Gaussian sum reapproximation applied to the probability of collision calculations

In predicting the collision of space debris, the propagated orbital uncertainty may not follow a Gaussian distribution if the initial orbital uncertainty is large or the propagation time is long. In this paper, a Gaussian mixture uncertainty propagation method developed by (Psiaki et al., 2015) is used to calculate the collision probability. The initial Gaussian distribution is fitted by the weighted Gaussian mixture components. The linear matrix inequality is optimized to prevent the covariance matrix of Gaussian mixture components from being too small, and an appropriate number of Gaussian mixture components is used to approximate the initial orbital covariance. At the same time, this paper provides a method to calculate the collision probability of two objects in which a Gaussian mixture is used to represent the distribution of orbital uncertainty. The linear method and the unscented Kalman filter (UKF) method for propagating the Gaussian covariance are analysed. The results of numerical simulations show that compared with the linear covariance propagation method, UKF method, and high-precision Monte Carlo covariance propagation method for space objects with a large initial orbital uncertainty, the Gaussian mixture method can be effectively applied to capture the non-Gaussian characteristics of the predicted nonlinear orbital dynamic uncertainty. Compared with the univariate splitting method, the advantage of this Gaussian mixture method is that it does not need to search for the most nonlinear direction. The accuracy of the collision probability calculation is improved from 1.460 × 10−3 to 1.663 × 10−3. A comparison of the computational burden between the Gaussian mixture algorithm and Vittaldev’s algorithm to achieve the same results is presented. The calculation burden of the Gaussian mixture method is approximately 3 times that of the univariate Gaussian method.