## A Combinatorial Approach to Planar Non-colliding Robot Arm Motion Planning (2000)

Venue: | In Proc. 41st FOCS |

Citations: | 101 - 14 self |

### BibTeX

@INPROCEEDINGS{Streinu00acombinatorial,

author = {Ileana Streinu},

title = {A Combinatorial Approach to Planar Non-colliding Robot Arm Motion Planning},

booktitle = {In Proc. 41st FOCS},

year = {2000},

pages = {443--453}

}

### Years of Citing Articles

### OpenURL

### Abstract

We propose a combinatorial approach to plan noncolliding motions for a planar robot arm. The approach works even with certain types of movable polygonal obstacles and flexible polygonal fences. This yields a very efficient deterministic algorithm for a category of robot arm motion planning problems with many degrees of freedom, for which the known general roadmap techniques have exponential complexity. The main result is an efficient algorithm for convexifying a simple (open or closed) polygonal path with rigid non-intersecting motions in the plane. It works by computing in O(n²) time a monotone mechanism with one degree of freedom, whose motion is controlled by the rotation of a single edge around one of its endpoints. As it moves, all the interdistances between pairs of points not joined by a bar are nondecreasing, thus guaranteeing non-collision. At most O(n²) such motions suffice to reach a convex configuration of the original linkage. At each step, recomputing the next motion from ...

### Citations

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Citation Context ...rithms for high degree of freedom problems such as the one we are interested in. See [SS], [Ca88], [BPRa] and [BPRb]. In practical applications, probabilistic roadmap algorithms are used instead, see =-=[Ka]-=-, [HA]. Under various conditions, problems about recon guration of linkages range in complexity from polynomial ([LW]) to NP- and even PSPACE-hard, see [HJW], [W92], [JP]. The particular problem of st... |

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Citation Context ... problems based on roadmaps work well on problems with bounded degrees of freedom, but yield exponential algorithms for high degree of freedom problems such as the one we are interested in. See [SS], =-=[Ca88]-=-, [BPRa] and [BPRb]. In practical applications, probabilistic roadmap algorithms are used instead, see [Ka], [HA]. Under various conditions, problems about recon guration of linkages range in complexi... |

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Citation Context ...anning problems based on roadmaps work well on problems with bounded degrees of freedom, but yield exponential algorithms for high degree of freedom problems such as the one we are interested in. See =-=[SS]-=-, [Ca88], [BPRa] and [BPRb]. In practical applications, probabilistic roadmap algorithms are used instead, see [Ka], [HA]. Under various conditions, problems about recon guration of linkages range in ... |

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Citation Context ... for high degree of freedom problems such as the one we are interested in. See [SS], [Ca88], [BPRa] and [BPRb]. In practical applications, probabilistic roadmap algorithms are used instead, see [Ka], =-=[HA]-=-. Under various conditions, problems about recon guration of linkages range in complexity from polynomial ([LW]) to NP- and even PSPACE-hard, see [HJW], [W92], [JP]. The particular problem of straight... |

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Citation Context ...this versatile data structure, a slight generalization of the one introduced by Pocchiola and Vegter[PV] in their study of the visibility complex and recently applied to kinetic geometric algorithms (=-=[AB00]-=-, [BE99]). Novelty. To the best of our knowledge, there are very few results on the motion planning of rigid frameworks when self-intersections are not allowed, and ours comes within a clean combinato... |

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Citation Context ...t least four equivalent ways, exhibiting their rich combinatorial properties. Some of these are specialized versions of Laman's 2n 3 count and Henneberg constructions from combinatorial rigidity (see =-=[W96]-=- or [GSS]). The proof of correctness of our approach derives from these properties, as well as from a generalization, from simple polygons to the wider class of pseudo-triangulation frameworks, of the... |

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Citation Context ...s of pseudo-triangulations which we present in this paper are likely to add to the applicability of this versatile data structure, a slight generalization of the one introduced by Pocchiola and Vegter=-=[PV]-=- in their study of the visibility complex and recently applied to kinetic geometric algorithms ([AB00], [BE99]). Novelty. To the best of our knowledge, there are very few results on the motion plannin... |

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Citation Context ...ic roadmap algorithms are used instead, see [Ka], [HA]. Under various conditions, problems about recon guration of linkages range in complexity from polynomial ([LW]) to NP- and even PSPACE-hard, see =-=[HJW-=-], [W92], [JP]. The particular problem of straightening bar-andjoint linkages and convexifying polygons has accumulated a distinguished history, with some approaches going back to a question of Erdos ... |

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Citation Context ..., [W92], [JP]. The particular problem of straightening bar-andjoint linkages and convexifying polygons has accumulated a distinguished history, with some approaches going back to a question of Erdos [=-=Er]-=-. See [To99] for a fascinating account. There are abundant connections with work done in the computational biology, chemistry and physics literature and motivated by topics such as protein folding or ... |

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Citation Context ...onstructions, Lovasz and Yemini covering with two trees [LY] and Maxwell's Theorem are to be found there. For oriented matroids, see [Bj93], although we won't need more than the circular sequences of =-=[GP] (-=-see also [St]). Notation and abbreviations. Our setting is the Euclidian plane. All index arithmetic is done mod n in the set [n] := f1; ; ng. We abbreviate \counterclockwise " as ccw and \one... |

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Citation Context ... only a constant number of joints at a time). Recent results in the mathematics literature aim at understanding the topology of the conguration space of closed chains, but they allow crossings, see ([=-=KM95]-=-. The universality result of [KM98] is also for the general case of self-intersecting rigid frameworks. Following the McGill University Workshop on Folding in Barbados, organized by Anna Lubiw and Sue... |

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Citation Context ...t a time). Recent results in the mathematics literature aim at understanding the topology of the conguration space of closed chains, but they allow crossings, see ([KM95]. The universality result of [=-=KM98]-=- is also for the general case of self-intersecting rigid frameworks. Following the McGill University Workshop on Folding in Barbados, organized by Anna Lubiw and Sue Whitesides in 1998, several result... |

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Citation Context ... In practical applications, probabilistic roadmap algorithms are used instead, see [Ka], [HA]. Under various conditions, problems about recon guration of linkages range in complexity from polynomial (=-=[LW]-=-) to NP- and even PSPACE-hard, see [HJW], [W92], [JP]. The particular problem of straightening bar-andjoint linkages and convexifying polygons has accumulated a distinguished history, with some approa... |

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Clusters of Stars, Proc
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Citation Context ...ovasz and Yemini covering with two trees [LY] and Maxwell's Theorem are to be found there. For oriented matroids, see [Bj93], although we won't need more than the circular sequences of [GP] (see also =-=[St]).-=- Notation and abbreviations. Our setting is the Euclidian plane. All index arithmetic is done mod n in the set [n] := f1; ; ng. We abbreviate \counterclockwise " as ccw and \one-degree-of-free... |