Control of underactuated mechanical systems is of interest as it allows for control authority over all of a system's degrees of freedom without requiring actuation of the full system. In addition to this, open-loop control of a system provides the advantage of applying to systems with unmeasurable states or where sensor integration is not feasible. Vibrational control is an open-loop control strategy that uses high-frequency, high-amplitude forcing to control underactuated mechanical systems. This thesis is concerned with exploring two underactuated mechanical systems that are controlled using vibrational inputs. The first system, a 3 degrees of freedom (DOFs) 2-link mechanism with 1 actuated DOF which is an example of a vibrational control system with 1 input and 2 unactuated DOFs, is used to review analytical results of stability analysis using the averaged potential. Theoretical and numerical results are presented for the achievable stable configurations of the system and the effects of changing the physical parameters on the achievable stable configurations are studied. The primary contribution of this effort is the development of an experimental apparatus where vibrational control is implemented. The second system is a 4DOF system composed of a 2DOF spherical pendulum supported by an actuated 2DOF cart used to study the effects of multiple vibrational inputs acting on a system. Theoretical and numerical analysis results are presented for three variants of harmonic forcing applied to the two actuated degrees of freedom: 1) identical input waveforms, except for the amplitudes, 2) identical input waveforms, except for the amplitudes and a phase shift, and 3) identical input waveforms, but at different frequencies and amplitudes. The equilibrium sets under open-loop vibrational forcing are determined for all three cases. A general closed-loop vibrational control scheme is presented using proportional feedback of the unactuated coordinates superposed with the zero-mean, $T$-periodic vibrational input. / M.S. / Underactuated mechanical systems are systems where the driven degrees of freedom are fewer than the total degrees of freedom of the system. These systems can be controlled using vibrational control which is an open-loop control strategy that uses high-frequency, high-amplitude forcing to control the states of a system. An open-loop control strategy is one in which there are no measurements of the system states required in the control scheme. This allows for control of systems where sensor integration is not feasible. This thesis is concerned with exploring vibrational control of two underactuated mechanical systems. The stability of the equilibrium sets of these systems is assessed using the averaged potential, which is an energy-like quantity used to determine stability of equilibria of systems with high-frequency inputs. Theoretical and numerical results are presented for both systems and the effects of physical parameters and variants of harmonic forcing on the achievable stable configurations of the systems are studied. The two main contributions of the thesis are the development of an experimental apparatus where vibrational control is physically implemented for one system and the outline of the closed-loop vibrational control scheme.

This dissertation presents novel vibrational control strategies for mechanical control-affine systems with high-frequency, high-amplitude inputs. Since these control systems use high-frequency, zero-mean, periodic inputs, averaging techniques are widely used in the analysis of their dynamics. By studying their time-averaged approximations, new properties of the averaged dynamics of this class of systems are revealed. Using these properties, the problem of input optimization of vibrational control systems was formulated and solved by transforming the problem to a constrained optimization one. Geometric control theory provides powerful tools for studying the control properties of control-affine systems. Using the concepts of vibrational and geometric controls and averaging tools, a closed-loop control strategy for trajectory tracking of a class of underactuated mechanical control-affine systems is developed. In the developed control law, the fact that for underactuated systems, the actuated coordinates together with the corresponding generalized velocities can be considered as generalized inputs for the unactuated dynamics plays the main role. Using the developed control method, both actuated and unactuated coordinates of the system are able to follow slowly time-varying prescribed trajectories on average. The developed control method is applied for altitude control of flapping wing micro-air vehicles by considering the sweeping (flapping) angle of the wings as the inputs. Using the feathering (pitch) angles of the wings as additional inputs, and using non-symmetric flapping, the control method is then extended for three-dimensional flight control of flapping wing micro-air vehicles. / Ph. D.

Nowadays, reactions on surfaces are attaining great scientific interest because of their diverse applications. Some well known examples are production of ammonia on metal surfaces for fertilizers and reduction of poisonous gases from automobiles using catalytic converters. More recently, also photoinduced reactions at surfaces, useful, textit{e.g.}, for photocatalysis, were studied in detail. Often, very short laser pulses are used for this purpose. Some of these reactions are occurring on femtosecond (1 fs=$10^{-15}$ s) time scales since the motion of atoms (which leads to bond breaking and new bond formation) belongs to this time range. This thesis investigates the femtosecond laser induced associative photodesorption of hydrogen, H$_2$, and deuterium, D$_2$, from a ruthenium metal surface. Many interesting features of this reaction were explored by experimentalists: (i) a huge isotope effect in the desorption probability of H$_2$ and D$_2$, (ii) the desorption yield increases non-linearly with the applied visible (vis) laser fluence, and (iii) unequal energy partitioning to different degrees of freedom. These peculiarities are due to the fact that an ultrashort vis pulse creates hot electrons in the metal. These hot electrons then transfer energy to adsorbate vibrations which leads to desorption. In fact, adsorbate vibrations are strongly coupled to metal electrons, textit{i.e.}, through non-adiabatic couplings. This means that, surfaces introduce additional channels for energy exchange which makes the control of surface reactions more difficult than the control of reactions in the gas phase. In fact, the quantum yield of surface photochemical reactions is often notoriously small. One of the goals of the present thesis is to suggest, on the basis of theoretical simulations, strategies to control/enhance the photodesorption yield of H$_2$ and D$_2$ from Ru(0001). For this purpose, we suggest a textit{hybrid scheme} to control the reaction, where the adsorbate vibrations are initially excited by an infrared (IR) pulse, prior to the vis pulse. Both textit{adiabatic} and textit{non-adiabatic} representations for photoinduced desorption problems are employed here. The textit{adiabatic} representation is realized within the classical picture using Molecular Dynamics (MD) with electronic frictions. In a quantum mechanical description, textit{non-adiabatic} representations are employed within open-system density matrix theory. The time evolution of the desorption process is studied using a two-mode reduced dimensionality model with one vibrational coordinate and one translational coordinate of the adsorbate. The ground and excited electronic state potentials, and dipole function for the IR excitation are taken from first principles. The IR driven vibrational excitation of adsorbate modes with moderate efficiency is achieved by (modified) $pi$-pulses or/and optimal control theory. The fluence dependence of the desorption reaction is computed by including the electronic temperature of the metal calculated from the two-temperature model. Here, our theoretical results show a good agreement with experimental and previous theoretical findings. We then employed the IR+vis strategy in both models. Here, we found that vibrational excitation indeed promotes the desorption of hydrogen and deuterium. To summarize, we conclude that photocontrol of this surface reaction can be achieved by our IR+vis scheme. / Heutzutage werden Reaktionen auf Oberflächen wegen ihrer vielfältigen Anwendungen intensiv untersucht. Einige der bekannten Beispiele sind die Herstellung von Ammoniak auf Metalloberflächen für die Kunstdüngerproduktion und die Reduktion giftiger Abgase in Autokatalysatoren. In letzter Zeit wurden auch photoinduzierte Reaktionen an Oberflächen eingehender untersucht, die z.B. für die Photokatalyse verwandt werden können. Häufig werden in diesen Untersuchungen sehr kurze Laserpulse benutzt. Einige der Reaktionen finden auf einer Femtosekunden-Zeitskala mbox{(1 fs =10$^{-15}$ s)} statt, da die Bewegungen einzelner Atome in derart kurzen Zeitspannen ablaufen (durch die der Bindungsbruch und das Knüpfen neuer Bindungen verursacht wird). Diese Arbeit untersucht die femtosekunden-laserinduzierte assoziative Photodesorption von Wasserstoff, H$_2$, und Deuterium, D$_2$, von einer Rutheniumoberfläche. Viele interessante Eigenschaften dieser Reaktion wurden in Experimenten entdeckt: (i) ein großer Isotopeneffekt in der Desorptionswahrscheinlichkeit von H$_2$ und D$_2$, (ii) die Desorptionsausbeute steigt nicht-linear mit der (vis) Laserfluenz an und (iii) eine Nicht-Gleichverteilung der Energie auf die einzelnen Freiheitsgrade. Diese Auffälligkeiten sind durch den Umstand verursacht, dass der ultrakurze vis-Laserpuls heiße Elektronen im Metall erzeugt. Die heißen Elektronen transferieren dann Energie in die Schwingungen des Adsorbats, was zur Desorption führt. Tatsächlich sind die Adsorbatschwingungen stark an die Elektronen gekoppelt, nämlich durch nicht-adiabatische Kopplungen. Dies bedeutet, dass durch Oberflächen neue Kanäle für den Energietransfer geöffnet werden, was die Kontrolle von Oberflächenreaktionen im Vergleich zu solchen in der Gasphase erschwert. In der Tat sind die Quantenausbeuten von photochemischen Oberflächenreaktionen bekannterweise klein. Eines der Ziele in der vorliegenden Arbeit ist es auf der Basis von theoretischen Simulationen Strategien vorzuschlagen, um die Photodesorptionsausbeute von H$_2$ und D$_2$ von Ru(0001) zu kontrollieren bzw. zu verbessern. Zu diesem Zweck schlagen wir ein gemischtes Kontrollschema für die Reaktion vor, bei dem zunächst die Adsorbatschwingungen vor dem vis-Puls durch einen infraroten (IR) Puls angeregt werden. Sowohl adiabatische als auch nicht-adiabatische Repräsentationen für photoinduzierte Desorptionsprozesse werden dabei benutzt. Die adiabatische Repräsentation ist in klassischen Molekulardynamik-Simulationen mit elektronischer Reibung verwirklicht. In einer quantenmechanischen Beschreibung werden nicht-adiabatische Repräsentationen innerhalb der Dichtematrixtheorie für offene Quantensysteme verwandt. Die zeitliche Entwicklung des Desorptionsprozesses wird in einem Zwei-Modenmodell reduzierter Dimensionalität mit einer Schwingungs- und einer Translationskoordinate des Adsorbats beschrieben. Die Potentiale für den elektronische Grundzustand und den angeregten Zustand sind abgeleitet aus quantenchemischen Rechnungen (textsl{first principles}). Die IR-getriebene Schwingungsanregung der Adsorbatmoden mit moderatem Wirkungsgrad wird mit (modifizierten) $pi$-Pulsen und/oder der Theorie der optimalen Kontrolle erreicht. Die Abhängigkeit der Desorption von der Fluenz wird mit Hilfe der elektronischen Temperatur des Metalls berechnet, welche im Rahmen des Zwei-Temperatur-Modells bestimmt wird. Dabei weisen unsere Ergebnisse eine gute Übereinstimmung mit experimentellen und früheren theoretischen Arbeiten auf. Daraufhin wandten wir die IR+vis Strategie in beiden Modellen an. Dadurch konnten wir zeigen, dass Schwingungsanregung in der Tat die Desorption von Wasserstoff und Deuterium begünstigt. Zusammenfassend stellen wir fest, dass die Photokontrolle dieser Oberflächenreaktion durch unser IR+vis Schema erreichbar ist.