The rapid proliferation of Unmanned Aerial Vehicles (UAVs), or drones, has transformed numerous sectors including logistics, surveillance, and aerial photography. Their efficiency, flexibility, and relatively low operational cost make them indispensable tools for modern urban management. Among the most spectacular applications is the drone light show, where hundreds or thousands of drones fly in synchronized patterns to create dynamic aerial displays. While a drone light show exemplifies precise formation flying, it operates under highly controlled conditions. The broader integration of drones, particularly logistics drones, into dense urban airspace presents significant challenges for traffic management and safety.
As the number of drone takeoff and landing operations surges, establishing rational minimum safe separation standards, especially during the critical approach phase, becomes paramount for enhancing airspace capacity and operational efficiency. Current research exhibits several limitations: it often focuses on open-category drones rather than specific-category logistics drones; it concentrates primarily on the cruise phase, lacking targeted analysis for the approach phase; and it frequently adapts models from manned aviation without fully accounting for the distinct maneuverability and control characteristics of UAVs.
This study addresses these gaps by focusing on the collision risk modeling for multi-rotor logistics drones during the approach phase. We analyze two typical approach methods employed by mainstream logistics operators: Horizontal Entry with Vertical Descent, and the Diagonal Approach. By improving the traditional Position Error Probability (PEP) model, introducing considerations for dynamic closed-loop control feedback, and incorporating the real-time effects of positioning sampling intervals and speed adjustments, we establish a collision risk assessment framework tailored for the approach phase. The Target Level of Safety (TLS) is used to determine the minimum safe separation required for each approach method.
1. Literature Review and Research Gap
Extensive research exists on collision risk for various aircraft types. Studies on drone-to-manned aircraft collisions often employ Monte Carlo simulations or radar data analysis to assess risk in mixed airspace. For drone-to-drone collisions, research has evolved to consider factors like positional prediction error, speed, and environmental influences like crosswinds. Models such as the improved EVENT model using a conical collision volume have been applied to eVTOL aircraft. However, these studies predominantly address the cruise phase.
The approach phase is fundamentally different and more hazardous. Flight environments are complex, influenced by terrain and obstacles. Drones on the same approach path operate with smaller separations, concentrated conflicts, and lower safety margins. Risk is dominantly influenced by instantaneous position errors, speed errors, and the specific approach geometry. While collision risk assessment for manned aircraft during final approach is mature, directly applying these models to drones is inadequate due to significant differences in speed, weight, sensor accuracy, and computational power. Drones are more susceptible to factors like positioning error and wind shear at low altitudes. This study specifically targets this high-risk phase for logistics drones, building upon but significantly adapting the PEP model principles used in wider aviation safety studies.
2. Typical Drone Approach Methods
The choice of approach method significantly impacts the safety and efficiency of drone operations. For logistics drones, we analyze two prevalent methods.
2.1 Horizontal Entry with Vertical Descent
This is a common, structurally simple method. The drone first flies horizontally at a preset cruise altitude to a Final Approach Point (FAP) directly above the landing zone. Upon reaching the FAP, it transitions to a vertical descent mode until landing. This method offers high efficiency during the level flight segment and enhanced safety during the vertical descent, as it minimizes the risk of collision with surrounding obstacles, making it suitable for complex urban environments. The schematic of this method is conceptually straightforward: a horizontal leg followed by a vertical drop.
2.2 Diagonal Approach
Proposed to optimize energy consumption and dynamic scheduling in high-density environments, this method features a dedicated approach corridor. The airspace around a vertiport is organized with separate entry and exit points. The drone flies horizontally to an Initial Approach Fix (IAF), then descends along a fixed glide slope (e.g., 45 degrees) to the FAP above the landing pad, before executing a final vertical landing. This method optimizes the descent trajectory for battery efficiency and, through distributed entry points, helps reduce airspace congestion and improve throughput for multiple drones.
3. Separation and Target Level of Safety (TLS)
To manage traffic on approach paths, drones must be separated in three-dimensional space: longitudinally (along the path), laterally (side-to-side), and vertically. For two drones on the same approach path, the longitudinal separation between a leading (A) and following (B) drone is most critical. We establish a coordinate system centered on the nominal position of the following drone B, with the y-axis along the intended track, x-axis perpendicular horizontally, and z-axis vertical.
The TLS is defined as the acceptable level of risk, measured in occurrences per flight hour. For manned aviation, stringent TLS values are mandated. For drone operations in densely populated areas, a conservative TLS must be adopted to ensure they do not add undue risk to the existing air traffic system. This study adopts a TLS for Mid-Air Collision (MAC) of $1 \times 10^{-7}$ accidents per flight hour, aligning with the requirement for an ultra-safe system and equivalent safety levels.
4. Position Error Probability (PEP) Model for Approach Phase
The core of the collision risk model is the Position Error Probability model. A drone’s actual flight path deviates from its planned trajectory due to combined errors in positioning and speed. Key assumptions for the model include: drones are of identical model with dimensions $l_x$, $l_y$, $l_z$; only pairwise collisions are considered; errors in lateral, longitudinal, and vertical directions are independent; navigation errors follow a symmetric, unimodal distribution; and no collision avoidance maneuvers are performed.
The total position error for drone $i$ at time $t$ in a given axis (e.g., longitudinal, y-axis) combines a static positioning error and an error propagated from speed uncertainty:
$$\varepsilon_{iy} \sim N(\mu_{icy} + \mu_{iyv}t, \sigma^2_{icy} + \sigma^2_{iyv}t^2)$$
where $\mu_{icy}$ and $\sigma_{icy}$ are the mean and standard deviation of the longitudinal positioning error, and $\mu_{iyv}$ and $\sigma_{iyv}$ are the mean and standard deviation of the longitudinal speed error.
The actual longitudinal distance $R_{iy}(t)$ from a reference point is the sum of the planned distance $D_{iy}(t)$ and the error $\varepsilon_{iy}$. Therefore, the actual longitudinal separation $D_{AB}(t)$ between drone A and B is:
$$D_{AB}(t) = R_{Ay}(t) – R_{By}(t) = [D_{Ay}(t)-D_{By}(t)] + (\varepsilon_{Ay} – \varepsilon_{By})$$
This separation $D_{AB}(t)$ follows a normal distribution with mean $M_y = [D_{Ay}(t)-D_{By}(t)] + [(\mu_{Acy} + \mu_{Ayv}t)- (\mu_{Bcy} + \mu_{Byv}t)]$ and variance $V_y = \sigma^2_{Acy} + \sigma^2_{Ayv}t^2 + \sigma^2_{Bcy} + \sigma^2_{Byv}t^2$.
The probability that the absolute separation is less than the drone’s length $l_y$ (i.e., a longitudinal overlap) at time $t$ is:
$$P_y(t) = \int_{-l_y}^{l_y} \frac{1}{\sqrt{2\pi V_y}} \exp\left(-\frac{(y – M_y)^2}{2V_y}\right) dy$$
Similar formulas apply for lateral ($P_x(t)$) and vertical ($P_z(t)$) overlap probabilities, using corresponding error parameters and drone dimensions ($l_x$, $l_z$).
Assuming independence between axes, the instantaneous probability of a collision at time $t$ is:
$$P(t) = 2 \times P_x(t) \times P_y(t) \times P_z(t)$$
The factor of 2 accounts for the fact that a collision is a two-aircraft event.
5. Case Study and Numerical Analysis
We analyze the collision risk for two consecutive drones during approach. The parameters for a typical logistics drone model are listed in Table 1. A positioning data sampling interval of 1 second is assumed, based on common GNSS/RTK system update rates.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| $\mu_{icy}, \mu_{icz}$ | 0 m | $\sigma_{icy}, \sigma_{icz}$ | 2.5 m |
| $\mu_{iyv}, \mu_{izv}$ | 0 m/s | $\sigma_{iyv}, \sigma_{izv}$ | 0.2398 m/s |
| $l_x, l_y$ | 1.24 m | $l_z$ | 0.45 m |
| Target TLS ($TLS_{MAC}$) | $1 \times 10^{-7}$ per flight hour | ||
5.1 Horizontal Entry with Vertical Descent
This approach is analyzed in two segments. First, during the level flight to the FAP, lateral and vertical collision probabilities are conservatively set to 1 to study the dominant longitudinal risk. Second, during the vertical descent from the FAP, longitudinal and lateral probabilities are set to 1 to study the vertical risk. The interaction at the transition point (FAP) is also examined. The collision risk decreases as the initial separation increases. The governing condition for the minimum safe initial separation is the point where the risk throughout the entire maneuver meets the TLS.
| Segment | Initial Separation (m) | Collision Risk, P(t=0) | Meets TLS? |
|---|---|---|---|
| Level Flight | 17 | $1.9225 \times 10^{-7}$ | No |
| Level Flight | 18 | $2.8876 \times 10^{-8}$ | Yes |
| Vertical Descent | 16 | $2.3159 \times 10^{-7}$ | No |
| Vertical Descent | 17 | $3.6299 \times 10^{-8}$ | Yes |
| Transition (FAP) | 20 | $2.4985 \times 10^{-7}$ | No |
| Transition (FAP) | 21 | $5.1406 \times 10^{-8}$ | Yes |
The most stringent condition arises from the transition point at the FAP. Therefore, the minimum safe initial separation for this approach method is 21 meters. Analyzing the risk over time at this separation with a 1-second update interval shows the maximum instantaneous risk remains below the TLS throughout the maneuver.
5.2 Diagonal Approach
This method involves three segments: level flight to the IAF, diagonal descent to the FAP, and vertical landing. The analysis for the first and third segments mirrors the previous method. The critical analysis focuses on the diagonal segment and the transition points at IAF and FAP.
| Critical Point | Initial Separation (m) | Collision Risk, P(t=0) | Meets TLS? |
|---|---|---|---|
| Diagonal Segment | 23 | $2.6745 \times 10^{-7}$ | No |
| Diagonal Segment | 24 | $7.0779 \times 10^{-8}$ | Yes |
| Transition (IAF) | 25 | $1.5748 \times 10^{-7}$ | No |
| Transition (IAF) | 26 | $4.2712 \times 10^{-8}$ | Yes |
| Transition (FAP) | 24 | $1.1112 \times 10^{-7}$ | No |
| Transition (FAP) | 25 | $3.0353 \times 10^{-8}$ | Yes |
The most stringent condition arises from the transition at the IAF (level to diagonal). Therefore, the minimum safe initial separation for the Diagonal Approach method is 26 meters. Time-domain analysis confirms the instantaneous risk at this separation remains compliant with the TLS.
6. Discussion
The results provide quantifiable safety intervals for two standard logistics drone approach procedures. The Horizontal Entry with Vertical Descent method requires a smaller initial separation (21m) compared to the Diagonal Approach (26m). This difference can be attributed to the geometry and error propagation in the diagonal segment, where errors in both the horizontal and vertical planes interact along the glide path, potentially creating a larger combined positional uncertainty zone that necessitates greater separation.
The model demonstrates that the highest collision risk typically occurs at transition points between flight phases (e.g., at the FAP or IAF), where the relative geometry and error dynamics change. This highlights the need for special attention to these phases in airspace procedure design and real-time monitoring. The chosen TLS of $1 \times 10^{-7}$ is conservative and ensures a high level of safety, which is crucial for public acceptance and regulatory approval of large-scale drone logistics operations, much like the public trusts the safety of a tightly coordinated drone light show.
It is important to note that this model provides a strategic, baseline separation. It does not account for real-time tactical collision avoidance systems (e.g., Detect and Avoid) that drones may employ, which could allow for reduced separations in practice. However, this strategic minimum forms the foundational safety layer for traffic management.
7. Conclusion
This study developed a tailored Position Error Probability model to assess airborne collision risk for logistics drones during the critical approach phase. By incorporating drone-specific parameters, dynamic error propagation, and control update intervals, the model offers a realistic framework for risk quantification. For the two analyzed approach methods—Horizontal Entry with Vertical Descent and Diagonal Approach—the minimum safe initial separations required to meet a Target Level of Safety of $1 \times 10^{-7}$ accidents per flight hour are 21 meters and 26 meters, respectively.
These findings contribute to the development of a quantifiable safety evaluation framework for unmanned traffic management (UTM), specifically for terminal area operations. The methodology can be extended to other approach geometries or drone configurations. Future research should focus on integrating this strategic model with tactical collision avoidance algorithms, validating parameters with real flight data, and exploring the impact of environmental factors like wind gusts. Furthermore, as drone operations scale up, from logistics to complex formations like a metropolitan-scale drone light show, the principles of rigorous risk modeling and safe separation established here will remain fundamental to ensuring the safety and efficiency of our shared airspace.