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View Full Version : Joint attitudes: (final?) round 3.

Herman J. Woltring
02-26-1990, 04:01 AM
"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.

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.