Discontinuously movable seven-link mechanisms via group-algebraic approach

Combining planar and spherical 4R chains generates three kinds of new discontinuously movable (DM) seven-link mechanisms. The mobility analysis is based on algebraic concepts of set theory. These mechanisms are called hybrid planar-spherical 7R, hybrid spherical-spherical 7R, and hybrid planar-planar 6R1P DM mechanisms. Their discontinuous mobility is explained employing the Lie group algebraic properties of the displacement set. Moreover, the same given spatial arrangement of joints can be linked in two ways constituting two distinct chains, which have a quite different mobility. One chain has two global degrees of freedom (d.o.f.), which disobey the general Grubler-Kutzbach mobility criterion. The other chain exhibits a singular pose, which is a bifurcation towards two distinct working modes of mobility with one d.o.f. Then the set of relative motions between any two specific links is not manifold but consists of the union of displacement one-dimensional manifolds. In general, when two or more subgroups are involved in a closed-loop chain, the discontinuous mobility can happen by changing the order of the kinematic pair linking