## 3 Velocity Obstacle Model for MASS
### Velocity Obstacle
#### 3.1.1 Relative Velocity Model  
Figure 1 illustrates the concepts of the VO model. At a time instant \(t\), suppose there is an MASS, denoted by \(A\), whose velocity vector is represented by \(v_{A}\). There is another ship \(B\) whose velocity vector is \(v_{B}\). Since the shapes of \(A\) and \(B\) cannot be known exactly, the VO model firstly abstracts the shape of \(A\) and \(B\) by circles of radius \(r_{A}\) and \(r_{B}\), respectively (\(r_{A}\) and \(r_{B}\) may be determined by the sizes of the ships). The velocity vectors are then originated at the centers of the circles. In Figure 1c, the problem is transformed to the configuration space of \(A\) by considering the relative motion between \(A\) and \(B\). The MASS \(A\) is converted to a point \(A^{\prime}\), and the circle \(B\) is expanded to a larger circle \(B^{\prime}\) whose radius is \(r_{A}+r_{B}\). The relative speed vector between \(A\) and \(B\) is calculated as: \(v_{AB}=v_{A}-v_{B}\).  
**Definition 1** (Collision cone of mobile ships).: _A collision cone of mobile ship \(\mathsf{CC}_{AB}\) is defined as a cone-shaped region, centered at \(A^{\prime}\) and bounded by the two tangent lines \(\lambda_{r}\) and \(\lambda_{f}\) from \(A^{\prime}\) to \(B^{\prime}\), as shown in Figure 1c._  
\[\mathsf{CC}_{A,B}=\left\{v_{AB}|\lambda_{AB}\cap B^{\prime}\
eq\emptyset\right\} \tag{1}\]  
_where \(\lambda_{l}\) and \(\lambda_{r}\) are tangent lines from \(A\) to \(B^{\prime}\) as shown in Figure 1c; \(\lambda_{AB}\) is the line extended from the relative velocity vector \(v_{AB}\). When the line of the relative velocity vector, i.e., \(\lambda_{AB}\) touches \(B^{\prime}\), this means the MASS \(A\) may collide with \(B\) after some time if their relative velocity is \(v_{AB}\). The selections of \(V_{AB}\) outside \(\mathsf{CC}_{AB}\) are safe for collision avoidance._  
If the detected object is a static object on the map, such as the shore-side and islands, the shapes of these objects are generally irregular. However, we can still extend the VO model by considering the relative velocity vector between the MASS and the static object. Since the object is static, the relative velocity \(v_{AB}=v_{A}\).  
Figure 1: Illustration of the Velocity Obstacle (VO) model. (**a**) Ship \(A\) is a Maritime Autonomous Surface Ship (MASS), and ship \(B\) is a conventional ship, which are moving in velocity vector \(v_{A}\) and \(v_{B}\), respectively; (**b**) represents the geometric shapes of the two ships by circles; (**c**) represents the collision region by relative velocity: if the relative velocity vector \(v_{AB}\) falls into the shaded area, the MASS \(A\) may collide with the ship \(B\); (**d**) represents the collision region by the absolute velocity vector of \(A\), i.e., if the velocity vector \(v_{A}\) ends in the shaded area, the MASS \(A\) may collide with \(B\); (**e**) the velocity obstacle \(VO_{B}\) for a short time interval.  
**Definition 2** (Collision cone of static objects).: _A collision cone between an MASS and a static object is defined as \(\text{CC}_{AB}\), which is a cone-shaped region, centered at \(A^{\prime}\) and bounded by the two tangent lines \(\lambda_{r}\) and \(\lambda_{f}\) from \(A^{\prime}\) to \(B^{\prime}\) as shown in Figure 2a._  
\[\text{CC}_{A,B}=\left\{v_{AB}|\lambda_{AB}\cap B^{\prime}\
eq\emptyset\right\} \tag{2}\]  
_where \(\lambda_{AB}\) is the line extended from the relative velocity vector \(v_{AB}\)._