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Page 102 Li et al. Intell Robot 2022;2(1):89–104 I http://dx.doi.org/10.20517/ir.2022.02

Table 5. Results of different working frequencies with the proposed controller
Max. Error (deg) Mean Error (deg)
Joint Time 1 2 3 4 1 2 3 4
0.25 0.71 0.0021 0.61 0.0016 0.27 0.0004 0.24 0.0003
0.15 0.78 0.0039 0.68 0.0021 0.37 0.0013 0.33 0.0007
0.50 0.58 0.0004 0.50 0.00025 0.23 0.0001 0.20 0.00008

Table 6. Results by tracking different PPO trajectories
Error Type Joint 4–5-6-7 th polynomial 5 th polynomial
Max. Error (deg) Joint 1 0.7113 0.6854
Joint 2 0.0021 0.0032
Joint 3 0.6116 0.5875
Joint 4 0.0016 0.0020
Mean Error (deg) Joint 1 0.2697 0.2707
Joint 2 0.0004 0.0013
Joint 3 0.2404 0.2447
Joint 4 0.0003 0.0009

Figure 12. The varying RMSEs for different trajectories: (A) Joints 1 and 3; and (B) joints 2 and 4.

Moreover, different working frequencies and trajectories are selected to evaluate the generalization ability of
the controller. The results are listed in Tables 5 and 6, respectively. Figure 12 shows the varying RMSE for
different trajectories.

Fromtheresults, itcanbeseenthattheproposedcontrollershowsgoodperformanceunderdifferentoperating
frequencies and different trajectories, meaning that the proposed control law can work effectively to track
different task trajectories and have good generalization capabilities.

6. CONCLUSIONS
In this work, an open-closed loop PD type iterative learning control method is proposed for parallel robots to
track repetitive work trajectories, thanks to its advantages of simple implementation and practicability in in-
dustrial engineering. According to the complexity and uncertainties of the working environment, two external
disturbances, i.e., repetitive and non-repetitive ones, are taken into account for the model-based control de-
sign. The designed controller is compared with the D-ILC law and evaluated along with a 4-dof parallel robot,
and the results show the better performance of the PD-ILC law compared with the classical D-ILC law. The
test results with and without disturbances also show the robustness in terms of the trajectory tracking errors.
In addition, different working frequencies and trajectories are adopted to evaluate the generalization capabili-