Biomedical Engineering Reference
In-Depth Information
seg
(
1
)
=
[2,
m
1
,
I
1
], seg
(
2
)
=
[4,
m
2
,
I
2
], seg
(
3
)
=
[6,
m
3
,
I
3
]
trq(
1
)
=
[0, 1,
T
1
],
trq(
2
)
=
[1, 2,
T
2
],
trq(
3
)
=
[2, 3,
T
3
],
frc(
1
)
=
[5,
F
,0]
The steps that follow are to create displacement and velocity lists that
contain all points and use the results to obtain system Lagrangian and the gen-
eralized forces that are necessary for the Lagrange equations. The following
is a pseudo-code for the derivation procedures:
For
p
1,
...
, 6, derive displacement and velocity expressions for point
(
p
) [Equations (8.16), (8.17)].
For
=
s
=
1,
...
, 3,
calculate
KE
and
PE
for
seg(s)
and
update
the
Lagrangian [Equations (8.5), (8.23), (8.25)].
For
t
=
1,
...
, 3, find the work done by torque
trq(t )
and update system
work
W
[Equation (8.34)].
For
f
1, find the work done by the force
frc(f )
and update
W
[Equation
(8.34)].
For
i
=
1,
...
, 4, apply Lagrangian equations for coordinate
q
i
[Equations
(8.8), (8.9)].
=
The intermediate results for the relevant parts of these steps are as follows:
1. Displacement vectors,
disp
(
1
)
=
[
q
1
,0]
disp
(
2
)
=
[
q
1
+
r
1
cos
(q
2
)
,
r
1
sin
(q
2
)
]
disp
(
3
)
=
[
q
1
+
l
1
cos
(q
2
)
,
l
1
sin
(q
2
)
]
disp
(
4
)
=
[
q
1
+
l
1
cos
(q
2
)
+
r
2
cos
(q
3
)
,
l
1
sin
(q
2
)
+
r
2
sin
(q
3
)
]
disp
(
5
)
=
[
q
1
+
l
1
cos
(q
2
)
+
l
2
cos
(q
3
)
,
l
1
sin
(q
2
)
+
l
2
sin
(q
3
)
]
disp
(
6
)
=
[
q
1
+
l
1
cos
(q
2
)
+
l
2
cos
(q
3
)
+
r
3
cos
(q
4
)
,
l
1
sin
(q
2
)
+
l
2
sin
(q
3
)
+
r
3
sin
(q
4
)
]
2. Velocity vectors for selected points,
velo
(
2
)
=
q
1
−
r
1
q
2
sin
(q
2
)
,
r
1
q
2
cos
(q
2
)
velo
(
4
)
q
3
sin
(q
3
)
,
l
1
q
2
cos(
q
2
)
+
r
2
q
3
cos (
q
3
)]
velo
(
6
)
=
[
q
1
−
l
1
q
2
sin
(q
2
)
−
l
2
q
3
sin
(q
3
)
−
r
3
q
4
sin
(q
4
)
,
l
1
q
2
cos
(q
2
)
+
l
2
q
3
cos
(q
3
)
+
r
3
q
4
cos
(q
4
)
]
=
[
q
1
−
˙
l
1
˙
q
2
sin
(q
2
)
−
r
1
˙
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