Improved Three-Step Input Shaping Control of Crane System

Transcription

1 WSEAS TRANSACTIONS o SYSTEMS Sirri Suay Gurlyuk, Ozgur Bahair, Yuus Turkka a Haka Usti Improv Thr-Stp Iput Shapig Cotrol of Cra Systm SIRRI SUNAY GÜRLEYÜK Elctric-Elctroic Egirig Dpt. Zogulak Karalmas Üivrsity Zogulak TURKEY gurlyuk@iam.com ÖZGÜR BAHADIR HEMA Iustry A.Ş. Zogulak TURKEY ozgurbahair67@gmail.com YUNUS TÜRKKAN MEB Karabük TURKEY turkkayuus@hotmail.com HAKAN ÜŞENTİ Vocatioal Istitutio of Eucatio Elctric Dpartmt Düzc Uivrsity Düzc TURKEY ustihaka@hotmail.com Abstract: Shapig comma iput is prform for rucig rsiual vibratios i motio cotrol of yamical structurs. Systm iputs ar chag so that th structur rachs to pla motio i miimum uratio without rsiual oscillatio. Shapig iput is obtai by covolvig sir iput with a impuls squc. Mai goal of th iput shapr sig is to fi impuls amplitus a thir istats from th yamical bhavior of ucotroll rspos. Zro Vibratio (ZV, Zro Vibratio a Drivativ (ZVD a Extra Issitiv (EI ar th most commo shapr typs. Improvig th robustss rspct to molig rrors rquirs mor impulss. Ths icras cotrol uratio a mak ifficult to solv costrait quatios irctly. Rquir paramtrs ca b trmi from th rivatio of costrait quatios. This papr prsts Thr-Stp (TS iput shapig tchiqu. A solutio spac for thr-impuls shapig is giv icluig both positiv a gativ shaprs. It is show hr that ZVD a EI shaprs ar th spcial solutio poits i TS shapig spac. Th uratio of th shapr lss or mor tha ZVD s ca b obtai i th fi spac. Som of th w shaprs giv bttr robustss tha ZVD. Exprimtal rsults from a prototyp plaar cra systm ar us to support th umrical rsults. Kywors: Vibratio, Iput shapig, Thr-stp shapr, Plaar cra, Robustss, Comma gratio Itrouctio Th vibratio is a sigificat problm i yamical systms that ar rquir to prform prcis motio i prsc of structural flxibility. Stp motors, robotic arms, flxibl maipulators a cra systms ar som xampls for this catgory. Oscillatory bhavior is s ot oly systm itslf but also i its loa or togthr. Exampls of such loas rag from shts to log pips a othr lastic matrials. Th prformac of prcisio motio ps o ampig capacity of th systm. Th ampig capability of a yamical systm ca b hac by passiv or activ ampig mthos. I th passiv approach, ISSN: Issu 6, Volum 7, Ju 008

2 WSEAS TRANSACTIONS o SYSTEMS Sirri Suay Gurlyuk, Ozgur Bahair, Yuus Turkka a Haka Usti oscillatio ampig is icras by ployig xtral amprs such as ashpots or viscous amprs []. Fback cotrol ca also b us as a activ approach i a wi ba of issitivity. Aothr approach is fforwar cotrol tchiqus. For xampl, rsiual vibratio of th systm is succssfully ruc by shapig th iput comma. Th arlist icaratio of this slf-caclig comma gratio was vlop i th 950's by O.J.M. Smith []. His posicast cotrol mtho ivolv brakig a comma of crtai magitu ito two smallr magitu commas, o of which is lay o-half prio of vibratio. Ufortuatly, his tchiqu was xtrmly ssitiv to molig rrors [3]. Sigr a Srig vlop rfrc commas that wr robust ough to b ffctiv o a wi rag of systms [4]. This w robust tchiqu is am as iput shapig. Iput shapig is implmt by covolvig a squc of impulss, a iput shapr, with a sir systm comma to prouc a shap iput that proucs slfcaclig comma sigal [5]. Iput shapr is sig by gratig a st of costrait quatios which limit th rsiual vibratio, maitai actuator limitatios, a sur som lvl of robustss to molig rrors [6]. Th procss has th ffct of placig zros ar th locatios of th flxibl pols of th oscillatory systm. I th iput shapr, th amplitus a tim locatios of th impulss ar trmi by solvig th st of costraits [6-0]. Most xistig cra cotrol systms ar sig to maximiz sp, i a attmpt to miimiz systm vibratio a achiv goo positioal accuracy i a miimum uratio [-4]. High stiffss ca b achiv by usig short rop or havy carrir ha. As a rsult, such cras ar usually havy with rspct to its payloa. This limits th sp of opratio of trasportatio, icrass siz of rivig motor a rgy cosumptio [5-7]. Th payloa to carrir wight ratio is also low ur this coitio. I cotrast, a tall cra systm with a light carrir ha provis may rturs. It rquirs lss matrial, is lightr i wight, has highr covyac sp, lowr powr cosumptio, rquirs smallr motors, is mor mauvrabl, is safr to oprat, has lss total cost a has highr payloa to carrir wight ratio. Covrsly, th cotrol of such systms with accurat positioig is ot asy. Troubls bgi u to prcis positioig rquirmts i may applicatio aras. Som ifficultis occur to obtai accurat mol of th systm [7]. This papr prsts xprimtal ivstigatios ito th vlopmt of cotrol schms for vibratio cotrol of a light a vry oscillatory cra systm rivig i high sps. It is prfrr i this stuy a fforwar cotrol tchiqu, iput shapig. Fback ati-sway cotrol schms ca ot b us cra systms, grally. Mai problm i th fback cotrol systms. Optical, lasr or proximity ssor masurmts ar ot asy to stup i most systms. Th rop lgth a high vibratio amplitus mak ifficult to fi prcisio masurmt. Classical a w iput shapig mthos ar compar rspct to thir prformac. Th iput shapr is sig o th basis of th yamic charactristics of th cra systm a us for pr-procssig th rfrc iput. Prformacs of th vlop cotrol schms ar valuat i trms of lvl of vibratio ructio, trackig capability i high sps a robustss to th molig rrors. Exprimtal rsults of th trolly travl a payloa acclratio with th cotrol schms ar prst. If th costraits ar miimum uratio a zro rsiual vibratio, th th solutio shapr is ZV shapr. Howvr, ZV shapr is ot wll o most systms bcaus it is ssitiv to molig rrors. Robustss ca b improv by aig mor impuls to th shapr. Th rsultig shapr is a thr stp shapr. Dirct solutio of amplitus a thir uratio is ot possibl u to th iaquat umbr of costrait quatios. ZVD shapr, as a spcial solutio, ca b obtai by sttig th rivatio of costrait quatios with rspct to th frqucy of th rsiual vibratio qual to zro. ZVD schm is lss ssitiv to timig rror a hc grally is mor robust i ral applicatios tha th ZV schm. Howvr, it rquirs a tim palty. ZVD shapr has uratio of o prio of ushap vibratio, whil ZV has oly a half prio [7, 8, -5]. Thr ar ifiit umbrs of solutio for TS shapig icluig ZVD shapr. Duratio of obtai shaprs for th slct spac is btw ZV a ZVD. This papr mostrats that TS shapig ca b improv to amp th rsiual vibratio wh th lgth of th comma is ot allow to b icras. Som w shaprs prouc mor robustss tha ZVD somwhat. I Sctio II, th rivatio of xprssios for TS shapig is giv. Sctio III isplays rspos a robustss comparisos for th w a classical shaprs. Fially, cocluig rmarks ar mphasiz i Sctio IV. Molig of Iput Shapig Liar yamical systms ar grally mol by sco orr iffrtial quatios. A amp oscillatory yamic systm mol ca b giv as: x x m + c + kx = Fu( t ( t t ISSN: Issu 6, Volum 7, Ju 008

3 WSEAS TRANSACTIONS o SYSTEMS Sirri Suay Gurlyuk, Ozgur Bahair, Yuus Turkka a Haka Usti Th trasfr fuctio of this sco-orr yamical mol is G( s = ω s + s + ω ( whr, ω is th uamp atural frqucy a ξ is th ampig ratio. Rlatioship btw th cofficits of trasfr fuctio a iffrtial quatio is giv as: ω = k m (3 c ξ = (4 km Stp or impuls rspos of th systm yils grally amp oscillatory bhavior [9, 6, 7]. Impuls rspos of a sco orr systm at tim t is: ω ( t t0 x( t = A si( ω ξ ( t t0 ξ (5 Whr A a t 0 ar th impuls amplitu a istat of th impuls, rspctivly. For impulss, th impuls rspos ca b xprss as x( t = M si( ω t + φ (6 whr ω = ω 0 ξ is amp frqucy, Th rsiual vibratio amplitu is obtai at th tim of th last impuls, t as V = C + S (7 t ( ξ, ω ( ξ, ω ( ξ, ω whr t (, ω = A Cos(ω t = t (, ω = A Si(ωt = C ξ (8 S ξ (9 Iput shapig limits rsiual vibratio by gratig a comma profil that ts to cacl its vibratio [6]. Th vibratio occurs u to th first part of th comma iput is compsat by vibratio occur u to th xt part of th comma iput. Ths squcs of impulss ar covolv with th sir systm comma. Th, th covolutio is us to riv th systm. Th sir comma is a stp iput thortically. Svral typs of iput shaprs hav b propos [4, 7]. Th shapr cotais impulss; all of thm hav positiv amplitus for positiv iput shapig. Iput shapig rquirs costraits to prouc a solutio. Eq.(8 a Eq.(9 shoul b iptly zro to achiv vibratio fr rspos aftr th last impuls [0, 8]. Th sum of amplitus of th impulss is rquir uity to sur that th shap comma proucs th sam st poit as ushap motio, A =. Th first = impuls is appli at tim zro, t =0. Impuls amplitus ar slct btw 0 a to obtai a positiv shapr. If th systm is solv for two impuls squc, ZV shapr is obtai. Amplitus a tim locatios of th ZV shapr is show i Eq.(0 as first a sco rows rspctivly. M = ( B cos( ωt + ( B si( ωt = = A A ZV = (0 0 t a Exact solutio of Eq.(0 for a giv systm is B = Aω ξ φ = cos ( ξ ( t t i A a t ar th magitus a tims at which th impulss occur [7]. ZV = + K 0 whr ξπ K + K π ω ξ K =. ( ISSN: Issu 6, Volum 7, Ju 008

4 WSEAS TRANSACTIONS o SYSTEMS Sirri Suay Gurlyuk, Ozgur Bahair, Yuus Turkka a Haka Usti Fig. shows th ZV shapr implmtatio by covolvig squc of two impulss with th sir systm comma. Fig. ZV shapr Robustss ca b improv by icrasig th umbr of impulss. Aig a impuls to ZV proucs TS shapr. Dirct solutio is impossibl owig to th isufficit costrait quatios. Although w hav oly thr quatios, thr-impuls amplitus a thir tim locatios rquir solvig fiv ukow paramtrs, A, A, A 3, t a t 3. A spcific aalytical solutio ca b obtai by makig th rivativ of th costrait quatios (8 a (9 rspct to atural frqucy of th systm qual to zro, C = At ω a = S = At ω = t t si( ω t cos( ω t = 0 = 0 ( (3 Howvr, icras th impuls umbr rsults a lay i rspos tim. Rsultig shapr is am ZVD. ZVD = + K + K 0 K K + K + K + K + K π π ω ω (4 Implmtatio of th ZVD shapr is show i fig.. rsiual vibratio of th oscillatory payloa. Th sarch shapr cotais oly positiv amplitus. TS shapr is slct bcaus it is lss ssitiv to molig rrors. I th cas of TS iput shapig, th cotrol systm sigr wats to avoi rivatio procss. Th it is rquir to writ th costrait quatios i matrix form. Thr costrait quatios for fiv ukow paramtrs ca b xprss p o thm as: 0 t t Cos( ω t Si( ω t t3 t3 Cos( ω t3 A 0 Si( ω = t3 A 0 A 3 (5 Th matrix form ca b rwritt for impuls amplitus that ach ps o t a t 3 []. A A = A 3 ( t + t3 Whr ( Cos( ω t Si( ω t Si( ω t Cos( ω t K t t3 K K 3 K K K Si( ω t Si( ω t t 3 K = ( ( ( 3 t Si ω t Cos t ω t K = ( ( 3 t Si ω t Cos ω t ( 3 3 (6 Fig.3 shows msh plot of amplitus rspct to scal tim locatios by π ω. 3 Fig. ZVD shapr TS Iput Shapig This papr valuats TS iput shaprs. Th propos shaprs ar riv from costraits that rquir zro ISSN: Issu 6, Volum 7, Ju 008

5 WSEAS TRANSACTIONS o SYSTEMS Sirri Suay Gurlyuk, Ozgur Bahair, Yuus Turkka a Haka Usti Fig.4 Positiv TS shapr rgio ZV a ZVD ar poit out as spcial locatios i this figur. For th miimum shapig uratio, th Rgio-I is slct as sarch spac i this stuy. All of th shaprs i this ara hav lss uratio tha ZVD. Impuls amplitus variatios i th slct ara ar msh as: (a Msh plot for A (b Msh plot for A (c Msh plot for A 3 Fig.3 A, A a A 3 amplitus rspct to ormaliz tim locatios Th usabl rgios for all positiv impuls amplitus ca b show i fig.4. A, A a A 3 prouc positiv valu i ths rgios []. ISSN: Issu 6, Volum 7, Ju 008

6 WSEAS TRANSACTIONS o SYSTEMS Sirri Suay Gurlyuk, Ozgur Bahair, Yuus Turkka a Haka Usti (a Msh plot for A for slct spac (b Msh plot for A for slct spac (c Msh plot for A 3 for slct spac Fig.5 A, A a A 3 amplitus msh plot for slct spac Our sarch focuss o som optimal rsults for iffrt prformac masurs. Miimum cotrol tim a issitivss to molig rrors ar mai optimizatio paramtrs. Th fastst is ZV a th most robust is ZVD. Ths ar th xtrm poits i th sarch ara. Th sarch ara proucs ifiit umbr of w shaprs. Duratios of ths shaprs ar shortr tha ZVD shapr. Thus, th first prformac masur is provi. Othr masur is robustss. It rquirs that th valu of th prct rsiual vibratio to molig rrors shoul b as small as possibl. I th othr wors, rsiual vibratio amplitu shoul b stay ur th fi lvl as wi as possibl for mor robustss whil oscillatio frqucy chags. Fig.6 shows th ssitivity curv alog th slct impuls tim variatio ara. Som shaprs i this ara satisfy th robustss critria partly. Ths ar mor robust tha ZVD ur th fi lvl of rsiual vibratio for highr frqucis. Fig.6 Rsiual vibratio msh rspct to impuls tim ix 3 Rsults Fig.7 shows th thr-stp iput shapr cotrol of th xprimtal plaar cra systm stup. This is a prototyp xprimtal systm. Som spcificatios ar giv i Tabl.. Tabl. Cra systm spcificatios Driv Systm Gar DC Motor Motor Cotrollr A3953SB H-Brig Cotrollr PIC6F877A Programmig Laguag PIC-C Compilr Startig Poit St Mtho Optical Ssor Supply Voltag 30V Rop Lgth 466mm Total Travl Lgth 36mm Travl Sp 0.6m/s Payloa Wight 8.8g Th payloa is cotroll usig prtrmi thr stps at fi tim itrvals. Startig tim t is slct zro, stat-i. Th cra is mov alog th A isplacmt. Systm is brak at th of motio, stat-ii. Cra is wait urig th t lay. Payloa rachs its maximum lvl. Cra is mov agai to A istac at this tim, stat-iii. Th comma of th fial stp, A 3, is giv aftr th last lay t 3, stat-iv. ISSN: Issu 6, Volum 7, Ju 008

7 WSEAS TRANSACTIONS o SYSTEMS Sirri Suay Gurlyuk, Ozgur Bahair, Yuus Turkka a Haka Usti Fig.9 Motor iput a motio variatio Fig.7 Thr-stp shap cra motio systm Fig.8 shows th xprimtal cra systm. I th systm, startig positio is st by usig th fix optical ssor as limit switch a movig covr attach to bltpully systm, at th right bottom. Th gar DC motor is show at th lft bottom. Th rop a payloa is at th mil bottom. Small circuit i th figur is H- brig rivr, Allgro Micro Systms A3953SB. Th motio lgth of trolly is tr by usig pakyboar to PIC cotrollr circuit. Payloa is slct as a 66z barig. Fig.8 Exprimtal stup ZVD shapr is sig as stp fuctios i vrtical lar form, ormally. Howvr, it is impossibl to fiish th motio at th tim of zro. Acclratio of th motor a th rigi boy motio of th cra carrir a a lay to th total oury. Thrfor, th motio is a ramp fuctio whil motor voltag is th stp fuctios. Fig. 9 shows th motor iput a motio rspct to tim. Our sarch is focus o som masurs as []. Rsiual Vibratio: Th chos cotrol tchiqu shoul ot prouc ay vibratio urig th cotroll motio. This must b vali for all iffrt lgth of oury for trolly.. Ssitivity: Ssitivity curv is a plot of prctag rsiual vibratio vrsus th ormaliz frqucy. Prctag rsiual vibratio is fi as th vibratio with shapig ivi by vibratio without shapig. Normaliz frqucy is fou as th actual frqucy ivi by molig frqucy. 3. Robustss: Robustss is fi as th with of ssitivity curv at a spcific lvl of accptabl vibratio. It is also kow as issitivity rag. 4. Tim Palty: Mor robustss rquirmt icrass th lgth of th iput shapr. It rsults a icrasig i uratio of cotrol. Duratio of th ZVD shapr is gratr a half prio tha ZV shapr. Trasctal atur of costrait quatios (8 a (9 rsults multipl solutio for thr stp iput shapig cotrol tchiqu. To mak th solutio tim optimal subct to th rsiual vibratio a robustss costraits, th shapr must b ma as short as possibl. 5. Tim Efficicy: It is quivalt to tim optimality i th ss of satisfyig all aitioal costraits. 6. Othr masurs: Iput shaprs ar sig bas o som aitioal st of sir prformac spcificatios. Ths spcificatios iclu costraits o quatitis such as ris tim, brakig mchaism charactristic, tc. Aothr ia that coms ito viw i this papr is th thought that ths spcificatios shoul b accptabl i practic. For xampl, ral systms always xhibit som lvl of rsiual vibratio. Thrfor, wh a costrait is plac o th rsiual vibratio amplitu, it is bttr to limit th vibratio to som low lvl rathr tha ISSN: Issu 6, Volum 7, Ju 008

8 WSEAS TRANSACTIONS o SYSTEMS Sirri Suay Gurlyuk, Ozgur Bahair, Yuus Turkka a Haka Usti rquir th vibratio to b itically zro. To achiv th thortical possibility of zro rsiual vibratio, som othr prformac critria must b sacrific []. Som assumptios ar tak to accout for applyig iput shapig cotrol to th cra systm.. All iitial coitios ar zro: Mol of th cotrol systm rquirs all of th iitial coitios shoul b zro. If th iitial coitios ar ot zro th it is har to form a mol for yamical bhavior of th cra systm.. Extral isturbacs shoul b omitt: If th systm has ay xtral isturbacs, thy shoul ot affct th mol yamics of th systm. 3. Impossibility of masurmt for fback cotrol: If th cra oscillatio ca ot b masur asily, oly fforwar cotrol systm ca b appli to th cra systm. isappar wh th propos shaprs approachs to th ZVD shapr. Th slct w shaprs ar ot bttr tha ZVD shaprs i crasig frqucy. Th rsiual vibratio stays ur a spcifi lvl alog a wi rag for icrasig frqucy. For xampl, maximum lvl of rsiual vibratio i.nw Shapr is ur 8.66% whil frqucy riss.653 tim of atural frqucy. Th uratio of this shapr is oly * T ZVD..Nw Shapr stays 4.343% alog th risig frqucy of.434 a its uratio is oly 0.79 * T ZVD. Ths ar oly two xampl poit i th Rgio-I of fig.4. This rgio givs lots of w shaprs which ar mor robust a shortr i uratio tha ZVD shapr. Rsposs of th ushap, som w shaprs, ZV a ZVD shaprs ar show togthr i fig.0. Th lgth of ZVD shapr is T ZVD =0.749 s. ZV shapr uratio is half of ZVD shapr. Fig. Ssitivity curv of th shaprs Th xprimtal tsts show that th prst mthos ar xtrmly robust to th variatio of loa mass. W a thr tims of wight to th mol wight. Thr is o sigificat oscillatio usig ths tchiqus. Fig.0 Rspos of th shaprs Th robustss curvs for thm ar giv i fig.. Th w shaprs ca b am as skw shaprs u to thir shaps. Th skw shapr cocpt is itrouc from vctor iagram. Th issitivity of th classical shaprs ca b wi by isplacig th vctors from th frqucy axis. Wh th vctors ar locat off th horizotal axis, th ssitivity curv is skw. It is ot symmtrical pig o th mol atural frqucy. I this cas, mai goal is to fi th maximum issitivity rag of th skw. Duratios of th propos shaprs ar lss tha ZVD shapr i th tir sarch rgio, Rgio-I i th fig.4. Skwig 4 Coclusio This stuy proposs a TS iput shapig tchiqu. W hav prst a w approach to th iput shapig gratio tchiqu. Th w shapr xts ZVD shapig tchiqu ito a graliz TS shapr mtho that is asily applicabl for ay yamical systm mol. It is mostrat hr that small lgth shap TS shapr, though havig highr amplitus, rsult i lss vibratio tha ZVD shapr spcially ovr rags of atural frqucis somwhat. It is mostrat i this stuy that importat amout of ructio i sttlig tim ca b obtai by usig TS shapig. I this stuy, w spcifically sarch th Rgio-I i fig.4. This spac provis wi ssitivity curv if th vibratio frqucy riss. Th Rgio-II ca giv mor robust rsults i crasig vibratio frqucy though its uratio logr tha ZVD shapr. Rgio-III ca also b sarch for som avatags as futur works. ISSN: Issu 6, Volum 7, Ju 008

9 WSEAS TRANSACTIONS o SYSTEMS Sirri Suay Gurlyuk, Ozgur Bahair, Yuus Turkka a Haka Usti Th maor avatags of th propos tchiqu ar: To show th ZVD shapr is ot uiqu solutio for positiv thr-stp iput shapr. Mor robust thr stp shapr ca b slct. Th uratio of th shapr ca b cras by usig th propos sarch rgio. Both fastr a mor robust TS iput shapr ca b grat i som icras molig frqucis. Usarch rgios i fig.4 ca b ivstigat for robustss although th uratio of th shapr icrass i futur works. Rfrcs: [] T-S. Yag, K.-S. Ch, C.-C. L a J.-F. Yi, Supprssio of Motio-Iuc Rsiual Vibratio of a Catilvr Bam by Iput Shapig, J. of Egirig Mathmatics, 54, 006, pp. 5. [] O.J.M. Smith, Posicast Cotrol of Damp Oscillatory Systms, Procigs of th IRE, 957, pp [3] M. J. Robrtso a W. E. Sighos, Clos-Form Dflctio-Limitig Commas, Amrica Cotrol Cofrc, Portla, USA, 005, pp [4] N.C. Sigr a W.P. Srig, Prshapig Comma Iputs to Ruc Systm Vibratio, Joural of Dyamic Systms, Masurmt a Cotrol,, 990, pp [5] W. Sighos, E. Crai a W. Srig, Covolv a Simultaous Two-Mo Iput Shaprs, IEE Proc. Cotrol Thory Appl., Vol. 44, No. 6, 997, pp [6] W. Sighos a N. Sigr, Effcts of Iput Shapig o Two-Dimsioal Tractory Followig, IEEE Tras. o Robotics a Automatio, Vol., No. 6, 996, pp [7] T. Chag, K. Gobol a E. Hou, Optimal Iput Shapr Dsig for High-Sp Robotic Workclls, J. Vibratio a Cotrol, Vol. 9, 003, pp [8] C. F. Cutforth a L. Y. Pao, Cotrol Usig Equal Lgth Shap Commas to Ruc Vibratio, IEEE Tras. o Cotrol Sys. Tch., Vol., No., 003, pp.6-7. [9] S.S. Gürlyük, Vibratio Ructio i a Stp motor Usig Optimal Cotrol Tim Itrvals a Amplitus, Acta Mchaica, Vol.77, No, 005, pp [0] S.S. Gürlyük, Optimal Uity-Magitu Iput Shapr Duratio Aalysis, Archiv of Appli Mchaics, Vol.77, No., 007, pp [] S.S. Gürlyük, R. Hacıoğlu a Ş. Cial, Thr- Stp Iput Shapr for Dampig Tubular Stp Motor Vibratios, J. of Mchaical Egirig Scic, Vol., No., 007, pp.-9. [] S.S. Gürlyük a Ş. Cial, Robust Thr-Impuls Squc Iput Shapr Dsig, J. of Vibratio a Cotrol, Vol. 3, No., 007, pp [3] S.A. Kaarachos a K. Sptzas, Noliar Vibratio Cotrol of a Flxibl-Lik Robot Arm Bas o a Taylor Sris Expasio of th Cotrol Law, WSEAS Trasactios o Circuits a Systms, Issu, Vol., April 003, pp [4] C. Aguilar, J.H. Sossa, M.S. Suárz a F. Flors, Lyapuov Bas Cotrol for Activ Vibratio Dampig i a Vrtical Ivrt Pulum, WSEAS Trasactios o Circuits a Systms, Issu 0, Vol. 3, Dcmbr 004, pp [5] L. Li a Q. Li, Vibratio Aalysis Bas o Full Multi-Boy Mol a th Activ Cotrol of Commrcial Vhicl Suspsio Systm, WSEAS Trasactios o Systms a Cotrol, Issu, Vol., Fbruary 00, pp [6] W. Sighos, L. Portr, M. Kiso a E. Kriikku, Effcts of hoistig o th iput shapig cotrol of gatry cras, Cotrol Egirig Practic, Vol. 8, 000, pp [7] Z. Moham, J.M. Martis, M.O. Tokhi, J.S. Costa a M.A. Botto, Vibratio cotrol of a vry flxibl maipulator systm, Cotrol Egirig Practic, Vol. 3, 005, pp [8] A. Poty, P. Mlchior, B. Orsoi, F. Lvro a A. Oustaloup, ZV a ZVD shaprs for xplicit fractioal rivativ systms, ICAR 003, Coimbra, Portugal, Ju30-uly3, 003, pp [9] M.N. Sahikaya, Iput shapig for vibratio-fr pistoig of flxibl systms, Proc. Ist. Mch. Egrs., Vol. 5 (, 00, pp ISSN: Issu 6, Volum 7, Ju 008

10 WSEAS TRANSACTIONS o SYSTEMS Sirri Suay Gurlyuk, Ozgur Bahair, Yuus Turkka a Haka Usti [0] G. Alıcı, S. Kapucu a S. Baysc, A robust motio sig tchiqu for flxibl-oit maipulatio systms, Robotica, Vol.4, 006, pp [] B.-J. Park, K.-S. Hog a C.-D. Huh, Timfficit iput shapig cotrol of cotair cra systms, It.Cof. o Cotrol Applicatios, USA, Spt. 5-7, 000, pp [] Y. Pig, Dyamic ivstigatio a optimal sig of a ovl flui couplig shock absorbr for ual ma of vibratio a impact safty of prcisio systms, Acta Mchaica, Vol.90, 007, pp.05 ISSN: Issu 6, Volum 7, Ju 008