Quaternion Regularization of Singularities of Astrodynamic Models Generated by Gravitational Forces (Review)

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

The article presents an analytical review of works devoted to the quaternion regularization of the singularities of differential equations of the perturbed three-body problem generated by gravitational forces, using the four-dimensional Kustaanheimo–Stiefel variables. Most of these works have been published in leading foreign publications. We consider a new method of regularization of these equations proposed by us, based on the use of two-dimensional ideal rectangular Hansen coordinates, two-dimensional Levi-Civita variables, and four-dimensional Euler (Rodrigues–Hamilton) parameters. Previously, it was believed that it was impossible to generalize the famous Levi-Civita regularization of the equations of plane motion to the equations of spatial motion. The regularization proposed by us refutes this point of view and is based on writing the differential equations of the perturbed spatial problem of two bodies in an ideal coordinate system using two-dimensional Levi-Civita variables to describe the motion in this coordinate system (in this coordinate system, the equations of spatial motion take the form of equations of plane motion) and based on the use of the quaternion differential equation of the inertial orientation of the ideal coordinate system in the Euler parameters, which are the osculating elements of the orbit, as well as on the use of Keplerian energy and real time as additional variables, and on the use of the new independent Sundmann variable. Reduced regular equations, in which Levi-Civita variables and Euler parameters are used together, have not only the well-known advantages of equations in Kustaanheimo–Stiefel variables (regularity, linearity in new time for Keplerian motions, proximity to linear equations for perturbed motions), but also have their own additional advantages: 1) two-dimensionality, and not four-dimensionality, as in the case of Kustaanheimo-Stiefel, a single-frequency harmonic oscillator describing in new time in Levi-Civita variables the unperturbed elliptic Keplerian motion of the studied (second) body, 2) slow change in the new time of the Euler parameters, which describe the change in the inertial orientation of the ideal coordinate system, for perturbed motion, which is convenient when using the methods of nonlinear mechanics. This work complements our review paper [1].

About the authors

Yu.N. Chelnokov

Institute of Precision Mechanics and Control Problems RAS

Author for correspondence.
Email: ChelnokovYuN@gmail.com
Russia, Saratov

References

  1. Chelnokov Yu.N. Quaternion and biquaternion methods and regular models of analytical mechanics (Review) // PMM, 2023, vol. 87, iss. 4, pp. 519–556.
  2. Levi-Civita T. Sur la regularization du probleme des trois corps // Acta Math., 1920, vol. 42, pp. 99–144. doi: 10.1007/BF02418577
  3. Stiefel E.L., Scheifele G. Linear and Regular Celestial Mechanics. Berlin: Springer, 1971. 350 p.
  4. Aarseth S.J., Zare K.A. Regularization of the three-body problem // Celest. Mech., 1974, vol. 10, pp. 185–205.
  5. Aarseth S.J. Gravitational N-Body Simulations. Cambridge: Univ. Press, 2003. 408 p.
  6. Hopf Н. Uber die Abbildung der dreidimensionalen Sphare auf die Kugelflache // Math. Ann., 1931, vol. 104, pp. 637–665.
  7. Hurwitz A. Mathematische Werke. Vol. 2. Basel: Birkhauser, 1933.
  8. Chelnokov Yu.N. Quaternion regularization in celestial mechanics and astrodynamics and trajectory motion control. II // Cosmic Res., 2014, vol. 52, no. 4, pp. 350–361. doi: 10.1134/S0010952514030022
  9. Deprit A. Ideal frames for perturbed keplerian motions // Celest. Mech., 1976, vol. 13, no. 2, pp. 253–263.
  10. Sundman K.F. Memoire sur le probleme des trois crops // Acta Math., 1912, vol. 36, pp. 105–179.
  11. Velte W. Concerning the regularizing KS-transformation // Celest. Mech., 1978, vol. 17, pp. 395–403.
  12. Vivarelli M.D. The KS-transformation in hypercomplex form // Celest. Mech., 1983, vol. 29, pp. 45–50.
  13. Vivarelli M.D. Geometrical and physical outlook on the cross product of two quaternions // Celest. Mech., 1988, vol. 41, pp. 359–370.
  14. Vivarelli M.D. On the connection among three classical mechanical problems via the hypercomplex KS-transformation // Celest. Mech. & Dyn. Astron., 1991, vol. 50, pp. 109–124.
  15. Shagov O.B. On two types of equations of motion of an artificial Earth satellite in oscillatory form // Mech. Solids, 1990, no. 2, pp. 3–8.
  16. Deprit A., Elipe A., Ferrer S. Linearization: Laplace vs. Stiefel // Celest. Mech. & Dyn. Astron., 1994, vol. 58, pp. 151–201.
  17. Vrbik J. Celestial mechanics via quaternions // Canad. J. Phys., 1994, vol. 72, pp. 141–146.
  18. Vrbik J. Perturbed Kepler problem in quaternionic form // J. Phys. A: Math. & General, 1995, vol. 28, pp. 193–198.
  19. Waldvogel J. Quaternions and the perturbed Kepler problem // Celest. Mech. & Dyn. Astron., 2006, vol. 95, pp. 201–212.
  20. Waldvogel J. Quaternions for regularizing celestial mechanics: the right way // Mech. & Dyn. Astron., 2008, vol. 102, no. 1, pp. 149–162.
  21. Saha P. Interpreting the Kustaanheimo–Stiefel transform in gravitational dynamics // Monthly Notices Roy. Astron. Soc., 2009, vol. 400, pp. 228–231. doi: 10.1111/j.1365-2966.2009.15437.x. arXiv:0803.4441
  22. Zhao L. Kustaanheimo–Stiefel regularization and the quadrupolar conjugacy // R&C Dyn., 2015, vol. 20, no. 1, pp. 19–36. doi: 10.1134/S1560354715010025
  23. Roa J., Urrutxua H., Pelaez J. Stability and chaos in Kustaanheimo–Stiefel space induced by the Hopf fibration // Monthly Notices Roy. Astron. Soc., 2016, vol. 459, no. 3, pp. 2444–2454. doi: 10.1093/mnras/stw780.arXiv:1604.06673
  24. Roa J., Pelaez J. The theory of asynchronous relative motion II: universal and regular solutions // Celest. Mech.&Dyn. Astron., 2017, vol. 127, pp. 343–368.
  25. Breiter S., Langner K. Kustaanheimo–Stiefel transformation with an arbitrary defining vector // Celest. Mech.&Dyn. Astron., 2017, vol. 128, pp. 323–342.
  26. Breiter S., Langner K. The extended Lissajous–Levi-Civita transformation // Celest. Mech.&Dyn. Astron., 2018, vol. 130, Art. no. 68. doi: 10.1007/s10569-018-9862-4
  27. Breiter S., Langner K. The Lissajous–Kustaanheimo–Stiefel transformation // Celest. Mech.&Dyn. Astron., 2019, vol. 131, Art. no. 9. doi: 10.1007/s10569-019-9887-3
  28. Ferrer S., Crespo F. Alternative angle-based approach to the KS-map. An interpretation through symmetry // J. Geom. Mech., 2018, vol. 10, no. 3, pp. 359–372.
  29. Chelnokov Yu.N. On regularization of the equations of the three-dimensional two body problem // Mech. Solids, 1981, vol. 16, no. 6, pp. 1–10.
  30. Chelnokov Yu.N. Regular equations of the three-dimensional two body problem // Mech. Solids, 1984, vol. 19, no. 1, pp. 1–7.
  31. Chelnokov Yu.N. 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).
  32. Chelnokov Yu.N. 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).
  33. Chelnokov Yu.N. Application of quaternions in the theory of orbital motion of an artificial satellite. I // Cosmic Res., 1992, vol. 30, no. 6, pp. 612–621.
  34. Chelnokov Yu.N. Application of quaternions in the theory of orbital motion of an artificial satellite. II // Cosmic Res., vol. 31, no. 3, 1993, pp. 409–418.
  35. Chelnokov Yu.N. Quaternion regularization and stabilization of perturbed central motion. I // Mech. Solids, vol. 28, no. 1, 1993, pp. 16–25.
  36. Chelnokov Yu.N. Quaternion regularization and stabilization of perturbed central motion. II // Mech. Solids, vol. 28, no. 2, 1993, pp. 1–12.
  37. Chelnokov Yu.N. Analysis of optimal motion control for a material points in a central field with application of quaternions // J. Comput.&Syst. Sci. Int., 2007, vol. 46, no. 5, pp. 688–713.
  38. Chelnokov Yu.N. Quaternion regularization in celestial mechanics and astrodynamics and trajectory motion control. I // Cosmic Res., 2013, vol. 51, no. 5, pp. 353–364. doi: 10.1134/S001095251305002X
  39. Chelnokov Yu.N. Quaternion regularization in celestial mechanics, astrodynamics, and trajectory motion control. III // Cosmic Res., 2015, vol. 53, no. 5, pp. 394–409.
  40. Chelnokov Yu.N. Perturbed spatial two-body problem: Regular quaternion equations of relative motion // Mech. Solids, 2019, vol. 54, no. 2, pp. 169–178.doi: 10.3103/S0025654419030075
  41. Chelnokov Yu.N. Quaternion equations of disturbed motion of an artificial earth satellite // Cosmic Res., 2019, vol. 57, no. 2, pp. 101–114. doi: 10.1134/S0010952519020023
  42. Chelnokov Yu.N. Quaternion methods and models of regular celestial mechanics and astrodynamics // Appl. Math.&Mech., 2022, vol. 43, no. 1, pp. 21–80. doi: 10.1007/s10483-021-2797-9
  43. Bordovitsyna T.V. Modern Numerical Methods in Problems of Celestial Mechanics. Moscow: Nauka, 1984. 136 p.
  44. Bordovitsyna T.V., Avdyushev V.A. Theory of Motion of Artificial Satellites of the Earth. Analytical and Numerical Methods. Tomsk: Tomsk Univ. Pub., 2007. 178 p.
  45. Fukushima T. Efficient orbit integration by linear transformation for Kustaanheimo–Stiefel regularization // Astron. J., 2005, vol. 129, no. 5, 2496. doi: 10.1086/429546
  46. Fukushima T. Numerical comparison of two-body regularizations // Astron. J., 2007, vol. 133, no. 6, 2815.
  47. Pelaez J., Hedo J.M., Rodriguez P.A. A special perturbation method in orbital dynamics // Celest. Mech.&Dyn. Astron., 2007, vol. 97, pp. 131–150. doi: 10.1007/s10569-006-9056-3
  48. Bau G., Bombardelli C., Pelaez J., Lorenzini E. Non-singular orbital elements for special perturbations in the two-body problem // Monthly Notices Roy. Astron. Soc., 2015, vol. 454, pp. 2890–2908.
  49. Amato D., Bombardelli C., Bau G., Morand V., Rosengren A.J. Non-averaged regularized formulations as an alternative to semi-analytical orbit propagation methods // Celest. Mech.&Dyn. Astron., 2019, vol. 131, no. 21. doi: 10.1007/s10569-019-9897-1
  50. Bau G., Roa J. Uniform formulation for orbit computation: the intermediate elements // Celest. Mech.&Dyn. Astron., 2020, vol. 132, no. 10. doi: 10.1007/s10569-020-9952-y
  51. Chelnokov Y.N., Loginov M.Y. New quaternion models of spaceflight regular mechanics and their applications in the problems of motion prediction for cosmic bodies and in inertial navigation in space // 28th St. Petersburg Int. Conf. on Integrated Navigation Systems, ICINS 2021, 9470806.
  52. Chelnokov Yu.N., Sapunkov Ya.G., Loginov M.Yu., Shchekutiev A.F. Forecast and correction of spacecraft orbital motion using regular quaternion equations and their solutions in Kustaanheimo–Stiefel variables and isochronic derivatives // PMM, 2023, vol. 87, iss. 2, pp. 124–156.
  53. Chelnokov Yu.N. Quaternion regularization of the eguations of the perturbed spatial restricted three-body problem: I // Mech. Solids, 2017, vol. 52, no. 6, pp. 613–639. doi: 10.3103/S0025654417060036
  54. Euler L. De motu rectilineo trium corporum se mutuo attrahentium // Nov. Comm. Petrop., 1765, vol. 11, pp. 144–151.
  55. Levi-Civita T. Traettorie singolari ed urbi nel problema ristretto dei tre corpi // Ann. Mat. Pura Appl., 1904, vol. 9, pp. 1–32.
  56. Levi-Civita T. Sur la resolution qualitative du probleme restreint des trois corps // Opere Math., 1956, no. 2, pp. 411–417.
  57. Kustaanheimo P. Spinor regularization of the Kepler motion // Ann. Univ. Turku, 1964, vol. 73, pp. 3–7. doi: 10.1086/518165
  58. Kustaanheimo P., Stiefel E. Perturbation theory of Kepler motion based on spinor regularization // J. Reine Anqew. Math., 1965, vol. 218, pp. 204–219.
  59. Hopf H. Uber die Abbildung der dreidimensionalen Sphare auf die Kugelflache // Math. Ann., 1931, vol. 104, pp. 637–665.
  60. Volk O. Concerning the derivation of the KS-transformation // Celest. Mech., 1973, vol. 8, pp. 297–305.
  61. Lidov M.L. Increasing the dimension of Hamiltonian systems. KS-transform, use of partial integrals // Cosmic Res., 1982, vol. 20, no. 2, pp. 163–176.
  62. Lidov M.L. Method for constructing families of spatial periodic orbits in the Hill problem // Cosmic Res., 1982, vol. 20, no. 6, pp. 787–807.
  63. Lidov M.L., Lyakhova V.A. Families of spatial periodic orbits of the Hill problem and their stability // Cosmic Res., 1983, vol. 21, no. 1, pp. 3–11.
  64. Poleshchikov S.M. Regularization of the canonical equations of the two-body problem using the generalized KS-matrix // Cosmic Res., 1999, vol. 37, no. 3, pp. 322–328.
  65. Stiefel E.L., Waldvogel J. Generalisation de la regularisation de Birkhoff pour le mouvement du mobile dans l’espace a trois dimensions // Comptes Rendus Hebdomadaires des Seances de Lacademie des Sciences. 1965, Paris.
  66. Stiefel E., Rossler M., Waldvogel J., Burdet C.A. Methods of regularization for computing orbits in celestial mechanics // NASA Contractor Rep. NASA CR-769, 1967, pp. 88–115.
  67. Birkhoff G.D. The restricted problem of three bodies // Rendiconti del Circolo Matematico di Palermo (1884–1940), 1915, vol. 39 (1), pp. 265–334.
  68. Waldvogel J. Die Verallgemeinerung der Birkhoff-Regularisierung fur das raumliche Dreikorperproblem // Bull. Astron. Ser. 3, 1967, vol. 2, no. 2, pp. 295–341.
  69. Andoyer H. Cours de Mecanigue Celeste. Paris: Gauthier-Vilars, 1923.
  70. Musen P. Application of Hansen’s theory to the motion of an artificial satellite in the gravitational field of the Earth // J. Geoph. Res., 1959, vol. 64, pp. 2271–2279.
  71. Broomberg V.A. Analytical Algorithms of Celestial Mechanics. Moscow: Nauka, 1980. 208 p.

Copyright (c) 2023 Ю.Н. Челноков

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies