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### Session E4a: Accurate GNSS Navigation in Challenging Environments

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**Application of Adaptive Kalman Filtering on Smartphone Positioning**

*Naman Agarwal, Kyle O’Keefe, Department of Geomatics Engineering, University of Calgary*

**Date/Time:** Thursday, Sep. 19, 2:58 p.m.

Peer Reviewed

Application of Adaptive Kalman Filtering on Smartphone Positioning

ABSTRACT

An Adaptive Kalman filter (AKF) is used to estimate smartphone GNSS pseudorange measurement variance. The filter is applied to a dataset from the 2021 Google Smartphone Decimeter challenge. The adaptive filter is compared to three other processing strategies: conventional weighted least-squares, a random-walk velocity Kalman filter, and an alternative KF implementation that uses Doppler to adapt process noise, all of which use a standard elevation and C/N_0 measurement variance model. The adaptively estimated measurement variance is compared to the true error variance and all four methods are evaluated in the position domain. The full paper will also include an assessment of fault detection and exclusion in all four cases.

INTRODUCTION

Precise smartphone positioning has emerged as a leading GNSS research area due to the smartphone’s widespread availability and potential mass market applications. While accurate measurement variance modelling is crucial for precise positioning, smartphone GNSS measurements present challenges in modelling due to factors such as low-cost cellphone-grade chipsets, poor antenna design, environmental effects, holding modes, and other variables. Least-squares estimation and conventional Kalman Filters require precise apriori knowledge of the uncertainty of measurements (R), to provide optimal performance. The standard way of determining R relies on intensive analysis of empirical data to generate models, usually based on satellite elevation angle and/or C/N_0. These models work well for receivers that operate in a relatively clean observation environment. When the environment changes, adaptation is required. Adaptive Kalman Filtering has largely until now has been implemented for professional-grade GNSS receivers. A few of the popular methods are:

1. Multiple Model-based Adaptive Estimation (MMAE) [1], [2]

2. A class of AKF based on Mehra’s variance component matrix estimation (VCME) method [3], [4], [5], [6], [7]. This VCME method is also called “Covariance Matching” or “Adaptive Sage Windowing Filter”.

3. AKF based on Helmert’s Variance Component Estimation (VCE) principle [8], [9], [10], [11].

4. Yang’s adaptive robust Kalman filter [12], [13], [14], [15].

Very few prior works have investigated the application of AKF to smartphone data [16], [17].

This work will investigate the applicability of AKF for smartphone positioning where the raw GNSS data has a larger and more variable variance compared to professional-grade receivers. The AKF method will be integrated with a Doppler-based prediction (DBP) technique [18], [19], which performs accurate process noise Q estimation, and therefore will allow us to solely focus on adapting the R matrix. We will show that the adapted R matches the real measurement noise covariance, and the proposed AKF’s position solution is superior to the other estimators. The benefits of using the AKF to reweight observations over removing them with Fault, Detection and Exclusion (FDE) will also be analyzed for the Smartphone GNSS case.

METHODOLOGY

The Adaptive Kalman Filter (AKF) implemented in this work is based on a modified VCE method derived in [8], [9], [10]. It integrates the Doppler-based prediction (DBP) technique [19], which performs state prediction using the Doppler-derived velocity, this has the advantage of continuously generating an accurate Q matrix that represents the uncertainty of the system’s dynamics. Since only one noise covariance matrix, Q or R can be accurately adapted at a time in the AKF and the success in correctly adapting either is highly dependent on the accuracy of presumed values of the other, the focus can be provided solely on adapting the R matrix. The rationale for using the VCE method for adaptive filtering is that it is capable of computing individual variance components and since in the proposed AKF only R is going to be adapted, the variance of each measurement can be estimated. Estimating the individual variance of each measurement offers a significant advantage, unlike other AKF methods, which necessitate tracking satellites entering and leaving at each epoch to maintain the order of the innovation/residual vector, the VCE method eliminates this requirement. The implemented VCE method uses multiple epochs of data in a sliding window to estimate and adapt the variance.

Another reasonable assumption made to enhance the VCE technique is that satellites belonging to the same constellation system will exhibit identical measurement variance if their elevation angle and carrier-to-noise ratio (C/N_0) values fall within a specified range. Satellites were grouped into bins of 5 degrees elevation angle and 5 dB C/N_0, and collectively used to compute the adaptive variance. This, along with the sliding window technique, facilitates a more robust and accurate estimation. The flowchart of the filter is shown below in Figure 1.

Figure 1: Flowchart of the proposed AKF

RESULTS

To investigate the performance of the proposed AKF and compare it with existing conventional filters, the algorithms were tested using the Google Smartphone Decimeter Challenge 2021 dataset [20]. The data was collected using a Samsung Galaxy S20 Ultra handset, which was mounted in a vehicle and driven around the San Francisco Bay Area on 29 April 2021. The dataset label is “2021-04-29-US-SJC-2.”

In conventional Kalman filtering, the measurement noise statistics (R) are typically determined through test analysis and prior knowledge of the observation type. This empirical technique often necessitates post-processing of the data, particularly when the nature of the environment and receiver quality is unknown. Since measurement variance heavily relies on the satellite’s elevation angle and C/N_0 ratio, multiple variance mapping functions have also been proposed. The variance function that has been utilized in this study is the following,

sigma^2=10^((-C/N_0)/10)/(sin^2 (theta)) ,where theta is elevation angle

Such variance function still requires to be scaled appropriately to fit the real statistic noise level of the dataset, and the scale factor is decided either based on prior knowledge or post-processing. Figure 2 exhibits the 3D plot of the standard deviation (SD) obtained from the function. It is evident from the figure that, satellites or bins associated with low elevation angle and C/N_0 ratio have the higher SD and vice versa.

Figure 2: Standard Deviation function based on Elevation Angle and C/N ratio

To validate the accuracy or fidelity of the measurement covariance (R) estimated using the proposed AKF, the true covariance of measurement noise was approximated using the ground truth file. The plots of adaptive SD and true SD are shown in Figures 3 and 4. Comparing the SD function shown in Figure 2 with the true SD plot, discrepancies are evident, particularly in bins with high elevation angles and C/N ratio, where high SD values are observed. Utilizing the SD mapping function under these conditions would lead to inaccurate R values and consequently result in a suboptimal position solution. Conversely, when comparing the VCE-based adaptive SD with the true SD, their plots exhibit similarities, indicating consistent variance scale and trend across a broad range.

Figure 3: Standard deviation of the measurement noise obtained after performing VCE-based adaption

Figure 4: Approximated true standard deviation of the measurement noise

The proposed AKF leads to improved positioning results detailed in Table 1 and Figures 5 and 6. The proposed AKF was compared to: (1) WLS: Weighted least squares, (2) VRWD: Conventional KF with velocity as random walk model along with Doppler-derived velocity in the measurement update, the state vector contains both position and velocity parameter and (3) DBP Filter: KF based on Doppler-based prediction technique [19], which performs state prediction using the Doppler-derived velocity. The position errors are significantly smaller for the proposed AKF compared to any other filter. All the filters apart from AKF use the variance mapping function shown in Figure 2.

Figure 5: ENU positioning results

Figure 6: Horizontal positioning results (left) and horizontal position trajectory (right)

SIGNIFICANCE AND FUTURE WORK

This work presented an Adaptive Kalman Filter for Smartphone GNSS positioning. Smartphone GNSS measurements are highly noisy and in a kinematic scenario, the ever-changing environment makes it quite difficult to model the R matrix in real-time such that it emulates the real noise statistics of the pseudorange measurements. Conventional filters have been observed to perform poorly in the real-time scenario for kinematic smartphone GNSS, as they either assign the values for the R matrix based on empirical analysis of the measurement errors or manually in an ad hoc fashion. The proposed AKF method aptly continuously adapts the R leading to a superior position performance. The future work will entail testing the proposed AKF with additional smartphone datasets. Specifically, we will investigate whether changes in phone orientation affect the variance estimated by the adaptive filter and whether the variance bin mapping observed in Figure 3 varies accordingly. Furthermore, an analysis will be conducted to assess the impact of adapting R on the Fault Detection and Exclusion (FDE) results of the AKF and other filters. Given that the AKF yields a more accurate R, theoretically, this should result in fewer Type I and Type II errors during hypothesis testing

REFERENCES

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[19] N. Agarwal and K. O’Keefe, “Use of GNSS Doppler for Prediction in Kalman Filtering for Smartphone Positioning,” IEEE J. Indoor Seamless Position. Navig., vol. PP, pp. 1–10, Nov. 2023, doi: 10.1109/JISPIN.2023.3337188.

[20] G. (Michael) Fu, M. Khider, and F. van Diggelen, “Google Smartphone Decimeter Challenge.” May 2021. [Online]. Available: Retrieved [Date Retrieved] from https://www.kaggle.com/c/google-smartphone-decimeter-challenge/data.

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