KINEMATIC OPTIMAL DESIGN OF A NEW ROLLING MILL: TWO SPs APPROACH

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1 KINEAIC OPIAL DESIGN OF A NEW ROLLING ILL: WO SPs APPROACH Jun-H Lee a Keum-Shk Hng b a Department f echancal Intellgent Systems Engneerng Pusan Natnal Unversty; 3 Jangjen-dng Geumjeng-gu Busan Krea. el: Fax: Emal: junh7@pusan.ac.kr b Schl f echancal Engneerng Pusan Natnal Unversty; 3 Jangjen-dng Geumjeng-gu Busan Krea. el: Fax: Emal: kshng@pusan.ac.kr Abstract: In ths paper the manpulalty analyss f a new parallel-type rllng mll Paramll n ts cnceptual desgn stage s nvestgated. Paramll uses tw Stewart platfrms (SPs n ppste drectn fr the generatn f degree-f-freedm mtns f each wrk-rll. he bjectve f ths new technlgy s t permt an ntegrated cntrl f the strp thckness strp shape par-crssng angle unfrm wear f rlls tensn f the strp. he frward/nverse knematcs prblems are frmulated. w man knematc parameters the sze f the base the penng angle made f tw neghbrng jnts fr a gven sze f the wrk-rll have been determned n the way that the frce mment transmssn frm the actuatrs t the wrk-rll s maxmzed. Cpyrght 3 IFAC Keywrds: Parallel manpulatr frward nverse knematcs Stewart platfrm rllng mll Jacan matrx manpulalty.. INRODUCION he cntnuus rllng s a mechancal prcess whereby the plastc defrmatn f a metal (a plate s acheved by passng t thrugh a seres f sts yeldng a thn sheet f the metal. Each st cnssts f tw sets f wrk-rll backup-rll. he purpse f a backup-rll s t supprt the wrk-rll durng the rllng. Bth rlls mve up dwn as a unt when the strp thckness s adjusted. Snce the mddle sectn f the strp s less defrmed than the sde edges due t the bendng f the rlls tw wrkrlls are blquely placed. he blque placement f tw wrk-rlls s called the par-crssng wth whch a unfrm thckness acrss the strp s acheved. adjust the par-crssng angle f the tw rlls tw addtnal hrzntal hydraulc cylnders are used. In the cnventnal mll the degree f freedm (DOF f a rll s three: heave (up dwn mtn parcrssng (yawng rllng. In the current rllng mll nce the rll gap the par-crssng angle are set up they cannt be mdfed durng the prcess. Only the rll velcty the lper angle fr adjustng the strp tensn can be changed. herefre an ntegrated (smultaneus cntrl f the strp thckness strp tensn strp shape unfrm wear f the rll s nt pssble. hs necesstates the develpment f a new rllng technlgy whch can prvde DOF mtns f the rll. he prpsed new rllng mll s based upn Stewart platfrm (SP type manpulatr (erlet ts manpulalty analyss has already been addressed n (Hng et al.. Hwever the wrk n (Hng et al. has cnsdered nly ne SP nt tw SPs tgether that wuld represent the real rllng prcess. In ths paper a knematc ptmal desgn f a new parallel-type rllng mll Paramll s nvestgated. Nte that a new rllng mll shuld prvde at least 5 DOF mtns t the wrk-rll: surge (strp tensn cntrl sway (even wear acrss the rll heave (strp thckness cntrl rllng (strp shape cntrl yawng (even thckness acrss the strp mtns. Even thugh the SP can prvde sx DOF mtns the tch mtn f the rll s nt cnsdered because the rll tself s rtatng. hs then wll make the lper mechansm whch s the current technlgy fr tensn cntrl unnecessary wll allw an ntegrated cntrl f the 5 mtns. he desgn prblem cnsdered n ths paper dscusses the determnatn f a st sze fr the platfrm s gven sze (.e. the sze f a wrk-rll. herefre the sze f a base jnts cnfguratn the length f a hydraulc cylnder have t be determned fr a gven length f the wrk-rll. he result f (Hng et al. that cnsdered nly ne

2 SP n the knematc ptmal desgn wuldn t be suffcent because the analyss n (Hng et al. dd nt cnsder the rllng frce mment generated between the tw wrk-rlls n cntact. A knematc cnstrant equatn n the cnfguratn where tw SPs are n cntact at the neutral pstn s frst derved. Frm the knematc cnstrant equatn a velcty-jacan matrx a subsequent frce-jacan matrx are derved. A manpulalty measure as a rat f the manpulalty ellpsd vlume the cndtn number f each velcty/frce-jacan matrx s defned. w man knematc parameters the sze f the base the penng angle f tw neghbrng jnts fr a gven sze f the wrk-rll have been determned n the way that the frce mment transmssn frm the actuatrs t the wrk-rll s maxmzed. he results wll be cmpared wth the results n (Hng et al... PARAILL SRUCURE. Paramll Cnfguratn Fg. shws the crdnate systems ntrduced fr the new Paramll whch cnssts f tw SPs n ppste drectns. hs paper deals wth the desgn f ne st because each st has the same structure. Crdnate systems fr the lwer SP are frst ntrduced. Let { } be the crdnate system that s attached t the base whch s a fxed crdnate system dentng X Y Z crdnates. he rgn f { } s O whch s the center f the base. Let { } be the crdnate system that s attached t the platfrm whch s a mvng crdnate system dentng x y z crdnates. he rgn f { } s whch s the center f the platfrm. Let B P ( = dente sx jnts n the base n the platfrm respectvely. Let b = O B p = P ( = be the pstn vectrs frm the rgns f the base platfrm t the crrespndng jnts respectvely. Let a = BP ( = be the leg vectrs frm the sx jnts n the base t the sx jnts n the platfrm. Let R d = O [ d X dy dz ] = be the rtatn matrx the translatn vectr between { } { } respectvely. A smlar ntatn can be ntrduced fr the upper SP. Nte that all varables fr the upper SP are wrtten n talc whle thse fr the lwer SP are wrtten n nrmal face. Let {} the crdnate system that s attached t the base f the upper SP whch s a fxed crdnate system dentng X Y Z drectns. Nte that {} has been rtated by R X ( π wth respect t { }.e. 8 degrees wth respect t X -axs f the lwer SP. Let O dente the center f the base f the upper SP. Let { } dente the crdnate system that s fxed t the platfrm f the upper SP. { } s a mvng crdnate system dentng x y z drectns whse rgn s. Let B P ( = dente the sx jnts n the base platfrm respectvely. Let p = P b = O B ( = be the pstn vectrs t the jnts. he actuatr vectrs n the upper SP are a = B P ( =. Let R [ d d ] d = O = x y dz be the rtatn matrx the translatn vectr frm the mvng crdnate { } t the fxed crdnate {}. Snce the mtns f tw SPs are nt ndependent r = O B s = O P ( = see Fg. are ntrduced as ntermedate varables fr clarfyng dependence. Let O O = O O d be the vectr frm the rgn f {}- crdnate t that f {}-crdnate. Let R [ t t t ] t = x y z be the rtatnal matrx the translatnal vectr frm {}-crdnate t {}- crdnate respectvely where t dentes the thckness f the strp. Let the rtatn matrx f the platfrm f the lwer SP wth respect t {}- crdnate be R = RZ ( γ RY ( β RX ( α where α β γ dente the fxed angle representatn. Smlarly let the rtatn matrx f the platfrm f the upper SP wth respectve t {}-crdnate be R = ( R = R ( γ R ( β R ( where Z α Y X β γ α dente the fxed angle representatn f the upper SP. Nte that α β γ α β γ represent the rllng tchng yawng angles f the crrespndng platfrm. Hence R can be determned frm R. Fg. shws the arrangement f jnts n the base platfrm fr cnnectng the hydraulc cylnders. Such an arrangement n Fg. can avd the knematc sngularty (Gsseln Angeles 99; Lee Park. Let the acute angle made f the center lne ts neghbrng jnt n the base platfrm be φ lb φ lp fr the lwer SP respectvely φ ub φ up fr the upper SP respectvely. B u 5 a a B l 5 B l b B u r P l 5 P u 5 s b p P l 4 B l 4 Y P u 4 B u 4 p Z O Z O {} P l {} P u Y X R z R X B l 3 B u 3 d {} y z R d y {} P l t B u x P u B u P l 3 P u 3 B B l l Fg.. he crdnate systems ntrduced fr the new Paramll. x P u P l

3 B l4 Y O B l3 X P l5 y P l4 φ lp P l3 P l x the heght f a st. Hence a becmes a = a + R + R t R + d =. ( r lp B l5 B u5 B l r ub B u r lb Base Y O φ lb Bu4 B u 3 Base B l φ ub B l (a Lwer Stewart platfrm B u B u X P u5 (b Upper Stewart platfrm P u φ up P u 4 P l P l Platfrm y r up Platfrm Fg.. he arrangement f jnts n the base platfrm fr cnnectng the hydraulc cylnders. 3. KINEAIC ANALYSIS f PARAILL 3. Knematc Cnstrants Snce Paramll nvlves tw SPs the cnfguratn n whch tw wrk-rlls are n cntact has t be cnsdered. he mtn f the upper wrk-rll s nt ndependent frm the mtn f the lwer ne because the cntact mtn has t be mantaned at all tme yeldng a clsed lp chan. Frst n Fg. the leg vectrs are derved as a = d + R = P u P u 3 P u x a f the lwer SP ( where the superscrpt n the left-h sde f p dentes { } crdnate system. In the stuatn that a msunderstng mght ccur the left-h superscrpt lke n R n p wll be added. Hwever f the crdnate system used s qute clear t wll be skpped. he r s vectrs n Fg. are r = d + R t+ R d + b = ( R s = d + R t + =. (3 Hence the leg vectrs a f the upper SP becme a = r + s =. (4 he substtutn f ( (3 nt (4 yelds: a = ( d + R t + R d + b + d + R t + R p R = ( d R + =. (5 he multplcatn f ( R at bth sdes f (5 yelds: ( a = d R + =. ( By re-arrangng terms ( becmes d = R + ( a =. (7 he dstance frm O t O s d = d + R t + d. (8 Fnally by substtutng ( (7 nt (8 the clsed lp chan cnstrant equatn s derved as fllws: d = a + R R t R + a = (9 where d dentes a cnstant vectr determned by 4. JACOBIAN ARICES 4. Velcty Jacan atrx Usng the cnstrant equatns ( ( the velcty Jacan matrx s derved as fllws: Frst the nner prduct f ( yelds: a a = a ( d + R =. ( Dfferentatng ( wth respect t tme usng d ( R =ω l R (Spng Vdyasagar 989 the dt fllwng expressn s derved. a a& = a ( d& + ωl R = a d& + ωl R a = ( where = [ ω ω ω ] ω l l X ly l dentes the angular Z velcty vectr f the lwer platfrm. Als the nner prduct f ( yelds: a a = a ( a + R + R t R R b + R d =. (3 he dfferentatn f (3 wth respect t tme yelds: a & [ a & b& ( p R p& ( t R t& a = a ( R + R R + R b + R Rb& + ( p + R + R p& d & ] = (4 Snce d & p & b& p& b& are cnstant vectrs &d = p& = &b = p & = b& = hlds. Rearrangng the terms n (4 yelds: a & [ a & ( p ( t R t& a = a + + ( R R R + R + R + ( p + R =. (5 d Usng ( =ω dt d ( R =ω u R where the subscrpt sts fr dt the par-crssng the fllwng expressn s derved. a & [ a & ( ω R p ( ω R t R t& a = a l + l + ( ωl R R + Rω R + R ω R b + ( ω l R + Rω = ( where = [ ω ω ω ] u ux uy uz ω s the angular velcty vectr f the upper platfrm ω = [ ω ω ω ] s the relatve angular velcty vectr f the tw wrk-rlls. ω l ω u ω satsfy the fllwng relatnshp (7 because the vectr frm O t O s a cnstant vectr wth n rtatnal mtn. ω ω + R R ω = l + R u R ωu ωl R x y u z = ω. (7-75 -

4 he substtutn f a& = d & + ωl R equatn (7 nt ( yelds: a & = a d & + ω ( R t R R R b a + R b + R a R ω R ( R l R b p a + a ( R t & =. (8 he matrx ntatn f ( (8 becmes L & l = & mvng t the ther sde & = L& l = J & vl (9 where J v s called the velcty Jacan matrx f Paramll & l = [ a & a a ] L a& & L & & = [ d& X d& Y d& Z ωlx ωly ωlz t& ] x t& y t& z ω x ω y ω z a L = = O a a a a A = R a a ( A a ( A a ( B a ( B a O a 3 3 Ra Ra 3 3 ( C a ( C a B = R t R R + R + R C = R R =. 4. Frce/ment Jacan atrx he frce/mment Jacan matrx relates the frces acng at the twelve legs wth the frce/mment resulted n the platfrms. herefre the rllng frce mment needed at tw wrk-rlls can be cnfgured by dentfyng the frce/mment Jacan matrx at a gven cnfguratn the twelve frces at ndvdual actuatrs. Nw ths relatnshp s derved by applyng the prncple f vrtual wrk as fllws: akng the varatns n bth sdes f (9 δ = Jvδl ( where δ = [ δd δd δd δα δβ δγ x y X z Y δ t δt δt δα Z δβ δγ ] δ = [ δ a Lδ δ Lδ a ] l. a fl l u L u Let = [ L f f f ] a f be the actuatng frces at the legs. Let F = F F F F F F [ X Y Z rllx rlly rllz ] [ ] = be the rllng X Y Z x y frce mment at the lwer wrk-rll respectvely. he applcatn f the prncple f vrtual wrk t ( yelds: z f δl = τ δ. ( he substtutn f ( nt ( yelds: ( f τ J v δl =. ( Snce ( δ l n ( s lnearly ndependent f = τ. (3 f = v J v Defnng J ( J (3 becmes τ = J f f (4 where J f s the frce/mment Jacan matrx that maps the actuatng frces at legs t the resultant frce mment at the end effectr. Snce J v n (9 J f n (4 relate the nput magntudes wth the utput velcty frce/mment they can be used as a tl fr analyzng the manpulalty f the structure. 5. KINEAIC OPIAL DESIGN 5. anpulalty Analyss One mght want t knw hw easly the mechansm can be manpulated n terms f transmttng velcty frce (Yshkawa 985; Zanganeh Angeles 997; Park Km 998. hat s t wuld be desrable t acheve the target jbs n the cnfguratn space wth mnmal effrts n the jnt space. analyze the nput-utput characterstcs f a gven mechansm the unt-nrm nputs are ften used. But these unt-nrm nputs may nt represent the actual peratng range f the mechansm because the maxmum velctes frces f ndvdual actuatrs may dffer. herefre ndvdual actuatr velcty frce must be nrmalzed (Hng Km. Nw the nrmalzatn f nput velctes nput frces are defned as & ˆ l W & l (5 = l fˆ = f W f. ( where = dag a& L & a& L a& W l ( a max max max max ( f L f f L f W f = dag lmax lmax umax umax repres ent the maxmum velctes frces generated n the twelve actuatrs ^ dentes the nrmalzed value. Here ntng that the arrangement f the twelve hydraulc cylnders s symmetrcal we can assume that the maxmum velctes frces f all actuatrs are equal. hus W l W can be assumed dagnal matrces wth prper weghtngs. he substtutn f (5 ( nt (9 (4 respectvely yelds: vl Fl ωl = ( J &ˆ l v v W l (7 l = ( J ˆ f rll f W f (8 Frll ω where v l ω l F l l dente the velcty angular velcty frce mment f the lwer wrk-rll; v rll ω F rll represent the relatve velcty relatve angular velcty rllng f - 7 -

5 frce rllng mment f the tw wrk-rlls. It s nted that the varables related t the lwer SP 3.e. vl ωl Fl l R have been ncluded fr cmputatnal purpse. In (7 (8 the varables that deserve attentn are v rll ω Frll whch are the rll gap velcty angular velcty f the par-crssng rllng frce rllng mment. Nw splttng the Jacan matrces f (7 (8 nt tw parts the relatve mtn f tw wrk-rlls the rllng frce mment are fnally derved as fllws v rll ω Frll : vrll Jˆ v &ˆ F = rll Jˆ l ω ˆ (9 = F fˆ Jω ˆ (3 J where 3 v rll ω Frll R ˆ ˆ ˆ ˆ 3 J v Jω JF J R the subscrpt dentes utput. Snce all analyses f the fur splt Jacan matrces n (9 (3 are the same we carry ut nly ne analyss as a representatve ne. he manpulalty can be defned as hw easly unfrmly the end-effectr s able t mve n an artrary drectn. he vlume f the utput manpulalty ellpsd (EV the cndtn number (CN fr a gven weghted Jacan matrx are defned as fllws (Ahn Hng 99: ν ν π σ max EV = σ CN = (3 ν Γ( + = σ mn where R s the dmensn f the manpulalty ellpsd Γ ( s the Gamma functn. Integratng lcal manpulaltes ver the gven wrkspace (Hng et al. the fllwng glbal manpulalty ndex can be defned. λ Ω ( rlb rub φlb φub φlp φup dω Λ = = 3 4 (3 dω Ω where Ω λ Λ represent the ttal wrkspace the lcal manpulalty measure the glbal manpulalty respectvely. In rder t cnfrm the crrectness f the prpsed methd the results n ths paper the results n (Hng et al. n whch ne SP was used are cmpared. Fg. shws the 3-D plts f the glbal frce mment manpulalty measures wth the use f ne SP see (Hng et al.. Fg. 3 shws the 3-D plts f the glbal frce mment manpulalty measures wth the use f tw SPs. It can be seen frm Fg. 3(a Fg. 4(a that the maxmum value f the glbal frce manpulalty n Fg. 4(a s greater than that f Fg. 3(a. he reasn fr ths s that the cndtn number s the same but the manpulalty ellpsd vlume fr the case f tw SPs s gger than the case f ne SP. Whch means that fr a gven level f desred rllng frce/mment the st sze may be cmpactly desgned wth the analyss f usng tw SPs. hs s very mprtant because ths wll allw the reductn f ttal prductn lne length. On the ther h frm Fg. 3(b Fg. 4(b t can be seen that the mment manpulalty s almst the same. he reasn fr ths can be traced frm that the rtatnal wrkspace fr the rllng prcess s relatvely small cmpared wth the translatnal wrkspace. 5. Determnatn f Knematc Parameters Let the dstance between O O at a statc equlbrum state f the wrk-rll be d = t = d =.8 n whch the rtatn matrx [ ] R R s the dentty matrx. he ranges f parameter values fr the pursut f ptmal values are set as fllws: < φ < mm r 85 mm where φ = φlb = φlp = φub = φup r = r lb = rub have been assumed. (Gsseln Angeles 99. (a 3D plt f the glbal frce-manpulalty. (b 3D plt f the glbal mment manpulalty. Fg. 3. anpulalty analyss usng tw SPs. Glbal mment manpulalty Base radus [m] Angle between jnts [degree] (a 3D plt f the glbal frce manpulalty. Glbal frce manpulalty Base radus [m] Angle between jnts [degree] (b 3D plt f the glbal mment manpulalty. Fg. 4. anpulalty analyss usng sngle SP

6 Fg. 5 shws the tw dmensnal cnturs f the cmned glbal frce mment manpulaltes n Fg. 3. he reasn fr the cmnatn f the tw manpulaltes s because t s dffcult t fgure ut the maxmum parameter values that satsfy bth the frce mment manpulales at the same tme. Nte that the tw manpulaltes have dfferent unts. herefre frst bth manpulaltes have been nrmalzed by ther peak values. It s fnally bserved that a cmmn range that maxmze bth manpulaltes s mm~8mm fr the radus f the base 7.5 ~ 35.5 fr the jnt angle. able shws the ptmal results btaned wth the use f sngle SP n (Hng et al. whle the results wth the use f tw SPs are gathered n able. By cmparng able able t s seen that the base can be made smaller thrugh the analyss f tw SPs whch wll allw the smaller sze f a st agan whch makes the entre prductn lne shrter. Fg. 5. D cnturs f the glbal frce manpulalty measure. able Fnal specfcatns btaned by knematc ptmzatn: Sngle SP case Platfrm radus ( r p Base radus ( r b Angle between jnt ( φ b = φ p nmum length f leg ( l mn axmum length f leg ( l max mm 9mm mm 9.3mm able Fnal specfcatns btaned by knematc ptmzatn: w SPs case Angle nmum Platfrm Base between jnt length f radus radus ( φ lb = φ lp leg ( r lp = r up ( r lb = r ub ( φ ub = φ up ( l mn axmum length f leg l ( max mm 8 mm 3 7. mm 73.3mm. CONCLUSIONS In ths paper a feaslty study n a new paralleltype rllng mll based upn tw SPs was nvestgated. Because an advantage f usng dfferent knematc parameters n the lwer upper SPs was nt seen bth SPs were assumed t have the same structure but used n ppste drectns. w man knematc parameters the sze f the base the angle made f tw neghbrng jnts fr a gven sze f the wrk-rll have been determned n such a way that the frce mment transmssn frm the actuatrs t the wrk-rll s maxmzed. It s the authrs desre that the results n ths paper prvde a gd gudelne fr a parallel-type rllng mll at the cnceptual desgn stage f the mll. Future wrks nclude anther type f mechansm nt based upn the SP dynamcs analyss an ntegrated cntrl f the strp thckness strp tensn rll speed strp shape par-crssng angle unfrm wear f the rll etc. ACKNOWLEDGEEN hs wrk was supprted by the Krea Research Fundatn KRF -4-E75. REFERENCES Ahn B. J. Hng K. S. Frce/ment ransmssnalty Analyss f a Parallel anpulatr Jurnal f the Krean Scety f Precsn Engneerng Vl. 3 N. 4 pp Gsseln C.. Angeles J. Sngularty Analyss f Clsed-Lp Knematc Chans IEEE ransactns n Rbtcs Autmatn Vl. N. 3 pp Hng K. S. Km J. G. anpulalty Analyss f a Parallel achne l: Applcatn t Optmal Lnk Parameter Desgn Jurnal f Rbtcs Systems Vl. 7 N. 8 pp Hng K. S. Lee S. H. Ch C. Optmal Desgn f a New Rllng ll Based upn Stewart Platfrm: axmzatn f knematc anpula lty (n Krean Jurnal f Cntrl Autmatn Systems Engneerng Vl. 8 N. 9 pp Lee. K. Park K. W. Wrkspace Sngularty Analyss f a Duble Parallel anpulatr IEEE/ASE ransactns n echatrncs Vl. 5 N. 4 pp erlet J. P. Parallel Rbts Kluwer Academc Publshers. Park F. C. Km J. W. anpulalty f Clsed Knematc Chans ASE ransactns Jurnal f echancal Desgn Vl. N. 4 pp Spng. W. Vdyasagar. Rbt Dynamcs Cntrl Jhn Wley & Sns 989. Yshkawa. anpulalty f Rbtc echansms Internatnal Jurnal f Rbtcs Research Vl. 4 N. pp Zanganeh K. E. Angeles J. Knematc Istrpy the Optmum Desgn f Parallel anpulatrs Internatnal Jurnal f Rbtcs Research Vl. N. pp