Identification of Real and Complex Solution Varieties and Their Singularities Defined by Loop Constraints of Linkages Using the Kinematic Tangent Cone

Abstract

The configuration space (c-space) of a mechanism is the real-solution variety of a set of loop closure constraints, which is therefore the chief object in kinematic analysis. Singularities of this variety (referred to as c-space singularities) are singular configurations of the mechanism. In addition, a mechanism may exhibit other kinematic singularities that are not visible from the differential geometry of the c-space (referred to as hidden singularities). Such situations were analyzed by investigating the local geometry of the c-space and its corank stratification. It has been shown recently that hidden singularities and shakiness can be attributed to the fact that complex solution branches intersect with the c-space, i.e., with real-solution branches. This paper employs the kinematic tangent cone to identify local solution branches. While the kinematic tangent cone is an established generally applicable concept, which gives rise to a computational (numeric and symbolic) algorithm, it has yet only been applied to analyzing the real-solution set. Application of the method is shown for several examples. Further, the algebraic aspects are briefly elaborated.