Biomedical Engineering Reference
In-Depth Information
5.9.4
Oscillating motion with boundary conditions and one state-
response constraint—PD solution
Adding one state constraint,
qð
0
:
76
Þ
52
:
44 rad(obtained from
Figure 5.16
), to
the optimization formulation in the previous section, predictive dynamics is
formulated as:
2
Minimize
Jðq; τ; tÞ
Subject to
I q
1
mg
l
2
cos
q
5
τ
;
_
qð
0
Þ
5
0
qð
0
Þ
5
0
(5.40)
qð
0
:
76
Þ
52
:
2
44
qðTÞ
52
2
:
40
; qðTÞ
52
2
:
85
;
T
1
:
79
5
2
π
#
#
π
2
10
#
τ
#
10
q
Solving the above optimization problem with the two performance measures,
the corresponding joint angle, velocity, and torque are obtained, as shown in
Figures 5.17
5.19
.
In
Figures 5.16
5.18
, min
max performance closely predicts joint angle and
velocity histories of the system. However, it has a bang-bang type prediction for
the joint torque. Torque-square performance only predicts the trends of joint angle
and velocity and fails to predict joint torque.
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 5.17
Joint angle prediction of the single pendulum, Case 3.
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