Biomedical Engineering Reference
In-Depth Information
Z
i
θ
i
Joint
i
α
i
Z
i-1
Joint
i
-1
X
i
a
i
O
i
d
O
i-1
i
Y
i
X
i-1
FIGURE 2.16
The relation between two coordinate systems with four parameters.
We shall always refer to
T
as having the following vectors
nsap
nsap
nsap
⎡
⎤
x
x
x
x
⎢
⎥
nsap
⎡
⎤
⎢
⎥
y
y
y
y
T
=
=
⎢
(2.48)
⎥
0001
⎢
⎥
⎣
⎦
z
z
z
z
⎢
⎥
0001
⎣
⎦
A single general transformation matrix between any two coordinate systems is
needed. A general motion (translation and rotation) between any two configura-
tions can be identified by four consecutive transformations taking the rigid body
from one configuration to its final destination. These transformations are illus-
trated in
Figure 2.16
and are listed as follows:
1.
A rotation of
angle about the OX axis
2.
A translation of
a
units along the OX axis
3.
A translation of
d
units along the OZ axis
4.
A rotation of
α
θ
angle about the OZ axis.
Since all transformations are about the world coordinate system, the transforma-
tion matrices are pre-multiplied. Remember that a basic homogeneous transformation
matrix characterizes each transformation. The final transformation can be written as
2
3
2
3
cos
θ
2
sin
θ
00
100
a
0100
0010
0001
4
5
4
5
sin
θ
cos
θ
00
T
ð
d
; θ; α;
a
Þ
5
T
z;θ
T
z;d
T
x;a
T
x;α
5
0
0
1
0
0
0
0
1
2
4
3
5
2
4
3
5
1000
0100
001
d
0001
10 00
0
cos
α
2
sin
α
0
0
0
00 01
sin
α
cos
α
2
4
3
5
cos
θ
2
cos
α
sin
θ
sin
α
sin
θ
a
cos
θ
sin
θ
cos
α
cos
θ
2
sin
α
cos
θ
a
sin
θ
T
ð
d
; θ; α;
a
Þ
5
0
sin
α
cos
α
d
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