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
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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