## 4 Results and Discussion
### Trajectory Planning for AUV  
In the first simulation test, we assess the effectiveness of the proposed algorithms within a 3D environment by utilizing 13 sensor nodes. These sensor nodes are randomly and uniformly distributed within a square area measuring 0.25 km \(\times\) 0.25 km \(\times\) 0.25 km, with a surface station positioned at the center of the sea surface. Figure 2 illustrates the trajectory planning processes of an AUV using three algorithms: boundary tour ACO (BTACO), boundary tour adaptive ACO (BTAACO), and boundary tour modified ACO (BTMACO). BTACO is a hybrid algorithm that combines traditional ACO with BT, whereas BTAACO is a hybrid algorithm that just uses adaptive parameters and lacks an inheritance method. The black dots in Figure 2 represent the sensor node locations, and the red dots represent the best location for the AUV to start collecting information from the sensor nodes. The sensor nodes send acoustic signals to the AUV within the transmission range marked with blue circles. Figure 2: The optimal coverage path generated by three algorithms with 13 sensor nodes distributed within 0.25 km \(\times\) 0.25 km \(\times\) 0.25 km: (**a**) BTACO, (**b**) BTAACO, and (**c**) BTMACO. As depicted in Figure 2a, the optimal tour generated by BTACO reveals that the sequence of sensor visits does not follow the nearest-neighbor approach, leading to a longer route for the AUV and consequently requiring more energy to complete its task of gathering information from the nodes. The shortcomings of this algorithm stem from its inability to adjust to the specific dynamics of the system, such as pheromone levels, among other factors. Conversely, the adaptive variant of ACO involves making modifications throughout the path selection process, influenced by dynamic elements such as the pheromone and heuristic exponential weights. This indicates that the algorithm possesses a level of flexibility, allowing it to alter its route according to real-time conditions to determine the optimal sequence of nodes to visit, ultimately conserving the energy of the AUV and extending its lifespan during navigation. As illustrated in Figure 2b, the AUV strategically avoids sensors located far from the starting point and in the midst of its route. Instead, it aims to visit neighboring sensors sequentially, which leads to the establishment of the shortest path and helps prevent excessive energy depletion of the AUV. This adaptive approach enhances its efficiency compared with the traditional BTACO in situations where the selection process is variable and energy management is essential. Furthermore, it is important to highlight that the method proposed in this study demonstrated exceptional performance in selecting the shortest path when compared with the alternative method. As shown in Figure 2c, the AUV travels the shortest distances for several transitions between a certain node and the next one, in an area close to the desired sensor node and close to the boundaries of the next sensor node. These organized transitions allow the AUV to visit sensors to collect data with the least possible energy consumption, especially when the sensor network is expanded. Additionally, we notice that the trajectory that was constructed is established in a ring format, which indicates that the BTMACO algorithm established the main tour perfectly with the best order of sensor nodes, but since the data transmission areas are within certain limits and from all directions, it led to a slight increase in energy consumption compared with BTAACO. On the other hand, with the increase in the search area and the number of sensors, the proposed method performed the best among its peers. This enhancement can be attributed to several factors: The revised approach advances performance by incorporating additional enhancements, such as the integration of more advanced trajectory planning techniques aimed at improving both energy efficiency and coverage.