机器人学状态估计详解：线性与非线性系统方法

机器人学

状态估计

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《STATE ESTIMATION FOR ROBOTICS》是一本专为机器人学领域读者编写的权威教材，作者为Timothy D. Barfoot。该书主要探讨了机器人状态估计的关键理论和技术，涵盖了从基础概率论到高级应用的深入内容。全书共分三个部分，分别是状态估计、三维空间运动以及实际应用。在状态估计部分，作者首先回顾了线性高斯系统状态估计的重要性，这是机器人导航和控制的基础，它利用概率密度函数（PDF）的概念来描述系统的不确定性。重点讲解了概率密度函数的定义，包括贝叶斯规则和推断过程，以及统计独立性和无相关性的概念。接着，作者详细介绍了高斯概率密度函数（Gaussian PDF），这是许多机器人状态估计算法的核心，如卡尔曼滤波。书中讨论了 Isserlis 定理，这是一种计算多变量高斯分布特定统计性质的方法，以及联合高斯PDF的处理，以及如何进行概率推理。在三维空间运动部分，内容转向了旋转和平移的数学表示，这在机器人定位和姿态估计中至关重要。作者深入剖析了李群理论，一种用于处理连续旋转和变换的数学框架，这对于理解机器人传感器数据和运动模型的转换非常关键。应用部分则涵盖了实际问题的解决策略，包括机器人位姿估计（如基于视觉或惯性测量单元的定位）、点云配准（将不同传感器数据融合，提高环境模型的精度）以及束调整（BA，通过优化方法调整相机位置和姿态，以最小化观测数据与模型之间的误差）。这些技术在自主机器人、无人车、无人机等领域有着广泛的应用。《STATE ESTIMATION FOR ROBOTICS》不仅介绍了基础的概率和统计概念，还深入浅出地阐述了如何将这些理论应用于机器人状态估计的实际问题。对于想要在这个前沿领域深化学习的工程师和研究人员来说，这本书提供了一个全面而实用的学习指南。通过阅读这本书，读者将能够掌握状态估计的理论工具，理解和实现机器人在复杂环境中的精确导航和控制。

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xvi Foreword

ments, I have of t en been faced with vehicles operating in three dimen-

sions. Accordingly, I h ave attempted to provide a det ai le d look at how

to app roach state estimation in three dimen si ons . In particular, I show

how t o treat rotations and poses in a s imple 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 w il l ﬁnd something useful; I

know I learned a great deal while creating them.

I have provided some historical notes in t h e margins through ou t 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 Wikipedi a as the source for this informati on . 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 coll aborations of

many fantastic graduate students along th e 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 e↵ectively in esti m ati on problems. Paul’s later work

led me to become interested in continuous-ti me es t im at ion. C hi Hay

Tong’s PhD thesis introduced me to the use of G au ssi an processes in

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

tails of the continuous-time met h ods 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 not es from my gradu at e course on state

estimation into this book.

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

studying rotations and referen ce frames in the context of dy nam ic s;

many of the tools he s howed m e transferred e↵ort l es sl y to state es-

timation. He also taught me t he importance of clean, unambiguous

notation.

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

drafts of this book, particu l arl y 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 applicat i ons ,

particularly state estimation and control.

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

tation, and velo c ity, 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 r obots behave in a certain way. But,

the ﬁrst ste p in doing so is often the process of determining the st ate .

Moreover, the diﬃculty of state estimation is often underestimated for

real-world p rob l ems , 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, advocat i n g 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 breakd own of the contents

of the book and provides some other suggested reading.

1.1 A Little Histor y

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

state estimation problem : how to determine a ship’s position w hi l e at

sea. Early attempts to develop primi t ive 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 beari ng) . A seriesFigure 1.1

Quadrant. A tool

used to measure

angles.

of instruments was then gradual ly invented that made it possible to

measure the angle between distant p oi nts (i.e., cross-sta↵, astrolabe ,

quadrant, sextant, theodolite) with increasing accuracy.

These instruments allowed latitude to be determined at sea fairly

readily usi ng celestial navigation. For example, in the Nort h er n 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 tim ep ie ce was the

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

haviours of key celestial bodies appear di↵erently at di↵erent locations

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

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

Harrison’s H4. The

ﬁrst clock able to

keep accurate time

at sea, enabling

determination of

longitude.

accurate portable timepiec e that e↵ectively 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 pionee re d

by Gauss, who developed t he t e chnique to

Carl Friedrich

Gauss (1777-1855)

was a German

mathematician

who contributed

signiﬁcantly to

many ﬁelds

including statistics

and estimation.

minimize the impact of measur eme nt error in the prediction of orbits.

Gauss reportedly used least squares to predict the position of t h e 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 o↵. This was likely correlated

with the dawn of the compute r age. In 1960, K alm an pub l is he d 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 i n fe r re d 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 widel y 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 ﬁl t er to aid in

the estimation of spacecraft traject or ie s on the Ranger, Mariner, and

Apollo programs. In part i cu l ar, 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 posi t i on

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 M arkov chains

1933 (Chapman)-

Kolmogorov

equations

1949 Wiener ﬁlter

1960 Kalman

(Bucy) ﬁlter

1965 Rauch-Tung-

Striebel

smoother

1970 Jazwinsk i

coins “Bayes

ﬁlter”

Many incremental improvements have been made to th e ﬁeld of state

estimation since these early milestones. Fas t er and cheaper computers

have all owed much more computationally complex techniques to be

implemented in p r act i cal systems. However, u ntil about 15 years ago,

it seemed that estim ati on was possibly waning as an active r es ear ch

area. But, something has happened to change th at ; exciting new s e ns ing

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

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

1.2 Sensors, Measurements, an d 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 der i ved 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, i t 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 t he best we can with the

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

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

Theodolite. A

better tool to

measure angles.

lite sensor that was developed in 1787 to al low triangulation across

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

and helped show t h at 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 fol l ow

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, bei n g, or

activated by stimuli received by an organism from outsid e.

Sometimes proprioceptive is used synonomously.

Merriam-Webster’s Dictionary.

4 Introduction

Typical interocep t i ve sensors are the accel er omet e r (measu r es tr an s-

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

odometer (measures angular rate). Typical exteroceptive sensors are

the camera (measures range/bear in g to a landmark or landmarks)

and time-of-ﬂight transmitter/receiver (e.g., laser rangeﬁnder, p se ud o-

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 interocep t i ve ones are

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

estimation concepts m ake use of both interoceptive and exteroceptive

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

ceptive) and an inertial measurement unit (three linear accelerom et e rs

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 u sed 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 probl e m and just as many

solutions. The goal is to understand which method s work well in which

situations, in order to pick the best tool for the j ob.

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 th e details of working in three-dimensional

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

both re cu rs ive state estimation techniques and batch methods (less

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

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

tion in this book. We contrast (full) Bayesian metho d s with maximum

a posteriori (MAP) methods, and attempt to make cl ear the di↵er-

ence between these when faced with nonlinear p rob l ems . The book also

connects continuous-time estimation with Gaussian process regression

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