Homogeneous Rigid Body Mechanics with Elastic Coupling

Abstract

Geometric algebra is used in an essential way to provide a coordinate- free approach to Euclidean geometry and rigid body mechanics that fully inte- grates rotational and translational dynamics. Euclidean points are given a homo- geneous representation that avoids designating one of them as an origin of coordi- nates and enables direct computation of geometric relations. Finite displacements of rigid bodies are associated naturally with screw displacements generated by bivectors and represented by twistors that combine multiplicatively. Classical screw theory is incorporated in an invariant formulation that is less ambiguous, easier to interpret geometrically, and manifestly more efficient in symbolic compu- tation. The potential energy of an arbitrary elastic coupling is given an invariant form that promises significant simplifications in practical applications.