6. Recursive Averages

To avoid memory and performance problems of keeping track of all previous measurements, use the recursive average formula

Markov Assumption

Updated estimate only depends on current observation and previous estimate

New measurements don’t affect the recursive average after a long period of time t. So we use exponential averaging.

but this also has similar drawbacks to the normal average.

We need control to solve the drawbacks of the exponential moving average

We’re going to use our own model of the system to incorporate our control input and expected output.

If we know the control input we can model what we expect the measurement to be.

We increase uncertainty in our estimate

Then we make a measurement to decrease the uncertainty

bel_bar is an integral, but can be a closed interval.

When the belief is gaussian, it takes the form of matrix operations over the mean and covariance matrix.

We use the predicted state and update it with a new estimation using Bayes’ Rule. It uses P(z|x) x belief

You’ll usually see Kalman Filter math expressed in the matrix notation below.

This is equivalent to the more readable version of it above, which just says that to advance the state:

State