Advertisement

Systematic Design of Optimal Low-Thrust Transfers for the Three-Body Problem

  • Shankar KulumaniEmail author
  • Taeyoung Lee
Article
  • 17 Downloads

Abstract

We develop a computational approach for the design of continuous low thrust transfers in the planar circular restricted three-body problem. The use of low thrust propulsion allows the spacecraft to depart from the natural dynamics and enables a wider range of transfers. We generate the reachable set of the spacecraft and use this to determine transfer opportunities, analogous to the intersection of control-free invariant manifolds. The reachable set is developed on a lower dimensional Poincaré section and used to design transfer trajectories. This is solved numerically as a discrete optimal control problem using a variational integrator, which preserves the geometric structure of the motion in the three-body problem. We demonstrate our approach with two numerical simulations of transfers in the Earth-Moon three-body system.

Keywords

Three body problem Reachability Optimal control 

Notes

Acknowledgments

This research has been supported in part by NSF under the grants CMMI-1243000 (transferred from 1029551), CMMI-1335008, and CNS-1337722. The authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest, or non-financial interest in the subject matter or materials discussed in this manuscript.

References

  1. 1.
    Bryson, A.E., Ho, Y.C.: Applied Optimal Control: Optimization, Estimation and Control. CRC Press (1975)Google Scholar
  2. 2.
    Chang, D.E., Chichka, D.F., Marsden, J.E.: Lyapunov-based transfer between elliptic Keplerian orbits. Discret. Cont. Dyn. Syst. Series B 2(1), 57–68 (2002)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Choueiri, E.Y.: New dawn for electric rockets. Sci. Am. 300(2), 58–65 (2009).  https://doi.org/10.1038/scientificamerican0209-58 CrossRefGoogle Scholar
  4. 4.
    Conley, C.C.: Low energy transit orbits in the restricted three-body problem. SIAM J. Appl. Math. 16(4), 732–746 (1968). http://www.jstor.org/stable/2099124 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dellnitz, M., Junge, O., Post, M., Thiere, B.: On target for venus – set oriented computation of energy efficient low thrust trajectories. Celest. Mech. Dyn. Astron. 95(1-4), 357–370 (2006).  https://doi.org/10.1007/s10569-006-9008-y MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Folta, D., Dichmann, D., Clark, P., Haapala, A., Howell, K.: Lunar cube transfer trajectory options. In: Proceedings of the AAS/AIAA Spaceflight Mechanics Meeting, p. 353. Williamsburg (2015)Google Scholar
  7. 7.
    Gómez, G., Koon, W., Lo, M.W., Marsden, J., Masdemont, J., Ross, S.: Invariant manifolds, the spatial three-body problem and space mission design. In: AAS/AIAA Astrodynamics Specialist Conference. American Astronautical Society, Quebec City (2001)Google Scholar
  8. 8.
    Grebow, D.J., Ozimek, M.T., Howell, K.C.: Design of optimal low-thrust lunar pole-sitter missions. J. Astronaut. Sci. 58(1), 55–79 (2011).  https://doi.org/10.1007/BF03321159 CrossRefGoogle Scholar
  9. 9.
    Haque, S.E., Keidar, M., Lee, T.: Low-thrust orbital maneuver analysis for cubesat spacecraft with a micro-cathode arc thruster subsystem. In: Proceedings of the Thirty-Third International Electric Propulsion Conference. Electric Rocket Propulsion Society, Washington DC (2013)Google Scholar
  10. 10.
    Holzinger, M., Scheeres, D.: Reachability analysis applied to space situational awareness. In: Advanced Maui Optical and Space Surveillance Technologies Conference (2009)Google Scholar
  11. 11.
    Kirk, D.E.: Optimal Control Theory: An Introduction. Courier Corporation (2012)Google Scholar
  12. 12.
    Komendera, E.E., Scheeres, D.J., Bradley, E.: Intelligent computation of reachability sets for space missions. In: IAAI (2012)Google Scholar
  13. 13.
    Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D.: Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. Chaos: Interdiscip. J. Nonlinear Sci. 10(2), 427–469 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D.: Dynamical systems, the three-body problem and space mission design. Marsden Books. http://www2.esm.vt.edu/~sdross/books/ (2011)
  15. 15.
    Kulumani, S., Lee, T.: Low-thrust trajectory design using reachability sets near asteroid 4769 castalia. In: Proceedings of the AIAA / AAS Astrodynamics Specialists Conference. Long Beach. http://arc.aiaa.org/doi/abs/10.2514/6.2016-5376 (2016)
  16. 16.
    Lanczos, C.: The Variational Principles of Mechanics, vol. 4. Courier Corporation (1970)Google Scholar
  17. 17.
    Llibre, J., Martínez, R., Simó, C.: Tranversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near l 2 in the restricted three-body problem. J. Diff. Equ. 58 (1), 104–156 (1985).  https://doi.org/10.1016/0022-0396(85)90024-5. http://www.sciencedirect.com/science/article/pii/0022039685900245 CrossRefzbMATHGoogle Scholar
  18. 18.
    Lo, M.W.: Libration point trajectory. Design 14(1–3), 153–164 (1997).  https://doi.org/10.1023/A:1019108929089 MathSciNetzbMATHGoogle Scholar
  19. 19.
    Lygeros, J.: On the relation of reachability to minimum cost optimal control. In: Proceedings of the 41st IEEE Conference on Decision and Control, 2002, vol. 2, pp. 1910–1915.  https://doi.org/10.1109/CDC.2002.1184805 (2002)
  20. 20.
    Lygeros, J.: On reachability and minimum cost optimal control. Automatica 40(6), 917–927 (2004).  https://doi.org/10.1016/j.automatica.2004.01.012. http://www.sciencedirect.com/science/article/pii/S0005109804000263 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numerica 2001(10), 357–514 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Mingotti, G., Topputo, F., Bernelli-Zazzera, F.: Earth–mars transfers with ballistic escape and low-thrust capture. Celest. Mech. Dyn. Astron. 110(2), 169–188 (2011).  https://doi.org/10.1007/s10569-011-9343-5 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ozimek, M.T., Howell, K.C.: Low-thrust transfers in the earth-moon system, including applications to libration point orbits. J. Guid. Control Dyn. 33(2), 533–549 (2010).  https://doi.org/10.2514/1.43179 CrossRefGoogle Scholar
  24. 24.
    Pellegrini, E., Russell, R.P.: On the computation and accuracy of trajectory state transition matrices. J. Guid. Control Dyn., 39.  https://doi.org/10.2514/1.G001920(2016)
  25. 25.
    Ross, S.D.: The interplanetary transport network. Am. Scient. 94(3), 230 (2006)CrossRefGoogle Scholar
  26. 26.
    Schmidt, G.R., Patterson, M.J., Benson, S.W.: The nasa evolutionary xenon thruster (next): The next step for us deep space propulsion. NASA Glenn Research Center. Document IAC-08-C4, vol. 4 (2008)Google Scholar
  27. 27.
    Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, Texts in Applied Mathematics, vol. 12. Springer, New York (2013)Google Scholar
  28. 28.
    Szebehely, V.: Theory of Orbits. The Restricted Problem of Three Bodies, p 1. Academic Press, New York (1967)zbMATHGoogle Scholar
  29. 29.
    Varaiya, P.: Reach set computation using optimal control. In: Verification of Digital and Hybrid Systems, pp. 323–331. Springer (2000)Google Scholar
  30. 30.
    West, M.: Variational integrators. Ph.D. thesis, California Institute of Technology (2004)Google Scholar

Copyright information

© American Astronautical Society 2019

Authors and Affiliations

  1. 1.Department of Mechanical & Aerospace EngineeringGeorge Washington UniversityWashingtonUSA

Personalised recommendations