Optimal Low-Thrust Trajectories to the Interior Earth-Moon Lagrange Point

  • Christopher Martin
  • Bruce A. Conway
  • Pablo Ibán̈ez
Conference paper


Minimum-time and hence minimum-fuel trajectories are found for a spacecraft using continuous low-thrust propulsion to leave low-Earth orbit and enter a specified periodic orbit about the interior Earth-Moon Lagrange point. The periodic orbit is generated with a new method that finds a periodic orbit as a solution to a numerical optimization problem, using an analytic approximation for the orbit as an initial guess. The numerical optimization method is then employed again to determine the low-thrust trajectory from low-Earth orbit to the specified periodic orbit. The optimizer chooses the thrust pointing angle time history and the point of arrival into the periodic orbit, in order to minimize the total flight time. The arrival position and velocity matching conditions are obtained from a parameterization of the orbit using cubic splines since no analytic description of the orbit exists.


Periodic Orbit Optimal Control Problem Initial Guess Collocation Point Shooting Method 
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.


  1. 1.
    Poincare, H., New Methods of Celestial Mechanics. Volume II - Methods of Newcomb, Gylden, Lindstedt, and Bohlin., NTIS, United States, 1967, Translation.Google Scholar
  2. 2.
    Ross, S., Cylindrical Manifolds and Tube Dynamics in the Restricted Three-Body Problem, Ph.D. thesis, California Institute of Technology, 2004.Google Scholar
  3. 3.
    Tripathi, R. K., Wilson, J., Cucinotta, F., Anderson, B., and Simonsen, L., “Materials trade study for lunar/gateway missions,” Advances in Space Research, Vol. 31, No. 11, 2003, pp. 2383 – 2388.CrossRefADSGoogle Scholar
  4. 4.
    Richardson, D., “Analytic construction of periodic orbits about the collinear points,” Celestial Mechanics, Vol. 22, No. 3, 1980, pp. 241 – 53.Google Scholar
  5. 5.
    Gomez, G. and Marcote, M., “High-order analytical solutions of Hill’s equations,” Celestial Mechanics and Dynamical Astronomy, Vol. 94, No. 2, 2006, pp. 197 – 211.MATHCrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Guibout, V. and Scheeres, D., “Periodic orbits from generating functions,” Advances in the Astronautical Sciences, Vol. 116, No. 2, 2004, pp. 1029–1048.Google Scholar
  7. 7.
    Gomez, G., Masdemont, J., and Simo, C., “Quasihalo orbits associated with libration points,” Journal of the Astronautical Sciences, Vol. 46, No. 2, 1998, pp. 135 – 176.MathSciNetGoogle Scholar
  8. 8.
    Howell, K. and Pernicka, H., “Numerical determination of Lissajous trajectories in the restricted three-body problem,” Celestial Mechanics, Vol. 41, No. 1–4, 1987–1988, pp. 107 – 24.Google Scholar
  9. 9.
    Rayman, M. D., Fraschetti, T. C., Raymond, C. A., and Russell, C. T., “Coupling of system resource margins through the use of electric propulsion: Implications in preparing for the Dawn mission to Ceres and Vesta,” Acta Astronautica, Vol. 60, No. 10–11, 2007, pp. 930 – 938.CrossRefADSGoogle Scholar
  10. 10.
    Russell, C., Barucci, M., Binzel, R., Capria, M., Christensen, U., Coradini, A., De Sanctis, M., Feldman, W., Jaumann, R., Keller, H., “Exploring the asteroid belt with ion propulsion: Dawn mission history, status and plans,” Advances in Space Research, Vol. 40, No. 2, 2007, pp. 193 – 201.CrossRefADSGoogle Scholar
  11. 11.
    Bryson, A. E and Ho Y. C, Applied optimal control: Optimization, estimation and control, Hemisphere Publishing Corporation, 1975.Google Scholar
  12. 12.
    Herman, A. L. and Conway, B. A., “Direct optimization using collocation based on high-order Gauss-Lobatto quadrature rules,” Journal of Guidance, Control, and Dynamics, Vol. 19, No. 3, 1996, pp. 592 – 599.MATHCrossRefGoogle Scholar
  13. 13.
    Burnett, D., Barraclough, B., Bennett, R., Neugebauer, M., Oldham, L., Sasaki, C., Sevilla, D., Smith, N., Stansbery, E., Sweetnam, D., and Wiens, R., “The Genesis Discovery Mission: return of solar matter to Earth,” Space Science Reviews, Vol. 105, No. 3–4, 2003, pp. 509 – 34.Google Scholar
  14. 14.
    Sabelhaus, P. A. and Decker, J., “James Webb space telescope: Project overview,” IEEE Aerospace and Electronic Systems Magazine, Vol. 22, No. 7, 2007, pp. 3 – 13.CrossRefGoogle Scholar
  15. 15.
    Schwarzschild, B., “WMAP spacecraft maps the entire cosmic microwave sky with unprecedented precision,” Physics Today, Vol. 56, No. 4, 2003, pp. 21 – 24.CrossRefGoogle Scholar
  16. 16.
    Tarragó, P., Study and Assessment of Low-Energy Earth-Moon Transfer Trajectories, Master’s thesis, Université de Liége, 2007.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Christopher Martin
    • 1
  • Bruce A. Conway
    • 2
  • Pablo Ibán̈ez
    • 3
  1. 1.Department of Aerospace EngineeringUniversity of IllinoisChicagoUSA
  2. 2.Department of Aerospace EngineeringUniversity of IllinoisUrbanaUSA
  3. 3.ETSI AeronáuticosTechnical University of Madrid (UPM)MadridSpain

Personalised recommendations