group, the identification

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71 given by (33) does not preserve the group action (30) . Therefore a new group action will be required, this problem is addressed in the following section. Further, equation (33) is the first introduction of the dual unit e in this work; a second role of the dual unit e will be employed in the biquaternion representation of the Euclidean group (see section 3.7). 3 . 5 The Restriction of the Group Action of the Spin Representation on re -L .e 3 .e 3 l Section 3.4 showed that the restriction on [ei,e 2 ,e 3 ] of the action of the group Spin + (1,0,3) on R 1 ' 4 constitutes a Euclidean motion. Although equation (4.30) of the group action includes the isotropic vectors (e 0 + e 4 ) and (-e 0 + e 4 ) , it is clear that their computational importance is non-existent. Furthermore, the introduction of the dual unit e (eqn.(4.33)) did not preserve the natural group action of the spin groups over the orthogonal space. Thus, it appears natural to search for another group action of Spin + (1,0,3) on [e 1 ,e 2 ,e 3 ] exclusively; the new action is required to produce the same effect on [e]_,e 2 ,e 3 ] as the original one. A guideline is evident, the new action cannot be of the form y ' = gyg" With ye[e 1 ,e 2 ,e 3 ] because this action consists only of successive application

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72 of the Clifford product. Since the Clifford product is linear, for y = 0 y ' = gOg~ = 0 Therefore this action cannot represent a translation. Consider the set {e, e^, &2' e 3 } '> due to the properties of e (eqn. (4 . 34) ) , this set can be regarded as an orthonormal basis of the degenerate space IR 1 ' 0 ' 3 . Then the mapping t : k3 ^ 1 , 0,3 1 (y) = 1 + ey (la) is evidently injective; hence 1* : IR 3 -*• 1 (IR 3 ) l*(y) = i(y) VyeIR 3 (lb) is bijective. Consider the following action, which resembles that proposed by Porteous [1981] and used by Selig [1986] a g : Spin + (l,0,3)xIR 1/0/ 3 Ri / 0 , 3 <*g(y) = [gi etg 1 /(4a 4 ) ] [1 + ey] [gi" eg 1 "t/(4a 4 )] = 1 + eCg^gi t/(2a 4 )] (2) Therefore (i*) -1 a g i*(y) : IR 3 -* IR 3 y' = g-^yg-^ t/(2a 4 ) (3) will be a Euclidean motion if, and only if, -t/(2a 4 ) = t (4) this condition requires a 4 = -1/2. Thus, for this specific group action, the representation of the Euclidean group is g = g x + etgi/2 (5) Using this result, the group action transforms into (i*) 1 Q!gi*(y) : IR 3 IR 3 y.' = g^yg^ + t (6) This is precisely the action proposed by Porteous and used by Selig. However, the particular value of a 4 depends upon the

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73 mapping i : IR 3 ->• 0 3/ and other mappings are certainly possible. 3 . 6 Invariants of the Representation of the Euclidean Group The invariants associated with the representation of the Euclidean group will be now indicated. If the representation provided by equation (4.29), or its equivalent equation (4.35), is used, the unique representation invariant is gigi" = 1 (eqn. (4.28b) ) However, if we use the representation provided by the equations (4.20), and (4.21), the substitution of (4.28a) leads to g = g x + (e 0 + e^g^ (la) and the substitution of (4.33) finishes the transformation into g = gi + eg 3 ( lb ) with the conditions = 1 (eqn. (4 . 28b) ) and gi"g3 + g3"9l = 0 (eqn. (4.21b) ) Hence, the invariants of this representation are precisely those given by (4.28b), and (4.21b).

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74 3 . 7 The Biauaternion Representation of the Euclidean Group Section 3.4 proved that the most general representation of the Euclidean group is of the form g = g x (e 0 + e 4 )tg 1 /(4a 4 ) (eqn.(4.29)) where gieR 3 ° with gigi“ = 1, teIR 3 , and a 4 eIR with a 4 f 0. In the process of proving proposition A. 4. 3, in appendix A, it is shown that if t = t^! + t 2 e 2 + t 3 e 3 , ( 1 ) and gi = age^ + a 1 e 2 e 3 + 33636 ! + a 3 e!e 2 , ( 2 ) then tgi = c 0 e 1 e 2 e 3 + + c 2 e 2 + c 3 e 3 , (3) where C 1 a 0 t l + a 3 t 2 ” a 2 fc 3 ( 4a ) c 2 = -a 3 t l + a 0 t 2 + a l t 3 ( 4t) ) c 3 = a 2 fc l " a l t 2 + *0^ ( 4c ) c 0 = a l fc l + a 2 fc 2 + a 3 t 3 ( 4d ) However, it is possible to rewrite (3) in the form tgi = e 1 e 2 e 3 (c 0 c 1 e 2 e 3 03636 ! c 3 eie 2 ) (5) Further, a simple calculation shows that c 0 ” c l e 2 e 3 “ c 2 e 3 e l “ c 3 e l e 2 ~ “( t l e 2 e 3 + t 2 e 3 e l + t 3 e l e 2) ( a 0 e
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75 g = gi + (e 0 + e 4 ) e 1 e 2 e 3 t*g 1 / (4a 4 ) Since (8) [ ( e o + e 4 )e 1 e 2 e 3 ] 2 = 0 (9a) and [ ( e o + e 4 ) e l e 2 e 3 ] e i = “ e i [ ( e 0 + e 4 )e 1 e 2 e 3 ] V i = 1,2,3, (9b) it is possible to introduce the substitution e = [ (e 0 + e 4 )e 1 e 2 e 3 ] (10) with the usual properties e 2 = 0 (eqn. (4.34a) ) ea = ae V aeIR 3 0 (eqn. (4 .34b) ) ea = -ae V aeIR 3 1 (eqn. (4.34c) ) Then the representation of the Euclidean group will be g = g x + et*g 1 /(4a 4 ) (11) with gigi= 1 (eqn. (4.28b) ) This expression can be regarded as a generalized biquaternion representation, and it exhibits the second role played, in this work, by the dual unit e. It can be easily proved that the set of elements of the form given by equation (8) together with the Clifford product forms a group. Certainly, the same is true for the set of elements of the form given by equation (11), but because of the introduction of the dual unit, the Clifford product becomes the usual biquaternion product (also called the dual quaternion product, Yang [1963]). The set of all elements of the form g = h! + eh 2 where h 1 ,h 2 e IR 3 ° = IH (12)

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76 is correctly called the ring of biquaternions or dual quaternions and is denoted here by 2 IH. Certainly the biquaternion representation, given by equation (11) , can be used to obtain the image of any point, of the Euclidean space, under a Euclidean mapping. Consequently, the biquaternion representation provides a representation of the Euclidean group in the extended sense of section 3.1. Consider the mapping i : 1R 3 -* 2 IH l (y) = i + V 2 e 2 + Y3 e 3) = 1 + 6 (y 1 e 2 e 3 + Y2 e 3 e l + Y3 e l e 2) = 1 + ey* (13) Evidently the map is injective; hence 1* ; IR 3 -* i (IR 3 ) t*(y) = i (y) V yelR 3 is bijective. Consider the action a g : 2 IH -* 2 IH “g(Y) = [91 + €t*g 1 /(4a 4 ) ] (1 + ey*) [g ± ~ + eg 1 "t*/(4a 4 )] = 1 + e[giY*gi" + t*/ (2a 4 ) ] ; (14) then the mapping (i*) _1 a g i*(y) : IR 3 — IR 3 y' = giyg! + t/(2a 4 ) (15) is a Euclidean mapping if, and only if, t/(2a 4 ) = t (16) This condition requires that a 4 = 1/2. Thus, for this specific group action, the representation of the Euclidean group is g = g x + e(t*g 1 )/2 (17)

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77 Again, it is necessary to realize that this particular value of a 4 depends upon the injection i : IR 3 -*• 2 IH, and other injections are certainly possible. Of course, if the starting point of this analysis is the representation given by 9 = 9 l + (e 0 + e 4 )g 3 (eqn.( 6 . 1 a)) with gigi =1 (eqn. (4.28b) ) gi“g3 + 93 gi = (eqn. (4.21b) ) then, following the same procedure, and introducing the dual unit (eqn. ( 10 )), the representation can be written as g = gi + [(e 0 + 64 ) 6 x 6263 ] (e 1 e 2 e 3 g 3 ) = gi + e ( e 1 e 2 e 3 g 3 ) (18) Since g 3 , e 1 e 2 e 3 eIR 3 1 , then exe 2 e 3 g 3 eIR 3 0 = IH . Calling 93* = e l e 2 e 393 (19) the representation becomes 9 = 9 i + eg 3 * ( 20 ) with 9l9l" = 1 (eqn. (4.28b) ) and 9l"93* + 93*"9l = 0 ( 21 ) These equations constitute the foundations of the kinematic mapping proposed by Ravani and Roth [1984], which associates a Euclidean mapping with a point of the dual four-dimensional sphere.

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78 3 . 8 The Representation of the Induced Line Transformation Throughout this study, the Euclidean space has been considered as an aggregate of points satisfying certain properties; accordingly, the Euclidean group was considered as a group of transformations acting upon this aggregate of points. This point of view is not unique, and no claim is made that it is the most appropriate for the study of spatial kinematics. Since the middle of last century (Klein [1939], Giering [1982]), it was well known that the Euclidean space can be deemed as a collection of straight lines (or even a collection of planes, a possibility which will not be considered here) . It is useful to let the Euclidean group act upon this set of lines, and to investigate its effects on the set. Following the fundamentals of chapter 2, a straight line passing through two points, M and N, of the Euclidean space is defined as MN = (PeE | S (P,M) = X
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79 p = (w, wxy) (3) where x stands for the usual vector product. Since any other vector w' = /iw, for some non-zero /xelR, also lies along the same line, it is possible to use MP = (MW, /xwxy) (4) It can be proved that there is a bijective correspondence between the collection of sets of six-dimensional vectors given by equation (4) , and the lines of the Euclidean space. Therefore, for a given p = (w,wxy) , the set ii p = {/xp | for some nonzero /xelR} is called the homogeneous Pliicker coordinates of the line represented by p. McCarthy [1986b] shows that when a Euclidean mapping given by y' = Ry + t (5) acts on the Euclidean space, the Plucker coordinates of the image of p = (w,wxy) are w' R 0 W (wxy) 1 TR R wxy where T stands for the skew-symmetric matrix representing the usual three-dimensional vector product; viz. (V) The transformation given by equation 6 is usually referred as the induced — by a Euclidean motion — line

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80 transformation (Klein [1939]). Using these fundamental results, McCarthy reformulates the equation into a three-dimensional dual-matrix equation, and uses this dual-matrix form to provide another illustration of the closure equation for spatial mechanisms. In what follows, it will be shown that these rather cumbersome representations of the induced line transformation are unnecessary. In fact, the same representation, of the Euclidean group, can be employed regardless of whether the action of the group is upon points or lines. This result is overlooked in the literature on kinematics. Let y = 6(M,0), and z =
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81 ctg : E(3)xl*(L 3 ) i * (L 3 ) a g (i*(P)) = [gi + (tgx/2) e ] { (y z) + [(zy -yz)/2]e} [gi" " (gi"t/2)e] = gi(y z)<3i~ + e {gx (zy yz)g ± ~ + t[gi(y z)g 1 ~] [g 1 (y z)g 1 “]t}/2 Finally, the induced line transformation is given by l* 1 agi* : L 3 L 3 i* -1 a g i*(p) = gi (y z)g ± ~ + e{g 1 {zy yz)g 1 ~ + t[g x (y z) g x “] [g x (y z)g 1 "]t}/2 3 . 9 Screw Representation of the Euclidean Group It has been known, since the last century (Ball [1900]), that a Euclidean motion can be represented, after selecting a reference system, by a twist applied to a screw. The screw is defined as an arbitrary line endowed with a pitch. Section 2.8, demonstrated that the component of the translation vector along the rotation axis is not a Euclidean invariant; therefore the pitch of the screw — the ratio of the magnitude of this component to the rotation angle — is not a Euclidean invariant. In order to underscore this point, a screw will be represented here by $ = (9, u, r, d) (1) where 9 = Rotation angle, or twist, with 0 < © < n .

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82 u = Unit vector associated to the rotation axis, such that the rotation is perceived, from the tip of u, as counterclockwise. If 0 = 0, according to this rule, u is undefined; in this case, u must lie along the translation vector. r = Perpendicular vector from the origin of the reference system to the line associated to the screw (r.u = 0) . d = Magnitude of the component of the translation vector along the axis of rotation. If 9 = 0, du is the total translation vector. This ordering separates those screw parameters which are indeed Euclidean invariants (9,u) , from those which depend upon the selection of the reference system (r,d) . The restrictions on 9 and u aim to avoid any multiplicity of the representation. This idea was outlined, for the pure rotation case, by Altmann [1986]. The set of all tetrads (9, u, r, d) satisfying the condition given in the definition above is denoted by S. From what has been pointed out, it follows that the mapping S ; E ( 3 ) S E(v^) = (9, u, r, d) = $ (2) is a bijection. This bijection permits to furnish the set S with a group structure, according to the rule $21 = $2 $1 = S [ S 1 ($ 2 ) 2 _ 1 ($ i )] This rule assigns to $ 2 $i the screw of the element of ( 3 )

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83 Euclidean group obtained by composing S -1 ($ 2 ) and Z 1 ($ 1 ), in this order. It is desired to obtain the four parameters of $ 21 in terms of the parameters of and $ 2 . This problem was solved by Halphen (Beggs [1983]), using descriptive geometry methods, a century ago. However, an analytical solution did not come out easily, even though Rodrigues (Gray [1980]) provided, in 1840, the correct equation for the composition of general spatial rotations. In fact, as late as 1963, Paul [1963] published a didactic paper about composition of finite rotations. With regard to the more general problem of analytically composing general spatial motions, Beatty [1966] acknowledged many geometrical proofs but none analytical. Then Beatty provides a vector and matrix algebra solution of the composition of spatial motions, and the determination of some characteristics of the resultant motion. The problem was partially solved by Roth [1967], who introduced the screw triangle as an spatial generalization of the planar pole triangle. This screw triangle permits finding the composite screw in terms of the two composing screws. However, the formuli exhibit there assume the origin of the coordinate system lying on one of the composing screws. Recently, Phillips and Zhang [1987] considered the relationship between the screw triangle and its infinitesimal counterpart the cylindroid.

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84 Here, the problem will be posed as follows: Given $1 = (0!, u lf r lf d x ) , and $ 2 = (0 2 , u 2 , r 2 , d 2 ) , find equations that provide $ 21 = (©21' u 21 , r 21' d 2l) in terms of $! and $ 2 . Moreover, the problem will be solved within the realm of the Clifford algebra R 3 . The results obtained in section 2.9, concerning the semi-direct structure of the Euclidean group with the rotation as the direct part, indicate that the parameters related to the rotation (0 23 , u 2 i) are independent of the parameters related to the translation. Therefore, they must be calculated first. Let u 3 = Aie 3 + X 2 e 2 + X 3 e 3 , and u 2 = /i 3 e 3 + M 2 e 2 + M 3 e 3 be the unit vectors associated to the rotation axes; then the elements of IR 3 0 = IH representing the rotations are gi = Cl + Sl(X 1 e 2 e 3 + X 2 e 3 e 3 + X 3 e 1 e 2 ) (4a) g 2 = C2 + S2(/iie 2 e 3 + M2 e 3 e l + M3 e l e 2)' (4b) where Ci = Cos(0j/2), and Si = Sin(0j/2) ; therefore, the element representing the composite rotation is 921 = 9291 = [ C2C1 S2Sl(X 1 /i 1 + X 2 /i 2 + X 3 /x 3 ) ] + e 2 e 3 [S2Cl/i 3 + C2S1X^ + S2S1 (/x 2 X 3 — /x 3 X 2 ) ] + e 3 e 3 [S2C1 /u 2 + C2S1X 2 + S2S1(^ 3 X-j_ — /i 3 X 3 ) ] + e 3 e 2 [S2Cl/i 3 + C2S1X 3 + S2S1 (/x 3 X 2 — /i 2 X^_) ] (5) From equation (5) , it follows that the angle of the composite rotation satisfies C21 = C2C1 S2Sl(X 1 /i 1 + X 2 /i 2 + X 3 /i 3 ) (6)

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85 Since by restriction 0 < 0 21 < n, the sine of the angle must be positive; hence S21 = [1 (C21) 2 ] ^ (7) and the unit vector associated to the composite rotation is u 2 l = { e]_ [ S2Cl/i]_ + C2S1A 2 . S2S1(/z 2 A 3 — ^3^2) ] + e 2 [S2Cl)Ll 2 + C2S1A 2 + S2Sl(/i 3 A]_ — Mx ^ 3 ) ] + e 3 [S2C1ju 3 + C2S1A 3 + S2Sl(jLi 1 A 2 M2 x l ) ] )/ S21 ( 8 ) The calculation of the parameters related to the translation basically involves the reconstruction, from the screw parameters, of the translation vector associated with the origin of the reference system, followed by the calculation of the corresponding vector of the composite motion, and a final decomposition. A simple calculation shows that the translation of the origin of the reference system due to the motion represented by $! and $ 2 are t x = r i gx^gx" + d^ (9a) t 2 = r 2 g 2 r 2 g 2 " + d 2 a 2 ( 9b ) Thus the translation vector of the origin of the reference system under the composite motion is t 2 l = g2 t r l gi r l
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86 Finally, the perpendicular vector from the origin of the reference system to the screw axis is obtained from t 21 ± = r 21 “ 921 r 2l92l” (12a) Premultiplying by <321~ • anc * postmultiplying by g 2 i/ equation (12a) becomes 92l” t 21' L 921 = < 321~ r 21^21 ~ r 21 (12b) Since r 21 is perpendicular to the rotation axis associated with g 2 i, one obtains 921 r 2l92l" = r 21 Cos0 21 + w (13a) and 921~ r 2l921 = r 21 Cos0 21 “ w (13b) with we [r 21 , u 2 ± The substitution of equations (13a) and (13b) into (12a) and (12b) yields t 21 ± = r 21 ( 1 ” Cose 2l) + w ( i4a ) and g2l" t 21 J -g21 = “ r 21 ( 1 Cose 2l) + w ( i4b ) Subtracting equation (14b) from (14a) , and solving for r 21 , one obtains r 2 1 = ( b 21" L “ < ?21 -t 21" L 92l)/ C 2 (1 “ Cos0 2l) 3 ( i5 ) These equations provide the composition laws for the group of tetrads of quantities representing the screws. Of course, if the reference system is selected such that the origin is located on one of the axes of the composing screws, these equations collapse to those provided by the screw triangle.

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CHAPTER 4 THE EUCLIDEAN GROUP AS A LIE GROUP Thus far, this thesis has dealt exclusively with the algebraic aspects of the Euclidean group and its action upon the Euclidean space. In this chapter the topological and metrical characteristics of the Euclidean group will evolve when the group is regarded as a Lie Group. It is important to recognize that there is an algebra associated with every Lie group. In the case of the Euclidean group, this algebra will be identified with the algebra of infinitesimal screws or screws for short. Furthermore, this algebra is endowed with symmetric bilinear forms which remain invariant under the action of the group into itself. This property provides the Euclidean group with the structure of a semi-Riemmanian manifold with a hyperbolic metric tensor. Even though the theory of Lie groups, differential manifolds and Riemannian geometry is relatively recent, the wealth of information it yields is extraordinary. It is therefore not possible here to give details of all the background material. Reference is however made to the standard works in the relevant subjects. 87

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88 4 . 1 Lie Groups Definition 1. A Lie group G (Boothby [1975]) is an algebraic group which simultaneously is a differentiable manifold, and such that the mappings GxG G (g2/9l) 92^1 and ~ rr-1 ( 1 ) ( 2 ) G -*• G g are differentiable mappings. A differentiable manifold is, loosely speaking, a set with sufficient topological structure that permits the development of a differential calculus on the set. A formal definition is provided below (Boothby [1975]). Definition 2. A differentiable manifold M is a Hausdorff space with a countable basis of open sets, such that each point of M has a neighborhood homeomorphic to an open set of IR n . Each pair (U,§), where U is an open set of M and $ is a homeomorphism of U to an open set of R n , is called a coordinate neighborhood. In addition, the differentiable manifold must have a family 11 = {(U a ,$ a )} of coordinate neighborhoods such that 1. The U a form a covering of M, 2. For any a,/? the coordinate neighborhoods are compatible; viz. either U a nU^ = (}), or the mappings $ Q! ( , $0)“1 : R n -+ R n and $^($ a ) _1 : IR n are diffeomorphisms. R n and

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89 3. If a coordinate neighborhood (V,'//) is compatible with every (U a ,$ a )e1/, then (V,^)e‘V. In order to prove that the Euclidean group E(3) is a Lie group, perhaps, the best approach (Bel infante and Kolman [1972]) is to consider firstly the general linear group GL(n) , which is the group of all non-singular nxn real matrices. This group is a manifold since the n 2 entries of an arbitrary matrix can be used as the image of an element of GL(n) under a homeomorphism from GL(n) to IR nxn . Furthermore, since the determinant function is continuous, the elements of the group GL (n) form an open subset. Moreover, the entries of a product of matrices are polynomials in the entries of the factors; whilst the entries of the inverse of a matrix are rational functions of the entries of the original matrix. Thus, the functions are analytical and therefore differentiable. Hence, GL(n) is a Lie group. The final step is to make use of a result (Warner [1983]) which states that closed algebraic subgroups of a Lie group are Lie subgroups. It has already been shown (see section 3.3) that E(3) = S0 + (l,0,3) is a subgroup of GL(4) , and that the conditions imposed on E(3) produce a closed subset of GL ( 4 ) . Hence, the result proves that E(3) is a Lie group .

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90 4.2 The Lie Algebra of the Euclidean Group An important property of Lie groups is the existence of an algebra of vector fields defined on the group, which is isomorphic to an algebra of tangent vectors belonging to the tangent space at the identity element (O'Neill [1983]). Since both algebras are isomorphic, any one of them is referred as the Lie algebra of the corresponding Lie group. If G is a Lie group, it is customary to represent its Lie algebra by g . . In particular, the Lie algebra of the euclidean group E(3) is represented by e(3). Here an explicit representation of e(3) is provided. Of course this representation can be obtained using anyone of the representations of the Euclidean group given in chapter 3 (Karger and Novak [1985]). Nonetheless, the spin representation is used here. An advantage of using this Clifford algebra based representation is that Clifford algebra is robust enough to handle concisely algebraic manipulations of the new algebra e(3) without having to resort to a new algebraic structure. Let an arbitrary motion of a rigid body be given by g = gi + etgi/2 (eqn. (3.5.5)) where g, g lf and t are functions of time; it was previously shown that g^ can be expressed in the form g^=C (0/2 ) +S (0/2) (U23©2®3 ^3 1® 3®1 ^12®1®2 ) ( • (3.4.36)) with u 23 2 + u 31 2 + u 12 2 = ^ (la)

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91 and t is given by t = tiei + t 2 e 2 + t 3 e 3 (lb) If g represents the identity mapping i in the Euclidean group, then 0 = 0°; thus g^ = 1, and t = 0, with u = u 23 e 2 e 3 + u 3 ie 3 ei + u^ 2 eie 2 unitary but otherwise unrestricted. The derivative of the motion, represented by g, at the identity i of the Euclidean group is given by g(i) = g x (i) e [t (i ) g x (t ) + t(i)g 1 (l)]/2 (2) where ' indicates the derivative with respect to time. Since 9l = 0(S(0/2))/2 + 0 (C(0/2) ) (u 23 e 2 e 3 + u 31 e 3 e 1 + u 12 e 1 e 2 )/2 + S (0/2) (u 23 e 2 e 3 + u 3; j_e 3 ei + u 12 e 1 e 2 ) (3a) and • • • • t — t]_S]_ + t 2 e 2 + t 3 e 3 , ( 3b) then 9l(t) e ( i )( u 23 e 2 e 3 + u 31 e 3 e l + u 12 e l e 2)/ 2 (4a) and t(i) = t 1 (l)e 1 + t 2 (i)e 2 + t 3 (i)e 3 (4b) Hence, finally g(t) = 0(i)(u 23 e 2 e 3 + u 31 e 3 e 1 + u 12 e 1 e 2 )/2 e(t 1 (i)e 1 + t 2 (i)e 2 + t 3 (i)e 3 )/2 (5) Incorporating the scalar terms into the components of the vector, the tangent space at the identity — i.e. the collection of the derivatives evaluated at the identity of all possible motions — will be

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92 e(3) = {p = p + ep* I p = Poie 2 e 3 + Po2 e 3 e l + P03 e l e 2 and p* = P 2 3 e i + P3l e 2 + Pl2 e 3 with p^jelR} (6) The selection of the suffixes conveys the close relationship between e(3) and the theory of screws with its characteristic screw coordinates. However, it is important to recognize that it reverses the suffixes suggested by equation (5). It can be proved that e(3) form a real vector space under the operations P + q = (p + ep*) + (q + eq*) = (p + q) + e(p* + q*) (7a) and Xp = X(p + ep*) = (Xp) + e(Xp*), (7b) where the addition and scalar multiplication on the right hand side are the usual componentwise addition and scalar multiplication in the respective vector spaces. Although the algebraic definition of the operation is deceptively simple, the approach following the theory of manifolds is a little more complicated (Boothby [1975], O'Neill [1983]). Furthermore, the vector space e(3) forms an algebra under the following operation /V* »v [p q] = (P + ep*) (q + eq*) (q + eq*) (p + ep*) = (pq qp) + e[(pq* q*p) (qp* p*q) ] (8) where the products in the right hand side of the equation represent the Clifford product as indicated in section 3.4. The algebraic closure of the set e(3) under this product is guaranteed firstly because p,qeR 3 °, and K 3 ° is a subalgebra

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93 of ER 3 , and secondly because proposition A. 4. 4, in the appendix A, assures that (pq* q*p) (qp* p*q)eIR 3 C [R 3 1 . Again, the closure proof from the standpoint of manifold theory is more complicated. This product is closely related to the dual vector product of infinitesimal screws (Brand [1947], Dimentberg [1965], Duffy [1985]). /v /v* Furthermore, the product [p q] satisfies the following properties: 1. Anti-symmetry /v* t\j [P P] = (PP " PP) + e[(PP* “ P*P) (PP* “ P*P) ] = 0 (9a) 2. Bilinear [(MiP + ^ 2 ^) r 3 = (MiP + M 2 q)r r(/*iP + M 2 ^) = Mi(pr rp) + M 2 (q r ~ rq) /V /V IV/ /V = Ml [P r] + M 2 [q r 3 ( 9b ) »V IV V [r (MiP + M 2 < 3 ) 3 = r(/iiP + M 2 < 5 ) “ (MiP + M 2
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94 4. Jacobi identity <\S >v IV /V IV IV IV IV IV [p [q r]] + [q [r p] ] + [r [p q] ] = P (qr rq) (qr rq)p + q(rp -pr) (rp -pr) q + r(pq -qp) (pq -qp)r = 0 (9e) Therefore the algebra defined on e(3) is a Lie algebra and the product in this algebra is also known as the Lie bracket (Warner [1983], O'Neill [1983]). There is a close relationship between a Lie group and its Lie algebra. Many results about Lie groups can be proved via their Lie algebras and conversely. For instance, there is a one-to-one correspondence between the subalgebras and ideals of a Lie algebra on the one hand and the subgroups and normal subgroups of the corresponding Lie group on the other hand. In particular, it was shown (see chapter 2) that the set of translations forms an abelian normal subgroup of E(3), which produces an abelian ideal of the Lie algebra e(3) , and it is denoted by t(3). Similarly, the subgroup of the rotations around an arbitrary fixed point produces a subalgebra of e(3) denoted by so (3) . Furthermore, since t(3) is abelian, then [t x t 2 ] = 0 V t 1 ,t 2 et(3) Thus t ( 3 ) is a solvable ideal (Sattinger and Weaver [1986]). It can be shown that t(3) is the maximal solvable ideal of e(3) , such an ideal is known as the radical (Herstein

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95 [1968]). Since t ( 3 ) is a proper ideal of e(3), e(3) itself is not solvable. An algebra is said to be semi-simple if it contains no abelian ideals (other than {0}). It can be shown that this is the case for so (3) . On the other hand, since t(3) is an abelian ideal of e(3) , e(3) itself is not semi-simple. Finally, the Levi decomposition (Sattinger and Weaver [1986]) shows that any Lie algebra is the semi-direct product of its radical and a semi-simple subalgebra. In the case of e(3) this semi-simple subalgebra is so (3) . These results complete the analogy between the structure of E(3) as a Lie group and e(3) as a Lie algebra. 4 . 3 The Adjoint Representation of the Euclidean Group It is well known that for any group G and any geG, the mapping 'Pg : G -+ G ^g(x) = gxg -1 VxeG (1) is a group isomorphism called the conjugation by g. Conjugation has the property that 'Pg (s) = geg -1 = gg -1 = e VgeG (2) Thus, the identity element is invariant under conjugation. Furthermore, if G is a Lie group, the mapping ^ g is a dif feomorphism; hence its differential map (Boothby [1975]) maps bijectively the tangent space at the identity, g , into itself.

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96 It can be proved (Karger and Novak [1985]) that the differential mapping of in the Euclidean group, denoted d'pg, is given by d* g : e ( 3 ) e (3) d^ g (p) = gpg" 1 V pee (3) V geE(3) (3) Developing this expression in terms of the Clifford algebra, one obtains d^g(P) = ^(p + ep*) = g (p + ep*)g _1 = [gi + etg 1 /2](p + ep*) [g x _ eg 1 "t/2] = gipgi" + e(giP*gi" + [tgxpgi" gipgi t]/2]j (4) The map d^ g is usually referred to as the adjoint map of g, and it is denoted by Adg. Consider now the mapping Ad : E (3 ) L[e (3) ] Ad(g) = Ad g , (5) where L[e(3)] represents the set of all linear automorphisms of e(3) into itself. It can be proved that L[e(3)] has a Lie algebra structure where the product is the commutator. In what follows, it will be shown that Ad is a group homomorphism; hence Ad is a representation of the Euclidean group . Let pee (3) be arbitrary, then [Ad (g*g) ] (p) = (g*g) ( p) ( g*g) -1 = g* (gpg -1 ) g* -1 = g*[Adg(p) ]g* -1 = Adg* [ Adg ( P )] Therefore Ad (g*g) = Ad (g*) Ad (g) (6) Furthermore [ Ad ( i ) ] (p) = lpl = p Thus

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97 Ad (1 ) = I e ( 3 ) (7) Accordingly, the mapping Ad : E(3) -*• L[e(3)] is indeed a group representation of the Euclidean group, and it is called the adjoint representation. It is important to notice that since the adjoint representation acts on the tangent vectors at the identity, this representation requires an interplay between the algebraic and topologic structures of the Euclidean group, regarded as a Lie group. Consequently, the adjoint representation could not have been devised based solely on the algebraic arguments of the previous chapter. 4 . 4 The Euclidean Group as a Semi-Riemannian Manifold In this section the Euclidean group is endowed with the structure of a semi-Riemannian manifold. The usual procedure (O'Neill [1983]) is to provide a bi-invariant metric in the Lie algebra of vector fields defined on the group. It has been already mentioned that this algebra of vector fields is isomorphic to the Lie algebra of tangent vectors at the group identity. The following development makes use of both interpretations . The first step consists of endowing the algebra of vector fields with a metric tensor that is invariant under left multiplication by an arbitrary element of the Euclidean group. Such a metric is called a left-invariant metric. O'Neill [1983] shows that a left-invariant metric on the

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98 algebra of vector fields is equivalent to a symmetric bilinear form defined upon g. . Furthermore, O'Neill [1983] proves that a left-invariant metric in a Lie group is right-invariant (hence bi-invariant) if, and only if, the metric is invariant under the action of the adjoint representation of the group G; i.e. Ad (G) -invariant , or equivalently 1 (Ad g x, Ad g y) = (gxg -1 , gyg -1 ) = (x,y) Vx,yegVgeG (1) In what follows, it is shown that the Killing form and the Klein form, also known as the reciprocal product of screws (Karger and Novak [1985]), are symmetric bilinear forms upon the vector space of tangent vectors at the identity of the Euclidean group. Moreover, they are Ad (E (3 )) -invariant; hence they can be used as bi-invariant metrics in the process of furnishing E(3) with a semi-Riemannian manifold structure. The proof uses the Clifford algebra IRi o,3* Let p = p + ep*, and q = q + eq* be arbitrary elements of e(3); then the Killing form is defined by (Belinfante and Kolman [ 1972 ] ) Ki(p,q) = Poiqoi + P02<302 + P03
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99 Hence, denoting
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100 the algebra of vector fields defined on E(3) 2 , or equivalently a bi-invariant and non-degenerate symmetric bilinear form in e(3) . Therefore any symmetric bilinear form to be used for inducing a semi-Riemannian manifold structure in E ( 3 ) must be of the form (p,q) : e ( 3 ) xe ( 3 ) IR /V /V /V /V *v (p,q) = Kl(p,q) + /xKi(p,q) for some /xelR ( 6 ) Lipkin [1985] shows that regardless of the value of n chosen in equation ( 6 ) , e(3) considered as an orthogonal space, is always isomorphic to IR 3 ' 3 . Thus, e(3) is called a neutral orthogonal space (Porteous [1981]), and can be orthogonally decomposed into three hyperbolic planes (Kaplansky [1969]). In what follows, it will be shown that the mutually orthogonal hyperbolic planes are independent of the value of /i chosen in equation ( 6 ) . /V Let { Pi , P 2 1 P 3 } be an arbitrary subset of e(3) such that 'V /V •'V* Ki (Pi, Pi) = 1 and Ki(pi,pj) =0 V i,je{l,2,3} i ? j and (P23 ) i = (P3l)i = (Pl2)i = 0 v ie(l/2,3) The existence of such a set is guaranteed by the existence theorem of orthonormal basis of orthogonal spaces (Porteous [1981]). In particular it applies to e(3), regarded as an 2 It is possible to define a semi-Riemannian manifold with a degenerate metric tensor; however, the theory of this class of manifolds is still in the very early stages, and many of the important properties of non-degenerate manifolds are not shared by their degenerate counterparts (Kupeli [1987]).

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101 orthogonal space, where the Killing form is the symmetric bilinear form. Consider now the following subspaces of e(3) Hi = [(Pi + e0), (0 + e qi )] (7a) H 2 = C (P 2 + e °) ' (° + ec l 2 ) 3 (7b) H 3 = [ (P3 + e °) / (° + e 93) ] (7c) where (q 2 3)i = (Poi)i/ (
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102 The characteristic equation for this matrix is P(M = |M Xlg | = [ (M *)(-M l] 3 = 0 ( 10a) which reduces to (X 2 Xu l) 3 = 0 (10b) Solving the quadratic equation contained in the parenthesis, it is evident that M has two eigenvalues of multiplicity three, and values = M + (M 2 + l) 3 * and X 2 = M “ (M 2 + 1) h (11) These results confirm the identification e(3) = IR 3 ' 3 . 4.5 Analysis of the Structure of the Set of Second Derivatives of the Euclidean Group at the Identity In this section the structure of the set of second derivatives of the Euclidean group at the identity will be analyzed. As in section 4.2, the elements of the Euclidean group will be represented by elements of the spin group; hence, the Clifford algebra R 1/0 ,3 w iH be used in this development. The methods are similar to those used in the analysis of the Lie algebra e(3) as an orthogonal space, and they are closely related to the study of tangent groups of Lie groups (Yano and Ishihara [1973]). It has been already shown that if the elements of the Euclidean group are represented by g = g x + etg 1 /2, (eqn. (3.5.5)) where g 1 =C(0/2) +S (0/2) (u 2 3e2 e 3 ^31®3®1 ^12®1®2^ (eqn. (3.4.36)) with

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103 and u 23 2 + u 31 2 + u 12 2 = 1 (eqn. (4.2.1a) t = t^e^ + t 2 e 2 + t 3 e 3 , (eqn. (4.2.1b) then the elements of the Lie algebra e(3) are given by 9(1) = 9(1) (u 23 e 2 e 3 + u 31 e 3 e 1 + u 12 e 1 e 2 )/2 • • • e(ti(i)e^ + t 2 (i)e 2 + t 3 (i)e 3 )/2 (eqn. (4.2.5) Forming the derivative of this expression with respect to time, evaluating the derivative at the identity, and • • imposing the conditions 6(1) =0 and t(i) = 0, one obtains g(i) = 9i(l) + et(l )/2 ( 1 ) where 9l(l) 9(1) (u 23 e 2 e 3 + u^e^-^ + u 12 e 1 e 2 )/2 (2) and t(t) = tiflje! + t 2 (l)e 2 + t 3 (l)e 3 (3) It can be shown that gi(t) represents one half of the angular acceleration of the moving reference frame, and therefore one half of the angular acceleration of the rigid body, with respect to a fixed reference frame; whilst t(i) represents the acceleration of the origin of the moving reference frame with respect to a fixed reference frame. • • Since g(l ) eR^ q, 3 depends on six independent variables, the subset of those elements of ^ 1 , 0 , 3 form a six-dimensional real vector space under componentwise addition and scalar multiplication. This vector space will be denoted by e'(3),

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104 and an arbitrary element of e'(3) will be represented by >Xl p« = p' + ep* ' ; this representation is analogous to that of the Lie algebra given by equation (2.6). It will now be shown that e' (3) has also the structure of an orthogonal space. This will be accomplished by defining the equivalent of the Killing and Klein's form on the space e'(3). *v»
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CHAPTER 5 APPLICATIONS In this chapter the theory developed previously is applied to one issue in spatial kinematics. The topic is an analysis and proof of the principle of transference. Since the principle is frequently used as a basis for analysis and synthesis of spatial mechanisms and manipulators, as well as applications in instantaneous kinematics, its analysis and proof constitutes, without doubt, a subject of contemporary importance. As far as the author is. aware, a complete proof of the principle has not appeared in the literature. In this chapter a thorough analysis of the principle and its complete proof are given 5 . 1 The Principle of Transference The history of the principle of transference is interesting. It was originally proved by Kotelnikov, a Russian mathematician, in the latter part of the last century. Indeed, Study [1903] acknowledged a theorem with the name (Uebertragunsprincip) , and he associated it with Kotelnikov 's name. However, it appears that the publication containing the proof was lost during the Russian revolution. The principle was stated by Dimentberg [1965] in the following terms 105

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106 Let us consider a certain collection of vectors r l , r 2 , • . . , whose origin is at a certain reduction point 0. Assume that, together with each of the vectors r^, we also consider a certain moment r^° attributed to it and referred to point 0, with the result that an additional set of moments ri°,r 2 °,..., referred to point 0, makes its appearance, so that we have two sets of two vectors, i.e., motors (r^, r^ 0 ) , (r 2 , r 2 °) . . . Each motor (r-^,r^°) referred to point 0, naturally defines a certain screw R-^ -its axis, vector and parameter. The set of motors (r^r^ 0 ), (r^,^ 0 )..., referred to the reduction point 0, determines a set of screws R 1 ,R 2 ,.... The ends of all vectors and moments with origins at point 0 form a sixdimensional point space, while the axes of the screws defined by them form a four-dimensional linear space, with a two-dimensional space of screws corresponding to each axis and, consequently, the screw space will be sixdimensional. Thus, with the aid of the reduction point, we establish correspondence between the space of vector doublets or motors (or point pairs) on the one hand and the space of screws on the other. To each motor in the first space there corresponds a screw in the second space. (Dimentberg [1965] pp. 58-59) One of the first applications of the principle to spatial kinematics appears to have been due to Keler who, in 1958, applied the principle to the analysis and synthesis of spatial mechanisms. However, his results went unnoticed until the early 70s (Keler [1970a], Keler [1970b]). In 1964, Yang and Freudenstein [1964] solved an RCCC spatial mechanism by dualizing the closure equations of a spherical four-bar mechanism, and subsequently solving the resulting system. The principle was located in a prominent place in the theory of applied kinematics when Rooney [1974], in his doctoral dissertation, employed the principle in the

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107 development of a powerful and comprehensive theory for the analysis of spatial mechanisms. There, one can find this explicit statement of the principle All valid laws and formulae relating to a system of intersecting unit line vectors (and hence involving real variables) are equally valid to an equivalent system of skew unit line vectors, if each real variable, a in the formulae is replaced by the corresponding dual variable a = a + ea, and each constant, k, is replaced by the dual constant, k = k+ e0 (i.e. it has zero secondary part) . Here k is not a parameter but a definite fixed number. (Rooney [1974], p. 88) Following this, Rooney attempted to prove the principle by resorting to the algebraic structure of the dual numbers. It seems that Rooney, himself, was not satisfied with his proof because, in a later publication (Rooney [1975]), he made no mention of it, but rather restated the principle as follows All valid laws and formulae relating to a system of intersecting unit line vectors (and hence involving real variables) are equally valid to an equivalent system of skew unit line vectors, if each real variable, a in the formulae is replaced by the corresponding dual variable a = a + ea 0 (Rooney [1975], p. 1092). No mention was made of the omission, and although Rooney indicated that the quote is taken from Kotelnikov's works, he also stated that the reference was lost. In his new contribution, Rooney attempted again to prove the principle, but he essentially only demonstrated the mechanics of the dualization process. A final mention of the principle

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108 of transference was made by Rooney [1978a] in his treatment of several representations of general spatial displacements. In 1981, the principle of transference was applied to the study of instantaneous invariants of rigid body motions (Hsia and Yang [1981]). Most recently, Selig [1986] published a partial proof of the principle of transference that employs the spin and biquaternion representation of the Euclidean group. This proof essentially describes a transition from the spin representation to the biquaternion representation as indicated in section 3.7. It did not address a key issue which is the relationship between the real and dual parts, and which is, perhaps, the most fruitful characteristic of the principle, when it is applied to spatial kinematics. In the next sections a novel complete proof of the principle of transference is given; the proof addresses the relationship between the real and dual parts. Furthermore, a new statement of the principle is explicitly stated. In addition, for the first time, the relationship between the principle of transference and the Hartenberg and Denavit notation (Denavit and Hartenberg [1955]) is explicitly stated. 5 . 2 Analysis of the Dualization Process The proof of the principle of transference begins with a group theoretical analysis of the dualization process

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109 described by Rooney [1975] and Duffy [1980]. The approach is perhaps a little abstract. However it has significant advantages with regards to the clarity and expeditiousness of the proof. Throughout this development, the isomorphism between the even Clifford algebra IR 3 0 and the quaternion skewfield (see appendix A) will be employed. Firstly, some important results concerning the representations of rotation around a fixed point and Euclidean mappings will be restated. It is well known (Porteous [1981]) that a rotation around a fixed point can be represented by g X = C ( 0/ 2 ) + S ( 9/ 2 ) u (1) where 0 is the rotation angle and u is a unit vector in the direction of the rotation axis. Moreover the mapping, : n 0 w u %](P) = 9i ( 2 ) where n 0 is the subgroup of the rotations around point 0, and IH U is the subgroup of unit quaternions, is an isomorphism. Analogously, in section 3.7, it was shown that, after choosing a coordinate system, a Euclidean mapping, v/>, can be represented 1 by g = gi + et*g^/2, (eqn. 3.7.17) •^It is important to recognize that the results concerning the principle of transference are valid only for this biquaternion representation of the Euclidean group, which is not the most general representation.

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110 where t* represents the displacement of the origin of the coordinate system, and g ^ is a unit quaternion of the form given by equation (1) . Furthermore, the mapping R e : E (3) 2 IH u R e (*) = g x + et* gi /2 (3) where 2 IH U denotes the subgroup of biquaternions whose real part is unitary, is also an isomorphism. Because of these results, it is not necessary to make an explicit distinction between the rotations around a fixed point and Euclidean mappings on the one hand, and their representations by means of quaternions and biquaternions on the other hand. Proposition 1. The mapping <5 : 1H U -* 2 IH u 6(g x ) = 6 [C (0/2) + S (8/2 ) u] = C ( 0/ 2 ) + S (0/2 ) u = [C (0/2 ) + S (0/2 ) u] + ed[-S (0/2) + C (0/2 ) u]/2 = gi + ego ( 4 ) where 0 = 0 + ed, is a group homomorphism. Proof: Consider two rotations around a common fixed point whose quaternion representations are g-L = C (0/2 ) + S (0/2 ) u (5a) h ± = C (a/2) + S (a/2 ) v (5b) Their corresponding dualized spatial displacements are <5(gi) = [C (0/2) + S (0/2 ) u] + ed u [-S(©/2) + C(0/2)u]/2 (6a) S (h x ) = [ C (a/2 ) + S (a/2 ) v] + ed v [-S(a/2) + C(a/2)v]/2 (6b) Moreover the composite rotation is

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Ill 9l h l = C(e/2)C(a/2) S(e/2)S(o/2) (u.v) + C (8/2 ) S (a/2 ) v + S (8/2) C(a/2)u + S(8/2)S(a/2) (uxv) (7) A straightforward calculation shows that fi(gihi) = 6(g 1 )
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112 Proof: Consider a rotation of 0 degrees around u; its quaternion representation is g ± = C (0/2 ) + S ( 0/ 2 ) u , (eqn. 1) and its corresponding dualized mapping will be represented by g x = g = [ C (0/2 ) + S (0/2 ) u] + ed[-S (0/2) + C(0/2)u]/2 (eqn. 4) where the dual variable 0 = 0 + ed. Comparing this expression with the biquaternion representation of a Euclidean mapping (eqn. 3.7.17) yields t*g x = d [ -S (0/2 ) + C (0/2 ) u] (9) where t* is associated with the displacement of the fixed point, considered as the origin. Solving for t*, one obtains t* = t*g 1 g 1 “ = d[-S (0/2) + C(0/2)u] [C(0/2) S (0/2 ) u] = d[-S(0/2)C(0/2) + S(0/2)C(0/2) (u.u) S 2 (0/2 ) u + C 2 (0/2 ) U S (0/2) C(0/2)UXU] = d[S 2 (0/2) + C 2 (0/2) ]u = du (10) Hence, the displacement of the fixed point is along the axis of rotation, and the mapping is a screw motion along an axis parallel to u and passing through the fixed point. It is important to recognize that the set of screw motions along arbitrary axes passing through a fixed point is a subset of the Euclidean group that is not closed under composition; thus, they do not form a subgroup. A simple example of this situation is given now. Consider the screw motions, with displacements along axes X and Z passing through a fixed point, regarded as the origin, which are given by the biquaternions

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113 g = [C (0/2) + S(6/2)k] + e [-S (0/2) + C(0/2)k]z/2 (11a) h = [C (a/2) + S(a/2)i] + e[-S(a/2) + C(a/2)i]x/2 (lib) Consider now the composite displacement gh = [C(0/2) C(a/2) +C(6/2) S (a/2) i + S (0/2 ) S (a/2 ) j + S (0/2) C(a/2) k] + € { [~zS (0/2) C(a/2) xC (0/2 ) S (a/2 ) ] + [-zS(0/2)S(a/2) + xC ( 0/ 2 ) C ( a/ 2 ) ] i + [zC(0/2)S(a/2) + xS(0/2)C(a/2) ] j + [zC(0/2)C (a/2 ) xS(0/2)S(a/2) ]k}/2 (12) Solving for the displacement of the origin, one obtains t* = xC(0) i + xS ( 0 ) j + zk (13) The unit vector along the rotation axis and the direction of the displacement of the origin are respectively C(0/2)S(a/2)i + S (0/2 ) S (a/2 ) j + S (0/2 ) C (a/2 ) k u = 1 (14a) [C 2 (0/2)S 2 (a/2) + S 2 (0/2) S 2 (a/2) + S 2 (0/2 ) C 2 (a/2 ) ] ^ and xC(0)i + xS(0)j + zk t = — (14b) (x 2 + z 2 ) * The scalar product of these vectors is given by U.t = [xC(0/2)S(a/2) + zS(0/2)C(a/2) ]/D (15a) where D = [C 2 (0/2) S 2 (a/2) + S 2 (0/2)S 2 (a/2) + S 2 (0/2)C 2 (a/2) (x 2 + z 2 )^ (15b) For 0 = a = 90 0 , u.t = [ (x + z)/3] * (16)

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114 It is thus evident that t is parallel to u, i.e. u.t = 1, only for special values of x and z. If one considers serial kinematic chains with the usual Hartenberg and Denavit convention (Denavit and Hartenberg [1955]), it is easy to recognize that the displacement vector t* given by equation (13) is precisely of the type obtained when one passes from one reference system to the successive system. 5 . 3 Statement and Proof of the Principle of Transference Consider the skeletal of a spatial kinematic chain shown in figure 5.1 labelled with the Hartenberg and Denavit notation (Denavit and Hartenberg [1955]). The axis Z^ is directed along the axis of the i-th pair, and the axis is directed along the common perpendicular to the axes and Z ± . The corresponding skeletal spherical chain (Duffy [1980]) is shown in figure 5.2. Since both the reference systems are right-handed, for clarity only the X and Z axes are drawn. Furthermore, in the equivalent spherical chain, two reference systems are equal if, and only if, pairs of corresponding axes coincide. In this development, the axes X and Z will be used for purposes of determining the equality of reference systems. It is well known that, in the spherical chain, the (i-l)-th reference system is made to coincide with the

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115 z i+l Figure 5.1 Skeletal Spatial Kinematic Chain.

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116 Figure 5.2 Skeletal Spherical Chain Associated with the Spatial Chain of Figure 5.1

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117 i-th reference system by performing first a rotation of degrees around the X-^_^ axis, followed by a rotation of 0^ degrees around the axis. Equivalently, using the quaternion representation and the symbolic notation ( x i' Y i' z i) = 9i^i { ^i— l , Y i— i , 2 j.— i } = [C(0i/2) + S(0i/2)k] t c ( a i-l,i/ 2 ) + S(a i _ 1/i /2)i]{X i _ 1 ,Y i . 1 ,Z i _ 1 } (1) Consider now a serial spatial chain with n pairs. Both, the spatial chain and its associated spherical chain, have n reference systems which will be numbered from 0 to n-1. It is convenient to add to the spatial and spherical chains an n-th reference system defined by {X n , Y n , Z n } = {X 0 , Y 0 , Z 0 } (2) In the spherical chain, there are two alternative paths which transform the zero-th reference system into the n-th reference system. One path is given by equation (2) . The other path is obtained by recursive application of equation ( 1 ) ( x n' Y n* z n) = (9n^n) ( x n-l' Y n-1' z n-l) = ( • = (9n h n) (9n-l h n-l) • • • (92 h 2) ( x l^ Y l> z l> = (9n h n) (9n-l h n-l) • • • (*?2 h 2) (9i h i) { x o» Y 0' z oH 3 ) Substituting equation (2) into equation (3) yields ( x o, Y 0 , Z 0 } = (g n h n ) (g n -i h n-i) — (g 2 h 2 ) (9i h i) {x 0 , y 0 , z 0 }

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118 or 1 =(9n h n) (9n-l h n-l) • • • (^2 h 2) (9l h l) ( 4 ) This is the closure equation 2 for the associated spherical chain; hence, it is related to the orientation of the reference systems of the original spatial chain. The homomorphism & : H u 2 IH u , given by equation (1.4) , when applied to equation (4) yields 1 + eO = 6(1) = S[ (g n h n ) (g n -i h n-l) . . . (g 2 h 2 ) (gih x ) ] = & (9n h n) 5 (9n-l h n-l) • • • 5 (92 h 2) s (9l h l) = S (g n ) 6 (h n ) 5 (g n _!) 6 (h n _!) . . . 6 (g 2 ) 6 (h 2 ) 6 ( 9l ) 5 (h x ) = [ (gn +e 90n) ( h n +eh On) 1 [ (^n-l+^O , n-l) ( h n-l +h O , n-l) 1 • • • [ (g2 +£ 902) ( h 2 +eh 02 ) ] [ (gi +£ goi) ( h l +eh Ol) ] ( 5 ) It is important to recognize that the homomorphism S assures this equation is satisfied, and nothing further is required. For completeness, a physical interpretation of the real and dual parts of this dual equation is now given. The real part of equation (5) yields the closure equation of the associated spherical chain, equation (4) . The dual part yields 0 = (gn h on + gon h n) (gn-i h n-i) • • • (g 2 h 2 ) (gi h i) + (g n h n) (gn-i h o,n-i + go , n-i h n-i) • • • (g 2 h 2 ) (gi h i) + ... + (gn h n) (gn-l h n-l) • • • (g 2 h 02 + g 02 h 2 ) (gi h i) + (g n h n) (gn-i h n-i) • • • (g 2 h 2 ) (gi h oi + goi h i) ( 6 ) 2 0ther alternative forms of the closure equation, also known as subsidiary equations, can be obtained by preor post-multiplying equation (4) by the adequate inverses of (gihf), see Duffy [1980].

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119 However, equation (3.7.17) shows that 9i h 0i + 9oi h i “ (t*i/2)g i h i (7) with t*i = XjCfeiJi + x-jSCejJj + Zik (8) where {i, j, k} are the unit vectors along the positive axes of the i-th reference system. Therefore, equation (6) can be expressed in the form 0 = (t*n/ 2 ) (g n h n ) (gn-ihn.i) . . . (g 2 h 2 ) (gih x ) + (9n h n) ( t *n-l/ 2 ) (9n-l h n-l) • • • (^2 h 2) (9l h l) + ••• + (9n h n) (9n-l h n-l) • • • (t*2/ 2 ) (92 h 2) (
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120 0 = t T = t* n (g n h n ) . . . (g 2 h 2 ) (gihi) (g 1 h 1 )~ 1 (g 2 h 2 ) 1 • • • (9n-l^n-l) 1 + (^n^n) (t* n -i) (Sn-l^n-l) * • • (g2 h 2) (9l h l) (gi h l)" 1 ( < ?2 h 2)" 1 * • • (gn-l h n-l)" 1 (gn h n)" 1 + ••• + (g n h n) ( 9 n-i h n-i) • • • (t* 2 ) (g2 h 2) (gi h i) (gi h i) _ 1 (g2 h 2 ) _1 • • • (gn-l^n-l) ^(gn^n) ^ + (gn^n) (gn-l^n-l) * * * (92^2 ) (t*x) (gihi) (gih 1 ) _1 (g 2 h 2 )" 1 . . . (g n -i h n-l) _1 (g n h n) -1 Thus 0 = tip = t* n + (g n h n ) t* n -i (gn^n) ^ + ••• + (gn^n) (gn-l^n-l) . . . (g3 h 3) t *2(g 3 h 3) _1 . • • (g n -i h n-l)“ 1 (gn h n)” 1 + (gn h n) (gn-l h n-l) • • • (g 2 h 2 ) t *l(g 2 h 2 )" 1 • • (gn-i h n-i)" 1 (gn h n)" 1 ( 10 ) The right hand side of the equation (10) represents the vector sum of the sides of the spatial polygon formed by the origins of the reference systems attached to the spatial chain together with the axes X and Z^, and their intersections. Thus the result is indeed zero and it is independent of the reference system in which the equation is represented. In particular equation (10) is expressed in terms of the n-th reference system. It is however possible to express the closure condition in many other reference systems. Thus the following result has been shown. Proposition 1 (Principle of Transference) . Any valid equation involving a finite product of unit quaternions of the form g^ = C(9j_/2) + S(9j_/2)k or h-^ = C(aj_/2) + S(aj_/2)i remains valid when the variables 9^ and are replaced by the dual variables 9^ and

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121 Since any non-zero quaternion can be reduced to a unit quaternion, this limitation is superfluous. This statement of the principle permits transfering a valid equation involving quaternions into a valid equation involving biquaternions. These biquaternions can be regarded as quaternions whose components are dual numbers. Moreover, it is important to consider these quaternion and biquaternion expressions as representations of Euclidean mappings, a rotation around a fixed point in the case of quaternions, and a spatial displacement in the case of biquaternions. Therefore equations relating quaternions or biquaternions indicate that an arbitrary element (point or line) of the Euclidean space will have the same image under either the mapping of the left hand side of the equation or the mapping of the right hand side of the equation. Furthermore, the principle can be translated into any other pair of representations of rotations around a fixed point and spatial displacements. For instance, consider the isomorphism between the quaternion representation and the three-dimensional orthogonal representation of the rotation, the last one denoted by 0(3) ; then *[C(0i/2) + S(0i/2)k] = •H CD 1 -S©i 0 ce^ 0 = R0 j_ (11) _0 0 1 _ and

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122 *[C(a=i_i,i/2) + S(a i _ 1/i /2)i] = 10 0 0 Ca i-i r i _Sa i-l,i 0 Sai-i,! Ca i-i,i = Rai-1 , i (12) Moreover, the isomorphism ' between the biquaternion representation and the 4x4 matrix representation, with k = 1 (see section 3.3), of spatial displacements yields * i {[C(e i /2) + S(e i /2)k] + ez i [-S(0 i /2) + C(0.j/2)]k 00^ -S0i 0 0 S©i C0i 0 0 0 0 0 0 1 Z 0 1 = R0 i , z i (13) and 'J>'{[C(a i _ 1/i /2)+S(a i _ 1/i /2)i]+ex i [-S(a i _ lfi /2)+C(Q! i _ 1/i /2)i] } 10 0 xi 0 Ca i-i,i " Sa i-l,i 0 0 Sa i-i,i Ca i-l,i 0 1 — Rc*i— i f i » ^i (14) Then the result can be extended to this pair of representations as indicated in the following diagram *-1 s o V ' 0(3) IH U 2 IH u Ei(4), and the result can be restated as follows Proposition 2. Any valid equation involving a finite product of matrices of the form R0^ or Rai_ lf i remains valid

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123 when the matrices are replaced by the matrices R©i,z^ and RQ: i-l,i' x i respectively. Proposition 1 and its equivalent proposition 2 deal with the dualization of the mapping. The dual equations used in the analysis of spatial mechanisms (Duffy [1980]) are obtained by applying the dualized equation (5) , or one of their equivalent forms called subsidiary equations, to particular vectors. These vectors are usually unitary, and frequently they are the unit vector of the reference systems associated with spatial chain (see Figure 5.1). In what follows, it is shown that the dualized equation (5) can be also applied to arbitrary unit vectors yielding an equation related to the image of a unit line vector whose direction is precisely that of the original unit vector. Assume u is an arbitrary unit vector; then u = (Cr x , Cr 2 , Cr 3 ) (15) with C t± 2 + Cr 2 2 + Cr 3 2 = 1 (16) The dualization of the unit vector and its condition yields u = u + eu 0 = (Ct^ ed^T-L, Cr 2 ed 2 Sr 2 , Cr 3 ed 3 Sr 3 ) (17) and ( CTi ed-^Sr 2 + (Cr 2 ed 2 Sr 2 ) 2 + (Cr 3 ed 3 Sr 3 ) 2 = 1 + eO (18) This last condition can be decomposed into real and dual parts, the real part is given by the original condition, equation (16), and the dual part is given by -2(d 1 Cr 1 Sr 1 + d 2 Cr 2 Sr 2 + d 3 Cr 3 Sr 3 ) = 0 (19)

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124 It is easy to recognize equation (15) as the Pliicker or quadratic identity. This condition ensures that u represents a unit line vector. The direction of the unit line vector u is precisely that of the unit vector u. Consider now an arbitrary unit quaternion g x = C(9/2) + S(e/2)v (20) Its dual is given by g = g x = [C (9/2) + S(9/2 )y] + et[-S(9/2) + C(0/2)v]/2 (21) It has already been shown (see sections 3.7 and 5.2) that g represents a screw motion along the axis given by v, and the magnitude of the associated linear displacement is t. Then g can be represented by g = g x + etvg 1 /2 (22) Finally applying to u the biquaternion equivalent of the induced line transformation introduced in section 3.8 one obtains u' = u' + eu 0 ' = [g x + etvg 1 /2][u + eu 0 ] [g^ eg 1 _ tv/2] = giMg! + efgiUog-L + [tv(g 1 ug 1 ") (giug-L ) tv]/2 } (23) It is easy to identify u' as the image of the line u under the screw motion represented by g. Thus, the following result has been proven Proposition 3 . Let u be an arbitrary unit vector and let g-L = C (0/2 ) + S(0/2)v be an arbitrary unit quaternion; then the action of the biquaternion g, obtained by dualizing the quaternion g^, over the unit line vector u, obtained by dualizing the unit vector u, is the line transformation

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125 induced by the Euclidean mapping represented by the biquaternion g. It is highly possible that this feature may well be advantageously employed in the analysis of spatial mechanisms .

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CONCLUSIONS The results obtained in this work demonstrate that a deeper study of the properties of the Euclidean group, together with their representations provides invaluable information that can be applied to the analysis of kinematic chains. The theory of group representations yields the required fundamentals for a unified theory of spatial kinematics, and a natural way to relate the distinct representations of the Euclidean group that are used in spatial kinematics. Furthermore, the analysis of the Euclidean group regarded as a Lie group produces important results concerning the nature of its Lie algebra considered as an orthogonal space, as well as the nature of the Euclidean group itself regarded as a semi-Riemannian manifold. The analysis and proof of the principle of transference, is a significant result that previously defied solution. It is also an illustrative example of the power of group theory when it is applied to kinematics. Some of the topics that deserve further examination are 1. The relation between the orthogonal structure of the Lie algebra of the Euclidean group and the classification of screw systems. 126

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127 2. A revision of the methods of analysis of spatial mechanisms based on the systematic application of the principle of transference under the light of the proof and the use of biquaternion representation. 3. After the description of the Euclidean group as a semi-Riemannian manifold, some problems of the theory of manipulators posed as "given an initial position of the end effector determine if the end effector can reach another pre-specif ied position", can be reformulated as problems in global geometry.

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APPENDIX A CLIFFORD ALGEBRAS Of the mathematical subjects that this work has required a prior knowledge, Clifford Algebras is, perhaps, the one with the most scattered information. This appendix attempts to gather the pertinent facts in concise form; therefore, no proofs are even sketched, and only the results pertinent to this work are quoted. The only departure from this policy occurs in the last section in which very specific results concerning the Clifford algebra of IR 3 are proved. Since the results are thought to be new, or not widely known, the complete development is given. In the study of Clifford algebras, it is specially important to be aware of small differences among the definitions of the Clifford product that could produce results apparently in conflict. The standard reference is Porteous [1981]; hence, only results taken from other sources are explicitly indicated as such. A. 1 Definition of Clifford Algebras and their Dimensions Definition 1. Let X be a finite-dimensional real orthogonal space; then the associative algebra A, with unit, containing isomorphic copies of X and IR as linear subspaces in such a way that u 2 = -(u,u) VueX 128

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129 and generated as an algebra by {1} and X is called the Clifford algebra of X. If X = a Clifford algebra of X is denoted by Proposition 1. Let v,weX; then (v, w) = (vw + wv) / 2 , where the products in the right hand side represents the product in the Clifford algebra, also called Clifford product. In particular v,weX are orthogonal if, and only if, vw = -wv. Proposition 2. Let veX; then v is invertible in A if, and only if, v is invertible with respect to the symmetrical bilinear form. Moreover v" 1 = -v/(v,v) Proposition 3. Let X be a finite-dimensional real orthogonal space with an orthonormal basis {e^ |l < i < n}, where dim X = n, and let A be a real associative algebra with unit 1, containing IR and X as linear subspaces. Then u 2 = (u,u) VueX if, and only if, ei 2 = -(e if ei) V 1 < i < n and e^ej + eje^ = 0 V 1 < i,j < n with i f j This result provides necessary conditions, upon an orthonormal basis of X, for A to be a Clifford algebra of X. Definition 2. An orthonormal subset of a real associative algebra A with unit 1 is a linearly independent

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130 subset S of mutually anticommuting elements of A, the square of any element of S being 0,1,-1. The previous definition is generated extending the concept of an orthonormal subset in an orthogonal space; if SCX, then S is also an orthonormal subset of X. Proposition 4. Let X be the linear span of an orthonormal subset S of the real associative algebra A. Then, there is a unique orthogonal structure for X such that for all ueS, u 2 = -(u,u). Furthermore, if S is of the type (r,p,q), X with this structure is isomorphic to R r 'P' < 3. If S also generates A, then A is a Clifford algebra of X. Definition 3. Let (e^ |l < i < n} be an ordered subset of elements of an associative algebra A. Let I be a naturally ordered subset of (1,2, ...,n}; then Proposition 5. Let A be a real associative algebra with unit 1 (identified with leIR) , and suppose that (e^ |l < i < n} is a subset of n elements of A generating A as an algebra such that then, the set {ffej | I c {1,2, — ,n}} spans A as a vector space. It is important to recognize that given an orthonormal basis {e 1 ,e 2 , ... ,e n } of an orthogonal vector space X, the k where ijel and IIe(j) = 1 where 4> is the null set e-^ej + eje^ elR V 1 < i,j < n;

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131 basis satisfies the conditions of the previous proposition, for e-^ej +eje^ = 0 V 1 < i,j < n with i f j, and e i e i + e i e i = 2ej[ 2 e { -2 , 0 , 2 } V 1 < i,j < n. Further, they generate A as an algebra; thus, they are a natural choice for constructing the Clifford algebra of X. Corollary 6. Let A be a Clifford algebra of a n-dimensional orthogonal space X; then dim A < 2 n . Proposition 7. Let A be the Clifford algebra for a n-dimensional non-degenerate orthogonal space (X = IR 0 '?'^ with n = p + q) ; then dim A = 2 n or 2 n_1 , the lower value being a possibility only ifp-q-l=0 mod 4, in which case n is odd and ITe^i^, . . . ,n} = +1 or -1 f° r an Y ^ as i c orthonormal frame of X. Proposition 8. 1 Let X be a real orthogonal space isomorphic to rI/P' 9, with n = p + q, and {e^ | 0 < i < p+q} be an orthonormal basis of X such that e^ 2 = 0 for i = 0 e-L 2 = +1 for 1 < i < p e^ 2 = -1 for p+l
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132 2. dim A = 3(2 n_1 ) if n is odd, and e 0 and e 0 e 1 ...e n are linearly dependent; a necessary condition for this is that p q 1 = 0 mod 4 . Proposition 9. Let A and B be two 2 n -dimensional Clifford algebras for an n-dimensional real orthogonal space X. Then A = B. Because of this result, one calls such an algebra a universal Clifford algebra for X. A. 2 Main Involution and Conjugation Definition 1. The algebra involution of A induced by the orthogonal involution -l x : X -+ X -l x u = -u V ueX will be denoted by * : A -+ A ~a = a V aeA and it is called the main involution of A; a is referred as the involute of a. The algebra anti-involutions of A induced by the orthogonal involutions l x and -l x will be denoted by ~:A-*A 'a = a V aeA and : A -* A “a = a” V aeA They are called respectively reversion and conjugation; a is called the reverse of a, and a” is called the conjugate of a.

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133 Proposition 1. The main involution induces a direct sum decomposition of A = A 0 © A 1 with A 0 = {aeA | a = a} and A 1 = {aeA | a = -a} Moreover, A 0 is a subalgebra of A, referred to as the even Clifford algebra of A. Proposition 2. 2 Assuming p + q > 1, then K r,p,q° = K r,p,q-1 when <3 ^ 1 and K r,p,q° = K r,p-l,q when P * 1 A. 3 The Clifford and Spin Groups Proposition 1. Let g be an element of A such that gxg _1 eX VxeX; then the map PX, g : X X P x , gX = gxg" 1 is an orthogonal automorphism of X. In particular, if geX, then Px,g i s the reflection of X in the hyperplane [K(g) Proposition 2. The set r(X) = {geA | g is invertible, and gxg -1 eX VxeX} is a subgroup of A, with respect to the Clifford product, and it is called the Clifford group of X. This definition of the Clifford group is equivalent to that given by Brooke [1980] r(X) = {geA | g is invertible, and g -1 xgeX VxeX} 2 This result is due to Brooke [1980].

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134 Proposition 3. The set r°(X) = T(X) n A 0 is a normal subgroup of the Clifford group, called the even Clifford group. Further, r(X)/r°(X) = Z 2 . Proposition 4. The sets IR + = {AelR | X > 0} and R* = {AeIR|X ^ 0} are normal subgroups of T(X), and of r°(X). The corresponding quotient groups are denoted by and called respectively Gi + (r) = T(X)/R + Pin (X) Gi(r) = T(X)/R* Projective Clifford group Gi+(r°) = r°(x)/R + Spin (X) Gi(r°) = r°(x)/iR* Even projective Clifford group Furthermore , IR + is a normal subgroup of R* with IR*/1R + = Z 2 . Corollary 5. Gi+fD/G^r) = z 2 G 1 +(r°)/G 1 (r°) = z 2 The following four results, attributed to Dieudonne [1955], provide the relationship between these subgroups of the Clifford algebra. A, of a non-degenerate orthogonal space X, and the group of orthogonal automorphisms of X. Proposition 6. Any orthogonal transformation t : X -*• X, of a non-degenerate finite-dimensional orthogonal space X, is expressible as the composite of a finite number of hyperplane reflections of X. The number of reflections is not greater than 2dim X, or if X is positive-definite dim X. Corollary 7. Any orientation preserving orthogonal transformation t : X -*• X, of a non-degenerate

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135 finite-dimensional orthogonal space X, is expressible as the composite of a finite even number of hyperplane reflections of X. The number of reflections is not greater than 2dim X, or, if X is positive-definite dim X. Proposition 8. Let X be a non-degenerate orthogonal space; then the map PX : r ( x ) °( x ) Px(9) = PX,g is a surjective homomorphism with ker p x = R*. Thus G^r) = T(X)/R* = 0(X) For the orientation preserving orthogonal automorphisms of X, there is a corresponding proposition. Proposition 9. Let X be a non-degenerate orthogonal space; then the map P X ; r° (X) SO (X) Px (g) = p x , g is a surjective homomorphism with ker p x = R*. Thus G^r 0 ) = r°(X)/R* = so(x) Hence Spin (X) /SO (X) = Z 2 This is one of the reasons why one refers to the spin groups as double covers of the orthogonal groups. In this initial characterization, the spin groups are quotient groups of the Clifford group. In what follows, the norm mapping will provide an easier characterization of the spin groups. Definition 1. Let A be a Clifford algebra of an orthogonal space X; then the norm mapping is defined by N : A -*• A N (a) = a~a VaeA

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136 Proposition 9. Let X be a non-degenerate orthogonal space; then 1. For any ger(X), N(g)elR, 2. N ( 1) = 1, 3. for any g,g'er(X), N(gg') = N(g)N(g'), and 4. for any geT(X), N(g) f 0 and N(g -1 ) = (N(g)) -1 . Proposition 10: The map * : (ger°(X) | |N(g) | = 1} Spin(X) * (g) = R + {g} is a group isomorphism. Due to this result, Spin(X) is identified with the domain of the mapping above, a subgroup of r°(X). Finally, the condition of the previous proposition is simplified for low-dimensional orthogonal spaces. Proposition 11. Let X be a non-degenerate orthogonal space with dim X < 5; then Spin (X) = {geA° | |N(g) | = 1} A. 4 The Clifford Algebra of 1R3The orthogonal space IR 3 , also denoted by IR 0 ' 0 ' 3 , is a three-dimensional vector space endowed with the symmetric bilinear form given by ( , ) : IR 3 xIR 3 -*• IR ((x 1 ,x 2 # x 3 ) , (y 1 ,y 2 ,y 3 )) = xiYi + x 2 y 2 + x 3 y 3 (i) If {e 1 ,e 2 ,e 3 } denote the canonical orthonormal basis of IR 3 , then a basis for IR 3 , the universal Clifford algebra of IR 3 , will be {e^; e 1 ,e 2 ,e 3 ; e 2 e 3 , e^ , e^ ; e 1 e 2 e 3 }.

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137 As already indicated, the main involution induces the linear decomposition of IR 3 given by IR 3 = IR3 0 © IR3 1 , ( 2 ) where IR 3 0 denotes the even Clifford algebra of IR 3 , and one of its bases is { e(j), e 2 e 3 , 6363^ , ) • According to proposition 2.2, IR 3 0 = IR 2 = IH; in fact, it is straightforward to prove that the corresponding algebra isomorphism is defined by $ : K3 0 !H $(e ^ ^ 12 3 ^ Finally, four important results concerning the behavior of elements in ER 3 will be proved. Since the results do not appear in the mathematical literature, their complete development is shown here. Proposition 1 . Let geD^ 1 be given by g = aoe^^ + a^ + a 2 e 2 + a 3 e 3 ( 4 ) Then gg~ = a 0 2 + a x 2 + a 2 2 + a 3 2 = g"g ( 5 ) In particular gg“ = 0 g = 0 (6) Proof: The conjugate of g is

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138 g = 30636263 a 2 e 2 a 3 e 3 ( 7 ) A lengthy but uninteresting series of computations prove the result. Proposition 2 . Let g,heIR 3 0 ; then g“h h“g = 0 o h = Xg for some XeIR (8) Proof: Assume h = Xg; then g~h h“g = g-(Xg) (Xg) "g = X(g~g g~g) = 0 Assume g~h h~g = 0 ; since IR 3 0 = IH, where IH is the skew field of the quaternions, it is possible to write g = a 0 + a and h = b 0 + b ( 9 ) where a 0 ,b 0 ,a and b are respectively the scalar and vector parts of g and h; then g” = a 0 a and h“ = b 0 b Thus and g“h = a 0 b 0 + a.b + a 0 b b 0 a axb ( 10 ) ( Ha) h _ g = a 0 b 0 + a.b + b 0 a a 0 b bxa (lib) where . and x stands for the usual dot and cross vector products respectively; hence g"h h g = 2(a 0 b b 0 a + bxa) =0 (12) Assuming that a and b are linearly independent, then a,b and bxa are linearly independent, and the condition implies l 0 = b 0 = 0 and bxa = 0 which is a contradiction to the linear independency of a and b. Therefore b = Xa for some XeIR, and bxa = 0 ; then g“h h~g =0 o a 0 Xa b 0 a = 0 <* (a 0 X b 0 ) a = 0 ( 13 )

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139 Thus bg = Xag and b = Xa. Notice that if a = 0, then trivially b = 0, and h = Xg. Further, if g = 0, or h = 0, then trivially g = Oh or h = Og respectively, and the result holds. Proposition 3. Let geIRg 0 with g f 0, and heIRg 1 ; then g“h + h“g = 0 h = tg for some teIR 3 (14) Proof: Assume h = tg; then g“ (tg) + (tg) “g = g"tg + g“t“g = g“tg g"tg = 0 Assume g“h + h”g = 0, and consider (g“h)“ = h“g; then g~h + h“g = (g"h) + (g~h)“ (15) Since g“heIR 3 1 , then g“h = kg + kg , where kg = bgeg + b 2 e 2 +bge 3 and kg = bgegegeg (16) Thus (g”h) “ = (kg + k 3 ) “ = kg + k 3 “ = -kg + k 3 (17) Hence 0 = g“h +h“g = kg + kg -kg + kg = 2kg <* kg = 0 (18) A simple calculation reveals that if g = a 0 e(j) + age 2 e 3 + a 2 e 3 eg + a 3 e 1 e 2 (19a) and, h = Cgegegeg + c 3 eg + c 2 e 2 + Cgeg, (19b) then kg = a 0 Cg 3gCg a 2 C 2 agCg (20) The problem is now formulated as follows: Given g and h as in the equations above, together with a 0 c 0 " a l c l ” a 2 c 2 “ a 3 c 3 = 0 ( 21 ) Then

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140 h = tg for some teIR 3 (22) Assume now, that t = t;^! + t 2 e 2 + t 3 e 3 (23) Then the condition (22) leads to a system of four equations with three unknowns C 1 = a 0 t l + a 3 t 2 " a 2 t 3 (24a) c 2 = ” a 3 t l + a 0 t 2 + a l t 3 (24b) c 3 = a 2 t l “ a l t 2 + a 0 t 3 (24c) eg = a^t^ + a 2 t 2 + a 3 t 3 ( 24d) Solving for the t^'s from the first three equations, one obtains t^=[c^(aQ^ + a^ 2 ) + c 2 (a 2 a^ ” a 0 a 3) + c 3( a 3 a l + a 0 a 2) 3/^ (25a) t 2 =[ c i( a i a 2 + a 0 a 3 ) + c 2 (a 0 2 + a 2 2 ) + c 3 (a 2 a 3 a 0 a x ) ]/5 (25b) t 3 =[ c i( a i a 3 a 0 a 2 ) + c 2 (a 2 a 3 + aga^ + c 3 (a 0 2 + a 3 2 )]/6(25c) where S = a 0 (ag 2 + a^ 2 + a 2 2 + a 3 2 ) (26) Since g f 0, then a 0 2 + a x 2 + a 2 2 + a 3 2 f 0. Assuming a 0 f 0, and substituting in the last equation, the system will have a solution if, and only if, c 0 =a 1 [c 1 (a 0 2 + a-L 2 ) + c^a^ a 0 a 3 ) + c 3 (a 3 a! + a 0 a 2 ) ]/& + a 2 C c l ( a l a 2 + a 0 a 3 ) + c 2 ( a 0 2 + a 2 2 ) + c 3 ( a 2 a 3 “ a 0 a l) 3/<* + a 3 [c 1 (a 1 a 3 a 0 a 2 ) + c 2 (a 2 a 3 + aga x ) + c 3 (a 0 2 + a 3 2 )]/<5(27) or, rearranging (ag 2 + a^ 2 + a 2 2 + a 3 2 ) (agCg a^c^ a 2 c 2 a 3 c 3 ) = 0 (28) Thus provided g f 0, the condition reduces to a 0 Cg a^c^ a 2 c 2 a 3 c 3 = 0 (eqn. (21))

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141 Hence, the result follows. If a 0 = 0, equations (21) and (24) become a l c l + a 2 c 2 + a 3 c 3 = 0 ( 29 ) and C 1 = a 3 t 2 “ a 2 t 3 ( 30a ) C 2 = ” a 3^1 *" a l^-3 (30b) C 3 = a 2 t]_ — a 1^2 (30c) c 0 = a l t l + a 2 t 2 + a 3 fc 3 ( 30d ) Solving for the t^'s using equations (30a, b,d), one obtains t ± = [ 00 ( 3 ^ 3 ) c 1 (a 1 a 2 ) c 2 (a 2 2 + a 3 2 )]/5 (31a) t 2 = [c 0 (a 2 a 3 ) + c 1 (a 1 2 + a 3 2 ) + c 2 (a 1 a 2 )]/
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142 C 1 = “ a 2 t 3 (33a) c 2 = a l t 3 (33b) c 3 = a 2 t 1 a]^t 2 (33c) c 0 = a l t l + a 2 t 2 (33d) The subsystem formed by (33c) and (33d) has a solution if/ and only if, its coefficient matrix is non-singular; viz. a l 2 + a 2 2 t 0 (34) Since g f 0, and by assumption a 0 = a 3 = = 0, then this condition evidently holds. Furthermore, solving (33a) for t 3 t 3 = c 1 /a 2 (35a) Assuming a 2 f 0, the system will have a solution if, and only if/ c 2 = a l(“ c l/ a 2 ) and rearranging, the expected condition is obtained a l c l + a 2 c 2 = 0 ( eqn . (33)) The assumption a 2 = 0, coupled with a 0 = a 3 = 0, simplifies equations (21) and (24) up to a-j^ = 0 (36) and o II i— i 0 (37a) c 2 = a l fc 3 (37b) c 3 = “ a l t 2 (37c) c 0 = a l t l (37d) However, since g f0 and a 0 = a 2 = a 3 = 0, then a;L f 0;

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143 hence c ± = 0. Thus, in this final case, the system has clearly a solution. Proposition 4. Let geIR 3 0 with = 0, and heIR 3 c IR 3 1 ; then gh hgeIR 3 C IR 3 1 . Proof: Let g = g 2 3 e 2 e 3 + 931 e 3 e l + < 3l2 e l e 2' and h = h]^ + h 2 e 2 + h 3 e 3 ; then gh = (g 23 h x + g 31 h 2 + g 12 h 3 ) e 1 e 2 e 3 + (g 31 h 3 gi 2 b 2) e l + (9l2 h l " 923 h 3) e 2 + (923 h 2 “ 931 h l) e 3 (38a) and hg = (g 2 3 h l +
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APPENDIX B A PHYSICAL INTERPRETATION OF THE ADJOINT TRANSFORMATION The invariance of a symmetrical bilinear form, defined on the Lie algebra e(3), under the adjoint transformation induced by an arbitrary element of the Euclidean group E(3) is central to the process of endowing the Euclidean group with the structure of a semi-Riemannian manifold (see Section 4.4). Although Loncaric [1985] indicates the relationship between invariance under right and left translations with the invariance with respect to an arbitrary selection of a body based and an inertial frame of reference respectively, here a simpler and more direct interpretation of the concept of invariance under the adjoint representation will be shown. In chapter 3, the representation of the Euclidean group was expressed in the form, g = g x + etg^/2 (eqn. (3.5.5)) where g 1 =C(0/2)+S(9/2) (u 2 3 e 2 e 3 + u^e^ + u 12 e 1 e 2 ) (eqn. (3.4.36)) with u 23 2 + u 31 2 + u 12 2 = 1 (eqn. (4.2. (la)) and t = t^e 3 + t 2 e 2 + t 3 e 3 ( eqn . (4.2. lb ) ) It is necessary to arbitrarily select a reference frame attached to the rigid body under consideration. The location 144

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145 of this moving frame with respect to another arbitrarily selected fixed reference frame determines an element of the Euclidean group. It was also shown that g^ is related to the orientation of the moving reference frame with respect to the fixed one, and t is related to the location of the origin, 0, of the moving reference frame with respect to the fixed reference frame. Then an arbitrary element of the Lie algebra e(3) ; i.e. a tangent vector at the identity, i, of the Euclidean group will be given by g(i) = G(i) (u 23 e 2 e 3 + u 31 e 3 e 1 + u 12 e 1 e 2 )/2 eft^Ue! + t 2 (i)e 2 + t 3 (l)e 3 )/2 (eqn. (4 . 2 . 5) ) The real part represents one half of the rotation rate of the moving frame with respect to the fixed frame. Furthermore, since the moving frame is attached to the rigid body, it also represents one half of the angular velocity of the rigid body. The dual part represents one half of the velocity of the origin of the moving frame, 0, with respect to the fixed frame. It is important to notice that since the tangent vector is computed at the identity, 1 , of the Euclidean group, both reference systems are instantaneously coincident. However, if a result generated by manipulating tangent vectors at the identity is to be regarded as a result of the rigid body as a whole, the arbitrary nature of the selection of the moving reference frame needs to be addressed. This

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146 problem is analogous to a problem which frequently occurs in abstract mathematics and which addresses the well-definiteness of a concept, when the definition makes use of an arbitrary choice. Assume now that a new moving reference frame with origin 6 and orthogonal directions (e^, e 2 , ©3} is chosen in the rigid body. Moreover assume that the transformation from the original reference system, 0, {e 3 , e 2 , ©3}* to the new one is accomplished by the element of the Euclidean group h = hi + edhi/2 (1) This implies that Aij_ei + X 2 ie 2 + X 2 j_s 3 = hi®ihi V i = 1 , 2,3 ( 2 a) and [h x + edh 1 /2][l + eO][h 1 “ eh 1 "d/2] = 1 + ed (2b) where d is the location of 0, in terms of the new system 6, (® 1 ' ® 2 ' ® 3 ) * Consider now h[g(i ) ]h _1 = [h x + edhi/2] (0(1) (u 23 e 2 e 3 + u 31 e 3 e 1 + u 12 e 1 e 2 )/2 e(t 1 (i)e 1 + t 2 (i)e 2 + t 3 (i ) e 3 )/2 } [h]^" eh! d/2] = h 1 {6(i) (u 23 e 2 e 3 + u 31 e 3 e x + u^e^) /2 }h 1 + e{h 1 [t 1 (t)e 1 + t 2 (i)e 2 + t 3 (i ) e 3 )/2]h 1 “ + [dh 1 {6(l ) (u 23 e 2 e 3 + u 31 e 3 e 1 + u 12 e l e 2) / 2 ) h i~ ~ hi(6 (i ) (u 23 e 2 e 3 + u 3 ie 3 e-L + Ui 2 eie 2 )/2 Jh^ d ]/2 ) ( 3 ) Observing that the angular velocity vector is independent of the moving reference system, as long as, it is attached to the rigid body, then the real part of the right

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147 hand side of equation (3) represents one half of the angular velocity of the rigid body in the new reference system. Furthermore, the dual part of the same expression corresponds (see proposition A. 4. 4) to one half of the sum of the absolute velocity of point 0 plus the relative velocity of point 6 with respect to the original moving reference system. Thus the result is the tangent vector at the identity of the Euclidean group when the moving reference system is determined by 0, {e lf e 2 , e 3 } . From these developments is evident that a result obtained by manipulating the elements of the Lie algebra e(3) will be a property of the rigid body if, and only if, the result is invariant under the adjoint transformation induced by an arbitrary element of the Euclidean group.

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149 Boothby, W.M. [1975], An Introduction to Differentiable Manifolds and Riemannian Geometry . Orlando: Academic Press, Inc. Bottema, 0. and Roth, B. [1979], Theoretical Kinematics , Amsterdam: North-Holland Publishing Company. Brackx, F. , Delanghe, R. , and Sommen, F. [1982], Clifford Analysis . Boston: Pitman Publishing Inc. Brand, L. [1947], Vector and Tensor Analysis . New York: John Wiley and Sons, Inc. Brockett, R.W. [1983], "Robotic Manipulators and the Product of Exponentials Formula," In Mathematical Theory of Networks and Systems . Ed. Fuhrmann P.A. , New York: Springer Verlag, pp. 120-129. Brooke, J.A. [1980], Clifford Algebras, Spin Groups and Galilei Invariance New Perspectives . Diss. The University of Alberta. Casey, J. and Lam, V.C. [1986], "A Tensor Method for the Kinematical Analysis of Systems of Rigid Bodies," Mechanism and Machine Theory , vol. 21, pp. 87-97. Chace, M. A. [1963], "Vector Analysis of Linkages," Journal of Engineering for Industry , vol. 85, pp. 289-296. Chevalley, C. [1946], Theory of Lie Groups, Vol. I , Princeton: Princeton University Press. Chisholm, J.S.R. and Common, A.K. [1985], Clifford Algebras and Their Applications in Mathematical Physics , Dordrecht: D. Reidel Publishing Company. Clifford, W.K. [1876], "On the Classification of Geometric Algebras," In Mathematical Papers W.K. Clifford , Ed. Tucker R. , London: Macmillan and Co. pp. 397-401. Clifford, W.K. [1882], "Preliminary Sketch of Biquaternions," In Mathematical Papers W.K. Clifford . Ed. Tucker, R. , London: Macmillan and Co., pp. 181-200. Crowe, M.J. [1985], A History of Vector Analysis , New York: Dover Publications, Inc. Crumeyrolle, A. [1974], Alaebres de Clifford et Spineurs , Tolouse: Universite Paul Sabatier.

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151 Herstein, I.N. [1968], Noncommutative Rings, The Carus Mathematical Monographs . vol . 15 , The Mathematical Association of America. Herstein, I.N. [1975], Topics in Algebra , 2nd ed. , Lexington Massachusetts: Xerox College Publishing. Herve, J.M. [1978], "Analyse Structurelle des Mecanismes par Groupe des Deplacements," Mechanisms and Machine Theory , vol. 13, pp. 437-450. Hestenes, D. [1966], Space-Time Algebra . New York: Gordon and Breach. Hestenes, D. [1971], "Vectors, Spinors, and Complex Numbers in Classical and Quantum Physics," American Journal of Physics . vol. 39, pp. 1013-1027. Hestenes, D. [1985], "A Unified Language for Mathematics and Physics," In Clifford Algebras and Their Applications in Mathematical Physics . Eds. Chisholm, J.S.R., and Common, A.K., Dordrecht: D. Reidel Publishing Company. Hestenes, D. [1986], New Foundations for Classical Mechanics , Dordrecht: D. Reidel Publishing Company. Hestenes, D. , and Sobczyk G. [1984], Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics . Dordrecht: D. Reidel Publishing Company. Hiller, M. and Woernle, C. [1984], "A Unified Representation of Spatial Displacements," Mechanism and Machine Theory , vol. 19, pp. 477-486. Ho, C.Y. [1966], "Tensor Analysis of Spatial Mechanisms," I.B.M. Journal of Research and Development , vol. 10, pp. 207-212. Hsia, L.M. and Yang, A.T. [1981], "On the Principle of Transference in Three-Dimensional Kinematics," J ournal of Mechanical Design , vol. 103, pp. 652-656. Hunt, K.H. [1978], Kinematic Geometry of Mechanisms , Oxford: Oxford University Press. Kaplansky , I. [1969], Linear Algebra and Geometry a Second Course . 2nd ed. , New York: Chelsea Publishing Company. Karger, A. and Novak, J. [1985], Space Kinematics and Lie Groups . Translated by M. Basch, New York: Gordon and Breach Publishers.

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152 Keler, M. [1970a], "Die Verwendung dual-komp lexer Grd/3en in Geometrie, Kinematik und Mechanik," Feinwerktechnik , vol. 74, pp. 31-41. Keler, M. [1970b], "Analyse und Synthese raumlicher, spharischer und ebener Getriebe in dual-komplexer Darstellung, " Feinwerktechnik . vol. 74, pp. 341-351. Klein, F. [1939], Elementary Mathematics from an Advanced Standpoint, Geometry . Translated from the third German edition by Hedrick E.R. and Noble C.A., New York: The Macmillan Co. Kupeli, D.N. [1987], "Degenerate Manifolds," Geometriae Dedicata, vol. 23, pp. 259-290. Lam, T.Y. [1973], The Algebraic Theory of Quadratic Forms , Reading Massachusetts: W.A. Benjamin, Inc. Lie, S. [1975], Sophus Lie's 1880 Transformation Group Paper , Translated by Ackerman, M. , comments by Hermann, R. , Brookline Massachusetts: Math Sci Press. Lin S.K. [1987], "Coordinate Transformations with Euler Parameters as a Quaternion -An Alternative Approach to Kinematics and Dynamics of Manipulators," In Proceedings of the IEEE International Conference on Robotics and Automation , pp. 33-38. Lipkin, H. [1985], Geometry and Mappings of Screws with Applications to the Hybrid Control of Robot Manipulators . Diss. University of Florida. Lipkin, H. and Duffy, J. [1985], "A Vector Analysis of Robot Manipulators," In Recent Advances in Robotics , Eds. Beni, G. and Hackwood, S., New York: John Wiley and Sons, pp. 175-241. Loncaric, J. [1985], Geometrical Analysis of Compliant Mechanisms in Robotics . Diss. Harvard University. McCarthy, J.M. [1986a], "Kinematic Mappings and Robotics," In IEEE 1986 International Conference on Robotics and Automation . pp. 1168-1173. McCarthy, J.M. [1986b], "Dual Orthogonal Matrices in Manipulator Kinematics," The International Journal of Robotics Research , vol. 5, pp. 45-51.

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154 Pengilley, C.J. and Browne, J.M. [1987], "Grassmann, Screws and Mechanics," In The Theory of Machines and Mechanisms. Proceedings of the 7th. World Congress , 1722 September 1987 Sevilla, Spain, Eds. Bautista, E., Garcia-Lomas, J. and Navarro, A., Oxford: Pergamon Press, pp. 183-186. Pennock, G.R., and Yang, A. T. [1984], "Applications of Dual-Number Matrices to the Inverse Kinematics Problem of Robot Manipulators," ASME Paper 84-DET-33. Phillips, J. [1982], Freedom in Machinery. Vol. 1: Introducing screw theory . Cambridge: Cambridge University Press. Phillips, R. and Zhang W.X. [1987], "The Screw Triangle and the Cylindroid," In The Theory of Machines and Mechanisms. Proceedings of the 7th. World Congress , 1722 September 1987 Sevilla, Spain, Eds. Bautista, E., Garcia-Lomas, J. and Navarro, A., Oxford: Pergamon Press, pp. 179-182. Popplestone, R.J. [1984], "Group Theory and Robotics," In Robotics Research : The First International Symposium . Ed. Brady, M. and Paul, R. , Cambridge: The MIT Press, pp. 56-64. Porteous, I.R. [1981], Topological Geometry , 2nd ed. , Cambridge: Cambridge University Press. Ravani, B. [1982], Kinematic Mappings as Applied to Motion Approximation and Mechanism Synthesis . Diss. Stanford University. Ravani, B. and Roth, B. [1984], "Mappings of Spatial Kinematics," Journal of Mechanisms. Transmissions, and Automation in Design , vol. 106, pp. 341-347. Riesz , M. [1958], Clifford Numbers and Spinors (Chapters I-IV^ . Lecture Series No. 38 . University of Maryland. Rooney, J. [1974], A Unified Theory for the Analysis of Spatial Mechanisms Based on Spherical Trigonometry . Diss. Liverpool Polythecnic. Rooney, J. [1975], "On the Principle of Transference," Proceedings of the Fourth World Congress on the Theory of Machines and Mechanisms. London: The Institution of Mechanical Engineers, pp. 1089-1094.

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157 Wittenburg, J. [1984], "Dual Quaternions in the Kinematics of Spatial Mechanisms," In Computer Aided Analysis and Optimization of Mechanical System Dynamics , Ed. Haug E.J., Dordrecht: D. Reidel Publishing Company, pp. 129-145. Woo, L. and Freudenstein, F. [1969], "Application of Line Geometry to Theoretical Kinematics and the Kinematic Analysis of Mechanical Systems," I.B.M. Technical report No. 320-2982. Yang, A. T. [1963], Application of Quaternion Algebra and Dual Numbers to the Analysis of Spatial Mechanisms , Diss. Columbia University. Yang, A.T. [1974], "Calculus of Screws," In Basic Questions of Design Theory . Ed. Spillers, W. R. , Amsterdam: North-Holland Pub. Co. pp. 265-281. Yang, A.T. and Freudenstein, F. [1964], "Application of Dual-Number Quaternion Algebra to the Analysis of Spatial Mechanisms," Journal of Applied Mechanics , vol. 31, pp. 300-308. Yano, K. and Ishihara, S. [1973], Tangent and Cotangent Bundles . New York: Marcel Dekker, Inc. Yuan, M.S.C., Freudenstein, F., and Woo, L.S. [1971a], "Kinematic Analysis of Spatial Mechanisms by Means of Screw Coordinates. Part 1-Screw Coordinates." Journal of Engineering for Industry , vol. 93, pp. 61-66. Yuan, M.S.C., Freudenstein, F., and Woo, L.S. [1971b], "Kinematic Analysis of Spatial Mechanisms by Means of Screw Coordinates. Part 2-Analysis of Spatial Mechanisms," Journal of Engineering for Industry , vol. 93, pp. 67-73.

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BIOGRAPHICAL SKETCH Jose Maria Rico Martinez was born in April 9, 1954, in Celaya, Mexico. He is the third member of a family of ten. His parents are Jose Rico Paniagua and Maria de los Angeles Martinez de Rico. Since 1975, Susana Perez Almanza and he have shared their lives, and they have two children, Susana and Jose. He completed his elementary education at the Colegio Vasco de Quiroga, Celaya, Mexico. In 1969, he finished his high school education at the Instituto Tecnologico de Celaya, Mexico. In 1974, he earned a B. Sc. degree, with honors, in industrial engineering at the same institution. He was awarded a scholarship by the Mexican Council of Science and Technology (CONACYT) , for the period (1974-1975) to attend the Instituto Tecnologico y de Estudios Superiores de Monterrey, Mexico where in 1977, he was awarded an M. Sc. degree in mechanical engineering. From 1975 to 1984, he was an instructor at the Mechanical Engineering Department at the Instituto Tecnologico de Celaya. He was awarded a combined scholarship for the period 1984-88 by the Mexican Ministry of Public Education and 158

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159 CONACYT , to study in the Mechanical Engineering Department at the University of Florida. After receiving a Ph. D. degree, he hopes to return to the Instituto Tecnologico de Celaya and resume his teaching career.

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Joseph) Dutfy^'cjl^ Graduate Research Professor of Mechanical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Gary K. /Matthew, Cochair Associate Professor of Mechanical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Ralpt^G. saifrptlge Professor of computer and Information Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. IC. ihlA K^rmit N. Sigmon Professor Ck A Associate Mathematics of

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. David A. Drake Professor of Mathematics This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 1988 iU^ja . Dean, College of Engineering Dean, Graduate School