## 2 Materials and Methods  
### General Framework  
It is essential for modeling a finger's movement to consider how the tendons are anchored to the bones and the geometry of the hand's articulations because these restrict the movements of each bone segment [26]. A simplified model of the human finger considering the location of the tendons is proposed in [PERSON]'s study [27]. In [PERSON]'s study, the range of motion (ROM) for each articulation is given by two factors: the tendon length when the finger is fully extended and the radius of each articulation. Regarding the total ROM of a human index finger in [PERSON]'s work [28], the functional ROMs found for each articulation were the following:  
* 0\({}^{\circ}\) to 100\({}^{\circ}\) for the metacarpophalangeal (MCP) joint. * 0\({}^{\circ}\) to 105\({}^{\circ}\) for the proximal interphalangeal (PIP) joint. * 0\({}^{\circ}\) to 85\({}^{\circ}\) for the distal interphalangeal (DIP) joint. Concerning the number DOFs for the flexion and extension of the index finger, it could be said that there are three DOFs in total due to the three articulations in the finger (MCP, PIP, and DIP). However, in [26], it is demonstrated that due to the finger's anatomy and the ligaments' position, both movements at the PIP and DIP joints are just one DOF and thus can be modeled as a conjoined movement. Therefore, with just two DOFs, the movements of interest for this work can be replicated: one DOF for the proximal phalanx movement and one for the conjoined motion of the middle and distal phalanges. The type of mechanism selected for this study is a closed-loop bar rigid mechanism with virtual centers; this implies that the finger's position is always constant according to the input of the mechanism. On the other hand, by using virtual centers, the hinges that allow for the rotation of the finger's phalanges can be located away from the biological rotation center of each articulation, thus interfering less with the natural ROM of the human fingers. This mechanism is selected because it contributes the most to the rehabilitation success factors mentioned in the introduction of this work; it allows for a comprehensive set of consistent and complex finger motions, which also makes the implementation of functional motion therapy possible. The rehabilitation mechanism can be divided into two separate sub-mechanisms. The first mechanism is shown in Figure 1, which moves the distal and middle phalanges, while the second mechanism, which moves the proximal phalanx, is represented in Figure 2. As can be seen, the first mechanism is much more important because it must move two phalanges at the same time. In order to generate the first design of the mechanism (before implementing the parametric model), a series of measurements are taken of a patient's finger. For taking these dimensions, a scanner is used. The procedure, shown in Figure 3a, consists of taking a scan of the patient's hand and an element of a known length, which is a scale mark of 50 mm. Then the scanned image is transferred to a video and image analysis software called Tracker. Once within the software, a digital scale is established using the element of known length, allowing the software to scale pixels into millimeters. Then vectors are added in the software to each element of interest within the index finger. In Figure 3a, one vector represents the proximal phalanx from the MCP to the PIP joints (in magenta), another vector is the middle phalanx from the PIP to DIP joints (in cyan) and another for the distal phalanx from the DIP joint to the tip of the finger (in blue). As the last step, the magnitudes of the vectors are extracted from the software. It is important to consider that the distal phalanx length is not directly the magnitude of the vector. According to the study by [PERSON] [24], the average length of soft tissue at the tip of the index finger is 3.84 \(\pm\) 0.59 mm, so in order to get the proper bone length, an offset of 3.84 mm is used to obtain the distal phalanx's correct length.