9.1 Introduction to Monomial Cubature Integration Rules

In the derivations for the EKF in Chapter 7, all nonlinear functions were expanded in a Taylor polynomial about a single point and the moment equations were then used to evaluate the integrals for each monomial term of the expansion. This constitutes an analytical method for evaluation of the integrals over the Gaussian-weighted nonlinear functions. This analytical approach was continued for the FDKF, where the Taylor polynomial is replaced by a multidimensional Stirling's interpolation formula. Stirling's interpolation approximation required the evaluation of the nonlinear functions at vector points equally spaced in all dimensions about the original Taylor polynomial expansion point. Although the FDKF has the same form as a sigma point Kalman filter, it was derived in a completely analytical way and can therefore also be thought of as part of the analytical linearization class of Kalman filters.

An alternative approach to evaluation of the Gaussian-weighted integrals is through the use of multidimensional numerical integration methods. In one dimension, these methods are classified as the well-known (to those who know them well) quadrature integration methods. For the multidimensional case, they have been labeled as “multi-dimensional quadrature” or “cubature” integration methods [1, 2]. In this book, we will use both terms interchangeably.

As noted in Chapter 5, the Gaussian prediction equations shown in (5.51) through ...

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