Gaussian Sum Reapproximation for Use in a Nonlinear Filter

Abstract

A new method has been developed to approximate one Gaussian sum by another. This algorithm is being developed as part of an effort to generalize the concept of a particle filter. In a traditional particle filter, the underlying probability density function is described by particles: Dirac delta functions with infinitesimal covariances. This paper develops an important component of a more general filter, which uses a Gaussian sum with “fattened” finite-covariance “blobs” (i.e., Gaussian components), which replace infinitesimal particles. The goal of such a filter is to save computational effort by using many fewer Gaussian components than particles. Most of the techniques necessary for this type of filter exist. The one missing technique is a resampling algorithm that bounds the covariance of each Gaussian component while accurately reproducing the original probability distribution. The covariance bounds keep the blobs from becoming too “fat” to ensure low truncation error in extended Kalman filter or unscented Kalman filter calculations. A new resampling algorithm is described, and its performance is studied using two test cases. The new algorithm enables Gaussian sum filter performance that is better than standard nonlinear filters when applied in simulation to a difficult seven-state estimation problem: the new filter’s root mean square error is only 60% higher than the Cramer–Rao lower bound, whereas the next best filter’s root mean square error is 370% higher.