# Robust Statistics for GNSS Positioning under Harsh Conditions: A Useful Tool?
## 5 Test and Results
### Experimentation under Real Harsh Conditions  
To experimentally address the performance of the MM-estimator for GNSS positioning, a data collection was performed for an automotive scenario. The test vehicle was equipped with a geodetic antenna (navXperience 3G+C) connected to a geodetic GNSS receiver (Javad Delta), as shown in Figure 6 (left). The experiment was carried out on 15 May 2019 (DOY 135, UTC 10:00-18:00), covering a distance of approximately 800 km from Koblenz, in west Germany, to Neustrelitz, in northeast Germany, as illustrated in Figure 6 (right). Along the route, a wide variety of GNSS harsh conditions were confronted: urban navigation, high-speed highways, national roads under forest foliage, bridge passing, etc. Thus, the capability of the MM estimator for dealing with corrupted observations can be consistently evaluated on real multipath and NLOS conditions. The onboard GNSS receiver allows for multi-constellation (GPS, GLONASS, and Galileo), multi-frequency (L1, L2, and L5) tracking, and the sampling rate was set at 2 Hz. The ground truth reference solution was based on a dual frequency GPS+GLONASS PPP solution derived from the CSRS-PPP service [57]. Unfortunately, the PPP solution results are unavailable for the most challenging situations (e.g., tunnel or bridge crossing), thus the LS and MM performance could not be assessed during these epochs. Figure 4: Loss-of-efficiency of the estimators as a function of the number of observations available. For the evaluation, the positioning performance of a classical LS solution was compared to the MM-estimator, which has been shown to be the most suitable among the robust methods for GNSS  
Figure 5: Surface (**left** column) and contour (**right** column) plot of the loss functions, projected in the east–north frame, for the LS (**top**), M-Huber (**middle**) and M-Tukey (**bottom**) estimates. The red diamond highlights the ground truth on the right column. Figure 6: Vehicle employed for the measurement campaign (**left**). Trajectory covered during the data collection, starting in Koblenz and finishing in Neustrelitz (**right**). positioning. The estimation of the positioning solution used GPS and Galileo observations on the L1 frequency, with an elevation mask of 5\({}^{\circ}\), and the clock offset of Galileo was considered independent from the GPS one (hence, the number of parameters of the state estimate is of dimension five). The number of observations and the Position Dilution of Precision (PDOP) over time are illustrated in Figure 7 (bottom left). The combination of GPS and Galileo grants the availability of radio-navigation for around 96.5% of time, with often satellite tracking losses due to signal reflection and blockage. Figure 7 (top left) depicts the three-dimensional squared positioning error over time for the LS and MM estimators. For the majority of the studied epochs--nominal opens-sky conditions for GNSS navigation--the performance of the LS and MM estimators is equivalent and the LoE of the MM is not even perceptible. This statement is supported with the results shown in Figure 7 (right), which illustrates the distribution of the positioning errors. For the bulk of the results, or errors under 5 m, LS and MM offer similar performance, with the MM even being slightly better. While LS presents a large population of positioning errors between 10 and 20 m, the MM estimator is capable of mitigating the vast majority of these errors. The largest positioning errors--for instance, shortly after 12:00 and around 15:30--cannot be mitigated by the MM estimator, which becomes as biased as the LS. This is due to a reduced satellite visibility combined with several satellites being contaminated for these epochs. To better illustrate the outlier rejection capability of the MM estimator, time spans \"A\" and \"B\" are highlighted using a gray shaded area in Figure 7 (top left). These periods **A** and **B** of 15 and 30 min duration, respectively, are shown in detail in Figure 8, including pictures taken from the automobile during these instances.