DOCSLIB.ORG

© 2020 Alexandra Q. Nilles DESIGNING BOUNDARY INTERACTIONS for SIMPLE MOBILE ROBOTS

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
© 2020 Alexandra Q. Nilles DESIGNING BOUNDARY INTERACTIONS for SIMPLE MOBILE ROBOTS © 2020 Alexandra Q. Nilles DESIGNING BOUNDARY INTERACTIONS FOR SIMPLE MOBILE ROBOTS BY ALEXANDRA Q. NILLES DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science in the Graduate College of the University of Illinois at Urbana-Champaign, 2020 Urbana, Illinois Doctoral Committee: Professor Steven M. LaValle, Chair Professor Nancy M. Amato Professor Sayan Mitra Professor Todd D. Murphey, Northwestern University Abstract Mobile robots are becoming increasingly common for applications such as logistics and delivery. While most research for mobile robots focuses on generating collision-free paths, however, an environment may be so crowded with obstacles that allowing contact with environment boundaries makes our robot more efficient or our plans more robust. The robot may be so small or in a remote environment such that traditional sensing and communication is impossible, and contact with boundaries can help reduce uncertainty in the robot's state while navigating. These novel scenarios call for novel system designs, and novel system design tools. To address this gap, this thesis presents a general approach to modelling and planning over interactions between a robot and boundaries of its environment, and presents prototypes or simulations of such systems for solving high-level tasks such as object manipulation. One major contribution of this thesis is the derivation of necessary and sufficient conditions of stable, periodic trajectories for \bouncing robots," a particular model of point robots that move in straight lines between boundary interactions. Another major contribution is the description and implementation of an exact geometric planner for bouncing robots. We demonstrate the planner on traditional trajectory generation from start to goal states, as well as how to specify and generate stable periodic trajectories. In addition, we demonstrate the utility of the planner for environment geometry analysis, with respect to the role of environment geometry and system design constraints on the reachability and stability of bouncing robot trajectories. We propose a general approach for the design of bouncing robot systems, as well as more general classes of wild robots. We start by identifying useful, robust open-loop motion strategies, then integrate sensors and information space reasoning to determine conditions for switching between these open-loop behaviors. This approach is demonstrated on the task of \cart-on-track" object manipulation, motivated by design constraints for robots at the micrometer length scale. We identify the parts of this approach most amenable to automation, and provide a collection of supporting software tools. The final contribution of the thesis is a chapter including qualitative design principles for automated robot design systems, as well as an example of a live-coding interface for the design of mobile robot motion patterns demonstrating some of these principles. We conclude with an outline of our vision for this area of research in the future. ii To my parents, and all my teachers. iii Acknowledgments They say a Ph.D. is a marathon, not a sprint; although I've never ran that far in one go, I'd say it's an accurate analogy. However, I think a more appropriate cliche is the one about how it takes a village - certainly, this thesis would never have been completed without a large village of incredibly supportive people. Of course I must thank my parents, who were \blessed" with two extremely precocious and stubborn children. Thank you for all your love and support, and for always encouraging me to “finish strong." Thank you to my brother Andrew, for all the laughs and for getting me out into the wilderness when I needed a break from the robots. I must also thank my friends and chosen family, especially Leah Moldauer, Alex Phillips, Halie Kastle, Lucy Orsi, Simona Mara Smith, Jacob Dixon, Ted Kim, and Eric Jones. Your love kept me sane through the toughest times. Special thanks to Jordan Hall, for your incredible patience through the emotional rollercoaster of 2020, and for always being able to make me laugh. I would also like to remember Maria Schmidt and David Vargas, two friends that I lost during the Ph.D. May they rest in power. I owe an immeasurable debt to my advisor, Professor Steve LaValle. From the beginning Steve has treated me (and all his students and collaborators) with deep respect. He has taught me a great deal about math, motion planning, and technical writing; he has also been a great teacher in how to stay honest, kind, and humble under pressure. His love of learning is infectious and inspirational, whether focused on technology, business, language, or new cuisines. Thank you to Steve, Anna, and your family for your grace over the years and for being incredible hosts. I hope to be able to visit Oulu again soon! Thank you to all the people in the Motion Strategy Lab that I have worked with over the years - no matter how brief our collaboration, I have learned something from everyone I have worked with. Israel Becerra visited as a postdoc and provided some much-needed structure to my project management when I was just starting out. Thanks to Samara Ren for all the delightful conversations, lovely illustrations, mathematical insights, and for helping me navigate the maze of research directions over several years of collaboration. I have been lucky to work with many undergraduate students over the years, many of which are better researchers than I and are constant inspirations to me. I wish to thank Justin Wasserman, Austin Born, John Born, Chris Horn, Thomas Driscoll, and Jon Park in particular for sticking with me on the weaselballs project, even when it seemed a wild goose chase. Deep thanks to Dr. Amy LaViers and the entire Robotics, Automation, and Dance (RAD) iv Lab, especially Umer Huzaifa, Ishaan Pakrasi, Catie Cuan, Anum Jang Sher, and everyone else in Amy's 598 class. The RAD Lab is a wonderfully self-aware community and helped me feel less alone for caring about what I care about. On a related note, thank you to Dance at Illinois for allowing me into your spaces; even though the credits didn't count toward my degree they were some of the most valuable classes I have ever taken. Thank you to the Department of Computer Science, especially Kathy Runck, Viveka Ku- daligama, Maggie Metzger Chappell, Julie Owens, and all of the administrative staff that helped me with the surprisingly difficult logistics of being a graduate student. Thanks to everyone who helped me resurrect the nearly-defunct graduate student organization, espe- cially Zane Ma, Deepak Kumar, Helen Wauck, Patrick Lin, and Shant Boodaghians; that was fun, and Chez Tom will live on forever in my heart. Thanks to the theory group in particular for being friendly and welcoming despite my relative lack of rigor! I must thank all my teachers and professors over the years. I would like to especially thank Yuliy Baryshnikov for extending support several times when I was in need, and for his patience with my still-ongoing progress toward publishable results. I would also like to acknowledge the late Dr. Alfred Hubler, who managed to reignite my love of experiments, as well as gave me a much-needed kick in the pants to start publishing. His loss was deeply felt. I would also like to extend thanks to Leonardo Bobadilla, Jason O'Kane, Andrea Censi, Dylan Shell, and Seth Hutchinson for your mentorship and support over the years, as well as everyone who has kindly welcomed me to the robotics community and made conferences a highly entertaining experience. Finally, my deepest gratitude to my committee { Nancy Amato, Sayan Mitra, and Todd Murphey { for your extra support in Steve's absence, your help with talk preparation, and all the constructive feedback and ideas for future work. Several local Champaign-Urbana institutions were instrumental to the completion of this work, including Golden Wok, Ozu Ramen, Aroma Cafe, Watson's Shack and Rail, Clark Bar, and 25 O'Clock Brewing Company. When I moved to the middle of Illinois, I never expected to find such a vibrant community of lovely people, and so much art, activism, and cooperative spirit. CU will always have a special place in my heart. Finally, I must give my love and appreciation to my co-op, the Community of Urbana- Champaign Cooperative Housing (COUCH), which housed and fed me for the entirety of my time in graduate school. I also found some of my closest friends there. While helping manage a cooperative business may have been somewhat distracting from my Ph.D. work, I would not trade the experience for anything. Living in community with many different people taught me how to be a dramatically more kind, responsible, communicative person. COUCH also taught me quite a lot about anti-racism and equity work, lessons I hope will benefit my communities for the rest of my life. v Table of Contents Chapter 1 Introduction . 1 1.1 Roadmap: Moving Up the Abstraction \Stack" . .2 1.2 Contributions . .4 Chapter 2 Background and Related Work . 6 2.1 Handling Uncertainty in Robotics, Control Theory and Artificial Intelligence6 2.2 Formal Methods in Robotics . .7 2.3 Mobile Robots . 10 2.4 Micro-Robotics . 13 2.5 Related Mathematical Problems . 15 2.6 Conclusion . 16 Chapter 3 Modelling . 17 3.1 Bouncing Robots . 18 3.2 Wild Bodies . 22 Chapter 4 Trajectories as Dynamical Systems . 25 4.1 What Behaviors are Useful? . 25 4.2 Trajectories in Regular Polygons . 26 4.3 Nondeterministic Trajectories . 37 4.4 Trajectories in Nonconvex Polygons . 44 4.5 Ergodic and Chaotic Trajectories . 45 Chapter 5 Motion Planning for Bouncing Robots . 46 5.1 Planning for a Perfect Robot .
Load more

Recommended publications
  • From Deterministic Dynamics to Thermodynamic Laws II: Fourier's
    FROM DETERMINISTIC DYNAMICS TO THERMODYNAMIC LAWS II: FOURIER'S LAW AND MESOSCOPIC LIMIT EQUATION YAO LI Abstract. This paper considers the mesoscopic limit of a stochastic energy ex- change model that is numerically derived from deterministic dynamics. The law of large numbers and the central limit theorems are proved. We show that the limit of the stochastic energy exchange model is a discrete heat equation that satisfies Fourier's law. In addition, when the system size (number of particles) is large, the stochastic energy exchange is approximated by a stochastic differential equation, called the mesoscopic limit equation. 1. Introduction Fourier's law is an empirical law describing the relationship between the thermal conductivity and the temperature profile. In 1822, Fourier concluded that \the heat flux resulting from thermal conduction is proportional to the magnitude of the tem- perature gradient and opposite to it in sign" [14]. The well-known heat equation is derived based on Fourier's law. However, the rigorous derivation of Fourier's law from microscopic Hamiltonian mechanics remains to be a challenge to mathemati- cians and physicist [3]. This challenge mainly comes from our limited mathematical understanding to nonequilibrium statistical mechanics. After the foundations of sta- tistical mechanics were established by Boltzmann, Gibbs, and Maxwell more than a century ago, many things about nonequilibrium steady state (NESS) remains un- clear, especially the dependency of key quantities on the system size N. There have been several studies that aim to derive Fourier's law from the first principle. A large class of models [29, 31, 12, 11, 10] use anharmonic chains to de- scribe heat conduction in insulating crystals.
    [Show full text]
  • Ergodic Theory Plays a Key Role in Multiple Fields Steven Ashley Science Writer
    CORE CONCEPTS Core Concept: Ergodic theory plays a key role in multiple fields Steven Ashley Science Writer Statistical mechanics is a powerful set of professor Tom Ward, reached a key milestone mathematical tools that uses probability the- in the early 1930s when American mathema- ory to bridge the enormous gap between the tician George D. Birkhoff and Austrian-Hun- unknowable behaviors of individual atoms garian (and later, American) mathematician and molecules and those of large aggregate sys- and physicist John von Neumann separately tems of them—a volume of gas, for example. reconsidered and reformulated Boltzmann’ser- Fundamental to statistical mechanics is godic hypothesis, leading to the pointwise and ergodic theory, which offers a mathematical mean ergodic theories, respectively (see ref. 1). means to study the long-term average behavior These results consider a dynamical sys- of complex systems, such as the behavior of tem—whetheranidealgasorothercomplex molecules in a gas or the interactions of vi- systems—in which some transformation func- brating atoms in a crystal. The landmark con- tion maps the phase state of the system into cepts and methods of ergodic theory continue its state one unit of time later. “Given a mea- to play an important role in statistical mechan- sure-preserving system, a probability space ics, physics, mathematics, and other fields. that is acted on by the transformation in Ergodicity was first introduced by the a way that models physical conservation laws, Austrian physicist Ludwig Boltzmann laid Austrian physicist Ludwig Boltzmann in the what properties might it have?” asks Ward, 1870s, following on the originator of statisti- who is managing editor of the journal Ergodic the foundation for modern-day ergodic the- cal mechanics, physicist James Clark Max- Theory and Dynamical Systems.Themeasure ory.
    [Show full text]
  • AN INTRODUCTION to DYNAMICAL BILLIARDS Contents 1
    AN INTRODUCTION TO DYNAMICAL BILLIARDS SUN WOO PARK 2 Abstract. Some billiard tables in R contain crucial references to dynamical systems but can be analyzed with Euclidean geometry. In this expository paper, we will analyze billiard trajectories in circles, circular rings, and ellipses as well as relate their charactersitics to ergodic theory and dynamical systems. Contents 1. Background 1 1.1. Recurrence 1 1.2. Invariance and Ergodicity 2 1.3. Rotation 3 2. Dynamical Billiards 4 2.1. Circle 5 2.2. Circular Ring 7 2.3. Ellipse 9 2.4. Completely Integrable 14 Acknowledgments 15 References 15 Dynamical billiards exhibits crucial characteristics related to dynamical systems. Some billiard tables in R2 can be understood with Euclidean geometry. In this ex- pository paper, we will analyze some of the billiard tables in R2, specifically circles, circular rings, and ellipses. In the first section we will present some preliminary background. In the second section we will analyze billiard trajectories in the afore- mentioned billiard tables and relate their characteristics with dynamical systems. We will also briefly discuss the notion of completely integrable billiard mappings and Birkhoff's conjecture. 1. Background (This section follows Chapter 1 and 2 of Chernov [1] and Chapter 3 and 4 of Rudin [2]) In this section, we define basic concepts in measure theory and ergodic theory. We will focus on probability measures, related theorems, and recurrent sets on certain maps. The definitions of probability measures and σ-algebra are in Chapter 1 of Chernov [1]. 1.1. Recurrence. Definition 1.1. Let (X,A,µ) and (Y ,B,υ) be measure spaces.
    [Show full text]
  • Propagation of Rays in 2D and 3D Waveguides: a Stability Analysis with Lyapunov and Reversibility Fast Indicators G
    Propagation of rays in 2D and 3D waveguides: a stability analysis with Lyapunov and Reversibility fast indicators G. Gradoni,1, a) F. Panichi,2, b) and G. Turchetti3, c) 1)School of Mathematical Sciences and Department of Electrical and Electronic Engineering, University of Nottingham, United Kingdomd) 2)Department of Physics and Astronomy , University of Bologna, Italy 3)Department of Physics and Astronomy, University of Bologna, Italy. INDAM National Group of Mathematical Physics, Italy (Dated: 13 January 2021) Propagation of rays in 2D and 3D corrugated waveguides is performed in the general framework of stability indicators. The analysis of stability is based on the Lyapunov and Reversibility error. It is found that the error growth follows a power law for regular orbits and an exponential law for chaotic orbits. A relation with the Shannon channel capacity is devised and an approximate scaling law found for the capacity increase with the corrugation depth. We investigate the propagation of a ray in a 2D wave guide whose boundaries are two parallel horizontal lines, with a periodic corrugation on the upper line. The reflection point abscissa on the lower line and the ray horizontal I. INTRODUCTION velocity component after reflection are the phase space coordinates and the map connecting two consecutive re- The equivalence between geometrical optics and mechanics flections is symplectic. The dynamic behaviour is illus- was established in a variational form by the principles of Fer- trated by the phase portraits which show that the regions mat and Maupertuis. If a ray propagates in a uniform medium of chaotic motion increase with the corrugation amplitude.
    [Show full text]
  • (2020) Physics-Enhanced Neural Networks Learn Order and Chaos
    PHYSICAL REVIEW E 101, 062207 (2020) Physics-enhanced neural networks learn order and chaos Anshul Choudhary ,1 John F. Lindner ,1,2,* Elliott G. Holliday,1 Scott T. Miller ,1 Sudeshna Sinha,1,3 and William L. Ditto 1 1Nonlinear Artificial Intelligence Laboratory, Physics Department, North Carolina State University, Raleigh, North Carolina 27607, USA 2Physics Department, The College of Wooster, Wooster, Ohio 44691, USA 3Indian Institute of Science Education and Research Mohali, Knowledge City, SAS Nagar, Sector 81, Manauli PO 140 306, Punjab, India (Received 26 November 2019; revised manuscript received 22 May 2020; accepted 24 May 2020; published 18 June 2020) Artificial neural networks are universal function approximators. They can forecast dynamics, but they may need impractically many neurons to do so, especially if the dynamics is chaotic. We use neural networks that incorporate Hamiltonian dynamics to efficiently learn phase space orbits even as nonlinear systems transition from order to chaos. We demonstrate Hamiltonian neural networks on a widely used dynamics benchmark, the Hénon-Heiles potential, and on nonperturbative dynamical billiards. We introspect to elucidate the Hamiltonian neural network forecasting. DOI: 10.1103/PhysRevE.101.062207 I. INTRODUCTION dynamical systems. But from stormy weather to swirling galaxies, natural dynamics is far richer and more challeng- Newton wrote, “My brain never hurt more than in my ing. In this article, we exploit the Hamiltonian structure of studies of the moon (and Earth and Sun)” [1]. Unsurprising conservative systems to provide neural networks with the sentiment, as the seemingly simple three-body problem is in- physics intelligence needed to learn the mix of order and trinsically intractable and practically unpredictable.
    [Show full text]
  • Jaime Bobadilla Molina.Pdf
    c 2013 Jaime Leonardo Bobadilla Molina MINIMALIST MULTI-AGENT FILTERING AND GUIDANCE BY JAIME LEONARDO BOBADILLA MOLINA DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science in the Graduate College of the University of Illinois at Urbana-Champaign, 2013 Urbana, Illinois Doctoral Committee: Associate Professor Samuel T. King, Chair Professor Steven M. LaValle, Director of Research Professor Tarek Abdelzaher Assistant Professor Dylan A. Shell, Texas A&M University ABSTRACT Advances in technology have allowed robots to be equipped with powerful sensors, complex actuators, computers with high processing capabilities, and high bandwidth communication links. This trend has enabled the development of sophisticated algorithms and systems to solve tasks such as navigation, patrolling, coverage, tracking, and counting. However, these systems have to deal with issues such as dynamical system identification, sensor calibration, and computation of powerful filters for state-feedback policies. This thesis presents novel techniques for tackling the above mentioned tasks. Our methods differ from traditional approaches since they do not require system identification, geometric map building, or state estimation. Instead, we follow a minimalist approach that takes advantage of the wild motions of bodies in their environment. The bodies move within regions connected by gates that enforce specific flows or provide simple sensor feedback. More specifically, five types of gates are proposed: 1) static gates, in which the flow direction of bodies cannot be changed during execution; 2) pliant gates, whose flow directions can be changed by gate-body collisions; 3) controllable gates, whose flow directions can be changed by powered actuators and sensor feedback; 4) virtual gates, in which the flow is affected by robot sensing and do not represent a physical obstruction; and 5) directional detection gates that do not change the flow of bodies, but simply detect bodies’ transitions from region to region.
    [Show full text]
  • Ballistic Electron Transport in Graphene Nanodevices and Billiards
    Ballistic Electron Transport in Graphene Nanodevices and Billiards Dissertation (Cumulative Thesis) for the award of the degree Doctor rerum naturalium of the Georg-August-Universität Göttingen within the doctoral program Physics of Biological and Complex Systems of the Georg-August University School of Science submitted by George Datseris from Athens, Greece Göttingen 2019 Thesis advisory committee Prof. Dr. Theo Geisel Department of Nonlinear Dynamics Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Stephan Herminghaus Department of Dynamics of Complex Fluids Max Planck Institute for Dynamics and Self-Organization Dr. Michael Wilczek Turbulence, Complex Flows & Active Matter Max Planck Institute for Dynamics and Self-Organization Examination board Prof. Dr. Theo Geisel (Reviewer) Department of Nonlinear Dynamics Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Stephan Herminghaus (Second Reviewer) Department of Dynamics of Complex Fluids Max Planck Institute for Dynamics and Self-Organization Dr. Michael Wilczek Turbulence, Complex Flows & Active Matter Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Ulrich Parlitz Biomedical Physics Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Stefan Kehrein Condensed Matter Theory, Physics Department Georg-August-Universität Göttingen Prof. Dr. Jörg Enderlein Biophysics / Complex Systems, Physics Department Georg-August-Universität Göttingen Date of the oral examination: September 13th, 2019 Contents Abstract 3 1 Introduction 5 1.1 Thesis synopsis and outline.............................. 10 2 Fundamental Concepts 13 2.1 Nonlinear dynamics of antidot superlattices..................... 13 2.2 Lyapunov exponents in billiards............................ 20 2.3 Fundamental properties of graphene......................... 22 2.4 Why quantum?..................................... 31 2.5 Transport simulations and scattering wavefunctions................
    [Show full text]
  • Universality in Statistical Measures of Trajectories in Classical Billiard Systems
    Applied Mathematics, 2015, 6, 1407-1425 Published Online July 2015 in SciRes. http://www.scirp.org/journal/am http://dx.doi.org/10.4236/am.2015.68132 Universality in Statistical Measures of Trajectories in Classical Billiard Systems Jean-François Laprise, Ahmad Hosseinizadeh, Helmut Kröger Department of Physics, Laval University, Québec, Canada Email: [email protected], [email protected], [email protected] Received 13 January 2015; accepted 27 July 2015; published 30 July 2015 Copyright © 2015 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract For classical billiards, we suggest that a matrix of action or length of trajectories in conjunction with statistical measures, level spacing distribution and spectral rigidity, can be used to distin- guish chaotic from integrable systems. As examples of 2D chaotic billiards, we considered the Bu- nimovich stadium billiard and the Sinai billiard. In the level spacing distribution and spectral ri- gidity, we found GOE behaviour consistent with predictions from random matrix theory. We stu- died transport properties and computed a diffusion coefficient. For the Sinai billiard, we found normal diffusion, while the stadium billiard showed anomalous diffusion behaviour. As example of a 2D integrable billiard, we considered the rectangular billiard. We found very rigid behaviour with strongly correlated spectra similar to a Dirac comb. These findings present numerical evi- dence for universality in level spacing fluctuations to hold in classically integrable systems and in classically fully chaotic systems. Keywords Classical Chaos, Dynamical Billiards, Random Matrix Theory, Level Spacing Fluctuations, Universality 1.
    [Show full text]
  • Arxiv:1505.01418V2 [Math.DS] 22 May 2021 Discovered by Bunimovich [Bun74, Bun92], Wojtkowski [Woj86], Markarian [Mar88] and Donnay [Don91]
    CONVEX BILLIARDS ON CONVEX SPHERES PENGFEI ZHANG Abstract. In this paper we study the dynamical billiards on a convex 2D sphere. We investigate some generic properties of the convex billiards on a general convex sphere. We prove that C1 generically, every periodic point is either hyperbolic or elliptic with irrational rotation number. Moreover, every hyperbolic periodic point admits some transverse homoclinic intersections. A new ingredient in our approach is Her- man's result on Diophantine invariant curves that we use to prove the nonlinear stability of elliptic periodic points for a dense subset of convex billiards. 1. Introduction The dynamical billiards, as a class of dynamical systems, were introduced by Birkhoff [Bir17, Bir27] in his study of Lagrangian systems with two degrees of freedom. A Lagrangian system with two degrees of freedom is isomorphic with the motion of a mass particle moving on a surface rotating uniformly about a fixed axis and carrying a fixed conservative field of force with it. If the surface is not rotating and the force vanishes, then the particle moves along geodesics on the surface. If the surface has boundary, then the resulting system is a billiard system. The classical results of dynamical billiards are closely related to geometrical optics, which has a much longer history. For example, the discovery of the integrability of elliptic billiards, according to Sarnak [Sar11], goes back at least to Boscovich in 1757. Surprisingly, the billiard dynamics is also related to the spectra property of Laplace{Beltrami operator on manifolds with a boundary. More precisely, Weyl's law in spectral theory gives the first order asymptotic distribution of eigenvalues of the Laplace{Beltrami operator on a bounded domain.
    [Show full text]
  • Learning the Arrow of Time
    Learning the Arrow of Time Nasim Rahaman1;2;3 Steffen Wolf1 Anirudh Goyal2 Roman Remme1 Yoshua Bengio2 1 Image Analysis and Learning Lab Ruprecht-Karls-Universität Heidelberg, Germany 2 Mila Montréal, Quebec, Canada 3 Max-Planck Institute for Intelligent Systems Tübingen, Germany Abstract We humans seem to have an innate understanding of the asymmetric progression of time, which we use to efficiently and safely perceive and manipulate our envi- ronment. Drawing inspiration from that, we address the problem of learning an arrow of time in a Markov (Decision) Process. We illustrate how a learned arrow of time can capture meaningful information about the environment, which in turn can be used to measure reachability, detect side-effects and to obtain an intrinsic reward signal. We show empirical results on a selection of discrete and continuous environments, and demonstrate for a class of stochastic processes that the learned arrow of time agrees reasonably well with a known notion of an arrow of time given by the celebrated Jordan-Kinderlehrer-Otto result. 1 Introduction The asymmetric progression of time has a profound effect on how we, as agents, perceive, process and manipulate our environment. Given a sequence of observations of our familiar surroundings (e.g. as photographs or video frames), we possess the innate ability to predict whether the said observations are ordered correctly. We use this ability not just to perceive, but also to act: for instance, we know to be cautious about dropping a vase, guided by the intuition that the act of breaking a vase cannot be undone. It is manifest that this profound intuition reflects some fundamental properties of the world in which we dwell, and in this work, we ask whether and how these properties can be exploited to learn a representation that functionally mimics our understanding of the asymmetric nature of time.
    [Show full text]
  • (NOT) JUST for FUN Be Sure to Visit Our Logic Section for Thinking Games and Spelling/Vocabulary Section for Word Games Too!
    (NOT) JUST FOR FUN Be sure to visit our Logic section for thinking games and Spelling/Vocabulary section for word games too! Holiday & Gift Catalog press down to hear him squeak. The bottom of A new full-color catalog of selected fun stuff is each egg contains a unique shape sort to find the available each year in October. Request yours! egg’s home in the carton. Match each chick’s 000002 . FREE eyes to his respective eggshell top, or swap them around for mix-and-match fun. Everything stores TOYS FOR YOUNG CHILDREN easily in a sturdy yellow plastic egg carton with hinged lid. Toys for Ages 0-3 005998 . 11.95 9 .50 Also see Early Learning - Toys and Games for more. A . Oball Rattle & Roll (ages 3 mo+) Activity Books Part O-Ball, part vehicle, these super-grabba- ble cars offer lots of play for little crawlers and B . Cloth Books (ages 6 mo .+) teethers. The top portion of the car is like an These adorable soft cloth books are sure to ☼My First Phone (ages 1+) O-ball, while the tough-looking wheels feature intrigue young children! In Dress-Up Bear, the No beeps or lights here: just a clever little toy rattling beads inside for additional noise and fun. “book” unbuttons into teddy bear’s outfit for the to play pretend! Made from recycled materials Two styles (red/yellow and (green/blue); if you day. The front features a snap-together buckle by PLAN toys, this phone has 5 colorful buttons order more than one, we’ll assort.
    [Show full text]
  • Quantum Chaos: an Introduction Via Chains of Interacting Spins-1/2
    Quantum chaos: an introduction via chains of interacting spins-1/2 Aviva Gubin and Lea F. Santos Department of Physics, Yeshiva University, 245 Lexington Ave, New York, NY 10016, USA This work introduces aspects of quantum chaos by focusing on spectral statistical properties and structures of eigenstates of quantum many-body systems. Quantum systems whose classical counterparts are chaotic have properties that differ from those of quantum systems whose classical counterparts are regular. One of the main signatures of what became known as quantum chaos is a spectrum showing repulsion of the energy levels. Here, we show how level repulsion may develop in one-dimensional systems of interacting spins-1/2 which are devoid of random elements and involve only two-body interactions. Chains of interacting spins-1/2 are prototype quantum many-body systems. They are used to study subjects as diverse as quantum computing, quantum phase transition, quantum transport, and quantum control. Nonetheless, they are simple enough to be accessible to students. In addition to the statistics of eigenvalues, we analyze also how the structure of the eigenstates may indicate chaos. We make available online all computer codes used to obtain the data for the figures. This should allow students and professors to easily reproduce our results and to further explore new questions. I. INTRODUCTION systems in the quantum limit. Main advances in this direc- tion were achieved by the periodic-orbit theory. This theory provides a way to compute the spectrum of a quantum system Classical chaos is related to the extreme sensitivity of the from its classical periodic orbits (trajectories that repeat them- dynamics of a system to its initial conditions, a concept that selves after a period).
    [Show full text]