Biomedical Engineering Reference
In-Depth Information
attitude of the final segment of the chain in the base
coordinate system R
0,
is obtained simply by the
multiplication of these elementary matrices, as follows:
0
TTT T T
0
1
i 1
−
n 1
−
=
.
. ...
. ...
n
1
2
i
n
This homogeneous matrix helps us calculate the
coordinates in the fixed coordinate system, by knowing the
coordinates of a point M in the coordinate system attached to
the final segment of the chain, using the relation:
1
|
.
|
with
|
To obtain the values of the joint parameters at each
instant, we can write the solution analytically if the number
of degrees of freedom is small enough (up to three).
Otherwise, we can use a numerical optimization instead,
which will minimize the following function expressing the
gap between the actual markers (fixed onto moving body
segments) and the corresponding markers considered to be
rigidly attached to the chain segments:
T
ˆ
[
]
ˆ
qPTqpWPTqp
ˆ
⎡
0
ˆ
ˆ
⎤
⎡
0
ˆ
ˆ
⎤
f
()
=−
()
⋅
⋅
⋅
()
−
⋅
⎣
⎦
⎣
⎦
n
n
where
q
is the vector (
n
components) of kinematic
parameters;
p
is the vector composed of homogeneous
coordinates (four components) of markers in the local
coordinate system of the segment (deemed constant, and
defined by recording a static posture),
P
is the vector
composed of homogeneous coordinates of markers in the base
coordinate system R
0
in motion and
0
n
Tq
represents
the geometric model of the kinematic chain, as a function of
the variable kinematic parameters.
()
