# An Extended Kalman Filter and Back Propagation Neural Network Algorithm Positioning Method Based on Anti-lock Brake Sensor and Global Navigation Satellite System Information

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## Abstract

**:**

## 1. Introduction

## 2. Positioning Scheme Based on ABS Sensor and GNSS Information Fusion

#### 2.1. Fusion Positioning System

#### 2.2. Fusion Positioning Method

_{abs}values. Second, judge the positioning validity status. Third take the vehicle’s starting position as the basic point, convert x

_{lo}, y

_{la}to absolute position x

_{gnss}(m), y

_{gnss}(m) and convert the high-precision RTK differential position data into the absolute position values x

_{rtk}, y

_{rtk}. The fusion position algorithm involves EKF and BPNN. The EKF is a nonlinear version of the Kalman filter which linearizes about an estimate of the current mean and covariance.

_{k}

_{1}, γ

_{k}

_{1}can be synthesized from the relative position, but this positioning method is easily suffers from interference from heading angle speed errors, resulting relative positioning failure. The purpose of BPNN is to obtain the heading angle speed error corresponding to different vehicle and heading angle speeds. The BPNN network is a kind of multilayer feed forward network with the error back propagation nature. There is no need to set up an initial dynamic or noise model and it will find a relationship among Δγ and u

_{k}

_{1}, γ

_{k}

_{1}through self-study. The concrete steps of the fusion position method are as follows: fuse ABS sensor data u

_{fl}, u

_{fr}, u

_{rl}, u

_{rr}, γ

_{abs}and λ

_{abs}to u

_{k1}and γ

_{k}

_{1}by Pre-EKF. Distinguish positioning valid status, if the status is valid, through After-EKF fuse u

_{k}

_{1}, γ

_{k}

_{1}and GNSS data x

_{gnss}, y

_{gnss}, θ

_{gnss}to the new positioning data x

_{k}

_{2}, y

_{k}

_{2}and θ

_{k}

_{2}. In the BPNN structure, the training sample output value Δγ is from θ

_{k}

_{2}and γ

_{k}

_{1}. u

_{k}

_{1}and γ

_{k}

_{1}serve as input values of the training samples to train the BPNN. When the GNSS positioning status is invalid, we can put u

_{k}

_{1}and γ

_{k}

_{1}into the well-trained BPNN and get Δγ. The relative location is synthesized through u

_{k}

_{1}with a corrected γ

_{k}

_{1}by Δγ in accordance with the dead reckoning method [31].

## 3. Dual Kalman Filtering-Based Positioning Research

_{k}

_{1}, γ

_{k}

_{1}and GNSS data. The purpose of the dual EKF design is to improve the position accuracy, and provide low-noise training and validation samples for the BP neural network algorithm. And nomenclature related to the structure of the car are presented in Table 2.

#### 3.1. Fusion Positioning System

**W**

_{1}**~N (0, Q**is the state noise, following a Gauss distribution with zero vector mean and covariance matrix

_{1})**Q**,

_{1}**V**

_{1}**~N (0, R**is the measurement noise, following a Gauss distribution with zero vector mean and covariance matrix

_{1})**R**.

_{1}#### 3.2. After-EKF Model

**W**

_{2}**~N (0, Q**is the state noise, following a Gauss distribution with zero vector mean and covariance matrix

_{2})**Q**

_{2}, V_{2}**~N (0, R**is the measurement noise, following a Gauss distribution with zero vector mean and covariance matrix

_{2})**R**.

_{2}**F**and measurement matrix

_{k}**H**can be derived by the derivatives of the functions f(∙) and h(∙). The results from such calculations are provided in Equations (15)–(18) respectively:

_{k}## 4. Study of Positioning Method Based on BP Neural Network

- ${\gamma}_{a}$: actual value of $\gamma $ corresponding to time ${t}_{k}$
- ${\gamma}_{b}$: expected value referring to the average value of $\gamma $ from time T to time 2T

#### 4.1. BP Neural Network Model

#### 4.2. Determination of the BP Neural Network Structure

_{i}and a

_{i}represent the original and normalized datum, respectively, A

_{max}and A

_{min}represent the maximum and minimum of A

_{i}, respectively.

## 5. Experimental Study and Analysis of Results

#### 5.1. Testing Program

**Q**and

_{1}**R**values of the Pre-EKF covariance matrix are adjusted to achieve the optimal filtering effect.

_{1}#### 5.2. Analysis of Positioning Effects in Case of GNSS Positioning Status Being Valid

**Q**and

_{2}**R**values of the After-EKF covariance matrix are adjusted to achieve the optimal filtering effect.

_{2}#### 5.3. Analysis of Positioning Effects in Case of Invalid of GNSS Positioning Status

- (1)
- 30 min sample data;
- (2)
- 15 min performance display data.

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Acknowledgements

## Conflicts of Interest

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Parameters | Parameters | ||
---|---|---|---|

u | vehicle speed | γ | heading angle speed |

x_{lo} | longitude of BDS | y_{la} | latitude of BDS |

x_{rtk} | longitude of RTK | y_{rtk} | latitude of RTK |

θ | heading angle | PS | positioning valid status |

${\psi}_{abs}$ | rotation angle of the steering wheel | ${\gamma}_{abs}$ | heading angle speed |

${u}_{fl}$ | speed of left front-wheel | ${u}_{fr}$ | speed of right front-wheel |

${u}_{rl}$ | speed of left rear-wheel | ${u}_{rr}$ | speed of right rear-wheel |

${u}_{k1}$ | vehicle speed | ${\gamma}_{k1}$ | heading angle speed |

${\lambda}_{k1}$ | tangent value of front-wheel steering angle | ||

${x}_{k2}$ | relative latitude-conversion | ${y}_{k2}$ | relative longitude-conversion |

${\theta}_{k2}$ | heading angle | ${u}_{k2}$ | vehicle speed |

${\gamma}_{k2}$ | heading angle speed | Δγ | heading angle speed error |

Parameters | Parameters | ||
---|---|---|---|

$\alpha $ | Front-wheel virtual steering angle | $\lambda $ | Tangent value of front-wheel steering angle |

l | Vehicle wheelbase | r | Steering radius of the vehicle |

b_{f} | Front-wheel track | b_{r} | Rear-wheel track |

${\alpha}_{f}$ | Deflection angle of the left front-wheel | ${\alpha}_{r}$ | Deflection angle of the right front-wheel |

r_{fl} | Left front-wheel | r_{fr} | Right front-wheel |

r_{rl} | Left rear-wheel | r_{rr} | Right rear-wheel |

Parameters | Values |
---|---|

Front track | 1.496 m |

Rear track | 1.490 m |

Wheelbase | 2.550 m |

Test speed | ~40 km/h |

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**MDPI and ACS Style**

Hu, J.; Wu, Z.; Qin, X.; Geng, H.; Gao, Z. An Extended Kalman Filter and Back Propagation Neural Network Algorithm Positioning Method Based on Anti-lock Brake Sensor and Global Navigation Satellite System Information. *Sensors* **2018**, *18*, 2753.
https://doi.org/10.3390/s18092753

**AMA Style**

Hu J, Wu Z, Qin X, Geng H, Gao Z. An Extended Kalman Filter and Back Propagation Neural Network Algorithm Positioning Method Based on Anti-lock Brake Sensor and Global Navigation Satellite System Information. *Sensors*. 2018; 18(9):2753.
https://doi.org/10.3390/s18092753

**Chicago/Turabian Style**

Hu, Jie, Zhongli Wu, Xiongzhen Qin, Huangzheng Geng, and Zhangbin Gao. 2018. "An Extended Kalman Filter and Back Propagation Neural Network Algorithm Positioning Method Based on Anti-lock Brake Sensor and Global Navigation Satellite System Information" *Sensors* 18, no. 9: 2753.
https://doi.org/10.3390/s18092753