Local Kinematic Analysis of Closed Loop Linkages—Mobility, Singularities, and Shakiness

Abstract

The mobility of a linkage is determined by the constraints imposed on its members. The geometric constraints define the configuration space (cspace) variety as the geometric entity in which the finite mobility of a linkage is encoded. The aim of a local kinematic analysis of a linkage is to deduce its finite mobility, in a given configuration, from the local cspace geometry. In this paper, a method for the local analysis is presented adopting the concept of the tangent cone to a variety. The latter is an algebraic variety approximating the cspace. It allows for investigating the mobility in regular as well as singular configurations. The instantaneous mobility is determined by the constraints, rather than by the cspace geometry. Shaky and underconstrained linkages are prominent examples that exhibit a permanently higher instantaneous than finite DOF even in regular configurations. Kinematic singularities, on the other hand, are reflected in a change of the instantaneous DOF. A cspace singularity as a kinematic singularity, but a kinematic singularity may be a regular point of the cspace. The presented method allows to identify cspace singularities. It also reveals the ithorder mobility and allows for a classification of linkages as overconstrained and underconstrained. The method is applicable to general multiloop linkages with lower pairs. It is computationally simple and only involves Lie brackets (screw products) of instantaneous joint screws. The paper also summarizes the relevant kinematic phenomena of linkages.