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Kjartan Halvorsen
01-13-2003, 10:10 PM
Dear Colleagues,

In Biomechanics, Euler angles is the dominant representation of joint
motion, as
exemplified by the standardization proposal referred to by Dr. Veeger.
(http://www.wbmt.tudelft.nl/mms/dsg/intersg/ISGproposal.pdf)
The strength of the representation lies primarily in the close connection
to anatomical nomenclature, which is valuable when interpreting 3D joint
motion. The weakness of the representation is mathematical; see van den
Bogert and Kwon's recent discussion. Herman Woltring fought for the
acceptance of the "attitude vector" as the standard for representing 3D
attitude (and rotations). See [1] and contemporary discussions on
biomech-l.

I would like to direct this forum's attention to a different representation
of 3D rotation [2], proposed in the computer graphics literature as a
convenient representation for ball-and-socket type of joints with large
range of motion. The representation is sometimes called "swing-and-twist",
and it has a straightforward interpretation and nice mathematical
properties (no gimbal lock, singularity only for rotations ("swings") of
180 degrees from the reference orientation).

A brief introduction to swing-and-twist: Consider the motion of the
shoulder joint. The motion is decomposed in
two rotations: a swing of the arm, which causes NO axial rotation of the
humerus, followed by an axial rotation of the humerus (the
twist part). The swing of the arm is represented by a rotation vector
that is constrained to lie in the plane normal to the longitudinal axis of
the humerus. The idea of a rotation vector may seem to imply that the
swing-and-twist representation is as difficult to envision (interpret) as
the attitude vector of Woltring. However, the swing axis lies in a fixed
plane, and 2D geometry is a lot easier to visualize and understand than 3D
(at least for most of us, it is).

Take a look at Grassia's paper [2], online at
http://www.cs.cmu.edu/~spiff/moedit99/expmap.pdf
I think it offers a nice compromise between the ease of interpretation of
Euler angles and the nice mathematical properties of the attitude vector.

Yours sincerely,

Kjartan Halvorsen

[1]
@article{biomech_woltring_94,
author = {H.J. Woltring},
title = {3-{D} attitude representation of human joints: A
standardization proposal},
journal = {Journal of Biomechanics},
year = {1994},
month = {},
volume = {27},
pages = {1399--1414},
}

[2]
@article{biomech_grassia_98,
author = {F.S. Grassia},
title = {Practical parameterization of rotations using the exponential
map},
journal = {Journal of Graphics Tools},
year = {1998},
month = {},
volume = {3},
pages = {29--48},
url = {http://www.cs.cmu.edu/~spiff/moedit99/expmap.pdf}
}

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The Department of Systems and Control
Uppsala University
http://www.syscon.uu.se/
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