SO(n) är rotationsgrupper som beskriver rotationer kring origo i motsvarande n-dimensionella rum. Vi betraktar här specifikt rotationsgrupperna SO(2) och SO(3), dessa kan representera en fast kropps rotation kring ett fixt centrum i två och tre dimensioner.   Att finna en representation av dessa grupper visar sig i två dimensioner vara lättförståeligt då vi endast behöver en rotationsvinkel för att representera SO(2). I tre dimensioner stöter vi däremot på svårigheter i form av singulariteter då vi försöker avbilda med hjälp av rotationsvinklar. För att få en bijektiv avbildning måste vi använda oss av Liegruppen SU(2).

This dissertation addresses rigid body attitude control with piecewise sinusoidal signals. We consider rigid-body attitude kinematics on SO(3) with a class of sinusoidal inputs. We present a new closed-form solution of the rotation matrix kinematics. The solution is analyzed and used to prove controllability. We then present kinematic-level orientation-feedback controllers for setpoint tracking and command following. Next, we extend the sinusoidal kinematic-level control to the dynamic level. As a representative dynamic system, we consider a CubeSat with vibrating momentum actuators that are driven by small $\epsilon$-amplitude piecewise sinusoidal internal torques. The CubeSat kinetics are derived using Newton-Euler's equations of motion. We assume there is no external forcing and the system conserves zero angular momentum. A second-order approximation of the CubeSat rotational motion on SO(3) is derived and used to derive a setpoint tracking controller that yields order O(ε2) closed-loop error. Numerical simulations are presented to demonstrate the performance of the controls. We also examine the effect of the external damping on the CubeSat kinetics. In addition, we investigate the feasibility of the piecewise sinusoidal control techniques using an experimental CubeSat system. We present the design of the CubeSat mechanical system, the control system hardware, and the attitude control software. Then, we present and discuss the experiment results of yaw motion control. Furthermore, we experimentally validate the analysis of the external damping effect on the CubeSat kinetics.

Cette thèse traite la problématique d'estimation de la pose (position relative et orientation) d'un corps rigide en mouvement dans l’espace 3D par fusion de données issues de capteurs inertiels et visuels. Les mesures inertielles sont fournies à partir d’une centrale inertielle composée de gyroscopes 3 axes et d’accéléromètres 3 axes. Les données visuelles sont issues d’une caméra. Celle-ci est positionnée sur le corps rigide en mouvement, elle fournit des images représentatives du champ visuel perçu. Ainsi, les mesures implicites des directions des lignes, supposées fixes dans l’espace de la scène, projetées sur le plan de l’image seront utilisées dans l’algorithme d’estimation de l’attitude. La démarche consistait d’abord à traiter le problème de la mesure issue du capteur visuel sur une longue séquence en utilisant les caractéristiques de l’image. Ainsi, un algorithme de suivi de lignes a été proposé en se basant sur les techniques de calcul du flux optique des points extraits des lignes à suivre et utilisant une approche de mise en correspondance par minimisation de la distance euclidienne. Par la suite, un observateur conçu dans l’espace SO(3) a été proposé afin d’estimer l’orientation relative du corps rigide dans la scène 3D en fusionnant les données issues de l’algorithme de suivi de lignes avec les données des gyroscopes. Le gain de l’observateur a été élaboré en utilisant un filtre de Kalman de type M.E.K.F. (Multiplicative Extended Kalman Filter). Le problème de l’ambigüité du signe dû à la mesure implicite des directions des lignes a été considéré dans la conception de cet observateur. Enfin, l’estimation de la position relative et de la vitesse absolue du corps rigide dans la scène 3D a été traitée. Deux observateurs ont été proposés : le premier est un observateur en cascade avec découplage entre l’estimation de l’attitude et l’estimation de la position. L’estimation issue de l’observateur d’attitude alimente un observateur non linéaire utilisant des mesures issues des accéléromètres afin de fournir une estimation de la position relative et de la vitesse absolue du corps rigide. Le deuxième observateur, conçu quant à lui directement dans SE(3) , utilise un filtre de Kalman de type M.E.K.F afin d’estimer la pose par fusion de données inertielles (accéléromètres, gyromètres) et des données visuelles. Les performances des méthodes proposées sont illustrées et validées par différents résultats de simulation / AbstractThis thesis addresses the problems of pose estimation of a rigid body moving in 3D space by fusing data from inertial and visual sensors. The inertial measurements are provided from an I.M.U. (Inertial Measurement Unit) composed by accelerometers and gyroscopes. Visual data are from cameras, which positioned on the moving object, provide images representative of the perceived visual field. Thus, the implicit measure directions of fixed lines in the space of the scene from their projections on the plane of the image will be used in the attitude estimation. The approach was first to address the problem of measuring visual sensors after a long sequence using the characteristics of the image. Thus, a line tracking algorithm has been proposed based on optical flow of the extracted points and line matching approach by minimizing the Euclidean distance. Thereafter, an observer in the SO(3) space has been proposed to estimate the relative orientation of the object in the 3D scene by merging the data from the proposed lines tracking algorithm with Gyro data. The observer gain was developed using a Kalman filter type M.E.K.F. (Multiplicative Extended Kalman Filter). The problem of ambiguity in the sign of the measurement directions of the lines was considered in the design of the observer. Finally, the estimation of the relative position and the absolute velocity of the rigid body in the 3D scene have been processed. Two observers were proposed: the first one is an observer cascaded with decoupled from the estimation of the attitude and position estimation. The estimation result of the attitude observer feeds a nonlinear observer using measurements from the accelerometers in order to provide an estimate of the relative position and the absolute velocity of the rigid body. The second observer, designed directly in SE (3) for simultaneously estimating the position and orientation of a rigid body in 3D scene by fusing inertial data (accelerometers, gyroscopes), and visual data using a Kalman filter (M.E.K.F.). The performance of the proposed methods are illustrated and validated by different simulation results

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In flexible fitting, the high-resolution crystal structure of a molecule is deformed to optimize its position with respect to a low-resolution density map. Solving the flexible fitting problem entails answering the following questions: (A) How can the crystal structure be deformed? (B) How can the term "optimum" be defined? and (C) How can the optimization problem be solved? In this dissertation, we answer the above questions in reverse order. (C) We develop PFCorr, a non-uniform SO(3)-Fourier-based tool to efficiently conduct rigid-body correlations over arbitrary subsets of the space of rigid-body motions. (B) We develop PF2Fit, a rigid-body fitting tool that provides several useful definitions of the optimal fit between the crystal structure and the density map while using PFCorr to search over the space of rigid-body motions (A) We develop PF3Fit, a flexible fitting tool that deforms the crystal structure with a hierarchical domain-based flexibility model while using PF2Fit to optimize the fit with the density map. Our contributions help us solve the rigid-body and flexible fitting problems in unique and advantageous ways. They also allow us to develop a generalized framework that extends, breadth-wise, to other problems in computational structural biology, including rigid-body and flexible docking, and depth-wise, to the question of interpreting the motions inherent to the crystal structure. Publicly-available implementations of each of the above tools additionally provide a window into the technically diverse fields of applied mathematics, structural biology, and 3D image processing, fields that we attempt, in this dissertation, to span. / text

Recent efforts to classify off-shell representations of supersymmetry without a central charge have focused upon directed, supermultiplet graphs of hypercubic topology known as Adinkras. These encodings of Super Poincare algebras, depict every generator of a chosen supersymmetry as a node-pair transformtion between fermionic bosonic component fields. This research thesis is a culmination of investigating novel diagrammatic sums of gauge-quotients by supersymmetric images of other Adinkras, and the correlated building of field theoretic worldline Lagrangians to accommodate both classical and quantum venues. We find Ref [40], that such gauge quotients do not yield other stand alone or "proper" Adinkras as afore sighted, nor can they be decomposed into supermultiplet sums, but are rather a connected "Adinkraic network". Their iteration, analogous to Weyl's construction for producing all finite-dimensional unitary representations in Lie algebras, sets off chains of algebraic paradigms in discrete-graph and continuous-field variables, the links of which feature distinct, supersymmetric Lagrangian templates. Collectively, these Adiankraic series air new symbolic genera for equation to phase moments in Feynman path integrals. Guided in this light, we proceed by constructing Lagrangians actions for the N = 3 supermultiplet YI /(iDI X) for I = 1, 2, 3, where YI and X are standard, Salam-Strathdee superfields: YI fermionic and X bosonic. The system, bilinear in the component fields exhibits a total of thirteen free parameters, seven of which specify Zeeman-like coupling to external background (magnetic) fluxes. All but special subsets of this parameter space describe aperiodic oscillatory responses, some of which are found to be surprisingly controlled by the golden ratio, [phi] = 1.61803, Ref [52]. It is further determined that these Lagrangians allow an N = 3 - > 4 supersymmetric extension to the Chiral-Chiral and Chiral-twistedChiral multiplet, while a subset admits two inequivalent such extensions. In a natural proiii gression, a continuum of observably and usefully inequivalent, finite-dimensional off-shell representations of worldline N = 4 extended supersymmetry are explored, that are variate from one another but in the value of a tuning parameter, Ref [53]. Their dynamics turns out to be nontrivial already when restricting to just bilinear Lagrangians. In particular, we find a 34-parameter family of bilinear Lagrangians that couple two differently tuned supermultiplets to each other and to external magnetic fluxes, where the explicit parameter dependence is unremovable by any field redefinition and is therefore observable. This offers the evaluation of X-phase sensitive, off-shell path integrals with promising correlations to group product decompositions and to deriving source emergences of higher-order background flux-forms on 2-dimensional manifolds, the stacks of which comprise space-time volumes. Application to nonlinear sigma models would naturally follow, having potential use in M- and F- string theories.