On the Problem of Mobile Robots Path Tracking

u,pyrigth ~ IFAC Motion Control for Intell igent Automation Perugia . Italy. October 27·29 . 1992

ON THE PROBLEM OF MOBILE ROBOTS PATH TRACKING S.V. GUSEV·

aDd LA. MAKAROV··

.. Sankl Petersburg State University. Department of Mathematics and Mechanics. Bibliotechnaya sq .• Sankl-Petersburgh 198904 . Russia .. Institute of Problems of Mechanical Engineering. Academy of Sciences of Russia. Bolsoj 61.. V . 0 .. Sankl-Petersburgh 199178. Russia Abstract: The problem of mobile robot desired tnjectory st3.billz3.tion is considered. It Us shown that this task ma.y be solved in the case when the plant Us a.tfected by both pa.ra.metric a.nd externa.! diBturba.nces. Preaented a.da.ptive controller couUste of two feedba.ck levels ud provides UII the desired precision of pa.th following. Key words: Mobile robot, nonllolonomic constraints, l\.onlinea.r control, a.da.ptive control.

Introd uction.

1

i\n interest to the mobile robots is increuing in recent time. )uch plants ma.y be considered aB the mec:h.:mic:U systems

"ith nonllolonomic constraints. The motion of constrained mechanical Iystems is deIcribed by the l)'Btem. of Euler-Lagra.nge dynamic equa.tions III'hlch. describe releva.n.t dynamical upec:ts of the plant mo:.ion and by the l)'Btem of constraint equation.s which give ~he lcinem.a.tic dependencies between dependent and inde~endent coordina.tes. In the caee of holonomic (integrable) :onstraints the con.straint equa.tions is written in the form )f algebraic equation.s defining the con.stra.int ma.ni!old in ~e conliguration Ipa.ce of the Iyste.m. The cluaical nonlolonomic (nonintegra.ble) constnWl.ts &.re written as differ!lI.tia.I one-fOrm! defined on the cOllBtraint m.a.nil'old in the I}'Btem. phue IIpa.ce. It ia known that the constraint multipliera ca.n be elimnued by uaing classica.! method described, for instance, n [1]. The algebraic formalization of thia result wu pro)oeeci for holonomic mechanical systems by McClamroch ~d W8Jl3 [2]. In this pa.per waa shown tha.t a. locally smooth illfeomorph..i.nn of the system Ita.te spa.ce into itsel! exists ruch. tha.t in the tra.nliormed va.ria.bles the dynamical equ~ ;iou of motion do not depend on the conltn.int multipli!no On the bue of these tnnsformed equa.tiollB the sta.biiza.tion control algorithm for independent coordina.teI! wu )ropoeed. Since va.ria.tion of the dependent cool'dina.tes of ;heholonomic mechanical synem is given by the algebrak ~1l&tiOIlB of conltn.ints then it is evident tha.t the sta.bili:r ;ion of independent coordina.te8 in the neighborhood of its leaired values is sufficient condition for the sta.bili:a.tion of Ifhole YeCtor of genen.lized coordina.tes. Applying thlJ a.pproa.ch to nonholonomic mech.a.nica.l sy.. emlllOme a.uthon Ita.te the tuk of independent coordlna.tes mooth ata.biliza.tion Clee works of Bloch. ud McClamroch 3, ud Butin and Ca.mpion [51). Unfortuna.teiy, in the :ue of m.echa.nical system with nonholonomic COllBtramtl :he ata.bili:Ltion of independent COOrdlnLtes does not imwes the atahi1i'ation of depenwt coordina.tea. Thus, in :he tub where the lta.billzLtion of whole con1igv.ra.tion vecor is neceuary, this general a.pproa.ch of coutralnt multi,lien ellmiJl&tiq is not nita.ble. The tuk of mobile robots )&th tra.cki.ug .is ex:LCtiy Ille!:. tuk. We shaD couider this problem for special lOAd of non,alonomic: medlaJlicall)'atema, &.&mely: for two-depee a!

freedom mobile robot. The lrinema.tic model of this robot was investigated in our previotul work [6]. In this C&le the nonholonomic constraint can be written in the following form (1) cos( t/J) - y sin(,p) = 0,

z

where z and y &.re the coordinates of the driving wheels LXiI center; 1jJ is the hea.ding angle. Ta.king into a.ccount tha.t the nonllolonomic constra.int ( 1) subjecting to the plant under consideration defines Imooth covector field defined on the con1igura.tion spa.ce, we CLl\. transfer the problem of pa.th tracking into the control problem. Indeed, it is easy to see tha.t the vector fields COIl(I/I)8/8z + Bin(1jJ)8/8y and 8/81/1 spa.n the diatribution on the constraint ma.ni1'old Ll\.d tha.t differential one-form COI( I/I)dz - sin ( 1/1 )dy, d.efined by constraint ( 1), a.nnihila.tes . thiB diJtribution. Thu the pa.th following problem. ca.n be restated u the problem. of the finding of BUch longitudinal Vi(t) and angula.r velocities ~(t) of the robot hull tht the control system described by the following Iystem of differential equa.tiollB

(2)

Iteen the desired pa.th of motion. Let M be the smooth C1ll'Ve d.efiJling the desired pa.th to be followed. This curve is IllppOIed to be given by a. smooth {unction acC') = (Zt(')'Yt(,», acC,) E CICRI - R2), Le. Mc { Zi(') I , E Rl}. Denote ~(,) the desired hea.ding ua1e of the robot, Le. the ua1e between a.beci8s~ a.xla ud ta.D.gent vector of cu.rrve M at point Zi(,». We aaume a.lJo tha.t the deriVLtive of the desired hea.ding angle ~(,) ill bounded

(3)

41

Berea.!ter we denote ~ the let of nooeptive numbers. For &rbitrary time panlmetri.lauoru(t) E el(R. - RI) of the desired pa.th III the motion ~(,(t» sa.tide. noUol~ nomic coutr&ints ( 2). Denote the 'Vector of robot Ca.rtaian coordinates .J(t) c col(:( C), y( e». We sh.all say that the control aim ia reali3ed if for a..given positive COIlBta.D.t ( the following inequlity

n - 139

d.ist(.t(t),M) < ,

(4)

GCSEV S.V .. MAKAROV LA

Aold. {or all t E ~. lA other worda, the control law should be designed in .uch ~ way th.a.t th.e tra.jectory of robot motion i! enclosed in c-neighborhood of desired path M. We .hall .how tbt the control aim ( 4) ~ -:..ieving i! possible even "hen the longitudinal velocity of t ._; robote Y,(t) is a.n ariitrary nonzero fu.nction of time. The finding of desired a.ngulu velocity of the robot v;.Ct) ia the tint .t&6e of control algorithm deaign which will be presented in this pa.per. The second tt~e of the control algorithm deaign i! to .ta.bilize the real longitudinal a.nd a.ngulu velocitie.a in lIllall neighborhood onts desired values which were obtalned on the first st&6e. The lut aim CLJ1 be a.chieved by using gene.ra.lized forces a.cting on th.e system a.nd provided by the a.ctua.tors.

y

z(t)

ChoOle tAe vector of generalized coordina.tea 11 .. coI(q/, lie) in euch a. way thILt its first component represents the path. length hLving been covered during tAe internl [0, tl, i.e. II/(t) = Y,(,)d.r a.nd the second one qe is the llea.ding a.ngle.p. Then th.e dynamics of couidered mecha.nical pJant driven by 2-a.ctUILtOt'S can be described by th.e following Euler-Lagrange equation:

~--L_ _

x

J:

A(q, 8)q + b(q, 4, 8) = u(t) + ,et),

2 (5)

where the vector 8 is a vector of th.e pla.nt Lhatra.ct dynamic pa.n.meten; A(q,8) is .ymmetric a.nd positive definite inertia matrix IUch that th.e inequality IIA- l (q, 8)11 < Cl, (Cl > 0) holde for all vectors q; b(q, 4, 8) is the vector of centripetal, Coriou." a.nd friction forces; u.(t) = col(ul, Ul) is the vector of gener&lized fol'Ce8 acting on the .ystem a.nd provided by the a..ctua.tOrB; {( t) i! the vector of external dilturba.ncea. We emphOl.l1ze tha.t due to the IIpecial choOle of the generalized co 0 rdina.tes , th.e dynamic equLtion does not depend on constraint multipliers.

In this pa.per we shall cOlUlider two claaael of diaturba.nces subjecting to the plant. We denote D!.(Cd the clll.ll8 of uniformly iounaed auiurOanccl

where

q

is pOllitive number. Introduce a. c1us of 4ut"r.

;oncu with ,ummd14 p",tuienmce, D~(€(,C() c

{{(t):

[ "here f( a.nd e( function

&re

a. -

The behavior of cOlUlidered mecha.nical 'Yltem is deecribed both by dynamical model ( 5) and by kinema.tic model ( 2). In this lection we uaume th. ..t th.e model of robot motion is determined by equations ( 2) only. The longitudinal velocity J7(t) islupposed to be known function. Let a.ngulu velocity of the robot hull turning y'*(t) be the control input varia.ble. Thls input signal must be determined to compenIa.te dilturba.nces in the c101ed-loop control Iystem ( 2) a.nd to ensure the control aim ( 4) achieving. To design a. control algorithm, providing the sta.biliza.tion of robot motion in a neighborh.ood of desired pLth M, we &re in need of a. time parametriza.tion , = .r(t) of the pa.th M determining the desired po.ition of the robot z.t(t) :=: ~(,(t)) a.nd its desired hea.ding a.ngle t/1,,(t) = .pt(.f(t)). The time parametriza.tion ,et) will be cholen in euch .. wa.y tb.t th.e equa.red dista.nce between the current robot position z(t) ud the point ~(,(t» of deaired tra.jectory M, defined by the function p(z(t),~(,(t))) .. ,,~(.(t)) - z(t)lIl, will be equal to a. given amall pomtive couta.nt r:

(8)

~ €( },

(7)

pOlitive numben; 4(4) is the Ha.VYlide'lI

""\c¥ c

The first stage of the control algorithm design: kinematic model.

R"I

A(lIe(t)lI- C()lIe(t)lIdt

J.')

Figure 1: Coordinate system for mobile robot model.

{O,1,

The time pa.n.metriza.tion ,et) to be found ia tAe .olution of algebraic equa.tion ( 8). To ..void di1!ic:ultia of tAe nonlineu algebraic equa.tion on-line .olving we can ue tAe fonowing
d

0 ~0

dtP +7,(P - r) = 0,

a < Ot

Coututl C( in definition. ( 6) ud ( 7)

u reference equILtion for time parametriza.tion

&re

not the lame.

In tAe cue when the vector of dynamic parameters 8 is unJmown tAe problem of deaired tra.jectory following ma.y be 80lved by me&ILI of ada.ptive controller. It will be DOwn tha.t in thia eue both external ud parametric disturba.ncee nbject.ing to the p1a.nt ma.y be nc:ceu!ully parried. .

(9)

,(t) deriv-

ma. lA expreaion ( 9) tAe number r ia tAe I&m.e poaitive consta.nt u in ( 8), 7, 18 tAe potitive parameter of algorithm. It is clea.r tha.t the equa.tiOJl ( 9) eu~ tAe function p(z(t),Zc(,(t))) expOJle:Jltial convergency to L given number r. Introduce the functiou ",(z(t),Zc(,(t») = a:g(~(,(t»­ z(t» ud 6( ..(t), ;(t), Zc(,(t))) - ",(t,,) - ;(t) !epreaen~

n -140

\ . TI-lE PROIH..Dl 0 1· .\10 13 I1...E ROUOTS ; "' TI-llRACKI.'\G

ing SO C4lled dir-ec',orJ Uld delliln,on u.glee correspondingly (lee Fig.l). The..a.. ta.king total de.riva.tive of the fU.II.ction p(z(t). Zt(.f(t))) along trajectorie8 of the system ( 2)

BD. BD BZt . -p= -Z+--4 tit 8: 8Zt 8,

In the cue when the input :tign.:1.I..s of the

V1(t) = y'-Ct) + Ct(:). Va(t) = V.:CI(t), z(t), fII(t))

d

we ca.n obtain the foDowi.a.g clliferential equa.tion determin. ing the time parametrintion of desired motion trajectory

; = KC08(W-tP.t) 1 .

[Vtcos(5 )

-1'pr ] ~..rp

(10)

wh.ere k = J(~):I + (~):I ia nppoeed to be a nouero va.riable for &ll •. Let the ilUtial condition '(0) = latisfies

'0

(11 ) The..a. the solution of eqution ( 10 ) is unique, exists a.nd yields such time paramet:ization "C t ) of the desired tnjec. tory M, tbt the point Zt (6(t)) move!! along the plS.th M synchronously with the motion of the robot z(t) a.nd, moreover, the dista.nce between these points te..a.ds to r. It is to be note tha.t this statement is valid in the case when the following inequality 11/1,(.(:)) -

wet, .(t))1 < 1'/2

(12)

holds for all t E ~ . It will be shown that inequality ( 12) nay be e..a..sured by prope:- choosing of robot a.ngulu velocity Moreover, SUc!l choice provides the stabilization of the robot hea.d.ing ugle I/I (t) in IS. small given neighborhood of the desired hea.din.g ugle 1/11 (IC t)). Indeed, tlLking total deriva.tive of the fU.II.ction6(z(t), fII(t), ~ ( I ( t))) along trajectories of the system C 2) a.nd C 10)

V:.

d tit

-6=

LIe

a.ll t E 14 we

the following equation ti tit 6 • 1'6 6 = 0,

1'6

>0

(13)

a reference equation for the deriving of desired a.ngulu 'elocity

.8

( 2) &re

(15 )

+ cc(t)

where ~Ct) u.d e.(t) are cfuturbances in the dOled. loop control .ywtem, then the propertie8 o{ preaented control aJgoritlun are formullS.tcd in the following theorem.. Theorem 1 Suppou that real robot po.ition %(t), hMdin; angle 1JJ(tl and tielired ro60t po~ition %:i(t) ,atiJ/y at the initial moment to = 0 the lnequalitieJ ( 11) and

(16) where po~itive e01UtGnt r

u the ~ame al in ( 9) anti numier

6. i, a po.itive c:onl1ant. Suppou ailo, that the vec:tor DJ dutu,.ianc:e, e(t) = col(e,(t), cc(t)) in the c:lo.eJ.loop c:Dntroi 'lI,tem ( !), ( 15) hdong, to the cia", of uniformly hounded di,turoanc:eJ:

Then for an1l po,itive c:on,:ant, , in ( 4) and

be found

c.

there Mn

(a) ,uffic:ientlv ,mall pOlitive eonlta12.t, Cl and C" in ( 17), C~ in ( 3), r in ( 9) and in ( 16),(6 ) ,uffieientlv lar; e (;012.,tant, "'rp > 0 in ( 9) and·15 > 0 in ( 13), ,uch that the c:Dntrol aim ( 4) u relllue4 And inequality

o.

(18) hold., Jor ,olutioru of the 'y,tem (!), ( 15), ( 1./) anti ( 10).

3

86 . 86 · 86 8Zt -.+ -'lb+--; Eh 81/J . 8Zt 8.

~d supposing that inequality ( 12) holds for

:t~tem

disturbed u followl

The second stage of the control algorithm design: dynamic model of motion.

In this .ection we .hall couider dynamic model of m~ bile robot motion ( 5). This model will be used for the computa.tion of gene.raliud force.a u( t) IUch tha.t, acting on the plut, these forces Ita.bilize the velocities of robot m~ tiOD. 9(t) in .mall given neighborhood of ita deaired value8

,,(t» .

pet) = colcv,·(t), V:(.(t), z(t), In the cue, when the vector of robot a.n.d loa.d pa.ra.meters , it mown and pet) is .. givu function of time, the problem. of control algorithm .ynthaia would be limply 80lved by c:hoa.iD.g

(14) would nu to I'tlUI the fonowing I'pecial future of the :)ntrol algorithm ( 14). The first term of this algorithm is le feedback 1&w with respect to the deuiQiDn ugle 6(.). 'he IeCOnd IlOAlilleu term .often! the tr&l1Iient proc:ees in le doed.loop.ywtem ( 2) ud ( 1-4). One Ca.JI. thow, th.&t if the value of real longitudinal vecity Y,(t) iI equal to the desired value Y,(t) v,*(t), u.d 19U1ar Yelodty Ye (t) is determined by the expreaion ( 14), Len the control aim ( 4) is realized ud inequality ( 12) iI wd for all t E ~ along trajectories of dOled ->'Item ( 2), ~e

=

U lI:

A(q, I)[F(t) + 7.e] + b(q, q, I),

(19)

where e(t) lI: pet) - q(t) 18 the vector of th.e velocities tr:I.cking errors a.n.d 7. is a pOIitive couta.n.t. This method of control 1&w deriving 18 lmown in robotics u 'computed to~ue method' ( Paul [8]) and., in the cue when {et) s D, this method eu1U'eI exponential tta.bllity of the IOlution

q(t) = pet). However, in couidered cue p - pet, .(t),z(t), ~(t» is not a. given function of time but the jaeiIJu.!&w deiJled by ( H). Ita total deriva.t.ive c:u. be wriHen u follOWl

10) ud ( 14).

n - 141

8.·

8"

' . 8'1' 8p . 8p . Bp P -= it + + 8z % + y'*(t, .(t), z(t), ,,(t))

GCSE\" S\' o J\lAIV\RO\' l.A

v...

",here wtead or i, .i &nd it neecb to lIubll~itute the eX')t'eSsione ( 10), ( 2) ud ( 14) cOrre!!pondinglv To o~ t ,', controller more swtll.ble for pra.ctice we sha.. lltve!tigate simplified version of control algorithm ( 19). Namely, we drop lIoph.istic:a.ted computll.tion of pC:) ud coneider the inner loop 8tructure with. the following control algorithm

" ="Y~A(q,e)e+b(q,q,e),

(20)

The algori tlun. C 14) &nd ( 20) jointly provide UB the twolevel algorithm of mobile robot motion sta.bilizatioIj" in doing 110 the force feedback ( 20) .ta.bili2el the kinem..tic feedbll.Ck ( 14). Now we consider the problem of UBefulness of the control algorithm ( 20) when the system is subjected to external diaturbucea {et) alao. In this cue the following theorem confirms the worlm.bility of the controlla.w ( 20).

Theorem 2 Su.ppOIt that initial rooot pOlition z(O), he4d-

waere 17(:, r~') U, :10 cwed crror jur.c:ion.. Exploiting property ( 23), we ca.n derive the followin.g form of the error function 7)(t, re:)~

= e(t) + Y(q, q, q)(r(t) -

8)

(25 )

It is easy to lIee, tha.t equa.tion ( 24 ) has the form of EtLlerL&gra.nge equa.tion LIld it. leit-ha.nd .ide depend. on the mown ertima.tion of pe.n.meters 1'(:) - not on th.e unknown vector of real dyn&m.ic panmeters 8. The error function 7)( t, 1'( t)) in ( 24) has the slLIIle function as the vectcr of disturba.nces in ( 5). Then, reca.lling results of previous pa..ragra.ph, we ca.n a.dva.nce the hypothesis the.t in the case when the error function belongs to the class of diBturba.ncea with summa-bIe protu bera.n.cea

,,(t,r(t) E D~({",C,,)

(26)

po,ition Zoi( 0) ,ati,fy ine~ualitif!l ( 11) , ( 16) Ilnd the initial vdocitie3 Jati3fy the following inequ.ality (21) Ilci(O) - ci"'(O)1I ~ 5•.

for all estima.tions r(t), t E 14 a.nd for sufficiently small pa.ra.meter l(, then the control algorith.m

SUppOIt IlUO, that utema l dilturbanecl {(t), luojeeting to the plant ( 5), belong to th.e elau of duturoaneu with .fUmmable protuheranceJ

is quite suita.ble to ensure a. good tracking of th.e ideal t:nv jectory p pet, .(t),z(t), T/JCt)) . For th.e proper using of results, presented in previous section, the estimation algorithm mut be designed in such a. way tha.t the ma.trix of kinetic energy A(q,r(t) would be non-lIingulu for all r(t). With that end in view, introduce in the IIpace of robot dynamic pan.mete.rs the closed convex neighborhood Q of th.e point such tha.t the following ineq uali ty (28 )

ing

~ng{e

T/J( 0) and

de8i~d

(22) Then jor any po,itivt: eon"tant4 f in ( -I), f.; in ( 18) and C( in ( !!) the~ ean be found (a ) "ufjicien'ly ,mall po,itive eorutant, 'r in (9),5. in ( 16),5. in (!1) and,( in (%!); (b) ,ufjieiently large po,i'ive c:on,'an', 1p in ( 9), "Y6 in ( IS) "Y. in ( %0); IUeh thllt the eont",l aim ( -0 i, r-ealized Jor all ,olutionl of the ,vstem ( !), ( S), ( !O ) and ( 1.0.

It impliell from Theorem 2 tha.t tULiformly bounded dilturbance1I u well u disturbance! with summable protubera.nCe1l m ..y be lucceasfully parried by chooling .ufficiently large ga.in "Ye. It will be shown tha.t the errors of the robot dyne.mic panmeten eatima.tioll ca.ue the a.ri.eing of ciaturbucea with lumma.ble protubere.nces in the cloeed-Ioop control .yatem- To .often this effect it .hould be designed. a. special algorithm of a.dju.sting of the pla.nt uiliown dy)tamic parametenl. Couider the cue when the vector of real dynamic p&ra.metenl 8 itlUlknown. The.n, in ge.nen.l, the controlla.w ( 20) is not fit to perform the tuk of the deaired pe.th fonowing. Ti.e problem of a.da.ptive controner .yntheaia a.n.e.. F'U'IJt of all, let 118 remind one property of dynamic equa.tion ( 6). It ia well known (Khoal.. ud K&na.de, Cralg, Slotine ud Li [9, 10, 11]) tha.t the dynamic structure ( 5) la linear in terml of .. nita.bly Hlected. vector of robot ud 1014 p&nmeters 8 i.e.

A(q,8)q + beg, q, 8)

== Y(q, q, q)8,

~rty it valid for

all values of dynunic parameters 8. Denote re:) the eatima.tion of the dynamic pa.rameters IteCtor 8. Rewrite the dynamic eq1l.&tion ( 5) u follOWll:

= 1£(t) + '7(', r(t»,

(24)

= "Y.A (q, r)e + b(q, q, r),

(2; )

=

e

holds for all q, r E Q &nd for some positive coust&nt (! , Denote 8Q the boundary of Q ud Tr(8Q) the t&ngellt spa.ce at a. point.,. E 8Q to the surface 8Q. Then introduce the opentor Pr, carrying out the ortogonal projection oi the velocity vector 1- on the ta.ngent space Tr(8Q ) in the cue when the point r belongs to the bounda.ry of the set Q a.nd the vector r(t) is directed outside of the set Q. Otherwi..te, Pr la identical opentor. ThUB, in a.d.ditioll to the requirement ( 25) th.e algorithm of the vector l' upde.ting .hould be designed in Rch. a. .....y tha.t the eatime.twne r(t) Ia.t.isfies the following incluion r(~) E

Q

(29)

for..n tEa.. It la to be noted tha.t, in accordlLllc:e with the model ( 24), the value ofthe errorf1lll.ction C&Zl be mea.surecL The COD-tin1l0U time parameter e.tima.tion algorithm will be proposed. in the form of gradient procedure for Iqua:e
{(t) E D!.(CC)'

(23)

"le.re matrix Y(q, 4, q) h.a.s proper dimeuion. Thia prop-

A{q, r(t»q+ 6(q,q, r(t»

u

(30)

ia included. in the error function ( 25). Hence, even if the atima.tiou a.re equal to the uhown vector of robot real. dynamic parameten r(t) == 8, the error fucuon '7(t, r(t)) ia not equal zero. Then to avoid tile dynamic panmeten ai:l.jutment C&.ued. by the intuenoe of external cU.turbuc:e.e the 4"4 .lone ahould be ued in the algorithm of

n - 142

GCSE\' S.\' ., :>.lAKARO\' LA .

dynamic pa.ns.mete~ estimation. Tha.t m~ tha.t pan.meters upda.ting algorithm. mnst be 'switched on' in a momat when the error function exceed! the upper threshold of external disturba.nc:es 111](t, T(t))1I ~ C( . In the cue when II'T( t, r( t»1I < C( the llpda.tin~ &lgorithm. mat be 'lIwitched off" , We shall show tha.t the estima.tion procedure sa.tiaiying the a.bove mentioned requi.remats ( 29) a.nd ( 26) ca.n be written in the following form r ( 1)

= -/3P~(t)

.-------------------~

Pla.nt

ELE

ACA

l'

NHC

[1.(ll1](t, r)lI- C,,) yT(q,riJ) 1](~. r l t))]

PUA

tJ ! )

,

= A(". 1'(t))q + b(g, 4, 1'(t)) -

u(t)

_ %,]1, '"

(32)

One Call see tll.:l.t t he exploiting 0: the projection ope:-a.tor p~ Jea.ds to requirement ( 29) ftiliillment since the lolution ret) of the 'O.?dating alg0rithm ( 31) is defined in the point r l t . ) E 8Q ay the following st:l.tement

f« •• ..-0 \/ -- P.1" [;.. ,r.. . -0 l]

=

=

where +-(:.+0 ) lim._ •• +o :" (t ) a.nd r (t._o) lim,_t._ o "'(1), The reqn.i~ment ( 26 ) !3 iultilied by virtue of t he followill.g lem.m.a.

Lemma 1 Suppou thc t en~Mal di3turbc!'1ceJ. lubjectin; to the plant ( S) heion; to the elau of uniformly hounded di,. tu~;anceJ ( 30) . Then

0.[01'1; the 30lutiof'IJ o/the 3Yltem ( 5), ( 31 ) and Jor any p03itive cnnltant C,,: C" > C( the error function ( U ) hdon;3 to the cla33 of oounded disturbances with ,ummaile protuberanees

L

Wnert c.,

II~(O ) = 213(C", -

,

----------1---------·

where fJ ie a. pOlitive pa.n.m.eter of the algorithm. We strees that value of the error function rK. t, 1') can be mesured in &.CCOrcia.nce to the following formula

1](t, 1'(t))

'

el!~

CC )

In a.ccorda.nce with the exprelsion ( 25), the error fUllctlon is compounded of two terms , The first term {(t) i.tI conditioned by the t!.X~al cU.turba.ncel. The IeCOnd term iI ca.uaed by the errors in the pa..r&meterl'l estima.tion. Recalling the Theorem 2 we CLIl .how tha.t the term of uniformly bounded diaturbuees ma.y be ncceufully nppre88ed by ch~ au!ic:iatly large pin 7. in the c:ozlU'Ol algorithm. Pre8ellted Lemma. .hoW! tha.t the problem of panmetric cU.turba.nc:a wud.ing off' ma.y be .ucceufully .olved alto. Indeed, c:hoomllg n1ficiently la.:ge c:oUta.J1t p, we c:&n fill requirement ( 26) for IIDf po.itive p&n.meter (". Hence, we can rta.te the following theorem ualogoua to the Theorem

2.

Theorem S Supptlle tAAt ",iDt pD.itiDD z( t), lI."tiin.l on.lle ,,(t) An.d 4uif"eti ptllitiDn. ~(t) 14t.Ui'; At tile initiAl moment to c: 0 inefu4liiiu ( 11), ( 18) laDd tile mitwl lIelDciiiu ,Atufy &Ae iDqwt, (%1). La &Ae iDitwl.Alue o! uiiftWicm .~tor T'(O) i£iD"" 'Cl &Ae ut Q. T&1m /tlr OD, pOI;,iee CD,,"."" ! in ( -I), ~ in ( 1&), C( in ( 30) GAll C,,: CIf > C( &An'! C&D ie founti (G) .vffi~mUr 61Mll pD.itioe cerutcnw r in ( .), 6.; in ( 16) GD46. m (%1);

Figure :!: The control algorith.m block di:l.gr:uIl,

(h ) 3ufficiently large pOJitive con31an.'s '1p In ( 9). '16 in ( l:J ) , "Ye in ( !7) and fJ in ( J: ) : 3ucn t.h.11: tile t:Dntrcl aim ( ,.) u rec.iued jor 4lilolutioTU oj the IYJtem ( !). ( 5) . ( 31). ( J!), ( !7). ( 14). T!t~ block dl&!;r:uIl of proPQlled a.daptive control ~~')­ rithm! ie shown. in Figure 2. Inner loop couatl of blow of Desired Motion Cakula.tion (DMC), Ada.ptive Control Aigorithm. (ACA) a.nd Pa.n.m.eten Updating Algorithm (PUA). The e.uenc:e of these blow ie given by equatiOD.JI ( 14). ( 27) a..nd ( 31) correspondingly. Both eqna.t.iolll of Nonholonomlc ConJtnWltl ( 2) (NHC) a.nd Euler-Lagra.nge eqUlLtiolll ( 5) (ELE) form the model of the pla.nt. Note, tb.a.t considered in. ( Slotine II.lld Li [11]) controller of rigid link ma.nipul&tor (holonomic mecha.nical system) hu the similar inner loop structure. But in. contra.diction to this kind of d3alic controller, our control scheme contains ~he block of desired motion clcla.tion (DMC-block) whic!t !la.:; the form of feedba.ck with reapec:t to dependent coordin.3.tee. It Ihould be pointed out tha.t the "impla.nta.tion" of DMCblock into uother kind of controDen (PID-like regula.tonr II.lld o then ) i.tI pOlaible a.nd lew to other controllchemes ( Fn.dk.ov, Guev a.nd Ma.b.rov [7]).

4

Simulation results.

The simula.tion reaulc. de8Crlbed in thia eection are designed to illutme the worb.biIity of preae.nted algorithm. of mobile robot &d.a.ptive cx)1ltrcl. We chooee the c:ircle u the desired PUA of robot motioD.. Figures 3, 4 ud 5 .how the motion tra.jectories &Ad tnLn· aient pl'OCeael in the c:loed loop control.,.te:m. On F'lg1ll'eI 31., 4a. ud Sa. duhed ud IOlid curva dra.w the desired &Ad real tra.jectoriea of robot motion a.ccordillsly. Figures 3b, 4b &Ad 5b dem~ ae trumellt proc:e.es in tile doeed loop control aywtem. Solid., dotted ud duh.ed curves c:omspoJld to pet), 6~(t) - .4(t) - .(t) &Ad 6(t). In the cue when dyu&mic p&l'&Dleten of the mobile robot are nppoeed to be DOWJI. we ue tile cx)lltrol algorithm ( 21)

with T'(t) • , Vt E JlamlC

a.. (igure 3). lA ti.e cue wko dy-

p&ra.meten are 1UUmown &Ad panmeten llpda.tiJlg

algoritlun ( 31) iJ ,twitched off' (i.e. r(t) E T'(O), Vt E 14),

n -143

0 ,' \ ruE PRO!3:...E..\l 0 ;: .\10!3I!...E I
:he tra..nsient process is essentially W'O~e owing to the pua.netric disturbances presence (figure 4). In a.ccordlll.ce with :he Lemm~ we ca.n suppress p~etric di!turblll.C'es by lS~g of pu:uneters updat~g a.lgoritlun ( 31). In th.e case "hen dynamic para.mete~ are unknown a.nd pa.r.uneters IIplatill.g algorithm ( 31) is 'switch.ed on', we can essentially mpro'lll:: tlte t~ient process (figure 5).

. )

[9] P.Kh.oala alld R.Ka.na.de, "Pa.rameter identinca.tion oi robot dynamics", Proceedinll of the IEEE Can.fer-enu on D~e"io" and Concro~ Fort La.uderdale, USA, 1986.

[10] J.J .Craig,

Conclusion .

Adaptiv~

Control of Mechanical Mani~'J.i:; .

tor-J, Adciison- Wesley: New- York, 1 ~8S .

~he th~ cOMeqnentiy compliC':1.ted pr'f)blems of mobile obots path. follow~g hve been considered. Algorith..m.s f path tracking based on kinematic and dynamic models f robot motion and a.d.a.pti ve control algorithm have been onstructed. Control a.lgoritlun tLking ~to account the dyna.mic model frobot motion h.a.! two-level structure. The firs~ level algo.thm ata.bilizea the desired tra.jectory ~ ter:ns of kinema.tic lodel of motion, the lIecone level algorithm IItabilizes the rs~ level feedback. The robustness oi the seconc level J.iorithm in the new class of disturballces with !umm3.ble rotuberancea has been investigated. Tb.e adaptive control algori~hm ha.s been prop oiled to ,Ive the path truking problem in the cue when the dy:un..ic pa..-~.:':lete!"S of the plall~ are llnknown. This algorithm ses the un.known pa.~eters estimation procedure.

ieferences )] J .WittenbuTg, DynamiCl of SYJteml of Rigid BOa.I~, B.G.Tellbner, Stuttg~, 1977.

2J N.H.McCIa.m.roch. !lie D. W:l.D.g, "Feedba.ck

(8] J.Y_S.Lllh. M.W.Walker a.nd R.P.C.Pa.ul, "Reaoiveci. l.ccelera.tion control of mechan.ic3l ma.rupulators". . EEE 7huuactio,,", on. Automatic C(}'UT"O~ v.::S. pp.o468-474,1988.

IIta.b~

tion a.nci tra.citin~ 0; constrainea rooots", IEEE J'NfU· actioTu on Automatic Contr-04 v.33, pp,419-o426, 1988.

3J A.M.Bloch a.nd N.H.McClamroch, "Control ofmecha.n-

ical systems with dassic:u nonh.oionomic constramta," Proeeeding4 of th.e :!.8th IEEE Con.ference on Deeuion IInd Contro~ Thmp:\., Florida, pp.201-20S, 19851.

41 A.M_Bloch, M.Reyha.nogiu, and N.H.McClamroch, "Control a.nd Sta.biliz~tion of Non.4oloJlomic Ca.piygm Dynamic Systems," Proeeeliinll of 30th IEEE Ca~ Jet'ence an. D~uit1n. and Can'ro~ BM/}aan., En,land, Dee.1991, pp.1l27-1132.

>1 G.Butill a.nd G.Campioll., "Ada.ptive Control of NonAoioJlomic Mecha.nica1 Syatem.", in Procee4inll aJ tAe lIt Europe4n Contro' Confermu, Grenobie, Fn.nc:e, 19511, p.1334-1338.

;j S.V.Guev a.nd lA.Ma.ka.rov, "Sta.biliza.tioll. of prognmmed motion of truLlport vehicle with a. tn.cki&ying c:husis", Veltnu LeninlrGd Univenitv: JlAti&-emGtiu, v.22(3), 1989.

1 A.L.Fra4kov,

S.V.Guev a.nd lA.Ma.ka.rov, "Rebut tpeed-gradient ada.ptive control algorithml !or ma.niJ>ul&.ton a.nd mobile robota", ProCftding. DJ tJu JOtA IEEE Ctln/ermee tin D"uion Gnd ConU'D~ BrigJlton, UK,1891.

ll-l44

[11J J.J.E_Slotine :l.D.d W.Li, "Ad3.ptive m:ULipubtor control: A cue study", IEEE 1hl1uG~'io1U on A ucomG,ic Contr-04 v_J3, pp.995-1003, 1988.