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  • Letter
  • Open Access

Designing robust trajectories by lobe dynamics in low-dimensional Hamiltonian systems

Naoki Hiraiwa1,2, Mai Bando1, Isaia Nisoli3, and Yuzuru Sato4,5,*

  • 1Department of Aeronautics and Astronautics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 8190395, Japan
  • 2Colorado Center for Astrodynamics Research, University of Colorado Boulder, 3775 Discovery Drive, Boulder, Colorado 80303, USA
  • 3Instituto de Matemática, Universidade Federal do Rio de Janeiro, Avenida Athos da Silveira Ramos, 149, Edifício do Centro de Tecnologia, Bloco C (Térreo), Cidade Universitária 21941-909, Brazil
  • 4RIES-MSC/Department of Mathematics, Hokkaido University, N12 W7 Kita-ku, Sapporo, Hokkaido 0600812, Japan
  • 5London Mathematical Laboratory, 14 Buckingham Street, London WC2N 6DF, United Kingdom

  • *ysato@math.sci.hokudai.ac.jp
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Phys. Rev. Research 6, L022046 – Published 24 May, 2024

DOI: https://doi.org/10.1103/PhysRevResearch.6.L022046

Abstract

Modern space missions with uncrewed spacecraft require robust trajectory design to connect multiple chaotic orbits by small controls. To address this issue, we propose a control scheme to design robust trajectories by leveraging a geometrical structure in chaotic zones, known as a lobe. Our scheme shows that appropriately selected lobes reveal possible paths to traverse chaotic zones in a short time. The effectiveness of our method is demonstrated through trajectory design in both the standard map and Hill's equation.

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Corrections

7 June, 2024

Correction: The postal codes for the first and third affiliations were incorrect and have been fixed.

Article Text

Supplemental Material

References (35)

  1. S. Creech, J. Guidi, and D. Elburn, Artemis: An overview of NASA's activities to return humans to the moon, in 2022 IEEE Aerospace Conference (AERO), Big Sky, MT, USA (IEEE, New York, 2022), pp. 1–7.
  2. T. Ikenaga, K. Yamanaka, S. Ueda, and N. Ishii, Study on the low-energy ballistic lunar transfer orbit for future cargo mission to gateway, in AIAA Scitech 2020 Forum (AIAA, Reston, VA, 2020).
  3. S. Bandyopadhyay, R. Foust, G. P. Subramanian, S.-J. Chung, and F. Y. Hadaegh, Review of formation flying and constellation missions using nanosatellites, J. Spacecr. Rockets 53, 567 (2016).
  4. V. Rom-Kedar and S. Wiggins, Transport in two-dimensional maps, Arch. Ration. Mech. Anal. 109, 239 (1990).
  5. J. E. Prussing and B. A. Conway, Orbital Mechanics (Oxford University Press, Oxford, UK, 1993).
  6. W. S. Koon, M. W. Lo, J. E. Marsden, and S. D. Ross, Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics, Chaos 10, 427 (2000).
  7. W. S. Koon, M. W. Lo, and S. D. Ross, Dynamical Systems, the Three-Body Problem and Space Mission Design (Marsden Books, 2011).
  8. J. S. Parker and R. L. Anderson, Low-Energy Lunar Trajectory Design (Wiley, Hoboken, NJ, 2014).
  9. B. P. McCarthy and K. C. Howell, Four-body cislunar quasi-periodic orbits and their application to ballistic lunar transfer design, Adv. Space Res. 71, 556 (2023).
  10. S. D. Ross, W. S. Koon, M. W. Lo, and J. E. Marsden, Design of a multi-moon orbiter, Adv. Astronaut. Sci. 114, 669 (2003).
  11. E. Ott, C. Grebogi, and J. A. Yorke, Controlling chaos, Phys. Rev. Lett. 64, 1196 (1990).
  12. K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A 170, 421 (1992).
  13. M. Yamaguti, Towards the Harnessing of Chaos (Elsevier, Amsterdam, 1994).
  14. H. Jaeger and H. Haas, Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication, Science 304, 78 (2004).
  15. T. Shinbrot, E. Ott, C. Grebogi, and J. A. Yorke, Using chaos to direct trajectories to targets, Phys. Rev. Lett. 65, 3215 (1990).
  16. E. M. Bollt and J. D. Meiss, Targeting chaotic orbits to the moon through recurrence, Phys. Lett. A 204, 373 (1995).
  17. C. G. Schroer and E. Ott, Targeting in Hamiltonian systems that have mixed regular/chaotic phase spaces, Chaos 7, 512 (1997).
  18. C. Senatore and S. D. Ross, Fuel-efficient navigation in complex flows, in Proceedings of the 2008 American Control Conference (IEEE, New York, 2008), pp. 1244–1248.
  19. A. G. Ramos, V. J. García-Garrido, A. M. Mancho, S. Wiggins, J. Coca, S. Glenn, O. Schofield, J. Kohut, D. Aragon, J. Kerfoot et al., Lagrangian coherent structure assisted path planning for transoceanic autonomous underwater vehicle missions, Sci. Rep. 8, 4575 (2018).
  20. K. Krishna, Z. Song, and S. L. Brunton, Finite-horizon, energy-efficient trajectories in unsteady flows, Proc. R. Soc. London, Ser. A 478, 20210255 (2022).
  21. G. Haller, Transport Barriers and Coherent Structures in Flow Data (Cambridge University Press, Cambridge, UK, 2023).
  22. J. Duan and S. Wiggins, Lagrangian transport and chaos in the near wake of the flow around an obstacle: A numerical implementation of lobe dynamics, Nonlinear Process. Geophys. 4, 125 (1997).
  23. C. Coulliette and S. Wiggins, Intergyre transport in a wind-driven, quasigeostrophic double gyre: An application of lobe dynamics, Nonlinear Process. Geophys. 8, 69 (2001).
  24. F. Lekien, S. C. Shadden, and J. E. Marsden, Lagrangian coherent structures in n-dimensional systems, J. Math. Phys. 48, 065404 (2007).
  25. K. Oshima and T. Yanao, Jumping mechanisms of Trojan asteroids in the planar restricted three- and four-body problems, Celest. Mech. Dyn. Astron. 122, 53 (2015).
  26. S. Naik, F. Lekien, and S. D. Ross, Computational method for phase space transport with applications to lobe dynamics and rate of escape, Regular Chaotic Dyn. 22, 272 (2017).
  27. B. V. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep. 52, 263 (1979).
  28. V. Szebehely, Theories of Orbits: The Restricted Problem of Three Bodies (Academic Press, New York, 1967).
  29. V. Rom-Kedar, A. Leonard, and S. Wiggins, An analytical study of transport, mixing and chaos in an unsteady vortical flow, J. Fluid Mech. 214, 347 (1990).
  30. R. S. MacKay, J. D. Meiss, and I. C. Percival, Resonances in area-preserving maps, Physica D 27, 1 (1987).
  31. J. D. Meiss, Thirty years of turnstiles and transport, Chaos 25, 097602 (2015).
  32. R. S. MacKay, J. D. Meiss, and I. C. Percival, Transport in Hamiltonian systems, Physica D 13, 55 (1984).
  33. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.6.L022046 for detailed explanations of the optimization in the standard map and Hill's equation and for the coordinate transformation for the periapsis Poincaré map in Delaunay elements.
  34. S. Wiggins, On the geometry of transport in phase space I. Transport in k-degree-of-freedom hamiltonian systems, 2≤k<∞, Physica D 44, 471 (1990).
  35. M. Dellnitz, O. Junge, W. S. Koon, F. Lekien, M. W. Lo, J. E. Marsden, K. Padberg, R. Preis, S. D. Ross, and B. Thiere, Transport in dynamical astronomy and multibody problems, Int. J. Bifurcation Chaos 15, 699 (2005).

Outline

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Phys. Rev. Research 6, L022046– Published 24 May, 2024
  • Received 10 January 2024
  • Accepted 1 April 2024
  • Corrected 7 June 2024
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