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TwoGlobally ConvergentAdaptiveSpeed Observers for Mechanical Systems JoseGuadalupe Romero* a, RomeoOrtegab, 5 a Laboratoire d’Informatique,deRobotique etde Microélectronique deMontpellier, 161 RueAda, 34090 Montpellier,France 1 0 b Laboratoire des Signauxet Systémes, Supélec, Plateaudu Moulon,91192 Gif-sur-Yvette, France 2 n a J Abstract 4 A globally exponentially stable speed observer for mechanical systems was recently reported in the literature, under the ] assumptions of known (or no) Coulomb friction and no disturbances. In this note we propose and adaptive version of this S observer, which is robust vis–à–vis constant disturbances. Moreover, we propose a new globally convergent speed observer D that, besides rejecting the disturbances, estimates some unknown friction coefficients for a class of mechanical systems that . containsseveral practical examples. h t a m Keywords: Mechanical systems,adaptiveobservers, robustness. [ 1 1 Introduction (ZRS) [11,14,16], we propose a new adaptive speed ob- v 9 serverthat,besidesrejectingthedisturbances,estimates 2 Thedesignofspeedobserversformechanicalsystemsisa some of the unknown friction coefficients. It should be 7 noted that in this case there are products of unmea- problemofgreatpracticalimportancethathasattracted 0 surablestates and unknownparameters,a situation for theattentionofresearchersforover25years—thereader 0 which very few results are available in the observer de- isreferredtotherecentbooks[1,3]foranexhaustivelist 1. ofreferences.Thefirstgloballyexponentiallyconvergent sign literature—even for the case of linear systems. A 0 speed observer for general, simple mechanical systems similartransformationhasbeenpresentedin[4,5],where 5 acoordinatetransformationthatremovesthequadratic was recently reported in [2], where the Immersion and 1 termsinvelocityisfoundto solvechallengingposition– Invariance (I&I) techniques developed in [1] were used. : feedbacktrackingproblemsforsurfaceshipsandmobile v Although the observer of [2] considers the case of sys- robots. i temswithnon–holonomicconstraints,itreliesontheas- X sumptionsofnofrictionandtheabsenceofdisturbances. r In[13] this speed observerwasredesignedto accommo- Thepaperisorganizedasfollows.Thetwoadaptiveob- a datethepresenceofknownCoulombfrictionanditwas servationproblemsaddressedinthepaperarepresented usedtodesignauniformlygloballyexponentiallystable in Section 2. The standing assumptions and a prelim- trackingcontrollerusing only positionfeedbackfor me- inary lemma are given in Section 3. In Section 4 we chanicalsystems (withoutnon–holonomicconstraints). presentanadaptiveobserverforsystemswithunknown frictionanddisturbances.InSection5the I&Iobserver In this paper we propose two new robust velocity ob- of [13] is redesigned to accommodate the possible pres- serversformechanicalsystems(withoutnon–holonomic ence ofdisturbances andknownfriction.Some physical constraints). First, we add an adaptation stage to the examplesaregiveninSection6.Thepaperiswrapped– observerof[2]torejectconstantinputdisturbances.Sec- upwithsomefuture researchinSection7. ond,formechanicalsystemswithzeroRiemannsymbols Notation. To avoid cluttering the notation, throughout ⋆ thepaperκandαaregenericpositiveconstants.I isthe Corresponding author J.G. Romero Tel. +33 4 67 14 95 n 68. n nidentitymatrixand0n sisann smatrixofzeros, Emailaddresses: [email protected] 0n×is an n–dimensional col×umn vect×or of zeros. Given (Jose GuadalupeRomero* ),[email protected] ai R, i n¯ := 1,...,n , we denote with col(ai) the ∈ ∈ { } (RomeoOrtega). n–dimensionalcolumnvectorwithelementsai.Forany Preprintsubmitted toAutomatica 6 January 2015 matrix A Rn n, (A) Rn denotes the i–th column, The problem is to design a globally convergent robust × i (A)i the i–∈throw and(A∈) the ij–th element. Thatis, adaptiveobserverforthemomentap.The maincontri- ij withe Rn, i n¯,theEuclideanbasisvectors,(A) := butionsofthe paper arethe following. i i Ae , (A∈)i := e∈A and (A) := e Ae . For x Rn, S i Rn×n, S =⊤iS⊤ >0,weidjenote⊤ithe Ejuclidean∈norm (i) For systems with ZRS design a new observer that is G|xi|∈v2en:=axfu⊤nxc,tiaonndfth:eRnweighRtewd–endoerfimnekxthke2Sd:i=fferxe⊤nStixa.l gtulorbbaalnlycecdonavnedrgseonmteinusnpkinteowofntfhreicptiroensecnoceeffiocfitehnetsdris-. → i operators (ii) If the friction is known robustify the observer of [2] torejectthe disturbanced. ∂f ⊤ ∂f ⊤ f := , f := , ∇ ∂x ∇xi ∂x (cid:18) (cid:19) (cid:18) i(cid:19) Remark2.1 The qualifier “some" in item (i) is essen- wherex Rp isanelementofthevectorx.Foramap- tial because, as will become clear later, except for the i pingg :R∈n Rm,itsJacobianmatrixis definedas case of constant inertia matrix, we will not be able to → considerthepresenceofunknownfrictioninallgeneral- izedcoordinates. ( g ) 1 ⊤ ∇ g := ... , Remark2.2 Notice that the objective is to observe ∇ only the momenta (equivalently, the velocity) not to   ( gm)⊤ ensure consistent estimation of the parameters d and  ∇  r:=col(r ).Asiswell–knownintheidentificationliter-   i whereg :Rn R isthe i-thelementofg. ature, a necessary condition for parameter convergence i → isthatthesignalssatisfyapersistencyofexcitationcon- dition[10].Inthisrespect,wenoticethatthesystem(1), 2 FORMULATIONOFTWOROBUSTSPEED (2),whichcanbe representedin the statespaceform OBSERVATION PROBLEMS q˙=v In the paper we consider n–degrees of freedom, per- v˙ =U(q,v) Rv+G(q)u+d − turbed, simple, mechanical systems described in port– r˙ =0 Hamiltonian(pH) form[15]by d˙=0, q˙ 0 I 0 0 n = H(q,p)+ u+ (1) "p˙ # " In R#∇ "G(q)# " d # for some U : Rn Rn Rn, with measurementq does − − × → not satisfy the observability rank condition [7] at zero withtotalenergyfunction H :Rn Rn R velocity, hampering the observation of the parameters × → r.1 1 H(q,p)= p M 1(q)p+V(q), (2) 2 ⊤ − Remark2.3 See Remark1in [12]fora physicalinter- pretationofthedisturbancesdthat,weunderscore,en- where q,p Rn are the generalized positions and mo- teratthe levelofthe momenta. menta, resp∈ectively, u Rm is the control input, G : Rn Rn m is the in∈put matrix, the inertia matrix × M :→Rn Rn n verifies M(q) = M (q) > 0 and 3 ASuitablepH Representation × ⊤ V : Rn →R is the potential energy function. As cus- → tomaryintheobserverliterature,itisassumedthatthe Asshownin [16],the changeofcoordinates control signal u(t) is such that trajectories exist for all t 0. ≥ (q,p) (q,T⊤(q)p), 7→ The systemis subjecttotwodifferentperturbations. whereT :Rn Rn n is afull rankfactorizationofthe × → - Unknown constantdisturbancesd=col(d ) Rn. inverseinertiamatrix,that is, i ∈ - Coulombfrictioncapturedby M 1(q)=T(q)T (q), (4) − ⊤ R=diag r ,r ,.,r Rn n, (3) 1 2 n × { }∈ 1 The authors thank Prof. Witold Respondek for this in- withunknown ri 0, i n¯. sightfulremark. ≥ ∈ 2 transforms(1)into (iii) Thereexistsa mappingQ:Rn Rn suchthat → q˙ 0 T(q) Q(q)=T−1(q). (9) = W(q,p) ∇ "p˙# " T⊤(q) J(q,p) R(q)#∇ Remark4.1 Mechanical systems verifying Assump- − − tion 4.1 have been extensively studied in analytical 0 0 + u+ , (5) mechanics and have a deep geometric significance— "T⊤(q)G(q)# "T⊤(q)d# stemming from Theorem 2.36 in [11]. They belong to the class of systems that are partially linearizing via change of coordinates studied in [16]—see that paper withnew HamiltonianW :Rn Rn R forsomeadditionalreferences. × → 1 W(q,p)= p2+V(q), 4.2 Assumptionson friction 2| | the jk element of the skew–symmetricmatrix J : Rn Todesignourrobustadaptiveobserver,besidesAssump- Rn Rn n givenby × tion 4.1, a restriction on the friction coefficients is im- × → posed.Namely,weassumethatthere ares,with s n, ≤ unknown coefficient and decompose the friction matrix J (q,p)= p [(T) ,(T) ], (6) jk − ⊤ j k R(3)as with [, ] the standard Lie bracket [14] and the trans- R=Rk+Ru formed· f·rictionmatrix whereRk,Ruaren ndiagonalmatricescontainingthe × knownandtheunknownfrictioncoefficientsrespectively. R(q):=T (q)RT(q) 0. (7) Asaworkingexampleconsiderthecasen=3ands=2 ⊤ ≥ with Remark3.1 One possible choice of the factorization (4) is the Cholesky factorization [6,8]. But, as will be- Rk =diag 0,r2,0 , Ru =diag r1,0,r3 { } { } comeclearbelow,otherchoicesmayprovemoresuitable forthe solutionofthe problem. Similarly,withanobviousdefinition,wedecomposethe transformedfrictionmatrix(7)into 4 ROBUST OBSERVER FOR A CLASS SYS- R(q)=R (q)+R (q). TEMS WITH FRICTION AND DISTUR- k u BANCES Tostreamlinethepresentationallfrictioncoefficientsare groupedinavectorr =col(r ) Rn withthe unknown Inthissectionwesolvetheproblemofrobustobservation andknowncoefficientsinvectoirs∈r Rsandr Rn s, of momenta in the presence of disturbances and some u ∈ k ∈ − respectively.Thus,forourworkingexample wehave unknown friction coefficients for a class of mechanical systems. r =col(r ,r ,r ), r =r , r =col(r ,r ). 1 2 3 k 2 u 1 3 4.1 Assumptionon M(q)and apreliminary lemma We also define a set of integersκ n¯ that contains the ⊂ indices of the unknown coefficients of r, which in the Assumption4.1 M−1(q) admits a factorization (4) exampleis κ= 1,3 . withafactorT(q)verifying { } Finally,we define amatrixC Rn s suchthat × ∈ T(q) , T(q) =0, i,j n¯. (8) i j ∈ C r =r . (10) h(cid:16) (cid:17) (cid:16) (cid:17) i ⊤ u Clearly,the matrixC verifies: Instrumental for the developments of this paper is the followingresult,whoseproofmaybe foundin[16]. rank C =s. • { } Forj κ,(C) =e . Lemma4.1 The followingstatements areequivalent: • ∈ j κj Inourexample (i) M(q)satisfiesAssumption4.1. (ii) The RiemannsymbolsofM(q)areallzero.2 1 0 C =0 0. 2 Seeequations(6)and(7)of[16]forthedefinitionofthese symbols. 0 1     3 The following assumption and lemma are instrumental 4.3 First robustmomentaobserver forourfuture developments. Proposition1 Considerthe system(1),(2)wherethe Assumption4.2 The i–th row of factor T(q) is inde- inertia matrix M(q) and the friction matrix R verify pendent ofq for i κ.3 Assumptions 4.1 and 4.2. The 2n+s dimensional I&I ∈ adaptivemomenta observer Lemma4.2 Under Assumption 4.2, there exists con- stotarnstzm=atcroilc(ezs)Yj ∈RnRwn×esh,ajve∈ n¯, such that, for all vec- p˙I =−T⊤(q)[∇V −G(q)u−dˆ] i n ∈ ( Y pˆ)rˆ [λQ(q)+R (q)]pˆ i i u k n − − i=1 R (q)z =( Y z )r . (11) X u j j u n Xj=1 r˙uI =( Yi⊤pî)(p˙I +λpˆ) i=1 Proof4.1 From(10)itfollowsthat X d˙ =T(q)pˆ I n pˆ=pI +λQ(q) Ru = eie⊤i (e⊤i Cru). (12) 1 s rˆ =r + ( pˆ L pˆ)e Xi=1 u uI 2λ ⊤ i i i=1 X Usingthe definitionofRu(q)we get dˆ=dI +q n pˆ =T−⊤(q)pˆ R (q)z=T (q) e e (e Cr ) T(q)z, u ⊤ i ⊤i ⊤i u n "Xi=1 # wgiivtehntihne(c9o)n,sYtiantRnn××sn,miatrn¯icgeisveLni igniv(e1n4)byan(d15λ),>Q0(qa) ∈ ∈ = T (q)e e T(q)ze Cr freeparameter,ensuresboundedness ofallsignalsand ⊤ i ⊤i ⊤i u i=1 X lim[pˆ(t) p(t)]=0. (16) n n t − →∞ = T (q)e e T(q) e z e Cr . (13)  ⊤ i ⊤i  j j ⊤i u forallinitialconditions (q(0),p(0)) Rn Rn. Xi=1 Xj=1 ∈ ×   Proof4.2 Let the observation and parameter estima- Hence,(11)followsswappingthe sums anddefining tionerrorsbe definedas n p˜=pˆ p Y := T (q)e e T(q)e e C j ⊤ i ⊤i j ⊤i − r˜ =rˆ r (17) Xi=1 u u− u = L e e C, j n¯ (14) d˜=dˆ d. (18) i j ⊤i ∈ − X Following the I&I adaptive observer procedure [1] we withmatricesL definedas i propose to generate the estimates as the sum of a pro- portionalandanintegralterm,thatis, L :=T (q)e e T(q), i n¯ (15) i ⊤ i ⊤i ∈ pˆ=p +p (q) I P rˆ =r +r (pˆ) (19) ItonlyremainstoprovethatthematricesYj andLiare u uI uP constant. Towards this end we refer to (14) and notice dˆ=d +d (q), (20) I P that,inviewofAssumption4.2,theterme T(q)iscon- ⊤i stantfori κwhiletheterme C isan1 szerovector wherethe mappings ∈ ⊤i × fori / κ.Completingthe proof. ∈ p :Rn Rn P → Remark4.2 As indicated in Remark 2.1, except for r :Rn Rs uP → the case when M (and, consequently, the factor T) are d :Rn Rn, P constant, to satisfy Assumption 4.2 we have to assume → thatsomeoftheelementsofr areknown.SeeSection6 andtheobserverstatesp ,d Rn andr Rswillbe forsomephysicalexamples. definedbelow.4 I I ∈ uI ∈ 3 Consequently,theunknownfrictioncoefficientsarelocated 4 The reason for the particular selection of the arguments in theserows. oftheproportional termswill becomeclear below. 4 First,westudythedynamicbehaviorofpãndcompute yields d˜˙= d T(q)p˜. (24) p˜˙ =p˙ + p T(q)p+T (q)[ V G(q)u] −∇ P I ∇ P ⊤ ∇ − We will now analyze the stability of the error model +[Rk(q)+Ru(q)](pˆ p˜) T⊤(q)(dˆ d˜), (22),(23)and(24)withtheaidoftheproperLyapunov − − − functioncandidate where we have invoked Assumption 4.1 that ensures— via(6)and(8)—thatJ(q,p)=0,andused(17)toobtain V(p˜,d˜,r˜ )= 1(p˜2+ d˜2+ r˜ 2). (25) thetermsinthesecondrow.Invoking(11)wecanwrite u 2 | | | | | u| n Takingits time-derivativewe obtain R (q)pˆ=( Y pˆ)r . u i i u i=1 n X V˙ = p˜ R(q)+λI p˜ p˜ ( Y pˆ)r˜ T (q)d˜ ⊤ n ⊤ i i u ⊤ − − − Hence,proposing h i (cid:2) Xi=1 (cid:3) λr˜ r +d˜ d T(q) p˜. (26) p˙I = pPT(q)pˆ T⊤(q)[ V G(q)u] − u⊤∇ uP ⊤∇ P −∇n − ∇ − (cid:2) (cid:3) Clearly,ifthemappingsr (pˆ)andd (q)solvethepar- Y pˆ rˆ R (q)pˆ+T (q)dˆ, (21) uP P i i u k ⊤ tialdifferentialequations(PDEs) − − (cid:16)Xi=1 (cid:17) n yields 1 r = ( Y pˆ) ∇ uP −λ i⊤ i (27) i=1 n X p˜˙ = [R(q)+ qpPT(q)]p˜ ( Yipî)r˜u+T⊤(q)d˜ ∇dP =In, − ∇ − i=1 n X onegets = [R(q)+λI ]p˜ ( Y pˆ)r˜ +T (q)d˜, (22) n i i u ⊤ − − Xi=1 V˙ =−p˜⊤[R(q)+λIn]p˜≤−λ|p˜|2. (28) whereto obtainthe secondequationswe haveselected From (25), (28) we conclude that p˜ and 2 d˜,r˜ . Doing some standard sig∈naLl ch∩asLin∞g it is u pP(q)=λQ(q), straig∈htfLor∞wardto provefromherethat(16)holds. withQ(q)givenin(9). Motivatedbytheconclusionaboveletusnowstudythe PDEs(27).Thesecondonehasthetrivialsolutiond = P Now,the time derivativeofr˜u is givenas q. Regarding the first one, it is clear that the elements ofthe mappingr (pˆ)mustbe ofthe quadraticform r˜˙ =r˙ + r pˆ˙ uP u uI ∇ uP =r˙ + r [p˙ + p T(q)p] 1 =r˙uI +∇ruP[p˙I +∇λ(pˆP p˜)]. (ruP(pˆ))i = 2λpˆ⊤Lipˆ, i∈s¯, uI ∇ uP I − Hence,choosing with constant, symmetric matrices Li Rn×n. Replac- ∈ ingthe expressionaboveinthe PDE (27)yields r˙ = r (p˙ +λpˆ), uI −∇ uP I pˆ L ⊤ 1 yields . n r˜˙u =−λ∇ruPp˜. (23)  .. =− Yi⊤pî. Finally,the time derivativeofd˜is givenas pˆ⊤Ls Xi=1     d˜˙=d˙ + d T(q)p Thatleadus tothe solution I P ∇ =d˙ + d T(q)(pˆ p˜). I P ∇ − L⊤1ej ... L⊤sej =−Yj, j ∈n¯. (29) Hence,choosing h i d˙ = d T(q)pˆ, It only remains to show that the resulting matrices L I P i −∇ 5 aresymmetric.From(29)weget whosesolutionis notobvious. −L⊤j =− L⊤j e1 L⊤j e2 ... L⊤j en 5 ROBUST OBSERVER FOR GENERAL h i PERTURBED MECHANICAL SYSTEMS = Y e Y e ... Y e 1 j 2 j n j WITH KNOWNFRICTION h i Clearly,the matrixL is symmetric ifandonly if j InthissectionweredesigntheI&Ispeedobserverof[13], see also [2], to ensure its global convergence in spite of e Y =e Y , i,k n¯, i=k. ⊤i k ⊤k i ∀ ∈ 6 the presence of the unknown disturbances d and known frictionforcesinallcoordinates. This factcanbe easilyverifiedusing (14) Proposition2 Consider the system (1), (2) with n e⊤kYi=e⊤k T⊤(q)eje⊤j T(q)eie⊤j C known frictionmatrixR.Thereexistsmoothmappings j=1 Xn A:R4n R 0 Rn Rn R4n+1 =e⊤i T⊤(q)eje⊤j T(q)eke⊤j C B:R4n×R≥0×Rn× Rn→ j=1 × ≥ × → X =e⊤i Yk. suchthatthe interconnectionof (1),(2)with X˙ =A(X,q,u) Replacing all the derivations above in p˙ , d˙ and r˙ (31) I I uI pˆ =B(X,q), givesthe equationsgivenin the propositioncompleting the proof. where X R4n R , pˆ Rn, ensures (16) holds for 0 Remark4.3 If Assumption 4.1 is not imposed a term allinitial∈conditi×ons≥ ∈ J(q,p)p appears in the error equation (22) and (23). Eventhoughthistermisquadraticintheunknownstate (q(0),p(0),X(0)) Rn Rn R4n R . 0 p, the properties of J(q,p) can be used to handle this ∈ × × × ≥ term in the first error equations—this is done in the Thisimpliesthat,inspiteofthepresenceoftheunknown secondobserverinthenextsection.However,thereisno disturbances d, (31) is a globally convergent momenta obviouswaytocreateasuitableerrortermforthesecond observerforthemechanicalsystemwithfriction(1),(2). errorequationstymingtherelaxationofAssumption4.1. Remark4.4 If the matrices Y are not constant the i Proof5.1 Theconstructionoftheobserverfollowsvery first PDE in (27) does not admit a solution, hence As- closely the one reported in [13] with the only difference sumption 4.2 is required. It can be shown that making of the inclusion of an adaptation law for the unknown r function ofq doesnotsolvethe problem,because a uP disturbanceparametersd.However,forthesakeofcom- quadraticfunction ofpˆwillappearinV˙. pleteness,a detailedderivationofallthe steps isgiven. Remark4.5 As showninProposition6of[16]the dy- Define the estimationerrors namicsofmechanicalsystemssatisfyingAssumption4.1 expressedinthe coordinates(Q,p)takethe form p˜=pˆ p − (32) Q˙ =p d˜=dˆ−d. p˙ =−R˜(Q)p−T˜⊤(Q)[∇V˜(Q)−G˜(Q)u+d], (30) Following the I&I adaptive observer procedure [1] we proposeto generatethe estimates as where (˜)(Q) := ()(QI(Q)), with QI : Rn Rn a left inverse·of Q(q), th·at is, Q(QI(z)) = z for→all z Rn. pˆ:= p +p (q,q|,|p) ∈ I P Althoughthe constructionofanobserverfor(30)when (33) the friction is knownis straightforward,the case of un- dˆ:= dI +dP(q,r) knownfrictionisfarfromtrivial.ApplyingtheI&Ipro- cedure used in Proposition 1 leads, for the definition of dP(q), to aPDE ofthe form where the mappings pP : Rn Rn Rn Rn and d Rn,andthe observerstate×s d ×Rn,p→ Rn and P I I S(Q)=T˜(Q), r ∈R aredefined suchthat(16)hold∈s. ∈ ∇ ∈ 6 We,therefore,studythedynamicbehaviorofpãndcom- Now, since the term in brackets in (38) is equal to zero pute if|p=pândq|=q,there existmappings p˜˙ =p˙I +∇qpPq˙+∇q|pPq|˙ +∇|ppP|p˙− ∆q,∆p :Rn×Rn×Rn →Rn×n J(q,p)p+T (q)[ V G(q)u]+R(q)p T (q)d. ⊤ ⊤ − ∇ − − verifying In[2]ithasbeenshownthatthemappingJ(q,p)defined ∆ (q,|p,0)=0, ∆ (q,|p,0)=0, (39) in(6)verifiesthe followingproperties: q p andsuchthat (P.i) J(q,p)is linearinthe secondargument,thatis (q,pˆ) (q|,|p)=∆ (q,q|,e )+∆ (q,|p,e ), (40) J(q,α1p+α2p¯)=α1J(q,p)+α2J(q,p¯) H −H q q p p where forallq,p,p¯ Rn,andα ,α R. (P.ii) Thereexistsam∈appingJ¯:R1n 2R∈n Rn×n satisfy- eq :=q|−q, ep :=|p−pˆ. (41) × → ing J(q,p)p¯=J¯(q,p¯)p. Substituting (36),(38)and(40)in(35),yields Hence,proposing p˜˙ =[J(q,p) ψI R]p˜ n − − p˙ := p q|˙ p |p˙ +J(q,pˆ)pˆ R(q)pˆ + ∆q(q,q|,eq)+∆p(q,|p,ep) T(q)p˜+T⊤(q)d˜. I q| P |p P −∇ −∇ − − (cid:16) (cid:17) T(q)⊤(q) V +v qpPT(q)pˆ+T⊤(q)dˆ, Themappings∆ ,∆ playtheroleofdisturbancesthat − ∇ −∇ q p (34) are dominated with a dynamic scaling and a proper choiceoftheobserverdynamics.For,definethedynam- togetherwithProperties(P.i) and(P.ii) yields icallyscaledoff–the–manifoldcoordinate p˜˙ =[J(q,p)+J¯(q,pˆ) R(q) qpPT(q)]p˜+T⊤(q)d˜. η = 1p˜, (42) − −∇ (35) r Itis clearthatif the mappingp solvesthe PDE P where r is a scaling factor to be defined. The dynamic qpP =[ψIn+J¯(q,pˆ)]T−1(q), behaviorofη is givenby ∇ 1 r˙ withψ >0adesignconstant,thep˜–dynamicsreducesto η˙ =(J R ψI)η+(∆q+∆p)Tη+ T⊤d˜ η, (43) − − r −r p˜˙ =[J(q,p) ψIn R(q)]p˜+T⊤(q)d˜. where, for brevity, the arguments of the mappings are − − omitted. Recalling that J(q,p) is skew–symmetric and R(q) 0 ≥ the unperturbed part of the error dynamics above, i.e. Mimicking[2]selectthe dynamics ofq|,|pas whend˜=0,is exponentiallystable. q|˙ =T(q)pˆ ψ e 1 q Similarly to [13], to avoid the solution of the PDE, the − |p˙ = T (q) V +v+J(q,pˆ)pˆ Rpˆ (44) dynamic scaling technique is used. Towards this end, ⊤ − ∇ − define the mapping ψ e +T (q)dˆ 2 p ⊤ − H(q,pˆ):=[ψIn+J¯(q,pˆ)]T−1(q). (36) where ψ1,ψ2 are some positive functions of the state definedlater.Using (44),togetherwith(41),weget anddefine p as P e˙ =T(q)ηr ψ e q 1 q − (45) pP(q,q|,|p):=H(q|,|p)q. (37) e˙p =∇qpPT(q)ηr−ψ2ep. Thechoiceaboveyields qpP = (q|,|p),whichmaybe Moreover,selectthe dynamics ofr as ∇ H writtenas ψ r r˙ = (r 1)+ ( ∆ T 2+ ∆ T 2), r(0) 1, (46) qpP = (q,pˆ) [ (q,pˆ) (q|,|p)]. (38) −4 − ψ k p k k q k ≥ ∇ H − H −H 7 with the matrix induced 2–norm.At this point we whichclearlysolvesthe PDE k·k makethe importantobservationthatthe set 1 r R:r 1 r qdP(q,r)=0. { ∈ ≥ } r − ∇ isinvariantforthedynamics(46).Hence,r(t) 1, t ≥ ∀ ≥ Itonlyremainsto study the termr˜r˙,whichisgivenby 0. Onother hand,taking the time-derivativeofd˜,weget r˜r˙ = ψr˜2+r˜r( ∆ T 2+ ∆ T 2). p q −4 ψ k k k k d˜˙=d˙ + d T(q)p+ d r˙ I ∇q P ∇r P Now, (39) ensures the existence of mappings ∆¯ , ∆¯ : =d˙I +∇qdP(q,r)T(q)(pˆ−rη)+∇rdPr˙ Rn Rn Rn Rn×n suchthat q p × × → andchoosing ∆ (q,|p,e ) ∆¯ (q,|p,e ) e q q q q q k k≤k k| | d˙I =−∇qdPT(q)pˆ−∇rdPr˙, (47) k∆p(q,|p,ep)k≤k∆¯p(q,|p,ep)k|ep|. the d˜–dynamicstakethe form Hence d˜˙= r qdPT(q)η (48) ∆pT 2+ ∆qT 2 T 2 ( ∆¯p 2 ep 2+ ∆¯q 2 eq 2). − ∇ k k k k ≤k k | k k | | | k | | Setting Wenowanalyzetheerrorsystem(42),(45),(46),(48)— with the coordinate r˜= (r 1). For, define the proper Lyapunovfunctioncandidat−e. ψ3 =κ rr˜ ψ = T 2 ∆¯ 2+κ V(η,e ,e ,r˜,z ):= 1 η 2+ e 2+ e 2+r˜2+ d˜2 . 4 4(1+ψ3)k k k qk q p a q p 2 | | | | | | | | rr˜ (cid:16) (4(cid:17)9) ψ5 = T 2 ∆¯p 2+κ, 4(1+ψ )k k k k Takingits time-derivativeweobtain 3 onegets ψ 1 V˙ 1 η 2 ψ r2 T(q) 2 e 2 1 q ≤− 4 − | | − − 2 k k | | (cid:18) (cid:19) (cid:18) (cid:19) V˙ κ(η 2+ e 2+ e 2+r˜2] (52) ψ 1r2 p 2 T(q) 2 e 2+r˜r˙+ ≤− | | | q| | p| 2 q P p − − 2 k∇ k k k | | (cid:18) (cid:19) forsomepositiveconstantκ. 1 +d˜⊤ r qdP T(q)η r − ∇ (cid:18) (cid:19) The proof is completed invoking the arguments of [13], selectingthe observerstateas Clearly,ifweset ψ =4(1+ψ ), ψ = 1r2 T(q) 2+ψ (50) X:=(q|,|p,pI,dI,r−1), 3 1 4 2 k k and defining A(X,q,u) from (34), (44), (46) and (47) and andsetting B(X,q) via(33). 1 ψ = r2 p 2 T(q) 2+ψ , 2 q P 5 2 k∇ k k k Remark5.1 Itisclearfromtheproofthatthekeystep where ψ ,ψ ,ψ are positive functions of the state de- 3 4 5 to reject the disturbances is to make the proportional finedbelow,one gets term of the parameter estimator, d a function of the P V˙ ψ η 2 ψ e 2 ψ e 2+r˜r˙ dynamicscalingfactorr, see(51). 3 4 q 5 p ≤− | | − | | − | | 1 +d˜ r d (q,r) T(q)η. ⊤ q P r − ∇ 6 PHYSICAL EXAMPLES (cid:18) (cid:19) To eliminate the cross term appearing in the last term Inthissectionwepresentthreephysicalmechanicalsys- ofthe righthandsideaboveweselect tems that satisfy the conditions of Proposition 1. Con- 1 sequently,robustadaptivespeedobservationispossible dP(q,r)= r2q, (51) forthem. 8 6.1 Constantinertia matrix andthedefinitionofallconstantsmaybe foundin[16]. TheCholeskyfactorizationisgivenas In the case of constant inertia matrix Assumption 4.1 is trivially satisfied, because the factor T can be taken 1 0 0 0 to be constant. Assumption 4.2 is also satisfied with √I dim(q) = n and C = In, hence all friction coefficients T(q)= √−1I a12 0 0 , canbe identified.  0 M21 1 S 1 0   − m a3 12 a3  Gsoilvveens(a9n)yiscognivsetannbtyfactor T, the vector field Q(q) that  0 qMm21 a13C12 0 a13   q  andthe vectorfield Q(q)thatsolves(9)is Q(q)=T 1q. − √Iq Finally,from(14)and(15)weget 1  a2(q1+q2)  n Q(q)= . M1 Yj = T⊤(q)eie⊤i T(q)eje⊤i j ∈n¯ −a2 m2C12+a3q3 Xi=1  a qM21S +a q  Li =[(T)i]⊤(T)i, i∈n¯. − 2q m 12 3 4 AmatrixC thatsatisfiesAssumption4.2is 6.2 Planarredundantmanipulatorwithoneelasticde- greeof freedom I 2 C = . "02 2# This is a 4-dof underactuated mechanical system de- × pictedin1.Theinverseinertiamatrixis givenby Hence, we can consider as unknown the frictions in the elasticcoordinater andthe revolutejointr . 1 2 Finally, q4 m 1 0 0 0 Y = I 1⊤  1 a2 0 0 L I −√I   q2 0 0 0 0 Y = 2⊤  a2 a2 0 0 q1 −√I 2 M  1 0  I 0 2 2 L1 ="0 0# ×  q3  02 2 02 2  × ×  Fig.1.Planarredundantmanipulatorwithoneelasticdegree   1 a2 of freedom. I −√I 0 2 2 L2 =−√a2I a22  ×  1I −I1 0 0  02×2  02×2   M−1(q)=∗ a122 + I1 −m1ℓS12 m1ℓC12, 6.3 2D-Spider cranegantrycart ∗ ∗ a123 0   1  ∗ ∗ ∗ a23  This is a 3-dof underactuated mechanical system de-   pictedinFig.2.The inertiamatrixis wherewedefined m +m 0 mL C S :=sin(q +q ), C :=cos(q +q ) r 3 3 12 1 2 12 1 2 M(q)= m +m mL S , √Mmℓ ∗ r 3 3 a2 := √m+M, a3 :=√M +m,  ∗ ∗ mL23    9 Fy Fromthe definitionofT(q)in(53)itis clearthat (xr, xy) mr C⊤ = 0 0 1 h i F x satisfiesAssumption4.2.Hence,we canconsiderasun- knownthe frictionparameterr . 3 Y q 3 L3 Finally,from(14)and(15)we get X m Y1⊤ = 0 0 c2 , h i 0 0 0 Fig. 2.2D-Spidercrane gantrycart L1 =0 0 0 . withinverse 0 0 c2   mr+mC32 mC3S3 C3   (mr+m)mr (mr+m)mr L−3mr We show several simulations of the proposed observer. M−1(q)= ∗ m(mr+rm+m−)mmCr32 −mSr3L3  Ttrhoel isnypsutetsm, thhaasttihs,eutw=ocfoolr(cFes,sFho)wanndincFonigs.ta2natsincpount-  ∗ ∗ mmrrL+23mm  matrixGofthe form x y   where,to simplify the notation,wehavedefined 1 0 G=0 1. S :=sin(q ),C :=cos(q ), 3 3 3 3 0 0   and the definition of all constants may be found in [9].   AnuppertriangularfactorizationofM 1(q)isgivenas Theparametersaretakenasmr =0.5kg forringmass, − m=1kgforpayloadmassandL =0.5mforthecable 3 length. We fix the control inputs as F = 1.535cos(t) x a 0 −bC3 and Fy = 7.67sin(t). The disturbances are taken as d = col(0.1,0.2,0.2) and the friction coefficients r = T(q)=0 a bS , (53) 3 − col(0,0,0.5),with r being unknown. 3 0 0 c      The transient behavior of the error signals p˜,r˜ and d˜ 3 wherewedefined the constants with the tuning parameter λ = 0.8 and different initial conditionsofd ,r andP areshowninFig.3andFig.4. I 3I I Asseenfromthefigures,besidestheconvergencetozero 1 1 m +m r a:= ,b:= ,c:= . ofthemomentaestimatepredictedbythetheory,wealso √m +m cL m mL2m r 3 r s 3 r observethattheestimatedparametersconvergetotheir truevalue—assessingthefactthatthesignalschosenfor We can check that the columns of T(q) satisfy (8) and the simulationarepersistentlyexciting. thus the system verifies Assumption 1. Taking the in- verseofT(q)weget To evaluate the effect ofthe tuning parameterλ onthe transient behavior we also show in Fig.5 and Fig.6 the transientbehavioroftheerrorsignalsfordifferentvalues 1 0 aL mC a 3 3 ofλ. T−1(q)=0 a1 aL3mS3, 0 0 1  Finally to illustrate the robustness of the adaptive ob-  c  server,wecarriedouta simulationconsideringthatthe   inputdisturbanced issubjecttostepchanges.Thetra- FromtheequationaboveitisclearthatamappingQ(q) 1 jectories of d (t) and its estimation, depicted in Fig. 7, thatsolves(9)is 1 clearlyillustratethetrackingcapabilityoftheproposed observer. 1q +aL mS a 1 3 3 Remark6.1 Itisinterestingtonotethatthestandard Q(q)= a1q2−aL3mC3. (lower triangular) Cholesky factorization of M−1(q)  1cq3  doesnotsatisfy (8).   10