Numerical algebraic geometry and kinematics

Abstract

Numerical algebraic geometry uses numerical methods, principally numerical tracking of paths defined by polynomial homotopies, to find and manipulate algebraic sets defined by systems of polynomial equations. Kinematics is the study of the geometrical aspects of mechanical motion. The kinematical problems arising in the analysis and design of most robots and mechanisms are essentially algebraic, because these devices are well-modeled as rigid bodies in contact along algebraic surfaces. In particular, the constraints imposed by the most common types of joints, such as simple hinges or ball-and-socket joints, are equivalent to containments of linear features (points, lines, and planes) that are maintained during rigid body motion of the parts. Kinematical studies have driven the development of numerical algebraic geometry and remain one of its most important application areas. Numerical algebraic geometry has proven to be particularly apt for the natural parameterizations presented by problems from kinematics. This extended abstract gives brief overviews of basic numerical algebraic geometry and kinematics.