Optimal robot localization in trees

The problem of localization, i.e. of a robot finding its pcrsition on a map, is an important task for autonomous mobile robots. It has applications in numerous areas of robotics ranging from aerial photography to autonomous vehicle exploration. In this paper we present a new strategy for a robot to find its position on a map where the map is represented as a geometric tree. Our strategy exploits to a high degree the self-similarities that may occur in the environment. We use the framework of competitive analysis to analyze the performance of our strategy. In particular, we show that the distance traveled by the robot is at most O(W) times longer than the shortest possible route to localize the robot, where n is the number of vertices of the tree. This is a significant improvement over the best known previous bound of O(n2f3). Moreover, since there is a lower bound of f2(@), our strategy is optimal up to a constant factor. Using the same approach we can also show that the problem of searching for a target in a geometric tree, where the robot is given a map of the tree and the location of the target but does not know its own position, can be solved by a strategy with a competitive ratio of O(W), which is again optimal up to a constant factor. *The first author is supported by DIMACS. DIMAW is au NSF Scienceand Technology Center funded under contract STC88-09648 and also receives support from the New Jersey Conunission on Science rmd Technology. The second author is supported by the Deutsche Forschungsgemeinscbaft ~der grant No. Ot 64/8-1, “DMrrete Probleme”, under grants of the Natural Sciences and Engineering Research Council of Canada, from the Information Technology Research Centre of Ontario, and by an NSERC international fellowship. tDIMACS, Rutgers University, Piscataway, NJ 08855-1179, email: romanik@dimacs. rutgers. edu t Iustitut fii Iuformatik, Universitiit Freibu.rg, Am Flughafen 17, Geb. 051, D-79I1o PYeiburg, Fed. Rep. of G. rmany, e-mail: sohuiere@inforrnat ik. uni-freiburg. de Permission to make digitsllhard copies of all or part of rhk material for personal or classroom use is grantad without fee provided that the copies are not ~de or dhibuted for profit or commercial advantage, the copyrigbt notice, the title of the publication and ita date appear, and notice is given that copyright is by permission of ~e ACM, Inc. To copy othe~iao, to republish, to post on servers or to redMribute to lists, requiresapeclfic permissionandlor fee. Computational Geometry’96, Philadelphia PA, USA Q 1996 ACM &89791 -g~-5/96/t)5. .$3.50 Sven Schuierer$