DescriptionIn their 1995 paper titled The Robot Localization Problem, Guibas, Motwani, andRaghavan present an idealized method for localizing a mobile robot equipped with a LiDAR and compass in a polygonal map. By constructing a notion of equivalence between visibility polygons, a partition on the map is formed by grouping related points into polygonal visibility cells, which are then organized into a searchable data structure. In this thesis, we make a series of modifications to their approach that makes it suitable for use in a live robotic system, accounting for angular uncertainty, sensor noise, and occlusions in the map. Rather than searching for exact correspondences, we build a robust fingerprint for each visibility cell by recording its set of visible map edges parametrized by (r, θ) pairs. When a query LiDAR scan is received, the lines in the LiDAR image are extracted and an approximate visibility fingerprint is constructed. By using the (r, θ) parametrization, we compress the search space, and decouple the 3 degrees of freedom search into a series of simpler 1D correlations. We then present the output and runtime of our implementation on a number of synthetic maps with added sensor noise and occlusions to demonstrate its viability in a live system.