ION GNSS 2012
Session B5: Next Generation GNSS Integrity

Title: RAIM Detector and Estimator Design to Minimize the Integrity Risk
Author(s): M. Joerger, F-C. Chan, S. Langel, and B. Pervan, Illinois Institute of Technology
Date/Time: Friday, September 21, 2012, 9:43 a.m.
Room: 204 (NCC)

This paper describes the design, analysis and evaluation of a new fault detection algorithm that outperforms both residual-based and solution separation methods. First, we establish the conditions of equivalence between full-state solution separation (FS), and residual-based (RB) approaches. Second, we identify limitations in FS and RB techniques, and in the single-state solution separation (SS), which is the basis for protection level equations used in the aviation community. Finally, we develop a new ´generalized´ receiver autonomous integrity monitoring (RAIM) algorithm that fully exploits measurement redundancy.
GNSS measurements are vulnerable to rare-event faults such as satellite failures. In response, in safety-critical applications, fault-detection algorithms such as RAIM can be implemented. Multiple RAIM methods have been developed over the past 25 years. Whereas equivalence between parity-based and RB methods has been proven, most other approaches have only been roughly compared.

The starting point of this paper is a complete and rigorous mathematical analysis of the conditions of equivalence of three of the most widely implemented RAIM methods: RB, FS, and SS RAIM. In this regard, we define the measurement fault vector as the product of a scalar fault magnitude with a unit fault direction vector. Non-zero elements of the fault vector identify the fault mode, i.e., the set of measurements affected by the fault. Each fault mode defines a hypothesis under which subset solutions for the SS methods can be derived as explained below.

The RB method is fully defined in the literature. The two solution separation approaches are derived from a solution separation vector, which is obtained by differencing a fault-free subset solution (estimated using all measurements except the ones assumed to be faulty) from the full-set solution (established using all measurements). The FS test statistic is the weighted norm of the solution separation vector (the weighting matrix is described in the paper). The SS test statistic does not exploit the full solution separation vector, but instead uses a state (or subset of states) of interest. The states of interest are the ones used to define hazardous conditions. For example, in GNSS-based aircraft navigation, estimates of the vertical and/or horizontal position coordinates are of primary concern (unlike the state corresponding to the receiver clock offset).

Of particular relevance in this work is the ability of RAIM to facilitate integrity risk evaluation. In the presence of a fault, the integrity risk is defined as a joint probability of the estimate error being larger than a specified limit (defining hazardous conditions), and of the detection test statistic being smaller than a threshold (set to limit the probability of false alarms). This joint probability can be evaluated as a product of probabilities if estimate error and test statistic are stochastically independent. This is the case for RB RAIM. In this work, we prove that it is also the case for FS and for SS.

It follows that the conditions of equivalence between the three methods are found by simply comparing their detection test statistics (state estimation is identical in all three methods). The paper shows that the three methods are equivalent if and only if, in the presence of single-measurement faults, the difference between the numbers of measurements and of states is one. We also demonstrate that RB and FS methods are only equivalent if the number of assumed faulty measurements is equal to the number of measurements minus the number of states. In general, RB RAIM is less efficient than FS because the RB test statistic is not tuned to the fault hypothesis. In contrast, the fault-free subset solution used to derive the FS test statistic carries knowledge of the fault mode.

In addition, we compare FS and SS in the presence of multi-measurement faults. Multi-measurement faults are likely to occur in future multi-constellation GNSS navigation systems because of the increased number of satellites. In this case, both fault magnitude and fault direction will impact the integrity risk. The fact that the SS test statistic is derived from a subset component of the full solution separation vector causes two conflicting effects. On the one hand, the impact of fault-free errors on the test statistic is smaller for SS than for FS. Therefore, the detection threshold for SS is smaller than for FS, which tends to lower the integrity risk of SS versus FS. On the other hand, fault observability (i.e., fault impact on test statistic) also diminishes using SS as compared to FS, which has the opposite impact on the integrity risk. We use illustrative examples to show that the integrity monitoring performance comparison varies with fault direction. To eliminate the dependency on fault direction, the two solution separation methods are further compared using worst-case fault directions, which maximize the integrity risk. Determining worst-case fault directions can be expressed as an eigenvalue problem, which is defined and solved in this paper for the two solution separation methods.

Finally, we introduce a new ´generalized RAIM´ (GRAIM) algorithm. In the existing RAIM methods, the test statistic is stochastically independent of the full estimate error vector. However, for the test statistic to enable integrity risk computation, it only needs to be independent of the subset of states of interest. In this work, we derive the GRAIM test statistic to only be independent of the states used to define hazard. The resulting GRAIM method exhibits increased detection performance while still enabling integrity risk computation. We show that the GRAIM test statistic follows a generalized non-central chi-square distribution, as opposed to RB and solution separation test statistics, which are non-centrally chi-square distributed.

Performance evaluations for RB, FS, SS and for GRAIM are carried out for an example application of aircraft precision positioning. The integrity risk is evaluated for single-measurement and multi-measurement faults, and for each algorithm´s worst-case fault magnitude and direction. Numerical results illustrate the theoretical analysis. Overall availability is evaluated at an example location, for GPS/Galileo satellite geometries simulated over a ten day period.

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