L'objectif de ce travail de thèse est de concevoir des algorithmes de commande et de navigation pour le suivi des cibles dynamiques au sol en utilisant des véhicules aériens. Les quaternions, qui fournissent une alternative aux représentations classiques de la dynamique des véhicules aériens, ont été choisis comme une base pour développer des contrôleurs robustes et des algorithmes de navigation agile, en raison de leurs avantages tels que l'absence de singularités et discontinuités, et leur simplicité mathématique lors de la manipulation des rotations. Les approches de commande explorées à l'aide de quaternions dans cette thèse commencent par le retour d'état, la passivité, et des contrôleurs basés sur l'énergie, jusqu'à des modes glissants, et des approches de saturation en trois dimensions. Ensuite, des stratégies de navigation autonomes et semi-autonomes pour quadrirotors ont été explorées. Un algorithme a été développé pour le pilotage d'un quadrirotor en utilisant des gestes d'un utilisateur portant un bracelet. Afin de faciliter le fonctionnement des multi rotors dans des scénarios défavorables, une stratégie de déploiement agressive a été proposée ou un quadrirotor est lancé à la main avec ses moteurs éteints. Finalement, des techniques de navigation autonomes pour le suivi des cibles dynamiques ont été conçues. Un algorithme de génération de trajectoire basée sur des équations différentielles a été introduit pour le suivi d'un véhicule terrestre tout en décrivant des cercles. Enfin un algorithme de planification de chemin distribué a été développé pour une flottille de drones, afin de suivre de façon autonome des cibles au sol, en résolvant un problème d'optimisation en ligne. / The objective of this thesis work is to design control and navigation algorithms for tracking of dynamic ground targets using aerial vehicles. Quaternions, which provide an alternative to the classical representations of aerial vehicle dynamics, have been chosen as a basement to develop robust controllers and agile navigation algorithm, due to their advantages such as the absence of singularities and discontinuities and their mathematical simplicity when handling rotations. The quaternion-based control approaches explored in this thesis range from state feedback, passivity, and energy-based controllers, up to sliding modes, and three-dimensional saturation approaches. Then, autonomous and semi-autonomous navigation strategies for quadrotors were explored. An algorithm has been developed for controlling a quadrotor using gestures from a user wearing an armband. To facilitate the operation of multirotors in adverse scenarios, an aggressive deployment strategy has been proposed where a quadrotor is launched by hand With its motors turned off. Finally, autonomous navigation techniques for tracking dynamic targets have been designed. A trajectory generation algorithm based on differential equations has been introduced to track a land vehicle while describing circles. Finally a distributed path planning algorithm has been developed for a fleet of drones to autonomously track ground targets by solving an online optimization problem.

The problem addressed in this research is the in-flight generation of trajectories for autonomous unmanned aircraft, which requires a method of generating pseudo-optimal trajectories in near-real-time, on-board the aircraft, and without external intervention. The focus of this research is the enhancement of a particular inverse dynamics direct method that is a candidate solution to the problem. This research introduces the following contributions to the method. A quaternion-based inverse dynamics model is introduced that represents all orientations without singularities, permits smooth interpolation of orientations, and generates more accurate controls than the previous Euler-angle model. Algorithmic modifications are introduced that: overcome singularities arising from parameterization and discretization; combine analytic and finite difference expressions to improve the accuracy of controls and constraints; remove roll ill-conditioning when the normal load factor is near zero, and extend the method to handle negative-g orientations. It is also shown in this research that quadratic interpolation improves the accuracy and speed of constraint evaluation. The method is known to lead to a multimodal constrained nonlinear optimization problem. The performance of the method with four nonlinear programming algorithms was investigated: a differential evolution algorithm was found to be capable of over 99% successful convergence, to generate solutions with better optimality than the quasi- Newton and derivative-free algorithms against which it was tested, but to be up to an order of magnitude slower than those algorithms. The effects of the degree and form of polynomial airspeed parameterization on optimization performance were investigated, and results were obtained that quantify the achievable optimality as a function of the parameterization degree. Overall, it was found that the method is a potentially viable method of on-board near- real-time trajectory generation for unmanned aircraft but for this potential to be realized in practice further improvements in computational speed are desirable. Candidate optimization strategies are identified for future research.

The goal of this project was to make a shirt with three embedded IMU sensors (Inertial Measurement Unit) that can measure a person’s movements throughout an entire workday. This can provide information about a person’s daily routine movements and aid in finding activities which can lead to work-related injuries in order to prevent them. The objective was hence to construct a sensor fusion framework that could retrieve the measurements from these three sensors and to create an estimate of the human body orientation and to estimate the angular movements of the arms. This was done using an extended Kalman filter which uses the accelerometer and magnetometer values to retrieve the direction of gravity and north respectively, thus providing a coordinate system that can be trusted in the long term. Since this method is sensitive to quick movements and magnetic disturbance, gyroscope measurements were used to help pick up quick movements. The gyroscope measurements need to be integrated in order to get the angle, which means that we get accumulated errors. This problem is reduced by the fact that we retrieve a correct long-term reference without accumulated errors from the accelerometer and magnetometer measurements. The Kalman filter estimates three quaternions describing the orientation of the upper body and the two arms. These quaternions were then translated into Euler angles in order to get a meaningful description of the orientations. The measurements were stored on a memory card or broadcast on both the local net and the Internet. These data were either used offline in Matlab or shown in real-time in the program Unity 3D. In the latter case the user could see that a movement gives rise to a corresponding movement on a skeleton model on the screen.

Les travaux de cette thèse s'inscrivent dans le contexte du traitement et de l'analyse des images couleur. Les premiers travaux pour traiter ces images consistaient à appliquer des traitements déjà existant en niveaux de gris marginalement sur les trois composantes constituant la couleur et le plus généralement dans l'espace RVB. Ces traitements ont été peu à peu améliorés notamment par l'utilisation d'espaces couleur d'avantage liés à la perception humaine mais aussi par des approches vectorielles. Dans ce travail de thèse nous nous plaçons dans la continuité de ces travaux et nous proposons une modélisation mathématique de la dimension vectorielle dans le but de manipuler les couleurs de manière globale. Trois formalismes sont présentés pour représenter la couleur : les complexes, les quaternions et les algèbres géométriques. Dans ce cadre, il est proposé de définir de nouveaux outils d'analyse couleur avec notamment une caractérisation numérique fréquentielle de chacun de ces modèles. Une étude approfondie de leurs utilisations permet de faire ressortir leurs propriétés ainsi que leurs principaux avantages et inconvénients à savoir : impossibilité des complexes à représenter les vecteurs couleurs qui par nature s'expriment en trois dimensions minimum contrairement aux quaternions et aux algèbres géométriques ; distinction entre objets manipulés (vecteurs couleur) et opérations effectuées sur ces objets (projections, rotations,...) pour les algèbres géométriques contrairement aux quaternions... Enfin nous avons montré que la transformée de Fourier quaternionique analyse la couleur avec une direction indiquée par un vecteur couleur, tandis que la transformée de Fourier définie au moyen de l'algèbre G3, plus générique, répartit l'information couleur sur des composantes fréquentielles indépendantes. L'utilisation de modèles algébriques pour représenter l'information couleur permet la définition et le développement d'un filtre spatial de détection de contours tenant compte de la dispersion dans l'espace couleur.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we study the notion of R-algebra. Roughly, they are structures that generalize some arithmetic properties of the body of complex numbers. The ?exibi- lity in this generalization is the non-requirement of properties such as commutativity, associativity and identity element existence. We focus primarily on the ?nite dimen- sional division R-algebras. As is well known, modulo isomorphisms exist exactly four of those R-algebras. In the development of the dissertation we will discuss in detail its main algebraic and geometric properties. / Nesse trabalho estudamos a noção de R-álgebra. A grosso modo, elas são es- truturas que generalizam algumas propriedades aritméticas do corpo dos números complexos. A ?exibilidade nessa generalização é a não exigência de propriedades como comutatividade, associatividade e existência de elemento identidade. Focamos principalmente nas R-álgebras de divisão de dimensão ?nita. Como é bem conhe- cido, módulo isomor?smos existem exatamente quatro dessas R-álgebras. No desen- volvimento da dissertação discutiremos detalhadamente suas principais propriedades algébricas e geométricas.

Orientadores: Reginaldo Palazzo Jr., Mercio Botelho Faria / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-08T06:25:10Z (GMT). No. of bitstreams: 1 Vieira_VandenbergLopes_D.pdf: 990187 bytes, checksum: 2212b8074f5503f78aa813ce4422cc4b (MD5) Previous issue date: 2007 / Resumo: Dentro do contexto de projetar sistema de comunicação digital em espaços homogêneos, em particular, em espaços hiperbólicos, é necessário estabelecer um procedimento sistemático para construção de reticulados O, como elemento base para construção de constelações de sinais geometricamente uniformes. E através desse procedimento que identificamos as estruturas algébrica e geométrica além de construir códigos geometricamente uniformes em espaços homogêneos. Propomos, a partir desses reticulados, a construção de grupos fuchsianos aritméticos Tp obtidos de tesselações hiperbólicas {p; q}, derivados de álgebras de divisão dos quaternios A sobre corpos de números K. Generalizamos o processo de identificação desses grupos em ordens dos quatérnios (reticulados hiperbólicos), associadas às constelações de sinais geometricamente uniformes, provenientes de grupos discretos. Esse procedimento permite rotular os sinais das constelações construídas por elementos de uma estrutura algébrica / Abstract: Within the context of digital communications system in homogeneous space in particular, in hyperbolic spaces, it is necessary to establish systematic procedure for the construction of lattices O ; as the basic entity for construction of eometrically uniforms signal constellations. By this procedure we identify the algebraic and geometric structures to construct geometrically uniforms codes in homogeneous spaces. We propose, from lattices, the construction of arithmetic fuchsian groups ¡p obtained by hyperbolic tessellations {p; q}, derived from division quaternion algebras A over numbers fields K. We generalize the process of identification of these groups in quaternion orders (hyperbolic lattices), which are associated with geometrically uniforms signal constellations, proceeding from discrete groups. This procedure allows us to realize the labelling of the signals belonging to such constellations by elements of an algebraic structure / Doutorado / Telecomunicações e Telemática / Doutor em Engenharia Elétrica

Orientador: Waldyr Alves Rodrigues Junior / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-15T23:09:32Z (GMT). No. of bitstreams: 1 Resende_AdrianaSouza_M.pdf: 17553204 bytes, checksum: a66cefe30e9957cc4351e03d3aec35b2 (MD5) Previous issue date: 2010 / Resumo: O presente trabalho tem a intenção de apresentar por intermédio de uma linguagem unificada alguns conceitos de cálculo vetorial, álgebra linear (matrizes e transformações lineares) e também algumas idéias elementares sobre os grupos de rotações em duas e três dimensões e seus grupos de recobrimento, que geralmente são tratados como "fragmentos" em várias modalidades de cursos no ensino superior. Acreditamos portanto que nosso texto possas ser útil para alunos dos cursos de graduação dos cursos de Engenharia, Física, Matemática e interessados em Matemática em geral. A linguagem unificada à que nos referimos acima é obtida com a introdução do conceitos das álgebras geométricas (ou de Clifford) onde, como veremos, é possível fornecer uma formulação algébrica elegante aos conceitos de vetores, planos e volumes orientados e definir para tais objetos o produto escalar, os produtos contraídos à esquerda e à direita, o produto exterior (associado, como veremos, em casos particulares ao produto vetorial) e finalmente o produto geométrico (Clifford), o que permite o uso desses conceitos para a solução de inúmeros problemas de geometria analítica no R ² e no R ³. Procuramos ilustrar todos estes conceitos com vários exemplos e exercícios com graus variáveis de dificuldades. Nossa apresentação é bem próxima àquela do livro de Lounesto, e de fato muitas seções são traduções (eventualmente seguidas de comentários) de seções daquele livro. Contudo, em muitos lugares, acreditamos que nossa apresentação esclarece e completa as correspondentes do livro de Lounesto / Abstract: This paper aims to present using an unified language a few concepts of vector calculus, linear algebra (matrices and linear transformations) and also some basic ideas about the groups of rotations in two and three dimensions and their covering group, which generally are treated as "fragments" in various types of courses in higher education. We believe therefore that our text should be useful to students of undergraduate courses like Engineering, Physics, Mathematics and people interested in Mathematics in general. The unified language that we refer to above is obtained by introducing the concept of geometric (or Clifford) algebra where, as we shall see, it is possible to give an elegant algebraic formulation to the concepts of vectors, oriented planes and oriented volumes, and to define to those objects the scalar product, the right and left contracted products, the exterior product (associated, as we shall see, in particular cases to the vector product) and finally the geometric (Clifford) product, and moreover, to use those concepts to solve may problems of analytic geometry in R ² and R ³. We illustrated all those concepts with several examples and exercises with variable degrees of difficulties. Our presentation is nearly the one in Lounesto's book, and in fact some sections are no more than translations (eventually with commentaries) from sections of that book. However, in many places, we believe that our presentation clarify nd completement the corresponding ones in Lounesto's book / Mestrado / Ágebra / Mestre em Matemática

Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2016-12-15T15:01:25Z No. of bitstreams: 2 Dissertação - Davi José dos Santos - 2016.pdf: 5567090 bytes, checksum: 5606aa47f640cc5cd4495d2694f38cda (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Jaqueline Silva (jtas29@gmail.com) on 2016-12-15T17:28:21Z (GMT) No. of bitstreams: 2 Dissertação - Davi José dos Santos - 2016.pdf: 5567090 bytes, checksum: 5606aa47f640cc5cd4495d2694f38cda (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2016-12-15T17:28:21Z (GMT). No. of bitstreams: 2 Dissertação - Davi José dos Santos - 2016.pdf: 5567090 bytes, checksum: 5606aa47f640cc5cd4495d2694f38cda (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-08-30 / This research with theoretical approach seeks to investigate inmathematics, octonions,which is a non-associative extension of the quaternions. Its algebra division 8-dimensional formed on the real numbers is more extensive than can be obtained by constructing Cayley-Dickson. In this perspective we have as main goal to answer the following question: "What number systems allow arithmetic operations addition, subtraction, multiplication and division? " In the genesis of octonions is the Irish mathematician William Rowan Hamilton, motivated by a deep belief that quaternions could revolutionize mathematics and physics, was the pioneer of a new theory that transformed the modern world. Today, it is confirmed that the complexs/quaternions/octonions and its applications are manifested in different branches of science such as mechanics, geometry, mathematical physics, with great relevance in 3D animation and robotics. In order to investigate the importance of this issue and make a small contribution, we make an introduction to the theme from the numbers complex and present the rationale and motivations of Hamilton in the discovery of quaternions/octonions. Wemake a presentation of the algebraic structure and its fundamental properties. Then discoremos about constructing Cayley-Dickson algebras that produces a sequence over the field of real numbers, each with twice the previous size. Algebras produced by this process are known as Cayley-Dickson algebras; since they are an extension of complex numbers, that is, hypercomplex numbers. All these concepts have norm, algebra and conjugate. The general idea is that the multiplication of an element and its conjugate should be the square of its norm. The surprise is that, in addition to larger, the following algebra loses some specific algebraic property. Finally, we describe and analyze certain symmetry groups with multiple representations through matrixes and applications to show that This content has a value in the evolution of technology. / Esta pesquisa com abordagem teórica busca investigar na matemática, os octônios, que é uma extensão não-associativa dos quatérnios. Sua álgebra com divisão formada de 8 dimensões sobre os números reais é a mais extensa que pode ser obtida através da construção de Cayley-Dickson. Nessa perspectiva temos comometa principal responder a seguinte questão: "Que sistemas numéricos permitemas operações aritméticas de adição, subtração, multiplicação e divisão?" Na gênese dos octônios está o matemático irlandêsWilliam Rowan Hamilton que, motivado por uma profunda convicção de que os quatérnios poderiam revolucionar a Matemática e a Física, foi o pioneiro de uma nova teoria que transformou o mundo moderno. Hoje, confirma-se que os complexos/quatérnios/octônios e suas aplicações se manifestam em diferentes ramos da ciências tais como a mecânica, a geometria, a física matemática, com grande relevância na animação 3D e na robótica. Com o propósito de investigar a importância deste tema e dar uma pequena contribuição, fazemos uma introdução ao tema desde os números complexos e apresentamos o raciocínio e motivações de Hamilton na descoberta dos quatérnios/octônios. Fazemos uma apresentação da estrutura algébrica, bem como as suas propriedades fundamentais. Emseguida discoremos sobre a construção de Cayley-Dickson que produz uma sequência de álgebras sobre o campo de números reais, cada uma com o dobro do tamanho anterior. Álgebras produzidas por este processo são conhecidas como álgebras Cayley-Dickson; uma vez que elas são uma extensão dos números complexos, isto é, os números hipercomplexos. Todos esses conceitos têm norma, álgebra e conjugado. A idéia geral é que o produto de um elemento e seu conjugado deve ser o quadrado de sua norma. A surpresa é que, além de maior dimensão, a álgebra seguinte perde alguma propriedade álgebrica específica. Por fim, descrevemos e analisamos alguns grupos de simetria, com várias representações através de matrizes e aplicações que demonstram que este conteúdo tem uma utilidade na evolução da tecnologia.

Most mobile video-recording devices of today, e.g. cell phones and music players, make use of a Rolling Shutter camera. The camera captures video by recording every frame line-by-line from top to bottom leading to image distortion when either the target or camera is moving. Capturing video by hand also leads to visible frame-to-frame jitter. This thesis presents algorithms for estimation of camera orientations using accelerometer and gyroscope. These estimates can be used to reduce the image distortion caused by camera motion using image processing. In addition an adaptive low pass filtering algorithm used to produce a smooth camera motion is presented. Using the smooth motion the frame-to-frame jitter can be reduced. The algorithms are implemented on the iPod 4 and two output videos are evaluated in a blind experiment with 30 participants. Here, videos are compared to those of competing video stabilization software. The results indicate that the iPod 4 application performs equal or better than its competitors. Also the iPod 4 accelerometer and gyroscope are compared to high end reference sensors in terms of bias and variance. / Det är vanligt att dagens mobiltelefoner använder en kamera med rullande slutare för videoinspelning. Dessa kameror fångar varje bild rad för rad från topp till botten vilket leder till bilddistorsion när antingen målet eller kameran rör sig. Inspelade videor upplevs även som skakiga eftersom kameran ej är stilla under inspelningen. I detta examensarbete presenteras algoritmer för skattning av kamerans orientering med hjälp av accelerometer och gyroskop. Skattningarna används sedan för att reducera bilddistorsionen som uppstår då kameran rör sig. En adaptiv algoritm för lågpassfiltrering av kamerans rörelse presenteras. Den mjuka rörelsen som produceras används sedan för att reducera skakigheten i filmerna. Algoritmerna implementeras på iPod 4 och de resulterande filmerna utvärderas i ett blindtest med 30 deltagare. Filmerna jämförs med versioner stabiliserade av konkurrerande mjukvara. Resultaten visar att iPod-applikationen producerar ett lika bra eller bättre resultat än konkurrenterna. Accelerometern och gyroskopet på iPod 4 jämförs även med referenssensorer i den högre prisklassen.

A pair (V, G) is called geometric structure, where V is a vector space and G is a subgroup GL(V), which is a set of transmission matrices. In this thesis we classify structures, which are based on properties of quaternions. Geometric structures based on quaternions are called triple structures. Triple structures are four structures with similar properties as quaternions. Quaternions are generated from real numbers and three complex units. We write quaternions in this shape a+bi+cj+dk.