## 2 Materials and Methods
### Finger Real Motion
#### 2.2.2 Results of the Analysis of the Fingers Real Motion
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**New Set of Variables** & **Eigenvalues** & **\% of the Total Variance** & **\% of the Cumulative Variance** \\ \hline PC1 & 2444.9 & 82.0531 & 82.05 \\ \hline PC2 & 516.5 & 17.3345 & 99.39 \\ \hline PC3 & 18.2 & 0.6125 & 100.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Eigenvalues and percentage of the variance of each principal component. Aside from the trajectories, we computed the range of motion of every finger. We defined the \(\mathbf{u_{f_{i}}}\) vector in the finger plane, where \(f\epsilon\{I,F,R,S\}\) corresponds to index (\(I\)), Middle (\(M\)), Ring (\(R\)), and Small (\(S\)), and the index \(i\) corresponds to the frame number from 1 (initial position) to \(N\) (last position). The computed vector corresponds to the subtraction between the \(xz\) position of the tip marker (I1, M1, R1, S1) and the \(xz\) position of the MCP marker (I4, M4, R4, S4), as defined in Equation (9). \[ \begin{array}{ccc}\text{Finger}&\text{Vector}\\ \text{Index}&\mathbf{u_{I_{i}}}=\text{I1}-\text{I4},\\ \text{Middle}&\mathbf{u_{M_{i}}}=\text{M1}-\text{M4},\\ \text{Ring}&\mathbf{u_{R_{i}}}=\text{R1}-\text{R4},\\ \text{Small}&\mathbf{u_{P_{i}}}=\text{S1}-\text{S4}.\end{array}  \tag{9}\]  
Then, we defined \(\hat{\mathbf{u}_{f_{i}}}\) as the normal vector of \(\mathbf{u_{f_{i}}}\) as Equation (10) shows. The vector \(\mathbf{w_{f}}\) represents the dot product between the initial position and the \(i^{th}\) position, in order to compute the angle between these normal vectors, which is expressed in Equation (11). \[\hat{\mathbf{u}_{f}}=\frac{\mathbf{u_{f}}}{\left|\mathbf{u_{f}}\right|}\,\mathbf{w_{f}}=\left[ \begin{array}{cc}\sum\limits_{j=1}^{j=2}v_{f1}\cdot v_{fj}\dots&\sum\limits_ {j=1}^{j=K}v_{f1}\cdot v_{fj}\dots&\sum\limits_{j=1}^{j=N}v_{f1}\cdot v_{fj} \end{array} \right], \tag{10}\]  
\[\gamma(i)\ =arc\cos\Big{(}\mathbf{w_{f}}\Big{)}. \tag{11}\]  
Figure 15 shows the trajectories of one subject from the index to small in the \(xz\) plane and the maximum and minimum angle of each finger's movement. The maximum angle for the small finger was chosen as \(\gamma_{max}\cong 76^{\circ}\) because several points were outside the trajectory and represented false measurements (Figure 13). As can be observed from Figure 14, the trajectories for the different fingers were similar in shape but with different maximum angle (\(\gamma_{max}\)) amplitudes. Table 4 shows the values of the angles of movement among the volunteers after several tests. Figure 14: Real trajectories of flexion–extension. To conclude this section on real finger motion. Figure 15a simultaneously illustrates the original prototype trajectory and index finger trajectory from one subject, which was chosen as the recorded real trajectory. The real finger trajectory did not have a regular elliptical shape, and it seems to have a more closed shape. In contrast, when the end-effector's prototype trajectory maintained an elliptical behavior, an offset between the two trajectories was noted. It seems feasible that the prototype trajectory follows the real trajectory more closely. Figure 15b shows the quadratic error between the two trajectories. It can be noticed that this error was not negligible and the error was used as an optimization criterion. The end-effector, located at the fingertip, was required to follow a specified curve with a minimum error computed between the obtained and desired trajectories.