Position detection of an autonomous robot by aperiodic tilings

We propose an aperiodic tiling suitable for a floor pattern to provide an autonomous robot with position information. The most direct approach to this problem would be to equip the robot with an optical sensor to detect and count its traversals across the lines of a rectangular grid painted on the floor. However, dirt and other sensing errors would result in missing counts and imprecise information of its absolute position. Our approach replaces the rectangular grid by aperiodic tilings similar to Penrose tilings. These patterns may have rotational symmetries, but no nontrivial translational symmetries. By scanning a small area the robot can gather information about its position, where the precision of the information is increasing as the scanned area increases, using an optical sensor. The position information is encoded into the local configuration of the tiles, i.e. into the way the tiles are put together locally. Scanning a larger neighbourhood will increase the precision of the coordinate information. Advantages of this approach include scalability and low cost. Further we discuss a practical implementation, an application to 3-D object model recovery from 2-D images using structured light, and an extension to three dimensional position detection, where the two dimensional tiles are replaced by three dimensional analogs