Relations Between Sparse-Grid Quadrature Rule and Spherical-Radial Cubature Rule in Nonlinear Gaussian Estimation

The recently emerging cubature Kalman filter and the sparse-grid quadrature filter approximate the numerical integrations for the mean and covariance in Gaussian filters using the spherical-radial cubature rule and the sparse-grid quadrature rule, respectively. This technical note reveals that 1) spherical rules can be obtained by the projection of sparse-grid quadrature rules; 2) the third- and some of the fifth-degree spherical-radial cubature rules can be directly constructed from sparse-grid quadrature rules.