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Multi-Robot Motion Planning: Theory and Practice

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Multi-Robot Motion Planning: Theory and Practice The Raymond and Beverly Sackler Faculty of Exact Sciences The Blavatnik School of Computer Science Multi-Robot Motion Planning: Theory and Practice Dissertation submitted in partial fulfillment of the requirements for the degree of \doctor in philosophy" in the School of Computer Science, Tel-Aviv University by Kiril Solovey This work has been carried out under the supervision of Prof. Dan Halperin March 2018 Acknowledgments I wish to express my endless gratitude to my advisor Prof. Dan Halperin. I had a great privilege to work with this kind, smart, professional, generous, and inspiring person. I thank him for instilling in me the passion for science, learning, and exploring the unknown. I would like to thank my dear friends and lab mates Michal Kleinbort, Oren Salzman, and Omri Perez, with whom I've had memorable moments, and who made my time at the lab extremely enjoyable. I also wish to thank all the members of the applied computational geometry group over the years and the algorithmic game theory group for their companionship. I also thank the administrative staff of the School of Computer Science, Tel Aviv University, and especially Gilit Zohar-Oren, Pnina Neria Barzilay, Larisa Slepyan-Eshed, Anat Amirav, and Mika Shahar. I wish to thank my parents for being remarkably supportive throughout my life and especially so for my years in academia. The last person I thank is my partner Nadia, without whom this life would have been mostly a dull journey through space. I would also like to thank the Clore Israel Foundation for supporting me during my Ph.D. studies. My work has also been supported in part by the Israel Science Foundation (grants nos. 1102/11 and 825/15), by the Blavatnik Computer Science Research Fund, and by the Blavatnik Interdisciplinary Cyber Research Center at Tel Aviv University. Thank you all, Kiril Preface This thesis presents a collection of seven published papers that were prepared during my Ph.D. studies. It consists of three journal publications and five peer-reviewed conference publications (see below). Journal Publications Kiril Solovey and Dan Halperin, \On the Hardness of Unlabeled Multi-Robot Motion Planning." • Special issue (invited), International Journal of Robotics Research, 35(14): 1750{1759, 2016. Also in Robotics: Science and Systems, finalist for best paper, and winner of best student paper, 2015. Kiril Solovey, Oren Salzman and Dan Halperin, \Finding a Needle in an Exponential Haystack: • Discrete RRT for Exploration of Implicit Roadmaps in Multi-Robot Motion Planning." Special issue (invited), International Journal of Robotics Research, 35(5): 501{513, 2016. Also in Workshop on Algorithmic Foundations of Robotics, 2014. Aviv Adler, Mark de Berg, Dan Halperin and Kiril Solovey (alphabetical order), “Efficient • Multi-Robot Motion Planning for Unlabeled Discs in Simple Polygons." Special issue (invited), Transactions on Automation Science and Engineering, 12(4): 1309{1317, 2015. Also in Workshop on Algorithmic Foundations of Robotics, 2014. Conference Publications Kiril Solovey and Michal Kleinbort, \The Critical Radius in Sampling-based Motion Planning" • In Robotics: Science and Systems, Carnegie Mellon University, Pittsburgh, PA, USA, 2018. Kiril Solovey and Dan Halperin, “Efficient Sampling-based Bottleneck Pathfinding over Cost • Maps." In International Conference on Intelligent Robots and Systems, Vancouver, BC, Canada, 2017. Kiril Solovey and Dan Halperin, \Asymptotically-Optimal Bottleneck Pathfinding with Applica- • tions to Fr´echet-Type Optimization." In European Symposium on Algorithms, 76:1{76:16, Aarhus, Denmark, 2016. Kiril Solovey, Oren Salzman and Dan Halperin, \New Perspective on Sampling-based Motion • Planning via Random Geometric Graphs." In Robotics: Science and Systems, University of Michi- gan, MI, USA, 2016. Kiril Solovey, Jingjin Yu, Or Zamir and Dan Halperin, \Motion Planning for Unlabeled Discs • with Optimality Guarantees." In Robotics: Science and Systems, Sapienza University of Rome, Italy, 2015. Abstract Motion planning is a fundamental problem in robotics concerned with allowing autonomous robots to efficiently navigate in environments cluttered with obstacles. Although motion planning was initially studied as a theoretical problem in computer science, it is applied nowadays in various fields such as computational biology, computer graphics, and most naturally robotics. Unfortunately, motion planning is notoriously challenging|both theoretically and practically|and many aspects of the problem are not sufficiently understood, even after 30 years of extensive multidisciplinary research. Perhaps the most challenging aspect of motion planning is the complex and the high-dimensional search space that it induces, which accounts for the geometric structure of the robot, the physical constraints that it needs to satisfy, and the environment in which it operates. This thesis is mainly devoted to the study of theoretical and practical issues of high-dimensional systems consisting of multiple robots. In particular, we consider multi-robot motion planning (MRMP), which is a natural extension of motion planning concerned with the collision-free coordination of multiple robots operating in a shared environment. We present a theoretical hardness result concerning MRMP, which applies to several variations of the problem, including the unlabeled setting which consists of identical and interchangeable robots. We show that it also applies in the standard labeled case. This is the strongest MRMP hardness result in that it pertains to the simplest possible robots over previous proofs: unit squares. We then show that by making relatively-mild simplifying assumptions on the problem one can drastically reduce its complexity. In particular, we consider two settings with such assumptions and devise algorithms that run in time that is (low-)polynomial both in the number of robots and the complexity of the environment. Notably, one of the approaches even comes up with solutions whose cost, i.e., the total length the robots traverse, is (theoretically) close to the optimum. We also consider a sampling-based approach for MRMP and present a new algorithm, which applies to more general settings of the problem, and outperforms the best previously known solutions in practice. The second part of the thesis is devoted to the study of sampling-based planners, with emphasis on their theoretical guarantees. In particular, we present a general approach to analyze such algorithms by establishing a novel connection to the a well-studied mathematical subject of random geometric graphs. We then present two papers that are devoted to sampling-based motion planning under bottleneck cost, which had only been lightly treated before them in the literature, despite its importance. Contents 1 Introduction 2 2 On the Hardness of Unlabeled Multi-Robot Motion Planning 19 3 Efficient Multi-Robot Motion Planning for Unlabeled Discs in Simple Polygons 31 4 Motion Planning for Unlabeled Discs with Optimality Guarantees 41 5 Finding a Needle in an Exponential Haystack: Discrete RRT for Exploration of Implicit Roadmaps in Multi-Robot Motion Planning 53 6 New Perspective on Sampling-based Motion Planning via Random Geometric Graphs 67 7 Sampling-Based Bottleneck Pathfinding with Applications to Fr´echet Matching 87 8 Efficient Sampling-based Bottleneck Pathfinding over Cost Maps 105 9 The Critical Radius in Sampling-based Motion Planning 113 10 Conclusion and Future Work 125 11 Bibliography 129 1 Introduction Remark: This notation, which appears throughout the introduction, indicates that the corresponding text describes a contribution of the author that is part of this thesis. Motion planning is a fundamental problem in robotics concerned with allowing autonomous robots to efficiently navigate in environments cluttered with obstacles. Although motion planning was initially studied as a theoretical problem in computer science [29], nowadays it is applied in various fields. Notably, it arises in coordination of multiple autonomous vehicles [23], steering surgical needles [9], and planning trajectories of spacecrafts in orbit [94], to name just a few examples. Unfortunately, motion planning is notoriously challenging|both theoretically and practically|and many aspects of the problem are not sufficiently understood, even after 30 years of extensive multidisciplinary research. The complexity of the problem can be attributed to its continuous and high-dimensional search space, which accounts for (i) the geometric structure of the robot, which may consist of several rigid bodies connected through joints, (ii) the physical constraints that it needs to satisfy, e.g., velocity constraints and limited steering angle as a car has, (iii) and the environment in which it operates, which may change with time, contain unexpected obstacles (as pedestrians jumping into the road), or be a priori unknown altogether. In its most basic form, motion planning consists of finding collision-free paths for a robot in a (two or three-dimensional) workspace cluttered with static obstacles. The spatial pose of the robot, or its configuration, is uniquely defined by its degrees of freedom (DoFs). The set of all configurations is C termed the configuration space of the robot, and decomposes into the disjoint sets of free and forbidden configurations, namely and , respectively. Thus, given start and target configurations, the F ⊆ C C n F T T P1 P1 S P2 S P2 (a) (b) Figure 1.1: Motion planning for a square robot translating in the plane. In (a) the start and target configu- rations S; T , are denoted by the blue and pink squares, respectively. The robot is confined to a room (black border) which contains two triangular obstacles (gray). In (b), the forbidden regions of the configurations space are drawn in gray, whereas the white region is the free space. Observe that the depicted path is entirely collision free. problem can be restated as the task of finding a continuous curve in connecting the two configurations. F See a simple example in Figure 1.1. It is important to note that the input to the problem consists of only a representation of the robot R and the workspace in which it operates, whereas the structure of the free space is implicitly induced F by those parameters.
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