# Virtual Streamline Traction: Formation Cooperative Obstacle Avoidance Based on Dynamical Systems
## 2 Obstacle-Avoidance Formulation  
Consider a USV operating in calm water without obstacles. Let the position of the USV at time \(t\) be \(\xi_{t}=(x_{t},y_{t})\) and the target point of its current voyage be \(\xi_{d}=(x_{d},y_{d})\). At this moment, the velocity amplitude of the USV is \(V\). The route is then a straight line between the two points with coordinates \(\xi_{t}\) and \(\xi_{d}\). The initial flow field \(u_{t}\) in the geodetic coordinate system is as follows:  
\[u_{t}=\left[\frac{V(x_{d}-x_{t})}{d}\quad\frac{V(y_{d}-y_{t})}{d}\right]^{T}, \tag{1}\]  
where \(d=\sqrt{\left(x_{t}-x_{d}\right)^{2}+\left(y_{t}-y_{d}\right)^{2}}\) is the distance between the position if the USV is at time \(t\) and the target point. When obstacles are encountered, they induce disturbances in the initial flow field, causing the navigation route to deviate from obstacles and eventually converge toward the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Solution** & **Instance** & **Disadvantage** \\ \hline Based on geometric analysis methods & Velocity obstacle method & Unable to adapt quickly to changes in dynamic environments \\ \hline Model predictive control (MPC) methods & Nonlinear MPC method & High computational requirements and workload; low computational efficiency \\ \hline Stochastic planning methods & Rapidly exploring random-tree algorithm & Low calculation efficiency in complex environments \\ \hline Methods of simulating force fields & Artificial potential field (APF) method & Many local minima problems \\ \hline Intelligent optimization algorithms & Neural networks and reinforcement learning & Sample inefficiency and generalizability problems; poor practicality \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Current mainstream solutions to the obstacle-avoidance problem. Figure 1: Flowing water avoiding rocks. target. In practical scenarios, employing sensors that detect obstacles with high precision can be challenging. To enhance planning efficiency, obstacles can be represented by their boundary volumes. This approach involves categorizing obstacles into standard convex shapes. For example, geographical features such as islands, reefs, and shorelines may be represented as triangles, rectangles, hexagons, ellipses, and circles, or combinations thereof. Similarly, moving boats and vessels can be simplified into circles and ellipses for ease of modeling. To facilitate a standardized approach to obstacle modeling, the representation of obstacles is uniformly defined as follows [30]:  
\[\Gamma(\xi)=\big{(}\frac{\mathrm{x_{t}-x_{0}}}{\mathrm{c}}\big{)}^{2a}+\big{(} \frac{y_{t}-y_{0}}{d}\big{)}^{2b}, \tag{2}\]  
where \(\xi_{0}=(x_{0},y_{0})\) is the center of the obstacle, \(c\) and \(d\) are the axial lengths of the obstacle, and \(a\) and \(b\) are the shape parameters of the obstacle. This method can construct boundary volumes for most obstacles. Extremely large obstacles can be represented as combinations of multiple obstacles to significantly increase the feasibility of obstacle avoidance routes. Let \(\overline{M}\) represent the disturbances caused by all obstacles to the initial flow field (the velocity vector associated with each point in the original flow field is modulated) [29]. The expression is as follows:  
\[\overline{M}=\sum_{k=1}^{N}W_{k}M_{k}, \tag{3}\]  
where \"\(N\)\" is the total number of obstacles, the letter \"\(k\)\" represents the \(k\)-th obstacle, \(W_{k}\) is the weight of the \(k\)-th obstacle, and \(M_{k}\) is the disturbance to the initial flow field \(u_{t}\) caused by the \(k\)-th obstacle.