MULTIPLE PARAMETER CONTINUATION: COMPUTING IMPLICITLY DEFINED k-MANIFOLDS

Henderson, Michael E.

doi:10.1142/s0218127402004498

Cited by 121 publications

(145 citation statements)

References 31 publications

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“…The methods that have been proposed essentially fall into two classes: simplicial methods (e.g. see [10] and references therein) and predictor-corrector methods (see [6,7,8,16]). Predictor-corrector methods show more favorable computational complexity in the case of low dimensional manifolds embedded in a high dimensional space, which is the case of interest for us.…”

Section: Multi-parameter Continuation Algorithmmentioning

confidence: 99%

“…From our standpoint, the main feature that distinguishes among the predictor-corrector methods cited above is how the manifold is represented. In [6], the manifold is represented as a set of overlapping neighborhoods, each defined through a chart, a center point, basis for the tangent space at the center point, and other relevant data. A triangulation of the manifold can be constructed a posteriori by appropriately connecting the center points of the neighborhoods, but this does not allow for adaptive construction of the triangulation.…”

Section: Multi-parameter Continuation Algorithmmentioning

confidence: 99%

“…Let us mention that multi-parameter continuation algorithms have already been proposed to compute numerical approximations for smooth manifolds implicitly defined by finite dimensional nonlinear equations [6,7,8,9,10]. In fact, the simplex-based multiparameter continuation algorithm that we propose to compute the manifold is strongly inspired by the work of [7,8].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

2015

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In this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite dimensional nonlinear operators. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite dimensional domain of the operator. The idea behind the construction of the smooth charts is to use the radii polynomial approach to verify the hypotheses of the uniform contraction principle over a simplex. The construction of the manifold is globalized by proving smoothness along the edge of adjacent simplices. We apply the method to compute portions of a twodimensional manifold of equilibria of the Cahn-Hilliard equation.

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Section: Multi-parameter Continuation Algorithmmentioning

confidence: 99%

Section: Multi-parameter Continuation Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

2015

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“…In many problems, however, it may be sufficient to explore only those configurations that are path-connected to a given point. To this end, the CUIK suite implements higher-dimensional continuation tools allowing to trace arbitrary manifolds [10]. While branch-and-prune meth- ods proceed by discarding non-valid configurations, continuation techniques march from a given point in all directions identifying new feasible configurations.…”

Section: Motivation and Outlookmentioning

confidence: 99%

The CUIK Suite: Analyzing the Motion Closed-Chain Multibody Systems

Porta

Ros

Bohigas

et al. 2014

IEEE Robot. Automat. Mag.

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Many situations in Robotics require the analysis of the motions of complex multibody systems. These are sets of articulated bodies arising in a variety of devices, including parallel manipulators, multifingered hands, or reconfigurable mechanisms, but they appear in other domains too, as mechanical models of molecular compounds or nanostructures. Closed kinematic chains arise frequently in such systems, either due to their morphology, or to geometric or contact constraints to fulfill during operation, giving rise to configuration spaces of an intricate structure. Despite appearing very often in practice, there is a lack of general software tools to analyze and represent such configuration spaces. Existing packages are either oriented to open chain systems, or to specific robot types, which hinders the analysis and development of innovative manipulators. This paper describes the CUIK suite, a software toolbox for the kinematic analysis of general multibody systems. The implemented tools can isolate the valid configurations, determine the motion range of the whole multibody system or of some of its parts, detect singular configurations leading to control or dexterity issues, or find collision-and singularity-free paths between configurations. The toolbox has applications in robot design and programming, and it is the result of several years of research and development within the Kinematics and Robot Design group at IRI, Barcelona. It is available under GPLv3 license from

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“…To this end, the CUIK suite implements higher-dimensional continuation tools allowing to trace arbitrary, implicitly-defined manifolds [6]. Note that while several packages provide state-of-the-art path planning methods, they are oriented to open-chain robots [11,22,21,23], or to particular classes of closed-chain devices [10].…”

Section: Introductionmentioning

confidence: 99%

An Open-Source Toolbox for Motion Analysis of Closed-Chain Mechanisms

Porta

Ros

Bohigas

et al. 2013

Computational Kinematics

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Many situations in Robotics require an effective analysis of the motions of a closed-chain mechanism. Despite appearing very often in practice (e.g. in parallel manipulators, reconfigurable robots, or molecular compounds), there is a lack of general tools to effectively analyze the complex configuration spaces of such systems. This paper describes the CUIK suite, an open-source toolbox for motion analysis of general closed-chain mechanisms. The package can determine the motion range of the whole mechanism or of some of its parts, detect singular configurations leading to control or dexterity issues, or find collision-and singularity-free paths between given configurations. The toolbox is the result of several years of research and development within the Kinematics and Robot Design group at IRI, Barcelona, and is available under GPLv3 license from http://www.iri.upc.edu/cuik.

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MULTIPLE PARAMETER CONTINUATION: COMPUTING IMPLICITLY DEFINED k-MANIFOLDS

Cited by 121 publications

References 31 publications

Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

The CUIK Suite: Analyzing the Motion Closed-Chain Multibody Systems

An Open-Source Toolbox for Motion Analysis of Closed-Chain Mechanisms

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