Predicting State Uncertainty Bounds Using Non-Linear Stochastic Reachability Analysis for Urban GNSS-Based UAS Navigation

Unmanned aircraft systems (UAS) generally use Global Navigation Satellite System (GNSS) measurements to estimate their state (position and orientation) for outdoor navigation. However, in urban environments, GNSS pseudorange measurements contain biases due to multipath effects and signal blockages by nearby buildings. For safe navigation in such environments, it is beneficial to predict the state uncertainty while accounting for the effect of measurement biases. Reachability analysis is a commonly used tool to predict the state uncertainty of a system. However, existing works do not account for the effect of measurement biases on state estimation, which consequently affects the predicted state uncertainty. Additionally, majority of the existing literature focuses on linear systems, whereas the dynamics of widely used practical systems are better captured by non-linear models. Thus, in this paper we present a non-linear stochastic reachability analysis to predict bounds on the state uncertainty while accounting for measurement biases. We derive the analysis for a fixed-wing UAS navigating using ranging measurements. In order to evaluate our predicted bounds for GNSS-based navigation, we simulate a 3D urban environment and the pseudorange biases due to multipath effects. We validate the predicted bounds for multiple trajectories in the simulated environment. Finally, we demonstrate the applicability of our predicted bounds towards ensuring safe UAS navigation in a shared airspace.