In Situ Measurement of Spindle Radial Error for Ultra-Precision Machining Based on Three-Point Method

The radial error is an important parameter to evaluate the performance of ultra-precision spindles. The three-point method has not yet been well applied in nanometer-scale measurement due to its disadvantages of harmonic suppression and the complicated error separation process. In order to verify that the three-point method can realize the nanometer-scale measurement of the radial error in the machining environment, an in situ measurement and evaluation system is established. Experiments are performed using the system, and a comparative experiment is conducted to verify the accuracy of the system. The average value and standard deviation of the measurement results are 23.096 nm and 0.556 nm, respectively. The in situ measurement result was in good agreement with the Donaldson reversal method using a commercially available spindle analyzer.


Introduction
Ultra-precision spindles have been widely used in advanced manufacturing and ultra-precision measurement, supporting development in semiconductor electronics, space exploration and other fields [1,2]. Radial error refers to the deviation of the instantaneous axis of the rotating spindle relative to the average axis in the radial direction. Since the spindle drives the tool to rotate, the radial error will directly affect the depth of cut, and then affect the face accuracy of the machined workpiece [3][4][5]. Therefore, the measurement of the radial error is a key link to design a spindle and evaluate the machining performance of the ultra-precision spindles [6]. Focused on the method of ultra-precision spindle radial error measurement, scholars have carried out a significant amount of research. The proposal of error separation technology is a major breakthrough in the field of spindle radial error measurement [7], promoting the development of spindle metrology. Several error separation techniques were then developed, including the multi-point method [7], multi-step method [8,9] and the Donaldson reversal method [10,11]. The multi-step method is rarely used in the field of ultra-precision measurement because of its complex operation steps.
After the three methods above were put forward, most of the scholars' research was carried out on the basis of these three measurement methods. Compared with the other two methods, the Donaldson reversal method is based on a simpler principle. Cui et al. [12] constructed a nanometer system for measuring the radial error of aerostatic ultra-precision based on the Donaldson reversal method. The effect of the cogging torque of the motor, the angle deviation, artifact eccentricity and spindle axial motion on the measuring accuracy of the spindle were studied. The accuracy of the measurement system and the validity of the Donaldson reversal method were confirmed. Chen et al. [13] measured the uncertainty of the rotary accuracy of an ultra-precision aerostatic spindle based on the Donaldson

Mathematical Model of the Circular Cross-Sectional Profile
The circular cross-sectional profile of shaft parts has a periodic character and can be decomposed into sinusoidal waves of different orders. These sine waves are superimposed in a defined pattern to obtain the cross-sectional profile.
The circular cross-sectional profile of shaft parts is shown in Figure 1, with the solid line indicating the actual profile and the dashed line indicating the ideal profile. It can be found that the shape of the cross-sectional profile of the shaft parts in rotation will exhibit deterministic periodic signal characteristics, which can be expanded by Fourier series as r(θ p ) = r 0 + ∞ ∑ k=1 (a k coskθ p + b k sinkθ p ) (1) where θ p is the polar angle; r 0 is the average radius of the circular cross-sectional profile of the workpiece; k is the number of harmonics; and a k and b k are Fourier coefficients of the roundness error profile. Since the sampling points are discrete in the actual sampling process, the Fourier coefficients can be calculated by Euler's criterion as 1

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( ) cos where N is the number of sampling points in each circle and n is the n th − of the total number of sampling points in each circle. In the actual sampling, a sensor probe with suitable accuracy is selected for data acquisition. The collected data include synchronous error and asynchronous error, and these errors can be separated. The synchronous error can be obtained by averaging the raw data of multiple turns, and the asynchronous error data is obtained by subtracting the synchronous error from the raw data, as shown in Figure 2. After Fourier transform, the synchronous error data are distributed in integer multiples of the spindle rotation frequency and its neighborhood, and the asynchronous error data are distributed in non-integer multiples of the rotation frequency interval, as shown in Figure 3. The lines marked with asterisks represent the frequency components of Since the sampling points are discrete in the actual sampling process, the Fourier coefficients can be calculated by Euler's criterion as where N is the number of sampling points in each circle and n is the n − th of the total number of sampling points in each circle. In the actual sampling, a sensor probe with suitable accuracy is selected for data acquisition. The collected data include synchronous error and asynchronous error, and these errors can be separated. The synchronous error can be obtained by averaging the raw data of multiple turns, and the asynchronous error data is obtained by subtracting the synchronous error from the raw data, as shown in Figure 2. Since the sampling points are discrete in the actual sampling process, the Fourier coefficients can be calculated by Euler's criterion as where N is the number of sampling points in each circle and n is the n th − of the total number of sampling points in each circle. In the actual sampling, a sensor probe with suitable accuracy is selected for data acquisition. The collected data include synchronous error and asynchronous error, and these errors can be separated. The synchronous error can be obtained by averaging the raw data of multiple turns, and the asynchronous error data is obtained by subtracting the synchronous error from the raw data, as shown in Figure 2. After Fourier transform, the synchronous error data are distributed in integer multiples of the spindle rotation frequency and its neighborhood, and the asynchronous error data are distributed in non-integer multiples of the rotation frequency interval, as shown in Figure 3. The lines marked with asterisks represent the frequency components of After Fourier transform, the synchronous error data are distributed in integer multiples of the spindle rotation frequency and its neighborhood, and the asynchronous error data are distributed in non-integer multiples of the rotation frequency interval, as shown in Figure 3. The lines marked with asterisks represent the frequency components of synchronous errors, and the lines with circular marks represent the frequency components of asynchronous 4 of 16 errors. The synchronization error data, combined with the data of other probes or other measurement steps, can be further separated into the radial error of the spindle and the roundness error of the standard workpiece. The separation method of the synchronization error will be further introduced below.
Micromachines 2023, 13, x FOR PEER REVIEW 4 of 17 synchronous errors, and the lines with circular marks represent the frequency components of asynchronous errors. The synchronization error data, combined with the data of other probes or other measurement steps, can be further separated into the radial error of the spindle and the roundness error of the standard workpiece. The separation method of the synchronization error will be further introduced below.

Error Separation Techniques
Roundness error separation methods generally include the multi-point method, multi-step method and Donaldson reversal method. Compared with the multi-step method, the three-point method and the reversal method have the advantages of fewer clamping times, thus introducing less error. Therefore, the measurement of radial error of the spindle generally uses the three-point method and the reversal method.

Three-Point Method
As shown in Figure 4, three sensor probes are placed at a certain angle to collect data synchronously. After removing the asynchronous error and the installation eccentricity of the standard workpiece, the data collected by the three sensors can be expressed as where  is the spindle rotation angle; α is the angle between sensor A and sensor B;  is the angle between sensor A and sensor C;

Error Separation Techniques
Roundness error separation methods generally include the multi-point method, multistep method and Donaldson reversal method. Compared with the multi-step method, the three-point method and the reversal method have the advantages of fewer clamping times, thus introducing less error. Therefore, the measurement of radial error of the spindle generally uses the three-point method and the reversal method.

Three-Point Method
As shown in Figure 4, three sensor probes are placed at a certain angle to collect data synchronously. After removing the asynchronous error and the installation eccentricity of the standard workpiece, the data collected by the three sensors can be expressed as where θ is the spindle rotation angle; α is the angle between sensor A and sensor B; β is the angle between sensor A and sensor C; m A (θ), m B (θ), m C (θ) respectively represent the readings of the three sensors after removing the asynchronous error and workpiece eccentricity; p(θ) is the roundness error of the standard workpiece; and x(θ) and y(θ) respectively represent the projected components of the radial error of the spindle on the x-axis and y-axis.
The radial error of the spindle and roundness error of the standard workpiece can be obtained by analyzing the weighted sum of each sub-formula of Formula (4) using the Fourier transform method. Micromachines 2023, 13, x FOR PEER REVIEW 5 of 17 The radial error of the spindle and roundness error of the standard workpiece can be obtained by analyzing the weighted sum of each sub-formula of Formula (4) using the Fourier transform method.

Donaldson Reversal Method
The measurement using the Donaldson reversal method is mainly divided into two steps: first, fix the sensor at the 0° position, and collect the first set of data; then, the measured standard workpiece and the sensor are rotated 180° at the same time to collect the second set of data. The measurement steps of the Donaldson reversal method are shown in Figure 5. Detailed technical details will be introduced in the comparative experiment. The average value of each point measured before the reversal is 1 T , and that after the reversal is 2 T :

Donaldson Reversal Method
The measurement using the Donaldson reversal method is mainly divided into two steps: first, fix the sensor at the 0 • position, and collect the first set of data; then, the measured standard workpiece and the sensor are rotated 180 • at the same time to collect the second set of data. The measurement steps of the Donaldson reversal method are shown in Figure 5. Detailed technical details will be introduced in the comparative experiment.  The radial error of the spindle and roundness error of the standard workpiece can be obtained by analyzing the weighted sum of each sub-formula of Formula (4) using the Fourier transform method.

Donaldson Reversal Method
The measurement using the Donaldson reversal method is mainly divided into two steps: first, fix the sensor at the 0° position, and collect the first set of data; then, the measured standard workpiece and the sensor are rotated 180° at the same time to collect the second set of data. The measurement steps of the Donaldson reversal method are shown in Figure 5. Detailed technical details will be introduced in the comparative experiment. The average value of each point measured before the reversal is 1 T , and that after the reversal is 2 T :  The average value of each point measured before the reversal is T 1 , and that after the reversal is T 2 : where C ij is the sample value of the j − th point in the i − th cycle before the reversal, and D ij is that after the reversal; i is the number of revolutions sampled; j is the number of sampling points per revolution; and θ j is the rotated angle of the spindle when the j − th point is sampled. T 1 and T 2 can also be expressed as where s is the radial error of the spindle and p is the roundness error of the standard workpiece. From Equation (13), s and p can be calculated:

Overall Structure of In Situ Measurement and Evaluation System
Based on the previously theoretical derivations, an in situ measurement and evaluation system is designed, and its overall structure diagram is shown in Figure 6. The system consists of a standard workpiece, fixture, digital capacitive micro-displacement sensors, spindle, circular grating, data acquisition card, computer and servo control system of the machine tool. The reason for choosing capacitive sensors is that compared to other sensors, capacitive sensors are more suitable for working in harsh environments, such as ultra-precision machining workshops. The sensor used in this study is the capacitive micro-displacement sensor designed by the Lion Precision Company. The sensor model is CPL190/C8-2. 0, the range is 5.0 × 10 4 nm, the resolution is 1.0 nm and the linearity is 0.15% F.S.
where s is the radial error of the spindle and p is the roundness error of the standard workpiece. From Equation (13), s and p can be calculated:

Overall Structure of In Situ Measurement and Evaluation System
Based on the previously theoretical derivations, an in situ measurement and evaluation system is designed, and its overall structure diagram is shown in Figure 6. The system consists of a standard workpiece, fixture, digital capacitive micro-displacement sensors, spindle, circular grating, data acquisition card, computer and servo control system of the machine tool. The reason for choosing capacitive sensors is that compared to other sensors, capacitive sensors are more suitable for working in harsh environments, such as ultra-precision machining workshops. The sensor used in this study is the capacitive microdisplacement sensor designed by the Lion Precision Company. The sensor model is CPL190/C8-2. 0, the range is 5.0×10 4 nm, the resolution is 1.0 nm and the linearity is 0.15% F.S. The system works as follows: firstly, the standard workpiece, fixture and sensor probes are fixed to the spindle; secondly, a servo control system is used to drive the spindle to rotate at a certain speed, three sensor probes are used to collect data, and two grating data are collected simultaneously; thirdly, the above five signals are imported into the computer and processed. Then, the radial error of the spindle and the roundness error of The system works as follows: firstly, the standard workpiece, fixture and sensor probes are fixed to the spindle; secondly, a servo control system is used to drive the spindle to rotate at a certain speed, three sensor probes are used to collect data, and two grating data are collected simultaneously; thirdly, the above five signals are imported into the computer and processed. Then, the radial error of the spindle and the roundness error of the standard workpiece can be obtained. Technical details will be given later in the experiments.

Design of Fixture
In order to improve the measurement accuracy of the three-point method, an integrated fixture is designed to ensure the accuracy of the sensor installation position. The fixture can be used for both three-point method measurement and Donaldson reversal method measurement.
The problem of harmonic suppression is the main disadvantage of the three-point method measurement, but it can be reduced or even avoided by choosing an appropriate installation angle for the three sensors. In this paper, angle α and β between the three sensors are calculated by referring to the methods from previous research [19,26]. Two sets of sensor arrangements applicable to the three-point method are selected: The fixture designed in this paper is shown in Figure 7.
In order to improve the measurement accuracy of the three-point method, an integrated fixture is designed to ensure the accuracy of the sensor installation position. The fixture can be used for both three-point method measurement and Donaldson reversal method measurement.
The problem of harmonic suppression is the main disadvantage of the three-point method measurement, but it can be reduced or even avoided by choosing an appropriate installation angle for the three sensors. In this paper, angle  and  between the three sensors are calculated by referring to the methods from previous research [19,26]. Two sets of sensor arrangements applicable to the three-point method are selected: The fixture designed in this paper is shown in Figure 7.

Design of Standard Workpiece
The key to the standard workpiece design is to ensure that the roundness of the standard workpiece is in the same order of magnitude as the radial error of the spindle, so as to ensure the high precision of the error separation.
The radial error of the ultra-precision spindle used in this paper is between 10 nm and 50 nm. Therefore, the roundness of the designed standard workpiece should also be guaranteed in the same order of magnitude through ultra-precision turning. The effective working area of the standard workpiece is a cylinder with a diameter of 20 mm, as shown in Figure 8. According to the performance of the ultra-precision lathe, the roundness of the effective working area of the standard workpiece is between 15 nm and 20 nm.

Design of Standard Workpiece
The key to the standard workpiece design is to ensure that the roundness of the standard workpiece is in the same order of magnitude as the radial error of the spindle, so as to ensure the high precision of the error separation.
The radial error of the ultra-precision spindle used in this paper is between 10 nm and 50 nm. Therefore, the roundness of the designed standard workpiece should also be guaranteed in the same order of magnitude through ultra-precision turning. The effective working area of the standard workpiece is a cylinder with a diameter of 20 mm, as shown in Figure 8. According to the performance of the ultra-precision lathe, the roundness of the effective working area of the standard workpiece is between 15 nm and 20 nm.

Determination of Sampling Points
The number of sampling points for the three-point method can be determined according to the following algorithm.
The sampling frequency can be expressed by

Determination of Sampling Points
The number of sampling points for the three-point method can be determined according to the following algorithm.
The sampling frequency can be expressed by where N is the number of sampling points per revolution and ω 0 is the angular velocity of the spindle.

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The frequency component of the signal we want to identify is f max , and the corresponding number of spectral lines is N f m . By Nyquist's sampling theorem, it is required that f s ≥ 2 f max .
An anti-alias low-pass filter is usually set to filter out the frequency components above f s 2 . When using a capacitive sensor for measurement, since the capacitive sensor probe pole plate has a certain width, it is equivalent to adding a moving average low-pass filter with a cut-off frequency of where f c is the cut-off frequency; D r is the diameter of the standard workpiece; D is the diameter of the capacitive sensor circular pole plate; n is speed of the spindle; and the unit of n is revolutions per minute.
In general, f c will be lower than the cut-off frequency of the anti-alias low-pass filter available, so N has to satisfy f s = N · n ≥ 2.56 f c (12) Therefore, N can be chosen as The diameter of the sensor pole plate used in this paper is 2 mm, and the diameter of the measured cylindrical workpiece is 20 mm. So according to Equation (13), N can be chosen as 512 points. The filtering effect of the capacitive sensor acts as an anti-aliasing low-pass filter.

In Situ Measurement and Evaluation Experiment of Radial Error Using Three-Point Method
The measurement is carried out in the ultra-precision machining workshop. The temperature of the workshop is kept at 20-24 • C, and the humidity is at 50-60%, which mainly changes with the seasons. During the in situ measurement and evaluation experiment, the temperature and humidity are almost constant. The measurement object is an ultraprecision air-bearing spindle independently developed and manufactured by the laboratory. The spindle is installed on an ultra-precision lathe in the machining workshop and the design value of the radial rotation accuracy of the spindle is 20 nm. The main body of the ultra-precision lathe is granite, and the lathe is installed on a shock-absorbing mechanism to achieve a good shock-absorbing effect and ensure the processing performance.
The in situ measurement and evaluation system is used to perform the experiment. The installation position of the three sensors is decided according to Equation (9). Figure 9 shows the detailed measurement process, which is critical to ensure the accuracy and stability of the in situ measurement and evaluation system. Compared with the traditional operation steps of three-point method of measurement, the in situ measurement and evaluation system is more operable while ensuring the installation accuracy.
First, as shown in Figure 9a, the standard workpiece is adsorbed on the air-bearing spindle by the pressure of the vacuum, and the geometric center of the standard workpiece is roughly aligned with the axis of the standard workpiece; second, as shown in Figure 9b, a dial indicator is used to make rough adjustments so that the eccentricity of the standard workpiece is within 1 µm; third, as shown in Figure 9c, the sensor probe is used for fine adjustments to make the eccentricity of the standard workpiece within 0.1-0.2 µm. The more accurate the centering, the higher the accuracy of the measurement results. Finally, the fixture and sensor probes are mounted in place, as shown in Figure 9d, and the measurement can be performed immediately. The measurement results will be presented and analyzed in Section 5. Figure 9b, a dial indicator is used to make rough adjustments so that the eccentricity of the standard workpiece is within 1 μm ; third, as shown in Figure 9c, the sensor probe is used for fine adjustments to make the eccentricity of the standard workpiece within 0.1-0.2 μm . The more accurate the centering, the higher the accuracy of the measurement results. Finally, the fixture and sensor probes are mounted in place, as shown in Figure  9d, and the measurement can be performed immediately. The measurement results will be presented and analyzed in Section 5.

Comparative Experiment of the Results
The comparative experiment is performed based on the Donaldson reversal method to validate the results measured by the in situ measurement and evaluation system. The spindle analyzer developed by Lion Precision and self-designed experimental equipment are used in the comparative experiment.
The measurement steps of the comparative experiment are as follows: first, the sensor probe is mounted in the 0° direction of the spindle for the first dataset acquisition, as shown in Figure 10a; after this, the fixture, sensor and standard workpiece are reversed 180° at the same time. Then, the sensor probe is in the 180° direction, acquiring the second set of data, as shown in Figure 10b. Before and after reversing, the speed of the spindle should be the same.

Comparative Experiment of the Results
The comparative experiment is performed based on the Donaldson reversal method to validate the results measured by the in situ measurement and evaluation system. The spindle analyzer developed by Lion Precision and self-designed experimental equipment are used in the comparative experiment.
The measurement steps of the comparative experiment are as follows: first, the sensor probe is mounted in the 0 • direction of the spindle for the first dataset acquisition, as shown in Figure 10a; after this, the fixture, sensor and standard workpiece are reversed 180 • at the same time. Then, the sensor probe is in the 180 • direction, acquiring the second set of data, as shown in Figure 10b. Before and after reversing, the speed of the spindle should be the same.
The measurement results are shown in Figure 11. The curves in Figure 11a-d show the synchronization error before reversal, the synchronization error after reversal, the roundness error of the standard workpiece and the radial error of the spindle, respectively. The radial error of the spindle is 25.135 nm, and the roundness error of the standard workpiece under testing is 20.515 nm. The measurement results based on the Donaldson reversal method are taken as the reference results. The measurement results obtained using the in situ measurement and evaluation system will be analyzed and discussed together with the reference results in Section 5. spindle analyzer developed by Lion Precision and self-designed experimental equipment are used in the comparative experiment.
The measurement steps of the comparative experiment are as follows: first, the sensor probe is mounted in the 0° direction of the spindle for the first dataset acquisition, as shown in Figure 10a; after this, the fixture, sensor and standard workpiece are reversed 180° at the same time. Then, the sensor probe is in the 180° direction, acquiring the second set of data, as shown in Figure 10b. Before and after reversing, the speed of the spindle should be the same. The measurement results are shown in Figure 11. The curves in Figure 11a-d show the synchronization error before reversal, the synchronization error after reversal, the roundness error of the standard workpiece and the radial error of the spindle, respectively. The radial error of the spindle is 25.135 nm, and the roundness error of the standard workpiece under testing is 20.515 nm. The measurement results based on the Donaldson reversal method are taken as the reference results. The measurement results obtained using the in situ measurement and evaluation system will be analyzed and discussed together with the reference results in Section 5.

Result and Discussion
For the convenience of the discussion, the sampling data when the spindle speed is 60 revolutions per minute is selected as the first example. The raw data obtained by the in situ measurement and evaluation system is processed to remove the asynchronous error and eccentricity. Then, the synchronous error data are obtained. Figure 12a-c shows the amplitude phase diagrams of the multi-turn sampling data of the three sensors with the eccentricity and synchronization error removed in the polar coordinate system, respectively. Figure 13a,b and c show the amplitude phase diagrams of the multi-turn sampling data of the three sensors with the eccentricity and synchronization error removed in the Cartesian coordinate system, respectively.

Result and Discussion
For the convenience of the discussion, the sampling data when the spindle speed is 60 revolutions per minute is selected as the first example. The raw data obtained by the in situ measurement and evaluation system is processed to remove the asynchronous error and eccentricity. Then, the synchronous error data are obtained. Figure 12a-c shows the amplitude phase diagrams of the multi-turn sampling data of the three sensors with the eccentricity and synchronization error removed in the polar coordinate system, respectively. Figure 13a-c show the amplitude phase diagrams of the multi-turn sampling data of the three sensors with the eccentricity and synchronization error removed in the Cartesian coordinate system, respectively.

Result and Discussion
For the convenience of the discussion, the sampling data when the spindle speed is 60 revolutions per minute is selected as the first example. The raw data obtained by the in situ measurement and evaluation system is processed to remove the asynchronous error and eccentricity. Then, the synchronous error data are obtained. Figure 12a-c shows the amplitude phase diagrams of the multi-turn sampling data of the three sensors with the eccentricity and synchronization error removed in the polar coordinate system, respectively. Figure 13a,b and c show the amplitude phase diagrams of the multi-turn sampling data of the three sensors with the eccentricity and synchronization error removed in the Cartesian coordinate system, respectively.  It is obvious that the repeatability of the data collected by the sensors is good. By comparing the data in the polar coordinate system, it can be observed that the sampled data of the three sensors show a shape characteristic with a fixed-phase deviation just as the marker in Figure 12 shows. The phase deviation is the same as 2  and 2  determined in Equation (9), reflecting the correctness and stability of the sampling results. It is obvious that the repeatability of the data collected by the sensors is good. By comparing the data in the polar coordinate system, it can be observed that the sampled data of the three sensors show a shape characteristic with a fixed-phase deviation just as the marker in Figure 12 shows. The phase deviation is the same as α 2 and β 2 determined in Equation (9), reflecting the correctness and stability of the sampling results.
According to the error separation principle of the three-point method, the radial error and roundness error can be calculated, as shown in Figures 14 and 15. The radial error of the spindle and the roundness error of the measured standard workpiece are 22.379 nm and 16.362 nm, respectively. Compared with the reference results, the radial error and roundness error differ from the reference results by 2.756 nm and 4.153 nm, respectively. According to the research results of Eric R. March [17,18], the reversal method has more theoretical advantages than the other methods, but it is also easier to couple more error components. The key to ensuring the measurement accuracy of the reversal method is to minimize the installation angle error and eccentricity error before and after the reversal, so the precision mechanism is generally used for reversal and installation positioning. In the comparative experiment, it is difficult to completely eliminate the error components introduced by the installation angle and eccentricity since no precision mechanism is used. As a result, the roundness error and radial error measured by the reversal method are larger and the measurement results of the in situ measurement and evaluation system are believed to be more accurate. According to the error separation principle of the three-point method, the radial error and roundness error can be calculated, as shown in Figures 14 and 15. The radial error of the spindle and the roundness error of the measured standard workpiece are 22.379 nm and 16.362 nm, respectively. Compared with the reference results, the radial error and roundness error differ from the reference results by 2.756 nm and 4.153 nm, respectively. According to the research results of Eric R. March [17,18], the reversal method has more theoretical advantages than the other methods, but it is also easier to couple more error components. The key to ensuring the measurement accuracy of the reversal method is to minimize the installation angle error and eccentricity error before and after the reversal, so the precision mechanism is generally used for reversal and installation positioning. In the comparative experiment, it is difficult to completely eliminate the error components introduced by the installation angle and eccentricity since no precision mechanism is used. As a result, the roundness error and radial error measured by the reversal method are larger and the measurement results of the in situ measurement and evaluation system are believed to be more accurate.  In order to validate the reliability and stability of the in situ measurement and evaluation system, measurements based on the system are performed at different spindle speeds. The measurement results are presented in Table 1, and a line graph is drawn based on the measurement results, as shown in Figure 16. According to the error separation principle of the three-point method, the radial error and roundness error can be calculated, as shown in Figures 14 and 15. The radial error of the spindle and the roundness error of the measured standard workpiece are 22.379 nm and 16.362 nm, respectively. Compared with the reference results, the radial error and roundness error differ from the reference results by 2.756 nm and 4.153 nm, respectively. According to the research results of Eric R. March [17,18], the reversal method has more theoretical advantages than the other methods, but it is also easier to couple more error components. The key to ensuring the measurement accuracy of the reversal method is to minimize the installation angle error and eccentricity error before and after the reversal, so the precision mechanism is generally used for reversal and installation positioning. In the comparative experiment, it is difficult to completely eliminate the error components introduced by the installation angle and eccentricity since no precision mechanism is used. As a result, the roundness error and radial error measured by the reversal method are larger and the measurement results of the in situ measurement and evaluation system are believed to be more accurate.  In order to validate the reliability and stability of the in situ measurement and evaluation system, measurements based on the system are performed at different spindle speeds. The measurement results are presented in Table 1, and a line graph is drawn based on the measurement results, as shown in Figure 16. In order to validate the reliability and stability of the in situ measurement and evaluation system, measurements based on the system are performed at different spindle speeds. The measurement results are presented in Table 1, and a line graph is drawn based on the measurement results, as shown in Figure 16.   The calculation of the statistical results in Table 1 shows that the average value of the radial error and roundness error are 23.096 nm and 16.386 nm, respectively. The standard deviation of the radial error and roundness error are 0.556 nm and 0.408 nm, respectively. The measurement results of the radial error and roundness error are approximately 2 nm and 4 nm smaller than the reference results, respectively. Considering that no precision mechanism was used for inversion in the comparative experiment, the deviations of the measurement results from the reference results are caused by the secondary clamping error in the comparative experiment. From this point of view, the measurement result of the in situ measurement and evaluation system is more accurate than that of the Donaldson reversal method. Environmental factors such as temperature change and vibration can also lead to the deviations, which can also contribute to the fluctuation in the measurement results shown in Figure 16. The control of environmental factors is extremely important to ensure the radial rotation accuracy of the spindles in actual machining.
Based on the analysis above, it is concluded that the in situ measurement and evaluation system can perform accurate in situ measurements of the radial error in the machining environment. The measurement accuracy of the in situ measurement and evaluation system can reach 20 nm. Since the operation of the system is simpler and fewer clamping errors are introduced, it has advantages over the Donaldson reversal method. The calculation of the statistical results in Table 1 shows that the average value of the radial error and roundness error are 23.096 nm and 16.386 nm, respectively. The standard deviation of the radial error and roundness error are 0.556 nm and 0.408 nm, respectively. The measurement results of the radial error and roundness error are approximately 2 nm and 4 nm smaller than the reference results, respectively. Considering that no precision mechanism was used for inversion in the comparative experiment, the deviations of the measurement results from the reference results are caused by the secondary clamping error in the comparative experiment. From this point of view, the measurement result of the in situ measurement and evaluation system is more accurate than that of the Donaldson reversal method. Environmental factors such as temperature change and vibration can also lead to the deviations, which can also contribute to the fluctuation in the measurement results shown in Figure 16. The control of environmental factors is extremely important to ensure the radial rotation accuracy of the spindles in actual machining.
Based on the analysis above, it is concluded that the in situ measurement and evaluation system can perform accurate in situ measurements of the radial error in the machining environment. The measurement accuracy of the in situ measurement and evaluation system can reach 20 nm. Since the operation of the system is simpler and fewer clamping errors are introduced, it has advantages over the Donaldson reversal method.

Conclusions
The three-point method is verified to realize nanometer-scale measurements in the machining environment. An in situ measurement and evaluation system is established and used to perform the experiment. The system simplifies the operation process of the ultra-precision measurement based on the three-point method. In particular, the innovation of the integrated fixture and experimental technical details is the key to improving the in situ measurement accuracy. The accuracy and advantages of in situ measurement and evaluation system are verified through a comparative experiment. The research has a reference function for the practical application of the three-point method. Funding: This work was financially supported by the National Natural Science Foundation of China (51991371).

Data Availability Statement:
The data presented in this study are available upon request from the corresponding author. The data are not publicly available because it is part of an ongoing study.