# Hysteresis Compensation and Trajectory Tracking Control Model for Pneumatic Artificial Muscles
## 4 Results and Discussion
### Validation Results of the FN-QUPI Model  
This section validates the applicability of the FN-QUPI model under different load conditions and input pressure signals. Ultimately, the high precision and accuracy of the model in describing the positional hysteresis of the PAM are demonstrated through the use of the mean absolute error (MAE) and root mean square error (RMSE) values. (1) The same load with varying frequency and amplitude sinusoidal excitation signals. When a load of 2 kg is applied to the end of the PAM, the sinusoidal pressure signal from Equation (19) is used to drive the PAM, resulting in the actual output displacement of the PAM. The input signal is then processed through the FN-QUPI model to predict the PAM's output displacement. The hysteresis curves corresponding to three amplitude levels at 0.05 Hz and 0.1 Hz are obtained. The comparison between the actual hysteresis curves and the predicted PAM hysteresis curves, as well as the real-time error analysis for the three cases, are shown in Figures 9a-c and 9d-f, respectively. \[u(t)=(\text{p}/2)\sin(c\pi t)+p/2,(p=0.5,0.8,1.3\text{ bar};c=0.1,0.2) \tag{21}\]  
The MAE and RMSE values for the displacement between the FN-QUPI model output and the actual output under the same load with different frequencies and amplitudes are shown in Table 1. The fitting error magnitude of the hysteresis model for the FN-QUPI model is on the order of \(10^{-6}\) m, indicating that the predicted output displacement by the FN-QUPI model aligns well with the experimental data, demonstrating high accuracy and excellent predictive capability. According to the real-time error graph, the error increases with the amplitude, and when the amplitude reaches 1.3 bar, the error is most noticeable during large PAM contractions. This phenomenon is determined by the flexible characteristics of pneumatic muscles. When the amplitude increases to approach the contraction limit of the pneumatic muscle, the nonlinearity of the displacement becomes more apparent, making predictions more challenging. Therefore, as the amplitude increases, the nonlinear behavior of the pneumatic muscle under high stress becomes more prominent,further affecting the prediction accuracy. However, despite the increase in error at larger amplitudes, the FN-QUPI model still exhibits strong fitting ability, accurately predicting the displacement output of the PAM under complex conditions. This further demonstrates the effectiveness of the FN-QUPI model in addressing nonlinear hysteresis phenomena, providing a reliable foundation for future control system optimization. (2) Different loads and different frequency sinusoidal excitation signals. Under the same amplitude conditions, the comparison between the predicted output displacement of the FN-QUPI model and the actual displacement of the PAM at different loads and frequencies, along with the real-time error graphs, are shown in Figure 10. In each row, the graphs from left to right correspond to frequencies of 0.05 Hz, 0.1 Hz, 0.5 Hz, and 1 Hz, respectively. From top to bottom, the loads correspond to 4 kg and 6 kg. The red curves on the far right represent the real-time error graphs for each corresponding condition. By analyzing the comparison graphs, it can be observed that at low frequencies (such as 0.05 Hz and 0.1 Hz), the predicted displacement of the FN-QUPI model almost completely overlaps with the actual displacement, with very minimal error. This indicates \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Frequency (Hz)** & \multicolumn{2}{c}{**0.05**} & \multicolumn{2}{c}{**0.1**} \\ \hline Amplitude (bar) & 0.5 & 0.8 & 1.3 & 0.5 & 0.8 & 1.3 \\ MAE (mm) & 0.0046 & 0.0167 & 0.0337 & 0.0013 & 0.0096 & 0.0201 \\ RMSE (mm) & 0.0064 & 0.0247 & 0.0392 & 0.0017 & 0.0142 & 0.0301 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Errors of PAM corresponding to different frequencies under a 2 kg load. Figure 9: Hysteresis curves of PAM corresponding to different frequencies and amplitudes under 2 kg load: (**a–c**) 0.05 Hz; (**d–f**) 0.1 Hz.