Improved Stochastic Modelling of Low-Cost GNSS Receivers Positioning Errors
Ahmed Radi, Sameh Nassar, Maan Khedr, University of Calgary, Canada; Roberto Molinari, Stéphane Guerrier, Pennsylvania State University, USA; Naser El-Sheimy, University of Calgary, Canada
Global Navigation Satellite System (GNSS) currently plays a crucial role in providing positioning services for various fields and applications such as autonomous driving, robotics applications, and Unmanned Aerial Vehicle (UAV), where accurate position information is required. To achieve this, a common approach is to integrate GNSS with an Inertial Navigation System (INS) to provide better performance in terms of accuracy and reliability when compared to stand-alone systems. Such an integration architecture usually relies on Bayesian filtering techniques such as Kalman Filtering (KF) to fuse INS sensor measurements with the GNSS position and velocity, where the GNSS velocity and position are used as the KF updates. In addition, the KF is used as well as a common estimation techniques for GNSS stand-alone kinematic positioning. Indeed, the above mentioned applications require high positioning accuracy which, in turn necessitates precise analysis of the residual noise characteristics of the GNSS positioning solutions and their quantitative models. Commonly, GNSS observations’ noise is assumed to be white to satisfy standard KF requirements and for simplicity reasons as well. In other words, successive GNSS observations are considered to be uncorrelated. However, such an assumption is not always optimal for modelling GNSS positioning errors and could, in turn, negatively affect the quality of the overall positioning accuracy of the integrated system.
The most recent research works utilized different standard stochastic modeling techniques, some of them are time domain and others are frequency domain, to study the characteristics of the error signals related to GNSS position solutions. For instance, spectral density and Maximum Likelihood Estimation (MLE) have been used to determine the GNSS positioning noise type and level of the East, North and Up (ENU) components (Zhang et al., 1997, Williams et al., 2004, Genrich and Bock, 2006). Also, some researches applied the Allan Variance (AV) technique (due to being simple to compute, simple to understand and straightforward (Allan, 1966)) for analysing geodesic time series (Friederichs, 2010, Malkin, 2011, Khelifa et al., 2011), and to identify the noise type of Global Positioning System (GPS) positioning residuals. (Niu et al., 2014) also used the AV for investigation the error characteristics related to GNSS positioning errors in order to build corresponding error models. However, few literatures investigate the stochastic error characteristics of GNSS positioning error for commercial low-cost GNSS receivers.
Moreover, a recently developed Wavelet Variance (WV)-based technique, namely the Generalized Method of Wavelet Moments (GMWM), proposed by (Guerrier et al., 2013) which makes use of the relation between the WV and the parameters of a process, estimating the latter by minimizing the distance between the empirical and model-based WVs. Such an approach, which is considered to be an improved version of the classical AV, combines on one hand the WV and on the other hand the generalized least square principle to finally estimate latent composite processes. Based on that, the GMWM have been used for the stochastic calibration of inertial sensors error signals (Stebler et al., 2012).
Mainly, this research is addressing two main challenges:
1- Explore the stochastic characteristics of GNSS positioning residual time series for low-cost GNSS receivers. As mentioned earlier, GNSS stochastic errors are commonly modeled by using relatively simple White Noise (WN) processes. Through this paper, we will analyse whether we need to use more complex stochastic models.
2- Investigates the recently developed GMWM for the stochastic modelling of the aforementioned GNSS error signal and compare the results to the highly cited AV approach, which is currently the main method to study the stochastic characteristics of different time series, to highlight the AV merits and demerits regarding identification and calculation of the error parameters meticulously.
Preliminary tests and results:
GNSS positioning solutions were analysed in Single Point Positioning (SPP) mode at different data rates, 1 and 10 Hz. Multiple data sets were collected using Swift Navigation Piksi v.2.3.1 low-cost GPS receiver. Static data sets were collected with length 6 and 2 hours at 1 and 10 Hz sampling rates, respectively. The Swift Navigation Piksi v.2.3.1 receiver was pointed at a pillar with precisely known coordinates on the rooftop of the ENF building at the University of Calgary. A high-performance GPS antenna was set on the top side of the pillar where the reflective signal could be minimized. Satellites with elevation angle below 15° were rejected, to minimize the multipath effect, in the GPS data processing. The aforementioned data sets were collected in static mode to ensure the capable of reflecting most GPS error sources except the effect of multipath, which is complex and varies in different environment. The positioning residual signal were calculated using the precisely known coordinates of rooftop pillar, then the positioning error at each epoch were transformed to local level coordinates, ENU. The time series of the positioning residual signals of SPP GPS solution for both data sets, 1 and 10 Hz, were plotted in the ENU directions. Results showed that the noise level for the vertical direction is much higher than the horizontal one as generally expected.
For the previous collected data, a signal trend was detected which represents a long-term evolution of the time series. In many applications, such spectral data analysis, trend is one of the most critical quantities that could affect signal under test and is necessary to be removed from the data. Moreover, a time series with a trend is called non-stationary. Once the trend is determined, a process known as “detrending” can be implemented which can be defined as the operation of removing the trend. As a result, identifying the trend and detrending the data are both essential in data analysis (Wu et al., 2007). Based on the above, linear and non-linear trend were detected for the aforementioned three-directional position residual signals. The linear trend, which is a straight line fitted to the error signal, was first removed. After that, the non-linear one was eliminated by fitting a low-order polynomial to the desired signal. In our case, polynomials of different orders have been fitted to all three-directional 1 and 10 Hz position residual signals and subtracted from them (further details about signal detrending will be illustrated in the full paper)
After the trend removal process, both aforementioned techniques, AV and GMWM, were used for identifying and characterising the different latent stochastic process and their related coefficients for the detrended GNSS position residual signals where precise models of the latter have been built.
Allan Variance Analysis:
Starting with AV, the characteristic log-log curve is first obtained by applying the AV algorithm to the three position error components, east, north, and up. The curve is then measured to determine the types and magnitudes of certain random errors possibly residing in the data according to its slope. Finally, the random errors are identified and modelled. Preliminary results show that AV curves in the east, north, and up directions for both data sets look similar. Moreover, the magnitude of the vertical component residual noise is slightly higher than the horizontal components. The reason for that could refer to having better position accuracy for the GPS in horizontal direction rather than the vertical one. Moreover, the noise structure is detected to be a combination of high frequency noise, i.e. WN, mixed with low flicker noise. To be more specific, a slope near the value of -1/2 fits the left-hand side of the AV curve which indicated that WN random process is the dominant noise term for short cluster length for all three directions for both data sets. On the other hand and for medium cluster length, an approximation for a zero slope line could be noticed which reflects the existence of flicker noise, which is an approximation to correlated noise. For more details, some perturbations could be noticed for all three position error components for medium cluster length which emphasize on the existence of noise correlation random processes. However, the AV tool could not precisely recognize the parameters of such correlated noise. As a result, flicker noise is assumed as an approximation for such correlated noise. A detailed list of the corresponding identified error coefficients will be highlighted in the full paper.
Generalized Method of Wavelet Moment Analysis:
Wavelet coefficients are first being calculated by applying a Haar wavelet filter to the data under test. Taking the average of such coefficients, estimated WV is being calculated. After that, the log-log characteristic curve for the standard WV is constructed. In order to estimate the model parameters, more than one model could be candidate and ranked using a certain Wavelet Information Criteria (WIC) to find the model that best describes the latent stochastic processes of the data sets under test.
The overall preliminary test results contribute that for low-cost GNSS receivers, only a WN process is not sufficient for accurate position residual signals’ modelling as the noise is not always independent between successive observations, especially for higher sampling rates. The results also stressed out that the GNSS error signal models are complicated where the corresponding error model structures were represented as a sum of WN and one or more 1st order Gauss-Markov processes which indicates the existence of short and relatively long correlated noise. Moreover, the results show that the GMWM generally outperforms the AV in terms of correlated noise identification and characterization which, consequently, gives it a great potential to be used as a prime method for time series analyzing.
The full paper will include definitions for the main random nature processes considered in this work for constructing different models. Full theoretical flowcharts will be illustrated for both utilized approaches, AV and GMWM, and the analytical performance of each methods will be demonstrated as well. GMWM analysis results for other data sets acquired from another low-cost GNSS receiver will be compared with AV results to show how the GMWM could detect some latent processes that AV could not with higher degree of accuracy. Moreover, tables for the exact values of the parameters related to the selected models will be introduced.
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