A new look into the kinematics and dynamics of finite rigid body rotations using Lie group theory

Abstract

A systematic theoretical approach is presented, in an effort to provide a complete and illuminating study on kinematics and dynamics of rigid bodies rotating about a fixed point. Specifically, this approach is based on some fundamental concepts of differential geometry, with particular reference to Lie group theory. This treatment is motivated by the form of the configuration space corresponding to large rigid body rotation, which is a differentiable manifold possessing group properties. First, the basic steps of the classical approach on the subject are briefly summarized. Then, some geometrical tools are presented, which are essential for supporting and illustrating the steps and findings of the new approach. Finally, the emphasis is placed on a thorough investigation of the problem of finite rotations. A key idea is the introduction of a canonical connection, matching the manifold and group properties of the configuration space. This proves to be sufficient and effective for performing the kinematics. Next, following the selection of an appropriate metric, the dynamics is also carried over. The present approach is theoretically more demanding than the traditional treatments in engineering but brings substantial benefits. In particular, an elegant interpretation is provided for all the quantities with fundamental importance in both rigid body kinematics and dynamics. Most importantly, this also leads to a correction of some misconceptions and geometrical inconsistencies in the field. Among other things, the deeper understanding of the theoretical concepts provides powerful insight and a strong basis for the development of efficient numerical techniques in problems of solid and structural mechanics involving large rotations.