Gaussian process quadratures in nonlinear sigma-point filtering and smoothing

This paper is concerned with the use of Gaussian process regression based quadrature rules in the context of sigma-point-based nonlinear Kalman filtering and smoothing. We show how Gaussian process (i.e., Bayesian or Bayes-Hermite) quadratures can be used for numerical solving of the Gaussian integrals arising in the filters and smoothers. An interesting additional result is that with suitable selections of Hermite polynomial covariance functions the Gaussian process quadratures can be reduced to unscented transforms, spherical cubature rules, and to Gauss-Hermite rules previously proposed for approximate nonlinear Kalman filter and smoothing. Finally, the performance of the Gaussian process quadratures in this context is evaluated with numerical simulations.