Isolating Multiple Faulty GNSS Measurements with the Jackknife Residual: An Incrementally Expanding Approach
Penggao Yan, Weisong Wen, and Li-Ta Hsu, Department of Aeronautical and Aviation Engineering, The Hong Kong Polytechnic University
Alternate Number 1
FDI for multiple faults is essential for satellite navigation: Fault detection and isolation (FDI) is essential for localization and navigation systems in some safety-critical applications (Joerger & Pervan, 2016; Osechas et al., 2012; Pervan et al., 1998). Fault detection is technology to check the occurrence of faults in the system (Gao et al., 2015), while fault isolation aims to separate faulty measurements from healthy measurements. In satellite navigation research, the FDI technique closely relates to integrity monitoring systems and is frequently designed to fulfill the integrity requirements of these systems. In the early stage of satellite navigation with limited satellites in operation, the FDI technique, such as the chi-squared test detector (Parkinson & Axelrad, 1988) and the multiple hypothesis solution separation detector (Pervan et al., 1998) in the conventional receiver autonomous integrity monitoring (RAIM) system (Walter & Enge, 1995), mainly focuses on the single fault case. However, with the growing number of satellites and constellations, the probability of simultaneous faults becomes nonnegligible, highlighting the urgent need for FDI techniques in handling multiple faults.
Deletion-based method suffers from swamping and masking effects: Mathematically speaking, the FDI technique for multiple faults aims to find a subset of K measurements that produces the largest reduction in the sum of square errors (SSE) when deleted (Rohlf, 1975). This mechanism is perfectly implemented by the solution separation method in the navigation community, which computes the positioning solution with a subset of measurements being excluded (the subset solution) and compares it with the positioning solution with the full set measurements (the full set solution) (Blanch, Walker, Enge, et al., 2015; Blanch et al., 2024). The maximum difference between the subset and full set solutions is used to separate the faulty measurements from nominal measurements, which can be found by enumerating all possible combinations of subsets (Blanch, Walker, Enge, et al., 2015). However, this exhaustive search algorithm is computationally intractable with a large number of measurements (Knowles & Gao, 2023). Therefore, several greedy search methods are proposed to accelerate the speed of FDI. Most of these greedy search methods are developed based on the deletion philosophy, i.e., removing subsets of measurements to reduce the properly chosen statistic. For example, Blanch, Walker, and Enge (2015) propose the greedy search method by removing the measurement with the largest normalized residual in an iterative process; Hsu et al. (2017) propose the iterative consistency check method by excluding the measurement that results in the largest reduction in the weighted SSE. Although these deletion-based methods produce competitive performances compared with the exhaustive search method, they are unavoidably affected by the swamping and masking effects (Kianifard & Swallow, 1989). Masking occurs when a faulty subset goes undetected. Swamping occurs when healthy measurements are incorrectly identified as faults. Especially, the swamping event could be extraordinarily dangerous because the exclusion operation reduces the redundancy of healthy measurements, subsequently increasing the risk of large positioning deviation.
Our contributions in this study: Apart from the deletion approach, an alternative approach for multiple faults isolation can be realized by incrementally expanding the measurement set until the faults are detected. Some practices on the linear regression model can be found in the statistic literature (Atkinson & Mulira, 1993; Hadi & Simonoff, 1993). It is proved that the expanding approach is more resistant to the masking and swamping effects than the deletion approach and even has a competitive computation load (Hadi & Simonoff, 1993). However, to the best of the authors’ knowledge, this expanding approach for multiple fault isolation is rarely seen in satellite navigation, which is worth exploring. Therefore, this paper aims to propose an expanding approach to detect and isolate multiple faulty measurements in pseudorange-based positioning systems. Specifically, the proposed algorithm starts with constructing a minimum basic subset, which has the minimum studentized residual computed based on full set measurements. Then we incrementally expand the basic subset with no-fault hypothesis testing by examining the ordered jackknife residual. Since the no-fault hypothesis testing is a multiple-testing problem, we further apply the Bonferroni correction (Bonferroni, 1936) to correct the threshold in testing. The proposed method is evaluated in a simulated experiment to isolate multiple pseudorange faults for a set of users distributed over the world during one day. In the setting of six simultaneous injected faults with the magnitude uniformly distributed in [5 m, 10 m], the proposed method exhibits a nearly 50 % reduction in the mean swamping event rate, compared to the deletion-based greedy search method (Blanch, Walker, & Enge, 2015). In addition, the max positioning error of the proposed method is around 35 % less than the deletion-based greedy search method. Similar results are observed when the magnitude of the injected faults is increased to the range of [10 m, 20 m].
The contributions of this study are two folds:
1. Propose a multiple fault isolation algorithm for pseudorange-based positioning systems by incrementally expanding the basic subset leveraging ordered jackknife residuals;
2. Exemplify the feasibility and outstanding performance of the proposed method compared to the conventional deletion-based method through a worldwide simulation experiment.
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