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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
Control design for a lower-limb paediatric therapy device using linear motor technology
Farouk Chrif
a,b, Tobias Nef
b, Max Lungarella
c, Raja Dravid
c, Kenneth J. Hunt
a,∗aInstituteforRehabilitationandPerformanceTechnology,DivisionofMechanicalEngineering,DepartmentofEngineeringandInformationTechnology, BernUniversityofAppliedSciences,CH-3400Burgdorf,Switzerland
bGerontechnologyandRehabilitationResearchGroup,ARTORGCenterforBiomedicalEngineeringResearch,UniversityofBern,CH-3008Bern,Switzerland
cDynamicDevicesAG,Technoparkstrasse1,CH-8005Zurich,Switzerland
a r t i c l e i n f o
Articlehistory:
Received11October2016
Receivedinrevisedform17February2017 Accepted16May2017
Keywords:
Rehabilitationrobots Paediatricrehabilitation Controlsystems Impedancecontrol
a b s t r a c t
Background:Rehabilitationrobotssupportdeliveryofintensiveneuromusculartherapyandhelppatients toimprovemotorrecovery.Thispaperdescribesthedevelopmentandevaluationofcontrolstrategies foranovellower-limbpaediatricrehabilitationrobot,basedonlinear-motoractuatortechnologyand theleg-pressexercisemodality.
Methods:Afunctionalmodelwasdesignedandconstructedandanoverallcontrolstrategywasdeveloped tofacilitatevolitionalcontrolofpedalpositionbasedonthecognitivetaskpresentedtothepatient, togetherwithautomaticcontrolofpedalforcesusingforcefeedbackandimpedancecompensation.
Results:Eachindependentdrivefortheleftandrightlegscanproduceforceupto288Nattheuser’sfoot.
Duringdynamictesting,theusermaintainedavariabletargetpositionwithroot-mean-squaretracking error(RMSE)of3.8◦withpureforcecontroland2.8◦withcombinedforce/impedancecontrol,onarange ofperiodicmotionof20–80◦.Withimpedancecompensation,accuracyofforcetrackingwasalsoslightly better(RMSEof9.3vs.9.8N,force/impedancevs.forcecontrolonly).
Conclusions:Thecontrolstrategyfacilitatedaccuratevolitionalcontrolofpedalpositionand,simulta- neously,accurateandrobustcontrolofpedalforces.Impedancecompensationshowedperformance benefits.Controlaccuracyandforcemagnitudearedeemedappropriateforrehabilitationofchildren withneurologicalimpairments,but,duetocurrentlevelsrequired,linearmotortechnologymaynotbe suitableforapplicationswherehigherforceisneeded.Furtherworkisrequiredtovalidatethedevice withinthetargetpopulationofimpairedchildrenandtodevelopappropriatepatient-interfacesoftware.
©2017TheAuthor(s).PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
Therecoveryandmaintenanceofmotorfunctionisonekeyaim ofrehabilitationinterventions.Robotictechnologyisincreasingly usedinclinicalrehabilitationenvironmentstofacilitatelongtrain- ingsessions,alargenumberofmovementrepetitions,andthereby toimprovetherapeuticoutcomes[1].
The field of rehabilitation robotics is developing rapidly.
Withfaster and morepowerful computers, new computational approaches and sophisticated electromechanical components, robots have become an important tool to improve the thera- peuticoutcomesin rehabilitation[2].Robots canaidtherapists intheimplementationofrehabilitationprogrammesbyenabling
∗Correspondingauthor.
E-mailaddress:kenneth.hunt@bfh.ch(K.J.Hunt).
repetitive,highqualitytask-specificmovements,byincreasingthe duration andintensityof rehabilitationsessionsand byprovid- ingalargevarietyofexercisemodalities[3].Furthermore,robotic systems provide the possibility of recording information about movementparameters(force,position,velocity,etc.)duringexer- cise,whichallowsthesubsequentinterpretationandanalysisof thetherapyperformanceandprogress[4,5].
Thecurrentgenerationofrehabilitationrobotsdifferintermsof mechanicaldesign,actuationtechnologyandcontrolarchitecture [1,6,7].Theycanbecategorizedwithrespecttotheirapplication focusasassistiveortherapeuticdevices:assistiverobotsareused toassistpatientsintheirdaily-livingactivities,whereastherapeutic robotsareusedtoimprovevariousneurophysiologicalaspectsof bodyfunction,andtheyaremainlyusedinclinicalenvironments [1].
Rehabilitationrobotscanbefurtherdelineatedwithrespectto theirmechanicaldesignaseitherend-effectororexoskeletonsys-
http://dx.doi.org/10.1016/j.bspc.2017.05.011
1746-8094/©2017TheAuthor(s).PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/
4.0/).
source: https://doi.org/10.24451/arbor.5549 | downloaded: 14.2.2022
Notationandabbreviations F measuredforce F* targetforce
Fsim simulated(nominal)force Fimp impedanceforce
pedalangle
* targetangle
i current
i* targetcurrent
s Laplace-transformcomplexvariable Cfb(s) forcefeedbackcontroller
Po(s) plantforforcecontroller Cimp(s) impedancecontroller Ci currentcontroller
Pi plantforcurrentcontroller IPC industrialPC
tems.End-Effectorrobotsimposeforcesonthedistalsegmentsof theupperorlowerlimbs[8],buttheycannotdirectlycontrolindi- vidualjointssincethecontactbetweenthepatientandtherobotis atlimbendpoints.Examplesofend-effectorrehabilitationrobots aretheG-EOSystem[9],MIT-Manus[10],theGait-Trainer[11], GENTELE/s[12]andBi-Manu-Track[13].Exoskeleton-basedrobots, ontheotherhand,useexternalstructuresattachedatseveralpoints acrossthepatient’slimbs.Thejointsoftheexoskeletonarealigned tothoseofthehumanbody[14],whichallowsdirectcontrolofthe joints[15].ExamplesofexoskeletonrobotsaretheLokomat[16], LOPES[17],ARMin[18],T-WREX[19],Dampace[20]andL-Exos [21].
Thedynamicleg-pressformofexercise,hithertoappliedmainly inthesportscontextformusculoskeletalconditioning[22,23],has potentialasanewmodalityforneuromuscularrehabilitationappli- cations.Duetothepossibilityofacompactdesign,andprovisionof asafe,semi-orfully-recumbentseatedposture,leg-pressdevices havepotentialforapplicationparticularlyinpaediatricrehabilita- tion.Examplesofleg-pressrehabilitationrobotsaretheLambda [24],LegoPress[25]andAllegro[26].
Themainaimofcontrolstrategiesforleg-pressdevicesisto provideoptimalexercisestopromoteneuroplasticityandthere- foreimprovemotorrecovery.Forrehabilitationroboticsingeneral, a variety of control strategies have been developed, and sev- eral research reviews have been done [27–30]. Rehabilitation controlstrategiescanbecategorizedintwomaingroups:(i)tra- jectorytracking controllersand (ii)assist-as-neededcontrollers (AAN)[29].Trajectorytrackingcontrollersarepositioncontrollers adaptedfromthoseappliedinindustrialrobots.Theyprovidepas- siverepetitiveexercise,wherethepatient’slimbismadetofollow apredefined trajectory. Inadvanced versions,knownas “adap- tivepositioncontrollers”,thecontrollerallowsfordeviationfrom thepredefinedtrajectorybasedonthemotionofthepatient[1].
Trajectorytrackingcontrollersareimportantintheearlyrehabili- tationstages,wherepassiveexerciseisneeded,butlacktheability tomotivatesincetheactiveparticipationofthepatientisnotof concernatthisstage[30].Ontheotherhand,assist-as-neededcon- trollersadjusttheamountofassistancegivenbytherobotbasedon thepatient’sreal-timecontributionandability.Comparedtotra- jectorytrackingcontrollers,AANcontrollersallowmorefreedom andvariabilityofmovement[31]andincreasetheparticipationand motivationofthepatient[32].OneofthemostappropriateAAN approacheswhichencouragesactiveparticipationofthepatient isimpedancecontrol[33,34].Impedancecontrolstrategiesallow deviationfromthepredefinedtrajectoryanddonotimposerigid movement.Thiscanregulatethedynamicrelationshipbetweenthe
motionofthepatient’slimbandtheforceappliedbytheactuator [35].Furthermore,impedancecontrolparameterscanbeadjusted dependingonthepatient’sabilitiesandneeds.Anothercommon AANapproachisa“tunnelcontroller”.Thiscreatesavirtualtunnel alongthereferencetrajectorywherethepatienttriestomaintain hislimbposition.Aslongasthelimbiswithinthevirtualtunnel, therobotwillapplynocorrectiveforces.Ifthelimbdivergesfrom thetunnel,therobotwillincreasetheappliedforcetopushthe limbbacktothedesiredtrajectory[36,37].Thesystemdescribed inthispaperappliesimpedancecontrol.
Theaimofthisworkwastodesign,constructandtestanovel lower-limb end-effector rehabilitation robot, based onthe leg- press exercisingapproach, witha target populationof children withneuromuscularimpairments.Thesystemwhichwasdevel- oped,asdescribedinthispaper,islegpresstrainingdevicewhich allowsactiveexerciseofthelowerlimbs.Thefeetareconnectedthe footplatesoftwoseparatepedalmechanisms.Thedeviceallows movement of the lower limbs in the sagittal plane,with flex- ion/extensionofthekneejoints.Thefocusinthepresentreportis onthedevelopmentandevaluationofforceandimpedancecontrol strategiesbasedonlinear-motoractuatortechnology.
2. Methods
2.1. Devicespecificationsanddesign
Themechanicaldesignandconstructionoftheprototypedevice isdepictedinFig.1.Since thefocusinthepresentworkis on controlstrategydevelopment,themechanicaldesigndetailsand specificationsareonlysummarisedinbriefhere.
Theprototypedevicecomprises aseatwithadjustable back- restandposition,footplatesattachedviaalevermechanismtotwo independentlinearelectricmotors,andavisualfeedbackscreen positionedatthefront.Thepatientsitsonthechairwiththeback- restadjustedasdesiredbetweenanalmostuprightpositionandan almostfullyrecumbentposition.Thefeetareplacedonfootplates attachedtoseparatepedalmechanisms.Themaximumrangeof motionof the footplates is defined by the strokeof thelinear motors.Toadapttherobotforpatientswithdifferentbodysizes andleglengths,andtogiveappropriatejointrangesofmotion,the distancebetweentheseatandthefootplatesissetbymovingthe chairbackorforward.Thevisualfeedbackscreenatthefrontpro- videsthepatientwithmotiontargetsandreal-timefeedbackofkey performancevariables(e.g.anglesandforces)forimplementation ofspecificneuromusculartrainingandassessmenttherapies.
Thetargetpopulationforthedeviceischildrenaged4–14years withbodymassofupto50kg.Thedevicewasrequiredtobecapable ofgeneratingatotalcontinuousforceonthefootplatescorrespond- ingto1.2×bodymass,i.e.acombinedleft+rightequivalentmassof
∼60kg.Thepedalsareactuatedbytwoindependentdrives(leftand rightlegs)eachofwhichiscapableofproducingacontinuousforce of354Nandapeakforceof1024N.Becauseofthepedalgeometry andavailableleverarms,thearrangementcangenerateacontin- uousforceof288Nateachfootplate.Thisgivesatotalcontinuous forcemagnitudeof288×2=576N,correspondingtoanequivalent bodymassof59kg,which,accordingtotheabovespecifications andgiventheabilityofthemotorstogenerateshort-termforces ofnearlythreetimesthecontinuouslevels,isdeemedappropriate fortherapyofchildrenwithimpairments.
Thetherapydevicewasrequiredtofacilitaterehabilitationexer- cises forchildren withneuromuscular impairments.Thedevice canbeflexiblyprogrammedforimplementationofspecifictrain- ingexercises,andwasalsodesignedtomeetthefollowinggeneral criteriaforneuromuscularandskeletalrehabilitation[38]:
Fig.1.Prototypeofthelower-limbend-effectorrehabilitationdevice:(a)principalcomponentsoftheoverallfunctionalmodel;(b)detailofthelinear-motoractuators.Legs atmaximumextension;(c)legsatmaximumflexion.
1.Flexibility and range of motion: the robot is based on bi- directionalelectricactuators,which allowstheapplicationof bothpassiveandactiveexercisestogiveflexibilityandarbitrary movementpatterns.
2.Strengthandmuscleendurance:therobotiscapableofproviding sufficientforcetoimproveandreinforcethepatient’smuscu- loskeletalcondition,basedonamultipleofmaximumbodymass.
3.Coordinationandagility:thevisualfeedbackmoduleprovides cognitivechallenges,requires engagementof thepatientand thus has potential to improve coordination using real-time, interactive sensor-controlled exercises and task-orientated training.
2.2. Overallcontrolstrategyandoutcomemeasures
Theprincipaltaskofthefeedbackcontrolsystemistomaintain atargetforceF*ateachfootplate,i.e.atthepointofinteraction betweenthehumanfootandtherobot’spedalmechanism.This isimplementedusinga closed-loopforcecontrolsystem(Fig.2 ),wheretargetforceF*(t)isacompletelyarbitrarilypre-specified profile.Theleftandrightsidesareseparatelycontrolledbyinde- pendentfeedbacksystemssuchthattargetforcesforthetwosides canbedifferent:forsimplicity,thecontrolstrategyispresented hereasiffora singleside,butin practiceitis implementedin duplicate.
Fig.2. Closed-loopforcecontrolsystem.
ThefeedbackcontrollerCfbcomparestarget(F*)andmeasured (F)forcesandcontinuouslycomputesatargetcurrenti*whichis transmittedtotheproprietarymotorcontrolunit;thisunit,inturn, hasaninternalfeedbackcontrollerwhichdynamicallymaintains thecurrentactuallysenttotheactuator(i)closetothetargeti*.The resultingmotoractuationgeneratestheforceFasmeasuredatthe footplate.Theoverallplanttobecontrolled,asviewedfromthe controllertransferfunctionCfb,isthereforethenominaltransfer functionPolinkingthecontrolsignali*andthecontrolledvariable F(seeEq.(3),andFig.2).
Thebasicforcecontrollerisembeddedwithinanoverallcon- trolarchitectureforthedevice,comprisingalsothevisualfeedback module(seeFig.3).The“controllermodule”inthefigureexplicitly showstheforcefeedbackcontrollerandalsothecurrentcontroller Ci which is internal to the motor control unit (the function Pi denotesthenotionalplantforthecurrentcontroller).
Asdescribedinthesequel(Section2.5),thetargetforceF*can beaugmentedusinganimpedancecontrollerblockCimp(Fig.4);
Fig.3.Overallcontrolarchitecture.Thecontrollermoduleregulatestheforcesatthefootplates:itcomprisesaforcefeedbackcontrollerCfb(and,optionally,animpedance controlelementCimp,Fig.4andEq.(12)),implementedwithinanindustrialPC(IPC);thecurrentcontrollerCiisembeddedinaseparatemotorcontrolunit.Theindicated functionsoftheuserinterfacemoduleareimplementedintheIPC.ThenominalplantfordesignofCfbisthetransferfunctionfromtargetcurrenti*tomeasuredforceF, i*→F:Po(s)=k/(s+1),Eq.(3)andFig.2.
Fig.4.Force-impedancecontrolstrategy.
thismodifiesthetargetforceF* basedonposition,velocity and accelerationofthepedalrotation(Eq.(12));inthiscase,thevari- ablelabelled“forceoffset”inFig.4correspondstothetargetforce profileF*generatedbythe“userinterfacemodule”inFig.3.Thus, theimpedancecontrollerCimpcontinuouslygeneratesvariationsin targetforceinrealtimearoundthebaselineprofile(forceoffset) whichispre-specifiedintheuserinterfacemodule.Bythismeans, theindividualcomponentsofmechanicalimpedanceactuallyfelt bythepatientatthehuman-machineinteractionpoints,i.e.stiff- ness,dampingandinertia,canbepurposelymodifiedbysetting theparametersoftheCimptransferfunctioninEq.(12),incommon withgenericimpedancecontrolstrategies[39].
AswellasgeneratingthetargetforceprofileF*,theuserinterface moduleprovidesamovementtaskbymeansofanarbitrarytarget angleprofile*,anditmanagesthevisualfeedbackscreen.Inits simplestform,thevisualfeedbackdisplaysthepositiontarget* togetherwiththeactualanglemeasuredbyasensormountedat theaxisofthepedalmechanism.Forsimplicity,intheexperiments reportedhereforevaluationoffeedbackcontrollerperformance,
thetargetangleprofilewaschosenasasinusoid,whilethetarget forcewaseitherkeptconstantorchangedinastepwisefashion tointroducetaskperturbations.Forclinicalapplications,alterna- tivepositionandforceprofilesshouldbeinvestigated.Thepatient isrequiredtokeepthepedalpositionascloseaspossibletothe targetbymeansofvolitionalcontrol,whiletheforcesappliedon the footplates are automaticallycontrolled as described above.
Thiscombinationofpatientinvolvementviathecognitiveposition controltaskand theautomaticforce/impedancecontrol system representsachallengingenvironmentforneuromusculartraining andrehabilitation.
Comparisonoftheperformanceofthedifferentcontrolstrate- gieswasbasedonroot-mean-squaretrackingerror(RMSE)forforce andangle:
RMSEF=
1
N
Ni=1
(Fsim(i)−F(i))2 (1),
RMSE=
1
N
Ni=1
(∗(i)−(i))2 (2),
whereiarethediscretetimeindicesoftheNevaluationdatapoints andFsimisthesimulatednominalforce.Eq.(1)givesaquantitative measureoftheaccuracyofautomaticforcetracking,whileEq.(2) measurestheaccuracyofhumanvolitionalcontrolofposition.Eval- uationoftheoutcomemeasureswasperformedoff-line,following real-timecontrollertests,usingMatlabsoftware(MathworksInc., USA).
Table1
Linearmotorcharacteristics.
LinMotP01-48x360/180x330
Stroke 330mm
Peakforce 1024N
Max.continuousforce 354N
Forceconstant 32N/A
Max.current(@72VDC) 32A
Max.velocity 2.1m/s
2.3. Actuators,sensorsandcontrollerhardware
The two actuators are linear electric motors (P01- 48x360/180x330, NTI AG LinMot, Switzerland). These are electromagneticdirectdrivesconsistingoftwoparts:theslider, whichdirectlyprovidestranslationalmotion;andthestator.The sliderisastainlesssteeltubefilledwithneodymiummagnets.The statorcontainsthemotorwindings,bearingsfortheslider,position capturesensorsandamicroprocessorcircuitformonitoringthe motor.Thelinear motorsarebi-directionaland thuscapableof generatingforcesinbothdirections.Peakforceis1024Nandthe maximum continuous forceis 354N (key motor specifications:
Table1).Themotors arecommanded byseparatecontrolunits (E1250-UC, NTIAG LinMot, Switzerland) which implement the currentcontrolfunctionCiindicatedinthecontrollermodule(Fig.
3).
Therobothastwoforcesensorsandtwoanglesensors,mounted asindicatedinthefigure(Fig.1(a)).Theforcesensors(KD140,1kN, TransmetraGmbH,Switzerland)aremountedbehindthefootplates todirectlymeasuretheforcesappliedbythepatient.Theanglesen- sors(GL60,ContelecAG,Switzerland)aremountedattherotational axisofthepedals.
Theforce/impedancecontrolstrategyinthecontrollermodule, togetherwiththeuserinterfacemodulefunctions(Fig.3),were implementedinareal-timeindustrialPC(IPC,CX5010,Beckhoff AutomationAG,Germany).Thefoursensorsignalswereinterfaced directlytotheIPC’sanalogueinputchannels.TheIPCsendsthe targetcurrentcommands(thecontrolsignali*)viaEthernettothe individualmotorcontrolunits.
During real-time feedback control tests, all relevant signals (force,position)aremonitoredandstoredusingdataacquisition andsignalprocessingsoftware(ScopeView,Beckhoff)runningin theIPC.Thesesignalsarethenprocessedoff-lineusingMatlabas notedabove.Positionsignalsarefedbacktotheuserinreal-time asshowninFig.3.
2.4. Feedbackcontrollerdesign
ThenominalplantPo(s)linkingtargetcurrentand measured forcewasmodelledasalinearfirst-ordertransferfunctionwith steady-stategainkandtimeconstant,expressedas
i∗→F: Po(s)= Bo(s) Ao(s)= k
s+1 = k/
s+1/ (3)
wherethepolynomialsAo(s)andBo(s)aredefinedasAo(s)=s+1/
andBo(s)=k/.
Modelparameterskandwereobtainedempiricallyusingstep responsesandsystemidentification.Duringopen-loopidentifica- tiontests,thepedalwasmechanicallyfixedusingarigidbarto preventmovementandstepinputswereappliedusingthetarget currenti*.Elevenmeasurementsweredonewithstepchangesin currentofmagnitude2Aand4A,distributedacrosstherangefrom 2to18A.Modelparameterswereestimatedforeachmeasurement usinglinearleast-squares.Thegainkwasfoundtovarybetween 16.1N/Aand 17.5N/A,andthe time-constant varied between 0.019sand0.026s.Sincetheseparameterrangesarequitenarrow,
thenominalmodelparametersweretakenastheaveragevalues fromtheelevenmeasurements,giving
Po(s)= 16.9
0.0226s+1 (4).
TheparametersofthelinearcompensatorCfb(s)wereobtained byfollowinganalgebraicpole-assignmentapproachtoobtaina closed-formanalyticalsolution,e.g.[40].Cfb(s)waschosentobea linear,timeinvariant,strictly-propertransferfunction,
Cfb(s)= G(s)
H(s)= g1s+g0
s(s+h0) (5)
whereg0,g1,andh0 arerealcoefficientstobedetermined.The polynomialsG(s)andH(s),usedinthefurthercontrollerderiva- tionbelow,canbeidentifiedasG(s)=g1s+g0andH(s)=s(s+h0).By virtueofthefactor1/sinCfb,thecompensatorcontainsintegral action.
Withthefeedback loopofFig.2,theresultingcharacteristic polynomialofthesystem,denoted(s),is
(s)=Ao(s)H(s)+Bo(s)G(s)=
s+1
(s+h0)s+k
(g1s+g0) (6) whichbyexpansiongives
(s)=s3+(h0+1 )s2+1
(kg1+h0)s+k
g0 (7).
Theclosed-looppolesaretherootsofthecharacteristicpolynomial ,which,fromEq.(7),hasdegree3.Thus,musttakethestructure (s)=s3+2s2+1s+0 (8).
Usingapole-assignmentapproach,thecoefficients0,1and2 arederivedbyarbitrarypositioningofthethreeclosed-looppoles inthes-plane.Here,theapproachtakenistosetonerealpoleat position−a,andtoplacetheothertwopolescorrespondingtothe standardsecond-ordertransferfunctionwithdampingratioand naturalfrequencyωn.Thus,
(s) =(s+a)(s2+2ωns+ω2n)
=s3+(2ωn+a)s2+(ω2n+2aωn)s+aω2n
(9) In thefollowing, critical damping =1 was employed and the parameterawassettotheopen-looppolevalueof1/,i.e.a=1/, suchthattheopen-looppolewasnotshiftedbythefeedback.Fur- ther,notingthat,aroundthecriticaldampingvalueof=1,ωnis relatedtothe10–90%closed-looprisetimetrbytheapproximation ωn=3.35/tr[41,p.196],asinglefeedbackdesignparameter,viz.tr, wasthenusedtoobtainadesirableperformance(ωn,asrequired inEq.(9),wascomputedasnotedaboveasωn=3.35/tr).
Theunknowncontrollerparametersg0,g1andh0areobtained bymatchingcoefficientsoflikepowerinEqs.(7)and(9),resulting in
h0 =2ωn+a−1 g0 =
kaω2n g1 =
k(ω2n+2aωn)−1
k(2ωn+a)+ 1 k
(10)
The experimental results described in the sequel (Section 3) were obtained using the controller tuning parameter value tr=0.55s(desiredclosed-looprisetime).Withthenominalplant parameters k=16.9and =0.0226, andwiththedesign choices a=1/=44.2and=1,thecontrollertransferfunction(5)canbe calculatedusingthesolution(10)as
Cfb(s)= g1s+g0
s(s+h0)=0.0543s+2.404
s(s+12.73) (11).
Fig.5.Statictestwithpedalmechanicallyfixedinoneposition;notestpersononthedevice.
Controllers wereimplemented in discrete time using a sample periodTs=0.05s.
Withthedesignchoicea=1/,thecharacteristicEq.(6)reveals thattheplantpolepolynomialAo(s)=s+1/mustalsoappearasa zeroofthecompensator,i.e.asafactorofG(s):sinceAoappearsin thefirsttermontheleftof(6)andalsointherighthandside(see Eq.(9),withs+a=s+1/),then,forasolutiontoexist,itmustalso beafactorofGinthesecondtermofthelefthandsideof(6).To seethatthisholdsinthesolutionofEq.(11),Gcanbewrittenas G(s)=0.0543(s+44.2):theterms+44.2isseentobeequaltos+1/.
Thisisaclassicalplant-polecancellationstrategy,applicablewhen thecancelledpolesarestableandwelldamped[42].
2.5. Impedancecontroller
The impedance control strategy (Fig.4)augments the force controller bymodifying target forceF* based onangle dynam- ics,i.e.basedonthepatient’svolitionalmovementofthepedals.
Thepurposeistomodifytheindividualcomponentsofmechanical impedanceperceivedbythepatientthroughthepedals.
Thebasicequationofthesecond-order dynamicrelationship betweenthepedalangleandtheforceFimpisgiveninthetime domainby:
Fimp(t)=ka¨+kv˙+kp (12) whereka,kv andkp aretheimpedanceparametersrepresenting thedesiredmodificationsofinertia,dampingandstiffnessofthe rotationalsystem,respectively.
ByLaplacetransformation,thefrequency-domainrepresenta- tionofFimpis
Fimp(s)=kas2(s)+kvs(s)+kp(s) (13)
andthetransferfunctionCimpisseentobe Cimp(s)= Fimp(s)
(s) =kas2+kvs+kp (14) TheactualmechanicalimpedancegivenbyCimp,definedasthe ratioofforceandangularvelocity,istherefore
Fimp(s)
s(s) =kas+kv+kp
s (15).
Intheresultsreportedbelow,theimpedancecontrollerparam- eterswereselectedaskp=0,kv=30andka=300,whichshould servetoincreasetheeffectivedampingandinertiaofthesystem.
3. Results
Closed-loop control results are reported herefor two types oftest:“static”and “dynamic”tests.Duringthestatictests,the pedalwasmechanicallyfixedinoneposition(constant),andthe responseoftheforcefeedbacksystemtoasquare-wavetargetforce F*from80to120Nwasstudied.Thisfixed-pedalconfigurationis thesameasthatusedfortheopen-loopidentificationexperiments (Section2.4),andwascarriedoutwithoutanytestpersonsitting onthedevice.
Indynamictests,ahealthytestperson,usingthevisualfeedback, wasrequiredtofollowasinusoidaltargetangle*offrequency0.16 Hzandamplitudeof30◦.Withoutimpedancecontrolcompensa- tion,thetargetforcewasconstantatF*=90N.Withimpedance control,theforceoffsetvariablewasconstantat90Nanddynamic modifications to F* were implemented as F*=forceoffset+Fimp usingtheimpedancecontrollerparametersnotedabove.
The static test gave accurate force control tracking with RMSEF=3.38Nandadynamicresponseclosetothenominalclosed- loopresponsewithrisetimeoftr=0.55s(Fig.5,middlegraph).The
Fig.6.Dynamictestwithforcecontroller,noimpedancecontrol.Testpersonvolitionallyfollowstheposition/angletargetprofile.
controlsignal(targetcurrenti*)wassmoothandwellbehaved(Fig.
5,lowergraph).
Thefirstdynamictestreportedherewascarriedoutwithout impedancecompensation.Thetestpersonwasabletofollowthe targetanglecloselywithRMSE=3.79◦(Fig.6,uppergraph).The forcecontrollerwasable,onaverage,tomaintaintheconstanttar- getforcewithRMSEF=9.77N(Fig.6,middlegraph).Themeasured forcevariedontherangeofapproximately65–120N.Thesevalues correspond,respectively,totheminimumandmaximumvalues oftheanglesignal,wheretheuserchangesthedirectionofthe movementandthemagnitudeofangularaccelerationpeaks.The target-currentcontrolsignalwasagainsmoothandwellbehaved (Fig.6,lowergraph).
The addition of the impedance control element resulted in improvedtrackingofthetargetanglesignalwithRMSE=2.82◦ (Fig. 7, upper graph; cf. no impedance, where RMSE=3.79◦).
Force-trackingaccuracy,withRMSEF=9.31N(Fig.7,middlegraph), wasslightlybetterthanthenon-impedancecase(RMSEF=9.77N).
Withimpedancecompensation,themeasuredforcewasonasub- stantiallysmallerrangeofapproximately71–108N(cf.65–120N withoutimpedancecontrol);peakcurrentswerecorrespondingly lower.Thecontrolsignali*wasagainwellbehaved,albeit with asomewhatgreaterdegreeofvariabilitywhencomparedtothe non-impedancecase(Fig.7,lowergraph),duetothecontinuous dynamiccompensationdrivenbychangesinangularvelocityand acceleration.
4. Discussion
Theaimofthisworkwastodesign,constructandtestanovel lower-limb end-effector rehabilitation robot, based onthe leg- pressexercisingapproach,usinglinear-motoractuatortechnology, andwithasmallandcompactdesigntargetedatpaediatricappli-
cations.Thefocusherewasonthedevelopmentandevaluationof forceandimpedancecontrolstrategies.
Forcecontrollerdesignwasbaseduponasimpleplantmodel obtainedasanaveragefrommultiplesteptestsacrosstheoper- atingrangeofthemotors;thesetestswereperformedwiththe pedalsfixedinonepositionandwithoutatestpersononthedevice.
Despitethissimpleandapproximateapproach,thefeedbackcon- trolperformanceprovedtobeaccurateandrobustacrosstherange ofconditionstested,withandwithoutinvolvementofatestper- son.
Theintegrationofimpedancecompensationgaveimprovedtar- getangletrackingbasedonvolitionalcontrolbythetestperson,and reducedthedifferencebetweentheminimumandpeakforces.The mechanicaldesignofthedevicehadcharacteristicsoflowiner- tiaanddampingatthepedals;increasingtheeffectiveinertiaand dampingviatheimpedancecompensation,resultinginturnina lowerforcerange,presumablymodifiedthemechanicalproperties inawaythatfacilitatedbetterandmoreaccuratepositioncontrol onthepartofthetestperson.
Employmentoflinearmotortechnologycontributedtothecom- pactdesigncharacteristicsandsimpleactuator/pedalarrangement.
Thesefeaturesareattributable,inpart,totheabsenceofanyneed for agearingmechanism betweenthemotorsand theactuated devicejoints.Theresults,furthermore,demonstratedahighdegree ofmotorcontrollability,givingaccurateforcecontrolinarange suitedtopaediatricrehabilitationapplications.
Alimitationofthelinearmotors,andanegativeconsequence ofthelackofgearing,isthatarelativelyhighcurrentisrequired.
Inthetestsreportedhere,motorcurrentwasjustover8Aeven thoughthemeasuredforceswerelessthanhalfofthemaximum continuousforcewhichisavailable.Thelevelofforcegenerated, andthecorrespondingcurrent,isappropriateinconsiderationof performancespecificationsfortherapyofchildrenwithneurologi-
Fig.7.Dynamictestwithforcecontrollerandimpedancecompensation.Testpersonvolitionallyfollowstheposition/angletargetprofile.
calimpairments,butscalinguptoable-bodiedchildrenortoadults wouldbeasignificantlimitationtotheemploymentoflinearmotor technology.
5. Conclusions
Theoverall controlstrategywasfoundtofacilitatevolitional controlofpedalpositionbasedonthecognitivetaskpresented tothetestperson.Simultaneously,accurateandrobustcontrolof pedalforceswasobservedbasedonforcefeedbackandimpedance compensation.
Theaccuracyofcontrolandthelevelofforcesgeneratedare deemedappropriateforrehabilitationofchildrenwithneurological impairments,butlinearmotortechnologyisunlikelytobesuitable forapplicationswherehigherforcelevelsareneeded.
Thenextstepinthisworkistocarryoutpilotclinicalevaluations tovalidatetheprototypedevicewithinthetargetpopulationof impairedchildren.Initialworkwillfocusonchildrenwithimpaired neuromuscularstrengthand coordinationsecondary tocerebral palsy.Performancewillbeevaluatedusingformalfeasibilitycrite- riaincludingtechnicalimplementation,patientacceptability,and responsiveness;samplesizewillbedeterminedbasedonapriori statisticalpoweranalysis.Furtherworkisalsonecessarytodevelop implementationsoftherapyprogrammesandmodesofoperation thataretask-orientated,attention-demandingandhighlymotivat- ingforchildren,e.g.usingconceptsfromseriousgamessoftware technology[43,44].Priortoclinicalevaluation,thedevicedesign shouldbeaugmentedtoembodyriskanalysisandcorresponding precautionstoensurepatientsafety.
Authors’contributions
All authors contributed to the study design and technical developments.FC,KJHandTNcontributedtotheanalysisandinter- pretationoftheexperimentaldata.FCwrotethemanuscript;KJH,
TN,MLandRDreviseditcriticallyforimportantintellectualcon- tent.Allauthorsreadandapprovedthefinalmanuscript.
Conflictofinterest
Theauthorsdeclarethattheyhavenoconflictofinterest.
Acknowledgements
ThisworkwassupportedinpartbytheSwissCommissionfor TechnologyandInnovation(CTI,GrantRef.16081.2PFLS-LS).
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