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). The latter are manifested by a drop of rank of the constraint Jacobian while the c-space is locally a smooth manifold. Another kinematic feature that is only due to the corank of the constraint Jacobian, i.e. of the joint screw system, is the shakiness of a mechanism. 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.