## 3 Random-Heading-Based Calibration of UAV Safety Intervals
### Conflict Probability Definition and Measurement  
For the determination of the conflict between two UAVs \(i\) and \(j\), the conflict distance is defined as \(d_{l}\), when the distance is \(D_{i,j}[k]\leq d_{l}\) between two UAVs, it is considered that the two UAVs are at the point of \(k\) conflict. Where \(d_{l}\) is determined by the distance of the safety zone when the two UAVs are at a tangent. In this paper, we simplify the position space of the UAVs and fully consider the flight attitude of the UAVs, taking into account the high-frequency attitude changes of the UAVs, such as their pitching motion, and in order to ensure that the wings of the UAVs did not go beyond the area of the safety zone of the UAVs in the process of attitude change. The proposed use of the outer sphere of the body of a UAV as the safety zone for UAVs is shown in Figure 6.  
The physical parameters of the UAV safety zone sphere are as follows:  
Assuming that the horizontal length of the UAV is x, the horizontal width is y, and the height is z, the radius \(\lambda\), the volume v, and the surface area s of the sphere model can be obtained as shown in Equations (10)-(12):  
\[\sqrt{(\frac{x}{2})^{2}+(\frac{y}{2})^{2}+(\frac{z}{2})^{2}}=\lambda \tag{10}\]  
\[V=\frac{4}{3}\tau\lambda^{3} \tag{11}\]  
\[S=4\pi\lambda^{2} \tag{12}\]  
By constructing a sphere model of the safety zone, the conflict problem between UAVs can be transformed into a geometric problem of analyzing the positional relationship of spheres in 3D space. To simplify the analysis process, the safety zone of UAV i is relocated to the position of UAV j. This adjusted protection area is defined as the combined sphere safety zone. The motion relationship between UAV i and UAV j can be expressed as a relative motion state; i.e., UAV i is regarded as a stationary entity under the global coordinate system, whereas UAV j has a specific relative velocity with respect to UAV i. In this setting, the UAVs within the combined sphere safety zone are regarded as potentially vulnerable. In this setting, UAV j within the safety of the combined sphere is treated as a target potentially attacked, while the other UAV i is abstracted as a prime, symbolic intruder. This strategy eliminates the complex spatial geometric relationships and dynamic interactions in traditional collision analysis, effectively focuses on the critical intrusion threat assessment, and thus provides a more intuitive and simplified mathematical model for UAS security policy formulation, as shown in Figure 7.  
Figure 6: Schematic diagram of the UAV safety zone: (**a**) three-dimensional figure; (**b**) front view; (**c**) top view. In summary, UAVs i and j are considered to be at risk of conflict when \(D_{ij}[k]\leq\lambda_{i}+\lambda_{j}\). For the calculation of the conflict probability between two UAVs, the concept of a conflict heading cone is introduced, and the definition of a conflict heading cone is as follows:  
Conflict heading cone: In the three-dimensional space from the UAV to the combined sphere safety zone as a ray, so that the ray and the combined sphere safety zone are at a tangent, tangent rays are combined into a cone side. This side surrounded by the cone includes the set of relative headings that will be in conflict in the three-dimensional space. A sectional view of the cone of conflicting headings is shown in Figure 8.  
Based on the above definition, a formula is derived for the conflict probability between two UAVs:  
\[P_{ij}[k]=\frac{l_{1}}{l_{0}}=\frac{S_{collision}}{S_{direction}} \tag{13}\]  
The relative heading of UAV i is spherically distributed in three-dimensional space, and the sphere contains the set of relative headings of UAV i. The conflict probability can be regarded as the ratio between the geometric surface areas.