Skip to main content
SpringerLink
Log in

Quaternion Methods and Regular Models of Celestial Mechanics and Space Flight Mechanics: The Use of Euler (Rodrigues–hamilton) Parameters to Describe Orbital (Trajectory) Motion. I: Review and Analysis of Methods and Models and Their Applications

  • Published:
Mechanics of Solids Aims and scope Submit manuscript

Abstract

The problem of regularization of the classical equations of celestial mechanics and space flight mechanics (astrodynamics) is considered, in which variables are used that characterize the shape and dimensions of the instantaneous orbit (trajectory) of the moving body under study, and Euler angles describing the orientation of the rotating (intermediate) coordinate system used, or the orientation of the instantaneous orbit, or orbital plane of the moving body in the inertial coordinate system. Singularity-type (divide-by-zero) singularities of these classical equations are generated by Euler angles and effectively eliminated by using four-dimensional Euler (Rodrigues–Hamilton) parameters and Hamiltonian rotation (rotation) quaternions.

The article presents a review and analysis of the models of celestial mechanics and astrodynamics, known to us, regular in the indicated sense, constructed using the Euler parameters and Hamilton rotation quaternions based on the differential equations of the perturbed three-body problem. The applications of these models in the problems of optimal control of the orbital motion of a spacecraft, which are solved using the Pontryagin maximum principle, are considered. It is shown that the efficiency of analytical research and numerical solution of boundary value problems of optimal control of the trajectory (orbital) motion of spacecraft can be dramatically increased through the use of regular quaternion models of astrodynamics.

There is also a review and analysis of publications that use dual Euler parameters and dual quaternions (Clifford biquaternions) to solve the problems of controlling the general spatial motion of a rigid body (spacecraft), which is a composition of rotational (angular) and translational (orbital) motions of a rigid body, equivalent to its screw motion, using the feedback principle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. 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 

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

    Google Scholar 

  3. 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 

  4. 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 

  5. 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 

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

    Google Scholar 

  7. 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 

  8. W. R. Hamilton, Lectures on Quaternions (Hodges and Smith, Dublin, 1853).

    Google Scholar 

  9. P. Kustaanheimo, “Spinor regularization of the Kepler motion,” Ann. Univ. Turku 73, 3–7 (1964).

    MathSciNet  MATH  Google Scholar 

  10. P. Kustaanheimo and E. Stiefel, “Perturbation theory of Kepler motion based on spinor regularization,” J. Reine Anqew. Math. 218, 204–219 (1965).

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  ADS  Google Scholar 

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

    MATH  Google Scholar 

  14. 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].

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

    MATH  Google Scholar 

  16. J. Pelaez, J. M. Hedo, and P. Rodriguez de Andres, “A special perturbation method in orbital dynamics,” Celest. Mech. Dyn. Astron. 97, 131–150 (2007). https://doi.org/10.1007/s10569-006-9056-3

    Article  MathSciNet  MATH  ADS  Google Scholar 

  17. G. Bau, H. Urrutxua, and J. Pelaez, “EDROMO: An accurate propagator for elliptical orbits in the perturbed two-body problem,” Adv. Astronaut. Sci. 152 (06), 379–399 (2014).

    Google Scholar 

  18. G. Bau, C. Bombardelli, J. Pelaez, and E. Lorenzini, “Non-singular orbital elements for special perturbations in the two-body problem,” MNRAS 454 (3), 2890–2908 (2015). https://doi.org/10.1093/mnras/stv2106

    Article  ADS  Google Scholar 

  19. P. Libraro, PhD Dissertation (Princeton University, Princeton, New Jersey, 2016).

  20. J. Roa and J. Kasdin, “Alternative set of nonsingular quaternionic orbital elements,” J. Gui. Contr. Dyn. 40 (11), 2737–2751 (2017). https://doi.org/10.2514/1.G002753

    Article  Google Scholar 

  21. D. Amato, C. Bombardelli, G. Bau, et al., “Non-averaged regularized formulations as an alternative to semi-analytical orbit propagation methods,” Celest. Mech. Dyn. Astron. 131, 21 (2019). https://doi.org/10.1007/s10569-019-9897-1

    Article  MathSciNet  MATH  ADS  Google Scholar 

  22. G. Bau and J. Roa, “Uniform formulation for orbit computation: the intermediate elements,” Celest. Mech. Dyn. Astron. 132, 10 (2020). https://doi.org/10.1007/s10569-020-9952-y

    Article  MathSciNet  MATH  ADS  Google Scholar 

  23. H. Andoyer, Cours de Mecanigue Celeste (Gauthier-Villars, Paris, 1923).

    MATH  Google Scholar 

  24. A. Deprit, “Ideal elements for perturbed Keplerian motions,” J. Res. National Bureau Stand. - B. Mat. Sci. 79B (1-2), 1-15 (1975). https://doi.org/10.6028/JRES.079B.001

    Article  MATH  Google Scholar 

  25. P. Musen, “Application of Hansen’s theory to the motion of an artificial satellite in the gravitational field of the Earth,” J. Geophys. Res. 64 (12), 2271–2279 (1959). https://doi.org/10.1029/JZ064i012p02271

    Article  ADS  Google Scholar 

  26. E. W. Brown and C. A. Shook, Panetary Theory (Cambridge Univ. Press, Cambridge, 1933).

    Google Scholar 

  27. P. Musen, “On stromgren’s method of special perturbations,” J. Astron. Sciences. 8, 48–51 (1961).

    Google Scholar 

  28. P. Musen, On the Application of Pfaff’s Method in the Theory of Variation of Astronomical Constants, NASA Technical Note D-2301 (Goddard Space Flight Center, Greenbelt, MD, 1964).

    Google Scholar 

  29. R. Broucke, H. Lass, and M. Ananda, “Redundant variables in celestial mechanics,” Astron. Astrophys. 13, 390–398 (1971).

    MATH  ADS  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Google Scholar 

  32. W. Clifford, “Preliminary sketch of biquaternions,” Proc. London Math. Soc. 4, 381–395 (1873).

    MathSciNet  MATH  Google Scholar 

  33. Yu. N. Chelnokov, “On integration of kinematic equations of a rigid body’s screw-motion,” J. Appl.Math. Mech. 44 (1), 19–23 (1980). https://doi.org/10.1016/0021-8928(80)90168-9

    Article  MathSciNet  MATH  Google Scholar 

  34. 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).

  35. 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).

  36. Yu. N. Chelnokov, “Quaternion regularization and stabilization of perturbed central motion. I,” Mech. Solids 28 (1), 16–25 (1993).

    Google Scholar 

  37. Yu. N. Chelnokov, “Quaternion regularization and stabilization of perturbed central motion. II,” Mech. Solids 28 (2), 1-12 (1993).

    Google Scholar 

  38. 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 

  39. 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 

  40. Yu. N. Chelnokov, “Regular quaternion models of perturbed orbital motion of a rigid body in the Earth’s gravitational field,” Prikl. Mat. Mekh. 83 (4), 562–585 (2019). https://doi.org/10.1134/S003282351902005X

    Article  Google Scholar 

  41. Yu. N. Chelnokov, “Regular quaternion models of perturbed orbital motion of a rigid body in the Earth’s gravitational field,” Mech. Solids 55 (7), 958–976 (2020). https://doi.org/10.3103/S0025654420070079

    Article  MATH  ADS  Google Scholar 

  42. 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 

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

  44. Yu. N. Chelnokov, “Optimal control of spacecraft motion in the newtonian gravitational field: application of quaterni-ons to describe orbit orientation,” Cosmic Res. 37 (4), 409–418 (1999).

    ADS  Google Scholar 

  45. 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). https://doi.org/10.1023/A:1012345213745

    Article  ADS  Google Scholar 

  46. 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 

  47. 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 

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

    Google Scholar 

  49. 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 

  50. R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics (AIAA Press, New York, 1987). https://doi.org/10.2514/4.861543

    Book  MATH  Google Scholar 

  51. Yu. V. Afanasyeva and Yu.N. Chelnokov, “The problem of rendezvous of a controlled space vehicle with an uncontrolled space vehicle moving along an elliptical keplerian orbit in the central newtonian gravitational field,” J. Comput. Syst. Sci. Int. 46 (3), 468-484 (2007). https://doi.org/10.1134/S106423070703015X

    Article  MathSciNet  MATH  Google Scholar 

  52. Yu. V. Afanasyeva and Yu.N. Chelnokov, “The problem of optimal control of the orientation of an orbit of a spacecraft as a deformable figure,” J. Comput. Syst. Sci. Int. 47, 621–634 (2008). https://doi.org/10.1134/S106423070804014X

    Article  MathSciNet  MATH  Google Scholar 

  53. 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 

  54. 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 

  55. 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 

  56. 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 

  57. 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 

  58. Y. G. Sapunkov and Y. 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 

  59. Y. G. Sapunkov and Y. 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 

  60. 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  MATH  ADS  Google Scholar 

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

    Google Scholar 

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

  63. 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 

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

    Google Scholar 

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

    Google Scholar 

  66. 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 

  67. 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 

  68. Yu. N. Chelnokov, “A screw method for the description of the motion of a rigid body,” in Collection of Research and Methodology Papers on Theoretical Mechanics, Issue 11 (Vysshaya Shkola, Moscow, 1981), pp. 129–138 [in Russian].

  69. Yu. N. Chelnokov, “One form of the inertial navigation equations,” Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 5, 20–28 (1981).

  70. A. P. Kotelnikov, Helical Calculus and Some of Its Applications to Geometry and Mechanics (Kazan, 1895) [in Russian].

    Google Scholar 

  71. A. P. Kotelnikov, “Screws and complex numbers,” Izv. Fiz.-Mat. Obshch. Imper. Kazan. Univ. Ser. 2, No. 6, 23–33 (1896).

  72. N. A. Strelkova, “Optimal in the speed of response kinematic control of screw displacement of a rigid body,” Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 4, 73–76 (1982).

  73. V. V. Malanin and N. A. Strelkova, Optimal Control of Orientation and Helical Motion of a Rigid Body (NITs “Regularn. i Khaotich. Dinamika,” Moscow-Izhevsk, 2004) [in Russian].

  74. D. Han, Q. Wei, and Z. Li, “Kinematic control of free rigid bodies using dual quaternions,” Int. J. Automat. Comput. 5 (3), 319–324 (2008). https://doi.org/10.1007/s11633-008-0319-1

    Article  Google Scholar 

  75. D. Han, Q. Wei, Z. Li, and W. Sun, “Control of oriented mechanical systems: a method based on dual quaternion,” IFAC Proc. Vols. 41 (2), 3836–3841 (2008). https://doi.org/10.3182/20080706-5-KR-1001.00645

  76. D. Han, Q. Wei, and Z. Li, “A dual-quaternion method for control of spatial rigid body. networking, sensing and control,” in 2008 IEEE Intern. Conf. Networking Sensing Control (IEEE, 2008), pp. 1–6. https://doi.org/10.1109/ICNSC.2008.4525172

    Book  Google Scholar 

  77. E. Ozgur and Y. Mezouar, “Kinematic modeling and control of a robot arm using unit dual quaternions,” Robot. Autonom. Syst. 77, 66–73 (2016).

    Article  Google Scholar 

  78. Yu. N. Chelnokov, “Biquaternion solution of the kinematic control problem for the motion of a rigid body and its application to the solution of inverse problems of robot-manipulator kinematics,” Mech. Solids 48, 31–46 (2013). https://doi.org/10.3103/S0025654413010044

    Article  ADS  Google Scholar 

  79. Yu. N. Chelnokov and E. I. Nelaeva, “Biquaternion solution of the kinematic problem on optimal nonlinear stabilization of arbitrary program movement of free rigid body,” Izv. Sarat. Univ. Nov. Ser., Ser.: Mat., Mekh., Inf. 16 (2), 198–206 (2016).

    Google Scholar 

  80. A. Perez and J. M. McCarthy, “Dual quaternion synthesis of constrained robotic systems,” J. Mech. Design. 126 (3), 425–435 (2004). https://doi.org/10.1115/1.1737378

    Article  Google Scholar 

  81. D. Han, Q. Wei, Z. Li, and W. Sun, “Control of oriented mechanical systems: a method based on dual quaternions,” IFAC Proc. Vols. 41 (2), 3836–3841 (2008). https://doi.org/10.3182/20080706-5-KR-1001.00645

  82. M. Schilling, “Universally manipulable body models – dual quaternion representations in layered and dynamic MMCs,” Auton. Rob. 30, 399–425 (2011). https://doi.org/10.1007/s10514-011-9226-3

    Article  Google Scholar 

  83. F. Zhang and G. Duan, “Robust integrated translation and rotation finite-time maneuver of a rigid spacecraft based on dual quaternion,” in AIAA Guid. Navig. Control Conf. 2011. Portland, Oregon. USA (AIAA, 2011), pp. 6396. https://doi.org/10.2514/6.2011-6396

  84. J. Wang and Z. Sun, “6DOF Robust adaptive terminal sliding mode control for spacecraft formation flying,” Acta Astron. 73, 76–87 (2012). https://doi.org/10.1016/j.actaastro.2011.12.005

    Article  Google Scholar 

  85. J. Wang, H. Liang, Z. Sun, et al., “Finite-time control for spacecraft formation with dualnumber based description,” J. Guid. Contr. Dyn. 35 (3), 950–962 (2012). https://doi.org/10.2514/1.54277

    Article  ADS  Google Scholar 

  86. J. Wang and C. Yu, “Unit dual quaternion-based feedback linearization tracking problem for attitude and position dynamics,” Syst. Control Lett. 62 (3), 225–233 (2013). https://doi.org/10.1016/j.sysconle.2012.11.019

    Article  MathSciNet  MATH  Google Scholar 

  87. N. Filipe and P. Tsiotras, “Rigid body motion tracking without linear and angular velocity feedback using dual quaternions,” in 2013 European Control Conference (ECC) (IEEE, 2013), pp, 329–334. https://doi.org/10.23919/ECC.2013.6669564

  88. U. Lee, PhD Dissertation (Univ. of Washington, 2014).

  89. N. Filipe, M. Kontitsis, and P. Tsiotras, “Extended Kalman filter for spacecraft pose estimation using dual quaternions,” J. Guid. Contr. Dyn. 38 (9), 1625–1641 (2015). https://doi.org/10.2514/1.G000977

    Article  ADS  Google Scholar 

  90. N. Filipe and P. Tsiotras, “Adaptive position and attitude–tracking controller for satellite proximity operations using dual quaternions,” J. Guid. Contr. Dyn. 38 (4), 566–577 (2015).

    Article  ADS  Google Scholar 

  91. U. Lee and M. Mesbahi, “Optimal powered descent guidance with 6-DoF line of sight constraints via unit dual quaternions,” in AIAA Guidance, Navigation, and Control Conference. 5–9 January 2015 Kissimmee, Florida (AIAA, 2015), 0319. https://doi.org/10.2514/6.2015-0319

  92. H. Gui and G. Vukovich, “Cite as dual-quaternion-based adaptive motion tracking of spacecraft with reduced control effort,” Nonlin. Dyn. 83 (1–2), 597–614 (2016).

  93. S. A. Akhramovich, V. V. Malyshev, and A. V. Starkov, “Mathematical model of drone motion in the biquaternion form,” Polet 4, 9–20 (2018).

    Google Scholar 

  94. S. A. Akhramovich and V. V. Malyshev, “Biquaternions application in the aircraft control problems,” in System Analysis, Control and Navigation. Proceedings (MAI, Moscow, 2018), pp. 117–120 [in Russian].

  95. S. A. Akhramovich and A. V. Barinov, “The system for controlling drone’s motion with predicting model in the biquaternion form,” in System Analysis, Control and Navigation. Proceedings (MAI, Moscow, 2018), pp. 120–122 [in Russian].

  96. C. Garcia, D. M. Prett, and M. Morari, “Model predictive control: theory and practice,” Automatica, No. 3, 335–348 (1989).

  97. Yu. N. Chelnokov, “Controlling the spatial motion of a rigid body using biquaternions and dual matrices,” Mech. Solids 56, 13–33 (2021). https://doi.org/10.3103/S0025654421010064

    Article  ADS  Google Scholar 

  98. Yu. N. Chelnokov, “Synthesis of control of spatial motion of a rigid body using dual quaternions,” Prikl. Mat. Mekh. 83 (5–6), 704–733 (2019). https://doi.org/10.1134/S0032823519050035

  99. Yu. N. Chelnokov, “Synthesis of control of spatial motion of a rigid body using dual quaternions,” Mech. Solids 55 (7), 977–998 (2020). https://doi.org/10.3103/S0025654420070080

    Article  MATH  ADS  Google Scholar 

  100. Yu. 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

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Yu. N. Chelnokov

Authors
  1. Yu. N. Chelnokov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu. N. Chelnokov.

Additional information

Translated by I. K. Katuev

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chelnokov, Y.N. Quaternion Methods and Regular Models of Celestial Mechanics and Space Flight Mechanics: The Use of Euler (Rodrigues–hamilton) Parameters to Describe Orbital (Trajectory) Motion. I: Review and Analysis of Methods and Models and Their Applications. Mech. Solids 57, 961–983 (2022). https://doi.org/10.3103/S0025654422050041

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0025654422050041

Keywords:

Search

Navigation