Biomedical Engineering Reference
In-Depth Information
3 top left) of the matrix
characterizing the attitude of segments S
i
and S
i-1
, the
rotation matrix corresponding to the relative motion between
S
i
and S
i-1
is given by:
From the rotation part (3
×
(
)
−
1
i
−
1
RRR
=
0
.
0
i
i
−
1
i
(
)
−
1
where
R
. Note
that, with a rotation matrix, the inverse is equal to the
transposed matrix, which is easily obtained by writing in
lines, the columns of the initial matrix.
0
1
R
represents the inverse of matrix
0
1
i
−
i
−
The nine elements of the rotation matrix are not
independent, since it is orthogonal and each column (or line)
represents a unit vector, which all give three degrees of
freedom. The composition of successive rotations is easily
expressed, by the multiplication of corresponding rotation
matrices, which makes it very practical to use.
The approach using
Euler angles
(or rotations about
mobile axes) involves deconstructing the 3D rotation of the
coordinate system into three successive elementary
rotations: the first rotation occurs about an axis of the
upstream coordinate system R
i-1
, the final rotation
taking place about a downstream coordinate system axis
R
i
and the intermediate rotation taking place about the
axis defined by the common perpendicular to two others,
called the floating axis. Nevertheless, the definition of
Euler angles is not unique and in the literature many
different conventions are used. These conventions depend on
axes on which the rotations occur (X, Y or Z), and their
sequence (the rotations are not commutative). It is therefore
impossible to compare the angular values obtained from this
approach if the axes or order of the sequence are not strictly
identical.
