Magnetometer Error Models of Low-Cost Land Vehicle Navigation System

This article implies a method of magnetometer error determination. Using solid-state magnetic sensors, vehicle odometer and GPS sensor, low-cost complex navigation system for land vehicles can be realized. The methods and algorithms of magnetometer calibration are shown. Results of magnetometer calibration experiments and possibility to detect magnetometer error parameters in process of car driving are analysed. To acquire vehicle heading output, navigation system described here is developed. DOI: http://dx.doi.org/10.5755/j01.eie.22.6.17225


I. INTRODUCTION
The design of low-cost integrated land vehicles navigation systems becomes a more and more actual problem.Core to the integration is concept of fusing measurements from GPS sensor and inertial measurement unit using linear or nonlinear estimation techniques [1].Nowadays, the magnetometers and odometers are also used in low-cost integrated land vehicles navigation systems for dead recognition (DR) attitude determination during GPS outage [2].
The proposed navigation system consists of a GPS, an odometer, a map and a magnetic course sensor (Fig. 1) and provides the position and heading angle.The odometer filter estimates the velocity of the land vehicle, the heading filter estimates the heading angle (true course) using information about GPS true course magnetic variation and corrected magnetic course, and the position filter estimates the position, velocity and the heading of the complex GPS and DR system.All filters use Kalman filtering for data preprocessing.
Manuscript received 14 March, 2016; accepted 25 September, 2016.In this paper we analyse only magnetic course sensor errors and their compensation with heading filter.

II. MAGNETOMETER TYPES AND STANDARD CALIBRATION METHODS
The magnetometers are used for absolute heading determination with reference to local magnetic north, where the heading is derived from the horizontal force of the magnetic field.If the magnetometer was aligned with the local horizontal plane, the heading ψ, would be calculated as (2).
where Mx and My represent the horizontal magnetic field components of the Earth [6].Nowadays, there are many types of magnetic field sensors, including fluxgate sensors, AMR sensors, search coil sensors, GMR sensors, Hall Effect sensors, and other magnetic field sensors.
One of the most popular magnetic field sensors is the anisotropic magneto resistive (AMR) sensor.AMR effect in ferromagnetic materials is the dependence of the electrical resistivity on the angle between the direction of the electrical current and the magnetization.When a magnetic field is applied to the sensor, magnetic field data will be received.AMR sensors can be used for the quantitative measurement of magnetic field, for instance, AMR sensors are used in electronic compasses [7].MEMS technology allows manufacturers to fabricate sensors with a size of a few millimetres [8].
Unfortunately, any slight distortion of the Earth's magnetic field can affect output of the magnetometer.It is crucial to remove unwanted distortions caused by the external magnetic field before the start and in the process of navigation.Otherwise, using these results, false heading (true course) will be determined, thereby affecting the whole inertial navigation solution.Furthermore, noises, drift error, sensor axis misalignment error must be taken into account.
There are several techniques to compensate distortions caused by the external magnetic field.The simplest way to calibrate a magnetometer for hard and soft iron distortions is to use ellipsoid fitting algorithm.
To perform this calibration method, 360 degrees' rotation magnetic field data on horizontal plane must be acquired.To compensate distortions, two scale factors Xsf and Ysf can be determined by (3) to change the ellipsoid response to a circle.Offset values Xoff and Yoff can then be calculated by ( 4) to center the circle around the 0,0 origin [2]: Using scale factors (Xsf, Ysf) and the zero offset values (Xoff, Yoff), calibrated magnetic field values Mx and My can be determinate by (5) [2].

III. MAGNETOMETER CALIBRATION EXPERIMENT
The main device used in the tests was Xsens MTi-G GPS/MEMS IMU navigation system.
MTi-G has thin film magneto resistive onboard magnetometer [9].Even this magnetic field sensor was considered as a primary sensor in the experiment to test it's ability to determine course.also GPS data (latitude, longitude, velocity) were collected.
The true course reference data were acquired from HOLUX GPS receiver using GPS data logger software (VisualGPS) and synchronized to Xsens MTi-G experimental data.Data processing was made by high-level scripting language MATLAB.MTi-G device was mounted to the roof rack of vehicle absolutely aligned with reference to vehicle body frame.It is worth to mention that all test drives were performed on asphalt surface road containing sections of parking zones and real-time traffic conditions.
First of all, the vehicle was driven along a circular route trajectory to collect magnetic field data for calibration.Despite the fact, that Xsens MTi-G supports specialized magnetometer auto-calibration procedure, ellipsoid fitting algorithm was used to calibrate magnetometer.
Figure 2 shows the result before and after magnetometer calibration using ellipsoid fitting algorithm.As we can see, after calibration process, magnetometer values are shifted to the origin.I) suggest that mainly hard iron distortions were observed.Figure 3 presents calculated magnetic course data while circular route trajectory was performed.The mean absolute error is 24.87°.Result reveals that potentially huge heading error could be possible, for instance, when performing bend road trajectory without magnetometer calibration.Also three experimental test drives were made -straight road trajectory test, right bend trajectory test and left bend trajectory test.Mercator (UTM) coordinate system.Position was estimated by using two methods: 1) GPS speed value and calibrated magnetometer readings, 2) another GPS data (Xsens GPS).The declination angle was compensated using information from National Oceanic and Atmospheric Administration homepage World Magnetic Model (WMM) data.In Riga declination angle is 7.41 degrees (16.12.2015).To compare true course results, linear interpolation was applied to approximate values of HOLUX GPS receiver data because of GPS update rate 1 Hz, while MTi-G magnetometer sampling frequency 100 Hz.Experimental results (Fig. 4) show that magnetometer error cause growing in time position error.

Calculated parameters values (shown in Table
In the same manner as before, navigation solution for right and left bend trajectory tests are displayed in Fig. 5 and Fig. 6.Clearly, we can see that a deviation of trajectory is observed when curves of the vehicle are driven.For magnetometer error compensation in driving process authors develop method which will be described in next part.

IV. MAGNETOMETER ERROR MODELS
As it was described above, the magnetic field shift Δx and Δy and scale factor for orthogonal magnetic field sensors ksx and ksy are important to course measurement precision.Error model analysis begins with field shift errors.In this case magnetic field Mx and My values are (6): where ψ is magnetometer determinate value of magnetic course, but φ is true value of magnetic field angle.The value of magnetic course is calculated according to (2).
Calculated value ψ include error Δψsh(φ).In Fig. 7 it is shown Δψsh(φ) graphic if Δx = -0.1 and Δy = -0.05.Formula for shift error compensation Δsc(φ) can be written in form of (7).This error of hard iron effects is known as single-cycle errors.Shift errors compensation for all values of angle φ are shown in Fig. 7 if Δx = -0.1 and Δy = -0.05.Corrected course can be determined by summing measured value of magnetic course ψ with compensation value for course φ calculated from GPS data ψGPS.
Very important is that from Δsh(φ) value for constant φ, using formula (7), the shift values Δx and Δy can be calculate in driving process.For this procedure iteration method can be used, with starting values Δx0 and Δy0, obtained from magnetometer data.In GPS outage mode compensation value Δsh(φ) can be known for all drive angles and can be summed with magnetometer data for course calculation.
Second factor which causes error is scale factor for orthogonal magnetic field sensors ksx and ksy.If ksx ≠ ksy, then Xmax ≠ Ymax.We can write that Xmax = ksx × Xmax0 and Ymax = ksy•Ymax0.In this case magnetic field orthogonal parts are (8):  In real use magnetometer one-cycle and two-cycle errors discard measurements together.Practical filter realization shows, that first shift error and then scale factor error must be compensated.Vehicles driving course is changing all time, and it is possible to use various headings for error calculation.In this case two equations for shift values Δx and Δy determination may be used.By using GPS data, it is possible to write equations for course error (n) determination, as follows: By solving equation system, we can extract shift values Δx (12) and Δy (13): sin( ) sin( ) .cos( ) sin( ) cos( ) sin( ) Estimated formulas are used for right bend trajectory test mentioned before; thereby we can calculate shift error compensation for all courses (Fig. 9).

V. CONCLUSIONS
Modelling and experimental results show that using solidstate magnetic sensors, vehicle odometer and GPS sensor low-cost complex navigation system for land vehicles can be realized.After calibration is made magnetometer error is growing rapidly and must be eliminated.In this article we provide a method for magnetometer error correction in kinematics by using GPS heading data.Experimental results show possibility to detect magnetometer error parameters in process of car driving and increase performance of low-cost integrated land vehicle navigation system.

Figure 4
Figure 4 illustrates estimated navigation solution for straight road trajectory test in Universal Transverse

Figure 8
Figure 8 shows this kind of course measuring error Δφm(φ) and compensation error compensation parameter Δφ(φ) for all angles φ, for ks = 1.091.This error due to soft iron effects is known as two-cycle error.Determination of error for one driving course gives possibility to calculate ks and all values of compensation curve for random driving angles.In real use magnetometer one-cycle and two-cycle errors discard measurements together.Practical filter realization shows, that first shift error and then scale factor error must be compensated.

TABLE I .
MAGNETOMETER CALIBRATION PARAMETERS.