## 3 Structural Optimization and Actuation Design
### Structural Optimization
#### 3.1.2 Establishment of Optimization  
It is important and necessary to optimize the dimension of the closed-chain mechanism [39]. Because of the dual-objective optimization, the special method of TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) was chosen for dimension optimization. The same trend of evaluation indicators facilitates a reasonable compromise between the two optimization objectives. Initially proposed by [PERSON] and [PERSON] in 1981 [40; 41; 42], two vectors are constructed:  
\[F^{*}=[f^{*}_{m},f^{*}_{n}]^{\rm T} \tag{7}\]  
\[F(\mathbf{X})=[f_{m}(\mathbf{X}),f_{n}(\mathbf{X})]^{\rm T} \tag{8}\]  
The vector represented by Equation (7) is regarded as an ideal point of the vector function represented by Equation (8). Then the smaller the deviation between \(F(\mathbf{X})\) and \(F^{*}\), the closer the objective function will be to the ideal point, i.e., the closer the design variable will be to the optimal solution.  
The distance equation of the TOPSIS Method is:  
\[F_{\rm d}=\left\|F(\mathbf{X})-F^{*}\right\|=\left\{\sum_{i=1}^{n}{\lambda_{j}[f_{ j}(x_{i})-f_{i}\}^{p}\right\}^{\rm \frac{1}{p}} \tag{9}\]  
where \(p\subset[1,\ +\infty)\). When \(p=1\), it is the Hamming distance or absolute distance; when \(p=2\), it is the Euclidean distance; when \(p=  
are determined by (1) the length of each linkage, (2) the angle between the line connecting the hinge points \(P_{2}\) & \(P_{6}\) and the \(X_{0}\)-axis direction, and (3) the position of the crank-gear-bar. Therefore, the optimization variables matrix \(\mathbf{X}\) can be concluded as being:  
\[\mathbf{X}=\left[x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8},x_{9},x_{10} \right]^{\mathrm{T}}=\left[l_{1},l_{2},l_{3},l_{4},l_{5},l_{6},l_{7},l_{8}, \beta_{1},\beta_{2}\right]^{\mathrm{T}} \tag{11}\]  
where \(l_{1}=l_{P_{1}P_{2}};l_{2}=l_{P_{2}P_{4}};l_{3}=l_{P_{2}P_{6}};l_{4}=l_{P_{2}P_ {6}};l_{5}=l_{P_{4}P_{6}};l_{6}=l_{P_{8}P_{10}};l_{7}=l_{P_{10}P_{12}};l_{8}=l_ {P_{12}P_{11}};\beta_{2}=\theta_{1}\). \(\beta_{1}\) is the angle between the initial position of the crank-gear-bar \(P_{2}P_{4}\) and the positive direction of the \(X_{0}\)-axis.  
(3) Constraint conditionsAccording to the conditions for the existence of a crank-rocker mechanism in the planar four-bar linkage \(P_{2}P_{4}P_{8}P_{6}\), the bars \(l_{2}\), \(l_{3}\), \(l_{4}\), and \(l_{5}\) have the following linear constraints:  
\[\mathbf{A_{1}X}\leq\mathbf{b}_{1} \tag{12}\]  
Considering the limitation of the processing technology of the component and the requirement for the non-  
By analyzing the data shown in Table 2, it can be proven that: (1) The optimization result satisfies the range of values of the design variables; (2) The bars \(l_{2}\), \(l_{3}\), \(l_{4}\), and \(l_{5}\) satisfy the constraints of the crank-rocker mechanism; (3) The dimension of the bars \(l_{1}\), \(l_{2}\), \(l_{3}\) avoid interference during assembly; (4) Optimized data meets the constraints on the kinematical chain. The optimized method is used to gain a set of ideal data.  
The graphical optimization result is shown in Figure 8a. To avoid interference during hopping, changes are made to the optimization result, and the solid hopping mechanism model is shown in Figure 8b.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Optimization Parameters** & \(l_{1}\)/mm & \(l_{2}\)/mm & \(l_{3}\)/mm & \(l_{4}\)/mm & \(l_{5}\)/mm \\
**Optimization Results** & 30.000 & 15.729 & 32.000 & 50.939 & 52.851 \\
**Optimization Parameters** & \(l_{6}\)/mm & \(l_{7}\)/mm & \(l_{8}\)/mm & \(\beta_{1}\)/(\({}^{\circ}\)) & \(\beta_{2}\)/(\({}^{\circ}\)) \\
**Optimization Results** & 34.061 & 75.000 & 25.000 & 276.180\({}^{\circ}\) & 134.768\({}^{\circ}\) \\ \hline \end{tabular}
\end{table}
Table 2: Optimization results of the hopping mechanism dimensions.  
Figure 8: Hopping mechanism. (**a**) Optimization result. (**b**) Solid model.  
Figure 7: The flow chart of the optimization.