The robot localization problem in two dimensions

We consider the following problem: given a simple polygon <italic>P</italic> and a star-shaped polygon <italic>V</italic>, find a point (or the set of points) in <italic>P</italic> from which the portion of <italic>P</italic> that is visible is congruent to <italic>V</italic>. The problem arises in the localization of robots using a range-finder—<italic>P</italic> is a map of a known environment, <italic>V</italic> is the portion visible from the robot's position, and the robot must use this information to determine its position in the map. We give a scheme that preprocesses <italic>P</italic> so that any subsequent query <italic>V</italic> is answered in optimal time <italic>O</italic>(<italic>m</italic> + log <italic>n</italic> + A), where <italic>m</italic> and <italic>n</italic> are the number of vertices in <italic>V</italic> and <italic>P</italic>, and A is the number of points in <italic>P</italic> that are valid answers (the output size). Our technique allows us to trade off smoothly between the query time and the preprocessing time or space. We also devise a data structure for output-sensitive determination of the visibility polygon of a query point inside a polygon <italic>P</italic>. We then consider a variant of the localization problem in which there is a maximum distance to which the robot can “see”—this is motivated by practical considerations, and we outline a similar solution for this case. We also show that a single localization query <italic>V</italic> can be answered in time <italic>O</italic>(<italic>mn</italic>) with no preprocessing.

[1]  Kurt Mehlhorn,et al.  Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry , 2012, EATCS Monographs on Theoretical Computer Science.

[2]  Richard Cole,et al.  Shape from Probing , 1987, J. Algorithms.

[3]  Leonidas J. Guibas,et al.  Visibility and intersection problems in plane geometry , 1989, Discret. Comput. Geom..

[4]  Leonidas J. Guibas,et al.  Optimal Point Location in a Monotone Subdivision , 1986, SIAM J. Comput..

[5]  Ingemar J. Cox,et al.  On The Congruence Of Noisy Images To Line Segment Models , 1988, [1988 Proceedings] Second International Conference on Computer Vision.

[6]  Chih-Ming Wang.  Location Estimation and Uncertainty Analysis for Mobile Robots , 1990, Autonomous Robot Vehicles.

[7]  Ingemar J. Cox,et al.  Blanche-an experiment in guidance and navigation of an autonomous robot vehicle , 1991, IEEE Trans. Robotics Autom..

[8]  David G. Kirkpatrick,et al.  Optimal Search in Planar Subdivisions , 1983, SIAM J. Comput..

[9]  A. Mishkin,et al.  Autonomous Navigation and Control of a Mars Rover , 1989 .

[10]  C. M. Wang,et al.  Location estimation and uncertainty analysis for mobile robots , 1988, Proceedings. 1988 IEEE International Conference on Robotics and Automation.

[11]  Leonidas J. Guibas,et al.  Ray Shooting in Polygons Using Geodesic Triangulations , 1991, ICALP.

[12]  Alain Fournier,et al.  Triangulating Simple Polygons and Equivalent Problems , 1984, TOGS.

[13]  J. B. Kruskal,et al.  Determining the 2- or 3-dimensional similarity transformation between a point set and a model made of lines and arcs , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[14]  Bernard Chazelle.  Triangulating a simple polygon in linear time , 1991, Discret. Comput. Geom..

[15]  C. N. Shen,et al.  Autonomous navigation to provide long-distance surface traverses for Mars Rover Sample Return Mission , 1989, Proceedings. IEEE International Symposium on Intelligent Control 1989.

[16]  Bernard Chazelle.  An Improved Algorithm for the Fixed-Radius Neighbor Problem , 1983, Inf. Process. Lett..