## 3 Improved Motion Model Based on PE-Match
### Improved Mutation Strategy  
One of the common issues with particle swarm optimization is the tendency to prematurely converge. As the number of iterations increases, the population diversity decreases, leading the particle swarm to easily get stuck in local optima. In contrast, the mutation strategy of the differential evolution algorithm [29] can introduce disturbances to particles during the search process, thus possessing strong global search capabilities. Therefore, integrating the mutation strategy can enhance the global convergence ability of the particle swarm optimization algorithm.  
The traditional mutation strategy is to weight the difference between two random individuals in the population and add it to the third random individual to obtain the mutation individual, as shown in the following Equation (6):  
\[\left\{\begin{aligned} h_{i}=q_{r1}+f\\ f=W(q_{r2}-q_{r3})\end{aligned}\right. \tag{6}\]where \(q_{r1}\), \(q_{i2}\), \(q_{r3}\) are three particle individuals randomly selected in the population, with \(r1\
eq r2\
eq r3\); \(h_{i}\) is the \(ith\) mutant individual; \(W\) is the scaling factor; and \(f\) is the perturbation. According to Equation (6), it can be seen that in the early stage of the search due to the randomness of the initial distribution of the particle swarm resulting in a large discrepancy between particles, the traditional mutation method applies a large perturbation to the particles. As the search proceeds, the particle swarm gradually converges, the difference between particles becomes smaller, and the perturbation becomes smaller.  
The particle swarm optimization algorithm has the advantage of fast convergence in the early stage, but its drawback is that it easily gets stuck in local optima in the later stage. Directly applying traditional mutation strategies to the individual best values of the particle swarm can result in the individual best values being minimally disturbed in the late stage of convergence, making it difficult for the particle swarm to escape from local optima.  
To address the aforementioned issue, this paper proposes a new mutation strategy, the core idea of which is to reduce disturbances to particles when the differences between particles are significant in the early convergence stage of the particle swarm, thereby accelerating the early convergence process. When the differences between particles are small in the later stages of convergence, the degree of disturbance to particles is increased to help the particle swarm escape from local optima. The new mutation strategy is shown in Equation (7), written as follows:  
\[\left\{\begin{array}{l}h_{i}=q_{r1}+f(D)\\ \\ f(D)=\frac{\tilde{Dimension}}{1+\exp^{(\frac{r}{100\cdot D})}}-\frac{\tilde{Dimension }}{2}\\ \\ D=\frac{\frac{\tilde{\sigma}^{2}-q_{r2}}{\max(q_{r2}-q_{r2})}}{n}\end{array}\right. \tag{7}\]  
In the formula, \(Dimension\) is the search range of particles, \(D\) is the difference degree of particles, and \(n\) is the number of perturbations. Assuming that the search range \(Dimension=100\), the result of perturbation \(f(D)\) is shown in Figure 6, and the perturbation \(f(D)\) of particles decreases as the degree of difference \(D\) of particles increases.