The integration of large Unmanned Aerial Vehicles (UAVs or drones) into controlled airspace alongside manned aircraft presents significant challenges for air traffic safety. In complex, high-density environments like terminal areas, intersecting flight paths dramatically increase the potential for mid-air collisions. Effective safety management in such mixed-traffic scenarios hinges on the ability to assess collision risk accurately and in real-time, providing a foundation for traffic flow management and strategic deconfliction decisions. Traditional collision risk models, while foundational, often rely on simplifying assumptions about the statistical nature of navigation errors, such as the Laplacian (double exponential) distribution. These assumptions can limit their accuracy and adaptability when confronted with the complex, non-Gaussian, and often asymmetric error characteristics observed in real-world trajectory data from systems like ADS-B. This paper addresses these limitations by proposing a novel methodology for real-time collision probability estimation that leverages historical trajectory data and nonparametric statistical modeling, offering a more flexible and precise tool for assessing the risks posed by UAV drone operations in shared airspace.
The core challenge in probabilistic collision risk modeling lies in accurately characterizing the uncertainty in an aircraft’s reported position, known as the Navigation System Error (NSE). The total system error, which defines the deviation of the true position from the intended flight path, comprises several components: Path Definition Error (PDE), Flight Technical Error (FTE), and the NSE. For risk assessment based on surveillance data like ADS-B, the NSE—the discrepancy between the reported navigation position and the true position—is the primary source of uncertainty to be modeled. The performance and reliability of the UAV drone’s navigation system directly feed into this error term, making its accurate representation paramount for credible risk analysis.
Classical analytical models, such as the Anderson and subsequent Aldis models, provide a framework for computing collision probabilities by convolving the probability density functions (PDFs) of the position errors of two aircraft. The Aldis model, in particular, offers a computationally efficient, closed-form solution for crossing-track scenarios by assuming independent Laplacian distributions for the along-track and cross-track error components. The horizontal overlap probability (HOP) in the Aldis formulation for aircraft 1 and 2 is given by:
$$HOP = 2\pi \lambda_{xy}^2 \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f_{A_2}(\xi) f_{C_2}(\eta) f_{A_1}(\xi \cos \theta – \eta \sin \theta – D_x(t)) f_{C_1}(\xi \sin \theta + \eta \cos \theta – D_y(t)) \, d\xi \, d\eta$$
where \(\lambda_{xy}\) is the combined horizontal dimension, \(f_{A_i}\) and \(f_{C_i}\) are the along-track and cross-track error PDFs for aircraft \(i\), \(\theta\) is the heading intersection angle, and \(D_x(t)\), \(D_y(t)\) describe the relative motion projected onto the coordinate axes. The vertical collision probability \(P_z\) is computed separately based on the altitude error. However, the inherent assumption of a Laplacian distribution for all error components is a significant constraint, as real-world NSE data from UAV drone and manned aircraft operations often exhibits skewness, multimodality, or heavier tails than the Laplacian model can capture. This mismatch can lead to systematic biases in the estimated risk. Furthermore, many models are designed for steady-state or fixed-time-window analysis, lacking the granularity to estimate risk dynamically at any arbitrary moment during a flight encounter, which is crucial for real-time monitoring.
To overcome these limitations, this work introduces a nonparametric approach centered on Kernel Density Estimation (KDE) for modeling the NSE. The proposed methodology follows a structured pipeline: data preprocessing, error feature extraction via B-spline smoothing, nonparametric error modeling with KDE, real-time probability calculation, and rigorous validation through Monte Carlo simulation.
Methodology for Nonparametric Risk Assessment
1. Trajectory Preprocessing and Error Decomposition
Historical ADS-B trajectory data, containing time-stamped positions (latitude, longitude, altitude), heading, and speed, forms the foundation. Data from both manned aircraft and the UAV drone are synchronized to a common time grid (e.g., 0.5-second intervals) using interpolation. The core task is to extract the NSE components—the deviation of the reported position from a plausible, smooth reference path. This is achieved using B-spline curve fitting. For a set of original track points \(\{X_i\}_{i=1}^N\), a smooth reference trajectory \(C(u)\) is obtained by solving:
$$\min_{\{P_j\}} \sum_{i=1}^{N} \left\| X_i – \sum_{j=1}^{M} N_{j,k}(u_i)P_j \right\|^2 + s \cdot R(C)$$
where \(N_{j,k}\) are the B-spline basis functions of order \(k\), \(P_j\) are the control points, \(s\) is a smoothing factor, and \(R(C)\) is a regularization term penalizing curvature. The optimal smoothing factor \(s\) is determined through an ablation study, comparing the collision probability error against a simulated baseline across multiple scenarios. A Wilcoxon signed-rank test is used to identify the factor that yields stable, minimal error without overfitting.
For each raw position \(X_i\), the corresponding point on the fitted curve \(C(u_i)\) and its tangent direction (heading) \(\theta_i\) are calculated. The navigation error vector is \(\Delta e_i = X_i – C(u_i)\). This vector is then decomposed into the critical error components relative to the flight direction:
$$
\begin{align*}
\text{Cross-Track Error (CTE)}_i &= -\Delta x_i \cdot \sin(\theta_i) + \Delta y_i \cdot \cos(\theta_i) \\
\text{Along-Track Error (ATE)}_i &= \Delta x_i \cdot \cos(\theta_i) + \Delta y_i \cdot \sin(\theta_i)
\end{align*}
$$
A similar process is applied to altitude data to obtain vertical navigation errors. This decomposition generates empirical samples of CTE, ATE, and vertical error for the UAV drone and the manned aircraft, which are used for the subsequent modeling step.
2. Nonparametric Error Modeling using Kernel Density Estimation
Instead of imposing a parametric form like the Laplacian distribution, the probability density function of each error component is estimated directly from the data using Kernel Density Estimation (KDE). For a set of error samples \(\{e_1, e_2, …, e_N\}\) (e.g., all CTE samples for a particular UAV drone type), the KDE estimate of the PDF is:
$$\hat{f}(e) = \frac{1}{Nh} \sum_{i=1}^{N} K\left( \frac{e – e_i}{h} \right)$$
where \(K(\cdot)\) is the kernel function (a Gaussian kernel is used here: \(K(u) = \frac{1}{\sqrt{2\pi}} \exp(-\frac{u^2}{2})\)), and \(h\) is the bandwidth, optimized using Silverman’s rule of thumb. This approach allows the model to faithfully represent asymmetries, multiple peaks, and tail behavior present in the actual UAV drone navigation error data. The resulting PDFs \(\hat{f}_{A_{UAV}}, \hat{f}_{C_{UAV}}, \hat{f}_{A_{manned}}, \hat{f}_{C_{manned}}, \hat{f}_{z}\) are used as direct replacements for the parametric functions in the collision probability calculus.
3. Real-Time Collision Probability Calculation
The proposed model integrates the nonparametric error PDFs into a framework for dynamic risk assessment. At any given time \(t\) during the encounter between the UAV drone and a manned aircraft, their planned positions (from flight plans or predicted paths), relative velocity, and geometry are known or estimated. The horizontal overlap probability is computed by evaluating a discretized version of the Aldis convolution integral, but using the KDE-derived PDFs:
$$HOP(t) \approx 2\pi \lambda_{xy}^2 \sum_{m} \sum_{n} \hat{f}_{A_2}(\xi_m) \hat{f}_{C_2}(\eta_n) \hat{f}_{A_1}(\xi_m \cos \theta – \eta_n \sin \theta – D_x(t)) \hat{f}_{C_1}(\xi_m \sin \theta + \eta_n \cos \theta – D_y(t)) \Delta \xi \Delta \eta$$
The vertical overlap probability \(P_z(t, h_z)\) is computed similarly using the KDE model for altitude error. The instantaneous probability of collision at time \(t\) is the product of the horizontal and vertical overlap probabilities, scaled by the inverse of the relative speed to yield a probability-per-unit-time metric. This calculation can be performed in real-time as the UAV drone progresses along its route, providing a continuous risk stream.
4. Validation via Monte Carlo Simulation
To validate the accuracy of the proposed KDE-based model, a high-fidelity Monte Carlo (MC) simulation is employed as a benchmark for “ground truth.” For each synchronized time step in a historical or simulated encounter, the reported positions of both aircraft are perturbed by random navigation errors. These errors are drawn from statistical distributions (e.g., Gaussian) whose parameters are derived from the actual ADS-B error statistics. This process is repeated millions of times (\(N_{total} \approx 10^{12}\)).
The simulated horizontal collision probability at time \(t\) is:
$$P^{MC}_{hor}(t) = \frac{N_{col,hor}(t)}{N_{total}}$$
where \(N_{col,hor}(t)\) is the count of simulations where the perturbed horizontal separation fell below the minimum horizontal separation threshold (e.g., the sum of wingspans). The total collision probability also considers vertical separation:
$$P^{MC}_{total}(t) = \frac{N_{col,both}(t)}{N_{total}}$$
The collision probability time series \(P^{MC}(t)\) generated from this exhaustive simulation serves as the reference against which the proposed KDE model and the traditional Laplacian model are compared.
Experimental Analysis and Results
Scenario and Parameter Configuration
The analysis focuses on an encounter scenario involving a large UAV drone (modeled after a Twin-Tailed Scorpion A) and a manned aircraft (Cessna 172) within a general aviation corridor. Historical ADS-B data from a specific flight date was used, showing a characteristic “converge-cross-diverge” pattern with a minimum horizontal separation of approximately 322 meters. Key parameters for the aircraft involved are summarized in Table 1.
| Parameter | UAV Drone (Twin-Tailed Scorpion A) | Manned Aircraft (Cessna 172) |
|---|---|---|
| Length | 10.5 m | 8.2 m |
| Wingspan | 20.2 m | 11.0 m |
| Height | 2.9 m | 2.7 m |
| Operational Speed | ~58 m/s (210 km/h) | Varies (~40-60 m/s) |
B-spline Smoothing Factor Selection
The ablation study for the B-spline smoothing factor \(s\) compared five values across multiple simulated encounter scenarios. The optimal factor was selected based on the minimum average absolute error between the model-predicted collision probability and the MC simulation baseline, coupled with statistical significance in the Wilcoxon test. The results, summarized in Table 2, identified \(s = 0.0005\) as the optimal choice, providing a stable balance between fitting the underlying path and filtering out noise without overfitting, a critical step for reliable error extraction for the UAV drone data.
| Smoothing Factor (s) | Mean Absolute Error | Wins in Pairwise Wilcoxon Tests | Effect Size (Cliff’s Delta) |
|---|---|---|---|
| 0.05 | 7.59e-9 | 0 | 0.833 |
| 0.005 | 4.97e-9 | 1 | 0.603 |
| 0.0005 | 3.81e-10 | 4 | 0.605 |
| 0.00005 | 1.38e-9 | 3 | 0.658 |
| 0.000005 | 1.16e-9 | 2 | 0.734 |
Model Accuracy Verification
The KDE-based model and the traditional Laplacian model were used to calculate collision probability time series for 100 randomly generated encounter scenarios with varying geometries and speeds. The absolute error of each model relative to the MC “truth” was computed for each scenario. The aggregated results, shown in Table 3 and Figure 1, demonstrate a clear superiority of the nonparametric approach.
| Model | Average Absolute Error | Improvement over Laplacian |
|---|---|---|
| Laplacian (Parametric) | 5.94e-9 | Baseline |
| KDE (Nonparametric) | 3.76e-9 | 36.7% |
Figure 1: The relative error ratio (model error as a percentage of the true probability) for the KDE and Laplacian models across the 100 test scenarios. The KDE model consistently shows lower and more stable error levels, confirming its enhanced robustness and accuracy in quantifying UAV drone collision risk.
The KDE model achieved an average error reduction of approximately 36.7% compared to the Laplacian model. More importantly, the error distribution for the KDE model was tighter, indicating greater reliability and consistency across diverse encounter conditions that a UAV drone might face. Applying the KDE model to the real historical encounter data yielded a peak instantaneous collision probability of \(4.2 \times 10^{-28}\) per second. When extrapolated to a standard metric, this corresponds to an event rate far below established Target Level of Safety (TLS) benchmarks for general aviation airspace (e.g., \(1 \times 10^{-7}\) per flight hour), confirming the inherent safety of the properly managed encounter but also validating the model’s ability to produce quantifiable, ultra-low-risk figures.
Sensitivity Analysis via Random Forest
To understand the dominant factors influencing the collision probability calculated for a UAV drone encounter, a sensitivity analysis was conducted using a Random Forest regressor. Four key input features were considered: relative distance (\(d\)), heading intersection angle (\(\theta\)), manned aircraft speed (\(v_1\)), and UAV drone speed (\(v_2\)). The target variable was the collision probability output from the KDE model for thousands of scenario variations. Feature importance was calculated using permutation importance. The results, visualized in Table 4, reveal the dominant drivers of risk.
| Feature | Importance Score (Normalized) | Interpretation |
|---|---|---|
| Relative Distance (d) | 1.00 | Most critical factor. Probability decays sharply with increasing separation. |
| Heading Angle (\(\theta\)) | 0.35 | Moderate influence. Affects the geometry and duration of the encounter. |
| Manned Aircraft Speed (\(v_1\)) | 0.06 | Minor influence when considered in isolation. |
| UAV Drone Speed (\(v_2\)) | 0.04 | Minor influence when considered in isolation. |
Partial Dependence Plots (PDPs) further elucidated these relationships. The PDP for relative distance confirmed a strong, monotonically decreasing relationship with collision probability, consistent across different speed levels. The PDP for the heading angle showed a more complex, non-linear interaction with speed, as the angle and speeds together determine the relative velocity vector. The primary insight is that spatial proximity (relative distance) is the overwhelming determinant of instantaneous collision risk for a UAV drone, followed by the encounter geometry encapsulated by the relative velocity vector. This underscores the fundamental principle that collision risk arises primarily from positional uncertainty in close proximity, guiding risk mitigation strategies to prioritize separation assurance.
Conclusion
This research developed and validated a novel framework for the real-time, quantitative assessment of airborne collision risk for UAV drones operating in mixed airspace. By replacing the restrictive parametric error assumptions of classical models with a flexible, data-driven nonparametric approach using Kernel Density Estimation, the proposed method achieves a significant improvement in modeling accuracy—approximately 36.7% more accurate than the standard Laplacian model—while maintaining the computational efficiency necessary for real-time application. The integration of B-spline-based error extraction ensures robust feature engineering from noisy trajectory data.
The methodology provides a dynamic risk portrait, capable of estimating the probability of collision at any arbitrary moment during a flight encounter, which is a critical capability for future autonomous traffic management systems. The sensitivity analysis using Random Forest quantitatively identified relative distance as the paramount factor governing collision risk, providing valuable guidance for operational planning and the design of separation minima for UAV drone integration.
This work demonstrates that leveraging historical performance data through advanced statistical modeling can substantially enhance the fidelity of safety assessments. The proposed KDE-based model offers a powerful and adaptable tool for regulators, air navigation service providers, and UAV drone operators to evaluate and mitigate collision risks in increasingly complex and congested airspace environments, paving the way for safer and more efficient integration of unmanned systems.