# Study of Urban Unmanned Aerial Vehicle Separation in Free Flight Based on Track Prediction
## 4 Case Application
### Calculation of Required Separation
#### 4.2.2 Collision Probability in Conflict Parameters  
The parameters involved in the calculation process, along with their symbols and values, are shown in Table 1.  
When calculating the required avoidance distance, it is necessary to assess extreme conflict situations, specifically, when two aircrafts encounter each other head-on and implement a horizontal avoidance maneuver. Let the maneuvering avoidance distance be \(d_{avoid}\), as shown in Figure 11.  
Each UAV has a speed of 12 m/s, with a closure rate of 24 m/s. The combined physical dimension of the two aircrafts, \(d_{\text{min}}\), is 5 m, which corresponds to the collision threshold. Considering that some UAVs are fixed-wing, for safety reasons, the method for calculating manned aircraft turning parameters is applied. The maximum tilt angle of the UAV is set to 25\({}^{\circ}\), and the coordinated turn load factor is taken as 1.1 g according to the ICAO DOC8168 [21]. The derivation of the turn rate is shown in Equations (17)-(19):  
\[LF=\frac{1}{\cos\theta}=\frac{1}{\cos 25^{\circ}}=1.1 \tag{17}\]  
\[R=\frac{v^{2}}{g\cdot\sqrt{LF^{2}-1}}=\frac{12^{2}}{10\cdot\sqrt{1.1^{2}-1}}= 31.42 \tag{18}\]  
\[\phi=\frac{v}{R}=\frac{12}{31.42}\cdot\frac{180}{\pi}=21.88 \tag{19}\]  
To calculate \(d_{avoid}\), it is necessary to identify when the two UAVs are at the closest point of approach (CPA), \({L_{1}}^{2}+{L_{2}}^{2}\geqslant{d_{\text{min}}}^{2}\), as shown in Equations (20)-(24):  
\[L_{1}=d_{avoid}-2\int_{0}^{t}v\cdot\cos\phi tdt=d_{avoid}-1.0967\sin\phi t \tag{20}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Prediction error & \(\epsilon\) & \(N(5.47,1.25^{2})\) \\ Collision threshold & \(d_{\text{min}}\) & 5 m \\ Load factor & LF & 1.1 g \\ Roll angle & \(\theta\) & 25\({}^{\circ}\) \\ Turning radius & R & 31.42 m \\ Turn rate & \(\phi\) & 21.88\({}^{\circ}\)/s \\ UAV dimensions & & 2.50 m \(\times\) 2.50 m \(\times\) 0.60 m \\ \hline \hline \end{tabular} Note: Based on research, considering communication delays, information processing, and decision-making time, and drawing on the study by [PERSON] et al. [28], the response time for the air traffic control or flight control module is set at 0.04 s.  
\end{table} 
Table 1: The parameters for collision probability calculation. \[L_{2}=2\int_{0}^{t}v\cdot\sin\phi tdt=1.0967-1.0967\cos\phi t \tag{21}\] \[L_{1}^{2}+L_{2}^{2}\geq d_{\min}^{2}\] (22) \[\left(d_{avoid}-1.0967\sin\phi t\right)^{2}+\left(1.0967-1.0967 \cos\phi t\right)^{2}\geq 5^{2}\] (23) \[d_{avoid}\geq 1.0967\sin\phi t+\sqrt{25-\left(1.0967-1.0967 \cos\phi t\right)^{2}} \tag{24}\]  
The \(d_{avoid}\) calculation result is 26 m. This value represents the required avoidance distance without considering track prediction error, response time, etc. Collision Probability Calculation  
As Table 1 shows, the single UAV track prediction error conforms to the normal distributions \(N(5.47,1.25^{2})\). So, the distance between the two UAVs conforms to the normal distributions \(N(10.94,1.77^{2})\). Furthermore, the response time for the air traffic control and flight control module corresponds to 1.92 m, and the required avoidance distance is 26 m. According to Equation (8), the probability density function for the two UAVs' actual required distance corresponding to the normal distribution is \(N(38.86,1.77^{2})\). This value represents the required separation considering track prediction error and response time. Based on these data, the collision probability in conflict can be calculated using Equation (10). Calculation of Required Separation  
By substituting the conflict frequency and the collision probability in conflict into Equation (11), the overall collision risk is obtained, as shown in Equation (25):  
\[Prob(\text{collision})=2\times 1.92\times 10^{-6}\times D^{2}\int_{D}^{+\infty} \frac{1}{\sqrt{2\pi}\times 1.77^{2}}e^{-\frac{\left(x-38.86\right)^{2}}{2 \times 1.77^{2}}}dx \tag{25}\]  
Let \(Prob(\text{collision})=TLS\), to determine the required separation for UAVs.