Kalman filters(in simple words)
I work on systems that incorporate Kalman filters, but I don't design them myself. So here's my "layman's description" of what's going on:

A Kalman filter maintains an internal model of some physical system. You take measurements of that physical system and feed them to the filter to update the model. You can then use data from that model in place of data from the physical system.

The problem is that the measurements (also called "observables") do not necessarily relate directly to the state information you're interested in. Furthermore, the measurements themselves might be corrupted by noise and other errors.

Therefore, the Kalman filter also models how the measurements relate to the true state of the system. It makes those same measurements on its model, and checks to see how well those measurements correlate with the actual measurements. This allows it to assign weighting factors to the measurements based on how "trustworthy" they are, before it uses them to update its internal state. It can also model things like bias and scale factor errors in the data.

In this way, it can be shown that the state of the internal model is the best possible estimate (in a strict mathematical sense) of the true state of the system.



In the systems that I work on, we're interested in the best possible estimate of the true position, velocity and orientation of a vehicle. The observables are things like acceleration and rotation rate from an IMU, magnetic field direction from an electronic compass, and satellite range and range rates from one or more GNSS receivers. The outputs of the Kalman filter are position, velocity and attitude of the vehicle.