Speaker
Prof.
Ricardo García-Pelayo
(Universidad Politécnica de Madrid)
Description
The increase of conjunctions between active satellites in Low Earth Orbit (LEO) and other objects, either space debris or other satellites, has made necessary to evaluate the risk posed by these conjunctions in order to decide if evasive maneuvering is needed. The calculation of the collision probability between a pair of objects must done by a precise and fast algorithm because of the enormous and growing LEO population and because numerical methods for collision avoidance maneuver optimization may need to evaluate this probability several times during its execution.
In this article, a series to compute the collision probability of two spheres under the assumptions of short-term encounter, which generally hold in LEO, has been derived. It is valid for both Gaussian and non-Gaussian distributions of the position of the spheres, and in the particular case of a Gaussian distribution the use of Hermite polynomials yields a simple form for the series.
The parameters that appear in our formula, or in others which address the same issue, are the axes of the projection of some incertitude ellipsoid on the collision plane and the projection on the same plane of the relative coordinates of the objects which might collide. A region of practical interest in this parameter space has been carefully defined based on satellites' real data, and a representative sampling set was chosen. On this sampling set a comparison between the new series and previous algorithms has been performed for the Gaussian case to measure the performance of the proposed method. The presented series is found to be faster than any other algorithm in every explored case. Numerical evidence suggests that if the series for the Gaussian case is truncated when the last term is smaller than the computed probability times a tolerance of 0.1, then the last term is an upper bound for the error. This article also presents very strong evidence for the case that the first two terms of the series are sufficient for the computation of the probability of collision, and that the absolute value of its second term is an upper bound for the error made when using it.
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Primary author
Prof.
Ricardo García-Pelayo
(Universidad Politécnica de Madrid)
Co-author
Mr
Javier Hernando-Ayuso
(The University of Tokyo)
