Manipulation of motion via dual entities

Abstract

The manipulation of motion and ancillary operations are important tasks in kinematics, robotics, and rigid and flexible multibody dynamics. Motion can be described in purely geometric terms, based on Chasles’ theorem. Representations and parameterizations of motion are also available, such as Euler motion parameters and the vectorial parameterization, respectively. Typical operations to be performed on motion involve the selection of local or global parameterizations and the derivation of the associated expressions for the motion tensor, velocity or curvature vector, composition of motions, and tangent tensors. Many of these tasks involve arduous, error-prone algebra. The use of dual entities has been shown to ease the manipulation of motion, yet this concept has received little attention outside of the fields of kinematics and robotics. This paper presents a comprehensive treatment of the topic using a notation that eliminates the bookkeeping parameter typically used in dual number algebra, thereby recasting all operations within the framework of linear algebra and streamlining the process. The manipulation of geometric entities is recast within this formalism, paving the way for the manipulation of motion. All developments are presented within the framework of dual numbers directly; the principle of transference is never invoked: The manipulation of rotation is a particular case of that of motion, as should be. The problem of interpolation of motion, a thorny issue in finite element applications, is also addressed.