Proceedings ArticleDOI
A combinatorial approach to planar non-colliding robot arm motion planning
Ileana Streinu
- pp 443-453
TLDR
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.read more
Citations
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Journal ArticleDOI
Locked and Unlocked Polygonal Chains in Three Dimensions
Therese C. Biedl,Erik D. Demaine,Martin L. Demaine,Sylvain Lazard,Anna Lubiw,Joseph O'Rourke,Mark H. Overmars,Steven M. Robbins,Ileana Streinu,Godfried T. Toussaint,Sue Whitesides +10 more
TL;DR: In this paper, it was shown that a simple closed polygonal chain can be made convex in 3D by O(n) basic O(m) moves, and that any simple closed chain that initially takes the form of a planar polygon can be constructed in three dimensions.
Proceedings Article
Counting triangulations and pseudo-triangulations of wheels.
TL;DR: An inequality #PT ≤ 3#T for the numbers of minimum pseudo-triangulations and triangulations of any point configuration with i interior is proved.
Proceedings Article
Locked and Unlocked Polygonal Chains in Three Dimensions
Therese C. Biedl,Erik D. Demaine,Martin L. Demaine,Sylvain Lazard,Anna Lubiw,Joseph O'Rourke,Mark H. Overmars,Steven M. Robbins,Ileana Streinu,Godfried T. Toussaint,Sue Whitesides +10 more
TL;DR: The main result is an algorithmic proof that any simple closed chain that initially takes the form of a planar polygon can be made convex in three dimensions.
Journal ArticleDOI
The non-solvability by radicals of generic 3-connected planar Laman graphs
J.C. Owen,S. C. Power +1 more
TL;DR: In this article, it was shown that planar embeddable 3-connected Laman graphs are generically non-soluble, i.e., they do not satisfy the vertex-edge count 2v - 3 = e together with corresponding inequality for each subgraph.
Journal ArticleDOI
Tight degree bounds for pseudo-triangulations of points
Lutz Kettner,David G. Kirkpatrick,Andrea Mantler,Jack Snoeyink,Bettina Speckmann,Fumihiko Takeuchi +5 more
TL;DR: It is demonstrated that every point set in general position has a minimum pseudotriangulation whose maximum face degree is four and that minimum pseudo-triangulations realizing these bounds can be constructed in O(n logn) time.
References
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Journal ArticleDOI
Probabilistic roadmaps for path planning in high-dimensional configuration spaces
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).
Book
The complexity of robot motion planning
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.
Journal ArticleDOI
On the “piano movers” problem. II. General techniques for computing topological properties of real algebraic manifolds
Jacob T. Schwartz,Micha Sharir +1 more
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.
Book
Handbook of convex geometry
Peter M. Gruber,Jörg M. Wills +1 more
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.
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