Modeling and Adaptive Control of a Coordinate Measuring Machine

Transcription

1 Modelng and Adaptve Contol of a Coodnate Measung Machne Â. Yudun Obak, Membe, IEEE Abstact Although tadtonal measung nstuments can povde excellent solutons fo the measuement of length, heght, nsde and outsde dametes, etc., a coodnate measuement machne (CMM), at least n pncple, can combne all these equements n a sngle vesatle nstument. As they can also be fully automated, lnked to a CAD system and/o bult nto the flexble manufactung cell, CMMs ae wdely used n today s ndusty. he am of ths pape s to desgn a stable contolle fo a 3D coodnate measung machne n the pesence of uncetantes and unknown dstubances. o acheve ths task, fst a smplfed model of the CMM wll be obtaned. Afte ths modelng step, dffeent types of adaptve contol technques ae appled to get a sutable and effcent contol system. he esults ae dscussed n detal n ode to gve nsght to some common poblems n contollng CMMs. I. INRODUCION HE man goal of ths pape s to desgn a stable contolle fo a 3D coodnate measung machne (CMM) n the pesence of uncetantes and unknown dstubances [9,, 3]. he desgn of the CMM s smla to some of the machne tools lke CNC s wth the dffeence beng the pesence of exta contol nputs [,, 3]. he pesence of these exta contols smplfes the poblem and we could fomulate t nto a mult-nput obotc contol poblem as dscussed n [4]. he contolle used s an adaptve one wth the nonlneates and the dstubances beng lumped nto just a sngle bounded tem. In [4], poblems of these types have been consdeed n a systematc way qute extensvely when all the states ae avalable fo feedback. he stuctue of ths contolle has the advantage that, apat fom gvng good pefomance n tackng n spte of the pesence of unknown dstubances and uncetantes, t also gves a obustly stable algothm. As opposed to the CNC machnes, the CMM poblem s dffeent n the sense that thee ae lesse numbe of contol nputs. hus, a full-state feedback adaptve contolle s used to a lneazed model of the CMM. If all the states ae not avalable fo feedback, then n geneal t s qute complcated to buld an obseve fo a nonlnea system. Howeve, n ou case the system equatons ae lneazed (except fo the dstubance tem) and we can theefoe have an adaptve obseve to estmate the unknown states (usng an MRAC fomulaton). Usng these estmates, one can then use essentally a smla contolle stuctue as done when all the states ae avalable fo feedback. hs consttutes some of the futue wok that needs to be accomplshed. In the followng sectons, fst the modelng of a CMM s pefomed. hen the contol methodologes dscussed above wll be llustated. he smulaton esults ndcate that the contolles pefom vey well. II. MODELING OF HE COORDINAE MEASURING MACHINE he schematc dagam of the coodnate measung machne s shown n Fgue. Wtng the mathematcal model of the 3D model shown wll be too cumbesome wth vey lttle beneft, f only we could get a smple model that descbes the essental featues faly accuately [, 4, 8]. Snce the ams of the CMM ae suppoted on a beangs, the dampng s educed consdeably. Fom the dagam and fom the actual machne one obseves that the dynamcs n the y and z dectons ae neglgble compaed to that of x decton at least as a easonable fst appoxmaton (ths can be vefed by gettng the bode magntude plot fom the nput of the moto on the y-axs to the measuement on the lnea encode on the x-axs whch would be much below the 0 db lne). D. Â. Yudun Obak s cuently wth the Industal Engneeng Depatment, Uludag Unvesty, Busa, 609, ukey (e-mal: obak@uludag.edu.t, obak@alum.mt.edu). Fg.. Schematc dagam of CMM.

2 he tansmsson of the toque fom the moto to the machne s though a belt dve, whch can be appoxmated as a spng wth a dampe. hs s a good appoxmaton at low fequences whle at hgh fequences the aboveappoxmated stffness of the belt n the longtudnal decton s less compaed to the one that s pependcula to ths decton. hs stffness n the pependcula decton s nonlnea and s not easy to model. On the othe hand, the bandwdth of the machne s easonably small ( Hz) that ths unmodeled nonlnea stffness can be appoxmated as a bounded dstubance. Notce futhe that because of the lage gea educton, the nonlneates and othe unmodeled effects ae lagely dmnshed on the moto sde. Wth all the above-mentoned ealstc assumptons the coodnate measung machne can be modeled as a twodmensonal model. Fo ths system, the man objectve s to desgn a stable contol algothm to contol the poston of the tp of the coodnate measung machne (CMM) and to tack a specfed tajectoy despte the pesence of unknown but bounded dstubances and paamete uncetantes. he appoach taken fo the soluton of ths poblem s that of a obust adaptve sldng contol technque [4]. III. DESCRIPION OF HE MODEL As mentoned n the pevous secton, the coodnate measung machne s modeled as a smplfed twodmensonal model as shown n the Fgue. In ths model the z-axs s completely elmnated and t s epesented on the x-y plane as masses M and m. he dstance l a of the mass m s a paamete of the poblem and thee s no dynamcs nvolved along the y decton due to ths mass m. Howeve, note that both the neta and the locaton of the cente of mass of the combned system (ncludng M and m) changes f l a s changed. heefoe, t s concluded that l a has sgnfcant effect on the dynamcs of the system. he spngs k and k epesent the belts of the CMM. he mass m s attached to the gd body M that s fee to tanslate and otate about ts cente of mass. he foce F at Q as shown n the fgue epesents the actuato. he lnea encodes ae pesent at A and at B. hee s also a otay encode on the moto, whch when tanslated nto ths model s epesentaton gves the measuement x 3. hs encode s a collocated senso whle the lnea encode at ethe A o B s a noncollocated senso. In fomulatng the equatons of the model, the followng assumptons ae made: he angle of otaton θ of the mass about the cente of mass s small so that sn θ θ and cosθ. he actuato moves hozontally and does not otate. he nput to the system s the foce F appled on the actuato. Fg.. wo-dmensonal model of a CMM wth the foce F actng at Q. he case n whch the spng constants k and k ae not necessaly equal s consdeed. he numbe of degees of feedom fo ths system s thee and the genealzed coodnates can be taken as x, θ and x 3 (o equvalently x, x and x 3 ). he equatons of moton fo the system shown n Fgue can be wtten fom the Lagangan (o fom Newton s laws) as ( M + m) x + ( c + c ) x + ( k + k ) x + ( c l c l ) θ + + ( kl k θ + ( c I CM + ( k l l l ) θ = ( c + c + c θ + l ) m a x 3 3 M + m) ) x 3 ( kl + ( k + k + k l ) x 3 + τ ) θ + ( c l c l ) x + k l ) x = ( c l cl ) x 3 + ( kl k l ) x3 + τ = τ ( x cx () 3 he unknown paametes fo ths system ae the coeffcents of the states and the devatves. he followng ae the paametes defned n the pesent stuaton: a = I CM a 6 = k + k a c + a = k l k a = c 7 l 3 = cl cl a 8 = kl + kl 4 = cl cl a = M + m 9 = c + c c a 0 = ma a + a + One can also epesent () n the standad fom as H + C + Kq = τ (3) whee H = dag( [ M + m I CM m a ]) c + c cl cl c c C = cl cl cl + cl cl + cl (4) c c c l + c l c + c + c ()

3 and k + k kl kl k k τ K = kl kl kl + kl kl + kl, τ = τ () k k kl + kl k + k τ 3 [ ] Hee the poston vecto q s defned as q = x θ x3. he nonlnea stffness and the dstubances ae lumped nto a sngle tem and ae epesented as a bounded functon d( q,, t) and the coespondng equaton of moton s modfed as H + C + Kq = τ + d( q,, t) (6) whee d ( q,, t) D. In ths pape, the goal, as mentoned eale, s to desgn a stable contolle that tacks a desed tajectoy q d fo the system epesented by (3) and (6) fo both the mult-nput and sngle-nput cases. hese ssues ae dscussed n the followng sectons n detal. IV. CONROL PROBLEM Fst a stable adaptve contolle (see [4] fo example) that does pefect tackng fo the system gven by (3) when all the thee contol nputs ae pesent and thee s no dstubance wll be desgned. hen, the adaptve law wll be modfed to contol the system gven by (6), whch wll nclude dstubance tems. Howeve, n ths case t wll be seen that pefect tackng s not acheved and thee wll be a dead zone due to the bounded dstubance. he above two methods ae also useful n the case of CNC machnes whee thee ae typcally many nputs. On the othe hand, fo the coodnate measung machne thee s only one nput at x 3 as shown n Fgue. Fo these types of systems a sldng adaptve contol technque s geneally used. In the followng subsectons each of these appoaches wll be dscussed n geate detal. A. Full State Feedback Mult-Input Contol n the Absence of Dstubances Let q d be the desed tajectoy that we wsh to tack and q be the efeence sgnal vecto. hen the eo s gven by q = q qd. By lettng λ as an postve constant, a suface s=s(t) can be defned as s = q + λ q =. Hee = q d λq s taken. hen s =. Now, let s choose a Lyapunov functon canddate V as V s s a = H + Γ a (7) whee a = aˆ a s the paamete eo, â s the estmate of the tue paamete a, and Γ s a postve defnte matx. hen V = s Hs + aˆ Γ a = s ( H H ) + a ˆ Γ a (8) = s ( τ H C Kq) + a ˆ Γ a Now let s choose the contol law as τ = Hˆ + Cˆ + Kˆ q K s whee Ĥ, Ĉ, and Kˆ d coespond to the estmates of the tue values of H, C and K espectvely, whch consttute the paametes of ou system, and K d s a postve defnte matx. In addton to the contol law f one defnes a Y accodng to H + C + Kq = Ya (9) one then obtans H ˆ + Cˆ + Kˆ q = Yaˆ (0) As a esult, V s a = ( Y K d s) + aˆ Γ a () If the adaptaton law s chosen as a ˆ = ΓY s, ths mples that V = s K d s 0 () Now V = s K d s (3) s bounded because s and ã ae bounded. Note that s = H ( Ya K d s) (compae (8) and ()) s also bounded. heefoe, V 0, V 0 and V s bounded mples that V 0 as t fom Babalat s lemma. heefoe s 0 snce Kd s postve defnte, whch mples that x 0 as t,.e., pefect tackng s acheved. In othe wods, n the absence of any dstubances, pefect tackng s acheved wth the above choce of contol and adaptve laws. he smulaton esults that llustate the above theoetcal esults ae descbed next. he values of the vaous paametes of the model ae tabulated n able. ABLE I PARAMEER VALUES FOR SIMULAION l a 0.4 m c 0 N s/m L. m c 0 N s/m M 00 kg c 4 N s/m m 00 kg k 000 N/m m a 0 kg k 00 N/m he contolle s wthout the bounded dstubance. heefoe, the contol and the adaptve laws mentoned eale ae used n the smulatons. he esults ae shown n Fgues 3 though. he desed tajectoy chosen was d ( sn 4t 0 sn t) q = 3, and the othe paametes used n the smulatons wee Γ = dag 0,,4,,,,9,7,0,, λ = 0, and K d = dag ([ ]) ([ 0,0,0] ). Notce that thee s vey good tackng by the above choce of contol and adaptve laws n all states. And the contol nput equed to acheve good tackng s qute hgh. he paametes do not convege to the tue values as expected because the nput s not suffcently ch o pesstently exctng.

4 Fg. 3. Compason of states (no dstubance). V = s s a a H + ˆ Γ = s ( q q ) aˆ a H H + Γ (6) = s ( q q q) aˆ a τ + d H C K + Γ he contol and the adaptaton laws ae chosen as τ = Yaˆ K d s and a ˆ = ΓY s, espectvely whee Y s defned as befoe n (9). hen, s V = s K d s + s d K dφ sat φ (7) If one chooses K d = kd I and defne sgn(s) as a vecto of the sgns of the ndvdual components, he/she obtans, V s kd s 0 (8) Hee the followng ae used: () the -nom of a vecto s the sum of the absolute values of ts components, and () the -nom s the standad Eucldean nom. As n the pevous secton, one can obseve that V s k s (9) s s bounded snce and ã ae bounded. Note that s = H ( Ya K d s + d) s also bounded. heefoe, V 0, V 0 and V s bounded mples that V 0 as t fom Babalat s lemma. d Fg. 4. Eo n tackng (no dstubance). In the followng subsecton, a stable contolle s desgned fo the case wth bounded dstubances. B. Full State Feedback Mult-Input Contol n the Pesence of Dstubance he appoach taken s smla to the pevous case except that the Lyapunov functon canddate s chosen as V s s a = a H + Γ (4) whee s s = s φ sat () φ wth satuaton functon sat( ) defned component wse. Hee φ s a postve scala. hen Fg.. Contol nputs (no dstubance). Snce K d s postve defnte, one sees that fom s, s φ. hs mples that n x φ λ as t,.e., a dead zone n adaptaton and hence n tackng s pesent. he smulaton esults wth bounded dstubance ae as shown n Fgues 6 though 8. he paametes used n these smulatons ae the same as befoe except that the bound on the dstubance s chosen as D = 0. Futhemoe, φ s chosen such that k d φ = D. Notce the ncease n eos compaed to the case wth no dstubance.

5 Fg. 6. Compason of states (wth dstubance). C. Full State Feedback Sngle-Input Adaptve Contol Now let s tun to the queston of contollng the CMM when thee s only one nput F as shown n Fgue. he equatons of moton ae essentally the same as () except that τ and τ ae zeo and τ 3 =F. As befoe, t s assumed that full state s avalable fo feedback. he output of nteest s x n ths case. he addtonal assumpton that smplfes the contol stuctue desgn s when the tansfe functon fom the nput F to the output x s mnmum phase. It s obseved that wth the nomnal values chosen as gven n able, ths mnmum phase assumpton s satsfed. he elatve ode of the tansfe functon fom F to x s thee (sx poles and thee zeos). hus an appoach smla to nput-output feedback lneazaton can be used to expess the states as the output and ts devatves. Snce the elatve ode s thee, t s seen that the output x needs to be dffeentated thee tmes to see the contol nput n the state equaton. he task then s to desgn a stable adaptve contolle fo ths thd ode system wth the states x, x and x and the mnmum phaseness wll guaantee global asymptotc stablty of the ntenal dynamcs whch conssts of the othe thee states. he appoach taken s agan the one descbed n [4]. he state equatons ae gven n the companon fom as hx + a f = u (0) = Dffeentatng the fst equaton (havng x ) n () and substtutng fo θ and x 3 fom the emanng two equatons, one obseves that the contol nput appeas n ths equaton and can theefoe be expessed n the fom gven n (0) fo sutable values of h, a s and f. Hee, u = F, s the contol nput. Fg. 7. Eo n tackng (wth dstubance). Defne e = x x d and s = e + λ e + λ e = x () whee x = x λe λ e. Now, d hs = hx hx = u = x a f h x Fg. 8. Contol nputs (wth dstubance). () If all the paametes ae known, then one can choose a contol law that gves a guaanteed convegence of s by defnng u as u = a f + hx ks (3) = hs choce of contol leads to the tackng eo dynamcs to h s + ks = 0. Fo the adaptve contol, the contol law s chosen as u = aˆ = f + hx ˆ ks whee â and ĥ ae the

6 estmates of the tue paametes. he tackng eo then becomes hs + ks = a f = + h x (4) h s = + + a f h x () p k h = Now by usng Lemma 8. of [4], the adaptaton law s: hˆ = γ sgn( h) sx (6) a ˆ = γ sgn( h) sf hs s obtaned by takng the Lyapunov functon canddate as V = h s + γ h + a (7) = hs then gves V = k s. hs choce of contol and adaptve laws gve global tackng convegence. he smulaton esults ae shown n Fgues 9 and 0. he gans fo vey good tackng ae vey hgh compaed to the mult-nput case and theefoe the contol nputs ae also hgh. he desed tajectoy chosen s x d = sn (t). Fg. 9. States and the eos. Fg. 0. Contol nputs. V. CONCLUSIONS In ths pape, fst the modelng of a coodnate measung machne s consdeed and t s assumed that all the states ae avalable fo feedback. Wth ths assumpton an adaptve contolle based on sldng contol achtectue s desgned fo the mult-nput case as descbed n [4]. Afte ths step, a obust contolle n the pesence of dstubances s desgned. It s obseved that the pefomance slghtly dmnshes n the second case. Howeve, the contolle assues obust stablty. he smulatons also ndcate that when thee s only one sngle nput n the contol stuctue, the pefomance s good only at the expense of vey lage contol nput sgnals. REFERENCES [] Y. Shen and X. Zhang, Modelng of Petavel fo ouch gge Pobes on Indexable Pobe Heads on Coodnate Measung Machnes, Intenatonal Jounal of Advanced Manufactung echnology, Vol. 3/4, pp. 06-3, Apl 997. [] G. Lee, J. Mou, and Y. Shen, Samplng Stategy Desgn fo Dmensonal Measuement of Geometc Featues usng Coodnate Measung Machnes, Intenatonal Jounal of Machne ools and Manufactue, Vol. 37, No. 3, pp , 997. [3] Y. Shen, Automated Dmensonal Inspecton Usng Coodnate Measung Machnes, Poceedngs of the 993 US/awan Jont Automaton and Poductvty Wokshop, pp , ape, awan, July 993. [4] Y. Shen, Compute-Integated Dmensonal Inspecton Usng Coodnate Measung Machnes, Poceedngs of Intenatonal Confeence on echnology ansfe and Compaatve Management, pp , Los Angeles, CA., August 993. [] Y. Shen and X. Zhang, Uncetanty Assessment n Measuement Results Usng Coodnate Measuement Systems, Poceedngs of 99 Intenatonal Mechancal Engneeng Congess and Exposton, San Fancsco, CA, Novembe 99. [6].-Z. Shen and C.-H. Menq, Automatc camea calbaton fo a multple-senso ntegated coodnate measuement system, IEEE ansactons on Robotcs and Automaton, Vol. 7, No. 4, pp. 0-07, August 00. [7] F. Peto,. Redace, P. Boulange, and R. Lepage, oleance contol wth hgh esoluton 3D measuements, Poceedngs of hd Intenatonal Confeence on 3-D Dgtal Imagng and Modelng, pp , 00. [8] L. Dejun, C. Rensheng, L. Zfang, and L. Xaochuan, Reseach on the theoy and the vtual pototype of 3-DOF paallel-lnk coodnate-measung machne, Poceedngs of the 8th IEEE Instumentaton and Measuement echnology Confeence, IMC 00, Vol., pp , 00. [9] S. N. Sptz and A. A. G. Requcha, Multple-goals path plannng fo coodnate measung machnes, Poceedngs of IEEE Intenatonal Confeence on Robotcs and Automaton, ICRA '00, Vol. 3, pp. 3-37, 000. [0] S. N. Sptz, A. J. Spyd, and A. A. G. Requcha, Accessblty analyss fo plannng of dmensonal nspecton wth coodnate measung machnes, IEEE ansactons on Robotcs and Automaton, Vol., No. 4, pp , August 999. [] A. Lmaem and H. A. Elmaaghy, Automatc path plannng fo coodnate measung machnes, Poceedngs of IEEE Intenatonal Confeence on Robotcs and Automaton, Vol., pp , 998. [] H. Zhuang and Y. Wang, A coodnate measung machne wth paallel mechansms, Poceedngs of IEEE Intenatonal Confeence on Robotcs and Automaton, Vol. 4, pp , 997. [3] M. R. Kateb,. Lee, and M. J. Gmble, otal contol of fast coodnate measung machnes, IEE Poceedngs on Contol heoy and Applcatons, Vol. 4, No. 6, pp , Novembe 994. [4] J-J. Slotne and W. L, Appled Nonlnea Contol, Pentce Hall, Inc. Englewood Clffs, New Jesey 0763, 99.