Optimal Estimation of a Subset of Integers with Application to GNSS

Brack, Andreas

doi:10.1515/arsa-2016-0011

Cited by 8 publications

(6 citation statements)

References 16 publications

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“…Alternatively, the framework of Generalized Integer Aperture (GIA) estimation, introduced by Brack in his series of works [15][16][17]40], extends the concepts on IAR and PAR to describe selective pull-in regions and their aperture. Thus, GIA estimators procure a joint subset selection, integer estimation, and test validation upon the aperture of the decision regions.…”

Section: Partial Ambiguity Resolution Strategiesmentioning

confidence: 99%

Precision-Aided Partial Ambiguity Resolution Scheme for Instantaneous RTK Positioning

et al. 2021

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The use of carrier phase data is the main driver for high-precision Global Navigation Satellite Systems (GNSS) positioning solutions, such as Real-Time Kinematic (RTK). However, carrier phase observations are ambiguous by an unknown number of cycles, and their use in RTK relies on the process of mapping real-valued ambiguities to integer ones, so-called Integer Ambiguity Resolution (IAR). The main goal of IAR is to enhance the position solution by virtue of its correlation with the estimated integer ambiguities. With the deployment of new GNSS constellations and frequencies, a large number of observations is available. While this is generally positive, positioning in medium and long baselines is challenging due to the atmospheric residuals. In this context, the process of solving the complete set of ambiguities, so-called Full Ambiguity Resolution (FAR), is limiting and may lead to a decreased availability of precise positioning. Alternatively, Partial Ambiguity Resolution (PAR) relaxes the condition of estimating the complete vector of ambiguities and, instead, finds a subset of them to maximize the availability. This article reviews the state-of-the-art PAR schemes, addresses the analytical performance of a PAR estimator following a generalization of the Cramér–Rao Bound (CRB) for the RTK problem, and introduces Precision-Driven PAR (PD-PAR). The latter constitutes a new PAR scheme which employs the formal precision of the (potentially fixed) positioning solution as selection criteria for the subset of ambiguities to fix. Numerical simulations are used to showcase the performance of conventional FAR and FAR approaches, and the proposed PD-PAR against the generalized CRB associated with PAR problems. Real-data experimental analysis for a medium baseline complements the synthetic scenario. The results demonstrate that (i) the generalization for the RTK CRB constitutes a valid lower bound to assess the asymptotic behavior of PAR estimators, and (ii) the proposed PD-PAR technique outperforms existing FAR and PAR solutions as a non-recursive estimator for medium and long baselines.

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Section: Partial Ambiguity Resolution Strategiesmentioning

confidence: 99%

Precision-Aided Partial Ambiguity Resolution Scheme for Instantaneous RTK Positioning

et al. 2021

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“…Depending on the index set I and the covariance matrix Qâ either of the two above introduced integer least-squares based estimators may be the better choice in terms of the probability of correct integer estimates. It is, however, possible to formulate the optimal estimator, which resolves a given subset I of the integer parameters a with the highest possible probability of correct integer estimates P (ǎ = a I ) within the class of partial integer estimators as defined in (4.2), see Brack (2016a). Let the integer estimatesǎ opt of a I be given by…”

Section: Optimal Partial Integer Estimationmentioning

confidence: 99%

Partial Carrier-Phase Integer Ambiguity Resolution for High Accuracy GNSS Positioning

Brack¹

2020

Preprint

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Global navigation satellite systems provide ranging based positioning and timing services. The use of the periodic carrier-phase signals is the key to fast and accurate solutions, given that the inherent ambiguities of the carrier-phase measurements are correctly resolved. The idea of partial ambiguity resolution is to resolve a subset of all ambiguities, which enables faster solutions but does not fully exploit the high precision of the carrier-phase measurements. Theory, methods, and algorithms for partial ambiguity resolution are discussed and analyzed with simulated and real data.

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“…For PAR, the index set I can assume any of the 2 n possibilities that result from either including each of the n ambiguities in I or not. The optimal estimator for a given subset I leading to the highest possible probability of correct estimates is discussed in [20] and given byž = argmax…”

Section: Reliable Model-driven Ambiguity Resolutionmentioning

confidence: 99%

Long Baseline GPS+BDS RTK Positioning with Partial Ambiguity Resolution

Brack¹

2017

Proceedings of the 2017 International Technical Meeting of the Institute of Navigation

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Reliable real-time kinematic positioning requires the occurrence of incorrect integer ambiguity estimates to be limited to a maximum tolerable rate. For long baselines this usually implies long convergence times due to the presence of atmospheric delay parameters. Several simulation studies have shown that partial ambiguity resolution (PAR) techniques are beneficial in terms of faster solutions, since it is more likely that a subset of all ambiguities can be reliably resolved rather than the full set, but they also result in a positioning precision that is inferior to the one after successful full ambiguity resolution. We analyze the impact of PAR on the positioning capabilities of dual-frequency single and combined GPS and BDS on a long baseline. It will be demonstrated in numerical simulations that the time to reach centimeter level positioning results can be expected to be clearly reduced when using PAR techniques, in particular for the combined system. This will be verified with one day of real global navigation satellite system data from an 88.5 km baseline in the area of Perth, Australia.

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Optimal Estimation of a Subset of Integers with Application to GNSS

Cited by 8 publications

References 16 publications

Precision-Aided Partial Ambiguity Resolution Scheme for Instantaneous RTK Positioning

Precision-Aided Partial Ambiguity Resolution Scheme for Instantaneous RTK Positioning

Partial Carrier-Phase Integer Ambiguity Resolution for High Accuracy GNSS Positioning

Long Baseline GPS+BDS RTK Positioning with Partial Ambiguity Resolution

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