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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Change history
11 February 2022
A Correction to this paper has been published: https://doi.org/10.1007/s42064-022-0136-2
References
Mereta, A., Izzo, D., Wittig, A. Machine learning of optimal low-thrust transfers between near-earth objects. Hybrid Artificial Intelligent Systems, 2017, 543–553.
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.
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.
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.
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.
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.
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.
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.
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.
Koon, W. S., Lo, M. W., Marsden, J. E., Ross, S. D. Celestial Mechanics and Dynamical Astronomy, 2001, 81(1/2): 63–73.
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.
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.
Parker, J. S., Anderson, R. L. Low-Energy Lunar Trajectory Design. Hoboken, NJ, USA: John Wiley & Sons, Inc., 2014. DOL10.1002/9781118855065.
Villac, B. F. Dynamics in the Hill problem with applications to spacecraft maneuvers. Ph.D. thesis, University of Michigan, 2003.
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.
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.
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.
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.
Haapala, A., Howell, K. C. Trajectory design using periapse Poincar’e maps and invariant manifolds. In: Proceedings of AIAA/AAS Astrodynamics Specialist Conference, 2011.
Van Der Maaten, L., Postma, E., Van den Herik, J. Dimensionality reduction: a comparative review. Journal itof Machine Learning Research, 2009, 10: 66–71.
Hornik, K., Stinchcombe, M., White, H. Multilayer feedforward networks are universal approximators. Neural Networks, 1989, 2(5): 359–366.
Villac, B. F., Scheeres, D. J. New class of optimal plane change maneuvers. Journal of Guidance, Control, and Dynamics, 2003, 26(5): 750–757.
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.
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.
Scheeres, D.J., Marzari, F. Spacecraft dynamics in the vicinity of a comet. Journal of the Astronautical Sciences, 2002, 50(1): 35–52.
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.
Arbib, M. A. The Handbook of Brain Theory and Neural Networks, 2nd edn. MIT Press, 2003.
Nielsen, M. Neural Networks and Deep Learning. Determination Press, 2015.
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.
MATLAB Statistics and Machine Learning Toolbox 11.2. The MathWorks Inc., Natick, MA, USA, 2017.
Bridle, J. S. Probabilistic interpretation of feedforward classification network outputs, with relationships to statistical pattern recognition. Neurocomputing, 1990: 227–236.
Golik, P., Doetsch, P., Ney, H. Cross-entropy vs. squared error training: a theoretical and experimental comparison. Interspeech, ISCA, 2013: 1756–1760.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42064-019-0050-4