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A Combinatorial Approach to Planar Noncolliding Robot Arm Motion Planning
 In Proc. 41st FOCS
, 2000
"... 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 ..."
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Cited by 100 (14 self)
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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 nonintersecting 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 noncollision. At most O(n²) such motions suffice to reach a convex configuration of the original linkage. At each step, recomputing the next motion from ...
Folding and Unfolding in Computational Geometry
"... Three open problems on folding/unfolding are discussed: (1) Can every convex polyhedron be cut along edges and unfolded at to a single nonoverlapping piece? (2) Given gluing instructions for a polygon, construct the unique 3D convex polyhedron to which itfolds. (3) Can every planar polygonal chain ..."
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Cited by 53 (4 self)
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Three open problems on folding/unfolding are discussed: (1) Can every convex polyhedron be cut along edges and unfolded at to a single nonoverlapping piece? (2) Given gluing instructions for a polygon, construct the unique 3D convex polyhedron to which itfolds. (3) Can every planar polygonal chain be straightened?
Polygonal Chains Cannot Lock in 4D
 In Proc. 11th Canad. Conf. Comput. Geom
, 2001
"... We prove that, in all dimensions d 4, every simple open polygonal chain and every tree may be straightened, and every simple closed polygonal chain may be convexified. These reconfigurations can be achieved by algorithms that use polynomial time in the number of vertices, and result in a polyno ..."
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Cited by 21 (2 self)
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We prove that, in all dimensions d 4, every simple open polygonal chain and every tree may be straightened, and every simple closed polygonal chain may be convexified. These reconfigurations can be achieved by algorithms that use polynomial time in the number of vertices, and result in a polynomial number of "moves." These results contrast to those known for d = 2, where trees can "lock," and for d = 3, where open and closed chains can lock. Smith Technical Report 063 (Major revision of the August 1999 version with the same report number.) Dept. of Computer Science, Smith College, Northampton, MA 01063, USA. frcocan, orourkeg@cs.smith.edu. Research supported by NSF Grant CCR9731804. Results first reported in [CO99]. i Contents 1 Introduction 1 1.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Straightening Open Chains in 4D 3 2.1 Algorithm 1a . . . . . . . . . . . . ....
Pseudotriangulations, rigidity and motion planning
 Discrete and Computational Geometry
, 2005
"... Abstract We propose a combinatorial approach to planning noncolliding trajectories for a polygonal barandjoint framework with n vertices. It is based on a new class of simple motionsinduced by expansive onedegreeoffreedom mechanisms, which guarantee noncollisions by moving all points away fr ..."
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Cited by 3 (0 self)
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Abstract We propose a combinatorial approach to planning noncolliding trajectories for a polygonal barandjoint framework with n vertices. It is based on a new class of simple motionsinduced by expansive onedegreeoffreedom mechanisms, which guarantee noncollisions by moving all points away from each other. Their combinatorial structure is captured by pointedpseudotriangulations, a class of embedded planar graphs for which we give several equivalent characterizations and exhibit rich rigidity theoretic properties.The main application is an efficient algorithm for the Carpenter's Rule Problem: convexify a simple barandjoint planar polygonal linkage using only nonselfintersecting planarmotions. A step of the algorithm consists in moving a pseudotriangulationbased mechanism along its unique trajectory in configuration space until two adjacent edges align. At thealignment event, a local alteration restores the pseudotriangulation. The motion continues for O(n3) steps until all the points are in convex position. 1 Introduction We present a combinatorial solution to the Carpenter's Rule Problem: how to plan noncolliding reconfigurations of a planar robot arm. The main result is an efficient algorithm for the problem of continuously moving a simple planar polygon to any other configuration with the same edgelengths and orientation, while remaining in the plane and never creating selfintersections along the way. This is done by first finding motions that convexify both configurations with expansive motions (which never bring two points closer together) and then taking one path in reverse. All of the constructions are elementary and are based on a novel class of planar embedded graphs called pointed pseudotriangulations, for which we prove a variety of combinatorial and rigidity theoretical properties. More prominently, a pointed pseudotriangulation with a removed convex hull edge is a onedegreeoffreedom expansive mechanism. If its edges are seen as rigid bars (maintaining their lengths) and are allowed to rotate freely around the vertices (joints), the mechanism follows (for a well defined, finite time interval) a continuous trajectory along which no distance between a pair of points ever decreases. The expansive motion induced by these mechanisms provide the building blocks of our algorithm.
Expansive Motions for dDimensional Open Chains
"... We consider the problem of straightening chains in d ≥ 3 dimensions, possibly embedded into higher dimensions, using expansive motions. For any d ≥ 3, we show that there is an open chain in d dimensions that is not straight and not selftouching yet has no expansive motion. Furthermore, for any ∆> 0 ..."
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We consider the problem of straightening chains in d ≥ 3 dimensions, possibly embedded into higher dimensions, using expansive motions. For any d ≥ 3, we show that there is an open chain in d dimensions that is not straight and not selftouching yet has no expansive motion. Furthermore, for any ∆> 0 and d ≥ 3, we show that there is an open chain in d dimensions that cannot be straightened using expansive motions when embedded into R d ×[−∆, ∆] (a bounded extra dimension). On the positive side, we prove that any open chain in d ≥ 2 dimensions can be straightened using an expansive motion when embedded into R d+1 (a full extra dimension). 1