# Path Planning of Mobile Robots with an Improved Grey Wolf Optimizer and Dynamic Window Approach
## 6 Experiments Simulation and Analysis  
### Algorithm Testing and Analysis  
To validate the performance of the PAGWO algorithm proposed in this study, we systematically selected a diverse set of benchmark functions commonly employed in optimization, including different problem types (as shown in Table 1), to ensure comprehensive evaluation. \(f_{1}\)-\(f_{6}\) are single-peak functions that test the algorithm's essential optimization ability and convergence speed; \(f_{7}\)-\(f_{11}\) are multimodal functions to evaluate the algorithm's global search capability, particularly its ability to find the global optimum in complex search spaces; \(f_{12}\) and \(f_{13}\) are low-dimensional fixed functions to ensure that the algorithm can balance global and local search; \(f_{14}\) and \(f_{15}\) are composite functions used to assess the robustness of algorithms in handling complex, multimodal problems. Subsequently, the algorithms are tested independently. The selected algorithms and parameters, as shown in Table 2, are tested using a population size of 50 and 1000 iterations. To ensure the validity of the test and maintain objectivity, each algorithm is independently tested 30 times, using the average, standard deviation, and optimal value as evaluation metrics. The test results are shown in Table 3. \begin{table}
\begin{tabular}{c c c c c} \hline
**F** & **Function** & **Dim** & **Range** & **Best** \\ \hline \(f_{1}\) & Sphere Model & 30/100 & \([-100,\,100]\) & 0 \\ \(f_{2}\) & [PERSON]’s problem 2.22 & 30/100 & \([-10,\,10]\) & 0 \\ \(f_{3}\) & [PERSON]’s problem 1.2 & 30/100 & \([-10,\,10]\) & 0 \\ \(f_{4}\) & [PERSON]’s problem 2.21 & 30/100 & \([-100,\,100]\) & 0 \\ \(f_{5}\) & Quartic Function & 30/100 & \([-1.28,\,1.28]\) & 0 \\ \(f_{6}\) & Shifted Zakharov Function & 10 & \([-100,\,100]\) & 300 \\ \(f_{7}\) & Generalized Rastrigin’s Function & 30/100 & \([-512,\,5.12]\) & 0 \\ \(f_{8}\) & Ackley’s Function & 30/100 & \([-32,\,32]\) & 0 \\ \(f_{9}\) & Generalized Griewank Function & 30/100 & \([-600,\,600]\) & 0 \\ \(f_{10}\) & Schaffer F7 Function & 10 & \([-100,\,100]\) & 600 \\ \(f_{11}\) & Shifted Levy Function & 10 & \([-100,\,100]\) & 900 \\ \(f_{12}\) & Kowalik’s Function & 4 & \([-5,\,5]\) & 0.0003 \\ \(f_{13}\) & Goldstein-Price Function & 2 & \([-2,\,2]\) & 3 \\ \(f_{14}\) & Composition Function 2 & 10 & \([-100,\,100]\) & 2400 \\ \(f_{15}\) & Composition Function 7 & 10 & \([-100,\,100]\) & 2700 \\ \hline \end{tabular}
\end{table} 
Table 1: International standard test function. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Algorithms** & **GA** & **SCA** & **PSO** & **GWO** & **PAGWO** \\ \hline \multirow{4}{*}{Parameter configuration} & \(p_{c}=0.8\); & \multirow{4}{*}{\(a=2\); \(Pop=50\); \(t_{m}=1000\)} & \(w=0.8\); & \multirow{4}{*}{\(p=0.3\); \(t_{1}=1.5\); \(Pop=50\); \(t_{m}=600\)} \\  & & & & \\ \cline{1-1}  & & & & \\ \cline{1-1}  & & & & \\ \hline \end{tabular}
\end{table} 
Table 2: Algorithmic parameter. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Function** & **Metric** & **GA** & **SCA** & **PSO** & **GWO** & **PAGWO** \\ \hline \multirow{4}{*}{\(f_{1}\)} & Mean & \(4.52\times 10^{3}\) & \(4.19\times 10^{-3}\) & \(4.46\times 10^{-6}\) & \(9.31\times 10^{-71}\) & \(4.22\times 10^{-156}\) \\  & Std & \(3.47\times 10^{3}\) & \(9.05\times 10^{-3}\) & \(4.27\times 10^{-6}\) & \(1.57\times 10^{-70}\) & \(2.31\times 10^{-155}\) \\  & Best & \(4.58\times 10^{2}\) & \(6.76\times 10^{-7}\) & \(3.19\times 10^{-7}\) & \(9.39\times 10^{-73}\) & \(3.75\times 10^{-165}\) \\ \hline \multirow{4}{*}{\(f_{2}\)} & Mean & \(3.62\times 10^{1}\) & \(5.70\times 10^{-6}\) & \(1.25\times 10^{-4}\) & \(4.71\times 10^{-41}\) & \(5.57\times 10^{-90}\) \\  & Std & \(1.14\times 10^{1}\) & \(1.42\times 10^{-5}\) & \(7.70\times 10^{-5}\) & \(4.97\times 10^{-41}\) & \(1.80\times 10^{-89}\) \\ \cline{1-1}  & Best & \(1.61\times 10^{1}\) & \(3.34\times 10^{-8}\) & \(2.53\times 10^{-5}\) & \(4.09\times 10^{-42}\) & \(1.39\times 10^{-92}\) \\ \hline \end{tabular}
\end{table}