## 5 Simulation Results and Analysis
These oscillations emerge from the smoothing procedure, applied to enhance the trajectory's fidelity, and the addition of intermediate points via interpolation to mitigate abrupt heading transitions. They commonly occur just after heading changes, where the speed limit is nearly equal to the vehicle's maximum speed. This strategy minimizes the deviation from the desired route, thereby reducing collision risks in complex situations. The average computation time for each RL algorithm remains below 0.5 s (except for TD3-HER) primarily because the main computational load arises from simulating the full 6-DOF vehicle model. At every step, the simulator must compute the forces, torques, and kinematics, which dominates the runtime regardless of the specific RL approach. Only the HER algorithms incur the extra overhead from re-labeling the transitions in their Hindsight Experience Replay buffer, thus extending the total processing time. Additionally, algorithms that converge on shorter paths reduce the number of steps that the simulator must execute, further curbing the per-episode runtime. The better the policy, the shorter the time needed for computation, as the step number deceases. As a result, most RL methods converge in broadly similar overall computation times. In the classical algorithms, the simulation time spent by the AUV in following the computed trajectory is not included, as this step merely verifies the correctness of the path planning and trajectory generation. Notably, the RRT* and APF algorithms exhibit exceptionally low average computation times (0.04 s and 0.05 s, respectively), making them approximately ten times more computationally efficient compared to the other methods analyzed. This efficiency arises primarily from their simplified computational frameworks: the APF leverages straightforward potential field calculations, while the RRT* quickly explores the state space using incremental sampling and local rewiring without the computational overhead of exhaustive search. However, it is important to highlight that the APF does not achieve a high success rate, contrasting with the RRT*, which consistently provides feasible paths within minimal computation times. Nevertheless, the RRT* tends to yield suboptimal routes characterized by increased distances and prolonged mission durations. On the other hand, the total computation times for PSO and the GA, while higher than those of the RRT* and APF, remain comparable to those of the RL-based approaches and yield trajectories closely approximating theoretical optimality regarding the path length and mission duration. The highest computation time is observed for the A* algorithm (approximately 1.70 s on average), primarily due to its systematic and exhaustive graph search process, requiring the evaluation of numerous candidate paths to guarantee the optimal solutions. Regarding the average direct path percentage, most of the RL algorithms either achieved the goal or ended very close to it--often circling around the target, as seen especially for SAC-HER (Figure 5g) and TD3-HER (Figure 5i). This factor is also a reflection of the policy optimality for RL algorithms. If the agents reach the target, the factor is set to 100 %. For the GA and PSO, the goal was reached in every trial (100%). Slightly lower values were observed for the RRT* and A* (99% and 97%, respectively). The lowest value for this metric was observed for the APF (86.3%), which was associated with its relatively low success rate and the frequent occurrence of the vehicle becoming trapped in local minima when encountering obstacles. The feasibility-constrained mission time in RL-based approaches is typically proportional to the feasibility-constrained distance metric. Among the classical algorithms, the RRT* produced the longest mission times (an average of 418.9 s), reflecting the conservative speed limits it applies to accommodate numerous path segments. Although the RRT* demonstrated robust exploration and a short computation time, it also yielded routes that required advanced smoothing procedures.