identification Parameter methods for real redundant manipulators Technology Journal of Applied Researchand

Availableonlineatwww.sciencedirect.com

Journal of Applied Research and Technology

www.jart.ccadet.unam.mx JournalofAppliedResearchandTechnology15(2017)320–331

Original

Parameter identification methods for real redundant manipulators

Claudio Urrea

, José Pascal

GrupodeAutomática,DepartmentofElectricalEngineering,UniversidaddeSantiagodeChile,Chile Received9April2016;accepted16February2017

Availableonline31July2017

Abstract

Thisworkpresentsthedevelopment,assessmentandcomparisonoffourtechniquesforidentifyingdynamicparametersinanindustrialredundant manipulatorrobotwith5degreesoffreedom.BasedontheLagrange–Eulerformulation,alinearmodeloftherobotwithunknownparameters isobtained.Then, theseparametersare identifiedusingthe following techniques:leastsquares,artificialAdalineneuralnetworks,artificial HopfieldneuralnetworksandextendedKalmanfilter.Theparametersidentifiedarevalidatedbyusingthemforcomputationallysimulatingthe performanceoftheredundantmanipulatorrobot,towhichareimposedreferencetrajectoriesdifferentfromtheonesusedintheestimation.To relatethetrajectoriesperformedbytheredundantmanipulatorrobotwiththeestimatedparameters,thefollowingerrorindexesarecalculated:

ResidualMeanSquare,ResidualStandardDeviationandAgreementIndex.Finally,todeterminethesensitivityofthemodelidentified–dueto thevariationsoftheestimatedparameters–anewsimulationisconductedontherobot,consideringthatitsparametersvaryinarestrictedrange.

Inaddition,theRMSerrorindexofthetrajectoriesperformedisdetermined.Afterthisstep,theparametersoftheredundantmanipulatorrobot weresuccessfullyidentifiedand,thus,itsmathematicalmodelwasknown.

©2017UniversidadNacionalAutónomadeMéxico,CentrodeCienciasAplicadasyDesarrolloTecnológico.Thisisanopenaccessarticleunder theCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: Robotdynamics;Systemidentification;Mathematicalmodel;Errorindices

1. Introduction

Fourcloselyrelatedtypesofproblemmaybedistinguished inthe systems theory, namely modeling, analysis,estimation and control (Kessman, 2011). Modeling and estimation are extremely important to system identification. Modeling is a critical step when applying the systems theory to real pro- cessesaimedatobtainingamathematicalmodelthatadequately describes a physical situation to be represented. To achieve this goal, several specifications should be considered, such as the limits andvariables of the system. Then, the relation- ships between these variables need to be specified based on previous knowledge as wellas on the assumptionsabout the uncertaintiesofthemodel,whichcharacterizeitsstructure.On the otherhand,after obtaining anobservable andidentifiable model, the estimation of its unknown variables is addressed

∗Correspondingauthor.

E-mailaddress:[email protected](C.Urrea).

PeerReviewundertheresponsibilityofUniversidadNacionalAutónomade México.

using an input/outputdataset. Basically, thereis adistinction between states andparameter estimation. Therefore, systems identificationconsistsinexperimentallydeterminingatempo- rarybehaviorofaprocessorsystem,basedonamathematical modelandusingmeasuredsignalstodeterminesuchabehavior.

Theerror(deviation)betweentherealprocessanditsmathemat- icalsystemhastobeminimizedtorepresenttherealprocessor systeminthebestpossibleway(Isermann&Münchhof,2011).

Intheidentificationprocess,datacharacterizingthedynamics of the real processor systemis first obtained.Then, aset of modelsischosenand, finally,thebestmathematicalmodel of thesetisselected.Itshouldbenotedthatamathematicalmodel maybedeficientduetoreasonssuchas(Ljung,1987):

- Thefailureinthenumericalprocedureconductedtofindthe bestmathematicalmodel,givenaspecificcriterion.

- Ill-chosencriterion.

- Nomathematicalmodelsatisfactorilydescribesthesystem.

- Theinformation(data)isnotsufficientfortheselectionofan appropriatemathematicalmodel.

http://dx.doi.org/10.1016/j.jart.2017.02.004

1665-6423/©2017UniversidadNacionalAutónomadeMéxico,CentrodeCienciasAplicadasyDesarrolloTecnológico.Thisisanopenaccessarticleunderthe CCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Purpose

Conduction of experiment

Data (measurements)

Physical foundations equations initial experiments conditions of operation

Yes

A priori knowledge

Data inspection and pre- processing

Application of identification

method

Model processing Model validation

Selected model

No

Fig.1.Systemidentificationscheme(Isermann&Münchhof,2011).

Insummary, the generalproblemof systemsidentification maybe expressedas:onceknown thestructureandthevalue ofsomeoftheinputandoutputvariablesofthesystemunder study,itisnecessarytodeterminethesetofparametersthatbetter adjuststothesemeasures(Anríquez,2011).Figure1showsthe generalschemeoftheidentificationofonesystem.

2. Methodsforparameteridentification

The methods for parameter identification applied to the redundantmanipulatorrobotareexplainedbelow:

2.1. Leastsquares

Amongtheparameterestimation methods, leastsquaresis widelyusedintheidentificationofmanipulatorrobots(Bona&

Curatella,2005),sinceitisbasedontheuseofalinearmodel inverse with respect to the parameters. This method allows estimating baseinertial parameters fromthe measurement or estimationof torques andjoint positions,optimizingthe root meansquareerrorofthemodelundertheassumptionthatmea- surementerrors are negligible. However, one problemin the applicationofthismethodisthenoisesensitivitypresentinthe measureddata (Kessman, 2011), becausenoise willlimit the accuracyoftheobtainedparametersaswellastheconvergence rateofthealgorithm.Toovercomethisproblem,itissuggested togenerateexcitationtrajectoriesortoapplyafiltertodata(Ha, Ko,&Kwon,1989).

The dynamic model may be written as a setof unknown parameterslinearequations,asshowninEq.(1):

τ=Φ(q,˙q,¨q)·λ (1)

whereτ correspondstothegeneralizedforcesvector,q tothe generalized coordinates vector, λ to the vector or unknown dynamicparametersandΦtotheobservationmatrixorregres- sor.Byapplyingtheidentificationmodeltothemodelshownin (1),thelinearequationsysteminλisobtainedas:

τ=Φ(q,˙q,¨q)·λ+ρ (2)

whereρistheresidualerrorvector.Theestimated ˆλvaluemay beobtainedas:

ˆλ=minρ² (3)

whichisthesameascalculating:

ˆλ=

Φ^TΦ₋₁

Φ^Tτ (4)

2.2. Adalineneuralnetworks

Aneuralnetworkconsistsofasetofsimpleprocessingunits, whichcommunicatebysendingsignalstooneanotherthrough weightedconnection.Adaline(adaptivelinearelement)isone typeofneuralnetwork.Foraone-layernetworkwithoneoutput andonelinearactivationfunction,theoutputisgivenbyEq.(5):

y=

n i=1

w_ix_i+θ (5)

Thisnetworkisabletorepresentalinearrelationbetween the values of theoutput and inputunits.The purpose of this networkistoproduceagivenvaluey=d^pintheoutputwhen thesetofvaluesx^p_i,i=0,1,...,n.isappliedtotheinputs.It isintendedtodeterminethecoefficients(weights) wi,i=0,1, ...,n.sotheinput–outputresponseiscorrectforalargenumber ofarbitrarilychosensignalsets.Inthiscase,the“deltarule”to adjustweightsisusedforAdaline,trainingthenetworkinsuch awaythatthebestadjustmentforeachsetofdatacomposedof theinputvaluesx^p_i anddesiredoutputsd^pisobtained.Foreach inputdatum,thenetworkoutputdiffersfromthedesiredvalue byd^p−y^p,wherey^pcorrespondstothecurrentoutputforthis pattern.The“deltarule”usesacostorerrorfunctionbasedon thisdifferencetoadjusttheweights.Theerrorfunctionisgiven bythemeansquarederror.Therefore,thetotalerrorEisdefined by:

E=

p

E^p=1 2



p

d^p−y^p2

(6)

wherethepindexextendsoverthesetofinputpatternsandE^p representstheerrorovertheppattern.Thisprocedureallowsfor

findingthevalueofallweightsthatminimizetheerrorfunction bymeansofthegradientdescentmethod:

Δpwi=−γ∂E^p

∂w_i (7)

whereγisaconstantofproportionalityand,also:

∂E^p

∂w_i = ∂E^p

∂y^p

∂y^p

∂w_i (8)

Duetothelinearrelationshipshowedin(5):

∂y^p

∂w_i =x_i (9)

and

∂E^p

∂y^p =−

d^p−y^p

(10)

Finally,weobtain:

Δ_pw_i=γδ^px_i (11)

whereδ^p= (d^p−y^p) isthedifferencebetweenthedesiredout- putandthecurrentoutputfortheppattern.

2.3. Hopfieldneuralnetworks

HopfieldneuralnetworksorHNNscanbeusedforestimating parameterson-line.In Hopfield’sformulation,thedynamicof neuroniisdescribedbytheordinarydifferentialequation(12) (Alonso,Mendonc¸a,&Rocha,2009):

dui

dt =−1 Ci

⎛

⎝ 1Ri

u_i(t)+

j

W_ijf_i(uj(t))+I_i

⎞

⎠ (12)

whereuiistheneuroninputi,fiisastrictlyincreasing,bounded, nonlinearcontinuousfunction,andCi,Ri,WijandIiareparam- etersthatcorrespondtoacapacitance,aresistance,theweight associatedtotheconnectionofthejneuronwiththeineuron, andtheineuronexternalinputorbias.InAbe’sformulation,it isassumedthatCi=1andRi=∞.

Theconsideredsystemislinearwithrespecttotheparame- ters,i.e.ithastheformshowninEq.(1),which(1)worksunder thefollowingassumptions:

(a) τ,Φ∈ C¹. (b) τ,Φarebounded.

(c) τ(t),Φ(t)areknownineachtinstant.

(d) λ ∈ ]−c,c[ⁿforsomec>0unknown.

GivenAbe’sformulation,consideringanetworkwithNneu- rons,uidenotesthetotalinputoftheineuron,itsderivativewith respecttotimeis:

dui(t) dt =−

⎛

⎝^N

j=1

W_ij(t)Sj(t)+I_i(t)

⎞

⎠ (13)

whereSj representsthestate (or output)of the jneuron. The input-staterelationfortheineuronisgivenby:

S_i(t)=αtanh u_i(t)

β

(14) whereα,β>0and,thus,∀i ∈ N,t≥t₀ S_i(t) ∈ ]−α,α[ .

Usingmatrixnotation,thenetworkisasfollows:

du

dt =− (W(t)S(t)+I(t)) (15)

S(t)=αtanh u(t)

β

(16) where S(t),I(t) ∈ R^N×1 and W ∈ R^N×N, under the state- spacerepresentationis:

dS dt =− 1

αβDα(S(t))(W (t)S(t)+I(t)) (17) where

D_α(S(t))=diag 

α²−S_i(t)² 

i∈N

(18) Thisanonlinearandnon-autonomousdynamicsystemwhose architectureiscompletelydeterminedbytheNnumberofneu- rons,withcharacteristicsgivenbyα,β,WandI.

ThisworkconsidersaHNNwhosestateS(t)overtimetis takingastheestimation ˆλ(t)oftheλparameter,inotherwords, ˆλ(t)=S(t);implicitly,thereisanetworkwithasmanyneurons asparameterstoestimate.Then,takingα=c,Eq.(14)guarantees thatthetrajectorygeneratedforthenetworkisinafeasibleregion of theestimation problem.Therefore,fromtherepresentation of states inEq. (17),it is consideredthat:W=Φ(t)^TΦ(t)and I=−Φ(t)^Tτ.Inthisway,thedynamicoftheHHNtobeusedas anon-lineestimationalgorithmisdefinedby:

d ˆλ dt(t)= 1

cβD_c( ˆλ(t))Φ(t)^T

τ−Φ(t) ˆλ(t)

(19) TheadvantageofusingHNNtoresolvetheestimationprob- lemisthatinvertingΦ(t)^TΦ(t),isnotnecessary,sincethelatter mightbeill-conditioned.Inaddition,givenitsstructure,prior knowledgeisnotrequiredforselecting ˆλ(t0),i.e.oftheinitial estimationofλ.

Finally,theλparameterisaglobally,uniformlyandasymp- totically stable balance point of this HNN if the following sustains:

I ⊂[t0,+∞[,

t∈I

ker(Φ(t))={0} (20)

2.4. ExtendedKalmanfilter

TheextendedKalmanfilterisbasedonthedirectdynamic model,whichisnonlinearwithrespecttostateandparameters.

Inthisalgorithm,physicalparametersareconsideredunderthe

useofanextendedstate.Kalmanfilterintendstoestimatethe x ∈ Rⁿstateofaprocesscontrolledindiscretetime(Welch&

Bishop,2006).Thealgorithmhastwostepsthatareexecutedin aniterativeway,namelyprediction,andcorrectionorupdateof thestate.However,whenasystemhasnon-linearcharacteristics, theformulationshouldbemodified.Thischangeinthealgorithm isknownasextendedKalmanfilter.Inthiscase,theprocesswill berepresentedbythefollowingnon-linearequations:

x_k =f(xk−1,u_k−1)+w_k−1 (21)

z_k =h(xk)+v_k (22)

wheref corresponds to a non-linear function that allows for knowingtherelationbetweenthepreviousstepstatevectork−1 andtheuinputwiththecurrentstatevectork,accordingtothe followingprocedure:

1. Predictionstage

1.1 Predictionofthestate:

ˆx⁻_k =f(xk−1,u_k−1) (23) 1.2 Predictionoferrorcovariancematrix:

P_k⁻=F_k−1P_k−1F_k^T₋₁+Q (24) 2. Correctionorupdatingstage

2.1 Kalmangaincalculation Kk=P_k⁻H_k^T

HkP_k⁻H_k^T +R₋₁

(25) 2.2 StateupdatewithmeasuresZk:

ˆxk=ˆx⁻_k +Kk

zk−h ˆx⁻_k

(26) 2.3 Covariancematrixupdate:

P_k= (I−K_kH_k) P_k⁻ (27) whereFk−1andHkcorrespondstheJacobianmatricesdescribed inEqs.(28)and(29),respectively:

F_k−1= ∂f

∂x



x_k−1,u_k−1

(28)

H_k = ∂h

∂x



ˆx⁻_k

(29) whereQandRaretheprocessandobservationnoisecovariance matrices,respectively.Inaddition,theyarediagonalandsym- metric.Ontheotherhand,ˆx⁻_k ∈ Rⁿarethestatesestimated aprioriatkinstant,giventheprocessbehaviorandˆxk ∈ IRⁿ theaposterioriestimatedstateatkinstant,giventhemeasurezk. 3. Modeloftheindustrialredundantmanipulatorrobot

Themanipulatorrobotusedinthisstudyisredundant,sinceit hasfivedegreesoffreedom(DOF),andaPRRRPconfiguration.

Figure2showsaschemeofmanipulatorrobotanditscoordinate assignment.Inaddition,itsDenavit–Hartenbergparametersare presentedinTable1.

Fig.2.AssignmentofcoordinatesystemsofthePRRRPmanipulatorrobot.

Table1

Devanavit–Hartenbergparameters.

Articulation θi di ai αi

1 0^◦ l1+d1 0 0^◦

2 θ2 0 l2 0^◦

3 θ3 0 l3 0^◦

4 θ4 0 l4 180^◦

5 0^◦ l5+d5 0 0^◦

TheLagrange–Euler formulationdescribesthebehaviorof adynamic systemin termsof workandenergy storedin the system.TheLagrangianLisdefinedas:

L(qi,˙qi)=T−U (30)

From this Lagrangian, the dynamic system equations of motionaregivenby:

d dt

∂L

∂˙qi −∂L

∂q_i =Qi (31)

whereListheLagrangianfunction,Tthekineticenergy,Uthe potentialenergy,qithegeneralizedcoordinates,andQithegen- eralizedforceappliedtoqi.Thegeneralformofamanipulator robotdynamicmodelisshowninAnnexA.

4. Simulationofthemethodsforparameter identification

Table2 showsthe dynamicparametersof eachlink of the redundantmanipulatorrobot(mass,length,centerofmassand inertia)(Urrea&Kern,2016).

Inthiswork,weconsiderthefollowingmethodsforidenti- fyingtheparametersoftheredundantmanipulatorrobot:least squares,Adalineneuralnetworks,Hopfieldneuralnetworksand extendedKalmanfilter.Abriefdescriptionofthesemethodsis presentedbelow:

Table2

Parametersoftheredundantmanipulatorrobotparameterswith5DOL.

Parameter Link1 Link2 Link3 Link4 Link5 Unit

l 0.524 0.2 0.2 0.2 0.14 [m]

lc – 0.0229 0.0229 0.0229 – [m]

m 1.228 1.023 1.023 1.023 0.5114 [kg]

Izz – 0.0058 0.0058 0.0058 – [kgm²]

4.1. Leastsquares

The least squares proceduretoestimate the parametersof eachlinkoftheredundantmanipulatorrobotmightbesumma- rizedinthefollowingsteps:

- Usingsomeformulation,generatearobotmodelthatislinear withrespecttotheinertialparameters.

- Reducingtheinertialparameterstoasetofbaseparameters.

- Determiningtheoptimaltrajectoryoftheparametersandopti- mizetheexcitationofthetrajectories.

Thereisanumberofmethodologiestodetermineboththe parametersinfluencingthesystemsdynamicandthesetofmin- imumparametersrequiredtoidentifythemodelsofsuchsystems (Choi,Yoon,Park,&Kim,2011;Mayeda,Yoshida,&Ohashi, 1989; Mayeda,Yoshida, &Osuka, 1990; Radkhah,Kulic, &

Croft, 2007). This work considers a numerical method that allowsforidentifyingthesetofindependentparametersofthe redundantmanipulatorrobot,basedontheanalysisofthekernel oftheregressor.

Foraredundantmanipulator robotwithjlinks,teninertial parametersareconsideredperlink:

ξj =

m_j p^T_g,j v^T_j T

(32) where mj,pg,j=

x_g,j y_g,j z_g,j_T

and vj=

I_xx,j I_yy,j I_zz,j I_xy,j I_xz,j I_yz,jT

are the mass, the position of the center of gravity, andthe vector withthe elementsoftheinertiatensorofthejlink,respectively.Alinear formmaybecomposedbygroupingtheparametersasfollows:

τ =Φ(q,˙q,¨q)·λ

¯ξ

⎛

⎜⎜

⎜⎜

⎜⎝ τ₁ τ2

... τn

⎞

⎟⎟

⎟⎟

⎟⎠=

⎛

⎜⎜

⎜⎜

⎜⎝

φ₁₁ φ₁₂ ... φ_1n 0 φ22 ... φ2n

...

0 0 ... φnn

⎞

⎟⎟

⎟⎟

⎟⎠

⎛

⎜⎜

⎜⎜

⎜⎝ λ₁

¯ξ λ₂

¯ξ ... λ_n

¯ξ

⎞

⎟⎟

⎟⎟

⎟⎠

(33)

where λj

¯ξ

=

m_j m_jp^T_g,j π^T_j v^T_j T

∈ R¹⁶ πj=mj

x²_g,j y²_g,j z²_g,j x_g,jy_g,j y_g,jz_g,j z_g,jx_g,jT

(34)

The regressor Φ is not always linearly independent or has a full rank, thus there might not be a unique solution.

Table3

Groupedparametersoftheredundantmanipulatorrobot.

Groupedparameters Inertialparameters

λ^∗₁ m1+m2+m3+m4

λ^∗₂ m2l²_c2+m3l²₂+m4l²₂+m5l²₂+I2zz

λ^∗₃ m3l²_c3+m4l²₃+m5l²₃+I3zz

λ^∗₄ m3lc3l2+m4l3l2+m5l3l2

λ^∗₅ m4l²_c4+m5l²₄+I3zz

λ^∗₆ m4lc4l2+m5l4l2

λ^∗₇ m4lc4l3+m5l4l3

λ^∗₈ m5

Nevertheless, itispossibletoreducethenumberof unknown parameters andtransform the systeminto: τ=Φ^*·λ^*, where λ^*isthevectorforregroupedparameters.Itishardtofindthe linearrelationbetweendependentandindependentcolumnsof theregressor,sincedynamicrelationshipsarecomplex.Tofind thisrelation,thekernelofthematrixisused,whichdenotesthe relationbetweencolumnsdependentandindependentfromthe regressorinlinearcombinations.Inthisway,thenewgrouped parametersλ^*andthefullrankmatrixΦ^*maybefound.The calculationisconductedinarecursiveway,startingfromthen jointuptothearticulation1.

Inthiscase,thevectorofregroupedparametersλ^*iscom- posedofeightelementstobeidentified,whicharepresentedin Table3.ThestructureoftheregressorΦ^∗ ∈ R^5×8isdescribed inAnnexB.

4.1.1. Excitationtrajectories

When asystemidentificationexperiment isdesigned,it is necessary toconsideratrajectory that sufficientlyexcites the redundantmanipulatorrobotparametersduringthemovement.

Otherwise,someparameterswillbeimpossibletoidentify, or will become too sensitive to noise (Swevers, Ganseman, De Schutter,&VanBrussel,1996).Meanwhile,sincetheregressor Φ^*dependsonjointcoordinatesanditstimederivatives,when themanipulatorrobotexecutesaparticulartrajectoryGDL·npts

equationswillbeobtained,generatinganover-determinedsys- tem.Thus,consideringthepointstraveledinthetrajectory,the systemisgivenbytheexpression:

τ =W·λ^∗ (35)

where τ is the torque/force vector ∈ R^GDL·npts, W is the ∈ R^{GDL·npts×np}systemobservationmatrix,λ^*isthevectorofthe groupedparameters∈ R^np,n_ptosisthenumberofsamplesand npcorrespondstothenumberofparametersgrouped.Then,to identifytheparametersitisnecessarytocalculate:

λ^∗=

W^TW₋₁

W^Tτ (36)

Parameters accuracy is closely related to the observation matrixW,sothepurposeistoobtainawell-conditionedmatrix thatallowsforenhancingtheestimationresults(Wu,Wang,&

You,2010).Thisimpliesfinding thedenominated “excitation trajectories”,whichcorrespondtoanoptimizationproblemthat

Fig.3.Referenceandoutputtrajectoriesoftheredundantmanipulatorrobotwith5DOL.

consistsinminimizingafunctionliketheonepresentedin(37) subjectedtotheconstraintsofEq.(38):

minf(q,˙q,¨q) (37)

q_min≤q≤q_max

˙qmin≤ ˙q≤ ˙qmax

¨qmin≤ ¨q≤ ¨qmax

(38)

where function f satisfies the criterion k(W )=(σmax/σmin), q,˙q,¨qarethegeneralizedcoordinatesofposition,velocityand acceleration,respectively,σ_maxisthehighestsingularvalueof Wandσ_minthelowestsingularvalueofW.

4.1.2. Trajectoryparameterization

Theprismaticjointsmotiondoesnotparticipateintherotary jointdynamics.Thus,withoutlossofgenerality,trajectoryopti- mizationisonlyappliedtorotaryjoints,sincetheyareinconflict withthe parameterestimation ofleast squares.The reverseis truefortheprismaticlinkwithlinearcharacteristics.Theexci- tationtrajectoriesareplannedusingapolynomialofdegreefive (Benimeli,2005)intheformofEq.(39)andtheelementstobe optimizedaretheircoefficients:a0i,a1i,...,a5i:

qi =a0i+a1it+a2it²+a3it³+a4it⁴+a5it⁵ (39) Referencesinusoidaltrajectoriesareappliedtothefirstand fifthlink(prismaticjoints).Allthereferenceandoutputtrajec- toriesof allthejoints ofthe redundantmanipulatorrobotare

showninFigure3.Anelectronicnoiseisobservedduetothe sensing(measuring)oftheoutputtrajectories.

4.2. Adalineneuralnetworks

Applying the linear model formulation to the parameters τ=Φ^*·λ^*(seeAnnexB),wehavethat:

F1=Φ^∗₁₁λ^∗₁+Φ^∗₁₈λ^∗₈

τ2=Φ^∗₂₂λ^∗₂+Φ^∗₂₃λ^∗₃+Φ^∗₂₄λ^∗₄+Φ^∗₂₅λ^∗₅+Φ^∗₂₆λ^∗₆+Φ^∗₂₇λ^∗₇ τ3=Φ^∗₃₃λ^∗₃+Φ^∗₃₄λ^∗₄+Φ^∗₃₅λ^∗₅+Φ^∗₃₆λ^∗₆+Φ^∗₃₇λ^∗₇ τ4=Φ^∗₄₅λ^∗₅+Φ^∗₄₆λ^∗₆+Φ^∗₄₇λ^∗₇

F5=Φ^∗₅₈λ^∗₈

(40)

Therefore, when using an Adaline network, the parame- terstobeidentifiedcoincidewiththenetworkweights.Below, one-layer neural networksaredesigned withtheir inputscor- responding tothe elementsof the regressorandtheir outputs totheelementsoftorque/forces.Figure4showstheschemeof thenetworksusedfortheidentificationoftheparametersofthe redundantmanipulatorrobot.

4.3. Hopfieldneuralnetworks

Forsatisfactorilyidentifyingtheparametersoftheredundant manipulatorrobot,allintervalsoft⊂ [0, +∞] shouldsatisfy the conditiongivenbyEq.(20).Figure5 showstheresponse

Φ^∗11 λ^∗1

F₁

τ

λ^∗8

λ^∗2

λ^∗3

λ^∗4

λ^∗5

λ^∗6

λ^∗7

Φ^∗18

Φ^∗22

Φ^∗23

Φ^∗24

Φ^∗25

Φ^∗26

Φ^∗27

a

b

Fig.4. Adalineneuralnetworkforidentification.(a)Network1.(b)Network2.

producedbytheHNN.Thelastestimatedvaluedisusedforthe comparison.

4.4. ExtendedKalmanfilter

In thismethod,use is made of the dynamic model of the redundantmanipulatorrobot,i.e.thejointcoordinatesthatmath- ematically depended on the force/torque applied tothem are used.Thedirectdynamicmodelissymbolicallycalculatedfrom the Lagrange–Euler formulation and the grouped parameters described in Table3,andthen isexpressed inthe state vari- ables presented in Eq. (41). The parameters to be identified areincluded inanextendedstateofthealgorithmwithatime derivativeofnullvalues,asshowninEq.(42):

x=

d1 ˙d1 θ2 ˙θ2 θ3 ˙θ3 θ4 ˙θ4 d5

˙d₅ λ^∗₁ λ^∗₂ λ^∗₃ λ^∗₄ λ^∗₅ λ^∗₆ λ^∗₇ λ^∗₈

_T (41)

˙x=

˙d1 ¨d1 ˙θ2 ¨θ2 ˙θ3 ¨θ3 ˙θ4 ¨θ4 ˙d5

¨d5 0 0 0 0 0 0 0 0

T

(42)

Fig.5.OnlineestimationofHopfieldneuralnetworks.

Fig.6.EvolutionoftheparametersestimatedbyextendedKalmanfilter.

Euler approximationis used forthe equations whose time derivativewascalculated,asseeninEq.(43):

˙x≈ x^k+1−x^k

t (43)

Therefore,fromtheequationabove,x^k+1=x^k+t·˙x^kis obtained,whosesamplingtimeist=0.02.Inturn,theinitial state vectorsused are initial errorcovariance matrix,process noisecovariancematrixandobservationcovariancematrix,as presentedinEq.(44):

x⁰=0,1·

1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1

_T

P⁰ ∈ R^18×18, P_ij=1 Q=diag

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1



R=10·diag

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

 (44)

Thereferencetrajectoriesusedinthisidentificationprocess arethesameappliedtotheleastsquaremethods,i.e.thosepre- sentedinFigure3.Theevolutionoftheparameterestimation

Table4

Estimatedparameters.

Parameter Leastsquares Adaline Hopfield Kalman

λ^∗₁ 4.2969 4.2027 4.2984 4.2966

λ^∗₂ 0.1094 0.1450 0.0822 0.0647

λ^∗₃ 0.0677 0.0500 0.0551 −0.0114

λ^∗₄ 0.0671 0.0652 0.0692 −0.0080

λ^∗₅ 0.0271 0.0259 0.0354 0.0742

λ^∗₆ 0.0260 0.0181 0.0578 0.0634

λ^∗₇ 0.0251 0.0459 0.0287 0.0859

λ^∗₈ 0.5114 0.6056 0.5136 0.5119

curvesisshowninFigure6.Inthiscase,thelastvalueof the estimationisconsideredforerrorcomparison.

5. Results

Table4showsthebaseparametersidentifiedforeachofthe studiedmethods.

5.1. Trajectoryvalidation

Tovalidatetheidentifiedparameters,theyareusedinperfor- mancesimulationsontheredundantmanipulatorrobot.Inthese simulations,validationtrajectoriesdifferentfromthoseusedin

Fig.7.Validationtrajectories.

the estimation are imposedto therobot. To relatethe trajec- toriesperformedbytheredundantmanipulatorrobotwiththe estimatedparametersof thesame,thefollowingerrorindexes arecalculated:ResidualMeanSquare(RMS),ResidualStandard Deviation(RSD)andAgreementIndex(AI);whicharemathe- maticallyrepresentedinEqs.(45)–(47),respectively.RMSand RSDcorrespondtonormalizedpercentagevaluesbetween0and 1;indexesthatexplainthedispersionanddeviationoftheseries, respectively.Inaddition,theAIindexindicatesthetrendofthe twoseriestobecompared;thevaluesthatadoptsmayfluctuate between0and1,where:1correspondstoaperfectmatchand 0 toabsence of agreement.Thetrajectories performedbythe redundantmanipulatorrobotareshowninFigure7,andtheir errorindexesinTable5.Itshouldbenotedthatthetrajectories presentedinFigure7arebasicallyoverlapped,andthattheval- uesoftheparametersdonotshowhighsensitivitytofollowing imposedtrajectories:

RMS=

n

i=1(oi−p_i)²

n (45)

RSD=

n

i=1(oi−p_i)²

_n

i=1(oi)² (46)

AI=1−

n

i=1(oi−p_i)²

_n

i=1o_i − p_i² o_i=oi−om

p_i=pi−om

(47)

Table5 Errorindexes.

Algorithm

Joint Leastsquares Adaline Hopfield Kalman

1 RMS 0.0000 0.0000 0.0000 0.0000

RSD 0.0000 0.0000 0.0000 0.0000

AI 1.0000 1.0000 1.0000 1.0000

2 RMS 0.0000 0.0000 0.0000 0.0001

RSD 0.0000 0.0001 0.0001 0.0002

AI 1.0000 1.0000 1.0000 1.0000

3 RMS 0.0000 0.0000 0.0000 0.0001

RSD 0.0000 0.0000 0.0000 0.0001

AI 1.0000 1.0000 1.0000 1.0000

4 RMS 0.0000 0.0000 0.0000 0.0001

RSD 0.0000 0.0000 0.0000 0.0002

AI 1.0000 1.0000 1.0000 1.0000

5 RMS 0.0000 0.0012 0.0000 0.0000

RSD 0.0000 0.0112 0.0003 0.0001

AI 1.0000 0.9998 1.0000 1.0000

where oi are the values observed, om the mean value of the observations,piarethepredictedvaluesandnisthetotalnumber ofobservations.

Todeterminethesensitivityoftheidentifiedmodel,duetothe variationsoftheestimatedparameters,theredundantmanipula- torrobotissimulated,consideringthatitsparametersvariedina restrictedrangeofvalues.Also,itsRMSerrorindexiscalculated betweenthetrajectoriesdescribed.Table6presentstherangeof valuesforthevariationsinthe±50%parameters.InFigure8,

Fig.8.Sensitivityofthemodeloftheredundantmanipulatorrobot.

Table6

Rangeofestimatedparameters.

Parameter Minimumvalue Maximumvalue

λ^∗₁ 2.1335 6.4004

λ^∗₂ 0.0547 0.1641

λ^∗₃ 0.0339 0.1016

λ^∗₄ 0.0334 0.1007

λ^∗₅ 0.0136 0.0407

λ^∗₆ 0.0130 0.0390

λ^∗₇ 0.0126 0.0377

λ^∗₈ 0.2557 0.7671

theevolutionoftheRMSindexof theperformedtrajectories, withrespecttoeachestimatedparameter,isillustrated.

6. Conclusions

Inthispaper,we presentedtheidentificationoftheparam- etersof anindustrial redundantrobotwith5 DOF.By means of the Lagrange–Euler formulation, the linear model with unknownparameterswasobtained.Toidentifytheseparameters, fourmethodswereused,assessedandcompared,namelyleast squares,Adalineneuralnetworks,Hopfieldneuralnetworks,and extendedKalmanfilter.

Theidentifiedparameterswerevalidatedbycomputationally simulatingtheperformanceoftheredundantmanipulatorrobot.

Tothisend,referencetrajectoriesdifferentfromthoseusedin theestimationwereimposedtotherobot.

Torelatethetrajectoriesdescribedbytheredundantmanipu- lator robotwith itsestimated parameters, the following error indexes were calculated: Residual Mean Square, Residual StandardDeviationandAgreementIndex.

Thesensitivityoftheidentifiedmodeltoparametervariation islow.

The parameters of the redundant manipulator robot were successfully identified and, therefore,the robotmathematical modelwasknown.Thankstothisachievement,itisnowpossi- bletodesign,simulate,compare,validateandimplementdiverse strategiesforcontrollingthisindustrialredundantrobotwith5 DOF.

Conflictofinterest

Theauthorshavenoconflictsofinteresttodeclare.

Acknowledgments

This work was supported by Proyectos Basales and the VicerrectoríadeInvestigación,DesarrolloeInnovaciónof the UniversidaddeSantiagodeChile,Chile.

AnnexA. Dynamicmodel

Thedynamicmodelofthemanipulatorrobotwasdeveloped in(Urrea&Kern,2016).Belowarepresentedthecomponentsof

theinertiamatrix,thecentrifugalforceandCoriolisforcevector andthegravityvector,respectively.

M₁₁=m₁+m₂+m₃+m₄+m₅ (A.1)

M₁₂=M₂₁=M₁₃ =M₃₁=M₁₄=M₄₁=M₂₅

=M₅₂=M₃₅ =M₅₃=M₄₅=M₅₄=0 (A.2)

M15=M51=−m5 (A.3)

M22=l²_c2m2+

l²₂+l²_c3+2l2lc3c3

 m3

+

l²₂+l²₃+ l²c4+2l2l₃c₃) m4+2 (l3l_c4c₄+l₂l_c4c₃₄) m4

+

l²₂+l²₃+l²₄+2l2l₃c₃ 

m₅+2 (l3l₄c₄+l₂l₄c₃₄) m5

+I_2zz+I_3zz+I_4zz (A.4)

M23=M32=

l²_c3+l2lc3c3

 m3

+

l²₃+l²_c4+l₂l₃c₃+ 2l3l_c4c₄) m4+ (l2l_c4c₃₄) m4

+ (l2l4cos34) m5+

l²₃+l²₄+l2l3c3+2l3l4c4

 m5

+I3zz+I4zz (A.5)

M₂₄=M₄₂=

l²_c4+l₃l_c4c₄+l₂l_c4c₃₄ 

m₄

+

l²₄+l3l4c4+l2l4c34

m5+I4zz (A.6)

M₃₃=l_c3²m₃+

l₃²+l²_c4+2l3l_c4c₄ 

m₄

+

l²₃+l²₄+2l3l4c4

m5+I3zz+I4zz (A.7)

M₃₄=M₄₃=

l²_c4+l₃l_c4c₄ 

m₄+

l₄²+l₃l₄c₄ 

m₅+I_4zz (A.8)

M44=l²_c4m4+l²₄m5+I4zz (A.9)

M55=m5 (A.10)

C11 =C15=0 (A.11)

C₁₂=−

l₂s₃· ˙θ3²+2l2s₃· ˙θ2˙θ₃

(lc3m₃+l₃m₄+l₃m₅) +

l₂s₃₄· ˙θ₃²−2 (l3s₄+l₂s₃₄) ˙θ3˙θ4

(lc4m₄+l₄m₅) +

l2s34· ˙θ2˙θ3−

˙θ₄²+2˙θ2˙θ4

(l2s34+l3s4) 

× (lc4m₄+l₄m₅) (A.12)

C₁₃=

l_c3m₃+l₃m₄+l₃m₅ l₂s₃· ˙θ2²

+

l₂s₃₄· ˙θ²2− 2

˙θ2+ ˙θ3

+ ˙θ4

l₃s₄˙θ4

(lc4m₄+l₄m₅) (A.13)

C₁₄= (lc4m₄+l₄m₅)(l3s₄+l₂s₃₄) ˙θ²₂ +

l3s4· ˙θ3²+2l3s4· ˙θ2˙θ3

(lc4m4+l4m5) (A.14)

G=

(m1+m2+m3+m4+m5) gz 0 0 0 −m5gz

T

(A.15) where s3=sinθ₃, s4=sinθ₄, c3=cosθ₃, c4=cosθ₄, s34=sin(θ3+θ₄) and c34=cos(θ3+θ₄); m1, m2, m3, m4, m5 andl1,l2,l3,l4,l5representthemassesandlengthof the first,second,third,fourthandfifthlink,respectively;lc2lc3and lc4expressthelengthsfromtheorigintothemasscenterofthe second,thirdandfourthlink,accordingly.Finally,l2zz,l3zzand l4zzrepresentthe momentsof inertiaof thesecond, thirdand fourthlinks,referredtothezaxisofeachjoint,respectively.

AnnexB. Linearmodelwithregroupedparameters BelowareshowntheelementsoftheregressorΦ^∗ ∈ R^5×8, obtained fromthelinearformulationwithgroupedparameters τ=Φ^*·λ^*:

τ =

F₁ τ₂ τ₃ τ₄ F₅_T

(B.1)

Φ^∗=

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

φ₁₁^∗ 0 0 0 φ^∗₁₅ 0 φ^∗₂₂ φ^∗₂₃ φ^∗₂₄ 0 0 0 φ^∗₃₃ φ^∗₃₄ 0

0 0 0 φ^∗₄₄ 0

0 0 0 0 φ^∗₅₅

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠

(B.2)

φ^∗₁₁= ¨d1+g (B.3)

φ^∗₁₅= ¨d1− ¨d5+g (B.4)

φ^∗₂₂= ¨θ2 (B.5)

φ^∗₂₃=

 ¨θ2+ ¨θ3

(2¨θ2+ ¨θ3)c3−(2˙θ2˙θ3+ ˙θ²₃)s3

T

(B.6)