Geometric Characterization of the Configuration Space of Rigid Body Mechanisms in Regular and Singular Points

Abstract

The kinematics of rigid body mechanisms is considered from a differential-geometric perspective. Geometric properties of a mechanism are intrinsically determined by the topology of its configuration space — the solution set of closure functions. The mechanism kinematics is usually characterized by the tangent space and tangent cone to the configuration space, i.e. by locally considering its topology. There are, however, mechanisms for which this is not sufficient. Generally, beside the topology, a complete picture of the kinematics needs both, the configuration space and the ideal generated by the closure functions. Tangent spaces/cones are differential-geometric objects associated to a variety. Two additional objects are introduced in this paper: the kinematic tangent space and the kinematic tangent cone. Three locally equivalent models for the mechanism kinematics are introduced. Due to their different mathematical nature the different models admit to apply specific mathematical tools. The analysis of model I is based on Lie group and screw algebraic methods, while model II and III are analyzed using methods from algebraic geometry. A computationally efficient algorithm for the construction of the kinematic tangent cone is presented. Its application is shown for several examples. A novel mechanism is presented of which the differential and local degree of freedom are different in regular points, so-called ‘paradox-in-the-small’.