## 3 Random-Heading-Based Calibration of UAV Safety Intervals  
### Stochastic Movement Model  
In order to explore in depth the high flexibility characteristics demonstrated by UAVs when flying at low altitude, this paper adopted a stochastic movement model approach in order to systematically model the dynamic behavior of UAVs. By accurately assessing the risk of collision between UAVs, and then establishing safety interval criteria, this ensures  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Category of Drones**} & \multirow{2}{*}{\begin{tabular}{c} **Maximum Take-Off** \\ **Weight (kg)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Maximum Flight** \\ **Altitude (m)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Maximum Level Flight** \\ **Speed (m/s)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Maximum Climbing** \\ **Speed (m/s)** \\ \end{tabular} } \\ \hline Category 1 & 0.798 & 3450 & 6.306 & 3.415 \\ Category 2 & 2.168 & 4700 & 9.731 & 4.014 \\ Category 3 & 4.073 & 5400 & 14.305 & 4.612 \\ Category 4 & 8.591 & 6800 & 18.413 & 5.345 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Performance indicators for each type of UAV.  
Figure 4: UAV clustering result map: (a) Category 1; (b) Category 2; (c) Category 3; (d) Category 4.  
that UAVs perform their tasks safely and efficiently in dense environments. Within the framework of the stochastic movement model, the trajectories of nodes (i.e., UAVs) are assumed to be free, random, and independent of each other, and their trajectories are not affected by other nodes, a setting that helps to simulate the dynamic behaviors of UAVs in complex environments. In this paper, a random heading movement model was used to finely simulate the movement process of UAVs, which can not only capture the characteristics of UAVs moving randomly in space, but also reflect their behavioral patterns in a highly flexible operating environment.  
In the stochastic-heading-based model, the UAV performs independent flight routes in low-altitude airspace \([0,B]^{3}\). At each instantaneous point, the UAV autonomously selects a heading in a given coordinate space and travels along the selected heading \(\theta\in[0,\pi),\phi\in[0,2\pi)\) at a constant rate. The movement of the UAV in each direction in 3D space presents an equal probability distribution, and the flight probability in any given direction remains the same, and thus a uniform probability density function is constructed \(f(\theta,\phi)\), which is described in Equation (6) below:  
\[f(\theta,\phi)=\frac{1}{4\pi},\,\theta\in[0,\pi),\phi\in[0,2\pi) \tag{6}\]  
where \(\theta\) denotes the polar angle and \(\phi\) denotes the azimuth angle.  
In order to simulate a wide range of airspaces, this paper adopted the encircling boundary model to deal with the case of a UAV impacting the boundary, which will appear on the opposite side with the same speed and heading when it impacts the boundary. The UAV motion model in this scenario is described as shown in Equations (7)-(9) below:  
\[X[k]=X[k-1]+v\cos\theta[k-1]-B\bigg{[}\frac{X[k-1]+v\cos\theta[k-1]}{B}\bigg{]} \tag{7}\]  
\[Y[k]=Y[k-1]+v\sin\theta[k-1]-B\bigg{[}\frac{Y[k-1]+v\sin\theta[k-1]}{B}\bigg{]} \tag{8}\]  
\[Z[k]=Z[k-1]+v\cos\phi[k-1]-B\bigg{[}\frac{Z[k-1]+v\cos\phi[k-1]}{B}\bigg{]} \tag{9}\]  
where \(X[k]\), \(Y[k]\), and \(Z[k]\) denote the equations of motion of the UAV along the \(x\)-axis, \(y\)-axis, and \(z\)-axis, and \(\lfloor\cdot\rfloor\) denotes the rounding functions. The boundary wrapping model allows drones to re-enter the designated airspace from the opposite boundary at the same angle, preventing them from exceeding the limits of the specified airspace.  
Figure 5 shows the two-dimensional random heading model.  
Figure 5: Two-dimensional schematic of the stochastic heading model.