METHOD FOR PRECISION CALIBRATION OF ROTARY SCALE ERRORS AND PRECISION DETERMINATION OF GEAR TOOTH INDEX ERRORS

Mechanical Systems and Signal Processing (1998) 12(6), 723–752 Article No. pg980172

METHOD FOR PRECISION CALIBRATION OF ROTARY SCALE ERRORS AND PRECISION DETERMINATION OF GEAR TOOTH INDEX ERRORS W. D. M Applied Research Laboratory and Department of Mechanical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802 , U.S.A.

(Received March 1998 , accepted after revisions July 1998) A method using the ‘closure principle’ is developed for precision calibration of rotary scale errors of gear measurement machines and precision calibration of gear tooth index errors (accumulated tooth-spacing errors) after ‘removal’ of rotary scale errors. The method uses the standard machine procedure for measuring gear tooth index errors applied to a spur gear artifact mounted between machine centers. Therefore, the rotary scale calibrations include consistent effects arising from eccentricities of machine gear-mounting centers and scale-mounting center relative to the instantaneous axis of table rotation, wobble of the instantaneous axis of table rotation, as well as scale graduation errors, etc. Gear-artifact index errors are referenced to the axis connecting the mounting centers located on the gear. Successful implementation of the method does not require super precision of the gear artifact. A method for obtaining approximate uncertainties (standard deviations) of both scale and index error calibrations is developed that utilises the same measurement data required for the scale and index error calibrations. The developed methods are illustrated by applications to scale and index error calibrations obtained from multiple sets of measurements. Typical standard deviations achieved are under 0.05 mm (1/10th of the wavelength of light). Good consistency is achieved between predicted and measured results. The general methods developed should be useful for other types of rotary axis calibrations. 7 1998 Academic Press

1. INTRODUCTION

Virtually all gears are designed and manufactured with the goal of transmitting exactly constant angular velocity ratios between rotating shafts. Imperfections in the contacting surfaces of gear teeth (together with tooth and gearbody elastic deformations) are a source of undesirable variability in the transmission of such angular velocity ratios by meshing gear pairs. Several manufacturing firms build precision measurement machines dedicated to the task of measuring such imperfections in the contact surfaces of gear teeth. These machines typically possess three linear axes and a rotary axis. The measurement accuracies achieved by these machines are directly dependent on the achievable positioning accuracies of the linear and rotary axes. As is the case with all machine elements, the rotary axes of gear measurement machines contain imperfections. Such imperfections include lack of concentricity between the mounting center of the rotary scale and the axis of rotation, lack of concentricity between the machine gear mounting centers and the axis of rotation, wobble in the instantaneous center of the axis of rotation, and errors in the graduations of the rotary scale. In the present paper, a measurement procedure is developed to measure the combined effects of 0888–3270/98/060723 + 30 $30.00/0

7 1998 Academic Press

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. . 

the consistent (i.e. repeatable) component of such rotary axis imperfections, and a very simple computational algorithm is provided that can be applied to correct for these rotary axis imperfections using the axis measurement data. A high-quality spur gear is used as the measurement artifact, thereby insuring that imperfections in the machine gear mounting centers are included in the corrections. The successful implementation of the method does not require super precision in the gear artifact. An additional algorithm is provided that corrects the measured index errors (accumulated or absolute tooth-spacing errors [1]) of the spur gear artifact for the above-described rotary-axis imperfections, thereby providing a method for the precise determination of gear tooth index errors. Finally, it is shown how inconsistencies in the measurement data can be used to estimate the uncertainties in both the calibrations of the rotary-axis imperfections, and the computed index errors of the gear artifact after correction for the rotary-axis imperfections. The tooth contact surfaces of a spur gear artifact with N teeth divide the pitch or base circle [1] of the gear into N nominally equal intervals. Thus, when the artifact is mounted on the rotary table center, the tooth contact surfaces also divide the rotary scale into N nominally equal intervals. The rotary scale calibration method to be described provides scale corrections at the intersections of each of these N nominally equal scale intervals. The method depends on the principle that the sum of the adjacent tooth-spacing errors, measured between the N tooth contact surfaces around the full circumference of the artifact gear, is exactly zero. Similarly, the artifact calibration method depends on the principle that the sum of the N scale-error increments around the full 360° of the rotary scale also is exactly zero. In both cases, a circle has been ‘closed’ [2]. Proof of the validity of the methods, and the formulae for their practical implementation, would appear to have some significant advantages in simplicity in comparison with existing methods [3–8] that have been applied to the calibration of angular encoders, precision polygons, indexing tables, etc. 2. MEASUREMENT OF GEAR TOOTH INDEX ERRORS AND ERROR MODEL

Both the rotary-axis calibration procedure and the artifact index error determination procedure can be carried out using measurements obtained by the standard method for measuring gear tooth index errors. To obtain such measurements, the artifact gear is centered on the rotary table and rigidly attached so that the gear and table are rotated as a rigid body as the measurements are taken. The number N of nominally equally spaced teeth, and the base circle or pitch circle radius of the artifact gear are supplied to the machine. Measurement of the position of each tooth is made in the direction of the tangent to either the base circle or pitch circle (perferably the base circle). In carrying out such measurements, the CNC controls of the machine sequentially rotate the table and artifact gear by angular increments of 360/N degrees, stopping after each such increment to allow the measurement probe to be positioned automatically at the correct radial location, after which the tooth measurement is taken and recorded (stored). The probe then is retracted, the table advanced by another 360/N degrees to the nominal location of the next tooth, another tooth measurement is taken, and so on until the positions of all N teeth have been measured. Thus, in principle, the probe deflection measures and records the absolute linear deviation (in a direction tangent to either the base circle or pitch circle) from the position each tooth would occupy if the teeth were exactly equally spaced. This deviation is commonly called either the index error or the accumulated tooth spacing error. We shall refer to it as the ‘absolute’ error [1] (with algebraically correct sign). The rotational position of the table at each of the N locations where a tooth measurement is taken is determined by the rotary scale. Thus, any error in the rotary scale

      

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results in a table positioning error that either algebraically adds or subtracts to the measured tooth error; that is, at any given tooth measurement location, the lineal contribution os = Rdu arising from an angular error of du radians in the rotary scale at that location provides an algebraically additive contribution to the actual absolute error ot in the position of the tooth. Hence, the measured error o* t of the tooth is the sum of the actual tooth error ot and the lineal contribution os from the scale error at that particular rotary table location: o* t = ot + os .

(1)

In what follows, a superscript asterisk denotes a measured quantity. Each of the three quantities in equation (1) describes an absolute (rather than adjacent) error. The central goal of the analysis presented herein is to develop a method for determining the absolute scale errors os as a function of angular scale location, and the absolute tooth spacing errors ot as a function of tooth number, utilizing only the measured absolute tooth spacing errors o* t (which contain the scale-error contaminations os ). The absolute scale errors os to be determined contain the above-mentioned contributions from lack of concentricities between both the rotary scale mounting center and gear mounting centre and the axis of rotation, wobble of the instantaneous center of the axis of rotation, and errors in the graduations of the rotary scale. In addition to the usual tooth spacing errors, the absolute tooth spacing errors ot to be determined contain contributions that may exist from eccentricity between the mounting centres on the gear and centre of the base circle used in tooth generation.

3. DETERMINATION OF ABSOLUTE ROTARY SCALE ERRORS

The number N of equispaced table locations at which the rotary scale is to be calibrated is determined by the number N of nominally equispaced teeth to be measured on the artifact gear. It will be seen, shortly, that to obtain the absolute rotary scale errors os at these N locations (and the absolute tooth spacing errors ot of the N teeth uncontaminated by the scale errors) in the most straightforward manner, it is required to make N complete sets of index error measurements on the artifact gear, where, for each such set, the artifact gear is rotated and held fixed relative to its position with respect to the table during the prior set, by ‘exactly’ one tooth spacing interval of the N teeth. The N equispaced teeth on the artifact gear, together with a ‘stop’ fixture rigidly attached to the table, can be used to provide accurate rotational positioning of the gear with respect to the table at each of the required N positions. The error contributions to the calibrations arising from imperfections in positioning of the artifact gear with respect to the table, and imperfections in the spacing of the artifact gear teeth are shown in Appendix A to be negligible under normal circumstances. Figure 1 illustrates one such position of the artifact gear with respect to the table, where the gear has been rotated clockwise by h = 2 tooth spacing intervals relative to its position during the first set of index measurements where the value of h was zero. The inner circle in Fig. 1 represents the rotary table and scale, with N scale locations at which calibrations are to be determined labelled i = 0, 1, . . . , N − 1. The outer circle in Fig. 1 represents the artifact gear, where the N outwardly projecting lines representing the tooth contact surfaces are labelled j = 0, 1, . . . , N − 1. For each integer value of h = 0, 1, . . . , N − 1 the absolute positions of all N tooth contact surfaces on the artifact gear are to be measured and recorded. For each value of h, the N scale locations labelled i = 0, 1, . . . , N − 1 are

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. . 

Figure 1. Relative positions of artifact gear tooth measurement surfaces j = 0, 1, . . . , N − 1 (spokes on outer circle) and locations on rotary scale i = 0, 1, . . . , N − 1 (spokes on inner circle) where these measurements are taken and the scale calibrations are determined. N complete index measurements of artifact gear are made where, after each complete index measurement, artifact gear is rotated relative to the rotary scale by one tooth interval. Index h designates location of artifact gear with respect to rotary scale for each of the N complete index measurements. js(i), i = 0, 1, . . . , N − 1 designates scale error increment between scale locations i and i + 1. jt (j), j = 0, 1, . . . , N − 1 designates (adjacent) tooth spacing error of artifact gear between tooth surfaces j and j + 1.

the nominal scale locations at which the measured positions of the contact surfaces of the N teeth on the gear are taken. Let o* t (j) denote the measured value of the absolute (accumulated) tooth spacing error of tooth j of the artifact gear (relative to some reference tooth). Measurements of o* t (j) are recorded with reference to the rotary scale location, which is designated by index i in Fig. 1. Moreover, tooth index j is related to scale index i by j = h + i,

(2)

where h denotes the number of tooth spacings illustrated in Fig. 1 by which the artifact gear has been rotated relative to the rotary scale for a particular index measurement set. Then, using equation (2), the measured value of the increment in artifact absolute tooth spacing error between scale locations i = l and i = l + m is seen to be I(l, m),o* t (h + l + m) − o* t (h + l),

(3)

where scale locations i = l and i = l + m are illustrated in Fig. 1. The measured values o* t (h + l + m) and o* t (h + l) contain contributions from both the actual tooth spacing errors of the artifact gear and the scale errors within the indicated measurement span.

      

727

Consider, now, the average value I (l, m) of I(l, m) averaged over all N possible values h = 0, 1, . . . , N − 1 of artifact position relative to the rotary scale: 1 N−1 s [o* (h + l + m) − o* t (h + l)]. N h=0 t

I (l, m),

(4)

Let j* t (j) denote a measured value of the artifact adjacent tooth spacing error between tooth j and tooth j + 1. Then, since the accumulated errors o* t are summations of adjacent errors j* t , it follows from equation (3) and Fig. 1 that m−1

I(l, m) = j* t (h + l) + j* t (h + l + 1) + · · · + j* t (h + l + m − 1) = s j* t (h + l + k), k=0

(5) and, by combining equations (3–5), that I (l, m) =

1 N−1 m−1 s s j* (h + l + k). N h=0 k=0 t

(6)

Let jt (j) denote the actual artifact adjacent tooth spacing error between tooth j and tooth j + 1, and js (i) denote the actual scale error increment between scale location i and scale location i + 1 (Fig. 1). Since, according to equation (1), tooth spacing errors and scale errors are additive, the measured adjacent tooth spacing error j* t (j) can be expressed, for any artifact position h relative to the rotary scale, as the superposition of the actual adjacent tooth spacing error jt (j) and the associated scale-error increment js (i) = js (j − h) for that particular measurement span; see equation (2) and Fig. 1: j* t (j) = jt (j) + js (j − h).

(7)

Substituting equation (7) into equation (6) and reversing the order of summation gives I (l, m) =

1 N−1 m−1 s s [j (h + l + k) + js (l + k)] N h=0 k=0 t m−1

= s

k=0

1 N−1 s [j (l + k) + jt (h + l + k)] N h=0 s

m−1

= s [js (l + k) + k=0

1 N−1 s j (h + l + k)] N h=0 t

m−1

= s js (l + k),

(8)

k=0

since, for every integer value of k, one must have N−1

s jt (h + l + k) 0 0.

(9)

h=0

Equation (9) states that starting with any tooth location, say j = l + k, the summation of the adjacent tooth spacing errors between all N tooth measurement surfaces of the artifact

. . 

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gear must be exactly zero; a circle has been ‘closed’ [2]. The summation of scale error increments js (l + k) on the right-hand side of equation (8) represents the difference in the absolute scale errors os between locations i = l and i = l + m (Fig. 1). Hence, one has I (l, m) = os (l + m) − os (l).

(10)

Combining equations (10) and (4) gives os (l + m) − os (l) =

1 N−1 s [o* (h + l + m) − o* t (h + l)]. N h=0 t

(11)

In summary, if the absolute (accumulated) tooth spacing errors of all N equispaced teeth of an artifact gear are measured and recorded for each of the N possible rotational displacements h = 0, 1, . . . , N − 1 of the artifact gear with respect to the rotary scale, and the average of the difference of these N measured values is taken between the generic scale locations i = l and i = l + m, then one obtains the actual scale error increment between these two scale locations (Fig. 1). The precise meaning of this statement is given by equation (11). The validity of this result does not depend, directly, on super precision of the spacing between the contact surfaces of the artifact gear teeth (see Appendix A). When the artifact absolute tooth-spacing error measurements o* t in equation (11) are taken, they are recorded as a function of scale location i and artifact rotational displacement h relative to the rotary scale. Thus, it is convenient to provide a modest but important change in notation that explicitly recognises this fact: o* t (i; h),o* t (h + i) = o* t (j),

(12)

where the second equality is a consequence of equation (2). Moreover, there is no need beyond this juncture to consider any value of i = l other than l = 0. Hence, our estimation formula for determination of the absolute scale error at the generic location m = i relative to that at l = i = 0, i.e. oˆ s' (i),os (i) − os (0),

(13)

is, according to equations (11) through (13), oˆ s' (i) =

1 N−1 s [o* (i; h) − o* t (0; h)], N h=0 t

i = 0, 1, . . . , N − 1

(14)

where o* t (i; h) is the measured value of the artifact absolute (accumulated) tooth spacing error recorded at rotary scale location i, and h is the rotational displacement of the artifact gear relative to the rotary scale for that measurement as illustrated in Fig. 1. The primes in equations (13) and (14) designate that oˆ s' (i) is an estimate of the difference in the two quantities shown in the right-hand side of equation (13). Equation (14) is to be used to compute estimates of the absolute scale errors at the N equispaced scale locations i = 0, 1, . . . , N − 1 that are illustrated in Fig. 1. 4. DETERMINATION OF ARTIFACT GEAR ABSOLUTE TOOTH SPACING ERRORS

The above-described analysis leading to equation (14) has allowed the utilisation of an imperfect artifact gear to determine the absolute scale errors defined by equation (13). By, in effect, reversing the roles of the scale and artifact gear, the same set of N complete index measurements made on the artifact gear can be used to ‘eliminate’ the effects of scale inaccuracies, thereby providing a more accurate measurement of the artifact gear absolute

      

729

(accumulated) tooth spacing errors. In this alternative application of the same analysis and measurement data, instead of the circle of artifact adjacent tooth spacing errors being ‘closed’ as in equation (9), the circle of scale ‘adjacent’ errors between the N scale measurement locations i = 0, 1, . . . , N − 1 illustrated in Fig. 1 is closed. The result provided by equation (14) expresses an average over all N rotational positions h = 0, 1, . . . , N − 1 of the artifact gear with respect to the position of the rotary scale. To apply this result to obtain the artifact errors, one requires the artifact counterpart to equation (13), which is the artifact absolute (accumulated) tooth spacing error of tooth j relative to that of tooth j = 0, i.e. oˆ t' (j),ot (j) − ot (0).

(15)

In carrying out the average provided by the right-hand side of equation (14) for the present application, both terms o* t (i; h) and o* t (0; h) must be ‘attached’ to specific teeth on the gear rather than to locations on the scale. The first of these two terms may be attached to tooth number j of the gear simply by using equation (2), i.e. i = j − h, while the second term may be attached to tooth j = 0, also by using equation (2), by setting i = −h. Hence, the artifact absolute (accumulated) tooth spacing errors, equation (15), may be computed from the two-dimensional array of measurements o* i = 0, 1, . . . , N − 1; t (i; h), h = 0, 1, . . . , N − 1 by oˆ t' (j) =

1 N−1 s [o* (j − h; h) − o* t (−h; h)], N h=0 t

j = 0, 1, . . . , N − 1,

(16)

where, as before, o* t (i; h) is the artifact absolute (accumulated) tooth spacing error measured at scale location i, and h is the position of the artifact gear relative to the rotary scale, both illustrated in Fig. 1. Notice in the first term o* t (j − h; h) in the right-hand side of equation (16) that when h is increased by one count, the scale observation point i = j − h must be decreased by one count in order that j remain the same value, i.e. that this first term remain fixed on the same artifact tooth j. In the case of the second term o* t (−h; h), when h is increased by one count, i = −h also must be decreased by one count in order that the second term remain fixed on tooth j = 0. In equation (16), our coordinate system for fixed values of j is attached to the artifact gear while the average over h = 0, 1, . . . , N − 1 is taken. For values of j Q h and all values of h q 0, negative values of i = j − h and i = −h, respectively, are encountered in the two terms o* t (i; h) in equation (16). This ‘problem’ is fixed by recognising that, as used in equation (16), the measured values o* t (i; h) of artifact absolute (accumulated) tooth spacing errors are periodic in i with period N, i.e. o* t (i + kN; h) = o* t (i; h),

k = 0, 21, 22, . . . .

(17)

The forms of this relationship required for use in equation (16) are o* t (j − h; h) = o* t (N + j − h; h)

(18)

o* t (−h; h) = o* t (N − h; h).

(19)

and

Utilisation of equation (16) removes the effects of the scale errors from the measured artifact absolute (accumulated) tooth spacing errors, thereby providing a method for obtaining very accurate tooth spacing error measurements of gears.

. . 

730

5. ESTIMATION OF UNCERTAINTY IN DETERMINATION OF ROTARY SCALE ERRORS

Up to now, the fact that the individual measurements o* t (i; h) contain errors associated with the measurement process, e.g. thermal instability, slippage, etc., has been ignored. The existence of such measurement errors creates errors in the estimates of both the scale errors oˆ s' (i) given by equation (14) and the artifact errors oˆ t' (j) given by equation (16). However, it will be shown below that the numerical data obtained in the computation of these estimates can be used to provide approximations to the uncertainties in both the scale error estimates oˆ s' (i), i = 0, 1, . . . , N − 1 and the artifact error estimates oˆ t' (j), j = 0, 1, . . . , N − 1. Uncertainties in the scale error estimates oˆ s' (i) are considered first. Every measurement o* t (i; h) of the artifact absolute tooth spacing error recorded at rotary scale location i, with rotational displacement h of the artifact gear with respect to the rotary scale, consists of the exact value ot (i; h) of this measured quantity plus a measurement error contribution et (i; h), i.e. o* t (i; h) = ot (i; h) + et (i; h).

(20)

Substituting this expression for o* t (i; h) into equation (14) allows one to decompose the estimate oˆ s' (i) of scale error at rotary scale location i into its true value, 1 N−1 s [o (i; h) − ot (0; h)], N h=0 t

o's (i),

(21)

and the error e's (i) in the scale error estimate oˆ s' (i), 1 N−1 s [e (i; h) − et (0; h)], N h=0 t

e's (i),

(22)

from which one obtains the decomposition of the scale error estimate oˆ s' (i) into its true value o's (i) and the error e's (i) in the estimate, i.e. oˆ s' (i) = o's (i) + e's (i).

(23)

It is desirable to obtain an estimate of the uncertainty in our estimate oˆ s' (i) of scale error given by equation (14). This uncertainty is governed by the statistical properties of the error e's (i) in our scale error estimate oˆ s' (i). Clearly, e's (i) is not directly observable. Define e's (i; h),et (i; h) − et (0; h)

(24)

which is the difference in the errors in the artifact measurements taken at scale location i and scale location i = 0, both measurements having been taken with the same rotational displacement h of the artifact gear with respect to the rotary scale. Substituting the definition given by equation (24) into equation (22) yields e's (i) =

1 N−1 s e' (i; h), N h=0 s

i = 0, 1, . . . , N − 1.

(25)

The desired metric of uncertainty in our estimate oˆ s' (i) of scale error is the standard deviation of the error e's (i) in oˆ s' (i). The foregoing results permit such an estimate to be obtained. The results given by equations (14) and (16) for estimates of absolute scale errors oˆ s' (i) and artifact absolute tooth spacing errors oˆ t' (j), respectively, enable one to obtain an approximation to the measurement error e's (i; h) defined by equation (24). As indicated

      

731

by equation (1), any exact measurement of an artifact error consists of the sum of the true artifact error and the true scale error. Thus, if one possessed the true values of artifact error and scale error for any particular measurement of artifact error, then the error in the artifact error measurement could be obtained by subtracting the sum of the true values of artifact error and scale error from the measurement of the artifact error. In the present situation, we do not possess the true values of artifact and scale errors, but we do possess estimates of these values given by equations (16) and (14), respectively. Thus, the sum of these estimates of artifact error and scale error can be subtracted from the measured values of artifact error to obtain an approximation to the measurement error defined by equation (24). This approximation to e's (i; h) is denoted by the caret symbol, i.e. eˆs' (i; h). Recognising that equation (24) defines the error in the difference of two measurements, o* t (i; h) − o* t (0; h), see equation (20), and, further, recognising the notation utilised in equation (12), it can be seen that the estimate of the artifact error to be subtracted from the measurement difference [o* is, utilising equation (16), t (i; h) − o* t (0; h)] [oˆ t' (h + i) − oˆ t' (h)]. Furthermore, recognising the definition provided by equation (13), the estimate of the scale error to be subtracted from the measurement difference [o* t (i; h) − o* t (0; h)] is oˆ s' (i) as given by equation (14). Hence, one obtains for the approximation eˆs' (i; h) to the measurement error e's (i; h), eˆs' (i; h),[o* t (i; h) − o* t (0; h)] − [oˆ t' (h + i) − oˆ t' (h)] − oˆ s' (i).

(26)

All terms in the right-hand side of equation (26) are obtainable from the artifact measurements o* t (j) by using equations (12), (16), and (14). In utilising the results of equation (16) in equation (26) for the case (h + i) q (N − 1), the fact that oˆ t' (j), j = 0, 1, . . . , N − 1 is periodic in tooth number j with period N must be used. Validity of the above approximation for the measurement error e's (i; h) is further confirmed in Appendix B, where it is shown explicitly that the total contribution to the right-hand side of equation (26) from the exact values of artifact absolute tooth spacing errors ot (j) and absolute scale errors os (i) is exactly zero, leaving only contributions arising from measurement errors. Our interest is in the error e's (i) in the estimate oˆ s' (i) of the absolute scale error at scale location i. Using the approximation eˆs' (i; h), equation (26), to e's (i; h), one can obtain an approximation eˆs' (i) to e's (i) by using equation (25): 1 N−1 s eˆ ' (i; h), N h=0 s

eˆs' (i),

i = 0, 1, . . . , N − 1.

(27)

From equation (26), it also is shown in Appendix B that eˆs' (i) is identically zero, although the individual terms in the summation in equation (27) are not identically zero. From the definition given by equation (24), e's (i; h) is the measurement error in measurement of the difference o* t (i; h) − o* t (0; h) in artifact absolute tooth spacing errors, where both of these measurements are made with the same rotational displacement h of the artifact with respect to the rotary scale. Since the individual terms e's (i; h) in the summation in equation (25) represent measurement errors for different rotational displacements h of the artifact with respect to the rotary scale, it is reasonable to assume that these errors, for different values of h, are statistically independent with zero expected values, [9] from which there follows E{e's (i; h)e's (i, k)} = 0,

k$h

(28)

. . 

732

where E{ · · · } denotes the mathematical expectation (expected value) of the quantity within the braces. Utilising the assumption given by equation (28), from equation (25), one obtains E{[e's (i)]2} =

=

1 N−1 N−1 s s E{e's (i; h)e's (i; k)} N2 h = 0 k = 0 1 N−1 s E{[e's (i; h)]2} N2 h = 0

6 $ N−1

%7

e's (i; h) N

=E s

h=0

2

,

(29)

where equation (28) was used in going to the second line. Equation (29) expresses E{[e's (i)]2} as the mathematical expectation of the sum of N random variables [e's (i; h)/N]2, which have been assumed to be statistically independent. It can be argued, then, from the law of large numbers [9] that the value of the sum in the right-hand side of equation (29), for large values of N, can be expected to be a good approximation to its expected value E{ · · · }, in the sense that the standard deviation of the sum should be a small fraction of its expected value (or realised value). Clearly, this argument becomes more valid with increasing values of N. For large values of N, therefore, N−1

E{[e's (i)]2} 1 s

h=0

$

%

2

e's (i; h) , N

N1.

(30)

Now the above argument is applied to the approximation eˆ 's (i) of e's (i) given by equation (27). Since eˆs' (i) is identically zero (Appendix B), application of equation (30) to the approximation eˆs' (i) of e's (i) given by equation (27) yields N−1

Var {e's (i)} 1 s

h=0

$

%

2

eˆs' (i; h) , N

i = 0, 1, . . . , N − 1;

N1

(31)

where Var { · · · } designates the variance [9] of the quantity within the braces. Since eˆs' (i; h) can be evaluated from the artifact measurements by using equation (26), the approximation to the variance of e's (i) given by equation (31) also can be evaluated from the artifact measurements. Thus, the square root of the quantity evaluated by equation (31) should provide a reasonable approximation to the uncertainty in the estimate of absolute scale error oˆ s' (i) given by equation (14), where this measure of uncertainty is the standard deviation of the error e's (i) in the estimate oˆ s' (i). Equation (31) evaluates this uncertainty as a function of rotary scale location i = 0, 1, . . . , N − 1. In Appendix C, it is shown that the approximation given by equation (31) is valid whenever N is large in comparison with unity and the measurement errors in the artifact tooth measurements are uncorrelated for different rotational displacements h of the artifact with respect to the rotary scale. Furthermore, by also assuming that the measurement errors in the individual artifact tooth measurements have the same common variance, it is shown that for moderate values of N, equation (31) should provide, on the average, slightly more accurate results if its right-hand side is increased by a factor of [1 − (2/N)]−1 [equation (C8)].

      

733

6. ESTIMATION OF UNCERTAINTY IN DETERMINATION OF ARTIFACT-GEAR ABSOLUTE TOOTH SPACING ERRORS

The development below for the approximation to the standard deviation of the error in the estimate oˆ t' (j), equation (16), of artifact absolute tooth spacing errors parallels the development given above for our approximation to the standard deviation of the error in the estimate oˆ s' (i), equation (14), of absolute scale errors. Substituting equation (19) into equation (16), and then applying the decomposition of the measurement terms o* t of the result into their true values ot and the measurement errors et as delineated by equation (20), there follows for the true value o't (j) of the artifact absolute tooth-spacing errors, 1 N−1 s [o (j − h; h) − ot (N − h; h)], N h=0 t

o't (j),

(32)

and for the error e't (j) in the estimate oˆ t' (j), 1 N−1 s [e (j − h; h) − et (N − h; h)], N h=0 t

e't (j),

(33)

from which one obtains the decomposition of the artifact error estimate oˆ t' (j) into its true value o't (j) and the error e't (j) in the estimate, i.e. oˆ t' (j) = o't (j) + e't (j).

(34)

The periodic property, equation (18), for the measurements o* t (j − h; h) also applies to the true values ot (j − h; h) of these measurements and to the measurement errors et (j − h; h), as does the periodic property, equation (19). Define e't (j; h),et (j − h; h) − et (N − h; h).

(35)

Substituting this definition into the definition given by equation (33) yields e't (j) =

1 N−1 s e' (j; h), N h=0 t

j = 0, 1, . . . , N − 1

(36)

which is the expression for the error in our estimate oˆ t' (j) of the artifact errors that is the counterpart to equation (25) for errors in our scale error estimates. Now an expression is required for the estimate eˆt' (j; h) of the measurement error e't (j; h) that is the counterpart to equation (26) for scale errors. Recognising that equation (35) defines the error in the difference of two measurements, [o* t (j − h; h) − o* t (−h; h)] = [o* t (j − h; h) − o* t (N − h; h)],

(37)

it can be seen using equation (2) and Fig. 1 that the estimate of the scale error contribution to the measurement difference, equation (37), is [oˆ s' (j − h) − oˆ s' (N − h)], where oˆ s' (i) is given by equation (14). Furthermore, recognising the definition given by equation (15), the estimate of the artifact error contribution to the measurement difference, equation (37), is, simply, the value of oˆ t' (j) given by equation (16). Therefore, the estimate of the measurement error in the difference of measurements defined by equation (37) is that quantity minus the sum of the estimates of scale error and artifact error contributions to that quantity, i.e. eˆt' (j; h),[o* t (j − h; h) − o* t (N − h; h)] − [oˆ s' (j − h) − oˆ s' (N − h)] − oˆ t' (j),

(38)

. . 

734

which is the artifact measurement error counterpart to the expression provided by equation (26) for estimation of uncertainties in scale error estimates. All terms in the right-hand side of equation (38) are obtainable from the artifact measurements o* t (j) by using equations (12), (14) and (16). In Appendix B, it is shown explicitly that the total contribution to the right-hand side of equation (38) from the exact values of artifact absolute tooth spacing errors ot (j) and absolute scale errors os (i) is exactly zero, leaving only contributions from measurement errors. Using the approximation eˆt' (j; h), equation (38), to e't (j; h), one can obtain an approximation eˆt' (j) to e't (j) by using equation (36): 1 N−1 s eˆ ' (j; h), N h=0 t

eˆt' (j),

j = 0, 1, . . . , N − 1.

(39)

Utilising equation (38), it also is shown in Appendix B that eˆt' (j) also is identically zero, although the individual terms in the summation in equation (39) are not identically zero. The individual terms e't (j; h) in the summation in equation (36) represent measurement errors for different rotational displacements h of the artifact gear with respect to the rotary scale. Hence, it is reasonable to assume that these errors, for different values of h, also are statistically independent with zero expected values, from which follows E{e't (j; h)e't (j; k)} = 0,

k $ h.

(40)

%7

(41)

Utilising this assumption, from equation (36),

6 $ N−1

E{[e't (j)]2} = E s

h=0

e't (j; h) N

2

,

which is the artifact uncertainty counterpart to equation (29) for scale error uncertainties. Reasoning as before from the law of large numbers, then for large values of N, N−1

E{[e't (j)]2} 1 s

h=0

$

%

2

e't (j; h) , N

N1

(42)

and applying this result to the approximation eˆt' (j) of e't (j) given by equation (39), one obtains N−1

Var {e't (j)} 1 s

h=0

$

%

2

eˆt' (j; h) , N

j = 0, 1, . . . , N − 1;

N1

(43)

since eˆj (j) is identically zero. Since eˆt' (j; h) can be evaluated from the artifact measurements by using equation (38), the square root of equation (43) may be used to obtain an approximation to the uncertainty in the estimate of the artifact absolute tooth spacing error oˆ t' (j) given by equation (16), where this measure of uncertainty is the standard deviation of the error e't (j) in the estimate oˆ t' (j). Equation (43) evaluates this uncertainty as a function of artifact tooth number j = 0, 1, . . . , N − 1. In Appendix C it is shown that the approximation given by equation (43) is valid provided N is large in comparison with unity and the measurement errors et (i; h) are uncorrelated for measurements made using different rotational displacements h of the artifact with respect to the rotary scale. In the same manner as for equation (31), it also is shown in Appendix C that equation (43) should provide, on the average, slightly more accurate results if its right-hand side is increased by a factor of [1 − (2/N)]−1 [equation (C13)].

      

735

Equations (31) and (43) provide variance approximations as a function of rotary scale location i and artifact tooth location j, respectively. Useful overall variance approximations of the errors in the absolute rotary scale error estimates oˆ s' (i), equation (14), and the absolute (accumulated) tooth-spacing error estimates oˆ t' (j), equation (16), may be obtained as averages over i = 1, 2, . . . , N − 1 and j = 1, 2, . . . , N − 1, respectively, of the variance approximations of the errors in these two estimates given by equations (31) and (43), i.e.

Average approximate variance {e's } =

1 N−1 s Var {e's (i)}, N − 1 i=1

(44)

Average approximate variance {e't } =

1 N−1 s Var {e't (j)}. N − 1 j=1

(45)

and

7. APPLICATION

The above-described method of rotary scale calibration has been used to provide rotary-axis corrections to the M&M Precision systems Model QC9000 gear measurement machine developed under a recently completed NIST Advanced Technology Program award. Rotary scale errors were computed using equation (14) together with their approximate standard deviations computed from the square root of equation (31). Gear artifact absolute tooth spacing errors were computed using equation (16) together with their approximate standard deviations computed from the square root of equation (43). The gear artifact used in this procedure had 240 teeth and a base circle diameter of 22.9135 cm. The number 240 is 2 × 5! and therefore is evenly divisible by 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, and 120, thereby providing a very large number of sets of equispaced teeth that could be used in the required measurement procedure. Three complete tests were run. The first and third tests utilised N = 8 equispaced artifact teeth, thereby providing scale calibrations and artifact calibrations at 360 6 8 = 45° intervals. The second test utilized N = 40 teeth, thereby providing both scale and artifact calibrations at 9° intervals. Thus, in all three tests, both scale and artifact calibrations were provided at the same 45° interval locations. The purpose of this overall procedure was to ‘bracket’ the 9° (second) test by two less time consuming tests that would provide comparisons with the more detailed second test at the common 45° interval locations. The three tests were conducted in a temperature-controlled environment (20.05°C); the second and third tests were conducted on the day following the first test. The artifact gear was not removed from the rotary table during the three tests, but was rotated with respect to the table N times during each test as required by the test procedure. The output of each test consisted of an N × N array of absolute (accumulated) tooth-spacing error measurements o* t (i; h), i = 0, 1, . . . , N − 1; h = 0, 1, . . . , N − 1 as defined by equation (12), from which the results provided by equations (14), (16), (31), and (43) were computed. Figure 2 displays the absolute scale errors computed by equation (14) for the three tests. The results for test 2, which were computed at 9° intervals, occur at the discontinuities in slope of the continuous line shown in Fig. 2. The scale errors shown in Fig. 2 are displayed as linear errors because the measurement probe provides a linear error output.

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. . 

Figure 2. Absolute scale errors computed by equation (14) for three independent tests. Tests 1 (+) and 3 (×) values shown were computed at 45° intervals; test 2 values (—), located at the discontinuities in slope, were computed at 9° intervals. Base circle radius of artifact gear used was 11.45675 cm. Differences in the computed scale errors are tabulated in Table 1; absolute values of the same differences are plotted in Fig. 4.

These and all other linear error metrics provided in this paper can be converted to angular errors in arc seconds (0) using the relationship du(arcsec) =

3600 × 360 × dy (mm) p × 229 135

= 1.800 × dy (mm),

(46)

where dy represents a linear error in units of micrometers.

Figure 3. Artifact gear absolute (accumulated) tooth-spacing errors computed by equation (16) for tests 1 (+), 2 (—) and 3 (×). Tests 1 and 3 values shown were computed at 45° intervals; test 2 values, located at the discontinuities in slope, were computed at 9° intervals. Base circle radius of artifact gear used was 11.45675 cm. Differences in the computed absolute tooth spacing errors are tabulated in Table 2; absolute values of the same differences are plotted in Fig. 5.

      

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T 1 Differences between pairs of absolute scale errors computed by equation (14) Scale location (°)

Test 2 minus test 1 (mm)

Test 3 minus test 1 (mm)

Test 3 minus test 2 (mm)

0 45 90 135 180 225 270 315

0 +0.02178 +0.03327 −0.01897 +0.00711 +0.00616 +0.04921 +0.07930

0 −0.02572 −0.02413 −0.00762 −0.00919 +0.00635 +0.02477 +0.00445

0 −0.04750 −0.05740 +0.01135 −0.01631 +0.00019 −0.02445 −0.07485

When the artifact absolute (accumulated) tooth spacing errors are computed by equation (16) for the same three tests, the results for test 2, which were computed at 9° intervals, occur at the discontinuities in slope of the continuous line as was the case in Fig. 2 (Fig. 3). At the scale and artifact locations in Figs 2 and 3, respectively, where all three tests were performed, it is, for the most part, difficult to distinguish between the three test values. Thus, differences between the three sets of test results are displayed in Tables 1 and 2 for the absolute scale errors and absolute (accumulated) tooth spacing errors shown in Figs 2 and 3, respectively. The maximum difference among the 21 differences in computed scale errors is 0.07930 mm, and the maximum difference among the 21 differences in computed artifact errors is 0.09022 mm. Using equation (46), these two worst-case values are equivalent to 0.14270 and 0.16240, respectively. One should keep in mind that each of the tabulated values contains the sum of the errors resulting from two tests, and in each of the above mentioned two worst-case values, the errors from the two tests are almost certainly additive. Each entry in Tables 1 and 2 can be regarded as the sum of the errors from two statistically independent tests. Hence, the standard deviation for the error in a single test at a ‘typical’ location can be computed from each table as the square root of (2 × 21)−1 times the sum of the squares of the 21 non-zero entries in each table. This computation yields a value of 0.02437 mm for the root mean square (rms) error in the individual scale error estimates of Table 1 and 0.03230 mm for the (rms) error in the individual artifact T 2 Differences between pairs of artifact absolute (accumulated) tooth spacing errors computed by equation (16) Scale location (°)

Test 2 minus test 1 (mm)

Test 3 minus test 1 (mm)

Test 3 minus test 2 (mm) (mm)

0 45 90 135 180 225 270 315

0 +0.03068 +0.05067 +0.01765 −0.00432 −0.04514 −0.05867 −0.03538

0 +0.02540 +0.04953 +0.05779 +0.07112 +0.04509 +0.00076 +0.01143

0 −0.00528 −0.00114 +0.04013 +0.07544 +0.09022 +0.05944 +0.02395

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. . 

Figure 4 Comparison of absolute values (r) of differences in scale errors computed by equation (14) for tests 1, 2 and 3 with approximate standard deviations (q) of these differences computed from the test data. Approximate standard deviations were computed [10] as the square roots of the sum of the approximate variances obtained by equation (31) for each test of a test pair. (a) Tests 2 and 3; (b) tests 1 and 3; (c) tests 1 and 2. Differences in computed scale errors are tabulated in Table 1; values of computed scale errors are plotted in Fig. 2.

(accumulated) tooth spacing error estimates of Table 2. Using equation (46), the corresponding rms angular error values are 0.043870 and 0.058140, respectively. It is of interest to compare the differences displayed in Tables 1 and 2 with the approximate values of standard deviation computed from the test data by utilising equations (31) and (43), respectively. In order to make this comparison, the measurement errors between the individual tests 1, 2, and 3 can be assumed to be statistically independent; hence, the standard deviation of each of the differences in test results displayed in Tables 1 and 2 would be obtained by taking the square root of the sum of the two variances associated with the two tests [10]. Thus, each of the differences displayed in Table 1 is to be compared with the approximate standard deviation of the difference computed as the square root of the sum of the corresponding two approximate variances, each approximate variance to be computed by equation (31) with the help of equation (26). Similarly, each of the differences displayed in Table 2 is to be compared with the approximate standard deviation of that difference computed as the square root of the sum of the corresponding two approximate variances, each approximate variance to be computed by equation (43) with the help of equation (38). The absolute values of the scale error differences displayed in Table 1 are plotted in Fig. 4 along with the approximate standard deviations of these differences computed from the test data by using equation (31) as described above. Similarly, the absolute values of the artifact error differences displayed in Table 2 are plotted in Fig. 5 along with the approximate standard deviations of these differences computed from the test data by utilising equation (43) as described above. To interpret the results shown in Figs 4 and 5, at each of the scale and artifact locations where the data are plotted, the absolute

      

739

difference denoted by D is to be considered as the absolute value of an error drawn from a statistical population whose approximate standard deviation denoted by q is shown at the same location. Thus, on average, the approximate values of standard deviation shown in these two figures should represent ‘typical’ values of individual error samples which are the absolute differences in the pairs of scale error estimates and artifact error estimates. The significant variability of the absolute differences in comparison to the rather small variability of the approximate standard deviation values shown in Figs 4 and 5 is to be expected. If it is assumed that the collection of absolute difference samples denoted by D in each of Figs 4 and 5 is drawn from a statistical population governed by the normal probability density function, then the absolute values of 68.3% of the samples should, on the average, be less than one standard deviation. The plots in Figs 4 and 5 each have a total of 21 samples. The closest fraction of these 21 samples to 68.3% is 14 out of 21. Examination of the numerical values from which Fig. 4 was plotted shows that 13 out of the 21 absolute difference samples shown there have values less than their corresponding approximate standard deviations shown in Fig. 4. The difference by only one sample from the 14 out of 21 predicted by the normal distribution is well within the expected variability of the sample. However, when the relatively small correction of [1 − (2/N)] − 1 to the right-hand side of equation (31), suggested by equation (C8), is applied to all of the numerical values of the approximate variances from which the standard deviation data shown in Fig. 4 was computed, exactly one additional value of approximate standard deviation is made larger than its corresponding absolute difference, i.e. at the 45° location in tests 1 and 2, thereby

Figure 5. Comparison of absolute values (r) of differences in artifact (accumulated) tooth spacing errors computed by equation (16) for tests 1, 2, and 3 with approximate standard deviations (q) of these differences computed from the test data. Approximate standard deviations were computed [10] as the square roots of the sum of the approximate variances obtained by equation (43) for each test of a test pair. (a) Tests 2 and 3; (b) tests 1 and 3; (c) tests 1 and 2. Differences in computed (accumulated) tooth spacing errors are tabulated in Table 2; values of computed (accumulated) tooth spacing errors are plotted in Fig. 3.

. . 

740

T 3 Overall approximate standard deviations of errors in absolute scale error estimates and artifact absolute (accumulated) tooth spacing error estimates computed by the square roots of equations ( 44) and ( 45) , respectively, by using equations ( 31) and ( 43) , as appropriate Scale errors Artifact errors

Test 1 (mm)

Test 2 (mm)

Test 3 (mm)

0.02288 0.02113

0.02154 0.01554

0.01651 0.01917

yielding 14 of 21 samples of absolute difference below their corresponding approximate standard deviations, exactly as predicted by the normal distribution. In the case of the numerical values from which Fig. 5 was plotted, only six of the 21 absolute difference samples shown there have values less than their corresponding approximate standard deviations. However, all 21 absolute difference samples in Fig. 5 have values less than three times their corresponding approximate deviations; this same behaviour is true for the absolute difference samples in Fig. 4 as well. Nevertheless, it would appear in this latter case that, on the average, equation (43) has somewhat underpredicted the approximate variances of the errors in the artifact error estimates computed by equation (16), even when the correction factor of [1 − (2/N)] − 1, suggested by equation (C13), is applied to its right-hand side. The above-described behaviour of the results shown in Figs 4 and 5 suggests that the square roots of equation (31) and (43), or equations (44) and (45), can serve as useful approximations to the standard deviations of the errors obtained in computation of absolute scale errors by equation (14) and artifact absolute tooth spacing errors by equation (16), thereby providing the means for assessing approximate uncertainties to the scale error and artifact error estimates obtained by these computations. The overall approximate standard deviations of the errors in the absolute scale error estimates shown in Fig. 2 and the absolute tooth spacing error estimates shown in Fig. 3, computed for each of the three tests by the square roots of equation (44) and (45), respectively, by using equations (31) and (43) in their right-hand sides as appropriate, are provided in Table 3. In a final test, the same artifact gear used in the above-described tests was removed from the M&M QC9000 gear measurement machine and placed between centers on a newly built M&M Model 3595 gear measurement machine that was located in an indoor manufacturing assembly area with no special temperature control. Estimates of absolute scale errors and artifact absolute tooth spacing errors were computed using equations (14) and (16), respectively, and the approximate variances of the errors in these estimates were computed by equations (31) and (43), respectively. Due to available time limitations, only N = 20 scale locations and N = 20 artifact teeth were utilised in this final test. The 20 teeth chosen on the artifact consisted of every second tooth of the 40 teeth used earlier on the QC9000 machine. Thus, this final test provided the opportunity to obtain a calibration of the same artifact gear on a different measurement machine possessing different rotary-axis errors, thereby providing a check on our procedure for removing the effects of rotary-axis errors from estimates of the artifact errors. The estimates of artifact absolute (accumulated) tooth spacing errors obtained by measurements taken on the Model 3595 machine are plotted in Fig. 6 along with those obtained earlier from measurements taken on the same 20 teeth of the QC9000 machine. Both sets of results shown in Fig. 6 were computed by equation (16). The QC9000 values shown in Fig. 6 are identical to the values for those same 20 teeth shown as the test 2 values in Fig. 3.

      

741

Figure 6. Comparison of absolute (accumulated) tooth spacing errors computed by equation (16) for measurements made on 20 equispaced teeth of same artifact gear by the M&M QC9000 (q) and M&M Model 3595 (e) machines. Base circle radius of artifact gear used was 11.45675 cm.

The maximum difference among the 20 pairs of values shown in Fig. 6 is 0.2374 mm, which occurs on the tooth located at 108°. From the 19 pairs of differing numerical values shown in Fig. 6, an rms difference of 0.1131 mm was obtained between the two sets of calibrations. Some, if not most, of the difference between the two sets of calibrations shown in Fig. 6 can be attributed to the fact that somewhat different tooth surface asperities were contacted by the measurement probe during the two sets of calibrations because of imperfections in probe positioning that arose from the required relocation of the artifact gear onto the Model 3595 measurement machine. The above-cited rms difference of 0.1131 mm between these two sets of calibrations is smaller than the rms roughness of the artifact ground tooth surfaces. Finally, one can conclude from the results presented in Figs 3 and 4 that the typical scale calibration errors illustrated in Fig. 4 are about a factor of 100 smaller than the typical accumulated tooth spacing errors, shown in Fig. 3, of the artifact gear used in the calibration procedure. This enormous leveraging in accuracy has been achievable through the use of the closure principle, equation (9), that yielded the fundamental calibration procedure described by equation (11), which was re-expressed using a more useful notation by equation (14), and later for the artifact gear calibration by equation (16). 8. SUMMARY

A method has been developed for precision calibration of the rotary scale errors of gear measurement machines and precision calibration of gear tooth index errors after removal of the effects of the rotary scale errors. Since the method uses the standard machine procedure for measuring gear tooth index errors (accumulated tooth spacing errors), it automatically includes in the rotary scale calibrations the consistent effects arising from eccentricities of machine gear-mounting centers and eccentricity of the scale mounting centre relative to the instantaneous axis of table rotation, consistent effects of wobble of the instantaneous axis of table rotation, as well as scale graduation errors, etc. The gear artifact index errors obtained in the calibration procedure are referenced to the axis connecting the mounting centers located on the gear.

742

. . 

Formulae were derived for estimating the approximate uncertainties in both the rotary scale calibrations and the index error calibrations, both as a function of the angular location of the rotary scale calibration or index error calibration. The metric of uncertainty used in both of these cases is the approximate standard deviation. These uncertainty estimates use exactly the same measurement data required for the rotary scale calibrations and index error calibrations. Three ‘independent’ tests using the developed methods showed good consistency both between the predicted calibrations and the predicted approximate standard deviations. In order to obtain the above-described results for N equispaced rotary scale locations and N equispaced artifact gear teeth, N complete sets of artifact-gear index error measurements are required, where each complete set of index error measurements is obtained after rotating the gear (always in the same direction) with respect to the measurement machine table by 360 6 N degrees beyond the gear position utilised in the previous complete set of index error measurements. Letting a particular value of h = 0, 1, . . . , N − 1 designate the position of the gear with respect to the table for one such set of measurements and i = 0, 1, . . . , N − 1 designate the rotary scale (table) location of a particular tooth measured during that set as illustrated in Fig. 1, using equation (12) one complete set of index error measurements has been designated for a particular value of h by o* t (i; h), i = 0, 1, . . . , N − 1 and the collection of N such complete sets of index error measurements by the square array of index error measurements o* t (i; h), i = 0, 1, . . . , N − 1; h = 0, 1, . . . , N − 1. Using this notation, equation (14) provides the estimate of rotary scale errors and equation (16) provides the estimate of artifact gear index errors. From the index error measurements o* t (i; h) and the above-described results computed by equations (14) and (16), combined with equations (26) and (31), one can obtain the approximate variance of each scale error estimate, and using these same quantities combined with equations (38) and (43), one can obtain the approximate variance of each index error estimate. Equations (44) and (45) provide average values of the approximate variances obtained by equations (31) and (43), respectively. The square roots of each of these approximate variances yields the approximate standard deviation of the same quantity. These expressions for approximate standard deviation allow the estimation of uncertainties in the rotary scale calibrations and index error estimates without the requirement of making additional measurements. ACKNOWLEDGMENTS

This work was supported by the National Institute of Standards and Technology (NIST) Advanced Technology Program (ATP) Cooperative Agreement 70NANB4H1526 with M&M Precision Systems Corporation. Support for statistical refinements and writing of the paper by the Technology Reinvestment Project Cooperative Agreement N00014-94-20013 between The Office of Naval Research and The American Society of Mechanical Engineers also is gratefully acknowledged. The application results presented were carried out on the M&M Model QC9000 gear measurement machine developed under the NIST ATP award. Design and building of the machine and the ‘stop’ fixture used in the calibrations were supervised by M. Evans and C. Dennison, and software used in the rotary table controls was prepared by M. Cowan, all of M&M Precision Systems Corporation. The artifact gear measurements used in the calibrations were made by R. L. Homan, and the computer program used to compute the calibration results from these measurements was prepared by R. T. Harvey, both of the Applied Research Laboratory of The Pennsylvania State University.

      

743

REFERENCES 1. D. W. S and R. H. E 1995 Encyclopedic Dictionary of Gears and Gearing. New York: McGraw-Hill. 2. W. R. M 1970 Foundations of Mechanical Accuracy. Bridgeport, Connecticut: The Moore Special Tool Company. 3. T. M and M. K 1989 Precision Engineering 11, 95–100. An automatic calibration system for angular encoders. 4. T. M and M. K 1985 Precision Engineering 7, 15–21. Automatic calibration system for polygon mirrors. 5. R. J, A. S and A. N 1980 Tokyo Institute of Technology Bulletin of the Research Laboratory of Precision Machinery and Electronics 45, 17–24. Calibration system for precision angle standards. 6. C. P. R 1975 National Bureau of Standards, NBSIR 75 –750 . The calibration of indexing tables by subdivision. 7. A. K, A. T and K. K 1970 Bulletin of the Japan Society of Precision Engineering 4, 41–42. A method of accurate measurement of the circular dividing error. 8. A. H. C 1954 British Journal of Applied Physics 5, 367–371. The calibration of circular scales and precision polygons. 9. E. P 1960 Modern Probability Theory and Its Applications. New York: John Wiley. 10. G. W. S and W. G. C 1989 Statistical Methods, 8th edn. Ames Iowa: Iowa State University Press. 11. T. M. A 1957 Mathematical Analysis. Reading, MA: Addison-Wesley.

APPENDIX A: EFFECTS OF ARTIFACT GEAR TOOTH SPACING ERRORS AND ITS IMPERFECT POSITIONING WITH RESPECT TO THE ROTARY TABLE Apart from errors introduced by imperfections in the measurement probe, thermal drift, slippage of artifact gear with respect to the rotary table, etc., it would at first appear that the calibration results produced by equations (14) and (16) should be exact. However, exceedingly small errors in the calibrations will be introduced by unavoidable imperfections in the positioning of the individual tooth contact surfaces of the artifact gear with respect to the rotary scale. It is shown below that the calibration errors introduced by these artifact gear tooth positioning errors normally will be negligible. For each value of h = 0, 1, . . . , N − 1 that designates the rotational position of the artifact gear with respect to the rotary table, a ‘stop’ fixture attached rigidly to the rotary table, in contact with a tooth surface, can be used to determine the desired rotational position of the artifact gear with respect to the rotary table (Fig. 1). (It is imperative that once the artifact gear is positioned with respect to the table for each value of h, absolutely no relative motion of the artifact gear with respect to the rotary table be permitted during the complete index error measurement of the gear.) There are two sources of errors in positioning of the individual teeth of the artifact gear with respect to the rotary table. The first source arises from imperfections in the ‘stop’ fixture, its elasticity, variations in the force used in positioning the tooth contact surfaces against the stop fixture, etc. The artifact gear positioning error from this source can be held to within a few micrometers, with care. The second source arises from the accumulated tooth spacing errors of the artifact gear. For the worst-case scale measurement location i, this second error contribution is never larger than the maximum peak-to-peak (i.e. total) accumulated tooth spacing error of the artifact gear. For the artifact gear used in our applications, this maximum value is about 6 mm, according to Fig. 3. Thus, using care and a good quality artifact gear, the maximum positioning error of any artifact tooth, in a direction tangent to the base circle or pitch circle, can be held to within about 10 mm.

. . 

744

Figure A1. Illustration of effects of tooth position errors ot on computation of scale error at scale location i utilising equations (14) or (A1). According to equation (A2) and the first mean value theorem for Riemann–Stieltjes integrals [11], the scale error determined by each of the sums in equation (A1) is the actual scale error at some point within the immediate vicinity of scale location i that is determined by the distribution of position errors of the artifact gear teeth measured at scale location i.

Consider, now, the effects on the scale calibration given by equation (14) of such artifact tooth positioning errors with respect to the rotary table. Equation (14) can be rewritten as oˆ s' (i) =

1 N−1 1 N−1 s o* s o* (0; h), t (i; h) − N h=0 N h=0 t

i = 0, 1, . . . , N − 1.

(A1)

Ideally, if there were no artifact tooth positioning errors, every one of the N artifact tooth measurements o* t (0; h), h = 0, 1, . . . , N − 1 would take place exactly at scale location i = 0, and every one of the N artifact tooth measurements o* t (i; h), h = 0, 1, . . . , N − 1 would take place exactly at scale location i (Fig. 1). However, because of the above-described two sources of small positioning errors of the artifact teeth, the N different individual measurements o* t (0; h), for h = 0, 1, . . . , N − 1 at i = 0 will take place at slightly different scale locations very near the correct location at i = 0; similarly for the N measurements o* t (i; h) at scale location i. It has been shown already by equations (8) through (10) that, in forming the average represented by the right-hand sides of equations (14) and (A1), the direct contribution to these right-hand sides from the artifact gear tooth-spacing errors vanishes identically, leaving only the contribution from the scale errors. Hence, each of the two sums in the right-hand side of equation (A1) represents forming the average value of the scale error at that nominal scale location i (or i = 0) over a very small range of actual scale locations determined by the positioning errors of the artifact teeth. For simplicity in understanding the effects of these artifact tooth positioning errors, we can represent these positioning errors ot by a probability density function p(ot ) as sketched in Fig. A1, where the total width of this probability density can be held to about 10 mm as described above. Thus, each of the sums in the right-hand side of equation (A1), in reality, is a local average value o¯ s (i) of the scale errors in the immediate vicinity of scale location i, which can be represented, using the probability density p(ot ) of tooth position errors by

g

o¯ s (i) = os (ot )p(ot ) dot

(A2)

      

745

where, by definition,

g

p(ot ) dot = 1,

(A3)

and where, in equation (A2), we have designated by the symbol os (ot ) the dependence of the scale error os on the location of its measurement determined by error ot in the position of the particular artifact tooth utilised in that measurement. To be mathematically precise, the above two integrations should be regarded as Stieltjes integrals [11], which include Dirac delta functions in the probability denstities p(ot ) that arise from the finite sums in equation (A1). It follows then from the first mean value theorem for Riemann–Stieltjes integrals [11] that this local average value o¯ s (i) of scale error must necessarily take on a value between the maximum and minimum values of the scale error within the scale locations determined by artifact gear tooth positioning errors. If the scale error is a continuous function in the neighborhood of scale location i, then the scale error computed by each of the sums in the right-hand side of equation (A1) must take on a value equal to the true value of the scale error somewhere within the (very small) scale range about scale location i that is determined by the range of artifact tooth positioning errors at scale location i. Therefore, from another (equivalent) point of view, the effect of the above-described tooth positioning errors is identical to that which would be caused by very small errors in the sampling locations of the exact scale errors, where these errors in sampling locations are of the order of typical artifact tooth positioning errors, which we have argued can be held to within about 10 mm. Thus, normally, the effects of artifact tooth positioning errors on the accuracy of scale errors determined by equation (14) are completely negligible. A similar argument can be used to show that the effects of scale errors on determination of the absolute (accumulated) tooth-spacing errors by equation (16) also should be very small. APPENDIX B: PROOFS THAT eˆs' (i), EQUATION (27), AND eˆt' (j), EQUATION (39), BOTH ARE IDENTICALLY ZERO Inserting the expressions for oˆ s' (i) and oˆ t' (j) given by equations (14) and (16), respectively, into equation (26), after replacing the index h in equations (14) and (16) by the ‘dummy’ index k, one obtains eˆs' (i; h) = o* t (i; h) − o* t (0; h) −

+

1 N−1 s [o* (h + i − k; k) − o* t (−k; k)] N k=0 t

1 N−1 1 N−1 s [o* s [o* (i; k) − o* t (h − k; k) − o* t (−k; k)] − t (0; k)] N k=0 N k=0 t

= o* t (i; h) − o* t (0; h) − +o* t (i; k) − o* t (0; k)].

1 N−1 s [o* (h + i − k; k) − o* t (h−k; k) N k=0 t (B1)

Each term in the right-hand side of equation (B1) represents a single measurement o* t that consists of the exact value of the quantity being measured plus a measurement error, as described by equation (20). However, as proved below, the total contribution to the

. . 

746

right-hand side of equation (B1) from the exact values of artifact absolute tooth spacing errors ot (j) and absolute scale errors os (i) is exactly zero, leaving only contributions arising from measurement errors. The errors ot (j) and os (i) denote, respectively, the artifact absolute tooth spacing error at artifact location j and absolute scale error at scale location i, where these locations are illustrated in Fig. 1. Consider, first, only the contributions to the right-hand side of equation (B1) from these exact values of artifact and scale errors. Using equation (1), the notation of equation (12), and Fig. 1, these exact value contributions to the individual terms in equation (B1) may be expressed as o* t (i; h) = ot (h + i) + os (i) o* t (0; h) = ot (h) + os (0) o* t (h + i − k; k) = ot (h + i) + os (h + i − k) o* t (h − k; k) = ot (h) + os (h − k) o* t (i; k) = ot (i + k) + os (i) o* t (0; k) = ot (k) + os (0).

(B2a–f)

Substituting equations (B2a–f) into equation (B1) gives eˆs' (i; h) = [ot (h + i) + os (i)] − [ot (h) + os (0)] 1 N−1 s {[o (h + i) + os (h + i − k)] − [ot (h) + os (h − k)] N k=0 t

−

+[ot (i + k) + os (i)] − [ot (k) + os (0)]},

(B3)

or, upon rearranging terms, eˆs' (i; h) = ot (h + i) − ot (h) −

+os (i) − os (0) −

1 N−1 s [o (h + i) − ot (h)] N k=0 t

1 N−1 s [o (i) − os (0)] N k=0 s

1 N−1 1 N−1 s os (h + i − k) + s o (h − k) N k=0 N k=0 s

−

1 N−1 1 N−1 s ot (i + k) + s o (k). N k=0 N k=0 t

−

(B4)

Observing that the summands in the first two lines of the above equation are independent of k, it follows immediately that each of the first two lines in the right-hand side of equation (B4) is identically zero. Furthermore, observing that the scale errors os (i) are periodic with period N, it follows that the two sums in the third line cancel each other, and from the same periodic property of the artifact errors ot (j), it follows that the fourth line also is identically zero. Hence, the contributions of the exact artifact and scale errors to the right-hand sides of equations (B1) and (26) are identically zero, leaving only measurement error contributions.

      

747

Therefore, using the notation of equation (20); one may replace each measurement o* t in the right-hand side of equation (B1) by its corresponding measurement error et without changing the value represented by the entire right-hand side. Making this change in notation and forming the average over h as indicated by equation (27) gives

eˆs' (i) =

1 N−1 s [e (i; h) − et (0; h)] N h=0 t

1 N−1 N−1 − 2 s s [et (h + i − k; k) − et (h − k; k)] N k=0 h=0 1 N−1 1 N−1 s s [e (i; k) − et (0; k)] N h=0 N k=0 t

−

(B5)

where, in the first of the two double summations, the order of summation has been interchanged. The summand in the second double summation is independent of h; hence, the average over h there has no effect on the average over k there. Thus, that second double summation is identical to the single summation in the first line; these two summations exactly cancel each other. There remains the first of the two double summations in equation (B5). However, the measurement errors et (i; k) are periodic in the scale index i with period N. From this fact, for every value of i and k, it follows that N−1

N−1

h=0

h=0

s et (h + i − k; k) = s et (h − k; k);

(B6)

hence, the first of the two double summations in equation (B5) is identically zero. Thus, the entire right-hand side of equation (B5) is identically zero, i.e. eˆs' (i) = 0,

i = 0, 1, . . . , N − 1.

(B7)

The proof that eˆt' (j) given by equation (39) is identically zero is analogous to the above proof. Substituting oˆ t* (N − h; h) for oˆ t* ( − h; h) in equation (16), as permitted by the periodic property, equation (19), then introducing the ‘dummy’ index k for the index h in the resulting form of equation (16), and in equation (14), and introducing these expressions into equation (38), gives after minor rearrangement eˆ 't (j; h) = o* t (j − h; h) − o* t (N − h; h) − +o* t (j − k;k) − o* t (N − k;k)],

1 N−1 s [o* (j − h; k) − o* t (N − h; k) N k=0 t (B8)

which is the counterpart for the present case to equation (B1). As was the case for equation (B1), the total contribution to the right-hand side of equation (B8) from the exact values of artifact absolute tooth spacing errors oˆ t (j) and absolute scale errors oˆ s (i) is exactly zero, as shown below, leaving only contributions arising from measurememt errors.

. . 

748

Using equation (1), the notation of equation (12), and Fig. 1, these exact contributions to the individual terms in equation (B8) are o* t (j − h; h) = ot (j) + os (j − h) o* t (N − h; h) = ot (N) + os (N − h) o* t (j − h; k) = ot (j − h + k) + os (j − h) o* t (N − h; k) = ot (N − h + k) + os (N − h) o* t (j − k; k) = ot (j) + os (j − k) o* t (N − k; k) = ot (N) + os (N − k).

(B9a − f)

Substituting equations (B9a–f) into equation (B8) and rearranging terms gives eˆt' (j; h) = ot (j) − ot (N) −

1 N−1 s [o (j) − ot (N)] N k=0 t

+os (j − h) − os (N − h) −

1 N−1 s [o (j − h) − os (N − h)] N k=0 s

1 N−1 1 N−1 s ot (j − h + k) + s o (N − h + k) N k=0 N k=0 t

−

1 N−1 1 N−1 s os (j − k) + s o (N − k), N k=0 N k=0 s

−

(B10)

which is the counterpart for the present case to equation (B4). Using the same reasoning as for equation (B4), every line in the right-hand side of equation (B10) is seen to be identically zero. Hence, the contributions of the exact artifact and scale errors to the righ-hand sides of equations (B8) and (38) are identically zero, leaving only measurement error contributions. Therefore, one may replace each measurement o* t in equation (B8) by its corresponding measurement error et without changing the value represented by the entire right-hand side. Making this change in notation and forming the average over h as indicated by equation (39) gives eˆt' (j) =

1 N−1 s [e (j − h; h) − et (N − h; h)] N h=0 t 1 N−1 N−1 s s [e (j − h; k) − et (N − h; k)] N2 k = 0 h = 0 t

−

1 N−1 1 N−1 s s [e (j − k; k) − et (N − k; k)], N h=0 N k=0 t

−

(B11)

which is the counterpart for the present case to equation (B5). The first summation and the last double summation in equation (B11) cancel each other, as in equation (B5), and

      

749

the two terms in the first double summation also cancel each other in the sum over h in the exact same manner as in equations (B5) and (B6), thereby yielding the result, eˆt' (j) = 0,

j = 0, 1, . . . , N − 1.

(B12)

APPENDIX C: PROOF OF VALIDITY OF APPROXIMATE VARIANCE FORMULAE, EQUATIONS (31) AND (43), FOR ERRORS IN SCALE ERROR AND ARTIFACT ERROR ESTIMATES, RESPECTIVELY Two critical assumptions are used in our proof of the validity of equations (31) and (43): the measurement errors et (i; h) are uncorrelated for measurements made utilising different rotational displacements h of the artifact with respect to the rotary scale; and the number of scale and artifact measurement locations N is large in comparison with unity. For convenience, it is assumed that each of the measurement errors et (i; h), i = 0, 1, . . . , N − 1, h = 0, 1, . . . , N − 1 has the same common variance st2 , and is uncorrelated with all of the others. First consider equation (22), which is an exact expression for the error e's (i) in the scale error estimate oˆ s' (i) given by equation (14). Equation (22) expresses the error e's (i) as a summation of 2N measurement errors et (i; h) and et (0; h). Utilising the above-mentioned assumptions and applying the theorem [9] that the variance of the sum of uncorrelated random variables is equal to the sum of their individual variances, one obtains, using equation (22), the variance of the true error e's (i): N−1

Var {e's (i)} = s

Var

h=0

=N

6

6 $

%

$

%7

et (i; h) e (0; h) + Var t N N

7

st2 s2 2s 2 + t = t, N2 N2 N

i $ 0.

(C1)

This value is to be compared with the expression, derived below, for the variance of the estimate eˆs'(i) of the true error e's (i), where eˆs'(i) is obtained by inserting equation (26) into equation (27). An expression for the quantity defined by equation (26) is given in Appendix B by equation (B1). Again, as shown in the proof following equation (B4), the total contribution to the right-hand side of equation (B1) from the exact values of the measured quantities o* t is zero, leaving only the measurement error contributions. Hence, recognising the decomposition of the measurements o* t into their true values and measurement errors, as described by equation (20), one again may replace the measured values o* t in equation (B1) by their corresponding measurement errors et . Carrying out this change in notation and examining the terms k = h within the summation in the resulting form of equation (B1) gives

0 1

eˆ s' (i; h) = 1 −

2 [e (i; h) − et (0; h)] N t

1 N−1 s [e (h + i − k; k) − et (h − k; k) + et (i; k) − et (0; k)]. N k=0 t

−

except k = h

(C2)

. . 

750

It follows from the derivation of equation (26), together with Fig. 1, that each of the individual terms et (,; ,) within the summation in equation (C2) is periodic in each of its two arguments with period N. Using this fact, it follows that for every pair of values i = 0, 1, . . . , N − 1 and h = 0, 1, . . . , N − 1 two terms within the summation for k = h + i cancel each other, and two terms for k = h − i cancel each other, yielding

0 1

eˆ 's (i; h) = 1 −

2 [e (i; h) − et (0; h)] N t

1 N−1 s [e (h + i − k; k) − et (h − k; k) + et (i; k) − et (0; k)] N k=0 t

−

except k = h, h + i, h − i

1 [e (i; h + i) − et (−i; h + i) + et (2i; h − i) − et (0; h − i)]. N t

−

(C3)

Furthermore, again from the fact that et (,; ,) is periodic in its first argument with period N, for i = N/2 the quantity within the last bracket in equation (C3) is seen to vanish identically. With this exception, the remaining terms in the right-hand side of equation (C3) represent the errors experienced in different measurements, except for the case i = 0 where eˆs' (0; h) = 0. Assuming these different measurement errors to be uncorrelated, and again applying the theorem that the variance of the sum of uncorrelated random variables is equal to the sum of their individual variances, each individual error term et being assumed to have the same variance st2 , we obtain, from equation (C3), for the variance of eˆs' (i; h),

0 1

Var {eˆs' (i; h)} = 1 −

2

2 (N − 3) 2 1 2st2 + 4st + 2 4st2 , N N2 N

i$0

(C4)

where the last added term in the right-hand side of equation (C4) is to be removed for the case i = N/2. Simplifying the above expression gives, whenever i $ 0,

F 2 G 2st2 1 − N , i $ N/2 Var {eˆs' (i; h)} = g G 2st2 1 − 2 − 22 , i = N/2. N N f

0 1 0 1

(C5)

Finally, using the fact that Var {eˆs' (i; h)/N} =

1 Var {eˆs' (i; h)} N2

(C6)

      

751

and the above-cited theorem pertaining to the variance of the sum of uncorrelated random variables, from equations (27), (C5), and (C6) wherever i $ 0, one has N−1

Var{eˆs' (i)} = s Var {eˆs' (i; h)/N} h=0

= N Var {e's (i; h)/N}

0 1

2 2s 2 F G Nt 1 − N , G =g G 2st2 2 2 1− − 2 , G N N N f

0

1

i $ N/2

i = N/2.

(C7)

This result, which was derived from the approximation eˆs' (i; h), given by equation (26), to the actual measurement error e's (i; h) defined by equations (20–24), is in asymptotic agreement with equation (C1) for large values of N. As a practical matter, for all but extremely small values of N, the two results on the right-hand side of equation (C7) are essentially the same. Hence, by comparing equations (C1) and (C7) and by utilising the approximations in the above analysis, one has Var {e's (i)} 1

1

0 1

Var {eˆs' (i)}

(C8)

2 1− N

which suggests that, for moderate values of N, equation (31) should provide, on the average, slightly more accurate results if its right-hand side is increased by a factor of [1 − (2/N)] − 1. The assumption that each measurement error term et (,; ,) in the above analysis possesses the same variance st2 has been convenient in deriving the results given by equations (C7) and (C8). However, by carefully comparing equations (24) and (C3), it can be seen by forming the variances of these two different expressions that it is only really required that N be large in comparison with unity and that for different rotational locations h of the artifact with respect to the rotary scale, i.e. different values of the ‘dummy’ variable k in equation (C3), the measurement errors et (,; k) be uncorrelated, in which case the contributions of the summation over k and of the last line in equation (C3) will be small in comparison with the contribution from the first line, which for large N, is essentially the same as equation (24). Proof of the validity of equation (43) is obtained in exactly the same manner as that obtained above, utilising the same approximations, and requiring as a practical matter only that N be large in comparison with unity and that the measurement errors be uncorrelated for different rotational locations h of the artifact with respect to the rotary scale. Equation (33) describes the error e't (j) in the estimates oˆ 't (j) of artifact absolute tooth spacing errors, which is the counterpart for the present case to equation (22) for errors in scale error estimates. Since equations (33) and (22) both have the same general forms, it follows from equation (C1) that the variance of the true error e't (j) is Var {e't (j)} =

2st2 , N

j $ 0.

(C9)

. . 

752

Turning now to the variance of the error estimate eˆt' (j), by substituting the measurement error et for the measurement o* t in each term of equation (B8) and examining the terms for k = h, one obtains

0 1

eˆt' (j; h) = 1 −

2 [e (j − h; h) − et (N − h; h)] N t

1 N−1 − s [et (j − h; k) − et (N − h; k) + et (j − k; k) − et (N − k; k)], (C10) N k=0 except k = h

which is the artifact counterpart to equation (C2). Examining the terms for k = h + j − N and k = h − j + N in the summation in equation (C10), it is found that, in each of these two cases, two terms within the summation cancel one another, leaving

0 1

eˆt' (j; h) = 1 −

2 [e (j − h; h) − et (N − h; h)] N t

1 N−1 s [e (j − h; k) − et (N − h; k) + et (j − k; k) − et (N − k; k)] N k=0 t

− except

k = h, h + j − N, h − j + N

1 [e (j − h; h + j − N) − et (2N − h − j; h + j − N) N t

−

+et (2j − h − N; h − j + N) − et (N − h; h − j + N)],

(C11)

which is the artifact counterpart to equation (C3). Furthermore, using the fact that et (,; ,) is periodic in its first argument with period N, for j = N/2 the last quantity within the brackets in equation (11) vanishes identically. Thus, the behaviour of equation (C11) is identical in form to that of equation (C3), from which one obtains, from equations (39) and (C7) whenever j $ 0,

F2st2 2 G N 1−N , j $ N/2 Var {eˆt' (j)} = g 2 G2st 1 − 2 − 22 , j = N/2 N N fN

0 1 0 1

(C12)

which is the artifact counterpart to equation (C7). Comparing equations (C9) and (C12) and recognizing for moderate and larger values of N that the correction of − 2/N 2 in equation (C12) is negligible, one has Var {e't (j)} 1

1

0 1

Var {eˆt' (j)}

(C13)

2 1− N

as was the case in equation (C8). Equation (C13) suggests that, for moderate values of N equation (43) should provide, on the average, slightly more accurate results if its right-hand side is increased by a factor of [1 − (2/N)] − 1.