# Robust TOA-Based UAS Navigation under Model Mismatch in GNSS-Denied Harsh Environments

^{*}

## Abstract

**:**

## 1. Introduction

## 2. State-Space TOA-Based Navigation Formulation

#### 2.1. Nominal State-Space Model

#### 2.2. Non-Nominal Mismatched State-Space Model

#### 2.2.1. Mismatched SSM

#### 2.2.2. Harsh Propagation Conditions

## 3. State-of-the-Art EKF-Based Solutions

#### 3.1. Standard Extended KF Solution

#### 3.2. Augmented-State EKF Solution

## 4. Robust Filtering under Model Mismatch

#### 4.1. The Idea behind Robust Estimation

#### 4.2. Standard Robust Regression EKF

#### 4.3. Robust Regression EKF for Mismatched Models

- (i)
- first, consider the augmented-state formulation in Section 3.2 to cope with the Tx mismatch;
- (ii)
- second, consider a RREKF-based solution; and
- (iii)
- within the M-estimator, instead of the MAD, use a mismatched three sigma rule to normalize the residuals, that is,$${\widehat{\sigma}}_{3s}=3\sqrt{{\sigma}_{r,i}^{2}+{\sigma}_{p,i}^{2}}.$$

#### 4.4. Computationally Efficient Solution: Robust Weighting Uncertainty Indicator for Robust EKF

## 5. Illustrative UWB-Based Indoor Navigation Example

- Standard EKF (SEKF) in Section 3.1.
- Augmented-state EKF (MEKF) in Section 3.2.
- Robust regression EKF (RREKF) from ([26] Chapter 7).
- Robust regression EKF for mismatched models (MRREKF) in Section 4.3.
- Robust covariance weighting EKF (MRCEKF) in Section 4.4.

#### 5.1. Simulation Scenario

#### 5.2. Results

- As expected, for a given percentage of contamination, if the outliers are not very strong ($\alpha \le 1$ m), the MEKF performs much better than the SEKF, but it breaks down rapidly as the amplitude of the outliers increases. Indeed, the MEKF tries to compensate the modeling error due to the mismatch, but the measurements, which are corrupted by outliers, are understood by the filter as a bias in the anchors’ position. In comparison, the SEKF is less affected by this behavior, with no clear performance breakdown, but in general this method provides a worse estimate when compared to the two robust estimators.
- W.r.t. the MEKF behavior, the standard RREKF estimator is relatively stable, thus correctly dealing with outliers, but its nominal performance is degraded because of the mismatch, therefore not bringing a suitable solution for the problem at hand. For low outlier magnitudes, $\alpha $, it is clear, when comparing the outcome of both MEKF and RREKF, that a method being able to cope with both mismatch and outliers should retain the best qualities of both estimators.
- Indeed, the compromise between the MEKF and RREKF is brought by the new MRCEKF and MRREKF, which are relatively stable and cope perfectly with contaminated observations and anchor position mismatch, up to a certain level of contamination. Notice that the interest of the MRREKF is clear with a large outlier amplitude. The MRCEKF outperforms the rest when the number of contaminated measurements is less or equal than 6 and $\epsilon \le 50\%$, and the MRREKF provides a good solution up to eight contaminated anchors and for $\epsilon =50\%$. It can be concluded, from the analysis of these results, that for $\epsilon <50\%$, $\alpha \le 3$ m and the number of anchors in multipath/NLOS conditions $\le 6$, the MRCEKF approach guarantees horizontal and vertical performances below the tens of centimeters, unlike the other standard approaches. For more extreme propagation conditions, the MRREKF is needed and we can not avoid the iterative M-estimate procedure within every Kalman filter iteration.

## 6. Realistic UWB/LTE TOA-Based Urban Navigation Example

#### 6.1. Realistic Urban Scenario Definition

- Anchors located in the agent’s reference anchor street have an outlier percentage of $\epsilon =10$, that is, mild multipath probability due to the narrow streets set-up.
- Anchors located in a corner of the trajectory have an outlier percentage of $\epsilon =25$, that is, a larger multipath conditions probability.
- The rest of the anchors not on the same agent’s reference anchor street have an outlier percentage of $\epsilon =50$, induced by the lack of direct visibility, i.e., NLOS conditions.

#### 6.2. Results

## 7. Conclusions and Outlook

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Amin, M.G.; Closas, P.; Broumandan, A.; Volakis, J.L. Vulnerabilities, threats, and authentication in satellite-based navigation systems [scanning the issue]. Proc. IEEE
**2016**, 104, 1169–1173. [Google Scholar] [CrossRef] - Medina, D.; Li, H.; Vilà-Valls, J.; Closas, P. Robust Statistics for GNSS Positioning under Harsh Conditions: A Useful Tool? Sensors
**2019**, 19, 5402. [Google Scholar] [CrossRef] [PubMed][Green Version] - Teunissen, P.J.G.; Montenbruck, O. (Eds.) Handbook of Global Navigation Satellite Systems; Springer: Basel, Switzerland, 2017. [Google Scholar]
- Del Peral-Rosado, J.A.; Raulefs, R.; Lopez-Salcedo, J.A.; Seco-Granados, G. Survey of Cellular Mobile Radio Localization Methods: From 1G to 5G. IEEE Commun. Surv. Tutor.
**2018**, 20, 1124–1148. [Google Scholar] [CrossRef] - Dardari, D.; Closas, P.; Djuric, P. Indoor tracking: Theory, Methods, and Technologies. IEEE Trans. Veh. Tech.
**2015**, 64, 1263–1278. [Google Scholar] [CrossRef][Green Version] - Groves, P.D. Principles of GNSS, Inertial, and Multisensor Integrated Navigation Systems, 2nd ed.; Artech House: Norwood, MA, USA, 2013. [Google Scholar]
- Liu, J.; Cai, B.G.; Wang, J. Cooperative Localization of Connected Vehicles: Integrating GNSS with DSRC Using a Robust Cubature Kalman Filter. IEEE Trans. Intel. Transp. Syst.
**2017**, 18, 2111–2125. [Google Scholar] [CrossRef] - Li, J.; Gao, J.; Zhang, H.; Qiu, T.Z. RSE-assisted lane-level positioning method for a connected vehicle environment. IEEE Trans. Intel. Transp. Syst.
**2019**, 20, 2644–2656. [Google Scholar] [CrossRef] - Wang, J.; Liu, J.; Kato, N. Networking and communications in autonomous driving: A survey. IEEE Commun. Surv. Tutor.
**2019**, 21, 1243–1274. [Google Scholar] [CrossRef] - Vetrella, A.R.; Opromolla, R.; Fasano, G.; Accardo, D.; Grassi, M. Autonomous Flight in GPS-Challenging Environments Exploiting Multi-UAV Cooperation and Vision-aided Navigation. In Proceedings of the AIAA Information Systems-AIAA Infotech @ Aerospace, Grapevine, TX, USA, 9–13 January 2017. [Google Scholar]
- Lau, M.; Steffens, M.; Mavris, D.N. Evaluating the Performance Impact of Cooperative Navigation for Unmanned Aerial Systems in GPS-Denied Environments. In Proceedings of the 2018 Modeling and Simulation Technologies Conference, Atlanta, Georgia, 8–12 January 2018. [Google Scholar]
- Kassas, Z.; Khalife, J.; Shamaei, K.; Morales, J. I hear, therefore I know where I am: Compensating for GNSS limitations with cellular signals. IEEE Signal Process. Mag.
**2017**, 34, 111–124. [Google Scholar] [CrossRef] - Shamaei, K.; Khalife, J.; Kassas, Z. Exploiting LTE signals for navigation: Theory to implementation. IEEE Trans. Wirel. Commun.
**2018**, 17, 2173–2189. [Google Scholar] [CrossRef][Green Version] - Del Peral-Rosado, J.A.; Lopez-Salcedo, J.A.; Zanier, F.; Seco-Granados, G. Position Accuracy of Joint Time-Delay and Channel Estimators in LTE Networks. IEEE Access
**2018**, 6, 25185–25199. [Google Scholar] [CrossRef] - Del Peral-Rosado, J.A.; Seco-Granados, G.; Kim, S.; Lopez-Salcedo, J.A. Network Design for Accurate Vehicle Localization. IEEE Trans. Veh. Technol.
**2019**, 68, 4316–4327. [Google Scholar] [CrossRef] - Maaref, M.; Khalife, J.; Kassas, Z. Lane-level Localization and Mapping in GNSS-challenged Environments by Fusing Lidar Data and Cellular Pseudoranges. IEEE Trans. Intell. Veh.
**2019**, 4, 73–89. [Google Scholar] [CrossRef] - Kassas, Z.; Maaref, M.; Morales, J.; Khalife, J.; Shamaei, K. Robust vehicular navigation and map-matching in urban environments with IMU, GNSS, and cellular signals. IEEE Intell. Transp. Syst. Mag.
**2020**, 12, 36–52. [Google Scholar] [CrossRef] - Dardari, D.; Conti, A.; Ferner, U.; Giorgetti, A.; Win, M.Z. Ranging with ultrawide bandwidth signals in multipath environments. Proc. IEEE
**2009**, 97, 404–426. [Google Scholar] [CrossRef] - Jiménez, A.R.; Seco, F. Comparing Decawave and Bespoon UWB Location Systems: Indoor/Outdoor Performance Analysis. In Proceedings of the IEEE International Conference on Indoor Positioning and Indoor Navigation (IPIN), Sapporo, Japan, 18–21 September 2016. [Google Scholar]
- Tiemann, J.; Schweikowski, F.; Wietfeld, C. Design of an UWB Indoor-Positioning System for UAV Navigation in GNSS-Denied Environments. In Proceedings of the IEEE International Conference on Indoor Positioning and Indoor Navigation (IPIN), Banff, AB, Canada, 7 December 2015. [Google Scholar]
- Tiemann, J.; Pillmann, J.; Boecker, S.; Wietfeld, C. Ultra-Wideband Aided Precision Parking for Wireless Power Transfer to Electric Vehicles in Real Life Scenarios. In Proceedings of the IEEE Vehicular Technology Conference (VTC-Fall), Montréal, QB, Canada, 18–21 September 2016. [Google Scholar]
- Pietrzyk, M.M.; Grun, T.V.D. Experimental Validation of a TOA UWB Ranging Platform with the Energy Detection Receiver. In Proceedings of the IEEE International Conference on Indoor Positioning and Indoor Navigation (IPIN), Zurich, Switzerland, 15–17 September 2010. [Google Scholar]
- Cetin, O. An Experimental Study of High Precision TOA based UWB Positioning Systems. In Proceedings of the IEEE International Conference on Ultra-Wideband (ICUWB), Syracuse, NY, USA, 17–20 September 2012. [Google Scholar]
- Pagès, G.; Vilà-Valls, J. UWB-based Indoor Navigation with Uncertain Anchor Nodes Positioning. In Proceedings of the ION GNSS+, Miami, FL, USA, 16–20 September 2019. [Google Scholar]
- Cebrian, A.; Bellés, A.; Martin, C.; Salas, A.; Fernández, J.; Arribas, J.; Vilà-Valls, J.; Navarro, M. Low-Cost Hybrid GNSS/UWB/INS Integration for Seamless Indoor/Outdoor UAV Navigation. In Proceedings of the ION GNSS+, Miami, FL, USA, 16–20 September 2019. [Google Scholar]
- Zoubir, A.M.; Koivunen, V.; Ollila, E.; Muma, M. Robust Statistics for Signal Processing; Cambridge University Press: Cambridge, UK, 2018. [Google Scholar]
- Vilà-Valls, J.; Closas, P. NLOS Mitigation in Indoor Localization by Marginalized Monte Carlo Gaussian Smoothing. EURASIP J. Adv. Signal Process.
**2017**, 62. [Google Scholar] [CrossRef] - Vilà-Valls, J.; Vincent, F.; Closas, P. Decentralized Information Filtering under Skew-Laplace Noise. In Proceedings of the 2019 53rd Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, USA, 29 July 2019. [Google Scholar]
- Anderson, B.; Moore, J.B. Optimal Filtering; Prentice–Hall: Englewood Cliffs, NJ, USA, 1979. [Google Scholar]
- Chaumette, E. Minimum Variance Distortionless Response Estimators For Linear Discrete State-Space Models. IEEE Trans. Autom. Control
**2017**, 64, 2048–2055. [Google Scholar] [CrossRef][Green Version]

**Figure 2.**2D (

**top**) and 3D (

**bottom**) trajectories and the corresponding estimates. The scenario is based on 5 anchors out of 8 having a random position bias of $\pm 0.5$ m (blue circles). The mixed line-of-sight/non-line-of-sight (LOS/NLOS) conditions are generated for a specific region of the trajectory and concerns all 8 anchors. They are modeled as a Markovian process with the following parameters, ${\sigma}_{{}_{{r}_{i},\mathrm{NLOS}}}=3$ m, $\epsilon =25\%$.

**Figure 3.**Average horizontal (

**top**) and vertical (

**bottom**) root mean sqaure error (RMSE) results using the standard extended Kalman filter (SEKF), the mismatch extended Kalman filter (MEKF), the robust regression extended Kalman filter (RREKF), and the mismatch robust weight-based extended Kalman filter (MRCEKF).

**Figure 4.**3D Image of the Urban Scenario with the mobile agent’s real trajectory (red), and the three LTE antennas (blue circles).

**Figure 5.**2D map of the urban scenario with the agent’s real trajectory (red), the simulated trajectory (yellow), and the LTE antennas with their labels (blue circles).

**Figure 7.**Example of the number of communications links with the mobile agent located near anchor 12. The anchors having a link are circled in red.

**Figure 8.**Horizontal RMSE results of the Monte Carlo simulation on 1000 runs. Each plot depicts a different anchor position bias $\Delta {\mathbf{p}}_{i}\in $ (

**a**) $[-1,+1]$ m, (

**b**) $[-3,+3]$ m, (

**c**) $[-5,+5]$ m and (

**d**) $[-10,+10]$ m.

**Figure 9.**Vertical RMSE results of the Monte Carlo simulation on 1000 runs. Each plot depicts a different anchor position bias $\Delta {\mathbf{p}}_{i}\in $ (

**a**) $[-1,+1]$ m, (

**b**) $[-3,+3]$ m, (

**c**) $[-5,+5]$ m and (

**d**) $[-10,+10]$ m.

Number of runs | 100 |

Range variance noise (m${}^{2}$) | 0.01 |

Number of anchors | 8 |

Number of mismatched anchors | ${L}_{e}=5$ |

Anchor position bias (cm) | $[-50,+50]$ |

Outlier percentage $\epsilon $ | 10–25–50 |

Outlier magnitude $\alpha $ | 1–5–10–30–60 |

Number of multipath/NLOS anchors | 2–4–6–8 |

Number of runs | 1000 |

UWB range variance noise (m${}^{2}$) | 0.01 |

LTE range variance noise (m${}^{2}$) | 30.25 |

Number of anchors | 35 (3 LTE + 32 UWB) |

Number of mismatched anchors | ${L}_{e}=32$ |

Anchor position bias (m) | $[-1,+1],[-3,+3],[-5,+5],[-10,+10]$ |

Outlier magnitude $\alpha $ | 30–50–100–150–200–300 |

© 2020 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/).

## Share and Cite

**MDPI and ACS Style**

Mortier, J.; Pagès, G.; Vilà-Valls, J. Robust TOA-Based UAS Navigation under Model Mismatch in GNSS-Denied Harsh Environments. *Remote Sens.* **2020**, *12*, 2928.
https://doi.org/10.3390/rs12182928

**AMA Style**

Mortier J, Pagès G, Vilà-Valls J. Robust TOA-Based UAS Navigation under Model Mismatch in GNSS-Denied Harsh Environments. *Remote Sensing*. 2020; 12(18):2928.
https://doi.org/10.3390/rs12182928

**Chicago/Turabian Style**

Mortier, Jan, Gaël Pagès, and Jordi Vilà-Valls. 2020. "Robust TOA-Based UAS Navigation under Model Mismatch in GNSS-Denied Harsh Environments" *Remote Sensing* 12, no. 18: 2928.
https://doi.org/10.3390/rs12182928