Cooperative RTK Positioning

• Cooperative RTK positioning is an advanced GNSS technique that integrates user cooperation and measurement differencing to significantly reduce reference station noise.
• It employs a unified estimation framework with Fisher Information Matrix and Cramér–Rao bounds to quantify performance and enhance positioning accuracy.
• Simulation results show that increasing aiding users and satellite visibility improves RMSE and integer ambiguity resolution, advancing real-time high-precision navigation.

Cooperative Real-Time Kinematic (C-RTK) positioning is an extension of classical differential GNSS methodologies that leverages user cooperation across large receiver networks to attenuate reference-station noise, approaching ideal positioning accuracy even with mixed-quality infrastructure. The approach is formalized through unified estimation frameworks that incorporate measurement differencing over networks, enabling theoretical and empirical quantification of estimator performance via Fisher Information Matrix (FIM) and Cramér–Rao bounds (CRB) parameterized by network size, reference noise, and satellite geometry (Calatrava et al., 9 Jan 2026).

1. GNSS Measurement Models and Differencing

Single-receiver GNSS observations are modeled via pseudorange ($\rho_r^s$) and carrier phase ($\Phi_r^s$) measurements for receiver $r$ and satellite $s$. Pseudorange is expressed as

$$\rho_r^s = \|p^s - p_r\| + T_r^s + I_r^s + c(dt_r - dt^s) + \epsilon_r^s,$$

while carrier phase is

$$\Phi_r^s = \|p^s - p_r\| + T_r^s - I_r^s + c(dt_r - dt^s) + \lambda N_r^s + \bar{\epsilon}_r^s,$$

with $p^s$, $p_r$ representing satellite and receiver positions, $T_r^s$, $I_r^s$ denoting tropospheric and ionospheric delays, $dt^s$, $dt_r$ clock biases, $\lambda$ the carrier wavelength, $N_r^s$ integer ambiguity, and $\epsilon_r^s$, $\bar{\epsilon}_r^s$ measurement errors.

In multi-user networks indexed by one surveyed base ("b") and $N$ user receivers, single-difference (SD) and double-difference (DD) operations suppress common-mode errors. SDs form per-user differences relative to the base receiver; DDs, constructed with respect to a pivot satellite, further eliminate satellite-dependent biases. Data stacking and differencing yield the network-observation vector, essential for cooperative estimation.

2. Linearized Measurement Equations and Network Modelling

Linearization around an a priori position yields for each receiver:

$$\Delta\rho_r \simeq H_r \Delta x_r + c\, dt_s + T_r + I_r + \epsilon_r,\ \Delta\Phi_r \simeq H_r \Delta x_r + c\, dt_s + T_r - I_r + \lambda N_r + \bar{\epsilon}_r,$$

where $H_r$ is the observation matrix constructed from line-of-sight unit vectors $E_r$, $\Delta x_r$ denotes update to the state vector, and $dt_s$ the satellite clock offset.

Stacking and applying SD and DD operations leads to the network linear Gaussian model:

$$y = [\Delta\tilde{\rho}_b^p;\, \Delta\tilde{\Phi}_b^p] \sim \mathcal{N}(B b + A a,\, \Sigma_b^p)$$

with $b$ as stacked user baseline positions, $a$ double-differenced integer ambiguities, $B$ and $A$ system matrices, and $\Sigma_b^p$ the full covariance post-differencing.

3. Estimation Framework and Statistical Bounds

The estimation problem is cast as mixed-integer least squares:

$$\{b^*, a^*\} = \arg\min_{b \in \mathbb{R}^{3N},\, a \in \mathbb{Z}^{N(K-1)}} \|y - B b - A a\|^2_{\Sigma_b^{p,-1}}.$$

Standard solution uses a three-step procedure: (i) float solution (real ambiguities), (ii) integer ambiguity resolution (ILS), and (iii) fixed estimation (fixing integer ambiguities and re-solving for $b$).

The FIM in float regime is

$$J_f = G^T R^{-1} G,\;\; G = [B\ A],\;\; R = \Sigma_b^p,$$

and the baseline-only CRB is the Schur complement $\mathrm{CRB}_f(b) = [J_f^{-1}]_{1:3N,1:3N}$.

Key assumptions include Gaussian, zero-mean noise with known covariances, identical satellite geometry across users, receiver noise described by $\Sigma_\rho = \sigma_n^2 W^{-1}$ and base-to-user noise ratio $\alpha = \sigma_{\mathrm{ref}}^2/\sigma_n^2$.

4. Fisher Information Matrix, Cramér–Rao Bounds, and Asymptotic Analysis

For homogeneous noise and geometry (Remark 3), $\Sigma_b^p$ manifests Kronecker and block-Toeplitz structure, permitting parameterized closed-form FIM and CRB. Definitions:

• $J_c = E^T W E$ (shared satellites),
• $J_o$ (exclusively seen satellites),
• $$\beta_1(\alpha) = \frac{\alpha N+1}{\alpha N+\alpha+1}$$,
• $$\beta_2(\alpha) = -\frac{\alpha}{\alpha N + \alpha + 1}$$,
• $\beta_4(\alpha) = \frac{1+\alpha(N-1)}{1+\alpha N}$,
• $\beta_5(\alpha) = -\frac{\alpha}{1+\alpha N}$.

FIM structure:

$$J(\omega) = \begin{bmatrix} \beta_1 J_c & \beta_2 1_N^T \otimes J_c \ \beta_2 1_N \otimes J_c & (I_N + \beta_2 J_c) \otimes J_c + (I_N + \beta_5 1_N) \otimes J_o \end{bmatrix}$$

CRB for the baseline follows as the inverse of the Schur complement:

$$\mathrm{CRB}_f(b) = \left[\beta_1 J_c - \beta_2^2 1_N^T \otimes \left(J_c (\cdots)^{-1} J_c \right)\right]^{-1}.$$

In the fixed ambiguity regime,

$$J_{\mathrm{fix}}(b) \approx B^T (\Sigma_b^p)^{-1} B \approx J_c \cdot (\alpha, M, ...) + J_o \cdot (\alpha, M, ...),$$

with corresponding fix-CRB.

Asymptotic analysis elucidates three principal regimes:

• Ideal-visibility ($J_o \rightarrow \infty$): $$\mathrm{CRB} \rightarrow J_c^{-1} / \beta_1(\alpha)$$, so cooperation always beneficial for $\alpha > 0$.
• Homogeneous-visibility ($J_o = 0$): Cooperation reduces to non-cooperative CRB: $(1+\alpha)J_c^{-1}$.
• Large $N$ ($N \rightarrow \infty$): $\beta_1 \rightarrow 1$, $\beta_2 \rightarrow 0$, yielding the ideal noiseless-reference bound: $\mathrm{CRB} \rightarrow J_c^{-1}$.

5. Algorithmic Workflow and Computational Complexity

The practical C-RTK algorithm consists of:

• Data collection: Each receiver and base acquires raw code and phase from all visible satellites.
• Formation of SDs and DDs via $D_b$ and $D_p$ operators.
• Assembly of matrices $B$, $A$, and $\Sigma_b^p$ (using known noise ratio $\alpha$, geometry $W$).
• Float stage: Solve normal equations $(Bb + Aa \approx y)$ for real-valued $a$, $b$.
• Integer ambiguity resolution (e.g. LAMBDA algorithm) to map $a_f \rightarrow \hat{a} \in \mathbb{Z}$.
• Fixed stage: Re-estimate $b$ conditional on $\hat{a}$ using weighted least squares.
• Optional iterative refinement via Gauss–Newton linearization.

Complexity scales as:

• Assembly: $O(N K^2)$,
• Computation of $J = G^T R^{-1} G$: $O((3N + NK)^3)$ nominally (${N^3}{K^3}$), but exploitable structure can reduce cost,
• Float solution inversion: $O(M^3)$ ($M = 3N + N(K-1)$),
• Integer search (LAMBDA): $O(Q N(K-1) \log Q)$ with $Q$ candidate solutions.

The $O((N K)^3)$ scaling motivates distributed or approximate algorithmic variants for large $N$ and moderate $K$ ($10$–$20$ satellites).

6. Simulation Results and Empirical Performance

Simulations with a base and $N_c=2$ "urban" users ($K_c=4$ satellites, GDOP $\approx 2.5$) supplemented by $N_o$ open-sky aiding users (tracking $K = K_c + K_o$ satellites, $K_o=0 \ldots 15$) use noise parameters $\sigma_\rho=1\,\mathrm{m}$, $\sigma_\Phi = \sigma_\rho/100$, identity $W$, and base-to-user ratio $\alpha \in \{0.3, 1, 3\}$. Monte Carlo results ($10^4$ runs) quantify RMSE and ambiguity fixing success rate $P_\mathrm{succ}$.

Key findings include:

• C-DGNSS (code-only): RMSE lies between non-cooperative bound $(1+\alpha)J_c^{-1}$ and ideal $J_c^{-1}$; even $N_o=1$ aiding user confers $10$–$20\%$ RMSE reduction, while increasing $K_o$, $N_o$ drives RMSE to ideal-visibility bound ($\beta_1(\alpha)^{-1} J_c^{-1}$).
• C-RTK (code + phase): Float RMSE matches C-DGNSS; fix RMSE approaches phase-only limit ($\ll$ code-only bound).
• Integer ambiguity resolution: For $\sigma_\rho \leq 1$ m, single-user RTK $P_\mathrm{succ}\approx70\%$, but increases to $>95\%$ for $N_o=5$; $N_o=25$ achieves near-perfect fixing at $\sigma_\rho=2$ m.

Empirical RMSE and $P_\mathrm{succ}$ trends rigorously validate the FIM/CRB theory and demonstrate asymptotic convergence to the ideal reference limit. Tabulated summaries delineate accuracy dependence on $N_o$, $K_o$, and $\alpha$.

Parameter	Effect on RMSE	Effect on $P_\mathrm{succ}$
$N_o$ (aiding users)	Decreases RMSE toward ideal	Increases fixing probability
$K_o$ (extra satellites)	Improves RMSE (visibility gain)	Accelerates $P_\mathrm{succ}$ improvement
$\alpha$ (base/user noise ratio)	Larger $\alpha$ increases RMSE	Cooperation mitigates $\alpha$ impact

7. Theoretical Significance and Practical Implications

C-RTK positioning, as formalized in the unified C-DGNSS/C-RTK framework, demonstrates that user cooperation can asymptotically restore accuracy to the ideal noiseless-reference regime, even when the reference station has significant noise or heterogeneous quality. Analytical FIM/CRB expressions clarify regimes where cooperation is most effective—primarily when aiding users access additional satellites not visible to all and as network size increases. For homogeneous visibility, cooperation yields limited benefit; expansion of network size and satellite visibility confers maximal accuracy gains. Simulation results substantiate these theoretical conclusions.

A plausible implication is that distributed C-RTK networks with strategic placement of open-sky aiding receivers and algorithmic optimization for large $N$ will enable robust, scalable, and infrastructure-independent high-precision positioning in GNSS-denied or urban scenarios (Calatrava et al., 9 Jan 2026).