# Path Planning for Conformal Antenna Surface Detection Based on Improved Genetic Algorithm
## 3 Improved Genetic Algorithm Combining the Historical Optimal Population
### Adaptive Crossover and Mutation Operators  
The crossover operation tries to retain the good genes of the parents by simulating the reproductive behavior found in nature. The crossover operation is done through crossover operators, usually including partially mapped crossover (PMX), sequential crossover, multi-parent partially mapped crossover (MPPMX), multi-parent sequential crossover, etc. [14]. These crossover operators generally feature measures to avoid repetition and have achieved good results in solving TSPs. In this paper, a sequential crossover operator with a good effect was used to perform the crossover operation, and the operator was slightly improved. The crossover process of the operator is shown in Figure 4. The improvement was mainly reflected in the generation of offspring 2, that is, the selected fragment of parent 2 was placed at the front of offspring 2 instead of at the same position as in parent 2. This kind of operation improves the diversity of the population to a certain extent. It also prevents the problem that the original sequential crossover operator can only produce two individuals that are exactly the same as the parents when the two parents are the same in a few cases. As an example, two identical parents, (1 2 3 4 5), were randomly selected for crossover at two positions: (1 \(\mid\) 2 3 4 \(\mid\) 5). The crossover results were (1 2 3 4 5) and (2 3 4 1 5), respectively, producing new individuals instead of the same individuals as the parents. The mutation operation changes the genetic information of organisms by simulating the variation of chromosomes. Embodied in the genetic algorithm, it can avoid iterations falling into a local optimal solution, and there is a certain probability of jumping out of the local optimal solution. The mutation operation is completed by mutation operators, usually including two-point exchange mutation, 2-opt, sliding mutation, partial reverse sequence mutation, center inverse mutation, etc. To overcome the limitations of using a single mutation operator, this work used a combination of multiple mutation operators to perform mutation operations. A schematic diagram of some mutation operations is shown in Figure 5. Among them, 2-opt mainly plays the role of eliminating crossover  
Figure 4: Schematic diagram of improved sequential crossover operator. between lines. First, the lines that are impossible to intersect are eliminated through the rapid repulsion test. Second, the intersecting lines are quickly located through the straddle test, so as to eliminate the intersection, as shown in Figure 6. Compared with the traditional method of calculating the intersection point of two straight lines and judging whether the intersection point is within the range of the line segment [22], this method has a small amount of calculation and faster operation speed. In addition to the implementation of crossover and mutation operators, the probability of crossover and mutation will also affect the iteration results. Taking the crossover operator as an example, a larger crossover probability will increase the computational cost, while a smaller crossover probability will cause the search to slow down. The research of [PERSON] et al. [23] showed that when solving medium- and small-scale TSPs, the linear decrease of the crossover probability and the linear increase of the mutation probability can produce better iterative results. This work built on this foundation by introducing  
Figure 5: Mutation operation. Figure 6: (**a**) The rapid repulsion test; (**b**) the straddle test. Bold **ad**, **cd**, etc. represent vectors. an adaptive adjustment strategy of crossover and mutation probability: when the fitness did not change for multiple generations, the change speed of crossover probability and mutation probability was accelerated, as shown in Formulas (3) and (4). \[P_{cross}=\left\{ \begin{array}{l}P_{cross}-\frac{1}{\frac{2}{GENEKATION}},\,if\, \,t<THRESHOLD/10\\ P_{cross}-\frac{2}{GENEKATION},\,if\,\,t\geq THRESHOLD/10\end{array} \right.