A Formula for Solution Separation without Subset Solutions for Advanced RAIM
Juan Blanch, Todd Walter, Per Enge, Stanford University
The safe operation of UAVs and autonomous vehicles in general will be partly dependent on an assured navigation solution. That is, the position error in the solution computed onboard should be within a known bound (the protection level) with very high probability. In radio-navigation, such position error bounds are easy to compute when the error distribution of the pseudorange errors is known. When some of the measurements are faulty, Receiver Autonomous Integrity Monitoring (RAIM) can be used to compute a guaranteed error bound. RAIM was developed for aviation applications, where the probability of fault of more than one measurement could be considered negligible. However, autonomous vehicles will be expected to operate in environments where this is no longer true. Multipath and erroneous clocks (in the case of terrestrial signals of opportunity) could cause very large delays, and with high probability.
Advanced RAIM (ARAIM) algorithms developed for aviation  can, in theory, be applied in situations where the probabilities of fault are arbitrarily high. These class of algorithms typically form a list of subsets corresponding to a fault mode (a fault mode being a combination of p simultaneous measurement faults), and assigns a probability to it. As the probabilities of fault of each measurement increases, more subsets need to be characterized. In order to compute a protection level, the receiver must compute the covariance of the position error corresponding to each subset. This is practical when no more than two simultaneous faults need to be considered, but it becomes impractical when the receiver must list all subsets with N-p measurements when p is three or more (N being the number of measurements). For example, for 30 range measurements, a target integrity of 10^-7, and a probability of fault of 1%, the receiver would need to form a list of 2 million subsets.
This paper will consider two problems that arise when large and heterogeneous probabilities of fault are assumed in ARAIM. The first one is the computation of the integrity risk arising from the fault modes that are not explicitly monitored. To address this problem, we present an exact and compact formula for the contribution to the integrity risk of all fault modes arising from the combination of a given number of measurement faults.
The second, and more significant problem, is the large amount of geometries that might need to be considered. Many techniques to reduce the number of subsets have been described ,,. In this paper, we investigate two possible ways of characterizing the subset geometries without having to compute their covariance explicitly. One way consists in finding an upper bound on the worst case geometry for a given subset size. We will show that although this problem is combinatorial in nature, it is possible to formulate an upper bound without listing all possible geometries. The other way consists in characterizing the distribution of the subset solution covariance. We will investigate whether this distribution has properties than can be leveraged.
 Working Group C, ARAIM Technical Subgroup, Milestone 3 Report, February 26, 2016. Available at: http://www.gps.gov/policy/cooperation/europe/2016/working-group-c/
 Walter, Todd, Blanch, Juan, Choi, Myung Jun, Reid, Tyler, Enge, Per, "Incorporating GLONASS into Aviation RAIM Receivers," Proceedings of the 2013 International Technical Meeting of The Institute of Navigation, San Diego, California, January 2013, pp. 239-249.
 Orejas, Martin, Skalicky, Jakub, "Clustered ARAIM," Proceedings of the 2016 International Technical Meeting of The Institute of Navigation, Monterey, California, January 2016, pp. 224-230.
 Orejas, Martin, Skalicky, Jakub, Ziegler, Ute, "Implementation and Testing of Clustered ARAIM in a GPS/Galileo Receiver," Proceedings of the 29th International Technical Meeting of The Satellite Division of the Institute of Navigation (ION GNSS+ 2016), Portland, Oregon, September 2016, pp. 1360-1367.