## 2 Human-Machine Coupling Model
### Large Deflection Analysis of VSFH  
Figure 3 shows a VSFH subjected to a combined end force and moment. The model is composed of \(n\) hinges with different stiffness connected by \(n-1\) rigid connecting blocks. The hinge with force and moment, in the end, is the \(n\)-th hinge. The length of the rigid connecting block along the direction \(\theta_{i}\) is denoted by \(w_{i}\) (\(w_{i}\geq 0\)). The moment at the end of the \(i\)-th hinge (I \(\leq n-1\)) can be calculated by the deflection of VSFH as follows:  
\[M_{i}=\sum\limits_{x=i+1}^{n}\{F_{0}[(a_{x}+w_{x-1})\sin\phi_{x}-b_{x}\cos \phi_{x}]\}+M_{0} \tag{8}\]  
Plugging (8) into (4) and rearranging terms, the curvature of the corresponding hinge can be calculated by  
\[k_{i}=\sqrt{\frac{2F_{0}}{E_{i}I_{i}}[\cos(\phi_{i}-\theta_{i})-\cos(\phi_{i} -\theta)]+\left(\frac{M_{i}}{E_{i}I_{i}}\right)^{2}} \tag{9}\]  
The curvature of the \(n\)-th hinge can be calculated according to (4). Figure 2: Large deflection of a cantilever beam subjected to a combined end force and moment. This section presents a simplified human-machine coupling model of a single finger and VSFH in three different flexion states (see Figure 4). The established model can improve the flexion synchronization between VSFH and finger, to avoid interference by optimizing the stiffness relationship between each hinge in VSFH. Meanwhile, the human-machine coupling model is used to accurately calculate the stiffness of each hinge to provide personalized passive stretching force for wearers. Since the width difference of human fingers is small, the hinge width \(b_{\text{h}}\) = 7 mm, and the thickness of the hinge is changed to obtain different stiffness. The length of the connection rope that connects VSFH to finger \(l_{\text{r}}\) is used to compensate \(\Delta L\) and optimize the assembly of VSFH. In the model, VSFH is composed of two hinges with different stiffness. Here, we choose the length of connecting block \(w_{1}\) = 0 to improve the working life of the flexible hinge. The 1-th and 2-th hinges mimic the flexion and extension of MCP and PIP joints,  
Figure 4: Human-machine coupling model based on human finger and V.  
Figure 3: Large deflection of a VSFH subjected to a combined end force and moment. respectively. The \(x\)-axis is aligned to the direction of the VSFH in the unstressed state and the center of the coordinate is placed at the fixed end of VSFH. In the unstressed state, VSFH which is treated as a straight cantilever beam is assembled at angle \(\theta_{\text{a}}\) (the angle between the VSFH and the horizontal line) on the fixed block of the dorsal palm. According to the measurement of finger phalanges, the required lengths of each hinge is defined as follows:  
\[l_{1}=l_{\text{j}}+L_{1} \tag{10}\]  
\[l_{2}=L_{2}+L_{3} \tag{11}\]  
where \(l_{\text{j}}\) is the horizontal distance from the fixed end of VSFH to the center of MCP joint. In the final state, the flexion angles of each joint and hinge reach the maximum. Furthermore, the flexion angles of each hinge correspond to that of finger joints, so that VSFH and finger can be flexed in equal proportion to achieve synchronous motion. The final flexion angles of both hinges in the human-machine coupling model are defined as follows:  
\[\theta_{1,\text{f}}=\theta_{\text{a}}+\varphi_{1} \tag{12}\]  
\[\theta_{2,\text{f}}=\varphi_{2} \tag{13}\]  
where \(\theta_{1,\text{f}}\) and \(\theta_{2,\text{f}}\) are the final flexion angles of the 1-th hinge and the 2-th hinge. In the original state, we assume that the finger is straight under the action of the stretching force generated by the deformation of VSFH, so that \(\varphi_{1}=\varphi_{2}=\varphi_{3}=0\). Meanwhile, the connection rope connects the free end of VSFH perpendicular to the finger.