Fast and Accurate Computation of Orbital Collision Probability for Short-Term Encounters
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.
References
[1] , “Probability of Collision Error Analysis,” Journal of Space Debris, Vol. 1, No. 1, 1999, pp. 21–35. doi:https://doi.org/10.1023/A:1010056509803
[2] , “Probability of Collision Between Space Objects,” Journal of Guidance, Control, and Dynamics, Vol. 23, No. 5, Sept.–Oct. 2000, pp. 769–772. doi:https://doi.org/10.2514/2.4611
[3] , “General Method for Calculating Satellite Conjunction Probability,” Journal of Guidance, Control, and Dynamics, Vol. 24, No. 4, July–Aug. 2001, pp. 716–722. doi:https://doi.org/10.2514/2.4771
[4] , “A Numerical Implementation of Spherical Object Collision Probability,” Journal of Astronautical Sciences, Vol. 53, No. 1, Jan.–March 2005, p. 103.
[5] , Spacecraft Collision Probability, AIAA, The Aerospace Press, El Segundo, CA, 2008.
[6] , “Including Velocity Uncertainty in the Probability of Collision Between Space Objects,” AAS/AIAA Spaceflight Mechanics Meeting, American Astronautical Soc. Paper 2012-247, Feb. 2012.
[7] , “Satellite Conjunction Monte–Carlo Analysis,” Proceedings of AAS/AIAA Spaceflight Mechanics Meeting, American Astronautical Soc. Paper 2009-233, Feb. 2009.
[8] , “A Parametric Analysis of Orbital Debris Collision Probability and Maneuver Rate for Space Debris,” NASA JSC-25898, Aug. 1992.
[9] , “Aerospace Support to Space Situation Awareness,” MIT Lincoln Laboratory Satellite Operations and Safety Workshop, Haystack Observatory, Chelmsford, MA, Oct. 2002.
[10] , “Collision Probability Analyses for Earth-Orbiting Satellites,” Advances in the Astmnautical Sciences, Vol. 96, July 1997, pp. 1033–1048.
[11] , “Solving a Class of Multivariate Integration Problems via Laplace Techniques,” Applicationes Mathematicae, Vol. 28, No. 4, 2001, pp. 391–405. doi:https://doi.org/10.4064/am28-4-2
[12] , “A Holonomic Systems Approach to Special Functions Identities,” Journal of Computational and Applied Mathematics, Vol. 32, No. 3, 1990, pp. 321–368. doi:https://doi.org/10.1016/0377-0427(90)90042-X
[13] , “D-Finiteness: Algorithms and Applications,” Kauers M., ISSAC 2005, Proceedings of the 18th International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2005, pp. 2–3.
[14] , Computation of Multivariate Normal and t Probabilities, Vol. 45, Springer–Verlag, Berlin, 2009.
[15] , “Satellite Collision Probability for Nonlinear Relative Motion,” Journal of Guidance, Control, and Dynamics, Vol. 26, No. 5, Sept.–Oct. 2003, pp. 728–733. doi:https://doi.org/10.2514/2.5127
[16] , “Collision Probability for Larger Bodies Having Nonlinear Relative Motion,” Journal of Guidance, Control, and Dynamics, Vol. 29, No. 6, 2006, pp. 1468–1472. doi:https://doi.org/10.2514/1.23509
[17] , “Beta Conjunction Analysis Tool,” Proceedings of AAS/AIAA Astrodynamics Specialist Conference, American Astronautical Soc. Paper 2007-393, Aug. 2007.
[18] , “Spacecraft Collision Probability for Long-Term Encounters,” Advances in the Astronautical Sciences, Vol. 116, No. 1, 2003, pp. 767–784.
[19] , “Evaluating the Short Encounter Assumption of the Probability of Collision Formula,” AAS/AIAA Spaceflight Mechanics Meeting, American Astronautical Soc. Paper 2012-248, Feb. 2012.
[20] , Probability, Random Variables, and Stochastic Processes, McGraw–Hill, New York, 2002.
[21] , “Calculating Collision Probability for Arbitrary Space-Vehicle Shapes Via Numerical Quadrature,” Journal of Guidance, Control, and Dynamics, Vol. 28, No. 6, 2005, pp. 1326–1328. doi:https://doi.org/10.2514/1.14526
[22] , “Multiple-Precision Evaluation of the Airy Ai Function with Reduced Cancellation,” edited by Nannarelli A., Seidel P.-M. and Tang P. T. P., 21st IEEE Symposium on Computer Arithmetic, IEEE Computer Soc., Washington, D.C., 2013, pp. 175–182.
[23] , “Reduced Cancellation in the Evaluation of Entire Functions and Applications to the Error Function,” SIAM Journal on Numerical Analysis, Vol. 45, No. 6, 2007, pp. 2564–2576. doi:https://doi.org/10.1137/060669589
[24] , An Introduction to Transform Theory, Academic Press, New York, 1971.
[25] , The Laplace Transform,
Princeton Mathematical Series , Princeton Univ. Press, Princeton, NJ, 1946.[26] , “Gfun: a Maple Package for the Manipulation of Generating and Holonomic Functions in One Variable,” ACM Transactions on Mathematical Software, Vol. 20, No. 2, 1994, pp. 163–177. doi:https://doi.org/10.1145/178365.178368
[27] , Analytic Combinatorics, 1 ed., Cambridge Univ. Press, New York, 2009.
[28] , Handbook of Floating-Point Arithmetic, Birkhäuser Boston, Cambridge, MA, 2010.
[29] , An Introduction to Probability Theory and Its Applications, Vol. 1, Wiley, New York, 1957.
