# Mobile Robot Path Planning Based on Fused Multi-Strategy White Shark Optimisation Algorithm
## 4 Experimental Simulation
### Algorithm Time Complexity Calculation
The new fitness function is as follows:  
\[L=\sum\
olimits_{i=1}^{end-1}\sqrt{\left(x_{i+1}-x_{i}\right)^{2}+\left(y_{i+ 1}-y_{i}\right)^{2}} \tag{20}\]  
\[\theta=\arccos(\frac{b^{2}+c^{2}-a^{2}}{2 bc}) \tag{21}\]  
\[S=\sum\
olimits_{i=1}^{end}\theta_{i} \tag{22}\]  
\[C=k\sum map(i) \tag{23}\]  
\[fit=\alpha\cdot\frac{1}{L}+\beta\cdot\frac{1}{S}-\gamma\cdot C-\delta\cdot P_ {R}-\epsilon\cdot P_{D} \tag{24}\]  
In the formula: L is the path length, \(\rm\theta\) is the angle corresponding to the turning point, S is the path smoothness, which is the sum of the angles corresponding to all turning points, C is the obstacle avoidance penalty, where k is a large constant, map (i) is the map information of the current node, and \(\alpha,\beta,\gamma,\delta,\epsilon\) are the corresponding weight coefficients. To validate the feasibility and effectiveness of the FMWSO algorithm in path planning, we first conducted a simulation experiment on a relatively simple 20 \(\times\) 20 grid map, setting the starting point at (0.5, 19.5) and the endpoint at (19.5, 0.5), with the iteration count set to 50. The FMWSO algorithm was compared with WSO, D*, A*, and GWO. The main parameters for D*, A*, and GWO are shown in the Table 4, while the parameters for the other algorithms are consistent with those in Table 1. To ensure the reliability of the results, each algorithm was run independently 15 times. The path planning simulation results and iteration count curves for the five algorithms on a 20 \(\times\) 20 grid map are shown in Figures 6(a) and 6(b), respectively. As shown in Figure 6(a), WSO and GWO generate paths with more turning points and less smoothness; FMWSO, which uses third-order Bezier curves, produces paths with fewer sharp turns and redundant segments, indicating that FMWSO performs better in terms of the smoothness-length trade-off compared to other algorithms. As shown in Figure 6(b): In the early stages of iteration, FMWSO starts below WSO, D*, and other algorithms, and the curve drops rapidly, converging to the optimal solution much faster than other algorithms, highlighting the positive effect of the chaotic elite pool strategy on population initialization; In the middle of the iteration, WSO, D*, and other algorithms exhibit oscillations or plateaus, \begin{table}
\begin{tabular}{l l} \hline \hline
**Algorithm** & **Main Parameter** \\ \hline D* & \(c(s,n)=1/\infty,\triangle c=1\) \\ A* & \(f(n)=g(n)+h(n)\) \\ GWO & a decreases linearly from 2 to 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Main parameter. while FMWSO continues to decline steadily. This is due to the synergistic effects of adaptive weights and rapid annealing, with the former ensuring a reasonable search rhythm, while the latter avoids getting stuck in local optima through probabilistic jumps. In the late stages of iteration, compared to the slight fluctuations and declines of algorithms like A* and GWO, the FMWSO curve flattens out early, significantly accelerating convergence. The predation mechanism of the white shark algorithm enhances local fine-grained search, while improved annealing accelerates the convergence process. Together, these two factors maximize the algorithm's late-stage development capabilities. Table 5 presents the average simulation results from 15 runs on a 20 \(\times\) 20 map. The FMWSO algorithm demonstrates a significant advantage in path planning. In terms of path length, its shortest path of 27.841 is the lowest among all algorithms, 2.08% shorter than the second-best A* and 8.43% shorter than WSO, indicating stronger global search capabilities that effectively reduce energy consumption and task duration in engineering scenarios. In terms of path quality, the smoothness (path standard deviation) of 0.392 is the lowest, making the path more linear and simplifying motion control. Additionally, the number of turning points is the lowest at 5, reducing mechanical wear and improving control precision. In terms of stability, the average path length of 28.404 is extremely close to the shortest path length.