Dynamic Modeling and Inverse Dynamic Analysis of Flexible Parallel Robots

Transcription

1 nthwb.com Dnamc Modlng and Invrs Dnamc nalss of Flbl Paralll Robots Du Zhaoca and Yu Yuqng Collg of Mchancal Engnrng & ppld Elctroncs chnolog,ng Unvrst of chnolog,ng, Chna Corrspondng author E-mal: bstract: hs papr prsnts a mthod for th dnamc modlng of paralll robots wth flbl lnks and rgd movng platform basd on fnt lmnt thor. h rlaton btwn lastc dsplacmnts of lnks s nvstgatd, takng nto consdraton th couplng ffcts of lastc moton and rgd moton. h knmatc and dnamc constrant condtons of lastc dsplacmnts of flbl paralll robots ar prsntd. h Knto- Elastodnamcs thor and moshnko bam thor ar mplod to drv th quatons of moton, consdrng th ffcts of dstrbutd mass, lumpd mass, rotar nrta, sharng dformaton, bndng dformaton, latral dformatons and all th dnamc couplng trms. h dnamc bhavor du to flblt of lnks s wll llustratd through numrcal smulaton. Compard wth th rsults of SMCEF softwar smulaton, th numrcal smulaton rsults show good cohrnc and th advantags of th mthod. h flblt of lnks s dmonstratd to hav sgnfcant mpact on sstm prformanc and stablt. mthod for th nvrs dnamc analss of flbl paralll robots s prsntd. Kwords: flbl paralll robot, flbl lnk, dnamc modlng, nvrs dnamc analss. Introducton h paralll robot has bn an advancd topc n robotcs rsarch. Paralll robots ar wdl usd n man applcatons (Lbrt, G. t al, 993), such as ntrtanmnt, hom srvcs, flng machns, submarns, assmblng robots, tc. Planar paralll robots ar good canddats for mcromnaturzaton nto mcrodvcs. Compard wth sral robots, paralll robots ar provdd wth a srs of advantags n trms of hav paload, small rror, hgh postonal accurac, as of control, and so on. Currntl, man ndustrs gnrall us sral robots n opratons. Rsults ar good, but both accurac and throughput could b sgnfcantl mprovd b usng paralll robots (Gabrl P., 3). Svral rsarchrs hav studd paralll robots, but most of th rsarchs hav bn rstrctd to th rgd paralll robot (Lu, K. t al, 993 ; Lu, K. t al, 994). Fw nvstgatons (Wang, X Y., 5 ; Wang, X Y. & Jams, K M., 6) hav bn concrnd wth th flbl paralll robot. h trnd towards hghr opratng spd and us of lss matral rqurs that som phnomna whch usd to b omttd hav to b takn nto account n dnamc analss, and t s ncssar to consdr lnks flblt and couplng ffcts of th flbl lnks lastc dsplacmnt. h dnamcs of flbl robots workng at hgh spd has bn studd b man rsarchrs (ook, W J., 99 ; Gaultr, P E. & Clghorn, W L., 989 ; Santosha K D. & Ptr E. 6), and a numbr of approachs hav bn dvlopd to prdct th lastc dnamc bhavor of flbl sral robots. Som rsarchrs hav proposd an ffcnt procdur for computr gnraton of smbolc modlng quatons for planar sral robots wth rgd and flbl lnks (Wang, D. & Vdasagar, M., 99 ; Ln, J. & Lws, F L., 994). caus of th complt rsultng from th prsnc of mult-closd loops and th lmtaton of computr faclts accssbl, ths mthod ma not b sutabl for paralll robots wth flbl lnks and rgd movng platform. In rcnt ars, paralll robots hav rcvd mor and mor attnton. lthough svral nvstgators hav usd fnt lmnt tchnqus to modl flbl paralll robots, th dd not nclud all th nfluncs blow (nosman, M. & L, V G., ; Fattah,. t al, 994 ): (a) Lumpd mass, rotar nrta. (b) Sharng dformaton, bndng dformaton, latral dformaton. (c) ll th dnamc couplng trms. ll now, a practcal mthod to nabl dsgnrs to prdct th lastc dnamc bhavor of paralll robots wth flbl lnks and rgd movng platform has not bn avalabl. It s blvd that a comprhnsv dnamc modl s crucal n th dsgn procss, n prformanc valuaton and for control purposs. h Knto-Elastodnamcs (KED) thor studs movng mchansms, takng nto account dformatons of th flbl lnks du to trnal and ntrnal loads. h lastc dformatons of th lnks pla a sgnfcant rol n ISS , pp. 5-5

2 hgh-spd oprtatons bcaus th lnks ar usuall lghtr n wght, and th ntrnal forcs ar gratr. h obctv of th nvstgaton n ths papr s to dvlop a smpl and ffcnt mthod for dnamc modlng and nvrs dnamc analss of flbl paralll robots. hs s achvd usng th KED thor (Frad, M. & Lukaswcz, S., ; Jrz, Z. & Plosa, S W., ; Yang, J. & Sadlr, P., ; Zhang, C. t al., 997) and consdrng th lastc dsplacmnt of lnks and dnamc couplng ffcts. h ffcts of dstrbutd mass, lumpd mass, rotar nrta, sharng dformaton, bndng dformaton and latral dformaton ar all takn nto account. h concpt of th knmatc and dnamc constrant condtons of lastc dsplacmnt for flbl paralll robots s usd to dcoupl movng platform s moton quatons from thos of th sub-chans. h poston rror and orntaton rror of movng platform rsultd from th lnks lastc dsplacmnt ar calculatd. asd on th dnamc modl, a mthod for th nvrs dnamc analss of flbl paralll robots s ntroducd.. Dnamc quatons.. Modl of flbl bam lmnt flbl paralll robot can b dvdd nto svral parts whch ar composd of qual cross-scton bam lmnts. h qual cross-scton bam lmnt s usuall usd to dscrb a lnks lastc dsplacmnt. two-nod fnt bam lmnt rprsntng a porton of lnk of th flbl paralll robots s shown n Fg.. h lmnt nodal dsplacmnts, or gnralzd coordnats, ar prssd n matr form as u = whr u, + 4 {,,,,,,, } u u u u u u u u () u ( =,,3and 4) ar aal dsplacmnts along th as, transvrs dsplacmnts along th as, and th rotar dsplacmnts and curvatur dsplacmnts wth rspct to th z as, of nods and, rspctvl. u 3 u u 4 u Fg.. Modl of a flbl bam lmnt L u 7 u6 u 5 u 8 h dsplacmnts of a pont n th lmnt nclud aal, transvrs, rotar and curvatur dsplacmnts, prssd as: S (, ), S (, ), S (, ) and t t 3 t S 4 (, t). hs can b dscrbd wth a st of ntrpolaton functons and gnralzd coordnats S (, t )= u ( =,) () (, ) ( ) = S 3 t Ν u 3 u (3) (, ) ( = S 4 t u u (4) 4 whr and ar th vctors of quntc Hrmt polnomals and a lnar polnomal, rspctvl. h lastc moton of a bam lmnt s shown n Fg.. O s a global rfrnc fram and s a local rfrnc fram attachd to th lft nd of lnk. So th as s alwas drctd along th as of lnk n ts rgd bod confguraton (shown n sold lns) whch s orntd b anglθ. It s assumd that th dformd poston (shown n dashd lns) transforms two nods, and, from thr rgd bod poston, and C to thr dformd confguraton, and C, rspctvl. ow w can drv th dnamc quatons of flbl paralll robots b applng th basc bam lmnt dvlopd abov... Kntc nrg and potntal nrg of th bam lmnt h bam lmnt dscrbd n Fg. contans a bam wth lumpd mass attachd to both ndponts. h lumpd mass phscall rprsnts th ffct of th actuator mass or paload on th dnamc rspons. hs should not b smpl nglctd. h total kntc nrg of th lmnt dpnds on gnralzd coordnats. hn, consdrng translatng kntc nrg, rotatng kntc nrg and th kntc nrg 3 of th lumpd mass attachd to th ndponts of lmnt, th total kntc nrg of th lmnt can b dscrbd as Whr + + = (5) 3 3 = ρ ( + & θ snθ + & θ 3) + u& m u& vl v L L + & θ u m u + && θu b u+ u Y + u& Z = ρθ& + u& F + u& mr u& I L 3 = ( m + m) v + m & θ L + m & θlvsn θ + ( J + J ) & θ + & θ u m u + u& m u& + u Y + & θu& bc u+ u& Zc + u& mj u& + & θu& ZJ ρ, and L ar dnst of matral, th ara of crossscton and th lngth of th lmnt, rspctvl. v s c c c 6

3 Du Zhaoca and Yu Yuqng: Dnamc Modlng and Invrs Dnamc nalss of Flbl Paralll Robots th absolut vloct of nod. θ s th angl of as of th local rfrnc fram wth rspct to as of th global rfrnc fram. I s th momnt of nrta of th cross-scton wth rspct to z as passng through th cntrod of th lmnt. m, m, J and J ar lumpd mass and lumpd rotar nrta at th ndponts of th lmnt, rspctvl. O Fg.. Elastc moton of a bam lmnt c α C ϕ θ h total potntal nrg V of th lmnt s composd of th bndng-sharng stran nrg V du to transvrs dformatons, tnsl-comprsson stran nrg V du to aal dformatons and transvrs dformatons whr C d W V V + V V θ = (6) λ λ λ V = EJ Z u ( Ω + k + k + k3 ) u V = u ( E χ + k) u whr E s tnsl modulus of lastct. J Z s ara momnt of nrta of th cross-scton wth rspct to z as..3. Dnamc quatons of lmnt h Lagrangan prncpl s mplod n drvng th moton quatons of th lmnt d( u& ) dt u+ V u = Q (7) substtutng quatons (5) and (6) nto quaton (7), and prformng th rqurd dffrntaton and algbrac manpulators, th moton quatons of th lmnt can b wrttn n matr form K = EJ ( Ω + k + k + k ) + Eχ + k λ λ λ Z 3 + && θ( b+ bc) & θ ( m+ mc) p = Y + Y Z & Z & F & θ && Z c c J M, C and K ar th mass matr, dampng matr and stffnss matr of th lmnt, rspctvl. Subcrpt rprsnts th lmntal proprt. f s th forc vctor rtd b adacnt lmnts. q s th trnal forc vctor. p s th nrta forc vctor of th lmnt. Ecpt for f and q, all th trms n quaton (8) ar gvn n rfrnc(zhang, C. t al., 997)..4. Knmatc constrant condtons S shown n Fg.3, a paralll robot s composd of a movng platform, naml th nd-ffctor, connctd to th bas wth svral ndpndnt knmatc chans. Each of ths chans contans man ndpntnt passv onts and actuatd onts. h rlaton btwn th actual confguraton (shown as a sold ln) and th nomnal confguraton (shown as a dashd ln) can b dscrbd usng th moton of pont P on movng platform, as shown n Fg. 4. ( δ, δ ) and ε ar, aal dsplacmnts and angular dsplacmnt of movng platform changng from th nomnal confguraton to actual confguraton, rspctvl. h local rfrnc fram P s attachd to movng platform. h poston vctor of th ndpont of sub-chan n actual confguraton can b wrttn as whr D D D = R PP D (9) ε δ R PP = ε δ, ) and, ) ( ( D D D D ar poston vctors of th ndpont of sub-chan n actual confguraton and M u&& + C u& + K u= p + f + q (8) whr M = m+ mc + mr + m J C = & θ ( b+ b ) c Fg. 3. brdgd gnral vw of paralll robots 7

4 Y O P D ε P O Fg. 4. Rlaton btwn th actual confguratonand nomnal confguraton nomnal confguraton, rspctvl. R PP s th transformaton matr from th global rfrnc fram to th local rfrnc fram P. h poston vctor D D D can b prssd as U D D DD = = U D D X () D D rprsnts th poston vctor of th whr ndpont of th sub-chan whch s changng from th nomnal confguraton to th actual confguraton. U and U ar, aal lastc dsplacmnts of th ndpont of th sub-chan, rspctvl. substtutng quaton (9) and () nto quaton (), th knmatc constrant condtons of lastc dsplacmnts ar obtand U U = ε ε δ δ D D ().5. Dnamc constrant condtons h poston vctor of pont P on movng platform can b prssd as ( β P P P δ P = P + δ β β ε () whr, P, ) and (,, β ) P P ar th poston vctors of pont P n th actual confguraton and th nomnal confguraton, rspctvl. h dnamc constrant condtons of th flbl planar paralll manpulator ar obtand usng wton-eulr formula. m whr P m P && P && & P = I P β & M = = = = f f + M snφ F F M ( f, f ) L o o o (3) = = cos φ m P, I P ar mass and rotar nrta of movng platform, rspctvl. f, f and M ar, aal ntrnal forcs and ntrnal momnt rtd on movng platform b sub-chan, rspctvl. Fo, Fo and M o ar, aal rsultant trnal forcs and th rsultant trnal momnt rtng on movng platform, rspctvl. Φ s th nclnaton angl of as of th global rfrnc fram wth rspct to th vctor connctng pont P wth th ndpont of sub-chan n th actual confguraton. L s th dstanc from pont P to th ndpont of sub-chan. s th numbr of sub-chans...6. Sstm dnamc quatons h paralll robot can b convnntl modld as a sstm n trms of a st of sstm gnralzd coordnats U wth rspct to th fd fram O. hrfor a propr transformaton btwn th lmnt and th corrspondng sstm coordnats s ssntal. hs s achvd b applng a lnar transformaton btwn th lmnt gnralzd coordnats u and th corrspondng sstm gnralzd coordnats U va th approprat transformaton matr R. hus whr u= R U (4) cosθ snθ snθ cosθ R = cosθ snθ snθ cosθ 8

5 Du Zhaoca and Yu Yuqng: Dnamc Modlng and Invrs Dnamc nalss of Flbl Paralll Robots R rprsnts th transformaton matr from th global rfrnc fram to a local rfrnc fram. hs matr s a functon of anglθ. U s th prsson of u n th global rfrnc fram. Emplong all th moton quatons of th basc bam lmnt, th knmatc constrant condtons and dnamc constrant condtons, and prssng n trms of th global coordnatsu, th sstm dnamc quatons ar obtand M U&& + C U& + K U = P+ Q (5) whr M, C and K ar th mass, th quvalnt dampng and th stffnss matr of sstm, rspctvl. M, C, K, P and Q ar th functons of θ, & θ and & θ. h dampng matr C n quaton (5) nvolvs onl th quvalnt dampng of th sstm. It s gnrall rcognzd that th actual dampng should contan th structural dampng. hrfor, nglctng structural dampng n th dnamc modl ma lad to an unrasonabl rspons. h dampng matr n quaton (5) should b modfd n ordr to nclud structural dampng n th modl whnvr prsnt. owards ths obctv, th ovrall sstm dampng matr s gvn as Whr C = C + C s (6) C = α M + α K s C, C s rprsnt th total dampng and structural dampng of th sstm, rspctvl. C s s drvd b usng th Ralgh proportonal dampng mthod, and α, α ar Ralgh proportonal dampng coffcnts. Substtutng C wth C n quaton (5), th gnral form of th sstm dnamc quatons ar obtand M U&& + C U& + K U = P+ Q (7) 3. Invrs dnamcs analss h rsarch on nvrs dnamcs of th flbl paralll robot s qut dffcult. h rlaton btwn th poston varabls, orntaton varabls and th ont varabls can not b dtrmnd from th knmatc analss. hs varabls must b obtand b solvng th dnamc quatons. ll th gnralzd coordnats cpt for th poston, orntaton, vloct and acclraton of th ndffctor rman unknown, and all th sstm mass matr, stffnss matr, dampng matr and gnralzd forcs ar th functons of gnralzd coordnats. So t s vr dffcult to solv th nvrs dnamc quatons. hr ar two solutons for nvrs dnamcs of flbl robots basd on th rol of th drvng forcs (momnts) n th nvrs dnamcs quatons. Gnrall, th drvng forcs ar consdrd as drv forc and constrand forc, rspctvl. For th convnnc of dscrpton, th solutons ar calld drvng forc mthod and drvng constrant mthod n ths papr. Drvng constrant mthod basd on th flbl multbod thor s an accurat mthod. h drvng forcs ar usd as constrants to snthsz th dnamc quatons, so th mthod s wdl usd for rgd bod sstm. For th flbl bod sstm, th moton dffrntal quaton should b ntgratd to calculat th drvng forcs vn f thr s no rdundant rgd bod frdom. h man drawback of ths mthod s th dffcult n dnamc modlng for mult-lnk robots. caus th dnamc modl of flbl paralll robots basd on th flbl multbod dnamc thor has not bn obtand, ths mthod can not b mplod n ths papr. h drvng forc mthod s an appromaton mthod. Frstl, th rgd bod moton varabls ar appromatl dtrmnd undr th assumpton that th actual tractor s carrd out b th rgd robot. Such an appromaton ma lad to dfnt rror n th rsult. Scondl, th rgd bod drvng forcs ar drvd from th rgdbod moton quatons. hrdl, ths rgd bod drvng forcs ar substtutd nto th dnamc quatons of th flbl robot to calculat th lastc moton varabls. Lastl, all ths varabls ar substtutd nto th quaton (8) to calculat th drvng forcs. k k r sr M rqr + ( ) qsqr r= r, s= qs q k r= M V + K q + = Q r r q M ( =, k) (8) M r, M sr and K r ar componnt of mass whr matr and stffnss matr, rspctvl. q s, q r ar th th th s and r gnralzd coordnat, rspctvl. V s th th sstm potntal nrg. Q s th componnt of th gnralzd forcs wth rspct to gnralzd coordnats. k s th numbr of gnralzd coordnats. In ordr to ncras th accurac of th rsults, modfd drvng forc mthod s prsntd hr. h obct s achvd b prformng tratv computatons. s w know, th poston and orntaton vctor S of th ndffctor s dtrmnd b ont varabls θ and lastc dformaton of lnks, thus S can b prssd as S = S( θ, U ) (9) 9

6 &&& and lastc dformaton Jont varabls ( θθθ,, ) ( UUU, &, &&) of lnks ar dtrmnd b ont drvng forcs (momnts) τ, thus F( τ) = W τ = f ( θ, &&& θ, θ, UUU, &, &&) () whr W s coffcnt matr. h smultanous quatons consttutd b quaton (9) and quaton () ar th unvrsal modl of nvrs dnamcs of flbl robots (Guo, J F., 99). Du to th complt of ths quatons, ths modl s not sutabl for thortcal analss. Gnrall, som assumptons ar prsntd to smplf th calculaton and obtan th appromat rsult. In ths papr, an tratv computng algorthm s prsntd to ncras th calculatng accurac. Frstl, th tractor of th ndffctor s assumd to b ralzd b th rgd robot, thus th rgd ont varabls θ (orgnal valu) and rgd bod drvng forcs (momnts) ar calculatd and thn th lastc dformaton U s obtand. Scondl, θ and U ar substtutd nto quaton (9) to calculat th poston and orntaton vctor S of th nd-ffctor. caus θ s drvd from th assumpton, ths wll rsult n th rror of th poston and orntaton vctor of th nd-ffctor. h rror can b prssd as Δ S = S ( θ, U ) S * ( θ, U ) () * * - * * * -( θ, U ) whr S s th poston and orntaton vctor calculatd n stp, and * * * S ( θ, U) = S ( θ, U). If th rror s lss than th gvn accurac ε, θ and U wll b consdrd as th rqurd valus, othrws lt S ( θ, U ) = S ( θ, U ) ΔS () * * * * * * - Rpat th abov-mntond tratv procss untl th rror ΔS s lss than th prscrbd accurac ε. Lastl, th ont varabls θ and lastc dformaton U ar obtand and thn th drvng forcs (momnts) ar achvd. flbl paralll robots for th purpos mntond abov. hrfor th numrcal ampl of a flbl planar 3- RRR paralll robot s prsntd hr. Gnrall, R stands for rvolut, thrfor RRR rprsnts a knmatc chan composd of, startng from th ground, thr rotatonal onts. Wth ths nomnclatur, th actuatd ont s undrlnd. hrfor, a 3-RRR robot s composd of thr RRR knmatc chans. tpcal 3-RRR planar robot s shown n Fg. 5. Each lnk s mad of stl wth mass dnst of 78kg/m3, lastc modulus of. Pa, Posson s rato of.3. h lngth of ach lnk s.m, and th crossscton s.3m.3m. h lumpd mass attachd to ach ndpont of th lnks s.4kg. h lngth of ach dg of th trangular movng platform s.4m, and pont P s th cntrod of movng platform. h mass of movng platform s.kg. h coordnats of th fd hng ar (-.3, ), (.5,.) and (., ), rspctvl. h moton of pont P s dscrbd as X =.4 cos( π t ) Y = sn( π t ) ϕ=. cos( π t ) ( t = ~ 4s) whr X, Y rprsnt, aal coordnats of th pont P, rspctvl. ϕ rprsnts th orntaton angl of movng platform. t rprsnts tm. In ordr to vrf th valdt of th mthod, th rsults ar compard wth SMCEF softwar smulaton rsults. h poston rrors and orntaton rrors of pont P, shown n Fgs. 6-8, ar prssd n a global fram. h orntaton rror can not b calculatd b usng SMCEF softwar, so thr s onl a numrcall calculatd rsult shown n Fg. 8. h rsults ar analzd as follows. Compard wth th rsults of SMCEF softwar smulaton, th numrcal smulaton rsults show good corrspondnc and dmonstrat th accurac of th mthod. h dffrnc btwn th softwar smulaton rsults and th numrcal smulaton rsults s lss than %. h dnamc charactrstcs of flbl paralll robots ar llustratd usng th modl. So th modl s usful for dnamc analss of flbl paralll robots. U 6 U 7 U 8 U 5 U 6 U 5 U 4 U 3 U 4 U U 4. umrcal smulaton and analss U 3 U U 3 U 9 U 8 caus of th complt of manufactur and control, t s a challngabl task to dsgn and produc a flbl paralll robot, naml all th lnks ar flbl and movng platform s rgd, fulflld th prdfnd rqurmnts and adquatl rvald th ntrnscal charactrstcs. So far thr has bn no practcabl U U U U 4 U U U 7 U 9 Fg. 5. Gnralzd coordnats of 3-RRR

7 Du Zhaoca and Yu Yuqng: Dnamc Modlng and Invrs Dnamc nalss of Flbl Paralll Robots X aal poston rror (m) m (s) Fg. 6. X aal poston rror of pont P Y aal poston rror (m) SMCEF rsult modl rsult 3 4 m (s) Fg. 7. Y aal poston rror of pont P orntaton angl (rad) SMCEF rsult modl rsult modl rsult s shown n Fgs. 6-8, th frquncs of oscllatons of poston rrors and orntaton rror ar rmarkabl, and th rrors of an poston ar dramatcall dffrnt from thos of othr postons. h rsults ar usful n slctng workng postons to fulfll gvn rqurmnts. Morovr, th actual tractor oscllats rgularl around th nomnal tractor. So th concluson can b drawn that th actual tractor s ssntall an lastc oscllaton. In ordr to shortn th papr, onl th drvng momnt of th st sub-chan of flbl paralll robot s shown n Fg. 9 (shown n ral lns). h drvng momnt of th rgd on (shown n dashd lns) s also prsntd. h drvng momnt of flbl robot oscllats around th drvng momnt of rgd robot. h oscllatng ampltud s about.m. h data and thr changng pattrn ar concdnt wth th dnamc charactrstcs of flbl robots. h rsults vrf th valdt of th nvrs dnamcs mthod prsntd n ths papr. 5. Conclusons () n approach to dnamc modlng of flbl paralll robots s prsntd usng th knmatc and dnamc constrant condtons of lastc dsplacmnts proposd n ths papr. It s applcabl for dnamc modlng of spatal paralll robots wth varous knds of knmatc pars. () h lnks flblt and th couplng ffcts of lastc moton and rgd moton hav a sgnfcant mpact on sstm prformanc and stablt. (3) h dnamc rspons of flbl paralll robots s drastcall dffrnt from th rspons of rgd ons. (4) Poston rrors and orntaton rror ar mportant prformanc crtra to masur th capact of flbl paralll robots. hs rrors can ndcat th moton stablt. (5) h analss of th poston rrors and orntaton rror s usful to control th vbraton of flbl paralll robots. (6) h mthod prsntd for th nvrs dnamc analss s vald and ffctv m (s) Fg. 8. Orntaton rror of pont P h mamum poston rrors and orntaton rror du to lastc dsplacmnt of lnks ar rmarkabl. h, aal poston rrors ar 6mm and 8mm, rspctvl, and th orntaton rror s -.5 rad. Morovr, th poston rrors of som postons ar lss, whl th orntaton rror s gratr. So t s ncssar to analz both th poston rrors and orntaton rror. Orntaton rror should b takn nto account n slctng th workng poston. Drvng momnt ( m) m (t) Fg. 9. h drvng momnt of flbl robot rgd robot flbl robot

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