# Bionic Design of a Miniature Jumping Robot
## 3 Jumping Mechanism Analysis and Optimization  
### Design and Parameter Optimization of the Jumping Leg Mechanism  
This section explores the effect of the angle of the variable \(C_{1}DG\) and the length of \(DG\) on the magnitude of the spring deformation, provided that the fixed rod \(DG\) and the length of \(DE\) are constants, to allow the double six-link mechanism to rotate to the limited position of maximum compression of the linear spring, as shown in Figure 6. To reduce the minimum torque required by the cam during compression, a cam with a small actuator stroke was first designed, followed by the calculation of the deformation of the spring when the rotating linkage is rotated clockwise to a critical state based on the profile of the cam, and the analysis of the variation law between the angle of the variable \(C_{1}DG\) and the length of \(DG\) and the deformation of the spring.  
Figure 5: Bouncing movement flow chart.  
Figure 6: Mathematical model of a six-link mechanism.  
In Figure 6, the six-rod mechanism \(AB_{1}C_{1}DE_{1}F_{1}G\) represents the initial state of the jumping leg mechanism without energy storage, \(AB_{2}C_{2}DE_{2}F_{2}G\) is the limit state of the jumping leg mechanism at maximum energy storage, the rotating linkage \(C_{1}E_{1}\) rotates the fixed angle \(\beta\) to reach the critical state, and the \(DG\) rod is a fixed rod. According to the relationship between two linear spring positions with symmetry, \(DE_{1}=DG=L_{1}\) and is a constant, and \(E_{1}F_{1}=GF_{1}=L_{2}\), \(L_{2}\) is a variable. According to the given cam profile, \(\beta=\pi/3\), let the initial length of the spring be \(L_{0}\), the limit length at the maximum of the deformation variable is \(L^{\prime}\), the angle between the rotating connecting rod \(C_{1}E_{1}\) and the fixed rod \(DG\) is the variable \(\alpha\), and the value of \(\alpha\) is in the range of (0, \(\pi\)). In the triangle \(DE_{1}F_{1}\), according to the cosine theorem, we have:  
\[cos\frac{\pi-\alpha}{2}=\frac{L_{1}^{2}+L_{0}^{2}-L_{2}^{2}}{2L_{0}L_{1}} \tag{4}\]  
In the triangle \(DE_{2}F_{2}\):  
\[cos\frac{\pi-\alpha-\frac{\pi}{3}}{2}=\frac{L_{1}^{2}+{L^{\prime}}^{2}-L_{2}^ {2}}{2L^{\prime}L_{1}} \tag{5}\]  
Simplifying the above two equations yields:  
\[\left\{\begin{aligned} L_{0}^{2}-2L_{1}sin\frac{\alpha}{2}L_{0}+L_{ 1}^{2}-L_{2}^{2}=0\\ {L^{\prime}}^{2}-2L_{1}cos\left(\frac{\pi}{3}-\frac{\alpha}{2}\right)L^{ \prime}+L_{1}^{2}-L_{2}^{2}=0\end{aligned}\right. \tag{6}\]  
From Equation (6), it follows that:  
\[L^{\prime}-L_{0}=L_{1}\Big{[}\cos\Big{(}\frac{\pi}{3}-\frac{\alpha}{2}\Big{)} -sin\frac{\alpha}{2}\Big{]}+\sqrt{L_{1}^{2}cos^{2}\Big{(}\frac{\alpha}{2}- \frac{\pi}{3}\Big{)}-\left(L_{1}^{2}-L_{2}^{2}\right)}-\sqrt{L_{1}^{2}sin^{2} \frac{\alpha}{2}-\left(L_{1}^{2}-L_{2}^{2}\right)} \tag{7}\]  
In Equation (7), \(\alpha\) and \(L_{2}\) are two independent variables that have no influence on one another. Using the control variables method, the mathematical analysis of this function leads to the conclusion that the smaller the angle of \(\alpha\), the larger the value of \(L^{\prime}-L_{0}\), and the larger the deformation of the spring, while \(L_{2}\) remains constant. If \(\alpha\) remains constant, the value of \(L^{\prime}-L_{0}\) tends to increase as \(L_{2}\) decreases. We thus propose a design principle that is compatible with this six-rod mechanism: the linkage length \(L_{1}\) remains constant, the initial spring length \(L_{0}\) and the limit length \(L^{\prime}\) satisfy the conditions for the use of the spring, and the size of \(\alpha\) or is \(L_{2}\) is reduced in the six-rod mechanism so that the size of the linkage is reduced and the deformation of the spring is quickly maximized. There is, however, a theoretical minimum value for \(L_{2}\), which is the minimum value when \(E_{1}F_{1}G\) coincides with the position of the dead point of the mechanism, when \(E_{1}F_{1}G=2L_{2}\).