Positioning technologies are widely used in automotive, aviation, rail, and maritime safety-critical applications. Therefore, the computation of the probability of positioning failure for vehicles, which is the probability that the position estimator is outside a safety region, is of interest for positioning safety analyses. Since parameter estimation and statistical hypothesis testing for model misspecifications are commonly employed in positioning algorithms, the resulting position estimator is conditioned on the statistical hypothesis testing outcome. Hence, the probability density function (PDF) of the vehicle position estimator that accounts for the dependence between the two inference concepts should be used in the computations. In this contribution, we propose a method to compute the probability of positioning failure using the PDF of the vehicle position estimator, which accounts for the aforementioned dependence and is based on rare event simulation techniques, specifically Importance Sampling and the Cross-Entropy method. We apply the proposed method to a satellite-based positioning scenario, in decimeter precision, of an automated vehicle. The results show that the proposed method enables extensive positioning safety-analyses giving insights that can be used in the development of positioning algorithms and deciding whether safety targets and/or requirements are met. Finally, we discuss some limitations of the method and propose several further improvements.

The analysis of positioning safety often employs a probability-based formulation. This approach quantifies the probability of positioning failure, which is the probability of the position estimator being outside a safety-region, and compares it against an application specific requirement. The design of positioning algorithms for safety-critical applications, such as automated/autonomous vehicles, should consider the dependence between parameter or state estimation and statistical hypothesis testing for model misspecifications in the evaluation of positioning safety. If this dependence is not considered, as this article shows, the conclusions drawn from the positioning safety analysis might be overly-optimistic. Therefore, this article focuses on the aforementioned dependence through a vehicle positioning scenario based on an Extended Kalman Filter (EKF) and the Detection, Identification, and Adaptation (DIA) method for misspecifications in the motion and measurement models. Grounded in the distributional theory for the DIA method, our positioning safety analysis utilizes the conditional probability density functions (PDFs) of the combined EKF and DIA position error, which are generally nonnormal. We compute the probability of vehicle positioning failure in two cases 1) when the dependence is considered and 2) when it is not, to quantify the over-optimism introduced by ignoring this dependence. Finally, we present our conclusions and recommendations.

GNSS-based positioning plays an important role in safety-critical applications (e.g., automotive, aviation, shipping, rail) where positioning safety is paramount. Safety analyses typically include a probability-based formulation, such as calculating the probability that the position estimator falls outside a safety region (probability of positioning failure). Once this probability is being computed, it can be compared to an application specific requirement to decide whether or not this requirement is met. In this context, we carry out a component-wise analysis of the probability of positioning failure for the Detection, Identification, and Adaptation (DIA) estimator. The probability of positioning failure is formulated based on the DIA estimator’s probability density function (PDF) which accounts for the dependence between parameter estimation and statistical hypothesis testing. The probability of positioning failure can be further expressed in terms of its conditional components via the law of total probability which enables the assessment of the Most Impactful Components (MICs). Knowledge about the MICs can be useful to determine the main contributors to the probability of positioning failure. Using a dual GNSS (GPS and Galileo) positioning model for an automated vehicle, we compute the MICs based on the conditional PDFs components of the DIA estimator. Furthermore, we determine the worst-case scenario by computing the maximum total probability of positioning failure over a range of magnitudes for the outliers in the observables and over different orientations of the vehicle. These types of analyses can be useful during the design stage of DIA estimators to verify whether the safety requirements for positioning are met. Lastly, we summarize our contributions and provide an outlook on future work.