Control design for a lower-limb paediatric therapy device using linear motor technology

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

Control

design

for

a

lower-limb

paediatric

therapy

device

using

linear

motor

technology

Farouk

Chrif

_,

_Tobias

_Nef

_,

_Max

_Lungarella

_,

_Raja

_Dravid

_,

_Kenneth

_J.

_Hunt

a_{InstituteforRehabilitationandPerformanceTechnology,DivisionofMechanicalEngineering,DepartmentofEngineeringandInformationTechnology,}

BernUniversityofAppliedSciences,CH-3400Burgdorf,Switzerland

b_{GerontechnologyandRehabilitationResearchGroup,ARTORGCenterforBiomedicalEngineeringResearch,UniversityofBern,CH-3008Bern,Switzerland} c_{DynamicDevicesAG,Technoparkstrasse1,CH-8005Zurich,Switzerland}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received11October2016

Receivedinrevisedform17February2017 Accepted16May2017

Availableonline9June2017 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 usedinclinicalrehabilitationenvironmentstofacilitatelong train-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:[email protected](K.J.Hunt).

repetitive,highqualitytask-specificmovements,byincreasingthe duration andintensityof rehabilitationsessionsand by provid-ingalargevarietyofexercisemodalities[3].Furthermore,robotic systems provide the possibility of recording information about movementparameters(force,position,velocity,etc.)during exer-cise,whichallowsthesubsequentinterpretationandanalysisof thetherapyperformanceandprogress[4,5].

Thecurrentgenerationofrehabilitationrobotsdifferintermsof mechanicaldesign,actuationtechnologyandcontrolarchitecture [1,6,7].Theycanbecategorizedwithrespecttotheirapplication focusasassistiveortherapeuticdevices:assistiverobotsareused toassistpatientsintheirdaily-livingactivities,whereastherapeutic robotsareusedtoimprovevariousneurophysiologicalaspectsof bodyfunction,andtheyaremainlyusedinclinicalenvironments [1].

Rehabilitationrobotscanbefurtherdelineatedwithrespectto theirmechanicaldesignaseitherend-effectororexoskeleton

sys-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/).

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],buttheycannotdirectlycontrol indi-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 potentialasanewmodalityforneuromuscularrehabilitation appli-cations.Duetothepossibilityofacompactdesign,andprovisionof asafe,semi-orfully-recumbentseatedposture,leg-pressdevices havepotentialforapplicationparticularlyinpaediatric rehabilita-tion.Examplesofleg-pressrehabilitationrobotsaretheLambda [24],LegoPress[25]andAllegro[26].

Themainaimofcontrolstrategiesforleg-pressdevicesisto provideoptimalexercisestopromoteneuroplasticityand there-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.Theyprovide pas-siverepetitiveexercise,wherethepatient’slimbismadetofollow apredefined trajectory. Inadvanced versions,knownas “adap-tivepositioncontrollers”,thecontrollerallowsfordeviationfrom thepredefinedtrajectorybasedonthemotionofthepatient[1]. Trajectorytrackingcontrollersareimportantintheearly rehabili-tationstages,wherepassiveexerciseisneeded,butlacktheability tomotivatesincetheactiveparticipationofthepatientisnotof concernatthisstage[30].Ontheotherhand,assist-as-needed con-trollersadjusttheamountofassistancegivenbytherobotbasedon thepatient’sreal-timecontributionandability.Comparedto tra-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.Thesystemwhichwas devel-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.Thepatientsitsonthechairwiththe back-restadjustedasdesiredbetweenanalmostuprightpositionandan almostfullyrecumbentposition.Thefeetareplacedonfootplates attachedtoseparatepedalmechanisms.Themaximumrangeof motionof the footplates is defined by the strokeof thelinear motors.Toadapttherobotforpatientswithdifferentbodysizes andleglengths,andtogiveappropriatejointrangesofmotion,the distancebetweentheseatandthefootplatesissetbymovingthe chairbackorforward.Thevisualfeedbackscreenatthefront pro-videsthepatientwithmotiontargetsandreal-timefeedbackofkey performancevariables(e.g.anglesandforces)forimplementation ofspecificneuromusculartrainingandassessmenttherapies.

Thetargetpopulationforthedeviceischildrenaged4–14years withbodymassofupto50kg.Thedevicewasrequiredtobecapable ofgeneratingatotalcontinuousforceonthefootplates correspond-ingto1.2×bodymass,i.e.acombinedleft+rightequivalentmassof ∼60kg.Thepedalsareactuatedbytwoindependentdrives(leftand rightlegs)eachofwhichiscapableofproducingacontinuousforce of354Nandapeakforceof1024N.Becauseofthepedalgeometry andavailableleverarms,thearrangementcangeneratea contin-uousforceof288Nateachfootplate.Thisgivesatotalcontinuous forcemagnitudeof288×2=576N,correspondingtoanequivalent bodymassof59kg,which,accordingtotheabovespecifications andgiventheabilityofthemotorstogenerateshort-termforces ofnearlythreetimesthecontinuouslevels,isdeemedappropriate fortherapyofchildrenwithimpairments.

Thetherapydevicewasrequiredtofacilitaterehabilitation exer-cises forchildren withneuromuscular impairments.Thedevice canbeflexiblyprogrammedforimplementationofspecific train-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’s muscu-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.Theleftandrightsidesareseparatelycontrolledby inde-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).

Thebasicforcecontrollerisembeddedwithinanoverall con-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

Fig.3.Overallcontrolarchitecture.Thecontrollermoduleregulatestheforcesatthefootplates:itcomprisesaforcefeedbackcontrollerCfb(and,optionally,animpedance

controlelementCimp,Fig.4andEq.(12)),implementedwithinanindustrialPC(IPC);thecurrentcontrollerCiisembeddedinaseparatemotorcontrolunit.Theindicated

functionsoftheuserinterfacemoduleareimplementedintheIPC.ThenominalplantfordesignofCfbisthetransferfunctionfromtargetcurrenti*tomeasuredforceF,

i*_→F:P

o(s)=k/(s+1),Eq.(3)andFig.2.

Fig.4.Force-impedancecontrolstrategy.

thismodifiesthetargetforceF* _{basedonposition,velocity and}

accelerationofthepedalrotation(Eq.(12));inthiscase,the vari-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.

Comparisonoftheperformanceofthedifferentcontrol strate-gieswasbasedonroot-mean-squaretrackingerror(RMSE)forforce andangle: RMSEF=











(Fsim(i)−F(i))2 (1),

RMSE=











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.Theangle sen-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-loop identifica-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),usedinthefurthercontroller deriva-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_



k

(g1s+g0) (6)

whichbyexpansiongives (s)=s3_+(h 0+ 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_+2ω ns+ω2n) =s3_+(2ωn+a)s2_+(ω2 n+2aωn)s+aω2 n (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(ω 2 n+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

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-domain representa-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,theimpedancecontroller param-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◦.Withoutimpedancecontrol compensa-tion,thetargetforcewasconstantatF*_{=90N.Withimpedance}

control,theforceoffsetvariablewasconstantat90Nanddynamic modifications to F* _{were implemented as F}*_{=forceoffset+F}

imp

usingtheimpedancecontrollerparametersnotedabove.

The static test gave accurate force control tracking with RMSEF=3.38Nandadynamicresponseclosetothenominal

Fig.6.Dynamictestwithforcecontroller,noimpedancecontrol.Testpersonvolitionallyfollowstheposition/angletargetprofile.

controlsignal(targetcurrenti*_{)wassmoothandwellbehaved(Fig.}

5,lowergraph).

Thefirstdynamictestreportedherewascarriedoutwithout impedancecompensation.Thetestpersonwasabletofollowthe targetanglecloselywithRMSE=3.79◦(Fig.6,uppergraph).The

forcecontrollerwasable,onaverage,tomaintaintheconstant tar-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,themeasuredforcewasona sub-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, andwithasmallandcompactdesigntargetedatpaediatric

appli-cations.Thefocusherewasonthedevelopmentandevaluationof forceandimpedancecontrolstrategies.

Forcecontrollerdesignwasbaseduponasimpleplantmodel obtainedasanaveragefrommultiplesteptestsacrossthe oper-atingrangeofthemotors;thesetestswereperformedwiththe pedalsfixedinonepositionandwithoutatestpersononthedevice. Despitethissimpleandapproximateapproach,thefeedback con-trolperformanceprovedtobeaccurateandrobustacrosstherange ofconditionstested,withandwithoutinvolvementofatest per-son.

Theintegrationofimpedancecompensationgaveimproved tar-getangletrackingbasedonvolitionalcontrolbythetestperson,and reducedthedifferencebetweentheminimumandpeakforces.The mechanicaldesignofthedevicehadcharacteristicsoflow iner-tiaanddampingatthepedals;increasingtheeffectiveinertiaand dampingviatheimpedancecompensation,resultinginturnina lowerforcerange,presumablymodifiedthemechanicalproperties inawaythatfacilitatedbetterandmoreaccuratepositioncontrol onthepartofthetestperson.

Employmentoflinearmotortechnologycontributedtothe com-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 performancespecificationsfortherapyofchildrenwith

neurologi-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.Performancewillbeevaluatedusingformalfeasibility crite-riaincludingtechnicalimplementation,patientacceptability,and responsiveness;samplesizewillbedeterminedbasedonapriori statisticalpoweranalysis.Furtherworkisalsonecessarytodevelop implementationsoftherapyprogrammesandmodesofoperation thataretask-orientated,attention-demandingandhighly motivat-ingforchildren,e.g.usingconceptsfromseriousgamessoftware technology[43,44].Priortoclinicalevaluation,thedevicedesign shouldbeaugmentedtoembodyriskanalysisandcorresponding precautionstoensurepatientsafety.

Authors’contributions

All authors contributed to the study design and technical developments.FC,KJHandTNcontributedtotheanalysisand inter-pretationoftheexperimentaldata.FCwrotethemanuscript;KJH,

TN,MLandRDreviseditcriticallyforimportantintellectual con-tent.Allauthorsreadandapprovedthefinalmanuscript.

Conflictofinterest

Theauthorsdeclarethattheyhavenoconflictofinterest. Acknowledgements

ThisworkwassupportedinpartbytheSwissCommissionfor TechnologyandInnovation(CTI,GrantRef.16081.2PFLS-LS). References

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