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A greedy global search algorithm for connecting unstable periodic orbits with low energy cost.

Application to the Earth–Moon system

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Abstract

A method for space mission trajectory design is presented in the form of a greedy global search algorithm. It uses invariant manifolds of unstable periodic orbits and its main advantage is that it performs a global search for the suitable legs of the invariant manifolds to be connected for a preliminary transfer design, as well as the appropriate points of the legs for maneuver application. The designed indirect algorithm bases the greedy choice on the optimality conditions that are assumed for the theoretical minimum transfer cost of a spacecraft when using invariant manifolds. The method is applied to a test case space mission design project in the Earth–Moon system and is found to compare favorably with previous techniques applied to the same project.

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Notes

  1. Unlike the methodology presented in Tsirogiannis (2012) the greedy global search algorithm has no transformation to a combinatorial optimization problem or combinatorial optimization part. The only common point is the use of the kd-tree, which in the present paper is not the core part since the Haapala and Howell (2012) methodology can be used, instead.

References

  • Bentley, J.L.: Multidimensional binary search trees used for associative searching. Commun. ACM 18, 509–517 (1975)

    Article  MATH  Google Scholar 

  • Boutonnet, A., Martens, W., Schoenmaekers, J.: SOURCE: a Matlab-orientated tool for interplanetary trajectory global optimization. Fundamentals, AAS/AIAA Space Flight Mechanics Meeting, Kauai, Hawaii (2013)

  • Conway, B.A.: Spacecraft Trajectory Optimization. Cambridge University Press, Cambridge, UK (2010)

    Book  Google Scholar 

  • Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge (2009)

    MATH  Google Scholar 

  • D’Amario, L.A.: Minimum impulse three-body trajectories. PhD Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts (1973)

  • Davis, K.E.: Locally optimal transfer trajectories between libration point orbits using invariant manifolds. PhD Thesis, University of Colorado, Boulder, Colorado (2009)

  • 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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Davis, K., Born, G., Deilami, M., Larsen, A., Butcher, E.: Transfers to Earth–Moon L3 Halo orbits. AIAA/AAS Astrodynamics Specialist Conference, Minneapolis, Minnesota (2012)

  • Davis, K.E.: Private communication (2012)

  • 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, 357–370 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Dunham, D.W., Farquhar, R.W.: Libration point missions, 1978–2002. In: Gòmez, G., Lo, M.W., Masdemont, J.J. (eds.) Libration Point Orbits and Applications: Proceedings of the Conference. World Scientific Publishing Company, Aiguablava, Spain (2003)

  • García Yárnoz, D., Sanchez, J. P., McInnes, C.R.: Easily retrievable objects among the NEO population. Celest. Mech. Dyn. Astron. 116, 367–388 (2013)

  • Gómez, G., Masdemont, J.: Some zero cost transfers between libration point orbits. In: AAS/AIAA Spaceflight Mechanics Meeting, Clearwater, Florida (2000)

  • Gómez, G., Jorba, A., Masdemont, J., Simó, C.: Study of the transfer between halo orbits. Acta Astronaut. 43, 493–520 (1998)

    Article  ADS  Google Scholar 

  • 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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Grebow, D.: Trajectory design in the Earth–Moon system and lunar south pole coverage. PhD Thesis, Purdue University, West Lafayette, Indiana (2010)

  • Haapala, A.F., Howell, K.C.: Representations of higher-dimensional Poincaré maps with application to spacecraft trajectory design. In: IAF 63rd International Astronautical Congress, Naples, Italy (2012)

  • Hamera, K., Mosher, T., Gefreh, M., Paul, R., Slavkin, L., Trojan, J.: An evolvable lunar communication and navigation constellation concept. In: IEEE Aerospace Conference. Number IEEE 1491, Big Sky, Montana (2008)

  • 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)

    Article  ADS  Google Scholar 

  • Hill, K., Parker, J.S., Born, G.H., Demandante, N.: A lunar \(L_2\) navigation, communication, and gravity mission. In: AIAA/AAS Astrodynamics Specialist Conference. Number AIAA 2006–6662, Keystone, Colorado (2006)

  • Howell, K.C.: Three dimensional, periodic, ‘halo’, orbits. Celest. Mech. 32, 53–71 (1984)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Howell, K.C., Hiday-Johnston, L.A.: Time-free transfers between libration point orbits in the elliptic restricted problem. Acta Astronaut. 32, 245–254 (1994)

    Article  ADS  Google Scholar 

  • Knuth, D.E.: The Art of Computer Programming, vol. 1: Fundamental Algorithms, 3rd edn. Addison-Wesley, Massachusetts (1997)

    MATH  Google Scholar 

  • 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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Lo, M.W., Williams, B.G., Bollman, W.E., Han, D.S., Hahn, Y.S., Bell, J.L., Hirst, E.A., Corwin, R.A., Hong, P.E., Howell, K.C., Barden, B., Wilson, R.: GENESIS mission design. J Astronaut. Sci. 49, 169–184 (2001)

    Google Scholar 

  • Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics. Wiley, New York (1995)

    Book  MATH  Google Scholar 

  • Samet, H.: Foundations of Multidimensional and Metric Data Structures. Morgan Kaufmann, Amsterdam (2006)

    MATH  Google Scholar 

  • Sweetser, T.H.: Estimate of the global minimum \(DV\) needed for Earth–Moon transfer. In: AAS/AIAA Spaceflight Mechanics Meeting, Houston, Texas (1991)

  • Szebehely, V.: Theory of Orbits. Academic Press, New York (1967)

    Google Scholar 

  • Trumbauer, E., Villac, B.: Search and representation strategies for automated trajectory design. In: AIAA/AAS Astrodynamics Specialist Conference, Minneapolis, Minnesota (2012)

  • Tsirogiannis, G.A.: A graph based methodology for mission design. Celest. Mech. Dyn. Astron. 114, 353–363 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  • Wall, B., Conway, B.A.: Near-optimal low-thrust Earth–Mars trajectories via a genetic algorithm. J Guid. Control Dyn. 28, 1027–1031 (2005)

    Article  ADS  Google Scholar 

  • 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)

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Acknowledgments

Thanks are due to two anonymous referees for constructive comments.

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Authors and Affiliations

  1. Department of Engineering Sciences, University of Patras, Patras, Greece

    G. A. Tsirogiannis & V. V. Markellos

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  1. G. A. Tsirogiannis
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  2. V. V. Markellos
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Correspondence to G. A. Tsirogiannis.

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Tsirogiannis, G.A., Markellos, V.V. A greedy global search algorithm for connecting unstable periodic orbits with low energy cost.. Celest Mech Dyn Astr 117, 201–213 (2013). https://doi.org/10.1007/s10569-013-9508-5

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  • DOI: https://doi.org/10.1007/s10569-013-9508-5

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