# Hysteresis Compensation and Trajectory Tracking Control Model for Pneumatic Artificial Muscles
## 4 Results and Discussion
### Validation Results of PAM Displacement Tracking Control  
After obtaining the QUPI inverse model parameters through system identification using the particle swarm optimization algorithm, the two composite control schemes were used to predict and compare the PAM's output displacement with the actual displacement. The model's adaptability was validated under different PAM input pressure signals, and the accuracy of the model was ultimately demonstrated through error values. The parameter values of the control law are shown in Table 3.  
Table 4, were selected and defined as Scheme 1 through Scheme 6. Based on these different input conditions, the performance of the two composite controllers was compared, with a focus on analyzing the performance of the two control methods under complex dynamic input conditions. Figure 11 shows the PAM trajectory tracking results of the two controllers under the six schemes. In Figure 11, the three subplots in (a), from top to bottom, respectively, represent the feedforward control pressure compensation, the pressure compensation of the two feedback controls, and the total feedforward-feedback compensation. Comparing the results of the (a) plots across the six schemes reveals that when using the same feedforward scheme, the feedforward control pressure compensation for both control strategies is identical. However, the performance of the two feedback control strategies differs significantly, directly leading to a notable difference in the final total compensation. Part (b) of Figure 11 contains two subplots. From top to bottom, they show the fitting of the final output displacement to the desired displacement for the two control strategies, as well as the real-time error of the fitting. The experiments show that although the tracking error of the PAM-driven system increases under high-frequency and step input conditions, the overall error remains within 1 mm, indicating that the system has strong robustness and adaptability under complex working conditions. By finely adjusting the control parameters, the system can further improve its tracking performance under extreme input conditions, reducing errors and enhancing response accuracy. In the PAM-driven system, due to its inherent hysteresis, simple feedback control often struggles to handle complex nonlinear behavior. However, by introducing the iterative learning algorithm, the system can gradually optimize the control output, compensate for hysteresis effects, and adapt to dynamic changes across multiple operations. This iterative learning algorithm leverages historical error data to continuously adjust the controller's input signal, making Control Method 2 exhibit higher control accuracy and a faster response speed compared to Control Method 1. The introduction of iterative learning significantly enhances the control system's adaptive capability, making it perform more effectively when faced with complex nonlinear systems. This controller can not only quickly correct errors and improve trajectory tracking accuracy but also achieve stable system responses in a shorter time, significantly boosting the overall performance of the control system. Under different input signals, Control Method 2 for pneumatic artificial muscles (PAMs) improved the error accuracy by approximately 10 times compared to Control Method 1 in both the MAE and RMSE values. The comparison of error values between the two control schemes is shown in Table 5. For the PAM flexible actuators, incremental PID-feedback control may exhibit certain limitations in handling complex nonlinear and dynamic hysteresis effects, making it difficult to achieve the desired control accuracy. However, by combining iterative learning with incremental PID-feedback control, the gradual error correction capability of iterative learning allows the control output to be continuously optimized over multiple iterations, progressively eliminating errors caused by the hysteresis effects and nonlinear characteristics.