# Virtual Streamline Traction: Formation Cooperative Obstacle Avoidance Based on Dynamical Systems
## 2 Obstacle-Avoidance Formulation
The expressions for \(W_{k}\) and \(M_{k}\) are as follows:  
\[W_{k}=\left\{\prod_{i=1,i\
eq k}^{N}\frac{1}{(\Gamma_{i}-1)}\frac{N=1}{(\Gamma _{i}-1)+(\Gamma_{k}-1)}N\
eq 1^{\prime},\right. \tag{4}\]  
\[M_{k}=I-\frac{n_{k}n_{k}^{T}}{|\Gamma_{k}|^{1/\rho_{k}}n_{k}^{T}n_{k}}, \tag{5}\]  
where \(I\) is the second-order identity matrix, \(-\frac{n_{k}n_{k}^{T}}{|\Gamma_{k}|^{1/\rho_{k}}n_{k}^{T}n_{k}}\) is the disturbance matrix (representing the disturbance to the initial flow field caused by the obstacle), \(\rho_{k}\) is the disturbance coefficient (adjusting the degree of disturbance caused by the obstacles to the initial flow field; the greater the disturbance coefficient \(\rho_{k}\), the more severe the disturbance caused by the obstacles), and \(n^{k}\) is a vector normal to the surface of the \(k\)-th obstacle, \(n^{k}=\left[\frac{\mathrm{d}\Gamma_{k}}{\mathrm{d}x}\quad\frac{\mathrm{d} \Gamma_{k}}{\mathrm{d}y}\right]^{T}\). Next, consider the presence of dynamic obstacles such as moving vessels and other mobile platforms. To avoid them, the original flow velocity \(u_{t}  
The route leads to the target point. When the USV is far from all obstacles, the value of \(\Gamma_{k}\) will gradually increase, and we can approximately consider \(\Gamma_{k}\) to approach infinity when the distance from the obstacle is sufficiently large (\(\Gamma_{k}\approx\infty\)). This implies that the disturbance matrix \(-\frac{n_{k}n_{t}^{T}}{|\Gamma_{k}|^{1/p_{k}}n_{t}^{T}n_{k}}\) in Equation (5) approaches 0. Therefore, the obstacles do not disturb the initial flow field. The total flow field value stands at \(\overline{u}_{t}=u_{t}\), satisfying the requirement that the route must end at the target point. \(\square\)  
The obstacles are impenetrable. To ensure the impenetrability of obstacles (the USV does not traverse the obstacle model), it is sufficient to demonstrate that the normal velocity disappears at the surface of the obstacle, i.e., \(n_{k}^{T}\overline{u}_{t}=0\)[31]. On the surface of the obstacle, \(\Gamma_{k}=1\) and \(n_{k}^{T}\overline{u}_{t}=n_{k}^{T}M_{k}u_{t}\). Because \(n_{k}^{T}M_{k}=n_{k}^{T}-\frac{n_{k}^{T}n_{k}n_{k}^{T}}{n_{k}^{T}n_{k}}=n_{k}^ {T}-n_{k}^{T}=0\), it follows that \(n_{k}^{T}\overline{u}_{t}=0\), the velocity vector of the USV at the surface of the obstacle, when projected onto the normal vector direction, becomes zero, meeting the requirement that the obstacles must be impenetrable. \(\square\)  
Therefore, the obstacle-avoidance method ensures that the vehicles eventually reach the target point, even though the obstacles are impenetrable. Local Minima and Solution  
To some extent, the aforementioned approach mitigates the various local minima problems that are intrinsic to APF, resulting in smooth obstacle-avoidance routes reminiscent of the way in which water flows around rocks. However, when the above approach is utilized in two-dimensional planes, two types of local minima problems persist: stationary points and trap areas. This section intends to discuss and solve these problems.