The standardization proposals recently made by Yeadon/Bogert/Legnani
(16 October) and endorsed by J. Dapena (19 Oct.), are in full
agreement with my original proposal of 28 September 1995 concerning
the use of Cardan angles to describe 3D whole-body and segment
orientation, and the use of the internationally accepted 3D coordinate
system XYZ. In fact, my three-dimensional 17-segment human body
model first published in 1980 ("A mathematical model for the
computational determination of parameter values of anthropomorphic
segments", J. of Biomechanics 13, 833-843) and then used repeatedly
in subsequent papers ("Neuromusculoskeletal control systems modeling -
a critical survey of recent developments", IEEE Transactions on
Automatic Control AC-25, 375-385, 1980, "HOMSIM: a simulator
of three-dimensional hominoid dynamics", CSIR - Special Report
SWISK 23, 1981, "Computerized optimization of sports motions: an
overview of possibilities, methods and recent developments", J. of
Sports Sciences 1, 3-12, 1983, etc.) already employs Cardan angles
for describing 3D whole-body and segment orientations and, as
spatial and segmental coordinate system, a Cartesian XYZ reference
frame with the X-axis pointing towards the PASSIVE observer (or
to the right of the subject IF THE PLANE OF FORWARD
PROGRthe right of the PASSIVE observer (or in the direction of the
subject moving forward in the sagittal plane), and the Z-axis normal
to the other two axes in a right-handed Cartesian coordinate system.

My reasons for selecting Cardan angles (also called Bryant angles)
were similar to those of Fred Yeadon and Jesus Dapena. While the
critical case (gimbal lock) occurs for Cardan angles at PHI2=PI/2+n*PI
(n=0,1,2,...), this is the case at THETA=n*PI (n=0,1,2,...) for Euler
angles. However, for most situations arising in biomechanics it is
more advantageous to have the critical case occurring at a larger
angle and not near THETA=0. This makes also possible the derivation
of particularly simple expressions for the elements of a Cardan
transformation matrix for the case where all three angles are near
zero, which can not be done if Euler angles are used. This was an
additional reason why I selected Cardan angles.

As far as the XYZ labelling of the spatial and segmental coordinate
axes is concerned, the overriding criterion for me was to adhere
to internationally accepted norms. As already pointed out in my
original e-mail of 28 Sept. 1995, this implies that the XYZ-axes
system described above and also agreed upon by Yeadon, Bogert,
Legnani, Dapena, and many others be used. Exceptions to this
standard should be possible for specific purposes.

I might also mention that I have sent summaries of the present
ISB-standardization e-mail discussions to Dr. R. Brand in his
capacity as editor of the Journal of Biomechanics. I am sure that he
also wants to contribute to the present discussion.

Herbert Hatze, Ph.D.
Professor of Biomechanics, University of Vienna