## 3 Structural Optimization and Actuation Design  
### Structural Optimization  
#### 3.1.1 Kinematics Analysis  
According to the selected configuration of the hopping mechanism in Figure 4d, the kinematics model of the hopping mechanism is depicted in Figure 6. Because the selected configuration of the hopping mechanism is symmetrical (the branched chain \(P_{2}\)-\(P_{4}\)-\(P_{8}\)-\(P_{10}\)-\(P_{12}\)-\(P_{6}\) is symmetric with the branched chain \(P_{1}\)-\(P_{3}\)-\(P_{7}\)-\(P_{9}\)-\(P_{11}\)-\(P_{5}\)), and only one of the branched chains is selected for kinematic analysis (the branched chain \(P_{2}\)-\(P_{4}\)-\(P_{8}\)-\(P_{10}\)-\(P_{12}\)-\(P_{6}\) is selected). Figure 5: The energy storage and release phases. \(\alpha\) in the horizontal axis represents the rotation angle of the crank. (**a**) The initial position of the energy storage phase, which means that the tension spring is in a natural state. (**b**) The general position of the energy storage phase. (**c**) The end position of the energy storage phase, which means that the energy storage of the tension spring is maximum. (**d**) The general position of the energy release phase. (**e**) The initial position of the energy storage phase. According to the Denavit-Hartenberg Method [38], the homogeneous transformation matrix from the \(i\)-1 coordinate system to the \(i\) coordinate system is:  
\[{}^{i-1}T_{i}=\left[ \begin{array}{cccc}C\theta_{i}&-S\theta_{i}&0&a_{i-1}\\ Ca_{i-1}S\theta_{i}&Ca_{i-1}C\theta_{i}&-S\alpha_{i-1}&-d_{i}S\alpha_{i-1}\\ Sa_{i-1}S\theta_{i}&Sa_{i-1}C\theta_{i}&Ca_{i-1}&d_{i}Ca_{i-1}\\ 0&0&0&1\end{array} \right] \tag{4}\]  
where \(C\theta_{i}\triangleq C(\theta_{i})\triangleq\cos(\theta_{i})\), \(S\theta_{i}\triangleq S(\theta_{i})\triangleq\sin(\theta_{i})\). In Figure 6, the branched chain \(P_{2}\)-\(P_{4}\)-\(P_{8}\)-\(P_{10}\)-\(P_{12}\)-\(P_{6}\) can be further subdivided into two branched chains, the main chain \(P_{2}\)-\(P_{4}\)-\(P_{8}\)-\(P_{10}\)-\(P_{12}\) and the secondary chain \(P_{2}\)-\(P_{6}\)-\(P_{8}\)-\(P_{10}\). The specific D-H parameters of the main and secondary chain are shown in Table A1 in Appendix A. According to Table A1 and Equation (4), the position vectors of each hinge in the main chain and the secondary chain in the fixed coordinate system are:  
\[\left\{ \begin{array}{l}{r_{P_{4}}}^{\mathrm{T}}=l_{P_{2}P_{4}}\Big{[}\begin{array} []{ccc}C\theta_{1}&S\theta_{1}&0\end{array}\Big{]}\\ {r_{P_{8}}}^{\mathrm{T}}={r_{P_{4}}}^{\mathrm{T}}+l_{P_{4}P_{8}}\Big{[}\begin{array} []{ccc}C(\theta_{1+2})&S(\theta_{1+2})&0\end{array}\Big{]}\\ {r_{P_{10}}}^{\mathrm{T}}={r_{P_{8}}}^{\mathrm{T}}+l_{P_{8}P_{10}}\Big{[} \begin{array}{ccc}C(\theta_{1+2+3})&S(\theta_{1+2+3})&0\end{array}\Big{]}\\ {r_{P_{12}}}^{\mathrm{T}}={r_{P_{10}}}^{\mathrm{T}}+l_{P_{10}P_{2}}\Big{[} \begin{array}{ccc}C(\theta_{1+2+3+4})&S(\theta_{1+2+3+4})&0\end{array} \Big{]}\\ {r_{P_{6}}}^{\mathrm{T}}=l_{P_{2}P_{6}}\Big{[}\begin{array}{ccc}C\theta_{1} &-S\theta_{1^{\prime}}&0\end{array}\Big{]}\\ {r_{P_{8}}}^{\mathrm{T}}={r^{\prime}}_{P_{8}}^{\mathrm{T}}+l_{P_{6}P_{8}}\Big{[} \begin{array}{ccc}C(\theta_{1^{\prime}-2^{\prime}})&-S(\theta_{1^{\prime}-2^ {\prime}})&0\end{array}\Big{]}\end{array} \right. \tag{5}\]  
where \(r_{\mathrm{A}}\) is the position vector of the hinge A in the main chain; \(r_{\mathrm{A}}\) = [A\({}_{\mathrm{x}}\) A\({}_{\mathrm{y}}\) A\({}_{\mathrm{z}}\)]T; \(r_{\mathrm{A}}^{\prime}\) is the position vector of the hinge A in the secondary chain; \(r_{\mathrm{A}}^{\prime}\) = [A\({}_{\mathrm{x^{\prime}}}\) A\({}_{\mathrm{y^{\prime}}}\) A\({}_{\mathrm{z^{\prime}}}\)]T; \(\theta_{i+j}=\theta_{i}+\theta_{j}\), \(\theta_{i-j}=\theta_{i}-\theta_{j}\). The hopping mechanism has the characteristics of symmetry and is combined with Equation (5); the expression for the position vector of each joint point in the branched chain \(P_{1}\)-\(P_{3}\)-\(P_{7}\)-\(P_{9}\)-\(P_{11}\)-\(P_{5}\) is:  
Figure 6: The kinematics model of the hopping mechanism.