Mobile Robotics. Mathematics, Models, and Methods

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1 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 undergraduate courses, the content of the book reflects current approaches to developing effective mobile robots. Professor 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 engineering 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 these principles. Complex subjects are reduced to implementable algorithms extracted from real systems wherever possible, to enhance the real-world relevance of the text. 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 undergraduate levels, conducting research in robot simulation, modeling, controls, position and estimation, motion planning, and human interfaces.

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3 Mobile Robotics Mathematics, Models, and Methods Carnegie Mellon University

4 32 Avenue of the Americas, New York NY , 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. w w w.ca mbr idge.org Information on this title: / A lon z o Kel ly 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 /. pages cm Includes bibliographical references and index. ISBN (hardback) 1. Mobile robots Textbooks. I. Title. TJ K dc ISBN Ha rdba ck 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.

5 Contents Preface page xiii 1 Introduction Applications of Mobile Robots Types of Mobile Robots Automated Guided Vehicles (AGVs) Service Robots Cleaning and Lawn Care Robots Social Robots Field Robots Inspection, Reconnaissance, Surveillance, and Exploration Robots Mobile Robot Engineering Mobile Robot Subsystems Overview of the Text Fundamentals of Wheeled Mobile Robots References and Further Reading Exercise Math Fundamentals Conventions and Definitions Notational Conventions Embedded Coordinate Frames References and Further Reading 2.2 Matrices Matrix Operations Matrix Functions Matrix Inversion Rank-Nullity Theorem Matrix Algebra Matrix Calculus Leibnitz Rule References and Further Reading Exercises 40 v

6 vi CONTENTS 2.3 Fundamentals of Rigid Transforms Definitions Why Homogeneous Transforms Semantics and Interpretations References and Further Reading Exercises Kinematics of Mechanisms Forward Kinematics Inverse Kinematics Differential Kinematics References and Further Reading Exercises Orientation and Angular Velocity 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 Kinematic Models of Sensors Kinematics of Video Cameras Kinematics of Laser Rangefinders References and Further Reading Exercises Transform Graphs and Pose Networks Transforms as Relationships Solving Pose Networks Overconstrained Networks Differential Kinematics Applied to Frames in General Position References and Further Reading Exercises Quaternions Representations and Notation Quaternion Multiplication Other Quaternion Operations Representing 3D Rotations Attitude and Angular Velocity References and Further Reading Exercises Numerical Methods Linearization and Optimization of Functions Vectors of Linearization Optimization of Objective Functions Constrained Optimization References and Further Reading Exercises Systems of Equations Linear Systems Nonlinear Systems 136

7 CONTENTS vii References and Further Reading Exercises Nonlinear and Constrained Optimization Nonlinear Optimization Constrained Optimization References and Further Reading Exercises Differential Algebraic Systems Constrained Dynamics First- and Second-Order Constrained Kinematic Systems Lagrangian Dynamics Constraints References and Further Reading Exercises Integration of Differential Equations Dynamic Models in State Space Integration of State Space Models References and Further Reading Exercises Dynamics Moving Coordinate Systems Context of Measurement Change of Reference Frame Example: Attitude Stability Margin Estimation Recursive Transformations of State of Motion References and Further Reading Exercises Kinematics of Wheeled Mobile Robots Aspects of Rigid Body Motion WMR Velocity Kinematics for Fixed Contact Point Common Steering Configurations References and Further Reading Exercises Constrained Kinematics and Dynamics Constraints of Disallowed Direction Constraints of Rolling Without Slipping Lagrangian Dynamics Terrain Contact Trajectory Estimation and Prediction References and Further Reading Exercises Aspects of Linear Systems Theory Linear Time-Invariant Systems State Space Representation of Linear Dynamical Systems Nonlinear Dynamical Systems Perturbative Dynamics of Nonlinear Dynamical Systems References and Further Reading Exercises 244

8 viii CONTENTS 4.5 Predictive Modeling and System Identification Braking Turning Vehicle Rollover Wheel Slip and Yaw Stability Parameterization and Linearization of Dynamic Models System Identification References and Further Reading Exercises Optimal Estimation Random Variables, Processes, and Transformation Characterizing Uncertainty Random Variables Transformation of Uncertainty Random Processes References and Further Reading Exercises Covariance Propagation and Optimal Estimation Variance of Continuous Integration and Averaging Processes Stochastic Integration Optimal Estimation References and Further Reading Exercises State Space Kalman Filters 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 Bayesian Estimation Definitions Bayes Rule Bayes Filters Bayesian Mapping Bayesian Localization References and Further Reading Exercises State Estimation Mathematics of Pose Estimation 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 397

9 CONTENTS ix 6.2 Sensors for State Estimation Articulation Sensors Ambient Field Sensors Inertial Frames of Reference Inertial Sensors References and Further Reading Exercises Inertial Navigation Systems Introduction Mathematics of Inertial Navigation Errors and Aiding in Inertial Navigation Example: Simple Odometry-Aided Attitude and Heading Reference System References and Further Reading Exercises Satellite Navigation Systems Introduction Implementation State Measurement Performance Modes of Operation References and Further Reading Exercises Control Classical Control Introduction Virtual Spring-Damper Feedback Control Model Referenced and Feedforward Control References and Further Reading Exercises State Space Control Introduction State Space Feedback Control Example: Robot Trajectory Following Perception Based Control Steering Trajectory Generation References and Further Reading Exercises Optimal and Model Predictive Control Calculus of Variations Optimal Control Model Predictive Control Techniques for Solving Optimal Control Problems Parametric Optimal Control References and Further Reading Exercises 492

10 x CONTENTS 7.4 Intelligent Control Introduction Evaluation Representation Search References and Further Reading Exercises Perception Image Processing Operators and Algorithms Taxonomy of Computer Vision Algorithms High-Pass Filtering Operators Low-Pass Operators Matching Signals and Images Feature Detection Region Processing References and Further Reading Exercises Physics and Principles Radiative of Sensors Radiative Sensors Techniques for Range Sensing Radiation Lenses, Filters, and Mirrors References and Further Reading Exercises Sensors for Perception Laser Rangefinders Ultrasonic Rangefinders Visible Wavelength Cameras Mid to Far Infrared Wavelength Cameras Radars References and Further Reading Exercises Aspects of Geometric and Semantic Computer Vision Pixel Classification Computational Stereo Vision Obstacle Detection References and Further Reading Exercises Localization and Mapping Representation and Issues Introduction Representation Timing and Motion Issues Related Localization Issues Structural Aspects Example: Unmanned Ground Vehicle (UGV) Terrain Mapping References and Further Reading Exercises 593

11 CONTENTS xi 9.2 Visual Localization and Motion Estimation Introduction Aligning Signals for Localization and Motion Estimation Matching Features for Localization and Motion Estimation Searching for the Optimal Pose References and Further Reading Exercises Simultaneous Localization and Mapping Introduction Global Consistency in Cyclic Maps Revisiting EKF SLAM for Discrete Landmarks Example: Autosurveying of Laser Reflectors References and Further Reading Exercises Motion Planning Introduction Introducing Path Planning Formulation of Path Planning Obstacle-Free Motion Planning References and Further Reading Exercises Representation and Search for Global Path Planning Sequential Motion Planning Big Ideas in Optimization and Search Uniform Cost Sequential Planning Algorithms Weighted Sequential Planning Representation for Sequential Motion Planning References and Further Reading Exercises Real-Time Global Motion Planning: Moving in Unknown and Dynamic Environments Introduction Depth-Limited Approaches Anytime Approaches Plan Repair Approach: D* Algorithm Hierarchical Planning References and Further Reading Exercise 690 Index 691

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13 Preface Robotics can be a very challenging and very satisfying way to spend your time. A profound moment in the history of most roboticists is first the moment a robot performed a task under the influence of his or her software or electronics. Although productive a pursuit of the study of robotics involves aspects of engineering, mathematics, and physics, its elements do not convey the magic we all feel when interacting with a responsive semi-intelligent device of our own creation. This book introduces the science and engineering of a particularly interesting class of robots mobile robots. Although there many are analogs to the field of robot manipulators, mobile robots are sufficiently different to justify their treatment in an entirely separate text. Although the book concentrates on wheeled mobile robots, most of its content is independent of specific the locomotion subsystem used. The field of mobile robots is changing rapidly. Many specialties are evolving in both the research and the commercial sectors. Any textbook offered in such evolving field will represent only a snapshot of field the as it was understood at the time of an publication. However, the rapid growth of the field, its several decades of history, and its pervasive popular appeal suggest that the time is now right to produce an early text that attempts to codify some of the fundamental ideas in a more accessible manner. Another indication of timeliness might be the fact that much useful information must be omitted. Many topics, such as perception, are treated only briefly, and others, including legged locomotion, calibration, simulation,human interfaces, and multirobot systems, are omitted completely. The goal of this book is to extract from both the underlying specialties and the depth of mobile robotics research literature a coherent exposition of the concepts, methods, and issues that rise to the forefront in practice, and to represent the core that is unique about this field. To that end, as much as possible of the material is restricted to two-dimensional wheeled vehicle motion and to structured environments. These assumptions produce a consistent exposition with just enough richness to relevant be and illustrative without overwhelming the reader with details irrelevant to the purpose. The book follows a logical progression, mimicking the order in which mobile robots are constructed. Each chapter represents new a topic or capability that depends on what came before, and the concepts involved span the fields of numerical methods, xiii

14 xiv PREFACE signal processing, estimation and control theory, computer vision, and artificial intelligence in that order. As of this writing, the Mars Science Laboratory Rover named Curiosity has recently arrived on Mars. It is our third mobile robotic mission to Mars and legacy the of the last (MER) mission is already historic. This book is not for everyone, but for those who are prepared and motivated, if you master the content of the text you have will a very good idea of what is going on inside the brain of a mobile robot, and you will be well prepared to make one of your own.

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