Abstract

Many widely used nonlinear estimation methods are based on quadrature numerical integration rules, such as the unscented transformation, the Gauss-Hermite quadrature rule, and the cubature rule. The conventional spherical-radial cubature rule and the unscented transformation are only the third-degree numerical rules that may not be accurate enough in some applications, such as long-term uncertainty propagation. The Gauss-Hermite quadrature rule is accurate to arbitrary degrees and all weights are positive. However, it suffers the curse-of-dimensionality problem. To balance the computational complexity and accuracy, the high-degree sparse-grid quadrature rule and the high-degree spherical-radial cubature rule have been proposed. However, the weights of these two rules may become negative, which can lead to negative definite covariance matrix. In this paper, two new highdegree compact quadrature rules, including a fifth-degree and a seventh-degree rule, are proposed to achieve all positive weights for a class of nonlinear estimation problems. In addition, for this class of problems the compact quadrature rules use less number of points than the sparse-grid quadrature rule and the high-degree cubature rule of the same accuracy degrees. A long-term space object orbit propagation problem and a target tracking problem are used to show the performance of the new rules.

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