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
~~
~~ |