# Design of a Two-DOFs Driving Mechanism for a Motion-Assisted Finger Exoskeleton
## 3 Kinematic Design
The link lengths D0-D and D-E need to be calculated to fulfil the desired motion trajectory of the PIP joint, according to the motion reference in Figure 4. Considering the finger in fully close configuration, one can identify that in this configuration, the longest distance is expected to be D0-E. This distance can be calculated as 63.3 mm from the reference motion data. For achieving this value, the link length MCP - D\({}_{0}\) has to be equal to 27.8 mm,  
Figure 4: Acquired trajectories of PIP, DIP, and FT markers (as in the set-up of Figure 1 for the finger movement of a healthy human (dots indicate the measured data from the video capture tracking). Figure 5: A kinematic scheme of the proposed design with its linkage structure. the link length D\({}_{0}-\)D has to be equal to 32.0 mm, and the link length D-E has to be equal to 58.1 mm. Joints E and A0 on body F have coaxial axes. To size the link lengths of the second linkage mechanism, a graphic/geometric synthesis dimensional synthesis has been carried out to obtain the desired motion of point C. The procedure is reported as a general formulation, for example, in [35]. The dimensional synthesis starts by calculating the desired relative motion of point C relative to A0 based on the experimental data in Figures 3 and 4 and on the relative position of the joints in the proposed linkage mechanism shown in the scheme of Figure 6. In this scheme, \(\gamma\) is the angle between the horizontal axis and line through A0, \(\alpha\) is the angle between the horizontal axis and line through C, and \(\eta\) is the difference between the angles. The relative coordinate system is located in A0 and has its p-axis along the line MCP - A\({}_{0}\), as shown in Figure 6.  
The u-axis is perpendicular to the p-axis with direction from top to bottom. \(\Delta\)c is the distance between A0 and C along the p-axis, and \(\Delta\)d is the distance between A0 and C along the u-axis. The coordinates of point PIP are known from the measured trajectory of the finger movement in Figure 4. The displacement of the PIP point and A0 is shown in Figure 7 where \(\varphi\) is the angle between the horizontal axis and line through PIP, while \(\psi\) is the angle of MCP from A0 to PIP. The distance between MCP and A0 is I\({}_{\text{A0}}\), and the distance between MCP and PIP is I\({}_{\text{PIP}}\). Parameters a and b represent the displacement between A0 and PIP. The coordinates of point A0 can be calculated by using the parameters a and b and the angle \(\gamma\). Since the angle \(\varphi\) and the length I\({}_{\text{PIP}}\) are known from the calculations of the reference trajectory in Figure 4, \(\psi\) can be calculated as  
\[\psi=\arctan\left(\frac{\text{b}}{\text{I}_{\text{PIP}}-\text{a}}\right) \tag{1}\]  
Therefore, referring to Figure 7, the angle \(\gamma\) is given by  
Figure 5.0  
The coordinates of A0 can be calculated as  
Figure 5.1  
Figure 5.2  
Figure 5.3 A scheme for determining C coordinates in the second linkage mechanism by considering the known PIP (proximal interphalangeal joint), DIP (distal interphalangeal joint), and FT (fingertip) coordinates. By squaring and summing Equations (3) and (4), it is possible to obtain  
Figure 5.4  
A similar procedure is used to calculate the global coordinates of C. Figure 6 shows a sketch for the displacement between FT and C, where \(\lambda\) is the angle between the horizontal axis and the line from DIP to C; \(\theta\) is the angle between the horizontal axis and the line from DIP to FT. The distance between DIP and FT is \(\mathrm{l_{FT}}\), and the distance between DIP and C is \(\mathrm{l_{C}}\). Parameter e is the displacement between FT and C on the line from FT to DIP, and f is the displacement between C and FT perpendicular to that line.