Computation of Probability of Positioning Failure: A Method for Positioning Safety Assessment in Safety-Critical Applications
S. Ciuban (TU Delft - Civil Engineering & Geosciences)
P.J.G. Teunissen – Promotor (TU Delft - Civil Engineering & Geosciences)
C.C.J.M. Tiberius – Copromotor (TU Delft - Civil Engineering & Geosciences)
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Abstract
Satellite-based Positioning, Navigation, and Timing (PNT) technologies, including Global Navigation Satellite Systems (GNSS) and emerging Low Earth Orbit (LEO) constellations, alongside Terrestrial Networked Positioning Systems (TNPS) and various other sensors for positioning (e.g., inertial measurement units, cameras, LiDAR), are used and of interest for safety-critical applications across automotive, aviation, rail, and maritime domains. An important positioning safety criterion for these applications is represented by the probability of positioning failure, defined as the probability that a position estimator falls outside an application-specific safety-region. Rigorous quantification of this probability, denoted PF, is essential to verify compliance with safety requirements and to support the design and evaluation of positioning algorithms and systems.
This thesis addresses the challenges associated with computing PF when the position estimator results from a combined parameter estimation and statistical hypothesis testing procedure for model misspecifications in the positioning model. A key challenge is posed by the multimodality of the probability density function (PDF) of the position estimator, which renders analytical integration methods intractable. Another key challenge is represented by the stringent requirements that PF must satisfy for safety-critical applications (e.g., below 10-5), which implies that the event of positioning failure F must be rare—rendering standard Monte Carlo techniques computationally too expensive. Therefore, a novel method is developed in this thesis which addresses these challenges and is grounded in rare event simulation techniques, specifically Importance Sampling and the Cross-Entropy method. This method enables the construction of a 'failure-tree' that decomposes PF into components conditioned on the hypothesis testing decisions, thereby supporting rigorous positioning safety analyses during the design stage of positioning algorithms, and systems, for safety-critical applications.
The positioning safety is assessed in several representative scenarios. The importance of accounting for estimation–testing dependence is emphasized in a scenario involving cooperative positioning of automated vehicles, where neglecting this dependence results in probabilities of positioning failure being underestimated by an order of magnitude. Furthermore, positioning safety analyses for Unmanned Aerial Vehicles (UAVs) across multiple European airspace regions reveal substantial variability in the probabilities of positioning failure due to changes in receiver-satellite geometry over time, highlighting the importance of comprehensive simulation-based assessments. Additionally, an example is shown in which the probability of positioning failure is computed while accounting for multidimensional model misspecifications (e.g., multiple simultaneous outliers, or faults, in the observations). Collectively, the contributions and findings of this thesis highlight a rigorous approach to computing probabilities of positioning failure and conducting positioning safety analyses.