Orbital Uncertainty Propagation Using Positive Weighted Compact Quadrature Rule

Abstract

In space surveillance, long-term orbital propagation is often required to track space objects when the measurement updates are scarce. It is a challenging uncertainty propagation problem and can be addressed by many different numerical integration rules, such as the unscented transformation, the Gauss–Hermite quadrature rule, and the cubature rule. The conventional cubature rule and the unscented transformation are only the third-degree numerical rules that may not be precise enough for the long-term uncertainty propagation. The Gauss–Hermite quadrature rule is accurate to arbitrary degrees. 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 in recent years. Unfortunately, the weights of these two rules may become negative, which can lead to a negative-definite covariance matrix and degrade the performance of uncertainty propagation. In this paper, two new compact quadrature rules together with an adaptive Gaussian-mixture model are proposed to propagate the uncertainty through the nonlinear orbital dynamics, which can achieve high degrees of propagation accuracy while maintaining all positive weights. High-Earth-orbit propagation and low-Earth-orbit propagation examples are used to demonstrate the superb performance of the proposed compact quadrature rules.