# Path Planning and Real-Time Collision Avoidance Based on the Essential Visibility Graph
## 3 Obstacle Avoidance Using EVG
Figure 9: Scenario #1. Distance over time between UAV 1 and UAV 2.

expected collision point according to the right of way rules. Figure 11 shows the distance between UAVs over time. As we can see, the collision avoidance system is able to guarantee a mutual distance not below the prescribed minimum value of 200 m.

### Scenario #3: Eight Converging UAVs

In this scenario, eight aircraft are uniformly distributed in the airspace as a circular pattern and follow trajectories all converging towards the center, which is the potential collision point. Figure 12

Figure 11: Scenario #2. Distance over time between UAV 1 and UAV 2. Figure 10: Scenario #2. Two opposite UAVs. UAV 1’s initial position \(\mathbf{A_{1}}(0)=(5\text{ km};0\text{ km})\), \(V_{1}(0)=V_{c}\), \(\psi_{1}=\pi\) rad. UAV 2’s initial position \(\mathbf{A_{2}}(0)=(-5\text{ km};0\text{ km})\), \(V_{2}(0)=V_{c}\), \(\psi_{2}(0)=0\) rad. \(t^{*}\) represents the predicted collision time instant. shows the UAVs' planned paths (dotted lines) and the actual trajectories followed (solid lines). As can be seen, each aircraft avoids a collision by re-planning its own nominal path making a turn maneuver to the right near the center of the scenario and reaching the destination point on the opposite side. Circular markers indicate the UAVs' positions at the time the collision is predicted. Note that the aircraft are placed all around a sort of fictitious traffic circle with a radius that depends on the number of vehicles, centered at the point where the collision is expected to occur. As previously said, due to the higher number of aircraft involved in this scenario, only the minimum distances between UAVs are summarized in Table 2. As we can see, they are always greater than the required minimum value. ### Scenario #4: Twenty Converging UAVs

In this scenario, twenty aircraft are uniformly distributed in the airspace as a circular pattern and follow trajectories all converging towards the center, which is the potential collision point. Figure 13 shows the UAVs' planned paths (dotted lines) and the actual trajectories followed (solid lines). Despite the even higher number of aircraft, also in this test case, the UAVs are able to avoid collisions. Once a potential collision point is detected, each aircraft re-plans its own path making a turn \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline
1 & - & 389 & 719 & 939 & 1017 & 939 & 719 \\
2 & - & 389 & 719 & 939 & 1017 & 939 & 719 \\
3 & & - & 389 & 719 & 939 & 1017 & 939 \\
4 & & & - & 389 & 719 & 939 & 1017 \\
5 & & & & - & 389 & 719 & 939 \\
6 & & & & & - & 389 & 719 \\
7 & & & & & & - & 389 \\
8 & & & & & & & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Scenario #3: Minimum distances between UAVs (m). Figure 12: Scenario #3: Eight UAVs uniformly distributed as a circular pattern converging to the center of the airspace and reaching the destination points on the opposite side. UAVs’ initial positions \(A_{i}=(5\cos(\alpha_{i});5\sin(\alpha_{i}))\) km, \(\alpha_{i}=(i-1)(2\pi/8)\)\(\forall i=1,\ldots,8\), \(V_{i}(0)=V_{c}\), \(\psi_{i}=\alpha_{i}+\pi\). Dotted lines represent the UAVs’ planned paths, whereas solid lines represent the actual trajectories. Circular markers indicate the UAVs’ positions at the time the collision is predicted. maneuver to the right compliant with the ICAO rules. Circular markers indicate the UAVs' positions at the time the collision is predicted. As in the previous scenario, the aircraft avoid a sort of fictitious traffic circle having a radius that depends on the number of vehicles, centered at the point where the collision is expected, before reaching their own destination points on the opposite side.