Generic mobility of rigid body mechanisms

In this paper the generic DOF of rigid body mechanisms is investigated. This is the DOF of almost all mechanisms that can be built from a given set of kinematic pairs in a certain arrangement, but with arbitrary link geometry. The generic DOF is the most likely DOF in the presence of link imperfections. Furthermore, we are interested in the generic DOF of different types of mechanisms, i.e. when the link geometries are arbitrary, but in accordance with a certain type (e.g. planar, spherical, spatial). A widely used mobility criterion is the extended Chebychev–Kutzbach–Grübler (CKG) formula. It is proven in this paper that the generic DOF of a mechanism comprising loops of specific motion types, such as planar, spherical, and spacial, is indeed given by the extended CKG formula. In particular, mechanisms with arbitrary link geometry have generically a DOF image, where image is the number of fundamental loops, and n is the total number of joint freedoms. That is, almost all mechanisms are trivial (not overconstrained). Moreover, such mechanisms have generically no configuration space singularities. The mobility and the classification of mechanisms as overconstrained, underconstrained, or kinematotropic is considered in view of the configuration space. It is pointed out that the local DOF cannot always be inferred from the number of constraints (overconstrained mechanisms) nor can the differential DOF always be deduced from the local DOF (underconstrained mechanisms).