## 2 Human-Machine Coupling Model
### Large Deflection Analysis of VSFH
According to the assembly angle \(\theta_{\text{a}}\), the original flexion angles of both hinges in the human-machine coupling model are defined as follows:  
\[\theta_{1,0}=\frac{\theta_{1,\text{f}}\theta_{\text{a}}}{\theta_{1,\text{f}}+ \theta_{2,\text{f}}} \tag{14}\]  
\[k_{i}=\sqrt{\frac{2F_{0}}{E_{i}I_{i}}[\cos(\phi_{i}-\theta_{i})-\cos(\phi_{i} -\theta)]+\left(\frac{M_{i}}{E_{i}I_{i}}\right)^{2}} \tag{9}\]0  
where \(\theta_{1,0}\) and \(\theta_{2,0}\) are the original flexion angles of the 1-th hinge and the 2-th hinge. Because the whole flexion angle of VSFH is equal to the assembly angle in the original state, the free end of VSFH is parallel to the direction of finger. Since VSFH cannot compensate for the elongation caused by the flexion of the joints, the length of the connection rope that is determined by the vertical distance from the free end of VSFH to the dorsal finger in the original state is crucial. In the coordinate system established in the model, the coordinates of the free end of VSFH (\(x_{2,0},y_{2,0}\)) are defined as follows:  
\[\left[ \begin{array}{c}x_{2,0}\\ y_{2,0}\\ 0\end{array} \right]=\left[ \begin{array}{cc}\cos\theta_{1,0}&-\sin\theta_{1, 0}&a_{1,0}\\ \sin\theta_{1,0}&\cos\theta_{1,0}&b_{1,0}\\ 0&0&0\end{array} \right]\left[ \begin{array}{c}a_{2,0}\\ b_{2,0}\\ 1\end{array} \right] \tag{16}\]  
According to the position of this point (\(x_{2,0},y_{2,0}\)), the length of the connection rope \(l_{\text{r}  
Based on the human-machine coupling model, in addition to optimizing the stiffness relationship between each hinge in VSFH, two additional conditionals should be met to improve the interactive performance between the hand exoskeleton and finger. Condition 1 is that \(l_{\text{r}}\) should be longer than \(\Delta L\) to avoid squeezing caused by inadequate compensation. \[\text{Condition 1}:\ l_{\text{r}}\geq\sum_{i=1}^{3}\Delta L_{i} \tag{19}\]  
Condition 2 is that the connection point between the connection rope and the finger should be on the distal interphalangeal (DP) to provide stretching force for each joint and big moment arm. \[\text{Condition 2}:\ l_{\text{j}}+L_{1}+L_{2}\leq d_{\text{r}}\leq l_{\text{j}}+ L_{1}+L_{2}+L_{3} \tag{20}\]  
MATLAB is needed to solve the integrated model to calculate the theoretical value of the assembly angle and the thickness of each hinge. In the process of solving the model, the lengths of each hinge are firstly decided through the (10) and (11). The assembly angle can be empirically set to a range of values and the original flexion angle of each hinge can be known through (12)-(15). The stretching force is set as an independent variable and the thicknesses needed for each hinge to reach the same angle under different forces are calculated through (5)-(9) to understand the relationship between the deformation of VSFH and stretching force. Finally, the position of free end of VSFH can be obtained through (16)-(18). The material of VSFH is polylactic acid (E = 2119 MPa). The cross-section of each hinge is rectangular so that I can be obtained by calculating the moment of inertia of the rectangular section. [PERSON] et al. [28] measured the functional range of motion (ROM) of the finger joints, which is 19\({}^{\circ}\)-71\({}^{\circ}\), 23\({}^{\circ}\)-87\({}^{\circ}\), and 10\({}^{\circ}\)-64\({}^{\circ}\) at MCP, PIP, and DIP joints, respectively. Meanwhile, [PERSON] et al. [24] proposed a hand exoskeleton that can provide passive stretching force through the linear spring, and the proportions of flexion angles are approximately 1:2:1.5 for the MCP, PIP, and DIP joints, respectively. Therefore, our expected joint angle of each joint and the size of the index finger of a healthy subject are listed in Table 1.  
Through the MATLAB solution, Figure 5 shows the position of the free end of VSFH under the action of different assembly angles and stretching forces. As the assembly angle increases, the length of the connection rope increases, resulting in the free end of VSFH being further from the fingertip.