Several algebraic properties for systems in which either or both the input and output vectors have elements with different physical units. The condition son linear transformation A for a physically consistent noncommensurate system, u=Ax, are given. Linear noncommensurate systems do no generally have eigenvalues and eigenvectors. The requirements for noncommensurate linear systems do not have a physically consistent singular value decomposition. The manipulator Jacobian maps possibly noncommnesurate robot joint-rate vectores into noncommensurate twist vectores. The inverse velocity problem is often solved through the use of the pseudo-inverse of the Jacobian. This solution is generally scale and frame dependent. The pseudo-inverse solution is physically inconsistent, in general, requiring the addition of elements of unlike physical units. For some manipulators there may exist pointsâ€”called decouple pointsâ€”at which the pseudo-inverse of the Jacobian is physically consistent for all frames at these points. In decouple frames, the pseudo-inverse is shown to be equivalent to the weighted generalized-inverse with identity metrics. An entire class of nonidentity metrics used with the weighted generalized-inverse are shown to give identical solutions to the pseudo-inverse solution at decouple points. At decouple points, the twist and wrench spaces can be decomposed into two metric-independent subspaces. This decomposition is accomplished with kinestatic filtering projection matrices. The Mason/Raibert hybrid control theory of robotics is shown to be useful only for frames located at decouple points and is not optimal in any objective sense. The current manipulability theory, which depends on the eigensystem of various functions of the Jacobian, is shown to be invalid. Two new classes of manipulators are introduced, self-reciprocal manipulators and decoupled manipulators. The twists of freedom of a self-reciprocal manipulator are reciprocal. The class of self-reciprocal manipulators consists of planar manipulators, spherical manipulators, and prismatic-jointed manipulators. Decoupled manipulators are show to decouple at every point. The manipulators of this class are planar manipulators, prismatic-jointed manipulators, and SCARA-type manipulators. Results that are generalized from decoupled manipulators often prove to b invalid for manipulators that do not decouple at every point.

Thesis:

Thesis (Ph. D.)--University of Florida, 1995.

Bibliography:

Includes bibliographical references (leaves 119-123).

General Note:

Typescript.

General Note:

Vita.

Statement of Responsibility:

by Eric M. Schwartz.

Record Information

Source Institution:

University of Florida

Holding Location:

University of Florida

Rights Management:

Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.

ALGEBRAIC PROPERTIES OF NONCOMMENSURATE SYSTEMS
AND THEIR APPLICATIONS IN ROBOTICS

By

ERIC M. SCHWARTZ

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

UNIVERSITY OF FLORIDA

1995

To My Wife Gabriella
To My Parents Marilyn and Seymour
To My Parents Marilyn and Seymour

ACKNOWLEDGMENTS

I will always be grateful for the opportunity I have had to work with Professor

Keith L. Doty. Our discussions on everything from robotics, to politics and religion

have made my work especially enjoyable. This dissertation would not have been

possible without his direction and support. He has encouraged and enhanced my

growth as an engineer and as a member of the species. I will never forget the summer

in Italy that he made possible and that through him I met my wife.

I would like to thank Professor Thomas E. Bullock for making the considerable

time we have spent working together in the controls area both informative and in-

teresting. I have learned a great deal from him and have very much enjoyed doing

so.

My thanks are offered to the Electrical Engineering Department for supporting me

through the first half of my graduate studies and to the Electronics Communication

Laboratory for supporting me through most of the second half.

I would like to express my gratitude to my friends in the Machine Intelligence

Laboratory, both past and present, who have allowed me to debate, listen, learn and

party with them. In particular, I, thank Kimberly Cephus for her technical advise,

moral support, and friendship.

Finally, I would like to thank my family, especially my parents Seymour and

Marilyn, for being proud of me and for doing whatever it was they did to make me

me. And my very special thanks go to my wife Gabriella for putting up with my long

times away from her, for pushing me along, and for choosing me for her husband.

TABLE OF CONTENTS

ACKNOWLEDGMENTS ............................. ii

LIST OF TABLES .................... ............. vi

LIST OF FIGURES ................................ viii

KEY TO SYMBOLS .................... ............ ix

ABSTRACT ..................... ............... xi

CHAPTERS

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

1.1 Noncommensurate Vector Spaces ................. ..... 3
1.2 The Pseudo- and Generalized-Inverses ................. 14
1.2.1 The Moore-Penrose Pseudo-Inverse ............... 15
1.2.2 The Weighted Generalized-Inverse ................ 16
1.3 Eigenvalues, Eigenvectors and SVD .................. 18

2 LINEAR NONCOMMENSURATE SYSTEMS ................ 19

2.1 Eigensystem In Noncommensurate Systems ............... 20
2.2 Conditions for Physically Consistent Eigensystems ........... 21

3 PHYSICAL CONSISTENCY OF JACOBIAN FUNCTIONS ........ 24

3.1 Inappropriate Uses of the Euclidean Norm in Robotics ....... 24
3.2 Physical Consistency of JrJ and JJ. ................ 28
3.2.1 Consistency of lu = Ax ...................... 32
3.2.2 Invalid use of Eigensystem and SVD of JJ ... ..36

4 INVERSE VELOCITY KINEMATICS . ..... 40

4.1 Physical Consistency of Jt ................... ..... 42
4.1.1 Rotations and Consistency of Jt . ... 42
4.1.2 Translations and Consistency of Jt . ..... .. 44
4.1.3 Consistency of Jt in All Frames . ..... 45
4.2 Invariance of Jt to Scaling . ..... ...... 47
4.3 Equivalent Generalized Inverses . ..... .. 54

5 MANIPULATOR MANIPULABILITY .....

6 DECOMPOSITION OF SPACES ........

6.1 Projections and Kinestatic Filters .....
6.2 Twist Decomposition ............
6.3 Wrench Decomposition ...........
6.4 Hybrid Control ...............
6.5 Decomposition with Ray Coordinate Twist
6.6 Space Decomposition at Decouple Point .
6.7 Self-Reciprocal Manipulators .

Space .
, .
,. ,

7 SUMMARY AND CONCLUSIONS . .

APPENDIX

A D-H PARAMETERS FOR VARIOUS MANIPULATORS. .

REFERENCES ...........................

BIOGRAPHICAL SKETCH ....................

LIST OF TABLES

D-H parameters for GE P50 manipulator. ................. 29

Physical units of Det[J'J] for various non-redundant manipulators. 30

Physical units of Det[JJr] for various redundant manipulators. ... 32

D-H parameters for

D-H parameters for

D-H parameters for

PR virtual manipulator .....

the SCARA manipulator ..

the PRP Small Assembly Robot

. . 44

. . 46

(SAR). ..... 55

6.1 D-H parameters for a non-planar RRR manipulator .

D-H parameters for

D-H parameters for

D-H parameters for

D-H parameters for

D-H parameters for

D-H parameters for

D-H parameters for

D-H parameters for

D-H parameters for

D-H parameters for

D-H parameters for

D-H parameters for

D-H parameters for

PR virtual manipulator. . ... 106

an RR manipulator. . ... 107

a general RRR manipulator. . ... 107

the Planar RRR manipulator. . .... 108

the Spherical RRR manipulator. ... 108

the Non-planar RRR manipulator. ... 109

the PPP orthogonal manipulator. ... 109

the PRP Small Assembly Robot (SAR). 110

the RPR manipulator. . ... 110

the RRRP-1 SCARA manipulator. ....... 111

the RRRP-2 manipulator. . .... 111

the RRRP-3 manipulator. . .... 112

5R GE-P50 manipulator. . .... 113

A.1

A.2

A.3

A.4

A.5

A.6

A.7

A.8

A.9

A.10

A.11

A.12

A.13

A.14 D-H parameters for

A.15 D-H parameters for

A.16 D-H parameters for

A.17 D-H parameters for

A.18 D-H parameters for

A.19 D-H parameters for

the 7R Redundant Anthropomorphic Arm 114

the 7R CESAR Research Manipulator. 115

the 7R K-1207 Robot Research Arm. ...... .115

the 7R PUMA-260+1 Spherical Wrist Manipulator. 116

the 3P-4R Redundant Spherical Wrist Robot. 117

the 2R-P-4R GP66+1 Manipulator. ... 117

LIST OF FIGURES

4.1 Peg-in-the-hole with PR virtual manipulator. . ... 45

4.3 Small Assembly Robot (SAR) ...................... .56

6.1 Decomposition of the twist space in frame i into decoupled subspaces. 81

KEY TO SYMBOLS

Symbol or
Variable Definition
t for Xt, the Moore-Penrose pseudo-inverse of X
# for X#, the weighted generalized-inverse of X
| I for jx| where x is a vector, |Ix = V
I IL for IXIMX where x is a vector, Ixi = V/x Q Mx
x for x x y, the vector cross product of vectors x and y
o for x G y, the inner (or dot) product of vectors x and y
o for X o Y, the klein (or reciprocal) product of screws X and Y
E for X E y, the direct sum of the subspaces X and Y
M M
ME for X E Y, the direct sum of the M-orthogonal subspaces X and y
possibly equal, often physically inconsistent
def
de defined as
S numerically equal to
(.)(i,) for matrix X(ij), element of X in i-th row, j-th column
()(.j) for matrix X(.,j), the j-th column of X
()(,.) for matrix X(i,.), the i-th row of X
[']r,c for [X]r,c,an r x c matrix with all units identical to the units of L
[O],c r x c matrix of zeros
[*]b for [X], matrix where the column vectors constitute a basis for X
[.]' the transpose operator
On zero vector of dimension n
a angle between successive joint axes projected on plane with common
normal used in D-H parameterization
A orthogonal 6 x 6 matrix that converts between ray and axis
coordinates
0 angle about a joint axes used in D-H parameterization
Ki cos(ai)
0i sin(ai)
r the generalized-force vector containing n joint forces and/or joint
torques corresponding to prismatic and/or revolute joints
w angular velocity 3-vector
A wrench coordinate transformation matrix
a perpendicular distance between successive joint axes used in D-H
parameterization

Symbol or
Variable Definition
B skew-symmetric 3 x 3 translation matrix of b
b translation 3-vector
Ci+j cos(0i + 0j)
TD defect manifold
d distance along joint axis used in D-H paramtetrization
EI matrix such that XE, is the column-reduced echelon form of X
f force 3-vecor
G twist coordinate transformation matrix
Ib body's inertia tensor at the center-of-mass expressed
in principal corrdinates-a diagonal matrix
Ij j x j identity matrix
J manipulator Jacobian that transforms joint rates into twists, V = Jq
J, first three rows of J, such that v = Jq
J, rows four through six of J, such that w = J,4
[L]r,c r x c units matrix with all units of length
n number of joints in manipulator
n moment of force 3-vector
Null[A] null space of matrix A, i.e., all x such that Ax = 0
Q joint-rates vector space
R 3x3 rotation matrix
1R radical subspace
R" commensurate m-space over reals
Range[A] range space of matrix A, i.e., all y such that y = Ax
S or Si rotation vector of screw i
So or Soi translation vector of screw i
S, change of units scaling matrix for joint rates
S, change of units scaling matrix for twists
si+j sin(0i + 0j)
T generalized (joint) forces vector space
[U],,c r x c unitless units matrix
units[.] the physical dimensions of the matrix inside the brackets
V twists in Plucker ray coordinates, V = [VT, w ]7
V twists screw space
Vf twists of freedom subspace
,Vf twists of nonfreedom manifold
v linear velocity 3-vector
W wrench in Plucker axis coordinates, W = [fr, n']T
W wrenches screw space
'We wrenches of constraint subspace
Wac wrenches of nonconstraint manifold
z unit vector in z direction ([0, 0, 1]')

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

ALGEBRAIC PROPERTIES OF NONCOMMENSURATE SYSTEMS
AND THEIR APPLICATIONS IN ROBOTICS

By

ERIC M. SCHWARTZ

May 1995

Chairman: Keith L. Doty
Major Department: Electrical Engineering

Several algebraic properties are given for systems in which either or both the input

and output vectors have elements with different physical units. The conditions on

linear transformation A for a physically consistent noncommensurate system, u = Ax,

are given. Linear noncommensurate systems do not generally have eigenvalues and

eigenvectors. The requirements for a noncommensurate system to possess a physically

consistent eigensystem are presented. It is also shown that noncommensurate linear

systems do not have a physically consistent singular value decomposition.

The manipulator Jacobian maps possibly noncommensurate robot joint-rate vec-

tors into noncommensurate twist vectors. The inverse velocity problem is often solved

through the use of the pseudo-inverse of the Jacobian. This solution is generally

scale and frame dependent. The pseudo-inverse solution is physically inconsistent, in

general, requiring the addition of elements of unlike physical units. For some manip-

ulators there may exist points-called decouple points-at which the pseudo-inverse

of the Jacobian is physically consistent for all frames at these points.

In decouple frames, the pseudo-inverse is shown to be equivalent to the weighted

generalized-inverse with identity metrics. An entire class of nonidentity metrics used

with the weighted generalized-inverse are shown to give identical solutions to the

pseudo-inverse solution at decouple points.

At decouple points, the twist and wrench spaces can be decomposed into two

metric-independent subspaces. This decomposition is accomplished with kinestatic

filtering projection matrices.

The Mason/Raibert hybrid control theory of robotics is shown to be useful only

for frames located at decouple points and is not optimal in any objective sense.

The current manipulability theory, which depends on the eigensystem of various

functions of the Jacobian, is shown to be invalid.

Two new classes of manipulators are introduced, self-reciprocal manipulators and

decoupled manipulators. The twists of freedom of a self-reciprocal manipulator are

reciprocal. The class of self-reciprocal manipulators consists of planar manipulators,

spherical manipulators, and prismatic-jointed manipulators. Decoupled manipula-

tors are shown to decouple at every point. The manipulators of this class are planar

manipulators, prismatic-jointed manipulators, and SCARA-type manipulators. Re-

sults that are generalized from decoupled manipulators often prove to be invalid for

manipulators that do not decouple at every point.

CHAPTER 1
INTRODUCTION

Optimum, according to Webster [58], means "best; most favorable." In real phys-

ical systems, to say a solution is optimum or optimal one must specify the criteria for

optimality.

The theory of hybrid control of manipulators developed by Mason in 1978 [41, 40]

and then tested and expanded by Raibert in 1981 [51] has been shown by Lipkin

and Duffy [37, 36] and others [1, 19] to be erroneous. Lipkin and Duffy explain that

the failure of Mason and Raibert's hybrid control theory (MRHCT) is in their use

of orthogonality. In MRHCT, the orthogonality of two vectors with terms of unlike

units is used when it is easily seen that the inner product of these vectors in not

invariant to scaling. Because so many authors continued to use MRHCT, Duffy [22]

found it necessary to write an editorial debunking this theory.

The problem with MRHCT, in this author's view, is that the terms of their optimal

solution were not sufficiently defined. An exploration of the meaning of their optimal

solution would have shown that the solution is based on minimizing the Euclidean

norms of two non-Euclidean vectors.

In 1989, Doty noticed and eventually published research [14, 19] that the Moore-

Penrose pseudo-inverse solution in the robotics inverse velocity problem gives results

that are dependent on the frames of reference. Doty's algebraic viewpoint, together

with Duffy and Lipkin's geometric results using screw theory, suggested a further

investigation of the possible non-invariance of solution techniques in several areas of

robotics and applied mathematics in general.

This dissertation is based in part on correcting the inappropriate use of the pseudo-

inverse in the field of robotics. Researchers such as Doty [18], Duffy [22], Lipkin

and Duffy [37, 36], Lipkin [35], Griffis [26], and Schwartz [54, 53] have shown the

fallacy of incorrectly applying optimization techniques to robotics problems without

a judicious investigation of the underlying metrics incorporated. This dissertation

intends to formalize and explain these problems and offer consistent solutions and

interpretations of these solutions.

Each of these problems involves solving a set of linear equations which by some

manipulation can be put in the form u = Ax, where A is nonsquare or singular. More

often than not, a multitude of robotics researchers including [12, 23, 29, 32, 38, 39,

43, 44, 45] have solved these problems by using the pseudo-inverse. The inconsistent

results generated through the use of the pseudo-inverse (without a metric or metrics)

are explained in this dissertation.

The robotics literature [10, 31, 46, 57, 59, 60] also makes use of the eigenvalues,

eigenvectors, or singular values of matrices whose eigenvalues and singular values are

not invariant to changes in scale or coordinate transformations, and are therefore not

true "eigensolutions". The eigensolution problem is also discussed in this dissertation.

The basic mathematics and terminology of robotics and screw theory necessary

for an understanding of the issues discussed will be introduced in this chapter. There

is no original work in this chapter other than some basic definitions with regard to

noncommensurate systems. Since a general understanding of the Euclidean vector

norm, the pseudo-inverse, the weighted generalized-inverse, eigenvalues, eigenvectors,

and singular value decomposition are paramount to understanding this dissertation,

these topics will also be presented and examples (with references) of their use in

robotic systems will be given in this chapter.

1.1 Noncommensurate Vector Spaces

Systems involving elements of different physical units are defined here as non-

commensurate systems. Robotics systems are noncommensurate when they deal with

both position and orientation or have both revolute and prismatic joints. A vector of

elements of unlike physical units is defined as a noncommensurate vector. (The non-

commensurate vector is also called a compound vector [14, 53] and non-homogeneous

vector [15].)

In robotics, the equation that relates joint velocities to twists (1.1) describes a

noncommensurate system,

V = J (1.1)

The manipulator joint-rate vector is

4 = [41, 42, .* qn] (1.2)

where n represents the total number of revolute and prismatic joints of the manipu-

lator. The manipulator's instantaneous twist vector,

V = [v', w'] (1.3)

is composed of the linear velocity v = [vs, vy, vz] and the angular velocity w =

[w,, wy, wz]'. The Jacobian J is a 6 x n matrix, where 6 is the number of coordinates
necessary to describe the position and orientation of a body in space.

The twist represents a noncommensurate vector since the units of v and w dif-

fer. When the manipulator has both revolute and prismatic joints, the joint-rate

vector is also noncommensurate and the manipulator is called a noncommensurate

manipulator.

The vector i'mV,,k represents the twist of a point p, fixed to frame k, and expressed

in frame i coordinates with respect to a fixed frame m. Since the Jacobian i'mJp,k has

columns that are also twists, the superscript i and m and the subscripts p and k

have the same interpretations as in i'mVp,k. When the subscripts p and k and the

superscript m are omitted in 'V and "J, it is understood that k is the end-effector

frame n of an n-jointed manipulator, m is the base frame (frame 0), and point p is at

the origin of frame i, the frame of expression (iV = i,0ViO).

To transform twists or Jacobians to representations in different frames, the twist

coordinate transformation matrix G is used,

'Gp'q [iRj iBpq R (1.4)
= 0]3,3 iRj

where [0]3,3 is a 3 x 3 matrix of zeros and iRj is a rotation transformation which

rotates a vector from frame j into frame i. Since rotation matrices are orthogonal,

the inverse is equal to the transpose, i.e.,

'R11 = 'R = 7 R, (1.5)

(By convention, the term orthogonall matrix" refers to matrices with orthonormal

columns [56].) The matrix 'Bp,, = [ibp,q x] is a skew symmetric matrix that represents
translation from point p to q expressed in frame i. The B matrix is the matrix-form

of the vector cross-product, i.e., Be = b x c, where b and c are arbitrary 3-vectors
and B is defined as
0 -'b, Zb,
'Bp, = b 0 -b (1.6)
-'b, 'ib 0
The vector bp,q = [ib, b, ib]y, is a position vector from point p to point q expressed

in frame i coordinates.

Since B is skew symmetric, it has the following properties:

Bq,p = -Bp,q (1.7)

(Bp,q)" = -Bp,q and (1.8)

RjjBp,q = 'Bp,qRj (1.9)

With the above equations it is easily shown that

(iGP)-1 =jGiq' (1.10)

Note that ('Gp)T JG'.
The expressions for the frame transformations of twists and Jacobians are

iVp, = (iG,) jVq,k and (1.11)

Jp,k = (iG ) Jq,k (1.12)

The shorthand notation 'Gj is used when the transformation has no translation and
the notation iGpq is used when the transformation has no rotation.
The twists that a manipulator can accomplish with joint-rate control in a given
configuration are know as the twists of freedom [5, 8, 22],

'iv = Range[iJ] (1.13)

where V represents a twist manifold and i is the frame of expression. The twist of
freedom manifold is a subspace.
It is important when writing vectors, matrices, and manifolds to make the frame
of expression clear. In this dissertation, the expression frame, if not explicitly written
as a leading superscript, will be otherwise described in the context of the discussion.
Note that throughout this dissertation, a calligraphic symbol (such as V) repre-
sents a manifold (or set) of vectors or screws. Therefore, X = {Xi} is the manifold
of vectors or screws Xi, for various i. The column vectors of the matrix [X]b con-
stitute a basis for X. The matrix E. converts the basis set, [X]b, to a matrix in
column-reduced echelon form [56], [X]bE,.
The application of a wrench W at the end-effector of a static serial manipulator
will induce a balancing generalized-force vector r,,

T = J'W (1.14)

where a wrench, W = [f', n']', is the noncommensurate 6-vector composed of the

two 3-vectors of forces f and moments n. A generalized-force vector, 7, is the n-vector

of joint torques (for revolute joints) and/or joint forces (for prismatic joints).
The matrix 'Wa,p = [ifp, in represent a wrench at point p expressed in frame

i, with the moments taken about point a. When the subscript a is omitted it is

understood that the point a is at the point p, so that iWp = iW,,,. When both

subscripts are omitted the origin of the frame is the point at which moments are
taken, i.e., 'W = 'W,
Wrenches transform via the wrench coordinate transformation matrix A,

W, = ( Apq) Wq (1.15)

where
B Rj []3,3 (1.16)
3 ~ B q 'Rj

Equations (1.7)-(1.9) can also be used to show that

('A"-' = A'p ,and (1.17)

iG?)' = 'Aj (1.18)

The wrenches applied at the end effector that require no joint forces for balancing
are know as the wrenches of constraint, 'We, and form a subspace,

W, = Null['J'] (1.19)

These wrenches will cause no joint motion when applied to a static manipulator.
Manipulators (of at least 6 joints) in configurations with Jacobian of rank 6 have

no constraint wrenches, i.e., some nonzero joint forces are required to balance every
possible wrench.
Notice that the above twists and wrenches are screws (defined below) expressed in
axis coordinates and ray coordinates [27, 30], respectively. The designations of Pliicker

ray coordinates and Pliicker axis coordinates are based on the original formulation of
screw theory by Ball in 1900 [5]. Ball defined lines in two ways, each independently
leading to coordinate system definitions: the join of two points lead to ray coordinates
and the meet of two planes lead to axis coordinates. These sets of identical but
reordered coordinates are know as the homogeneous Plicker line coordinates. The

distinction is only necessary when lines or screws in different Plficker coordinates are
used simultaneously, as is the case with the traditional algebraic descriptions of twists
and wrenches previously defined.
A screw $ is defined as a line with an associated pitch h. For example, the
motion defined by a physical screw being advanced into a pre-threaded hole can be
characterized by the following screw (in axis coordinates),

$ [= hSS = ] (in axis coords), (1.20)

where the line passes through the coordinate system origin. (A more general de-
scription is given in (1.24) below.) The vector S is a commensurate 3-vector in the
direction of linear motion and the rotation is about this axis using the right-hand-rule.
For every 0 radians of rotation, the screw advances by hO in the S direction.
A screw may also be defined as a linear combination of unlimited lines [5, 25]. An
unlimited line L is defined with two vectors: a unit vector S in the direction of the

line and a vector r from the coordinate system origin to any point on the line,

Lais r = [ (in axis coords). (1.21)

Lines also have the property that S O So = S 0 (r x S) = 0. The ray coordinate
version of this same line is

LraY= [ ] = [S ](in ray coords). (1.22)
r X S SIo

A linear combination of two lines in axis (ray) coordinates creates a screw in axis

(ray) coordinates,

r$ axis = yaxis 2Laxis 7Y1(r X Si) + 72(r2 X S2) r SOr (1.23)
71 S1 + 7 72 S2 -r I S(

For screws, Sr G So, = hr, where hr the pitch of the resultant screw. Therefore screws
are not lines except in the special case when the pitch is zero. The resultant screw
can be written as
axis (r, x S,)+ hrSr
r S r (1.24)

The differences in equations (1.20) and (1.24) are due to different coordinate
system definitions. If r, = 0, i.e., the coordinate system origin is on the line of
rotation, the two equations are identical. A general screw can always be converted to
a "pure screw" as in (1.20) by a twist coordinate transformation for axis coordinate
screws or a wrench coordinate transformation for ray coordinate screws. For example,
a twist coordinate transformation will transform the pure axis coordinate screw into
a general axis coordinate screw,

hw] [ R BR hw hRw + BRw(12
[w ]= [01, R W Rw (1.25)

Note that coordinate translations (B) do not affect the angular velocity vector-
the bottom component in the right-hand-side of equation (1.25). Although rotations
affect both parts of the screw, if there is no translation, the rotation will not affect
the apparent purity of a screw viewed in each of the frames.
As stated above, the pitch of a screw can be found simply by

So OS
h = SI2 IS -0. (1.26)

If S is the zero vector, the screw is said to have infinite pitch and (1.25) is replaced
by
G[ v R BR v Rv (1.27)
03 [0]3,3 R 03 03

Note that the translation B has no affect on the resulting screw representation. If

the pitch is zero, So is zero and the screw represents a pure rotation.

The translation that will move a general axis screw to a pure axis screw is

So xS
b = ,1 ISl 0. (1.28)

where B can be found from b with (1.6).

All rigid body motion is instantaneously equivalent to a screw motion twist [9].

The twist defined previously, V = [vo, w], is equal to a linear velocity vo (referenced

to some origin 0) and an angular velocity w, a free vector [25] and is here defined as a

screw in Pliicker axis coordinates [48, 49]. A twist can also be represented in Plicker

ray coordinates, Vray = [w, vo].

Similarly, a wrench is instantaneously equivalent to a force and moment on a rigid

body. The Pliicker ray coordinates of a wrench, W = [f, no], is equal to a force f in

the direction of the wrench and a moment n referenced about origin 0. A wrench can

also be represented in Plucker axis coordinates, Wa's = [mo, f].

Unless otherwise noted, twists will be expressed in Pliicker axis coordinates and

wrenches will be expressed in Plicker ray coordinates.

The matrix A [36] transforms a screw or line in axis coordinates to a screw or line

in ray coordinates and a screw or line in ray coordinates to a screw or line in axis

coordinates,

$ray = A$as (1.29)

$axis = A$ray (1.30)

A = ,[013,3 [0 (1.31)
/3 [013 ,

The matrix A is an unitary matrix (and therefore also an orthogonal matrix) with

the properties

A = A-1 A = A' AA = (1.32)

y

The matrix A is an example of a more general transformation, defined as a correlation

[27] that maps an axis screw to a ray screw (or a ray screw to an axis screw). A

collineation maps a ray screw to a ray screw (or an axis screw to an axis screw).

The reciprocal or Klein product [5, 22] of any two screws in identical axis or ray

coordinates-twists Vi and V2, for example-is defined as

VI o V2 = V1( AV2 = VAV2 (1.33)

= v1i 0 2 +2 02 (1.34)

where 0 represents the Euclidean inner or dot product.

The Klein product of a screw in axis coordinates, and a screw in ray coordinates

V and W is

VoW=V W=V'W=v f +w On (1.35)

Notice that no A matrix is needed in the expansion of the Klein product of a twist

and a wrench, whereas the A matrix is needed in the expansion of the Klein product

of two twists or two wrenches. The Klein product of a twist and wrench of the end-

effector of a serial manipulator gives the instantaneous virtual power (work) [36] that

the manipulator end-effector contributes to the environment.

A well known important characteristic of the reciprocal product is that it is in-

variant to coordinate transformations. This is shown in the following theorem and

proof [5]. The proof is given to provide the reader an understanding of the notation

and mathematics involved.

Theorem 1 The reciprocal product of a manipulator twist and wrench expressed at the

same point and in the same coordinate system is invariant to coordinate transforma-

tions.

Proof

'Vp,q = (iGi ')-j1 iVq (1.36)

= (iGj') jGpq 'V,q as shown in (1.10) (1.37)
'Vo = ( 'G G Vq 0o 'Wp (1.38)
= j p,q iV o (iG''0Gp) 'W (1.39)
= jG' p,q (A ) Wp ,, as shown in (1.18) (1.40)

= Vq,k o Wq as shown in (1.11) and (1.15). (1.41)

The twist or screw motion created by a single revolute joint i is a pure rotation,

-Vre = o, o, o, O, 0, = (1.42)

where the above equation is expressed in the frame of the previous joint i 1, and
z is the vector [0, 0, 1]7. The twist coordinate transformation matrix enables this
screw to be expressed in different coordinate frames-as shown in (1.25). To express
this screw in various coordinate frames, the twist coordinate transformation matrix
may be employed as shown in (1.25). When the frame is translated to a frame j that
is located by vector b from the frame i 1, the screw motion is
SVrev = b (1.43)

S B] (1.44)

= [ -b'il (1.45)

where B is the screw symmetric matrix of (1.6) corresponding to the translation
vector b = [b., by, bj].
When the frame is rotated to a frame k with no translation from frame i 1, the
screw motion is

The twist or screw motion created by a single prismatic joint i is a pure translation,

i 03 (1.47)
i-1V 10s 0, 0 d 0, 0, 0, = (1.47)

Again the twist coordinate transformation matrix can be employed to express this

screw in various coordinate systems. An arbitrary coordinate transformation kGP'q1
rotates the twist to frame k while the translation has no affect for any p and q,

That translation has no affect on this twist was verified by symbolically performing

the multiplication kGP1 i'-lVPis in (1.27).
Screws can be added to form new screws. In this manner the motion of the end-

effector (or any other point) of a serial manipulator may be found by a summation
of the screws of each of the joints,

V = VI+V2+--.+V, (1.49)

= 41$1 + 42$2 + + 'n$n (1.50)

= [$1, $2, *'', $n]q (1.51)

= J (1.52)

where q is the vector of manipulator joint rates of (1.2) and (1.52) is identical to
(1.1). To perform the addition of screws, it is first necessary to reference them to the

same coordinate frame and point via the appropriate screw coordinate transformation
matrices, e.g., the summation of the screws in (1.50) is actually accomplished with
the equation
iV = E4iiGi'",n (1.53)
j=1
Any twist can be constructed by six or less independent screws each representing

either a prismatic or a revolute motion. Therefore a virtual manipulator can always
be constructed to instantaneously accomplish any twist. Griffis [25] defines a virtual

manipulator as any imaginary serial manipulator "whose joint displacements and

speeds uniquely describe any permissible twist (Vf) and any permissible position and

orientation of its end-effector." The permissible end-effector wrenches (We) together

with the twists completely describe the instantaneous kinematics of a real or virtual

manipulator end-effector.

Theorem 2 below, given in [5], shows that the reciprocal product of any twist of

freedom and any wrench of constraint must be zero. The proof is shown to give the

reader an insight to the concept of reciprocity.

Theorem 2 The Klein or reciprocal product of Vf and We, is zero, i.e.,

V o We = 0 VV, E Vf and VWe We (1.54)

Proof

Vf = J4 VVf E Vf and some {q} (1.55)

(Vf) = 4J (1.56)

(VM) W = FJTW (1.57)

Now let W be a constraint wrench We E We, so that

(Vyf)We = 4JTWe (1.58)

But J'We = 0 by definition in (1.19), so

(Vy)TWe = 0 (1.59)

But by the definition of the Klein product in (1.35),

(Vf)TWe = Vf o W (1.60)

so that Vf o We = 0.
E

This means that the manipulator can do no work with any wrench of constraint

or, alternatively, can not move with the screw motion of any ray coordinate constraint

wrench interpreted as an axis coordinate twist.

The reciprocity relationship between Vf and We has been inadvertently (and in-

appropriately) used by researchers to characterize the entire space through the use of

the direct sum decomposition of the 6-space of position and orientation.

The fundamental theorem of linear algebra [56] states that

~'m = Range[A] E Null[A'] (1.61)

where m is the number of rows of A. The symbol ( represents the direct sum and

implies that Range[A] n Null[A'] = {0} and Rm = Range[A] U Null[A']. Applying
this theorem to robotics by letting A be the Jacobian can be misleading,

'6 I Range[J] Null[J] (1.62)

Since J has physical meaning, with terms not all of the same units, the implication
of this theorem applied to robotics is that the total space is a combination of axis

coordinate (twists) and ray coordinate (wrenches). The subspaces Range[J] and
Null[J'] are noncommensurate. What does it mean to decompose a vector (or screw)

into the sum of an axis coordinate vector and ray coordinate vector? This problem

will be addressed in Chapter 6.

1.2 The Pseudo- and Generalized-Inverses

The Moore-Penrose pseudo-inverse and the weighted generalized-inverse can both

be used to solve linear equations. Of course each of the solutions is based on different

optimality conditions for their solutions.

1.2.1 The Moore-Penrose Pseudo-Inverse

The Moore-Penrose pseudo-inverse gives a unique minimum norm least-squares

solution to a linear equation,

u = Ax, (1.63)

for example. The pseudo-inverse of A (A E R(mxn)), is denoted At and has the

following properties [6, 34]:

AAtA = A (1.64)

AtAAt = At (1.65)

(AAt)' = AAt (1.66)

(AtA)' = AtA (1.67)

The pseudo-inverse can be found through a full-rank factorization of A, A = FC,

where F E R(mxr) has full column rank r and C E (rxn) has full row rank r. The
pseudo-inverse of A can be expressed as

At = C'(FrAC')-IF' (1.68)

= CT(CC')- (F'F)-1F' (1.69)

SCtFt (1.70)

The unique minimum norm least-squares solution to (1.63) is therefore

= Atu (1.71)

The solution xz, is a least-squares solution in that the residual (if any), lu Axl,
is minimized, where I -I is the Euclidean vector norm (see equation (1.72)). The
solution x, is minimum norm since any other solutions xl to Ax = u has Euclidean

norm |Il > Ixs|.
A least-squares solution is obtained if (1.64) and (1.66) are true and the solution
is minimum norm if (1.64) and (1.67) are true [6].

It is a fortunate fact that the least-squares solution and the minimum norm solu-

tion are identical for linear systems and equal to the pseudo-inverse solution.

The Euclidean norm of a vector x E R" (also known as the square root of the

Euclidean inner-product of x with itself) is defined as

XiJ = + 2 X12
n
1X12 = < x, >= x = x = (1.72)
i=1

If matrix A has full row rank or full column rank, (1.68)-(1.70) has the simplified

solutions

At = A7(AA')- A full row rank, and (1.73)

At = (AA)-1A' A full column rank. (1.74)

These equations are derived directly from (1.69), substituting F = Ir when A has full

column rank and C = Ir when A has full row rank. (Matrix Ir is the r x r identity

matrix.)

1.2.2 The Weighted Generalized-Inverse

The weighted generalized-inverse gives a unique minimum Ms-norm least M,,-

squares solution to a linear equation. The weighted generalized-inverse of A (called

the generalized-inverse throughout the rest of this dissertation), is denoted A# and

has the following properties [6, 19]:

AA#A = A (1.75)

A#AA# = A# (1.76)

(MAA#)' = MAA# (1.77)

(MA#A)' = M,A#A (1.78)

The matrices M, and M, are metrics. A metric is a symmetric positive definite

matrix.

The generalized-inverse of A [6, 7, 19], with the same full-rank factorization

A = FC discussed previously, is

A# = M'ICT(F'MuAM-IC')-1F'M, (1.79)

= [Mx1C'(CM 1C')-1] [(F'MuF)-'F'M] (1.80)

= C#F# (1.81)

where F# and C# are defined by (1.81) and the bracketed expressions in (1.80).

The unique minimum M,-norm least Ms-squares solution to (1.63) is therefore

x, = A#u (1.82)

The solution xz, is a least Mu-squares solution in that the residual (if any),

Iu AxIM., is minimized, where I[IM is defined below in (1.83). The solution x, is min-
imum Ms-norm since any other solutions xl to Ax = u has Ms-norm xiaIM, > IxslM.

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