Robotics: Science and Systems VI

Robotics

Robotics Science and Systems VI

edited by Yoky Matsuoka, Hugh Durrant-Whyte, and Jos´e Neira

The MIT Press Cambridge, Massachusetts London, England

c 2011 Massachusetts Institute of Technology  All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. For information about special quantity discounts, please email special [email protected] or write to Special Sales Department, The MIT Press, 55 Hayward Street, Cambridge, MA 02142. Printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Robotics: Science and Systems Conference (6th : 2010 : Zaragoza, Spain) Robotics : science and systems VI / edited by Yoky Matsuoka, Hugh Durrant-Whyte, and Jos´e Neira. p. cm. “This volume contains the 40 papers presented at Robotics: Science and Systems (RSS) 2010, held at the University of Zaragoza in Spain, from June 27 to June 30, 2010”—Pref. Includes bibliographical references. ISBN 978-0-262-51681-5 (pbk. : alk. paper) 1. Robotics—Congresses. I. Matsuoka, Yoky. II. Durrant-Whyte, Hugh F., 1961- III. Neira, Jos´e. VI. Title. TJ210.3.R6435 2011 629.8’92—dc22 2011007422

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Contents Preface

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ix

Organizing Committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Program Committee Sponsors

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Biophysically Inspired Development of a Sand-Swimming Robot Ryan D. Maladen, Yang Ding, Paul B. Umbanhowar, Adam Kamor, and Daniel I. Goldman .

1

Passive Torque Regulation in an Underactuated Flapping Wing Robotic Insect P. S. Sreetharan and R. J. Wood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

Color-Accurate Underwater Imaging Using Perceptual Adaptive Illumination Iuliu Vasilescu, Carrick Detweiler, and Daniela Rus . . . . . . . . . . . . . . . . . . . . .

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Probabilistic Lane Estimation Using Basis Curves Albert S. Huang and Seth Teller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Reinforcement Learning to Adjust Robot Movements to New Situations Jens Kober, Erhan Oztop, and Jan Peters . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

Analysis and Control of a Dissipative Spring-Mass Hopper with Torque Actuation M. Mert Ankaralı and Uluc¸ Saranlı . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

On Motion and Force Control of Grasping Hands with Postural Synergies D. Prattichizzo, M. Malvezzi, and A. Bicchi . . . . . . . . . . . . . . . . . . . . . . . . . .

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Back-drivable and Inherently Safe Mechanism for Artificial Finger Koichi Koganezawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

Segmentation and Unsupervised Part-based Discovery of Repetitive Objects Rudolph Triebel, Jiwon Shin, and Roland Siegwart . . . . . . . . . . . . . . . . . . . . . .

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Scale Drift-Aware Large Scale Monocular SLAM Hauke Strasdat, J. M. M. Montiel, and Andrew J. Davison

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73

Preliminary Results in Decentralized Estimation for Single-Beacon Acoustic Underwater Navigation Sarah E. Webster, Louis L. Whitcomb, and Ryan M. Eustice . . . . . . . . . . . . . . . . . .

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A Non-invasive, Real-Time Method for Measuring Variable Stiffness Giorgio Grioli and Antonio Bicchi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Consistent Data Association in Multi-robot Systems with Limited Communications Rosario Arag¨ue´ s, Eduardo Montijano, and Carlos Sag¨ue´ s . . . . . . . . . . . . . . . . . .

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Singularity-invariant Leg Rearrangements in Doubly-planar Stewart-Gough Platforms J´ulia Borr`as, Federico Thomas, and Carme Torras . . . . . . . . . . . . . . . . . . . . . . 105 On the Kinematic Design of Exoskeletons and Their Fixations with a Human Member Nathana¨el Jarrass´e and Guillaume Morel . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Assessing Optimal Assignment under Uncertainty: An Interval-based Algorithm Lantao Liu and Dylan A. Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 LQG-MP: Optimized Path Planning for Robots with Motion Uncertainty and Imperfect State Information Jur van den Berg, Pieter Abbeel, and Ken Goldberg . . . . . . . . . . . . . . . . . . . . . . 129 The Smooth Curvature Flexure Model: An Accurate, Low-dimensional Approach for Robot Analysis Lael U. Odhner and Aaron M. Dollar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Multi-priority Cartesian Impedance Control Robert Platt Jr., Muhammad Abdallah, and Charles Wampler

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Variable Impedance Control: A Reinforcement Learning Approach Jonas Buchli, Evangelos Theodorou, Freek Stulp, and Stefan Schaal . . . . . . . . . . . . . 153 A Fast Traversal Heuristic and Optimal Algorithm for Effective Environmental Coverage Ling Xu and Tony Stentz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Stochastic Complementarity for Local Control of Discontinuous Dynamics Yuval Tassa and Emo Todorov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Distributed Optimization with Pairwise Constraints and Its Application to Multi-robot Path Planning Subhrajit Bhattacharya, Vijay Kumar, and Maxim Likhachev . . . . . . . . . . . . . . . . . 177 PLISS: Detecting and Labeling Places Using Online Change-Point Detection Ananth Ranganathan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 A Constant-Time Algorithm for Vector Field SLAM Using an Exactly Sparse Extended Information Filter Jens-Steffen Gutmann, Ethan Eade, Philip Fong, and Mario Munich . . . . . . . . . . . . . 193 Efficient Probabilistic Planar Robot Motion Estimation Given Pairs of Images Olaf Booij, Ben Kr¨ose, and Zoran Zivkovic . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Efficient Non-parametric Surface Representations Using Active Sampling for Push Broom Laser Data Mike Smith, Ingmar Posner, and Paul Newman . . . . . . . . . . . . . . . . . . . . . . . . 209 Sensor Placement for Improved Robotic Navigation Michael P. Vitus and Claire J. Tomlin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Task-driven Tactile Exploration Kaijen Hsiao, Leslie Pack Kaelbling, and Tom´as Lozano-P´erez . . . . . . . . . . . . . . . . 225 On the Role of Hand Synergies in the Optimal Choice of Grasping Forces Marco Gabiccini and Antonio Bicchi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233  vi

Dynamic Constraint-based Optimal Shape Trajectory Planner for Shape-Accelerated Underactuated Balancing Systems Umashankar Nagarajan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Design and Optimization Strategies for Muscle-like Direct Drive Linear Permanent Magnet Motors Bryan P. Ruddy and Ian W. Hunter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Study of Group Food Retrieval by Ants as a Model for Multi-robot Collective Transport Strategies Spring Berman, Quentin Lindsey, Mahmut Selman Sakar, Vijay Kumar, and Stephen Pratt . . 259 Incremental Sampling-based Algorithms for Optimal Motion Planning Sertac Karaman and Emilio Frazzoli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Stochastic Modeling of the Expected Time to Search for an Intermittent Signal Source Under a Limited Sensing Range Dezhen Song, Chang-Young Kim, and Jingang Yi . . . . . . . . . . . . . . . . . . . . . . . 275 Closing the Learning-Planning Loop with Predictive State Representations Byron Boots, Sajid M. Siddiqi, and Geoffrey J. Gordon . . . . . . . . . . . . . . . . . . . . 283 Belief Space Planning Assuming Maximum Likelihood Observations Robert Platt Jr., Russ Tedrake, Leslie Kaelbling, and Tom´as Lozano-P´erez . . . . . . . . . . 291 Motion Planning under Bounded Uncertainty Using Ensemble Control Aaron Becker and Timothy Bretl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Remotely Powered Propulsion of Helical Nanobelts Gilgueng Hwang, Sinan Haliyo, and St´ephane R´egnier . . . . . . . . . . . . . . . . . . . . 307 A Molecular Algorithm for Path Self-Assembly in 3 Dimensions Rebecca Schulman and Bernard Yurke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

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Preface This volume contains the 40 papers presented at Robotics: Science and Systems (RSS) 2010, held at the University of Zaragoza in Spain, from June 27 to June 30, 2010. A record of 239 paper were submitted to RSS 2010. RSS takes pride in having a rigorous reviewing process: the 218 members of the program committee wrote more than 1, 000 high-quality reviews so that each paper received at least four. The authors were invited to rebut the reviews, and after further discussion between the program committee members and the 16 area chairs, the reviews were finalized. Final acceptance and presentation category (20 poster presentations, 20 podium presentations) was decided during the full-day area chair meeting in Los Angeles, making the acceptance rate of RSS be under 17%. The selected papers cover a wide range of topics in robotics spanning mechanisms, kinematics, dynamics and control, human-robot interaction and human-centered systems, distributed systems, mobile systems and mobility, manipulation, field robotics, medical robotics, biological robotics, robot perception, and estimation and learning in robotic systems. The conference spanned three and a half days. There were four invited talks by leaders in fields that inspire robotics: • Prof. Christopher M. Bishop from Microsoft Research Cambridge and the University of Edinburgh gave a talk titled “Third Generation Machine Intelligence”. • Prof. Deborah M. Gordon from Stanford University gave a talk titled “Interaction Networks as Distributed Algorithms in Ants”. • Prof. Larry Matthies from the Jet Propulsion Laboratory and the University of Southern California gave a talk titled “Robotic Solar System Exploration: Progress and Challenges”. • Prof. Philip H. S. Torr from Oxford Brookes University gave a talk titled “Towards Global Energy Models for Scene Understanding”. There were two early carrer spotlight talks this year, given by rising stars in the robotics community: • Russ Tedrake from Massachusetts Institute of Technology gave a talk titled “Dynamic walking on rough terrain and flying like a bird: a computational approach to exploiting nonlinear dynamics”. • Pieter Abbeel from University of California at Berkeley gave a talk titled “Apprenticeship Learning for High-Performance Robot Control”.

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The Workshop Chair, Kevin Lynch (Northwestern University), together with a group of very competent reviewers, selected twelve workshops that were extremely well attended: more than 260 people attended the eight Sunday workshops, and more than 110 attended the four Monday morning workshops. The workshops were the following: • OMNIVIS Omnidirectional Vision, Camera Networks and Non-classical Cameras organized by Ryad Benosman, Christopher Geyer, and Olivier Koch; • Towards Closing the Loop: Active Learning for Robotics organized by Rub´en Mart´ınezCant´ın, Jan Peters and Andreas Krause; • RGB-D: Advanced Reasoning with Depth Cameras organized by Xiaofeng Ren, Dieter Fox, Jana Kosecka and Kurt Konolige; • Predictive Models in Humanoid Gaze Control and Locomotion organized by Paolo Dario, Alain Berthoz, Jose Santos-Victor and Atsuo Takanishi; • Strategies and Evaluation for Mobile Manipulation in Household Environments organized by Antonio Morales, Mario Prats, Siddhartha Srinivasa and Radu Bogdan Rusu; • Motion Planning: from Theory to Practice organized by Kris Hauser, Ron Alterovitz, Kostas Bekris and Juan Cortes; • Grasp Acquisition: How to Realize Good Grasps organized by Jeff Trinkle, Patrick van der Smagt and Thomas Wimboeck; • Learning for Human-Robot Interaction Modeling organized by Mohamed Chetouani and Adriana Tapus; • Enabling Technologies for Image-Guided Robotic Interventional Procedures organized by Gregory Fischer and Robert Webster; • Representations for Object Grasping and Manipulation in Single and Dual Arm Tasks organized by Danica Kragic, Bruno Siciliano and Vronique Perdereau; • Good Experimental Methodology in Robotics and Replicable Robotics Research organized by Fabio Bonsignorio, John Hallam and Angel del Pobil; • Non-Smooth Contact Modeling in Robotic Simulation and Control organized by Katsu Yamane and Abderrahmane Kheddar.

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RSS 2010 was a big success thanks to the efforts of many people. We gratefully acknowledge the enormous effort and time spent by the program committee and the 16 area chairs, whose joint expertise covered an extraordinary broad swath of the robotics landscape. The area chairs were: Alin Albu-Schffer (DLR, Germany), Jos´e Castellanos (Universidad de Zaragoza), Peter Corke (CSIRO, Australia), Gregory Dudek (McGill University), Vincent Hayward (Universit´e Pierre et Marie Curie), David Hsu (National University of Singapore), Dan Koditschek (University of Pennsylvania), Jana Kosecka (George Mason University), Danica Kragic (Royal Institute of Technology, Sweden), James Kuffner (Carnegie Mellon University), Jan Peters (Max Planck Institute for Biological Cybernetics), Sanjiv Singh (Carnegie Mellon University), Cyrill Stachniss (University of Freiburg), Russ Tedrake (Massachusetts Institute of Technology), Stefan Williams (University of Sydney) and Katsu Yamane (Disney Research). Our warmest thanks also to the publicity chair, Katherine Kuchenbecker (University of Pennsylvania), who did a great job in attracting the largest yet number of authors and attendees to an RSS conference. The local arrangements chairs, Jos´e Neira, Juan D. Tard´os and Luis Montano (University of Zaragoza), chose an amazing venue and organized everything beautifully, making sure that attendees got the best value for their money. Enormous thanks also to our Webmaster, David Ribas (University of Girona), who designed and maintained an elegant and ergonomic webpage. We also want to thank the conference support staff the Instituto de Investigaci´on en Ingenier´ıa de Arag´on for handling local details, even on a Sunday. Last but not least, we would like to thank the team of twenty three volunteers that were always enthusiastically at hand when something needed to be done. RSS 2010 was possible because of industrial and institutional sponsoring; thanks to Willow Garage, Google, the Spanish Ministry of Science and Innovation, the Regional Government of Arag´on, Heartland Robotics, Aldebaran Robotics, Barrett Technology and Robotnik for providing funds for the general conference. Thanks to Springer for funding the best student paper award, and to Willow Garage for funding the best open source code award. We would also like to thank our technical sponsors: IEEE Robotics and Automation Society, the Association for the Advancement of Artificial Intelligence, the International Foundation of Robotics Research and the Robotics Society of Japan. Finally, we would like to thank again the robotics community for adopting RSS and its philosophy. RSS 2010 had a record attendance of 315 researchers from 25 countries in North America, Europe, Asia and Australia. The attendance rate of 8 persons per presented paper confirms RSS as one of the highest quality single-track mainstream robotics conferences. We enthusiastically look forward to yet more exciting meetings in the years to come. The online version of these proceedings (including color and links) can be found at: http://www.roboticsproceedings.org/rss06/index.html Yoky Matsuoka, University of Washington Hugh Durrant-Whyte, University of Sydney Jos´e Neira, University of Zaragoza July 2010  xi

Organizing Committee General Chair

Yoky Matsuoka, University of Washington

Program Chair

Hugh Durrant-Whyte, University of Sydney

Local Arrangement Co-Chairs

Jos´e Neira, Universidad de Zaragoza Juan Tard´os, Universidad de Zaragoza Luis Montano, Universidad de Zaragoza

Publicity Chair

Katherine Kuchenbecker, University of Pennsylvania

Publications Chair

Jos´e Neira, Universidad de Zaragoza

Workshop Chair

Kevin Lynch, Northwestern University

Web Master

David Ribas, Universitat de Girona

Area Chairs

Alin Albu-Schffer, DLR, Germany Jos´e Castellanos, Universidad de Zaragoza Peter Corke, CSIRO, Australia Gregory Dudek, McGill University Vincent Hayward, Universit´e Pierre et Marie Curie David Hsu, National University of Singapore Dan Koditschek, University of Pennsylvania Jana Kosecka, George Mason University Danica Kragic, Royal Institute of Technology, Sweden James Kuffner, Carnegie Mellon University Jan Peters, Max Planck Institute for Biological Cybernetics Sanjiv Singh, Carnegie Mellon University Cyrill Stachniss, University of Freiburg Russ Tedrake, Massachusetts Institute of Technology Stefan Williams, University of Sydney Katsu Yamane, Disney Research

RSS Foundation Board President

Sebastian Thrun, Stanford University

Directors

Oliver Brock, Technische Universitt Berlin Dieter Fox, University of Washington Lydia Kavraki, Rice University Sven Koenig, University of Southern California John Leonard, Massachusetts Institute of Technology Daniela Rus, Massachusetts Institute of Technology Stefan Schaal, University of Southern California Gaurav Sukhatme, University of Southern California Jeff Trinkle, Rensselaer Polytechnic Institute

Treasurer

Wolfram Burgard, University of Freiburg

Secretary

Nick Roy, Massachusetts Institute of Technology  xiii

Program Committee Abbeel, Pieter Adams, Martin Akella, Srinivas Andrade-Cetto, Juan Antonelli, Gianluca Argyros, Antonis Arras, Kai Oliver Asano, Fumihiko Asfour, Tamim Atkeson, Chris Bagnell, James Bailey, Tim Barkby, Stephen Bennewitz, Maren Bergbreiter, Sarah Bingham, Brian Birchfield, Stan Blaschko, Matthew Borst, Christoph Bosse, Michael Brennan, Sean Brock, Oliver Brooks, Alex Bryson, Mitch Buehler, Martin Burschka, Darius Byl, Katie Carloni, Raffaella Censi, Andrea Chestnutt, Joel Christensen , Henrik Civera, Javier Clark, Jonathan Cowan, Noah Cutkosky, Mark Davison, Andrew Deisenroth, Marc Dellaert, Frank Dissanayake, Gamini Douillard, Bertrand Etienne , Burdet Eustice, Ryan

Everett, Hazel Fairfield, Nathanial Fermuller, Cornelia Ferrie, Frank Fitch, Robert Fox, Dieter Frisoli, Antonio Geyer, Chris Ghrist, Robert Gillespie, Brent Goldman, Daniel Gomez, Manuel Gould, Steve Grizzle, Jessy Grocholsky, Ben Grollman, Dan Grosse-Wentrup, Moritz Gruppen, Rod Guerrero Campo, Jos´e Hafner, Roland Harders, Matthias Hauser, Kris Hirai, Shinichi Hirche, Sandra Hoburg, Warren Hollinger, Geoff Hosoda, Koh Howard, Andrew Huebner, Kai Hyon, sang-ho Iida, Fumiya Inamura, Tetsunari Jadbabaie, Ali Jakuba, Michael Jenkin, Michael Jenkins, Chad Jensfelt, Patric Johnson-Roberson, Matthew Kagami, Satoshi Kajita, Shuuji Kersting, Kristian  xv

Kikuuwe, Ryo Kim, Sangbae Kober, Jens Kolter, Zico Kootstra, Gert Kroemer, Oliver Kuipers, Benjamin Kulic, Dana Kurniawati, Hanna Kyrki, Ville Lacroix, Simon Lampert, Christoph Lane, David LaValle, Steven Lee, Dongheui Leonard, John Leonardis, Alex Lepetit, Vincent Li, Tsai-Yen Lien, Jyh-Ming Likachev, Maxim Lilienthal, Achim Lopes, Manuel L´opez-Nicol´as, Gonzalo MacDonald, Bruce MacLean, Karon Macnab, Chris Mahon, Ian Mansard, Nicolas Mart´ınez-Cant´ın, Rub´en Matsuoka, Yoky Merino, Luis Metta, Giorgio Michael, Nathan Mochiyama, Hiromi Moll, Mark Montesano, Luis Montiel, Jos´e M. M. Morales, Antonio Morimoto, Jun Morse, Bryan Muelling, Katharine

Murillo, Ana Murrieta, Rafael Negahdaripour, Shahriar Neira, Jos´e Nenchev, Dragomir Neumann, Gerhard Newman, Paul Nguyen-Tuong, Duy Niemeyer, Gunter Nishiwaki, Koichi Nuske, Steve O’Kane, Jason Ogata, Tetsuya Okada, Masafumi Olson, Edwin Ott, Christian Oztop, Erhan Palli, Gianluca Papanikolopoulos, Nikolaos Paz, Lina Petillot, Yvan Pfaff, Patrick Piater, Justus Pinies, Pedro Pizarro, Oscar Plagemann, Christian Plaku, Erion Platt, Robert Posner, Ingmar Pradalier, C´edric Pratichizzo, Domenico

Pratt, Jerry Rajan, Kanna Ramos, Fabio Redon, Stephane Rekleitis, Ioannis Riedmiller, Martin Roberts, Jonathan Robuffo Giordano, Paolo Rock, Stephen Rodr´ıguez-Losada, Diego Roman, Chris Roumeliotis, Stergios Roy, Nicholas Sag¨ue´ s, Carlos Sanz, Pedro Saranli, Uluc Saripalli, Srikanth Saxena, Ashutosh Schaal, Stefan Schiele, Bernt Sentis, Luis Shibata, Tomohiro Shimoda, Shingo Shiriaev, Anton Sim, Robert Simeon, Thierry Singh, Surya Singh, Hanumant Smart, Bill Sol`a, Joan Spenko, Matt Spletzer, John

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Srinivasa, Siddhartha Stasse, Olivier Stilman, Mike Sugihara, Tomomichi Sukhatme, Gaurav Sukkarieh, Salah Tard´os, Juan Taylor, CJ Ting, Jo-Anne Tipaldi, Diego Torres-Mendez, Luz Ude, Ales Ueda, Jun van den Berg, Jur van der Stappen, Frank Vanderborght, Bram Velagapuddi, Pras Venture, Gentiane Villani, Luigi Vona, Marsette Wang, Zhikun Whitcomb, Louis Wingate, David Wyeth, Gordon Yim, Mark Yokokohji, Yasuyoshi Zha, Hong-Bin Zhang, Hong Zillich, Michael Zlot, Robert

Sponsors The organizers of Robotics Science and Systems 2010 gratefully acknowledge the following conference sponsors: • Gold Sponsors:

• Silver Sponsors:

• Bronze Sponsors:

• Awards Sponsors:

• Institutional Sponsors:

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• Technical Sponsors:

• Organized by:

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Biophysically,nspired'evelopmentRfa 6Dnd-6wimminJ5Rbot Ryan D. Maladen∗ , Yang Ding† , Paul B. Umbanhowar‡ , Adam Kamor† and Daniel I. Goldman∗†

∗ Bioengineering

Program, † School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332–0250 of Mechanical Engineering, Northwestern University, Evanston, IL 60208 email: [email protected]

‡ Department

Abstract— Previous study of a sand-swimming lizard, the sandfish, Scincus scincus, revealed that the animal swims within granular media at speeds up to 0.4 body-lengths/cycle using body undulation (approximately a single period sinusoidal traveling wave) without limb use [1]. Inspired by this biological experiment and challenged by the absence of robotic devices with comparable subterranean locomotor abilities, we developed a numerical simulation of a robot swimming in a granular medium (modeled using a multi-particle discrete element method simulation) to guide the design of a physical sand-swimming device built with off-the-shelf servo motors. Both in simulation and experiment the robot swims limblessly subsurface and, like the animal, increases its speed by increasing its oscillation frequency. It was able to achieve speeds of up to 0.3 body-lengths/cycle. The performance of the robot measured in terms of its wave efficiency, the ratio of its forward speed to wave speed, was 0.34±0.02, within 8 % of the simulation prediction. Our work provides a validated simulation tool and a functional initial design for the development of robots that can move within yielding terrestrial substrates.

I. I NTRODUCTION There is a need for robots that can move within complex material like sand, rubble, and loose debris. For example such robots could help locate hazardous chemical leaks [2], function as self propelled inspection devices [3], and search for victims in disaster sites [4, 5, 6]. Limbless robots that use their bodies to move appear better suited to navigate complex terrains than traditional wheeled [7, 8, 9, 10] and legged [11, 12, 13, 14, 15, 16, 17] robots which are often impeded by the size or shape of their appendages which can result in entrapment or failure. Previous terrestrial limbless robots utilized serpentine locomotion to move on the surface of media. Of these, most were tested on rigid surfaces [18, 19, 20, 21, 22] with only a few developed for and tested in unstructured environments [23, 24, 25]. Advances in creating high performing flying and swimming devices [22, 26, 27] in aerial and aquatic domains and wheeled/tracked vehicles on relatively structured terrestrial terrain have occurred mainly because the respective fields of aerodynamics, fluid dynamics, and terramechanics [28, 26] provide accurate models of locomotor-media interaction which are used in turn to design improved wings, fins, wheels, and legs. However, a major hurdle arises when one attempts to design robots to move on and within complex flowing particulate environments (e.g. sand, soil, and leaf-litter) that can display both solid and fluid-like behavior in response to stress. In such materials, comparable and comprehensive val-

1

idated analytic continuum theories at the level of the NavierStokes equations [29] for fluids do not exist. However, it is possible to understand the interaction between the locomotor and the media by using numerical and physical modeling approaches [30, 31, 32]. In the absence of theory, the biological world is a fruitful source of principles of movement that can be incorporated into the design of robots that navigate within complex substrates. Many desert organisms like scorpions, snakes, and lizards burrow and swim effectively in sand [33, 34, 35, 36, 37] to escape heat and predators, and hunt for prey [38, 39]. It has been hypothesized that many of these animals have evolved morphological adaptations like marked body elongation and limb reduction to deal with deformable terrain [40, 41]. Our recent high speed x-ray imaging study investigating the subsurface locomotion of the sandfish Scincus scincus, a small (∼ 10FPORQJVQRXWWDLOWLS OizardWKDWLnhabitsWhe Saharan desert [1] (Fig. 1), reveals that once within the media the animal no longer uses limbs for propulsion but “swims” forward by propagating a sinusoidal traveling wave posteriorly from head to tail. Motivated by the subsurface locomotion of the sandfish, the present work utilizes a numerical simulation of a sandfish inspired undulator as a design tool to build an appendageless sand-swimming device. The robot is driven by a simple open loop controller which, like the animal kinematics, varies the joint position trajectories to create a sinusoidal wave that travels posteriorly along the device. The robot swims within a model laboratory granular medium of plastic particles and displays locomotion features similar to the organism and predicted by the numerical robot simulation. II. P REVIOUS WORK A. Biological(xperiment The biological experiments presented in [1] model the subsurface undulatory motion of the sandfish with a posteriorly traveling single-period sinusoidal wave 2π (1) (x + vw t) λ with x the position along the sandfish, y the body displacement from the midline of the animal, A the amplitude, λ the wavelength and vw = f λ the wave speed where f is the wave frequency. The spatial characteristics, A and λ, did y = A sin

A

B

C

Fig. 1. (A) The sandfish Scincus scincus, a sand-swimming lizard that inhabits the Saharan desert, (B) burying into granular media (0.3 mm spherical glass beads), and (C) swimming subsurface where the x-ray image shows the body (light area) and opaque markers fixed to limbs and midline. Red dashed line indicates tracked midline.

not vary significantly with media preparation and their ratio was approximately 0.2 implying that the animal increased its forward velocity by increasing its oscillation frequency. A measure of undulatory performance is the wave efficiency, η, the ratio between the forward speed of the swimmer, vx , and the velocity of the wave traveling down its body, vw , or equivalently the slope of the velocity-frequency relationship for velocity measured in wavelengths per second. Typical wave efficiencies of undulatory organisms moving in fluids at low Reynolds number (such as nematodes in water) are 0.25 [42, 43, 44], whereas η ≈ 0.8 − 0.9 for organisms undulating (creeping) along solid – air interfaces [45, 46, 47]. Locomotion with η = 1 is equivalent to movement within a rigid tube. For the sandfish swimming in glass beads, η ≈ 0.5 independent of particle size and media preparation (i.e. packing density). B. Resistive Force Theory for*ranulaU0edia An empirical resistive force theory (RFT) was developed to predict wave efficiency η for undulatory subsurface granular locomotion [1]. The RFT, inspired by theory used to predict swimming speeds of microorganisms in fluids [42], partitions the body of the organism into infinitesimal segments each of which generates thrust and experiences drag when moving through the medium. These segmental forces are integrated over the entire body, and, by setting the net forward force to zero (assuming a constant average velocity), η is solved for numerically.

2

Unlike fluids, in granular media no validated theory exists in the regime relevant to sand-swimming to estimate the force on individual segments moving through the medium. Previously, Maladen et al. [1] obtained these forces empirically by dragging a rod (representative segment) through the media the animal was tested in. With these forces as input and by propagating a sinusoidal traveling wave along the body, the RFT shows that translational motion within granular media without limb use is possible. Also, the RFT accurately predicts that the sandfish swims with η ≈ 0.5 within a granular media of 0.3 mm glass particles (comparable in size and density to desert sand [48]). While the RFT qualitatively describes some features of sand-swimming, it is based on several assumptions: e.g. the measured drag force on a rod is representative of the average force on a segment of the sandfish, the forces generated by a segment are localized, and the center of mass of the animal does not oscillate laterally. Since the assumptions of the RFT have not been rigourously tested and applying the RFT to different treatments (particle friction, particle size, body design, etc.) require force laws to be measured for each condition, we instead use numerical simulation techniques as a general robotic design tool. A numerical simulation approach, once validated against experiment, can provide an understanding of body generated drag and thrust forces from the particle perspective and can be used to generate empirical drag laws for input into the RFT. Our numerical simulation is a flexible design tool that accurately predicts robot performance and allows easy variation of physical and design parameters such as particle-particle friction and number of segments. III. N UMERICAL S IMULATION OF S AND -S WIMMING ROBOT A. Development and9alidation To design a sand-swimming robot, we developed a numerical simulation of a laboratory scale device with a finite number of discrete, rigid segments to test if it could swim within granular medium. The simulation couples a numerical model of the robot to a model of the granular medium. We model the granular material using a multi-particle discrete element method (DEM) simulation [30]. To compute the robot-particle and particle-particle interaction forces we calculate the normal force [49], Fn , and the Coulombic tangential force, Fs , acting at each contact with Fn = kδ 3/2 − Gn vn δ 1/2 Fs = μFn ,

(2)

where δ is the virtual overlap between particles or between particle and robot segment, vn is the normal component of relative velocity, and k = 2 × 105 kg s−2 m−1/2 and Gn = 5 kg s−1 m−1/2 are the hardness and viscoelastic constant. μ quantifies the particle-particle (μpp = 0.08) or body-particle (μbp = 0.27) friction coefficient depending on which elements are in contact. μbp was measured between the robot skin

Acceleration (g)

and plastic particles used in the physical experiments. To in a horizontal plane at fixed depth in this study, and there reduce the required torque in the physical experiments and to was excellent agreement between experiment and simulation decrease the computational time, we used a granular medium in preliminary studies. The simulated robot was sized for easy composed of 4.7×105 spherical plastic particles with diameter testing of the corresponding physical device in the laboratory. 6 mm and density 1.03 ± 0.04 g/cm3 in experiment, and Since the sandfish does not use its limbs to move subsurface 3.2 × 105ELGLVSHUVH VSKHUHPL[WXUHPP 6 and RFT had shown that body undulation was sufficient for and density 1.06 g/cm3 in simulation. The 35 particle deep propulsion [1], the simulated robot did not include limbs. bed of particles in experiment and 24 particle deep bed of No tapering along the device was considered. The simulated particles in simulation were held in a container with horizontal robot consisted of 49 cuboidal segments interconnected and cross section of 188 × 62 particle diameters. To validate the actuated by virtual motors (vertical cylinders) of the same simulated medium and obtain the values of μpp , k, and Gn height (Fig. 3). Depending on the number of segments (N ) given above, we dropped an aluminum ball (diameter 6.35 cm employed, every 48/N motor was driven with an open loop and mass 385 g) into the plastic particles with varying impact signal to generate a sinusoidal wave traveling posteriorly from velocity (0.5 − 3 m/s) in both experiment and simulation and head to tail while the remaining motors were immobilized to set grain interaction parameters to best match the measured form a straight segment of length 48 / N cm. To approxiand simulated penetration force during the impact collision as mate a sinusoidal traveling wave, the angle between adjacent a function of time (Fig. 2). With parameters determined from segments is modulated using impact at v = 1.4 m/s, the force profile fit well at other impact velocities. In additional experiments, we directly measured μpp β(i, t) = β0 ξ sin(2πξi/N − 2πf t), (3) and the coefficient of restitution (determined by Gn with fixed th k) for the plastic particles and found them to be within 5% and with β(i, t) the motor angle of the i motor at time t, β0 10% of the fitted values respectively. For simplicity we used the angular amplitude which determines A/λ, ξ the number the same normal force parameters for both particle-particle of wavelengths along the body (period), and N the number of motors. and body-particle interactions. WM integrates the equations of motion of the coupled links and the DEM calculates the resultant forces from both the particle-particle and body-particle interactions. For each time 2 cm step, the net force from particles on each segment is passed to 4 WM, and velocity and position information transferred back to DEM. Roll and pitch were not modeled.

A

2

motor i-1

D

βi

3 cm

motor i+1

motor i

B

0.05 Time (s)

Experiment Simulation 0.1

[

0

[

0

Fig. 2. Validation of the multi-particle discrete element method (DEM) simulation of the granular medium using measured acceleration of a sphere during vertical impact after free-fall. Acceleration vs. time in simulation (blue dashed trace) and experiment (red solid trace) agree well. The impact velocity for this representative run is 1.4 m/s. Acceleration is in units of g, the acceleration due to gravity. (Left inset) Aluminium ball instrumented with accelerometer resting on 6 mm plastic particles. (Right inset) Ball and particles in simulation.

To model the sand-swimming device we used the commercial multi-body simulator software package Working Model (WM) 2D (Design Simulation Technologies). Modeling the device in a 2D simulation environment is sufficient to capture the dynamics since the sand-swimming robot moves roughly

3

C

[

Fig. 3. Simulation of a sand-swimming robot. (A,B) Side and top view of the robot modeled with 49 inter-connected motor segments and one head segment. The angle between adjacent motors (βi ) is modulated using Eqn. 3 to reproduce the sandfish’s sinusoidal traveling wave kinematics. (C) Top view of the device submerged in 6 mm particles with particles above the robot rendered transparent. (D) Rendering of the simulated robot for the same parameters used in robot experiment (see Fig. 4). The brackets ( [ ) indicate a single robot segment.

Using Eqn. 3 the simulated robot with 7 total segments moved forward within 6 mm plastic particles and increased its forward speed linearly with oscillation frequency (Fig. 7). The wave efficiency was η = 0.36 ± 0.02, less than that of the sandfish lizard. Motivated by these results we built a physical instantiation of the scaled model. IV. S AND S WIMMING ROBOT

5cm

B tail A

head

A. Design and Control The basic mechanical design of our device was adapted from previously developed snake robots [25] which consisted of repeated modules (motors) each with a single joint that permit angular excursions in a plane and connected via identical links. In our design, each module consists of a servomotor attached to an aluminium bracket and is connected to adjacent motors via the brackets. The wire bundle that routes power and control signals to each motor was run atop each module over the length of the device and strain relieved at the last (tail) segment. For convenience and to maintain a reasonable size, our device employed 6 standard size servomotors and a dummy segment (the head) with the same weight and form factor as the motor segments for a total of 7 segments (Fig. 4A). The simulation found that the peak torque required to swim subsurface at a depth of 4 cm was 0.7 N m. To verify this finding we dragged an object with the same form factor as a motor through the 6 mm plastic medium at 0.25 m/s. The measured force at a depth of 4 cm was 3.2 N. Since the maximum torque occurs at the middle motor (0.23 m to either end) we estimated the maximum possible total force along an effective segment extending from the middle servo to either the tail or the head (length 0.23 m) to be 18 N with a corresponding maximum torque of 2.0 Nm. We selected a servomotor that exceeded both torque estimates, see Table I. Servomotors are powered in parallel from a 7.4 V, 30 A supply. The pulse width based control signal for each motor is generated in LabVIEW using Eqn. 3 as a multiplexed signal, output from a PCI-card (NI-6230), and connected to the clock input of a decade counter (CD4017BC) which functions as a demultiplexer and distributes a control pulse to each motor every 20 ms. Since the robot operates in a granular medium it is critical to encase it in a material that prevents particles from getting between the motor segments but allows the device to easily undulate. After testing several materials we found that a 2layer encasement consisting of an outer Lycra spandex sleeve with a single seam (located at the top of the device) enclosing an inner thin latex sleeve that fit tautly around the motors was satisfactory (Fig. 4). B. Experimental Methods We tested the robot in a container of the same dimensions as used in the robot simulations and filled with 6 mm plastic particles prepared in an as-poured state [50]. Overhead video (100 fps) was collected for each condition tested. To facilitate subsurface tracking the first and last module were fixed with

4

head

tail

B

tail

head

C Fig. 4. Prototype of the sand-swimming robot. (A) basic construction (servomotors and aluminium brackets with power wires running along the top of the device). The robot has a double layer skin: (B) tight fitting thin latex inner layer, and (C) Lycra spandex outer layer. Balls on narrow masts on the head and tail segments allow subsurface motion tracking.

a mast with a visible marker. The wire bundle was run up the mast on the last segment and tethered above the container. The kinematics of the subsurface motion of the robot were also obtained using x-ray imaging for a representative condition (f = 0.25 Hz, A/λ = 0.2), see Fig. 5A-C. For each test the top of the robot was submerged 4 cm below the surface and the surface leveled. Due to the servomotor angular velocity limits the maximum oscillation frequency was 1 Hz. For each frequency, 1 − 2 cycles of motion were collected.

0.48 × 0.028 × 0.054 m3 0.83 kg HSR-5980SG 2.94 N m 6 7

6 mm plastic beads segment S1

TABLE I P HYSICAL ROBOT CHARACTERISTICS

A

B

C S4

head

D

E

motion

S4

S1

motion

S7

motion 3 cm

F

tail

S7

segment S7

head

B

35

tail

S1 S1

A 65 y (cm)

Dimensions Mass Motor Motor Torque Number of Motors Total Segments

2 cm

Fig. 5. Subsurface swimming in experiment and simulation. (A-C) Sequential x-ray images of the robot swimming in 6 mm particles, and (D-F) robot swimming in simulation. Segments from head to tail are denoted as S1 to S7.

C. Robot Performance To calibrate the device we placed it on a rigid surface and used video to track the position of the segments from which we determined the mapping between the maximum relative segment angle β0 (Eqn. 3) and A/λ. Within the granular material, the forward velocity of the device monotonically increased with increasing oscillation frequency (Fig. 7) for A/λ = 0.2 and a single period wave. The slope of this relationship (η) was 0.34 ± 0.02. For the same parameters the simulation predicted η = 0.36 ± 0.02. V. D ISCUSSION Like the sandfish, the robot swims within granular media by propagating a traveling sinusoidal wave posteriorly from head to tail without limb use. The physical device demonstrates that subsurface locomotion in granular media using a relatively low degree of freedom device and a open loop control scheme is possible. However, the robot does not move forward as fast (normalized by body-length) or with the same wave efficiency as the animal. In the biological experiments, η for a range of granular material preparations and bead size was approximately 0.5. The robot in both experiment and simulation performed below this value. We hypothesized that the number of segments (for a fixed length device) affected both η and the forward speed of the device. Increasing the number of segments in the robot simulation caused the device to move forward faster and with greater wave efficiency until N ∼ 15 where η

5

5 cm

0 0 x (cm) 15

Fig. 6. Subsurface swimming in experiment and simulation. (A) Robot submerged in a container filled with 6 mm plastic particles. Masts with spherical markers are attached to the first and last module. (B) Kinematics of the first and last segment of the robot in experiment (green circles) and simulation (blue triangles).

plateaued (Fig. 8). Interestingly, the maximum η ≈ 0.5 is the same as measured in the animal experiment. We utilized our previously developed RFT to predict the performance of the sand-swimming device with parameters set to match those for the plastic particles used in the robot experiment. We estimated η = 0.56 for a smooth profiled undulator which corresponds to the numerical robot simulation prediction for N > 15 (gray band, Fig. 8). Increasing N allowed the device to better match a sinusoidal wave and increased η This suggests that deviation from the smooth form of a traveling sinusoidal wave reduces performance. A seven segment robot operates below the minimum N required to achieve maximum η. As a design criterion, N is important when the length of the device is fixed as increasing the number of motors beyond the critical N requires motors with smaller dimensions but capable of producing the same torque. We used the numeric robot simulation to measure the time varying torque required to move within the medium. As expected, the torque was approximately sinusoidal for all motors and the torque amplitude generated by the central motors (3 and 4) was larger than the torque from the motors nearest the ends, see Fig. 9. As noted earlier, the maximum torque in the simulation of 0.7 N m was well below the maximum of the motors used in experiment (see Table 1).

0.7 Torque (N m)

η = 0.36 0.3 η = 0.34

0

0

Frequency (Hz)

A

0

−0.7 0 0.7

Torque amplitude (Nm)

-1 Speed (vx /λ) (s )

Simulation Experiment

1

Fig. 7. Forward velocity vs. oscillation frequency for the robot in experiment (green circles) and simulation (blue triangles) (A/λ = 0.2). The slope of the dashed (simulation) and solid (experiment) fit lines gives the wave efficiency η.

2

4

6

B

0.35

tail 1

0

Time (s)

head 3

6

5

6

4

3 Motor

2

1

Fig. 9. Motor torque for the simulated 7 segment, 6 motor robot (f = 1 Hz) (A) varying with time. (B) Torque amplitude vs. motor position; orange (solid curve), green (dotted curve) , and black (dash-dot curve) correspond to motor 6 (tail), 4, and 1 (head) with motor position 1 denoting segment number 2 in Fig. 5 and 6.

0.7

η

0.35

N= 5

0 0

10

N= 15

N= 48

20 30 40 Number of Segments

50

Fig. 8. Wave efficiency increases with number of segments for a fixed length robot in simulation (blue dashed curve)(f = 1 Hz and A/λ = 0.2). The red, black, and cyan triangles correspond to 5, 15, and 48 segment robots respectively. The green square corresponds to the seven segment physical robot, and the grey line indicates η predicted by the RFT solved for a continuous body profile (see text for details).

Also, the fluctuations in torque at frequencies higher than the oscillation frequency of the robot were small in comparison to the torque amplitude. VI. F UTURE W ORK A sand swimming robot combined with a proven simulation tool opens many avenues for further research. Of immediate interest is testing the RFT prediction that an optimal spatial form (ratio of amplitude to wavelength) maximizes forward speed of an undulatory sand swimmer [1]. The effect of the predicted optimal kinematics can also be evaluated by measuring the mechanical cost of transport. In conjunction with the numerical simulation the robot can test the effect of motion profiles (wave shapes) on performance. Since the

6

sandfish uses the same kinematics to move in a variety of media, duplicating the animals control methods and sensing modalities in a robot could lead to more effective locomotion. The sandfish has a non-trivial shape which suggests changing the morphological characteristics of the robotic device. For example, the cross sectional shape of the sandfish (flat belly and rounded top) have been hypothesized to aid rapid burial into granular media [51]. Our simulated and physical robot can be used to explore the influence of this morphology along with body taper on performance. The robotic simulation can also tune parameters like skin friction and body compliance to identify optimal values which could then be tested with our robot. VII. C ONCLUSION Motivated by biological experiments revealing rapid subsurface sand-swimming in the sandfish lizard, we have used numerical simulation as a design tool to build an undulatory sand-swimming device. We used our robot simulation to test whether a device with a finite number of segments (7) could advance using a simple open loop (traveling wave sinusoid) control scheme and calculated the motor torque requirements for the robot. We then built and tested a prototype of the device to validate the biological observations and predictions from the RFT [1] and simulations that limbless body undulations were sufficient to propel the robot forward. Our findings show that the device can swim, and that it translates faster by increasing its oscillation frequency just as the sandfish does. The design tools (numerical model and robot) we developed can generate testable hypotheses of neuromechanical control [52] and improve our understanding of how organisms exploit the solid and fluid-like properties of granular media, enabling the

construction of robots that can locomote effectively within complex environments. ACKNOWLEDGMENTS We thank Nick Gravish and Chen Li for help with the physics experiments, and we thank Daniel Cohen and Andrew Slatton for help with development of the numerical simulation. We also acknowledge our funding from The Burroughs Wellcome Fund Career Award at the Scientific Interface, NSF Physics of Living Systems grant PHY-0749991, and the Army Research Laboratory (ARL) Micro Autonomous Systems and Technology (MAST) Collaborative Technology Alliance (CTA) under cooperative agreement number W911NF08 − 2 − 0004. R EFERENCES [1] R. Maladen, Y. Ding, C. Li, and D. Goldman, “Undulatory Swimming in Sand: Subsurface Locomotion of the Sandfish Lizard,” Science, vol. 325, no. 5938, p. 314, 2009. [2] C. Humphrey and J. Adams, “Robotic Tasks for Chemical, Biological, Radiological, Nuclear and Explosive Incident Response,” Advanced Robotics, vol. 23, no. 9, pp. 1217–1232, 2009. [3] J. McKean, S. Buechel, and L. Gaydos, “Remote sensing and landslide hazard assessment,” Photogrammetric engineering and remote sensing, vol. 57, no. 9, pp. 1185–1193, 1991. [4] G. Metternicht, L. Hurni, and R. Gogu, “Remote sensing of landslides: An analysis of the potential contribution to geo-spatial systems for hazard assessment in mountainous environments,” Remote sensing of Environment, vol. 98, no. 2-3, pp. 284–303, 2005. [5] A. Ashcheulov, I. Gutsul, and V. Maevski, “Device for monitoring the radiation temperature in coal mines,” Journal of Optical Technology, vol. 67, no. 3, p. 281, 2000. [6] W. Marcus, C. Legleiter, R. Aspinall, J. Boardman, and R. Crabtree, “High spatial resolution hyperspectral mapping of in-stream habitats, depths, and woody debris in mountain streams,” Geomorphology, vol. 55, no. 1-4, pp. 363–380, 2003. [7] R. Ritzmann, R. Quinn, and M. Fischer, “Convergent evolution and locomotion through complex terrain by insects, vertebrates and robots,” Arthropod structure and development, vol. 33, no. 3, pp. 361–379, 2004. [8] R. Siegwart, P. Lamon, T. Estier, M. Lauria, and R. Piguet, “Innovative design for wheeled locomotion in rough terrain,” Robotics and Autonomous systems, vol. 40, no. 2-3, pp. 151–162, 2002. [9] J. Wong, “On the study of wheel-soil interaction,” Journal of Terramechanics, vol. 21, no. 2, pp. 117–131, 1984. [10] P. Arena, P. Di Giamberardino, L. Fortuna, F. La Gala, S. Monaco, G. Muscato, A. Rizzo, and R. Ronchini, “Toward a mobile autonomous robotic system for Mars exploration,” Planetary and Space Science, vol. 52, no. 1-3, pp. 23–30, 2004. [11] C. Li, P. B. Umbanhowar, H. Komsuoglu, D. E. Koditschek, and D. I. Goldman, “Sensitive dependence of the motion of a legged robot on granular media,” Proceedings of the National Academy of Science, vol. 106, no. 9, pp. 3029–3034, 2009. [12] C. Li, P. Umbanhowar, H. Komsuoglu, and D. Goldman, “The effect of limb kinematics on the speed of a legged robot on granular media,” Experimental Mechanics, pp. 1–11, 2010. [13] U. Saranli, M. Buehler, and D. Koditschek, “RHex: A simple and highly mobile hexapod robot,” The International Journal of Robotics Research, vol. 20, no. 7, p. 616, 2001. [14] R. Playter, M. Buehler, and M. Raibert, “BigDog,” in Unmanned Ground Vehicle Technology VIII, ser. Proceedings of SPIE, D. W. G. Grant R. Gerhart, Charles M. Shoemaker, Ed., vol. 6230, 2006, pp. 62 302O1– 62 302O6. [15] A. Hoover, E. Steltz, and R. Fearing, “RoACH: An autonomous 2.4 g crawling hexapod robot,” in IEEE Intelligent Robots and Systems Conference Proceeding, 2008, pp. 26–33. [16] A. Saunders, D. I. Goldman, R. J. Full, and M. Buehler, “The rise climbing robot: body and leg design,” in Unmanned Systems Technology VIII, G. R. Gerhart, C. M. Shoemaker, and D. W. Gage, Eds., vol. 6230. SPIE, 2006, p. 623017.

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 27

(8)

We use f B = (f1B , f2B , . . . , fnB ) to denote the projection of f onto b. Given a fixed basis curve b, the vector μ is effectively an estimate of f B . In this sense, there are two estimates: the estimate μ of f B , and the estimate ˆf of f , where ˆf is defined in terms of μ and b as in Eq. (8).

which each observation is associated with the curve that best “explains” that observation. Other techniques, such as jointcompatibility branch and bound [12], may also be used. If no tracked curve is likely to have generated the observation according to the current estimates, then the observation is used to initialize a new curve for tracking.

A. Observations

C. Curvature prediction

We model an observation z of f as a subset of f , corrupted by zero-mean i.i.d. noise. Note that z is typically a partial observation of f , since only a portion of a painted road boundary may be observed at any given moment. For simplicity, we first consider the case where z is a full observation of f , and later consider partial observations. If we project z onto b and refer to the piecewise linear approximation of z as Z = (z1 , z2 , . . . , zn ) , then each zi can be expressed as: ¯i zi = bi + (fiB + vi )b (9) B¯ = bi + z bi

As the vehicle travels, it may receive observations that extend beyond the end of its current curve estimate. To incorporate these observations, the observation and curve estimate are first extended (Fig. 4b). We use a method described in previous work [8], briefly summarized here. Once we have observed and estimated one portion of a curve, we can reliably predict the shape of the nearby unobserved parts of the curve. To do so, we fit a first-order Markov model of road curvature to a dataset containing the manually annotated coarse geometry of existing public roadways. Given the signed curvature at one point in a road, the model predicts a distribution over the curvature at a further point along the road. This model can then be used to extend both the curve estimates and observations as the vehicle travels.

i

where we model the noise terms v = (v1 , v2 , . . . , vn ) as jointly Gaussian   with zero mean and covariance R, such that v ∼ N 0, R . The projection zB = (z1B , z2B , . . . , znB ) of z onto b can then be described as: ziB = fiB + vi

(10)

zB = Af B + v

(11)

and:

(a) Curve estimate (left) and a non- (b) Predicted extensions of both overlapping observation (right) curves

where the observation matrix A = In×n is a trivial linear transformation. If z is a partial observation of f , then it can still be related to f by choosing an observation matrix A of size m × n, where m ≤ n, and A has an m × m identity sub-matrix and is zero elsewhere. For example, if the basis curve has three control points and the observation curve spans only the first 2 of these control points, then the corresponding observation matrix A2×3 is:

1 0 0 A2×3 = (12) 0 1 0 B. Data association Not all observations correspond to the desired curve f . To avoid corrupting the curve estimate, only true observations should be incorporated, and non-observations ignored or used to update a different curve estimate. Determining if an observation corresponds to a tracked curve is the well-known data association problem [14]. We use a standard χ2 outlier rejection approach. Define the scalar random variable y as the following Mahalanobis distance [2]: y = (zB − Aμ) (R + AΣA )−1 (zB − Aμ)

(13)

If z is an observation of the curve f , then y obeys a χ2 distribution with m degrees of freedom. Observations that fall in the extremities of this distribution are rejected as outliers. When simultaneously estimating and tracking multiple curves, we apply a gated greedy matching procedure, in

(c) Final curve estimate Fig. 4. A curvature prediction model allows us to associate observations with existing curve estimates when there is little or no overlap. The short line segments perpendicular to each curve indicate 1-σ uncertainty.

D. Update Once an observation z has been associated with an existing curve estimate, the algorithm revises that estimate using the update step of a standard Kalman filter [2]. As mentioned earlier, approximation errors resulting from projection onto a basis curve are minimized when the basis curve geometry matches the true curve geometry. Therefore, we reparameterize the curve estimate, so that the basis curve coincides with the newly updated maximum likelihood curve estimate. Since this estimate is a variation of the current basis curve, reparameterization consists of offsetting the basis curve by the mean vector, then setting the mean vector to zero (Sec. IV-B). We also re-sample the basis curve control points to maintain nearly uniform control point spacing. Finally, we note that propagating the curve estimate forward through time consists of a simple identity transform, a consequence of expressing the curve geometry in a coordinate system fixed to the local environment. Algorithm update curve estimate shows a full update cycle. It

 28

˜ (b, μ, Σ) over a single takes as input the initial belief N curve, and an observation. If the observation passes the data association test, then it is used to generate the updated curve ˜ (b+ , μ+ , Σ+ ). Fig. 4 illustrates an update step for estimate N a single curve estimate. Algorithm update curve estimate ˜ (b, μ, Σ) Input: an initial belief N ˜ (b, zB , R) Input: an observation N Input: an observation matrix A ˜ (b+ , μ+ , Σ+ ) Output: the new belief N 1: Augment the initial estimate and observation as necessary via curve prediction 2: Let m be the dimensionality of zB 3: Let k be the 0.95 percentile of the χ2m distribution if (zB − Aμ) (R + AΣA )−1 (zB − Aμ) > k then Reject observation as an outlier ˜ (b, μ, Σ) 6: return N 4: 5:

12:

K ← ΣA (AΣA + R)−1 ˜ ← μ + K(zB − Aμ) μ ˜ ← (I − KA)Σ Σ ¯ ˜ B b+ ← B + diag(μ) μ+ ← 0 ˜ Σ+ ← Σ

13:

˜ (b+ , μ+ , Σ+ ) return N

7: 8: 9: 10: 11:

where each μci describes the mean offset of a centerline control point, and each μhi describes the mean half-width of the lane estimate at the control point. It is sometimes useful to change the basis curve upon which a lane distribution has been defined, while incurring minimal changes to the actual distribution. Choosing a new mean and covariance in the cases of re-sampled and offset basis curves follows the same procedure as in Sec. IV-B, with minor modifications. When the new basis curve is a variation of the original basis curve, the width components of the mean lane do not change. When the new basis curve is a re-sampling of the original basis curve, the re-sampling matrix H must account for re-sampling the width components in addition to the centerline offset values. B. Observation model, data association, and update A full boundary observation of f is a curve, which we describe with the matrix of control points Z = (z1 , z2 , . . . , zn ) , where each zi can be written: B ¯i + afhi + vi )b zi = bi + (fci (17) ¯i = bi + ziB b

VI. L ANE ESTIMATION The boundary curves of a single lane are highly correlated, and information about one boundary gives valuable information about the other. We represent a lane as a piecewise linear centerline curve whose width varies along the curve, and describe its control points with the matrix F = (f 1 , f 2 , . . . , f n ) , where each f i is defined as f i = (fxi , fyi , fhi ) . Using the convention that the normal vectors of a curve point “left”, two points f li and f ri on the left and right boundaries, respectively, can be described as:

fxi + fhi f¯xi fxi − fhi f¯xi f li = f ri = (14) fyi + fhi f¯yi fyi − fhi f¯yi where ¯f i = (f¯xi , f¯yi ) is the normal vector to the centerline curve at point i. A. Lane distributions As with zero-width curves, a basis curve can be used to represent and approximate lanes. We describe the projection f B of f onto b as: B B B , fh1 , fc2 , fh2 , . . . , fcn , fhn ) f B = (fc1

A normal distribution over the projections of f onto b defines a distribution over lanes. We use such a distribution, parameterized by μ and Σ, to represent a belief over the true geometry of f . The mean estimate can be represented by a ˆ = (ˆf 1 , ˆf 2 , . . . , ˆf n ) , where each matrix of control points F ˆ ˆ ˆ control point f i = (fxi , fyi , fˆhi ) can be expressed as: ⎡ˆ ⎤ ⎡ ⎤ fxi bxi + μci¯bxi ˆf i = ⎣fˆyi ⎦ = ⎣byi + μci¯byi ⎦ (16) ˆ μ hi fhi

(15)

B B where fc1 , . . . , fcn is the projection of the centerline of f onto b. Thus, the projection of a lane onto b is simply the projection of its centerline augmented by its half-width terms.

where a has value +1 or −1 for an observation of the left or right boundary, respectively, and we model  the noise terms v = (v1 , v2 , . . . , vn ) jointly as v ∼ N 0, R . Collectively, the offset vector zB = (z1B , z2B , . . . , znB ) can then be expressed as: zB = Af B + v

(18)

where the elements of the observation matrix A are chosen to satisfy Eq. (17). If z is a partial observation of the boundary, such that it projects onto only m control points of b, then A has size 2m × 2n, similar to the case for zero-width curves. Data association and update steps can be approached in the same way as for zero-width curves. Given a lane distribution and observation as expressed above, we can apply a χ2 test to determine if z is an observation of f . When estimating multiple lanes, we use a gated greedy assignment procedure to assign observations to lanes. Once an observation has been associated with a lane estimate, the standard Kalman update steps are used to update the mean and covariance. After the updated estimates have been computed, we once again reparameterize the distribution such that the basis curve coincides with the updated maximum likelihood estimate, to minimize approximation error in future update steps. Fig. 5 shows a full update cycle, where an observation of a lane boundary is used to both update and extend the lane.

 29

(a) A lane estimate

(b) A boundary observation

(c) The updated estimate Fig. 5. A boundary observation is used to update the lane estimate. The middle (black) curve marks the lane centerline, and the outer (blue) curves mark the left and right boundary marginal distributions. Short line segments along the curves mark control points, and the length of these segments indicate 1-σ uncertainty. Note that both boundaries are updated even though only one boundary is observed.

C. Initial estimate We initialize a lane estimate by independently estimating many zero-width curves as described in Sec. V, while periodically searching for curve pairs that are sufficiently long, parallel, and separated by an appropriate distance. Once a suitable pair of boundary curves is identified, they are used to initialize a lane estimate. The initial lane basis curve b is chosen by projecting one boundary curve onto the other and scaling the offset vector by 0.5 (an approximation of the medial axis), and both curve estimates are then reparameterized with b. ˜ (b, μ , Σl ) Referring to the left and right curve estimates as N l ˜ (b, μr , Σr ), we treat them as independent observations and N of the same lane, and express them jointly as: μl Al B (19) z= = f + v = Az f B + v μr Ar where Al and Ar are the transformation matrices relating a lane to its left and right boundary observations (Sec. VI-B), of the unobserved true lane onto b, and f B is the  projection  v ∼ N 0, Σz is a noise term described by:

Σl 0 (20) Σz = 0 Σr Using the information filter [14], we can see that the initial distribution parameters best representing the information provided by the boundary curves can be expressed as: Σ0 μ0

−1 −1 = (A z Σz Az )  −1 = ΣAz Σz z

(21)

VII. E XPERIMENTS To quantitatively assess the performance of our system, we evaluated it against ground truth across two datasets containing data from a forward-facing wide-angle camera (Point Grey Firefly MV, 752x480 @22.8 Hz), and a Velodyne HDL-64E laser range scanner. As input to our system, we used visionand LIDAR-based road paint and curb detection algorithms described in previous work [7].

The first dataset consists of 30.2 km of travel in 182 minutes, and can be characterized by wide suburban lanes, no pedestrians, generally well-marked roads, few vehicles, and a bright early morning sun. The vehicle also traverses a short 0.4 km dirt road and a 1.7 km stretch of highway. The second dataset consists of 13.6 km of travel in 58 minutes through a densely populated city during afternoon rush hour. This dataset can be characterized by roads of varying quality, large numbers of parked and moving vehicles, and many pedestrians. To produce ground truth, we annotated high-resolution georegistered ortho-rectified aerial imagery with lane geometry. The vehicle’s GPS estimates during a data collection provide an initial guess as to the vehicle’s pose; these were corrected by manually aligning sensor data (i.e., camera and LIDAR data) with the aerial imagery at various points in the data collection. The result is a dataset containing ground truth lane geometry relative to the vehicle at every moment of travel. We emphasize that our algorithm uses only local sensor data – GPS and the ground truth map were used only for evaluation purposes. We compare the results of our algorithm, which we refer to as the basis curve (BasCurv) algorithm, with our previous work in the DARPA Urban Challenge [7], which we refer to as the evidence image (EvImg) algorithm. The evidence image algorithm can be used as a standalone lane estimation system by using the output of the first of its two stages, which performs lane detection from sensor data only. Both algorithms use the same features as input. For computational speed, our implementation of the basis curve algorithm used diagonal covariance matrices when estimating lane boundaries, and block-diagonal covariance matrices (2 × 2 blocks) for lane estimation. This introduces additional approximation errors, but yielded good performance in our experiments. After each observation update, basis curves are re-sampled to maintain a uniform (1 m) control point spacing. Parameters such as covariances and data association thresholds were determined experimentally. The basis curve algorithm was implemented in Java and runs at real-time speeds. A. Centerline error The centerline error of a lane estimate at a given point on the estimate is defined as the shortest distance from the estimated lane centerline point to the true centerline of the nearest lane. Fig. 6 shows the 50th and 90th percentile values for the centerline error of the two algorithms as a function of distance from the vehicle. The basis curve algorithm has significantly lower error at all distances. Fig. 7 shows the centerline error as a function of true lane width. The evidence image algorithm assumes a fixed lane width of 3.66 m, and its performance degrades as the true lane width departs from this assumed value. Since the basis curve algorithm jointly estimates lane width and centerline geometry, it remains relatively invariant to changes in lane width.

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Median centerline lateral error as a function of true lane width.

B. Lookahead distance and time The lookahead distance and lookahead time metrics measure how much farther the vehicle can travel before reaching the end of its current lane estimate, assuming constant speed. Lookahead distance is computed by measuring the distance from the vehicle to the farthest point ahead on the current lane estimate, and lookahead time is computed by dividing the lookahead distance by the vehicle’s instantaneous speed. Fig. 8 aggregates lookahead statistics over both datasets, and shows the lookahead distance and lookahead time cumulative distributions for the two algorithms. In all cases, the basis curve algorithm outperforms the evidence image algorithm. For example, the basis curve algorithm provided some lane estimate forward of the vehicle for 71% of distance traveled, compared to 36% for the evidence image algorithm. C. Qualitative results Fig. 9 shows the output of the basis curve lane estimation algorithm in a variety of challenging scenarios. In (a) and (b), tree shadows and an erroneous curb detection are detected and rejected as outliers, leaving the lane estimates intact. In (b), the median strip separating opposite lanes of traffic is correctly excluded from the lane estimates. In (c) and (d), correctly detected road paint is successfully excluded from

VIII. D ISCUSSION Overall, the basis curve algorithm provides lane estimates of equal or better accuracy to those produced by the evidence image approach, and does so more often and with a greater lookahead. We attribute this to the data association and outlier rejection properties of the basis curve algorithm, and to the joint estimation of lane width and centerline geometry. The evidence image algorithm performs no outlier rejection, attempting to fit lanes to both true lane boundary detections and false detections such as tree shadows and non-boundary road paint; nor does it estimate lane width. We have formulated the lane estimation problem in such a way that standard estimation and tracking algorithms based on the Kalman filter can be used for complex lane geometries. In doing so, we gain the advantages of the Kalman filter, and also invite all of its shortcomings. Cases where outliers appear very similar to inliers, such as long shadows nearly parallel to the road, can cause the lane estimates to diverge. Another failure mode arises when one road marking appears initially to be the correct boundary, but the true lane boundary then comes into view. In this case, our method will converge upon the first marking as the lane boundary, since the unimodal nature of the Kalman filter will prevent it from assigning substantial weight to the true boundary. These difficulties are similar to those studied in other estimation domains, and it should also be possible to apply lessons learned in those domains to lane estimation with basis curves. One approach is particle filtering, which has been successfully applied in many estimation and tracking problems to model complex distributions and enable multihypothesis tracking. However, the high dimensionality of the lane estimates would require careful treatment. Finally, in using a Gaussian noise model, we are simplifying and approximating the true system errors. This has proved successful in practice, although more careful study is required to understand the extent to which our simplifications and approximations are valid, and when other models may be more appropriate. IX. C ONCLUSION This paper introduced the notion of basis curves for curve estimation, and described an application to the lane estimation

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a.

b.

c.

d.

e.

f.

g. Fig. 9. Lane estimation in a variety of environments. Column 1: Camera images. Column 2: Detections of road paint (magenta) and curbs (green). Column 3: Lane centerline estimates (black) and boundary curve estimates (blue) projected into the image. Column 4: Synthesized overhead view of lane estimates.

problem. A detailed evaluation of our method’s performance on a real-world dataset, and a quantitative comparison against ground truth and a previous approach, shows distinct advantages of the basis curve algorithm, particularly for estimating lanes using partial observations, for handling noisy data with high false-positive rates, and for jointly estimating centerline geometry and lane width. R EFERENCES [1] N. Apostoloff and A. Zelinsky. Vision in and out of vehicles: Integrated driver and road scene monitoring. Int. Journal of Robotics Research, 23(4-5):513–538, Apr. 2004. [2] Y. Bar-Shalom and X.-R. Li. Estimation with Applications to Tracking and Navigation. John Wiley & Sons, Inc., 2001. [3] M. Bertozzi and A. Broggi. GOLD: a parallel real-time stereo vision system for generic obstacle and lane detection. IEEE Transactions on Image Processing, 7(1):62–80, Jan. 1998. [4] M. Bertozzi, A. Broggi, and A. Fascioli. Vision-based intelligent vehicles: State of the art and perspectives. Robotics and Autonomous Systems, 1:1–16, 2000. [5] A. Blake and M. Isard. Active Contours. Springer-Verlag, 1998. [6] E. Dickmanns and B. Mysliwetz. Recursive 3-D road and ego-state recognition. IEEE Trans. Pattern Analysis and Machine Intelligence, 14(2):199–213, Feb. 1992.

[7] A. S. Huang, D. Moore, M. Antone, E. Olson, and S. Teller. Finding multiple lanes in urban road networks with vision and lidar. Autonomous Robots, 26(2-3):103–122, Apr. 2009. [8] A. S. Huang and S. Teller. Lane boundary and curb estimation with lateral uncertainties. In Proc. IEEE Int. Conf. on Intelligent Robots and Systems, St. Louis, Missouri, Oct. 2009. [9] Z. Kim. Robust lane detection and tracking in challenging scenarios. IEEE Trans. Intelligent Transportation Systems, 9(1):16–26, Mar. 2008. [10] Y. Matsushita and J. Miura. On-line road boundary modeling with multiple sensory features, flexible road model, and particle filter. In Proc. European Conference on Mobile Robots, Sep. 2009. [11] J. C. McCall and M. M. Trivedi. Video-based lane estimation and tracking for driver assistance: Survey, system, and evaluation. IEEE Transactions on Intelligent Transport Systems, 7(1):20– 37, Mar. 2006. [12] J. Neira and J. D. Tardos. Data association in stochastic mapping using the joint compatibility test. IEEE Trans. Robotics and Automation, 17(6):890–897, Dec 2001. [13] C. Thorpe, M. Hebert, T. Kanade, and S. Shafer. Vision and navigation for the Carnegie-Mellon Navlab. IEEE Transactions on Pattern Analysis and Machine Intelligence, 10(3):362–373, May 1988. [14] S. Thrun, W. Burgard, and D. Fox. Probabilistic Robotics. MIT Press, 2005. [15] Y. Wang, E. K. Teoh, and D. Shen. Lane detection and tracking using B-Snake. Image and Vision Computing, 22(4):269 – 280, 2004.

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Reinforcement Learning to Adjust Robot Movements to New Situations Jens Kober

Erhan Oztop

Jan Peters

MPI for Biol. Cybernetics, Germany Email: [email protected]

ATR Comput. Neuroscience Labs, Japan Email: [email protected]

MPI for Biol. Cybernetics, Germany Email: [email protected]

Abstract—Many complex robot motor skills can be represented using elementary movements, and there exist efficient techniques for learning parametrized motor plans using demonstrations and self-improvement. However, in many cases, the robot currently needs to learn a new elementary movement even if a parametrized motor plan exists that covers a similar, related situation. Clearly, a method is needed that modulates the elementary movement through the meta-parameters of its representation. In this paper, we show how to learn such mappings from circumstances to meta-parameters using reinforcement learning. We introduce an appropriate reinforcement learning algorithm based on a kernelized version of the reward-weighted regression. We compare this algorithm to several previous methods on a toy example and show that it performs well in comparison to standard algorithms. Subsequently, we show two robot applications of the presented setup; i.e., the generalization of throwing movements in darts, and of hitting movements in table tennis. We show that both tasks can be learned successfully using simulated and real robots.

I. I NTRODUCTION In robot learning, motor primitives based on dynamical systems [1], [2] allow acquiring new behaviors quickly and reliably both by imitation and reinforcement learning. Resulting successes have shown that it is possible to rapidly learn motor primitives for complex behaviors such as tennis-like swings [1], T-ball batting [3], drumming [4], biped locomotion [5], ball-in-a-cup [6], and even in tasks with potential industrial applications [7]. The dynamical system motor primitives [1] can be adapted both spatially and temporally without changing the overall shape of the motion. While the examples are impressive, they do not address how a motor primitive can be generalized to a different behavior by trial and error without re-learning the task. For example, if the string length has been changed in a ball-in-a-cup [6] movement1 , the behavior has to be re-learned by modifying the movements parameters. Given that the behavior will not drastically change due to a string length variation of a few centimeters, it would be better to generalize that learned behavior to the modified task. Such generalization of behaviors can be achieved by adapting the meta-parameters of the movement representation2 . In machine learning, there have been many attempts to use meta-parameters in order to generalize between tasks [8]. 1 In this movement, the system has to jerk a ball into a cup where the ball is connected to the bottom of the cup with a string. 2 Note that the tennis-like swings [1] could only hit a static ball at the end of their trajectory, and T-ball batting [3] was accomplished by changing the policy’s parameters.

Figure 1: This figure illustrates a 2D dart throwing task. The situation, described by the state s corresponds to the relative height. The meta-parameters γ are the velocity and the angle at which the dart leaves the launcher. The policy parameters represent the backward motion and the movement on the arc. The meta-parameter function γ(s), which maps the state to the meta-parameters, is learned. Particularly, in grid-world domains, significant speed-up could be achieved by adjusting policies by modifying their metaparameters (e.g., re-using options with different subgoals) [9]. In robotics, such meta-parameter learning could be particularly helpful due to the complexity of reinforcement learning for complex motor skills with high dimensional states and actions. The cost of experience is high as sample generation is time consuming and often requires human interaction (e.g., in cart-pole, for placing the pole back on the robots hand) or supervision (e.g., for safety during the execution of the trial). Generalizing a teacher’s demonstration or a previously learned policy to new situations may reduce both the complexity of the task and the number of required samples. For example, the overall shape of table tennis forehands are very similar when the swing is adapted to varied trajectories of the incoming ball and a different targets on the opponent’s court. Here, the human player has learned by trial and error how he has to adapt the global parameters of a generic strike to various situations [10]. Hence, a reinforcement learning method for acquiring and refining meta-parameters of pre-structured primitive movements becomes an essential next step, which we will address in this paper. We present current work on automatic meta-parameter acquisition for motor primitives by reinforcement learning. We focus on learning the mapping from situations to meta-

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parameters and how to employ these in dynamical systems motor primitives. We extend the motor primitives of [1] with a learned meta-parameter function and re-frame the problem as an episodic reinforcement learning scenario. In order to obtain an algorithm for fast reinforcement learning of metaparameters, we view reinforcement learning as a rewardweighted self-imitation [11], [6]. As it may be hard to realize a parametrized representation for meta-parameter determination, we reformulate the rewardweighted regression [11] in order to obtain a Cost-regularized Kernel Regression (CrKR) that is related to Gaussian process regression [12]. We compare the Cost-regularized Kernel Regression with a traditional policy gradient algorithm [3] and the reward-weighted regression [11] on a toy problem in order to show that it outperforms available previously developed approaches. As complex motor control scenarios, we evaluate the algorithm in the acquisition of flexible motor primitives for dart games such as Around the Clock [13] and for table tennis. II. M ETA -PARAMETER L EARNING FOR M OTOR P RIMITIVES The goal of this paper is to show that elementary movements can be generalized by modifying only the meta-parameters of the primitives using learned mappings. In Section II-A, we first review how a single primitive movement can be represented and learned. We discuss how such meta-parameters may be able to adapt the motor primitive spatially and temporally to the new situation. In order to develop algorithms that learn to automatically adjust such motor primitives, we model meta-parameter self-improvement as an episodic reinforcement learning problem in Section II-B. While this problem could in theory be treated with arbitrary reinforcement learning methods, the availability of few samples suggests that more efficient, task appropriate reinforcement learning approaches are needed. To avoid the limitations of parametric function approximation, we aim for a kernel-based approach. When a movement is generalized, new parameter settings need to be explored. Hence, a predictive distribution over the metaparameters is required to serve as an exploratory policy. These requirements lead to the method which we derive in Section II-C and employ for meta-parameter learning in Section II-D. A. Motor Primitives with Meta-Parameters In this section, we review how the dynamical systems motor primitives [1], [2] can be used for meta-parameter learning. The dynamical system motor primitives [1] are a powerful movement representation that allows ensuring the stability of the movement, choosing between a rhythmic and a discrete movement and is invariant under rescaling of both duration and movement amplitude. These modification parameters can become part of the meta-parameters of the movement. In this paper, we focus on single stroke movements which appear frequently in human motor control [14], [2]. Therefore, we will always focus on the discrete version of the dynamical

system motor primitives in this paper (however, the results may generalize well to rhythmic motor primitives and hybrid settings). We use the most recent formulation of the discrete dynamical systems motor primitives [2] where the phase z of the movement is represented by a single first order system z˙ = −τ αz z.

(1)

This canonical system has the time constant τ = 1/T where T is the duration of the motor primitive and a parameter αz , which is chosen such that z ≈ 0 at T . Subsequently, the internal state x of a second system is chosen such that positions q of all degrees of freedom are given by q = x1 , the velocities by q˙ = τ x2 = x˙ 1 and the accelerations by ¨ = τ x˙ 2 . The learned dynamics of Ijspeert motor primitives q can be expressed in the following form x˙ 2 = τ αx (βx (g − x1 ) − x2 ) + τ Af (z) , x˙ 1 = τ x2 .

(2)

This set of differential equations has the same time constant τ as the canonical system and parameters αx , βx are set such that the system is critically damped. The goal parameter g, a transformation function f and an amplitude matrix A = diag (a1 , a2 , . . . , aI ), with the amplitude modifier a = [a1 , a2 , . . . , aI ] allow representing complex movements. In [2], the authors use a = g −x01 , with the initial position x01 , which ensures linear scaling. Other choices are possibly better suited for specific tasks, see for example [15]. The transformation function f (z) alters the output of the first system, in Equation (1), so that the second system in Equation (2), can represent complex nonlinear patterns and is given by N f (z) = n=1 ψn (z) θ n z. (3) Here, θn contains the nth adjustable parameter of all degrees of freedom, N is the number of parameters per degree of freedom, and ψn (z) are the corresponding weighting functions [2]. Normalized Gaussian kernels are used as weighting functions given by   2 exp −hn (z − cn )  . (4) ψn =  N 2 exp −h (z − c ) m m m=1 These weighting functions localize the interaction in phase space using the centers cn and widths hn . As z ≈ 0 at T , the influence of the transformation function f (z) in Equation (3) vanishes and the system stays at the goal position g. Note that the degrees of freedom (DoF) are usually all modeled independently in the second system in Equation (2). All DoFs are synchronous as the dynamical systems for all DoFs start at the same time, have the same duration and the shape of the movement is generated using the transformation f (z) in Equation (3), which is learned as a function of the shared canonical system in Equation (1). One of the biggest advantages of this motor primitive framework [1], [2] is that the second system in Equation (2), is linear in the shape parameters θ. Therefore, these parameters can be obtained efficiently, and the resulting framework is wellsuited for imitation [1] and reinforcement learning [6]. The

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resulting policy is invariant under transformations of the initial position x01 , the goal g, the amplitude A and the duration T [1]. These four modification parameters can be used as the meta-parameters γ of the movement. Obviously, we can make more use of the motor primitive framework by adjusting the meta-parameters γ depending on the current situation or state s according to a meta-parameter function γ(s). The state s can for example contain the current position, velocity and acceleration of the robot and external objects, as well as the target to be achieved. This paper focuses on learning the metaparameter function γ(s) by episodic reinforcement learning. Illustration of the Learning Problem: As an illustration of the meta-parameter learning problem, we take a 2D dart throwing task with a dart on a launcher which is illustrated in Figure 1 (in Section III-B, we will expand this example to a robot application). Here, the desired skill is to hit a specified point on a wall with a dart. The dart is placed on the launcher and held there by friction. The motor primitive corresponds to the throwing of the dart. When modeling a single dart’s movement with dynamical-systems motor primitives [1], the combination of retracting and throwing motions would be represented by one movement primitive and can be learned by determining the movement parameters θ. These parameters can either be estimated by imitation learning or acquired by reinforcement learning. The dart’s impact position can be adapted to a desired target by changing the velocity and the angle at which the dart leaves the launcher. These variables can be influenced by changing the meta-parameters of the motor primitive such as the final position of the launcher and the duration of the throw. The state consists of the current position of the hand and the desired position on the target. If the thrower is always at the same distance from the wall the two positions can be equivalently expressed as the vertical distance. The meta-parameter function γ(s) maps the state (the relative height) to the meta-parameters γ (the final position g and the duration of the motor primitive T ). The approach presented in this paper is applicable to any movement representation that has meta-parameters, i.e., a small set of parameters that allows to modify the movement. In contrast to [16], [17], [18] our approach does not require explicit (re)planning of the motion. In the next sections, we derive and apply an appropriate reinforcement learning algorithm.

Algorithm 1: Meta-Parameter Learning Preparation steps: Learn one or more motor primitives by imitation and/or reinforcement learning (yields shape parameters θ). Determine initial state s0 , meta-parameters γ0 , and cost C 0 corresponding to the initial motor primitive. Initialize the corresponding matrices S, Γ, C. Choose a kernel k, K. Set a scaling parameter λ. For all iterations j: Determine the state sj specifying the situation. Calculate the meta-parameters γj by: Determine the mean of each meta-parameter i γi (sj ) = k(sj )T (K + λC)−1 Γi , Determine the variance σ 2 (sj ) = k(sj , sj ) − k(sj )T (K + λC)−1 k(sj ), Draw the meta-parameters from a Gaussian distribution γj ∼ N (γ|γ(sj ), σ2 (sj )I). Execute the motor primitive using the new meta-parameters. Calculate the cost cj at the end of the episode. Update S, Γ, C according to the achieved result.

policy gradient approaches and natural gradients3 . Reinforcement learning of the meta-parameter function γ(s) is not straightforward as only few examples can be generated on the real system and trials are often quite expensive. The credit assignment problem is non-trivial as the whole movement is affected by every change in the meta-parameter function. Early attempts using policy gradient approaches resulted in tens of thousands of trials even for simple toy problems, which is not feasible on a real system. Dayan & Hinton [19] showed that an immediate reward can be maximized by instead minimizing the Kullback-Leibler divergence D(π(γ|s)R(s, γ)||π  (γ|s)) between the rewardweighted policy π(γ|s) and the new policy π  (γ|s). Williams [20] suggested to use a particular policy in this context; i.e., the policy π(γ|s) = N (γ|γ(s), σ2 (s)I), where we have the deterministic mean policy γ(s) = φ(s)T w with basis functions φ(s) and parameters w as well as the variance σ 2 (s) that determines the exploration  ∼ N (0, σ 2 (s)I). The parameters w can then be adapted by reward-weighted regression in an immediate reward [11] or episodic reinforcement learning scenario [6]. The reasoning behind this reward-weighted regression is that the reward can be treated as an improper probability distribution over indicator variables determining whether the action is optimal or not. C. A Task-Appropriate Reinforcement Learning Algorithm

B. Problem Statement: Meta-Parameter Self-Improvement The problem of meta-parameter learning is to find a stochastic policy π(γ|x) = p(γ|s) that maximizes the expected return ˆ ˆ J(π) = p(s) π(γ|s)R(s, γ)dγ ds, (5) S

G

where R(s, γ) denotes all the rewards following the selection of the meta-parameter γ according to a situation described T by state s. The return of an episode is R(s, γ) = T −1 t=0 r t with number of steps T and rewards r t . For a parametrized policy π with parameters w it is natural to first try a policy gradient approach such as finite-difference methods, vanilla

Designing good basis functions is challenging, a nonparametric representation is better suited in this context. There is an intuitive way of turning the reward-weighted regression into a Cost-regularized Kernel Regression. The kernelization of the reward-weighted regression can be done straightforwardly (similar to Section 6.1 of [21] for regular supervised learning). Inserting the reward-weighted regression solution w = (ΦT RΦ + λI)−1 ΦT RΓi , and using the Woodbury formula (ΦT RΦ + λI)ΦT = ΦT R(ΦΦT + λR−1 ), we 3 While we will denote the shape parameters by θ, we denote the parameters of the meta-parameter function by w.

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(b) Policy after 2 updates: R=0.1

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Figure 2: This figure illustrates the meaning of policy improvements with Cost-regularized Kernel Regression. Each sample consists of a state, a meta-parameter and a cost where the cost is indicated the blue error bars. The red line represents the improved mean policy, the dashed green lines indicate the exploration/variance of the new policy. For comparison, the gray lines show standard Gaussian process regression. As the cost of a data point is equivalent to having more noise, pairs of states and meta-parameter with low cost are more likely to be reproduced than others with high costs. transform reward-weighted regression into a Cost-regularized Kernel Regression −1  ¯ i = φ(s)T w = φ(s)T ΦT RΦ + λI ΦT RΓi γ  −1 = φ(s)T ΦT ΦΦT + λR−1 Γi , (6) where the rows of Φ correspond to the basis functions φ(si ) = Φi of the training examples, Γi is a vector containing the training examples for meta-parameter component γ i , and λ is a ridge factor. Next, we assume that the accumulated rewards Rk are strictly positive Rk > 0 and can be transformed into costs by ck = 1/Rk . Hence, we have a cost matrix C = R−1 = diag(R1−1 , . . . , Rn−1 ) with the cost of all n data points. After replacing k(s) = φ(s)T ΦT and K = ΦΦT , we obtain the Cost-regularized Kernel Regression ¯ i = γ i (s) = k(s)T (K + λC)−1 Γi , γ which gives us a deterministic policy. Here, costs correspond to the uncertainty about the training examples. Thus, a high cost is incurred for being further away from the desired optimal solution at a point. In our formulation, a high cost therefore corresponds to a high uncertainty of the prediction at this point. In order to incorporate exploration, we need to have a stochastic policy and, hence, we need a predictive distribution. This distribution can be obtained by performing the policy update with a Gaussian process regression and we directly see from the kernel ridge regression −1

σ 2 (s) = k(s, s) + λ − k(s)T (K + λC)

k(s),

where k(s, s) = φ(s) φ(s) is the distance of a point to itself. We call this algorithm Cost-regularized Kernel Regression. The algorithm corresponds to a Gaussian process regression where the costs on the diagonal are input-dependent noise priors. Gaussian processes have been used previously for reinforcement learning [22] in value function based approaches while here we use them to learn the policy. If several sets of meta-parameters have similarly low costs the algorithm’s convergence depends on the order of samples. The cost function should be designed to avoid this behavior and to favor a single set. The exploration has to be restricted to safe meta-parameters. T

D. Meta-Parameter Learning by Reinforcement Learning As a result of Section II-C, we have a framework of motor primitives as introduced in Section II-A that we can use for reinforcement learning of meta-parameters as outlined in Section II-B. We have generalized the reward-weighted regression policy update to instead become a Cost-regularized Kernel Regression (CrKR) update where the predictive variance is used for exploration. In Algorithm 1, we show the complete algorithm resulting from these steps. The algorithm receives three inputs, i.e., (i) a motor primitive that has associated meta-parameters γ, (ii) an initial example containing state s0 , meta-parameter γ 0 and cost C 0 , as well as (iii) a scaling parameter λ. The initial motor primitive can be obtained by imitation learning [1] and, subsequently, improved by parametrized reinforcement learning algorithms such as policy gradients [3] or Policy learning by Weighting Exploration with the Returns (PoWER) [6]. The demonstration also yields the initial example needed for meta-parameter learning. While the scaling parameter is an open parameter, it is reasonable to choose it as a fraction of the average cost and the output noise parameter (note that output noise and other possible hyper-parameters of the kernel can also be obtained by approximating the unweighted meta-parameter function). Illustration of the Algorithm: In order to illustrate this algorithm, we will use the example of the 2D dart throwing task introduced in Section II-A. Here, the robot should throw darts accurately while not destroying its mechanics. Hence, the cost corresponds to the error between desired goal and the impact point, as well as the absolute velocity of the endeffector. The initial policy is based on a prior, illustrated in Figure 2(a), that has a variance for initial exploration (it often makes sense to start with a uniform prior). This variance is used to enforce exploration. To throw a dart, we sample the meta-parameters from the policy based on the current state4 . After the trial the cost is determined and, in conjunction with 4 In the dart setting, we could choose the next target and thus employ CrKR as an active learning approach by picking states with large variances. However, often the state is determined by the environment, e.g., the ball trajectory in the table tennis experiment (Section III-C) depends on the opponent.



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Figure 3: This figure shows the performance of the compared algorithms averaged over 10 complete learning runs. Costregularized Kernel Regression finds solutions with the same final performance two orders of magnitude faster than the finite difference gradient (FD) approach and twice as fast as the reward-weighted regression. At the beginning FD often is highly unstable due to our attempts of keeping the overall learning speed as high as possible to make it a stronger competitor. The lines show the median and error bars indicate standard deviation. The initialization and the initial costs are identical for all approaches. However, the omission of the first twenty rollouts was necessary to cope with the logarithmic rollout axis. the employed meta-parameters, used to update the policy5 . If the cost is large (for example the impact is far from the target), the variance of the policy is large as it may still be improved and therefore needs exploration. Furthermore, the mean of the policy is shifted only slightly towards the observed example as we are uncertain about the optimality of this action. If the cost is small, we know that we are close to an optimal policy and only have to search in a small region around the observed trial. The effects of the cost on the mean and the variance are illustrated in Figure 2(b). Each additional sample refines the policy and the overall performance improves (see Figure 2(c)). If a state is visited several times and different meta-parameters are sampled, the policy update must favor the meta-parameters with lower costs. Algorithm 1 exhibits this behavior as illustrated in Figure 2(d). III. E VALUATION In Section II, we have introduced both a framework for meta-parameter self-improvement as well as an appropriate reinforcement learning algorithm used in this framework. In this section, we will first show that the presented reinforcement learning algorithm yields higher performance than off-the shelf approaches. Hence, we compare it on a simple planar cannon shooting problem [23] with the preceding reward-weighted regression and an off-the-shelf finite difference policy gradient approach. 5 In the dart throwing example we have a correspondence between the state and the outcome similar to a regression problem. However, the mapping between the state and the meta-parameter is not unique. The same height can be achieved by different combinations of velocities and angles. Averaging these combinations is likely to generate inconsistent solutions. The regression must hence favor the meta-parameters with the lower costs. CrKR can be employed as a regularized regression method in this setting. The proposed reinforcement learning method only requires a cost associated with the outcome of the trial. In the table tennis experiment (Section III-C), the state corresponds to the position and velocity of the ball over the net. We only observe the cost related to how well we hit the ball. After a table tennis trial, we do not know which state would have matched the employed meta-parameters, as would be required in a regression setting.

The resulting meta-parameter learning framework can be used in a variety of settings in robotics. We consider two scenarios here, i.e., (i) dart throwing with a simulated robot arm, a real Barrett WAM and the JST-ICORP/SARCOS humanoid robot CBi, and (ii) table tennis with a simulated robot arm and a real Barrett WAM. Some of the real-robot experiments are still partially work in progress. A. Benchmark Comparison In the first task, we only consider a simple simulated planar cannon shooting where we benchmark our Reinforcement Learning by Cost-regularized Kernel Regression approach against a finite difference gradient estimator and the rewardweighted regression. Here, we want to learn an optimal policy for a 2D toy cannon environment similar to [23]. The setup is given as follows: A toy cannon is at a fixed location [0.0, 0.1] m. The meta-parameters are the angle with respect to the ground and the speed of the cannon ball. In this benchmark we do not employ the motor primitives but set the parameters directly. The flight of the canon ball is simulated as ballistic flight of a point mass with Stokes’s drag as wind model. The cannon ball is supposed to hit the ground at a desired distance. The desired distance [1..3] m and the wind speed [0..1] m/s, which is always horizontal, are used as input parameters, the velocities in horizontal and vertical directions are predicted (which influences the angle and the speed of the ball leaving the cannon). Lower speed can be compensated by a larger angle. Thus, there are different possible policies for hitting a target; we intend to learn the one which is optimal for a given cost function. This cost function consists of the sum of the squared distance between the desired and the actual impact point and one hundredth of the squared norm of the velocity at impact of the cannon ball. It corresponds to maximizing the precision while minimizing the employed energy according to the chosen weighting. All approaches performed well in this setting, first driving the position error to zero and, subsequently, optimizing the impact

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(a) The dart is placed on the launcher.

(b) The arm moves back.

(c) The arm moves forward on an arc.

(d) The arm stops.

(e) The dart is carried on by its momentum.

(f) The dart hits the board.

Figure 4: This figure shows a dart throw in a physically realistic simulation.

(a) The dart is placed in the hand.

(b) The arm moves back.

(c) The arm moves forward on an arc.

(d) The arm continues moving.

(e) The dart is released and the arm follows through.

(f) The arm stops and the dart hits the board.

Figure 5: This figure shows a dart throw on the real JST-ICORP/SARCOS humanoid robot CBi. velocity. The experiment was initialized with [1, 10] m/s as initial ball velocities and 1 m/s as wind velocity. This setting corresponds to a very high parabola, which is far from optimal. For plots, we evaluate the policy on a test set of 25 uniformly randomly chosen points that remain the same throughout of the experiment and are never used in the learning process but only to generate Figure 3. We compare our novel algorithm to a finite difference policy gradient (FD) method [3] and to the reward-weighted regression (RWR) [11]. The FD method uses a parametric policy that employs radial basis functions in order to represent the policy and adds Gaussian exploration. The learning rate as well as the magnitude of the perturbations were tuned for best performance. We used 51 sets of uniformly perturbed parameters for each update step. The FD algorithm converges after approximately 2000 batch gradient evaluations, which corresponds to 2, 550, 000 shots with the toy cannon. The RWR method uses the same parametric policy as the finite difference gradient method. Exploration is achieved by adding Gaussian noise to the mean policy . All open parameters were tuned for best performance. The RWR algorithm converges after approximately 40, 000 shots with the toy cannon. For the Cost-regularized Kernel Regression (CrKR) the inputs are chosen randomly from a uniform distribution. We use Gaussian kernels and the open parameters were optimized by cross-validation on a small test set prior to the experiment. Each trial is added as a new training point if it landed in the desired distance range. The CrKR algorithm converges after approximately 20, 000 shots with the toy cannon. After convergence, the costs of CrKR are the same as for RWR and slightly lower than those of the FD method. The CrKR method needs two orders of magnitude fewer shots than the FD method. The RWR approach requires twice the shots of CrKR demonstrating that a non-parametric policy, as employed by CrKR, is better adapted to this class of problems than a parametric policy. The squared error between the actual and desired impact is approximately 5 times higher for the

finite difference gradient method, see Figure 3. B. Dart-Throwing Now, we turn towards the complete framework, i.e., we intend to learn the meta-parameters for motor primitives in discrete movements. We compare the Cost-regularized Kernel Regression (CrKR) algorithm to the reward-weighted regression (RWR). As a sufficiently complex scenario, we chose a robot dart throwing task inspired by [23]. However, we take a more complicated scenario and choose dart games such as Around the Clock [13] instead of simple throwing at a fixed location. Hence, it will have an additional parameter in the state depending on the location on the dartboard that should come next in the sequence. The acquisition of a basic motor primitive is achieved using previous work on imitation learning [1]. Only the meta-parameter function is learned using CrKR or RWR. The dart is placed on a launcher attached to the end-effector and held there by stiction. We use the Barrett WAM robot arm in order to achieve the high accelerations needed to overcome the stiction. See Figure 4, for a complete throwing movement. The motor primitive is trained by imitation learning with kinesthetic teach-in. We use the Cartesian coordinates with respect to the center of the dart board as inputs. The parameter for the final position, the duration of the motor primitive and the angle around the vertical axis are the meta-parameters. The popular dart game Around the Clock requires the player to hit the numbers in ascending order, then the bulls-eye. As energy is lost overcoming the stiction of the launching sled, the darts fly lower and we placed the dartboard lower than official rules require. The cost function is the sum of ten times the squared error on impact and the velocity of the motion. After approximately 1000 throws the algorithms have converged but CrKR yields a high performance already much earlier (see Figure 6). We again used a parametric policy with radial basis functions for RWR. Designing a good parametric policy proved very difficult in this setting as is reflected by the poor performance of RWR.

 38



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Figure 6: This figure shows the cost function of the dartthrowing task for a whole game Around the Clock in each rollout. The costs are averaged over 10 runs with the errorbars indicating standard deviation. This experiment is also being carried out on two real, physical robots, i.e., a Barrett WAM and the humanoid robot CBi (JST-ICORP/SARCOS). CBi was developed within the framework of the JST-ICORP Computational Brain Project at ATR Computational Neuroscience Labs. The hardware of the robot was developed by the American robotic development company SARCOS. CBi can open and close the fingers which helps for more human-like throwing instead of the launcher employed by the Barrett WAM. See Figure 5 for a throwing movement. Parts of these experiments are still in-progress.



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Figure 7: This figure shows the cost function of the table tennis task averaged over 10 runs with the error-bars indicating standard deviation. The red line represents the percentage of successful hits and the blue line the average cost. At the beginning the robot misses the ball 95% of the episodes and on average by 50 cm. At the end of the learning the robot hits almost all balls. to update the policy if the robot has successfully hit the ball. Figure 9 illustrates different positions of the ball the policy is capable of dealing with after the learning. Figure 7 illustrates the costs over all episodes. Preliminary results suggest that the resulting policy performs well both in simulation and for the real system. We are currently in the process of executing this experiment also on the real Barrett WAM. IV. C ONCLUSION & F UTURE W ORK

C. Table Tennis In the second evaluation of the complete framework, we use it for hitting a table tennis ball in the air. The setup consists of a ball gun that serves to the forehand of the robot, a Barrett WAM and a standard sized table. The movement of the robot has three phases. The robot is in a rest posture and starts to swing back when the ball is launched. During this swingback phase, the open parameters for the stroke are predicted. The second phase is the hitting phase which ends with the contact of the ball and racket. In the final phase the robot gradually ends the stroking motion and returns to the rest posture. See Figure 8 for an illustration of a complete episode. The movements in the three phases are represented by motor primitives obtained by imitation learning. The meta-parameters are the joint positions and velocities for all seven degrees of freedom at the end of the second phase (the instant of hitting the ball) and a timing parameter that controls when the swing back phase is transitioning to the hitting phase. We learn these 15 meta-parameters as a function of the ball positions and velocities when it is over the net. We employed a Gaussian kernel and optimized the open parameters according to typical values for the input and output. As cost function we employ the metric distance between the center of the paddle and the center of the ball at the hitting time. The policy is evaluated every 50 episodes with 25 ball launches picked randomly at the beginning of the learning. We initialize the behavior with five successful strokes observed from another player. After initializing the meta-parameter function with only these five initial examples, the robot misses ca. 95% of the balls as shown in Figure 7. Trials are only used



In this paper, we have studied the problem of metaparameter learning for motor primitives. It is an essential step towards applying motor primitives for learning complex motor skills in robotics more flexibly. We have discussed an appropriate reinforcement learning algorithm for mapping situations to meta-parameters. We show that the necessary mapping from situation to meta-parameter can be learned using a Cost-regularized Kernel Regression (CrKR) while the parameters of the motor primitive can still be acquired through traditional approaches. The predictive variance of CrKR is used for exploration in onpolicy meta-parameter reinforcement learning. We compare the resulting algorithm in a toy scenario to a policy gradient algorithm with a well-tuned policy representation and the reward-weighted regression. We show that our CrKR algorithm can significantly outperform these preceding methods. To demonstrate the system in a complex scenario, we have chosen the Around the Clock dart throwing game and table tennis implemented both on simulated and real robots. Adapting movements to situations is also discussed in [16] in a supervised learning setting. Their approach is based on predicting a trajectory from a previously demonstrated set and refining it by motion planning. The authors note that kernel ridge regression performed poorly for the prediction if the new situation is far from previously seen ones as the algorithm yields the global mean. In our approach we employ a cost weighted mean that overcomes this problem. If the situation is far from previously seen ones, large exploration will help to find a solution.

 39

(a) The robot is the rest posture.

(b) The arm swings back.

(c) The arm strikes the ball.

(d) The arm follows through and decelerates.

(e) The arm returns to the rest posture.

Figure 8: This figure shows a table tennis stroke on the real Barrett WAM.

(a) Left.

(b) Half left.

(c) Center high.

(d) Center low.

(e) Right.

Figure 9: This figure shows samples of the learned forehands. Note that this figure only illustrates the learned meta-parameter function in this context but cannot show timing and velocity and it requires a careful observer to note the important configuration differences resulting from the meta-parameters. Future work will require to sequence different motor primitives by a supervisory layer. This supervisory layer would for example in a table tennis task decide between a forehand motor primitive and a backhand motor primitive, the spatial meta-parameter and the timing of the motor primitive would be adapted according to the incoming ball, and the motor primitive would generate the trajectory. This supervisory layer could be learned by an hierarchical reinforcement learning approach [24] (as introduced in the early work by [25]). In this framework, the motor primitives with meta-parameter functions could be seen as robotics counterpart of options [9] or macro-actions [26]. R EFERENCES [1] A. J. Ijspeert, J. Nakanishi, and S. Schaal, “Learning attractor landscapes for learning motor primitives,” in Advances in Neural Information Processing Systems 16, 2003. [2] S. Schaal, P. Mohajerian, and A. J. Ijspeert, “Dynamics systems vs. optimal control — a unifying view,” Progress in Brain Research, vol. 165, no. 1, pp. 425–445, 2007. [3] J. Peters and S. Schaal, “Policy gradient methods for robotics,” in Proc. Int. Conf. Intelligent Robots and Systems, 2006. [4] D. Pongas, A. Billard, and S. Schaal, “Rapid synchronization and accurate phase-locking of rhythmic motor primitives,” in Proc. Int. Conf. Intelligent Robots and Systems, 2005. [5] J. Nakanishi, J. Morimoto, G. Endo, G. Cheng, S. Schaal, and M. Kawato, “Learning from demonstration and adaptation of biped locomotion,” Robotics and Autonomous Systems, vol. 47, no. 2-3, pp. 79–91, 2004. [6] J. Kober and J. Peters, “Policy search for motor primitives in robotics,” in Advances in Neural Information Processing Systems 22, 2009. [7] H. Urbanek, A. Albu-Schäffer, and P. van der Smagt, “Learning from demonstration repetitive movements for autonomous service robotics,” in Proc. Int. Conf. Intelligent Robots and Systems, 2004. [8] R. Caruana, “Multitask learning,” Machine Learning, vol. 28, pp. 41–75, 1997. [9] A. McGovern and A. G. Barto, “Automatic discovery of subgoals in reinforcement learning using diverse density,” in Proc. Int. Conf. Machine Learning, 2001.

[10] K. Mülling, “Motor control and learning in table tennis,” Master’s thesis, University of Tübingen, 2009. [11] J. Peters and S. Schaal, “Reinforcement learning by reward-weighted regression for operational space control,” in Proc. Int. Conf. Machine Learning, 2007. [12] C. E. Rasmussen and C. K. Williams, Gaussian Processes for Machine Learning. MIT Press, 2006. [13] Masters Games Ltd., “The rules of darts,” online http://www.mastersgames.com/rules/darts-rules.htm, July 2010. [14] G. Wulf, Attention and motor skill learning. Champaign, IL: Human Kinetics, 2007. [15] D.-H. Park, H. Hoffmann, P. Pastor, and S. Schaal, “Movement reproduction and obstacle avoidance with dynamic movement primitives and potential fields,” in Proc. Int. Conf. Humanoid Robots, 2008. [16] N. Jetchev and M. Toussaint, “Trajectory prediction: learning to map situations to robot trajectories,” in Proc. Int. Conf. Machine Learning, 2009. [17] D. B. Grimes and R. P. N. Rao, “Learning nonparametric policies by imitation,” in Proc. Int. Conf. Intelligent Robots and System, 2008. [18] D. C. Bentivegna, A. Ude, C. G. Atkeson, and G. Cheng, “Learning to act from observation and practice,” Int. Journal of Humanoid Robotics, vol. 1, no. 4, pp. 585–611, 2004. [19] P. Dayan and G. E. Hinton, “Using expectation-maximization for reinforcement learning,” Neural Computation, vol. 9, no. 2, pp. 271–278, 1997. [20] R. J. Williams, “Simple statistical gradient-following algorithms for connectionist reinforcement learning,” Machine Learning, vol. 8, pp. 229– 256, 1992. [21] C. M. Bishop, Pattern Recognition and Machine Learning. Springer Verlag, 2006. [22] Y. Engel, S. Mannor, and R. Meir, “Reinforcement learning with gaussian processes,” in Proc. Int. Conf. Machine Learning, 2005. [23] G. Lawrence, N. Cowan, and S. Russell, “Efficient gradient estimation for motor control learning,” in Proc. Int. Conf. Uncertainty in Artificial Intelligence, 2003. [24] A. Barto and S. Mahadevan, “Recent advances in hierarchical reinforcement learning,” Discrete Event Dynamic Systems, vol. 13, no. 4, pp. 341 – 379, 2003. [25] M. Huber and R. Grupen, “Learning robot control using control policies as abstract actions,” in NIPS’98 Workshop: Abstraction and Hierarchy in Reinforcement Learning, 1998. [26] A. McGovern, R. S. Sutton, and A. H. Fagg, “Roles of macro-actions in accelerating reinforcement learning,” in Grace Hopper Celebration of Women in Computing, 1997.

 40

Analysis and Control of a Dissipative Spring-Mass Hopper with Torque Actuation M. Mert Ankaralı

Uluc¸ Saranlı

Dept. of Electrical and Electronics Eng., Middle East Technical University, 06531 Ankara, Turkey

Dept. of Computer Engineering, Bilkent University, 06800 Ankara, Turkey

[email protected]

[email protected]

Abstract— It has long been established that simple springmass models can accurately represent the dynamics of legged locomotion. Existing work in this domain, however, almost exclusively focuses on the idealized Spring-Loaded Inverted Pendulum (SLIP) model and neglects passive dissipative effects unavoidable in any physical robot or animal. In this paper, we extend on a recently proposed analytic approximation to the stance trajectories of a dissipative SLIP model to analyze stability properties of a planar hopper with a single rotary actuator at the hip. We first describe how a suitably chosen torque controller can compensate for damping losses, maintaining the same energy level across strides and hence reducing the return map to a single dimension. We then identify and characterize equilibrium points for this return map under a fixed leg placement policy and show that “uncontrolled” asymptotic stability is feasible for this energy-regulated system. Subsequent presentation of simulation evidence establishes that the predictions of this approximate model are consistent with the exact plant model. The paper concludes with the application of our energy-regulation scheme to the design of a task-level gait controller that uses explicit leg placement commands in conjunction with the hip torque.

I. I NTRODUCTION Long term practical utility of mobile robots in unstructured environments critically depends on their locomotory aptitude. In this context, the performance of ground mobility that can ultimately be achieved by legged platforms is superior to any other alternative as evidenced by numerous examples in nature as well as a number of very successful dynamically stable autonomous legged robots that have been built to date [10, 25, 26, 30, 37]. Unfortunately, even on flat ground, legged morphologies do not enjoy the simplicity of models supported by the conveniently constrained and continuous modes of ground interaction observed in wheeled and, to some extent, tracked vehicles. Even the most basic legged behaviors such as walking and running require hybrid dynamic models whose analysis and control involve difficult challenges [14, 20, 23]. In the world of quasi-static locomotion with multi-legged robots, one can recover some of this simplicity through active or structural suppression of second order dynamics [39], but these methods are not directly applicable to dynamically dexterous modes of locomotion such as running. One of the most significant discoveries in this context was most likely the recognition of similar center of mass (COM) movement patterns in running animals of widely different sizes

and morphologies [1, 6, 7, 9, 24]. This led to the development of the simple yet accurate Spring-Loaded Inverted Pendulum (SLIP) model to describe such behaviors [21, 34]. Significant research effort was devoted to both the use of this model as a basis for the design of fast and efficient legged robots [10, 19, 27, 30] as well as its analysis to reveal fundamental aspects of associated locomotory behaviors [20]. The present paper falls into the latter category and contributes by investigating the previously unaddressed question of how the presence of passive damping affects the behavioral characteristics of running with the SLIP model. Our treatment of this question is based on the use of analytic approximations to the otherwise non-integrable stance dynamics of the model. A number of such approximations have already been proposed in the literature. In particular, [35] uses a Hamiltonian formulation of the SLIP dynamics with an iterative application of the mean-value theorem to obtain an accurate return map for symmetric steps. More recently, [16] presents an analytically simple approximation to the stance dynamics of a conservative SLIP equipped with a linear spring. This approach is based on a linearization of gravity around mid-stance, similar in form to the solution proposed in [28] but involving a much more carefully formulated derivation. Inaccuracies of these approximations in the presence of nonsymmetric gravitational effects were partially addressed in [5] using explicit corrections. In contrast to the lossless models adopted in all these approaches, a new return map for the lossy SLIP model with viscous damping in the leg was proposed in [4], providing a basis for our present inquiry. Despite the availability of methods to analyze stability properties of locomotory behaviors in the absence of closed-form expressions for a Poincar´e map [2, 3], a number of different possible approaches become available once a sufficiently accurate analytic return map is available. For example, [18] investigates in depth stability properties of a SLIP model attached to a rigid body by neglecting the effects of gravity, which allows for the derivation of suitable closed-form expressions for stride trajectories. A similar but less rigorous stability analysis was provided in [16] with comparisons to previous numerical results in [36] as well as biological data. In contrast to the lack of feedback control in these characterizations, [33] studies the stability of the SLIP model under a novel leg placement

 41

body reaches its maximum height during flight with y˙ = 0. Another important event, not illustrated in the figure, is bottom, corresponding to the point of maximal leg compression during stance. Table I details the notation used throughout the paper.

descent

Fig. 2.

k y

c

θ

Fig. 1 illustrates the Torque-actuated Dissipative SpringLoaded Inverted Pendulum (TD-SLIP) plant we investigate in this paper. It consists of a fixed orientation (2-DOF) planar rigid body with mass m, connected to a massless, fully passive leg with linear compliance k, rest length r0 and linear viscous damping c, through an actuated rotary joint with torque τ . The system alternates between stance and flight phases during running, with the flight phase further divided into the ascent and descent subphases. Fig. 2 illustrates the three important events that define transitions between these phases: touchdown, where the leg comes into contact with the ground, liftoff, where the toe takes off from the ground and finally apex, where the

apex

liftoff ascent

A single TD-SLIP stride with definitions of transition states TABLE I

System States, Event States and Control Inputs x, y, x, ˙ y˙ Cartesian body position and velocities r, θ, r, ˙ θ˙ Leg length, leg angle and velocities τ Hip torque command during stance ya , x˙ a Apex height and velocity θtd , r˙td , θ˙td Touchdown leg angle, polar velocities tb , rb , θb Bottom time, leg length and angle tlo , rlo , θlo , r˙lo , θ˙lo Liftoff time, leg length, angle and velocities pθ Angular momentum around the toe Kinematic and Dynamic Parameters m, g Body mass and gravitational acceleration k, r0 , c Leg stiffness, rest length and damping

During flight, the body obeys ballistic flight dynamics



x ¨ 0 = y¨ −g and the massless leg can be arbitrarily positioned. In contrast, during stance, the toe remains stationary on the ground while the body mass feels forces generated by both the passive spring-damper pair and the hip torque. The stance dynamics of the planar SLIP model in polar leg coordinates with respect to the toe location take the form d dt

x Fig. 1. TD-SLIP : Dissipative spring-mass hopper with rotary hip actuation

stance

N OTATION USED THROUGHOUT THE PAPER

A. System Dynamics and the Apex Return Map

r

[rlo , θlo , r˙lo , θ˙lo ]

touchdown

[θtd , r˙td , θ˙td ]

II. T HE T ORQUE -ACTUATED D ISSIPATIVE SLIP M ODEL

τ

[ya , x˙ a ]k+1

[ya , x˙ a ]k

apex

control strategy that only relies on easily obtainable state measurements. Another inquiry on how to achieve effective control of the SLIP model, now extended to a spatial setting, is provided in [8], focusing on lateral motions of the model. In conjunction with these studies primarily focused on running behaviors, similar analytically motivated contributions were also made to the structurally different walking behaviors [15, 22, 40], providing evidence that the same dynamic model can provide a unified description for both walking and running [17]. Our contributions in the present paper have a number of important differences from existing work. Firstly, our plant model is dissipative, impairing the accuracy of most existing analytic approximations and associated predictions. Secondly, in contrast to the usual energy regulation mechanisms in the literature through adjustments of the leg length or changing stiffness, our model uses only a single torque actuator at the hip relative to a virtual body with fixed orientation to compensate for energy losses. These changes are motivated by being much more realistic from an implementation point of view, as evidenced by the successful use of similar actuation mechanisms in the Scout quadrupeds [26] and the RHex hexapod [30] as well as a number of other monopedal platforms [12, 32]. Finally, our approximate solutions to the return map also take into account the effect of gravity on the angular momentum for steps that are non-symmetric with respect to the gravitational vertical, increasing the practical applicability of associated stability results.



mr˙ mr 2 θ˙





=

mrθ˙2 − mg cos θ − k(r − r0 ) − cr˙ mgr sin θ + τ



. (1)

A very useful abstraction for the analysis and control of cyclic TD-SLIP trajectories is provided by the apex return map, defined as a Poincar´e map from one apex point to the next. In the following sections, we will use this map to study stability properties of TD-SLIP, and later adopt it as a tasklevel gait representation for a closed-loop running controller. The apex return map can be formulated as P := Pa ◦Ps ◦Pd by composing three individual submaps Pd , Ps , Pa for the descent, stance and ascent phases, respectively. The descent and ascent maps are trivial and are given by    −x˙ a r˙td  (2) ) = R(π/2 − θ td r0 θ˙td 2g(ya − r0 cos θtd )     2 ya rlo cos θlo + y˙ lo /(2g) = (3) Pa : x˙ a x˙ lo 

Pd :

 42

where x˙ lo and y˙ lo are liftoff velocities in Cartesian coordinates and R denotes the standard 2D rotation matrix. Unfortunately, the dynamics of (1) are not integrable in closed form. Consequently, we will use an analytical approximation for the stance map, which we describe in the next section.

The final step in completing the stance map requires finding the time of liftoff. Only one of the two liftoff conditions described in [4] is applicable in the context of the present paper since we do not allow control of the liftoff leg length. Consequently, the liftoff time is solely determined by the ˙ lo ) = 0, for solution to the equation k(r0 − r(tlo )) − c r(t which a sufficiently accurate analytical approximation can be found by approximating the exponential coefficient in the radial solution of (6) by its value at a specific instant during decompression. In particular, noting that the compression and decompression times are roughly equal, we use e−ζ ωˆ 0 t ≈ e−ζ ωˆ 0 2tb , where tb denotes the bottom time, easily found by solving (7). Under this assumption, we have

B. An Approximate Stance Map for the Unforced TD-SLIP A new analytical approximation to the dynamics of a dissipative SLIP model was proposed in [4]. However, this method assumes the presence of radial leg actuation, either in the form of a controllable leg stiffness, or the regulation of touchdown and liftoff leg lengths. In this section, we briefly review their method and extend it to support the hip torque actuation of our model. The approximation proposed in [4], which, in turn, is based on the methods described in [16], relies on two key assumptions: 1). The angular travel throughout stance is relatively small and remains close to the vertical, allowing linearization of the gravitational potential in the Lagrangian with subsequent conservation of the angular momentum pθ := mr2 θ˙ and 2). the radial compression is small with r0 − r r0 , allowing a truncated Taylor expansion of related terms. As described in [4], under these conditions and assuming, for now, that τ = 0, the radial component of (1) reduces to

tlo

−

ω02 , (5) e−ζ ωˆ 0 t (A cos(ωd t) + B sin(ωd t)) + F/ˆ  where ˆ 0 := ω02 + 3ω 2 , ζ := c/(2mˆ ω0 ), ωd :=  we have ω 2 2 2 ω ˆ 0 1 − ζ , F := −g+r0 ω0 +4r0 ω and A and B determined by touchdown states as

=

ω02 , := r0 − F/ˆ := (r˙td + ζ ω ˆ 0 A)/ωd .

r(t) ˙

=

M e−ζ ωˆ 0 t cos(ωd t + φ) + F/ˆ ω02 , −M ω ˆ 0 e−ζ ωˆ 0 t cos(ωd t + φ + φ2 ) ,

(6) (7)

with M , φ and φ2 determined through trigonometric identities. At this point, the angular trajectories can be determined using the constant angular momentum. An additional linearization of the term 1/r2 leads to an analytical solution for the rate of change of the leg angle as ˙ θ(t)

ˆ 02 ) − = 3ω − 2ωF/(r0 ω −ζ ω ˆ0t 2ωM e cos(ωd t + φ)/r0 ,

(8)

integrated to yield the angular trajectory θ(t)

=

(9) cos(ωd t + φ + φ3 ) − cos(φ + φ3 )).

θtd + X t + −ζ ω ˆ0 t

Y (e

with X, Y and φ3 computed accordingly as in [4].

(11)

C. Stance Map for the Torque Controlled TD-SLIP

Simple differentiation and further simplification yields radial TD-SLIP trajectories as r(t)

⎤ r(tlo ) θ(tlo ) ⎥ ⎥ . r(t ˙ lo ) ⎦ ˙ lo ) θ(t

where the right hand side is a function of touchdown states. Note, however, that these derivations completely ignore the presence of the hip torque. In the next section, we propose a new method to incorporate the effects of the hip torque through a fixed correction on the angular momentum value pθ in a way similar to the one used in [5] for gravity corrections.

=

A B

(2π − arccos(k(r0 − F/ˆ ω02 )/(M M e−ζ ωˆ 0 γtb )) (10) φ − φ4 )/ωd ,

which yields the stance map as ⎡ ⎤ ⎡ rlo ⎢ θlo ⎥ ⎢ ⎥ ⎢ Ps : ⎢ ⎣ r˙lo ⎦ = ⎣ θ˙lo

r¨ + (c/m)r˙ + (ω02 + 3ω 2 )r = −g + r0 ω02 + 4r0 ω 2 , (4)  where we define ω0 := k/m and ω := pθ /(mr02 ). Solutions to this simple second-order ODE can be found as r(t)

≈

Hip actuation in legged systems can serve a number of different purposes. Among both biological [1] and robotic [13, 19, 27] systems, its most common uses involve retraction of legs in flight and control of body posture with legs in stance. Interestingly, the use of hip actuation to provide thrust has not been studied as extensively in the robotics literature. In addition to a few direct experimental inquiries [12, 32] and indirect uses in multi-legged platforms [26, 30], it has received limited attention in [2] in the form of an active spring. In the present paper, we propose an open-loop hip actuation regime that enforces the ramp torque profile  τ0 (1 − ttf ) if 0 ≤ t ≤ tf (12) τ (t) = 0 if t > tf during stance, with τ0 and tf chosen prior to touchdown. This open-loop profile has three important advantages. Firstly, its simple functional dependence on time allows us to easily incorporate its effects into the derivations of the previous section. Second, if we choose tf to be the liftoff time, we have τ (tlo ) = 0, which prevents premature leg liftoff due to the action of the hip and ensures a structural match to the trajectories of the unforced system. Finally, its unidirectional action ensures that no negative work is done during stance. Inspection of the TD-SLIP dynamics of (1) shows that the hip torque directly acts on the angular dynamics and only indirectly effects radial motion. Consequently, we hypothesize

 43

that an average correction to the constant angular momentum pθ of Section II-B can capture the effects of the hip torque on system trajectories. Normally, the angular momentum during stance can be formulated as  t  t τ (η)dη + mgr(η) sin θ(η)dη, (13) pθ (t) = pθ (0) + 0

0

by integrating the angular dynamics. Adopting the method proposed in [5], we compute a corrected angular momentum pˆθ = pθ (0) + Δpτ + Δpg .

(14)

where Δpτ and Δpg incorporate the time averaged effects of the leg torque and gravitational acceleration, respectively. Assuming tf = tlo , we have   tlo  η1 tlo 1 Δpτ := . (15) τ (η2 )dη2 dη1 = τ0 tlo 0 3 0 However, even with available analytic approximations, derivation of an exact closed-form expression for Δpg is not feasible. Instead, we use a linear approximation to the integrand r(η) sin θ(η) using its values at the touchdown and liftoff, resulting in mgtlo (16) (2r0 sin θtd + rlo sin θlo ) . 6 Estimated values for the liftoff time tlo , leg angle θlo and leg length rlo are provided by the unforced approximations of the previous section. Substituting pˆθ for the constant angular momentum in all derivations of Section II-B, we obtain a new approximation that takes into account the effects of both the hip torque and gravity on the stance trajectories. Note that the corrections we propose have an iterative character since both (15) and (16) use prior estimates of tlo and θlo . Consequently, starting from the unforced approximations, it is possible to iteratively apply these corrections to obtain more accurate predictions at the expense of analytic simplicity. Our simulations show that more than a single iteration is only needed for extreme conditions such as the angle of attack being very close to the touchdown leg angle, causing a bounce-back. Δpg :=

and keep it constant across subsequent strides. The underlying idea is that since damping losses are proportional to the total energy level of the system, constant energy input will give rise to trajectories at a stable energy level. Unfortunately, in our 2-DOF model, such a strategy results in a two dimensional return map, for which, analytical solution and characterization of equilibrium points is not feasible. Another possibility, which we adopt in the present paper, is to use the hip torque to compensate for all dissipative effects within a single step, ensuring conservation of energy in the apex return map and hence reducing its dimension by one. Note that the total energy dissipated within a single TD-SLIP step is given by (17) Eloss = Ec + Ek , where Ec represents damping losses with  tlo cr˙ 2 (η) dη , Ec :=

and Ek := (rlo − r0 )2 /2 captures the leftover energy in the leg spring when it lifts off before it is fully extended due to damping. Fortunately, our analytic approximations provide closed form expressions for both of these components. In particular, damping losses can be approximately computed as Ec

Unlike previous stability studies of lossless spring-mass hoppers, fully passive self-stabilization with a fixed touchdown angle and no active control is not possible with the TDSLIP model since damping losses will eventually drain out all energy in the system. Consequently, active hip thrust must be employed to sustain locomotion. Recall that our choice of the hip torque in (12) incorporates two parameters: τ0 and tf . We have already shown that choosing tf = tlo is advantageous in preventing early liftoff and ensuring structural correspondence of system trajectories to our analytical approximation. The simplest possible strategy for the remaining parameter τ0 , very close in spirit to the radial actuation strategy adopted by Raibert’s runners [27] and its subsequent analysis in [21], is to choose a particular value

= −

−c/M 2 ω ˆ0 (ζ cos(2(φ + φ2 ) + φ3 ) + 1 (19) 4ζ e−2ζ ωˆ 0 tlo (ζ cos(2ωd tlo + 2(φ + φ2 ) + φ3 ) + 1)) ,

while Ek only depends on the previously computed rlo . In contrast, the energy supplied by the hip torque is  tlo t ˙ (1 − ) θ(t)dt , (20) Eτ = τ0 tlo 0 for which our analytical approximations can also be used to obtain closed-form expressions. We omit the details here for space considerations. Since both (17) and (20) can be obtained in closed form as a function of initial conditions and the choice of touchdown angle θtd , we can easily find the desired torque magnitude τ0 by solving

III. S TABILITY OF AN E NERGY-R EGULATED TD-SLIP A. Compensation of Damping Losses

(18)

0

Eτ

=

Eloss .

(21)

As noted above, this choice of torque results in successive apex states having the same energy, at least while working within our approximate apex return map. Naturally, additional corrections would be needed to apply these ideas to the exact plant model since inaccuracies of our approximations would invalidate this conservation. Nevertheless, we use this active compensation regime to reduce the dimension of our analytic apex return map, allowing us to easily identify its equilibrium points and characterize their stability. B. Equilibrium Points with a Fixed Leg Placement Policy In this section, we use our analytic approximations to identify and characterize equilibrium points of the one dimensional “energy-regulated” return map on the apex height ya arising from the use of a fixed touchdown angle policy with

 44

2

θtd = β and the energy-regulating hip torque described in Section III-A. All results in this section will be presented in non-dimensional versions of relevant variables, defined as

Mean-Square % Error in E¯a

6 5

ζ0

y¯a

4

y¯a ¯ x˙ a ¯a E k¯

:= ya /r0 √ := x˙ a / gr0

3

:= Ea /(mgr0 )

2

:= kr0 /(mg) √ := c/(2 mk) .

1 4

Finally, in order to facilitate comparison with earlier studies, we use kinematic and dynamic parameters that roughly match those of an average human with m = 80kg and r0 = 1m. Fig. 3 shows two families of return maps for β = 20◦ and β = 32◦ , respectively, together with the dependence of equilibrium points on the energy level of the system. These results show that the TD-SLIP exhibits asymptotically stable behavior under the fixed touchdown angle, energy-regulated regime, with the location of the equilibrium point depending on the chosen energy level. We can also observe that as the fixed touchdown angle β increases, the energy range for which stable fixed points exist increases as well. This is rather natural since the torque actuation at the hip can only supply energy through the angular momentum, which directly increases the angular span during stance. Increasing the touchdown angle admits a larger angular span for stance, allowing stable fixed points to form at higher energy levels as well. 2 1.8

y¯a

1.6 1.4 1.2 1 1

1.2

1.4

1

1.5

2

1.6

1.8

2.5

3

y¯a

2

1

2

3

E¯a

4

3.5 3

y¯a

2.5 2 1.5 1

y¯a

3.5

5

10

E¯a

15

Fig. 3. Apex height return map (left) and associated equilibrium points (right) for the TD-SLIP model as a function of different (dimensionless) energy levels, generated with the proposed analytical approximations. The top ¯ = 40, ζ0 = 0.07, whereas the bottom plots are obtained with β = 20◦ , k ¯ = 40, ζ0 = 0.07 in dimensionless coordinates. Solid plots use β = 32◦ , k and dashed lines in the right figure indicate stable and unstable equilibrium points respectively. Shaded regions correspond to kinematically infeasible configurations.

Having established the presence of stable equilibrium points for the torque-controlled SLIP model, Fig. 4 shows a comparison of fixed points predicted by our analytic approximations,

6

E¯a

8

10

1.5

1

0.5

0

4

6

E¯a

8

10

Fig. 4. Left: Comparison of stable equilibrium points predicted by our analytic approximation (solid line) with those obtained by numeric simulation of TD-SLIP dynamics (plus signs) for β = 28◦ and different apex energy ¯a ∈ [2, 10]. Shaded region in the middle illustrates the levels in the range E stable domain of attraction for the simulated plant model. Right: Percentage mean-square error between initial and steady-state dimensionless energy levels for the simulated plant.

with those that arise within simulations of the exact TDSLIP model. In order to make direct comparisons possible, we started TD-SLIP simulations from a large range of initial ya and Ea values, with a fixed touchdown angle and an energy regulation controller similar to the one presented Section IIIA, but now taking the energy level of the very first step as an overall regulation goal. This modification was necessary since using the approximations to locally enforce energy conservation at every step would slowly cause prediction errors to accumulate, either draining all energy out of the system, or causing it to diverge. We then checked whether the system converges to a stable equilibrium point in apex coordinates after 25 steps up to a tolerance of 10−4 . Shaded region in the middle of the left plot of Fig. 4 illustrates the resulting domain of attraction, while the plus marks in the same plot illustrate the associated set of fixed points. Note, also, that the domain of attraction exhibited by the simulation almost exactly covers the region between the unstable and stable fixed points predicted by our approximations. There is also an almost exact match between the fixed points predicted by our approximations and those obtained from simulation. The cavities to the right of the region of attraction arise from the presence of the “gap” region in the return map, resulting from kinematic constraints that require the apex height to be sufficiently large to allow leg placement. The reason for this can be clearly seen in the bottom right plot of Fig. 3, where parts of the return map overlap with the kinematically infeasible gray region on the bottom. This means that some initial conditions at high energy levels will lead to apex states for which leg placement at an angle of β is impossible. This gap was also observed by previous studies [16], and is reproduced by both our analytical approximations, and the simulated plant. The right plot in Fig. 4 shows the mean and standard deviations of the percentage mean-square energy difference between the initial and steady state apex points for the simulated plant. The fact that this difference is consistently below 0.3% shows that our approximations are capable of very accurately modeling energy losses and successfully predict fixed points of

 45

the exact TD-SLIP plant. It is worth noting, also, that accuracy also increases significantly with increasing energy levels.

Center of Mass Trajectory

C. Parameter Dependence of Equilibrium Points Equilibrium points that arise from our fixed touchdown angle, energy-regulated regime naturally depend on the kinematic and dynamic parameter choices. Fig. 5 illustrates the dependence of stable fixed points on each individual parameter (the touchdown angle β, the dimensionless leg stiffness k¯ or leg damping ζ0 ) with the remaining two parameters kept constant. The leftmost figure mirrors our observations in the previous section, namely that the range of stable energy levels increase with larger touchdown angles. β = 28◦ , ζ0 = 0.07

k¯ = 40, ζ0 = 0.07 2

2

1.8

1.8

1.6

1.6

1.6

1.4

1.4

y¯a

2

β

1.2

1 0.8

1.2

1

5

10

E¯a

15

20

0.8

introduced in the beginning of the stance phase, resulting in associated virtual footfalls appearing behind the actual toe location. Towards the end of the stance phase, the hip torque approaches zero and brings the virtual footfall and actual toe locations together. This qualitative structure is observed for all steady-state trajectories of the TD-SLIP model and is remarkably consistent with biological data presented in [38]. Even though we do not yet have any quantitative basis in which any predictive claims can be made, we think that this correspondence may provide evidence towards the use of hip torque as an additional source of energy used by biological runners, improving the predictive accuracy and utility of dynamic models of running.

1.4

k¯

ζ0

1

5

10

E¯a

15

20

0.8

5

10

E¯a

15

Leg Force Directions

Fig. 6. COM trajectory for a single steady-state stride of the TD-SLIP ¯ = 40, ζ0 = 0.07, running at approximately 3m/s (0.96 in model with k dimensionless units). Comparison of ground reaction force directions during stance to biological data presented in [38] reveals a remarkable qualitative match.

β = 28◦, k¯ = 40

1.8

1.2

Leg length

20

Fig. 5. Dependence of stable equilibrium points on variations of the ¯ (middle) and leg damping ratio touchdown angle β (left), leg spring stiffness k ζ0 (right). Arrows indicate increasing directions for each varied parameter.

The dependence of equilibrium points on the leg stiffness, illustrated in the middle figure shows that increasing spring constants cause an increase in the range of stable energy levels. This is also natural since an increased stiffness corresponds to shorter stance times, resulting in decreased damping losses and a corresponding decrease in the necessary torque input. Finally, we observe that the impact of the damping coefficients on the equilibrium points is not as pronounced, providing evidence that our compensation strategy successfully balances damping losses. Nevertheless, increasing the amount of damping causes a slight decrease in the range of stable energy levels.

IV. F EEDBACK C ONTROL OF TD-SLIP RUNNING A. Deadbeat Control by Inversion of the Apex Return Map

D. Correspondence of the Model to Biological Data A recent quantitative comparison of ground reaction force data from a variety of running animals to those predicted by a simple, passive spring-mass model shows that despite the very good correspondence of vertical force components between biological data and the idealized SLIP model, there are some discrepancies in how well horizontal forces can be predicted [38]. In this section, we report on an interesting property of the torque-actuated TD-SLIP morphology: It seems to be capable of qualitatively reproducing ground reaction force profiles very similar to those observed in biological systems. Fig. 6 illustrates the body trajectory for a single stride of steady-state running with the TD-SLIP model, together with a depiction of “virtual footfalls” in the direction of instantaneous ground reaction force vectors throughout the stance phase. As a result of the ramp torque profile we use for supplying energy to the system, large backward horizontal forces are

The presence of a sufficiently accurate analytic formulation of the apex return map naturally motivates its inversion to obtain a controller for stabilizing the system around a desired operating point [ya∗ , x˙ ∗a ] in apex state coordinates. A similar approach was adopted in a number of studies [8, 29, 31], but never in the context of a lossy model or torque actuation. In this section, we describe a deadbeat gait controller for TD-SLIP as an application of our approximations, and show that it is capable of very accurately regulating the apex states of a running TD-SLIP and improves on both the accuracy and stability of previous attempts to control a similar, torqueactuated model in [12]. An explicitly specified desired apex state will require a nonzero change in the energy level of the system. Using a strategy similar to the energy-conserving torque controller of Section III-A, we will use the hip torque to supply the requested energy input to the system in a single step. Similar to (21), this energy is given by 1 m((x˙ ∗a )2 − x˙ 2a ) + mg(ya∗ − ya ) + Eloss , (22) 2 which can easily be solved to determine the ramp torque magnitude τ0 , assuming, once again, that tf = tlo .



46

Eτ

=

Once the desired torque profile is determined, the return map has only one remaining degree of control freedom: the touchdown angle θtd . A deadbeat controller can be formulated as a one dimensional minimization problem in the form θtd = argmin −π 2
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Consistent Data Association in Multi-robot Systems with Limited Communications Rosario Arag¨ue´ s

Eduardo Montijano

Carlos Sag¨ue´ s

Departamento de Inform´atica e Ingenier´ıa de Sistemas, Instituto de Investigaci´on en Ingenier´ıa de Arag´on, Universidad de Zaragoza, Mar´ıa de Luna, 50018 Zaragoza, Spain [email protected], [email protected], [email protected]

Abstract— In this paper we address the data association problem of features observed by a robot team with limited communications. At every time instant, each robot can only exchange data with a subset of the robots, its neighbors. Initially, each robot solves a local data association with each of its neighbors. After that, the robots execute the proposed algorithm to agree on a data association between all their local observations which is globally consistent. One inconsistency appears when chains of local associations give rise to two features from one robot being associated among them. The contribution of this work is the decentralized detection and resolution of these inconsistencies. We provide a fully decentralized solution to the problem. This solution does not rely on any particular communication topology. Every robot plays the same role, making the system robust to individual failures. Information is exchanged exclusively between neighbors. In a finite number of iterations, the algorithm finishes with a data association which is free of inconsistent associations. In the experiments, we show the performance of the algorithm under two scenarios. In the first one, we apply the resolution and detection algorithm for a set of stochastic visual maps. In the second, we solve the feature matching between a set of images taken by a robotic team.

I. I NTRODUCTION During the last years, there has been an increasing interest in multi robot systems. In these scenarios, a team of robots cooperatively perform some task in a more efficient way than a single robot would do. In this paper, we address the data association problem in multi robot scenarios. It consists of establishing correspondences between different measurements or estimates of a common element. It is of high interest in localization, mapping, exploration, and tracking applications [1]. The Nearest Neighbor (NN), and the Maximum Likelihood (ML), are widely used methods which associate each observation with its closest feature in terms of the Euclidean or the Mahalanobis distance [23], [15], [12]. Other popular method is the Joint Compatibility Branch and Bound (JCBB) [18], which considers the compatibility of many associations simultaneously. The Combined Constraint Data Association [2] builds a graph where the nodes are individually compatible associations and the edges relate binary compatible assignments. Over this graph, a Maximal Common Subgraph problem is solved for finding the maximum clique in the graph. Scan matching and Iterative Closest Point (ICP) [6] are popular methods for comparing two laser scans. Other methods, like the Multiple Hypothesis Tracking, and the Joint Probabilistic Data Association, maintain many association

hypothesis instead of selecting one of them. And exist many variations of these techniques that combine RANSAC [9] for higher robustness. In solutions based on submaps, one of them is usually transformed into an observation of another. The local submaps are merged with the global map following a sequence [22], or in a hierarchical binary tree fashion [5]. All the mentioned data association approaches, operate on elements from two sets. One set usually contains the current observations, and the other one consists of the feature estimates. These sets may be two images, two laser scans, or two probabilistic maps. Lately, many localization, mapping, and exploration algorithms for multi robot systems have been presented. However, they have not fully addressed the problem of multi robot data association. Some solutions have been presented for merging two maps [21], [23] that do not consider a higher number of robots. Many approaches rely on broadcasting all controls and observations measured by the robots. Then, the data association is solved like in a single robot scenario, using scan matching and ICP for laser scans [14], [10], [16], [20], or NN, ML, and visual methods for feature-based maps [12], [17]. In these methods, the problem of inconsistent data associations is avoided by forcing a cycle-free merging order. This limitation has also been detected in the computer vision literature. In [8] they approach an inconsistent association problem for identifying equal regions in different views. They consider a centralized scenario, where each 2 views are compared among them in a 2-by-2 way. Then, their results are arranged on a graph where associations are propagated and conflicts are solved. The work in [7], from the target tracking literature, simultaneously considers the association of all local maps. It uses an expectation-maximization method for both computing the data association and the final global map. The main limitation of this work is that the data from all sensors needs to be processed together, what implies a centralized scheme, or a broadcast method. All the previous methods rely on centralized schemes, full communication between the robots, or broadcasting methods. However, in multi robot systems, distributed approaches are more interesting. They present a natural robustness to individual failures since there are no central nodes. Besides, they do not rely on any particular communication scheme, and they are robust to changes in the topology. On the other hand, distributed algorithms introduce an additional level of

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complexity in the algorithm design. Although the robots make decisions based on their local data, the system must exhibit a global behavior. In this paper, we address the data association problem for distributed robot systems. Each of our robots posses a local observation of the environment. Instead of forcing a specific order for associating their observations, we allow the robots compute its data association with each of its neighbors in the graph. Although this scenario is more flexible, it may lead to inconsistent global data associations in the presence of cycles in the communication graph. These inconsistencies are detected when chains of local associations give rise to two features from one robot being associated among them. These situations must be correctly identified and solved before merging the data. Otherwise, the merging process would be wrong and could not be undone. In this paper, we approach a distributed data association, under limited communications. Instead of comparing any 2 local observations among them, only the local observations of neighboring robots can be compared. Besides, there is no central agent that has knowledge of all the local associations and each robot exclusively knows the associations computed by itself. Then, each robot updates its local information by communicating with its neighbors. We present an algorithm where, finally, each robot is capable of detecting and solving any inconsistent association that involves any of its features. II. P ROBLEM D ESCRIPTION We consider a robotic team composed of n ∈ N robots. The n robots have communication capabilities to exchange information with the other robots. However, these communications are limited. Let Gcom = (Rcom , Ecom ) be the undirected communication graph. The nodes are the robots, Rcom = {1, . . . , n}. If two robots i, j can exchange information then there is an edge between them, (i, j) ∈ Ecom . Let Ni be the set of neighbors of robot i, Ni = {j | (i, j) ∈ Ecom }. Each robot i has observed a set Si of mi features, i Si = {f1i , . . . , fm }. i

(1)

It can compute the data association between its own set Si , and the sets of its neighbors Sj , with j ∈ Ni . However, these data associations are not perfect. There may appear inconsistent data associations relating different features from the same set Si (Fig. 1). If the robots merge their data as soon as they solve the local data association, inconsistent associations cannot be managed since the merging cannot be undone. The goal of our algorithm is to detect and resolve these inconsistent associations before executing the merging. In order to make the reading easy, along the paper we will use the indices i, j and k to refer to robots and indices r, r , s, s , to refer to features. The rth feature observed by the ith robot will be denoted as fri . Given a matrix A, the notation [A]r,s will correspond to the component (r, s) of the matrix

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Fig. 1. Robots A, B, C and D associate their features comparing their maps in a two-by-two way. Robot A associates its feature f1A with f2D and with f1B ; robot B associates f1B with f1C ; robot C associates f1C with f1D (solid lines). As a result, there is a path (dashed line) between f1D and f2D . This is an inconsistent situation. Finding this path would require the knowledge of the whole association graph.

whereas Aij will denote the block (i, j) when the matrix is defined by blocks. Let F be a function that computes the data association between any two sets of features, Si and Sj , and returns an association matrix Aij ∈ Nmi ×mj where  1 if fri and fsj are associated, [Aij ]r,s = 0 otherwise, for r = 1, . . . , mi and s = 1, . . . , mj . We assume that F satisfies the following conditions. Assumption 2.1: When F is applied to the same set Si , it returns the identity, F (Si , Si ) = Aii = I. Assumption 2.2: The returned association Aij has the property that the features are associated in a one-to-one way, mi  r=1

[Aij ]r,s ≤ 1 and

mj 

[Aij ]r,s ≤ 1,

s=1

for all r = 1, . . . , mi and s = 1, . . . , mj . Assumption 2.3: Robots i and j associate their features in the same way. Given two sets Si and Sj it holds that F (Si , Sj ) = Aij = ATji = (F (Sj , Si ))T . Most of the existing data association methods satisfy these assumptions. If F is applied to all pairs of sets of features Si , Sj , for i, j ∈ {1, . . . , n}, its results can be represented by an undirected graph Gmap = (Vmap , Emap ). Each node in Vmap is a feature fri , for i = 1, . . . , n, r = 1, . . . , mi . There is an edge between two features fri , fsj iff [Aij ]r,s = 1. If F were providing the ground truth data association, then Ggt map would exclusively contain disjoint cliques, identifying features observed by multiple robots [8]. Since F is not perfect, Gmap is a perturbed version of Ggt map that includes additional spurious edges, while it misses others. In addition, due to communication restrictions, the data associations available to our robots are just a subset of Gmap . We name G = (V, E) this available association graph. It has

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the same set of nodes than Gmap , V = Vmap . It has an edge between two features fri , fsj only if the edge exists in Gmap and the robots i and j are neighbors in the communication graph. The goal of our algorithm is to detect and resolve inconsistencies in this graph G ⊆ Gmap in a decentralized fashion. Definition 2.4: An association set is a set of features such that they form a connected component in G. Such set is an inconsistent association or a conflictive set if there exists a path in G between two or more features from the same robot. A feature is inconsistent or conflictive if it belongs to an inconsistent association. Along this paper,  we name msum the number of features in G, msum = ni=1 mi . We name d the diameter of G, the length of the longest path between any two nodes in G. The diameter satisfies d ≤ msum . We name W ∈ Nmsum ×msum the adjacency matrix of G, ⎤ ⎡ W11 . . . W1n ⎥ ⎢ . .. (2) W = ⎣ ... ⎦, . .. Wn1

...

Wnn

where Wij is the block within W for the associations between robots i and j,  Aij if j ∈ {Ni ∪ i}, (3) Wij = 0 otherwise. III. D ECENTRALIZED D ETECTION OF I NCONSISTENT A SSOCIATIONS Considering definition 2.4 we observe that in order to detect an inconsistent association it is required to compute the paths that exist among the elements in G. As the following lemma states [4], given a graph G, the powers of its adjacency matrix contains the information about the number of paths existing between the nodes of G: Lemma 3.1 (Lemma 1.32 [4]): Let G be a weighted graph of order |V | with un-weighted adjacency matrix W ∈ {0, 1}|V |×|V | , and possibly with self loops. For all i, j ∈ {1, . . . , |V |} and t ∈ N the (i, j) entry of the tth power of W , W t , equals the number of paths of length t (including paths with self loops) from node i to node j. The computation of the powers of W requires, a priori, the information about the whole matrix. We show now that this computation can also be done in a decentralized manner. Let each robot i ∈ Rcom maintain the blocks within W t associated to its own features, Xij (t) ∈ Nmi ×mj , j = 1, . . . , n, t ≥ 0,, which are initialized as  I, j = i, Xij (0) = (4) 0, j = i, and are updated, at each time step, with the following algorithm  Xij (t + 1) = Wik Xkj (t), (5) k∈{Ni ∪i}

with Wik as defined in (3). It is observed that the algorithm is fully distributed because the nodes only use information about its direct neighbors in the communication graph.

Theorem 3.2: Let [W t ]ij ∈ Nmi ×mj be the block within W t related to the associations between robot i and robot j. The matrices Xij (t) computed by each robot i using the decentralized algorithm (5) are exactly the sub-matrices [W t ]ij , Xij (t) = [W t ]ij ,

(6)

for all i, j ∈ {1, . . . , n} and all t ∈ N. Proof: The proof is done using induction. First we show that eq. (6) is satisfied for t = 0. In this case we have that W 0 = I, thus for all i, j ∈ {1, . . . , n}, [W 0 ]ii = I and [W 0 ]ij = 0, which is exactly the initial value of the variables Xij (eq. (4)). Now we have that for any t > 0, n   Wik [W t−1 ]kj = Wik [W t−1 ]kj , [W t ]ij = k∈{Ni ∪i}

k=1

/ {Ni ∪ i}, because Wik = 0 for k ∈ Assuming that for all i, j ∈ {1, . . . , n} and a given t > 0, Xij (t − 1) = [W t−1 ]ij is true, then  Xij (t) = Wik Xkj (t − 1) = k∈{Ni ∪i}

=



Wik [W t−1 ]kj = [W t ]ij .

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k∈{Ni ∪i}

Then, by induction, Xij (t) = [W t ]ij is true for all t > 0. Corollary 3.3: The variables Xij (t) contain the information about all the paths of length t between features observed by robots i and j. Proof: By direct application of Lemma 3.1. Analyzing the previous algorithm the first issue to deal with is how to simplify the computation of the matrices in order to avoid high powers of W. In the case we are studying it is just required to know if there is a path between two elements in G and not how many paths are. This means that in this situation it is enough that [Xij (t)]r,s > 0 in order to know that features fri and fsj are connected by a path. Another issue is to decide when the algorithm in (5) must stop. Since the maximum length of a path between any two nodes in a graph is its diameter, then after d iterations the algorithm should stop. However, in general situations the robots will not know neither d nor msum , which makes this decision hard to be made a priori. ¯ Definition 3.4: We will say that two matrices W and W ¯ , if for all r of the same dimensions are equivalent, W ∼ W and s it holds ¯ ]r,s > 0 and [W ]r,s = 0 ⇔ [W ¯ ]r,s = 0. [W ]r,s > 0 ⇔ [W In practice any equivalent matrix to the Xij (t) will provide the required information, which allows us to simplify the computations simply by changing any positive value in the matrices by 1. Moreover, the equivalency is also used to find a criterion to stop the algorithm: Proposition 3.5: For a robot i, let ti be the first time instant, t, such that Xij (t) ∼ Xij (t − 1) for all j = 1, . . . , n. Then robot i can stop to execute the algorithm at time ti .

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¯ ij (t) be the components in Xij (t), such Proof: Let X that [Xij (t − 1)]r,s = 0 and [Xij (t)]r,s > 0. The cardinal, ¯ ij (t)|, represents the number of features fsj ∈ Sj such that |X the minimum path length in G between them and one feature fri ∈ Si is t. At time  ti , Xij (ti ) ∼ Xij (ti − 1) ∀j for the ¯ ij (ti )| = 0 because no component first time, and then nj=1 |X has changed its value from zero to a positive. This means that there is no path of minimum distance ti linking any feature fri with any other feature in G. By the physical properties of a path, it is obvious that if there are no features at minimum distance ti , it will be impossible that a feature is at minimum distance ti + 1 and all the paths that connect features of robot i with any other feature have been found. Corollary 3.6: All the robots end the execution of the iteration rule (5) in at most in d + 1 iterations. Proof: Recalling that the maximum distance between two nodes n in¯ G is the diameter of the graph, denoted by d, then j=1 |Xij (d + 1)| = 0 for all i = 1, . . . , n. If a robot j at time t does not receive the information Xij (t) from robot i then it will use the last matrix received, because robot i has already finished computing its paths and Xij (t) ∼ Xij (t − 1). It remains to analyze which features are conflictive and which are not. Each robot has the information about all the association paths of its features and the features of the rest of the robots in the network in the different variables Xij (ti ). The robots detect all the conflictive features using two simple rules. A feature fri is conflictive if and only if one of the following conditions are satisfied: (i) There exists other feature fri , with r = r , such that [Xii (ti )]r,r > 0;

(8)

(ii) There exist features fsj and fsj , s = s , such that [Xij (ti )]r,s > 0 and [Xij (ti )]r,s > 0.

(9)

In conclusion, the proposed algorithm will be able to find all the inconsistencies in a finite number of iterations. The algorithm is decentralized and it is based only on local interactions between the robots. Each robot only needs to know its local data associations. It updates its information based on the data exchanged with its neighbors. When the algorithm finishes, each robot i can extract from its own matrices Xij (ti ) all the information of any conflict that involves any of its features. If the robot has any conflictive feature, it also knows the rest of features that belong to the conflictive set independently of the robot that observed such features. In the next section we present an additional technique to solve the inconsistencies found by the algorithm. IV. D ECENTRALIZED R ESOLUTION OF I NCONSISTENT A SSOCIATIONS The resolution of inconsistent associations consists of deleting edges from G so that the resulting graph is conflict-free. Definition 4.1: Let c denote the number of conflictive sets in G. We say a conflictive set C is detectable by a robot i

if there exists a r ∈ {1, . . . , mi } such that fri ∈ C. The set of robots that detect a conflictive set C is R ⊆ Rcom . The number of features from each robot i ∈ R involved in C is m ˜ i . We say G is conflict-free if c = 0. All the edges whose deletion transforms G into a conflictfree graph, belong to any of the c conflictive sets of G. Since the conflictive sets are disjoint, they can be considered separately. From now on, we focus on the resolution of one of the conflictive sets C. The other conflictive sets are managed in the same way. The resolution problem consists of partitioning C into a set of disjoint conflict-free components Cq such that ∪ Cq = C, and Cq ∩ Cq = ∅, q



for all q, q = 1, 2, . . . . The number of such conflict-free components is a priori unknown and it will be discussed later in this section. Obtaining an optimal partition that minimizes the number of deleted edges is complicated. If there were only two inconsistent features fri , fri , it could be approached as a maxflow min-cut problem [19]. However, in general there will be more inconsistent features, m ˜ i ≥ 2, within C associated to a robot i ∈ R. Besides, there may also be m ˜ j ≥ 2 inconsistent features belonging to a different robot j ∈ R. The application of [19] separately to any pair of inconsistent features does not necessarily produce an optimal partition. It may happen that a single edge deletion simultaneously resolves more than one inconsistent association. Therefore, an optimal solution should consider multiple combinations of edge deletions, what makes the problem computationally intractable, and imposes a centralized scheme. We propose a resolution algorithm that is not optimal but is efficient and is proven to be correct. Besides, it allows a decentralized computation. Proposition 4.2: Let R be the set of robots that detect C. Let i be the robot with the most features involved in C, i = arg max m ˜ i.

(10)

i∈R

The number of conflict-free components in which C can be decomposed is lower bounded by m ˜ i . Proof: Each conflict-free component can contain, at most, one feature from a robot i ∈ R. Then there must be, at ˜i = m ˜ i components. least, maxi∈R m The resolution algorithm constructs m ˜ i conflict-free components using a strategy close to a BFS tree construction. Initially, each robot i detects the conflictive sets for which it is the root using its local information Xi1 (ti ), . . . , Xin (ti ). The root robot for a conflictive set is the one with the most inconsistent features involved. In case two robots have the same number of inconsistent features, the one with the lowest robot id is selected. Then, each robot executes the resolution algorithm (Algorithm 1). The root robot creates m ˜ i components and initializes each component Cq with one of its features f i ∈ C. Then, it tries to add to each component Cq the features directly associated to f i ∈ Cq . Let us consider that fsj has been assigned to Cq . For all fri such that [Wij ]r,s = 1, robot j sends a component

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Algorithm 1 Decentralized resolution algorithm - Robot i 1: – Initialization 2: for each conflictive set C for which i is root (i = i ) do 3: create m ˜ i components 4: assign each inconsistent feature fri ∈ C to a different component Cq 5: send component request to all its neighboring features 6: end for 7: – Algorithm 8: for each component request from fsj to fri do 9: if (b) or (c) then 10: [Wij ]r,s = 0 11: send reject message to j 12: else if (d) then 13: assign fri to the component 14: send component request to all its neighboring features 15: end if 16: end for 17: for each component reject from fsj to fri do 18: [Wij ]r,s = 0 19: end for

request message to robot i. When robot i receives it, it may happen that (a) fri is already assigned to Cq ; (b) fri is assigned to a different component; (c) other feature fri is already assigned to Cq ; (d) fri is unassigned and no feature in i is assigned to Cq . In case (a), fri already belongs to the component Cq and robot i does nothing. In cases (b) and (c), fri cannot be added to Cq ; robot i deletes the edge [Wij ]r,s and replies with a reject message to robot j; when j receives the reject message, it deletes the equivalent edge [Wji ]s,r . In case (d), robot i assigns its feature fri to the component Cq and the process is repeated. Theorem 4.3: Let us consider that each robot i ∈ Rcom executes the decentralized resolution algorithm (Algorithm 1) on G, obtaining G , (i) after t = n iterations no new features are added to any component Cq and the algorithm finishes; (ii) each obtained Cq is a connected component in G ; (iii) Cq is conflict free; (iv) Cq contains at least two features; for all q ∈ {1, . . . , m ˜ i } and all conflictive sets. Proof: (i) The maximal depth of a conflict-free component is n since, if there were more features, at least two of them would belong to the same robot. Then, after at most n iterations of this algorithm, no more features are added to any component Cq and the algorithm finishes. (ii) There is a path in G between any two features belonging to a conflictive set C. Therefore, there is also a path in G between any two features assigned to the same component Cq . Since the algorithm does not delete edges from G within a component (case (a)), then Cq it is also connected in G . Since

none feature can be assigned to more than one component (case (b)), the components are disjoint. Therefore, Cq is a connected component in G . (iii) By construction, two features from the same robot are never assigned to the same component Cq (case (c)). Therefore, each component is conflict-free. (iv) Each conflictive set has more than one feature. Because of Assumptions 2.1 and 2.2, each feature and its neighbors are conflict free. Therefore, each component Cq contains, at least, its originating feature, and a neighboring feature. Thus, it has at least two features. Corollary 4.4: After executing Algorithm 1, the size of each conflict set C is reduced by at least 2 m ˜ i , where m ˜ i ≥ 2. When the algorithm finishes, each original conflictive set C has been partitioned into m ˜ i conflict-free components. It may happen that a subset of features remains unassigned. These features may still be conflictive in G . The detection algorithm can be executed on the subgraph defined by this smaller subset of features. Proposition 4.5: Consider each robot i iteratively executes the detection (Section III) and the resolution (Section IV) algorithms. Then, in a finite number of iterations, all conflictive sets disappear. Proof: After each execution of the resolution algorithm, the size of each conflict set C is reduced by, at least, 2 m ˜ i ≥ 4 (Corollary 4.4). Then, in a finite number of iterations, it happens that |C| < 4. A set with 3 features fri , fri , fsj cannot be conflictive; this would require the existence of edges (fri , fsj ) and (fri , fsj ), what is impossible (Assumption 2.2). A set with 2 features cannot be conflictive (Assumptions 2.1 and 2.2), and a set with a single feature cannot be inconsistent by definition. Therefore, there will be no remaining inconsistencies or conflictive sets. The main interest of the presented resolution algorithm, is that it is fully decentralized and it works on local information. Each robot uses its own Xij (ti ) for detecting the root robot of each conflictive set. During the resolution algorithm, the decisions and actions taken by each robot are based on its local associations Wij , and the components assigned to its local features. Moreover, each robot is responsible of deleting the edges from its local association matrices Wij , with j ∈ {1, . . . , n}. In addition, the presented algorithm works in finite time. Let us note that although we presented the algorithm for a single conflictive set, all conflictive sets are managed in parallel. V. E XPERIMENTS A. Data association of stochastic maps The presented algorithm is of high interest in multi-robot exploration scenarios with limited communications. Each robot has explored a section of the environment and it has built a stochastic map. When the exploration finishes, the local maps are merged into a global map of the environment. Each robot can compute the data association between its local map, and the maps of its neighbors. However, it does not know the data

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associations obtained by other robots in the team. If they start the merging process using only its local associations, and there is any inconsistency, at some point a robot will be forced to fuse two or more of its features into a single one. To avoid this situation, they execute the presented algorithm for resolving any inconsistent association before merging the maps.

(a) One cycle Fig. 2.

(b) Four cycles

(c) Complete

Communication graphs between the 9 robots after the exploration.

The behavior of the algorithm is analyzed under real data. We use a data set [11] with bearing information obtained with vision (Sony EVI-371DG). The landmarks are vertical lines extracted from the images. The measurements are labeled so that we can compare our results with the ground-truth data association. We select 9 subsections of the whole path for the operation of 9 different robots (Fig. 3 (a)). A separate SLAM is executed on each subsection, producing the 9 local maps (Fig. 3 (b)). The local data associations are computed using the JCBB [18] since it is very convenient for clutter situations like the considered scenario (Fig. 3 (b)). The JCBB is applied to the local maps of any pair of neighboring robots. We analyze the performance of the algorithm under 3 communication graphs (Fig. 2). Table I gives statistics about the number of inconsistencies found considering the different network topologies in Fig. 2. We show the obtained associations compared to the ground truth results. The number of association sets is the number of connected components of W t . The number of good links (true positives) are obtained associations between 2 features which are true (ground truth). The missing links (false negatives) are associations that are in the ground truth information, but have been not detected. And spurious links (false positives) are associations found between features that are different according to the ground truth. The sixth row, c, is the number of conflictive sets. The next row in the table shows the total number of features which have been associated to any other feature from other local map. The last row gives information about how many of those features are conflictive. The amount TABLE I I NITIAL ASSOCIATIONS BETWEEN THE 9 Comm. graph Association sets (ground truth) Association sets Good links (true positives) Missing links (false negatives) Spurious links (false positives) Conflictive sets (c) Number of features msum Conflictive features

(a) 242 182 160 82 22 3 138 16

LOCAL MAPS

(b) 284 218 190 94 28 5 144 24

(c) 400 290 228 172 62 8 154 51

of missing and spurious associations obtained is very high

(a) Ring

(b) Pyramidal

(c) Star-Ring

(d) Complete

Fig. 4.

Formations used in the experiments

for the three network topologies. This is the expected result for many real scenarios, where the landmarks are close to each other, and where the only available information are their cartesian coordinates. As a result, the conflictive features are more than a 10% of the total. In communication graphs with more cycles (Fig. 2 (b)(c)), there are more conflictive features. In the three cases, after a single execution of the detection and the resolution algorithms, all the inconsistencies are solved (Table II, 1st row). An interesting result is that, although our algorithm cannot distinguish between good and spurious edges, in practice a high number of the deleted edges (last row) are spurious ones. TABLE II M ANAGEMENT OF THE INCONSISTENCIES Comm. graph Iterations Initial conflictive sets Deleted links Good deleted links (true positives) Spurious deleted links (false positives)

(a) 1 3 6 2 4

(b) 1 5 10 2 8

(c) 1 8 34 12 22

B. Matching sets of images We have also tested the performance of our proposal with a set of images. A team of robots equipped with cameras and limited communication capabilities is a typical situation in which not all the images are available to execute a global matching. A solid set of matches is required independently of the task the team is performing (visual SLAM, formation control, etc.). In the proposed experiment 6 robots moving in formation are considered. Each robot acquires one image with its onboard camera and extracts SURF features [3]. The local matching is only applied to pairs of images which are connected in the communication graph. For the local matching the epipolar constraint combined with RANSAC is imposed [13]. The detection and resolution of inconsistencies is analyzed for four different typical communication graphs (Fig. 4). Although the epipolar constraint discards most part of the wrong matches, some spurious associations are still found.

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(a) Section of the dataset used in the experiments. (b) Local maps acquired by 9 robots exploring the region in (a).

Figure 5 shows an example of an inconsistency found by our algorithm in the case of the formation (d). The figure also shows how the Algorithm 1 solves such inconsistency; the result is a subset of the initial matches with more connected components than before but without any conflict. More general results about the experiment can be seen in Table III. Since for this experiment there is no ground truth available, the number of missing and spurious links found in the local matching cannot be provided. Some interesting conclusions are extracted with the obtained results. First of all, the number of association sets and the number of features involved msum are increased with the number of edges in Gcom . The more local associations there are, the more matches the algorithm will have available. With respect to the inconsistencies, they grow with the number of cycles in Gcom because each cycle can generate inconsistencies independently of the rest of the communication network. The size of the cycles also influence on the conflicts, cycles of smaller length will cause more inconsistencies because the number of local associations required to find a conflict is also smaller.

vision system because they include features from the same image which are far away from each other. Table IV shows the statistics on how are the solutions to the conflicts provided by our algorithm. Although ground truth is not available in this experiment it is easily observable when the inconsistent matches have been separated correctly in the used images and when not (Fig. 5). We have manually classified the conflicts as well solved and wrongly solved. We consider a conflict as well solved when it does not contain any spurious edge in any of the associations that the algorithm returns. The spurious and the good links that the algorithm deletes, also manually counted, are shown in fifth and sixth rows of table IV respectively. TABLE IV M ANAGEMENT OF THE INCONSISTENCIES Formation Initial conflictive sets Inconsistencies well solved Inconsistencies wrongly solved Deleted links Good deleted links (true positives) Spurious deleted links (false positives)

TABLE III I NITIAL ASSOCIATIONS FOR THE DIFFERENT FORMATIONS Formation Association sets Conflictive sets (c) Number of features msum Conflictive features

(a) 528 2 1335 14

(b) 590 22 1643 144

(c) 605 31 1743 204

(a) 2 1 1 2 1 1

(b) 22 20 2 26 2 24

(c) 31 24 7 46 19 27

(d) 72 55 17 131 43 88

VI. C ONCLUSIONS

(d) 632 72 2069 521

Another important aspect to analyze is how the resolution of the conflicts is performed. We have observed that some of the inconsistencies are caused by features that are very close to each other in one image and due to the RANSAC threshold the miss-match appears. These inconsistencies are introduced because of the extractor parameters but the errors they can cause are negligible. However, there are also some other inconsistencies which can generate problems to the

We have presented a new technique to match several sets of features observed by a team of robots in a consistent way under limited communications. Local associations are found only within robots that are neighbors in the communication graph. After that a fully decentralized method to compute all the paths between local associations is carried out, allowing the robots to detect all the inconsistencies related with their observations. For every conflictive set detected, in a second step the method is able to delete local associations to break the conflict using only local communications. The whole method is proved to finish in a finite amount of time finding and solving all the inconsistent associations. Experimental results show the performance of the method in two different scenarios with

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Fig. 5. Example of one inconsistency found by the algorithm. The inconsistency is represented by the whole set of depicted links. It is observed that if all the links are considered, features of the same image are matched. After executing Algorithm 1 the inconsistency is solved. In this example the root of the inconsistency is the top-middle image. The dark line with circles is the link deleted by the algorithm. Solid lines represent one of the conflict free components and dashed lines the second. For clarity, the rest of the SURF features are only shown in the top middle image.

great interest in robotic tasks. To the best of our knowledge this is the first algorithm that is able to do it. ACKNOWLEDGMENT This work was supported by project Ministerio de Ciencia e Innovacion DPI2009-08126, and grants MEC BES-200714772 and AP2007-03282. The data set used in one of the experiments was provided by U. Frese and J. Kurlbaum. The authors gratefully acknowledge them for providing data and support. R EFERENCES [1] T. Bailey and H. Durrant-Whyte. Simultaneous localization and mapping: part II. IEEE Robotics & Automation Magazine, 13(3):108–117, 2006. [2] T. Bailey, E. M. Nebot, J. K. Rosenblatt, and H. F. Durrant-Whyte. Data association for mobile robot navigation: a graph theoretic approach. In IEEE Int. Conf. on Robotics and Automation, pages 2512–2517, San Francisco, USA, April 2000. [3] H. Bay, T. Tuytelaars, and L. Van Gool. Surf: Speeded up robust features. In European Conference on Computer Vision, pages 404–417, 2006. [4] F. Bullo, J. Cort´es, and S. Mart´ınez. Distributed Control of Robotic Networks. Applied Mathematics Series. Princeton University Press, 2009. Electronically available at http://coordinationbook.info. [5] C. Cadena, F. Ramos, and J. Neira. Efficient large scale SLAM including data association using the Combined Filter. In European Conference on Mobile Robotics, ECMR, pages 217–222, Mlini/Dubrovnik, Croatia, September 2009. [6] A. Censi. An accurate closed-form estimate of ICP’s covariance. In IEEE Int. Conf. on Robotics and Automation, pages 3167–3172, Roma, Italy, April 2007. [7] R. W. Deming and L. I. Perlovsky. Concurrent multi-target localization, data association, and navigation for a swarm of flying sensors. Information Fusion, 8(3):316 – 330, 2007. [8] V. Ferrari, T. Tuytelaars, and L. Van Gool. Wide-baseline multiple-view correspondences. In IEEE Computer Society Conference on Computer Vision and Pattern Recognition, volume 1, pages 718–725, Madison, USA, June 2003. [9] M. A. Fischler and R. C. Bolles. Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography. Communications of the ACM, pages 381–395, 1981.

[10] D. Fox, J. Ko, K. Konolige, B. Limketkai, D. Schulz, and B. Stewart. Distributed multirobot exploration and mapping. In Proceedings of the IEEE, volume 94, pages 1325–1339, July 2006. [11] U. Frese and J. Kurlbaum. A data set for data association, June 2008. http://www.sfbtr8.spatial-cognition.de/insidedataassociation/. [12] A. Gil, O. Reinoso, M. Ballesta, and M. Julia. Multi-robot visual SLAM using a rao-blackwellized particle filter. Robotics and Autonomous Systems, 58(1):68–80, 2009. [13] R. Hartley and A. Zisserman. Multiple View Geometry in Computer Vision. Cambridge University Press, Cambridge, 2000. [14] A. Howard. Multi-robot simultaneous localization and mapping using particle filters. International Journal of Robotics Research, 25(12):1243– 1256, 2006. [15] M. Kaess and F. Dellaert. Covariance recovery from a square root information matrix for data association. Robotics and Autonomous Systems, 57(12):1198 – 1210, 2009. [16] K. Konolige, J. Gutmann, and B. Limketkai. Distributed map-making. In Workshop on Reasoning with Uncertainty in Robotics, Int. Joint Conf. on Artificial Intelligence, Acapulco, Mexico, August 2003. [17] H. S. Lee and K. M. Lee. Multi-robot SLAM using ceiling vision. In IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, pages 912–917, St. Louis, USA, October 2009. [18] J. Neira and J. D. Tard´os. Data association in stochastic mapping using the joint compatibility test. IEEE Transactions on Robotics and Automation, 17(6):890–897, 2001. [19] C. H. Papadimitriou and K. Steiglitz. Combinatorial Optimization: Algorithms and Complexity, chapter 6.1 The Max-Flow, Min-Cut Theorem, pages 120–128. Dover, 1998. [20] M. Pfingsthorn, B. Slamet, and A. Visser. A scalable hybrid multirobot SLAM method for highly detailed maps. In U. Visser, F. Ribeiro, T. Ohashi, and F. Dellaert, editors, RoboCup 2007: Robot Soccer World Cup XI, volume 5001 of Lecture Notes in Artificial Intelligence, pages 457–464. Springer Verlag, Berlin Heidelberg New York, 2008. [21] S. Thrun and Y. Liu. Multi-robot SLAM with sparse extended information filers. In Int. Symposium of Robotics Research, pages 254–266, Sienna, Italy, October 2003. [22] S. B. Williams and H. Durrant-Whyte. Towards multi-vehicle simultaneous localisation and mapping. In IEEE Int. Conf. on Robotics and Automation, pages 2743–2748, Washington, DC, USA, May 2002. [23] X. S. Zhou and S. I. Roumeliotis. Multi-robot SLAM with unknown initial correspondence: The robot rendezvous case. In IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, pages 1785–1792, Beijing, China, October 2006.

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z1 z2 z3 z4 z5 z6 z t1 t2 t3 t4 t5 t6 t

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y1 t1 y2 t2 y3 t3 y4 t4 y5 t5 y6 t6 yt

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Bt (ut−1 − ut−1 ) + Vt mt , zt = h(xt , 0) + Ht (xt − xt ) + Wt nt ,



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¯ t = xt − x

¯ t = ut − u

ut ,

¯zt = zt −

h(xt , 0),



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mt ∼ N (0, Mt ), nt ∼ N (0, Nt ),

 

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t=0

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˜− ˜ t−1 + Bt u ¯ t−1 x t = At x



Pt− = At Pt−1 ATt + Vt Mt VtT ,



0HDVXUHPHQW XSGDWH Kt = Pt− HtT (Ht Pt− HtT + Wt Nt WtT )−1



˜t = x ˜− ˜− x zt − H t x t + Kt (¯ t )



Pt = (I − Kt Ht )Pt− .



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S = C Lt =

xt = f (xt−1 , ut−1 , 0) + At (xt−1 − xt−1 ) +

xt ,

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St =

T T −(Bt+1 St+1 Bt+1 + D)−1 Bt+1 St+1 At+1 T T C + At+1 St+1 At+1 + At+1 St+1 Bt+1 Lt .

 

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 ˜ t−1 + Bt Lt−1 x ˜ t−1 + Kt Ht x ¯ t + Wt nt − = At x  ˜ t−1 + Bt Lt−1 x ˜ t−1 ) Ht (At x  ˜ t−1 + Bt Lt−1 x ˜ t−1 + Kt = At x ¯ t−1 + Bt Lt−1 x ˜ t−1 + Vt mt ) + Wt nt − Ht (At x  ˜ t−1 + Bt Lt−1 x ˜ t−1 ) Ht (At x ˜ t−1 + Bt Lt−1 x ˜ t−1 + Kt Ht At x ¯ t−1 + = At x ˜ t−1 , Kt Ht Vt mt + Kt Wt nt − Kt Ht At x

X goal d φ v

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¯ t−1 ¯t At x x Bt Lt−1 = + ˜t ˜ t−1 Kt Ht At At + Bt Lt−1 − Kt Ht At x x



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qt ∼ N (0, Qt ).



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ˆ 0 = 0, y P R0 = 0 0



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 ¯t xt I 0 x x = + t ,  ˜t ut ut 0 Lt x Rt = Ft Rt−1 FtT + Gt Qt GTt ,

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The Smooth Curvature Flexure Model: An Accurate, Low-dimensional Approach for Robot Analysis Lael U. Odhner and Aaron M. Dollar Yale University Email: {lael.odhner, aaron.dollar}@yale.edu Abstract— This paper presents a new and comprehensive method for modeling robots having highly flexible members such as flexure joints. An accurate model of large deformation bending is important for precisely describing the configuration of the flexible member. Additionally, the accuracy of the Jacobian and Hessian of the forward kinematics are critically important at large angles for predicting the deformation and the stiffness of the joint under load. The model introduced here is based on the assumption that the curvature of a beam in bending is smooth, and thus can be approximated by low-order polynomials. This produces a parameterized description of flexure motion that can be used as a joint model when expressed in Denavit-Hartenberg form, as a transformation from one rigid link to the next in a serial manipulator. We will show that with only three parameters, this model faithfully reproduces the elastic deformation of a flexure hinge predicted by the continuum model, even for large angles, without requiring numerical integration or many finite elements. It can also be used to compute the compressive buckling load of the flexure as predicted by the continuum model.

I.

INTRODUCTION

Highly flexible members have been frequently considered in the context of robotic hardware. A number of studies into the behavior of flexible link robots have been conducted, often for the purposes of controlling for undesirable dynamic effects related to working with long, thin links (e.g. [1-3]). A smaller number of efforts have dealt with the beneficial aspects of highly flexible links, such as providing a large number of degrees of freedom for manipulation tasks [4] or low stiffness for grasping and assembly purposes [5]. A related application is the use of highly flexible members as joints between rigid links, typically referred to as flexures. The contrast between these two applications is shown in Fig. 1. These features are commonly used to allow motion in monolithic structures, and have been used as joints in a number of different robotic mechanisms, particularly in compliant hands [6-8]. The benefits of flexure-based joints include having no sliding parts (and therefore no friction or stick-slip effects), no backlash, and are able to compliantly deform in response to unplanned collisions, making them ideal for robots that must operate in unstructured environments [9]. Another major benefit of flexures is the simplicity and potential lower cost compared to standard revolute joints, which require bearings for smooth, accurate motion. As robots become more common as commercial products, flexures are

Rigid Links Pin Joints

Flexible Links Pin Joints

Rigid Links Flexure Joints

Figure 1. A comparison between traditional rigid robot manipulators (left), flexible link robots (center), and flexure-joint robots, in which flexible links act as hinges (right).

likely to be used with increasing frequency due to their compatibility with inexpensive polymeric fabrication processes such as multi-shot injection molding [10] and shape deposition manufacturing [11]. One drawback to flexure-based robot mechanisms is the complex mechanical behavior that they exhibit compared to pin joints. A pin joint has one degree of freedom, whereas the elastic deformation of a beam in bending has infinitely many degrees of freedom. Moreover, a flexure hinge in a robot often bends to angles up to 90 degrees or more, so classical smalldeflection beam bending models are inapplicable. As a result, there is no canonical parametric model for planar flexure hinges suitable for robot analysis. This paper presents a model that can fill this role. In order to apply the rich set of tools available for serial manipulator design, control, and analysis, one must have a model of elastic behavior that is accurate and computationally simple. The design specification for a good flexure model can be broken into three functional requirements: 1. It should be possible to compute both the shape of the robot and the elastic energy associated with deformation as a function of a small set of generalized coordinates, as one might describe a jointed mechanism using the internal joint angles. 2. It should be possible to compute the force in generalized coordinates resulting from a force on the robot at any point using the Jacobian of that point’s coordinates, as well as the local equilibrium position resulting from such a force. 3. It should be possible to compute the stiffness in generalized coordinates resulting from a force on the robot at a point by the Hessian of that point’s coordinates, as well as any buckling modes the robot has (configurations/loads having zero stiffness in some direction).

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Many models of flexible robot components meet some, but not all of these requirements. One common approach is to model flexure elements as having constant curvature [12]. Flexures have also been approximated as a single pin joint halfway between the ends of the flexure [13]. Both of these models capture the relative rotation between rigid bodies on the manipulator, making them useful for inverse kinematic computation and form closure grasp analysis. However, because these models have fewer degrees of freedom than a real flexure, they are too rigid and under-predict the deformation of a loaded flexure. Another family of models, called pseudo-rigid body models, consists of one or several joints placed to approximate the flexure’s center of rotation, connected in parallel with nonlinear springs fit by regression to the exact force-deflection profile [14]. These models can be used to find flexure deflection under load, but they are unsuitable for manipulator analysis because the linkage geometries used to approximate the beam bending change based on the direction of applied force, and thus are not purely kinematic descriptions of flexure behavior. Another approach is based on assuming some set of superimposed deformational modes [15]. This technique has been applied to flexible links (e.g. [16]) and continuum manipulators [17]. Modal models for flexures have been proposed based on analytically calculated small-deformation solutions, as well as finite element solutions [18,19]. However, modal models are specific to the behavior they are designed to model, and none of the currently available models in the robotics literature accurately capture largedeformation flexure behavior. The flexure model discussed in this paper is a modal model which approximates the curvature of a flexure using a polynomial basis. The polynomial coefficients define the relative position and orientation of two bodies connected by the flexure, as well as the elastic energy stored in the flexure itself. This model meets all three of the functional requirements introduced above, while avoiding the need to use numerical integration or to break the beam into many finite elements. It predicts not only the deflection of a flexure under load, but also second-order kinematic effects such as buckling and the change in flexure stiffness resulting from compressive or tensile loads. These second-order effects are particularly useful in the study of grasping and manipulation, where grasp stability may depend on the elastic stability of the manipulator itself [20]. Previous study of this model has dealt with stiffness prediction [21]. This paper is intended as a more general discussion of this model’s applicability to robotic systems. The remainder of this paper is divided into three sections. Section II is an overview of the flexure model, describing how the parameters define the shape and the energy function of a flexure hinge. Section III examines the shape of the flexure when an arbitrary load and moment is applied at one end. The results are compared to exact large deformation beam solutions. Additional results are shown comparing the smooth curvature model to finite element models of a sample mechanism. Section IV demonstrates the second-order kinematic accuracy of the model by comparing classical continuum buckling models to the discrete buckling predicted by the proposed parameterized model.

Figure 2. A comparison of the moment profile in small- and largedeflection cantilevered beam bending.

II.

THE SMOOTH CURVATURE MODEL

A. Motivation In 1694, Jacob Bernoulli proposed (and solved) the problem of finding the shape of a pre-bent cantilevered beam of length L that would bend into a straight line when loaded with an arbitrarily large force P at the tip [22]. Today the curve is known as the clothoid or Euler spiral. The curvature κ(s) of the clothoid curve varies linearly with the arc length s from the base of the cantilever,

κ (s) =

P ⎛ s⎞ ⎜1 − ⎟ EI ⎝ L ⎠

(1)

Here E is the elastic modulus and I is the constant planar moment of the beam area. The clothoid is also a passable approximate solution the more useful problem of finding the deformed shape of an initially straight beam loaded at one end with a large load. This can be seen by examining the nonlinear deformation of a cantilevered beam, as shown in Fig. 2. For a small end load (left), the bending moment will be almost exactly proportional to the distance from the tip of the flexure, as expected. A large end load (right) will produce a non-linear deformation profile, but the moment, plotted as a function of arc length, is still roughly linear. The curvature is directly proportional to bending moment in the beam, τ(s),

κ (s) =

τ ( s) EI

(2)

For some range of large loads, then, the curvature of a beam can be approximated as some constant plus a linear function of arc length. Horn discussed this approximation in the context of spline curves [23]. The accuracy of this model could be further improved by noting that while the curvature may not be exactly linear, it is certainly smooth, and so might be described with a basis of n Legendre polynomials, G0(s)…Gn-1(s). The curvature is expressed as a weighted sum of the bases,

κ ( s, q) = q 0 G 0 ( s) + q1 G1 ( s) + K + q n −1 G n −1 ( s)

(3)

The coefficients, q0…qn-1, used as a generalized coordinate vector q for describing the deformation of the flexure, are central to the proposed flexure model. Two particular cases will be considered here, corresponding to the models with 2

 138

and 3 parameters, whose basis functions are Legendre polynomials, translated and scaled to be orthogonal on [0, L],

κ 2 ( s, q ) = q 0

1 2s − L + q1 L L2

κ 3 ( s, q ) = q 0

1 2s − L 6s − 6sL + L + q1 + q2 L L2 L3

(4) 2

Proximal Coordinate Frame

2

(5)

Models of this type, which we will call smooth curvature models, can be used to predict the shape of the flexure in bending, as well as the elastic bending energy. These derivations follow in the next two sections.

Figure 3. A kinematic model of a flexure consists of a transformation mapping the coordinate frame at the proximal side of the joint to the coordinate frame at the distal side of the joint.

B. Flexure Shape Joints in a serial robot manipulator are often represented in Denavit-Hartenberg notation, that is, as a parameterized axial, radial and angular transformation from one joint to another. The analogous transformation for a pin joint is a rotation about the joint axis. In the case of a flexure, this transformation corresponds to the translation and rotation from one end of the flexure to the other, as shown in Fig. 3 [12]. This could be written as a matrix, for instance,

⎡cos(ϕ tip ) − sin(ϕ tip ) T = ⎢⎢ sin(ϕ tip ) cos(ϕ tip ) ⎢⎣ 0 0

x tip ⎤ y tip ⎥⎥ 1 ⎥⎦

(6)

The three quantities characterizing this transformation are the flexure tip displacement (xtip, ytip), and the relative angle from the base of the flexure to the tip of the flexure, φtip. They can be written as functions of the polynomial coefficient vector, q. The angular profile, φ(s, q), is the integral of the curvature,

ϕ ( s, q) = ∫ κ ( s, q )ds ϕ 2 ( s, q ) = q 0

s s 2 − sL + q1 L L2

s 2 − sL s 2 s 3 − 3s 2 L + sL2 ϕ 3 ( s, q ) = q 0 + q 1 + q2 2 L L L3

(7)

L

(11)

0

L

y tip = ∫ sin(ϕ ( s, q ))ds

L

U (q) =

EI κ ( s, q) 2 ds 2 ∫0

(12)

0

These expressions are transcendental. The second order solution can be solved in terms of Fresnel integrals by

(13)

If the 2 parameter curvature is used, U(q) evaluates to a weighted sum of the squared parameters.

EI 2

2

⎛ q 0 4 q1 ( 2 s − L ) ⎞ EI ⎛ 2 q12 ⎞ ⎜ q0 + ⎟ (14) ⎜⎜ + ⎟ = ds ⎟ L L2 2 L ⎜⎝ 3 ⎟⎠ ⎠ 0 ⎝

L

∫

Because the polynomial basis is orthogonal under convolution over the interval [0, L], there are no cross-terms in this expression. The expression for energy given 3 parameters differs only in the addition of a single term:

U 3 (q) =

(10)

The tip position of the flexure can be found by integrating the cosine and sine of the angular profile,

x tip = ∫ cos(ϕ ( s, q ))ds

C. Elastic Energy Having found the shape of the hinge as a function of q, we now turn to finding an expression for the elastic energy in the flexure. The energy stored in an Euler-Bernoulli beam is proportional to the integral of the squared curvature [15]:

U 2 (q) =

At the end of the flexure (s=L), the tip angle is equivalent to q0 irrespective of the model order. This is a happy side effect of using orthogonal polynomials, as all the higher, non-constant terms must integrate to zero:

ϕ tip = ϕ ( L, q) = q 0

completing the square and using trigonometric addition identities, but it contains discontinuities, and is not practically useful. Further, this strategy does not generalize to the 3 parameter model. Instead, an interpolating approximation was used, so that (11) and (12) can be analytically approximated within some reliable error bounds. In this paper, Chebyshev interpolation [24] was used to approximate the sine and cosine functions as polynomials. There is a trade-off between the domain of interpolation and the computational cost, so maximum flexure rotation was limited to be less than 108°. Alternatively, Gaussian quadrature could be used, essentially interpolating the entire integrand as a polynomial.

(8) (9)

Distal Coordinate Frame

EI ⎛ 2 q12 q 22 ⎞ ⎜ q0 + ⎟ + 2 L ⎜⎝ 3 5 ⎟⎠

(15)

D. Summary The smooth curvature model for flexure hinges has been introduced, based on the observation that the curvature of a flexure can be approximated using a low-dimensional basis of orthogonal polynomials. The position and orientation of the flexure tip can be found relative to its base using only a few model parameters, and these can be used to represent the flexure as a joint in Denavit-Hartenberg form. The elastic energy stored in the bent beam is a weighted sum of the squared flexure parameters.

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The remaining sections will demonstrate that this model can be used to satisfy the two other requirements for a flexure joint model, that is, that the model accurately predicts the equilibrium position of the flexure when an arbitrary force and moment are applied, and that the model accurately predicts variable stiffness effects and buckling due to compressive loads. III.

DEFLECTION UNDER LOAD

A. Jacobian Analysis of the Forces on a Robot Accurate descriptions of the force exerted on a robot and the resulting deflection are central to many problems in the control and analysis of robot manipulators. The net generalized force F on a manipulator experiencing a force fp at some point p is given by the Jacobian of that point’s coordinates, and the gradient of the potential energy function U(q),

F = ∇ q ( p) T f

p

+ ∇ q (U (q ))

(16)

Figure 4. A flexure, loaded at the end by a force and a moment. This figure shows the direction of load, θ.

deformation beam bending 1 . The equilibrium configurations were compared to the exact solution obtained by numerically integrating the large-deformation Euler-Bernoulli equation,

~ s ) ⎤ ⎡ P (cos(θ ) cos(ϕ (~ ⎡τ~ (~ s )) − sin(θ ) sin(ϕ (~ s )))⎤ ⎢ ⎥ ⎢ϕ (~ ⎥ ~ ~ τ (s ) d ⎢ s )⎥ ⎢ ⎥ = ⎥ x (~ s )⎥ ⎢ d~ s ⎢~ cos(ϕ (~ s )) ⎥ ⎢~ ~ ⎥ ⎢ ~ sin(ϕ ( s )) ⎣ y ( s )⎦ ⎣⎢ ⎦⎥

The generalized force balance equation is only realistic if the generalized coordinates faithfully represent all of the motions that the robot is capable of making. For a flexure hinge, it is important that the motion of flexure tip, as described in the previous section, is accurate, so that the forces and moments transmitted from one link to the next result in a physically realistic deformation of the flexure hinges.

This is a restatement of (2), (7), (11) and (12) in differential form, after applying the substitutions from (17). This equation was solved using a Runge-Kutta solver. The tip moment M, force angle θ, and tip angle φtip, were specified, and the integral from the tip of the flexure to the based was computed. The tip force, P, was found using a bisection search such that the boundary conditions at both ends of the flexure were simultaneously satisfied.

This section considers two tests for benchmarking the ability of the smooth curvature flexure model to predict deformation under load. The first test compares the exact deflection of a cantilevered flexure (using numerically computed elastica curves [25]) to the tip position predicted by the smooth curvature model. The second test computes the deflection of a two-link finger from a tendon-driven elastic gripper developed by the authors. The two flexure hinges in the finger are modeled with finite element flexure models, and with the 3 parameter smooth curvature model, showing that the two models agree despite the vastly reduced parameter space of the smooth curvature model.

Each force-moment combination (P, θ, M) that was computed for the exact beam equation was applied to the tip of the smooth curvature flexure model, using the generalized force balance from (16). The Jacobian of the tip coordinates xtip, ytip and φtip was derived from (10), (11) and (12),

∂ϕ tip

⎧1, i = 0 =⎨ ∂q i ⎩0, i ≠ 0

B. Tip Deflection of a Loaded Flexure A straightforward method of examining the accuracy of a flexible beam is to clamp one end and examine the deflection of the other end when subject to an arbitrary moment M and force P exerted at an angle θ, as shown in Fig. 4. To ensure proper scaling of the results, a non-dimensional form of the beam bending equations should be used, based on these substitutions:

~ PL2 ~ ML ~ τL ~ s ~ x ~ y P= ,M = ,τ = ,s = ,x= ,y= EI EI EI L L L

(18)

(17)

The dimensionless beam bending equations are equivalent to modeling a beam as having length 1, and an elastic modulus and cross-sectional moment equal to 1.The only parameter that is unaffected by this scaling is the beam’s angular profile, φ(s). Results will be computed for the case when the tip angle, φtip, is equal to 90°, a prototypical test case in the study of large-

∂x tip ∂q i ∂y tip ∂q i

(19)

L

∫

= − sin(ϕ ( s, q)) 0

L

∫

= cos(ϕ ( s, q )) 0

∂ϕ tip ∂q i

∂ϕ tip ∂q i

ds

ds

(20)

(21)

As in (11) and (12), Chebyshev interpolation was used to produce analytical approximations of (20) and (21). The derivatives of the energy function are much simpler, and can be found from (15),

1 This is a generalization of the rectangular elastica problem posed by Bernoulli, the problem of finding the shape of a cantilevered beam bent at a right angle by a force at the tip [25].

 140

0 ⎤ ⎡q 0 ⎤ ⎡1 0 EI ⎢ 0 1 / 3 0 ⎥⎥ ⎢⎢ q1 ⎥⎥ ∇ qU (q) = ⎢ L ⎢⎣0 0 1 / 5⎥⎦ ⎢⎣q 2 ⎥⎦

(22)

The force balance, computed from (19)-(22), was set to zero to form a system of nonlinear equations, which was solved numerically. Three force-moment combinations were used to compare the two models. These combinations, labeled A, B, and C, are shown in Fig. 5. In case A, a pure bending moment was applied, sufficient to bend the flexure to an angle of 90°. Case B was a pure force of a magnitude sufficient to bend the flexure to 90°. Case C was the most complex load, consisting of a moment equal and opposite to the moment applied in combination 1, counteracted by a force so that the flexure tip angle remained at 90°. The prediction errors from the smooth curvature models were computed for values of θ ranging from 20 degrees to 105 degrees (as in Fig. 4). The lower bound of 20 degrees was chosen because the magnitude of the load required to bend a beam to 90° in case B has a vertical asymptote at θ=0. Thus, the behavior of the flexure becomes increasingly unrealistic in this case. The upper bound of 105 degrees was chosen because the elastica curves generated with the Runge-Kutta solver could not predict buckled configurations, and as the force on the flexure tip becomes increasingly compressive (i.e. θ > 90°), good reference comparisons could not be made. Instead, compressive loads were compared to finite element models in the following subsection. The positional error of the flexure tip was found, that is, the norm of the vector from the predicted flexure tip to the tip of the numerically computed elastica curve. This is plotted in nondimensional form, meaning that the error is given as a fraction of the flexure length. The angular error is also shown. The errors for the 2 parameter model are shown in Fig. 6, and the errors for the 3 parameter model are shown in Fig. 7. These plots show a number of significant results. First, the errors in case A (pure moment loading) were very small for both the 2 and 3 parameter smooth curvature models. The exact shape one would expect for a beam having a constant bending moment is an arc, a shape that can be exactly reproduced with both models. Thus, the error is correspondingly small. The errors observed in loading cases B and C indicate that the 3 parameter model is significantly more accurate, especially when the flexure is loaded by an opposed force and moment. Most importantly, in every case, the 3 parameter smooth curvature model was within a positional accuracy of 1 percent of the beam length, and an angular accuracy bound of 1°. C. Finite Element Comparisons One purpose of the smooth curvature model of particular interest to the authors is to enable efficient analysis of manipulators having multiple flexure joints. In previous work, the authors have developed robot hands incorporating polymeric elastic flexure joints [8,9]. These hands are made up of 2-link, tendon-driven fingers, represented in Fig. 8A. In order to evaluate the usefulness of the smooth curvature model for multi-link manipulators, a finite element model was constructed using an object-oriented Matlab library created by

 141

Figure 5. The three loading cases used to test the smooth curvature model.

Figure 6. Prediction errors for the 2 parameter smooth curvature model. Cases A, B, and C correspond to the cases in Fig. 5.

Figure 7. Prediction errors for the 3 parameter smooth curvature model. Cases A, B, and C correspond to the cases in Fig. 5.

TABLE I.

FEM VS. SMOOTH CURVATURE FINGER MODEL RESULTS

Case FEM, Tendon Force (i) SC, Tendon Force (ii) FEM, Tendon + Pad Force (iii) SC, Tendon + Pad Force (iv)

Pad x -0.0048 -0.0047 0.0615 0.0615

Pad y 0.1012 0.1012 0.1046 0.1046

Pad angle 147.6553 147.5760 103.0476 103.0168

A. Flexure-based robot finger Tendon

Smooth Curvature Model

B. Tendon force

Finite Element Model

C. Tendon force + pad force Figure 9. A single tendon-driven joint, modeled using the smooth curvature model (left) and finite rotational elements (right).

Figure 8. A comparison between FEM and smooth curvature (SC) models: A. The FEM model, no tendon force vs 10 N tendon force. B. SC model (i) vs FEM (ii), 10 N tendon force. C. SC model (iii) vs. FEM (iv), 10 N force + 0.5 N force on distal link.

the authors [26]. This model represents each flexure in a fashion similar to the finite element model proposed in [1], consisting of 16 small rotational links. The transformation representing each joint in Denavit-Hartenberg form was the composition of the many resulting rotations and translations making up each finite element. A model of the finger was also constructed using our 3 parameter smooth curvature flexures. This model uses the joint transformation from (6) and the energy function from (15) to describe the joint behavior in terms of three generalized coordinates per joint, for a total of 6. The finite element model, by comparison, had a total of 32 generalized coordinates. Both models were subjected to two different loading conditions: in the first condition, the finger was actuated with a single tendon connected to the distal link, as in Fig. 8B. The second condition, depicted in Fig. 8C, included the same tendon force and a horizontal force applied to the center of the pad on the distal link. In both cases, the generalized force balance was computed using (16), computing the kinematics of the tendon and the center of the distal pad using a composition of geometric joint and link transformations. The results of the test are shown in Fig. 8 and Table I, which describes the agreement between the smooth curvature and finite element models as to the position and orientation of the center of the distal link. The results show that the displacement of the distal link pad as calculated by the smooth curvature model is within 0.1% of the position predicted by the finite element model. The angular agreement is similarly within a tenth of a degree in both cases. Attempts at visual comparison between the FEM and smooth curvature models by overlaying the two were unsuccessful, because they were almost indistinguishable to the eye.

D. Summary For both a single flexure hinge and a two-joint manipulator, the smooth curvature model has been shown to accurately predict the deformation of a manipulator under a wide range of forces and moments. This is interesting and new because most methods of accurately solving large-deflection beam bending problems involve numerical integration, or the breaking down of a beam into many finite elements. Because the smooth curvature model achieves a useful degree of accuracy with only three parameters per joint, calculation of dynamics and statics for control or motion planning is a much simpler process. IV.

STIFFNESS AND BUCKLING

A. Stiffness of a Loaded Elastic Structure One major difference between the well-studied problem of flexible-link manipulators and the newer field of flexurejointed manipulators is the relatively increased importance of buckling in flexure joints. The flexure joints shown in Fig. 9 show a typical tendon/flexure actuation scheme. The tendon exerts a force in tension, which is balanced by a compressive force in the flexure, unless a parallel load path exists. This compressive force is quite large, and can easily approach the Euler buckling load of a thin flexure (the load at which the lateral stiffness of the flexure is zero). Unlike structural columns, buckling does not represent a necessarily undesirable effect. The fact that the flexure is buckled just means that its rotational stiffness is very low. Most pin joints, for example, have zero rotational stiffness and this is not an impediment to their use in robots. However, this change in stiffness as a function of load is critically important in some robotic tasks. For example, a change in joint stiffness will affect computed torque control models. Additionally, some tasks such as grasping and manipulation rely on the elastic stability of the

 142

whole system; a buckling mode could be harmless, or it could correspond to configuration in which a grasped object twists out of its gripped position [20]. As a result of all these concerns, it is important that a flexure model provide a reasonable model of elastic buckling. This section briefly describes the ability of the smooth curvature model to predict elastic buckling in a flexure using the Hessian of the flexure kinematics. As a proxy case for comparing the continuum behavior of a flexure to the smooth curvature model, the smooth curvature model will be used to predict buckling in compression by finding the smallest compressive load for which the generalized stiffness matrix is singular. This result can be compared to Euler’s buckling load formula. The 2 and 3 parameter models produce successively better approximations of buckling.

⎡ 1 / 3 − 1 / 12⎤ ∇ q ∇ q ( x tip ) = L ⎢ ⎥ ⎣− 1 / 12 1 / 30 ⎦

∇ q ∇ q (U (q)) =

∂q i ∂q j

L

= −∫ 0

∂φ ∂φ ds ∂q i ∂q j

(29)

1 12 EI 1 − 2 3P2 L 30

=0

(30)

(31)

This is only 0.75% larger than the true value reported by the continuum model in (23). This exercise can be repeated for the three parameter model, to find the predicted buckling load, P3,

EI 1 − 2 P3 L 3 1 12 1 60

1 12

1 60

EI 1 − 2 3P3 L 30

0

0

EI 1 − 5 P3 L2 210

=0

(32)

The resulting buckling load prediction is within 0.02% of the value predicted by Euler’s beam buckling formula,

(25)

(26)

⎡1 0 ⎤ ⎡ 1 / 3 − 1 / 12⎤ ⎢ ⎥ − PL ⎢ ⎥ ⎣0 1 / 3⎦ ⎣− 1 / 12 1 / 30 ⎦

P2 L2 = 2.4860... EI

Stiffness is a function of the Hessian of the coordinates where force is applied, and of the Hessian of the energy, U(q). When this stiffness matrix has an eigenvalue that is zero or negative, it buckles. In other words, there will exist some eigenvector δq, which, when applied to the robot as a perturbation, will produce a destabilizing force. This could also be thought of as a test for the convexity of the total energy in the robot.

∂ 2 x tip

(28)

The smallest root of this polynomial is the most physically meaningful, as it represents the load at which the unconstrained flexure will buckle,

(24)

When the flexure is straight, q0=q1=q2=0. In this configuration, (25) can be simplified, because the sine term disappears and the cosine term approaches one,

EI L

EI 1 − P2 L2 3 1 12

The concept analogous to elastic buckling in a generalized coordinate model has to do with the generalized stiffness matrix obtained by taking the gradient of the generalized force balance from (16) with respect to q,

L ⎡ ∂ 2 xtip ∂φ ∂φ ∂ 2φ ⎤ = − ∫ ⎢cos(φ ( s)) + sin(φ ( s)) ⎥ ds ∂qi ∂q j ∂qi ∂q j ∂qi ∂q j ⎥⎦ 0 ⎢ ⎣

⎡1 0 ⎤ ⎢ ⎥ ⎣0 1 / 3⎦

The buckling load of the 2 parameter model, P2, is the value of P for which the determinant of K is zero, indicating that the matrix has an eigenvalue equal to zero,

(23)

We will derive the generalized stiffness matrix for the smooth curvature model, when the flexure is loaded in the -x direction with a force, P. Thus, the contact point p from (24) above is the scalar xtip, as described in (8). The Hessian elements can be calculated from the Jacobian of xtip in (20),

EI L

The coordinate Hessian and the energy Hessian can be substituted back into (24),

2

K = ∇ q ∇ q ( p ) T f p + ∇ q ∇ q (U (q ))

(27)

The stiffness due to potential energy can be found by taking the Hessian of the energy function derived in (14),

K=

B. Continuum vs. Discrete Buckling A continuum structure is said to buckle when it has zero (or negative) stiffness in some direction, so that a small perturbation to the structure’s shape is met by a destabilizing force, rather than a restoring force. The compressive load Pcrit at which a clamped-free beam should buckle is given by Euler’s well-known formula [27]:

Pcrit L2 ⎛ π ⎞ = ⎜ ⎟ = 2.4674... EI ⎝2⎠

These are polynomials, which can be evaluated to compute the Hessian of xtip with respect to the generalized coordinates. For the 2 parameter model, the Hessian is a 2 by 2 matrix,

P3 L2 = 2.4677... EI

(33)

C. Discussion The results of this brief study indicate that the smooth curvature model has no difficulty predicting the stiffness of a straightened flexure hinge as a function of load. A more general argument, too long to be presented here, would also cover the stiffness of a constrained flexure, such as one which is pinned or clamped at the distal end. Although the effect of constraints on the generalized stiffness is lengthier to describe,

 143

[5]

the 3 parameter model can successfully describe these cases. Finally, it is worth mentioning that although stiffness prediction of a flexure undergoing large deformation has not been presented here, the smooth curvature model will predict the stiffness matrix for large deformations, as we have shown in [21]. V.

[6]

[7]

CONCLUSIONS

A. In Summary In this paper, we have presented a model for flexible links that is accurate for large deformations, so that it can be used for the special case of flexible links as flexure hinges. These models are compatible with all of the standard tools used for manipulator analysis, because they are in a form where the shape of the joint and the elastic energy of the joint can be entirely described by a set of generalized coordinates. We have demonstrated that a flexure can be described to a high level of accuracy using only three parameters – arguably the minimum number of parameters capable of describing a flexure with three independent end conditions, xtip, ytip and φtip. This model is useful for “zeroth” order descriptions (shape and energy), first order descriptions (local deformation and force), and for second order descriptions (buckling configurations and stiffness) of mechanical behavior of flexible members undergoing large deformations under loads.

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[10]

[11]

[12]

[13]

[14]

B. Future Directions The smooth curvature flexure model presents several obvious directions for further study. First, this model does not take into account axial deformation, which is important in accurately analyzing parallel mechanisms. We are confident that this method can be extended to describe such mechanisms through the addition of one or two axial deformation modes. Second, it is worth noting that although models of planar beam bending are quite useful, many flexures admit a great deal of out-of-plane motion. This behavior can be characterized by modal models similar to the one presented here. We are working on extensions of smooth curvature models to three dimensions, including possible techniques for coping with the greatly increased complexity of describing non-commutative spatial rotations. REFERENCES [1]

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= =

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=

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) W

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= J1 (I − Jˆ2 J2 ) = J1 .



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1 2



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∗ x ¨1 = Ω−1 1 (f1 − f1 ),

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ˆ2 N1 β. +J2 N1 N 7KH ODVW WHUP RI WKLV HTXDWLRQ FDQ EH HOLPLQDWHG E\ QRWLQJ ˆ2 N1 J 2 N1 N

= =

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1

=

J2 N1 − J2 N1 (W 2 J2 N1 )+ W 2 J2 N1 / 1 1 W − 2 W 2 J2 N1 0 1 1 1 −W 2 J2 N1 (W 2 J2 N1 )+ W 2 J2 N1

=

0.

=

∗ J2 M −1 J1T Λ1 Ω−1 1 (f1 − f1 ) −1 T +J2 M −1 J2|1 Λ2|1 Ω2 (f2 − f2∗ ).

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7KHUHIRUH WKH UHVXOWLQJ FORVHGORRS LPSHGDQFH DW x2 LV / 0 ∗ ˙ ˙ x ¨2 = J2 Jˆ2 Ω−1 2 (f2 − f2 ) − J2 q˙ + J2 q˙ 0 / ∗ ˙ +(I − J2 Jˆ2 )J2 J1+ Ω−1 1 (f1 − f1 ) − J1 q˙ .  (TXDWLRQ  FDQ EH XQGHUVWRRG E\ UHFRJQL]LQJ WKDW J2 Jˆ2 DQG (I − J2 Jˆ2 ) DUH RUWKRJRQDO ZHLJKWHG SURMHFWLRQ PDWULFHV J2 Jˆ2 (I − J2 Jˆ2 ) = 0 7KH ÀUVW WHUP LV SURMHFWHG WKURXJK J2 Jˆ2 DQG GHVFULEHV WKH FRQWULEXWLRQ RI WKH VHFRQGSULRULW\ LPSHGDQFH WR WKH FORVHGORRS DFFHOHUDWLRQ RI x2  7KH ODVW WHUP LV SURMHFWHG WKURXJK (I − J2 Jˆ2 ) DQG GHVFULEHV WKH HIIHFW WKDW WKH ÀUVWSULRULW\ LPSHGDQFH REMHFWLYH KDV RQ WKH FORVHG ORRS EHKDYLRU DW WKH VHFRQGSULRULW\ 325 1RWLFH WKDW WKH VHFRQG SULRULW\ LPSHGDQFH RSHUDWHV FRPSOHWHO\ LQGHSHQGHQWO\ LQ WKH UDQJH VSDFH RI J2 Jˆ2  0XOWLSO\LQJ ERWK VLGHV RI (TXDWLRQ  E\ J2 Jˆ2  ZH KDYH ∗ ¨2 = J2 Jˆ2 Ω−1 J2 Jˆ2 x 2 (f2 − f2 ).

∗ ˙ Ω−1 1 (f1 − f1 ) − J1 q˙

∗ ˙ ˙ ZKHUH α = DQG β = Ω−1 2 (f2 − f2 ) − J2 q 'LIIHUHQWLDWLQJ f DQG g DQG FRQVWUXFWLQJ WKH /DJUDQJLDQ ZH KDYH  hT J1 + q¨T J2T W J2 − β T W J2 = 0,

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N1 J2T W J2 M −1 J1T Λ1 α = 0.

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DQG J2|1 = J2 NM , ZKHUH NA LV WKH LQHUWLDZHLJKWHG QXOO VSDFH RI WKH ÀUVW SULRULW\ REMHFWLYH NM = I − M −1 J1T (J1 M −1 J1T )−1 J1 .

 148

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ˆ2 J + J1 M −1 τ +M N 1 +M Jˆ2 J2 M −1 τ ˆ2 N1 β +M N1 N

DQG (TXDWLRQ  LV LGHQWLFDO WR (TXDWLRQ  :KHQ H[WHUQDO ORDGV DUH DSSOLHG WR ERWK 325V VXFK WKDW τ = J1T f1 + J2T f2  WKHQ ORDGV DSSOLHG WR WKH VHFRQGSULRULW\ 325 f2  DUH SUR MHFWHG WKURXJK Λ1 J1 M −1 J2T RQWR WKH ÀUVWSULRULW\ 325

−τ.



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= =

N1 (I − Jˆ2 J2 )N1 (N1 − N1 Jˆ2 J2 )N1 (N1 − Jˆ2 J2 )N1 (I − Jˆ2 J2 )N1

=

ˆ2 N1 . N

= =

x ¨2

ˆ2 J + (Λ−1 f ∗ + J˙1 q) J2 M −1 τ − J2 N ˙ 1 1 1 −1 ∗ −J2 Jˆ2 (Λ f + J˙2 q) ˙ + J˙2 q˙

=

1

2

∗ ˙ ˙ J2 M −1 τ − (I − J2 Jˆ2 )J1+ (Λ−1 1 f1 + J1 q) −1 ∗ −J2 Jˆ2 Λ f2 + (I − J2 Jˆ2 )J˙2 q. ˙

=

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1

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ˆ2 N1 γ. +M N

Λ1 J1 M −1 τ = f1 + Λ1 J1 M −1 J2T f2 .



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