## 5 Improved NSGA-II Algorithm
### Calculation of Crowding Distance  
In the NSGA-II algorithm, the confirmation of crowding distance (CD) represents a crucial stage in the maintenance of population diversity [44]. The crowding distance is initially set to 0 for each solution, and the population is subsequently ranked in accordance with the specified optimization objectives. For the boundary solutions of each objective, the crowding distance is set to infinity, thereby ensuring that these boundary solutions are retained. The crowding distance value is then updated by calculating and normalizing the neighboring differences of each solution on that objective. With regard to both optimization objectives, the crowding distance is defined as the perimeter of the rectangle formed by the adjacent \(i-1\) and \(i+1\), as illustrated in Figure 8.  
\[CD_{int}=\frac{f_{m}(x_{i+1})-f_{m}(x_{i-1})}{f_{m}(x_{max})-f_{m}(x_{min})},i=1,2,\cdots\cdots,(l-1) \tag{37}\]  
\[CD_{i}=\sum_{m=1}^{M}CD_{im} \tag{38}\]  
Figure 8 illustrates that \(CD_{i}\) is diminished in size and more densely populated in the vicinity of the individual in question. In order to maintain the diversity of the population, it is necessary to compare the crowding degree operators in order to guarantee that the algorithm converges on a uniform Pareto front. Consequently, as a consequence of the non-dominated ordering and crowding calculation, each individual (i) in the population is characterized by two attributes: The Pareto rank and the crowding degree (CD) are two key attributes. In the event that two individuals possess disparate non-dominated rankings, the individual with the lower Pareto rank is selected. Conversely, if two individuals are in the same rank, the individual with the higher crowding degree is chosen, thereby promoting diversity within the population and averting the potential issue of premature convergence.  
Figure 9 shows the algorithm flow of Improved Non-dominated Genetic Algorithm (INSGA-II) based on elite strategy with the following steps:  
Figure 8: Calculate crowding distance.  
Figure 7: Pareto curve.  
1. Determine the time interval range \(t_{i}\in[D_{i},G_{i}]\) for each path point based on the maximum velocity \(v_{m}\) of each joint, where \(D_{i}\) and \(G_{i}\) represent the upper and lower time limits. In this paper, we take 1 s as the lower limit and 3 s as the upper limit. The initial parent population, designated as \(H_{0}\), is generated using the set of good points in the time interval range. The population size is set to N to ensure that the initial solution set is uniformly distributed. 2. The adaptation value of each population target is calculated according to Equations (5) and (6). Following the non-dominated sorting process, the initial generation of the offspring population, designated as \(I_{1}\), is generated through the application of three fundamental genetic algorithm operations: selection, crossover, and mutation. 3. From the second generation onwards, a parent selection method based on linear ranking was used to identify parent individuals for fusion with the offspring population, thereby generating a population \(P_{i}\) with a population size of 2N. The new population adaptation value was then calculated. 4. The elite selection strategy entails non-dominated sorting of 2N individuals in the population \(P_{i}\) according to fitness, with the crowding degree of individuals in the same dominance layer calculated according to Equation (38). New individuals are then selected based on dominance relationships and crowding degree values, forming the parent population \(H_{i+1}\) with a population number N. 5. Repeat steps 2-3 until the termination condition is satisfied, and output the Pareto optimal solution set.