Arduino_EKF_Library

This is a compact Extended Kalman Filter (EKF) library for Teensy4.0/Arduino system (or real time embedded system in general).

• It's not using Eigen (small source code - more simple to understand).
• It's not using C++ Standard Library/std (for embedded consideration).
• If you set SYSTEM_IMPLEMENTATION to SYSTEM_IMPLEMENTATION_EMBEDDED_NO_PRINT in konfig.h, the code is platform agnostic (not using any library beside these C header files: stdlib.h, stdint.h, stdbool.h, string.h, and math.h).
• There's no malloc/new/free dynamic memory allocation for real time application (but using heavy stack local variables, so you need to run it through static memory analyzer if you are really concerned about implement this in mission critical hard real time application).

The Background

The Extended Kalman Filter is a nonlinear version of Kalman Filter (KF) used to estimate a nonlinear system. In actuality, EKF is one of many nonlinear version of KF (because while a linear KF is an optimal filter for linear system; as this paper conclude, there is no general optimal filter for nonlinear system that can be calculated in finite dimension). In essence, EKF is (one of many) approximation to the optimal filter for nonlinear system. The strengh of EKF is its simplicity, its generality, and its low calculation overhead. That's why I made this library for any student who want to learn the structure of EKF, the computer code implementation of it, and how to use the filter for a nontrivial problem.

This library is made with specific goal for educational purpose (I've made decision to sacrifice speed to get best code readability I could get) while still capable of tackling real-time control system implementation (the code is computed in around 80-100 us for non trivia application! See Some Benchmark section below). I strongly suggest you to learn the linear kalman filter first before delve deeper into EKF.

First we start with the nomenclature:

The EKF algorithm can be descibed as:

How to Use

The EKF code is self contained and can be accessed in the folder ekf_engl (this is the template project). Inside you will find these files:

• matrix.h/cpp : The backbone of all my code in this account. This files contain the class for Matrix operation.
• ekf.h/cpp : The source files of the EKF Class.
• konfig.h : The configuration file.
• ekf_engl.ino : The arduino main file (this is only the template file).

For custom implementation, typically you only need to modify konfig.h and *.ino files. Where basically you need to:

1. Set the length of X, U, Z vectors and sampling time dt in konfig.h, depend on your model.
2. Implement the nonlinear update function f(x), measurement function h(x), jacobian update function JF(x), jacobian measurement function JH(x), initialization value P(k=0), and Qn & Rn constants value in the *.ino file.

After that, you only need to initialize the EKF class, set the non-zero initialization matrix by calling EKF::vReset(X_INIT, P_INIT, QInit, RInit) function at initialization, and call EKF::bUpdate(Y,U) function every sampling time.

To see how you can implement the library in non-trivial application, I've implemented 2 example:

1. ekf_example1_pendulum. This example simulate the damped pendulum. See the README file inside the folder to get more information.
2. ekf_example2_imu. This example process IMU (Inertial Measurement Unit) data using sensor MPU9250. See the README file inside the folder to get more information.

*For Arduino configuration (SYSTEM_IMPLEMENTATION is set to SYSTEM_IMPLEMENTATION_EMBEDDED_ARDUINO in konfig.h): The code is tested on compiler Arduino IDE 1.8.10 and hardware Teensy 4.0 Platform.

*For PC configuration (SYSTEM_IMPLEMENTATION is set to SYSTEM_IMPLEMENTATION_PC in konfig.h): The code is tested on compiler Qt Creator 4.8.2 and typical PC Platform.

Important note: For Teensy 4.0, I encounter RAM limitation where the MATRIX_MAXIMUM_SIZE can't be more than 28 (if you are using double precision) or 40 (if using single precision). If you already set more than that, your Teensy might be unable to be programmed (stack overflow make the bootloader program goes awry?). The solution is simply to change the MATRIX_MAXIMUM_SIZE to be less than that, compile & upload the code from the compiler. The IDE then will protest that it cannot find the Teensy board. DON'T PANIC. Click the program button on the Teensy board to force the bootloader to restart and download the firmware from the computer.

next step: implement memory pool for Matrix library!

Some Benchmark

The computation time needed to compute one iteration of EKF::bUpdate(Y,U) function are:

1. ekf_example1_pendulum (2 state, no input, 2 output): 14 us to compute one iteration (single precision math) or 15 us (double precision). The result, plotted using Scilab (you can see at the beginning, the estimated value is converging to the truth despite wrong initial value):

2. ekf_example2_imu (4 state, 3 input, 6 output): 86 us to compute one iteration (single precision math) or 107 us (double precision). The result, displayed by Processing script based on FreeIMU project:

You can also see the video in the ekf_example2_imu folder.

Closing Remark

I hope you can test & validate my result or inform me if there are some bugs / mathematical error you encounter along the way! (or if you notice some grammar error in the documentation).

I published the code under CC0 license, effectively placed the code on public domain. But it will be great if you can tell me if you use the code, for what/why. That means a lot to me and give me motivation to expand the work (⌒▽⌒)