Reaching a goal with directional uncertainty

We study two problems related to planar motion planning for robots with imperfect control, where, if the robot starts a linear movement in a certain commanded direction, we only know that its actual movement will be confined in a cone of angle a centered around the specified direction. First, we consider a single goal region, namely the "region at infinity", and a set of polygonal obstacles, modeled as a set S of n line segments. We are interested in the region from where we can reach infinity with a directional uncertainty of a. We prove that the maximum complexity of is O(n/a5). Second, we consider a collection of k polygonal goal regions of total complexity m, but without any obstacles. Here we prove an O(k3m) bound on the complexity of the region from where we can reach a goal region with a directional uncertainty of a. For both situations we also prove lower bounds on the maximum complexity, and we give efficient algorithms for computing the regions.