Biomedical Engineering Reference
In-Depth Information
2.5
2.5
2
2
Joint 1
Joint 2
1.5
1.5
1
1
0.5
0.5
0
0
Joint 2
Joint 1
−
0.5
−
0.5
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.5
tim
e (s)
tim
e (s
)
Optimal and
semi-optimal
optimal and
semi-optimal
joint angles
joint angles
1.6
2
1.4
Joint 1
Joint 1
1
1.2
1
0
0.8
0.6
−
1
0.4
Joint 2
0.2
Joint 1
−
2
0
−
0.2
−
3
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.5
tim
e (s)
tim
e (s
)
optimal and semi-optimal
joint velocities
optimal and semi-optimal
joint velocities
3
3
2
Joint 1
2
Joint 1
1
1
Joint 2
Joint 2
0
0
−
1
−
1
−
2
2
−
3
−
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.5
tim
e (s)
tim
e (s
)
optimal and semi-optimal
joint torques
optimal and semi-optimal
joint torques
(a) The motion from S to E1
(b) The motion from S to E2
T2
T3
0.5
E2
0.4
E1
S
0.3
0.2
T4
T1
0.1
teaching information
start position
end position
optimal trajectory
semi-optimal trajectory
0
0
0.1
0.2
0.3
0.4
0.5
x
- position [m]
(c) The trajectories in the task space
Figure 16.7 Comparison of the semioptimal solutions of the diffusion-based approach with the optimal ones.
Here, (c) shows the robot's end-effector trajectories in the task space, while (a) and (b) show two examples of the
time responses for the motions from S point to E1 and E2 points as given in (c), respectively. It is clear that the
solutions of our diffusion-based approach are almost the same as those that are obtained by solving the complex
two-point boundary problem.
ð
T
f
J
¼
1
2
:::
T
:::
d
t
(16
:
14)
0
which shows that human implicitly plans the PTP movements in the task space. Here x is the
position vector of the human arm's end-point. The optimal trajectory with zero boundary velocities
and accelerations can be obtained based only on the arm's kinematic model as
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