# Gesture Recognition of Sign Language Alphabet Using a Magnetic Positioning System
## 2 Materials and Methods
### Reconstruction of the Hand Gesture
For the angle \(a_{\text{MCP}}^{\text{I}}\), the following relation holds (a modification of an analogous relation in ([PERSON] and [PERSON], 2018)):  
Figure 8: (**a**) A picture of the magnetic glove. Transmitting nodes are numbered. The PSoC used to supply nodes 1 and 2, together with its own battery, is attached to the glove on the back of the hand. Custom PCBs are attached to the phalanges. (**b**) The complete hand reconstruction resulting from the kinematic model. CMC–carpometacarpol joints; MCP—metacarpophalangeal joints; PIP—proximal interphalangeal joints; DIP—distal interphalangeal joints; and IP—interphalangeal joint. Lengths of the phalanges are indicated with \(L\). \[\alpha_{\text{MCP}}^{\text{I}}\equiv\alpha\left(\theta_{\text{CMC,a}}^{\text{I}} \right)=\alpha_{0}+\frac{\pi}{18}\frac{\theta_{\text{CMC,a}}^{\text{I}}-\theta _{\text{CMC,a, min}}^{\text{I}}}{\theta_{\text{CMC,a, max}}^{\text{I}}-\theta_{\text{CMC,a, min}}^{\text{I}}}, \tag{3}\]  
where \(\alpha_{0}\) is the twist angle of the thumb measured when it is fully extended during the calibration phase. Minimum and maximum values \(\theta_{\text{CMC,a, min}}\) and \(\theta_{\text{CMC,a, max}}\), are reported in Table 3.  
In order to determine the length of the phalanges for each finger, during the calibration procedure, the distances between nodes 3-7 and node 1 are measured, then all the lengths \(L_{j}^{i}\) are obtained through a set of proportionality coefficients determined from average anthropometric data [43]. The detailed procedure is illustrated in [38], where nodes 3-7 were considered to be mounted on the middle of the distal phalanx of each finger. In this work, node 3 is positioned approximately at 2/3 of the distal phalanx of the thumb, while nodes 4-7 are positioned at 3/4 of the distal phalanges. Hence, the calibration described in [38] has been performed using an updated set of proportionality coefficients reported in Table 4. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Joint** & \(\mathbf{a}\) & \(\mathbf{d}\) & \(\mathbf{\theta}\) & \(\mathbf{\alpha}\) \\ \hline
5 & DIP & \(L_{3}^{i}\) & 0 & \(\theta_{\text{DIP,f}}^{i}\) & 0 \\
4 & PIP & \(L_{2}^{i}\) & 0 & \(\theta_{\text{IP,f}}^{i}\) & 0 \\
3 & MCP,f & 0 & 0 & \(\theta_{\text{MCP,f}}^{i}\) & \(\pi/2\) \\
2 & MCP,a & \(L_{1}^{i}\) & 0 & \(\theta_{\text{MCP,a}}^{i}\) & \(-\pi/2\) \\
1 & CMC,f & 0 & 0 & \(\theta_{\text{CMC,f}}^{i}\) & \(\pi/2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: MDH parameters for fingers II–V. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Thumb** & **Index** & **Middle** & **Ring** & **Pinkie** \\ \hline \(\theta_{\text{CMC,a}}\) & \([0^{\circ},125^{\circ}]\) & N.D. & N.D. & N.D. & N.D. \\ \(\theta_{\text{CMC,f}}\) & \([0^{\circ},70^{\circ}]\) & \([0^{\circ},5^{\circ}]\) & 0 & \([0^{\circ},10^{\circ}]\) & \([0^{\circ},15^{\circ}]\) \\ \(\theta_{\text{MCP,a}}\) & N.D. & \([-30^{\circ},30^{\circ}]\) & \([-22^{\circ},22^{\circ}]\) & \([-22^{\circ},22^{\circ}]\) & \([-25^{\circ},25^{\circ}]\) \\ \(\theta_{\text{MCP,f}}\) & \([0^{\circ},60^{\circ}]\) & \([-40^{\circ},90^{\circ}]\) & \([-40^{\circ},90^{\circ}]\) & \([-40^{\circ},95^{\circ}]\) & \([-40^{\circ},95^{\circ}]\) \\ \(\theta_{\text{PIP,f}}\) & N.D. & \([0^{\circ},120^{\circ}]\) & \([0^{\circ},110^{\circ}]\) & \([0^{\circ},120^{\circ}]\) & \([0^{\circ},135^{\circ}]\) \\ \(\theta_{\text{DIP,f}}\) & \([-10^{\circ},90^{\circ}]\) & \([-5^{\circ},80^{\circ}]\) & \([-5^{\circ},80^{\circ}]\) & \([-5^{\circ},80^{\circ}]\) & \([-5^{\circ},90^{\circ}]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Lower and upper bound of angles \(\theta\).