Surveillance for a simply connected region: A one-center disk-covering problem

In this paper we address a surveillance task carried out by a single agent that is modeled as a disk. The surveillance problem can be formulated mathematically as a one-center disk-covering problem with the velocity control in which the center of the disk is adjusted continuously to cover a simply connected region for surveillance in R². Different from existing methods that require the global environmental information on the geometric shape of the uncovered surveillance region and the condition on the convexity of the surveillance region, in the proposed method we need only the local environmental information and the connectivity of the surveillance region. The kinematic movement of the disk center is determined in terms of the intersected arc segments between the disk and the surveillance region. In each intersected arc segment, a virtual force is calculated and the magnitude of the force is made proportional to the length of the arc segment. The movement direction of the disk is determined by a normal vector that originates from the origin of the center and heading towards the center point of the arc formed by the leftmost and rightmost points of amongst all intersected arc segments. The magnitude of the overall virtual force is the sum of projection of each virtual force on the movement direction. In this paper we prove that the movement of the disk center will stop only when the surveillance region is fully covered by the disk.