Necessary and sufficient mobility conditions for single-loop overconstrained nH mechanisms

• Local approximation of the closure conditions of nH mechansims
• Formulation of necessary conditions for finite mobility of nH mechansims
• Numerical estimation of a sufficient approximation order for finite mobility

For overconstrained single-loop mechanisms with helical joints (nH mechanisms) necessary conditions for the finite mobility are derived using a local approximation of the loop closure conditions by means of a Taylor-series expansion of the 1st-order closure conditions. The loop closure conditions and their derivatives are formulated in terms of the screw coordinates of the helical joint axes, whereby the series terms are recursively expressed. Although it has been shown by means of the Artin approximation theorem that there exists a finite local approximation order mmax that guarantees finite mobility of the mechanism, the value of this approximation order is in general unknown. For a numerical estimation for mmax an algorithm for the solution of the m nonlinear mobility conditions has been developed to predict the sufficient local approximation order mnum for various nH mechanisms. The procedure enables the synthesis of overconstrained nH mechanisms.