Biomedical Engineering Reference
In-Depth Information
1.0
0.5
Torque square
Min-max
Forward dynamics
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Time (s)
FIGURE 5.14
Joint angle prediction of the single pendulum, Case 2.
5.9.3
Oscillating motion with boundary conditions—PD solution
Oscillating pendulum makes the motion more complex. The predictive dynamics
approach is examined in this case by extending the final time to
T
79 s (more
than one and one-half period). The optimization formulation is similar
1
:
5
to
Equation (5.33)
except for the final conditions.
Minimize
Jðq
; τ;
tÞ
mg
l
Subject to
I
q
€
1
2
cos
q
5
τ
(5.39)
qð
0
Þ
5
0
; _
qð
0
Þ
5
0
qðTÞ
52
2
:
40
; _
qðTÞ
52
2
:
85
;
T
1
:
79
5
2
π
#
q
#
π
10
#
τ
#
10
2
Note from the results of the previous section that the performance measure of
minimum total time or a constant is not appropriate for predicting the natural
swinging motion of the single pendulum. Thus, only torque squares and min
max
formulations are tested as performance measures for the present case. The opti-
mized joint angle, velocity, and applied torque are given in
Figures 5.14
5.16
.
For the oscillating motion, the predictive dynamics fails to predict joint
angle, velocity, and torque histories with only the boundary conditions specified.
Although an energy-related performance measure is chosen, predictive dynamics
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