Biomedical Engineering Reference
In-Depth Information
In order to determine the position and orientation of the C coordinate system
with respect to the
A
coordinate system, we multiply the two matrices
2
4
3
5
0
107
1 005
0 010
0 001
2
A
T
B
B
T
C
5
(2.28)
To determine how the thumb is seen by coordinate system
A
, we calculate
A
A
T
B
B
T
C
C
T
x
Q
5
x
Q
5
75
:
5
11
(2.29)
2
From
Figure 2.8
, it can be observed that indeed the vector describing the point
on the thumb has coordinates (7, 5.5,
1) with respect to the
A
-coordinate system.
2
2.6
Directed transformation graphs
Consider the coordinate frames depicted in
Figure 2.9
. Each graph from one
frame to another represents a transformation matrix and is denoted by the
T-
matrix. The direction of the arrow indicates subscript and superscript, respec-
tively, of the
T-
matrix, i.e., the transformation from frame 0 to frame 1 is denoted
by
0
T
1
. A graph from frame 1 to frame 2 is represented by
1
T
2
.
Applying a sequence of transformations such as
0
T
1
followed by
1
T
2
yields a
graph from frame 0 directly to frame 2 represented by the transformation matrix
0
0
T
1
1
T
2
5
T
2
(2.30)
Similarly, applying another transformation from frame 2 to frame 3 can be
represented by a directed graph from frame 0 to frame 3 and characterized by
0
0
T
1
1
T
2
2
T
3
5
T
3
(2.31)
Frame 1
0
T
1
1
T
2
Frame 2
1
T
3
3
T
0
2
T
3
Frame 0
Frame 3
FIGURE 2.9
Four directed transformation graphs.
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