The Detection, Identification, and Adaptation (DIA)-estimator integrates parameter estimation and hypothesis testing for model misspecifications. This contribution presents a positioning safety analysis approach grounded in the DIA-estimator framework, with a particular emphasis on multidimensional model misspecifications, such as simultaneous outliers in the observations. While recent work has focused on the performance of the detection and identification of multidimensional model misspecifications, we turn our attention to how they affect the probability density function (PDF) of the DIA-estimator and, consequently, the probability of positioning failure–an indicator relevant for safety-of-life applications (e.g., automotive, aviation, rail, maritime). This work formulates and quantifies the probability of positioning failure and its conditional components. A representative simulation-based study is presented for a UAV equipped with a GPS receiver configured to achieve performance comparable to Technical Standard Order (TSO)-certified receivers. The analysis is carried out for two scenarios: a fixed GPS satellite geometry at a single time snapshot, and for a varying GPS satellite geometry over a 24-hour period over an authorized UAV airspace region in the Netherlands using real satellite ephemeris data. Together, these scenarios provide insights into the structure of the DIA-estimator’s PDF, such as multimodality and orientation with respect to the chosen positioning safety region, and support comprehensive evaluation of positioning safety. Although the current focus is on GPS-based positioning, the presented approach is general and can be extended to include multisensor configurations, additional GNSS constellations, and applied to other safety-critical applications, which are subjects of future work.

Unmanned Aerial Vehicles (UAVs) support, or are planned to support, a wide range of operations, including emergency response, environmental research, urban air mobility, and (commercial) air transportation, where positioning safety is paramount. This contribution presents a framework for assessing positioning safety of UAVs by computing the probability of positioning failure, rather than conservative upper bounds, while accounting for time-varying positioning models. In contrast to existing studies, we (i) explicitly adopt UAV safety regions and target probability of positioning failure requirements as specified by the European Union Agency for the Space Programme (EUSPA) for Specific Assurance and Integrity Levels (SAIL) 3 (10^-4/hour) and 4 (10^-5/hour), and (ii) use representative positioning models for the UAV GPS receiver which are consistent with Technical Standard Order (TSO) specifications. For the computation of the probability of positioning failure, we use a method based on rare event simulation techniques while accounting for the dependence between parameter estimation and statistical hypothesis testing. We apply the framework to simulation-based positioning safety analysis across authorized European airspace regions in eight countries using real GPS satellite orbit data. The probability of positioning failure is computed over a 24-hour period, then connected to per-hour requirements using one-hour moving averages, and compared against the EUSPA SAIL 3 and 4 requirements. The time-dependent analysis further reports best-case and worst-case probabilities of positioning failure and quantifies sensitivity to key hypothesis-testing design parameters, such as the level of significance. This analysis can help UAV operators and regulators verify compliance with EUSPA safety standards, supporting management of safe UAV operations.

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.