Dual Lie Algebra Representations of Rigid Body Motion With Dual Cayley Maps: An Overview

Abstract The main objective of this research is to develop a new minimal parameterization technique for the displacement and motion of rigid bodies using hypercomplex dual algebra. Our study is based on the properties of dual tensors and dual quaternions, more precisely, their Lie groups and algebras. Based on the higher-order modified fractional Cayley transforms, for the first time, a complete and unitary parameterization framework, which gives the possibility of developing direct unitary solutions for the calculation of the leading entities of kinematic representation of displacement and motion of rigid body: Euler dual vector, higher-order Rodrigues dual vector, dual quaternions, dual sin family parameters and dual orthographic projection. The representation is coordinate-free and in a closed-form.