A synthesis method for 1-DOF mechanisms with a cusp in the configuration space

Abstract

Significant progress has been made in the study of singularities of mechanisms. This research has, however, exclusively focused on situations where different motion branches intersect, i.e. bifurcation points of the configuration space (c-space). Other types of singularities have not been studied due to lack of mechanisms examples. In particular, mechanisms exhibiting cusp singularities in their c-space are almost unknown, besides a planar linkage presented by Connelly and Servatius, which served as an example where the common definition of rigidity fails. In this paper, a method for the synthesis of spatial 1-degree-of-freedom (1-DOF) cusp mechanisms is presented. This method consists in synthesizing the mechanical generator of a spatial curve with specific characteristics and then appropriately connecting this module with its mirrored version. Several examples are presented including a kinematotropic linkage, which is characterized by a singularity that is a cusp (1 DOF motion) and a bifurcation of a curve and a surface (2-DOF motion). It is discussed that all available methods for the local analysis of singularities fail at cusp singularities. The presented synthesis method allows for constructing mechanisms that shall initiate the research into the study of cusp singularities.