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Herman J. Woltring
02-14-1990, 06:43 AM
Dear Biomch-L readers,

The abstract below has been accepted for presentation at the previously announ-
ced, 4th International Biomechanics Seminar in Gothenburgh/Sweden on 26 and 27
April, 1990. In view of various comments about this proposal, both from Dr
Grood in Cincinnati/Ohio and elsewhere, Ed Grood has agreed to a debate on the
pro's and con's of various 3-D joint attitude parametrizations. For the sake
of clarity, this text deviates slightly from the accepted text.

Following earlier items on Biomch-L last year, I hope that this debate may be
one of many to follow. It is for this kind of activities that an email discus-
sion list can be much more efficient than, say, Letters to the Editors in formal
journals. This is not to say that such letters are not useful; rather, they
might become the result of more interactive debates, whether at conferences,
during laboratory visits, or from behind a terminal. Ed, it's up to you, now!

Herman J. Woltring
Eindhoven, The Netherlands.



3-D ATTITUDE REPRESENTATION : A NEW STANDARDIZATION PROPOSAL (1)

Herman J. Woltring, Eindhoven, The Netherlands

Grood & Suntay (Journal of Biomechanical Engineering 1983, 136) have proposed a
`sequence independent, oblique co-ordinate system' in which the current orienta-
tion (i.e., position and attitude) is thought to be reached from a predefined
reference orientation via an ordered sequence of rotations ijk of three elemen-
tary, helical displacements a b o u t (PHI.) and a l o n g (D.) the axes i,
j, and k of an electrogoniometric linkage system. The terminal axes i and k
are imbedded in the body segments comprising the joint, and they are identical
to prior selected, Cartesian co-ordinate axes defined in these segments; the
intermediate or `floating' axis j is normal to the two imbedded axes, and iden-
tical to the `line of nodes' in classical handbook descriptions of Euler/Cardan
angles.

Although t e m p o r a l sequence dependency of Cardan/Eulerian rotation
conventions is avoided in such predefined electrogoniometric systems, a similar
effect is now imposed by the g e o m e t r i c a l choice of imbedded and
floating axes. Thus, different numerical results may be obtained for current
joint attitudes (given identical segment co-ordinate systems), and adverse
effects such as g i m b a l - l o c k (i.e., for some PHIj, either the sum
or the difference of PHIi and PHIk is undefined) and C o d m a n ' s P a r a -
d o x (i.e., both {PHIi,PHIj,PHIk} and {PHIi+PI,PI-PHIj,PHIk+PI} (N.B.: -PI
works also) are valid solutions; cf. A.E. Codman, The Shoulder, Boston 1934)
continue to occur.

Instead of defining joint orientation or movement in terms of an ordered
sequence of three helical displacements, it seems more appropriate to view
the current orientation in terms of a s i n g l e helical displacement,
to be decomposed into orthogonal components in either body segment's co-ordi-
nate system which, apart from a sign inversion, appear to be identical. For
attitude representation (position representation is more complicated), one
can define an attitude `vector' THETA = theta * N, where N is the unit direction
vector about which the helical, scalar rotation theta occurs. This vector
THETA, while not a true vector as rotations are not additive, is symmetrical
in its three components and not affected by gimbal-lock or Codman's Paradox.
Unlike N, THETA is well-determined from noisy measurements even for small theta.
A further advantage is that the helical representation corresponds approximately
with the mean value of all valid, Cardanic representations once Codman's Paradox
is accounted for.

(1) This is a paper under the CAMARC project. CAMARC, for "Computer Aided
Movement Analysis in a Rehabilitation Context", is a project under the Advanced
Informatics in Medicine action of the Commission of the European Communities,
XIII-F/CEC), with academic, public-health, industrial, and independent partners
from Italy, France, U.K. and The Netherlands. Its scope is pre-competitive.