Multi-UAV Cooperative Localization in Pseudolite-Augmented GNSS-Denied Regions: An Anomaly-Resilient Adaptive Kalman Filter with Group Covariance Compensation

Highlights

What are the main findings?

• We propose DGCC-AKF, a distributed adaptive Kalman filter that decouples heterogeneous UAV-to-base-station (U-B) and UAV-to-UAV (U-U) innovation channels into independent adaptive groups, with a bounded factor that alleviates the noise underestimation caused by Covariance Intersection (CI)-induced covariance conservatism.
• Under observation fault periods, DGCC-AKF achieves up to about 75% accuracy improvement over single-UAV EKF and about 56% over distributed AKF while compressing the worst-case (95th-percentile) error by up to about 3.7× and remaining the only scheme that stays both accurate and stable across all UAVs and fault types.

What are the implications of the main findings?

• The framework is validated on a 2800 km² semi-physical testbed using measured fixed-wing trajectories exceeding 300 km/h, providing a strong baseline toward field deployment for resilient multi-UAV navigation in pseudolite-augmented GNSS-denied regions such as remote sensing corridors and disaster-response zones.
• The grouped channel decoupling mechanism generalizes to other distributed cooperative localization scenarios with heterogeneous observation types, providing a transferable design pattern for adaptive multi-source filtering under anomalies.

Abstract

In complex low-altitude environments, unmanned aerial vehicles (UAVs) require reliable positioning, yet Global Navigation Satellite System (GNSS) signals are vulnerable to occlusion and interference. Pseudolite-augmented cooperative localization, which combines ground base-station signals with inter-UAV relative observations, can complement GNSS in such environments. However, two practical issues remain in real-world deployment: UAV-to-base-station (U-B) and UAV-to-UAV (U-U) observations have markedly different error statistics that a unified noise adjustment cannot handle, and the conservative covariance estimates produced by Covariance Intersection (CI) fusion bias the innovation-based adaptive noise estimation in distributed architectures. To address these issues, this paper proposes a Distributed Group Covariance Compensation Adaptive Kalman Filter (DGCC-AKF) for collaborative enhancement of UAV regional localization. DGCC-AKF establishes a group adaptive mechanism that independently adjusts the noise covariance matrices of U-B and U-U observations, enabling observation-type-level adaptive weighting that suppresses anomalous U-B or U-U measurements at the group level. In addition, a bounded covariance compensation factor is incorporated to alleviate the CI-induced conservatism in the adaptive noise estimation. The proposed method is evaluated on a 2800 km² semi-physical testbed based on the Ground-based High-precision Local Positioning System (GH-LPS) pseudolite network using measured U-B observations and high-dynamic (>300 km/h) flight trajectories collected from a fixed-wing platform across three independent flight sessions. Results demonstrate that under observation fault periods, the proposed method improves 3D positioning accuracy by up to about 75% over single-UAV extended Kalman filter (EKF). Compared with two advanced algorithms in this field, variational Bayesian adaptive Kalman filter (VBAKF) and maximum correntropy criterion Kalman filter (MCC-EKF), it is the only scheme that remains accurate and stable across all UAVs and fault types. The framework provides a practical step toward field deployment for resilient multi-UAV cooperative navigation in pseudolite-augmented GNSS-denied regions.

1. Introduction

Unmanned aerial vehicles (UAVs) have become indispensable platforms for low-altitude applications such as environmental monitoring, emergency disaster response, infrastructure inspection, and cooperative surveillance [1]. With the progressive deregulation of low-altitude airspace and the continuous development of the low-altitude economy, UAV operations are expanding from single-platform missions to multi-UAV cooperative tasks over large operational areas [2,3,4]. This trend imposes higher requirements on UAV navigation and positioning systems, which must simultaneously achieve high accuracy, robust reliability, and stringent real-time performance [5].

However, for complex low-altitude application scenarios, existing positioning technologies still struggle to meet all of these requirements simultaneously. Although the Global Navigation Satellite System (GNSS) is currently the most mature and widely used positioning technology, it is susceptible to occlusion, interference, and spoofing in low-altitude environments. In urban canyons, tunnels, and mining areas, GNSS suffers from significant service outages, making it insufficient to independently meet UAV positioning demands [6]. Pseudolite-based regional navigation systems serve as an important complement to GNSS by transmitting radio positioning signals through ground base stations. They can provide continuous, stable, and high-frequency positioning services in GNSS-limited environments, demonstrating good applicability for low-altitude UAV operations [7,8]. However, such systems rely heavily on the spatial deployment and observation quality of ground base stations. Issues such as degraded geometric configurations, short-term base station failures, and multipath and non-line-of-sight (NLOS) errors can still significantly degrade positioning performance [9,10].

As the scale and complexity of UAV remote sensing missions continue to grow, multi-UAV collaborative operation has emerged as a promising paradigm for improving spatial coverage. In this context, UAV cooperative localization technology offers an effective solution to the above-mentioned technical challenges [11,12]. Specifically, UAV swarms can acquire additional observation constraints via inter-UAV ranging measurement and inter-node information interaction. When integrated with cooperative localization algorithms, this framework enables efficient information fusion and state consistency maintenance. Consequently, the overall positioning performance of the UAV swarm remains stable even under unfavorable geometric configurations of ground base stations or degraded measurement quality. Ruan et al. [13] proposed a UAV cooperative localization scheme based on coalition formation games, effectively addressing observation data loss. In simulation experiments, the horizontal positioning accuracy reached up to 8 m with a success rate exceeding 99%. Ouyang et al. [14] designed an extended colored Kalman filter based on the Allan variance, improving UAV cooperative localization accuracy by 36.67% compared to the conventional extended Kalman filter (EKF). Qiu et al. [15] derived an efficient closed-form solution for angle-of-arrival (AOA)-based cooperative localization and proposed an iterative constrained least squares method. Simulation results showed that the positioning accuracy reached 0.78 m, and increasing the number of cooperating agents significantly improved localization accuracy. Xu et al. [16] designed an improved bat algorithm for the non-convex optimization problem in cooperative localization. Simulation results demonstrated that the positioning accuracy can converge to the Cramér–Rao lower bound. Guo et al. [17] proposed a block Kalman filter to address the high complexity of multi-UAV cooperative localization. Simulation experiments showed that incorporating cooperative observations significantly improved positioning accuracy under poor geometric configurations of ground base stations. More recently, Li et al. [18] integrated inertial navigation with data link ranging through a sliding-window factor graph to enable distributed relative pose estimation for UAV formations in GNSS-denied environments. Wu et al. [19] further developed ST-DCL, a cooperative localization framework that combines a dynamic-weighted multidimensional scaling optimizer with a spatio-temporal graph neural network to model dynamic network topologies in drone swarms.

Existing studies have preliminarily verified the capability of cooperative localization to improve regional navigation system performance. However, the majority of these works rely on simulation validation, where observation error characteristics are known and stable. Among the few studies that include real-flight validation, the largest reported outdoor evaluation, to the best of our knowledge, covers approximately 1.6 km² (Xu et al. [20]), with UAV dynamics also remaining limited. In typical regional UAV missions, operational scenarios span hundreds of square kilometers and involve fixed-wing platforms operating at speeds exceeding 300 km/h. Under such conditions, the quality of observations often fluctuates rapidly with environmental changes and is susceptible to occlusion, multipath, and NLOS propagation [21,22,23]. Moreover, such high-dynamic operation imposes stringent real-time demands on the per-epoch positioning update. While graph-optimization-based [24] and learning-based [25] cooperative-localization methods offer strong accuracy potential, their elevated computational complexity and latency constrain their use in this regime.

In summary, Adaptive Kalman filtering (AKF) within the filter-based family preserves high computational efficiency while offering an effective mechanism for dynamically adjusting observation weights in response to such strongly time-varying observation quality. Wang et al. [26] proposed an adaptive optimal selection-robust hybrid AKF to suppress observation outliers in BeiDou Navigation Satellite System (BDS)/ground-based combined positioning. Experimental results showed that this method improved positioning accuracy by 60% in complex urban environments. Juston et al. [27] proposed a robust error-state Sage–Husa AKF framework, which handles unknown measurement noise by introducing quaternion dynamics and an improved fuzzy logic system, achieving a 30% improvement in positioning accuracy over conventional methods. Wang et al. [28] designed an AKF by incorporating dynamic motion model switching and distance-based noise covariance adaptation, achieving a positioning accuracy of 0.31 m in complex urban environments. Hadjiloizou et al. [29] designed a Maximum Correntropy Criterion Kalman Filter (MCC-EKF) for indoor quadrotor multi-sensor fusion, achieving robust pose estimation in the inertial measurement unit (IMU)/ultra-wideband (UWB)/vision observations. Xue et al. [30] developed an adaptive variational Bayesian Kalman filter (VBAKF) for UAV BDS/5G hybrid positioning, jointly estimating the UAV state and the measurement noise distribution to maintain high-precision performance during urban shuttle flights.

However, the above AKF studies all target single-UAV positioning, and directly extending AKF to multi-UAV cooperative localization scenarios faces two critical challenges. First, UAV-to-base-station (U-B) and UAV-to-UAV (U-U) observations differ significantly in their link conditions and error characteristics [31,32]; applying a unified noise adjustment to both observation types would noticeably degrade positioning accuracy. Second, AKF estimates the observation noise by subtracting the state covariance propagated into the observation space from the innovation covariance. The Covariance Intersection (CI) fusion algorithm commonly used in distributed cooperative localization [33], however, produces overly conservative (inflated) covariance estimates [34]. When these inflated covariance values are used as a priori information, the AKF systematically underestimates the observation noise, thereby losing its ability to properly detect and respond to anomalous observations. To the best of our knowledge, no existing AKF or cooperative localization study has jointly addressed both the U-B/U-U observation heterogeneity and the CI-induced covariance conservatism within a distributed cooperative architecture, motivating the development of the present work.

To bridge the gap in existing multi-UAV cooperative localization research regarding the strongly time-varying nature of observation quality in real scenarios, this paper introduces an adaptive filtering mechanism and focuses on resolving the challenges in cooperative localization. The main contributions are summarized as follows:

1.

To address the markedly different error statistics of U-B and U-U observations in distributed cooperative localization, we propose a block innovation channel decoupling mechanism that independently regulates the noise covariance matrices of heterogeneous observation groups, enabling observation-type-level adaptive weighting and suppressing cross-contamination between observation types at the group level.

2.

To alleviate the underestimation of observation noise caused by CI-induced covariance conservatism in distributed cooperative architectures, we introduce a bounded covariance compensation factor into the innovation-based noise estimation. The decoupling mechanism and the compensation factor together form DGCC-AKF, a Distributed Group Covariance Compensation Adaptive Kalman Filter for cooperative enhancement of UAV regional localization.

3.

We construct a large-scale semi-physical cooperative localization evaluation framework based on the Ground-based High-precision Local Positioning System (GH-LPS) regional pseudolite navigation system independently developed by our team [35], covering approximately 2800 km² with measured fixed-wing flight trajectories at speeds exceeding 300 km/h across three independent flight sessions. Error characteristic analysis and anomalous observation pattern mining are performed on the measured data, substantially extending the experimental basis of existing studies that predominantly rely on simulation with limited scenario scales and weak UAV dynamics.

The remainder of this paper is organized as follows. Section 2 presents the distributed cooperative localization model based on AKF, followed by the group adaptive mechanism and CI compensation principle. Section 3 introduces the hardware and site information of the physical experiments, analyzes the quality and error characteristics of the measured data, and verifies the effectiveness of the proposed method through both trajectory-based simulation and semi-physical experiments. Section 4 concludes the paper.

2. Methodology

This section presents the mathematical formulation of DGCC-AKF. We first establish the AKF-based distributed cooperative localization framework, and then introduce two key improvements: the group adaptive noise adjustment mechanism and the CI covariance compensation strategy. The complete algorithm flow is illustrated in Figure 1.

2.1. AKF-Based Distributed Cooperative Localization

Cooperative localization architectures are mainly classified into centralized and distributed types. Distributed cooperative localization has received widespread attention due to its decentralized nature, high scalability, and high robustness [36]. Depending on the state maintenance approach, distributed cooperative localization methods can be further divided into local state estimation and global state estimation. The latter maintains joint state information in a unified state space, preserving cooperative observation constraints more completely [37]. When the UAV swarm is small, it offers higher accuracy. As the swarm grows, strategies such as coalition grouping [38] and consensus variable construction [39,40] can be employed to suppress computational complexity growth. Therefore, this paper adopts a global state distributed cooperative localization scheme.

Each UAV obtains ranging observations from ground base stations and cooperating UAVs. Taking UAV i as an example, the U-B and U-U observation equations at time k are:

$${\rho }_k^{ij}=r_k^{ij}+c(\delta t_k^i-\delta t_k^j)+T_{\mathrm{trop},k}^{ij}+{\epsilon }_k^{ij}$$

(1)

$${\rho }_k^{i{b}_i}=r_k^{i{b}_i}+c(\delta t_k^i-\delta t_k^{b_i})+T_{\mathrm{trop},k}^{i{b}_i}+{\epsilon }_k^{i{b}_i}$$

(2)

where ${\rho }_k^{ij}$ and ${\rho }_k^{i{b}_i}$ are the range observations between UAV i and UAV j, and between UAV i and base station $b_i$, respectively. $r_k^{ij}$ and $r_k^{i{b}_i}$ are the corresponding true geometric distances; c is the speed of light. $\delta t_k^i$ and $\delta t_k^j$ are the receiver clock biases of UAV i and UAV j. $\delta t_k^{b_i}$ is the clock bias of base station $b_i$; $T_{\mathrm{trop},k}^{ij}$ and $T_{\mathrm{trop},k}^{i{b}_i}$ are the tropospheric delay errors. ${\epsilon }_k^{ij}$, ${\epsilon }_k^{i{b}_i}$ include multipath/NLOS errors, hardware delay residuals, and observation noise.

Among the above observation errors, the base station clock bias can be eliminated through the precise time synchronization technique proposed by our team [9]. Tropospheric delay errors can be accurately modeled. The clock bias between the UAV and ground base stations is removed through inter-station single differencing, while the clock bias between UAVs can be estimated via bidirectional measurements:

$$\Delta {\rho }_k^{i{b}_i}={\rho }_k^{i{b}_{\mathrm{ref}}}-{\rho }_k^{i{b}_i}=(r_k^{i{b}_{\mathrm{ref}}}-r_k^{i{b}_i})+({\epsilon }_k^{i{b}_{\mathrm{ref}}}-{\epsilon }_k^{i{b}_i})$$

(3)

$$\delta t_k^{ij}={\displaystyle \frac{{\rho }_k^{ij}-{\rho }_k^{ji}}{2c}}-{\displaystyle \frac{{\epsilon }_k^{ij}-{\epsilon }_k^{ji}}{2c}}$$

(4)

where $\Delta {\rho }_k^{i{b}_i}$ is the single-difference observation of UAV i (all subsequent U-B observations refer to this single-difference observation). $b_{\mathrm{ref}}$ is the reference base station index. $\delta t_k^{ij}$ is the clock bias between UAV i and UAV j. Equation (4) assumes that the bidirectional measurements ${\rho }_k^{ij}$ and ${\rho }_k^{ji}$ are approximately time-aligned within one ranging exchange, so that the true geometric range terms cancel.

Therefore, only the UAV motion state information needs to be estimated. The state vector of UAV i is modeled as:

$${\mathit{x}}_k^i={\left[\delta {\mathit{p}}_k^1,\cdots ,\delta {\mathit{p}}_k^L,\delta {\mathit{v}}_k^1,\cdots ,\delta {\mathit{v}}_k^L\right]}_{6L\times 1}^T$$

(5)

where the state vector is formulated in the error-state domain. $\delta {\mathit{p}}_k^i\in {\mathbb{R}}^3$ and $\delta {\mathit{v}}_k^i\in {\mathbb{R}}^3$ denote the position error and velocity error of UAV i at time k. L is the number of UAVs participating in cooperative computation.

In the distributed cooperative localization architecture, each UAV maintains an independent local filter. It performs state prediction independently, updates the state using its own observations, and then broadcasts its estimated state and covariance matrix to neighboring UAVs through communication, achieving global information integration via a fusion algorithm. The state prediction method is identical for all UAVs. Taking UAV i as an example:

$${\widehat{\mathit{x}}}_{k/k-1}^i={\mathsf{\Phi }}_k{\widehat{\mathit{x}}}_{k-1/k-1}^i$$

(6)

$${\mathit{P}}_{k/k-1}^i={\mathsf{\Phi }}_k{\mathit{P}}_{k-1/k-1}^i{\mathsf{\Phi }}_k^T+{\mathit{Q}}_k$$

(7)

where ${\widehat{\mathit{x}}}_{k/k-1}^i$ and ${\mathit{P}}_{k/k-1}^i$ denote the predicted state vector and covariance matrix of UAV i at time k, respectively. ${\widehat{\mathit{x}}}_{k-1/k-1}^i$ and ${\mathit{P}}_{k-1/k-1}^i$ are the posterior state estimate and covariance matrix of UAV i at time $k-1$. ${\mathit{Q}}_k$ is the process noise matrix. ${\mathsf{\Phi }}_k$ is the state transition matrix.

The observation matrix is obtained by linearizing the error-corrected observation equations. The observation matrix ${\mathit{H}}_k^i$ of UAV i at time k is:

$${\mathit{H}}_k^i={\left[{\mathit{H}}_{k,\mathrm{row}}^{i1};\cdots ;{\mathit{H}}_{k,\mathrm{row}}^{iL};{\mathit{H}}_{k,\mathrm{row}}^{i{b}_1};\cdots ;{\mathit{H}}_{k,\mathrm{row}}^{i{b}_n}\right]}_{m_k^i\times 6L}$$

(8)

$${\mathit{H}}_{k,\mathrm{row}}^{ij}={\left[{\mathbf{0}}_{1\times 3(i-1)},{\displaystyle \frac{{\mathit{p}}_{k,0}^i-{\mathit{p}}_{k,0}^j}{{\rho }_{k,0}^{ij}}},{\mathbf{0}}_{1\times 3(j-i-1)},-{\displaystyle \frac{{\mathit{p}}_{k,0}^i-{\mathit{p}}_{k,0}^j}{{\rho }_{k,0}^{ij}}},{\mathbf{0}}_{1\times (6L-3j)}\right]}_{1\times 6L}$$

(9)

$${\mathit{H}}_{k,\mathrm{row}}^{i{b}_i}={\left[{\mathbf{0}}_{1\times 3(i-1)},{\displaystyle \frac{{\mathit{p}}_{k,0}^i-{\mathit{p}}^{b_{\mathrm{ref}}}}{{\rho }_{k,0}^{i{b}_{\mathrm{ref}}}}}-{\displaystyle \frac{{\mathit{p}}_{k,0}^i-{\mathit{p}}^{b_i}}{{\rho }_{k,0}^{i{b}_i}}},{\mathbf{0}}_{1\times (6L-3i)}\right]}_{1\times 6L}$$

(10)

where ${\mathit{H}}_{k,\mathrm{row}}^{ij}$ and ${\mathit{H}}_{k,\mathrm{row}}^{i{b}_i}$ are the observation row vectors for U-U and U-B links, respectively (assuming $j>i$). ${\mathit{p}}_{k,0}^i$ and ${\mathit{p}}_{k,0}^j$ are the a priori positions of UAV i and UAV j. ${\mathit{p}}^{b_i}$ and ${\mathit{p}}^{b_{\mathrm{ref}}}$ are the true positions of base station $b_i$ and the reference base station $b_{\mathrm{ref}}$. ${\rho }_{k,0}^{ij}$, ${\rho }_{k,0}^{i{b}_i}$, and ${\rho }_{k,0}^{i,b_{\mathrm{ref}}}$ are the a priori ranges between the corresponding nodes. $m_k^i$ is the total number of observations available to UAV i at time k. $b_n$ is the number of observable base stations.

The observation vector ${\mathit{z}}_k^i$ of UAV i at time k is:

$${\mathit{z}}_k^i={\left[z_k^{i1},\cdots ,z_k^{iL},z_k^{i{b}_1},\cdots ,z_k^{i{b}_n}\right]}_{m_k^i\times 1}^T$$

(11)

$$z_k^{ij}={\rho }_k^{ij}-{\rho }_{k,0}^{ij}$$

(12)

$$z_k^{i{b}_i}=({\rho }_k^{i{b}_{\mathrm{ref}}}-{\rho }_k^{i{b}_i})-({\rho }_{k,0}^{i{b}_{\mathrm{ref}}}-{\rho }_{k,0}^{i{b}_i})$$

(13)

where $z_k^{ij}$ is the U-U observation residual between UAV i and UAV j. $z_k^{i{b}_i}$ is the U-B single-difference observation residual between UAV i and base station $b_i$.

Using the observation vector ${\mathit{z}}_k^i$ and the a priori covariance matrix ${\mathit{P}}_{k/k-1}^i$, the innovation covariance is estimated via an exponential moving average (EMA) to improve statistical robustness over single-epoch sample estimation:

$${\widehat{\mathit{S}}}_k^i=(1-\alpha ){\widehat{\mathit{S}}}_{k-1}^i+\alpha {\mathit{v}}_k^i{\left({\mathit{v}}_k^i\right)}^T$$

(14)

where ${\mathit{v}}_k^i={\mathit{z}}_k^i-{\mathit{H}}_k^i{\widehat{\mathit{x}}}_{k/k-1}^i$ is the innovation vector and $\alpha \in (0,1]$ is the EMA coefficient that controls the sensitivity to new observations.

$${\widehat{\mathit{R}}}_k^i={\widehat{\mathit{S}}}_k^i-{\mathit{H}}_k^i{\mathit{P}}_{k/k-1}^i{\left({\mathit{H}}_k^i\right)}^T$$

(15)

$${\mathit{R}}_k^i={\displaystyle \frac{\mathrm{tr}\left({\widehat{\mathit{R}}}_k^i\right)}{m_k^i}}·{\mathit{I}}_{m_k^i\times m_k^i}$$

(16)

where ${\widehat{\mathit{S}}}_k^i$ is the innovation covariance matrix estimated via exponential moving average; ${\widehat{\mathit{R}}}_k^i$ is the observation noise matrix estimate obtained through innovation covariance matching; ${\mathit{R}}_k^i$ is the observation noise matrix used for measurement update.

Each UAV independently performs local measurement updates in the same manner:

$${\mathit{K}}_k^i={\mathit{P}}_{k/k-1}^i{\left({\mathit{H}}_k^i\right)}^T{\left({\mathit{H}}_k^i{\mathit{P}}_{k/k-1}^i{\left({\mathit{H}}_k^i\right)}^T+{\mathit{R}}_k^i\right)}^{-1}$$

(17)

$${\widehat{\mathit{x}}}_{k/k}^i={\widehat{\mathit{x}}}_{k/k-1}^i+{\mathit{K}}_k^i\left({\mathit{z}}_k^i-{\mathit{H}}_k^i{\widehat{\mathit{x}}}_{k/k-1}^i\right)$$

(18)

$${\mathit{P}}_{k/k}^i=\left(\mathit{I}-{\mathit{K}}_k^i{\mathit{H}}_k^i\right){\mathit{P}}_{k/k-1}^i{\left(\mathit{I}-{\mathit{K}}_k^i{\mathit{H}}_k^i\right)}^T+{\mathit{K}}_k^i{\mathit{R}}_k^i{\left({\mathit{K}}_k^i\right)}^T$$

(19)

where ${\mathit{K}}_k^i$ is the Kalman gain matrix; ${\widehat{\mathit{x}}}_{k/k}^i$, ${\mathit{P}}_{k/k}^i$ denote the updated local state vector and covariance matrix of UAV i at time k.

After all UAVs complete their local measurement updates, they broadcast their current localization results to neighboring UAV nodes. To reduce communication overhead, the position and velocity corrections along with the covariance matrix are transmitted. The fusion message ${\mathit{M}}_k^i$ can be expressed as:

$${\mathit{M}}_k^i=\left\{i,k,{\mathit{P}}_{k/k}^i,{\widehat{\mathit{x}}}_{k/k}^i\right\}$$

(20)

When UAV i receives the estimation results from neighboring UAVs, information fusion is required to refine its current state. Since multiple UAVs inevitably share common information sources, the state estimates from their local filters are correlated, which poses a risk of filter divergence. To address this issue, we employ the CI method, which performs multi-node information fusion by constructing a convex combination in the information domain as follows:

$${\left({\mathit{P}}_{k/k,\mathrm{CI}}^i\right)}^{-1}=\sum _{j\in {\mathcal{N}}_i\left(k\right)\cup \left\{i\right\}}{\omega }_k^j({\mathit{P}}_{k/k}^j{)}^{-1}$$

(21)

$${\widehat{\mathit{x}}}_{k/k,\mathrm{CI}}^i={\mathit{P}}_{k/k,\mathrm{CI}}^i\sum _{j\in {\mathcal{N}}_i\left(k\right)\cup \left\{i\right\}}{\omega }_k^j{\left({\mathit{P}}_{k/k}^j\right)}^{-1}{\widehat{\mathit{x}}}_{k/k}^j$$

(22)

$$\sum _{j\in {\mathcal{N}}_i\left(k\right)\cup \left\{i\right\}}{\omega }_k^j=1,{\omega }_k^j\ge 0$$

(23)

where ${\mathit{P}}_{k/k,\mathrm{CI}}^i$ is the global posterior covariance of UAV i after fusion; ${\widehat{\mathit{x}}}_{k/k,\mathrm{CI}}^i$ is the global error-state estimate after fusion; ${\omega }_k^j$ is the weight of UAV j’s information in this fusion step; and ${\mathcal{N}}_i\left(k\right)$ denotes the set of neighboring UAVs that can communicate with UAV i at time k.

To balance consistency and real-time performance, we replace the grid search method used in traditional CI approaches with Fast Covariance Intersection [41]:

$${\omega }_k^j={\displaystyle \frac{{\left(\mathrm{tr}\left({\mathit{P}}_{k/k}^j\right)\right)}^{-1}}{{\sum }_{r\in {\mathcal{N}}_i\left(k\right)\cup \left\{i\right\}}{\left(\mathrm{tr}\left({\mathit{P}}_{k/k}^r\right)\right)}^{-1}}}$$

(24)

2.2. Group Covariance Compensation Adaptive Kalman Filter

The AKF-based distributed cooperative localization scheme described above requires two assumptions: (1) the observation error characteristics from all sources are statistically consistent and the corresponding noise components are uncorrelated; (2) the a priori state covariance matrix is sufficiently accurate without significant anomalies.

However, in real-world applications, neither assumption holds. First, the state covariance matrix after CI fusion inherently exhibits conservatism relative to the true value. Using it directly in AKF computation significantly degrades the algorithm’s adaptive adjustment capability. Second, as shown in Figure 2, U-B and U-U observations differ fundamentally in their link construction and propagation environments. U-B observations are inter-station single-difference observations, while U-U observations are direct ranging measurements. Whereas the baseline AKF in Equation (16) approximates ${\mathit{R}}_k^i$ as a trace-uniform scaled identity for simplicity, the proposed scheme retains the full block structure dictated by the heterogeneous observation sources, which can be expressed as:

$${\mathit{R}}_k^i={\left[\begin{array}{cc}{\sigma }_{\mathrm{uu},i}^2{\mathit{I}}_{(L-1)}& \mathbf{0}\\ \mathbf{0}& {\sigma }_{\mathrm{ub},i}^2\left({\mathit{I}}_{b_n}+{\mathbf{1}}_{b_n}{\mathbf{1}}_{b_n}^T\right)\end{array}\right]}_{m_k^i\times m_k^i}$$

(25)

where ${\sigma }_{\mathrm{ub},i}^2$ and ${\sigma }_{\mathrm{uu},i}^2$ denote the per-link ranging noise variances of UAV i with any base station and any neighboring UAV, respectively. The structured term $\mathit{I}+\mathbf{1}{\mathbf{1}}^T$ in the U-B block captures the cross-correlations introduced by inter-station single differencing in Equation (3), since all U-B single-differences share the same reference-base-station noise, whereas U-U observations are independent bidirectional rangings and remain diagonal.

To address these issues, this section proposes a group adaptive noise adjustment mechanism and a CI compensation strategy. The core objective of the group adaptive noise adjustment is to achieve independent regulation of noise weights for the two heterogeneous observation types and group-level suppression of anomalous observations through block decoupling of U-B and U-U observations. At time k, the observation vector, observation matrix, and observation noise matrix of UAV i can be partitioned by observation source as follows:

$${\mathit{z}}_k^i={\left[\begin{array}{c}{\mathit{z}}_{k,\mathrm{uu}}^i\\ {\mathit{z}}_{k,\mathrm{ub}}^i\end{array}\right]}_{m_k^i\times 1},{\mathit{H}}_k^i={\left[\begin{array}{c}{\mathit{H}}_{k,\mathrm{uu}}^i\\ {\mathit{H}}_{k,\mathrm{ub}}^i\end{array}\right]}_{m_k^i\times 6L},{\mathit{R}}_k^i={\left[\begin{array}{cc}{\mathit{R}}_{k,\mathrm{uu}}^i& \mathbf{0}\\ \mathbf{0}& {\mathit{R}}_{k,\mathrm{ub}}^i\end{array}\right]}_{m_k^i\times m_k^i}$$

(26)

where ${\mathit{z}}_{k,\mathrm{ub}}^i$, ${\mathit{H}}_{k,\mathrm{ub}}^i$, and ${\mathit{R}}_{k,\mathrm{ub}}^i$ denote the U-B observation vector, observation matrix, and noise matrix, respectively. ${\mathit{z}}_{k,\mathrm{uu}}^i$, ${\mathit{H}}_{k,\mathrm{uu}}^i$, and ${\mathit{R}}_{k,\mathrm{uu}}^i$ denote the corresponding U-U quantities. Following the EMA-based estimation in Equation (14), the AKF separately adjusts the observations from different sources:

$$\left\{\begin{array}{c}{\widehat{\mathit{S}}}_{k,\mathrm{ub}}^i=(1-\alpha ){\widehat{\mathit{S}}}_{k-1,\mathrm{ub}}^i+\alpha {\mathit{v}}_{k,\mathrm{ub}}^i{\left({\mathit{v}}_{k,\mathrm{ub}}^i\right)}^T\hfill \\ {\widehat{\mathit{S}}}_{k,\mathrm{uu}}^i=(1-\alpha ){\widehat{\mathit{S}}}_{k-1,\mathrm{uu}}^i+\alpha {\mathit{v}}_{k,\mathrm{uu}}^i{\left({\mathit{v}}_{k,\mathrm{uu}}^i\right)}^T\hfill \end{array}\right.$$

(27)

To alleviate the conservatism injected by CI fusion into ${\mathit{P}}_{k/k-1}^i$, a bounded compensation coefficient is applied to the propagated covariance term when subtracting it from the innovation covariance, yielding the per-group observation noise estimate:

$$\left\{\begin{array}{c}{\widehat{\mathit{R}}}_{k,\mathrm{ub}}^i={\widehat{\mathit{S}}}_{k,\mathrm{ub}}^i-{\lambda }_{k,\mathrm{ub}}^i·{\mathit{H}}_{k,\mathrm{ub}}^i{\mathit{P}}_{k/k-1}^i{\left({\mathit{H}}_{k,\mathrm{ub}}^i\right)}^T\hfill \\ {\widehat{\mathit{R}}}_{k,\mathrm{uu}}^i={\widehat{\mathit{S}}}_{k,\mathrm{uu}}^i-{\lambda }_{k,\mathrm{uu}}^i·{\mathit{H}}_{k,\mathrm{uu}}^i{\mathit{P}}_{k/k-1}^i{\left({\mathit{H}}_{k,\mathrm{uu}}^i\right)}^T\hfill \end{array}\right.$$

(28)

where ${\widehat{\mathit{S}}}_{k,\mathrm{ub}}^i$ and ${\widehat{\mathit{S}}}_{k,\mathrm{uu}}^i$ are the innovation covariance matrices for U-B and U-U observations estimated via EMA, ${\mathit{v}}_{k,\mathrm{ub}}^i$ and ${\mathit{v}}_{k,\mathrm{uu}}^i$ are the corresponding innovation sub-vectors, and ${\lambda }_{k,\mathrm{ub}}^i$, ${\lambda }_{k,\mathrm{uu}}^i\in [{\lambda }_{min},{\lambda }_{max}]$ are bounded covariance compensation coefficients. When U-B or U-U observations exhibit anomalies, the system selectively inflates only the corresponding $\widehat{\mathit{R}}$. The prior compensation values ${\lambda }_{k,\mathrm{ub}0}^i$ and ${\lambda }_{k,\mathrm{uu}0}^i$ are computed as:

$$\left\{\begin{array}{c}{\lambda }_{k,\mathrm{ub}0}^i={\displaystyle \frac{\mathrm{tr}\left({\widehat{\mathit{S}}}_{k,\mathrm{ub}}^i-{\widehat{\mathit{R}}}_{k-1,\mathrm{ub}}^i\right)}{\mathrm{tr}\left({\mathit{H}}_{k,\mathrm{ub}}^i{\mathit{P}}_{k/k-1}^i{\left({\mathit{H}}_{k,\mathrm{ub}}^i\right)}^T\right)}}\hfill \\ {\lambda }_{k,\mathrm{uu}0}^i={\displaystyle \frac{\mathrm{tr}\left({\widehat{\mathit{S}}}_{k,\mathrm{uu}}^i-{\widehat{\mathit{R}}}_{k-1,\mathrm{uu}}^i\right)}{\mathrm{tr}\left({\mathit{H}}_{k,\mathrm{uu}}^i{\mathit{P}}_{k/k-1}^i{\left({\mathit{H}}_{k,\mathrm{uu}}^i\right)}^T\right)}}\hfill \end{array}\right.$$

(29)

where ${\widehat{\mathit{R}}}_{k-1,\mathrm{ub}}^i$ and ${\widehat{\mathit{R}}}_{k-1,\mathrm{uu}}^i$ are the estimated observation noise covariance matrices for U-B and U-U observations from the previous epoch.

Considering that ${\mathit{P}}_{k/k-1}^i$ may also be affected by process noise matrix mismatch, anomalous observation geometry, and unreasonable motion model settings, the prior compensation coefficients require additional smoothing. The compensation parameters are increased only when the covariance matrix remains persistently large:

$$\left\{\begin{array}{c}{\lambda }_{k,\mathrm{ub}}^i=min\left({\lambda }_{max},max\left({\lambda }_{min},\gamma {\lambda }_{k,\mathrm{ub}0}^i+(1-\gamma ){\lambda }_{k-1,\mathrm{ub}}^i\right)\right)\hfill \\ {\lambda }_{k,\mathrm{uu}}^i=min\left({\lambda }_{max},max\left({\lambda }_{min},\gamma {\lambda }_{k,\mathrm{uu}0}^i+(1-\gamma ){\lambda }_{k-1,\mathrm{uu}}^i\right)\right)\hfill \end{array}\right.$$

(30)

where $\gamma \in (0,1]$ is the compensation coefficient smoothing factor. ${\lambda }_{k-1,\mathrm{ub}}^i$ and ${\lambda }_{k-1,\mathrm{uu}}^i$ are the compensation coefficients from the previous epoch, initialized to 1. ${\lambda }_{min}$ and ${\lambda }_{max}$ define the adjustment range of the compensation factor to prevent numerical instability caused by excessive adjustment. In the experiments, the parameters were set to $\gamma =0.3$, ${\lambda }_{min}=0.5$, and ${\lambda }_{max}=1.0$. The small smoothing factor $\gamma$ suppresses transient fluctuations in the prior compensation coefficient, while the bounded range $[{\lambda }_{min},{\lambda }_{max}]$ ensures that the propagated covariance term is neither over-suppressed nor entirely re-introduced, keeping the innovation matching within a physically meaningful regime.

Using the grouped noise matrix estimates from Equation (28), the observation noise matrix at time k is constructed as:

$$\left\{\begin{array}{c}{\widehat{\sigma }}_{k,\mathrm{ub}}^{2,i}=max\left(\eta {\sigma }_{0,\mathrm{ub}}^2,{\displaystyle \frac{\mathrm{tr}\left({\widehat{\mathit{R}}}_{k,\mathrm{ub}}^i\right)}{2m_{k,\mathrm{ub}}^i}}\right)\hfill \\ {\widehat{\sigma }}_{k,\mathrm{uu}}^{2,i}=max\left(\eta {\sigma }_{0,\mathrm{uu}}^2,{\displaystyle \frac{\mathrm{tr}\left({\widehat{\mathit{R}}}_{k,\mathrm{uu}}^i\right)}{m_{k,\mathrm{uu}}^i}}\right)\hfill \end{array}\right.$$

(31)

$$\left\{\begin{array}{c}{\mathit{R}}_{k,\mathrm{ub}}^i={\widehat{\sigma }}_{k,\mathrm{ub}}^{2,i}\left({\mathit{I}}_{b_n}+{\mathbf{1}}_{b_n}{\mathbf{1}}_{b_n}^T\right)\hfill \\ {\mathit{R}}_{k,\mathrm{uu}}^i={\widehat{\sigma }}_{k,\mathrm{uu}}^{2,i}{\mathit{I}}_{L-1}\hfill \end{array}\right.$$

(32)

$${\mathit{R}}_k^i={\left[\begin{array}{cc}{\mathit{R}}_{k,\mathrm{uu}}^i& {\mathbf{0}}_{(L-1)\times b_n}\\ {\mathbf{0}}_{b_n\times (L-1)}& {\mathit{R}}_{k,\mathrm{ub}}^i\end{array}\right]}_{m_k^i\times m_k^i}$$

(33)

where ${\widehat{\sigma }}_{k,\mathrm{ub}}^{2,i}$ and ${\widehat{\sigma }}_{k,\mathrm{uu}}^{2,i}$ are the adaptively estimated per-link noise variances at epoch k, and ${\mathit{R}}_{k,\mathrm{ub}}^i$, ${\mathit{R}}_{k,\mathrm{uu}}^i$ are the resulting measurement-update noise matrices following the structure of Equation (25). ${\sigma }_{0,\mathrm{ub}}^2$ and ${\sigma }_{0,\mathrm{uu}}^2$ denote the per-link initial variances, and $\eta =0.5$ lower-bounds the adapted variance to prevent collapse below half of its initial level. $m_{k,\mathrm{ub}}^i$ and $m_{k,\mathrm{uu}}^i$ denote the numbers of U-B and U-U observations of UAV i at epoch k. The factor 2 in the U-B denominator of Equation (31) reflects the doubled diagonal trace of the structured covariance $\mathit{I}+\mathbf{1}{\mathbf{1}}^T$, in which each diagonal element equals 2.

3. Experiments

This section validates the effectiveness of the DGCC-AKF method through two sets of experiments. The evaluation is conducted in a semi-physical manner: the UAV flight trajectories and U-B observations are directly measured from a GH-LPS-based fixed-wing platform across three independent flight sessions. The U-U observations are generated from the measured trajectories using noise statistics derived from the analyzed U-B error characteristics. This design enables a controlled comparison of multiple cooperative algorithms under identical real-world flight dynamics and U-B noise conditions; hardware-level inter-UAV ranging effects such as timing synchronization and antenna directionality are not modeled at this stage.

3.1. Experimental Setup and Hardware Equipment

The experimental data were collected in Zhoukou City, Henan Province, China, covering an area of approximately 2800 km². The experiment was designed with the scenario of regional detection and emergency disaster relief, in which three UAVs were planned to accomplish a search mission over the target area under GNSS-denied conditions. A total of six GH-LPS ground base stations were deployed within the experimental area, comprising one master station and five slave stations. The master station was equipped with an omnidirectional antenna for precise time synchronization with the slave stations and a transmitting antenna for broadcasting navigation signals. Each slave station was equipped with a panel antenna for time synchronization with the master station and a transmitting antenna for signal transmission. The inter-station distances ranged from 10 km to 56 km. The coordinates of all base stations were precisely calibrated using a Qianxun SRmini receiver (Qianxun Spatial Intelligence Inc., Shanghai, China), achieving centimeter-level accuracy. The experimental scenario, ground base station deployment, and UAV flight trajectory design are shown in Figure 3.

A fixed-wing aircraft was employed as the carrier platform, equipped with a GH-LPS receiver and a POLA-V12E integrated navigation system (Beijing Ocean Strategy Technology Co., Ltd., Beijing, China) (including a GNSS receiver). The aircraft operated at a flight speed exceeding 300 km/h and a flight altitude of 4500 m. The reference trajectory was provided by the integrated navigation system at an observation frequency of 10 Hz. The hardware assembly is shown in Figure 4.

Limited by experimental conditions, only a single UAV was available for the test. To verify the cooperative localization performance, we designed a semi-physical simulation scheme: the same UAV was flown along three pre-planned routes in separate flight sessions, and observation data from each flight were assigned unique UAV identifiers to construct multi-UAV flight datasets. The U-B observation quality is analyzed first, with the results presented in Figure 5.

In terms of observation accuracy, the mean root mean square (RMS) of the U-B single-difference observation errors across the three flights was 2.3 m. Notably, UAV1 exhibited an observation anomaly lasting approximately 10 s within the epoch range [1200, 1300], during which the error increased to over 25 m. Regarding the number of observations, since the UAVs flew from the periphery of the experimental field inward, the U-B observation count during the first 200 s of each flight was low and unstable, and at certain epochs the number of available observations was insufficient to meet the minimum requirements for positioning.

3.2. Algorithm Evaluation Scheme

We compare DGCC-AKF against two recent adaptive Kalman filter paradigms most frequently reported in UAV and integrated navigation literature over the past three years. The first is the Variational Bayesian Adaptive Kalman Filter (VBAKF) [30,42], which infers the unknown observation noise covariance jointly with the state through variational Bayes. The second is the Maximum Correntropy Criterion Kalman Filter (MCC-EKF) [29,43], which replaces the quadratic measurement cost with a maximum-correntropy cost to down-weight heavy-tailed residuals. Both baselines preserve their published mechanisms and are integrated into our distributed cooperative architecture for fair comparison. All algorithms and statistical analyses in this study were implemented in Python (version 3.13.5).

To isolate the contribution of each algorithmic component, we design a factorial ablation across four mechanisms: distributed cooperation, AKF noise adjustment, CI compensation, and group decoupling.

Table 1. Factorial ablation design of six algorithm variants (“✓”/“×” indicates whether the corresponding mechanism is enabled).

Variant	Full Name	Coop.	AKF	CI Comp.	Group
S-EKF	Single-UAV EKF	×	×	×	×
D-EKF	Distributed EKF	✓	×	×	×
D-AKF	Distributed AKF	✓	✓	×	×
DCC-AKF	Distributed CI-compensated AKF	✓	✓	✓	×
DG-AKF	Distributed grouped AKF	✓	✓	×	✓
DGCC-AKF	Distributed grouped CI-compensated AKF	✓	✓	✓	✓

lists the six resulting variants and their factor settings.

Performance is evaluated under $N_{\mathrm{MC}}=100$ independent Monte Carlo runs sharing the same random seed across schemes. The following metrics are reported.

3D positioning RMSat each epoch t across the $N_{\mathrm{MC}}$ runs:

$${\mathrm{RMS}}_{3D}\left(t\right)=\sqrt{{\displaystyle \frac{1}{N_{\mathrm{MC}}}}\sum _{k=1}^{N_{\mathrm{MC}}}{∥{\widehat{\mathit{p}}}_t^{\left(k\right)}-{\mathit{p}}_t∥}_2^2},$$

(34)

where ${\widehat{\mathit{p}}}_t^{\left(k\right)}$ and ${\mathit{p}}_t$ are the estimated and reference 3D positions at epoch t in Monte Carlo run k.

Mean and 95th-percentile (P95) reported in tables and figures denote the mean and 95% quantile of ${\mathrm{RMS}}_{3D}\left(t\right)$ over the epochs in the evaluation period.

Bootstrap CI, Wilcoxon signed-rank test, and Cohen’s $d_z$ are computed on $N_{\mathrm{MC}}$ paired samples per algorithm–condition pair, where each sample is the RMS of one Monte Carlo run aggregated over the epochs in the evaluation period.

95% confidence interval (CI₉₅) of the mean is obtained by bias-corrected and accelerated (BCa) bootstrap with $B=2000$ resamples [44].

Cohen’s $d_z$ for the paired RMS difference between two schemes: $d_z=\overline{d}/s_d$, where $\overline{d}$ and $s_d$ are the mean and standard deviation of paired differences [45]. The magnitude of $d_z$ is categorized by the conventional thresholds as n.s. ($|d_z|<0.2$), small ($0.2\le |d_z|<0.5$), medium ($0.5\le |d_z|<0.8$), or large ($|d_z|\ge 0.8$). Throughout this paper, $d_z$ is reported as the effect of a baseline relative to DGCC-AKF, so a positive value indicates that DGCC-AKF attains the lower error. In all result tables, an asterisk (∗) marks a statistically significant difference from DGCC-AKF ($p_{\mathrm{Holm}}<0.01$); a dagger (†) marks the cases where a baseline outperforms DGCC-AKF; and the em dash (—) marks the reference method DGCC-AKF itself, against which all baselines are compared, so that no effect size is defined for it.

Pairwise statistical significance is assessed by the paired Wilcoxon signed-rank test, with the resulting p-values corrected by the Holm–Bonferroni procedure [46] to control the family-wise error rate within each evaluation period. A difference is regarded as statistically significant when the corrected p-value satisfies $p_{\mathrm{Holm}}<0.01$.

3.3. Simulation of Typical Observation Degradation Scenarios Based on Measured Trajectories

Analysis of the measured data reveals that observation count reduction and quality degradation can occur simultaneously, making it difficult to isolate their individual effects on positioning. To address this, controlled-variable simulations were first conducted to separately evaluate the DGCC-AKF scheme under two conditions: observation loss and observation quality degradation. In the simulations, the flight trajectory and motion state of each UAV were derived from real measured data, while the U-B and U-U observations were synthetically generated with parameters configured based on the statistical characteristics of real observations analyzed in the previous subsection.

We fixed the U-B and U-U observation accuracy, and simulated three typical observation count loss scenarios in practical applications by adjusting the number of available observations per epoch: (1) insufficient ground base stations limited by environmental and cost constraints; (2) random observation loss from airframe occlusion, base station or receiver malfunctions, or packet loss on the inter-UAV data link; (3) unavailable low-elevation satellite/base station observations due to occlusion by buildings, trees and other obstacles. The experimental scenarios are detailed in

Table 2. Unmanned aerial vehicle (UAV) observation quality configuration schemes.

Scenario	Base Station Count	Random Loss Ratio	Elevation Mask	U-B/U-U Error (m)
Experiment 1	6	0%	0°	2.0/1.5
Experiment 2	4	0%	0°	2.0/1.5
Experiment 3	6	20%	0°	2.0/1.5
Experiment 4	6	0%	10°	2.0/1.5

.

The DGCC-AKF filter is initialized with the settings summarized in

Table 3. DGCC-AKF filter initial settings.

Parameter	Symbol	Value
Initial position uncertainty (std, per axis)	${\sigma }_{p,0}$	10 m
Initial velocity uncertainty (std, per axis)	${\sigma }_{v,0}$	5 m/s
Position process noise (std, per axis)	${\sigma }_{q,p}$	10 m
Velocity process noise (std, per axis)	${\sigma }_{q,v}$	0.5 m/s
Initial U-B observation noise (std)	${\sigma }_{0,\mathrm{ub}}$	2 m
Initial U-U observation noise (std)	${\sigma }_{0,\mathrm{uu}}$	1.5 m

.

All values in Table 3 are reported as one-axis standard deviations, with the corresponding covariance matrices constructed as ${\sigma }^2·\mathit{I}$. The filter prior ${\sigma }_{p,0}$ and ${\sigma }_{v,0}$ are kept fixed across all Monte Carlo runs, reflecting a realistic deployment in which the true initial-error magnitude is unknown to the filter.

The four experiments primarily test the dependence of each algorithm on the number of available base stations; the observation noise statistics remain stable across them. The ablation variants of DGCC-AKF are therefore expected to perform identically. To keep the comparison focused, we evaluate the positioning performance of five representative schemes: S-EKF, D-EKF, DGCC-AKF, VBAKF, and MCC-EKF. Figure 6 reports the 3D positioning RMS over the post-convergence epochs. Subplots in the same row correspond to the same observation condition listed in

Table 4. Statistical comparison of the 3D positioning accuracy of the three UAVs across the four observation scenarios.

Scenario	Metric	S-EKF	D-EKF	VBAKF	MCC-EKF	DGCC-AKF
	P95 (m)	2.95	2.52	2.94	2.88	2.45
Exp. 1	CI95 (m)	[2.26, 2.30]	[1.95, 2.01]	[2.20, 2.26]	[2.47, 3.04]	[1.89, 1.95]
	Cohen’s $d_z$	large *	large *	large *	large *	—
	P95 (m)	5.91	4.81	5.56	8.67	4.74
Exp. 2	CI95 (m)	[3.92, 4.06]	[3.27, 3.41]	[3.67, 3.82]	[4.08, 4.92]	[3.23, 3.36]
	Cohen’s $d_z$	large *	large *	large *	large *	—
	P95 (m)	4.84	3.93	3.84	9.06	3.76
Exp. 3	CI95 (m)	[3.51, 3.62]	[2.92, 3.00]	[2.97, 3.09]	[4.09, 5.38]	[2.84, 2.93]
	Cohen’s $d_z$	large *	large *	large *	large *	—
	P95 (m)	40.11	7.87	7.53	15.45	8.34
Exp. 4	CI95 (m)	[12.41, 17.07]	[3.96, 4.45]	[4.24, 4.64]	[5.80, 7.08]	[3.96, 4.44]
	Cohen’s $d_z$	large *	n.s.†	large *	large *	—

.

As shown in Figure 6 and Table 4, the single-UAV S-EKF is the most sensitive to the degradation of base-station observations. Since it relies only on U-B ranges, its accuracy drops sharply when fewer stations are available. The degradation is most severe in Experiment 4, where its P95 error rises to 40.11 m and its CI₉₅ widens to [12.41, 17.07] m. In contrast, the four cooperative schemes all resist this degradation well, because the inter-UAV ranges provide additional constraints that compensate for the missing base-station observations.

Among all the compared schemes, DGCC-AKF delivers the best overall positioning accuracy. D-EKF is only slightly worse than DGCC-AKF. In Experiment 1, for example, their CI₉₅ intervals are [1.95, 2.01] m and [1.89, 1.95] m, respectively. This indicates that cooperation is the dominant factor in mitigating the shortage of base stations for regional navigation, whereas the CI compensation and the grouped adaptive filtering embedded in DGCC-AKF act as auxiliary refinements that further improve accuracy on top of the cooperative baseline. VBAKF is slightly worse than both in every scenario; its CI₉₅ in Experiment 1 is [2.20, 2.26] m. This is because VBAKF re-estimates the noise covariance at every epoch through variational inference. When no anomaly is present, this constant re-estimation brings no benefit. Instead, it adds estimation noise to the covariance and causes a small loss of accuracy.

MCC-EKF is clearly affected in its convergence speed when the base-station geometry weakens. Its P95 reaches 8.67 m, 9.06 m, and 15.45 m in Experiments 2–4, with correspondingly wider CI₉₅ intervals. The cause is that, at the start of positioning, the large but legitimate innovations produced by the initial position error are mistaken for outliers and down-weighted by the correntropy criterion, which slows convergence. As the number of stations decreases, the innovations during this phase grow larger, and the convergence becomes even slower.

To verify the effectiveness of the grouped adaptive mechanism and CI compensation under observation anomalies, we inject additional ranging errors in Experiment 1. During epochs [1000, 1500] the error is added to the U-U observations of UAV3, and during epochs [2000, 2500] to the U-B observation between UAV3 and BS1. During epochs [3000, 3500], both are injected at the same time. The same composite error is used in all three windows. It superimposes three terms on the affected range. The first is a bias that ramps from 0 to 2 m over the first half of the window and then stays at 2 m. The second is zero-mean Gaussian noise with a standard deviation of 8 m. The third is a 20 m outlier that occurs with a probability of 5%. This experiment compares the two state-of-the-art (SOTA) baselines against the six DGCC-AKF ablation variants. The global positioning error RMS of UAV3 is shown in Figure 7, and the mean RMS over the normal, U-U fault, U-B fault, and concurrent fault periods is shown in Figure 8. The corresponding statistical comparison across the four fault periods is summarized in

Table 5. Statistical comparison of the 3D positioning accuracy of UAV3 (the fault-injected UAV) across the four fault periods.

Period	Metric	S-EKF	D-EKF	D-AKF	DCC-AKF	DG-AKF	VBAKF	MCC-EKF	DGCC-AKF
	P95 (m)	4.33	3.62	3.98	3.89	3.87	5.30	12.44	3.84
Normal	CI95 (m)	[3.26, 3.35]	[2.75, 2.84]	[2.92, 3.01]	[2.82, 2.92]	[2.88, 2.97]	[3.60, 3.69]	[4.25, 5.22]	[2.80, 2.89]
	Cohen’s $d_z$	large *	large *†	large *	large *	large *	large *	large *	—
	P95 (m)	2.90	5.58	5.66	5.46	2.95	3.55	2.77	3.03
U-U fault	CI95 (m)	[2.48, 2.57]	[4.66, 4.87]	[4.59, 4.86]	[4.45, 4.72]	[2.55, 2.65]	[2.94, 3.04]	[2.21, 2.41]	[2.59, 2.70]
	Cohen’s $d_z$	medium *†	large *	large *	large *	large *	large *	large *†	—
	P95 (m)	6.27	4.02	3.03	2.96	3.01	5.19	3.26	2.92
U-B fault	CI95 (m)	[5.20, 5.39]	[3.28, 3.38]	[2.54, 2.65]	[2.45, 2.56]	[2.52, 2.65]	[4.35, 4.46]	[2.59, 2.89]	[2.43, 2.55]
	Cohen’s $d_z$	large *	large *	large *	n.s.	large *	large *	n.s.	—
	P95 (m)	7.55	7.94	11.06	10.56	7.95	7.58	8.51	8.02
Concurrent	CI95 (m)	[6.44, 6.71]	[6.34, 6.72]	[7.30, 7.91]	[6.95, 7.55]	[5.39, 5.87]	[6.44, 6.74]	[5.46, 6.22]	[5.40, 5.87]
	Cohen’s $d_z$	large *	large *	large *	large *	large *	large *	n.s.	—

.

During the normal period, the five cooperative ablation schemes are all clearly more accurate than the single-UAV S-EKF (3.17 m), keeping the mean 3D RMS of UAV3 between 2.58 and 2.76 m. DGCC-AKF reaches 2.63 m, essentially the same as D-EKF. The other four ablation variants and the two SOTA baselines are all worse than DGCC-AKF. VBAKF is the worst at 3.43 m. MCC-EKF (3.23 m) is limited by slow convergence; its error does not settle until about the 25th second, which raises its P95 to 12.44 m, the largest of all methods.

During the U-U fault period, D-EKF, D-AKF, and DCC-AKF degrade sharply, with mean errors of 4.71 m, 4.66 m, and 4.52 m, about 85% higher than S-EKF (2.50 m), and CI₉₅ intervals all above 4.4 m. In contrast, DG-AKF, DGCC-AKF, VBAKF, and MCC-EKF all suppress the fault and keep the mean error within 2.3–2.9 m. DGCC-AKF reaches 2.60 m with a P95 of 3.03 m. VBAKF is slightly worse than DGCC-AKF (2.93 m), while MCC-EKF is slightly better (2.30 m), the latter benefiting from the large number of base stations in this experiment. The grouped adaptation is decisive here: by isolating the U-U channel, DG-AKF and DGCC-AKF adjust only the affected weights and leave the healthy U-B observations untouched.

During the U-B fault period, DGCC-AKF is the best of all methods, with a mean error of 2.49 m and the lowest P95 (2.92 m) and CI₉₅ ([2.43, 2.55] m). It improves on S-EKF (5.23 m) by 52% and on D-EKF (3.31 m) by 25%. VBAKF is again notable, as its accuracy deteriorates to 4.39 m, only slightly better than S-EKF. Together with its degradation in the normal period, this shows that the repeated online re-estimation of the state covariance $\mathit{P}$ and the noise covariance $\mathit{R}$ in VBAKF not only introduces extra estimation noise but can also destroy the inherent conservativeness of CI fusion, which in turn harms accuracy.

During the concurrent fault period, where U-U and U-B errors are injected at the same time, DGCC-AKF still gives the lowest mean error (5.60 m). The most notable change is that the two scalar adaptive schemes, D-AKF and DCC-AKF, degrade markedly to 7.46 m and 7.11 m, falling behind even S-EKF (6.63 m) and D-EKF (6.51 m). This reflects the risk of applying a single unified adjustment to the U-U and U-B observations: neither channel can be down-weighted correctly, which clearly lowers accuracy. The grouped schemes avoid this by estimating each channel separately; relative to D-AKF and DCC-AKF, DGCC-AKF reduces the error by about 25% and 21%, respectively.

In terms of overall performance across all periods, DGCC-AKF attains the lowest mean 3D RMS of all methods, 2.94 m, improving on S-EKF, VBAKF, D-AKF, D-EKF, DCC-AKF, MCC-EKF, and DG-AKF. To evaluate the parameter sensitivity and generalization capability of DGCC-AKF, we performed a parameter sweep over its key hyperparameters:

Figure 9 presents the 3D positioning RMS of UAV3 when each of the four parameters ($\alpha$, $\gamma$, ${\lambda }_{min}$, $\eta$) is swept over $[0.1,0.7]$ with the others fixed at their defaults. DGCC-AKF shows low sensitivity to all four parameters. Under the normal, U-U fault, and U-B fault conditions, the RMS variation caused by sweeping any single parameter stays at the centimeter level. Under the concurrent U-U and U-B fault condition, $\alpha$ has the strongest effect on accuracy, yet this effect remains within 5% of the overall positioning error. These results indicate that DGCC-AKF is robust to parameter variation, so default settings can be retained across the tested conditions without per-scenario retuning.

3.4. Semi-Physical Simulation Verification Using Multi-Group Measured U-B Observations

This subsection further employs measured U-B observation data for experimental verification. The inter-UAV observations were still generated through simulation with error characteristics consistent with the settings in the previous subsection, and a U-U observation anomaly was injected into the UAV3 observations during epochs [2500, 3000], configured in the same way as in the simulation experiment. The initial position was obtained using the first-epoch observations and a least-squares algorithm. The average 3D positioning errors of the three UAVs are as follows.

As shown in Figure 10 and

Table 6. Statistical comparison of the 3D positioning accuracy of the three UAVs over all epochs and the first 200 s.

UAV	Period	Metric	S-EKF	D-EKF	D-AKF	DCC-AKF	DG-AKF	VBAKF	MCC-EKF	DGCC-AKF
UAV1	All epochs	P95 (m)	20.01	13.39	12.68	12.86	12.63	13.28	12.19	12.34
		CI95 (m)	[11.44, 11.48]	[7.31, 7.33]	[6.96, 6.98]	[6.97, 7.00]	[6.93, 6.96]	[6.86, 6.88]	[6.61, 6.63]	[6.87, 6.90]
		$d_z$	large *	large *	large *	large *	large *	n.s.	large *†	—
	First 200 s	P95 (m)	23.21	10.44	10.05	10.12	10.07	9.44	9.71	10.01
		CI95 (m)	[13.95, 14.02]	[8.04, 8.08]	[7.27, 7.31]	[7.31, 7.35]	[7.21, 7.25]	[6.60, 6.63]	[6.54, 6.56]	[7.12, 7.16]
		$d_z$	large *	large *	large *	large *	large *	large *†	large *†	—
UAV2	All epochs	P95 (m)	71.43	62.97	58.56	58.44	58.65	73.60	88.12	57.60
		CI95 (m)	[28.45, 28.67]	[21.12, 21.48]	[20.43, 20.75]	[20.49, 20.82]	[20.36, 20.69]	[24.43, 24.67]	[29.98, 30.34]	[20.21, 20.61]
		$d_z$	large *	large *	medium *	medium *	large *	large *	large *	—
	First 200 s	P95 (m)	103.25	69.75	67.68	67.71	67.64	83.45	111.33	67.57
		CI95 (m)	[41.23, 41.53]	[30.62, 31.13]	[29.58, 30.05]	[29.70, 30.19]	[29.50, 30.00]	[35.34, 35.67]	[43.45, 43.95]	[29.54, 30.02]
		$d_z$	large *	large *	medium *	medium *	large *†	large *	large *	—
UAV3	All epochs	P95 (m)	11.92	11.33	11.32	11.31	11.20	12.41	12.24	11.15
		CI95 (m)	[5.97, 5.98]	[5.10, 5.14]	[5.17, 5.21]	[5.13, 5.17]	[4.96, 4.99]	[6.21, 6.24]	[5.65, 5.69]	[4.89, 4.93]
		$d_z$	large *	large *	large *	large *	large *	large *	large*	—
	First 200 s	P95 (m)	17.46	13.44	13.72	13.65	13.52	13.71	16.18	13.44
		CI95 (m)	[8.51, 8.54]	[6.85, 6.90]	[6.83, 6.88]	[6.82, 6.87]	[6.81, 6.86]	[7.19, 7.23]	[7.35, 7.39]	[6.76, 6.81]
		$d_z$	large *	large *	large *	large *	large *	large *	large *	—

, over all epochs DGCC-AKF achieves the lowest mean 3D error of the eight methods on UAV2 and UAV3 (10.91 m and 3.87 m, respectively). On UAV1 the two baselines are slightly more accurate, but the gap is small, and the difference between VBAKF and DGCC-AKF is not significant, since their CI₉₅ intervals ([6.87, 6.90] m and [6.86, 6.88] m) overlap. Because the base-station observation quality in this experiment remains generally stable, the overall improvement in positioning performance across the full time period comes mainly from cooperative localization rather than from the adaptive mechanism, so the five cooperative schemes perform similarly. Even so, the advantage of DGCC-AKF over the two SOTA baselines is consistent: on UAV2 its P95 (57.60 m) is 21.7% and 34.6% lower than VBAKF (73.60 m) and MCC-EKF (88.12 m), and on UAV3 it is 10.2% and 8.9% lower.

During the first 200 s the UAVs fly far from the base-station area, so the number of visible stations is small and fluctuates frequently. S-EKF therefore performs poorly, while cooperative localization improves the accuracy markedly. This phase also exposes the weakness of the two SOTA baselines on UAV2, where the geometry is worst: MCC-EKF yields the largest error of all methods (P95 111.33 m, even above the 103.25 m of S-EKF), and VBAKF (P95 83.45 m) lags clearly behind DGCC-AKF. DGCC-AKF holds the UAV2 P95 at 67.57 m, which is 19.0% and 39.3% lower than VBAKF and MCC-EKF and 34.5% lower than S-EKF.

Two localized observation anomaly conditions exist in this experiment: a naturally-occurring U-B observation anomaly on UAV1 (epochs [1200, 1300], observation error > 20 m) and the injected U-U observation anomaly on UAV3 (epochs [2500, 3000]) described above. The positioning accuracy during these two anomaly periods was further analyzed.

As shown in Figure 11 and

Table 7. Statistical comparison of the 3D positioning accuracy of the three UAVs during the U-U and U-B observation fault periods.

UAV	Fault	Metric	S-EKF	D-EKF	D-AKF	DCC-AKF	DG-AKF	VBAKF	MCC-EKF	DGCC-AKF
UAV1	U-B fault	P95 (m)	47.26	30.88	13.59	13.95	13.05	9.39	6.77	12.90
		CI95 (m)	[22.89, 22.89]	[16.84, 17.08]	[8.68, 9.06]	[9.25, 9.64]	[8.34, 8.70]	[7.73, 7.94]	[5.52, 5.66]	[8.31, 8.61]
		$d_z$	large *	large *	large *	large *	n.s.	medium *†	large *†	—
	U-U fault	P95 (m)	4.88	4.73	4.42	4.38	4.26	4.28	4.00	4.23
		CI95 (m)	[3.94, 3.94]	[4.03, 4.10]	[3.61, 3.65]	[3.60, 3.64]	[3.62, 3.65]	[3.44, 3.45]	[3.34, 3.37]	[3.57, 3.60]
		$d_z$	large *	large *	medium *	medium *	large *	large *†	large *†	—
UAV2	U-B fault	P95 (m)	24.59	16.90	10.94	11.29	10.30	22.56	10.84	10.17
		CI95 (m)	[22.08, 22.08]	[9.05, 9.53]	[6.26, 6.58]	[6.53, 6.86]	[6.16, 6.47]	[19.64, 20.31]	[9.34, 9.83]	[6.03, 6.35]
		$d_z$	large *	large *	n.s.	large *	large *	large *	large *	—
	U-U fault	P95 (m)	3.85	4.09	3.76	3.68	3.81	3.79	3.21	3.75
		CI95 (m)	[2.54, 2.54]	[3.01, 3.09]	[2.63, 2.68]	[2.59, 2.64]	[2.48, 2.52]	[2.37, 2.41]	[2.16, 2.18]	[2.43, 2.46]
		$d_z$	large *	large *	large *	large *	large *	medium *†	large *†	—
UAV3	U-B fault	P95 (m)	5.16	9.42	5.74	5.82	5.74	6.78	6.48	5.70
		CI95 (m)	[4.27, 4.27]	[6.00, 6.07]	[4.66, 4.73]	[4.74, 4.81]	[4.65, 4.71]	[5.53, 5.55]	[5.30, 5.32]	[4.61, 4.67]
		$d_z$	large *†	large *	large *	large *	large *	large *	large *	—
	U-U fault	P95 (m)	3.53	4.81	5.41	5.11	3.23	3.20	3.70	3.18
		CI95 (m)	[2.15, 2.15]	[3.98, 4.11]	[4.52, 4.72]	[4.25, 4.44]	[2.22, 2.27]	[2.22, 2.26]	[2.26, 2.38]	[2.16, 2.21]
		$d_z$	n.s.	large *	large *	large *	large *	medium *	medium *	—

, when the U-B observations of UAV1 are severely corrupted, S-EKF and D-EKF are the most affected: on the faulted UAV1 their median 3D errors reach 13.68 m and 11.62 m, and the S-EKF P95 rises to 47.26 m. On UAV1 the six adaptive schemes rank, from best to worst by median 3D error, as MCC-EKF (6.02 m) < DGCC-AKF (7.01 m) < DG-AKF (7.62 m) < VBAKF (7.92 m) < D-AKF (8.05 m) < DCC-AKF (8.61 m). Notably, on UAV2, although no anomaly is injected in this window, S-EKF, VBAKF, and MCC-EKF still degrade markedly—the P95 of VBAKF reaches 22.56 m and that of MCC-EKF 10.84 m—because the visible base stations here barely meet the minimum positioning requirement and leave no redundancy. By contrast, DGCC-AKF attains the highest accuracy on UAV2, with a median error 75.8% below that of S-EKF and 73.8% below that of VBAKF. Across the three UAVs, DGCC-AKF is the only method that combines accuracy and stability, remaining best or second-best under almost every condition with a narrow confidence interval.

As shown in Figure 12 and Table 7, when the U-U observations of UAV3 are severely corrupted, the cooperative schemes without grouping degrade the most on the faulted UAV3: the median errors of D-AKF, DCC-AKF, and D-EKF rise to 4.60 m, 4.29 m, and 4.01 m—more than twice the 1.90 m of S-EKF—and their P95 reaches 5.41 m, 5.11 m, and 4.81 m. The grouped schemes DG-AKF and DGCC-AKF, together with the two baselines, stay near 2 m. DGCC-AKF reaches a median of 2.04 m and a CI₉₅ of [2.16, 2.21] m, about 56% and 52% below D-AKF and DCC-AKF. Its median is only marginally above MCC-EKF (2.02 m), but its P95 (3.18 m) is clearly below that of MCC-EKF (3.70 m), so DGCC-AKF bounds the worst-case error more tightly.

In summary, the simulation and semi-physical experiments show that DGCC-AKF combines high accuracy with high stability and copes well with changes in the number of base stations and with degraded observation quality. When base stations are sufficient and observations are stable, the cooperative schemes perform similarly and DGCC-AKF is on par with the best. Under observation loss or anomalies, it still maintains a superior overall accuracy. The two SOTA baselines are far less stable: MCC-EKF surpasses DGCC-AKF only when base-station observations are abundant, while VBAKF becomes unreliable whenever the visible-station set changes frequently or the U-B fault is correlated.

4. Conclusions

This paper proposed a Distributed Group Covariance Compensation Adaptive Kalman Filter (DGCC-AKF) to enhance UAV positioning accuracy in GNSS-denied regional navigation scenarios. DGCC-AKF integrates three complementary strategies: distributed cooperative localization for observation redundancy, a grouped adaptive mechanism for independent noise adjustment of heterogeneous U-B and U-U observations, and a CI compensation strategy for alleviating the conservative covariance estimation inherent in distributed fusion.

Experimental validation was conducted on a large-scale testbed covering over 2800 km² in Zhoukou City, China, using fixed-wing aircraft at speeds exceeding 300 km/h. Two sets of experiments—simulation based on measured trajectories and semi-physical verification using real U-B observations—yield the following key findings. First, under observation loss conditions, cooperative localization is the dominant factor mitigating performance degradation, while the grouped adaptive mechanism and CI compensation provide supplementary optimization. Second, under observation anomaly conditions, the grouped adaptive mechanism is essential for preventing cooperative contamination; without it, cooperative schemes can perform worse than single-UAV positioning, yielding more than double the error of the affected UAV under U-U anomalies in the semi-physical experiment. Third, the CI compensation strategy is critical for maintaining AKF effectiveness in distributed architectures, as AKF without CI compensation yields higher errors than schemes with CI compensation even during normal observation periods. Fourth, benchmarked against two recent state-of-the-art adaptive filters, VBAKF and MCC-EKF, DGCC-AKF is the only scheme that stays both accurate and stable across all UAVs and fault types; the two baselines are inconsistent, each degrading severely under at least one condition—VBAKF under correlated U-B faults and frequent base-station switching, and MCC-EKF when base-station observations are sparse. Overall, DGCC-AKF combines distributed collaboration, group-based adaptation, and CI compensation to deliver both high accuracy and high stability; under observation faults, it improves the 3D positioning accuracy by up to about 75% over single-UAV EKF while bounding the worst-case error more tightly than the competing schemes.

These contributions carry several practical implications for low-altitude UAV positioning. First, the underlying GH-LPS regional navigation system extends the operating envelope of GNSS-based positioning. It provides continuous service in areas where GNSS coverage is unstable or denied, complementing satellite navigation in low-altitude and remote scenarios. Second, the introduction of inter-UAV cooperation substantially reduces dependence on dense ground base-station deployment. As shown by Scenario 2 in Section 3.3, four base stations are sufficient to support cooperative UAV positioning across an operational area exceeding 2000 km². This translates into significantly lower infrastructure cost compared with denser pseudolite networks. Third, the DGCC-AKF algorithm delivers metre-level 3D positioning accuracy on fixed-wing platforms operating above 300 km/h, even under observation anomalies. This accuracy level is compatible with the operational requirements of regional remote sensing, emergency response, and similar high-dynamic UAV missions.

Some limitations should be acknowledged. From a deployment standpoint, our evaluation is only partially physical: the U-B observations and flight trajectories come from a real GH-LPS testbed, but the U-U ranges are still generated from the measured trajectories due to flight-test constraints. As a result, the simulated U-U observations may not fully capture every characteristic of real inter-UAV ranging. In particular, synchronization residuals, communication latency, packet loss, and antenna directivity could introduce additional biases or intermittency on the inter-UAV link. Among these, packet loss is partly reflected through the random observation-loss scenario, while the others are not explicitly modeled. These factors are expected to have a limited effect in the present scenario, and a more complete treatment is left for future work. In addition, owing to cost constraints, the cooperative dataset was constructed from only three flight sessions of a single UAV platform, and the applicability of the proposed method to larger UAV swarms remains to be validated. Accordingly, future work will target fully physical multi-UAV cooperative flight validation on larger swarms, together with more detailed modeling of these inter-UAV link factors.

From a methodological standpoint, the proposed group adaptive mechanism currently regulates noise weights at the observation-type level, rather than at the individual-link level. We plan to extend it to link-level adaptive covariance estimation for finer-grained anomaly handling. Beyond filter-based methods, graph-optimization-based and learning-based cooperative-localization paradigms also hold strong potential. The former enables batch refinement and re-linearization under highly non-linear measurement models. The latter captures complex non-parametric dependencies in dense swarm topologies. We will further investigate the integration of these paradigms with the proposed grouped adaptive mechanism. The goal is to combine their accuracy potential with the real-time and deterministic profile of recursive filtering.