# Automatic Docking Trajectory Design-Based Time-Varying-Radius Dubins for Unmanned Surface Vessel
## 2 Preliminaries and System Description
### System Model  
Considering the motivation given by the facts and challenges, we propose a novel algorithm for automatic docking path design. The total automatic docking system comprises of two main parts, the USV and the dockyard, as presented in Figure 3. It is worth noting that both of them are mobile, i.e., the dockyard floats with the current and the USV is guarded by its own controller. As we know, the advantage of GPS-RTK is the high positioning accuracy of a few centimeters on the surface of marine environments. Herein, the fixed and mobile GPS-RTK stations are equipped on the dockyard and USV, respectively. Moreover, two long-distance wireless transceivers are also deployed in the USV and dockyard. As a result, the relative positions and heading angles between the USV and mobile dockyard can be measured precisely at all times. In practice, the heading value of the dockyard is synchronized to that of the USV with a long-distance wireless transceiver. The self-made RTK device is shown in Figure 4. The self-made RTK device is composed mainly of STM32 One-chip Computer, Wireless LoRa Transceiver and RTK Chip. The relative positioning accuracy can reach 10 cm, and the relative positioning distance between the USV and dockyard can reach 10 km; thus, it meets the needs of long-distance precise positioning between the USV and dockyard. On the basis of the position and heading of the mobile USV and dockyard, the optimal automatic docking trajectory can be achieved through the path planning strategy.  
For path planning with the starting point, end point and tangent angle, to the best of our knowledge, the maximum number of possible shortest paths is six when transitioning from one Dubins state to another Dubins state [20], as shown in Figure 5[25; 26]. We can divide the six possible shortest paths into two categories based on the straight segments. The first category starts with a left (\(L\)) or right (\(R\)) turn, followed by a straight segment (\(S\)), and ending with a left or right turn (denoted by \(LSR\), \(LSL\), \(RSL\) and \(RSR\), respectively). The another category uses the turning instead of straight segment (denoted by \(RLR\) and \(LRL\)). In this case, the optimal path based on the Dubins curve needs to satisfy the maximum turning angle.  
In this paper, we apply the Dubins curve to design our automatic docking path. Furthermore, the traditional Dubins curve is switched to segmented mode in this paper, and each segmented Dubins curve has a different minimal turning radius. The trajectory is limited to an area of \(W\times L\). Suppose that the actual speed \(V\) of the speed of the USV is limited between the minimum \(V_{min}\) and maximum \(V_{max}\) values. For simplicity, it is assumed that the speed of the USV can reach the current value, i.e., the acceleration and  
Figure 4: Self-made RTK device.  
Figure 3: The framework of the automatic docking trajectory.  
deceleration times are ignored in this paper. Furthermore, it is assumed that the minimal turning radius \(R_{i}\) is proportional to the USV's velocity \(V_{i}\) under the effect of the radius influence factor \(k\).