Serial Manipulator Kinematics with Dual Quaternions and

Grassmannians

by Pasquale Gervasi

B.Eng. (McGill University), 1995

Department of Mechanical Engineering Centre for Intelligent Machines

McGill University Montréal, Québec, Canada

September 1999

A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfilment of the requirements of the degree of

Master of Engineering

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Abstract

Inverse kinematics (IK) of serial manipulators is related to reflection-free Euclidean

spaces where distances and orientations remain invariant. These direct isometries can

be derived from simpler forms known as quaternions and dual quaternions. Combining

them with Hermann Grassmann's Extension Principle, the IK of serial manipulators

is formulated entirely in projective geometry free of any metric. The only rules

governing this geometry is the preservation of ratios and incidence. Actually, the

holonomicity of these IK problems can be described using incidence relations alonc.

The algebraic constraints, derived from incident relations. define a manifestly

holonomic system as opposed to general holonomic systems that need only satisfy

Frobenius' Theorem for integrability using Pfafian forms. The solution to these al-

gebraic equations will require an introduction to algebraic geomet ry and commutative

algebra. Definitions of baçic geometric and algebraic objects along with a study of

their respective properties are included. Pertinent theorerns are proved and illustrated

with simple examples to establish a dictionary between algebra and geometry. By t his means, kinematic analysis is conveniently subjected to the theories of algebraic

geometry and commutative algebra.

More precisely, the inverse kinematics (IK) of a 6R serial manipulator (6Rsm)

is formulated in the homogeneous projective space of dual quaternions (DQ). This

leads to a robust algorithm because the 8-space of DQ together with the Grassman-

nian extension ensures that only valid solutions, which satisfy al1 of the constraint

equations, are admitted. Numerical esamples, based on two real6Rsm architectures,

are presented to illustrate the efficacy of the new algorithm. Its fiber (solutions) and

critical loci (singularities) are described.

Résumé

Le modèle géométrique inverse (MGI) des manipulateurs sériels est lié à l'espace

euclidien des déplacements dans lequel les distances et les orientations demeurent

invariantes. Ces isométries directes peuvent être obtenues à partir de formes plus

simples comme les quaternions et les quaternions duaux. En les associant avec le

principe d'extension d'Hermann Grassrnann, le (MGI) des manipulateurs sériels est

entièrement exprimé en terme de géométrie projective, donc sans nécessité d'une

métrique. Les seules lois régissant cette géométrie sont la préservation des rapports

et des incidences. En fait. l'holonomicité du (MGI) peut etre représentée en utilisant

seulement les relations d'incidence.

Les contraintes algébriques issues des relations d'incidence définissent un système

manifestement holonome par opposition à un système holonome général qui ne res-

pecte que le Théorème de Frobenius d'intégration en utilisant les formes différentielles

de P faff. La résolution de ces équations algébriques nécessitera l'introduction de

notions de géométrie algébrique et d'algèbre commutative. Les définitions de base de

la géométrie et des objets algébriques ainsi qu'une étude de leurs propriétés respectives

sont données. Des théorèmes bien adaptés a u problèmes sont démontrés et illustrés

à l'aide d'exemples simples pour établir un lexique de référence entre l'algèbre et la

géométrie. De cette manière la modélisation géométrique est formulée en terme de

géométrie algébrique et d'algèbre commutative.

Plus précisément, le (MGI) d'un manipulateur sériel à six articulations rotoïdes

est exprimé dans l'espace de projection homogène des quaternions duaux (QD). Il en résulte un algorithme robuste car l'espace de dimension huit des (QD) ajouté à

l'extension de Grassmann assure que seules les solutions valides, qui tiennent compte

de toutes les équations de contraintes, sont obtenues. Des exemples numériques,

reposant sur l'analyse des architectures de deux manipulateurs réelles sériels à six

articulations rotoïdes sont présentés pour illustrer l'efficacité du nouvel algorithme.

Leurs solutions et leurs singularités sont aussi décrites.

Acknowledgement s

I am eternally grateful to Professor Paul J. Zsombor-Murray for supplying an infinite

amount of enthusiasm, support, and patience; without it 1 could never have completed

rny studies. But his most humanitarian attribute must be the art of arriving at

deviously simple ideas inspired by the need to transfer that knowledge to others.

I'd like to estend rny gratitude to my parents Celestino and Lauretta Genrasi.

Their love and support was always there and contributed greatly to the cornpletion

of rny work. A special thanks goes out to David Delnick, a persona1 friend and full-

time tennis partner? who showed me that as long as you have footwork any shot is

conceivabIe.

Claim of Originality

The inverse kinematic problem(1KP) of the 6R serial rnanipulator is solved in a geom-

etry where one can identify an invariant property between al1 conics thus rendering

their respective shaye superfluous. The statement may seem irrelevant but it does

portray the events which are to follow quite accurately? not to mention my daim of

originality.

At present, formulations of the IKP are dependent on the ability to measure

distance in the space where the manipulator is embedded (ie. a "measuring stick"

which enables one to tell the difference between an ellipse and a circle). The following

arguments will show that the solution to the IKP does not require a measure. In other

words one need not tiifferentiate among different conics but need only acknowledge a

conic "section". The derivation of the algebraic constraints in this newer geometry.

namely projective geometry, requires a novel interpretation of the quaternion and

incidence relations.

The handling of any set of algebraic constraints adheres to the laws of algebraic

geometry and comrnu tative algebra w hich will be presented.

An algorithm based on these mathematical tools is proposed. It quickly and

safely retrieves all solutions to the IKP.

Table of Contents

. . Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Claim of Originality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Tables . . + . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . x

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . si

List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1. The Denavit-Hartenberg Notation . . . . . . . . . . . . . . . . . . 2

1.1.2. A Semi-Graphical Method . . . . . . . . . . . . . . . . . . . . . . 8

1.1.3. The Monovariate Polynomial Approach . . . . . . . . . . . . . . . 10

1.2. Systematic Discussion of Geometry and Its Foundations . . . . . . . . 13

Chapter 2. Dual Quaternions, A Map, Fiber, Critical Locus . . . . . . . . . 17

T.4BLE OF CONTENTS

2.1. Repositioning the Quaternion . . . . . . . . . . . . . . . . . . . . . . 17

2.2. Dual Quaternions as General Displacements . . . . . . . . . . . . . . 19

Chapter 3 . Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1

3.1. Grassmannians in the Plane . . . . . . . . . . . . . . . . . . . . . . . 21

3.2. Grassmannians in Space . . . . . . . . . . . . . . . . . . . . . . . . . 22

Chapter 4 . Synopsis of the IKP of a 6R Serial Manipulator . . . . . . . . . . 24

Chapter 5 . Converting a DH Table to Dual Quaternions . . . . . . . . . . . . 26

Chapter6 . TheAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 7 . Esamples 32

ri. l.1. Example 1: Fanuc Arc Mate . . . . . . . . . . . . . . . . . . . . . . . 32

- , .l. 1. First Elirnination Step . . . . . . . . . . . . . . . . . . . . . . . . 33

7.1.2. Second Elirnination Step . . . . . . . . . . . . . . . . . . . . . . . 35

7.1.3. The Characteristic Polynomial . . . . . . . . . . . . . . . . . . . . 36

7.1.4. The Extension Step . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7.2. Example 2: Diestro . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 8 . Conclusion 40

Appendix A . Algebraic Geometry Sr Commutative Algebra . . . . . . . . . . 41

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Varieties and Ideals 41

2 Sums. Products. and Intersections of Ideals . . . . . . . . . . . . . . . 43

2 . . Sums of Tdeals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 2 . 2 . Products of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . 44

L2.3. Intersections of Ideals . . . . . . . . . . . . . . . . . . . . . . . . 44

vii

TABLE OF CONTENTS

A.3. A Discussion on Kinematic Analysis as Application of Algebraic Geometry

and Commutative Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

.\A. Projective Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . 49

List of Figures

Coordinate frames of a Puma robot . . . . . . . . . . . . . . . . . 4

rLvis and displacement vectors of a Puma robot . . . . . . . . . . . - 3

Key vectors associated with solving the IKP of a Puma robot . . . 7

List of Tables

DH Parameters of the Fanuc Arc Mate Manipulator . . . . . . . . 32

IK Solutions of the Fanuc Arc Mate . . . . . . . . . . . . . . . . . 37

DH Parameters of Diestro . . . . . . . . . . . . . . . . . . . . . . 38

List of Symbols

Symbol Description

Six space of Euclidean displacements A n element of D6 Three space of reflection-free rotations Three space of point displacements Quaternion Dual quaternion n dimensional projective space n dimensional manifold A projection map Variety - see Appendix A Ideal - see Appendix .A Grobner basis of V - see Appendis A

List of Abbreviations

Symbol

6Rsm DH DKP EE FF FE( IK IKP P R sm

Description

Six revolute serial manipulator Denavit-Hartenberg Direct kinemat ics pro blern End effector Base frame Forward kinematics Inverse kinematics Inverse kinemat ics problem Prismatic joint Revolute joint Serial manipulator

Chapter 1

Introduction

There are two types of kinematic problems associated with robotic manipulators,

namely the direct kinematic problem(DKP) and the inverse kinematic problem ( I K P ) .

The DKP of serial manipulators (sm) is known to be quite simple, but the IKP is

more complex because it leads to a system of non-linear algebraic equations. The

IKP solution is usuall- if not always. associated with transformations? called direct

isometries, whose invariant properties are found in metric geometry. In the literature.

algorithms for solving the IK of a 6Rsm use these invariants to derive the algebraic

constraints. Shen the elimination process begins but not wvithout more metric con-

straints. By outlining the critical steps of two such algorithms one can argue that

although a metric geometry, finds a solution its diversity in invariant properties leads

to a myriad of definitions and different types of constraints. The objective, therefore,

is to solve the IKP using another kind of geornetry with fewer invariant properties.

Affine geometry and projective geometry are two such geometries, with the latter hav-

ing fewer invariants than the former. The separation of these other geometries from

metric geometry will constitute a starting point in preparing the reader to understand

the rules which govern their structure.

1.1. BACKGROUND

1.1. Background

The kinematic analysis of serial manipulators begins by uniquely determining

its architecture using Denuvit- Hartenberg(DH) notation. The DH parameters also

determine the manipulator's configuration or posture by defining a frame for each

link. The frame associated with the 1 s t link is comrnonly referred to as the pose

of the end-effector(EE) . Fonuard kznematics(FK) or direct kinematics(DK) involves

finding the pose of the end-effector given the DH parameters and actuator variables

values. The IKP consists of finding the configuration of the manipulator given the

pose of the end-effector.

1.1.1. The Denavit-Hartenberg Notation. The DH parameters can un-

ambiguously represent a kinematic model of a robotic manipulator. The model is a

kinematzc chain consisting of rigid bodies or links coupled by joints called kinematic

pairs. .-\ kinematic pair constrains the relative motion between two rigid bodies.

There are two basic types, namelx upper and lower kinematic pairs. üpper kinematic

pairs are those articulations where contacts between rigid bodies occur along a line

or a t a point common to both bodies as in roller bearings element. Lower kinematic

pairs constrain the relative motion to act along a surface common to both bodies.

One can derive five of the six1 different types of lower kinematic pairs by combining

two basic types:

a Revolute joint? R, allows two rigid bodies to rotate relative to each other

about the aues of the revolute; defining a common cylinder between the links.

a Prismatic joint, P, allows the two bodies to move along a common surface of

contact k n o m as a prism of arbitrary cross section. The tn70 bodies can only

move in pure translation parallel to edges of the prisrn.

' ~ h e helical joint, Le., a screw-in-out motion, is the special case which canno: be so deriveci.

1.1. BACKGROUND

The kinernatic chain of a serial rnanipulator is simple, every link is coupled to, at

most, two other links. The chah is coosidered open when there is a link, located at

either chah end, which is connected to only one other !ink as opposed to a closed chain

where every link is joined to two other links to create a fznkage. The 6Rsm consists

of a simple kinematic open chah with six R joints. The following is a procedure to

determine the relative location of the axes which in turn attach frarnes to the links.

Links are numbered 0 ,1 , . . . ,7 , where link O is the base and link 7 is the EE.

The ith R joint couples the (i - l ) l h link with the ith link.

A frame .Fi is attached to the (i - l ) th link.

a Zi mis, of frarne F i , is coincident with the ith joint aues.

a S, is the common perpendicular between Zi-l and Z,, directed frorn the former

to the latter. If Zi-l and 2, intersect then me adopt the right hand rule. If

Zi-l and Zi are parallel then &Yi can be chosen to go through the origin of

Fi-,*

0 Define ai to be the distance between Zi and Zi+l. From the definition of Si+i

it is easy to see that ai is nonnegative

Define bi to be the signed distance between iC, and S,+l measured along the

positive 2, direction.

Define ai to be the twist angle separating Z, and Zi+I and measured about the

positive direction

Define the actuator angle di separating ,Y, and .Y,+l and measured about the

positive 2,.

Fig. 1.1 illustrates the coordinate frames of the Puma robot. Xote that the orientation

of the last frame (i.e., F7) is chosen arbitrarily and its origin is usually placed a t the

overation point P of the EE. The IK algorithms illustrated here require a complete

1.1. BACKGROUND

mathematical description of a link's frame Fi+1. The following rotation matLu Qi

and the position vector g carry the frame Fi into Fi+i.

FIGURE 1.1. Coordinate frames of a Puma robot

COS Bi - COS cri sin Bi sin ai sin Bi ai COS Bi Qi = [ sin,, COS a, cos & filiai COS 0 ] 4 = [ nt 13. ] (1.1.1)

O siIl CI* COS ai

The successive application of this transformation, or direct isometry. from the base

to the EE can be represented by

where Q and p are the

base frame. Constraints

orientation and origin of the EE

eqn. (1.1.2) and eqn. (1.1.3), are

frame with respect to the

used to solve the IKP of a

6Rsm. One should also note that each element of the rnatrix Qi and the vector a, is

4

1.1. BACKGROUND

a linear expression in the terms [cos (Oi), sin(ei)]. By introducing six vectors

we can now Say that the product &Qi is hezulznear or. simply, multi-lznear in

vectors {xi}:.

FIGURE 2.2. Xxis and displacement vectors of a Puma robot

They represent an algebraic systern of twelve nonlinear equations, nine of them for

orientation and three for position. However, since Qi is orthogonal. we only have

three independent equations from eqn. (1.1.2) and three from eqn. (1.1.3) to solve

for the s k unknowns, namely, O*, for i = 1 . . .6. We need a few more definitions

before continuing. First re-express eqn. (1.1.1) in two forrns

rny Qi [ :[ ] [ pi qi ui ] (1.1.4)

From eqn. (1.1.1) one should note that the third row of Qi, oi, is independent of

Bi: later it will be shown how this observation helps eliminate one of the unknown

angles. In Fig. 1.2 the unit vector ei defines the direction of the ich joint ais. Its

cornponents in Fi are

1.1. BACKGROUND

by definition from the DH notation. Thus we have

Qioi E Q T U ~ = e

After introducing the following definition

a, = Qibi

Xext, we deriw six equations free of Os by first introducing the following equations

where 06 is the last row of Qs matrix. So we obtain

where the RHS of eqn. (1.1.7) is bilinear in xl and x* and the LHS is trilinear in

XQ, and xj. If we separate the LHS and the RHS of both equations, we obtain the

following

However, not al1 of the six scalar equations, free of Os, are independent because LHS

and the RHS of the second pair have to fulfill the following

1.1. BACKGROUND

FIGURE 1.3. Key vectors associated with solving the IKP of a Puma robot

and so we have five independent equations in the two vector eqns. (1.1.6) and (1.1.7).

There exists three methods used to eliminate four of the five unknoms from this

system. Li, Woernle and Hiller (1991) as well as Raghavan and Roth (1990, 1993)

approached the problem numerically so as to obtain a 1 6 ' ~ degree mono-variate poly-

nomial in terms of the half-angle of the fith unknown joint angle. The 3rd method by

Angeles, and Etemadi Zangane (1992), used a semi-graphical approach to obtain two

bivariate equations in two unknowns, from which they generated a set of contours.

1.1. BACKGROUND

1.1.2. A Semi-Graphical Method. From eqns. (1.1.8) it is possible to ob-

tain the three vector equations

f = g

h = i

f x h = g x i

To obtain a 4th vector equation, the condition that the reflection of the vector h

ont0 a plane perpendicular to vector f is equal to the reflection of i ont0 a plane

perpendicular to g was used so

and since I l f I l 2 = 1 1 g I l 2 we have

The two additional scalar equations are derived as

LHS of al1 four vector equations are trilinear in x3, q and x5 and RHS are bilinear

in xl and xz. LHS of the both scalar equations are bilinear in x d , xo and the RHS

are linear in XI. This system is then separated into two groups of scalar equations:

O the first g o u p contains the first two scalar equations of each of the four vector

equations

O the second group is composed of the third scalar equations of each vector

equation together with the two remaining scalar equations.

First group is represented in the following f o m

1.1. BACKGROUND

where xlz is bilinear in xl and x* and b is trilinear in x1, and xs. The six equations

of the second group are represented as follows

where d is bilinear in and x5. From these two groups we obtained sLu linear

homogeneous equations of the form

where

cos e3 '3' [";hl

and D is a 6 x 3 matrix with entries bilinear in xi and xs. After partitioning D into

two 3 x 3 matrices Du and Di we obtained two conditions for the singularity of the

matrix D, namely

and, since the non-trivial solution y3 has to satisfy both of the above equations, the

third equation is imposed, namely

det (Du - Di) f3 (O4, 9,) = O

The set of contours cl, c?, and q is defined by f i ? f2, and f3. The intersections of al1

three contours are valid solutions of the problem, while the points of intersection of

oniy two contours are spurious solutions.

1.1. BACKGROUND

1.1.3. The Monovariate Polynomial Approach.

Procedure of Raghavan and Roth. This procedure begins with the equation

which is similar to the equation

used by Angeles and Etemadi Zanganeh. Here C is a 6 x 9 matrix whose entries are

linear in xs and with x45 defined as

After using the relations for si and c, in the form ri = tan (1). - for i = 4,s: and after

multiplying LHS and RHS of eqn. (1.9) with (1 + ri)(l + ri) ,

is obtained. After introducing the same trigonornetric identities for i = 3 and rnulti-

plying the first four scalar equarions by (1 + r i ) eqn. (1.10 becomes

C " X ~ ~ = O

where C" is a 6 x 9 niatriu. The determiriant of any 6 x 6 sub-matrix of matrix

C" is an gth degree polynomial in r3. With the application of dialytic elimination [l]

both r4 and rj are removed. .A new system of equations is then obtained in the form

S%5 = O12

where S is s 12 x 12 matri.. To obtain nontrivial solutions,

det (S) = O

must be satisfied and from there one obtains a degree mono-variate polynornial

in ~ 3 .

1.1. BACKGROUND

Procedure of Li et al. Lee's procedure differs sornewhat from the previous one.

multiplies a rotational m a t i u

Ci =

The rnatriv Ui is obtained using the identities

where ~i = tan($-), si = sin(B), and ci = cos(6,). Shen two new vectors. narnely F and h. are introduced

where

such that Qi = C i h i . Then the four vector eqns. ( M a - 1.8d) are multiplied by T3

to obtain the following set

u3T = T3g (Llla)

u36 = T3i (1.llb)

U3 (T x k) = T3 (g x i ) (1.11~)

U3 [(f - f ) h - 2 ( f -h)T] = T3[(g g ) i - 2 ( g i)g] ( l . l id)

1.1. BACKGROUND

The twelve equations above together with two additional scalar equations eqns. (1.8e

& 1.8f) form the set of fourteen scalar equations. By using the fact that the third

scalar equations in al1 four vector equations remain the same, because of the form of

matrices T3 and U3, ive can write the following

al1 of whicb are free of B3. Theri. by multiplying both sides of eqns. (1.12a - 1.12e)

and eqns. (1.8e Sr 1.8f) with r3 we obtain six new scalar equations, which will give

us a set of twenty scalar equations. This set can be cast in form

where the vector 0 is trilinear in ~3~ XI and xz and rnatrix A is a 20 x 16 matrix

and it is a function of design parameters only. A and B are then partitioned as

After solving for x from A, we obtain

These equations can be presented in the form

which after expressing

(Ci - Ai) ~ i 7 - 3 +

si and c, in the form of T,, we obtain

2 B i ~ 2 ~ 3 + (Ai + Ci) 7-3 + (F , - Di) T: + 2Eir2 + Di = O

12

1.2. SYSTEM.4TIC DISCUSSION OF GEOMETRY AND ITS FOUNDATIONS

Then with the multiplication of the above equations with r2 ive will obtain 4 more

equations resulting in the system of 8 equations which can be represented in the form

and whose nontrivial solution is obtained by setting

This will give us a polynomial octic in xl, ie., of 16th order in tan %, which is the

mono-variate polynomial in rnind. Other elimination methods have been introduced

y others but al1 use eqns. (1.1.2 Sr 1.1.3) as the basis for the constraints. Hence,

elimination proceeds, by definition, in a metric geometry; the geometry of direct

isometries. It is argued that the elirnination procedure can be simplzfied if one adopted

a more general geometry.

1.2. Systematic Discussion of Geomet ry and Its Foundat ions

It will be illustrated. using DQs and Grassmannians, that the special structure

of a metric geometry is not necessary to solve the IKP. It is special because metric

geometry is a specific case of other kinds of g e o m e t ~ . Paraphrasing Klein [2]? to

extract these other geometries we muçt proceed as "chemists" who apply stronger

reagents to isolate more of the valuab!.e ingredients from raw material. Our reagents

will be affine, followed by projective transformations. It is by means of this process

that an understanding of projective geometry, mhere the formulation of the IE; of the

6Rsm will take place, will be gained.

We begin wit h the following general point transformation:

1.2. SYSTEMATIC DISCUSSION OF GEOMETRY AND ITS FOUNDATIONS

If it is a direct isometry, the following two conditions are required

QQ= = hx3

det (Q) = +i

where

If we relax the above conditions such that

Q Q ~ = x ~ I ~ ~ ~

det (Q) = &A

then we include two more transformations, namely reflections and sirnilarity transfor-

mations with the origin as center. The result is a geometqi in rvhich invariant p rop

erties include the fixed distance between two points given (xi, y,, z i ) , and ( 1 2 , y2 , z?),

d(xi - x2)? + (gI - y?)* + (zI - z ? ) ~ X remains unchanged (i.e. independent of the

location of the points). Similarly. one can say the same about angle measures sub-

tended by three such points or that five points in a plane define a conic with particular

principal auis and inter-focal length. The totality of these geometric properties be-

longs to metric geomet ry.

If we begin again with eqn. (1.1) but this time unconstrain the relations among

the coefficients, we obtain the group of affine transformations. Notice that metric ge-

ometry is therefore a subgroup of affine transformations. The aggregate of concepts

and theorems which rernain invariant under affine transformations form a s n e geom-

etry. The notion of distance and angle disappear as do the distinctions between circle

and ellipse. But one can distinguish between h i t e and infinite space and every-thing

which depends on these tnro spaces such as parallelism of two lines or distinction of

conics among ellipses, hyperbolas, parabolas, etc.

1.2. SYSTEMATIC DISCUSSION OF GEOMETRY ,4ND ITS FOUXDATIONS

Proceeding to projective transformations, we begin by introducing linear frac-

tional transformations

which includes affine transformations as a subgroup. Here, the role of infinity and

the concepts connected with it in affine geornetry disappear. There is only one proper

conic because al1 others can be derived from it.

By introducing homogeneous coordinates (6 : 7 : C : r) such that

eqn. (1.5) can be written as

The origin in the homogeneous space of {{, il, C, T ) is removed or nonexistent; at least

one homogeneous coordinate must be non-zero. The group of projective transfor-

mations is termed collineations because three collinear points remain collinear after

transformation by eqn. (1.6). A collineation is an affine transformation if it sends the

plane a t infinity into itself, which means that, to every point with r = O, there cor-

responds another with r' = O. From eqn. (1.6) this happens when a4 = b4 = c4 = 0.

By dividing eqn. (1.6) by p ' ~ ' and replacing al : d4 by ai! bi : d4 by bi, and so on

we obtain eqn. (1.1). In order t o obtain the g o u p of similarity transformations from

the projectivities, we leave invariant not only the plane at i n h i t y (i.e. r = 0) but

also the imaguiary sphericai circle (i.e. c2 + q2 + c2 = O). TO prove this, let T = O

1.2. SYSTEMATIC DISCUSSION OF GEOMETRY AND ITS FOUND.4TIONS

Hence we can quickly point out that the six conditions which Ieave the imaginary

spherical circle invariant define the group of similarity transformations. Finally, the

direct isometries used to solve the IKP are just particular cases of projectivities. It

turns out that as long as one preserves the isotropy of Euclidean space. the IKP can

be formulated entirely in a projective space.

Chapter 2

Dual Quaternions, A Map, Fiber, Critical Locus

2.1. Rep osit ioning the Quaternion

Conventional notions of the quaternion are attributed to A. Cayley's [3] view that

the rotation of a vector, by means of a quaternion, is in general a bilinear operation.

But Sir W. R. Hamilton, who invented quaternions: saw them to be much more

than just an artificial method in reconstructing known results. Hamilton used the

word quaternion' to denote a new mathematical method "as an extension, or im-

provement, of Cartesian geometry, which the artifices of coordinate aues. are got rid

of, al1 directions in space being treated on precisely the same terms. It is therefore,

except in some degraded forms, possessed of the perfect isotropy of Euclidean space

[4] ." Such was his view that he questioned the necessity to have coordinate aues, i. e.,

the columns in the orthogonal m a t r ~ u Q, when expressing a rotation of Euclidean

space. According to his biographer, Peter Guthrie Tait, Hamilton refrained from try-

ing to convince his fellow mathematicians of this intrinsic nature of the quaternion -

represented by the novel use of the three imaginaries i, j, and k - because he feared his

- -

' meaning a set of four

2.1. REPOSITIONING THE QUATERNION

method nrouldn't be recognized by those steeped in Cartesianism [5]. It is believed

that these arguments just ik a short review of the quaternion as seen by its originator.

Geometrically, the quaternion q can be viewed as the quotient of two vectors in

space

where W-' = IIvI12. HOW Hamilton derived the representation of a vector using the

imaginary quantity fl is an interesting story worth reading (4). For our purposes

we will only mention the result. Given the vectors

v2 = x'i + y'j + z'k

vl = xi + yj + ik

Q = v 4

= (EX' + yyl+ 22') + (z'y - ylz)i + (X'Z - zlx)j + ( y ' ~ - xfy)k

The three distinct space units i, j, and k foliow the multiplication rules i2 = j2 = . . k2 = -1, 1j = -ji = k, jk = -kj = i' ki = -ik = j. It is the use of these units

which allows one to represent a rotation without artifices coordinate axes. Using

the notation of the previous section, the columns of Q represent the new coordinate

aves resulting from the rotation of the inertial frame; but one can assume to know

everything about the rotation using only a quaternion q. If the rotation were to

happen normal to the vector one can write the transformation as

In this case q is a linear operator which rotates vl into vz and therefore its axis iis

perpendicular to both vectors. Rotating a vector about an arbitrary avis is expressed

as a bilinear operation

2.2. DUAL QUATERNIONS AS GENEML DISPLACEMENTS

which was popularized by Cayley [3], even though Hamilton knew of the results

independently [3] [4]. In essence, Cayley views rotation, by means of a quaternion,

as a bilinear operation, whereas Hamilton views it as a linear operation. By adopting

insights frorn both camps we obtain a quaternion which represents general rotations

while its multiplication is limited to a linear operation.

2.2. Dual Quaternions as General Displacements

A quaternion, whether Cayley's or Hamilton's, represents a rotation. -4 general

displacement may also include the displacement of the origin represented by a position

vector d. By multiplying d by a quaternion q

we have another quaternion y. A spatial displacement can now be represented by a

pair of quaternions or, more conciselyf by a dual quaternion

One can find a detailed account of this representation in [6] , pp. 521-324. The eight

terms which make up 4 are the eight homogeneous coordinates known as the Study

soma. The soma maps the set of Euclidean displacements into a six-dimensional

manifold in a seven dimensional projective space [7i. Consider the toroidal 6-space,

76, of six R-joint displacernents, 8, where 0 is an element of 76. Consider also the

6-space of Euclidean displacements D E D6 = S03 x A3 where D is an element of V6,

the product of the 3-space of rotations S03 and the 3-space of point displacements A3.

This represents an end effector (EE) displacement. @ : 76 -t V6 rnaps the six joint

angles 19 into an EE pose D via the product of six dual quaternions B * &4&&j5&.

4 is a point which represents a screw in a homogeneous projective 8-space. a ( 0 )

transforms a rigid-body reference frame Fi -t via joint angle Bi, while D is

evpressed as & ? which transforms the base hame (FF) + F7 ddirectly into EE

2.2. DUAL QUATERNIONS AS GENERAL DISPLACEMENTS

frame. Recall that

.t . . where e2 = O, ii =j = kk = -1, IJ = -JI = k, ki = -ik = j, and jk = -kj = i.

The Euler-Rodrigues parameters are {ri :): while {yi :)O = -a(ro + rli + r j + r3k)/2

accounts for position vector a directed from origin Oi of frame Fi to Oj of frarne

Fj. Sets ul eigiit cuurdiiiates Leiulig tu Study's soma Si aucl are heuce scraws il iiiey

belong to the varieties V(E?='=,yiri = 0) and V1(Cf==,r? # O). -1 genenc fiber of map

+ consists of a number of points, counted with multiplicity, in 76, which map into

the same point in Si. .A fiber of S admits 16 solutions, fewer if it belongs to a critical

locus^ ie., where the Jacobian matri- J is singular. For a 6R serial manipulator J

is a 6 x 6 matrix, with columns arranged in frame sequence from h e d (FF) to end

effector (EE) frarne, of Plücker line coordinates of the six R-joint âues with respect

to the operating point P on EE. If the rank of J, p < 6, then the lines beiong to a

complex ( p = 5), congruence ( p = 4) or series (p = 3). Throughout the critical loci.

the axes assume such configurations. But a fiber (solution set) containing critical

loci (singularities) may also contain noncritical elements (nonsingular solutions) and

a good algorithm should find the 161h-order univariate polynomial or detect if @ is

singular. If it is singular, then the algorithm must separate the solution set into its

critical and noncritical elements. This requires the facility to decompose a higher-

dimensional projective space into incident linear subspaces. For this reason it is useful

to examine Klein's [2] Grassmannian treat ment of incidence relations of simplect ic

segments of planes, lines and points in the plane and these three as well as the

tetrahedral simplex in 3-space.

Chapter 3

Grassmannians

The proposed algorithm makes extensive use of Grassmannians. Their meaning, as far

as the algorithm is concerned, can be best explained by constructing Grassmannians

in the 2D plane and in 3D space.

3.1. Grassmannians in the Plane

Consider 3 x 3 matrices with r o m of homogeneous point coordinates.

If the rank of E2 is p = 3, then the determinant is 2! times the area of the triangle on

the three points. If p = 2, three 2 x 2 subdeterminants, taking any two rows, are the

coordinates of the line on the three collinear points. If p = 1: al1 three coordinates

are the same point and multiple combinations of each other. Note that if p = 2 (or

p = 1) one rnay manipulate the matriv to obtain one (or two) row(s) of (O : O : 0).

Example: Line Equation as an Incidence Relation

In the plane, one can view three collinear points as spanning a triangle with zero area

det (E2) = O

3.2. GRASSMANNIANS IN SPACE

Incidence relations are not restricted to linear configurations.

Example: The Circle Equation as an Incident Relation

Find the circle on three given points.

It is interesting to note that the determinant multiplying the term (x2 + y2) is 2! times

the area of the triangle defined by the three points. If the three points are colliriear

then this term disappears and we are left with a line equation!

3.2. Grassmannians in Space

In space. points have four homogeneous coordinates: so? consider 4 x 4 matrices. If

p = 4, the determinant is 3! times the volume of the tetrahedron on the four points.

If p = 3. four 3 x 3 determinants. taken from any three rom. give the coordinates

of the plane on three points. If p = 2, from any two rows, six 2 x 2 matrices give

the Plücker ray coordinates of the line on two points, while if p = 1. there are four

versions of the same point. It will now be shown how the deficient r om provide

the key to incident relations. For example, three points don't define a plane if they

are collinear or coincident. If they don't define a plane, they can only produce the

nonexistent coordinates (0 : 0 : 0 : 0).

Example: Incident Relations of a Line on a Plane

Incidence relations of a line 1: = (poi : po2 : : P23 : p 3 ~ : p12) and a plane

7r = (qo : 91 : Q : q3) are summarized below.

3.2. GRASSMANNIANS IN SPACE

Lemma 1 The intersection L n n = (xo : xi : z2 : x3) either defines a unique point

in P3 O f L C 7r.

Proof By Bezout's Theorem the point is unique because varieties L and a are linear.

The only nonexistent invalid point is {O : O : O : O) because even if n is parallel to L:

they nevertheless intersect a t a unique valid point at infinity (a'. e . ;co = O and xi + x: + x: # O). Therefore. to determine L c n one "finds a point" with the invalid

coordinates by evaluating a singular 4 x 4 "point-equation" determinant. The first row

contains the linearly-dependent, variable plane coordinates of a fourth plane which

is on the point usually defined at the intersection of the three given planes whose

coordinates populate the three lower rows. Recall that the coefficients of the linear

equation that results from expanding the first row minors are the coordinates of the

point of intersection. In what follows Ci are the fourth variable plane coordinates, q*

are the coordinates of n, and q: and qr are the coordinates of any two planes on L: so

as to generate its axial Plücker coordinates.

Co Cl C2 C3

770 711 t72 q3

We expand on first row minors, collect terms,

by the t hree numerical rows, qiy qi, qy, are x i ,

note that the coefficients generated

and axially convert the ray Plücker

coordinates of L for the lower two rows. With the K T C I ~ ~ C ~ ~ T - ~ ~ ~ , this produces four

sufficient incident relationships. 3

Zi = C(1 - Gij)i),pij = O : i = O.. .3 , pij = -pji j = O

As explained by Klein, [2], the Grassmannian Eztension Princzple, [8] , illustrated

above for 2- on 3-dimensional subspaces incidence, both on a vector h p a c e , is valid

in higher dimension too.

Chapter 4

Synopsis of the IKP of a 6R Serial Manipulat or

Applying the tools introduced earlier and including theories in algebraic geornetry

and commutative algebra, as outlined in Appendix -4, formulation and solution of

two examples will be treated as follows.

a Converting a DH table to dual quaternions

The algorithm

(i) The constraint equations

(ii) First elimination step

(iii) Second elimination st ep

(iv) The characteristic polynomial

(v) The extension step

There exists a simple geometric problem which reveals the algorithm's intent. The

intersection of two circles in the plane is a problem in high school mathematics but it

is not obvious that the problem is equivalent to the intersection of a paraboloid and

two planes in 3D space. Let us begin with the general homogeneous equation of two

CHAPTER 3. SYNOPSIS OF THE IKP OF A 6R SERIAL hL4NIPULATOR

circles

A ( X ~ + y2) + BXZ + Cyz + 0z2 = O

R(x2 + ZJ~) + Btxz + C'yz + D'Z' = O

Now ive introduce a polynomial map

The above map obviously constrains the new coordinates qi, for z = 1. . -4:

Substituting the monomials of eqns. (4.1) and (4.2) according to eqns. (4.3) produces

In sumrnary. eqn. (4.3) transforms the 2D plane to a paraboloid, eqn. (4.4)' in 3D

space and turns circles7 eqns. (4.1) & (4.2), into 3D planes, eqns. (4.5) & (4.6). By

estending the space of solutions to a manifold, having the same dimensions, ernbedded

in a higher space, nonlinear constraints can be made linear. The extension can also

lead to a clearer understanding of the original problem. In the circle example above.

one can easily show that two concentric circles map into a line a t cc in the new 3D

space.

Chapter 5

Converting a DH Table to Dual Quaternions

Constraint equations arise as a product of dual quaternions, six of which are obtained

from joint coordinates and Denavit-Hartenberg (DH) parameters. The seventh r e p

resents the given pose of EE. The figure belotv illustrates how a row in the DH table

effects a frame change.

Frame Fi is transformed to Fi+1 by a dual quaternion composed of four primitive

ones. The first is a pure translation bi along Zi:

The second is a pure rotation about Zi by 6,:

CHAPTER 5. CONVERTING A DH TABLE TO DUAL QUATELVIONS

so Pei = cos(ei/2) + sin(Bi12)k. Similarly, @ai = 1 - (a,/2)Ei, @ai = cos(ai/2) + sin(ai / a ) i Finally,

Now we convert a rotation mat rk Q and vector a

(5.7)

= d into a dual quaternion. The

d --(rO + rl i + rzj + r3k)

2 ( 5 . 8 ~ )

thereby expressing a dual quaternion Q E E which transforms FF into EE. Now we

illustrate the IK algorithm with two examples. ie., the Fanuc Arc Mate and Diestro.

The second is given a pose which will require treatment of a fiber with singular and

non-singular values. Design parameters, here and elsewhere, are from Angeles (91.

Furthermore one should note that the group structure of a general displacement also

manifests itself in these special dual quaternions. Furthermore we have the conjugate

of a dual quaternion such that

g = s + v , j = s - î .

= s2 + 1 1 ~ 1 1 ~

where s is a dual nurnber and v is a dual vector. If 4 lies on the Study soma, 56,

then

= ri + r: + r: +r3 # O

Chapter 6

The Algorit hm

The algorithm is divided into five steps:

Step 1. The constraint equations.

Step 2. First elimination step.

Step 3. Second elimination step.

Step 4. The characteristic polynomial.

Step 5 . The extension step.

Step 1. The constraint equations

Constraint equations arise as a product of dual quarternions. qi? six of which are

obtained from joint coordinates and Denavit-Hartenberg (DH) parameters and the

seventh, qEE, is given by the pose of the end-effector:

to produce seven homogeneous equations:

CHAPTER 6. THE ALGORITHM

Step 2. First Elimination Step

Introduce a polynomial map M3:

Substitute the expressions above into I and compute the Grassmannian:

where q = [rio.. . . . rl i ]T. The implicit representation of M 3 is:

Substitute expressions for x into I(M3) to obtain the projection:

CHAPTER 6. THE ALGORITHM

Step 3. Second Elimination Step

Introduce a polynornial map MI':

Substitute the expressions above into S and compute the Grassrnannian:

where < = [b, . . . . &IT. The impiicit representation of M 2 consists of 20 homoge-

neous equations in y.

S tep 4. The Characteristic Polynomial

Substitutiog expressions for y into I(M2) results in a projection

Each generator of IT = ( t l , . . . t20) c k[c3, s3] factors into two 1 6 ' ~ order univariates

ti = hgi; h is the characteristic univariate and the other is different for each generator

ti. Geometrically, we can express hgi as the union of two hypersurfaces:

CHAPTER 6 . THE ALGORITHM

t herefore:

Step 5. Extension Step

Looking at both polynomial maps introduced earlier we see that:

x2 : 26 = CJ : 3.1:

xg : xfj = Cs : Sg,

22 : 2 3 = C6 : s6 9

Y 0 : Y l = Cl :si7

Y'l : y5 = C2 : s2

Chapter 7

Examples

7.1. Example 1: Fanuc Arc Mate

We will solve the IK of the Fanuc Arc Mate for the pose

O 1 0 R = [ O 0 1 ] d = [ :;: ] (mm)

1 0 0 1540

The DH parameters are given in TABLE 7.1. Now the pose generated by the actuators

must be the one given as X4i&&&&&, = iEE, where X indicates that only the ratios of

eight homogeneous coordinates are preserved. Using eqns. (5.8a - 5 . 8 ~ ) the right-hand

side is

= [(1 - €630) - (1 + e205)i - (1 - el4O)j - (1 + ~565)k]/2

Then it is easy to show that the product of the sis dual quaternions will form

eight hexalinear polynomial expressions in

TABLE 7.1. DH Parameters of the Fanuc Arc Mate Manipulator

i bi (mm) Bi ai (mm) <ri

1 810 Oi 200 90"

7.1. EXAMPLE 1: F.4NUC ARC MATE

LVe now rnultiply both sides of eqn. (7.1) by a,,

to produce seven homogeneous equations that form an ideal as defined in Appendk

A.

The algorithm will introduce projective maps which will bring the variety V(I) ont0

T c Pi. The characteristic univariate is T.

7.1.1. First Elimination Step. First one must decide which three angles should

be eliminated first and then introduce the appropriate polynomial map. The following

rnap represents the decision to eliminate 8.4, 05, and $1

where c, = cos % and si = sin 2. The polynomial map descnbes a parametrization of

a 3-dimensional manifold M 3 in a 7-dimensional projective space P'. The complete

elimination of &, 95 and B6 requires a description of MI in terms of homogeneous

equations in the polynomial ring k[xo, . . . , x7] (see eqn. 4.4). It is very difficult to

decide this by just looking a t the rnap even though some are readily apparent such as

We need to compute a basis, I(M3) c k[xo, . . . , xi], which is a set of generators that

can span al1 poiynomials that vanish on M 3 . .A Grobnet basis, was computed using

7.1. EXAMPLE 1: FANUC ARC MATE

the grobner package in MapleV

The polynomial map, eqns. (7.3): also transforms I into a hornogeneous systern of

seven Iinear equations in [la, . . . , xi]. Solving for [xo, . . . , x7] by forming a Grass-

rnannian using I

where = [etao, . . . , q7IT and f = [ f . . . f7IT. Each expression xi has the non-zero

2 3 2 factor (s: + C : ) ~ ( S ; + c2) (s3 + ~ 3 ) ~ in common. Once factored out we then substitute

expressions for x into MJ to obtain the projection

The projective variety S implies an ideal whose generators are Is = (sl , . . . , sg) C

k[c l , sl c?? s2, c3, s 3 ] Note that these generators do not necessarily form a bais . We

show here the first generator of Is.

7.1. EXAMPLE 1: FANUC ARC MATE

7.1.2. Second Elimination Step. Similar to the first elimination step we intro-

duce the following polynomial map:

which describes a 2-dimensional manifold M? c Pa. The implicit representation

of M I is found by computing a basis I ( M I ) c k[ya,. . . y8] which consists of 20

homogeneous equations. The redundancy is necessary in order to span any polynornial

which vanishes on M 2 under the rules of the polynornial ring k[yo,. . . , ys]. The new

polynomial niap transforms S into nine hyperplanes LIi c Pa that intersect in a point

y E P8 for a given 03 . Again ive construct the first order Grassmannian to find the

intersection point

where U = [Ul, U2, l j3, LI4, U5, Us, Li7, Us] and q- = [ i lo. . . . pl. Note that we require

only eight independent hyperplanes, Liï, to define a point y E Pa. Below we have the

expression for y 1 :

7.1. EXAMPLE 1: FANUC ARC MATE

7.1.3. The Characteristic Polynomial. Substitut ing expressions for y into I(M2)

results in a projection

Each generator t i of IT = (tl , . . . , t20) c k[c3, s3] factors into two 1 6 ~ ~ order univari-

ates t i = hgi; h is the characteristic univariate. We can prove this geometrically by

expressing ti = hgi as the union of two projective varieties

Therefore,

The result is the hornogeneous univariate

7.1.4. The Extension Step. Using the above ratios we can extend, or solve for,

the complete set of unknowns by expressing the five pairs of ratios {(s : si)) in t e m

7.2. EXAMPLE 2: DIESTRO

of (Q, s 3 ) Looking at both polynomial maps introduced earlier ive see that

One solution of eqn. (7.4), is (c3 : s3) = (1 : O). Extending to the other angles using

eqn. (7.5)

To understand what happened with the last ratio let us look at x

which says that to solve for 04, 05, and 06, one must use the coordinate x6. -4 closer

look at the first polynomial map reveals that

The other real solutions to the IK of the Fanuc Arc Mate are computed similarly to

finally arrive at the results of Table 6.2.

TABLE 7.2. IK Solutions of the Fanuc Arc Mate

Solution 1& 2

3 4

dl 90" 90"

75.1566"

92 90"

16.0095" 15.3252"

03

O" 153.403" 150.851"

0 4 180" 180"

15.2657"

0 5

180" 100.588"

-103.353"

8 s O" O"

176.393"

7.2. EXAMPLE 2: DIESTRO

TABLE 7.3. DH Parameters of Diestro

1 i 1 bi (mm) di a, (mm) B;t50

7.2. Exarnple 2: Diestro

We will solve the IKP of the Diestro for the pose

,=[O O "1 d=L; r J ( rn rn ) 1 O

The DH parameters of the manipulator are given in TABLE 7.3. In this second ex-

ample the solution contains both critical and non-critical values. The singularity was

detected as we proceeded with the algorithm to obtain the characteristic polynornial

and found that projecting M 2 ont0 PL gave the zero function; more precisely,

which describes a one-parameter set of solutions. Extending to the complete solution.

(ci : sI) = (2c3 : -(c3 - s3)) (7.1 a )

(c2 : s2) = (c3 - ~3 : -(ss + 3 ~ 3 ) ) (7.1 b)

(4 : s4) = (s3 + c3 : Sc3 - s3) (7. ic )

2 (CS : s = ((c3 - s3) : (s3 + c3)(s3 + 3 ~ ~ ) ) (7. ld )

(cg : 36) = ( ~ 3 + c3 C 3 - ~ 3 ) (7. le)

This curve describes the critical values of the fiber. the non-critical values are found

by satisfjmg incidence; recall that the excluded origin (O : . . . : O) in projective space

defines incidence. Non-critical values exîst if there are values for B3 such that

7.2. EXAMPLE 2: DIESTRO

It turns out that if

then eqn. (7.2) is satisfied. From the above equation there are only two real solutions,

namely û3 = 90' and d3 = -90'. Extending these solutions will require eqns. (7.la

- 7.le) and the projection ni ( M 3 ) . Recall that the first elimination step resulted in

compute the Grobner basis

The above equat ions have both critical and non-critical solutions

Eqn. (7.3) represent the singular solutions because they satisfy eqns. (7.h -7.le) and

therefore eqn. (7.4) represent the non-singular solutions.

Chapter 8

Conclusion

Projective geometry fully described the IK, finding fibers with finite or infinite solu-

tions, aithout formulation singularit. The IK was formulated with dual quaternions,

the eight hornogeneous coordinates of the 6-space of rigid body displacement. Thus

reforrnulated, other problerns in kinematics may be better solved as well. Though

we ask questions about objects rnanipulated in metric space. answers may corne frorn

projective geometry where? Jacob Steiner thought. [2], that order is brought to chaotic

theorems and everything arranges itself there naturally.

Appendix A

Algebraic Geometry & Commutative Algebra

A.1. Varieties and Ideals

This section is abstracted from [IO] pp. 5-36.

We begin by defining geometric objects called d n e varieties.

DEFINITION A.l.1. Let k be a field, and let JI, . . . , fs be polynomials in k [ z l : . . . . x,]. Then zue set

1 ) = ((al , . . . , a n ) € k n : f i ( a1 , . . . , a n ) = O V l S i l s }

We cal1 V(f1, . . . , f,) the atfine variety defined by f i , . . . , f,.

For example. V(x2 + y2 - 1) is a unit circle centered at the origin. Polynomials

f 1, . . . . fs also define an algebraic object.

DEFINITION -4.1.2. 1 c k[xi l . . . lx,] zs an ideal if O E 1, if f,g E I then

f + g E 1, and if f E I and h E k[xl,. . . ,x,], then h f E 1.

DEFINITION A.1.3. Let f 1 , . . . , f, be polynomials in k [ q : . . . , x,]. Then we set

A. 1. V.4RIETIES AND IDEALS

The crucial fact is that (fi,. . . , f,) is an ideal.

LEMMA -4.1.1. If f i , . . . , f, E k [ x l , . . . , x,], then (f l , . . . , f,) zs an ideal

of k[zl, . . . , x,] - We wzll cal1 ( f i , . . . , f,) the ideal generated by f . . . , f,.

An ideal is similar to a subspace. Both are closed under addition and multiplica-

tion. For a subspace, we multiply by scalars, for an ideal, by polynomials. Further.

notice that the ideal generated by polynomials Il, . . . , f, is similar to basis vectors

u l , . . . , un. An affine variety I.* = V(fi.. . . , f,) C kn says that I l , . . . ,j, vanishes on

Io. But are t here other polynomials t hat vanish on C*'? Consider V = V (y - z2. r - x3)

and W = V(r - xy. y2 - xz). Does V = LI.'?

f i s , because the two polynomials of V span the two polynornials of W . But how

do we find or express al1 other such polynomials that vanish on the same geometric

object called V or GV? Consider the set of all polynomials that vanish on a given

varie t y.

DEFINITION A.1.4. Let V c kn be an aflne variety and

I (V) = { f E k [ x l . . . . ,x,] : f (a l , . . . .a,) = O

V ( a l , . . . ,an) E V }

LEMMA A.1.2. ÿ f i ? . . . , fJ E k [ q , .. . .x,], then ( f i , . .. . fs) C I(V(fl?. .. .f,))'

although equalit y need not occur.

Which States that:

need not be true. How does one find the polynomials g,, . . . , gt that generate

I(V( fi:. . . , fs))? The Hilbert Basis Theorem states that the generating set is finite

so g,, . . . ! gt is a basis of 1. Finally, one may find such a basis, called a Grübner

[IO]. We can think of (9,:. . . , g,) as a basis for spanning any polynomial or any set

of polynomials vanishing on V.

A.2. Sums, Products, and Intersections of Ideals

This section is abstracted from [IO] pp.180-188.

Sum. Product. and Intersection are operations on a pair of ideals to produce another

ideal.

A.2.1. Sums of Ideds.

DEFINITION -4.2.1. If I and J are ideals of the ring k[zl, . . . x,]: then the sum

of I and J . denoted I + J , 2s the set

Geometrically, this is what happens. Let I = (x2 + y) and J = ( z ) be ideals in

R3. Then 1 + J = (x2 + y, 2) contains both x2 + y and z. That is, it must be the

intersection of V ( I ) and V ( J ) .

THEOREM A.2.1. If I and J are ideals in k[xl, . . . , x,], then V(I + J ) = V ( I ) fi

V( J )

A.2.2. Products of Ideals. A product of two ideals corresponds to the union

of the variet ies:

V(f1Y . , f r ) UV(g1,. . - ,g,) =

V ( . f i g j , l 5 2 5 r , l 5 j 5 S )

For instance, the variety V ( x t . gr) corresponds to the product of the generators of the

ideais, (2, y) and (2) in k j x , y, ZJ, which in turn implies V(xz, gz) = V(x, y) U V(z).

Algebraically!

DEFINITION A.2.2. If I and J are two ideals in k [ x l , . . . ,x,], then their prod-

uct, denoted by I J , is the ideal generuted by al2 polynomials fg awhere f E I and

g E J .

Cornputing a set of generators for the product of ideals I J . given sets of generators

for I and J , is now shown.

PROPOSITION A.2.1. Let I = (flT.. . , f,) and J = (g,, . . . , g,). The I J i s gen-

erated by the set of al1 products of both generators:

I J = ( f i g j : l < ~ < r , l < j s s )

A.2.3. Intersections of Ideals.

DEFINITION A.2.3. The intersection I n J of two ide& is the set of polynomials

belonging t o both I and J .

N.B., I J c (1 n J ) because I J implies polynomials of the form fg where f E I

and g E J and therefore f g E I n J. However, I J can be strictly contained in I n J .

Let I = J = (z, y), then I J = (x2, xy, is strictly contained in I n J = I = J =

(x, y) (x E I ri J , but x I J ) . An algorithm to compute I n J requires a theorem

and some new notation. If we have f ( t ) E k[ t ] and I E k[xl, . . . , x,] then f I is the

ideal in k[xl, . . . , x,, t ] generated by the set of polynomials f h : h E I . Notice that

each ideal f and I lie in different rings; in h c t I E k[xl,. . . , x,] is not an ideal in

k[xl, . . . , x,, t] since multiplying a polynomial f E I by t will make it a polynomial

in k[xl, . . . , x,: t] $ k[xl, . . . , xn] and, therefore, I is not an ideal in k[xl, . . . , r,, t]

by definition. From now on we will write a polynomial h E k[xl , . . . , x,] as h ( x ) but

g ( x , t ) E k[xl ,... ,t,,t]. So fI = f ( t ) I = (f(t)h(x) : h(x) E 1).

LEMMA -1.2.1.

a If I 2s generated as an ideal in k[xl, . . . . r,] by pl(x), . . . , p r ( z ) , then f ( t ) I =

( 1 ~ 1 7 . - * f p r ) -

I f g ( x , t ) E f I and a E k then g(z, a) E I

Finally, we can state the following theorem:

THEOREM -4.2.2. Let 1, J be ideals in k[xl, . . . . x,]; then,

The above theorem leads to an algorithm for computing the Grobner b a i s of

the intersection of two ideals: if I = (fi , . . . , f,) and J = (gl, . . . , g,) are ideals in

k[xl , . . . , x,] , t hen consider:

First compute the Grobner basis of the above ideal in k[xl, . . . ,x,, t ] , with respect

to a lexicographical monomial ordering where t is greater than xi. Elements of this

basis not in terms of t form a basis for I n J. Using Elimination Theory it can be

proven that a basis for I n J can always be computed in this way. Geometrically,

THEOREM A.2.3. If I and J are ideals in k[xl, . . . , x,], then V(I n J ) = V(I ) U

V( J )

PROPOSITION -4.2.2. If V and W' are varieties with V c W , then W = V u

(n) . Where CV - V is the smallest dgebmic van'ety cont~inzng CV - V ; ÇV - V

is called the Zariski closure of W - V .

A.3. A Discussion on Kinematic Analysis as Application of Algebraic Geometry and Commutative Algebra

This section is abstracted from [IO] pp. 190-206.

Although the ideas presented above may seem abstract, their relevance to the study

of kinematics will be illustrated with a simple example. Obviously. one can only study

manifestly holonomic systems when applying the theorems introduced here. In other

words, one models the system as a set of polynomial constraints f1 = O,. . . , f, = O

which defines an ideal I ; we write

But this ideal, using the .\lgebra -us.- Geometry dictionary. defines a variety (i. e . .

a geornetric object):

At this point one can compute I(W) (i.e., a Grobner basis) using lericographical

ordering to eliminate variables and arrive at the best algebraic representation of the

system. But an interesting alternative is to remove those constraints which define the

metric geometry in the system and interpret them differently. In kinematic analysis

one usually must deal with equations like:

sin (O)? + COS (0)" 1

q;++Q:+q;+q; = 1

which are independent of the design parameters of the system. By removing them

from the original definition of the ideal I and having them define an ideal J we can

still arrive a t the desired result but through a dserent algorithm. First of al1 the

ideal J would look like:

Then the ideal quotient of I by J is computed as

where W = V(L) is the variety describing the solution set of the original system.

The ideal quotient, in this example, interprets the polynomials defining J as non-

zero conditions. In general, there exists a computational way of removing unwanted

solutions if one can bundle them into a variety. The following is a simple but non-

trivial example. Consider the variety:

The variety is sketched below. Y is union of two curves. It is apparent that the z-axis'

V(x, g), vanishes on V ; so, how do we express V - V(x, y) algebraically? According

to eqn. (1.1) we need to compute

The above can be written as

where I = (xz - y', x3 - YZ). TO compute I : x we first calculate I n (2) using

Theorem -4.22 with lexicographical order r > y > x (z'z - zy2 x4 - xyz 13Y - x z 2 \

1 : x = *

Similarly, if we compute I : (y) we find that I : (y) = I : (x); therefore,

CC' = V(xz - y2?x3 - yz,x2y - z2) is a curve parametrized by ( t3. t\tt). So the

equation x2y - z2 may be redundant but it weeds out al1 of the z-auis except the

origin. So I.V is the Zariski closure of V - V(x, y)

Establishing a language between algebra and

metric operations to their algebraic equivalent.

CVe finish this section with the ALGEBRA -us.-

lished in [IO].

(2 . e., CC' = V - V(x, y) ).

geometry allows one to map geo-

GEOMETRY dictionary as pub-

ALGEBRA GEOMETRY radical ideals variet ies

I -+ I(V) t

V ( 4 v

addition of ideals intersection of varieties I + J + n V( 4

JI(v) + J(ÇV) t t' n cv p;oduct of ideais union of varieties

I J + v(I) u v( J )

,/I(v)v(w) t V U CV intersection of ideals union of varieties

I n J + V ( I ) u V( J )

I(V) n I(w) + V u W quotient of ideals difference of varieties

I : J + V ( I ) - V( J )

I(V) : I(W) t v- - bV elimination of variables projection of varieties JI^ k[x1+1, . . . ,xn] t-t rt (V(0 )

A.4. PROJECTIVE ALGEBRAIC GEOMETRY

A.4. Projective Algebraic Geometry

This section is abstracted from [IO] pp. 349-390.

Projective spaces extend systems of polynomial equations to completion. Con-

sider the trivial example of an affine variety, its partial solutions and their projection

niap.

( z y - 1 = 0 , x z - 1 = 0 ) + y - z = O + { ( l / a , a , a ) : a f O )

A qualification is required to exclude the spurious member a = O from the one pa-

rameter solution family (x, y, 2 ) . Using w as a fourth homogeneous coordinate, we

have

(xy- w2 = O , x z - w 2 = O ) -t y - - = O + {(l : a 2 : a 2 : a ) }

The Projection Eztension Theorem, proven in Cox, Little, and O'Shea [IO], says that

systems of homogenous polynomial equations always extend to completion..

REFERENCES

Salmon, G., 1885, "Higher Algebra" , Chelsea Publishing Co., New York. Klein, F., 1948, "Elementary Mat hematics fkom an Advanced Standpoint (Geome- try)", Translated by Hedrick & Noble, Dover. Shuster, M. D., 1993, "A Survey of Attitude Representations"! Journal of the As- tronautical Sciences, Vol. 4, No. 4, pp. 439-517. Tait. P. G., 1900, "Quaternions", Scientific Papers, Cambridge University Press, Vol. 2, pp.445-456. Tait, P. G.. 1900, "On The Intrinsic Nature of the Quaternion Method", Scientàfic Pape~s, Cambridge University Press, Vol. 2! pp.392-398. Bottema, 0. and Roth, B., 1990, "Theoretical Kinematics" , Dover. Husty, M., 1994, "An Algorithm for Solving the Direct Kinematics of Stewrt-Gough- type Platforms," Dept. of Mechanical Engineering and Centre for Intelligent Ma- chines Technical Report, McGill University, Montreal. Grassmann, H., 1995, "A New Branch of Mathematics", Tkanslated by Kannenberg, Open Court. Angeles, J., 1996, ''Fundamentals of Robotic Mechanical Systems" , Springer-Verlag. Cox. D., Little, J. and O'Shea, D., 1996, "Ideals, Varieties, and Algorithms", S pringer- Verlag. Clifford, W. K., 1873, "Preliminary Sketch of Bi-Quaternions," Ptoc. London Math- ernatical Society, Vol. 4, pp. 381-395. Daniali, H. R. M., Zsombor-Murray, P. J., and Angeles, J., 1993, "Dual Numbers", Dept. of Mechanical Engineering and McGill Research Centre for Intelligent Ma- chines, McGill University, Montreal. Ravani. B. and Roth, B., 1984, "Mappings of Spatial Kinematics" ? Journal of Mech- anisms, Transmissions, and Automation in Design, Vol. 106, pp. 341-346.

Yaglom, 1.. 1996, "Felix Kein and Sophus Lie", Birkhauser.