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Analytical formulation of impulsive collision avoidance dynamics


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.

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  1. Although this condition is required for the present theoretical development, the analytical formulas here derived work also for the case of parallel velocities.

  2. According to the present non-dimensionalisation that means the conjunction duration is small compared to the orbital period.

  3. Typical collision avoidance manoeuvres involve velocity changes of the order of m/s, completely negligible when compared to their orbital velocity.


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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.

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Correspondence to Claudio Bombardelli.

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Bombardelli, C. Analytical formulation of impulsive collision avoidance dynamics. Celest Mech Dyn Astr 118, 99–114 (2014).

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