Cooperative Differential GNSS Positioning: Estimators and Bounds

TL;DR

This work develops a unified estimation framework for cooperative differential GNSS, enabling C-DGNSS and C-RTK to mitigate reference-station noise through large-scale user cooperation. By deriving parameterized Fisher information matrices and CRB-based bounds as functions of network size, satellite geometry, and base-station noise ratio $\alpha$, the authors characterize regimes where cooperation restores ideal, noise-free-reference accuracy. The analysis introduces a two-cluster visibility model and closed-form expressions for the FIM blocks, revealing that cooperation yields substantial gains for users with limited visibility and can asymptotically achieve the ideal bound as the cooperative network scales. Simulations corroborate the theoretical insights, showing improved 3D positioning and faster ambiguity resolution in C-RTK, supporting practical deployment of cooperative GNSS services with low-cost reference stations.

Abstract

In Differential GNSS (DGNSS) positioning, differencing measurements between a user and a reference station suppresses common-mode errors but also introduces reference-station noise, which fundamentally limits accuracy. This limitation is minor for high-grade stations but becomes significant when using reference infrastructure of mixed quality. This paper investigates how large-scale user cooperation can mitigate the impact of reference-station noise in conventional (non-cooperative) DGNSS systems. We develop a unified estimation framework for cooperative DGNSS (C-DGNSS) and cooperative real-time kinematic (C-RTK) positioning, and derive parameterized expressions for their Fisher information matrices as functions of network size, satellite geometry, and reference-station noise. This formulation enables theoretical analysis of estimation performance, identifying regimes where cooperation asymptotically restores the accuracy of DGNSS with an ideal (noise-free) reference. Simulations validate these theoretical findings.

Figures (6)

• Figure 1: Overview of the cooperative DGNSS architecture considered in this work. Nearby users $r\in{1,\ldots,N}$ transmit code ($\boldsymbol{\rho}_r$) and carrier-phase ($\boldsymbol{\Phi}_r$) measurements to a CPC, which jointly processes them with base-station DGNSS corrections. Cooperation is centralized, base-station–anchored, and does not involve inter-user ranging.
• Figure 2: Overview of the derivation flow in Sec. \ref{['sec:method']} leading to a unified observation model and performance bound for C-DGNSS and C-RTK, summarized in Tables \ref{['tab:crb_models']} and \ref{['tab:crb_model_summary']}. The resulting unified formulation enables the theoretical analysis in Sec. \ref{['sec:bounds']}, which constitutes the main contribution of this work. Solid arrows indicate section flow, while dashed arrows denote result reuse across derivations.
• Figure 3: Heatmaps of the covariance matrix for pseudorange measurements, $\tilde{\boldsymbol{\Sigma}}_{b,\rho}^{p,\alpha}$, under the assumptions of Remark \ref{['remark:3']} (cf. \ref{['eq:cov_crtk_remark_1']}), with $\mathbf{W} = \mathrm{I}$, $\sigma_\rho = 1$ m, $N=3$, and $\alpha = 1$. An analogous pattern is obtained for $\tilde{\boldsymbol{\Sigma}}_{b,\Phi}^{p,\alpha}$. $[k,:]$ and $[:,k]$ denote row and column slices. See calatrava_cdgnss_2025 for the case under $\alpha=1$. See Supplemental Material (Sec. S1-D) for a detailed explanation of the element-wise noise contributions in C-RTK.
• Figure 4: Impact of cooperative network parameters on 3D positioning RMSE in C-DGNSS. Results are shown for $K_o = \{8,\,14,\,19\}$. We report $\mathbf{CRB}_{\mathrm{DGNSS}}$\ref{['eq:bound:noncoop']}, $\mathbf{CRB}_{\mathrm{Ideal\,DGNSS}}$\ref{['eq:bound:ideal']}, the asymptotic benchmark $\beta_1(\alpha)^{-1}$\ref{['eq:proof_1']}, and, for the C-DGNSS model in Table \ref{['tab:crb_model_summary']}, $\mathbf{CRB}_{\mathrm{C\text{-}DGNSS}}$\ref{['eq:bound_cdgnss']} and $\mathbf{WLS}_{\mathrm{C\text{-}DGNSS}}$\ref{['eq:cdgnss_estimator']}.
• Figure 5: Impact of cooperative network parameters on 3D positioning RMSE and ambiguity resolution in C-RTK. Results are reported for $N_{o} \in \{1, 5, 25\}$. We show (left) the bounds $\mathbf{CRB}_{\mathrm{C\text{-}RTK,float}}$ in \ref{['eq:crb_float']} and $\mathbf{CRB}_{\mathrm{C\text{-}RTK,fix}}$ in \ref{['eq:crb_freq']}, together with the empirical $\mathbf{WLS}_{\mathrm{C\text{-}RTK}}$ performance, and (right) the success probability $\hat{P}_{\mathrm{succ}}$ in \ref{['eq:success_rate']}. For reference, the corresponding non-cooperative baselines are included in all cases.

Theorems & Definitions (4)

• Definition 1: C-DGNSS Model
• Definition 2: C-RTK Model
• Remark 1: C-DGNSS/C-RTK Model Covariance Structure
• Remark 2: Solvability condition