## 5 Improved NSGA-II Algorithm
### Solving Pareto Optimal Solution Set  
In this study, the woodworking manipulator and worktable are taken as the research object. Four path points, designated A(a), B, C(c), and D, are set in the space. The actuator ends of the woodworking manipulator and worktable overlap at the two points B and D, which are used to simulate the machining operation at these two points. Consequently, the manipulator and the worktable can be regarded as woodworking equipment with seven joints. The values of the joint variables of the joints are shown in Table 3. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Path Points** & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline A(a) & 0.800 m & 1.200 m & 3.141 rad & 0.000 rad & 0.400 m & 0.500 m & 0.000 rad \\ B & 0.578 m & 0.868 m & 1.897 rad & 1.047 rad & 0.236 m & 0.359 m & \(-1.402\) rad \\ C(c) & 0.668 m & 0.567 m & 1.468 rad & 1.711 rad & 0.344 m & 0.407 m & \(-0.785\) rad \\ D & 0.459 m & 0.964 m & 2.355 rad & 0.754 rad & 0.325 m & 0.310 m & \(-0.098\) rad \\ \hline \end{tabular}
\end{table} 
Table 3: Seven joint positions and angle of path points. Figure 11: Objective function comparison. Figure 12: Effect of multi-objective iterative optimization. The enhanced NSGA-II algorithm is used to identify the optimal Pareto solution set within the context of multi-objective optimization. The INSGA-II algorithm has been simulated, and the algorithm's parameters have been configured as follows: The population size is 200, the upper bound is 3, the lower bound is 1, the crossover rate is 0.9, the variance rate is 0.1, the indexes of hybridization and variance distributions are both 20, and the number of iterations is 50. The optimization objective is to minimize both the shortest time and the smallest jerk. The optimization objective function is \(S_{1}\) and \(S_{2}\), and the decision variables are three trajectory times (\(t_{1}\), \(t_{2}\), and \(t_{3}\)). The velocity (\(v\)), acceleration (\(\alpha\)), and jerk (\(j\)) constraints are imposed on the individual joints to obtain the Pareto frontier curves after trajectory optimization through the NSGA-II algorithm. These results are shown in Figure 13.  
As illustrated in Figure 12, point A represents the time-first optimal solution, point C denotes the jerk-first optimal solution, and point B signifies the balanced optimization good point. As the point of reference is approached, the time required is reduced, and the joint jerk increases. Conversely, as the point of reference is approached, the time required is increased, and the joint jerk decreases. The time and jerk are two coupled and contradictory motion indexes that restrict each other's optimal performance. In order to eliminate the influence of different features on parameter selection and to facilitate the processing and comparison of data, the trajectory running time and jerk are normalized and scaled to a target range of \([0,1]\). This normalization allows for a fair comparison of the advantages and disadvantages of different solutions. The normalization transformation formula is as follows:  
\[S^{*}=\frac{S-\min(S)}{\max(S)-\min(S)} \tag{39}\]  
where \(S^{*}\) is the normalized \([0,1]\) range function value, and \(S\) is the original function value. In order to select a solution from the Pareto curve that is applicable to actual working conditions, it is necessary to introduce weight parameters \(\omega_{1}\) and \(\omega_{2}\) to define a weight function as follows:  
\[S_{c}=\omega_{1}S_{1}^{*}+\omega_{2}S_{2}^{*} \tag{40}\]  
where \(S_{c}\) is the normalized weight function, \(S_{1}^{*}\) is the normalized running time, and \(S_{2}^{*}\) is the normalized total jerk. Figure 13: Time-jerk Pareto optimal solution set based on INSGA-II algorithm. Figures 14-17 illustrate the displacement, velocity, acceleration, and jerk curves of the manipulator and worktable. The displacement, velocity, and acceleration curves of each joint are characterized by smooth, continuous, and uninterrupted behavior, and the jerk is also continuous and uninterrupted.