## 2 Materials and Methods
### Kinematic Model
#### 2.3.1 Direct Model
Similarly to the loops shown in Equations (1) and (2), the kinematic chains in Equations (3) and (4) can be followed in Figure 1.  
\[\vec{R}^{\prime} = \vec{R}_{1}+\vec{R}_{2}+\vec{R}_{3} \tag{3}\] \[\vec{R}_{d} = \vec{R}_{2}+\vec{R}3 \tag{4}\]  
In the vector loop form, one of the vectors is fully defined from the start (constant): \(\vec{R}_{1}\), there is one input vector: \(\vec{R}_{5}\) and there are eight unknown vectors, which are the ones that have elements that need a solution (it could be magnitude or direction); \(\vec{R}_{2}\), \(\vec{R}_{3}\), \(\vec{R}_{4}\), \(\vec{R}_{6}\), \(\vec{R}_{7}\), \(\vec{R}_{8}\), \(\vec{R}_{d}\), \(\vec{R}^{\prime}\). At the beginning of the analysis, the constant relationship given by the \(\alpha\) angles must be considered: between \(\theta_{8}\) and \(\theta_{3}\), and \(\theta_{6}\) and \(\theta_{2}\). The said relationship is shown in Equations (5) and (6). \[\theta_{8} = \theta_{3}+\alpha_{d} \tag{5}\] \[\theta_{6} = \theta_{2}+\alpha_{m} \tag{6}\]  
Since the \(\alpha\) angles are constant, then \(\theta_{8}\) and \(\theta_{6}\) can be found solely from \(\theta_{3}\) and \(\theta_{2}\). The initial values for \(\vec{R}_{2}\), \(\vec{R}_{3}\), \(\vec{R}_{4}\), \(\vec{R}_{6}\), \(\vec{R}_{7}\) and \(\vec{R}_{8}\) can be obtained since we know that, in this initial position of the mechanism, the finger is horizontal, which means that the angles for \(\vec{R}_{2}\) and \(\vec{R}_{3}\) are \(0^{\circ}\). If the initial vector loops are solved, then we can obtain all the magnitudes for the vectors since they are constant through all of the movement of the mechanism (except for \(\vec{R}_{5}\), which is an input and has the angle constant but the magnitude changes). The unknown magnitudes are: \(r_{2}\), \(r_{4}\), \(r_{7}\), and \(r_{8}\), and since the angles \(\theta_{2}\) and \(\theta_{3}\) are known for the initial position, then \(\theta_{8}\) and \(\theta_{6}\) can be found using Equations (5) and (6) for the initial position. To find the magnitude of \(\vec{R}_{2}\), the parametric model is used, and it is given by Equation (7). The parametric model is explained in further detail in Section 2.2. Figure 1.0  
For the magnitude of \(\vec{R}_{8}\), a quadratic regression is used, in which the independent variable is the length of the distal phalanx (\(r_{3}\)). Equation (8) is obtained from the previously mentioned regression in which the \(R^{2}\) had a value of 0.998. During the regression, the value of \(r_{8}\) is changed, so the final angle of flexion for the distal phalanx is \(85^{\circ}\). This is performedfor each of the \(r_{3}\) inputs varying from 8 to 40 mm using increments of 1 mm. The distal phalanx length range used in the regression covers most of the population, according to various studies [30; 31; 32]. Figure 1.1  
Finally, the magnitudes \(r_{4}\) and \(r_{7}\) could be found from the initial position and the already known vectors using the transformation of Equations (1) and (2) into [PERSON]'s identity form since the magnitude of \(\vec{R_{5}}\) at this moment is zero, then \(r_{5}\) in Equations (9) and (10) is zero. Figure 1.2 Figure 1.3  
Now, given all the magnitudes, the angles can be calculated using the Freudein-setin equation and semi-angular identities to transform the cosines and sines into tangent functions obtaining the following equations, as a result:  
Figure 1.4  
Figure 1.5  
Figure 1.6  
Figure 1.7  
In Equations (12) and (14), all the values for \(A\), \(A_{c2}\), \(B\), \(B_{c2}\), etc., can be seen in Appendix A. Finally, to find the four remaining unknowns: \(X^{\prime}\), \(Y^{\prime}\), \(X_{m}\), \(X_{m}\), the Equations (A7)-(A10) are used (contained in Appendix A).