# Research on Path Planning Multiple Mobile Robots Based on the LAPGWO Algorithm
## 4 Results
### Path Simulation Comparison
#### 4.2.3 Determination of Key Parameters  
The value range of the chaotic interval determines whether the algorithm can have a strong global search ability in the early stage and focus the search later to improve the search accuracy. The value of the repulsion gain coefficient determines whether it can ensure that appropriate repulsion can be provided for the robot in different scenarios, thus guaranteeing the safety and efficiency of the movement. The perturbation intensity coefficient ensures the algorithm can avoid excessive perturbation, guide the algorithm to jump out of the local optimum, and find the global optimum solution. This paper's selection of parameters is mainly based on a large amount of experimental data.  
Figure 5: Simple Warehouse Map.  
To verify the influence of the above key parameters on searchability, this paper will first elaborate on why the range of each parameter is selected and then use the central composite design (CCD) method to find the optimal combination of each parameter.  
The chaotic interval of the grey wolf population is improved by adding the logistic chaotic mapping. However, the range of the chaotic interval is not the larger, the better. Because for the logistic mapping equation \(X_{n+1}=\mu X_{n}(1-X_{n})\), when \(\mu\in(3.57-4)\), the system is in the chaotic region, and the Lyapunov exponent \(>0\). The system enters the divergence region when \(\mu>4\). The chaotic interval diagram is shown in Figure 6.  
The selection of the ranges of the gain coefficient \(\rho(t)\) and the perturbation intensity coefficient \(\eta\) is shown in Figure 7a,b.  
Through a large amount of experimental data, the approximate parameter interval range has been obtained. However, considering the need to find the optimal parameter combination, this paper will use the central composite design (CCD) method to find the optimal parameters, and the results are shown in Table 4.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Experimental Number** & \(\mu\) & \(\rho(t)\) & \(\eta\) & **Type** & **Path Length** & **Computation Time** & **Path Smoothness** \\ \hline
1 & 3.60 & 20 & 0.10 & Cubic Point & 17.10 & 3.41 & 0.24 \\ \hline
2 & 3.60 & 35 & 0.27 & Cubic Point & 16.44 & 2.42 & 0.16 \\ \hline
3 & 3.60 & 40 & 0.30 & Cubic Point & 17.57 & 3.27 & 0.25 \\ \hline \end{tabular}
\end{table}
Table 4: Test results of key parameters.  
Figure 6: Distribution map of the chaotic interval.  
Figure 7: The selection range of parameters. (**a**) The selection range of the gain coefficient \(\rho(\mathrm{t})\). (**b**) The selection range of the perturbation intensity coefficient \(\eta\).  
This paper will adopt the method of controlling variables. Under two scenarios and with other variables remaining unchanged, different values will be successively selected for the three parameters to determine the optimal value of each parameter. The specific results are shown in Table 4. For the convenience of result comparison, this paper will normalize the three evaluation indicators of \(\mu\), \(\rho(t)\), and \(\eta\) and present the results in Figure 5. According to the results, the combination of optimal parameters will be determined.  
The abscissa of Figure 8 is the experimental number, and each experimental label corresponds to a set of parameter combinations. The blue solid line represents the path length, the red dashed line represents the computation time, and the green dashed line represents the path smoothness. According to the data in the CCD table and Figure 8, different parameter values impact the algorithm's performance differently. The combination with the experimental number 16 has relatively low values regarding path length, computation time, and path smoothness and is, thus, determined as the optimal parameter combination. Through testing, the following optimal parameter values are obtained