# Lecture Notes for A Mathematical Introduction to Robotic Manipulation

###### Mathematical Introduction toRobotic Manipulation 24
Chapter 6 Parallel Manipulators SummerSchoolMath. MethodsinRoboticsTUBS.DEÔççÔJulyòýýÀ Ô LectureNotes Chapter â Parallel Manipulators for Introduction AMathematicalIntroductionto Configuration Space and RoboticManipulation Singularities Singularity Classification Dynamics of By Parallel ∗ j Manipulators Z.X.Li andY.Q.Wu Control of Parallel ∗ Manipulators Dept. ofECE,HongKongUniversityofScienceTechnology j SchoolofME,Shanghai JiaotongUniversity ò¥JulyòýýÀChapter 6 Parallel Manipulators SummerSchoolMath. MethodsinRoboticsTUBS.DEÔççÔJulyòýýÀ ò Chapter â Parallel ChapterâParallelManipulators Manipulators Introduction Ô Introduction Configuration Space and Singularities ò Con�gurationSpaceandSingularities Singularity Classification Dynamics of ç SingularityClassi�cation Parallel Manipulators Control of ¥ DynamicsofParallelManipulators Parallel Manipulators   ControlofParallelManipulatorsChapter â Parallel Manipulators 6.1 Introduction ç ◻Samplesofparallelmanipulators: ÔDoF: Chapter â Parallel Manipulators Introduction Configuration Space and Singularities òDoF: Singularity Classification Dynamics of Parallel Manipulators Control of çDoF: Parallel ManipulatorsChapter â Parallel Manipulators 6.1 Introduction ¥ ◻Samplesofparallelmanipulators: ¥DoF: Chapter â Parallel Manipulators Introduction Configuration Space and Singularities  DoF: Singularity Classification Dynamics of Parallel Manipulators Control of âDoF: Parallel ManipulatorsChapter â Parallel Manipulators 6.2 Conﬁguration Space and Singularities   klimbs,withSE(ò)astaskspace. Chapter â Parallel Limbi: Manipulators Introduction θ =(θ ,...,θ )∈E i iÔ in i i Configuration Space and g ∶E ↦SE(ò)∶ θ ↦ g(θ ) i i i i i Singularities k Singularity ˙ ˙ Classification n= n V =J (θ )θ =⋯=J (θ )θ Q i st Ô Ô Ô k k k Dynamics of i=Ô Parallel Manipulators AmbientSpace: Control of Parallel Manipulators E=E ×⋯×E Ô k LoopequationsorStructureconstraints: g (θ )=⋯=g (θ ) Ô Ô k kChapter â Parallel Manipulators 6.2 Conﬁguration Space and Singularities â De�ne Chapter â Parallel k−Ô Manipulators H ∶E↦SE(ò)×⋯×SE(ò)=SE (ò) Introduction ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ k−Ô Configuration −Ô −Ô Space and θ ↦(g (θ )g (θ ),...,g (θ )g (θ )) Ô Ô ò Ô Ô k ò k Singularities Singularity Con�gurationSpace(CS) Classification Dynamics of Parallel Q=θ ∈ ESH(θ)=I Manipulators Control of JacobianofH atθ ∈ Q: Parallel ⎡ ⎤ Manipulators J (θ ) −J (θ ) ý ⋯ ý Ô Ô ò ò ⎢ ⎥ ⎢ ⎥ ⋮ ý −J (θ ) ý ⋮ ç(k−Ô)×n ç ç D H ≜J(θ)=⎢ ⎥∈R θ ⋮ ⋮ ⋱ ý ⎢ ⎥ ⎢ ⎥ J (θ ) ý ⋯ ý −J (θ ) ⎣ Ô Ô k k ⎦Chapter â Parallel Manipulators 6.2 Conﬁguration Space and Singularities Þ Proposition: ç(k−Ô)×n If∀θ ∈ Q,J(θ)∈R isofconstantrankç(k−Ô),thenQisa dišerentiablemanifoldofdimensiond=n−ç(k−Ô). Chapter â Parallel Manipulators De�nition: Introduction IfJ(θ)isoffullrank,constraintsH aresaidtobelinearly Configuration independent. Space and Singularities Gru¨blerFromulaforpredictingdimensionofQ: Singularity Classification n=Numberofjoints Dynamics of Parallel th Manipulators f =DoFofthei joint i Control of m=Numberoflinks Parallel Manipulators n n ⎧ ⎪ ⎪ çm− (ç−f)=ç(m−n)+ f ⇒ (planar) Q Q ⎪ i i ⎪ ⎪ ⎪ i=Ô i=Ô d=⎨ n ⎪ ⎪ ⎪ ⎪â(m−n)+ f ⇒ (spatial) Q ⎪ i ⎪ ⎩ i=ÔChapter â Parallel Manipulators 6.2 Conﬁguration Space and Singularities — ◇Example:PlanarmechanismDelta manipulator Chapter â Parallel a n=¥,f =Ô,m=ç Manipulators i Introduction d=ç(ç−¥)+¥=Ô Configuration Space and Singularities Singularity Classification Dynamics of Parallel Manipulators b n= ,f =Ô,m=¥ i Control of Parallel Manipulators d=ç(¥− )+ =òChapter â Parallel Manipulators 6.2 Conﬁguration Space and Singularities À c n=À,f =Ô,m=Þ i Chapter â Parallel d=ç(Þ−À)+À⋅Ô=ç Manipulators Introduction Configuration Space and Singularities Singularity d n=Þ×ç=òÔ,f =Ô,m= ×ç+Ô=Ôâ i Classification d=â×(Ôâ−òÔ)+òÔ=−À() Dynamics of Parallel Manipulators Control of Parallel ManipulatorsChapter â Parallel Manipulators 6.2 Conﬁguration Space and Singularities Ôý De�nition:CSSingularity Apoint θ ∈Qisacon�g. spacesingularityifD H dropsrank. θ Chapter â ♢Review:Dišerentialforms(independentof Parallel Manipulators coordinates) Introduction n ∂h ∗ Configuration θ ∶(θ ,...,θ )∈ E,h∶E↦R,dh= dθ ∈ T E Q Ô n i θ Space and ∂θ i i=Ô Singularities Singularity d Classification dh∶ T E↦R,v↦dh(v)= V h(θ(t)) θ dt Dynamics of t=ý Parallel ˙ whereθ(ý)= θ,θ(ý)=v Manipulators Control of Parallel ∂ Manipulators dθ ( )= δ , dθ ∧dθ =−dθ ∧dθ i ij i j j i ∂θ j dθ ∧dθ ∶T E×T E↦R i j θ θ (dθ ∧dθ )(v,w) i j =dθ (v)dθ (w)−dθ (w)dθ (v) i j i jChapter â Parallel Manipulators 6.2 Conﬁguration Space and Singularities ÔÔ Givenh ,h : Ô ò ∂h ∂h ∂h Ô Ô Ô ⋯ Chapter â ∂θ ∂θ ∂θ Ô ò n Parallel dh= ∂h ∂h ∂h ò ò ò Manipulators ⋯ ∂θ ∂θ ∂θn Ô ò Introduction Principalminorsofò×ò: Configuration Space and Singularities ∂h ∂h ∂h ∂h Ô ò Ô ò ( ⋅ − ⋅ )dθ ∧dθ Singularity Ô ò Classification ∂θ ∂θ ∂θ ∂θ Ô ò ò Ô Dynamics of ∂h ∂h ∂h ∂h Ô ò Ô ò Parallel ( ⋅ − ⋅ )dθ ∧dθ Ô ç Manipulators ∂θ ∂θ ∂θ ∂θ Ô ç ç Ô Control of ∂h ∂h ∂h ∂h Ô ò Ô ò Parallel us,dh ∧dh = ( ⋅ − ⋅ )dθ ∧dθ Q Ô ò i j Manipulators ∂θ ∂θ ∂θ ∂θ i j j i ijChapter â Parallel Manipulators 6.2 Conﬁguration Space and Singularities Ôò De�nition: Chapter â Parallel h andh aresaidtobelinearlyindependentifdh (θ)anddh (θ) Manipulators Ô ò Ô ò Introduction arelinearlyindependentatθ ∈ E Configuration Proposition: Space and Singularities h ,h linearlyindependent⇔dh ∧dh S ≠ý Ô ò Ô ò θ Singularity Proposition: Classification Dynamics of Asetoffunctionsh ,i=Ô,...,narelinearlyindependentiš i Parallel Manipulators dh ∧dh ∧⋯∧dh S ≠ý Ô ò m θ Control of Parallel ManipulatorsChapter â Parallel Manipulators 6.2 Conﬁguration Space and Singularities Ôç ◇Example:¥barmechanism θ =(θ ,θ ,θ )∈ E Chapter â Ô ò ç Parallel Manipulators Loopequations: Introduction Configuration Space and Singularities Singularity ò H ∶E↦R Classification Dynamics of h (θ) l sinθ +l sinθ −l sinθ Ô Ô Ô ò ò ç ç Parallel θ ↦H(θ)= ≜  Manipulators l cosθ +l cosθ −l cosθ −δ h (θ) Ô Ô ò ò ç ç ò CSSingularities: Control of Parallel dh (θ)∧dh (θ)=l l sin(θ −θ )dθ ∧dθ Manipulators Ô ò Ô ò Ô ò Ô ò −l l sin(θ −θ )dθ ∧dθ Ô ç Ô ç Ô ç −l l sin(θ −θ )dθ ∧dθ ò ç ò ò ò ç dh (θ)∧dh =ý⇔sin(θ −θ )=ý,ij Ô ò i jChapter â Parallel Manipulators 6.2 Conﬁguration Space and Singularities Ô¥ Assume: l −l l ,l l Ô ò ç ò ç Singularities ParameterRelation ParameterValue Chapter â p =(ý,ý,π) l +l +l ≜ δ δ = δ Parallel Ô Ô ò ç ¥ ¥ Manipulators p =(ý,ý,ý) l +l −l ≜ δ δ = δ ò Ô ò ç ç ç Introduction p =(ý,π,π) l −l +l ≜ δ δ = δ ç Ô ò ç ò ò Configuration p =(ý,π,ý) l −l −l ≜ δ δ = δ Ô Ô ò ç Ô Ô Space and Case 1 Case 2 Singularities Singularity l l l lT l TT T 1 1 2 2 3 1 2 2 Classification 3 T 1 Dynamics of T Parallel 3 l ManipulatorsG 3 G 4 3 Control of Case 3 Case 4 Parallel l Manipulators 2 l l 1 1 T 1 T T 1TT 2 3 2 T 3 G 1 l l 3 2 G 2 l 3Chapter â Parallel Manipulators 6.2 Conﬁguration Space and Singularities Ô  ◇Example:SNUmanipulator ˆ ˆ ξ θ ξ θ i,Ô i,Ô i,  i,  Chapter â g(θ )=e ⋯e g(ý),i=Ô,ò,ç i i i Parallel   Manipulators −Ô ∧ (g dg) =QAd ˆ ξ dθ i ˆ i,j i,j i ξ θ ξ θ Introduction i,j i,j i,  i,  e ⋯e g (ý) i j=Ô Configuration ∗ Space and ∈se (ç)∶ MaurerCartanform Singularities Singularity Classification Dynamics of   Parallel −Ô ˙ Manipulators V = Ad ξ θ Q ˆ ˆ i,j i,j ξ θ ξ θ i,j i,j i,  i,  e ⋯e g(ý) i j=Ô Control of Parallel ˙ Manipulators =J(θ )θ ,i=Ô,ò,ç i i iChapter â Parallel Manipulators 6.2 Conﬁguration Space and Singularities Ôâ Chapter â Loopconstraint: Parallel Manipulators Introduction Configuration g (θ )=g (θ )=g (θ ) Ô Ô ò ò ç ç Space and ω Singularities −Ô ∧ −Ô ∧ θ,Ô (g dg ) −(g dg ) Ô ò J dθ −J dθ Ô Ô ò ò Ô ò ω = ⋮ ≜ =  θ Singularity −Ô ∧ −Ô ∧ J dθ −J dθ ω (g dg ) −(g dg ) Ô Ô ç ç Ô ç θ,Ôò Classification Ô ç Dynamics of ω ∧⋯∧ω =ý, athomecon�g. θ,Ô θ,Ôò Parallel Manipulators Control of Parallel ManipulatorsChapter â Parallel Manipulators 6.2 Conﬁguration Space and Singularities ÔÞ ◻RelationbetweenQandδ: −Ô ˜ Q=h (ý):Aòdimensionaltorus Ô Chapter â Parallel Manipulators Morsefunction: ˜ ˜ h ∶ Q↦R∶ θ ↦h (δ) Introduction ò ò −Ô Configuration ˜ =l c +l c −l c ,Q=h (δ) Ô Ô ò ò ç ç Space and ò Singularities ˜ De�nition: q∈Qisacriticalpoint Singularity ˜ ˜ ˜ Classification ofh if∀v∈ T Q,⟨dh ,v⟩S =ý. ò p ò q Dynamics of ˜ δ =h (q)iscalledacriticalvalue. ò Parallel Manipulators Control of −Ô ˜ Parallel AsQ=h (ý),⟨dh ,v⟩S =ý Ô ý Ô Manipulators ⇒dh ∧dh S =ý Ô ò q ⇒qisaCSSingularity.Chapter â Parallel Manipulators 6.2 Conﬁguration Space and Singularities Ô— ◻Morseeory: a −Ô ¯ ˜ ˜ Let a b and Q = h (−∞,a = q ∈ QSh (q) ≤ a contains ò ò Chapter â a b a ˜ ˜ ˜ ˜ Parallel no critical points of h , then Q is dišeomorphic to Q . (Q is a ò Manipulators b ˜ deformationretractofQ )⇒ δ shouldbelieina,b Introduction ò ˜ Configuration IfD h (q)isnondegenerate,thenqisanisolatedcriticalpoint. ò q Space and Singularities ˜ ParameterizeQby(θ ,θ ) Ô ò Singularity Classification ∂θ ∂h ∂h ç Ô Ô Dynamics of ⇒ θ = θ (θ ,θ ), =− ,i=Ô,ò ç ç Ô ò Parallel ∂θ ∂θ ∂θ i i ç Manipulators ò ò ˜ ˜ ⎡ ∂ h ∂ h ⎤ ò ò Control of ⎢ ò ⎥ ∂θ ∂θ ∂θ ò Ô ò Parallel ˜ ˜ ⎢ Ô ⎥ h (θ ,θ )=h (θ ,θ ,θ (θ ,θ ))⇒ D h = ò Ô ò ò Ô ò ç Ô ò ò ò ò ⎢ ˜ ˜ ⎥ Manipulators ∂ h ∂ h ò ò ⎢ ⎥ ò ∂θ ∂θ ∂θ ⎣ Ô ò ⎦ ò ò ò ⎡ ⎤ l c l s s l l c c Ô Ô ò Ô Ô Ô ò Ô ò ⎢ ⎥ −l c − + Ô Ô ç ç c ⎢ ç l c l c ⎥ ç ç ç ç =⎢ ⎥ ò ò l c ⎢ l l c c l s s ⎥ Ô ò Ô ò ò ò ç ò ò −l c − + ⎢ ç ò ò ç ⎥ c l c ç l c ç ç ç ç ⎣ ⎦Chapter â Parallel Manipulators 6.3 Singularity Classiﬁcation ÔÀ ◻Con�gurationspaceversusgeometric Chapter â parameterδ: Parallel Manipulators Introduction ParameterValue DescriptionofQ MorseIndex Configuration δ = δ asinglepoint M =ò Space and ¥ i Singularities δ ∈(δ ,δ ) Unitcircle ç ¥ Singularity δ = δ Figure— M =Ô ç i Classification δ ∈(δ ,δ ) Twoseparatecircles ò ç Dynamics of δ = δ Figure— M =Ô Parallel ò i Manipulators δ ∈(δ ,δ ) Unitcircle Ô ò Control of δ = δ Asinglepoint M =ý Ô i Parallel δ ∈(ý,δ ),(δ ,∞) Emptyset Manipulators Ô ¥Chapter â Parallel Manipulators 6.3 Singularity Classiﬁcation òý ◻ParametrizationSingularity: Consider ç ò ò ò H ∶R ↦R∶(x ,x ,x )↦x +x +x −Ô Ô ò ç Chapter â Ô ò ç Parallel −Ô Manipulators Q=H (ý)∶ unitsphere Introduction Localcoordinates: ψ (x) Configuration ò x Ô Ô ψ ∶ Q↦R ∶x↦ =  Space and x ò ψ (x) ò Singularities T ψ dropsrankonQ⇔∃v∈T Qs.t.⟨dψ ,v⟩=ý p p i Singularity Classification However,⟨dH,v⟩=ý Dynamics of ⇒dψ ,dψ ,dH arelinearlydependent. Ô ò Parallel Manipulators ⇒ dH∧dψ ∧dψ =ý Ô ò Control of Parallel ∂H ∂H ∂H Manipulators =( dx + dx + dx )∧dx ∧dx Ô ò ç Ô ò ∂x ∂x ∂x Ô ò ç ∂H = dx ∧dx ∧dx =òx dx ∧dx ∧dx Ô ò ç ç Ô ò ç ∂x ç ⇒x =ý, Pointsoftheequator arePsingularity. çChapter â Parallel Manipulators 6.3 Singularity Classiﬁcation òÔ ∂H Alternatively,p∈ QisaPsingularityiš of(∗)dropsrank ∂x p ∂H ∂H dx + x =ý (∗) Chapter â a p ∂x ∂x Parallel a p Manipulators wherex =(x ,x ),x =x . a Ô ò p ç Introduction ò ImplicitFunctioneorem⇒∃ψ ∶R ↦Rs.t. x =ψ(x ). p a Configuration Space and Proposition: Singularities Letψ ∶ E↦RbeasetoflocalcoordinatefunctionsonQ. Apoint i Singularity Classification p∈QisaPsingularityiš Dynamics of dh ∧⋯∧dh ∧dψ ∧⋯∧dψ S =ý Ô m Ô n−m p Parallel Manipulators ActuatorSingularity: Control of Parallel ψ ∶(θ ,...,θ )↦ θ Ô n a Manipulators dh ∧⋯∧dh ∧dθ ∧⋯∧dθ S =ý Ô m a,Ô a,n−m p EndešectorSingularity: ψ ∶ Q↦ SE(ò)∶ θ ↦x dh ∧⋯∧dh ∧dx ∧⋯∧dx S =ý Ô m Ô n−m pChapter â Parallel Manipulators 6.3 Singularity Classiﬁcation òò ◻IrregularPsingularity: Ifp∈QisalsoaCSsingularity,then Chapter â Parallel Manipulators dh ∧⋯∧dh ∧dψ ∧⋯∧dψ S =ý Ô m Ô n−m p Introduction Configuration holdsautomatically. Qisnotamanifold, gains Space and Singularities dimensionbyÔormore,thus: Singularity Nominallyactuated⇒underactuated Classification Dynamics of Parallel Manipulators Control of Parallel Manipulators ◻Redundantactuation: ò S =U ∪U ∪U (x ,x ) (x ,x ) (x ,x ) Ô ò ò ç Ô ç PSingularity: x =ý∩x =ý∩x =ý=∅ ç Ô òChapter â Parallel Manipulators 6.3 Singularity Classiﬁcation òç ◻Singularityclassi�cation: Asingularityofredundantlyactuatedmanipulator Chapter â Parallel n−m+l Manipulators lredundantactuators: θ =(θ ,...,θ )∈R ,lý a a,Ô a,n−m+l Introduction p∈ Q: Asingularityif Configuration Space and Singularities dh ∧⋯∧dh ∧dθ ∧⋯∧dθ S =ý,Ô≤i ≤⋯≤i ≤n−m+l Ô m a,i a,i p Ô n−m Ô n−m Singularity Classification Ešect: eliminateAsingularity Dynamics of Parallel Manipulators T 2 l l Actuators Control of 2 2 ParallelT 2 l Manipulators l 3 3 l 1 l T T 1 3 1T c 3 T 1 c Eliminate Asingularity byT Eliminate Asingularity by T 3 1Chapter â Parallel Manipulators 6.3 Singularity Classiﬁcation ò¥ ◻Strati�edstructureofAsingularities: Chapter â SingularsetQ ⊂Q:allAsingularities Parallel s Manipulators Introduction Q =p∈ QSdh ∧⋯∧dh ∧dθ ∧⋯∧dθ S =ý, s Ô m a,i a,i p Ô n−m Configuration Space and dh ∧⋯∧dh S ≠ý,Ô≤i ⋯i ≤n−m+Ô Ô m p Ô n−m Singularities Singularity Annihilationspace Classification Dynamics of Parallel T V =v∈ T QS⟨v,dθ ⟩=⟨v,dh⟩=ý, p p a,j i Manipulators ,i=Ô,⋯,m,j=Ô,⋯,n−m+l Control of Parallel Manipulators Degreeofde�ciency: d=dim(T V) p d≤d =min(n−m,m−l) ýChapter â Parallel Manipulators 6.3 Singularity Classiﬁcation ò  Strati�edstructure Chapter â Q =p∈Q Sdim(T V)=k,k=Ô,⋯,k sk s p max Parallel Manipulators k k max max Introduction Δ = T V, Q = Q ,Δ = Δ  p s  s  sk sk sk Configuration p∈Q k=Ô k=Ô sk Space and Singularities De�nition: Singularity Classification p∈Q isa�rstordersingularityištheredoesnotexistv∈ Δ sk sk Dynamics of thatisalsotangenttoQ .Otherwise,pisasecondorder sk Parallel Manipulators singularity. Control of De�nition: Parallel Manipulators Asecondordersingularityp∈Q isdegenerateiš∃constant sk ¯ rankk (k k)subdistribution Δ ⊂ Δ s.t. Ô Ô sk sk Ô ¯ Ô Δ (p)⊂T Q , sk p sk Ô ¯ ò Δ (p)isinvolutive. sk ÔChapter â Parallel Manipulators 6.3 Singularity Classiﬁcation òâ Chapter â Parallel Manipulators Introduction Configuration Space and Singularities Singularity Classification Dynamics of Parallel Manipulators Control of Parallel ManipulatorsChapter â Parallel Manipulators 6.3 Singularity Classiﬁcation òÞ ◻DegeneracyofPsingularity: Chapter â Degenerate:allowcontinuousmotionwith�xedparameters Parallel Manipulators Nondegenerate:allowinstantaneousmotionwith�xed Introduction parameters Configuration Space and Singularities Singularity Classification Dynamics of Parallel Manipulators Control of Parallel Manipulators Fig. 19. Nondegenerate Psingularity. Fig. 20. Degenerate Psingularity.Chapter â Parallel Manipulators 6.3 Singularity Classiﬁcation ò— ◻ConditionsfordegenerateAsingularity: Basisof Δ : Chapter â sk Parallel Manipulators Δ =spanY,Y =Y ,⋯,Y sk Ô k Introduction Configuration Space and Annihilationvector: Singularities Singularity ˙ ý θ k Classification a v = = =Yα∈ Δ ,α∈ R s ˙ sk ˙ θ θ p p Dynamics of Parallel Manipulators θ ∈ Q :degenerateAsingularityiš s sk Control of Parallel h(θ +εv )−h(θ )=ý,i=Ô,⋯,m i s s i s Manipulators Conditionsoncoe›cientsofTaylorseries ò ò ∂ h ∂ h i i T T  (Yα,Yα)= α Y  Yα=ý,⋯,i=Ô,⋯,m ò ò ∂θ ∂θ θ θ s sChapter â Parallel Manipulators 6.3 Singularity Classiﬁcation òÀ ◻Classi�cationdiagram: Chapter â Parallel Manipulators Introduction Configuration Space and Singularities Singularity Classification Dynamics of Parallel Manipulators Control of Parallel Manipulators Fig. 15. A hierarchic diagram of singularities, Asing.: actuator singularity. Esing.: endeffector singularity. Psing.: parametrization singularity. N. Degenerate: Nondegenerate.On Mechanism Classification and Quotient Kinematics Machines 机构分类与商联机构的微分流型理论 Z.X. Li Y.J. Lou,  Y.Q. Wu  H. Wang Dept. of ECE, Hong Kong Univ. of Science  Tech. CoMEDivision , HIT Shenzhen Graduate School School of MechEng., Shanghai JiaotungUniversity 0. Background No knowledge can be certain, if it is not based upon the mathematical sciences or upon some other knowledge which is itself based upon the . mathematical sciences (Leonard Da Vinci 14521519) (“Let no man who is not a mathematician read the elements of my work”) A mathematical theory for machine design ( or classification of mechanism) . F. Klein (18491925) S. Lie (1873) Mechanism Characters: E. Cartan (1930), H. Weyl (192325) “Motion pattern” of its endeffector. . F. Reuleaux (1875) R(p,ω) C(p,ω) T(u) S(p) H (p,ω) PL(u) ρ7 7 7 •R. Ball (1876): Screw Theory (Lie algebra se(3) of SE(3)) ˆ ξθ exp:se(3)→SE(3):ξθ→e or h→exp(h) Table of Contents: 0. Background 1. Subgroup motion generators (or mechanisms) 2. Submanifold motion generators 3. Quotient motion generators 4. Quotient kinematics machines 5. Conclusion1. Subgroup motion generators ¯ ½∙ ¸ ¾ ¯ R p 3 ¯ SE(3)= R∈SO(3),p∈R ¯ 0 1 ¯ © ª 3×3 T ¯ SO(3)= R∈R R R =I,detR=1 Properties of a Lie group G: Lie algebra : ⎧⎡⎤⎡⎤ ¯⎫ ∧ ¯ x 0−z y ⎨⎬ ¯ 3×3 T 3 ⎣⎦ ¯ ⎣⎦ y so(3),T SO(3) = , z 0−x∈R (x,y,z)∈R e ¯ ⎩⎭ ¯ z−y x 0 ¯ ½∙ ¸ ¾ ¯ ωˆ v 3 ¯ se(3),T SE(3)= v,ω∈R e ¯ 0 0 Euler (17071783) Chasles (17931880)7 k G eˆ Basis of • Canonical coordinates of 2nd kind: i i=1 (θ ,...,θ )→exp(eˆθ )···exp(eˆθ ) =g∈G 1 k 1 1 k k −1 • Conjugate subgroups of SE(3): I G := gGg g Classification of subgroup of SE(3)A mech. M is said to have the motion type of G, if open nbdU e s.t. M U =G U ∃ 3∩∩ e e e Subgroup motion generators: SE(3) T(3) SO(3) SO(3) Delta: T(3) H4:X(z)2. Submanifoldmotion generators A submfd M⊂SE(3) • not necessary closed under group operation U(0,x,y)=R(o,x)·R(o,y)6 M ·N, M⊂T(3), N⊂SO(3) • Category I submfd: U(o,x,y)T(z) T(y)U(o,z,x) ABB IRB940 (Tricept) G ·G , G∩G =φ, G subgps of SE(3) • Category II submfd: 1 2 1 2 i PL(ω )·S(o) S(N )·S(o) j j Cat.I II submfds are of the form: ˆ ˆ ξθξθ 1 1 k k M =e ··· e θ∈(−²,²) i “纯机构”7 3. Quotient motion generators  (mechanisms with parasitic motions) . H⊂G (⊂SE(3)) H gH G/H =gH g∈G • • e : or g∼g if g ·g∈H 1 2 1 2 G : Quotient space of G : mfd of dim k =dim G−dim H . M⊂G of dim k is a quotient mechanism, or a complement of H in G, M∈G/H, if M ·H =G (near e) or T M +H =G e orφ: M⊂G−→G/H, g→gH is a local diﬀeo near e E.g.1: Cspace of a cylinder: E.g.2: Cspace of a 5axis machine: M∈SE(3)/R(o,z) Q = SE(3)/H Z Z H =C(o,z) Y X X YE.g.3: M∈PL(z)/R(o,z) P (z,x)R(p,z) R(p ,z)R(p ,z) T(y)R(p,z) a 1 2 E.g.4: Translational quotient modules Cartesian Tricept/Spherical SE(3)/S(p) Scara …… Elbow M = T(3) 1 M =U(o,x,y)T(z) ESEC TY 6 Cylindrical M =R R T(z) 2 z z M = R R R M =T(z)R T(x) 3 z x x 5 z M =T(x)R R 4 z yˆ ˆ ξθξθ 1 1 k k M =e ··· e θ∈(−²,²) i : canonical Rep. of G/H （纯机构） E.g.5: Canonical Rep. of X(z)/C(o,z) T(x)T(y), T(x)H (p,z), H (p,z)T(y), H (p ,z)H (p ,z) ρρρ 1ρ 2 1 2 4 realizations Classification of canonical representations for quotient modules of dim 4 5:E.g.6: Quotient mechanisms with parasitic motions † M =Z3∈SE(3)/PL(z) , 1 † 2 :M =∩ I 2mπ (PL(x)S(ly))⇒ 1 m=0 R ( ) z 3 T M =eˆ ,eˆ ,eˆ ,T M⊕pl(z) =se(3)⇒ e 1 3 4 5 e 1 M∈SE(3)/PL(z) 1 , not a subgroup z Parasitic motion: x xˆφ+yˆψ zˆσ R(φ,ψ,σ)=e e : tilting torsion angles y Parameterize M by : (z,φ,ψ): 1 Torsion free: σ =σ(z,φ,ψ) Parasitic motion in PL(z): σ =0 x=x(z,φ,ψ) (to be compensated) y =y(z,φ,ψ) Canonical Rep. G/H Noncanonical Rep. (with parasitic motions)E.g.6: Quotient mechanisms with parasitic motions Nonzero torsion σσ ψ ψ L z = z =0 φφ 10 σ σ ψ ψ 2L 3L z = z = φ φ 10 10 σσ ψψ 4L 5L z = z = φφ 10 10 3S·S PKM L: maximal achievable heightE.g.6: Quotient mechanisms with parasitic motions Nonzero xtranslation (y similar) x x ψψ L z = z =0 φ 10φ x x ψψ 2L 3L z = z = φφ 10 10 x x ψ ψ 4L 5L z = z = φ φ 10 10 3S·S PKM L: maximal achievable heightApplications to Compliant Mechanism Modeling:4. Quotient Kinematics Machines (QKMs) • SKM Delta • PKM • HKM H + 4 • QKM Definition: Quotient Kinematics Machine A New Synthesis Theory is Required Motion type Q⊂SE(3) M 1 Quotient modules M : Motion type M M : Motion type M 1 1 2 2 M⊂Q M⊂Q O 1 2 1 g 1 Acting in unison O 2 O g 2 QKM Q (M ,M ) 1 2 −1 Motion type M ·M =Q 2 1 (Equality deﬁned in a nbhd. of e) M 2 −1−1 Note: M ·M :g g∈SE(3)g attainable byM 2 2 i i 1 1 =g∈SE(3)g attainable by M w.r.t. M 1 2yy G (H,G/H) Subgroup QKM: ½ SKM H PKM ½ G SKM Canonical Rep. PKM G/H ½ SKM Noncanonical Rep. PKM G/H Quotient QKM(5‐axis machines): (G/H,H /H ) 1 0 Z∈SE(3)/PL(z) 3 T (z) 2Various 5axis QKM topologies: 1. SE(3)/R(o,z) (SE(3)/PL(z),PL(z)/R(o,z)) 1T2R+2T 2. SE(3)/R(o,z) (SE(3)/X(z),X(z)/R(o,z)) 2R+3T 3. SE(3)/R(o,z) (SE(3)/SO(3),SO(3)/R(o,z)) 3T+2R 4. SE(3)/R(o,z) (SE(3)/C(o,z),C(o,z)/R(o,z)) 2T2R+1T 5. X(x)·X(y) (X(x)/T (z),X(y)/T(z)) 1T1R+2T1R 2 6. X(x)·X(y) (X(x),X(y)/T(3)) 3T1R+R ½ SKM H PKM ½ G SKM G/H Canonical Rep. 1 PKM G/H ½ SKM H /H 1 0 Noncanonical Rep. PKMy Properties of QKM: SKM PKM QKM Stiﬀness accuracy × X X Workspace volume X × X Modularity Structure variety × X XX Existence of parasitic motion (compensatable);4. Conclusions •A rigorous mathematical foundation for mechanism classification (Our response to Da Vinci’s dream); •Extension from subgroup to submanifold and quotient motion generators; •Systematic formulation of quotient kinematics machines; •Design tools for QKM designs are neededKey References: A. Lie groups in Rigid Body and Robot Kinematics: 1 R. Brockett, Robot manip. and product of exponential form. In Math. Theory of Networks and Systems, Springer, 1984. 2 R. Murray, Z.X. Li and S.S. Sastry, A math. intro. to robotic manipulation, CRC Press, 1994. 3 J. Selig, Geometric methods in robotics, Springer, 1996. B. Screw Theory for mechanism synthesis: 1 X.W. Kong and C. Gosselin. Type synthesis of parallel mechanisms. Springer, 2007. ….. C. Lie group theory in mechanism synthesis and analysis: 1 J.M. Herve and F. Sparacino. Structural synthesis of parallel robots generating spatial translation. ICRA, pp. 808813, 1991. 2 C.C. Lee and J.M. Herve. Translational parallel manipulator with doubly planar limbs. MMT, 41(4):359486, 2006. 3 J. Meng, G.F. Liu, Z.X. Li. A geom. theory for analysis and synthesis of sub6 dof parallel manip.. IEEE TRO, 23(4):625 649, 2007. D. Ingenious Parallel Mechanisms: th 1 R. Clavel. Delta, a fast robot with parallel geometry. In Proc. 18 Intl. Symposium on Industrial Robots, pp. 91100, 1988. 2 Gough, V. E., Contr. to discussion of papers on research in Automobile Stability, Control and Tyre performance, Proc. Auto Div. Inst. Mech. Eng., pp. 392394, 19561957
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