# Dynamic Modeling of McKibben Muscle Using Empirical Model and Particle Swarm Optimization Method
## 7 Discussion  
The performance of dynamic modeling of McKibben muscle was evaluated based on comparison test of empirical modeling and hysteresis data. The hysteresis data was obtained through the experimental study, while the empirical model was modified based on the existing model presented in literature. Empirical modeling consists of two parts: the first part is the stating modeling of the contracting force model, while the second part is the dynamic modeling of the inherent hysteresis. Before performing the empirical modeling, all the initial parameters were identified and measured. Then, the simulation model was performed with parametric identification using PSO algorithm. Based on simulation analysis, the contracting force of static empirical modeling was able to characterize the quasi-static behavior within the linear region of McKibben muscle's model. The nonlinear region of the McKibben muscle which is at zero pressure condition was represented by using elasticity model. When the McKibben muscle was stretched beyond its nominal length with zero pressure intake, the McKibben consisted only of a braided mesh sleeve and the rubber tube mechanism. The force dynamic of the proposed empirical model for quasi-static dependency can be seen in Figure 5. The result shown that the proposed model was able to generate an appropriate force dynamic comparable with the one obtained through experimental study. The experimental data was obtained from the experimental test at minimum load condition. Furthermore, the model also demonstrated comparable hysteresis behavior during cyclic compression and expansion as shown in Figure 7. The model was compared with hysteresis data in Figure 2. As describe in the literature, the extracted hysteresis data was dependent on the contraction velocity of the McKibben muscle. The modified dynamic friction model was proven to be able to capture the quasi-rate behavior of McKibben muscle which follows the sliding regime.  
The random wave of input pressure was used to test the effect of the nonlocal memory or historic dependency on the proposed empirical model. This random input data consisted of various contraction patterns during cyclic contraction and expansion of the McKibben muscle. By employing random input data, the ability of the model to redefine its path towards required value can be clarified. In addition, the reliability and adaptability of the dynamic model of hysteresis also can be verified. The ability to convert the present path into the desired path is what defines the history dependency. The influence of the nonlocal memory causes the dynamic modeling of McKibben muscle complicated and difficult to be achieved. The proposed model testing while considering quasi-static, quasi-rate and historic dependencies were shown in Figure 8. These tests were accomplished by generating the random wave of input pressure with different excitation frequencies (0.5-2 Hz), and were then performed with various static loading (0-300 N). Based on the result, the proposed model was able to simulate the desired contracting length precisely with all sources of dependencies. The performance of the proposed empirical model was evaluated using a correlation test. Table 3 shows the correlation test of the proposed empirical model at all dependencies. The correlation test indicated the model performance with high mean correlation value and low standard deviation. The model was performed very well at lower frequencies with correlation values of (0.944 \(\pm\) 0.009) at 0.5 Hz and (0.897 \(\pm\) 0.007) at 1 Hz. For higher frequencies, the result was still significant with slightly decreasing correlation values of (0.843 \(\pm\) 0.016) at 1.5 Hz and (0.752 \(\pm\) 0.027) 2 Hz. Although the modification model and correction function were presented, there were some errors remaining due to the high nonlinearities of McKibben muscle that cannot be captured. These remaining errors could easily be eliminated with intelligent control strategies.