Group Theory Can Explain the Mobility of Paradoxical Linkages

Abstract

The present contribution shows that a closer look into the applications of group theory to the kinematics of spatial linkages provides a concise, simple and correct explanation of the mobility of several linkages classified as overconstrained and paradoxical. The analysis readily yields the Bennett linkage as well as families of R-C-R-C and H-C-H-C linkages, all of them overconstrained and paradoxical. Moreover, the analysis not only yields the geometric characteristics of the linkages but also the relationships between the displacements of the different kinematic pairs. Furthermore, these relationships easily provide, when the motion of the linkages starts from a special configuration, the input-output equations. The study of paradoxical linkages are important to robot kinematics because they are inherently more rigid that trivial linkages and they might provide better architectures for serial or parallel manipulators.