Visually Guided Robotics Using Conformal Geometric Computing

Classical Geometry, as conceived by Euclid, was a plataform from which Mathematics started to build its actual form. However, since the XIX century, it was a language that was not evolving as the same pase as the others branches of Physics and Mathematics. In this way, analytic, non-Euclidean and projective geometries, matrix theory, vector calculus, complex numbers, rigid and conformal transformations, ordinary and partial differential equations, to name some, are different mathematical tools which are used nowadays to model and solve almost any problem in robotic vision, but the presence of the classical geometric theory in such solutions is only implicit. However, over the last four decades a new mathematical framework has been developed as a new lenguage where not only the classical geometry is included, but where many of these mathematical systems will be embedded too. Instead of using different notation and theory for each of those systems, we will simplify the whole study introducing the CGA, a unique mathematical framework where all those systems are embedded, gaining in principle clarity and simplicity. Moreover, incidence algebra operations as union and intersection of subspaces, are also included in this system through the meet and join operations. In this regard, CGA appears promising for dealing with kinematics, dynamics and projective geometric problems in one and only one mathematical framework. In this chapter we propose simulated and real tasks for perception-action systems, treated in a unify way and using only operations and geometrical entities of this algebra. We propose applications to follow geometric primitives or ruled sufaces with an arm’s robot for shape understanding and object manipulation, as well as applications in visual grasping. But we believe that the use of CGA can be of great advantage in visually guided robotics using stereo vision, range data, laser, omnidirectional or odometry based systems.