Incremental Construction of the Generalized Voronoi Diagram, the Generalized Voronoi Graph, and the Hierarchical Generalized Voronoi Graph

1 Overview This work introduces a new incremental construction procedure for the generalized Voronoi diagram (GVD), the locus of points equidistant to two obstacles. This procedure is incremental in that it requires line of sight information (see Figure 4) to construct the GVD. Note that this notion of incremental is di erent from previous de nition of incremental where the GVD is constructed by inserting one obstacle at a time into the environment. Finally, unlike other GVD construction techniques, this procedure does not place any special restrictions on the types of obstacles; that is, obstacles need not be polygons nor convex sets. The incremental construction procedure was originally developed for sensor based planning of highly articulated robots. Sensor based planning incorporates sensor information into a robot's planning procedure, in contrast to classical planning which assumes full knowledge of the environment prior to planning. When a robot has no a priori information about the environment, the robot must employ an incremental motion planner because most environments do not contain one vantage point from which a robot can \see" the entire world, and thereby construct a plan or representation from such a single vantage point. Although the incremental construction procedure was designed for when the robot has no a priori knowledge of the environment, it is also useful when full or partial knowledge of the environment is available to the robot. The GVD serves as a roadmap structure for a mobile robot operating in the plane. Roadmaps are geometric structures that capture the full topology of a robot's environment and have three key properties: accessibility, connectivity, and departability. Motion planning is achieved by planning a path onto the GVD (accessibility), planning a path in the GVD to the vicinity of the goal (connectivity) and then planning a path to the goal (departability). The GVD is useful in motion planning because the bulk of the motion planning occurs in a one-dimensional subset of the robot's two-dimensional environment. When a robot incrementally constructs the entire GVD for an environment, it has essentially explored that environment. The incremental construction procedure is not limited to mobile robot applications, but rather it applies