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Uncertainty of dimensional measurments obtained from self-initialized instruments

Dissertation
Author: Vincent Lee
Abstract:
Precision dimensional measurement instruments often contain sensors that can only measure displacement of a moving body from some reference position. In order to measure the length of an object they often require a calibrated artifact to initialize their measurement sensors so that they may provide an absolute measurement instead of displacement. Instruments which can realize a null value, i.e. zero length, don't require one; however instruments which can't need to reference an object of known size. These calibration artifacts also serve as part of the chain of metrological traceability. The group of instruments presented in this dissertation can self-initialize by deriving their own calibrated artifact. These instruments rely on a unique artifact geometry, which is uncalibrated, to determine a length value via a series of displacement measurements provided by the self-initializing instrument. All of the self-initializing instruments described in this dissertation rely on a precision sphere coupling with a three point kinematic seat (TPKS) as the mechanical interface between the instrument and un-calibrated artifact. The combination of the TPKS and sphere are deterministic in nature in defining a point in space, e.g. the location of the center of the sphere relative to the body of the TPKS. In practice, high precision spheres are inexpensively available, and testing has shown that the locational repeatability of the sphere/TPKS coupling to be in the range of the surface roughness of the spheres, thus allowing nanometer-level repeatability. The combination of this feature and the displacement measurement sensors in these instruments allow the instrument to directly measure length without resorting to a measure of extension. The Laser Ball Bar instrument, an instrument which pioneered the self-initialization method for length measurement instruments, and can't realize a null value measurement for initialization, is functionally decomposed to better understand the requirements for self-initialization. Two instruments that fulfill these requirements will be presented as case studies of how a self-initialized instrument may be designed, and constructed. Measurement uncertainty with these instruments, using self-initialization, and initialization with an independently calibrated artifact will be explored. A complete uncertainty analyses are provided for both instruments using both the self-initialization mode and the calibrated artifact mastering mode of operation; and the predicted results are compared to experimental measurement data. This dissertation: (1) Derives and/or explains the geometric conditions which enable self-initialization in an instrument (2) Describes two novel instruments that are capable of self-initialization (3) Provides an uncertainty analyses for these instruments when they are self-initialized and when they are initialized using a master artifact (4) Compares and contrasts the achievable uncertainty for each mode of use, and (5) Provides conditions under which lower uncertainty is achievable using self-initialization.

TABLE OF CONTENTS Page TITLE PAGE .................................................................................................................................. i ABSTRACT ................................................................................................................................... ii DEDICATION ............................................................................................................................... iii ACKNOWLEDGMENTS .............................................................................................................. iv LIST OF TABLES ....................................................................................................................... vii LIST OF FIGURES .....................................................................................................................viii CHAPTER 1 OVERVIEW OF RESEARCH ................................................................................................... 1 Introduction ........................................................................................................ 1 Background of Research ................................................................................... 4 Metrological Traceability and Measurement Uncertainty ................................ 13 Motivation and Scope of Research ................................................................. 16 Outline of Dissertation ..................................................................................... 18 2 SELF-INITIALIZED ONE DIMENSIONAL MEASUREMENT INSTRUMENTS ...................... 20 Introduction ...................................................................................................... 20 The Self-Initialized Instrument System ............................................................ 21 Functional Requirements Necessary for Self-Initialization in One Dimension 28 Measurement Uncertainty of Self-initialization ................................................ 29 Concluding Remarks ....................................................................................... 34 3 A MACHINE FOR MEASURING BALL BARS UPTO 3 METERS IN LENGTH ..................... 35 Introduction ...................................................................................................... 35 Instrument Design ........................................................................................... 36 Initializing the 1-DMM ...................................................................................... 39 Measuring Ball Bars ........................................................................................ 43 Uncertainty of Ball Bar Measurements ............................................................ 44 Ball bar Measurement Results ........................................................................ 55 Discussion of Results ...................................................................................... 59 Concluding Remarks ....................................................................................... 62 4 UNCERTAINTY OF ABBE OFFSET ERROR CORRECTIONS IN ONE DIMENSION ......... 64 Introduction ...................................................................................................... 64 In Situ Abbe Offset Error Estimation ............................................................... 65 Uncertainty of Abbe Offset Error Estimations in 1-Dimension ........................ 69 Assigning a Correlation Coefficient ................................................................. 72 Experimental Setup ......................................................................................... 73 Results ............................................................................................................. 76 Discussion ....................................................................................................... 80 Uncertainty in estimating yaw, and displacement of POI ................................ 81

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Concluding Remarks ....................................................................................... 83 5 INSTRUMENT FOR GAUGEING LARGE RING SHAPED OBJECTS .................................. 85 Introduction ...................................................................................................... 85 Instrument Design ........................................................................................... 87 Initialization of Angular Encoders .................................................................... 93 Concluding Remarks on Angle initialization .................................................... 99 Uncertainty Analysis of Circular Measurement ............................................... 99 Experimental Results ..................................................................................... 103 Discussion of Results .................................................................................... 109 Concluding Remarks ..................................................................................... 109 6 CONCLUDING REMARKS ................................................................................................... 110 APPENDICES .......................................................................................................................... 113 CALIBRATION CERTIFICATE FOR A GAUGE BLOCK .............................. 114 TYPE B UNCERTAINTY ANALYSIS EXAMPLE .......................................... 116 REFERENCES ........................................................................................................................ 121

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TABLE OF TABLES Table Page Table 3-1: Assumed conditions for Type B uncertainty analysis ................................................... 47 Table 3-2: Properties and standard uncertainties for ball bar ....................................................... 53 Table 3-3: Uncertainty of 1-DMM initialization ............................................................................... 54 Table 3-4: Expected ball bar measurement uncertainties (k=2).................................................... 56 Table 3-5: Measurements obtained using mastered gauging method. ......................................... 57 Table 3-6.Nominal dimensions of 3-BBBs used for initializing 1-DMM ......................................... 58 Table 3-7. Measurements obtained using masterless gauging. .................................................... 58 Table 3-8: Initialization lengths derived for each 3-ball ball bar using self-initialization ................ 59 Table 4-1: Correlation coefficients for noise versus beam spacing, and condition ....................... 79 Table 4-2: Machine and measurement system parameters .......................................................... 82 Table 5-1: Values for influencing quantities for initializing angle with a calibrated artifact............ 96 Table 5-2: Estimated measurement uncertainty for measuring a 546mm ring ........................... 102 Table 5-3: Measurement results from repeat mountings onto calibration fixture ........................ 103 Table 5-4: Angle measured by angle encoders compared to CMM ............................................ 106 Table 5-5: Measurement results from repeat initializations using initialization fixture; results for 10 initializations and measurements................................................................... 107 Table 5-6: Repeated measurements of the OD on a 546.12 mm ring ........................................ 108

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TABLE OF FIGURES Figure Page Figure 1-1: Position, distance, and extension are used to quantify the dimensions of a machine part. ............................................................................................................... 2 Figure 1-2: Initializing a length comparator, and measuring a part of unknown size ...................... 4 Figure 1-3 : Qualifying a CMM stylus with a sphere of knonw size ................................................. 6 Figure 1-4: Simple CMM assembly showing a 2D coordinate system (left), two points probed at opposing ends determine part‟s lateral dimension, after probe radius is compensated (right) ..................................................................................................... 6 Figure 1-5: Laser Ball Bar (LBB) concept and component layout [7, 10] ........................................ 8 Figure 1-6: Quasi Kinematic seat detail; points of contact exposed, and ball mounted .................. 8 Figure 1-7: Self-initialization procedure of LBB [12] ........................................................................ 9 Figure 1-8: One Dimension Measuring Machine (1-DMM) [8] ...................................................... 11 Figure 1-9: Initialization and ball bar measurement sequence for 1-DMM currently installed at NIST ........................................................................................................................... 12 Figure 1-10: Traceability chain for a measurement value to the fundamental SI Unit of measure for length [18] .............................................................................................. 14 Figure 2-1: Sphere and three point kinematic seat coupling ......................................................... 22 Figure 2-2: Three-ball kinematic seat [29] ..................................................................................... 23 Figure 2-3: Laser Ball Bar (LBB) concept and component layout [7, 10] ...................................... 24 Figure 2-4: Quasi Kinematic seat detail; points of contact exposed, and ball mounted ................ 24 Figure 2-5: Laser ball bar (LBB), critical components displayed ................................................... 25 Figure 2-6: Calibration artifact for LBB; distance between seats 1, 2, & 3 are unknown .............. 26 Figure 2-7: First step constrains LBB, followed by "zeroing" the displacement sensor ................ 26 Figure 2-8: The distance between two TPKS is measured during step 2 ..................................... 27 Figure 2-9: Initialize using the distance recorded in step 2 ........................................................... 27 Figure 2-10: LBB initialization fixture, non-co-linearity of TPKS exaggerated .............................. 34 Figure 3-1: Top view of new 1-DMM ............................................................................................. 36 Figure 3-2: Yaw of sled .................................................................................................................. 37

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Figure 3-3: Final design and layout of 1-DMM instrument ............................................................ 38 Figure 3-4: Self-initialization procedure for the 1-DMM ................................................................. 40 Figure 3-5: Chart displaying lines of constant uncertainty............................................................. 42 Figure 3-6: Rectangular distribution for expected error ................................................................. 48 Figure 3-7: Yaw motion of sled as it travels along the guide way. ................................................ 50 Figure 3-8: New 1-DMM fully constructed ..................................................................................... 55 Figure 3-9. 3-ball ball bar used for initializing 1-DMM ................................................................... 58 Figure 3-10: Measurement results, mastered, and self-initialized (masterless) ............................ 60 Figure 3-11: Sagging of 3-BBB when supported below neutral axis ............................................. 61 Figure 4-1: Abbe offset error with hand held measurement instruments (micrometer vs. dial caliper) ....................................................................................................................... 64 Figure 4-2: An example of how angular displacements induce error motions during translational displacement of guide way .................................................................... 66 Figure 4-3: Angular displacement of carriage induces error motions to the functional point of interest ....................................................................................................................... 67 Figure 4-4: Variation of uncertainty due to Abbe offset as a function of α and r ............................ 71 Figure 4-5: Detail of component layout for 1 Dimensional Measuring Machine [6] ....................... 74 Figure 4-6: Test setup of interferometers on surface plate ........................................................... 75 Figure 4-7: 1-DMM's displacement estimate compared using independent laser interferometer ............................................................................................................. 76 Figure 4-8: Yaw of carriage versus carriage position .................................................................... 77 Figure 4-9: Sample correlation data, 38mm spacing unshielded .................................................. 78 Figure 4-10: Correlation between interferometer Axis 1 & 2 vs. position of 1-DMM's carriage .... 79 Figure 4-11: Displacement of POI, 1-DMM measurement vs. independent laser interferometer ............................................................................................................. 80 Figure 5-1: Bar gauge with a micrometer head for size feedback ................................................. 86 Figure 5-2: Measuring a ring using 3 point contact ....................................................................... 88 Figure 5-3: Front view of instrument, encoders and styli shown ................................................... 88 Figure 5-4 : Large ring gauge measuring a ring. ........................................................................... 89

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Figure 5-5: Measuring taper of a cone .......................................................................................... 90 Figure 5-6: Setting the stylus standoff distance using a gage-block stack.................................... 92 Figure 5-7: Large Ring Gauge fully assembled and resting on initialization fixture. ..................... 93 Figure 5-8: Initializing with an artifact of known length, and using law of cosine to resolve an initial angle (length of arms need to be known) ......................................................... 94 Figure 5-9: Large ring gauge can mount onto calibration fixture in two deterministic positions ... 97 Figure 5-10: Three step initialization and calibration procedure ................................................... 97 Figure 5-11: Plot of repeatability of multiple mountings on calibration fixture ............................. 104 Figure 5-12: Arms of instrument experience torsion due to off axis loading ............................... 105 Figure 5-13: Plot of repeatability of multiple initializations using calibration fixture .................... 107 Figure 5-14: Ring measurement repeatability, measuring a ring with a nominal OD of 546.12mm. ............................................................................................................... 108 Figure 7-1: Calibration certificate for a 1 inch gauge block (Courtesy of Mitutoyo) .................... 114 Figure 7-2: Rectangular distribution of error band ....................................................................... 118

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CHAPTER ONE 1 OVERVIEW OF RESEARCH This section provides an introduction to this dissertation‟s topic by providing relevant background material, outlining the scope of the research, and its relationship to practical applications. Introduction

All length measurement instruments in some fashion, relate the physical boundaries of objects which they measure to the fundamental unit of measure, the meter. Many of these instruments provide a measurement result by making physical contact with objects. These points of contact are linked to a graduated scaled, via mechanical framework, to define their position relative to other points or boundaries on the object. The graduations on the scales are calibrated to ensure that each measurement accurately represents the correct proportion of the meter. Swyt [1, 2] outlined four dimensional measurement types, they are: „position – location of an object in space displacement – the change in location of an object over a time interval distance – the difference in location of two objects at the same time extension – the distance between points on opposing-face boundaries of an object” Objects subjected to be measured can be a simple geometry such as a sphere, or may be an object which contains other features, such as a rectangular block that contain multiple features such as holes, pockets, and slots. A measurement instrument is used to assign a dimension to the object and its features by utilizing at least one of the four dimensional measurement types [3]. Referring to the following figure (Figure 1-1) as an example, extension is used to define the dimensions of the block, the slot, and the hole by measuring the extents of their boundaries. Distance is used to define the relative location of each “object”, and their boundaries, to each other. The position of the center of the hole and slot, relative to a coordinate system defined by reference surfaces on the body are derived from measures of extension and distance. The results from these measurement types assign an absolute dimension, in units of length, to each feature.

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Figure 1-1: Position, distance, and extension are used to quantify the dimensions of a machine part. To perform absolute dimensional measurements, for instruments with sensors that can only measure displacement, a means to initialize an instrument‟s measurement system to a known value is required. The term initialization means to “set to a value” [4]. For a dimensional measurement instrument, initialization “sets” the instrument to a known reference value, which displacement is measured from. Initialization essentially performs what is defined under definition 3.11 in the International Vocabulary of Basic General Terms in Metrology (VIM), as “a adjustment of a measurement system”, which reads [3]: “set of operations carried out on a measuring system so that it provides prescribed indications corresponding to given values of a quantity to be measured.” This definition is followed by three important notes: “NOTE 1: Types of adjustment of a measuring system included zero adjustment of a measuring system (definition 3.12 in the VIM), offset adjustment, and span adjustment (sometimes called gain adjustment). NOTE 2: Adjustment of a measuring system should not be confused with calibration, which is a prerequisite for adjustment. P Dt E = position = extension = distance P E P P E E E Dt

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NOTE 3: After an adjustment of a measuring system, the measuring system must usually be recalibrated.” “Note 1” defines the type of „measurement system adjustments‟ and reflects the adjustment (offset adjustment) that initialization accomplishes. “Note 2” specifically outlines that the actions outlined in “Note 1” are not calibration. For instruments which are subject to initialization or adjustment of its measurement system, the displacement sensor(s) contained by the instrument are considered to be calibrated such that each graduation is a known proportional representation of a fundamental unit of measure. The process of initialization or scale offset adjustment is often confused by even the most technically experienced individuals as calibration, which has its own definition in the VIM under definition 2.39, which is: “operation that, under specified conditions, in a first step, establishes a relation between the quantity values with measurement uncertainties provided by measurement standards and corresponding indications with associated measurement uncertainties provided by measurement standards and corresponding indications with associated measurement uncertainties and, in a second step, uses this information to establish a relation for obtaining a measurement result from an indication” As for “Note 3” indicating that an instrument “must usually be recalibrated” after an adjustment to its measurement system, I believe this to be true for gain adjustments, since a gain adjustment may alter the calibration of the measurement system. In the case of an offset or zero adjustment, recalibration shouldn‟t be necessary since the scaling of the measurement system is preserved. There are two possible initialization techniques self-initialization, and initialization with an independently calibrated artifact. A measurement instrument such as a dial caliper can butt their measurement surfaces together to provide a zero-length value, and thus can self-initialize or “zero” its displacement measurement sensor, since it can realize a null measurement value. Others type of instruments, such as a gauge block comparator, which can‟t perform such a feat will require a calibrated artifact of known dimensions to serve as a reference standard to set its displacement sensor to a known initial value [5]. More recently developed instruments which

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contain measurement points that aren‟t able to touch each other, or otherwise realize a zero- length condition, are still able self-initialize [6-9]. However these instruments utilize an initialization method, via an un-calibrated artifact that has no metrological traceability to a fundamental unit of measure. Without metrological traceability, a measurement uncertainty can‟t be confidently assigned to a measured value, thus compromising any assurance one would have in the results provided by the instrument. In order to maintain traceability these instruments must rely on alternate methods such as external calibration of their displacement sensors.

Background of Research

Traditional Measurement Instruments Instruments which are unable to self-initialize reference a calibrated artifact prior to measuring a part of unknown size. For example, consider a comparator type instrument which comprises of a rigid frame, an anvil, and a calibrated displacement sensor; the sensor‟s sensitive direction is normal to the surface of the anvil (Figure 1-2).

Figure 1-2: Initializing a length comparator, and measuring a part of unknown size One way to initialize the comparator is to bring the anvil and the tip of the sensor into physical contact so that a null measurement value can be realized. However since the sensor has a limited range of travel and is unable to bring its tip into contact with the anvil, it would be unable to perform a zero adjustment of the measurement system onto itself. Therefore a part of known 0 . 0000 mm M P M Master gauge block ( master part ) is used to set sensor output to “ 0 ” Gauge block , unknown length is place under sensor . Displacement of gauge indicates its size relative to master . - 0 . 0012 mm Anvil Anvil

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size (the master part) is used to initialize the instrument by setting a known offset from the anvil‟s surface to the sensor‟s tip. This procedure is often referred to as “mastering” an instrument. When the master part is placed in between the anvil and the sensor‟s tip, the output value of the displacement sensor is “zeroed”; analogous to accounting for a tare weight for weight measurements. Other parts of unknown size may now be measured by placing them in between the anvil and the sensor‟s tip, as long as the displacement limits of the sensor aren‟t exceeded (Figure 1-2). By measuring a calibrated master part of known size and dimensional uncertainty, the size of the new parts and the uncertainty of that size can be estimated. Another example of a measurement system which requires a calibrated artifact for initialization of its measurement sensor is the coordinate measuring machine (CMM). A CMM measures a part by physically touching various points along its surface(s), typically using a touch probe which carries a spherically tipped stylus. The points which the stylus contacts are transferred to a measurement scale via a mechanical link to a reference coordinate system. The most common modern CMM embodies a 3-D Cartesian coordinate system to provide the coordinates of the stylus center at each of the points which are probed. However, in order to accurately determine the location of the contact points on the part, the dimensions of the stylus need to be known (typically its tip radius). This is important since a CMM actually tracks the position of the probe holder assembly; which is set by its manufacturer. The relative position of the stylus‟s tip to the probe assembly is set by the end user since there are a seemingly infinite number of stylus geometries available. Qualification of a stylus is typically performed by touching a precision sphere of known size and form error, at specific locations, which afterwards the tip radius of the stylus may be determined. During this qualification process the CMM measures the sphere without compensating for the stylus radius or offset. The coordinates of the probed points are best fit to spherical geometry and the relative coordinates of the best-fit sphere center and the known location of the physical sphere center are used to determine the relative position of the stylus tip center and the probe holder. The radius of the best fit sphere will be larger than the

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actual radius of the physical sphere since it passes through the stylus center, not the contact points (Figure 1-3).

Figure 1-3 : Qualifying a CMM stylus with a sphere of knonw size However, if the physical sphere‟s size is known, the radius of the stylus tip can be resolved by calculating the difference between measured sphere size and the actual size. With the styli‟s radius known, the location of the point of contact at a given part may be resolved to the probe assembly, by accounting for the stylus‟s offset and tip radius, and then to the reference coordinate system (Figure 1-4).

Figure 1-4: Simple CMM assembly showing a 2D coordinate system (left), two points probed at opposing ends determine part‟s lateral dimension, after probe radius is compensated (right) r M r f r M = calibrated size of sphere r f = best fit sphere radius by CMM PART r Probe assembly Measurement scales CMM Reference Frame r

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From a conceptual standpoint, to avoid the stylus qualification process, the most ideal stylus radius is zero, so that a calibrated artifact needs not be referenced. With a stylus of zero radii (a null value) any point measured on a part can be directly related to the scale without any intermediary length to transfer that point. Sine a stylus of zero radiuses isn‟t physical possible, one must rely on a calibrated artifact to qualify the stylus. For these two examples, a calibrated artifact was necessary to initialize each instrument to some known value to enable an absolute dimensional measurement. This is due to the nature of their sensors inability to self-initialize to a null value or null point. Instruments which are able to self-initialize to a null value by bringing their measurement points of contact together, have been limited to “C-frame” shaped instruments such as a dial caliper, and short range micrometers. By butting the measurement points of these instruments together and “zeroing” the sensor‟s output indicator, initialization has been completed. Self-Initialized Instruments Novel instruments which do not rely on master artifacts for measurement scale initialization have been developed in recent years. However, these instruments don‟t self-initialize by realizing a null measurement value, like the dial caliper or micrometer. They rely on an artifact or fixture to capture a displacement measurement from the instrument, directly convert it to a length, which may then be reused to initialize the same instrument. A physical embodiment of such an instrument is the Laser Ball Bar (LBB) system, by Ziegert et al.; illustrated in the following figure (Figure 1-5) [7, 10].

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Figure 1-5: Laser Ball Bar (LBB) concept and component layout [7, 10] The LBB is an instrument consisting of three concentric precision telescopic tube assemblies which houses a set of interferometer optics, and two precision spheres, on each end of the instrument. As the tube extends, the laser interferometer system measures the relative displacement of the optics, i.e. the change in distance between the sphere centers. The spherical ball ends on this instrument permit precision interface to a specially designed three-point kinematic seat (TPKS) [11]. Each of these TPKS‟s has three precision points which make contact with the ball (Figure 1-6). These three contacts constrain all translation motion of the center of the sphere, but do not restrict any rotation of the sphere.

Figure 1-6: Quasi Kinematic seat detail; points of contact exposed, and ball mounted 3 Contact Points

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This feature allows the LBB to interface with measurement points with excellent repeatability. To initialize the laser interferometer to measure an absolute distance between the sphere centers, an un-calibrated artifact, with three TPKS‟s, is used in combination with the following procedure (Figure 1-7).

Figure 1-7: Self-initialization procedure of LBB [12] In the procedure outlined in Figure 1-7, the first measurement of displacement, when Ball B is moved from seat 2 to seat 3, also was a measurement of the previously unknown distance between points 2 and 3 on the artifact. This distance, measured from the center of seat 2 to the center of seat 3, is in turn used to initialize the displacement measurement sensors of the LBB in STEP 1 Set ball A and B of laser ball bar into seats 1 and 2 ; “zero” the instrument STEP 2 Unseat ball B from seat 2 and place onto seat 3 while extending LBB . Record the distance displaced during movement ; this measured length will be used to initialize the laser encoder . Calibration Artifact STEP 3 Unseat ball A from seat 1 and seat ball A onto seat 2 . Enter the value recorded from STEP 2 to initialize the instrument . 1 2 3 1 2 3 1 2 3 LBB Kinematic Seats L A B A B A B

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step 3, so that now the instrument always provides absolute distance between the ball centers of the instrument. The calibration artifact used in the LBB measurement system doesn‟t require any independent calibration. That is, the distance between seats 1, 2 and 3, of the calibration artifact in is unknown and may change over time. To self initialize the LBB all that is required is the following:  The measurement instrument measures the artifact in more than one position  The artifact remains dimensionally stable during the short time that is needed to manipulate the instrument on the calibration artifact.  The laser interferometer displacement measurement is accurate and repeatable  Interface with the balls of the LBB and the kinematic seats are repeatable. While it would be possible to initialize the LBB using an externally calibrated master artifact, such as a bar with two sockets a known distance apart, utilizing self-initialization has some key advantages. As already mentioned, a calibrated artifact is not necessary, by doing so the expense of acquiring and periodically recertifying a precision artifact is eliminated. Calibration of precision artifacts will always include a set of defined conditions, where the calibration is valid; usually 20°C for simple parts [13]. In the case of the LBB, it is not unusual for it to be used in environments of varying temperature. Using an artifact which has a calibration that is only valid at a single temperature to initialize an instrument, such as the LBB, would yield an erroneous adjustment of its measurement system; unless the length of the artifact is corrected for thermal expansion. By utilizing a self-initialization method, the initialization of the LBB may be performed under the same operating conditions where the measurement is taking place. A properly designed artifact for self-initialization only needs to remain dimensionally stable during the amount of time that it takes to perform the self-initialization procedure. Another example of a self-initialized instrument is the One Dimensional Measuring Machine (1-DMM) used for measuring fixed length ball bars, by Ziegert et al [7]. As the name

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would suggest, a ball bar is comprised of a fixed-length bar or rod with two or more balls attached along its length, the simplest of these is made of a rod with a precision ball attached to each end. The following figure illustrates the 1-DMM (Figure 1-8) [8].

Figure 1-8: One Dimension Measuring Machine (1-DMM) [8] This instrument is constructed by using a granite straight edge, with two air bearing supported sleds straddling it. Each sled contains a kinematic seat and retro-reflector, with another kinematic seat fixed to the center mount; these seats serve as the measurement points for the ball bar. Laser interferometers are used to track the displacement of each sled as it glides across the granite; travel is limited by the fixed center mount and the metrology frame. When the 1-DMM is initially powered on, the machine has no knowledge of the relative position of the sleds to each other, or to the fixed center mount. In order to utilize the 1-DMM as a measurement machine, a precise datum needs to be set. Initialization of the 1-DMM‟s displacement measuring system could be accomplished by using a ball bar of known length, as a reference or master-part, or self- initialization may be used. The unique design of the 1-DMM was created with self-initialization ability in mind. The three kinematic seats and their arrangement allows for simultaneous initialization of the 1-DMM, and measurement of the ball bar. This procedure is outlined following figure (Figure 1-9).

Full document contains 136 pages
Abstract: Precision dimensional measurement instruments often contain sensors that can only measure displacement of a moving body from some reference position. In order to measure the length of an object they often require a calibrated artifact to initialize their measurement sensors so that they may provide an absolute measurement instead of displacement. Instruments which can realize a null value, i.e. zero length, don't require one; however instruments which can't need to reference an object of known size. These calibration artifacts also serve as part of the chain of metrological traceability. The group of instruments presented in this dissertation can self-initialize by deriving their own calibrated artifact. These instruments rely on a unique artifact geometry, which is uncalibrated, to determine a length value via a series of displacement measurements provided by the self-initializing instrument. All of the self-initializing instruments described in this dissertation rely on a precision sphere coupling with a three point kinematic seat (TPKS) as the mechanical interface between the instrument and un-calibrated artifact. The combination of the TPKS and sphere are deterministic in nature in defining a point in space, e.g. the location of the center of the sphere relative to the body of the TPKS. In practice, high precision spheres are inexpensively available, and testing has shown that the locational repeatability of the sphere/TPKS coupling to be in the range of the surface roughness of the spheres, thus allowing nanometer-level repeatability. The combination of this feature and the displacement measurement sensors in these instruments allow the instrument to directly measure length without resorting to a measure of extension. The Laser Ball Bar instrument, an instrument which pioneered the self-initialization method for length measurement instruments, and can't realize a null value measurement for initialization, is functionally decomposed to better understand the requirements for self-initialization. Two instruments that fulfill these requirements will be presented as case studies of how a self-initialized instrument may be designed, and constructed. Measurement uncertainty with these instruments, using self-initialization, and initialization with an independently calibrated artifact will be explored. A complete uncertainty analyses are provided for both instruments using both the self-initialization mode and the calibrated artifact mastering mode of operation; and the predicted results are compared to experimental measurement data. This dissertation: (1) Derives and/or explains the geometric conditions which enable self-initialization in an instrument (2) Describes two novel instruments that are capable of self-initialization (3) Provides an uncertainty analyses for these instruments when they are self-initialized and when they are initialized using a master artifact (4) Compares and contrasts the achievable uncertainty for each mode of use, and (5) Provides conditions under which lower uncertainty is achievable using self-initialization.