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STATE ESTIMATION FOR
ROBOTICS
A Matrix-Lie-Group Approach
Timothy D. Barfoot
Copyright
c
2016
In preparation for publication by Cambridge University Press
Send errata to <tim.barfoot@utoronto.ca>
Draft compiled on January 14, 2016
Revision History
1 Sept 2015 First draft released
15 Dec 2015 Added a new section to Chapter 3 on recurive
discrete-time smoothers and their relationship
to the batch solution; fixed a few typos
17 Dec 2015 Fixed a few typos in the new section on
smoothers
14 Jan 2016 Added historical note regarding Stanley
Schmidt’s role in EKF to Chapter 4
iii
Contents
Notation xi
Foreword xiii
1 Introduction 1
1.1 A Little History 1
1.2 Sensors, Measurements, and Problem Definition 3
1.3 How This Book is Organized 4
1.4 Relationship to Other Books 5
Part I Estimation Machinery 7
2 Primer on Probability Theory 9
2.1 Probability Density Functions 9
2.1.1 Definitions 9
2.1.2 Bayes’ Rule and Inference 10
2.1.3 Moments of PDFs 11
2.1.4 Sample Mean and Covariance 11
2.1.5 Statistically Independent, Uncorrelated 12
2.1.6 Shannon and Mutual Information 12
2.1.7 Cram´er-Rao Lower Bound and Fisher Information 13
2.2 Gaussian Probability Density Functions 14
2.2.1 Definitions 14
2.2.2 Isserlis’ Theorem 15
2.2.3 Joint Gaussian PDFs, Their Factors, and Inference 18
2.2.4 Statistically Independent, Uncorrelated 19
2.2.5 Linear Change of Variables 20
2.2.6 Product of Gaussians 22
2.2.7 Sherman-Morrison-Woodbury Identity 23
2.2.8 Passing a Gaussian Through a Nonlinearity 24
2.2.9 Shannon Information of a Gaussian 28
2.2.10 Mutual Information of a Joint Gaussian PDF 29
2.2.11 Cram´er-Rao Lower Bound Applied to Gaussian PDFs 30
2.3 Gaussian Processes 31
2.4 Summary 32
2.5 Exercises 33
v
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