## 3 Structure Optimization
### Optimization Results  
According to the NSGA-II algorithm, the Pareto solution set and plane projection are obtained, as shown in Figure 4. The blue dot * represents the Pareto solution. The red dotted line indicates the first-order natural frequency before optimization, and the brown dotted line indicates the weight before optimization. The black arrow indicates the optimization direction of each optimization objective. Because the deformation before optimization is large, all points in the Pareto solution set meet the optimization objective. According to the analysis, it can be seen that there are six optimal solutions meeting each optimization objective. The values are listed in Table 5. Because small first-order natural frequency will cause low-order vibration, so the solution with higher first-order natural frequency is preferred as the optimization value. Therefore, No. 6 is the optimization value in Table 5. The optimized size is rounded to facilitate the processing of parts, and the preten of bolt connector \(F_{0}\), the thickness of the clamped part \(c\) and the width \(t\) are 1700 N, 9 mm and 30 mm, respectively. The comparisons between the optimized design and the initial design are listed in Table 6. It can be seen that the maximum deformation \(L\) is reduced by 67.23%, and the first-order natural frequency is increased by 8.4%. The improvement is relatively obvious. The weight \(M\) increases slightly, but it can be ignored due to the very small impact on the robot.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Models**} & \multicolumn{3}{c}{**Goodness of Fit**} \\ \cline{2-5}  & \multicolumn{1}{c}{\(R^{2}\)} & \multicolumn{1}{c}{**C.V**} \\ \hline \(L=\left(5.16-7.86x_{1}+0.446x_{2}+0.994x_{3}+6.44x_{1}^{2}+0.187x_{2}^{2}\right.\) & 0.9946 & 0.0902 \\ \(-0.359x_{3}^{2}-1.24x_{1}x_{2}-0.408x_{1}x_{3}+0.027x_{2}x_{3}\right)\times 1 0^{-5}\) & & \\ \(f=465.64-41.46x_{1}+35.75x_{2}+0.06x_{3}-6.55x_{1}^{2}+0.064x_{2}^{2}+\) & & \\ \(0.4x_{3}^{2}+1.9x_{1}x_{2}-0.033x_{1}x_{3}-0.067x_{2}x_{3}\) & & \\ \(M=2.09-0.076x_{1}-0.099x_{2}-5.3\times 10^{-4}x_{3}-0.01x_{1}^{2}+0.0014x_{2}^{2}+\) & & \\ \(1.33\times 10^{-7}x_{3}^{2}+4.63\times 10^{-3}x_{1}x_{2}\) & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: The fitted models.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**No.** & **c/mm** & **t/mm** & **F\({}_{0}\)/N** & **L/m** & **M/kg** & _f/Hz_ \\ \hline
1 & 8 & 33 & 2500 & \(5.67\times 10^{-5}\) & 1.105 & 501.6 \\
2 & 8 & 27 & 2500 & \(5.97\times 10^{-5}\) & 0.801 & 430.69 \\
3 & 8 & 27 & 1500 & \(4.36\times 10^{-  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Optimization Objective** & **Initial Design Scheme** & **Optimized Design Scheme** & **Rate of Change** \\ \hline \(L/\)m & \(1.01\times 10^{-4}\) & \(3.31\times 10^{-5}\) & 67.23\% reduction \\ \(M/\)kg & 0.983 & 0.993 & 1.02\% increase \\ \(f/\)Hz & 451.35 & 489.26 & 8.4\% increase \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison between the optimized design scheme and the initial design scheme.  
Figure 4: Pareto solution set and plane projection. (**a**) Pareto solution; (**b**) plane projection of maximum deformation and weight; (**c**) plane projection of maximum deformation and first-order natural frequency; (**d**) plane projection of weight and first-order natural frequency.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No.** & _c/mm_ & _t/mm_ & \(\mathbf{F_{0}/N}\) & _L/m_ & _M/kg_ & _f/Hz_ \\ \hline
1 & 9.32 & 27.36 & 1547 & \(3.32\times 10^{-5}\) & 0.871 & 457.92 \\
2 & 8.27 & 29 & 1558 & \(4.25\times 10^{-5}\) & 0.9 & 459.62 \\
3 & 8.96 & 28.68 & 1578 & \(3.49\times 10^{-5}\) & 0.919 & 468.27 \\
4 & 8.85 & 29.04 & 1548 & \(3.78\times 10^{-5}\) & 0.932 & 470.59 \\
5 & 8.83 & 29.39 & 1577 & \(4.05\times 10^{-5}\) & 0.95 & 474.98 \\
6 & 8.84 & 29.73 & 1723 & \(3.17\times 10^{-5}\) & 0.977 & 478.93 \\ \hline \hline \end{tabular}
\end{table}
Table 5: The optimal solution set.