A novel geometrical derivation of the Lie product

The Lie product involves two screws, namely, a driver screw (so-called screwer in this paper) and a driven screw (termed screwen in this paper). Thus, in a spatial motion, the screwer performs a displacement of the screwen. Under this context, the Lie product represents the change of the screwen due to the infinitesimal motion performed by the screwer. To this end, this paper sets forth an integrated formulation that brings together a general geometric interpretation of the Lie product and a novel and simple derivation of the Lie product in terms of the involved screws. Moreover, since the formulation is purely geometric, a great deal of physical insight into the description of the motion problem is gained. As a result, the screw partial derivatives, in terms of the involved Lie products, are obtained for three frequently encountered types of kinematic joints, namely, helical, revolute and prismatic. Since the derivation of the Lie product is completely general, this derivation should provide a sound foundation for further applications of the Lie product.