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In the mammalian auditory system, specifically in the cochlea of the inner ear, the Basilar Membrane (BM) and hair cells are responsible for transducing incoming acoustic waves into electrical signals. These acoustic signals are carried as traveling waves by the BM and propagate from the base of the cochlea toward its apex where the helicotrema is located. An impressive feature of the mammalian auditory system is to prevent the propagated waves from reflecting which allows mammals to hear sounds without any reflection or overlap. This extraordinary characteristic of the inner ear is the main inspiration for this work. In the present study, the dynamic behavior of a beam structure with one or more attached spring-damper (SD) systems as passive absorbers is studied when excited by a harmonic force. The location of the spring-damper system divides the beam into two dynamic regions: one which exhibits non-reflecting traveling waves and the other with standing waves. In this work, the separation of traveling and standing waves is studied analytically, numerically, and experimentally. To the best of the author's knowledge, this is the first time in the literature that traveling and standing wave separation in a beam is realized experimentally using a single-point excitation and a spring-damper. Experimental results are used to validate the models of the system. Moreover, a parametric study is performed to gain a better understanding of the effect of different parameters on the quality of the generated waves in the structure. Furthermore, the effect of attaching the second spring-damper to the system is presented. Adding the secondary SD system results in increasing the excitation frequency range so that wave separation can be achieved. The results of this work can be used in various applications such as vibration suppression, energy absorption, particle transportation, and in exploring possible explanations for the BM and helicotrema functions in the cochlea. Doctor of Philosophy In the inner ear of the mammalian auditory system, the sound waves travel inside the cochlea where they are converted to electrical signals sent to the brain. A fascinating characteristic of the mammalian auditory system is that the sound waves traveling in the cochlea do not reflect when they reach its apex where the helicotrema is located. Therefore, we are able to hear sounds without any reflection or overlap. This work is inspired by the biological behavior of the inner ear and studies the dynamic behavior of a simple structure such as a beam with one (or two) attached spring-damper(s). In this work, the attached spring-damper system(s) prevents the waves traveling from the source to the beam's boundary from reflecting. This is similar to what happens in the inner ear. The location of the spring-damper divides the beam into two dynamic regions, one which exhibits non-reflecting traveling waves and the other with standing waves. The wave separation and parameters affecting the wave quality and its reflective or non-reflective features are studied analytically, numerically, and experimentally. To the best of the author's knowledge, the experiments carried out to generate the aforementioned wave types coexisting with each other on the beam are one of a kind. Furthermore, the results of this study showed a very good agreement between the experimental and theoretical results. The outcomes of this work can potentially be used in exploring possible explanations for the function of the cochlea and helicotrema and various applications such as particle transportation and suppression of unwanted vibrations.

Activities of daily living (ADL) such as writing, eating, and object manipulation are challenging for patients suffering from pathological tremors. Pathological tremors are involuntary, rhythmic, and oscillatory movements that manifest in limbs, the head, and other body parts. Among the existing treatments, mechanical loading through wearable rehabilitation devices is popular for being non-invasive and innocuous to the human body. In particular, a few exoskeletons are developed to actively mitigate pathological tremors in the forearm. While these forearm exoskeletons can effectively suppress tremors, they still require significant improvements in ergonomics to be implemented for ADL applications. The ergonomics of the exoskeleton can be improved via design and motion control pertaining to human biomechanics, which leads to better efficiency, comfort, and safety for the user. The wrist is a complicated biomechanical joint with two coupled degrees of freedom (DOF) pivotal to human manipulation capabilities. Existing exoskeletons either do not provide tremor suppression in all wrist DOFs, or can be restrictive to the natural wrist movement. This motivates us to explore a better exoskeleton solution for wrist tremor suppression. We propose TAWE - a wearable exoskeleton that provides alleviation of pathological tremors in all wrist DOFs. The design adopts a 6-DOF rigid linkage mechanism to ensure unconstrained natural wrist movements, and wearability features without extreme tight-binding or precise positioning for convenient ADL applications. When TAWE is equipped by the user, a closed-kinematic chain is formed between the exoskeleton and the forearm. We analyze the coupled multibody dynamics of the human-exoskeleton system, which reveals a few robotic control problems - (i) The first problem is the identification of the unknown wrist kinematics within the closed kinematic chain. We realize the real-time wrist kinematic identification (WKI) based on a novel ellipsoidal joint model that describes the coupled wrist kinematics, and a sparsity-promoting Extended Kalman Filter for the efficient real-time regression; (ii) The second problem is the exoskeleton motion control for tremor suppression. We design a robust adaptive controller (IO-RAC) based on model reference adaptive control and inverse optimal robust control theories, which can identify the unknown model inertia and load, and provide stable tracking control under disturbance; (iii) The third problem is the estimation of voluntary movement from tremorous motion data for the motion planning of exoskeleton. We develop a lightweight and data-driven voluntary movement estimator (SVR-VME) based on least square support vector regression, which can estimate voluntary movements with real-time signal adaptability and significantly reduced time delay. Simulations and experiments are carried out to test the individual performance of robotic control algorithms proposed in this study, and their combined real-time performance when integrated into the full exoskeleton control system. We also manufacture the prototype of TAWE, which helps us validate the proposed solutions in tremor alleviation exoskeletons. Overall, the design of TAWE meets the expectations in its compliance with natural wrist movement and simple wearability. The exoskeleton control system can execute stably in real-time, identify unknown system kinematics and dynamics, estimate voluntary movements, and suppress tremors in the wrist. The results also indicate a few limitations in the current approaches, which require further investigations and improvements. Finally, the proposed exoskeleton control solutions are developed based on generic formulations, which can be applied to not only TAWE, but also other rehabilitation exoskeletons. Doctor of Philosophy Activities of daily living (ADL) such as writing, eating, and object manipulation are challenging for patients suffering from pathological tremors, which affect millions of people worldwide. Tremors are involuntary, rhythmic, and oscillatory movements. In recent years, rehabilitation exoskeletons are developed as non-invasive solutions to pathological tremor alleviation. The wrist is pivotal to human manipulation capabilities. Existing exoskeletons either do not provide tremor suppression in all wrist movements, or can be restrictive to natural wrist movements. To explore a better solution with improved performance and ergonomics, we propose TAWE - a wearable exoskeleton that provides tremor alleviation in full wrist motions. TAWE adopts a high-degree-of-freedom mechanism to ensure unconstrained natural wrist movements, and wearability features for convenient ADL applications. The coupled dynamics between the forearm and TAWE leads to a few robotic control problems. We propose novel real-time robotic control solutions in the identification of unknown wrist kinematics, robust adaptive exoskeleton control for tremor suppression, and voluntary movement estimation for motion planning. Later, simulations and experiments validate the TAWE prototype and its exoskeleton control framework for tremor alleviation, and reveal limitations in the current approaches that require further investigations and improvements. Finally, the proposed exoskeleton control solutions are developed based on generic formulations, which can be applied to not only TAWE, but also other rehabilitation exoskeletons.

Modeling the dynamics of complex multibody systems, such as those representing the motion of animals, can be accomplished through well-established geometric methods. In these formulations, motions take values in certain types of smooth manifolds which are coordinate-free and intrinsic. However, the dimension of the full configuration manifold can be large. The first study in this dissertation aims to build low-dimensional models models from motion capture data. This study also expands on the so-called learning problem from statistical learning theory over Euclidean spaces to estimating functions over manifolds. Experimental results are presented for estimating reptilian motion using motion capture data. The second study in this dissertation utilizes reproducing kernel Hilbert space (RKHS) formulations and Koopman theory, to achieve some of the advantages of learning theory for IID discrete systems to estimates generated over dynamical systems. Specifically, rates of convergence are determined for estimates generated via extended dynamic mode decomposition (EDMD) by relating them to estimates generated by distribution-free learning theory. Some analytical examples illustrate the qualitative behavior of the estimates. Additionally, a examination of the numerical stability of the estimates is also provided in this study. The approximation methods are then implemented to estimate forward kinematics using motion capture data of a human running along a treadmill. The final study of this dissertation contains an examination of the continuous time regression problem over subsets and manifolds. Rates of convergence are determined using a new notion of Persistency of Excitation over flows of manifolds. For practical considerations, two approximation methods of the exact solution to the continuous regression problem are introduced. Characteristics of these approximation methods are analyzed using numerical simulations. Implementations of the approximation schemes are also performed on experimentally collected motion capture data. Doctor of Philosophy Modeling the dynamics of complex multibody systems, such as those representing the motion of animals, can be accomplished through well-established geometric methods. However, many real-world systems, including those representing animal motion, are difficult to model from first principles. Machine learning, on the other hand, has proven to be extremely powerful in its ability to leverage "big data" to generate estimates from typically independent and identically distributed (IID) data. This dissertation expands on the so-called learning problem from statistical learning theory over Euclidean spaces to those over manifolds. This dissertation consists of three studies, the first of which aims to build low-dimensional models models from motion capture data. Using the distribution-free learning theory, estimates discussed in this dissertation minimize a proxy of the expected error, which cannot be calculated in closed form. This dissertation also includes a study into approximations of the so-called Koopman operator. This study determined that the rate of convergence of the estimate to the true operator depends on the reduced dimensionality of the embedded submanifold in the high-dimensional ambient input space. While most of the current work on machine learning focuses on cases where the samples used for learning or regression are generated from an IID, stochastic, discrete measurement process, this dissertation also contains a study of the regression problem in continuous time over subsets and manifolds. Additionally, two approximation methods of the exact solution to the continuous regression problem are introduced. Each of the aforementioned studies also includes several analytical results to illustrate the qualitative behavior of the approximations and, in each study, implementations of the estimation schemes are performed on experimentally collected motion capture data.

A power grid network is an interconnected network of coupled devices that generate, transmit and distribute power to consumers. These complex and usually large-scale systems have high dimensional models that are computationally expensive to simulate especially in real time applications, stability analysis, and control design. Model order reduction (MOR) tackles this issue by approximating these high dimensional models with reduced high-fidelity representations. When the internal description of the models is not available, the reduced representations are constructed by data. In this dissertation, we investigate four problems regarding the MOR and data-driven modeling of the power networks model, particularly the swing equations. We first develop a parametric MOR approach for linearized parametric swing equations that preserves the physically-meaningful second-order structure of the swing equations dynamics. Parameters in the model correspond to variations in operating conditions. We employ a global basis approach to develop the parametric reduced model. We obtain these local bases by $mathcal{H}_2$-based interpolatory model reduction and then concatenate them to form a global basis. We develop a framework to enrich this global basis based on a residue analysis to ensure bounded $mathcal{H}_2$ and $mathcal{H}_infty$ errors over the entire parameter domain. Then, we focus on nonlinear power grid networks and develop a structure-preserving system-theoretic model reduction framework. First, to perform an intermediate model reduction step, we convert the original nonlinear system to an equivalent quadratic nonlinear model via a lifting transformation. Then, we employ the $mathcal{H}_2$-based model reduction approach, Quadratic Iterative Rational Krylov Algorithm (Q-IRKA). Using a special subspace structure of the model reduction bases resulting from Q-IRKA and the structure of the underlying power network model, we form our final reduction basis that yields a reduced model of the same second-order structure as the original model. Next, we focus on a data-driven modeling framework for power network dynamics by applying the Lift and Learn approach. Once again, with the help of the lifting transformation, we lift the snapshot data resulting from the simulation of the original nonlinear swing equations such that the resulting lifted-data corresponds to a quadratic nonlinearity. We then, project the lifted data onto a lower dimensional basis via a singular value decomposition. By employing a least-squares measure, we fit the reduced quadratic matrices to this reduced lifted data. Moreover, we investigate various regularization approaches. Finally, inspired by the second-order sparse identification of nonlinear dynamics (SINDY) method, we propose a structure-preserving data-driven system identification method for the nonlinear swing equations. Using the special structure on the right-hand-side of power systems dynamics, we choose functions in the SINDY library of terms, and enforce sparsity in the SINDY output of coefficients. Throughout the dissertation, we use various power network models to illustrate the effectiveness of our approaches. Doctor of Philosophy Power grid networks are interconnected networks of devices responsible for delivering electricity to consumers, e.g., houses and industries for their daily needs. There exist mathematical models representing power networks dynamics that are generally nonlinear but can also be simplified by linear dynamics. Usually, these models are complex and large-scale and therefore take a long time to simulate. Hence, obtaining models of much smaller dimension that can capture the behavior of the original systems with an acceptable accuracy is a necessity. In this dissertation, we focus on approximation of power networks model through the swing equations. First, we study the linear parametric power network model whose operating conditions depend on parameters. We develop an algorithm to replace the original model with a model of smaller dimension and the ability to perform in different operating conditions. Second, given an explicit representation of the nonlinear power network model, we approximate the original model with a model of the same structure but smaller dimension. In the cases where the mathematical models are not available but only time-domain data resulting from simulation of the model is at hand, we apply an already developed framework to infer a model of a small dimension and a specific nonlinear structure: quadratic dynamics. In addition, we develop a framework to identify the nonlinear dynamics while maintaining their original physically-meaningful structure.

As the robotic industry has matured, the study of animal motion has given rise to many robot designs. Researchers from multiple areas, such as biomechanics, control theory, and machine learning, have spent their energy and efforts making robots more realistic. The intent is that the automatic system can replace real animals and even perform certain tasks in harsh, or even dangerous environments. However, animal motions encompass a wide range of motion that depends on body geometries and various animal behaviors. From human walking to lizards crawling, from dogs running to horses pacing, many studies of motion only focus on one species or a few behaviors. An ever-increasing collection of papers are published that study animal motions for different species and motion regimes, and these are often based on video footage and motion capture data. This is particularly true for human motion research. While there are huge volumes of data acquired from motion capture and video, not many researches as of yet are using dynamical system analysis techniques such as dynamic mode decomposition, extended dynamic mode decomposition, or even Koopman method to understand and compare the motion across different species. Thus, the goal of this thesis is to further develop the methods mentioned above to analyze and characterize animal motion. The algorithms derived should apply regardless the shape of the body or the number of degrees of freedom for the joins. Using strategies from statistical learning theory and Koopman operator theory, several methods are derived and compared. The analysis culminates in a motion approximation, that subsequently could be used in robotic control to emulate an animal motion as much as possible. Master of Science As the robotic industry has matured, the study of animal motion has given rise to many robot designs. Researchers from multiple areas, such as biomechanics, control theory, and machine learning, devote their energy and efforts to making the robots more realistic, so that the automatic system can replace real animals and even perform certain tasks in harsher, or even dangerous environment. However, animal motion itself encompasses a wide range of motions that depend on body geometries and various behaviors. From human walking to lizards crawling, from dogs running to horses pacing, many studies of motion only focus on one species or a few behaviors. Many animal motion studies are often based on video footage and motion capture data. This is particularly true for human motion research when researchers are trying to create the gait pattern in medical research. While there are huge volumes of data acquired from motion captures and videos, not many researches as of yet are using dynamical system analysis techniques to understand and compare the motion across different species. Thus, the goal of this thesis is to use dynamical system analysis techniques and further develop the methods to analyze and characterize animal motion. Regardless of the shape of the body or the join types at different locations on the body, strategies from the theories of machine learning and dynamical system analysis are used to derive algorithms which should be applied to all animal motions. Several methods are derived and compared. The analysis culminates in a motion approximation, that subsequently could be used in robotic control to emulate an animal motion as realistic as possible.