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On target for Venus — set oriented computation of energy efficient low thrust trajectories

  • Michael Dellnitz
  • Oliver Junge
  • Marcus Post
  • Bianca Thiere
Conference paper

Abstract

Recently new techniques for the design of energy efficient trajectories for space missions have been proposed that are based on the circular restricted three body problem as the underlying mathematical model. These techniques exploit the structure and geometry of certain invariant sets and associated invariant manifolds in phase space to systematically construct energy efficient flight paths. In this paper, we extend this model in order to account for a continuously applied control force on the spacecraft as realized by certain low thrust propulsion systems. We show how the techniques for the trajectory design can be suitably augmented and compute approximations to trajectories for a mission to Venus.

Keywords

Set oriented numerics Dynamical system Earth venus transfer Three body problem Low thrust trajectory Invariant manifold Reachable set Space mission design 

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References

  1. Abraham, R., Marsden, J.: Foundations of mechanics, 2nd edn Addison-Wesley, Reading, MA (1978)MATHGoogle Scholar
  2. Colonius, F., Kliemann, W.: The dynamics of control. Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA. With an appendix by Lars Grüne (2000)Google Scholar
  3. Dellnitz, M., Hohmann, A.: The computation of unstable manifolds using subdivision and continuation. In: Broer, H., van Gils, S., Hoveijn, I., Takens, F. (ed.) Nonlinear dynamical systems and chaos, pp. 449–459, Birkhäuser, PNLDE 19, (1996)Google Scholar
  4. Dellnitz, M., Hohmann, A.: A subdivision algorithm for the computation of unstable manifolds and global attractors. Numerische Mathematik, 75, 293–317 (1997)CrossRefMathSciNetMATHGoogle Scholar
  5. Dellnitz, M., Junge, O.: On the approximation of complicated dynamical behavior. SIAM J. Numer. Anal. 36(2), 491–515 (1999)CrossRefMathSciNetGoogle Scholar
  6. Dellnitz, M., Junge, O., Thiere, B.: The numerical detection of connecting orbits. Discrete Contin. Dyn. Syst. Ser. B, 1(1), 125–135 (2001)MathSciNetMATHCrossRefGoogle Scholar
  7. Deuflhard, P., Bornemann, F.: Scientific computing with ordinary differential equations, Texts in Applied Mathematics, Vol. 42. Springer-Verlag, New York (2002)Google Scholar
  8. Deuflhard, P., Pesch, H.-J., Rentrop, P.: A modified continuation method for the numerical solution of nonlinear two-point boundary value problems by shooting techniques. Numer. Math. 26(3), 327–343 (1976)CrossRefMathSciNetMATHGoogle Scholar
  9. ESA: Venus Express mission definition report. European Space Agency ESA-SCI, 6, 36–40 (2001)Google Scholar
  10. Fabrega, J., Schirmann, T., Schmidt, R., McCoy, D.: Venus Express: the first european mission to Venus. Int. Astronautical Congress, IAC-03-Q.2.06, 1–11 (2003)Google Scholar
  11. Gerthsen, C., Vogel, H.: Physik. Springer, Berlin (1993)Google Scholar
  12. Gómez, G., Jorba, À., Simó, C., Masdemont, J.: Dynamics and mission design near libration points. Vol. III, World Scientific Monograph Series in Mathematics, Vol. 4 World Scientific Publishing Co. Inc., River Edge, NJ (2001)Google Scholar
  13. Hairer, E., Nørsett, S.P., Wanner, G.: Solving ordinary differential equations. I. Non-stiff problems. Series in Computational Mathematics, Vol.8, 2nd edn. Springer-Verlag, Berlin, (1993)Google Scholar
  14. Koon, W., Lo, M., Marsden, J., Ross, S.: Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. Chaos, 10, 427–469 (2000a)CrossRefADSMathSciNetMATHGoogle Scholar
  15. Koon, W., Lo, M., Marsden, J., Ross, S.: Shoot themoon. In: AAS/AIAA Astrodynamics specialist conference, Florida, 105, 1017–1030 (2000b)Google Scholar
  16. Koon, W., Lo, M., Marsden, J., Ross, S.: Constructing a low energy transfer between jovian moons. Contemp. Math. 292, 129–145 (2002)MathSciNetGoogle Scholar
  17. Lo, M., Williams, B., Bollman, W., Han, D., Hahn, Y., Bell, J., Hirst, E., Corwin, R., Hong, P., Howell, K., Barden, B., Wilson, R.: Genesis mission design. J. Astronautical Sci. 49, 169–184 (2001)Google Scholar
  18. McGehee, R.: Some homoclinic orbits for the restricted 3-body problem. PhD thesis, University of Wisconsin, (1969)Google Scholar
  19. Meyer, K., Hall, R.: Hamiltonian mechanics and the n-body problem. Applied Mathematical Sciences, Springer-Verlag, Berlin (1992)Google Scholar
  20. Stoer, J., Bulirsch, R.: Introduction to numerical analysis, Vol. 12. Springer-Verlag, New York (2002)MATHGoogle Scholar
  21. Szebehely, V.: Theory of orbits—the restricted problem of three bodies. Academic Press, New York (1967)Google Scholar
  22. von Stryk, O.: Numerical solution of optimal control problems by direct collocation. In:Bulirsch, R., Miele, A., Stoer, J., Well, K.-H. (ed.) Optimal control—calculus of variations, optimal control theory and numerical methods, Internat. Ser. Numer. Math., pp. Vol. 111, 129–143. Birkhäuser, Basel (1993)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  • Michael Dellnitz
    • 1
  • Oliver Junge
    • 2
  • Marcus Post
    • 1
  • Bianca Thiere
    • 1
  1. 1.Faculty of Computer Science, Electrical Engineering and MathematicsUniversity of PaderbornPaderbornGermany
  2. 2.Center for Mathematical SciencesMunich University of TechnologyGarchingGermany

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