Mathematical aspects of GPS RAIM

This paper presents and proves two new theorems that provide deeper understanding of receiver autonomous integrity monitoring (RAIM) methods. The theorems answer the following two questions about RAIM and the least squares navigation solution. (i) Given one set of GPS measurements (a "snapshot"), is it ever possible to determine the navigation error from the parity vector? (ii) Consider a set of measured satellite ranges. All the measurements are subject to noise and one of the measurements is biased by an unknown amount. Three navigation solutions may be obtained. (A) All the measurements are used. (B) Solution A is corrected using standard RAIM techniques for estimating the bias and its effect on the navigation solution. (C) The biased measurement is excluded and a navigation solution is obtained with the remaining measurements. The first theorem reveals the surprising result that the navigation solution is independent of the parity vector, given any single set of measurements. It indicates that accumulation of measurements is essential for an accurate integrity solution. Based on this result, NAVSYS Corporation has developed the NAVSAFE software package which verifies navigation precision by accumulating sets of measurements. The second theorem states that Solution B and Solution C are equivalent. The NAVSAFE software package uses Solution B to correct biased measurements with excellent success. The proof presented in this paper gives sound theoretical foundation to the results observed in practice. Both theorems are proved by employing the singular value decomposition of the observation matrix.<>