# Hysteresis Compensation and Trajectory Tracking Control Model for Pneumatic Artificial Muscles
## 4 Results and Discussion
### Validation Results of the FN-QUPI Model
(**a**) Actual hysteresis curves at 0.05 Hz; (**b**) Comparison of hysteresis curves at 0.05 Hz; (**c**) Real-time errors at 0.05 Hz; (**d**) Actual hysteresis curves at 0.1 Hz; (**e**) Comparison of hysteresis curves at 0.1 Hz; (**f**) Real-time errors at 0.1 Hz. that the model is highly capable of capturing the dynamic characteristics of the PAM in low-frequency environments, demonstrating excellent accuracy and adaptability. As the frequency increases, especially at 0.5 Hz and 1 Hz, although the model still accurately tracks the displacement variations of the PAM, the error graphs show a noticeable trend of increasing error, particularly at the higher frequency of 1 Hz, where this error increase becomes more prominent. A detailed analysis of the causes for the increased error reveals that it mainly occurs during the rapid contraction and release phases of the PAM, where the nonlinear effects of the PAM system become more complex. Overall, the FN-QUPI model demonstrates excellent dynamic response capability and the precise tracking of PAM displacement. Figure 10: Hysteresis curves of PAM under the same amplitude corresponding to different frequencies and loads. (**a**) Fitting diagram of hysteresis at different frequencies under a 4 kg load; (**b**) Error diagram of hysteresis at different frequencies under a 4 kg load; (**c**) Fitting diagram of hysteresis at different frequencies under a 6 kg load; (**d**) Error diagram of hysteresis at different frequencies under a 6 kg load. As the load gradually increases, the displacement response of the PAM exhibits a decreasing trend, and the hysteresis phenomenon becomes more pronounced. However, the FN-QUPI model is still able to fit these changes well, demonstrating its strong adaptability and modeling capabilities. Overall, the prediction accuracy of the FN-QUPI model is higher, and the error is smaller when the load is lighter. However, as the load increases, especially under the combination of high frequency and high load, the error tends to increase. This indicates that the PAM system exhibits stronger nonlinear effects under complex working conditions, which challenges the prediction accuracy of the model. Nevertheless, the FN-QUPI model still shows strong robustness in most cases. The MAE and RMSE error values between the predicted displacement from the FN-QUPI model and the actual displacement of the PAM under different frequencies and loads with the same amplitude are shown in Table 2.  
As the table indicates, the error magnitude is also in the order of \(10^{-6}\) m, meaning the predicted output displacement of the model aligns very closely with the experimental data. Especially under lower frequency and lighter load conditions, the error is even smaller, further validating the accuracy of the model. This result not only proves the precision of the FN-QUPI model within the normal operating range of pneumatic artificial muscles (PAM) but also highlights the model's effectiveness in dynamic system modeling. The FN-QUPI model is able to precisely capture the nonlinear dynamics of the PAM system, particularly under low-frequency and low-load conditions, where its high prediction accuracy showcases the model's deep understanding and control of complex dynamic processes. Although the error increases under conditions of high load and high frequency, the FN-QUPI model still tracks the actual displacement changes of the PAM well, demonstrating its powerful ability to handle nonlinearities. This result further validates the advantages of the FN-QUPI model in managing complex nonlinear systems, indicating its wide application prospects and practical value, especially in scenarios requiring high-precision modeling and real-time control.