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Homotopy continuation methods, intrinsic localized modes, and cooperative robotic workspaces

dc.contributor.authorBrake, Daniel Abram, author
dc.contributor.authorPutkaradze, Vakhtang, advisor
dc.contributor.authorMaciejewski, Tony, advisor
dc.contributor.authorMarconi, Mario, committee member
dc.contributor.authorBates, Dan, committee member
dc.contributor.authorShipman, Patrick, committee member
dc.date.accessioned2007-01-03T08:26:18Z
dc.date.available2007-01-03T08:26:18Z
dc.date.issued2012
dc.description.abstractThis dissertation considers three topics that are united by the theme of application of geometric and nonlinear mechanics to practical problems. Firstly we consider the parallel implementation of numerical solution of nonlinear polynomial systems depending on parameters. The program written to do this is called Paramotopy, and uses the Message Passing Interface to distribute homotopy continuation solves in another program called Bertini across a supercomputer. Paramotopy manages writing of Bertini input files, allows automatic re-solution of the system at points at which paths failed, and makes data management easy. Furthermore, parameter homotopy nets huge performance gains over fresh homotopy continuation runs. Superlinear speedup was achieved, up to hard drive throughput capacity. Various internal settings are demonstrated and explored, and the User's Manual is included. Second, we apply nonlinear theory and simulation to nanomechanical sensor arrays. Using vibrating GaAs pillars, we model Intrinsic Localized Modes (ILMs), and investigate ILM-defect pinning, formation, lifetime, travel and movement, and parameter dependence. Intrinsic Localized Modes have been analyzed on arrays of nonlinear oscillators. So far, these oscillators have had a single direction of vibration. In current experiments for single molecule detection, arrays made of Gallium Arsenide will be innately bidirectional, forced, dissipative. We expand previous full models to bidirectionality, and simulate using ODE solvers. We show that small regions of a very large parameter space permit strong ILM formation. Additionally, we use Hamiltonian mechanics to derive new simplified models for the monodirectional ILM travel on an infinite array. This monodirectional ILMs of constant amplitude have unrealistic behavior. Permitting the amplitude of the ILM to vary in time produces much more realistic behavior, including wandering and intermittent pinning. The final set of problems concerns the application of numerical algebraic geometric methods to untangle the phase space of cooperating robots, and optimize configuration for fault tolerance. Given two robots in proximity to each other, if one experiences joint failure, the other may be able to assist, restoring lost workspace. We define a new multiplicity-weighted workspace measure, and use it to solve the optimization problem of finding the best location for an assistance socket and separation distance for the two robots, showing that the solution depends on robot geometry, which link is being grasped, and the choice of objective function.
dc.format.mediumborn digital
dc.format.mediumdoctoral dissertations
dc.identifierBrake_colostate_0053A_11500.pdf
dc.identifierETDF2012500289MATH
dc.identifier.urihttp://hdl.handle.net/10217/71548
dc.languageEnglish
dc.language.isoeng
dc.publisherColorado State University. Libraries
dc.relation.ispartof2000-2019
dc.rightsCopyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright.
dc.subjectsensors
dc.subjecthomotopy continuation
dc.subjectintrinsic localized mode
dc.subjectkinematics
dc.subjectparallel computing
dc.subjectrobotics
dc.titleHomotopy continuation methods, intrinsic localized modes, and cooperative robotic workspaces
dc.typeText
dcterms.rights.dplaThis Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
thesis.degree.disciplineMathematics
thesis.degree.grantorColorado State University
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy (Ph.D.)

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