## 3 Modeling of Coupled Hysteresis
### Stiffness-Dependent Submodule  
Neural networks have strong nonlinear mapping abilities to comprehensively and accurately describe highly nonlinear systems. In this paper, the stiffness-dependent submodule \(g(\cdot)\) is fitted by the T-S fuzzy neural network (TSFNN) that has good generalization capacity for modeling the nonlinear systems. The structure of \(g(\cdot)\) is shown in the following Figure 7.  
This structure of \(g(\cdot)\) in Figure 7 is referred to the TSFNN structure [39] based on if-then rules. The fuzzification process is achieved in the first layer. Each input element corresponds to three fuzzy variables, and membership relations are represented by the Gaussian membership function calculated in Equation (9):  
\[\begin{cases}L_{ij}^{1}(t)=\mu_{ij}(t)=\exp\left\{-\left(I_{i}(t)-m_{ij}\right) ^{2}/2\sigma_{ij}^{2}\right),\;(i=1,2,\ldots,n;\;j=1,2,3),\\ I_{i}(t)\in\{h_{1}(t),h_{2}(t),h_{3}(t),p_{0}(t)\}.\end{cases} \tag{9}\]where input elements \(I_{1}(t),\ I_{2}(t),\cdots,I_{n}(t)\) are composed of \(\{h_{1}(t),h_{2}(t),h_{3}(t),p_{0}(t)\}\), and there are \(n=4\) input nodes in total. Figure 7 shows that every input matches with three membership functions \(\mu_{ij}(i=1,2,\ldots,n;j=1,2,3)\). \(L^{1}_{ij}(t)=\mu_{ij}(t)\) shows outputs of the first layer. \(\sigma_{ij}\) and \(m_{ij}\) are antecedent parameters representing the width and center of Gaussian function, respectively. There are \(N_{1}=3n\) nodes in the first layer. The complexity of the stiffness-dependent submodule is mainly determined by the number of fuzzy variables. Fewer fuzzy variables will reduce model accuracy, but more fuzzy variables will reduce the real-time performance of the model, and more training samples will be required. The output \(L^{2}_{d}(t)\) of the second layer is calculated as shown in Equation (10):  
\[L^{2}_{d}(t)=\prod_{i=1}^{n}\mu_{ij}(t),\ (j\in\{1,\ 2,\ 3\};d=1,\ 2,\ \cdots,N_{2}) \tag{10}\]  
Then, \(L^{2}_{d}(t)\) is normalized in the third layer with the output \(L^{3}_{d}(t)\):  
\[L^{3}_{d}(t)=\overline{\omega}_{h}=L^{2}_{d}(t)/\sum_{d=1}^{3^{n}}L^{2}_{d}(t ),\ (d=1,\ 2,\ \cdots,N_{3}) \tag{11}\]  
The node number in the third layer is equal to that in the second layer, meaning that \(N_{2}=N_{3}=3^{n}\).  
The fourth layer and the fifth layer in this paper constitute the consequent network part with three parallel subnetwork outputs corresponding to estimated length variations of different contractile PAMs. Each node in the consequent network part carries out a selection of inputs of this layer according to the if-then fuzzy logic rule. Thus, each subnetwork has the same number of nodes as the third layer (\(N_{4}=N_{3}\)). As shown in Equation (12), \(L^{4}_{dk}(t)\) (\(d=1,\ 2,\ \cdots,\ N_{4}\)) defines the output of the kth subnetwork in the fourth layer, where \(p^{k}_{di}(t)\) represents the consequent parameter and \(k\) indicates different subnetworks, respectively.  
\[L^{4}_{dk}(t)=\overline{\omega}_{h}\sum_{i=1}^{n}p^{k}_{di}A_{i}(t),\ (d=1,\ 2,\ \cdots,\ N_{4};\ k=1,\ 2,\ 3) \tag{12}\]  
Figure 7: Structure of the stiffness-dependent submodule.  
The final outputs \(L_{k}^{5}(t)\) of the SDCHM are obtained in the fifth layer with the calculation shown in Equation (13):  
\[L_{k}^{5}(t)=\hat{y}_{k}(t)=\sum_{d=1}^{N_{4}}L_{d}^{4}(t),\ (k=1,2,3) \tag{13}\]  
In the above stiffness-dependent submodule \(g(\cdot)\), \(m_{ij},\sigma_{ij}\), and \(p_{di}^{k}\) should be identified by experimental data.