Learning combinatorial map information from permutations of landmarks

This paper considers a robot that moves in the plane and that is able to sense only the cyclic order of landmarks with respect to its current position. No metric information is available regarding the robot or landmark positions; moreover, the robot does not have a compass or odometers (i.e., knowledge of coordinates). We carefully study the information space of the robot, and establish its capabilities in terms of mapping the environment and accomplishing tasks, such as navigation and patrolling. When the robot moves exclusively inside the perimeter of the set of landmarks, the information space may be succinctly characterized as an order type that provides information powerful enough to determine which points lie inside the convex hulls of subsets of landmarks. In addition, if the robot is allowed to move outside the perimeter of the set of landmarks, the information space is described with a swap cell decomposition, that is, an aspect graph in which each aspect is a cyclic permutation of landmarks. Finally, we show how to construct such decomposition through its dual, using two kinds of feedback motion commands based on the landmarks sensed.