## 5 Simulation Results and Analysis  
To compare the performance of RL-based models with that of classical path-finding algorithms, test data were generated in the form of 100 randomly selected target locations on the map presented in Figure 1d. To ensure consistency, each trial started from the center of the map, with the AUV oriented eastward (\(090^{\circ}\)) and an initial velocity of \(\frac{m}{s}\). The evaluation of the efficiency of the tested methods was carried out based on the following metrics:  
* **The average success rate** represents the average value for successful simulations. For each individual simulation, the success rate can take a value of either 0 or 1. If the vehicle reaches an area within a 10 m radius of the target point without colliding with any obstacles, this metric is assigned a value of 1. Otherwise, it is assigned a value of 0. * **Feasibility-constrained distance**--the average length of the traveled path, calculated based on 100 test cases. For each individual simulation, this metric is computed as the sum of the Euclidean distances between consecutive points along the simulated trajectory. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Algorithm** & **Time Complexity** & **Memory Usage** & **Hardware** \\ \hline A* & \(O(n^{2}\log n)\) & Moderate & CPU \\ RRT* & \(O(k\log k)\) & Moderate & CPU \\ GA/PSO & \(O(P\times f_{\text{evaluate}})\) & Varies & CPU \\ SAC/TD3 & \(O(\text{steps}\times\text{NN update})\) & Large (replay buffer) & GPU \\ TQC & Similar to TD3 but multi-critic & Larger & High-end GPU \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Approximate time complexity and hardware demands of the selected methods in this study. * **Total average speed--**the average speed, calculated as the sum of the average speeds from each individual simulation divided by 100. For a single trial, the average speed is determined as the total Euclidean distance traveled along the trajectory divided by the total episode duration. * **Reward** refers to the mean reward value (applicable to RL algorithms). * **Average standard deviation in the heading difference--**represents the standard deviation in the heading differences between consecutive time samples. * **The average computation time** refers to the processing time, which, for RL-based methods, is measured from the start of the simulation episode to its completion. In classical algorithms, it represents the total time required for path planning by the respective algorithm, smoothing the path, and determining the trajectory. * **The average direct path %** indicates the average percentage of direct path completion for the evaluated algorithms. For a single simulation, this percentage is calculated based on the ratio of the distance obtained from the projection of the final point of the trajectory onto the straight line connecting the start and target points to the total length of the computed trajectory. * **Feasibility-constrained mission time--**the total duration of the mission in seconds. For classical algorithms, this metric corresponds to the time required to execute the planned trajectory using the mathematical AUV model. An objective evaluation of the path length is significantly challenging in simulations where a collision has occurred in the obtained trajectory. To address this, an approach was adopted in which the feasibility-constrained distance metric was assigned a value corresponding to the highest result among all algorithms that successfully reached the target, plus an additional 25% of this value. The same principle is applied to the feasibility-constrained mission time metric. This approach aims to emphasize the impact of collisions within these metrics in a way that depends on the length of the path and the mission time in the \"worst-case\" scenario. This is justified by the fact that when the start and target points are farther apart, the discrepancies in the path length and the mission time between different algorithms become more pronounced.