Measurement Techniques, Vol. 37, No. 9, 1994

D Y N A M I C A S P E C T S O F P R E C I S I O N L I N E A R - D I S P L A C E M E N T

M E A S U R E M E N T

L. Z. Dich UDC 53.083 +681.325

A qualitative analysis is presented for dynamic linear-displacement measurement. When measurements are

made with errors in the submicron and nanometer ranges, it may be necessar 3' to correct for dynamic errors

whose origin is geometrical.

Long experience with precision length measurements and certain methods of performing them indicates that the tasks

characteristic of that area in metrology are usually considered as static ones. There are various standardization documents

dealing with the general conditions for making linear measurements [1] and the conditions for particular measurements [2, 3]

that consider reducing the effects of factors such as vibration and rapid temperature change that cause dynamic errors. That

approach has been realizable and probably justified because in most cases one provides high length-measurement accuracy

traditionally in operations of test character.

However, developments in engineering and particularly in microelectronics make it necessary to build industrial

equipment in which the position of the tool or workpiece must be monitored with errors in the submicron hundreds of

millimeters, displacement speeds of I m/sec or more, and accelerations in the mobile parts of 10-50 m/sec 2 [4, 5]. Naturally,

such equipment must contain means of measuring length and displacement with analogous characteristics. Although the main

difficulty lies with the designer in producing instruments of that type, the exceptionally tight specifications for such development

in my opinion make it impossible to carry it through without using theoretical and practical experience accumulated in

metrology and without involving national metrological services. It is problematic to consider measuring displacements in

precision and high-throughput equipment on the basis of static measurements, on account of the high speeds and accelerations

that are realized. Here I consider features that characterize dynamic displacement measurement.

The formulation of length measurement as a task in deriving a displacement is influenced by features of the equipment

and of the length-signal conversion. Most precision length converters are incremental, i�9 are devices in which the output

signal is formed by accumulating information on the changes in the input signal (displacement), and in general the formal

features are such [6, 7] that this mode of conversion could be called dynamic, but nevertheless this has not as a rule been

considered in relation to incremental converters, nor has the comparison of instantaneous values for the input and output

signals, and in the analysis of errors it is assumed that the information is read out only at the end of the displacement, which

may be unlimitedly remote.

There are two features of displacement measurement as such. The first is that one determines the displacement of the

point of interest in accordance with the purpose of the measurement, which is virtually always impossible (it is impossible for

example to measure dimensions directly in the machining zone or to measure the position of the cutting edge of a tool in situ

or to determine the position of the light spot in laser drilling or to find the coordinates of the measurement tip in contact

�9 measurements at the point of contact and so on). These problems mean that differences arise in the space of the converter and

the object, and one records the displacements of certain other remote points, which one identifies with the displacements of the working point.

The second feature is organically related to the first and is that material bodies of finite size are displaced during the

measurement, and the displacement in a real situation is not only translational but also has rotational components�9 This means

Translated from Izmeritel'naya Tekhnika, No. 9, pp. 23-25, September, 1994.

0543-1972/94/3709-1011512.50 �9 Plenum Publishing Corporation 1011

that the displacements of different points on the moving sensor element become different, so one has to consider what is the

relation between the displacements of what points in the moving body and the number of the output of the converter and what displacements of the latter point and the readings of the converter relate to the displacements of the working point.

If the moving bodies can be considered as absolutely rigid (and this is so if the dynamic perturbations during the displacement are small, e.g., the movement occurs at low velocities and accelerations or the measurement conditions enable

one to allow the dynamic perturbations to decay and then take the reading), then the response to this question can be derived

from the concepts of solid-body kinematics. It can be shown [8] that there is a locus for the points in an instrumental convener

that is uniquely determined (comparator straight line) and whose displacements can be related unambiguously to the readings

of the converter, while for these readings to be identified with the displacements of the working point, the latter should lie on

the comparator straight line. That requirement in essence repeats the content of the Abb6 principle in its initial formulation [9] but with the difference that the comparator straight line and the material carrier of the unit of length in the converter may be linked very indirectly. The position of the comparator straight line in space is defined by the mutual disposition of the mobile

and fLxed components in the converter and may vary during the displacement measurement. The change dN in the convener

reading for a small displacement is proportional to the scalar product VcI,(r)dr, in which AcI,(r) is the gradient in the parameter

governing the change in the information signal (for example, the gradients of the light-wave phase in an interferometer), while

dr is the displacement vector for the point to which the converter readings are linked. For a finite displacement, the change

in convener reading is proportional to the integral

A, 'r ( V ~ ( r ) d r , (1)

which is calculated along the displacement path for that point. If the converter is considered as a set of absolutely rigid bodies,

any point on the comparator straight line can be considered as rigidly coupled to the converter even if it does not physically

belong to the latter. These features are important in displacement measurement dynamics. With a speed of 1 m/sec and an acceleration of

5g, the accelerations over such a short time are comparable with shock acceleration and involve considering the finite rigidly

of all the components and correction for the deformations on acceleration.

Two cases can be distinguished:

1. The stationary approximation: the elastic-perturbation propagation time in the elements is negligibly small by

comparison with the permissible processing time, and no deformation waves arise whose periods are comparable with the sizes

of the elements. Then one can assume that the velocity and acceleration for each of the moving points will be determined only

by the kinematic parameters. If one considers a time interval in which the acceleration and deformation may be taken as

constant, the structure can again be considered as a set of absolutely rigid bodies, and the displacement can be calculated from

(1) on the basis that the position of the comparator line is dependent not only on the current coordinate but also on the current

deformation or acceleration: VcI, = VcI,[r(t), ~(t)]. The position of the comparator line in space may alter because of the varying acceleration, which can lead to errors

that are dynamic. For example, estimates have been made for standard displacement sensors based on diffraction gratings [10],

which show that an acceleration of 5g can turn the comparator line through 40". Over a displacement of 10 ram, which

corresponds to the acceleration time up to a velocity of 1 m/sec or complete retardation, there is an additional error that can

be calculated on the basis that the nominal comparator line is turned through about 1 ~ relative to the direction of displacement

and thus gives an error of 0.04 #m, which is very important in the nanometer and submicron error range.

2. The nonstafionary case: in most general form, there are elastic waves in the elements whose periods are comparable

with the sizes of them. There are differences in displacement between the points due not only to kinematic rotations but also

to differences between the velocities and accelerations due to the elastic forces. If the structure has a point for which the

equation of motion in space is known, one can determine the displacement of another point in space on the basis of the dynamic

theory' of elasticity, i.e., by solving the Lam6 equation

a2u(r,t) (~.+2~)V2u(r,t)-i-F(r,Q=P Ot2 (2)

in which u is the displacement vector at point r, F(r, t) the bulk forces, )~ and/z the Lam~ constants, and a density.

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To make it formally possible to realize this approach, it is necessary to determine correctly the point P whose

displacements correspond to the sensor readings in the presence of deformation. That point can be located by operating with

points in the sensor that physically participate in forming the output information and constitute the signal recording zone. In an optimal sensor, for example, this can only be points within the light beams. It is not obvious in advance however that point

P exists and is unique for a general formulation. The following intuitive arguments may be important in resolving this topic.

The signal recording zone S can be represented as a set of parts dS i having small sizes and within which the deformations can be neglected. One can also assume that each such part is subject to a small displacement Ar i, which leads to the recording of

a change AcI' i in the phase of the periodic signal: A~ i = V~i'Ar i. The resultant and actually recorded change in phase for the

entire recording zone is

[ l ( r ) s i n S ~ ( r ) d S s (3)

/ , 4 0 ) = a r c I ~ ~ l ( r ) cosA~(r )dS ' S

in which I(r) is a weighting factor determining the intensity of the contributions from the various parts of the recording zone

to the signal. Each part dS i by virtue of its lack of deformation can be put into correspondence with a comparator line l i, and the

set of these defines a certain lineated surface, which is not necessarily continuous. One can assume that if I(r), V,I,(r) and Ar(r)

are continuous, the set of dS i will contain a part j such that z~Oj = < z%~ > , the position change in this case be related to the

change < 2xI, > in the sensor readings, and which in turn can be compared with the comparator line lj containing the required

point P. However, this does not exhaust the complexity and does not allow one to pass directly to solving (2). It may be that

such arguments give several comparator lines, and that none of the points on these lines will coincide with material points in

the device. That fact is not an obstacle for an absolutely rigid model, but if the rigidity is finite, it remains uncertain what

values of the physical parameters X, t~, and P should be assigned to the medium outside the structure, and how one should

proceed if there are several comparator lines. Possibly, the difficulty related to localizing the comparator line may be overcome

if one assumes formally that P on the comparator line and area dSj are linked by an infinitely thin and absolutely rigid rod.

. It is quite likely that the dynamic treatment for a linear displacement converter does not make it possible to put the

readings into correspondence with the motion of any actual point or to construct an acceptable geometrical model containing

such a point. Then the readings on the converter will in that case be those of a black box and should be related directly to the

displacement of the working point, while the error of measurement should be determined either on the basis of a priori

information, with the extrapolation possibly of features from existing mechanical systems, or else empirically. However, it is

difficult to imagine a device capable of producing for example a test signal as the instantaneous displacement with nanometer

error over a given distance or as a harmonic oscillation of given frequency and amplitude having the same order of error.

I suggest that in precision dynamic measurements one should consider a further source of errors additional to the known

sources of dynamic errors occurring in any measurement device that have been adequately described in the literature: it is

geometrical in origin and is characteristic of means of length measurement. I have not been able to fred in the publications

accessible to me any arguments on this topic, so I suggest the following formulation of the dynamic treatment for precision

linear measurements.

I envisage an incremental linear-displacement sensor having any structure and finite rigidity, which contains a working

point and whose mobile elements may move with variable accelerations along any paths within the tolerance corridor. The

length measure with which the sensor operates is ideal and the error in measuring the displacement of the working point is zero

when the units in the sensor are displaced along an ideally straight path with zero acceleration. It is necessary to give a clear-cut

geometrical interpretation for the readings, i.e., to define or construct a geometrical object whose motion can be compared

unambiguously with the converter readings on the basis that the converter contains elastic waves (or else to show that such an

object cannot be defined in the general case). When the required geometrical object exists, it is necessary to derive relationships

between the equations of motion for the points in the object and the working point on the basis of the actual physical properties.

The above arguments cannot be considered rigorous. They rather illustrate the difficulty of the topics arising in the

area called nanometrology, and it is not suggested that they fully illuminate all the problems and cannot replace fundamental

theoretical studies. I consider that such studies are certainly necessary, and to perform them requires the combination of efforts

of qualified specialists within appropriate scientific programs, as is being done in certain countries [11, 12]. The purpose of

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such research should be to set up methods of theoretical analysis for mechanical devices in the nanometer displacement-error range and thus to define methods and means of experimental research on the accuracy characteristics of such devices and relate them to the existing test scheme for means of length measurement.

REFERENCES

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5. 6. 7.

8.

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All-Union State Standard 8.050.73: The State Systems of Measurements: Normal Conditions for Performing Linear and Angular Measurements [in Russian]. All-Union State Standard 8.367.79: The State Systems of Measurements: Plane-Parallel Standard End Length Measures of Classes 1 and 2 and Working Classes of Accuracy 00 and 0 of Length up to 1000 mm: Methods and Means of Checking [in Russian]. MI 2079-90: Plane-Parallel End Length Measures of Classes 3, 4, and 5 for Lengths up to 100 mm: Methods and

Means/of Checking [in Russian]. E. E. Onegin, Precision Engineering for Microelectronics [in Russian], Radio i Svyaz', Moscow (1986). Progress in Precision Engineering, Springer-Verlag, Braunschweig (1991). K. P. Shirokov et al., Izmer. Tekh., No. 12, 9 (1975). V. A. Granovskii, Dynamic Measurements [in Russian], Energoatomizdat, Leningrad (1984). L. Z. Dich, in: Lectures at the School of Methods of Calculating Accuracy in Engineering and Instrumentation [in Russian], Inst. Problem Mashinovedeniya RAN, Moscow (1993), p. 57. E. AblM, Gesammelte Abhandlungen, Band II, Jena (1906), p. 206.

E. P. Malamed, et al., Opt.-Mekh. Prom., No. 6, 28 (1978). H. Kuzmann, Metrologia, No. 28, 443 (1991/1992). K. Nakajama, et al., Metrologia, No. 28, 483 (1991/1992).

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