Advanced Textbooks in Control and Signal Processing

Series Editors

Professor Michael J. Grimble, Professor of Industrial Systems and Director

Professor Michael A. Johnson, Professor Emeritus of Control Systems and Deputy Director Industrial Control Centre, Department of Electronic and Electrical Engineering, University of Strathclyde, Graham Hills Building, 50 George Street, Glasgow G1 1QE, UK Other titles published in this series:

Genetic Algorithms

K.F. Man, K.S. Tang and S. Kwong Introduction to Optimal Estimation

E.W. Kamen and J.K. Su

Discrete-time Signal Processing

D. Williamson

Neural Networks for Modelling and Control of Dynamic Systems

M. N?rgaard, O. Ravn, N.K. Poulsen

and L.K. Hansen

Fault Detection and Diagnosis in Industrial Systems

L.H. Chiang, E.L. Russell and R.D. Braatz Soft Computing

L. Fortuna, G. Rizzotto, M. Lavorgna, G. Nunnari, M.G. Xibilia and R. Caponetto Statistical Signal Processing

T. Chonavel

Discrete-time Stochastic Processes

(2nd Edition)

T. S?derstr?m

Parallel Computing for Real-time Signal Processing and Control

M.O. Tokhi, M.A. Hossain and

M.H. Shaheed

Multivariable Control Systems

P. Albertos and A. Sala Control Systems with Input and Output Constraints

A.H. Glattfelder and W. Schaufelberger Analysis and Control of Non-linear Process Systems

K.M. Hangos, J. Bokor and

G. Szederkényi

Model Predictive Control (2nd Edition) E.F. Camacho and C. Bordons Principles of Adaptive Filters and Self-learning Systems

A. Zaknich

Digital Self-tuning Controllers

V. Bobál, J. B?hm, J. Fessl and

J. Machá?ek

Control of Robot Manipulators in Joint Space

R. Kelly, V. Santibá?ez and A. Loría Receding Horizon Control

W.H. Kwon and S. Han

Robust Control Design with MATLAB?D.-W. Gu, P.H. Petkov and

M.M. Konstantinov

Control of Dead-time Processes

J.E. Normey-Rico and E.F. Camacho Modeling and Control of Discrete-event Dynamic Systems

B. Hrúz and M.

C. Zhou

Bruno Siciliano ? Lorenzo Sciavicco Luigi Villani ? Giuseppe Oriolo Robotics

Modelling, Planning and Control 123

Bruno Siciliano, PhD

Dipartimento di Informatica e Sistemistica Università di Napoli Federico II

Via Claudio 21

80125 Napoli

Italy Lorenzo Sciavicco, DrEng Dipartimento di Informatica e Automazione Università di Roma Tre

Via della Vasca Navale 79

00146 Roma


Luigi Villani, PhD

Dipartimento di Informatica e Sistemistica Università di Napoli Federico II

Via Claudio 21

80125 Napoli

Italy Giuseppe Oriolo, PhD

Dipartimento di Informatica e Sistemistica Università di Roma “La Sapienza”

Via Ariosto 25

00185 Roma


ISBN 978-1-84628-641-4 e-ISBN 978-1-84628-642-1

DOI 10.1007/978-1-84628-642-1

Advanced Textbooks in Control and Signal Processing series ISSN 1439-2232

A catalogue record for this book is available from the British Library

Library of Congress Control Number: 2008939574

? 2009 Springer-Verlag London Limited

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Series Editors’Foreword

The topics of control engineering and signal processing continue to?ourish and develop.In common with general scienti?c investigation,new ideas,concepts and interpretations emerge quite spontaneously and these are then discussed, used,discarded or subsumed into the prevailing subject paradigm.Sometimes these innovative concepts coalesce into a new sub-discipline within the broad subject tapestry of control and signal processing.This preliminary battle be-tween old and new usually takes place at conferences,through the Internet and in the journals of the discipline.After a little more maturity has been acquired by the new concepts then archival publication as a scienti?c or engineering monograph may occur.

A new concept in control and signal processing is known to have arrived when su?cient material has evolved for the topic to be taught as a specialised tutorial workshop or as a course to undergraduate,graduate or industrial engineers.Advanced Textbooks in Control and Signal Processing are designed as a vehicle for the systematic presentation of course material for both popular and innovative topics in the discipline.It is hoped that prospective authors will welcome the opportunity to publish a structured and systematic presentation of some of the newer emerging control and signal processing technologies in the textbook series.

Robots have appeared extensively in the artistic?eld of science?ction writing.The actual name robot arose from its use by the playwright Karel ˇCapek in the play Rossum’s Universal Robots(1920).Not surprisingly,the

artistic focus has been on mechanical bipeds with anthropomorphic person-alities often termed androids.This focus has been the theme of such cine-matic productions as,I,Robot(based on Isaac Asimov’s stories)and Stanley Kubrick’s?lm,A.I.;however,this book demonstrates that robot technology is already widely used in industry and that there is some robot technology which is at prototype stage rapidly approaching introduction to commercial use.Currently,robots may be classi?ed according to their mobility attributes as shown in the?gure.

viii Series Editors’Foreword


The largest class of robots extant today is that of the?xed robot which does repetitive but often precise mechanical and physical tasks.These robots pervade many areas of modern industrial automation and are mainly con-cerned with tasks performed in a structured environment.It seems highly likely that as the technology develops the number of mobile robots will signif-icantly increase and become far more visible as more applications and tasks in an unstructured environment are serviced by robotic technology.

What then is robotics?A succinct de?nition is given in The Chamber’s Dic-tionary(2003):the branch of technology dealing with the design,construction and use of robots.This de?nition certainly captures the spirit of this volume in the Advanced Textbooks in Control and Signal Processing series entitled Robotics and written by Bruno Siciliano,Lorenzo Sciavicco,Luigi Villani and Giuseppe Oriolo.This book is a greatly extended and revised version of an earlier book in the series,Modelling and Control of Robot Manipulators(2000, ISBN:978-1-85233-221-1).As can be seen from the?gure above,robots cover a wide variety of types and the new book seeks to present a uni?ed approach to robotics whilst focusing on the two leading classes of robots,the?xed and the wheeled types.The textbook series publishes volumes in support of new disciplines that are emerging with their own novel identity,and robotics as a subject certainly falls into this category.The full scope of robotics lies at the intersection of mechanics,electronics,signal processing,control engineer-ing,computing and mathematical modelling.However,within this very broad framework the authors have pursued the themes of modelling,planning and control.These are,and will remain,fundamental aspects of robot design and operation for years to come.Some interesting innovations in this text include material on wheeled robots and on vision as used in the control of robots. Thus,the book provides a thorough theoretical grounding in an area where the technologies are evolving and developing in new applications.

The series is one of textbooks for advanced courses,and volumes in the series have useful pedagogical features.This volume has twelve chapters cov-ering both fundamental and specialist topics,and there is a Problems section at the end of each chapter.Five appendices have been included to give more depth to some of the advanced methods used in the text.There are over twelve pages of references and nine pages of index.The details of the citations and index should also facilitate the use of the volume as a source of reference as

Series Editors’Foreword ix well as a course study text.We expect that the student,the researcher,the lecturer and the engineer will?nd this volume of great value for the study of robotics.

Glasgow Michael J.Grimble August2008Michael A.Johnson


In the last25years,the?eld of robotics has stimulated an increasing interest in a wide number of scholars,and thus literature has been conspicuous,both in terms of textbooks and monographs,and in terms of specialized journals dedicated to robotics.This strong interest is also to be attributed to the inter-disciplinary character of robotics,which is a science having roots in di?erent areas.Cybernetics,mechanics,controls,computers,bioengineering,electron-ics—to mention the most important ones—are all cultural domains which undoubtedly have boosted the development of this science.

Despite robotics representing as yet a relatively young discipline,its foun-dations are to be considered well-assessed in the classical textbook literature. Among these,modelling,planning and control play a basic role,not only in the traditional context of industrial robotics,but also for the advanced scenarios of?eld and service robots,which have attracted an increasing interest from the research community in the last15years.

This book is the natural evolution of the previous text Modelling and Con-trol of Robot Manipulators by the?rst two co-authors,published in1995,and in2000with its second edition.The cut of the original textbook has been con?rmed with the educational goal of blending the fundamental and techno-logical aspects with those advanced aspects,on a uniform track as regards a rigorous formalism.

The fundamental and technological aspects are mainly concentrated in the ?rst six chapters of the book and concern the theory of manipulator structures, including kinematics,statics and trajectory planning,and the technology of robot actuators,sensors and control units.

The advanced aspects are dealt with in the subsequent six chapters and concern dynamics and motion control of robot manipulators,interaction with the environment using exteroceptive sensory data(force and vision),mobile robots and motion planning.

The book contents are organized in12chapters and5appendices.

In Chap.1,the di?erences between industrial and advanced applications are enlightened in the general robotics context.The most common mechanical

xii Preface

structures of robot manipulators and wheeled mobile robots are presented. Topics are also introduced which are developed in the subsequent chapters.

In Chap.2kinematics is presented with a systematic and general approach which refers to the Denavit-Hartenberg convention.The direct kinematics equation is formulated which relates joint space variables to operational space variables.This equation is utilized to?nd manipulator workspace as well as to derive a kinematic calibration technique.The inverse kinematics problem is also analyzed and closed-form solutions are found for typical manipulation structures.

Di?erential kinematics is presented in Chap.3.The relationship between joint velocities and end-e?ector linear and angular velocities is described by the geometric Jacobian.The di?erence between the geometric Jacobian and the analytical Jacobian is pointed out.The Jacobian constitutes a fundamen-tal tool to characterize a manipulator,since it allows the determination of singular con?gurations,an analysis of redundancy and the expression of the relationship between forces and moments applied to the end-e?ector and the resulting joint torques at equilibrium con?gurations(statics).Moreover,the Jacobian allows the formulation of inverse kinematics algorithms that solve the inverse kinematics problem even for manipulators not having a closed-form solution.

In Chap.4,trajectory planning techniques are illustrated which deal with the computation of interpolating polynomials through a sequence of desired points.Both the case of point-to-point motion and that of motion through a sequence of points are treated.Techniques are developed for generating trajectories both in the joint space and in the operational space,with a special concern to orientation for the latter.

Chapter5is devoted to the presentation of actuators and sensors.After an illustration of the general features of an actuating system,methods to control electric and hydraulic drives are presented.The most common proprioceptive and exteroceptive sensors in robotics are described.

In Chap.6,the functional architecture of a robot control system is illus-trated.The characteristics of programming environments are presented with an emphasis on teaching-by-showing and robot-oriented programming.A gen-eral model for the hardware architecture of an industrial robot control system is?nally discussed.

Chapter7deals with the derivation of manipulator dynamics,which plays a fundamental role in motion simulation,manipulation structure analysis and control algorithm synthesis.The dynamic model is obtained by explicitly tak-ing into account the presence of actuators.Two approaches are considered, namely,one based on Lagrange formulation,and the other based on Newton–Euler formulation.The former is conceptually simpler and systematic,whereas the latter allows computation of a dynamic model in a recursive form.Notable properties of the dynamic model are presented,including linearity in the pa-rameters which is utilized to develop a model identi?cation technique.Finally,

Preface xiii the transformations needed to express the dynamic model in the operational space are illustrated.

In Chap.8the problem of motion control in free space is treated.The distinction between joint space decentralized and centralized control strategies is pointed out.With reference to the former,the independent joint control technique is presented which is typically used for industrial robot control. As a premise to centralized control,the computed torque feedforward control technique is introduced.Advanced schemes are then introduced including PD control with gravity compensation,inverse dynamics control,robust control, and adaptive control.Centralized techniques are extended to operational space control.

Force control of a manipulator in contact with the working environment is tackled in Chap.9.The concepts of mechanical compliance and impedance are de?ned as a natural extension of operational space control schemes to the constrained motion case.Force control schemes are then presented which are obtained by the addition of an outer force feedback loop to a motion control scheme.The hybrid force/motion control strategy is?nally presented with reference to the formulation of natural and arti?cial constraints describing an interaction task.

In Chap.10,visual control is introduced which allows the use of infor-mation on the environment surrounding the robotic system.The problems of camera position and orientation estimate with respect to the objects in the scene are solved by resorting to both analytical and numerical techniques. After presenting the advantages to be gained with stereo vision and a suit-able camera calibration,the two main visual control strategies are illustrated, namely in the operational space and in the image space,whose advantages can be e?ectively combined in the hybrid visual control scheme.

Wheeled mobile robots are dealt with in Chap.11,which extends some modelling,planning and control aspects of the previous chapters.As far as modelling is concerned,it is worth distinguishing between the kinematic model,strongly characterized by the type of constraint imposed by wheel rolling,and the dynamic model which accounts for the forces acting on the robot.The peculiar structure of the kinematic model is keenly exploited to develop both path and trajectory planning techniques.The control problem is tackled with reference to two main motion tasks:trajectory tracking and con?guration regulation.Further,it is evidenced how the implementation of the control schemes utilizes odometric localization methods.

Chapter12reprises the planning problems treated in Chaps.4and11 for robot manipulators and mobile robots respectively,in the case when ob-stacles are present in the workspace.In this framework,motion planning is referred to,which is e?ectively formulated in the con?guration space.Several planning techniques for mobile robots are then presented:retraction,cell de-composition,probabilistic,arti?cial potential.The extension to the case of robot manipulators is?nally discussed.

xiv Preface

This chapter concludes the presentation of the topical contents of the text-

book;?ve appendices follow which have been included to recall background


Appendix A is devoted to linear algebra and presents the fundamental

notions on matrices,vectors and related operations.

Appendix B presents those basic concepts of rigid body mechanics which

are preliminary to the study of manipulator kinematics,statics and dynamics.

Appendix C illustrates the principles of feedback control of linear systems

and presents a general method based on Lyapunov theory for control of non-

linear systems.

Appendix D deals with some concepts of di?erential geometry needed for

control of mechanical systems subject to nonholonomic constraints.

Appendix E is focused on graph search algorithms and their complexity in

view of application to motion planning methods.

The organization of the contents according to the above illustrated scheme

allows the adoption of the book as a reference text for a senior undergrad-

uate or graduate course in automation,computer,electrical,electronics,or

mechanical engineering with strong robotics content.

From a pedagogical viewpoint,the various topics are presented in an in-

strumental manner and are developed with a gradually increasing level of di?-

culty.Problems are raised and proper tools are established to?nd engineering-

oriented solutions.Each chapter is introduced by a brief preamble providing

the rationale and the objectives of the subject matter.The topics needed for a

pro?cient study of the text are presented in the?ve appendices,whose purpose

is to provide students of di?erent extraction with a homogeneous background.

The book contains more than310illustrations and more than60worked-

out examples and case studies spread throughout the text with frequent resort

to simulation.The results of computer implementations of inverse kinemat-

ics algorithms,trajectory planning techniques,inverse dynamics computation,

motion,force and visual control algorithms for robot manipulators,and mo-

tion control for mobile robots are presented in considerable detail in order to

facilitate the comprehension of the theoretical development,as well as to in-

crease sensitivity of application in practical problems.In addition,nearly150

end-of-chapter problems are proposed,some of which contain further study

matter of the contents,and the book is accompanied by an electronic solu-

tions manual(downloadable from

containing the MATLAB R code for computer problems;this is available free of charge to those adopting this volume as a text for courses.Special care has

been devoted to the selection of bibliographical references(more than250)

which are cited at the end of each chapter in relation to the historical devel-

opment of the?eld.

Finally,the authors wish to acknowledge all those who have been helpful

in the preparation of this book.

With reference to the original work,as the basis of the present textbook,

devoted thanks go to Pasquale Chiacchio and Stefano Chiaverini for their

Preface xv contributions to the writing of the chapters on trajectory planning and force control,respectively.Fabrizio Caccavale and Ciro Natale have been of great help in the revision of the contents for the second edition.

A special note of thanks goes to Alessandro De Luca for his punctual and critical reading of large portions of the text,as well as to Vincenzo Lippiello, Agostino De Santis,Marilena Vendittelli and Luigi Freda for their contribu-tions and comments on some sections.

Naples and Rome Bruno Siciliano July2008Lorenzo Sciavicco

Luigi Villani

Giuseppe Oriolo


1Introduction (1)

1.1Robotics (1)

1.2Robot Mechanical Structure (3)

1.2.1Robot Manipulators (4)

1.2.2Mobile Robots (10)

1.3Industrial Robotics (15)

1.4Advanced Robotics (25)

1.4.1Field Robots (26)

1.4.2Service Robots (27)

1.5Robot Modelling,Planning and Control (29)

1.5.1Modelling (30)

1.5.2Planning (32)

1.5.3Control (32)

Bibliography (33)

2Kinematics (39)

2.1Pose of a Rigid Body (39)

2.2Rotation Matrix (40)

2.2.1Elementary Rotations (41)

2.2.2Representation of a Vector (42)

2.2.3Rotation of a Vector (44)

2.3Composition of Rotation Matrices (45)

2.4Euler Angles (48)

2.4.1ZYZ Angles (49)

2.4.2RPY Angles (51)

2.5Angle and Axis (52)

2.6Unit Quaternion (54)

2.7Homogeneous Transformations (56)

2.8Direct Kinematics (58)

2.8.1Open Chain (60)

2.8.2Denavit–Hartenberg Convention (61)

xviii Contents

2.8.3Closed Chain (65)

2.9Kinematics of Typical Manipulator Structures (68)

2.9.1Three-link Planar Arm (69)

2.9.2Parallelogram Arm (70)

2.9.3Spherical Arm (72)

2.9.4Anthropomorphic Arm (73)

2.9.5Spherical Wrist (75)

2.9.6Stanford Manipulator (76)

2.9.7Anthropomorphic Arm with Spherical Wrist (77)

2.9.8DLR Manipulator (79)

2.9.9Humanoid Manipulator (81)

2.10Joint Space and Operational Space (83)

2.10.1Workspace (85)

2.10.2Kinematic Redundancy (87)

2.11Kinematic Calibration (88)

2.12Inverse Kinematics Problem (90)

2.12.1Solution of Three-link Planar Arm (91)

2.12.2Solution of Manipulators with Spherical Wrist (94)

2.12.3Solution of Spherical Arm (95)

2.12.4Solution of Anthropomorphic Arm (96)

2.12.5Solution of Spherical Wrist (99)

Bibliography (100)

Problems (100)

3Di?erential Kinematics and Statics (105)

3.1Geometric Jacobian (105)

3.1.1Derivative of a Rotation Matrix (106)

3.1.2Link Velocities (108)

3.1.3Jacobian Computation (111)

3.2Jacobian of Typical Manipulator Structures (113)

3.2.1Three-link Planar Arm (113)

3.2.2Anthropomorphic Arm (114)

3.2.3Stanford Manipulator (115)

3.3Kinematic Singularities (116)

3.3.1Singularity Decoupling (117)

3.3.2Wrist Singularities (119)

3.3.3Arm Singularities (119)

3.4Analysis of Redundancy (121)

3.5Inverse Di?erential Kinematics (123)

3.5.1Redundant Manipulators (124)

3.5.2Kinematic Singularities (127)

3.6Analytical Jacobian (128)

3.7Inverse Kinematics Algorithms (132)

3.7.1Jacobian(Pseudo-)inverse (133)

3.7.2Jacobian Transpose (134)

Contents xix

3.7.3Orientation Error (137)

3.7.4Second-order Algorithms (141)

3.7.5Comparison Among Inverse Kinematics Algorithms (143)

3.8Statics (147)

3.8.1Kineto-Statics Duality (148)

3.8.2Velocity and Force Transformation (149)

3.8.3Closed Chain (151)

3.9Manipulability Ellipsoids (152)

Bibliography (158)

Problems (159)

4Trajectory Planning (161)

4.1Path and Trajectory (161)

4.2Joint Space Trajectories (162)

4.2.1Point-to-Point Motion (163)

4.2.2Motion Through a Sequence of Points (168)

4.3Operational Space Trajectories (179)

4.3.1Path Primitives (181)

4.3.2Position (184)

4.3.3Orientation (187)

Bibliography (188)

Problems (189)

5Actuators and Sensors (191)

5.1Joint Actuating System (191)

5.1.1Transmissions (192)

5.1.2Servomotors (193)

5.1.3Power Ampli?ers (197)

5.1.4Power Supply (198)

5.2Drives (198)

5.2.1Electric Drives (198)

5.2.2Hydraulic Drives (202)

5.2.3Transmission E?ects (204)

5.2.4Position Control (206)

5.3Proprioceptive Sensors (209)

5.3.1Position Transducers (210)

5.3.2Velocity Transducers (214)

5.4Exteroceptive Sensors (215)

5.4.1Force Sensors (215)

5.4.2Range Sensors (219)

5.4.3Vision Sensors (225)

Bibliography (230)

Problems (230)

xx Contents

6Control Architecture (233)

6.1Functional Architecture (233)

6.2Programming Environment (238)

6.2.1Teaching-by-Showing (240)

6.2.2Robot-oriented Programming (241)

6.3Hardware Architecture (242)

Bibliography (245)

Problems (245)

7Dynamics (247)

7.1Lagrange Formulation (247)

7.1.1Computation of Kinetic Energy (249)

7.1.2Computation of Potential Energy (255)

7.1.3Equations of Motion (255)

7.2Notable Properties of Dynamic Model (257)

7.2.1Skew-symmetry of Matrix˙B?2C (257)

7.2.2Linearity in the Dynamic Parameters (259)

7.3Dynamic Model of Simple Manipulator Structures (264)

7.3.1Two-link Cartesian Arm (264)

7.3.2Two-link Planar Arm (265)

7.3.3Parallelogram Arm (277)

7.4Dynamic Parameter Identi?cation (280)

7.5Newton–Euler Formulation (282)

7.5.1Link Accelerations (285)

7.5.2Recursive Algorithm (286)

7.5.3Example (289)

7.6Direct Dynamics and Inverse Dynamics (292)

7.7Dynamic Scaling of Trajectories (294)

7.8Operational Space Dynamic Model (296)

7.9Dynamic Manipulability Ellipsoid (299)

Bibliography (301)

Problems (301)

8Motion Control (303)

8.1The Control Problem (303)

8.2Joint Space Control (305)

8.3Decentralized Control (309)

8.3.1Independent Joint Control (311)

8.3.2Decentralized Feedforward Compensation (319)

8.4Computed Torque Feedforward Control (324)

8.5Centralized Control (327)

8.5.1PD Control with Gravity Compensation (328)

8.5.2Inverse Dynamics Control (330)

8.5.3Robust Control (333)

8.5.4Adaptive Control (338)

Contents xxi

8.6Operational Space Control (343)

8.6.1General Schemes (344)

8.6.2PD Control with Gravity Compensation (345)

8.6.3Inverse Dynamics Control (347)

8.7Comparison Among Various Control Schemes (349)

Bibliography (359)

Problems (360)

9Force Control (363)

9.1Manipulator Interaction with Environment (363)

9.2Compliance Control (364)

9.2.1Passive Compliance (366)

9.2.2Active Compliance (367)

9.3Impedance Control (372)

9.4Force Control (378)

9.4.1Force Control with Inner Position Loop (379)

9.4.2Force Control with Inner Velocity Loop (380)

9.4.3Parallel Force/Position Control (381)

9.5Constrained Motion (384)

9.5.1Rigid Environment (385)

9.5.2Compliant Environment (389)

9.6Natural and Arti?cial Constraints (391)

9.6.1Analysis of Tasks (392)

9.7Hybrid Force/Motion Control (396)

9.7.1Compliant Environment (397)

9.7.2Rigid Environment (401)

Bibliography (403)

Problems (404)

10Visual Servoing (407)

10.1Vision for Control (407)

10.1.1Con?guration of the Visual System (409)

10.2Image Processing (410)

10.2.1Image Segmentation (411)

10.2.2Image Interpretation (416)

10.3Pose Estimation (418)

10.3.1Analytic Solution (419)

10.3.2Interaction Matrix (424)

10.3.3Algorithmic Solution (427)

10.4Stereo Vision (433)

10.4.1Epipolar Geometry (433)

10.4.2Triangulation (435)

10.4.3Absolute Orientation (436)

10.4.43D Reconstruction from Planar Homography (438)

10.5Camera Calibration (440)

xxii Contents

10.6The Visual Servoing Problem (443)

10.7Position-based Visual Servoing (445)

10.7.1PD Control with Gravity Compensation (446)

10.7.2Resolved-velocity Control (447)

10.8Image-based Visual Servoing (449)

10.8.1PD Control with Gravity Compensation (449)

10.8.2Resolved-velocity Control (451)

10.9Comparison Among Various Control Schemes (453)

10.10Hybrid Visual Servoing (460)

Bibliography (465)

Problems (466)

11Mobile Robots (469)

11.1Nonholonomic Constraints (469)

11.1.1Integrability Conditions (473)

11.2Kinematic Model (476)

11.2.1Unicycle (478)

11.2.2Bicycle (479)

11.3Chained Form (482)

11.4Dynamic Model (485)

11.5Planning (489)

11.5.1Path and Timing Law (489)

11.5.2Flat Outputs (491)

11.5.3Path Planning (492)

11.5.4Trajectory Planning (498)

11.5.5Optimal Trajectories (499)

11.6Motion Control (502)

11.6.1Trajectory Tracking (503)

11.6.2Regulation (510)

11.7Odometric Localization (514)

Bibliography (518)

Problems (518)

12Motion Planning (523)

12.1The Canonical Problem (523)

12.2Con?guration Space (525)

12.2.1Distance (527)

12.2.2Obstacles (527)

12.2.3Examples of Obstacles (528)

12.3Planning via Retraction (532)

12.4Planning via Cell Decomposition (536)

12.4.1Exact Decomposition (536)

12.4.2Approximate Decomposition (539)

12.5Probabilistic Planning (541)

12.5.1PRM Method (541)