Deeper understanding of the homography decomposition for vision-based control

The displacement of a calibrated camera between two images of a planar object can be estimated by decomposing a homography matrix. The aim of this document is to propose a new method for solving the homography decomposition problem. This new method provides analytical expressions for the solutions of the problem, instead of the traditional numerical procedures. As a result, expressions of the translation vector, rotation matrix and object-plane normal are explicitly expressed as a function of the entries of the homography matrix. The main advantage of this method is that it will provide a deeper understanding on the homography decomposition problem. For instance, it allows to obtain the relations among the possible solutions of the problem. Thus, new vision-based robot control laws can be designed. For example the control schemes proposed in this report combine the two final solutions of the problem (only one of them being the true one) assuming that there is no a priori knowledge for discerning among them