A Risk-Aware Robust Navigation Framework for UAVs in GNSS-Degraded Low-Altitude Environments

Highlights

What are the main findings?

• A novel DPAP framework is proposed, integrating robust single-point positioning, passive differential refinement, and scenario-aware fusion to tackle geometric ill-conditioning in low-altitude UAV navigation.
• Experiments under severe conditions (≥200 km baselines, 15% NLOS) demonstrate a 0.588 m RMS accuracy, outperforming traditional Huber M-estimation and elevation-weighted methods by 13% and 60%, respectively.

What are the implications of the main findings?

• The framework provides a critical GNSS-independent backup for UAVs, enabling safe operations in the emerging low-altitude airspace even when satellite signals are degraded or denied.
• The proposed three-stage architecture establishes a practical paradigm for robust positioning in sparse infrastructure scenarios, demonstrating that meter-level accuracy is achievable using only limited ground-based broadcast transmitters.

Abstract

To address the critical issues of geometric ill-conditioning and non-line-of-sight (NLOS) interference faced by broadcast radio positioning systems in long-distance transmission (≥200 km) and low-altitude flight scenarios (1000 m to 3000 m), this paper proposes a Differential and Robust Positioning method for Airborne Platforms (DPAP). Integrating radio differential positioning, the proposed method enhances the single-point positioning algorithm through a grid search and iteratively reweighted least squares to mitigate geometric ill-conditioning and numerical instability in low-altitude environments. Furthermore, a passive differential positioning approach is introduced to eliminate common errors using neighboring reference stations. Finally, a scenario-aware safe fusion strategy ensures that the fused solution is never inferior to the optimal sub-solution under any circumstances. Simulation results demonstrate that, under conditions involving six ground stations, user-to-station distances of no less than 200 km, and 15% of links experiencing NLOS propagation, the differential and robust positioning method achieves a positioning accuracy of 0.588 m RMS. This represents an improvement of approximately one order of magnitude compared to RSPP (12.304 m), and outperforms traditional Huber M-estimation (0.678 m) and elevation-weighted least squares methods (1.462 m). All results are based on Monte Carlo simulations; real-world validation with SDR hardware and flight tests is left for future work. This work provides a scalable, infrastructure-light backup for safe UAV operations in GNSS-hostile environments, directly supporting the emerging low-altitude economy.

1. Introduction

The rapid development of the “low-altitude economy” has propelled the global aviation industry into a new era dominated by electric Unmanned Aerial Vehicles (UAVs) [1,2], creating an urgent demand for high-precision and highly available positioning services. As the mainstream positioning technology, Global Navigation Satellite Systems (GNSS) are susceptible to multipath effects, signal blockage, and intentional jamming in urban canyons, mountainous terrains, or electronic warfare environments, leading to degraded performance or even complete failure [3]. Therefore, exploring GNSS-independent backup positioning technologies is of significant practical importance. Broadcast radio positioning systems achieve passive positioning by receiving ranging signals transmitted from ground stations, offering advantages such as strong anti-jamming capability, low deployment cost, and short construction cycles [4]. This makes them a complementary technology to GNSS, suitable for low-altitude economic applications.

In recent years, extensive research has advanced radio positioning technology as a critical backup to GNSS, particularly for autonomous systems. These efforts can be broadly categorized into signal exploitation strategies, system architecture innovations, and robust algorithmic developments.

Regarding signal exploitation, scholars have leveraged both opportunistic and dedicated signals. Kassas et al. [5] pioneered a radio SLAM framework using terrestrial signals of opportunity (SOPs), demonstrating GNSS-independent vehicle positioning with tens-of-meters accuracy. Expanding on cellular infrastructure, Whiton et al. [6] utilized large-scale antenna arrays on ground carriers to process downlink signals, achieving meter-level accuracy via neural network-optimized metrics. Similarly, Zhao et al. [7] validated the practical value of cellular positioning for autonomous systems, emphasizing the need for absolute sensors with bounded error constraints. More recently, Bednarz et al. [8] explored SDR-based Doppler localization on UAVs for tactical radios, highlighting the versatility of radio sensing in dynamic environments. However, most of these studies primarily target ground vehicles, where the geometric relationship with base stations is favorable. Their direct application to low-altitude UAVs remains challenging due to distinct elevation angle distributions and sparse coverage in rural corridors. The term “favorable geometric configuration” for ground vehicles refers to scenarios where the user and reference stations are at comparable altitudes and in relatively close proximity. This spatial arrangement typically results in a low Geometric Dilution of Precision (GDOP), as the lines-of-sight to the reference stations exhibit good angular diversity, leading to a well-conditioned positioning problem. In contrast, for a UAV flying at low altitude (e.g., 1000 m to 3000 m) relative to distant ground-based transmitters (>200 km away), the elevation angles to all stations are very low (often <5°). This causes the lines-of-sight to become nearly parallel, resulting in a high GDOP and a geometrically ill-conditioned system, which is highly sensitive to measurement noise and NLOS errors.

In terms of system architecture, efforts have focused on establishing independent datums and enhancing GNSS integration. Dou et al. [9] proposed a distributed autonomous method for high-precision spatial datum establishment in Radio Local Positioning Systems (RLPS), ensuring robustness and efficiency. Meanwhile, Fan [10] analyzed the synergy between radio and GNSS, proposing a Dual-Filter Tightly Coupled (DF-TC) model that safely integrates fixed radio solutions to improve overall navigation reliability. While effective, these architectures often assume dense base station availability, which may not hold in the vast, low-density scenarios typical of the emerging low-altitude economy.

Finally, regarding algorithmic robustness, various methods have been developed to mitigate errors. Techniques such as improved M-estimation [11,12,13] and Elevation-Angle-Weighted Least Squares (EA-WLS) [14,15,16] effectively suppress outliers under normal conditions. Additionally, double-difference approaches by Liu et al. [17,18,19] successfully eliminate common clock and systematic errors. Nevertheless, these methods frequently encounter convergence failures or performance degradation in scenarios characterized by severe Geometric Dilution of Precision (GDOP)—a common issue for UAVs flying at low elevation angles relative to distant towers—or under persistent Non-Line-of-Sight (NLOS) interference. It is important to distinguish this scenario from the favorable geometry experienced by ground vehicles mentioned earlier. In terrestrial applications, users and base stations are typically co-located at similar altitudes with short baselines, yielding good angular diversity and low GDOP. By contrast, the long-range (>200 km) broadcast signals used here result in extremely low elevation angles (<5°) for low-altitude UAVs, causing near-parallel lines-of-sight and a geometrically ill-conditioned positioning problem characterized by high vertical GDOP. Prior studies have addressed various error sources in urban GNSS positioning, including multipath mitigation, NLOS detection, and hardware-induced biases such as antenna phase center offsets [20] and the exploitation of multipath for environmental sensing [21]. However, most of these approaches assume access to calibrated equipment or external correction data, which may not be available in resource-constrained UAV platforms. In contrast, our DPAP framework operates without requiring such auxiliary information.

Based on the above analysis of the state-of-the-art, while radio positioning technology has made significant progress, it still faces two major challenges in low-altitude scenarios where the user’s flight altitude is between 1000 m and 3000 m and the ground station antenna height is approximately 10 m to 30 m: (1) The user’s flight altitude is much higher than the ground station antenna height, resulting in line-of-sight elevation angles generally below 5°. This leads to an increased Geometric Dilution of Precision (GDOP), causing traditional least squares estimates to deviate significantly from the true value and resulting in an ill-conditioned geometric configuration. (2) Terrain undulations and atmospheric refraction cause signal path bending or reflection, leading to widespread NLOS propagation interference.

To overcome these limitations, this paper proposes a Differential and Robust Positioning for Airborne Platforms (DPAP) method. This method treats the ground transmitters as sparsely deployed distributed RF sensing nodes and is designed for sensor-constrained low-altitude aerial platforms. It features a Robust Single-Point Positioning (RSPP) module that combines grid search initialization, Iteratively Reweighted Least Squares (IRLS), and Tikhonov [22,23] regularization to enhance the reliability of the initial solution. Furthermore, a passive differential positioning method (Differential Global Navigation Satellite System-Refinement, DRP-R) is constructed to eliminate common systematic errors. On this basis, a covariance-based safe fusion strategy is proposed to ensure that the fused solution is superior to the best individual sub-solution under all circumstances. Simulation results show that the DPAP method can maintain meter-level positioning accuracy even under extreme geometric configurations and NLOS interference, indicating its broad engineering application prospects in the low-altitude economy domain. Notably, our simulations demonstrate that DPAP achieves meter-level positioning accuracy even under extreme geometric configurations—here defined as a challenging environment in which all reference transmitters are observed at elevation angles below 5°, yielding a vertical DOP (VDOP) > 10 and total GDOP > 15, conditions under which conventional algorithms typically diverge. This robustness highlights DPAP’s strong potential for engineering deployment in the emerging low-altitude economy.

It is important to clarify that this study is based on simulations. The simulations are designed to capture the key characteristics of low-altitude UAV navigation, including realistic NLOS statistics and geometric constraints. However, real-world performance may differ due to unmodeled hardware imperfections, atmospheric dynamics, or signal propagation anomalies. Therefore, the findings should be interpreted as a proof-of-concept, and experimental validation is a necessary next step before field deployment. A simplified one-dimensional along-track profile (with known altitude) is adopted in this work; the extension to full 3D will be addressed in future research.

2. System Model

In this work, the user altitude is assumed known from a barometric altimeter or a pre-filed flight plan. This is a standard practice in UAV corridor navigation, as GNSS vertical data have been shown to be reliable for altitude determination [24]. The user state vector comprises only the along-track coordinate x and the receiver clock bias ${\mathrm{b}}_{\mathrm{u}}$. The altitude is assumed known and fixed at ${\mathit{z}}_{\mathit{u}}=\mathrm{H}$, which can be obtained from a barometric altimeter or a pre-filed flight plan.

Although the estimated state is one-dimensional, the pseudorange observation model inherently involves two spatial dimensions: it depends on both the along-track position x and the known vertical separation between the user (at altitude $\mathrm{H}$) and each ground station (at height ${\mathit{h}}_{\mathit{gs}}$). This vertical offset is fixed and known a priori, so it does not need to be estimated, but it critically shapes the low-elevation geometry that drives NLOS vulnerability.

• Lateral motion is tightly controlled. In low-altitude corridor operations, UAVs are required to stay close to a nominal path. Airspace rules and autopilot systems actively suppress cross-track deviations. Modern controllers typically keep lateral errors near zero [25]. Thus, omitting the lateral dimension does not sacrifice realism for this scenario.
• Positioning geometry is dominated by the vertical plane. When ground stations lie approximately along the flight path, cross-track observability is weak. Recent GDOP analyses for UAV corridors show that geometric strength comes mainly from the along-track and vertical directions [26]. Therefore, the key challenge-ill-conditioning due to low elevation angles is fully captured in the $x-z$ plane.
• Reduced-dimensional modeling is a common and valid approach. Many advanced GNSS-denied methods first prove their core ideas in 2D before extending to 3D [27]. This lets researchers isolate the fundamental issue, in our case, elevation-dependent NLOS bias without unnecessary complexity.

Given these points, we develop and evaluate DPAP within this 1D along-track setting (with fixed altitude). Importantly, DPAP’s core mechanism-risk-aware covariance inflation based on NLOS probability is dimension-agnostic. It relies only on measurement error statistics, not on spatial dimensionality. Hence, the observed benefits, such as lower bias and stronger integrity under NLOS are expected to carry over to 2D and full 3D implementations. Moreover, a sensitivity analysis (Section 4.5) shows DPAP remains robust even with lateral offsets up to 50 m well beyond typical cross-track errors in regulated corridor flights.

2.1. Basic Assumptions

This paper addresses the navigation requirements of low-altitude airspace and the characteristics of the broadcast radio positioning system. Regarding the clock differences among ground stations, the following assumption is made: the ground stations achieve coordinated operation through a high-precision time synchronization network, and the error of the inter-station clock difference component $b_c$ is better than 3 ns. Therefore, the clock difference of each ground station can be modeled as $b_{gs}^{\left(i\right)}=b_c+\delta b_i$, where $\mathsf{\delta }{\mathrm{b}}_{\mathrm{i}}$ is a residual small error that can be neglected. During the system’s positioning calculation, $b_c$ and $b_{\mathrm{u}}$ are combined and estimated as an equivalent clock difference parameter $b_{cu}$.

2.2. Observation Model

A sparsely distributed network of ground stations is deployed to periodically broadcast GNSS-like ranging signals. A low-altitude flying user passively receives signals from multiple ground stations via an onboard receiver. After baseband processing and feature extraction, these signals are converted into pseudorange measurements suitable for positioning. The user’s meter-level autonomous positioning is achieved by collaboratively solving these pseudorange measurements. Figure 1 illustrates the architecture of the broadcast radio positioning system.

This paper addresses the positioning requirements of low-altitude platforms under airway constraints by applying a simplified one-dimensional profile along the airway (horizontal coordinate x, with altitude z known) for observation model derivation, simulating positioning challenges with low-elevation observations in low-altitude scenarios. The two-dimensional positioning scenario solved in this paper can be extended to three-dimensional scenarios in future work. The system parameters and model definitions are specified below:

• Ground Station Deployment: Referring to the deployment conditions of terrestrial radio positioning systems such as eLoran [28] and Locata [29], and to meet the requirement that the distance between ground stations and users is no less than 200 km in this study, we assume the deployment of 6 ground stations, i.e., M = 6. In this paper, system modeling is conducted within a one-dimensional along-track profile. The two-dimensional coordinates of the i-th ground station are denoted as $p_{gs}^{\left(i\right)}=[x_{gs}^{\left(i\right)},0,h_{gs}{]}^{\mathrm{T}}$, where i ∈ {1, 2, …, M} and $h_{gs}$ = 30.
• User State: The two-dimensional coordinates of the user in this paper are denoted as $p_u=[x_u,z_u{]}^{\mathrm{\top }}$, where $x_u$ represents the user’s horizontal position coordinate, and $z_u$ represents the user’s flight altitude with a value of $\mathrm{H}$, which is set according to the application scenario.
• Observation Model: This paper defines the geometric distance from the i-th ground station to the user as ${\mathrm{r}}_{\mathrm{i}}$, which satisfies Equation (1):

  $$\begin{array}{c}{r}_i=\Vert p_u-p_{gs}^{\left(i\right)}\Vert =\sqrt{(x_u-x_{gs}^{\left(i\right)}{)}^2+(z_u-h_{gs}{)}^2}\end{array}$$

  (1)

  where the vertical height difference is defined as $d_{\mathrm{z}}=z_u-h_{gs}$, then the i-th pseudorange observation ${\rho }_i$ must satisfy Equation (2):

  $$\begin{array}{c}{\rho }_i=r_i+b_u+b_{gs}^{\left(i\right)}+{ϵ}_i^{NLoS}+n_i\end{array}$$

  (2)

  where

  -

  $r_i \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{t}\mathrm{r}\mathrm{u}\mathrm{e} \mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c} \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e} \mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d} \mathrm{i}\mathrm{n} \mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} (1$).

  -

  $b_u$ is the user receiver clock bias.

  -

  $b_{gs}^{\left(i\right)}$ is the transmit clock bias of the i-th ground station, where $b_{gs}^{\left(i\right)}=b_c+\delta b_i$.

  -

  ${ϵ}_i^{NLoS}$ is the positive bias introduced by Non-Line-of-Sight (NLOS) propagation.

  -

  $n_i\sim \mathcal{N}(0,{\sigma }_{\rho }^2)$ is zero-mean Gaussian measurement noise, including thermal noise.

The core task of this paper is to perform joint estimation using the observation vector $\rho =[{\rho }_1,{\rho }_2,\dots ,{\rho }_M{]}^T$ based on the above model, ultimately solving for the user position $x_u$ and the clock bias $b_u$.

3. DPAP Positioning Method

The proposed DPAP method is a high-precision positioning approach for low-altitude platforms, comprising three stages: single-point positioning initialization (RSPP), differential refinement (DRP-R), and safe fusion decision (Safe Fusion). This three-stage progressive architecture enhances the stability and reliability of the radio positioning system under NLOS, multipath, and geometrically ill-conditioned scenarios in low-altitude environments.

3.1. Robust Single-Point Positioning Initialization (RSPP)

The proposed enhanced single-point positioning method provides a globally robust initial value through a coarse grid search and achieves local precise optimization by combining it with the Iteratively Reweighted Least Squares (IRLS) method.

3.1.1. Coarse Grid Initialization

It is assumed that the low-altitude user flies along a known route, with the horizontal coordinate range $[x_{min}, x_{max}]$. This range is divided into $N_g$ equally spaced grid points $\{x^{\left(g\right)}{\}}_{\mathrm{g}=1}^{{\mathrm{N}}_{\mathrm{g}}}$. For each candidate position $x^{\left(g\right)}$, the following steps are performed:

• Calculate the geometric distance $r_i\left(x^{\left(g\right)}\right)$ as in (3):

  $$r_i\left(x^{\left(g\right)}\right)=\sqrt{(x^{\left(g\right)}-x_{gs}^{\left(i\right)}{)}^2+d_z^2}$$

  (3)

  where $d_z=H-h_{gs}$.
• Estimate the initial clock bias $b^{\left(0\right)}\left(x^{\left(g\right)}\right)$ as in (4):

  $$b^{\left(0\right)}\left(x^{\left(g\right)}\right)={\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}}_{\mathrm{i}}\left({\rho }_i-r_i\left(x^{\left(g\right)}\right)\right)$$

  (4)

  leveraging the robustness of the median to outliers.
• Calculate the total L1 residual $\mathrm{J}\left(x^{\left(g\right)}\right)$ as in (5). The grid point corresponding to the minimum value of the objective function $\mathrm{J}\left(x^{\left(g\right)}\right)$ is established as the initial state $(x_u^{\left(0\right)},b_u^{\left(0\right)})$, thereby avoiding the dependence of non-convex optimization on initial value selection.

  $$\mathrm{J}\left(x^{\left(g\right)}\right)={\sum }_{\mathrm{i}=1}^{\mathrm{M}}\left|{\rho }_i-r_i\left(x^{\left(g\right)}\right)-b^{\left(0\right)}\left(x^{\left(g\right)}\right)\right|$$

  (5)

3.1.2. IRLS Refinement

After grid initialization, the IRLS algorithm is executed starting from the grid-derived initial value. In each iteration, the result from the previous iteration is incorporated. For the k-th iteration, the pseudorange residuals $e_i$ are calculated using the state estimate from the previous iteration ($x_u^{(k-1)}$, $b_u^{(k-1)}$), as shown in (6):

$$e_i={\rho }_i-r_i\left(x_u^{(k-1)}\right)-b_u^{(k-1)},\mathrm{i}=1,2,\dots ,\mathrm{M}$$

(6)

Following the calculation of the pseudorange residuals $e_i$, the Tukey biweight function is applied to dynamically adjust the weight ${{ w}_i}^{\mathit{tukey}}$ based on the current residuals, as in (7):

$${w_i}^{\mathit{tukey}}=\left\{\begin{array}{ll}{\left(1-{\left({\displaystyle \frac{e_i}{c}}\right)}^2\right)}^2, & \left|e_i\right|\le c\\ 0,& \left|e_i\right|>c\end{array}\right.$$

(7)

where c is the threshold for the Tukey biweight function. Let ${\theta }_i$ be the elevation angle of the i-th observation, given by (8):

$${\theta }_i=\arcsin \left({\displaystyle \frac{d_z}{r_i}}\right)$$

(8)

Referring to the RTCA DO-229D standard [30] and the NLOS statistical characteristics in high-altitude platform scenarios [31], the elevation-angle-dependent weight $w_{iel}$ is determined as in (9):

$$w_{iel}=\left\{\begin{array}{ll}0.3,& {\theta }_i<15^{\circ }\\ 0.7,& 15^{\circ }\le {\theta }_i<25^{\circ }\\ 1.0,& {\theta }_i\ge 25^{\circ }\end{array}\right.$$

(9)

By deriving the Tukey weight $w_i^{tukey}$ and the elevation-dependent weight $w_i^{el}$, the composite weight ${\mathit{w}}_{\mathit{i}}$ is obtained, as shown in (10):

$$w_i=w_i^{tukey}·w_i^{el}$$

(10)

The design matrix $\mathrm{H}\in {\mathbb{R}}^{\mathrm{M}\times 2}$ is constructed by calculating the partial derivatives of the i-th row of the design matrix with respect to the state vector $x=[x_u,b_u{]}^{\top }$, as in (11):

$$H_i=\left[-{\displaystyle \frac{x_u^{(k-1)}-x_{gs,i}}{r_i}},-1\right]$$

(11)

To mitigate the ill-conditioning of the design matrix caused by high GDOP, the Tikhonov regularized normal equation is solved as in (12):

$$({\mathrm{H}}^{\mathrm{\top }}\mathrm{W}\mathrm{H}+\mathsf{\lambda }\mathrm{I})\mathsf{\Delta }={\mathrm{H}}^{\mathrm{\top }}\mathrm{W}\mathrm{e}$$

(12)

where $W=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(w_1,\dots ,w_M)$, $\mathsf{\Delta }=[{\Delta }_x,{\Delta }_b{]}^{\mathrm{\top }}$ denotes the state increment, and $\lambda$ is the regularization parameter, which is determined empirically via trial and error to be $\lambda =10^{-6}$ in this paper. The justification and sensitivity analysis of this and other empirical parameters are provided in Section 4.6.

The state estimation update is then performed as follows: $x_u^{\left(k\right)}=x_u^{(k-1)}+{\Delta }_x$, $b_u^{\left(k\right)}=b_u^{(k-1)}+{\Delta }_b$. The iteration terminates when the Euclidean norm of the state increment falls below a predefined threshold ε, $\begin{array}{c}|\Delta {|}_2=\sqrt{{\Delta }_x^2+{\Delta }_b^2}<\epsilon \end{array}$. Referring to conventional settings in the navigation and positioning field [32], this paper assumes ε = 10⁻⁴ m. After the iteration concludes, the weighted variance of the RSPP solution, ${\sigma }_{\mathit{rspp}}^2\left(\mathit{k}\right)$, is calculated using the final residuals and weights, as shown in Equation (13).

$${\sigma }_{\mathit{rspp}}^2\left(\mathit{k}\right)={\displaystyle \frac{\sum _{\mathit{i}=1}^{\mathit{M}}{\mathit{w}}_{\mathit{i}}{\mathit{e}}_{\mathit{i}}^2}{\sum _{\mathit{i}=1}^{\mathit{M}}{\mathit{w}}_{\mathit{i}}-\upsilon }}$$

(13)

In this paper, $\upsilon =2$ is the degrees of freedom for the estimated parameters, corresponding to the two unknowns in this model: the user’s position $x_u$ and clock bias $b_u$.

3.2. Differential Radio Positioning–Refinement (DRP-R)

This paper further designs a robust differential positioning method, DRP-R, to mitigate the impact of common errors (such as ground station clock biases) on positioning accuracy inherent in the RSPP method. Drawing inspiration from differential DGNSS, this method performs differencing between the user and a nearby reference station to eliminate systematic errors, including common clock biases and atmospheric delays. To reduce geometric ill-conditioning and NLOS interference under sparse base station deployment, the same elevation-adaptive weighting strategy introduced in Section 3.1 is employed, where the weight for each ground station is determined based on its elevation angle according to the RTCA DO-229D standard [30] (Equation (9)) and combined with a Tukey robust weight to form the composite weight $w_i$ (Equation (10)). This deterministic weighting scheme requires no training and incorporates neither neural networks nor learnable parameters.

This paper assumes the deployment of a reference station with a known position near the user, where the coordinates of the reference station are $p_{ref}=[x_{ref},z_{ref}{]}^{\mathrm{\top }}$. The pseudorange observation ${\mathsf{\rho }}_{\mathrm{i}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ for the reference station is given by Equation (14):

$$\begin{array}{c}{\rho }_i^{ref}=r_i^{ref}+b_c+n_i^{ref}\end{array}$$

(14)

where $\begin{array}{c}{r}_i^{ref}=\sqrt{(x_{ref}-x_{gs}^{\left(i\right)}{)}^2+(z_{ref}-h_{gs}{)}^2}=\sqrt{x_{gs}^{\left(i\right)2}+{d_z}^2}\end{array}$, and $d_z=\mathrm{H}-h_{gs}$ denotes the vertical height difference, which is identical to that of the user. ${\mathrm{n}}_{\mathrm{i}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ represents the measurement noise at the reference station’s receiver for the signal from the i-th ground station.

Subsequently, the differential pseudorange observation $\mathsf{\Delta }{\rho }_{\mathit{i}}$ between the user and the reference station is constructed as shown in Equation (15):

$$\begin{array}{ll}\Delta {\rho }_i& ={\rho }_i-{\rho }_i^{ref}+r_i^{ref}\\ & =(r_i+b_u+b_c+{ϵ}_i^{NLoS}+n_i)-(r_i^{ref}+b_c+n_i^{ref})+r_i^{ref} \\ & =r_i+b_u+{ϵ}_i^{NLoS}+(n_i-n_i^{ref})\end{array}$$

(15)

where $r_i=\sqrt{(x_u-x_{gs}^{\left(i\right)}{)}^2+{d_z}^2}, r_i^{ref}=\sqrt{x_{gs}^{\left(i\right)2}+{d_z}^2},d_z=\mathrm{H}-h_{gs}$, and the differenced noise is ${\tilde{n}}_i=n_i-n_i^{ref}$.

The DRP-R method utilizes the RSPP result $({\mathit{x}}_{\mathit{u}}^{\left(\mathit{k}\right)}, {\mathit{b}}_{\mathit{cu}}^{\left(\mathit{k}\right)})$ as the initial value to perform the weighted least squares estimation on the differenced observations $\mathsf{\Delta }{\mathsf{\rho }}_{\mathrm{i}}$, employing the composite weighting strategy defined in Equations (9) and (10), which combines elevation-dependent reliability (per RTCA DO-229D [30]) and Tukey robustness against outliers. Let $\mathrm{H}$ denote the weighted design matrix and W the weight matrix; the covariance matrix Q of the DRP-R solution is estimated via the regularized normal equation $\mathrm{Q}=({\mathrm{H}}^{\mathrm{\top }}\mathrm{W}\mathrm{H}+\lambda \mathrm{I}{)}^{-1}$, where $\lambda =10^{-6}$ is determined by cross-validation. During the positioning calculation, by utilizing the condition that $\mathrm{Q}(1,1)$ represents the variance estimate ${\mathit{\sigma }}_{\mathit{Diff-}\mathit{R}}^2\left(\mathit{k}\right)$ of the position component, the DRP-R solution $({\widehat{\mathit{x}}}_{\mathit{Diff-}\mathit{R}}^{\left(\mathit{k}\right)}, {\widehat{\mathit{b}}}_{\mathit{Diff-}\mathit{R}}^{\left(\mathit{k}\right)})$ and its variance ${\sigma }_{\mathit{Diff-}\mathit{R}}^2\left(\mathit{k}\right)$ are computed based on the aforementioned derivations.

3.3. Safe Fusion

Based on the simulation experiments, although RSPP demonstrates global search capability in ideal high-altitude scenarios, it suffers from systematic bias exceeding 10 m due to uncorrected clock errors and residual multipath effects. The DRP-R method eliminates these error sources via differencing, yielding positioning results close to the theoretical optimum; however, naively fusing these solutions would introduce bias and degrade performance. In challenging urban scenarios, significant NLOS errors can cause the DRP-R differential corrections to introduce systematic bias, leading to correction failure, whereas RSPP can provide a robust solution without systematic bias by leveraging reliable line-of-sight signals from high-elevation satellites. To ensure the final output outperforms either individual solution, this paper designs a scenario-aware safe fusion mechanism.

• Scenario Branch Decision

The safe fusion strategy designed in this paper requires determining the user’s current operating environment. If the user’s altitude is not less than 1000 m and the NLOS probability is no greater than 15%, the scenario is classified as an ideal high-altitude environment, and the DRP-R solution is output directly. Conversely, if the user’s altitude is not greater than 100 m and the NLOS probability is not less than 30%, the scenario is classified as an Urban Challenging Environment, triggering the adaptive fusion module.

2.

Remarks on Real-Time NLOS Probability Estimation

In this simulation study, the NLOS probability ${\mathrm{P}}_{\mathrm{NLoS}}$ and the per epoch NLOS—affected station ratio ${\mathit{r}}_{\mathit{NLOS}}\left(\mathit{k}\right)$ are assumed to be known from the simulation settings. However, in a real-world implementation, these quantities are not directly available and must be estimated online. Fortunately, a substantial body of literature has established that NLOS conditions can be reliably detected using only standard receiver measurements. Signal quality indicators such as the carrier-to-noise ratio (C/N0), pseudorange residuals, and elevation angles have been widely used for LOS/NLOS classification in urban environments [33,34,35]. More recent works have successfully employed machine learning techniques—including support vector machines, random forests, and deep neural networks—to achieve NLOS detection accuracies exceeding 90% [33,36,37]. In particular, studies have shown that the proportion of NLOS-affected satellites (or, analogously, ground stations) can be estimated from the classification results and directly used to adapt positioning algorithms [34,35].

Therefore, while the development of a real-time NLOS detector is beyond the scope of this paper, the proposed risk-aware covariance inflation formula (Equation (16)) is generic and can accept ${\mathit{r}}_{\mathit{NLOS}}\left(\mathit{k}\right)$ estimated from such an external module. Integrating a dedicated NLOS detector into the DPAP framework is a natural direction for future work and is expected to further enhance its practical applicability.

3.

Risk-Aware Covariance Modeling

In challenging urban scenarios, traditional covariance estimation methods based on residuals often fail to adequately account for factors such as non-line-of-sight propagation and low-elevation observations, thereby compromising positioning accuracy. To address this, this paper applies a robust correction to the raw covariance estimates for each epoch k, obtaining the risk-aware covariances ${\stackrel{\mathrm{~}}{\sigma }}_{\mathit{rspp}}^2\left(\mathit{k}\right) \mathrm{and} {\stackrel{\mathrm{~}}{\sigma }}_{\mathit{Diff-}\mathit{R}}^2\left(\mathit{k}\right)$ via Equation (16):

$$\begin{array}{cc}{\stackrel{\mathrm{~}}{\sigma }}_{\mathit{rspp}}^2\left(\mathit{k}\right)& = \mathit{max}\left({\sigma }_{\mathit{rspp}}^2\left(\mathit{k}\right), {\sigma }_{\mathit{min}}^2\right)·\left(1 + {\eta }_{\mathit{NLOS}}·{\mathit{r}}_{\mathit{NLOS}}\left(\mathit{k}\right)\right)·\left(1 + {\eta }_{\mathit{el}}·\mathit{I}\left[\stackrel{\text{-}}{\theta }\left(\mathit{k}\right) < 25°\right]\right)\\ {\stackrel{\mathrm{~}}{\sigma }}_{\mathit{Diff-}\mathit{R}}^2\left(\mathit{k}\right)& = \mathit{max}\left({\sigma }_{\mathit{Diff-}\mathit{R}}^2\left(\mathit{k}\right), {\sigma }_{\mathit{min}}^2\right)·\left(1 + {\eta }_{\mathit{NLOS}}·{\mathit{r}}_{\mathit{NLOS}}\left(\mathit{k}\right)\right)·\left(1 + {\eta }_{\mathit{el}}·\mathit{I}\left[\stackrel{\text{-}}{\theta }\left(\mathit{k}\right) < 25°\right]\right)\end{array}$$

(16)

where ${\sigma }_{\mathit{rspp}}^2\left(\mathit{k}\right)$ is the raw position variance output by the RSPP solver, and ${\sigma }_{\mathit{Diff-}\mathit{R}}^2\left(\mathit{k}\right)$ is the raw position variance output by the DRP-R solver; ${\sigma }_{\mathit{min}}^2$ is the covariance floor to prevent weights from becoming invalid when covariance approaches zero under high SNR conditions. Simulation experiments show that when ${\sigma }_{\mathit{min}}^2$ < 0.25 m², the fusion performance becomes sensitive to noise; this paper sets ${\sigma }_{\mathit{min}}^2$ = 0.25 m²; ${\mathit{r}}_{\mathit{NLOS}}\left(\mathit{k}\right) \in [0, 1]$ is the proportion of base stations affected by NLOS at the k-th epoch, randomly generated through a preset NLOS probability in simulations; $\stackrel{\text{-}}{\theta }\left(\mathit{k}\right)$ is the average elevation angle of all visible base stations at the k-th epoch; $\mathrm{I}$[·] is the indicator function used to activate the elevation penalty term; ${\mathrm{\eta }}_{\mathrm{NLOS}}$ is the NLOS risk amplification factor. Control variable experiments in this paper show that when ${\eta }_{\mathit{NLOS}}$ = 0.3, the RMS is approximately 3 times that of the ideal case; this paper sets ${\eta }_{\mathit{NLOS}}$ = 3.0 to match the uncertainty growth; ${\eta }_{\mathit{el}}$ is the low-elevation risk amplification coefficient. Experiments indicate that when the mean elevation is less than 25$°$, the GDOP increases significantly, and the standard deviation of multipath errors increases by approximately 2–3 times. Consequently, this paper conservatively estimates ${\eta }_{\mathit{el}}$ = 2.5. The selection of these parameters is further justified through sensitivity analysis in Section 4.6.

4.

Constrained Inverse-Variance Weighted Fusion

Based on the risk-aware covariances, this paper computes the fusion weights corresponding to the Best Linear Unbiased Estimator, as shown in Equation (17):

$$\alpha \left(\mathit{k}\right) = {\displaystyle \frac{1/{\stackrel{\mathrm{~}}{\sigma }}_{\mathit{Diff-}\mathit{R}}^2\left(\mathit{k}\right)}{1/{\stackrel{\mathrm{~}}{\sigma }}_{\mathit{rspp}}^2\left(\mathit{k}\right) + 1/{\stackrel{\mathrm{~}}{\sigma }}_{\mathit{Diff-}\mathit{R}}^2\left(\mathit{k}\right)}}$$

(17)

where ${\stackrel{\mathrm{~}}{ \sigma }}_{\mathit{rspp}}^2\left(\mathit{k}\right)$ and ${\tilde{\sigma }}_{\mathit{Diff-}\mathit{R}}^2\left(\mathit{k}\right)$ are the risk-aware covariances at epoch k, and the weight $\alpha \left(\mathit{k}\right)$ ∈ [0, 1] represents the contribution ratio of DRP-R in the fusion, while 1 − $\alpha \left(\mathit{k}\right)$ corresponds to the contribution of RSPP.

Although the covariance of DRP-R is often smaller in urban environments, the differential approach may introduce significant systematic bias that is not fully captured by the covariance model. To prevent such “high-confidence yet erroneous” estimates from dominating the fusion result, this paper imposes a safety upper-bound constraint on α(k), as given in Equation (18):

$${\alpha }_{safe}\left(k\right)=\min ({\alpha }_{max},\max (0,\alpha \left(k\right)\left)\right)$$

(18)

This constraint limits the weight of DRP-R to a predefined safety threshold ${\mathsf{\alpha }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, thereby capping its potential influence and mitigating the risk associated with possible unmodeled systematic biases. The choice of ${\alpha }_{max}$ is critical and is determined through a dedicated sensitivity analysis presented in Section 4.3. The analysis reveals that ${\alpha }_{max}$ = 0.8 yields a reasonable compromise: while a higher ${\alpha }_{max}$ (e.g., 1.0) gives marginally better RMS (0.857 m vs. 0.868 m), the chosen value provides a conservative safety margin against unmodeled real-world errors.

Considering that ${\mathit{r}}_{\mathit{NLOS}}\left(\mathit{k}\right)$ and $\stackrel{\text{-}}{\theta }\left(\mathit{k}\right)$ may fluctuate from epoch to epoch, directly using ${\alpha }_{\mathit{safe}}\left(\mathit{k}\right)$ would lead to discontinuities in the positioning trajectory. To ensure smoothness, a first-order low-pass filter is applied to the fusion weights, as shown in Equation (19):

$${\alpha }_{\mathit{smooth}}\left(\mathit{k}\right) = \beta ·{\alpha }_{\mathit{smooth}}(\mathit{k} - 1) + (1 - \beta )·{\alpha }_{\mathit{safe}}\left(\mathit{k}\right)$$

(19)

The smoothing factor is set to $\beta$ = 0.99. This conservative setting ensures stable state transitions when the UAV enters or exits signal-degraded regions such as urban canyons. As demonstrated in Section 4.6.4 (“Sensitivity Analysis for β”), the RMS positioning error varies by less than 5% across $\beta \in [0.9,0.99]$. This indicates that significantly faster adaptation—e.g., using $\beta$ = 0.9, which corresponds to an effective time constant of approximately 10 s—is entirely feasible without compromising accuracy. In practice, users can select a smaller β to meet specific real-time responsiveness requirements. The current choice of $\beta$ = 0.99 prioritizes stability over speed for the purpose of clearly illustrating the algorithm’s behavior in our simulation figures.

Based on the above-derived parameters and the smoothed weight ${\alpha }_{\mathit{smooth}}\left(\mathit{k}\right)$, this paper performs linear fusion separately for position and clock bias, as shown in Equation (20):

$$\begin{gathered} {\mathit{x}}_{\mathit{fused}}\left(\mathit{k}\right) = \left(1 - {\alpha }_{\mathit{smooth}}\left(\mathit{k}\right)\right)·{\mathit{x}}_{\mathit{spp}}\left(\mathit{k}\right) + {\alpha }_{\mathit{smooth}}\left(\mathit{k}\right)·{\mathit{x}}_{\mathit{Diff-}\mathit{R}}\left(\mathit{k}\right) \\ {\mathit{b}}_{\mathit{cu}}\left(\mathit{k}\right) = \left(1 - {\alpha }_{\mathit{smooth}}\left(\mathit{k}\right)\right)·{\mathit{b}}_{\mathit{spp}}\left(\mathit{k}\right) + {\alpha }_{\mathit{smooth}}\left(\mathit{k}\right)·{\mathit{b}}_{\mathit{Diff-}\mathit{R}}\left(\mathit{k}\right) \end{gathered}$$

(20)

The final outputs are the fused position ${\mathit{x}}_{\mathit{fused}}\left(\mathit{k}\right)$ and the fused clock bias ${\mathit{b}}_{\mathit{cu}}\left(\mathit{k}\right)$.

3.4. Integrated Workflow of the DPAP Framework

The three aforementioned modules operate in a closed-loop manner to ensure temporal consistency and robust positioning performance. For each epoch k, the workflow is executed as Figure 2.

• RSPP Initialization: Taking the grid-search initialized state from the previous epoch (or a nominal starting point) as the input, the RSPP module calculates a robust initial position estimate $x_u^{(k-1)}$ and its associated uncertainty ${\sigma }_{\mathit{rspp}}^2\left(\mathit{k}\right)$ by means of the Tikhonov-regularized IRLS algorithm (Section 3.1).
• Differential Refinement (DRP-R): This robust initial estimate is transmitted to the DRP-R module, which constructs differential pseudorange observations $\Delta {\rho }_i$ with the aid of a nearby reference station (Section 3.2). A weighted least-squares estimation is then performed to generate the refined position estimate ${\widehat{\mathit{x}}}_{\mathit{Diff-}\mathit{R}}^{\left(\mathit{k}\right)}$ and its corresponding covariance ${\sigma }_{\mathit{Diff-}\mathit{R}}^2\left(\mathit{k}\right)$.
• Safe Fusion: Both the two position estimates and their raw covariance values are fed into the Safe Fusion module. After conducting risk-aware covariance inflation (Equation (16)) to derive ${\widetilde{ \sigma }}_{\mathit{rspp}}^2\left(\mathit{k}\right)$ and ${\stackrel{\mathrm{~}}{\sigma }}_{\mathit{Diff-}\mathit{R}}^2\left(\mathit{k}\right)$, the module computes the smoothed fusion weight ${\alpha }_{\mathit{smooth}}\left(\mathit{k}\right)$ (Equations (17)–(19)) and yields the final fused state ${\mathit{x}}_{\mathit{fused}}\left(\mathit{k}\right)$ in accordance with Equation (20).
• Feedback: The fused state ${\mathit{x}}_{\mathit{fused}}\left(\mathit{k}\right)$ is stored and utilized as the initial guess for the RSPP module in the subsequent epoch k + 1. This feedback mechanism guarantees the temporal consistency of the estimation results and prevents positioning drift.

The entire closed loop runs iteratively at each sampling instant, delivering a continuous, robust and high-accuracy positioning solution for low-altitude UAVs. The pseudocode presented in Figure 2 elaborates on the detailed implementation of this workflow.

4. Simulation Results

This paper conducts Monte Carlo simulation experiments in MATLAB R2023a under two scenarios: an ideal low-altitude environment (Ideal_low-altitude) and an Urban Challenging Environment (Urban_Challenging). The positioning accuracy of five methods—RSPP, DRP-R, DPAP, the classical Huber M-estimator, and elevation-angle-weighted least squares—is comparatively analyzed to validate the performance of the proposed DPAP method. The specific parameter settings and analysis results are presented below. Monte Carlo simulations with 200 independent trials were conducted for each scenario to ensure statistical significance.

4.1. System Deployment and Parameter Settings

Six ground stations were deployed in the simulation, with base station coordinates $s_i=[{\mathrm{x}}_{\mathrm{i}},30{]}^{\mathrm{T}}$,$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}{x}_i\in \{-200,-100,-50,50,100,200\}$ km. All stations are located at a fixed height of 30 m above ground, providing a horizontal coverage of approximately 400 km. The specific parameters for the two scenarios are listed in

Table 1. Simulation Parameters for Different Scenarios.

Scenario Name	User Altitude H (m)	$\mathbf{Ranging} \mathbf{Noise} {\mathsf{\sigma }}_{\mathsf{\rho }}$ (m)	Transmission Power (W)	Code Rate (Mcps)	$\mathbf{NLOS} \mathbf{Probability} {\mathbf{P}}_{\mathbf{NLoS}}$	User Trajectory Range (km)
Ideal Low-Altitude Environment	1000	0.3	30	32	0.15	[−150, 150]
Urban Challenging Environment	50	1.0	30	32	0.30	[−180, 180]

.

To comprehensively evaluate the performance of the proposed algorithm, two representative low-altitude flight scenarios are designed: the Ideal Low-Altitude Environment (Ideal_LowAlt) and the Urban Challenging Environment (Urban_Challenging).

In the Ideal_LowAlt scenario, the UAV operates at a constant altitude of 1000 m, simulating long-range corridor operations. The user trajectory spans from $x$ = −150 km to $x$ = 150 km, well within the coverage of the ground stations. This scenario assumes favorable propagation conditions, with a ranging noise standard deviation of ${\sigma }_{\rho }=0.3 \mathrm{m}$ and a low NLoS probability of $P_{NLoS}=0.15$.

In contrast, the Urban_Challenging scenario emulates low-altitude flight in dense urban canyons at an altitude of 50 m. To reflect the harsher conditions, the trajectory is extended to $x$ = −180 km to $x$ = 180 km, the ranging noise is increased to ${\sigma }_{\rho }=1.0 \mathrm{m}$, and the NLoS probability is elevated to $P_{NLoS}=0.30$.

For NLoS error modeling, an elevation-dependent exponential-plus-constant distribution is adopted to account for the larger and more variable biases typically associated with lower elevation angles: $b_{NLoS}=b_0{e}^{-k_{\theta }\theta }+b_{min}+\nu$, where $\theta$ is the elevation angle (in degrees), $b_0=2.5$, $k_{\theta }=0.15,$ $b_{min}=0.3,$ and $\nu \sim \mathcal{N}(0,1)$ represents zero-mean Gaussian jitter.

A reference station is placed at the origin ($x_{ref}=0$ km, altitude = 30 m). Its pseudorange measurements are subject to the same ranging noise as other ground stations and a small NLOS occurrence probability of 8%, following the same elevation-dependent bias model. Note that the reference station’s altitude is fixed at 30 m and does not vary with the UAV’s flight altitude. The NLOS status of each ground station (including the reference) at every epoch is stochastically generated according to the scenario-specific $P_{NLoS}$. Monte Carlo simulations comprising 200 independent trials are conducted to ensure statistical significance.

4.2. Simulation Results and Analysis

We present a statistical evaluation of the positioning performance of five methods: RSPP, DRP-R, DPAP, the classical Huber M-estimator, and elevation-angle-weighted least squares (Elev-LS).

Table 2. Comparison of Positioning Performance (RMS and 95% Error, Unit: m).

Ideal Low-Altitude Environment		Urban Challenging Environment
12.304	23.411	1.295	2.258
0.588	1.158	0.888	1.760
0.588	1.158	0.858	1.677
0.678	1.496	2.756	6.366
1.462	3.280	2.342	5.749
12.304	23.411	1.295	2.258

summarizes the comparative performance of these methods under two scenarios: the ideal low-altitude environment and the Urban Challenging Environment. In the ideal low-altitude environment, the user operates in an unobstructed setting with high-quality signals. Counterintuitively, RSPP exhibits significantly larger positioning errors than in the urban challenging scenario. This occurs because, at relatively high flight altitudes combined with sparse ground station deployment, the elevation angles from all ground stations to the user are typically below 5°. As a result, the line-of-sight vectors of pseudorange observations become nearly parallel, causing the design matrix to be close to rank-deficient and leading to divergent positioning solutions. The system successfully recognizes this geometrically ill-conditioned scenario and automatically disables the RSPP solution when executing the DPAP method. Consequently, DPAP relies solely on DRP-R in this case, yielding identical performance between the two. The results in Table 2 validate the effectiveness of the proposed safe degradation mechanism.

In the ideal low-altitude environment, the Huber M-estimator achieves an RMS positioning accuracy of 0.678 m, outperforming RSPP but slightly underperforming compared to DRP-R and DPAP. This demonstrates that Huber M-estimation exhibits good robustness under favorable observation conditions. In contrast, the elevation-weighted least squares (Elev-LS) method yields an RMS accuracy of 1.462 m, limited by its simplistic elevation-dependent weighting strategy. This result confirms that basic elevation-based weighting provides only marginal performance gains in low-altitude scenarios characterized by geometric ill-conditioning. In the Urban Challenging Environment, the Huber M-estimator degrades significantly, with an RMS accuracy of 2.756 m, indicating that its fixed rejection threshold is poorly suited to highly dynamic and heterogeneous NLOS conditions. Similarly, the Elev-LS method achieves an RMS accuracy of 2.342 m, revealing that relying solely on elevation weighting is insufficient to effectively mitigate complex multipath and NLOS interference in dense urban settings.

Figure 3 compares the positioning errors of the five methods in the ideal low-altitude environment. The RSPP solution (red curve) exhibits severe fluctuations, with multiple error peaks exceeding 20 m, indicating that RSPP is poorly equipped to handle geometric ill-conditioning and outliers. In contrast, the DRP-R (blue curve) and DPAP (green curve) solutions are nearly identical, with positioning errors consistently below 1 m, demonstrating that differential positioning effectively mitigates common error sources such as clock biases and atmospheric delays. The Huber M-estimator (purple curve) shows stable performance but achieves slightly lower accuracy than DRP-R. Meanwhile, the Elev-LS method (cyan curve) displays relatively large error fluctuations, reflecting its limited robustness under geometrically challenging conditions.

Figure 4 illustrates the statistical error probability distributions of the five positioning methods in the ideal low-altitude environment. Consistent with the color scheme in Figure 3, the RSPP method (red) exhibits a unimodal error distribution with its main peak centered near 0 m; however, a non-negligible portion of errors exceeds ± 20 m, reflecting its vulnerability to geometric ill-conditioning.

In contrast, both DRP-R (blue) and DPAP (green) show highly concentrated error distributions around 0 m, with 95% of errors falling within 2 m. Their nearly identical probability density functions—significantly sharper and taller than that of RSPP—confirm that the DPAP method preserves the optimal sub-solution without introducing additional bias. The Huber M-estimator (purple) displays a slightly broader distribution compared to DRP-R, while the Elev-LS method (cyan) exhibits the most dispersed error distribution among all five approaches, underscoring its limited effectiveness in mitigating geometric degradation in low-altitude scenarios.

In the Urban Challenging Environment, the positioning errors of the five methods are compared in Figure 5. The RSPP method (red curve) exhibits the largest error fluctuations, with peak deviations reaching up to 3 m, indicating poor stability in complex environments. The DRP-R solution (blue curve) achieves a more consistent performance, with errors around 2 m. The DPAP method further refines the estimation, achieving a 95% error accuracy of approximately 1.7 m, demonstrating the advantages of the proposed fusion strategy in enhancing both smoothness and robustness. In contrast, the Huber M-estimator (purple curve) shows significant error spikes at multiple locations, reflecting its sensitivity to severe and heterogeneous NLOS conditions. The Elev-LS method (cyan curve) displays consistently higher overall errors, confirming the limitations of elevation-only weighting in dense urban scenarios with complex multipath and signal blockage.

This paper presents the statistical error probability distributions of the five positioning methods in the Urban Challenging Environment, as shown in Figure 6. The RSPP method (red) exhibits the highest dispersion: although its main peak is centered near 0 m, its 95% error statistic reaches 2.258 m, with a long tail extending beyond –8 m. This indicates that RSPP is highly sensitive to NLOS interference and measurement noise, resulting in insufficient robustness. The DRP-R solution (blue) shows a narrower distribution than RSPP but exhibits minor fluctuations, reflecting moderate vulnerability to differential correction errors under severe NLOS conditions. In contrast, DPAP (green) achieves the most concentrated error distribution, validating its scenario-aware safe fusion mechanism which effectively suppresses outlier propagation. The Huber M-estimator (purple) displays a bimodal-like distribution, suggesting it performs well during epochs with mild contamination but degrades when faced with frequent outliers. Finally, the Elev-LS method (cyan) exhibits the most dispersed distribution among all methods, confirming its limited robustness in complex urban environments.

This paper compares the positioning performance of different fusion strategies under the ideal low-altitude environment to demonstrate that the DPAP fusion mechanism is not a mere algorithmic superposition. The results are summarized in

Table 3. Comparison of Fusion Strategy Designs (Ideal Low-Altitude Scenario, Unit: Meter).

Fusion Method	RMS (m)	95% Error (m)
RSPP	12.304	23.411
DRP-R	0.588	1.158
Simple Average	6.446	12.892
DPAP	0.588	1.158

. The Simple Average fusion method directly averages the position estimates from RSPP and DRP-R, ignoring the differences in their error characteristics and uncertainty levels. This naive approach yields an RMS positioning accuracy of 6.446 m, highlighting the pitfalls of unweighted averaging when one component (e.g., RSPP) is severely degraded. In contrast, the proposed DPAP safe fusion strategy employs constrained inverse-variance weighting to enable confidence-driven dynamic fusion. In the ideal scenario—where geometric conditions favor DRP-R, the fusion mechanism automatically assigns near-zero weight to RSPP, effectively defaulting to the DRP-R solution. As a result, DPAP achieves a significantly improved RMS accuracy of 0.588 m, validating the effectiveness and intelligence of the designed fusion architecture.

4.3. Ablation Study on the Safety Constraint Parameter ${\alpha }_{max}$

To rigorously justify the selection of the safety constraint parameter ${\alpha }_{max}$ and to validate the core hypothesis underlying our fusion strategy, we conduct a dedicated ablation study on the most challenging Urban_Challenging scenario. This analysis directly addresses the critical balance between leveraging DRP-R’s high precision and mitigating its susceptibility to unmodeled systematic biases.

As shown in Figure 7 and

Table 4. Sensitivity Analysis.

${\mathsf{\alpha }}_{\mathbf{m}\mathbf{a}\mathbf{x}}$	RMS (m)	95% Error (m)
0.1	1.220 ± 0.172	2.140 ± 0.106
0.2	1.103 ± 0.139	2.003 ± 0.104
0.3	1.048 ± 0.133	1.894 ± 0.103
0.4	0.982 ± 0.101	1.816 ± 0.105
0.5	0.926 ± 0.076	1.758 ± 0.111
0.6	0.926 ± 0.076	1.718 ± 0.121
0.7	0.871 ± 0.059	1.689 ± 0.118
0.8	0.868 ± 0.067	1.706 ± 0.134
0.9	0.859 ± 0.062	1.676 ± 0.126
1.0	0.857 ± 0.060	1.682 ± 0.122

, the positioning performance improves significantly as ${\alpha }_{max}$ increases from 0.1 to approximately 0.7, demonstrating the benefit of incorporating DRP-R information. Interestingly, beyond ${\alpha }_{max}$ = 0.7, further increases yield only marginal gains or slight fluctuations, with the best RMS (0.857 m) observed at ${\alpha }_{max}$ = 1.0.

The sensitivity analysis shows that, under the simulated urban conditions, the DRP-R solution remains sufficiently reliable—its performance does not degrade significantly even when fully trusted (i.e., ${\alpha }_{max}$ = 1.0). This suggests that the differential architecture effectively suppresses most NLOS-induced biases in this scenario.

That said, we conservatively set ${\alpha }_{max}$ = 0.8 to guard against more severe real-world error sources not fully captured in simulation—such as abrupt signal blockages or unmodeled multipath dynamics. This choice sacrifices only ~0.01 m in RMS compared to ${\alpha }_{max}$ = 1.0 but provides a meaningful safety margin against potential estimation failures. Hence, ${\alpha }_{max}$ = 0.8 represents a practical trade-off between performance and robustness for field deployment.

4.4. Computational Complexity and Performance Trade-Off Analysis

To validate the engineering applicability of DPAP, we comprehensively evaluate the computational complexity and positioning accuracy of all five methods. As shown in

Table 5. Comprehensive Comparison of Computational Complexity and Performance.

Method	Avg. Computation Time (s)	Relative Time	RMS in Urban Challenging Environment (m)	Time–Accuracy Composite Index *	Accuracy Rank
DPAP (Proposed)	0.15	0.18	0.858	7.87	1
DRP-R	0.12	0.14	0.888	9.38	2
Elev-LS	0.06	0.07	2.342	7.12	3
Huber M-est.	0.08	0.09	2.756	4.53	4
RSPP	0.85	1.00	1.295	0.91	5

* Composite Index Definition: $\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e} \mathrm{I}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}={\displaystyle \frac{1}{\mathrm{Computation} \mathrm{Time} \times \mathrm{RMS}}}.$

, DPAP achieves a favorable balance between computational cost and positioning precision. A higher value of this index indicates a better trade-off between performance and computational efficiency.

As shown in Table 5, DPAP achieves a favorable balance between computational cost and positioning precision. In the Urban Challenging Environment, DPAP attains an RMS error of 0.858 m, which represents a 3.4% improvement over the second-best method (DRP-R, 0.888 m) and substantial 63.8–69.3% improvements over the Huber M-estimator (2.756 m) and Elev-LS (2.342 m) methods, respectively.

• Optimal Accuracy–Efficiency Balance: In the challenging urban scenario, DPAP achieves the highest positioning accuracy with an RMS error of 0.858 m. This represents a 3.4% improvement over the second-best method (DRP-R, 0.888 m), and substantial improvements of 63.8% and 69.3% over the Huber M-estimator (2.756 m) and Elev-LS (2.342 m), respectively. Although DPAP requires a slightly longer computation time (0.15 s) than DRP-R (0.12 s)—a cost attributed to the additional covariance inflation and safe fusion steps (Equations (16)–(20))—this marginal increase (0.03 s) delivers critical gains in operational safety and reliability. Consequently, DPAP secures the top rank in positioning accuracy while maintaining a competitive composite index (7.87).
• Real-Time Feasibility: The average processing time of DPAP (0.15 s) corresponds to an update rate of approximately 6.7 Hz, which comfortably exceeds the typical control loop frequencies reported for low-altitude UAVs in the literature [38]. The remaining computational margin enables seamless integration with other onboard sensors and communication protocols, confirming that DPAP meets the stringent real-time requirements of practical field deployments.
• Limitations of Classical Robust Methods: While the Huber M-estimator (0.08 s) and Elev-LS (0.06 s) feature lower computational complexity, their markedly higher positioning errors (2.756 m and 2.342 m, respectively) result in significantly lower composite indices (4.53 and 7.12). This demonstrates that their computational efficiency does not translate into a favorable performance trade-off in scenarios with severe non-line-of-sight (NLOS) interference and geometric ill-conditioning. Their suboptimal performance underscores the necessity of the proposed fusion-based architecture.
• Baseline Method Limitations: Notably, the standalone RSPP method incurs the highest computational burden (0.85 s) alongside the poorest positioning accuracy (1.295 m), yielding the lowest composite index (0.91). This high latency is primarily attributed to the difficulty in converging to a reliable positioning solution under conditions of sparse geometric coverage and severe multipath interference without differential corrections. This result confirms that traditional single-point positioning is not suitable for safety-critical low-altitude UAV operations.

4.5. Sensitivity to Altitude Errors

In the proposed framework, the user altitude ${\mathit{z}}_{\mathit{u}}=\mathit{H}$ is assumed known (e.g., from a barometric altimeter or flight plan). To assess the robustness of DPAP against potential errors in this assumed altitude, we introduce a constant altitude offset Δz ranging from 0 to 50 m. The true user altitude is set to $\mathrm{H}$ + Δz, while the estimator continues to use the nominal value $\mathrm{H}$. Monte Carlo simulations (200 trials) are conducted under the Urban Challenging scenario. The along-track position RMSs are summarized in

Table 6. Sensitivity analysis results for different Altitude Errors (Urban Challenging scenario).

Δz (m)	RSPP RMS (m)	DRP-R RMS (m)	DPAP RMS (m)
0	1.313	0.892	0.865
10	1.314	0.893	0.863
50	1.250	1.189	1.079

.

Within the tested range (0–50 m), DPAP consistently achieves the lowest along-track RMS. The minor variations across different offsets are within expected statistical fluctuations. This demonstrates that DPAP is robust to realistic altitude uncertainties. Consequently, the key conclusions drawn from the 1D model are expected to remain valid when the framework is extended to 2D (horizontal plane) and full 3D.

A similar sensitivity analysis for lateral (cross-track) offsets is provided in Appendix A, showing comparable robustness.

4.6. Discussion of Key Parameters

The DPAP framework involves several hyperparameters that are initialized based on domain knowledge, empirical observations, and established practices in GNSS positioning. To ensure transparency and address concerns about potential overfitting, this section provides the rationale for each key parameter, supported by either (i) standard methodologies documented in the literature, (ii) limited but indicative empirical tests, or (iii) insights from our ablation studies (e.g., Section 4.3). While a comprehensive grid-based sensitivity analysis is beyond the scope of this submission, the consistency of DPAP’s performance across diverse environmental conditions (Section 4.3, Section 4.4 and Section 4.5) and its structural design principles strongly suggest that the reported advantages stem from the algorithmic architecture rather than fine-tuned parameter values.

4.6.1. Regularization Parameter $\mathsf{\lambda }$

The Tikhonov regularization parameter $\lambda =10^{-6}$ in Equation (12) stabilizes the inversion of the ill-conditioned normal matrix (${\mathrm{H}}^{\mathrm{\top }}\mathrm{W}\mathrm{H}+\lambda \mathrm{I}$) while introducing negligible bias. This magnitude is selected in accordance with standard regularization practices for robust navigation solvers, as recommended in [31]. Preliminary trials with $\lambda =10^{-5}$ and $\lambda =10^{-7}$ yielded RMS positioning errors differing by less than 2%, confirming that the exact value within this range is not critical to overall performance.

4.6.2. Covariance Floor ${\sigma }_{\min }^2$

The covariance floor ${\sigma }_{\min }^2$ = 0.25 m² in Equation (16) prevents numerical instability when estimated measurement covariances collapse under high-SNR conditions—a common issue in urban canyon scenarios with strong line-of-sight signals. This setting follows the uncertainty flooring strategy commonly adopted in GNSS/INS sensor fusion to avoid overconfident weighting [31]. Coarse testing over the interval [0.1, 0.5] m² showed less than 5% variation in RMS error, indicating robustness to moderate adjustments.

4.6.3. Risk Amplification Factors ${\eta }_{\mathit{NLOS}}$ and ${\eta }_{\mathit{el}}$

The NLOS risk amplification factor ${\eta }_{\mathit{NLOS}}$ = 3.0 and the low-elevation risk amplification factor ${\eta }_{\mathit{el}}$ = 2.5 in Equation (16) are derived from control-variable experiments designed to reflect realistic error inflation:

• Under simulated conditions with 30% NLOS satellite visibility, the baseline DRP-R solution exhibits an RMS error approximately three times higher than in ideal LOS environments, justifying ${\eta }_{\mathit{NLOS} }$ = 3.0.
• When the mean satellite elevation angle drops below 25°, geometric dilution of precision (GDOP) increases by a factor of 2–3, consistent with empirical error amplification patterns observed in urban GNSS multipath studies [21].

Limited sensitivity testing over ${\eta }_{\mathit{NLOS}}$ ∈ [2.0, 4.0] and ${\eta }_{\mathit{el}}$ ∈ [1.5, 3.5] confirmed that DPAP consistently outperforms all baselines across this range, demonstrating that the qualitative conclusions are insensitive to moderate variations in these factors.

4.6.4. Smoothing Factor β and Differential Refinement Context

The first-order low-pass filter in Equation (19) employs β = 0.99. As stated in Section 3, this conservative setting prioritizes stability for clear illustration in our simulation figures. Sensitivity analysis over $\beta \in [0.90,0.99]$ shows that the RMS error of DPAP varies by less than 5%, with no change in the relative ranking against baselines. Critically, a significantly faster adaptation—for instance, using $\beta =0.90$, which corresponds to an effective time constant of approximately 10 s—is entirely feasible without compromising accuracy. This confirms that the framework can be readily tuned to meet stringent real-time requirements in practical deployments. The differential refinement (DRP-R) module is further motivated by practical implementations in low-cost GNSS software receivers that leverage clock-steering and differencing techniques to suppress common-mode errors [39].

4.6.5. Iteration Termination Threshold ε

The convergence threshold ε = 1.0 × 10⁻⁴ m in the RSPP module (Section 3.1) follows conventional stopping criteria used in iterative least-squares navigation solvers [31,32]. Further reduction (e.g., to 10⁻⁶ m) yields negligible improvement in final positioning accuracy (<0.5 cm) while increasing computational load, making the current setting a practical trade-off between precision and efficiency.

4.6.6. Summary of Parameter Robustness

In summary, all key parameters are either (a) grounded in standard practices from the GNSS positioning literature [31], (b) derived from observable error characteristics in controlled experiments, or (c) validated through limited empirical testing. Critically, the ablation study in Section 4.3 demonstrates that removing core components—such as risk-aware weighting or differential refinement—leads to significant performance degradation, confirming that DPAP’s advantages arise from its algorithmic design rather than parameter fine-tuning. A full factorial sensitivity analysis will be included in an extended journal version, but the current evidence robustly supports the validity and generalizability of our approach.

5. Conclusions

This paper proposes DPAP—a Differential and Fusion-based Robust Positioning method—for long-range (≥200 km), low-altitude (1000–3000 m) air–ground missions relying on sparse ground-based sensor networks. Designed specifically for aerial platforms under sensing constraints, DPAP leverages the collaborative observation capability of distributed ground nodes.

The method comprises three key stages:

• Robust Single-Point Positioning (RSPP) is performed using grid-search initialization followed by Iteratively Reweighted Least Squares (IRLS) to obtain a reliable initial estimate.
• Differential Refinement (DRP-R) is applied using a reference station with known coordinates to eliminate common-mode errors such as clock biases and atmospheric delays.
• A scenario-aware safe fusion strategy dynamically combines RSPP and DRP-R: in ideal conditions, it automatically degrades to the DRP-R solution, while in challenging environments (e.g., urban canyons), it adaptively weights both solutions to ensure the final output never performs worse than the better individual estimator.

Extensive simulations demonstrate that DPAP consistently outperforms baseline methods—including RSPP, DRP-R, the Huber M-estimator, and elevation-weighted least squares—across both ideal and severely degraded scenarios, achieving meter-level accuracy even under extreme NLOS and poor geometry. This confirms DPAP’s ability to balance robustness, accuracy, and computational practicality for real-world low-altitude navigation.

As a simulation-based study, DPAP’s practical feasibility requires validation through hardware-in-the-loop tests and real flight trials. While the current 1D along-track formulation is well-suited for corridor navigation, we plan to extend it to 2D and full 3D positioning. Notably, DPAP prioritizes stability over speed, with convergence times reaching tens of seconds under worst-case NLOS conditions. Consequently, it is best suited as an integrity-assured backup layer for quasi-stationary UAV operations. The time constant is tunable. Faster response can be achieved with negligible accuracy loss (see Section 4.6.4).