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Mobile Robotics Mathematics, Models, and Methods Mobile Robotics offers comprehensive coverage of the essentials of the field suitable for both students and practitioners. Adapted from the author's graduate and undergrad- uate courses, the content of the book reflects current approaches to developing effective mobile robots. Professor Alonzo Kelly adapts principles and techniques from the fields of mathematics, physics, and numerical methods to present a consistent framework in a notation that facilitates learning and highlights relationships between topics. This text was developed specifically to be accessible to senior-level undergraduates in engineer- ing and computer science, and includes supporting exercises to reinforce the lessons of each section. Practitioners will value the author’s perspectives on practical applications of implementable algorithms extracted from real systems wherever possible, to enhance the real-world relevance of the text. these principles. Complex subjects are reduced to Alonzo Kelly holds undergraduate degrees in aerospace engineering and computer science, and graduate degrees in robotics. Dr. Kelly worked in the aerospace industry for ten years before returning to academia. As a professor at the Robotics institute at Carnegie Mellon University, he teaches mobile robotics at the graduate and under- graduate levels, conducting research in robot simulation, modeling, controls, position estimation, motion planning, and human interfaces.

Mobile Robotics Mathematics, Models, and Methods Alonzo Kelly Carnegie Mellon University

32 Avenue of the Americas, New York NY 10013-2473, USA Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107031159 © Alonzo Kelly 2013 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2013 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication data Kelly, Alonzo. Mobile robotics : mathematics, models and methods / Alonzo Kelly. pages cm Includes bibliographical references and index. ISBN 978-1-107-03115-9 (hardback) 1. Mobile robots–Textbooks. TJ211.415.K39 2013 629.8′932–dc23 2013022113 I. Title. ISBN 978-1-107-03115-9 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication, and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

Contents Preface 1 Introduction 1.1 1.2 1.3 Applications of Mobile Robots Types of Mobile Robots 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 Automated Guided Vehicles (AGVs) Service Robots Cleaning and Lawn Care Robots Social Robots Field Robots Inspection, Reconnaissance, Surveillance, and Exploration Robots Mobile Robot Engineering 1.3.1 1.3.2 1.3.3 1.3.4 1.3.5 Mobile Robot Subsystems Overview of the Text Fundamentals of Wheeled Mobile Robots References and Further Reading Exercise 2 Math Fundamentals 2.1 2.2 Notational Conventions Embedded Coordinate Frames References and Further Reading Conventions and Definitions 2.1.1 2.1.2 2.1.3 Matrices 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.2.7 2.2.8 2.2.9 Matrix Operations Matrix Functions Matrix Inversion Rank-Nullity Theorem Matrix Algebra Matrix Calculus Leibnitz’ Rule References and Further Reading Exercises v page xiii 1 2 2 2 3 4 4 5 6 7 7 8 9 11 11 12 12 13 17 21 21 21 24 25 28 29 31 39 40 40

vi 2.3 2.4 2.5 2.6 2.7 2.8 C O N T E N T S Forward Kinematics Inverse Kinematics Differential Kinematics References and Further Reading Exercises Definitions Why Homogeneous Transforms Semantics and Interpretations References and Further Reading Exercises Fundamentals of Rigid Transforms 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 Kinematics of Mechanisms 2.4.1 2.4.2 2.4.3 2.4.4 2.4.5 Orientation and Angular Velocity 2.5.1 2.5.2 2.5.3 2.5.4 2.5.5 2.5.6 Kinematic Models of Sensors 2.6.1 2.6.2 2.6.3 2.6.4 Transform Graphs and Pose Networks 2.7.1 2.7.2 2.7.3 2.7.4 2.7.5 2.7.6 Quaternions 2.8.1 2.8.2 2.8.3 2.8.4 2.8.5 2.8.6 2.8.7 Representations and Notation Quaternion Multiplication Other Quaternion Operations Representing 3D Rotations Attitude and Angular Velocity References and Further Reading Exercises Orientation in Euler Angle Form Angular Rates and Small Angles Angular Velocity and Orientation Rates in Euler Angle Form Angular Velocity and Orientation Rates in Angle-Axis Form References and Further Reading Exercises Kinematics of Video Cameras Kinematics of Laser Rangefinders References and Further Reading Exercises Transforms as Relationships Solving Pose Networks Overconstrained Networks Differential Kinematics Applied to Frames in General Position References and Further Reading Exercises 3 Numerical Methods 3.1 3.2 Linearization Optimization of Objective Functions Constrained Optimization References and Further Reading Exercises Linearization and Optimization of Functions of Vectors 3.1.1 3.1.2 3.1.3 3.1.4 3.1.5 Systems of Equations 3.2.1 3.2.2 Linear Systems Nonlinear Systems 41 41 42 43 55 56 57 57 61 66 69 69 70 70 75 77 79 81 81 82 82 83 89 90 90 90 93 95 97 102 103 103 104 105 107 109 111 114 114 116 116 117 120 124 130 130 131 131 136

C O N T E N T S 3.3 3.4 3.5 References and Further Reading Exercises Nonlinear Optimization Constrained Optimization References and Further Reading Exercises 3.2.3 3.2.4 Nonlinear and Constrained Optimization 3.3.1 3.3.2 3.3.3 3.3.4 Differential Algebraic Systems 3.4.1 3.4.2 3.4.3 3.4.4 3.4.5 3.4.6 Integration of Differential Equations 3.5.1 3.5.2 3.5.3 3.5.4 Dynamic Models in State Space Integration of State Space Models References and Further Reading Exercises Constrained Dynamics First- and Second-Order Constrained Kinematic Systems Lagrangian Dynamics Constraints References and Further Reading Exercises 4 Dynamics 4.1 4.2 4.3 4.4 Aspects of Rigid Body Motion WMR Velocity Kinematics for Fixed Contact Point Common Steering Configurations References and Further Reading Exercises Moving Coordinate Systems Context of Measurement 4.1.1 Change of Reference Frame 4.1.2 Example: Attitude Stability Margin Estimation 4.1.3 4.1.4 Recursive Transformations of State of Motion References and Further Reading 4.1.5 4.1.6 Exercises Kinematics of Wheeled Mobile Robots 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 Constrained Kinematics and Dynamics Constraints of Disallowed Direction 4.3.1 Constraints of Rolling Without Slipping 4.3.2 Lagrangian Dynamics 4.3.3 4.3.4 Terrain Contact Trajectory Estimation and Prediction 4.3.5 References and Further Reading 4.3.6 4.3.7 Exercises Aspects of Linear Systems Theory Linear Time-Invariant Systems 4.4.1 State Space Representation of Linear Dynamical Systems 4.4.2 Nonlinear Dynamical Systems 4.4.3 4.4.4 Perturbative Dynamics of Nonlinear Dynamical Systems References and Further Reading 4.4.5 4.4.6 Exercises vii 138 139 140 140 146 150 150 151 151 154 157 162 166 167 168 168 168 172 172 173 173 174 175 180 182 186 186 187 187 191 195 200 201 201 202 207 211 217 220 224 225 226 227 234 239 240 244 244

viii 4.5 C O N T E N T S Predictive Modeling and System Identification 4.5.1 4.5.2 4.5.3 4.5.4 4.5.5 4.5.6 4.5.7 4.5.8 Braking Turning Vehicle Rollover Wheel Slip and Yaw Stability Parameterization and Linearization of Dynamic Models System Identification References and Further Reading Exercises 5 Optimal Estimation Variance of Continuous Integration and Averaging Processes Stochastic Integration Optimal Estimation References and Further Reading Exercises 5.1 5.2 5.3 5.4 Characterizing Uncertainty Random Variables Transformation of Uncertainty Random Processes References and Further Reading Exercises Random Variables, Processes, and Transformation 5.1.1 5.1.2 5.1.3 5.1.4 5.1.5 5.1.6 Covariance Propagation and Optimal Estimation 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 State Space Kalman Filters 5.3.1 5.3.2 5.3.3 5.3.4 5.3.5 5.3.6 5.3.7 5.3.8 Bayesian Estimation 5.4.1 5.4.2 5.4.3 5.4.4 5.4.5 5.4.6 5.4.7 Introduction Linear Discrete Time Kalman Filter Kalman Filters for Nonlinear Systems Simple Example: 2D Mobile Robot Pragmatic Information for Kalman Filters Other Forms of the Kalman Filter References and Further Reading Exercises Definitions Bayes’ Rule Bayes’ Filters Bayesian Mapping Bayesian Localization References and Further Reading Exercises 6 State Estimation 6.1 Mathematics of Pose Estimation 6.1.1 6.1.2 6.1.3 6.1.4 6.1.5 6.1.6 6.1.7 6.1.8 Pose Fixing versus Dead Reckoning Pose Fixing Error Propagation in Triangulation Real Pose Fixing Systems Dead Reckoning Real Dead Reckoning Systems References and Further Reading Exercises 245 245 247 250 253 256 259 268 269 270 270 270 272 279 289 294 295 296 296 301 307 315 315 316 316 319 321 327 338 344 344 345 346 346 349 353 358 365 369 369 370 370 371 372 376 384 385 396 396 397

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