Port-Hamiltonian Based Control of the Sun-Earth 3D Circular Restricted Three-Body Problem: Stabilization of the <i>L</i><sub>1</sub> Lagrange Point

Authors

Abstract

In this paper, we use Port-Hamiltonian framework to stabilize the Lagrange points in the Sun-Earth three-dimensional Circular Restricted Three-Body Problem (CRTBP). Through rewriting the CRTBP into Port-Hamiltonian framework, we are allowed to design the feedback controller through energy-shaping and dissipation injection. The closed-loop Hamiltonian is a candidate of the Lyapunov function to establish nonlinear stability of the designed equilibrium, which enlarges the application region of feedback controller compared with that based on linearized dynamics. Results show that the Port-Hamiltonian approach allows us to successfully stabilize the Lagrange points, where the Linear Quadratic Regulator (LQR) may fail. The feedback system based on Port-Hamiltonian approach is also robust against white noise in the inputs.

Citation

Journal: Modern Mechanical Engineering
Year: 2020
Volume: 10
Issue: 03
Pages: 39–49
Publisher: Scientific Research Publishing, Inc.
DOI: 10.4236/mme.2020.103005

BibTeX

@article{Yan_2020,
  title={{Port-Hamiltonian Based Control of the Sun-Earth 3D Circular Restricted Three-Body Problem: Stabilization of the &amp;lt;i&amp;gt;L&amp;lt;/i&amp;gt;&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; Lagrange Point}},
  volume={10},
  ISSN={2164-0181},
  DOI={10.4236/mme.2020.103005},
  number={03},
  journal={Modern Mechanical Engineering},
  publisher={Scientific Research Publishing, Inc.},
  author={Yan, Haotian},
  year={2020},
  pages={39--49}
}

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References

Musielak, Z. E. & Quarles, B. The three-body problem. Rep. Prog. Phys. 77, 065901 (2014) – 10.1088/0034-4885/77/6/065901
Pilbratt, G. L. et al. HerschelSpace Observatory. A&A 518, L1 (2010) – 10.1051/0004-6361/201014759
Gardner, J. P. et al. The James Webb Space Telescope. Space Sci Rev 123, 485–606 (2006) – 10.1007/s11214-006-8315-7
Farquhar, R. The flight of ISEE-3/ICE - Origins, mission history, and a legacy. AIAA/AAS Astrodynamics Specialist Conference and Exhibit (1998) doi:10.2514/6.1998-4464 – 10.2514/6.1998-4464
Stone, E. C. et al. Space Science Reviews 86, 1–22 (1998) – 10.1023/a:1005082526237
Sweetser, T. H. et al. ARTEMIS Mission Design. The ARTEMIS Mission 61–91 (2012) doi:10.1007/978-1-4614-9554-3_4 – 10.1007/978-1-4614-9554-3_4
Shirobokov, M., Trofimov, S. & Ovchinnikov, M. Survey of Station-Keeping Techniques for Libration Point Orbits. Journal of Guidance, Control, and Dynamics 40, 1085–1105 (2017) – 10.2514/1.g001850
Meyer, K. R. & Schmidt, D. S. The stability of the Lagrange triangular point and a theorem of Arnold. Journal of Differential Equations 62, 222–236 (1986) – 10.1016/0022-0396(86)90098-7
G�mez, G., Jorba, A., Masdemont, J. & Sim�, C. Study of the transfer from the Earth to a halo orbit around the equilibrium pointL 1. Celestial Mech Dyn Astr 56, 541–562 (1993) – 10.1007/bf00696185
Richardson, D. L. Halo Orbit Formulation for the ISEE-3 Mission. Journal of Guidance and Control 3, 543–548 (1980) – 10.2514/3.56033
Cielaszyk, D. & Wie, B. New approach to halo orbit determination and control. Journal of Guidance, Control, and Dynamics 19, 266–273 (1996) – 10.2514/3.21614
Ming, X. & Shijie, X. Trajectory and Correction Maneuver During the Transfer from Earth to Halo Orbit. Chinese Journal of Aeronautics 21, 200–206 (2008) – 10.1016/s1000-9361(08)60026-6
Yamato, H. & Spencer, D. B. Transit-Orbit Search for Planar Restricted Three-Body Problems with Perturbations. Journal of Guidance, Control, and Dynamics 27, 1035–1045 (2004) – 10.2514/1.4524
LYAPUNOV, A. M. The general problem of the stability of motion. International Journal of Control 55, 531–534 (1992) – 10.1080/00207179208934253
van der Schaft, A. & Schumacher, H. An Introduction to Hybrid Dynamical Systems. Lecture Notes in Control and Information Sciences (Springer London, 2000). doi:10.1007/bfb0109998 – 10.1007/bfb0109998
van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. FnT in Systems and Control 1, 173–378 (2014) – 10.1561/2600000002
Putting energy back in control. IEEE Control Syst. 21, 18–33 (2001) – 10.1109/37.915398
Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica 38, 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
Ortega, R., van der Schaft, A., Castanos, F. & Astolfi, A. Control by Interconnection and Standard Passivity-Based Control of Port-Hamiltonian Systems. IEEE Trans. Automat. Contr. 53, 2527–2542 (2008) – 10.1109/tac.2008.2006930
Liu, C. & Dong, L. Physics-based control education: energy, dissipation, and structure assignments. Eur. J. Phys. 40, 035006 (2019) – 10.1088/1361-6404/ab03e8
Liu, C. & Dong, L. Stabilization of Lagrange points in circular restricted three-body problem: A port-Hamiltonian approach. Physics Letters A 383, 1907–1914 (2019) – 10.1016/j.physleta.2019.03.033
LaSalle, J. Some Extensions of Liapunov’s Second Method. IRE Trans. Circuit Theory 7, 520–527 (1960) – 10.1109/tct.1960.1086720