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Maneuvering through 3D structures nimbly is pivotal to the advancement of legged locomotion. However, few methods have been developed that can generate 3D gaits in those terrains and fewer if none can be generalized to control dynamic maneuvers. In this thesis, foot placement planning for dynamic locomotion traversing irregular terrains is explored in three dimensional space. Given boundary values of the center of mass' apexes during the gait, sagittal and lateral Phase Plane trajectories are predicted based on multi-contact and inverted pendulum dynamics. To deal with the nonlinear dynamics of the contact motions and their dimensionality, we plan a geometric surface of motion beforehand and rely on numerical integration to solve the models. In particular, we combine multi-contact and prismatic inverted pendulum models to resolve feet transitions between steps, allowing to produce trajectory patterns similar to those observed in human locomotion. Our contributions lay in the following points: (1) the introduction of non planar surfaces to characterize the center of mass' geometric behavior; (2) an automatic gait planner that simultaneously resolves sagittal and lateral feet placements; (3) the introduction of multi-contact dynamics to smoothly transition between steps in the rough terrains. Data driven methods are powerful approaches in absence of accurate models. These methods rely on experimental data for trajectory regression and prediction. Here, we use regression tools to plan dynamic locomotion in the Phase Space of the robot's center of mass and we develop nonlinear controllers to accomplish the desired plans with accuracy and robustness. In real robotic systems, sensor noise, simplified models and external disturbances contribute to dramatic deviations of the actual closed loop dynamics with respect to the desired ones. Moreover, coming up with dynamic locomotion plans for bipedal robots and in all terrains is an unsolved problem. To tackle these challenges we propose here two robust mechanisms: support vector regression for data driven model fitting and contact planning, and trajectory based sliding mode control for accuracy and robustness. First, support vector regression is utilized to learn the data set obtained through numerical simulations, providing an analytical solution to the nonlinear locomotion dynamics. To approximate typical Phase Plane behaviors that contain infinite slopes and loops, we propose to use implicit fitting functions for the regression. Compared to mainstream explicit fitting methods, our regression method has several key advantages: 1) it models high dimensional Phase Space states by a single unified implicit function; 2) it avoids trajectory over-fitting; 3) it guarantees robustness to noisy data. Finally, based on our regression models, we develop contact switching plans and robust controllers that guarantee convergence to the desired trajectories. Overall, our methods are more robust and capable of learning complex trajectories than traditional regression methods and can be easily utilized to develop trajectory based robust controllers for locomotion. Various case studies are analyzed to validate the effectiveness of our methods including single and multi step planning in a numerical simulation and swing foot trajectory control on our Hume bipedal robot. / text

This thesis studies deformation quantization and its application to contact interactions and systems with dissipation. We consider the subtleties related to quantization when contact interactions and boundaries are present. We exploit the idea that discontinuous potentials are idealizations that should be realized as limits of smooth potentials. The Wigner functions are found for the Morse potential and in the proper limit they reduce to the Wigner functions for the infinite wall, for the most general (Robin) boundary conditions. This is possible for a very limited subset of the values of the parameters -- so-called fine tuning is necessary. It explains why Dirichlet boundary conditions are used predominantly. Secondly, we consider deformation quantization in relation to dissipative phenomena. For the damped harmonic oscillator we study a method using a modified noncommutative star product. Within this framework we resolve the non-reality problem with the Wigner function and correct the classical limit. / iii, 188 leaves ; 29 cm

Grand-canonical inverse-temperature calculations of a single mode Bose-Hubbard model are presented, using the Gaussian phase space representation. Simulation of 100 particles is achieved in the ground state, having started with a low-particle-number thermal state. A preliminary foray into a three-mode lattice is made, but the sampling error appears to be too large for the simple approach taken here to be successful in larger systems. The quantum (real-time) dynamics of a one-dimensional Bose gas with two-particle losses are investigated. The Positive-P equations for this system are unstable, and this causes Positive-P simulations to `die' after a certain amount of time. Gauges are used to (sometimes partially) stabilise the equations. The effects on simulation times of various gauges, branching methods, and non-square diffusion matrix factorisations on simulation times are investigated. Despite the absence of repulsive inter-particle interactions, it is observed that $g^{(2)}$ rises above 1 at a finite particle separation. A phase space method for spin systems is introduced, based on SU(2) coherent states. This is essentially a spin analogue of the Positive-P method. The system of stochastic differential equations arising out of this method require weighted averages to be taken, and the weights can vary exponentially, leading to inefficient sampling. For the case of the Ising model, a transform is made to a set of equations which relaxes (in a dummy time variable) to the partition function at a given temperature, and allows unweighted ensemble averages to be taken. This allows accurate simulations to be achieved at a range of temperatures, with nearest-neighbour correlation functions agreeing with theory. This represents a proof of principle for the use of stochastic phase space methods in spin systems, and furthermore the method should be suited to open spin systems, at least for a small number of qubits.

Thesis (Ph. D.)--Washington State University, May 2008. / Includes bibliographical references.

Simulating the nonequilibrium behavior of interacting quantum systems is an important way to understand results of experimental quantum simulators, motivate new materials, and refine new quantum algorithms. However, this is a challenging task due to the exponential difficulty of such systems, which motivates dimensional reduction methods, such as semiclassical limits. This work extends semiclassical phase space methods to spin systems with no clear classical limit with the cluster truncated Wigner approximation (cTWA), and improves on Schrieffer-Wolff low energy effective dynamics with variational adiabatic generators. The cTWA was used to compute nonequilibrium dynamics in spin chains, finding behavior such as signatures of many body localization; rapid thermalization and preservation of fluctuations; effective thermodynamic classical behaviors; and signatures of quantum chaos and butterfly velocities, in 1d spin 1/2 chains. Variational Schrieffer-Wolff methods were used to find efficient non-perturbative dressings for the Hubbard model and find effective quasiparticle dynamics and nonthermal states in quantum chaotic spin chains. These methods are potentially effective tools to separate essential quantum behavior from classical behavior, and can be used to diagnose quantum thermalization behavior in interacting quantum systems.

Electrocardiogram (ECG) time-domain signals contain important information about the heart. Several techniques have been proposed for creating a two-dimensional visualization of an ECG, called a Cardioid, that can be used to detect heart abnormalities with computer algorithms. The derivative method is the prevailing technique, which is popular for its low complexity, but it can introduce distortion into the Cardioid plot without additional signal processing. The Hilbert transform is an alternative method which has unity gain and phase shifts the ECG signal by 90 degrees to create the Cardioid plot. However, the Hilbert transform is seldom used and has historically been implemented with a computationally expensive process. In this thesis we show a low-complexity method for implementing the Hilbert transform as a finite impulse response (FIR) filter. We compare the fundamental differences between Cardioid plots generated with the derivative and Hilbert transform methods and demonstrate the feature-preserving nature of the Hilbert transform method. Finally, we analyze the RMS values of the transformed signals to show how the Hilbert transform method can create near 1:1 aspect ratio Cardioid plots with very little distortion for any patient data.

An investigation into the complex function of muscle using non-invasive imaging and novel analytical approaches. / The human body is inherently complex as seen through the structural organization of muscle in terms of its contractile subunit organization and scaling, innervation patterns, and vascular organization. However, the functional complexity of muscle such as its state of oxygenation, metabolism or blood-flow has been less well explored. Thus in an effort to improve our understanding of muscle, blood oxygenation level dependent (BOLD) magnetic resonance imaging of the lower leg, at rest and during a variety of weighted plantar-flexion paradigms, at 40% maximal voluntary contraction, was employed. Prior to experimentation, on 11 healthy subjects, an ergometer and electromyogram (EMG), suitable for use within the MRI, were constructed to allow for concurrent exercise and image acquisition. After collecting muscle BOLD data, four novel techniques were using to describe muscle function. The first technique used the fractal dimension, a measure of complexity, conveying the rate of variation of muscle blood flow at rest. This technique was able to determine differences between the muscles of lower leg, which have varying distributions of muscle fibre types based on function. The second exploratory technique was the use of the phase space, which provides insight into state/state-transitions of a system over time. The phase space representation of the BOLD signal provided novel insight into the muscle activation state. It demonstrated that muscle has more than the two blood flow states of reduced levels at rest and increased levels when exercising. The third technique involved using a signal saturation (SAT) region, proximal to the imaging region, to mitigate the arterial in-flow effects to more accurately represent muscle activation. By observing the correlation between the ideal reference and recorded signal, the acquisition with the arterial suppression improved the assessment of what regions in the muscle were active in the range borderline activation, which has the highest uncertainty. The final outlook on muscle behaviour involved using measures of fatigue from the collected EMG data to develop novel metrics of fatigue based on the BOLD signal. Concurrent BOLD and EMG of the anterior compartment of the lower leg during a plantar-flexion block design, demonstrated that the change in blood-flow between rest and contracted states is an excellent indicator of muscle fatigue. The primary outlook of this thesis is to provide a unique data collection and analytic framework to describe muscle behaviour, which was achieved using non-invasive measures with a complex outlook. / Thesis / Master of Applied Science (MASc) / The human body is complex, and an incredible amount of research has been done to better understand it. Specifically, muscle is built and works in a complex way to allow us to move and perform everyday tasks. There are many diseases that affect how a muscle works, which is why there is a need to describe muscle performance when it is healthy and unhealthy. In this research, muscle behaviour is explored by taking pictures of the leg. From these pictures the blood flow in the calf and shin was measured both when staying still and when performing exercise. Four new techniques were created to describe the blood flow in the leg. The first technique measured how complex the muscle activity is, while staying still. If blood-flow changes a lot in a short amount of time, it is complex. This showed that the different components of muscle, either used for stamina or power, receive blood differently. The second technique used a different way of looking at the muscle to show that there are many different rates and amounts of blood-flow in the muscle. It revealed that muscle has more than the two blood flow options of 1) the normal level when resting and 2) the increased level when exercising. The third technique involved using an image filter to get a clean image of the muscle without the blood vessels affecting or misrepresenting the image. It was able to show what muscle regions were involved in exercise more accurately than before. The final technique involved measuring muscle electricity and blood flow at the same time, to find out when the muscle was exhausted. It demonstrated that muscle, when exhausted, showed larger changes in blood flow when going from resting to exercising. Overall, this research described how muscle performs in healthy individuals using new techniques. These techniques can now be used to compare healthy muscle to damaged/diseased muscle to determine how the muscle is recovering or to diagnose muscular disease.

This dissertation presents useful nonlinear models for fluid forcing on a moving body in two distinct contexts, and methods for analyzing the geometric structure within those and other mathematical models. This manuscript style dissertation presents three works within the theme of understanding fluid forcing and geometric structure. When a bluff body is free to move in the presence of an incoming bluff body wake, the average forcing on the body is dependent on its position relative to the upstream bluff body. This position-dependent forcing can be conceptualized as a stiffness, much like a spring. This work presents an updated model for the quasi-steady fluid forcing of a wake and extends the notion of wake stiffness to consider a nonlinear spring. These results are compared with kinematic experimental results to provide an example of the application of this framework. Fluid force models also play a role in understanding the behavior of passive aerodynamic gliders, such as gliding animals or plant material. The forces a glider experiences depend on the angle that its body makes with respect to its direction of motion. Modeling the glider as capable of pitch control, this work considers a glider with a fixed angle with respect to the ground. Within this model, all trajectories in velocity space collapse to a 1-dimensional invariant manifold known as the terminal velocity manifold. This work presents methods to identify the terminal velocity manifold, investigates its properties, and extends it to a 2-dimensional invariant manifold in a 3-dimensional space. Finally, in the search for manifolds such as the terminal velocity manifold, this dissertation introduces a new diagnostic for identifying the low dimensional geometric structure of models. The trajectory divergence rate uses instantaneous vector field information to identify regions of large normal stretching and strong normal convergence between nearby invariant manifolds. This work lays out the mathematical basis of the trajectory divergence rate and shows its application to approximate a variety of structures including slow manifolds and Lagrangian coherent structures. This dissertation applies nonlinear theoretical and numerical techniques to analyze models of fluid forcing and their geometric structure. The tools developed in this dissertation lay the groundwork for future research in the fields of flow-induced vibration, plant and animal biomechanics, and dynamical systems. / Ph. D. / When an object moves through a fluid such as air or water, the motion of the surrounding fluid generates forces on the moving object, affecting its motion. The moving object, in turn, affects the motion of the surrounding fluid. This interaction is complicated, nonlinear, and hard to even simulate numerically. This dissertation aims to analyze simplified models for these interactions in a way that gives a deeper understanding of the physics of the interaction between an object and a surrounding fluid. In order to understand these interactions, this dissertation looks at the geometric structure of the models. Very often, there are low-dimensional points, curves, or surfaces which have a very strong effect on the behavior of the system. The search for these geometric structures is another key theme of this dissertation. This dissertation presents three independent studies, with an introduction and conclusion to discuss the overall themes. The first work focuses on the forces acting on a cylinder in the wake of another cylinder. These forces are important to understand, because the vibrations that arise from wake forcing are important to consider when designing bridges, power cables, or pipes to carry oil from the ocean floor to offshore oil platforms. Previous studies have shown that the wake of a circular cylinder acts like a spring, pulling harder on the downstream cylinder the more it is moved from the center of the wake. In this work, I extend this idea of the wake as a spring to consider a nonlinear spring, which keeps the same idea, but provides a more accurate representation of the forces involved. The second work considers a simple model of gliding flight, relevant to understanding the behavior of gliding animals, falling leaves, or passive engineered gliders. Within this model, a key geometric feature exists on which the majority of the motion of the glider occurs, representing a 2-dimensional analogy to terminal velocity. In this work, I study the properties of this influential curve, show several ways to identify it, and extend the idea to a surface in a 3-dimensional model. The third study of this dissertation introduces a new mathematical quantity for studying models of systems, for fluid-body interaction problems, ocean flows, chemical reactions, or any other system that can be modeled as a vector field. This quantity, the trajectory divergence rate, provides an easily computed measurement of highly attracting or repelling regions of the states of a model, which can be used to identify influential geometric structures. This work introduces the quantity, discusses its properties, and shows its application to a variety of systems.

No / Bifractional displacement operators, are introduced by performing two fractional Fourier transforms on displacement operators. They are shown to be special cases of elements of the group G , that contains both displacements and squeezing transformations. Acting with them on the vacuum we get various classes of coherent states, which we call bifractional coherent states. They are special classes of squeezed states which can be used for interpolation between various quantities in phase space methods. Using them we introduce bifractional Wigner functions A(α,β;θα,θβ)A(α,β;θα,θβ), which are a two-dimensional continuum of functions, and reduce to Wigner and Weyl functions in special cases. We also introduce bifractional Q-functions, and bifractional P-functions. The physical meaning of these quantities is discussed.