We present a comprehensive study on cooperative detection and coverage surveillance using fixed-wing unmanned aerial vehicles (fixed-wing UAV) flying in the same circular orbit. Our work addresses the spatiotemporal dynamics inherent in multi-fixed-wing UAV collaborative area coverage, a critical challenge for applications such as border patrol, disaster monitoring, and military reconnaissance. While single fixed-wing UAV operations are constrained by limited endurance and sensor range, deploying multiple fixed-wing UAVs in a coordinated manner can significantly enhance area coverage efficiency and detection probability. However, most existing approaches rely on simple geometric coverage models and fail to fully exploit the temporal dimension of cooperative sensing. In this paper, we derive analytical expressions for the effective coverage zone of two fixed-wing UAVs sharing the same circular trajectory, considering both joint detection probability and allowable monitoring time intervals. Our proposed methodology provides a rigorous theoretical foundation for optimal deployment, leading to a measured coverage improvement of 37.13% over single-fixed-wing UAV operation.
Problem Formulation and Modeling
We consider a scenario where multiple homogeneous fixed-wing UAVs patrol along a circular orbit of radius \(r\) at constant altitude. Each fixed-wing UAV is equipped with an onboard detection sensor (e.g., radar or electro-optical payload). The instantaneous detection probability for a single fixed-wing UAV against a ground target at horizontal distance \(d\) is modeled as:
\[
p(d) = \begin{cases}
1, & d \le d_{\min}, \\
f(d), & d_{\min} < d \le d_{\max}, \\
0, & d > d_{\max},
\end{cases}
\]
where \(f(d)\) is a monotonically decreasing function satisfying \(f(d_{\min})=1\) and \(f(d_{\max})=0\). In this work we adopt a linear form: \(f(d) = 1 – \frac{d – d_{\min}}{d_{\max} – d_{\min}}\). For a set of \(n\) fixed-wing UAVs, the joint detection probability at a target point \(A\) is:
\[
p_n(A) = 1 – \prod_{k=1}^{n} \left(1 – p(d_k)\right),
\]
where \(d_k\) is the horizontal distance from target \(A\) to the \(k\)-th fixed-wing UAV. A target is considered “effectively detected” if the joint probability exceeds a predefined threshold \(p_c\). During continuous patrol along the circular trajectory, some regions are always within the detection zone (strong-effect region), while others are only intermittently covered (weak-effect region). We define the effective coverage zone as the set of points whose temporal coverage ratio \(\eta(A)\) (fraction of time during which the joint detection probability exceeds \(p_c\)) is at least a threshold \(\eta_c\). Mathematically:
\[
S_{\text{eff}} = \{ A \mid \eta(A) \ge \eta_c \}.
\]
Single Fixed-Wing UAV Effective Coverage
For a single fixed-wing UAV flying a circular orbit of radius \(r\), the strong-effect region is a concentric circle of radius \(R_q\) centered at the orbit center \(O\). The boundary point \(T\) satisfies \(p(T)=p_c\). From the geometry:
\[
f(r + R_q) = p_c,
\]
which yields \(R_q\) directly for the given detection function. The effective coverage region (radius \(R_{\text{eff}}\)) is determined by the temporal threshold \(\eta_c\). As the fixed-wing UAV moves along the orbit, the target point accesses the detection zone only when within distance \(d_c\) (where \(p(d_c)=p_c\)). The fraction of time that target \(A\) at radius \(R_{\text{eff}}\) is covered equals the angle subtended by the detection arc divided by \(2\pi\). Setting this equal to \(\eta_c\), we obtain the system:
\[
\begin{cases}
d_{AT_1}^2 = (R_{\text{eff}} + r)^2 + (R_{\text{eff}} \sin\alpha)^2, \\
f(d_{AT_1}) = p_c, \\
\alpha = \pi(1 – \eta_c),
\end{cases}
\]
which is solved numerically via bisection. The resulting \(R_{\text{eff}}\) decreases as \(r\) increases.
Two Fixed-Wing UAVs in the Same Orbit: Cooperative Deployment
We now consider two identical fixed-wing UAVs flying on the same circular orbit of radius \(r\). They are symmetrically placed at opposite points on the circle (angular separation \(180^\circ\)) and move in the same direction. This configuration maximizes the instantaneous joint detection probability. The strong-effect region remains circular. Its radius \(R_q\) satisfies:
\[
f\left(\sqrt{R_q^2 + r^2}\right) = 1 – \sqrt{1 – p_c}.
\]
The effective coverage region for the cooperative pair is also a concentric circle of radius \(R_{\text{eff}}\). The boundary condition is derived by considering the two critical points \(T_1\) and \(T_2\) on the boundary where the joint probability equals \(p_c\) and the arc between them corresponds to the uncovered time fraction \(1-\eta_c\). Setting \(\alpha = \pi(1-\eta_c)/2\), we obtain:
\[
\begin{cases}
d_{AT_1}^2 = R_{\text{eff}}^2 + r^2 – 2rR_{\text{eff}}\cos\left(\frac{\pi}{2} – \alpha\right), \\
d_{BT_1}^2 = R_{\text{eff}}^2 + r^2 – 2rR_{\text{eff}}\cos\left(\frac{\pi}{2} + \alpha\right), \\
1 – \left(1 – f(d_{AT_1})\right)\left(1 – f(d_{BT_1})\right) = p_c.
\end{cases}
\]
These equations are solved numerically to find \(R_{\text{eff}}\). The ratio of \(R_{\text{eff}}\) to the single-fixed-wing UAV case quantifies the benefit of cooperative deployment. Table 1 lists the symbols used in the model.
| Symbol | Definition |
|---|---|
| \(r\) | Radius of the circular flight path |
| \(d_{\min}, d_{\max}\) | Minimum and maximum detection range limits |
| \(p_c\) | Detection probability threshold for effective detection |
| \(\eta_c\) | Temporal coverage ratio threshold |
| \(R_q\) | Radius of strong-effect region |
| \(R_{\text{eff}}\) | Radius of effective coverage region |
| \(d_{AT}, d_{BT}\) | Distances from target to fixed-wing UAVs A and B |
Simulation Results and Analysis
We performed extensive simulations to validate our theoretical derivations. The fixed-wing UAV parameters were set as: \(d_{\min}=420\) km, \(d_{\max}=600\) km, \(f(d)=3 – d/200\) (linear form scaled appropriately), \(p_c=0.8\), \(\eta_c=0.8\), flight radius \(r=100\) km, speed 100 m/s, minimum turn radius 2 km. A grid of 1200×1200 km² was discretized into 1 km² cells. For each cell, the instantaneous and cumulative detection probabilities over one full orbit were computed for three scenarios: (i) single fixed-wing UAV, (ii) two fixed-wing UAVs in the same orbit with same direction (180° separation), and (iii) two fixed-wing UAVs in the same orbit with opposite directions. Table 2 summarizes the simulation parameters.
| Parameter | Value |
|---|---|
| \(d_{\min}\) | 420 km |
| \(d_{\max}\) | 600 km |
| Detection function \(f(d)\) | \(3 – d/200\) |
| \(p_c\) | 0.8 |
| \(\eta_c\) | 0.8 |
| Orbit radius \(r\) | 100 km |
| Grid resolution | 1 km² per cell |
The area coverage percentage (fraction of total grid cells with \(\eta(A) \ge \eta_c\)) was measured. Results are shown in Table 3.
| Deployment Strategy | Coverage (%) | Improvement over single UAV (%) |
|---|---|---|
| Single fixed-wing UAV | 22.30 | – |
| Two fixed-wing UAVs, same orbit, same direction (180° separation) | 59.43 | +37.13 |
| Two fixed-wing UAVs, same orbit, opposite directions | 52.86 | +30.56 |
The cooperative configuration with both fixed-wing UAVs flying in the same direction and separated by 180° achieves the highest coverage area of 59.43%, representing a 37.13% increase compared to a single fixed-wing UAV. When the two fixed-wing UAVs fly in opposite directions, coverage drops to 52.86% because the joint detection geometry becomes less favorable over time. Furthermore, we analyzed the sensitivity of coverage to the angular separation between the two fixed-wing UAVs for both same-direction and opposite-direction cases. The optimal angular separation for same-direction flight is exactly 180°, yielding the maximum coverage. For opposite-direction flight, the optimal separation shifts slightly to 262.8° (i.e., 97.2° measured from one fixed-wing UAV), but the resulting coverage (52.86%) is still less than the same-direction case.
Our analytical results for \(R_{\text{eff}}\) as a function of orbit radius \(r\) agree well with simulation data. For the same-direction cooperative case, \(R_{\text{eff}}\) decreases monotonically with \(r\), but remains significantly larger than the single-fixed-wing UAV baseline for reasonable \(r\) values. The findings confirm that the proposed temporal-probability-based cooperative model effectively captures the benefit of multi-fixed-wing UAV coverage.
Conclusion
We have presented a rigorous framework for analyzing cooperative detection and coverage surveillance using two fixed-wing UAVs flying along the same circular orbit. By combining joint detection probability with a temporal coverage threshold, we derived analytical expressions for the strong-effect and effective coverage zone radii. Simulation results demonstrate that the optimal co-orbital deployment (two fixed-wing UAVs at 180° separation, same direction) achieves a 37.13% increase in effective coverage area compared to a single fixed-wing UAV. Our approach provides a solid theoretical basis for mission planning in dynamic monitoring scenarios. Future work will extend the model to more than two fixed-wing UAVs, heterogeneous sensor suites, and non-circular trajectories, aiming to enhance robustness and scalability for real-world fixed-wing UAV operations.