# Research on Path Planning Multiple Mobile Robots Based on the LAPGWO Algorithm
## 3 Methodology
### LAPGWO Algorithm
#### 3.1.2 Improvement and Optimization Based on Logistic Chaotic Mapping  
The GWO algorithm uses randomly generated data in the initial map as the position information of the initial population. This random generation method makes it challenging to ensure that the population can cover the entire map, and the grey wolf population cannot be evenly distributed. Therefore, in multi-modal or complex optimization problems, it is easy to miss the global optimal solution. Chaotic mapping has strong randomness and ergodicity, and the chaotic sequence generated by the logistic mapping has even stronger randomness and sensitivity to the initial values, making it suitable for scenarios that require a high perturbation intensity. For this reason, in this paper, the principle of logistic chaotic mapping is selected to generate a uniform chaotic sequence, which is then introduced into the individual search space to develop the individual position sequence in the initial stage of the GWO algorithm. This aims to increase the diversity of the initial population's position distribution and improve the GWO algorithm's convergence efficiency. The mathematical expression is as follows:  
\[X_{(i+1)}=\mu\cdot X_{i}\cdot(1-X_{i}) \tag{8}\]  
Among them, \(X_{i}\in[0,1]\) represents the state value of the \(i\)-th iteration, and \(\mu\in[3.57,4]\) represents the chaotic interval. After combining with the GWO algorithm, the initial positions of the population are generated by the logistic chaotic sequence, replacing the traditional uniform random initialization. The expression is as follows:  
\[X_{t}^{0}=Ib+X_{c}^{(k)}\cdot(ub-Ib) \tag{9}\]  
Among them, \(X_{t}^{0}\) represents the initial position of the \(t\)-th wolf; \(X_{c}^{(k)}\) represents a specific term in the chaotic sequence \(X_{(i+1)}\) generated by the logistic mapping iteration,represents the \(k\)-th iteration, and \(lb\) and \(ub\) represent the lower and upper bounds of the search, respectively.  
The search range can be dynamically adjusted by adjusting the value of \(\mu\). In the early search stage, increasing the value of \(\mu\) can enhance the global exploration ability. In the later stage, reducing the value of \(\mu\) can focus on the local area. Meanwhile, during the grey wolves' position-updating process, logistic chaotic perturbation is introduced to avoid premature convergence. The improved position-updating expression is as follows:  
\[X_{(i+1)}=\frac{X_{1}(i)+X_{2}(i)+X_{3}(i)}{3}+\eta\cdot X_{c}^{(i)} \tag{10}\]  
When allocating tasks, the robots' trajectories in completing them are comprehensively taken into account. The task allocation for each robot should preferably conform to the robots' spatiotemporal kinematic trajectories. The next task point should preferably be in the same direction as the line connecting the completed task point, forming a joint optimization with the path's linearity.  
The spatiotemporal constraint \(S_{p}\) for the coherence of the robot's path is as follows:  
\[S_{p}=\frac{||V_{n}\cdot d_{k}||}{||V_{n}||\cdot||d_{k}||}\,e^{-\beta\cdot L_{ d}} \tag{11}\]  
Among them, \(\eta\) is the perturbation intensity coefficient, and \(X_{c}^{(i)}\) is the chaotic value at the current moment. When dealing with high-dimensional problems, the value of \(\eta\) can be adjusted downward to avoid excessive oscillations. In the early search stage, the perturbation term \(\eta\cdot X_{c}^{(i)}\) and the relatively large \(\mu\) can jointly enhance the global exploration ability. In the later stage of the search, although the value of \(\mu\) is reduced to focus on the local area, the perturbation term \(\eta\cdot X_{c}^{(i)}\) will keep running to prevent the algorithm from becoming trapped in the local optimum. It injects new changes into the search process, guiding the algorithm to break out of the local optimum and continue to search for the global optimum.