EQUIVALENCE OF EULERS AND CAYLEY-KLEINS VARIABLES IN THE KINEMATICS OF A RIGID BODY

To increase the accuracy and speed of the numerical analysis of motion of a rigid body or a multibody system (especially if the region of possible phase states of the system is not known in advance), one has to replace the angular variables by variables whose kinematic equations do not degenerate in the entire phase space. Direction cosines (associated with the Poisson equations) [1, 2], Cayley-Kleinʹs variables [3-5], Rodrigues-Hamiltonʹs variables [5, 6], Eulerʹs variables [7], and Rodriguesʹ variables [1] can serve as examples. The Poisson equations are overdelermined. Therefore, because of round-off errors, numerical integration of these equations can lead to violation of the condition for the basis vectors to be orthonormal. In this sense, all other kinds of variables seem to be more preferable. However, the properties of Eulerʹs variables [8], Cayley-Kleinʹs variables, and Rodriguesʹ variables have been investigated less thoroughly than, for example, the properties of quaternions [2, 4, 6, 7, 9]. Although all these groups of variables are very close to each other in their nature, traditionally only quaternions are widely used in practice. However, other variables also have advantages. For example, Eulerʹs variables permit one to use conventional vector operations in a 3D space, Cayley-Kleinʹs variables make it possible lo use only addition and multiplication of matrices, and Rodriguesʹ variables provide the possibility of operating with non-normalized quaternions. The literature devoied to this topic is rather extensive and has a long history. Analysis of this literature reveals a lot of various applications of these kinematic characteristics. The present paper does not claim a discovery of new properties of the variables under consideration. We develop an approach which permits one lo establish an isomorphism between groups S0(3), SU(2) and quaternions so as to avoid tedious transformations in deriving formulas and equations of the rigid body kinematics. This approach is based on using a special sort of skew-Hermitian matrices. As a result, the exposition of the corresponding theory becomes comprehensive and compact.