Sparse Gauss-Hermite Quadrature Filter with Application to Spacecraft Attitude Estimation

A novel sparse Gauss-Hermite quadrature filter is proposed using a sparse-grid method for multidimensional numerical integration in the Bayesian estimation framework. The conventional Gauss-Hermite quadrature filter is computationally expensive for multidimensional problems, because the number of Gauss-Hermite quadrature points increases exponentially with the dimension. The number of sparse-grid points of the computationally efficient sparse Gauss-Hermite quadrature filter, however, increases only polynomially with the dimension. In addition, it is proven in this paper that the unscented Kalman filter using the suggested optimal parameter is a subset of the sparse Gauss-Hermite quadrature filter. The sparse Gauss-Hermite quadrature filter is therefore more flexible to use than the unscented Kalman filter in terms of the number of points and accuracy level, and it is more efficient than the conventional Gauss-Hermite quadrature filter. The application to the spacecraft attitude estimation problem demonstrates better performance of the sparse Gauss-Hermite quadrature filter in comparison with the extended Kalman filter, the cubature Kalman filter, and the unscented Kalman filter.