## 5 Robot Motion Simulation Experiment
### Trajectory Optimization Model Solving  
The variation range of the optimization time interval is determined according to the motion scheme, \(T_{\min}=[0.5,\,1.5,\,1.0,\,1.0,\,1.5,\,0.5]^{T}\), \(T_{\max}=[1.5,\,2.5,\,2.0,\,2.0,\,2.5,\,1.5]^{T}\). The stability constraint \(\psi>const=0.35\). Therefore, comparisons are made in two cases. Case 1: stability constraints are not considered. Case 2: stability constraints are considered (\(\psi>0.35\)). Then, the improved NSGA-II algorithm is run, and the two situations are compared and analyzed, as shown in the Figure 8.  
The Pareto-optimal solutions based on time and energy are solved by the improved NSGA-II optimization algorithm in Figure 8. The selection principle of the reference points is the average of the maximum and minimum values of the Pareto-optimal solutions. It can be found that the reference points in two cases are basically near points (8.1, 1634.9) and (9.5, 1320.9), according to the many optimization results. We set the weight vector as \([0.4,\,0.5]^{T}\) in the Euclidean distance between the Pareto solution and the reference point. The threshold in Euclidean distance-dominated ranking is \(\delta=0.001\). The solutions of interest to the decision-maker are shown in the Figure 9. Then, 10 groups of optimal solutions are selected according to the population size, as shown in Tables 3 and 4. Figure 8: Comparison of two cases. (**a**) Optimization case 1. (**b**) Optimization case 2. Figure 7: The robot motion scheme. The optimal solution is selected according to the principle that the accelerations at the starting point and the end point are as small as possible; that is, the range of T\({}_{1}\) and T\({}_{6}\) is as large as possible. Therefore, the data of group 8 in Table 3 and group 7 in Table 4 are selected for analysis. The analysis results are shown in Table 5. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Serial Number** & \(\mathbf{T_{1}}(s)\) & \(\mathbf{T_{2}}(s)\) & \(\mathbf{T_{3}}(s)\) & \(\mathbf{T_{4}}(s)\) & \(\mathbf{T_{5}}(s)\) & \(\mathbf{T_{6}}(s)\) & \(\mathbf{T_{all}}(s)\) & \(\mathbf{E}(J)\) \\ \hline
1 & 1.2780 & 1.9981 & 1.4066 & 1.1603 & 1.6607 & 1.3595 & 8.8632 & 1233.8 \\
2 & 1.2460 & 1.9310 & 1.3428 & 1.2604 & 1.7834 & 1.3281 & 8.8917 & 1233.3 \\
3 & 1.2637 & 2.0259 & 1.4131 & 1.1594 & 1.7355 & 1.3137 & 8.9113 & 1232.9 \\
4 & 1.2951 & 2.0069 & 1.4217 & 1.1844 & 1.6918 & 1.3372 & 8.9371 & 1216.5 \\
5 & 1.2380 & 2.0180 & 1.4084 & 1.2698 & 1.7691 & 1.3068 & 9.0101 & 1199.1 \\
6 & 1.2898 & 2.0062 & 1.3376 & 1.1794 & 1.8908 & 1.3522 & 9.0560 & 1186.9 \\
7 & 1.2999 & 1.9775 & 1.3980 & 1.2375 & 1.8474 & 1.3625 & 9.1228 & 1174.6 \\
8 & 1.2925 & 1.9080 & 1.3222 & 1.4640 & 1.7820 & 1.3618 & 9.1305 & 1171.7 \\
9 & 1.2660 & 1.9884 & 1.3046 & 1.3985 & 1.8742 & 1.3101 & 9.1418 & 1164.5 \\
10 & 1.2420 & 1.8977 & 1.4467 & 1.3512 & 1.8872 & 1.3200 & 9.1448 & 1162.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Optimal solutions of case 2. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Optimal Solution** & \begin{tabular}{c} **Optimization** \\ **Case 1** \\ \end{tabular} & \begin{tabular}{c} **Optimization** \\ **Case 2** \\ \end{tabular} & \
\begin{tabular}{c} **Variation** \\ **Change** \\ \end{tabular} \\ \hline Energy-optimal solution \(E(J)\) & 1436.6 & 1174.6 &
\begin{tabular}{c} Down \(\downarrow\) \\ Up \(\uparrow\) \\ \end{tabular} & 18.20\% \\ Time-optimal solution \(T_{all}(s)\) & 8.1983 & 9.1228 & Up \(\uparrow\) & 11.28\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Analysis of the two cases. Figure 9: Reference point and optimal solutions of two cases. (**a**) Optimization case 1. (**b**) Optimization case 2. It can be seen from Table 5 that the optimal solution in optimization case 2 increases by 11.28% compared with that in optimization case 1; however, the optimal solution in energy reduces by 18.20%.