ContentslistsavailableatScienceDirect
Biomedical
Signal
Processing
and
Control
j ou rn a l h o m e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / b s p c
Hunt–Crossley
model
based
force
control
for
minimally
invasive
robotic
surgery
A.
Pappalardo
a,
A.
Albakri
b,
C.
Liu
b,∗,
L.
Bascetta
a,
E.
De
Momi
a,
P.
Poignet
b aDepartmentofElectronics,InformationandBioengineering(DEIB),PolitecnicodiMilano,PiazzaLeonardodaVinci32,20133Milano,Italy bDepartmentofRobotics,LIRMM(UMR5506),UM-CNRS,161RueAda,34095Montpellier,Francea
r
t
i
c
l
e
i
n
f
o
Articlehistory:
Received18October2015
Receivedinrevisedform18March2016 Accepted7May2016
Keywords: Forcecontrol Softtissueinteraction Activeobserver Hunt–Crossleymodel
a
b
s
t
r
a
c
t
Inminimallyinvasivesurgery(MIS)thecontinuouslyincreasinguseofroboticdevicesallowssurgical operationstobeconductedmorepreciselyandmoreefficiently.Safeandaccurateinteractionbetween robotinstrumentsandlivingtissueisanimportantissueforbothsuccessfuloperationandpatientsafety. Humantissue,whichisgenerallyviscoelastic,nonlinearandanisotropic,isoftendescribedaspurely elas-ticforitssimplicityincontactforcecontroldesignandonlinecomputation.However,theelasticmodel cannotreproducethecomplexpropertiesofarealtissue.Basedoninvitroanimaltissuerelaxationtests, weidentifytheHunt–Crossleyviscoelasticmodelasthemostrealisticonetodescribethesofttissue’s mechanicalbehavioramongseveralcandidatemodels.AforcecontrolmethodbasedonHunt–Crossley modelisdevelopedfollowingthestatefeedbackdesigntechniquewithaKalmanfilterbasedactive observer(AOB).Bothsimulationandexperimentalstudieswerecarriedouttoverifytheperformanceof developedforcecontroller,comparingwithotherlinearviscoelasticandelasticmodelbasedforce con-trollers.ThestudiesandcomparisonsshowthattheHunt–Crossleymodelbasedforcecontrollerensures comparablerisetimeintransientresponseasthecontrollerbasedonKelvin–Boltzmannmodelwhichis reportedasthemostaccuratedescriptionforrobot-tissueinteractioninrecentliterature,butitcauses muchlessovershootandremainsstablefortaskswithfasterresponsetimerequirements.
©2016ElsevierLtd.Allrightsreserved.
1. Introduction
Minimallyinvasivesurgery(MIS)isreplacingtraditionalopen
surgicalprocedureswhichnormallyinvolvelargeincisiontoaccess
thepatient’s body.MIS isperformed usinglonginstrumentsto
enterthepatientbodythroughsmallincisionsandleadstodirect
advantagesincludinglesspain,hemorrhagingandtrauma,reduced
riskofinfections,shortenedhospitalstay,andhencelessburdenfor
boththepatientandsocialhealthcaresystem[1–3].
Withthedevelopmentoftechnologyintheareaofrobotics,
roboticdeviceshavefoundtheirwaytotheoperatingroom(OR)
andleadtothenewconceptofminimallyinvasiveroboticsurgery
(MIRS).Many roboticsurgical devices have beendeveloped for
MISoperationsintheliterature[4–9]withthemostwidelyused
andbestknownexampleofdaVincirobotfromIntuitiveSurgical
∗ Correspondingauthor.Tel.:+33685817659.
E-mailaddresses:pappalardo.antonio87@gmail.com(A.Pappalardo), Abdulrahman.Albakri@lirmm.fr(A.Albakri),Chao.Liu@lirmm.fr(C.Liu), luca.bascetta@polimi.it(L.Bascetta),elena.demomi@polimi.it(E.DeMomi), Philippe.Poignet@lirmm.fr(P.Poignet).
(Sunnyvale,CA,USA).Themovementsoftheroboticdevices,
com-mandedbythesurgeon,canmimicthemotionofhumanhands
insidepatientbodytoaccomplishsaferandmoreprecise
opera-tions.However,aspointedoutin[10],thesurgeoncannotretain
thehapticfeelingoftheinteractionwithtissueduetolackofdirect
contactwiththeworkingsiteandhencetheamountofapplied
forceontissuesurfacecannotbeaccuratelycontrolled.
Executionofpropercontactforceisnecessaryandeven
essen-tialformanysurgicaloperations[11–13].Fortaskslikesuturing
andpre-tensioning,excessiveforceleadstotissuedamageandtoo
lowforcecannotmakethetaskssuccessful.Forhapticteleoperation
withforcecommand,operatingtransparencycanbeobtainedonly
whenthedesiredinteractionforceisaccuratelygeneratedbetween
robotictoolandtissue.Moreover,usingforcecontrolallowsto
per-formthesameoperationwithhigherprecisionanddexterityby
reducinghumanerrors[14].
In the MIRS scenario, one has to control the contact force
betweenroboticsurgicaltoolsandsofttissues(i.e.muscles,organs,
veins,arteries,etc).Todesigntheforcecontroller,aproper
tool-tissue contact model is required. In literature, however, force
controlmethodsaremainlybasedonpureelasticcontactmodel
whichiseasytoimplementandappliestomosthardcontactcases.
http://dx.doi.org/10.1016/j.bspc.2016.05.003 1746-8094/©2016ElsevierLtd.Allrightsreserved.
Evensomeexistingworksonforcecontrolforsurgicalapplications
assumeanelasticcontactmodel[15].Unfortunately,thismodelis
notsuitabletodescribethecontactwithsofthumantissueswhich
exhibitscomplexpropertiessuchasnonlinearity,viscoelasticity,
anisotropy,etc[16].
Overthelastdecades,severalcompliantcontactmodelshave
beenproposedtodescribethebehaviorbetweencontactbodies
[17].Combination oflinear springsanddampers isone wayto
describetheviscoelasticbehaviorofthecontactforce,although
theseapproachesshowphysicalinconsistenciesintermsofpower
exchangeduringcontact[18].Suchkindoflinear contact
mod-elsincludeMaxwellmodel,Kevin–Voigtmodel,Kelvin–Boltzmann
model,etc.Recentworksshowthatrobot-tissueforcecontrolbased
onKelvin–Boltzmannmodeloutperformscontrolmethodsbased
ontraditionalelasticmodelintermsofrisetimeandstability[19].
Ontheotherhand,nonlinearmodels,suchastheHunt–Crossley
model,areexpectedtobemoreaccuratefordescribing thereal
behaviorduringthecontactwithsofthumantissues[18,20].
How-ever,sofarnonlinearmodelshaveonlybeenusedforparameter
estimation[18,21,22],butnotforforcecontroldesign.
ThispaperpresentsthefirstattempttoadopttheHunt–Crossley
(HC)modelfor thecontroldesignofcontactforcebetweensoft
tissue and robotictool in literature.The controldesign utilizes
theactiveobserver(AOB)techniquewhichhelpstocompensate
parameterandmodelingmismatchesduringcontrol.Stabilityof
theHC model based force controlsystem is analyzed, and the
advantageofthisapproachwithrespecttothemethodsbasedon
linearcontactmodelsisshownthroughbothnumericalsimulation
andinvitroexperimentalstudies.
Therestofthepaperisorganizedasfollows:Section2
intro-ducesthebriefbackgroundonsofttissuecontactmodelsreported
inliterature;Section3presentsacomparisonstudybetweenlinear
andnonlinearviscoelasticcontactmodelsthroughinvitro
relax-ationtests,Section4describesthedevelopmentofaforcecontrol
methodbasedontheidentifiedHunt–Crossleymodel,Sections5
and 6show theperformance comparisonsbetween linear
con-tactmodelbasedforcecontrollersandHunt–Crossleymodelbased
forcecontrollerthrough simulationsand experiments,Section7
summarizestheworkreportedinthispaper.
2. Softtissuecontactmodels
Severalmodelshavebeendevelopedinliteraturetodescribe
theviscoelasticbehaviorofsofttissues[23].Themostcomplete
studyonviscoelastictissuemodelisaddressedin[16],where a
quasi-linearviscoelastic(QLV)modelisproposedtorepresentthe
stress–strainrelationshipasfollows
F(t)=G(t)
e[ε(0)]
I + t 0 G(t−)
∂
e()∂
d II (1)whereF(t)denotesthecontactforce;e,εdenotesthe
instanta-neouselasticstress and strainrespectively;G(t) isthe reduced
relaxationmodulus.Thismodelingfunctioniscomposedoftwo
parts:thefirstpart(I)istheinstantaneousstressresponseandthe
second(II)givesthestressrelatedtothepasthistory[16].Although
accurateforoff-lineanalysis,thismodeliscomplexanddifficultto
beusedforcontactforcecontroldesign.
One simple and intuitive way to describe the interaction
between robotic tools and soft tissues is to analytically build
theforce–displacementrelationship.Analyticalmodelsareusually
presentedasacombinationofspringsanddampers[16],andare
definedbythefollowingcomponents:theexertedforcebythe
tis-sue,Fe(t),whenastrainisapplied;theindentation(orpenetration),
x(t),computedastheamountofdisplacementofthetissuefromthe
restposition;thevelocityofthedeformation ˙x(t);theelasticand
dampingcoefficientsKandbrespectively.
Followingthismodeling method,severallinearmodelshave
beendeveloped.The firstmodel,oftenusedin traditionalforce
control[24],istheelasticmodel(Fig.1(a))describedby
Fe(t)=Kx(t). (2)
TheMaxwell(MW)modelisrepresentedbytheseriesofaspring
andadamper(Fig.1(b))andisexpressedas
Fe(t)=b ˙x(t)−˛ ˙Fe(t) (3)
where ˙Fe(t)isthederivativeoftheexertedforceand˛=b/K.
TheKelvin–Voigt(KV)modelconsistsofaspringinparallelwith
adamper(Fig.1(c))andisdescribedby
Fe(t)=Kx(t)+b ˙x(t). (4)
AnotherviscoelasticmodelistheKelvin–Boltzmann(KB)model
whichisobtainedbyaddingaspringinseriestoKelvin–Voigtmodel
(Fig.1(d))anditscharacteristicequationisgivenby
Fe(t)=Kx(t)+ ˙x(t)− ˙Fe(t) (5)
whereK=k1k2/(k1+k2),=bk2/(k1+k2),=b/(k1+k2)withk1,k2
andbdenotingtheelasticanddampingcoefficientsrespectively.
Theabovelinearmodelsmayapplytocontactswithobjectsof
linearandhomogeneousproperties,butphysicallimitationscanbe
observedwhencontactwithsofttissueisconsidered.Asillustrated
inFig.2,duringthecontactbetweenarigidtoolandthesofttissue,
twophasescanbeidentified:thefirstone,correspondingto
load-ing,takesplaceatstartingcontacttimet0andendsattMaxwhen
themaximumdisplacementinthesofttissuexmaxisreached;the
secondone,correspondingtounloading,takesplacefromtmaxto
theinstanttFinalwhenthetoolandthesofttissueseparate.
Com-biningtheloadingandunloadingbehavior,anhysteresisloopcan
bedefinedastheforce–displacementrelationshipforsofttissue
duringthecontact.
The power flow during the contact is calculated by P(t)=
Fe(t) ˙x(t)asplottedontherightsideofeachsubfigureinFig.3.For
linearviscoelasticmodels,thedissipatedenergyH,represented
bytheareaenclosedbythehysteresisloop,canbecomputedas
thealgebraicsumoftheenergiesH1,H2andH3,whichare
calcu-latedasintegrationsofpowerflowfordifferentperiodsasshownin
Fig.3(a)[18].Linearviscoelasticmodelsshowthesamebehaviors
Fig.2.Stress–straincurve(hysteresis)forselectedsectionoflaryngealmuscle[25].
asinFig.3(a)withonlydifferenceinshapes.Itcanbeseenthat
linearviscoelasticmodelsshowinconsistenciesagainstphysical
intuitionintwoaspects[20,18].Thefirstoneconcernsthe
con-tactforcewhenthesofttissuesurfacejuststartstodeform(time
t0).Physicallytheinteractionforceshouldstartfromzeroandthen
buildupovertime,butanon-zeroforceisobservedatthe
begin-ningduetothedampingterms[26].Thesecondiscausedbythe
tensileforces requiredtoseparatethebodies incontact. Inthe
unloadingphasethedisplacementx(t)decreasestozeroandthe
velocity ˙x(t)isnegative.Forthisphase,linearviscoelasticmodels
indicatetheexistenceofapositiveenergyH3meaningthatpower
stillflowsfromtheend-effectortothetissueafterseparation,which
isphysicallyimpossible[18].
Toaddresstheaforementionedproblemswithlinear
viscoelas-tic models,Hunt and Crossley proposed in [20] a new model.
Replacingthespring/damperlinearcombinationin(4)witha
non-linearone,theobtainedmodelequationisgivenby
F(t)=Kxˇ(t)+ xˇ˙x(t) (6)
whereˇisapositivescalarandnormallyrangesfrom1.1to1.3
forsofttissues.ThemaincharacteristicoftheHunt–Crossley(HC)
modelistheeffectofdeformationdepthonthedampingterm.With
theHCmodel,thecontactforceevolvesfromzeroreachingthe
maximumvalueduringtheloadingphaseandreturnstozero
dur-ingtheunloadingphase.Accordingly,aphysicallycorrectpower
flowisseenasinFig.3(b).
3. Softtissuecontactmodelidentificationandparameter estimation
Starting from the models described in the previoussection
(Table1),experimentaltestsbasedoninvitrotissuesampleshave
beencarriedoutinordertoassesstheiraccuracyindescribingthe
contactforce-deformationrelationship.
Table1 Candidatemodels.
Model Constitutivelaw
Elastic F(t)=Kx(t)
MW F(t)=b ˙x(t)−˛ ˙F(t)
KV F(t)=Kx(t)+b ˙x(t)
KB F(t)=Kx(t)+ ˙x(t)− ˙F(t)
HC F(t)=Kxˇ(t)+ xˇ(t) ˙x(t)
3.1. Relaxationtestdesign
Inordertoanalyzetherelationshipbetweentissuedeformation andreactingforce,itiscommontoperformrelaxationtestsonin vitrospecimens.Therelaxationtestconsistsofgeneratingafixed deformationalongthedirectionnormaltothetissuesurfaceand measuringthecorrespondingforceexertedbythetissue.Forceand deformationdepthdataaresavedtooff-lineestimatethe param-etersofcandidatemodelsandthustocalculatethereconstructed forcebasedondifferentmodelsusingrecordeddeformationdata. Inthis relaxation test, a lamb’s heart isused asthe invitro specimenand aRAVEN-IIsurgicalrobotisusedtogeneratethe tissuedeformationasexplainedindetailinnextsubsection.The testhasbeenrunfourtimestoverifyconsistencyoftheobserved tissueinteractionproperty.Toidentifythemostaccurate (realis-tic)modeltodescribesofttissue interaction,bothtransientand overallforcereconstructionerrorsareconsideredasselection crite-ria.Theexperimentresultsareevaluatedbycomparingthemean forceerror(MFE)andcorrespondingstandarddeviation(STD)of thereconstructedforcesforeachmodel,andKruskal–Wallistest (p<0.05)isusedtoinvestigatethesignificantdifferencesbetween differentmodel-basedforcereconstructionerrors.
3.2. Relaxationtestexecution
TheRaven-IIsurgicalrobotfromAppliedDexterity(Seattle,WA, USA)was usedfor therelaxation tests asshown in Fig.4. The
robotplatformisrunningaReal-TimeLinuxsystemwithan
over-allsystemworkingfrequencyof1kHz.Thisrobotiscomposedof
twocable-actuatedarmsandeachhassevendegreesoffreedom,
whichcreatesasphericalmechanismthatallowsdexteroustool
Fig.4. RelaxationtestusingRavenIIrobot.
manipulationwithinalargeworkspacesimilartothatofmanual
laparoscopy[28].
Intherelaxationtests,theinvolvedjointistheprismaticone
oftherightarm.AsshowninFig.4,thisarmwassetinvertical
configuration(alongzaxis)andaPIDpositioncontrolwasused
togeneratetheverticalmotiontodeformthetissuesample.The
motiondatawererecordedbytheprismaticjointencoderwhich
isofhighaccuracyandnoexternalpositionsensorwasused.The
exertedforcedatawerecollectedbyanATIMINI45forcesensor
(ATIIndustrialAutomation,Apex,NC)mountedonthewristjoint
oftherobotbya3Dprintedplasticsupportwithasphericaltip.The
datawererecordedwithasamplingperiodTsof1ms(1kHz).
Fourrelaxationtestswereperformed.Beforetherelaxationtest,
therobotinstrumentwascontrolledtomovedownfromfreespace
(nocontact)until thedetectedcontactforceincreasedby 0.5N
abovethebaselinevalue(0N)toensurerealcontactbetweenthe
robottipandthetissue.ThenthePIDpositioncontrolwas
trig-geredtogenerateadisplacementdepthof15mmfromtheinitial
surfacewiththerampuptimeof0.5sandloadingtimeof20s.In
ordertoobservethehysteresisbehavior,theunloadingwasalso
donewitharampdowntimeof1s.Fig.5(a)showstherecorded
displacement-forcehistoryofonetest(Trial3).
3.3. Parameterestimation
Basedonthesaveddataforpositionx(t)andforcemeasurement
F(t),off-lineleastsquaresmethodisusedtoestimatethe
parame-tersK,b,,˛,ofthelinearcontactmodels(elastic,MW,KV,KB).
˙x(t)and ˙F(t)arecalculatedbasedonfilteredpositionandforcedata
usingmovingaveragetechnique.
ForthenonlinearHCmodel,theestimatedparametersK, ,ˇ
arecalculatedthroughnonlinearleastsquaresmethodusingthe
Levenberg–Marquardtalgorithm[29,30].Theestimated
parame-tersforthecandidatemodelsinthefourtestsaresummarizedin
Table2.
3.4. Modelidentificationbasedonreconstructedforce
Withtheestimatedparameters,thecontactforcecanbe
recon-structedbasedontherecordedpositioninformationusingdifferent
model.Sinceallfourtestsgeneratedsimilarresults,onlythe
recon-structedforcescorrespondingtoTrial3areshowninFig.5.
GraphicalinspectionofthereconstructedforcesasinFig.5(b)
showsthatallmodelsgiverisetothesamedescriptionsforsteady
statecontactforcesexcepttheMaxwellmodel,whichpresentsa
totallyunrealisticreconstructed force(zerostatic contactforce)
andthereforeisnotconsideredinthefurthercomparisonand
anal-ysis.AsshowninFig.5(c),thepureelasticmodelclearlyshowsthe
worstreconstructedforce.Thereconstructedtransientresponses
Table2
Valuesoftheestimatedparameters.
Model Parameter Mean STD STD/mean[%]
Elastic K 628.92 14.11 2.24 MW b 349.58 13.39 3.83 ˛ 0.27 0.01 3.70 KV K 629.11 14.11 2.24 b 112.72 9.10 8.07 KB K 636.05 16.21 2.55 156.92 4.10 2.61 0.052 0.0046 8.85 HC K 1,732.23 275.82 15.92 41,891.57 1,314.90 3.14 ˇ 1.23 0.03 2.44
Fig.5.ReconstructedforcesfortheTrial3:(a)rampdeformationandforcemeasurement,(b)reconstructedforces,(c)zoomedinreconstructedtransientresponses,(d) reconstructedhysteresisbehaviors.
Table3
AveragevaluesofMFE[N]andSTD[N]forthefourrelaxationtests.
Model MFE STD
Elastic 0.80 0.98
KV 0.72 0.73
KB 0.71 0.71
HC 0.65 0.58
forthelinearviscoelasticmodels(KV,KB)aresimilarandshowa leadinphasecomparedtotherealmeasuredforceatthe begin-ningofcontact.TheHunt–Crossleymodelshowsaquiteaccurate reconstructedtransientresponseastherealforce.
TheMFEandSTDofreconstructedforcesaccordingtodifferent contactmodelsarecalculatedandsummarizedinTable3.Asshown
inthistable,theHunt–Crossleymodelexhibitsthesmallestmean
errorandstandarddeviation.Kruskal–Wallistest(p<0.05)withthe
nullhypothesis“thereconstructionerrorswithdifferentinteraction
modelsfollowthesamedistribution”hasbeenconductedandthetest
resultsareshowninTable4.
FromTable4itisseenthatthenullhypothesisisfalsewithall
p-valuesp1%andthereforethefourinteractionmodelsdonot
producethesamedistributionsofreconstructedforceerrorswith
theHunt–Crossleymodelpresentingthesmallestmeanerror.To
furtherclarifyifthereisanymodelwhichmaygenerate
reconstruc-tionerrorwiththesamedistributions asHunt–Crossleymodel,
Kruskal–Wallisanalysishasalsobeenconductedassummarized
inTable5.
AboveanalysesshowthatHunt–Crossleymodelgives
signif-icantlysmallerreconstructedcontactforceerrorwithrespectto
theotherlinearinteractionmodels.Itshouldbenotedthatthe
aboveanalysesareforthewholerelaxationtestdurationincluding
boththetransientresponsereconstructionandthesteadystate
reconstruction.Fromthezoomedintransientcontactresponseas
inFig.5(c),itisseenthatallreconstructedforcesbasedondifferent
Table4
p-valueforthefourrelaxationtests.
Test1 Test2 Test3 Test4
p-value 7.7451e−12 6.6636e−05 3.3294e−12 3.6771e−13
modelsconvergetothesamevalueinlessthan1s(sameforall fourtests),henceonlyreconstructionerrorsbetween0and0.8s (transientresponse)areconsideredtoavoidmisleadingresultsdue tothelargeamountofdataforsteadystatewithalmostthesame values(after0.8s)suchthatthesignificanceofvalidationtestsis clearer.Thenotchedboxwhiskermethodisusedtoillustratethe reconstructedforceerrorsinallfourtestsduring0–0.8sasshown
inFig.6.Itisseenfromthisfigurethatthereisstrongevidence
(95%)thattheforcereconstructionerrorofHunt–Crossleymodel
hasa differentmedianfromothermodelssince itsbox’snotch
doesnotoverlapwithanyothermodel.Anditisclearlyshownthat
themedianforHunt–Crossleymodelissmallerthanothermodels
indicatingthatHunt–Crossleymodelprovidesmoreaccurateforce
reconstructioninthevalidationtest.
Moreover,thereconstructedloadingandunloadingbehaviors
ofthecandidatemodelsarealsoshowninFig.5(d).Thelinear
vis-coelasticmodels(KV,KB)predictnegativecontactforcesin the
unloadingphase andtheHunt–Crossleymodelshowsarealistic
hysteresisbehavior,which isconsistentwiththephysical
inter-pretationsasintroducedinSection2.
Therefore,throughtheinvitrorelaxationtests,Hunt–Crossley
model hasbeen identifiedas themost accurateone among all
candidatemodelstodescribetheinteractionbetweentherobot’s
end-effectorandthesofttissue.
Table5
Kruskal–Wallisanalysisforeachtest.
HC(p-value) Test1 Elastic 3.4060e−10 KV 1.5193e−10 KB 1.8489e−06 Test2 Elastic 0.0108 KV 1.3866e−07 KB 8.8199e−07 Test3 Elastic 1.2584e−04 KV 8.3806e−12 KB 2.3771e−10 Test4 Elastic 0.0178 KV 1.0167e−08 KB 3.2448e−13
Fig.6.Evaluationofforcereconstructionerrors(0–0.8s)forallfourtestwiththeirp-valuesrespectively.Theverticalboxesshowmedianvalues(inred)andquartiles(25% asboxlowerlimitand75%asboxupperlimit)ofthereconstructureerrordatasets.ThehorizontallinesrepresentsignificantdifferencesbetweenHunt–Crossleyandother modelsdeterminedthroughcomparisons(Kruskal–Wallistest,p<0.05).(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtotheweb versionofthisarticle.)
4. Hunt–Crossleymodelbasedforcecontrol
Forsurgicalapplications,linearcontrolmethodsarenormally preferredsincethereexist matureanalysisanddesign toolsfor linearsystems.Also,clearandsimplerelationcanbeestablished betweensystemperformancespecificationsandthecontrol param-eters. On the contrary, in general no uniform tools exist for nonlinearsystemcontroldesign,anditisdifficulttorelate con-trollerdesigntopre-definedperformancespecifications,e.g.rise time,phase/gainmargin,etc.
Tosynthesizetheforcecontrollerusinglineardesigntoolsbased onHunt–Crossleymodel,thismodelhastobelinearized.Inthis work,theforcecontrolisdesignedresortingtostatefeedback reg-ulationwithastochasticKalmanfilterbasedactiveobserver(AOB), whoseroleistoestimatethestatesofthesystemandcompensate formodeling errors(e.g.linearization)and systemdisturbances throughanextra“active”state.
4.1. LinearizationofnonlinearHunt–Crossleymodel
Consideringthatduringcontactthetissuedeformationisfast (transientstate)andthenkeepsalmostconstant(“steady”state),
as can beseen from Fig. 5(b), it is reasonable to linearizethe
Hunt–Crossleymodelaroundthe“steady”statewhichisthe
equi-libriumpositionxsforagivendesiredforcecommandFdasseenin
Fig.7
UsingTaylorexpansion,thecorrespondingfirst-orderlinearized
Hunt–Crossleymodelatx=xshasthefollowingexpression
Fe(x, ˙x)≈Fe(xs, ˙xs)+
∂F
e∂x
xs, ˙xs (x−xs)+∂F
e∂
˙x xs, ˙xs ( ˙x − ˙xs). (7)Foragivendesiredforce,Fd,fromEq.(6)theconstant
equilib-riumpositionxscanbecalculatedfrom
Fd=Fe(xs, ˙xs)=Kxsˇ+ xˇs˙xs=Kxˇs, (8) givingriseto xs= ˇ
Fd K. (9)Thepartialderivativeterms
∂
Fe/∂
xand∂F
e/∂x˙satxsand ˙xsin(7)canthenbecalculatedusing(9)as
∂Fe ∂x
xs, ˙xs =Kˇxsˇ−1+ ˇxˇ−1s ˙xs= ˜ K KˇFd K (ˇ−1)/ˇ ∂Fe ∂ ˙x xs, ˙xs = xˇs = ˜ Fd K . (10)
Substituting(10), (9) into (7), thelinearized function ofthe
Hunt–CrossleymodelforthegivendesiredforceFdcanbeobtained
as
Fe(x, ˙x)= ˜Kx+ ˜ ˙x+(1−ˇ)Fd. (11)
Theneglectedhighordertermswouldbeconsideredas
model-ingerrorsandhandledbytheactivestateofAOB.
4.2. Robotdynamicsandcontrolsystemformulation
Thejointspacedynamicsofarobot,withndegreesoffreedom
andjointvariablesdenotedbyq,canbewritteninthefollowing
form[24]
M(q)¨q+V(q, ˙q)+G(q)=c−JT(q)Fe (12)
whereM(q)istherobotinertiamatrix,V(q, ˙q)representsCoriolis
andcentrifugalforces,G(q)isthegravityforcevectorandc
repre-sentsthegeneralizedtorqueinput.Fedenotestheinteractionforces
duetocontactwiththeenvironmentandJ(q)istheJacobianmatrix
mappingjointspacevelocity ˙qtoCartesianspacevelocity ˙Xas
˙X =J(q) ˙q. (13)
TherobotdynamicsinCartesianspacecanbeequallyobtained
as[27,24]
Mx(q) ¨X+Vx(q, ˙q)+Gx(q)=Fc−Fe (14)
whereXrepresentstheCartesiancoordinates,Fcistheend-effector
forceduetojointactuationas
c=JT(q)Fc (15)
andthematricesin(14)aregivenby
Mx(q) =J−T(q)M(q)J−1(q)
Vx(q, ˙q) =J−T(q)V(q, ˙q)−Mx(q)˙J(q) ˙q
Gx(q) =J−T(q)G(q).
Assumingthattheabovematricescanbeenaccuratelyidentified
andaforcesensorisavailabletomeasureFe,therobotdynamics
(14)canbelinearizedusingthefollowinginverse-dynamicscontrol
input
Fc=Mx(q)U+Vx(q, ˙q)+Gx(q)+Fe. (16)
whereUrepresentstheauxiliary controlsignal.Incases where
thesystemdynamicsparameterscannotbeaccuratelycalibrated,
adaptive compensation technique couldbe used in thecontrol
design.
Substituting(16)in(14),thesystembecomes
U= ¨X (17)
presentingadecoupledunitymasssystemalongeach Cartesian
dimension.Byaddinganinnervelocitydampingloopwith
diago-nalmatrixKv,thesystemstabilitycanbeimproved[31].Thenew
dynamicsequationisgivenby
U= ¨X+Kv˙X. (18)
Nowthatthesystemisdecoupled,eachdegreeoffreedomcan
beindependentlyexpressedas
Ui= ¨Xi+Kv˙Xi (19)
where ¨Xiand ˙Xirepresenttheaccelerationandthevelocityalong
oneaxis,andUiistheauxiliarycontrolinput.Thegeneralsystem
blockdiagramispresentedinFig.8.
DifferentiatingthelinearizedHunt–Crossleysofttissuemodel
in(11)andnotingthatFdisconstant,oneobtains
˙Fe= ˜K ˙Xi+ ˜ ¨Xi, ¨Fe= ˜K ¨Xi+ ˜ តXi (20)
andaccordingtothedecoupleddynamicsequation(18),itfollows
˙Ui= តXi+Kv¨Xi. (21)
Fig.8. SystemdiagramforoneCartesianaxis.
FromEqs.(19),(20)and(21),therelationshipbetweencontact
forceFeandcontrolinputUialongoneaxiscanbewrittenas
¨Fe+Kv˙Fe= ˜ ˙Ui+ ˜KUi. (22)
Basedon(22),thetransferfunctionG(s)describingtherelation
betweenthecontrolinputUiandFecanbegivenby
G(s)= Fei(s) Ui(s) = ˜ s+ ˜K s2+K vs. (23)
Thesystemin(22)canberepresentedinstatespaceformaswell
basedonitstransferfunction(23).Usingtheobservablecanonical
form[33],oneobtains
˙ x1 ˙ x2
=
−Kv 1 0 0
x1 x2
+
˜ ˜ K u=Ax+Bu y(t) =[ 1 0 ]
x1 x2
=Cx. (24)
Forpracticalimplementation,system(24)isoftendiscretized
usingzeroorderhold(ZOH)withasampletimeTs.Thenthediscrete
statespacerepresentationisgivenby
x(k) =x(k−1)+ u(k−1)+ω(k)
y(k) =Cx(k)+(k) (25)
where , arethediscrete statematrix andthediscrete input
matrixrespectively,andtwoadditionaltermsω,areintroducedto
denotethemodelandmeasurementuncertaintiesinrealpractice.
Thediscretestatematriceswereobtainedas[32]
=eATs=I+ATs+A 2T2 s 2! + A3T3 s 3! +... =
Ts 0 eATsBd=BTs+ABT 2 s 2!+A 2 BT 3 s 3!+... (26)whereA,Barethematrixdefinedin(24)andIistheidentitymatrix.
Thecontrollawcanbedesignedfollowingthestatefeedback
regulationmethodas
u(k)=−Lx(k)+r(k) (27)
wherer(k)isthereferenceforce(Fd)andListhestatefeedbackgain
which canbecalculatedaccordingtopoleplacementtechnique
usingtheAckermann’sformula[32].
4.3. ActiveobserverbasedonKalmanfilter
Itshouldbenotedthatcontrollaw(27)needsfullsystemstate
informationwhichisdifficulttoachievesincethesystemstatex(k)
in(25)maynothaveclearphysicalinterpretationandcannotbe
directlymeasured.Inthiscase,anobserverisusuallythenatural
solution.Theroleoftheclassicalobserveristocomputethe
esti-matedsystemstate ˆx(k)basedonthesystemequationsandoutput
measurements.Kalmanfilterisapopulartechniquetoachievethis
Fig.9.Forcecontrolarchitecture.
Inrealapplication,thereexistsanerrore(k)betweenthe
esti-matedstateandtheactualstateasfollows
e(k)=x(k)− ˆx(k). (28)
Onecannoticethatwhenstatefeedbackisperformedin
com-binationwithanobserver,theerrorin(28)isintroducedinthe
systemasanundesiredextrainputasfollows
Lˆx(k)=Lx(k)−Le(k). (29)
TraditionalKalmanfilterscannotsolvethissystemstate
obser-vation/reconstructionerrorproblem.Inthiswork,aKalmanfilter
basedactiveobserver(AOB)techniqueisusedtohandlethestate
observationerror.DifferentfromtraditionalKalmanfilters,thegoal
oftheactiveobserver,asshowninFig.9,istoestimateand
com-pensatethiserrorbyusingafeedforwardtermbasedonanextra
state,alsocalledasactivestatep(k),givenby[34,35]:
p(k)=Le(k). (30)
Accordingto[35],astochasticapproachisusedtomodelp(k)as
p(k)−p(k−1)= (k) (31)
where (k)isazero-meanGaussianrandomvariableandisusedin
Kalmanequation,bytheAOBalgorithm,asawhitenoisetodescribe
theevolutionofp(k)[35].
RewritingEq.(25)bytakingintoaccounttheextrastatep(k),
thediscretestatespacedescriptionbecomes
x(k) p(k) =
0 1
x(k−1) p(k−1) +
0 u(k−1)+
ω(k) (k) y(t) =Ca
x(k) p(k) +(k) (32)
withtheaugmentedcontrolinput
u(k−1)=r(k−1)−[ L 1 ]
x(k−1) p(k−1) andCa=[ C 0 ].
Thentheclosedloopsystemcanbewrittenas
x(k) p(k) =
− L 0 0 1
x(k−1) p(k−1) +
0 r(k−1)+
ω(k) (k) (33)
andtheactiveobserverbasedon(33)isgivenby
ˆx(k) ˆp(k) =
− L 0 0 1
ˆx(k−1) ˆp(k−1) +
0 r(k−1)+K(k)(y(k)− ˆy(k)) (34)
Fig.10.Looptransferfunction.
where ˆy(k)=Ca
− L 0 0 1
ˆx(k−1) ˆp(k−1) +
0 r(k−1) .
TheKalmanfiltergain K(k)is calculated usingthefollowing
equations K(k) =P1(k)CaT[CaP1(k)CaT+R(k)]−1 P1(k) =aP(k−1)Ta+Q (k) P(k) =P1(k)−K(k)CaP1(k) (35) with a=
0 1 , Q (k)=
Qx(k) 0 0 Qp(k)
wherethematrixaistheaugmentedopenloopmatrix,Q(k)isthe
processcovariancematrixandR(k)isthemeasurementcovariance
noise.ThecovarianceQp(k)isrelatedtotheeffectivenessofactive
state ˆp(k)tocompensatetheerrors.Thehigherthisvalueis,the
moreeffectivetheerrorcompensationwillbe.
4.4. Systemstability
Toevaluatethesystemstability,themostcommoncriteriaused
arethephaseandgainmarginsthatcanindicatebothabsoluteand
relativestability.Forthesystemconsideredhere,onecancompute
thelooptransferfunction(LTF)astherelationshipbetweeninput
u(k)andtheoutputY,asshowninFig.10.Thecorrespondingstate
spacerepresentationisgivenby[35]
ˆx(k) e(k) =
−ϒ L K(k)C ϒ L −K(k)C
ˆx(k−1) e(k−1) +
K(k)C (I−K(k)C) u(k−1) (36) Y=L 0
ˆx(k) e(k) (37) whereϒ=I−K(k)C.
Thetransferfunctionof(36)canbeobtainedas
HLTF=[L 0][I−z−1]−1z−1 (38)
where,arethestatetransitionandcommandmatricesof(36).
GivenHLTF,onecanplottheBodediagramandcomputethegain
andphasemarginofthesystemin(36),asillustratedinnext
sec-tion.Itisworthwhiletonoticethatthisstabilityanalysisapplies
whenthesystemworksclosetotheequilibriumsincealinearized
approximationoftheHunt–Crossleymodelaroundtheequilibrium
Table6
StabilityanalysisoftheHunt–Crossleymodelbasedforcecontrolsystem.
Model Pole Gm[dB] ϕm[◦] HC −2 29.2 64.3 −5 29.0 63.0 −8 28.8 61.7 5. Numericalstudies
Inthissection,firstlystabilityoftheHunt–Crossleymodelbased forcecontrolsystemasdevelopedinlastsectionisevaluated.The performanceoftheforcecontrollersdesignedonthebasisof dif-ferentcontactmodelsarethencompared.
5.1. StabilityanalysisforHCmodelbasedforcecontrol
Using the average values of the estimated parameters (see Table2),theHunt–Crossleycontactmodelisgivenbythefollowing equation
Fe(t)=2·103 x1.23(t)+4·104 x1.23(t) ˙x(t) (39)
andtheextradampingin(18)issetasKv=0.05.
FortheAOBforcecontroldesign,thelinearizedHunt–Crossley
equationin(11)isused.ThediscretizationtimeisequaltoTs=1ms
and the covariance matrices in (35) are selected as Qx=10−6,
Qp=10−2,R=0.005.Thedesiredforceinputr(k)(Fd)isdefinedasa
stepfunctionof5Nat0.5swithaninitialvalueof0.5Ntobe
con-sistentwiththeexperimentalsetupinnextsection.Threedifferent
situationsarestudiedwithslowertofasterrisetimerequirements,
correspondingtoclosedloopsystempolessetat2rad/s,5rad/sand
8rad/srespectively.
Gain(Gm)andphase(ϕm)marginsgreaterthan10dBand30◦,
respectively,areusuallyrequiredtoguaranteegoodsystem
robust-nessagainstexternaldisturbances.Table6showstheresultsofthe
stabilityanalysisforHunt–Crossleymodelbasedforcecontrol
sys-tem.Inallthreecasesthesystemisstableandretainsgoodgain
andphasemargins.Itisalsoevidenthowfasterpolesdecreasethe
systemgainandphasemargins.
5.2. Simulationcomparisonwithlinearmodelbasedcontrol
TheperformanceoftheforcecontrollersdevelopedusingAOB
basedonelasticandKelvin–Boltzmanncontactmodelsarestudied
inthissubsection.NonlinearHunt–Crossleymodelisusedasthe
contactmodelinthissimulationstudytoapproximatethereal
tool-tissueinteractionasjustifiedinSection3.
ThevaluesofthemodelparametersK,andaresetaccording
totheestimatedaveragevaluesinTable2,andthecorresponding
contactmodelequationsare
Felastic(t)=628.92x(t)
FKB(t)=636.05x(t)+156.92 ˙x(t)−0.052 ˙F(t)
. (40)
Table7
Transientresponsesofthethreecontrolforce.
Risetime(s)
Pole HC KB Elastic
−2 1.21 1.11 1.28
−5 0.47 0.28 0.58
−8 0.23 0.08 0.36
InFig.11,theperformanceofallthreeforcecontrolsystems,
basedondifferentcontactmodelswithdifferentclosed-looppoles,
are illustrated for direct graphical comparison. The simulation
resultsoftheforcecontrolmethodsunderstudyaresummarized
inTable7.
Among the three developed force control methods, theone
basedonKBmodelshowsthefastestrisetimeinallcases,butthe
downsideisthatwhenafastpoleisassignedovershootappearsin
theexertedforceasseeninFig.11(c),whichshouldbeminimized
forsurgicalapplications.Incomparison,theHCmodelbasedcontrol
systemshowsslightlylowerrisetimeandtheelasticonepresents
theslowestresponseinallcases.
Remark1. Theelasticmodelrepresentsthemostoftenused
con-tactmodelforforcecontrolinliterature.TheKelvin–Boltzmann
modelisshowntobesuperiortoelasticandotherlinear
viscoelas-ticcontactmodelsinforcecontrolforsofttissuesbytherecent
works [19].Therefore, forthesimulationstudiesof thissection
andtheexperimentalstudiesinnextsection,onlytheelasticand
Kelvin–Boltzmannmodelbasedcontrollersareconsideredfor
per-formancecomparisonwiththeHunt–Crossleymodelbasedcontrol
method.
Remark 2. The performance of the linearized Hunt–Crossley
modelbasedcontrollermaybeaffectedbytheneglectednonlinear
terms,whichareunknownandtime-varyingandthusaredifficult
togetadeterministicdescription.WiththeKalmanfilterbased
AOB,thestochasticallydrivenactivestatep(k)aimstocompensate
theeffectofneglectednonlineartermsinrealtime[34,35].
Sim-ulationstudiesarecarriedouttotesttherangeofvalidityofthe
controllerbasedonlinearizedHunt–Crossleymodel(around5Nas
anexample)withAOB.ItshouldbenotedthatformostMIS
proce-duresthemeanforceappliedduringtissuegraspsisaround5–11N
andevensmallerfortissuesuturingandotherdelicateoperations
[38,39].Sointhisevaluation,thedesiredinteractionforcehasbeen
chosen as2N,10N,15Nand 20N withthesystempolesetto
p=5rad/s(mediumpole).Thecorrespondingcontrolperformances
areillustratedinFig.12,whereitshowsthatforthecontrollerbased
onaHunt–Crossleymodellinearizedaround5Ntheforcecontrol
performancesaresatisfactorywithdesiredforcecommandranging
from2to15N.WhenFd=20N,anotableovershootisobserved
(Fig.12(f))indicatingthatthecontrolperformancedegradeswhen
thesystemworkstoofarawayfromthelinearizationpoint,which
isanaturalandunderstandablefact.
Fig.12.Forcecontrolperformanceswithdifferentreferenceforces.
Fig.13.Experimentresultswithpoleatp=2rad/s.
6. Experimentalstudies
Inthissection,theexperimentalresultsfor theforcecontrol
methodsbasedonthelinearizedHunt–Crossley,theelasticand
thelinearviscoelasticKelvin–Boltzmannmodelsarereportedand
compared.
6.1. Systemsetup
Theplatformusedfor theexperimentaltestsistheRaven-II
robotsystemasintroducedinSection3andillustratedinFig.4.A
lamb’sheartwasusedasthesofttissuesamplefortheexperimental
studies.
Astepinputof5Nissetasthereferenceforcecommandthat
shouldexertonalamb’sheartsurface,in linewiththe
simula-tionstudiesof lastsection. Asmallinitialcontact forcearound
0.5Nisexertedontissuesurfacetoensurearealcontactbetween
robotandtissueatthebeginningofexperiment.Onlymotionalong
thenormaldirection(zaxis)isallowedfortherobot.Theextra
dampingtermKvinthecontrollerwassettoKv=0.05,andbased
onthesimulationstudiesthecovariancematricesinAOBhavebeen
selectedasfollows
Qx(k)=10−6, Qp(k)=10−3, R(k)=0.005
6.2. Experimentalforcecontrolresults
Figs.13–15showtheperformanceofthecontrolmethodsbased
onElastic,KB,HCmodelsandfordifferentclosed-looppoles(p=2,
5,8rad/s)throughinvitroexperimentaltests.Toevaluatethe
con-trolperformancequantitatively,thecontactforceovershoot,rise
time(RT)andtheMFE/STDvaluesofforcetrackingerror(0–6s)for
eachexperimentarecalculatedandsummarizedinTable8.
Fromtheexperimentalresultsitcanbeobservedthatthe
elas-ticmodelbasedcontrolalwayspresentstheslowestresponsesand
Kelvin–Boltzmannmodel based controlshows thesmallest
ris-ingtime.TheHunt–Crossleymodelbasedcontrolshowsslightly
slower but comparable responses. Both Kelvin–Boltzmann and
Hunt–Crossleymodelbased controlshowsimilarforcetracking
performanceintermsofMFEandSTD,whiletheelasticmodelbased
controlpresentsabiggerforcetrackingerror.
Concerningovershoot,whichisacriterionofparticular
Fig.14.Experimentresultswithpoleatp=5rad/s.
Fig.15.Experimentresultswithpoleatp=8rad/s.
model based control shows overshoot even for the slow
sys-tempolep=2rad/s.TheKelvin–Boltzmannmodelbasedcontrol
remainedstableforthefastpolep=8rad/sbuttheovershootis
over 50%(2.72N)of thedesired force(5N).Comparatively,the
Hunt–Crossleymodelbasedcontrolshowedverysmallovershoot
for allthree situations. In fact, for ourexperimentalsetup,the
Hunt–Crossleymodelbasedcontrolpresentedsimilarperformance
for even faster closed-looppoles, but Kelvin–Boltzmannmodel
basedcontrolgeneratedtoobigovershootsuchthattheRaven-II
robotsystementered“E-Stop”protectionmode[36]forpolesfaster
thanp=8rad/s.
6.3. Discussion
Includingtheexperimentsintroducedindetailinlast
subsec-tion,theforcecontrolexperimentshavebeenconductedseveral
timesusingdifferentanimalsamples(beefmuscleandlambheart).
Theaveragecontrolperformanceresultsfortheexperimentsbased
onlambheartaresummarizedinTable9andtheonesbasedon
beefmuscleinTable10.Theaveragedoverallperformancevalues
ofallexperimentsconductedaresummarizedinTable11.From
Tables9–11,itisseenthatallexperimentsleadtoverysimilar
con-clusionsofthecontrolperformancesbasedonthethreeinteraction
modelsundercomparison.
Theelasticmodelbasedcontrolshowstheslowestresponses,
andovershootisobservedevenforslowclosed-looppoles.These
areconsistentwiththerecentworkof[37]comparingthe
elas-ticmodelbasedandKelvin–Boltzmannmodelbasedforcecontrol
methods. The force control based on Kelvin–Boltzmann model
alwaysgivesthefastestresponsesfordifferentclosed-looppoles
Table8
Experimentalresults.
Model Pole RT[s] Overshoot[N] MFE[N] STD[N]
Elastic −2 1.71 0.81 0.86 1.59 KB 1.02 0.67 0.57 1.14 HC 1.34 0.23 0.68 1.30 Elastic −5 0.62 0.72 0.33 1.14 KB 0.22 0.40 0.11 0.62 HC 0.40 0.23 0.21 0.81 Elastic −8 0.35 1.02 0.19 0.91 KB 0.09 2.72 0.002 0.58 HC 0.16 0.40 0.10 0.60 Table9
Averageresultsforexperimentsusinglambheart.
Model Pole RT[s] Overshoot[N] MFE[N] STD[N]
Elastic −2 1.49 0.69 0.89 1.70 KB 1.05 0.46 0.56 1.14 HC 1.26 0.24 0.68 1.31 Elastic −5 0.52 0.73 0.32 1.19 KB 0.20 0.51 0.11 0.66 HC 0.38 0.23 0.21 0.84 Elastic −8 0.32 0.94 0.19 0.94 KB 0.05 2.19 0.002 0.55 HC 0.13 0.43 0.10 0.64 Table10
Averageresultsforexperimentsusingbeefmuscle.
Model Pole RT[s] Overshoot[N] MFE[N] STD[N]
Elastic −2 1.72 1.33 0.92 1.84 KB 0.86 0.59 0.57 1.18 HC 1.30 0.64 0.68 1.40 Elastic −5 0.61 1.34 0.36 1.33 KB 0.20 0.49 0.11 0.66 HC 0.41 0.49 0.21 0.86 Elastic −8 0.39 1.37 0.19 0.99 KB 0.11 1.79 0.002 0.52 HC 0.20 0.78 0.10 0.65
but, asthedownside,ispronetolargeovershootespecially for fastpoles.Thisgreatlylimitsitsapplicabilityfor taskswithlow overshootrequirements.TheHunt–Crossleymodelbasedcontrol presentsslightlyslowerresponses,butalwaysshowsverysmall overshootsandthereforeitisapplicabletoawiderrangeofforce
Table11
Averageresultsofallexperiments.
Model Pole RT[s] Overshoot[N] MFE[N] STD[N]
Elastic −2 1.60 0.95 0.87 1.76 KB 0.97 0.51 0.55 1.16 HC 1.27 0.40 0.66 1.34 Elastic −5 0.55 0.98 0.32 1.24 KB 0.20 0.50 0.11 0.66 HC 0.39 0.33 0.21 0.84 Elastic −8 0.34 1.11 0.19 0.96 KB 0.08 2.03 0.002 0.54 HC 0.15 0.56 0.10 0.64
controltaskswithrequirementsonbothresponsetime(efficiency) andovershoot(safety).
Experimental studies alsoreveal that theKelvin–Boltzmann modelbasedcontrollerleadstochatteringcontactforces.Thiscould beexplainedbythemodelEq.(5),wherethederivativeofcontact
forceisusedwhosemeasurementisusuallyverynoisy.
Amainlimitationtoourexperimentalstudyisthatthe
hard-waresafetyprotectionofRavenIIsystempreventsusfromfurther
exploringtheperformanceof each controllerwithharshertask
requirements,e.g.toobserveoscillationorevendivergenceinorder
tobetterevaluatethecontrolsystemstability.Thismaybe
allevi-atedbycarryingouttheexperimentalstudyonageneralpurpose
industrialrobot.Also,sofartheexperimentswereconductedbased
oninvitrotissuesamples,invivoanimaltestswillbetterevaluate
differentcontactmodelbasedforcecontrolmethods.
Both simulation and experimental studies confirm that
the Hunt–Crossley model based force control system
pro-vides comparable response time and force control accuracy as
Kelvin–Boltzmannmodel,butitpresentssmootherperformance
withmuchlessovershootthanElasticandKBmodelbased
con-trolsystems.Thisisparticularlyimportantforsurgicalapplications,
whereovershootshouldbeminimizedforsafetyreasonandprompt
responseshouldbegeneratedwhenacommandismade.Therefore,
theHunt–Crossleymodelstandsasthemoresuitablecandidateto
developtheforcecontrolschemeforroboticinstrumentandsoft
tissueinteraction.
7. Conclusion
Inthecontextofminimallyinvasiveroboticsurgery,thecontrol
oftheinteractionforcebetweenroboticinstrumentandsofttissue
isanimportantresearchtopicfrombothsafetyandefficiencypoint
ofview.
Mostforcecontrolmethodsinliterature,includingsomeworks
forrobotictool-tissueforcecontrol,assumethat theinteraction
betweentherobotandtheoperatingenvironmentcanbeestimated
byapureelasticmodel.Tobetterdescribethecomplexviscoelastic
propertiesofsofttissuecontact,moresophisticatedmodelshave
beenproposed.TheKelvin–Boltzmannmodelisoneofthelinear
viscoelasticmodelsthatbestdescribestheseproperties[19].To
improvemodelingaccuracy,thenonlinearpropertiesofsofttissues
mustbeconsideredaswell,anditisnecessarytoexplorenonlinear
modelsliketheHunt–Crossleymodel,whichhasbeenreportedas
amorerealisticoptionfromthephysicalpointofview.
Inthiswork,Hunt–Crossleyhasbeenusedforthefirsttimein
lit-eratureasthebasistodesigntheinteractionforcecontroller.The
designisbasedonstatefeedbackregulationandactiveobserver
(AOB)techniques.Bothsimulationandinvitroexperimental
stud-ieswerecarriedouttoevaluatethedevelopedcontrolmethodand
compareitsperformancewithotherlinearcontactmodelbased
controlmethods.TheHunt–Crossleymodelbasedforcecontroller
isshowntobesuperiortoothermodelbasedmethodsconsidering
themainevaluationcriteriainthescenarioofroboticsurgerysuch
asresponsetime,controlaccuracyandovershoot.
Acknowledgement
This work was supported by French Centre National de la
RechercheScientifique(CNRS)incollaborationwithPolitecnicodi
Milano,Italy.
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