With rapid development in logistics and low-altitude economies, major corporations have deployed urban delivery drones for last-mile distribution. Current segregated operational models face limitations in airspace utilization efficiency, necessitating multi-altitude operations along shared corridors. This creates critical safety challenges requiring rigorous vertical collision risk assessment. We develop a precise vertical separation framework for delivery UAVs by accurately modeling position errors from operational data.
The collision risk model considers two delivery drones on reciprocal courses at adjacent flight levels. We represent drone A as a cuboid collision box with dimensions $2λ_x × 2λ_y × 2λ_z$ and drone B as a point mass. When drone A penetrates the vertical separation layer with relative velocity components $(u, v, w)$, the collision risk $N_z$ is:
$$N_z = 2S_L P_y(0) P_z(S_z) \left( \frac{λ_x}{S_x} + \frac{\sqrt{u^2 + v^2 + w^2}}{w} \cdot \frac{2λ_x λ_y λ_z + λ_x^2 λ_y}{S_x} \right) E_0$$
where $S_L$ = vertical separation loss frequency, $P_y(0)$ = lateral overlap probability, $P_z(S_z)$ = vertical overlap probability at separation $S_z$, $E_0$ = longitudinal proximity rate, and $S_x$ = longitudinal separation.
We analyzed 25,000 operational records from urban delivery UAV flights at 0.2s resolution during cruise phases. Monte Carlo simulations determined relative velocities by modeling drone kinematics with uniformly distributed pitch angles $α, β ∼ U[-25^\circ, 25^\circ]$ and yaw angles $θ, φ ∼ U[0, 360^\circ]$. Velocity magnitudes followed $V ∼ U[0.95V_{nom}, 1.05V_{nom}]$:
| Parameter | Value | Unit |
|---|---|---|
| Nominal cruise speed ($V$) | 12.0 | m/s |
| Longitudinal relative velocity ($u$) | 7.53 | m/s |
| Lateral relative velocity ($v$) | 7.52 | m/s |
| Vertical relative velocity ($w$) | 3.44 | m/s |
| Drone dimensions ($λ_x, λ_y, λ_z$) | 2.5, 2.5, 0.6 | m |
Vertical position errors exhibited complex distributions inadequately captured by single distributions. Using Bayesian Information Criterion (BIC), we developed a hybrid model combining normal and Laplace distributions:
$$f_z(z) = m \cdot \mathcal{N}(z|0,σ^2) + (1-m) \cdot \mathcal{L}(z|0,b)$$
$$\text{BIC} = -2\ln(L) + k\ln(G_s)$$
where $m=0.53$, $σ=0.52$ m, $b=0.63$ m, and $G_s=5,287$ samples. The hybrid distribution outperformed single distributions across error ranges:
| Model | Central Region BIC | Tail Region BIC |
|---|---|---|
| Normal distribution | 12,418 | 16,892 |
| Laplace distribution | 14,726 | 11,573 |
| Hybrid model | 13,572 | 12,831 |
Vertical overlap probability incorporates this hybrid error model:
$$P_z(S_z) = \int_{-λ_z}^{λ_z} \left[ m \cdot \frac{1}{\sqrt{2π(σ_A^2+σ_B^2)}} \exp\left(-\frac{(z – S_z)^2}{2(σ_A^2+σ_B^2)}\right) + (1-m) \cdot \frac{1}{4b_Ab_B} \exp\left(-\frac{|z – S_z|}{b_A} – \frac{|z – S_z|}{b_B}\right) \right] dz$$
Lateral overlap probability combines normal navigation errors and large deviations from faults/weather:
$$P_y(0) = (1-n) \int_{-λ_y}^{λ_y} \frac{1}{\sqrt{2π(σ_{y1A}^2+σ_{y1B}^2)}} \exp\left(-\frac{y^2}{2(σ_{y1A}^2+σ_{y1B}^2)}\right) dy + n \int_{-λ_y}^{λ_y} \frac{1}{\sqrt{2π(σ_{y2A}^2+σ_{y2B}^2)}} \exp\left(-\frac{y^2}{2(σ_{y2A}^2+σ_{y2B}^2)}\right) dy$$
with $σ_{y1}=1.58$ m (normal), $σ_{y2}=2.6$ m (large deviations), and occurrence rate $n=0.003$. Longitudinal proximity rate $E_0=0.01$ was derived from traffic density $λ=30$ drones/hour.
Target Level of Safety (TLS) requirements determine minimum vertical separation:
| TLS (per flight hour) | Vertical Separation |
|---|---|
| $2.5 × 10^{-9}$ | 13.4 m |
| $1.0 × 10^{-6}$ | 10.2 m |
For urban delivery UAV operations, we recommend 13.4 m vertical separation when adopting the conservative TLS of $2.5 × 10^{-9}$. This framework enables safe high-density operations for delivery drones while maintaining airspace efficiency. The position error modeling approach can be adapted to various delivery UAV configurations by updating kinematic parameters and error distributions.