In many practical situations, measurements are characterized by interval uncertainty – namely, based on each measurement result, the only information that we have about the actual value of the measured quantity is that this value belongs to some interval. If several such intervals – corresponding to measuring the same quantity – have an empty intersection, this means that at least one of the corresponding measurement results is an outlier, caused by a malfunction of the measuring instrument. From the purely mathematical viewpoint, if the intersection is non-empty, there is no reason to be suspicious. However, from the practical viewpoint, if the intersection is too narrow – i.e., almost empty – then we should also be suspicious, and mark this as an possible additional outlier case. In this paper, we describe a natural way to formalize this idea, and an algorithm for detecting such additional possible outliers.

Reliable confidence domains for positioning with Global Navigation Satellite System (GNSS) and inconsistency measures for the observations are of great importance for any navigation system, especially for safety critical applications. In this work, deterministic error bounds are introduced in form of intervals to assess remaining observation errors. The intervals can be determined based on expert knowledge or - as in our case - based on a sensitivity analysis of the measurement correction process. Using convex optimization, bounding zones are computed for GPS positioning, which satisfy the geometrical constraints imposed by the observation intervals. The bounding zone is a convex polytope. When exploiting only the navigation geometry, a confidence domain is computed in form of a zonotope. We show that the relative volume between the polytope and the zonotope can be considered as an inconsistency measure. A small polytope volume indicates bad consistency of the observations. In extreme cases, empty sets are obtained which indicates large outliers. We explain how shape and volume of the polytopes are related to the positioning geometry. Furthermore, we propose a new concept of Minimum Detectable Biases. Using the example of the Klobuchar ionospheric model and Saastamoinen tropospheric model, we show how observation intervals can be determined via sensitivity analysis of these correction models for a real measurement campaign. Taking GPS code data from simulations and real experiments, a comparison analysis between the proposed deterministic bounding method and the classical least-squares adjustment has been conducted in terms of accuracy and reliability. It shows that the computed polytopes always enclose the reference trajectory. In case of large outliers, large position deviations persist in the least-squares solution while the polytope algorithm yields empty sets and thus successfully detects the cases with outliers.

Confidence domains for Global Navigation Satellite System (GNSS) positioning and inconsistency measures of the observations are of great importance for any navigation system, especially for safety critical applications. In this work, deterministic error bounds are derived from a sensitivity analysis of the observation correction models and introduced in form of intervals to assess remaining observation errors. Using convex optimization, bounding zones are computed for GPS positioning using geometrical constraints imposed by the observation intervals. The bounding zone is a convex polytope. We show that the volume of the polytope is an inconsistency measures rather than confidence measures, since small polytopes indicates bad consistency of the observations. In extreme cases empty sets are obtained which indicates large outliers. We explain how the shape and the volume of the polytope are related to the positioning geometry and how observations of maximum impact can be revealed. In a first attempt, a point position can by associated to the solution set by its barycenter. However, we show that this assignment has based to be interpreted with care. Furthermore, we propose a new concept of Minimum Detectable Biases (MDB) on the geometric relations. Taking GPS data from simulations and real experiments, a comparison analysis between the proposed deterministic bounding method and the classical least-squares adjustment has been conduct in terms of accuracy and reliability. This helps validating that our proposed deterministic bound methods shows high internal and external reliability compared to the probabilistic approaches and that it provides rigorous inconsistency measures.

Guaranteed protection levels of the Global Navigation Satellite System (GNSS) are of great importance, especially for the safety critical application such as: landing approach and navigation of autonomous vehicles. In order to guarantee the computed protection levels, reliable outlier detection and exclusion algorithms must be apply. In the past 30 years, di erent algorithms have been investigated based on statistical hypothesis testing. Thanks to their out-performance, the residual based test statistics and the solution separation have gained most interest compared to other algorithms. However, statistic-based fault detection and exclusion algorithms do not guarantee a safe navigation when the underlying assumptions on error probability density functions may not be ful lled. In addition, interval-based fault detection techniques have been investigated in literature e.g: qrelax intersection. This technique guarantees the protection level but it su ers from low accuracy in the multiple fault situation. In this work, we propose a fault detection and exclusion technique based on deterministic observation intervals. The inconsistency of the positioning problem is indicated by the size of the polytope obtained from the intersection of the observation intervals. For the optimal case of no observation noise and no outliers, the polytope is a zonotope. Thus, we will use the normalized relative volume between a nominal polytope (zonotope) and the

Global Navigation Satellite Systems (GNSS) deliver absolute position and velocity, as well as time information (P, V, T). However, in urban areas, the GNSS navigation performance is restricted due to signal obstructions and multipath. This is especially true for applications dealing with highly automatic or even autonomous driving. Subsequently, multi-sensor platforms including laser scanners and cameras, as well as map data are used to enhance the navigation performance, namely in accuracy, integrity, continuity and availability. Although well-established procedures for integrity monitoring exist for aircraft navigation, for sensors and fusion algorithms used in automotive navigation, these concepts are still lacking. The research training group i.c.sens, integrity and collaboration in dynamic sensor networks, aims to fill this gap and to contribute to relevant topics. This includes the definition of alternative integrity concepts for space and time based on set theory and interval mathematics, establishing new types of maps that report on the trustworthiness of the represented information, as well as taking advantage of collaboration by improved filters incorporating person and object tracking. In this paper, we describe our approach and summarize the preliminary results.

Reliable computation of confidence domains for point position, and inconsistency check of the observations is of great importance for any navigation system, especially for safety critical applications. In this work, non-stochastic error bound for reaming observation errors are introduced as intervals. Using concepts from interval mathematics four different deterministic bounding methods will be applied on GPS positioning, namely: Least Squares Adjustments based on Interval Analysis (LSA-IA), extension of LSA-IA by means of Zonotopes, Linear Programing (LP) applied to the navigation equation linearized by Taylor expansion, and finally solving the non-linear navigation equation by Set Inversion Via Interval Analysis (SIVIA). These methods provide different type of confidence regions and inconsistency measures. Each method will be investigated alone, then conducting a comparison study of the mathematical properties of these methods, and providing geometrical interpretation with respect to the navigation problem (e.g. form and orientation of the bounding area w.r.t. line-of-sight and number of satellites in view). Thanks to a reduced position scatter and the implicit inconsistence measure, the LP approach seems especially promising for further investigation towards an alternative integrity concept.

Traditional statistical data processing techniques (such as Least Squares) assume that we know the probability distributions of measurement errors. Often, we do not have full information about these distributions. In some cases, all we know is the bound of the measurement error; in such cases, we can use known interval data processing techniques. Sometimes, this bound is fuzzy; in such cases, we can use known fuzzy data processing techniques. However, in many practical situations, we know the probability distribution of the random component of the measurement error and we know the upper bound on the measurement error's systematic component. For such situations, no general data processing technique is currently known. In this paper, we describe general data processing techniques for such situations, and we show that taking into account interval and fuzzy uncertainty can lead to more adequate statistical estimates.