## 3 Results and Discussions  
In this work, the design and fabrication of a finger exoskeleton are made. The purpose of the exoskeleton is to transmit movement to the index finger of a rehabilitation patient. The type of therapy that the system assists with is ROM or movement therapy. During the design of the mechanism, a parametric model is used, which could be considered the most significant contribution of this work. The parametric model is based on  
Figure 4: Mechanism’s 3D model for 2 DOF motion transmission of a human finger. Figure 5: Three-dimensional-printed prototype for Subject 1.  
previous studies [24]. With the finger proportions, a program is created in Python. This program produces a.txt file, later imported to SolidWorks and used as a reference for the mechanism dimensions. After inputting the experimental anatomical values from the parametric program in Python, the experimental and theoretical values for the middle and proximal phalanges were obtained, along with the absolute and relative errors giving Table 2 as a result. Due to document limitations and to keep it concise, only the inputs and numerical results are shown for subject 1 as an example. Nevertheless, further down this section, all the plots comparing the different trajectories for each subject can be seen, so a comparison between the different mechanisms can be made. For the first DOF mechanism, which is in charge of the conjoined movement of the index and middle finger, the dimensions that were used to find the distal tip theoretical trajectory for subject 1 are the following: \(r_{1}\) = 67.7 mm, \(r_{2}\) = 23.8 mm, \(r_{3}\) = 17 mm, \(r_{4}\) = 61.5 mm, \(r_{5}\) with input varying from 0 to 37 (0.1 increments), \(r_{6}\) = 38.3 mm, \(r_{7}\) = 85.3 mm, \(r_{8}\) = 19.18 mm, \(\theta_{1}\) = 333.95\({}^{\circ}\), and \(X_{1}\) = 60.82. In the case of the second DOF, which moves the proximal phalanx, equations similar to those in Section 2.3.1 were used. These equations needed, as input, the following values to find the position of the tip of the proximal phalanx: \(X_{p}\) and \(Y_{p}\): \(r_{9}\) = 71.72 mm, \(r_{1}\)0 = 42.5 mm, \(r_{1}\)1 = 64.13 mm, \(r_{1}\)3 = 27.3 mm, \(r_{1}\)2 with input varying from 0 to 34 (0.1 increments) mm, \(\theta_{12}\) = 0\({}^{\circ}\), \(\theta_{9}\) = 71.12\({}^{\circ}\), and \(\theta_{p}\) = 108.29\({}^{\circ}\). Similarly to the direct models, the inverse kinematic needs a series of constants and an input to give its output. Therefore, for the first DOF mechanism, the following values are needed as inputs to solve the inverse kinematic model: \(X^{\prime}\) as an input taken from the direct kinematic model, \(r_{1}\) = 67.7 mm, \(r_{2}\) = 23.8 mm, \(r_{3}\) = 17 mm, \(r_{7}\) = 85.3 mm, \(r_{8}\) = 19.18 mm, \(\theta_{1}\) = 333.95\({}^{\circ}\), \(\alpha_{d}\) = 88.821\({}^{\circ}\), and \(\alpha_{m}\) = 89.884\({}^{\circ}\). On the inverse kinematic model for the second DOF, the input would be the coordinates of the proximal phalanx tip, along with the lengths of the links within the mechanism for this second DOF. This specific mechanism's input values are the following: \(X_{p}\) as an input taken from the direct kinematic model, \(Y_{p}\) as an input taken from the direct kinematic model, \(r_{9}\) = 71.72 mm, \(r_{1}\)0 = 42.5 mm, \(r_{1}\)1 = 64.13 mm, \(r_{1}\)3 = 27.3 mm, \(\theta_{12}\) = 0\({}^{\circ}\), \(\theta_{9}\) = 71.12\({}^{\circ}\), and \(\theta_{p}\) = 108.29\({}^{\circ}\). A significant amount of time could be saved by producing the parametric model. In the case of the mechanism's 3D model, thanks to this parametric model, it is not necessary for a human designer to manually change every dimension of the mechanism to fit the patient's hand, and instead, this is done by the algorithm proposed in this work. This algorithm consists of a program written in Python, which produces a.txt file as an output. Then a SolidWorks assembly searches and uses the values on this.txt file as references for all the mechanisms' different elements.