Quaternion regularization in celestial mechanics and astrodynamics and trajectory motion control. I

Chelnokov, Yu. N.

doi:10.1134/s001095251305002x

Cited by 23 publications

(12 citation statements)

References 6 publications

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“…In the first part of this work [2], the problem of reg ularizating differential equations was considered for a perturbed three dimensional two body problem: How best to eliminate a singularity that arises in two body problem equations when the second body collides with the central body (distance r between them being equal to zero). Such a singularity at the origin of the coordi nates creates both theoretical and practical (comput ing) difficulties.…”

Section: Eliminating Singularities In Equations Of Celestial Mechanicmentioning

confidence: 99%

“…The same is true of models describing in angular variables the instantaneous ori 1 This work is a review based on materials of the plenary session report Quaternion Regularization in Trajectory Motion Control and Astrodynamics, presented at the 10th National Meeting on Fundamental Problems of Theoretical and Applied Mechanics [1] (Section I, "General and Applied Mechanics"). entation of an orbit or the orbital plane of a celestial body or spacecraft.…”

Section: Eliminating Singularities In Equations Of Celestial Mechanicmentioning

confidence: 99%

“…(7.9), (7.10), (7.12), (7.13), and (7.16) in the form of regular Levi-Civita equations for a planar two body problem. Note also that the above subsystem of equations can be formally derived if we set in the regular Kustaanheimo-Stiefel equations (1.4)-(1.6) presented in [2].…”

Section: Regular Equations Of a Three Dimensional Three Body Problem mentioning

confidence: 99%

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Quaternion regularization and trajectory motion control in celestial mechanics and astrodynamics: II

Chelnokov

2014

Cosmic Res

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Problems of regularization in celestial mechanics and astrodynamics are considered, and basic regular quaternion models for celestial mechanics and astrodynamics are presented. It is shown that the effec tiveness of analytical studies and numerical solutions to boundary value problems of controlling the trajectory motion of spacecraft can be improved by using quaternion models of astrodynamics. In this second part of the paper, specific singularity type features (division by zero) are considered. They result from using classical equations in angular variables (particularly in Euler variables) in celestial mechanics and astrodynamics and can be eliminated by using Euler (Rodrigues-Hamilton) parameters and Hamilton quaternions. Basic regu lar (in the above sense) quaternion models of celestial mechanics and astrodynamics are considered; these include equations of trajectory motion written in nonholonomic, orbital, and ideal moving trihedrals whose rotational motions are described by Euler parameters and quaternions of turn; and quaternion equations of instantaneous orbit orientation of a celestial body (spacecraft). New quaternion regular equations are derived for the perturbed three dimensional two body problem (spacecraft trajectory motion). These equations are constructed using ideal rectangular Hansen coordinates and quaternion variables, and they have additional advantages over those known for regular Kustaanheimo-Stiefel equations.

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Section: Eliminating Singularities In Equations Of Celestial Mechanicmentioning

confidence: 99%

Section: Eliminating Singularities In Equations Of Celestial Mechanicmentioning

confidence: 99%

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Quaternion regularization and trajectory motion control in celestial mechanics and astrodynamics: II

Chelnokov

2014

Cosmic Res

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“…Various aspects of quaternion regularization of differential equations of the perturbed spatial two-body problem by using the KS variables have been studied by Velte [1] , Vivarelli [2][3][4] , Shagov [5] , Deprit et al [6] , Vrbik [7][8] , Waldvogel [9][10] , Saha [11] , Zhao [12] , Roa et al [13][14] , Roa and Pelaez [15] , Breiter and Langner [16][17][18] , Ferrer and Crespo [19] , and also by the author of this paper, Chelnokov [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36] .…”

Section: Introductionmentioning

confidence: 99%

“…Section 3 describes the regular equations of the perturbed spatial two-body problem in quaternion osculating (i.e., slowly changing) elements, corresponding to the KS variables and their first derivatives with respect to fictitious time. These equations were derived by Chelnokov [28,30] from quaternion regular equations in the KS variables by the method of variation of arbitrary constants of integration. This section also covers regular equations of the perturbed spatial two-body problem derived by Chelnokov [21] for the case when the essence of transformations of the original equations of this problem stays the same, but the KS variables are not used.…”

Section: Introductionmentioning

confidence: 99%

Quaternion methods and models of regular celestial mechanics and astrodynamics

Chelnokov

2022

Appl. Math. Mech.-Engl. Ed.

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This paper is a review, which focuses on our work, while including an analysis of many works of other researchers in the field of quaternionic regularization. The regular quaternion models of celestial mechanics and astrodynamics in the Kustaanheimo-Stiefel (KS) variables and Euler (Rodrigues-Hamilton) parameters are analyzed. These models are derived by the quaternion methods of mechanics and are based on the differential equations of the perturbed spatial two-body problem and the perturbed spatial central motion of a point particle. This paper also covers some applications of these models. Stiefel and Scheifele are known to have doubted that quaternions and quaternion matrices can be used efficiently to regularize the equations of celestial mechanics. However, the author of this paper and other researchers refuted this point of view and showed that the quaternion approach actually leads to efficient solutions for regularizing the equations of celestial mechanics and astrodynamics.This paper presents convenient geometric and kinematic interpretations of the KS transformation and the KS bilinear relation proposed by the present author. More general (compared with the KS equations) quaternion regular equations of the perturbed spatial two-body problem in the KS variables are presented. These equations are derived with the assumption that the KS bilinear relation was not satisfied. The main stages of the quaternion theory of regularizing the vector differential equation of the perturbed central motion of a point particle are presented, together with regular equations in the KS variables and Euler parameters, derived by the aforementioned theory. We also present the derivation of regular quaternion equations of the perturbed spatial two-body problem in the Levi-Civita variables and the Euler parameters, developed by the ideal rectangular Hansen coordinates and the orientation quaternion of the ideal coordinate frame.This paper also gives new results using quaternionic methods in the perturbed spatial restricted three-body problem.

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Quaternion regularization in celestial mechanics, astrodynamics, and trajectory motion control. III

Chelnokov

2015

Cosmic Res

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Quaternion regularization in celestial mechanics and astrodynamics and trajectory motion control. I

Cited by 23 publications

References 6 publications

Quaternion regularization and trajectory motion control in celestial mechanics and astrodynamics: II

Quaternion regularization and trajectory motion control in celestial mechanics and astrodynamics: II

Quaternion methods and models of regular celestial mechanics and astrodynamics

Quaternion regularization in celestial mechanics, astrodynamics, and trajectory motion control. III

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