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Signature Redacted Efficiency and Abstraction in Task and Motion Planning by William Vega-Brown Submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 2020 ©Massachusetts Institute of Technology 2020. All rights reserved. Author ................. Signatureredacted Depar(ment of Mechanical Engineering January 8, 2020 Certified by Signature redacted ........... A' Nicholas Roy Professor of Aeronautics and Astronautics A I Thesis Supervisor Certified by................. Signature redacted V John Leonard Professnerin Accepted by............ MASSCHUETSNSTTUT redcte~dcnsantno MASS SL NS ITUTE W\icolas Hadjiconstantinou TEC G IProfessor of Mechanical Engineering FEB 0 5 2020 Chairman, Committee on Graduate Students LIBRARIES 77 Massachusetts Avenue Cambridge, MA 02139 MITLibrades http:"-ibrariesmit.du/sk DISCLAIMER NOTICE Due to the condition of the original material, there are unavoidable flaws in this reproduction. We have made every effort possible to provide you with the best copy available. Thank you. The images contained in this document are of the best quality available. ii Efficiency and Abstraction in Task and Motion Planning by William Vega-Brown Submitted to the Department of Mechanical Engineering on January 8, 2020, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering Abstract Modern robots are capable of complex and highly dynamic behaviors, yet the decision- making algorithms that drive them struggle to solve problems involving complex behaviors like manipulation. The combination of continuous and discrete dynamics induced by contact creates severe computational challenges, and most known practical approaches rely on hand-designed discrete representations to mitigate computational issues. However, the relationship between the discrete representation and the physical robot is poorly understood and cannot easily be empirically verified, and so many planning systems are brittle and prone to failure when the robot encounters situations not anticipated by the model designer. This thesis addresses the limitations of conventional representations for task and motion planning by introducing a constraint-based representation that explicitly places continuous and discrete dynamics on equal footing. We argue that the chal- lenges in modelling problems with both discrete and continuous dynamics can be reduced to a trade-off between model complexity and empirical accuracy. We pro- pose the use of abstraction to combine models that balance those two constraints differently, and we claim that by using abstraction we can build systems that reliably generate high-quality plans, even in complex domains with many objects. Using our representation, we construct and analyze several new algorithms, pro- viding new insight into long-standing open problems about the decidability and com- plexity of motion planning. We describe algorithms for sampling-based planning in hybrid domains, and show that these algorithms are complete and asymptotically optimal for systems that can defined by analytic constraints. We also show that the reachability problem can be decided using polynomial space for systems described by polynomial constraints satisfying a certain technical conditions. This class of sys- tems includes many important robotic planning problems, and our results show that the decision problem for several benchmark task and motion planning languages is PSPACE-complete. Thesis Supervisor: Nicholas Roy Title: Professor of Aeronautics and Astronautics Committee Chair: John Leonard Title: Professor of Mechanical Engineering iii iv Acknowledgments This work would not have been possible without the help of many people. I thank my advisor, Nick Roy. His insight has been invaluable, and his guidance helped shape both the ideas in this thesis and the way I approach research. I thank my committee: John Leonard, for his willingness to guide me through the hidden paths of my department, as well as for his help and support in all my years at MIT; Leslie Kaelbling, for many conversations about planning in robotics, for helping me to frame my work, and for her guidance in better understanding the planning literature; Dan Koditschek, for his many technical and theoretical insights, for sharing the resources of his lab, and especially for for steering me towards the tools of semialgebraic and o-minimal geometry; Alberto Rodriguez, for his help and advice in our meetings; and Marc Toussaint, for helping to expand my understanding of what a planning problem is and what planning algorithms can do. I thank my colleagues and fellow students: Rohan Paul, whose ideas and insights helped shape both the problems I tackled and the wayI approached them; Vasileios Vasilopoulos and Turner Topping, for their tireless efforts in putting the ideas in this thesis to work on the Minitaur; Charlie Richter, who taught me the importance of doing one thing at a time; as well as all the members of the Robust Robotics Group. I thank Laura Hallock, whose patience as I walked this long and winding road was nothing less than saintly, and whose support-both emotional and practical-meant more to me than she knows. Finally, I thank my parents, without whom I could be neither where I am nor who I am. I am eternally grateful for all you have done for me. v vi Author's Note The version of this document you are reading was submitted to the MIT libraries as a printed copy. The most recent version of this document, which includes color figures as well as digital tools like hyperlinks and cross-references, is available online at https://people.csail.mit. edu/wrvb/. vii viii Contents 1 Introduction 1 1.1 The challenges of deliberative planning .. ... ... ... ... ... 3 1.2 Planning in robotics ... .... .... .... ..... .... ... 4 1.2.1 Complete algorithms for path planning .. ..... ..... 5 1.2.2 Algorithms with weaker guarantees ... .... .... ... 7 1.2.3 Factored discretization .. ... .... ... ... .... ... 8 1.2.4 Discrete planning .. ... ... ... ... ... .. ... ... 9 1.2.5 Combined Task and Motion Planning .... .... .... .. 11 1.3 Representation, Efficiency, and Verification .... .... .... ... 16 1.4 Statement of contributions .... .... .... ..... .... ... 19 1.5 Structure of this document .... ..... .... ..... ..... 20 2 The Continuous Constraint Contract Representation 23 2.1 States and variables .... .... .... .... ..... .... ... 25 2.2 A ctions .. .... .... .... .... ..... .... .... .... 26 2.3 Axioms, knowledge, and constraints .. ..... .... .... .... 29 2.4 Planning problems .. ..... .... .... .... .... .... 30 2.5 Conclusions .... ...... ..... .... ..... ..... ... 31 3 Randomized Algorithms for Planning 35 3.1 Asymptotic Optimality under Piecewise-Analytic Differential Constraints 36 3.2 Dem onstration .... ... ......... ........ ...... 44 3.3 Compact discretizations with factored graphs ... ...... .... 45 ix 3.3.1 Discretizing C3 ... ..... ..... ..... ..... 45 3.4 A Proof that SRGG is Asymptotically Optimal .. ..... .. 51 3.4.1 Construction of a sequence of paths .. ..... ..... 51 3.4.2 Construction of balls on the intersections between strata . 52 3.4.3 Construction of balls on an arbitrary leaf .... .... 52 3.4.4 Bounding the cost of the path returned . .... ..... 55 3.4.5 Proofs of propositions . .... .... .... .... .. 56 4 Planning with Abstraction 69 4.1 Admissible angelic semantics . ... .. .... ... ... ... 7 0 4.1.1 Problem Formulation .. ... .. ..... ..... .... 7 1 4.1.2 Angelic Semantics .. ... .. .. .. ... .. .. ... .. 7 3 4.1.3 Admissible Abstractions . .... ... ... ... ... ... 7 7 4.1.4 Abstractions in C 3 ... ... .. .... ... ... ... 8 5 4.2 Abstract planning algorithms .. .. .. ... .... ... ... .. 8 6 4.2.1 Angelic A* .... ... ... .. .. ... .... ... ... 8 7 4.2.2 Acyclic Angelic A* .. ... ... ... ... ... .... 9 5 4.2.3 Approximate Angelic A* .. ... .. ... .. .. ... .. 9 8 4.3 Demonstration . ... ... ... ... ... .. .. ... .. .. 102 4.4 Analysis . ... .. ... .. ... .. .. ... .. .. ... .. ... 105 4.5 Notes and Extensions . ... ... ... .. .. .. ... .. ... 1 0 8 5 Exact Algorithms for Robot Planning 111 5.1 Transition systems ... ... ... ... 112 5.2 Undecidability of Semialgebraic Transition Systems . ... ... ... 115 5.3 Uniform Accessibility, Decidability, and Complexity ... ....... 118 5.3.1 Stratification .... ........ ........ ....... 118 5.3.2 Uniform accessibility ..... ........ ....... ... 119 5.3.3 Uniform stratified accessibility ...... ........ .... 121 5.3.4 Deciding stratifiable semialgebraic transition systems . .. .. 122 5.4 Relationships between representations .... .... ..... ... .. 127 x 5.4.1 Analytic functions and semianalytic sets . 128 5.4.2 Duality for analytic representations . .. 130 5.4.3 Computability and approximation .... 134 5.5 Im plications .................... 136 6 Conclusions 139 6.1 Summary of contributions ................... ..... 139 6.2 Open problems .. ............ ............. ... 140 6.3 A vision for verifiable performant systems .... ........ ... 141 Bibliography 143 xi xli List of Figures 3.1 Exampleplanning domain . .. .. .. .. .. .. .. .. .. .. .. 38 3.2 Comparison . .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 45 3.3 Coupled actionsubgraphs .. .. .. .. .. .. .. .. .. .. .. 48 4.1 Angelic constraints .. .. .. .. .. .. .. ... .. .. .. .. .. 74 4.2 Convexangelic bounds . .. .. .. ... .. .. .. .. .. .. .. .. 82 4.3 Door puzzle .. .. .
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