# 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:

- 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.
- Upon completion, an algorithm determines the ellipse that best fits the distorted circle.
- 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:

- Centering the ellipse by removing the hard-iron bias from the measurements
- Rotating the ellipse to align with the nominal x and y-axes
- Stretching the ellipse along its major and minor-axes to form a unit-circle
- 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 ParameterValueUnitCenter-0.128, 0.126 [G] Major/Minor axes0.225, 0.198 [G] Soft-Iron Scale Factor0.882 [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**

Tablelists 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 DataData with Added Iron SourceHeading [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

Tablereflect 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.