Biomedical Engineering Reference
In-Depth Information
Single Pendulum
0.0
10.0
Joint angle
Joint Angular velocity
-0.5
5.0
-1.0
-1.5
0.0
-2.0
-5.0
-2.5
-3.0
0.0
-10.0
0.5
1.0
Time (sec)
1.5
2.0
FIGURE 5.10
Forward dynamics solved by ADAMS.
5.9.2
Simple swing motion with boundary conditions—PD solution
The swinging motion without oscillation is studied with the initial and final con-
ditions. The total time is randomly selected as
T
43 s (less than first half
period of oscillation), and the single pendulum starts at rest in the horizontal posi-
tion and ends up with the final conditions that are obtained from
Figure 5.10
as
qðTÞ
52
0
:
5
=s
. The swinging motion is driven by the
external torque and gravity. Besides boundary condition and total travel time
T
,
neither external torque nor joint angle is known. Therefore, the predictive dynam-
ics problem is formulated as in
Equation (5.31)
to reveal the natural swing motion
of the single pendulum.
2
:
40 rad and
qðTÞ
52
5
:
86 rad
Minimize
Jðq
; τ;
tÞ
mg
l
Subject to
I
q
€
2
cos
q
5
τ
1
(5.34)
qð
0
Þ
5
0
; qð
0
Þ
5
0
qðTÞ
52
2
:
40
; qðTÞ
52
5
:
86
; T
5
0
:
43
2
π
#
q
#
π
10
#
τ
#
10
2
Treating q and
τ
as design variables, four performance measures are tested as
follows
ð
T
t
5
0
τ τ
J
1
ðq
; τ;
tÞ
5
dt
(5.35)
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