Robust Super-Twisting Sliding Control of PAM- actuated Manipulator Based on Perturbation Observer

Figures & data

Figure 1. The single link robotic PAM arm

Figure 2. Schematic diagram of observer-based STSMC PAM manipulator

Table 1. List of parameters assigned to PSO algorithm

Parameters of PSO Technique	Value
The inertia coefficient $w$	$1.4$
The personal acceleration coefficient $C_1$	$2$
The social acceleration coefficient $C_2$	$2$
The swarm size (population size)	30
The number of iteration	60

Figure 3. Open-loop response for the Single-Arm Manipulator

Table 2. The robotic arm parameters

Parameters	Value
$m$	$20kg$
$l$	$0.5m$
$I$	$1.667kg.m^2$
$g$	9.81 $m/s^2$
$r$	0.05 $m$
$P_{0t}$	400 $kPa$
$P_{0b}$	400 $kPa$

Table 3. The optimal values of design parameters for SMO-STSMC and STSMC based on the PSO algorithm

Design Constants for SMO-STSMC	Value	Design Constants for STSMC	Value
$c_1$	0.6370	${\lambda }_1$	2.4911
$c_2$	2.4997	$c_3$	2.4931
$\lambda$	2.9774	$c_4$	1.5016
$k$	35.7621	-	-
$\beta$	35.4726	-	-
$\tau$	0.0224	-	-

Figure 4. Evolution of the objective function versus the iterations using SMO-STSMC

Figure 5. Evolution of the optimized parameters ($\lambda$, $c_1$ and $c_2$) versus iterations.

Figure 6. Evolution of the optimized parameters ($\mathrm{k}$ and $\beta$) versus the iterations.

Figure 7. Evolution of the optimized

Figure 8. Evolution of the objective function versus the iterations by using STSMC

Figure 9. Evolution of the optimized parameters (${\lambda }_1$, $c_3$, and $c_4$) versus iterations.

Figure 10. Angle response without uncertainty

Figure 11. Angular velocity without uncertainty

Figure 12. Control action without uncertainty

Figure 13. Angle error signal without uncertainty

Figure 14. Surface variable without uncertainty

Figure 15. Estimated uncertainty without uncertain

Figure 16. Angle response with 5% uncertainty in parameters

Figure 17. Angular velocity response with 5% uncertainty in parameters

Figure 18. Control action with 5% uncertainty in parameters

Figure 19. Angle error signal with 5% uncertainty in parameters

Figure 20. Surface curve with 5% uncertainty in parameters

Figure 21. Estimated uncertainty with 5% uncertainty in parameters

Figure 22. Angle response under disturbance

Figure 23. Angular velocity response under disturbance

Figure 24. Control signals due to both controllers

Figure 25. The error responses due to both controllers (Under disturbance)

Figure 26. Responses of Sliding Variables

Figure 27. Disturbance Estimation

Figure 28. Angle response (with combined effect)

Figure 29. Angular velocity response (with combined effect)

Figure 30. Control signals due to both controllers (with combined effect)

Figure 31. The error responses due to both controllers (with combined effect)

Figure 32. Responses of Sliding Variables

Figure 33. Disturbance-Uncertainty Estimation

Table 4. The numerical reports of the dynamic performance of proposed controllers

Controller Type	State Variable	RMSE
Without Uncertainty and Disturbance
SMO-STSMC	$q$	0.0584
STSMC	$q$	0.0604
With Uncertainty (+5% in parameters)
SMO-STSMC	$q$	0.0586
	Estimation error of SM-Observer	0.0059
STSMC	$q$	0.0694
With Disturbance
SMO-STSMC	$q$	0.0584
	Estimation error of SM-Observer	0.0089
STSMC	$q$	0.0605
With Disturbance and Uncertainty (+5% in parameters)
SMO-STSMC	$q$	0.0586
	Estimation error of SM-Observer	0.0067
STSMC	$q$	0.0701