Statistical-Geometric Degeneracy in UAV Search: A Physics-Aware Asymmetric Filtering Approach

TL;DR

This work proposes a physically-grounded solution, the AsymmetricHuberEKF, which explicitly incorporates the non-negative physical prior of NLOS biases via a derived asymmetric loss function, and shows that standard symmetric filters correspond to a degenerate case of this framework where the physical constraint is relaxed.

Abstract

Post-disaster survivor localization using Unmanned Aerial Vehicles (UAVs) faces a fundamental physical challenge: the prevalence of Non-Line-of-Sight (NLOS) propagation in collapsed structures. Unlike standard Gaussian noise, signal reflection from debris introduces strictly non-negative ranging biases. Existing robust estimators, typically designed with symmetric loss functions (e.g., Huber or Tukey), implicitly rely on the assumption of error symmetry. Consequently, they experience a theoretical mismatch in this regime, leading to a phenomenon we formally identify as Statistical-Geometric Degeneracy (SGD)-a state where the estimator stagnates due to the coupling of persistent asymmetric bias and limited observation geometry. While emerging data-driven approaches offer alternatives, they often struggle with the scarcity of training data and the sim-to-real gap inherent in unstructured disaster zones. In this work, we propose a physically-grounded solution, the AsymmetricHuberEKF, which explicitly incorporates the non-negative physical prior of NLOS biases via a derived asymmetric loss function. Theoretically, we show that standard symmetric filters correspond to a degenerate case of our framework where the physical constraint is relaxed. Furthermore, we demonstrate that resolving SGD requires not just a robust filter, but specific bilateral information, which we achieve through a co-designed active sensing strategy. Validated in a 2D nadir-view scanning scenario, our approach significantly accelerates convergence compared to symmetric baselines, offering a resilient building block for search operations where data is scarce and geometry is constrained.

Figures (7)

• Figure 1: Schematic illustration of the NLOS localization problem and the proposed active sensing solution in a post-disaster scenario. The UAV acts as an aerial base station trying to localize a trapped User Equipment (UE). Grey polygons represent rubble obstacles blocking the Line-of-Sight (LOS). At time $t_1$, the NLOS signal reflection causes a significantly positively biased range measurement (outer arc). By actively maneuvering to $t_2$ (a "crossing" geometry), the UAV obtains a second, differently biased measurement. The intersection of these biased geometric constraints generates "bilateral information," significantly constraining the estimated target region despite the persistent NLOS conditions.
• Figure 2: Visual comparison of different loss functions versus the residual $r$. The standard L2 loss (blue, dashed) penalizes all errors quadratically. The symmetric Huber loss (orange, dotted) is robust to outliers in both directions. Our proposed one-sided Huber loss (green, solid) is asymmetric: it correctly treats large positive residuals (indicative of NLOS) with linear penalty, while penalizing negative residuals quadratically, thus retaining sensitivity to potentially LOS measurements.
• Figure 3: Active planning dramatically accelerates convergence in the canonical scenario ($\sigma_r=1.5$ m). Note the steep initial RMSE drop of all active methods compared to the slow, gradual decline of passive ones. Our lightweight Proposed (Reactive) system (black, solid) shows highly competitive initial convergence, while the computationally intensive FIM-based methods achieve the best final accuracy, highlighting a key performance-versus-complexity trade-off.
• Figure 4: Note how the Proposed methods correctly converge to a higher effective RTT bias ($\approx 5.5$ m) due to persistent NLOS, unlike the standard Huber-EKF which incorrectly converges to the hardware-only bias (1.5 m). For AoA, all methods converge to the true bias ($-3^\circ$), with our methods achieving faster stabilization.
• Figure 5: Empowered by our filter, different planners reveal distinct strategic personas. The Passive planner (orange) acts as a methodical but inefficient "lawnmower." The Reactive planner (black) behaves like an aggressive "hunter," taking a direct, efficient path to the target. In contrast, the FIM planner (pink) performs as a careful "scientist," executing a complex orbit to meticulously gather information and minimize final uncertainty.

Theorems & Definitions (11)

• Proposition 4.1: Analytical Solution for NLOS Bias
• Proposition 4.2: Properties of the One-sided Huber Loss
• Proposition A.1: Analytical Solution for NLOS Bias
• Proposition A.2: Marginalized Cost Equivalence
• Proposition A.3: Properties of the One-sided Huber Loss
• Proposition A.4: Equivalence of Implementation Parameters
• Lemma B.1: Form of the Expected Robust Curvature Matrix (ERCM)
• Proposition B.1: Bilateral Information and Restricted Strong Convexity
• Proposition B.2: Necessity of Bilateral Information for Fast Convergence
• Proposition C.1: Degeneracy from Unilateral Perspective and Biased Residuals