The constraint-wrench analysis of mechanisms, with focus on parallel robots, is the subject of this paper. Although the method proposed here can be generalized for parallel robots with multiple-loop kinematic chains, here, single-loop chains are targeted. To this end, a novel representation of the constraints imposed by the kinematic pairs is introduced. With this representation, the constraint matrix of a mechanism is readily derived. For the calculation of the constraint wrenches, by means of the constraint matrix and based on the Newton–Euler formulation, a new procedure is introduced. As a case study, the constraint wrench analysis of the McGill Schönflies Motion Generator (SMG), while undergoing a test cycle adopted by the industry, is conducted.