"I don't know what you mean by `glory'," Alice said. Humpty Dumpty smiled con-

temptuously. "Of course you don't know - till I tell you. I meant `there's a

nice knock-down argument for you!'." "But `glory' doesn't mean `a nice knock-

down argument'," Alice objected. "When I use a word," Humpty Dumpty said in a

rather scornful tone, "it means just what I choose it to mean - neither more

nor less." "The question is," said Alice, "whether you can make words mean so

many different things." "The question is," said Humpty Dumpty, "which is to be

master - that's all."

Lewis Carrol (1872), Through

the Looking Glass, Chapter VI.

Dear Biomch-L readers,

If we substitute `independent' or `commutative' for `glory,' there's not much

new under the sun.

(a) Commutation. In mathematics, c o m m u t a t i v i t y stands for the

property (a o b) = (b o a) of a binary operation (a o b), for all valid a

and b, where a and b are the entities upon which the binary operation

o performs its function (e.g., +, -, *, /).

If a and b are scalars or vectors, + and - are commutative operators,

whereas - (and, for scalars, /) are not. If a and b are matrices, + is

commutative while - and * are not; furthermore, / is generally undefined,

although one may define certain classes of operations that more-or-less

correspond to scalar division.

(b) If a and b are functions of certain scalar parameters, say a1 and b1,

respectively, the t e m p o r a l `commutation' as originally envisaged

by Ed Grood and others [e.g., Bernard Roth, Finite Position Theory Applied

to Mechanism Design, Journal of Applied Mechanics, Sept. 1967, p. 600, left

column], is nothing else but the temporal order in which these parameters

are changed from their reference values (zero or one, usually) to their

final or current settings. This has nothing to do with mathematical com-

mutativity, as proposed in one of my previous postings in this debate, nor

with `similarity of matrices,' as Roth would have it.

(c) Independence. I don't know what Ed means with `independent'. From a sta-

tistical point of view, dependence and correlation of error sources are

important when we are close to or at gimbal-lock. Using the formulae in

Woltring et al, J. of Biomechanics 1985 on the Finite Helical Axis and

Finite Centroid, in combination with formulae (2.32) in J. Wittenburg,

Dynamics of Rigid Bodies (B.G. Teubner, Stuttgart/FRG, 1977), the `radial

s.d.' or root-sum-of-squares of the s.d.'s for the three Cardanic angles

is a function of PHIj due to the PHIj-dependent correlation between the

SIGMA. errors, and can be derived as

SIGMA(PHIj) = sqrt[SIGMAi**2 + SIGMAj**2 + SIGMAk**2]

:= SIGMA(PHIj=0) * sqrt[{1 + 2/cos(PHIj)**2}/3]

As before, PHIj denotes rotation anbout the floating axis, with gimbal-lock

when |PHIj| = PI/2. If PHIj is close to gimbal-lock, SIGMA is much larger

than when PHIj = 0 degrees. The closer we approach the gimbal-lock situa-

tion, more erratic the calculated angles become. If a physician wishes to

interpret joint angle graphs for hip or shoulder, e.g., in complex sportive

movement, these adverse effects should preferably be avoided.

(d) The utility of `orthogonal attitude components' is even more apparent once

we become interested in both kinematics and kinetics. Positions and

translational and rotational velocities, accelerations, forces, and moments

are all vectors, only attitudes are not. Should we really decompose all

these vectorial entities in terms of the non-vectorial behaviour of rota-

tions/attitudes? I think that it is much better to try and find a rota-

tional representation which maximally approaches the vectorial properties

of these other, biomechanically highly relevant entities. Again, I see

no reason why we should impose or require trajectorial properties for our

attitude parametrizations.

Given the fact that there are different, Cardanic commutations, and the con-

comitant asymmetries between the `floating' and `imbedded' axes (i.e., the

floating axis follows from the vector product of the two imbedded ones, while

neither imbedded axes is generally derivable as the vector product of the two

other axes), I think that a symmetric representation like the helical decompo-

sition continues to be a better candidate.

After Ed's reply to this 3rd round, I believe that the debate should be

suspended --- unless others on the list wish to enlighten (or to confuse) the

readership with their views. At any rate, I have been enjoying the exchange

of views, and it will certainly help me in preparing a somewhat more formal

presentation on this issue in April at, I hope, both sides of the Atlantic.

If Ed should finally agree with me, I would be delighted to ask him as a co-

author ...

Herman J. Woltring

Research Associate ,

Eindhoven University of Technology, The Netherlands.