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Andrew Karduna
04-06-1998, 07:59 AM
This posting is with regard to the sequence dependent nature of Euler/Cardan Angles,
specifically as it relates to the scapula. I have read the debate of this topic in the
biomch-l archives and do not wish to simply repeat that discussion. So although I
welcome general responses, I am specifically concerned about the description of
scapular rotations with respect to the thorax.

If we accept Woltring's argument that Euler Angles are not temporally sequence
dependent, but are geometrically sequence dependent, then as long as everyone is
using the same geometric sequence, we can directly compare angles between studies.
Since there currently isn't an accepted standard that can be applied to all joints,
different investigators often use different sequences.

Many studies have demonstrated that in general, different sequences of Euler
Angle rotations result in different angle magnitudes (eg, Blankevoort et al, 1988, Cole
et al, 1993, Woltring, 1994). Clearly, based on these data, the physical interpretation
of a motion may be dependent on the sequence selected. For example, in figure 4 of
Woltring, J Biomech, 1994, at the end of a gait cycle, the knee appears to be in
adduction for some sequences and abduction for others.

The standardization proposed by Cole et al (1993) suggests an order of
flexion/extension - abduction/adduction - internal/external rotation. Since the
geometry of the scapula is such that there is no "long axis," this description will not work.

I have heard other suggestions, for the selection of a sequence - for example the first
rotation should be the largest, the second rotation should be the smallest (to avoid
gimbal lock) and the third rotation is what is left.

The three Euler Angle rotations and their corresponding axis at the neutral position of
the scapula are typically presented in the literature as:

- Internal/external rotation or protraction/retraction (superior/inferior axis)
- Upward rotation (anterior/posterior axis)
- Anterior/posterior tilting or tipping (medial/lateral axis)

What follows is one possible rational for a sequence:

First Rotation -
Internal/external rotation - The next rotation (either tilting or upward rotation) will
always be about a true horizontal axis, as they are described clinically.

Second Rotation -
Posterior Tilting - Since upward rotation is the typically the largest rotation, it may
result in a gimbal lock position, especially if the AC joint is used as a digitized landmark.

Third Rotation -
Upward Rotation - Final rotation

Assuming that each rotation is about a separate axis (ie, Cardan Angles), there are five
other sequences, each with their own possible rationale (eg, internal/external rotation -
upward rotation- posterior tilting, which is often seen in the literature). Depending on
which sequence is chosen, I have found large discrepancies in scapular angles for data
I have collected. For example, during elevation of the humerus in the scapular plane,
one sequence typically shows a progression of scapular external rotation, another
shows little motions and yet a third shows scapular internal rotation.

So finally, after all of this rhetoric I come to my questions:

1) Is there any FUNDAMENTAL (mechanical, clinical or mathematical) motivation
for the selection of a particular sequence that would make it the "best" sequence?

2) How does this relate to the description scapular rotations?

- Andy

__________________________________________________ ________

Andrew Karduna, Ph.D.
Assistant Professor, Department of Physical Therapy
Director, Biomechanics Laboratory
219 N Broad St, 8th Floor
Allegheny University of the Health Sciences