User guide
Charnwood Dynamics Ltd. Coda cx1 User Guide – Advanced Topics III - 3
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4
See: Euler, Leonhard. De Immutatione Coordinatarum, Caput IV, Appendix de
Superficiebus, Introductio in Analysin Infinitorum'. (Lausanne, France 1748).
In fact Euler’s original work showed that a rigid body may be re-oriented by a sequence
of three rotations about just two axes: R
X
about the x-axis, followed by R
Y
, followed by a
second rotation, R
`
X
, about the displaced x`-axis.
5
A single, displaced-axis projection angle may be represented by notation:
[dist-axis]
P
[plane /
prox-axis]
.
A projection angle set contains three such projections and there are, in all, 8 such sets:
(1) (
x`
P
X-Y
,
x`
P
X-Z
,
y`
P
Y-Z
); (2) (
x`
P
X-Y
,
x`
P
X-Z
,
z`
P
Y-Z
); (3) (
x`
P
X-Y
,
z`
P
X-Z
,
z`
P
Y-Z
);
(4) (
y`
P
X-Y
,
x`
P
X-Z
,
y`
P
Y-Z
); (5) (
y`
P
X-Y
,
z`
P
X-Z
,
y`
P
Y-Z
); (6) (
y`
P
X-Y
,
z`
P
X-Z
,
z`
P
Y-Z
);
(7) (
x`
P
X-Y
,
z`
P
X-Z
,
y`
P
Y-Z
); (8) (
y`
P
X-Y
,
x`
P
X-Z
,
z`
P
Y-Z
).
Note that
z`
P
X-Y
,
y`
P
X-Z
,or
x`
P
Y-Z
are NOT useful projections and so do not participate
above.
6
Grood, E. S., and W. J. Suntay, 1983. A joint coordinate system for the clinical
description of three-dimensional motions. Journal of Biomechanical Engineering 105,
136 - 144.
7
Crawford, N. R., G. T. Yamaguchi and C. A. Dickman, 1996. Methods for determining
spinal flexion /extension, lateral bending, and axial rotation from marker coordinate data:
Analysis and refinement. Human Movement Science 15, 55 - 78.
8
The projection angle sets listed in note 5 above correspond to the Euler angle
sequences described here with their compound rotation matrices:
(1) (
x`
P
X-Y
,
x`
P
X-Z
,
y`
P
Y-Z
) ≅ (θ, φ, ψ) for R
xyz
; (2) (
x`
P
X-Y
,
x`
P
X-Z
,
z`
P
Y-Z
) ≅ (θ, ψ , φ) for
R
xzy
;
(3) (
x`
P
X-Y
,
z`
P
X-Z
,
z`
P
Y-Z
) ≅ (ψ, θ, φ) for R
zxy
; (4) (
y`
P
X-Y
,
x`
P
X-Z
,
y`
P
Y-Z
) ≅ (φ, θ, ψ) for
R
yxz
;
(5) (
y`
P
X-Y
,
z`
P
X-Z
,
y`
P
Y-Z
) ≅ (φ, ψ, θ) for R
yzx
; (6) (
y`
P
X-Y
,
z`
P
X-Z
,
z`
P
Y-Z
) ≅ (ψ, φ, θ) for
R
zyx
.
The underscored projection angle in each set exactly matches the last Euler angle of the
corresponding sequence (in bold) and other angles are approximately equal if less than
~30
0
.
Projection angle sets (7) and (8) (the only ones which utilise all three distal axis vectors)
do not correspond to any Euler sequence at all.
9
Paul, R. P., 1982. Robot Manipulators, pp. 45 - 71. Cambridge: The MIT Press.
10
Readers are referred to the internet postings of the Biomechanics Forum and, in
particular, the excellent discussions between Hermann Woltring, Ed Grood, and others,
(‘ANGLES3D TOPIC’) on the relative merits of a variety of schemes for the
representation of joint angles between rigid body segments. The problem of ‘gimbal-
lock’ is fully aired.