# Research on Path Planning Multiple Mobile Robots Based on the LAPGWO Algorithm
## 4 Results
### Comparison of Convergence Curves  
Four standard test functions are selected in the experiment, and the detailed information is shown in Table 2. Among them, the unimodal function is \(F_{1}\sim F_{2}\), which is used to test the local search ability of the algorithm; the multimodal function is \(F_{3}\sim F_{4}\), which is used to test the algorithm's convergence and global search ability. The convergence curves of the four algorithms are shown in Figure 2.  
As can be seen from Table 3, compared with the other four algorithms, the LAPGWO algorithm obtains an ideal average value, which indicates that the LAPGWO algorithm has better optimization performance; at the same time, the obtained standard deviations are all relatively small, indicating that the improved algorithm has good stability and enhanced local search ability. The LAPGWO algorithm successfully reaches its theoretical optimal value, that is, 0, in the tests of functions [list the functions here if there are specific ones]. This shows that, compared with other algorithms, the average value obtained by the LAPGWO algorithm in the test functions is the best, reflecting that the LAPGWO algorithm has stronger convergence, higher optimal accuracy, and greater stability. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Setting Values** \\ \hline Perturbation intensity coefficient \(\eta\) & 0.1–0.3 \\ Gravitational gain coefficient \(\zeta\) & 0.5–2 \\ Correction coefficient \(\gamma\) & 0.1–0.5 \\ Repulsive force gain coefficient \(\rho(t)\) & 20–40 \\ Adaptive potential field weight \(\lambda\) & 0.35 \\ Nonlinear factor \(\psi\) & 0.6 \\ Influence range of obstacles \(d_{o}\) & 2–3 m \\ Safety distance threshold between robots \(d_{s}\) & 1.5 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Standard Test Function** & **Dimension** & **Search** & **Minimum** \\ \hline \(F_{1}(x)=\sum\limits_{i-1}^{d}\left(\sum\limits_{j}^{i}x_{j}\right)^{2}\) & 30 & \([-100,100]\) & 0 \\ \hline \(F_{2}(x)=\sum\limits_{i-1}^{d}\left[100\left(x_{i+1}-x_{i}^{2}\right)^{2}+(x_ {i}-1)^{2}\right]\ \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multicolumn{2}{c}{**GWO**} & \multicolumn{2}{c}{**LM-GWO**} & \multicolumn{2}{c}{**DSA-EGA**} & \multicolumn{2}{c}{**LAPGWO**} & \multicolumn{2}{c}{**CWSO**} \\ \cline{2-10}  & **Average Value** & **Standard** & **Average Value** & **Standard** & **Average Value** & **Standard** & **Average Value** & **Standard** & **Average Value** & **Standard** & **Average Value** & **Standard** & **Deviation** \\ \hline \(F_{1}(x)\) & \(6.45\times 10^{-3}\) & \(7.123\)-\(04\) & \(9.47\times 10^{-9}\) & \(2.65\times 10^{-30}\) & \(3.354\times 10^{7}\) & \(3.151\times 10^{1}\) & \(0\) & \(0\) & \(-1.46\times 10^{-13}\) & \(1.22\times 10^{-7}\) \\ \hline \(F_{2}(x)\) & \(2.97\times 10\) & \(2.86\times 10\) & \(2.68\times 10\) & \(2.63\times 10\) & \(2.91\times 10\) & \(2.86\times 10\) & \(2.59\times 10\) & \(2.51\times 10\) & \(0.000197\) & \(0\) \\ \hline \(F_{3}(x)\) & \(2.80\) & \(4.10\) & \(0\) & \(0\) & \(0\) & \(0\) & \(0\) & \(0\) & \(0\) & \(0\) \\ \hline \(F_{4}(x)\) & \(4.29\times 10^{-4}\) & \(5.89\times 10^{-4}\) & \(1.29\times 10^{-271}\) & \(0\) & \(3.72\times 10^{-4}\) & \(0\) & \(0\) & \(0\) & \(8.83\times 10^{-16}\) & \(0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance comparison. From the convergence curves in Figure 3, it can be seen that, compared with the other four algorithms, the convergence curve of the LAPGWO algorithm shows a rapid decline in the early stage of the search, indicating that the strategy of using the logistic chaotic map to initialize the population can effectively improve the convergence rate in the initial stage of algorithm iteration; in the middle and late stages of the search, the LAPGWO algorithm can more accurately lock the global optimal solution.