Analysis of estimation and compensation of angular errors of linear motion/Tiesiaeigio judejimo kampiniu paklaidu ivertinimo ir kompensacijos analize.

Barakauskas, A. ; Kasparaitis, A. ; Kausinis, S. 等

1. Introduction

Linear motion modules are widely used in various purpose machines and mechanisms. One of their main functional characteristics is accuracy which is especially urgent for metrological and precision technological equipment.

A moving module's minor angular fluctuations about the axes perpendicular to the main movement are ranked as major errors of linear motion. Those errors are estimated as single-argument functions, e.g. of a shift in the main movement direction, or multiple-argument functions, e.g. involving also the ambient temperature or temperature of guides, or deviation thereof from normal temperature. Their value is dependent on the design peculiarities and production precision of linear motion modules, first of all, on the precision of guides, types of bearings and their mounting system, drive gear parameters and the effect of the variable parameters of the environment.

If Abbe principle requirements are ignored which is a frequent case in practice, the above-said errors manifest themselves by processing and measurement errors or by worsening the other functional characteristics of the equipment. Therefore, an attempt to remove or minimize roll and yaw axis rotations effect is made. To this end the design may be improved and the production precision may be increased. However, in many cases this method is expensive and its potentials are limited.

Promising methods include computational error compensation methods (in metrological equipment) or module movement correction methods (in technological equipment), which do not require any material increase in complexity of the equipment and production precision. To realize those methods, mathematical models that depend on the abovementioned movement errors are created. To this end, components of the abovementioned linear motion error should be determined and estimated.

Accuracy of the two main computational methods of angular fluctuation error estimation, statistical and active, their peculiarities and scope of application will be discussed hereinafter. Experimental tests have been carried out using precise line scale calibrator, precise coordinate measuring machines and specially created test benches.

2. Error estimation and their compensation methods

2.1. Statistical estimation of errors

The essence of this method consists in measuring, statistical processing, storing of errors, and elimination of the error influence during operation of the equipment.

This is realized by the following stepsio

1. Selection of errors the systematic part of which is stored and compensated or the influence of which is corrected.

2. Selection of simple functions dependent on few parameters, whereby the stored errors are approximated.

Errors dependent on a single argument, e.g., displacement of the measuring module, are approximated by single-valued parametrical functions. Errors dependent on several arguments, e.g., displacement, temperature and direction of measurement, are approximated by more complex multiple-valued functions [1].

3. Determination of errors and calculation of parameters of the error-approximating functions in the course of measurements.

4. Creation and input of programs into the measuring equipment processor's memory, which programs are used at the time of each measurement for the calculation of compensatory or corrective values in accordance with which the measurement results are compensated or software is corrected.

It is considered that error [DELTA].sub.[??]] consists of a systematic component which is defined by function [F.sub.Q] approximating this error, and random approximation error [E.sub.Q]

[[DELTA].sub.Q] = [F.sub.Q] ([xi],[theta])+[E.sub.q] (1)

where [xi] = ([[xi].sub.1],[[xi].sub.2],..., [[xi].sub.k]) are arguments (coordinates, temperature, etc.), registered during each measurement; [theta] = ([[theta].sub.1],[[theta].sub.2],...,[[theta].sub.k]) are parameters of function [F.sub.Q].

To approximate error [[DELTA].sub.Q] the following is selected: the main arguments [[xi].sub.j], approximation function [F.sub.Q], values of its parameters [theta] are easily calculated by the equipment's meters.

First of all, significance of errors is established, i.e. whether it is purposeful to compensate or correct the effect of those errors. To estimate the abovementioned an expert method is used, by comparing the mean error, the difference of its maximum [[delta].sub.imax] and minimum [[delta].sub.imin] values [[DELTA].sub.max] = [[delta].sub.imax] - [[delta].sub.imin] with root-mean-square deviation [S.sub.PV] of error values [[delta].sub.ij] from error mean values [[delta].sub.i].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

where [[delta].sub.ij] is value of the error j-th realization at i-th point i = 1,2,...,M ; j = 1,2,...,N ; [[delta].sub.i] is value of mean error at i-th [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Compensation or correction is practically meaningful when [DELTA].sub.max] [greater than or equal to] [+ or -] 1 * [S.sub.PV].

The number of arguments [xi] of the approximation function corresponding to the number of factors that have a maximum effect on the error, is determined by an empirical error distribution analysis in accordance with histograms defined by the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

where [delta] is the error under investigation; [[delta].sub.1], [[delta].sub.2],...,[[delta].sub.N] are its observable values; [1.sub.A](x) is indicator function of set A

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is operator-selected

distribution of error S value interval.

If the histogram has more than one peak, it shows that the error value is dependent on the influence of displacement and other factors. Arguments of approximation functions and variations of ambient temperature of the equipment operating environment in the course of error calibration, dependence of the trajectory of motion on its direction occurring, e.g. due to mechanical hysteresis, etc., may have and be such influence.

In pursuance of maximum precision of error evaluation the approximation function and its parameters must minimize the root-mean-square deviation of the error values from the approximation function.

Parametric functions of systematic error approximation are selected in accordance with their variation nature and their correlativity value analyzed by means of an empiric correlation function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

which is calculated in accordance with measurement results S(q), or in accordance with their spectral density estimated by the following periodogram

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

here [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is mean error; [[delta].sub.k] is results of experiment; v is frequency.

Previous research [2] has shown that the error pe riodogram of precision motion roller bearing blocks has maximums within the high and low frequency ranges, whereas in case of aerostatic bearing blocks (b) - in the low frequency range.

The type of approximation functions and their parameters are selected in accordance with the aforesaid. Angular errors of linear motion are rather precisely approximated by algebraic polynomials, algebraic polynomial splines and trigonometric polynomials.

An expert method is the simplest way to estimate the approximation accuracy in accordance with the root-mean-square deviation of error values from the approximation function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

by comparing it with the root-mean-square deviation of error values from mean errors [S.sub.PV] (Eq. (2)).

Optimality of error approximation can be estimated in a more objective way by determining whether the residual errors are distributed in accordance with Gaussian law. This is verified by using Kolmogorov and Smirnov criterion. If necessary, ratio values of standard asymmetry [??] and excess [??] are calculated.

After designating the empiric distribution function of error [delta] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

and it's real distribution function by

F (x )= P{[delta],x} (8)

Kolmogorov and Smirnov statistics will be calculated in accordance with the following formula:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

If F(x) [equivalent to] [PHI](x;a;[[sigma].sup.2]) where [PHI](X;a;[[sigma].sup.2]) is Gaussian distribution, the mean value of which is a and dispersion is [[sigma].sup.2, then distribution of random value [D.sub.N] is known and does not depend on values a and [[sigma].sup.2]. Let's designate thus distributed random value by gN. Using Kolmogorov and Smirnov criterion to verify whether the distribution is Gaussian, the value of probability P{[[xi.sub.N])[D.sub.N]} is calculated, which is called Kolmogorov and Smirnov criterion value.

If this value is lesser than 0.05, it is a hypothesis that the residual error has been distributed in accordance with Gaussian distribution.

In case of average Kolmogorov and Smirnov criterion values (from 0.05 to 0.75), standard coefficients of asymmetry a and excess c should be additionally calculated

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

here [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is a statistical estimate of the i-th sequence, when k = 1,2,... ; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is a statistical estimate of root-mean-square deviation [sigma] ; E[xi], D[xi] = E[([xi] - E[xi]).sup.2] - respectively, average and dispersion of the random value.

If the residual error is distributed in accordance with distribution [PHI](x;a;[[sigma].sup.2]) then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

whereas distributions of random values [??] and [??] in case of large N, are approximate to distribution [PHI](x;0;1).

Therefore, if values a and c do not fall into interval [-2, 2], then the hypothesis that residual error S is distributed in accordance with normal law, should be rejected.

2.2. Active error estimation method

This method is usually applied to real time compensation or correction of the influence of minor angular fluctuations of moving modules on accuracy. The essence of this method consists in the fact that during operation of the equipment angular fluctuations of the moving module about the two axes perpendicular to the direction of the main motion are measured, and the so-called Abbe errors or errors of comparison resulting from those fluctuations are calculated and compensated or corrected.

First of all, significance of the errors is determined in a manner similar to the statistical method.

Abbe error correction value [k.sub.v] on the vertical plane is set to [l.sub.v]

[k.sub.v] = [a.sub.v][l.sub.v] (12)

here [a.sub.v] is minor angular fluctuations of the moving module about y axis, [l.sub.v] is distance between the displacement measuring line and the line of other purpose function.

Abbe error correction value [k.sub.h] on the horizontal plane is set analogically.

This method's compensation or correction error is related to precision of the evaluation of both components of Eq. (1). Minor angular fluctuations of moving modules in precision equipment are measured by laser interferometers with special optical devices.

Minor angular fluctuations of a moving module are calculated in accordance with the difference of displacement of its two points by dividing the above difference by the distance between laser beams. Therefore, errors of measurement of displacement of those points will directly influence the readings of the above-mentioned minor fluctuations of the module [3]. The most significant of them are:

--estimation and compensation error of air temperature deviation within the whole measurement length,

--estimation and compensation error of air pressure deviation,

--air moisture estimation and compensation error,

--carbon dioxide estimation and compensation error,

--air turbulence error,

--uncertainty of Edlen formula,

--laser wavelength stability in vacuum.

The main sources of angular fluctuation measurement errors are:

--displacement measurement error and variation of its value under the effect of variable environmental factors;

--error of determining distance between laser beams [4];

--error resulting from elastic deformations of the movable module.

An optic diagram of the Abbe error measuring system is shown in Fig. 1.

[FIGURE 1 OMITTED]

Displacement measured with an ideal interferometer under ideal conditions is expressed by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)

here [lambda] is a laser wavelength; [DELTA][phi] is phase difference between measuring and reference beams of the interferometer.

Laser interferometer displacement readings L are dependent on air refraction index n and are defined by the following dependence

L = n[L.sub.id] (14)

The refraction index depends on measured environmental parameters: pressure p , temperature T , relative air humidity RH and C[O.sub.2] concentration. It is estimated in accordance with modified Edlen's equations [5]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)

here p, T , RH are measured pressure, temperature and relative air humidity values, respectively.

All components of this point displacement measuring error are directly dependent on displacement value q. Therefore, it is logical that the error of minor angular fluctuations of the moving module will depend on the magnitude of the measured displacement and may be essential in large displacement measuring or other technological systems [6, 7]. This would by far reduce the efficiency of such error compensation method.

3. Comparative results of experimental tests

The methods under consideration are compared by using the measuring data of angular fluctuations of a precise lined scale calibrator carriage within the 3500 mm length.

The comparator's basis comprises a 4 meter long massive fine-structured granite traverse placed on four pneumatic supports on the horizontal plane for damping of high frequency vibrations and a carriage moving on air bearings along the traverse on six high-precision guideways (Fig. 2). The carriage is designed so that it moves on rigidly mounted aerostatic bearings that are tightly adjusted by springy aerostatic bearings mounted on the opposite side of the carriage's guide-ways. The carriage is pulled on the granite guide-ways by means of the program-controlled friction gear. The carriage and gear system design eliminates the influence of the drive on the precision of linear movement. The carriage consists of the two components, i.e. a power component and a precise component. The drive is connected to the first component, whereas all measuring systems designed for measurement of the carriage's movement and for detection of the calibrated scale lines are connected to the precise component. Compensation of Abbe errors is implemented in the additional optical interferometer system with the error calculation software. Environmental correction of the air refraction index is realized by means of calculation using Edlen's equation. Tests were conducted in a thermo stable laboratory. The following ambient conditions were maintained in the laboratory during the tests: temperature at + 20 [+ or -] 0.15[degrees]C, humidity 40 [+ or -] 0.10%, air pressure changed within the range of 1000 [+ or -] 10 hPa. To avoid the formation of intensive air flows and turbulences in the room, the air was fed into and evacuated from the room in many places.

To obtain data for approximation, the angular yaw and pitch fluctuations of the carriage were measured several times with a Zygo ZMI 2000 model heterodyne laser interferometer while the carriage was moving in different directions.

The calibrator is described in detail and its diagrams are presented [8, 9].

[FIGURE 2 OMITTED]

Graphic results of six realizations of angular fluctuation measurements by means of a double-frequency laser interferometer are presented in Fig. 3.

[FIGURE 3 OMITTED]

4. Analysis of precision of the statistical error estimation method

First of all, purposefulness of the compensation of those errors was determined by comparing the mean error, the difference of maximum [[delta].sub.imax] and minimum [[delta].sub.imin] values with root-mean-square deviation [S.sub.PV] of error values

[[delta].sub.ij] from mean errors [[delta].sub.i] After evaluating the periodogram, the algebraic polynomial spline has been accepted as a function of approximation of mean error values dependence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (16)

here m is a spline sequence, m= 1,2.; v is number of modules [u.sub.1], [u.sub.2],..,[u.sub.v];d is defectiveness of spline,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Approximation has also been performed by means of algebraic polynomials of the first and third order. Diagrams of mean values of six realizations 1 and approximation by polynomials of the first 2 and third 3 order and by a spline of the second order 4 are presented in Fig. 4.

[FIGURE 4 OMITTED]

After analyzing measurement data the following results have been obtained.

The systematic error component determined by the difference of minimum and maximum mean values is 2.1".

The following values of root-mean-square deviation were obtained after calculation:

--error values versus mean values [S.sub.PV] = 0.12";

--error values versus approximation function values [S.sub.PF] = 0.14";

--approximation function values versus mean values [S.sub.fv] = 0.18".

Spread of measurement results defined by rootmean-square deviation [S.sub.PV] occurs due to the following two reasons:

--instability of linear motion of the measured point,

--uncertainty of measurement of angular fluctuations.

Those two values are not precisely known.

Linear motion instability of a measured point of linear motion modules of aerostatic bearing can be approximately estimated by variation of spacing in aerostatic bearings. It has been experimentally found that while in motion a module varies within the limits of 0.4 Lim [9]. The root-mean-square deviation of the module's random angular fluctuations resulting from the above variations is [S.sub.a] = 0.069". Then, standard uncertainty of measurement of angular fluctuations will be [S.sub.m] = 0.098".

It has been experimentally established that the spread of angular fluctuations varies while the carriage is located at different distances from the laser interferometer measuring that carriage [10]. The spread of measurement results increases with the increase of this distance. Fig. 5 illustrates that the estimate of the root-mean-square deviation of the spread of measurement results of the carriage's angular fluctuations on the vertical plane is dependent on the carriage's distance from the laser interferometer q calculated in accordance with the results of six realizations presented in Fig. 3.

In the case under consideration this variation is conveniently expressed by the following parametric function D(q) obtained through approximation of root-meansquare deviation values.

[FIGURE 5 OMITTED]

While the carriage is sliding from the minimum to the maximum 3500 mm distance from the laser interferometer, the mean value of root-mean-square deviation [S.sub.PV] of the spread of measurement results varies from 0.113" to 0.215". The main cause for the spread of measurement results are the above-mentioned laser interferometer's length measurement error components dependent on the length of measurement. Random angular fluctuations of a moving module that are estimated by variation of the spacing in aerostatic bearings are identical in both positions. After eliminating the influence of the random-moving module's angular fluctuations on the measurement results, it was found out that root-mean-square deviation [S.sub.m] of the spread of measurement results, which deviation depends on variation of the laser interferometer readings, varied from 0.082" to 0.198" with the change of the measurement distance.

The value of elastic deformations of the module and the measurement errors or errors of other nature resulting from the above deformations and other parameters depend on the excitation of the moving module, its structural parameters, etc. Their calculation methods and sample analysis are presented in [11].

In the course of experimental tests of the structure the researchers found out relative vibrations of the carriage displacement and minor angular fluctuations measurement module in relation to the raster position measurement microscope. Results of the measurement at working speed are presented in Fig. 6.

The estimate of root-mean-square deviation of those vibrations is [S.sub.x] =1.7*[10.sup.-2] [micro]m.

The effect of this factor like the effect of the laser inter-beam distance determination error will be identical for both error evaluation methods under consideration.

To compare the methods the summary error of each of them will be found out by summation of individual error components. Whereas individual random errors Si are distributed in accordance with Gaussian law they are summed up geometrically according to the formula

[FIGURE 6 OMITTED]

S = 1/N [N summation over I=1] [S.sup.2.sub.i] (17)

In the case of statistical method of estimation of angular fluctuations individual summary error components of the method will be:

--root-mean-square deviation of the motion uniqueness (Sa = 0.069"),

--root-mean-square deviation of values of the approximation function versus mean values ([S. sub.fv] = = 0.165").

In the case of active method the above components will be:

--root-mean-square deviation of the effect of relative vibrations on conversion to angular fluctuations of the carriage ([Ss.ub.v] = 0.092"),

--root-mean-square deviation of the angular fluctuation measurement error ([S.sub.m] varies from 0.082" to 0.198")

Then the root-mean-square deviation of the error of the statistical error estimation method is constant, equal to S = 0.2". The root-mean-square deviation of the error of the active error estimation method varies within the measurement limits from 0.123" to 0.22".

Absolutely similar results were obtained in the course of research into linear motion granite and steel guide modules of coordinate measuring machines.

The root-mean-square deviation of motion uniqueness of the tested precise linear motion roller bearing modules is [S.sub.a] = 0.262". Then the root-mean-square deviation of the error of the statistical error estimation method is equal to S = 0.322". The root-mean-square deviation of the error of the active error estimation method remains the same as that of aerostatic bearing modules, i.e. varies from 0.125" to 0.22".

5. Conclusions

The investigation involves two computational methods of estimation and compensation of angular errors of linear motion--statistical and active precision.

It is suggested to estimate purposefulness of applying statistical error compensation in accordance with the systematic-to-random total error component ratio. This method is purposeful in the cases where the systematic error is not lesser than the root-mean-square deviation of error values from the mean error.

The number of arguments of an approximation function corresponding to the number of factors having maximum effect on the error, can be easily determined according to empiric error distribution histograms.

It is suggested that residual error distribution in accordance with Gaussian law should be accepted as an objective criterion of approximation accuracy. This is verified by using Kolmogorov and Smirnov criterion and, if necessary, also by calculating standard asymmetry and excess ratio values.

When both the methods are used, with the increase in displacement, due to the effect of external factors, there is an increase in the random error component of measurements of angular errors of linear motion, i.e. a spread of results of those measurements. The effect of those errors is eliminated in the case of applying the statistical method of error estimation and compensation, and the effect fully shows itself when the active method of error estimation and compensation is used. If the abovementioned strict regulation of the environmental conditions is realized the effect of this error becomes essential in the case where measurement displacement exceeds 1-1.5 m.

From the point of view of precision and economy the statistical method is regarded as the optimum computation method of estimation and compensation of angular errors of linear motion, when the movement is stable and is measured and compensated on large runs. E.g., this is characteristic of aerostatic bearing guides, where runs are larger than 1-1.5 m.

When for movement is a greater characteristic instability of the trajectory, e.g. in the case of roller bearing guides and, especially, when runs are small, the higher level of accuracy of estimation and compensation of angular errors of linear motion by the computational method is achieved by using the active method.

Received August 17, 2009

Accepted October 05, 2009

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A. Barakauskas *, A. Kasparaitis **, S. Kausinis, R. Lazdinas ****

* JSC "Precizika Metrology", Zirmunu 139, 09120 Vilnius, Lithuania, E-mail: a.barakauskas@precizika.lt

** Vilnius Gediminas Technical University, Basanaviciaus 28, 03224 Vilnius, Lithuania, E-mail: a.kasparaitis@precizika.lt

*** Kaunas University of Technology, A. Mickeviciaus 37-118, 44244 Kaunas, Lithuania, E-mail: saulius.kausinis@ktu.lt

**** Vilnius Gediminas Technical University, Basanaviciaus 28, 03224 Vilnius, Lithuania, E-mail: r.lazdinas@gmail.com