Hierarchical Motion Planning Using a Spatial Index

We investigate the problem of constructing a shortest path of a point{like robot between two conngurations in the euclidean plane cluttered with (intersecting) convex polygonal obstacles. One common approach is to construct the visibility graph and search within this graph in a total time of O(n 2). We show that in general it is not necessary to construct the entire visibility graph. In contrast, we develop two hierarchical motion{planning techniques based on the monotonous bisector tree and the visibility graph, which are shown to be more eecient in scenes of low object density. We show that in our setting the visibility graph can be incrementally constructed in time O(n 2 log n), where n is the complexity of the scene. A shortest path can then be constructed in time O(l 4 log l+n log l), if a shortest path of length l exists. The dependency on n can be further reduced by the use of a spatial data structure complying a query assumption to O(l 4 log l + l 2 log n). If l is not small compared to the diameter D S of the scene, the algorithm still respects an O(n 2 (log n+log D S)) upper bound. If no path exists, then similar estimations for the diameter of a set of obstacles certifying the failure can be stated. We report on experimental results verifying the eeciency of the methods.