Thesis (Ph. D.)--University of Texas at Austin, 2009. / Title from PDF title page (University of Texas Digital Repository, viewed on Sept. 9, 2009). Vita. Includes bibliographical references and index.

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1971. / Vita. Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisor: Philip J. Morrison. Vita. Includes bibliographical references.

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Recent work has established that a group theoretical viewpoint of completely integrable dynamical systems with N degrees of freedom yields an algorithm that provides new information concerning the symmetry transformation group structure of this class of dynamical systems. The work presented here rests heavily on the results presented in reference and it is recommended that the reader consult this reference for a more rigorous discussion of the results given in this thesis.

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Energy-shaping techniques are used to expand the range of autonomous motion of unmanned aerial systems without prohibitively {color{black}increasing the computational cost of the resultant controller}. Passivity-based control presents a method to implement a static, nonlinear state feedback control law that stabilizes the motion of an aircraft with a large region of attraction. {color{black} The energy-based control scheme is applied to both multirotor and fixed-wing aircraft}. Multirotor aircraft dynamics are cast into a port-Hamiltonian System and the concept of trajectory tracking using canonical feedback transformation is implemented to construct a cross-track controller. Fixed-wing aircraft dynamics are cast in port-Hamiltonian form and a passivity-based nonlinear control law for steady, wings-level flight of a fixed-wing aircraft to a specified inertial velocity (speed, course, and climb angle) is constructed. Results in simulations and experiments suggest robustness, and a large region of attraction of the controller. The control law extended to support time-varying inertial velocity tracking that incorporates banking to turn. The results are extended by including a line-of-sight guidance law and varying the direction as a function of position relative to a desired path, rather than as a function of time. The control law and the associated proof of stability follow similarly to that of the time-varying directional stabilization problem. The results are supported with simulations as well as experimental flight tests. / Doctor of Philosophy / This dissertation presents an alternative but intuitive approach to regulate unmanned aerial vehicles' flight that would allow for more maneuverability {color{black} than conventional methods}. This scheme relies on modifying the energy of the system to achieve the desired motion and leverages the properties of the aircraft rather than eliminating them and imposing different properties. This approach is applied to both fixed-wing and aircraft and quadcopters. Simulations and experimental flights have show the efficacy of this approach compared to other more established methods.

The problem considered here is one of finding the minimal Hamiltonian chain of a graph. A single chain must traverse all 𝑛 vertices of a graph with the minimal distance. The proposed procedure reduces a large problem into several smaller problems and uses a branch and bound algorithm to find the minimal Hamiltonian chain of each partitioned subproblem. The graph is decomposed and partitioned into subproblems with the use of necessary conditions for the existence of a Hamiltonian chain. This process is only applicable to graphs with relatively few incident edges per vertex. The branch and bound algorithm makes use of concepts developed by Nicos Christofides. Hamiltonian chains are derived by using minimal spanning trees. / Master of Science

Dans cette thèse, on s'intéresse à deux aspects différents des systèmes dynamiques intégrables. La première partie est dévouée à l'étude de trois familles de systèmes hamiltoniens intégrables : les systèmes de pliage de Kapovich et Millson sur les espaces de modules de polygones 3D de longueurs de côtés fixées, les systèmes de Gelfand-Cetlin introduits par Guillemin et Sternberg sur les orbites coadjointes du groupe de Lie U(n), et une famille de systèmes définie par Nohara et Ueda sur la variété grassmannienne Gr(2,n). Dans chaque cas on montre que les fibres singulières de l'application moment sont des sous-variétés plongées et on en donne des modèles géométriques sous la forme de variétés quotients. La deuxième partie poursuit une étude initiée par Zung et Minh sur les actions totalement hyperboliques de Rn sur des variétés compactes de dimension n, qui apparaissent naturellement lors de l'étude des systèmes non-hamiltoniens intégrables dont toutes les singularités sont non-dégénérées. On s'intéresse au flot engendré par l'action d'un vecteur générique de Rn. On donne une définition d'indice pour ses singularités qu'on relie à la théorie de Morse classique, et on utilise ce flot pour obtenir des résultats sur le nombres d'orbites de dimension donnée. Une étude plus poussée est effectuée en dimension 2, et en particulier sur la sphère S2, où les orbites de l'action dessinent un graphe plongé dont on analyse la combinatoire. On termine en construisant explicitement des exemples d'actions hyperboliques en dimension 3 sur la sphère S3 et dans l'espace projectif RP3. / In this thesis, we are interested in two different aspects of integrable dynamical systems. The first part is devoted to the study of three families of integrable Hamiltonian systems: the systems of bending flows of Kapovich and Millson on the moduli spaces of 3D polygons with fixed side lengths, the Gelfand-Cetlin systems introduced by Guillemin and Sternberg on the coadjoint orbits of the Lie group U(n), and a family of integrable systems defined by Nohara and Ueda on the Grassmannian Gr(2,n). In each case we prove that the fibers of the momentum map are embedded submanifolds for which we give geometric models in terms of quotients manifolds. In the second part we carry on with a study initiated by Zung and Minh of the totally hyperbolic actions of R^n on compact n-dimensional manifolds that appear naturally when investigating integrable non-hamiltonian systems with nondegenerate singularities. We study the flow generated by the action of a generic vector in Rn. We define a notion of index for its singularities and we use this flow to obtain results on the number of orbits of given dimension. We investigate further the 2-dimensional case, and more particularly the case of the sphere S2, where the orbits of the action draw an embedded graph of which we analyse the combinatorics. Finally, we provide explicit examples of totally hyperbolic actions in dimension 3, on the sphere S3 and on the projective space RP3.