# Calibration of Magnetometers with GNSS Receivers and Magnetometer-Aided GNSS Ambiguity Fixing

## Abstract

**:**

## 1. Introduction

## 2. Measurement Models

#### Measurement Model for Magnetic Field Sensor

- transformation from local geomagnetic frame into local geographic navigation-frame, depending on the magnetic declination $\delta \nu $,
- transformation from local geographic navigation-frame into body-fixed frame, depending on the attitude of sensor’s platform (roll $\phi $, pitch $\theta $ and heading $\psi $),
- transformation from body-fixed frame into sensor-fixed frame, depending on misalignment errors of sensors (roll offset $\Delta \phi $, pitch offset $\Delta \theta $ and heading offset $\Delta \psi $).

## 3. Calibration of Magnetic Flux Sensors

#### 3.1. Estimation of Magnetometer Biases

#### 3.2. Attitude Determination with Three GNSS Receivers

- Carrier phase measurement $\lambda {\phi}_{r}^{k}$,⊕ carrier phase can be tracked with millimeter accuracy,⊖ carrier phase is period with $\lambda =19$ cm and requires ambiguity resolution,
- Pseudorange measurement ${\rho}_{r}^{k}$,⊕ pseudorange is an unambiguous range measurement,⊖ pseudorange measurement is more sensitive to multipath,⊖ pseudorange measurement can only be tracked with meter-level accuracy.

#### 3.2.1. Modeling of Differential GNSS Measurements

#### 3.2.2. Joint Estimation of Baselines, Pseudorange Multipaths and Ambiguities

#### 3.2.3. Integer Ambiguity Fixing Using Prior Information on Baseline Coordinates

${\stackrel{\u02c7}{\overrightarrow{b}}}_{1r}^{\phantom{\rule{0.222222em}{0ex}}\mathrm{n}}({N}_{1r}^{1l},\dots {N}_{1r}^{kl})$ | baseline estimate for partially fixed integer ambiguities, |

${\sigma}_{{\stackrel{\u02c7}{l}}_{1r}}^{2}$ | variance of length of baseline estimate assuming correct partial ambiguity fixing, |

${\sigma}_{{\overline{l}}_{1r}}^{2}$ | variance of prior information on baseline length, |

${\gamma}_{l}^{2}$ | upper bound on the squared normalized baseline length error, |

${\gamma}_{\phi ,\theta ,\psi}^{2}$ | upper bound on the sum of squared baseline residuals. |

## 4. Fast Initialization of GNSS Attitude Ambiguity Fixing with Calibrated Magnetometers

#### Analysis of Benefit of Magnetometer-Based Attitude Information for GNSS Integer Ambiguity Fixing

- Estimation of float solution of baselines and ambiguities by least-squares estimation using single epoch measurements,
- Normalization of baseline estimates with prior information on baseline length and respective adjustment of float ambiguities,
- Integer ambiguity fixing with sequential tree search and integer decorrelation using magnetometer-based attitude information and baseline length prior information.

**several orders of magnitude**by the magnetometer based heading/pitch angle with an accuracy of ${10}^{\circ}$.

## 5. Measurement Results

- Calibration with Multi-GNSS (GPS + GLONASS)/INS tightly coupled attitude information instead of GPS-only attitude estimate, enabling higher reliability due to inertial sensors and faster calibration due to higher update rate,
- Use of three instead of two GNSS receivers for full 3D attitude information,
- Estimation of 3D magnetic flux in North-East-Down frame instead of 1D magnetic flux in the North-only direction, enabling use also in areas with systematic distortions of magnetic field and/or close to magnetic poles,
- Use of the newest sensor generation: LEA M8T Multi-GNSS receiver of u-blox (Thalwil, Switzerland), Taoglas AGGP.35F dual-band GNSS antenna (Enniscorthy, Ireland), and MPU 9250 inertial sensor (San Jose, CA, USA).

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Benefit of magnetometer-based prior attitude information for single epoch integer ambiguity fixing: the accuracy of the prior heading information is shown on the x-axis and the accuracy of the prior pitch information is provided in the legend.

**Figure 3.**Benefit of magnetometer-based prior attitude information for single epoch integer ambiguity fixing: the probability varies over time due to the changing satellite constellation.

**Figure 5.**Magnetic flux measurements in sensor-fixed frames: the measured magnetic flux in x- and y- directions depends on the heading and is quite noisy. The least-squares estimate of the magnetic flux is much less noisy, as the estimation combines the measurements from 40 s to determine the 3D magnitude of the magnetic flux and the 3D biases.

**Figure 6.**Heading determination with calibrated magnetometer in comparison to tightly-coupled GNSS/INS heading: the latter one has an accuracy of 0.25 degrees and serves as a reference. As the magnetometer-based heading is noisy, a filtered version with a time constant of 0.1 s is also shown.

**Figure 7.**Detailed analysis of heading performance for two sections with moderate to high rotational dynamics: the filtered magnetometer-based heading deviates by less than 10 degrees from the GNSS/INS tightly coupled heading.

simulated measurements | single frequency double difference |

pseudoranges and carrier phase measurements | |

on L1 (${f}_{c}=1575.42$ MHz) of 27 Galileo satellites | |

using nominal Walker constellation [12] | |

(satellite altitude: $23,222$ km, orbital inclination: ${56}^{\circ}$) | |

receiver position | longitude $\lambda =11.{568578}^{\circ}$ E, latitude $\phi =48.{150889}^{\circ}$ N |

baseline vector | length of 1 m, random attitude angles |

noise statistics | phase noise: ${\sigma}_{\phi}=2$ mm |

code noise: $\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}{\sigma}_{\rho}=2$ m including multipath | |

accuracy of prior information on baseline length | ${\sigma}_{{\overline{l}}_{1r}}=2$ cm |

accuracy of magnetometer based attitude information | variable accuracies for both heading and pitch angles |

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**MDPI and ACS Style**

Henkel, P. Calibration of Magnetometers with GNSS Receivers and Magnetometer-Aided GNSS Ambiguity Fixing. *Sensors* **2017**, *17*, 1324.
https://doi.org/10.3390/s17061324

**AMA Style**

Henkel P. Calibration of Magnetometers with GNSS Receivers and Magnetometer-Aided GNSS Ambiguity Fixing. *Sensors*. 2017; 17(6):1324.
https://doi.org/10.3390/s17061324

**Chicago/Turabian Style**

Henkel, Patrick. 2017. "Calibration of Magnetometers with GNSS Receivers and Magnetometer-Aided GNSS Ambiguity Fixing" *Sensors* 17, no. 6: 1324.
https://doi.org/10.3390/s17061324