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Quaternion and Biquaternion Methods and Regular Models of Analytical Mechanics (Review)

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Abstract

This paper is an analytical review in which we present quaternion and biquaternion methods for describing motion, models of the theory of finite displacements and regular kinematics of a rigid body based on the use of four-dimensional real and dual Euler (Rodrigues–Hamilton) parameters. These models, in contrast to the classical models of kinematics in Euler–Krylov angles and their dual counterparts, do not have division-by-zero features and do not contain trigonometric functions, which increases the efficiency of analytical research and numerical solution of problems in mechanics, inertial navigation, and motion control.

The problem of regularization of differential equations of the perturbed spatial two-body problem, which underlies celestial mechanics and space-flight mechanics (astrodynamics), is discussed using the Euler parameters, four-dimensional Kustaanheimo–Stiefel variables, and Hamilton quaternions; this problem consists in eliminating the singularities (division by zero) which are generated by the Newtonian gravitational forces acting on a celestial or cosmic body and which complicate the analytical and numerical study of the motion of a body near gravitating bodies or its motion along highly elongated orbits. The history of the regularization problem and the regular Kustaanheimo–Stiefel equations, which have found wide application in celestial mechanics and astrodynamics, are presented. We present the quaternion methods of regularization, which have a number of advantages over Kustaanheimo–Stiefel matrix regularization, and various regular quaternion equations of the perturbed spatial two-body problem (for both absolute and relative motion), which are useful for predicting and correcting the orbital motion of celestial and cosmic bodies.

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Notes

  1. Difficulties associated with near-collisions also arise in problems of classical celestial mechanics, in particular, in the study of the motion of a comet in the sphere of action of a large planet (Ed. note by V.A. Brumberg).

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Funding

This work was supported by the Russian Scientific Foundation (project no. 22-21-00218).

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

    Yu. N. Chelnokov

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Chelnokov, Y.N. Quaternion and Biquaternion Methods and Regular Models of Analytical Mechanics (Review). Mech. Solids 58, 2450–2477 (2023). https://doi.org/10.3103/S0025654423070051

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