Conformal Geometric Algebra for Robotic Vision

Abstract

In this paper the authors introduce the conformal geometric algebra in the field of visually guided robotics. This mathematical system keeps our intuitions and insight of the geometry of the problem at hand and it helps us to reduce considerably the computational burden of the problems. As opposite to the standard projective geometry, in conformal geometric algebra we can deal simultaneously with incidence algebra operations (meet and join) and conformal transformations represented effectively using spinors. In this regard, this framework appears promising for dealing with kinematics, dynamics and projective geometry problems without the need to resort to different mathematical systems (as most current approaches do). This paper presents real tasks of perception and action, treated in a very elegant and efficient way: body---eye calibration, 3D reconstruction and robot navigation, the computation of 3D kinematics of a robot arm in terms of spheres, visually guided 3D object grasping making use of the directed distance and intersections of lines, planes and spheres both involving conformal transformations. We strongly believe that the framework of conformal geometric algebra can be, in general, of great advantage for applications using stereo vision, range data, laser, omnidirectional and odometry based systems.