## 5 Experimental Results and Discussion
### Implementation of the Decoupling Inverse Compensator SDIHC  
According to Figure 8, the implementation of the SDIHC also requires three parts: the inverse stiffness-dependent submodule identification, decoupling submodule identification, and the inverse hysteresis submodule identification. Part 1: The inverse stiffness-dependent submodel is identified. The network structure of the \(g^{-1}(\cdot)\) depicted in the Figure 8 is identical to the TSFNN shown in Figure 6. To identify the inverse stiffness-dependent submodule, inputs and outputs of \(S=\{\textbf{{INPUT}},\textbf{{OUTPUT}}\}\) are taken as the corresponding outputs and inputs of \(g^{-1}(\cdot)\). During training of the inverse stiffness-dependent submodule, uniform optimization methods and training parameters like the iteration number and the population size of particles are employed for training the stiffness-dependent submodule. Part 2: The decoupling submodule is identified. According to the identification of the actuator coupling submodule, parameters in the decoupling submodule have been confirmed. We adopted the Jacobi iterative method [40] to solve the linear simultaneous Equation (15) for programming and computation simplification. Thus, the following iterative formula is used, and \(m\) represents the following iterations:  
\[\mathbf{v}_{d}^{(m+1)}=\mathbf{B}_{I}\mathbf{v}_{d}^{(m)}+\mathbf{D}^{-1} \mathbf{b} \tag{28}\]  
Figure 11: RMSE of the SDCHM and TSFNN model corresponding to different pressure in the extensor PAM. (**a**) \(p_{0}=0.1\) bar; (**b**) \(p_{0}=1.5\) bar; (**c**) \(p_{0}=1.7\) bar. with  
\[\left\{ \begin{aligned} \mathbf{v}_{d}=[v_{1d},v_{2d},v_{3d}],\\ \mathbf{B}_{j}=\mathbf{D}^{-1}(\mathbf{D}-\mathbf{A}),\\ \mathbf{D}=\text{diag}(G_{11},G_{22},G_{33}),\\ \mathbf{A}=\begin{bmatrix}G_{11}&G_{21}&G_{31}\\ G_{12}&G_{22}&G_{32}\\ G_{13}&G_{23}&G_{33}\\ \mathbf{b}=[h_{1d},h_{2d},h_{3d}].\end{bmatrix}\end{aligned} \right. \tag{29}\]  
When the error of two adjacent iteration results \(\left|v_{id}^{(m+1)}-v_{id}^{(m)}\right|\leq 0.01,\ (i=1,2,3)\), the iteration is suspended, and outputs of the decoupling submodel are finally confirmed. **Proof of Convergence for Equation (28). ** The Jacobi iterative method declared that, assuming the Jacobi iterative matrix is \(\|\mathbf{B}_{j}\|<1\), the Jacobi iterative method converges to any initial value. According to the identified actuator coupling submodule,  
\[\left\{ \begin{aligned} \mathbf{D}=\text{diag}(1,1,1),\\ \mathbf{A}=[1&0.1521&0.5368;0.1124&1&0.6321;0.0356&0.2153&1].\end{aligned} \right.\end{aligned}\right. \tag{30}\]  
Thus, \(\mathbf{B}_{j}=\mathbf{D}^{-1}(\mathbf{D}-\mathbf{A})=[0&-0.1521&-0.5368;-0.1124&0&-0.6321;\\ -0.0356&-0.2153&0]\) and \(\|\mathbf{B}_{j}\|=-0.016<1\). The Jacobi iterative method is convergent for Equation (27). \(\square\)  
Part 3: The inverse hysteresis submodule is identified. The GPI model has an analytical inverse solution, and based on the determined parameters of the SDCHM model in the above section, the inverse hysteresis submodule can be completed following Equations (15)-(20). Figure 12 illustrates the relationship between desired outputs and compensation results with the SDIHC and its control groups. In the first line of Figure 12, the desired outputs of \(y_{1d}=6\sin(0.04\pi t)+33\) mm, \(y_{2d}=y_{3d}=33\) mm with \(p_{0}=0.7\) bar are tracked. In the second line of Figure 12, the desired outputs of \(y_{2d}=6\sin(0.04\pi t)+33\) mm, \(y_{1d}=y_{3d}=33\) mm with \(p_{0}=1.5\) bar are tracked; and in the last line, the desired outputs of \(y_{3d}=6\sin(0.04\pi t)+33\) mm, \(y_{1d}=y_{2d}=33\) mm with \(p_{0}=2.3\) bar are tracked. Yellow lines in Figure 12 indicate the SDIHC compensator results. As control groups, experiments of individual hysteresis compensator without the decoupling and inverse stiffness-dependent submodules are also conducted, and these are represented by blue lines. Tangerine lines denote the ideal linearity between the desired outputs and reference inputs, and black lines show PID control results.