New kinematic parameters of finite rotation of a solid

Abstract

A new family of kinematic (global and local) orientation parameters of a solid is presented and described. All kinematic parameters are obtained by the method of mapping the variables onto the corresponding oriented subspace (hyperplane). In particular, the method of stereographic projection of a point of a five-dimensional sphere S6 ⊂ R6 onto the oriented hyperplane R5 is presented for the classical direction cosines of the angles determining the orientation of two coordinate systems. A family of global kinematic parameters obtained by the method of mapping of five-dimensional kinematic Hopf parameters given in the space R5 onto the four-dimensional oriented subspace R4 is described. The theorem of the homeomorphism of two topological spaces (the four-dimensional sphere S4 ⊂R5 with one deleted point and the oriented hyperplane R4) is used to establish the correspondence between five- and four-dimensional kinematic parameters defined in the corresponding spaces. It is also shown what global four-dimensional orientation parameters, quaternions defined in the subspace R4 are associated with the classical local parameters, i.e., the Rodrigues and Gibbs three-dimensional finite rotation vectors. The projection method is used to obtain the kinematic differential equations (KDE) of rotation corresponding to the five- and four-dimensional orientation parameters. All above-introduced kinematic orientation parameters of a solid permit efficiently solving the classical Darboux problem by using the corresponding KDE, i.e., determining the body current angular position in the space R3 from the known (measured) angular velocity of the object rotation and its initial position in space.