## 2 Materials and Methods
### Finger Real Motion
#### 2.2.2 Results of the Analysis of the Fingers Real Motion  
The recorded gesture focused on the flexion-extension movement leading to the fingertip trajectory. This finger movement occurs in the finger plane, which is normal to all joint axes. Consequently, the fingertip trajectory were defined only by the x and z coordinates (as shown in Figure 10). A series of three-dimensional curves, as shown in Figure 11, was obtained using the flexion-extension movement in the \(xyz\) space, which corresponded to the recorded motion for a volunteer's index finger. All subjects were asked to move their finger with a fixed abduction-adduction. Despite this virtual constraint and the low-speed exercise, the obtained fingertip trajectory presented a reduced variation along the y-axes, probed through principal component analysis (PCA). The PCA reduces the dimensionality of a dataset of interrelated variables while retaining the dataset's variation. A new set of variables is generated, a linear combination of the original variables. This new set of variables is called the principal components [42]. Figure 8: Location of the markers and the digital reconstruction of the hand. Figure 9: (**a**) Initial position to record. (**b**) Hand in closed fist configuration. The PCA indicates which variable is the most valuable for clustering the data. In this article, the original set of variables are the \(xyz\) coordinates of the \(n\) point of the flexion-extension movement, which means that three principals components exist. The principal component is obtained by calculating the average values for all the variables, as shown in Equation (8). The original variables are defined as \(p\)-dimensional vectors that need to be projected onto a \(q\)-dimensional subspace. \[\overline{x}=\frac{\sum_{i=1}^{n}x_{i}}{n}\overline{y}=\frac{\sum_{i=1}^{n}y_{ i}}{n}\overline{z}=\frac{\sum_{i=1}^{n}z_{i}}{n} \tag{8}\]  
The PCA identifies the directions (principal components) along which the variation in the data is maximal. The direction of PC1 represents the most significant variation among the data, PC2 is the second most important direction and is uncorrelated to PC1 and PC3. PC3 represents the less important direction, also uncorrelated with the other principal components. Principal components are normalized eigenvectors of the covariance matrix. These are ordered according to how much of the variation present in the data they contain. The dimensionality of the three-dimensional data can be reduced to two-dimensional data using the first two principal components [43; 44]. Table 3 shows the covariance data matrix's eigenvalues and the percentage of each principal component total variance. The eigenvalues measure the amount of variation retained by each principal component [44]. Figure 11: Flexion-extension movement in the space for the index finger of a volunteer. Figure 10: Finger trajectory in the \(xz\) plane. As can be observed in Table 3, the first two components explained 99.39% of all variability, as shown in Figure 12. This proves that the \(y\)-axis presents a reduced variation in the flexion-extension movement of the fingers. Due to this, one can conclude that the flexion-extension movement can be assumed in the \(xz\) plane. The abduction-adduction movement of the finger was not considered. The finger motion, from the index to small, was analyzed. A similarity was observed between all fingertip trajectories, as shown in Figure 13a. An example of graphs obtained for all fingers of one subject is depicted in Figure 13b. The left side of Figure 14 shows all the trajectories recorded in the \(xyz\) space. The right side shows one movement in the \(xz\) plane to improve the visualization of each finger's trajectories. Figure 12: Principal components representation. (**a**) Percentage of the total variance. (**b**) The data represented in the space of the three principal components. Figure 13: Trajectories for flexion–extension from index to small, (**a**) \(xyz\) spatial representation, (**b**) \(xz\) planar representation of one participant.