Kinematical analysis of overconstrained and underconstrained mechanisms by means of computational algebraic geometry

We investigate and explain several exceptional phenomena appearing in mechanism kinematics. The starting point for the kinematical analysis of a mechanism is the formation of the relevant constraint map defining the constraint equations for the coordinates of the particular system. The constraint equations define the configuration space of the mechanism, which reveals the essential kineamtic characteristics. But in some cases the properties of the map, and not the configuration space itself, are important. This is true for example for so called under- and overconstrained mechanisms for which the standard formulation of constraints gives usually not enough or too many constraints when considering the dimension of their configuration space. These concepts also naturally lead to the concept of kinematotropic mechanisms which posses motion modes of different dimension. In this context the concept of a kinematotropy as a motion between such modes is introduced in this paper. We present a general approach to the kinematic analysis of mechanisms using the theory of algebraic geometry and tools of computational algebraic geometry. The configuration space is considered as a real algebraic variety defined by the constraints. The phenomena and needed theory are explained and several illustrative examples are given. In particular the underconstrained phenomenon is explained by considering the real and complex dimension of the configuration space variety.