# Mobile Robot Path Planning Based on Fused Multi-Strategy White Shark Optimisation Algorithm
## 4 Experimental Simulation
### Algorithm Time Complexity Calculation  
In In terms of time complexity analysis, the time complexity of the WSO algorithm is primarily determined by three key parameters: population size N, maximum iteration count TMAX, and individual dimension d. During the population random initialization phase, the time complexity is O(N \(\times\) d); during the main iteration phase, the time complexity is O(N \(\times\) T\({}_{\text{max}}\)\(\times\) d). In the FMWSO algorithm, during the initialization phase of the elite population using the chaotic elite pool strategy, the time complexity is O (N \(\times\) d); during the iteration phase, the adaptive weights are dynamically adjusted based on the iteration progress, and the weight calculation for each individual is an independent operation. The total time complexity for this phase is O(N \(\times\) T\({}_{\text{max}}\)), which is a lower-order term in the iteration process and does not affect the overall complexity level of the algorithm. In the core iteration stage, an improved simulated annealing mechanism is embedded into the individual position update process of WSO. Each position update requires adjusting the d-dimensional parameters based on the annealing criteria. In each iteration round, the d-dimensional position update for each individual involves fusion operations, resulting in a time complexity of O(N \(\times\) T\({}_{\text{max}}\)\(\times\) d), which is on the same level as the complexity of the original WSO core iteration, as the fusion operations do not alter the dominant term of \"population \(\times\) iteration \(\times\) dimension\". In the post-processing stage, a third-order Bessel curve is used to smooth the optimized path, requiring the calculation of control points and the generation of a continuous curve. The complexity of the smoothing operation for each node is O(1). Since the number of path nodes m is typically much smaller than N \(\times\) d, the time complexity of this stage is O(m), which can be simplified to a lower-order term and has no impact on the overall complexity. In summary, the dominant terms of the time complexity in each stage of FMWSO are consistent with those of the original WSO, and the overall complexity remains O(N \(\times\) T\({}_{\text{max}}\)\(\times\) d), indicating that the improved strategy enhances algorithm performance without increasing time complexity. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Function** & **Norm** & **PSO** & **SA** & **WSO** & **HWSO** & **IWSO** & **FMWSO** \\ \hline \multirow{4}{*}{\(f_{1}\)} & optimal & 3.403 \(\times\) 10\({}^{-1}\) & 1.755 \(\times\) 10\({}^{-1}\) & 4.167 \(\times\) 10\({}^{-8}\) & 2.985 \(\times\) 10\({}^{-31}\) & 2.352  
\(\rm R_{min}\) is the minimum turning radius; Equation (17) represents the constrained motion direction constraint: \(\rm v_{i}\) is the motion direction vector of segment \(\rm i\), \(\rm\phi_{i}\) is the robot's current orientation; \(\rm\theta_{max}\) is the maximum allowable steering angle; \(\rm P_{R}\) and \(\rm P_{D}\) are the minimum turning radius penalty and constrained motion direction penalty, respectively, and \(\rm k_{R}\) and \(\rm k_{D}\) are the penalty coefficients. In path planning experiments, the fitness function typically uses path length, which is simple but ignores turning points in the path. During robot movement, turning not only consumes a lot of energy but also reduces search efficiency. To address this issue, a path evaluation function was designed based on the above kinematic model. This function comprehensively evaluates path length, path smoothness, obstacle avoidance penalties, minimum turning radius, and restricted movement directions in path planning.