Both PID and Kalman filters are used extensively in Robotics. I gather, Kalman filters for sensing and PID for control. What is the mathematical, statistical or other relationship between them?
There isn't really a relationship between them, unless you're asking how the dynamics feed each other when combined into a system.
If that's the case, then I would suggest maybe starting with a review of the Luenberger observer. It's typically taught as the "dual" of the state feedback control problem - state feedback controls attempt to drive the system states to zero, and the observer design "rephrases" the problem to make feedback error the state and then attempts to drive the feedback error to zero with the same "pole placement" mechanic.
The filter has some error dynamics, and that dynamic error will get sent to the PID controller. The PID controller acts on the feedback + error dynamics and the resulting control signal gets sent to the plant.
I suggested starting with the Luenberger observer because IMO it's easier to look at with regards to error dynamics than the Kalman filter (try this, starting at equation 25). The process for evaluating the effect of the error is otherwise the same - filter generates feedback error until the filter states converge, and that error is fed to the controller.
This may be way off your intent with your question too, though. Please let me know if this is along the lines of what you're asking.
The Kalman filter optimizes the estimate of the system state in the presence of measurement errors and model errors. PID tries to drive a particular model/measurement system to a chosen state.
Allow me to summarize and add to Chuck's great answer.
You use a filter to make sense out of your raw sensor readings. They mathematically reduce noise, or coalesce multiple data points into a single useful measurement.
The difference between the actual measurement and the desired value is called error. The job of a PID controller is to minimize the error, keeping your robot as close to the target as possible. For example, a line-following robot may filter multiple light sensor values to determine how far away is is from the line. It uses a PID to adjust steering to minimize its distance from the line.