## 5 Experimental Results and Discussion
### SDCHM Model Identification and Validation

Table 3: Optimized coefficients \(G_{ij}\). Figure 9: Identification results and errors of the inherent hysteresis model. (**a**) Identification results for GPI\({}_{1}\); (**b**) identification results for GPI\({}_{2}\); (**c**) identification results for GPI\({}_{3}\). nonlinearities unable to be completely established in the first two submodels. As mentioned in Figure 6, inputs to the stiffness-dependent submodule consist of virtual displacements generated from the actuator coupling submodule and \(p_{0}(t)\) in the extensor PAM. Therefore, the sample set \(S=\{\textbf{{INPUT}},\textbf{{OUTPUT}}\}\) is constructed with the following structure:  
\[\textbf{{INPUT}}= \begin{bmatrix}H_{1}[t]&0&0&H_{1}[t]\\ 0&H_{2}[t]&0&0\\ 0&0&H_{3}[t]&0\\ p_{0}[t]&p_{0}[t]&p_{0}[t]\end{bmatrix}  \tag{23}\]  
with  
\[ \begin{cases}H_{1}[t]=\sum\limits_{i=1}^{3}\left[G_{i1}\cdot\text{GPI}_{i}[p _{i}][t]\right],\\ H_{2}[t]=\sum\limits_{i=1}^{3}\left[G_{i2}\cdot\text{GPI}_{i}[p_{i}][t]\right],\text{ with }t=1,2,\cdots,N_{1}\\ H_{3}[t]=\sum\limits_{i=1}^{3}\left[G_{i3}\cdot\text{GPI}_{i}[p_{i}][t]\right].\end{cases}  \tag{24}\]  
where \(H_{i},\ (i=1,2,3)\) is obtained from the output of the actuator coupling submodule. Contractile PAMs are inflated with pressure \(p_{i}[t]=\frac{A_{s}(t)}{2}\sin(0.04\pi t)+\frac{A_{s}(t)}{2}\), \((i=1,\ 2,\ 3)\) in Equation (21), and \(N_{1}=f_{r}\cdot t_{f1}\cdot T_{s}\), where the sampling frequency is \(f_{r}=20\) Hz, the number of cycles is \(t_{f1}=2\), and the period is \(T_{s}=50s\). The output sample set, **OUTPUT**, is shown in the following Equation (25):  
\[\textbf{{OUTPUT}}= \begin{bmatrix}y_{1}^{1}[t]&y_{1}^{2}[t]&y_{1}^{3}[t]&y_{1}^ {4}[t]\\ y_{2}^{2}[t]&y_{2}^{2}[t]&y_{2}^{3}[t]&y_{2}^{2}[t]\\ y_{3}^{3}[t]&y_{3}^{3}[t]&y_{3}^{3}[t]\end{bmatrix} ,\text{ with }t=1,2, \cdots,N_{1} \tag{25}\]  
In Equation (25), \(y_{1}[t]\), \(y_{2}[t]\), and \(y_{3}[t]\) are the experimental displacements of contractile PAM I, PAM II, and PAM III, respectively. Each column in matrix (25) represents experimentally collected outputs of contractile PAMs in the soft manipulator corresponding to every column input of Equation (24), and superscripts mark the number of columns. In order to highlight the influence of manipulator stiffness on the manipulator outputs, experiments are repeated under different inflation pressures \(p_{0}\), and \(p_{0}\) increases from 0 to 2.5 bar, with an increment of 0.1 bar. In order to sufficiently train the stiffness-dependent submodel, among the experimental data collected above, the data corresponding to pressure \(p_{0}\in[0\,\text{bar},0.2\,\text{bar},0.4\,\text{bar},0.6\,\text{bar},0.8\, \text{bar},1.0\,\text{bar},1.2\,\text{bar},1.4\,\text{bar},1.6\,\text{bar},1.8\, \text{bar},2.0\,\text{bar},2.2\,\text{bar},2.5\,\text{bar}]\) are selected as the training set. Meanwhile, data corresponding to pressure \(p_{0}\in[0.1\,\text{bar},0.5\,\text{bar},0.7\,\text{bar},1.1\,\text{bar},1.5\, \text{bar},1.7\,\text{bar},2.3\,\text{bar}]\) are selected as the validation set, and the rest of the samplings are utilized as the test set. Hence, 104,000 training samples in total are employed to train the stiffness-dependent submodule, and 56,000 samples and 48,000 samples act as the validation and test set, respectively. By providing the TSFNN with a diverse dataset that comprehensively covers the normal operating pressure ranges of PAMs, its generalization ability can be enhanced. Furthermore, through training and optimization with large-scale sample data, the computational efficiency and response speed of the TSFNN can be improved. During the training process of the stiffness-dependent submodule, the particle swarm optimization method is used for antecedent parameter identifications due to its excellent global and local search capacity, and the LS method is utilized to complete consequent parameter identifications to improve the training efficiency of the stiffness-dependent submodule.