# Development of a Wearable Finger Exoskeleton for Rehabilitation
## 3 Exoskeleton Hand Analysis  
### Position Analysis  
In several mechanisms, position analysis has the goal to determine the position of all the links in different phases of an operative cycle [39].  
The position analysis of exoskeleton components is developed using an analytic method through a geometric approach. In Figure 3a, a schematic diagram is shown, where  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Link/Phalanx** & **Joint** & & **Length (m)** & **Range Movement** \\ \hline \multirow{4}{*}{Proximal L1} & & Index & 0.0238 & \\  & & Middle & 0.0254 & \\  & & Ring & 0.0220 & 26\({}^{\circ}\) \\  & & Little & 0.0181 & \\  & & Thumb & 0.0288 & \\ \hline \multirow{4}{*}{Media L2} & & Index & 0.0187 & \\  & & Middle & 0.0224 & \\  & & Ring & 0.0212 & 43\({}^{\circ}\) \\  & & Little & 0.0163 & \\  & & Thumb & N/ A & \\ \hline \multirow{4}{*}{Distal L3} & & Index & 0.0224 & \\  & & Middle & 0.0237 & \\ \cline{1-1}  & & Ring & 0.0214 & 43\({}^{\circ}\) \\ \cline{1-1}  & & Little & 0.0202 & \\ \cline{1-1}  & & Thumb & 0.0289 & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Length and range of movements of the hand phalanges [16].  
Figure 2: (**a**) Proposed exoskeleton (general view) and (**b**) exploded view of the single DOF.  
the lengths of the exoskeleton links are known. \(L_{1}\) represents the element that joins the linear actuator with element \(L_{2}\). The length of \(L_{1}\) is variable according to the linear displacement of the actuator; \(L_{2}\) represents the section of the mechanism that moves the proximal phalanx; \(L_{3}\) is the element that allows the angular displacement of link \(L_{4}\)-\(L_{5}\); \(L_{4}\)-\(L_{5}\) is the element that moves to link \(L_{8}\); \(L_{9}\) is the element linked to link \(L_{4}\)-\(L_{5}\) with link \(L_{8}\); \(L_{8}\) represents the section of the mechanism that moves the middle phalanx; \(F_{1}\) is the force applied by the linear actuator to move the mechanism. Figure 3b, represents the final position achieved after the linear actuator reaches its final position.  
To solve for \(\theta_{2}\), from Figure 3a, the initial position of element \(L_{2}\) is obtained, using the values of \(L_{2}\), \(L_{1}\), \(D_{2}\), and \(d_{1}\) the input data are:  
\[R_{1}=\sqrt{d_{1}^{2}-D_{2}^{2}} \tag{1}\]  
\[\sin\theta_{2}=\frac{d_{1}}{R_{1}} \tag{2}\]  
\[cos\theta_{2}=\sqrt{1-\left(\sin\theta_{2}\right)^{2}} \tag{3}\]  
\[\theta_{2}=atan2\left(\frac{d_{1}}{\sqrt{d_{1}^{2}+D_{2}^{2}}},\sqrt{1-\frac {d_{1}^{2}}{d_{1}^{2}+D_{2}^{2}}}\right) \tag{4}\]  
Figure 3: (**a**) Initial position for proposed mechanism from a lateral view and (**b**) analysis of the final position of the mechanism.  
where Equation (16) describes the angular displacement of element \(L_{3}\). Using \(R_{5}\), \(L_{3}\) and \(L_{4}\) the angular displacement for the link \(L_{3}\) and \(L_{4}\) can be obtained as:  
\[\theta_{21}=atan2\left(\sqrt{1-\left(\frac{-L_{3}^{2}+R_{5}^{2}+L_{4}^{2}}{2R_{5 }L_{4}}\right)^{2}},\frac{-L_{3}^{2}+R_{5}^{2}+L_{4}^{2}}{2R_{5}L_{4}}\right) \tag{19}\]  
where \(\theta_{21}\) is the angle between segments \(C_{2}C_{3}\) and \(C_{3}C_{5}\). Using the triangle formed by elements \(R_{8}\), \(d_{4}\), and \(L_{5}\), the angular position for links \(L_{8}\) and \(L_{9}\) can be expressed by:  
\[R_{7}=\sqrt{L_{7}^{2}+R_{8}^{2}-2L_{7}R_{8}\cos(90-\theta_{18})} \tag{20}\]  
\[R_{8}=\sqrt{L_{5}^{2}+d_{4}^{2}-2L_{5}d_{4}\cos(180-\theta_{14}-\theta_{16}- \theta_{17})} \tag{21}\]  
\[\theta_{19}=atan2\left(\sqrt{1-\left(\frac{-R_{8}^{2}+R_{7}^{2}+L_{7}^{2}}{2R _{7}L_{7}}\right)^{2}},\frac{-R_{8}^{2}+R_{7}^{2}+L_{7}^{2}}{2R_{7}L_{7}}\right) \tag{22}\]  
\[\theta_{20}=atan2\left(\sqrt{1-\left(\frac{-L_{9}^{2}+L_{8}^{2}+R_{7}^{2}}{2R _{7}L_{8}}\right)^{2}},\frac{-L_{9}^{2}+L_{8}^{2}+R_{7}^{2}}{2R_{7}L_{8}}\right) \tag{23}\]  
\[\theta_{22}=\theta_{19}+\theta_{20} \tag{24}\]