Parallel Manipulators

serial and parallel robot manipulators and parallel manipulator reverse displacement analysis
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libbyConway,Malaysia,Researcher
Published Date:14-07-2017
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Chapter 6 Parallel Manipulators SummerSchool-Math. MethodsinRoboticsTU-BS.DEÔç-çÔJulyòýýÀ 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,HongKongUniversityofScience&Technology j SchoolofME,Shanghai JiaotongUniversity ò¥JulyòýýÀChapter 6 Parallel Manipulators SummerSchool-Math. MethodsinRoboticsTU-BS.DEÔç-çÔJulyòýýÀ 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: Ô-DoF: Chapter â Parallel Manipulators Introduction Configuration Space and Singularities ò-DoF: Singularity Classification Dynamics of Parallel Manipulators Control of ç-DoF: 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 â-DoF: 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:Planarmechanism&Delta 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)∶ Maurer-Cartanform 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):Aò-dimensionaltorus 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— ◻Morseeory: 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)isnon-degenerate,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 areP-singularity. ç