The visibility-Voronoi complex and its applications

We introduce a new type of diagram called the VV(c)-diagram (the visibility-Voronoi diagram for clearance c), which is a hybrid between the visibility graph and the Voronoi diagram of polygons in the plane. It evolves from the visibility graph to the Voronoi diagram as the parameter c grows from 0 to ∞. This diagram can be used for planning natural-looking paths for a robot translating amidst polygonal obstacles in the plane. A natural-looking path is short, smooth, and keeps--where possible--an amount of clearance c from the obstacles. The VV(c)-diagram contains such paths. We also propose an algorithm that is capable of preprocessing a scene of configuration-space polygonal obstacles and constructs a data structure called the VV-complex. The VV-complex can be used to efficiently plan motion paths for any start and goal configuration and any clearance value c, without having to explicitly construct the VV(c)-diagram for that c-value. The preprocessing time is O(n2 logn), where n is the total number of obstacle vertices, and the data structure can be queried directly for any c-value by merely performing a Dijkstra search. We have implemented a CGAL-based software package for computing the VV(c)-diagram in an exact manner for a given clearance value and used it to plan natural-looking paths in various applications.

[1]  Micha Sharir,et al.  Retraction: A new approach to motion-planning , 1983, STOC.

[2]  Ron Wein,et al.  High-Level Filtering for Arrangements of Conic Arcs , 2002, ESA.

[3]  David M. Mount,et al.  An output sensitive algorithm for computing visibility graphs , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[4]  Menelaos I. Karavelas.  Segment Voronoi diagrams in CGAL , 2004 .

[5]  Chee-Keng Yap,et al.  A "Retraction" Method for Planning the Motion of a Disc , 1985, J. Algorithms.

[6]  Dan Halperin,et al.  Advanced programming techniques applied to Cgal's arrangement package , 2007, Comput. Geom..

[7]  Mark H. Overmars,et al.  Eurographics/ACM SIGGRAPH Symposium on Computer Animation (2004) , 2022 .

[8]  Jur P. van den Berg,et al.  The visibility--voronoi complex and its applications , 2005, EuroCG.

[9]  Hans Rohnert.  Moving a disc between polygons , 2005, Algorithmica.

[10]  Jirí Matousek,et al.  Piecewise linear paths among convex obstacles , 1995, Discret. Comput. Geom..

[11]  Dan Halperin,et al.  Code Flexibility and Program Efficiency by Genericity: Improving Cgal's Arrangements , 2004, ESA.

[12]  Lydia E. Kavraki,et al.  Probabilistic roadmaps for path planning in high-dimensional configuration spaces , 1996, IEEE Trans. Robotics Autom..

[13]  J. Sack,et al.  Handbook of computational geometry , 2000 .

[14]  Stephen G. Kobourov,et al.  Drawing with Fat Edges , 2001, Graph Drawing.

[15]  D. T. Lee,et al.  Generalization of Voronoi Diagrams in the Plane , 1981, SIAM J. Comput..

[16]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[17]  Joseph O'Rourke,et al.  Handbook of Discrete and Computational Geometry, Second Edition , 1997 .

[18]  Steven Fortune,et al.  Voronoi Diagrams and Delaunay Triangulations , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[19]  Howie Choset,et al.  Complete sensor-based coverage with extended-range detectors: a hierarchical decomposition in terms of critical points and Voronoi diagrams , 2001, Proceedings 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems. Expanding the Societal Role of Robotics in the the Next Millennium (Cat. No.01CH37180).

[20]  Michel Pocchiola,et al.  The visibility complex , 1993, SCG '93.

[21]  Joseph S. B. Mitchell,et al.  Shortest Paths and Networks , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[22]  Yunhui Liu,et al.  Finding the shortest path of a disc among polygonal obstacles using a radius-independent graph , 1995, IEEE Trans. Robotics Autom..

[23]  Micha Sharir,et al.  On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles , 1986, Discret. Comput. Geom..

[24]  Mark H. Overmars,et al.  Useful cycles in probabilistic roadmap graphs , 2004, IEEE International Conference on Robotics and Automation, 2004. Proceedings. ICRA '04. 2004.

[25]  Mark H. Overmars,et al.  Clearance based path optimization for motion planning , 2004, IEEE International Conference on Robotics and Automation, 2004. Proceedings. ICRA '04. 2004.

[26]  Pankaj K. Agarwal,et al.  Polygon decomposition for efficient construction of Minkowski sums , 2000, Comput. Geom..