Theory of Applied Robotics

Transcription

1 Theory of Applied Robotics

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3 Reza N. Jazar Theory of Applied Robotics Kinematics, Dynamics, and Control Second Edition 123

4 Prof.RezaN.Jazar School of Aerospace, Mechanical, and Manufacturing Engineering RMIT University Melbourne, Victoria Australia ISBN e-isbn DOI / Springer New York Dordrecht Heidelberg London Library of Congress Control Number: c Springer Science+Business Media, LLC 2006, 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Cover illustration c Konstantin Inozemtsev Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 Dedicated to my wife, Mojgan and our children, Vazan and Kavosh.

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7 I am Cyrus, king of the world, great king, mighty king, king of Babylon, king of Sumer and Akkad, king of the four quarters. I ordered to write books, many books, books to teach my people, I ordered to make schools, many schools, to educate my people. Marduk, the lord of the gods, said burning books is the greatest sin. I, Cyrus, and my people, and my army will protect books and schools. They will fight whoever burns books and burns schools, the great sin. Cyrus the great

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9 Preface to the Second Edition The second edition of this book would not have been possible without the comments and suggestions from my students, especially those at Columbia University. Many of the new topics introduced here are a direct result of student feedback that helped me refine and clarify the material. My intention when writing this book was to develop material that I would have liked to had available as a student. Hopefully, I have succeeded in developing a reference that covers all aspects of robotics with sufficient detail and explanation. The first edition of this book was published in 2007 and soon after its publication it became a very popular reference in the field of robotics. I wish to thank the many students and instructors who have used the book or referenced it. Your questions, comments and suggestions have helped me create the second edition.

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11 Preface This book is designed to serve as a text for engineering students. It introduces the fundamental knowledge used in robotics. This knowledge can be utilized to develop computer programs for analyzing the kinematics, dynamics, and control of robotic systems. The subject of robotics may appear overdosed by the number of available texts because the field has been growing rapidly since However,the topic remains alive with modern developments, which are closely related to the classical material. It is evident that no single text can cover the vast scope of classical and modern materials in robotics. Thus the demand for new books arises because the field continues to progress. Another factor is the trend toward analytical unification of kinematics, dynamics, and control. Classical kinematics and dynamics of robots has its roots in the work of great scientists of the past four centuries who established the methodology and understanding of the behavior of dynamic systems. The development of dynamic science, since the beginning of the twentieth century, has moved toward analysis of controllable man-made systems. Therefore, merging the kinematics and dynamics with control theory is the expected development for robotic analysis. The other important development is the fast growing capability of accurate and rapid numerical calculations, along with intelligent computer programming. Level of the Book This book has evolved from nearly a decade of research in nonlinear dynamic systems, and teaching undergraduate-graduate level courses in robotics. It is addressed primarily to the last year of undergraduate study and the first year graduate student in engineering. Hence, it is an intermediate textbook. This book can even be the first exposure to topics in spatial kinematics and dynamics of mechanical systems. Therefore, it provides both fundamental and advanced topics on the kinematics and dynamics of robots. The whole book can be covered in two successive courses however, it is possible to jump over some sections and cover the book in one course. The students are required to know the fundamentals of kinematics and dynamics, as well as a basic knowledge of numerical methods.

12 xii Preface The contents of the book have been kept at a fairly theoretical-practical level. Many concepts are deeply explained and their use emphasized, and most of the related theory and formal proofs have been explained. Throughout the book, a strong emphasis is put on the physical meaning of the concepts introduced. Topics that have been selected are of high interest in the field. An attempt has been made to expose the students to a broad range of topics and approaches. Organization of the Book The text is organized so it can be used for teaching or for self-study. Chapter 1 Introduction, contains general preliminaries with a brief review of the historical development and classification of robots. Part I Kinematics, presents the forward and inverse kinematics of robots. Kinematics analysis refers to position, velocity, and acceleration analysis of robots in both joint and base coordinate spaces. It establishes kinematic relations among the end-effecter and the joint variables. The method of Denavit-Hartenberg for representing body coordinate frames is introduced and utilized for forward kinematics analysis. The concept of modular treatment of robots is well covered to show how we may combine simple links to make the forward kinematics of a complex robot. For inverse kinematics analysis, the idea of decoupling, the inverse matrix method, and the iterative technique are introduced. It is shown that the presence of a spherical wrist is what we need to apply analytic methods in inverse kinematics. Part II Dynamics, presents a detailed discussion of robot dynamics. An attempt is made to review the basic approaches and demonstrate how these can be adapted for the active displacement framework utilized for robot kinematics in the earlier chapters. The concepts of the recursive Newton-Euler dynamics, Lagrangian function, manipulator inertia matrix, and generalized forces are introduced and applied for derivation of dynamic equations of motion. Part III Control, presents the floating time technique for time-optimal control of robots. The outcome of the technique is applied for an openloop control algorithm. Then, a computed-torque method is introduced, in which a combination of feedforward and feedback signals are utilized to render the system error dynamics. Method of Presentation The structure of presentation is in a "fact-reason-application" fashion. The "fact" is the main subject we introduce in each section. Then the reason is given as a "proof." Finally the application of the fact is examined in some "examples." The "examples" are a very important part of the book because they show how to implement the knowledge introduced in "facts." They also cover some other facts that are needed to expand the subject.

13 Preface xiii Prerequisites Since the book is written for senior undergraduate and first-year graduate level students of engineering, the assumption is that users are familiar with matrix algebra as well as basic feedback control. Prerequisites for readers of this book consist of the fundamentals of kinematics, dynamics, vector analysis, and matrix theory. These basics are usually taught in the first three undergraduate years. Unit System The system of units adopted in this book is, unless otherwise stated, the international system of units (SI). The units of degree (deg) or radian ( rad) are utilized for variables representing angular quantities. Symbols Lowercase bold letters indicate a vector. Vectors may be expressed in an n dimensional Euclidian space. Example: r, s, d, a, b, c p, q, v, w, y, z ω, α, ², θ, δ, φ Uppercase bold letters indicate a dynamic vector or a dynamic matrix. Example: F, M, J Lowercase letters with a hat indicate a unit vector. Unit vectors are not bolded. Example: î, ĵ, ˆk, ê, û, ˆn Î, Ĵ, ˆK, êθ, ê ϕ, ê ψ Lowercase letters with a tilde indicate a 3 3 skew symmetric matrix associated to a vector. Example: ã = 0 a 3 a 2 a 3 0 a 1, a = a 1 a 2 a 2 a 1 0 a 3 An arrow above two uppercase letters indicates the start and end points of a position vector. Example: ON = a position vector from point O to point N

14 xiv Preface A double arrow above a lowercase letter indicates a 4 4 matrix associated to a quaternion. Example: q 0 q 1 q 2 q 3 q = q 1 q 0 q 3 q 2 q 2 q 3 q 0 q 1 q 3 q 2 q 1 q 0 q = q 0 + q 1 i + q 2 j + q 3 k The length of a vector is indicated by a non-bold lowercase letter. Example: r = r, a = a, b = b, s = s Capital letters A, Q, R, andt indicate rotation or transformation matrices. Example: cα 0 sα 1 cos α sin α 0 Q Z,α = sin α cos α 0, G T B = sα 0 cα Capital letter B is utilized to denote a body coordinate frame. Example: B(oxyz), B(Oxyz), B 1 (o 1 x 1 y 1 z 1 ) Capital letter G is utilized to denote a global, inertial, or fixed coordinate frame. Example: G, G(XY Z), G(OXY Z) Right subscript on a transformation matrix indicates the departure frames. Example: T B = transformation matrix from frame B(oxyz) Left superscript on a transformation matrix indicates the destination frame. Example: G T B = transformation matrix from frame B(oxyz) to frame G(OXY Z) Whenever there is no sub or superscript, the matrices are shown in a bracket. Example: cα 0 sα 1 [T ]= sα 0 cα

15 Preface xv Left superscript on a vector denotes the frame in which the vector is expressed. That superscript indicates the frame that the vector belongs to; so the vector is expressed using the unit vectors of that frame. Example: G r = position vector expressed in frame G(OXY Z) Right subscript on a vector denotes the tip point that the vector is referred to. Example: G r P = position vector of point P expressed in coordinate frame G(OXY Z) Left subscript on a vector indicates the frame that the angular vector is measured with respect to. Example: G B v P = velocity vector of point P in coordinate frame B(oxyz) expressed in the global coordinate frame G(OXY Z) We drop the left subscript if it is the same as the left superscript. Example: B Bv P B v P Right subscript on an angular velocity vector indicates the frame that the angular vector is referred to. Example: ω B = angular velocity of the body coordinate frame B(oxyz) Left subscript on an angular velocity vector indicates the frame that the angular vector is measured with respect to. Example: Gω B = angular velocity of the body coordinate frame B(oxyz) with respect to the global coordinate frame G(OXY Z) Left superscript on an angular velocity vector denotes the frame in which the angular velocity is expressed. Example: B 2 G ω B 1 = angular velocity of the body coordinate frame B 1 with respect to the global coordinate frame G, and expressed in body coordinate frame B 2 Whenever the left subscript and superscript of an angular velocity are the same, we usually drop the left superscript. Example: Gω B G Gω B

16 xvi Preface If the right subscript on a force vector is a number, it indicates the number of coordinate frame in a serial robot. Coordinate frame B i is set up at joint i +1.Example: F i = force vector at joint i +1 measured at the origin of B i (oxyz) At joint i there is always an action force F i, that link (i) applies on link (i +1), and a reaction force F i, that link (i +1) applies on link (i). On link (i) there is always an action force F i 1 coming from link (i 1), and a reaction force F i coming from link (i +1). Action force is called driving force, and reaction force is called driven force. If the right subscript on a moment vector is a number, it indicates the number of coordinate frames in a serial robot. Coordinate frame B i issetupatjointi +1.Example: M i = moment vector at joint i +1 measured at the origin of B i (oxyz) At joint i there is always an action moment M i,thatlink(i) applies on link (i +1), and a reaction moment M i, that link (i +1) applies on link (i). On link (i) there is always an action moment M i 1 coming from link (i 1), and a reaction moment M i coming from link (i+1). Action moment is called driving moment, and reaction moment is called driven moment. Left superscript on derivative operators indicates the frame in which the derivative of a variable is taken. Example: G d G dt x, d B r P, dt B d G dt Br P If the variable is a vector function, and also the frame in which the vector is defined is the same as the frame in which a time derivative is taken, we may use the following short notation, G d G r P = G ṙ P, dt and write equations simpler. Example: B d B o r P = B o ṙ P dt G G d v = G r(t) = G ṙ dt If followed by angles, lowercase c and s denote cos and sin functions in mathematical equations. Example: cα =cosα, sϕ =sinϕ

17 Preface xvii Capital bold letter I indicates a unit matrix, which, depending on the dimension of the matrix equation, could be a 3 3 or a 4 4 unit matrix. I 3 or I 4 are also being used to clarify the dimension of I. Example: I = I 3 = An asterisk F indicates a more advanced subject or example that is not designed for undergraduate teaching and can be dropped in the first reading. Two parallel joint axes are indicated by a parallel sign, (k). Two orthogonal joint axes are indicated by an orthogonal sign, (`). Two orthogonal joint axes are intersecting at a right angle. Two perpendicular joint axes are indicated by a perpendicular sign, ( ). Two perpendicular joint axes are at a right angle with respect to their common normal.

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19 Contents 1 Introduction HistoricalDevelopment RobotComponents Link Joint Manipulator Wrist End-effector Actuators Sensors Controller Robot Classifications Geometry Workspace Actuation Control Application Introduction to Robot s Kinematics, Dynamics, and Control F Triad UnitVectors ReferenceFrameandCoordinateSystem VectorFunction ProblemsofRobotDynamics PreviewofCoveredTopics RobotsasMulti-disciplinaryMachines Summary Exercises I Kinematics 29 2 Rotation Kinematics RotationAboutGlobalCartesianAxes SuccessiveRotationAboutGlobalCartesianAxes GlobalRoll-Pitch-YawAngles RotationAboutLocalCartesianAxes SuccessiveRotationAboutLocalCartesianAxes... 50

20 xx Contents 2.6 EulerAngles LocalRoll-Pitch-YawAngles LocalAxesVersusGlobalAxesRotation GeneralTransformation ActiveandPassiveTransformation Summary KeySymbols Exercises Orientation Kinematics Axis-angleRotation F EulerParameters F DeterminationofEulerParameters F Quaternions F SpinorsandRotators F ProblemsinRepresentingRotations F Rotationmatrix F Angle-axis F Eulerangles F Quaternion F Eulerparameters F CompositionandDecompositionofRotations Summary KeySymbols Exercises Motion Kinematics RigidBodyMotion HomogeneousTransformation InverseHomogeneousTransformation CompoundHomogeneousTransformation F ScrewCoordinates F InverseScrew F CompoundScrewTransformation F ThePlückerLineCoordinate F TheGeometryofPlaneandLine F Moment F AngleandDistance F PlaneandLine F ScrewandPlückerCoordinate Summary KeySymbols Exercises

21 Contents xxi 5 Forward Kinematics Denavit-HartenbergNotation Transformation Between Two Adjacent Coordinate Frames ForwardPositionKinematicsofRobots SphericalWrist AssemblingKinematics F CoordinateTransformationUsingScrews F NonDenavit-HartenbergMethods Summary KeySymbols Exercises Inverse Kinematics DecouplingTechnique InverseTransformationTechnique F IterativeTechnique F ComparisonoftheInverseKinematicsTechniques F ExistenceandUniquenessofSolution F InverseKinematicsTechniques F Singular Configuration Summary KeySymbols Exercises Angular Velocity AngularVelocityVectorandMatrix F TimeDerivativeandCoordinateFrames RigidBodyVelocity F VelocityTransformationMatrix DerivativeofaHomogeneousTransformationMatrix Summary KeySymbols Exercises Velocity Kinematics F RigidLinkVelocity ForwardVelocityKinematics JacobianGeneratingVectors InverseVelocityKinematics Summary KeySymbols Exercises Numerical Methods in Kinematics LinearAlgebraicEquations MatrixInversion

22 xxii Contents 9.3 NonlinearAlgebraicEquations F Jacobian Matrix From Link Transformation Matrices Summary KeySymbols Exercises II Dynamics Acceleration Kinematics Angular Acceleration Vector and Matrix RigidBodyAcceleration F AccelerationTransformationMatrix Forward Acceleration Kinematics InverseAccelerationKinematics F RigidLinkRecursiveAcceleration Summary KeySymbols Exercises Motion Dynamics ForceandMoment RigidBodyTranslationalKinetics RigidBodyRotationalKinetics MassMomentofInertiaMatrix Lagrange sformofnewton sequations LagrangianMechanics Summary KeySymbols Exercises Robot Dynamics RigidLinkNewton-EulerDynamics F RecursiveNewton-EulerDynamics RobotLagrangeDynamics F Lagrange Equations and Link Transformation Matrices RobotStatics Summary KeySymbols Exercises III Control Path Planning CubicPath PolynomialPath

23 Contents xxiii 13.3 F Non-PolynomialPathPlanning ManipulatorMotionbyJointPath CartesianPath F RotationalPath Manipulator Motion by End-EffectorPath Summary KeySymbols Exercises F Time Optimal Control F MinimumTimeandBang-BangControl F FloatingTimeMethod F Time-OptimalControlforRobots Summary KeySymbols Exercises Control Techniques OpenandClosed-LoopControl ComputedTorqueControl LinearControlTechnique ProportionalControl IntegralControl DerivativeControl SensingandControl PositionSensors SpeedSensors AccelerationSensors Summary KeySymbols Exercises References 853 A Global Frame Triple Rotation 863 B Local Frame Triple Rotation 865 C Principal Central Screws Triple Combination 867 D Trigonometric Formula 869 Index 873