# Design of a Two-DOFs Driving Mechanism for a Motion-Assisted Finger Exoskeleton
## 4 Mechanism Synthesis of the Second Linkage
The aim is to design a mechanism that matches the desired motion trajectory of point C.  
The positions of point C with respect to the u-p coordinate system are calculated by Equations (11) and (12). For the dimensional synthesis, the positions of A0, B0 and the lengths \(\overline{\text{A}_{0}\text{A}}\) and \(\overline{\text{AC}}\) are required as well as three reference positions of C within its desired motion trajectory, according to step 1 in Figure 10. A0 and B0 are set as frame joints. A0 has the coordinates \((0;0)\) with respect to the u-p-coordinate system. B0 is designed with the coordinates \((-8;-17.8)\). The input parameters have been iterated to find a mechanism that fulfills the desired path of C well. After few iterations, sufficient input parameters were found. Positions 1 to 15 of the calculation of point C have been considered for the synthesis, as it gets more difficult to approximate a longer motion path. Furthermore, the identified range of motion is sufficient for rehabilitation purposes. Figure 9: Schemes for a dimensional synthesis: (**a**) the problem to be solved; (**b**) graphical solution as based on the approach that is described in [33]. The initially chosen link lengths and point coordinates are shown in Figure 11, similar to Figure 4. With this information, positions B2 and B3 can be determined as intersecting points with the known segments of \(\overline{\text{B}_{0}\text{B}}\) and \(\overline{\text{BC}}\), as mentioned in step 2. As a result of iterations of the synthesis, \(\overline{\text{B}_{0}\text{B}}\) has been measured to have a length of 46 mm, and \(\overline{\text{BC}}\). has been measured to have a length of 53 mm. As input from the positions of C, the first position is chosen as C1, position C2 is in the middle, and position C3 refers to the last used configuration during the finger closing motion. According to step 3, pose 1 is the reference for the synthesis. A0 with respect to position \(\overline{\text{B}_{3}\text{C}_{3}}\) is transferred into position \(\overline{\text{B}_{1}\text{C}_{1}}\), resulting in \(\left(\text{A}_{0}\right)_{3}^{1}\). \(\left(\text{A}_{0}\right)_{3}^{1}\) is A0 in pose 3 transferred to pose 1. This means that the triangles C3-B3-A0 and C1-B1-\(\left(\text{A}_{0}\right)_{3}^{1}\) are identical. In the same manner, the point \(\left(\text{A}_{0}\right)_{2}^{1}\) can be identified. This corresponds to step 4. The triangles to find the position \(\left(\text{A}_{0}\right)_{3}^{1}\) are given in Figure 12a. The synthesized mechanism and the calculated positions of C are shown in Figure 12b. The previously calculated coordinates for C are marked as crosses, and the computed trajectory of the mechanism is marked as dots. The figure also shows that the trajectory matches well with the desired motion for point C.  
The position of A1 is the center point of a circle through the positions A0, \(\left(\text{A}_{0}\right)_{2}^{1}\) and \(\left(\text{A}_{0}\right)_{3}^{1}\), as mentioned in step 5. The resulting position is A1, as pose 1 has been used as a reference. The link lengths can be determined with the obtained position of A1. Since the given data in step 1 have already been identified by a prior iteration, step 7 can be skipped. The complete kinematic design is obtained by combining both linkages. The resulting kinematic design of the whole exoskeleton mechanism is shown in Figure 13. The numerical values are summarized in Table 2, defined according to the scheme in Figure 5. The calculated positions of A0 and C are indicated with plus (+) and cross (x) marks, respectively. Figure 10: A flowchart for the mechanism synthesis according to the scheme in Figure 9.  
Figure 11: Initial problem of the synthesis for the second linkage with the selected three points C1, C2, C3 along the desired motion trajectory of point C.  
Figure 12: A scheme for determining the second linkage: (**a**) Triangles of the relative positions C0; (**b**) synthesized mechanism and resulting trajectory passing through the selected points C1, C2, C3.