Generic Mobility of Rigid Body Mechanisms: On the Existence of Overconstrained Mechanisms

In this paper the generic DOF of 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. In particular, this is the most likely DOF in the presence of link imperfections. The local DOF of a mechanism is the dimension of its configuration space. The differential DOF is the number of linearly independent velocity constraints. It is pointed out, that the local DOF can not always be inferred from the number of constraints (overconstrained mechanisms) nor can the differential DOF always be deduced from the local DOF. It is proven, that the generic DOF of a mechanism is given by the Chebychev-Kutzbach-Gru¨bler formula $δ=Σαfa−g(j−b+1)$⁠, with b and j the number of bodies and joints, and f_α the DOF of joint α. Consequently, almost all mechanism are trivial, i.e. not overconstrained.