The big idea: blending a guess with a measurement

Imagine a rover driving down a straight corridor. You know roughly how fast it is going, so you can predict where it will be one second from now. You also have a noisy position sensor that measures where it is. Which do you believe?

The Kalman filter’s answer: both, weighted by how much you trust each one.

• If your prediction is very uncertain and the sensor is good, lean towards the sensor.
• If the sensor is noisy and your model is good, lean towards the prediction.

That weight is called the Kalman gain, written \(K\). It is a number between 0 and 1 (a matrix in the general case):

The bracket is the innovation: how surprised you are by the measurement.

Two noisy guesses, one better answer

Suppose the prediction says the rover is at 10.0 m with variance \(4\ \text{m}^2\), and the sensor says 12.0 m with variance \(1\ \text{m}^2\). The best combined estimate is

and the new variance is \((1-K)\cdot 4 = 0.8\ \text{m}^2\), smaller than either input. Combining information always reduces uncertainty. That is the whole filter in one line; the rest of the course makes it work for many variables at once and for things that move.

The loop

A Kalman filter runs forever in a two-step loop:

1. Predict: push the estimate forward with the motion model, and grow the uncertainty.
2. Update: when a measurement arrives, pull the estimate towards it and shrink the uncertainty.

Keep this picture in mind. Every equation in Lesson 4 belongs to one of those two steps.