Theory of applied robotics : kinematics, dynamics, and control

836 Pages·2022·18.156 MB·English

by  Reza N. Jazar

Reza N. Jazar
Theory of
Applied Robotics
Kinematics, Dynamics, and Control
Third Edition
Theory of Applied Robotics
Reza N. Jazar
Theory of Applied Robotics
Kinematics, Dynamics, and Control
Third edition
123
RezaN.Jazar
SchoolofEngineering
RMITUniversity
Melbourne,VIC,Australia
ISBN978-3-030-93219-0 ISBN978-3-030-93220-6 (eBook)
https://doi.org/10.1007/978-3-030-93220-6
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Preface to the Third Edition
Ten years have passed since the second edition of Theory of Applied Robotics has been
published.Thesecondeditionofthebookhasbeenstronglyacceptedinacademiaforresearch
and teaching. Many universities and colleges adopted this book as a teaching text or as a
reference.ThebookalsohadgreatsuccessinChineseuniversities,especiallyafteritsChinese
translation was officially published by Springer in 2018. During the past 10 years, I have
received many constructive comments from all around the world, from colleagues, friends,
instructors,researchers,andstudents.Inthisthirdedition,Iusedthosecommentstoimprove
thetextaswellasremovetyposandnumericalerrors.Further,Itriedtomakethethirdedition
moreeducationalandclearertomakethebookmoresuitableforself-training.
Thekeyconceptofrobotics’kinematicsanddynamicsistheunderstandingofthecoordinate
frametransformation,atopicthatTheoryofAppliedRoboticsiscoveringinthebestpossible
way. We begin this topic by analyzing rotation about a principal axis of global and body
coordinateframes.Rotationaboutanarbitraryaxisisthenextstep.Understandingofrotation
about an arbitrary axis is the best way to study kinematics of robots. We will show how to
break such a complicated rotation into a series of principal rotations to develop the required
mathematical relations by a series of simple steps. To make it possible, we will show how
we may introduce extra dummy coordinate frames to simplify moving from initial to final
coordinateframebyamodularandstructuralprocess.Auxiliarycoordinateframesisaunique
conceptthathasbeenaddedtothekinematicknowledgebodyofrobotkinematicsbythisbook.
Readersofthisbookwillbecomemastersinusingthisscientifictrickinkinematicanalysis.
The robot kinematics will be continued by covering the first and second time derivatives
of rotation transformation mathematics. The first derivative introduces the complicated and
strange concept of angular velocity. Angular velocity is a complicated concept because we
donothaveanyuniquemathematicalquantitywhosetimederivativewillbeangularvelocity.
Angularvelocityisavectorialquantity,butitisnottimederivativeofanotherscalarorvectorial
physical quantity. This fact makes angular velocity appear as a new individual concept. It,
however, will be connected to a combination of time derivative of rotation transformation
matrices. The appearance of rotation transformation matrices in the definition of angular
velocitygivesustheabilitytodefinetimederivativeoperationsindifferentcoordinateframes.
Therefore,wewillseethatthereareseveraldifferentvelocitiesallmathematicallycorrectbut
notallsensible.Thesecondderivativeandintroductionofaccelerationkinematicsmakethis
conceptmorecomplicatedsuchthatwewillbeabletodefineandcalculategreateraccelerations
than those we work with. The freedom of taking a vectorial physical quantity from one
coordinate frame and taking derivative in another coordinate and still expressing in a third
coordinateframehelpedmediscoveranewaccelerationcalledRaziacceleration(Jazar2011;
Harithuddinetal.2015).
My intention in this book is to explain robotics in a manner I would have liked to be
explained to me as a student. This book can now help students by being a great reference
that covers all aspects of robotics and that provides students with detailed explanations and
information.
ix
x PrefacetotheThirdEdition
OrganizationoftheBook
Thetextisorganizedinsuchamannerthatitcanbeusedforteachingorself-study.Chapter1
“Introduction,”containsgeneralpreliminarieswithabriefreviewofthehistoricaldevelopment
andclassificationofrobots.
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
isintroducedandutilizedforforwardkinematicsanalysis.Theconceptofmodulartreatment
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
inversematrixmethod,andtheiterativetechniqueareintroduced.Itisshownthatthepresence
ofasphericalwristiswhatweneedtoapplyanalyticmethodsininversekinematics.
Part II, Derivative Kinematics, explains how the derivatives of vectors are calculated and
how they are related to each other. It covers angular velocity, velocity, and acceleration
kinematics. Definitions of derivatives and coordinate frames are covered in this part. It is
fascinatingtounderstandthatderivativeisaframe-dependentoperation.
Part III, 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 recursive Newton-Euler dynamics, Lagrangian function, manipulator inertia matrix, and
generalizedforcesareintroducedandappliedforderivationofdynamicequationsofmotion.
Part IV, Control, presents the floating time technique for time-optimal control of robots.
Theoutcomeofthetechniqueisappliedtoanopen-loopcontrolalgorithm.Then,acomputed-
torquemethodisintroduced,inwhichacombinationoffeedforwardandfeedbacksignalsare
utilizedtorenderthesystemerrordynamics.
MethodofPresentation
Thestructureofpresentationisina“fact-reason-application”fashion.The“fact”isthemain
subject we introduce in each section. Then the reason is given as a “proof.” Finally, the
applicationofthefactisexaminedinsome“examples.”The“examples”areaveryimportant
part of the book because they show how to implement the knowledge introduced in “facts.”
Theyalsocoversomeotherfactsthatareneededtoexpandthesubject.
LeveloftheBook
This book has evolved from nearly a decade of research in nonlinear dynamic systems and
teaching undergraduate- and graduate-level courses in robotics. It is addressed primarily to
thelastyearofundergraduatestudyandthefirst-yeargraduatestudentinengineering.Hence,
it is an intermediate textbook. This book can even be the first exposure to topics in spatial
kinematicsanddynamicsofmechanicalsystems.Therefore,itprovidesbothfundamentaland
advancedtopicsonthekinematicsanddynamicsofrobots.Thewholebookcanbecoveredin
twosuccessivecourses;however,itispossibletojumpoversomesectionsandcoverthebook
inonecourse.Thestudentsarerequiredtoknowthefundamentalsofkinematicsanddynamics,
aswellasabasicknowledgeofnumericalmethods.
Thecontentsofthebookhavebeenkeptatafairlytheoretical-practicallevel.Manyconcepts
aredeeplyexplainedandtheiruseemphasized,andmostoftherelatedtheoryandformalproofs
havebeenexplained.Throughoutthebook,astrongemphasisisputonthephysicalmeaning
oftheconceptsintroduced.Topicsthathavebeenselectedareofhighinterestinthefield.An
attempthasbeenmadetoexposethestudentstoabroadrangeoftopicsandapproaches.
PrefacetotheThirdEdition xi
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
thefirstthreeundergraduateyears.
UnitSystem
Thesystemofunitsadoptedinthisbookis,unlessotherwisestated,theinternationalsystem
ofunits(SI).Theunitsofdegree(deg)orradian(rad)areutilizedforvariablesrepresenting
angularquantities.
Symbols
• Lowercaseboldlettersindicateavector.Vectorsmaybeexpressedinanndimensional
Euclidianspace.Example:
r , s ,d, a ,b, c
p , q , v ,w, y , z
ω ,α , (cid:4) , θ , δ ,φ
(cid:129) Uppercaseboldlettersindicateadynamicvectororadynamicmatrix.Example:
F , M , I , L
(cid:129) Lowercaseletterswithahatindicateaunitvector.Unitvectorsarenotbolded.Example:
ıˆ , jˆ , kˆ , eˆ , uˆ , nˆ
Iˆ , Jˆ , Kˆ , eˆ , eˆ , eˆ
θ ϕ ψ
(cid:129) Lowercaseletterswithatildeindicatea3×3skewsymmetricmatrixassociatedtoa
vector.Example: ⎡ ⎤ ⎡ ⎤
0 −a a a
3 2 1
a˜ =⎣ a 0 −a ⎦ , a=⎣a ⎦
3 1 2
−a a 0 a
2 1 3
(cid:129) An arrow above two uppercase letters indicates the start and end points of a position
vector.Example:
−−→
ON =apositionvectorfrompointO topointN
(cid:129) A double arrow above a lowercase letter indicates a 4 × 4 matrix associated to a
quaternion.Example:
⎡ ⎤
q −q −q −q
0 1 2 3
←→q =⎢⎢⎣qq1 qq0 −qq3 −qq2 ⎥⎥⎦
2 3 0 1
q −q q q
3 2 1 0
q =q +q i+q j +q k
0 1 2 3
xii PrefacetotheThirdEdition
(cid:129) Thelengthofavectorisindicatedbyanon-boldlowercaseletter.Example:
r =|r| , a =|a| , b=|b| , s =|s|
(cid:129) CapitallettersA,Q,R,andT indicaterotationortransformationmatrices.Example:
⎡ ⎤
⎡ ⎤ cα 0−sα −1
cosα −sinα 0 ⎢ ⎥
QZ,α =⎣sinα cosα 0⎦ GTB =⎢⎣s0α 10 c0α 00..52⎥⎦
0 0 1
0 0 0 1
(cid:129) CapitalletterB isutilizedtodenoteabodycoordinateframe.Example:
B(oxyz) B(Oxyz) B (o x y z )
1 1 1 1 1
(cid:129) Capital letter G is utilized to denote a global, inertial, or fixed coordinate frame.
Example:
G G(XYZ) G(OXYZ)
(cid:129) Rightsubscriptonatransformationmatrixindicatesthedepartureframes.Example:
T =transformationmatrixfromframeB(oxyz)
B
(cid:129) Leftsuperscriptonatransformationmatrixindicatesthedestinationframe.Example:
GT =transformationmatrixfromframeB(oxyz)
B
toframeG(OXYZ)
(cid:129) Wheneverthereisnosuborsuperscript,thematricesareshowninabracket.Example:
⎡ ⎤
cosα 0−sinα −1
⎢ ⎥
[T]=⎢⎣ 0 1 0 0.5⎥⎦
sinα 0 cosα 0.2
0 0 0 1
(cid:129) Left superscript on a vector denotes the frame in which the vector is expressed. That
superscriptindicatestheframethatthevectorbelongsto,andsothevectorisexpressed
usingtheunitvectorsofthatframe.Example:
Gr=positionvectorexpressedinframeG(OXYZ)
(cid:129) Rightsubscriptonavectordenotesthetippointthatthevectorisreferredto.Example:
Gr =positionvectorofpointP
P
expressedincoordinateframeG(OXYZ)
(cid:129) Leftsubscriptonavectorindicatestheframethattheangularvectorismeasuredwith
respectto.Example:
Gv =velocityvectorofpointP incoordinateframeB(oxyz)
B P
expressedintheglobalcoordinateframeG(OXYZ)
Wedroptheleftsubscriptifitisthesameastheleftsuperscript.Example:
Bv ≡ Bv
B P P