Quaternion and Biquaternion Methods and Regular Models of Analytical Mechanics (Review)

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Abstract

The work is of a survey analytical nature. The first part of the work presents quaternion and biquaternion methods for describing motion, models of the theory of finite displacements and regular kinematics of a rigid body based on the use of four-dimensional real and dual Euler (Rodrigues–Hamilton) parameters. These models, in contrast to the classical models of kinematics in Euler–Krylov angles and their dual counterparts, do not have division-by-zero features and do not contain trigonometric functions, which increases the efficiency of analytical research and numerical solution of problems in mechanics, inertial navigation, and motion control.
The problem of regularization of differential equations of the perturbed spatial two-body problem, which underlies celestial mechanics and space flight mechanics (astrodynamics), is discussed using the Euler parameters, four-dimensional Kustaanheimo–Stiefel variables, and Hamilton quaternions: the problem of eliminating singularities (division by zero), which are generated by the Newtonian gravitational forces acting on a celestial or cosmic body and which complicate the analytical and numerical study of the motion of a body near gravitating bodies or its motion along highly elongated orbits. The history of the regularization problem and the regular Kustaanheim–Stiefel equations, which have found wide application in celestial mechanics and astrodynamics, are presented. We present the quaternion methods of regularization, which have a number of advantages over Kustaanheimo–Stiefel matrix regularization, and various regular quaternion equations of the perturbed spatial two-body problem (for both absolute and relative motion). The results of a comparative study of the accuracy of numerical integration of various forms of regularized equations of celestial mechanics and astrodynamics in Kustaanheimo–Stiefel variables and Newtonian equations in Cartesian coordinates are presented, showing that the accuracy of numerical integration of regularized equations in Kustaanheimo–Stiefel variables is much higher (by several orders of magnitude) than the accuracy of numerical integration Newtonian equations.

About the authors

Yu.N. Chelnokov

Institute of Precision Mechanics and Control Problems of the RAS

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

References

  1. Euler L. Problema Algebraicum ob Affectiones Prorsus Singulares Memorabile // Novi Comm. Acad. Sci. Imper. Petrop., 1770, vol. 15, pp. 75–106.
  2. Rodrigues O. Des lois geometriques qui regissent les deplacements d’un systems olide dans l’espase, et de la variation des coordonnee sprovenant de ses deplacement sconsideeres independamment des causes qui peuvent les produire // J. des Math. Pureset Appl., 1840, vol. 5, pp. 380–440.
  3. Whittaker E.T. A Treatise on the Analytical Dynamics. Cambridge: Univ. Press, 1927.
  4. Lurie A.I. Analytical Mechanics. Moscow: Fizmatlit, 1961. 824 p.
  5. Chelnokov Yu.N. On integration of kinematic equations of a rigid body’s screw-motion // Appl. Math.&Mech., 1980, vol. 44, no. 1, pp. 19–23.
  6. Chelnokov Yu.N. On one helical method for describing the motion of a rigid body // in: Sat. Sci. Meth. Art. on Theoret. Mech. Moscow: Higher School, 1981. Iss. 11. pp. 129–138.
  7. Chelnokov Yu.N. One form of the equations of Inertial navigation // Mech. Solids., 1981, vol. 16, no. 5, pp. 16–23.
  8. Hamilton W.R. Lectures on Quaternions. Dublin: Hodges and Smith, 1853. 382 p.
  9. Branets V.N., Shmyglevsky I.P. Application of Quaternions in Problems of Orientation of a Rigid Body. Moscow: Nauka, 1973. 320 p.
  10. Chelnokov Yu.N. Quaternion and Biquaternion Models and Methods of Rigid Body Mechanics and Their Applications. Geometry and Kinematics of Motion. Moscow: Fizmatlit, 2006. 511 p.
  11. Zhuravlev V.F. Fundamentals of Theoretical Mechanics. Moscow: Fizmatlit, 2008. 304 p.
  12. Clifford W. Preliminary sketch of biquaternions // Proc. London Math. Soc., 1873, no. 4, pp. 381–395.
  13. Kotelnikov A.P. Screw Calculus and Some of Its Applications to Geometry and Mechanics. Kazan: 1895. 215 p.
  14. Kotelnikov A.P. Screws and complex numbers // Izv. Phys.-Math. Society at Kazan Univ., 1896, Ser. 2, no. 6, pp. 23–33.
  15. Kotelnikov A.P. Theory of vectors and complex numbers // in: Some Applications of Lobachevsky’s Ideas in Mechanics and Physics. Moscow: Gostekhizdat, 1950. pp. 7–47.
  16. Branets V.N., Shmyglevsky I.P. Introduction to the Theory of Strapdown Inertial Navigation Systems. Moscow: Nauka, 1992. 280 p.
  17. Gibbs J.W. Scientific Papers. N.Y.: Dover, 1961.
  18. Gibbs J.W. Vector Analysis. N.Y.: Scribners, 1901.
  19. Stiefel E.L., Scheifele G. Linear and Regular Celestial Mechanics. Berlin: Springer, 1971. 350 p.
  20. Bellman R. Introduction to Matrix Analysis. N.Y.: McGraw-Hill. 1960.
  21. Ickes B.F. A New method for performing digital control system attitude computations using quaternions // AIAA J., 1970, no. 8, pp. 13–17.
  22. Plotnikov P.K., Chelnokov Yu.N. Application of quaternion matrices in the theory of finite rotation of a rigid body // in: Sb. Sci.&Meth. Art. on Theoret. Mech. Moscow: Higher School, 1981. Iss. 11, pp. 122–129.
  23. Dimentberg F.M. Theory of Screws and Its Applications. Moscow: Nauka, 1978. 328 p.
  24. Chelnokov Yu.N. On the stability of solutions to the biquaternion kinematic equation of the helical motion of a rigid body // in: Sb. Sci.&Meth., Art. on Theoret. Mech. Moscow: Higher School, 1983. Iss. 13, pp. 103–109.
  25. Chelnokov Yu.N. Study of some algorithmic problems of determining the orientation of an object by strapdown inertial navigation systems. Abstract of the dissertation for the degree of Cand. tech. Sciences. Leningrad Electrotechnical Institute named after V.I. Ulyanov (Lenin). Leningrad: 1974. 20 p.
  26. Chelnokov Yu.N. Quaternion and biquaternion methods in problems of rigid body mechanics and material systems. Abstract of the dissertation for the degree of Doctor of Physical and Mathematical Sciences. Institute for Problems in Mechanics of the Academy of Sciences of the USSR. Moscow: 1987. 36 p.
  27. Chelnokov Yu.N. Quaternion Models and Methods of Dynamics, Navigation and Motion Control. Moscow: Fizmatlit, 2011. 560 p.
  28. Velte W. Concerning the regularizing KS-transformation // Celest. Mech., 1978, vol. 17, pp. 395–403.
  29. Vivarelli M.D. The KS-transformation in hypercomplex form // Celest. Mech., 1983, vol. 29, pp. 45–50.
  30. Vivarelli M.D. Geometrical and physical outlook on the cross product of two quaternions // Celest. Mech., 1988, vol. 41, pp. 359–370.
  31. 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.
  32. 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.
  33. Deprit A., Elipe A., Ferrer S. Linearization: Laplace vs. Stiefel // Celest. Mech.&Dyn. Astron., 1994, vol. 58, pp. 151–201.
  34. Vrbik J. Celestial mechanics via quaternions // Canad. J. Phys., 1994, vol. 72, pp. 141–146.
  35. Vrbik J. Perturbed Kepler problem in quaternionic form // J. Phys. A: Math.&General, 1995, vol. 28, pp. 193–198.
  36. Waldvogel J. Quaternions and the perturbed Kepler problem // Celest. Mech.&Dyn. Astr., 2006, vol. 95, pp. 201–212.
  37. Waldvogel J. Quaternions for regularizing Celestial Mechanics: the right way // Celest. Mech.&Dyn. Astr., 2008, vol. 102, no. 1, pp. 149–162.
  38. 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
  39. Zhao L. Kustaanheimo-Stiefel regularization and the quadrupolar conjugacy // R.&C. Dyn., 2015, vol. 20, no. 1, pp. 19–36. doi: 10.1134/S1560354715010025
  40. 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
  41. 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.
  42. Breiter S., Langner K. Kustaanheimo-Stiefel transformation with an arbitrary defining vector // Celest. Mech.&Dyn. Astron., 2017, vol. 128, pp. 323–342.
  43. 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
  44. Breiter S., Langner K. The extended Lissajous–Levi-Civita transformation // Celest. Mech.&Dyn. Astron., 2019, vol. 131, art. no. 9. doi: 10.1007/s10569-018-9862-4
  45. 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.
  46. 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.
  47. Chelnokov Yu.N. Regular equations of the three-dimensional two body problem // Mech. Solids, 1984, vol. 19, no. 1, pp. 1–7.
  48. Chelnokov Yu.N. Quaternion Methods in Problems of Perturbed Motion of a Material Point. Pt. 1. General Theory. Applications to Problem of Regularization and to Problem of Satellite Motion. Moscow: VINITI, 1985. no. 8628-B.
  49. Chelnokov Yu.N. Quaternion Methods in Problems of Perturbed Motion of a Material Point. Pt. 2. Three-Dimensional Problem of Unperturbed Central Motion. Problem with Initial Conditions. Moscow: VINITI, 1985. no. 8629-B.
  50. 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.
  51. Chelnokov Yu.N. Application of quaternions in the theory of orbital motion of an artificial satellite. II // Cosmic Res., 1993, vol. 31, no. 3, pp. 409–418.
  52. Chelnokov Yu.N. Quaternion regularization and stabilization of perturbed central motion. I // Mech. Solids, 1993, vol. 28, no. 1, pp. 16–25.
  53. Chelnokov Yu.N. Quaternion regularization and stabilization of perturbed central motion. II // Mech. Solids, 1993, vol. 28, no. 2, pp. 1–12.
  54. 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.
  55. 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
  56. 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
  57. Chelnokov Yu.N. Quaternion regularization in celestial mechanics, astrodynamics, and trajectory motion control. III // Cosmic Res., 2015, vol. 53, no. 5, pp. 394–409.
  58. 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
  59. 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
  60. 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
  61. Bordovitsyna T.V. Modern Numerical Methods in Problems of Celestial Mechanics. Moscow: Nauka, 1984. 136 p.
  62. Bordovitsyna T.V., Avdyushev V.A. Theory of Motion of Artificial Satellites of the Earth. Analytical and Numerical Methods. Tomsk: Tomsk Univ., 2007. 178 p.
  63. Fukushima T. Efficient orbit integration by linear transformation for Kustaanheimo-Stiefel regularization // The Astronomical J., 2005, vol. 129, no. 5. 2496. doi: 10.1086/429546
  64. Fukushima T. Numerical comparison of two-body regularizations // The Astronomical J., 2007, vol. 133, no. 6. 2815.
  65. 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
  66. Baù 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.
  67. Amato D., Bombardelli C., Baù 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
  68. Baù 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
  69. 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 Syst., 2021, 9470806.
  70. 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.
  71. 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
  72. Euler L. De motu rectilineo trium corporum se mutuo attrahentium // Nov. Comm. Petrop., 1765, vol. 11, pp. 144–151.
  73. Levi-Civita T. Traettorie singolari ed urbi nel problema ristretto dei tre corpi // Ann. Mat. Pura Appl., 1904, vol. 9, pp. 1–32.
  74. Levi-Civita T. Sur la regularization du probleme des trois corps // Acta Math., 1920, vol. 42, pp. 99–144. doi: 10.1007/BF02418577
  75. Levi-Civita T. Sur la resolution qualitative du probleme restreint des trois corps // Opere Math., 1956, no. 2, pp. 411–417.
  76. Kustaanheimo P. Spinor regularization of the Kepler motion // Ann. Univ. Turku, 1964, vol. 73, pp. 3–7. doi: 10.1086/518165
  77. Kustaanheimo P., Stiefel E. Perturbation theory of Kepler motion based on spinor regularization // J. Reine Anqew. Math., 1965, vol. 218, pp. 204–219.
  78. Brumberg V.A. Analytical Algorithms of Celestial Mechanics. Moscow: Nauka, 1980. 208 p.
  79. Musen P. On Stromgren’s method of special perturbations // J. Astron. Sci., 1961, vol. 8, pp. 48–51.
  80. Musen P. // NASA TN D-2301, 1964, pp. 24.
  81. Hopf Н. Uber die Abbildung der dreidimensionalen Sphare auf die Kugelflache // Math. Ann., 1931, vol. 104, pp. 637–665.
  82. Sundman K.F. Memoire sur le probleme des trois crops // Acta Math., 1912, vol. 36, pp. 105–179.
  83. Bohlin K. Note sur le probleme des deux corps et sur une integration nouvelle dans le problem des trois corps // Bull. Astron., 1911, vol. 28, pp. 113–119.
  84. Burdet C.A. Theory of Kepler motion: The general perturbed two body problem // Zeitschrift fur angewandte Math. und Phys., 1968, vol. 19, pp. 345–368.
  85. Burdet C.A. Le mouvement Keplerien et les oscillateurs harmoniques // J. fur die reine und angewandte Math., 1969, vol. 238, pp. 71–84.
  86. Study E. Von der Bewegungen und Umlegungen // Math. Annal., 1891, vol. 39, pp. 441–566.

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