# DIA-Estimator and Multidimensional Model Misspecifications: GNSS-based Positioning Safety Analysis for UAVs

## TL;DR

This paper analyzes how errors in GPS data affect the safety of UAV positioning using a method called the DIA-estimator.

## Contribution

It introduces a new approach to quantify positioning failure probability under multidimensional model errors.

## Key findings

- The DIA-estimator's probability density function shows multimodal behavior under model misspecifications.
- Positioning failure probabilities were evaluated for UAVs using real GPS satellite data over 24 hours.
- The method supports safety analysis for applications like aviation and can be extended to other systems.

## Abstract

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.