Biomedical Engineering Reference
In-Depth Information
sum of
R
m
+
c
. The vector
R
m
is the GRS coordinates of the tracking marker,
m
T
1
, while
c
is a constant vector joining
m
to
c
, as defined earlier.
7.2.1 Example of a Kinematic Data Set
7.2.1.1 Calibration — Calculation of [Marker to Anatomical] Matrix.
Let
us now look at an example of numerical data to see how the various trans-
formations are calculated. The leg segment in Figure 7.2 will be used as an
example. Recall that there are three tracking markers on this segment plus
two calibration markers whose coordinates were digitized during the cali-
bration period when the subject stood steady in the anatomical position. In
Figure 7.2, the subject would be facing the
ve X
direction, and the left leg
was being analyzed. Table 7.1 gives the
X
,
Y
, and
Z
coordinates in the GRS,
averaged over one second.
In this calibration position, the ankle
+
=
(m
T
2
+
m
C
1
)/
2,
X
a
=
2
.
815,
Y
a
=
22
.
685
.
The coordinates of the knee
10
.
16,
Z
a
=
=
(m
T
3
+
m
C
2
)/
2,
X
k
=
6
.
67,
Y
k
=
41
.
89,
Z
k
=
20
.
965. The COM of the leg
=
0
.
567
×
knee
+
0
.
433
×
ankle,
X
c
=
5
.
001,
Y
c
=
28
.
151,
Z
c
=
21
.
710.
Now we have to locate the anatomical
x
,
y
, and
z
-axes. Let the line joining
the ankle to the knee be the
y
axis, and the line joining the lateral malleolus
to the medial malleolus be an interim
z
axis (because it is not exactly normal
to the
y
axis but will be corrected later). These two axes now form a plane,
and the
x
axis, by definition, is normal to the
yz
plane and, therefore, is the
“cross product” of
y
and
z
,or
x
an
=
(y
an
×
z
an
)
. Using the subscript “
an
”to
indicate anatomical axes:
z
an
=
(m
C
1
−
m
T
2
)
:
x
z
=−
0
.
21,
y
z
=
0
.
21,
z
z
=
7
.
67
y
an
=
(
knee
−
ankle
)
:
x
y
=
3
.
855,
y
y
=
31
.
73,
z
y
=−
1
.
72
x
an
=
(y
an
×
z
an
)
:
x
x
=
243
.
575,
y
x
=−
29
.
207,
z
x
=
7
.
126
One final correction must be done to our anatomical model.
z
an
is the
line joining the medial to lateral malleolii and is approximately 90
◦
from the
long axis
y
an
. To ensure that all three anatomical axes are at right angles to
TABLE 7.1
Marker Coordinates during Standing Calibration
Marker
Location
X
(cm)
Y
(cm)
Z
(cm)
m
T
1
Midleg
9.39
30.02
21.90
m
T
2
Lateral malleolus
2.92
10.10
18.85
m
T
3
Fibular head
5.05
41.90
15.41
m
C
1
Medial malleolus
2.71
10.22
26.52
m
C
2
Medial condyle
8.29
41.88
26.52
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