A Sensitivity-based Method for Fault Detection and Identification

Receiver Autonomous Integrity Monitoring (RAIM) for GNSS is required for safety and liability critical applications. The existing FDI (fault detection and identification) algorithms for RAIM are usually based on statistic testing, which is carried out with a statistic computed from the residuals of measurements. In this paper, a scheme which combines the conventional statistic test procedure with the sensitivity matrix and the correlation coefficients between the w-test statistics is presented. Simulated faults are introduced into real GNSS data to evaluate the performance of the proposed FDI by comparison with the conventional FDI. The results show that by taking into account of the sensitivity matrix and the correlation coefficients, correct fault detection and identification rates are increased and at the same time, the computation time is reduced. INTRODUCTION Global Navigation Satellite Systems (GNSS) have been used for many applications, and therefore it is important to guarantee the integrity of the GNSS positioning solutions under a variety of operating environments. Some ground based integrity monitoring systems, such as WAAS and LAAS, can send warning message to a user when the system has some problems. However such integrity information is relevant for a large area for satellite system failures, and does not provide any integrity for the user’s local errors, such as local interferences/multipath etc. As a result, Receiver Autonomous Integrity Monitoring (RAIM) is more preferred approach to provide the integrity information at a local user level. With the RAIM, horizontal protection level (PHL) and vertical protection level (VPL) can be predicted based on satellite geometry, standard deviation of measurements and false alarm rate and power of test. PHL and VPL indicate how much the users can trust the positioning solution (i.e. external reliability). An ideal RAIM method provides the users not only those PHL and VPL, but also the ability to detect fault measurement sources timely, and if possible, to identify and removes them from GNSS receiver navigation solutions. There has been extensive literature on RAIM and FDI (Fault Detection and Identification). Brown and McBurney (1987) suggests that an effective technique to take a set of n measurements from all satellites in view and to solve for a set of n solutions by leaving one of the measurements out of each solution. A failure would be declared if this maximum separation distance exceeds a threshold value. The procedure is time consume if multiple faults exist. In order to identify a fault, the statistical redundancy based w-test (Baarda, 1968) procedure for fault identification was mostly generally used and the correlation problem relating to misidentification is also discussed in Tiberius (1998). The misidentification mainly results from the correlation of wtest statistics. A high correlation coefficient for a pair of measurements implies that it is hard to distinguish which measurement is contaminated. Thereby, good measurement is removed by masking the true fault measurement, resulting in weak geometry and biased positioning results. In this paper, the reasons for misdetection and misidentification are discussed. Conventional FDI scheme is improved by combining the statistic test procedure with the sensitivity matrix and the correlation coefficients between the w-test statistics. The purpose of this study is to decrease fault misdetection and misidentification rates and improve computational efficiency. THE LEAST SQUARES SOLUTION Assume the number of the visible satellites is n, and then the GNSS linearized measurement equation can be represented as ( ) l Ax n ε = + + (1) where l is the measurement vector, A is the design matrix (i.e the direction cosines) of the state vector (position and clock corrections) x . (n ε + ) is measurement error vector, in which n is a zero mean Gaussian noise vector and ε is a vector of uncompensated measurement faults. The following least squares formulae are used for the estimation of the unknown state vector and measurement residuals 1 1 1 1 ˆ ( ) T T x A A A l G l − − − − = Σ Σ = Σ (2) where Σ is the covariance matrix of the measurements and 1 1 ( ) T T G A A A − − = Σ . It is unavoidable that measurements are contaminated by errors, so the estimated state vector is thus biased: 1 ˆ ( ) x x G n ε − = + Σ + (3) ( ) error x G n ε − = Σ + (4) The matrix 1 G − Σ maps the measurement errors onto to the state vector . The residual vector is estimated by 1 1 1 ˆ ( ( ) ) T T v Ax l I A A A A l − − − = − = − − Σ Σ (5) Suppose the measurement on the i th satellite has a fault, i.e.: (0, ,0, ,0, ,0) T T b ε = and the measurements errors are independent, i.e. the covariance matrix Σ is a diagonal matrix, the fault causes a navigation solution error as follows: (0, ,0, ,0, ,0) error x G b − = Σ (6) In the above and following equations the influence of noise is neglected just for convenience of discussions. The Horizontal and Vertical Radial Error (HRE and VRE) of a navigation solution as 2 2 1 2 2 1 2 i errorN errorE i i i HRE x x b G G − = + = Σ + (7) 2 1 2 3 i errorU i i VRE x b G − = = Σ (8) where errorN x and errorE x denote the horizontal position errors in north and east directions, and errorU x is the vertical error caused by the fault. i Σ is the variance of the i th measurement. RESIDUAL SENSITIVITY MATRIX AND GLOBAL TEST STATISTIC For convenience, define the matrix in equation (5) 1 1 1 ( ( ) ) T T I A A A A − − − − − Σ Σ as S − ( ) v Sl S n ε = − = − + (9) The row or column in S corresponds to how a variation in the measurement from each satellite would affect the residuals. As shown in the above equation, measurement errors are mapped onto the residual vector by the matrix S , which is called the (residual) sensitivity matrix. The sensitivity matrix S is a symmetric and idempotent matrix and the sum of any row and column in this matrix equals zero. The square root of the sum of the squares of the elements of any row or column is equal to the square root of the associated diagonal element. Since the state vector, x , has four elements, The rank of S is 4 n − where n is the number of satellites. As shown in Parkinson and Alxerad (1998), the sum of the squared least squares residuals, 1 T T V V − = Σ has a Chi-square distribution with 4 n − degrees of freedom (DOF). If the test statisticT exceeds the normalized ChiSquare threshold cs T , Faults are declared. Above test is also called global test. The he normalized Chi-Square threshold cs T is calculated by inverse Chi-Square cumulative distribution function (cdf) (1 | ) cs T F DOF α − = − (10) where F is a cumulative distribution function. The global testing can be used to identify fault. Taking a subset of measurements from all satellites in view to calculate the sum of the squared residuals for this subset measurements, which can be done by sequentially leaving one of the measurements out of each solution and a fault would be declared if one of the test statistics smaller than the defined threshold. However, this procedure results in a big computational burden and furthermore the statistic test is not always valid because of small diagonal elements of S matrix. Assuming that the measurement noises are independent and normally distributed with zero mean, the fault has an impact on the test statistic 1 2 1 T i i ii T V V b S − − = Σ = Σ (11) where ii S is the i th diagonal element of the matrix S . If ii S is quite small, the fault is less mapped onto residuals and the above test statistic, thus the test statistic will not exceed threshold and faults cannot be detected. Furthermore, the undetected fault would result in a significantly positional error because the fault is mainly absorbed into the state vector instead of the measurement residuals (Sturza, 1988). This can be shown by ratios of