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Algebraic properties of noncommensurate systems and their applications in robotics

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Title:
Algebraic properties of noncommensurate systems and their applications in robotics
Creator:
Schwartz, Eric Michael, 1959- ( Dissertant )
Doty, Keith L. ( Thesis advisor )
Bullock, Thomas E. ( Thesis advisor )
Staudhammer, John ( Reviewer )
Crane, Carl D. ( Reviewer )
Yeralan, Sencer ( Reviewer )
Phillips, Winfred M. ( Degree grantor )
Holbrook, Karen A. ( Degree grantor )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1995
Language:
English
Physical Description:
xii, 124 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Coordinate systems ( jstor )
Coordinate transformations ( jstor )
Distance functions ( jstor )
Eigenvalues ( jstor )
Ellipsoids ( jstor )
Jacobians ( jstor )
Mathematical vectors ( jstor )
Matrices ( jstor )
Robotics ( jstor )
Wrenches ( jstor )
Dissertations, Academic -- Electrical Engineering -- UF
Electrical Engineering thesis, Ph. D
Robots -- Control systems ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
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 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 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.

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University of Florida
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University of Florida
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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.
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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

4.2 SCARA manipulator ............................ 46

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

kVre= 0, 0 0 0, 0, = [ 03 (1.46)
0, 0, 0, 0 0i i-1









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,

kvpris = k ~ic i-lvpris = [ R-l0 (1.48)
i-1 03 (1.48)

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.

The M-norm of vector a,

lalM = +la 1M ,
n
Iai2M = =aOMa= yai(Ma)i (1.83)
i=1

In addition to the positive definite requirement for a metric, a metric must also make

the corresponding square of the M-norm physically consistent [15], e.g., a Ma must

be physically consistent for any a if M is to be considered a valid metric.

A least-M, squares solution is obtained if (1.75) and (1.77) are true and the

solution is minimum Ms-norm if (1.75) and (1.78) are true [7].

It is a fortunate fact that the least Mu-squares solution and the minimum M,-

norm solution are identical for linear systems and equal to the generalized-inverse

solution given in (1.79)-(1.81).

In order for solutions to be invariant to coordinate transformations [19] in both

the spaces defined by u and x, the metrics must transform via a specific congruence









transformation [56],

M' = G'MG (1.84)

If u' = G-ju, then the metric for u' must be M,, = GM,uGu. This will insure that

the Mu-norm is invariant, Iu'| = IuI The metric M, must also transform via a

congruence transformation, Mx, = GxMGx, where x' = G~,x.

1.3 Eigenvalues, Eigenvectors and SVD

Eigenvalues and eigenvectors of an n x n matrix A are defined [34] by the equation,

Ae(i)= A()e(i) (1.85)

where the i eigenvalues and eigenvectors are represented by (i) and e(i), respectively.

The singular value decomposition (SVD) of an m x n matrix A of rank r is defined

[34, 56] by the equation
A = UEV' (1.86)

where E is an m x n matrix with the singular values of A (cri) on the main diagonal,

U is an m x m orthogonal matrix, and V is an n x n orthogonal matrix.

The columns of U are the eigenvectors of AA7 and the columns of V are the

eigenvectors of A'A. The r singular values are the nonnegative square roots of the

nonzero eigenvalues of both AA' and A'A, i.e., the eigenvalues of AA' and A'A are

equal to the square of the singular values of A.

The eigenvalues are preserved for similarity transformations, B = S-1AS, and

the eigenvectors of B are S-'e(i). Eigenvalues are not preserved under congruence

transformations, B = STAS (unless S is a rotation, in which case ST = S-1 and B is

also a similarity transformation).















CHAPTER 2
LINEAR NONCOMMENSURATE SYSTEMS


For linear noncommensurate system, u = Ax, the requirements on the structure of

A are determined in this section, where A is an n x m matrix, x is a noncommensurate

m-vector, and u is a noncommensurate n-vector. Upon expanding u = Ax, it is found

that

i = a;jxj (2.1)
j=l,m
so that

units[aij]units[xj] = units[ui] (2.2)

Using two terms in the sum of (2.1) for two elements of u, we get

zi = aijxj + aikxk (2.3)

zi = ajxj + alkXk (2.4)

for all i, j, k, 1, where units[z,] = units[u;]. Solve (2.3) for xk and substitute the result

into (2.4) to get
zi i = (aikaij aijaik) (2.5)
aik aik
Therefore, for physically consistency,

units[aunitsfaij][aunits units[aijunits[alk] (2.6)

or

units[ik = units[k] (2.7)
aij alj
In other words, if m 2 columns and n 2 rows are eliminated, the determinant

of the remaining 2 x 2 matrix must be physically consistent for the system to be

noncommensurate.









Using three terms in the sum of (2.1) for three elements of u, we get three equations

similar to (2.3) and (2.4). Solving these equations leads to a condition similar to that

shown in (2.6), i.e., if m 3 columns and n 3 rows are eliminated, the determinant

of the remaining 3 x 3 matrix must be physically consistent for the system to be

noncommensurate.

By induction, the above technique shows that for all i > 2, and i < m < n

or i < n < m, if m i columns and n i rows are eliminated, the determinant

of the remaining i x i matrix must be physically consistent for the system to be

noncommensurate.

Another requirement on matrix A is found by viewing A as a matrix of column

vectors A(.,),

u= "xiA(., (2.8)
i=l,m
It is evident that the units of any two columns of A must be proportional. This is

an alternate way to express the results of (2.7) and a simple way to visually deduce

whether or not a matrix is a candidate noncommensurate linear system matrix.

All linear systems are either commensurate, noncommensurate, or physically in-

consistent. Commensurate and noncommensurate systems are physically consistent

systems. For commensurate systems, all elements of the A matrix have identical

units.

2.1 Eigensystem In Noncommensurate Systems

As was mentioned at the start of Section 1.1, many researchers make 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. These

are therefore not true "eigensolutions" in the sense that they may only subjectively

characterize a manipulator configuration based on a particular observer (with a choice

of scale and coordinate frame of reference) as opposed to a more relevant objective

characterization of a manipulator configuration.









2.2 Conditions for Physically Consistent Eigensvstems

When does a matrix A have physically consistent eigenvalues and eigenvectors?

Let A be an n x n matrix,
all a12 aln
S a21 a22 2n (2.9)
A= (2.9)

an1 an2 ann
and let the domain of A be X' where X" is a space with physical units. The X"-space

can be characterized as follows. Let P be an n-vector of possibly distinct physical

units

P/= [1 /2 "' n ]r (2.10)

Any x E X" is equivalent to an item-wise multiplication of # and y, y E 9", i.e.,

a = P/3y=[1 = yI1 #2Y2 *-.. fn] (2.11)

x e X" (2.12)

y E (2.13)

so that + A X" and each eigenvector of A from (1.85), e(i), is an element of X'-

space.
Substituting (2.9) into (1.85) and performing the matrix multiplication results in

the following equations:
alle(i) + al2e2(i) + + anen() = (i)el()
a21ea(i) a22e2(i) +- + a2nen(i) = A(i)e2(i)
(2.14)

anlel(0 + a2e2(i) + + ae(i) = Ae(i)
Recognizing that only quantities with identical physical units may be added leads to

the following theorem.

Theorem 3 The equation Ax = Ax is physically consistent if and only if

units[akjunits[xj] = units[A]units[xk], for all j and k.









Proof

By hypothesis, 1Cjl akjXj = AXk for all k. Recognizing that only identical physi-
cal units can be added together, we immediately conclude that units[akjunits[x] =

units[A]units[xk], for all j and k.

Now, assume units[akj units[xj] = units[A]units[xk] for all j and k. Clearly, the

equation E'j= akjxj = Axk is physically consistent for all k, i.e., Ax = Ax is physi-

cally consistent.



Observe that units[akjlunits[xj] = units[A]units[xk] implies that units[A] = units[aii],

for all i. Hence, any matrix with a physically consistent eigenvalue equation must

have diagonal elements with the same physical units and all its eigenvalues must have
those same units.

A simple test for a physically consistent eigensystem is the validity of the below

equation for each element in matrix A,

units[akj]units[ajk] = units[ai] (2.15)

Since the singular values of A are the nonnegative square roots of the nonzero
eigenvalues of both AAT and A'A, a test on these matrix products (similar to the

tests discussed above for the eigensystem of A) will determine if the SVD of A is

physically consistent. The conditions for the physical consistency of the SVD of A

are stated in Corollary 1 below.

Corollary 1 The singular value decomposition of A, A = UEVr, is physically con-

sistent if and only if units[bkj]units[xjl = units[A]units[xk], for all j and k, where

B = AAr or B = AA, and Bx = Ax.









Proof

This follows directly from Theorem 3 and the properties of SVD, i.e., the eigensys-

tem tests on the matrices AAT and A'A determine the singular values and orthogonal

matrices U and V'. Therefore the test of Theorem 3 and (2.15) can be directly ap-

plied to AA' and A'A to determine if the SVD of A is physically consistent.



Let j in Corollary 1 be equal to k. Then units[bkk] = units[A], and all diagonal

elements of B must have the same physical units. If A is an n x m matrix, then each

diagonal term of B is

( "=,kj ,for B =AAr
bkk = or for all k. (2.16)
E =Z ajk ,for B = ATA
Therefore, all the elements in the k-th row or k-column A must have identical units,

for B = AA' or B = A'A, respectively. But since units[bkk] = units[bjj] for all j

and k, all the elements of A must have the same units. Therefore, singular value

decomposition is only valid for commensurate systems, i.e.,

Theorem 4 Noncommensurate system never have a physically consistent singular value

decomposition.


The major results of this chapter are summarized below. The requirements on

A for all physically consistent linear noncommensurate systems, u = Ax, were given

in (2.6) and (2.7). The requirements for A to have a physically consistent eigen-

system were given in (2.15). And, finally, it was shown that physically consistent

linear noncommensurate systems do not have a physically consistent singular value

decomposition. Only commensurate systems have a physically consistent SVD.














CHAPTER 3
PHYSICAL CONSISTENCY OF JACOBIAN FUNCTIONS

The manipulator Jacobian is used by many researchers in ways which result in
physically inconsistent results. Several of these will be discussed in this chapter.

3.1 Inappropriate Uses of the Euclidean Norm in Robotics

A multitude of researchers [3, 31, 45, 59, 60] have characterized a robot configu-

ration or condition in terms of the scalar quantity of the Euclidean norm. This will
be shown to be invalid, in general. One or more non-Euclidean metrics are often
necessary [14, 17, 19, 53, 54] to find a physically consistent (non-Euclidean) norm.
Although this may not seem obvious at first glance, consider the following examples.
The twist vector V-defined in (1.3)-is composed of the translational velocity
vector v and the angular velocity vector w. The square of the Euclidean norm is often
inappropriately applied to the twist vector,

IV12=V V = VV V (3.1)

But the expression V G V is physically inconsistent since

IV2 ?Lvov+wo w (3.2)

and v has units of [length/time] while w has units of [angle/time]. This is like adding

apples to oranges, generally inappropriate without a metric on the worth of an apple
compared to an orange ([length/time] compared to [angle/time]).
For example, if vT = [1 1 1] and w' = [11 1]rd, then V12 1 6. Changing

the scale from cm to mm will change the result to 1Vi2 ? 303 5 6!
If we define a metric for twists, My, we can use the Me-norm -defined in (1.83)-
to get a measure of twists, IV12M = VQMV. The metric M, must be positive definite

24








and make IV|2 physically consistent. A metric My can be selected such that this
norm describes the kinetic energy, K, of a rigid body,

1 1
K = -V 0 MV = -VMV (3.3)
2 2

The kinetic energy metric expressed at the center-of-mass with axes aligned with
the body's principal axes is the principal mass-inertia matrix of a rigid body,

MKE mbI [0]3,3 (3.4)
S[013,3 Ib J
where mb is the body's mass and Ib is the body's inertia tensor at the center-of-mass
expressed in principal coordinates-a diagonal matrix. We must express this metric in
the same frame as the twists- see (1.84)-(or express the twists at the body's center-
of-mass aligned with the body's principal axis). Transforming the metric MKE to the
frame of expression of the twist results in the metric

S[ mbI3 mRBR (3.5)
v -MbREBr? Rr(Ib + mbBrB)R
Mv = G-MKEG [ mbRBR R(h + mbBB)R (3.5)

where G, is defined in (1.4), with R = iR,, B = 'Bp,, i is the expression frame for the
twist V = iV, and p is the frame of the principal axes of the body. The lower right
3 x 3 matrix in (3.5) is the inertia matrix of the body in the twist frame. If there
is no rotation between the twist frame and the principal frame, the inertia matrix is
I' = (Ib + mbBrB). This inertia matrix could have also been determined using the
parallel axis theorem [42].
The metric of (3.5) is the twist inertia matrix of a rigid body composed of the
zero-order mass-moment (mass), the first-order mass-moment (momentum), and the
second-order mass-moment (inertia).
For a second example of the problem of using Euclidean norms in robotic appli-
cations, let us look at the generalized-force vector 7 of the manipulator joints. The
square of the Euclidean norm of r is


I7TI2 712 + 2 + 2 + 2


(3.6)









If all the joints of the manipulator are revolute or all are prismatic, (3.6) is physically

consistent (but this measure of the sum of the square of joint torques is probably

of little value since the driving component of some joints is generally quite different

from other joints). But, if the manipulator has both revolute and prismatic joints -

i.e., the manipulator joints angles or velocities form a noncommensurate vector-this

equation sums physically inconsistent force-squared and moment-squared terms.

Let us view the Euclidean norms of V and 7 from a different perspective namely,

by looking at the manipulator Jacobian defined in (1.1). Of course if the manipulator

has 6 joints and J has full rank, then J-1 can be found and the solution to (1.1) is

qs = J-1 V (3.7)

To solve for 4 when J is not a square matrix, many researchers use the pseudo-inverse

Jt-see (1.68)-(1.70)-and the equation

Jt V (3.8)

For a full row rank matrix J, the pseudo inverse is

Jt = Jr(JJr)-1 J full row rank. (3.9)

Equation (3.9) is often used with redundant manipulators (manipulators with more

than 6 joints). For manipulators with less than 6 joints, the pseudo-inverse for a full

column rank matrix is often used,

Jt = (J' )-1J J full column rank. (3.10)

Note that the pseudo-inverse in one case involves the term JJT and in the other

case involves the term J'J. There is often a units problem (physical inconsistency)

with both of these terms. One of these terms also appears in each of the norm of a

twist V and the norm of the generalized-force vector 7,, as will be shown below.








From equations (3.1) and (1.1), we get the Euclidean norm of twist V of

I|V2 ? V7 V = (J 4)(J 4) = 4- (J'J) 4 (3.11)

A similar technique will be used to find an alternate form of the Euclidean norm of
the generalized-force vector.
The static wrench defined in (1.14) is repeated here for convenience, r, = JSW,
where r, is the n-vector of generalized-forces-joint torques (for revolute joints)
and/or joint forces (for prismatic joints)-induced by an end-effector wrench W, and
J is the manipulator Jacobian. A wrench W = [f7, n'*] is composed of the 3-vectors
of force f and moment n.
The term JJ' again appears in the Euclidean norm of r,. Equation (3.6) can be
rewritten using (1.14) as

IT712 T, 7-, = W (JJT) W (3.12)

Let us now look at the physical consistency of these Euclidean norms by perform-
ing a units analysis on JTJ and JJr.
The units of a manipulator Jacobian matrix is found simply by noting that the
units of the range of J is equal to the units of V and is not dependent on the structure
of the manipulator. Therefore the units of elements in a Jacobian column have one
of the following two forms [13, 16, 53]:

If manipulator joint i is revolute, the i-th column of the Jacobian has the units

units[J(.,] = [[L]1 for revolute joints (3.13)
[U]3,1]
If manipulator joint i is prismatic, the i-th column of the Jacobian has the units

units[J(.,] = U]3,1 for prismatic joints (3.14)
[013,1 J
The [']j,k in the above equations corresponds to a j x k matrix whose elements have
units of L for units of length or U for unitless. The [0]j,k term identifies a matrix
whose elements are equal to zero (and says nothing about the elements' units).









3.2 Physical Consistency of J'J and JJr

If all n joints manipulator are revolute, the units of J'J is

units[J'J] [ [L2 + U],, ] for n revolute joints, (3.15)

i.e., each term sums a length-squared term with a unitless term. Since the Euclidean

norm of V in (3.11) requires the product (JrJ), the Euclidean norm of V is obviously

physically inconsistent, as shown in (3.2).

For noncommensurate manipulators, if the i-th and j-th joints of a manipulator

are revolute, then the (i,j)-th element of the matrix J7J is physically inconsistent

with units of

units[(J'J)(,j] = L2 + U for i-th and j-th joints revolute. (3.16)

If the i-th joint is revolute and the j-th joint is prismatic, then the (i, j)-th element

of the matrix JrJ is physically consistent with units of

units[(J'J)(i,j)] = L for i-th joint revolute, j-th joints prismatic. (3.17)

If the i-th and j-th joints are both prismatic, then the (i,j)-th element of the matrix
J'J is physically consistent with units of

units[(J'J)(i,j)] = U for i-th and j-th joints prismatic. (3.18)

Similarly, the Euclidean norm of 4 is also physically inconsistent for noncommen-
surate manipulators, i.e.,

2l + +.. + (3.19)

making a noncommensurate vector of joint rates with units of (L2 + U)/T2, where T

represents time units.

The Euclidean norm of V and the matrix J J are physically consistent for an

all prismatic-jointed manipulator since the entire JrJ matrix is unitless and V =

[v'T, 0, 0, 0]', i.e., the angular velocity is zero.









Table 3.1. D-H parameters for GE P50 manipulator.
Joint Type d a 0 a
R 0 0 01 7r/2
R a2 0 02 0
R a3 0 03 0
R 0 0 04 7r/2
R 0 0 05 0


The General Electric P50 manipulator (with 5 revolute joints) has Denavit-Hart-

enberg parameters given in Table 3.1 and a frame 2 Jacobian

0 0 0 a3s3 0
0 a2 0 -a3C3 0
2= -a2c2 0 0 0 -a3c4 (3.20)
52 0 0 0 83+4
C2 0 0 0 -C3+4
0 1 1 1 0

The matrix 2Jr 2J d 2(J7J) has elements with inconsistent physical units such as the

(4,4) term whose calculated value is 1 + aj.
The determinant of 2(JrJ) for the P50 manipulator has terms that sum elements

with units of L4 with L6. The determinant of JrJ for a variety of manipulators was

calculated in various frames and generally found to be physically inconsistent. A

summary appears in Table 3.2. This table also shows the units of the determinant
for each of the manipulators in various frames. (Refer to Appendix A for the D-H

parameters for each manipulator in this table. This appendix also has the Jacobian

and the determinant of JrJ in a particular frame or frames for each of the manipu-

lators.) The frame "general" corresponds to any nonzero translation. Pure rotations

have no affect on the value of J'J since

(J')'J' = (GJ)'(GJ) (3.21)

= J'G'GJ (3.22)

G' = G-1 for rotations (no translation), (3.23)

= (J')'J' = J7J for rotations (no translation). (3.24)









Table 3.2. Physical units of Det[JTJ] for various non-redundant manipulators.
Manipulator Coordinate Units of
Description Frame Det [JJ]
PR Virtual 0,1,2 U
PR Virtual general U + L2
Planar RRR All L4
Non-planar RRR 0,1,2,general U + L2 + L4
General RRR 0,1,2,general U + L2 + L4
PPP Orthogonal All U
SAR (PRP) 0,1,2 U
SAR (PRP) 3,general U + L2
RPR 0,1,2,3 U + L2
RPR general U + L2 + L4
SCARA(RRRP) Any L4
RRRP-2 0 L2
RRRP-2 1,2,3,4,general L2 + L4
RRRP-3 0,1 U + L2
RRRP-3 2,3,4,general U + L2 + L4
P50 (5R) 0,1,2,3,4,5 L4 + L6
P50 (5R) t L4
6-jointed, Det[J] 0 Any frame L6-2p

Although the physical consistency of J'J assures the physical consistency of the

determinant of JrJ, the inverse of this statement is not always true. For instance,

the RRRP-2 manipulator in frame 0 has physically inconsistent terms in O(JrJ), but

Det[0(J-J)] = a2S2 is physically consistent.

It will be shown in Section 6.6 that the physical consistency of the determinant

of J'J assures that Jt is physically consistent.

Frames in which Jt is physically consistent are called decoupled frames. The

reason these frames are called decoupled frames will be made clear in Chapter 6.

Definition 1 A frame is called a decouple frame of a manipulator if the pseudo-inverse

of the manipulator Jacobian in this frame is physically consistent.

The determinant of J for a manipulator with six joints can always be calculated

since J is 6 x 6 for these robots. The physical dimensions of Det[J] (always physically

consistent) is L3-p, where p is the number of prismatic joints up to three. (Any more








than three prismatic joints will mean the manipulator always has Det[J] = 0.) The
determinants of J'J and JJ' therefore have physical dimensions L2(3-p) and are equal
since
Det[A]Det[B] = Det[AB] (3.25)

for all square matrices A and B with identical matrix dimensions. Equation (3.25)
also guarantees the equality Det[J-J] = Det[JJ7] = (Det[J])2.
The determinant of JTJ is zero for manipulators with more than six, joints since
JTJ can have at most rank 6, the maximum rank of J (not rank n). So instead we
look at the matrix JJ' for redundant manipulators.
The units of JJ' for an all revolute joint manipulator is

units[JJ = [L23,3 [L]3,3 for all revolutejoints. (3.26)
[L] 3,3 [U ',3

The units of this matrix are physically consistent, as is the case for an all prismatic-
jointed manipulator where

units[JJ' = [U],3 [013 for all prismatic joints. (3.27)
[013,3 [013,3 J

For a noncommensurate manipulator, the JJ' units matrix of

units[JJ'] [ [L2+U]3,3 []3,3 for noncommensurate manipulator, (3.28)

is physically inconsistent.
The determinant of JJ' is frame independent (i.e., invariant to both rotations
and translations) since for J' = GJ,

Det[J'(J')'] = Det[GJ(GJ)'] = Det[GJJ'G'] (3.29)
= Det[G] Det[JJ'] Det[G'] (3.30)

= Det[JJ'] (3.31)

and the determinant of the twist coordinate transformation matrix G is one.









Table 3.3. Physical units of Det[JJT] for various redundant manipulators.
Manipulator Coordinate Units of
Description Frame Det[JJ']
6-jointed, Det[J] / 0 Any frame L6-2p
Anthropomorphic Arm (7R) Any L6
Puma-260 +1 (7R) Any L6
CESAR (7R) Any L6
K-1207 (7R) Any L6
3P-4R Any U
GP66 +1 (2R-P-4R) Any L4 + L6

(The determinant of JJP for manipulators with less than six joints is of course

zero since the rank of J and thus the rank of JJ' is less than six for these robots.)

The determinant of JJT for several redundant manipulators was calculated and

the physical consistency of the determinants corresponded to the physical consistency

discussed above for the matrix JJ7 in all cases. Table 3.3 shows the units of the de-

terminant for each of the manipulators. See Appendix A for the Denavit-Hartenberg

parameters of each of these manipulators, the Jacobian in a particular midframe, and

the determinant of JJP in this frame.

3.2.1 Consistency of ju = AxI

A generalization of some of the above results for the physical consistency of the

Euclidean norm will be shown in this section. For a linear set of equations u = Ax,

Theorem 5 and Corollary 2 (both below) show that the physical consistency (or incon-

sistency) of the Euclidean norm of u can be determined by the physical consistency

(or inconsistency) of ArA.

Theorem 5 Ifu = Ax, where A is an m x n matrix (m >_ n), then the for the following

statements S1 through S3, Sl implies S2 and S2 implies S3, so that Sl implies S3.

Sl The equation lul2 = u 0 u = u8u is physically consistent (inconsistent).









S2 The nonzero elements in a given column of A have identical units (not all iden-
tical units), i.e.,

If aik = 0 and ajk 7 0, then units[aik] = units[ajk],

for k E {1,2,...,n} and i,j E {1,2,...,m}. (3.32)


S3 The matrix A'A is physically consistent (inconsistent).


In other words, Theorem 5 tells us that the physical consistency of the Euclidean

norm of u implies that all elements in a given column of A have identical units (or

are equal to zero) and that ATA is physically consistent.


Proof

This proof is split up into two parts: the first proof shows that S1 implies S2; the

second proof shows that S2 implies S3. Then by transitivity, S1 implies S3.
The following hold throughout these proofs: i,j E {1,2,...,m} and k,h E
{1,2,... ,n}.

Assume S1 to prove S2.

Since u'u is physically consistent, units[ui] = units[uj] = units[u].

Since u = Ax, ui = E =1 aikxk.

Since ui is physically consistent, units[aikxk] = units[aihXh].

Since units[ui] = units[uj], units[E=1 aikxk] = units[E=1 ajkxk].

But units[EC, aikxk] = units[a;kxk], so that units[a;kxk] = units[ajkxk].

Therefore, units[aik] = units[ajk] and all terms in column k of A have
identical units. This proves S2 given S1.









Assume S2 to prove S3.

Given that units[aik] = units[ajk].

Let B = ArA, so that bhk = EZ= aihaik-

Since all elements in a column k of A are identical (units[aik] = units[ajk]),

units[bhk] = units[aihaik] so that each element bhk of B = ArA is physically

consistent. This proves S3 given S2.



Corollary 2 below follows directly from the above theorem when the Euclidean

norm of x is physically consistent.

Corollary 2 If u = Ax, where A is an mx n matrix (m > n), and the Euclidean norm

of x is physically consistent, then the three statements in Theorem 5 are equivalent
and are equivalent to the statement

S4 All elements of A must have (must not have) identical units.


Proof
To prove the corollary, it is only necessary to show that with the added condition

of a physically consistent |Ix, statement S3 of Theorem 5 implies S4 of the corollary

and S4 implies S1 of the theorem.

Throughout this corollary, let i,j E {1,2,.. ., m} and k,h E {1,2,... ,n}.

Assume xzx, and ATA are physically consistent.

Since u = Ax, ui = =1 aikxk.

Since xTx is physically consistent, units[xk] = units[xh] = units[x].

Then units[ui] = units[aikx] = units[aihx], and units[aik] = units[aih]. This

means that all elements in the i-th row of A have identical units.









The diagonal elements of B = A7A are bkk = n=l aikaik. Since B is physically

consistent, units[aik] = units[ajk]. This means that all elements in the k-th

column of A have identical units.

Since all elements in any row or any column of A have identical units, then all

elements of A have identical units. This proves statement S4.

Finally, I will show that statement S4 implies S1. Since the elements of x have

identical units and S4 tells us that the elements of A have identical units, then

the equation u = Ax forces the elements of u to have identical units. Therefore,

u has a physically consistent Euclidean norm. This proves statement S1.



A theorem similar to Theorem 6 (offered without proof) can be constructed with

the following conditions relating the physical consistency of 1x12, the units of all

elements in each row of A, and the physical consistency of AA'.


Theorem 6 Ifu = Ax, where A is an m x n matrix (m < n), then the for the following

statements Sl through 53, Sl implies S2 and S2 implies S3, so that S1 implies S3.

S1 The equation 1x|2 = x 0 x = xTx is physically consistent (inconsistent).

S2 The nonzero elements in a given row of A have identical units (not all identical

units), i.e.,

If aki $ 0 and akj 5 0, then units[aki] = units[akjl,

for k E {1,2,...,m} and i,j E {1,2,...,n}. (3.33)


S3 The matrix AA' is physically consistent (inconsistent).









In other words, Theorem 6 tells us that the physical consistency of the Euclidean

norm of x implies that all elements in a given row of A have identical units (or are

equal to zero) and that AAT is physically consistent.

Corollary 3 follows directly from the above theorem when the Euclidean norm of

u is physically consistent (and is also offered without proof).

Corollary 3 Ifu = Ax, where A is an mx n matrix (m < n), and the Euclidean norm

of u is physically consistent, then the three statements in Theorem 6 are equivalent

and are equivalent to the statement

S4 All elements of A must have (must not have) identical units.


The implications of these two theorems and two corollaries are that noncom-

mensurate systems generally need be dealt with in a more considered manner than

commensurate systems which has often not been the case in robotics. Since the ma-

trices ATA and AAr are used in the pseudo-inverse solution x, = Atu, for full column

rank A or full row rank A, respectively, the above theorems can be used to determine

the general validity of these results. (The validity is not absolutely determined by the

physical consistencies of these matrix products as was evidenced in the fact that the

RRRP-2 has a physically inconsistent o(JTJ) but a physically consistent Det[O(JTJ)]

and OJt.)

In the robotics inverse velocity problem, solving V = Jq for q, given V, through use

of the pseudo-inverse gives physically inconsistent results due to the non-Euclidean

nature of the twist and (sometimes) joint spaces. This physical inconsistency is

apparent in the physical inconsistency of JrJ or JJ'.

3.2.2 Invalid use of Eigensvstem and SVD of JJ'

Since the pseudo-inverse for redundant manipulators of equation (3.9) contains

the matrix JJr, many researchers have used this factor in solving (1.1) for the joint









rates or to characterize a manipulator configuration [2, 3, 12, 23, 29, 32, 39, 44,

45, 52, 57, 59, 60]. Yoshikawa [59, 60], for example, was the first of many to use

Det(JJr) as a manipulability measure for a manipulator in a given configuration.

Further, Yoshikawa (and others including [31, 46]) defined a manipulability ellipsoid

with principal axes in the direction of the eigenvectors of JJr. Each ellipsoid axis

was given the length of 1/A(i), where A ) is an "eigenvalue" of JJT.

Recall that Theorem 3 in Section 2.2 gives the requirements for meaningful eigen-

values and eigenvectors. Even though JJ' is physically consistent for an all revolute

joint manipulator (see the units matrix of 3.26), this matrix does not have an invari-

ant eigensystem since (2.15) requires that the units of each term on the main diagonal

of the matrix must be identical where in fact they are [L2, L2, L2, U, U, U].

The matrix JJT for most noncommensurate manipulators also does not have

meaningful eigensystems since the matrix is itself physically inconsistent. An excep-

tion to the general physical inconsistency of JJr for noncommensurate manipulators

occurs with the 3P-4R Redundant Spherical Wrist Robot with D-H parameters given

in Table A.18 when expressed in a particular set of frames.

The matrix JJT for the 3P-4R manipulator is generally physically inconsistent

as expected. But in any frame with origin located at the center of the spherical

wrist (the origin of frames 4, 5, 6, and 7), the matrix JJ' is physically consistent

and unitless. The eigenvalues and eigenvectors of JJT are therefore well defined by

the rules given in Theorem 3 and (2.15) and are dimensionless. The eigenvalues are

[1, 1,12,0.873,1.912] and are invariant to rotation of the frame (with this origin). The

invariance of eigenvalues to rotations can be deduced from the well known theorem

that similarity transformations preserve eigenvalues, i.e., if Ae = Ae, then SAS-e' =

Ae' for full rank S. The twist coordinate transformation matrix G acts like S in the

similarity transformation derived below:












JJ'e = Ae

GJJ'e = AGe

e = GYe'

GJJ'G'e' = AGG e' (3.34)

GT = G-1 for rotations (no translation),

GJJG-le' = AGG-le' (3.35)

J' = GJ

J'(J')'e' = Ae'

=. A invariant to rotations. (3.36)


Notice that if translations are allowed, the congruence transformations of (3.34) re-

sults. Since GGr $ I6, translations (and congruence transformations) do not preserve

eigenvalues.

Even though JJ7 for the 3P-4R manipulator in frames located at the intersec-

tion of the spherical joint axes appears to have physically meaningful eigenvalues and

eigenvectors, the interpretation of this manipulability ellipsoid is problematic since

the eigenvectors appears to be unitless (not the necessary wrenches that should be

expected for the wrench manipulability ellipsoid discussed in Chapter 5). Moreover,

as was stated in Theorem 4, noncommensurate systems never have a physically con-

sistent SVD.

The matrix JJT for an all prismatic-jointed manipulator (with at most three

degrees-of-freedom and no orientation capabilities) also has a meaningful eigensystem

but these limited manipulators will not be discussed.

Therefore, since JJ' does not have eigenvalues or eigenvectors (except for the

special cases mentioned above), the above configuration characterization theory is






39

invalid. (Several of the commonly used manipulability ellipsoids are shown in [17] to

be physically inconsistent.)

It will be shown later, in Section 5, that the use of metrics on the appropriate

noncommensurate twist and joint spaces (discussed in the next chapter) does not

change the fact that the manipulability ellipsoid theory violates the eigensystem and

SVD theorems of Section 2.2.














CHAPTER 4
INVERSE VELOCITY KINEMATICS

Several authors [14, 19, 35, 53, 54] have discussed the inappropriateness of using

the pseudo-inverse in solving for the joint rates given a desired twist vector since this
inverse utilizes the Euclidean norms of both the joint-rate vector and the twist vector.
But the twist is not a Euclidean space (and neither is the joint-rate vector when the
manipulator is composed of both revolute and prismatic joints). This problem has
been addressed in these above papers and extensively in [19] by using the (weighted)
generalized-inverse along with metrics on both the twist (M,) and joint rates (Mq).
From (1.68)-(1.70) and (1.79)-(1.80), the pseudo-inverse and generalized-inverse
of the manipulator Jacobian [19] are

jt ? CT(FTJCT)-1Fr (4.1)

? CT(CCT)-1(FTF)-1FT (4.2)

SCtFt, (4.3)

and

J# = M ^C'(F',JM-1C')- F'M (4.4)

= [M-C'(CM-1C')-1] [(F'rMF)-1FM,] (4.5)

= C#F#, (4.6)

respectively. A full-rank factorization of J, J = FC, is used in the above equations,
where F E R(6xr) has full column rank r C 3 (rTX) has full row rank r, and n is
the number of joints in the manipulator.









Two special cases of the generalized-inverse of a Jacobian are obtained when J is

either full row rank or full column rank, i.e.,

J# = M-1J'(JM-1J')-1 J full row rank (4.7)

J# = (J'M,J)-J'rM, J full column rank, (4.8)


where (4.7) is found by letting F = 16 and (4.8) is found by letting C = I, in (4.5).

As stated earlier, the metrics must be positive definite, and for invariance to coor-

dinate transformations and scaling, the metrics must transform according to (1.84),

i.e.,

M,, = G'M,G, for V' = G,V, (4.9)

M,, = G MqG, for q' = G,q. (4.10)

If the desired twist is in the range of the Jacobian, then no metric on the twists

is necessary since the residual V Jq is zero, i.e.,

J# = [M,-C'(CM-1C')-1] [(FF)-1Fr] V E Range[J] (4.11)


This equation is found by substituting M, = I6 in (4.5).
If the Jacobian has full column rank, then no metric on joint rates is necessary

and (4.8) may be used.

If the conditions of both (4.11) and (4.8) are valid-i.e., V is in the range of J

and J has full column rank-then neither metric is needed and the generalized-inverse

is equal to the pseudo-inverse,

J# = Jt V E Range[J] and J full column rank. (4.12)


But, since all manipulators (including redundant manipulators) have singular config-

urations [4], and at singular configurations there exist V's not in the range of J, every

manipulator has configurations in which a twist metric is needed.









For redundant manipulators, where J has full row rank except in singular con-

figurations, the generalized-inverse is independent of the twist metric and (4.7) may

be used. Furthermore, if all joints are revolute (or all are prismatic) the metric on

the joint space is not needed for physical consistency-and the pseudo-inverse can

be used-but the metric is needed for invariance to coordinate transformations and

scaling.

For noncommensurate manipulators with J full row rank, the pseudo-inverse will

generally be physically inconsistent (and not invariant to coordinate transformations

and scaling) since the minimum norm 141 is physically inconsistent.

Sections 4.1-4.2 will discuss the situations in which the pseudo-inverse solution is

physically consistent, invariant to scaling, and invariant to rigid body transformations.

4.1 Physical Consistency of Jt

Although the pseudo-inverse of the manipulator Jacobian may be physically con-

sistent in a given frame, there may be other frames in which Jt is not physically

consistent. (This was suggested by equations (3.9), (3.10), (3.11), all of which have

the terms J'J or JJ' embedded in them, and Section 3.2 which discussed the possible

physical inconsistencies of these matrices.)

4.1.1 Rotations and Consistency of Jt

Theorem 7 shows that if the pseudo-inverse is physically consistent in a given

frame then it will remain physically consistent under any rigid body rotation.

Theorem 7 If the pseudo-inverse of J in frame i (iJt) is physically consistent, then

for every rigid body rotation from frame i to frame j the pseudo-inverse of J in frame

j (jjt) is physically consistent.








Proof
Let iV and jV be twists such that frame j is a rotation of frame i (no translation),
iV = jGi iV.
Assume that the pseudo-inverse of iJ is physically consistent. The pseudo-inverse
of the Jacobian in frame i is

Jt = C'(CC)-1(F'F)-1F' (4.13)

where J = FC is a full-rank factorization, F full column rank and C full row rank.
The pseudo-inverse of the Jacobian in frame j is

jt = (JGi)t= [(jGiF)C]t (4.14)

= C'(CC')-i(F' jG Gi F)-1F'r G (4.15)

= C'(CC')-l(F'F)-1F"'Gj (4.16)

= Jt Gj (4.17)

where (4.16) follows from (4.15) since jGa = (jGi)-' = 'Gj for the case under discus-
sion of jGi a rotation (with no translation). It is now only necessary to prove that
iJt iG is physically consistent.
Partition the pseudo-inverses in frames i and j into two n x 3 matrices, W and
X, and Y and Z, respectively,

ijt = [W X] (4.18)

Jft = [Y Z] = [WR XR] (4.19)

where R = iRj. Since iJt operates on iV = [v', w']', then each component in a row of
W (or a row of X) must have like units or have zero value. Since R is dimensionless,
the units of the elements in a row of Y (or Z) are identical to the units of the elements









Table 4.1. D-H parameters for PR virtual manipulator.
Joint Type d a 0 a
P dl 0 0 0
R d2 0 02 0

in a row of W (or X) and are therefore of consistent physical dimension. Therefore

jJt is physically consistent.



Decouple frames are therefore actually decouple points, points at which the pseudo-

inverse of the manipulator Jacobian (with respect to any frame at the decouple point)

is physically consistent. The reason this point is called a decouple point will be made

clear in Chapter 6.

Definition 2 A point is called a decouple point of a manipulator if the pseudo-inverse

of the manipulator Jacobian in any frame located at this point is physically

consistent.

4.1.2 Translations and Consistency of Jt

A rigid body translation may cause a physically consistent Jt to become physically

inconsistent. An example will demonstrate this fact.

The virtual manipulator [25] associated with the peg-in-the-hole problem [19, 37]

after insertion has begun is shown in Figure 4.1. This PR manipulator has the

Denavit-Hartenberg parameters given in Table 4.1.

The Jacobian in frame 2 is

2 0 0 1 0 0 0 (4.20)
1 000001

and the pseudo-inverse in this frame, 2jt = 2J, is physically consistent.

In an arbitrarily translated frame (no rotation) the Jacobian is 'J = (2Gt,2) 2j,

where
2Gt,2 [ Ipx] 1 (4.21)
1013,O3 13 1

























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

and p = [p,, py ,p]'. The Jacobian in this arbitrarily translated frame is

S0 0 1 0 0 O (4.22)
SPy -P, 0 0 0 1 '( )

and the pseudo-inverse is

[ 0 0 1 0 0 0
tjt -P~ 0 0 1 (4.23)
1+p++p2 l+p22+p 1+p +p2

Note the physical inconsistency in the denominator of the terms in tJt. The physical
inconsistency of this virtual manipulator model of the peg-in-the-hole problem is an

alternative demonstration for the non-validity of the Mason-Raibert hybrid control
techniques stated in published research [19, 22, 24].

4.1.3 Consistency of Jt in All Frames

The SCARA manipulator (Selective Compliant Articulated Robot for Assembly)
[11] in Figure 4.2, with Denavit-Hartenberg parameters in Table 4.2 has a frame 2










Table 4.2. D-H parameters for the SCARA manipulator.
Joint Type d a 0 a
R 0 al 01 0
R 0 a2 02 0
R 0 0 03 0
P d4 0 0 0


Figure 4.2. SCARA manipulator.


Jacobian of
als2 0 0 0
a2 +ac2 a2 0 0
0 0 01
2J0 0 0 0 (4.24)
0 0 00

1 1 10
Translating the frame of expression of the manipulator by an arbitrary vector p,

results in a Jacobian, tJ = (2G',2) 2J, whose pseudo-inverse is

1 0 0 0 0
als2 als2
_a2+alc2 1 0 0 0 a2py+alc2py+alpxs2
tgt = a2s 22 a2 a1a22 (4.25)
-1. 0 0 0 -c2ps 2.2-82s p)
a2S2 a2 as2 S
0 0 1 0 0 0
Since this pseudo-inverse is physically consistent, the pseudo-inverse in any trans-

lated or rotated frame (see Theorem 7) will be physically consistent for the SCARA

manipulator.









The planar RRR manipulator, with its three revolute joints identical to the first

three joints of the SCARA, also has a physically consistent pseudo-inverse in any

frame. These two manipulators are often used as example manipulators to demon-

strate new algorithms [2, 60]. Perhaps this is not appropriate, given their aforemen-

tioned special properties.

4.2 Invariance of Jt to Scaling

When the pseudo-inverse of the manipulator Jacobian is physically inconsistent,
terms of unlike physical units are summed. If the parameters in this manipulator

were re-scaled, perhaps from British to SI units, the physically inconsistent terms

will cause the resulting pseudo-inverse to give a different result.

It has been argued that the problem of physical inconsistencies can be "factored
out" by scaling the problem. The fallacy of this statement will presently be shown.
A change of units scaling matrix is a diagonal matrix that converts a physically

consistent vector with physical units into a vector with similar physical units or no

units. For example, if V = [v',wr]r, units[vx] = units[v,] = units[v,] = m/s, and
units[wx] = units[wy] = units[wz] = rad/s, then S, is a change of units scaling matrix
if
S[aI3 0 (4.26)
S=[ 0 a, \ '
where, for example, av = (100cm/m)(60s/min) and a, = (60s/min). The scaled

twist, V' = [avv, a,ww]r, has similar units to V, i.e., each element of v and av has
units of L/T and each element of w and aw has units of 1/T.

A manipulator joint-rate vector j should have the change of units joint-rate scaling
matrix

= Diag[el, e2, e] where e i { a, if joint i is revolute
S, = Diag[e1, e2,..., where e< = j i j i p (4.27)
a,, if joint i is prismatic '

where the scalar physical unit transformations a, and a, are the identical to those

used in (4.26).








Any scaling of a physical unit for a single element of a noncommensurate vector
must be identically scaled in all other elements of the noncommensurate vector. For
instance, in the example discussed above, the time units were necessarily converted

from seconds to minutes in both a. and a,.
The change of units scaling matrix S, is also normalizing if only the units-not

the numerical value-of the noncommensurate vector is changed, i.e., for the twist
example above a, = (s/m) and a, = s. A normalizing units scaling matrix is
numerically equal to the identity matrix, e.g., S, N 16.
Scaling will now be applied to the inverse velocity problem. The twist vectors are
scaled with the change of units diagonal scaling matrix S, and the joint-rate vectors
are scaled with the change of units diagonal scaling matrix Sq [15] such that

V, = S, V (4.28)

4 = SqI (4.29)

The scaled version of the mapping of joint rates to twist of (1.1) is

V, = SV = S.JSq .Sq = J s (4.30)

where the scaled Jacobian is

J = SJS1 (4.31)

To obtain the pseudo-inverse of J,, first get the full rank factorization J = FC so
that J, = FsC = (SF)(CS,-). Equation (4.2) is then used replacing all F's with
F,'s and all C's with C,'s so that

(J.)t ? S.lC'(CS-C')-l(F'SF)-F'S, (4.32)

The scaled joint-rate solution is thus

qrs (J)ttV, (4.33)

(J )tSV (4.34)

SS,-lC(CS1-2CT)-1 ((F-SF)-lF'rSV (4.35)









and the unsealed joint-rate solution is


:, = Slq,, ? S (J)tS (4.36)

S C(CS 2 (FS ,F)-F'SV (4.37)


Compare (4.37) with the generalized-inverse solution of q, = J#V obtained using

(4.5), i.e.,

= M C'(CM 1CT)-(Fr'MF)-lF'MV (4.38)

It is evident that the two scaling matrices act as metrics where S' and Sq2 in (4.37)

correspond to the metrics M, and M, in (4.38), respectively. Since S2, and S2 are both

positive definite and symmetric, they need only meet the additional requirements that

V 0 SV and q D Seq2 are physically consistent in order for the = symbol in (4.37) to

become an equal sign.

When the desired twist V is in the range of J, the solution 4, = JtV is always

physically consistent. If Jt is physically inconsistent, the inconsistencies are canceled

out when Jt is multiplied by V.

The RRRP-2 manipulator has a physically consistent pseudo-inverse in frame 0

and physically inconsistent pseudo-inverse in frame 2,

0 0 0 0 0 1
C1C2+3 S1C2+3 s2+3 sl(als2+3+a2s3) -cl(a1s2+3+a2s3) 0
Ojt a2s3 a23 a2 S3 23 2s3 (4.39)
-C1C2+3 -S1C2+3 -S2+3 --s182+3 alClS2+3 0
a2s3 a283 a2 3 a2s3 a2s3
L C1 2 1C2 a ai151 -ailctls2 0
S3 S3 S3 S3 S3

0 0 -(.,+2 C2) 82. 2 0
2jt a3 2 (4.40)
0 -_=1 0 0 0 1
a253 a2
0 0 0 0 0
S3
where = 1 +a + a2c2 + 2aa2c2 is physically inconsistent. When the desired twist is

in the range of J, the solution in each of the frames are identical and physically con-

sistent. For instance, the twist for an arbitrary joint-rate vector, q = [i,, q2 3, q4',









in each of frames 0 and 2 are

Cl(a293S2 + 94S2+3) 94S3
si(a24932 + 94S2+3) a292 034
O = -al(q2 + 93) a02C23 C2+344 2V -4l(al + a2c2) (.
V s(+ V =4 (4.41)
S1(92 + 93) 52 91
-cl(92 + 93) C241
91 92 + 93

where 2V = 2GoOV. Substituting OV and (4.39) into 4, = JtV, and substituting 2V

and (4.40) into s, = JtV, both the solutions are 4, = 4 = [41, 92, 93, 44]T In frame 2,

the physically inconsistent terms in 2Jt cancel when multiplied by any V E Range[J].

For any twist not in the range of J, the solution is frame dependent. In frame 0

the solution is independent of scaling; in frame 2 the solution is not independent of

scaling. For example, let the configuration be defined by


01 = 0.lrad 02 = 0.2rad 03 = 0.3rad d4= 4m (4.42)


and let

aI = 0.3m a2 = m (4.43)

Now consider the equivalent desired twists

2.4m 1.329m
0.2m 0.4640-
s s
-7m -1.240m
0d = rd 2G0V 0.1967rad (4.44)
s s
-6rad 0.9805rad
1ra 6.030rad
S S
not in the range of J. The solution for 0Vd is


sa = oJ = 1.000 a, 4.759 1.271 4.496m (4.45)

The resulting actual twist obtained by substituting this joint-rate vector into

Va = 0J4s is

m m rad m rad rad
S= 2.396 0.2404-, -7.000 0.6020m, -6.000 1.000 ,(4.46)
5 s s 5 ss









which in frame 2 coordinates is 2Vsa = 2GooVsa,

m m m rad rad rad
2V = [1.329, 0.4640-, -1.280-, 0.1987-, 0.9801-, 6.030- (4.47)
S s s S S S

The solutions found in frame 0 will now be compared with those found in frame

2. The solution for 2Vd is
o.9686(o.6301m2+1m4)
o.6103m2 s+lm4
4. 2Jt2j2 ? 4.759/s
b 1.271/s in frame 2, (4.48)
4.496m/s
q(b b [0.9805, 4.759, 1.271, 4.496]' using units of m and s. (4.49)

The joint-rate solution 4,b in (4.48) is physically inconsistent. The resulting actual
twist obtained by using qb in 2V4b = 2J4,b is

2Vb [1.329, 0.4641, -1.255, 0.1948, 0.9610, 6.030] (4.50)

which transformed to frame 0 is

oVb = OG22Vsb [2.396, 0.2404, -7.000, 0.6020, -6.000, 0.9805] (4.51)

These twists are different from the desired twists in (4.44).
If the twists are scaled according to (4.26) and the joint rates are scaled according
to (4.27), where ao, = 100cm/m and a, = 1, then the numerical solution in frame 2
equals

4q [0.9686, 4.759, 1.271, 449.6]' using units of cm and s. (4.52)

The resulting actual twist obtained by using s in 2Vsc = 2Jsc is

2Vc g [132.8, 46.48, -124.0, 0.1924, 0.9493, 6.030]7 (4.53)

which transformed to frame 0 is

OVe = oG22V_ ~Y [239.6, 24.04, -700.0, 0.6020, -6.000, 0.9686] (4.54)









Notice that the results of (4.49) and (4.52) differ, i.e., qsb # 4c. The first joint-rate

components differ by more than 10%, the second and third joint rates are numerically

identical, and the fourth joint-rate component (corresponding to the prismatic joint)

in (4.52) is (as expected) 100 times the fourth component in (4.49). Since only terms in

the first row of 2Jt in (4.40) are physically inconsistent, then only the first component

of the joint-rate solution is adversely affected by scaling; the other components are

scaled appropriately.

The solutions qsb and qc& are as "near" as they are only because the specified twist

vector is "nearly" in the range of J, i.e., the desired twist of (4.44) is "almost the

same" (whatever that means!) as

m m m rad rad radio
V= 2.501-, 0.2510-, -7.951-, 0.4992 -4.975 1.000- (4.55)
s s s s S S

2v= m m m rad rad rad
2= 1.182-, -1.821-, -1.280 0.1987-, 0.9801-, 5.000 1 (4.56)
S S s S S S
which are in the range of J.

The resulting actual twists Vsb and Vc, are not equal, are both different from the

desired twist Vd, and are both also different form the physically consistent result found

in Vsa.

For the special cases of unitless J, Jt is physically consistent.


Theorem 8 If J in some frame is unitless, then Jt in this frame is physically consis-

tent.



Proof

Since the pseudo-inverse does not introduce any units not already in J, then Jt

can have only the units of J and the inverse of the units of J or any combination of

the two. Therefore, if J is unitless, then Jt is unitless.

E









For example, the Jacobians expressed in frames 1 and 2 for the SAR (PRP)

manipulator,
0 0 s2 0 0 0 '
0 0 -c2 1 0 0
0 0 -c2 100
1 0 0 2 0 0 1 (4.57)
0 0 0 '0 0 0
0 0 0 0 1 0
0 1 0 0 0 0
are unitless and the pseudo-inverses, Vt = 1T and 2Jt = 2r, are physically consistent.

Of course the inverse of Theorem 8-i.e., if J in some frame is not unitless, then

Jt in this frame is not physically consistent-is not true. For example, the RRRP-2

manipulator has a physically consistent inverse in frame 0, yet the frame 0 Jacobian

is not unitless.

Assume that the qr = JtV is scaleable. Then rewriting (4.36), the scaled inverse

velocity equation,

q (S,' (J)tS) V (4.58)

it is apparent that (S,-(J8)tS,) acts like Jt in the unsealed equation qr = JtV. When

(J,)t is physically consistent, the = can be replaced by an = since scaleability means
that r = qs. In this case,

jt = S-'(J,)tS, when (J,)t physically consistent. (4.59)

Theorem 9 below must be used to verify this equation.


Theorem 9 If D and E are physically consistent invertable diagonal matrices, then A

is physically consistent if and only if DAE is physically consistent.


Proof

Let B = DAE, where dii and ejj are the diagonal elements of the diagonal ma-

trices D and E, respectively. Then bij = diaijejj. Since there is no addition in the

equation for bij and no dii or ejj is zero, then bij is physically consistent if and only









if aij is physically consistent. Therefore, B = DAE is physically consistent if A is

physically consistent. The other direction of the proof follows directly from the fact

that D-1 and E-1 are diagonal matrices and A = D-1BE-1 has the same form as

B = DAE.



Theorem 9 and (4.59) tell us that if (J,)t is physically consistent, then Jt is

physically consistent. Conversely, solve (4.59) for (J,)t,

SqJtS- = (J,)t when Jt physically consistent, (4.60)

to show that if Jt is physically consistent, so is any scaling (J,)t of Jt. These results

lead us directly to the fact that


Fact 1 If Jt is physically consistent, the solution 4 = JtV is independent of scaling

for all V.


If Jt is not physically consistent, then (4.60) is not valid, and the pseudo-inverse

solution to the inverse velocity problem is not scaleable.

The results of this section can be summarized as follows. A real physical system

is always scaleable, e.g., V = Jq can always be scaled. The inverse velocity solution,

q 1 JtV, is scaleable for all twists if and only if Jt is physically consistent; in this
case q, = JtV. If Jt is physically consistent, i.e., the frame of expression has its

origin at a decouple point, then scaling will not affect the resulting joint rates and

the solution q, is independent of scaling.

4.3 Equivalent Generalized Inverses

If an identity metric is assumed in a particular frame, the pseudo-inverse is equal

to the generalized inverse. But in addition, there are other metrics that also give the

same result.









Table 4.3. D-H parameters for the PRP Small Assembly Robot (SAR).
Joint Type d a 0 a
P di 0 0 0
R 0 0 02 7r/2
P d3 0 0 0

Using Theorem 10 below, all metrics which result in identical joint velocities can

be found. Theorem 10 stems from Theorem 2.2 in [19] and the facts that JJ# = FF#

and J#J = C#C. The proof of these Theorems is given in [19].


Theorem 10 All statements in the left column are equivalent statements and all state-

ments in the right column are equivalent statements [19]:

JJ# = (JJ#)' J#J = (J#J)' (4.61)

MJJ# = JJ#M, MJ#J = J#JMq (4.62)

Jt = J# jt = J# (4.63)

MJJt = JJtM, MqJtJ = JtJMq (4.64)


If we assume (4.63), that the pseudo-inverse is equal to the generalized inverse,
then the left equation of (4.64) may be used to solve for all equivalent twist metrics,

MJJt JJt M = 0 (4.65)

For example, the PRP Small Assembly Robot (SAR) shown in Figure 4.3, with

Denavit-Hartenberg parameters given in Table 4.3, has a pseudo-inverse in frame 2

of
S0 1 0 0 00
t 2 = 0 0 0 0 1 0 (4.66)
00 1 0 0 0























Figure 4.3. Small Assembly Robot (SAR).

Any metric of the form in (4.67) that is also positive definite will cause the

generalized-inverse to equal the pseudo-inverse, i.e.,

min 0 0 m14 0 m16
0 m22 n23 0 m25 0
= 0 m23 m33 0 m35 0 .
M, = (.67)
m14 0 0 m44 0 m46 6
0 m25 m35 0 m55 0
m16 0 0 m46 0 m66

The important result of this section is that if a pseudo-inverse is physically con-

sistent, then there are a set of metrics which give identical results when using the

generalized-inverse, i.e., for every decouple point of a manipulator, a class of metrics

exist for which the pseudo-inverse and generalized-inverse of the Jacobian are equal.














CHAPTER 5
MANIPULATOR MANIPULABILITY

As was discussed in Section 3.2.2, the matrices JJ' and J'J do not have physically

consistent eigenvalues, eigenvectors, or a SVD. A few authors [17, 20, 31, 46, 60] have

used other Jacobian functions-some Jacobian functions incorporating metrics-in

manipulability definitions. In this section several of these manipulability ellipsoids

will be introduced and their eigensystems will be explored.

There are three basic types of manipulability ellipsoids. Each of these arise from

setting the square of the a Euclidean or non-Euclidean norm to less then or equal to 1.

The manipulability ellipsoid discussed previously is called the wrench manipulability

ellipsoid (or force manipulability ellipsoid) since this ellipsoid is defined as

ITr2 = W7(JJ)WW < 1 (5.1)

The "eigenvalues," A;, and "eigenvectors," ei, of JJ' are used to create the ellipsoid

with each principal axis in the direction of an ei and axis length equal to A

singular value decomposition of J can be used to deduce these same quantities (see

Section 1.3).

As discussed previously in Section 3.2.2, this analysis is faulty due to the failure

of JJ' to have a physically meaningful eigensystem (see Theorem 3).

It was proposed in [20] that incorporating a metric to replace the Euclidean norm

of T might correct this problem. The resulting equation if a metric is used to determine

the M,-norm of T, is

IT" = W(JM7T)W 1 (5.2)

It will now be shown that the ellipsoid defined by the eigenvalues and eigenvectors of

JMrJ does not meet the requirements for a physically consistent eigensystem.

57








The physical units of M, found by forcing r 0 M,-T to be physically consistent,
are

units[M,] = C where C = [cij] and (5.3)
F2L2
U joints i and j revolute
Cij = L either joint i or j revolute, the other prismatic (5.4)
L2 joints i and j prismatic.
The units variable -, is equal to the desired units of Ir.12

With the above units for M, the resulting units matrix for JMJ' is

___ -[L2 ]3,3 [L]33 1
units[JM.J'] = [L]3,3 [] ] (5.5)
F2L2 [L]3,3 1U]3,3

The units matrix for JMJ' is a scalar multiple of the units matrix of JJ7 for
manipulators with all revolute joints-given in (3.26). Therefore, by Theorem 3,
the wrench manipulability ellipsoid with metric M, is also based on a physically
inconsistent eigensystem.
It should be pointed out that no metric is needed for a physically consistent I||
if all the joints are of identical type, therefore the above result could have been
immediately deduced.
A units analysis of the Mq metric used to make Iq42 physically consistent leads
to the units matrix

T2
units[Mq] = 7,- C where C = [cij] and (5.6)
SL2 ,joints i and j revolute
cij = L joint i or j revolute, other prismatic (5.7)
U ,joints i and j prismatic,
where the units variable 7, is equal to the desired units of Iq\2 The units matrix
M-1 is therefore
-1-
units[M-1] = C' C ,where C = [cij] and (5.8)
U ,joints i and j revolute
c = L joint i or j revolute, other prismatic (5.9)
L2 ,joints i and j prismatic.








This units matrix differs by a scalar constant from the units matrix of Mr. Therefore,
metrics derived for joint rates can be inverted and then used for joint torques, i.e.,
M' = Mq1
The twist manipulability ellipsoid was defined originally [59] as

14|2 = V ((Jt)it) V < 1 (5.10)

The twist manipulability ellipsoid can alternatively be defined with a generalized-
inverse and/or with a joint-rate metric as

IQ2 = vT ((J#)'J#) V 1 ,(5.11)

Iqs,1 = VT ((Jt)'MJt) V < or (5.12)

MI4 = ((J#)'MJ#) V < 1 (5.13)

Since noncommensurate manipulators generally have physically inconsistent Jt and
thus can not have physically consistent eigensystems, only all revolute-jointed manip-
ulators will be analyzed for the definitions in (5.10) and (5.12). The units analysis
below for revolute joints using Jt and (5.12) is equivalent to the units analysis of any
manipulator using J# and (5.13).
Each of the n rows of Jt has the units

111
units[Jt](i.) = [ L, U, U, U, U for all revolute joints. (5.14)

(Notice that the rows of this Jt are ray coordinate screws as opposed to the axis
coordinate screws of the columns of J.) Therefore, the units of (Jt)TJt for an all
revolute-jointed manipulator are

units[(J)Jt] = 1 [ 3U],33 [L]3,3 for all revolute joints. (5.15)
L( ] [1L]3,3 [L 2]3,3 '

And since for an all revolute joint manipulator the metric M, is entirely composed of
identical units, the units of (Jt)TMqJt are proportional to the units of (Jt)TJt. By
Theorem 3, the matrix (Jt)rJt does not have a physically meaningful eigensystem.






60

Replacing the pseudo-inverse of J with the weighted generalized inverse of J does

not change the fact that the matrix (J#)'MqJ# does not have a physically meaningful

eigensystem. (The physical units of J# are a scalar multiple of the units of Jt when

Jt is physically consistent.) But the matrix (J#)'J# is physically consistent even for

noncommensurate manipulators.

The dynamic-manipulability ellipsoid [17, 20, 60] is derived from the manipulator

dynamics equation

7 = M(q)q + h(q, q) + g(q) (5.16)

where 7 represents the generalized-force vector at the joints, M(q) is a positive definite

mass matrix, 4 is the joint acceleration, h(q, q) represents the Coriolis and centrifugal

forces, and g(q) represents the gravitational forces. Solving for i results in

4 = M-1 [r h(q, 4) g(q)] (5.17)

where the dependency in M(q) on q has been dropped for simplicity of notation.

The development here follows from [20] and is given here to demonstrate the

method with which manipulability matrices have been derived. Differentiating V =

Jq with respect to time results in

S= J+ q (5.18)

Again to simplify the notation, define A as the frame acceleration,

A=Jq=V-Jq (5.19)

and r as

r = -h(q, q)- g(q) (5.20)

Substituting (5.20) into (5.17) and the result into (5.19) yields

A = JM-'I (5.21)








Solving for r we get
f, (JM-)tA (5.22)

or
= (JM-')#A (5.23)

The Mr-norm of f, (using only the generalized inverse since the pseudo-inverse may
be physically inconsistent) is

AIs2M. = A' ([(JM-1)#]'M,(JM-1)#) A (5.24)
= ~' (J[(JM-1)#]rTM(JM-1)#J) i (5.25)

If J has full column rank, then

(JM-1)# = MJ# J full column rank (5.26)

f, = MJ#A J full column rank. (5.27)

and

17 2M = (MJ#A)'M,(MJ#A) J full column rank (5.28)
= A' ((J#)'M'MTMJ#) A J full column rank (5.29)

The dynamic-manipulability ellipsoid is found using (5.29) so that

I|f,2 = A7 ((J#)rM'TMMJ#) A < 1 J full column rank, (5.30)

and the ellipsoid is found from eigensystem of (J#)'TMT'MMJ#. As discussed pre-
viously, a metric M,- can be used for Mr. If M, = M so that |11q is the kinetic
energy of the manipulator, then (5.30) reduces to

IFs2M, = A ((J#)'MJ#) A 1
J full column rank and Mr = M-1. (5.31)

The ellipsoid found from the eigensystem J#TMJ# (J full column rank) is physically
consistent but does not meet the criteria of a valid eigensystem in (2.15), since the









units of this matrix are proportional to the units of (5.15). (Notice that the matrix

defining the dynamic manipulability ellipsoid is identical to the matrix defining the
twist manipulability ellipsoid.)

Let us look a little further J#'MJ#, the definition for the dynamic manipulability

ellipsoid as originally developed in [59]. Expanding (5.31) by substituting (5.21) for

A yields

|i, = (Jr(J#)TMJ#J) J full column rank, Mr = M-1. (5.32)

But for full column rank J, J#J = I, and (5.32) to the trivial equation

|f,| = 'M4 J full column rank, Mr = M-1, (5.33)

and the ellipsoid is dependent only on the metric. But since M has the units of Mq
and Mq does not satisfy the conditions necessary for a valid eigensystem for noncom-
mensurate manipulators, again the dynamic manipulability ellipsoid is shown to have
an invalid eigensystem. Note that although Mq is unitless for commensurate manip-
ulators and thus Mq has a valid eigensystem, the dynamic manipulability ellipsoid
does not have a valid eigensystem even for commensurate manipulators.

For the case when J does not have full column rank, (5.24) is used to define the
ellipsoid [60]. But again, a units analysis of the matrices shows that the eigensystem
requirements are violated. This is also true for the expanded version of this ellipsoid
determined by (5.25) when the manipulator is noncommensurate; but, if the manipu-
lator is commensurate, each term of the matrix determining the ellipsoid has identical

units and the eigensystem is physically meaningful.
To summarize, none of the manipulability ellipsoids possess geometric invariance.
The wrench manipulability ellipsoid defined by the eigensystem of matrix JMrJ7 is

not valid for any manipulator. The twist manipulability ellipsoid originally defined
by the eigensystem of (Jt)rJt and subsequently modified to (Jt)rMqJt and then to

(J#)'MJ#, is not valid for any manipulators. The dynamic-manipulability ellipsoid,









defined by the eigensystem of matrix [(JM-1)#*Mr(JM-1)#, is not a physically
consistent eigensystem even for the case when J has full column rank. If J has

full column rank and Mr = M-', this matrix product reduces to (J#)'MJ# which

also does not have a valid eigensystem. An expansion of the dynamic-manipulability

equation leads to J' ((J#)'rMJ#) J = M,, which has a valid eigensystem if the

manipulator is commensurate.
Although the existing manipulability theory has been shown to be invalid in all

cases for manipulators with six or more joints, for manipulators with six or fewer

joints, the scalar manipulability measure, Det[JrJ], is physically meaningful at de-

couple points. At decouple points, the manipulability measure is physically consistent

(see equation (6.105)). Thus, when a decoupled coordinate frame is used, the manip-
ulability of these manipulators in one configuration can be meaningfully compared to
the manipulability at other configurations.














CHAPTER 6
DECOMPOSITION OF SPACES

Griffis recently introduced a special six dimensional spring for use as a wrist

placed on a 6-jointed manipulator [26]. He thus created a wrench space via small

displacements (or twists) creating a K-orthogonal complement to the twists of free-

dom, which he called the twists of compliance. With this technique Griffis and Duffy

[28] showed that independent position and force control can be accomplished for a

two-dimensional example and that the twists of compliance are in fact K-orthogonal

complements to the twists of freedom. Without adding such a wrist, this chapter

explores several techniques for twist and wrench space decomposition.

Let us assume that a twist space referenced to a particular coordinate system is

decomposed into two manifolds, and one of these manifolds is the twists of freedom

subspace, Vf = Range[J], as previously defined in (1.13). The other manifold is the

twists of nonfreedom, ,Vn, introduced by Lipkin and Duffy [36] in their important
article on the nature of twists and wrenches as screws.

The twists of nonfreedom are the twists that are not possible to accomplish in

a given configuration. Lipkin and Duffy [36] define this as a "subspace which is the

orthogonal complement of the twists of freedom, although Duffy later repudiates this

notion in [22]. But since Vf is a noncommensurate space, the orthogonal complement

of Vf is not an appropriate manifold to introduce since it does not have the physical

dimensions of a twist manifold. This manifold would have the strange property of

dependence on the units of expression of Vf. The wrenches of constraint subspace,

We, when viewed as a unitless vector space in R6, is recognized as the orthogonal

complement of an assumed unitless version of Vf. But We only in special cases









appear to have the physical units of twist vectors, which is necessary for the manifold

V,f to be meaningful. (To be fair, [36] defines twists of nonfreedom in the context

of an example that appears to have a unitless basis for We, which could therefore

be viewed as an appropriate twist subspace. This dissertation defines wrenches of

constraint in a manner consistent to the definition given in [36].)

6.1 Projections and Kinestatic Filters

In commensurate systems, the pseudo-inverse and generalized-inverse can be used

to separate various spaces into two disjoint spaces [34, 56]. In noncommensurate

systems, care must be taken when using the pseudo-inverse. If the pseudo-inverse

is physically inconsistent, projections using this inverse are also generally physically

inconsistent.

All types of projections for the various manipulator spaces are derived below using
the generalized-inverse, although in cases of a physically consistent pseudo-inverse,

the generalized-inverse may be replaced by the pseudo-inverse.

The twist space projection is found through the following series of equations:

V = J4 (6.1)

41 = J#Vd (6.2)

V J = J4, (6.3)

V, = JJ#Vd (6.4)

where the s subscript is for "solution", the "d" subscript is for "desired', and the "r"

subscript is for "resulting."

The joint-rate space projection, obtained by substituting (6.1) into (6.2), is

4, = J#J4d (6.5)

The wrench space projection is found through the following series of equations:

T = JW (6.6)






66

W, = J#'Td (6.7)

W, = J# J'd = (JJ#)Trd (6.8)

The generalized-force space projection, obtained by substituting (6.7) into (6.6),

is

r = JTJ#Trd = (J#J)YTd (6.9)

The various projection matrices are the four kinestatic filters [19],


P, = JJ# P, = J#J P, = (JJ#)' P = (J#J) (6.10)

The various spaces can now be decomposed into disjoint spaces using the above

projection matrices and (6.4), (6.5), (6.8), and (6.9),

V = Null[JJ#i] E Range[JJ#] (6.11)
Mq
Q = Null[J#J] &' Range[J#J] (6.12)
M-1
W = Null[(JJ#)] D Range[(JJ#)r] (6.13)
M-1
T = Null[(J#J)'] M Range[(J#J)] (6.14)

My
where the symbol E means that the two subspaces on either side of this symbol are

M,-orthogonal. The normal direct sum (E) means that the two spaces are orthogonal
(in the Euclidean sense). Notice that the above decompositions do not follow from the

fundamental theorem of linear algebra, 'm = Null[Ar] E Range[A], where the range

and null operators operate on a matrix and its transpose. For the metric-dependent

decompositions, the range and null operators operate on the same matrix.

The above decomposition equations can be simplified by applying some facts about

the full rank decomposition of the Jacobian, J = FC and J# = C#F# of (4.5),

JJ# = FCC#F# = FF# (6.15)

J#J = C#F#FC = C#C (6.16)









and some facts about the null and range space operators,

Null[JJ#] = Null[FF#] = Null[F#] (6.17)

Range[JJ#] = Range[FF#] = Range[F] (6.18)

Null[J#J] = Null[C#C] = Null[C] (6.19)

Range[J#J] = Range[C#C] = Range[C#] (6.20)

Each of the statements in (6.17)-(6.20) can be proven in a manner similar to that

shown below for (6.17).

Let FF#x = 0. Multiply both sides by F# to give F#FF#x = 0. But by the

property of the generalized-inverse given in (1.76), F#FF# = F#, so that F#x = 0.

Therefore, Null[FF#] = Null[F#].

These simplifications lead to the below simplified decomposition equations:

V = Null[J#] ~' Range[J] (6.21)
Mq
Q = Null[J] ( Range[J#] (6.22)
Mw1
W = Null[J'] E Range[(J#)'] (6.23)
MV
T = Null[(J#)T] E Range[J] (6.24)

and the even simpler decomposition equations:

V = Null[F#] e' Range[F] (6.25)
Mq
Q = Null[C'] Range[C#] (6.26)


M;-1
W = Null[F'] M& Range[(F#)*r] (6.27)

T = Null[(C#)'] D Range[C] (6.28)

Each metric will give a different decomposition. If the metric has the required

property (that it transforms via a congruence transformation, (1.84)), then the frame

of expression has no bearing on the decomposition.









The below two facts allow us, in some cases, to apply the above metric dependent
decompositions, which use the generalized-inverse, to a metric independent decom-
position, which uses the pseudo-inverse.

Fact 2 If Jt = J# for some metric M, and some metric Mq, then Jt is physically
consistent.

Fact 3 If Jt is physically consistent, then Jt = J# for some metric M, and some
metric Mq.

If the pseudo-inverse is used instead of the generalized-inverse by choosing change
of unit identity scaling metrics for M, and Mq, the decomposition is frame dependent
and only valid if the pseudo-inverse is physically consistent. The decomposition for
physically consistent Jt is

V = Null[JJt] Range[JJt] (6.29)

Q = Null[JtJ] E Range[JtJ] (6.30)

W = Null[(JJt)'] E Range[(JJt)'] (6.31)

T = Null[(JtJ)r] e Range[(JtJ)'] (6.32)

From Theorem 10 and the fact that Jt = j# for some metric (since Jt is assumed
physically consistent), JJt = (JJt)T and Jt = (JtJ)-. Therefore the above decom-
positions simplify to

V = W = Null[JJt] E Range[JJt] (6.33)

Q = T = Null[JtJ] e Range[JtJ] (6.34)

when Jt is physically consistent. The spaces V and W are decomposed identically as
are the spaces Q and T.









The above decomposition can be further simplified by using the below equations:


JJt

JtJ

Null[JJt]



Range[JJt]

Null[JtJ]

Range[JtJ]


= FFt

= CtC

= Null[FFt] = Null[Ft] = Null[(F'F)-1F']

= Null[F'] = Null[C'F'] = Null[J']

= Range[J]

= Null[J]

= Range[CtC] = Range[Ct] = Range[C'(CC')-1]

= Range[C'] = Range[C'F'] = Range[J] .


The space decompositions for frames in which Jt is physically consistent are there-
fore

V = W = Null[J'] e Range[J] (6.43)

Q = T = Null[J] G Range[J'] (6.44)

Equations (6.43) and (6.44) appear to be direct applications of the fundamental
theorem of linear algebra; this is a deceptive notion. The reader should remember
the limited scope of these equations-i.e., they are only valid in frames in which Jt

is physically consistent-and their rather involved derivations.
This decomposition will be explored further in the subsequent sections.

6.2 Twist Decomposition

In order to demonstrate the problem with defining a twist of nonfreedom manifold

as a subspace, two examples will be shown. One example will show when these twists
constitute a subspace and the other will show when they do not form a subspace.
First consider the SCARA manipulator of Figure 4.2. The SCARA Jacobian
expressed in frame 2 coordinates was given in (4.24). The column-reduced echelon


(6.35)

(6.36)

(6.37)

(6.38)

(6.39)

(6.40)

(6.41)

(6.42)









form of the wrench of constraint subspace in this frame, We = Null[2J7], is

[2W E= 0 0 0 1 0 0 (6.45)


where E, is the matrix that converts [Wc]b to column-reduced echelon form. Note that

these wrenches might also be interpreted as twists of nonfreedom with no discrepancy

with units,
[2v] [0 0 0 1 0 (6.46)
i2 LbE= 0 0 0 1 0

The SAR (PRP) manipulator of Figure 4.3 has the Jacobian and wrench of con-

straint subspace basis vectors expressed in frame 3 coordinates of

0 d3 0 0 -_ 0
1 0 0 0 0 0
0 0 1 [3 ,]0 0 0
3j 0 0 1 [W E. 0 0 0 (6.47)
0 0 0 0 0 1
0 1 0 0 1 0
0 0 0 1 0 0

Note that these basis wrenches cannot be interpreted as twists of nonfreedom since

the second basis vector does not have the units of a twist (an axis coordinate screw).

Therefore, for this manipulator expressed in frame 3 coordinates, the concept of twists

of nonfreedom as described previously (as a subspace) is untenable.
A slightly modified definition of twists of nonfreedom is therefore necessary and

is given below.


Definition 3 Twists of nonfreedom are twists that the manipulator cannot fully gen-

erate in a given configuration,


Vnf = V Vf (6.48)

The meaning of the above equation might need explanation. The manifold Vnf include

all the twists of V except those twists in Vf. This is not the orthogonal complement

of Vf, which (as stated previously) is physically inconsistent for screws.









The nonfreedom twist manifold might also be defined as

Vf = {Vf : Vf E V and Vf i Vj} (6.49)

In general, the manifold Vq is not a subspace. Typically, two twists of nonfreedom

might sum to a twist of freedom or a nonfreedom twist.

For example, two nonfreedom twists for the SAR manipulator expressed in frame

3 coordinates are

0, 0
0 0
1m 0
Vd 3V, = rad (6.50)
1rad nf -1_
s s
0 0
0 0

The sum of these two nonfreedom twists is the twist of freedom [0, 0, 1m, 0, 0, 0]'.

The difference of these two nonfreedom twists is the nonfreedom twist

[0, 0, 1M, 2 0, 0]-.
Since VnY is not, in general, a subspace, a direct sum decomposition of twists of

freedom and twists of nonfreedom is not typically possible, i.e.,

V 7 Vf E, (6.51)

In the special cases when We can be interpreted entirely as twists, the twist space

can be decomposed as the direct sum decomposition, V = Vf & Vi, where V, are the

subspace of inaccessible twists defined below.

Definition 4 Inaccessible twists constitute the screw subspace of twists such that

vi C ,n (6.52)


and the inner product Vi Vf vi vf + wi ) wf (which is generally physically

inconsistent) is physically consistent for any Vi E Vi and any Vf E Vf. The

subspace Vi may not exist.









If Vi = Vf, then the twist space is uniquely decomposed by the direct sum decom-

position

V =Vf Vi if i =Vnf. (6.53)

6.3 Wrench Decomposition

Assume that a wrench space referenced to a particular coordinate system is decom-

posed into two manifolds. One of these manifolds equals the wrenches of constraint,

We = Null[Jr], as previously defined in (1.19). Since We is the null space of a matrix,

it must be a subspace. The other manifold is the wrenches of nonconstraint manifold

[36], Wn,.

The wrenches of nonconstraint, when applied at the end effector of a manipulator,

require some nonzero joint forces for static balancing or will cause some motion of the

manipulator. Lipkin and Duffy [36] define wrenches of nonconstraint in an analogous

fashion to the twists of nonfreedom, i.e., according to [36], the wrench of noncon-

straint manifold is the orthogonal complement of the wrench of constraint subspace.

But the orthogonal complement of We has physical dimensions of a twist manifold.

Furthermore, when We is viewed as a unitless vector space in R6, the orthogonal com-

plement is a unitless version of Vf. But the axis screw vectors of Vf, only in special

cases appear to have the physical units of wrench vectors, a necessary requirement

for the manifold Wn, to be meaningful.

For example, the orthogonal complement of 3Wy for the SAR manipulator is the

Jacobian, 3J, given in (6.47). The second basis vector of 3J in (6.47) is obviously

not a wrench (a ray coordinate screw), so this subspace cannot describe wrenches of

nonconstraint.

To avoid the above problems, a slight modification of the definition of wrenches

of nonconstraint is given below.









Definition 5 Nonconstraint wrenches are wrenches that will produce a nonzero power

with some twist of freedom [25],


Wc = W W (6.54)

The manifold of nonconstraint wrenches are all the wrenches of W except those

wrenches in W,. This is not, in general, the orthogonal complement of We, which (as

stated previously) is physically inconsistent for screws.

Wrenches of nonconstraint might also be defined as


We = {Wnc: Wc E W and Wc V We} (6.55)

Note that Wn, is a manifold that, in general, is not a subspace, so that no di-

rect sum decomposition of wrenches of constraint and wrenches of nonconstraint is

generally possible, i.e.,

W # Wc WC (6.56)

In the special cases when the twists of Null[WT] possess a meaningful interpretation

as Wc,, the wrench space can be decomposed via the direct sum decomposition,

W = We E Wd, where Wd are the subspace of driving wrenches defined below.

Definition 6 Driving wrenches constitute the screw subspace of wrenches such that


Wd C We (6.57)


and the inner product Wd 0 Wnc fd 0 fnc + nd 0 n,, (which is generally

physically inconsistent) is physically consistent for any Wd E Wd and any Wnc E

W,c. The subspace Wd may not exist.


If Wd = Wc, then the wrench space is uniquely decomposed by the direct sum
decomposition


W = Wd e Wc if Wd = Wn.


(6.58)









If both twists of nonfreedom and wrenches of nonconstraint are subspaces (and

thus are identically the inaccessible twists and the driving wrenches, respectively),

then a hybrid control is accomplished by decomposing the desired twist into twists of

freedom and twists of nonfreedom and the desired wrench into wrenches of constraint

and wrenches of nonconstraint, and then filtering out the inaccessible twists and

constraint wrenches. This assures that the control inputs will be entirely composed

of twists of freedom and driving wrenches.

6.4 Hybrid Control

The hybrid control algorithms of Mason [40, 41] and Raibert [51] inherently assume

a decomposition essentially equivalent to

6 Range[J] e Null[Jr] = V1f We (6.59)

This theory splits the hybrid control problem into "natural" and "artificial" con-

straints at what is now commonly know as the "center of compliance" or "compliance

center" [2, 25, 60] (called a constraint frame in [11, 51]). A center of compliance is

defined as a point through which pure forces produce only pure translations and pure

couples produce only pure rotations about that point. This point may or may not

exist, or may exist at more than one point.

When the coordinate reference frame origin is located at the center of compliance,

the MRHCT (Mason and Raibert's hybrid control theory) states that the diagonal

selection matrices [11, 51] are used to determine the appropriate action for each loop

of the hybrid position and force control., i.e., each joint is used to control either a

position component (twist) or a force component (wrench).

The MRHCT calls these two subspaces orthogonal complements, which these sub-

spaces appear to be if the screw spaces were instead commensurate six dimensional

vector subspaces as in (1.61). But they are not orthogonal complement screw sub-

spaces.









An example will now demonstrate the MRHCT [1, 2, 19]. The task at hand is

to place a peg into a hole as shown in Figure 4.1. (In this example, the virtual PR

manipulator of the figure is not involved.) The "natural" and "artificial" constraints,

taken together (since the distinction between the two is sometimes open to interpre-

tation), with respect to frame 2 are v. = vy = 0, f, = 0, and nz = 0. Both the twist

and wrench selection matrices are diagonal matrices, both with elements of either 0

or 1. This leads to the twist selection matrix, 2p,, and the wrench selection matrix

2pW, i.e.,

000000' 100000
000000 010000
2 0 0 1 0 00 2 0 0 0 0 00
P = 0P o = (6.60)
0 0 0 0 0 0 w 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 1 0 0 0 0 0 0

The selection matrices are always related by the equation

P = 16 -P (6.61)

The hybrid control then filters the specified twist, V,, and wrench, W,, with the
selection matrices as follows:

2V = 2Pv 2W = 2P 2W (6.62)

This guarantees that the twist 2V E 2Vf and 2W E 2W, in frame 2.

It is apparent that the selection matrices, P, and P,, act as filters on twists and

wrenches. In fact, P, and Pw are projection matrices,

2 2B 2B [2B(2B B)-2B-] (6.63)
2p 2C2Ct = 2c [2C(2C2C)-12C-] (6.64)









where B represents a basis for the twists of freedom and C represents a basis for the

wrenches of constraint,

0 0 1 0 0 0
0 0 0 1 0 0
1 0 2= 0 0 0 0(6.65)
0= 0 0 0 1 0
00 0 0 0 1
01 0 0 0 0

In frame 2 the MRHCT seems to work. But in a frame t (see Figure 4.1), arbitrarily

translated from frame 2, the MRHCT fails. In this frame the projection matrices,

P, = tt and P, = tWC'WIt, are physically inconsistent, i.e.,

P._ 00 0 p py 00 0 0 0
-PPL d, 0 0 0 +-p- 0 0 0 p0 3
pV 0 0 1 0 00 0 0 0 0 0 0
0 0 000 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 1 0
0E 0 0 0 0 -P 0 0 0 0
2 =. 000 -. & 000^'
(6.66)

where 7 1 + p2 + p, a physically inconsistent quantity.

6.5 Decomposition with Ray Coordinate Twist Space

Recently several authors [1, 24] have expanded a discussion on isotropic subspaces

begun in [52] and greatly enhanced in [37]. These articles have attempted a differ-

ent decomposition using four manifolds, two of which are the twists of freedom and

wrenches of constraint. Manipulate the twists space via the A matrix so that the

twists of freedom and wrenches of constraint subspaces are both defined using ray

coordinate screws, i.e.,

Vyay = AVy = Range(AJ) (6.67)

The radical manifold, ?, is the screw manifold of the common elements in V'y and

We,


Rv = vfy n c


(6.68)









The defect manifold, V, is the manifold not covered by V; Y and We,


(V7a U We) U D = $6 (6.69)

where $6 is the full 6-dimensional ray coordinate screw space.
Let us investigate how each of these manifolds relate to the others. As shown

in Theorem 1, Vf and We are reciprocal subspaces. Since V;a is the ray coordinate
version of Vf, then V and We are also reciprocal subspaces. This theorem leads to

the corollary below which states that the radical manifold is a self-reciprocal subspace.
The proof for the theorem below is based in part on Theorem 1 which states that
coordinate transformations do not affect the reciprocal product.

Corollary 4 The radical screw subspace "R is self-reciprocal,

ri o rj = 0 V ri,rj E (6.70)


Proof

Since r E 7, r E V}ay, and r E We, and all V'a E Vay and W, E We are reciprocal

(VTa o We = 0) by Theorem 2, then ri o rj = 0 for all i and j.



Since the screw subspace 7R is self-reciprocal, the screws in this subspace are self-
reciprocal and mutually reciprocal. The theorem below also shows that each column
of a manipulator Jacobian is self-reciprocal.

Theorem 11 For revolute and/or prismatic jointed manipulators, each column of a
manipulator Jacobian is self-reciprocal.









Proof

If the i-th joint in a manipulator is revolute, the i-th column of the manipulator

Jacobian in frame i 1 is [,0, 0,0,0, 1]'. If the i-th joint in a manipulator is pris-

matic, the i-th column of the manipulator Jacobian in frame i 1 is [0,0,1,0,0, 0].

Since both these screws are self-reciprocal and reciprocity is invariant to coordinate

transformations, then regardless of the frame, the i-th column of the Jacobian is self-

reciprocal.



The radical is always a subspace since it is the intersection of two subspaces. But

V Uay U We is generally not a subspace as is shown in the below example.

The P50 manipulator with 02 = 03 = r/2 and 04 = 0 has

O '0 0 0 0 0 0
-1 0 0 0 -1 0
0 1 1 1 0 0
[V}ab= 0 'a2 0 0 0 [c]b= 1
0 a3 a3 0 0 0
a3 0 0 0 0 0
(6.71)

Summing the fifth screw of VaY and 7 times the only screw of We results in the

vector [0, -1, 0, 7, 0, 0]', for all 7, where units[7] = L. This screw is not in

V}7y U We for any nonzero 7. Therefore V7a U We is not a screw subspace.

Similarly, the defect manifold is generally not a screw subspace, since 2' = $6 -

(V} U We), although [24, 37] both claim that the defect is a subspace. For example,

the SAR manipulator in frame 2 has twist of freedom and wrench of constraint basis

sets of
0 0 0 0 0 0 1 '
0 1 0 0 0 0
ray 2J 0 0 0 2 0 0 0
[2VfYb=AJ= 0 0 0 W ]b 0' 1 ,'
1 0 0 0 0 0
0 0 1 L 1 0 0
(6.72)






79

so that the radical basis set is

[2R]b = {[0, 0, 0, 0, 0, 1]} (6.73)

The defect manifold contains all screws

[2D]b = {[/3., f l 6, 8y, z]} (6.74)

with nonzero 7. This is not a subspace, although [24] claims that a basis can be

selected for the defect, [2D]b = {[0, 0, 7, 0, 0, 0] }.

In frame 3, the SAR manipulator has twist of freedom and wrench of constraint

basis sets of
0 0 0 0 -0
0 1 0 0 0 0
[3y 0 0 0 [3Wb 1 0 0
[ V0 d3 0 0 0 1
1 0 0 0 1 0
0 0 1 0 0 0
(6.75)
so that the radical basis set is empty, i.e., [3R]b = 0. The defect manifold is also

empty for the SAR manipulator in frame 3.
It is apparent now that the decomposition theory of [24, 37] is not unique and the

claims made are generally invalid. Therefore a new technique for screw and wrench
space decomposition is presented in the next section and the results of the previous

sections of this chapter are tied together.

6.6 Space Decomposition at Decouple Point

In Section 6.2, it was shown that in some cases the twist space can be decomposed

uniquely via a (Euclidean) direct sum decomposition, (see (6.53)) and in other cases
not. In this section, the conditions for which this decomposition is possible are found.

When the wrenches of constraint are put in column-reduced echelon form,

[Wc]bE,, some of the columns may appear unitless. Since wrenches are screws, unit-
less columns will only exist in columns that have zeros in the force or moment posi-

tions. Each unitless column of [Wc]bE, represents one of the following two types of









wrenches: the wrench is a pure force, i.e.,


Force = [fr, fy, f/, 0, 0, O]) (6.76)

or the wrench is a pure moment with respect to a frame on the wrench (screw) axis,

i.e.,

Moment = [0, 0, 0, nx, ny, n]. (6.77)

Group these apparently unitless columns into [W-]bE,, the wrenches of constraint

with either zero force or zero moment. The columns of [Wc]bE, that are not unitless

are called the nonzero force and nonzero moment wrenches of constraint, [Wnz]bE,.

If [W]bE, = [Wj]bE,, then the manipulator twist space decouples as shown in

Theorem 12 below.


Theorem 12

'V = 'Vf E 'Vnf <- 'We = 'IWX


Proof

First prove that, in a given frame, there exists a direct sum decomposition of V if
We = WVf; and then prove that, in a given frame, if there is a direct sum decomposition

of V, then We = Wz.

If We = Wz, the column-reduced echelon form basis vectors of [WV]bE, have no

units and can therefore be used for a basis of Vi. But since the dimension of We

plus the dimension of Vf is six and Wc = Wz, then V,f = V;. Therefore [Vnf]bE, =
[WclbE,,. This proves one half of the theorem.

The second half of the theorem is proven as follows. If the decomposition V =

Vf ) Vnf is assumed, then the projection involved is Euclidean, i.e.,


Vf = Range[JJt] = Range[J] ,


(6.78)
















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

and


Vf = Null[JJt] = Null[Jt] = Null[J] ,


(6.79)


where Jt must be physically consistent from the assumption. But We = Null[J'] by

definition. Since Null[Jr] can be interpreted as both a twist (of nonfreedom) and a

wrench (of constraint), then We = Wf.



The twist space decomposition, when possible, is shown schematically in Fig-

ure 6.1. Conditions for this decomposition are given in Theorem 12 above and The-

orem 13 below.

The above proof leads to a corollary that a subspace, V,, of V containing the twists

of freedom, V, 2 Vf, always has a direct sum decomposition V, = Vf E Vi, i.e.,


[V4]E, = [W ]bE ,


(6.80)


where VY does not exist (is empty) if there are no wrenches of constraint with zero

force or zero moment in the chosen frame.


Corollary 5

,S = V e 'Vi (6.81)

where iVf C iv, C iV.









Proof

If 'Wf = 'We, then the proof of this corollary is identical to the proof of The-

orem 12 and 'V, = iV. Otherwise, if 'WI C 'We, then the proof again follows the

reasoning of the proof of Theorem 12, although the dimensions of the space 'V, is

reduced from 6 (the dimensions of iV) to Dim[Vf] + Dim['Wy].



To continue this discussion of twist space decomposition, separate the twists of

freedom into linear velocities of freedom and angular velocities of freedom, and sepa-

rate the wrenches of constraint into forces of constraint and moments of constraint,



V f= [ We[ (6.82)

In a given frame i, if all ifc are orthogonal to all 'vf and all inc are orthogonal to all
iwf, then the manipulator decouples and the twist space can be uniquely decomposed

into twists of freedom and twists of nonfreedom subspaces.

Theorem 13

ife f = 0, V' e, iVf
iV = zVf if inc Ow -w= 0, Vinc, 'wf


Proof

Assume iV = 'V e iVnf and remember from (1.35) that 'Vj o We = 0. From
Theorem 12, iWe = iWz, which implies that either if% = 0 or inc = 0 for each wrench

in 'We. In the case f = 0,


iVo iWC = iwOin = 0 (6.83)

and the right-hand side of the theorem is proven. In the case 'nc = 0, then


'Vf o'We = ZvSf o = ,


(6.84)









completing the proof that the right-hand side of the theorem follows from the left-

hand side.

The other direction of the proof proceeds as follows. The right-hand side of the

theorem implies that iW, = iWz, and then the proof of Theorem 12 will suffice.



For example, the wrenches of constraint in column-reduced echelon form of the

PR virtual manipulator of Figure 4.1, expressed in frame 2 is

0001
0010
0 0 1 0
[2W]E = [Null[2J ]]b = 0 1 0 0 (6.85)
1000
1 0 0 0
L0 0 0 0 J
0000

where (4.20) gives the Jacobian of this manipulator. The conditions on the right

hand side of Theorem 12 are satisfied since the above matrix is also [2W ]bE,. The

conditions on the right hand side of Theorem 13 are also met since 2fc 0 2v1 = 0

and 2nc 0 2Wf = 0 for all 2fc, 2Vf, 2n, and 2wf. Therefore, both Theorem 12 and

Theorem 13 tell us that the decomposition of the twist space into unique disjoint

subspaces is valid in this frame.

The Jacobian of the PR manipulator expressed in the translated frame t was given

in (4.22). The wrenches of constraint in column-reduced echelon form are

-1 0 0 .-
Pu Py
0 0 0 1
S 0 0 1 0 (6.86)
S0 0 1 0
0 1 0 0
1 0 0 0
where py 0. If p, = 0, the first column of [tWc]bE, is replaced by

[0, 1/p,, 0, 0, 0, 1] and the last column by [1, 0, 0, 0, 0, 0].









The requirement of the right hand side of Theorem 12 is violated by the above

['Wcf]bE,. Also, both conditions on the right hand side of Theorem 13 are violated

by the wrench in the first column of (6.86).

Theorem 12 and Theorem 13 lead to a similar unique decomposition of the wrench

space. The wrench space can sometimes be split into two disjoint subspaces, the

wrenches of constraint and the wrenches of nonconstraint, 'Wc. But first define a

subspace Vj in a manner similar to the definition of Wz, i.e., [VI]bE, are the twists

of freedom with either zero linear velocity or zero angular velocity. This leads to

Theorem 14 below.


Theorem 14

W = 'Wc E 'We = Vf = VJ


Proof

First prove that there exists a direct sum decomposition of W if Vf = Vj; and

then prove that if there is a direct sum decomposition of W, then Vf = Vj.

If Vf = Vj, the column-reduced echelon form basis vectors of [Vf]bE, have no units
and can therefore be used for a basis of Wd. But since the dimension of Vf plus the

dimension of We is six and Vf = Vj, then Wc, = Wd. Therefore [Wc]JbE, = [V]]bEv.

This proves one half of the theorem.

The second half of the theorem is proven as follows. If the decomposition W =

We E W,, is assumed, then the projection involved is Euclidean, i.e.,

We = Null[JJt] = Null[Jd] (6.87)

and

Wc = Range[JJt] = Range[J] (6.88)

where Jt must be physically consistent from the assumption. But Vf = Range[J] by






85

definition. Since Range[J] can be interpreted as both a wrench (of nonconstraint)

and a twist (of freedom), then Vf = Vj.



Finally, Theorem 15 below shows the equivalence of the decomposition of the

twist and wrench spaces when Jt is physically consistent i.e., the unique Euclidean

decomposition of the twists space results in the unique Euclidean decomposition of

the wrench space, and vice-versa.

Theorem 15 If Jt is physically consistent, the following are equivalent statements:

2'f = 'V; (6.89)

iWe = 'CW (6.90)

iW = 'Wc EWc (6.91)

iV = t'Vf Vc (6.92)

iv = 'W = 'Vf D W = Range[J] e Null[J] (6.93)



Proof

If Vf = Vj < We = Wz, Theorem 12 and Theorem 14 can be used to prove the

equivalence of the rest of the statements. From Theorem 12,

W, = W1 V = Vf E V, f = Vf E We (6.94)

From Theorem 14,


Vf = VJ 4V W = W E W,c = Wc @V (6.95)

Since the right-hand-side decomposition of these two equations are identical, (6.89)

and (6.90) are equivalent statements.

*









If the twist and wrench screw spaces are uniquely decomposable in a chosen frame,

then a rotation of the frame of expression on the disjoint subspaces will preserve

disjointedness since 'Gj = 'Aj. But a translation of the frame of expression will not

preserve the decomposition of the subspaces. In fact, only special manipulators have

the two unique subspaces (twists of constraint and wrenches of freedom) for twist

and wrench space decompositions in all configurations. (These manipulators will

be discussed in Section 6.7.) Generally, the set of twists that a manipulator cannot

achieve, Vf, is not a subspace of twists so no unique Vf can be found; and generally,

the set of wrenches that a manipulator can apply, W,,, is not a subspace of wrenches

so no unique Wn can be found.

The SCARA and the planar RRR manipulator discussed earlier are special manip-

ulators that decouple the twist and wrench spaces into two disjoint subspaces in all

frames of expression. For the SCARA manipulator in a frame arbitrarily translated

from frame 2, the column-reduced echelon form twists of freedom and the column-

reduced echelon form wrenches of constraint are
0 0 0 1- '0 0
0 0 1 0 0 0
f],E,, 0 1 0 00 0
bE= 0 0 0 0 E 0 1 (6.96)
0 0 0 0 10
1000 00

Since each of the column-reduced echelon form twists of freedom have zero linear

velocity or zero angular velocity, the manipulator decouples. It also decouples since

each of the column-reduced echelon form wrenches of constraint have zero force. Since

both of the constraint wrenches have zero force, this manipulator can apply a force

to the environment in any direction as long as the manipulator is not in a singular

configuration.

The terms decouple frame and decouple point are defined in Section 3.2 and

Section 4.1.1, respectively. The pseudo-inverse of the manipulator Jacobian in a









frame located at a decouple point (a decouple frame) is physically consistent. Some

new meaning of decouple points can now be presented.

Theorem 15 is based on the condition that Jt is physically consistent, i.e., the

frame of expression is located at a decouple point. All of the statements in this the-

orem are therefore the requirements necessary for a manipulator space, with respect

to a particular frame, to decouple. If the frame of expression is at a decouple point,

the twist and wrench spaces decouple identically as shown in (6.43) and (6.93).

Raibert and Craig [51] define a "constraint frame" as a frame in which the natural

and orthogonall" artificial constraints can be independently specified. A constraint

frame or a compliant frame [2] is a frame in which the twist and wrench spaces

decouple entirely into subspaces, and therefore twists and wrenches may be uniquely

decomposed into constraint and freedom components. For the SCARA and the planar

RRR manipulators, all frames are compliant frames.

The author of this paper prefers the term decouple point to describe a point at

which a frame can be placed that will allow the twist and wrench spaces to be uniquely

decomposed. This is also a point at which the pseudo-inverse is physically consistent.

In fact, at a decouple point, the fundamental theorem of algebra for commensurate

systems is meaningful for this noncommensurate system. As was shown in the Chap-

ter 4, any rotations of the frame at this point will not affect the decoupled nature of

the spaces.

When the frame of expression is not located at a decouple point, the twist and

wrench spaces cannot be uniquely decomposed by a direct sum. But, a part of the

twist or wrench spaces may be uniquely decomposable so that

Subspace[iV] = i iVi = TiV z iV (6.97)

Subspace['W] = 'Wc 'Wd = iW c Vf W (6.98)

For any frame i, a wrench coordinate transformation 'Ati exist that will convert

any single wrench of constraint with nonzero force and nonzero moment to a wrench









with a zero moment and the same force. This particular wrench coordinate transfor-

mation consists of a translation vector of


S- n f (6.99)
P- fl2

and no rotation. Note that this transformation will also generally convert other

wrenches that had zero moments to wrenches with nonzero moments.

Therefore, for all manipulators with Jacobian of rank less then six (i.e., a non-

empty wrench of constraint subspace), there exists a frame that makes at least one

of the constraint wrenches into an element of W,-, and thus W, : 0 in some frame.

For example, a P50 manipulator in frame 3 coordinates has the Jacobian and

column-reduced echelon form wrench of constraint basis of

0 a2S3 0 0 0 0
0 a3 + a23 a3 0 0 0
82+3+4
3= -a2C2 a3C2+3 0 0 0 0 [3Wc]bE, 4(a2c2a3 2+3)
82+3 0 0 0 S4
84
C2+3 0 0 0 -C4 1
0 1 11 0 0
(6.100)

Note that frame 3 is not a decouple frame. But (6.99) can be used with (1.16) to

find a frame where the manipulator does decouple,

(a2C2 + a3C2+3)4 (a2C2 + a3C2+3)C4 0 T (6.101)
p =- O (6.101)
82+3+4 32+3+4
82+3+4
twe = 3At,3[3W bE, = 0, 0, ( 2+3+4 0, 0, (6.102)
(a2C2 + a3C2+3).S4J

The physically consistent determinant of JTJ in frame t is

Det[tJ7 tJ] = (a2a353S2+3+4)2 (6.103)


A non-planar RRR manipulator with Denavit-Hartenberg parameters given in

Table 6.1 has a frame 2 Jacobian and column-reduced echelon form wrenches of




Full Text
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