Application of the transformation of variables technique for uncertainty mapping in nonlinear filtering

This paper addresses the impact nonlinear observations of state variables have on uncertainty accuracy associated with state estimation algorithms. The transformation of variables technique is applied to exactly map probability density functions (PDFs) between domains completely spanned by different combinations of basis vectors. The technique allows for proper generation of the likelihood density when converting from measurement to state variable space and for association of a present state distribution with prior observation data. The exact mapping of probability distribution functions between domains and proper characterization of prior knowledge allows for Bayesian estimation to be appropriately carried out. A Bayes filter utilizing the technique is developed which uses the technique to map the uncertainty in time for generation of the prior density and in space for generation of the likelihood density. The filter is compared with conventional nonlinear filtering techniques in multiple scenarios to demonstrate the utility and insight offered for object tracking and parameter estimation applications.