## 2 Materials and Methods
### Kinematic Model  
#### 2.3.1 Direct Model  
In order to know the coordinates of the index finger's tip, the forward kinematic model of the mechanisms needs to be solved. For the case of the system proposed in this study, there are two independent mechanisms with one autonomous DOF for each one. From hereafter, in order to simplify this work, only one DOF of the mechanism is analyzed; in this case, only the mechanism that moves the middle and distal phalanges is subject to a kinematic analysis. The mechanism for the second DOF (which moves the proximal phalanx) is very similar to the first mechanism, with the only difference being the links' length and the cams' diameter; therefore, the steps for analyzing this mechanism are very similar to the ones shown below. In Table 1, all the variables and constants for the kinematic chains of the first DOF mechanism are shown. In Figure 1, all the kinematic chains that are needed to be solved for the distal-middle phalanx movement are shown. The variables that are needed to describe the finger's tip are the coordinates \(X^{\prime}\) and \(Y^{\prime}\), which are referenced to the origin at the starting position of the linear actuator. From Table 1, it can be seen that there are two input variables: \(r_{3}\), which is the input for the parametric model (the distal phalanx length), and \(r_{5}\), which is the magnitude of the \(\vec{R}_{5}\) vector that changes as the actuator moves. In regards to the constants shown in Table 1, \(r_{1}\), \(r_{6}\), and \(\theta_{1}\) are fixed and given by the original mechanism's design. Similarly, \(\theta_{5}\) \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Variables solved using** & \(\theta_{2}\) & \(\theta_{3}\) & \(\theta_{4}\) & \(\theta_{6}\) & \(\theta_{7}\) & \(\theta_{8}\) & \(\vec{K}_{d}\) & \(\vec{R}^{\prime}\) \\ \hline
**Variables solved by the** & \(r_{2}\) & \(r_{4}\) & \(r_{7}\) & \(r_{8}\) & & & \\ \hline
**Input variables** & \(r_{3}\) & \(r_{5}\) & & & & & \\ \hline
**Constants** & \(\vec{R}_{1}\) & \(r_{6}\) & \(\theta_{1}\) & \(\theta_{5}\) & \(\alpha_{m}\) & \(\alpha_{d}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variables and constants for mechanism 1.  
is given by the orientation of the actuator, which, in this case, is a motor connected to a screw; therefore, being linear. Lastly, angles \(\alpha_{d}\) and \(\alpha_{m}\) are constant since they represent the angle between the phalanges and vectors \(\vec{R}_{6}\) and \(\vec{R}_{8}\), respectively, which the designer also gives. It is crucial for generating the kinematic models to consider where the angles for each vector are being measured. Considering the representation in Figure 1, all vector angles were referenced to the positive \(+x\)-axis of the coordinate system where they originated from, or, in the case of vectors that originated elsewhere, their angle is considered to be measured from the same axis as the previous vector they are connected to. It could be said that there are two main kinematic chains for the mechanism. These two main kinematic chains are represented in closed vector loop form in Equations (1) and (2). The path of these chains can be traced in Figure 1.  
\[\vec{R}_{5}+\vec{R}_{4} = \vec{R}_{1}+\vec{R}_{6} \tag{1}\] \[\vec{R}_{5}+\vec{R}_{7} = \vec{R}_{1}+\vec{R}_{2}+\vec{R}_{8} \tag{2}\]  
Besides the two main kinematic chains for the mechanism, there are two more. These other two kinematic chains are the ones that are used to represent the fingertip's coordinates referenced to the general origin \(O\) and the origin at the PIP joint \(O_{m}\) shown in Figure 1. The closed vector loop equations for these chains are (3) for the fingertip's coordinates with respect to \(O\), and (4) with respect to \(O_{m}\).