Expanding the Twin Pulse Swing Free Profile

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1 8 IEEE Ieraioal Coferece o Roboics ad Auomaio Pasadea, CA, USA, May 19-3, 8 Expadig he Twi Swig Free Profile David Bowlig, Gregory Sarr, Joh Wood ad Ro Lumia Deparme of Mechaical Egieerig The Uiversiy of New Mexico, Albuquerque, New Mexico 87131, USA dbowlig,sarr,jw,lumia)@um.edu Absrac The wi-pulse moio profile is sill perhaps he simples profile i exisece for reducig residual vibraio. Is primary drawback is ha of arrow badwidh, i.e., i is effecive for aeuaig oly a very arrow rage of uecies abou is desig uecy. This has led o he iclusio of more pulses o improve robusess o uecy variaio. However, oe cos of addig pulses is exedig he duraio of he resulig moio. This paper preses several echiques whereby he basic wi-pulse moio profile ca be expaded o reduce is uecy sesiiviy while sill preservig is advaages. These echiques iclude ieioally mis-aligig he ull pois of ideal wi pulses ad a covolved boxcar fucio, he addig a rapezoidal profile compoe, fially culmiaig i wha we erm a misaliged boxcar / rapezoidal / wi pulse M- BC:TZ:TP) moio profile. This expaded profile ad a basic wi-pulse profile were boh evaluaed o a e-pedula sysem wih aural uecies varyig over a rage of early 1%. Simulaio resuls showed a overall reducio of 97% wih he expaded wi-pulse profile, ad oably all e uecies were aeuaed; oly oe uecy was aeuaed wih he sadard wi-pulse profile. Experimes usig a liear rail raspor sysem ad he e pedula device validaed he simulaio. These expaded profiles reai he simpliciy ad shor duraio of he wi-pulse wih icreased robusess o uecy variaios. I. INTRODUCTION Posicas corol was he firs use of pulses o elimiae uwaed oscillaios i lighly-damped sysems [1], [], ad was origially developed as a compesaor o be used i cojucio wih a exisig corol sysem. The use of posicas corol was he expaded by Sarr [3] o he realm of profile geeraio o creae profiles ha elimiaed or reduced uwaed residual vibraios. Sighose, Siger ad Seerig [9], [13] expaded Sarr s moio work by iroducig he cocep of ipu-shapig i which a base profile is covolved wih a series of wo o five pulses. Boh approaches creae profiles ha firs excie he cacel ou a rage of uecies i he sysem ha is beig drive. Sarr s origial work solved he problem by rampig up he velociy i wo seps, separaed by half he period of he aural uecy of he sysem. Siger ad Seerig approached he problem by covolvig a wi pulse kerel wih a posiio profile. Their pulses were also separaed by half he period of he aural uecy of he sysem. This reduces o Sarr s work, sice he derivaive or velociy of a covolved wi pulse is a wo sep velociy profile as used by Sarr. The wi-pulse TP) moio profile used by Sarr was sesiive o errors i he aural uecy of he drive This work is suppored by DOE Gra #DE-FG5-4NA559 awarded o he UNM Maufacurig Egieerig Program sysem. Covoluio based TP ipu-shapig IS) reduced he sesiiviy o uecy errors ad allowed for muliple modes by icreasig he umber of pulses. The curre maximum umber of pulses used by Seerig, Siger, ad Sighose is five as used i he hree-hump exra isesiive EI) shaped moio. This paper preses a geeral approach of decreasig he sesiiviy of he origial TP moio by maipulaig he shape ad widh of he pulses i he classic TP profile. II. OVERVIEW OF EXISTING PULSE-BASED PROFILES Profiles covolved wih a series of pulses ca creae moio ha will o resul i residual vibraio i a drive sysem Figure 1). Acually, pulse-based profiles firs iroduce vibraios io a sysem, he cacel ou he vibraios whe he fial series of pulses is applied. Figure shows a series of uecy specra produced by a hree-hump EI shaper. The figure shows he resulig specra as successive pulses i a EI shaper are applied o a sysem. The firs pulse creaes a fla specrum ha excies all uecies. As each addiioal pulse is added, he widh of he och icreases ad he iude of he specrum decreases. If his pulse rai is covolved wih a sep i posiio, a profile is geeraed ha will cacel ou all residual vibraio over a give uecy rage. This is he basic implemeaio of TP ad IS moio from a sigal processig view) as firs described by Sarr Figure 1). A) B) Time SF C) D) Time ω SF Time Fig. 1: Swig Free moio wih SF = π/ω. A) Posiio profile; B) Velociy profile; C) Acceleraio profile; D) specrum wih ull poi a ω = rad/sec /8/$5. 8 IEEE. 9

2 Twi Combied Specrum afer 1 s pulse ime ime ime ime ime Specrum afer d pulse Specrum afer 3 rd pulse Specrum afer 4 h pulse Specrum afer 5 h pulse * = ime ime ime = Noch Fig. : Ipu Shapig excies a sysem wih is firs pulse. As each successive pulse is applied o he sysem, he uecy specrum iude is reduced ad he och wides. III. MINIMAL PULSE-BASED PROFILES A. Twi TP) Profile Sarr s origial work was based o exciig ad he cacelig he eergy i a sysem usig wo pulses. However, a pulse Dirac fucio) cao be physically realized i sofware or hardware. The pulses mus have some physical widh, usually limied by he uecy or samplig period of sofware ad hardware. Whe widh is added o he pulses, he shape of he uecy respose of he TP profile sars o chage. The uecy specrum of he origial TP moio ad he shape of he pulse iself are muliplied ogeher o creae a fial combied uecy specrum. B. Twi / Moio Profiles TP:BC profile) This combiig of he specra ca be advaageous. If he pois i he uecy specrum where he iude goes o zero ull or och uecy) are aliged, he overall sesiiviy of he profiles ca be reduced. For example, if boxcar fucios are used for he pulses, he ull pois ca be aliged o decrease he sesiiviy or icrease he effecive widh) of he och i he uecy domai. Thus he slope of he specrum a he och is reduced a he ceer uecy Figures 3, 4). Whe desigig a sadard wi pulse profile he desired uecy o be caceled,, is aliged wih he firs ull uecy of he profile. The profile i Figure 3 has he desired cacellaio uecy aliged wih he secod ull poi of he TP specrum. The uecy specrum of he TP : BC profile is give by, 1 TP:BC) = e k 1)jπω ) ω e k+1)jπω +e jπω) ω 1, ω ) where k = The wi pulse ull poi umber o alig wih he boxcar ull poi = Noch uecy 1) Combied Specra of Twi ad Fig. 3: Resul of combiig a wi pulse TP) wih a boxcar TP:BC)whe ull pois are aliged. The secod ull poi k =) is aliged wih he boxcar ull poi. db w r/s) Ceer.7 r/s) 5% 6 db) och.54 r/s).6 r/s).6 r/s) Swig Free Double Swig Free combied wih boxcar.1 r/s).5 r/s).4 r/s) 1% 4 db) och Fig. 4: Close up Semi-log plo of combied wi pulse wih a boxcar TP:BC) of Figure 3. The widh of he boxcar, ad he space bewee he wo pulses of he TP : BC profile are give by ) ad 3): boxcartp:bc) = π ) ierpulsetp:bc) = π k 1). 3) The oal ime required for he profile legh of he kerel) is he sum of equaios ) ad 3): kereltp:bc) = π k +1). 4) 93

3 For he firs ull poi of a wi pulse ad boxcar fucio o alig, he widh of he boxcar mus be wice he disace bewee he pulses. This creaes a FIR filer kerel) wih a pyramid shape Figure 5). This kerel profile closely resembles he firs improvemes of Siger ad Seerig IS, wih hree pulses. Twi Swig Free) + = Twi Wih Double Twi Swig Free) Double Fig. 5: Resul of combiig a wi pulse of widh = TP ad a boxcar of widh = BC = TP. 1 M TP:BC) = e jπω ) ) δω ω +1 e jπω ) ω+δω 1) j + δ ) 5) I should be oed ha he offse δ ca be added o he boxcar ad subraced from he wi pulse referred o as he M TP : BC mehod) or subraced from he boxcar ad added o he wi pulse referred o as he M BC : TP mehod). If he laer mehod is chose, he uecy specrum of 6) is creaed. 1 M BC:TP) = e jπω ) ) ω+δω +1 e jπω ) ω+δω 1) j δ ) 6) The legh of he kerel is also affeced by he mehod. The M TP : BC mehod creaes he shores kerel bu has a larger iude, resulig i higher acceleraio. The legh of boh kerels is show i equaios 7) ad 8). The shape of he kerel for he M TP : BC profile is show i Figure 7. The primary differece bewee he mehods is he order i i which he ull pois appear i he uecy domai. For he M TP : BC he wi pulse ull poi has he lower uecy. C. Mis-aliged Twi- / Profiles M-TP:BC profiles) The och widh of he Twi / moio profile ca be icreased furher by slighly mis-aligig he ull pois of he wi pulse ad he boxcar Figure 6). The ull pois of he boxcar ad wi pulse ca be slighly offse by δ o each side of he desired och uecy. The uecy specrum of he offse profile is defied by equaio 5). Give a desired error for he uecy specrum iude, he correspodig offse δ ca be foud by ieraig over 5) usig a algorihm such as he bisecio mehod [5]. ω + δω ) π ω + δω π kerelm TP:BC) = π3 δ ) ω δω kerelm BC:TP) = π3 + δ ) ω δω Mag 7) 8) Twi swig free) ω δω ω + δω π ω δω π ω + δω π 3 δω ) ω δω Time Fig. 7: Kerel shape for a mis-aliged M-TP:BC) profile Twi pulse wih miss aliged boxcar δω Icreased Noch Widh Fig. 6: Combiig a wi pulse wih small mis aliged boxcar o icrease uecy och M-TP:BC). D. / Velociy Ramp BC:VR) Moio Profile If we le he widh of he boxcar fucios ad he wi pulse be equal i legh hey combie o form a sigle boxcar fucio wih a legh wice ha of he wi pulse. This is a sraighforward pulse-based profile. A simple boxcar ca creae he same ull poi effec i he uecy domai, bu 94

4 also has he same robusess issues bu o quie as severe) as he wi pulse profile. The advaage of he wi pulse is ha he duraio of he kerel ca be less ha ha of a boxcar. If he iude of he acceleraio of he wo profiles is kep cosa, he he wi pulse degeeraes o a sigle boxcar Figure 8). A) B) * = > dp * dp bc = bc = Fig. 8: Compariso of whe a wi pulse ad boxcar degeeraes o a sigle boxcar; A) Resul of combiig a wi pulse of widh = TP =.4π ad a boxcar of widh = BC =.4π.; B) Resul of combiig a wi pulse of widh TP = BC =.4π. There is a advaage of usig he boxcar kerel o geerae profiles. The boxcar profile is he sadard profile used i mos commercial moio corol sysems. This profile is he firs half of wha is usually referred o as a rapezoidal velociy profile. Give he och or aural uecy ad he desired slew velociy v he acceleraio commad a ca be compued for a ramp velociy profile. = π 9) v = a Combie, elimiaig, ad he solvig for a resuls i. a = v 1) π The duraio of he acceleraio secio of he profile is give by. BC:VR) = v a = π 11) E. / Trapezoidal BC:TZ) Swig-Free Profile Eve hough he / Velociy Ramp BC:VR) kerel is easily implemeed i mos commercial moio corol sysems, he sesiiviy of he arrow och i he uecy specrum makes he sigle boxcar profile impracical i mos applicaios. The robusess ca be improved by usig he combiig echique described i secio III-C. The rapezoidal uecy specrum acceleraio followed by a deceleraio) ca be combied wih he boxcar uecy specrum. If he desired profile sars a res, slews a disace he comes o a sop rapezoidal profile) he overall profile ca be used o wide he och i he uecy specrum. The rapezoidal profile ca be broke dow io wo Dirac pulses oe posiive ad oe egaive) covolved wih a boxcar fucio. As wih he wi pulse ad boxcar TP:BC) example i secio III-B, he ull pois i he uecy specrum ca be aliged o creae he improved specrum i Figure 9. I he wi pulse aligme echique he firs ull poi of he wi pulse ad boxcar specrum were forced o alig. I he BC:TZ profile he same echique is used bu he ull poi mus be picked o saisfy he disace ad velociy cosrais of a rapezoidal profile. The disace ad slew speed desired for he rapezoidal profile may require ha he drive objec cycles hrough may half cycles of is aural period before i is brough o res. The firs ull uecy of he boxcar ca sill be used, bu he fac ha he sysem goes hrough may cycles requires ha ay ull poi of he rapezoidal pulse ca be used. The firs ull poi of he boxcar will be aliged wih he rapezoidal ull poi ha creaes a slew velociy ha comes closes o machig he desired slew velociy. I mos cases, he ull poi will be picked ha creaes he larges velociy ha does o exceed he desired slew velociy. The followig are he goverig equaios for he ull poi aligme of a rapezoidal profile: BC = π TZ = π k BC:TZ) = BC + TZ = π1+k) d v slew = TZ, 1) where d = Disace o ravel v slew = Slew velociy of rapezoidal profile BC = Duraio of boxcar TZ = Duraio bewee rapezoidal pulses BC:TZ = Toal ime of rapezoidal profile k = Null poi umber of rapezoidal profile. Equaio 1) ca be solved for k; ) dω k = ceil. πv slew 13) Whe compuig k he desired slew velociy v slew ) is used. The value of k is he rouded up o he eares ieger. Oce k is compued, he duraios of he rapezoidal profile TZ ad boxcar BC profile ca be compued. Lasly he acual slew velociy for he profile is compued also usig k, v acual slew = d πk. 14) F. Mis-aliged / Trapezoidal M-BC:TZ) Swig-Free Profile The mis-aligme echique used for he wi-pulse profile ca also be used for he rapezoidal profile wih he same wideed-uecy specrum och. The resuls for such a profile ca be see i Figure 1. The formulaio o compue he duraios of rapezoidal pulses ad boxcar profiles is similar o ha for he aliged rapezoidal profile. Oly he 95

5 Trapezoidal Profile Double Twi Trapezoidal Profile Twi * = Trapezoidal Profile Trapezoidal Profile Wih Aligme Fig. 9: Trapezoidal profile wih uecy specrum of he pulses ad boxcar aliged. offse δ mus be added ad subraced from he aural period of he sysem i 1). π BCM BC:TZ) = δ π TZM BC:TZ) = +δ k M BC:TZ) = BCM BC:TZ) + TZM BC:TZ) = πωk+1) δωk 1)) ω δω d v slew = TZM BC:TZ) Posiio Velociy Acceleraio Trapezoidal pulses Trapezoidal s wih Noch Widh 15) Fig. 1: Trapezoidal profile wih uecy specrum of he pulses ad boxcar slighly miss-aliged o icrease widh of uecy och. As wih he wi pulse mis-alig echique, he δ offse ca be added o he rapezoidal ad subraced from he boxcar referred o as he M BC : TZ mehod), or subraced from he rapezoidal ad added o he boxcar referred o as he M BC : TZ mehod). The M TZ : BC mehod creaes he profile wih he shores duraio, bu also has a larger iude, hus higher acceleraio. The followig are he goverig equaios for he ull poi mis-aliged rapezoidal profile: π BCM TZ:BC) = +δ π TZM TZ:BC) = δ k M TZ:BC) = BCM TZ:BC) + TZM TZ:BC) = πωk+1)+δωk 1)) ω δω d v slew = TZM TZ:BC). 16) These equaios ca he be solved for k ad he profile imes compued: ) d ω δw) k = ceil. 17) πv slew Whe compuig k he desired slew velociy v slew ) is used. Oce k is compued he acual slew velociy is compued usig 16). The uecy specrum iude for he mis-aliged rapezoidal profiles is defied by 18). As wih he mis-aliged, wi pulse profiles, he quaiy δ ca be compued by ieraig over he equaios uil he desired error is achieved a. M BC:TZ) = M TZ:BC) = jω δω) 4π e jπω ω δω ) + e jπωk e ) ω+δω 1 ) ) jπωk+1) δωk 1)) ω +δω jω δω) 4π e jπω ) ω+δω + e jπωk ) ω δω 1 ) ) jπωk+1)+δωk 1)) ω e +δω 18) 19) G. Mis-aliged / Trapezoidal / Twi M- BC:TZ:TP) Moio Profile The mis-aligme echique ca be ake oe sep furher by combiig he wi pulse, boxcar, ad rapezoidal profiles ogeher. Whe wo base profiles are combied, wo possible resulig profiles exis. Whe hree base profiles are combied, six resulig profiles are possible. The umber of resulig profiles ca be reduced o wo if he rapezoidal profile is always he ceer profile. This ca be jusified by viewig he uecy specrum of he hree profiles. The uecy specrum of he rapezoidal profile will usually have he arrowes och. This arrow och is a resul of he disace ad ime required o raverse he rapezoidal profile. For log duraio moves, he sysem will go hrough may cycles. The more cycles required o raverse he profile, he more compac he lobes become i he uecy specrum. This compacio of he specrum creaes arrow oches. For each complee cycle of he sysem here will be a ull poi o he lef of 96

6 he desired och uecy. This requires he aligme of high order ull pois wih he firs ull pois of he wi pulse ad boxcar. Placeme of he wider oches slope of he uecy specrum ear he ull poi) of he wi pulse ad boxcar profiles o each side of he rapezoidal profile helps o icrease he overall widh of he och. The riple mis-aligme echique ca be hough of as he wi pulse boxcar echique described above wih he rapezoidal profile ceered a he och uecy. The kerel legh of he riple mis-aligme is he he legh of he TP:BC profile equaios 7) ad 8)) wih he duraio of he rapezoidal profile added. A) Swig Twi Free ad B) M BC:TZ:TP) = π3 δω) ω δw + πk ) Trapezoidal, Twi Swig Free,, ad M TP:TZ:BC) = π3 + δω) ω δw + πk 1) The uecy specrum iude for he TP:TZ:BC profiles is defied by ). As wih he mis-aliged boxcar, wi pulse ad mis-aliged boxcar rapezoidal profiles, a δ offse ca be compued by ieraig over he equaios uil he desired error a is achieved: Fig. 11: A) Mis-aliged wi pulse wih boxcar M-TP:BC) uecy specrum compared o wi pulse wih boxcar ad rapezoidal M-TP:TZ:BC) wih equal δω s; B) Mis-aliged wi pulse wih boxcar M-TP:BC) uecy specrum compared o wi pulse wih boxcar ad rapezoidal M-TP:TZ:BC) wih δω s adjused o creae equal iude aeuaed. M TP:TZ:BC) = M BC:TZ:TP) = jω+δω) e jπω ) ) δω ω +1 e jπω ) ) ω+δω 1 1 e jπω k)) δω ω, ) jω+δω) e jπω ) ) δω ω +1 e jπω ) ) ω δω 1 IV. EXPERIMENTAL RESULTS A experime was coduced usig a muli-pedulum device. The leghs of he 1 pedula were arraged o have a uecy spa of 3.36 rad/s o 3.66 rad/s. The uecy rage correspods o a he masses o he pedula beig beig evely spaced bewee 8.8 i o 34. i. 1 e jπω δω+ω k)). 3) Figure 11 is a compariso of a M-TP:BC profile wih a M- TP:TZ:BC profile wih he same δ offse. I his compariso we see ha he addiio of he rapezoidal specrum reduces he respose by over 1db. The effec of he riple mis-aligme profile ca be hough of as wo filers i series, reducig he residual vibraios i a sequeial fashio. The example specra i Figure 11A) shows a TP:BC profile ad a M-TP:TZ:BC profile. Durig he ramp-up acceleraio phase of he profile, oly he TP:BC porio of he profile is fel by he sysem. A his poi i he profile he residual vibraio will aeuaed by 48db. Durig he ramp-dow deceleraio phase of he profile, he residal vibraio will be aeuaed a exra 1db o 6db due o he added aligme of he rapezoidal par of he profile. Figure 11 also shows he differece of he a M-TP:BC profile ad a M-TP:TZ:BC profile wih δ offses adjused o creae he same iude of filerig a he uecy och. Te Pedulums wih Differe Naural Frequecies Fig. 1: Muliple pedula of differe leghs he comb) beig drive a he Uiversiy of New Mexico by he RTU i he MTTC roboics lab. The pedula were drive a disace of 1.7 meers usig wo differe profiles. The firs profile was a classic wi pulse profile wih a slew velociy of.7 m/s show i Figure

7 The secod profile was a expaded wi pulse M-BC:TZ:TP) moio profile wih a desired slew velociy of.7 m/s which resuled i a acual velociy see Equaio 16, k= ) of.4735 m/s Figure 14). The pulses i boh profiles were cosruced usig filers such ha he area uder he s acceleraio profile) was equal o he slew velociies. Boh profiles were cosruced o have uecy oches ceered a 3.5 rad/s. The M-BC:TZ:TP) profile was desiged wih a δw of.5 rad/s above ad below he och uecy. From Figures 13 ad 14 i ca be see ha he acceleraio profiles of boh mehods have he same form. Bu by usig he mehods preseed herei he arrow och of he classic wi pulse profile has bee sigificaly wideed aroud 3.5 rad/s. Boh profiles were iiially esed o a simulaio of he pedula. The resuls of he simulaios are show i Figure 15. From he figure i ca be see ha he M-BC:TZ:TP) profile elimiaed virually all residual vibraios over he uecy rage of he pedula, while he classic wi pulse profile oly elimiaed he residual vibraio of he pedula ear 3.5 rad/s. The simulaio showed a reducio i residual oscillaios of 97% usig he M-BC:TZ:TP) profile as compared o he classic wi pulse profile for he aural uecy rage of he pedula used i he simulaio of 3.36 rad/s o 3.66 rad/s). The classic wi-pulse profile produced a maximum oscillaio of.374 degs whereas he he M-BC:TZ:TP) profile had a maximum oscillaios of.1 degs. Whe he acual pedula were drive wih he profiles i Figures 13 ad 14, he resuls mached ha of he umerical simulaios. Whe he classic wi pulse sysem came o res oly oe of he pedula had virually o residual oscillaios while he remaiig pedula had a oiceable moio. The experime also showed he sesiiviy of uexpaded wi pulse profiles. The profile was creaed based o a compued =3.5 rad/s for he ceer pedulum. The pedulum ha came o res was he shores pedulum which had a compued aural uecy of 3.66 rad/s. This offse of he wi pulse uecy och is also appare i he uecy specrum of Figure 13. The och i he specrum is o exacly a 3.5 rad/s. The specrum i he figure was creaed from he acual profile used i he experime. The liear rail raspor sysem used o drive he pedula has a posiio-ime updae resoluio of 1/6 secod. The ime required for a wi pulse o och ou 3.5 rad/s see Equaio 3, k = 1)isπ/ = π/3.5 =.8976 which is o evely divisible by 1/6 coribuig o he sesiiviy of implemeig wi pulse profiles. The expaded wi pulse M-BC:TZ:TP) experime closely mached he umerical simulaio wihou ay sesiiviy issues. Whe he M-BC:TZ:TP) sysem came o res all of he pedula came o res wih virually o residual oscillaios. V. CONCLUSION This paper demosraes how he origial swig free wi pulse profile ca be modified o icrease sigificaly is robusess. The mehod is based o exploiig he uecy Accel m/s ) Vel m/s) Pos m) Magiude db) Time secs) Time secs) Time secs) r/s) Fig. 13: Acceleraio, velociy, posiio, ad uecy respose of classic swig free wi pulse) profile. Accel m/s ) Vel m/s) Pos m) Magiude db) Time secs) Time secs) Time secs) r/s) Fig. 14: Acceleraio, velociy, posiio, ad uecy respose of Mis-aliged / Trapezoidal / Twi M-BC:TZ:TP) moio profile. respose of all he compoes used o cosruc a wi pulse profile. The echique icludes ieioally mis-aligig he ull pois of ideal wi pulses ad a covolved boxcar fucio, he addig a rapezoidal profile compoe, fially culmiaig i wha we erm a mis-aliged boxcar / rapezoidal / wi pulse M-BC:TZ:TP) moio profile. This expaded profile ad a basic wi-pulse profile were boh evaluaed o a e-pedula sysem wih aural uecies from 3.36 o 3.66 rad/s. Simulaio of he profiles showed a reducio i residual oscillaios of 97% usig he M- BC:TZ:TP) profile as compared o he classic wi pulse pro- 98

8 Agle deg) Agle deg) Moio D uraio Moio D uraio Time sec) Time sec) Fig. 15: Compariso of resuls of simulaed classic swig free wi pulse) profile upper plo) ad expaded wi pulse M-BC:TZ:TP) Moio Profile lower plo). Boh plos show he swig agles of each of he 1 pedula of he comb i Figure 1. The leghs of he 1 pedula were arraged o have a uecy spa of 3.36 rad/s o 3.66 rad/s. file. Experimes usig a liear rail raspor sysem ad he e-pedula device validaed he echiques. These expaded profiles reai he simpliciy ad shor duraio of he wipulse wih icreased robusess o uecy variaios. VI. ACKNOWLEDGEMENT We d like o hak Dave Vick for his help i he experimeal validaio of he classic ad expaded wi pulse profiles. REFERENCES [1] Smih O. J. M., Posicas Corol of Damped Oscillaory Sysems Proceedig of he IRE, Sepember 1957, pg [] Tallma G.H., Smih O. J. M. Aalog Sudy of Dead-Bea Posicas Corol IRE Trasacios of Auomaic Corol, March 1958, pg 14-1 [3] Sarr, P, G. Swig-free raspor of suspeded objecs wih a pah coroller robo maipulaor, ASME Joural of Dyamics Sysems Measureme ad corol, March 1985, vol 17, pg 97-1 [4] Bowlig D., Sarr G., Wood J., Lumia R. Wide Bad Suppressio of Moio-Iduced Vibraio, IEEE Ieraioal Coferece o Roboics ad Auomaio, 7 [5] Khalsa, S, Dharam High Performace Moio Corol Trajecory Commads Based O The Covoluio Iegral ad Digial Filerig, PCIM Iellige Moio Coferece, Ocober 199, pg [6] Gursel A., Sadei K., ad Seda B. Swig-free rasporaio of suspeded objecs wih robo maipulaors, Roboica, 1999, vol 17, pg [7] Siger, N., ad Seerig W., Preshapig Commad Ipus o Reduce Sysem Vibraio, ASME Joural of Dyamics Sysems Measureme ad corol, March 199, vol 11, pg 76-8 [8] Sighose, W., Dereziski, S., Siger, N. Ipu Shapers for Improvig he Throughpu of Torque-Limied Sysems, Proceedig of he 1994 Coferece o Corol Applicaios, 1994 [9] Sighose, W., Siger, N. Iiial Ivesigaios io he Effec of Ipu Shapig o Trajecory Followig, Proceedig of he 1994 America Corol Coferece, 1994 [1] Sighose, W., Bohlke, K., Seerig, W. Fuel-Efficie Shaped Commad Profiles for Flexible Spacecraf, Proceedig of he 1995 AIAA Guidace, Navigaio, ad Corol Coferece, 1995 [11] Sighose, W., Seerig, W., Siger, N. The Effec of Ipu Shapig o Coordiae Measurig Machie Repeaabiliy, Proceedig of he 1995 IFToMM World Cogress o he Theory of Machies ad Mechaisms, 1995 [1] Sighose, W.,Chuag T. Reducig Deviaios from Trajecory Compoes wih Ipu Shapig, 1995 America Corol Coferece, 1995 [13] Sighose, W., Siger, N., Seerig, W.Compariso of Commad Shapig Mehods for Reducig Residual Vibraio, Proceedigs of he 1995 Europea Corol COferece, 1995 [14] Pao, L., Sighose, W.Uiy -Magiude Ipu Shapers ad heir Relaio o Time-Opimal Corol, Proceedig of he 1996 IFToMM World Cogress, 1996 [15] Crai, E., Sighose W., SeerigDerivaio ad Properies of Covolved ad Simulaeous Two-Mode Ipu Shapers, Proceedig of he 1996 IFToMM World Cogress, 1996 [16] Sighose W., Crai, E., Seerig Covolved ad Simulaeous Two-Mode Ipu Shapers, IEE Proceedig of Corol Theory ad Applicaios, Nov 1997, vol 144 No 6, pg [17] Sighose W., Porer, L., Tule, T., SeerigVibraio Reducio Usig Muli-Hump Ipu Shapers, Trasacios of he ASME, Jue 1997, vol 119, 1997, pg 3-36 [18] Pao, L., Lau, M. Ipu Shapig Desig o Accou for Uceraiy i Boh ad Dampig i Flexible Srucures, Proceedig of he America Corol Coferece, Philadelphia, PA, Jue 1998 [19] Ecoomou, D., Mavroidis, C., Aoiadis, I.Robus vibraio suppressio i flexible sysems usig ifiie impulse respose digial filers Joural of Guidace, Corol, ad Dyamics ) 4 vol. 7 o. 1 [] Ecoomou, D, Mavroidis, C, Aoiadis, I, Lee, C Maximally robus ipu residual precodiioig for residual vibraio suppressio usig low-pass FIR digial filers Joural Of Dyamic Sysems Measureme Ad Corol-Trasacios of The ASME; Mar, V.14, No.1, P [1] Glossiois, G. N., Aoiadis, I. A. Digial Filer Based Maximally Robus ad Time Opimal Vibraio Free Moio Srucures wih Desely Pack Modes [] Craig, J., Joh Iroducio o Roboics Mechaics ad Corol [3] Smih W., Seve Digial Sigal Processig, 3 [4] Lyos G. Richard Udersadig Digial Sigal Processig, 4 [5] James M.L., Smih G.M., Wolford J.C Applied Numerical Mehods for Digial Compuaio, 1985 [6] Oppeheim, Ala ad Schafer, Roald Discree-Time Sigal Processig, Secod Ediio, 1999 [7] Bracewell, Roald The Fourier Trasform ad is Applicaio, 99