Comments on "Series for Collision Probability in Short-Encounter Model"

Denis Arzelier (1) , Florent Bréhard (2) , Mioara Joldes (2)

1 LAAS-ROC - Équipe Recherche Opérationnelle, Optimisation Combinatoire et Contraintes
2 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande

An analytic solution based on a series expansion of the collision probability of two spheres under the assumption of short-term encounters has recently been proposed in \cite{Garcia-PelayoJGCD2016}. Relying on the analycity of the Gaussian distribution of the relative positions and its Taylor expansion, this approach yields a series expansion for the classical 2D formulation of the collision probability, where the terms are obtained as a product of Hermite polynomials. Unfortunately, this work cannot be considered as a new contribution to the field since the method itself is nothing but a crude variety of the method published shortly before in \cite{serraJGCD2016} (see also the early conference version \cite{Serra2014}) and referenced in \cite{Garcia-PelayoJGCD2016}. Indeed, this note shows that the series expansion formulation of the probability collision given in \cite{Garcia-PelayoJGCD2016} is exactly the one exposed in \cite{serraJGCD2016} without a preconditionner and therefore suffers from numerical issues clearly identified in this article and due to the so-called cancellation phenomenon. Classic and real-case numerical examples further demonstrate that the results of \cite{Garcia-PelayoJGCD2016} are of limited interest.

Mots clés: collision probablity; space encounters; d-finite functions

Domaines: Probabilités [math.PR]

Denis Arzelier, Florent Bréhard, Mioara Joldes. Comments on "Series for Collision Probability in Short-Encounter Model". Journal of Guidance, Control, and Dynamics, 2020, 43 (5), ⟨10.2514/1.G004560⟩. ⟨hal-02123054⟩