The state of a dynamical system and its uncertainty, as defined by its probability density function (PDF), are valuable for numerous fields in science and engineering. There have been numerous methods proposed to estimate and quantify this uncertainty. In as-trodynamics, space situational awareness (SSA) is a major area that relies on uncertainty quantification to estimate a space object's state and its associated uncertainty. This data is invaluable for making informed decisions regarding probability of collision, tracking, and catalog maintenance. A new method for uncertainty quantification based on orthogonal polynomials and the application of Liouville's theorem is developed. The method identifies the region of extreme probability at the time of interest and populates that region with structured points. The associated PDF is computed based on the a-priori PDF of the initial conditions and/or the nominal values of the system parameters (e.g. drag coefficient). High dimension orthogonal polynomials are used to approximate the PDF at the target time. Having an analytical expression for the propagated PDF enables rigorous probabilistic analysis. The present method is applied to several problems to compute the probability of collision between two objects. Numerical experiments show orders of magnitude improvement in computational cost versus classical Monte Carlo Methods. The new approach is easy to implement, extensible to higher dimensions, computationally efficient and provides a rigorous approach to address probability of collision problems in SSA.
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