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

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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.


  • [1] Alfriend K. T., Akella M. R., Frisbee J., Foster F. L., Lee D.-J. and Wilkins M., “Probability of Collision Error Analysis,” Journal of Space Debris, Vol. 1, No. 1, 1999, pp. 21–35. doi: CrossrefGoogle Scholar

  • [2] Akella M. R. and Alfriend K. T., “Probability of Collision Between Space Objects,” Journal of Guidance, Control, and Dynamics, Vol. 23, No. 5, Sept.–Oct. 2000, pp. 769–772. doi: LinkGoogle Scholar

  • [3] Patera R. P., “General Method for Calculating Satellite Conjunction Probability,” Journal of Guidance, Control, and Dynamics, Vol. 24, No. 4, July–Aug. 2001, pp. 716–722. doi: LinkGoogle Scholar

  • [4] Alfano S., “A Numerical Implementation of Spherical Object Collision Probability,” Journal of Astronautical Sciences, Vol. 53, No. 1, Jan.–March 2005, p. 103. CrossrefGoogle Scholar

  • [5] Chan F. K., Spacecraft Collision Probability, AIAA, The Aerospace Press, El Segundo, CA, 2008. LinkGoogle Scholar

  • [6] Coppola V. T., “Including Velocity Uncertainty in the Probability of Collision Between Space Objects,” AAS/AIAA Spaceflight Mechanics Meeting, American Astronautical Soc. Paper  2012-247, Feb. 2012. Google Scholar

  • [7] Alfano S., “Satellite Conjunction Monte–Carlo Analysis,” Proceedings of AAS/AIAA Spaceflight Mechanics Meeting, American Astronautical Soc. Paper  2009-233, Feb. 2009. Google Scholar

  • [8] Foster J. L. and Estes H. S., “A Parametric Analysis of Orbital Debris Collision Probability and Maneuver Rate for Space Debris,” NASA JSC-25898, Aug. 1992. Google Scholar

  • [9] Alfano S., “Aerospace Support to Space Situation Awareness,” MIT Lincoln Laboratory Satellite Operations and Safety Workshop, Haystack Observatory, Chelmsford, MA, Oct. 2002. Google Scholar

  • [10] Chan F. K., “Collision Probability Analyses for Earth-Orbiting Satellites,” Advances in the Astmnautical Sciences, Vol. 96, July 1997, pp. 1033–1048. Google Scholar

  • [11] Lasserre J. B. and Zeron E. S., “Solving a Class of Multivariate Integration Problems via Laplace Techniques,” Applicationes Mathematicae, Vol. 28, No. 4, 2001, pp. 391–405. doi: CrossrefGoogle Scholar

  • [12] Zeilberger D., “A Holonomic Systems Approach to Special Functions Identities,” Journal of Computational and Applied Mathematics, Vol. 32, No. 3, 1990, pp. 321–368. doi: CrossrefGoogle Scholar

  • [13] Salvy B., “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. Google Scholar

  • [14] Genz A. and Bretz F., Computation of Multivariate Normal and t Probabilities, Vol. 45, Springer–Verlag, Berlin, 2009. CrossrefGoogle Scholar

  • [15] Patera R. P., “Satellite Collision Probability for Nonlinear Relative Motion,” Journal of Guidance, Control, and Dynamics, Vol. 26, No. 5, Sept.–Oct. 2003, pp. 728–733. doi: LinkGoogle Scholar

  • [16] Patera R. P., “Collision Probability for Larger Bodies Having Nonlinear Relative Motion,” Journal of Guidance, Control, and Dynamics, Vol. 29, No. 6, 2006, pp. 1468–1472. doi: LinkGoogle Scholar

  • [17] Alfano S., “Beta Conjunction Analysis Tool,” Proceedings of AAS/AIAA Astrodynamics Specialist Conference, American Astronautical Soc. Paper  2007-393, Aug. 2007. Google Scholar

  • [18] Chan F. K., “Spacecraft Collision Probability for Long-Term Encounters,” Advances in the Astronautical Sciences, Vol. 116, No. 1, 2003, pp. 767–784. Google Scholar

  • [19] Coppola V.T., “Evaluating the Short Encounter Assumption of the Probability of Collision Formula,” AAS/AIAA Spaceflight Mechanics Meeting, American Astronautical Soc. Paper  2012-248, Feb. 2012. Google Scholar

  • [20] Papoulis A. and Pillai S. U., Probability, Random Variables, and Stochastic Processes, McGraw–Hill, New York, 2002. Google Scholar

  • [21] Patera R. P., “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: LinkGoogle Scholar

  • [22] Chevillard S. and Mezzarobba M., “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. CrossrefGoogle Scholar

  • [23] Gawronski W., Müller J. and Reinhard M., “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: CrossrefGoogle Scholar

  • [24] Widder D. V., An Introduction to Transform Theory, Academic Press, New York, 1971. Google Scholar

  • [25] Widder D.V., The Laplace Transform, Princeton Mathematical Series, Princeton Univ. Press, Princeton, NJ, 1946. Google Scholar

  • [26] Salvy B. and Zimmermann P., “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: CrossrefGoogle Scholar

  • [27] Flajolet P. and Sedgewick R., Analytic Combinatorics, 1 ed., Cambridge Univ. Press, New York, 2009. CrossrefGoogle Scholar

  • [28] Muller J.-M., Brisebarre N., de Dinechin F., Jeannerod C.-P., Lefèvre V., Melquiond G., Revol N., Stehlé D. and Torres S., Handbook of Floating-Point Arithmetic, Birkhäuser Boston, Cambridge, MA, 2010. CrossrefGoogle Scholar

  • [29] Feller W., An Introduction to Probability Theory and Its Applications, Vol. 1, Wiley, New York, 1957. Google Scholar