Modeling the Kinematics and Dynamics of Compliant Contact

In this presentation, we discuss an intuitive model of the kinematics and dynamics of spatial compliant contact. The kinematic model gives a relation between the relative velocity of two rigid bodies and the velocities of two points on their surfaces that

ModelingtheKinematicsandDynamicsofCompliantContact

VincentDuindamUniversityofTwente

P.O.Box217,7500AEEnschede

TheNetherlandsv.duindam@

Abstract

Inthispresentation,wediscussanintuitivemodelofthekinematicsanddynamicsofspatialcompliantcontact.Thekinematicmodelgivesarelationbetweentherelativeve-locityoftworigidbodiesandthevelocitiesoftwopointsontheirsurfacesthathavesmallestdistance.Thedynamicmodelusesa6Dspatialspringelementtomodeltheelasticforcesbetweenthebodiesresultingfromthedeformationofthesurfacesduringthecontactsituations.Simulationshowstheapplicationofthismodelfortwoobjectsbouncingonthe oorandoneachother.

1ProblemDescription

Inthemodelingofwalkingmachines,wewouldliketohavearelativelysimplemodeloftheinteractionbetweenthefeetandthegroundthatstillcapturestheforcesandtorquesinalldirections.Currentcontactmodelsareeithertoosimple(e.g.onlyintwodimensions,orbasedoninstantaneousdis-sipationonimpactasdescribedforexamplein[1])ortoocomplex(e.g.FEM)forourpurposesandingeneralnotveryintuitive.

Theroboticsframeworkoftwistsandwrenchestomodelgeneralizedvelocitiesandforcesinsixdimensionscanbeusedtoobtainasimple,intuitivecontactmodelthatstillcanhandlethefull3Dbehavior.

2KinematicModel

Thekinematicpartmodelsthevelocitiesofthetwoclosestpointsp1andp2(oneoneachbody),asshowninFigure1.Itiseasilyobtainedfromtwotrivialobservations:(a)Thetwosurfacenormalsatthecontactpointsareequalbutopposite(whenexpressedinthesamecoordinateframe).Inequationform:

g1= H2

1g2(1)

(b)Thepointsareseparatedbyalinewithlength anddirectiong1(wecouldalsochooseg2ofcourse).

H2

1

p2 p1= g1(2)

where = g1,H2

1

p2 p1 Thekinematicsmodelisthenobtainedsimplybytakingthe

timederivativesof(1)and

(2).

StefanoStramigioli

UniversityofTwente

P.O.Box217,7500AEEnschede

TheNetherlandss.stramigioli@

Figure1:Con gurationforkinematicsmodeling.

3DynamicModel

Whentwoobjectscollide(whichisexactlywhen changessign),thecontactforcescausethesurfacestodeformandthevelocitiesoftheobjectstochange.Thesecontactforces(andtorques)canbemodeledbya6Dspatialspring[2]at-tachedbetweenthebodies.Thespringcompressesastheobjectscollideandpenetrateeachotherandgeneratesforcesandtorquesopposingthepenetratingmotionandcausingtheobjectstobounce.

Apartfromtheelasticforcesmodeledbythespring,energydissipationcanbeeasilymodeledbya6Ddamperelement.References

[1]T.Futura,T.Tawara,Y.Okumura,M.Shimizu,K.Tomiyama,Designandconstructionofaseriesofcom-pacthumanoidrobotsanddevelopmentofbipedwalkcon-trolstrategies,RoboticsandAutonomoussystems,Vol.37,p.81–100,2001.

[2]S.Stramigioli,ModelingandIPCControlofInter-activeMechanicalSystems–ACoordinate-FreeApproach,Springer-Verlag,2001.