Abstract

The finite degree of freedom (DOF) of a mechanism is determined by the number of independent loop constraints. In this paper a method is introduced to determine the maximal number of loop closure constraints (which is independent of a specific configuration) of multi-loop linkages and is applied to calculate the finite DOF. It rests on an algebraic condition on the joint screws and the corresponding computational algorithm to determine the maximal rank of the constraint Jacobian in an arbitrary (possibly singular) reference configuration, making use of the analytic condition that minors of certain rank and their higher derivatives vanish. Unlike the Lie group methods for estimation the DOF of so-called exceptional linkages, this method does not rely on partitioning kinematic loops into partial kinematic chains, and it is applicable to multi-loop linkages. The DOF computed with this method is at least as accurate as the DOF computed with the Lie group methods. It gives the correct DOF for any (possibly overconstrained) linkage where the constraint Jacobian has maximal rank in regular configurations. The so determined maximal rank has further significance for classifying linkages as being exceptional or paradoxical, but also for detecting singularities and shaky linkages.

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