Group Title: new approach to motor calculus and rigid body dynamics with application to serial open-loop chains /
Title: A new approach to motor calculus and rigid body dynamics with application to serial open-loop chains /
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Title: A new approach to motor calculus and rigid body dynamics with application to serial open-loop chains /
Physical Description: v, 122 leaves : ill. ; 28 cm.
Language: English
Creator: Lovell, Gilbert H., 1957-
Publication Date: 1986
Copyright Date: 1986
 Subjects
Subject: Dynamics, Rigid   ( lcsh )
Vector fields -- Mathematical models   ( lcsh )
Mechanical Engineering thesis Ph. D
Dissertations, Academic -- Mechanical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (Ph. D.)--University of Florida, 1986.
Bibliography: Bibliography: leaves 118-121.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Gilbert H. Lovell III.
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Bibliographic ID: UF00097403
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000896444
notis - AEK5071
oclc - 015359963

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A NEW APPROACH TO MOTOR CALCULUS
AND RIGID BODY DYNAMICS
WITH APPLICATION TO SERIAL OPEN-LOOP CHAINS





By

GILBERT H. LOVELL III


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

1986














ACKNOWLEDGMENTS

I would like to express my gratitude to all those who

have helped make this work possible.

I am especially indebted to my advisor, Dr. Joseph

Duffy, who has been a constant source of inspiration.

The many discussions with the faculty and students of

the Center for Intelligent Machines and Robotics have been

most helpful. Special thanks are due to Dr. Ralph

Selfridge, Dr. Gary Matthew, Dr. Renatus Ziegler, Sabri

Tosunoglu, Resit Soylu, Harvey Lipkin, and Mark Thomas.

Finally, I am grateful for the excellent typing by

Ms. Carole Boone.














TABLE OF CONTENTS


PAGE

ACKNOWLEDGMENTS .......................................... i

ABSTRACT....................................................



CHAPTER

I INTRODUCTION............................................. 1

II FUNDAMENTALS OF MOTOR CALCULUS......................6

2.1 The Motor ..................................... 6
2.2 The Bivector Representation ....................9
2.3 The Velocity Motor and the Wrench.............10
2.4 Algebraic Operations ..........................13
2.5 Some Linear Algebra Concepts ..................16
2.5.1 Coordinates and Subspaces..............16
2.5.2 On the Reciprocal Product..............18
2.5.3 Dyadics, Induced Inner Products and
Norms .................................. 24
2.6 Motor Differentiation and Integration .........28
2.7 Differentiation in a Moving Frame.............32
2.8 The Acceleration Motor ........................34
2.9 Identities.....................................35

III GEOMETRICAL CONSIDERATIONS........................ 38

3.1 The Pitch and Central Axis....................38
3.2 On Representing Lines, Rotors and "Vectors"...42
3.3 A Useful Decomposition ........................45
3.4 A Note on the Dual Operator and Dual Numbers..47
3.5 Streamlines of a Motor ........................50
3.6 The Unit Motor ................................50
3.7 The Screw..................................... 52
3.8 Geometry of Motor Operations .................. 55
3.8.1 The Reciprocal Product .................55
3.8.2 The Cross Product.......................58
3.8.3 The Sum ................................ 59
3.8.4 The Linear Combination .................62









IV ELEMENTARY DYNAMICS................................73

4.1 The Wrench and the Velocity Motor Revisited...73
4.2 The Momentum Motor ............................75
4.3 The Law of Momentum for a Particle System..... 76
4.4 Impulse, Conservation of Momentum.............79
4.5 Power, Work, and Energy for Rigid Bodies......80

V ELEMENTARY KINETOSTATICS...........................83

5.1 Constraint Between Bodies .....................83
5.1.1 Characterizing Constraint..............83
5.1.2 Static Equilibrium......................85
5.2 The Screw Pair................................. 86
5.3 Open-Loop Chains ..............................90
5.3.1 Modeling ...............................90
5.3.2 Velocity and Acceleration Analysis..... 90
5.3.3 Statics ................................94

VI DYNAMICS OF RIGID BODIES ...........................97

6.1 The Intertia Dyadic............................97
6.1.1 Derivation .............................97
6.1.2 Dyad Expansion ........................100
6.1.3 The Inertia Inner Product..............104
6.2 Equations of Motion for a Rigid Body.........106
6.3 Dynamics of an Open-Loop Chain...............108
6.4 Discussion ....................................113

VII CONCLUSIONS.......................................116

REFERENCES... ...........................................118

BIOGRAPHICAL SKETCH .................................... 122















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

A NEW APPROACH TO MOTOR CALCULUS
AND RIGID BODY DYNAMICS
WITH APPLICATION TO SERIAL OPEN-LOOP CHAINS

By

GILBERT H. LOVELL III

May 1986

Chairman: Dr. Joseph Duffy
Major Department: Mechanical Engineering

Motor calculus is a mathematical system that is quite

analogous to vector calculus and is particularly well-suited

for rigid body dynamics. This work presents a new approach

to motor calculus and its application to dynamics. As

opposed to the common bivector definition, a motor is here

defined to be a special kind of vector field. It is

demonstrated that this novel motor definition is both useful

for investigating the mathematical structure of motor

calculus and for applying motor calculus to rigid body

dynamics. New results include simple derivations and

representations of the inertia dyadic for a rigid body and

of the equations of motion for a serially connected open-

loop kinematic chain.















CHAPTER I
INTRODUCTION

Motor calculus is an elegant mathematical system that

is particularly well-suited to rigid body kinematics and

dynamics. Briefly, a motor is a geometric entity that can

be identified with the following physical entities: the

velocity distribution of a rigid body, the moment

distribution of a force system, and the momentum

distribution of a mass system. These vector fields are

respectively called the velocity motor, the wrench and the

momentum motor. Rigid body dynamics can be formulated as

the study of the algebraic and differential relationships

among these three kinds of vector fields; these

relationships are studied by means of motor calculus.

The basic ideas of motor calculus received considerable

attention in the late 19th century. Perhaps the single most

important work of that time was a treatise by R. S. Ball

entitled The Theory of Screws (1900). A screw is defined as

a line together with a real number called the pitch. By

adjoining another real number, called the magnitude, to the

screw, we obtain an entity that can be identified with a

motor. It was Clifford (1873), in fact, who defined the

motor as a "magnitude associated with a screw."









Ball considered, as did many other authors at that time

(perhaps most notably Plucker, 1866), the dynamics of a

rigid body from the standpoint of line geometry; the screw

is considered the fundamental entity. The magnitude

associated with a screw was incorporated through the

coordinates used to represent the screw--the screw

coordinates as Ball called them. Ball derived many of the

dynamical relations of an initially quiescent body in terms

of screw coordinates (he, in effect, did not consider

inertial forces and moments due to velocity).

In motor calculus the motor as opposed to the screw is

considered to be the fundamental entity. By introducing

various algebraic and differential operations for motors,

von Mises (1924a and 1924b) developed motor calculus in such

a way that it is quite analogous to vector calculus. He

also illustrated the ease with which motor calculus can be

applied to the dynamics of a rigid body in general (he

included the inertial forces and moments due to velocity).

For some forty years after von Mises's works, screw

theory and motor calculus received little attention. The

revival of these subjects, much of which has been in the

area of instantaneous kinematics of rigid bodies, was

initiated by Phillips and Hunt (1964), and there is

continuing interest at the present time (for a good review

of recent contributions, see Hunt, 1978; Bottema and Roth,

1979; and Phillips, 1984). Significant recent developments









in motor calculus and its application to rigid body dynamics

are due to Dimentberg (1965), Yang (1967, 1969, 1971, 1974),

Woo and Freudenstein (1971), Pennock and Yang (1982), and

Featherstone (1983, 1984).

It was my original intention to consider the use of

motor calculus in developing efficient algorithms for

dynamic analysis of serially connected open-loop kinematic

chains (it should be pointed out that this has been

investigated most recently by Featherstone, 1984). It turns

out, however, that the emphasis of this work is more

theoretical than originally intended. In the course of the

research it became apparent that there were many areas of

motor calculus proper and its application to rigid body

dynamics that could be treated more rigorously than had been

done previously.

This dissertation is essentially a rigorous development

of motor calculus and its application to rigid body

dynamics. An important new concept introduced is the

definition of a motor as a special kind of vector field.

This is in contrast to the common bivector definition (here,

we use the bivector as a particular representation for a

motor). The motivation for the vector field definition is

that it is more tractable as regards analytical

considerations of motor calculus and as regards the

application of motor calculus to dynamics. Aside from

investigating motor calculus proper and describing









elementary dynamics via motor calculus, we also consider the

dynamics of serially connected open-loop kinematic chains

using some of the new concepts developed.

We now briefly describe the contents each of the

chapters.

In Chapter II we define the motor as a vector field

with a special property, and we then proceed to develop the

fundamentals of motor calculus. With the new motor

definition, addition of motors is defined simply as

pointwise addition of the corresponding vector fields and we

show that the derivative of a motor is given by pointwise

differentiation of the vectors in the corresponding vector

field. It is believed that some of the results obtained for

motor dyadics and motor derivatives and integrals are new.

The dyad expansion that is given for a motor dyadic will be

seen to have novel and important applications in the

formulation of the dynamic equations for a rigid body, and

it is shown that by considering the set of all motors as a

normed space enables limiting operations for motor functions

to be treated in a more general and rigorous manner.

In Chapter III we consider various geometrical

properties of motors and motor operations. We, in effect,

consider some of the connections between motor calculus and

screw theory. These geometrical aspects of motor calculus

are often useful when motor calculus is applied to rigid

body kinematics and dynamics. Aside from using the new










motor definition to describe geometric properties, a novel

derivation of the so-called circle representation of a two-

system is given, which is believed to be simpler than any

other found in the literature.

In Chapter IV we consider elementary dynamics using the

motor calculus developed in Chapter II. Here, the new motor

definition is particularly useful for casting the laws of

motion in terms of motors.

In Chapter V we consider elementary kinetostatics using

material from both Chapters II and III. We consider in

particular the static and velocity/acceleration analysis of

an open-loop chain. The static analysis for an open-loop

chain constitutes a generalization of that found in Lipkin

and Duffy (1982) and the equations given here for

velocity/acceleration analysis of an open-loop chain are the

same as those given in Featherstone (1984).

In Chapter VI we tie together many of the results of

Chapters II-V for application to rigid body dynamics. The

new definition of a motor enables a relatively simple

construction of the inertia dyadic for a rigid body.

Further, the dyad expansion representation of the inertia

dyadic given here is novel and for the first time it relates

directly to the classical principal screws of inertia due to

Ball (1900). Finally, the equations of motion for an open-

loop chain are derived. As far as the author is aware, both

the derivation and the form of these equations are the

simplest to appear in the literature thus far.















CHAPTER II
FUNDAMENTALS OF MOTOR CALCULUS

In this chapter motor calculus is developed using the

novel vector field definition of the motor. We assume

familiarity with elementary vector calculus, linear algebra,

and calculus.



2.1 The Motor

We denote vectors and vector quantities by bold lower

case letters (e.g., v), and, as is customary in engineering,

we shall usually confound these two terms. The association

or lack thereof of a vector with a point or line determines

whether the vector is considered free or bound. Whether a

vector is being considered a free, point, or line vector

will often not be made explicit (this is also common in

engineering). In fact, the same vector may have different

associations in different contexts. The reason for this

convention is that the association (or lack thereof) is

often clear without making formal algebraic distinctions.

For example, suppose that vectors u and v are associated

with two distinct points. Then, we assume that such an

expression as u = v implies that u and v have the same

magnitude and direction. Or, the sum u + v is obtained by

assuming u and v are associated with the same point (i.e.,










assuming that u and v are translated in a parallel sense to

the same point so that they can be added), and then this sum

may, depending on the context, be a free, point, or line

vector.

A vector field is a set of vectors that is indexed by

all points in space. A vector at some point P in a vector

field v is denoted Vp. Here, v can be interpreted as an

arbitrary vector in the vector field to which it belongs.

We now define the motor as a special kind of vector field

and give some associated definitions and properties.

DEFINITION. A motor is a vector field m for which

there is a vector i such that for every pair of points P, Q,

we have



mQ mp = x PQ (2.1)



where PQ is a vector directed from P to Q.

The vector in is called the principal vector* of the

motor, and, as regards the above definition, this vector is

considered free. In mechanics, however, it is often

convenient to consider the principal vector bound.

THEOREM 2.1. The principal vector of a motor is

unique.





* This term is borrowed from Dimentberg (1965) and its
meaning here is equivalent to his definition.










PROOF. Suppose f and m' are principal vectors of the

motor m and let P, Q be chosen as distinct points such that

PQ i = PQ a' = 0 (i.e., PQ is perpendicular to m and

m'). From eqn. (2.1), we must have



Sx PQ = m' x P ,



or, on rearranging,



(i m') x PQ = 0



Cross multiplying both sides with PQ and using the identity

a x (b x c) = (a c)b (a b)c, we obtain


(2.2)


Since PQ : 0 (P and Q are distinct by assumption) and



PO ( ') = PQ PQ *



= 0 + 0


eqn. (2.2) becomes


if fi' = 0


(PQ PQ)(m m') [PQ (m m')]PO = 0










Thus f = f'. Q.E.D.

We denote motors by bold capital letters (e.g., M, N)

and the set of all motors by M. The notation M:m or m:M

indicates the vector field m corresponding to M. When we

write upper case letters for motors (M, N, L, etc.), we

shall always assume that vectors in their corresponding

vector fields are denoted by corresponding lower case

letters (m, n, 1, etc.). Thus, if the vector field v is a

motor, and we wish to denote this motor by M (i.e., M:v),

then we have m = v (where we assume that m and v are

associated with the same point).

Two motors M, N are said to be equal, M = N, if and

only if m = n (i.e., the vector fields are equal pointwise).

It follows from Theorem 2.1 that m = n.

Finally, the zero motor M = 0 is defined as the zero

vector field m = 0. It follows from eqn. (2.1) that f = 0.



2.2 The Bivector Representation

DEFINITION. A bivector corresponding to a motor M is

the ordered pair of vectors (m, mp) where P can be any

point.

THEOREM 2.2. A motor is characterized by any one of

its bivectors and any bivector characterizes a unique motor.

PROOF. This follows immediately from eqn. (2.1) and

Theorem 2.1. Q.E.D.









By virtue ot this theorem, any motor can be represented

by a necessarily unique bivector at a given point. We

denote the set of all bivectors at a point P by Bp.

Clearly, M and Bp have the same cardinal number.* We

indicate the bivector of a motor M at a point P by (M)p;

thus, (M)p = (i, mp).

Two bivectors (M)p, (N)p are said to be equal, (M) =

(N)p, if and only if m = n and mp = np. It follows that

(M)p = (N)p if and only if M = N.

Finally, the zero bivector is defined as (0,0). It

follows that (0)p = (0,0).



2.3 The Velocity Motor and the Wrench

We introduce here two of the three kinds of motors with

which we shall be concerned in rigid body mechanics. As

mentioned in the introduction, applying motor calculus to

rigid body dynamics comprises the study of the algebraic

and differential relationships among wrenches, velocity

motors and momentum motors. We introduce the velocity

motor and wrench here so that we can discuss some of their

mathematical properties in subsequent sections of this

chapter and in the next chapter. We reserve the

introduction of the momentum motor for Chapter IV. (It





* That is, there exists a one-one correspondence between M
and Bp.










turns out that the definition of the momentum motor is quite

similar to that of the wrench.)

DEFINITION. The velocity distribution of one frame

relative to another is called a velocity motor.

Let v be the velocity distribution of frame V' relative

to frame E. That v is in fact a motor is verified by

differentiating (with respect to time) a vector joining two

fixed points P,Q in E'. On the one hand we have



SPQ = Q x PQ
dt


where w is the so-called angular velocity of E' relative to

E, and on the other hand


d --
t PQ = VQ v


Thus,



VQ Vp = x PQ



The velocity motor V:v has principal vector 7 = w.

DEFINITION. The moment distribution of a force system

is called a wrench.*




* Ball (1900) also used the term wrench but not in the same
sense as here. We consider Ball's definition in Chapter IV.










Let m be the moment distribution of a force system,

composed of forces fl, f2, ***. fn (these force vectors can

be considered point vectors). In order to show that m is a

motor, let P,Q be arbitrary points and let rp, rQ be vectors

directed from P and Q to the point of application of fi.

Then, we have



mQ mp = r x f r P x


i i
= I (r r ) x f.
Q P 1


= QP x f.
1


= (f fi) x PQ



Thus, the wrench M:m has principal vector fi = If..

Because of the nature of the cross product, the forces

can be slid along their lines of action without altering the

moment distribution. Thus, if desired, the forces of a

force system can be considered line vectors as regards the

wrench of this force system.

We, of course, can be more general and say that the

moment distribution of any set of point (or line) vectors is

a motor. We shall use this fact when we consider the

momentum motor in Chapter IV.









2.4 Algebraic Operations

For every pair of motors M, N and real scalars a, B

(or, more concisely, for all M, N e M and a, a E R), we

define scalar multiplication and addition/subtraction by



aM BN : am n ;



the cross product by



M x N : mx n + mx n



and the reciprocal product by



M o N = m np + p n ,



where P is any point. In addition, we define -M:-m (or,

equivalently, -M = (-l)M).

In order to verify that the vector fields am $n and

Sx n + m x are in fact motors, we must show that they

have principal vectors (such that eqn. (2.1) is satisfied).

For every pair of points P,Q we have



(amQ Bn) (amp Bnp) = a(mQ mp) 8(nQ np)


= a x pQ BrT x PQ


= (am Bnff) x pQ









and



(x x no + m x ) -



(x x np + m x n) = m x (nQ np) + (mQ mp) x



= x (fi x PQ) + (m x PQ) x



= (m x ) x PQ ,



which verify that am m and m x K are the principal

vectors of aM aN and M x N respectively.

The reciprocal product M o N is independent of the

choice for P since for every point Q, it can be shown using

eqn. (2.1) that


S* nQ + mQ m = m np + mp n


We note that since M is closed under scalar

multiplication and addition (i.e., aM + SN e M for all M,

N e M and a, 8 e R), M is a vector space.*

We now define analogous operations to those above on

the set of bivectors Bp such that an ismorphism** is




* That all the axioms defining a vector space are satisfied
is easily verified.

** An isomorphism is a one-one correspondence that
preserves significant relations between two sets.









established between Bp and M. For all M, N E M and a,

B e R, the operations are defined by



a(ff, mnp) 8(i, np) = (am Sf, amp + 8np)



(i, mp) x (S, np) = (m x f, m x np + mp x ) ,



(i, mp) o (r, np) = m np + mp *



Then, we have an isomorphism for which



(aM BN)p = a(M)p B(N)p



(M)p x (N)p = (M x N)p



(M)p o (N)p = M o N


Note that Bp is a vector space and that (') P:M BP is an
onto, invertible linear transformation.
-1
Finally, we define (*)p as the inverse of (*)p.
p p
It is interesting to note that with the above

definition of motor addition, we can express a wrench as a

sum of motors each of which is the moment distribution of a

single force. This then would provide an alternative method

for verifying that the moment distribution of a force system

is a motor. We need only prove that the moment distribution










of a single force is a motor. The sum then is a motor since

M is closed under addition.



2.5 Some Linear Algebra Concepts

2.5.1 Coordinates and Subspaces

THEOREM 2.3. dim M = dim B = 6.*

PROOF. Since there exists an invertible linear

transformation of M onto Bp, namely, (*)p, we have that

dim M = dim Bp. Now, let {el, e2, e3} be a linearly

independent set of vectors. Then, the set {(el, 0),

(e2, 0), (e3, 0), (0, el), (0, e2), (0, e3)} is clearly a

basis for Bp. Thus dim Bp = 6. Q.E.D.

An isomorphism (with respect to addition and scalar

multiplication) between M and R6 can be established once a

basis is selected for M. Let {M1, M2, ..., M} be a basis

for M. Then the scalars al, a2, ..., a6 such that

6
M = aiMi are called the coordinates of M with respect to
i=l
this basis. We use the common notation [M] to denote a

column vector of coordinates (i.e., a coordinate vector):





* dim M and dim Bp refer to the dimensions of these vector
spaces; the dimension of a vector space is the number of
elements in a basis.









a1

a2

"3
[M] =
a4

a5

a6


By assuming the standard definitions for addition and scalar

multiplication of matrices (here, [M] is a 6 x 1 matrix), we

establish an isomorphism between M and R Since
6
(M6)p a I(M i) and {(Ml)P, (M2)P' "" (M6)P is a
i=l
basis for Bp, we have that al, a2, ..., a6 are also the

coordinates of (M)p with respect to this basis of bivectors.

Let El, E2, ..., Eg be defined such that (El ) =

(el, 0), (E2)p = (e2, 0), (E3)P = (e3, 0), (E4)p = (0, el),

(E5)P = (0, e2), (E6)P = (0, e3) where {el, e2, e3} is an

orthonormal set of vectors. Then, {E1, E2, ..., E6} is a

basis, which we call a standard basis at point P. Such a

basis, it turns out, is often convenient to use and is

commonly used in the literature.* When we refer a motor M

to a standard basis we shall often simply write







* Featherstone (1984) defined the term standard basis
equivalently. We shall show in Chapter III that there is a
rectangular set of coordinate axes associated with every
standard basis.








m
[M] =[



instead of writing the six coordinates.

We denote the subspace spanned by any set S of motors

by . If N is an n-dimensional subspace of M and

{MI, M2' ..., Mn} is a basis for N, then we have that
M2, ..., Nn> = N where for simplicity we have dropped the

set brackets. We define the coordinates of a motor M e N

with respect to the basis for N in a like manner as above;

thus,



al

a2

[Mn] =

an


n
where M = oa.M.. On selecting a basis for N, an
i=l1
isomorphism is established between N and R.



2.5.2 On the Reciprocal Product

The reciprocal product is a special kind of bilinear

scalar product on M. It is particularly useful in rigid

body dynamics (in ways analogous to the usefulness of the

dot product in particle dynamics). Here, we give some

definitions and properties concerning this product.










DEFINITIONS. Two motors, M, N are said to be

reciprocal to each other (or, simply, reciprocal) if M o N =

0. A motor M is self-reciprocal if M o M = 0.

Unlike the dot product for vectors, the reciprocal

product is not positive definite; that is, there exist

motors that are self-reciprocal. For example, if (M)p =

(I, 0) or (M)p = (0, m), then M oM = 0. The reciprocal

product, however, is non-degenerate; that is, when the

reciprocal product is expressed in matrix form,



M o N = [M]TQ[N]



the 6 x 6 matrix Q has full rank (or equivalently, is non-

singular) for all bases that induce this matrix form. Now,

let Qij denote the i,jth entry of Q. Then,



Q. = M. M.
ij 1 3


where {Mi M2,

of [M], [N] refer.

0 0

0 0

0 0
Q =
1 0

0 1

0 0


M6} is a basis to which the coordinates

Referring to a standard basis, we have

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

1 0 0 0
1000










for which IQI = -1 (thus verifying that 0 has full rank). A

diagonal form for Q is obtained by referring to a

co-reciprocal basis; that is, a basis for which


M. o M. = 0
1 3


for


i j


Let {el, e2, e3} be an orthonormal set. Then, the set {M1,

M2, ..., M6} such that


(MI)p = (el, el),



(M3)p = (e2, e2),



(M5)P = (e3, e3),


(M2 ) = (e -e ),



(M4)P = (e2, -e2),



(M6 ) = (e3, -e3)


is a particular co-reciprocal basis for which


0 0 0

-2 0 0

0 2 0

0 0 -2

0 0 0

0 0 0
000


0 0

0 0

0 0

0 0

2 0

0 -2










In this diagonal form, we see that the signature* of Q is

zero and that, again, Q has full rank.

Ball (1900) worked almost exclusively with co-

reciprocal bases. (In fact, the terms reciprocal and

co-reciprocal are due to Ball.) The screw coordinates that

Ball used are equivalent to the coordinates of a motor that

is referred to a co-reciprocal basis. Most of the equations

in Ball's work are expressed in terms of these coordinates.

It must be emphasized that it is considered desirable in

this dissertation to obtain equations and expressions in the

more general coordinate-free form. A special kind of motor

dyadic defined by von Mises (1924a) (which is covered in the

next subsection) enables an elegant means of expressing many

of Ball's results concerning the dynamics of a rigid body in

coordinate-free form. The importance of the reciprocal

product and co-reciprocal bases is immediately manifest in

this dyadic (as we shall see).

As regards the co-reciprocal basis given above, which

was used to obtain the diagonal form for Q, we note that

each motor in this basis is not self-reciprocal. We can

make a general statement concerning this.






* The signature of d bilinear form is defined as the number
of positive eigenvalues minus the number of negative
eigenvalues of its representing matrix (for our case Q).
The signature, like the rank, is independent of the choice
of basis.










THEOREM 2.4. Let the set of non-zero motors {M M2,

..., M6} be co-reciprocal. Then, this set is linearly

independent if and only if each motor in the set is not

self-reciprocal.

PROOF. We must show that al = a2 = ... = a6 = 0 is the

only solution for alM1 + a2M2 + ... + a6M6 = 0. Now, for

some j,



M. o (aiMI + a2M2 + ... a6M6) = M o 0



Or,



a.M. o M. = 0
3 3 3


If Mj is not self-reciprocal, then it follows that aj = 0.

Conversely, suppose that Mj is self-reciprocal. Assume

that the set {M1, M2' ..., M6} is linearly independent.

Now, if we choose this basis to obtain the matrix expression

for the reciprocal product, that is [MITQ[N], we find that

Q, which is a diagonal matrix, has a zero diagonal entry

corresponding to Mj o Mj. This implies that 0 is less than

full rank, which contradicts the assumption of linear

independence. Q.E.D.

(Note that this theorem can be generalized to any non-

degenerate scalar product that is not positive definite.)










DEFINITIONS. Let N c M be a subspace. Then, the

reciprocal subspace NR is the set of all motors reciprocal

to every motor in N. The two subspaces N, NR are said to be

reciprocal to each other (or, simply reciprocal). The

subspace N is said to be self-reciprocal if N c NR.

That NR is a subspace follows from the bilinearity of

the reciprocal product.

Since the reciprocal product is a special kind of non-

degenerate scalar product, theorems concerning these

products in general apply to the reciprocal product

inparticular. We state two such theorems whose proofs can

be found in (Lang, 1966).*

THEOREM 2.5. Let NR be reciprocal to N. Then,

dim NR + dim N = 6.

THEOREM 2.6. Every subspace of M has a co-reciprocal

basis.

In Lang, these theorems result from a rather general

treatment of non-degenerate scalar products. A less

abstract approach can be found in (Sugimoto and Duffy,

1982), where they exploit a one-one correspondence between

so-called reciprocal and orthogonal screw systems (these

screw systems are what we would call subspaces).








* See Lang's Theorem 5 on p. 129 and Theorem 9 on p. 135.









It is important to note that a subspace N and its

reciprocal subspace NR need not be complements;* this is

because of the existence of self-reciprocal motors.

Consider, for example, the subspace N = where M is self-

reciprocal. It then follows that N is self-reciprocal; that

is, Nc NR. Thus N and NR are not complements since N n NR

= N {0}. Finally, note that a self-reciprocal subspace

can have dimension no greater than three. For, if N is

self-reciprocal, then we must have dim N < dim NR (as

N c NR), and Theorem 2.5 would be contradicted it dim N > 3.



2.5.3 Dyadics, Induced Inner Products and Norms of
Symmetric Dyadics

We represent linear transformations on M by means of

dyadics. Motor dyadics were originally employed by

von Mises (1924a and 1924b) and, as we shall see in

Chapter VI, are well-suited for use in dynamics.

DEFINITIONS. A dyad MN comprises two motors and

represents a linear transformation such that tor every motor

L,


(MN)L = M(N o L)

n
A dyadic is a sum of dyads A = MiNi defined by
i=l





* N and NR would be complements, by definition, it N n NR =
{0).









n
AL = M(Ni o L)
i=l


We shall call a dyadic symmetric if it is symmetric

with respect to the reciprocal product; that is, for every

pair M, N, we have



M o AN = N o AM



(Von Mises defined symmetric dyadics likewise.) We shall be

primarily concerned with symmetric dyadics, and we now give

several associated definitions and properties of symmetric

dyadics.

If the bilinear form M o AN of a symmetric dyadic A

is positive definite (i.e., M o AM > 0 if and only if

M $ 0), then A is said to induce a Euclidean inner product

(*I*) defined by (MIN) = M o AN. The induced Euclidean

norm I q is defined by IIMII = (MIM)1/2. All inner

products that we shall consider are Euclidean.

We call the eigenvectors of a dyadic eigenmotors.

THEOREM 2.7. If a symmetric dyadic A induces an inner

product (-*I), then there exists a co-reciprocal, linearly

independent set of six eigenmotors [{M, M2, ..., M6}.

Moreover, with X1, X2, ..., 6 denoting the corresponding

eigenvalues, the dyadic can be expressed in the form


6 X.
A = Mo M.M (2.3)
i=1 1 1










PROOF. Since A is symmetric with respect to the

reciprocal product, A is also symmetric with respect to

(*I*); for, (MIAN) = M o (AN) = (AM) o AN = (AMIN). Thus,

there exists an orthogonal set of six eigenmotors (i.e., the

set is orthogonal with respect to (*I*)), and the

corresponding eigenvalues must be real. In order to show

that the set of eigenmotors {M1, M2, ..., M6} is co-

reciprocal, we note that

(MiIM.) = M.oAM.

= M.oX.M.
i J3

= X (M.oM.)
3 1 3



Now, since (I**) is positive definite, we must have

Xj > 0. Since (MiMj ) = 0 for i / j, we have also that

Mi o Mj = 0.

In order to prove the latter part of the theorem, we
6 X.
merely verify that the eigenmotors of M. MiM. are the
M oM. 1 0
i=l 1 1
same as those of A:


6 X. X.
S( M )M = M (M0oM)
i=l M. M. i i M j M. j j
i= 1 1 J 3


=j .M.









Now, that Mi o Mi / 0 can be proven two ways. Since {MI,

M2' ..." M6} is co-reciprocal and linearly independent,

by Theorem 2.6 of subsection 2.5.2 we have that the

motors cannot be self-reciprocal. Alternatively, Mi o Mi


1 1 1
(Mi o X Mi) Y (Mi o AM^) = (MilMi) > 0. Q.E.D.

6 X.
We call the expression MM. a dyad expansion
M.oM. 1 1
i=l 1 1
for A. This is similar to the spectral decomposition of a

linear transformation; if we were to define the dyad MiMi

such that (MiMi)N = Mi(MiIN), then the spectral

6 X.
decomposition of A would be A = M 1 M..
i=l (MiMi) i 1
It is interesting to note that if A is symmetric with

respect to (.**), then A always has a spectral

decomposition; yet, if A is symmetric with respect to the

reciprocal product and does not induce an inner product,

then A might not have a dyad expansion. This is because

eigenmotors of A can never be self-orthogonal; but it is

possible that some eigenmotors be self-reciprocal (consider

the dyadic MM where M is self-reciprocal).*

Theorem 2.7 constitutes a generalization of some

results of Featherstone (1984, see p. 57). It is

interesting to note that Featherstone obtained the dyad

expansion


* That A induce an inner product is only a necessary
condition for A to have a dyad expansion. It is not
sufficient (consider the dyadic MM where M is an element of
a co-reciprocal basis for M, all elements of which can be
taken as eigenmotors of MM).










6 (AM )(AMi)
A = ( M.)
i= (MiMi)




The dyads, however, in this dyadic can in fact be replaced

with the simpler expression in eqn. (2.3).

Finally, because of the form of eqn. (2.3), an

expression for the inverse A-1 is known immediately. Now,

it is easy to show that A-1 induces an inner product and

that the eigenmotors of A can also be taken as the

eigenmotors of A-1. Thus, we have

-1
6 1
A-I = 1 M.iMi (2.4)
i=l 1 1




2.6 Motor Differentiation and Integration

The existence of norms on M makes M a special kind of

metric space: a normed space. We assume that limiting

operations on M are with respect to Euclidean norms, and we

assume that limiting operations .for vectors are with

respect to the norm that gives the magnitude of a vector

(i.e., I|v = (v v)1/2), which is also Euclidean. We

consider motors as functions of real variables only. Thus,

with t, a e R, lim M(t) = L means that for every e > 0,
t+a
there exists a 6 > 0 such that 0 < It al < 6 implies i L -

M(t)II < C. Because normed, finite-dimensional vector









spaces are topologically closed,* we have that all limits

for motor functions are motors (hence, L is a motor).** The

notation lim m(t) denotes the vector field such that for
t+a
every fixed point P in the frame of reference, the

corresponding vector in this vector field is lim m (t).
t+a
Before we give the relationship between lim M(t) and
t+a
lim m(t), we first need to consider a few definitions and
t+a
properties.

DEFINITIONS. A motor M(t) is fixed in a frame if for

every fixed point P in this frame, mp(t) is constant. A set

of motors is fixed in a frame if each motor in the set is

fixed.

We shall assume for the remainder of this section that

all motors are referred to the same frame. Thus, if a motor

is fixed, it is fixed with respect to this frame.

Let {Ml, M2' ..., M6} be a fixed basis, and let al(t),

a2(t), ..., a6(t) be the coordinates of M(t). We shall

assume the equivalence of convergence in norm and coordinate

convergence; that is, lim M(t) = 0 if and only if lim ai(t)
t+a t+ a







* A set that contains its limit points is said to be
topologically closed or, simply, closed.


** See Lemma 54.2 of Voyevodin (1983, p. 173), which states
that any finite-dimensional subspace of a normed space is
closed.









= 0 for each i.* It follows that, on referring to a fixed

standard basis, we have

lim m(t)

[lim M(t)] = .
t+a lim mp(t)
Lt+a j

We now give the relationship between lim M(t) and
t+a
lim m(t). For simplicity, we suppress some notation so that
t+a
lim M = lim M(t), lim m = lim m(t), etc.
t+a t+a
THEOREM 2.7. For the motor function M(t), the

existence of lim M or lim m implies the existence of the

other and



lim M : lim m



for which the principal vector is lim fi.

PROOF. Suppose lim M exists, and let M be referred to

a standard basis. Then,


lim m
[lim M] =
lim m


Since P is arbitrary, we have







* For a general norm, the proof ot this is a bit
involved. For a Euclidean norm (which is all that we
consider), however, the proot is straightforward (Halmos,
1958, p. 175).









lim M : lim m,



for which the principal vector is lim m.

Suppose lim m exists. For every pair of points P, Q,

which we assume are fixed, we have



mQ mp = i x PQ



Taking the limit of both sides,



lim mQ lim mp = lim(i x PQ)



Since PQ is arbitrary and fixed, we have


lim(m x PQ) = (lim rn) x PQ


Thus lim m is a motor with principal vector lim m. Since



lim fn
=m [lim M]
lim m



we have lim m : lim M. Q.E.D.
am
We use the notation in order to indicate that
at
differentiation is taken with respect to fixed points;
thus m lm m(t)-m(a)
thus lim -.
at t-a
t+a
COROLLARY. For the motor function M(t), the existence
d O m
of either N M or am implies the existence of the other and
dt at









d am
-M
dt a" t

dm
for which the principal vector is d-
dt"
PROOF. Apply the theorem to the motor function

M(t)-M(a) m(t)-m(a)
: Q.E.D.
t-a t-a Q.E.D.

We shall always assume that limits and derivatives of

motor functions exist. In addition, we shall assume that

motor functions are integrable. With these assumptions, we

give the following theorem without proof.

THEOREM 2.8. For the motor function M(t), we have

a a
f M(-)dT : f m(T)d ,
a a

for which the principal vector is f m(T)dr.
a
The important results of this section, as regards the

remainder of this work, can be summarized as follows:


~ am
d dm P
dt P dt' at



( f MdT)p = ( / md, mpdT)
a a a




2.7 Differentiation in a Moving Frame

The relationship between the time derivatives of a

motor taken with respect to two frames is analogous to that
d d'
for a vector. If M and M are derivatives taken with
dt dt










respect to and ' and V the velocity motor of E' relative

to E, then

d d'
SM = M + V x M (2.5)
dt dt

Before we verify this, we need to introduce two

operations associated with vector fields: the material
dm I dm
derivative -m, and the gradient Vm. The notation dm
dt E dt V
denotes the derivative with respect to Z of a vector m

associated with a fixed point in V' (this vector field is

not necessarily a motor). The gradient Vm at a point P is a

linear operator defined by

|m -mp-(Vm)PQ = 0 (2.6)
lim r =0 (2.6)
Q+P |PIQ

where P can be approached in any direction by Q. We have

immediately that Vm = mx (where we define the operator mx

in the obvious way) since m is a motor; for, with this

substitution in eqn. (2.6), the numerator is zero for all

pairs of points P,Q. Thus the gradient Vm is independent of

the choice for P and is determined by the principal vector

m.





* In vector analysis, the curl of a vector field v, which
is denoted by.Vv, is defined by the operator equation

(Vv)x = Vv + (Vv)T.

Now, for the motor M, (Vm)T = -Vm since Vm = Ex. Thus, we
have that Vxm = 2ii.









Now, by the chain rule of differential calculus, we

obtain


dm m m
dt E at


(Note that -m dm- ) On rearranging and substituting for
at dt 2
Vm, we have


am 3'm ~ ~
Z'- t + v x m
-t at


which is the vector field form of eqn. (2.5).



2.8 The Acceleration Motor

The acceleration motor of a frame Z' relative to E is

defined as the time derivative of the velocity motor V

relating these two frames:


d : av
dt at


It is important to note that -L is not the acceleration

of the point with which it is associated. Or, more

precisely, it is not the total acceleration, which is
dv
given by the material derivative t- F,. The acceleration

motor V comprises what is called the local acceleration

field of E' relative to E. The relation between the local

and total acceleration can be determined from eqn. (2.7) of

the previous section:









av dv ~
St z- + v x
Btdt I'


Algebraic


2.9 Identities

Identities:


MoN = N o M



M x N = -(M x N)



L o (M + N) = L 0 M + L N


L x (M + N) = L x


M + Lx N


(XM) o N = X(M o N)



(XM) x N = X(M x N)



The algebraic identities can be proven using algebraic

properties of the corresponding vector fields. For example,



M x N: mx n + m x n



= -(n x m) (n x m)


= -(n x m + n x m) : -(N x M)









Differential Identities:



d (XM) = M + X ,
dt


d
d (M + N)


d
S(M o N)


d
(M x N)
dt


= A + i ,



+ oN + M o



= x N + Mx N


The differential identities can be proven using the

definition of derivative and the algebraic identities.

(Note that we need not resort to the differential properties

of the corresponding vector fields; this is because we have

defined limits for motors with respect to a metric on M.)

For example,


d (M x N)
dt t=a


= lim M(t)xN(t)-M(a)xN(a)
a t-a
t+a


S M(t)xN(t)+M(t)xN(a)-M(t)xN(a)-M(a)xN(a)
= lim -
t-a
t+a

lim [M(t)-M(a)]xN(a) + lim M(t)x[N(t)-N(a)]
t-a a t-a
t+a t a


= A x N + M x





37




The following additional identity proves very useful in

rigid body dynamics (see Chapter VI):



L o M x N = Lx Mo N
















CHAPTER III
GEOMETRICAL CONSIDERATIONS



3.1 The Pitch and Central Axis

DEFINITION. The pitch of a non-zero motor M is given

by


m*m
if m i 0
m~m
h =
m if m= 0




A non-zero motor is said to be proper if h is finite

and improper if h is infinite. The product m mp is

independent of the choice of point P as can be verified from

eqn. (2.1). In fact, m = M o M (for all motors); so
1 MoM
we can alternatively write h for proper motors.
2 p
m*m
THEOREM 3.1. For every proper motor, there exists a

unique line such that mp = hm for every point P on the line.

PROOF. We assert that the equation of the line such

that mp = h, P being a point on the line, is given by


mxnm
r = + km k E R
m*m










where O is an arbitrary point assumed not on the line and

r = OP.

By definition of a motor, we have



ip = mO + E x r



Substituting for r and simplifying,


mxmO
mp = m0 + i x (- + km)
m*m



= mO + (hm mO + 0)



= hm ,



which verifies the assertion. Now, r= -- for k = 0.
m*m
Since points 0 and P are distinct (by assumption), we

mxm
have 0, which in turn implies i x mO ? 0. Thus,
m*m
mO n hE (for, mO = h. implies m x mO = 0). Thus the given

line is unique. Q.E.D.

The line for a proper motor such that mp = hf for

points P on the line is called the central axis of the

motor.

THEOREM 3.2. Let M be proper. Then, every non-zero

motor in has the same central axis and pitch. Moreover,

only motors in have this central axis and pitch

combination.




40




PROOF. Let P be on the central axis of M. Then,


mp = h-


(3.1)


where h is the pitch of M. Multiplying by a non-zero scalar

X,


Xmp = h X


and noting that



(XM)p = (Xi, Xmp)



we conclude that P is also on the axis of XM and that the

pitch of XM is also h.

For the latter part of the theorem, suppose motor N has

the same central axis and pitch as M. Then,


np = hni


(3.2)


Since i and ii are parallel, there exists a scalar B such

that ? = im. Thus, from eqns. (3.1) and (3.2), we also have

np = 8mp. Now,


(N)p = (i, np)










= (Bi, amp)



= (BM)p



Therefore, N = BM e . Q.E.D.

Thus, we have that one-dimensional subspaces of proper

motors characterize central axis and pitch combinations.

For improper motors, there is no central axis in the

sense defined above. We note that an improper motor M

comprises a constant vector field (i.e., for every pair of

points P, Q, mp = mQ). Since a = O and h = -, there can be

no points P such that mp = hm. We now resort to projective

geometry in order to define the central axis of an improper

motor.

Since one-dimensional subspaces of proper motors

characterize central axis and pitch combinations, we

likewise define the central axis of an improper motors

(which by definition has infinite pitch) as some geometric

entity characterized by the one-dimensional subspace to

which the motor belongs. In projective geometry, this new

type of central axis is known as a line at infinity or a

non-Euclidean line, as opposed to a Euclidean line, which is

a line in Euclidean space.

By Theorem 3.2 and the above definition of the central

axis for a improper motor, we can state the following:









THEOREM 3.3. There exists a one-one correspondence

between one-dimensional subspaces of M and all central axis

and pitch combinations.



3.2 On Representing Lines, Rotors and "Vectors"

By definition, we have that for every non-Euclidean

line, there exists a motor (which is improper) that has this

line as a central axis. We must, however, establish an

analogous fact regarding Euclidean lines.

For some arbitrary Euclidean line, let v be a non-zero

vector bound to the line and r be directed from an arbitrary

point in space to any point on the line. Then, the moment

distribution r x v is a motor with principal vector v. (If

v is a force, then this motor is a wrench.) Let M:r x v.

Then, f = v and m = r x v. The pitch h of M is clearly zero

since i m = 0. Thus, we have that the given line is the

central axis of M since for any point P on the line mp = hm

(=0), and we can now state the following:

THEOREM 3.4. For every line (Euclidean and non-

Euclidean), there exists some motor that has this line as a

central axis.

We can thus represent every line (Euclidean and non-

Euclidean) by a motor. Therefore, we can, in turn,

represent a line by motor coordinates' or by a bivector.

When a motor M is of zero or infinite pitch, then the

coordinates of M with respect to a standard basis, that is,











[M] = mm



constitute the well-known Plucker coordinates of the line*

that is the central axis of M.

DEFINTITION. A rotor is a motor of zero pitch.

The term rotor is adopted from Clifford (1873), which

he defined in the sense of a line vector (i.e., a vector

bound to a Euclidean line), and the definition we use is

equivalent to Clifford's. The discussion preceding

Theorem 3.4 shows that there is a unique motor of zero pitch

with a given principal vector and central axis (here, we can

consider the principal vector bound to the central axis),

and it turns out that our definition for addition of rotors

(which, of course, is simply addition of motors of zero

pitch) is also equivalent to Clifford's.

Rotors can thus be used to represent any Euclidean line

and/or line vector. (In particular, rotors can be used to

represent forces when forces are interpreted as line

vectors.)

Now, what we call an improper motor is what Clifford

defined a vector to be. Thus, a "vector" is the set of all



* The Plucker coordinates of a line are called homogeneous
coordinates. Coordinates of a geometric entity, say, aI,
a2, ... a are homogeneous when Xal, Xa', ..., Xan
represent the same geometric entity where X is an arbitrary
non-zero real scalar.









vectors indexed to points in space that have the same

magnitude and direction (i.e., a constant vector field).

Addition of "vectors," as Clifford defined it, is equivalent

to addition of improper motors.

As Clifford pointed out, the set of "vectors" is closed

under addition whereas the set of rotors is not.* Clifford

designated a motor as that which may be a rotor, "vector,"

or the new entity that can result from the addition of two

rotors. This new entity is what we call a motor of non-

zero, non-infinite pitch. We shall show in the next section

that such a motor can be decomposed into a rotor and an

improper motor (i.e., a "vector").

We close this section with a geometric description of a

standard basis. Let {E1, E2, ..., E6 be such a basis for

which



(El)O = (el, 0), (E2)0 = (e2, 0), (E3)0 = (e3, 0)



(E4)O = (0, el), (E5) = (0, e2), (Eg)O = (0, e3)



We note that E1, E2, E3 are rotors and E4, E5, Eg are

improper motors. The central axes of E1, E2, E3 constitute




* We tacitly assume here that the zero "vector" and zero
rotor are defined; these would both be the same as the zero
motor. Note, however, that we shall maintain the convention
that the zero motor is neither proper nor improper (or,
equivalently, the pitch is undefined).









a rectangular set of lines that intersect at O (see

Fig. 3.1). If we designate the directions of these lines by

el, e2, e3, we then have a Cartesian coordinate system with

origin O. Clearly, the first three coordinates of a motor M

with respect to this standard basis are the vector

coordinates of f in this Cartesian system and the last three

coordinates are the vector coordinates of mO in this same

Cartesian system.



3.3 A Useful Decomposition

Let M be proper with pitch h. We define



MO:m hE and M :m



so that we have



M = MO + hM (3.3)



The superscripts indicate the pitches. We can show

simultaneously that the pitch of MO is in fact zero and that

the central axis of M0 is the same as that of M: If P is a

point on the central axis of M, then



m = mp hm = Om ,


where m0:M0.


















e2


el


e3






Fig. 3.1. The Cartesian Coordinate System of a Standard
Basis


N"'I










We thus have that every proper motor can be decomposed

into a rotor and an improper motor (i.e., a "vector" in the

sense of Clifford as mentioned in the proceeding section).

Now, from the previous section we have that



MO:r x im



where r is directed from an arbitrary point in space to the

central axis of MO. Thus, we have



M:r x i + hi (3.4)



which enables us to construct a proper motor given the

central axis, pitch, and principal vector.

Finally, we note that with respect to a standard basis,

[M0] comprises the Plucker coordinates of the central axis

of M.



3.4 A Note on the Dual Operator and Dual Numbers

It is perhaps of interest here to consider a special

operator for motors introduced by Clifford (1873), which he

designated by w so defined that


and w(wM) = 0


WM = Mm









for every proper motor M. It is easy to show that the rank

of the linear transformation represented by w is three; the

null space comprises all improper motors (and the zero

motor). (That wN = 0 for improper N follows from the fact

that N = w((n, v))-1 where v can be any vector and P any

point. The motor ((n, v))pl is, of course, necessarily

proper since n $ 0.) With respect to a standard basis, the

matrix of this transformation is



0 0 0 0 0 0

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 1 0 0 0



We can equivalently define w as a special kind of

number such that for all a, 0 E R and every proper motor M,



(a + sw + w2)M = aM + BM (3.5)



From this, we can conclude w2 = 0 (chose a = B = 0) and wO =

0 (choose B = 0). (Note that since w2M = ((wM), we have

that wN = 0 for any improper motor N.) Thus, multiplication

of a motor by the number w is equivalent to operating on it

by the operator w.







Numbers ot the form a + Bw were called dual numbers by

Study (1903) and were applied by him in linear geometry.

The term dual operator is often used in recent literature

for w when the operator interpretation is used.

The algebra of dual numbers as induced trom eqn. (3.5)

is a commutative ring. Some authors who have investigated

properties of dual number algebra include Dimentberg (1965),

Velakamp (1976), and Rooney (1975). The motivation for the

study of dual number algebra lies primarily in its

application to line geometry and rigid body kinematics

(Study, 1903; Brand, 1947; Dimentberg, 1965; Yang, 1969,

1974; Yang ana Freucenstein, 1964; Veldkamp, 1976; Keler,

1973, 1979). The application ot dual number algebra to

rigid body dynamics, however, has received much less

attention (DimentDery, 1965; Yany, 1967, 1969, 1971, 1974;

Pennock and Yang, 1983).

Dual number algebra is not a particularly useful tool

for rigid body dynamics. Although equations ot motion can

be expressed in dual number storm (see tne cited works of

Dimentberg and Yang), such a form does not, in my opinion,

provide a catalyst for explication or dynamic properties.

It is for this reason that we will not exploit dual number

algebra.









3.5 Streamlines of a Motor

By use of the decomposition in eqn. (3.3), we can

easily describe the geometric nature of the vector field of

a motor. Let M be proper and let r = OP where O is any

point and P is on the central axis of M. Then we have m =

r x fi (where m0:M0). The streamlines of m0 comprise

concentric circles about the central axis and in planes

normal to the central axis. One such circle is illustrated

in Fig. 3.2. By adding hi to m0 (note hm + m :M), the

streamlines for m become concentric helices of pitch h about

the central axis. One such helix is illustrated in

Fig. 3.3. The streamlines of mm(=m) are simply straight

lines parallel to the central axis.



3.6 The Unit Motor

We reserve the symbol S for unit motors. A unit motor

S is defined such that |1| = 1 ( Is = 1) if S is proper

(improper).

It is important to note that a unit motor is not

defined with respect to a norm. The reason for so defining

unit motors is that they are convenient for geometric

purposes and for use in mechanics. (Von Mises defined unit

motors similarly.)
















K

central /
axis









Fig. 3.2. Streamline of a Rotor


r x Yi + lhim


Fiq. 3.3. Streamline of a Motor (h#O and hl)









3.7 The Screw

DEFINITION (Synthetic). A screw is a central axis and

pitch combination of some motor.

DEFINITION (Analytic). A screw is a one-dimensional

subspace of M.

The terms synthetic and analytic are taken from

projective geometry. The synthetic definition of the screw

is the same as that given by Ball (1900). We shall,

however, usually confound the two definitions by exploiting

Theorem 3.3. The synthetic definition is convenient for

visualization purposes, and the analytic definition is

convenient for algebraic purposes.

In analogy with motors, we say that a screw is proper

(improper) if its pitch is finite (infinite).

We can now define a proper motor in terms of a screw.

It is in this manner that Clifford defined the motor: "Just

as a vector . is magnitude associated with direction,

and as a rotor . is magnitude associated with an axis;

so this new quantity, which is the sum of two or more rotors

. is magnitude associated with a screw." It is in

fact implicit in Clifford's definition of motor (and rotor)

that a sense is assumed for the central axis. Thus, the

central axis, pitch, and the magnitude and sense (with

respect to the central axis) of the principal vector

determines a proper motor.









We can now give Ball's definitions of twist and

wrench. A "twist about a screw" is equivalent to a velocity

motor. The twist is the magnitude of the angular velocity

(or linear velocity if the velocity motor is improper). A

"wrench on a screw" is the same as our definition of

wrench. Ball's wrench is the magnitude of the net force of

the force system (or the couple of the force system if the

representing wrench is improper). (As with Clifford, a

sense is assumed for the central axes of screws.)

By the use of unit motors, we can make the above notion

of "magnitude associated with a screw" precise. Every motor

can be expressed as a scalar multiple of a unit motor:

M = S where X = I\I (X = |m| for improper M) and S =

(S can be chosen as any unit motor if X = 0). Thus, S

represents the screw of M together with a direction, where

we assume the direction of S is given by s (s) if S is

proper (improper). We assume there is no definite direction

if X = 0. We call X the magnitude of the motor, which,

obviously, may be positive or negative (we note in

particular that the magnitude of each unit motor S may be

either 1). If M is a wrench, then X is the magnitude of a

force (torque) if M is proper (improper); if M is a velocity

motor, then X is the magnitude of an angular (linear)

velocity if M is proper (improper).

The following theorem is useful in kinematics.









THEOREM 3.5. A unit motor S is fixed in a frame E if

and only if

(i) the central axis of S is fixed in ;

(ii) the pitch of S is constant;

(iii) the direction of S is constant.

PROOF. The proof is immediate for improper S. If S is

proper, then conditions (i) and (iii) determine SO, and S

which in turn, determines S". Both S and So must be

fixed. That the pitch h is constant implies that S = SO +

hS" is fixed.

The proof of the converse is immediate. Q.E.D.

DEFINITIONS. The screws , are said to be

reciprocal to each other (or, simply, reciprocal) if M o N =

0 (i.e., the motors M, N are reciprocal). A motor system is

a subspace of M. A screw system is the set of screws

corresponding to a motor system. The order of a motor

system and its corresponding screw system is the dimension

of the motor system. A one-, two-, ..., n-system is a screw

system of order one, two, ..., n.

The term screw system is due to Ball and the term motor

system is due to Everett (1875).

Motor systems are convenient for characterizing

permissible relative motion between rigid bodies in

constrained rigid body systems (see Chapter V).









3.8 Geometry of Motor Operations

In this section, we present some geometric properties

associated with the reciprocal product, cross product, sum,

and linear combination. All the results obtained for the

first three operations are well-known. The derivation of

the circle representation for a two-system in subsection

3.8.4, however, is believed to be simpler than any other

derivations found in the literature.

In the following, we assume that the pitches of the

motors M1, M2, and M (and the unit motors Si' S2, and S) are

hl, h2, and h.



3.8.1 The Reciprocal Product

Let M1 = XS1 and M2= X2S2 be proper, the central axes

of which are illustrated in Fig. 3.4. The common normal

line intersects the central axes at points P1 and P2. We

specify the relative position of one axis relative to the

other by the vector de = P P2, where lel = 1 and d > 0, and

by the angle a, which is subtended by l and s2 and is

measured in a right hand sense about e.

Consider



Mi o M2 = 2S1 o S2


0 h + h0)
12(S hlS) o (S + h2S 2
= I 12s + s22


















s


Fig. 3.4. Central Axes of S1 and S2












= X1X2(S8 o S2 + hlS1 o S + h2S2o S + hhS2 o S2



Now,


0 0 0
1 2 1


0
o (S2 )
2 1


= (, 0) o (s2, de x s2)



= -de s x s2



= -de esinal2



= -d sinal


CO 0 ~
S1 S2 = S2



= COSa.


S2 o S1
2 s


= 2


12



* s


= cosa2


S o S20
s1 2 S


= 0









Substituting, we obtain



M1 o M2 = X 2[(hl + h2)cosa d sina] (3.6)



which is the well-known expression for the reciprocal

product of two proper motors.*

Using eqn. (3.6), we can deduce the following:

THEOREM 3.6. Suppose the proper motors M1, M2 (screws

, ) have intersecting central axes. Then, the

motors (screws) are reciprocal if and only if one or both of

the following hold:



(i) hi = -h2'

(ii) the central axes are perpendicular.



It is clear that every pair of improper motors (screws)

is reciprocal. If M1 is proper and M2 improper, then it is

also clear that M1 o M2 = 0 if and only if m1 m2 = 0.



3.8.2 The Cross Product

THEOREM 3.7. Let M1, M2 be proper motors with non-

parallel central axes. Then, M1 x M2 is a proper motor




* Ball called the expression
1[(hl + h2) cosa d sina]
the virtual coefficient of the screws ad S2
the virtual coefficient of the screws and .









whose central axis is the common normal of the central axes

of M1 and M2.

PROOF. The product M1 x M2 is proper since m1 x m2

/ 0. Now, since



l x m2 1 m-il= ml x m2 m2 = 0



and



M1 x M2 o M1 = M1 x M2 M2 = 0 ,



we have by Theorem 3.6 that the central axis of M1 x M2 is

the common normal to the central axes of M1 and M2. Q.E.D.

THEOREM 3.8. The product M1 x M2 is improper if and

only if either condition holds:



(i) M1 is improper and m x m2 / 0,

(ii) M1 and M2 are proper motors whose central axes

are distinct and parallel.



PROOF. The proof is immediate. Q.E.D.



3.8.3 The Motor Sum

THEOREM 3.9. If the proper motors M M2 have non-

parallel central axes (or parallel central axes such that

ml / -"2)' then the sum M = M1 + M2 is a proper motor whose




60




central axis intersects the common normal line (or every

common normal line) of the central axes of M1 and M2.

PROOF. Let the common normal line (or some common

normal line) be represented by the rotor S. Then, we must

have



Es 1 = m2 = 0



and by Theorem 3.6, we have



S M1= S M2 = 0



Now, since mi + m2 / 0 (thus M is proper), we have that the

central axis of M is perpendicular to central axis of S;

for,



s (m + n2) = s ml + s m2



=0


In addition, since


S o (M1 + M2) = S o M1 + S o M2


= 0









it follows from Theorem 3.6 that the central axis of M also

intersects the central axis of S. Q.E.D.

THEOREM 3.10. Let M1, M2 be proper motors whose

central axes are parallel and ml = -m2. Then, the sum M =

M1 + M2 is



(i) improper if the central axes of M1 and M2 are

distinct or hi / h2, or, equivalently, /

;

(ii) zero if the central axes of M1 and M2 are the

same and h1 = h2, or, equivalently, = .



PROOF. For both (i) and (ii) we have m = ml + m2 = 0.

For (i), 2 implies that M = M1 + M2 / 0; thus, M

is improper. For (ii), = and mi = -m2 implies

that M1 + M2 = 0.

THEOREM 3.11. If two proper motors MI, M2 have non-

parallel, coplanar central axes, then the central axis of

the sum M = M1 + M2 passes through the point of intersection

of the central axes of Ml and M2 if and only if hi = h2.

Moreover, h = hl(= h2) where h is the pitch of M, and the

central axis of M lies in the plane of the central axes of

M1 and M2.

PROOF. Let P be the point of intersection of the

central axes of M1 and M2. Then, we have









(M1 + M2)P = (M1)P + (M2)P



= (i1, h 1 ) + (m"2 h2m2)



= (mi + m2, hlm~ + h22 2)



Now, M + N is proper since ml + m2*f 0. Point P is on the

central axis of M if and only if (m1 + m2) x (hl ml + h2m2)

= 0, in which case h1 = h2 so that ml + m2 = hl(m1 + ml)

(= h2(f1 + i2)) and h = hl(= h2).

That the three central axes are coplanar follows from

Theorem 3.9. Q.E.D.



3.8.4 The Linear Combination

In this subsection, we study the geometry of the linear

combination of two motors; that is, we study the geometry of

the two-system. (We are assuming, of course, that the two

motors constitute a linearly independent set.)

THEOREM 3.12. Every pair of distinct screws in a two-

system determines the two-system.

PROOF. Let and be distinct screws in a two-

system. Then, n = {0} implies that {S1, S2} is

linearly independent and thus forms a basis for the motor

system that corresponds to the two-system. Q.E.D.

(Note that this theorem cannot be extended to screw-

systems of order greater than two: it is possible that ,









, ..., be distinct and that {Sl, S2, ..., Sn is

linearly dependent for n > 2.)

DEFINITION. The ruled surface comprising the central

axes of a two-system is called a cylindroid.

The cylindroid was discovered by Hamilton (1830) and

named by Cayley.*

A two-system is characterized by its cylindroid and the

distribution of pitch for the lines on the cylindroid.

We classify two-systems according to the following

scheme:



CLASS 1. All screws are proper.

CLASS 2. All screws are improper.

CLASS 3. Only one screw is improper.



THEOREM 3.13. The above classification scheme is

exhaustive and each class is non-empty.

PROOF. Let and be distinct screws in a two-

system. That this classification scheme is exhaustive

follows from the fact that if and are improper,

then all screws in the two-system are improper.





* In the footnote on p. 20 of Ball's work, he writes, "The
name cylindroid was suggested by Professor Cayley in 1871 in
reply to a request I made when in ignorance of the previous
work of both Plucker and Battaglini, I began to study this
surface."









We must show that each class is non-empty. Clearly,

there exist two-systems that belong to Class 2. Now,

suppose and are proper. If the central axes of

these screws are non-parallel, then and determine

a two-system that comprises proper screws (i.e., a Class 1

two-system). If the central axes of and are

parallel and 91 = -s2, then the one improper screw of the

Class 2 two-system determined and is .

Q.E.D.

We note that every pair of distinct screws in a Class 1

two-system must have non-parallel central axes; for, a two-

system that has a pair of distinct screws whose central axes

are parallel must also have an improper screw (see Theorem

3.10). We also note that the proper screws of a Class 3

two-system must have parallel central axes: the central axes

of a proper motor and the sum of this motor and any improper

motor are parallel. By Theorem 3.9, we also have that the

central axes of the proper screws of a Class 3 two-system

are coplanar.

DEFINITION. A two-system is called degenerate if it

contains at least one improper screw.

Thus, Class 2 and Class 3 two-systems are degenerate,

and Class 1 two-systems are non-degenerate.

The cylindroids of degenerate two-systems are simple

geometrically. The central axes of a Class 2 two-system all









lie in the so-called plane at infinity.* The central axes

of the proper screws of a Class 3 two-system are either all

the same or lie in a plane.

The cylindroids of non-degenerate two-systems, however,

are much more complex, and we devote the remainder of the

subsection to non-degenerate two-systems. We shall

ultimately obtain the so-called circle representation of a

non-degenerate two-system, which was discovered by Lewis

(1880) and used to great advantage by Ball (1900). The

circle representation quite elegantly illustrates the

geometric properties of non-degenerate two-systems as can be

evidenced in Ball's work. Since the circle representation

applies only to non-degenerate two-systems (this is the

motivation for the above definition), we shall assume

henceforth that "two-system" means "non-degenerate two-

system" and that cylindroids are of two-systems (i.e., non-

degenerate ones).

THEOREM 3.14. All lines on a cylindroid intersect a

common normal line.

PROOF. Every pair of lines on a cylindroid has a

unique common normal line. It follows from Theorem 3.9 that

there exists a unique common normal line for all lines on

the cylindroid. Q.E.D.





* In projective geometry all lines at infinity and all
points at infinity lie in the plane at infinity.









The common normal line of a cylindroid is called the

central axis of the cylindroid.

LEMMA. To each line on a cylindroid there exists a

corresponding unique line on the cylindroid that is

perpendicular to it.

PROOF. Let and be distinct screws of a

two-system. A screw in this two-system whose central

axis is perpendicular to the central axis of is

. This perpendicular central axis is

unique; for, otherwise, there would be two parallel central

axes, which is impossible. Q.E.D.

If and are distinct screws of a two-system,

every screw in this two-system can be given by

where c4 = cost and s. = sin, and for which pe[0, i). (This

follows since it is the ratio s :c. that determines the

screw, and all possible ratios are given with pe[0,r).)

THEOREM 3.15. If the cylindroid of a two-system does

not lie in a plane, then there exists a unique pair of lines

on the cylindroid that intersect at right angles.

PROOF. Let and be chosen from a two-system

such that their central axes are perpendicular (that there

exists a pair follows from the Lemma). Now, all pairs of

screws with perpendicular central axes are given by
s S2> and <-s S1 + c S2> where EI[0,P). By Theorem 3.6,

the necessary and sufficient condition that such a pair of

screws also have perpendicular central axes is









(c S1 + s S2) o (-s S1 + c S2) = 0



Distributing the product and substituting 2h1 = S1 o S1'

2h2 = S2 o S2, d = -S1 o S2, s2 = 2sc and c2 = c s2

yields



(h2 hI) s2p dc2 = 0



Now, it is not possible that both h2 h1 = 0 and d = 0;

for, if h1 = h2 and d = 0, then, by Theorem 3.11, all the

central axes would lie in a plane. Thus, there is a unique

solution with p 0 E [0, ) and therefore a unique pair of

screws with central axes that intersect at right angles.

Q.E.D.

The two screws on a cylindroid that have intersecting

perpendicular axes are called the principal screws of a

cylindroid. This pair, of course, is unique for cylindroids

that do not lie in a plane. If the cylindroid does lie in a

plane, then any pair of perpendicular central axes can be

chosen for the principal screws.

We now obtain the equation of the cylindroid and the

distribution of pitch for a two-system. Let a right-handed

Cartesian coordinate system be located such that the x- and

y-axes are the principal axes of the cylindroid of the two-

system (thus, the z-axis is the central axis). Let and

be the principal screws where ,1 and 32 determine the










positive x- and y-axis directions. Any screw in the

two-system is given by = where E[0O, ir).

The position of the central axis of can be specified by

*, which is the angle subtended by s and i1, and by the

distance z, which locates the intersection of the central

axis with the z-axis (see Fig. 3.5).

The equation of the cylindroid can be obtained by

taking the reciprocal product of S with the rotor Sr =
0 0
-sS1 + c S0 whose axis is perpendicular to both the z-axis

and the central axis of S and passes through the origin.

Using eqn. (3.6) we have



S o S (h + 0) cos 3- z sin 3
2 2


= z



We also have











0 0
where we have substituted h = S o S, h = S2 o S2 and 0 =

S 0 S S 0 0. Thus, the equation of the cylindroid is
given by2
given by




69


















S2



y









x


Fig. 3.5. Position of Central Axis on Cylindroid









z = -s c (hI h2).



The distribution of pitch is given by


1
h = S o S



= (cI S + s S2) o (cS1 + ~S2)


2 2
= c2h + s h2
i 1 2


where we have substituted h S1 o S, h = S2 o S2,

and 0 = Sl 0 S2.
1 1
Employing the identities s25 = s c (1 + c2) =
2 1 2
c and -(1 c2) = s the equation of the cylindroid

and the distribution of pitch can be expressed as



z = (h h2)s2 (3.7)




1 1
h = (h h2 + (h h2)c (3.8)



In the hz plane, it is evident that h vs. z is a

circle. Assuming without loss of generality that hi > h2, a

circle representation of the two-system is illustrated in

Fig. 3.6. For hi h2, each point on the circle represents

a unique screw in the two-system; for hi = h2, the circle






















































Fig. 3.6. Circle Representation of Two-System









degenerates to a point, which represents all screws in the

two-system (the cylindroid lies in a plane). By

consideration of the circle representation, the following

facts are evident:



(i) The principal screws assume the extreme values of

pitch.

(ii) The length of the cylindroid (i.e., along the

z-axis) is |hl h21

(iii) The plane containing the principal axes (i.e.,

the x- and y-axes) bisects the length.















CHAPTER IV
ELEMENTARY DYNAMICS

In this chapter we express basic principles of dynamics

in terms of motors. We assume familiarity with the basic

definitions and principles of analytical mechanics. We only

consider finite systems of mass particles (which is the

realm of analytical dynamics), because to consider

continuous systems would introduce unnecessary complexities.

Our primary concern is rigid body mechanics, and it is

sufficient for our purposes to consider that a rigid body

comprises a finite number of particles.



4.1 The Wrench and Velocity Motor Revisited

In this section we consider some additional definitions

and properties associated with the wrench and the velocity

motor.

DEFINITIONS. A pure force (rotation) is a wrench

(velocity motor) of zero pitch and a couple (translation) is

a wrench (velocity motor) of infinite pitch.

We have that any wrench can be expressed as a sum of

pure forces, each pure force of which corresponds to a force

in the force system of the wrench, or the wrench can be

decomposed into a force and couple (see Section 3.3).









Before we can give physical meaning to addition of

velocity motors, we need the following

THEOREM 4.1. Let V21 be the velocity motor of frame Z2

relative to E1 and V32 be the velocity motor of frame 3

relative to E2. Then, the velocity motor of 3 relative to

E1 is V21 + V32.

PROOF. This follows immediately from the principle of

relative velocities for points. Q.E.D.

Thus any velocity motor can be expressed as a sum of

velocity motors, and, in particular, it can be decomposed

into a rotation and translation (see Section 3.3).

DEFINITION. Two force systems are said to be

equivalent if they have the same wrench.*

All the common reduction procedures for equivalent

force systems are easily deduced using motor properties

(e.g., that forces are transmissible and that any two forces

acting on intersecting lines can be replaced with a single

force, the resultant, on a line through the point of

intersection).








* This is, in fact, equivalent to the standard definition
for equivalent force systems (the term equipollent is also
sometimes used). In dynamics texts the definition is
usually given as follows: two force systems are said to be
equivalent if the sum of the forces for each system are the
same and the sum of the moments at any point for each system
are the same.









4.2 The Momentum Motor

DEFINITION. The angular momentum distribution of a

particle system is called a momentum motor.*

Like the wrench, the momentum motor is a moment

distribution of a set of vectors, namely, of the linear

momentum vectors (or, simply, the linear moment) of the

particle system -(we can consider these moment point

vectors). Consider a system of mass particles with masses

ml, m2, .., mn located at points PI' P2' ... Pn. Let ri be

directed from an arbitrary point to Pi. Then, the momentum

motor of this system is given by


n i
H : r x miV.
i=l 1


where vpi is the velocity of Pi. The principal vector is

given by


n
m = mivp
i=l 1


which, of course, is the linear momentum of the system.

Denoting the center of mass by C, the velocity of the center

of mass by vC, and the total mass of the system by m (thus,
n
m = mi), we have
i=l


* It appears that Clifford (1878) introduced the momentum
motor; he called it the momentum of twist. Dimentberg
(1965) erroneously states that Kotelnikov (1895) was the
first to define the momentum motor.









h = mC



We say that the momentum motor of a particle system

represents the total momentum of the system. A bivector of

the momentum motor comprises the linear momentum and angular

momentum (at a point) of the system:



(H)p = (mvc, hp)



4.3 Law of Momentum for a Particle System

DEFINITION. An external (internal) wrench acting on a

particle system is the wrench of a force system comprising

of external (internal) forces.

We shall assume the strong law of action and reaction:

to every force, there is an equal and opposite reaction

force with the same line of action.* As is customary in

elementary dynamics texts, we assume that this is equivalent

to Newton's third law.** In terms of motors, we have that

to every pure force, there is an equal and opposite reaction





* This is in contrast to the weak law of action and
reaction, which states only that forces and their reaction
forces be equal and opposite (they need not be colinear).

** According to C. Truesdell, Newton did not actually
assert this law. "In Newton's own statement of his third
law, there is no explanation of what kinds of 'bodies' he
had in mind or what he meant by their actions on each
other," Essays in the History of Mechanics, Springer-Verlag,
New York, 1968, p. 270.









pure force. Since any wrench can be expressed as a sum of

pure forces we can again restate this:

NEWTON'S THIRD LAW. To every wrench, there is an equal

and opposite reaction wrench.

For particle systems for which Newton's third law

holds, the law of moment of momentum also holds. If H is

the momentum motor of a system and M the external wrench

acting on the system, then this law is given by



M = (4.1)



where the derivative is taken with respect to an inertial

frame (it is, of course, in the vector field form in which
3h
this law is usually given: m = -). Now, if f denotes the
at
net force of the force system and mvC the total linear

momentum, then we can deduce Newton's second law by simply

equating the principal vectors of eqn. (4.1):



f = m (4.2)



Newton's second law is also known as the law of linear

momentum. Since the law of linear momentum can be deduced

from the law of moment of momentum in analytical mechanics









(as we have shown here),* we call eqn. (4.1) the law of

momentum for a particle system.

Equation (4.1) does not necessarily characterize the

dynamics of a particle system; that is, the trajectories of

the particles cannot in general be determined given the

external wrench as a function of time. We shall not

investigate this in detail since it is with rigid bodies

that we are primarily concerned, and, as we shall show in

Chapter IV, eqn. (4.1) does characterize the dynamics for a

rigid body. Suffice it to say, that for an arbitrary

particle system, we cannot in general determine the particle

velocities at an instant given the particle positions and

the momentum motor of the system at that instant. Consider,

for example, any two particle systems for which the

particles have equal and opposite linear moment, both

vectors of which are parallel to the line on which the

particles lie. The momentum motor of all such two particle

system is the zero motor.

Of course, eqn. (4.1) can be applied separately to each

particle in a particle system, in which case the motion can

be completely described. In fact, we need only apply

Newton's second law (i.e., eqn. (4.2)) to each particle







* For the dynamics of deformable bodies, these two laws
are, in general, not dependent.





79




(note that the velocity of the center of mass of a single

particle system is simply the particle velocity).



4.4 Impulse, Conservation of Momentum

On integrating eqn. (4.1) over a time interval [to, t],

we obtain


t
f MdT = H(t) H(tO) = AH (4.3)
t
0
t
where f MdT is the impulse motor imparted to the system.
t
o
Thus, we have that the impulse motor is equal to the change

in the momentum motor AH. If M = 0 then H will remain

constant (i.e., AH = 0); this is the law of conservation of

momentum in motor form.

We say that the impulse motor represents the total

impulse imparted to a particle system. A bivector of the

impulse motor comprises the linear impulse and the angular

impulse imparted to the system:


t t t
( f MdT)p = ( f fdT, f mpdr)
t t t


where f is the net force acting on the system. With (H)p =

(mvC, hp), it follows from eqn. (4.3) that


t
f fdT = mvC t
t o








t
f mpdT = hp I t
t o
0

from which we can deduce the conservation laws for linear

and angular momentum:


t
mC It = 0


t
hp \t = 0


when f = m = 0.

It follows that the total momentum is conserved if and

only if both the linear momentum and angular momentum (at

any point; hence, all points) is conserved.

Finally, we note that if a constant pure force acts on

a particle system, then the angular momentum for points on

the central axis of the pure force is conserved, and if a

couple acts on the system, then the linear momentum is

conserved.



4.5 Power, Work, and Energy for Rigid Bodies

Let a rigid body comprise n mass particles with masses

mi and located at points Pi. Let V be the velocity motor of

the rigid body frame relative to some inertial frame and let

M be the wrench of the torce system acting on the body. We

assume that the force system comprises n forces fi (some of

which may be zero) acting on the particles. Then, the power









W, which is the time rate of change of the work W, imparted

to the body by the force system is given by



W = fi (Vo + x OPi)



= ( fi) VO + v. ( OPi x fi)



=MoV



We say that M o V is the power imparted to the body by the

wrench M. Now, it can be shown (see any text on elementary

dynamics) that for rigid bodies the internal forces

impart no power to the body, and that the power imparted

to the body (by the external forces) is equal to the time

rate of change of kinetic energy T, which is given by

T = mvPi. So, we have

2 1 2
i ivP
dt P.
1


Now, let H be the momentum motor of the system. Again

choosing an arbitrary fixed point 0 in the body frame, we

have that



I m2 i = I(m ip.) ( + x OP.)
1 1


= (I mivPi) V0 + * ( JOPi x (mivpi))
1









= VO + ho



= Ho V


1
Thus, T = H o V is the kinetic energy and


M V = (lH o V) (4.4)


Finally, on integrating eqn. (4.4) over a time interval

[to, t], we obtain the principle of work and energy:

1 t
SMo VdT = (H o V) t
t o

t t
where f M o VdT ( = Wdr = W ) is the work imparted
t t o
0 0
over the time interval.

This section should illustrate the central importance

of the reciprocal product as regards the dynamics of rigid

bodies, as this product is intimately related to the concept

of power. The concept of power is important for constrained

systems: if the total power imparted to a system by the

contraint forces is always zero (i.e., for any possible

motion), then there exists a formulation ot the equations of

motion for the system for which the contraint forces are not

present. Such constraints that contribute no power

(positive or negative) to the system are known as workless

constraints. We shall give the definition ot workless

constraints in terms ot motors in the next chapter.















CHAPTER V
ELEMENTARY KINETOSTATICS

Phillips (1984, p. 4) describes the term kinetostatics

as ". . the study of angular and linear velocities and

thus the kinematics of mechanism on the one hand, this being

in close conjunction with the study of forces and couples

and thus of massless mechanism on the other." It is

important to point out that it is instantaneous kinematics

to which Phillips is alluding (which is the study of

kinematic properties at an instant, that is, the relations

of velocities and higher derivatives in mechanisms and

machines). The intimate relation between instantaneous

kinematics and statics is, perhaps, best illustrated through

the use of motor calculus. We consider some elementary

aspects of these subjects and, in particular, we consider

the instantaneous kinematics and statics of open-loop

kinematic chains (or, simply, open-loop chains).



5.1 Constraint Between Bodies

5.1.1 Characterizing Constraint

If a body is constrained to move in a particular manner

relative to another body, then we shall assume that, at any

instant, the constraint between the bodies is completely

characterized by a motor system of permissible relative









velocity motors (Ball, 1900, equivalently characterized

motion freedom via screw systems). It would be more general

to consider arbitrary sets of velocity motors, as opposed to

just subspaces; instantaneous kinematic properties would

then become much more involved. This, however, is beyond

the scope of this work, and it turns out that in many

mechanisms and machines (including robots), the constraints

can be sufficiently modeled by motor systems.

We shall assume, in addition, that constraints are

workless, which we now define in terms of motors.

DEFINITION. The constraint between a body 1 and body 2

is said to be workless if M o V12 = 0 where M is the wrench

that body 1 exerts on body 2 via the constraint and V12 is

any velocity motor of body 1 relative to body 2 belonging to

the motor system that characterizes the constraint.

We note that -M is the wrench that body 2 exerts on

body 1 (by Newton's third law) and that V21 = -V2. Since

(-M) o V21 = M o V12 we have that the definition is

symmetric.

We call the wrench exerted via a workless constraint a

constraint wrench. It is clear that a constraint wrench

must belong to a reciprocal motor system. It is thus

possible to equivalently characterize a constraint by a

reciprocal motor system, which comprises permissible

constraint wrenches.









5.1.2 Static Equilibrium

DEFINITION. A body is said to be in static equilibrium

if the body's velocity motor with respect to any inertial

frame is a constant translation (or the zero motor).

It follows that each particle in the body has a

constant velocity so that the net force on each particle is

zero. Thus, the net wrench on the body is also zero.

DEFINITION. A body is said to be grounded (or, it is

simply called ground) if it is in static equilibrium and

maintains static equilibrium regardless of any applied

wrench. A body is said to be constrained to ground if its

motion is constrained relative to a grounded body.

Consider a rigid body constrained to ground and

stationary relative to ground (i.e., the relative velocity

motor is zero). Let the constraint be characterized by the

motor system N and let M' be an external wrench to be

applied and Mc be the resulting constraint wrench also

applied to the body. Since the body is initially stationary

relative to ground, it follows that the body is in static

eqilibrium before M' is applied. From the principle of

virtual work, we have that the necessary and sufficient

condition that the body maintain static equilibrium is


M' E NR


(5.1)









for, this is equivalent to saying that M' imparts no power

during a so-called virtual displacement (i.e., M' o V = 0

for all V e N). Since the net wrench exerted on a body must

be zero if the body is in static equilibrium, we can

determine the constraint wrench from



Mc = -M' (5.2)





5.2 The Screw Pair

We shall consider the screw pair illustrated in

Fig. 5.1 to be the most fundamental joint between two

bodies. The screw pair is in fact the most general single

degree of freedom lower pair* (Waldron, 1972). Common

multiple degree of freedom joints such as the cylindric,

ball and socket, planar, and universal can be constructed as

special serial combinations of screw pairs (see Duffy,

1980).

As can be seen in the figure, uniform rotation

accompanies uniform translation: a rotation of 6 about the

axis of the screw pair accompanies a translation of he along

the axis where h is the pitch of the screw pair. Should

h = m, however, then we define the screw pair to permit only





* A lower pair comprises mating surfaces (of the joint
between two bodies) that are surfaces of revolution.











































X = he


Fig. 5.1. The Screw Pair









translation for which the pair axis can be taken as any line

parallel to the translation or the pair axis can be taken as

a line at infinity. For h -, we note that for every value

of e, the screw pair permits a screw (i.e., a one-

dimensional subspace) of velocity motors for which the pitch

is also h and the central axis is identical to the axis of

the screw pair. Since the screw axis is fixed in the two

bodies that the screw pair connects, it follows that the

screw is fixed in both bodies. Thus, the screw of velocity

motors is independent of the value of e (and is hence the

same at every instant). For h = -, the screw of velocity

motors at any instant is clearly improper, which is also

fixed in both bodies. If desired, we can take the central

axis of this improper screw, which is a line at infinity, as

the central axis of the screw pair.

Two practically important special cases of the screw

pair are the revolute pair for which h = 0 and the prismatic

pair for which h = , both joints of which are illustrated

in Fig. 5.2.

Since a screw pair can be characterized by a screw, it

can be characterized by any motor in the screw. In

particular, it is very convenient to represent screw pairs

by unit motors. Let S represent a screw pair. Then, if the

pair is of non-infinite pitch (as in Fig. 5.1), then we

assume the direction of S is such that the angle of rotation

(8 in Fig. 5.1) is measured in a right sense about 9; if the


























>- e


--


Fig. 5.2. The (a) Revolute and (b) Prismatic Pair









pair is prismatic (as in Fig. 5.2b), then we assume the

direction of S is such that the length of translation (Z in

Fig. 5.2b) is measured in a positive sense in the direction

of s. The velocity motor can then be given by qS where q is

the measure of displacement that is, the joint displacement

of the screw pair (q = 6 in Fig. 5.1 and q = in

Fig. 5.2b).

Finally, we note that by Theorem 3.5, a unit motor that

represents a screw pair is fixed in both bodies connected by

the pair.



5.3 Open-Loop Chains

5.3.1 Modeling

We shall assume that the joints connecting successive

bodies (or links) in an open-loop chain are screw pairs.

Let an open-loop chain comprise n links numbered from

the base outward as illustrated in Fig. 5.3. We number the

base (or the ground link) 0. The joints are denoted by the

unit motors Si where the ith joint connects links i-i and i,

and the corresponding joint displacement is given by qi

where qi = 8i (qi = Zi) if Si is proper (improper).



5.3.2 Velocity and Acceleration Analysis

Obtaining the equations that are necessary for velocity

and acceleration analysis of open-loop chains is

straightforward using motor calculus.



















































Fig. 5.3. The Open-Loop Chain




92




Let Vj denote the velocity motor of the jth link

relative to the ground link (thus, V0 = 0). Then, by

Theorem 4.1, we have



Vj = Sq (5.2)
i=l


which can be expressed in the recursive form



Vj = Vj-1 + Sjj (5.3)



Differentiating both sides of eqn. (5.2) (with respect

to the ground frame), we obtain



V = i =(Sqi )


Now, since Si is fixed in both links i-i and i, we have that

S = V i- x S. = V. x S. (the time derivatives of Si with

respect to links i-i and i are both zero); we choose

S. = V. x Si for the following. Thus, we have

1 1 1 Vi ii
V = (Siqi + V. x Sq.)
i=1i


.. 1

= [Siq + ( Skk) x Sq ]
i=l k=l











= ISq + Sk x qiqk
i=l 1i= k=l


Differentiating both sides of eqn.

recursive form


V. = Vi + Sjq. + Vj x S.q
j j-1 j j J J J


(5.3) we obtain the


(5.5)


Summarizing,



V = Sq (5.2)
i=l1




= ISq + Sk Sq q (5.4)
3 i=l i=l k=l


which are explicit expressions in the first and second time

derivatives of the joint displacements, and


Vj = Vj-1 + Sq ,





V. = V + Sq. + V x Sq
j] -1 j r ] o j


which are recursive expressions.


(5.3)





(5.5)


(5.4)









5.3.3 Statics

We shall assume that open-loop chains are fully

actuated; that is, associated with each joint is an

actuator. (Most industrial robots would fall into this

category.) Each actuator in a fully actuated open-chain can

create a wrench "in-between" the links joined by the

associated joint. We now make these notions precise and

obtain the equations of equilibrium for the open-loop chain.

We denote the screw of the ith actuator by Sa and we
1
assume that this screw is fixed in the ith link. The wrench

that the ith actuator exerts on the ith link is given by

TiSi, and the wrench that is exerted on the (i-l)th link is

-TiSi (hence, the actuator creates a wrench "in-between" the

two links). We note that Ti is the magnitude of a force or

torque (which we allow to be negative) depending on whether

Sa is proper or improper.

Suppose that on each link of an n-link open-loop chain

an external wrench Mi is exerted and that we require the

chain be in static equilibrium; that is, each link be in

static equilibrium (we assume, of course, that the base is

grounded).

Then, applying condition (5.1) to link i, we obtain


n R
TiS + I M
p=i P




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