Algebraic Geometry and Kinematics

Abstract

In this overview paper we show how problems in computational kinematics can be translated into the language of algebraic geometry and subsequently solved using techniques developed in this field. The idea to transform kinematic features into the language of algebraic geometry is old and goes back to Study. Recent advances in algebraic geometry and symbolic computation gave the motivation to resume these ideas and make them successful in the solution of kinematic problems. It is not the aim of the paper to provide detailed solutions, but basic accounts to the used tools and examples where these techniques were applied within the last years. We start with Study’s kinematic mapping and show how kinematic entities can be transformed into algebraic varieties. The transformations in the image space that preserve the kinematic features are introduced. The main topic are the definition of constraint varieties and their application to the solution of direct and inverse kinematics of serial and parallel robots. We provide a definition of the degree of freedom of a mechanical system that takes into account the geometry of the device and discuss singularities and global pathological behavior of selected mechanisms. In a short paragraph we show how the developed methods are applied to the synthesis of mechanical devices.