Semi-Riemannian Geometry and Rolling

Abstract

The main goal of this thesis is modelling rolling without slipping or twisting. To this end we must gain a better understanding of semi-Riemannian geometry. The central object of study in a semi-Riemannian geometry is the semi-Riemannian manifold which is a smooth manifold on which we have defined a notion of length. This length is not necessarily positive, but can in fact also be negative. Semi-Riemannian geometry is in fact the natural context to model rolling in. Now we want to model rolling without slipping or twisting. So we need some way to encode these conditions. We do this in terms of distribution. Lastly we take a look at control systems. The main goal of this part is to show that the no slipping distribution, which models rolling without slipping, is bracket generating. By that we mean that given two objects, we can achieve any configuration we like by rolling without slipping.