## 4 Results and Discussion
### Trajectory Planning for AUV
Modifications may involve hybrid methods, including the use of pheromone update learning to forecast optimal paths, along with innovative strategies for managing node visitation sequences. By inheriting a specific sensor network order from the leading member within the population, the system aims to explore a greater number of locations within the search space. Even with advancements in the processing phase, this ensures the optimal selection of the sensor visitation order, which has a significant impact on energy conservation, particularly when accounting for variables such as the distance between nodes, battery longevity, and data transmission. To further assess the proposed model's performance against its competitors, we broadened the search space by incorporating additional sensor nodes. In this scenario, a grid network of 36 sensor nodes was established within a 0.6 km \(\times\) 0.6 km \(\times\) 0.6 km area, with a surface station located at the center of the sea surface. As illustrated in Figure 3a, the BTACO method shows limitations, particularly as the network grid expands. It often converges to suboptimal solutions due to the static characteristics of its search process, frequently adhering to fixed routes and lacking the necessary adaptability for large, dynamic settings. Conversely, BAACO modifies its trajectory in response to current conditions, leading to greater improvements in efficiency; as depicted in Figure 3b, its trajectory demonstrates more consistency compared with BTACO. Although this flexibility enhances performance, BTAACO still encounters difficulties when the network grid significantly increases, as it does not inherently generate the best possible tour but instead adjusts the existing path based on prevailing conditions. On the other hand, BTMACO represents a significant improvement, especially in larger networks with expanded search areas. This algorithm incorporates the dynamic restructuring of the tour during the search process and produces a smooth trajectory for the AUV, as shown in Figure 2(c). As the network grid expands, the modified version (BTMACO) does not simply adapt to existing conditions but seeks to generate new tours dynamically throughout the exploration process. This restructuring allows the algorithm to find optimal paths wherein the travel distance is reduced along with the energy consumption being minimized for the AUV. The performance differences among these methods become apparent as the search area expands and the network grid develops. In these kinds of situations, BTMACO shines, as it keeps changing its search approach. AUVs follow a dynamic path instead of following a fixed route or making only minor changes. This structured dynamism helps the modified version outperform its peers; it avoids the pitfalls of getting stuck in suboptimal paths (common in standard ACO) or overrelying on current network conditions (as in BTAACO). BTMACO achieves better results by seeking out new energy-efficient tours, even as the complexity of the network increases. Figure 4 shows the rate of convergence for finding the path length with generation. As shown in Figure 4, BTMACO seeks to explore the search space deeply as the smallest path length constricted in the first phase (MACO). This is because MACO dynamically updates the pheromone level and a new tour is generated depending on the dominant tour. In the second phase, the hybrid method (i.e., BT) explores the best locations of the transmission areas of the sensor nodes, which in turn leads to minimizing the energy consumption of the AUV by minimizing its journey length. Another test applied to the proposed algorithm with different settings and objectives was conducted, and we found that the proposed BTMACO algorithm makes the AUV consume less energy among the competitors, about 11,509.6 kJ, which is less than the 11,759.6 kJ and 12,184.4 kJ consumed by the AUV when using the BTAACO and BTACO algorithms, respectively.