Collaborative positioning with GNSS as a technique for global high-precision positioning in real time

SergeiDolin1✉EmailSergeydolin@icloud.com

Research Institute of Strategic DevelopmentSiberian State University of Geosystems and Technologies630108NovosibirskRussia

Sergei Dolin^z,2

^z,2 Research Institute of Strategic Development, Siberian State University of Geosystems and Technologies, Novosibirsk, 630108, Russia; Email: Sergeydolin@icloud.com ORCID: 0000-0002-0686-2272

Abstract

The development of precise real-time positioning systems for global and continuous navigation is crucial for a wide range of applications. However, current GNSS-based methods fail to meet the demands of modern users for real-time navigation, as they have limitations in terms of range and data transmission. The RTK and N-RTK techniques, while efficacy in certain scenarios, are limited by their range and require substantial data transfer. On the other hand, this problem does not exist in Precise Point Positioning (PPP). However, the process of Ambiguity Resolution (AR) poses significant challenges for real-time applications. Despite these challenges, post-processing solutions are generally less problematic than real-time solutions. PPP-AR, with its various implementations, allows for a fixed solution in both static and kinematic modes, and its potential real-time application in precise positioning presents more practical and significant use cases. This paper explores the technique of a collaborative positioning approach that combines PPP-AR with RTK, aiming to overcome the limitations of high-precision real-time coordinate determination methods. The proposed approach is designed to enhance the capabilities of existing technologies in this field. This technology can contribute to the establishment of a novel, highly accurate dynamic ground-based infrastructure. Pseudo-kinematic experiments showed a level of accuracy of 10–20 cm for each coordinate regardless of the distance between receivers during real-time operation. The findings and analysis provided herein offer valuable insights and essential in demonstrating a prospective approach to integrating two precise techniques. This implementation has significant potential for advanced business solutions requiring real-time coordinate determination.

Keywords

PPP ‧ RTK ‧ SSR ‧ Collaborative Positioning ‧ Real-Time ‧ Precise Positioning

Introduction

Modern high-precision positioning applications based on GNSS in real-time typically use three main methods: Precise Point Positioning (PPP) (Zumberge et al., 1997; Ge et al., 2008; Teunissen, 2012), the relative method with short or long baselines in real-time (RTK) (Li et al., 2014; Sioulis, Tsakiri and Stathas, 2015; Pehlivan, Bezcioglu and Yilmaz, 2019) and Network RTK (N-RTK) (Teunissen and Khodabandeh, 2015). Each method in this list has its own characteristics and limitations that hinder the rapid determination of coordinates at any location on the Earth's surface. High-precision global positioning is achievable only with PPP or PPP-AR, but these methods are not fast; high-precision fast positioning is possible with network or relative solutions, but these solutions have distance limitations. Thus, a key idea in recent times is the unification of these methods into a single approach to eliminate all restrictions.

A popular and reliable solution is known as PPP-RTK (Wübbena, Schmitz and Bagge, 2005). PPP-RTK is an extension of the PPP model that enables users to achieve greater accuracy and faster convergence rates than the classical PPP method. This enhancement is achieved through the application of ambiguity resolution to GNSS observations. Unlike the traditional method of transmitting corrections to the Observation State Representation (OSR), PPP-RTK transmits SSR (State-Space Representation) corrections. This enables more detailed corrections, including high-precision orbit and clock products, ionospheric and tropospheric delays, and uncalibrated phase delays.

Another significant area of research is collaborative positioning (Alam and Dempster, 2013; Banville et al., 2014). Collaborative positioning is a promising technology for precise navigation and may become the primary method of positioning in the future. In wireless sensor networks, various methods of collaborative positioning are employed to enhance navigation (Teunissen and Montenbruck, 2017). These solutions are particularly effective when GNSS-based positioning is unavailable. They are essential for ensuring safety, reliability, and integrity. These implementations typically leverage existing telecommunications infrastructure to share information among users operating within a specific area, which limits their applicability for global positioning.

One of the earliest proposals to combine collaborative and satellite positioning technology was presented at the conference: the 27th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS + 2014) (Huang et al., 2014), the Earth on the Edge: Science for a Sustainable Planet conference (Kealy et al., 2014), and the idea was also discussed at the Fig. 2019 Working Week (Chris Rizos et al., 2019). The core concept was that each GNSS user could transmit real-time corrections to the positioning of other users, thereby forming a ground-based dynamic infrastructure for high-precision positioning.

This study proposes a high-precision positioning method that combines PPP and RTK, along with a new method for distributing GNSS status information between user navigation equipment (UNE). This new approach is referred to as collaborative positioning (CP). Additionally, the study considers an approach to the ambiguity resolution strategy in PPP within the context of CP. Recognizing the need for global and highly accurate real-time positioning, this research emphasizes a new approach to GNSS user collaboration. In the dynamic GNSS system, users can adjust each other's locations, providing highly accurate navigation without the need for permanent base stations.

Following this introduction, the article will present a technique for collaborative positioning, along with an implemented prototype of a collaborative positioning service; it will also present an ambiguity resolution strategy in PPP based on the CP approach and the implementation of a variational Bayesian-based robust adaptive Kalman filter; and finally, it will present experimental results confirming the effectiveness of the proposed implementation.

Methodology

In this section, describe the collaborative positioning model including the algorithm of user interaction in the CP system, PPP-AR model, the relative and moving-baseline observation model, as well as the algorithm of the variational Bayesian-based robust adaptive Kalman filter.

Collaborative positioning

Positioning model

To move from fixed continuously operating reference stations (CORS) to a dynamic infrastructure, we need to review the existing receiver types. The following scheme of interaction between UNEs is presented in Fig. 1. Currently, 'Base station' and 'Rover' are not considered fixed receiver types but rather the current roles or statuses of the receivers. Additionally, a new 'Candidate' status has been introduced. 'Rover' serves as the default status. The system assigns this status to the UNE immediately upon activation or when it determines that the positioning accuracy is poor or unknown. After receiving this status, the UNE sends its approximate coordinates and a request for RTK corrections to the cloud data center (CDC). The UNE receives 'Candidate' status if the standard deviation of its coordinates is low and the user consents to share their high-precision coordinates with others. The CDC assigns 'Base station' status to a 'Candidate' if any nearby rover requires corrections. 'Base station' transmits a stream of OSR corrections – measurement data and high-precision coordinates in RTCM-3 format (Martin Schmitz, 2012) to the rover. PPP-AR is the default method for all users, regardless of type.

Fig. 1

– Scheme of interaction between roles in CP

The proposed structure of GNSS roles enables the formation of a dynamic, independent infrastructure capable of supporting high-precision navigation and possessing self-replicating properties. Such infrastructure should be implemented as an online service using cloud architecture. This approach allows for scalable computing power and mitigates critical failures that could lead to a complete shutdown of the service, thanks to load balancing algorithms and the integration of backup software modules. In the present study, the prototype CP service consists of three components: database, cloud service, and consumer sector. The database includes information about the approximate coordinates of the phase center of the UNE, the type of navigation device, dynamic status, GNSS status, and standard deviations. The cloud service consists of a front-end application based on RTKLIB (T. Takasu, 2013) and a back-end proprietary application that performs analysis and management of the online CP service. This system requires UNE equipment equipped with an OEM module for dual-frequency phase/code measurements and a computational block for the client application. Figure 2 shows the prototype diagram of the CP service.

Fig. 2

– Scheme of prototype of CP service

Here

${{\mathbf{\tilde {x}}}_{k|k}}={{\mathbf{\hat {x}}}_{k|k}}+\left( {X\sim N(\mu ,{\sigma ^2})} \right)$

is approximate coordinate of UNE, where

$\mu =100\,m,\sigma =10\,m$

denote expected value and square of standard deviation, respectively. Estimation of coordinate include in

${{\mathbf{\hat {x}}}_{k|k}}$

, at time

is evolved from the state at

. Measurement vector denote as

${{\mathbf{z}}_k}$

;

${\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{x} }}$

is reference coordinates from ITRF2020-u2023, only for CORS stations. All of users transmit approximate coordinate was needed for analysis of geometry of collaborative web.

To analyze the need for increased accuracy in the UNE for the "Rover" status, the Bertrand’s test is used to evaluate the convergence of the solution. Then the geometric average of standard deviations from PPP and RTK is compared. If the UNE is in motion, an additional criterion is applied to account for the dynamic conditions. The third criterion evaluates the compatibility of the solutions obtained from PPP and RTK in moving baseline mode, using the following equations:

${\rho _n} \equiv \ln \left( n \right)\left( {n\left( {\frac{{{{\bar {\sigma }}_{IFL{C_{n - 1}}}}}}{{{{\bar {\sigma }}_{IFL{C_n}}}}} - 1} \right) - 1} \right)$

${\bar {\sigma }_{IFL{C_t}}}>{\bar {\sigma }_{RE{L_t}}}$

$\begin{gathered} \left| {{X_{MB}} - {X_{IFLC}}} \right|>2\sqrt {\sigma _{{{X_{MB}}}}^{2}+\sigma _{{{X_{IFLC}}}}^{2}} \hfill \\ \left| {{Y_{MB}} - {Y_{IFLC}}} \right|>2\sqrt {\sigma _{{{Y_{MB}}}}^{2}+\sigma _{{{Y_{IFLC}}}}^{2}} \hfill \\ \left| {{Z_{MB}} - {Z_{IFLC}}} \right|>2\sqrt {\sigma _{{{Z_{MB}}}}^{2}+\sigma _{{{Z_{IFLC}}}}^{2}} \hfill \\ \end{gathered}$

Here

is data-points,

is time of measurements,

is all of measurements,

$\bar {\sigma }=\sqrt[n]{{\prod\limits_{{i=1}}^{n} {{\sigma _i}} }},n=3$

, where

$i=\left\{ {1,2,3} \right\} \equiv x,y,z$

, denotes the geometry average of standard deviations, IFLC, REL and MB is Ionosphere-Free Linear Combination (IFLC) for PPP, Relative method and Moving-baseline accordingly.

If all criteria are met and the ambiguity in PPP has been set to a float solution, the solution is reinitialized. This process involves replacing the dispersions from PPP with those from RTK in the covariance matrix and updating the coordinates in the state vector at the time when all conditions are satisfied. Other parameters are updated based on their correlations. The reinitialization is performed in the following manner:

${{\mathbf{x}}_{k|k - 1}}={\left[ {\begin{array}{*{20}{c}} {{{\mathbf{r}}_{REL}}}&{{{\mathbf{v}}_{IFLC}}}&{cd{t_{IFLC}}}&{ZT{D_{IFLC}}}&{{G_{IFLC,N}}}&{{G_{IFLC,E}}}&{{{\mathbf{A}}_{IFLC}}} \end{array}} \right]^T}$

${{\mathbf{P}}_{k|k - 1}}={\text{diag}}({\sigma ^2}_{{{X_{REL}}}},{\sigma ^2}_{{{Y_{REL}}}},{\sigma ^2}_{{{Z_{REL}}}},{\sigma ^2}_{{V{X_{IFLC}}}},{\sigma ^2}_{{V{Y_{IFLC}}}},{\sigma ^2}_{{V{Z_{IFLC}}}},...,)$

${{\mathbf{\hat {x}}}_{k|k}}={{\mathbf{\hat {x}}}_{k|k - 1}}+{{\mathbf{P}}_{k|k - 1}}({{\mathbf{z}}_k} - {{\mathbf{H}}_k}{{\mathbf{\hat {x}}}_{k|k - 1}})$

Here

${{\mathbf{x}}_{k|k - 1}}$

${{\mathbf{P}}_{k|k - 1}}$

is new priori state vector and variance covariance matrix at time

is evolved from the state at

$k - 1$

, accordingly. The state vector

${{\mathbf{x}}_{k|k - 1}}$

include:

${{\mathbf{r}}_{REL}}$

– vector of coordinate from relative method;

${{\mathbf{v}}_{IFLC}}$

– vector of velocities from PPP;

$cd{t_{IFLC}}$

– estimate of receiver clock;

$ZT{D_{IFLC}}$

– the wet zenith tropospheric delay;

${G_{IFLC,N}},{G_{IFLC,E}}$

– gradient parameters, for North and East direction, accordingly;

${{\mathbf{A}}_{IFLC}}$

– vector of float ambiguity. The re-estimate of

${{\mathbf{\hat {x}}}_{k|k}}$

contains:

${{\mathbf{z}}_k}$

is the measurement vector;

${{\mathbf{H}}_k}$

is the observation matrix.

PPP-AR model

Due to the approach used to obtain corrections, an ambiguity resolution model, known as the uncalibrated phase delay model, was chosen. Unlike the integer clock model, the authors of the paper (Ge et al., 2008) suggested that, instead of defining satellite clocks, outdated IGS clock estimates could be utilized for the uncalibrated phase delay model. The equations for measurements based on the IFLC for PPP are presented as follows:

$P_{{r,IFLC}}^{{S,\Theta }}={C_i}P_{{r,i}}^{{S,\Theta }}+{C_j}P_{{r,j}}^{{S,\Theta }}=\rho _{r}^{{S,\Theta }}+c(d{t_r} - d{t^{S,\Theta }})+T_{r}^{{S,\Theta }}+DCB_{{i,j}}^{{S,\Theta }} - \xi _{r}^{{S,\Theta }}+ \in _{{{P_{IFLC}}}}^{{S,\Theta }}$

$L_{{r,IFLC}}^{{S,\Theta }}={C_i}L_{{r,i}}^{{S,\Theta }}+{C_j}L_{{r,j}}^{{S,\Theta }}=\rho _{r}^{{S,\Theta }}+c(d{t_r} - d{t^{S,\Theta }})+T_{r}^{{S,\Theta }}+A_{{r,IFLC}}^{{S,\Theta }}+{b_r}+{b^{S,\Theta }} - \xi _{r}^{{S,\Theta }}+\varepsilon _{{{P_{IFLC}}}}^{{S,\Theta }}$

with

${C_i}=\frac{{f_{i}^{{{W^2}}}}}{{f_{i}^{{{W^2}}} - f_{j}^{{{W^2}}}}};{C_j}= - \frac{{f_{j}^{{{W^2}}}}}{{f_{i}^{{{W^2}}} - f_{j}^{{{W^2}}}}};$

$\xi _{r}^{{S,\Theta }}=\frac{{2\mu }}{{{c^2}}}\ln \frac{{\left| {{{\mathbf{r}}^{S,\Theta }}} \right|+\left| {{{\mathbf{r}}_r}} \right|+\left| {{\mathbf{r}}_{r}^{{S,\Theta }}} \right|}}{{\left| {{{\mathbf{r}}^{S,\Theta }}} \right|+\left| {{{\mathbf{r}}_r}} \right| - \left| {{\mathbf{r}}_{r}^{{S,\Theta }}} \right|}}$

Here

and

is index of satellite and rover accordingly,

$\Theta$

is the system index: GPS (G), Galileo (E). The frequencies

${f_1},{f_2}$

correspond to L1, L2 for GPS and E1, E5a for Galileo.

$T_{r}^{{S,\Theta }},{G_N},{G_E}$

are zenith troposphere delay and the gradient parameters with the mapping function (Boehm et al., 2006). Differential code biases compute as

$DCB_{{i,j}}^{{S,\Theta }}=d_{i}^{{S,\Theta }} - d_{j}^{{S,\Theta }}$

where

$d_{i}^{{S,\Theta }}$

and

$d_{j}^{{S,\Theta }}$

is uncalibrated code delay, as an

${b_r}$

and

${b^{S,\Theta }}$

is uncalibrated phase delay. The term

$\xi _{r}^{{S,\Theta }}$

is the Shapiro signal propagation delay and it introduces a general relativistic correction into the geometric range. The term

$A_{{r,IFLC}}^{{S,\Theta }}$

denote is float ambiguity which can be resolved using the Melbourne-Wübbena equation with uncalibrated phase delays. Next, we will see exactly how the ambiguity resolution process takes place in IFLC measurements.

The Melbourne-Wübbena (MW) equation is a necessary addition to existing measurement equations (Li et al., 2013; Chen et al., 2021). To resolve ambiguities in IFLC, the following algorithm must be followed:

Calculate the single differences of ionospheric-free float ambiguities. The Kalman filter computes the difference between the IF float ambiguities and their variance-covariance matrix.

The calculated MW ambiguity is rounded to a wide-lane ambiguity and then fixed.

After fixing the wide-lane ambiguity, the float narrow-lane ambiguities are derived from their relationship with the ionospheric-free and fixed wide-lane ambiguities.

In the end, we can resolve the IF ambiguity and update the other parameters, including position, zenith delays, and remnant unfixed ambiguities, based on their correlation with the fixed ambiguities.

$A_{{r,IFLC}}^{{S,\Theta }}=A_{{r,IFLC}}^{{{S_i},\Theta }} - A_{{r,IFLC}}^{{{S_j},\Theta }}$

$A_{{MW}}^{{S,\Theta }}=\lambda _{{WL}}^{{S,\Theta }}\left( {\frac{{L_{{r,i}}^{{S,\Theta }}}}{{{\lambda ^{{S_i},\Theta }}}} - \frac{{L_{{r,j}}^{{S,\Theta }}}}{{{\lambda ^{{S_j},\Theta }}}}} \right) - \left( {\frac{{f_{i}^{{S,\Theta }}P_{{r,i}}^{{S,\Theta }}+f_{j}^{{S,\Theta }}P_{{r,j}}^{{S,\Theta }}}}{{f_{i}^{{S,\Theta }}+f_{j}^{{S,\Theta }}}}} \right)$

$N_{{r,WL}}^{{S,\Theta }}=\left\lfloor {\frac{{A_{{MW}}^{{S,\Theta }}+UPD_{{WL}}^{{S,\Theta }}}}{{\lambda _{{WL}}^{{S,\Theta }}}}+0.5} \right\rfloor$

$N_{{r,NL}}^{{S,\Theta }}=\frac{{\left( {A_{{r,IFLC}}^{{S,\Theta }}+\frac{{\lambda _{j}^{{S,\Theta }}}}{{\gamma _{j}^{{S,\Theta }} - 1}}N_{{r,WL}}^{{S,\Theta }}+UPD_{{NL}}^{{S,\Theta }}} \right)}}{{\lambda _{{NL}}^{{S,\Theta }}}}$

${{\mathbb{R}}^n} \to {{\mathbb{Z}}^n}|\left\{ {A_{{r,IFLC}}^{{S,\Theta }} \to N_{{r,IFLC}}^{{S,\Theta }}} \right\}:N_{{r,IFLC}}^{{S,\Theta }}=\lambda _{{NL}}^{{S,\Theta }}N_{{r,NL}}^{{S,\Theta }} - \frac{{\lambda _{j}^{{S,\Theta }}}}{{\gamma _{j}^{{S,\Theta }} - 1}}N_{{r,WL}}^{{S,\Theta }}$

Here wavelength denote as

$\lambda$

and

${\lambda _{NL}}=\frac{c}{{f_{i}^{{S,\Theta }}+f_{j}^{{S,\Theta }}}};{\lambda _{WL}}=\frac{c}{{f_{i}^{{S,\Theta }} - f_{j}^{{S,\Theta }}}}$

. The coefficient

$\gamma _{j}^{{S,\Theta }}={\left( {\frac{{f_{i}^{{S,\Theta }}}}{{f_{j}^{{S,\Theta }}}}} \right)^2}$

The algorithm eliminates ambiguity in the IFLC and can be used in real time. In experiments, used GPS and Galileo measurements with correction data for all processing methods, obtained from the French Space Agency CNES (National Space Research Center), a contributor to the International GNSS Service (IGS), which provides real-time services (RTS). CNES transmits ultra-rapid ephemerides and clock data, as well as differential code and phase biases. This information enables high-precision positioning with PPP and PPP-AR based on undifferenced phase measurements (Laurichesse, 2011; Loyer et al., 2012; Katsigianni, Loyer and Perosanz, 2019). The SSR parameters from CNES are transmitted via the BKG NTRIP caster under the product name SSRA00CNE1.

Relative and Moving-Baseline observation model

The CP technology uses the relative method in three modes: static, kinematic, and moving-baseline. This method used for re-initializing the extended Kalman filter in the PPP method. Re-initialization typically takes between 10 seconds and 2 minutes, depending on the distance between the "Rover" and the "Candidate." Mode selection depends on two main factors: the distance to the rover and the dynamic status, Fig. 3. The distance determines the method used to account for atmospheric delays, while the dynamic status affects the method for estimating the base station coordinates. For baselines shorter than 25 km, atmospheric delays are eliminated through double differencing. For longer baselines (> 25 km), these delays are estimated. By default, the base station coordinates and their standard deviations are derived from PPP-AR.

Fig. 3

– Scheme of mode selection

Since the relative method requires less time for convergence, it enables more effective position refinement when used alongside the PPP method. However, it should be noted that this combined approach does not support ambiguity resolution via double differencing, as these ambiguities differ from those in IFLC. Nevertheless, transferring coordinates and their deviations from the relative method’s estimates significantly reduce the convergence time in PPP, thereby accelerating ambiguity resolution.

Variational Bayesian-based robust adaptive Kalman filter

To achieve reliable positioning using collaboration technology and a reliable PPP solution, an adaptive Kalman filter based on a variational Bayesian approach is used (Sarkka and Nummenmaa, 2009; Huang and Zhang, 2012; Huang et al., 2018; Zhang et al., 2018; Pan et al., 2021). Conventional robustness approaches for PPP are suboptimal because they assume uniform observation accuracy across all measurement types – an assumption that does not hold in PPP models, where observations exhibit varying levels of precision (Zhang et al., 2018). Moreover, standardized residuals computed using traditional methods often fail to accurately represent the true residuals of heterogeneous observations, leading to biased state estimates.

To solve this problem, it is necessary to use the ability of the matrix of equivalent weights to reduce the outliers of observations by introducing a reliable function of equivalent weights based on classification. This function leverages Student’s t-test, where the test statistic is derived from standardized residuals of observations within the same category. The general form of this function is expressed as a factor function, enabling more precise control over measurement anomalies.

$\Xi _{i}^{\theta }=\left\{ {\begin{array}{*{20}{c}} 1 \\ {\frac{{T_{i}^{\theta }}}{{{t_0}}}\left( {\frac{{{t_1} - {t_0}}}{{{t_1} - T_{i}^{\theta }}}} \right)} \\ \infty \end{array}} \right.\begin{array}{*{20}{c}} {}&{T_{i}^{\theta } \leqslant {t_0}({a_0},\tau )} \\ {}&{{t_0}({a_0},\tau )<T_{i}^{\theta } \leqslant {t_1}({a_1},\tau )} \\ {}&{T_{i}^{\theta }>{t_1}({a_1},\tau )} \end{array}$

where

$\Xi _{i}^{\theta }$

denotes variance inflation factor,

$\theta$

type of observation

$\theta =1,2$

here 1 is pseudorange and 2 is phase-range,

is the number of type of observations.

${t_0}$

and

${t_1}$

are the student’s test coefficients at significance

${a_0}$

and

${a_1}$

with freedom degree

$\tau ={n^\theta } - 1$

$T_{i}^{\theta }=\frac{{\left| {{\mathbf{\bar {v}}}_{i}^{\theta } - \frac{1}{{{n^\theta }}}\sum\limits_{\begin{subarray}{l} k=1 \\ k \ne i \end{subarray} }^{{{n^\theta }}} {{\mathbf{\bar {v}}}_{k}^{\theta }} } \right|}}{{\sqrt {\frac{1}{{{n^\theta }}}\sum\limits_{\begin{subarray}{l} k=1 \\ k \ne i \end{subarray} }^{{{n^\theta }}} {{{\left( {{\mathbf{\bar {v}}}_{i}^{\theta } - \frac{1}{{{n^\theta }}}\sum\limits_{\begin{subarray}{l} k=1 \\ k \ne i \end{subarray} }^{{{n^\theta }}} {{\mathbf{\bar {v}}}_{k}^{\theta }} } \right)}^2}} } }}$

Here

$T_{i}^{\theta }$

is the student’s statistic. The standardized residual of the

$ith$

measurements of the

$\theta th$

type of observations denotes

$\bar {v}_{i}^{\theta }$

and compute according (Zhang et al., 2018). Assuming there are no observation errors and dynamic model errors, the predicted residual vector

$\varsigma$

should be white noise sequences and subject to the normal distribution. Each predicted residual vector cutting on two vectors of type of observations such a normal distribution as

$\bar {v}_{i}^{\theta }=\frac{{v_{i}^{\theta }}}{{\sqrt {\hat {\sigma }_{\theta }^{2}{{\mathbf{R}}_{v_{i}^{\theta }}}} }}$

$\begin{gathered} {\varsigma _P}\sim {\rm N}\left( {0,{{\mathbf{R}}_P}+{\mathbf{H}}_{P}^{T}{{\mathbf{P}}_{{x_P}}}{{\mathbf{H}}_P}} \right) \hfill \\ {\varsigma _L}\sim {\rm N}\left( {0,{{\mathbf{R}}_L}+{\mathbf{H}}_{L}^{T}{{\mathbf{P}}_{{x_L}}}{{\mathbf{H}}_L}} \right) \hfill \\ \end{gathered}$

$\hat {\sigma }_{\theta }^{2}=\frac{{\varsigma _{\theta }^{T}{\mathbf{R}}_{{{\varsigma _\theta }}}^{{ - 1}}{\varsigma _\theta }}}{{{n_\theta }}}\begin{array}{*{20}{c}} {}&\begin{gathered} \left( {\theta =P,L} \right) \hfill \\ \left( {{{\mathbf{R}}_{{\varsigma _\theta }}}={{\mathbf{R}}_\theta }+{\mathbf{H}}_{\theta }^{T}{{\mathbf{P}}_{{x_\theta }}}{{\mathbf{H}}_\theta }} \right) \hfill \\ \end{gathered} \end{array}$

Besides robustness, another critical aspect is the choice of prior distribution for the measurement process model. This study uses the algorithm proposed by (Huang et al., 2018), to estimate the measurement noise covariance matrix. This algorithm leverages the inverse Wishart probability density function (PDF) as a prior and minimizes the Kullback-Leibler (KL) divergence between the approximate distribution and the true posterior distribution. For a symmetric positive definite random matrix

${{\mathbf{{\rm A}}}_{n \times n}}$

, the inverse Wishart PDF is formulated as follows:

$PD{F_{Inverse - Wishart}}=fx({\mathbf{A}};{\mathbf{\Psi }},v)=\frac{{{{\left| {\mathbf{\Psi }} \right|}^{v/2}}}}{{{2^{vp/2}}{\Gamma _n}\left( {\frac{v}{2}} \right)}}{\left| {\mathbf{A}} \right|^{ - (v+n+1)/2}}{e^{ - 0.5tr({\mathbf{\Psi }}{{\mathbf{A}}^{ - 1}})}}$

Here

is a degrees of freedom,

${\mathbf{\Psi }}$

denote inverse scale matrix,

${\Gamma _n}\left( \cdot \right)$

is a

- variant gamma function. If the condition

${\mathbf{{\rm A}}}\sim PD{F_{Inverse - Wishart}}$

is fulfilled, then

${\rm E}\left[ {{{\mathbf{A}}^{ - 1}}} \right]=(v - n - 1){{\mathbf{\Psi }}^{ - 1}}$

when

$v>n+1$

. Since the covariance matrix of Gaussian PDF, its prior distribution

$\rho \left( {{{\mathbf{P}}_{k|k - 1}}|{{\mathbf{z}}_{1:k - 1}}} \right)$

are chosen as inverse Wishart PDF

$\rho \left( {{{\mathbf{P}}_{k|k - 1}}|{{\mathbf{z}}_{1:k - 1}}} \right)=PD{F_{Inverse - Wishart}}\left( {{{\mathbf{P}}_{k|k - 1}};{{\hat {t}}_{k|k - 1}},{{{\mathbf{\hat {T}}}}_{k|k - 1}}} \right)$

Where

${\hat {t}_{k|k - 1}},{{\mathbf{\hat {T}}}_{k|k - 1}}$

denotes degree of freedom and inverse scale matrix respectively and defined as follows

${\hat {t}_{k|k - 1}}=n+\tau +1$

${{\mathbf{\hat {T}}}_{k|k - 1}}=\tau {{\mathbf{P}}_{k|k - 1}}$

here

is the number of rows of the state vector;

$\tau$

is the tuning parameter

$\tau \in \left[ {2,6} \right]$

. The full measurement processing algorithm is shown in the Fig. 4.

Fig. 4

– Algorithm of variational Bayesian-based robust adaptive Kalman filter

The source data contains

${{\mathbf{\hat {x}}}_{k - 1|k - 1}}$

is the state estimation vector,

${{\mathbf{P}}_{k - 1|k - 1}}$

is the estimation error covariance matrix,

${{\mathbf{F}}_{k - 1}}$

is the state transition matrix,

${{\mathbf{H}}_k}$

is the observation matrix,

${{\mathbf{z}}_k}$

is the measurement vector,

${{\mathbf{R}}_{k - 1}}$

is the measurement noise covariance matrix and also

$n,\tau ,N$

is the number of estimated parameters, tuning parameter and fixed-point iterations, respectively. While the KL divergence will never reach zero, the number of fixed-point iterations can be adjusted. As demonstrated by (Huang et al., 2018), this parameter critically influences both estimation accuracy and computational efficiency. Increasing iterations generally improves accuracy but at the cost of longer computation time. The required number of iterations also depends on the dimensions of the state and measurement vectors. Higher-dimensional systems typically require more iterations due to the increased complexity of evaluating additional parameters. However, in practice, the solution converges after 10–15 iterations, with further iterations yielding diminishing returns.

Results and analysis

To confirm the validity of the proposed technology, experiments were implemented. Pseudo-kinematic experiments and kinematic tests were performed. Pseudo-kinematic means a kinematic mode for processing measurements at a stationary point. Reference points were used from IGS and the Regional Reference Frame Sub-Commission for Europe (EUREF). For all the reference points used, the coordinates were taken from ITRF2020-u2023, calculated on the date of the measurements. Measurements and SSR corrections (CNES – SSRA00CNE1) were from IGS-RTS. The mount points were chosen to increase the distance to the candidate/base station, specifically at distances of 79 km, 125 km, and 317 km. All measurements were processed in real-time, split into 4-hour intervals using GPST (GPS Time). After the estimation, the precision was calculated as the average root mean square error.

$ARMSE=\frac{1}{s}\sum\limits_{{i=1}}^{s} {\left( {\sqrt {\frac{1}{n}\sum\limits_{{j=1}}^{n} {{{({x_j} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{x} )}^2}} } } \right)}$

Here

is the number of four-hour measurement samples;

is the number of measurements in each sample;

$jth$

measurement;

$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{x}$

is the reference coordinate in ITRF2020-u2023 on a date of measurement.

Table 1
Pseudo-kinematic experiment with reference stations
BORJ-WSRT (79 km)
	External Convergence ARMSE, m			Internal Convergence ARMSE, m			Average FIX, %
	E	N	U	E	N	U	Average FIX, %
CP	0.1449	0.1323	0.3863	0.0823	0.0537	0.1249	78.06
PPP-AR	0.2672	0.2915	0.6983	0.1280	0.0924	0.1686	14.54
RTK	0.1558	0.1259	0.2679	0.0244	0.0187	0.0791	80.18
KARL-FFMJ (125 km)
	External Convergence ARMSE, m			Internal Convergence ARMSE, m			Average FIX, %
	E	N	U	E	N	U	Average FIX, %
CP	0.0771	0.1044	0.2002	0.0454	0.0289	0.0688	77.93
PPP-AR	0.2204	0.2136	0.8729	0.1437	0.1075	0.1582	14.74
RTK	0.1038	0.0761	0.1566	0.0378	0.0293	0.1156	82.73
WGTN-MQZG (317 km)
	External Convergence ARMSE, m			Internal Convergence ARMSE, m			Average FIX, %
	E	N	U	E	N	U	Average FIX, %
CP	0.1548	0.1088	0.1740	0.0459	0.0297	0.0721	74.04
PPP-AR	0.1683	0.2319	0.7240	0.1704	0.1134	0.1955	27.43
RTK	0.1103	0.1243	0.3018	0.0887	0.0476	0.1797	59.47

The results presented in Table 1 demonstrate the effectiveness of the CP method compared to the PPP-AR and RTK methods. CP shows a lower error (Average Root Mean Square Error, ARMSE) than PPP-AR for all components: East (E), North (N), and Up (U). For example, for the BORJ-WSRT station (79 km), the error in the vertical component (U) for CP is 0.3863 m, while for PPP-AR it is 0.6983 m. When considering only the horizontal components (E, N), CP achieves accuracy comparable to RTK but is inferior to RTK in the vertical component (U). However, this difference varies as the distance between the RTK base stations and the rover increases. CP achieves a high percentage of fixed solutions (over 70%), which is comparable to RTK and exceeds that of PPP-AR. These results demonstrate the reliability of CP in ambiguity resolution in real-time scenarios.

Kinematic tests with a car are currently a difficult task, as this implementation provides for an online collaborative positioning service that connects two or more participants. This service is currently under development, and previous tests were conducted and presented in the preprint (Dolin and Lipatnikov, 2024). The kinematic tests were conducted using a car equipped with a geodetic-class receiver installed on its roof. The tests were conducted while driving through the city of Novosibirsk. The route was designed so that the car passed under bridges during the measurement process. This setup aimed to demonstrate the effectiveness of collaborative positioning compared to the PPP method in scenarios where the signal is lost and it is necessary to restore the solution after passing an obstacle. To assess accuracy, a track was constructed using the relative method based on the fundamental astronomical and geodetic network point NSK1 during post-processing.

Discussion

The proposed Collaborative Positioning (CP) method offers a promising approach to high-precision GNSS navigation by combining the global coverage of PPP with the rapid initialization of RTK. Experimental results demonstrate that CP achieves centimetre-level accuracy (e.g., an ARMSE of 0.077–0.386 m in pseudo-kinematic tests) while maintaining a high ambiguity resolution success rate (> 70%), even at baseline distances exceeding 300 km. This performance bridges the gap between traditional PPP, which has slow convergence, and RTK, which has a limited range, providing a scalable solution for various real-time applications. In contrast to RTK and N-RTK, CP does not depend on static CORS networks. Instead, it leverages a dynamic user network where devices transition between "Rover," "Candidate," and "Base Station" roles. This method aligns with a trend suggesting a paradigm shift: future infrastructure may increasingly rely on consumer devices rather than professional stations. The proliferation of affordable dual-frequency GNSS receivers (e.g., smartphones like the Xiaomi Mi 8) and declining costs of geodetic equipment enable crowdsourced high-precision positioning. Moreover, this trend will lead to adaptive automation. CP’s self-organizing network architecture autonomously optimizes user roles and corrections. Users benefit from high accuracy without needing technical expertise, as the system dynamically adapts to the situation in real-time. The collaborative positioning service being developed will also incorporate coordinate transformation capabilities (Lipatnikov, 2025), linking dynamic and static coordinate systems and bringing the millimeter-level accuracy obtained in GNSS to practical applications.

However, technique has a limitation with network density dependency. CP performance depends with count of users. Sparse networks may experience slower convergence or reduced accuracy, necessitating hybrid approaches e.g., integrating CORS where available. Also, current CP implementation requires dual-frequency phase/code measurements, though ongoing modernization of GNSS chips may soon mitigate this constraint.

Funding

This work was supported by the Research Project GEOTECH-KVANT-3.

Author Contribution

S.D. conceived the study, designed the experiments, collected and analyzed the data, wrote the main manuscript text, and prepared all figures and tables. S.D. also reviewed and edited the manuscript.As the sole author of this manuscript, I, Sergei Dolin (S.D.), am responsible for all aspects of the research and manuscript preparation.

Acknowledgement

This work was sponsored by the Public law company "Roskadastr, Research Project GEOTECH-KVANT-3.

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Sergei Dolin

is currently a researcher at the Scientific Research Institute for Strategic Development and a senior lecturer in the department of Space and Physical Geodesy at the Siberian State University of Geosystems and Technologies. His research interests relate to the theories of high-precision GNSS and collaborative positioning with GNSS.

Yes