# Underwater Vehicle Path Planning Based on Bidirectional Path and Cached Random Tree Star Algorithm
## 2 Related Works  
As previously discussed, the optimal kinodynamic motion planning using incremental sampling-based methods (RRT*) [30] algorithm has emerged as the leading approach for path planning in robotics, owing to its superior performance and other state-of-the-art methodologies. Consequently, recent research endeavors have focused on leveraging the RRT* methodology to expedite the convergence and search processes. Our research is in line with pertinent studies utilizing the RRT* and its derivatives. Figure 1 illustrates the structure of the RRT* algorithm. In the pursuit of enhancing rapidly exploring random tree star (RRT*) performance, two fundamental methods have been proposed. The first approach concentrates on refining the sampling process, while the second involves path optimization strategies. [PERSON] and [PERSON] proposed an artificial potential field approach using the potential function to guide samples towards the local minima, which correspond to the optimal path [31]. The merits of this method are that it efficiently reduces sample dispersion and accelerates convergence. [PERSON] et al. introduced an RRT* heuristic function that learns estimators online to guide motion tree expansion for path optimization [32]. This learning approach generates available guidance, enabling efficient exploration of the optimum solution. [PERSON] et al. employed a node-rejection criterion to enhance computational efficiency, where the number of iterations for a single state is affected by the sampling domain [33]. [PERSON] and [PERSON] conducted collaborative research into keypoint suppression and bounded sampling techniques by applying the RRT*-adjustable boundary (RRT*-AB) method [34]. Graph clipping is a strategy that reduces the model size by utilizing heuristic functions. It retains samples that optimize existing solutions while eliminating others during the exploration process. [PERSON] and [PERSON] present a computational and asymptotically optimal approach to the algorithm, generating RRT* and PRM*, significantly improving sampling-based methods' efficiency. Figure 1: Schematic diagram of optimal kinodynamic motion planning using incremental sampling-based methods (RRT*) [30]. The meaning of \(X_{init}\) is initial position. The meaning of \(X_{min}\) is the point closest to the initial position. The meaning of \(X_{near}\) is the point adjacent to the initial position. The meaning of \(X_{new}\) is the new location node explored according to the algorithm. The meaning of \(X_{rand}\) is that there is a probability of the most new node. However, these approaches might exaggerate the heuristic expenses and result in inaccuracies when utilizing graph compression techniques, primarily occurring at the search tree's beginning [35]. By merging RRT-connect and RRT* approaches, [PERSON] et al. devised a proficient randomized motion planning algorithm that attains a theoretical optimum more rapidly than its RRT* counterpart [36]. [PERSON] et al. proposed a novel path-planning algorithm consisting of potentially guided intelligent bidirectional RRT* (PIB-RRT*) and potentially guided bidirectional RRT*PB-RRT* [37]. [PERSON] et al. proposed a fast path-planning method for underwater robots that improves RRT* by combining target and search algorithms to realize quick path planning [38]. The aforementioned algorithm mentioned above has achieved remarkable success in various domains. However, due to the sparse visual spatial distribution of submerged impediments, the elevated likelihood of arbitrary emergence of novel impediments, the fluctuating submerged topography, and other factors, there is a requirement for more efficient path-planning algorithms that connect the initial and objective points. [PERSON] et al. [39] devised a virtual force method. The idea of the method is to simulate the environment as an electrostatic field and a real-world fluid to specify the effects of targets and obstacles on the robot, and then plan the channel, but it is easy to fall into the local minima. [PERSON] et al.