Mechanism mobility and a local dimension test

The mobility of a mechanism is the number of degrees of freedom (DOF) with which it may move. This notion is mathematically equivalent to the dimension of the solution set of the kinematic loop equations for the mechanism. It is well known that the classical Grübler–Kutzbach formulas for mobility can be wrong for special classes of mechanisms, and even more refined treatments based on displacement groups fail to correctly predict the mobility of so-called “paradoxical” mechanisms. This article discusses how recent results from numerical algebraic geometry can be applied to the question of mechanism mobility. In particular, given an assembly configuration of a mechanism and its loop equations, a local dimension test places bounds on the mobility of the associated assembly mode. A publicly available software code makes the idea easy to apply in the kinematics domain.

Research Highlights
► Mobility, degrees of freedom (DOFs), and dimension of the solution set of a mechanism's loop equations are equivalent concepts.
► A mechanism can have several assembly modes with different mobilities, which correspond to local dimension.
► Thus, a local dimension test from numerical algebraic geometry gives mobility information.
► An analysis of higher derivatives can distinguish between finite and infinitesimal DOFs.