# Automatic Docking Trajectory Design-Based Time-Varying-Radius Dubins for Unmanned Surface Vessel
## 1 Introduction  
Currently, unmanned surface vehicles (USVs) [1] are developing rapidly. They are widely used in different fields, such as marine data collection, and maritime target search and rescue [2; 3; 4]. As a new type of surface mobile platform that can reach a defined target location, USVs can complete tasks partially or fully autonomously. For example, in our project, a self-made USV is used as an unmanned rescue device at sea, which is used to rescue a drowning person and transport them back to the mobile dockyard. The well-designed diagram for the USV and its dockyard is described in Figure 1.  
Path planning for USVs involves finding the shortest collision-free trajectory that reaches the destination; this has been extensively researched. Indeed, there are many scholars working in the field of path planning and tracking of USVs [5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15]. [PERSON] et al. [5] propose a second path planning method for USVs. The method uses the nonlinear least squares method to obtain the motion constraint of USVs. Then, the method analyzes the three-point geometric relationship of the optimized path to ensure that the curvature radius of the optimized path meets the requirement. [PERSON] et al. [6] propose a model predictive trajectory tracking method for USVs. The method uses the path following algorithm based on the Lyapunov model to follow the leader. Then, the method proposes the trajectory state estimator to ensure the followers pass through water channels. [PERSON] et al. [7] propose a 3D-point cloud-based water obstacle tracking and detection method for USVs. The method uses a stereo camera to obtain a 3D-point cloud in front of the USV. Then, the method uses fast approximate semantic segmentation to filter the point cloud. Finally, the method uses a depth appearance model to track the obstacles. [PERSON] et al. [8] propose a fuzzy trajectory control method for USVs under uncertainty. The method designs a reference route for the USV by control techniques and logical virtual vessel. Then, the method uses the online construction of fuzzy approximator to deal with the uncertainty of conditions. [PERSON] et al. [9]propose a USV trajectory reconstruction method based on automatic identification system. The method uses the kinematic model of USVs to eliminate the anomalous data of automatic identification system. Then, the method respectively uses empirical mode decomposition and Fermat spiral fitting for data denoising and curve fitting. [PERSON] et al. [10] propose an integral sliding-mode adaptive neural scheme for path tracking of USVs under unknown disturbances. The scheme uses a finite-time predictor-based guidance law to ensure that the USVs converge to a predefined path. Then, the scheme designs the control law through the neural network and integral sliding-mode approach to ensure that the USVs move along a predefined path. [PERSON] et al. [11] propose a dynamic-model-based adaptive path tracking control method for USVs. The method replaces the fully nonlinear dynamic model with a three-degree-of-freedom model. Then, the method incorporates the adaptive element of the tracking controller to improve the stability of the control process. [PERSON] et al. [12] propose a trajectory tracking fuzzy controller based on event triggering. The controller reduces the computational complexity and network communication burden by performing updates at the moment of event triggering. Path planning is a critical issue in the effective application of USVs. USV path planning is mainly divided into global and local path planning methods. [PERSON] et al. [13] propose a USV path planning algorithm based on A* and a dynamic windows algorithm. The algorithm uses the A* algorithm to plan the global path. Then, the algorithm sets the objective function of the dynamic windows algorithm according to the environmental changes in order to avoid dynamic obstacles. [PERSON] et al. [14] propose a USV path planning algorithm based on focused search.