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Equilibrium configurations of the tethered three-body formation system and their nonlinear dynamics

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Abstract

This paper considers nonlinear dynamics of tethered three-body formation system with their centre of mass staying on a circular orbit around the Earth, and applies the theory of space manifold dynamics to deal with the nonlinear dynamical behaviors of the equilibrium configurations of the system. Compared with the classical circular restricted three body system, sixteen equilibrium configurations are obtained globally from the geometry of pseudo-potential energy surface, four of which were omitted in the previous research. The periodic Lyapunov orbits and their invariant manifolds near the hyperbolic equilibria are presented, and an iteration procedure for identifying Lyapunov orbit is proposed based on the differential correction algorithm. The non-transversal intersections between invariant manifolds are addressed to generate homoclinic and heteroclinic trajectories between the Lyapunov orbits. (3,3)-and (2,1)-heteroclinic trajectories from the neighborhood of one collinear equilibrium to that of another one, and (3,6)- and (2,1)-homoclinic trajectories from and to the neighborhood of the same equilibrium, are obtained based on the Poincaré mapping technique.

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References

  1. Montenbruck, O., Kirschner, M., D’Amico, S., et al.: E/IVector separation for safe switching of the GRACE formation. Aerospace Science and Technology 10, 628–635 (2006)

    Article  MATH  Google Scholar 

  2. D’Amico, S., Montenbruck, O.: Proximity operations of formation flying spacecraft using an eccentricity/inclination vector separation. Journal of Guidance, Control and Dynamics 29, 554–563 (2006)

    Article  Google Scholar 

  3. Penzo, P.A., Ammann, P.W.: Tethers in Space Handbook. (2nd edn.) NASA Office of Space Flight, Washington, D. C., May (1989)

    Google Scholar 

  4. Johnson, L., Estes, R.D., Lorenzini, E.C., et al.: Propulsive small expendable deployer system experiment. Journal of Spacecraft and Rockets 37, 173–176 (2000)

    Article  Google Scholar 

  5. Mazzoleni, A.P., Hoffman, J.H.: Nonplanar spin-up dynamics of the ASTOR tethered satellite system. In: Proc. of AAS/AIAA Space Flight Mechanics Meeting, Santa Barbara, USA, Feb. 11–14, AAS 01-193 (2001)

  6. Lorenzini, E.C.: A three-mass tethered system for microg/ variable-g applications. Journal of Guidance, Control and Dynamics 10, 242–249 (1987)

    Article  Google Scholar 

  7. Misra, A.K., Modi, V.J.: Three-dimensional dynamics and control of tether-connected n-body systems. Acta Astronautica 26, 77–84 (1992)

    Article  Google Scholar 

  8. Misra, A.K., Amier, Z., Modi, V.J.: Attitude dynamics of three-body tethered systems. Acta Astronautica 17, 1059–1068 (1988)

    Article  MATH  Google Scholar 

  9. Keshmiri, M., Misra, A.K., Modi, V.J.: General formulation for n-body tethered satellite system dynamics. Journal of Guidance, Control and Dynamics 19, 75–83 (1996)

    Article  MATH  Google Scholar 

  10. Kalantzis, S., Misra, A.K., Modi, V.J., et al.: Order-n formulation and dynamics of multibody tethered systems. Journal of Guidance, Control and Dynamics 21, 277–285 (1998)

    Article  Google Scholar 

  11. Misra, A.K., Nixon, M.S., Modi, V.J.: Nonlinear dynamics of two body tethered satellite systems: Constant length case. The Journal of the Astronautical Sciences 49, 219–236 (2001)

    MathSciNet  Google Scholar 

  12. Misra, A.K.: Equilibrium configurations of tethered three-body systems and their stability. The Journal of the Astronautical Sciences 50, 241–253 (2002)

    Google Scholar 

  13. Gómez, G., Llibre, J., Martínez, R., et al.: Dynamics and Mission Design Near Libration Points. 1. Fundamentals: The Case of Collinear Libration Points. World Scientific, Singapore (2001)

    Book  Google Scholar 

  14. Meyer, K.R., Hall, R.: Hamiltonian Mechanics and the n-Body Problem. Springer-Verlag, Applied Mathematical Sciences, USA (1992)

    Google Scholar 

  15. Koon, W.S., Lo, M.W., Marsden, J.E., et al.: Dynamical Systems, the Three-Body Problem and Space Mission Design. Springer, New York (2007)

    Google Scholar 

  16. Koon, W.S., Lo, M.W., Marsden, J.E.: The genesis trajectory and heteroclinic connections. In: Proc. of AAS/AIAA Astrodynamics Specialist Conference, Girdwood, Alaska, AAS Paper 99–451 (1999)

  17. Koon, W.S., Lo, M.W., Marsden, J.E., et al.: Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. Chaos 10, 427–469 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Xu, M., Xu, S.: Study on stationkeeping for halo orbits at EL1: Dynamics modeling and controller designing. Transactions of the Japan Society for Aeronautical and Space Sciences 55, 274–285 (2012)

    Article  Google Scholar 

  19. Xu, M., Xu, S.: J2 Invariant relative orbits via differential correction algorithm. Acta Mech. Sin. 23, 585–595 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Xu, M., Wang, Y., Xu, S.: On the existence of J2 invariant relative orbit from the dynamical system point of view. Celestial Mechanics and Dynamical Astronomy 112, 427–444 (2012)

    Article  MathSciNet  Google Scholar 

  21. Xu, M., Xu, S.J.: Nonlinear dynamical analysis for displaced orbits above the planet. Celestial Mechanics and Dynamical Astronomy 102, 327–353 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Barden, B.T., Howell, K.C.: Application of dynamical systems theory to trajectory design for a libration point mission. Journal of the Astronautical Sciences 45, 161–178 (1997)

    MathSciNet  Google Scholar 

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Correspondence to Ming Xu.

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The project was supported by the National Natural Science Foundation of China (11172020), Talent Foundation supported by the Fundamental Research Funds for the Central Universities, Aerospace Science and Technology Innovation Foundation of China Aerospace Science Corporation, and the National High Technology Research and Development Program of China (863) (2012AA120601).

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Xu, M., Zhu, JM., Tan, T. et al. Equilibrium configurations of the tethered three-body formation system and their nonlinear dynamics. Acta Mech Sin 28, 1668–1677 (2012). https://doi.org/10.1007/s10409-012-0160-1

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  • DOI: https://doi.org/10.1007/s10409-012-0160-1

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