Beyond Basins of Attraction: Quantifying Robustness of Natural Dynamics

Properly designing a system to exhibit favorable natural dynamics can greatly simplify designing or learning the control policy. However, it is still unclear what constitutes favorable natural dynamics and how to quantify its effect. Most studies of simple walking and running models have focused on the basins of attraction of passive limit-cycles and the notion of self-stability. We instead emphasize the importance of stepping beyond basins of attraction. We show an approach based on viability theory to quantify robust sets in state-action space. These sets are valid for the family of all robust control policies, which allows us to quantify the robustness inherent to the natural dynamics before designing the control policy or specifying a control objective. We illustrate our formulation using spring-mass models, simple low dimensional models of running systems. We then show an example application by optimizing robustness of a simulated planar monoped, using a gradient-free optimization scheme. Both case studies result in a nonlinear effective stiffness providing more robustness.Comment: 15 pages. This work has been accepted to IEEE Transactions on Robotics (2019

Taming the beast: diffusion method in nonlocal gravity

We present a method to solve the nonlinear dynamical equations of motion in gravitational theories with fundamental nonlocalities of a certain type. For these specific form factors, which appear in some renormalizable theories, the number of field degrees of freedom and of initial conditions is finite.Comment: 10 pages. v2: erratum on number of initial conditions in the scalar field theory included. arXiv admin note: text overlap with arXiv:1803.0784

Affinity for Scalar Fields to Dissipate

The zero temperature effective equation of motion is derived for a scalar field interacting with other fields. For a broad range of cases, involving interaction with as few as one or two fields, dissipative regimes are found for the scalar field system. The zero temperature limit constitutes a baseline effect that will be prevalent in any general statistical state. Thus, the results found here provide strong evidence that dissipation is the norm not exception for an interacting scalar field system. For application to inflationary cosmology, this provides convincing evidence that warm inflation could be a natural dynamics once proper treatment of interactions is done. The results found here also may have applicability to entropy production during the chiral phase transition in heavy ion collision.Comment: 21 pages, 4 figures (uses epsf, RevTeX). Just minor changes. Version in press Physical Review D (2001

Viability in State-Action Space: Connecting Morphology, Control, and Learning

Wie können wir Robotern ermöglichen, modellfrei und direkt auf der Hardware zu lernen? Das maschinelle Lernen nimmt als Standardwerkzeug im Arsenal des Robotikers seinen Platz ein. Es gibt jedoch einige offene Fragen, wie man die Kontrolle über physikalische Systeme lernen kann. Diese Arbeit gibt zwei Antworten auf diese motivierende Frage. Das erste ist ein formales Mittel, um die inhärente Robustheit eines gegebenen Systemdesigns zu quantifizieren, bevor der Controller oder das Lernverfahren entworfen wird. Dies unterstreicht die Notwendigkeit, sowohl das Hardals auch das Software-Design eines Roboters zu berücksichtigen, da beide Aspekte in der Systemdynamik untrennbar miteinander verbunden sind. Die zweite ist die Formalisierung einer Sicherheitsmass, die modellfrei erlernt werden kann. Intuitiv zeigt diese Mass an, wie leicht ein Roboter Fehlschläge vermeiden kann. Auf diese Weise können Roboter unbekannte Umgebungen erkunden und gleichzeitig Ausfälle vermeiden. Die wichtigsten Beiträge dieser Dissertation basieren sich auf der Viabilitätstheorie. Viabilität bietet eine alternative Sichtweise auf dynamische Systeme: Anstatt sich auf die Konvergenzeigenschaften eines Systems in Richtung Gleichgewichte zu konzentrieren, wird der Fokus auf Menge von Fehlerzuständen und die Fähigkeit des Systems, diese zu vermeiden, verlagert. Diese Sichtweise eignet sich besonders gut für das Studium der Lernkontrolle an Robotern, da Stabilität im Sinne einer Konvergenz während des Lernprozesses selten gewährleistet werden kann. Der Begriff der Viabilität wird formal auf den Zustand-Aktion-Raum erweitert, mit Viabilitätsmengen von Staat-Aktionspaaren. Eine über diese Mengen definierte Mass ermöglicht eine quantifizierte Bewertung der Robustheit, die für die Familie aller fehlervermeidenden Regler gilt, und ebnet den Weg für ein sicheres, modellfreies Lernen. Die Arbeit beinhaltet auch zwei kleinere Beiträge. Der erste kleine Beitrag ist eine empirische Demonstration der Shaping durch ausschliessliche Modifikation der Systemdynamik. Diese Demonstration verdeutlicht die Bedeutung der Robustheit gegenüber Fehlern für die Lernkontrolle: Ausfälle können nicht nur Schäden verursachen, sondern liefern in der Regel auch keine nützlichen Gradienteninformationen für den Lernprozess. Der zweite kleine Beitrag ist eine Studie über die Wahl der Zustandsinitialisierungen. Entgegen der Intuition und der üblichen Praxis zeigt diese Studie, dass es zuverlässiger sein kann, das System gelegentlich aus einem Zustand zu initialisieren, der bekanntermassen unkontrollierbar ist.How can we enable robots to learn control model-free and directly on hardware? Machine learning is taking its place as a standard tool in the roboticist’s arsenal. However, there are several open questions on how to learn control for physical systems. This thesis provides two answers to this motivating question. The first is a formal means to quantify the inherent robustness of a given system design, prior to designing the controller or learning agent. This emphasizes the need to consider both the hardware and software design of a robot, which are inseparably intertwined in the system dynamics. The second is the formalization of a safety-measure, which can be learned model-free. Intuitively, this measure indicates how easily a robot can avoid failure, and enables robots to explore unknown environments while avoiding failures. The main contributions of this dissertation are based on viability theory. Viability theory provides a slightly unconventional view of dynamical systems: instead of focusing on a system’s convergence properties towards equilibria, the focus is shifted towards sets of failure states and the system’s ability to avoid these sets. This view is particularly well suited to studying learning control in robots, since stability in the sense of convergence can rarely be guaranteed during the learning process. The notion of viability is formally extended to state-action space, with viable sets of state-action pairs. A measure defined over these sets allows a quantified evaluation of robustness valid for the family of all failure-avoiding control policies, and also paves the way for enabling safe model-free learning. The thesis also includes two minor contributions. The first minor contribution is an empirical demonstration of shaping by exclusively modifying the system dynamics. This demonstration highlights the importance of robustness to failures for learning control: not only can failures cause damage, but they typically do not provide useful gradient information for the learning process. The second minor contribution is a study on the choice of state initializations. Counter to intuition and common practice, this study shows it can be more reliable to occasionally initialize the system from a state that is known to be uncontrollable

Applied Safety Critical Control

There is currently a clear gap between control-theoretical results and the reality of robotic implementation, in the sense that it is very difficult to transfer analytical guarantees to practical ones. This is especially problematic when trying to design safety-critical systems where failure is not an option. While there is a vast body of work on safety and reliability in control theory, very little of it is actually used in practice where safety margins are typically empiric and/or heuristic. Nevertheless, it is still widely accepted that a solution to these problems can only emerge from rigorous analysis, mathematics, and methods. In this work, we therefore seek to help bridge this gap by revisiting and expanding existing theoretical results in light of the complexity of hardware implementation. To that end, we begin by making a clear theoretical distinction between systems and models, and outline how the two need to be related for guarantees to transfer from the latter to the former. We then formalize various imperfections of reality that need to be accounted for at a model level to provide theoretical results with better applicability. We then discuss the reality of digital controller implementation and present the mathematical constraints that theoretical control laws must satisfy for them to be implementable on real hardware. In light of these discussions, we derive new realizable set-invariance conditions that, if properly enforced, can guarantee safety with an arbitrary high levels of confidence. We then discuss how these conditions can be rigorously enforced in a systematic and minimally invasive way through convex optimization-based Safety Filters. Multiple safety filter formulations are proposed with varying levels of complexity and applicability. To enable the use of these safety filters, a new algorithm is presented to compute appropriate control invariant sets and guarantee feasibility of the optimization problem defining these filters. The effectiveness of this approach is demonstrated in simulation on a nonlinear inverted pendulum and experimentally on a simple vehicle. The aptitude of the framework to handle a system's dynamics uncertainty is illustrated by varying the mass of the vehicle and showcasing when safety is conserved. Then, the aptitude of this approach to provide guarantees that account for controller implementation's constraints is illustrated by varying the frequency of the control loop and again showcasing when safety is conserved. In the second part of this work, we revisit the safety filtering approach in a way that addresses the scalability issues of the first part of this work. There are two main approaches to safety-critical control. The first one relies on computation of control invariant sets and was presented in the first part of this work. The second approach draws from the topic of optimal control and relies on the ability to realize Model-Predictive-Controllers online to guarantee the safety of a system. In that online approach, safety is ensured at a planning stage by solving the control problem subject for some explicitly defined constraints on the state and control input. Both approaches have distinct advantages but also major drawbacks that hinder their practical effectiveness, namely scalability for the first one and computational complexity for the second one. We therefore present an approach that draws from the advantages of both approaches to deliver efficient and scalable methods of ensuring safety for nonlinear dynamical systems. In particular, we show that identifying a backup control law that stabilizes the system is in fact sufficient to exploit some of the set-invariance conditions presented in the first part of this work. Indeed, one only needs to be able to numerically integrate the closed-loop dynamics of the system over a finite horizon under this backup law to compute all the information necessary for evaluating the regulation map and enforcing safety. The effect of relaxing the stabilization requirements of the backup law is also studied, and weaker but more practical safety guarantees are brought forward. We then explore the relationship between the optimality of the backup law and how conservative the resulting safety filter is. Finally, methods of selecting a safe input with varying levels of trade-off between conservativeness and computational complexity are proposed and illustrated on multiple robotic systems, namely: a two-wheeled inverted pendulum (Segway), an industrial manipulator, a quadrotor, and a lower body exoskeleton.</p