Celestial Mechanics and Dynamical Astronomy

, Volume 118, Issue 2, pp 99–114 | Cite as

Analytical formulation of impulsive collision avoidance dynamics

  • Claudio Bombardelli
Original Article


The paper deals with the problem of impulsive collision avoidance between two colliding objects in three dimensions and assuming elliptical Keplerian orbits. Closed-form analytical expressions are provided that accurately predict the relative dynamics of the two bodies in the encounter b-plane following an impulsive delta-V manoeuvre performed by one object at a given orbit location prior to the impact and with a generic three-dimensional orientation. After verifying the accuracy of the analytical expressions for different orbital eccentricities and encounter geometries the manoeuvre direction that maximises the miss distance is obtained numerically as a function of the arc length separation between the manoeuvre point and the predicted collision point. The provided formulas can be used for high-accuracy instantaneous estimation of the outcome of a generic impulsive collision avoidance manoeuvre and its optimisation.


Collision avoidance Space debris Perturbation theory Orbit propagation COLA manoeuvre Relative dynamics Iridium-Cosmos collision 



The study has been supported by the research project “Dynamic Simulation of Complex Space Systems” supported by the Dirección General de Investigación of the (no longer existing) Spanish Ministry of Science and Innovation through contract AYA2010-18796. The author would like to thank the two reviewers as well as Noelia Sánchez-Ortiz (Deimos Space) and Pierluigi Righetti (Eumetsat) for their useful suggestions.


  1. Akella, M., Alfriend, K.: Probability of collision between space objects. J. Guid. Control Dyn. 23(5), 769–772 (2000)ADSCrossRefGoogle Scholar
  2. Alfano, S.: Aerospace support to space situational awareness. MIT Lincoln Laboratory Satellite Operations and Safety Workshop, Haystack Observatory, Chelmsford, Massachusetts (2002)Google Scholar
  3. Alfano, S.: Collision avoidance maneuver planning tool. In: 15th AAS/AIAA Astrodynamics Specialist Conference, pp. 7–11 (2005)Google Scholar
  4. Bombardelli, C, Baù, G.: Accurate analytical approximation of asteroid deflection with constant tangential thrust. Celest. Mech. Dyn. Astron. 114, 279–295 (2012)Google Scholar
  5. Bombardelli, C., Bau, G., Pelaez, J.: Asymptotic solution for the two-body problem with constant tangential thrust acceleration. Celest. Mech. Dyn. Astron. 110(3), 239–256 (2011)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. Chan, F.K.: Spacecraft Collision Probability. Aerospace Press, Beijing (2008)Google Scholar
  7. Chan, K.: Collision probability analyses for earth-orbiting satellites. In: 2001 Flight Mechanics, Symposium, vol 1 (2001)Google Scholar
  8. Conway, B.A.: Near-optimal deflection of earth-approaching asteroids. J. Guid. Control Dyn. 24(5), 1035–1037 (2001)ADSCrossRefGoogle Scholar
  9. COPUOS: Towards long-term sustainability of space activities: overcoming the challenge of space debris. Tech. rep., a/AC.105/C.1/2011/CRP.14. Committee on the Peaceful Uses of Outer Space. Available on line at (2011-02-08)
  10. Izzo, D.: Optimization of interplanetary trajectories for impulsive and continuous asteroid deflection. J. Guid. Control Dyn. 30(2), 401–408 (2007)ADSCrossRefMathSciNetGoogle Scholar
  11. Kahle, R., Hahn, G., Kuhrt, E.: Optimal deflection of neos en route of collision with the earth. Icarus 182(2), 482–488 (2006)ADSCrossRefGoogle Scholar
  12. Kim, E.H., Kim, H.D., Kim, H.J.: Optimal solution of collision avoidance maneuver with multiple space debris. J. Space Oper. 9(3), 19–31 (2012)Google Scholar
  13. Krag, H., Flohrer, T., Lemmens, S.: Consideration of space debris mitigation requirements in the operation of LEO missions. Paper 1257086. In: Proceedings of the 12th International Conference on Space Operations, Stockholm, 11-15 June 2012, AIAA (2012)Google Scholar
  14. Newman, K., Frigm, R., McKinley, D.: It’s not a big sky after all: justification for a close approach prediction and risk assessment process. Adv. Astronaut. Sci. 135(2), 1113–1132 (2009)Google Scholar
  15. Patera, R.P.: General method for calculating satellite collision probability. J. Guid. Control Dyn. 24(4), 716–722 (2001)Google Scholar
  16. Pelaez, J., Hedo, J., de Andres, P.: A special perturbation method in orbital dynamics. Celest. Mech. Dyn. Astron. 97(2), 131–150 (2007)ADSCrossRefzbMATHGoogle Scholar
  17. Righetti, P., Sancho, F., Lazaro, D., Damiano, A.: Handling of conjunction warnings in eumetsat flight dynamics. J. Aerosp. Eng. 3(2), 39 (2011)Google Scholar
  18. Sanchez-Ortiz, N., Grande-Olalla, I., Pulido, J.A., Merz, K.: Collision risk assessment and avoidance manoeuvres—the new coram tool for esa. In: 64 th International Astronautical Congress, Beijing, China (2013)Google Scholar
  19. Valsecchi, G., Milani, A., Gronchi, G., Chesley, S.: Resonant returns to close approaches: analytical theory. Astron. Astrophys. 408(3), 1179–1196 (2003)ADSCrossRefGoogle Scholar
  20. Vasile, M., Colombo, C.: Optimal impact strategies for asteroid deflection. J. Guid. Control Dyn. 31(4), 858–872 (2008)ADSCrossRefGoogle Scholar
  21. Yamanaka, K., Ankersen, F.: New state transition matrix for relative motion on an arbitrary elliptical orbit. J. Guid. Control Dyn. 25(1), 60–66 (2002)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Research Associate, ETSI AeronauticosTechnical University of MadridMadridSpain

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