Celestial Mechanics and Dynamical Astronomy

, Volume 125, Issue 3, pp 287–307 | Cite as

A two-level perturbation method for connecting unstable periodic orbits with low fuel cost and short time of flight: application to a lunar observation mission

Original Article


The proposed method connects two unstable periodic orbits by employing trajectories of their associated invariant manifolds that are perturbed in two levels. A first level of velocity perturbations is applied on the trajectories of the discretized manifolds at the points where they approach the nominal unstable periodic orbit in order to accelerate them. A second level of structured velocity perturbations is applied to trajectories that have already been subjected to first level perturbations in order to approximately meet the necessary conditions for a low \(\varDelta \text {V}\) transfer. Due to this two-level perturbation approach, the number of the trajectories obtained is significantly larger compared with approaches that employ traditional invariant manifolds. For this reason, the problem of connecting two unstable periodic orbits through perturbed trajectories of their manifolds is transformed into an equivalent discrete optimization problem that is solved with a very low computational complexity algorithm that is proposed in this paper. Finally, the method is applied to a lunar observation mission of practical interest and is found to perform considerably better in terms of \(\varDelta \text {V}\) cost and time of flight when compared with previous techniques applied to the same project.


Space mission design Unstable periodic orbits Perturbed trajectories of invariant manifolds Halo orbits Lunar mission 



The authors would like to thank an anonymous reviewer for constructive comments.


  1. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, NY (2009)MATHGoogle Scholar
  2. Davis, K.E.: Locally optimal transfer trajectories between libration point orbits using invariant manifolds. PhD Thesis, University of Colorado, Boulder, Colorado (2009)Google Scholar
  3. Davis, K.E., Anderson, R.L., Scheeres, D.J., Born, G.H.: The use of invariant manifolds for transfers between unstable periodic orbits of different energies. Celest. Mech. Dyn. Astron. 107, 471–485 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. Davis, K.E., Anderson, R.L., Scheeres, D.J., Born, G.H.: Optimal transfers between unstable periodic orbits using invariant manifolds. Celest. Mech. Dyn. Astron. 109, 241–264 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. Davis, K., Born, G., Deilami, M., Larsen, A., Butcher, E.: Transfers to Earth–Moon L3 halo orbits. In: AIAA/AAS Astrodynamics Specialist Conference, Minneapolis, Minnesota, August (2012)Google Scholar
  6. Davis, K., Born, G., Butcher, E.: Transfers to Earth–Moon \(\text{ L }_3\) halo orbits. Acta Astronaut. 88, 116–118 (2013)ADSCrossRefGoogle Scholar
  7. Davis, K., Parker, J., Butcher, E.: Transfers from Earth to EarthMoon \(\text{ L }_3\) halo orbits using accelerated manifolds. Adv. Space Res. 55, 868–1877 (2015)CrossRefGoogle Scholar
  8. Gómez, G., Jorba, A., Masdemont, J., Simó, C.: Study of the transfer from the Earth to a halo orbit around the equilibrium point \(\text{ L }_1\). Celest. Mech. Dyn. Astron. 56, 541–562 (1993)ADSCrossRefMATHGoogle Scholar
  9. Gómez, G., Jorba, A., Masdemont, J., Simó, C.: Study of the transfer between halo orbits. Acta Astronaut. 43, 493–520 (1998)ADSCrossRefGoogle Scholar
  10. Gómez, G., Masdemont, J.: Some zero cost transfers between libration point orbits. AAS/AIAA Spaceflight Mechanics Meeting, Clearwater, Florida, January (2000)Google Scholar
  11. Gómez, G., Koon, W.S., Marsden, J.E., Masdemont, J., Ross, S.D.: Connecting orbits and invariant manifolds in the spatial restricted three-body problem. Nonlinearity 17, 1571–1606 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. Grebow, D.: Generating periodic orbits in the circular restricted three-body problem with applications to lunar south pole coverage. M.S. Thesis, Purdue University, West Lafayette, Indiana (2006)Google Scholar
  13. Grebow, D.: Trajectory design in the Earth–Moon system and lunar south pole coverage. PhD Thesis, Purdue University, West Lafayette, Indiana (2010)Google Scholar
  14. Hamera, K., Mosher, T., Gefreh, M., Paul, R., Slavkin, L., Trojan, J.: An evolvable lunar communication and navigation constellation concept. In: IEEE Aerospace Conference, Big Sky, Montana, May (2008)Google Scholar
  15. Hiday-Johnston, L.A., Howell, K.C.: Transfers between libration point orbits in the elliptic restricted problem. Celest. Mech. Dyn. Astron. 58, 317–337 (1994)ADSCrossRefGoogle Scholar
  16. Hill, K., Parker, J.S., Born, G.H., Demandante, N.: A lunar \(\text{ L }_2\) navigation, communication, and gravity mission. In: AIAA/AAS Astrodynamics Specialist Conference, Keystone, Colorado, August (2006)Google Scholar
  17. Howell, K.C.: Three dimensional, periodic, ’halo’, orbits. Celest. Mech. 32, 53–71 (1984)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D.: Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. Chaos 10, 427–469 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D.: Dynamical Systems, the Three-Body Problem and Space Mission Design. Springer-Verlag, New York (2007)MATHGoogle Scholar
  20. Mingotti, G., Topputo, F., Bernelli-Zazzera, F.: Low-energy, low-thrust transfers to the Moon. Celest. Mech. Dyn. Astron. 105, 61–74 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  21. Mitzenmacher, M., Upfal, E.: Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, New York (2005)CrossRefMATHGoogle Scholar
  22. Parker, J.S., Born, G.H.: Modeling a low-energy ballistic lunar transfer using dynamical systems theory. J. Spacecr. Rockets 45, 1269–1281 (2008)ADSCrossRefGoogle Scholar
  23. Shampine, L.F., Gordon, M.K.: Computer Solution of Ordinary Differential Equations: The Initial Value Problem. W.H. Freeman, San Francisco (1975)MATHGoogle Scholar
  24. Strange, N., Landau, D., McElrath, T., Lantoine, G., Lam, T., McGuire, M., Burke, L., Martini, M. and Dankanich, J.: Overview of mission design for NASA asteroid redirect robotic mission concept. In: Proceedings of the 33rd International Electric Propulsion Conference, Washington, DC, October (2013)Google Scholar
  25. Surovik, D.A., Scheeres, D.J.: Adaptive reachability analysis to achieve mission objectives in strongly non-Keplerian systems. J. Guid. Control Dyn. 38, 468–477 (2015)ADSCrossRefGoogle Scholar
  26. Surovik, D.A., Scheeres, D.J.: Abstraction predictive control for chaotic spacecraft orbit design. In: 5th IFAC Conference on Nonlinear Model Predictive Control, Seville, Spain, September (2015)Google Scholar
  27. Sweetser, T.H.: Estimate of the global minimum \(DV\) needed for Earth–Moon transfer. In: AAS/AIAA Spaceflight Mechanics Meeting, Houston, Texas, February (1991)Google Scholar
  28. Szebehely, V.: Theory of Orbits. Academic Press, New York (1967)Google Scholar
  29. Trumbauer, E., Villac, B.: Heuristic search-based framework for onboard trajectory redesign. J. Guid. Control Dyn. 37, 164–175 (2014)ADSCrossRefGoogle Scholar
  30. Tsirogiannis, G.A.: A graph based methodology for mission design. Celest. Mech. Dyn. Astron. 114, 353–363 (2012)ADSMathSciNetCrossRefGoogle Scholar
  31. Tsirogiannis, G.A., Markellos, V.V.: A greedy global search algorithm for connecting unstable periodic orbits with low energy cost. Celest. Mech. Dyn. Astron. 117, 201–213 (2013)ADSMathSciNetCrossRefGoogle Scholar
  32. Welch, C.M., Parker, J.S., Buxton, C.: Mission considerations for transfers to a distant retrograde orbit. J. Astronaut. Sci. 62, 101–124 (2015)ADSCrossRefGoogle Scholar
  33. Wilson, R.: Derivation of differential correctors used in GENESIS mission design. Technical Report JPL IOM 312.I–03–002, Jet Propulsion Laboratory, Pasadena, California (2003)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.ASML Netherlands B.V.VeldhovenThe Netherlands
  2. 2.The Colorado Center for Astrodynamics ResearchUniversity of ColoradoBoulderUSA

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