A global approach for the redefinition of higher-order flexibility and rigidity

TL;DR

This paper introduces a global method inspired by Sabitov's algorithm to redefine higher-order flexibility and rigidity, resolving classical dilemmas exemplified by mechanisms like the double-Watt and Tarnai's Leonardo structure.

Contribution

It presents a novel global approach for analyzing higher-order flexibility and rigidity, overcoming limitations of local mobility analysis and providing iterative configuration computation.

Findings

Successfully applied to double-Watt mechanisms and Tarnai's Leonardo structure

Determined all configurations of a 3-RPR manipulator with third-order flexion

Developed a joint-bar framework of flexion order 23

Abstract

The famous example of the double-Watt mechanism given by Connelly and Servatius raises some problems concerning the classical definitions of higher-order flexibility and rigidity, respectively, as they attest the cusp configuration of the mechanism a third-order rigidity, which conflicts with its continuous flexion. Some attempts were done to resolve the dilemma but they could not settle the problem. As cusp mechanisms demonstrate the basic shortcoming of any local mobility analysis using higher-order constraints, we present a global approach inspired by Sabitov's finite algorithm for testing the bendability of a polyhedron, which allows us (a) to compute iteratively configurations with a higher-order flexion and (b) to come up with a proper redefinition of higher-order flexibility and rigidity. The presented approach is demonstrated on several examples (double-Watt mechanisms and Tarnai's Leonardo structure). Moreover, we determine all configurations of a given 3-RPR manipulator with a third-order flexion and present a corresponding joint-bar framework of flexion order 23.