Instantaneous, Dual-Frequency, Multi-GNSS Precise RTK Positioning Using Google Pixel 4 and Samsung Galaxy S20 Smartphones for Zero and Short Baselines

The recent development of the smartphone Global Navigation Satellite System (GNSS) chipsets, such as Broadcom BCM47755 and Qualcomm Snapdragon 855 embedded, makes instantaneous and cm level real-time kinematic (RTK) positioning possible with Android-based smartphones. In this contribution we investigate the instantaneous single-baseline RTK performance of Samsung Galaxy S20 and Google Pixel 4 (GP4) smartphones with such chipsets, while making use of dual-frequency L1 + L5 Global Positioning System (GPS), E1 + E5a Galileo, L1 + L5 Quasi-Zenith Satellite System (QZSS) and B1 BeiDou Navigation Satellite System (BDS) code and phase observations in Dunedin, New Zealand. The effects of locating the smartphones in an upright and lying down position were evaluated, and we show that the choice of smartphone configuration can affect the positioning performance even in a zero-baseline setup. In particular, we found non-zero mean and linear trends in the double-differenced carrier-phase residuals for one of the smartphone models when lying down, which become absent when in an upright position. This implies that the two assessed smartphones have different antenna gain pattern and antenna sensitivity to interferences. Finally, we demonstrate, for the first time, a near hundred percent (98.7% to 99.9%) instantaneous RTK integer least-squares success rate for one of the smartphone models and cm level positioning precision while using short-baseline experiments with internal and external antennas, respectively.

A mixed-integer least-squares formulation of the GNSS snapshot positioning problem

This paper presents a formulation of snapshot positioning as a mixed-integer least-squares problem. In snapshot positioning, one estimates a position from code-phase (and possibly Doppler-shift) observations of global navigation satellite system (GNSS) signals without knowing the time of departure (timestamp) of the codes. Solving the problem allows a receiver to determine a fix from short radio-frequency snapshots missing the timestamp information embedded in the GNSS data stream. This is used to reduce the time to first fix in some receivers, and it is used in certain wildlife trackers. This paper presents two new formulations of the problem and an algorithm that solves the resulting mixed-integer least-squares problems. We also show that the new formulations can produce fixes even with huge initial errors, much larger than permitted in Van Diggelen's widely-cited coarse-time navigation method.

Theory of best integer equivariant estimation for contaminated normal and multivariate t-distribution with applications

<p>Best integer equivariant (BIE) estimators provide minimum mean squared error (MMSE) solutions to the problem of GNSS carrier-phase ambiguity resolution for a wide range of distributions. The associated BIE estimators are universally optimal in the sense that they have an accuracy which is never poorer than that of any integer estimator and any linear unbiased estimator. Their accuracy is therefore always better or the same as that of Integer Least-Squares (ILS) estimators and Best Linear Unbiased Estimators (BLUEs).</p><p>Current theory is based on using BIE for the multivariate normal distribution. In this contribution this will be generalized to the contaminated normal distribution and the multivariate t-distribution, both of which have heavier tails than the normal. Their computational formulae are presented and discussed in relation to that of the normal distribution. In addition a GNSS real-data based analysis is carried out to demonstrate the universal MMSE properties of the BIE estimators for GNSS-baselines and associated parameters.</p><p> </p><p><strong>Keywords: </strong>Integer equivariant (IE) estimation · Best integer equivariant (BIE) · Integer Least-Squares (ILS) . Best linear unbiased estimation (BLUE) · Multivariate contaminated normal · Multivariate t-distribution . Global Navigation Satellite Systems (GNSSs)</p>

A General Solution to the Rank Deficient Integer Least Squares and its Application to GNSS Positioning

<p>The contribution will present a general solution for the estimation of rank deficient integer parameters. A procedure will be presented that allows the computation of integer estimable function for any integer rank deficient least squares problem. The procedure is then applied to GNSS estimation problems. In the framework of undifferenced and uncombined GNSS models, the specific solution to some rank deficient integer least squares model will be presented, namely: the choice of pivot ambiguities in a network of receivers, GLONASS positioning, codeless positioning in the presence of ionospheric delay, satellite specific pseudorange biases estimation in the presence of ionospheric delay. It will been shown how the developed theory generalize previous results and ad hoc solutions present in the literature. Numerical results from real GNSS data will be presented too.</p>

A Minimum Size of the Search Cube in the MAFA-ILS Method

The Modified Ambiguity Function Approach – Integer Least Squares (MAFA-ILS) is one of the modern method for the precise real-time GNSS positioning, for many applications such as geodesy or surveying engineering. Contrary to the classic approach, in the MAFA-ILS method the solution is sought in the coordinate domain. However, to obtain a proper solution, the search region cannot be too small. On the other hand, to have an effective solution – in the computational load sense – this region cannot be too big. In this case the determination of the minimum size of appropriate search region is not a trivial task. The paper presents a certain solution of this problem.

Optimal Estimation of a Subset of Integers with Application to GNSS

The problem of integer or mixed integer/real valued parameter estimation in linear models is considered. It is a well-known result that for zero-mean additive Gaussian measurement noise the integer least-squares estimator is optimal in the sense of maximizing the probability of correctly estimating the full vector of integer parameters. In applications such as global navigation satellite system ambiguity resolution, it can be beneficial to resolve only a subset of all integer parameters. We derive the estimator that leads to the highest possible success rate for a given integer subset and compare its performance to suboptimal integer mappings via numerical studies. Implementation aspects of the optimal estimator as well as subset selection criteria are discussed.