# Bioinspired Divide-and-Conquer Design Methodology for a Multifunctional Contour of a Curved Lever
## 4 Curved Lever Contour Design
### Partial Contour 1 (PC1): Minimizing the Required Force for Torque Variation  
The PC1 should not change the position of the spring follower when the pivot position varies along the pivot guide. The CVGC varies the compensation torque when the rotation angle of the cam is zero. Since the cam follower side of the lever always contacts the cam follower, the translation and rotation of the lever occur simultaneously when the pivot position changes, as illustrated in Figure 5a. Figure 5: _Cont._  
In a global coordinate system, the contour design is complex due to the changes of relative angles and distances between the lever and the spring follower. However, the positions of points that determine the rigid lever shape do not change in the lever coordinate system. Therefore, the lever design problem can become easier when in a lever coordinate system. To calculate the coordinate transformation, representative pivot positions are determined by dividing the pivot guidelines equally into n sections. Nine representative pivot positions were selected, including the top and bottom pivot positions. The unit length for a section and the rotation angles at each pivot position can be determined using (2) and (3) as follows:  
\[D_{x}=\frac{{{}^{G}X_{9}^{ pivot}-{{}^{G}X_{1}^{ pivot}}}}{8},\ D_{y}=\frac{{{}^{G}Y_{9}^{ pivot}-{{}^{G}Y_{1}^{ pivot}}}}{8} \tag{2}\]  
\[\varphi_{i}=\frac{\pi}{2}+\sin^{-1}\frac{{{}^{G}X_{i}^{ pivot}}}{{B_{i}}} \tag{3}\]  
where \({{}^{G}X_{i}^{ pivot}}\) and \({{}^{G}Y_{i}^{ pivot}}\) indicate the position of the \(i^{th}\) pivot in the global coordinates; \(D_{x}\) and \(D_{y}\) are the \(x\) and \(y\) components of the unit length, respectively; \(r_{cam}\) is the radius of the cam follower. Then, the translation and transformation matrices at the \(i^{th}\) selected pivot positions can be calculated using (4) and (5), respectively, as follows:  
\[\left[ \begin{array}{c}{{}^{G}X_{i}^{ pivot}}\\ {{}^{G}Y_{i}^{ pivot}}\end{array} \right]=\left[ \begin{array}{cc}{{}^{G}X_{1}^{ pivot}}&+(i-1)D_{x}\\ {{}^{G}Y_{1}^{ pivot}}&+(i-1)D_{y}\end{array} \right] \tag{4}\]  
\[\frac{{}^{L}}{G}TF_{\text{PC1,}i}=\left[ \begin{array}{cccc}{C_{-\varphi_{i} }}&{-S_{-\varphi_{i}}}&0&-{{}^{G}X_{i}^{ pivot}}\\ {S_{-\varphi_{i}}}&{C_{-\varphi_{i}}}&0&-{{}^{G}Y_{i}^{ pivot}}\\ 0&0&1&1\\ 0&0&0&1\end{array} \right] \tag{5}\]  
where the left subscript \(G\) in (4) is the lever and global coordinate system, the left subscript \(G\) and the superscript \(L\) represent the global coordinates and lever coordinates, respectively, and \(C\) and \(S\) are the cosine and sine, respectively. Using the transformation matrices, the fixed spring follower positions are transformed into a lever coordinate system and nine circles are generated, as shown in Figure 5b. The PC1 should be designed to have a shape that makes contact with all nine circles at the same time. For the next step, the common circumferential lines and points between adjacent circles are computed to find contact points between the lever and circles. Except for the  
Figure 5: Design processes of the PC1. The units of \(x\)- and \(y\)-axes are mm: (**a**) Parameters to calculate transformation matrices at each representative pivot position; (**b**) spring followers are represented in the lever coordinate system; (**c**) common circumferential lines and points and principle to choosing the representative contact points; (**d**) the representative contact points of each spring follower and Hermite spline-based PC1. first and last circles, the \(i^{th}\) circle has two common circumferential lines and contact points between the \(i-1^{th}\) and \(i+1^{th}\) circles of spring followers. To assign one tangential line and point to a circle, the line and contact point between the \(i-1^{th}\) circle are selected. As a result, each of the nine circles has a corresponding contact point and slope of the tangential line at that point, as illustrated in Figure 5c.