# Robust-Extended Kalman Filter and Long Short-Term Memory Combination to Enhance the Quality of Single Point Positioning

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*Applied Sciences*: Invited Papers in Electrical, Electronics and Communications Engineering Section)

## Abstract

**:**

## 1. Introduction

## 2. Single Point Positioning Technology

## 3. Robust Extended Kalman Filter

#### 3.1. Robust Extended Kalman Filter Model

**Step 1**: Linearizing equations

**Step 2**: Reforming the filter

**Step 3**: Updating

#### 3.2. MM Estimation Theory

#### 3.3. Iterative Reweighted Least Squares Algorithm

- Find an initial estimate ${\widehat{X}}_{0}$$$\begin{array}{c}\hfill {\widehat{X}}_{0}={\left({G}_{0}^{T}{G}_{0}\right)}^{-1}{G}_{0}^{T}y\end{array}$$
- Estimate the vector for initial residuals of the observations ${r}_{0}$:$${r}_{0}=y-{G}_{0}{\widehat{X}}_{0}$$
- Define the initial scale value ${s}_{0}$ [33]$${s}_{0}=1.4826median\left(\left|{r}_{0}\right|\right)$$
- Estimate the initial diagonal weight matrix by MM-Estimation.$${W}_{0}(i,i)=\left\{\begin{array}{cc}1\hfill & \left|\frac{{r}_{0}\left(i\right)}{{s}_{0}}\right|\le \alpha \hfill \\ 0\hfill & \left|\frac{{r}_{0}\left(i\right)}{{s}_{0}}\right|>\alpha \hfill \end{array}\right.$$
- While (j is $jth$ iteration)
- (a)
- Update the value of the matrix ${G}_{j}$ at the $jth$ iteration with ${\widehat{X}}_{j-1}$
- (b)
- Solve the estimated state ${\widehat{X}}_{j}$ using the weighted least-squared method$${\widehat{X}}_{j}={\left({G}_{j}^{T}{W}_{j-1}{G}_{j}\right)}^{-1}{G}_{j}^{T}{W}_{j-1}y$$
- (c)
- Calculate the HPL (in Section 3.4)
- (d)
- If $\u2225{\widehat{X}}_{j}-{\widehat{X}}_{j-1}\u2225<0.001$; break
- (e)
- If $\u2225{\widehat{X}}_{j}-{\widehat{X}}_{j-1}\u2225>0.001$; continue
- (f)
- Update the estimated residuals$${r}_{j}=y-{G}_{j}{\widehat{X}}_{j}$$
- (g)
- Calculate the scale value$${s}_{j}=1.4826median\left(\left|{r}_{j}\right|\right)$$
- (h)
- Recalculate the diagonal weighted matrix using MM-Estimation:$${W}_{j}(i,i)=\left\{\begin{array}{cc}1\hfill & \left|\frac{{r}_{j}\left(i\right)}{{s}_{j}}\right|\le \alpha \hfill \\ 0\hfill & \left|\frac{{r}_{j}\left(i\right)}{{s}_{j}}\right|>\alpha \hfill \end{array}\right.$$
- (i)
- Go to step (a)

- End
- If $HPL\le HAL$ then the estimated positions are accepted, (HAL: Horizontal Alert Limit)If not, they are rejected.

#### 3.4. RAIM Algorithm

## 4. Experimental Results of Applying Robust-EKF

#### 4.1. Applying Robust-EKF

**Step 1**: Linearizing equations

**Step 2**: Reforming filter

**Step 3**: Updating

#### 4.2. Experimental Results

- Scenario #1: Navigation solution based on GPS data,
- Scenario #2: Navigation solution based on Galileo data,
- Scenario #3: Navigation solution based on GLONASS data,
- Scenario #4: Navigation solution based on GPS/Galileo/GLONASS data,
- Scenario #5: Navigation solution based on robust-EKF GPS/Galileo/GLONASS data.

## 5. De-Noising Filter Method and Experimental Results

#### 5.1. De-Noising Filter Model

#### 5.2. Experimental Results

## 6. Conclusions and Perspectives

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Weighting functions [22].

Name | Weight Function |
---|---|

Huber | $w\left({r}_{i}\right)=\left\{\begin{array}{cc}1& |{r}_{i}|\le \alpha \\ \alpha /|{r}_{i}|& |{r}_{i}|>\alpha \end{array}\right.$ |

Bi-Tukey | $w\left({r}_{i}\right)=\left\{\begin{array}{cc}1& |{r}_{i}|\le \alpha \\ 0& |{r}_{i}|\ge \alpha \end{array}\right.$ |

Bi-Square | $w\left({r}_{i}\right)=\left\{\begin{array}{cc}[1-(|{r}_{i}{|/\alpha )]}^{2}\hfill & |{r}_{i}|\le \alpha \hfill \\ 0\hfill & |{r}_{i}|>\alpha \hfill \end{array}\right.$ |

RMS Error (m) | GPS | GAL | GLO | GPS/GAL/GLO | Robust-EKF GPS/GAL/GLO |
---|---|---|---|---|---|

RMS-E | 1.42 | 1.34 | 31.00 | 8.11 | 0.74 |

RMS-N | 1.46 | 1.2 | 6.65 | 2.74 | 0.75 |

RMS-U | 3.94 | 3.08 | 25.7 | 9.50 | 1.82 |

3D-RMS | 4.43 | 3.56 | 40.81 | 12.73 | 2.1 |

RMS Error (m) | Robust-EKF | rEKF-LSTM |
---|---|---|

RMS-E | 0.82 | 0.39 |

RMS-N | 0.83 | 0.32 |

RMS-U | 2.04 | 0.36 |

3D-RMS | 2.35 | 0.62 |

RMS Error (m) | GPS | GAL | GLO | GPS/GAL/GLO | Robust-EKF GPS/GAL/GLO | rEKF-LSTM |
---|---|---|---|---|---|---|

RMS-E | 1.43 | 1.21 | 30.71 | 7.08 | 0.82 | 0.39 |

RMS-N | 1.58 | 1.19 | 6.18 | 2.27 | 0.83 | 0.32 |

RMS-U | 3.32 | 3.07 | 21.10 | 8.18 | 2.04 | 0.36 |

3D-RMS | 3.95 | 3.50 | 37.76 | 11.05 | 2.35 | 0.62 |

Base Station | RMS Error (m) | GPS | GAL | GLO | GPS/GAL/GLO | Robust-EKF GPS/GAL/GLO | rEKF-LSTM |
---|---|---|---|---|---|---|---|

RMS-E | 1.12 | 1.36 | 17.76 | 3.40 | 0.72 | 0.73 | |

AJAC | RMS-N | 1.65 | 1.25 | 9.30 | 4.57 | 0.88 | 0.46 |

RMS-U | 2.52 | 2.42 | 3.40 | 4.75 | 1.33 | 0.45 | |

3D-RMS | 3.21 | 3.04 | 20.33 | 7.42 | 1.75 | 0.97 | |

RMS-E | 1.14 | 1.38 | 17.39 | 3.42 | 0.67 | 0.75 | |

GRAC | RMS-N | 1.58 | 1.28 | 11.87 | 5.85 | 0.91 | 0.54 |

RMS-U | 2.89 | 3.02 | 12.74 | 4.76 | 2.29 | 1.66 | |

3D-RMS | 3.49 | 3.56 | 24.6 | 8.28 | 2.55 | 1.89 | |

RMS-E | 1.59 | 1.07 | 33.12 | 6.62 | 0.68 | 0.51 | |

LMMF | RMS-N | 1.36 | 0.96 | 11.81 | 1.82 | 0.63 | 0.30 |

RMS-U | 3.38 | 2.20 | 4.32 | 8.17 | 1.80 | 0.47 | |

3D-RMS | 3.96 | 2.63 | 35.43 | 10.67 | 2.02 | 0.76 |

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**MDPI and ACS Style**

Tan, T.-N.; Khenchaf, A.; Comblet, F.; Franck, P.; Champeyroux, J.-M.; Reichert, O.
Robust-Extended Kalman Filter and Long Short-Term Memory Combination to Enhance the Quality of Single Point Positioning. *Appl. Sci.* **2020**, *10*, 4335.
https://doi.org/10.3390/app10124335

**AMA Style**

Tan T-N, Khenchaf A, Comblet F, Franck P, Champeyroux J-M, Reichert O.
Robust-Extended Kalman Filter and Long Short-Term Memory Combination to Enhance the Quality of Single Point Positioning. *Applied Sciences*. 2020; 10(12):4335.
https://doi.org/10.3390/app10124335

**Chicago/Turabian Style**

Tan, Truong-Ngoc, Ali Khenchaf, Fabrice Comblet, Pierre Franck, Jean-Marc Champeyroux, and Olivier Reichert.
2020. "Robust-Extended Kalman Filter and Long Short-Term Memory Combination to Enhance the Quality of Single Point Positioning" *Applied Sciences* 10, no. 12: 4335.
https://doi.org/10.3390/app10124335