The hierarchical generalized Voronoi graph (HGVG) is a new roadmap developed for sensor-based exploration in unknown environments. This paper defines the HGVG structure: a robot can plan a path between two locations in its work space or configuration space by simply planning a path onto the HGVG, then along the HGVG, and finally from the HGVG to the goal. Since the bulk of the path planning occurs on the one-dimensional HGVG, motion planning in arbitrary dimensioned spaces is virtually reduced to a one-dimensional search problem. A bulk of this paper is dedicated to ensuring the HGVG is sufficient for motion planning by demonstrating the HGVG (with its links) is an arc-wise connected structure. All of the proofs in this paper that lead toward the connectivity result focus on a large subset of spaces in R³, but wherever possible, results are derived in R^m. In fact, under a strict set of conditions, the HGVG (the GVG by itself) is indeed connected, and hence sufficient for motion planning. The chief advantage of the HGVG is that it possesses an incremental construction procedure, described in a companion paper, that constructs the HGVG using only line-of-sight sensor data. Once the robot constructs the HGVG, it has effectively explored the environment, because it can then use the HGVG to plan a path between two arbitrary configurations.

We present techniques for fast motion planning by using discrete approximations of generalized Voronoi diagrams, computed with graphics hardware. Approaches based on this diagram computation are applicable to both static and dynamic environments of fairly high complexity. We compute a discrete Voronoi diagram by rendering a three-dimensional distance mesh for each Voronoi site. The sites can be points, line segments, polygons, polyhedra, curves and surfaces. The computation of the generalized Voronoi diagram provides fast proximity query toolkits for motion planning. The tools provide the distance to the nearest obstacle stored in the Z-bu er, as well as the Voronoi boundaries, Voronoi vertices and weighted Voronoi graphs extracted from the frame bu er using continuation methods. We have implemented these algorithms and demonstrated their performance for path planning in a complex dynamic environment composed ofmorethan 140,000 polygons.

Exact cellular decompositions are structures that globally encode the topology of a robot's free space, while locally describing the free space's geometry. These structures have been widely used for path planning between two points, but can be used for mapping and coverage of robot free spaces. In this paper, we define exact cellular decompositions where critical points of Morse functions indicate the location of cell boundaries. Morse functions are those whose critical points are non-degenerate. Between critical points, the structure of a space is effectively the same, so simple control strategies to achieve tasks, such as coverage, are feasible within each cell. In this paper, we derive a general framework for defining decompositions in terms of critical points and then give examples, each corresponding to a different task. All of the results in this paper are derived in an m-dimensional Euclidean space, but the examples depicted in the figures are two- and threedimensional for ease ...

This paper prescribes an incremental procedure to construct roadmaps of unknown environments. Recall that a roadmap is a geometric structure that a robot uses to plan a path between two points in an environment. If the robot knows the roadmap, then it knows the environment. Likewise, if the robot constructs the roadmap, then it has effectively explored the environment. This paper focuses on the hierarchical generalized Voronoi graph (HGVG), detailed in the companion paper in this issue. The incremental construction procedure of the HGVG requires only local distance sensor measurements, and therefore the method can be used as a basis for sensor-based planning algorithms. Simulations and experiments using a mobile robot with ultrasonic sensors verify this approach. 1.

Abstract — Proximity query is an integral part of any motion planning algorithm and takes up the majority of planning time. Due to performance issues, most existing planners perform queries at fixed sampled configurations, sometimes resulting in missed collisions. Moreover, randomly determining collision-free configurations makes it difficult to obtain samples close to, or on, the surface of C-obstacles in the configuration space. In this paper, we present an efficient and practical local planning method in contact space which uses “continuous collision detection ” (CCD). We show how, using the precise contact information provided by a CCD algorithm, a randomized planner can be enhanced by efficiently sampling the contact space, as well as by constraining the sampling when the roadmap is expanded. We have included our contact-space planning methods in a freely available stateof-the-art planning library- the Stanford MPK library. We have been able to observe that in complex scenarios involving cluttered and narrow passages, which are typically difficult for randomized planners, the enhanced planner offers up to 70 times performance improvement when our contact-space sampling and constrained sampling methods are enabled.

This article develops an algorithm for a group of guards statically positioned in a nonconvex polygonal environment with holes. Each guard possesses a single searchlight, a ray sensor which can rotate about the guard’s position but cannot penetrate the boundary of the environment. A point is detected by a searchlight if and only if the point is on the ray at some instant. Targets are points which move arbitrarily fast. The objective of the proposed algorithm is to compute a schedule to rotate a set of searchlights in such a way that any target in an environment will necessarily be detected in finite time. This is known as the Searchlight Scheduling Problem and was described originally in 1990 by Sugihara et al. We take an approach known as exact cell decomposition in the motion planning literature. The algorithm operates by decomposing the searchlights ’ joint configuration space and the environment, and then by constructing a so-called information graph. Searching the information graph for a path between desired states yields a search schedule. We also introduce a new problem called the φ-Searchlight Scheduling Problem in which φ-searchlights sense not just along a ray, but over a finite field of view. We show that our results for searchlight scheduling can be directly extended for φ-searchlight scheduling. Proofs of completeness, complexity bounds, and computed examples are presented.

Emerging applications for networked and cooperative robots motivate the study of motion coordination for groups of agents. For example, it is envisioned that groups of agents will perform a variety of useful tasks including surveillance, exploration, and environmental monitoring. This paper deals with basic interactions among mobile agents such as “move away from the closest other agent” or “move toward the furthest vertex of your own Voronoi polygon.” These simple interactions amount to distributed dynamical systems because their implementation requires only minimal information about neighboring agents. We characterize the close relationship between these distributed dynamical systems and the disk-covering and sphere-packing cost functions from geometric optimization. Our main results are: (i) we characterize the smoothness properties of these geometric cost functions, (ii) we show that the interaction laws are variations of the nonsmooth gradient of the cost functions, and (iii) we establish various asymptotic convergence properties of the laws. The technical approach relies on concepts from computational geometry, nonsmooth analysis, and nonsmooth stability theory.

Sensor-based planning for rod-shaped robots is necessary for the realistic deployment of noncircularly symmetric robots into unknown environments. Whereas circularly symmetric robots have two-dimensional Euclidean configuration spaces, planar rod robots posses three degrees-of-freedom, two for position and one for orientation, and hence have a three-dimensional configuration space, (2). In this work, we define the rod hierarchical generalized Voronoi graph (rod-HGVG) which is a roadmap of the rod's configuration space. Prior work in Voronoibased roadmaps use a retraction of the robot's free space to define the roadmap; here, we break a part the robot's free space into regions where fragments of the roadmap are defined and then connect the fragments. The primary advantage of the rod-HGVG is that it is defined in terms of workspace distance measurements, which makes it amenable to sensor-based planning. This paper also includes a numerical procedure that generates the rod-HGVG edge fragments using only information that is within line of sight of the rod robot. It is worth noting that this procedure does not require an explicit definition of configuration space, i.e., this procedure constructs a roadmap of rod configuration space without ever constructing the configuration space itself. Index Terms---Generalized Voronoi graph, retractions, rodshaped robots, sensor-based motion planning, Voronoi diagrams. I.

The hierarchical generalized Voronoi graph (HGVG) is a new roadmap developed for sensor based exploration in unknown environments. This paper defines the HGVG structure: a robot can plan a path between two locations in its workspace or configuration space by simply planning a path onto the HGVG, then along the HGVG, and finally from the HGVG to the goal. Since the bulk of the path planning occurs on the one-dimensional HGVG, motion planning in arbitrary dimensioned spaces is virtually reduced to a one-dimensional search problem. A bulk of this paper is dedicated to ensuring the HGVG is sucient for motion planning by demonstrating the HGVG (with its links) is an arc-wise connected structure. All of the proofs in this paper that lead toward the connectivity result focus on a large subset of spaces in R³, but wherever possible, results are derived in R^m. In fact, under a strict set of conditions, the HGVG (the GVG by itself ) is indeed connected, and hence sufficient for motion planning. T...

During the last couple of years much attention has been given to an important ecological problem, namely forest wildfire [1]. In a recent report [5] it has been shown that, starting with very simple assumptions, a Voronoi diagram structure [4], [3] naturally emerges as the landscape geometry generated by the spread out of forest wildfire. In this paper, we provide a tool that may help people locate refuge zones inside a forest fire using a Voronoi growth model. Forest fires is a complex problem with several open questions. In this short communication, we discuss a subproblem that is a component in a more complex analysis of forest fires. Here we assume homogeneous vegetation on a regular topography without a prevailing wind. As a result of this assumption, the fire wavefronts radially spread at a constant velocity from the ignition point sites, fP i g. This assumption, although restrictive, is quite reasonable in many locations. For example, in the Planalto region near Brasilia (Brazil...