Accelerometer and Magnetometer Calibration: Part 1 - Theory

Introduction

In this small series of posts I'll be going into the art of calibrating sensors and mainly the calibration of 3-axis accelerometers and 3-axis magnetometers. The question that arises might be why there even is a need of calibration and the best answer is that all sensors have manufacturing differences, differences in the PCB design, components close by creating distortions and so on, and to counter these effects calibration is mainly used. An example of how an ideal measurements of accelerometer and magnetometer looks versus measurements corrupted with gain errors and bias errors is shown bellow in Figure 1. The errors have been  some what over emphasized. This is the result you'd expect if, as an example, you put a magnetometer on a table and rotated it one turn while keeping it flat to the table.

Figure 1: Ideal sensor measurements versus corrupted measurements. Crosses denote measurement points and the red circle is the center of the ellipsoid.

This really rises three questions: 1) how to model the ideal case, 2) which errors can occur and 3) how to model the case with errors. All these problem have quite simple solutions however by just looking at the data it can be hard to identify them.
     I will warn the average person, there will be a lot of math and I will not explain the background theories. I will assume that you know about vector and matrix algebra, derivatives and how to calculate with ellipses.

Modeling

1) How to model the ideal case?
Before we start trying to make a model we need to figure out what we are measuring. Both the magnetic field and the gravitational field are, in each point on Earth, a vector that has a direction and a magnitude and what we are measuring is the x, y and z components of these two vectors respectively. In the ideal case we assume that the magnitude is constant for all directions, there is no shifting (bias) and with this we can create an ideal measurement model. A vector with constant magnitude is most simply modeled as a sphere with the base of the vector in its center and the tip being able to touch any point, this gives the following equation of a sphere with the radius R:

\begin{equation*} x^2 + y^2 + z^2 = R^2 \end{equation*}  This is how we would want our measurements to look, but unfortunately this is usually not the case as we will see in the next part.

2) Which errors can occur?

Before we start looking at different errors we have to make some assumptions. We assume that the errors are uncorrelated, meaning that for example errors in the x-axis does not effect the errors in the y- or z-axis and that the noise is Gaussian (White). With this we can start identifying the different errors from the right picture in Figure 1. In this figure we can see that the center of the ellipse has been shifted, we will call this bias errors, and that the circle is not round any more but elliptical, we will call this gain errors, plus on top of this there is some random noise.

3) How to model the errors?
To model this let start with the radius of the circle. If you remember the old trigonometry classes, the way to represent a sphere with gains on x, y and z is called an ellipsoid and the general equation for an ellipse with its center in the origin have the equation:

\begin{equation*} \left (\frac{x}{a} \right )^2 + \left (\frac{y}{b} \right )^2 + \left (\frac{z}{c} \right )^2 = 1 \end{equation*}

Here I have included the $R^2$ in the a, b and c constants (more information on ellipsoids) and with this equation we can represent gains in each of the axises in the sensor.

     Now, lets start looking at the offset of the center of the ellipsoid. If we look back to our old equation solving classes then we know that to shift a function $f(x)$ a distance $b$ to the right we can write that as $f(x-b)$. Remember that we assumed the error to be uncorrelated and knowing this we can rewrite the ellipsoid equation to incorporate bias. In this equation I have renamed the gains from $a, b, c$ to $g_x, g_y, g_z$ respectively and I call the biases $b_x, b_y, b_z$:

\begin{equation*}\left (\frac{x - b_x}{g_x} \right )^2 + \left (\frac{y - b_y}{g_y} \right )^2 + \left (\frac{z - b_z}{g_z} \right )^2 = 1 \end{equation*}

This is the complete measurement model of both the accelerometer and the magnetometer, where $x, y, z$ are the measurements corrupted by gain ($g_x, g_y, g_z$) and bias ($b_x, b_y, b_x$) errors respectively.
    Using this equation I will, in the next part, show how to use Least-Square solving methods for finding the gains ($g_x, g_y, g_z$) and biases ($b_x, b_y, b_x$) respectively plus how to handle the random noise on top of the model.