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Lecture Notes for A Mathematical Introduction to Robotic Manipulation

Lecture Notes for A Mathematical Introduction to Robotic Manipulation 28
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libbyConway,Malaysia,Researcher
Published Date:14-07-2017
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Chapter 4 Robot Dynamics and Control SummerSchool-Math. MethodsinRoboticsTU-BS.DEÔç-çÔJulyòýýÀ Ô LectureNotes Chapter ¥ Robot Dynamics and for Control AMathematicalIntroductionto Lagrangian Equations RoboticManipulation Inertial Properties of Rigid Body Dynamics of an Open-chain By Manipulator ∗ j Z.X.Li andY.Q.Wu Newton-Euler Equations ∗ Coordinate- Dept. ofECE,HongKongUniversityofScience&Technology invariant j SchoolofME,Shanghai JiaotongUniversity algorithms for robot dynamics ò¥JulyòýýÀ Lagrange’s Equations with ConstraintsChapter 4 Robot Dynamics and Control SummerSchool-Math. MethodsinRoboticsTU-BS.DEÔç-çÔJulyòýýÀ ò Chapter ¥ Robot Chapter¥RobotDynamicsandControl Dynamics and Control Ô LagrangianEquations Lagrangian Equations ò InertialPropertiesofRigidBody Inertial Properties of Rigid Body ç DynamicsofanOpen-chainManipulator Dynamics of an Open-chain Manipulator ¥ Newton-EulerEquations Newton-Euler Equations Coordinate-   Coordinate-invariantalgorithmsforrobotdynamics invariant algorithms for robot â Lagrange’sEquationswithConstraints dynamics Lagrange’s Equations with ConstraintsChapter 4 Robot Dynamics and Control 4.1 Lagrangian Equations ç y ◻ASimpleExample: F y F m F x Chapter ¥ Robot Dynamics and Control mg x ♢Review: Lagrangian Equations Newton’sEquation: LagrangianEquation: Inertial d ∂L ∂L Properties of − =F x Rigid Body ¨ mx=F x ˙ dt ∂x ∂x ⇔ Dynamics of d ∂L ∂L an Open-chain my¨=F −mg y − =F Manipulator y ˙ dt ∂y ∂y Momentum:P =mx˙ Newton-Euler x Equations Lagrangianfunction: P =my˙ y ∂L ∂L Coordinate- L=T−V,P = ,P = x y ∂x˙ ∂y˙ d d invariant P =F , P =F −mg x x y y algorithms for dt dt Kineticenergy: robot Ô ò ò ò dynamics ˙ ˙ T= m(x +y ) ò Lagrange’s Potentialenergy: Equations with Constraints V=mgyChapter 4 Robot Dynamics and Control 4.1 Lagrangian Equations ¥ ◻Generalizationtomultibodysystems: q ,i=Ô,...,n: generalizedcoordinates Chapter ¥ i Robot y Kineticenergy: Dynamics and Control m ç T=T(q,q˙) q ç Lagrangian Potentialenergy: Equations V=V(q) Inertial Properties of m ò Lagrangian: Rigid Body q ò Dynamics of L(q,q˙)=T(q,q˙)−V(q) an Open-chain Manipulator τ ,i=Ô,...,n: externalforceonq i i m Ô Newton-Euler LagrangianEquation: Equations q Ô Coordinate- x invariant d ∂L ∂L algorithms for − = τ ,i=Ô,...,n i robot dt ∂q˙ ∂q i i dynamics Lagrange’s Equations with ConstraintsChapter 4 Robot Dynamics and Control 4.1 Lagrangian Equations   ◇Example:Pendulumequation y Generalizedcoordinate: Chapter ¥ Ô θ∈S Robot Dynamics and Kinematics: Control x Lagrangian x=lsinθ,y=−lcosθ Equations ˙ ˙ ⇒ x˙=lcosθ⋅θ,y˙=lsinθ⋅θ Inertial Properties of Kineticenergy: θ Rigid Body Ô Ô ò ò ò ò ˙ ˙ ˙ ˙ Dynamics of T(θ,θ)= m(x +y )= ml θ an Open-chain ò ò Manipulator Potentialenergy: Newton-Euler mg V =mgl(Ô−cosθ) Equations Lagrangianfunction: Coordinate- invariant Ô ∂L ∂L ò ò ˙ ˙ algorithms for L=T −V = ml θ −mgl(Ô−cosθ),⇒ =ml θ, =−mglsinθ robot ˙ ò ∂θ ∂θ dynamics Equationofmotion: Lagrange’s d ∂L ∂L Equations with ò ¨ − = τ⇒ml θ +mglsinθ= τ Constraints ˙ dt ∂θ ∂θChapter 4 Robot Dynamics and Control 4.1 Lagrangian Equations â ◇Example:DynamicsofaSphericalPendulum Chapter ¥ Robot Dynamics and ⎡ ⎤ lsinθcosϕ Control ⎢ ⎥ r(θ,ϕ)=⎢ lsinθsinϕ ⎥ Lagrangian ⎢ ⎥ Equations ⎣ −lcosθ ⎦ Inertial Ô Ô ò ò ò ò ò ˙ ˙ Properties of T= mYr˙Y = ml (θ +(Ô−cos θ)ϕ ) Rigid Body ò ò θ Dynamics of V =−mglcosθ an Open-chain Manipulator Ô ò θ ò ˙ ˙ L(q,q˙)= ml (θ +(Ô−cos )ϕ )+mglcosθ Newton-Euler ò Equations Coordinate- mg invariant ϕ algorithms for robot dynamics Lagrange’s Equations with ConstraintsChapter 4 Robot Dynamics and Control 4.1 Lagrangian Equations Þ Chapter ¥ ⎧ Robot d ∂L d ∂L ⎪ ò ò ò ò ⎪ ˙ ¨ ˙ Dynamics and ⎪ = (ml θ)=ml θ, =ml sinθcosθϕ −mglsinθ ⎪ ⎪ Control ˙ dt dt ∂θ ⎪ ∂θ ⎨ Lagrangian d ∂L d ∂L ⎪ ò ò ò ò ò ⎪ Equations ˙ ˙ ¨ ˙ ⎪ = (ml sin θϕ)=ml sin θϕ+òml sinθcosθθϕ, =ý ⎪ ⎪ ˙ ⎪ dt dt ∂ϕ ∂ϕ Inertial ⎩ Properties of Rigid Body ò ò ò ˙ ¨ ml ý −ml s c ϕ θ mgls ý θ θ Dynamics of θ   + + =  ò ò ò an Open-chain ¨ ˙˙ ý ý ml s ý ϕ òml s c θϕ θ θ θ Manipulator Newton-Euler Equations Coordinate- invariant algorithms for robot dynamics Lagrange’s Equations with ConstraintsChapter 4 Robot Dynamics and Control 4.2 Intertial Properties of Rigid Body — ◻Kineticenergyofarigidbody: Chapter ¥ Robot Dynamics and r Control B Lagrangian Equations r a Inertial Properties of g Rigid Body ab A Dynamics of an Open-chain Manipulator Newton-Euler Volumeoccupiedbythebody: V Equations Massdensity: ρ(r) Coordinate- invariant Mass: m= ρ(r)dV algorithms for ∫ V robot Ô dynamics Masscenter: r ≜ ρ(r)rdV ∫ Lagrange’s m V Equations with Relativetoframeatthemasscenter: r=ý ConstraintsChapter 4 Robot Dynamics and Control 4.2 Intertial Properties of Rigid Body À InA-frame Chapter ¥ Robot Ô Ô ò ò T ò Dynamics and ˙ ˙ ˙ ˙ ˙ ˙ T= ρ(r)Yp+RrY dV= ρ(r)(YpY +òp Rr+YRrY )dV Control ∫ ∫ ò V ò V Lagrangian Ô Ô ò T ò Equations ˙ ˙ ˙ ˙ = mYpY +p R ρ(r)rdV+ ρ(r)YRrY dV ∫ ∫ ò V ò Inertial Properties of ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ Rigid Body =ý Dynamics of an Open-chain Manipulator Ô ò ˙ Newton-Euler ρ(r)YRrY dV ∫ Equations ò V Coordinate- Ô Ô Ô T ò ò ò ˙ invariant = ρ(r)YR RrY dV = ρ(r)YωˆrY dV = ρ(r)YˆrωY dV ∫ ∫ ∫ algorithms for ò ò ò robot Ô Ô Ô dynamics T ò T ò T = ρ(r)(−ω ˆr ω)dV = ω ‹ (−ρ(r)ˆr) dV�ω ≜ ω Iω ∫ ∫ Lagrange’s ò ò ò Equations with ConstraintsChapter 4 Robot Dynamics and Control 4.2 Intertial Properties of Rigid Body Ôý where ⎡ ⎤ I I I xx xy xz ⎢ ⎥ Chapter ¥ ò ⎢ ⎥ I I I I =− ρ(r)ˆr dV ≜ xy yy yz Robot ⎢ ⎥ ∫ ⎢ ⎥ Dynamics and I I I xz yz zz ⎣ ⎦ Control with Lagrangian ò ò Equations I = ρ(r)(y +z )dxdydz,I =− ρ(r)xydxdydz xx xy ∫ ∫ Inertial Properties of Rigid Body Ô Ô Ô Ô ò b T b b T ò b T b b Dynamics of ˙ ˙ T= mYpY + (ω ) I ω = mYR pY + (ω ) I ω an Open-chain ò ò ò ò Manipulator Ô mI ý b T b Newton-Euler = (V )  V b Equations ý I ò ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ Coordinate- b invariant M algorithms for robot b −Ô b T b T b dynamics ˆ ˙ ˙ ForV =g ⋅g˙,with ωˆ =R ⋅Randv =R p,M isthe Lagrange’s GeneralizedinertiamatrixinBframe. Equations with ConstraintsChapter 4 Robot Dynamics and Control 4.2 Intertial Properties of Rigid Body ÔÔ b ◇Example:M forarectangularobject z m Chapter ¥ ρ= Robot lωh Dynamics and Control ò ò I = ρ(y +z )dxdydz xx ∫ y Lagrangian V Equations h ω l x ò ò ò ò ò Inertial = ρ (y +z )dxdydz h ∫ ∫ ∫ Properties of h ω l l − − − ò ò ò Rigid Body Ô m Dynamics of ç ç ò ò w = ρ‹ (lω h+lωh )�= (ω +h ) an Open-chain Ôò Ôò Manipulator h ω l ò ò ò Newton-Euler I =− ρxydv=−ρ xydxdydz Equations xy ∫ ∫ ∫ ∫ h ω l V − − − ò ò ò Coordinate- l invariant h ω ò algorithms for ò ò y ò robot = −ρ x W dydz=ý ∫ ∫ dynamics h ω − − ò l ò ò − ò Lagrange’s Equations with ConstraintsChapter 4 Robot Dynamics and Control 4.2 Intertial Properties of Rigid Body Ôò ⎡ m ò ò ⎤ (w +h ) ý ý ⎢ ⎥ Ôò ⎢ ⎥ m ò ò mI ý ç×ç I =⎢ ý (l +h ) ý ⎥,M=  Ôò ý I ⎢ ⎥ m ò ò ⎢ ⎥ Chapter ¥ ý ý (w +l ) ⎣ Ôò ⎦ Robot Dynamics and Control g ý g (t) ò −Ô Lagrangian ˆ V =g ⋅g˙ , M A Ô Ô Ô Ô B Equations Ô T Inertial T= V M V Ô Ô Properties of Ô ò Rigid Body g (t) Ô g ý V =Ad V Dynamics of Ô g ò ý an Open-chain Manipulator Newton-Euler Ô Ô Ô T T T T Equations T= (Ad V ) M (Ad V )= V Ad M Ad V ≜ V M V g ò Ô g ò Ô g ò ò ò ý ý ò g ý ò ý ò ò ò Coordinate- invariant ◻Munderchangeofframes: algorithms for robot dynamics T Lagrange’s M =Ad M Ad ò Ô g g ý ý Equations with ConstraintsChapter 4 Robot Dynamics and Control 4.2 Intertial Properties of Rigid Body Ôç ◇Example:Dynamicsofaò-dofplanarrobot Chapter ¥ Robot ⎡ ⎤ I ý ý xx i Dynamics and ⎢ ⎥ ý I ý Control I =⎢ ⎥,i=Ô,ò yy i i ⎢ ⎥ ý ý I ⎣ ⎦ Lagrangian zz i Equations Inertial Properties of l 2 Ô Ô Rigid Body ò T ˙ T(θ,θ)= m Yv Y + ω I ω Ô Ô Ô Ô Ô θ 2 Dynamics of ò ò r 2 an Open-chain Ô Ô Manipulator ò T + m Yv Y + ω I ω l ò ò ò ò 1 ò y ò ò Newton-Euler Equations r1 θ 1 x Coordinate- invariant ý ý algorithms for ý ý robot ω = ω = Ô ò dynamics ˙ ˙ ˙ θ θ +θ Ô Ô ò Lagrange’s Equations with ConstraintsChapter 4 Robot Dynamics and Control 4.2 Intertial Properties of Rigid Body Ô¥ x i y P = :Masscenter γ:Distancefromjointitomasscenter i i i ý Chapter ¥ ˙ ˙ Robot x =−r s θ x =r c Ô Ô Ô Ô Ô Ô Ô Dynamics and ⇒ Control ˙ ˙ y =r s Ô Ô y =r c θ Ô Ô Ô Ô Ô Lagrangian ˙ ˙ ˙ Equations x =l c +r c x =−(l s +r s )θ −r s θ ò Ô Ô ò Ôò Ô ò Ôò ò ò Ô Ô ò Ôò ⇒ Inertial ˙ ˙ ˙ y =l s +r s Ô Ô ò Ôò y =(l c +r c )θ +r c θ Properties of ò Ô Ô ò Ôò Ô ò Ôò ò Ô Rigid Body Ô Ô Ô Ô ò ò ò ò ò ò ò ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ T(θ,θ)= m (x +y )+ I θ + m (x +y )+ I (θ +θ ) Dynamics of Ô z ò z Ô Ô Ô ò ò Ô ò Ô ò an Open-chain ò ò ò ò Manipulator Ô ˙ α+òβc δ+βc θ ò ò ˙ ˙ Ô = θ θ    Newton-Euler Ô ò ˙ δ+βc δ ò θ Equations ò ò ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ Coordinate- M(θ) invariant ò ò ò ò algorithms for α=I +I +m r +m (l +r ),β=m l r ,δ=I +m r ,L=T z z Ô ò ò Ô ò z ò Ô ò Ô Ô ò ò ò robot dynamics ˙ ¨ ˙ −βs θ −βx θ θ τ Ô ò ò Ô Ô ⇒ M(θ) +  =  Lagrange’s τ ¨ ˙ ˙ ò θ βs θ ý θ Equations with ò ò Ô ò ConstraintsChapter 4 Robot Dynamics and Control 4.3 Dynamics of Open-chain Manipulator Ô  ◻Dynamicsofopen-chainmanipulator: De�nition: Chapter ¥ ˆ ˆ ξ θ ξ θ Ô Ô i i Robot L :frameatmasscenteroflinki,g (θ)= exp ⋯exp g (o) i sl sl i i Dynamics and Control θ 1 Lagrangian θ2 θ3 Equations l l 1 2 L L 2 3 Inertial Properties of Rigid Body r r 1 2 l 0 Dynamics of L 1 an Open-chain Manipulator r 0 S Newton-Euler Equations ˙ ⎡ ⎤ Coordinate- θ Ô ⎢ ⎥ invariant ⎢ ⎥ ⋮ algorithms for ⎢ ⎥ ˙ robot ⎢ ⎥ b b † † † θ ˙ i ˙ ⎢ ⎥ dynamics V =J (θ)θ=ξ ξ ⋯ ξ ý⋯ý =J(θ)θ i sl sl Ô ò i i i ˙ ⎢ ⎥ θ i+Ô ⎢ ⎥ Lagrange’s ⋮ ⎢ ⎥ Equations with ⎢ ⎥ ˙ Constraints ⎣ θ ⎦ nChapter 4 Robot Dynamics and Control 4.3 Dynamics of Open-chain Manipulator Ôâ ˆ ˆ † −Ô ξ θ ξ θ j+Ô j+Ô i i ξ = Ad (e ⋯e g (ý))ξ , j≤i sl j j i Ô Ô Chapter ¥ b T b b T T b ˙ ˙ ˙ T(θ,θ)= (V ) M V = θ J (θ)M J(θ)θ Robot i i sl i sl i i i i Dynamics and ò ò n Control Ô T ˙ ˙ ˙ T(θ)= T(θ,θ)= θ M(θ)θ, Q Lagrangian i ò Equations i=Ô n Inertial Ô T b Properties of ˙ ˙ M(θ)= J (θ)M J(θ)= M (θ)θ θ Q Q i ij i j i i Rigid Body ò i i,j=Ô Dynamics of an Open-chain h(θ):HeightofL, V(θ)=mgh(θ), V(θ)= mgh(θ) Q i i i i i i i Manipulator i=Ô Newton-Euler Lagrange’sEquation: Equations d ∂L ∂L Coordinate- − = Γ, i= Ô,...,n, i invariant ˙ dt ∂θ ∂θ algorithms for i i robot n n dynamics ⎛ ⎞ d ∂L d ˙ ¨ ˙ ˙ = M θ = M θ +M θ Q Q Lagrange’s ij j ij j ij j ˙ dt dt⎝ ⎠ Equations with ∂θ i j=Ô j=Ô ConstraintsChapter 4 Robot Dynamics and Control 4.3 Dynamics of Open-chain Manipulator ÔÞ n ∂M ∂M ∂L Ô ∂V kj ij ˙ ˙ ˙ ˙ = θ θ − M = θ Q Q k j ij k ∂θ ò ∂θ ∂θ ∂θ i i i k j,k=Ô k Chapter ¥ Robot n n ∂M ∂M ij Ô kj ∂V Dynamics and ¨ ˙ ˙ ˙ ˙ ⇒ M θ + Œ θ θ − θ θ‘+ = Γ Control Q Q ij j j k k j i ∂θ ò ∂θ ∂θ i i k j=Ô j,k=Ô Lagrangian Equations n n ∂V k ¨ ˙ ˙ Inertial ⇒ M θ + Γ θ θ + = Γ Q Q ij j k j i ij Properties of ∂θ i j=Ô j,k=Ô Rigid Body Dynamics of ∂M ∂M Ô ∂M ij kj ik k an Open-chain Γ = Œ + − ‘ ij Manipulator ò ∂θ ∂θ ∂θ k j i Newton-Euler Equations ò ˙ ˙ ˙ , i≠j θ ⋅θ :Coriolis force θ :Centrifugalforce i j i Coordinate- invariant n n ∂M ∂M Ô ∂M ij kj k ik algorithms for ˙ ˙ ˙ De�ne: c (θ,θ)= Γ θ = Œ + − ‘θ Q Q ij k k robot ij ò ∂θ ∂θ ∂θ dynamics k j i k=Ô k=Ô Lagrange’s ¨ ˙ ˙ ⇒ M(θ)θ +C(θ,θ)θ +N(θ)= τ Equations with ConstraintsChapter 4 Robot Dynamics and Control 4.3 Dynamics of Open-chain Manipulator Ô— PropertyÔ: Chapter ¥ T T T ˙ ˙ ˙ ˙ Robot Ô M(θ)=M (θ),θ M(θ)θ≥ ý,θ M(θ)θ= ý⇔ θ= ý Dynamics and n×n Control ˙ ò M−òC∈R isskewsymmetric Lagrangian Equations Proof: Inertial Properties of ˙ ˙ Rigid Body (M−òC) =M −òc (θ) ij ij ij Dynamics of n ∂M ∂M ∂M ∂M an Open-chain ij ij kj ik ˙ ˙ ˙ ˙ Manipulator =Q θ − θ − θ + θ k k k k ∂θ ∂θ ∂θ ∂θ k k j i k=Ô Newton-Euler Equations n ∂M ∂M kj ik ˙ ˙ = θ − θ Coordinate- Q k k invariant ∂θ ∂θ i j k=Ô algorithms for robot T ˙ ˙ dynamics Switchingiandjshows(M−òC) =−(M−òC) Lagrange’s Equations with ConstraintsChapter 4 Robot Dynamics and Control 4.3 Dynamics of Open-chain Manipulator ÔÀ ◇Example:Planarò-DoFRobot(continued) Chapter ¥ m (θ)= α+òβcosθ ,m = δ ÔÔ ò òò Robot Dynamics and m (θ)=m (θ)= δ+βcosθ Ôò òÔ ò Control ˙ ˙ ˙ ˙ ˙ Lagrangian c (θ,θ)=−βsinθ ⋅θ ,c (θ,θ)=−βsinθ (θ +θ ) ÔÔ ò ò Ôò ò Ô ò Equations ˙ ˙ ˙ Inertial c (θ,θ)= βsinθ ⋅θ ,c (θ,θ)= ý òÔ ò Ô òò Properties of ⎧ Ô ∂M ∂M ∂M Ô ∂M Rigid Body ÔÔ ÔÔ ÔÔ ÔÔ ⎪ Ô ⎪ Γ = ( + − )= =ý ⎪ ÔÔ ⎪ ⎪ Dynamics of ò ∂θ ∂θ ∂θ ò ∂θ ⎪ Ô Ô Ô Ô ⎪ an Open-chain ⎪ ⎪ ⎪ Ô ∂M ∂M ∂M Ô ∂M Manipulator ⎪ ÔÔ Ôò òÔ ÔÔ ò ⎪ ⎪ Γ = ( + − )= =−βsinθ ò ⎪ ÔÔ ⎪ Newton-Euler ⎪ ò ∂θ ∂θ ∂θ ò ∂θ ò Ô Ô ò ⎨ Equations ⎪ Ô ∂M ∂M ∂M Ô ∂M Ôò ÔÔ Ôò ÔÔ ⎪ Ô ⎪ Γ = ( + − )= =−βsinθ Coordinate- ⎪ ò ⎪ Ôò ⎪ ò ∂θ ∂θ ∂θ ò ∂θ invariant ⎪ Ô ò Ô ò ⎪ ⎪ algorithms for ⎪ ⎪ Ô ∂M ∂M ∂M ∂M Ô ∂M ⎪ robot Ôò Ôò òò Ôò òò ⎪ ò ⎪ Γ = ( + − )= − =−βsinθ ⎪ ò dynamics Ôò ⎪ ⎩ ò ∂θ ∂θ ∂θ ∂θ ò ∂θ ò ò Ô ò Ô Lagrange’s Equations with ConstraintsChapter 4 Robot Dynamics and Control 4.3 Dynamics of Open-chain Manipulator òý ⎧ Ô ∂M ∂M ∂M ∂M Ô ∂M òÔ òÔ ÔÔ òÔ ÔÔ ⎪ Ô ⎪ Γ = ( + − )= − = βsinθ ⎪ ò òÔ ⎪ ⎪ ò ∂θ ∂θ ∂θ ∂θ ò ∂θ ⎪ Ô Ô ò Ô ò ⎪ Chapter ¥ ⎪ ⎪ ⎪ Robot Ô ∂M ∂M ∂M Ô ∂M ⎪ òÔ òò òÔ òò ò ⎪ ⎪Γ = ( + − )= =ý Dynamics and ⎪ òÔ ⎪ ⎪ Control ò ∂θ ∂θ ∂θ ò ∂θ ò Ô ò Ô ⎨ Lagrangian ⎪ Ô ∂M ∂M ∂M Ô ∂M ⎪ òò òÔ Ôò òò Ô ⎪ Equations ⎪Γ = ( + − )= =ý òò ⎪ ⎪ ò ∂θ ∂θ ∂θ ò ∂θ ⎪ Ô ò ò Ô ⎪ Inertial ⎪ ⎪ ⎪ Properties of ⎪ Ô ∂M ∂M ∂M Ô ∂M òò òò òò òò ò ⎪ ⎪ Γ = ( + − )= =ý Rigid Body ⎪ òò ⎪ ò ∂θ ∂θ ∂θ ò ∂θ ⎩ ò ò ò ò Dynamics of ˙ ˙ an Open-chain −òβsinθ ⋅θ −βsinθ ⋅θ ò ò ò ò ˙ M−òC= Manipulator ˙ −βsinθ ⋅θ ý ò ò Newton-Euler Equations ˙ ˙ ˙ −òβsinθ ⋅θ −òβsinθ (θ +θ ) ò ò ò Ô ò − ˙ Coordinate- òβsinθ ⋅θ ý ò Ô invariant algorithms for ˙ ˙ ý βsinθ (òθ +θ ) ò Ô ò robot = ⇐ skew-symmetric dynamics ˙ ˙ −βsinθ (òθ +θ ) ý ò Ô ò Lagrange’s Equations with Constraints