RAIM and Failure Mode Slope: Effects of Increased Number of Measurements and Number of Faults

This article provides a comprehensive analysis of the impact of the increasing number of measurements and the possible increase in the number of faults in multi-constellation Global Navigation Satellite System (GNSS) Receiver Autonomous Integrity Monitoring (RAIM). Residual-based fault detection and integrity monitoring techniques are ubiquitous in linear over-determined sensing systems. An important application is RAIM, as used in multi-constellation GNSS-based positioning. This is a field in which the number of measurements, m, available per epoch is rapidly increasing due to new satellite systems and modernization. Spoofing, multipath, and non-line of sight signals could potentially affect a large number of these signals. This article fully characterizes the impact of measurement faults on the estimation (i.e., position) error, the residual, and their ratio (i.e., the failure mode slope) by analyzing the range space of the measurement matrix and its orthogonal complement. For any fault scenario affecting h measurements, the eigenvalue problem that defines the worst-case fault is expressed and analyzed in terms of these orthogonal subspaces, which enables further analysis. For h>(m-n), where n is the number of estimated variables, it is known that there always exist faults that are undetectable from the residual vector, yielding an infinite value for the failure mode slope. This article uses the range space and its complement to explain: (1) why, for fixed h and n, the failure mode slope decreases with m; (2) why, for a fixed n and m, the failure mode slope increases toward infinity as h increases; (3) why a failure mode slope can become infinite for h≤(m-n). A set of examples demonstrate the results of the paper.