United States Department of Commerce

Technology Administration

National Institute of Standards and Technology

NIST Technical Note 1400

Issues, Concepts, and Standard Techniques in

Assessing Accuracy of Coordinate Measuring Machines

Dennis A. Swyt
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NIST Technical Note 1400

Issues, Concepts, and Standard Techniques in

Assessing Accuracy of Coordinate Measuring Machines

Dennis A. Swyt

Precision Engineering Division Manufacturing Engineering Laboratory

National Institute of Standards and Technology

Gaittiersburg, MD 20899

February 1993

U.S. Department of Commerce

Ronald H. Brown, Secretary

National Institute of Standards and Technology

Raymond G. Kammer, Acting Director
National Institute of Standards U.S. Government Printing Office For sale by the Superintendent

and Technology Washington: 1993 of Documents

Technical Note 1400 U.S. Government Printing Office

Natl. Inst. Stand. Technol. Washington, DC 20402

Tech. Note 1400

87 pages (Feb. 1 993)

CODEN: NTNOEF
Preface

Prepared for use by the Department of Energy's Oak Ridge Facilities, this report deals with a variety of issues, concepts, and standard techniques in assessing the accuracy of

measurements of modern laser-based computer-controlled coordinate measuring machines.

The report brings together the substance of three closely related works by the author dealing

successively with the issues, concepts, and techniques.

The first work and chapter of this report, after defining dimensions and tolerances of

manufactured parts and showing a major trend to tightening of tolerances, outlines briefly

some of the issues in assessing conformity of dimensions of manufactured parts to design

tolerances in terms of five topical areas [1]. These five topics are: 1) mathematicization

of geometric dimensions and tolerances; 2) evaluation of the computational aspects of

computerized measuring systems; 3) the evaluation of the physical measurement aspect of

measuring machines; 4) assignment of uncertainty to measurement results; and 5) estimation

of manufacturing deviations relative to tolerances in the presence of measurement

uncertainty.

The second work and chapter of this report presents a new scheme for the analysis of the

dimensional capabilities of measuring machines and machine tools aimed at the realization

of higher degrees of precision in the manufacture of dimensioned parts [2]. Successive

sections deal with the range and types of dimensional measurements, their relation to the

basic unit of measure, the means for making such measurements, and the errors associated

with them. Application of these principles is illustrated by a detailed analysis of the use of a laser-based coordinate measuring machine for the measurement of a physical part. Also

included is an indication of how the identical analysis may be applied to a machine tool

which shapes such parts.

The third work and chapter of this report compares U.S., German, British, and Japanese

standards for evaluation of coordinate measuring machines [3]. It also shows that using a

variety of special artifacts the U.S. standard alone specifies separate tests for each of the

four modes of length measurements (displacement, position, distance, and extension) and

that, depending on the number and orientations of faces probed, a bi-directional step gage

of the type suggested in the other standards can be used for tests of each of these.

ni

Contents

Page

Preface iii

1 Issues in Assessment of Conformity of Dimensions of Manufactured Parts to Design

Tolerances 2

1.1 Introduction 2

1.2 Dimensions of Manufactured Discrete Parts 3

1.3 Tolerances on the Dimensions of Discrete Parts 3

1.4 Problems in Assessment of Conformity to Tolerances 4

1.4.1 Design: Mathematicization of Geometric Dimensions

and Tolerances 5

1.4.2 Data Taking: CMM Measurements 6

1.4.3 Computation: Fit Criteria and Algorithm Testing 7

1.4.4 Evaluation: Measurement Uncertainty and Confidence 7

1.5 Conclusion 8

2 New Concepts of Precision Dimensional Measurement for Modern Manufacturing. . 8

2.1 Introduction 8

2.2 Scope of Dimensional Measurements 9

2.2.1 Range of Dimensional Measurements 9

2.2.1.1 Macro-Scale 9

2.2.1.2 Mid-Scale 9

2.2.1.3 Micro-Scale 9

2.2.1.4 Nano-Scale 9

2.2.2 Types of Dimensional Measurements 10

2.2.2.1 Position 10

a. Geometrical Concept 10

b. Physical Concept 10

c. Measurement Concept 10

2.2.2.2 Displacement 11

a. Geometrical Concept 11

b. Physical Concept 11

c. Measurement Concept 11
2.2.2.3 Distance 12

a. Geometrical Concept 12

b. Physical Concept 12

c. Measurement Concept 12

2.2.2.4 Extension 13

a. Geometrical Concept 13

b. Physical Concept 13

c. Measurement Concept 13

2.3 Basis of Dimensional Measurements 14

2.3.1 Realization of the Unit of Length 14

2.3.1.1 Formal Definition of the Meter 15

2.3.1.2 Derived Nature of the Meter 15

2.3.1.3 Operational Realization of the Meter 15

2.3.2 Transfer of the Metric to Physical Objects 16

2.3.2.1 Visible Light of Known Wavelength 16

2.3.2.2 Displacement Interferometry 16

2.3.2.3 Coordinate Measurement System 17

2.3.2.4 Physical Object to be Measured 19

a. Distance-Type Features of Objects 19

b. Extension-Type Features of Objects 19

2.3.3 Standards-Imposed Limits to Accuracy 19

2.3.3.1 Limit of Realization of the Meter 20

2.3.3.2 Limit of Displacement Interferometry 20

2.3.3.3 Limit of Coordinate Measurement 21

2.3.3.4 Limit of Physical Artifact Standards 21

2.4 Assessing Dimensional Measurements 22

2.4.1 Statistical Characterization of Error 23

2.4.1.1 Precision vs. Accuracy 23

a. Precision 23

b. Accuracy 23

2.4.1.2 Error or Total Uncertainty 24

2.4.1.3 Additive-Multiplicative Representation 24

a. Added and Multiplicative Components 25

b. Manufacturers' Statements of Accuracy 25

VI
2.4.2 Formal-Theoretical Characterization of Error 25

2.4.2.1 Fundamental Axioms of Measurement 26

a. Rank-Order Operator 26

b. Unit 26

c. Zero 26

d. Scale 27

2.4.2.2 Analysis of Error in Terms of the Axioms 28

a. Error of the Zero 28

b. Error of the Unit 28

c. Error of the Scale 28

d. Errors of Zero, Unit, and Scale 29

2.5 Principal Specific Sources of Errors 29

2.5.1 Error in Displacement Interferometry 29

2.5.1.1 Error of Wavelength in Vacuum k^ 30

2.5.1.2 Error of Index of Refraction of Medium n 30

a. Error of the Index of Ambient Air 30

b. Error of the Index of Compensated Standard Air . 30

c. Error of Index of Refraction in the Dead Path ... 30

2.5.1.3 Error in Fringe Fractioning 31

a. Error in Electronic Subdivision 31

b. Error from Polarization Mixing 31

c. Error from Thermal Drift 31

2.5.1.4 Error in Alignment of Interferometer 31

2.5.2 Error in Coordinate-Position Measurement 32

2.5.2.1 Error in Relation to Displacement Scales 32

a. Misalignment of Mirror Translation 32

b. Abbe Error 33

2.5.2.2 Error Related to Carriage Motion 33

2.5.2.3 Errors Related to Reference-Frame Geometry 33

2.5.2.4 Error Related to the Probe 34

a. Positional Error of Probe-as-Sensor 34

b. Positional Error of Probe-as-Measuring-Machine . 34

2.5.3 Error in Distance Measurement 34

2.5.3.1 Distance Error from Feature Probing 35

2.5.3.2 Misalignment of Object and Coordinate Axes 35

2.5.3.3 Error From Object Environment 35

a. Error From Object Thermal Expansion 35

b. Error From Object Distortion 36

Vll
2.5.4 Error in Extension Measurement 36

2.5.4.1 Error in Location of a Single Boundary 36

a. Error in Reducing a 3D Surface to a Point 36

b. Error in Compensation of Probe-Object Interaction 37

2.5.4.2 Error in Location of Two Boundaries 37

2.5.5 Compounding of Error in Measurement Types 38

2.6 Error Budget Example 39

2.6.1 Error in Displacement Measurement AD 41

2.6.1.1 Fringe-Fraction Error 6Djs[ 41

2.6.1.2 Index-of-Refraction Error 8Dn 41

a. Machine-Path Atmosphere 42

b. Interferometer-Optics Thermal Drift 42

c. Dead-Path Compensation 42

2.6.1.3 Interferometer-Axis Alignment Error SDetj 42

2.6.1.4 Sum of Displacement-Specific Errors 6D 43

2.6.1.5 Vacuum-Wavelength Error bk^ 43

2.6.1.6 Total Displacement Error AD(6D,6A,o) 43

2.6.2 Error in Position Measurement AP 44

2.6.2.1 Misalignment of Coordinate and Displacement 44

2.6.2.2 Carriage Abbe Tilt 44

2.6.2.3 Probe Setting 44

2.6.2.4 Sum of Position-Specific Errors 6p 45

2.6.2.5 Total Position Error AP(6P,AD) 45

2.6.3 Error in Distance Measurement Ad 45

2.6.3.1 Object-Axis/Probe-Path Alignment 46

2.6.3.2 Object Thermal Expansion 46

2.6.3.3 Object Point Definition 46

2.6.3.4 Sum of Distance-Specific Errors 6d 47

2.6.3.5 Total Distance Error Ad(6d,Ap) 47

2.6.4 Error in Extension Measurement Ae 47

2.6.4.1 Boundary-Location Errors 47

a. Location of Arbitrary Boundary 48

b. Compensated Boundary Location 48

2.6.4.2 Average-Material-Boundary Location 48

2.6.4.3 Sum of Extension-Specific Errors 6e 49

2.6.4.4 Total Extension Error Ae(6e,Ad) 49

2.6.5 Summary and Analysis of Errors 49

2.6.5.1 Summary of Numerical Results of Analysis 50

vni
2.6.5.2 Analysis of Results 50

a. Hierarchy of Accuracies Among the Types 50

b. Additive-Multiplicative Errors 51

2.6.5.3 Axiom-Type Matrix Analysis for the CMM Example ..51

a. Errors in the Zero of Each Type 51

b. Errors in the Unit of Each Type 52

c. Errors in the Scale of Each Type 53

2.6.5.4 Application of Type-Axiom Error Matrix to

Manufacturing Operations 53

a. "Machine Tool" vs. "Measuring Machine" 53

b. Example of Extension Errors in Milling 53

2.7 Conclusion 54

2.8 Appendix: Errors Due to Angular Misalignments of System Elements . . 54

2.8.1 Misalignment Error Specific to Displacement 55

2.8.2 Misalignment Error Specific to Coordinate Position 55

2.8.3 Misalignment Error Specific to Object-Distance 56

2.8.4 Combined Misalignment Error in Distance 56

2.8.5 Explicit Form for Misalignment Error 56

3 Comparison of National Standards for Evaluation of CMM's 57

3.1 Introduction 57

3.2 Standard-Based Performance Tests of CMM's 58

3.2.1 U.S. Standard B89.1.12M 58

3.2.1.1 Measuring Repeatability 59

3.2.1.2 Linear Displacement Accuracy 59

3.2.1.3 Volumetric Performance 60

3.2.1.4 Bi-Directional Length Measurement Capability 60

3.2.1.5 Point-to-Point Probing Performance 61

3.2.2 Relation of B89.1.12M Tests to Dimensions of Parts Measured . . 61

3.3 Intercomparison of Standards for CMM Performance Testing 62

3.4 Conclusion 65

3.5 Appendix: Tests Specific to Length-Measurement Types 66

3.6 Appendix: Use of a Step Gauge in Four Measurement-Quantity Modes 67

4 References 69

5 Appendix: NIST Policy on Statements of Uncertainty in Measurements 74

IX
List of Tables

Table 1 Tasks and associated variabilities in comparing point-sample

measurements of manufacturing deviations with tolerances 5

Table 2 Quantity and nature of the length of path to be measured for

each of the four dimensional measurement types 14

Table 3 The principal functional components of a coordinate measuring

machine (as well as of a machine tool) 18

Table 4 Comparison of the value and uncertainty of vacuum wavelength

of helium-neon lasers of various degrees of stabilization 20

Table 5 Type, range, and accuracies of some dimensional standards calibrated by the

NIST 22

Table 6 The matrix of errors by measurement axiom and dimensional

type in general form 27

Table 7 Error budget example: analysis of measurement of extension of a 100 mm long object with an interferometer-based CMM 40

Table 8 Summary of total errors by dimensional type in the example of laser-based

CMM measurement of extension of 100 mm block 50

Table 9 Total error by dimensional type at ranges of 10, 100 and 1000

mm in the example of the laser-interferometer-based CMM

measurement of extension of a block 51

Table 10 The axiom-type error matrix for the example of the laserinterferometer-based CMM measurement of extension 52

Table 11 Comparison of the principal standards for CMM performance evaluation in

terms of the artifacts they specify and the type of length measurement to

which each artifact-based test corresponds 63
List of Figures

Figure 1. An engineering drawing representation of principal geometrical

dimensions of manufactured discrete part 2

Figure 2. Log-log plot of tolerance versus dimension for discrete-part

products in normal, precision, and ultraprecision regimes 3

Figure 3. Lobing in response of CMM probe giving rise to directional errors in coordinates of points on part being measured 6

Figure 4. Sources of misalignment errors in displacement-measuring

interferometers 55

Figure 5. Alignment of mirror normals and light direction 55

Figure 6. Alignment of mirror translation and mirror normals 55

Figure 7. Alignment of part axis and mirror translation 56

Figure 8. An engineering drawing representation of the principal

geometrical dimensions of a manufactured discrete part 58

Figure 9. B89.1.12M performance test for measuring repeatability 59

Figure 10. B89.1.12M performance test for displacement accuracy 59

Figure 11. B89.1.12M test for volumetric performance 60

Figure 12. B89.1.12M test for bi-directional length accuracy 60

Figure 13. B89.1.12M test for point-to-point probing 61

Figure 14. Example features of an automobile engine block by length

dimensional type 62

Figure 15. Use of step gage in four modes of length measurement 67

XI
Acknowledgments

The author wishes to acknowledge the extensive contribution and evaluation of ideas for the

publications upon which the chapters of this report are based by members of NIST's

Manufacturing Engineering Laboratory: for Chapter 1, Drs. Ted Doiron, Ted Hopp and

Steve Phillips; for Chapter 2, Drs. Tyler Estler, David Gilsinn, Steve Phillips, John Simpson,

Jack Stone, Clayton Teague and Ted Vorburger; and for Chapter 3, Dr. Steve Phillips.

xn
Issues, Concepts and Standard Techniques in

Assessing Accuracy of Coordinate Measuring Machines

Dennis A. Swyt

Precision Engineering Division

National Institute of Standards and Technology

Abstract

This report deals with a variety of issues, concepts and standard techniques in assessing the

accuracy of measurements of modern laser-based computer-controlled coordinate measuring

machines. It outlines technical issues in assessing conformity of dimensions of manufactured

parts to design tolerances with such CMM's, uses a new system of length-dimensional types

(displacement, position, distance, and extension) as the basis for an error-budget analysis of

CMM measurement capabilities and, based on that system, intercompares United States,

German, British, and Japanese CMM performance standards, showing that the U.S. standard

provides separate test of a CMM's capability to measure of each length-dimensional type

and the part features associated with them.

Key Words: accuracy; CMM's; coordinate measuring machines; coordinate metrology;

displacement; distance; extension; length; manufacturing tolerances; measurement;

performance evaluation; position; standards; testing
1 Issues in Assessment of Conformity of Dimensions of Manufactured

Parts to Design Tolerances

1.1 Introduction

In the manufacture of discrete parts, from automobile engine blocks to quantum-well

electronic devices, a principal determinant of quality is conformity of geometrical dimensions

of the part as manufactured to those of the part as designed. The degree of this conformity

is determined by the relationship of measured deviations of geometrical dimensions to

specified tolerances.

Today the historical system of manufacture — which involves relatively low precision, hard

gaging of part elements, graphic representations of tolerances, and a willingness to accept

high scrap rates — is being supplanted by a new system of manufacture — which involves

high-precision manufacture, computer-based coordinate-metrology, mathematical

representations of tolerances, and demand for making every part right the first time. This

new system has raised major issues for research and standardization across the spectrum of

design specification, precision manufacture, and measurement-based quality assessment.

After defining dimensions and tolerances of

manufactured parts and showing a major

trend to tightening of tolerances, this

chapter outlines briefly some of the issues

in assessing conformity of dimensions of

manufactured parts to design tolerances in

terms of five topical areas: 1)

mathematicization of geometric dimensions

and tolerances; 2) evaluation of the

computational aspects of computerized

measuring systems; 3) the evaluation of the

physical measurement aspect of measuring

machines; 4) assignment of uncertainty to

measurement results; and 5) estimation of

manufacturing deviations relative to

tolerances in the presence of measurement

uncertainty.

D B F

^

4

A

C

Figure 1. An engineering drawing

representation of principal geometrical

dimensions of manufactured discrete part.
1.2 Dimensions of Manufactured Discrete Parts

Whether aircraft, automobiles, computers, or integrated circuits, discrete-part products are able to function and to function properly because the components of these products have

specific geometrical shapes, sizes, and related dimensions.

Figure 1 is an engineering drawing representation of an automobile engine which illustrates

some of the principal types of geometrical dimensions of discrete parts. These types of

geometric features include: A, representing a length or width of the engine block; B,

representing the diameter of a cylinder bore; C, representing the distance between the

centers (axes) of cylinder holes; D, representing the depth of a blind hole; E, representing

the location of the center (axis) of a hole relative to a plane; and F, representing the (true)

position of a feature relative to the origin of a part coordinate system defined by, for

example, reference holes in the oil pan plane.

1.3 Tolerances on the Dimensions of Discrete Parts

As defined by U.S. ANSI Standard Y14.5

on "Geometric Dimensioning and

Tolerancing," a tolerance is a number given

in an engineering drawing which specifies

the total amount by which a dimension is permitted to vary [4]. Represented here by

±7", a tolerance defines a range of

acceptability of the magnitude of the

difference between the dimension of a partas-made (Dm) and the dimension of the

part-as-designed (Djy), that is, I D^ - D^ I

Figure 2 shows tolerance-versus-dimension

for selected examples of the wide variety of

outputs of the discrete -parts manufacturing

industry, spanning a range of dimensions

from 1 m to 10 nm and a range of

tolerances from 1 millimeter to 0.5

nanometers [5].

Tolerances vs Dimensions of

Current State-of-Art Mfd Parts

nm

'Normal'

Auto Door Ass'y

'Precisfon' Auto Piston Geometry

'UltraprecJslon'

O

/C Llnewlcl(hs

Mag Read-Write Heads

Te^ecommun '.Optical Fibers ^ Mlcropatteri Position

o Quantum Well Devices

Figure 2. Log-log plot of tolerance versus dimension for discrete-part products in

normal, precision, and ultraprecision regimes.

At the large-dimension end of the

precision-manufacturing spectrum is the state-of-the-art in automobile manufacture where

a 1 mm tolerance on the fit of 1 m size passenger-car door assemblies is achieved only by

the Japanese. Next, also in state-of-the-art automobile manufacturing, is the 7-8 nm
tolerance on metal-matrix composite pistons for a new-generation engine under development

by a leading Japanese car company. In the intermediate-dimension range is the 0.25 jim

accuracy of a slotting/slicing machine for use in the fabrication of the new-generation thinfilm magnetic read/write heads for computer memory systems. Toward the lower dimensions

are the 0.025 p,m (25 nm) tolerances on 125 nm diameter optical fibers for

telecommunications systems and the 0.010 jim (10 nm) tolerances representing current

demands in electron-beam micropatterning equipment in Japan. Finally at the smallestdimension end of the precision-manufacturing spectrum shown are 0.0005 jxm (0.5 nm)

variations on the 10 nm size dimensions of devices such as quantum-well lasers as well as quantum-wire and quantum-dot devices under development.

There is an ongoing trend toward dramatically tighter dimensional tolerances in

manufactured goods — first reported to the international production community in the early

1980's — which corresponds to a roughly factor-of-three decrease in the size of tolerances

every 10 years [6]. According to the projection of this trend, between 1980 and 2000 state- of-the-art tolerances will decrease: from 7.5 nm to 1 \im in the most-demanding "normal"

machine-tool-based production; from 0.075 nm to 0.01 p.m in the most-demanding

"precision," e.g., diamond turning machine production; and from 0.005 ^im (5 nm) to <

0.001 [im (< 1 nm) in the most-demanding atom-, electron-, or x-ray "machining"-based

production. It is the trend to such tightened tolerances, combined with the move in many

inspection applications towards use of computerized coordinate measuring machines, that

has given rise to major current metrological problems in measurement-based quality

assessment of precision-manufactured discrete-part products.

1.4 Problems in Assessment of Conformity to Tolerances

Shown schematically in Table 1 the principal aspects of the overall problem of determining

with a known degree of confidence whether the dimensions of a manufactured part conform

within specified tolerances in an engineering design. As indicated in Table 1, with each of

the functional tasks involved in design, fabrication, point-sample-based measurement, and

part evaluation, there is an associated variability which in the best situation is appropriately

quantified.

According to this scheme, a bottom-line evaluation of a part consists of the ability to state with confidence C that a part's material deviations D, known with measurement uncertainty

U, are within design-specified tolerances T, when the deviations have been determined by

means of a point-sample measuring machine which provides coordinate positions with errors X, of a number of points distributed at various locations on the surface of the part, when

those points are used in conjunction with a software implementation of a fitting algorithm

with associated variability Y.
Table 1. Principal tasks and their associated variabilities in comparing pointsample measurements of manufacturing deviations with design tolerances

Function Tasks Variability Symt

Design Choose Ideal Form

Specify Dimensions Design Tolerances T

Fabrication Choose Process

Shape Material Mfg Deviations D

Data-Taking Choose Number/Location of Points

Probe Surface at Those Points Coordinate Errors X

Computation Choose Fitting Algorithm

Compute Substitute Geometry Fitting Variation Y

Evaluation Calculate Manufacturing Deviations Uncertainty U

Decide If Within Tolerances Confidence C

1.4.1 Design: Mathematicization of Geometric Dimensions and Tolerances

In the part design phase of Table 1, the basic tasks are to choose an ideal geometric form

for the part and to specify its dimensions and tolerances. Current practice and standards

on geometric dimensioning and tolerancing, represented by ANSI Y14.5, deal with graphical,

engineering-drawing representation of geometry, dimensions, and tolerances. The first major

issue associated with the assessment of part manufacturing deviations relative to tolerances

by means of computerized point-sample coordinate measuring systems arises with the

tolerances themselves. Absent are the formal representations required for unambiguous

interpretation and computation [7]. The required "mathematicization" of dimensioning and

tolerancing standards is an area of active research underlying new standards [8]. Problems

include, for example: achievement of context-free definitions of concepts such as size, which

do not involve embedded notions such as diametrically-opposed points or fitted cross sections

[9]; and means for establishing simultaneously formal and common-sense bases for concepts

such as the various types of datum to which part features are referred [10].
1.4.2 Data Taking: CMM Measurements

Given that a manufacturing process has been chosen

which is likely to produce minimum deviations from

design values and the part fabricated, in the next step — that of data taking — the main tasks are to choose

the number and distribution of points about the part

surface at which coordinates are to be measured and

to actually probe the part. Two issues arise.

The first issue is appropriate selection of the number

and location of sample points on the surface of the

part being measured. Approaches include, for

example, equidistant, equiangular, and stratified.

Some standards provide guidelines for sampling

[11]. Active research on the design of sampling

strategies as a function, for example, of part geometry,

process-specific part form errors, and systematic and

random probing errors is being carried out as potential input to U.S. standardization on dimensional

measurement inspection techniques [12].

The second issue is taking into account the residual

measurement errors associated with each individual

probed point. Recent work has led to standard

techniques for characterizing these specific functional

aspects of coordinate measuring machines [13].

These residual errors, which are vectors of varying

direction and magnitude specific to each probed point, are complex functions of the coordinate measuring

machine geometry, location in its working volume,

type of probe, and direction of approach of the probe

to the part surface [14]. Ongoing research includes

that on the characterization of the errors of CMM

Figure 3. Lobing in response of

CMM probe which gives rise to

directional errors in coordinates of

points on part being measured (full

scale of top and bottom plots are 8

|im and 14 \im respectively).

probes which can have systematic, direction-dependent

errors such as the lobing shown in figure 3 [15], with an ideal probe having a circular, i.e., non-direction-dependent error pattern of zero radius. As can be inferred by comparing

figure 1 with figure 3, the cumulative effect of such lobed, direction-dependent errors on the

apparent dimension of a part being measured depends systematically upon which of the six

different types of geometric features is being probed, the number of points probed, and the

specific directions of approach to the part relative to the probe's lobing pattern.
1.4.3 Computation: Fit Criteria and Algorithm Testing

In the computation phase of the scheme of table 1, the basic tasks are to choose an

algorithm for achieving a best fit of the set of points measured on the surface of the part

and to compute the best-fit substitute geometric feature. Conventional fitting criteria

include, for example, least squares, mini-max, maximum inscribed and minimum

circumscribed. The issue here is that computed dimensions of the substitute feature,

including its size, location and orientation, depend on the substitute geometric element

selected, the best-fit criteria applied, and the software implementation of the fitting

algorithm — with no established theory of tolerances to rationalize the choice. The area is one of active research with implementations of algorithms based on specific criteria, such

as a minimum deviation zone, being evaluated against simulated measurement points for

specific substitute geometries such as cylinders [16]. Also under development are standardized tests for the evaluation of the ability of various substitute-geometry fitting

algorithms to detect particular types of form errors [17].

1.4.4 Evaluation: ]V[easurement Uncertainty and Confldence

Finally, in the evaluation phase of the assessment of part manufacturing deviations relative

to tolerances by means of computerized point-sample coordinate measuring systems as shown in table 1, there are the tasks of actually comparing the substitute geometry with the

design geometry to determine whether the part's manufacturing deviations are within

specified design tolerances, given the overall uncertainty in measurement. The conventional

approach in dealing with assessing manufacturing deviations relative to tolerances in the

presence of measurement uncertainty is to require that the accuracy of measurements be

some specified fraction of tolerances, typically one-fourth (as in the case of the U.S.

Department of Defense prescription) or one-tenth (as in the case of the widely applied "gage

maker's rule"). However, there is a major issue here in ascertaining conformance of

manufactured parts to design tolerances in that one does not know how to calculate the

confidence factors of the uncertainties in the overall measurement based on coordinate and

sampling errors [5].

The main issue is compounded by secondary issues. First is the absence of a statistically

sound and widely accepted measure of measurement uncertainty. At present there is a move away from the long-established system of characterizing uncertainty in terms of a random component (in terms of one-, two-, or three-standard deviations) and a systematic

component (in terms of an estimated residual after model-based systematics are removed)

long used in the United States [18] to a quadrature-addition, one-sigma representation

being pursued by the international metrology organizations [19], with the two approaches

seen to have more or less utility depending on the application [20]. In any case, the

question of calculating confidence in measurement uncertainty is still open.
1.5 Conclusion

Achievement of quality in design of discrete parts requires specification of tolerances on

manufacturing deviations which can be achieved to an expected degree of confidence. With

the press of tightening tolerances and the use of point-sampling coordinate measuring

machines, achievement of a known confidence is a problem. Solution of that problem

requires resolution of a wide range of issues involving mathematicization of dimensioning

and tolerancing standards, development of point sampling strategies, analysis of fitting

algorithms, testing of software implementations, and development of error propagation and

confidence estimation techniques. While aspects of the overall problem are being addressed

within the research and standards communities, much work remains to be done.

2 New Concepts of Precision Dimensional Measurement for Modem

Manufacturing

2.1 Introduction

A hallmark of modern products is the high precision of the dimensions of their functional

parts — high, that is, compared to that of their less-modern contemporaries. Since the

Industrial Revolution, such has been the case. Modern products of the last century included

certain factory-produced small arms, the precision of whose parts allowed them to be

interchangeably assembled in the then-new system of mass production. Modern products

of this century are video cassette recorders, the precision of whose parts allows read-write

heads to be aerodynamically flown over the recording medium at microscopic altitudes.

Since it is modern products which sell best, for U.S. manufacturing firms seeking to compete

better against evermore sophisticated foreign producers, the ability to realize precision

higher than that of their competitors is essential. Precision, however, is relative not only to

the capabilities of the competition, but also to the size of the functional parts in question.

Comparable degrees of high precision are a common end for a wide range of manufacturers,

from producers of the very large, such as commercial aircraft, to producers of the very small,

such as nanoelectronic devices.

This chapter presents a new scheme for the analysis of the dimensional capabilities of

measuring machines and machine tools aimed at the realization of higher degrees of

precision in the manufacture of dimensioned parts. Successive sections deal with the range

and types of dimensional measurements, their relation to the basic unit of measure, the

means for making such measurements, and the errors associated with them. Application of

these principles is illustrated by a detailed analysis of the use of a laser-based coordinate

measuring machine for the measurement of a physical part. Also included is an indication

of how the identical analysis may be applied to a machine tool which shapes such parts.
2.2 Scope of Dimensional Measurements

Manufacture of today's most modern products requires the ability to carry out precise

dimensional measurements over a wide scope, one which includes subtly different types and

astonishingly different scales. Dimensional measurements of all types and scales, however,

are expressed in terms of the basic unit of length.

2.2.1 The Range of Dimensional Measurements

The range over which precision dimensional measurements for the characterization of

manufactured goods are made today spans nearly 12 orders of magnitude, segmented into

relatively well recognized regimes.

2.2.1.1 The Macro-Scale

On the large-size end of the range are products which are not so much manufactured as constructed, that is, large-scale products often assembled on site rather than manufactured

on line. Macro-scale products such as ships, aircraft, and spacecraft can have longest

dimensions of the order of 100 meters.

2.2.1.2 The Mid-Scale

In the mid-size of the range are products the parts of which are manufactured and

assembled in the familiar factory environment, including a host of commonplace products

from automobiles to machine screws. Mid-scale products have characteristic dimensions of

the order of multiple- to fractional meters.

2.2.1.3 The Micro-Scale

On the conventional small-size end of the range are products which are manufactured within

special machines such as optical, electron-beam, and x-ray lithography systems, including less

familiar products such as ultra-large-scale-integration (ULSI) microelectronic devices.

Micro-scale products have characteristic dimensions often of less than a micrometer.

2.2.1.4 The Nano-Scale

Just emerging now is technology for the manufacture of products of the "nano-scale,"

including nanoelectronic (as opposed to microelectronic) transistor-type devices 100 times

smaller than those in present commercial production and mechanical devices such as ultraminiature pumps capable of implantation into blood vessels and automatically metering out

drugs. Nano-scale structures and devices have features composed of small numbers of atoms
and molecules and have characteristic dimensions of the order of nanometers (10"^ m, or 40

billionths of an inch). Achievement of high precision in dimensional measurements at the

nanometer scale demands utmost attention to physical-scientific, engineering, and

metrological principles [21].

2.2.2 Types of Dimensional Measurements

In the characterization of today's most modern manufactured products, precision

dimensional measurements are carried out not only over a wide range of dimensions but also over a range of types, each of which, while intimately related to the others, has a uniquely

distinct role in measurement.

2.2.2.1 Position

a. Geometrical Concept of Position: In geometry, position is the location of a point relative

to either: another point, such as the origin of a one-dimensional coordinate system; a line,

such as an axis of a two-dimensional coordinate system; or a plane, such as the reference

surface in a three-dimensional coordinate system. For a three-dimensional cartesian system,

the formally explicit expression for geometric position is a vector quantity:

p = Pixx,yy,zz) (1)

where x, y, and z represent the coordinates along unit vectors x, y, z relative to the

(suppressed null-vector) origin of the coordinate system.

b. Physical Concept of Position: (In physics, position is the location in space of a physical

body relative to a point of observation.) In the rigid-body approximation, where the

configuration of that body is fully specified by six generalized coordinates (three linear and

three rotational) corresponding to its six degrees of freedom, position is given by the three

linear coordinates. Note that since both a rigid body and a reference frame have the same

six degrees of freedom, locating a body relative to a reference frame is fully equivalent to

locating one rigid body relative to another. In a three-dimensional cartesian system, the

formally explicit expression for physical position is a vector quantity:

P = P{XX, YY, ZZ) (2)

where X, Y, and Z represent the coordinates of the location of one physical body relative

to the location of another physical body which acts as the reference frame of the coordinate

system.

c. Measurement Concept of Position: In measurements, position is the assigned numerical

value of the length of path between the single point which describes the location of a

10
physical body and the origin, axis, or plane of the one-, two-, or three-dimensional coordinate

system expressed in terms of the standard unit of measure. In a three-dimensional cartesian

system, the formally explicit expression for measurement position is a vector quantity:

P = P ( XX, YY, ZZ) (3)

where X, Y, and Z, expressed in meters, represent the coordinates of the location of one

physical body relative to the location of another physical body which acts as the reference

frame of the coordinate system. Measurements of position are made, for example, in

precision surveying of geodetic coordinates relative to control points and in measurements

of the locations of points on objects by means of coordinate measuring machines.

2.2.2.2 Displacement

a. Geometrical Concept of Displacement: In geometry, there is nothing fully equivalent to

displacement. However, the physical notion of displacement does involve the geometrical

notion of translation, where translation is the transformation of a point at one position into a point at a different position by means of the operation:

t = Pi-Po (4)

where t is the vector of translation and p^ and pf are the initial and final position vectors of

the point.

b. Physical Concept of Displacement'. In physics, displacement is the change in location in

space of a physical body with time, that is, its movement from one location to another, and

is described by:

T= Pf(tO -P,(t2), (5)

where T is the translation representing the change in position with time of a single body

from position i*o(ti) at time one to position Z'fCta) at time two.

c. Measurement Concept of Displacement: In measurements, displacement is the assigned

numerical value of the length of path between the location of a physical body at an initial time and its location at a final time expressed in terms of the standard unit of measure. In a three-dimensional cartesian system, the formally explicit expression for measured

displacement D, expressed in meters, would be a vector quantity:

D = Pf (to - P„ (to, (6)

11
where the difference between Po(ti) and Pf(t2) represents the change in location, between

times one and two, of one physical body relative to the location of another physical body

(which acts as the reference frame of the coordinate system). Measurements of

displacement are made, for example, in the characterization of the travel of a stage by

means of a laser interferometer, where the stationary parts of the interferometer comprise

the body which defines the reference frame.

2.2.2.3 Distance

a. Geometrical Concept of Distance: In geometry, distance is the separation between two

points. In a three-dimensional cartesian system, the formally explicit expression for

geometric distance would be a scalar quantity:

d = \ p^ (x^x, y^, z^z) - p2 {x^K, ynf, z^^ I

,

(7)

where x, y, and z represent the values of measured coordinates relative to the usually

suppressed null-vector origin of the coordinate system.

b. Physical Concept of Distance: In physics, distance is the separation in space of the

locations of two objects. In a three-dimensional cartesian system, the formally explicit

expression for physical distance would be a scalar quantity:

d = I P,(XyK Y,Y, Z,Z) - P2(X:^ Y^, Z-^) I

,

(8)

where d is the distance between two bodies located at positions P^ and P-i respectively.

c. Measurement Concept of Distance: In measurements, distance is the assigned numerical

value of the length of path between the location of one physical body and the location of

another, expressed in the standard unit of measure. A formally explicit expression for

measured distance, expressed in meters, would be a scalar quantity:

d = I P, (to - P2 (to I

,

(9)

where Pi(tO and P2(tO represent the locations of two different bodies at the same time.

Measurements of distance are made, for example, in the center-to-center separation of

engine cylinders and microelectronic circuit elements and in the calibration of the spacings

of the successive graduations of various types of what are usually called "length scales,"

ranging from hundred-meter-long survey tapes to micrometer-long "pitch" standards for

optical- and electron-microscope measurement systems.

12
2.2.2.4 Extension

a. Geometrical Concept ofExtension: In geometry, extension is the line segment between two

points which lie on a surface which is topologically equivalent to a sphere such that all points

on the line segment near the surface are either interior to or exterior to the surface.

Equivalently, extension is the line segment connecting two points on a surface which have

exterior normals with components that are anti-parallel. A formally explicit expression for

physical extension e is: e = \ rbi(i|r)-r,2(i|r + 180<') I, (10)

where rbi(ijr) and rb2(i|r + 180') represent the locations of points on a surface having exterior

normals 180* apart.

b. Physical Concept of Extension: In physics, extension is the amount of physical space that

is occupied by, or enclosed by, a material object. A formally explicit expression for physical

extension is: E = \ /?bi(i|r)-/?b2(ik + 180^) I, (11)

where /?bi(^) and /?b2(^ + 180*) represent respectively the locations of the opposite-facing

boundaries of the physical object which, for convenience, may be expressed in terms of a vector radius from the centroid of the body.

c. Measurement Concept ofExtension: In measurements, extension is the assigned numerical

value of the length of path between the location of one boundary of a material object

relative to the location of another opposite-facing boundary, expressed in terms of the

standard unit of measure. A formal expression for measured extension is:

e = l Pbi(i|r)-Pb2(1r + 180'') I (12)

where e is the extension and Pbi and Pb2 represent the locations of opposite-facing

boundaries of an object. Extension-type dimensions of objects include height, width,

diameter, and also "length." Measurements of extension are made in general to achieve

physical fit between mating parts. Prototypical examples of extension measurements are those in the automotive industry wherein the outside diameter of piston rings and the inside

diameter cylinder walls are measured to insure slip fit. For analogous reasons, extension

measurements are also made in the microelectronics industry on critical elements of

microcircuit devices. Table 2 below summarizes key aspects of the various types of

dimensional measurements just described.

13
Table 2. Description of tlie quantity and nature of the length of path to be measured for each of the four dimensional measurement

types

Position

Dimension Quantity The Length of Path Is That Between:

Vector P The location in space of a single object and an origin

of coordinates (equivalent to a second, reference object)

Displacement Vector D The change in location of one object from time tj to time

Distance Scalar d The locations of two objects

Extension Scalar e The locations of opposite-facing boundaries of an object

In familiar terms, each type of "length" relates to a different, but interrelated, concern. For

example, the "length" of an automobile (i.e., its extension) describes how big a parking space

is needed, the "length" of a racetrack (i.e., its distance) describes how far it is between start and finish lines, and the "length" of a stage's travel (i.e., its displacement) describes how far

its carriage can move.

2.3 Basis of Dimensional Measurements

What "lengths" of all scales and types have in common is that each is measured and

expressed in terms of a unit of length. For that unit to be the internationally standard unit

of length, a formal definition must be operationally realized and transferred.

2.3.1 Realization of the Unit of Length

The function of a unit of length is to provide a metric for the physical space in which

dimensional measurements are to be made. The function of a standard unit of length is to

provide a metric which can be used in common in international science and trade.

14
2.3.1.1 Formal Definition of the Meter

By treaty, the standard of length used in international science and trade is the meter. The

meter is part of the International System of Units (SI) and is referred to as the fundamental,

or SI, unit of length. Since 1983, the formal SI definition of the meter as the unit of length

has been:

'T/ie meter is the length ofpath travelled by light in vacuum in the time interval

1 I 299 792 458 of a second" [22].

No longer based on lines scribed on a platinum-iridium "meter" bar (as it was from 1887

until 1960), the meter today is the distance which light travels in free space in

(approximately) three billionths of a second.

2.3.1.2 Derived Nature of the Meter

The new definition of the meter makes it in effect a derived quantity governed by the lengthtime relation:

L = c-T, (13)

where L is the defined unit of length, c is the defined constant speed of light, and T is the

defined unit of time. The second, the SI unit of time, is defined as a fixed number of counts

of the frequency of oscillation of specific atomic states of a cesium-beam atomic clock.

2.3.1.3 Operational Realization of the Meter

The meter is operationally realized through a chain of intercomparisons of the frequencies

of various microwave-, infrared-, and visible -wavelength lasers, based on the relationship of

the frequency of electromagnetic waves and their wavelength:

f • Ao = c , (14)

where f is frequency, X^ is vacuum wavelength, and c is value of the speed of light (that is, 2.99792458 • 10* m/sec). Based on this relationship, the ultimate limit on the uncertainty

associated with the realization of the meter is that of the cesium clock, which is 10^^ At

present, for dimensional measurements access in principle to the frequency cesium-beam

atomic clock is through a chain of intercomparisons of wavelengths of microwave, infrared

and visible sources by which the frequency of visible-wavelength lasers is determined.

15
2.3.2 Transfer of the Metric to Physical Objects

In practical dimensional measurements, the means by which the SI unit of length as the

metric of physical space is transferred to measurements of physical objects ~ from

automobile piston rings to microelectronic circuit elements ~ is through a system which

successively involves visible light of known wavelength, displacement interferometry using

that light, coordinate measurement systems based on that interferometry, and measurements

on physical objects using such coordinate measurement systems.

2.3.2.1 Visible Light of Known Wavelength

For dimensional measurements, the practical primary standard of length today is the vacuum

wavelength of the 633-nm red-orange line of an iodine-absorption-stabilized helium-neon

laser. One may realize the Sl-unit-based wavelength, to a particular accuracy, by buying a commercial helium-neon laser stabilized to a corresponding degree.

2.3.2.2 Displacement Interferometry

The means by which propagating laser light, which defines the metric of physical space, is coupled to dimensional measurements in that space is through interferometric measurements

of displacement. The prototypical displacement interferometer is the two-beam, Michelson

interferometer. The Michelson interferometer consists of a laser light source, beam splitter,

fixed reflector (either a plane mirror or retroreflector), moving reflector and detector. In this

type of interferometer the translation of the moveable reflector, which must be parallel to

the axis of propagation of the laser light, is measured in terms of whole and fractional

numbers of fringes generated by the interference of the beams in the stationary and moving

arms of the interferometer.

The typical displacement interferometer of today employs the heterodyne principal for

measuring the change in optical phase between the two beams by having one frequencyshifted with respect to the other. A widely used commercial heterodyne interferometer

employs a two-frequency laser and the displacement of a plane mirror or retroreflector is measured by means of two channels of detectors, doublers, counters, subtracters and a

calculator [23].

By whatever means the phase difference between the reference and moving-mirror beams

is measured, the operational link between the Sl-unit-carrying wavelength of visible light and

a displacement measurement is through the interferometer equation:

D = Li - L2 = NXo / 2n, • cosG, (15)

16
where D is the displacement, L, - L2 is the change in length (i.e., extension) of the

interferometer cavity, X^ is the vacuum wavelength of the laser light used, N is the real

number describing the counted integer-order and measured fringe-fraction of the

interference, n^ is the index of refraction of the medium, and 6 is the angle between the

optical axis of the interferometer and the direction of propagation of the incident light. In

multi-pass fringe-fractioning interferometers:

D = MAo / 2 nm K • cosO, (16)

where M is the counted integer-order and measured fringe -fraction of the interference, m

is the number of passes of the light through the interferometer, and K is the number of

electronic sub-divisions of the interference fringes (typically a power of two ranging from 16

to 1024 depending on the system).

For the high-resolution commercial heterodyne interferometer system described above, the

limit of resolution, that is, the least count in displacement measurement is about k /400 or

1.5 nm. Under development now at NIST is a specially optimized heterodyne interferometer

intended to realize a least count of 20 times higher, that is, A, /8000 or 0.07 nm.

Note that, as described here, in a laser displacement-interferometry system, it is the value

of the vacuum wavelength of the laser light which provides the metric of physical space as the fundamental unit of length. It is the overall interferometer system doing the subdivision

of fringes which provides the "scale," that is, the metric-less subdivisions of that unit. In

common usage, however, displacement-measuring devices, such as interferometer systems, are often called "scales" and perform both of the functions described here: subdividing the axes (that is, providing the scale for an axis) and carrying the metric (that is, providing the

standard unit of length, relative to which the "scales" have somehow been appropriately

calibrated).

2.3.2.3 Coordinate Measurement System

After the wavelength of laser light and displacement interferometry, the next link to

dimensional measurements of physical objects is a coordinate measurement system. A

coordinate measurement system is necessary for the measurement of "position," that is, the

location of a single point in space.

To establish a coordinate system, one must establish the means for realizing the geometry

for physical space, including: axes (one, two, or three depending on the dimensionality of the

system desired; an origin (the null-vector point from which radiate the axes of the coordinate

system); a scale along each of the axes (that is, the graduation of each axis into subdivisions);

and a metric (that is, a single measure of distance in the space of the reference frame to

which the graduations of the axes relate).

17
In modern, automation-oriented manufacturing, the means by which the geometry of the

coordinate measurement system is realized and precision dimensional measurements are made, is the coordinate measuring machine (CMM). A coordinate measuring machine

embodies a coordinate system by means of its essential functional components, which are

its frame, scales, carriages and probes as outlined in table 3.

Today's high-performance three-dimensional coordinate measuring machines take on a wide

variety of forms depending on the specific way in which each of these functional components

is embodied. Frames can be of the bridge or cantilever-arm type. Scales can be ruled-glass

optical encoders or enclosed-path laser interferometers. Carriages can be translating and

rotating tables and moving bridges. Probes can be touch-fire or analog mechanical-contact

types or capacitive, optical, or machine-vision non-mechanical-contact types.

Table 3. The principal functional components of a coordinate measuring machine

(as well as of a machine tool)

Frame The frame is the geometry-generating structure, which is the means

for embodying the origin, axes, and angular relationships which

comprise a coordinate system.

Scales The scales are the displacement-measuring devices, which are the

means for realizing the metric-based graduations along each axis of

the frame.

Probe The probe is the sensor system, which is the means for linking the

boundary of the object to be measured to the frame and scales; in a machine tool, the "probe" is a material-moving element rather than

a material-locating element as it is in a measuring machine.

Carriage The carriage is the complete motion-generating system, which is the

means for translating the object to be measured relative to the

probe, frame, and scales.

Whatever particular form a coordinate measuring machine takes on, its fundamental function

is to provide measurements of locations in space, that is, of position. However, just as a manufacturer wants to buy holes not drills, users of CMM's in manufacturing want

measurements of dimensions of objects, such as those of their products, not coordinates of

locations in space.

18
2.3.2.4 Physical Object to be Measured

The terminus of the series of linkages, which allow dimensional measurements on physical

objects in terms of the SI unit of length, is the object itself. Now there are two different

kinds of features on objects corresponding respectively to the distance-type and the

extension-type of dimensional measurements described above.

a. Distance-Type Features ofObjects'. Distance-type features, include, for example, the centerto-center spacings of cylinders of an automobile or spacing of left-edges of successive

microcircuit elements. While, in fact, always to some degree extended, distance-type features are treated as extensionless points. Thus, the distance between automobile cylinders is measured in terms of their centroids. Similarly, the distance between the moon and the

earth is center to center. Measurements of distance type features are inherently, in effect,

point to point.

b. Extension-Type Features of Objects: Extension-type features include, for example, the

diameters of the automobile cylinders or the widths of the microcircuit elements. This type

of feature must be treated as made up of boundaries that must be approached from

opposite directions as in calipering, in which one locates two points, one on each and

measures the "distance" between them. However, it is in the essential nature of extension

measurements that these boundaries are opposite facing. As will be seen below, it is the

practical and theoretical difficulty in locating properly such boundaries that makes extension

the most difficult type of dimensional measurement.

In sum, successful measurement of, for example, the diameter of an automobile cyhnder or

the width of a microelectronic element in terms of the SI unit of length requires dimensional

measurements of each type in succession: displacement, position, distance, and, finally,

extension. The measure of success in this succession of measurements at each level and in

total is accuracy achieved.

2.3.3 Standards-Imposed Limits to Accuracy

The accuracy in practical dimensional measurements depends on the availability of standards

for the calibration of the measuring machines by which measurements are made. Attainable

accuracy is ultimately limited by national standards laboratories in measurements involving a realization of the meter, displacement interferometry, reference coordinate measuring

machines, and calibrated artifacts.

19
2.3.3.1 Limit of Realization of the Meter

The ultimate limit to accuracy in practical dimensional measurements is the accuracy with

which national standards laboratories can realize the definition of the meter. At present that

limit is the uncertainty with which the visible wavelength of the iodine-stabilized HeNe laser

is known, that is, 10 ' compared to the 10-^^ of the atomic clock. The meter as the unit-oflength metric of that space can be accessed directly then through the vacuum wavelength of

the HeNe laser. Table 4 below shows the effective wavelength and uncertainty associated

with HeNe lasers stabilized by various means.

Table 4. Comparison of the value and associated fractional uncertainty of the

vacuum wavelength of helium-neon lasers of various degrees of

stabilization

Wavelength Value

632. 991 nm

632. 9914

632. 99139

632. 991398

Uncertainty

10 -^^ 10

-^ 10-8

10-'

Type of Stabilization

Free-running

Any one of a number of

opto-electronic techniques

Iodine Absorption Cell

As indicated in table 4, the fractional uncertainties ( AX/A, ) associated with commercially

available HeNe lasers range from the 10"^ of a free-running device, through the 10"'' specified

stability of certain commercial lasers, and lO"* short-term stability of some such devices, up

to the 10"^ accuracy of the visible wavelength of the iodine-stabilized laser as defined in the

documents associated with the definition of the meter and realized in laboratory devices.

2.3.3.2 Limit of Displacement Interferometry

After the ultimate limit to accuracy of dimensional measurements imposed by the realization

of the meter in terms of the wavelength of visible light, the next limit to that accuracy is the

degree to which measurements of displacement can be made by means of optical-wavelength

interferometry.

Based on work by national standards laboratories, the limit to the accuracy of interferometric

displacement measurements at a displacement of a meter is estimated to be: lO-'' in air

uncompensated for pressure, temperature, and humidity; 10'' in fully compensated ambient

air; and 10^ in standard dry air. Based on results by a variety of workers, a reasonable

estimate for the practical limit for displacement interferometry in vacuum is lO"^**.

20
2.3.3.3 Limit of Coordinate Measurement

The next factor to limit the accuracy of dimensional measurements after the realization of

the meter and displacement interferometry is the ability to embody a coordinate system in a measuring machine. At present, the highest-accuracy coordinate measurement is achieved

by national standards laboratories on special-purpose one- and two-axis measuring machines.

At the United States National Institute of Standards and Technology, a machine for the

calibration of line scales over the range 1 jim to 1 m has associated v^ith it a total

uncertainty of measurement Ux given by:

Ut = 10 nm -h 10-^ • s,

,

(17)

where s^ is the separation of any two graduations on the scale. At one meter, the positional error of this machine is approximately I-IO"^ [24]. For a two-axis machine used for the

calibration of grid plates up to 600 mm x 600 mm, the corresponding positional error in

routine use is 0.5 jim, corresponding to about 5 -10 "^ [25].

Currently unmet is the need in the microelectronic industry for an accuracy of 50nm

accuracy over 250 mm, that is, 2-10 "^ [26]. Because of a trend to even more stringent

requirements, for the regime of nanotechnology, there is under development at NIST a "Molecular Measuring Machine"[27], which is a planar 50-by-50 mm xy-coordinate

measuring machine (with a 100 jim z-axis) which has a design-goal positional capability of:

Ut = 1 nm = lO^ @ 70 mm. (18)

2.3.3.4 Limit of Physical Artifact Standards

The limiting error in many practical dimensional measurements is the accuracy of artifact

standards (that is, physical objects), dimensions of which have been calibrated by a national

standards laboratory or by a secondary laboratory referenced to it. Accuracies

representative of various types of artifact standards calibrated by NIST are shown in the

table 5 below.

Note that because of their idiosyncratic nature and the intractability of dealing with the very

object-specific boundary-location errors, NIST calibrates extension-type objects, such as integrated-circuit photomask linewidths [28], only when the national interest justifies the

major long-term theoretical and experimental effort required.

21
Table 5. The type, range, and accuracies of some dimensional standards calibrated

by the National Institute of Standards and Technology

Dimensional Std Type Range Accuracy

Survey Tapes d 1 mm to 300 m 5.0 ^m + lO** • d

Gauge Blocks d 1 mm to 0,5 m 0.025 Jim to 0.15 nm

Line Scales d 1 ^im to 1 m 0.01 Jim + 10-^ • d

Photomask Lines e 0.5 nm to 20 nm 0.05 Jim

Polymer Spheres e 0.3 ^m to 3 ^m 0.01 Jim

Thin-Film Steps d 0.02jim to 10 ^m 0.003 Jim to 0.2 jim

Note also that while NIST provides measurement services for calibrations of distance and

extension (which involve associating the unit of length with locations on physical objects), it does not calibrate displacement or position (which involve associating the unit of length with

a device-based process [29].

2.4 Assessing Dimensional Measurements

As suggested earlier, manufacture of modern products demands dimensional measurements

of not only exceptional precision, that is, the closeness together, but also of exceptional

accuracy, that is, closeness to a true value [30].

Especially in high-technology products including, for example, microwave resonators for

communication systems and x-ray optics for microelectronic lithography, accuracy is required

because the dimension of device features is dictated not just by design convention but by the

laws of physics. As a result, dimensions of features must not only be very close to a value

but to the value. This section looks at the assessment of dimensional measurements in terms

of statistical characterization of error, formal-theoretical characterization of error, compounding of error, and standards-imposed limits of error.

22
2.4.1 Statistical Characterization of Error

Physical measurement is itself a process, like manufacturing, which produces output. The

output of a measurement process is numbers, that is, the measurement results. The quality

of these numbers can be assessed and characterized by means of measurement statistics in

terms of precision and accuracy. The manufacture of modern products demands

dimensional measurements that are of both high precision and high accuracy, however one

chooses to describe that combined state.

2.4.1.1 Precision vs. Accuracy

a. Precision: As a concept, precision conveys the notion of closeness together. As such, it

is the principal figure of merit for the achievement of higher quality in the sense of lower

variability. Precision is conventionally represented numerically in terms of the standard

deviation from the mean for a number of measurements presumed to be randomly drawn

from a large population describable by a Gaussian distribution function. The standard

deviation for a single such measurement a, is given by:

ai = [ Si (H, - LfKn - 1) V^, (19)

where C„ = Sj^n and the summations Ej are over the number of measurements n. Where

Oj is a measure of the precision of one measurement, o^ is a measure of the precision of the

mean of n measurements and is given by:

a^ = gJ n'^. (20)

Precision is given in terms of either one-, two-, or three-sigma values for single

measurements, corresponding respectively to confidences of 68.3%, 95.5%, and 99.7%.

Historically national standards laboratories typically have used 3o^ for characterizing

dimensional standards which they calibrate, while instrument manufacturers often used 2a^

for characterizing dimensional-measurement devices which they sell.

b. Accuracy: As a concept, accuracy conveys the notion of closeness to a true value. As

such, accuracy is the principal figure of merit for the achievement of interchangeability of

components and physical operation in high-technology devices. In those situations, actual

not relative dimensions determine functional performance. Accuracy is conventionally

represented numerically in terms of an estimate of the maximum that a measurement would

likely be different from a true or standard value. In metrological terms, accuracy is the

estimate of the degree to which measurements are free from systematic error and is represented by the term s.e.

23
2.4.1.2 Error or Total Uncertainty

The metrological term for a combined measure of precision and accuracy is uncertainty or

error. Total uncertainty includes a fully-described estimate of systematic error and fully

described measure of precision. NIST, along with a number of the national standards

laboratories of other countries, up until the present have typically used the three-sigma of

the mean in the assignment of a total uncertainty given by Uj = s.e. + 3a^.

Obviously, informed judgment must be made in how to sum and combine systematic and

random error contributions. For example, the magnitudes of systematic errors may be added

algebraically, that is, as signed quantities, if the signs are definitely known or arithmetically,

that is, without signs, if not. Random errors may be added in quadrature if known to be

uncorrelated or arithmetically if not. Throughout this chapter, the term "error" is used

synonymously with uncertainty Uj as described above and is denoted by the Greek character

delta, upper case (A) for total-type and lower (6) for type-specific K

2.4.1.3 Additive-Multiplicative Representation

Dimensional-measurement errors from different sources can be either additive in nature,

that is, appearing as incremental off-set, or they may be multiplicative, that is, appearing as a length proportional. Total error can be of the form:

AC = A + B • C, (22)

where A is the additive error and B the multiplicative.

^ Note Added at Time of This Report: The approach to the expression of uncertainty

recommended by the International Bureau of Weights and Measures (BIPM) which has been

recently adopted by major international standards organizations is to: associate with each

component of uncertainty a standard deviation of either Type A, evaluated by statistical

procedures, or Type B, evaluated by other means; combine these uncertainty components

in a sum of squares; and, if desired, to multiply by a coverage factor k, typically having a

value from 2 to 3. As described in the appendix of section 4, the U.S. National Institute of

Standards and Technology has now adopted the BIPM approach. Unless documented user's

needs dictate otherwise, the "accuracy" of NIST measurements will be given on a 2o basis,

i.e., the estimated uncertainty will be a combined uncertainty with a coverage factor of 2.

24
a. Added and Multiplicative Components: Given a system in which errors from a number of sources are to be combined to estimate the total error in a length-based dimensional

measurement of a given type, then:

AC = (S6(!^) + (SSdBi) • C, (23)

where A now is the sum Ej over individual additive contributions SC^i and B the sum Sj over

individual multiplicative contributions hi^^, where Aj and Bj can be comprised of either or both systematic errors and random errors.

b. Manufacturers* Statements of Accuracy: Many commercial producers of measuring

machines conventionally quote an accuracy for the performance of their products in additivemultiplicative form, as in, for example:

AL = 2 Jim + [ L(mm)/500mm ] jim, (24)

which means that over the 1 m-travel of the machine, the accuracy of single measurements

of position would range from a minimum of 2 nm to a maximum of 4 jim.

2.4.2 Formal-Theoretical Characterization of Error

Simple assessments and statements of accuracy which put errors into the simple additivemultiplicative form lump two types of additive errors, constant off-sets and non-linearities,

and provide an inadequate basis for identification of the sources of error. For diagnostic

purposes, the better approach is to consider the most general case, that of a dimensional

measuring system that produces output which includes linear and non-linear terms in the

form:

(!,bs = a -h (1-f-b) •(!. + c( (!,"), (25)

where Cobs is the measured value, C, is the true length, a and b are constants and c(C,") is the

sum of all non-linear terms. The error in measured values AC is then given by:

AC = Cob, - C. = a -H b • C, -h c(C,"). (26)

Each of the error components a, b, and c describes a specific type of error source and is a measure of the degree to which the measurement system has realized a fundamental

requirement of measurement.

25
2.4.2.1 Fundamental Axioms of Measurement

For dimensional measurements to yield physically meaningful results, they must conform to

axiomatically fundamental requirements given in the Philosophical Foundations of Physics by

Rudolf Carnap [31] and provided a more accessible account by Simpson [32]. These

axiomatic requirements are that a measuring system must be able to rank order objects

along the dimension of measurement and to reproduce properly the unit, the zero, and the

scale of that dimension.

a. The Rank-Order Operator: The rank-order operator is the overall procedure by which

objects or processes being measured are ordered and assigned values of the quantity being

measured. In length-based dimensional measurements, the ordering operator is equivalent

to a comparison of two objects involving, in effect, a translation in physical space. Each type

of dimensional measurement, however, requires its owm particular realization of that

ordering operator. As a result, displacement, position, distance, and extension each

constitutes its own measurement dimension which must be operationally tied to the

dimension of physical space.

b. The Unit: The unit is the rank-order greater of two objects or states of the physical

system assigned a defining numerical value of the quantity being measured. The unit is one

of the two points required to specify what is axiomatically a linear system. In length-based

dimensional measurements which conform to international standards, the unit is the SI unit

of length. In practical dimensional measurements today, the unit of length is an

internationally accepted value for a well-defined wavelength of an iodine-stabilized HeNe

laser as discussed above in section 2.3.1.

c. The Zero: The zero is the rank-order lesser of the two objects or states of the physical

system assigned a defining numerical value of the quantity being measured. In effect, the zero is the other of the two points required to specify the linear system. Implied by the

definition of the unit of length is that the zero of measurement has an assigned numerical

value of zero (that is, I C^ I = 0) and that the zero of length of dimensional measurements

is the length of the path in space which light traverses in the zero interval of time.

In practical dimensional measurements, the zero corresponds to the physical zero-vector

origin of coordinates which must be practically realized in each of the different dimensional

types. For example, in laser interferometry, the zero of displacement is the initial opticalpath difference between the reference beam and moving-mirror beam, one which must

remain constant during subsequent translation of the mirror in order to prevent loss of origin

of displacement. In coordinate measuring machines, the zero of position is the initial location of the tip of the probe relative to the interferometer reflector, which must remain

constant as the probe is moved about to prevent loss of origin of position.

26
d. The Scale'. The scale is the set of equal graduations into which the difference between

the unit and the zero is divided. In effect, the scale is the condition of linearity under which

differences in dimensions are equal, that is, the difference between one pair of objects is the

same as the difference between another pair. Operationally, the scale is the means for

measuring in increments smaller than the unit.

In practical dimensional measurements, the scale is the combination of a physical transducer

and a linearization algorithm by which linear interpolations between known points can be

carried out. For example, in displacement interferometry, the scale is generated by

interpolating between fringes using an assumed sinusoidal function of waveforms for the

linearization algorithm. Non-linear transducers such as capacitance gauges - when used in

conjunction with suitable mathematical models of their responses ~ can also be used to

generate the required linear scale.

In sum, the axioms of measurement define the requisite elements of a measurement system,

each of which must be operationally realized to achieve meaningful results. The first axiom — the rank-order operation ~ defines the dimension of measurement, such as temperature,

time, voltage, or, in this case, each of the four types of length. The other three axioms are the means by which numerical values are associated with measurements of that dimension,

table 6 below shows explicitly the form of the axiom errors for each of the dimensional

types.

Table 6. The matrix of errors by measurement axiom and dimensional type in general

form

Dimensional Error of Error of Error of

Type the Zero the Unit the Scale

Displacement ao bo-D Cd(D")

Position ap bp -P Cp(P")

Distance ac bd -d Cd(d")

Extension ae b, -e Ce(e")

27
2.4.2.2 Analysis of Error in Terms of the Axioms

Associated with each of the latter three axioms of measurement is a specific type of error with specific, fixable causes. Errors of the unit, the zero, and the scale correspond

respectively to the terms in eq (26) with a equal to the error of the zero, h to the error of

the unit, and c(f,") to the error of the scale, that is, the sum of all non-linear terms.

a. Error of the Zero: Errors of the zero occur for measurement systems which have a properly linear relation of measured value to true value, {obs to {„ but have a non-zero value

at C, = 0, that is:

Cobs = C. + a (27)

and ACz = a.

An error of the zero arises, for example, due to a shift in the origin of coordinates in the course of a dimensional measurement.

b. Error ofthe Unit: Errors of the unit occur for measurement systems which - while having

a properly linear relation of measured value to true value ~ have a non-unitary slope, that

is:

hy>. =(l+b) -C. (28)

and Afu = b • Cf

Error of the unit corresponds, for example, to an erroneous value of wavelength in

displacement interferometry or a constant misalignment of coordinate and object axes.

c. Error of the Scale: Errors of the scale occur for measurement systems which have a nonlinear relation of measured value to true value, that is:

AC, = c(V). (29)

Error of the scale represents failure of the interpolation scheme to divide the difference

between the unit and the zero into strictly equal intervals, that is, the degree to which the

system is nonlinear. The measurement system is linear, if given four objects (i.e., four

displacements, positions, distances, or extensions) for which the true difference between pairs

is the same, the measured differences are also the same. Error of the scale is, by definition,

the inequality in those differences, that is:

A, = ( Cobsl - <!ob,2) - ( <!ob»3 - fob.4 ) (30)

28
when ( dn - i!,2 ) = ( ^,3 - ^.2 ) •

That the error of the scale corresponds to a non-zero value of the coefficient c(C,'') can be

seen when the non-linear response of a dimensional measurement system is given a specific

form, such as the quadratic:

U. = a + (1+b) •{, + c(!.2. (31)

Evaluation of eq (30) at four values of f„ denoting the differences in the pairs of true values

by Aj, and substitution into eq (26) leads to the result:

A(!. = c • ({., - y • Ai

(32)

An error of the scale arises, for example, from to faulty interpolation of fringes in

interferometry or uncompensated nonlinearities in LVDT or capacitive displacement

measuring devices.

d. Errors ofZero, Unity and Scale: In summary, the axiom-specific errors for the general case

that Cobs = a + (1+b) • {, + c(C,'') are given by:

AC = AC, -I- A(!„ -h A(!, = a + b • C, + c(C,'>) (33)

where: a corresponds to A{„ the error in realizing the zero; b corresponds to Af^, the error

in realizing the unit; c corresponds to AC,, the error in realizing the scale; and H is any one

of the four dimensional measurement types.

2.5 Principal Specific Sources of Errors

Errors in dimensional measurements are particular to each of the different types and, since

each succeeding type is dependent on that which precedes it, total error increases as one

proceeds from the displacement, position, and distance to the extension of objects.

2.5.1 Error in Displacement Interferometry

The type of dimensional measurement most directly linked to the SI unit of length and the

basis for all subsequent types of dimensional measurements is displacement by laser

interferometry, that is, the measurement of a linear change of location in space with time

of a single object, in this case, the moving mirror of the interferometer. The measure of the

limit of this ability is displacement uncertainty or error, 6D. Errors in measurement of

displacement by laser interferometry are associated with each of the terms in eq (15).

29
2.5.1.1 Error of Wavelength in Vacuum Xo

The uncertainty in the wavelength in vacuum of the laser light depends on the type of

stabilization used in its construction. As indicated in table 4 above, uncertainty in the

vacuum wavelength of the 633 nm red-orange line of the helium-neon (HeNe) lasers widely

used in dimensional metrology range from 10"^ for unstabilized lasers down to 10"^° for

iodine-stabilized ones.

2.5.1.2 Error of Index of Refraction of Medium n

The uncertainty in the wavelength in medium of the laser light depends on the index of

refraction of the medium through which the light propagating in the interferometer passes.

a. Error of the Index of Ambient Air: Since index of refraction of a gas is a function of its temperature, pressure, humidity, and composition, without compensation for actual variations

in those parameters the upper limit of uncertainty for interferometric displacement

measurements in ordinary ambient air can be large: for example, an index error of 10"* would result from any one of the following variations: a 1 degree Centigrade change in

temperature, a 0.33 kPa (2.5 mm Hg) change in atmospheric pressure, or an 80% change

in relative humidity [33].

b. Error of the Index of Compensated Standard Air: Errors associated with the index of

refraction of the ambient medium can be reduced by compensation achieved by

measurement of the index of the actual ambient air or by calculation of the index of

standard dry air. The Edlen formula is an internationally agreed upon equation for the

calculation of the index of refraction of standard air as a function of wavelength, air

temperature, air pressure, and relative humidity [34].

With compensation, the lower limit of uncertainty for practical laser displacement

interferometry in ambient air is estimated to be about 1.2 • 10'^ [35]. The uncertainty in

the Edlen formula itself for the index of refraction of standard dry air is estimated to be 5

• 10® [36]. Operation of an interferometer in vacuum eliminates this source of error.

c. Error of Index of Refraction in the Dead Path: So-called "dead path" error is due to

improper compensation for the difference in the lengths of the optical paths of the two

interfering beams at the zero-displacement position of the system when index-altering

environmental changes occur during the course of the displacement measurement. For

commercial systems which use sensor-based index compensation, dead-path errors are largely

compensated for in software, leaving a residual error of 1.4 • 10'' times the dead-path

distance.

30
2.5.1.3 Error in Fringe Fractioning

Errors in realizing a scale by means of displacement interferometry arise from various sources in the process of the generation, subdivision, and counting of the interference

fringes. Representative of such errors are those which arise in widely-used displacement

interferometers of the polarization beam-splitting type. The principal sources of error in this

type of system are those associated with electronic subdivision, polarization mixing, thermal

drift, and dead path [37].

a. Error in Electronic Subdivision: The inherent half-wavelength resolution of

interferometers, corresponding to the spacing of the alternating light and dark of the fringes, can be extended by a variety of electronic fringe-interpolation schemes, each of which has

its own limiting resolution which contributes an additive, least-count error. Operating at 633

nm, certain commercial polarization heterodyne interferometer systems have least counts in

electronic-subdivision, depending on whether used with retro-reflectors or plane mirrors, of

A,/32 (approximately 20 nm) and X/64 (approximately 10 nm) respectively.

b. Error From Polarization Mixing: Less-than-perfect separation of the polarization states of the interfering beams in polarization-type interferometers, due to leakage of one

component into the other, produces a non-linear error in displacement measurement with

such systems. Such error varies as a function of change in optical path with a periodicity of

the wavelength of the laser source with an amplitude specific to an individual

interferometers. For one commercial linear interferometer system, the peak-to-peak phase

error was found to be 5.4'' corresponding to approximately 5 nm.

c. Error From Thermal Drift: Changes in temperature within the optical components of the

interferometer system, which produce a differential change in path optical length between

the interfering beams, give rise to drift type errors in displacement measurement. This

environmentally induced error is less for temperature-controlled systems and those which are

specifically designed to deal with this source of error. An example of the latter has a quoted

thermal error of 40 nm/C, which is 12 times better than a conventional plane-mirror system

with its 0.5 jim/C°[12].

2.5.1.4 Error in Alignment of Interferometer

Finally, constant angular misalignment between the interferometer and the incident light

gives rise to an error (derived from the basic interferometer equation):

6De = D • (-672), (34)

where 6De is misalignment contribution to the displacement error, D is the measured

displacement, and 66 is the small angle of misalignment between the axis of interferometer

31
(the normal to the parallel faces of the interferometer mirrors) and the axis of propagation

of the incident laser light.

In sum, this section has looked at sources of errors in the type of dimensional measurement

most directly linked to the SI unit of length, those of laser displacement interferometry by

the best available form of that technique, polarization-type heterodyne interferometry.

2.5.2 Error in Coordinate-Position Measurement

After displacement, the type of dimensional measurement next most directly linked to the

SI unit of length and inherent in subsequent types is position, that is, measurement of the

location of a single point relative to a coordinate system. The measure of the limit of the

ability to measure position is position error, 6P. Errors in position measurements are

associated with each of the functional components of the coordinate measurement system

as embodied by a coordinate measuring machine: the scales, frame, carriage, and probe.

2.5.2.1 Error in Relation to Displacement Scales

Lack of commonality of origin and co-linearity of axes of the coordinate-position

measurement system and the displacement system, which supplies the metric and scale

subdivisions, gives rise to two types of errors specific to positional measurement.

a. Misalignment of Mirror Translation'. Constant angular misalignment between the axis of

the displacement measurement and the axis of the coordinate system gives rise to "cosine

error," which is governed by the relationship:

P = D / cos a (35)

where D is the displacement, P is the coordinate position being assigned by the

measurement and a is the angle of misalignment between the two. For a displacement

generated by an interferometer system, a is the angle between the axis of translation of the

moving mirror and the optical axis of the interferometer (i.e., the normal to the parallel

faces of the reference and moving mirrors). For a small angle, the cosine error contribution

to coordinate-position error, a multiplicative error, is given by:

6Pe = D • ( -a2/2). (36)

Cosine error arises, in effect, when the axis of the displacement vector and axis of the

coordinate vector are rotated with respect to each other but have a common origin. [Note

that the errors due respectively to misalignment of interferometer and light (eq (34)) and

misalignment of mirror translation and interferometer (eq (36)) are of the same sign while

32
that due to misalignment of object and mirror translation (eq (40)) is opposite (appendix,

section 2.8).]

b. Abbe Error: Abbe error can arise when the axes of the displacement vector and the

coordinate vector do not share a common origin and they rotate with respect to each other,

that is, tilt, during the course of the displacement. Abbe error is governed by the

relationship:

D = P + O, • sin <|) (37)

where D is the measured displacement, P is the coordinate being assigned by the

measurement, O, is the lateral distance between the displacement and coordinate axes

(called the Abbe offset), and (j) is the angle of tilt which occurs during the course of the

displacement measurement.

For small angles, the Abbe error contribution to coordinate-position error, which is an

additive, is given by:

6P3 = O3 • 4>. (38)

2.5.2.2 Error Related to Carriage Motion

Carriage-associated position-measurement error occurs in measuring machines in which the

motion-generating system (which translates the object relative to the reference frame) is the

same as the displacement-measuring system (which measures that translation) as, for

example, in a machine which uses the same lead screw both to move a carriage and to

measure its position. For such machines, carriage-associated errors include backlash and

hysteresis, both of which are direction-dependent and result in loss of origin of the

coordinate system.

2.5.2.3 Errors Related to Reference-Frame Geometry

Limitations in the mechanical structure of a coordinate measuring machine give rise to

positional errors associated with the machine's coordinate axes and their angular

relationships. Since the reference frame of a coordinate measuring machine must consist

of one highly planar reference surface for each coordinate axis with each of these planes

angularly oriented with respect to the others in a constant and well-known way, the errors associated with reference frame geometry are complex.

Characterization of the positional errors in a 3D CMM requires dealing with 21 degrees of

freedom based on 6 degrees per axis (3 translational plus 3 angular, including roll, pitch, and

yaw) plus the three angular orientations of the axes with respect to each other. Specialized

33
matrix-based models and measurement algorithms for characterizing positional errors in

coordinate measuring machines and machine tools have been developed [38,39] and

standardized techniques adopted [40].

2.5.2.4 Error Related to the Probe

Limitations in the mechanism for linking the object to be measured to the machine's

coordinate measurement system give rise to positional errors specifically associated with the

machine's probe. Probe-specific errors are of two types, depending on whether the probe

is designed to carry out only functions of a probe as a simple sensor or a probe as a measuring machine.

a. Positional Error of Probe-as-Sensor: Positional errors of a probe-as-sensor arise to the

degree that it fails to carry out its primary function, that of detecting ~ in a binary sense of

"here" or "not here" ~ the boundary of an object, such as to locate it relative to the

coordinate system of the machine. Positional errors of probe-as-sensor include what in

touch-trigger mechanical-contact probes is called "pre-travel," that is, the displacement of the

probe tip after contact with the object but before the probe signals that contact. Such error

arises, for example, from variations in bending of the probe structure and the threshold

response of the mechanical-to-electrical transducer.

b. Positional Error of Probe-as-Measuring-Machine: Positional errors of a probe-asmeasuring-machine arise in, for example, "analog probes," which are designed not only to

carry out the primary function of linking machine and object, but also to provide an output

signal proportional to the displacement of the probe tip on contact with the object.

Positional errors of analog probes, being as they are miniature measuring machines attached

to a larger machine include: 1) errors of the probe as a sensor; 2) errors of the probe as a measuring machine (including those associated with scale, frame, and carriage); and 3) errors associated with the orientation of the probe as one coordinate system with respect to

another.

In sum, this section has looked at sources of errors in measurements of position, which after

displacement, is the next most directly linked to the SI unit of length. As indicated, position error deals with the ability to measure location of points in space and are associated with

each of the functional components of a coordinate measuring machine by which such

position measurements are made.

2.5.3 Error in Distance Measurement

After displacement and position, the type of dimensional measurement next most directly

linked to the SI unit of length is that of distance, that is, measurement of the separation of

34
two successively located point-like features on material objects. The measure of the limit

of the ability to measure distance is distance error, 6d. Errors specific to measurements of

distance are associated with the probing, angular alignment, and environmental-dependence

of separation of distance-type features of the object.

2.5.3.1 Distance Error From Feature Probing

Errors of feature probing arise from variations in the size, shape, and material of the

individual point-, line-, and plane-like features, the separation of which is being measured

in a distance measurement. Such variations limit the ability to reproducibly locate features

with a probe. Errors associated with such variations are assessed in terms of the

repeatability with which the probe of a particular coordinate measuring machine can be set on a particular feature.

2.5.3.2 Misalignment of Object and Coordinate Axes

Misalignment between the object being measured, and a coordinate axis of the measuring

machine gives rise to cosine error governed by the relationship:

d = I Pi - P2 I • cos Y, (39)

where Pi and P2 are the measured coordinate positions of the point-like features, d is the

distance being measured, and y is the angle of misalignment between the coordinate axis and

that of the object itself. For a small angle, the cosine error contribution to distance error

is given by:

6d, = d • (+y72). (40)

2.5.3.3 Error From Object Environment

Apart from the effects of changes in the environment on displacement-measuring devices

and the measuring machine itself, such environmental changes give rise to errors associated

with the object itself, including those associated with thermal expansion, mechanical

distortion, and contamination.

a. Error From Object Thermal Expansion: Distance error due to thermal expansion of the

object can be due to different temperatures for the whole object or different temperatures

for two positions on the object. In either case, variation in the dimensions of the object with

temperature is governed by the equation:

35
p = -(M/Q/ AT, (41)

where p is the coefficient of linear thermal expansion of the material and AC/{o is the

fractional change in the "length" of material for a difference in temperature AT. Note that

since thermal expansion is defined in terms of the change in length (either between distancetype features or extension-type boundaries), it is inherently a distance-type property given

by:

Sdx = p

• AT. (42)

b. Error From Object Distortion: Distance error can also arise from any other factor which,

in effect, distorts the object by altering the relative separation of features of the object.

Such factors include any local or overall mechanical distortion of the object due, for

example, to fixturing of the object on the measuring machine; and any gain or loss of

material on one feature with respect to the second due, for example, to contamination or wear.

In sum, this section has looked at sources of errors in measurements of distance which, after

displacement and position, is the next most directly linked to the SI unit of length, such

errors being associated with measurement of the separation of two point-like features of

material objects.

2.5.4 Error in Extension Measurement

The type of dimensional measurements operationally farthest removed from the primary

realization of the SI unit of length is that of extension. The measure of the limit of the

ability to measure extension is extension error, 6e. Most specific to the measurement of

extension-type features of objects, such as the inside diameters of holes and the outside

diameters of plugs, is error associated with locating a single boundary and error associated

with locating one boundary relative to another one.

2.5.4.1 Error in Location of a Single Boundary

a. Error in Reducing a 3D Surface to a Point: Errors occur in measuring the location of a

real boundary because a material object is not an ideal geometrical surface being intersected

by a line such as to define a point. Instead that boundary is a rough three-dimensional

surface being contacted by a finite-area probe. Estimates of the contribution to boundarylocation errors for specific object-probe combinations may be made based on measurements

of surface roughness, which involves statistical averaging over surface irregularities.

Boundary-location errors due to surface roughness can be comparable to the roughness

itself, which for a highly-polished steel surface, for example, can be of the order of 0.05 iim.

36
b. Error in Compensation ofProbe-Object Interaction'. Errors occur in measuring the position

of a single boundary as a result of any uncompensated off-set between the location of the

probe and the location of the surface which the probe signals. For example, in locating the

boundary of a machined part with a mechanical stylus, single-boundary-location errors occur due to incomplete compensation for both: the finite radius of the probe and its finite

penetration into the part. Analogous errors occur in locating the boundary of a microcircuit

element with the beam of an electron microscope.

Compensation for probe-object interaction, such as the deformation of a part by a mechanical stylus and penetration into the bulk of a material by an electron beam, can only

be done by fundamental theoretical modelling of the specific probe and object involved. Of

necessity, such modelling has been done, for example, for high-accuracy optical- and

electron-microscope calibrations of photomask linewidth standards for use in the

microelectronics industry [41J.

In sum, error in boundary location is a vector having a magnitude (associated with, for

example, the finite roughness of the surface, the finite extent of the probe and the finite

depth of penetration of the object by the probe) and a direction, positive or negative,

relative to the axis of coordinate measurement.

2.5.4.2 Error in Location of Two Boundaries

In measurements of extension, errors in the location of one boundary combine as vectors

with errors in the location of the second boundary - that is, one of opposite orientation ~

and the results are non-canceling, as indicated by:

K = 6Pbi + 6Pb2, (43)

where 6Pbi is the error in the location of the first boundary and 6Pb2 is the error in the

location of the second.

For the case that the boundary-location error is the same for the two boundaries, that is, where the magnitude of 6Pbi equals that of 6Pb2, the error in extension measurement due

to the combination of the two boundary-location errors is: K= ^ 26Pb, (44)

where the minus sign is for outside-caliper measurements (as in the diameter of a plug) and

the plus sign is for inside-caliper measurements (as in the diameter of a hole) when the

boundary-location errors are due, in effect, to "penetration" (positive or negative) of the

object by the probe.

37
An example of a positive boundary-location error is that due to the finite radius of a mechanical probe (that is, the extension of the probe tip along the axis of measurement).

An example of a negative boundary-location error is probe pre-travel (that is, the finite

displacement of the probe along the axis of measurement after mechanical contact with the

object, but before the probe transducer crosses a threshold and signals).

Additive boundary-location errors are unique to extension-type measurements. In

measurements of displacement, position, and distance, the boundaries involved face in the

same direction and boundary-location errors cancel, that is:

6PD.P.ord = 6Pb - 6Pb = 0. (45)

2.5.5 Compounding of Error in Measurement Types

A final aspect of assessing accuracy in practical dimensional measurements involves a look

at the cascading nature of the errors in the succession of the dimensional measurement

types. This cascading, that is, accumulation, of errors is shown formally below, where A

represents the total uncertainty of the type of measurement and 6 represents the typespecific error in that total.

In this notation, the total error in displacement measurement AD is a function of 6D, which

is all the displacement-specific errors described in section 2.5.1, plus AAo, which is the error

specific to the metric-carrying vacuum wavelength of laser light source, and is given by:

AD = AD(6D,6Xo). (46)

Similarly, the total error in position measurement Ap a function of 6p, which is all the

position-specific errors described in section 2.5.2, plus AD, which is the total error in

displacement measurement, and is given by:

Ap = Ap(6p,AD) (47) = Ap(6p,6D,6Ao).

And again, the total error in distance measurement, Ad, a function of the total error in

position measurement, plus all the distance-specific errors described in section 2.5.2 above,

and is given by:

Ad = Ad(6d,Ap) (48) = Ad(6d,6p,6D,6Xe)

38
Finally, the total error in extension measurement, Ae, a function of the total error in distance

measurement, plus the extension-specific errors described in section 2.5.4 above, and is given

by:

Ae = Ae(6e,Ad) (49) = Ae(6e,6d,6p,6D,6A,o)

As is indicated by this formal analysis, the accuracy of each type of dimensional

measurement depends on - and is limited by - the total error of the preceding type, as well

as by the sum of its own type-specific errors.

In sum, measurements of extension are the most difficult in which to achieve accuracy

because they inherit all the errors of the successive displacement, location, and distance

measurements upon which they are based and, addition, are subject to intractable boundarylocation errors to which the others are not.

2.6 Error Budget Example

All of the considerations of accuracy assessment discussed above can be taken into

systematic account in the assessment of the errors in a particular dimensional measurement

system by compilation of an error budget for the measuring system in question. In this

example, a simple sum-over-errors is used to estimate the total worst-case error; other

means for estimating the most likely errors have been described [42].

The example considered here is measurement of an extension-type object on a laserinterferometer-based single-axis coordinate measuring machine. In this example, the overall

system consists of a stabilized HeNe laser, a commercial heterodyne interferometer system, a machine frame with a moving carriage, a touch-fire probe, and a 100 mm long steel rod,

the length of which is to be measured, all within a normal atmospheric environment.

Table 7 below shows a compilation of the contributions to the overall accuracy of

measurement of the specified object with the hypothetical coordinate-measuring-machine

39
Table 7. Error budget example: analysis of measurement of extension of a 100 mm

long object with an interferometer-based coordinate measuring machine

Error Contribution by Source &«A. (bi

Interferometer Least Count 6N,e 0.005 Jim

Beam-Polarization Mixing 6Np, 0.002 nm

Machine-Path Atmosphere 6np3 10 ppm

Compensated 6npc 0.15 ppm

Optics Thermal Drift 6nio 0.02 Jim

Dead-Path Compensated 6ndp 0.015 Jim

D-vs-X Alignment 66, 0.05 ppm

Sum of D-Specific Errors 6D 0.042 Jim 0.20 ppm

Vacuum-Wavelength Error ^K 0.02 ppm

Displacement Error Total AD 0.042 ^m + 0.22 ppm

P vs. D Alignment 6?, 0.05 ppm

Carriage Abbe Tilt 6Pp 0.24 Jim

Probe Setting 6Pp 0.50 Jim

Sum of p-Specific Errors 6P 0.74 Jim 0.05 ppm

Displacement Error Total AD 0.042 Jim 0.22 ppm

Position Error Total A£ 0.782 um + 0.27 ppm

O vs. P Alignment bd^ 1.5 ppm

Thermal Expansion 6dp 1.0 ppm

Point-Feature Definition fid. 0.10 Jim

Sum of d-Specific Errors 6d 0.10 Jim 2.5 ppm

Position Error Total Ap 0.782 Jim 0.27 ppm

Distance Error Total Ad 0.882 um + 2.77 ppm

Arbitrarily-Located Bndry 6ebx 1.0 Jim

Compensated 6eb, 0.1 Jim

Shape-Averaged Boundary fiCbm 0.2 Jim

Sum of e-Specific Errors 6e 0.3 Jim

Distance Error Ad 0.882 Jim 2.77 ppm

Extension Error Total Ae 1.182 um + 2.77 ppme

40
system described. The organization of the error budget follows the sequence of the type

scheme used throughout the chapter. In the table, bft^i and bUg, represent, respectively,

additive and multiplicative contributions to the type-specific errors. The values are estimates

of the relative size of error contributions from the various sources. In the following sections,

each of the individual contributions to the type-specific errors for this hypothetical example

are discussed in turn.

2.6.1 Error in Displacement Measurement AD

In this example the errors in the laser-interferometric measurement of the displacement

derive from displacement-specific errors associated with each of the parameters in the basic

interferometer equation plus the error associated with the vacuum wavelength of the laser

light.

2.6.1.1 Fringe-Fraction Error 6Dn

The contributions to displacement-specific measurement error are due to errors in

determination of the change in the order of interference with the translation of the

interferometer reflector:

N = i -I- f

,

(50)

where N is a real number which is the sum of the counted order i and measured fringe

fraction f. Practically, the total error in N is associated with error in interpolating between

fringes, that is: 6N = 6f. (51)

In this example, the contributors to error in fringe-fractioning are the 5 nm due to the least

count of interferometer and the 2 nm due to the mixing of the polarization states of the

interfering beams.

2.6.1.2 Index-of-Refraction Error 5Dn

Contributions to displacement-specific measurement error arise from variations in the indices

of refraction along the optical paths of the interfering beams due to changes in

environmental conditions. The variation in phase between the two beams as a function of

variations in the individual indices of refractions in each of the segments of the referenceand moving-arm beams is given by:

6A(t) = (2n/K) [ Si 6nri -Hn-^i 6n„i • d^, ], (52)

41
where Ej indicates a sum over the i number of optical-path segments nj • H, in each of the arms. In this example, the principal segments of the overall optical path include the CMM

machine-axis path along which the moveable reflector is translated, the dead path, and the

interferometer-optics path, that is, the glass through which the light passes.

a. Machine-Path Atmosphere: The first and major source of index-of-refraction errors is the

variation of the atmospheric air along the displacement path. The following show the

changes in temperature, pressure, and relative humidity, each of which gives rise to a displacement error of 10"^:

bn^j = 1 ppm / 1 "C

Sn^p = 1 ppm / 0.33 kPa (2.5 mm Hg) (53)

bn^u = 1 ppm / 80% ARH.

In this example, a ±0.5 °C change in temperature, a ±25 mm change in pressure, and a ±10% change in relative humidity combine to produce a multiplicative error of about 10

ppm. With measurement of the changes and compensation for them, this is reduced to 0.15

ppm.

b. Interferometer-Optics Thermal Drift: The second source of index of refraction error is change in the temperature of the glass components of the interferometer through which the

light in the two beams passes. In this example, optics thermal drift error is 0.02 |im.

c. Dead-Path Compensation: A last source of index of refraction error is change in

temperature, pressure, and humidity of the dead path (the difference in the interferometer

arms at the starting point of the machine-axis path). In this example, compensation has

reduced dead-path error to 0.015 jim.

2.6.1.3 Interferometer-Axis Alignment Error fiDgti

The contribution to displacement-specific measurement error due to the angular

misalignment 6,i between the axis of the interferometer and the axis of propagation of the

incident light is, for a small angle of misalignment, given by:

6Deu = D • (- e,72). (54)

For an angular alignment corresponding to a lateral displacement of one-tenth the spot size

of a 3 mm beam at the 1 meter travel of the measuring machine stage, 0,j = 3x10'^ radians

and the resulting error 6Deti is 0.05 ppm.

42
2.6.1.4 Sum of Displacement-Specific Errors dD

In this example, the total displacement-specific error 6D is the sum of the errors due to

fringe-fractioning, index of refraction, and angular misalignment of translation-propagation axes and is given by:

6D = Ej 6Di

(E6D;^) + (E6DBi) • H (55)

0.042 tim -I- 0.20 ppm • D .

At a displacement of 100 mm (corresponding to the extension of the object being measured),

the sum of the displacement-specific errors is:

6D100 = 0.062 Jim = 0.62 ppm @ 100 mm (56)

2.6.1.5 Vacuum-Wavelength Error bX^

The contribution to the total displacement measurement error AD due to the error in the

value of the vacuum wavelength of the laser is:

6D,o = (6X0 / Xo) • D

,

(57)

which for the highly stabilized commercial laser of this example is 2*10'*. Thus,

6Dio = 0.02 ppm • D. (58)

2.6.1.6 Total Displacement Error AD(ftD,5Xo)

As indicated, combining the displacement-specific error 6D with the error in the vacuumwavelength of the laser source 6X0 yields the total displacement error AD over the 1 meter

travel of the measuring machine stage, that is: AD = bD + 6X0

(S6D^) + (SSDei) • C + (6kjk,) • H (59)

0.042 \Lm + 0.22 ppm • D

43
For a displacement measurement corresponding to length of the 100 mm long object, the

total error in the displacement measurement is: AD = 0.064 \im = 0.64 ppm @ 100 mm. (60)

2.6.2 Error in Position Measurement AP

In this example the errors in the coordinate-measuring-machine measurement of position

derive from errors associated with the geometrical relationships among the scale, frame,

carriage and probe of the CMM and the setting ability of the probe itself.

2.6.2.1 Misalignment of Coordinate and Displacement

The contribution to position-specific measurement error due to the angular misalignment

(J) between the axis of translation of the mirror and the axis of the interferometer is, for a

small angle, given by:

6P4, = D • (- <|)2/2). (61)

For an angular misalignment of (t),i= S-IO"' radians, the resulting error is:

6P^ = 0.05 ppm • P. (62)

2.6.2.2 Carriage Abbe Tilt

The contribution to position-specific measurement error due to a difference in angle of tilt of a non-zero-length probe between two positions (Abbe error) is, for a small angle:

6P, = O. • p. (63)

In this example, for a distance between the axis of the probe and the axis of the

displacement measurement of 50 mm and for a change in tilt of 1 arc second, the Abbe

error contribution to coordinate-position error, which is an additive, is:

6P3 = 0.24 Jim. (64)

2.6.2.3 Probe Setting

The contribution to position-specific measurement error due to probe setting is a measure

of the ability of an individual probe to reproducibly locate unvarying, uni-directionally

44
approached, probed features. Since a location measurement requires the probe to trigger

twice (once at the location of the origin and once at the location being measured, an

estimate of the error in probe-setting based on the 2a setting of a good-quality commercial

probe operated in the touch-trigger mode is:

6Pp = 2 • 0.25 Jim = 0.50 jim. (65)

2.6.2.4 Sum of Position-Specific Errors dp

In this example, the total position-specific error 6P is the sum of the errors due to

coordinate-scale misalignment, Abbe error, and probe setting.

6P =2 6Pi

0.74 Jim -I- 0.05 ppm • d (66)

0.745 Jim @ 100 mm

2.6.2.5 Total Position Error AP(6P,AD)

Combining the position-specific errors, 6p, with the total displacement error, AD, over the

1 meter travel of the measuring machine stage yields:

AP = 6P + AD = 0.782 jxm + 0.27 ppm -(87)

For a position corresponding to length of the 100 mm long object, the total error in the

position measurement is:

APioo = 0.809 Jim = 8.1 ppm @ 100 mm (68)

2.6.3 Error in Distance Measurement Ad

In this example the errors in the coordinate-measuring-machine measurement of distance

derive from distance-specific errors associated with the object alignment, temperature, and

distance-type features, plus the error in position measurement.

45
2.6.3.1 Object-Axis/Probe-Path Alignment

The contribution to distance-specific error due to the angular misalignment y between the

axis of the object and the axis of coordinate measurement, that is, the path of the probe, is, a small angle, given by:

Sd^ = d • ( + yV2). (69)

For an angular misalignment of 0.1° (corresponding to a lateral distance of less than 0.2 mm

over the 100 mm length of the object):

Sd^ = 1.5 ppm • d . (70)

Note that this error due to misalignment of object and coordinate axis (eq (40)) is opposite

in sign to that due to misalignment of interferometer and light (eq (34)) and misalignment

of mirror translation and interferometer (eq (36)).

2.6.3.2 Object Thermal Expansion

The contribution to distance-specific error due to the difference in temperature AT between

the object to be measured and a reference temperature (such as that of the CMM) is determined by the linear thermal expansion coefficient p for the object according to the

thermal expansion equation:

Sd^T = p • AT • d (71)

For the a steel object in this example, p is 10 ppm/°C, which for temperature difference AT

of 0.1 "C gives:

Sd^T = 1-0 ppm X d. (72)

2.6.3.3 Object Point Deflnition

The contribution to distance-specific measurement error due to object point definition is a measure of the ability to associate with the object features which can be characterized as

points, the position of which points can be reproducibly located by a variability-free probe.

In this example, the estimate of a practical lower limit for error associated with locating

points on real objects (twice that for defining a single point) is estimated to be:

6d. = 0.10 ^m. (73)

46
2.6.3.4 Sum of Distance-Specific Errors 6d

In this example, the total distance-specific error 6d is the sum of the errors due to

coordinate-scale misalignment, Abbe error, and probe setting.

fid = E 6dj = 0.10 tim + 2.5 ppm • d. (74)

At a distance of 100 mm corresponding to the extension of the object being measured, the

sum of the distance-specific errors is:

6dioo = 0.35 Jim = 3.5 ppm @ 100 mm. (75)

2.6.3.5 Total Distance Error Ad(6d,Ap)

Combining the sum of the distance-specific errors, 6d, with the total position error, Ap, over

the 1 meter travel of the measuring machine stage yields:

Ad = 6d + Ap = 0.882 nm + 2.77 ppm x d. (76)

For a distance corresponding to the length of the 100 mm long object, the total error in the

distance measurement is:

Ad = 1.109 Jim = 1.1 ppm @ 100 mm. (77)

2.6.4 Error in Extension Measurement Ae

In this example, the errors in the coordinate-measuring-machine measurement of extension

derive from extension-specific errors associated with boundary location and definition, plus

the error in distance measurement.

2.6.4.1 Boundary-Location Errors

Boundary-location error in extension measurements represents the systematic failure of the

probe to signal the proper location of the material boundary of the object and is of the

form:

6e = :p 2 • 6P,„ (78)

where the minus sign is for outside-caliper measurements, the plus sign is for inside-caliper

measurements, and 6Pbx is positive for location of the arbitrary boundary behind the material

boundary.

47
a. Location ofArbitrary Boundary. Various phenomena give rise to the direction-dependent

boundary-location errors inherent to extension measurements which individually contribute

to the aggregate effect and 6Pbx. In this example, there is the probe radius (that is, the finite

extension of the probe tip along the axis of measurement) and the probe pre-travel (that is, the finite displacement of the probe along the axis of measurement after mechanical contact

with the object is made, but before the probe transducer crosses a threshold and signals).

Assuming that the gross effects of probe radius and pre-travel are eliminated by precharacterization of the probe, there remain the extension-specific sources of error due, for

example, to deformation of the object by mechanical contact and variation of pretravel of

the probe with the direction of approach to the object.

Given a three-dimensional touch-trigger probe with a specified pre-travel variation over 360*

in the X-Y plane of ± 0.5 nm, the extension-specific error is taken to be:

6eb, = 2 • 0.5 nm = 1.0 ^im. (79)

b. Compensated Boundary Location'. The contribution to extension-specific error due to

location of an arbitrary boundary can in some situations be compensated by an empirical

procedure while in the majority of cases a theoretical model of the probe-object interaction

is required. In the example being followed here, the probe is assumed to have a residual

direction-dependent variation in pretravel of:

6eb, = 0.1 nm. (80)

2.6.4.2 Average-Material-Boundary Location

Extension-specific error due to average-boundary location is associated with local variation

in the position of the material boundary of the object such that it has no single extension.

In the example being followed here, the steel block is assumed to have a surface-finish peakto-valley of 0.1 \im (0.067 Jim RMS) which when mechanically probed leads to a contribution

to extension-specific error:

6eb. = 0.2 nm. (81)

48
2.6.4.3 Sum of Extension-Specific Errors 6e

In this example, the total extension-specific error 6D is the sum of the errors associated with

mislocation of a detected boundary relative to a material boundary and mislocation of a material boundary relative to some simple geometrical shape used to describe the object.

6e = S fiCi = 6eb, + 6e^^ (82) = 0.2 Jim + 0.1 Jim = 0.3 jim

Note that extension-specific error is a purely additive error, independent of the magnitude

of the extension.

2.6.4.4 Total Extension Error Ae(5e,Ad)

Combining the sum of the extension-specific errors 6e with the total distance error Ad yields

the total extension error:

Ae = 6e -f- Ad = 1.182 jim -I- 2.77 ppm • d (83)

For the 100 mm long object, the total extension error, computed and rounded to a

significant figure, is: Aeioo = 1.459 \im

1.5 nm = 15 ppm @ 100 mm. (84)

2.6.5 Summary and Analysis of Errors

This error-budget analysis ~ carried out for the case of a commercial-quality displacementinterferometer-equipped coordinate measuring machine used to measure the length of a

polished, parallel-faced, 100 mm long, steel block ~ illustrates issues specific to the example

as well as ones associated with precision dimensional measurements in general. The specific

numerical results may be summarized and then analyzed in terms of their relative typehierarchical, additive-multiplicative, and axiom-specific behaviors.

49
2.6.5.1 Summary of Numerical Results of Analysis

Table 8 below summarizes results specific to the hypothetical example being examined.

Tabulated are errors specific to the type and total errors for each type, on both an

incremental (±6C) and a fractional (±6C/C) basis. For comparison, also shown (in italics)

is the error in the vacuum wavelength of the laser which provides the reference to the SI

unit of length.

Table 8. Summary of total errors by dimensional type in the example of the laserinterferometer-based CMM measurement of extension of a 100 mm block

Dimensional Type-Specific Error Total-Error in Type

Type Add 6C Mult 6(!/{ Add A{ Mult A(!/«

Extension 0.3 nm 1.462 jim 14.62 ppm

Distance 0.35 ^m 3.5 ppm 1.102 p.m 11.02 ppm

Position 0.75 jim 7.5 ppm 0.812 nm 8.12 ppm

Displacement 0.06 jim 0.6 ppm 0.062 jxm 0.62 ppm

Wavelength 0.002 nm 0.02 ppm

2.6.5.2 Analysis of Results

The particular results of the CMM/extension-measurement example illustrate a number of

more general patterns in dimensional measurements including: a hierarchy of accuracies,

dominance by additive errors and multiplicative errors at the short end and long end of the

range respectively, and the minimal contribution of laser-wavelength error to total error.

a. Hierarchy of Accuracies Among the Types: The last column, which gives the total error A H/H by type, illustrates the inherent loss of accuracy as one moves through the progression

of types of dimensional measurements. Note that the accuracy of the displacement

measurements is down by an order of magnitude from that of the reference wavelength. The

accuracy of position is down by another order from there, while the accuracy of distance and

extension down another factor of 2 or 3 further from there. These trends in these

relationships are general. On the extremes, displacement measurements always have the

best and more readily obtained accuracy and position and distance are intermediate, while

extension measurements always have the lowest accuracy and the most dearly bought.

50
b. The Additive-Multiplicative Errors: The second and third columns, which give the typespecific errors, 611 and bd/d respectively, illustrate the additive vs. multiplicative character

of errors in the different types. On the extremes, the extension-specific errors are purely

additive, which means they dominate at short lengths. In contrast, the displacement-specific

errors are almost purely multiplicative, which means they dominate at long ranges. For the

hypothetical CMM of the example, these trends are further illustrated by table 9 which

shows errors in ranges of 10 and 1000 mm. Note especially in the table the near-pure

additive 1.2 nm (121 ppm) error in extension at 10 mm compared to the near-pure

multiplicative 0.26 ppm (0.26 jim) error in displacement at 1000 mm.

Table 9. Total error by dimensional type at ranges of 10, 100, and 1000 mm in the

example of the laser-interferometer-based CMM measurement of extension

of a block

Dimension Error @ 10 mm Error @ 100 mm Error @ 1000 mm

Extension 1.2 ^xm /121 ppm 1.5 p,m / 15 ppm 3.9 p.m / 4 ppm

Distance 0.9 jim / 91 ppm 1.2 p,m / 11 ppm 3.6 |j,m / 3.6 ppm

Position 0.08 Jim / 8 ppm 0.11 jim / 1 ppm 0.35 jim / 0.35 ppm

Displcmnt 0.04 [im I 4 ppm 0.06 \i.m I 0.6 ppm 0.26 jxm/ 0.26 ppm

2.6.5.3 Axiom-Type Matrix Analysis of Results for the CMM Example

The error-budget for the CMM measurement of the 100 mm block serves to illustrate the use of the axiom-type matrix of errors for the analysis of the performance of the overall

system of machine, object, and techniques for the achievement of precision dimensions.

Table 10 shows in the type vs. axiom matrix format of table 6 the results of the CMMextension error budget summarized in table 7. Also shown, in italics, are examples of the

matrix elements not illustrated by the CMM-extension example.

a. Errors in the Zero of Each Type: As indicated in table 10, each of the types of

dimensional measurements in the CMM-block example has associated with it a type-specific error of the zero (SC^) corresponding to a shift of the effective origin of that type. In the

displacement measurement, there are the uncompensated change in the index of refraction

of the dead path, the movement of the effective optical-path location of the interferometer

reference due to thermal drift, and the uncertainty in the location of the zero due to the

least-count of the interferometer system. In the position measurement, there are the non51
Table 10. The axiom-type error matrix for the example of the laser-interferometerbased CMM measurement of extension

Dimension Error of Zero Error of Unit Error of Scale

Displacement Dead-Path

Thermal-Drift

Least-Count

Position

Distance

Extension

Probe-Tilt Abbe

Probe-Setting

Point-Definition

Arbitrary Bndry

Matl-Av-Bndry

Backlash

Laser Wavelength

Mirror-Path Index

Cosine I vs. L

Cosine T vs. I

Polarization Mixing

Grad in MP Index

Bending of MP Ways

Grad in Table Temp

Cosine O vs. T Bending of Object

Object T-Expansion

Thickness-Dependent Approach to Probe

Probe 'Penetration'^ Resolution Limit

cumulative change in the location of the position origin due to the tilt of the probe relative

to the displacement axis (Abbe error) and the uncertainty in its location due to variations

in probe-setting. In the distance measurement, there is the uncertainty in objects geometry

which limits reduction to centroids, that is, points. Finally, in the extension measurements,

there are the arbitrariness of the boundary located relative to the material boundary and the

variation of the location of the material boundary about its mean. While not in the example,

backlash corresponds to an error of the zero of extension because it only appears with the

bi-directional approach to the object.

b. Errors in the Unit of Each Type: Similarly, as indicated in table 10, each of the types has

associated with it a type-specific error of the unit {hft^), that is, error proportional to the first order of the dimension measured. In displacement, there are the error in the value of the

vacuum wavelength of the laser, in the index of refraction of the path of the moveable

mirror, and the angle of misalignment between the axis of the interferometer (I) and the

direction of propagation of the incident light (L). In position measurement, there is the

angle of misalignment between the axis of translation of the mirror (T) and the axis of the

interferometer (I). In distance measurement, there is angle of misalignment between the

object (O) and the probe-path, which for a rigid-body system is the same as the axis of

translation of the mirror (T). While not discussed in the example, an error in the unit of

extension can also arise due to any dependence of the effective "penetration" of the probe

(positive or negative) on the thickness of the object.

52
c. Errors in the Scale of Each Type: Finally, as indicated in table 10, for each dimensional

type, there are type-specific nonlinearities corresponding to errors of the scale (6^1). In the

displacement measurement, there is the intermixing of the reference- and measuring-beams

due to their imperfect separation by polarization states which leads to a non-linearity in the

interpolation between fringes. While not in the example, other conditions lead to errors of

the scale. In displacement measurements, these include a gradient in the index of refraction

along the interferometer mirror path (MP) as well as any lateral bending of the mechanical

ways which support the mirror along that path; in position measurements, a gradient in the

temperature of the part of the machine which supports the object to be measured; in

distance measurements a bending of the object; and in extension measurements, an approach to the resolution limit of the probe.

In sum, this analysis of the hypothetical case of the use of a laser-interferometer-based

coordinate measuring machine for measurement of the length of a part has shown the nature

and types of the errors that can arise in that specific example. What the example illustrates

is that the use of the axiom-type matrix of errors provides as a complete, self-consistent

scheme for systematically partitioning and analyzing such errors. The approach is, however,

more generally useful.

2.6.5.4 Application of Type-Axiom Error Matrix to Manufacturing Operations

The axiom-type matrix just applied to the characterization of the dimension-measurement

process applies equally to the characterization of the dimension-generation process, that is, the manufacture-to-design of a dimensionally specified product.

a. ''Machine Toot' versus "Measuring Machine": Machine tools for the generation of

dimensioned forms and measuring machines for the characterization of dimensioned forms

share the elements of frame, carriage, scale, and "probe," where the former's probe is for

changing the location of material and the latter's "probe" is for ascertaining where the

material is located. This commonality is shared by all such "machine tools" — whether

metal-cutting lathes, ion-beam milling machines, or photolithographic step-and-repeat

cameras — and all such measuring machines — whether they be coordinate measuring

machines, measuring microscopes, or micrometer calipers. As such machine tools and

measuring machines must conform to the same axiom-type system to carry out their essential

functions.

b. Example ofExtension Errors in Milling: All the errors indicated in table 10 for the CMM

measurement of a block are also manifest in machine-tool-based manufacture of

dimensionally specified products.

Consider briefly, for example, the use of a laser-interferometer-based milling machine for

the manufacture of a multiple-finned turbine-blade fan where the part design calls for the

53
machining of equally spaced fins of identical tapering cross-section. In general the machine

will be subject to all the sources of error identified in table 10. Even under the conditions

that the machine can generate a cutter path that conforms ideally to the part design (that

is, can generate displacement, position, and distance perfectly), errors of extension can still occur.

Under certain conditions of tool-force and material-stiffness, deflection of the fin by the

cutter will vary in proportion to the thickness of the fin at the point of cutting, resulting in error in the material form even for an errorless cutter-path. Thus, while the spacing

between fins is perfect (corresponding to no errors of the unit, the zero and the scale in

distance), the error in the cross-section of each fin would have a "cooling-tower" shape

corresponding to errors in each of the zero, unit, and scale of extension and fully analogous

to the arbitrary boundary, thickness-dependent probe-penetration, and hmit-of-resolution

errors indicated in table 10.

2.7 Conclusion

This chapter on precision dimensional measurements for modern manufacturing has

presented a new scheme for the assessment of dimensional error, both in measurement

processes and in manufacturing processes. The scheme is based on a matrix of dimensional

types and measurement axioms which forms a complete, self-consistent system for assessing

dimensional error. The scheme has been illustrated by a case study of a coordinate

measuring measurement of an extended object, with the type-axiom related errors indicated

as being similarly manifest in manufacturing processes such as the milhng of turbine blades.

This type-axiom matrix should provide a useful means for the exposure of error, the

diagnosis of its cause, and the means of practical elimination through more effective design

of machines, processes, and techniques for the manufacture of products with precisely

dimensioned forms.

2.8 Appendix: Errors Due to Angular Misalignments of System Elements

Accuracy of measurement of dimensions of physical objects by means of a coordinate

measurement system based on a laser displacement interferometer, such as the prototypical

interferometer shown schematically in a plan-view representation here, depends upon precise

alignment of the axes of the laser, interferometer, translating mirror, and the object.

Errors in measurements of displacement, position, and distance (and/or extension) arise

respectively from misalignment of pairs of axes — interferometer and laser, translating

mirror and interferometer, and object and translating mirror, as described individually and

in combination below.

54
Error Specific to

N Mirror Normals

2.8.1 Misalignment

Displacement

For light of wavelength ko

propagating in a medium of

index of refraction n in a

direction K at angle 6 to the

axis of the interferometer

(defined by the normals N

to the reference and

moveable reflectors), the

change in spacing of the

interferometer mirrors, As, is related to the observed

change in fringe order

number. Am, by the interferometry equation:

K Light Propagation

Figure 5. Alignment

of mirror normals

and light direction.

Laser Source

Figure 4. errors in

Fixed Mirror c

^

Reflector

Translation

« > Object

1/

D

Detector

Light Propagation

Sources of misalignment

displacement-measuring

interferometers.

D = As = Am • kJ2n • cos6 (Al)

Note that misalignment between the axis of the interferometer and the axis of propagation

of the incident light constitutes an error of the metric or unit of displacement relative to that

of the light.

2.8.2 Misalignment Error Specific to Coordinate Position

For a moveable or "measuring" reflecting surface (plane mirror or

retroreflector) which translates at an angle (f) relative to the axis N

of the parallel-plate interferometer, the translation T is related to the

change in spacing As of the interferometer by the "cosine-error"

equation:

P • cos<t) = T • cos(t) = As

T Mirror Translation

<^

(A2) N Mirror Normals

Note that misalignment of the axis of translation of the measuring Figure 6. Alignment

element of the interferometer relative the axis of the interferometer of mirror translation

constitutes an error of the metric or unit of position (coordinate) and mirror normals.

relative to that of the displacement.

55
2.8.3 Misalignment Error Specific to Object-Distance

Finally, given an object with features with an axis O oriented at an —

angle y relative to the direction of translation of the measuring iranslation Axis

mirror of the interferometer, the relation of the distance between

features on the object and the translation of the mirror is given by

the equation:

d = T • cosy . (A3)

^

O Object Axis

Note that the misalignment of the axis of the object and that of the Figure 7. Alignment

translation of the measuring mirror constitutes an error in the metric of part axis and

or unit of distance relative to that of position. mirror translation.

2.8.4 Combined Misalignment Error in Distance

For a system in which there are all three misorientations described above,

the relation of the distance between features on the object and the observed change in the

fringe order number is given by:

d = (Am • ko cosy) / (2 • n • cos6 • cos<t>).

2.8.5 Explicit Form for Misalignment Error

(A4)

Given the sign convention that the observed distance dobs is the algebraic sum of the true

distance d,n,e plus the error term Ad, then:

Ad = dobs - d,rue = d-d(e,(t),Y)

= (Am • Ao/2n) • [ 1 - {cosY/(cose • cos(|))}]

(A5)

Under the conditions that the angles 0, <|), and y are each small, successive substitution of

the approximations cosx ~ 1 - x72 and 1/(1 - x^) » 1 + x^ into eq (A5) leads to:

Ad « (Am • kjln) • [ 1 - {(1 - y^l2)l{\ - 672) (1 - (t>V2)}]

« (Am • Xo/2n) • [ 1 - {(1 - y72)(l + 672) (1 + (|)V2)}] ,

(A6)

56
which with multiplication and retention of terms of the lowest order yields the final result:

Ad « (Am • XJln) • ( + 6^ + 4)2 - y^)l2 (A7)

Note that since the first and second terms are negative and the third term is positive, the

overall error in distance due to angular misalignments can be either positive or negative and

the measured distance can be either greater than or less than the actual.

3 A Comparison of National Standards for the Performance Evaluation of

Coordinate Measuring Machines in Terms of Length-Based Dimensional

Quantities

3.1 Introduction

A principal means to higher quality in the manufacture of products having dimensioned parts

is higher accuracy in both the machines which form the parts and the machines which

measure them. The dimensions of parts are of different types, each comprising a different

length-based dimensional quantity.

Since modern manufacturing depends on interchangeability of parts traded both domestically

and internationally, accuracy in these various types of part dimensions consists not only of

conformity of the dimensions to design but conformity of the units of the dimensions to a

standard. The international standards of length is the meter, which is defined in terms of

the propagation of light in free space. Practical realization of the formal definition is by

means of laser interferometry and another length-based dimensional quantity, displacement.

This chapter compares U.S., German, British, and Japanese standards for evaluation of

coordinate measuring machines and shows that using a variety of special artifacts the U.S.

standard alone specifies separate tests for each of the four modes of length measurements

(displacement, position, distance, and extension) and that, depending on the number and

orientations of faces probed, a bi-directional step gage of the type suggested in the other

standards can be used for tests of each of these.

Figure 8 is an engineering drawing representation of an automobile engine which illustrates

some of the principal types of geometrical dimensions of discrete parts. These types of

geometric features include: A, representing a length or width of the engine block; B,

representing the diameter of a cylinder bore; C, representing the distance between the

centers (axes) of cylinder holes; D, representing the depth of a blind hole; E, representing

the (true) position of a feature relative to the origin of a part coordinate system defined by,

for example, reference holes in the oil pan plane; and F, representing the location of the

center (axis) of a hole relative to a plane.

57
D

^ ^

A

C

Figure 8. An engineering drawing

representation of the principal geometrical

dimensions of a manufactured discrete part.

Each of the various part features shown in

figure 8 corresponds to a different type

length-based measurement quantity:

displacement, position, distance, and

extension. For example, in this case of the

automobile engine, there are extension-type

features such as the length of the block and

the diameter of cylinders; there are distancetype features such as the spacing of axes of

the cylinders; and there are position-type

features such as the locations of cylinders

relative to reference features such as dowel

holes. Features such as the depths of blind

bore holes are most akin to a displacement.

The significance of this approximate

correspondence between types of features and types of length measurements is that in both

the manufacture and measurement of dimensioned parts such as these, each of the lengthbased dimensional quantities involved is susceptible to type-specific errors which propagate

and compound in a way which has been described in detail in chapter 2 [43] and outlined

in the appendix (section 3.5).

For the evaluation of the accuracy of either machine tools or measuring machines, there are two general approaches relative to these part and length measurement types: first, overall

evaluation in terms of a machine's ability to produce/measure part dimensions of the

extension type; and evaluation of ability to produce/measure the aspects of each of the types — displacement, position, distance, and extension. Representative of the former approach

is the German/European VDWDE standard for the evaluation of coordinate measuring

machines; representative of the latter is the U.S. ANSI B89.1.12.

3.2 Standard-Based Performance Tests of CMM's

Two industry standards commonly referenced by U.S. manufacturers of commercial

coordinate measuring machines are the U.S. ASME B89.1.12 [44] and German VDIA^E

2617 [45]. Other major national standards for CMM's are the Japanese JIS B7440 [46]

and the British BS 6808 [47].

3.2.1. U.S. Standard B89.1.121V1

American National Standard ASME B89.1.12M-1990 specifies five principal tests for the

evaluation of the dimensional measurement capabilities of CMM's. These five performance

tests are called: Measuring Repeatability, Linear Displacement Accuracy, Volumetric

Performance, Bi-Directional Length Measurement Capability, and Point-to-Point Probing

58
Performance.

3.2.1.1 Measuring Repeatability

The first B89.1.12M machine performance test is measuring

repeatability and deals with the capability of the CMM to properly

measure position.

In this test a precision reference ball is located at the mid-point of

travel of each of the machine's three linear axes. Four widely

spaced non-planar contacts are made with the ball. From this set of

four readings are calculated coordinates of the center of the ball. Ten determinations of these center coordinates are so made. The

range of the ball center coordinate, that is, the maximum minus the minimum, is determined

for each axis.

Figure 9. B89.1.12M

test for measuring

repeatability.

The measuring repeatability is, then, this spread in the calculated locations of the center of

the ball for the 10 repeats. Thus this "measuring repeatability" performance test gives an

estimate, on a per axis basis, of the variability in measurement of position at a single

location:

AP'•^^ X = I

' P* xmax - P* xmin I' ' (1) V /

3.2.1.2 Linear Displacement Accuracy

<> D,

The second B89.1.12M machine performance test, linear displacement accuracy, deals in the words of the standard with the conformance of the machine's scales to the international standard of

length.

In this test, either a step gage or a laser interferometer may be used.

For each axis, the step gage or laser interferometer is oriented along

a line parallel to that axis through the center of the work zone. Measurements are made for at least 10 equal intervals, which are

specified to be greater than 25 mm and less than one-tenth the axis

length. The accuracy for a given step is the difference between the

step gage calibration or laser reading and the mean corrected

machine reading. The hnear displacement accuracy is then the sum of the magnitudes of

the maximum positive and maximum negative errors.

Figure 10. B89.1.12M

test for displacement

accuracy.

59
Thus this "linear displacement accuracy" performance test gives an estimate of the maximum

error in the ability to measure from any location to any other location in the full travel and

in the measurement of displacement-type features (see appendix, section 3.5):

AD = I AD„„+ I + I AD„,,. I

3.2.1.3 Volumetric Performance

(2)

0=0

(Brigurell.B89.1.12M

test for volumetric

performance.

The third B89.1.12M test, volumetric performance, deals with the

ability of the machine to properly measure relative distance

=KD (indirectly intercomparing the length scales on the various machine

axes and testing for imperfections of machine geometry).

In this test, a fixed-value, uncalibrated ball bar with a length 100 mm

shorter than the machine axis is used. At least four points are probed on each ball to locate its center and the distance between the

center of the two balls is calculated. For CMM's with nearly cubic

workzones, the process is repeated at each of twenty orientations of

the ball bar located at edges, face diagonals and body diagonals of

the cube. The "tolerance" on "volumetric performance" is the

-0

absolute value of the range of observed values for the 20 orientations.

Thus this "volumetric performance" test gives an estimate of the meiximum distance error anywhere within the arbitrary-metric space of the machine:

Ad = I d„3, - d„i„ (3)

3.2.1.4 Bi-Directional Length Measurement Capability

0.

Figure 12. B89.1.12M

test for bi-directional

length accuracy.

The fourth B89.1.12M machine performance test, bi-directional

length measurement capability, deals with the ability of the machine

to properly measure extension in rectilinear modes.

In this test, a single gage block of calibrated length of the order of

25 mm is located where parts are typically measured within the

workzone. The length of the gage block is measured at each of four

orientations of the block, three parallel to the axes of the machine

and one parallel to an off-axis diagonal. The tolerance on bidirectional length measurement capability is the meiximum deviation

of the observed values from the known value of the length of the

gage block.

60
This "bi-directional length measurement capability" performance test thus gives an estimate

of the maximum error in such a measurement of extension of rectilinear-type objects:

Aev = I e„K. obs - e'true ' max (4)

3.2.1.5 Poiiit-to-Point Probing Performance

The fifth B89.1,12M machine performance test, point-to-point probing performance, length

measurement capability, deals with the ability of the machine to properly measure locations

in radial directions. According to the standard, the test is devised to establish the magnitude of errors contributed by

probing sequences for probes used in the point-to-point

mode.

In this test, a roundness-calibrated precision reference ball is located within the workzone. The ball is probed at each of

49 points, 12 equally spaced on each of four circles of specified elevations plus one at the upper pole. From the set of

49 points, a sphere center is calculated and a radius determined for each measurement point.

max

Figure 13. B89.1.12M test for point-to-point probing.

The point-to-point performance is the maximum radius minus

the minimum radius. This is functionally equivalent to the

range of deviations of the 49 radii from the actual, but

unknown, radius. Thus this "point-to-point probing" performance test gives an estimate of the maximum error in a measurement of the radial extension

of an object in any orientation:

AeR = I eRmax - eRmin (5)

3.2.2 Relation of B89.1.12M Performance Tests to Dimensions of Parts

Measured

Each of the five B89.1.12M performance tests just described assesses the ability of a coordinate measuring machine to accurately perform one of the four fundamental types of

dimensional measurements, displacement, position, distance, and extension, three of which

correspond to dimensions of physical objects, such as the features of an automobile engine

block shown schematically here.

I The "measuring repeatability" performance test assesses the ability of the CMM to

repeatably measure position of features such as that of a cylinder hole relative to the part

61
P(m) d (m) 2R (m)

CD^Z)

L(m)

D->(m)

coordinate system of the block (shown here as a cor- ner and in practice defined by features such as dowel

holes and pan-plate surfaces).

H The "volumetric performance" test assesses the

ability of the CMM to uniformly measure relative

distance such as that between the centers of successive

cylinder holes independent of their orientation.

Figure 14. Example features of an

automobile engine block by length

dimensional type.

M The "bi-directional length measurement accuracy"

performance test assesses the ability of the CMM to

accurately measure extension of features such as the

length of the block. (This is the "rectilinear" type of

extension where L(left) and R(right) simply denote any pair of opposite-facing plane parallel

boundaries).

tm The "point-to-point probing" performance test assesses the ability of the CMM to

accurately measure radial extension type features such as the internal diameter of cylinder

holes, the external diameter of the mating pistons, as well as the locations of planes to

centers such as feature F in figure 8. (This is the "radial" type of extension where L(left)

and C(center) simply denote any location on the surface of a sphere relative to the center).

H Finally, the "linear displacement accuracy" performance test assesses the ability of the

CMM to accurately measure displacement itself, relative to the international standard of

length, as well as displacement-, position-, distance-, and extension-type features on objects.

3.3 Intercomparison of Standards for CMM Performance Testing

The principal standards dealing with performance evaluation of coordinate measuring

machines are those of Germany (VDWDE 2617), Britain (BS 6808), and Japan (JIS B7440)

as well as that of the United States (ASME B89.1.12) just discussed. Table 11 summarizes

the test artifacts each of these standards specifies and the measurement quantity to which

each artifact-based test corresponds.

For assessment of what it terms "length measuring uncertainty," the German VDIA^DE 2617

standard allows a step gauge or gauge blocks used in either one of two modes: bi-directional

(here called the high-option, corresponding to what is here called an extension-type mode);

or uni-directional (here called the low-option, corresponding to what is here called a displacement-type mode).

For assessment of what it terms "length measurement uncertainty," the British BS 6808

standard allows either a step gage or gauge blocks required to be used in a bi-directional

62
(extension-type) mode; the standard explicitly forbids use of the uni-directional

(displacement) mode for assessing length measurement uncertainty.

Table 11. Comparison of the Principal Standards for CMM Performance Evaluation in

Terms of the Artifacts They Specify and the Type of Length Measurement

to Which Each Artifact-Based Test Corresponds

Artifact

Specified

Length

Mode

VDIIVDE

2617

BS

6808

JIS

B7440

ASME

B89.1.12

Step Gage

Gage Block

Extension ' (L-to-R)High

Option

Required Required Required

Step Gage

Gage Block

Displacement

(L-to-L)

Low

Option

Forbidden Required Option 2

Ball

Bar

Distance

(C-to-C)

... ... ... Required

Laser

Interferometer

Displacement

(L-to-L)

Reqd for

Axes > 1 m

... ... Option 1

Reference

Ball

Position

(C-to-O)

Required ... ... Required

Reference

Ball

Extension ^ (L-to-C)Required Required

^ This is the "rectilinear" type of extension where L(left) and R(right) simply denote any

pair of opposite-facing plane parallel boundaries.

2 This is the "radial" type of extension where L(left) and C(center) simply denote any

location on the surface of a sphere relative to the center.

For assessment of what it terms "length measuring accuracy," the Japanese JIS B7440

standard allows either a step gauge or gauge blocks required to be used in a bi-directional

(extension) mode. In addition, for assessment of "Hnear displacement accuracy," the

Japanese standard allows either a step gage or "block gauges" used in a uni-directional

(displacement) mode.

63
For assessment of what it terms "bi-directional length measurement accuracy," the American

ASME 689.1.12(1990) standard requires a gauge block used in a bi-directional (extension)

mode. In addition, the American standard specifies a number of other tests. For assessment

of "linear displacement accuracy," the American standard allows use of either a laser

interferometer (a displacement-measuring device) or a step gauge used in a uni-directional

(displacement) mode. For assessment of "volumetric performance," this standard suggests use of a ball bar, a device which provides a fixed but uncalibrated distance. For assessment

of "position repeatability," it requires use of an uncalibrated precision reference ball, which

provides for a well-defined location (i.e., position). And, finally, for assessment of "point-topoint probing," it requires use of a roundness-calibrated precision reference ball (which

allows a measure of non- or omni-directional extension). In general, the test of CMM

performance required by the majority of these standards deals with the evaluation of the

capability to measure extension; only the U.S. standard specifies tests for all four unit-oflength-based quantities.

As indicated by the table, while described by slightly different nomenclature (the British

standard calling it "length measurement uncertainty," the Japanese "length measuring

accuracy," and the American "bi-directional length measuring accuracy"), three of these four

national standards specify either a step gauge, gauge blocks, or end standards used in a bidirectional mode for measurement of the extension.

The German standard does not require, but allows as an option, bi-directional

measurements, for assessing "length measuring capability". Since that standard defines

"length" as only "the distance between parallel planes", independent of whether they are the

same- or opposite-facing, bi-directional (extension) and uni-directional (displacement)

measurements are each allowed.

In addition to tests of extension-measurement capabilities, the Japanese, U.S., and German

standards have functional equivalents of the U.S.'s separate "linear displacement accuracy"

test, the U.S. standard explicitly allowing the optional use for the test of either laser

interferometer systems or artifacts such as step gauges used in a uni-directional mode.

Only the U.S. standard specifies separate tests related to each of the unit-of-length-based

measurement quantities which a CMM has the capability to measure, including distance and

position as well as extension and displacement. For assessment of what it calls "volumetric

performance," the U.S. standard requires use of gage blocks or a ball bar, a device which

provides a fixed but uncalibrated distance. For assessment of "position repeatability," it requires use of a precision reference ball, which provides for a well-defined location relative

to an origin or reference datum (i.e., position). And, finally, for assessment of "point-topoint probing," it requires use of a precision reference ball, which — while its radius is not

calibrated — allows one to set an upper limit on omni-directional or radial extension errors,

in contrast to the bi-directional or longitudinal extension described above.

64
As shown in the appendix, section 3.6, however, the bi-directional step gage referred to in

other national standards can be used for measurements in each of the four length modes.

In either case, however, that of application of the U.S. standard with its variety of reference

artifacts (end standards, ball bars, and precision reference spheres) or use of a bi-directional

step gage as outlined in the appendix, one may begin to separate sources of error by type

as a basis for remedial efforts or improved machine-system design.

3.4 Conclusion

With modern computer-controlled coordinate measuring machines and the parts they inspect

being widely traded internationally, evaluation of the performance of CMM's has become

an important matter for their producers and users. As a result, standards bodies in four

major industrial nations — the United States, Japan, Germany, and Great Britain — have

independently developed documentary standards for the evaluation of the performance of

CMM's.

This chapter intercompares the tests prescribed by each of these standards in terms of a

previously reported taxonomy of length-based measurements which: 1) makes distinctions

among extension, distance, position, and displacement types of length measurement; and 2)

shows that these types are ordered such that — in CMM measurements — errors propagate

from the simplest type, displacement, through position and distance, to the most complex

type, extension, which is the type most often thought of as "length."

The principal result of this intercomparison is to show that while the principal measure of

accuracy in CMM's is a test of ability to measure an extension type of feature (with 3 of the

standards requiring such a bi-directional test and the fourth allowing a uni-directional

displacement-type test as an acceptable alternative), only the U.S. standard requires tests for

each of the four length-measurement types. In doing so, it has in common with the other

standards the most important measure of CMM accuracy while also providing a more

comprehensive and unambiguous basis for a separation-of-variables analysis of overall system

performance.

65
3.5 Appendix: Relations of Errors Specific to Types of Length Measurements

This appendix outlines briefly how each of the various types of length-based dimensional

quantities is susceptible to type-specific errors which, in certain cases, propagate and

compound in succession as described in detail in reference [1]. The example here is that of

a coordinate measuring machine equipped with laser interferometers as scales and a touchfire probe used to measure an extension-type feature such as the length of a block.

The first and most basic measurement this CMM must be able to make is that of

displacement, where the machine accurately measures the change in location of its own

moving element relative to its internal scale, in this case, a laser interferometer system. The

errors in this measurement are shown formally as: AD = AD (6D,6A.), (Al)

which says that the total error in the displacement measurement AD is a function of 6A, the error in the vacuum wavelength of the laser, plus 6D, errors specific to the displacement

measurement, such as improper dead-path compensation and variations in the index of

refraction along the path of the moving mirror.

The second basic measurement this CMM must be able to make is that of position, where

the machine accurately measures the location of a feature, that is the center of that feature,

relative to an origin, that is, a reference datum, fixed relative to the frame of the machine.

The errors in this measurement are shown formally as:

AP = AP (6P, AD), (A2)

which says that the total error in the position measurement AP is a function of AD, the total error in displacement measurement, plus 6P, errors specific to the position measurement,

such as Abbe error due to carriage tilt and non-directional variation in probing.

The third basic measurement this CMM must be able to make is that of distance, where the

machine accurately measures the separation of the centers of two features on an object,

whether artifact standard or manufactured part. The errors in this measurement are shown

formally as:

Ad = Ad (6d, AP), (A3)

which says that the total error in the distance measurement Ad is a function of AP, the total error in position measurements, plus 66, errors specific to the distance measurement, such

as the cosine error of misalignment between the axis of the measuring machine and that of

the object.

66
Finally, the fourth and last basic measurement this CMM must be able to make is that of

extension, where the machine accurately measures the separation of two opposite-facing

boundaries of a feature, such as that of the end faces of a block. The errors in this

measurement are shown formally as:

Ae = Ae (6d, Ae), (A4)

which says that the total error in the extension measurement Ae is a function of Ad, the total error in distance measurement, plus 5e, which are errors specific to the extension

measurement, such as those due to improper compensation for the finite extent of the

probe.

3.6 Appendix: Use of a Step Gauge in Four Measurement-Quantity Modes

1L1R 4L4R

P

d

Figure 15. Use of step gage in

displacement, position, distance,

and extension modes of length

measurement.

This appendix shows how, depending on the number

and relative orientations of the boundaries of features

involved: 1) which the four pure types of length-based

measurements various features of step gages most

closely correspond; and 2) how a bi-directional step

gauge may be used to perform performance tests corresponding to these types.

First, a feature of the gage which has as its boundaries, for example, the left face of the first step

(IL) and the left face of another step (e.g., 4L)

corresponds to the blind-hole D in figure 8 and is here

called a displacement-type feature.

This use of the term displacement is in the same sense

that the U.S. B89 standard uses measurement of such

features to assess what it calls "linear displacement

accuracy," with a uni-directional approach to feature

boundaries. (It should be noted, however, that measurements of displacement-type features can be susceptible to errors associated with those of position per se, such as errors due to

Abbe offset combined with stage pitch).

Second, a feature of the gage which has as its boundaries, for example, the left face of the

first step when that face is treated as a reference datum or origin of coordinates (IL) to the

center of another step ( V2[4R4-4L] ) is here called a radial-extension type feature. This

feature corresponds to determination of the separation of a boundary from a center (and

vice versa), and corresponds to feature F in figure 8. Errors associated with measurement

of such features are evaluated by means of the B89 "point-to-point" performance test.

67
Third, a feature of the gage which has as its boundaries the center of the first step (

[ y2[lL

+ IR] ) and the center of another step ( e.g., y2[4L+4R] ) is what is here called a distancetype feature, since it involves the separation of the centers of two features. This corresponds

to feature C in figure 8. Note, however, that when one of the features is formally designated an origin of part coordinates, the location of the second center relative to the first is then

a position-type feature.

Finally, a feature which has as its boundaries a face of one step (IL) and an opposite face

of another step (e.g., 4R), corresponding to feature A in figure 8, is an extension-type

feature.

68
4 References

[1] Swyt, D. A., "Issues in Assessment of Conformity of Dimensions of Manufactured

Parts to Design Tolerances," from the Proceedings of the U.S.-Indo-Japanese

Conference on Quality in Design for Manufacture, Bangalore, India, January 1993.

[2] Swyt, D. A., "New Concepts of Precision Dimensional Measurement for Modern

Manufacturing," from the book Manufacturing and Automation Systems. Control and

Dynamic Systems. Vol. 45, Academic Press, 1992.

[3] Swyt, D. A, "A Comparison of National Standards for the Performance Evaluation

of Coordinate Measuring Machines," from the NIST internal report NISTIR 4978,

National Institute of Standards and Technology, November 1992.

[4] ANSI Y14.5M-1982, "Dimensioning and Tolerancing: American National Standard

Engineering Drawings and Related Documentation Practices," The American Society

of Mechanical Engineers, New York, NY.

[5] Swyt, D. A, "Challenges to NIST in Dimensional Metrology: The Impact of

Tightening Tolerances in the U.S. Discrete-Part Manufacturing Industry," NISTIR 91-

4747, U.S. Department of Commerce, National Institute of Standards and

Technology, January 1991.

[6] Current Status and Future Trends of Ultraprecision Machining and Ultrafine

Materials Processing, N. Taniguchi, Annals of the CIRP (International Institute of

Production Research), Vol. 2, No. 2, 1983.

[7] Feng, S. C, and Hopp, T. E., "A Review of Current Geometric Tolerancing Theories

and Inspection Data Analysis Algorithms," NISTIR 91-4509, U.S. Department of

Commerce, National Institute of Standards and Technology, February 1991.

[8] Tipnis, V. A, "Research Needs and Technological Opportunities in Mechanical

Tolerancing," Report of International Workshop, CRTD-15, Center for Research and

Technology Development, American Society of Mechanical Engineers, New York,

NY, October 1988.

[9] Srinivasan, V., "A Geometer Grapples with Tolerancing Standards," Research Report

RC 16815(74405), IBM Manufacturing Research Department, Yorktown Heights,

NY, April 1991.

[10] Hopp, T. E., "The Mathematics of Datums," American Society of Precision

Engineering Newsletter, Chapel Hill, NC, September 1990.

69
[11] British Standard Guide to Assessment of Position, Size, and Departure from Nominal

Form of Geometric Features, BS 7172-1989, British Standards Institution, 1989.

[12] Brown, C. W., "Dimensional Inspection Techniques Specification (DITS),"

Proceedings of Quality Expo Time, Chicago, IL, April 1991.

[13] American National Standard, "Methods for Performance Evaluation of Coordinate

Measuring Machines," ASME B89.1.12M-1990, American Society of Mechanical

Engineers, June 1991.

[14] Swyt, D. A., "New Concepts in Precision Manufacturing," in Control and Dynamic

Systems. Vol. 45, Part 1, Academic Press, 1992.

[15] Phillips, S., et al., "Properties of Touch Trigger Probes," Proceedings of the

Measurement Science Conference, Anaheim, CA, January 1992.

[16] Elmarghy, W. H., et al., "Determination of Actual Geometric Deviations Using

Coordinate Measuring Machine Data," ASME Manufacturing Review, Vol. 3, No. 1,

March 1990.

[17] Algeo, M. E. A., and Hopp, T. E., "Form Error Models of the NIST Algorithm

Testing System," NISTIR 92-4740, U.S. Department of Commerce, National Institute

of Standards and Technology, January 1992.

[18] Eisenhart, C, "Realistic Evaluation of the Precision and Accuracy of Instrument

Calibration Systems," Journal of Research of the National Bureau of Standards, Vol.

67C, No. 2, April-June 1963.

[19] Taylor, B. N., and Kuyatt, C. E., "Status Report on the ISO, lEC, OIML, and BIPM

Guide to the Expression of Uncertainty in Measurement," Proceedings of

Measurement Science Conference, Anaheim, CA, January 1992.

[20] Ku, H. H., "Uncertainty and Accuracy in Physical Measurements," NIST Special

Publication SP-805, U.S. Department of Commerce, National Institute of Standards

and Technology, December 1990.

[21] Teague, E. C, "Nanometrology," Proc. of Engineering Foundations Conference on

Scanning Probe Microscopes: STM and Beyond, Santa Barbara, January 1991,

published by The American Physical Society.

[22] "Documents Concerning the New Definition of the Metre," Metrologia 19, C. SpringerVerlag, 163-177, 1984.

70
[23] Quenelle, R. C, and Wuerz, L. J., "A New Microcomputer-Controlled Laser

Dimensional Measurement and Analysis System," Hewlett-Packard Journal, 2-22, 1983.

[24] Beers, J. S., and Lee, K. B., "Interferometric Measurement of Length Scales at the

National Bureau of Standards," Precision Engineering Journal, Vol. 4, No. 4, 205-214,

1982.

[25] Doiron, T. D., "High Precision Gaging With Computer Vision Systems," Elsevier,

Industrial Metrology 1, 43-54, 1990.

[26] Scace, R. L, "Foreign Trip Report to Japan," Center for Electronics and Electrical

Engineering, NIST, Gaithersburg, MD, Report No. 720-46, 1989.

[27] Teague, E. C, "The NIST Molecular Measuring Machine Project: Metrology and

Precision Engineering Design," Journal of Vacuum Science and Technology B7, 1890-

1902, 1989.

[28] Vezzetti, C. F., Varner, R. N., and Potzick, J. E., "Bright-Chromium Linewidth

Standard, SRM 476, for Calibration of Optical Microscope Linewidth Measuring

Systems," NIST Special Publication, 260-114, 1991.

[29] Layer, H. P., and Estler, W. T., "Traceability of Laser Interferometric Length

Measurements," NBS Technical Note 1248, 1988.

[30] Eisenhart, C, "Realistic Evaluation of the Precision and Accuracy of Instrument

Calibration Systems," Journal of Research of the National Institute of Standards and

Technology, Vol.67C, No.2, April-June 1963.

[31] Carnap, R., "Philosophical Foundations of Physics," An Introduction to the Philosophy

of Science, ed., Martin Gadner, Basic Books, Inc., New York, London, 1966.

[32] Simpson, J. A, "Foundations of Metrology," Journal of Research of National Institute

of Standards and Technology, Vol. 86, No. 3, 1981.

[33] Steinmetz, C. R., "Sub-Micron Displacement Measurement Repeatability on Precision

Machine Tools With Laser Interferometry," SPIE Conference on Optical Mechanical

and Electro-Optical Design of Industrial Systems, Vol. 959, 178-0192, 1988.

[34] Jones, F. E., "The Refractivity of Air," Journal of Research of the National Institute

of Standards and Technology, Vol. 86, No. 1, 1981.

[35] Estler, W. T., "Laser Interferometry in Length Measurement," NIST Special

Publication, 1990.

71
[36] Estler, W. T., "High-Accuracy Displacement Interferometry in Air," Applied Optics,

Vol. 24, No. 6, 808-815, 1985.

[37] Steinmetz, C. R., "Sub-Micron Displacement Measurement Repeatability on Precision

Machine Tools With Laser Interferometry," SPIE Conference on Optical Mechanical

and Electro-Optical Design of Industrial Systems, Vol. 959, 178-0192, 1988.

[38] Zhang, G., Veale, R., Charlton, T., Borchardt, B., and Hocken, R., "Error

Compensation of Coordinate Measuring Machines," Annals of the CIRP, Vol. 34, 445-

448, 1985.

[39] Duffie, N. A., and Malmberg, S. J., "Error Diagnosis and Compensation Using

Kinematic Models and Position Error Data," Annals of the CIRP, Vol. 36, 355-358,

1987.

[40] "Methods for the Performance Evaluation of Coordinate Measuring Machines,"

American National Standard B89.1.12, American Society of Mechanical Engineers,

New York, NY, 1991.

[41] Postek, M. T., "Scanning Electron Microscope-Based Metrological Electron Microscope

System and New Prototype Scanning Electron Microscope Magnification Standard,"

Scanning Microscopy, Vol. 3, No. 4, 1087-1099, 1989.

[42] Donaldson, R., "Error Budgets," in Technology of Machine Tools, Vol. 5, Ser/ 9.14,

1980.

[43] Swyt, D. A., "New Concepts in Precision Dimensional Measurement for Modern

Manufacturing," Control and Dynamic Systems. Vol. 45, Manufacturing and

Automation Systems: Techniques and Technologies, Part 1, Academic Press, San

Diego, CA, 1992.

[44] American National Standard ASME B89. 1.12-1990, "Methods for the Performance

Evaluation of Coordinate Measuring Machines," American National Standards

Institute, New York, NY, 1990.

[45] German Standard VDWDE 2617, "Accuracy of Coordinate Measuring Machines,

Characteristic Parameters and Their Checking," German Electrical Engineers (Verband

Deutscher Elektrotechniker, VDE) and the Association of German Engineers (Verein

Deutscher Ingenieure, VDI), available from Beuth Verlag, D-1000 Berlin, December

1989.

[46] Japanese Industrial Standard JIS B 7440, "Test Code for Accuracy of Coordinate

Measuring Machines," Japanese Standards Association, Tokyo, Japan, 1987.

72
[47] British Standard BS 6808, "Coordinate Measuring Machines, Part 2: 1987, Methods for

Verifying Performance," British Standards Institute, London, UK, 1987.

73
Appendix: NIST Policy on Statements of Uncertainty Associated

With Measurement Results^

A measurement result is complete only when accompanied

by a quantitative statement of its uncertainty. This policy

requires that NIST measurement results be accompanied by

such statements and that a uniform j^proach to expressing

measurement uncertainty be followed.

1. Background

Since the early 1980's, an international consensus has been

developing on a uniform approach to the expression of

imcertainty in measurement. Many of NIST's sister national standards laboratories as well as a number of in:^)ortant metrological organizations, including the Western European

Calibration Cooperation (WECC) and EUROMET, have

adopted the approach recommended by the International Committee for Weights and Measures (CIPM) in 1981 [1] and reaffirmed by the CIPM in 1986 [2].

Equally important, the CIPM approach has come into use in a significant number of areas at NIST and is also becoming

accepted in U.S. industry. For example, the National

Conference of Standards Laboratories (NCSL) is using it to develop a Recommended Practice on measurement

imcertainty for NCSL member laboratories.

The CIPM approach is based on Recommendation INC-1

(1980) of the Working Group on the Statement of

Uncertainties [3]. This group was convened in 1980 by the

International Bureau of Weights and Measures (BIPM) in response to a request by the CIPM. More recently, at the request of the CIPM, a joint BIPM/IEC/ISO/OIML working

group developed a comprehensive reference document on

the general application of the CIPM approach titled Guide

to the Expression of Uncertainty in Measurement [4] (lEC:

International Electrotechnical Commission; ISO:

International Organization for Standardization; OIML:

International Organization of Legal Metrology). The

development of the Guide is providing further impetus to the worldwide adoption of the CIPM approach.

2. Policy

All NIST measurement results are to be accompanied by

quantitative statements of uncertainty. To ensure that such

statements are consistent with each other and with present

international practice, this NIST policy adopts in substance

the approach to expressing measurement uncertainty

recommended by the International Committee for Weights

and Measures (CIPM). The CIPM ^proach as adapted for use by NIST is:

1) Standard Uncertainty: Represent each component of

imcertainty that contributes to the uncertainty of the measurement result by an estimated standard deviation u,, termed standard uncertainty, equal to the positive square root of the estimated variance u].

2) Combined Standard Uncertainty: Determine the combined standard uncertainty u,. of the measurement

result, taken to represent the estimated standard deviation

of the result, by combining the individual standard

uncenainties w, (and covariances as appropriate) using the usual "root-sum-of-squares" method, or equivalent

established and documented methods.

Commonly, u^ is used for reporting results of

determinations of fundamental constants, fundamental

metrological research, and international comparisons of

realizations of SI units.

3) Expanded Uncertainty: Determine an expanded

uncertainty U by multiplying u^ by a coverage factor k: U = ku^. The purpose of U is to provide an interval y

- U to y + U about the result y within which the value of Y, the specific quantity subject to measurement

and estimated by y, can be asserted to lie with a high

level of confidence. Thus one can confidently assert that y-U<Y<y-\-U,ox equivalently, Y = y ±U.

lA
Use expanded uncertainty U to report the results of all NIST measurements other than those for which u^ has

traditionally been employed. To be consistent with

airrent international practice, the value of ^ to be used

at NIST for calculating U is, by convention, k = 2. Values of k other than two are only to be used for

specific {^plications dictated by established and

documented requirements.

4) Reporting Uncertainty: Report U together with the coverage factor k used to obtain it, or report u^.

When reporting a measurement result and its uncertainty,

include the following information in the report itself or by referring to a published document:

- A list of all components of standard uncertainty,

together with their degrees of freedom where

appropriate, and the resulting value of w^. The

components should be identified according to the method used to estimate their numerical values: A. those which are evaluated by statistical methods,

B. those which are evaluated by other means.

- A detailed description of how each component of

standard imcertainty was evaluated.

- A description of how k was chosen when k is not taken equal to two.

It is often desirable to provide a probability

interpretation, such as a level of confidence, for the

interval defmed by U or u^. When this is done, the basis for such a statement must be given.

Additional guidance on the use of the CIPM ^proach at NIST may be found in Guidelines for Evaluating and

Expressing the Uncertainty of NIST Measurement Results

[5]. A more detailed discussion of the CIPM approach is given in the Guide to the Expression of Uncertainty in Measurement [4]. Classic expositions of the statistical evaluation of measurement processes are given in references

[6-8].

3. Responsibilities

a. Operating Unit Directors are responsible for compliance

with this policy.

b. The Statistical Engineering Division, Computing and

Applied Mathematics Laboratory, is responsible for providing technical advice on statistical methods for evaluating and expressing the uncertainty of NIST

measurement results.

c. NIST Editorial Review Boards are responsible for ensuring that statements of measurement uncertainty are included in NIST publications and other technical outputs

under their jurisdiction which report measurement results and that such statements are in conformity with this policy.

d. The Calibrations Advisory Group is responsible for ensuring that calibration and test reports and other technical outputs imder its jurisdiction are in compliance with this policy.

e. The Standard Reference Materials and Standard

Reference Data programs are responsible for ensuring that technical outputs imder their jurisdiction are in compliance

with this policy.

f. Authors, as part of the process of preparing manuscripts

and other technical outputs, are responsible for formulating

measurement uncertainty statements consistent with this policy. These statements must be present in drafts submitted

for NIST review and approval.

4. Exceptions

It is imderstood that any valid statistical method that is technically justified imder the existing circumstances may be

used to determine the equivalent of w,, u^, or U. Further, it

is recognized that international, national, or contractual agreements to which NIST is a party may occasionally

require deviation from this policy. In both cases, the report

of uncertainty must document what was done and why

.

5. References Cited

[1] CIPM, BIPM Proc.-Verb. Com. Int. Poids et Mesures

49, 8-9, 26 (1981) (French); P. Giacomo, "News

from the BIPM," Metrologia 18, 41-44 (1982)

(English).

[2] CIPM, BIPM Proc. Verb. Com. Int. Poids et Mesures

54, 14, 35 (1986) (French); P. Giacomo, "News

from the BIPM," Metrologia 24, 45-51 (1987)

(English).

75
[3] R. Kaarls, "Rapport du Groupe de Travail sur I'Expression des Incertitudes au Comit6 International des Poids et Mesures," Proc.-Verb. Com. Int. Poids

1 et Mesures 49, A1-A12 (1981) (French); P. Giacomo,

"News from the BIPM," Metrologia 17, 69-74

(1981) (English). [Note that the final English version

of Recommendation INC-1 (1980), published in an

internal BIPM report, differs slightly from thai given

in the latter reference but is consistent with the

authoritative French version given in the former

reference.]

[4] ISO, Guide to the Expression of Uncertainty in Measurement, prepared by ISO Technical Advisory

Group 4 (TAG 4), Working Group 3 (WG 3), to be

published (1993). ISO/TAG 4 has as its sponsors BIPM, lEC, IFCC (International Federation of

Clinical Chemistry), ISO, lUPAC (International

Union of Pure and Applied Chemistry), lUPAP

(International Union of Pure and Applied Physics),

and OIML. Although the individual members of

WG 3 were nominated by BIPM, lEC, ISO, or OIML, it is expected that the Guide will be published

by ISO in the name of all seven organizations. The

most recent draft of the Guide is available from the Calibration Program.

[5] B. N. Taylor and C. E. Kuyatt, Guidelines for

Evaluating and Expressing the Uncertainty of NIST

Measurement Results, NIST Technical Note 1297,

prepared under the auspices of the NIST Ad Hoc

Committee on Uncertainty Statements (U.S.

Government Printing Office, Washington, DC, 1992).

[6] C. Eisenhart, "Realistic Evaluation of the Precision

and Accuracy of Instrument Calibration Systems," J. Res. Natl. Bur. Stand. 67C, 161-187 (1963).

Reprinted, with corrections, in Precision Measurement

and Calibration: Statistical Concepts and Procedures,

NBS Special Publication 3(X), Vol. I, H. H. Ku,

Editor (U.S. Government Printing Office,

Washington, DC, 1969), pp. 21-48.

[7] J. Mandel, The Statistical Analysis of Experimental

Data, Interscience-Wiley Publishers (New York, NY,

1964, out of print; corrected and reprinted, Dover

Publishers, New York, NY, 1984).

[8] M. G. Natrella, Experimental Statistics, NBS

Handbook 91 (U.S. Government Printing Office,

Washington, DC, 1963); reprinted October 1966 with

corrections.

"Statements of Uncertainty Associated with Measurement Results," NIST

Administrative Manual, Subchapter 4.09, Appendix E, National Institute of

Standards and Technology, Gaithersburg, MD 20899, December 1992.

76 * U.S. G.P.0.:1993-341-832:81013
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