Alternative Integrity Measures Based on Interval Analysis and Set Theory

摘要

Confidence domains for Global Navigation Satellite Systems (GNSSs) positioning and consistency measures of the observations are of great importance for any navigation system, especially for safety-critical applications. Integrity of GNSS applies to a variety of highly demanding tasks, like e.g., precision landing approaches. Consequently, the quality and trust that we put into navigation solutions must be extremely high: Integrity measures this performance, i.e. the ability of the navigation system to timely warn the user when error thresholds, the so-called alert limits are transgressed. Integrity for aviation applications can be provided by ground-based augmented systems or satellite-based augmented systems, which are highly complex and expensive systems. An alternative is Receiver Autonomous Integrity Monitoring (RAIM), which evaluates the integrity risk (IR) inside the GNSS receiver itself. The IR evaluation is necessary when designing a navigation system to meet the predefined integrity requirements. Currently, there is a big interest in developing RAIM algorithms, especially for autonomous driving applications. The IR evaluation involves both assessing the fault detection and exclusion capability and quantifying the impact of undetected faults on the position estimation. There is a variety of RAIM algorithms based on statistical hypothesis tests acting on the measurement or position domain. The residual-based and solution separation RAIMs have gained more interest than the other has and they have been widely implemented over the last three decades. In the past years, new integrity approaches were proposed based on interval mathematics namely set inversion and least-squares. This thesis presents a newly developed deterministic bounding method based on convex optimization. In this deterministic bounding method, the observation interval bounds (OIBs) are applied to the observation equations that represent constraints for the parameters that have to be satisfied. As a result, a convex poly-tope is obtained. Consistency measures are obtained by comparing the non-regular polytope with a regular polytope (zonotope) computed solely from OIB and the current geometry. In this thesis, a primal-dual polytope algorithm is used to estimate all possible solutions. The obtained polytope represents exactly the feasible region of the positioning solution. The developed method guarantees the internal reliability that is presented by the minimum detectable bias. Besides, fault detection and exclusion algorithms based on the polytope consistency measures are developed. In addition, a guaranteed protection level is developed that corresponds to a one-relaxed zonotope. Our newly developed polytopic method will be investigated in all aspects and compared to the above-mentioned RAIM algorithms showing the benefits and the drawbacks of each method. To perform all the above-mentioned interval-based methods the observation interval bounds need to be estimated. Three main methods could be used to determine the OIBs, which will be investigated: a probabilistic approach with predefined IR, sensitivity analysis of the correction models, and expert knowledge. This work provides the theoretical framework, the mathematical properties, and the geometrical interpretation of the polytope concerning the positioning problem. A study of the form and orientation of the polytope w.r.t. line-of-sights, number of satellites in view, and random observation errors, as well as biases, is conducted. Also, a simulation study of a simple but didactic positioning system for better understanding the geometry of the polytopes and its relation to the navigation geometry is performed. Moreover, an intensive Monte Carlo simulation to study the newly developed RAIM algorithm is performed. Then, real GPS pseudo-range observations from a kinematic test drive will be analyzed, and a comparison between different RAIM algorithms are performed in terms of precision, accuracy, internal reliability, integrity, continuity, and availability.