Measurement of Positional Deviation of Numerically Controlled Axes

Transcription

1 XXXII. Semnar ASR '7 Instruments and Control, Farana, Smutný, Kočí & Babuch (eds) 7, VŠB-TUO, Ostrava, ISBN Measurement of Postonal Devaton of Numercally Controlled Axes KUREKOVÁ, Eva, HAAJ, Martn, OEB, Tomáš & TVRDOŇOVÁ, Martna doc. Ing., PhD., Katedra automatzáce, nformačnej a prístrojovej technky, SjF STU, nám Slobody 7, Bratslava, 8 3, Slovenská republka, eva.kurekova@stuba.sk doc. Ing., PhD., martn.halaj@stuba.sk Ing., PaedDr., tomas.loebl@stuba.sk martna.tvrdonova@stuba.sk Abstract: Nowadays many modern producton machnes requre postonng n thousandths of mllmeter. Therefore a bg attenton must be pad to prepare such a controllng software that drves ndvdual axes of the producton machne whch mnmses devatons between the desred poston and actually reached poston. The so called postonal devaton (dfference between the actual and target poston) belongs to the mportant crtera that descrbe the performance of numercally controlled axes. The procedure for determnaton of such devaton s descrbed n the nternatonal standard ISO 3-:997. Ths standard provdes calculaton of the postonal devaton only n several dscrete (measurng) ponts. Moreover t does not consder effects of the measurng nstrument on the obtaned results. Thus the new methodology must be adopted that enables estmaton of the postonal devaton n any pont of the axs travel, together wth the uncertanty of such estmate. Obtaned results can be ncorporated nto a control system n the form of correctons enhancng postonng possbltes of ndvdual axes. Keywords: postonal devaton, measured data, standard ISO 3-:997, numercally controlled axs Introducton Testng of the postonal devaton of the numercally controlled axs (ether rotary or longtudnal) s ruled by the nternatonal standard ISO 3-:997 []. Ths standard provdes gude for desgn of the test, testng condtons and also evaluaton procedure for processng the measured data. In general, the testng procedure s based on repeated measurements of the actual poston of the tested axs n several dscrete ponts (target postons), located equally along the axs travel (see Fgure ). The several parameters dealng wth postonal devaton can be measured and calculated accordng to such a scheme. The evaluaton of measured data accordng to the standard gves just estmaton of the devce performance n several dscrete ponts (measurement ponts P ). But the course of ndvdual parameters among the measurement ponts s just roughly estmated accordng to the standard, gvng no warranty on correctness of the results n between ponts. 9

2 Fgure Scheme of the measurement n ndvdual ponts Related terms Terms connected wth measurement of postonng numercally controlled axes, as ntroduced by the standard [] (for sake of clarty only undrectonal approach s consdered): - axs travel maxmum travel, lnear of rotary, over whch the movng component can move under numercal control (par..), - measurement travel part of the axs travel whch s used for data capture, selected so that the frst and the last target postons may be approached bdrectonally (par..), - target poston - poston to whch the movng part s programmed to move (par..3). Desgnaton P ( = to m), - actual poston measured poston reached by the movng part on the j-th approach to the -th target poston (par..4). Desgnaton P j ( = to m, j = to n), - devaton of poston (postonal devaton) actual poston reached by the movng part mnus the target poston (par..5). Desgnaton x j, magntude calculated as x j = P j P, - mean undrectonal postonal devaton arthmetc mean of the postonal devatons obtaned by a seres of n undrectonal approaches to a poston P (par..). Desgnaton x, magntude calculated as x = n x j n j= 3

3 - estmator of the undrectonal standard uncertanty of postonng at a poston estmator of the standard uncertanty of the postonal devatons (par..5). Desgnaton s, magntude calculated as n ( xj x ) s = n j = - undrectonal systematc postonal devaton of an axs the dfference between the algebrac maxmum and mnmum of the mean undrectonal postonal devatons at any poston P (par..9). Desgnaton E, - undrectonal accuracy of postonng of an axs (par..). Desgnaton A, magntude calculated as A = max [ x + s ] mn [ x s ]. 3 Evaluaton of measured data accordng to the standard The above mentoned standard ntroduces evaluaton of the measured data that s amed namely at determnng the maxmum postonal devaton over the whole axs (measurement) travel. The evaluaton of results covers calculaton of the parameters related to the postonal devaton (see part ) n each of the measurement ponts P, covered also by the devaton boundares x +3 s ; x - s (respectvely x + s ; x -3 s n reversal drecton), see Fgure. Fgure Example of the graphcal presentaton of evaluated postonal devaton accordng to the standard [] (results plotted for undrectonal postonng) Note that the devaton boundares are calculated only from measured data, consderng ther varaton as caused only by a tested machne and not nfluenced by the measurement nstrument tself. Ths s vald only when the accuracy of measurement nstrument s much better than expected postonal devatons. But for modern numercally controlled axes, operatng wth resolutons of. mm, ths s not always the case and the used measurng nstrument tself can sgnfcantly affect the measurement results. As you can observe from Fgure, the standard assumes lnear course of the postonal devaton among the measurement (testng pont), as well as lnear course of the devaton boundares. 3

4 4 Evaluaton of the postonal devaton n any pont of the axs travel When consderng the above presented measurement scheme, followng postonal devatons are measured n ndvdual measurement ponts: P : Δ, Δ,..., Δ j,..., Δ n P : Δ, Δ,..., Δ j,..., Δ n () M P m : Δ m, Δ m,..., Δ mj,..., Δ mn where to m s the runnng number of the measurement pont (see Fgure ), Δ j are postonal devatons* (see part ), j to n s the runnng number of the measurement of postonal devaton n a gven measurement pont. It s assumed that the same number n of measurements of the postonal devaton s performed n each measurement pont. (*Remark: the postonal devaton that s desgnated n the standard [] as x j, s for sake of better clarty and understandablty desgnated by dfferent symbol Δ j ). If we want to obtan the estmates of the postonal devatons also n other ponts than the measurement ones, we must approxmate course of estmates. The least squares method s sutable for such approxmaton. The curve n a form of polynomal of the thrd order wll be placed over the ponts (P, Δ ), (P, Δ ),..., (P, Δ ),..., (P m, Δ m ): Δ = a + b P + c P + d P 3 () where Δ s the postonal devaton of the target poston and actual poston n any pont P and P <P ; P m >, a, b, c, d are unknown parameters of the polynomal. Besdes that we want to determne also expanded uncertanty U of estmate of the postonal devaton Δ n any pont P. To be able to do ths, we must determne the estmates of polynomal parameters a, b, c and d, ther uncertantes and covarances among them. Thus we leave the evaluaton accordng to the Fgure and we get evaluaton provdng results accordng to the Fgure 3. To do so, we need to ntroduce a completely new approach to evaluaton of the measured data. The postonal devaton for each measurement j = to n n each partcular pont P, wth = to m, can be calculated as the dfference between the target poston and the measured actual poston (see part above) []: ' Δ = P j P (3) j where P s the target (programmed) poston, ' P j s the actual (measured) poston. The actual (measured) poston P ' j j mer. ' P j comprses two components: = P + δ (4) where P j δ mer. s the poston ndcated by the measurng nstrument, s the measurement error n the partcular pont, n our case estmated as the maxmum permssble error of the measurng nstrument [8]. Ths means that the error remans constant n any pont. 3

5 Fgure 3 Estmaton of the postonal devaton n any pont of the axs travel Accordng to the prevous calculatons of estmates of the postonal devaton (as the arthmetc means), the set of equatons for j = to n postonal devatons n m ponts (target postons) can be expressed as [5]: Δ + δ mer. = P P Δ = P P + δ mer (5) M Δ m = Pm P + δ mer. m where P s the estmate (obtaned as an arthmetc mean) of the actual (measured) postons n any gven pont P. The set of equatons descrbng the postonal devatons (5) can be wrtten n a matrx form: x = P - P + δ mer (6) where x P P x = ( Δ Δ, ) T,, Δm T ( P, P,..., P m ) P = P = (P, P,..., P m ) T s the vector of the estmates of ndvdual postonal devatons (dmenson m), s the vector of the estmates of measured actual postons (dmenson m), s the vector of target postons (dmenson m), s the unt vector (dmenson m) and When takng the matrx notaton nto account, the covarance matrx U(x) can be wrtten n a form (assumng P as non-random vector, P and δ are ndependent): U(x) = U( P ) - U(P) + u (δ) T (7) 33

6 The uncertanty of the target poston s zero n our case (no nfluence on value of the target poston) so that the covarance matrx U(x) gets the form [4]: U(x) = U( P ) + u (δ) T (8) After specfyng the ndvdual terms (because P a P j are ndependent): u P U(x) = u = ( ) u ( P ) u ( P ) m ( P ) + u ( δ ) u ( δ ) u ( δ ) u ( δ ) u ( P ) + u ( δ ) u ( δ ) u + u (δ) T = ( δ ) u ( δ ) u ( P ) + u ( δ ) The uncertantes u(p ), where =,,..., m, n the matrx (8) are evaluated by the type A method form measured data: u( P ) n = ( P, j P ) n( n ) j= Ths procedure yelds to the estmates of unknown parameters a, b, c, d, uncertantes of those estmates and covarances among them. 5 The experment The expermental measurement of postonal devaton was performed at the laboratores of Department of Producton Technology, Faculty of Mechancal Engneerng, Slovak Unversty of Technology n Bratslava [3]. The prototype of the unversal plasma cuttng head [] was employed for measurement of ndvdual postonal devatons (see Fgure 4., - Fgure s courtesy of the Department of Producton Technology, Faculty of Mechancal Engneerng, Slovak Unversty of Technology n Bratslava). As the plasma cuttng head contans totally 6 numercally controlled axs, the experment was restrcted to one axs and the others were n standstll. The schematc representaton of the measured axs together wth the precse lnear encoder (used as a reference measurng nstrument) s shown n Fgure 5. m (9) () Fgure 4 Unversal plasma cuttng head Fgure 5 Schematc representaton of the measured axs 34

7 The prelmnary results, evaluated accordng to the standard so far, are shown n Fgures 6 to 8. One can observe that the course of approach does not sgnfcantly affect the obtaned results. Due to a lack of tme the evaluaton accordng to a new methodology has not been performed yet but t s expected n a short tme. The presented methodology s adapted from the methods used for calbraton of measurng nstruments; therefore ts correctness has already been proved [6], [7]. Approach n a postve sense Approach n a negatve sense,6,5 Postonal devaton n mm,4,3,, Runnng number of measurement Fgure 6 Postonal devaton measured when approachng to ndvdual target ponts from both senses of movement Non-senstvty,,5 Non-senstvty n mm,, ,5 -, -,5 Runnng number of measurement Fgure 7 Non-senstvty of the CNC axs 6 Conclusons The presented methodology gves the opportunty to estmate the postonal devaton n any pont of the axs travel, no matter whether rotatonal or longtudnal. Moreover t provdes the estmate of the postonal devaton wth the respectve uncertanty of such estmate. Ths gves the desgner or programmer the possblty to buld approprate correctons nto the control program or the adequate desgn correctons can be performed n the desgn of the machne. The presented evaluaton of measured data accordng to the standard shows smlar behavor of the controlled axes when approachng to the desred poston from both sdes (postve and negatve). 35

8 Postonal devaton n a postve sense x+3s Postonal devaton n a negatve sense x+3s Postonal devaton n a postve sense x-3s Postonal devaton n a negatve sense x-3s,6 Postonal devaton n mm,5,4,3,, Runnng number of measurement Fgure 8 Boundares of the postonal devaton n both senses of moton Acknowledgements: The study was supported by the project 6 th FP EU BoSm and Slovak Mnstry of Educaton. 7 References [] EN ISO 93. Weldng and related processes. Bratslava s. (n Slovak). [] ISO 3-. Test code for machne tools Part : Determnaton of accuracy and repeatablty of postonng of numercally controlled axes pp. (n Englsh). [3] GROS, P. 5. An advanced plasma cuttng unt for both-sde bevellng on metal parts. In: Strojnícky časops = Journal of Mechancal engneerng. Vol. 56, No. 6. 5, pp ISSN [4] KUBÁČEK,. Foundatons of Estmaton Theory. The st edton. Amsterdam, Oxford, New York, Tokyo: Elsever, 988. pp. 38. ISBN [5] PAENČÁR, R., GROS, P., HAAJ, M. 6. Evaluaton of the postonal devaton of numercally controlled axes. In: Strojnícky časops = Journal of Mechancal engneerng. Vol. 57, No.. 6. pp. -. ISSN [6] TEREK, M., HRNČIAROVÁ, Ľ. 4. Statstcal qualty control. The st edton, Bratslava: IURA EDITION, 4. pp. 34. ISBN (n Slovak). [7] VDOEČEK, F. 4. Automaton and dagnostcs. In Proceedngs of the nternatonal conference Strojné nžnerstvo 4. The st edton, Bratslava: Vydavateľstvo STU. 4. pp. S-97 to S-4. ISBN (n Czech). [8] WIMMER, G., PAENČÁR, R., WITKOVSKÝ, V.. Stochastcké modely merana. The st edton. Bratslava : Grafcké štúdo - Jurga,. pp. 5. ISBN X. 36