A DUAL VECTORS BASED FORMALISM FOR PARAMETRIZATION OF RIGID BODY DISPLACEMENT AND MOTION

Abstract

This paper reveals a set of dual vectors based methods for rigid body displacement and motion parameterization. When parameterization methods are designed, a very important objective is to obtain is a reduced number of algebraic equations and fewer variables for a more compact notation. This feature is achieved by acknowledging that parameterization of the rigid body motion is a problem strongly connected with the definition and properties of proper orthogonal dual tensors. Tensor analysis expresses the invariance of the laws of physics with respect to the change of basis and change of frame operations. First, we propose a method for computing orthogonal dual tensors based on the dual vectors derived from the motion laws of both points and lines attached to the rigid body. The rigid body motion parameterization using dual vectors gives the possibility of constructing new computational methods for the screw axis (SA) and instantaneous screw axis (ISA) motion parameters. These methods are based on two results: the computation of SA is equivalent to the computation of the logarithm of an orthogonal dual tensor, the computation of ISA is equivalent with the computation of an algebraic entity entitled ”the velocity dual tensor”.