Isometric visualization of configuration spaces of two degrees of freedom mechanisms

Abstract

This paper presents a novel geometrical representation of the dynamical properties of two degrees-of-freedom mechanisms, as a surface called an isometric visualization of the mechanism. The kinematic properties of the mechanism are represented by the surface topology. The dynamical properties of the mechanism are represented by a metric induced on the surface by the mechanism's kinetic energy. Isometric visualization enables an intuitive study of the dynamical properties of a mechanism, since arc length on the isometric visualization surface is determined by the kinetic energy metric. In particular, free motions of the mechanism are represented by curves of minimal arc length in its isometric visualization surface, called geodesics. We consider two degrees of freedom open kinematic chain mechanisms, with prismatic and revolute joints. For these mechanisms, we present formulas for the isometric visualization of all planar mechanisms, and for two classes of spatial mechanisms. Then we render a catalog of isometric visualizations of several basic mechanisms, and present tools for extracting information on the mechanism's dynamics from its isometric visualization. These tools include a measure for the stability of a free motion, based on the Gaussian curvature of the mechanism's isometric visualization surface. This new representation of mechanism configuration spaces has potential uses in mechanism and controller design.