This thesis introduces a design combining an emerging area in robotics with a well established mathematical research topic: swarm intelligence and Voronoi tessellations, respectively. The main objective for this research is to design an economical and robust swarm system to achieve distributed control. This research combines swarm intelligence with Voronoi tessellations to localize a source and create formations. Extensive software coding must be implemented for this design, such as the development of a discrete centroidal Voronoi tessellation (CVT) algorithm. The ultimate purpose of this research is to advance the existing Mobile Actuator and Sensor Network (MASnet) platform to eventually develop a cooperative robot team that can sense, predict, and nally neutralize a diusion process. Previous work on the MASnet platform has served as a foundation for this research. While growing closer to the MASnet goal, results also provide stimulating discoveries for mathematical and swarm research areas.

Statistical aspects of random mosaics have not been heretofore given enough attention. This thesis deals with the derivation of estimators and statistical tests in a three-dimensional Poisson-Voronoi mosaic model. The first chapter compiles elementary results in the fields of point processes, random closed sets and particle processes. These are used in a second chapter to deduce geometric properties of random mosaics. The third chapter introduces the statistical research itself, estimators and model tests. Horvitz- Thompson estimator is introduced in order to correct statistics calculated on a reduced sample. Own results are tried in a computer simulation and compared to existing research in the last chapter. Mainly, the quality of estimators and the power of proposed tests is observed. 1

Uma tendência crescente entre os pesquisadores da Robótica Móvel é a elaboração de sistemas robóticos descentralizados denominados enxames de robôs, nos quais a ação conjunta de cada agente leva à execução de tarefas de maneira mais robusta que quando realizada por um único robô. Um acréscimo adicional à robustez é conveniente em tais sistemas para que eles sejam de maior confiabilidade no mundo real. Neste trabalho, uma rede neural hierárquica desenvolvida para o aprendizado em tempo real inicialmente elaborada para o aprendizado de navegação de um único robô será estendida para controlar um enxame de robôs. O sistema realiza um balanceamento da influência de comportamentos implementados previamente em um robô de acordo com conhecimentos obtidos através da interação do mesmo com o ambiente. Cada robô possui sua própria rede neural, adquirindo seu conhecimento tanto independentemente quanto com o compartilhamento de informações com outros robôs. Espera-se que o uso de tal arquitetura permita uma adaptação mais rápida dos robôs ao ambiente, permitindo uma mudança em tempo real de seus parâmetros de acordo com as peculiaridades do ambiente no qual os robôs estão inseridos. A tarefa de escolta de um robô pelos demais é adotada para a avaliação de desempenho do modelo de rede neural proposto. Dois comportamentos são ponderados pela rede neural hierárquica: o de manutenção de uma distância preestabelecida a um agente e um outro de cobertura de área baseado em Diagramas Centroidais de Voronoi. Os testes foram feitos nos ambientes Player/Stage e indicam que a rede neural hierárquica torna os robôs capazes não apenas de aprender à medida que interagem com ambiente como de utilizar este conhecimento em tempo real para realizar a escolta de forma bem sucedida / A growing trend among Mobile Robotics researchers is developing robot swarms, in which a decentralized robot team solves tasks by combining simple behaviors. It is convenient to have mechanisms to increase a robot systems robustness. In this work, a neural network inspired in behavioral analysis is used to make robots from a swarm to learn how to act propoerly. This network combines two innate behaviors and, according to its experience, learns with the robots mistakes how to make this combination. Each robot has access to its own independent neural network, and can share its knowledge with neighboring robots. It is expected that such architecture learns by itself when to stimulate or supress each behaviors influence as it interacts with the environment. The task chosen to evaluate the proposed system is the escorting of a mobile agent. Two behaviors are balanced to achieve an escorting behavior: maintenance of a minimum distance between a robot and the escort target and an area coverage method based on Centroidal Voronoi Tessellations. Tests were meade using the Player/Stage simulator, and they show that the robots not only are capable of adapting themselves but also are able to use the stored knowledge to improve their effectiveness in doing the desired task

Uma tendência crescente entre os pesquisadores da Robótica Móvel é a elaboração de sistemas robóticos descentralizados denominados enxames de robôs, nos quais a ação conjunta de cada agente leva à execução de tarefas de maneira mais robusta que quando realizada por um único robô. Um acréscimo adicional à robustez é conveniente em tais sistemas para que eles sejam de maior confiabilidade no mundo real. Neste trabalho, uma rede neural hierárquica desenvolvida para o aprendizado em tempo real inicialmente elaborada para o aprendizado de navegação de um único robô será estendida para controlar um enxame de robôs. O sistema realiza um balanceamento da influência de comportamentos implementados previamente em um robô de acordo com conhecimentos obtidos através da interação do mesmo com o ambiente. Cada robô possui sua própria rede neural, adquirindo seu conhecimento tanto independentemente quanto com o compartilhamento de informações com outros robôs. Espera-se que o uso de tal arquitetura permita uma adaptação mais rápida dos robôs ao ambiente, permitindo uma mudança em tempo real de seus parâmetros de acordo com as peculiaridades do ambiente no qual os robôs estão inseridos. A tarefa de escolta de um robô pelos demais é adotada para a avaliação de desempenho do modelo de rede neural proposto. Dois comportamentos são ponderados pela rede neural hierárquica: o de manutenção de uma distância preestabelecida a um agente e um outro de cobertura de área baseado em Diagramas Centroidais de Voronoi. Os testes foram feitos nos ambientes Player/Stage e indicam que a rede neural hierárquica torna os robôs capazes não apenas de aprender à medida que interagem com ambiente como de utilizar este conhecimento em tempo real para realizar a escolta de forma bem sucedida / A growing trend among Mobile Robotics researchers is developing robot swarms, in which a decentralized robot team solves tasks by combining simple behaviors. It is convenient to have mechanisms to increase a robot systems robustness. In this work, a neural network inspired in behavioral analysis is used to make robots from a swarm to learn how to act propoerly. This network combines two innate behaviors and, according to its experience, learns with the robots mistakes how to make this combination. Each robot has access to its own independent neural network, and can share its knowledge with neighboring robots. It is expected that such architecture learns by itself when to stimulate or supress each behaviors influence as it interacts with the environment. The task chosen to evaluate the proposed system is the escorting of a mobile agent. Two behaviors are balanced to achieve an escorting behavior: maintenance of a minimum distance between a robot and the escort target and an area coverage method based on Centroidal Voronoi Tessellations. Tests were meade using the Player/Stage simulator, and they show that the robots not only are capable of adapting themselves but also are able to use the stored knowledge to improve their effectiveness in doing the desired task

In this thesis, we consider stationary random mosaics with a focus on the Poisson-Voronoi mosaic and the Poisson-Delaunay mosaic. We consider properties of stationary random mosaics in R2, such as mean value results of the typical cell. Further, we simulate various mean value results of the typical cell, a random neighbor of the typical cell, and the zero cell for the Poisson-Voronoi mosaic in R2. Some theory of point processes is introduced that is needed for random mosaics, including Palm theory, marked point processes, and the Pois point process. Finally, we consider an incremental flip-based algorithm for generating the Voronoi mosaic.

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Comprendre l’origine microscopique du ralentissement de la dynamique au voisinage de la transition vitreuse reste l’un des problèmes fondamentaux de la physique de la matière condensée. Au cours de ce travail, nous introduisons un nouveau modèle de liquide, appelé liquide de Voronoï, et dont les interactions sont directement reliées aux propriétés géométriques des tessellations de Voronoï. Pour cette classe de liquides, les interactions sont à plusieurs corps et agissent de telle sorte que le système est toujours sous tension tout en restant stable. Le but de ce travail est d’étudier un mélange binaire du liquide de Voronoï et de voir de quelles façons ces interactions exotiques affectent le scénario habituel de la transition vitreuse. Tout au long de ce travail, nous caractérisons le liquide de Voronoï bidisperse théoriquement et par le biais des simulations numériques. Nous proposons également des comparaisons avec des liquides de Lennard-Jones surfondus bien décrit dans la littérature. / Understanding the origin of the important slowing down of the dynamics near glass transition is still one of the remaining fundamental problems of condensed matter physics. During this work we introduced a brand-new model of liquids named Voronoi liquid, whose interactions are directly related to the geometrical properties of Voronoi tessellations. For these class of liquids interactions are intrinsically manybody and act in such a way that the liquid is always under tension but remains stable. The aim of this work is to use a binary mixture of the Voronoi liquid to see to what extend these exotic interactions may affect the classical scenario of glass transition. Throughout this work we characterize theoretically and by mean of numerical simulation the bidisperse Voronoi liquid. Comparisons with well-known Lennard-Jones glass formers are systematically performed.

Orientador: Prof. Dr. Daniel Miranda Machado / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional, 2014. / Este trabalho tem como objetivo de pesquisa fazer um estudo sobre a aplicação da geometria das transformações (em especial das isometrias) e da pavimentação do plano com polgonos regulares em atividades que fazem uso da Geometria Dinâmica como recurso para visualização e manuseio de objetos geometricos. São apresentados inicialmente os pressupostos teoricos dos conceitos envolvidos com algumas de suas principais denic~oes e propriedades. Em seguida, o trabalho de M.C.Escher e exposto para exemplicar o uso da parte teorica nas artes com o intuito de motivar a apreciação do conteudo estudado. Uma situação-problema e colocada como meio para desenvolver a pesquisa atraves do software Geogebra que foi utilizado para auxiliar a execucão das atividades propostas. Atraves do desenvolvimento do presente estudo, foi possvel constatar que ha um interesse muito maior dos alunos e um consenso por parte de todos de que a utilizac~ao de recursos visuais e tecnologicos no ensino de matematica e mais agradavel e estimulante. / This work aims to make a research study on the application of geometry transformations (especially isometries) and paving the plane with regular polygons in activities that make use of Dynamic Geometry as a resource for viewing and handling of geometric objects. The theoretical assumptions of the concepts involved with some of its main denitions and properties are initially presented. Then the work M.C.Escher is exposed to exemplify the use of the theoretical part in the arts in order to motivate the assessment of the content studied. A problem situation is placed as a means to develop the research through the Geogebra software that was used to assist the implementation of the proposed activities. Through the development of this study, it was found that there is a much greater student interest and consensus by all that the use of visual and technological resources in teaching mathematics is more enjoyable and stimulating.

Made available in DSpace on 2014-07-29T16:02:16Z (GMT). No. of bitstreams: 1 Dissertacao ana paula faria machado.pdf: 253816 bytes, checksum: 17cd09106048916ae2c14d5f373c0dff (MD5) Previous issue date: 2008-02-15 / In this dissertation we define geometrically uniform codes in Euclidean spaces, presenting some symmetrical properties of these codes. Build geometrically uniforme partitions and starting these we define generalized coset codes. We generalize the concept of geometrically uniform codes to hyperbolic spaces. We show a characterization of generalized coset codes through the concept of G-linear codes. / Nesta dissertação definimos códigos geometricamente uniformes em espaços euclideanos, apresentando algumas propriedades simétricas destes códigos. Construímos partições geometricamente uniformes e a partir destas definimos classes laterais de códigos generalizadas. Estendemos aos espaços hiperbólicos o conceito de códigos geometricamente uniformes. Mostramos também uma caracterização de classes laterais generalizadas através do conceito de códigos G-lineares.

Origami-based approaches for the folding of thick materials for specific application to large deployable space arrays is explored in this work. The folding approaches presented utilize strain energy, spatial kinematics, membranes, compliant mechanisms, and or in combination together to fold finite-thickness materials viewed through the lens of origami-based engineering. Novel architectures and behaviors of mechanisms are developed to achieve packaging efficiency, deployment, and self-stiffening. A method for the folding of monolithic thick-sheet materials is developed by incorporating compliant mechanisms into the material itself to strategically add degrees of freedom. The design and characterization of the compliant mechanisms with consideration to stress, material selection, and stiffness is given. Other folding approaches developed include a bistable vertex and a double-membrane method.The folding approaches derived are applied to larger tessellations and folding patterns. The fold patterns developed and used lend themselves well to large reconfiguration and the combination of the folding approaches with the patterns create opportunities to fabricate products out of thick, functional materials. Of specific interest is the application of these approaches and patterns to the field of deployable space arrays. Spatial kinematics, computational dynamics, physical tests, and systems engineering are used to develop an array architecture that is self-deployable, self-stiffening, and retractable. This architecture is shown to open the design space of large deployable arrays by increasing packaging efficiency and mass.The method, approaches, and architectures developed by this dissertation contribute to the fields origami-based engineering and deployable space arrays. While a focus of this work is the advancement of space technologies, the depth of the analyses provided are transferable to other origami-based and compliant-mechanism disciplines.