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Flight in Non-spherical Gravity Fields

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Optimal Space Flight Navigation

Part of the book series: Control Engineering ((CONTRENGIN))

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Abstract

Spaceflight involving orbital transfers around irregularly shaped bodies or in the gravity field of several large bodies is fundamentally different from the flight in the gravity field of a single spherical body, which was covered in the previous chapters. The primary reason for this difference is that the spacecraft is no longer in a time-invariant gravity field of the two-body problem, but instead encounters a time-dependent field due to the relative motion of the multiple large bodies with respect to one another, or due to the changing position of the spacecraft relative to a rotating, non-spherical body.

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Notes

  1. 1.

    A different state-space representation is obtained by replacing the generalized momentum vector, p, with the velocity, \(\dot {\mathbf {q}}\):

    $$\displaystyle \begin{aligned} \mathbf{f}(\mathbf{X})=\left\{\begin{array}{c}\dot{\mathbf{q}}\\\ddot{\mathbf{q}} \end{array}\right\}=\left\{\begin{array}{c}\dot{\mathbf{q}}\\{\mathbf{K}}^T\mathbf{K}\mathbf{q}-\nabla_q(U)+2\mathbf{K}\dot{\mathbf{q}} \end{array}\right\} \end{aligned}$$

    whose Jacobian is given by

    $$\displaystyle \begin{aligned} {\mathbf{f}}^{\prime}(\mathbf{X})=\left(\begin{array}{cccc}\mathbf{0} &&& \mathbf{I}\\\varOmega_{\mathbf{qq}} &&& 2\mathbf{K} \end{array}\right) \end{aligned}$$

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1974)

    MATH  Google Scholar 

  2. Anderson, R.L., Lo, M.W.: Role of invariant manifolds in low-thrust trajectory design. J. Guid. Control. Dyn. 32, 1921–1930 (2009)

    Article  Google Scholar 

  3. Bai, X., Junkins, J.L.: Modified Chebyshev-Picard iteration methods for station-keeping of translunar halo orbits. Math. Probl. Eng. 2012, 1–18 (2012)

    MathSciNet  MATH  Google Scholar 

  4. Bando, M., Scheeres, D.J.: Attractive sets to unstable orbits using optimal feedback control. J. Guid. Control. Dyn. 39, 2725–2739 (2016)

    Article  Google Scholar 

  5. Battin, R.H.: An Introduction to the Mathematics and Methods of Astrodynamics. AIAA Education Series, Reston (1999)

    Google Scholar 

  6. Breakwell, J.V., Kamel, A.A., Ratner, M.J.: Station-keeping for a translunar communication station. Celest. Mech. 10, 357–373 (1974)

    Article  MathSciNet  Google Scholar 

  7. Byrnes, D.V.: Application of the pseudostate theory to the three-body lambert problem. J. Astronaut. Sci. 37, 221–232 (1989)

    MathSciNet  Google Scholar 

  8. Celletti, A., Pucacco, G., Stella, D.: Lissajous and halo orbits in the restricted three-body problem. J. Nonlinear Sci. 25, 343–370 (2015)

    Article  MathSciNet  Google Scholar 

  9. Conway, B.A. (ed.): Spacecraft Trajectory Optimization. Cambridge University Press, New York (2010)

    Google Scholar 

  10. Dunham, D.W., Davis, S.A.: Optimization of a multiple Lunar-Swingby trajectory sequence. J. Astronaut. Sci. 33, 275–288 (1985)

    Google Scholar 

  11. Farquhar, R.W.: The control and use of libration-point satellites. Goddard space flight center, Tech. Rep. NASA TR R-346 (September 1970)

    Google Scholar 

  12. Farquhar, R.W., Kamel, A.A.: Quasi-periodic orbits about the translunar libration point. Celest. Mech. 7, 458–473 (1973)

    Article  Google Scholar 

  13. Ghorbani, M., Assadian, N.: Optimal station-keeping near Earth-Moon collinear libration points using continuous and impulsive maneuvers. Adv. Space Res. 52, 2067–2079 (2013)

    Article  Google Scholar 

  14. Gomez, G., Howell, K.C., Simo, C., Masdemont, J.: Station-keeping strategies for translunar libration point orbits. In: Proceedings of AAS/AIAA Spaceflight Mechanics Meeting, AAS Paper 98-168, Monterey (1998)

    Google Scholar 

  15. Gomez, G., Masdemont, J., Simo, C.: Quasihalo orbits associated with libration points. J. Astronaut. Sci. 46, 135–176 (1998)

    MathSciNet  Google Scholar 

  16. Gomez, G., Koon, W.S., Lo, M.W., Marsden, J.E., Masdemont, J., Ross, S.D.: Connecting orbits and invariant manifolds in the spatial restricted three-body problem. Nonlinearity 17, 1571–1606 (2004)

    Article  MathSciNet  Google Scholar 

  17. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)

    Book  Google Scholar 

  18. Howell, K.: Three-dimensional, periodic, ‘Halo’ orbits. Celest. Mech. 32, 53–71 (1984)

    Article  MathSciNet  Google Scholar 

  19. Howell, K., Barden, B., Lo, M.: Application of dynamical systems theory to trajectory design for a libration point mission. J. Astronaut. Sci. 45, 161–178 (1997)

    MathSciNet  Google Scholar 

  20. Jones, B.L., Bishop, R.L.: H 2 optimal halo orbit guidance. J. Guid. Control. Dyn. 16, 1118–1124 (1993)

    Article  Google Scholar 

  21. Kulkarni, J.E., Campbell, M.E., Dullerud, G.E.: Stabilization of spacecraft flight in halo orbits: an H ∞ approach. IEEE Trans. Control Syst. Technol. 14, 572–578 (2006)

    Article  Google Scholar 

  22. Li, C., Liu, G., Huang, J., Gao, T., Guo, Y.: Stationkeeping control for libration point orbits using NMPC. In: Proceedings of AAS/AIAA Astrodynamics Specialist Conference, AAS Paper 15-692, Vail (2015)

    Google Scholar 

  23. Lian, Y., Gomez, G., Masdemont, J.J., Tang, G.: Station keeping of real Earth-Moon libration point orbits using discrete-time sliding mode approach. Commun. Nonlinear Sci. Numer. Simul. 19, 3792–3807 (2014)

    Article  MathSciNet  Google Scholar 

  24. Macdonald, M., McInnes, C.R.: Analytical control laws for planet-centred solar sailing. J. Guid. Control. Dyn. 28, 1038–1048 (2005)

    Article  Google Scholar 

  25. Marinca, V., Herisanu, N.: Nonlinear Dynamical Systems in Engineering. Springer, Berlin (2012)

    MATH  Google Scholar 

  26. Meyer, K.R., Hall, G.R.: Introduction to Hamiltonian Dynamical Systems and the N-body Problem. Springer, New York (1992)

    Book  Google Scholar 

  27. Moritz, H.: Advanced Physical Geodesy. Abacus Press, New York (1980)

    Google Scholar 

  28. Nazari, M., Anthony, W., Butcher, E.A.: Continuous thrust stationkeeping in Earth-Moon L 1 halo orbits based on LQR control and Floquet theory. In: Proceedings of AAS/AIAA Astrodynamics Specialist Conference, AIAA Paper 2014-4140, San Diego, CA (2014)

    Google Scholar 

  29. Otten, M., McInnes, C.R.: Near minimum-time trajectories for solar sails. J. Guid. Control. Dyn. 24, 632–634 (2001)

    Article  Google Scholar 

  30. Pavlak, T.A., Howell, K.C.: Strategy for optimal, long-term stationkeeping of libration point orbits in the Earth-Moon system. In: Proceedings of AAS/AIAA Astrodynamics Specialist Conference, AIAA Paper 2012-4665, Minneapolis (2012)

    Google Scholar 

  31. Richardson, D.L.: Analytic construction of periodic orbits about the collinear points. Celest. Mech. 22, 241–253 (1980)

    Article  MathSciNet  Google Scholar 

  32. Richardson, D.L.: Halo orbit formulation for the ISEE-3 mission. J. Guid. Control. Dyn. 3, 543–548 (1980)

    Article  Google Scholar 

  33. Shirobokov, M., Trofimov, S., Ovchinnikov, M.: Survey of station-keeping techniques for libration point orbits. J. Guid. Control. Dyn. 40, 1085–1105 (2017)

    Article  Google Scholar 

  34. Slotine, J.E., Li, W.: Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs (1991)

    MATH  Google Scholar 

  35. Szebehely, V.: Theory of Orbits: The Restricted Problem of Three Bodies. Academic, New York (1967)

    MATH  Google Scholar 

  36. Tewari, A.: Modern Control Design with MATLAB and Simulink. Wiley, Chichester (2002)

    Google Scholar 

  37. Tewari, A.: Atmospheric and Space Flight Dynamics–Modeling and Simulation with MATLAB and Simulink. Birkhäuser, Boston (2006)

    MATH  Google Scholar 

  38. Thurman, R., Worfolk, P.A.: Geometry of halo orbits in the circular restricted three-body problem. Technical Report GCG95, University of Minnesota, Minneapolis (1996)

    Google Scholar 

  39. Werner, R.A., Scheeres, D.J.: Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 CASTALIA. Celest. Mech. Dyn. Astron. 65, 313–344 (1996)

    MATH  Google Scholar 

  40. Wertz, J.R. (ed.): Spacecraft Attitude Determination and Control. Kluwer Academic Publishers, Dordrecht (1978)

    Google Scholar 

  41. Yang, H., Bai, X., Baoyin, H.: Rapid generation of time-optimal trajectories for asteroid landing via convex optimization. J. Guid. Control. Dyn. 40, 628–641 (2017)

    Article  Google Scholar 

  42. Zhang, C., Zhao, Y.: Low-thrust minimum-fuel optimization in the circular restricted three-body problem. J. Guid. Control. Dyn. 38, 1501–1509 (2015)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Aerospace Engineering, Indian Institute of Technology, Kanpur, IIT-Kanpur, India

    Ashish Tewari

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Tewari, A. (2019). Flight in Non-spherical Gravity Fields. In: Optimal Space Flight Navigation. Control Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-03789-5_6

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  • DOI: https://doi.org/10.1007/978-3-030-03789-5_6

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  • Publisher Name: Birkhäuser, Cham

  • Print ISBN: 978-3-030-03788-8

  • Online ISBN: 978-3-030-03789-5

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