Geometric Algebra for Multibody Dynamics—Part I: Elements of Geometric Algebra

Abstract Multibody systems are characterized by two distinguishing features: system components undergo finite motions and these components are connected by mechanical joints that impose restrictions on their relative motion. Clearly, kinematics plays a fundamental role in the analysis of these systems: it is required to describe the arbitrary motion of system components and is needed once again to evaluate the finite relative motion of these components where they are connected by mechanical joints. Hence, it is not surprising that numerous kinematics formulations have been used to describe multibody systems, in an effort to achieve simplicity of the formulation, freedom from singularities, and computational efficiency. Geometric algebra is a mathematical framework that provides an elegant, singularity-free description of geometrical entities such as points, lines, and planes. These geometric entities can be moved in space via reflections, translations, rotations, and motions operations that are based on the “geometric product,” a singularity-free mathematical operation that involves algebraic manipulations only. Although it has received little attention in multibody dynamics, geometric algebra is widely used in computer graphics. The goal of this paper is to demonstrate that geometric algebra is a sound basis for describing the kinematics of multibody systems and for performing all required operations on the relevant geometric entities.