## 2 Related Work
### Motion Planning  
Different from path planning, motion planning considers the short-term velocity control of the robot more. The robot uses on-board sensors to observe the nearest fraction of an unknown environment for iterative re-computation of a short-horizon trajectory. Many of the related techniques, such as the dynamic window [20] the curvature velocity [10], and the lane curvature [21] approaches treat the obstacles as static. On the other side, approaches like velocity obstacles [12; 22; 23; 24] or collision cones [25] assume a deterministic knowledge about the obstacle velocity and a moderate rate of its change.  
(1) Environment model: Exiting methods use different methods to model the environments. Occupancy Grid Map (OGM) [26], VO [12; 22; 23; 24], and state-time space [27] are representative methods. The occupancy grid method represents the environment with grids. The presence of mobile obstacles is represented by grid occupancy. The mobile objects can be represented by the dynamic change of the occupancy grid map. On the other hand, the VO model represents a moving object with nearly constant velocity by a velocity vector. The geometric relationship between the robot and the surrounding mobile objects can be inferred by the relationships among the velocity vectors. The feasible motion vectors of the robot can therefore be inferred from the geometric relationship. State-time space is a tool to formulate trajectory planning in dynamic workspace problems. It permits us to study the different aspects of dynamic trajectory planning, i.e., moving obstacles and dynamic constraints, in a unified way. A near-time-optimal approach that searches the solution trajectory over a restricted set of canonical trajectories was presented in [27]. Using the state-time space model, it is possible to transform the problem of finding the time-optimal canonical trajectory to finding the shortest path in a directed graph embedded in the state-time space.  
(2) Motion control algorithms: Based on the environment model, motion control algorithms have been widely investigated. Model Predictive Control (MPC) [28] is the one that has been increasingly applied to vehicle navigation. It performs the path planning process at every time instant, then applies the initial control related to the chosen trajectory to the vehicle. MPC has a set of variants, such as min-max MPC, which conducts predictive control under min-max constraints, and nonlinear MPC [29], which uses a nonlinear trajectory tracking system to ensure that the actual state converges to the nominal state. Obstacle avoidance via boundary following is a standard method employed by many obstacle-avoidance algorithms. In this method, the robot moves towards the target location until a threat of collision with an obstacle is detected. Then, to avoid the collision, the robot bypasses the obstacle by following its boundary while temporarily putting aside the main objective. After passing the obstacle, the robot switches back to the main moving objective. In the case that the obstacle information is fully known, including the location, velocity, and shapes, the VisBugclass of algorithms [30] has been proposed, which navigate toward a visible edge of an obstacle inside the detection range.  
However, existing motion planning algorithms generally have not taken the motion smoothness and risk of ship overturn into consideration, since most of the existing works are designed for underground robots or UGVs. In addition, the remote information collected by AIS enables the MASS to conduct the on-line motion plan with consideration of far away ships. This enables the dynamic programming approach to the online motion planning, which has been rarely reported in existing studies.