# Robust Statistics for GNSS Positioning under Harsh Conditions: A Useful Tool?

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## Abstract

**:**

## 1. Introduction

## 2. Robust Statistics Principles

#### 2.1. Dictionary of Robust Statistics Terms

#### 2.2. Robust Estimates for Regression Problems

#### 2.2.1. Huber and Tukey Families of Loss Functions

#### 2.2.2. M-Estimator

#### 2.2.3. S-Estimator

#### 2.2.4. MM-Estimator

- (1)
- Compute an initial consistent S-estimate of $\mathbf{x}$, namely ${\widehat{\mathbf{x}}}_{0}$, with a high breakdown point but possibly low normal efficiency.
- (2)
- Compute an M-estimate of the scale of the residuals ${s}_{M}\left(\mathbf{r}\left({\widehat{\mathbf{x}}}_{0}\right)\right)$ using the high breakdown point estimate ${\widehat{\mathbf{x}}}_{0}$.
- (3)
- Compute the regression M-estimate initialized at ${\widehat{\mathbf{x}}}_{0}$, considering the robust scale estimate ${s}_{M}\left(\mathbf{r}\left({\widehat{\mathbf{x}}}_{0}\right)\right)$ and using a recursive IRLS approach.

## 3. Robust Statistics for GNSS Positioning

Algorithm 1: IRLS procedure for robust GNSS SPP. |

## 4. Loss-of-Efficiency in Robust PVT Solvers

## 5. Test and Results

#### 5.1. Simulated Environment

#### 5.2. Experimentation under Real Harsh Conditions

## 6. Outlook and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Illustration of the loss (

**left**), score (

**middle**) and weighting (

**right**) functions for different classical and robust estimators. Here, the families of Huber and Tukey functions are depicted with parameters $a=1.345$ and $c=4.685$, respectively.

**Figure 3.**RMSE positioning error for $\epsilon \in \{10,30,40\}\%$ contamination data (each column) and for $n\in \{10,40\}$ (single- and multi-constellation cases, respectively) pseudorange observations (each row).

**Figure 4.**Loss-of-efficiency of the estimators as a function of the number of observations available.

**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**).

**Figure 7.**Number of GPS+Galileo satellites tracked and PDOP (

**bottom left**). Squared positioning errors for the LS and MM estimators over time, and highlight on time spans $\mathbf{A}$ and $\mathbf{B}$ (

**top left**). Histogram of positioning errors for LS and MM-estimator (

**right**).

**Figure 8.**Illustration of time span “A” where multiple bridges are present (

**top left**) and positioning errors during such time (

**top right**). Illustration of time span “B” for navigation under dense foliage (

**bottom left**) and the associated positioning errors (bottom right).

Simulation parameters | |
---|---|

Number of satellites n | {10,40} |

Percentage of outliers $\epsilon $ | {0,10,30,40} |

Outlier magnitude $\alpha $ | {1,3,6,10,30,60,100} |

Robust parameters | $a=1.345,b=0.5,c=4.685$ |

Single-constellation scenario setup | |

UTC time | $15/05/2017\phantom{\rule{4pt}{0ex}}09:30$ |

Location | Koblenz, Germany |

(5021${}^{\prime}$56${}^{\u2033}$ N, 735${}^{\prime}$55${}^{\u2033}$ E) | |

PDOP | $1.72$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Medina, D.; Li, H.; Vilà-Valls, J.; Closas, P.
Robust Statistics for GNSS Positioning under Harsh Conditions: A Useful Tool? *Sensors* **2019**, *19*, 5402.
https://doi.org/10.3390/s19245402

**AMA Style**

Medina D, Li H, Vilà-Valls J, Closas P.
Robust Statistics for GNSS Positioning under Harsh Conditions: A Useful Tool? *Sensors*. 2019; 19(24):5402.
https://doi.org/10.3390/s19245402

**Chicago/Turabian Style**

Medina, Daniel, Haoqing Li, Jordi Vilà-Valls, and Pau Closas.
2019. "Robust Statistics for GNSS Positioning under Harsh Conditions: A Useful Tool?" *Sensors* 19, no. 24: 5402.
https://doi.org/10.3390/s19245402