# Calibration of Magnetometers with GNSS Receivers and Magnetometer-Aided GNSS Ambiguity Fixing
## 5 Measurement Results  
This section describes the verification of the proposed magnetometer calibration and the achievable heading accuracy in a test drive using a tightly coupled GNSS/INS reference system. The method and results differ in the following aspects from our earlier paper [5]:  
* 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).  
The MPU 9250 [13] includes a three-axis silicon monolithic Hall-effect magnetic sensor with magnetic concentrator. The magnetic flux measurements are provided with a resolution of 0.6 \(\upmu\)T. Three multi-sensor modules were mounted on the roof of a vehicle, whereas each multi-sensor module included the above sensors. The inertial sensor provides 3D acceleration, 3D angular rate and 3D _magnetometer_ measurements in the sensor-fixed frame with an update rate of 100 Hz. The distances between the GNSS antennas were around 1 m. The Multi-GNSS (GPS + GLONASS)/INS tightly coupled attitude determination included some pre-processing (synchronization, cycle slip correction [14; 15] calibration of inertial sensor biases) and carrier phase integer ambiguity fixing.  
Figure 4 shows the track of the test drive. It was split into two phases, i.e., a first phase for calibrating the magnetometer, and a second phase for testing the calibration of the magnetometer. In both phases, the track includes rotational dynamics.  
Figure 5 shows the accuracy of the magnetic flux measurements and of our measurement model, i.e., the raw measurements are compared to estimated measurements derived from the least-squares estimate of the magnetic flux in the North-East-Down frame and the sensor biases. Obviously, both tracks fit quite well. The systematic variations over time in both subfigures are caused by changes in the attitude (especially heading). The estimated measurements are less noisy as the precise Multi-GNSS/INS tightly coupled attitude is used and as constant magnitudes of the magnetic field in North-East-Down directions are assumed over the considered period of 40 s.  
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 6 shows the heading using calibrated magnetometer measurements in comparison with the tightly-coupled Multi-GNSS/INS heading. The latter one has an accuracy of 0.25 degrees and serves as reference. The figure also includes a filtered, calibrated magnetometer heading with a time constant of 0.1 s to reduce the noise of the magnetometer. The filtered magnetometer-based heading deviates by less than 10 degrees from the GNSS/INS tightly coupled heading as shown in the enlarged sections in Figure 7.  
Figure 4: Maps with track of vehicle, split into two processing steps.  
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.