# Bioinspired Divide-and-Conquer Design Methodology for a Multifunctional Contour of a Curved Lever
## 3 Curved Lever Contour Design
### Three Types of Functional Requirements of Levers for CVGC  
According to the abovementioned principles, the CVGC mechanism has been developed to achieve a compact and circular shape as shown in Figure 4. The lever has the role of amplifying the spring force and transmitting it to the cam in the limited design space. To maximize the performance of CVGC, the multiple functions that the lever should satisfy are: (1) minimizing the required force of the torque variation, (2) maximizing the variable range of the compensation torque, and (3) avoiding interference between the lever and other mechanical elements. Figure 3: Operating concepts of CVGC for compensating a gravitational torque by mass: (**a**) Configuration and motions of CVGC. The white circled digits indicate the orders of motions describing the spring deformation due to the cam rotation. The black circled digits represent the order of forces that account for the compensating torque in the cam generated by the spring force. \(\theta\) is the rotation angle of the cam; (**b**) principle of torque variation. The movement of the pivot can change the amplification ratio (B/A or B\({}^{\prime}\)/A\({}^{\prime}\)). B/A is larger than B\({}^{\prime}\)/A\({}^{\prime}\), so it generates a larger compensation torque at the cam axis, and \(x\) denotes the distance between pivot position and cam follower. For the first function, minimizing the required force for the torque variation is essential to achieve high energy efficiency in the variable mechanism. When the variation is automated by an actuator, a large force is required which increases the actuator's size and weight and decreases the energy efficiency. Since the fundamental goal of CVGC is to create a compact mechanism, the first function should be satisfied by the lever design. The required force to change the pivot position can be calculated by the Lagrangian mechanics as follows:  
\[F_{variation}(x)=-\frac{\partial PE_{spring}(x)}{\partial x} \tag{1}\]  
\(x\), \(PE_{spring}(x)\), and \(F_{variation}(x)\) denote the generalized coordinates, the force, and potential energy profile of the mechanism, respectively. According to (1), if the energy variation is zero with respect to the change of pivot position, the required force can be zero in ideal frictionless conditions. The amount of spring deformation can be determined by the position of the spring follower. As shown in Figure 4, the rotation and translation of the lever occur simultaneously when the pivot changes. From the lever design perspective, the lever contour should not change the position of the spring follower when the pivot position varies. Next, the second function of the lever is important because a wide variable range of compensation torque obviously allows CVGC to compensate for variable weight. The variable range of the compensation torque is calculated as the difference between the highest and lowest compensation torques. As illustrated in Figure 4b, for the CVGC mechanism that varies the compensation torque by changing the pivot position, it generates the highest torque when the pivot is placed at the top position of the pivot guide and generates the lowest torque when the pivot is placed at the bottom position. According to [15], the maximum deformations of the spring at each pivot position of the CVGC mechanism could determine the amount of compensation torque. Therefore, to maximize the variable range of the torque, the lever contour should maximize the spring's deformation at the top pivot position as this is directly related to the highest torque. At the same time, this contour should minimize the spring's deformation at the bottom pivot position to minimize the lowest compensation torque. The design point to be considered in this process is that this lever contour must not compress the spring beyond its allowable deformation length. As a final function of a lever, the lever contour should be designed to prevent interference with other mechanical elements.