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

The work is of a survey analytical nature. The first part of the work presents 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: the problem of eliminating 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 Kustaanheim–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). The results of a comparative study of the accuracy of numerical integration of various forms of regularized equations of celestial mechanics and astrodynamics in Kustaanheimo–Stiefel variables and Newtonian equations in Cartesian coordinates are presented, showing that the accuracy of numerical integration of regularized equations in Kustaanheimo–Stiefel variables is much higher (by several orders of magnitude) than the accuracy of numerical integration Newtonian equations.