# Dynamic Modeling of McKibben Muscle Using Empirical Model and Particle Swarm Optimization Method
## 6 Results
### Empirical Model Testing  
Empirical model testing was carried out to evaluate the proposed model adaptability while considering quasi-static, quasi-rate, and historic dependencies. The model performance was measured based on the correlation test evaluation. The quasi-static model testing was performed by introducing static load disturbance into the model. This generates isotonic relation of the McKibben muscle model. Contrarily, the quasi-rate model testing was performed by introducing different excitation frequency to generate the input pressure signal. This caused the contraction length, velocity, and acceleration of the McKibben muscle to change significantly. This phenomenon affects frictional behavior as described in sliding regime. Meanwhile, the historic dependence model testing was performed by generating input pressure with random wave signals. The random signals created different pattern of minor hysteresis loops with nonlocal memory behavior effect. Figure 7 shows the superimposed of proposed model with hysteresis data while considering quasi-static, quasi-rate, and historic dependencies. This result shown that the empirical modeling with modified dynamic friction model was able to perform at all dependencies. This performance also can be verified based on the model output comparison with experimental data as shown in Figure 8. It shows that the proposed model adapted with changes in quasi-static and quasi-rate dependencies; thus being able to accurately simulate the contracting length of the McKibben muscle. Furthermore, the proposed model was able to correct its path even though there were sudden changes in input pressure. The changes occurred due to the random wave signal, which produces minor hysteresis loops. Table 3 shows the correlation test of the desired trajectories and model outputs from empirical modeling. The result shown significantly high correlation values produced at lower frequencies with (0.944 \(\pm\) 0.009) at 0.5 Hz and (0.897 \(\pm\) 0.007) at 1 Hz. Even though  
Figure 6: Model simulation: (**a**) existing model and (**b**) proposed model.  
the correlation values were slightly decreased at higher frequencies with (0.843 \(\pm\) 0.016) at 1.5 Hz and (0.752 \(\pm\) 0.027) at 2 Hz, the result was still significant. This indicated that the proposed empirical modeling was able to compensate all source of dependencies. However, there was still some error produced by the proposed model. This error cannot be avoided due to the nonlinearities inherited by the McKibben muscle were varied and complicated. These results were still not the best solution that can be produced by the proposed empirical model because the model parameters can still be improved by the PSO method.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Quasi-Static\(\backslash\)Rate** & **0.5 Hz** & **1 Hz** & **1.5 Hz** & **2 Hz** \\ \hline
5 kg & 0.929 & 0.894 & 0.815 & 0.732 \\
10 kg & 0.937 & 0.903 & 0.825 & 0.739 \\
15 kg & 0.952 & 0.883 & 0.847 & 0.763 \\
20 kg & 0.955 & 0.898 & 0.839 & 0.741 \\
25 kg & 0.941 & 0.895 & 0.853 & 0.797 \\
30 kg & 0.948 & 0.904 & 0.861 & 0.799 \\ \(\overline{x}\pm\sigma\) & 0.944 \(\pm\) 0.009 & 0.897 \(\pm\) 0.007 & 0.843 \(\pm\) 0.016 & 0.752 \(\pm\) 0.027 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Correlation test at all dependencies.  
Figure 7: Hysteresis based on empirical model and experimental tests at all dependencies.  
Figure 8: Empirical model testing at all dependencies (0.5–2.0 Hz).