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.