Representing dynamics in the eccentric Hill system using a neural network architecture

Abstract

This paper demonstrates how artificial neural networks can be used to alleviate common problems encountered when creating a large database of Poincaré map responses. A general architecture is developed using a combination of regression and classification feedforward neural networks. This allows one to predict the response of the Poincaré map, as well as to identify anomalies, such as impact or escape. Furthermore, this paper demonstrates how an artificial neural network can be used to predict the error between a more complex and a simpler dynamical system. As an example application, the developed architecture is implemented on the Sun-Mars eccentric Hill system. Error statistics of the entire architecture are computed for both one Poincaré map and for iterated maps. The neural networks are then applied to study the long-term impact and escape stability of trajectories in this system.

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References

  1. [1]

    Mereta, A., Izzo, D., Wittig, A. Machine learning of optimal low-thrust transfers between near-earth objects. Hybrid Artificial Intelligent Systems, 2017, 543–553.

    Google Scholar 

  2. [2]

    Das-Stuart, A., Howel, K., Folta, D. A rapid trajectory design strategy for complex environments leveraging attainable regions and low-thrust capabilities. In: Proceedings of the 68th International Astronautical Congress-IAC17C1.7.3, Adelaide, Australia, 2017.

    Google Scholar 

  3. [3]

    Shah, V., Beeson, R. Rapid approximation of invariant manifolds using machine learning. In: Proceedings of the AIAA/AAS Astrodynamics Specialist Conference, AAS17-784, Stevenson, WA, 2017.

    Google Scholar 

  4. [4]

    Zou, A. M., Kumar, K. D. Adaptive attitude control of spacecraft without velocity measurements using Chebyshev neural network. Acta Astronautica, 2010, 66(5-6): 769–779.

    Article  Google Scholar 

  5. [5]

    Furfaro, R., Simo, J., Gaudet, B., Wibben, D. Neural-based trajectory shaping approach for terminal planetary pinpoint guidance. In: Proceedings of AAS/AIAA Astrodynamics Specialist Conference, 2013.

    Google Scholar 

  6. [6]

    Furfaro, R., Linares, R. Waypoint-based generalized ZEM/ZEV feedback guidance for planetary landing via a reinforcement learning approach. In: Proceedings of the 3rd IAA Conference on Dynamics and Control of Space Systems, Moscow, Russia, 2017.

    Google Scholar 

  7. [7]

    Sanchez, C, Izzo, D., Hennes, D. Optimal real-time landing using deep networks. In: Proceedings of the 6th International Conference on Astrodynamics Tools and Techniques, Darmstadt, Germany, 2016.

    Google Scholar 

  8. [8]

    Peng, H., Bai, X. L. Artificial neural network-based machine learning approach to improve orbit prediction accuracy. Journal of Spacecraft and Rockets, 2018. 55(5): 1248–1260.

    Article  Google Scholar 

  9. [9]

    Zhong, R., Xu, S. Neural-network-based terminal sliding-mode control for thrust regulation of a tethered space-tug. Astrodynamics, 2018. 2(2): 175–185.

    Article  Google Scholar 

  10. [10]

    Koon, W. S., Lo, M. W., Marsden, J. E., Ross, S. D. Celestial Mechanics and Dynamical Astronomy, 2001, 81(1/2): 63–73.

    MathSciNet  Article  Google Scholar 

  11. [11]

    Broschart, S. B, Chung, M. J., Hatch, S. J., Ma, J. H., Sweetser, T. H., Weinstein-Weiss, S. S., Angelopoulos, V. Preliminary trajectory design for the Artemis lunar mission. Advances in the Astronautical Sciences, 2009. 135(2): 1329–1343.

    Google Scholar 

  12. [12]

    Hatch, S., Chung, M. K., Kangas, J., Long, S., Roncoli, R., Sweetser, T. Trans-lunar cruise trajectory design of GRAIL (gravity recovery and interior laboratory) mission. In: Proceedings of the AIAA/AAS Astrodynamics Specialist Conference, Reston, Virigina: American Institute of Aeronautics and Astronautics, 2010.

    Google Scholar 

  13. [13]

    Parker, J. S., Anderson, R. L. Low-Energy Lunar Trajectory Design. Hoboken, NJ, USA: John Wiley & Sons, Inc., 2014. DOL10.1002/9781118855065.

    Google Scholar 

  14. [14]

    Villac, B. F. Dynamics in the Hill problem with applications to spacecraft maneuvers. Ph.D. thesis, University of Michigan, 2003.

    Google Scholar 

  15. [15]

    De Smet, S., Parker, J. S., Scheeres, D. J. Harnessing the Sun’s gravity for LEO to GEO transfers. In: Proceedings of the 26th International Symposium on Space Flight Dynamics, Matsuyama, Japan, 2017.

    Google Scholar 

  16. [16]

    De Smet, S., Scheeres, D. J., Parker, J. S. Dynamics and stability of Sun-driven transfers from Low Earth to Geosynchronous Orbit. Journal of Guidance, Control, and Dynamics, 2018. 41(9): 2002–2010.

    Article  Google Scholar 

  17. [17]

    Villacand, B. F., Scheeres, D. J. Escaping trajectories in the Hill three-body problem and applications. Journal of Guidance Control and Dynamics, 2003. 26(2): 224–232.

    Article  Google Scholar 

  18. [18]

    Paskowitz, M. E., Scheeres, D. J. Robust capture and transfer trajectories for planetary satellite orbiters. Journal of Guidance, Control, and Dynamics, 2008, 29(2): 342–353.

    Article  Google Scholar 

  19. [19]

    Haapala, A., Howell, K. C. Trajectory design using periapse Poincar’e maps and invariant manifolds. In: Proceedings of AIAA/AAS Astrodynamics Specialist Conference, 2011.

    Google Scholar 

  20. [20]

    Van Der Maaten, L., Postma, E., Van den Herik, J. Dimensionality reduction: a comparative review. Journal itof Machine Learning Research, 2009, 10: 66–71.

    Google Scholar 

  21. [21]

    Hornik, K., Stinchcombe, M., White, H. Multilayer feedforward networks are universal approximators. Neural Networks, 1989, 2(5): 359–366.

    Article  Google Scholar 

  22. [22]

    Villac, B. F., Scheeres, D. J. New class of optimal plane change maneuvers. Journal of Guidance, Control, and Dynamics, 2003, 26(5): 750–757.

    Article  Google Scholar 

  23. [23]

    De Smet, S., Scheeres, D. J., Parker, J. S. Dynamics and Stability of Sun-driven transfers from LEO to GEO. In: Proceedings of AIAA/AAS Astrodynamics Specialist Conference, 2017.

    Google Scholar 

  24. [24]

    Villac, B. F., Scheeres, D.J. On the concept of periapsis in Hill’s problem. Celestial Mechanics and Dynamical Astronomy, 2004, 90(1): 165–178.

    MathSciNet  Article  Google Scholar 

  25. [25]

    Scheeres, D.J., Marzari, F. Spacecraft dynamics in the vicinity of a comet. Journal of the Astronautical Sciences, 2002, 50(1): 35–52.

    Google Scholar 

  26. [26]

    De Smet, S., Scheeres, D. J., Parker, J. S. Leveraging artificial neural networks to systematically explore solar gravity driven transfers in the Martian system. The Journal of the Astronautical Sciences, 2019, DOI:10.1007/s40295-018-00149-w.

    Google Scholar 

  27. [27]

    Arbib, M. A. The Handbook of Brain Theory and Neural Networks, 2nd edn. MIT Press, 2003.

    Google Scholar 

  28. [28]

    Nielsen, M. Neural Networks and Deep Learning. Determination Press, 2015.

    Google Scholar 

  29. [29]

    Villac, B. F., Scheeres, D. J. Third-body-driven vs. one-impulse plane changes. The Journal of the Astronautical Sciences, 2009, 57(3): 545–559.

    Article  Google Scholar 

  30. [30]

    MATLAB Statistics and Machine Learning Toolbox 11.2. The MathWorks Inc., Natick, MA, USA, 2017.

    Google Scholar 

  31. [31]

    Bridle, J. S. Probabilistic interpretation of feedforward classification network outputs, with relationships to statistical pattern recognition. Neurocomputing, 1990: 227–236.

    Google Scholar 

  32. [32]

    Golik, P., Doetsch, P., Ney, H. Cross-entropy vs. squared error training: a theoretical and experimental comparison. Interspeech, ISCA, 2013: 1756–1760.

    Google Scholar 

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Acknowledgements

This research has been performed through funding provided by Advanced Space. This work utilized the RMACC Summit supercomputer, which is supported by the National Science Foundation (awards ACI-1532235 and ACI-1532236), the University of Colorado Boulder, and Colorado State University. The Summit supercomputer is a joint effort of the University of Colorado Boulder and Colorado State University.

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Correspondence to Stijn De Smet.

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Stijn De Smet is a researcher in astrodynamics. Dr. De Smet began his professional career at Delft University of Technology where he studied from 2009 to 2014, graduating with B.S. and M.S. degrees in aerospace engineering. In 2015, he started the doctoral program in aerospace engineering at the University of Colorado at Boulder, under Jeffrey S. Parker. In 2016, he joined the Celestial and Spaceflight Mechanics Laboratory, led by Daniel J. Scheeres, from which he graduated with a Ph.D. degree in 2018. His research focuses on low thrust trajectory design and on astrodynamics in challenging dynamical environments using machine learning techniques.

Daniel J. Scheeres is a University of Colorado Distinguished Professor and is the A. Richard Seebass Endowed Chair Professor in the Ann and H.J. Smead Department of Aerospace Engineering Sciences at the University of Colorado Boulder. Prior to this he held faculty positions in aerospace engineering at the University of Michigan and Iowa State University, and was a senior member of the Technical Staff in the Navigation Systems Section at the California Institute of Technologys Jet Propulsion Laboratory. He was awarded Ph.D. (1992), M.S. (1988) and B.S. (1987) degrees in aerospace engineering from the University of Michigan, and holds a B.S. in Letters and Engineering from Calvin College (1985). Scheeres is a member of the National Academy of Engineering, and a Fellow of both the American Institute of Aeronautics and Astronautics and the American Astronautical Society. He was awarded the Dirk Brouwer Award from the American Astronautical Society in 2013 and gave the John Breakwell Lecture at the 2011 International Astronautical Congress. Asteroid 8887 is named Scheeres in recognition of his contributions to the scientific understanding of the dynamical environment about asteroids.

Jefrey S. Parker is the chief technology officer at Advanced Space, a GNC service-providing company in Boulder, Colorado. He is the lead mission designer on a Mars mission and the PI on several GNC-related SBIR programs. Previously, from 2012 to 2016, he was an assistant professor of astrodynamics at the University of Colorado at Boulder, where he studied advanced concepts in mission design and navigation. From 2008 to 2012, Dr. Parker spent five years at the Jet Propulsion Laboratory, where he helped to navigate missions such as GRAIL and Chandrayaan to the Moon.

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De Smet, S., Scheeres, D.J. & Parker, J.S. Representing dynamics in the eccentric Hill system using a neural network architecture. Astrodyn 3, 301–324 (2019). https://doi.org/10.1007/s42064-019-0050-4

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Keywords

  • periapse
  • Poincaré map
  • artificial neural networks
  • eccentric Hill system