机器人学状态估计：SLAM与AI基础

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"《机器人状态估计》是一本关于机器人学数学基础和SLAM（Simultaneous Localization and Mapping，同时定位与建图）的书籍，适合AI初学者和对机器人学感兴趣的读者。作者是Timothy D. Barfoot，由剑桥大学出版社于2017年正式出版。此书在不断更新修订，提供了对机器人状态估计领域的深入理解和实践指导。" 本书旨在介绍机器人状态估计的基础知识，这是机器人学和人工智能领域中的关键概念。状态估计是让机器人理解其自身位置和环境的重要手段，对于自主导航和感知至关重要。书中涵盖了概率理论、传感器测量、问题定义以及与其他相关书籍的对比。 Part I - "Estimation Machinery" 部分，作者首先介绍了概率论的基础知识，这是理解状态估计算法的基石。概率密度函数（PDFs）的定义、贝叶斯规则及其在推断中的应用、矩的计算，以及样本均值的概念，这些都为后续的统计推断和滤波算法打下基础。 2.1.1节详细阐述了概率密度函数的定义，它是描述随机变量概率分布的关键工具。2.1.2节讲解了贝叶斯规则，它是处理不确定性信息的核心方法，特别是在状态估计中，通过结合先验知识和观测数据来更新对机器人状态的估计。2.1.3节则讨论了矩的概念，它们用于描述分布的形状和中心趋势。最后，2.1.4节介绍了样本均值，它是估计总体均值的常用统计量。随着这些基础知识的铺垫，读者可以逐渐进入更复杂的主题，如滤波算法（如卡尔曼滤波、扩展卡尔曼滤波和粒子滤波），这些算法在处理实时传感器数据和解决定位问题时扮演着核心角色。书中还会涉及误差分析、传感器融合、系统模型建立以及优化理论等内容，这些都是构建精确和鲁棒的机器人状态估计系统的关键。此书不仅对理论进行了深入探讨，还提供了实例和练习，帮助读者将理论应用于实际问题，这对于初学者来说是非常宝贵的。通过对《机器人状态估计》的学习，读者将能够掌握机器人自主导航所需的关键技术，从而在AI和机器人学的道路上更进一步。

xvi Foreword

ments, I have often been faced with vehicles operating in three dimen-

sions. Accordingly, I have attempted to provide a detailed look at how

to approach state estimation in three dimensions. In particular, I show

how to treat rotations and poses in a simple and practical way using

matrix Lie groups. The reader should have a background in under-

graduate linear algebra and calculus, but otherwise, this book is fairly

standalone. I hope readers of these pages will ﬁnd something useful; I

know I learned a great deal while creating them.

I have provided some historical notes in the margins throughout the

book, mostly in the form of biographical sketches of some of the re-

searchers after whom various concepts and techniques are named; I

primarily used Wikipedia as the source for this information. Also, the

ﬁrst part of Chapter 6 (up to the alternate rotation parameterizations),

which introduces three-dimensional geometry, is based heavily on notes

originally produced by Chris Damaren at the University of Toronto In-

stitute for Aerospace Studies.

This book would not have been possible without the collaborations of

many fantastic graduate students along the way. Paul Furgale’s PhD

thesis extended my understanding of matrix Lie groups signiﬁcantly

by introducing me to their use for describing poses; this led us on

an interesting journey into the details of transformation matrices and

how to use them eﬀectively in estimation problems. Paul’s later work

led me to become interested in continuous-time estimation. Chi Hay

Tong’s PhD thesis introduced me to the use of Gaussian processes in

estimation theory, and he helped immensely in working out the de-

tails of the continuous-time methods presented herein; my knowledge

in this area was further improved through collaborations with Simo

S¨arkk¨a from Aalto University while on sabbatical at the University

of Oxford. Additionally, I learned a great deal by working with Sean

Anderson, Patrick Carle, Hang Dong, Andrew Lambert, Keith Leung,

Colin McManus, and Braden Stenning; each of their projects added to

my understanding of state estimation. Colin, in particular, encouraged

me several times to turn my notes from my graduate course on state

estimation into this book.

I am indebted to Gabriele D’Eleuterio, who set me on the path of

studying rotations and reference frames in the context of dynamics;

many of the tools he showed me transferred eﬀortlessly to state es-

timation. He also taught me the importance of clean, unambiguous

notation.

Finally, thanks to all those who read and pointed out errors in the

drafts of this book, particularly Marc Gallant and Shu-Hua Tsao, who

found many typos, and James Forbes, who volunteered to read and

provide comments.

Introduction

Robotics inherently deals with things that move in the world. We live

in an era of rovers on Mars, drones surveying the Earth, and, soon,

self-driving cars. And, although speciﬁc robots have their subtleties,

there are also some common issues we must face in all applications,

particularly state estimation and control.

The state of a robot is a set of quantities, such as position, orien-

tation, and velocity, that, if known, fully describe that robot’s motion

over time. Here we focus entirely on the problem of estimating the state

of a robot, putting aside the notion of control. Yes, control is essential,

as we would like to make our robots behave in a certain way. But,

the ﬁrst step in doing so is often the process of determining the state.

Moreover, the diﬃculty of state estimation is often underestimated for

real-world problems, and thus it is important to put it on an equal

footing with control.

In this book, we introduce the classic estimation results for linear

systems corrupted by Gaussian measurement noise. We then exam-

ine some of the extensions to nonlinear systems with non-Gaussian

noise. In a departure from typical estimation texts, we take a detailed

look at how to tailor general estimation results to robots operating in

three-dimensional space, advocating a particular approach to handling

rotations.

The rest of this introduction provides a little history of estimation,

discusses types of sensors and measurements, and introduces the prob-

lem of state estimation. It concludes with a breakdown of the contents

of the book and provides some other suggested reading.

1.1 A Little History

About 4,000 years ago, the early seafarers were faced with a vehicular

state estimation problem: how to determine a ship’s position while at

sea. Early attempts to develop primitive charts and make observations

of the sun allowed local navigation along coastlines. However, it was

not until the ﬁfteenth century that global navigation on the open sea

became possible with the advent of key technologies and tools. The

mariner’s compass, an early form of the magnetic compass, allowed

2 Introduction

crude measurements of direction to be made. Together with coarse

nautical charts, the compass made it possible to sail along rhumb lines

between key destinations (i.e., following a compass bearing). A seriesFigure 1.1

Quadrant. A tool

used to measure

angles.

of instruments was then gradually invented that made it possible to

measure the angle between distant points (i.e., cross-staﬀ, astrolabe,

quadrant, sextant, theodolite) with increasing accuracy.

These instruments allowed latitude to be determined at sea fairly

readily using celestial navigation. For example, in the Northern Hemi-

sphere, the angle between the North Star, Polaris, and the horizon

provides the latitude. Longitude, however, was a much more diﬃcult

problem. It was known early on that an accurate timepiece was the

missing piece of the puzzle for the determination of longitude. The be-

haviours of key celestial bodies appear diﬀerently at diﬀerent locations

on the Earth. Knowing the time of day therefore allows longitude to

be inferred. In 1764, British clockmaker John Harrison built the ﬁrstFigure 1.2

Harrison’s H4. The

ﬁrst clock able to

keep accurate time

at sea, enabling

determination of

longitude.

accurate portable timepiece that eﬀectively solved the longitude prob-

lem; a ship’s longitude could be determined to within about 10 nautical

miles.

Estimation theory also ﬁnds its roots in astronomy. The method of

least squares was pioneered

by Gauss, who developed the technique to

Carl Friedrich

Gauss (1777-1855)

was a German

mathematician

who contributed

signiﬁcantly to

many ﬁelds

including statistics

and estimation.

minimize the impact of measurement error in the prediction of orbits.

Gauss reportedly used least squares to predict the position of the dwarf

planet Ceres after passing behind the Sun, accurate to within half a

degree (about nine months after it was last seen). The year was 1801,

and Gauss was 23. Later, in 1809, he proved that the least-squares

method is optimal under the assumption of normally distributed errors.

Most of the classic estimation techniques in use today can be directly

related to Gauss’ least-squares method.

The idea of ﬁtting models to minimize the impact of measurement

error carried forward, but it was not until the middle of the twenti-

eth century that estimation really took oﬀ. This was likely correlated

with the dawn of the computer age. In 1960, Kalman published two

Rudolf Emil

K´alm´an

(1930-2016) was a

Hungarian-born

American electrical

engineer,

mathematician,

and inventor.

landmark papers that have deﬁned much of what has followed in the

ﬁeld of state estimation. First, he introduced the notion of observability

(Kalman, 1960a), which tells us when a state can be inferred from a

set of measurements in a dynamic system. Second, he introduced an

optimal framework for estimating a system’s state in the presence of

measurement noise (Kalman, 1960b); this classic technique for linear

systems (whose measurements are corrupted by Gaussian noise) is fa-

mously known as the Kalman ﬁlter, and has been the workhorse of

estimation for the more than 50 years since its inception. Although

used in many ﬁelds, it has been widely adopted in aerospace appli-

There is some debate as to whether Adrien Marie Legendre might have come up with

least squares before Gauss.

1.2 Sensors, Measurements, and Problem Deﬁnition 3

cations. Researchers at the National Aeronautics and Space Adminis-

tration (NASA) were the ﬁrst to employ the Kalman ﬁlter to aid in

the estimation of spacecraft trajectories on the Ranger, Mariner, and

Apollo programs. In particular, the on-board computer on the Apollo

11 lunar module, the ﬁrst manned spacecraft to land on the surface of

the Moon, employed a Kalman ﬁlter to estimate the module’s position

above the lunar surface based on noisy radar measurements.

Early Estimation

Milestones

1654 Pascal and

Fermat lay

foundations

of probability

theory

1764 Bayes’ rule

1801 Gauss uses

least-squares

to estimate

the orbit of

the planetoid

Ceres

1805 Legendre pub-

lishes “least-

squares”

1913 Markov chains

1933 (Chapman)-

Kolmogorov

equations

1949 Wiener ﬁlter

1960 Kalman

(Bucy) ﬁlter

1965 Rauch-Tung-

Striebel

smoother

1970 Jazwinski

coins “Bayes

ﬁlter”

Many incremental improvements have been made to the ﬁeld of state

estimation since these early milestones. Faster and cheaper computers

have allowed much more computationally complex techniques to be

implemented in practical systems. However, until about 15 years ago,

it seemed that estimation was possibly waning as an active research

area. But, something has happened to change that; exciting new sensing

technologies are coming along (e.g., digital cameras, laser imaging, the

Global Positioning System) that pose new challenges to this old ﬁeld.

1.2 Sensors, Measurements, and Problem Deﬁnition

To understand the need for state estimation is to understand the na-

ture of sensors. All sensors have a limited precision. Therefore, all mea-

surements derived from real sensors have associated uncertainty. Some

sensors are better at measuring speciﬁc quantities than others, but even

the best sensors still have a degree of imprecision. When we combine

various sensor measurements into a state estimate, it is important to

keep track of all the uncertainties involved and therefore it is hoped

know how conﬁdent we can be in our estimate.

In a way, state estimation is about doing the best we can with the

sensors we have. This, however, does not prevent us from, in parallel,

improving the quality of our sensors. A good example is the theodo- Figure 1.3

Theodolite. A

better tool to

measure angles.

lite sensor that was developed in 1787 to allow triangulation across

the English Channel. It was much more precise than its predecessors

and helped show that much of England was poorly mapped by tying

measurements to well-mapped France.

It is useful to put sensors into two categories: interoceptive

and ex-

teroceptive. These are actually terms borrowed from human physiology,

but they have become somewhat common in engineering. Some deﬁni-

tions follow

in·tero·cep·tive [int-@-r¯o-’sep-tiv], adjective: of, relating to, or being

stimuli arising within the body.

ex·tero·cep·tive [ek-st@-r¯o-’sep-tiv], adjective: relating to, being, or

activated by stimuli received by an organism from outside.

Sometimes proprioceptive is used synonomously.

Merriam-Webster’s Dictionary.

4 Introduction

Typical interoceptive sensors are the accelerometer (measures trans-

lational acceleration), gyroscope (measures angular rate), and wheel

odometer (measures angular rate). Typical exteroceptive sensors are

the camera (measures range/bearing to a landmark or landmarks)

and time-of-ﬂight transmitter/receiver (e.g., laser rangeﬁnder, pseudo-

lites, Global Positioning System (GPS) transmitter/receiver). Roughly

speaking, we can think of exteroceptive measurements as being of the

position and orientation of a vehicle, whereas interoceptive ones are

of a vehicle’s velocity or acceleration. In most cases, the best state

estimation concepts make use of both interoceptive and exteroceptive

measurements. For example, the combination of a GPS receiver (extero-

ceptive) and an inertial measurement unit (three linear accelerometers

and three rate gyros; interoceptive) is a popular means of estimat-

ing a vehicle’s position/velocity on Earth. And, the combination of a

Sun/star sensor (exteroceptive) and three rate gyros (interoceptive) is

commonly used to carry out pose determination on satellites.

Now that we understand a little bit about sensors, we are prepared

to deﬁne the problem that will be investigated in this book:

Estimation is the problem of reconstructing the underlying state of a system given a

sequence of measurements as well as a prior model of the system.

There are many speciﬁc versions of this problem and just as many

solutions. The goal is to understand which methods work well in which

situations, in order to pick the best tool for the job.

1.3 How This Book Is Organized

The book is broken into three main parts:

I: Estimation Machinery

II: Three-Dimensional Machinery

III: Applications

The ﬁrst part, Estimation Machinery, presents classic and state-of-the-

art estimation tools, without the complication of dealing with things

that live in three-dimensional space (and therefore translate and ro-

tate); the state to be estimated is assumed to be a generic vector.

For those not interested in the details of working in three-dimensional

space, this ﬁrst part can be read in a standalone manner. It covers

both recursive state estimation techniques and batch methods (less

common in classic estimation books). As is commonplace in robotics

and machine learning today, we adopt a Bayesian approach to estima-

tion in this book. We contrast (full) Bayesian methods with maximum

a posteriori (MAP) methods, and attempt to make clear the diﬀer-

ence between these when faced with nonlinear problems. The book also

connects continuous-time estimation with Gaussian process regression

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