Analytical determination of principal twists in serial, parallel and hybrid manipulators using dual vectors and matrices

The determination of principal twists of the end-effector of a multi-degree-of-freedom manipulator plays a central role in their analysis, design, motion planning and determination of singularities. Most approaches to obtain principal twists and the distributions of twists, such as the well-known classical results of cylindroid and hyperboloid, are based on geometric reasoning and involve intuitive choice of coordinate systems. In this paper, we present a formal algebraic approach to obtain the principal twists of any multi-degree-of-freedom serial, parallel or hybrid manipulator, by making use of the algebra of dual numbers, vectors and matrices. We present analytical expressions for the principal twists and the pitches for any manipulator of arbitrary degree-of-freedom. A consequence of our approach is that we can obtain analytical expressions for the screws along which a manipulator can lose or gain degrees-of-freedom at a singularity. The theoretical results are illustrated with the help of examples of parallel and hybrid manipulators.