1、COPYRIGHT American Society of Mechanical EngineersLicensed by Information Handling ServicesECHSPAPER 90 m 0759670 0082326 7 m ASME B89 TECHNICAL PAPER Space Plate Test Recommendations for Coordinate Measuring Machines The American Society of Mechanical Engineers 345 East 47th Street, New York, N.Y.
2、10017 COPYRIGHT American Society of Mechanical EngineersLicensed by Information Handling ServicesASME B89 TECHvPAPER 90 I O759670 0082327 9 M Date of Issuance: February 15, 1991 No part of this document may be reproduced in any form, in an electronic retrieval system orotherwise, without the priorwr
3、itten permission of the publisher. Copyright O 1991 by THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS All Rights Reserved Printed in U.S.A. COPYRIGHT American Society of Mechanical EngineersLicensed by Information Handling ServicesASME B89 TECHMPAPER 90 W 0759b70 0082328 O W FOREWORD This technical pa
4、per is the finalreport of Task ForceE - Ball Plates, of ASME Working Group B89.1.12, Coordinate Measuring Machines (CMM). Committee members of Task Force E, and authors of this technical paper are: Stanley R. Drake William S. Gehner Kenneth E. Moser, Chairman Benjamin R. Taylor This paper has been r
5、eviewed and approved by Working Group B89.1.12. iii COPYRIGHT American Society of Mechanical EngineersLicensed by Information Handling ServicesASME B89 TECH*PAPER 90 W 0759b70 0082329 2 m CONTENTS Foreword 1 Purpose 2 Space Plate Recommendations 3 Profiles of X. Y. and Z Deviations 4 Space Plate Des
6、ign Recommendations Section A - Four-Point Method of Data Analysis Al General A2 Derivation of Three-Dimensional Transformation Equations in Orthogonal Coordinate Systems . A3 Derivation of the Equations for H. J. Ky T. W. P Constants from Four Specified Points Section B - Least Squares Fit Method o
7、f Data Analysis B1 General Section C - Point-to-Point Method of Data Analysis C1 General Section D - Space Plate Design Recommendations . Figures A-1 The Four-Point Method . C-1 Profiles of Space Plate Deviations C-2 Layout of Balls on Space Plate D-1 Space Plate D-2 Item 1 D-3 Item 2 D-4 Items 3 an
8、d 10 . D-5 Item 4 D-6 Item 5 D-7 Item 6 D-8 Item 7 D-9 Item 11 . Tables A-1 Auxiliary Positions of Reference Data . A-2 Final Position of Reference Data . A-3 Listing of X. Y . and Z Deviations B-1 Final Position of Reference Data B-2 Final Position of Measured Data B-3 Listing of X. Y. and Z Deviat
9、ions C-1 Distances From Ball 1 C-2 Distances From Ball 6 C-3 Distances From Ball 30 D-1 Material List iii 1 1 2 2 3 5 6 11 16 36 4 18 32 37 38 39 39 40 40 41 41 42 8 9 10 .13 14 15 33 34 35 36 V COPYRIGHT American Society of Mechanical EngineersLicensed by Information Handling ServicesASME B89 TECH*
10、PAPER 90 0759670 0082330 9 SPACE PLATE TEST RECOMMENDATIONS 0 FOR COORDINATE MEASURING MACHINES SPACE PLATE TEST RECOMMENDATIONS FOR COORDINATE MEASURING MACHINES e 1 PURPOSE The purpose of this technical paper is to give guid- ance for analyzing the performance capabilities of CMMs using space plat
11、e technology. It is intended as a reference document to the ANSI/ASME B89. l. 12 Standard, “Methods for Performance Evaluation of Coordinate Measuring Machines. 2 SPACE PLATE RECOMMENDATIONS A three-dimensional space plate or similar artifact is an economical and practical technique for esti- mating
12、 the measuring uncertainties of CMMs and isolating the sources of errors. Conventional tech- niques to determine the measurement capability of a CMM are complex. Analysis of a CMM shows that there are 21 degrees-of-freedom for a measurement within the volume swept by a probe. It is possible to measu
13、re the individual parameters of geometry of a measuring machine such as pitch, yaw, roll, or or- thogonality errors between the three axes. These er- rors can then be combined to determine an overall volumetric error between any two points in the spec- ified working volume of the C”. In lieu of this
14、 time-consuming approach, the three-dimensional space plate is recommended for consideration in de- termining the measuring capability of CMMs. How- ever, space plate technology may not be applicable for very large CMMs. The limiting factors would be the weight of the plate and the capability to cer
15、tify the XYZ coordinates. A CMM may be calibrated or evaluated by posi- tioning the space plate on the CMM in the desired orientation, measuring the XYZ coordinates of the balls and then comparing these coordinates with the certified XYZ coordinates traceable to National In- stitute of Standards and
16、 Technology (NIST). If the distance between corresponding pairs of points differs between the two sets, the differences are attributed to errors in the CMM. In order to make this com- parison, the two sets of data are superimposed in the e e same coordinate system and the XYZ coordinates of matched
17、pairs of points are merely subtracted. This gives the XYZ components of the differences between the two sets of data. The superimposition may be done either physically or mathematically. In the physical approach, the plate is “tapped in” or adjusted on the CMM until certain specified balls produce t
18、he same coordinates as on the space plate. The problem with this approach is that the “tapping in” process is very difficult, tedious and, at best, a compromise. In the mathematical approach, no special orien- tation is required for the space plate on the CMM, The data may be superimposed mathematic
19、ally ac- cording to some specified scheme. This approach is preferred because it saves time and effort and does not by itself introduce any additional errors. Com- puter software has been developed to perform this required mathematical evaluation of the data. Three techniques are listed here for con
20、sideration. (a) Four-point method (b) Least squares fit method (c) Point-to-point method 2.1 Four-Point Method In the four-point method, an auxiliary set of co- ordinate axes is established in both sets of data through certain specified balls. By transforming both sets of data into these coordinate
21、systems, the data are, in effect, Superimposed. A full elucidation of the math- ematical steps used in the FORTRAN program is con- tained in Section A. Data analyzed by the four-point method are found in Tables A-1, A-2, and A-3. Table A-1 is data from a CMM with an auxiliary set of coordinate axes
22、fitted through the origin at ball 1 , X-axis parallel to balls 1 and 6, and XY plane through balls 1 , 6, and 30. Table A-2 is a set of reference data with the same set of auxiliary axes. Therefore, Table A-3 is a listing of the X, Y, and Z deviations for the corresponding balls. I n COPYRIGHT Ameri
23、can Society of Mechanical EngineersLicensed by Information Handling ServicesASME B89 TECH*PAPER 90 I 0757670 0082333 O R 2.2 Least Squares Fit Method In the least squares fit method, the position of the reference data is adjusted so that the sum of the squares of the undirected distances between cor
24、re- sponding points in both sets of data is a minimum. In the least squares fit method, the coordinate axes do not necessarily become superimposed. A full elu- cidation of the mathematical steps used in the FOR- TRAN program is found in Section B. Data analyzed by the least squares fit method are fo
25、und in Tables B-1, B-2, and B-3. The reference data in Table B-1 have been adjusted to the data in Table B-2 such that the sum of the squares of the undirected distances between corresponding data points in both sets of data is a minimum. Table B-3 is a listing of the X, Y, and Z deviations for the
26、corresponding balls. 2.3 Point-To-Point Method In the point-to-point method, the straight line dis- tances are calculated from one of the balls to all the other balls in both the reference data and the meas- ured data. These distances between corresponding pairs of balls are then merely subtracted.
27、In effect, this superimposes the two data sets and is analogous to the four-point method, except that only one axis (a line through the two balls) instead of three is su- perimposed. A full elucidation is presented in detail in Section C. Data analyzed by the point-to-point method are found in Table
28、s C-1, C-2, and C-3, where the straight line distances are calculated for the measured and reference distances from balls 1, 6, and 30, respec- tively. In each case, the differences between the meas- ured and reference distances are shown as deviations. 3 PROFILES OF X, Y AND Z DEVIATIONS Statistica
29、l analysis of the X, Y, and Z deviations from the four-point method can now be performed. These statistical calculations will give an indication of the CMMs total measuring errors within the vol- SPACE PLATE TEST RECOMMENDATIONS FOR COORDINATE MEASURING MACHINES Ume of the space plate at the time th
30、e data were generated. This is not all-inclusive from a practical viewpoint. All possible errors caused by the 21 degrees-of-free- dom of the CMM, including random errors, are in- cluded together. There is a large percent of dimensions of piece parts that are measured along a single axis or in a sin
31、gle plane. When measurements with the smallest uncertainty are required, it is recommended that the uncertainties be applied with this logic in mind. Large uncertainties that were arrived at from the volumetric approach should not be applied to these single axis and plane measurements. Some profiles
32、 of X, Y, and Z deviations are pre- sented with computer-generated examples. In all ex- amples presented, it is assumed that there are no errors in the 21 degrees-of-freedom of the CMM, except for those in the profile being exemplified (Fig. C-1). With the space plate in the horizontal position, all
33、 of the profiles demonstrated are an accurate repre- sentation of the CMMs geometry errors. This is true whenever the cluster of balls is parallel to any one of the three orthogonal planes of the CMM. However, with the space plate in the tilted position, some of the profiles are slightly changed by
34、an interaction of the other axes. By reviewing the profiles in both the horizontal and tilted positions, it is clear that the sources of errors can be singled out more than 90% of the time. As technicians become more experienced in reviewing these profiles and familiar with the dif- ferent machine d
35、esigns, they will become more effi- cient in identifying the geometry errors associated with that particular CMM. When observing these profiles, it is important to know the organization of the balls on the space plate, which is illustrated in Fig. C-2. These ball numbers correspond to the numbers sh
36、own on the sample pro- files (Fig. C-l). 4 SPACE PLATE DESIGN RECOMMENDATIONS Recommendations for design and construction of the space plate are illustrated in Section D. 2 COPYRIGHT American Society of Mechanical EngineersLicensed by Information Handling ServicesASME B89 TECH*PAPER 90 W O759670 008
37、2332 2 W SPACE PLATE TEST RECOMMENDATIONS FOR COORDINATE MEASURING MACHINES SECTION A FOUR-POINT METHOD OF DATA ANALYSIS AI GENERAL In the four-point method, the reference data set is mathematically superimposed onto the CMM data set in such a way that a set of auxiliary axes established in each dat
38、a set through four specified balls is su- perimposed. The desired deviations between the two sets of data are then obtained by merely subtracting the XYZ coordinates of corresponding balls. The total straight line three-dimensional distance D between corresponding balls is then found from D=1/(X-X)2
39、+(Y-Y)2+(Z-Z)2 (1) Where X, Y, Z are the original coordinates from the CMM data and X, Y, Z are the coordinates of the final position of the reference data after trans- formation. (All subscripts have been omitted for clarity.) The problem of finding the XYZ coordinates is best understood by referri
40、ng to Fig. A-1 where the CMM data set is represented by rectangle 1 and the reference data set by rectangle 2, (The Z axis has been omitted for clarity but with no loss of generality.) The XYZ coordinates on which all the data are plotted are understood to be the coordinate axes of the CMM, for it i
41、s in this coordinate system that D and its XYZ components are desired. The auxiliary axes that are put through the four specified balls are shown by the dotted axes XA, YA and XB, YB. When the reference data set is trans- formed onto the CMM set, the rectangle 2 will be moved up onto rectangle 1 so
42、that OB coincides with OA and the XB axis coincides with the XA axis. (In three dimensions, the XB,YB plane will also coincide with the XA,YA plane.) The three-dimensional equations used to transform data from one coordinate system to another are de- rived in this section. These equations require th
43、e six constants H,J,K,T,W,P to be known. These con- stants describe the position and angular orientation of the coordinate system into which the data are to be transformed. In the four-point method, this po- sition and angular orientation is dictated by four spec- ified balls. The derivation of the
44、equations for these six constants using these four points is given in this section. In order to move rectangle 2 up onto rectangIe 1 it is not sufficient to merely transform the reference data into the XI3,YB system and the CMM data into the =,YA system. While this does superimpose the two data sets
45、, they would no longer reside at the position of rectangle 1, but rather at the origin of the XY system such that OB and OA coincide at O. Since it is required that the deviations D and the XYZ com- ponents be measured relative to the CMM axes, the final position of the two superimposed sets must be
46、 at the original position of rectangle 1. To do this, the reference data set must be trans- formed into a coordinate system oriented as shown by the dotted axes X,Y. These transformed values, when plotted on the XY coordinate system, will place rectangle 2 on top of rectangle 1 as desired. If the si
47、x constants that define the position and angular ori- entation of the XY system are designated H,J,K,T,W,P, theoverallprobleminthefour- point method reduces to one of finding these six con- stants. This is done in the following manner. The four points Xl,Yl,Zl,O are established on the XYZ axes. The
48、point O must be at the origin; the other three can be at any position along the axes. (These are not the four specified balls for the XA,YA and XB,YB systems.) Using the four specified balls, the six transformation constants for transforming rec- tangle 1 into the XA,YA system are calculated. With t
49、hese, the coordinates of the four points Xl,Yl,Zl,O are forward transformed. When these transformed values are plotted on the XYZ coordinate system they become the points XTl,YTl,ZTl,OT and form the coordinate system XT,YT,ZT, positioned as shown in Fig. A-l. Again using the four specified balls, the six trans- formation constants that would transform rectangle 2 into the XB,YB system are calculated. Using these constants an inverse transformation is now carried 3 COPYRIGHT American Society of Mechanical EngineersLicensed by Information Handling ServicesASME B89 TEC
