1、BRITISH STANDARD BS 7172:1989 Guide to Assessment of position, size and departure from nominal form of geometric features UDC 658.5.011.54:519.688:681.3.06:62 4:531.7BS7172:1989 This British Standard, having been prepared under the directionof the Advanced Manufacturing Technology Standards Policy C
2、ommittee, waspublished under the authorityofthe Board of BSI andcomes intoeffect on 31July1989 BSI 11-1999 The following BSI references relate to the work on this standard: Committee reference AMT/8 Draft for comment 88/98386 DC ISBN 0 580 173593 Committees responsible for this British Standard The
3、preparation of this British Standard was entrusted by the Advanced Manufacturing Technology Standards Policy Committee (AMT/-) to Technical Committee AMT/8, upon which the following bodies were represented: Advanced Manufacturing Technology Research Institute British Telecommunications plc Cranfield
4、 Institute of Technology Department of Trade and Industry (National Engineering Laboratory) Department of Trade and Industry (National Physical Laboratory) Gauge and Tool Makers Association Institution of Production Engineers Ministry of Defence University of Manchester, Institute of Science and Tec
5、hnology Amendments issued since publication Amd. No. Date of issue CommentsBS7172:1989 BSI 11-1999 i Contents Page Committees responsible Inside front cover Foreword ii 1 Scope 1 2 Definitions 1 3 Symbols and abbreviations 2 4 Outline of guide 3 5 Parametrization of geometric elements 4 6 Measuremen
6、t procedure 5 7 Data pre-processing 9 8 Establishing a reference 10 9 Departure from nominal form 10 10 Information to be provided by an assessment 11 11 Numerical considerations 12 Appendix A Formulae for distance of a point to a geometric element 14 Figure 1 A distribution of points on a line 6 Fi
7、gure 2 A distribution of points in a plane 7 Figure 3 A “chess board” distribution of points in a plane 7 Figure 4 A distribution of points on a lobed circle 7 Figure 5 A distribution of points on a sphere 8 Figure 6 A distribution of points on a cylinder 8 Figure 7 A distribution of points on a con
8、e 9 Figure 8 Example of a profile with two equal maximum inscribed circles, centred at A and B 12 Figure 9 Example of a profile with a locally maximum inscribed circle, centred at A 13 Table 1 Minimum numbers of points 9BS7172:1989 ii BSI 11-1999 Foreword This British Standard has been prepared unde
9、r the direction of the Advanced Manufacturing Technology Standards Policy Committee. This standard is intended for use by manufacturers of coordinate measuring machines (CMMs) and software writers within the CMM industry. It covers the assessment of geometric form of workpieces measured with a CMM.
10、It gives information and guidance to promote the better use of CMMs for this purpose by the adoption of reliable software that provides well presented comprehensive information. This British Standard contains mathematical concepts and notation; therefore it is assumed that the execution of its guida
11、nce is entrusted to appropriately qualified and experienced personnel. A British Standard does not purport to include all the necessary provisions of a contract. Users of British Standards are responsible for their correct application. Compliance with a British Standard does not of itself confer imm
12、unity from legal obligations. Summary of pages This document comprises a front cover, an inside front cover, pages i and ii, pages1 to 17 and a back cover. This standard has been updated (see copyright date) and may have had amendments incorporated. This will be indicated in the amendment table on t
13、he inside front cover.BS7172:1989 BSI 11-1999 1 1 Scope This British Standard provides information and guidance to manufacturers of coordinate measuring machines (CMMs), and particularly software writers within the CMM industry. It contains recommendations for determining the position, size and depa
14、rture from nominal form of geometric features, given measurements of coordinates of points on a workpiece. The features covered correspond to the following geometric elements:lines, planes, circles, spheres, cylinders and cones. This standard is concerned with software implementations of algorithms
15、based on sound mathematical and computational principles, rather than automated versions of manual or graphical assessment procedures. This standard does not provide detailed guidance on methods for treating data gathered in an unstable environment. This standard does not cover secondary attributes,
16、 i.e. measures derived from departures from form of the above geometric elements, such as parallelism, concentricity or orthogonality. This standard primarily relates to CMMs that operate with a right-handed Cartesian coordinate system. It also relates to other measurement systems that provide such
17、coordinates. Some of the guidance provided also applies to other coordinate systems. 2 Definitions For the purposes of this British Standard the following definitions apply. 2.1 algorithm a step-by-step description in mathematical or other unambiguous terms of a process for solving a particular prob
18、lem, e.g. the determination of the parameters describing a geometric feature 2.2 centroid the point described by coordinates which are the arithmetic means of the corresponding coordinates of the data points 2.3 circularity measure of departure from nominal form for a mathematical circle 2.4 cylindr
19、icity measure of departure from nominal form for a mathematical cylinder 2.5 data point either a raw data point or a raw data point that has been processed in some way 2.6 departure from nominal form overall measure of the deviation of a workpiece from nominal form NOTEThe departure from nominal for
20、m is defined as the spread, or in terms of the spread. 2.7 deviation the straight-line (Euclidean) distance of a data point from the reference, measured normal to the geometric element. The distance is regarded as positive or negative according to which side of the element the point lies. Where appr
21、opriate it is negative if the point lies in the material of the workpiece, and positive otherwise. In the case of a line in two dimensions or a plane all points on one side are taken to have a positive deviation and those on the other a negative deviation. The deviation for the ith data point is den
22、oted by e i 2.8 direction cosines (of a line) the cosines of the angles between a line and the Cartesian axes 2.9 geometric element line, plane, circle, cylinder, cone or sphere 2.10 geometric feature (part of) an object nominally in the shape of one of the geometric elements 2.11 line a straight li
23、ne in two or three dimensions 2.12 measurement procedure a strategy for obtaining a representative set of points on a workpiece 2.13 nominal form the ideal geometric object of which the geometric feature under test is a machined or otherwise manufactured manifestation, e.g. sphere 2.14 normal a line
24、 passing through a point on a curve or surface and perpendicular to the tangent line or plane at the pointBS7172:1989 2 BSI 11-1999 NOTEA normal conventionally points out of the material. 2.15 outlier a data point that is not regarded as a member of a set of data points representative of the geometr
25、ic feature NOTEAn outlier may arise from malfunction of the CMM or an error on the part of its operator. 2.16 parameters algebraic variables representing the size and position of a geometric element, e.g. the radius and centre coordinates of a circle 2.17 parameter values numerical values of paramet
26、ers 2.18 parametrization a choice of algebraic variables to represent a geometric element 2.19 pre-processing operations upon measured data intended to render it more suitable for purposes of assessment of form 2.20 range of the deviations the difference between the largest and smallest signed devia
27、tions, i.e. max ie i min ie i 2.21 raw data measured coordinates of points on the boundary or surface of the geometric feature 2.22 reference a computed geometric element to be used as a basis for assessment 2.23 representative set of points a set of points that, for the purposes of the assessment,
28、adequately represent the geometric feature 2.24 residual a measure of the error of fit of a reference or trial reference at a point. At the ith point the residual is denoted by res i 2.25 root mean square (r.m.s.) deviation the square root of the quotient of the sum of the squares of the deviations
29、and the number of degrees of freedom v, i.e. (C i e i 2 /v) 2.26 software, software implementation a computer implementation of an algorithm 2.27 spread a measure of the scatter of the deviations NOTETwo useful definitions are the range of the deviations and the root mean square deviation. A further
30、 measure of spread is a suitable multiple of the root mean square deviation. 2.28 uniform pseudorandom number generator an algorithm or software for producing a sequence of numbers that, according to statistical tests, appear to be samples from a rectangular distribution 2.29 workpiece the object or
31、 component under test, containing the geometric feature being assessed 3 Symbols and abbreviations For the purposes of this British Standard the following symbols and abbreviations apply. Several meanings are given to some of the symbols and the specific meaning is implied in each case by the contex
32、t in which the symbols are used. a Direction cosine for x. b Direction cosine for y. c Direction cosine for z. c Constant in circle equation. C Circle. C Cylinder. C Cone. d Distance of a point from a geometric element. e i Deviation of the ith data point from a reference. E Objective function used
33、in computing a reference. f Parameter in circle equation. f Intermediate variable in distance formulae. F Measure of departure from nominal form. g Parameter in circle equation. g Intermediate variable in distance formula.BS7172:1989 BSI 11-1999 3 4 Outline of guide In order to obtain a reliable ass
34、essment of geometric form in any particular case, the corresponding geometric element should first be represented, i.e.parametrized, in a mathematically sound way. Recommended parametrizations are given in clause5. It is recommended that the assessment process itself be carried out in four stages: a
35、) apply an appropriate measurement procedure, i.e. a strategy for obtaining a representative set of measurements on the workpiece (see clause 6); b) (optionally) pre-process the data, i.e. replace the measured data by modified values in order, for example, to smooth the data, to remove inappropriate
36、 points or to compensate for environmental effects (see clause 7); c) compute the reference (e.g. an approximating circle in terms of its centre coordinates and radius), to give position and size (see clause 8); d) assess, in terms of the reference, the departure from nominal form (see clause 9). On
37、ce the assessment has been carried out, it is recommended that the software provides the information detailed in clause 10. To avoid unnecessary numerical inaccuracies during the assessment, software writers should adopt the recommendations given in clause 11. G Point on L or P closest to the centro
38、id of the data points. h Distance between two parallel planes. h Height of frustum of cone. i Subscript for data point. l Length of generator of frustum of cone. L Straight line. n Number of parameters necessary to describe a geometric element, which normally is also the same as the mathematical min
39、imum number of points required to define the element. n c Number of approximately uniformly spaced parallel planes. n p Number of measurements on or near a plane. N Number of measured points on a workpiece. P Plane. q Number of lobes on a nominally circular feature. r Radius of circle, sphere or cyl
40、inder. r Radial coordinate in a cylindrical or spherical coordinate system. r 1 , r 2 Radii of ends of frustum of cone. r i Distance of ith data point from centre of reference circle. res i Residual evaluated at the ith data point. s Difference between number of measurements on successive planes in
41、a measurement procedure for a cone. s Distance from the surface of a cone to a point on its axis. S Sphere S Set of data points. t Parameter proportional to distance. u, v, w Intermediate variables in distance formulae. x First Cartesian coordinate. Arithmetic mean of values of x i(= C ix i i/N). x
42、0 , x 1 , x 2 x-coordinates of locating points for a straight line or axis of a cylinder or a cone. x p x-coordinate of a general point on a geometric feature when computing distance from the point to a geometric element. y Second Cartesian coordinate. Arithmetic mean of values of y i(= C i y i i/N)
43、. x y y 0 , y 1 , y 2 y-coordinates of locating points for a straight line or axis of a cylinder or a cone. y p y-coordinate of a general point on a geometric feature when computing distance from the point to a geometric element. z Third Cartesian coordinate. Arithmetic mean of values of z i(= C i z
44、 i i/N). z 0 , z 1 , z 2 z-coordinate of locating points for a straight line or axis of a cylinder or a cone. z p z-coordinate of a general point on a geometric feature when computing distance from the point to a geometric element. Bearing angle for cylindrical or spherical coordinates. v Number of
45、degrees of freedom, given by N n. Azimuth angle for spherical coordinates. Apex angle of cone (equal to twice the angle between the cones generator and axis). zBS7172:1989 4 BSI 11-1999 NOTEAppendix A gives mathematical formulae for the distance of a point to a geometric element described by one or
46、other of the parametrizations given in clause 5. These formulae should be useful to the software writer in preparing algorithms for assessing departure from nominal form when using the recommended parametrizations. 5 Parametrization of geometric elements 5.1 General This clause is concerned with the
47、 manner in which the position and, where relevant, orientation and size of each geometric element considered in this standard should be described in mathematical terms. For a reliable assessment to be carried out the workpiece should be adequately represented by a set of measured data points in a Ca
48、rtesian coordinate system (see 6.2). The geometric element that is to act as a reference for the data is described in terms of this system. The description consists of assigning numerical values to parameters that define the geometric element. It is possible to parametrize each of the geometric elem
49、ents in more than one way. The parametrizations given here are recommended as being generally applicable. They have the property that small changes in the geometric element usually result in correspondingly small changes in the parameter values. Certain other parametrizations may be equally sound, although it should be noted that the use of some parametrizations can yield unreliable results. Example. It is possible to parametrize a cone in terms of position of the vertex, direction of the axis and the angle that the cone generator makes with the
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