A Theoretical Framework for Collaborative Estimation of Distances Among GNSS users
Alex Minetto, Politecnico di Torino, Italy, Prof. Fabio Dovis, Politecnico di Torino, Italy
Background and Introduction
Techniques for estimation of the baseline between two GNSS receivers has been investigated in the framework of relative positioning. They can exploit a large variety of passive exteroceptive sensors (i.e. LIDAR, UWB) or active exchange of navigation data transmitted over communication channels (i.e. VANET, V2X, V2B).
In previous works , , , the effectiveness of collaboration between ground communicating receivers for relative distance estimation has been discussed by means of data exchanged within different user-to-user configurations.
Such papers introduced and analyzed two main techniques for the baseline estimation: Absolute Position Difference (APD) and Pseudorange Difference (PRD). The concepts typically exploit the baseline measurement facing the problem of relative positioning but neglecting any potential integration focused to the absolute position problem. As mentioned in the work of Alam et al. , absolute cooperative positioning cannot benefit from data derived from Global Satellite Navigation System (GNSS) (i.e. position solutions) due to the redundancy of errors.
Such a piece of information can be useful not only in the framework of differential positioning but it could be exploited as a mean to improve the estimation of the absolute position of the user. The baseline integration in navigation algorithms (i.e. Inter-Agent Range) has indeed been discussed in  and  with promising results.
Collaborating agents are expected to continuously measure the pseudoranges from all the satellites in tracking. Given a reasonable proximity assumption, a subset of trackable satellites is visible to both the receivers and such an assumption can be extended to all the pairs of a set of receivers (namely multi-agent environment).
The inter-agent ranging allows to compute the length of the baseline between two interconnected receivers which share their raw pseudorange measurements and their steering vectors towards a single selected shared satellite. It exploits the triangular geometry described by the agents and the chosen satellite to solve the measure of the baseline as the unknown of Carnot Theorem (law of cosines). Furthermore, the algorithm has been designed to guarantee the privacy of collaborating agents avoiding reciprocal localization. It is supposed to be partially related to GNSS data about absolute position, but it integrates new ranging measurements from satellites combining them effectively.
The preliminary results obtained showed that the integration of this additional ranging measurements can help navigation algorithms in harsh environment supporting the continuity of positioning. Furthermore, preliminary experimental tests by using real receivers also show benefits in terms of precision of the PVT solution.
In this work, the impact of geometry of the cooperating systems is inspected for the estimation of the baseline between two GNSS receivers exploiting Inter-Agent Range (IAR) estimation algorithm .
The need of an exhaustive theoretical framework for IAR requires the definitions of proper error propagation models. The paper will introduce the concept of co-elevation as the combination of the elevation of the satellites as seen by each of the users, and a minimum error region for the identification of the best satellite to select for the IAR calculation will be properly justified.
As far as the mathematical model is concerned, the optimal combination of measurements leading to a minimum error in the estimated IAR will be discussed for different geometries of the available constellations. Furthermore, the theoretical results will be validated by means of a test campaign exploiting raw pseudorange measurements provided by mass market GNSS receivers.
The aim of this paper is the definition of a mathematical framework for the IAR algorithm which assesses the feasibility of this approach as a valid alternative to the different techniques discussed so far.
In this work it is investigated how the ranging errors on the pseudoranges propagate to the estimation of the baseline measure obtained by means of IAR algorithm. The mathematical framework of the problem is presented by means of an analytic error function which allows to calculate the resulting error considering the limiting condition of the geometry. The implementation of the IAR algorithm within a simulation toolbox will be proposed to validate the theoretical achievements exploiting raw GNSS data obtained from open-access receivers.
Geometrical constraints are very well-known issues in navigation systems. The relative positions of navigating agents and landmarks (i.e. satellites) affect the precision of the positioning solution. Likely, the integration of alternative measurements (i.e. relative distance between navigating items) in GNSS context requires the definitions of optimal geometrical conditions. The theoretical analysis presented here shows that properly chosen geometries can mitigate the resulting error induced by uncertainties on raw measurements (i.e. pseudoranges, carrier phase ranging) involved in the collaborative computation.
Given the main formula for the IAR, three sources of uncertainties are identified:
• Steering vectorsuncertainties which are the outcome of PVT loop relying on approximated position (for the linearized solution) or propagated estimations (in more advanced predictive routines).
• Pseudoranges errors of the collaborating receivers. They are typically modelled as a composition of independent terms (e.g. Ionospheric error, Tropospheric error, etc.). In this study the two error are assumed independent.
• Numerical errors due to finite precision computation. In simulation environment, additional uncertainties (i.e. cancellation errors, roundoff accumulation) are introduced, especially when very small values are considered (i.e. included angle between steering vectors).
Numerical errors are characterized by restrained discrete oscillations around the true value of IAR such that their contribution can be neglected. The steering uncertainties produce an error on the included angle which modifies the apparent geometry of the system. This distortion is undistinguishable from the one introduced by pseudorange errors so that they can be jointly modeled as a single error term. The distribution of pseudorange errors is hence adapted to include steering errors.
The error function described in this paper is given as exact formula and as specific approximation for limit angles which are typical of a GNSS scenario. The raw data (i.e. pseudoranges and steering vectors) coming from the collaborating agents are considered as the parameters for the evaluation of such an error function.
The geometrical aspects are investigated in a rotated local reference frame (Earth, North, Up) of one of the agents (identified as aided agents which starts the algorithm), whose local plane is aligned to the baseline in analysis. This solution allows to explore all the overall sample space of possible satellite relative position. In this frame, the co-elevation coincides with the elevation of the satellite with respect to the target agent. By building a relative local reference frame it is possible to identify the relative optimal sky regions that allow to minimize the estimation error.
Relevant and anticipated results
Given the high ratio between the distance of the satellites and the module of baseline vector of interest, the angle included between the steering vectors is small enough to reduce the analysis to a limit function which only depends on the single difference between pseudoranges and difference of their estimated uncertainties.
The best co-elevation which minimizes the error projections on the baseline can be identified for all the possible configurations. A few limit cases are inspected to justify the behavior of the error function. A specific approximation in case of null included angle holds for GNSS applications inducing negligible uncertainties, but it is not suitable for low altitude landmarks (i.e. pseudolites, UAV reference stations).
The analytic form of the error propagation is hence defined, thus providing a low complexity evaluation of the reliability of IAR measurements. It is expected to validate the distribution of optimal satellites locations through the analysis of simulated cooperation between real receivers.
The work theoretically demonstrates that designing the overall system on purpose or identifying the maximum co-elevated satellite, it is always possible to minimize the error for a given constellation in visibility. This provides a support to the preliminary experimental results presented in  and a step forward to the evaluation of the further error propagation models to Hybrid PVT algorithms.
The paper provides a criterion for the identification of the preferable shared satellite in real application. The effectiveness is demonstrated presenting results that integrate the estimated relative distance into a hybrid navigation module.
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