Isometric visualization of configuration spaces of two-degrees-of-freedom mechanisms

In this work we introduce an intuitive global representation of the dynamical properties of two-degrees-of-freedom mechanisms as a surface in R3. The surface, which we formally call an isometric visualization, represents the mechanismʹs configuration-space such that the Euclidean metric of the surface matches the kinetic-energy metric of the mechanism. Since a freely moving mechanism follows a path which minimizes the integral of its kinetic energy, the free motions of the mechanism appear on the isometric visualization surface as curves of minimal length, called geodesics. The geodesic curves can be located on the isometric visualization surface using intuitive criteria, allowing a qualitative study of the mechanism dynamics. We describe closed-form formulas for the isometric visualization surface of planar and spatial two-degrees-of-freedom open chain mechanisms. Then render these surfaces for spatial two-degrees-of-freedom chains whose joint axes are either parallel or perpendicular to each other. We also present several tools for inferring global properties of a mechanismʹs dynamics from its isometric visualization. These properties include a notion of stability of a free motion, and dynamical equivalence of mechanisms.