Formalizing SE(3) with generalized screws in 3D geometric algebra

Abstract In a previous work by Delafosse ( Phys. Scr. 99 056102 2024), a new formalism for Screw Theory based on Geometric Algebra (GA) was introduced to extend the mathematical notion of screw while preserving both geometrical clarity and algebraic efficiency. These generalized screws facilitate the handling of affine geometry, and thus are expected to allow for a natural treatment of finite rigid motions (the SE(3) group), whereas most current mathematical frameworks are either inefficient (like matrix representations) or physically obscure (like dual quaternions). This article aims at providing students and researchers with new geometrical insight into finite motions and their composition by deriving a simple and efficient description thereof in the generalized screw formalism. Notably, the motion composition law assumes a compact form. Finally, the equivalence between this formulation and dual quaternions is discussed, shedding a new light on the geometrical meaning of these hypercomplex numbers.