Theory of Applied Robotics: Kinematics, Dynamics, and Control [1 ed.] 9780387324753, 0387324755

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Theory of Applied Robotics Kinematics, Dynamics, and Control

Theory of Applied Robotics Kinematics, Dynamics, and Control

Reza N. lazar DepartmentofMechanical Engineering Manhattan College Riverdale, NY

ｾ

Springer

Reza N. lazar Department of Mechanical Engineering Manhattan College Riverdale, NY 1047 1

Theory of Applied Robotics: Kinemat ics, Dynamics, and Control Library of Congress Control Number: 2006939285 ISBN-IO: 0-387-32475-5 ISBN-13: 978-0-387-32475-3

e-ISBN-IO : 0-387-68964-8 e-ISBN-13: 978-0-387-68964-7

Printed on acid-free paper.

© 2007 Springer Science+Business Media , LLC 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 opinio n as to whethe r or not they are subject to proprietary rights. 9 8 7 6 5 432 1 springer.com

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

Everything depends on "What you like to do, what you have to do, and what you do." Happiness means "what you do is what you like to do." Success means "what you do is what you have to do."

Preface

This book is designed to serve as a text for engineering students. It introduces th e fundamental knowledge used in robotics. This knowledge can be utili zed to develop computer programs for analyzing the kinematic s, dynamics , and control of robo tic syste ms. Th e subject of robo tics may appear overdosed by the number of available texts because the field has been growing rapidl y since 1970. However, the topic rem ains alive with modern developments , which are closely relat ed to t he classical material. It is evident that no single text can cover the vast scope of classical and modern mat erials in robotics. Thus the demand for new books arises because t he field continues to progress. Another fact or is the t rend toward analyt ical unification of kinem ati cs, dyn amics, and cont rol. Classical kinematics and dyn amics of robots has its root s in the work of great scientists of the past four centuries who est ablished the methodology and und erstanding of the behavior of dynamic syst ems. The development of dynamic science, since th e beginning of the twenti eth cent ury, has moved toward an alysis of cont rollable man-m ade systems. Therefore, merging t he kinematics and dynamics with control theory is t he expected development for robotic analysis. The other important developmen t is th e fast growing cap abilit y of accurate and rapid num erical calculations, along with intelligent computer programming.

Level of the Book This book has evolved from nearly a decade of resear ch in nonlinear dynamic syste ms, and teaching undergraduate-gradu ate level courses in robotics. It is addressed prim arily to t he last year of undergradu ate st udy and the first year graduate student in engineering. Hence, it is an intermediat e textbook. Thi s book can even be th e first exposure to topics in spatial kinemat ics and dynamics of mechanical syste ms. Therefore, it provides both fundamental and advanced topi cs on the kinemati cs 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 t he book in one course. The students ar e required to know the fund amentals of kinem atic s and dynamics, as well as a basic knowledge of numerical methods.

x

Preface

Th e contents of th e book have been kept at a fairly theoretical-pract ical level. Many concepts are deeply explained and their use emphasized, and most of the relat ed th eory and formal proofs have been explained. Throughout th e book , a strong emphasis is put on the physical meaning of the concepts introduced. Topics that have been selected are of high interest in th e field. An at tempt has been made to expose th e students to a broad range of top ics and approaches.

Organization of the Book Th e text is organized so it can be used for teaching or for self-study. Chapter 1 "Int roduct ion," contains general preliminaries with a brief review of t he historical development and classification of robots . Part I "Kinematics," presents the forward and inverse kinematics of robots . Kinemati cs analysis refers to position , velocity, and accelerat ion analysis of robots in both joint and base coordinate spaces. It establishes kinemati c relations among the end-effecter and th e joint variables. The method of Denavit-Hartenberg for representing body coordinate frames is introduced and utilized for forward kinemati cs analysis. The concept of modular tre atment of robots is well covered to show how we may combine simple links to make the forward kinemati cs of a complex robot . For inverse kinematics analysis, th e idea of decoupling, th e inverse mat rix method, and the iterative technique are introduced. It is shown th at the presence of a spherical wrist is what we need to apply analytic methods in inverse kinematics. Part II "Dynamics," presents a det ailed discussion of robot dynamics. An attempt is made to review t he basic approaches and demonstrate how th ese can be adapted for the act ive displacement framework utilized for robot kinemati cs in th e earlier chapte rs. The concepts of th e recursive Newton-Eul er dynamics, Lagrangian function, manipul ator inerti a matrix, and genera lized forces are introduced and applied for derivation of dynamic equations of motion. Part III "Cont rol," presents the floating time technique for tim e-optimal control of robots. Th e out come of the technique is applied for an openloop control algorithm. Then, a compute d-t orque method is int roduced, in which a combinat ion of feedforward and feedback signals are utilized to render the syste m error dyn amics. Method of Presentation Th e structure of present ation is in a "fact-reason-application" fashion . Th e "fact" is th e main subject we introduce in each sect ion. Th en the reason is given as a "proof." Fin ally the application of the fact is examined in some" examples." The "examples" are a very impor tant part of the book because the y show how to implement th e knowledge introdu ced in "facts." Th ey also cover some other facts that are needed to expand the subject .

Preface

xi

Prerequisites Since the book is written for senior undergradu ate and first-year graduate level students of engineering, the assumption is th at users are familiar with matrix algebra as well as basic feedback control. Prerequisit es for readers of this book consist of the fundamentals of kinematic s, dynamics, vector analysis, and matrix t heory. These basics are usually taught in the first three und ergradu at e years. Unit System Th e system of unit s adopted in this book is, unless otherwise stated, th e international system of units (SI). The units of degree (deg) or radian (rad) are ut ilized for variables representing angular quantities. Symbols • Lowercase bold letters indicate a vector. Vectors may be expressed in an n dimensional Euclidian space. Example: s q

d

a

p

V

w

b y

w

a

€

(J

ｾ

r

c z cjJ

• Uppercase bold let ters indicat e a dynami c vector or a dynamic matri x, such as and Jacobian. Example: F

M

J

• Lowercase letters with a hat indicat e a unit vector . Unit vectors are not bolded. Example: i j

J j

k k

• Lowercase letters with a tilde indicat e a 3 x 3 skew symmet ric matrix associated to a vector. Example:

• An arr ow above two upp ercase letters indicat es th e start and end point s of a position vector. Example: --+

ON = a position vector from point 0 to point N

xii

Preface • A double arrow above a lowercase letter indicat es a 4 x 4 matrix associated to a quat ernion. Exampl e: qO -ql ql qo q2 q3 [ q3 -q2

f--+

q

q

=

qo + ql i

-q2 -q3 qo ql

-q3] q2 -ql qo

+ q2j + q3k

• The length of a vector is indicat ed by a non-bold lowercase letter . Example:

= [r]

r

a= [a]

b = [b]

s

= lsi

• Capital letters A, Q , R , and T indicat e rot at ion or t ransformation matrices. Example: - sin 0: cos 0:

COS 0:

Qz ,o: =

[

ｓｩｾ

ｏＺ

o

0] o

,

GT

B

=

[ CO: 0 0 1 so: 0 0 0

1

- so:

o

co:

o

-1]

0.5 0.2

1

• Capital let ter B is ut ilized to denote a bod y coordina te frame. Exampl e:

B(oxy z)

B (Oxy z)

• Capital letter G is utilized to denote a global, inertial, or fixed coordin ate frame . Example: G(XYZ)

G

G(OXYZ)

• Right sub script on a transformation matrix indicates t he departu re frames. Example:

T B = transformation mat rix from frame B(ox yz) • Left sup erscript on a transformation matrix indicat es t he dest ination fram e. Exampl e:

GTB

t ransformation matrix from frame B(oxy z) to frame G(OXYZ)

• Whenever th ere is no sub or superscript , the matrices are shown in a bracket. Exampl e:

[T] =

｛ ｃｾ 0ｾ so:

o

0

-so:

o CO:

o

-1 ] 0.5 0.2 1

Preface xiii • 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:

c r = position vector expressed in frame G(OXYZ) • Right subscript on a vector denotes the tip point that the vector is referred to . Example: Crp

position vector of point P expressed in coordinate frame G(OXYZ)

• Right subscript on an angular velocity vector indicat es the frame that the angular vector is referr ed to. Example: WB

= angular velocity of the body coordinate fram e B (oxy z )

• Left subscript on an angular velocity vector indicates the frame that the angular vector is measured with resp ect to . Example: CWB

angular velocity of the body coordinate fram e B(oxyz) with respect to the global coordinate frame G(OXYZ)

• Left superscript on an angular velocity vector denotes the frame in which the angular velocity is expressed. Example: angular velocity of the body coordinate frame B 1 with respect to the global coordina te frame G, and expressed in body coordinate frame B 2 Wh enever the subscript and superscript of an angular velocity are the same , we usually drop the left superscript. Example: CWB

_C

= CWB

Also for position, velocity, and accelerat ion vectors , we drop the left subscripts if it is the same as the left superscript. Example:

• If the right subscript on a force vector is a number, it indicates t he number of coordinate frames in a serial robot. Coordinate frame B, is set up at joint i + 1. Example:

F,

= force vector at joint i + 1 measured at the origin of Bi(oxyz)

xiv

Pr eface At joint i there is always an action force F i , t ha t 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 act ion 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 t he right subscript on a moment vector is a number , it indic ates the number of coordina t e frames in a serial robo t . Coordinate frame B, is set up at joint i + 1. Ex ample: M,

= mom ent

vect or at joint i+1 measured at t he origin of Bi(oxy z)

At joint i t here is always an act ion moment M j , t hat link (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 act ion mom ent M i - 1 coming from link (i -1) , and a react ion moment - M , coming from link (i + 1). Action moment is called driving moment , and reaction moment is called driven mome nt. • Left supe rscript on derivative operators indi cat es the fram e in which the derivative of a vari able is t aken . Ex ample:

Cd

- x dt If t he variable is a vector function , and also t he fram e in which the vector is defined is the same as the frame in which a time derivative is t aken , we may use t he following short not at ion,

and write equa t ions simpler. Ex ample:

• If followed by angles, lower case c and s denote cos and sin functions in mathemat ical equations. Example: co: = coso:

sip

= sin