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The Power Crust
, 2001
"... The power crust is a construction which takes a sample of points from the surface of a threedimensional object and produces a surface mesh and an approximate medial axis. The approach is to first approximate the medial axis transform (MAT) of the object. We then use an inverse transform to produce ..."
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Cited by 267 (7 self)
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The power crust is a construction which takes a sample of points from the surface of a threedimensional object and produces a surface mesh and an approximate medial axis. The approach is to first approximate the medial axis transform (MAT) of the object. We then use an inverse transform to produce the surface representation from the MAT.
MorseSmale Complexes for Piecewise Linear 3Manifolds
, 2003
"... We define the MorseSmale complex of a Morse function over a 3manifold as the overlay of the descending and ascending manifolds of all critical points. In the generic case, its 3dimensional cells are shaped like crystals and are separated by quadrangular faces. In this paper, we give a combinatori ..."
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Cited by 133 (28 self)
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We define the MorseSmale complex of a Morse function over a 3manifold as the overlay of the descending and ascending manifolds of all critical points. In the generic case, its 3dimensional cells are shaped like crystals and are separated by quadrangular faces. In this paper, we give a combinatorial algorithm for constructing such complexes for piecewise linear data.
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 82 (17 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
A Framework for Using the Workspace Medial Axis in PRM Planners
, 2000
"... Probabilistic roadmap planners have been very successful in path planning for a wide variety of problems, especially applications involving robots with many degrees of freedom. These planners randomly sample the configuration space, building up a roadmap that connects the samples. A major problem is ..."
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Cited by 80 (4 self)
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Probabilistic roadmap planners have been very successful in path planning for a wide variety of problems, especially applications involving robots with many degrees of freedom. These planners randomly sample the configuration space, building up a roadmap that connects the samples. A major problem is finding valid configurations in tight areas, and many methods have been proposed to more effectively sample these regions. By constructing a skeletonlike subset of the free regions of the workspace, these heuristics can be strengthened. The skeleton provides a concise description of the workspace topology and an efficient means of finding points with maximal clearance from the obstacles. We examine the medial axis as a skeleton, including a method to compute an approximation to it. The medial axis is a twoequidistant surface in the workspace. We form a heuristic for finding difficult configurations using the medial axis, and demonstrate its effectiveness in a planner for rigid objects in a three dimensional workspace.
On the probabilistic foundations of probabilistic roadmap planning
 In Proc. Int. Symp. on Robotics Research
, 2005
"... Probabilistic roadmap (PRM) planners [5, 16] solve apparently difficult motion planning problems where the robot’s configuration space C has dimensionality six or more, and the geometry of the robot and the obstacles is described by hundreds of thousands of triangles. While an algebraic planner woul ..."
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Cited by 63 (11 self)
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Probabilistic roadmap (PRM) planners [5, 16] solve apparently difficult motion planning problems where the robot’s configuration space C has dimensionality six or more, and the geometry of the robot and the obstacles is described by hundreds of thousands of triangles. While an algebraic planner would be overwhelmed by the high cost of computing an exact representation of the free space F, defined as the collisionfree subset of C, a PRM planner builds only an extremely simplified representation of F, called a probabilistic roadmap. This roadmap is a graph, whose nodes are configurations sampled from F with a suitable probability measure and whose edges are simple collisionfree paths, e.g., straightline segments, between the sampled configurations. PRM planners work surprisingly well in practice, but why? Previous work has partially addressed this question by identifying and formalizing properties of F that guarantee good performance for a PRM planner using the uniform sampling measure (e.g.,
Soft robotics: Biological inspiration, state of the art, and future research
, 2008
"... Publication details, including instructions for authors and subscription information: ..."
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Cited by 52 (2 self)
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Approximating the Medial Axis from the Voronoi Diagram with a Convergence Guarantee
 Algorithmica
, 2004
"... The medial axis of a surface in 3D is the closure of all points that have two or more closest points on the surface. It is an essential geometric structure in a number of applications involving 3D geometric shapes. Since exact computation of the medial axis is difficult in general, efforts continue ..."
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Cited by 46 (8 self)
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The medial axis of a surface in 3D is the closure of all points that have two or more closest points on the surface. It is an essential geometric structure in a number of applications involving 3D geometric shapes. Since exact computation of the medial axis is difficult in general, efforts continue to improve their approximations. Voronoi diagrams turn out to be useful for this approximation. Although it is known that Voronoi vertices for a sample of points from a curve in 2D approximate its medial axis, similar result does not hold in 3D. Recently, it has been discovered that only a subset of Voronoi vertices converge to the medial axis as sample density approaches infinity. However, most applications need a nondiscrete approximation as opposed to a discrete one. To date no known algorithm can compute this approximation straight from the Voronoi diagram with a guarantee of convergence. We present such an algorithm and its convergence analysis in this paper. One salient feature of the algorithm is that it is scale and density independent. Experimental results corroborate our theoretical claims.
Realtime path planning for virtual agents in dynamic environments
 PROC. OF IEEE VR
, 2007
"... We present a novel approach for realtime path planning of multiple virtual agents in complex dynamic scenes. We introduce a new data structure, Multiagent Navigation Graph (MaNG), which is constructed from the first and secondorder Voronoi diagrams. The MaNG is used to perform route planning and ..."
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Cited by 42 (13 self)
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We present a novel approach for realtime path planning of multiple virtual agents in complex dynamic scenes. We introduce a new data structure, Multiagent Navigation Graph (MaNG), which is constructed from the first and secondorder Voronoi diagrams. The MaNG is used to perform route planning and proximity computations for each agent in real time. We compute the MaNG using graphics hardware and present culling techniques to accelerate the computation. We also address undersampling issues for accurate computation. Our algorithm is used for realtime multiagent planning in pursuitevasion and crowd simulation scenarios consisting of hundreds of moving agents, each with a distinct goal.
Finding narrow passages with probabilistic roadmaps: The small step retraction method
 in Proc. IEEE/RSJ Int. Conf. on Intelligent Robots & Systems
, 2005
"... Abstract: Probabilistic Roadmaps (PRM) have been successfully used to plan complex robot motions in configuration spaces of small and large dimensionalities. However, their efficiency decreases dramatically in spaces with narrow passages. This paper presents a new method – smallstep retraction – tha ..."
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Cited by 37 (4 self)
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Abstract: Probabilistic Roadmaps (PRM) have been successfully used to plan complex robot motions in configuration spaces of small and large dimensionalities. However, their efficiency decreases dramatically in spaces with narrow passages. This paper presents a new method – smallstep retraction – that helps PRM planners find paths through such passages. This method consists of slightly “fattening ” robot’s free space, constructing a roadmap in fattened free space, and finally repairing portions of this roadmap by retracting them out of collision into actual free space. Fattened free space is not explicitly computed. Instead, the geometric models of workspace objects (robot links and/or obstacles) are “thinned ” around their medial axis. A robot configuration lies in fattened free space if the thinned objects do not collide at this configuration. Two repair strategies are proposed. The “optimist ” strategy waits until a complete path has been found in fattened free space before repairing it. Instead, the “pessimist ” strategy repairs the roadmap as it is being built. The former is usually very fast, but may fail in some pathological cases. The latter is more reliable, but not as fast. A simple combination of the two strategies yields an integrated planner that is both fast and reliable. This planner was implemented as an extension of a preexisting singlequery PRM planner. Comparative tests show that it is significantly faster (sometimes by several orders of magnitude) than the preexisting planner. 1.
Samplingbased roadmap of trees for parallel motion planning
 IEEE Trans. Robot
, 2005
"... Abstract — This paper shows how to effectively combine a samplingbased method primarily designed for multiple query motion planning (Probabilistic Roadmap Method PRM) with samplingbased tree methods primarily designed for single query motion planning (Expansive Space Trees, RapidlyExploring Rand ..."
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Cited by 36 (10 self)
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Abstract — This paper shows how to effectively combine a samplingbased method primarily designed for multiple query motion planning (Probabilistic Roadmap Method PRM) with samplingbased tree methods primarily designed for single query motion planning (Expansive Space Trees, RapidlyExploring Random Trees, and others) in a novel planning framework that can be efficiently parallelized. Our planner not only achieves a smooth spectrum between multiple query and single query planning but it combines advantages of both. We present experiments which show that our planner is capable of solving problems that cannot be addressed efficiently withPRM or singlequery planners. A key advantage of our planner is that it is significantly more decoupled thanPRM and samplingbased tree planners. Exploiting this property, we designed and implemented a parallel version of our planner. Our experiments show that our planner distributes well and can easily solve highdimensional problems that exhaust resources available to single machines and cannot be addressed with existing planners. Index Terms — Motion planning, samplingbased planning, parallel