# Virtual Streamline Traction: Formation Cooperative Obstacle Avoidance Based on Dynamical Systems
## 4 Disturbance-Based USB Formation Scheme
### Formation Obstacle Avoidance  
The formation structure, as generated by the aforementioned scheme, allows each vehicle to maintain its capability for obstacle avoidance within its respective disturbed flow field during the process of formation obstacle avoidance. Upon encountering obstacles, the formation can disassemble and later reassemble, facilitated by the traction effect of the streamlines. A significant concern addressed is the avoidance of collisions between vehicles within the formation, as illustrated in Figure 8.  
As shown in Figure 8, when multiple USVs appear on the same side of an obstacle during obstacle avoidance, the streamline density on that side may become excessively high because of the contracting force within the streamlines. The streamlines do not intersect. However, as USVs are not point masses but occupy volume, they may collide with each other when the streamlines are excessively dense. As stated in Section 4.1, vehicles within the formation perceive each other as obstacles based on their hierarchy. This approach yields satisfactory results in small clusters. However, in larger clusters, the formation is already established, and too many USVs may attempt to avoid the obstacle on the same side. Consequently, some USVs may become trapped between other USVs and the obstacle, as illustrated in Figure 9. This will result in unne  
Figure 8: Avoidance of inter-vehicle collisions. To differentiate between inter-vehicle-collision and obstacle avoidance, a new approach was developed for inter-vehicle disturbances. In Equations (5) and (6), it can be shown that  
\[\overline{u}=Mu=u+s. \tag{11}\]  
Therefore,  
\[ \begin{array}{l}s=Mu-u\\ =(M-I)u\\ =(I-\frac{m\pi}{|\Gamma|^{1/\rho}n^{T}n}-I)u\\ =-\frac{m\pi^{T}}{|\Gamma|^{1/\rho}n^{T}n}u\end{array}  \tag{12}\]  
When \(\Gamma=1\), vector \(s=-\frac{m\pi^{T}}{n^{T}n}u\). Geometrically, \(\frac{nm^{T}}{n^{T}n}u\) is the projection of \(u\) onto \(n\). Accordingly, an improved expression for inter-vehicle disturbance \(M_{j}\) is proposed:  
\[M_{j}=I-\frac{W_{j}n_{j}n_{j}^{T}}{E_{j}n_{j}^{T}n_{j}} \tag{13}\]  
This equation gives the disturbing effect of the \(j\)-th USV on the current one, where \(I\) is the identity matrix, \(n_{j}=\left[\frac{x_{j}-x_{j}}{d_{j}}\quad\frac{y-y_{j}}{d_{j}}\right]^{T}\) is the vector from the \(j\)-th USV to the current one, and \(d_{j}\) is the distance from the current USV to the \(j\)-th one. The modulation function satisfies \(E_{j}=\frac{\ln(d_{j}-D_{j}+\epsilon)}{\ln\epsilon}\), where \(\epsilon\) is the modulation coefficient, and \(D_{j}\) is the safety distance between vehicles. In addition, the disturbed flow field is weighted by redefined inter-vehicle weights \(W_{j}\) given by  
\[W_{j}=\prod_{i=1}^{N}\frac{(\Gamma_{i}-1)}{(\Gamma_{i}-1)+e^{\eta(d_{j}-D_{j} )}}\bullet\prod_{j=1,j\
eq j}^{J}\frac{e^{\eta(d_{j}-D_{j})}}{e^{\eta(d_{j}-D _{j})}+e^{\eta(d_{j}-D_{j})}} \tag{14}\]  
These weights are assigned to inter-vehicle disturbances, with \(\eta\) being a parameter. The total inter-vehicle disturbance of the USVs is  
\[\overline{M_{j}}=\prod_{j=1}^{J}M_{j} \tag{15}\]  
The total value of the initial flow field formula becomes  
\[u_{ji}=\overline{M_{j}}u_{t} \tag{16}\]  
Figure 9: Diagram of inter-vehicle-collision-avoidance trap. Furthermore, the weight of obstacle disturbance relative to inter-vehicle disturbance is added to the inter-obstacle disturbance \(M_{k}\):  
\[M_{k}=I-\frac{W_{kj}n_{k}n_{k}^{T}}{\left|\Gamma_{k}\right|^{1/\rho_{k}}n_{k}^{T }n_{k}}+\frac{W_{kj}i\tau_{k}n_{k}^{T}}{\left|\Gamma_{k}\right|^{1/\sigma_{k}}n _{k}^{T}n_{k}\mathcal{C}_{k}} \tag{17}\]  
\[W_{kj}=\prod_{j=1}^{I}\frac{e^{\eta(d_{j}-D_{j})}}{e^{\eta(d_{j}-D_{j})}+( \Gamma_{k}-1)} \tag{18}\]  
To verify the feasibility of the aforementioned method, the following assertions were shown to hold:  
(1) The route can reach the target point without causing inter-vehicle collisions. Proof.