Robot-Dynamics-and-Control

Chapter¥ManipulatorDynamicsIntroductionLagrange’sEquationsDynamicsofOpen-chainManipulatorsCoordinateInvariantAlgorithmsLagrange’sEquationswithConstraintsReferencesChapter4ManipulatorDynamics1LectureNotesforAGeometricalIntroductiontoRoboticsandManipulationRichardMurrayandZexiangLiandShankarS.SastryCRCPressZexiangLiÔandYuanqingWuÔÔECE,HongKongUniversityofScience&TechnologyJulyÔ¥,òýÔýChapter¥ManipulatorDynamicsIntroductionLagrange’sEquationsDynamicsofOpen-chainManipulatorsCoordinateInvariantAlgorithmsLagrange’sEquationswithConstraintsReferencesChapter4ManipulatorDynamics2Chapter¥ManipulatorDynamicsÔIntroductionòLagrange’sEquationsçDynamicsofOpen-chainManipulators¥CoordinateInvariantAlgorithms Lagrange’sEquationswithConstraintsâReferencesChapter¥ManipulatorDynamicsIntroductionLagrange’sEquationsDynamicsofOpen-chainManipulatorsCoordinateInvariantAlgorithmsLagrange’sEquationswithConstraintsReferences4.1IntroductionChapter4ManipulatorDynamics3Definition:DynamicsPhysicallawsgoverningthemotionsofbodiesandaggregatesofbodies.Chapter¥ManipulatorDynamicsIntroductionLagrange’sEquationsDynamicsofOpen-chainManipulatorsCoordinateInvariantAlgorithmsLagrange’sEquationswithConstraintsReferences4.1IntroductionChapter4ManipulatorDynamics3Definition:DynamicsPhysicallawsgoverningthemotionsofbodiesandaggregatesofbodies.◻Ashorthistory:Aristotle(384BC-322BC)“Everythinghappensforareason.”Chapter¥ManipulatorDynamicsIntroductionLagrange’sEquationsDynamicsofOpen-chainManipulatorsCoordinateInvariantAlgorithmsLagrange’sEquationswithConstraintsReferences4.1IntroductionChapter4ManipulatorDynamics3Definition:DynamicsPhysicallawsgoverningthemotionsofbodiesandaggregatesofbodies.◻Ashorthistory:Aristotle(384BC-322BC)“Everythinghappensforareason.”G.Galilei(1564-1642)ExperimentswithcannonballsfromthetowerofPisa.Chapter¥ManipulatorDynamicsIntroductionLagrange’sEquationsDynamicsofOpen-chainManipulatorsCoordinateInvariantAlgorithmsLagrange’sEquationswithConstraintsReferences4.1IntroductionChapter4ManipulatorDynamics3Definition:DynamicsPhysicallawsgoverningthemotionsofbodiesandaggregatesofbodies.◻Ashorthistory:Aristotle(384BC-322BC)“Everythinghappensforareason.”G.Galilei(1564-1642)ExperimentswithcannonballsfromthetowerofPisa.I.Newton(1642-1726)Lawsofmotion.(Continuesnextslide)Chapter¥ManipulatorDynamicsIntroductionLagrange’sEquationsDynamicsofOpen-chainManipulatorsCoordinateInvariantAlgorithmsLagrange’sEquationswithConstraintsReferences4.1IntroductionChapter4ManipulatorDynamics4L.Euler(1707-1783)Lawsofmotionfromparticlestorigidbodies.Chapter¥ManipulatorDynamicsIntroductionLagrange’sEquationsDynamicsofOpen-chainManipulatorsCoordinateInvariantAlgorithmsLagrange’sEquationswithConstraintsReferences4.1IntroductionChapter4ManipulatorDynamics4L.Euler(1707-1783)Lawsofmotionfromparticlestorigidbodies.J.Lagrange(1736-1813)CalculusofVariationandthePrinciplesofleastaction.Chapter¥ManipulatorDynamicsIntroductionLagrange’sEquationsDynamicsofOpen-chainManipulatorsCoordinateInvariantAlgorithmsLagrange’sEquationswithConstraintsReferences4.1IntroductionChapter4ManipulatorDynamics4L.Euler(1707-1783)Lawsofmotionfromparticlestorigidbodies.J.Lagrange(1736-1813)CalculusofVariationandthePrinciplesofleastaction.W.Hamilton(1805-1865)QuaternionsandHamilton’sPrinciple.†EndofSection†Chapter¥ManipulatorDynamicsIntroductionLagrange’sEquationsDynamicsofOpen-chainManipulatorsCoordinateInvariantAlgorithmsLagrange’sEquationswithConstraintsReferences4.2LagrangeEquationsChapter4ManipulatorDynamics5◻Asimpleexample:Newton’sEquation:LagrangianEquation:m¨x=Fxm¨y=Fy−mgMomentum:Px=m˙xPy=m˙yddtPx=Fx,ddtPy=Fy−mgxymmgFFxFyFigure4.1Chapter¥ManipulatorDynamicsIntroductionLagrange’sEquationsDynamicsofOpen-chainManipulatorsCoordinateInvariantAlgorithmsLagrange’sEquationswithConstraintsReferences4.2LagrangeEquationsChapter4ManipulatorDynamics5◻Asimpleexample:Newton’sEquation:LagrangianEquation:m¨x=Fxm¨y=Fy−mgMomentum:Px=m˙xPy=m˙yddtPx=Fx,ddtPy=Fy−mgxymmgFFxFyFigure4.1⇔ddt∂L∂˙x−∂L∂x=Fxddt∂L∂˙y−∂L∂y=FyLagrangianfunction:L=T−V,Px=∂L∂˙x,Py=∂L∂˙yKineticenergy:T=Ôòm(˙xò+˙yò)Potentialenergy:V=mgyChapter¥ManipulatorDynamicsIntroductionLagrange’sEquationsDynamicsofOpen-chainManipulatorsCoordinateInvariantAlgorithmsLagrange’sEquationswithConstraintsReferences4.2Lagrange’sEquationsChapter4ManipulatorDynamics6◻Generalizationtomultibodysystems:xymÔqÔmòqòmçqçFigure4.2qi,i=Ô,...,n:generalizedcoordinatesKineticenergy:T=T(q,˙q)Potentialenergy:V=V(q)Lagrangian:L(q,˙q)=T(q,˙q)−V(q)τi,i=Ô,...,n:externalforceonqiLagrangianEquation:ddt∂L∂˙qi−∂L∂qi=τi,i=Ô,...,nChapter¥ManipulatorDynamicsIntroductionLagrange’sEquationsDynamicsofOpen-chainManipulatorsCoordinateInvariantAlgorithmsLagrange’sEquationswithConstraintsReferences4.2Lagrange’sEquationsChapter4ManipulatorDynamics7◇Example:PendulumequationxymgθFigure4.3lGeneralizedcoordinate:θ∈SÔKinematics:x=lsinθ,y=−lcosθ˙x=lcosθ⋅˙θ,˙y=lsinθ⋅˙θKineticenergy:T(θ,˙θ)=Ôòm(˙xò+˙yò)=Ôòmlò˙θòPotentialenergy:V=mgl(Ô−cosθ)Lagrangianfunction:L=T−V=Ôòmlò˙θ−mgl(Ô−cosθ),⇒∂L∂˙θ=mlò˙θ,∂L∂θ=−mglsinθEquat