Legged locomotion arises from intermittent contact between limbs and terrain. Since it emerges from a closed-loop interaction, reductionist study of body mechanics and terrestrial dynamics in isolation have failed to yield comprehensive strategies for forward- or reverse-engineering locomotion. Progress in locomotion science stands to benefit a diverse array of engineers, scientists, and clinicians working in robotics, neuromechanics, and rehabilitation. Eschewing reductionism in favor of a holistic study, we seek a systems-level theory tailored to the dynamics of legged locomotion.

Parsimonious mathematical models for legged locomotion are hybrid, as the system state undergoes continuous flow through limb stance and swing phases punctuated by instantaneous reset at discrete touchdown and liftoff events. In their full generality, hybrid systems can exhibit properties such as nondeterminism and orbital instability that are inconsistent with observations of organismal biomechanics. By specializing to a class of intrinsically self-consistent dynamical models, we exclude such pathologies while retaining emergent phenomena that arise in closed-loop studies of locomotion.

Beginning with a general class of hybrid control systems, we construct an intrinsic state-space metric and derive a provably-convergent numerical simulation algorithm. This resolves two longstanding problems in hybrid systems theory: non-trivial comparison of states from distinct discrete modes, and accurate simulation up to and including Zeno events. Focusing on models for periodic gaits, we prove that isolated discrete transitions generically lead the hybrid dynamical system to reduce to an equivalent classical (smooth) dynamical system. This novel route to reduction in models of rhythmic phenomena demonstrates that the closed-loop interaction between limbs and terrain is generally simpler than either taken in isolation. Finally, we show that the non-smooth flow resulting from arbitrary footfall timing possesses a non-classical (Bouligand) derivative. This provides a foundation for design and control of multi-legged maneuvers. Taken together, these contributions yield a unified analytical and computational framework -- a hybrid dynamical systems theory -- applicable to legged locomotion.

In this dissertation, we detail the Recursive Multi-Frame Planar Parallax (RMFPP) algorithm, a recursive extension of Irani et al.'s Multi-Frame Planar Parallax (MFPP) batch algorithm that allows real-time reconstruction of distant static scenes using computer vision, with expected error that increases only linearly with depth. We present an overview and comprehensive derivation of the theoretical foundation on which the RMFPP algorithm is built, including the seminal planar-parallax work by Sawhney. We derive a recursive cost function that preserves more of the problem's nonlinearity than does the cost function in the MFPP algorithm, which allows a more accurate recursive procedure. In order to obtain a recursive algorithm, we remove the geometry-refining optimization that is present in the MFPP algorithm; however, we empirically show that our algorithm degrades gracefully in the presence of geometric error. We present results using both synthetic and real imagery that show that the RMFPP algorithm is at least as accurate as the original MFPP batch algorithm in many circumstances, is preferred to both fixed- and dynamic-baseline two-frame methods, and is suitable for real-time use.

In the modernization of infrastructure systems such as energy, transportation, and healthcare systems we are seeing the convergence of three research domains: Cyber–Physical Systems (CPS), Big Data, and the Internet of Things (IoT). Indeed, new CPS technologies are are being deployed to create large sensor-actuator networks which produce massive quantities of data often in real-time which is, in turn, being used to inform everyday decision-making of the entities that engage with these large-scale infrastructure systems. As a consequence, such systems are quickly evolving into societal-scale cyber-physical systems.

The result of this increasing connectivity and interdependence is two–fold: more and more data is being collected, transmitted, and stored, and more and more actuation modalities are available, allowing new ways to influence the behavior of infrastructure systems. These new and pervasive sensing/actuation modalities present new opportunities for improving efficiency, yet they expose novel vulnerabilities. In energy CPS, for instance, smart metering technologies increase the availability of streaming data thereby enabling monetization of energy savings. Such savings can be realized by employing novel machine learning algorithms to customize offerings to consumers. On the other hand, the availability of this fine-grained consumer/system data and the increased number of access points to the broader system expose new privacy and security risks. Hence, there is a inherent efficiency-vulnerability tradeoff. This tradeoff is becoming more pronounced due to greater dependence on CPS technologies and the push towards more human-centric operations, i.e. integration of human decision-making and preferences into the closed-loop behavior of the system.

Beginning with the problem of modeling the non-cooperative agents that interact with these large-scale sociotechnical systems and thus, compete over scarce resources, we analyze the of the outcome of their strategic interactions. In particular, we create a characterization of Nash equilibria—termed differential Nash equilibria—in games on non-convex strategy spaces that is amenable to computation. We show that such non-degenerate differential Nash equilibria are structurally stable and generic thereby robust to small modeling errors and measurement noise. Introducing a planner tasked with coordinating these decision-makers, we leverage this characterization in the construction of a utility learning and incentive design algorithm. We provide convergence results in both the case where agents play according to Nash and where they play using a myopic update rule.

Narrowing our focus to the demand-side of the smart grid, we consider that the planner will capitalize on new sensing/actuation modalities in the design incentives thereby exposing the efficiency-vulnerability tradeoff. We consider privacy risks introduced by smart metering technologies that produce streaming energy consumption data. On one hand the data has utility in the sense that it can help improve operations, yet on the other it exposes the consumer and the power company to greater privacy risk. We propose a solution that combines economic and statistics tools, i.e. privacy-aware service contracts in which service is differentiated according to privacy and consumers select based on their needs and wallet. We argue that the power company has an incentive to invest in security or purchase insurance because of inefficiencies that arise due to information asymmetries and we design insurance contracts accordingly. We provide a number of qualitative insights that have the potential to be useful for informing policy and regulations in the energy ecosystem. Finally, we conclude with an overview of the contributions and a discussion of future

research directions for the near and far terms.

The contributions are the first steps towards an emerging systems theory of societal-scale cyber-physical systems in which there are many tightly

coupled human-CPS decision-making loops and socioeconomic factors intricately woven into the fabric.

Every good collaboration is built on solid mutual understanding. Without understanding their machines' behavior, human operators cannot plan around them. Yet increasing automation is distancing them from active understanding. This dissertation will apply cognitive science to build automation that boosts human understanding.

The need for transparency is urgent for safety supervising tasks. Humans' environmental awareness and expansive understanding of safety can save robots from unforeseen edge cases. But only if those humans can also think through the robot's ongoing activity. Actions can be optimized to evidence safety or clearly anticipate faults, enabling supervisors to develop evidence-based appropriate trust. This work explores how observing action allows both humans and robots to construct better working models of the other.

In research on assured autonomy we focus on how machines can autonomously guarantee safety. Yet there will always remain a modeling gap that we require human collaborators to help fill: that's why after decades of autopilot experience and improvements, we still require two human pilots to validate ongoing safe operation. This thesis contends that safe robotics must work to inform these safety collaborators; that choices don't only function to complete objectives but are also evidence that other agents ultimately judge.

Characterizing how agents judge can empower our machines to choose actions to win correct judgements. First we will exposit how to learn humans' safety concerns from data despite noisy dynamics and demonstrations. After learning humans' concerns, we typify how they perceive and forecast danger. Building on cognitive science we present a model of human safety forecasting structured by reachability analysis. This structure induces data-efficient learning on small datasets so we can learn each supervisor's idiosyncratic ways of thinking -- enabling designers to conform their intelligent systems like a glove to a hand.

We build on these models of human safety judgement to support that judgement through machine choices. After learning each supervisor's unique alarms, respecting that safe set lets robot teams decrease supervisory false positives. Extending this approach to anticipate safety concerns ahead of the decision point, we optimize motion as evidence to reject the null hypothesis of danger.

The approaches in this dissertation contributes a mathematical lens for further inquiries into human risk-taking, safety negotiation, and technology learning. By employing the formalisms of intelligent safety to sketch human safety behavior, we imbue machines with a ``theory of mind'' that is essential to fluent collaboration for our societal systems.

As the capabilities of robotic systems increase, we move closer to the vision of ubiquitous robotic assistance throughout our everyday lives. In transitioning robots and autonomous systems from traditional factory and industrial settings, it is critical that these systems are able to adapt to uncertain environments and the humans who populate them. In order to better understand and predict the behavior of these humans, Inverse Reinforcement Learning (IRL) uses demonstrations to infer the underlying motivations driving human actions. The information gained from IRL can be used to improve a robot’s understanding of the environment as well as to allow the robot to better interact with or assist humans.

In this dissertation, we address the challenge of incorporating safety into the application of IRL. We first consider safety in the context of using IRL for assisting humans in shared control tasks. Through a user study, we show how incorporating haptic feedback into human assistance can increase humans’ sense of control while improving safety in the presence of imperfect learning. Further, we present our method for using IRL to automatically create such haptic feedback policies from task demonstrations.

We further address safety in IRL by incorporating notions of safety directly into the learning process. Currently, most work on IRL focuses on learning explanatory rewards that humans are modeled as optimizing. However, pure reward optimization can fail to effectively capture hard requirements, such as safety constraints. We draw on the definition of safety from Hamilton-Jacobi reachability analysis to infer human perceptions of safety and to modify robot behavior to respect these learned safety constraints. We also extend this work on learning constraints by adapting the framework of Maximum Entropy IRL in order to learn hard constraints given nominal task rewards, and we show how this technique infers the most likely constraints to align expected behavior with observed demonstrations.