# Design of a Two-DOFs Driving Mechanism for a Motion-Assisted Finger Exoskeleton
## 4 Mechanism Synthesis of the Second Linkage  
The dimensional synthesis of the second linkage can be outlined as the procedure of determining the remaining lengths of a 4-bar mechanism that guides a point on the coupler curve. One possibility for such a synthesis is the graphical method based on the Burmester theory as described in [37]. This synthesis approach has been selected since it can quickly address the desired features in terms of replicating a desired motion trajectory as given by the reference motion trajectory in Figure 4. The expected accuracy for joints (\(\pm 4\) degrees) does not justify the use of more complex and time-consuming synthesis methods such as numerical optimization. The proposed procedure allows an approximation of the desired movement from the measured trajectory in Figure 4 by defining three points on the coupler curve that are reached precisely by the mechanism. The synthesis can be conducted when two lengths of a mechanism and two positions of joints are known. Figure 9a shows the initial problem with all given parameters. The position of joints A0 and B0, lengths \(\overline{A_{0}B_{0}}\), \(\overline{B_{0}B}\), \(\overline{BC}\), and three positions of joint C are given in the initial problem. From the input, the position of the missing joint can be determined. When the positions of all joints are established, the lengths can be determined. Curve k in Figure 9a shows the original movement of joint C, and curve kc in Figure 9b is the movement after the synthesis. With the synthesis in Figure 9b, joint A can be determined, giving the missing lengths \(\overline{A_{0}A}\), \(\overline{AC}\), and \(\overline{AB}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **a** & **b** & **e** & **f** \\ \hline Length [mm] & 4 & 17 & 15 & 19 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design parameters from kinematic calculations in Figures 5 and 6. A pose of a mechanism consists of the positions and the orientations of all links. For example, for pose 1 of the mechanism in Figure 4, joint A is in position A1, joint B is in position B1, and joint C is in Position C1. The graphical procedure for the dimensional synthesis is given in the flow-chart of Figure 10 according to the following steps:  
1. The given parameters in Figure 9a are the positions of C1, C2, C3, and B0 as well as the lengths of \(\overline{\text{B}_{0}\text{B}}\) and \(\overline{\text{BC}}\);
2. With the information from step 1, joint B is known in pose 1, 2, and 3, as joint B moves on the curve kB. Therefore, the poses of \(\overline{\text{B}_{1}\text{C}_{1}}\), \(\overline{\text{B}_{2}\text{C}_{2}}\), and \(\overline{\text{B}_{3}\text{C}_{3}}\) are known. At this point, joint A is missing to complete the mechanism sizing:
3. Next, one link in one pose is chosen as a reference (in this example, link BC in pose 1);
4. Now, joint A\({}_{0}\) is virtually moved around \(\overline{\text{B}_{1}\text{C}_{1}}\) to create A0 that corresponds to position 2 in pose 1 (by moving A0 with respect to the coordinate system of pose 2 into pose 1). The transformation can be done by creating a triangle between C2, A0, and B2 and moving the triangle into the pose of C1 and B1. The resulting position is A\({}_{0}\) from pose 2 in pose 1, written as \((\text{A}_{0})^{1}_{2}\);
5. The same as step 4 is done with A\({}_{0}\) from pose 3 in pose 1, resulting in \((\text{A}_{0})^{1}_{3}\);
6. Finally, A\({}_{1}\) is found by the intersection of the perpendicular bisectors between three positions of A\({}_{0}\). The resulting mechanism guides joint C along curve kC. Curve kC matches curve k in the points C1, C2, and C3 [33];
7. With all known positions of the joints, the link lengths can be determined. 8. Other link lengths/positions can be chosen to improve the matching of curves k and kc. 9. The dimensional synthesis is applied to the second linkage mechanism by using the relative movement of C with respect to A0.