On singularities of real algebraic sets and applications to kinematics
The description of mechanisms via algebraic equations has made kinematics one of the most important fields of application for algebraic geometry. Mechanism science gives rise to important questions that can be answered using algebraic methods and, in turn, also promotes the development of new computational tools in (real) algebraic geometry. In the past, mechanism science had no problems in coming up with very exotic singularities in natural examples and with demands on computational algebraic geometry that pushes their algorithms to the limits. Singularities is a traditional topic of algebraic geometry that also plays an important role in the kinematics of mechanisms. However, full understanding of the behavior of a mechanism in a kinematic singularity of its configuration variety requires a detailed investigation. It not only depends on the type of singularity but also on features of the mechanism. On one end of the spectrum, a singularity may render the mechanism completely unusable, on the other end, it may not affect its kinematics at all. This article presents algorithmically feasible criteria for the identification of non-manifold points in the real part of an affine variety. The necessary theoretical results from real algebraic geometry are described in detail and illustrated with concrete examples and code. If the variety under investigation is a mechanism's configuration variety, the non-manifold points are called CS-singularities. The author demonstrates the usefulness of his approach by showing that all singularities of the well-known four-bar linkage are CS-singularities and that the singularities of the delta robot (a parallel robot with industrial applications and three degrees of freedom) are either isolated or CS as well.
scientific article; zbMATH DE number 7435591
scientific article