367 research outputs found
Polygonal Chains Cannot Lock in 4D
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.Comment: Major revision of the Aug. 1999 version, including: Proof extended to
show trees cannot lock in 4D; new example of the implementation straightening
a chain of n=100 vertices; improved time complexity for chain to O(n^2);
fixed several minor technical errors. (Thanks to three referees.) 29 pages;
15 figures. v3: Reference update
Liftings and stresses for planar periodic frameworks
We formulate and prove a periodic analog of Maxwell's theorem relating
stressed planar frameworks and their liftings to polyhedral surfaces with
spherical topology. We use our lifting theorem to prove deformation and
rigidity-theoretic properties for planar periodic pseudo-triangulations,
generalizing features known for their finite counterparts. These properties are
then applied to questions originating in mathematical crystallography and
materials science, concerning planar periodic auxetic structures and ultrarigid
periodic frameworks.Comment: An extended abstract of this paper has appeared in Proc. 30th annual
Symposium on Computational Geometry (SOCG'14), Kyoto, Japan, June 201
Allocation of Two Dimensional Parts Using a Shape Reasoning Heuristic.
A technique is outlined for the allocation of irregular parts onto arbitrarily shaped resources. Placements are generated by matching complementary shapes between the unplaced parts and the remaining areas of the stock material. The part and resource profiles are characterized to varying levels of detail using geometric features . Information contained in the features is used at each stage of processing to intelligently select and place parts on the resource. Techniques for the efficient handling of complex profiles and other practical implementation issues are described. The utility of the proposed approach is verified using diverse problems from a marine fabrication facility. The formulation and performance of the method is contrasted to previously published works
Efficient motion planning using generalized penetration depth computation
Motion planning is a fundamental problem in robotics and also arises in other applications including virtual prototyping, navigation, animation and computational structural biology. It has been extensively studied for more than three decades, though most practical algorithms are based on randomized sampling. In this dissertation, we address two main issues that arise with respect to these algorithms: (1) there are no good practical approaches to check for path non-existence even for low degree-of-freedom (DOF) robots; (2) the performance of sampling-based planners can degrade if the free space of a robot has narrow passages. In order to develop effective algorithms to deal with these problems, we use the concept of penetration depth (PD) computation. By quantifying the extent of the intersection between overlapping models (e.g. a robot and an obstacle), PD can provide a distance measure for the configuration space obstacle (C-obstacle). We extend the prior notion of translational PD to generalized PD, which takes into account translational as well as rotational motion to separate two overlapping models. Moreover, we formulate generalized PD computation based on appropriate model-dependent metrics and present two algorithms based on convex decomposition and local optimization. We highlight the efficiency and robustness of our PD algorithms on many complex 3D models. Based on generalized PD computation, we present the first set of practical algorithms for low DOF complete motion planning. Moreover, we use generalized PD computation to develop a retraction-based planner to effectively generate samples in narrow passages for rigid robots. The effectiveness of the resulting planner is shown by alpha puzzle benchmark and part disassembly benchmarks in virtual prototyping
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