Two Globally Convergent Adaptive Speed Observers for Mechanical Systems

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by  Jose Guadalupe Romero

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. [phone] 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⊤pˆi)(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,westudythedynamicbehaviorofp˜andcompute 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˜ ( Yipˆi)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⊤pˆi.
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,studythedynamicbehaviorofp˜andcom- Now, since the term in brackets in (38) is equal to zero
pute if|p=pˆandq|=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