## 2 BIT* Algorithm Procedure
### Algorithm Steps
\begin{bmatrix}Q_{E}\leftarrow\left\{(v_{m},x_{m})\in Q_{E}\mid g^{T}(v)+c^{ \prime}(v,x_{m})\geq g^{T}(x_{m})\right\}\end{bmatrix} \)
17 else
18\( \begin{bmatrix}Q_{E}\leftarrow\emptyset,\;Q_{v}\leftarrow\emptyset\end{bmatrix} \)
19 UntilSTOP
20 Return\((T(V,E))\)
```  
**Algorithm 1**Pseudo-code of the BIT* algorithm. In the BIT* algorithm, the following two key formulas play a crucial role in guiding the search process and ensuring the performance of the algorithm:  
Connection Radius: In order to maintain good connectivity of the tree while ensuring search efficiency, the BIT* algorithm uses a dynamically adjusted connection radius \(r_{\text{BIT*}}\). This radius determines the maximum distance at which a new sample point can attempt to connect with existing vertices in the tree. The formula for calculating the connection radius \(r_{\text{BIT*}}\) is as follows:  
\[r_{\text{BIT*}}=\min\left(1,\sqrt[n]{\frac{\gamma_{n}\cdot\ln(|V|+|X_{\text{ unconn}}|)}{2^{n}\cdot(|V|+|X_{\text{unconn}}|-m)}}\right), \tag{1}\]  
where \(n\) represents the dimensionality of the state space, \(\gamma_{n}\) denotes the volume of an \(n\)-dimensional unit hypersphere, \(m\) signifies the cardinality of the sample batch, \(|V|\) corresponds to the number of vertices currently in the tree, and \(|X_{\text{unconn}}|\) indicates the number of unconnected states. This formulation draws upon the theoretical underpinnings of RRT* and is tailored to accommodate the batch sampling strategy inherent to BIT*. The connection radius \(r_{\text{BIT*}}\) exhibits a monotonic decrease as the tree expands, a property that underpins the algorithm's asymptotic optimality. Intuitively, a larger connection radius is adopted during the initial stages of the search, facilitating rapid exploration of the state space. As the search progresses, the radius progressively diminishes, enabling more nu anced path refinement. Furthermore, by constraining new samples to connect solely with existing vertices within the \(r_{\text{BIT}}\), radius, the algorithm not only ensures the effective integration of new samples into the tree structure but also curtails superfluous edge evaluations, thereby enhancing computational efficiency. Heuristic Function Estimate: BIT* incorporates the heuristic search concept from the A* algorithm, employing a heuristic function \(f(x)\) to evaluate the priority of each state \(x\). This function considers both the actual accumulated cost \(g(x)\) from the start to state \(x\) and the estimated cost \(h(x)\) from state \(x\) to the goal. The heuristic function \(f(x)\) is defined as follows:  
\[f(x)=g(x)+h(x), \tag{2}\]  
where \(g(x)\) represents the actual cost of reaching state \(x\) from the start state \(x_{\text{start}}\) along the path in the current tree. \(h(x)\) denotes the estimated cost from state \(x\) to the goal state set \(X_{\text{goal}}\), typically using the Euclidean distance or other distance metrics that satisfy the admissibility and consistency requirements. \(h(x)\) must satisfy admissibility, meaning the estimated cost cannot exceed the actual cost. By comparing the \(f\) values of nodes (or candidate edges), the algorithm can prioritize paths that are most likely to improve the current solution during vertex expansion and edge selection. This heuristic-based guidance strategy allows BIT* to significantly accelerate the search speed while maintaining asymptotic optimality, especially in high-dimensional or complex environments. BIT*+TD3 Deep Reinforcement Learning Algorithm: A Solution Model for Path Planning of Unmanned Surface Vehicles in Hybrid Intelligent Systems  
The BIT* algorithm efficiently generates initial paths from a starting point to a goal point in static environments by dynamically adjusting its sampling region and heuristic search strategies. This initial path serves as the input for TD3, providing it with a reference trajectory and state information, thereby expediting the learning process.