Biomedical Engineering Reference
In-Depth Information
3. Lagrangian
L
=
KE
−
PE
2
m
1
˙
q
2
cos
(q
2
)
2
q
2
sin
(q
2
)
2
+
r
1
˙
1
1
q
2
)
2
L
=
q
1
−
r
1
˙
+
2
I
1
(
˙
m
1
g
r
1
sin
(q
2
)
+
2
m
2
˙
q
3
sin
(q
3
)
2
1
−
q
1
−
l
1
˙
q
2
sin
(q
2
)
−
r
2
˙
q
3
cos
(q
3
)
2
l
1
˙
m
2
g
I
1
sin
(q
2
)
1
q
3
)
2
+
q
2
cos
(q
2
)
+
r
2
˙
+
2
I
2
(
˙
−
r
2
sin
(q
3
)
+
2
m
3
˙
1
+
q
1
−
l
1
˙
q
2
sin
(q
2
)
−
l
2
˙
q
3
sin
(q
3
)
q
4
sin
(q
4
)
2
+
l
1
˙
−
r
3
˙
q
2
cos
(q
2
)
+
l
2
˙
q
3
cos
(q
3
)
q
4
cos
(q
4
)
2
m
3
g
l
1
sin
(q
2
)
1
q
4
)
2
+
r
3
˙
+
2
I
3
(
˙
−
+
l
2
sin
(q
3
)
r
3
sin
(q
4
)
+
The Lagrangian
L
is simplified by expanding the previous expression
and collecting terms,
2
( q
1
)
2
(m
1
+
m
2
+
m
3
)
+
1
L
=
q
2
)
2
m
1
(r
1
)
2
m
3
(l
1
)
2
1
m
2
(l
1
)
2
2
(
˙
+
I
1
+
+
2
( q
3
)
2
m
2
(r
2
)
2
+
I
2
+
m
3
(l
2
)
2
+
2
( q
4
)
2
m
3
(r
3
)
2
+
I
3
1
1
+
−
q
1
q
2
sin
(q
2
)(m
1
r
1
+
m
2
l
1
+
m
3
l
1
)
−
q
1
q
3
sin
(q
3
)
·
(m
2
r
2
+
m
3
l
2
)
−
q
1
q
4
(m
3
r
3
)
+
q
2
q
3
l
1
cos
(q
3
−
q
2
)
·
(m
2
r
2
+
m
3
l
2
)
+
q
2
q
4
l
1
cos
(q
4
−
q
2
)(m
3
r
3
)
+˙
m
2
g
l
1
sin
(q
2
)
q
3
˙
q
4
l
2
cos
(q
4
−
q
3
)(m
3
r
3
)
−
m
1
gr
1
sin
(q
2
)
−
r
2
sin
(q
3
)
−
m
3
g
l
1
sin
(q
2
)
r
3
sin
(q
4
)
+
+
l
2
sin
(q
3
)
+
4. External work
W
,
W
=
T
1
(q
2
−
0
)
+
T
2
(q
4
−
q
3
)
+
T
3
(q
4
−
q
3
)
+
F (q
1
+
l
1
cos
(q
2
)
+
l
2
cos
(q
3
))
5. Finally, the equations of motion are for the generalized coordinates
q
1
,
q
2
,
q
3
,
q
4
, respectively,
(a)
F
=
q
1
(m
1
+
m
2
+
m
3
)
−
(
q
2
sin
(q
2
)
·
q
2
)
2
cos
(q
2
)
˙
+¨
(m
1
r
1
+
m
2
l
2
+
m
3
l
1
)
−
(
˙
q
3
sin
(q
3
)
·
(m
2
r
2
+
q
3
)
2
cos
(q
3
)
+¨
m
3
l
2
)
−¨
q
4
m
3
r
3
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