STATE ESTIMATION FOR ROBOTICS.pdf

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Acronyms and Abbreviations

Notation

Foreword

Introduction

A Little History

Sensors, Measurements, and Problem Definition

How This Book Is Organized

Relationship to Other Books

Part I Estimation Machinery

Primer on Probability Theory

Probability Density Functions

Definitions

Bayes' Rule and Inference

Moments

Sample Mean and Covariance

Statistically Independent, Uncorrelated

Normalized Product

Shannon and Mutual Information

Cramér-Rao Lower Bound and Fisher Information

Gaussian Probability Density Functions

Definitions

Isserlis' Theorem

Joint Gaussian PDFs, Their Factors, and Inference

Statistically Independent, Uncorrelated

Linear Change of Variables

Normalized Product of Gaussians

Sherman-Morrison-Woodbury Identity

Passing a Gaussian through a Nonlinearity

Shannon Information of a Gaussian

Mutual Information of a Joint Gaussian PDF

Cramér-Rao Lower Bound Applied to Gaussian PDFs

Gaussian Processes

Summary

Exercises

Linear-Gaussian Estimation

Batch Discrete-Time Estimation

Problem Setup

Maximum A Posteriori

Bayesian Inference

Existence, Uniqueness, and Observability

MAP Covariance

Recursive Discrete-Time Smoothing

Exploiting Sparsity in the Batch Solution

Cholesky Smoother

Rauch-Tung-Striebel Smoother

Recursive Discrete-Time Filtering

Factoring the Batch Solution

Kalman Filter via MAP

Kalman Filter via Bayesian Inference

Kalman Filter via Gain Optimization

Kalman Filter Discussion

Error Dynamics

Existence, Uniqueness, and Observability

Batch Continuous-Time Estimation

Gaussian Process Regression

A Class of Exactly Sparse Gaussian Process Priors

Linear Time-Invariant Case

Relationship to Batch Discrete-Time Estimation

Summary

Exercises

Nonlinear Non-Gaussian Estimation

Introduction

Full Bayesian Estimation

Maximum a Posteriori Estimation

Recursive Discrete-Time Estimation

Problem Setup

Bayes Filter

Extended Kalman Filter

Generalized Gaussian Filter

Iterated Extended Kalman Filter

IEKF Is a MAP Estimator

Alternatives for Passing PDFs through Nonlinearities

Particle Filter

Sigmapoint Kalman Filter

Iterated Sigmapoint Kalman Filter

ISPKF Seeks the Posterior Mean

Taxonomy of Filters

Batch Discrete-Time Estimation

Maximum A Posteriori

Bayesian Inference

Maximum Likelihood

Discussion

Batch Continuous-Time Estimation

Motion Model

Observation Model

Bayesian Inference

Algorithm Summary

Summary

Exercises

Biases, Correspondences, and Outliers

Handling Input/Measurement Biases

Bias Effects on the Kalman Filter

Unknown Input Bias

Unknown Measurement Bias

Data Association

External Data Association

Internal Data Association

Handling Outliers

RANSAC

M-Estimation

Covariance Estimation

Summary

Exercises

Part II Three-Dimensional Machinery

Primer on Three-Dimensional Geometry

Vectors and Reference Frames

Reference Frames

Dot Product

Cross Product

Rotations

Rotation Matrices

Principal Rotations

Alternate Rotation Representations

Rotational Kinematics

Perturbing Rotations

Poses

Transformation Matrices

Robotics Conventions

Frenet-Serret Frame

Sensor Models

Perspective Camera

Stereo Camera

Range-Azimuth-Elevation

Inertial Measurement Unit

Summary

Exercises

Matrix Lie Groups

Geometry

Special Orthogonal and Special Euclidean Groups

Lie Algebras

Exponential Map

Adjoints

Baker-Campbell-Hausdorff

Distance, Volume, Integration

Interpolation

Homogeneous Points

Calculus and Optimization

Identities

Kinematics

Rotations

Poses

Linearized Rotations

Linearized Poses

Probability and Statistics

Gaussian Random Variables and PDFs

Uncertainty on a Rotated Vector

Compounding Poses

Fusing Poses

Propagating Uncertainty through a Nonlinear Camera Model

Summary

Exercises

Part III Applications

Pose Estimation Problems

Point-Cloud Alignment

Problem Setup

Unit-Length Quaternion Solution

Rotation Matrix Solution

Transformation Matrix Solution

Point-Cloud Tracking

Problem Setup

Motion Priors

Measurement Model

EKF Solution

Batch Maximum a Posteriori Solution

Pose-Graph Relaxation

Problem Setup

Batch Maximum Likelihood Solution

Initialization

Exploiting Sparsity

Chain Example

Pose-and-Point Estimation Problems

Bundle Adjustment

Problem Setup

Measurement Model

Maximum Likelihood Solution

Exploiting Sparsity

Interpolation Example

Simultaneous Localization and Mapping

Problem Setup

Batch Maximum a Posteriori Solution

Exploiting Sparsity

Example

Continuous-Time Estimation

Motion Prior

General

Simplification

Simultaneous Trajectory Estimation and Mapping

Problem Setup

Measurement Model

Batch Maximum a Posteriori Solution

Exploiting Sparsity

Interpolation

Postscript

References

Index

STATE ESTIMATION FOR ROBOTICS Timothy D. Barfoot Copyright c 2017 Cambridge University Press is the Ocial Publisher This Unocial Version Compiled on May 13, 2017 Send errata to

Revision History 13 May 2017 Version best matching published ﬁrst edition iii

Contents Acronyms and Abbreviations Notation Foreword 1 1.1 1.2 1.3 1.4 2 2.1 Introduction A Little History Sensors, Measurements, and Problem Deﬁnition How This Book Is Organized Relationship to Other Books Part I Estimation Machinery Primer on Probability Theory Probability Density Functions 2.1.1 Deﬁnitions 2.1.2 Bayes’ Rule and Inference 2.1.3 Moments 2.1.4 2.1.5 2.1.6 Normalized Product 2.1.7 2.1.8 Cram´er-Rao Lower Bound and Fisher Information Sample Mean and Covariance Statistically Independent, Uncorrelated Shannon and Mutual Information 2.2 Gaussian Probability Density Functions Isserlis’ Theorem Joint Gaussian PDFs, Their Factors, and Inference Statistically Independent, Uncorrelated 2.2.1 Deﬁnitions 2.2.2 2.2.3 2.2.4 2.2.5 Linear Change of Variables 2.2.6 Normalized Product of Gaussians 2.2.7 2.2.8 Passing a Gaussian through a Nonlinearity 2.2.9 2.2.10 Mutual Information of a Joint Gaussian PDF 2.2.11 Cram´er-Rao Lower Bound Applied to Gaussian PDFs Sherman-Morrison-Woodbury Identity Shannon Information of a Gaussian 2.3 Gaussian Processes 2.4 2.5 Summary Exercises xi xiii xv 1 1 3 4 5 7 9 9 9 10 11 12 12 13 14 14 15 15 16 18 20 20 22 23 24 28 30 30 32 33 33 v

vi 3 3.1 3.2 3.3 3.4 3.5 3.6 4 4.1 4.2 4.3 Linear-Gaussian Estimation Batch Discrete-Time Estimation 3.1.1 Problem Setup 3.1.2 Maximum A Posteriori 3.1.3 Bayesian Inference 3.1.4 Existence, Uniqueness, and Observability 3.1.5 MAP Covariance Recursive Discrete-Time Smoothing 3.2.1 Exploiting Sparsity in the Batch Solution 3.2.2 Cholesky Smoother 3.2.3 Rauch-Tung-Striebel Smoother Recursive Discrete-Time Filtering 3.3.1 Factoring the Batch Solution 3.3.2 Kalman Filter via MAP 3.3.3 Kalman Filter via Bayesian Inference 3.3.4 Kalman Filter via Gain Optimization 3.3.5 Kalman Filter Discussion 3.3.6 Error Dynamics 3.3.7 Existence, Uniqueness, and Observability Batch Continuous-Time Estimation 3.4.1 Gaussian Process Regression 3.4.2 A Class of Exactly Sparse Gaussian Process Priors 3.4.3 Linear Time-Invariant Case 3.4.4 Relationship to Batch Discrete-Time Estimation Summary Exercises Iterated Extended Kalman Filter IEKF Is a MAP Estimator Nonlinear Non-Gaussian Estimation Introduction 4.1.1 Full Bayesian Estimation 4.1.2 Maximum a Posteriori Estimation Recursive Discrete-Time Estimation 4.2.1 Problem Setup 4.2.2 Bayes Filter 4.2.3 Extended Kalman Filter 4.2.4 Generalized Gaussian Filter 4.2.5 4.2.6 4.2.7 Alternatives for Passing PDFs through Nonlinearities 4.2.8 Particle Filter 4.2.9 4.2.10 Iterated Sigmapoint Kalman Filter 4.2.11 ISPKF Seeks the Posterior Mean 4.2.12 Taxonomy of Filters Batch Discrete-Time Estimation 4.3.1 Maximum A Posteriori 4.3.2 Bayesian Inference 4.3.3 Maximum Likelihood 4.3.4 Discussion Sigmapoint Kalman Filter Contents 37 37 37 39 44 46 50 51 52 53 55 58 59 63 68 69 70 71 72 74 74 77 83 87 88 88 91 91 92 94 96 96 97 100 103 105 106 107 116 118 123 126 127 127 128 135 137 142

Contents 4.4 4.5 4.6 5 5.1 Batch Continuous-Time Estimation 4.4.1 Motion Model 4.4.2 Observation Model 4.4.3 Bayesian Inference 4.4.4 Algorithm Summary Summary Exercises Biases, Correspondences, and Outliers Handling Input/Measurement Biases 5.1.1 Bias E↵ects on the Kalman Filter 5.1.2 Unknown Input Bias 5.1.3 Unknown Measurement Bias 5.2 Data Association 5.2.1 External Data Association 5.2.2 Internal Data Association Handling Outliers 5.3.1 RANSAC 5.3.2 M-Estimation 5.3.3 Covariance Estimation Summary Exercises Part II Three-Dimensional Machinery Primer on Three-Dimensional Geometry Vectors and Reference Frames 6.1.1 Reference Frames 6.1.2 Dot Product 6.1.3 Cross Product Rotations 6.2.1 Rotation Matrices 6.2.2 Principal Rotations 6.2.3 Alternate Rotation Representations 6.2.4 Rotational Kinematics 6.2.5 Perturbing Rotations Poses 6.3.1 Transformation Matrices 6.3.2 Robotics Conventions 6.3.3 Frenet-Serret Frame Sensor Models 6.4.1 Perspective Camera 6.4.2 6.4.3 Range-Azimuth-Elevation 6.4.4 Inertial Measurement Unit Summary Exercises Stereo Camera 5.3 5.4 5.5 6 6.1 6.2 6.3 6.4 6.5 6.6 vii 143 143 146 146 147 148 149 151 152 152 155 157 159 160 160 161 162 163 166 168 168 171 173 173 174 174 175 176 176 177 178 184 188 192 193 194 196 199 199 206 208 209 211 212

viii Matrix Lie Groups 7 7.1 Geometry Contents Special Orthogonal and Special Euclidean Groups 7.1.1 7.1.2 Lie Algebras 7.1.3 Exponential Map 7.1.4 Adjoints 7.1.5 Baker-Campbell-Hausdor↵ 7.1.6 Distance, Volume, Integration 7.1.7 7.1.8 Homogeneous Points 7.1.9 Calculus and Optimization 7.1.10 Identities Interpolation 7.2 Kinematics 7.3 7.4 7.5 8 8.1 8.2 8.3 7.2.1 Rotations 7.2.2 Poses 7.2.3 Linearized Rotations 7.2.4 Linearized Poses Probability and Statistics 7.3.1 Gaussian Random Variables and PDFs 7.3.2 Uncertainty on a Rotated Vector 7.3.3 Compounding Poses 7.3.4 Fusing Poses 7.3.5 Propagating Uncertainty through a Nonlinear Camera Model Summary Exercises Part III Applications Pose Estimation Problems Point-Cloud Alignment 8.1.1 Problem Setup 8.1.2 Unit-Length Quaternion Solution 8.1.3 Rotation Matrix Solution 8.1.4 Transformation Matrix Solution Point-Cloud Tracking 8.2.1 Problem Setup 8.2.2 Motion Priors 8.2.3 Measurement Model 8.2.4 EKF Solution 8.2.5 Batch Maximum a Posteriori Solution Pose-Graph Relaxation 8.3.1 Problem Setup 8.3.2 Batch Maximum Likelihood Solution 8.3.3 8.3.4 Exploiting Sparsity 8.3.5 Chain Example Initialization 215 215 215 217 219 226 230 237 240 246 246 254 255 255 258 261 265 266 267 271 273 280 285 292 293 295 297 297 298 298 302 316 319 319 320 321 322 325 329 329 330 333 333 334

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