Biomedical Engineering Reference
In-Depth Information
s
y
52
sin
q
1
cos
ðq
2
1
q
3
1
q
4
Þ
(2.66)
s
z
52
sin
ð
q
2
1
q
3
1
q
4
Þ
(2.67)
a
x
5
sin
q
1
(2.68)
a
y
52
cos
q
1
(2.69)
a
z
5
0
(2.70)
p
x
5
cos
q
1
ð
2
3 sin
ð
q
2
1
q
3
1
q
4
Þ
1
9 cos
q
2
1
11 cos
ð
q
2
1
q
3
ÞÞ
(2.71)
p
y
5
sin
q
1
ð
2
3 sin
ð
q
2
1
q
3
1
q
4
Þ
1
9 cos
q
2
1
11 cos
ð
q
2
1
q
3
ÞÞ
(2.72)
(2.73)
e.
To determine the orientation of the 4
th
coordinate system with respect to the
hip, we substitute
q
1
5
p
z
5
9 sin
q
2
1
11 sin
ðq
2
1
q
3
Þ
1
3 cos
ðq
2
1
q
3
1
q
4
Þ
0 into
0
0,
q
2
5
0,
q
3
5
0, and
q
4
5
T
4
and the
T
T
orientation is identified as
n
5
001
,
s
52
100
, and
.
In order to determine the position of the point
Q
, defined with respect to
the 4
th
coordinate system, we shall use the extended vector equation as
T
a
5
0
10
2
5
0
4
1
1
0
T
4
ð
q
Þ
(2.74)
T
The position of a point on the foot is given by
v
Q
5
000
.Forthe
initial posture of the lower limb, the joints are
q
1
5
0,
q
2
5
0,
q
3
5
0, and
q
4
5
0.
In order to calculate the new posture given the change in joint variables,
we substitute into
0
T
4
2
3
0
1
0
11
2
4
5
0
0
10
2
90
90
0
T
4
ð
0
;
;
2
;
0
Þ
5
(2.75)
1
0
0
12
0
0
0
1
The position is calculated as
90
;
2
90
;
v
Q
ð
0
;
0
Þ
5
12
0
11
(2.76)
2
and the orientation is calculated as
T
n
5
100
(2.77)
T
s
5
001
(2.78)
T
a
5
0
10
(2.79)
2
as shown in
Figure 2.24
.
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