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32
A SingleQuery BiDirectional Probabilistic Roadmap Planner with Lazy Collision Checking
, 2001
"... This paper describes a nev probabilistic roadmap (PRM) path planner that is: (1) singlequery instead of precomputing a roadmap covering the entire free space, it uses the tvo input query configurations as seeds to explore as little space as possible; (2) hidirectional it explores the robotis free ..."
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Cited by 120 (4 self)
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This paper describes a nev probabilistic roadmap (PRM) path planner that is: (1) singlequery instead of precomputing a roadmap covering the entire free space, it uses the tvo input query configurations as seeds to explore as little space as possible; (2) hidirectional it explores the robotis free space by concur rently building a roadmap made of tvo trees rooted at the query configurations; (3) adaptive it makes longer steps in opened areas of the free space and shorter steps in cluttered areas; and (4) lazy in checking collision it delays collision tests along the edges of the roadmap until they are absolutely needed. Experimental results shov that this combination of techniques drastically reduces planning times, making it possible to handle difficult problems, including multirobot problems in geometrically complex environments.
On Delaying Collision Checking in PRM Planning  Application To MultiRobot Coordination
 INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH
, 2002
"... This paper describes the foundations and algorithms of a new probabilistic roadmap (PRM) planner that is: singlequery  instead of precomputing a roadmap covering the entire free space, it uses the two input query configurations to explore as little space as possible; bidirectional  it explo ..."
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Cited by 81 (15 self)
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This paper describes the foundations and algorithms of a new probabilistic roadmap (PRM) planner that is: singlequery  instead of precomputing a roadmap covering the entire free space, it uses the two input query configurations to explore as little space as possible; bidirectional  it explores the robot's free space by building a roadmap made of two trees rooted at the query configurations; and lazy in checking collisions  it delays collision tests along the edges of the roadmap until they are absolutely needed. Several observations motivated this strategy: (1) PRM planners spend a large fraction of their time testing connections for collision; (2) most connections in a roadmap are not on the final path; (3) the collision test for a connection is most expensive when there is no collision; and (4) any short connection between two collisionfree configurations has high prior probability of being collisionfree. The strengths of singlequery and bidirectional sampling techniques, and those of delayed collision checking reinforce each other. Experimental results
Spheres, Molecules, and Hidden Surface Removal
, 1996
"... We devise techniques to manipulate a collection of loosely interpenetrating spheres in threedimensional space. Our study is motivated by the representation and manipulation of molecular con gurations, modeled by a collection of spheres. We analyze the sphere model and point toitsfavorable properties ..."
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Cited by 47 (12 self)
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We devise techniques to manipulate a collection of loosely interpenetrating spheres in threedimensional space. Our study is motivated by the representation and manipulation of molecular con gurations, modeled by a collection of spheres. We analyze the sphere model and point toitsfavorable properties that make it more easy to manipulate than an arbitrary collection of spheres. For this special sphere model we present e cient algorithms for computing its union boundary and for hidden surface removal. The e ciency and practicality of our approach are demonstrated by experiments on actual molecule data.
On Fat Partitioning, Fat Covering and the Union Size of Polygons
, 1993
"... The complexity of the contour of the union of simple polygons with n vertices in total can be O(n 2) in general. A notion of fatness for simple polygons is introduced, which extends most of the existing fatness definitions. It is proved that a set of fat polygons with n vertices in total has unio ..."
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Cited by 28 (1 self)
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The complexity of the contour of the union of simple polygons with n vertices in total can be O(n 2) in general. A notion of fatness for simple polygons is introduced, which extends most of the existing fatness definitions. It is proved that a set of fat polygons with n vertices in total has union complexity is O(nloglogn), which is a generalization of a similar result for fat triangles [19]. Applications to several basic problems in computational geometry are given, such as efficient hidden surface removal, motion planning, injection molding, etc. The result is based on a new method to partition a fat simple polygon P with n vertices into O(n) fat convex quadrilaterals, and a method to cover (but not partition) a fat convex quadrilateral with O(1) fat triangles. The maximum overlap of the triangles at any point is two, which is optimal for any coveting of a fat simple polygon by a linear number of fat triangles.
Robust Geometric Computing in Motion
 INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH
, 2002
"... Transforming a geometric algorithm into an effective computer program is a difficult task. This transformation is particularly made hard by the basic assumptions of most theoretical geometric algorithms concerning complexity measures and (more crucially) the handling of robustness issues, namely is ..."
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Cited by 25 (2 self)
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Transforming a geometric algorithm into an effective computer program is a difficult task. This transformation is particularly made hard by the basic assumptions of most theoretical geometric algorithms concerning complexity measures and (more crucially) the handling of robustness issues, namely issues related to arithmetic precision and degenerate input. The paper starts with a discussion of the gap between the theory and practice of geometric algorithms, together with a brief review of existing solutions to some of the problems that this dichotomy brings about. We then turn to an overview of the CGAL project and library. The CGAL project is a joint effort by a number of research groups in Europe and Israel to produce a robust software library of geometric algorithms and data structures. The library is now available for use with significant functionality. We describe the main goals and results of the project. The central part of the paper is devoted to arrangements (i.e., space subdivisions induced by geometric objects) and motion planning. We concentrate on the maps and arrangements part of the CGAL library. Then we describe two packages developed on top of CGAL for constructing robust geometric primitives for motion algorithms.
Dynamic Data Structures for Fat Objects and Their Applications
, 1999
"... We present several efficient dynamic data structures for pointenclosure queries, involving convex fat objects in R2 or R3. Our planar structures are actually fitted for a more general class of objects (fi; ffi)covered objects which are not necessarily convex, see definition below. These stru ..."
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Cited by 23 (10 self)
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We present several efficient dynamic data structures for pointenclosure queries, involving convex fat objects in R2 or R3. Our planar structures are actually fitted for a more general class of objects (fi; ffi)covered objects which are not necessarily convex, see definition below. These structures are more efficient than alternative known structures, because they exploit the fatness of the objects. We then apply these structures to obtain efficient solutions to two problems (i) Finding a perfect containment matching between a set of points and a set of convex fat objects. (ii) Finding a piercing set for a collection of convex fat objects, whose size is optimal up to some constant factor.
New Results on Binary Space Partitions in the Plane
 COMPUT. GEOM. THEORY APPL
, 1994
"... We prove the existence of linear size binary space partitions for sets of objects in the plane under certain conditions that are often satisfied in practical situations. In particular, we construct linear size binary space partitions for sets of fat objects, for sets of line segments where the ra ..."
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Cited by 21 (7 self)
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We prove the existence of linear size binary space partitions for sets of objects in the plane under certain conditions that are often satisfied in practical situations. In particular, we construct linear size binary space partitions for sets of fat objects, for sets of line segments where the ratio between the lengths of the longest and shortest segment is bounded by a constant, and for homothetic objects. For all cases we also show how to turn the existence proofs into efficient algorithms.
3D Vertical Ray Shooting and 2D Point Enclosure, Range Searching, and Arc Shooting Amidst Convex Fat Objects
 COMPUT. GEOM. THEORY APPL
, 1995
"... We present a new data structure for a set of n convex simplyshaped fat objects in the plane, and use it to obtain efficient and rather simple solutions to several problems including (i) vertical ray shooting  preprocess a set K of n nonintersecting convex simplyshaped flat objects in 3space, ..."
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Cited by 20 (4 self)
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We present a new data structure for a set of n convex simplyshaped fat objects in the plane, and use it to obtain efficient and rather simple solutions to several problems including (i) vertical ray shooting  preprocess a set K of n nonintersecting convex simplyshaped flat objects in 3space, whose xyprojections are fat, for efficient vertical ray shooting queries, (ii) point enclosure  preprocess a set C of n convex simplyshaped fat objects in the plane, so that the k objects containing a query point p can be reported efficiently, (iii) boundedsize range searching  preprocess a set C of n convex fat polygons, so that the k objects intersecting a `nottoolarge' query polygon can be reported efficiently, and (iv) boundedsize segment shooting  preprocess a set C as in (iii), so that the first object (if exists) hit by a `nottoolong' oriented query segment can be found efficiently. For the first three problems we construct data structures of size O(s (n) log 3 n)...
Motion Planning for Multiple Robots
, 1998
"... We study the motionplanning problem for pairs and triples of robots operating in a shared workspace containing n obstacles. A standard way to solve such problems is to view the collection of robots as one composite robot, whose number of degrees of freedom is d, the sum of the numbers of degrees o ..."
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Cited by 19 (1 self)
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We study the motionplanning problem for pairs and triples of robots operating in a shared workspace containing n obstacles. A standard way to solve such problems is to view the collection of robots as one composite robot, whose number of degrees of freedom is d, the sum of the numbers of degrees of freedom of the individual robots. We show that it is sufficient to consider a constant number of robot systems whose number of degrees of freedom is at most d \Gamma 1 for pairs of robots, and d \Gamma 2 for triples. (The result for a pair assumes that the sum of the number of degrees of freedom of the robots constituting the pair reduces by at least one if the robots are required to stay in contact; for triples a similar assumption is made. Moreover, for triples we need to assume that a solution with positive clearance exists.) We use this to obtain an O(n d ) time algorithm to solve the motionplanning problem for a pair of robots; this is one order of magnitude faster than what the st...