Improved multidirectional Gaussian Mixture Models applied to probability of collision of resident space objects

Brown, Chase Patrick
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As the number of Resident Space Objects (RSOs) in Earth Orbit continues to rise by not only increased trackability but also an unprecedented number of commercial launches, conjunction assessment (CA) remains a paramount issue. Maintaining accuracy in probability of collision calculations is specifically of interest because any disparity will cause operators and analysts to lose trust, dismantling the entire CA system. Methods of state uncertainty propagation remain the most tractable way of controlling computational efficiency vs. accuracy. Gaussian Mixture Models (GMMs) have recently been used as an approach to maintain accuracy while decreasing the amount of computation time when compared to a Monte Carlo approach. These GMMs are able to represent the initial probability distribution function (pdf) as a weighted combination of individual Gaussian distributions. When propagated through a nonlinear function, such as the orbital equations of motion, higher order effects are maintained. How that initial pdf is split into a convolution of pdfs is the focus of this and current research. Multidirectional GMMs allow for the pdf to be split along directions of highest nonlinearity in a recursive manner. This study improves on this method by evaluating the directions at every split of every Gaussian mixture and also taking into account the weight of that Gaussian mixture. Equinoctial elements are also explored as a potential element space to perform the splitting due their ability to maintain linearity during propagation. Results show that these improved methods are able to capture the majority of the nonlinear effects very well with relatively few GMs, and therefore can generate accurate Pc calculations, but fail to converge to the exact Monte Carlo value as more mixtures are added in a reasonable time. This is still of use to get within 5% of the Monte Carlo value with very few propagations in highly nonlinear encounters