Astrodynamics

Astrodynamics and Orbital Mechanics

By using a result published for the first time in 1995, in the Ph.D. thesis, regarding the motion in central force fields with respect to arbitrary rotating reference frames, new results in Orbital Mechanics are obtained:

  • the closed form vectorial solution to the Kepler problem in rotating reference frames
    • the result is obtained by introducing a Sundman-like regularization
    • in this way, the approaches of Levi-Civita (1914) and Kustaanheimo-Stieffel (1964) are generalized to arbitrary rotating frames
  • the closed form solution to the spacecraft relative motion in gravitational force fields
    • is obtained by using a similar approach to the Kepler problem in rotating frames
    • no linearization/approximation assumptions are made
    • the Hill-Clohessy-Witshire equations (1963) and Tshauner Hempel equations (1964) are generalized
    • the result stands not only for close-proximity formation flying, but for any relative distances
    • a quaternionic version of the result was also published
    • applications: spacecraft formation flying, rendez-vous missions, ground-track accurate determination
  • the closed form solution to the relative motion in arbitrary central force fields
    • generalizes the aforementioned approach from gravitational-only fields to arbitrary central force fields
    • application: closed form solution to the relative motion in the equatorial plane of an oblate planet
  • the closed form unified solution to the Kepler problem
    • even if it might be considered a classic subject, the result unifies the approaches to the Kepler problem, by offering a comprehensive approach to all the possible situations which may arise, depending on the initial conditions
  • the closed form solution to the Foucault Pendulum problem
    • despite some claims that there does not exist a closed form solution to the Foucault pendulum problem, this solution is determined in a time-explicit form for the classical Foucault Pendulum, and then generalized to a wide class of problems, named “Foucault-Pendulum-like”. The approach is purely tensorial, therefore it is coordinate-free.
  • closed form solution to the relative orbital motion around oblate planets
    • by assuming an averaged expression for the perturbing potential due to the Earth oblateness factor J2, the relative motion is modeled with the help of a closed form soluton, which was determined also by using the tensor instrument developped in 1995.
  • hypercomplex eccentric anomaly and the relative motion
    • by making use of the hypercomplex numbers/functions, the solution to the relative motion in gravitational fields is presented in a unified way, and reduced to a 1st order linear differential equation. This reduction is not made by linearizing procedures, but by regularization procedures, therefore its solution is exactly the solution to the nonlinear model of the motion.
  • super-integrability of the relative orbital motion
    • despite its apparent difficulty, the relative motion in gravitational fields proves to be maximally-superintegrable
    • the result may be used in KAM-like perturbation theories, since there exist a closed form parameterization for the manifold associated to the unperturbed problem, and it is possible to model the unperturbed motion with well-defined tori.

References

  1. Condurache D., Reprezentari simbolice. Aplicatii in teoria semnalelor si studiul sistemelor dinamice, ISBN 973-97101-8-2, Nord-Est, Iasi, 1996.
  2. Condurache D., Noi procedee simbolice in studiul sistemelor dinamice, Universitatea Tehnica Gheorghe Asachi Iasi, 1995.
  3. Condurache, D., Martinusi, V., A Quaternionic Exact Solution to the Relative Orbital Motion Problem, AIAA Journal of Guidance, Control, and Dynamics, vo. 33, no.4, 2010, pp.1035-1047.
  4. Martinusi, V., Condurache, D., Remarks on the Hamiltonian of A Particle in A Rotating Reference Frame, Bul. Inst. Polit.Iasi, 3-4, Sect. Mathematics, Theoretical Mechanics, Physics, 2009.
  5. Condurache, D., Martinusi, V., Hypercomplex Eccentric Anomaly in the Unified Solution to the Relative Orbital Motion, Advances in the Astronautical Sciences, Vol.135, 2010, pp. 281-300.
  6. Condurache, D., Martinusi, V., Exact Solution to the Relative Orbital Motion in Eccentric Orbits, Solar System Research,Volume 43, Issue 1, 2009, pp.41-52.
  7. Martinusi, V., Gurfil, P., Solutions and Periodicity of Satellite Relative Motion Under Even Zonal Harmonics Perturbations, Celestial Mechanics and Dynamical Astronomy, Vol. 111, No. 4, 2011, pp. 387-414.
  8. Condurache, D., Martinusi, V., Foucault Pendulum-like problems: A Tensorial Approach, International Journal of Non-  linear Mechanics, vol. 43, issue 8, 2008, pp. 743-760.
  9. Condurache D., Martinusi, V., A complete Closed Form Solution to the Kepler Problem, Meccanica, Vol. 42, no.5, 2007, pp. 465-476.
  10. Condurache D., Martinusi, V., Relative Spacecraft Motion in a Central Force Field, AIAA Journal of Guidance, Control, and Dynamics, vol.30, no.3, 2007, pp. 873-876.
  11. Condurache D., Martinusi, V., Kepler’s Problem in Rotating Reference Frames. Part I: Prime Integrals. Vectorial Regularization, AIAA Journal of Guidance, Control and Dynamics, Vol. 30, no.1, 2007, pp. 192-200.
  12. Condurache D., Martinusi, V., Kepler’s Problem in Rotating Reference Frames. Part II: Relative Orbital Motion, AIAA Journal of Guidance, Control and Dynamics, Vol. 30, no.1, 2007, pp. 201-213.
  13. Condurache D., Martinusi, V., Vectorial Regularization and Temporal Means in Keplerian Motion, Journal of Nonlinear Mathematical Physics, Vol.13, No.3, 2006, pp.420-440.
  14. Condurache D., Martinusi, V., A Novel Hypercomplex Solution to Kepler’s Problem, PADEU, Astronomy Department. of the Eötvös University., vol. 19, 2007, pp. 65-80.
  15. Condurache D., Martinusi, V., A Closed Form Vectorial Solution to the Relative Orbital Motion, PADEU, Astronomy Department. of the Eötvös University., vol. 19, 2007, pp. 49-64.
  16. Condurache D., Martinusi, V., A Short Solution to the Keplerian Ballistic Problem Using the Velocity Hodograph, Bul. Inst. Polit. Iasi, LII(LVI), 1-2, Sect. Mathematics, Theoretical Mechanics, Physics, 2007.
  17. Condurache D., Martinusi, V., The Two Body Problem in Rotating Reference Frames, Bul. Inst. Polit. Iasi, LII(LVI), 3-4, Sect Mathematics, Theoretical Mechanics, Physics, 2005.
  18. Condurache D., Martinusi, V., Kepler’s Problem in Non-Inertial Reference Frames: A Vectorial Regularization, Bul. Inst. Polit. Iasi, LI (LV), l-2, Sect Mathematics, Theoretical Mechanics, Physics, 2005, pp. 45-61.
  19. Condurache D., Martinusi, V., The Solution to Kepler’s Problem in Non-Inertial Reference Frames, Bul. Inst. Polit. Iasi, LI(LV), l-2, Sect Mathematics, Theoretical Mechanics, Physics, 2005, pp. 43-55.
  20. Condurache D., Martinusi, V., About Some Temporal Means in Keplerian Motion, Bul. Inst. Polit. Iasi, L(LIV), 3-4, Sect Mathematics, Theoretical Mechanics, Physics, 2004, pp. 79-92.
  21. Condurache, D., Martinusi, V., Super-integrability in the unperturbed relative orbital motion problem, AIAA/AAS Astrodynamics Specialist Conference, Toronto, Canada, 2-5 August 2010.
  22. Condurache, D., Martinusi, V., A General Method to Study the Motion in A Non-inertial Reference Frame, 3rd International Conference “Computational Mechanics and Virtual Engineering” COMEC, Brasov, Romania, October 2009.
  23. Condurache, D; Martinusi, V., Analytic Solution to the Relative Orbital Motion Around an Oblate Planet, AIAA Guidance, Navigation and Control Conference and Exhibit, Chicago, Illinois, 10-13 Aug. 2009 (paper AIAA 2009-6098).
  24. Condurache, D; Martinusi, V., Hypercomplex Eccentric Anomaly in the Unified Solution to the Relative Orbital Motion, AAS/AIAA Astrodynamics Specialist Conference, Pittsburgh, Pennsylvania, 9-13 Aug. 2009 (paper AAS-09-321).
  25. Condurache, D.; Martinusi, V., Exact solution to the relative orbital motion in a central force field, IEEE/AIAA 2nd International Symposium on Systems and Control in Aerospace and Astronautics, Shenzhen, China, 10-12 Dec. 2008, DOI: 10.1109/ISSCAA.2008.4776296.
  26. Condurache, D.; Martinusi, V., A Quaternionic Exact Solution to the Relative Orbital Motion, AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Honolulu, Hawaii, 18-21 Aug. 2008, AIAA Paper 2008-6764
  27. Condurache, D., Martinusi, V., Exact Solution to the Relative Orbital Motion in Eccentric Orbits, International Conference “Analytical Methods of Celestial Mechanics”, Sankt-Petersburg, Russia, 2007.
  28. Condurache, D., Martinusi, V, A General Method to Study the Motion in a Noninertial Reference Frame, The 3rd International Conference on ²Computational Mechanics and Virtual Engineering² COMEC 2009
  29. Condurache, D., Martinusi, V., A Generalized Solution to the Relative Orbital Motion in a Central Force Field, 2st International Conference ²Computational Mechanics and Virtual Engineering ² COMEC 2006, Braşov, 2006.
  30. Condurache, D., Martinusi, V., A Quaternionic Procedure in the Study of the Keplerian Relative Orbital Motion, 2st International Conference ²Computational Mechanics and Virtual Engineering ² COMEC 2006, Braşov, 2006.
  31. Condurache, D., Martinuşi, V., A Novel Hypercomplex Solution to Kepler`s Problem, CMDA 2006 – International Workshop on Actual Problems in Celestial Mechanics and Dynamical Astronomy, Babeş-Bolyai University Cluj-Napoca, Romania, 2006.
  32. Condurache, D., Martinuşi, V., A Closed Form Vectorial Solution To the Relative Orbital Motion, CMDA 2006 – International Workshop on Actual Problems in Celestial Mechanics and Dynamical Astronomy, Babeş-Bolyai University Cluj-Napoca, Romania, 2006.
  33. Condurache, D., Martinuşi, V.,. An Exact Solution to Satellite Relative Orbital Motion, 1st International Conference ²Computational Mechanics and Virtual Engineering ² COMEC 2005, Braşov, 2005.
  34. Condurache, D., The Two Body Problem in Non-Inertial Reference Frames, Plenary Session, The 2nd International Symposium of Theoretical and Applied Mechanics “D.I.Mangeron”, Iaşi, Romania, 2005.
  35. Condurache, D., Martinuşi, V., A Vectorial Regularization of the Keplerian Motion, The 2nd. International Symposium of Theoretical and Applied Mechanics “D.I.Mangeron” Conference Proceedings, Bul. Inst. Polit. Iaşi, LI(LIV), 2005.
  36. Condurache, D., Martinuşi, V., About the Rectilinear Keplerian Motion, The 2nd International Symposium of Theoretical and Applied Mechanics “D.I.Mangeron” Conference Proceedings, Bul. Inst. Polit. Iaşi, LI(LIV), 2005.
  37. Condurache, D., Martinusi, V., Closed Form Solution to the Spacecraft Orbital Motion around a Precessing Planet, National Conference on Mechanics of Solids, Pitesti, Romania, 2008.

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