Bayesian Estimation and Tracking: A Practical Guide

Chapter 11

The Sigma Point Class: The Gauss–Hermite Kalman Filter

Practical methods for one-dimensional Gauss–Hermite integration has its origins in the work of Wilf [1], as noted in the Numerical Recipes book [2], and the short but excellent paper by Ball [3]. Unlike the methods addressed in the preceding chapters, Gauss–Hermite integration is a quadrature method that uses orthogonal polynomials instead of simple polynomials.

Consider the general multidimensional integral (5.60), rewritten here in more explicit form

(11.1)

Changing variables to remove the factor of 1/2 from the exponent, let

(11.2)

or

(11.3)

Thus, (11.1) becomes

(11.4)

with

(11.5)