A brief explanation of RNA and its general structure on dierent levels is given. The standard continuous Frenet frame is explained. A discrete version of the Frenet frame is explained in detail and constructed for a piecewise linear curve. The results of the application of the discrete Frenet frame to RNA is shown in the form of several distributions. An analysis of these distributions is conducted and gives some results regarding tiny structures in RNA. / <p>Master Degree Project thesis</p>

Proteins in nature fold to one dominant native structure. Despite being a heavily studied field, predicting the native structure from the amino acid sequence and modeling the folding process can still be considered unsolved problems. In this thesis I present a new approach to this problem with methods borrowed from theoretical physics. In the first part I show how it is possible to use a discrete Frenet frame to define the discrete curvature and torsion of the main chain of the protein. This method is then extended to the side chains as well. In particular I show how to use the discrete Frenet frame to produce a statistical distribution of angles that works in similar fashion as the commonly used Ramachandran plot and side chain rotamers. The discrete Frenet frame displays a gauge symmetry, in the choice of basis vectors on the normal plane, that is reminiscent of features of Abelian-Higgs theory. In the second part of the thesis I show how this similarity with Abelian-Higgs theory can be translated into an effective energy for a protein. The loops of the proteins are shown to correspond to solitons so that the whole protein can be constructed by gluing together any number of solitons. I present results of simulating proteins by minimizing the energy, starting from a real line or straight helix, where the correct native fold is attained. Finally the model is shown to display the same phase structure as real proteins.

This thesis work presents a model predictive control (MPC) based trajectory planner forhigh speed lane change and low speed parking scenarios of autonomous four wheel steering(4WS) vehicle. A four wheel steering vehicle has better low speed maneuverabilityand high speed stability compared with normal front wheel steering(FWS) vehicles. TheMPC optimal trajectory planner is formulated in a curvilinear coordinate frame (Frenetframe) minimizing the lateral deviation, heading error and velocity error in a kinematicdouble track model of a four wheel steering vehicle. Using the proposed trajectory planner,simulations show that a four wheel steering vehicle is able to track different type ofpath with lower lateral deviations, less heading error and shorter longitudinal distance. / I detta avhandlingsarbete presenteras en modellbaserad prediktiv kontroll (MPC) -baseradbanplaneringsplan f¨or h¨oghastighetsbanan och l°aghastighetsparametrar f¨or autonomtfyrhjulsdrift (4WS). Ett fyrhjulsdrivna fordon har b¨attre man¨ovrerbarhet med l°ag hastighetoch h¨oghastighetsstabilitet j¨amf¨ort med vanliga fr¨amre hjulstyrningar (FWS). MPC-optimalbanplanerare ¨ar formulerad i en kr¨okt koordinatram (Frenet-ram) som minimerar sidof¨orl¨angningen,kursfel och hastighetsfel i en kinematisk dubbelsp°armodell av ett fyrhjulsstyrda fordon.Med hj¨alp av den f¨oreslagna banaplaneraren visar simuleringar att ett fyrhjulsstyrfordonkan sp°ara olika typer av banor med l¨agre sidof¨orl¨angningar, mindre kursfel ochkortare l¨angsg°aende avst°and.

La plupart des méthodes de surveillance des systèmes sont basées sur l'analyse et la caractérisation de grandeurs physiques qui sont par nature tridimensionnelles. Tracées dans un repère euclidien à trois dimensions, ces grandeurs parcourent en fonction du temps une trajectoire dont les caractéristiques géométriques sont représentatives de l'état du système surveillé. Les techniques classiques de surveillance des systèmes étudient les grandeurs mesurées composante par composante, sans prendre en compte leur nature tridimensionnelle et les propriétés géométriques de leur trajectoire. Une part importante de l'information est ainsi ignorée. Dans le cadre de ce travail de recherche, on se propose de développer une méthode d'analyse et de traitement de grandeurs à trois composantes permettant de mettre en évidence les spécificités géométriques des données et de fournir une information complémentaire pour la surveillance des systèmes. La méthode proposée a été appliquée à deux cas différents : la surveillance des creux de tension des réseaux de puissance triphasés et la surveillance des défauts de roulement des machines électriques tournantes. Dans ces deux cas, les résultats obtenus sont prometteurs et montrent que les indicateurs géométriques estimés mènent à de l'information complémentaire qui peut être utile pour la surveillance des systèmes. / Most methods for condition monitoring are based on the analysis and characterization of physical quantities that are three-dimensional in nature. Plotted in a three-dimensional Euclidean space as a function of time, these quantities follow a trajectory whose geometric characteristics are representative of the state of the monitored system. Usual techniques of condition monitoring study the measured quantities component by component, without taking into account their three-dimensional nature and the geometric properties of their trajectory. A significant part of the information is thus ignored. In this research work, we would therefore like to develop a method for the analysis and processing of three-component quantities capable of highlighting the special geometric features of such data and providing complementary information for condition monitoring. The proposed method has been applied to two different cases: voltage dips monitoring in three-phase power networks and bearing faults monitoring in rotating electrical machines. In this two cases, the results obtained are promising and show that the estimated geometric indicators lead to complementary information that can be useful for condition monitoring.