## 5 Simulation Verification  
### Model Building  
The proposed method's feasibility was demonstrated by implementing it on a PUMA560 manipulator model in MATLAB, utilizing the data presented in Table 1. The model of the PUMA560 manipulator is shown in Figure 5:  
Considering the working performance and limits of each joint of the manipulator, the kinematic constraints are given in the Table 7:  
To fulfill the kinematic constraints of the manipulator, penalty functions were introduced on each objective function to ensure that the motion trajectory meets the kinematic constraints. The fitness function is designed as shown in Equation (27):  
\[\begin{split} f(1)&=(\sum\limits_{i=1}^{6}\sum \limits_{j=1}^{u}\left|P_{i,j}-P_{i,j+1}\right|)\times E_{a}+E_{b}\\ f(2)&=(\sum\limits_{i=1}^{9}t_{i})*E_{a}+E_{c}\end{split} \tag{27}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{5}{c}{**Test Function**} \\ \cline{2-5}  & **ZDT1** & **ZDT2** & **ZDT3** & **ZDT6** \\ \hline Traditional NSGA-II & 0.5318 & 0.2209 & 0.5274 & 0.3330 \\ MOPSO & 0.6992 & 0.1534 & 0.5930 & 0.3062 \\ MOEA/D & 0.7126 & 0.4083 & 0.5597 & 0.3885 \\ SPEA2 & 0.6027 & 0.3009 & 0.5564 & 0.3590 \\ KNSGA-II & 0.7175 & **0.5189** & **0.6375** & **0.4400** \\ AONSGA-II & 0.5908 & 0.2471 & 0.5744 & 0.3460 \\ Improved NSGA-II & **0.7181** & 0.4427 & 0.5986 & 0.3875 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of HV Mean Values.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{5}{c}{**Test Function**} \\ \cline{2-5}  & **ZDT1** & **ZDT2** & **ZDT3** & **ZDT6** \\ \hline Traditional NSGA-II & 0.5318 & 0.2209 & 0.5274 & 0.3330 \\ MOPSO & 0.6992 & 0.1534 & 0.5930 & 0.3062 \\ MOEA/D & 0.7126 & 0.4083 & 0.5597 & 0.3885 \\ SPEA2 & 0.6027 & 0.3009 & 0.5564 & 0.3590 \\ KNSGA-II & 0.7175 & **0.5189** & **0.6375** & **0.4400** \\ AONSGA-II & 0.5908 & 0.2471 & 0.5744 & 0.3460 \\ Improved NSGA-II & **0.7181** & 0.4427 & 0.5986 & 0.3875 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of IGD Mean Values.  
In the Equation (27), \(u\) represents the number of control points, and \(f(1)\) represents the joint increment of the six joints of the manipulator on the current motion trajectory, \(P_{i,j}\) represents the joint increment of the th joint based on the th control point of the quintic NURBS curve, \(f(2)\) represents the total running time of the manipulator on the current trajectory, and \(E_{a}\), \(E_{b}\), and \(E_{c}\) are penalty functions for each fitness function. When each joint exceeds the kinematic constraint, the penalty function is activated:  
\[\begin{array}{ll}E_{a}=&\sum\limits_{i=1}^{6}\left(\min P_{i}/p_{i,\min}+\max P _{i}/p_{i,\max}+\max\left|P_{i}\right|/V_{i,\max}\right.\\ &\left.+\max\left|P^{n}{}_{i}\right|/a_{i,\max}+\max\left|P^{(3)}{}_{i}\right| /I_{i,\max}\right)\end{array} \tag{28}\]  
The data of \(p_{\min}\), \(p_{\max}\), \(V_{\max}\), \(a_{\max}\) and \(J_{\max}\) from Tables 1 and 7, \(E_{b}\) and \(E_{c}\), are added once to the adaptation thresholds of \(1000^{\circ}\) and \(5\) s, respectively, otherwise \(E_{a}=1\), \(E_{b}=E_{c}=0\).