Distance-Optimal Navigation in an Unknown Environment Without Sensing Distances

Abstract

Abstract-This paper considers what can be accomplished using a mobile robot that has limited sensing. For navigation and mapping, the robot has only one sensor, which tracks the directions of depth discontinuities. There are no coordinates, and the robot is given a motion primitive that allows it to move toward discontinuities. The robot is incapable of performing localization or measuring any distances or angles. Nevertheless, when dropped into an unknown planar environment, the robot builds a data structure, called the Gap Navigation Tree, which enables it to navigate optimally in terms of Euclidean distance traveled. In a sense, the robot is able to learn the critical information contained in the classical shortest-path roadmap, although surprisingly it is unable to extract metric information. We prove these results for the case of a point robot placed into a simply connected, piecewise-analytic planar environment. The case of multiply connected environments is also addressed, in which it is shown that further sensing assumptions are needed. Due to the limited sensor given to the robot, globally optimal navigation is impossible; however, our approach achieves locally optimal (within a homotopy class) navigation, which is the best that is theoretically possible under this robot model.

Optimal Gap Navigation for a Disc Robot

“…The robot has an omnidirectional gap sensor [10,18], which is able to detect and track two types of discontinuities in depth information: discontinuities from far to near and discontinuities from near to far (in the counterclockwise direction along ∂ E). Figure 1(c) shows a robot in an environment in which the gap sensor detects some near-to-far and far-to-near gaps.…”

“…Let V = (v n , v n−1 , ..., v 0 ) be the sequence of intervals v i ⊂ ∂ C obtained by transforming the interval sequence U from ∂ E to ∂ C , element by element. The following lemma uses the definition of a generalized bitangent from [18]. Proof.…”

“…Therefore, the resulting global path is optimal. ⊓ ⊔ Constructing a complete GNT: In [18], it has been proved that the construction of the GNT for a point robot will terminate (Lemmas 2 and 3 in [18]). Incompleteness of the GNT is caused by any non-primitive leaves (leaves that correspond to the portions of the environment that have not been perceived by the robot).…”

“…Instead, it observes the world using mainly a gap sensor, introduced in [18], which allows it to determine the directions of discontinuities in depth (distance to the boundary) and move toward any one of those directions. Under this model, but for a point robot, a combinatorial filter called the Gap Navigation Tree (GNT) was introduced that encodes precisely the part of the shortest-path visibility graph that is needed for optimal navigation [18]. The learned data structure corresponds exactly to the shortest path tree [6] from the robot's location.…”

“…2. The robot first learns the GNT by executing a learning phase, which is described in [9,18], and remains unchanged in this paper. The process involves iteratively chasing "unknown" gaps, causing each to split or disappear.…”