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Session C1a: Applications of GNSS Measurements from Smartphones 

Best Integer Equivariant (BIE) Estimator for Smartphone RTK in Urban Canyon Environments
Ananya Vishwanath, Rhea Joyce Zambra, Department of Geomatics Engineering, University of Calgary; Robert Odolinski, University of Otago; Mohammed Khider, Google Inc.; Hongzhou Yang, Department of Geomatics Engineering, University of Calgary
Date/Time: Wednesday, Sep. 18, 8:57 a.m.

Recent developments in the GNSS capabilities of Android smartphones improve the feasibility of achieving centimeter-level accuracy using real-time kinematic (RTK) positioning. This includes the availability of dual-frequency multi-constellation pseudorange and carrier phase measurements. However, smartphone tracking of GNSS signals is generally of an inferior quality when compared with high-end receivers, primarily due to the reliance on low-cost global navigation satellite system (GNSS) chipsets and omnidirectional antennas. These antennas are more susceptible to interference and result in low C/N0 ratios, especially in urban environments. Additionally, signals in such environments suffer from constant cycle slips, larger multipath errors, and noise due to the frequent blocking of line-of-sight (LOS) signals. While conventional methods that use carrier phase measurements rely on ambiguity resolution for achieving centimeter-level position accuracy, the correct fixed solution is contingent on achieving a near 100% success rate, i.e. the probability of correct integer estimation [1]. The least-squares ambiguity decorrelation adjustment (LAMBDA) method [2] is typically chosen to solve the integer least-squares (ILS) problem for ambiguity resolution. On the other hand, if the success rate is too low, usually the float solution is preferred over the ambiguity fixed solution.
Several studies have investigated the quality of smartphone observations in open-sky conditions. In 2014, Pesyna et al. [3] showed that attaching an auxiliary external antenna to a smartphone could reduce multipath sensitivity observed in their internal antennas. In 2018, Zhang et al. [4] demonstrated that smartphones exhibited a C/N0 ratio 10 dB-Hz lower than that of geodetic receivers, and elevation-dependent weighting failed to compensate for the variation in smartphone signal-to-noise ratio (SNR). Heßelbarth et al. [5] explored antenna calibration to achieve centimeter-level positioning accuracy while Paziewski et al. [6] showed the presence of contaminating factors in phase residuals, that inhibited integer resolution and failed to maintain the time constant properties of the ambiguities.
Exploring integer ambiguity resolution for RTK smartphone positioning, Paziewski et al. [6] also achieved centimeter-level accuracy in static conditions using a short baseline phone-to-phone RTK setup. They utilized GPS L1 observations and the modified LAMBDA (MLAMBDA) method for ambiguity resolution. For a similar setup, Yong et. al [7] demonstrated precise instantaneous smartphone positioning with a near 100% ILS success rate using external and internal antennas. They utilised multi-GNSS (GPS, Galileo, BDS and QZSS) dual-frequency observations from smartphones. Recently, Odolinski et al. [8] achieved centimeter-level accuracy using internal antennas in newer Android smartphones that are equipped with dual-frequency multi-constellation capabilities. The approach utilized ILS for single-epoch and multi-epoch models. Tao et al. [9] proposed a new quality control model in smartphone RTK, which achieved sub-meter level accuracy in open sky kinematic test and meter level in suburban and urban areas. This was due to the frequent occurrence of cycle slips which led to repeated ambiguity re-initializations, rendering the ambiguity-fixed solutions unreliable.
From the previous research, it is evident that while centimeter-level RTK positioning is possible with smartphones, it heavily relies on correct fixing of integer ambiguities. This poses a significant challenge, particularly in urban environments, where it is difficult to maintain a signal lock for a duration sufficient enough to achieve a correct fix with a near 100% success rate. This paper explores the potential of utilizing the best integer equivariant (BIE) estimator as an alternative to the float or ILS counterparts. Introduced by Teunissen [10], BIE assumes the data to be normally distributed, and performs an infinite weighted sum across the entire integer space. In practice, it focuses on the ellipsoidal region around the float solution as the searchable integer space, with the radius determined by the corresponding covariance matrix. The weights range from zero to one, and their values are determined by the vector of GNSS observations and its probability density function (PDF) [10]. For lower success rates, BIE’s behaviour is similar to the float solution, and similar to the ILS solution for close to 100% success rates. In scenarios where the GNSS model strength improves but success rates remain below 100%, BIE-based solutions result in smaller positioning errors compared to both float and fixed solutions [1]. Unlike validation tests conducted by ILS to maximize success rates, BIE employs mean squared error (MSE) as a performance criterion. The unbiased nature of BIE as shown by Teunissen [10] makes MSE a valid performance criterion, providing a reliable measure of the estimator’s closeness to the target value. Odolinski et al. [11] demonstrated the potential of BIE for low-cost receivers by achieving superior positioning results compared to its float and fixed counterparts, particularly for low to medium success rates. The static measurements were collected in open sky conditions, capturing dual-frequency and multi-constellation data for both short and long baselines using low-cost receivers. Yong et al. [12] employed a similar methodology to assess BIE for instantaneous positioning with Pixel 4 smartphones over short baselines. The experiments utilized both external and internal antennas, and the positioning accuracy of BIE outperformed the float and fixed solutions. Zhang et al. [13] attained sub-meter level accuracy using BIE for low-cost receivers in kinematic conditions within an urban environment.
While BIE has demonstrated potential for static smartphone measurements in open sky conditions, similar setups have not been tested in urban environments. Therefore, this paper aims to evaluate the effectiveness of BIE for static and kinematic phone-to-phone measurements in urban canyons. Additionally, while evaluation of smartphone observations has been performed under static, open-sky conditions, these measurements have not been thoroughly assessed in urban environments. Thus, the quality of raw smartphone measurements collected in an urban setup is also evaluated in this paper.
To effectively address the goals outlined in this research, several action items have been planned. These include designing smartphone experiments, evaluating smartphone GNSS observation quality in urban environments, and implementing the BIE method in the RTK module for comparison with ILS. More details on these tasks are discussed as follows.
1. Smartphone Experiments Design
Two Google Pixel 7 Pro smartphones equipped with dual-frequency, multi-constellation capabilities will be utilized in the phone-to-phone setup. Smartphone measurements will be collected using both internal and external antennas across varying urban environments in Calgary, including suburban, urban, and deep urban areas. Configurations will involve different smartphone orientations [7] and varying levels of signal obstruction (particularly concerning antennas positioned at the back of smartphones). The experiments will utilize these diverse setup configurations to conduct static and kinematic measurements with short and long baselines in urban environments. A geodetic base station on the roof of the University of Calgary campus will be used, while a Novatel Pwrpak7 SPAN OEM7 system will serve as ground truth.
2. Smartphone GNSS Observation Quality Assessment in Urban Canyon
The collected data in urban canyons will be used to develop a robust smartphone GNSS data evaluation module. The evaluation will include an analysis of signal strength, double-differenced pseudorange and carrier phase residuals, multipath, as well as satellite elevation and geometry, across various constellations and frequencies. The distribution of observations will also be evaluated for its tail characteristics [14]. Moreover, the classification of smartphone LOS, non-LOS (NLOS) and multipath signals will be investigated. The study will explore the possibility of utilizing these metrics as features in a machine learning model to assign weights to observations based on their quality.
3. BIE Implementation in the RTK Module
The BIE estimator will be implemented, and its performance will be compared to the float solution, as well integer fixed solution using ILS. Based on the tail distributions observed from the quality assessment, the BIE-weighting for the estimator will be studied [15]. This will be done for the proposed phone-to-phone setup, for both single and multi-epoch models. Single and dual frequency modes, as well as single and multi-constellation scenarios will be investigated. Variations in the elevation angle and C/N0 mask settings will also be explored during analysis. The evaluation criteria for comparing the performance of BIE and ILS will focus on several key factors: positioning accuracy, standard deviation in East, North and Up directions, time to first fix (TTFF), and success rate for accurate position estimation across different datasets. Considering the hardware limitations smartphones impose, the computational complexity of the approaches for each experiment will also be examined.
Overall, this paper aims to assess the effectiveness of the BIE estimator in phone-to-phone-based RTK positioning in urban canyons, particularly in comparison with conventional float and integer fixed solutions. Additionally, the paper endeavours to perform a quality assessment of raw smartphone measurements in both static and kinematic modes in urban canyon environments. This includes investigating the classification of smartphone GNSS signals at reception. Various experiments will be conducted to analyze the quality of smartphone data under these conditions, with a strict evaluation criterion applied to assess the performance of BIE in comparison with float and ILS-fixed solutions.
References
[1] R. Odolinski and P. J. G. Teunissen, “Best integer equivariant estimation: performance analysis using real data collected by low-cost, single- and dual-frequency, multi-GNSS receivers for short- to long-baseline RTK positioning,” J Geod, vol. 94, no. 9, p. 91, Sep. 2020, doi: 10.1007/s00190-020-01423-2.
[2] T. P. J. G, “The Least-Square Ambiguity Decorrelation Adjustment?: A Method for Fast GPS Integer Ambiguity Estimation,” J. Geodesy, vol. 70, no. 1, pp. 65–82, 1995.
[3] K. M. Pesyna, R. W. Heath, and T. E. Humphreys, “Centimeter Positioning with a Smartphone-Quality GNSS Antenna”.
[4] X. Zhang, X. Tao, F. Zhu, X. Shi, and F. Wang, “Quality assessment of GNSS observations from an Android N smartphone and positioning performance analysis using time-differenced filtering approach,” GPS Solut, vol. 22, no. 3, p. 70, May 2018, doi: 10.1007/s10291-018-0736-8.
[5] A. Heßelbarth and L. Wanninger, “Towards centimeter accurate positioning with smartphones,” in 2020 European Navigation Conference (ENC), Nov. 2020, pp. 1–8. doi: 10.23919/ENC48637.2020.9317392.
[6] J. Paziewski, M. Fortunato, A. Mazzoni, and R. Odolinski, “An analysis of multi-GNSS observations tracked by recent Android smartphones and smartphone-only relative positioning results,” Measurement, vol. 175, p. 109162, Apr. 2021, doi: 10.1016/j.measurement.2021.109162.
[7] C. Z. Yong et al., “Instantaneous, Dual-Frequency, Multi-GNSS Precise RTK Positioning Using Google Pixel 4 and Samsung Galaxy S20 Smartphones for Zero and Short Baselines,” Sensors, vol. 21, no. 24, Art. no. 24, Jan. 2021, doi: 10.3390/s21248318.
[8] R. Odolinski, H. Yang, L.-T. Hsu, M. Khider, G. (Michael) Fu, and D. Dusha, “Evaluation of the Multi-GNSS, Dual-Frequency RTK Positioning Performance for Recent Android Smartphone Models in a Phone-to-Phone Setup,” presented at the 2024 International Technical Meeting of The Institute of Navigation, Long Beach, California, Feb. 2024, pp. 42–53. doi: 10.33012/2024.19575.
[9] X. Tao, W. Liu, Y. Wang, L. Li, F. Zhu, and X. Zhang, “Smartphone RTK positioning with multi-frequency and multi-constellation raw observations: GPS L1/L5, Galileo E1/E5a, BDS B1I/B1C/B2a,” J Geod, vol. 97, no. 5, p. 43, May 2023, doi: 10.1007/s00190-023-01731-3.
[10] P. Teunissen, “On the Computation of the Best Integer Equivariant Estimator,” Artificial Satellites, vol. 40, pp. 161–171, Jan. 2005.
[11] R. Odolinski and P. J. G. Teunissen, “On the Best Integer Equivariant Estimator for Low-cost Single-frequency Multi-GNSS RTK Positioning,” presented at the 2020 International Technical Meeting of The Institute of Navigation, San Diego, California, Feb. 2020, pp. 499–508. doi: 10.33012/2020.17158.
[12] C. Z. Yong, K. Harima, E. Rubinov, S. McClusky, and R. Odolinski, “Instantaneous Best Integer Equivariant Position Estimation Using Google Pixel 4 Smartphones for Single- and Dual-Frequency, Multi-GNSS Short-Baseline RTK,” Sensors, vol. 22, no. 10, Art. no. 10, Jan. 2022, doi: 10.3390/s22103772.
[13] Z. Zhang, H. Yuan, X. He, B. Li, and J. Geng, “Best Integer Equivariant Estimation With Quality Control in GNSS RTK for Canyon Environments,” IEEE Transactions on Aerospace and Electronic Systems, vol. 59, no. 4, pp. 4105–4117, Aug. 2023, doi: 10.1109/TAES.2023.3236916.
[14] L. Heng, G. X. Gao, T. Walter, and P. Enge, “Statistical Characterization of GPS Signal-In-Space Errors”.
[15] R. Odolinski and P. J. G. Teunissen, “Best integer equivariant position estimation for multi-GNSS RTK: a multivariate normal and t-distributed performance comparison,” J Geod, vol. 96, no. 1, p. 3, Dec. 2021, doi: 10.1007/s00190-021-01591-9.



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