Biomedical Engineering Reference
In-Depth Information
=
(
A
x
,A
y
,A
z
)
and B
=
(
B
x
,B
y
,B
z
)
and the cross product is a vector
A
C defined as C
B
)
. C is perpendicular to A and B in the direction
defined by the right-hand rule and has a magnitude
=
(
A
×
sin
θ
where
θ
is
the angle between A and B. The vector C is calculated by the expansion of
the determinant where i, j, and k are unit vectors in the x, y, z axes:
=|
A
B
|
i
j
k
C
=
A
×
B
=
A
x
A
y
A
z
B
x
B
y
B
z
C
=
(
Ay B z
−
AzBy
)
i
−
(
AxBz
−
AzBx
)
j
+
(
AxBy
−
Ay B x
)
k
=
iCx
+
jCy
+
kCz
(7.6)
7.2
MARKER AND ANATOMICAL AXES SYSTEMS
The following description outlines the steps that are necessary to transform
the
x
,
y
,
z
marker coordinates from the GRS to the anatomical axes of the seg-
ments of the person whose movement is being analyzed. Figure 7.2 presents
the axis systems involved for a given segment whose COM is at
c
and whose
axes
x
-
y
-
z
are as shown. The GRS has axes
X
-
Y
-
Z
, and they are fixed
for any given camera arrangement. The second axis system,
x
m
−
y
m
−
z
m
,is
the marker axis system for each segment, and this can vary from laboratory
to laboratory. Even within a given laboratory, each experiment could have a
different arrangement of markers. For a correct 3D analysis, there must be
at least three independent markers per body segment, and there must not be
common markers between adjacent segments. The markers on each segment
must not be collinear, which means they must not be in a straight line. They
must form a plane in 3D space; as shown in Figure 7.2, the three tracking
markers
m
T
1
,
m
T
2
, and
m
T
3
define the tracking marker plane. This plane is
assumed to contain the
x
m
and
z
m
axes such that all three markers are in the
+
ve x
m
and the
+
ve z
m
quadrant. One point on this marker plane is arbitrar-
ily chosen as the origin of the marker axes system; here
m
T
1
is chosen and is
labeled
m
. The line from
m
T
1
to
m
T
3
defines the
ve z
m
axis;
y
m
is normal
to the tracking plane, and
x
m
is orthogonal to the plane defined by
y
m
−
+
z
m
to
form a right-hand system.
The purpose of the anatomical calibration process is to find the relation
between the marker axes,
x
m
−
z
m
, and the anatomical axes,
x
-
y
-
z
.This
process requires the subject to assume a well-defined position; usually the
anatomical position is used. At this time, extra calibration markers may be
placed temporarily on the segment to define well-known anatomical points
from which the segment's anatomical axes can be defined. For example, for
y
m
−
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