Geometric Error Correction in Coordinate Measurement

Transcription

1 Ata Poltehnia Hungaria Vol. 4, No., 27 Geometri Error Corretion in Coordinate Meaurement Gula Hermann BrainWare Ltd. Völg uta 3/A, H-2 Budapet Hungar Abtrat: Software ompenation of geometri error in oordinate meauring i hot ubjet beaue it reult the dereae of manufaturing ot. The paper give a ummar of the reult and ahievement of earlier work on the ubjet. In order to improve thee reult a method i adapted to apture imultaneoul the new oordinate frame in order ue eat tranformation value at direte point of the meauring volume. The interpolation tehnique ued have the draw bak that the ould not maintain the orthogonalit of the rotational part of the tranformation matrie. The paper give a tehnique baed quaternion whih avoid thi problem and lead to better reult. Introdution Three dimenional oordinate metrolog i a firml etablihed tehnique in indutr. Their univeral appliabilit and high degree of automation aount for it ue in the lat 3 ear. In order to full-fill it tak to verif the geometr of produt on the bai of the meaured reult CMM- mut be in prinipall be an order of magnitude more aurate than the mahine tool ued to manufature the part. Over the lat 5 ear one an oberve enormou enhanement in poitioning and meauring aura. The main portion of thi enhanement i the reult of improved knowledge about high preiion mahine deign [8]. A fundamental priniple wa reognied b profeor Abbe alread in the 89 about the alignment of the diplaement meauring tem with the ditane to be meaured. Another fundamental priniple i the eparation of the trutural and meauring funtion in a mahine. Alread in the 88 meauring equipment wa built in with the meauring tem wa attahed to a eparate metrolog frame. The third important fator, to be onerned, i the thermal ditortion of the metrolog tem. A hort overview of novel ontrution for high preiion oordinate meauring mahine i given in [8]. 47

2 G. Hermann Geometri Error Corretion in Coordinate Meaurement A mehanial aura i otl, wherea repeatabilit i not epenive, oftware tehnique were ued from the beginning to ompenate for the temati error in order to keep manufaturing ot low. One of the earliet paper on error ompenation of oordinate meauring mahine i b Zhang et al. [25]. The deribe the ompenation of a bridge tpe indutrial three-oordinate meauring mahine, whih reulted in an aura improvement b approimatel a fator. The mahine onit of onl tranlational ai and the infiniteimal rotation error are deribed b the rotation matri where the trigonometri funtion are replaed b the firt term in their Talor erie. The orretion vetor are determined at equall paed point in the meauring volume and are tored in the memor of the omputer in the form of look-up table. The orretion vetor at intermediate point are alulated impl b linear interpolation. An analtial quadrati model for the geometri error of a mahine tool wa developed b Ferreire and Liu [6] uing rigid bod kinemati. The introdued the notion of hape and joint tranform. The former deribe the tranformation between the oordinate tem on the ame link and latter the tranformation aro a joint. To repreent the tranformation the introdued the ue of homogeneou tranformation in matri form. A quadrati epreion wa developed for the ae where the individual joint error var linearl with the movement along the joint or ai. The global error deription wa obtained b onatenating thee matrie. Duffie and Yang [2] invented a method to generate the kinemati error funtion from volumetri error meaurement. To repreent the diplaement error a vetorial approah wa followed. The rotational error were deribed b matrie in whih, taking into aount that the angular error are mall, the oine and the ine term an be approimated with the lowet order term of their Talor erie. The model wa ued to deribe the error of a meauring probe negleting the rotation. The tranlational error omponent were approimated b ubi polnomial. To find the oeffiient leat quare fit wa applied. Teeuwen [2] deribed the error motion of the kinemati omponent of a oordinate meauring mahine b uing homogeneou tranformation and onatenating thee tranformation to alulate the reulting global error. Auming that the rotational error are ver mall, he negleted the eond order term. Hereb he ould enure the ommutativit of the matrie, but at the ame time the orthonormalit of thee matrie wa lot, whih mean that the do not repreent a pure rotation anmore. The error motion of the probe diplaement were alo handled b tranformation matrie. In order to etablih the error map in the form of orretion vetor the variou error omponent were meaured on a emi automati wa at direte point of the meauring volume. To obtain a ontinuou deription of the orretion vetor, between thee point, regreion wa ued to etablih a pieewie polnomial repreentation. 48

3 Ata Poltehnia Hungaria Vol. 4, No., 27 Ruijl [5] ha built a high preiion oordinate meauring mahine with a meauring unertaint of 5 nm in a 4 mm meauring volume. The mahine ha a novel ontrution, where the air bearing table perform the meauring motion in all the three prinipal diretion. It wa derived that if the meauring tem are aligned with the entre of the probe tip the relationhip between the poition of the meauring tem and the ontat point on the workpiee i unique. Thi mean that the funtional point i the entre of the probe tip and hene it i poible to ompl with the Abbe priniple. The nano meauring mahine of SIOS [7] i baed on the ame priniple. Thi mahine i urrentl applied a the tage for a long range anning miroope. Kim et al. [2] have ontruted an unuual mahine. One attempt to get rid of the paralla error of orthogonal tpe oordinate meauring mahine i the appliation of the o-alled multilateration. It i to meaure the diagonal ditane of the probe uing traking laer interferometer with retro-refletor. The paper deribe a heme of multilateration baed on a ingle volumetri interferometer tem. The volumetri interferometer generate two pherial wavefront from the probe b uing diffration point oure. The emanated wavefront interfere within the meauring volume, while two dimenional arra of photodetetor mounted on the mahine frame apture the interferometri intenit field. Phae information i ued, from whih the oordinate of the probe are determined. A eond interferometer i intalled to meaure the and poition of the mahine table. Kim and Chung [] alo applied infiniteimal matri tranformation to orret the poition error due to geometri imperfetion and tranient thermal error of a mahine tool to improve on mahine meaurement aura. Thermal error were derived from the thermal drift of the pindle in the three priniple diretion. The tati and tranient thermal error and their ompenation are diued b Kruth et al. [3]. Capturing temperature ditribution of the mahine truture the thermal deformation an be alulated uing the linear thermal epanion oeffiient of the individual mahine omponent. However the determination of enor poition in a umberome trial and error tak. Reentl in a paper Tan and hi oauthor [9] deribe the appliation of neural network for the error ompenation of ingle-ai, a gentr and X-Y tage. The advantage of uing neural network i in the following: the ould be ued to approimate an ontinuou mapping. Thi mapping an be ahieved b learning. Parallel proeing and nonlinear interpolation an alo be performed. Uing thi tehnique the author ould improve the poitioning aura depending on the onfiguration invetigated b a fator between two and three. 49

4 G. Hermann Geometri Error Corretion in Coordinate Meaurement 2 Overview of the Error and their Soure When onidering the mehanial aura of oordinate meauring devie three primar oure of quai-tati error an be identified: Geometri error due to the limited aura of the individual mahine omponent uh a guidewa and meauring tem. Error related to the final tiffne of thoe omponent, mainl b moving part. Thermal error a epanion and bending of guidewa due to uniform temperature hange and temperature gradient. Geometri error are aued b out of traightne of the guidewa, imperfet alignment of the ai and flatne error. Deformation in the metrolog frame introdue meauring error. During meaurement the deformation of the metrolog frame i aued b the probing fore. It effet an be predited with relativel high preiion if the probing fore i known and therefore it eail inorporated into the model. The tati deformation of the table i aued be gravit fore. It manifet itelf a a ontribution to out of flatne error. That mean it an be handled on a imilar wa. The larget deformation of the metrolog frame are thermall indued. The main oure of the thermal diturbane are: heating and ooling oure in the environment, like lighting, air onditioning, people around the mahine et., heat generated b the mahine itelf, thermal memor: heat tored in the mahine omponent from a previou thermal tate. The ompenation of thermall indued error i rather umberome, beaue of the ompleit of the problem [3]. Baed on reult from the literature a linear thermal ompenation model an be ued. 3 Geometri Error Model A oordinate meauring mahine i a multiai mahine oniting of a hain of tranlational and/or rotational kinemati omponent. The geometri deviation of a CMM i originating from the geometri deviation of it omponent. In order to diu a general model the error model of the omponent are diued. 5

5 Ata Poltehnia Hungaria Vol. 4, No., 27 The meauring loop of a oordinate meauring mahine in general i given in Figure. Depending on the onfiguration of the mahine all meauring motion are performed b the probe or b the table, or divided between the two. probe - workpiee interation Probe tem Workpiee Table Diplaement meaured b the meauring tem Metrolog frame Figure Shema of the metrolog loop A linear tage of preiion mahiner i epeted to travel along a traight line and top at a predefined poition. However in the pratied the atual path deviate from the traight line due to the geometri error of the guidewa and it reult alo in angular error a it i given in Fig. 2. Vertial traightne error Y Yaw Roll Z Pith X Horiontal traightne error Figure 2 Repreentation of the i deviation of a tranlational kinemati omponent 5

6 G. Hermann Geometri Error Corretion in Coordinate Meaurement 52 For eah ai a tranformation matri an be ued to deribe in homogeneou oordinate the deviation from the ideal motion. The general form of a tranformation i given b: + err T φ Where, and are the tranlational and, and are the rotational omponent and repetivel are hort for in and o. In ae of the oordinate table the angular error are ver mall, and all the error are poition dependent the following approimation an be made: T err Analog reult an be derived for the and ai. Rotational omponent an preented on the ame wa and reult in rather imilar matri. Figure 3 Repreentation of the i deviation of a rotational kinemati omponent Thee omponent are alled guided element. The guided moving element are linked b oalled onneting element, whih an be repreented b matrie with imilar truture with onl ontant element. The reulting error matri an be obtained b multipling the individual matrie in the equene a the follow eah other in the kinemati hain. Z T X T Y T Z R X R ε Y R ε

7 Ata Poltehnia Hungaria Vol. 4, No., 27 A traditional o-ordinate meauring mahine onit of three tranlational omponent, and, and a probe i attahed to the end of the omponent. Uuall the probe an be onidered a a ontant tranlational tranformation. In ae of a meauring probe it error omponent an be handled on the imilar wa a it wa done in ae of a arriage and a rotational element. 4 Error of a Coordinate Meauring Mahine with three tranlational omponent In order to illutrate the appliation of the tehnique deribed in the previou paragraph, let u onider a oordinate meauring mahine with three tranlational omponent, given in Fig. 4. Z Y X Figure 4 The invetigated oordinate meauring mahine The arriage onit of two taked table; eah of them turn ret on vauum preloaded air bearing. The poition are determined b inremental two optoeletroni meauring tem, having,5 μm reolution. Thi reult in a lightweight ontrution, whih in turn enure fat and aurate poitioning. 53

8 G. Hermann Geometri Error Corretion in Coordinate Meaurement 54 The probe i attahed to a pinole running in an air buhing. A ounter weight minimie the fore needed for lifting the probe. The diplaement i meaured with a linear optoeletroni ale having a reolution of,5 μm. Figure 5 The frame of the CMM The poition dependent tranformation matrie for the table T table and the probe T probe oordinate frame are given below: + * table Y X T φ + * l ref probe Z Z Y X T ε ε ε Here again, and and ε, ε, ε tand for the tranlational and, and and φ, φ, φ for the rotational error of repetivel the table and the probe. The ontant value X, Y and Z are the offet oordinate ditane between the mahine oordinate tem and the oordinate tem attahed to the workpiee to be meaured. Z ref i the probe referene point and Z l i the probe length. The oordinate value aptured b the CMM are: Z l mahine frame Y Z X probe frame table frame

9 Ata Poltehnia Hungaria Vol. 4, No., 27 T mea mea mea mea T table + T probe If for aura reaon one ma not ubtitute α for inα and for oα than the omponent of eah matri deribing the tranformation hould be aptured in imultaneoul. In the ubequent paragraph a uitable method i outlined. 5 Determination of the Geometri Error b Meaurement For the alibration of oordinate meauring mahine Zhang et al. [23] propoed to determine the angular error b meauring the diplaement error along two parallel line to the ai of motion but eparated b a ditane in the appropriate orthogonal diretion. In a more reent paper Zhang and Fu [24] deribe the alibration of optial CMM- uing an unalibrated reverible grid plate in three poition. In the initial poition the plate i aligned with the mahine oordinate tem Net it i revered about Y ai of the mahine. In the third poition the grid i rotated 9º about the Z ai. To determine the ale error one of the mahine ai hould be alibrated b a laer interferometer. A imple meauring tehnique wa invented b Fan et al. [3] to determine the motion aura of a linear tage. The idea i baed on the fat that the poition and orientation of a rigid bod an be determined b appropriatel eleted i point. The meaure the diplaement of thee point and alulate from them the rotational and tranlational error omponent. The diplaement in the motion diretion and the angular error pith and aw perpendiular to thi diretion are meaured b three laer interferometer. The roll and the traightne error are aptured b an optial etup ontaining two quadrant photo detetor. Taking again into onideration that the angular error are mall their value are replaed b their tangent. The invention initiated the development a dual and triple beam interferometer. The above mentioned author publihed a paper [5] about the meaurement to determine the aura of a high preiion wafer tage. Therefore 6-DOF error of it poitioning aura are ignifiant. An improved verion of the above deribed tem wa ued. The moving part of it i an L-haped mirror and on top of one leg a long right angle mirror. The tationar part onit of four laer 55

10 G. Hermann Geometri Error Corretion in Coordinate Meaurement head, two beam plitter and two quadrant photo detetor. The laer head ue four laer Doppler ale, three of whih are parallel to eah other. The upper two laer beam an be refleted b the long right-angle mirror and the lower one b the Y leg of the L haped plane mirror. The fourth laer beam i aligned in the X- ai and refleted b the X mirror. Comparing the four linear meaurement b four laer doppler ale, the X and Y poitioning error of the moving table and it pith and aw error an be determined. The upper two refleting beam are plit and eah plit beam i reeived b a quadrant photo detetor. Comparing thee ignal the vertial traightne and the roll error an be derived at the ame time. In order to minimie the oine error among the three diplaement meaurement and to enure angular aura of the pith and aw meaurement the parallelim of thee beam hould be preiel adjuted. The tem take into onideration the quarene alignment and the flatne error of the plane mirror. In their paper Gao et al. [7] deribe the meaurement traightne and rotational error motion of a ommeriall available linear airbearing tage atuated b a linear motor. The pith and aw error were meaured b an autoollimator. For the roll error meaurement two apaitive diplaement probe an the flat urfae in the XZ plane. The probe with their ening ai in the Y diretion were aligned with a ertain paing. The roll error i obtained b dividing the differene of the output of the two probe b the paing between them. The horiontal and vertial traightne error were meaured b uing the traightne kit of a laer interferometer. The etup to detet motion error of the linear tage ue two laer interferometer [6] and three apaitive enor [4] i given in Fig. 4. The tationar part onit of a ingle and a dualbeam laer interferometer and three apaitive enor perpendiular to eah other. Both tranlational and rotational error an be derived out of the diplaement value aptured b the tranduer. Mirror Capaitive enor Single beam interferometer Dual beam interferometer Figure 6 Set up for determining the motion error of a linear tage 56

11 Ata Poltehnia Hungaria Vol. 4, No., 27 Z P 3 d 2 P P 2 d 3 P 5 P 4 d P 6 X,Y,Z Y X Figure 7 The meauring point on the urfae of the artifat and their relation to eah other Where d, d 2 and d 3 repreent the ditane between the laerbeam, repetivel the apaitive enor, parallel to the oordinate ai. B epanding the following determinant the equation of the table boundar plane an be determined. Thee equation an be ued to ompute the origo of the new oordinate tem and it prinipal ai. Having thee value one an diretl draw up the tranformation matri. d d tg d 3tg d 3 d d tg tg tg tg 6 tg tg 6 tg tg 6 tg 6 Error Compenation Sheme The error matrie are aptured at direte point of the meauring volume. To be able to model the behavior of the mahine in between the matrie hould be interpolated. Componentwie interpolation lead to a non-orthonornal rotational 57

12 G. Hermann Geometri Error Corretion in Coordinate Meaurement part whih hould be avoided. We ma onider thee matrie a a Lie group and tr to do the interpolation uing tehnique from their theor. However thi i a tediou job. Let u deompoe the matri into a rotational part and a diplaement vetor deribing the motion of the mahine omponent a follow: r r2 r3 T r2 r22 r23 + r3 r32 r33 Here the poition dependent diplaement vetor point an be interpolated b a pieewie polnomial pae urve while the rotational matri remain to be interpolated. However there are imple tehnique available baed on the appliation of quaternion. Quaternion [] wa invented b Sir William Hamilton in 843. He realied that four number are needed to deribe a rotation followed b a aling. One number deribe the ie of aling, one the number of degree to be rotated and the lt two number give the plane in whih the vetor hould be rotated. Quaternion onit of a alar part ЄR and v,,єr 3 : q [,v] [,,,] + i + j + k where í 2 j 2 k 2 ijk -, ij k and ji -k If q i a quaternion with q [o,inn] and p i a quartenion p [,r] then p qpq - i p rotated 2 about the ai n. Given a tranformation matri M the orreponding unit quaternion i an be alulated in two tep: firt we mut find whih i equal to: ± M + M 22 + M 33 + M 2 Now the other value follow: M M M M 4 M 4 M 4 Where the M- are the determinant of the repetive ubmatrie. A o-alled pherial linear quaternion interpolation Slerp an ued to ompute the intermediate quaternion. The quaternion generated b Slerp are unit

13 Ata Poltehnia Hungaria Vol. 4, No., 27 quaternion, whih mean that the repreent pure rotation matrie. The formula for Slerp i: o Ω Slerp q q o q, q, h q in h Ω + q in Ω where tand for the inner produt defined a in hω q q' ' + ' + ' + ' An even better moother interpolation an be formulated whih i the pherial ubi equivalent of a Beiér urve. Thi i alled Squad and thi defined b: Let q,..., q n point on the unit phere. Find the ubi pline whih interpolate the point in the given equene. Thi an be ahieved b the following formula: Squad q i, q i+, i, i+, h SlerpSlerpq i, q i+, h, Slerp i, i+, h, 2h-h where i are i q i ep - log q i - q i+ + log q - i q i- 4 where logq and ep q are defined a follow: if q [o,inv] then logq [,v] and if q [,v] then epq [o,inv] The uggeted proedure to find the intermediate rotation matrie onit of the following tep: Convert the matrie aptured b the proedure deribed in paragraph 4. in quaternion Find the Squad interpolation of thee point Convert the quaternion pline bak into matri form On thi wa a pherial pline repreenting pure rotation of the objet will be obtained. The tranlational error an be handled b finding a patianel interpolation pline funtion uing for eample leat quare fit. In the poeion of thee funtion one an eail reontrut the real oordinate value of the meaured point. In mot of the pratial appliation, due to the mall angular error pherial linear interpolation i uffiient. In thee ae the poition originating from traightne and perpendiularit error are dominant. 59

14 G. Hermann Geometri Error Corretion in Coordinate Meaurement In Figure 8 the omponent of the error vetor alulated on the above deribed wa are given for the meauring mahine preented in paragraph 4. Similar reult are available for the and the ai. Conluion The paper preent a new approah to the ompenation of geometri error in oordinate meauring mahine. It onit of a meauring proedure whih apture imultaneoul the i error omponent of moving rigid bod. The tranformation matrie obtained on thi wa are interpolated b uing quaternion repreentation. Hereb the orthonormalit of the rotation matrie are maintained. Simulation value with randoml generated error omponent howed that the intermediate value lead to aura improvement. Aknowledgment The author gratefull aknowledge the upport obtained within the frame of National Reearh Fund OTKA T Referene [] E. B. Dam, M. Koh, M. Lillholm: Quaternion: Interpolation and Animation, Tehnial Report DIKU-TR-98/5, Department of Computer Siene, Univerit of Copenhagen [2] N. A. Duffie, S. M. Yang: Generation of Parametri Kinemati Error- Corretion Funtion from Volumetri Error Meaurement, Annal of the CIRP Vol. 34//985, pp [3] K. C. Fan, M. J. Chen, W. M. Huang: A Si-Degree-of-Freedom Meaurement Stem for the Motion Aura of Linear Stage, Int. J. Mah. Tool Manufat. Vol. 38, No. 3, pp

15 Ata Poltehnia Hungaria Vol. 4, No., 27 [4] K. C. Fan, M. J. Chen: A 6-Degree-of-Freedom Meauring Stem for the Aura of X-Y Stage, Preiion Engineering 242 pp [5] G. Farin: Curve and Surfae for Computer Aided Geoemtri Deign: A Pratial Guide, Aademi Pre, Boton, 3 rd ed. 993 [6] P. M. Ferreira, C. R. Liu: An Analtial Quadrati Model for the Geometri error of a Mahine Tool, Journal of Manufaturing Stem, Vol. 5, No., pp [7] W. Gao, Y. Arai, A. Shibua, S. Kiono, C. H. Park: Meaurement of multi-degree-of-freedom error motion of a preiion linear air-bearing tage, Preiion Engineering 326 pp [8] G. Hermann: Deign onideration for a Modular High Preiion Coordinate Meauring Mahine, ICM 26, IEEE International Conferene on Mehatroni, Jul 3-5, 26, Budapet, Hungar, pp.6-65 [9] G. Hermann: Volumetri Error Corretion in Coordinate Meaurement. 4 th Serbian-Hungarian Joint Smpoium on Intelligent Stem SISY 26 September 29-3, Subotia, Serbia, pp , ISBN [] P. S. Huang, J. Ni: On-line Error Compenation of Coordinate Meauring Mahine, International Journal of Mahine Tool and Manufaturing 995, No. 3, pp [] K. D. Kim, S. C. Chung: Aura Improvement of the On-Mahine Inpetion Stem b Corretion of Geometri and Tranient Thermal Error, Tranation of NAMRI/SME Vol. XXXI, 23, pp [2] S. W. Kim, H. G. Rhee, Ji-Young Chu: Volumetri Phae Meaurement Interferometer for Three Dimenional, Preiion Engineering 2723, pp [3] J.-P. Kruth, P. Vanherk, C. Van den Bergh: Compenation of Stati and Tranient Thermal Error on CMM, Annal of the CIRP Vol. 5//2, pp [4] Lion Preiion: [5] T. A. M. Ruijl: Ultra Preiion Coordinate Meauring Mahine, PhD. Thei TU Delft 2 [6] SIOS: Metehnik GmbH: Miniatur interferometer mit Planpiegelreflektor SP 2-2/5/2 [7] SIOS: Metehnik GmbH: Nano meauring mahine [8] A. H. Sloum: Preiion Mahine Deign, Englewood Cliff, NJ: Prentie Hall, 992 6

16 G. Hermann Geometri Error Corretion in Coordinate Meaurement [9] K. K. Tan, S. N. Huang, S. Y. Lim, Y. P. Leow, H. C. Liaw: Geometri Error Modeling and Compenation Uing Neural Network, IEEE Tranation on Stem, Man and Cberneti, Vol. 36, No. 6, Nov. 26, pp [2] J. W. M. C. Teeuwen, J. A. Soon, P. H. J. Shelleken: A General Method for Error Deription of CMM Uing Polnomial Fitting Proedure, Annal of the CIRP Vol. 38//989, pp [2] S. M. Wang, K. F. Ehmann: Meaurement Methode for Poition Error of a Multi-ai mahine Part I: Priniple and Senitivit Anali, International Journal of Mahine Tool and Manufaturing [22] S. M. Wang, K. F. Ehmann: Meaurement Methode for Poition Error of a Multi-ai mahine Part II: Appliation and and Eperimental Reult, International Journal of Mahine Tool and Manufaturing [23] G. Zhang, R. Ouang, B. Lu: A Diplaement Method for Mahine Geometr Calibration, Annal of the CIRP Vol. 37//988, pp [24] G. X. Zhang, J. Y. Fu: A Methode for Optial CMM Calibration Uing a Grid Plate, Annal of the CIRP Vol. 49//2, pp [25] G. Zhang, R. Veale, T. Charlton, B. Borhardt, R. Hoken: Error Compenation of Coordinate Meauring Mahine, Annal of the CIRP Vol. 34//985, pp