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

Abstract

This 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.