Theory of applied robotics : kinematics, dynamics, and control PDF
Preview Theory of applied robotics : kinematics, dynamics, and control
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 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2007, 2010,2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whetherthewhole orpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorageand retrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafter developed. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublicationdoesnot imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsand regulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbookarebelieved tobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsortheeditorsgiveawarranty, expressedorimplied,withrespecttothematerialcontainedhereinorforanyerrorsoromissionsthatmayhavebeen made.Thepublisherremainsneutralwithregardtojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland v IamCyrus,kingoftheworld,greatking,mightyking, kingofBabylon,kingofSumerandAkkad,kingofthefourquarters. Iorderedtowritebooks,manybooks,bookstoteachmypeople, Iorderedtomakeschools,manyschools,toeducatemypeople. Marduk,thelordofthegods,saidburningbooksisthegreatestsin. I,Cyrus,andmypeople,andmyarmywillprotectbooksandschools. Theywillfightwhoeverburnsbooksandburnsschools,thegreatsin. Cyrusthegreat to: Vazan Kavosh Mojgan 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
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