Singularities Analysis of Basic Kinematic Chains and Complex Multiloop Planar Linkages

Abstract This paper offers a general approach for the singularity analysis of arbitrary complex multiloop planar linkages. A complex linkage is regarded as one composed of the input links and one or more zero degree-of-freedom basic kinematic chains. A complex linkages under a different input condition may decompose some different types of basic kinematic chains, and different linkages may correspond to the same basic kinematic chains. A general treatment for the singularity analysis of complex multiloop planar linkages is presented, i.e, firstly, decompose a complex linkage into basic kinematic chains (BKCs), then, based on the velocity matrix, deduce the analytical singularity conditions for these BKCs, and last, obtain the singular positions of linkages by the numerical solution method, or by the corresponding singular geometric configurations derived from singularity equations, i.e. the graphic solution method. These basic kinematic chains then served as the modules or building blocks of singularity analysis for complex linkages and manipulators. Meanwhile, three important conclusions for singularity analysis have been drawn: (1)The analytical singularity conditions for a BKC are not related with its assembly configurations; (2) The singular positions for a linkages depend on the type of BKC s and the parameters of the links; (3). When only one basic kinematic chain reaches a singular configuration, the complex linkage is at a singular position. Thus, the method of singularity analysis for a limited number of basic kinematic chains can be expanded to all complex planar linkages and parallel manipulators, and the singularity analysis of a complex linkage can simplified to singularity analysis of their BKCs. This paper also offers and discusses a formula to estimate the maximum of singular positions of a complex multiloop linkage. The approach is demonstrated in a Stephenson complex linkages with six-bar and two-loop.