非线性控制：机器人与无人机系统集成方法

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"《非线性控制下的机器人与无人驾驶航空器》是一本探讨高级自动化技术领域的专著，着重于处理复杂系统中的控制理论在机器人和无人航空器（如无人机）上的应用。该书主要关注非线性控制方法，这是现代机器人和无人驾驶系统设计中的关键技术，因为实际操作环境往往包含不可预测的动态特性和多变的外部因素。非线性控制理论在机器人领域中扮演着核心角色，它允许系统在面对非线性模型和不确定性的挑战时，仍能保持稳定性和精确操作。这包括但不限于PID（比例-积分-微分）控制器的扩展，滑模控制、自适应控制以及模型预测控制等高级控制策略。这些控制技术在处理机器人的关节运动、路径规划、姿态控制以及飞行器的姿态和位置保持等方面至关重要。书中提到的集成方法，可能涉及将不同的控制算法或传感器数据融合，以实现更高效的决策和执行。作者Ranjan Vepa可能会探讨如何利用MATLAB®和Simulink®这样的工具进行仿真和实时控制系统的开发，这两者是模拟和设计复杂系统过程的常用软件平台。然而，读者需要注意的是，尽管书中可能包含这些软件的使用，但并不构成The MathWorks对特定教学方法或软件使用的官方认可或赞助。本书由CRC Press出版，作为Taylor & Francis Group的一部分，强调了内容的权威性和可靠性，尽管如此，作者和出版社无法对所有引用的数据和信息承担绝对的责任，因为它们基于现有的公开资料和公认的来源。版本日期为2016年6月22日，国际标准书号(ISBN)为1-4987-6704-0（精装版），体现了对最新研究和技术的持续关注。《非线性控制下的机器人与无人驾驶航空器》提供了一个深入理解非线性控制技术在现代机器人和无人航空器领域应用的宝贵资源，适合从事控制工程、机器人学、航空航天工程等相关领域的研究人员和工程师阅读。"

Preface

During the last two decades, much progress has been made in the application of nonlinear

differential geometric control theory, rst to robotic manipulators and then to autonomous

vehicles. In fact, robot control is simply a metaphor for nonlinear control. The ability to trans-

form complex nonlinear systems sequentially to simpler prototypes, which can then be con-

trolled by the application of Lyapunov’s second method, has led to the development of some

novel techniques for controlling both robot manipulators and autonomous vehicles without the

need for approximations. More recently, a synergy of the technique of feedback linearization

with classical Lyapunov stability theory has led to the development of a systematic adaptive

backstepping design of nonlinear control laws for systems with unknown constant parame-

ters. Another offspring of the Lyapunov-based controllers is a family of controllers popularly

known as sliding mode controls. Currently, sliding mode controls have evolved into second-

and higher-order implementations, which are being applied extensively to robotic systems.

Some years ago, the author embarked on a comprehensive programme of research to

bring together a number of techniques in an attempt to formulate the dynamics and solve

the control problems associated with both robot manipulators and autonomous vehicles, such

as unmanned aerial vehicles (UAVs), without making any approximations of the essentially

nonlinear dynamics. A holistic approach to the two elds have resulted in new application

ideas such as the morphing control of aerofoil sections and the decoupling of force (or ow)

and displacement control loops in such applications. A number of results of several of these

studies were also purely pedagogical in nature. Pedagogical results are best reported in the

form of new learning resources, and for this reason, the author felt that the educational out-

comes could be best communicated in a new book. In this book, the author focuses on control

and regulation methods that rely on the techniques related to the methods of feedback linear-

ization rather than the more commonly known methods that rely on Jacobian linearization.

The simplest way to stabilize the zero dynamics of a nonlinearly controlled system is to use,

when feasible, input–output feedback linearization. The need for such a book arose due to the

increasing appearance of both robot manipulators and UAVs with operating regimes involv-

ing large magnitudes of state and control variables in environments that are not generally very

noisy. The underpinning themes which serve as a foundation for both robot dynamics and

UAVs include Lagrangian dynamics, feedback linearization and Lyapunov-based methods

of both stabilization and control. In most applications, a combination of these fundamental

techniques provides a powerful tool for designing controllers for a range of application tasks

involving tracking, coordination and motion control. Clearly, the focus of these applications

is primarily on the ability to handle the nonlinearities rather than dealing with the environ-

mental disturbances and noise which are of secondary importance. This book is of an applied

nature and is about doing and designing control laws. A number of application examples

are included to facilitate the reader’s learning of the art of nonlinear control system design.

xvi PrefaCe

Thebook is not meant to supplant the many excellent books on nonlinear and adaptive control

but is designed to be a complementary resource. It seeks to present the methods of nonlinear

controller synthesis for both robots and UAVs in a single, unied framework.

The book is organized as follows: Chapter 1 deals with the application of the Euler–

Lagrange method to robot manipulators. Special consideration is given to rapidly determining

the equations of motion of various classes of manipulators. Thus, the manipulators are classi-

ed as parallel and serial, as Cartesian and spherical and as planar, rotating planar and spatial,

and the methods of determining the equations of motion are discussed under these categories.

The denition of planar manipulators is generalized so that a wider class of manipulators can

be included in this category. The methods of deriving the dynamics of the manipulators can

be used as templates to derive the dynamics of any manipulator. This approach is unique to

this book. Chapter 2 focuses on the application of the Lagrangian method to UAVs via the

method of quasi-coordinates. It is worth remembering that the use of the Lagrangian method

for deriving the equations of motion of a UAV is not the norm amongst ight dynamicists.

Moreover, the chapter introduces the velocity axes, as the synthesis of the ight controller

in these axes is a relatively easy task. The concept of feedback linearization is introduced

in Chapter 3, while the classical methods of phase plane analysis of the stability of nonlin-

ear systems and their features are discussed in Chapter 4 in the context of Lyapunov’s rst

method. Chapter 5 presents an overview of the methods of robot and UAV control. Chapter6

is dedicated to introducing the concepts of stability, and Chapter 7 is exclusively about

Lyapunov stability with an enunciation of Lyapunov’s second method. The methodology of

computed torque control is the subject of Chapter 8, and sliding mode controls are introduced

in Chapter9. Chapter 10 discusses parameter identication, including recursive egression,

while adaptive and model predictive controller designs are introduced in Chapter11. In a

sense, linear optimal control, a particular instance of the Lyapunov design of controllers, is

also covered in the section on model predictive control, albeit briey. Chapter12 is exclusively

devoted to the Lyapunov design of controllers by backstepping. Chapter 13 covers the applica-

tion of feedback linearization in the task space to achieve decoupling of the position and force

control loops, and Chapter 14 is devoted to the applications of nonlinear systems theory to the

synthesis of ight controllers for UAVs.

It is the author’s belief that the book will not be just another text on nonlinear control but

serve as a unique resource to both the robotics and UAV research communities in the years to

come and as a springboard for new and advanced projects across the globe.

First and foremost, I thank Jonathan Plant, without his active support, this project would

not have been successful. I also thank my colleagues and present and former students at the

School of Engineering and Material Science at Queen Mary University of London for their

assistance in this endeavour. In particular, I thank Professor Vassili Toropov for his support

and encouragement.

I thank my wife Sudha for her love, understanding and patience. Her encouragement and

support provided the motivation to complete the project. I also thank our children Lullu, Satvi

and Abhinav for their understanding during the course of this project.

Ranjan Vepa

London, United Kingdom

MATLAB

is a registered trademark of The MathWorks, Inc. For product information, please

contact:

The MathWorks, Inc.

3 Apple Hill Drive

Natick, MA 01760-2098 USA

Tel: 508-647-7000

Fax: 508-647-7001

E-mail: info@mathworks.com

Web: www.mathworks.com

xvii

Author

Ranjan Vepa, PhD, earned a PhD (applied mechanics) from Stanford University (California),

specializing in the area of aeroelasticity under the guidance of the late Professor Holt Ashley.

He is currently a lecturer in the School of Engineering and Material Science, Queen Mary

University of London, where since 2001, he has also been the director of the Avionics

Programme. Prior to joining Queen Mary, Dr. Vepa was with the NASA Langley Research

Center, where he was awarded a National Research Council Fellowship and conducted

research in the area of unsteady aerodynamic modeling for active control applications.

Subsequently, he was with the Structures Division of the National Aeronautical Laboratory,

Bangalore, India, and the Indian Institute of Technology, Chennai, India.

Dr. Vepa is the author of ve books: Biomimetic Robotics (Cambridge University Press,

2009); Dynamics of Smart Structures (Wiley, 2010); Dynamic Modeling, Simulation and

Control of Energy Generation (Springer, 2013) and Flight Dynamics, Simulation, and

Control: For Rigid and Flexible Aircraft (CRC Press, 2014). Dr. Vepa is a member of the

Royal Aeronautical Society, London; the Institute of Electrical and Electronic Engineers,

New York and the Royal Institute of Navigation, London. He is also a Fellow of the Higher

Education Academy and a chartered engineer.

In addition, Dr. Vepa is studying techniques for automatic implementation of structural

health monitoring based on observer and Kalman lters. He is involved in the design of crack

detection lters applied to crack detection and isolation in aeroelastic aircraft structures such

as nacelles, casings, turbine rotors and rotor blades for health monitoring and control. Elastic

wave propagation in cracked structures is being used to develop distributed lters for struc-

tural health monitoring. Feedback control of crack propagation and compliance compensation

in cracked vibrating structures are also being investigated. Another issue is the modeling of

damage in laminated composite plates, nonlinear utter analysis and the interaction with

unsteady aerodynamics. These research studies contribute to the holistic design of vision-

guided autonomous UAVs, which are expected to be used extensively in the future.

Dr. Vepa’s research interests also include the design of ight control systems, aerodynam-

ics of morphing wings and bodies with applications in smart structures, robotics, biomedi-

cal engineering and energy systems, including wind turbines. In particular, his focus is on

the dynamics and robust adaptive estimation and control of linear and nonlinear aerospace,

energy and biological systems with parametric and dynamic uncertainties. The research in

the area of the aerodynamics of morphing wings and bodies is dedicated to the study of aero-

dynamics and its control, including the use of smart structures and their applications in the

ight control of air vehicles, jet engines, robotics and biomedical systems. Other applications

are in wind turbine and compressor control, maximum power point tracking, ow control over

smart aps and the control of biodynamic systems.

CHAPTeR ONe

Lagrangian methods and

robot dynamics

Introduction

The basis of the Newtonian approach to dynamics is the Newtonian viewpoint, that motion

is induced by the action of forces acting on particles. This viewpoint led Sir Isaac Newton to

formulate his celebrated laws of motion. In the late 1700s and early 1800s, a different view

of dynamic motion began to emerge. According to this view, particles do not follow trajecto-

ries because they are acted upon by external forces, as Newton proposed. Instead, amongst

all possible trajectories between two points, they choose the one which minimizes a specic

time integral of the difference between the kinetic and the potential energies called the action.

Newton’s laws are then obtained as a consequence of this principle, by the application of varia-

tional principles in minimizing the action integral. Also, as a consequence of the minimization

of the action integral, the total potential and kinetic energies of systems are conserved in the

absence of any dissipative forces or forces that cannot be derived from a potential function.

The alternate view of particle motion then led to a newer approach to the formulation and

analysis of the dynamics of motion. It was no longer required to isolate each and every particle

or body and forces acting on them, within a system of particles or bodies. The system of par-

ticles could be treated in a holistic manner without having to identify the forces of interaction

between the particles or bodies.

The variational approach seeks to derive the equations of motion for a system of particles

in the presence of a potential force eld as a solution to a minimization problem. The inde-

pendent variable in the problem will clearly be time, and the dependent functions will be the

three-dimensional (3D) positions of each particle. The aim is to nd a function L such that the

paths of the particles between times t

and t

extremize the integral:

ILxyzxyz

()

,,,,,;





. (1.1)

The integral I will be referred to as the action of the system and the function L as the

Lagrangian. In fact, we can show that when the Lagrangian L is dened as

LTVmvVxyzt=-=-

()

,,; ,

(1.2)

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