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A complete error calibration model with 12 independent parameters is established by analyzing the three-axis magnetometer error mechanism. The said model conforms to an ellipsoid restriction, the parameters of the ellipsoid equation are estimated, and the ellipsoid coefficient matrix is derived. However, the calibration matrix cannot be determined completely, as there are fewer ellipsoid parameters than calibration model parameters. Mathematically, the calibration matrix derived from the ellipsoid coefficient matrix by a different matrix decomposition method is not unique, and there exists an unknown rotation matrix

Accurate measurement of the geomagnetic field has been widely applied in geophysical research, space magnetic measurement, military defense, mineral resources exploration, and drilling practice [

The “two-step” algorithm directly calibrates the magnetometer error without an external heading reference. The principle of the two-step algorithm is that when the magnitude of the true geomagnetic field is invariable at the calibration spot, the ideal locus of the three-axis magnetometer output lies on a spherical surface, whereas the actual locus of the magnetometer measurement outcome lies on an ellipsoid under the disturbance of error. The first step of the two-step algorithm transforms the non-linear observation equation based on the square of the magnetic field magnitude into a linear equation of the new unknown parameter by a mathematical transformation and acquires the unknown parameter by a batch least squares solution. In the second step, the biases and the scale factors can be derived from an inverse analytical solution [

It is proposed in literature [

The transition matrix defined in [

Because of the influence of the manufacturing processes and environment, a three-axis magnetometer has manufacturing, installation and environmental errors. The manufacturing error contains a zero-deviation _{0} and sensitivity error _{s}; installation error includes non-orthogonal error _{nor} and misalignment error
_{s} and hard iron error _{p}. On account of all the errors listed above, the relationship between the true geomagnetic field _{e} and the three-axis magnetometer output _{m} is demonstrated in _{n}

The different types of errors are specifically shown in Subsections 2.1–2.6.

Sensitivity represents the proportional relationship between the magnetometer input and output. Sensitivity error results from the inconsistency between the amplification of the measuring electronic circuit and its nominal value, which can be expressed as:

If sensitivity is the only error that affects the sensor, the _{m} = _{s} · _{e}.

There is often a small voltage bias in the sensor output signal, even when there is no magnetic intensity. In such conditions, the sensor produces non-zero output, which can be expressed as:

If zero deviation is the only error that affects the sensor, the _{m} = _{e} + _{0}.

Non-orthogonal error is caused by the limitation of machining, which leads the _{m}:

If non-orthogonal error is the only error that affects the sensor, _{m} = _{nor} · _{e}.

During the installation, the sensor _{m} and the body orthogonal coordinate system

If misalignment is the only error that affects the sensor, the

Ferromagnetic materials produce an inductive magnetic field under the influence of a geomagnetic field and the induced electric current around the sensor. It also varies with the attitude of the magnetometer and the position in the magnetic field, which can be expressed as:

The _{ij}_{xy}_{m} = _{s} · _{e}.

This type of error results from permanent magnets and magnetic hysteresis, that is, remanence of magnetized iron materials. Mathematically, this error is equivalent to zero-deviation:

If hard iron is the only error that affects the sensor, _{m} = _{e} + _{p}.

Ignoring the measurement noise in _{e} can be derived as:

When an ideal magnetometer rotates in the calibration spot, the geomagnetic field magnitude remains constant, and the output locus of the three-axis magnetometer is a sphere. In contrast, a non-ideal magnetometer output locus would be an ellipsoid because of the influence of error. The square of _{e} is derived from ^{T}

Generally, non-diagonal elements of the soft iron matrix _{s}, the non-orthogonal error and misalignment error of the soft iron error, are minuscle, therefore, the error matrix ^{T}G

where,
_{0}.

The principle of ellipsoid fitting is finding an ideal ellipsoid where the sum of squares of the algebraic distances calculated from the measured data point to the ellipsoid is minimized [

In order to assure that the fitting result is an ellipsoid, Nikos contends that parameter σ should guarantee that matrix ^{2} = 1 can be imposed for the convenience of calculation, which would not lose its generality and can be expressed in the matrix form:

By introducing a Lagrange multiplier λ, we can obtain the simultaneous equation from

In order to obtain the extremum under constraints, the first-order derivative could be derived from

Therefore, the extremum under an ellipsoid constraint could be obtained:

It has been proved in [_{0} is derived from further calculation. It could be concluded from the comparison between

‖_{e}‖ is not necessarily the actual magnitude of the geomagnetic field because its magnitude only affects the components on the _{p} = _{0} − _{0}) to _{e} = _{m} − _{0}) − _{0} − _{0}) = _{m} − _{0}). Therefore, it is not necessary to use _{0}. The 3 × 3 matrix

Therefore, the key to deriving calibration matrix

This paper provides a constant intersection angle method to solve the rotation matrix

Similar to the classical ellipsoid fitting method, the constant intersection angle method also requires calibration in the same location in order to satisfy the condition that the magnitude and direction of the geomagnetic field and gravitation field are invariant. The intersection angle between the geomagnetic field vector and gravitation field vector at the same measurement spot is constant [_{e} and gravity acceleration vector _{g}, which ‖_{e}‖ · _{g} · cosθ, is invariable and can be calculated from the model of geomagnetic field and gravitation field. In the said equation, θ is the intersection angle between the geomagnetic field vector and gravitation field vector, and its value can be interpreted to have approximately the same effect as ‖_{e}‖ in _{e} and gravitation field vector _{g} at the same measurement spot is constant, even if there is no transformation into a geodetic coordinate system.

To begin, calibration matrix _{e}‖_{2} _{c} can be derived as below, by introducing _{0} into _{e} = _{c}, and

Recasting

The matrix containing As for the set of _{g} and _{c}, is expressed as

The least squares solution ^{T} · Φ)^{−1} · Φ^{T} ·

A high rotation matrix accuracy plays an important role in further correcting the heading error, which is calibrated by the classical ellipsoid fitting method. The measurement noise, however, are inevitable and have influence on the estimator of rotation matrix

When the condition number of matrix φ in

Where

^{T}^{T}^{2}^{T}^{T}^{−1} φ^{T}^{−}^{T}S^{−1}^{−1}φ^{T}^{T}R_{m}‖ to obtain rotation matrix _{m} is recast by matrix _{0} than _{m}. Introducing it into _{e}_{c}

During magnetometer calibration, to ensure the quality of the measurement data set, a proper uniform distribution of the data over all directions must be accomplished [_{raw} matrix, which is composed of 46 sample points. The collection method is shown in _{1} _{2} and ∠_{1} _{3} between any two adjacent data points are equal. Secondly, 46 data points are collected for the acceleration data, of which the magnitude is 9.8. As demonstrated in _{i}_{j}_{i}_{i}_{j}_{j}

Then assuming that error matrix
_{ellipse} = _{raw} · M′, which is distributed on the disturbing ellipsoid. Then, the Gaussian white noise of ^{2}) is added into _{ellipse}, where 0.002 is the standard deviation of the noise.

After determining the ellipsoid coefficient matrix, the preliminary calibration matrix _{raw} that is free from the influence of the error matrix, the heading error calibrated by matrix

The constant intersection angle method is not necessary for the additional calibration procedure mentioned in reference [

When the standard deviation of the Gaussian white noise is 0.003, and the magnitude of the simulation data is one, the heading error shown is shown in

In order to compare the calibration results between the classical ellipsoid fitting method and the constant intersection angle method proposed earlier, the nonmagnetic turntable experiment adopts a self-developed magnetic compass, which is mounted on the nonmagnetic turntable. The magnetic compass adopts a PIC24HJ128GP504 microprocessor (Microchip Technology, Chandler AZ, USA), equipped with a fluxgate sensor as the magnetic sensing element, and a Honeywell QA-T160 accelerometer (Honeywell Conglomerate Company, Morristown, NJ, USA) is used. The output heading angle of the optical–electrical encoder is regarded as the true heading reference, where the error is less than 3′. As shown in _{b}-axis, taking the downward direction as the _{b}-axis, and setting the _{b}-axis in conformity with the left-hand rule, the navigation coordinate frame (_{b}-axis points northward.

The experiment was performed in a Beijing suburb, where the magnetic environment in the vicinity of the magnetometer is better than that in the laboratory in the city. The principle of the constant intersection angle is adopted to ensure that the data points are distributed evenly on the sphere. The data collection is composed of 46 sets of data spread on the _{b}-axis can intersect with the north direction at angles of 6°, 66°, 126°, 186°, 246° and 306°; Where the roll angle is 0°, and the pitch angle is respectively 60° and −60°, the nonmagnetic turntable is rotated clockwise such that the magnetometer _{b}-axis can intersect with the north direction at angles of 6°, 42°, 78°, 114°, 150°, 186°, 222°, 258°, 294° and 330°. Where both the roll angle and the pitch angle are 0°, the nonmagnetic turntable is rotated clockwise such that the magnetometer _{b}-axis can intersect with the north direction at angles of 6°, 36°, 66°, 96°, 126°, 156°, 186°, 216°, 246°, 276°, 306°, and 336°. Where the roll angle is 0°, and the pitch angle is 90° and −90°, the _{b}-axis is vertically upward and downward, respectively. The accelerometer data should be collected simultaneously at the same 46 positions, because the method is attitude-dependent.

Then, magnetometer preliminary calibration can be conducted through the ellipsoid assumption method. The heading angle ψ could be is derived

θ and γ respectively refer to the pitch angle and roll angle, which are determined by the three-axis accelerations _{x}, g_{y}_{z}

As a type of inertial sensor, accelerometers are reliable, independent, and immune to environmental impacts. The primary errors of accelerometers are bias, sensitivity error, and non-orthogonal error. Such errors are similar to the magnetometer errors, as observed in _{0} is the bias of accelerometer, _{e} is the actual acceleration of gravity, and

The output trajectory of the three-axis accelerometer accords with the ellipsoid hypothesis as well. There are nine unknown parameters consisting of three sensitivity errors, three non-orthogonal error and three bias. Thus, the error parameters of the accelerometer can all be derived from the coefficients of the ellipsoid equation. The experiment is performed on the turntable shown in ^{2} to 4.6323 × e^{−4} m/s^{2}, and the mean square root error drops from 0.0545 m/s^{2} to 1.0358 × e^{−4} m/s^{2}; thus, the maximum error of the accelerometer can be reduced by a factor of approximately 1,000.

The calibration matrix _{m} and obtain _{c} calibrated by the classical ellipsoid fitting method. However, calibration matrix _{e} = _{c} is obtained. The heading error is shown in

As mentioned in Section 4.2, the accuracy of rotation matrix ^{T}R

The green line refers to the heading error that the rotation matrix

This study thoroughly analyses the three-axis magnetometer measurement errors and establishes a complete error model including 12 independent parameters, which is more universal and conforms to the ellipsoid restriction. However, the calibration matrix derived from the ellipsoid coefficient matrix by a different matrix decomposition method is not unique, and there exists an unknown rotation matrix

This paper is supported by “Projects from National Natural Science Foundation of China” (No.61273082) and “Fundamental Research Funds for the Central Universities” (No.FRF-TP-09-017B). Furthermore, the authors would also like to thank all who have helped with this study.

The authors declare no conflict of interest.

Sensor coordinate system non-orthogonal error.

Body orthogonal coordinate system and virtual orthogonal coordinate system.

Collecting data according to the equivalent intersection angle principle.

Constant intersection angle between the geomagnetic field vector and gravitation field vector.

(a) Heading error when the standard deviation of the noise is 0.002; (b) Heading error when the standard deviation of the noise is 0.003; and (c) Heading error when the standard deviation of the noise is 0.005.

Nonmagnetic turntable.

The accelerometer error before calibration and after calibration.

(

Comparison experiment about the noise suppression methodology.