Fast and Accurate Computation of Orbital Collision Probability for Short-Term Encounters

Romain Serra 1 Denis Arzelier 1, 2 Mioara Joldes 1, 2 Jean-Bernard Lasserre 1, 2 Aude Rondepierre 3 Bruno Salvy 4
1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes [Toulouse]
4 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : This article provides a new method for computing the probability of collision between two spherical space objects involved in a short-term encounter under Gaussian-distributed uncertainty. In this model of conjunction, classical assumptions reduce the probability of collision to the integral of a two-dimensional Gaussian probability density function over a disk. The computational method presented here is based on an analytic expression for the integral, derived by use of Laplace transform and D-finite functions properties. The formula has the form of a product between an exponential term and a convergent power series with positive coefficients. Analytic bounds on the truncation error are also derived and are used to obtain a very accurate algorithm. Another contribution is the derivation of analytic bounds on the probability of collision itself, allowing for a very fast and - in most cases - very precise evaluation of the risk. The only other analytical method of the literature - based on an approximation - is shown to be a special case of the new formula. A numerical study illustrates the efficiency of the proposed algorithms on a broad variety of examples and favorably compares the approach to the other methods of the literature.
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Romain Serra, Denis Arzelier, Mioara Joldes, Jean-Bernard Lasserre, Aude Rondepierre, et al.. Fast and Accurate Computation of Orbital Collision Probability for Short-Term Encounters. Journal of Guidance, Control, and Dynamics, American Institute of Aeronautics and Astronautics, 2016, 39 (5), pp.1009-1021. ⟨10.2514/1.G001353⟩. ⟨hal-01132149⟩

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