Improving Real-Time Position Estimation Using Correlated Noise Models †
<p>Map of GPS receiver locations for the datasets considered. A single GPS receiver was located at different times at the labelled locations: Dunedin (New Zealand) and North Stradbroke Island, Port Douglas and Thursday Island (Australia). The map demonstrates the spread of the four testing sites, indicating that the observed correlation patterns cannot be attributed solely to an unfortunate “bad” geographic location. The partial autocorrelation functions, <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>ρ</mi> <mo>^</mo> </mover> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>h</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, for latitude, longitude and altitude are shown on the left for the four geographical locations as a function of the lag, <span class="html-italic">h</span>, measured in seconds.</p> "> Figure 2
<p>(<b>a</b>–<b>c</b>) First 600 s of GPS time-series residuals taken at Port Douglas, illustrating the rapid initial convergence towards the “true” value; (<b>d</b>–<b>f</b>) corresponding later GPS time-series residuals illustrating the time-correlated deviations from the “true” value (zero). The apparent position quantization due to rounding is also visible.</p> "> Figure 3
<p>Residuals of latitude, longitude and altitude time-series at Port Douglas, Australia. The grey dashed lines guide the eye and highlight the mean-reverting nature of the time-series.</p> "> Figure 4
<p>Fitted Ornstein–Uhlenbeck (OU) parameters to GPS time-series from Dunedin, New Zealand (grey), North Stradbroke Island, Australia (blue), Port Douglas, Australia (green), and Thursday Island, Australia (red). (<b>a</b>) Estimates of <math display="inline"><semantics> <mi>θ</mi> </semantics></math> show no significant variation over geographic locations; however, the difference between the types of coordinates is apparent, <math display="inline"><semantics> <mi>θ</mi> </semantics></math> being much smaller for altitude (alt) than for either latitude (lat) or longitude (lon). (<b>b</b>) Estimates of parameter <math display="inline"><semantics> <msup> <mi>σ</mi> <mn>2</mn> </msup> </semantics></math> are depicted and show a similar pattern to <math display="inline"><semantics> <mi>θ</mi> </semantics></math> in (<b>a</b>), although, here, the altitude is much larger than the other two coordinates. Qualitatively, this can be understood by the larger variability of the altitude residuals (see <a href="#sensors-20-05913-f003" class="html-fig">Figure 3</a>).</p> "> Figure 5
<p>(<b>a</b>–<b>c</b>) The residuals are plotted for latitude (<b>a</b>), longitude (<b>b</b>) and (<b>c</b>) altitude as measured by a stationary receiver at Port Douglas. (<b>d</b>–<b>f</b>) The time-series of the inferred coordinates by an unscented Kalman filter algorithm equipped with OU noise model. The dataset plotted in (<b>a</b>–<b>c</b>) is used by this algorithm. Shaded regions represent the 90% credible intervals. (<b>g</b>–<b>i</b>) The same as the middle row, but using a Brownian model with process noise <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Σ</mi> <mi>d</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> m<sup>2</sup> within the UKF.</p> "> Figure 6
<p>(<b>a</b>–<b>c</b>) Log scores, <span class="html-italic">S</span>, for latitude (<b>a</b>) longitude (<b>b</b>) and altitude (<b>c</b>) inferences from the Port Douglas data using the KF with the OU noise model. (<b>d</b>–<b>f</b>) The same as above, but for the KF with iid Gaussian noise model.</p> "> Figure 7
<p>Performance of the predictions of the Unscented Kalman Filter with (<b>a</b>) the OU noise model and (<b>b</b>) iid Gaussian noise. Note that the <math display="inline"><semantics> <mover> <mi>S</mi> <mo>¯</mo> </mover> </semantics></math> scale in (<b>b</b>) is <math display="inline"><semantics> <mrow> <mn>3</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>3</mn> </msup> </mrow> </semantics></math> bigger than in (<b>a</b>).</p> "> Figure 8
<p>The time-averaged scores, <math display="inline"><semantics> <mover> <mi>S</mi> <mo>¯</mo> </mover> </semantics></math>, are plotted (<b>a</b>) for the raw data with the standard deviation given by the stationary limit and (<b>b</b>) for the same dataset as in (<b>a</b>) except that the uncertainty is multiplied by the appropriate Dilution-Of-Precision (DOP) variable. (<b>c</b>) <math display="inline"><semantics> <mover> <mi>S</mi> <mo>¯</mo> </mover> </semantics></math> from the unscented Kalman filter with OU noise model and (<b>d</b>) the unscented Kalman filter with iid Gaussian noise model. The colours are consistent with <a href="#sensors-20-05913-f004" class="html-fig">Figure 4</a>. The scales of the ordinates are different in each figure; most importantly, in (<b>d</b>), it is <math display="inline"><semantics> <mrow> <mn>3</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>3</mn> </msup> </mrow> </semantics></math> bigger than in the other figures.</p> "> Figure 9
<p>Estimates of the OU rate parameter <math display="inline"><semantics> <mi>θ</mi> </semantics></math> for the (<b>a</b>) latitude, (<b>b</b>) longitude and (<b>c</b>) altitude time-series of a stationary receiver at Port Douglas.</p> ">
Abstract
:1. Introduction
2. Data Analysis
2.1. Noise Correlations
2.2. Noise Models
3. Position and Uncertainty Estimation Methods
3.1. Earlier Work
3.2. Kalman Filter with Higher Order Noise
3.2.1. Uncorrelated Gaussian Noise
3.2.2. Ornstein–Uhlenbeck Noise
4. Quantifying the Filtering Algorithm Performance
5. Results
6. Discussion and Future Work
7. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A. Augmented-State UKF Algorithm
Algorithm A1: Augmented-state UKF [33,34,37]. |
References
- Kuhlmann, H. Kalman-filtering with coloured measurement noise for deformation analysis. In Proceedings of the 11th FIG Symposium on Deformation Measurements, Santorini, Greece, 25–28 May 2003. [Google Scholar]
- Li, L.; Kuhlmann, H. Real-time deformation measurements using time-series of GPS coordinates processed by Kalman filter with shaping filter. Surv. Rev. 2012, 44, 189–197. [Google Scholar] [CrossRef]
- Jo, K.; Chu, K.; Sunwoo, M. Interacting Multiple Model Filter-Based Sensor Fusion of GPS with In-Vehicle Sensors for Real-Time Vehicle Positioning. IEEE Trans. Intell. Transp. Syst. 2012, 13, 329–343. [Google Scholar] [CrossRef]
- Kalman, R.E. A New Approach to Linear Filtering and Prediction Problems. J. Fluids Eng. 1960, 82, 35–45. [Google Scholar] [CrossRef] [Green Version]
- Bryson, A.; Johansen, D. Linear filtering for time-varying systems using measurements containing colored noise. IEEE Trans. Autom. Control 1965, 10, 4–10. [Google Scholar] [CrossRef]
- Bucy, R. Optimal filtering for correlated noise. J. Math. Anal. Appl. 1967, 20, 1–8. [Google Scholar] [CrossRef] [Green Version]
- Johnson, D.J. Application of a colored noise Kalman filter to a radio-guided ascent mission. J. Spacecr. Rocket. 1970, 7, 277–281. [Google Scholar] [CrossRef] [Green Version]
- Soundy, A.; Panckhurst, B.; Molteno, T. Enhanced noise models for GPS positioning. In Proceedings of the 6th International Conference on Automation, Robotics and Applications (ICARA), Queenstown, New Zealand, 17–19 February 2015; pp. 28–33. [Google Scholar] [CrossRef]
- Petovello, M.G.; O’Keefe, K.; Lachapelle, G.; Cannon, M.E. Consideration of time-correlated errors in a Kalman filter applicable to GNSS. J. Geod. 2009, 83, 51–56. [Google Scholar] [CrossRef]
- Jiang, P.; Zhou, J.; Zhu, Y. Globally optimal Kalman filtering with finite-time correlated noises. In Proceedings of the 49th IEEE Conference on Decision and Control (CDC), Atlanta, GA, USA, 15–17 December 2010; pp. 5007–5012. [Google Scholar] [CrossRef]
- Uhlenbeck, G.E.; Ornstein, L.S. On the Theory of the Brownian Motion. Phys. Rev. 1930, 36, 823–841. [Google Scholar] [CrossRef]
- Jolliffe, I.T.; Stephenson, D.B. Forecast Verification: A Practitioner’s Guide in Atmospheric Science; Wiley: Chichester, UK, 2003. [Google Scholar]
- Bröcker, J.; Smith, L.A. Scoring Probabilistic Forecasts: The Importance of Being Proper. Weather Forecast. 2007, 22, 382–388. [Google Scholar] [CrossRef]
- Boero, G.; Smith, J.; Wallis, K.F. Scoring rules and survey density forecasts. Int. J. Forecast. 2011, 27, 379–393. [Google Scholar] [CrossRef]
- Gneiting, T.; Ranjan, R. Comparing Density Forecasts Using Threshold- and Quantile-Weighted Scoring Rules. J. Bus. Econ. Stat. 2011, 29, 411–422. [Google Scholar] [CrossRef] [Green Version]
- Martin, A.D.; Molteno, T.C.A.; Parry, M. Measuring the performance of sensors that report uncertainty. In Proceedings of the 21st Electronics New Zealand Conference, Hamilton, New Zealand, 20–21 November 2014. [Google Scholar] [CrossRef]
- Martin, A.D.; Soundy, A.W.R.; Panckhurst, B.J.; Brown, C.P.; Schumayer, D.; Molteno, T.C.A.; Parry, M. Real-time uncertainty quantification using correlated noise models for GNSS positioning. In Proceedings of the 2017 IEEE SENSORS, Glasgow, UK, 29 October–1 November 2017; pp. 1–3. [Google Scholar] [CrossRef]
- Akaike, H. A new look at the statistical model identification. IEEE Trans. Autom. Control 1974, 19, 716–723. [Google Scholar] [CrossRef]
- Mao, A.; Harrison, C.G.A.; Dixon, T.H. Noise in GPS coordinate time-series. J. Geophys. Res. Solid Earth 1999, 104, 2797–2816. [Google Scholar] [CrossRef] [Green Version]
- Kim, D.M.; Suk, J. GPS output signal processing considering both correlated/white measurement noise for optimal navigation filtering. Int. J. Aeronaut. Space Sci. 2012, 13, 499–506. [Google Scholar] [CrossRef]
- Bos, M.S.; Fernandes, R.M.S.; Williams, S.D.P.; Bastos, L. Fast error analysis of continuous GNSS observations with missing data. J. Geod. 2013, 87, 351–360. [Google Scholar] [CrossRef] [Green Version]
- Bos, M.S.; Montillet, J.P.; Williams, S.D.P.; Fernandes, R.M.S. Introduction to Geodetic Time Series Analysis. In Geodetic Time Series Analysis in Earth Sciences; Montillet, J.P., Bos, M.S., Eds.; Springer: Berlin/Heidelberg, Germany, 2020; pp. 29–52. [Google Scholar] [CrossRef] [Green Version]
- Johnson, H.O.; Agnew, D.C. Monument motion and measurements of crustal velocities. Geophys. Res. Lett. 1995, 22, 2905–2908. [Google Scholar] [CrossRef]
- Langbein, J. Noise in two-color electronic distance meter measurements revisited. J. Geophys. Res. Solid Earth 2004, 109. [Google Scholar] [CrossRef]
- Langbein, J. Improved efficiency of maximum likelihood analysis of time-series with temporally correlated errors. J. Geod. 2017, 91, 985–994. [Google Scholar] [CrossRef] [Green Version]
- Williams, S.D.P.; Bock, Y.; Fang, P.; Jamason, P.; Nikolaidis, R.M.; Prawirodirdjo, L.; Miller, M.; Johnson, D.J. Error analysis of continuous GPS position time-series. J. Geophys. Res. Solid Earth 2004, 109. [Google Scholar] [CrossRef] [Green Version]
- Hackl, M.; Malservisi, R.; Hugentobler, U.; Wonnacott, R. Estimation of velocity uncertainties from GPS time-series: Examples from the analysis of the South African TrigNet network. J. Geophys. Res. Solid Earth 2011, 116. [Google Scholar] [CrossRef]
- Santamaría-ómez, A.; Bouin, M.N.; Collilieux, X.; Wöppelmann, G. Correlated errors in GPS position time-series: Implications for velocity estimates. J. Geophys. Res. Solid Earth 2011, 116. [Google Scholar] [CrossRef]
- Masson, C.; Mazzotti, S.; Vernant, P. Precision of continuous GPS velocities from statistical analysis of synthetic time-series. Solid Earth 2019, 10, 329–342. [Google Scholar] [CrossRef] [Green Version]
- Caron, F.; Duflos, E.; Pomorski, D.; Vanheeghe, P. GPS/IMU data fusion using multisensor Kalman filtering: Introduction of contextual aspects. Inf. Fusion 2006, 7, 221–230. [Google Scholar] [CrossRef]
- Olivares-Pulido, G.; Teferle, F.N.; Hunegnaw, A. Markov Chain Monte Carlo and the Application to Geodetic Time Series Analysis. In Geodetic Time Series Analysis in Earth Sciences; Montillet, J.P., Bos, M.S., Eds.; Springer: Berlin/Heidelberg, Germany, 2020; pp. 53–138. [Google Scholar]
- Kaczmarek, A.; Kontny, B. Identification of the Noise Model in the Time Series of GNSS Stations Coordinates Using Wavelet Analysis. Remote Sens. 2018, 10, 1611. [Google Scholar] [CrossRef] [Green Version]
- Wan, E.A.; van der Merwe, R. The unscented Kalman filter for nonlinear estimation. In Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium, Lake Louise, AB, Canada, 4 October 2000; pp. 153–158. [Google Scholar] [CrossRef]
- Julier, S.; Uhlmann, J. Unscented filtering and nonlinear estimation. Proc. IEEE 2004, 92, 401–422. [Google Scholar] [CrossRef] [Green Version]
- Chan, Y.T.; Hu, A.G.C.; Plant, J.B. A Kalman Filter Based Tracking Scheme with Input Estimation. IEEE Trans. Aerosp. Electron. Syst. 1979, AES-15, 237–244. [Google Scholar] [CrossRef]
- Chang, S.Y.; Mills, G.; Latif, S. Application of Kalman Filter with Time-Correlated Measurement Errors in Subsurface Contaminant Transport Modeling. J. Environ. Eng. 2012, 138, 771–779. [Google Scholar] [CrossRef]
- Martin, A.; Molteno, T. Automated weighing by sequential inference in dynamic environments. In Proceedings of the 2015 6th International Conference on Automation, Robotics and Applications (ICARA), Queenstown, New Zealand, 17–19 February 2015; pp. 274–278. [Google Scholar] [CrossRef] [Green Version]
- van der Merwe, R.; Wan, E.A. The Square-Root Unscented Kalman Filter for State and Parameter-Estimation. In Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, Salt Lake City, UT, USA, 7–11 May 2001; pp. 3461–3464. [Google Scholar]
- Wang, K.; Li, Y.; Rizos, C. Practical Approaches to Kalman Filtering with Time-Correlated Measurement Errors. IEEE Trans. Aerosp. Electron. Syst. 2012, 48, 1669–1681. [Google Scholar] [CrossRef]
- Wang, X.; Ni, W. An improved particle filter and its application to an INS/GPS integrated navigation system in a serious noisy scenario. Meas. Sci. Technol. 2016, 27, 95005. [Google Scholar] [CrossRef] [Green Version]
- Constantinou, A.; Fenton, N. Solving the Problem of Inadequate Scoring Rules for Assessing Probabilistic Football Forecast Models. J. Quant. Anal. Sport 2012, 8, 1. [Google Scholar] [CrossRef]
- Wendel, J.; Trommer, G.F. An Efficient Method for Considering Time Correlated Noise in GPS/INS Integration. In Proceedings of the 2004 National Technical Meeting of the Institute of Navigation, San Diego, CA, USA, 26–28 January 2004; pp. 903–911. [Google Scholar]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Martin, A.; Parry, M.; Soundy, A.W.R.; Panckhurst, B.J.; Brown, P.; Molteno, T.C.A.; Schumayer, D. Improving Real-Time Position Estimation Using Correlated Noise Models. Sensors 2020, 20, 5913. https://doi.org/10.3390/s20205913
Martin A, Parry M, Soundy AWR, Panckhurst BJ, Brown P, Molteno TCA, Schumayer D. Improving Real-Time Position Estimation Using Correlated Noise Models. Sensors. 2020; 20(20):5913. https://doi.org/10.3390/s20205913
Chicago/Turabian StyleMartin, Andrew, Matthew Parry, Andy W. R. Soundy, Bradley J. Panckhurst, Phillip Brown, Timothy C. A. Molteno, and Daniel Schumayer. 2020. "Improving Real-Time Position Estimation Using Correlated Noise Models" Sensors 20, no. 20: 5913. https://doi.org/10.3390/s20205913