Two-Wheeled Welding Mobile Robot for Tracking a Smooth Curved Welding Path Using Adaptive Sliding-Mode Control Technique

Transcription

1 Intnational wo-whld Jounal Wlding of Contol, Mobil Automation, Robot fo acking and Systms, a Smooth vol. Cuvd, no. 3, Wlding pp , Path Using Jun Adaptiv 7 Sliding-Mod 83 wo-whld Wlding Mobil Robot fo acking a Smooth Cuvd Wlding Path Using Adaptiv Sliding-Mod Contol chniqu Ngo Manh Dung, Vo Hoang Duy, Nguyn hanh Phuong, Sang Bong Kim*, and Myung Suck Oh Abstact: In this pap, a nonlina contoll basd on adaptiv sliding-mod mthod which has a sliding sufac vcto including nw boundizing function is poposd and applid to a twowhld wlding mobil obot (WMR). his contoll maks th wlding point of WMR achiv tacking a fnc point which is moving on a smooth cuvd wlding path with a dsid constant vlocity. h mobil obot is considd in viw of a kinmatic modl and a dynamic modl in Catsian coodinats. h poposd contoll can ovcom unctaintis and xtnal distubancs by adaptiv sliding-mod tchniqu. o dsign th contoll, th tacking o vcto is dfind, and thn th sliding sufac vcto including nw boundizing function and th adaptation laws a chosn to guaant that th o vcto convgs to zo asymptotically. h stability of th dynamic systm is shown though th Lyapunov mthod. In addition, a simpl way of masuing th os by potntiomts is intoducd. h simulations and xpimntal sults a shown to pov th ffctivnss of th poposd contoll. Kywods: Adaptiv contol, Lyapunov function, nonlina contol, sliding-mod contol, wlding mobil obot. 1. INRODUCION Wlding automation has bn widly usd in sval manufactuing filds, and on of th most complx applications to manufactuing filds is a wlding systm basd on autonomous obots. Som spcial wlding obots can povid sval bnfits in ctain wlding applications. Among thm, a wlding mobil obot usd in lin wlding application can gnats th pfct movmnts at a ctain tavl spd, which can poduc a consistnt wld pntation and wld stngth. In pactic, som vaious obotic wlding systms hav bn dvlopd cntly. Kim, t al. [1] dvlopd a th dimnsional las vision systm fo an intllignt shipyad wlding obot to dtct th wlding position and to cogniz th 3D shap of th wlding nvionmnts. Jon, t al. [] psntd th sam tacking and motion contol of a two-whld wlding mobil obot fo lattic wlding; th contol Manuscipt civd May 3, 6; visd Fbuay 1, 7; accptd Fbuay 1, 7. Rcommndd by Editoial Boad mmb Sangdok Pak und th diction of Edito Ja-Bok Song. his wok was suppotd by Pukyong National Univsity Rsach Foundation Gant in 3. Ngo Manh Dung, Vo Hoang Duy, Nguyn hanh Phuong, Sang Bong Kim, and Myung Suck Oh a with th Collg of Engining, Pukyong National Univsity, San 1, Yongdang-dong, Nam-gu, Busan , Koa (-mails: {vndungtuyn, vhduy, phuongkoa}@yahoo.com, {kimsb, msoh}@pknu.ac.k). * Cosponding autho. is spaatd into th diving motions: staight locomotion, tuning locomotion, and toch slid contol. Kam, t al. [3] poposd a mobil wlding obot fo staight wlding path using body positioning snsos and sam tacking snso. Both of contolls poposd by Jon and Kam hav bn succssfully applid to th pacticd fild. Chung, t al. [4] poposd a sliding mod contol fo a mobil obot tacking a smooth-cuvd path vn in th systm with known boundd distubanc. In pactic, it is vy difficult to know th boundd distubanc of th WMR bcaus th pssu of wlding ac maks nonlina distubanc to th WMR. A long wlding pow cabl and CO gas tub a connctd fom wlding systm to th WMR of GMAW (gas mtal ac wld) pocss. Futhmo, most of pvious sliding mod contol mthods a applid to mobil obot fo tacking a fnc path but cannot liminat th o ppndicula to its hading diction although th angula o of th mobil obot achivs to zo fistly. hfo, in thi simulation and xpimnt sults, th postu of th mobil obot is chosn so that th o ppndicula to its hading diction convgs to zo bfo th angula o convgs to zo. o solv ths poblms, a nonlina contoll using adaptiv sliding-mod mthod which has a sliding sufac vcto including nw boundizing function is poposd and applid to th WMR fo tacking a smooth cuvd wlding path. h nw boundizing function is applid to dsign sliding

2 84 Ngo Manh Dung, Vo Hoang Duy, Nguyn hanh Phuong, Sang Bong Kim, and Myung Suck Oh 1 7 Wlding wi fd of th wlding systm h WMR Contol box of th WMR + - C Wlding systm Gas cylind touch-snso; wlding toch; 3,4 lft and ight whl diving motos; wlding point of WMR; 6 toch-slid-diving moto; 7 toch-slid Fig. 1. Configuation of th WMR. sufac vcto fo sliding mod mthod. h sliding mod mthod with this sliding sufac vcto nfocs all os to convg to zo vn though th angula o convgs to zo fistly. Unctaintis and xtnal distubancs of a systm a stimatd by adaptiv tchniqu. h stability of th dynamic systm is shown though th Lyapunov mthod. In addition, a simpl way of masuing th os by potntiomts is intoducd. h simulations and xpimntal sults a shown to pov th ffctivnss of th poposd contoll.. WMR SYSEM In this sction, th kinmatic and dynamic modls of th WMR a considd with nonholonomic constaints systm. h WMR is modld und th following assumptions: (1) h adius of wlding cuv is sufficintly lag than th tuning adius of th WMR, () h obot has two diving whls fo body motion, and thos a positiond on an axis passd though th obot s gomtic cnt, (3) wo passiv whls a installd in font and a of th bottom of mobil platfom fo its balanc, and thi motion can b ignod in th dynamics, (4) h mass cnt and th otation cnt of th WMR a assumd to b sam, () A toch slid is contolld by toch-slid-diving moto and locatd so as to coincid with th axis though th cnt of two diving whls, (6) A magnt is st up at th bottom of th obot s cnt to avoid slipping, (7) h unctaintis and xtnal distubanc a assumd to b unknown and boundd, and also thi divativs a assumd to b zo. h modl of th WMR as shown in Fig. has nomnclatus as th following: ( x, y) : Coodinats of th WMR s cnt [m] φ : Hading angl of th WMR [ad] v : Lina vlocity of th WMR s cnt [m/s] ω : Angula vlocity of th WMR s cnt [ad/s] ω w, ω lw : Angula vlocitis of th ight and th lft whls [ad/s] x, y ) : Coodinats of th wlding point [m] ( w w φ v w w : Hading angl of th wlding point [ad] : Lina vlocity of th wlding point [m/s] ω w : Angula vlocity of th wlding point [ad/s] x, : Coodinats of th fnc point [m] v y ω φ b l τ τ y y y w y w, Wlding point Rfnc point R( x, y, φ ) W( xw, y w, φ w ) och hold Y WMR x w Fig.. WMR configuation. 1 x : Dsid constant wlding vlocity [m/s] : Angula vlocity of th fnc point [ad/s] : Rfnc hading angl [ad] : Distanc btwn diving whl and th symmtic axis [m] : Radius of diving whl [m] : och hold lngth [m] : Contol input vcto [Kgm] τ lw : oqus of th motos acting on th ight and th lft whls [Kgm] x b φ C( x, y, φ ) X 3 φ l fnc wlding path φ x

3 wo-whld Wlding Mobil Robot fo acking a Smooth Cuvd Wlding Path Using Adaptiv Sliding-Mod 8 m c : Mass of th body without th diving whls [Kg] m w : Mass of ach diving whl with its moto [Kg] I w : Momnt of intia of whl and its moto about th whl axis [Kgm ] I m : Momnt of intia of whl and its moto about th whl diamt axis [Kgm ] I c : Momnt of intia of th body about th vtical axis though th mass cnt of th WMR [Kgm ] M (q) : Symmtic, positiv dfinit intia matix V( q, q) : Cntiptal and coiolis matix B (q) : Input tansfomation matix A (q) : Matix latd with th nonholonomic constaints λ : Constaint foc vcto : acking o vcto u : Contoll vcto..1. Kinmatic modl of th WMR Consid a obot systm having an n-dimnsional configuation spac with gnalizd coodinat vcto q = [ q 1,, q n ] and subjct to m constaints of th following fom: Aqq ( ) =, (1) m n wh A(q) R is th matix associatd with th constaints. As a sult, th kinmatic modl und th nonholonomic constaints in (1) can b divd as follows: q = J(q)z, () wh J(q ) is a n ( n m) full ank matix satisfying J (q)a (q) =, and z R is vlocity vcto. n m Fistly, th postu of mobil obot fo th cnt point of WMR, C ( x, y) n th Catsian spac in Fig., is dfind as q = [ xyφ,, ]. (3) If th mobil obot has nonholonomic constaint that th diving whls puly oll and do not slip, A(q) in (1) can b xpssd into [ φ φ ] A(q) = sin cos. (4) Fom (3)-(4), n = 3 and m = 1. h vlocity vcto in () is dfind as [ v ω]. z = () In th kinmatic modl of (), J(q) is wittn as cosφ J(q) = sinφ. (6) 1 h lationship btwn v, ω nd th angula vlocitis of two diving whls is givn by ωw 1/ b/ v. ω = lw 1/ b/ ω (7) Scondly, th kinmatic quation of th wlding point W ( x w, yw ) fixd on th toch hold can b divd fom th WMR s cnt C ( x, y) in Fig. as following [6]: xw = x lsinφ yw = y+ lcosφ φw = φ. h divativ of (8) yilds x w cosφ lcosφ l sinφ v y w = sinφ lsinφ + l cos φ, ω φw 1 (8) (9) wh l is contolld by toch-slid-diving moto. h coodinats ( x, y ) and th fnc hading angl φ of th fnc point R, which is moving on th fnc wlding path with th dsid constant vlocity of v, satisfis th following quations: x = v cosφ y = v sinφ φ = ω. (1) In Fig., th o vcto = [ 1,, 3] is dfind as th diffnc btwn th wlding point of WMR and th fnc point. h lationship of th o vcto btwn th global coodinat and th WMR s coodinat can b xpssd as follows: cosφ sinφ x x sin cos. 1 φ φ y y = 3 1φ φ h fist divativ of o vcto yilds 1 + l v cos sin. 1 3 v = 1 + v 3 l ω 3 ω (11) (1)

4 86 Ngo Manh Dung, Vo Hoang Duy, Nguyn hanh Phuong, Sang Bong Kim, and Myung Suck Oh.. Dynamic modl of th WMR h dynamic quations of th mchanical systm und nonholonomic constaints in (1) can b dscibd by Eul-Lagang fomulation as follows []: M(q)q + V(q, q)q = B(q)τ A (q)λ, (13) n n wh M(q) R is a symmtic positiv dfinit n n intia matix; V(q, q) R, a cntiptal and n Coiolis matix; B(q) R, an input tansfomation matix; A(q) R, a matix of nonholonomic m n constaints; τ R, a contol input vcto; and m λ R, a constaint foc vcto. Fo simplicity of analysis, it is assumd that = n m. Diffntiating (), substituting this sult in (13) and multiplying by J, th constaint matix A (q)λ is liminatd. Dynamics in platfom systm of th nonholonomic mobil obot with th constaint in (1) is as follows []: J MJz + J (MJ + VJ)z = J Bτ. (14) ( n m) 1 wh z R is fnc input vcto. Fom (16), (17) and (18), (19) is obtaind. f = u ( z z ) (19) In this pap, whn q = [ x, y, φ] is takn, that is, n = 3, m = 1 and =. h followings a obtaind fom (16)-(19). V md c φ b = [ τ τ ] md c φ b, = c + w + c + m, = [ v ω] f = [ f1 f] () ( mb + I ) + I ( ) w mb I 4b 4b M =, ( mb I ) ( mb + I ) + I w 4b 4b w lw c w τ =, m= m + m, I m d m b I I z,. Multiplying by ( B), follows: J (14) can b wittn as 3. ADAPIVE SLIDING-MODE CONROLLER DESIGN wh M(q)z + V(q, q)z = τ, (1) 1 ( n m ), 1 ( n m) M(q) = (J B) J MJ R V(q, z) = (J B) J ( MJ + VJ) R. In this pap, th bhavio of th wlding mobil obot in th psnc of xtnal distubancs 1 τ d R is considd. h al dynamic quation of th wlding mobil obot with th xtnal distubancs can b divd fom (1) as follows: M(q)z + V(q,q)z + τ = τ. (16) d It is assumd that th distubanc vcto can b xpssd as a multipli of matix M (q) as th following: Ou objctiv is to dsign a contoll so that th wlding point W tacks th fnc point R at a dsid constant vlocity of wlding v. So th dsignd contoll maks th WMR achiv as t. In this pap, th WMR is contolld in two cass: fixd toch slid and contollabl toch slid. In th scond cas, th lngth of th toch is contolld by toch-slid-diving moto Contoll dsign fo fixd toch In this cas, th l is qual to zo. o dsign an adaptiv sliding mod contoll, th sliding sufacs a dfind as follows: s + k s = =, s 3 + k3 + k3ψ ( 3) (1) τ = M(q)f, (17) d ( n m) 1 wh f R is th vcto of unctainty and th xtnal distubanc of systm. By a fdback linaization of th systm, th ( n m) 1 contoll vcto u R is dfind by computdtoqu mthod as follows []: ε ε 1 ψ ( 3 ) ε ε 3 τ = M(q)z + V(q,q)z + M(q)u, (18) Fig. 3. Chaactistic of ψ ( ) function.

5 wo-whld Wlding Mobil Robot fo acking a Smooth Cuvd Wlding Path Using Adaptiv Sliding-Mod 87 wh k1, k and k 3 a positiv constant valus; a boundizing function ψ ( ) is dfind as follows: 1 if 3 ε ψ( 3) = 1 if 3 ε no chang if ε < 3 < ε, () wh ε is positiv constant valu. No chang mans th valu of ψ ( ) function continuously kps its valu at point 3 = ± ε o 3 = ±ε bfo 3 nts into ε < 3 <. ε On th sliding sufac vcto ( s = ), th followings a obtaind fom (1): 1 = k 1 1, (3) = k k ψ ( ). (4) In (3), if 1 is positiv, 1 is ngativ, and vic vsa. hus, th quilibium point of 1 convgs to zo as t. It mans th o 1 as t along th sliding sufac ( s 1 = ). Bcaus th configuation of th WMR has th wlding point which is fixd on th axis though th two contact points btwn diving whls and floo, th o can b liminatd via 3. But, in th cas of 3 =, th valu of is constant bcaus WMR is unning paalll with fnc tajctoy. Fo that sult, th o 3 and its divativ in (4) cannot b zo in od to liminat th o whn th o. In cas of 3 > ε, fistly, ψ ( ) is inactiv until 3 ε. So (4) bcoms 3 = k 3. Similaly with (3), 3 convgs into ( ε ε ). Scondly, whn <, ψ ( ) is activ in 3 < ε. So (4) bcoms 3 ε 3 = 3 3. k k his sult maks o 3 b changd accoding to th valu of o. Basd on th spcific configuation of th WMR, this phnomnon focs. Whn =, (4) bcoms 3 = k 3. So 3 as t. h convgnc to zo of o vcto is shown claly in Figs. 1, 11 and 14 of th simulation sults sction. h following pocdu is to dsign an adaptation law vcto ρ ˆ and a contoll vcto u which mak th sliding sufac b stabilizd and convg to zo as t. Fistly, th adaptation law is poposd as th following: ρ ˆ = ξ s(), t () wh ρ ˆ = ˆ ρ1 ˆ ρ is an stimat valu of f = f f ξ ; = 11 ξ is positiv dfinit ξ matix which is dnotd as an adaptation gain. h stimation o is dfind as follows: ρ = f ρˆ ρˆ = f ρ. (6) Scondly, th contoll vcto u is chosn as follows: u = u 1 u ( v sin 3 1ω + l ) ω+ ( + l) ω v sin 3( ω ω) = k1[ + l] ω v+ v cos 3 + k( ω ω) + k3ψ( 3)( v sin 3 l 1ω + Qs + P sgn( s). (7) Fom (1), th (7) can b wittn as follows: ( + l ) ω + ( + l) ω v 3sin3 u = k sgn( ), k 3+ k3ψ ( 3) Qs P s (8) q11 ˆ ρ1 wh Q = and P = a positiv q ˆ ρ dfinit matics. hom: h abov contoll vcto u and adaptation law vcto ρ ˆ with th assumption (7) mak th sliding sufacs in (1) b stabilizd and convg to zo as t. Fom (1)-(4), this implis that th o vcto as t. Poof: Bcaus th wlding vlocity is constant, v =. Fom (1), th fist and th scond divativs of 1 and 3 yild 1 v cos 3 + ( + l) ω v, = 3 ( ω ω ) 1 ( + l ) ω + ( + l) ω v 3sin3 = 3 ( v v ). ( ω ω ) (9) (3) Using (19) and (8), (3) can b wittn as follows:

6 88 Ngo Manh Dung, Vo Hoang Duy, Nguyn hanh Phuong, Sang Bong Kim, and Myung Suck Oh 1 ( + l ) ω+ ( + l) ω v 3sin3 = +, f u (31) 3 1 k1 1 sgn( ). = f Qs P s 3 k3 + k3ψ ( 3) (3) Fom (1) and (3), th fist divativ of th sliding sufacs yilds s k = + = f Qs P sgn( s ). (33) k3 k3ψ ( 3) + h following Lyapunov function is chosn as 1 1 V = s s+ ρ ξρ. (34) Its divativ yilds V = s s + ρ ξρ. (3) h divativ of (6) and th assumption (7) yilds ρˆ = f ρ = ρ. (36) hfo, th divativ of Lyapunov function is as follows: V = s Qs+ s f s P sgn( s) + ( f ρˆ ) ξρ (37) = s Qs ( s ρˆ s ρˆ) s Qs. Sinc V and V is ngativ smi-dfinit, by Babalat s lmma, s as t. Fom (), this implis that ρˆ has constant valu as t. Fom (1), (3), and (4), this implis that as t. Whn, th wlding point of th WMR achivs tacking a fnc point which is moving on a smooth wlding path at a spcifid constant vlocity. 3.. Contoll dsign fo contollabl toch In th wlding application fild, th wlding vlocity is vy slow, which is 7.mm / s. hfo, if th xists an o, th WMR taks a long tim to convg th os vcto to zo. o solv this poblm, a sliding contollabl toch hold is placd fo fixd toch hold in th pvious cas. A toch-slid-diving moto is usd to div th sliding toch. In this cas, th lngth l is changabl so ( l ). Futhmo, a nw updat law l fo sliding contollabl toch is dsignd. h sam concpt with th pvious cas, a nw sliding sufac vcto s and a nw contoll vcto u a obtaind fom (1)-(8) with ( l ). o dsign an updat law fo sliding contollabl Rfnc valus By snso Eq. (4) d dt toch, a Lyapunov function candidat fo o as follows: 1 Vm =, (38) V = = ( ω + v sin l ). (39) m z 1 3 h updat law fo sliding contollabl toch is achivd by V as follows: m l = v sin + k ω. (4) Eq. (1) Eq. (8) o avoid ovshot contolling of, th maximum spd of sliding toch has to b limitd by satuation function blow l if l < δ ( v ) sat() l = δ( v) if l > δ( v), u s Eq. (4)-(41) Eq. () Eq. (19) (41) wh δ ( v ) is positiv valu which dpnds on v is chosn by dsign. Claly, V, Vm, V and V m with th poposd contoll vcto u, adaptation law vcto ρ ˆ and updat law l a satisfid. So th o vcto as t. Fo tacking a fnc wlding path, basd on th o vcto which is divd fom touch snso, th angula vlocitis of lft and ight whls of th WMR a obtaind by th following block diagam Fig SIMULAION AND EXPERIMENAL RESULS o vify th ffctivnss of th poposd contolls, th simulation has bn don fo th two cass of fixd toch and contollabl toch with a smooth cuvd fnc wlding path. Fig. shows th fnc cuv wlding path with staight lin of L 1 = 11mm, ac cuv lin of ( R 1 = 9.mm, 4 ), staight lin of L = 36mm, ac cuv lin of f ρ ˆ τ Eq. (17) d τ z Eq. (18) Eq. (16) l l ρˆ ωw ωlw Eq. (7) z Ral position of th wlding point Eq. (3) Eq. () q q Eq. (8) Fig. 4. Block diagam fo tacking a fnc wlding path.

7 wo-whld Wlding Mobil Robot fo acking a Smooth Cuvd Wlding Path Using Adaptiv Sliding-Mod 89 4 (3, 434) (4, 46) (36, 434) 4 (8, 38) R=9., 4 Y coodinat (mm) 3 3 (, 31) R1=9., 4 (, ) X coodinat (mm) Fig.. Smooth cuvd fnc wlding path. ( R = 9.mm, 4 ) and staight lin of L 3 = 38mm. h simulation sults show th o vcto convgs to zo fast in th cas of contollabl toch than in th cas of fixd toch. Futhmo, th xpimnt has bn don fo th contollabl toch cas Hadwa of th whol systm Fig. 6 shows configuation of th contol systm. h contol systm is basd on th intgation of two micocontoll PIC18F4s: on is usd fo two svo DC moto contol signal of lft whl and ight whl. Anoth is usd fo svo toch slid contoll and main cnt pocsso unit (CPU). h th svo contolls a contolld by CPU. h main contoll functionalizd as mast links to th th svo contolls via IC communication. h two A/D pots of th CPU a connctd with th two potntiomts fo snsing th os as considd in Sction 4.. wo micocontolls PIC18F4s a opatd with th clock fquncy 4MHz. h svo DC moto has 16-bit gist fo th captu modul which is usd fo civing signal fom moto-ncod and PWM modul fo contolling th PWM of DC moto. h sampling tim of contol Lft Whl Gabox Lft whl Moto Encod o o ouch snsos 1. Lina potntiom t. Angula potntiomt Fig. 7. Expimntal wlding mobil obot. systm is 1ms. h xpimntal WMR is shown in Fig. 7 and its dimnsions a shown in abl Masumnt of th os 1,, 3 In od to masu th componnts of th o vcto, a simpl masumnt schm using potntiomts is shown in Fig. 8. wo olls a placd at points O and O 3. wo snsos fo masuing th os a ndd. hat is, thy a on lina snso fo masuing d s and on otating snso fo masuing th angl btwn th toch and th tangnt lin of th wall at th wlding point. Fom Fig. 8, th lation of th componnts of o vcto can b xpssd as follows: 1 = s sin 3, = ds + s cos 3, = ( OO, O E) π, (4) wh O and O 3 a th cnt points of oto O and O 3 spctivly, O 1 is th cnt point of OO 3, E is th point on toch hold, s is th adius of th oll, d s is th lngth masud by th lina potntiomt, and 3 is th angl masud by th otating potntiomt. In Fig. 8, th wlding path is a lin. Whn th wlding path is a cuv, (4) Rfnc wlding path 1 R(x R,y R ) v R Svo Contoll of lft w hl m oto PIC18F4 Svo Contoll of ight w hl m oto IC Comm. Main CPU PIC18F4 Svo contoll of och slid diving moto Display & Kypad O O 3 d s 3 O 1 v E s E(x E,y E ) Roll Right Whl Gabox Right whl Moto Encod Encod och slid diving moto Fig. 6. Configuation of th contol systm. Gabox och slid och Fig. 8. Schm fo masuing th o vcto.

8 9 Ngo Manh Dung, Vo Hoang Duy, Nguyn hanh Phuong, Sang Bong Kim, and Myung Suck Oh [(6mms dgs )], th adaptation gain vcto ξ = [(1s ); ( 1s )], k 4 =.s and δ v ) =.ms. wo initial o vctos a considd as follows: 1 = (.4mm, 4.9mm, 6dg) and = (4.6mm, 6 mm, 6dg). h WMR s paamts and th initial valus fo th simulation a givn in abls 1 and. ( Fig. 9. ouch snso. is also valid if th distanc O O 3 is sufficintly small and th adius of th wlding path is nough lag Simulation and xpimntal sults h wlding spd in this application is v = 7. mm/s. h sampling tim of contol systm is 1ms. h dsignd paamts a as follows: k 1 = s, k = 6s, k 3 = 1s, =3 ε, Q = [(3s ); ( 1s )], th initial stimatd valu ρ ˆ = abl 1. Paamt valus of th WMR. Paamt Valu Unit b.1 m. m m. Kg w I w Kgm l.14 m m c 1 Kg I.81 Kgm c I m Kgm abl. Initial valus fo th simulation and xpimnt. Paamt Valu Unit x. m x.4 m w v m/s φ π / ad l.14 m y. m y.19 m w ω ad/s φ 84 o 96 dg ω ad/s Simulation sults fo th cas of fixd toch In this cas, th tacking o vcto has bn simulatd with two diffnt initial o vctos 1 and as shown in Figs. 1 and 11. Fig. 1 shows that fistly, ψ ( ) in (4) is un-activ and th o 3 intnds to convg to boundd o o ε ε = Scondly, whn, limit ( ) ( 3 3 ). 3 ε ψ ( ) is activ in th boundd ( ε ε ) at this tim. So 3 is changd in od to convg to zo. Whn, th pat [ k3ψ ( 3) ], too. hfo, o 3 is dcasd as is dcasd. Finally, whn, 3. acking o vcto ( mm) 1 ( mm) 3 (dg) im (s) Fig. 1. acking o vcto with initial o vcto 1. acking o vcto ( mm) 1 ( mm) 3 (dg) im (s) Fig. 11. acking o vcto with initial o vcto.

9 wo-whld Wlding Mobil Robot fo acking a Smooth Cuvd Wlding Path Using Adaptiv Sliding-Mod Rfnc wlding path.4.4 Rfnc wlding path Y coodinat (m).3.3 h WMR Y coodinat (m).3.3 h WMR. ajctoy of wlding point. ajctoy of wlding point X coodinat (m) X coodinat (m) Fig. 1. Movmnt of th WMR with fixd toch by initial o vcto. Fig. 11 shows that fistly, th is whn 3 convgs to boundd limit ( ε ε ). Du to th chaactistic of ψ ( 3 ), 3 is changd in od to convg to zo. In this cas if th o 3 > ε, ψ ( 3 ) is un-activ. So 3 appoachs into boundd limit ( ε ε ) again. his phnomnon focs th o vcto to b zo as t. Fig. 1 shows th movmnt of th WMR along th fnc wlding path with th initial o vcto. h poposd contoll maks th o vcto as t and th WMR tack th whol fnc wlding path vy wll and tightly. In Fig. 1, it naly taks about sconds to convg th vcto o to zo. So this disadvantag is ovcom by contollabl toch Simulation and xpimntal sults fo th cas of contollabl toch h WMR with contollabl toch movs along th fnc wlding path with th sam initial o vcto as shown in Fig. 13. In th cas of contollabl toch, th o vcto convgs to zo fast than in th cas of fixd toch as shown in Fig. 14. Aft about sconds, all th os convg to zo duing th wlding pocss. Fig. 1 shows th simulation and xpimnt sults fo tacking o vcto with initial o vcto 1 duing 1 sconds at bginning. It shows that th xpimnt sult of th tacking o vcto is boundd along th simulation sult. Fig. 16 shows th angula vlocity of th cnt of th WMR ω fo tacking staight lin, ac lin, Fig. 13. Movmnt of th WMR with toch contollabl by initial o vcto. acking os vcto ( mm) 1 ( mm) 3 (dg) im (s) Fig. 14. acking o vcto with initial o vcto 1. acking o vcto Sim ulation sults Expim ntal sults im (s) Fig. 1. Simulation and xpimntal sults with initial o vcto 1 fo 1 sconds at bginning. staight lin, ac lin and staight lin of th fully fnc wlding path. h angula vlocity ω has a littl lag chatting duing tacking th ac lin

10 9 Ngo Manh Dung, Vo Hoang Duy, Nguyn hanh Phuong, Sang Bong Kim, and Myung Suck Oh 1 Fo acking a staight lin 8 Angula vlocity (dg/s) - -1 Fo acking a cuv lin Contol vcto u Contol input u [dg/ s ] Contol input u 1 [ mm / s ] im (s) Fig. 16. Angula vlocity of cnt of th WMR im (s) Fig.. Contoll vcto u. angula vlocitis of ight and lft whls Angula vlocity of lft whl (dg/ s ) Angula vlocity of ight whl (dg/ s ) im (s) Fig. 17. Angula vlocitis of ight and lft whls of th WMR. Contol vcto U Contol input u1[ mm/ s] Sim ulation sults Contol input u [dg/ s] im (s) Fig. 1. Expimntal sults of contoll vcto u fo 1 sconds at bginning. Lina vlocity of wlding point mm/s im (s) Fig. 18. Lina vlocity of wlding point. Sliding sufac vcto Sliding sufac Sliding sufac s 1 [ mm / s ] s [dg/ s ] im (s) Fig. 19. Sliding sufac vcto s. Lina vlocity (mm/s) and toch lngth (mm) 1 och lngth (mm) Lina vlocity (mm/s) im (s) Fig.. Lina vlocity of toch slid l [mm/s] and toch lngth l [mm]. Estimatd valu of distubanc 1 ˆ ρ1[ mm / s ˆ ] ρ [dg/ s ] im (s) Fig. 3. Estimatd valus of distubancs ˆ ρ 1 [mm / s ] and than duing tacking on th staight lin bcaus th sliding mod is mo ffctiv on th ac lin than on th staight lin. Fig. 17 shows th angula vlocitis of two whls of th WMR. It shows that th vibation of angula vlocitis is lag fo tacking an ac lin than fo ˆ ρ [dg/ s ].

11 wo-whld Wlding Mobil Robot fo acking a Smooth Cuvd Wlding Path Using Adaptiv Sliding-Mod 93 tacking a staight lin. So Fig. 18 shows that th vibation of wlding spd is lag fo tacking th ac lin than fo tacking th staight lin. h lina vlocity of wlding point in Fig. 18 is v w = v ±.6mm/s. In th pactic of wlding fild, th o of wlding vlocity aound 1.mm/s is accptabl. Fig. 19 shows that th valu of sliding sufac vcto s convgs to th avag valu of zo vy fast duing th wlding pocss. h vcto s hav som stang pulss at th position wh th fnc wlding lin is changd fom staight lin to cuv o vic vsa but it convgs vy apidly to an avag valu of zo. Figs. and 1 show that th simulation and xpimntal sults of contol vcto u a boundd and thi avag valu convgs to zo. Fig. shows that th lina vlocity of contollabl toch is zo and th lngth of toch slid is constant whn o vcto convgs to zo. It mans that th contollabl toch acts as th fixd toch whn o vcto convgs to zo. Bcaus th sliding sufac vcto has an avag valu of zo, by adaptation law in (), th stimatd valus of distubanc convg to thi constant valu as shown in Fig. 3.. CONCLUSIONS A nonlina contoll basd on adaptiv slidingmod mthod which has a sliding sufac vcto including nw boundizing function to nhanc th tacking pfomancs of th WMR has bn intoducd. h contoll vcto is obust and unsnsitiv in spit of unctaintis and xtnal distubancs. o achiv th contoll vcto u, adaptation law vcto and updat law of th WMR a considd in viw of a kinmatic modl and a dynamic modl. h o configuation is dfind and thn sliding sufac vcto including nw boundizing function is chosn. h o vcto of systm asymptotically convgs to zo as asonabl as dsid. A simpl way of masuing th os is also poposd. h stability of th systm is considd in th sns of Lyapunov mthod. h simulation and xpimntal sults show that th poposd contoll can b applicabl and implmntd in th pactical fild. REFERENCES [1] M. Y. Kim, K. W. Ko, H. S. Cho, and J. H. Kim, Visual snsing and cognition of wlding nvionmnt fo intllignt shipyad wlding obots, Poc. of IEEE Intllignt Robots and Systms, vol. 3, pp ,. [] Y. B. Jon, S. S. Pak, and S. B. Kim, Modling and motion contol of mobil obot fo lattic typ of wlding, KSME Intnational Jounal, vol. 16, No. 1, pp ,. [3] B. O. Kam, Y. B. Jon, and S. B. Kim, Motion contol of two-whld wlding mobil obot with sam tacking snso, Poc. of IEEE Industial Elctonics, vol., pp , 1. [4]. L. Chung, H.. Bui,.. Nguyn, and S. B. Kim, Sliding mod contol of two-whld wlding mobil obot fo tacking smooth cuvd wlding path, KSME Intnational Jounal, vol. 18, no. 7, pp , 4. [] J. M. Yang and J. H. Kim, Sliding mod contol fo tajctoy tacking of nonholonomic whld mobil obots, IEEE ans. on Robotics and Automation, vol. 1, no. 3, pp , [6]. H. Bui,. L. Chung,.. Nguyn, and S. B. Kim, A simpl nonlina contol of a twowhld wlding mobil obot, Intnational Jounal of Contol, Automation, and Systm, vol. 1, no. 1, pp. 3-4, 3. [7] D. K. Chwa, J. H. So, P. J. Kim, and J. Y. Choi, Sliding mod tacking contol of nonholonomic whld mobil obots, Poc. of th Amican Contol Confnc, pp ,. [8] X. Yun and Y. Yamamoto, Intnal dynamics of a whld mobil obot, Poc. of IEEE Intllignt Robots and Systms, pp , [9]. Fukao, H. Nakagawa, and N. Adachi, Adaptiv tacking contol of a nonholonomic mobil obot, IEEE ans. on Robotics and Automation, vol. 16, no., pp ,. [1]. C. L, C. H. L, and C. C. ng, Adaptiv tacking contol of nonholonomic mobil obot by computd toqu, Poc. of IEEE Dcision and Contol, pp , [11] Y. Kanayama, Y. Kimua, F. Miyazaki, and. Noguchi, A stabl tacking contol mthod fo a nonholonomic mobil obot, Poc. of IEEE Intllignt Robots and Systms Wokshop, Japan, Vol. 3, pp , [1] J. J. E. Slotin and W. Li, Applid Nonlina Contol, Pntic-Hall Intnational, Inc., pp. 1-1, 1991.

12 94 Ngo Manh Dung, Vo Hoang Duy, Nguyn hanh Phuong, Sang Bong Kim, and Myung Suck Oh Ngo Manh Dung was bon in Vitnam on Januay 7, H civd th B.S. dg in th Faculty of Elctical and Elctonics Engining, Hochiminh City Univsity of chnology, Vitnam in H civd th M.S. and Ph.D. dgs in th Dpt. of Mchanical Engining, Pukyong National Univsity, Busan, Koa in 4 and 7. H is a Lctu of th Faculty of Elctical and Elctonics Engining, Hochiminh City Univsity of chnology, Vitnam. His filds of intsts a nonlina contol, pow lctonic and wlding automation pocss. Sang Bong Kim was bon in Koa on August 6, 19. H civd th B.S. and M.S. dgs fom National Fishis Univsity of Busan, Koa, in 1978 and 198. H civd th Ph.D. dg in okyo Institut of chnology, Japan in Aft thn, h is a Pofsso of Dpt. of Mchanical Engining, Pukyong National Univsity, Busan, Koa. His sach has bn on obust contol, biomchanical contol, mobil obot contol, bipd obot and quadupd obot. Vo Hoang Duy was bon in Vitnam on Mach 1, 197. H civd th B.S. and M.S. dgs in th Faculty of Elctical and Elctonics Engining, Hochiminh City Univsity of chnology, Vitnam in 1997 and 3. H is cuntly a Ph.D. studnt in th Dpt. of Mchanical Engining, Pukyong National Univsity, Busan, Koa. His filds of intsts a nonlina contol, mobil obot contol and quadupd obot. Nguyn hanh Phuong was bon in Vitnam on Apil 4, H civd th B.S. and M.S. dgs in th Faculty of Elctical and Elctonics Engining, Hochiminh City Univsity of chnology, Vitnam in 1998 and 3. H is cuntly a Ph.D. studnt in th Dpt. of Mchanical Engining, Pukyong National Univsity, Busan, Koa. His filds of intsts a nonlina contol, mobil obot contol and bipd obot. Emission. Myung Suck Oh was bon in Koa on Novmb 7, 194. H civd th M.S. and Ph.D. dgs fom National Fishis Univsity of Busan, Koa in 1983 and H is a Pofsso of th Dpt. of Mchanical Engining, Pukyong National Univsity, Busan, Koa. His sach has bn on Wlding Engining and Acoustic