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Herman J. Woltring
11-08-1990, 09:57 AM

In reply to Fabio Catani's Biomch-L posting last Monday on how to assess 3-D
ICR's, I have the following comments. Some of this is (to my knowledge) new
information, and it seems interesting to announce this in a non-refereed,
electronic bulletin rather than to wait for a comprehensive review process.

(a) Utility:

An instantaneous, 3-D centre of rotation for a moving rigid body (e.g., a
distal segment w.r.t. a proximal segment in the case of 3-D joint motion)
may be useful in a dynamics context for the following reasons:

- If you are looking at net joint moments without regarding individual force
and moment contributions in ligaments, muscles, and contact points, the
numerical values and the graphical results will strongly depend on the
chosen reference point at which the net moment M is assessed. Cappozzo in
Rome/Italy has amply demonstrated this (Bristol ESB Meeting, 1987).
- It is sometimes proposed that it is better to use as a reference point one
which is stationary w.r.t. the anatomy. While this is possible within one
segment, such a point will generally not be stationary w.r.t. the other
segment. Thus, which of the two segments comprising a joint should be
chosen?
- The net power transmitted through a joint can be assessed from the rotatio-
nal and translational contributions as M'omega + F'v, where F is the net
force, and omega and v the rotation and translation velocity vectors of the
joint (i.e., distal w.r.t. proximal or vice-versa). Of these entities, M
and v are position-dependent, and v is minimal at the Instantaneous Helical
Axis. The common practise to neglect the linear term F'v is appropriate
if v == 0 (planar movement at the IHA) which includes the case of a fixed
rotation axis or centre. While this may be the case for the healthy hip
joint, it is certainly debatable for other joints. Note: while the net
power is invariant with the position along the IHA, the net moment varies.
- If interest is oriented to assessing the efficacy of individual muscles, the
muscle's moment arm can be suitably assessed with respect to the IHA or ICR
as defined below.

(b) Definiton and Calculus/Estimation:

A 3-D ICR can be defined by looking for the point with minimal, instantaneous
displacement. Given the general rigid-body equation

y(t) = R(t).x + p(t), R(t) orthonormal, x arbitrary but fixed

one could try and find the point x for which the absolute velocity |dy/dt| and
acceleration |d(dy/dt)/dt| are minimal in some weighted sense. For example,
one might choose from all points with minimal velocity that particular one which
has minimal acceleration.

The class of points with minimal velocity is defined by the Instantaneous
Helical Axis about and along which the body is instantaneously translating
and rotating; it is defined by the projection s = p + omega * dp/dt / omega^2
of p onto it, and by its direction vector omega, where * denotes the external
vector product, and where omega = (omx,omy,omz)' follows from Poisson's equa-
tion,

[ 0 -omz omy]
A(omega) == [ omz 0 -omx] = dR/dt . R' (' denotes transposition).
[-omy omx 0 ]

The point q on the IHA with minimal acceleration then follows by minimizing
|d(dy/dt)/dt|^2 as a function of the parameter r in the equation y = s+r.omega
for an arbitrary point y(r) on the IHA. [N.B.: when working this out, be sure
not to reverse the sequence of differentiating and substitution; like the Car-
danic rotations of this spring's joint angle debate, these operations are not
commutative.]

The measurement system and protocol must be sufficiently accurate to allow
reliable assessment of both first and second derivatives. However, these data
are required anyhow if one plans to assess inverse dynamics including inertial
effects. Thus, skin motion artefacts should be minimized if external markers
are utilized, and the marker distribution should be sufficiently non-collinear.

Note also that the 3-D ICR is ill-defined unless the rotational velocity
OMEGA == omega * d(omega)/dt / |omega|^2 with which the IHA changes its own
direction is sufficiently large. Thus, the 3-D ICR is ill-defined if the
movement is nearly planar. Presumably, though, the net moment will in that
case have a negligible tendency to rotate the IHA into an other direction
(this might be an interesting research question).

It turns out that the ICR as defined above coincides with the so-called central
point c about which the IHA itself executes an instantaneous, helical movement
with rotation velocity vector OMEGA (cf. Suh & Radcliffe, Kinematics and Mecha-
nisms Design, Wiley 1978, Ch. 10). Fischer (1907) and Chao & An (ESB Nijmegen/
NL, 1982) have proposed that the central point may be a useful descriptor of
spatial movement.

The formal proof of the equivalence between ICR and c is quite tedious and
error-prone; fortunately, the recent computer-algebra posting on this list has
resulted in access to the Maple package (Maple Software, Waterloo, Canada) at
the Computer Algebra Netherlands Expertise Centre in Amsterdam by means of
which this relation could be verified [N.B.: SURFnet users in The Netherlands
can obtain access to this system; for details, send a note to can@can.nl].

I hope that this answers Dr Catani's question to some extent (a paper is in
preparation, which hopefully will contain some experimental results).

Sincerely -- Herman J. Woltring, Eindhoven/NL.