A New Mobility Formula for Spatial Mechanisms

Abstract

A new formula for predicting the mobility of spatial mechanisms is introduced. Instead of counting rigid links and the constraints between them, as is done in the usual Gru¨bler-Kutzbach formulae, we count vertices and edges in a polyhedral model of the mechanism. It is well known that the conventional formula underpredicts the mobility of certain exceptional classes of mechanisms, and in particular, does not easily accommodate compound spatial mechanisms that contain planar or spherical sub-mechanisms. The new approach provides a correct mobility whenever the conventional formula does and accounts for planar and spherical sub-mechanisms in a simple manner. Additionally, we present simple modifications to correctly model certain mechanisms that have remote spherical centers. We illustrate the method on compound mechanisms constructed from scissors linkages.