# Automatic Docking Trajectory Design-Based Time-Varying-Radius Dubins for Unmanned Surface Vessel
## 1 Introduction
Based on the A* algorithm, the algorithm sets the local search area, including the best path. Then, the algorithm focuses on searching the local search region to improve the overall search efficiency. The proposed method uses a quadtree representation of the marine environment to efficiently compute the nodes of the visibility graphs. [PERSON] et al. [15] propose a multi-USV path tracking control strategy in an uncertain environment. Based on the feedback control, the strategy designs a path tracking controller for a single USV. Then, the strategy uses a speed coordination controller to implement the formation path tracking of USVs. Although there are numerous studies on the path planning of USVs, their algorithms have different strengths and weaknesses. Moreover, automated launch and recovery of USVs have been one of most important problems recently. [PERSON] et al. [16] analyze the application cases of the USV in different scenarios, such as pipeline inspection and marine animal monitoring. However, there is  
Figure 1: Mechanical structure of the USV and dockyard. little research on path planning of USV automatic docking. [PERSON] et al. [17] propose a dock identification algorithm based on 3D LiDAR scans. The algorithm removes data points that are not relevant to the dock. Then, the algorithm uses clustering-based image segmentation to determine the location of the dock. [PERSON] et al. [18] investigate the parallel docking problem for USVs, and propose an adaptive control algorithm. However, this method requires an expensive radar or stereo vision system, so it is difficult to implement in real-time environments. [PERSON] et al. [19] propose a unified framework for USV path planning and tracking. The framework implements path planning by genetic algorithm. Then, the framework optimizes the path of the USV by the B-spline technique. Finally, the framework implements path tracking by deep reinforcement learning. The traditional USV is a vehicle with constant speed and a fixed minimum turning radius, which is limited by the minimum turning radius. Therefore, the traditional USV models are often referred to as Dubins vehicles [20]. When information such as the leaving and arrival angle of the Dubins vehicle is known, the shortest path of the Dubins vehicle at two points can be calculated. [PERSON] et al. [21] propose a Dubins-curve-based path planning algorithm for the task of USVs. The algorithm establishes the relationship model between the movement angle and velocity of the USV. Then, the algorithm uses the genetic algorithm to obtain the task execution order and path of the USV. [PERSON] et al. [22; 23] utilize Dubins path planning to solve the Dubins traveling salesman problem (DTSP) and Dubins interval problem (DIP) for fixed-wing aircraft. Facing the unique application scenarios, there is no path planning method for automatic docking in the existence of mobile dockyard. Our main contribution is to provide automatic docking trajectory design for unmanned surface vehicles. As far as we know, this paper is the first to investigate the path tracking problem of mobile dockyards in real sea environments. This paper is organized as follows: Section 2 outlines the dynamic and kinematic model of the USV. Then, the section provides the mathematical model for automatic docking of USV. Varying-radius-Dubins-based trajectory design is proposed to address the above problem in Section 3. Extensive simulations and experimental results for automatic docking trajectory planning are verified in Section 4. Section 5 is the conclusion.