## 5 Experimental Results and Discussion
### SDCHM Model Identification and Validation
The loss function is shown in Equation (26):  
\[loss=\frac{\sum\limits_{t=1}^{N_{1}}\left[\left(y_{1}[t]-\hat{y}_{1}[t] \right)^{2}+\left(y_{2}[t]-\hat{y}_{2}[t]\right)^{2}+\left(y_{3}[t]-\hat{y}_{3 }[t]\right)^{2}\right]}{N_{1}} \tag{26}\]where \(g_{1}[t],g_{2}[t]\), and \(g_{3}[t]\) are model results regarding the length variations of PAM I, PAM II, and PAM III. Based on the above analysis, Algorithm 1 gives the identification process of the proposed SDCHM. The neural network training and experiments discussed in this paper are all conducted on a computer equipped with a 3.40 GHz \(\times\) 64 Intel Core i7 processor, 24 GB of RAM, and a GeForce RTX 3050 GPU. ```
Inputs: The training datasets of pressure and displacements of the soft manipulator Outputs: The displacement outputs of the manipulator from the SDCHM model Set: number of generalized play operators \(N=10\); maximum iterations \(ite=50\); the population size of particles \(L=20\); learning factors \(c_{1}=c_{2}=0.5\); Initialize parameters: Parameters in the inherent hysteresis submodule such as \(x^{k}(r_{i}^{k})=0,\ (i=1,2,\cdots,N;k=1,2,3)\), \([a_{1}^{k},a_{2}^{k},a_{3}^{k}]=[0,0,0],\ (k=1,2,3)\), \([b_{1}^{k},b_{2}^{k},b_{3}^{k}]=[0,0,0],\ (k=1,2,3)\); parameters in the actuator coupling submodule such as \(G_{ij}=1,\ (i=1,2,3;j=1,2,3)\); parameters in the stiffness-dependent submodule such as \(m_{ij}\sim N(0,1)\), \(\sigma_{ij}\sim N(0,1)\), and \(p_{di}^{k}\sim N(0,1)\); 1: Optimize parameters of \(x^{k}(r_{i}^{k})\), \([a_{1}^{k},a_{2}^{k},a_{3}^{k}]\), and \([b_{1}^{k},b_{2}^{k},b_{3}^{k}]\) in the inherent hysteresis submodule by LS method with input and output data of each contractile PAM in the soft manipulator, and outputs of actuator coupling submodule and stiffness-dependent submodule are set as 1. 2: Obtain the identified hystereesis submodule output and input into the actuator coupling submodule;
3: Calculate the actuator coupling submodule output according to Equation (8);
4: Optimize parameters of \(G_{ij},\ (i=1,2,3;j=1,2,3)\) in the actuator coupling submodule by LS method with objective function \(\min\left\{[C_{ij}G_{ij}-d_{ij}]^{T}[C_{ij}G_{ij}-d_{ij}]\right\}\), and output of the stiffness-dependent submodule is set as 1;
5: Obtain the identified actuator coupling submodule output and input into the stiffness-dependent submodule;
6: While \(i\leq ite\), \(i\gets i+1\) Calculate the stiffness-dependent submodule output according to Equations (9)-(13);  Compute loss function \(loss=\frac{\sum\limits_{1}^{N_{1}}{[(y_{1}[t]-g_{1}[t])^{2}+(y_{2}[t]-g_{2}[ t])^{2}+(y_{3}[t]-g_{3}[t])^{2}]}}{N_{1}}\);  Optimize the consequent parameters by the LS method;  Compute loss function again \(loss=\frac{\sum\limits_{1}^{N_{1}}{[(y_{1}[t]-g_{1}[t])^{2}+(y_{2}[t]-g_{2}[ t])^{2}+(y_{3}[t]-g_{3}[t])^{2}]}}{N_{1}}\);  Optimize the antecedent parameters by the particle swarm optimization method;  Update the parameters. End while Return training results of the SDCHM model
```  
**Algorithm 1**Training process of the SDCHM model  
To validate the identified SDCHM model, the stiffness-dependent and coupled hysteresis outputs of the pneumatic manipulator corresponding to pressure in the contractile PAM \(p^{*}=0.7\sin(0.04\pi t)+0.7\) bar and \(p_{0}=0.1,\ 1.5,\ 1.7\) bar are predicted, respectively. To further evaluate the performance of the SDCHM, the results of the TSFNN model whose structure is the same to that of the stiffness-dependent submodule are employed as control groups. For example, Figure 10 displays the comparisons among the proposed model results, TSFNN model results, and the experimental data with \(p_{0}=0.1\) bar. Figures in the first line of Figure 10 are the manipulator outputs corresponding to pressures of \([p_{1},\ p_{2},\ p_{3}]=[p^{*},\ 0,\ 0]\), and the figures in the middle two rows of Figure 10 are manipulator outputs under inputs of \([p_{1},\ p_{2},\ p_{3}]=[0,\,p^{*},\ 0]\) and \([p_{1},\ p_{2},\ p_{3}]=[0,\,0,\,p^{*}]\), respectively.