 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12854 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4810 |
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| Yu.N. Chelnokov, "Quaternion Regularization of Singularities of Astrodynamics Models Generated by Gravitational Forces (Review)," Mech. Solids. 58 (8), 2855-2883 (2023) |
| Year |
2023 |
Volume |
58 |
Number |
8 |
Pages |
2855-2883 |
| DOI |
10.3103/S0025654423080071 |
| Title |
Quaternion Regularization of Singularities of Astrodynamics Models Generated by Gravitational Forces (Review) |
| Author(s) |
Yu.N. Chelnokov (Institute of Precision Mechanics and Control Problems, Russian Academy of Sciences, Saratov, 410028 Russia, ChelnokovYuN@gmail.com) |
| Abstract |
The article presents an analytical review of works devoted to the quaternion regularization
of the singularities of differential equations of the perturbed two-body problem generated by gravitational forces, using the four-dimensional Kustaanheimo–Stiefel variables. Most of these works have
been published in leading foreign publications. We consider a new method of regularization of these
equations proposed by us, based on the use of two-dimensional ideal rectangular Hansen coordinates,
two-dimensional Levi-Civita variables, and four-dimensional Euler (Rodrigues–Hamilton) parameters. Previously, it was believed that it was impossible to generalize the famous Levi-Civita regularization of the equations of plane motion to the equations of spatial motion. The regularization proposed
by us refutes this point of view and is based on writing the differential equations of the perturbed spatial
problem of two bodies in an ideal coordinate system using two-dimensional Levi-Civita variables to
describe the motion in this coordinate system (in which the equations of spatial motion take the form
of equations of plane motion) and based on the use of the quaternion differential equation of the inertial orientation of the ideal coordinate system in the Euler parameters, which are the osculating elements of the orbit, as well as on the use of Keplerian energy and real time as additional variables, and
on the use of the new independent Sundmann variable. Reduced regular equations, in which Levi-Civita variables and Euler parameters are used together, not only have the well-known advantages of
equations in Kustaanheimo–Stiefel variables (regularity, linearity in new time for Keplerian motions,
proximity to linear equations for perturbed motions), but also their own additional advantages: 1) two-dimensionality, and not four-dimensionality, as in the case of Kustaanheimo–Stiefel, a single-frequency harmonic oscillator describing in new time in Levi-Civita variables the unperturbed elliptic
Keplerian motion of the studied (second) body, and 2) slow change in the new time of the Euler
parameters, which describe the change in the inertial orientation of the ideal coordinate system, for
perturbed motion, which is convenient when using the methods of nonlinear mechanics. This work
complements our review paper [1]. |
| Keywords |
space flight mechanics (astrodynamics), perturbed spatial two-body problem, regularization of singularities generated by gravitational forces, ideal coordinate system, equations of orbital motion, Kustaanheimo–Stiefel variables, Euler (Rodrigues–Hamilton) parameters, Hansen coordinates, Levi-Civita variables, quaternion |
| Received |
04 September 2023 | Revised |
11 October 2023 | Accepted |
15 October 2023 |
| Link to Fulltext |
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