Issue #1/2022

ANALYSIS OF THE ANGULAR ERRORS COMPONENTS WHEN PROCESSING DATA BY THE READING HEAD

**B.G.Turukhano, N.Turukhano, V.V.Korotaev, A.S.Vasiliev, S.N.Khanov, O.G.Ermolenko, D.L.Konstantinov**ANALYSIS OF THE ANGULAR ERRORS COMPONENTS WHEN PROCESSING DATA BY THE READING HEAD

10.22184/1993-8578.2022.15.1.66.78

INTRODUCTION

Analysis of reading head error components

This work studied the scanning errors of the reading head when scanning code and readout limbs with electrooptic angle encoders and an angular measuring machine with increased accuracy using a reading head with enhanced accuracy.

The electron-beam lithography processes for obtaning the code and readout limbs have no resolution limitations and appear to be most promising [1].

The advantage of this method of fabricating the code and reading limbs is not only in their high resolution but also in the ability to automate the reading process using a personal computer, which significantly increases productivity when generating and analysing the recorded information.

The other method [2] also produces coding and reading limbs with high nanoscale resolution.

As nowadays the resolution of code and readout limbs can reach nanoscale levels in order to study the topology elements of angle measuring structures, it is necessary to pay special attention to the construction of electrooptic information reading system itself using angle sensors of electrooptic converters and angle measuring machine with reading head.

With the given reading head parameters, the limiting accuracy will be no more than 0.96''.

Selection and calculation of the main parameters of the angle measuring machine reading head elements

For high accuracy determination of the angular parameters and other characteristics of the working samples and photomasks of optical limbs, circular scales, rasters and multi-digit code disks, it is necessary to make a numerical estimation of the basic components of the reading head error, which is a part of the angle-measuring machine. To determine the error value of the reading components, it is necessary to determine and specify the basic parameters of the reading head assemblies.

The main and critical factor that defines the reading head accuracy is the linear magnification of the entire electrooptic part. For most tasks the optical disc error should not exceed ±0.065 µm for angle measuring structures with a minimum diameter of 90 mm, and the edge roughness size in the range of 0.1 to 0.5 µm. The required error can be achieved by using digital image processing methods of the stroke defect, which occupies at least three photosensitive elements on the surface of the matrix radiation receiver. By specifying the element size of the radiation matrix receiver and the defect size, we determine the required linear magnification β of the reading head by the formula:

, (1)

where Px - horizontal dimension of the photosensitive element of the radiation receiver, µm; Ws - minimum dimension of the stroke defect, µm.

The analysis has shown that the optimum solution is to use a Sony ICX834 CCD-based receiver with a 3.1 µm sensing element. An optical system with a linear magnification of 93x has to be implemented in order to provide the required accuracy.

At a given linear magnification and spatial resolution of 4330 × 2854 pixels of the radiation receiver, an observable linear field of 144×93 µm can be obtained in a single frame.

As far as the required errors are of sub-micron magnitude, it should be considered that the optical system has a limit of resolution that will distinguish individual defects of a given size. The resolution limit Δ, according to the Abbe criterion, is defined by the formula:

, (2)

where λ is the operating wavelength; АОб is the numerical aperture of the lens.

For a large value of linear magnification, the numerical aperture equals, approximately, a unit. The operating wavelength should be as short as possible and must match the sensitivity of the radiation receiver. In the case of a silicon radiation receiver the maximum sensitivity and quantum efficiency is at a wavelength of 450 nm.

Taking these values into account, the limiting resolution will be 0.22 µm. Therefore, the minimum possible size resolvable by the reading head optical system will be a stoke defect of 0.2 µm. According to expression (1), for the selected receiver, the useful linear magnification will be β = 50× and the linear field size 290 × 190 µm. Another important component of reading head uncertainty is the dynamic uncertainty arising due to rotation of the angle structure during measurement. The dynamic error will be determined by the amount of blurring of the optical image which, in turn, depends on the speed and exposure time of the photosensitive elements.

To calculate the rotation speed of the angle measuring structure in relation to the reading head, we take its maximum diameter to be 165 mm. The linear path length of the outer radius of the angle measuring structure will be SУИМ = 520 mm. In view of linear magnification of optical system 50х, observed linear field size H · V = 290 × 190 µm, and also maximum possible frame rate fps = 7 fps, we conclude that for the given linear field it is necessary to adhere to the ratio = 1800 frames.

The time required to capture this number of frames is = 260 s. In this case the average speed required to read the track at the maximum possible radius of the angle-measuring structure will be v = 2 mm/s and the angular rotation speed is ω = 0.024 rad/s = 1.4 deg/s.

Errors caused by inclining of the photoreceiver unit reading head

One of the sources of the reading head error is the systematic error due to inclination of the matrix photodetector installation relative to the angle-measuring structure being monitored.

In this case it is necessary to take into account the error in the receiver positioning relative to the OX and OY axes. Rotation of the matrix around the OZ axis (parallel to the plane of the observed structure) will not produce an error. Coordinates of the strokes arising at inclination of the photodetector unit can be calculated using the rotation matrix:

, (3)

where (xц, yц) – are the coordinates of the stroke in the absence of tilting of the photodetector unit; MR – is the rotation matrix which we calculate using the Tait-Brian angles. In our case the rotation around the OZ axis produces no effect on the accuracy of the stroke coordinates. Then the resulting rotation matrix after multiplication will look like:

. (4)

According to (4), expression (3) can be written as:

. (5)

The error caused by inclination of the photodetector head will be determined from the expression:

, (6)

where Px is the horizontal pixel size of the matrix photodetector, µm; β is the linear magnification of the reading head optical system.

By setting the photodetector angle limits α = 1 and β = 50×, we obtain δXRПОИ = 0.06'' for a maximum working area diameter of 90 mm.

Random error component caused by photodetector reading head noise

Given the requirements for high accuracy in measuring the stroke-face defect and the resulting high linear magnification of the optical system, one of the factors influencing the total error of the reading head will be the photodetector noise. Occurrence of noise in the image will result in a random redistribution of the stroke image energy and will create an error as a result of the stroke boundary detection.

The photodetector noise of the reading head has been calculated using the photon transfer method. This model is linear, so all the noise components add up. The total noise in the CCD sensor is the sum of the following components:

, (7)

where К is the camera magnification; – dark noise variation, – quantization noise variation; µy is the number of photons accumulated during the exposure time; µy.dark is the number of dark photons.

The value (µy – µy.dark) can be found from the expression for the signal-to-noise ratio SNR:

. (8)

The variation in quantization noise is calculated as follows:

, (9)

where µp.sat is the saturation capacity of a pixel, i.e. the maximum number of photoelectrons it can hold while maintaining a non-linear response, k is the ADC resolution. Taking into account (9), expression (7) takes a form:

, (10)

Hence, the δy value can be found.

For the given data δy = 14.78 electrons and noise variation = 218.58 electrons, corresponding to = 370.47 photons hitting the photodetector. The voltage corresponding to the radiation receiver noise is mV.

The relationship between the calculated value of the noise voltage and the deviation of the signal taken from the photodetector, in ADC units is expressed by the following relation:

, (11)

where Umax is the maximum voltage value at the photodetector equal to 3.3 V. Then δUADC = 140, which corresponds to a similar deviation in pixel brightness values in the image, makes 0.0085 of its full digit capacity.

For the developed processing algorithm, the error component caused by photodetector reading head noise will be δXПОИ = 0,012''.

The error component of the stroke image processing algorithm

The processing algorithm of the image obtained by the reading head of the system is aimed at detection of defects and deformations in elements of the limb topology, which include roughness of the stroke edge, chipping and nicks on the stroke, micro-scratches and dirt on the limb surface.

Based on the given conditions, the following operations must be included in the algorithm so as to determine the defect size: finding each individual stroke, calculating the nominal straight line relative to which the defect of the stroke will be calculated and calculating the defect for the detected stroke.

Stroke detection is complicated by the fact that at high magnification of the reading head optical system some strokes, depending on the working tracks they are located on, do not fall completely into the system field of view, or the strokes of the track do not fit within the height of the field of view. In order to be able to correctly identify a defective stroke and its position on the limb, it is necessary to observe the condition of at least two complete images of the strokes on the processed image. This condition can be met by using the image stitching method.

Stitching a large image sequence is a difficult and unreasonable operation since, while maintaining image resolution, the output merged image will occupy a huge amount of memory which would be difficult both to process by and to store in a storage device. It is therefore recommended that the stitching be done for minimal areas of the limb, sufficient to determine the spatial position of the individual stroke and its defect.

It is useful to further detect the stroke using the one-dimensional S(x) representation of the recorded image f(x, y). Figure 2 illustrates the result of representing the original image of a stroke in the form of a one-dimensional signal. The regions of interest containing the strokes must then be found in the one-dimensional representation of the image.

In the detected areas the stroke parameters are defined, including its centre position defect. The detection process does not determine accuracy of the calculated flaw or stroke position and has no effect on the resulting error. A high level of brightness and uniformity of illumination of the limb area within a given field of view of the reading head is required to increase probability of correct stroke detection.

To further analyse parameters of the limb element topology, it is necessary to calculate the nominal lines with respect to which the stroke faults will be calculated. For this purpose it is necessary and sufficient to determine two points through which only one line can be drawn. Finding points for drawing the nominal line of the stroke boundary is done on the basis of determining the stroke start points at the top and bottom of the frame. This is done by representing the image of the upper part of the frame based on the summation separately of the lines from 1 to N/2 and from N/2 to N. As a result, two energy distributions of the stroke images in the upper and lower part of the frame are formed (Fig.3).

After finding the nominal straight stroke edge, the positions of the start and end points of each stroke at the top and bottom of the frame are determined (Fig.4). For this purpose it is enough to find the derivative and determine the centre in the area of maximum and minimum peaks of the signal:

, (12)

where , is the derivative for the signal functions , at the top and bottom of the frame, respectively.

Finding the centre of the derivative in a given signal area is expediently implemented on the basis of a weighted summation algorithm:

. (13)

In order to eliminate the influence of the background component, it is useful to use a weighted summation algorithm in window [Ast, Asp]:

, (14)

where .

A nominal line for the n–th stroke is plotted through the resulting pairs of coordinates of the centres of the derived signals.

With respect to the found nominal lines, the magnitude of the defect and the boundary deformation of each line is calculated, which is defined as the standard deviation of the absolute difference between the coordinates of the line boundaries with respect to the nominal line in each N-th line of the image:

. (15)

For the developed algorithm, the error value of the algorithm will be determined with the error of determining the central point of the derivative of the n-th signal by the weighted summation method and will be equal to:

, (16)

where Px is a pixel size of matrix radiation receiver measured horizontally, μm; β is linear magnification of optical system of the reading head; 0.01 is an error of the weighted summation method. Taking into account values of linear magnification and size of photosensitive element δXdip = 0.0026''.

Component of the blurring error due to motion of the limb during circular reading

Movements of the controlled angular measuring structure with angular velocity ω cause blurring of the digital image of the observed strokes. According to geometric optics, in the first approximation, we can write the value of the increase in the size of the stroke image as a linear blurring value Δl, which, at the exposure time of the photodetector tэксп for the observed stroke as a vertical line, is equal to:

, (17)

where lh is a stroke size measured horizontally.

If the blurring effect is not compensated for, then, given that the energy centre algorithm is used to determine the stroke boundaries, there is a systematic error δXсмаз which is equal to:

. (18)

It follows from the above expression that δXсмаз depends on the exposure time and the stroke speed (the speed of the angular measuring structure).

The error of the stroke boundary coordinates (18) can also be expressed by variation of the exposure time δtэксп:

. (19)

Variation of the exposure time interval δtэксп is often caused by an operational factor affecting the electronic components of the system that generates temporal electrical pulses of the exposure time reference. Disturbances in timing characteristics are primarily caused by instabilities due to changes in temperature, supply voltage, and other factors. For the appropriate electronic components, the manufacturers provide a calibration chart to correct for the effect of temperature on the output signal. The value of the variation of the exposure time of the radiation receiver will then be expressed as:

, (20)

where ku, kt are correction factors for changes in output characteristics of operational amplifier and digital-to-analogue converter when the temperature changes by ΔT.

Given (20), expression (19) can be written as:

. (21)

From this expression, it can be seen that the higher the travel speed, the greater the effect of the exposure instability on the stroke boundary coordinate error.

Let us estimate the limiting value of the error due to image blurring and exposure time at a given angular velocity of limb rotation. At minimum exposure time of 0.04 ms the magnitude of error component of the image blurring due to limb motion at circular readout will make δXэксп = 0,012'', at maximum exposure time of 84.8 ms the error will estimate 8.3 µm. Thus, to minimize the error from image blurring, it is necessary to perform image registration at the minimum exposure.

Effects of the control object deformation (wedge-shaped and non-planar limb surfaces) and inaccuracy of its mounting

An important source of error in checking elements of limb topology is the shape of the limb, namely its deviations such as wedge-shaped and non-planar surfaces. Inaccuracies in the positioning of the limb, namely eccentricity and non-perpendicularity of the axis of rotation produce still another source of the error.

The wedge shape of the limb substrate, similar to a classical prism, causes the beam to change orientation at the exit from the substrate. As the limb rotates, the beam will describe a tapered surface synchronously with the rotation of the substrate. The measurement error caused by the wedge shape of the limb is found from the relation [17]:

, (22)

where θ is the wedge-shape tolerance, z is the working distance of the lens.

When given, z = 0,25° mm, β = 50×, and the value of θ is limited by the requirements of this assignment to 120 angular seconds. Then 0.06''.

Similarly, you can find the error due to non-perpendicularity of the limb surface from the ratio:

, (23)

where ρ is the perpendicularity tolerance, equal to about 3 µm for any section of the limb with a diameter d = 100 mm. With these data δX = 0.86''.

Thus, the systematic component of the measurement error due to the deformation of the limb equals:

δXs = δX< + δX = 0,92''.

As with eccentricity and optical system aberrations, the error caused by the angular measuring spindle run-out is systematic and varies periodically. This component of the error can be compensated for at the scale alignment stage and is eliminated by the introduction of suitable correction factors which are derived from the linear and angular coordinate processing units on the current spatial position of the limb relative to the reading head.

Total error of the main error components of the reading head

Using these error components, it is possible to calculate the total error of the reading head using the known procedure. The expression for the total error of the machine when controlling the magnitude of the defect of the elements of the topology of the angle-measuring structure is as follows:

. (24)

Calculation and study of the enhanced accuracy error due to the above components showed that the total error does not exceed 0.038 µm.

CONCLUSIONS

In order to meet the required enhanced accuracy requirements using a reading head for analysis of the images of the topology elements of angle-measuring structures, an optoelectronic system based on a matrix CCD photodetector with a photosensitive element size of at least 3.1 µm and a resolution of 4330 × 2854 pixels and an ADC resolution of at least 12 digits must be designed. The system should operate in reflected light at an operating wavelength of 450 nm. With a linear magnification value of β = 50× and a resolution limit of 0.2 µm, smaller defects will not be distinguishable. The size of the linear field of view at a given magnification will be 290 × 190 µm. The average speed required to read the track at the maximum possible radius of the angle measuring structure will be v = 2 mm/s and the angular velocity of rotation is ω = 1.4 deg/s. With the given reading head parameters, the margin of error will be no more than 0,96'' (0.038 µm).

PEER REVIEW INFO

Editorial board thanks the anonymous reviewer(s) for their contribution to the peer review of this work. It is also grateful for their consent to publish papers on the journal’s website and SELibrary eLIBRARY.RU.

Declaration of Competing Interest. The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Analysis of reading head error components

This work studied the scanning errors of the reading head when scanning code and readout limbs with electrooptic angle encoders and an angular measuring machine with increased accuracy using a reading head with enhanced accuracy.

The electron-beam lithography processes for obtaning the code and readout limbs have no resolution limitations and appear to be most promising [1].

The advantage of this method of fabricating the code and reading limbs is not only in their high resolution but also in the ability to automate the reading process using a personal computer, which significantly increases productivity when generating and analysing the recorded information.

The other method [2] also produces coding and reading limbs with high nanoscale resolution.

As nowadays the resolution of code and readout limbs can reach nanoscale levels in order to study the topology elements of angle measuring structures, it is necessary to pay special attention to the construction of electrooptic information reading system itself using angle sensors of electrooptic converters and angle measuring machine with reading head.

With the given reading head parameters, the limiting accuracy will be no more than 0.96''.

Selection and calculation of the main parameters of the angle measuring machine reading head elements

For high accuracy determination of the angular parameters and other characteristics of the working samples and photomasks of optical limbs, circular scales, rasters and multi-digit code disks, it is necessary to make a numerical estimation of the basic components of the reading head error, which is a part of the angle-measuring machine. To determine the error value of the reading components, it is necessary to determine and specify the basic parameters of the reading head assemblies.

The main and critical factor that defines the reading head accuracy is the linear magnification of the entire electrooptic part. For most tasks the optical disc error should not exceed ±0.065 µm for angle measuring structures with a minimum diameter of 90 mm, and the edge roughness size in the range of 0.1 to 0.5 µm. The required error can be achieved by using digital image processing methods of the stroke defect, which occupies at least three photosensitive elements on the surface of the matrix radiation receiver. By specifying the element size of the radiation matrix receiver and the defect size, we determine the required linear magnification β of the reading head by the formula:

, (1)

where Px - horizontal dimension of the photosensitive element of the radiation receiver, µm; Ws - minimum dimension of the stroke defect, µm.

The analysis has shown that the optimum solution is to use a Sony ICX834 CCD-based receiver with a 3.1 µm sensing element. An optical system with a linear magnification of 93x has to be implemented in order to provide the required accuracy.

At a given linear magnification and spatial resolution of 4330 × 2854 pixels of the radiation receiver, an observable linear field of 144×93 µm can be obtained in a single frame.

As far as the required errors are of sub-micron magnitude, it should be considered that the optical system has a limit of resolution that will distinguish individual defects of a given size. The resolution limit Δ, according to the Abbe criterion, is defined by the formula:

, (2)

where λ is the operating wavelength; АОб is the numerical aperture of the lens.

For a large value of linear magnification, the numerical aperture equals, approximately, a unit. The operating wavelength should be as short as possible and must match the sensitivity of the radiation receiver. In the case of a silicon radiation receiver the maximum sensitivity and quantum efficiency is at a wavelength of 450 nm.

Taking these values into account, the limiting resolution will be 0.22 µm. Therefore, the minimum possible size resolvable by the reading head optical system will be a stoke defect of 0.2 µm. According to expression (1), for the selected receiver, the useful linear magnification will be β = 50× and the linear field size 290 × 190 µm. Another important component of reading head uncertainty is the dynamic uncertainty arising due to rotation of the angle structure during measurement. The dynamic error will be determined by the amount of blurring of the optical image which, in turn, depends on the speed and exposure time of the photosensitive elements.

To calculate the rotation speed of the angle measuring structure in relation to the reading head, we take its maximum diameter to be 165 mm. The linear path length of the outer radius of the angle measuring structure will be SУИМ = 520 mm. In view of linear magnification of optical system 50х, observed linear field size H · V = 290 × 190 µm, and also maximum possible frame rate fps = 7 fps, we conclude that for the given linear field it is necessary to adhere to the ratio = 1800 frames.

The time required to capture this number of frames is = 260 s. In this case the average speed required to read the track at the maximum possible radius of the angle-measuring structure will be v = 2 mm/s and the angular rotation speed is ω = 0.024 rad/s = 1.4 deg/s.

Errors caused by inclining of the photoreceiver unit reading head

One of the sources of the reading head error is the systematic error due to inclination of the matrix photodetector installation relative to the angle-measuring structure being monitored.

In this case it is necessary to take into account the error in the receiver positioning relative to the OX and OY axes. Rotation of the matrix around the OZ axis (parallel to the plane of the observed structure) will not produce an error. Coordinates of the strokes arising at inclination of the photodetector unit can be calculated using the rotation matrix:

, (3)

where (xц, yц) – are the coordinates of the stroke in the absence of tilting of the photodetector unit; MR – is the rotation matrix which we calculate using the Tait-Brian angles. In our case the rotation around the OZ axis produces no effect on the accuracy of the stroke coordinates. Then the resulting rotation matrix after multiplication will look like:

. (4)

According to (4), expression (3) can be written as:

. (5)

The error caused by inclination of the photodetector head will be determined from the expression:

, (6)

where Px is the horizontal pixel size of the matrix photodetector, µm; β is the linear magnification of the reading head optical system.

By setting the photodetector angle limits α = 1 and β = 50×, we obtain δXRПОИ = 0.06'' for a maximum working area diameter of 90 mm.

Random error component caused by photodetector reading head noise

Given the requirements for high accuracy in measuring the stroke-face defect and the resulting high linear magnification of the optical system, one of the factors influencing the total error of the reading head will be the photodetector noise. Occurrence of noise in the image will result in a random redistribution of the stroke image energy and will create an error as a result of the stroke boundary detection.

The photodetector noise of the reading head has been calculated using the photon transfer method. This model is linear, so all the noise components add up. The total noise in the CCD sensor is the sum of the following components:

, (7)

where К is the camera magnification; – dark noise variation, – quantization noise variation; µy is the number of photons accumulated during the exposure time; µy.dark is the number of dark photons.

The value (µy – µy.dark) can be found from the expression for the signal-to-noise ratio SNR:

. (8)

The variation in quantization noise is calculated as follows:

, (9)

where µp.sat is the saturation capacity of a pixel, i.e. the maximum number of photoelectrons it can hold while maintaining a non-linear response, k is the ADC resolution. Taking into account (9), expression (7) takes a form:

, (10)

Hence, the δy value can be found.

For the given data δy = 14.78 electrons and noise variation = 218.58 electrons, corresponding to = 370.47 photons hitting the photodetector. The voltage corresponding to the radiation receiver noise is mV.

The relationship between the calculated value of the noise voltage and the deviation of the signal taken from the photodetector, in ADC units is expressed by the following relation:

, (11)

where Umax is the maximum voltage value at the photodetector equal to 3.3 V. Then δUADC = 140, which corresponds to a similar deviation in pixel brightness values in the image, makes 0.0085 of its full digit capacity.

For the developed processing algorithm, the error component caused by photodetector reading head noise will be δXПОИ = 0,012''.

The error component of the stroke image processing algorithm

The processing algorithm of the image obtained by the reading head of the system is aimed at detection of defects and deformations in elements of the limb topology, which include roughness of the stroke edge, chipping and nicks on the stroke, micro-scratches and dirt on the limb surface.

Based on the given conditions, the following operations must be included in the algorithm so as to determine the defect size: finding each individual stroke, calculating the nominal straight line relative to which the defect of the stroke will be calculated and calculating the defect for the detected stroke.

Stroke detection is complicated by the fact that at high magnification of the reading head optical system some strokes, depending on the working tracks they are located on, do not fall completely into the system field of view, or the strokes of the track do not fit within the height of the field of view. In order to be able to correctly identify a defective stroke and its position on the limb, it is necessary to observe the condition of at least two complete images of the strokes on the processed image. This condition can be met by using the image stitching method.

Stitching a large image sequence is a difficult and unreasonable operation since, while maintaining image resolution, the output merged image will occupy a huge amount of memory which would be difficult both to process by and to store in a storage device. It is therefore recommended that the stitching be done for minimal areas of the limb, sufficient to determine the spatial position of the individual stroke and its defect.

It is useful to further detect the stroke using the one-dimensional S(x) representation of the recorded image f(x, y). Figure 2 illustrates the result of representing the original image of a stroke in the form of a one-dimensional signal. The regions of interest containing the strokes must then be found in the one-dimensional representation of the image.

In the detected areas the stroke parameters are defined, including its centre position defect. The detection process does not determine accuracy of the calculated flaw or stroke position and has no effect on the resulting error. A high level of brightness and uniformity of illumination of the limb area within a given field of view of the reading head is required to increase probability of correct stroke detection.

To further analyse parameters of the limb element topology, it is necessary to calculate the nominal lines with respect to which the stroke faults will be calculated. For this purpose it is necessary and sufficient to determine two points through which only one line can be drawn. Finding points for drawing the nominal line of the stroke boundary is done on the basis of determining the stroke start points at the top and bottom of the frame. This is done by representing the image of the upper part of the frame based on the summation separately of the lines from 1 to N/2 and from N/2 to N. As a result, two energy distributions of the stroke images in the upper and lower part of the frame are formed (Fig.3).

After finding the nominal straight stroke edge, the positions of the start and end points of each stroke at the top and bottom of the frame are determined (Fig.4). For this purpose it is enough to find the derivative and determine the centre in the area of maximum and minimum peaks of the signal:

, (12)

where , is the derivative for the signal functions , at the top and bottom of the frame, respectively.

Finding the centre of the derivative in a given signal area is expediently implemented on the basis of a weighted summation algorithm:

. (13)

In order to eliminate the influence of the background component, it is useful to use a weighted summation algorithm in window [Ast, Asp]:

, (14)

where .

A nominal line for the n–th stroke is plotted through the resulting pairs of coordinates of the centres of the derived signals.

With respect to the found nominal lines, the magnitude of the defect and the boundary deformation of each line is calculated, which is defined as the standard deviation of the absolute difference between the coordinates of the line boundaries with respect to the nominal line in each N-th line of the image:

. (15)

For the developed algorithm, the error value of the algorithm will be determined with the error of determining the central point of the derivative of the n-th signal by the weighted summation method and will be equal to:

, (16)

where Px is a pixel size of matrix radiation receiver measured horizontally, μm; β is linear magnification of optical system of the reading head; 0.01 is an error of the weighted summation method. Taking into account values of linear magnification and size of photosensitive element δXdip = 0.0026''.

Component of the blurring error due to motion of the limb during circular reading

Movements of the controlled angular measuring structure with angular velocity ω cause blurring of the digital image of the observed strokes. According to geometric optics, in the first approximation, we can write the value of the increase in the size of the stroke image as a linear blurring value Δl, which, at the exposure time of the photodetector tэксп for the observed stroke as a vertical line, is equal to:

, (17)

where lh is a stroke size measured horizontally.

If the blurring effect is not compensated for, then, given that the energy centre algorithm is used to determine the stroke boundaries, there is a systematic error δXсмаз which is equal to:

. (18)

It follows from the above expression that δXсмаз depends on the exposure time and the stroke speed (the speed of the angular measuring structure).

The error of the stroke boundary coordinates (18) can also be expressed by variation of the exposure time δtэксп:

. (19)

Variation of the exposure time interval δtэксп is often caused by an operational factor affecting the electronic components of the system that generates temporal electrical pulses of the exposure time reference. Disturbances in timing characteristics are primarily caused by instabilities due to changes in temperature, supply voltage, and other factors. For the appropriate electronic components, the manufacturers provide a calibration chart to correct for the effect of temperature on the output signal. The value of the variation of the exposure time of the radiation receiver will then be expressed as:

, (20)

where ku, kt are correction factors for changes in output characteristics of operational amplifier and digital-to-analogue converter when the temperature changes by ΔT.

Given (20), expression (19) can be written as:

. (21)

From this expression, it can be seen that the higher the travel speed, the greater the effect of the exposure instability on the stroke boundary coordinate error.

Let us estimate the limiting value of the error due to image blurring and exposure time at a given angular velocity of limb rotation. At minimum exposure time of 0.04 ms the magnitude of error component of the image blurring due to limb motion at circular readout will make δXэксп = 0,012'', at maximum exposure time of 84.8 ms the error will estimate 8.3 µm. Thus, to minimize the error from image blurring, it is necessary to perform image registration at the minimum exposure.

Effects of the control object deformation (wedge-shaped and non-planar limb surfaces) and inaccuracy of its mounting

An important source of error in checking elements of limb topology is the shape of the limb, namely its deviations such as wedge-shaped and non-planar surfaces. Inaccuracies in the positioning of the limb, namely eccentricity and non-perpendicularity of the axis of rotation produce still another source of the error.

The wedge shape of the limb substrate, similar to a classical prism, causes the beam to change orientation at the exit from the substrate. As the limb rotates, the beam will describe a tapered surface synchronously with the rotation of the substrate. The measurement error caused by the wedge shape of the limb is found from the relation [17]:

, (22)

where θ is the wedge-shape tolerance, z is the working distance of the lens.

When given, z = 0,25° mm, β = 50×, and the value of θ is limited by the requirements of this assignment to 120 angular seconds. Then 0.06''.

Similarly, you can find the error due to non-perpendicularity of the limb surface from the ratio:

, (23)

where ρ is the perpendicularity tolerance, equal to about 3 µm for any section of the limb with a diameter d = 100 mm. With these data δX = 0.86''.

Thus, the systematic component of the measurement error due to the deformation of the limb equals:

δXs = δX< + δX = 0,92''.

As with eccentricity and optical system aberrations, the error caused by the angular measuring spindle run-out is systematic and varies periodically. This component of the error can be compensated for at the scale alignment stage and is eliminated by the introduction of suitable correction factors which are derived from the linear and angular coordinate processing units on the current spatial position of the limb relative to the reading head.

Total error of the main error components of the reading head

Using these error components, it is possible to calculate the total error of the reading head using the known procedure. The expression for the total error of the machine when controlling the magnitude of the defect of the elements of the topology of the angle-measuring structure is as follows:

. (24)

Calculation and study of the enhanced accuracy error due to the above components showed that the total error does not exceed 0.038 µm.

CONCLUSIONS

In order to meet the required enhanced accuracy requirements using a reading head for analysis of the images of the topology elements of angle-measuring structures, an optoelectronic system based on a matrix CCD photodetector with a photosensitive element size of at least 3.1 µm and a resolution of 4330 × 2854 pixels and an ADC resolution of at least 12 digits must be designed. The system should operate in reflected light at an operating wavelength of 450 nm. With a linear magnification value of β = 50× and a resolution limit of 0.2 µm, smaller defects will not be distinguishable. The size of the linear field of view at a given magnification will be 290 × 190 µm. The average speed required to read the track at the maximum possible radius of the angle measuring structure will be v = 2 mm/s and the angular velocity of rotation is ω = 1.4 deg/s. With the given reading head parameters, the margin of error will be no more than 0,96'' (0.038 µm).

PEER REVIEW INFO

Editorial board thanks the anonymous reviewer(s) for their contribution to the peer review of this work. It is also grateful for their consent to publish papers on the journal’s website and SELibrary eLIBRARY.RU.

Declaration of Competing Interest. The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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