Biomedical Engineering Reference
In-Depth Information
A graph from frame 3 to frame 0 is represented by
3
T
0
; therefore, the multipli-
cation of a transformation matrix times its inverse yields the identity matrix
0
T
3
3
T
0
5
I
(2.32)
It is also noted here that a transformation between any two frames can be
obtained independent of the route followed, i.e., the transformation from frame 0
to frame 2 can be obtained by
0
0
T
1
1
T
2
5
T
2
(2.33)
or by
0
0
T
3
3
T
2
5
T
2
(2.34)
where the transformation
3
2
T
3
Þ
2
1
(2.35)
The order in which matrices are multiplied is important since matrix multipli-
cation is not commutative, i.e., in general
A
T
2
5
ð
T
B
B
B
T
C
A
T
B
. In fact, great care
must be given to the order of multiplication. Consider two coordinate systems,
X
1
Y
1
Z
1
being the world coordinate system, and
X
2
Y
2
Z
2
being the body reference
frame. Two rules must be followed in applying the order of multiplication of
transformation matrices. These rules are given without proof:
T
C
6¼
1.
A transformation taking place with respect to the world reference frame
(
X
1
Y
1
Z
1
) necessitates the
pre-multiplication
of the previous transformation
matrix by an appropriate basic homogeneous transformation matrix.
2.
A transformation taking place with respect to the body's own reference frame
(
X
2
Y
2
Z
2
) necessitates the post-multiplication of the previous transformation
matrix by an appropriate basic transformation matrix.
2.6.1
Example: multiple transformations
A digital human lives in a computer-aided engineering environment. This human
will be requested to perform some tasks, i.e., to grasp and move objects. In order to
identify objects in the workspace, a virtual camera is embedded in the human's head,
which will function as his eyes. This camera will determine the position and orienta-
tion of an object in space and will return a homogeneous transformation matrix. The
virtual camera senses the position of a Joystick
J
shown in
Figure 2.10
. The virtual
camera's coordinate system is represented by
c
1
,
c
2
,and
c
3
as shown in Figure 2.10.
The camera identifies the position and
onfiguration of the shoulder and the joystick.
The homogeneous transformation matrix of the joystick
c
J
as seen by the
is represented by
C
camera
C
T
J
as
2
3
010 5
001 4
100
4
5
C
T
J
5
(2.36)
15
000 1
2
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