## 5 Experimental Results and Discussion
### SDCHM Model Identification and Validation  
Since the SDCHM includes three components, identification procedures for the inherent hysteresis submodule, actuator coupling submodule, and stiffness-dependent submodule are conducted in three steps in order. In the first step, outputs of the actuator coupling submodule and stiffness-dependent submodule are set to 1. Parameters including \(x_{i}^{k}(r_{i}^{k})\), \([a_{1}^{k},a_{2}^{k},a_{3}^{k}]\), and \([b_{1}^{k},b_{2}^{k},b_{3}^{k}]\) in three GPI models are identified. Since the GPI is utilized to describe the inherent hysteresis behavior of the single PAM, pressure-displacement experimental data of each contractile PAM used for identification are collected separately. Taking the identification of GPI\({}_{1}\) for instance, the input pressure signal for the manipulator is set as Equations (21) and (22):  
\[\left\{ \begin{array}{ll}p_{1}(t)=\frac{A_{s}(t)}{2}\sin(0.04\pi t)+\frac{A_ {s}(t)}{2},\\ p_{2}(t)=p_{3}(t)=p_{0}(t)=0.\end{array} \right. \tag{21}\]  
and  
\[A_{s}(t)= \begin{cases}1.0\text{ bar},&\quad t\in[0s,50s),\\ 1.5\text{ bar},&\quad t\in[51s,100s),\\ 2.0\text{ bar},&\quad t\in[101s,150s),\\ 2.5\text{ bar},&\quad t\in[151s,200s).\end{cases}  \tag{22}\]  
Input signal \(p_{1}(t)\) and corresponding output \(y_{1}(t)\) of the PAM I, experimentally measured with the experimental setup in Figure 2 with the sampling frequency of 50 Hz, are adopted to identify parameters in GPI\({}_{1}\), and they can be found using least square (LS) optimization. The number of generalized play operators is chosen to be \(N=10\), which decides the model's accuracy and complexity. Table 2 reflects the influence of \(N\) on model accuracy and complexity, where \(Err=\left(\sum\limits_{t=1}^{H}\lvert y_{1}(t)-\hat{y}_{1}(t)\rvert/\sum\limits_{t= 1}^{H}\lvert y_{1}(t)-y_{1,mean}(t)\rvert\right)\times 100\%\), and \(y_{1,mean}\) computes the mean value of \(y_{1}(t)\). This indicates that as \(N\) increases, the \(Err\) value decreases, and it tends to saturate when \(N\geq 10\). By repeating the above process, other parameters to be identified in GPI\({}_{2}\) and GPI\({}_{3}\) can be found. The final identification results of the inherent hysteresis model are presented in Figure 9.  
In the second step, parameter identifications of the actuator coupling submodule focus on coupling coefficients \(G_{ij}(i=1,2,3;j=1,2,3)\). The stiffness-dependent submodule is still set to 1. \(G_{11}=G_{22}=G_{33}=1\) are set to consider the cascade connection between GPI\({}_{i}\) and \(G_{ii}\), (\(i=1\), 2, 3). \(G_{ij}(i\
eq j)\) stands for the effect of the PAM \(i\) on the PAM \(j\). Identifications for each parameter \(G_{ij}(i\
eq j)\) are the same, so we only state the identification procedure of parameter \(G_{12}\) in detail. First, we apply the sinusoidal signal in Equation (21) into the identified model GPI\({}_{1}\) to obtain the output \(\
u_{1}=\text{GPI}_{1}[p_{1}](t)\); then, \(G_{12}\) can be determined by the LS method to obtain the minimum result of the objective function \(\min\)\(\left\{\left[\mathbf{C}_{21}G_{21}-\mathbf{d}_{12}\right]^{T}\left[\mathbf{C}_{ 21}G_{21}-\mathbf{d}_{12}\right]\right\}\), in which \(\mathbf{C}_{21}=[\
u_{1}[1],\
u_{1}[2],\cdots,\
u_{1}[N_{c}]]\), \(\mathbf{d}_{12}=[y_{2}[1],y_{2}[2],\cdots,y_{2}[N_{c}]]\). \(N_{c}\) indicates the number of experimental data involved in optimization and relies on signal period and sampling frequency. All optimized coefficients \(G_{ij}(i=1,2,3;j=1,2,3)\) are listed in Table 3.  
In the third step, identification of the stiffness-dependent submodule is, in fact, intended to train the TSFNN in Figure 7. The stiffness-dependent submodule is employed to model the output stiffness dependence of the pneumatic soft manipulator and other \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(G_{ij}\) & \(i=1\) & \(i=2\) & \(i=3\) \\ \hline \(j=1\) & 1 & 0.1521 & 0.5368 \\ \(j=2\) & 0.1124 & 1 & 0.6321 \\ \(j=3\) & 0.0356 & 0.2153 & 1 \\ \hline \hline \end{tabular}
\end{table}