A unvrsal sauraon conrollr sgn for mobl robos K.D. Do,,Z.P.Jang an J. Pan Dparmn of Elcrcal an Compur Engnrng, Polychnc Unvrsy, NY, USA. Emal: uc@mch.uwa.u.au, zjang@conrol.poly.u Dparmn of Mchancal Engnrng, Th Unvrsy of Wsrn Ausrala, Nlans, WA 697, Ausrala. Emal: pan@mch.uwa.u.au ABSTRACT: W propos a global m-varyng unvrsal conrollr o achv sablzaon an racng smulanously for mobl robos wh saura npus. Th conrollr synhss s bas on Lyapunov s rc mho an bacsppng chnqu. Numrcal smulaons ar prov o vala h ffcvnss of h propos conrollr. I. INTRODUCTION Ovr h las ca, a lo of nrs has bn vo o sablzaon an racng problms of nonholonomc mchancal sysms nclung whl mobl robos [, 3, 4, 5, 6, 9,,, 4, 6, ]. Th man ffculy of solvng hs problms s u o h fac ha h moon of h sysms n quson o b conroll has mor grs of from han h numbr of conrol npus unr nonholonomc consrans. Th ncssary conon of Broc s horm [7] shows ha any connuous m nvaran fbac conrol law os no ma h null soluon of h whl mobl robos asympocally sabl n h sns of Lyapunov. Sablzaon an racng of whl mobl robos hav hus bn ofn ra sparaly from ffrn vwpons. Sablzaon of whl mobl robos has bn solv by mployng wo man nonlnar conrol approachs. Th frs, na by Bloch an McClamroch [3], uss sconnuous fbac whl h scon mploys mvaryng connuous fbac, whch was frsly su by Samson [6]. Svral smooh fbac conrol laws wr propos by Pom [] wh slow asympoc convrgnc for h cas of rgulaon. Ths conrollrs wr hn mprov by an alrnav conrol schm by M Closy an Murray [6]. Tracng conrol of whl mobl robos has also rcv consrabl anon from h conrol communy [, 3, 4, 9,, 8, ]. In hs rfrncs, bas on Barbala s lmma [8] an h popular bacsppng mho [3] svral m-varyng conrollrs wr vlop o globally follow spcal pahs such as crcls an sragh-lns. In [8, ], only cran rfrnc rajcors can b rac such as hos gnra by h rfrnc mobl robo of whch forwar or angular vlocy canno convrg o zro. In [, 5], a hgh-gan ynamc conrollr was propos o solv boh sablzaon an racng bas on a ransformaon of h mobl robo no a so-call sw form. A rcn rsul on vlopng conrollrs for boh ass o achv praccal sably s gvn n [4]. Saura sablzaon an racng of whl mobl robos hav bn rarly arss n h lraur [9, ]. Mol-bas conrollrs wr propos n [] by applyng passvy an normalzaon o achv sablzaon an racng sparaly. In [9], a mvaryng saura conrollr was propos o solv sablzaon an racng of mobl robos. Howvr, svral racng cass wr no covr. Frsly, h rfrnc forwar vlocy mus b posv. Sconly, h rfrnc angular vlocy mus convrg o zro whn h rfrnc forwar vlocy ns o zro. In hs papr, w prsn a compl soluon o h long sanng opn problm of smulanous sablzaon an racng for a whl mobl robo. Mor spcfcally, w propos a unvrsal saura conrollr ha rmovs assumpons rqur n aformnon paprs o achv sablzaon an racng smulanously. No swchng s n. Th conrol vlopmn s surprsngly smpl an sably analyss s bas on Barbala s lmma an Lyapunov s rc mho. Noaons. Th sngl bar nos h absolu valu of, h oubl bar nos h Euclan norm. A connuous funcon α : R R s of class K f s ncrasng an vanshs a h orgn. For any x >, h sauraon funcon σ ( x s fn as x x f x x, σ ( x = f, x x x > x x f x < x II. PROBLEM FORMULATION Th nmac mol of a whl mobl robo movng n a plan s scrb as [5, 7] = vcos( θ, = vsn( θ, ( θ = ω whr v an ω ar h forwar an angular vlocs, ( xy, an θ ar h poson an ornaon of h mobl robo n h plan, rspcvly. W assum ha h rfrnc nmac mol for ( o b rac s gvn as (
= v cos( θ, = v sn( θ, (3 θ = ω whr noaons n (3 hav smlar manng o hos n (. In hs papr, w propos h frs unvrsal conrollr ha smulanously solvs sablzaon an racng problms of whl mobl robos wh vlocy npus subjc o h followng consrans: sup v( vmax, vmax > sup v(, sup ω( ωmax, ωmax > sup ω( (4 an unr h followng assumpons: Assumpon. Th rfrnc rajcory, x an y, an rfrnc vlocs, v an ω ar boun an ffrnabl wh boun rvav v,.. vmn v vmax < vmax, ωmn ω ωmax < ωmax, (5 v v. max Assumpon. On of h followng conons hols: C. v = ω =. v <. C. ( ( ω( ω( C3. ( τ τ δv(, δv, ω ( τ δ ω(, δ ω, v > < C4. > <. Rmar. Whn h conon C hols, h racng problm bcoms rgulaon/sablzaon on. Th conon C3 covrs h cas of sragh ln racng whl h parng problm blongs o h conon C. Th crcular racng blongs o h cas whn h conon C4 hols. Rmar. In h rcn wor [9], srongr conons han C3 an C4 ar rqur. Frsly, h rfrnc forwar vlocy s posv. Sconly, h rfrnc angular vlocy s no allow o b nonzro whn h rfrnc forwar vlocy s zro or approachs o h orgn. W us h followng poson, ornaon an vlocy rrors [5, 9] as x cos( θ sn( θ x x y sn( θ cos( θ y y =, (6 θ θ θ v = v v, ω = ω ω. From (, (3 an (6 w hav h racng rror ynamcs ( = v v cos( θ ωy ωy, = v sn( θ ωx ωx, (7 θ = ω. Th racng an rgulaon problms of mobl robos bcom h on of sablzng h sysm (7. In parcular, w ar nrs n sgnng xplc xprssons for v an ω such ha lm X( =, 3 T for all an X( R wh X = x, y, θ. In cas of C4, s shown ha µ ( X ( γ X (, < wh γ ( bng a class- K funcon, an µ a posv consan. III. CONTROL DESIGN Frs, w nrouc h followng coorna ransformaon ( z = θ asn y ( / x y (8 wh ( = λ( v λ( sgn( vmn cos( λ3 (9 whr h consans λ, 3 ar such ha < an wll b spcf n h sably sup ( analyss n Scon 4. Th choc of (9 s nrpr as follows. Whn v mn, ha s, v sasfs h prssn xcaon (PE conon or v ( v >, (Conon C3 h rm, mn λ( sgn( v mncos( λ3, vanshs an h rm λ v s sgn o sablz h clos loop racng rror sysm. Whn v mn =, ha s, v =, (Conon C, or lm v( =,(Conon C,hrm λ ( sgn( v mn cos( λ 3 s us o sablz h clos loop racng rror sysm. I s sn ha (8 s wll fn an convrgnc of z an ympls convrgnc of θ. Unr h coorna ransformaon (8, h racng rror ynamcs (7 s rwrn as = v v ( π/ π ωy ωy px, =vy/ π ωx ωx py, z = x/ π ω π y vy/ πωx ( ( ( π vy x v v pz π π y π whr, for noaonal smplcy, w us as (,an
π y px =v ( cos( z sn( z, π π π y py = v sn( z ( cos( z, π π y( xpx ypy pz = py, π π π, (. = x y π = x y W propos h followng unvrsal conrollr v = σ ( x /, x ωyy π ω = ω ω whr vy ω = y ωx π x π y π vy x, v v π π π π ω = ( 3σ ( z, z pz π x All posv sgn consans, 3an x, y, z ar o b chosn n h nx scon. W now prsn our man rsul whos proof s gvn n Scon 4. Thorm. Unr Assumpons an, h unvrsal saura conrollr ( solvs h racng an sablzaon conrol problm formula n Scon wh a suabl choc of h sgn consans, x, y, z, λ. an, 3. Corollary. Th racng rror ynamcs ( wh h saura conrollr ( s locally xponnally sabl a h orgn unr Assumpon an conons C3 an C4 of Assumpon. Proof of hs corollary follows rcly from h proof of Thorm. IV. PROOF OF THEOREM Subsung ( no ( yls h followng racng rror clos loop sysm = σ ( x / ( / x ωyy π v π π ( ω ω ω y px, vy (4 = ωx ωx ωx py, π z = σ ( z. 3 z W wll prov Thorm n h followng orr. Frsly w show ha h conrol law ( sasfs h consran (4. Sconly w prov ha ( x, y, z s boun, hn ha h clos loop sysm (4 s asympocally sabl a h orgn. IV. Bounnss of v an ω ( ( (3 From ( an (3, hr xs sgn consans, λ,,, an,suchha x y z v( v x ω y vmax, (5 ω( ω v v ω ω v v (6 ( ( 4 5 ω. x y v 3 z max W now prsn wo chncal lmmas o smplfy h proof of Thorm. Lmma. Th racng rror clos loop sysm (4 posssss h followng: Thr xss a m nsan such ha lm z( =, <, (7 3( z( z(, <. Thr xss a noncrasng funcon χ of ( x(, y(, z( such ha ( x (, y( χ, <, x( x. (8 3 Thr xss a noncrasng funcon χ3 of x (, y (, z ( such ha ( ( x(, y( χ3,. (9 ProofofLmma. By ffrnang V =.5z along h soluon of h las quaon of (4 yls (7 raly. Consr h followng quarac funcon ( V =.5 x y ( whos m rva along h soluon of h frs wo quaons of (4 sasfs = xσ ( x / x ωxyy π ( v x( π / π xpx vy / π ypy. Whn x x, subsung ( no ( yls =( x y ω v x x v ( z vy ( y v ( z. π W choos h sgn consans λ,, x an y such ha x y ω v. (3 I s sn ha hr always xs λ,, x an y (3 sasfs for boun v an ω. Subsung (3 no ( yls = x v ( z y v ( z vy / π (4 whch s furhr quvaln o = v V z v y / π. (5 ( (
Subsung (7 no (5 hn ngrang boh ss of (5 yls (8 raly. 3 I s asly sn from (. Lmma. Th soluon of h followng ffrnal nqualy ( ξ p ( ξ q ( ξ ( ηξ η σ (6 wh ξ, η, η, σ >, p (, q ( an ( p ( q ( < sasfs: ξ ( π (7 whr π s a noncrasng funcon of ξ (. ProofofLmma. I s asy o show ha (6 s quvaln o η δ ξ η ξ δ p ( η δ ξ η ( ( ( ( η δ (( η δ ξ η q ( ( η δ ξ η σ ( ( (8 whr δ s an arbrarly small posv consan. W nrouc h consan δ o covr h cas of η =. Subsung κ = ( η δ ξ η (9 no (8 yls σ ( (( ( κ η δ p( η δ q( κ (3 whch rsuls n ( η δ/ σ ( η δ/ σ ( κ( κ( η η η δ p( τ τ (3 whr η q( τ τ.snc ( ( ( p q <,h rgh han s of (3 s boun. From (3 an (9, w hav (7 raly. IV. Proof of Thorm. Snc lm z( = (s Lmma, w only n o show ha lm ( x(, y( =. Furhrmor ( x(, y( s boun whn as provn n Lmma. Thrfor w only consr < <. To prov lm ( x(, y( = for < <, w a h followng quarac funcon V =.5y.5( x ωy (3 Dffrnang (3 along h soluons of h frs wo quaons of (4 yls vy =xσ ( x / x ωyy π x y (33 ω y M M Ω whr for noaonal smplcy, w hav fn M = ω x σ ( y ( x v ω v y ( π / π y ωxy ( ωvxy ωvy/ π 3 ω x ωyσx x ω ωy ωxy ω ( ω ω yp y xp x ωω x ωxp y ωω y ωyp x ωω xy ωyp y. (, M = x y x y, (35 Ω= x 4ε ( ω ε ω (36 Afr a lnghy by smpl calculaon by complng h squars an nong ha χ x xσ ( x x (37 x w arrv a p( y p( xσ ( x ( x p y x y (38 3( p3( yσ y( y ( µ V µ whr χ p ( = (4 ω ω v ω ( ( v, ( ( ε εω ω ω ω v ( v, ( (39 p ( = v v (.5 v ( = ω εω εω ω ω ω ω.5 ω εω ω p ( (4 (4 ω (, p3( = ω ( ε (4 for som < ε <,an µ an µ ar noncrasng funcons of z (. W ar now n a poson o slc h sgn paramrs, λ an,, 3 such ha h clos loop sysm (4 s globally asympocally sabl a h orgn. W proc cas by cas accorng o Assumpon. For ach cas, w choos a subs of λ an. Thn h sgn paramrs blong o h subs ha s conan n all of h rv subss. As scuss n Scon, w frs choos
( v λ v λ sgn( mn <, (43 λ >, λ3 >. No ha hs prmary choc guarans sup ( < as rqur n Scon. Sconly, w choos λ an such ha * p p ( > (44 Bfor gong furhr o h choc of λ an,lus scuss ach cas of Assumpon. Cas of v = ω = In hs cas, by nong ha p( = p( = p3( = (45 w hav 3( x 3 x ( µ V µ f x > x, 3( x 3 x ( µ V µ f x < x, (46 x µ V µ 3( f x. ( 3 x ( µ V µ for any Thrfor ( 3 x.isno har o show ha hr xss a noncrasng funcon µ 3 of ( V (, µ, µ, 3 such ha V ( µ 3. Ingrang boh ss of (46 an applyng Barbala s lmma [9] yl lm x( =. To prov lm y( =,applynglmman[9]o h x -ynamc quaon yls lm ( ωyy / πv ( π / π (47 ( ω ω ω y px =. Snc v = ω =, lm px = lm ω =,(47s quvaln o lm ωy =. (48 From (46 w hav 3( V ( µµ µ 3 3 ( 3 µµ 3 µ V s whch mpls ha ( 3 crasng. Snc V s boun from blow by zro, V ns o a fn nonngav consan pnng on x (, y (, θ (. Ths mpls ha h lm of ( y ( xss an s a fn ral numbr, l y.if ly was no zro, hr woul xs a squnc of ncrasng m nsan { τ} wh = τ, such ha boh of h lms ( τ an ( τ y ( τ ar no zro, whch s mpossbl ly bcaus of (48. Hnc mus b zro. Thrfor w conclu from (48 ha y ( as for any λ >, =,3 by nong ha sgn( v mn = n hs cas.wfnhsubsof λ,, x an y sasfyng λ (3, (43 an (44 as Ξ. Cas of ( ( ω( ω( v < Snc 3( x 3 x p( V p( V ( µ V µ f x> x, 3( x 3 x p( V p( V ( µ V µ f x<x, (49 3( x p ( V p ( V µ V µ f x, ( 3 x w hav 3( V p( V p( V ( µ V µ, x R. (5 By nong ha lm p( = lm p ( =, p3( (5 applyng Lmma o (5 yls ha V s boun. I s no har o show from (5 ha V s crasng. Thrfor, by usng h sam argumns as n h prvous Subscon, w hav lm ( x(, y( =. Th subs of λ,, x an y ha sasfs all λ rqurmns n hs cas s h sam as Ξ. 3 Cas of v ( τ τ δ ( v In hs cas, w rwr (38 p( p( xσ x( x p( y χ (5 ( µ V µ ( 3 whch s furhr yls * p x p( x p( y χ δ χ ( µ V µ ( 3 (53 whr δ s an arbrarly small posv consan. Subsung (3 no (53 yls 3( ρ ( V ( µ V µ (54 whr * p x p( ρ( = mn, p (. (55 χ δ ω χ ω From (54, s sn ha hr xs noncrasng funcons ϕ an ϕ of ( x(, y(, z( such ha ( ( x(, y( ϕ ϕ (56 as long as w choos h sgn consans, λ an, such ha ρτ ( τ ϕ (, ϕ >. (57 Snc x(, y(, z(,global asympoc racng s achv. ϕ pns on (
W fn h subs of λ,, x an y sasfyng (3, λ (43, (44 an (57 as Ξ. 4 Cas of ω ( τ τ δ ω (. Sably analyss of hs cas s smlar o ha of h cas C3. Howvr, from (5 an (56, can b shown ha hr xss a fn m nsanc,, such ha x( x. Ths mpls ha hr xss a posv consan ϕ npnnc of h nal conon x (, y (, z ( such ha (56 hols for all ( <, whch mans ha xponnal racng s achv. Th subs of λ,, x an y ha sasfs all λ rqurmns n hs cas s h sam as Ξ. V. CONCLUSIONS A unvrsal conrollr has bn oban n hs papr o solv smulanously rgulaon an racng problms for whl mobl robos wh saura npus. Th propos conrollr s abl o globally asympocally forc h mobl robo o follow any rfrnc rajcory gnra by a suabl vrual robo. Whn h rfrnc angular vlocy sasfs PE conon, w achv h xponnal sably of h clos loop sysm a h orgn afr a consrabl m pro. ACKNOWLEDGEMENTS Ths rsarch suppor parally by a S Gran from h Ohmr Insu for Inrscplnary Sus an by h NSF Grans ECS-9376 an INT-998737. Th suy of h frs auhor s suppor by IPRS an UPA from h Unvrsy of Wsrn Ausrala. REFERENCES [] A. Asolf, Dsconnuous conrol of nonholonomc sysms, Sysms an Conrol Lrs, vol. 7, 996, pp. 37-45. [] A. Bhal, D.M. Dawson, W.E. Dxon an Y. Fang, Robus racng an rgulaon conrol for mobl robos. Procngs of h IEEE Confrnc on Conrol an Applcaon, 999, pp. 5-55. [3] A.M. Bloch an N.H. McClamroch, Conrol of mchancal sysms wh classcal nonholonomc consrans, Procngs of h IEEE Confrnc on Dcson an Conrol, 989, pp. -5. [4] A.M. Bloch an S. Draunov, Sablzaon an racng n h nonholonomc ngraor va slng mo, Sysms an Conrol Lrs, vol. 9, 996, pp. 9-99. [5] C. Canuas W, B. Sclano an G. Basn (Es., Thory of Robo Conrol, Sprngr, Lonon, 996. [6] C. Samson, Conrol of chan sysms-applcaon o pah followng an m-varyng pon sablzaon of mobl robos, IEEE Transacons on Auomac Conrol, vol. 4, no., 995, pp. 64-77. [7] G. Campon, G. Basn an B. Anra-Novl, Srucural proprs an classfcaon of nmac an ynamc mols of mobl robos, IEEE Transacons on Robocs an Auomaon, vol., no., 996, pp. 47-6. [8] H.K. Khall, Nonlnar sysms. 3 r., Prnc-Hall, NJ,. [9] I. Kolmanovsy an N.H. McClamroch, Dvlopmns n nonholonomc conrol problms, IEEE Conrol Sysms Magazn, vol. 5, 995, pp.-36. [] J. B. Pom, Explc sgn of m-varyng sablzng conrol laws for a class of conrollabl sysms whou rf, Sysms an Conrol Lrs, vol. 8, 99, pp. 467-473. [] J. Luo an P. Tsoras, Exponnal convrgn conrol laws for nonholonomc sysms n powr form, Sysms an Conrol Lrs, vol. 35, 998, pp. 87-95. [] J-M. Yang an J-H. Km, Slng mo conrol for rajcory racng of nonholonomc whl mobl robos, IEEE Transacons on Robocs an Auomaon, vol. 5, no. 3, 999, pp. 578-587. [3] M. Krsc, I. Kanllaopoulos, an P.V. Kooovc, Nonlnar an aapv conrol sgn, Nw Yor: Wly, 995. [4]P. Morn an C. Samon, Praccal sablzaon of a class of nonlnar sysms: Applcaon o chan sysms an mobl robos, 39 h IEEE Confrnc on Dcson an Conrol, Syny,, pp. 44-5. [5] R. Olfa-Sabr, Nonlnar conrol of unracua mchancal sysms wh applcaon o robocs an arospac vhcls, Massachuss Insu of Tchnology,. [6] R.T. M Closy an R.M. Murray, Exponnal sablzaon of nonlnar rflss conrol sysms usng homognous fbac, IEEE Transacons on Auomac Conrol, vol. 4, no. 5, 997, pp. 64-68. [7] R.W. Broc, Asympoc sably an fbac sablzaon, n R.W. Broc, R.S. Mllman an H.J. Sussmann (s, Dffrnal gomrc conrol hory, 983, pp. 8-9. [8] T. Kuao, H. Naagawa an N. Aach, Aapv racng conrol of nonholonomc mobl robo, IEEE Transacons on Robocs an Auomaon, vol. 6, no. 5,, pp. 69-65. [9] T.C. L, K.T. Song, C.H. L an C.C. Tng, Tracng conrol of uncycl-moll mobl robos usng a sauraon fbac conrollr, IEEE Transacons on Conrol Sysms Tchnology, vol. 9, no.,, pp. 35-38. [] Z.P. Jang an H. Njmjr, A rcursv chnqu for racng conrol of nonholonomc sysms n chan form, IEEE Transacons on Auomac Conrol, vol. 44, 999, pp. 65-79. []Z.P. Jang an H. Njmjr, Tracng conrol of mobl robos: a cas suy n bacsppng, Auomaca, vol. 33, 997, pp. 393-399. [] Z.P. Jang, E. Lfbr an H. Njmjr, Saura sablzaon an racng of a nonholonomc mobl robo, Sysms an Conrol Lrs, vol. 4,, pp. 37-33.