_B89_Technical_Paper_1990.pdf

ASME 889 TECH*PAPER 90 0759670 0082326 7 ASME 889 TECHNICAL PAPER&Space Plate Test Recommendations for Coordinate Measuring Machines The American Society of Mechanical Engineers._ _ 345 East 47th Street,New York,N.Y.10017-ASME 889 TECH*PAPER 90 0759670 0082327 9 Date of Issuance:February 16,1991 No part of this document may be reproduced in any form,in an electronic retrieval system orotherwise,without the prior written permission of the publisher.Copyright 1991 by THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS All Rights Reserved Printed in U.S.A.ASME B89 TECH*PAPER 90 0759670 0082328 0 FOREWORD This technical paper is the final report of Task Force E Ball Plates,of ASME Working Group B89.l.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 reviewed and approved by Working Group B89.l.12.iii ASME 889 TECH*PAPER 90 0759670 0082329 2 CONTENTS Foreword.iii 1 Purpose.1 2 Space Plate Recommendations.1 3 Profiles of X,Y,and Z Deviations.2 4 Space Plate Design Recommendations.2 Section A-Four-Point Method of Data Analysis Al General.3 A2 Derivation of Three-Dimensional Transformation Equations in Orthogonal Coordinate Systems.5 A3 Derivation of the Equations for H,J,K,T,W,P Constants from Four Specified Points.6 Section B-Least Squares Fit Method of Data Analysis Bl General.11 Section C-Point-to-Point Method of Data Analysis Cl 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 and 10.,.D-5 Item 4D-6 Item 5D-7 Item 6D-8 Item 7D-9 Item 11.-.Tables 16 36 4 18 32 37 38 39 39 40 40 41 41 42 A-1 Auxiliary Positions of Reference Data.8 A-2 Final Position of Reference Data.9 A-3 Listing of X,Y,and Z Deviations.10 B-1 Final Position of Reference Data.13B-2 Final Position of Measured Data.14 B-3 Listing of X,Y,and Z Deviations.15 C-1 Distances From Ball 1.33 C-2 Distances From Ball 6.34 C-3 Distances From Ball 30.35 D-1 Material List.O P=tan/A)AO P=180+tan(B/A)For the second rotation(angle W),about the new position of the Y axis,it can be shown that angle W in degrees will be as follows:ForA=O andB=O W=-(S ign of C)*90 AO and BO W=-t a n C/d m (12)For the third rotation(angle T),about the new position of the X axis,let D=X4-X1 E=Y4-Y1(13)F=Z4-Z1 and from a transformation through the first two an-gles P and W let G=D(cos P)(sin W)+E(sin P)(sin W)+F(cos W)(14)H=E(COS P)-D(.&P)It can be shown that the angle T in degrees will then be as follows:For H=O T=(Sign of G)*90 HO T=tan(G/H)*(15)H where N is the number of bal1s.k each data set.To minimize Q,the partial derivative of Eq.(5)is taken with respect to each of the six unknown trans-formation constants H,J,K,T,W,P and set to O.These six relationships are aQ/aH=o aQ/aJ=o aQ/aK=o aQ/aT=o aQ/aw=o aQ/aP=o 11 ASME B87 TECH*PAPER 90 I The evaluation of the first three of these reveals they are linear in H,J,K,and therefore can be solved for H,J,K in closed form.The results:H=(CU-AlCX-A4CY-A7CZ)J/N J=(CV-A2CX-A5CY-A8CZ)/N (7)K=(CW-A3CX-A6CY-A9CZ)J/N Evaluation of the last three equations of Eq.(6)reveals they are nonlinear in T,W,P and therefore cannot be solved in closed form.It can be shown that they will reduce to 64)(A71+(S51(Ag)+(S6)(Ag)(S1)(Bl)+(S2)(B2)+(S3)(B3)+(S7)(B7)+(S8)(B8)+(S9)(B9)=O (8)-(S7)(A4)-(S8)(A5)-(S9)(A=O+(S4)(B4)+(SS)(B5)+(S6)(B6)(S2)(A1)+(S5)(A4)+(S8)(A7)-(Sl)(A2)-(S4)(A5)-(S7)(A8)=O where 0757670 0082341 3 I SPACE PLATE TEST RECOMMENDATIONS FOR COORDINATE MEASURING MACHINES S1=(CX)(CU)-N(CXU)S2=(CX)(CV)-N(CXV)S3=(CX)(CW)-N(CXW)S4=(CY)(CU)-N(CYU)S5=(CY)(CV)-N(CYV)S6=(CY)(CW)-N(CYW)S7=(CZ)(CU)-N(CZU)S8=(CZ)(CV)-N(CZV)S9=(CZ)(CW)-N(CZW)B1=-(cos P)(sin W)B2=-(sin P)(sin W)B4=+(cos P)(sin T)(cos W)B5=+(sin P)(sin T)(cos W)B6=-(sin T)(sin W)B8=+(sin P)(cos T)(cos W)B9=-(cos T)(sin W)B3=-(COS W)B7=+(COS)(COS T)(COS W)Since equations in Eq.(8)are nonlinear,an iterative method such as Browns must be used,and T,W,P can be obtained only in numerical form.12 ASME B89 TECH%PAPER 90 H 0759670 0082342 5 m SPACE PLATE TEST RECOMMENDATIONS FOR COORDINATE MEASURING MACHINES TABLE B-I FINAL POSITION OF REFERENCE DATA Fitted Results From Least Squares Fit Method Final Position of Measured Data Y Data Pt.Ball N o.1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 i 4 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 X 0.000048 5.600672 11.200497 16.800526 22.400406 28.000371 0.000636 5.600730 11.200380 16.800998 22.400410 28.000327 0.000647 5.601026 1 1.20051 8 16.800367 22.400550 28.000405 0.000631 5.600558 11.200914 16.800616 22.400350 28.000480 0.000868 5.600654 11.200752 16.800529.22.400669 28.000589-0.000014 0.000482 0.000505 0.000509 0.000547 0.000208 4,500567 4.500263 4.500590 4.500916 4.500777 4.50