Motion Planning for a Convex Polygon in a Polygonal Environment

Abstract. We study the motion-planning problem for a convex m -gon P in a planar polygonal environment Q bounded by n edges. We give the first algorithm that constructs the entire free configuration space (the three-dimensional space of all free placements of P in Q ) in time that is near-quadratic in mn , which is nearly optimal in the worst case. The algorithm is also conceptually simple. Previous solutions were incomplete, more expensive, or produced only part of the free configuration space. Combining our solution with parametric searching, we obtain an algorithm that finds the largest placement of P in Q in time that is also near-quadratic in mn . In addition, we describe an algorithm that preprocesses the computed free configuration space so that reachabilityqueries can be answered in polylogarithmic time.

[1]  Micha Sharir,et al.  Planning a purely translational motion for a convex object in two-dimensional space using generalized Voronoi diagrams , 2016, Discret. Comput. Geom..

[2]  Mark de Berg,et al.  On lazy randomized incremental construction , 1994, STOC '94.

[3]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[4]  Micha Sharir,et al.  Computing envelopes in four dimensions with applications , 1994, SCG '94.

[5]  Sivan Toledo,et al.  On critical orientations in the kedem-sharir motion planning algorithm , 1997, CCCG.

[6]  Micha Sharir,et al.  An efficient and simple motion planning algorithm for a ladder moving in two-dimensional space amidst polygonal barriers (extended abstract) , 1985, SCG '85.

[7]  Micha Sharir,et al.  On translational motion planning in 3-space , 1994, SCG '94.

[8]  Leslie G. Valiant,et al.  Parallelism in Comparison Problems , 1975, SIAM J. Comput..

[9]  Mark H. Overmars,et al.  Spheres, molecules, and hidden surface removal , 1994, SCG '94.

[10]  Klara Kedem,et al.  A Convex Polygon Among Polygonal Obstacles: Placement and High-clearance Motion , 1993, Comput. Geom..

[11]  Sivan Toledo,et al.  Extremal polygon containment problems , 1991, SCG '91.

[12]  Jean-Daniel Boissonnat,et al.  A practical exact motion planning algorithm for polygonal objects amidst polygonal obstacles , 1988, Proceedings. 1988 IEEE International Conference on Robotics and Automation.

[13]  Micha Sharir,et al.  Algorithmic Techniques for Geometric Optimization , 1995, Computer Science Today.

[14]  W ShorPeter,et al.  Applications of random sampling in computational geometry, II , 1989 .

[15]  Leonidas J. Guibas,et al.  Computing a face in an arrangement of line segments , 1991, SODA '91.

[16]  Micha Sharir,et al.  The overlay of lower envelopes in three dimensions and its applications , 1995, SCG '95.

[17]  Micha Sharir,et al.  Efficient randomized algorithms for some geometric optimization problems , 1996, Discret. Comput. Geom..

[18]  John Canny,et al.  The complexity of robot motion planning , 1988 .

[19]  Kenneth L. Clarkson,et al.  Applications of random sampling in computational geometry, II , 1988, SCG '88.

[20]  Pankaj K. Agarwal,et al.  Ray shooting and other applications of spanning trees with low stabbing number , 1992, SCG '89.

[21]  Nimrod Megiddo,et al.  Applying parallel computation algorithms in the design of serial algorithms , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[22]  Micha Sharir,et al.  Largest Placement of One Convex Polygon Inside Another , 1998, Discret. Comput. Geom..

[23]  Sivan Toledo,et al.  Applications of parametric searching in geometric optimization , 1992, SODA '92.

[24]  Steven Fortune.  A Fast Algorithm for Polygon Containment by Translation (Extended Abstract) , 1985, ICALP.

[25]  Micha Sharir,et al.  On the number of critical free contacts of a convex polygonal object moving in two-dimensional polygonal space , 1987, Discret. Comput. Geom..

[26]  Roberto Tamassia,et al.  Efficient Point Location in a Convex Spatial Cell-Complex , 1989, SIAM J. Comput..

[27]  Micha Sharir,et al.  Davenport-Schinzel sequences and their geometric applications , 1995, Handbook of Computational Geometry.