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This thesis aims at investigating numerical methods through which the accuracy in landing of a Mars entry-descent-landing (EDL) vehicle can be improved. The methods investigated include the collocation method and model predictive control (MPC). The primary control variable utilized in this study is the bank angle of the spacecraft, which is the angle between the lift vector and the vertical direction. Modulating this vector affects the equations of system of equations and the seven state variables, namely altitude, velocity, latitude, longitude, flight path angle, heading angle and total time taken. An optimizer is implemented which utilizes the collocation method, through which the optimal bank angle is found at every discretized state along the trajectory which are equally separated through a definite timestep, which is a function of the end time state. A 3-sigma wind disturbance model is introduced to the system, as a function of the altitude, which introduces uncertainties to the system, resulting in a final state deviating from the targeted location. The trajectory is split into two parts, for better control of the vehicle during the end stages of flight. The MPC aims at reducing the end state deviation, through the implementation of a predictor-corrector algorithm that propagates the trajectory for a certain number of timesteps, followed by running the optimizer from the current disturbed state to the desired target location. At the end of this analysis, a new set of optimal bank angle are found, which account for the wind disturbances and navigates the EDL vehicle to the desired location. / M.S. / Landing on Mars has always been a process of following a set of predetermined instructions by the spacecraft, in order to reach a calculated landing target. This work aims to take the first steps towards autonomy in maneuvering the spacecraft, and finding a method by which the vehicle navigates itself towards the target. This work determines the optimal control scheme a Mars reentry vehicle must have through the atmosphere to reach the target location, and employs method through which the uncertainty in the final landing location is mitigated. A model predictive controller is employed which corrects the disturbed trajectory of the vehicle at certain timesteps, through which the previously calculated optimal control is changed so as to account for the disturbances. The control is achieved by means of changing the bank angle of the spacecraft, which in turn affects the lift and drag experienced by the vehicle. Through this work, a method has been demonstrated which reduces the uncertainty in final landing location, even with wind disturbances present.

In pursuit of producing robots capable of achieving the dexterity exhibited by animals in nature, roboticists have begun to explore the application of robotic tails. This thesis will explore the design, optimization, construction, and implementation of an articulated serpentine robotic tail. Numerous serpentine tail prototypes have been designed and tested; however, they have not yet been integrated with a mobile base. The main challenges preventing the incorporation of serpentine tails with mobile bases include: (1) the large size and inflexible packaging associated with the actuation unit for the tail, (2) the relatively low power to weight ratios of the existing serpentine tail systems, and (3) the complexity of optimizing the tails physical parameters. Therefore, to address these issues, a novel layout for a serpentine robotic tail actuation unit along with a design optimization methodology for the tail are proposed. The actuation unit will feature a power dense and modular design which allows for flexibility in packaging. Simulation results along with experimental data gathered using a prototype of the design will be reviewed in order to quantify the performance of the actuation unit. Following, a design optimization methodology which uses a modified direct collocation technique will be presented. The optimization allows for the simultaneous optimization of both a trajectory and the physical structure of a tail. Representative results of this technique will be presented and compared against more traditional methods for design optimization. To conclude the on-going and future work for both the actuation unit and optimization methodology will be stated. / Master of Science / Robotic tails largely fall into two categories based on their construction. These two categories are pendulum and serpentine structure. Pendulum structure tails consist of a long rigid rod with a weight attached to the end of it which can be swung to assist in controlling the orientation of the base which it is attached to. Serpentine tails are characterized by their ability to articulate and move in three dimensions similar to cat or monkey tails. The non-rigid structure of the tail opens up many new possibilities for their use. However, these possibilities come at the cost of design complexity. To date this complexity has led to designs for serpentine tails which are too heavy or unwieldy to be easily added to a mobile base. Additionally, the complexity of the tail structure itself make it difficult to optimize the design as has been done previously with pendulum designs. In an effort to overcome these challenges this thesis presents a novel design for a tail actuation unit and design optimization methodology. The actuation unit design is more power dense and provides greater flexibility in its layout than previous designs. This makes it much easier to adapt to and integrate with a mobile base. This will be demonstrated through the creation of a prototype tailed quadruped featuring the new actuation unit. The optimization methodology will use a technique known as direct collocation which has previously been developed for optimal path planning. This technique accommodates the complexities of serpentine tail designs and allows for the parameters such as length and weight of the tail to be optimized. The conclusion of the thesis will present the on-going and future work for both the actuation unit and optimization technique.

The solutions for a class of Neutral Functional Di erential Equations (NFDE) with weakly singular kernels are studied. Using singular expansion techniques, a representation of the solution of the NFDE is obtained by studing an associated Volterra Integral Equation. We study the Collocation Method as a projection method for the approximation of solutions for Volterra Integral Equations. Particulary, the possibility of achieving higher order ap- proximations is discussed. Special attention is given to the choice of the projection space and its relation to the smoothness of the approximated solution. Finally, we study the identification problem for a parameter appearing in the weakly singular operator of the NFDE. / Ph. D.

As simulation plays an increasingly central role in modern science and engineering research, by supplementing experiments, aiding in the prototyping of engineering systems or informing decisions on safety and reliability, the need to quantify uncertainty in model outputs due to uncertainties in the model parameters becomes critical. However, the statistical characterization of the model parameters is rarely known. In this thesis, we propose a variational approach to solve the stochastic inverse problem of obtaining a statistical description of the diffusion coefficient in an elliptic partial differential equation, based noisy measurements of the model output. We formulate the parameter identification problem as an infinite dimensional constrained optimization problem for which we establish existence of minimizers as well as first order necessary conditions. A spectral approximation of the uncertain observations (via a truncated Karhunen-Loeve expansion) allows us to estimate the infinite dimensional problem by a smooth, albeit high dimensional, deterministic optimization problem, the so-called 'finite noise' problem, in the space of functions with bounded mixed derivatives. We prove convergence of 'finite noise' minimizers to the appropriate infinite dimensional ones, and devise a gradient based, as well as a sampling based strategy for locating these numerically. Lastly, we illustrate our methods by means of numerical examples. / Ph. D.

L'objectif de ce travail est la résolution des équations intégrales singulières à noyau Cauchy. On y traite les équations singulières de Cauchy de première espèce par la méthode des approximations successives. On s'intéresse aussi aux équations intégrales à noyau de Cauchy de seconde espèce, en utilisant les polynômes trigonométriques et les techniques de Fourier. Dans la même perspective, on utilise les polynômes de Tchebychev de quatrième degré pour résoudre une équation intégro différentielle à noyau de Cauchy. Ensuite, on s'intéresse à une autre équation intégro-différentielle à noyau de Cauchy, en utilisant les polynômes de Legendre, ce qui a donné lieu à développer deux méthodes basées sur une suite de projections qui converge simplement vers l'identité. En outre, on exploite les méthodes de projection pour les équations intégrales avec des opérateurs intégraux bornés non compacts et on a appliqué ces méthodes à l'équation intégrale singulière à noyau de Cauchy de deuxième espèce / The purpose of this thesis is to develop and illustrate various new methods for solving many classes of Cauchy singular integral and integro-differential equations. We study the successive approximation method for solving Cauchy singular integral equations of the first kind in the general case, then we develop a collocation method based on trigonometric polynomials combined with a regularization procedure, for solving Cauchy integral equations of the second kind. In the same perspective, we use a projection method for solving operator equation with bounded noncompact operators in Hilbert spaces. We apply a collocation and projection methods for solving Cauchy integro-differential equations, using airfoil and Legendre polynomials

Ce travail traite de deux problèmes de grande importances en géodésie physique. Le premier porte sur la détermination du géoïde sur une zone terrestre donnée. Si la terre était une sphère homogène, la gravitation en un point, serait entièrement déterminée à partir de sa distance au centre de la terre, ou de manière équivalente, en fonction de son altitude. Comme la terre n'est ni sphérique ni homogène, il faut calculer en tout point la gravitation. A partir d'un ellipsoïde de référence, on cherche la correction à apporter à une première approximation du champ de gravitation afin d'obtenir un géoïde, c'est-à-dire une surface sur laquelle la gravitation est constante. En fait, la méthode utilisée est la méthode de collocation par moindres carrés qui sert à résoudre des grands problèmes aux moindres carrés généralisés. Le seconde partie de cette thèse concerne un problème inverse géodésique qui consiste à trouver une répartition de masses ponctuelles (caractérisées par leurs intensités et positions), de sorte que le potentiel généré par eux, se rapproche au maximum d'un potentiel donné. Sur la terre entière une fonction potentielle est généralement exprimée en termes d'harmoniques sphériques qui sont des fonctions de base à support global la sphère. L'identification du potentiel cherché se fait en résolvant un problème aux moindres carrés. Lorsque seulement une zone limitée de la Terre est étudiée, l'estimation des paramètres des points masses à l'aide des harmoniques sphériques est sujette à l'erreur, car ces fonctions de base ne sont plus orthogonales sur un domaine partiel de la sphère. Le problème de la détermination des points masses sur une zone limitée est traitée par la construction d'une base de Slepian qui est orthogonale sur le domaine limité spécifié de la sphère. Nous proposons un algorithme itératif pour la résolution numérique du problème local de détermination des masses ponctuelles et nous donnons quelques résultats sur la robustesse de ce processus de reconstruction. Nous étudions également la stabilité de ce problème relativement au bruit ajouté. Nous présentons quelques résultats numériques ainsi que leurs interprétations. / This work focuses on the study of two well-known problems in physical geodesy. The first problem concerns the determination of the geoid on a given area on the earth. If the Earth were a homogeneous sphere, the gravity at a point would be entirely determined from its distance to the center of the earth or in terms of its altitude. As the earth is neither spherical nor homogeneous, we must calculate gravity at any point. From a reference ellipsoid, we search to find the correction to a mathematical approximation of the gravitational field in order to obtain a geoid, i.e. A surface on which gravitational potential is constant. The method used is the method of least squares collocation which is the best for solving large generalized least squares problems. In the second problem, We are interested in a geodetic inverse problem that consists in finding a distribution of point masses (characterized by their intensities and positions), such that the potential generated by them best approximates a given potential field. On the whole Earth a potential function is usually expressed in terms of spherical harmonics which are basis functions with global support. The identification of the two potentials is done by solving a least-squares problem. When only a limited area of the Earth is studied, the estimation of the point-mass parameters by means of spherical harmonics is prone to error, since they are no longer orthogonal over a partial domain of the sphere. The point-mass determination problem on a limited region is treated by the construction of a Slepian basis that is orthogonal over the specified limited domain of the sphere. We propose an iterative algorithm for the numerical solution of the local point mass determination problem and give some results on the robustness of this reconstruction process. We also study the stability of this problem against added noise. Some numerical tests are presented and commented.

Le point de départ de la thèse est une interrogation sur l'importance de la variation du sémantisme en français médiéval entre les différents genres textuels au cours du Moyen Âge. L'étude a été menée sur plusieurs dénominations féodales dont il s’agit de déterminer le contenu sémantique et son éventuelle évolution du XIIe au XVe siècle. La recherche s'appuie sur un corpus comportant des textes de genres suivants : chansons de geste, romans, chroniques et textes documentaires. L'analyse du sémantisme des dénominations étudiées se fait par le biais de l'étude des occurrences relevées au cours du dépouillement exhaustif, automatique ou manuel, de la totalité des textes de notre corpus. Deux angles sont adoptés lors de l'analyse : l'étude du co-texte linguistique des occurrences et l'étude du contexte narratif de leur apparition. En raison des dénominations étudiées, qui appartiennent pleinement au vocabulaire de la féodalité, l'analyse linguistique ne se sépare pas des travaux d’historiens et de littéraires concernant l'évolution de la société féodale et les textes de notre corpus. Au terme de la thèse sont apparus nettement des régularités dans les évolutions des emplois et du contenu sémantique des dénominations étudiées, ce qui démontre la nécessité d'intégrer la question des genres textuels dans les études sémantiques. Au-delà de résultats pour ces dénominations, il s’agit l'expérimentation d'une méthode pour la sémantique du français médiéval. / The starting point of our thesis was the examination on the importance of semantic variations in Medieval French throughout different text genres during the medieval period. It was based on the study of several feudal names; the aim of the research was to point out their semantic structures and eventual change from the 12th to the 15th century. The corpus that we used consisted of texts of following genres: Epic texts, chivalry romances, chronicles and documentary texts. We analysed all texts by manual or automatic counting of all examples of the investigated names. Two perspectives were adopted: the study of the linguistic co-text of the names and that of narrative context of their use. Given the nature of the studied names, which are integrated in the feudal vocabulary, the linguistic analysis was completed by historical and literary studies on the feudal society and the texts of the corpus. The research pointed out clearly a number of regularities in the uses, semantic structures and linguistic change of the studied names; it confirms the relevance to consider the variation of text genre in semantic studies. Beyond the results for the examined names, our thesis suggests a method for Medieval French semantic studies.

Many types of radial basis functions, such as multiquadrics, contain a free parameter called shape factor, which controls the flatness of RBFs. In the 1-D problems, Fornberg et al. [2] proved that with simple conditions on the increasingly flat radial basis function, the solutions converge to the Lagrange interpolating. In this report, we study and extend it to the 1-D Poisson equation RBFs direct solver, and observed that the interpolants converge to the Spectral Collocation Method using Polynomial. In 2-D, however, Fornberg et al. [2] observed that limit of interpolants fails to exist in cases of highly regular grid layouts. We also test this in the PDEs solver and found the error behavior is different from interpolating problem.