## 5 Conflict Risk Assessment between Drones and Manned Aircraft
In this research, Monte Carlo simulation experiments are conducted. In the simulation experiment, the ratio of the number of correctly predicted collisions to the number of real collisions between the drone and manned aircraft is the correct detection rate. The ratio of the number of incorrectly predicted collisions to the number of non-collisions is the false alarm rate. \[Per_{cd}=\frac{\sum\limits_{\tau=1}^{Num}q_{\tau}=1}{Rel_{col}} \tag{11}\]  
\[Per_{fa}=\frac{Pre_{col}-\left(\sum\limits_{\tau=1}^{Num}q_{\tau}=1\right)}{Num -Rel_{col}} \tag{12}\]In the above formulas, \(Num\) is the total number of simulation tests; \(Per_{cd}\) is the correct detection rate; \(Per_{fu}\) is the false alarm rate. \(q_{r}=1\) indicates the correct prediction of real collisions in the \(r\)-th simulation test, which means that both the predicted results and the real results between the drone and manned aircraft are collisions; \(Rel_{col}\) is the number of real collisions in the simulation test; and \(Pre_{col}\) is the number of collisions predicted in the simulation test. Finally, the collision probability between the drone and the manned aircraft at the closest encountering point and the estimated time to reach the closest encountering point are calculated. The following assumptions are made:  
* The drone and the manned aircraft are regarded as a particle with a direction of movement;
* During the movement, the drone and the manned aircraft are independent of each other, and the influence of the wake factor is excluded;
* The effects of bad weather such as wind, rain, and thunderstorms are ignored;
* The horizontal and vertical movements of the drone are independent;
* The speed of the drone and the manned aircraft remain consistent at the closest moment. Based on the above assumptions, it can be judged whether there is a conflict risk according to the predicted position points of the manned aircraft and the predicted position distribution of the drone. The drones that do not have a conflict risk are screened out. For the remaining drones with conflict risks, assuming that the drone and the manned aircraft are predicted in the next \(k\) seconds from the current time \(n\), the minimum distance between the predicted position of the manned aircraft and the drone at the same time is defined as the closest encountering point. For any time \(n+s\) within the time \(n+1\)-\(n+k\), calculate the distance (\(L^{\mu}_{n+s}\)) between the predicted position of the manned aircraft at the time of \(n+s\) for each point in the predicted position distribution of the drone trajectory. \[ \begin{array}{c}L^{\mu}_{n+s}=\sqrt{\left(x_{n+s}-u^{\mu}_{n+s}\right)^{2}+ \left(y_{n+s}-v^{\mu}_{n+s}\right)^{2}+\left(z_{n+s}-w^{\mu}_{n+s}\right)^{2} }\\ \mu=1,2,\ldots,j\\ s=1,2,\ldots,k\end{array}  \tag{13}\]  
where \(x_{n+s}\), \(y_{n+s}\), \(z_{n+s}\) are the predicted longitude, latitude and altitude, respectively, for manned aircraft at time \(n+s\); \(u^{\mu}_{n+s}\), \(v^{\mu}_{n+s}\), \(w^{\mu}_{n+s}\) are the predicted longitude, latitude and altitude, respectively, for the drone at time \(n+s\); \(j\) is the number of quartiles. The minimum distance between the drone and the manned aircraft at time \(n+s\) (\(L_{n+s}\)) is calculated as:  
\[ \begin{array}{c}L_{n+s}=\min(L^{\mu}_{n+s})\\ \mu=1,2,\ldots,j\end{array}  \tag{14}\]  
According to the obtained minimum distance between the drone and the manned aircraft at the time \(n+s\), the minimum distance between the drone and the manned aircraft from time \(n+1\)-\(n+k\) (\(L_{n+l}\)) are obtained. \[ \begin{array}{c}L_{n+l}=\min(L_{n+s})\\ s=1,2,\ldots,k\end{array}  \tag{15}\]  
Thus, \(n+l\) is the timestamp when the drone and the manned aircraft meet at the nearest point, and \(l\) is the time when the drone and the manned aircraft are expected to reach the nearest encountering point from the current time. The procedure is illustrated using the following example.