Accelerometer Calibration

This page describes a method for calibrating low-g (~3g) three axis MEMS accelerometers. The method involves taking measurements at six different orientations, and then, solving for the 12 calibration parameters using least squares method. This calibration will account for the sensor's zero-g bias, cross-axis interaction, sensitivity changes, and package misalignment. Example and limitations are also provided.

Why calibrate?

Accelerometers need to be calibrated because offsets, axis misalignments, and sensitivity changes are introduced after production. These are caused by:

Without calibration, the device will appear tilted when it really is not, and give incorrect acceleration readings.

Calibration Model

We assume that raw measurements (rx, ry, rz) can be converted to calibrated values (cx, cy, cz) by multiplying a 4-by-4 matrix:

This is a 3D affine transformation which means that it does linear transformation (rotation, shear, scaling) and translation (offset).

Translation

This corrects for "0g-offset" (or "0g-level") which is the accelerometer reading at 0g. 0g-offset is also called "constant bias", "bias error", "long term bias", "measurement bias" or just "offset". An ideal accelerometer has offset value 0 because it should read 0 acceleration when there is no acceleration. In reality, the device outputs a small non-zero value even at 0g, so the offset must be subtracted when the device is being used. This value changes with temperature and the amount it changes is usually denoted in the datasheet as "Zero-G Level Change vs. Temperature". This term should not be confused with the displacement of the sensor from the center of gravity of the device the sensor is mounted on.

Scale

This compensates for sensitivity of the device. An ideal accelerometer has scale value 1 (meaning that the sensitivity denoted in the datasheet is correct). This value changes with temperature and the amount it changes is usually denoted in the datasheet as "Sensitivity Change vs. Temperature"

Skew

This takes care of the non-orthogonality of MEMS elements in each of the axis directions. Sometimes referred as "Axes Misalignment" and "Cross Coupling". Also related to "Cross-Axis Sensitivity".

Rotation

This accounts for the misalignment of die positioning on the package substrate. This corrects any package misalignements between accelerometer axes and device body axes (sensor board).

In summary, calibration method using this model will compensate for both the device cross-axis sensitivity and the misalignment between the accelerometer sensing axes and the board axes.

Calibration Method

Calibration parameters, a to l, are determined by least squares:

(eq. 1)
(eq. 2)

where

Linear transformation has 9 degrees of freedom and translation has 3 degrees of freedom, so it has 12 degrees of freedom in total (12 parameters: a to l). As a result, at least 12 equations are needed to solve for the 12 unknowns. 1 orientation results in 3 measurements (x, y, and z) so the device needs to be in at least 4 different orientations for the equation to be solvable.

Steps

The following describes six point calibration:

  1. Collect 5 to 10 seconds of raw data while the sensor's +Z axis is pointing up.
  2. Repeat for the other 5 faces and record them in each row of A matrix:
  3. Solve for x using (eq2) where b is as follows:

Once calibration parameters, x, are solved, raw measurements can be converted to calibrated parameters using (eq1)

where [Ax Ay Az] are raw measurements and [bx by bz] are calibrated values. Calibrated values are in units G.

Limitations

  1. Low-g accelerometers only
    Gravity is used as the reference, so this method works for low-g accelerometers (~3g) but not for high-g accelerometers (~200g). A larger controlled acceleration is needed to calibrate high-g accelerometers.
  2. Location dependent
    Gravity varies by 0.7% over the Earth's surface due to Earth's rotation, altitude, and the Earth not being a perfect sphere. For example,
    3. Calibration method described here will cancel out the gravitational field and assume whatever the gravitational acceleration is at the calibration site to be "1g". If, however, an absolute estimate in m/s/s is needed, then the gravitational field at the calibration site must be recorded. This value can then be multiplied to convert from "1g" to m/s/s.
  3. Temperature dependent
    MEMS elements are sensitive to temperature changes. Calibration parameters are needed for every temperature that might arise in practice. IMUs usually contain a temperature sensor, so this technique can be extended to include temperature dependence by performing calibration at two or more temperatures and interpolating the fitted calibration parameters to the actual temperature. Also note that two accelerometers of the same product may have different temperature characteristics.
  4. Other considerations

Sources