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Probabilistic Collision Avoidance for Long-term Space Encounters via Risk Selection

  • Romain SerraEmail author
  • Denis Arzelier
  • Mioara Joldes
  • Aude Rondepierre

Abstract

This paper deals with collision avoidance between two space objects involved in a long-term encounter, assuming Keplerian linearized dynamics. The primary object is an active spacecraft - able to perform propulsive maneuvers - originally set on a reference orbit. The secondary object - typically an orbital debris - is passive and represents a threat to the primary. The collision avoidance problem addressed in this paper aims at computing a fuel-optimal, finite sequence of impulsive maneuvers performed by the active spacecraft such that instantaneous collision probability remains below a given threshold over the encounter and that the primary object goes back to its reference trajectory at the end of the mission. Two successive relaxations are used to turn the original hard chance-constrained problem into a deterministic version that can be solved using mixed-integer linear programming solvers. An additional contribution is to propose a new algorithm to compute probabilities for 3-D Gaussian random variables to lie in Euclidean balls, enabling us to numerically validate the computed maneuvers by efficiently evaluating the resulting instantaneous collision probabilities.

Keywords

Collision Avoidance Convex Polyhedron Chance Constraint Reference Orbit Polyhedral Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Alfano, S.: A numerical implementation of spherical objet collision probability. Journal of Astronautical Sciences 53(1) (January-March 2005)Google Scholar
  2. 2.
    Bertsimas, D., Tsitsiklis, J.N.: Introduction to linear optimization. AIAA. Athena Scientific, Belmont, Massachusetts (1997)Google Scholar
  3. 3.
    Blackmore, L., Li, H., Williams, B.: A probabilistic approach to optimal robust path planning with obstacles. In: American Control Conference, Minneapolis, MA (June 2006)Google Scholar
  4. 4.
    Blackmore, L., Ono, M., Williams, B.: Chance-constrained optimal path planning with obstacles. IEEE Transactions on Robotics 27(6), 1080–1094 (2011)CrossRefGoogle Scholar
  5. 5.
    Chan, F.K.: Spacecraft Collision Probability. AIAA. The Aerospace Press (2008)Google Scholar
  6. 6.
    Chevillard, S., Mezzarobba, M.: Multiple-Precision Evaluation of the Airy Ai Function with Reduced Cancellation. In: Nannarelli, A., Seidel, P.-M., Tang, P.T.P. (eds.) 21st IEEE SYMPOSIUM on Computer Arithmetic, Los Alamitos, CA, April 2013, pp. 175–182. IEEE Computer Society (2013)Google Scholar
  7. 7.
    Gawronski, W., Müller, J., Reinhard, M.: Reduced cancellation in the evaluation of entire functions and applications to the error function. SIAM Journal on Numerical Analysis 45(6), 2564–2576 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Gurobi Optimization, Inc., Gurobi Optimizer Reference Manual (2014), http://www.gurobi.com
  9. 9.
    Lasserre, J.B., Zeron, E.S.: Solving a class of multivariate integration problems via laplace techniques. Applicationes Mathematicae (2001)Google Scholar
  10. 10.
    Mueller, J.B., Larsson, R.: Collision avoidance maneuver planning with robust optimization. In: 7th International ESA Conference on Guidance, Navigation and Control Systems (2008)Google Scholar
  11. 11.
    Patera, R.P., Peterson, G.E.: Space vehicle maneuver method to lower collision risk to an acceptable level. Journal of Guidance, Control and Dynamics 26(2) (March-April 2003)Google Scholar
  12. 12.
    Richards, A.R., Schouwenaars, T., How, J.P., Feron, E.: Spacecraft trajectory planning with avoidance constraints using mixed-integer linear programming. Journal of Guidance, Control and Dynamics 25(4) (August 2002)Google Scholar
  13. 13.
    Salvy, B.: D-finiteness: Algorithms and applications. In: Kauers, M. (ed.) ISSAC 2005: Proceedings of the 18th International Symposium on Symbolic and Algebraic Computation, Beijing, China, July 24-27, pp. 2–3. ACM Press (2005) Abstract for an invited talkGoogle Scholar
  14. 14.
    Salvy, B., Zimmermann, P.: Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software 20(2), 163–177 (1994)CrossRefzbMATHGoogle Scholar
  15. 15.
    Sánchez-Ortiz, N., Belló-Mora, M., Klinkrad, H.: Collision avoidance manoeuvres during spacecraft mission lifetime: Risk reduction and required δv. Advances in Space Research 38(9), 2107–2116 (2006)CrossRefGoogle Scholar
  16. 16.
    Serra, R., Arzelier, D., Lasserre, J.-B., Joldes, M., Rondepierre, A.: A new method to compute the probability of collision for short-term space encounters. In: Proceedings of AAS/AIAA Astrodynamics Specialist Conference, San Diego, California, USA (August 2014)Google Scholar
  17. 17.
    Slater, G.L., Byram, S.M., Williams, T.W.: Collision avoidance for satellites in formation flight. Journal of Guidance, Control, and Dynamics 29(5), 1140–1146 (2006)CrossRefGoogle Scholar
  18. 18.
    Tschauner, J., Hempel, P.: Optimale beschleunigungs-programme fur des rendezvous manover. Astronautica Acta 5-6, 296–307 (1964)Google Scholar
  19. 19.
    Widder, D.V.: An introduction to transform theory. Academic Press New York (1971)Google Scholar
  20. 20.
    Yamanaka, K., Ankersen, F.: New state transition matrix for relative motion on an arbitrary elliptical orbit. Journal of Guidance, Control and Dynamics 25(1) (January 2002)Google Scholar
  21. 21.
    Zeilberger, D.: A holonomic systems approach to special functions identities. Journal of Computational and Applied Mathematics 32(3), 321–368 (1990)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Romain Serra
    • 1
    Email author
  • Denis Arzelier
    • 2
  • Mioara Joldes
    • 1
  • Aude Rondepierre
    • 3
  1. 1.CNRS, LAASUniversité de Toulouse, INSA, LAASToulouseFrance
  2. 2.CNRS, LAAS, Université de Toulouse, LAASToulouseFrance
  3. 3.Institut de Mathématiques de Toulouse, INSAUniversité de ToulouseToulouseFrance

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