The over-extended Kalman filter - don't use it!

Abstract

Radar range rate measurements are not always used in target tracking filters because they are highly nonlinear in Cartesian space. A linear approxi- mation of range rate composed of its partial derivatives with respect to the track state vector is sometimes used in the measurement equation of an Extended Kalman filter. Unfortunately, this naive linearization can de- grade the filter's position estimates. The origins of this phenomenon are investigated and found to lie in the functional relationship induced by the linearization between the position elements of the track state vec- tor and the range rate innovation. An alternative lin- earization of range rate that is not a function of the position elements is derived. It is shown that the new linearization improves position estimate for some tra- jectories. An ordinary Kalman filter's gains are com- pared to those of the usual and alternative extended Kalman filters analytically and via simulation. The results show that the alternative linearization leads to a filter having the same position gains as an ordinary Kalman filter, and an additional gain on the track's radial velocity. This new extended Kalman filter can improve a tracking system's velocity estimates without risk to its position estimates.