Fundamental equation of mechanism kinematic geometry: Mapping curve in se(3) to counterpart in SE(3)

Abstract

The most general continuous rigid motion can be described by a curve in SE(3) of whose tangent vector the left or right invariant representation is a curve in se(3). The integral operation between them has not been achieved because there is no general solution for a system of the first-order linear differential equations with variable coefficients in mathematics. This paper develops a matrix equation (termed as the fundamental equation) based on the geometric properties of a pair of conjugate axodes equivalent to a curve in se(3) in algebras to achieve this integral operation. Moreover, the first-order and second-order derivations of the fundamental equation are derived. Furthermore, an algebraic method representing the body and spatial velocity twists as vector functions of dimensions and input parameters of mechanisms is founded on the theory of reciprocal screws. After that, this method and the fundamental equation are validated by the numerical examples of Bennett, spherical and planar four-bar linkages. Finally, this work presents a notion of generalized-involute which is a useful tool in the study of gear tooth profile and cam profile.