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IssuesArchive of Issues2022-5pp.961-983

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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. I: Review and Analysis of Methods and Models and Their Applications," Mech. Solids. 57 (5), 961-983 (2022)
Year 2022 Volume 57 Number 5 Pages 961-983
DOI 10.3103/S0025654422050041
Title 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
Author(s) Yu.N. Chelnokov (Institute of Precision Mechanics and Control Problems of the Russian Academy of Sciences, Saratov, 410028 Russia, ChelnokovYuN@gmail.com)
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.
Keywords regularization, equations of celestial mechanics and astrodynamics (space flight mechanics), two-body problem, orbit orientation, Euler (Rodrigues–Hamilton) parameters, orientation quaternion, optimal control of orbital motion, spacecraft, spatial motion of a rigid body, biquaternion
Received 14 October 2021Revised 05 December 2021Accepted 07 December 2021
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