With the rapid advancement of automation and artificial intelligence technologies, traditional logistics systems are evolving into intelligent networks where delivery drones play a pivotal role. The flexibility and labor-saving advantages of delivery UAVs (Unmanned Aerial Vehicles) have accelerated their adoption in last-mile logistics by companies like Amazon, DHL, and SF Express. However, the inherent high-speed operation and trajectory uncertainties of delivery drones create significant collision risks during batch operations. Such collisions may result in ground impacts causing severe casualties and property damage. This study quantifies collision probabilities between logistics UAVs by developing a dual-error probabilistic model incorporating positioning and velocity uncertainties, providing critical safety insights for urban airspace operations.
Conflict Zone Theory and Safety Framework
The foundation of delivery UAV collision risk assessment lies in conflict zone theory, where a conflict occurs when two UAVs violate minimum separation standards simultaneously in horizontal or vertical dimensions. We adopt the equivalent safety level proposed by the European Union Aviation Safety Agency (EASA), setting the acceptable collision risk threshold at $1.5 \times 10^{-8}$ accidents per flight hour. To model delivery UAV encounters, we define a spherical protection zone around each drone with radius $R$ derived from physical dimensions:
$$R = \max\left( \frac{\lambda_{1x} + \lambda_{2x}}{2}, \frac{\lambda_{1y} + \lambda_{2y}}{2}, \frac{\lambda_{1z} + \lambda_{2z}}{2} \right)$$
where $\lambda_{i\mu}$ represents the length of UAV $i$ ($i=1,2$) in dimension $\mu$ ($\mu = x,y,z$). This spherical approximation enables efficient 3D collision detection during dynamic flight maneuvers.
Dual-Error Collision Probability Model
The collision probability model integrates two critical uncertainty sources affecting delivery UAV operations:
- Positioning Error: Navigation system inaccuracies cause deviations between actual and intended positions. We model this error as a Gaussian random variable $\varepsilon_g \sim \mathcal{N}(\mu_g, \sigma_g^2)$ with $\mu_g = 0$.
- Velocity Error: Propulsion system variations induce speed fluctuations, modeled as $\varepsilon_v \sim \mathcal{N}(\mu_v, \sigma_v^2)$ with $\mu_v = 0$.
For two delivery drones on intersecting paths with track angle $\theta$ in the horizontal plane, the instantaneous collision probability at relative time $t_{\Delta}$ is:
$$P = \int_{-R}^{R} \frac{1}{\sqrt{2\pi}\sigma_{12}^g \sigma_{12}^v} \exp\left(-\frac{(L – L_b)^2}{2(\sigma_{12}^g)^2 (\sigma_{12}^v)^2}\right) dL$$
where $L$ denotes the actual separation considering errors, $L_b$ is the nominal separation under wind effects, and $\sigma_{12}^g$, $\sigma_{12}^v$ are composite errors derived through coordinate transformation:
$$\sigma_{12}^g = \sqrt{(\sigma_{xg}^{12})^2 + (\sigma_{yg}^{12})^2 + (\sigma_{zg}^{12})^2}$$
The total collision probability $CP_B$ across conflict duration $T$ integrates all possible entry time differences $t_{\Delta}$:
$$CP_B = \sum_{t_{\Delta}=0^{t_{\text{max}}} \frac{1}{T} \int_0^T P dt$$
Additionally, we incorporate delivery UAV reliability and human factors into the total operational risk $ER$:
$$ER = 2 \times 3,600 \times CP_B \times u_1 \times (1 – u_2)$$
where $u_1 = 1 – \prod_{i=1}^n (1 – p_i’)$ represents system failure probability (Table 1), and $u_2$ denotes human reliability.
| Failure Type | Impact | Probability |
|---|---|---|
| Battery Failure | Complete loss of power/thrust | $7.42 \times 10^{-4}$ |
| Motor Failure | Lift deficiency | $1.24 \times 10^{-4}$ |
| Propeller Failure | Uncontrolled descent | $4.95 \times 10^{-6}$ |
| Cargo Bay Failure | Payload detachment | $8.81 \times 10^{-5}$ |
| Control System Failure | Loss of manual override | $2.13 \times 10^{-6}$ |
| Total System Failure ($u_1$) | $9.68 \times 10^{-4}$ |
Wind Impact on Delivery Drone Operations
At low altitudes, wind significantly influences delivery UAV trajectory deviations. The aerodynamic force in direction $\mu$ ($\mu = x,y,z$) is calculated as:
$$F_{\mu} = \frac{1}{2} c \rho S_{\mu} (V_w^{\mu})^2$$
where $c = 0.2$ is the drag coefficient, $\rho = 1.293 \text{ kg/m}^3$ is air density, $S_{\mu}$ is the projected area, and $V_w^{\mu}$ is the wind velocity component. The resulting displacement $S_{\mu}$ during time $t$ is:
$$S_{\mu} = \frac{1}{2} \left( \frac{F_{\mu}}{m} \right) t^2$$
This wind-induced deviation directly affects the nominal separation $L_b$ in our collision probability model.
Collision Risk Analysis Across Flight Scenarios
Using parameters from the DJI M600 delivery drone (Table 2), we analyzed collision probabilities under varying track angles $\theta$ and wind directions.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Mass ($m$) | 12 kg | Length ($\lambda_x$) | 1.668 m |
| Width ($\lambda_y$) | 1.518 m | Height ($\lambda_z$) | 0.759 m |
| Max Speed | 18 m/s | Cruise Speed | 13 m/s |
| Min Speed | 3 m/s | Spherical Radius ($R$) | 0.843 m |
Minimum Safe Separation Analysis
The relationship between collision probability and track angle $\theta$ reveals critical risk thresholds (Figure 5 equivalent). Collision probability exhibits a bimodal distribution with maxima at $\theta = 30^\circ$ and $\theta = 140^\circ$, and minima near $90^\circ$. The minimum safe separation distance $L_{\text{min}}$ ensuring $ER \leq 1.5 \times 10^{-8}$ across all scenarios is:
$$L_{\text{min}} = 90.71 \text{ m}$$
This separation distance accommodates delivery UAV operations in parallel ($\theta = 0^\circ$), crossing ($\theta = 30^\circ – 150^\circ$), and head-on ($\theta = 180^\circ$) encounter geometries at 13 m/s cruise speed.
Wind Direction Sensitivity
Wind effects on collision probability vary significantly with track angle $\theta$ (Figure 7 equivalent):
- $\theta = 30^\circ, 60^\circ, 90^\circ$: Collision probability remains stable with wind direction variations ($\Delta CP < 12\%$). The $90^\circ$ crossing scenario shows minimal sensitivity ($\Delta CP < 3\%$).
- $\theta = 140^\circ$: High sensitivity to wind direction with probability fluctuations exceeding 40%. Critical risk occurs when crosswinds exceed 15° relative to UAV heading.
These findings indicate that delivery drone operations at acute and right crossing angles are robust to wind variations, while obtuse-angle encounters require stringent wind condition monitoring.
Conclusions
This study establishes a dual-error collision probability model for delivery UAVs that incorporates positioning inaccuracies, velocity variations, wind effects, and system reliability. Our analysis demonstrates that a minimum separation of 90.71 m maintains collision risk below the $1.5 \times 10^{-8}$ accidents/flight-hour safety threshold across all operational scenarios. The delivery UAV collision probability varies non-monotonically with track angle, showing peak sensitivity at 30° and 140° intersections. Wind direction significantly influences collision risk only at obtuse crossing angles ($\theta \approx 140^\circ$), suggesting that delivery route planning should avoid such geometries in variable wind conditions. These results provide critical input for urban air traffic management systems supporting large-scale delivery drone operations. Future work will integrate onboard collision avoidance systems into the risk framework to enable dynamic separation adjustments.