The Extended Kalman Filter: An Interactive Tutorial for Non-Experts
In working with autopilot systems like OpenPilot and Pixhawk I have frequently come across references to something called an Extended Kalman Filter (EKF). Googling this term led me to several different web pages and reference papers, most of which I found too difficult to follow. 
So I decided to create my own tutorial for teaching and learning about the EKF from first principles. This tutorial assumes only high-school-level math and introduces concepts from more advanced areas like linear algebra as needed, rather than assuming you already know them. Starting with some simple examples and the standard (linear) Kalman filter, we work toward an understanding of actual EKF implementations at end of the tutorial.
Part 1: A Simple Example
Imagine a airplane coming in for a landing. Though there are many things we might worry about, like airspeed, fuel, etc., the most obvious thing to focus on his the plane’s altitude (height above sea level). As a very simple approximation, we can think of the current altitude as a fraction of the previous altitude. For example, if the plane loses 2% of its altitude each time we observe it, then its altitude at the current time is 98% of its altitude at the previous time:
altitudecurrent_time = 0.98 * altitudeprevious_time
Engineers use the term recursive to refer to a formula like this where a quantity is defined in terms of its previous value: to compute the current value, we must “recur” back to the previous. Eventually we recur back to some initial “base case”, like a known starting altitude.
Try moving around the slider above to see how the plane’s altitude changes for different percentages.
Next: Dealing with Noise
 Two notable exceptions are Kalman Filtering for Dummies and the the Wikipedia page, from which I have borrowed here. For a more pictorial approach, take a look at this tutorial. If you’re already comfortable with math notation, you might want to check out this tutorial before spending any more time here.
This tutorial has also benefited from helpful comments by members of the OpenPilot and DIY Drones communities (esp. Tim Wilkin, who corrected a number of inaccuracies in my explanations).