# Robust Statistics for GNSS Positioning under Harsh Conditions: A Useful Tool?
## 1 Introduction  
Global navigation satellite systems (GNSS) play a fundamental role on prospective applications of intelligent transportation systems (ITS), as the main source of positioning information [1]. Besides, GNSS provides timing synchronization to critical applications such as the power grid or the stock market [2]. However, GNSS performance can be easily degraded by natural phenomena and signal reflection. Navigation in urban scenarios results particularly challenging due to the presence of severe multipath effects, inducing large errors in the observed pseudorange measurements. Most positioning techniques are based on maximum likelihood (ML) estimation, since it provides optimal solutions under the assumption of Gaussian distributed observation noise. Although this assumption is generally fulfilled for nominal GNSS open-sky conditions, positioning on signal-degraded scenarios constitutes a challenge for ML estimators such as the least-squares (LS) [3]. Thus, the GNSS community has devoted great efforts towards the development of resilient navigation solutions [4]. One of the most popular approaches is based on solution separation--also known as consistency-checking--where a statistical test is applied to the estimated residuals to verify whether the Gaussian assumption is fulfilled. Otherwise, combinations of subsets excluding one observation are computed and the statistical test applied again. This procedure is repeated until a fault-free subset is found. Advanced receiver autonomous integrity monitoring (ARAIM) is possibly the most well-known representative of the solution separation approach, becoming thede facto navigation method for vertical guidance in the aviation domain [5; 6]. Numerous other works have adapted consistency-checking navigation algorithms for single point positioning (SPP) in signal-degraded scenarios [7; 8; 9; 10]. Unfortunately, with the deployment and growing availability of GNSS constellations, solution separation methods present challenging scalability issues since their computation complexity constitutes a combinatorial problem with the number of observations and outliers, eventually becoming an intractable problem. Robust statistics provides an alternative framework for the definition of navigation methods resilient against multiple erroneous observations. Originally suggested for general data analysis in the early 1970s [11; 12; 13], robust estimators have experienced substantial research growth and their use has extended to manifold fields: signal processing [14; 15; 16], biomedical [17; 18], power systems [19], etc. Within the scope of GNSS, robust methods have been successfully applied to enhance receivers with anti-jamming capabilities based on the so-called Robust Interference Mitigation [20; 21; 22; 23; 24; 25; 26]. The application of robust estimators to compute position, velocity, and time (PVT) solutions in satellite-based navigation has also appealed numerous authors, both for memory-less SPP [27; 28; 29; 30] and for recursive estimation [31; 32; 33]. In that PVT context, the performance of robust techniques has been demonstrated on both simulated and real data, and this paper attempts to characterize those estimators in terms of quantities relevant to the robust statistics literature. This paper focuses on the SPP problem, thus purposely does not consider precise point positioning (PPP) or real-time kinematic (RTK) approaches, which typically involve more complex estimates and the application of different methodologies [34; 35; 36] to the ones investigated here. This work introduces the principles of robust statistics for regression problems and presents three of the most popular robust methods: M-, S-, and MM-estimators. Besides, a comprehensive guide on the implementation of such techniques for solving the GNSS SPP problem is detailed. Moreover, the specific challenges on the application of robust estimators for GNSS positioning are discussed.