# Robust Statistics for GNSS Positioning under Harsh Conditions: A Useful Tool?
## 5 Test and Results
### Simulated Environment
In Figure 3, the fraction of outliers \(\varepsilon\) grows from left to right, with \(\varepsilon=10\%\) (left column), \(\varepsilon=30\%\) (middle column), and \(\varepsilon=40\%\) (right column). A common element across all cases is, the absolute lack of robustness of the LS-estimates, whose RMSE is driven by the large errors present in the corrupted observations. Looking at the left column, M, S, and MM estimates evidence robustness at \(\varepsilon=10\%\), neglecting the effects of outliers regardless of their magnitude and number of constellations. The latter is interesting, since it appears that, even for the single constellation case (Figure 3, top left), ten observations provide enough redundancy for the estimation of four parameters and spotting a single corrupted measurement. For the middle column \(\varepsilon=30\%\), the S and MM estimators remain nearly unaffected by the outliers, indicating that their breakdown point \(\varepsilon^{*}\geq 30\%\) for \(n\geq 10\). On the other hand, the M estimator breaks down for the single constellation case. Finally, let us examine the right column with \(\varepsilon=40\%\). It is clear that all robust methods break down before such high fraction of contamination for the single constellation case. On the contrary, the S and MM estimators are capable of successfully bounding the effects of outliers for the multi-constellation scenario, where the large number of measurements provide with sufficient data redundancy. It becomes evident that robust methods, especially the MM estimator, represent a promising alternative to traditional ML or LS-based GNSS positioning. Especially for a near future, in which multiple GNSS constellations will be fully deployed and a large number of observations will be made available, robust methods can assure great resilience against satellite faults at a cost of minimal efficiency loss, as shown below. The relative efficiency of the estimators with respect to the LS is studied for the nominal case--e.g., when no outlying observations are present. Figure 4 depicts the LoE of the estimators, as defined in Section 4. The S-estimator is clearly the least efficient among the evaluated methods, and its efficiency even decreases with the number of observations. Contrarily, the MM-estimator exhibits the closest performance to the LS and it can be considered as an efficient estimator. Notice that the MM efficiency scales with the number of observations, making it an appealing option for prospective multi-GNSS scenarios. Overall, one can conclude that the MM-estimator results the most interesting among the robust methods compared, offering robustness (high breakdown point) while maintaining a high efficiency at the nominal Gaussian distribution of errors. Figure 3: RMSE positioning error for \(\varepsilon\in\{10,30,40\}\%\) contamination data (each column) and for \(n\in\{10,40\}\) (single- and multi-constellation cases, respectively) pseudorange observations (each row). To gain understanding on how robust estimators actually perform for GNSS-positioning, a test scenario with a single fault was studied. Moreover, this example serves as pictorial support for the discussion on convex (based on monotone loss functions) against nonconvex (based on redescending loss functions) minimization. Making use of the single-constellation geometry of Figure 2, satellite observations are assumed to present a low variance noise (0.12 m\({}^{2}\)) and a large bias inferred to satellite \(G17\). This drives to a position error of approximately 6 m in the west direction, while the north and down directions remain mostly unaffected. Figure 5 depicts the surface (on the left column) and contour (on the right column) of the LS, M-Huber, and M-Tukey loss functions (first, second, and third rows, respectively) for the aforementioned test scenario. Besides, the ground truth solution is marked with a red diamond on the right column.