Bayesian Estimation and Tracking: A Practical Guide

9.2 The Unscented Kalman Filter

9.2.1 Background

The unscented Kalman filter (UKF) has its origins in a pair of papers presented at the 1995 American Control Conference in Seattle, Washington [4, 5]. In both papers, a new method for implementing a linearized approximation to nonlinear state estimation was presented that, unlike the EKF, did not require the explicit calculation of Jacobians. Their method of “approximating a Gaussian distribution efficiently by a discrete distribution [of vector points] allows a nonlinear transformation to be applied to each of the points independently.” Their method was ad hoc in that they specified a set of vector points that are symmetric about the mean of a multidimensional Gaussian distribution. The points were selected so that when the set of points are used as the input to a nonlinear transformation, a weighted sum of the resulting output points produced a better estimate of the transformed mean than that produced by an EKF.

In Refs [6–8], Julier presents his method in much more detail where, in Ref. [7], he called his method a “distribution approximation filter” and demonstrated its superiority to the EKF for highly nonlinear functional transformations. In Ref. [8] Julier claimed “that it is easier to approximate a probability distribution than it is to approximate an arbitrary nonlinear function or transformation.” However, in Ref. [9], Lefebvre, et al. “derives the exact same estimator by linearizing the process and measurement functions ...