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.
REFERENCES
V. K. Abalakin, E. P. Aksenov, E. A. Grebenikov, et al., Reference Manual in Celestial Mechanics and Astrodynamics (Nauka, Moscow, 1976) [in Russian].
G. N. Duboshin, Celestial Mechanics: Methods of the Theory of Motion of Artificial Celestial Bodies (Nauka, Moscow, 1983) [in Russian].
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
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
Yu. N. Chelnokov, Quaternion Models and Methods in Dynamics, Navigation, and Motion Control (Fizmatlit, Moscow, 2011) [in Russian].
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
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
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
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
V. N. Branets and I. P. Shmyglevskii, Application of Quaternions in Problems of Attitude Control of a Rigid Body (Nauka, Moscow, 1973) [in Russian].
V. N. Branets and I. P. Shmyglevskii, Introduction to the Theory of Strapdown Inertial Navigation Systems (Nauka, Moscow, 1992) [in Russian].
Yu. N. Chelnokov, Quaternion and Biquaternion Models and Methods of Mechanics of Solids and Their Applications (Fizmatlit, Moscow, 2006) [in Russian].
V. Ph. Zhuravlev, Foundations of Theoretical Mechanics (Fizmatlit, Moscow, 2008) [in Russian].
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).
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).
Yu. N. Chelnokov, “Application of quaternions in the theory of orbital motion of an artificial satellite. I,” Cosmic Res. 30 (6), 612–621 (1992).
Yu. N. Chelnokov, “Application of quaternions in the theory of orbital motion of an artificial satellite. II,” Cosmic Res. 31 (3), 409–418 (1993).
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).
Yu. N. Chelnokov, “Application of quaternions to space flight mechanics,” Giroskop. Navig., No. 4 (27), 47–66 (1999).
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].
A. Deprit, “Ideal rames for perturbed keplerian motions,” Celest. Mech. 13 (2), 253–263 (1976).
Yu. N. Chelnokov, “Application of quaternions in theory of orbital motion of artificial satellite. II,” Cosmic Res. 31 (6), 409–418 (1992).
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).
S. M. Onishchenko, Hypercomplex Numbers in Inertial Navigation Theory (Naukova Dumka, Kyiv, 1983) [in Russian].
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
V. A. Brumberg, Analytical Algorithms of Celestial Mechanics (Nauka, Moscow, 1980) [in Russian].
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
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
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
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).
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).
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
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
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
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
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
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
M. Kopnin, “On the task of rotating a satellite’s orbit plane,” Kosm. Issl. 3 (4), 22–30 (1965).
V. N. Lebedev, Computation of Motion of a Spacecraft with Small Traction (VTs AN SSSR, Moscow, 1967) [in Russian].
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).
G. L. Grodzovskii, Yu. N. Ivanov, and V. V. Tokarev, Mechanics of Space Flight, Optimization Problems (Nauka, Moscow, 1975) [in Russian].
D. E. Okhotsimskii and Yu. G. Sikharulidze, Foundations of Space Flight Mechanics (Nauka, Moscow, 1990) [in Russian].
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).
R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics (AIAA Press, New York, 1987).
Yu. N. Chelnokov, “On regularization of the equations of the three-dimensional two body problem,” Mech. Solids 16 (6), 1–10 (1981).
Yu. N. Chelnokov, “Regular equations of the three-dimensional two body problem,” Mech. Solids 19 (1), 1–7 (1984).
E. L. Stiefel and G. Scheifele, Linear and Regular Celestial Mechanics (Springer, Berlin, 1971).
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
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by I. K. Katuev
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0025654422600787