Magnetic-Alignment

Overview

A so-called “magnetic-alignment” procedure enables estimation of the hard and soft-iron disturbances in the system. As these disturbances are fixed in the body, the corrections must be applied in the body-frame. The procedure works as follows:

  1. The magnetic-field is measured and recorded as the system undergoes a 360+ degree rotation about the z-axis. Ideally this is done when the system is level.
  2. Upon completion, an algorithm determines the ellipse that best fits the distorted circle.
  3. Ellipse parameters (related to the hard and soft-iron disturbances) are saved in the firmware and used to correct the magnetic-field measurements.

In most cases an ellipse describes magnetic-field distortions quite well. The ellipse parameters relate to the magnetic disturbances as follows:

  • The center of the ellipse is equal to the hard-iron values
  • The angle the major-axis of the ellipse makes with a nominal x-axis is equal to the soft-iron angle (which forms the matrix \(R_{SI}\))
  • The major and minor-axis lengths forms the scaling matrix \(S_{SI}\)

The formula for the corrected magnetic measurements works by:

  1. Centering the ellipse by removing the hard-iron bias from the measurements
  2. Rotating the ellipse to align with the nominal x and y-axes
  3. Stretching the ellipse along its major and minor-axes to form a unit-circle
  4. Rotating the unit-circle back into its nominal orientation

Note: as mentioned earlier, this correction is only done in the XY-plane and cannot correct the raw magnetometer signal. It is only done to determine the system heading.

Example

Magnetic-field information was collected as the system underwent a 360 degree rotation about the z- axis (Figure). This was performed twice, once in a disturbance-free environment (no iron added to the system) and once with additional iron added to the system. The data in each case was processed and a best-fit ellipse FN computed (dashed lines). In the disturbance-free case, the data and the fit were close to circular. In the case with additional iron, however, the circle was clearly distorted and shifted away from the origin.

Magnetic-Field Measurement in an Environment with and without Iron-Based Disturbances

For the measurements taken in the presence of additional iron, the estimation procedure produced the following best-fit ellipse parameters:

Best-Fit Ellipse Parameters

Ellipse Parameter Value Unit
Center -0.128, 0.126 [G]
Major/Minor axes 0.225, 0.198 [G]
Soft-Iron Scale Factor 0.882 [N/A]
Angle to Major-Axis -48.497 [deg]

In the correction equation (above), \(R_{SI}\) is the rotation matrix and corrects for a rotation of the magnetic-field due to soft-iron effects:

\[R_{SI} = \begin{bmatrix} { { \begin{split} cos{ \begin{pmatrix} { \eta } \end{pmatrix} } sin{ \begin{pmatrix} { \eta } \end{pmatrix} } 1 \end{split} } \hspace{5mm} { \begin{split} -sin{ \begin{pmatrix} { \eta } \end{pmatrix} } cos{ \begin{pmatrix} { \eta } \end{pmatrix} } 1 \end{split} } \hspace{5mm} { \begin{split} 0 0 1 \end{split} } } \end{bmatrix}\]

Where \(\eta\) is the angle from the nominal x-axis to the semi-major axis. \(S_{SI}\) (the scale-factor matrix) corrects for the stretching caused by the soft-iron:

\[S_{SI} = \begin{bmatrix} { { \begin{split} {1/a} 0 0 \end{split} } \hspace{5mm} { \begin{split} 0 {1/b} 0 \end{split} } \hspace{5mm} { \begin{split} 0 0 1 \end{split} } } \end{bmatrix}\]

\(a\) and \(b\) are the lengths of the semi-major and semi-minor axes.

For the data-set described above, the values for \(R_{SI}\) and \(S_{SI}\), resulting from the best-fit ellipse parameters, are:

\[R_{SI} = \begin{bmatrix} { { \begin{split} {0.66266} {-0.74892} 0 \end{split} } \hspace{5mm} { \begin{split} {0.74892} {0.66266} 0 \end{split} } \hspace{5mm} { \begin{split} 0 0 1 \end{split} } } \end{bmatrix}\]

and

\[S_{SI} = \begin{bmatrix} { { \begin{split} {4.45226} 0 0 \end{split} } \hspace{5mm} { \begin{split} 0 {5.04689} 0 \end{split} } \hspace{5mm} { \begin{split} 0 0 1 \end{split} } } \end{bmatrix}\]

Applying these correction factors to the raw magnetic-field measurements results in the unit-circle shown in Figure.

Corrected Magnetic Field Readings

Note: the nodes located at 45 degree increments around the circle are points where additional data was collected to test the heading calculation (described in the next section).

Results

Table lists the heading computed from test data using the above equations relating heading to corrected magnetic-field.

Heading Results from Magnetically Clean and Distorted Readings

True Heading
[deg]
 Disturbance-Free Data Data with Added Iron Source
Heading [deg] Error [deg] Heading [deg] Error [deg]
0 359.69 -0.31 0.013 0.013
45 45.19 0.19 44.82 -0.18
90 89.96 -0.04 90.15 0.15
135 135.05 0.05 135.08 0.08
180 180.57 0.57 180.68 0.68
225 225.64 0.64 225.62 0.62
270 270.63 0.63 270.48 0.48
315 315.30 0.30 315.09 0.09
360 359.79 -0.21 0.10 0.10

Note: the raw results reported a systematic error of approximately 2.0 degrees on all heading values. This was due to a misalignment of the test-fixture relative to true-north. The values presented in Table reflect this 2.0 degree correction. The systematic error is visible in Figures with data-clusters that do not fall on the x and y-axes.