# Introduction

As detailed in *IMU Processing - part 1*, the calibration method for accelerometers was not sufficient to guarantee a good precision. Indeed error reached up to 0.15g error on some axis, that corresponds to 8.5deg.

# Spherical model calibration

The goal of the calibration is to shape a sphere from a distorted one namely the uncalibrated measures of the accelerometer when rotated around its 3 axes.

As we look to shape a sphere, we need the maths detailed in this paper: https://jekel.me/2015/Least-Squares-Sphere-Fit/. Because we are not confident with only a few measures (that correspond to the number of unknown parameters to calibrate), we need to surdetermined our system with a lot of measures using Least Square.

They are several ways to solve it:

- Singular Value Decomposition: described here that compute the pseudo inverse matrix formed by our surdetermined system that then give the unknown parameters. This method is 'lighter' enough to run on microcontroller, and gives good results

- Optimization Solver: minimize the criterion defined by the raw points to match the sphere model by varying sphere equation parameter. There are many ways to solve optimization problem, and the only ones I found that are well documented require a lot of integration to run on micro controller.

# Advanced model calibration

After digging the web, I finally landed on a more advanced error model for MEMS sensors. This model also takes into account axis misalignment and axis non orthogonality, more info on the paper. Luckily the method has been implemented as well, more detail on the next paragraph.

# Integration approach

Finally i decided to not perform the sensor calibration onboard, instead raw values will be processed on a computer running solvers. The obtained results will be then used by the microcontroller (Arduino Due in my case).

The calibration workflow is currently not frozen, but here are some ideas:

- get the calibration results and directly compile the code (the simplest one, and currently used)
- try to fetch a results file located on SBC at ROS node startup (unsure how it is possible from the ucontroller)
- dynamically send raw data and fetch results on demand through a ROS service for example (way more longer to implement)

As told before an excellent integration work of the selected method as been done by albertopretto on this github repo. It relies on the Ceres solver, the Google open source solver (used in SLAM cartographer). The implementation only cover gyro and accelero, I decided to use the sphere model to calibrate magnetometer and solve the least square problem with Ceres.

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