Post on 17-Feb-2018
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..
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