Introduction to the Extended Kalman Filter

Introduction to EKF

Additional Resources:

Taylor series Wikipedia page

## Summary

The Kalman Filter is applicable to problems with linear motion and measurement functions. This is limiting, as much of the real world is nonlinear.

A nonlinear function can be used to update the mean of a function,

\mu \xrightarrow{f(x)} \mu'

but not the variance, as this would result in a non-Gaussian distribution which is much more computationally expensive to work with. To update the variance, the Extended Kalman Filter linearizes the nonlinear function f(x) over a small section and calls it F. This linearization, F, is then used to update the state's variance.

P \xrightarrow{F} P'

The linear approximation can be obtained by using the first two terms of the Taylor Series of the function centered around the mean.

F = f(\mu) + \frac{\delta f(\mu)}{\delta x}(x-\mu)