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Quaternion Methods and Regular Models of Celestial and Space Flight Mechanic: Using Euler (Rodrigues–Hamilton) Parameters to Describe Orbital (Trajectory) Motion. II: Perturbed Spatial Restricted Three-Body Problem

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Abstract

The article considers the problem of regularizing the features of the classical equations of celestial mechanics and space flight mechanics (astrodynamics), which use variables that characterize the shape and size of the instantaneous orbit (trajectory) of the moving body under study, and Euler angles that describe the orientation of the used rotating (intermediate) coordinate system or the orientation of the instantaneous orbit, or the plane of the orbit of a moving body in an inertial coordinate system. Singularity-type features (division by zero) of these classical equations are generated by Euler angles and complicate the analytical and numerical study of orbital motion problems. These singularities are effectively eliminated by using the four-dimensional Euler (Rodrigues–Hamilton) parameters and Hamiltonian rotation quaternions. In this (second) part of the work, new regular quaternion models of celestial mechanics and astrodynamics are obtained that do not have the above features and are built within the framework of a perturbed spatial limited three-body problem (for example, the Earth, the Moon (or the Sun) and a spacecraft (or an asteroid)): equations of trajectory motion written in non-holonomic or orbital or ideal coordinate systems, for the description of the rotational motion of which the Euler (Rodrigues–Hamilton) parameters and quaternions of Hamilton rotations are used. New regular quaternion equations of the perturbed spatial restricted three-body problem are also obtained, constructed using two-dimensional ideal rectangular Hansen coordinates, Euler parameters and quaternion variables, as well as using complex compositions of Hansen coordinates and Euler parameters (Cayley-Klein parameters). The advantage of the proposed orbital motion equations constructed using the Euler parameters over the equations constructed using the Euler angles is due to the well-known advantages of the quaternion kinematic equations in the Euler parameters included in the proposed equations over the kinematic equations in the Euler angles included in the classical equations.

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REFERENCES

  1. V. K. Abalakin, E. P. Aksenov, E. A. Grebenikov, et al., Reference Manual in Celestial Mechanics and Astrodynamics (Nauka, Moscow, 1976) [in Russian].

    Google Scholar 

  2. G. N. Duboshin, Celestial Mechanics: Methods of the Theory of Motion of Artificial Celestial Bodies (Nauka, Moscow, 1983) [in Russian].

    Google Scholar 

  3. Yu. N. Chelnokov, “Quaternion methods and regular models of celestial mechanics and space flight mechanics: the use of Euler (Rodrigues–Hamilton) parameters to describe orbital (trajectory) motion. 1: review and analysis of methods and models and their applications,” Mech. Solids 57, 961–983 (2022). https://doi.org/10.3103/S0025654422050041

    Article  ADS  MATH  Google Scholar 

  4. Yu. N. Chelnokov, “Analysis of optimal motion control for a material point in a central field with application of quaternions,” J. Comp. Syst. Sci. Int. 46 (5), 688–713 (2007). https://doi.org/10.1134/S1064230707050036

    Article  MATH  Google Scholar 

  5. Yu. N. Chelnokov, Quaternion Models and Methods in Dynamics, Navigation, and Motion Control (Fizmatlit, Moscow, 2011) [in Russian].

    Google Scholar 

  6. Yu. N. Chelnokov, “Quaternion regularization in celestial mechanics and astrodynamics and trajectory motion control. I,” Cosmic Res. 51 (5), 350–361 (2013). https://doi.org/10.1134/S001095251305002X

    Article  ADS  Google Scholar 

  7. Yu. N. Chelnokov, “Quaternion regularization in celestial mechanics, astrodynamics, and trajectory motion control. III,” Cosmic Res. 53 (5), 394–409 (2015). https://doi.org/10.1134/S0010952515050044

    Article  ADS  Google Scholar 

  8. Yu. N. Chelnokov, “Quaternion regularization of the equations of the perturbed spatial restricted three-body problem: I,” Mech. Solids 52 (6), 613–639 (2017). https://doi.org/10.3103/S0025654417060036

    Article  ADS  Google Scholar 

  9. Yu. N. Chelnokov, “Quaternion regularization of the equations of the perturbed spatial restricted three-body problem: II,” Mech. Solids 53 (6), 633–650 (2018). https://doi.org/10.3103/S0025654418060055

    Article  ADS  Google Scholar 

  10. V. N. Branets and I. P. Shmyglevskii, Application of Quaternions in Problems of Attitude Control of a Rigid Body (Nauka, Moscow, 1973) [in Russian].

    MATH  Google Scholar 

  11. V. N. Branets and I. P. Shmyglevskii, Introduction to the Theory of Strapdown Inertial Navigation Systems (Nauka, Moscow, 1992) [in Russian].

    MATH  Google Scholar 

  12. Yu. N. Chelnokov, Quaternion and Biquaternion Models and Methods of Mechanics of Solids and Their Applications (Fizmatlit, Moscow, 2006) [in Russian].

    Google Scholar 

  13. V. Ph. Zhuravlev, Foundations of Theoretical Mechanics (Fizmatlit, Moscow, 2008) [in Russian].

    Google Scholar 

  14. Yu. N. Chelnokov, Quaternion Methods in Problems of Perturbed Motion of a Material Point. Part 1. General Theory. Applications to Problem of Regularization and to Problem of Satellite Motion, Available from VINITI, No. 8628-B (Moscow, 1985).

  15. Yu. N. Chelnokov, Quaternion Methods in Problems of Perturbed Motion of a Material Point. Part 2. Three-Dimensional Problem of Unperturbed Central Motion. Problem with Initial Conditions, Available from VINITI, No. 8629-B (Moscow, 1985).

  16. Yu. N. Chelnokov, “Application of quaternions in the theory of orbital motion of an artificial satellite. I,” Cosmic Res. 30 (6), 612–621 (1992).

    ADS  Google Scholar 

  17. Yu. N. Chelnokov, “Application of quaternions in the theory of orbital motion of an artificial satellite. II,” Cosmic Res. 31 (3), 409–418 (1993).

    Google Scholar 

  18. Yu. N. Chelnokov, “Construction of optimum control and trajectories of spacecraft flight by employing quaternion description of orbit spatial orientation,” Cosmic Res. 35 (5), 499–507 (1997).

    ADS  Google Scholar 

  19. Yu. N. Chelnokov, “Application of quaternions to space flight mechanics,” Giroskop. Navig., No. 4 (27), 47–66 (1999).

  20. A. F. Bragazin, V. N. Branets, and I. P. Shmyglevskii, “Description of orbital motion using quaternions and velocity parameters,” in Abstracts of Reports at the 6th All-Union Congress on Theoret. and Applied Mechanics (Fan, Tashkent, 1986), pp. 133 [in Russian].

  21. A. Deprit, “Ideal rames for perturbed keplerian motions,” Celest. Mech. 13 (2), 253–263 (1976).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Yu. N. Chelnokov, “Application of quaternions in theory of orbital motion of artificial satellite. II,” Cosmic Res. 31 (6), 409–418 (1992).

    Google Scholar 

  23. Yu. N. Chelnokov, “The use of quaternions in the optimal control problems of motion of the center of mass of a spacecraft in a Newtonian gravitational field: I,” Cosmic Res. 39, 470–484 (2001).

    Article  ADS  Google Scholar 

  24. S. M. Onishchenko, Hypercomplex Numbers in Inertial Navigation Theory (Naukova Dumka, Kyiv, 1983) [in Russian].

    MATH  Google Scholar 

  25. Yu. N. Chelnokov, “Quaternion regularization and trajectory motion control in celestial mechanics and astrodynamics: II,” Cosmic Res. 52, 304–317 (2014). https://doi.org/10.1134/S0010952514030022

    Article  ADS  Google Scholar 

  26. V. A. Brumberg, Analytical Algorithms of Celestial Mechanics (Nauka, Moscow, 1980) [in Russian].

    MATH  Google Scholar 

  27. Yu. N. Chelnokov, “The use of quaternions in the optimal control problems of motion of the center of mass of a spacecraft in a Newtonian gravitational field: II,” Cosmic Res. 41, 85–99 (2003). https://doi.org/10.1023/A:1022359831200

    Article  ADS  Google Scholar 

  28. Yu. N. Chelnokov, “The use of quaternions in the optimal control problems of motion of the center of mass of a spacecraft in a Newtonian gravitational field: III,” Cosmic Res. 41, 460–477 (2003). https://doi.org/10.1023/A:1026098216710

    Article  ADS  Google Scholar 

  29. Yu. N. Chelnokov, “Optimal reorientation of a spacecraft’s orbit using a jet thrust orthogonal to the orbital plane,” J. Appl. Math. Mech. 76 (6), 646-657 (2012). https://doi.org/10.1016/j.jappmathmech.2013.02.002

    Article  MathSciNet  MATH  Google Scholar 

  30. I. A. Pankratov, Ya. G. Sapunkov, and Yu. N. Chelnokov, “About a problem of spacecraft’s orbit optimal reorientation,” Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform. 12 (3), 87–95 (2012).

    Google Scholar 

  31. I. A. Pankratov, Ya. G. Sapunkov, and Yu. N. Chelnokov, “Solution of a problem of spacecraftтaщs orbit optimalreorientation using quaternion equations of orbital systemof coordinates orientation,” Izv. Saratov Univ. (N. S.) Ser. Math. Mekh. Inform. 13 (1), 84–92 (2013).

    MATH  Google Scholar 

  32. Ya. G. Sapunkov and Yu. N. Chelnokov, “Investigation of the task of the optimal reorientation of a spacecraft orbit through a limited or impulse jet thrust, orthogonal to the plane of the orbit. Part 1,” Mekh. Avt. Upr. 17 (8), 567–575 (2016). https://doi.org/10.17587/mau.17.567-575

    Article  Google Scholar 

  33. Ya. G. Sapunkov and Yu. N. Chelnokov, “Investigation of the task of the optimal reorientation of a spacecraft orbit through a limited or impulse jet thrust, orthogonal to the plane of the orbit. Part 1,” Mekh. Avt. Upr. 17 (9), 633–643 (2016). https://doi.org/10.17587/mau.17.663-643

    Article  Google Scholar 

  34. Y. G. Sapunkov and Y. N. Chelnokov, “Optimal rotation of the orbit plane of a variable mass spacecraft in the central gravitational field by means of orthogonal thrust,” Autom. Remote. Control 80, 1437–1454 (2019). https://doi.org/10.1134/S000511791908006X

    Article  MathSciNet  MATH  Google Scholar 

  35. Ya. G. Sapunkov and Yu. N. Chelnokov, “Pulsed optimal spacecraft orbit reorientation by means of reactive thrust orthogonal to the osculating orbit. I,” Mech. Solids 53, 535–551 (2018). https://doi.org/10.3103/S0025654418080083

    Article  ADS  Google Scholar 

  36. Ya. G. Sapunkov and Yu. N. Chelnokov, “Pulsed optimal spacecraft orbit reorientation by means of reactive thrust orthogonal to the osculating orbit. II,” Mech. Solids 54, 1–18 (2019). https://doi.org/10.3103/S0025654419010011

    Article  ADS  Google Scholar 

  37. Ya. G. Sapunkov and Yu. N. Chelnokov, “Quaternion solution of the problem of optimal rotation of the orbit plane of a variable-mass spacecraft using thrust orthogonal to the orbit plane,” Mech. Solids 54, 941–957 (2019). https://doi.org/10.3103/S0025654419060098

    Article  ADS  MATH  Google Scholar 

  38. M. Kopnin, “On the task of rotating a satellite’s orbit plane,” Kosm. Issl. 3 (4), 22–30 (1965).

    Google Scholar 

  39. V. N. Lebedev, Computation of Motion of a Spacecraft with Small Traction (VTs AN SSSR, Moscow, 1967) [in Russian].

  40. M. Z. Borshchevskii and M. V. Ioslovich, “On the problem of rotating the orbital plane of a satellite by means of reactive thrust,” Kosm. Issl. 7 (6), 8–15 (1969).

    Google Scholar 

  41. G. L. Grodzovskii, Yu. N. Ivanov, and V. V. Tokarev, Mechanics of Space Flight, Optimization Problems (Nauka, Moscow, 1975) [in Russian].

    Google Scholar 

  42. D. E. Okhotsimskii and Yu. G. Sikharulidze, Foundations of Space Flight Mechanics (Nauka, Moscow, 1990) [in Russian].

    Google Scholar 

  43. S. A. Ishkov and V. A. Romanenko, “Forming and correction of a high-elliptical orbit of an earth satellite with low-thrust engine,” Cosm. Res. 35 (3), 268–277 (1997).

    ADS  Google Scholar 

  44. R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics (AIAA Press, New York, 1987).

    MATH  Google Scholar 

  45. Yu. N. Chelnokov, “On regularization of the equations of the three-dimensional two body problem,” Mech. Solids 16 (6), 1–10 (1981).

  46. Yu. N. Chelnokov, “Regular equations of the three-dimensional two body problem,” Mech. Solids 19 (1), 1–7 (1984).

  47. E. L. Stiefel and G. Scheifele, Linear and Regular Celestial Mechanics (Springer, Berlin, 1971).

    Book  MATH  Google Scholar 

  48. Y. N. Chelnokov, “Quaternion methods and models of regular celestial mechanics and astrodynamics,” Appl. Math. Mech. (Engl. Ed.) 43 (1), 21–80 (2022). https://doi.org/10.1007/s10483-021-2797-9

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  1. Institute of Precision Mechanics and Control Problems of the Russian Academy of Sciences, 410028, Saratov, Russia

    Yu. N. Chelnokov

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Correspondence to Yu. N. Chelnokov.

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Translated by I. K. Katuev

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Chelnokov, Y.N. Quaternion Methods and Regular Models of Celestial and Space Flight Mechanic: Using Euler (Rodrigues–Hamilton) Parameters to Describe Orbital (Trajectory) Motion. II: Perturbed Spatial Restricted Three-Body Problem. Mech. Solids 58, 1–25 (2023). https://doi.org/10.3103/S0025654422600787

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