User guide
Charnwood Dynamics Ltd. Coda cx1 User Guide – Advanced Topics III - 3
CX1 USER GUIDE - COMPLETE.doc 26/04/04
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Any co-ordinate vector v will be mapped to its image v` under matrix multiplication by any
of the above matrices. This is true for the unit vectors u
X
(= [1, 0, 0]
T
), u
Y
and u
Z
,
representing the axes of the distal EVB, which are mapped to u
X
`, u
Y
` and u
Z
`. Crucially,
a second rotation about a different axis maps u
X
` to u
X
`` etc.., and a third maps u
X
`` to
u
X
```etc.. Moreover, the components of the final image vectors (u
X
```etc..) depend on the
order in which the rotations are applied.
In terms of matrix algebra we would say that matrix multiplication is non-commutative.
Thus, if
U
D
=
[
u
X
T
, u
Y
T
, u
Z
T
]
is the distal
EVB
matrix (whose columns are the axial unit
vectors u
X
, etc..), we have the following results for compound mappings:
(1)
R
z
R
y
R
x
U
D
=
U
D
xyz
(2)
R
y
R
z
R
x
U
D
=
U
D
xzy
(3)
R
z
R
x
R
y
U
D
=
U
D
yxz
(4)
R
x
R
z
R
y
U
D
=
U
D
yzx
(5)
R
y
R
x
R
z
U
D
=
U
D
zxy
(6)
R
x
R
y
R
z
U
D
=
U
D
zyx
and, in general, the final images of the rotated distal EVB are not the same.
Note that
R
z
R
y
R
x
U
D
=
R
z
(R
y
(R
x
U
D
))
=
U
D
xyz
is the result of a rotation about the
(proximal) X axis, followed by rotation about the Y axis, and finally about the Z axis.
Now, if we are given a set of 3 Euler angles (θ, φ, ψ) said to represent the orientation of a
rigid segment we have six possible interpretations available to us (assuming the axes to
be orthogonal), only one of which is correct. It is essential that we know which axis
sequence was used if we are to obtain the correct interpretation.
Choosing the Axis Sequence
For any re-orientation of a distal segment (with EVB image U
D
```) there are six Euler
decompositions available, some more appropriate than others perhaps. Ideally, and to
avoid confusion, everyone should adopt the same convention, using the ‘best’ axis
sequence for the job if such exists.
The criteria for this choice are many: clinical relevance; robustness; intuitive ease of use;
correspondence with alternative schemes; generality of application etc.. In fact one
preferred sequence gained universal approval for lower limb analysis some years ago, but
is nicely justified by a recent appraisal in the context of spinal movement.
Crawford, Yamaguchi and Dickman (1996)
7
have illustrated an ideology for deciding the
Euler sequence based upon notions of symmetry, complementarity, and the
correspondence of Euler Angles with Projection Angles (a useful idea in itself).
Crawford et al. showed that certain sets of easily visualised projection angles relate
closely to the Euler angles of a given sequence
8
to the extent that the last angle in that
sequence is identical to the corresponding projection angle and, while the angles remain
‘small’ (< 30
o
), all of the angles will be comparable. The beauty of this correspondence is
that it allows us to visualise desirable attributes of projections before inferring the optimal
Euler sequence.