## 2 BIT* Algorithm Procedure
### A Performance-Enhanced Variant of the BIT* Algorithm  
#### 3.2.1 Underwater Energy Consumption Equation  
This research innovatively introduces the BIT* algorithm to generate initial paths and integrates hydrodynamic calculations to model the energy equation, thereby optimizing path planning for underwater engineering vehicles and addressing the issue of insufficient endurance. We have designed a weighting function that considers both distance and energy consumption to evaluate path selection. We utilize the BIT* algorithm to compute initial paths, and subsequently, we encode the BIT* path information into the state space of TD3. TD3, in turn, based on the initial path and perceived environmental information, outputs action commands and further incorporates a reward mechanism to guide TD3 to navigate along the BIT* path while simultaneously avoiding obstacles.  
Within the TD3 algorithm, we have streamlined underwater linear and steering kinematics to enhance computational speed, reduce computational time, and resolve the issue of vessel endurance. We have incorporated considerations of the BIT* path length and the number of turns, subsequently transforming these factors into an evaluation function,  
\[\text{f}(x,y)=\omega_{1}x+\partial\omega_{2}y, \tag{5}\]  
where \(x\) represents the distance of underwater linear motion, \(y\) denotes the number of turns, \(\delta\) is a steering angle correction parameter, \(\omega 1\) is the weight assigned to linear motion, and \(\omega 2\) is the weight assigned to turning maneuvers. Equation (5) is grounded in fluid dynamics principles and offers a more accurate estimation of the USV's energy consumption during motion. Specifically, this equation decomposes the total energy consumption of the USV into two parts: linear motion energy consumption and turning energy consumption. The linear motion energy consumption is proportional to the distance traveled by the USV \((x)\), and the coefficient \(w_{1}\) reflects the energy consumption of linear motion per unit distance. The turning energy consumption is related to the number of turns of the USV \((y)\) and the angle of each turn (corrected by the parameter \(\delta\)), and the coefficient \(w_{2}\) reflects the energy consumption of a unit turning action. This decomposition method enables Equation (5) to more accurately reflect the differences in energy consumption of the USV under different paths. For example, a longer path with fewer turns may be more energy-efficient than a shorter path with more turns. By adjusting the weighting coefficients \(w_{1}\) and \(w_{2}\), the relative importance of linear motion energy consumption and turning energy consumption can be adjusted according to the specific USV model and mission requirements.  
We further integrate the results of hydrodynamic numerical computations to refine the energy equation. These computations offer estimations for crucial parameters such as the weight associated with linear motion distance, turning angles, and their respective weights. Drawing upon these parameters, we have established an energy equation that accounts for BIT* path characteristics through regression fitting methodologies. Specifically, we utilize the total length of the BIT* path as a metric for linear motion distance and employ indicators like cumulative heading angle change or path curvature to quantify the turning cost.