Results 1  10
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21
Highlevel filtering for arrangements of conic arcs
 In Proc. ESA 2002
, 2002
"... Abstract. Many computational geometry algorithms involve the construction and maintenance of planar arrangements of conic arcs. Implementing a general, robust arrangement package for conic arcs handles most practical cases of planar arrangements covered in literature. A possible approach for impleme ..."
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Cited by 33 (9 self)
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Abstract. Many computational geometry algorithms involve the construction and maintenance of planar arrangements of conic arcs. Implementing a general, robust arrangement package for conic arcs handles most practical cases of planar arrangements covered in literature. A possible approach for implementing robust geometric algorithms is to use exact algebraic number types — yet this may lead to a very slow, inefficient program. In this paper we suggest a simple technique for filtering the computations involved in the arrangement construction: when constructing an arrangement vertex, we keep track of the steps that lead to its construction and the equations we need to solve to obtain its coordinates. This construction history can be used for answering predicates very efficiently, compared to a naïve implementation with an exact number type. Furthermore, using this representation most arrangement vertices may be computed approximately at first and can be refined later on in cases of ambiguity. Since such cases are relatively rare, the resulting implementation is both efficient and robust. 1
Accurate Minkowski Sum Approximation of Polyhedral Models
 In Pacific Conference on Computer Graphics and Applications
, 2004
"... We present an algorithm to approximate the 3D Minkowski sum of polyhedral objects. Our algorithm decomposes the polyhedral objects into convex pieces, generates pairwise convex Minkowski sums and computes their union. We approximate the union by generating a voxel grid, computing signed distance on ..."
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Cited by 31 (3 self)
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We present an algorithm to approximate the 3D Minkowski sum of polyhedral objects. Our algorithm decomposes the polyhedral objects into convex pieces, generates pairwise convex Minkowski sums and computes their union. We approximate the union by generating a voxel grid, computing signed distance on the grid points and performing isosurface extraction from the distance field.
Hybrid Motion Planning: Coordinating Two Discs Moving Among Polygonal Obstacles in the Plane
 Algorithmic Foundations of Robotics V
, 2002
"... The basic motionplanning problem is to plan a collisionfree motion for an object moving among obstacles between free initial and goal positions, or to determine that no such motion exists. The basic problem as well as numerous variants of it have been intensively studied over the past two decades ..."
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Cited by 18 (3 self)
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The basic motionplanning problem is to plan a collisionfree motion for an object moving among obstacles between free initial and goal positions, or to determine that no such motion exists. The basic problem as well as numerous variants of it have been intensively studied over the past two decades yielding a wealth of results and techniques, both theoretical and practical. In this paper, we propose a novel approach to motion planning, hybrid motion planning, in which we integrate complete solutions along with probabilistic roadmap (PRM) methods in order to combine their strengths and oset their weaknesses. We incorporate robust tools, that have not been available before, in order to implement the complete solutions. We exemplify our approach in the case of two discs moving among polygonal obstacles in the plane. The planner we present easily solves problems where a narrow passage in the workspace can be arbitrarily small. Our planner is also capable of providing correct nontrivial \no" answers, namely it can, for some queries, detect the situation where no solution exists. We envision our planner not as a total solution but rather as a new tool that cooperates with existing planners. We demonstrate the advantages and shortcomings of our planner with experimental results.
Boolean operations on 3D selective Nef complexes: Data structure, algorithms, and implementation
 IN PROC. 11TH ANNU. EURO. SYMPOS. ALG., VOLUME 2832 OF LNCS
, 2003
"... ..."
Generalized penetration depth computation
 In SPM ’06: Proceedings of the 2006 ACM symposium on Solid and physical modeling
, 2006
"... Penetration depth (PD) is a distance metric that is used to describe the extent of overlap between two intersecting objects. Most of the prior work in PD computation has been restricted to translational PD, which is defined as the minimal translational motion that one of the overlapping objects must ..."
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Cited by 9 (2 self)
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Penetration depth (PD) is a distance metric that is used to describe the extent of overlap between two intersecting objects. Most of the prior work in PD computation has been restricted to translational PD, which is defined as the minimal translational motion that one of the overlapping objects must undergo in order to make the two objects disjoint. In this paper, we extend the notion of PD to take into account both translational and rotational motion to separate the intersecting objects, namely generalized PD. When an object undergoes rigid transformation, some point on the object traces the longest trajectory. The generalized PD between two overlapping objects is defined as the minimum of the longest trajectories of one object under all possible rigid transformations to separate the overlapping objects. We present three new results to compute generalized PD between polyhedral models. First, we show that for two overlapping convex polytopes, the generalized PD is same as the translational PD. Second, when the complement of one of the objects is convex, we pose the generalized PD computation as a variant of the convex containment problem and compute an upper bound using optimization techniques. Finally, when both the objects are nonconvex, we treat them as a combination of the above two cases, and present an algorithm that computes a lower and an upper bound on generalized PD. We highlight the performance of our algorithms on different models that undergo rigid motion in the 6dimensional configuration space. Moreover, we utilize our algorithm for complete motion planning of polygonal robots undergoing translational and rotational motion in a plane. In particular, we use generalized PD computation for checking path nonexistence.
Boolean Operations on 3D Selective Nef Complexes: Optimized Implementation and Experiments
, 2005
"... Nef polyhedra in ddimensional space are the closure of halfspaces under boolean set operations. In consequence, they can represent nonmanifold situations, open and closed sets, mixeddimensional complexes and they are closed under all boolean and topological operations. We implemented a boundary ..."
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Cited by 9 (0 self)
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Nef polyhedra in ddimensional space are the closure of halfspaces under boolean set operations. In consequence, they can represent nonmanifold situations, open and closed sets, mixeddimensional complexes and they are closed under all boolean and topological operations. We implemented a boundary representation of threedimensional Nef polyhedra with efficient algorithms for boolean operations. These algorithms were designed for correctness and can handle all cases, in particular all degeneracies. The implementation is released as Open Source in the Cgal release 3.1. In this paper, we present experiments in order to (i) evaluate the practical runtime complexity, (ii) illustrate the effectiveness of several important optimizations, and (iii) compare our implementation with the Acis CAD kernel.
Improved Construction of Vertical Decompositions of ThreeDimensional Arrangements
 In Proc. 18th Annu
, 2002
"... We present new results concerning the refinement of threedimensional arrangements by vertical decompositions. First, we describe a new outputsensitive algorithm for computing the vertical decomposition of arrangements of n triangles in O(n log n + V log n) time, where V is the complexity of the de ..."
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Cited by 7 (3 self)
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We present new results concerning the refinement of threedimensional arrangements by vertical decompositions. First, we describe a new outputsensitive algorithm for computing the vertical decomposition of arrangements of n triangles in O(n log n + V log n) time, where V is the complexity of the decomposition. This improves significantly over the best previously known algorithms. Next, we propose an alternative sparser refinement, which we call the partial vertical decomposition and has the advantages that it produces fewer cells and requires lower degree constructors. We adapt the outputsensitive algorithm to efficiently compute the partial decomposition as well. We implemented algorithms that construct the full and the partial decompositions and we compare the two types theoretically and experimentally. The improved outputsensitive construction extends to the case of arrangements of n wellbehaved surfaces with the same asymptotic running time. We also extended the implementation to the case of polyhedral surfaces  this can serve as the basis for robust implementation of approximations of arrangements of general surfaces.
A Simple Algorithm for Complete Motion Planning of Translating Polyhedral Robots
 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH
, 2005
"... We present an algorithm for complete path planning for translating polyhedral robots in 3D. Instead of exactly computing an explicit representation of the free space, we compute a roadmap that captures its connectivity. This representation encodes the complete connectivity of free space and allows u ..."
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Cited by 7 (4 self)
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We present an algorithm for complete path planning for translating polyhedral robots in 3D. Instead of exactly computing an explicit representation of the free space, we compute a roadmap that captures its connectivity. This representation encodes the complete connectivity of free space and allows us to perform exact path planning. We construct the roadmap by computing deterministic samples in free space that lie on an adaptive volumetric grid. Our algorithm is simple to implement and uses two tests: a complex cell test and a starshaped test. These tests can be efficiently performed on polyhedral objects using maxnorm distance computation and linear programming. The complexity of our algorithm varies as a function of the size of narrow passages in the configuration space. We demonstrate the performance of our algorithm on environments with very small narrow passages or no collisionfree paths.
A simple path nonexistence algorithm using cobstacle query
 In Proc. of WAFR
, 2006
"... Abstract: We present a simple algorithm to check for path nonexistence for a robot among static obstacles. Our algorithm is based on adaptive cell decomposition of configuration space or Cspace. We use two basic queries: free cell query, which checks whether a cell in Cspace lies entirely inside ..."
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Cited by 5 (1 self)
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Abstract: We present a simple algorithm to check for path nonexistence for a robot among static obstacles. Our algorithm is based on adaptive cell decomposition of configuration space or Cspace. We use two basic queries: free cell query, which checks whether a cell in Cspace lies entirely inside the free space, and Cobstacle cell query, which checks whether a cell lies entirely inside the Cobstacle region. Our approach reduces the path nonexistence problem to checking whether there exists a path through cells that do not belong to the Cobstacle region. We describe simple and efficient algorithms to perform free cell and Cobstacle cell queries using separation distance and generalized penetration depth computations. Our algorithm is simple to implement and we demonstrate its performance on 3 DOF robots. 1
A Simple Method for Computing Minkowski Sum Boundary in 3D Using Collision Detection
"... Abstract: Computing the Minkowski sum of two polyhedra exactly has been shown difficult. Despite its fundamental role in many geometric problems in robotics, to the best of our knowledge, no 3d Minkowski sum software for general polyhedra is available to the public. One of the main reasons is the d ..."
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Cited by 5 (2 self)
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Abstract: Computing the Minkowski sum of two polyhedra exactly has been shown difficult. Despite its fundamental role in many geometric problems in robotics, to the best of our knowledge, no 3d Minkowski sum software for general polyhedra is available to the public. One of the main reasons is the difficulty of implementing the existing methods. There are two main approaches for computing Minkowski sums: divideandconquer and convolution. The first approach decomposes the input polyhedra into convex pieces, computes the Minkowski sums between a pair of convex pieces, and unites all the pairwise Minkowski sums. Although conceptually simple, the major problems of this approach include: (1) The size of the decomposition and the pairwise Minkowski sums can be extremely large and (2) robustly computing the union of a large number of components can be very tricky. On the other hand, convolving two polyhedra can be done more efficiently. The resulting convolution is a superset of the Minkowski sum boundary. For nonconvex inputs, filtering or trimming is needed. This usually involves computing (1) the arrangement of the convolution and its substructures and (2) the winding numbers for the arrangement