Instantaneous Best Integer Equivariant Position Estimation Using Google Pixel 4 Smartphones for Single- and Dual-Frequency, Multi-GNSS Short-Baseline RTK

Abstract

High-precision global navigation satellite system (GNSS) positioning and navigation can be achieved with carrier-phase ambiguity resolution when the integer least squares (ILS) success rate (SR) is high. The users typically prefer the float solution under the scenario of having a low SR, and the ILS solution when the SR is high. The best integer equivariant (BIE) estimator is an alternative solution since it minimizes the mean squared errors (MSEs); hence, it will always be superior to both its float and ILS counterparts. There has been a recent development of GNSSs consisting of the Global Positioning System (GPS), Galileo, Quasi-Zenith Satellite System (QZSS), and the BeiDou Navigation Satellite System (BDS), which has made precise positioning with Android smartphones possible. Since smartphone tracking of GNSS signals is generally of poorer quality than with geodetic grade receivers and antennas, the ILS SR is typically less than one, resulting in the BIE estimator being the preferred carrier phase ambiguity resolution option. Therefore, in this contribution, we compare, for the first time, the BIE estimator to the ILS and float contenders while using GNSS data collected by Google Pixel 4 (GP4) smartphones for short-baseline real-time kinematic (RTK) positioning. It is demonstrated that the BIE estimator will always give a better RTK positioning performance than that of the ILS and float solutions while using both single- and dual-frequency smartphone GNSS observations. Lastly, with the same smartphone data, we show that BIE will always be superior to the float and ILS solutions in terms of the MSEs, regardless of whether the SR is at high, medium, or low levels.

An Improved Ambiguity Resolution Algorithm for Smartphone RTK Positioning

“…A unimodal transformation and a search-and-shrink scheme based on the integer leastsquares (ILS) principle find the optimal ambiguity integers with a real-time computational load [27]. Until recently, the method of best integer equivariance (BIE) has proven to be a better replacement [28], where the AR performance is optimized in the sense of minimizing the mean square error [29][30][31]. However, these methods, including BIE, naturally consider the input float ambiguity estimations to be unbiased integers.…”

“…This does not fit in the application of smartphone AR since the existence of IPB, carrier phase multipath effects, and antenna offsets lead to non-negligible ambiguity biases. As a result, the unimodal transformation and the search-and-shrink scheme are inaccurate and likely to produce a set of incorrect ambiguity integers [31]. Although PAR coupled with improved ambiguity validation strategies, such as protection-level, are proposed [32,33], they are generally not sufficiently efficient to identify the correct ambiguity integer set for smartphones due to the volume of such ambiguity biases and the significant measurement noises.…”

GNSS Observation Generation from Smartphone Android Location API: Performance of Existing Apps, Issues and Improvement

“…The results indicated that the L5/E5a/B2a signals could generally obtain higher IAR fix-rate and positioning accuracies than the L1/E1/B1 signals. Yong et al [ 35 ] compared the best integer equivariant (BIE) estimator to the integer least squares (ILS) and float contenders using GNSS data collected by Google Pixel 4 smartphones for short-baseline RTK positioning. The results indicated that the BIE estimator will always give a better RTK positioning performance than that of the ILS and float solutions while using both single- and dual-frequency smartphone measurements for the combination of GPS + Galileo + QZSS + BDS.…”