Poznań University of Technology Institute of Mechanical Technology Poland,

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1 PAWEŁ SWORNOWSKI Pznań University f Technlgy Institute f Mechanical Technlgy Pland, swrnwski@wp.pl THE INFLUENCE OF INACCURACY OF CALCULATING ALGORITHMS USED IN THE CMMS ON MEASUREMENT RESULTS FINAL REPORT The authr has devised a test fr evaluating the accuracy f calculatin algrithms and used it t establish the accuracy f the sftware f furteen CMMs ffered by wrld tp manufacturers. Using a minimum number f measuring pints, the authr calculated varius gemetrical elements with arbitrary shape deviatins and the relatinships between them, and cmpared the btained results with the reference results. The input data were entered frm the keybard t the cmputer wrking with a CMM. Therefre the errr f the final calculatin result was intrduced nly by the sftware and the cmputer, and did nt include the errrs f the CMM. The cmputer and the sftware were treated as a black bx. The same test set was used fr all sftware. Keywrds: crdinate measuring technique, evaluatin sftware, cmputer simulatin. INTRODUCTION Frm time t time, every Crdinate Measuring Machine (CMM) shuld be calibrated (with cmplex r analytical methds) t establish its accuracy. During calibratin we usually assume that the next element in the measuring system, the cmputer r, strictly speaking, the calculating algrithm, is nt a surce f additinal errrs. It seems, hwever, that it is an unfunded assumptin and that the s-called sftware quality shuld be established befre the calibratin f the CMM is perfrmed. The standard DIN 880 [] unequivcally defines the way f determining basic elements and the relatinship between them. Hwever, almst all CMM manufacturers use different sftware fr cntrl and fr prcessing f input data. The authr aims t shw that with the CMM sftware used at present there is a cnsiderable chance t btain results with relatively large errrs, especially when the number f measuring pints is lw. The authr devised his wn test fr CMM sftware. The test allws every user t test his sftware and draw his wn cnclusins. The authr used his test t examine sftware in industrial cnditins. The measurement systems were nt the authr s prperty. Therefre it was necessary t wrk ut a methd which wuld require neither purchasing the sftware t be tested, nr the installatin f sphisticated reference sftware. The authr financed the research himself. In the test the substitute elements were calculated using a small number f measuring pints because f limited access t the tested sftware. Besides, in this case the differences in the results are visible. T achieve a mre detailed picture f sftware accuracy, the same set was tested fr mre pints (up t pints). The results btained in this part f the experiment, hwever, are beynd the scpe f the presented paper. The tested sftware came frm firms which have been perating n the Plish market since 997 (Table ). Table. List f the tested CMM sftware (in alphabetical rder). CMMs sftware Cmment Calyps * Gepak-Csms * Metrmeasure *

2 Metrsft * PC-DMIS ** Prelude Inspectin * Quinds *Fr DOS Quinds **Fr Windws Sftis *Fr DOS Sftis Tutr * Umess * **Fr Windws Zettmes * Zettmes **New versin *CMM sftware tested t 00 **CMM sftware tested in 005. DESCRIPTION OF THE TEST In rder t determine the value f errrs generated by the measuring sftware, the authr prpses a methd f calibrating the measuring sftware with a D gemetrical mdel standard. This is a virtual standard, which takes the frm f a set f arbitrary pints. Therefre the errrs which are determined depend nly n the type f cmputer and n the calculatin algrithms. The measurement f a circle, plane and straight line is simulated with the minimum number f measuring pints. Whether the small number f pints wuld affect the accuracy f the real measurement is f secndary imprtance here. The btained results are cmpared with the reference results, which were calculated with the Least Mean Square (LMS) methd in accrdance with the standard DIN In spite f the wellknwn deficiencies f this methd, it is the reference pint f the whle test. All differences between the reference results and the btained results are treated as calculatin errrs. In this way, the authr receives a true image f the accuracy f the sftware tested. T avid prejudice, the names f the manufacturers have been encded. The letters A t O were used instead f cmpany names. The test has been cnducted n the basis f elements f the ISO standard. The nly difference is that the input data are entered in the cmputer thrugh the keybard. This allws the peratr t intrduce his wn reference data instead f using cmmercial reference sftware. Mrever, since the CMM is nt wrking while the test is cnducted, it des nt influence the measurement results. (Fig. ). The virtual D gemetrical mdel cmprises the mst cmmn gemetrical elements with arbitrary shape deviatins. This allws creating the measuring prblem in an arbitrary way. The peratr chses the number and the distributin f measuring pints. The pints are nt cllected in a real measurement, but exist in virtual frm. Only the cmputer and the sftware, therefre, intrduce an errr t the final measurement result. The prpsed apprach is extremely cheap and flexible. It allws testing varius types f CMM sftware. We may determine hw a given piece f sftware calculates a straight line, a plane, a circle, a sphere, a cylinder, and hw it calculates the relatinships between these elements. The test invlves typical measurement tasks withut the s-called exceptinal cases. If we calculate a virtual gemetrical element and cmpare the result with the result given by the reference sftware, we get reliable infrmatin abut the real accuracy level f the sftware we use (Fig. ). Reference calculatins were executed n the PC cmputer (Pentium III prcessr, Windws XP). Technical parameters f the tested cmputer and reference cmputer were cmparable. The test has been carried ut fr different cases f sftware. T eliminate mistakes, all tests were perfrmed twice. In dubtful cases, the whle starting prcedure was repeated, including the resetting f the cmputer.

3 CMM CMM cut-ff switch "BLACK BOX" utput RESULTS input Virtual D-standard KEYBOARD Fig.. The blck scheme f the Crdinate Measuring Technique. Reference data set frm keybrd (virtual D-standard) Reference sftware (LMS - DIN 880) Test sftware Reference results Test results Reference cmputer CMM cmputer Cmparisn Fig.. The blck scheme f the cmparisn algrithm. The experiment was cnducted in tw phases. In the first phase we tested the cnfrmity f defining plane figures and slids in varius pieces f sftware with the criterin f the minimum number f measuring pints. In the secnd phase we attempted t determine the accuracy f particular algrithms. Phase : The surface f almst all measured bjects can be described by typical gemetrical elements: a pint, straight line, plane, circle, sphere, cylinder and cne. T determine all the abve elements (which are parts f a measured bject) we use the crdinates f pints which belng t these elements. The crdinates f the pints are either calculated r measured. Fr each element the smallest number f necessary pints is defined. This is different, depending n whether the element is determined mathematically (fewer pints are necessary) r whether it is calculated n the basis f measurements (mre pints are needed). Sme tested sftware ffers the ptin f using the mathematical minimum number f pints fr calculatins. It seems pintless, hwever, since many mre pints are needed t reliably recreate a given shape []. Our test has shwn that nly ne calculating algrithm fulfils the requirement f the minimum measurement number f

4 pints fr all elements. On the ther hand, as many as three algrithms d nt fulfil this requirement fr all six gemetrical elements. The algrithms use the minimal mathematical number f pints instead. Phase : The authr has created a D gemetrical mdel t be used in the tests. Fr the purpse f the test the authr has selected measurement tasks which are mst cmmn in mechanical engineering. The simulated shape deviatins are in the 6 8 accuracy class. The test includes the determinatin f the fllwing equatins and relatinships (Fig. ):. equatin f circle I ( n shape deviatins),. equatin f circle II (three-lbing),. caxiality f circles I and II (Fig. 6a),. equatin f circle III ( vality), 5. equatin f circle IV ( lcal flatness ), 6. distance between circles III and IV (Fig. 6b), 7. distance between planes I and II ( shape deviatin f plane I- warp; f plane II - cncavity) (Fig. 6c), 8. angle between planes I and III ( shape deviatin f plane I - warp; ideal plane III) (Fig. 6d), 9. intersectin pint f straight line and plane II ( shape deviatin f plane II - cncavity) (Fig. 6e). The crdinates f pints belnging t particular gemetrical elements with deviatins (in the 6 8 accuracy class) are entered in the cmputer wrking with the CMM. The final result shws nly the errrs generated by the cmputer and its sftware. The cmputer and sftware are treated as a black bx. The same test is used in all cases. The minimum measurement number f pints is used fr all gemetrical elements. The distributin f particular gemetrical elements in the D gemetrical mdel is shwn in Figs. -6, and the crdinates f these elements are given in Tables t 9. I Circle III C 0 80 y 0 L G z K x J H Circle IV 00 Circle I 0 Plane ABCDEF = Plane I Plane GHIJKL = Plane II Plane EFKL = Plane III D Circle II F A E B Fig.. Distributin f gemetrical elements in D space. In measurements, the mst cmmn gemetrical element is a circle (the shaft-hle element). Therefre in the D gemetrical mdel there is ne ideal circle and three circles with different shape deviatins (three-lbing, vality and lcal flatness). The angle distributin f the measuring pints is the same fr all fur circles (Fig. a). The distributin f fur measuring pints fr a given shape deviatin is presented in Fig. b, Fig. c and Fig. d. Circles I and II are situated in plane XY, and circles III and IV lie in plane XZ.

5 a) b) um c) d) 0 um 0 um Fig.. The distributin f measuring pints in circles I-IV: a) clse t ideal, b) three-lbing, c) vality, d) flatness. a) b) Plane II Plane I c) d) Plane III Line Fig. 5. The distributin f measuring pints in: a) plane I, b) plane II, c) plane III, d) line. Table. The crdinates f measuring pints f circle I [mm] Pint #! # # # X Y Z Table. The crdinates f measuring pints f circle II [mm] Pint # # # #

6 X Y Z Table. The crdinates f measuring pints f circle III [mm] Pint # # # # X Y Z Table 5. The crdinates f measuring pints f circle IV [mm] Pint # # # # X Y Z Table 6. The crdinates f measuring pints f plane I [mm] Pint # # # # X Y Z Table 7. The crdinates f measuring pints f plane II [mm] Pint # # # # X Y Z Table 8. The crdinates f measuring pints f plane III [mm] Pint # # # # X Y Z Table 9. The crdinates f measuring pints f a line [mm] Pint # # # X Y Z a) Circle II Circle I

7 b) c) Circle III Plane II Distance Circle IV Plane I d) e) Plane III Angle Plane I Plane II Crss pint Line Fig. 6. The determinatin f relatinships: a) caxiality f circles I and II, b) distance between circles III and IV, c) distance between planes I and II, d) angle between planes I and III, e intersectin pint f straight line and plane II.. TEST RESULTS Results were gruped in fur agreed upn intervals (I-crrect, II-incrrect t µm, IIIincrrect<µm, µm> and IV-incrrect>µm). The graphical representatin f the test results fr circles I-IV is given in Fig. 7. The influence f the type f shape deviatin n calculatin results is visible. The examined sftware determines the equatin f circle II (with three-lbing) unexpectedly well. On the ther hand, the determinatin f the (almst ideal) circle I is a serius prblem (% f errrs are larger than µm fr radius and 8% fr centre). The differences between circle I and circle III is the lcatin f pint. The difference is big and is shuld be reflected in the final results. Hwever, we see frm Fig. a and Fig. c that it has been neglected. The shift f the centres f circles I IV leads t an errr in determining the caxiality f circles I and II and in determining the distance between circles III and IV (Fig. 8). Similarly, large errrs appear when the width L f the D gemetrical mdel is calculated frm the distance between planes I-III (Fig. 6c). During the actual measurement the peratr des nt knw whether planes I and II are ideal, r whether they have any shape deviatins. If planes I and II are parallel, all the tested sftware yields the crrect result, i.e. L=00mm. A prblem arises when there is a shape deviatin in ne (r bth) f the planes. In this case the sftware must determine a substitute plane, using its (usually unpublished) fitting algrithm. The sftware determines substitute planes PI* and PII* (Fig. 9). Pint # plays the key rle in determining plane I (Fig. 6c). It is this pint that causes the shift f the whle plane I* in the D space. Since the planes determined in this way are nt parallel, it is nt pssible, in the mathematical sense, t determine the distance between them. Therefre the calculatin prcedure shuld end. Hwever, nly ne sftware did nt cntinue calculatins after analyzing the parallelism f substitute planes PI* and PII*. As many as pieces f sftware cntinued calculatins (!).

8 a) b) Circle I: centre Circle I: radius < um;um> % >um % <um;um> % >um 6% t um % 5 t um % c) d) <um;um> t um 7% Circle II: centre >um % <um;um> >um % Circle II: radius 89% t um 9% 57% e) f) Circle III: centre >um 8% <um;um> >um 8% Circle III: radius 5 t um 8% 6% <um;um> % t um g) h) Circle IV: centre Circle IV: radius <um;um> % >um >um 8% % t um 5% 6% <um;um> t um 9% Fig. 7. Measurement results: a) centre f circle I, b) radius f circle I, c) centre f circle II, d) radius f circle II, e) centre f circle III, f) g) radius f circle III, g) centre f circle IV, h) radius f circle IV. a) b) Caxiality Distance between circles III-IV <um;um> >um 9% % <um;um> 8% >um 8% 6% t um 5 t um 8% Fig. 8. Relatinships between circles: a) caxiality f circles I and II, b)distance between circles III and IV.

9 There appears the questin, then, what did the sftware calculate? It seems that mst f the algrithms pass frm ne calculatin ptin t anther, withut infrming the peratr abut the change. The peratr is cnvinced that the result is the distance between planes, but what he actually gets may be e.g. the distance between tw pints (Fig. 9). a) b) Distance between planes I-II Plane II* A L B % 00 Plane I* >um 5% <um;um> 8% t um 8% Fig. 9. The determinatin f the distance between planes PI and PII: a) the psitin f substitute planes PI* and PII* btained frm calculatins, b) the graphical presentatin f results. Such behaviur f the sftware tested made the authr perfrm additinal calculatins in rder t determine the used calculatin algrithm Therefre the fllwing distances were calculated (Fig.0): the distance between the substitute plane PI* and the centre f plane PII* (pint), the distance between substitute plane PII* and the centre f plane PI* (pint), the distance between the centres f substitute planes PI* and PII* (pint-pint). These three measuring ptins appeared in seven cases f sftware and six cases f sftware apply ther unknwn calculatin prcedures. It must be nted that when we measure the distance between tw nn-parallel planes, we get a value which is the result f using anther calculatin ptin. The measurement results fr the angle between plane I and plane III are the mst interesting (Fig. ). Pint # (Fig. 6d) in plane I is essential fr calculatins. Because f the psitin f this pint, the substitute plane calculated n the basis f all fur pints will nt lie n plane YZ. Sme calculatin algrithms ignre pint # in their calculatins (and use the remaining three pints,, which lie exactly n plane YZ). The questin arises as t which result is mre accurate. Is it the ne which is btained in accrdance with standard DIN 880, r the ne that ignres pint # in the calculatins? Which calculating methd is better? Is it the ne which perfrms calculatins using all fur measuring pints, r the ne which cnducts an initial analysis f mutual psitins f all pints in the -D space?

10 Crdinate f 8 measure pints Delimitatin f supplementary planes PI* and PII* Chice f ptin: Distance between PI* and PII* Parallel analysis are parallel RESULT (precise) are nt parallel Decisin end f calculatins NO RESULT apprximate calculatin Distance between plane PI* and pint A OR Distance between plane PII* and pint B OR Distance between pints A and B OR Different prcedure 6 KOMPUTER RESULTS (apprximate) x Results number Fig. 0. Blck scheme f determining the distance between tw planes. a) b) Angle between planes I-III Plane III (> 60') 7% % Angle Plane I* (< 0'';60' >) 5 (t 0'') 9% Fig.. The determinatin f the angle between planes I-III: a) the psitin f the calculated plane PIII and the calculated substitute plane PI*; b) the graphical representatin f results. Other irregularities appear when we attempt t determine the cmmn part f plane PII and a straight line (Fig. ). This pint is shifted alng axis Y, which is astnishing. Since all the three pints f the line lie n plane XZ, it is impssible t shift the straight line alng axis Y. Hwever, the shape deviatin f plane PII causes the determinatin f substitute plane PII*, which is n lnger parallel t plane YZ. The testing sftware shws that there are significant differences in calculatins f the cmmn part f the plane and the straight line. Fur kinds f sftware yielded astnishing results fr axis Y (+98; +990; +75; +887), which is the evidence f big irregularities in the calculating algrithms. Only three algrithms calculated the psitin f this pint crrectly.

11 Pint P Line >um 7% Plane II* <um;um> % t um Fig.. The determinatin f the cmmn part f the plane and the straight line: a) the psitin f the calculated plane PIII and the calculated substitute plane PI*; b) the graphical representatin f results. The whle test has shwn that we may expect a wide dispersin f the results we get in real measuring cnditins, if we use the criterin f minimum measuring pints. Only three algrithms have less than f errrs (Fig. a) and these algrithms imprve the verall assessment f errrs fr the whle test (Fig. b). a) b) 0 8 Test results (LMS methd) 6 >um 7% <um;um> % 8% A B C D E F G H I J K L M N Sftware t um % Fig..The graphical representatin f the errr values: a) fr particular cases f CMM sftware, b) fr the whle test. Frm Figure we might draw the cnclusin that sme algrithms are better than thers. It must be stressed, hwever, that this figure shws the agreement f the results with the LMS methd. Whether the LMS methd is the best fr all tasks, is a different issue. If we used anther reference methd, it culd turn ut that e.g. the sftware dented by H, which uses ther calculatin algrithms, will be better than the prduct dented by B. The verall cnclusin f the test, hwever, is that with high prbability we will btain different results frm the same set f input data. T illustrate the results f the sftware test, the sftware has been gruped in tw classes, accrding t the accuracy during the test. Class I (sftware prducts B, F, J) These prducts have sme irregularities, which were visible during measurements with a small number f measuring pints. Usually the errrs invlved were in the interval <-µm, +µm>; disprprtinately big errrs als appeared, althugh they were very rare (the interval >µm in Fig. ). The sftware in class I is prduced by firms which have been dealing with crdinate measurements fr a lng time and which have gained a lt f experience. The prducts f these firms are tp quality. The remaining prducts rate Class II. Figure illustrates the differences between the tw classes; 86.% f results in class I are in the range f accurate results (in accrdance with LMS), and nly 7% are in the <-µm, +µm> errr interval. In class II, n the ther hand, the results are spread mre

12 evenly in the fur errr intervals. The sftware frm categry II may yield relatively high errrs, especially if there are few measuring pints and if the pints are unevenly distributed in the gemetrical element. 0 86,% (LMS methd) 8 6 0, 5%,% 7, 8,7,9% Class I Class II crrect t um <um;um> >um Interval Fig.. Graphical presentatin f the test results..additional TEST The next step was t verify the results fr an increased number f measuring pints with the same shape deviatins. The tw prgrams representing thse tw classes ( B and H sftware) were tested (Fig. a). a) b) c) Y R R X 60 5 Y * R* R R 8 X Y 0 X Fig. 5. The example f the psitining f the measuring pints in the circle with vality: a) basic fur pints, b) psitining f pint *, c) psitining f the pints. The number f measuring pints was increased frm t 0 ( 0 ). The measured crdinates f the pints (in the example the fur pints) are input int the cmputer calculating the suitable center and radius f the circle. Hwever, the calculatins are apprximate and the user has n knwledge f the kind f applied algrithm. The errr may be minimized by increasing the number f measuring pints. In the test, the circle equatin was pinted ut with the pints. The psitining f thse pints n the circle with vality deviatin is shwn in Fig. 5a. Next, the number f pints was increased t 8, 6... and 0. A further increase f the number is impssible because the maximum number f pints may nt be exceeded. Figure 5a presents hw the fur pints are psitined n the circle. T increase the number f pints, the angle between the radii R and R was divided, and the measuring pint * was placed in the crssing f the bisectr and the circle (Fig. 5b). The rest f the pints were psitined in the same way (Fig. 5c). Figure 6 presents the results as a circle equatin pinted ut with chsen CMM prgrams fr the increased number f pints.

13 a) b) Circles I-VI: centre Cirles I-VI: radius Centre displacement [mm] 0,006 0,005 0,00 0,00 0,00 0, I II III IV V VI Radius displacement [mm] 0,00 9,998 9,995 9,99 9,989 9,986 9, I II III IV V VI Number f measuring pints n Number f measuring pints n Fig. 6. The influence f the number f measuring pints n the values f: a) centres I IV, b) radiuses I IV. 9, 98,8 "B" sftware fr pints "H" sftware fr pints "B" sftware fr 0 pints "H" sftware fr 0 pints 00 89,9 80 [%] , 5, , 7,8 0,9 0 0,6 0,5 0,7 0um t um <um;um> >um Interval Fig. 7. Graphical presentatin f the test results. Figure 7 illustrates the differences between the tw CMMs sftware with and 0 measuring pints. It is visible that the extensin f the number f measuring pints t 0 in a fundamental manner changes the results f the test. The reslutin became smaller by the difference in perfrmance between CMMs B and H sftware. A similar result appears in remaining CMMs sftware reckned f being class II. As a resume, t execute the measurement with the scanning methd practically ne btains apprximate results with the adaptatin f the LMS methd r its mdificatin. 5. CONCLUSIONS The analysis f accuracy f calculatin algrithms f CMMs, cnducted in the 90s, shwed a high level f errrs in the btained measurement results []. The prblem f CMM accuracy has been nticed again fairly recently by a number f researchers [-]. The results f the test cnducted by the authr n currently-used calculating algrithms shw that the prblem f CMM accuracy is still unslved. The CMM sftware may significantly influence the accuracy f the whle measurement system. Until nw, the manufacturers f CMM have been giving the MPE E (maximum permissible errr f indicatin f a CMM fr size measurement) [], which refers nly t the CMM. The cmputer and the sftware, r the calculatin algrithm, which are nw intrinsic parts f the CMM measuring system, have been treated as factrs which d nt intrduce any errrs. Hwever, n the basis f the test which invlved different algrithms it can be said that the cmputer and sftware is the surce f significant errrs, which in unfavurable cnditins can be much higher than the MPE E given by the manufacturers. The calculatin algrithms used in sftware ffered by different manufacturers are nt identical. In many cases the algrithm used is nt the classical Gaussian algrithm (LMS). Manufactures intrduce mdificatins f this algrithm; hwever, they d nt infrm the

14 users abut them. It must be stressed that there are n knwn fitting algrithms which are ideal, particularly fr measuring gemetrical elements with shape deviatins. Therefre the tendency t mdify the basic fitting algrithms is understandable. Hwever, sme mdificatins are better suited t detect sme kinds f deviatins and are abslutely useless fr detecting thers. It seems that we cannt describe all gemetrical elements with varius shape deviatins with ne fitting functin; we will always btain an apprximatin nly. Therefre it seems necessary t wrk ut new fitting algrithms using a specified lcatin and number f measurement pints, suited t particular shape deviatins. Sme sftware ffers mre than ne methd fr finding the circle equatin. The centre and the radius f a circle may be determined using the fllwing methds: LMS, MZ, MIC and MCC. In this case, hwever, the peratr must decide which methd t chse and what criteria t use in making the decisin. Fr a relatively small number f measuring pints it is difficult t chse the apprpriate fitting methd, since fitting methds becme stable nly fr a large number f measuring pints. The minimal numbers applied tday d nt ensure it; they must be increased cnsiderably and related t the kind f the shape deviatin. The shape deviatins f the real details are cmplicated and their descriptin is difficult, which means the need f applicatin f a large number f measuring pints. The accuracy f the measurement als depends n the psitining f thse pints. ACKNOWLEDGEMENTS The authr wuld like t thank everybdy wh helped him cllect the surce materials fr his tests. REFERENCES. Waldele F., Bittner B., Dreschner R., Elligsen R.: Test vn Sftware fur Krdinatenmessgerate. VDI-Z, /99, pp Cx M. G., Harris P. M.: Guidelines t Help Users Select and use Sftware fr their Metrlgy Applicatins. Natinal Physical Labratry (UK), Reprt CMSC 0/ Emmet L., Frme P.: Sftware Supprt fr Metrlgy Best Practice Guide. N. : The Develpment f Virtual Instruments. Adelard and Crwn cpyright Brinkley D.: Guide t the Develpment f Sftware fr Metrlgy SSfM Sftware Supprt fr Metrlgy Best Practice Guide N. : Guidance n Develping Sftware fr Metrlgy. Lgica (UK). 5. Cx M. G., Frbes A. B., Harris P. M.: Sftware Supprt fr Metrlgy Best Practice Guide N. : Discrete Mdelling Wichmann B.: Measurement Gd Practice Guide N 5: Sftware in Scientific Instruments. Natinal Physical Labratry (UK), Crwn cpyright Barker R. M., Harris P. M., Parkin G. I.: Sftware Supprt fr Metrlgy Best Practice Guide N. 7: Develpment and Testing f Spreadsheet Applicatins. Natinal Physical Labratry (UK), Lin Y.-J., Damdharan K., Shakarji C.: Standardised Reference Data Sets Generatin fr Crdinate Measuring Machine (CMM) Sftware Assessment. The Internatinal Jurnal f Advanced Manufacturing Technlgy, Springer-Verlag Lndn Limited, pp Cx M. G.: A Discussin f Appraches fr Determining a Reference Value in the Analysis f Key- Cmparisn Data. Natinal Physical Labratry (UK), Reprt CISE / Ratajczyk. E.: Crdinate Measuring Technique. Oficyna Wydawnicza Plitechniki Warszawskiej, Warszawa 005. (in Plish). Swrnwski P.: The Optimum Number and Distributin f Measuring Pints fr the Circle with the Shape Deviatin. Cmmittee f Mechanical Engineering, Plish Academy f Sciences, Pznan Divisin, Archives f Mechanical Technlgy and Autmatizatin, vl., n., 00, pp Swrnwski P.: Influence f Methd Fitting n Result f Measurement. Cmmittee f Mechanical Engineering, Plish Academy f Sciences, Pznan Divisin, Archives f Mechanical Technlgy and Autmatizatin, vl., n, 00, pp DIN 880 (986): Crdinate metrlgy. Gemetrical fundamental principles and terms. (in German). ISO 060: Gemetrical Prduct Specificatins (GPS) - Acceptance and reverificatin tests fr crdinate measuring machines (CMM).