Risk-Averse Performance-Specified State Estimation
E. Aghapour, F.S. Rahman, J.A. Farrell, Department of Electrical and Computer Engineering, University of California, Riverside
The use of GNSS-based navigation has increased drastically for automobile, aerial and unmanned vehicles in recent decades. Although standard GNSS provides positioning accuracy of 10 meters, certain applications will be enhanced by reliable full-state estimation consistent with sub-meter positioning accuracy: connected vehicle technology, driver assistance, autonomous driving. Positioning accuracy is hindered due to the presence of outliers in GNSS measurements that may be caused for example by multipath, overhead foliage, or non-line-of-sight signals .
In general, land vehicle navigation applications are signal rich: images contain many features, IMU's are viable, and GNSS comprises many separate systems each of which oversupplies the number of satellites necessary for state estimation. To achieve a specified level of state estimation accuracy, the full set of measurements is typically not required. Therefore, if the full set of measurements was used, then the state estimate would have been exposed to unnecessary risk, while the computed covariance would show that the estimator is over-performing relative to the specification. In reality, outliers are likely to have been included, making the state estimate incorrect and the covariance overly confident in that incorrect estimate Therefore, outlier accommodation is both possible (due to redundancy) and important.
The literature discusses various outlier detection techniques building on fundamental ideas , , , , , . The RAIM techniques are based on computing a party vector from the measurement residual , , ,  assuming that there is enough measurement redundancy to discriminate the outlier source. While many RAIM approaches assume that there is only one outlier, multiple outlier detection has also been well developed , , , , and . Extended RAIM (eRAIM)  incorporates an Inertial Measurement Unit (IMU) and Kalman filter based estimation into RAIM.
Data redundancy, quantified by the number of degrees-of-freedom (DOFs), is critical to successful outlier accommodation. Both RAIM and eRAIM are based on measurements from a single epoch, limiting data redundancy. Redundancy can be enhanced both by adding additional sensors or by solving the estimation problem using all sensor data within a sliding temporal window. Herein we consider the sliding-window approach. We build on theoretical and computational methods developed within the control , robotics , simultaneous localization and mapping (SLAM) , , , and receding horizon estimation , , ,  literatures. The resulting full nonlinear Maximum A Posteriori (MAP) estimator including GNSS and IMU, without outlier accommodation, is presented in . Application of residual-space outlier detection methods to the sliding-window MAP problem of  is presented in .
As is motivated in , , the outlier detection problem is fundamentally unobservable, when all measurements have the potential to be affected by outliers. Therefore, outlier detection methods such as those reviewed above are built on outlier hypothesis assumptions, resulting in tests to choose the most likely assumption. When the number of possible assumptions is too low, the actual outlier scenario may not be included, but the required level of computations increases with the number and complexity of the assumed fault scenarios.
Recently new methods for outlier accommodation without explicit detection have been presented in the literature. The Least Soft-thresholded Squares (LSS) approach, building on l1-regularization that was presented in , , , , . A version of the LSS approach adapted to the problem of  is presented in . Alternatively,  works within an optimization setting to find the largest set of measurements self-consistent with the assumed model. Finally,  works within an optimization setting to choose the set of measurements that achieve a performance specification with minimum risk. The performance specification is phrased in terms of the posterior information matrix. The method is able to quantify when the performance specification is feasible and to quantify the risk associated with the achieved level of performance. The risk is quantified by the norm of the covariance normalized residual vector. Reference  works in a general theoretical setting, with generic academic examples.
The main contribution of this paper is that for the first time, the risk-averse performance-specified approach of  will be applied to and explained for the GNSS state estimation problem, including both the theoretical derivation and experimental results. The experimental results utilize real-world Doppler and differential pseudorange data, considering the risk associated with both sets of measurements. The performance specification is meter level position accuracy.
 P. Teunissen and O. Montenbruck. “Springer Handbook of Global Navigation Satellite Systems.” Springer, 2017.
 J. Neyman and E. S. Pearson, “The testing of statistical hypotheses in relation to probabilities a priori.” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 29, pp. 492-510, 1933.
 A. S. Willsky. "A survey of design methods for failure detection in dynamic systems." Automatica 12.6, pp 601-611, 1976.
 E. Y. Chow and A. S. Willsky. "Issues in the development of a general design algorithm for reliable failure detection." 1980.
 E. Y. Chow and A. S. Willsky. "Analytical redundancy and the design of robust failure detection systems." IEEE Transactions on automatic control 29.7, pp 603-614, 1984.
 R. A. Fisher,” Statistical Methods for Research Workers.” Edinburgh, UK: Oliver and Boyd, pp. 43, 1925.
 R. J. Patton and J. Chen, “A review of parity space approaches to fault diagnosis for aerospace systems.” Journal of Guidance, Control and Dynamics, vol. 17(2), pp. 278-285, 1994.
 R. G. Brown, “A baseline RAIM scheme and a note on the equivalence of three RAIM methods,” Proceedings of National Technical Meeting. ION, pp. 127–137, 1992.
 R. G. Brown. "Solution of the two?failure GPS RAIM problem under worst?case bias conditions: parity space approach." Navigation 44.4, pp. 425-431, 1997.
 R. G. Brown and P. Hwang. "From RAIM to NIORAIM A New Integrity Approach to Integrated Multi-GNSS Systems." Inside GNSS, 2008.
 M. A. Sturza. “Navigation System Integrity Monitoring Using Redundant Measurements,” Navigation, Journal ION, vol. 35, no. 4, 1988-89.
 R. J. Patton. "Fault detection and diagnosis in aerospace systems using analytical redundancy." Computing & Control Engineering Journal 2.3, pp. 127-136, 1991.
 J. E. Angus. "RAIM with multiple faults." Navigation 53.4 -249-257, 2006.
 S. Weisberg, “Applied Linear Regression.” 2nd edition, New York, Wiley, 1985.
 S. Hewitson and J. Wang, “Extended Receiver Autonomous Integrity Monitoring (eRAIM) for GNSS/INS Integration,” Journal of Surveying Engineering, vol. 136, no. 1, pp. 13–22, 2010.
 A. Jazwinski. "Limited memory optimal filtering." IEEE Transactions on Automatic Control, pp. 558-563, 1968.
 P. E. Moraal and J. W. Grizzle. "Observer design for nonlinear systems with discrete-time measurements." IEEE Transactions on automatic control 40.3, pp 395-404, 1995.
 M. Kaess, A. Ranganathan, and F. Dellaert. "iSAM: Incremental smoothing and mapping." IEEE Transactions on Robotics 24.6, pp 1365-1378, 2008.
 F. Dellaert and M. Kaess. "Square Root SAM: Simultaneous localization and mapping via square root information smoothing." The International Journal of Robotics Research 25.12, pp 1181-1203, 2006.
 R. M. Eustice, H. Singh, and J. J. Leonard. "Exactly sparse delayed-state filters for view-based SLAM." IEEE Transactions on Robotics 22.6, pp 1100-1114, 2006.
 K. R. Muske, J. B. Rawlings, and J. H. Lee. "Receding horizon recursive state estimation." American Control Conference, 1993.
 D. Q. Mayne and H. Michalska. "Adaptive receding horizon control for constrained nonlinear systems." Decision and Control, 1993.
 G. Zimmer. "State observation by on-line minimization." International Journal of Control 60.4, pp 595-606, 1994.
 E. L. Haseltine and J. B. Rawlings. "Critical evaluation of extended Kalman filtering and moving-horizon estimation." 44.8, pp 2451-2460, 2005.
 S. Zhao, Y. Sheng. "Differential GPS aided inertial navigation: A contemplative real time approach." IFAC Proceedings Volumes 47.3, pp 8959-8964, 2014.
 P. F. Roysdon and J. A. Farrell, “GPS-INS outlier detection & elimination using a sliding window filter.” American Control Conference, IEEE, pp. 1244-1249, 2017.
 L. Carlone, A. Censi, and F. Dellaert. "Selecting good measurements via ? 1 relaxation: A convex approach for robust estimation over graphs." Intelligent Robots and Systems (IROS), 2014.
 E. Aghapour and J.A. Farrell, "Performance specified state estimation with minimum risk." Submitted to American Control Conference (ACC), 2018.
 J. Wright, John. "Robust face recognition via sparse representation." IEEE transactions on pattern analysis and machine intelligence 31.2, pp 210-227, 2009.
 X. Mei, Xue and H. Ling. "Robust visual tracking using ? 1 minimization." Computer Vision, 2009.
 D. Wang, H. Lu and M. H. Yang. "Robust visual tracking via least soft-threshold squares." IEEE 26.9, pp 1709-1721, 2016.
 P. J. Huber. "Robust Statistics. New York: John Wiley and Sons." HuberRobust statistics, 1981.
 A. M. Leroy, and P. J. Rousseeuw. "Robust Regression and Outlier Detection." Wiley Series in Probability and Mathematical Statistics, New York: Wiley, 1987.