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Proceedings ArticleDOI

A combinatorial approach to planar non-colliding robot arm motion planning

Ileana Streinu1
12 Nov 2000-pp 443-453
TL;DR: A combinatorial approach to plan noncolliding motions for a polygonal bar-and-joint framework based on a novel class of one-degree-of-freedom mechanisms induced by pseudo triangulations of planar point sets that yields very efficient deterministic algorithms for a category of robot arm motion planning problems with many degrees of freedom.
Abstract: We propose a combinatorial approach to plan noncolliding motions for a polygonal bar-and-joint framework. Our approach yields very efficient deterministic algorithms for a category of robot arm motion planning problems with many degrees of freedom, where the known general roadmap techniques would give exponential complexity. It is based on a novel class of one-degree-of-freedom mechanisms induced by pseudo triangulations of planar point sets, for which we provide several equivalent characterization and exhibit rich combinatorial and rigidity theoretic properties. The main application is an efficient algorithm for the Carpenter's rule problem: convexify a simple bar-and-joint planar polygonal linkage using only non self-intersecting planar motions. A step in the convexification motion consists in moving a pseudo-triangulation-based mechanism along its unique trajectory in configuration space until two adjacent edges align. At that point, a local alteration restores the pseudo triangulation. The motion continues for O(n/sup 2/) steps until all the points are in convex position.
Citations
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Proceedings ArticleDOI
12 Nov 2000
TL;DR: It is proved that the linkage can be continuously moved so that the arcs become straight, the cycles become convex, and no bars cross while preserving the bar lengths.
Abstract: Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles become convex, and no bars cross while preserving the bar lengths. Furthermore, our motion is piecewise-differentiable, does not decrease the distance between any pair of vertices, and preserves any symmetry present in the initial configuration. In particular this result settles the well-studied carpenter's rule conjecture.

185 citations


Cites background from "A combinatorial approach to planar ..."

  • ...At this workshop, a bond between several linkage openers began: Therese Biedl, Martin Demaine, Hazel Everett, Sylvain Lazard, Anna Lubiw, Joseph O’Rourke, Mark Overmars, Steven Robbins, Ileana Streinu, Godfried Toussaint, Sue Whitesides, and the second author....

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  • ...Streinu [31] claims that a polygonal arc can be opened by a sequence of at most O(n2) motions, where each motion is given by the mechanism of a single pseudo-triangulation....

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  • ...Since the motions of a mechanism are described by algebraic equations, Streinu’s algorithm leads to a finite algorithm for a digital computer, at least in principle....

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  • ...It remains to be seen how a practical implementation competes with our approach; in any case, as an algorithm for a direct realization of the motion by a mechanical device, Streinu’s algorithm appears attractive....

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  • ...Recently, Streinu [31] has found a class of such mechanisms, so-calledpseudo-triangulations....

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Journal ArticleDOI
TL;DR: This work overviews both the combinatorial perspective and the geometric perspective of edge flips in planar graphs, highlighting the similarities and differences of the two settings.
Abstract: We review results concerning edge flips in planar graphs concentrating mainly on various aspects of the following problem: Given two different planar graphs of the same size, how many edge flips are necessary and sufficient to transform one graph into another? We overview both the combinatorial perspective (where only a combinatorial embedding of the graph is specified) and the geometric perspective (where the graph is embedded in the plane, vertices are points and edges are straight-line segments). We highlight the similarities and differences of the two settings, describe many extensions and generalizations, highlight algorithmic issues, outline several applications and mention open problems.

132 citations


Cites background or methods or result from "A combinatorial approach to planar ..."

  • ...For pointed or minimum pseudotriangulations, it has been shown [26, 91 ] that a pseudoflip always results in the deletion and insertion of exactly one edge....

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  • ...Streinu [ 91 ] showed that all minimum pseudotriangulations are pointed....

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  • ...It has been shown [5, 26, 91 ] that the graph resulting after a pseudoflip is still a pseudotriangulation of the given point set....

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Journal ArticleDOI
TL;DR: This work presents kinetic data structures for detecting collisions between a set of polygons that are moving continuously, and describes an algorithm for maintaining a pseudo-triangulation of a point set that changes only quadratically many times if points move along algebraic arcs of constant degree.
Abstract: We present kinetic data structures for detecting collisions between a set of polygons that are moving continuously. Unlike classical collision detection methods that rely on bounding volume hierarchies, our method is based on deformable tilings of the free space surrounding the polygons. The basic shape of our tiles is that of a pseudo-triangle, a shape sufficiently flexible to allow extensive deformation, yet structured enough to make detection of self-collisions easy. We show different schemes for maintaining pseudo-triangulations as a kinetic data structure, and we analyze their performance. Specifically, we first describe an algorithm for maintaining a pseudo-triangulation of a point set, and show that the pseudo-triangulation changes only quadratically many times if points move along algebraic arcs of constant degree. In addition, by refining the pseudo-triangulation, we show triangulations of points that only change about O(n7/3) times for linear motion. We then describe an algorithm for maintaining...

125 citations


Additional excerpts

  • ...Streinu uses pseudo-triangulations of points to plan non-colliding motions for polygonal frameworks (Streinu 2000)....

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BookDOI
01 Jan 2004
TL;DR: In this article, the main mathematical part of the text contains only few citations and references to related material, and these additional bits of information are provided in the last section of each chapter, 'Notes and References'.
Abstract: Preface The body of this text is written. It remains to find some words to explain what to expect in this book. A first attempt of characterizing the content could be: In words: The questions posed and partly answered in this book are from the intersection of graph theory and discrete geometry. The reader will meet some graph theory with a geometric flavor and some combinatorial geometry of the plane. Though, the investigations always start in the geometry of the plane it is sometimes appropriate to pass on to higher dimensions to get a more global understanding of the structures under investigation. This is the in Chapter 7, for example, when the study of triangulations of a point configuration leads to the definition of secondary polytopes. David Hilbert said: Im großen Garten der Geometrie kann sich jeder nach seinem Geschmack einen Strauß pflücken. I like to think of this book as a collection which makes up a kind of bouquet. A bouquet of problems, ideas and results, each of a special character and beauty, put together with the intention that they supplement each other to form an interesting and appealing whole. The main mathematical part of the text contains only few citations and references to related material. These additional bits of information are provided in the last section of each chapter, 'Notes and References'. On average the bibliography of a chapter contains about thirty items. This is far from being a complete list of the relevant literature. The intention is to just indicate the most valuable literature so that these sections can serve as entry points for further studies. The text is supplemented by many figures to make the material more attractive and help the reader get a sensual impression of the objects. In some cases, I have confined the presentation to results which fall behind today's state of the art. I wanted to emphasize the main ideas and stop before technical complexity starts taking over. This strategy should make the mathematics accessibility to a relatively broad audience including students of computer science, students of mathematics, instructors and researchers. The book can serve different purposes. It may be used as textbook for a course or as a collection of material for a seminar. It should also be helpful to people who want to learn something about specific themes. They may concentrate on single chapters because all the chapters are self-contained …

120 citations

Book ChapterDOI
TL;DR: In this paper, the authors introduce the polytope of pointed pseudo-triangulations of a point set in the plane, which is defined as the polytoope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances.
Abstract: We introduce the polytope of pointed pseudo-triangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances. Its 1-skeleton is the graph whose vertices are the pointed pseudo-triangulations of the point set and whose edges are flips of interior pseudo-triangulation edges.

120 citations

References
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Journal ArticleDOI
01 Aug 1996
TL;DR: Experimental results show that path planning can be done in a fraction of a second on a contemporary workstation (/spl ap/150 MIPS), after learning for relatively short periods of time (a few dozen seconds).
Abstract: A new motion planning method for robots in static workspaces is presented. This method proceeds in two phases: a learning phase and a query phase. In the learning phase, a probabilistic roadmap is constructed and stored as a graph whose nodes correspond to collision-free configurations and whose edges correspond to feasible paths between these configurations. These paths are computed using a simple and fast local planner. In the query phase, any given start and goal configurations of the robot are connected to two nodes of the roadmap; the roadmap is then searched for a path joining these two nodes. The method is general and easy to implement. It can be applied to virtually any type of holonomic robot. It requires selecting certain parameters (e.g., the duration of the learning phase) whose values depend on the scene, that is the robot and its workspace. But these values turn out to be relatively easy to choose, Increased efficiency can also be achieved by tailoring some components of the method (e.g., the local planner) to the considered robots. In this paper the method is applied to planar articulated robots with many degrees of freedom. Experimental results show that path planning can be done in a fraction of a second on a contemporary workstation (/spl ap/150 MIPS), after learning for relatively short periods of time (a few dozen seconds).

4,977 citations

Book
29 Jun 1988
TL;DR: John Canny resolves long-standing problems concerning the complexity of motion planning and, for the central problem of finding a collision free path for a jointed robot in the presence of obstacles, obtains exponential speedups over existing algorithms by applying high-powered new mathematical techniques.
Abstract: The Complexity of Robot Motion Planning makes original contributions both to robotics and to the analysis of algorithms. In this groundbreaking monograph John Canny resolves long-standing problems concerning the complexity of motion planning and, for the central problem of finding a collision free path for a jointed robot in the presence of obstacles, obtains exponential speedups over existing algorithms by applying high-powered new mathematical techniques.Canny's new algorithm for this "generalized movers' problem," the most-studied and basic robot motion planning problem, has a single exponential running time, and is polynomial for any given robot. The algorithm has an optimal running time exponent and is based on the notion of roadmaps - one-dimensional subsets of the robot's configuration space. In deriving the single exponential bound, Canny introduces and reveals the power of two tools that have not been previously used in geometric algorithms: the generalized (multivariable) resultant for a system of polynomials and Whitney's notion of stratified sets. He has also developed a novel representation of object orientation based on unnormalized quaternions which reduces the complexity of the algorithms and enhances their practical applicability.After dealing with the movers' problem, the book next attacks and derives several lower bounds on extensions of the problem: finding the shortest path among polyhedral obstacles, planning with velocity limits, and compliant motion planning with uncertainty. It introduces a clever technique, "path encoding," that allows a proof of NP-hardness for the first two problems and then shows that the general form of compliant motion planning, a problem that is the focus of a great deal of recent work in robotics, is non-deterministic exponential time hard. Canny proves this result using a highly original construction.John Canny received his doctorate from MIT And is an assistant professor in the Computer Science Division at the University of California, Berkeley. The Complexity of Robot Motion Planning is the winner of the 1987 ACM Doctoral Dissertation Award.

1,538 citations

Journal ArticleDOI
TL;DR: In this paper, a decision method for finding a continuous motion connecting two given positions and orientations of the whole collection of bodies is presented. But it is not shown that this problem can be solved in polynomial time.

909 citations

Book
26 Mar 1993

879 citations

Book
01 Jan 1993
TL;DR: This handbook should be useful for mathematicians working in other areas, as well as for econometrists, computer scientists, crystallographers, physicists and engineers who are looking for geometric tools for their own work.
Abstract: One aim of this handbook is to survey convex geometry, its many ramifications and its relations with other areas of mathematics. As such it should be a useful tool for the expert. A second aim is to give a high-level introduction to most branches of convexity and its applications, showing the major ideas, methods and results. This aspect should make it a source of inspiration for future researchers in convex geometry. The handbook should be useful for mathematicians working in other areas, as well as for econometrists, computer scientists, crystallographers, physicists and engineers who are looking for geometric tools for their own work. In particular, mathematicians specializing in optimization, functional analysis, number theory, probability theory, the calculus of variations and all branches of geometry should profit from this handbook.

484 citations