IEEE Robotics & Automation Magazine - December 2022 - 74

Simulation results of the controller in [8] for the circulartracking
task are shown in Figure 6(b)-(e). In phases 2 and 4,
the position errors are the largest compared with the results
in phases 1 and 3, which are consistent with the conclusion
in simulation 2. Figure 6(d) and (e) show the torque changes
in different phases. Feedforward torque has a large jump at
the phase-changing time, which is affected by the changes of
external torques, and some values are larger than 1 N.rad per
0.65
0.6
0.65
0.65
0.45
0.4
0.35
0.3
0.25
-0.2
-0.1
X (m)
(a)
0.6
First Trajectory
Third Trajectory
0.5
Second Trajectory
0.6
Last Three Trajectories
0.5
0.4
0.4
0.1
0.2
step so that they will bring instability to system actuators. In
the proposed method, the trajectories quickly converge to
the desired values in all the phases except for phase 4, and
the smoothness of the final trajectory is similar to that in Figure
6(c). On the other hand, compared with Figure 6(d) and
(e), continuity and uniformity of the feedforward force in
Figure 6(h) and (i) are much better, especially at the phaseshifting
moments between phases 1 and 2, and phases 3 and
0.3
0.3
-0.1
X (m)
(b)
8
6
4
2
-2
Phase 1
Phase 2
01 2
Time (s)
(d)
Phase 3
Phase 4
34
External Torque
Feedforward Torque
Feedback Torque
State Switching Point
6
4
2
-2
-4
0.1
-0.2
-0.1
X (m)
(c)
External Torque
Feedforward Torque
Feedback Torque
State Switching Point
0.1
0.2
Phase 1
Phase 2
Phase 3
Phase 4
01 2 34
Time (s)
(e)
Figure 6. A simulation of the trajectory and torque adaptation in the MF field in Table 1. (a) The force field in four phases. (b) and (c)
Trajectories in the first and last three periods in [8]. (d) and (e) Joint and external torques in the final period in [8]. (Continued)
74 * IEEE ROBOTICS & AUTOMATION MAGAZINE * DECEMBER 2022
Torque (N.rad)
Y (m)
Y (m)
Torque (N.rad)
Y (m)

IEEE Robotics & Automation Magazine - December 2022

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - December 2022

Contents
IEEE Robotics & Automation Magazine - December 2022 - Cover1
IEEE Robotics & Automation Magazine - December 2022 - Cover2
IEEE Robotics & Automation Magazine - December 2022 - Contents
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