René Bartkowiak

Numerical Synthesis of Overconstrained Mechanisms Based on Screw Theory

An approach for the synthesis of overconstrained single-loop mechanisms with helical joints using screw theory is presented. By expressing higher-order derivatives of the screw axes with respect to the joint coordinates a local approximation of the closure conditions is obtained. The approximation is used to define conditions for the finite mobility of overconstrained mechanisms.

Sufficient Conditions for the Mobility of Overconstrained Mechanisms

A single-loop mechanism with n≤6 helical joints which has a finite mobility with a degree of freedom f≥1 is overconstrained. Here the case f=1 is considered. For the finite mobility of an overconstrained mechanism it is sufficient, that the first-order closure condition is fulfilled in an open neighborhood of the actual position of the joint axes. Since the explicit functional dependencies of the closure condition are unknown, a Taylor-series expansion yields necessary conditions for the mobility of the mechanism. According to Alexandrov (1998) it is sufficient for the finite mobility of the mechanism if all higher-order mobility conditions up to a maximum, but unknown, order m are fulfilled. Special solutions for necessary mobility conditions are discussed which guarantee that all higher-order necessary conditions are fulfilled. By this these special solutions are sufficient for finite mobility. Some of this special mobility conditions are presented.