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Prediction and Correction of the Orbital Motion of Spacecraft using Regular Quaternion Equations and Their Solutions in the Kustaanheimo–Stiefel Variables and Isochronic Derivatives

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Abstract

The regular quaternion equations of the orbital motion of a spacecraft (SC) proposed by us earlier in four-dimensional Kustaanheimo–Stiefel variables (KS variables) are considered. These equations use as a new independent variable a variable related to real time by a differential relation (Sundman time transformation) containing the distance to the center of gravity. Various new regular quaternion equations in these variables and equations in regular quaternion osculating elements (slowly changing variables) are also constructed, in which the half-generalized eccentric anomaly, widely used in celestial mechanics and space-flight mechanics, is used as a new independent variable. Keplerian energy and time are used as additional variables in these equations. These equations are used to construct quaternion equations and relations in variations of KS variables and their first derivatives and in variations of Keplerian energy and real time; the isochronous derivatives of the KS variables and of their first derivatives and the matrix of isochronous derivatives for the elliptical Keplerian motion of the spacecraft are found, which are necessary for solving the problems of predicting and correcting its orbital motion. The results of a comparative study of the accuracy of the numerical integration of the Newtonian equations of the spatial restricted three-body problem (the Earth, the Moon, and a spacecraft) in Cartesian coordinates and the regular quaternion equations of this problem in KS variables are presented, which show that the accuracy of the numerical integration of regular quaternion equations is much higher (by several orders) than the accuracy of numerical integration of the equations in Cartesian coordinates. This substantiates the expediency of using regular quaternion equations of the orbital motion of a spacecraft and the quaternion equations and relations in the variations constructed in the paper on their basis for the prediction and correction of the orbital motion of a spacecraft.

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Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct thisparticular research were obtained.

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Authors and Affiliations

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

    Yu. N. Chelnokov, Ya. G. Sapunkov & M. Yu. Loginov

  2. AO TsNIIMash, Korolev, Moscow region, Russia

    A. F. Schekutev

Authors
  1. Yu. N. Chelnokov
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  2. Ya. G. Sapunkov
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  3. M. Yu. Loginov
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  4. A. F. Schekutev
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Correspondence to Yu. N. Chelnokov or A. F. Schekutev.

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Translated by E. Seifina

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Chelnokov, Y.N., Sapunkov, Y.G., Loginov, M.Y. et al. Prediction and Correction of the Orbital Motion of Spacecraft using Regular Quaternion Equations and Their Solutions in the Kustaanheimo–Stiefel Variables and Isochronic Derivatives. Mech. Solids 58, 2478–2503 (2023). https://doi.org/10.3103/S0025654423070063

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