Integrity Risk and Continuity Risk for Fault Detection and Exclusion Using Solution Separation ARAIM

M. Joerger, S. Stevanovic, F-C. Chan, S. Langel, B. Pervan

Abstract:	The increased number of redundant ranging signals in future multi-constellation GNSS will improve RAIM-based integrity monitoring performance, but will also increase the probability of satellite faults, thereby increasing the continuity risk. In response, in this paper, a solution separation (SS) approach to fault detection and exclusion (FDE) is developed. The first part of the paper proves that for single-measurement faults, SS detection test statistics are projections of the parity vector on failure mode lines. It follows that the SS detection boundary can be represented as a polytope in parity space. To further analyze this result, we design a method that provides a piecewise linear detection boundary, which minimizes the integrity risk while limiting the probability of false alarms. This optimal detection region varies with navigation system parameters, but for realistic, practical requirements, the optimal detection region is shown to approach the SS polytope-shaped boundary. The second part of the paper introduces complete integrity risk and continuity risk equations for SS fault detection and exclusion. Probability bounds are developed, which express the reduction of continuity risk using exclusion at the cost of an increase in integrity risk. The integrity risk bound, also given in the form of protection levels, is designed to enable integrity risk evaluation in practical applications where computation resources are limited. In parallel, a continuity risk bound is derived, which provides the means to determine detection and exclusion thresholds that satisfy the continuity risk requirement. Parity space representations reveal the shape of the SS exclusion zones, and reaffirm the convenience of using normally-distributed SS test statistics for risk evaluation, especially in high-dimensional parity space. Finally, the SS FDE integrity and continuity risk bounds are implemented to establish worldwide availability maps for an example aircraft approach application using Advanced RAIM (ARAIM) based on GPS and Galileo measurements. OLD VERSION ------------------------------------------------------------- This paper describes the derivation, analysis and evaluation of a new fault-detection algorithm, which is designed to minimize the integrity risk using receiver autonomous integrity monitoring (RAIM). The proposed detection method introduces new test statistics in addition to the conventional solution separations used in RAIM. This augmented set of test statistics provides the means to define a piecewise-linear approximation of the optimal detection region. The piecewise-linear region is numerically determined by solving an integrity-risk minimization problem. This formulation can be modified to include a non-least squares (NLS) estimator as in [1], so that ultimately, both the integrity monitor’s detector and estimator are designed to minimize the integrity risk. Global navigation satellite system (GNSS) measurements are vulnerable to rare-event faults including satellite failures, which represent major integrity threats in safety-critical applications. In response, fault-detection algorithms such as RAIM can be implemented. RAIM exploits redundant ranging signals to achieve self-contained fault detection at the user receiver. With the modernization of GPS, the full deployment of GLONASS, and the emergence of Galileo, an increased number of redundant measurements becomes available, which has recently drawn a renewed interest in RAIM. In particular, RAIM can help alleviate requirements on ground monitors. For example, researchers in the European Union and in the United States are investigating Advanced RAIM (ARAIM) for worldwide vertical guidance of aircraft. RAIM does not only aim at detecting faults but also at evaluating the integrity risk. Integrity risk evaluation includes both assessing the fault detection capability and quantifying the impact of undetected faults on state estimate errors. More precisely, the integrity risk is defined as the joint probability (a) of the estimate error being larger than a specified limit (named alert limit in aviation applications) above which the positioning information is deemed hazardous, and (b) of the detection test statistic being smaller than a threshold, which is set according to a continuity requirement to limit the probability of false alarms. Hence, both the detector and the estimator influence RAIM performance. This research builds upon previous work in [1], which explores the potential of using NLS estimators to reduce the integrity risk at the cost of lower accuracy performance. This concept aims at improving the combined availability of accuracy and integrity, and is motivated by the fact that, in safety-critical applications, integrity requirements are more demanding than accuracy requirements. In parallel, this earlier work also shows that a solution separation-based detector provides better fault-detection capability than residual-based RAIM. Residual-based RAIM uses a single test statistic, which is the norm of the residual vector weighted by the inverse of the measurement noise covariance matrix. In contrast, solution separation considers as many test statistics as fault hypotheses: the test statistics are derived from solution separations, obtained by differencing the fault-free subset solution (estimated using all measurements except the ones assumed to be faulty) from the full-set solution (established using all measurements). Also, solution separation directly focuses on the state of interest (i.e., the state used to define hazardous conditions, for example, the vertical position coordinate in aircraft navigation). Therefore, the efficiency of solution separation RAIM stems from the fact that the least-squares solution separation test statistics are tailored to the fault hypotheses, and to the state of interest. This remains true when considering multiple simultaneous faults, which are more likely to occur in multi-constellation GNSS. We can identify two limitations of existing solution separation approaches. First, an upper bound on the continuity risk is typically evaluated assuming that all test statistics are independent, although they are actually correlated. Second, an upper bound on the integrity risk is evaluated because for each fault hypothesis, only the corresponding solution separation is exploited, without taking credit for its potential detectability using the other test statistics. The main reason for these two approximations is to avoid evaluating the correlation between the solution separation test statistics. In this work, a new integrity monitoring method is developed that provides three major improvements. We start by assuming a least squares estimator. First, the new method addresses the challenge of evaluating the correlation between test statistics. The solution separation test statistics are all arranged in a single vector, which follows a multivariate Gaussian distribution. The vector’s mean and covariance matrix are fully defined and the multivariate Gaussian cumulative distribution function can be evaluated numerically. Second, the fault-detection performance can be further improved by including additional test statistics. For example, in a two-dimensional parity space, the solution separation detection boundary is a convex polygon with a number of segments equal to (or lower than) twice the number of test statistics. In this research, the piecewise linear contour is further refined to approach the optimal detection boundary by considering additional test statistics in the parity space. The new test statistics are projections of the parity vector along multiple directions, which can be taken at regular angular intervals in the parity space. The detection thresholds for each of the test statistics are determined to minimize the integrity risk under the constraint of satisfying the continuity risk requirement. This convex, constrained minimization problem is numerically solved using a quickly-converging modified Newton method. An analytical expression for the gradient of the objective function (which is the integrity risk) over the detection thresholds is provided. Third, the detector is combined with the NLS estimator described in [1] to obtain an optimized RAIM algorithm. The vector of test statistics is augmented with the estimate error for the state of interest. The resulting vector’s covariance matrix is fully populated because the NLS estimate error and the test statistics are correlated. The integrity risk minimization problem is carried out over both the detection thresholds and the NLS estimator parameters introduced in [1]. Requirements in terms of computational resources are discussed. Finally, a performance analysis is carried out for an example application of positioning during aircraft precision approach. Overall availability is evaluated at multiple locations, for GPS/Galileo satellite geometries simulated over a ten day period. As compared to standard solution separation ARAIM, the new RAIM method is computationally more intensive in finding the optimal estimator-detector combination, but it achieves substantial integrity risk reduction while limiting the accuracy performance degradation. [1] Joerger, M., Chan, F.-C., Langel, S., and Pervan, B., “RAIM Detector and Estimator Design to Minimize the Integrity Risk,” Proceedings of the 25th International Technical Meeting of The Satellite Division of the Institute of Navigation (ION GNSS 2012), Nashville, TN, September 2012.
Published in:	Proceedings of the 26th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2013) September 16 - 20, 2013 Nashville Convention Center, Nashville, Tennessee Nashville, TN
Pages:	2702 - 2722
Cite this article:	Joerger, M., Stevanovic, S., Chan, F-C., Langel, S., Pervan, B., "Integrity Risk and Continuity Risk for Fault Detection and Exclusion Using Solution Separation ARAIM," Proceedings of the 26th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2013), Nashville, TN, September 2013, pp. 2702-2722.
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