A recursive overbounding filter for safety-critical navigation systems

Abstract

Ensuring the safety of navigation systems in safety-critical applications—such as aviation, autonomous vehicles, and space operations—requires the ability to monitor the trustworthiness of the navigation solutions, which is defined as the system's integrity. It includes the navigation system's ability to provide timely warnings when navigation errors exceed predefined safety thresholds. In the absence of true knowledge of the system/sensor uncertainty, a statistical technique called overbounding is often utilized to construct a conservative characterization of the navigation error distribution, particularly in the tail regions where rare but hazardous events reside, to ensure safety. However, traditional overbounding methods relying on Gaussian distributions can become overly conservative when the true error distributions exhibit heavy tails, thereby sacrificing the availability of the navigation system. This dissertation addresses this challenge by developing a Bayesian filtering framework that integrates tight overbounding models of heavy-tailed error distributions into recursive navigation filters, enabling both safety assurance and improved availability. The work in this dissertation focuses on the fault-free integrity monitoring in the presence of heavy-tailed sensor error distribution. A novel overbounding methodology called Gaussian-Pareto overbounding is utilized in this work to handle the heavier-than-Gaussian tails of the sensor error distribution. Gaussian-Pareto overbounding approach has demonstrated potential in achieving tight overbounding of heavy-tailed error distributions using with limited data, thereby, offering the promise in speeding up the certification process and increading the availability of the safety-critical navigation systems. However, it is mathematically complex to directly integrate the Gaussian-Pareto overbounds into the recursive navigation filters/estimators, which limits their direct application. To bridge this gap, a recursive Bayesian overbounding filter is proposed in this dissertation that is able to take the Gaussian-Pareto overbound as the input sensor error and maintain bounded tail probabilities of the system output errors. This estimator leverages Gaussian-Pareto distributions and the Dvoretzky-Kiefer-Wolfowitz inequality to provide conservative overbounds for sensor noise constructed from finite data. To enable efficient propagation of Gaussian-Pareto distributions, we develop a method that employs Gaussian Mixture Models to conservatively approximate these distributions. Given that convolutions with Gaussian mixtures can result in an exponential growth of the mixture terms, we devise a culling algorithm that periodically prunes the Gaussian mixture while ensuring conservative overbounding. Simulation case studies in both vertical and horizontal integrity monitoring during an aircraft landing phase of flight are performed to evaluate the proposed algorithm. Simulation results consistently demonstrate that the proposed approach can achieve smaller protection levels than conventional Kalman filters using Gaussian overbounds. These results highlight the potential of the proposed filtering framework to improve the navigation system's availability in the safety-critical applications.