概率机器人学基础：递归状态估计与高斯滤波

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"这是一份关于概率机器人的专业讲义，由Sebastian Thrun、Wolfram Burgard和Dieter Fox等专家编写。该讲义深入探讨了机器人领域中的不确定性处理以及概率方法在其中的应用。" 在这本讲义中，作者们首先介绍了机器人学中普遍存在的不确定性问题（1.1节）。在机器人操作和探索环境中，由于传感器限制、动态环境变化以及模型不精确性，不确定性是无法避免的。不确定性可能源于机器人自身的状态（如位置和姿态），也可能源于对环境的理解。接着，1.2节提出了概率机器人学的概念，这是一个利用概率理论来处理和建模这些不确定性的方法。这种方法允许机器人通过收集数据并更新其对世界的信念，来逐步改进其决策和行为。 1.3节讨论了概率机器人学的影响。通过概率模型，机器人可以更好地适应不确定环境，提高任务完成的准确性和效率。此外，这种方法还为解决复杂的机器人问题，如自主导航、目标检测和跟踪提供了新的视角。 1.4节给出了讲义的结构概览，即“RoadMap”，它将引导读者逐步深入到更具体的概率估计和滤波技术。1.5节则列出了相关文献，供进一步研究。接下来，2.0章节详细阐述了递归状态估计（Recursive State Estimation）。这是概率机器人学的核心概念，涉及如何通过新数据不断更新机器人对自身状态的估计。2.2节介绍了概率的基本概念，包括概率分布和随机变量。2.3节则讨论了机器人与环境的交互，包括状态、环境互动和概率生成法则，以及由此产生的信念分布。 2.4节重点介绍了贝叶斯滤波器，这是一种用于动态系统状态估计的统计方法。通过24页的例子和数学推导（2.4.2和2.4.3节），读者可以理解贝叶斯滤波器的工作原理及其在处理不确定性中的作用。2.4.4节介绍了马尔可夫假设，这是许多滤波算法的基础，它简化了状态转移的概率模型。 2.5节探讨了表示和计算方面的问题，这在实际应用中至关重要，因为复杂的概率模型往往需要高效的计算策略。2.6节总结了本章要点，并在2.7节中推荐了相关文献，以便读者深入学习。 3.0章节开始介绍高斯滤波器，特别是卡尔曼滤波器（3.2节），它是处理线性高斯系统的标准工具。3.2.1节定义了线性高斯系统，而3.2.2节详细阐述了卡尔曼滤波器算法，这是概率机器人学中最常用的滤波方法之一。这份讲义提供了一个全面的概率机器人学框架，涵盖了从基础的概率理论到高级的滤波算法，为理解和实施概率机器人技术提供了坚实的基础。

4 Chapter 1

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Figure 1.1 The basic idea of Markov localization: A mobile robot during global localiza-

tion.

Introduction 5

This example illustrates the probabilistic paradigm in the context of a speciﬁc per-

ceptual problem. Stated probabilistically, the robot perception problem is a state es-

timation problem, and our localization example uses an algorithm known as Bayes

ﬁlter for posterior estimation over the space of robot locations. Similarly, when select-

ing actions, probabilistic approaches considers the full uncertainty, not just the most

likely guess. By doing so, the probabilistic approach trades off information gathering

(exploration) and exploitation, and act optimally relative to the state of knowledge.

1.3 IMPLICATIONS

What are the advantages of programming robots probabilistically, when compared to

other approaches that do not represent uncertainty explicitly? Our central conjecture

is nothing less than the following:

A robot that carries a notion of its own uncertainty and that acts accordingly

is superior to one that does not.

In particular, probabilistic approaches are typically more robust in the face of sensor

limitations, sensor noise, environment dynamics, and so on. They often scale much

better to complex and unstructured environments, where the ability to handle uncer-

tainty is of even greater importance. In fact, certain probabilistic algorithms are cur-

rently the only known working solutions to hard robotic estimation problems, such as

the kidnapped robot problem, in which a mobile robot must recover from localization

failure; or the problem of building accurate maps of very large environments, in the

absence of a global positioning device such as GPS. Additionally, probabilistic algo-

rithms make much weaker requirements on the accuracy of models than many classical

planning algorithms do, thereby relieving the programmer from the unsurmountable

burden to come up with accurate models. Viewed probabilistically, the robot learning

problem is a long-term estimation problem. Thus, probabilistic algorithms provide a

sound methodology for many ﬂavors of robot learning. And ﬁnally, probabilistic al-

gorithms are broadly applicable to virtually every problem involving perception and

action in the real world.

However, these advantages come at a price. Traditionally, the two most frequently

cited limitations of probabilistic algorithms are computational inefﬁciency, and a

need to approximate. Probabilistic algorithms are inherently less efﬁcient than non-

probabilistic ones, due to the fact that they consider entire probability densities. The

need to approximate arises from the fact that most robot worlds are continuous. Com-

puting exact posterior distributions is typically infeasible, since distributions over the

6 Chapter 1

continuum possess inﬁnitely many dimensions. Sometimes, one is fortunate in that

the uncertainty can approximated tightly with a compact parametric model (e.g., dis-

crete distributions or Gaussians); in other cases, such approximations are too crude

and more complicated representations most be employed. Recent research has suc-

cessfully led to a range of are computationally efﬁcient probabilistic algorithms, for a

range of hard robotics problems—many of which are described in depth in this book.

1.4 ROAD MAP

This book attempts to provide a comprehensive and in-depth introduction into proba-

bilistic robotics. The choice of material is somewhat biased towards research carried

out at Carnegie Mellon University, the University of Bonn, and afﬁliated labs. How-

ever, we have attempted to include in-depth descriptions of other, important proba-

bilistic algorithms. The algorithms described here have been developed for mobile

robots; however, many of them are equally applicable to other types of robots. Thus,

the coverage of the material is by no means complete; probabilistic ideas have recently

become extremely popular in robotics, and a complete description of the ﬁeld would

simply not ﬁt into a single book. However, we believe that the choice of material is

representative for the existing body of literature.

The goal of the book is to provide a systematic introduction into the probabilistic

paradigm, from the underlying mathematical framework to implementation. For each

major algorithm, this book provides

a complete mathematical derivation,

pseudo-code in a C-like language,

discussions of implementation details and potential pitfalls, and

empirical results obtained in ﬁelded systems.

We believe that all four items are essential for obtaining a deep understanding of the

probabilistic paradigm. At the end of each chapter, the book also provides biblio-

graphical notes and a list of questions and exercises.

The book has been written with researchers, graduate students or advanced undergrad-

uate students in mind, specializing in robotics or applied statistics. We have attempted

to present the material in a way that requires a minimum of background knowledge.

Introduction 7

However, basic knowledge of probability theory will almost certainly help in under-

standing the material. The various mathematical derivations can easily be skipped at

ﬁrst reading. However, we strongly recommend to take the time and study the mathe-

matical derivations, as a profound mathematical understanding is will almost certainly

lead to deep and important insights into the working of the probabilistic approach to

robotics.

If used in the classroom, each chapter should be covered in one or two lectures; how-

ever, we recommend that the study of the book be accompanied by practical, hands-on

experimentation as directed by the questions and exercises at the end of each chapter.

This book is organized in four major parts.

The ﬁrst part, Chapters 2 through 5, discuss the basic mathematical framework

that underlies all of the algorithms described in this book. Chapters 2 through 4

introduce the basic probabilistic notation and describes a collection of ﬁlters for

probabilistic state estimation. Chapter 5 discusses speciﬁc probabilistic models

that characterize mobile robot perception and motion.

The second part, which comprised Chapters 7 to ??, describes a range of per-

ceptual algorithms, which map sensor measurements into internal robot beliefs.

In particular, Chapter 7 describes algorihtms for mobile robot localization, fol-

lowed by algorithms for map acquisition described in Chapters ??. This part also

contains a chapter on learning models.

The third part, in Chapters ?? to ??, introduces probabilistic planning and action

selection algorihtms.

Finally, Chapter ?? describes two robot systems that were controlled by proba-

bilistic algorithms. These robots were deployed in museums as interactive tour-

guide robots, where they managed to navigate reliably without the need to modify

the museums in any way.

The book is best read in order, from the beginning to the end. However, we have

attempted to make each individual chapter self-explanatory.

1.5 BIBLIOGRAPHICAL REMARKS

The term “robot” was invented in 1921 by the Czech novelist Karel

Capek [42], to describe a willing,

intelligent and human-like machines that make life pleasant by doing the type work we don’t like to do. In

the Fourties, Asimov coined the term “robotics”and postulated the famous three laws of robotics [1, 2],

8 Chapter 1

Robotics, as a scientiﬁ c discipline has been an active ﬁ eld of research foer several decades. In the early

years, most of the research focused on

Major trends in robotics:

Seventies: classical (deliberate) approach, accurate models, no uncertainty, no sensing. Still: very

hard problem. Reif: planning problem NP hard, but only doubly exponential algorithms known.

Canny: ﬁ rst single exponential planning algorithm. Latombe: Many impressive randomized planning

algorithms. Important difference: randomization used for search, not for representing uncertainty.

Planning algorithm for special topologies: Schwartz, and Sharir. Some take sensor data into account.

Koditschek navigation function for feedback control, generalizes Khatib’s potential ﬁ elds, which suf-

fer from local minima (navigation functions don’t).

Mid-Eighties: reactive approach, rejection of models, pure reliance on sensors, uncertainty (if any)

handled by feedback control. Relies on sensor data carrying sufﬁ cient information for action selec-

tion, therefore uncertainty not an issue. Typically conﬁ ned to small environments, where everything

of importance can be perceived by the sensors. Nevertheless, some impressive results for control of

legged robots. Brooks, Mataric, Steels, Connell, and many others. Biologically inspired robotics.

Smithers.

Mid-Nineties: Hybrid approaches, reactive at low level (fast decision cycle), deliberate at higher

levels. Uses sensing mostly at low level, and models at high level. Gat, Arkins, Balch, Firby, Simmons

and many others. harvests best of both worlds: robust through reaction, but can do more powerful

tasks due to deliberation.

Late Ninties: Probabilistic robotics, different way to integrate models and sensor data.

Probabilistic robotics can be traced back to Sixties to advent of Kalman ﬁ lters, which has been used exten-

sively in robotics. First serious advance is Smith and Cheeseman in mid-80s, who proppose an algorithm for

concurrent mapping and localization based on Kalman ﬁ lters that is now in widespread use. Durrant-Whyte,

Leonard, Castellanos, and many others.

Main activity in past ﬁ ve years. Main advances: more ﬂexible representations (beyond Kalman ﬁ lters),

more efﬁ cient algorithms. Statistical approaches for solving hard correspondence problems.

What is new in probabilistic robotics, relative to approaches above?

seamlessly blends models and perception in a novel way

sound mathematical theory, clear assumptions, therefore it’s easier to predict failure modes

applies to all levels: lowest to highest, since uncertainty arises everywhere

currently the best known solutions in a range of hard robotics problems

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