Origami, the Japanese art of paper folding, has come a long way from its traditionalroots. It’s now being used in modern engineering and design. In this thesis, I explored multi-stable origami structures. These structures can hold multiple stable shapes, which could have a big impact on various technologies. I aim to break down the complex ideas behind these structures and explain their potential applications in a way that’s easy to understand. In this research, I looked at the history of origami and recent developments in computational design to create and study multi-stable origami structures. I used computer tools like parametric modeling software and finite element analysis to come up with new origami designs. These tools helped me create, improve, and test these designs with a level of accuracy and speed that hadn’t been possible before. The process begins with the formulation of design principles rooted in the fundamental geometry and mechanics of origami. Leveraging mathematical algorithms and optimization techniques, diverse sets of origami crease patterns are generated, each tailored to exhibit specific multi-stable behaviors. Through iterative refinement and simulation-driven design, optimal solutions are identified, leading to the realization of intricate origami morphologies that defy traditional design constraints. Furthermore, the technological implications of multi-stable origami structures are explored across a spectrum of applications. In robotics, these structures serve as foundational building blocks for reconfigurable mechanisms capable of adapting to dynamic environments and tasks. In aerospace engineering, they enable the development of lightweight, deployable structures for space exploration and satellite deployment. In architecture, they inspire innovative approaches to adaptive building envelopes and kinetic facades, enhancing sustainability and user experience. In summary, this thesis presents a comprehensive exploration of multi-stable origami structures, from their generation through computational design methodologies to their application across diverse technological domains. By pushing the boundaries of traditional design paradigms and embracing the synergy between art, science, and technology, this research opens new frontiers for innovation and creativity in the realm of origami-inspired engineering.

Human-robot interactions can often be formulated as general-sum differential games where the equilibrial policies are governed by Hamilton-Jacobi-Isaacs (HJI) equations. Solving HJI PDEs faces the curse of dimensionality (CoD). While physics-informed neural networks (PINNs) alleviate CoD in solving PDEs with smooth solutions, they fall short in learning discontinuous solutions due to their sampling nature. This causes PINNs to have poor safety performance when they are applied to approximate values that are discontinuous due to state constraints. This dissertation aims to improve the safety performance of PINN-based value and policy models. The first contribution of the dissertation is to develop learning methods to approximate discontinuous values. Specifically, three solutions are developed: (1) hybrid learning uses both supervisory and PDE losses, (2) value-hardening solves HJIs with increasing Lipschitz constant on the constraint violation penalty, and (3) the epigraphical technique lifts the value to a higher-dimensional state space where it becomes continuous. Evaluations through 5D and 9D vehicle and 13D drone simulations reveal that the hybrid method outperforms others in terms of generalization and safety performance. The second contribution is a learning-theoretical analysis of PINN for value and policy approximation. Specifically, by extending the neural tangent kernel (NTK) framework, this dissertation explores why the choice of activation function significantly affects the PINN generalization performance, and why the inclusion of supervisory costate data improves the safety performance. The last contribution is a series of extensions of the hybrid PINN method to address real-time parameter estimation problems in incomplete-information games. Specifically, a Pontryagin-mode PINN is developed to avoid costly computation for supervisory data. The key idea is the introduction of a costate loss, which is cheap to compute yet effectively enables the learning of important value changes and policies in space-time. Building upon this, a Pontryagin-mode neural operator is developed to achieve state-of-the-art (SOTA) safety performance across a set of differential games with parametric state constraints. This dissertation demonstrates the utility of the resultant neural operator in estimating player constraint parameters during incomplete-information games.

Shape memory alloys (SMAs) are a class of smart materials that can recover their predetermined shape when subjected to an appropriate thermal cycle. This unique property makes SMAs attractive for actuator applications, where the material’s phase transformation can be used to generate controlled motion or force. The actuator design leverages the one-way shape memory effect of NiTi (Nickel-Titanium) alloy wire, which contracts upon heating and recovers its original length when cooled. A bias spring opposes the SMA wire contraction, enabling a cyclical actuation motion. Thermal actuation is achieved through joule heating by passing an electric current through the SMA wire. This thesis presents the design of a compact, lightweight SMA-based actuator, providing controlled and precise motion in various engineering applications. A design of a soft actuator is presented exploiting the responses of the shape memory alloy (SMA) to trigger intrinsically mono-stable shape reconfiguration. The proposed class of soft actuators will perform bending actuation by selectively activating the SMA. The transition sequences were optimized by geometric parameterizations and energy-based criteria. The reconfigured structure is capable of arbitrary bending, which is reported here. The proposed class of robots has shown promise as a fast actuator or shape reconfigurable structure, which will bring new capabilities in future long-duration missions in space or undersea, as well as in bio-inspired robotics.

Local interactions drive emergent collective behavior, which pervades biological and social complex systems. These behaviors are scalable and robust, motivating biomimicry: engineering nature-inspired distributed systems. But uncovering the interactions that produce a desired behavior remains a core challenge. In this thesis, I present EvoSOPS, an evolutionary framework that searches landscapes of stochastic distributed algorithms for those that achieve a mathematically specified target behavior. These algorithms govern self-organizing particle systems (SOPS) comprising individuals with strictly local sensing and movement and no persistent memory. For aggregation, phototaxing, and separation behaviors, EvoSOPS discovers algorithms that achieve 4.2–15.3% higher fitness than those from the existing “stochastic approach to SOPS” based on mathematical theory from statistical physics. EvoSOPS is also flexibly applied to new behaviors such as object coating where the stochastic approach would require bespoke, extensive analysis. Across repeated runs, EvoSOPS explores distinct regions of genome space to produce genetically diverse solutions. Finally, I provide insights into the best-fitness genomes for object coating, demonstrating how EvoSOPS can bootstrap future theoretical investigations into SOPS algorithms for challenging new behaviors.

In this thesis, I investigate a subset of reinforcement learning (RL) tasks where the objective for the agent is to achieve temporally extended goals. A common approach, in this setting, is to represent the tasks using deterministic finite automata (DFA) and integrate them in the state space of the RL algorithms, yet such representations often disregard causal knowledge pertinent to the environment. To address this limitation, I introduce the Temporal-Logic-based Causal Diagram (TL-CD) in RL.TL-CD encapsulates temporal causal relationships among diverse environmental properties. We leverage the TL-CD to devise an RL algorithm that significantly reduces environment exploration requirements. By synergizing TL-CD with task-specific DFAs, I identify scenarios wherein the agent can efficiently determine expected rewards early during the exploration phases. Through a series of case studies, I empirically demonstrate the advantages of TL-CDs, particularly highlighting the accelerated convergence of the algorithm towards an optimal policy facilitated by diminished exploration of the environment.

When solving analysis, estimation, and control problems for Partial Differential Equations (PDEs) via computational methods, one must resolve three main challenges: (a) the lack of a universal parametric representation of PDEs; (b) handling unbounded differential operators that appear as parameters; and (c), enforcing auxiliary constraints such as Boundary conditions and continuity conditions. To address these challenges, an alternative representation of PDEs called the `Partial Integral Equation' (PIE) representation is proposed in this work. Primarily, the PIE representation alleviates the problem of the lack of a universal parametrization of PDEs since PIEs have, at most, $12$ Partial Integral (PI) operators as parameters. Naturally, this also resolves the challenges in handling unbounded operators because PI operators are bounded linear operators. Furthermore, for admissible PDEs, the PIE representation is unique and has no auxiliary constraints --- resolving the last of the $3$ main challenges. The PIE representation for a PDE is obtained by finding a unique unitary map from the states of the PIE to the states of the PDE. This map shows a PDE and its associated PIE have equivalent system properties, including well-posedness, internal stability, and I/O behavior. Furthermore, this unique map also allows us to construct a well-defined dual representation that can be used to solve optimal control problems for a PDE. Using the equivalent PIE representation of a PDE, mathematical and computational tools are developed to solve standard problems in Control theory for PDEs. In particular, problems such as a test for internal stability, Input-to-Output (I/O) $L_2$-gain, $\hinf$-optimal state observer design, and $\hinf$-optimal full state-feedback controller design are solved using convex-optimization and Lyapunov methods for linear PDEs in one spatial dimension. Once the PIE associated with a PDE is obtained, Lyapunov functions (or storage functions) are parametrized by positive PI operators to obtain a solvable convex formulation of the above-stated control problems. Lastly, the methods proposed here are applied to various PDE systems to demonstrate the application.

Walking and mobility are essential aspects of our daily lives, enabling us to engage in various activities. Gait disorders and impaired mobility are widespread challenges faced by older adults and people with neurological injuries, as these conditions can significantly impact their quality of life, leading to a loss of independence and an increased risk of mortality. In response to these challenges, rehabilitation, and assistive robotics have emerged as promising alternatives to conventional gait therapy, offering potential solutions that are less labor-intensive and costly. Despite numerous advances in wearable lower-limb robotics, their current applicability remains confined to laboratory settings. To expand their utility to broader gait impairments and daily living conditions, there is a pressing need for more intelligent robot controllers. In this dissertation, these challenges are tackled from two perspectives: First, to improve the robot's understanding of human motion and intentions which is crucial for assistive robot control, a robust human locomotion estimation technique is presented, focusing on measuring trunk motion. Employing an invariant extended Kalman filtering method that takes sensor misplacement into account, improved convergence properties over the existing methods for different locomotion modes are shown. Secondly, to enhance safe and effective robot-aided gait training, this dissertation proposes to directly learn from physical therapists' demonstrations of manual gait assistance in post-stroke rehabilitation. Lower-limb kinematics of patients and assistive force applied by therapists to the patient's leg are measured using a wearable sensing system which includes a custom-made force sensing array. The collected data is then used to characterize a therapist's strategies. Preliminary analysis indicates that knee extension and weight-shifting play pivotal roles in shaping a therapist's assistance strategies, which are then incorporated into a virtual impedance model that effectively captures high-level therapist behaviors throughout a complete training session. Furthermore, to introduce safety constraints in the design of such controllers, a safety-critical learning framework is explored through theoretical analysis and simulations. A safety filter incorporating an online iterative learning component is introduced to bring robust safety guarantees for gait robotic assistance and training, addressing challenges such as stochasticity and the absence of a known prior dynamic model.

Soft robotics has garnered attention for its substantial prospective in various domains, such as manipulation and interactions with humans, by offering competitive advantages against rigid robotic systems, including inherent compliance and variable stiffness. Despite these benefits, their theoretically infinite degrees of freedom and prominent nonlinearities pose significant challenges in developing dynamic models and guiding the robots along desired paths. Additionally, soft robots may exhibit rigid behaviors and potentially collide with their surroundings during path tracking tasks, particularly when possible contact points are unknown. In this dissertation, reduced-order models are used to describe the behaviors of three different soft robot designs, including both linear parameter varying (LPV) and augmented rigid robot (ARR) models. While the reduced-order model captures the majority of the soft robot's dynamics, modeling uncertainties notably remain. Non-repeated modeling uncertainties are addressed by categorizing them as a lumped disturbance, employing two methodologies, $H_\infty$ method and nonlinear disturbance observer (NDOB) based sliding mode control, for its rejection. For repeated disturbances, an iterative learning control (ILC) with a P-type learning function is implemented to enhance trajectory tracking efficacy. Furthermore,for non-repeated disturbances, the NDOB facilitates the contact estimation, and its results are jointly used with a switching algorithm to modify the robot trajectories. The stability proof of all controllers and corresponding simulation and experimental results are provided. For a path tracking task of a soft robot with multi-segments, a robust control strategy that combines a LPV model with an innovative improved nonlinear disturbance observer-based adaptive sliding mode control (INASMC). The control framework employs a first-order LPV model for dynamic representation, leverages an improved disturbance observer for accurate disturbance forecasting, and utilizes adaptive sliding mode control to effectively counteract uncertainties. The tracking error under the proposed controller is proven to be asymptotically stable, and the controller's effectiveness is is validated with simulation and experimental results. Ultimately, this research mitigates the inherent uncertainty in soft robot modeling, thereby enhancing their functionality in contact-intensive tasks.

As the explorations beyond the Earth's boundaries continue to evolve, researchers and engineers strive to develop versatile technologies capable of adapting to unknown space conditions. For instance, the utilization of Screw-Propelled Vehicles (SPVs) and robotics that utilize helical screws propulsion to transverse planetary bodies is a growing area of interest. An example of such technology is the Extant Exobiology Life Surveyor (EELS), a snake-like robot currently developed by the NASA Jet Propulsion Laboratory (JPL) to explore the surface of Saturn’s moon, Enceladus. However, the utilization of such a mechanism requires a deep and thorough understanding of screw mobility in uncertain conditions. The main approach to exploring screw dynamics and optimal design involves the utilization of Discrete Element Method (DEM) simulations to assess interactions and behavior of screws when interacting with granular terrains. In this investigation, the Simplified Johnson-Kendall-Roberts (SJKR) model is implemented into the utilized simulation environment to account for cohesion effects similar to what is experienced on celestial bodies like Enceladus. The model is verified and validated through experimental and theoretical testing. Subsequently, the performance characteristics of screws are explored under varying parameters, such as thread depth, number of screw starts, and the material’s cohesion level. The study has examined significant relationships between the parameters under investigation and their influence on the screw performance.

Tire blowout often occurs during driving, which can suddenly disturb vehicle motions and seriously threaten road safety. Currently, there is still a lack of effective methods to mitigate tire blowout risks in everyday traffic, even for automated vehicles. To fundamentally study and systematically resolve the tire blowout issue for automated vehicles, a collaborative project between General Motors (GM) and Arizona State University (ASU) has been conducted since 2018. In this dissertation, three main contributions of this project will be presented. First, to explore vehicle dynamics with tire blowout impacts and establish an effective simulation platform for close-loop control performance evaluation, high-fidelity tire blowout models are thoroughly developed by explicitly considering important vehicle parameters and variables. Second, since human cooperation is required to control Level 2/3 partially automated vehicles (PAVs), novel shared steering control schemes are specifically proposed for tire blowout to ensure safe vehicle stabilization via cooperative driving. Third, for Level 4/5 highly automated vehicles (HAVs) without human control, the development of control-oriented vehicle models, controllability study, and automatic control designs are performed based on impulsive differential systems (IDS) theories. Co-simulations Matlab/Simulink® and CarSim® are conducted to validate performances of all models and control designs proposed in this dissertation. Moreover, a scaled test vehicle at ASU and a full-size test vehicle at GM are well instrumented for data collection and control implementation. Various tire blowout experiments for different scenarios are conducted for more rigorous validations. Consequently, the proposed high-fidelity tire blowout models can correctly and more accurately describe vehicle motions upon tire blowout. The developed shared steering control schemes for PAVs and automatic control designs for HAVs can effectively stabilize a vehicle to maintain path following performance in the driving lane after tire blowout. In addition to new research findings and developments in this dissertation, a pending patent for tire blowout detection is also generated in the tire blowout project. The obtained research results have attracted interest from automotive manufacturers and could have a significant impact on driving safety enhancement for automated vehicles upon tire blowout.