## 1 Introduction
This paper investigates the case when an MASS is sailing in a waterway when there are other MASS and conventional vessels present, which will be a general scenario of MASS application in the near future. We consider the following motion control problem:  
**Problem 1**. : _Suppose the waterway map is given, which is denoted by \(\mathcal{M}\), indicating the feasible area where the MASS can sail. The MASS starts from \(S\in\mathcal{M}\), and the destination of the MASS is given as \(T\in\mathcal{M}\). We consider a group of conventional ships, and other MASS are sailing in the same map \(\mathcal{M}\) at their own will. The considered MASS plans its velocity and direction (denoted by a vector \(\mathbf{V}_{t}\)) in a time-interval manner to optimize its motion. The interval length is denoted by \(\Delta_{t}\). The goal of the MASS is to reach the target as early as possible, while maintaining sailing safety and smoothness._  
We assume the MASS can obtain surrounding ship information in real time, including the position \(\{\mathbf{Y}_{i},i=1,\cdots,N\}\), velocity \(\{\mathbf{V}_{i},i=1,\cdots,N\}\), and identification of surrounding ships \(\{\mathbf{C}_{i},i=1,\cdots,N\}\) via radar, AIS, and camera sensing systems. Based on this information, the MASS adjusts its motion plan in every successive time interval by adjusting \(\mathbf{V}_{t}\) to \(\mathbf{V}_{t+1}=\mathbf{V}_{t}+\mathbf{A}_{t}\), where \(\mathbf{A}_{t}\) is the acceleration vector at time \(t\). In each interval, the MASS can only adjust its velocity under the constraint of acceleration limits and turning limitations. The MASS needs also to make sure it is free from collisions with surrounding ships and the map obstacles. Although the short-term motion planning problem has been widely studied for robots and UGVs [8; 9; 10; 11; 12], the problem of usingthe neighborhood extendand remote information to approach optimal motion control by dynamic programming has not been reported in existing works. The main contributions of this paper are as follows:  
1. We exploit the Velocity Obstacle (VO) model to represent the collision avoidance constraints with surrounding ships. The VO model for multiple mobile and static objects is presented. 2. A multiple objective optimization problem is formulated for the short-term motion planning problem. The motion smoothness, risk of collision, and sailing efficiency are jointly optimized. 3. A greedy interval-based motion control method is proposed, but we show that the greedy algorithm may fail to avoid collisions in future intervals. We therefore present a way-blocking metric to evaluate the future collision risks, which enables a greedy-risk interval-based motion-planning algorithm. 4. Then, by assuming the other ships are sailing at a constant velocity, the VOs in future time intervals can be calculated by the given state of the MASS. A dynamic programming-based method to seek the optimal motion trajectory is proposed for the sailing safety, efficiency, and smoothness. 5. Extensive simulations are conducted to show the performances of the greedy, greedy-risk, and DP methods. It is shown that the DP method has better performance than the other two methods in both collision avoidance and sailing efficiency. The remaining sections of the paper are organized as follows. Related works are presented in Section 2. The VO model is presented in Section 3. The greedy and greedy-risk algorithms are presented in Section 4. The multiple interval dynamic programming-based motion optimization method is presented in Section 5. The simulation results are presented in Section 6. The paper is concluded in Section 7 with remarks.