Friction Compensation for Passive Systems Based on the LuGre Model1

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Cop\Ti ght " II' AC Lagrangian and Ham i ltoni an M ethod s for Nonlinear Control. Sev ille. Spa in. ~ 003

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FRICTION COMPENSATION FOR PASSIVE SYSTEMS BASED ON THE LUGRE MODEL I Anton Shiriaev ' Ande rs Robertsson .• Rolf J ohansson ••

• T in AIaersk Institul e. Cni/'(-,.sity of So uth ern Denmark. CUI1l pll81'ej 33. DA'- 3.!.'IU Odense AI

DE.VA IARK .. Deparlmen t of Au tomatic Contro l. Lun d /lIstitlltf' of Techn ology. P. O. B o:r 11 8. SE-221 UU L und S W EDEN

Abst ract : The pa ppr suggests a llwt hod for ('ompensat ing the impact of fri ct ion for the colltrol nonlinear syst ems. \\'here the original cont roller (derived \yithout taking int o ac(,ount friction ) resul ts from t he p assivity rela tion . The illustra tiw exa mple is given. Copy right :r 2003 fFAC

1. L,TRODUCTIO:\

l'pnd ers t hp set

This pa per is devoted t o the problem of friction compensation in no nlinpa r control s~' s t e lll s . It is ass ullled tha t a s~'s t e m dnla mics \y it ho ut fri ction is cow red b~' the equ a tio n d

d ( 1'

= F (J') + C(.1') l1.

y

=

H (J')

l O= {,T: q ,T) = O}

of t he syste m ( 1) g loball y asy mpt oti(,ally st a ble, proy ided th a t t hc fun ct ion (; is a C I_Sllloot h a nd o(.l; )y > 0 'rI y -# 0: a nd tha t t he' sYstelll ( 1 ) is ::el'O-statr- dctec lablf. (1 ' -detectab le J,

( 1)

\\' 11('rc a state \'ect or ,7 ' E R ", a colltrol action 11 E RI . and F . C, H a n ' smooth H'ctor field s of a ppropriat e dimc nsions, "'hile, in addition, it is assumed t hp S\'stelll ( 1) is pass in' \\'it h a pro per sllIooth st orage fun ct ion" , lllin" = 0, i, p, the time deriYa tin' of " a long a m' solution of ( 1 ) sat isfics the follo\\'ing cl iff<:> rellti al rela t ion

!i dl "

-< ,yT 11

As one ('a n exp ect , a pJ'('selH 'C of friction in the act ua tor , tha t is \\' he n the ' trtl P' control action a pplipd to thp syste m ( 1) has tlie form T

=

- o( y)

=

11 -

F.

(5)

lI1 a~' prewnt fro m the st ab ilization of thf' set (-1 ) a nd disregard t he expected behayior of the d osC'd loop syste m,

(:2 )

It is \yell knmYll. see for exa mple ( B~Tn es El al.. 1991 : Shiriaey a nd Frad km' , 2001 ), that the feed back cont roller 11

( -1 )

The pro blem of modeling the friction F is the Yast a rea of research , a nd ma n\' different model;.; for this phcnome non a re 3ya il able in lit prature , In this paper it \\'ill Iw ass ulncd that the fri ('t ion is modeled based on the LuCre model. seC' (Ca nud as de " ' it et al.. 1995) , Thc LuCre fi ct ion model is a first-order cl ~' n a llli ca l S\'stPlIl of thc form

(3)

I T hE' ,,'or k h a.~ bE'ell s lI ppo rt ed b~' t hE' D an is h TE'ch n ica l RE'search Cou nc il. t ilE' grant 26-01 -016-1. and t hf> l! o l'j f>1 Fo und at ion.

159

holds, that is, J'(t) C01l\'erges to t he compact set I defined in (-1 ), •

F i ~' , 0-0-' -;:

l' -

u

(6)

gjr(l')

Proof. The dyn amics of t he error between the 't rue' int ernal friction st ate a nd it s est imat e

Here;: is the internal state of thp mod el: I' i" relatin> velocitv betweell t\\"o surfaces ill contact, and it i" assumed that it is available, The function gjr(c) has us uallv the forlll no > 0,

(\ 1

sc\tisties the equation

> n,

r'

In t 11(' fmt her deve loplIlPnt, \\'l' \\'ill not take imo consideration the exact form of gjr' while only the property that gjrl /') is positin> ami s triC'tI~' separated from 0 , will be USl'd,

=

I /, j -0-0 - - - ( -

(12 ) Its tillH' deri,'atiw is, then, d di

11' = I" + pC(= , ," + pc (

S .l/ ( (u + F, ) - F ) + pc = - YO(Y) - '/I ( 0-0(' +"

=

- o(y)

U

=

\\'here the friction estimate obser\'er

d clt

t

/' , /' -

( T o -' -

-

' -: -

= '-yo(y)

t

-o(y) ,

is detined

u\'

(

::(0)

Yjr( I' )

=

,

- PO-o-- e.cJjr(l')

ic l

-

J\' )

0-0-, -(' -

+ pr (

- 0 - 0Il'l - - ,- ( ' - J\')

9j,(I)

1'1 0 - 01 --(' -

-

9/r(l') -

(

ir i

J\')

0-0-'- ' -(' -

r:

+ )

gjrl l')

+

'-v--' (\

an -.-

J\' ,

\

9j,\I')

+ pc iI- !

J')

o-0gj,1,.(1l ' ) ( -

RESULT

To compensate fo r the frictioll it is suggest ed to modif\' controller (3) as follows U

(

0-1' )

= -yo(y) - yo-or - y0-1 ~IAr:\

( 11 )

Let us consider a Lyaplllloy functioll candidate for t hl' S~'stl'1ll a uglllPntl'd by t hl' frict ion Illode! (6) and the obser\'E~ r (7) as

Thl' lIlaill contribution of thl' pap er is the moditicatioll ofthp controller (3) to llP\\' onl' that enables to g UHrantP(> asvIllptotic stabilit~, of the set (-1 ), In course of doing this, nOlle of propt~rties of the Illodel (6) ha\'(' beell uspd except the fact that it s stat e is bounded , and the dn1alllica l system (6) itself i" tillle-inva riant and has a \'('rv particular st ructure, It is \\'orth lllentioning a nother ap proach to deal with c1osch ' related problem of posi t iOll tracking deYCloped for t he mechanical s~'stelll s in (Canudas dl' Wit and I\: ell \" H197 ),

2,

J\'

,fJj,\l')

J'] '

J

(7) 0

SimplE' calculat iOIlS sho\\'s that

.)

-ne - 3E

He re t h(' ob"en'er ga in J\' could be seell as a variable to 1)(' detinl'd, The next stat clllE'nt shcJ\\'s a good choice for the function A',

)

~J ~ = - ( V(\( - 2y'n

Based on this, we can cont illue

Th wT'Cm 1, Given the syst em ( 1 ), (5), ( G), consid er t he controller u

= -OIY) - F

(8)

\\'hen' the friction eq imat e F is defined b\' (7) \\"ith J\'

=

0-( -~

[

1 -

and p >

/,]

0-1---

P

.11

The fir st t",o t erms in the pre\'ious line is no npo"itin', let LI S consider thE' last n\'o t erms ill details, haw

( 9)

gjrll')

n, then a long an\' solutio n

[,1'(t), ;:(t),

"'e

: (f) ]

of thE' closed loop s\'st elll the limit rel a tio n /-

lim " (,l'(t)) -'<.

= ()

(1() )

160

}'

r-,-o-l,l) \ Il- l [O-llJo---y-lJoY-P\ , g jr(1') ,

= P.rJfr ,( l') ](2 -;- [.rJfr(I')Y

claO !I' )

Indced. ('hoose an~' initial condition .1'0 . ':0 . 50 and consider the corrcsponding soIl! t iOll

a 1Y ] ]\- ...,...

T

2 11' 1

2 ,

~

=0

.dt ). :.:(t ).

=b .)

cl(lao ~"-',-.) 91 "

t

('

A" directly S(,(']1. th(' hlllction 11' is not propcr on the state spac(' of thp O\'('rall S\'St<'ll1. that includes the state wctor J' of (1). tlH' friction state .:. ami th(' friction obserH'r state .:.

As known the minimum of thc POl\"llOllIial 11]\-:2 ~ h]\- -

\\'ith

(J

(.

> () is achi('Yed at

But by assumption the function \ - is proper on R". \\'hik Le]lIlna 1 pro\'ides the bound(~dll(,ss of th(' friction state .:(t) along thp solution. tll('refor(' the in('
ao [1 + a ] -'I-"-] Y ( 13) 211

f!

Yf' ( I')

and equals to

n~Il {

(f

]\-:2

-L

Id":

:.: (t . :':0 )· 5(t . ':o f

of the dos(>d loop syst elll ( 1) . (G). (8 ) . ( -;-) \\'ith the obserwr gain ( 13 ). Along this solution the nO]lJIcgatiw function IT" satisfies the rplation (lcl). amI non-increasing.

(a]aO~Y - aOY)-

=

::(t r = : .1'(t,,1'0 ).

ll'

(.1' (t ) . :.: (t ). .: (t ))

-:::: 11' (.1'0. ':0·

-+- ('}

':0)

in fact guarantees th(' bOlllldedness of th(' solutioll i ./'(t ). :.:(t). ': (t) ]. Then it has nOIl-e lllpt\· cOlllpact ~-limit set ~; u . and Oll this set

In our ('asE'

d 11dt

= o.

-

It is equi\-alent

two idelltities

10

_ clPYf r (l')

clao II' I

.

aOa]

')

yo (y) -+- - - . l r = 0

( 15 )

(I

au ~l_l'_!_ Yf r (l')

alld

(PC_

a]

y):2

=0

011

the set ~,o

The relation (15 ) implies that

y

( lG )

= O.

\\'hile the relation ( lG). combined \\'ith (15). states that 011 the set ~: 0 either

(=

O.

or

I'

=0

III am' case t h(' condit ion lJ = 0 and :.: ero-state detectability (1- -detec tability ) condition illlpl\' that th(' \'alup of \- (.r(t)) tPllds to zero. i. e.

Therefore. if the obsern' r gaill ]\- is chosell as in ( 13), or the sallle as ill (0 ). t hC'll the time dC'riYatiYe of the function 11 - along the solution of the dosC'd loop S\'stE'll] look;.; as

q.r (t) ) -

D.

t-

~x

This fini shes t he proof of TheorclII 1. •

.!!... d/ 11- = - .IJO(IJ . )-

R ema rk 1. In fact \y(' haw' prowll slightly mOH' than it is stated ill Theore lll 1. :\anlPh·. it is shO\\'1l that ill the' case if for

To anah'ze t hC' bchayior of the do,.;ed loop S\'stelll solutions hased on t lIP diff('[ential relation ( le! ). it is \\'orth to mentioll 011(' propert\· of the LuGre frict ion model (G)

lUll)

solution

[J' ( ~)'

:.:(t ). ': (t) ]

of th e clos ed loop system ( 1 J. ( 6 ) . (S J. (7 ) tl/(' v.:-lin7lt set ~. 0 consists of solutions . .for u-!tieh tllf relatil'e l'elocitlj 1' ( t) betwee n tu'O sllljaCt s in cOl/ta ct has at le(1.';t Ol/C t/lW- mom ent T. u-!/(l'( l'( T) i O. th en

Lemma 1. For any initial condition ':0. the solution .:(t ) of the system (G) is bounded .•

lim

/ - - :x.

(:.:(t ) - ':(t )) = O.

i. e. th e f stil7lat e :(t ) con/'ugrs to thr internal of th e LuCre fridion TIIodel. Otit erwisr. th fY can difje r only by II emu,tant. that ('or/'( sponds to unknown stiction mluf .•

Equipped \\'ith this fact onE' can readih' prOH' the feedback controller (8 ) as\"lnptoti('all~' stabilize t he set

.~tat f .:(t)

lo = {.r: 1"(.J) = 0}

161

3. EXA:\IPLE: STABILIZATIO:\ OF PERIODIC ORBITS FOR I'\"ERTIA " "HEEL PE.\"D ULU :\I

To this end. consider o lll~" the s ub s ~"ste ll1 (19 ) a nd ma k(' fped back transformat iOIl T

.1.1 Passit'ity Ba IJed Controller Derired without Taking in to A ccount th e Friction

31 rh

"'ly s in (8Il

fJ2

=

sin (fjl

) -

T

- 111)9 sin ( qIJ + I'

(22)

To o \)sern' t his one can d lPck that t he total e n e rg~' of (22) E (2:2 J( (JI . lj I! =

(1T)

(18)

~I Ij~

T

(.1 1 - rll )y ( coSI/I + 1) (2 3)

has Cl global lllinimum equ al to 0 at the pquilibriulll q1 = r. . ""hile tile reader call eas ily check th at a m ' subsd of the c~'lindrical phase space [If I· lil ] E SI x RI ,,"hen' th(' en erg~' E(2'2)((J1.lh) is fixC'd to be constant. corresponds to a ('ycle of the' unforccd s~ "s t ell1 (22) . except two critical cases "'hen such a set cont a ins one of the dowlI\\'arej or upright C'q uilibria. Let ('hoose a constant E •. and stabilize a c~"c:le of the unfor('ed S,"stelll (22) corresponding this energ,\' IE'Ye!. To do this, consider thE' Lyapuno\' function candidate

It s tim(' d eriya ti\'(~ a long a n\' solution ifJ l(t). (il(t) ] of (22) is

C ha nging the yariables RI' R2 to np,," ones. as done in (Spong et al.. 200l a) . RI'

Irlg

-'I > nl.

Here II is thc le ngt h of the pE'udulum: I c l is the position of the ('ent er of mass of the pendululll: nil is the mass of th e pendulum: rll2 is the mass of the dis(': .11 • oh arE' inertia of the pendulum a nd the disc aro und their centers of masses: 9 is the acceleration due to gray ity.

=

(2 1)

i'\Ild ih equilibriulll at (/1 = ;0 bE'colll(,s n p utrall~' sta blt' in the L\,apllllo\' seIlS(' proyidcd that

"'hen' HI is the a ngle that the pendulum makes with the ,"erticalline: 82 is the angle that descritws the position of the s~' mm ct ri(' disc: alld

ql

= -

= (.11

The ma the matica l model of this system deriyecl in (Sp ong et al .. 2001b : Spong et ul.. 2001a). is

+ d 12 8'2 = d 21 BI + d 22 82 = T

l'

The n (19 ) t akes the forll1

The I nertia Wh eel Pendulum is a ph~"sical penduIUlll ,,"ith a symmetric disk attached to the elld . The dis(' is coutrolled b~' a DC-motor that can change thp angular acceleration of the disc. ,,"hik the ph~'sical pe ndulum it self is frt'ch" rot a ting. The details alld the descr iption of hanh"arc ('a ll 1)(' foulld ill (Spollg et al.. 2001 b: Spong r-:t ul.. 200la ).

rillB I

= - JI g sin (qJ! -

;:t ' "=

(}I - (}2

brings the s.'"stem ( 1T). (18) into simplified decclllpled form

(E(22 )( fJI(t ).qdt)) - E.) 4dt) ·1'(25)

So thp Syst E'1ll (22) is passiw "'ith thE' storage function'" . the input 1.' a nd the o utput

( El )

(20 ) \,"ith

31

= (rill - d 22 ) and

32

Lf/lln/a

= 2d'.!.2·

2. Suppose t he subset of the phase space

of (22)

The problem is: To tonstl'uct and 8tabili::e an oijcillato l~lj behal'ior of th e pendulum around its upright eq uilibrium fjl = ;0. does not collt ain an ('q uili bri Ulll of t hp unforced ("' ith I' = 0) s ubs~"stE' m (22 ), ThE'n for thE' cloSE'd loop s\"st em (22 ) ,,"it h t he controller

To soh 'p tl1{' problem \n' ,,"ill procE'f'd in t\\"o steps:

• .~·r8tly. based on the id ea of p ot E'ntial shaping \YE-' d eriw the controller. that generates p er iod ic osci ll a tion a nd stabilizes it only for the SUbSyst E'lll (19): • Mcondl.tJ. it will be prOH'n that this controller prE'seryes the state:' of the s('('ond subsystt'll1 (20 ) bouncied.

I'(t)

=-

ky(t)

(28)

,,"it h any k > () and .IJ defined in (2G ). the C\'dl'. that lin's 011 thE' s('t ' 0' is orbit a lly asnnptoticalh- stablP. Furt hE'r lllore. a long a n\' sol ution

162

[q] (t). 4] (t) ] of the closed loop S\'stell1 that COl]wrges to the set 10 the function l'(qdt) . 4dt )) decreases to zero expollclItialh-. •

the closed loop s\'stem (22). (28) t ha t C'onyerges t o tlw set \ CJ' the intq!;ra l in (32) is bounded .•

Proof. The secon d pa rt of (32) is Proof. The first part of Lemma follO\\'s from the simple arguments briefl~' melltiolled ill IntroduC'tion. Let us check that tllf' fUlIction \' decreases cxponelltialh-. Consider an~' solution [qdt ). 4dt ) ~ of t he closed loop system. The controller (28) lea ds to the differential equality

.I t

{kli d s ) (E(22 )(CJd s ).li] (s)) -

L) }rls

(j

IS finitt' for any t ch](' to tll(' fact s that (f] (t) COllyerges to some periodic funct iOIl. i.('. rema ins boundt'd. \\' hil{' as shO\\'n in Lf' lllllla 2 the funct ion

\\'hich. in tum. implies the relation tE'lld::; to zero exponl'nt ially. \ '(CJ' (t).

4, (t )) = \ '(q] (0). ri, (0 )) x "'Xp { -

k!

1;1,,)d., }

To prow' that the first IHHt of (3:2 ) t hat is (29)

i I

jt -,\1g SillCJ J\8) riS :i < -'- x.

boullded .

'v' t

(33)

I

I ()

If the solution [q, (t ). q, (t) ] conwrges to the (:ycic. then rij (t ) approaches a nOlltriyial periodic function in time, Therefort' tht' relation (29) guaralltees that l' dccreases exponelltially to zero .•

IS

\\'(' obserw that it follO\\'s from (22) that I

I

' Sill q1l8) ds = --::-::--1_ _ /' {.:f,liI (s) - /'(s)} ds ( cH - m )g. ./

Let us cOllle t he second part and pro\'E' that the controller (21 ) . (28) guarantees the boundedness of am' solution [q2( t). rh(t )] of the second subsystem (20).

u

()

I

':12l12(t) =

T

= - JIg

sinqdt) -1'(t)

= -JIg sinq, (t) + ky (t) = - :Ug sin q] (t) + k4, (t)

X ./ kq] (::;) (E(22)((/1(8). 4] (8)) - E . ) cl.'; (30 )

(E('2'2)(q] (t). 4dtl) - E . )

To prow the boundednf'ss of [q'2(t). rj2(t l ]. \\'e need in fact to check onl~' the boundedness of 42(t ). The ya riable CJ2 (t) on the C'ylilldrical phase space is ah\'ays bounded.

The right hand side of the pre\'iOllS formula is bo und ed , Therefore thl' inE'qu a lit~, (33) holds. Lemma 3 is pro\'en .• Let us summarize t hp res ult

Th eo rem 2, Consider the IIwrtial "'hpel Pcncillhlln . see the equations ( 17)- ( 18) and choose the ('ollt roller

If o ne integrates (30 ) from 0 t o t. t hell an equiyale nt fonn is

;'].1

(j

I

42(t ) - 4'2(0 )

=

T(s)ds

., =

(3 1)

o

- JIg sine] + kih (E(2'2 )(H,.8] ) -

",here k

> n. JI

>

171

L)

and the fUllction

£ (22 )(8 j . 8 ]) isdefin E'd in (23). SllPPOSE' the sllbset

of t hE' phase space of (22 )

and the bo ulIdedness of 42 (t) fo1l0\\' 5 the boulIdedness of the int egral

.I t

o

.I I

T(s)d.s

=

{-jIg sillqd s) -

does not C'olltain an equilibriulll of thc unforced (\\'ith t· = 0) s ub s~' s t em (22). Then the c\'Cle. th at liws on the set \ o. is orbi~all\' as\'l11ptoticall~' stable fo r thE' phase.spacee, . H, : of the pendulum . \\'hile the rest 'fh. £12 , of the syst em yariables are bounded along t he closed loop sYst em sol ut ions that approaches \0 ill the part .•

(32 )

0

- k4 Il;;) (E I 22I (qd 8 ).4](s)) - L)}ds Lemma 3. Suppose the s ubset of the phase spa ce of (22 ) \ odoes not cOlltain an equilibriulll of the ullfOI'ced ,;u bs~,~tem (22 ). Thell for am' so lut ion of

'8,.e]

163

.J.z Passi1·ity Based Controller Augment ed by Frict ion CompeTlsator

" 'here p > 0, k > 0 , JI > liws on the set

\0 =

,\ss umillg a presencc of frier ion in the IIIE'rtia \\'heel Pelldulum. that has a structure of the LuGrE' model. we get all extend(>(l dnlalllics of the system compare to tll(' IHP\'ioush' considercd etjuatiolls (17 )- ( Itj)

m, Then the cycle, that

{ [(Jj,4 j J : E ml( (JI.ci Il

=

E.},

is orbit ally asymptotically stable in the phase plane : (J], 41 ), •

-1. CO:\CLUSIO:\S (35) ..

..

(rn B] + 11'22 (-)']

= T ~

F

This paper deals \\"it h t 11<' probkm of a frictioll compensation ill llonlillear cOlltrol s\"stems, It is assumed that the friction has a particular structure , the so called LuGre Inodel. \\"hile it is also assumed that the origillal COllt roller deriyed for the s,\'st em "'it hout frier iOIl, is based 011 t 11(' passi\'it\' relatioll, ThE' main result of the paper suggest:; a friet iOIl obs(~ rye r and modificat iOIl of t Ill' control ill such a "'a~' that t 11(' o\"('rall closed loop "-,,stelll presern's asymptotic stability of the desired attrartiw set that "'as obtain originall~' by the controller whell tll(' frictioll is llot takell illto account.

(3G)

where

Illtrod ucing

\"clria blps

lIE''''

we get the equiyalent syst em

J] ch = ~ illgsin ( (JIl J2

ch =

F

d

dt

(To'::

-

T ~

+F

~ T

F

(3tj )

+ (T1:: +

"

(37 )

( (j2 ~ (J] ) -

~

(ch

0']

BHnes C.I .. A . lsidori and J.C. Willems. Pass iyit~·, feedback equi\'alence. and the global st a bilization of minimum pha.se non linear svstems . IEEE Transa ctions on Automatic COTltrol. 36:1228 - 1240. 1£)91 Canlldas de Wit C .. H . Olsson , K.J. Astrolll and P. Lis c hill s k~·. A lle\\' mod el for control of Systellls with friction , IEEE Transactions on Automatic Control. 40:-119 --125. 1995. Canudas de " -it C. and R. Kelh·. Passiyity based control d esign for robot s \\'ith dn1amical friction. In: th e Proceedings of th e 5th lASTED Int ern ational Confarnce, :-lexico . pp. 8-1- 88,

4d

!rh~41 1

ao

REFERE.'\CES

,

,::

(39)

9/r'((J2 - (j])

The main result of the paper. Theorem 1. implies the following stat ement L emma 4, Consider the subsystcm (37), the LuGr(' fri ction model (39). tllf' friction obserwr

1997.

d dt

-

wit h \\"it h ~ (0) 1\'

Shiriaey AS and A.L. Fradkm·. Stabilization of im'ariant spt s for nonlinf'ar S~'stellls \\'ith applications to cOlltrol of oscillation,.;. Ini FI/IOtional Journal of Robu st and _Vanlin ear Control. 11 :215 .. 240. 2001. Spong :\1.\Y .. D .J . Block and 1\:..1. "\strOlll. The :-lechatronics Control Kit for Edu ca tioll and Re::
= 0 and

= _ ao [1 -'-- a] (J

(To [ 1 - (TI

P

the ga in

(h ~ ql 9/r (4'] ~

]

(id

4'] ~ ih ] 9/r((/'2 ~ 4] )

y ( ..11 )

X

and t 11(' feed back cont roller defined by T

= - .119 sin q] -'-= ~ .11!J sil}(ll +

I,- Y - F

...,..I,-Ih(t} (E(2'2Mdt},Q](t )) -

(-12 )

L) - i, 164