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A Methodology for the Computation of Constrained Orbits and its Application to the Design of Solar System Trajectories

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Abstract

The purpose of this paper is the development of a procedure for the determination of constrained trajectories of spacecraft in the Solar System and its application to the computation of orbits, such as heteroclinic connections in the Sun-Earth-Moon system, that cannot be computed using only a two or three-body problem approach. Starting from a nominal trajectory computed in a simplified force model, the aim is to compute a new one, close to the original, but satisfying more realistic equations of motion. The formulation adopted allows for the existence of discontinuities in velocity and specific constraints for the states of the orbit, including boundary conditions. The continuity requirement of the orbit is accomplished in position by a differential correction process. The discontinuities in velocity are minimized using an optimization procedure. Some significant applications are presented, mainly regarding libration point orbit missions. It will be shown how using the initial input, the constraints and weights, an orbit, initially determined in the Circular Restricted Three–Body Problem, can be refined in order to fulfill the equations of motion of the Restricted n–Body Problem and specific scopes. In all the cases, an accurate initial guess is a key point.

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Notes

  1. Here we simplify the notation in such a way that φ x ≡ φ(t i+1;t i , X i )|x .

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Acknowledgments

This work has been supported by the Spanish grants MTM2006–05849 (E.M.A., G.G.) and MTM2009–06973 and 2009SGR859 (J.J.M.) and by the Astronet Marie Curie fellowship MCRTN–CT–2006–035151 (E.M.A.). We also acknowledge the use of the UPC Applied Math cluster system for research computing (see http://www.ma1.upc.edu/eixam/index.html).

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Authors and Affiliations

  1. IFAC–CNR, Via Madonna del Piano 10, 50019, Sesto Fiorentino, FI, Italy

    Elisa Maria Alessi

  2. IEEC & Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007, Barcelona, Spain

    Gerard Gómez

  3. IEEC & Departament de Matemàtica Aplicada I, ETSEIB, Universitat Politècnica de Catalunya, Diagonal 647, 08028, Barcelona, Spain

    Josep J. Masdemont

Authors
  1. Elisa Maria Alessi
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  2. Gerard Gómez
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  3. Josep J. Masdemont
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Correspondence to Elisa Maria Alessi.

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Alessi, E.M., Gómez, G. & Masdemont, J.J. A Methodology for the Computation of Constrained Orbits and its Application to the Design of Solar System Trajectories. J of Astronaut Sci 59, 477–501 (2012). https://doi.org/10.1007/s40295-014-0004-2

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