A V E R A G I N G R A N D O M

R A S T E R G R A T I N G S

E. E. K o l o t i l i n

ERRORS OF L I N E A R T R A N S D U C E R S '

UDC 621.9.085.088 : 621.383.4

Digital measuring systems, f eedback circuits of machines with a programmed control, and other devices use photoelectr ic transducers with raster gratings which are suitable for automat ic testing of displacements over a length of 1000 mm or more with an error of a few microns. The pulse and phase transducers which use raster gratings with graduation spacings of 0.02 to 1 mm have been developed by foreign firms and several Soviet organizations [1-3].

Figure 1 shows an optical schemat ic of a transducer with a raster grating. The displacement values are read simultaneously over a whole group of graduations within the field of vision of the photoelectr ic receiver (sometimes several hundred graduations). Therefore, at each point along the measured displacement the transdueer's error de- pends on the spacing errors of al l the graduations in a group, and the errors of separate graduations are averaged out.

Below we examine the averaging of the basic spacing errors between the raster grating graduations in l inear transducers. The effectiveness of averaging, i .e . , the degree to which the transducer's error is reduced as compared with the grating errors,is normal ly considered to depend only on one factor, namely, on the number g of graduations within the f ield of vision of the photoelectr ic receiver , and i t is assumed that this number is very large. However, such an evaluat ion in inadequate and in cer tain cases leads to false conceptions about the effect of averaging on the reduction of spacing errors between the graduations of a transducer's grating.

Long raster rules are usually engraved in our country on mechanica l dividing machines with correction de- vices for compensating systematic errors. It is, however, much more difficult to deal with graduation-spacings random errors which become predominant in the engraving of precision rules.

The engraving of long raster rules is character ized by two basic random errors:

Random instrumentation errors related to the corresponding errors in the operation of the dividing machine mechanisms;

Random temperature errors due to the deviat ion of temperature from its normal value during the engraving.

Avera$in$ of Random Instrumentation Errors. Investigation of the engraving of rules on the type TLF m e - chanical dividing machine has shown that at the in i t ia l engraving section (~ 100 ram) the instrumentation errors are fairly large and have a transient nature, whereas on the remaining length they become stationary [4]. The ini t ia l transient engraving section should be e l imina ted to raise the precision in manufactu;ing rules. In this case the in- strumentation errors are character ized by the constant dispersion D x. Let us find the dispersion of the accumulated transducer errors, which is re la ted to this type of raster errors.

We understand the accumulated transducer's error at point k to be the deviat ion from its normal value of the displacement from the in i t ia l position to the k- th position, with both positions corresponding to the same luminous- flux level at the output of the raster grating system. This error can be represented by the following expression

g g g ' E ' E ' E ~7/~:~ - - ~ [X(kq_p) r - - X(p__l)i] - - - - X [Xpr- -X(p_l ) i l = g �9 [x(k+r,), r - Xpr], (1)

g P=I p = l p = l

where g is the number of graduation spacings in the averaging zone (the photoelectr ic receiver 's field of vision), the subscripts r and i refer to the raster rule and the index grating, and Xp represents the accumulated error of the p- th graduation spacing in the averaging area (see Fig. 1).

Translated from Izmer i t e l ' naya Tekhnika, No. 4, pp. 18-20, April, 1971. Original ar t ic le submitted April 15, 1969.

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Direct-ion of the relative dis- Photoelectric receiver ~ placement of gratings

Raster / ~ \ I n d e x ~ ' ~ " ~ ~ rule " grating

\Condenser Lamp

Fig. 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,mm

L 'av :!

Fig. 2

It will be seen from this formula that the spacing errors of graduations in the index grating do not affect in

this case the accumulated transducer's error. g g g

p = l p = l p = l

g g

where "~ Kxp and ~ K~k+p nominally denote summations of correlation instants of graduation errors taken in p = l

o= l

pairs of any combination within one group of graduations covered by the photoelectric receiver's field of vision and g

lying respectively at the beginning of the raster rule (k=O) and at a distance of k graduations from it; ~ Kxp, xk+ p p = l

denotes a summation of the correlation instants of graduation errors taken in pairs of any combinations, but ne- cessarily from different groups (from those enumerated above).*

Taking into consideration the properties of correlation matrices with errors of a stationary nature, it is possible to write, for the above summations which represent parts of these matrices, the following expressions:

*In the above expression and henceforth the subscript r is omitted, but it is understood that all the errors (with the exception of those with the subscript E) refer to the raster rules.

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g g g--I

~ I(xp --= ~ [(xkJr_p= ~ [l((Xl' Xl+p) (g-- P) ], p=l o=l p=l

g g--I

<,, (g--I n 1)1 P = I p=l--g

Then

g - - 1 ~'--1

Dxkx=~ gDx+2 [K(q, Xl+p)(g--p) l - - , ~ [ K ( x l , x~+l+p) (g-- !p l ) ] . (:3) p = [ p== 1 - - g

Separate terms of the summations in (3) are mult ipl ied by given factors corresponding to the ordinates of the

c0rrelation function Kx(A~ ).

In the first case, moreover, the summation of the correlation function is made over its ini t ia l section which

has a length approximarely equal to the photoelectric receiver's field of vision, whereas in the second case it is

made over a section which has double that length and is located symmetrical ly with respect to the abscissa A1 = kt.

The length of the photoelectric receiver's field of vision in actual transducers normally does not exceed 5-10

ram. Investigation of the correlation function of the stationary, instrumentat ion,scale-division errors has shown that

the function has an adequately smooth characteristic [4]. * Therefore, taking into consideration the short length of

summation sections, it is possible to assume that

K (x~, Xl+p) m Da; K(xl, xk+l+p) ~ Kx(kt).

In this case

~ , i [K(xl q + p ) ( g - - P ) ] - - g(g-- 1) D~ ' 2

p=l

g-1 t uo

[ g ( ~ ' , %1-1+p) ( g - 1 P ~)1 = ~" Kx(,~). f a= l - - g

Dxk2= 2 [ D x -- Kx(kf)J. (4)

However, (4) corresponds to an expression which could have been obtained directly from (t) for g = 1, i .e . , by

considering the displacement for one graduation. Therefore, it is possible to arrive at the conclusion that averaging

for stationary instrumentation errors is not effective and does not lead to an appreciable reduction of dispersions in

the corresponding transducer errors.

It should be noted that the transient instrumentation errors of the grating graduation spacings increase in each application with a relative smoothness over the ini t ial section, whereas they become constant over the remaining

length. On the basis of such a nature of variations, it is possible to assume that the averaging is also ineffective for the latter type of errors.

Averaging of Random Temperature Errors. In certain cases the graphs of temperature errors for a set of rules

can be represented in the form of a bunch of incl ined straight lines. If l inear approximation is not desirable for a

full application, it nevertheless is permissible over the length of an averaging section, and this is determined by the following circumstances.

The engraving of raster rules on a dividing machine is carried out at a speed not exceeding seven graduations

per minute, With a graduation spacing t = 0.04-1 mm the engraving of graduations over an averaging section will

last from 1 to 30 rain. With a system of thermostatic control in the premises of a dividing machine, and taking into consideration the thermal inertia of the medium, machine units, and rule blanks, it is possible to assume that the temperature during this period will change insignificantly and the corresponding section of the temperature error graphs can be approximated with sufficient precision by a straight l ine (Fig. 2).

*In analyzing the instrumentation errors we did not take into account the small periodic errors (with a period length of 1 ram) which are due to the beating of the dividing machine 's driving screw. However, it has been shown in [2] that the effect of the systematic component of these errors can be el iminated by an appropriate selection of the field of vision length. On the other hand, the random component of this type of errors is sufficiently small.

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In this case the value of the averaged temperature error is equal to the temperature error in the middle of the averaging section of the raster rule xt(lav), and the graph of the accumulated transducer's temperature errors re- produces virtually without any changes the graph of the temperature errors of the rule (with the exception of the initial and end sections of lengths gt/2 = 3-5 ram). Thus, for the actually used length of the photoelectric receiver's field of vision the averaging of the random temperature errors of the transducer's grating spacings is also ineffective.

The above examples show that in evaluating the effectiveness of averaging the spacing errors of raster gratings it is necessary to take into account not only the number of graduations in the photoelectric receiver's field of vision, but also the nature of the error's longitudinal distribution and their correlation links. In particular, if the correlation function of the instrumentation errors correspond to the type obtained in engraving the scale by means of the TLF mechanical dividing machine, a sufficiently effective averaging can be obtained by making the length of the photo- electric receiver's field of vision comparable to the length of the initial section which has a strong positive corre- lation, i.e., to ~ 100 ram. However, this entails considerable design difficulties and leads to a reduction in the length of the tested displacement. Therefore, it is better in the case of transducers which use gratings with errors of the above nature to raise precision by applying correlation devices.

The above approach to evaluating the effectiveness of averaging errors in the location of separate elements of a transducer's scale can probably also be utilized in analyzing the precision of certain other types of measuring devices in which several scale elements are read simultaneously and there exists an averaging effect.

L I T E R A T U R E C I T E D

1. V.F. Shan'gin and Yu. A. Shatalov, "Digital displacement transducer with raster gratings," in: Computer Techniques [in Russian], No. 5, Mashgiz, Moscow (1966).

2. A .V. Mironenko, Photoelectric Measurement Systems [in Russian], ~nergiya, Moscow (1967). 3. E.E. Kolotilin, Pribory i Sistemy Upravleniya, No. 5 (1967). 4. E.E. Kolotilin, Izmeri tel ' . Tekh., No. 9 (1968).

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