# GPS Data Correction Based on Fuzzy Logic for Tracking Land Vehicles

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Data Acquisition System

#### 2.2. Approximation Data

#### 2.3. Fuzzy System Design

#### 2.4. Kinematic Model of Car and Tuning of UKF

## 3. Results

#### 3.1. Analysis of Results with Our Own Dataset

#### 3.2. Analysis of Results with Public Dataset

## 4. Discussion

## 5. Conclusions

## 6. Recommendations

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**Fuzzy system design, (

**a**) Fuzzy system 1 (training): Latitude; (

**b**) Fuzzy system 2 (training): Longitude; (

**c**) Testing both fuzzy systems.

**Figure 7.**Fuzzy systems testing, (

**a**) Fuzzy system 1 (testing): Latitude; (

**b**) Fuzzy system 2 (testing): Longitude.

**Figure 10.**Reference (green) vs. FPC response (blue) vs. UKF (magenta). (

**a**) route 1, (

**b**) route 2, (

**c**) route 3, (

**d**) route 4.

**Figure 11.**Error: reference vs. fuzzy systems response vs. UKF. (

**a**) route 1, (

**b**) route 2, (

**c**) route 3, (

**d**) route 4.

Route | Distance (m) | Time (s) | Velocity (m/s) |
---|---|---|---|

(a) 1 | 282.45736 | 1020 | 0.276918 |

(b) 2 | 282.9798 | 840 | 0.336880 |

(c) 3 | 151.8607 | 480 | 0.316376 |

(d) 4 | 104.3988 | 420 | 0.248568 |

Route | Training Data | Validation Data | Total |
---|---|---|---|

1 | 751 | 250 | 1001 |

2 | 645 | 215 | 860 |

3 | 356 | 118 | 474 |

4 | 412 | 102 | 514 |

Fuzzy System | MF Input Lat | MF Input Lon | MF Output | Fuzzy Rules | RMSE (Train) |
---|---|---|---|---|---|

Latitude | 5 gaussian type 2 | 5 gaussian type 2 | linear | 25 | $4.29\times {10}^{-7}$ |

Longitude | 3 gaussian type 2 | 3 gaussian type 2 | linear | 9 | $1.1\times {10}^{-4}$ |

# | Q | R |
---|---|---|

1 | $\left(\left[0.1,\text{}0.1,\text{}\mathrm{rad}\left(350\right),\text{}0.1\right]\right)\times {10}^{2}$ | $\left(\left[0.1,-0.1\right]\right)^2$ |

2 | $\left(\left[0.1,\text{}0.1,\text{}\mathrm{rad}\left(350\right),\text{}0.1\right]\right)\times {10}^{3}$ | $\left(\left[0.1,-0.1\right]\right)\times {10}^{3}$ |

3 | $\left(\left[0.1,\text{}0.1,\text{}\mathrm{rad}\left(350\right),\text{}0.1\right]\right)\times {10}^{3}$ | $\left(\left[0.05,-0.05\right]\right)\times {10}^{3}$ |

4 | $\left(\left[0.001,\text{}0.001,\text{}\mathrm{rad}\left(350\right),\text{}0.001\right]\right)\times {10}^{3}$ | $\left(\left[0.025,-0.025\right]\right)\times {10}^{3}$ |

5 | $\left(\left[0.0001,\text{}0.0001,\text{}\mathrm{rad}\left(350\right),\text{}0.0001\right]\right)\times {10}^{3}$ | $\left(\left[0.025,-0.025\right]\right)\times {10}^{3}$ |

6 | $\left(\left[\mathbf{0.001},\mathbf{0.001},rad\left(\mathbf{350}\right),\mathbf{0.001}\right]\right)\times {\mathbf{10}}^{\mathbf{3}}$ | $\left(\left[\mathbf{0.025},\mathbf{-}\mathbf{0.025}\right]\right)\times {\mathbf{10}}^{\mathbf{3}}$ |

7 | $\left(\left[0.001,\text{}0.001,\text{}\mathrm{rad}\left(350\right),\text{}0.001\right]\right)\times {10}^{3}$ | $\left(\left[0.025,-0.025\right]\right)\times {10}^{3}$ |

8 | $\left(\left[0.001,\text{}0.001,\text{}\mathrm{rad}\left(350\right),\text{}0.001\right]\right)\text{}\times {10}^{4}$ | $\left(\left[0.025,-0.025\right]\right)\text{}\times {10}^{4}$ |

9 | $\left(\left[0.001,\text{}0.001,\mathrm{rad}\left(350\right),\text{}0.001\right]\right)\text{}\times {10}^{5}$ | $\left(\left[0.025,-0.025\right]\right)\text{}\times {10}^{5}$ |

10 | $\left(\left[0.001,\text{}0.001,\text{}\mathrm{rad}\left(350\right),\text{}0.001\right]\right)\times {10}^{6}$ | $\left(\left[0.025,-0.025\right]\right)\times {10}^{6}$ |

Route | UKF: RMSE (m) | Fuzzy (FPC): RMSE (m) |
---|---|---|

1 | $1.989\times {10}^{-4}$ | $1.490\times {10}^{-4}$ |

2 | $7.539\times {10}^{-4}$ | $2.289\times {10}^{-4}$ |

3 | $4.865\times {10}^{-4}$ | $2.926\times {10}^{-4}$ |

4 | $2.698\times {10}^{-4}$ | $2.510\times {10}^{-4}$ |

UKF | ||

Route | Mean (m) | $\mathrm{Variance}\text{}({m}^{2}$) |

1 | $1.366\times {10}^{-2}$ | $4.764\times {10}^{-5}$ |

2 | $5.744\times {10}^{-2}$ | $1.297\times {10}^{-3}$ |

3 | $3.755\times {10}^{-2}$ | $1.039\times {10}^{-4}$ |

4 | $1.736\times {10}^{-2}$ | $1.233\times {10}^{-4}$ |

FPC (Fuzzy system) | ||

Route | Mean (m) | $\mathrm{Variance}\text{}({m}^{2}$) |

1 | $1.300\times {10}^{-2}$ | $4.983\times {10}^{-5}$ |

2 | $5.829\times {10}^{-2}$ | $1.315\times {10}^{-3}$ |

3 | $3.766\times {10}^{-2}$ | $9.634\times {10}^{-5}$ |

4 | $1.852\times {10}^{-2}$ | $1.114\times {10}^{-4}$ |

RMSE Sensor vs. Ref (m) | RMSE: Fuzzy (FPC) vs. Ref (m) |
---|---|

$5.51\times {10}^{-4}$ | $5.250\times {10}^{-4}$ |

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**MDPI and ACS Style**

Correa-Caicedo, P.J.; Rostro-González, H.; Rodriguez-Licea, M.A.; Gutiérrez-Frías, Ó.O.; Herrera-Ramírez, C.A.; Méndez-Gurrola, I.I.; Cano-Lara, M.; Barranco-Gutiérrez, A.I. GPS Data Correction Based on Fuzzy Logic for Tracking Land Vehicles. *Mathematics* **2021**, *9*, 2818.
https://doi.org/10.3390/math9212818

**AMA Style**

Correa-Caicedo PJ, Rostro-González H, Rodriguez-Licea MA, Gutiérrez-Frías ÓO, Herrera-Ramírez CA, Méndez-Gurrola II, Cano-Lara M, Barranco-Gutiérrez AI. GPS Data Correction Based on Fuzzy Logic for Tracking Land Vehicles. *Mathematics*. 2021; 9(21):2818.
https://doi.org/10.3390/math9212818

**Chicago/Turabian Style**

Correa-Caicedo, Pedro J., Horacio Rostro-González, Martin A. Rodriguez-Licea, Óscar Octavio Gutiérrez-Frías, Carlos Alonso Herrera-Ramírez, Iris I. Méndez-Gurrola, Miroslava Cano-Lara, and Alejandro I. Barranco-Gutiérrez. 2021. "GPS Data Correction Based on Fuzzy Logic for Tracking Land Vehicles" *Mathematics* 9, no. 21: 2818.
https://doi.org/10.3390/math9212818