Unscented Kalman filter revisited — Hermite-Gauss Quadrature approach

Kalman filter is a frequently used tool for linear state estimation due to its simplicity and optimality. It can further be used for fusion of information obtained from multiple sensors. Kalman filtering is also often applied to nonlinear systems. As the direct application of bayesian functional recursion is computationally not feasible, approaches commonly taken use either a local approximation - Extended Kalman Filter based on linearization of the non-linear model - or the global one, as in the case of Particle Filters. An approach to the local approximation is the so called Unscented Kalman Filter. It is based on a set of symmetrically distributed sample points used to parameterise the mean and the covariance. Such filter is computationally simple and no linearization step is required. Another approach to selecting the set of sample points based on decorrelation of multivariable random variables and Hermite-Gauss Quadrature is introduced in this paper. This approach provides an additional justification of the Unscented Kalman Filter development and provides further options to improve the accuracy of the approximation, particularly for polynomial nonlinearities. A detailed comparison of the two approaches is presented in the paper.