An improved three-point method based on a difference algorithm

Precision Engineering 63 (2020) 68–82

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Precision Engineering journal homepage: www.elsevier.com/locate/precision

An improved three-point method based on a difference algorithm Ran Huang, Wei Pan ∗, Changhou Lu ∗, Yixin Zhang, Shujiang Chen Key Laboratory of High-efficiency and Clean Mechanical Manufacture of MOE, School of Mechanical Engineering, Shandong University, Jinan 250061, Shandong Province, China

ARTICLE

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ABSTRACT

Keywords: Three-point method Difference algorithm Roundness error Error separation

The reason for the rounding error in a measurement system under the condition of a limited sampling period is analysed, and a three-point method based on the difference algorithm (D3P method) is proposed. The effect of the rounding error is reduced by differential approximation of the roundness error and reconstruction of the transfer function. The generalized three-point method (G3P method), the three-point method based on the first-order difference algorithm (I-D3P method) and the three-point method based on the second-order difference algorithm (II-D3P method) are compared and analysed by simulation. The results show that the accuracy of the D3P method is greatly improved compared with that of the G3P method under the restricted sampling period and that the D3P method retains high precision in the case of failed error separation of the G3P method. An experimental system was constructed to realize error separation, and the experimental results show that the D3P method has higher accuracy than the G3P method over a small sampling period.

1. Introduction

scholars in the field; this method has been continuously improved and perfected. Shi et al. [14] proposed a hybrid three-probe method in the context of the time domain to reduce the effect of harmonic suppression, where several conventional three-probe measurements are performed for optimizing individual harmonic coefficients. Zhang et al. [9] use the four-point method to solve the problem that a small measuring uncertainty of the probe might cause quite large errors in the final results of measurements. In Lei et al. [15], a matrix threepoint method based on least squares is proposed to eliminate the influence of harmonic suppression, and its effectiveness is simulated and analysed. Gao et al. [8] proposed a mixed method that can completely separate the roundness error and the spindle error and can measure high-frequency components regardless of the probe distance. Muralikrishnan et al. [16] and Ping et al. [17] perform a comparison simulation of selected three-point methods. Zhang et al. [18] present software for correcting the errors caused by the measurement error of the angular positions of the sensors and the sensitivity difference between three sensors. Although the three-point method simulations or experiments mentioned in these reports can suitably achieve the error separation requirements, these three-point methods do not limit the sampling period or enable the default selection of a more ideal sampling period. In practice, due to the inevitable error in the mounting position of the sensor, if the measuring position points of each sensor do not

Static pressure bearings are widely used in high-precision machining because of their high stiffness, high rotation accuracy and favourable damping. Liang et al. [1] propose an elliptical inner surface forming method based on motion synthesis; the basic theory of motion forming of an elliptical inner surface is studied, and the machining scheme and forming principle of the surface are described in detail by using the active control method of the static pressure spindle. To realize highprecision elliptical inner surface machining, it is necessary to obtain the axial motion trajectory of the static pressure spindle accurately. Since the actual measurement signal contains the roundness error (roundness error refers to the amount of change of the actual circle to its theoretical circle measured within the same cross-section of the swing body) of the spindle, it is clearly not feasible to use the measurement signal directly as the axial trajectory of the spindle, which requires that the roundness error and the axial trajectory be separated from the measurement signal. The three-point method error separation technology has been widely used in online detection and separation of motion error and work piece shape error [2–7]. The method measures the spindle position information by placing three sensors in three different positions around the spindle. A modified method for measuring spindle information by using multiple probes is the multi-probe method [8–12]. In 1977, Whitehouse [13] presented the three-point method error separation technology, which has aroused the attention and research of the majority of

∗ Corresponding authors. E-mail addresses: [email protected] (R. Huang), [email protected] (W. Pan), [email protected] (C. Lu), [email protected] (Y. Zhang), [email protected] (S. Chen).

https://doi.org/10.1016/j.precisioneng.2020.01.008 Received 13 December 2018; Received in revised form 30 January 2020; Accepted 31 January 2020 Available online 5 February 2020 0141-6359/© 2020 Elsevier Inc. All rights reserved.

Precision Engineering 63 (2020) 68–82

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where 𝑐1 , 𝑐2 and 𝑐3 are the linear superposition coefficients. To eliminate the contribution of the motion error from 𝑟 (𝜃), the values of 𝑐1 , 𝑐2 and 𝑐3 must satisfy the following formula: 𝑐1 cos 𝜑1 + 𝑐2 cos 𝜑2 + 𝑐3 cos 𝜑3 = 0

(3)

𝑐1 sin 𝜑1 + 𝑐2 sin 𝜑2 + 𝑐3 sin 𝜑3 = 0

(4)

By solving Eqs. (3) and (4), the general solution of 𝑐1 , 𝑐2 and 𝑐3 can be derived: 𝑐1 = 𝜎

( ) sin 𝜑1 − 𝜑3 ( ) sin 𝜑3 − 𝜑2 ( ) sin 𝜑2 − 𝜑1 𝑐3 = 𝜎 ( ) sin 𝜑3 − 𝜑2

𝑐2 = 𝜎

Fig. 1. Schematic diagram of the three-point method.

where 𝜎 ≠ 0. Thus, Eq. (2) can be simplified as follows: ( ) ( ) ( ) 𝑟 (𝜃) = 𝑐1 ℎ 𝜃 + 𝜑1 + 𝑐2 ℎ 𝜃 + 𝜑2 + 𝑐3 ℎ 𝜃 + 𝜑3

coincide, then there must be an inherent error in the test system, which is called the rounding error. Although suitable sampling position points of each sensor can be maintained by increasing the sampling period (for example, the measuring instrument can be used to determine the position angle of three sensors as 0.00◦ , 60.29◦ and 180.17◦ ), to meet the requirements, the sampling period must be at least 36 000. This approach is acceptable for offline measurement systems or measurement systems that do not require high real-time performance, but it is clearly not feasible for control systems with stringent real-time requirements. How to solve the problem of error separation accuracy caused by rounding error in the case of a limited sampling period is the key research object of this paper. The rest of this paper is organized as follows. In Section 2, the principle of the generalized three-point error separation method (G3P method) is introduced. In Section 3, first, harmonic suppression is introduced, and then, the influence of the rounding error on the measurement accuracy is analysed. In Section 4, a three-point method based on differential decomposition is deduced (D3P method), and a three-point method based on first-order (I-D3P method) and secondorder (II-D3P method) difference equations is written. In Section 5, first, the error separation accuracy of the G3P method, I-D3P method and II-D3P method is compared by computer simulation analysis, and then, the error separation effect between the G3P method and the I-D3P method and the II-D3P method is compared experimentally, last, the uncertainty analysis was conducted through a Monte Carlo simulation under the distribution angle of the sensors obtained from the experiment. Section 6 gives the conclusions.

where 𝑝𝑚 = 𝑓 𝑖𝑥

=

𝑚=1

( ) 𝑐𝑚 ℎ 𝜃 + 𝜑𝑚 + 𝑔𝑥 (𝜃)

𝑚=1

( ) 𝑐𝑚 cos 𝜑𝑚 + 𝑔𝑦 (𝜃)

(8)

, 𝑚 ∈ {1, 2, 3}

(11)

Clearly, ℎ(𝑛) is a periodic sequence, and the period is 𝑁. By using the discrete Fourier transform (DFT) and its property of linearity and time shifting, we can obtain the frequency-domain expression of Eq. (11): (12)

𝑅 (𝑘) = DFT (𝑟 (𝑛)) = 𝐻 (𝑘) 𝑊 (𝑘)

Here, 𝐻(𝑘) is the discrete Fourier transform of ℎ(𝑛). The transfer function is designated 𝑊 (𝑘). 2𝜋i

𝑊 (𝑘) = 𝑐1 𝑒 𝑁

3 ∑

𝑘𝑝1

2𝜋i

+ 𝑐2 𝑒 𝑁

𝑘𝑝2

2𝜋i

+ 𝑐3 𝑒 𝑁

𝑘𝑝3

(13)

Then, (14)

𝐻 (𝑘) = 𝑅 (𝑘) ∕𝑊 (𝑘)

Using the inverse discrete Fourier transform (IDFT) of 𝐻(𝑘), the estimated values of the roundness error can be derived: ℎ̂ (𝑛) = IDFT (𝐻 (𝑘)) = IDFT (𝑅 (𝑘) ∕𝑊 (𝑘))

(15)

Substitute Eq. (15) into Eq. (1); then, the estimated values of the motion error can be derived: } { [ ] [ ] 1 𝑔̂𝑥 (𝑛) = ( ) 𝑑1 (𝑛) − ℎ̂ 1 (𝑛) sin 𝜑2 − 𝑑2 (𝑛) − ℎ̂ 2 (𝑛) sin 𝜑1 sin 𝜑2 − 𝜑1

𝑐𝑚 𝑑𝑚 (𝜃) 3 ∑

𝑚

𝛥

if 𝛿𝑚 = 0 (𝑚 ∈ {1, 2, 3}), then ( ) ( ) ( ) 𝑟 (𝑛) = 𝑐1 ℎ 𝑛 + 𝑝1 + 𝑐2 ℎ 𝑛 + 𝑝2 + 𝑐3 ℎ 𝑛 + 𝑝3

Here, 𝑔𝑥 (𝜃) and 𝑔𝑦 (𝜃) are the x and y components, respectively, of the motion error to be determined. ℎ(𝜃) represents the artefact roundness error. The three signals are linearly superposed:

𝑚=1 3 ∑

(𝜑 )

𝜑 𝛿𝑚 = 𝑚 − 𝑝𝑚 , 𝑚 ∈ {1, 2, 3} (9) 𝛥 where 𝑓 𝑖𝑥(⋅) is the rounding function. Let 𝛥 be normalized at 1; then, Eq. (7) is further simplified as ( ) ( ) ( ) 𝑟 (𝑛) = 𝑐1 ℎ 𝑛 + 𝑝1 + 𝛿1 + 𝑐2 ℎ 𝑛 + 𝑝2 + 𝛿1 + 𝑐3 ℎ 𝑛 + 𝑝3 + 𝛿1 (10)

As shown in Fig. 1, there are three displacement sensors mounted on the outside of the spindle, as indicated by S1, S2 and S3, with the angles between the sensors and 𝑋-axis being 𝜑1 , 𝜑2 and 𝜑3 , respectively. Let the corresponding output signals be 𝑑1 (𝜃), 𝑑2 (𝜃) and 𝑑3 (𝜃). ( ) 𝑑1 (𝜃) = ℎ 𝜃 + 𝜑1 + 𝑔𝑥 (𝜃) cos 𝜑1 + 𝑔𝑦 (𝜃) sin 𝜑1 ( ) (1) 𝑑2 (𝜃) = ℎ 𝜃 + 𝜑2 + 𝑔𝑥 (𝜃) cos 𝜑2 + 𝑔𝑦 (𝜃) sin 𝜑2 ( ) 𝑑3 (𝜃) = ℎ 𝜃 + 𝜑3 + 𝑔𝑥 (𝜃) cos 𝜑3 + 𝑔𝑦 (𝜃) sin 𝜑3

3 ∑

(6)

In reality, data from a sensor are discrete. Assume that the sampling period is 𝑁, the sampling interval is designated 𝛥 = 2𝜋∕𝑁. Therefore, Eq. (6) can be written in discrete form: ( ) ( ) 𝑟 (𝑛𝛥) = 𝑐1 ℎ 𝑛𝛥 + 𝑝1 𝛥 + 𝛿1 𝛥 + 𝑐2 ℎ 𝑛𝛥 + 𝑝2 𝛥 + 𝛿1 𝛥 (7) ( ) + 𝑐3 ℎ 𝑛𝛥 + 𝑝3 𝛥 + 𝛿1 𝛥

2. Principle of the three-point method

𝑟 (𝜃) =

(5)

( ) 𝑐𝑚 sin 𝜑𝑚

𝑔̂𝑦 (𝑛) =

𝑚=1

1 ( ) sin 𝜑2 − 𝜑1

{ [

(16) } ] [ ] 𝑑2 (𝑛) − ℎ̂ 2 (𝑛) cos 𝜑1 − 𝑑1 (𝑛) − ℎ̂ 1 (𝑛) cos 𝜑2 (17)

(2) 69

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When 𝐴1 = 𝐴2 = 𝐴3 , each sensor sampling position is in the same set. Thus far, Eq. (6) can be rewritten as ( ) ( ) ( ) ( ) 𝑟 𝜃̃ = 𝑐1 ℎ 𝜃̃ + 𝑝1 𝛥 + 𝑐2 ℎ 𝜃̃ + 𝑝2 𝛥 + 𝑐3 ℎ 𝜃̃ + 𝑝3 𝛥 (27) where 𝜃̃ = 𝜃 − 𝛿1 𝛥. Discretize Eq. (27): ( ) ( ) ( ) 𝑟 (𝑛) ̃ = 𝑐1 ℎ 𝑛̃ + 𝑝1 + 𝑐2 ℎ 𝑛̃ + 𝑝2 + 𝑐3 ℎ 𝑛̃ + 𝑝3

Clearly, Eq. (28) is equivalent to Eq. (11), at which point the measurement system does not have a rounding error caused by the sensor mounting position. Unfortunately, it is difficult to ensure that 𝐴1 = 𝐴2 = 𝐴3 in the actual sensor installation; to keep 𝐴1 = 𝐴2 = 𝐴3 , we need to choose a larger number of 𝑁. For example, when 𝜑1 = 0.00◦ , 𝜑2 = 45.59◦ , and 𝜑3 = 135.67◦ , 𝑁 must be at least 36 000 to satisfy 𝐴1 = 𝐴2 = 𝐴3 ; and in practical applications, to ensure that the fast Fourier transform (FFT) does not increase the calculation complexity, we generally take 𝑁 as positive integer power of 2, which makes it more difficult to choose a value of 𝑁 that meets the requirements. Assuming that the roundness error ℎ(𝜃) of the spindle has a qderivative, the following Taylor expansion can be written for ℎ(𝑛 + 𝑝𝑚 + 𝛿𝑚 ): ( ) 𝑞 ℎ(𝑖) 𝑛 + 𝑝𝑚 ( )𝑖 ( )𝑞 ( ) ( ) ∑ ℎ 𝑛 + 𝑝𝑚 + 𝛿𝑚 = ℎ 𝑛 + 𝑝𝑚 + 𝛿𝑚 + 𝑜[ 𝛿𝑚 ] (29) 𝑞! 𝑖=1

Fig. 2. Individual sensor measuring positions.

where ( ) ℎ̂ 1 (𝑛) = ℎ̂ 𝑛 + 𝑝1 ( ) ℎ̂ 2 (𝑛) = ℎ̂ 𝑛 + 𝑝2

(18) (19)

3. Error analysis

Clearly, the transfer function 𝑊 (𝑘) expression in Eq. (13) does not contain the factor of (𝛿𝑚 )𝑖 , so there is a rounding error in the estimation ̂ of motion error 𝑔̂𝑥 (𝑛), 𝑔̂𝑦 (𝑛) and roundness error estimate ℎ(𝑛) in Eqs. (15), (16) and (17). This condition is also the root cause of the error of the measurement system due to the mounting position of the sensor when there is an integer 𝑖 ≠ 𝑗(𝑖, 𝑗 ∈ 1, 2, 3), which results in 𝐴𝑖 ≠ 𝐴𝑗 . Since the rounding error caused by the mounting position of the sensor is inevitable, how to reduce the influence of this error is the key point to be resolved in this paper.

3.1. Suppressed harmonics Define 𝑤 (𝑘) = 𝑐1 + 𝑐2 𝑒i𝑘(𝜑2 −𝜑1 ) + 𝑐3 𝑒i𝑘(𝜑3 −𝜑1 )

(20)

Then, 𝑊 (𝑘) and 𝑤(𝑘) have the following relationship: 𝑊 (𝑘) = 𝑒2𝜋i𝜑1 𝑤 (𝑘)

(21)

Assume that the greatest common divisor among 𝜑2 − 𝜑1 , 𝜑3 − 𝜑1 and 2𝜋 is 𝛷. Then, there are two positive integers 𝑘1 and 𝑘2 , resulting in the following equations: 2𝜋 𝛷 2𝜋 𝛷

( ) 𝜑2 − 𝜑1 = 2𝑘1 𝜋

(22)

( ) 𝜑3 − 𝜑2 = 2𝑘2 𝜋

(23)

Then, ) ( ) 2𝜋 ( 𝜑3 − 𝜑1 = 2 𝑘1 + 𝑘2 𝜋 𝛷

4. Three-point method based on a difference equation The following differential expressions exist for the q-derivative of ℎ(𝑛 + 𝑝𝑚 ): (1) Forward difference equation: ( ) ( ) ( ) (𝑞) ℎ 𝑛 + 𝑝𝑚 = ℎ(𝑞−1) 𝑛 + 𝑝𝑚 + 1 − ℎ(𝑞−1) 𝑛 + 𝑝𝑚 𝑞 ( ) ∑ ( ) = ℎ 𝑛 + 𝑝𝑚 + 𝑞 + (−1)𝑖 C𝑖𝑞 ℎ 𝑛 + 𝑝𝑚 + 𝑞 − 𝑖

(24)

𝑖=1

( ) 𝑤 2𝜋𝑙∕𝜆 + 𝑘0 =𝑐1 + 𝑐2 𝑒i(2𝜋𝑙∕𝛷+𝑘0 )(𝜑2 −𝜑1 ) + 𝑐3 𝑒i(2𝜋𝑙∕𝛷+𝑘0 )(𝜑3 −𝜑1 ) =𝑐 + 𝑐 𝑒i[2𝜋𝑙(𝜑2 −𝜑1 )∕𝛷+𝑘0 (𝜑2 −𝜑1 )] 1

𝑞 ( ) ∑ ( ) = ℎ 𝑛 + 𝑝𝑚 + (−1)𝑖 C𝑖𝑞 ℎ 𝑛 + 𝑝𝑚 − 𝑖

(31)

𝑖=1

2

(25)

4.1. Three-point method based on the backward first-order difference equation In the I-D3P method, the approximate decomposition of ℎ(𝑛+𝑝𝑚 +𝛿𝑚 ) is carried out by using the backward first-order difference equation; the following equation can be derived: ( ) ( ) ( ) ℎ 𝑛 + 𝑝𝑚 + 𝛿𝑚 = ℎ 𝑛 + 𝑝𝑚 + 𝛿𝑚 ℎ′ 𝑛 + 𝑝𝑚 ( ) ( ) ( ) (32) = ℎ 𝑛 + 𝑝𝑚 + 𝛿𝑚 [ℎ 𝑛 + 𝑝𝑚 − ℎ 𝑛 + 𝑝𝑚 − 1 ] ( ) ( ) ( ) = 1 + 𝛿𝑚 ℎ 𝑛 + 𝑝𝑚 − 𝛿𝑚 ℎ 𝑛 + 𝑝𝑚 − 1

Eq. (25) means that 𝑤(𝑘) repeats with an integral periodicity of 2𝜋∕𝛷. Eqs. (3) and (4) suggest that when 𝑘 = ±1, 𝑤(𝑘) = 0. Hence, it can be derived that when 𝑘 = 2𝜋𝑙∕𝛷 ± 1, 𝑤(𝑘) = 0; thus, 𝑊 (𝑘) = 0. Accordingly, these harmonics are suppressed and insolvable. 3.2. Rounding error

Substituting Eq. (32) into Eq. (10), the following equation can be derived:

Each sensor measuring position is as shown in Fig. 2. Assume that each sensor sampling position set is 𝐴𝑚 . Then, 𝐴𝑚 = {𝜃𝑚𝑘 |𝜃𝑚𝑘 = 𝑘𝛥 + 𝛿𝑚 , 0 ≤ 𝑘 < 𝑁} (𝑚 ∈ {1, 2, 3})

(30)

(2) Backward difference equation: ( ) ( ) ( ) (𝑞) ℎ 𝑛 + 𝑝𝑚 = ℎ(𝑞−1) 𝑛 + 𝑝𝑚 − ℎ(𝑞−1) 𝑛 + 𝑝𝑚 − 1

According to Eq. (20), when 𝑘 = 2𝜋𝑙∕𝛷 + 𝑘0 , 𝑤(𝑘) can be written as

+ 𝑐3 𝑒i[2𝜋𝑙(𝜑3 −𝜑1 )∕𝛷+𝑘0 (𝜑3 −𝜑1 )] =𝑐1 + 𝑐2 𝑒i𝑘0 (𝜑2 −𝜑1 ) + 𝑐3 𝑒i𝑘0 (𝜑3 −𝜑1 ) ( ) =𝑤 𝑘0

(28)

𝑟 (𝑛) =

(26)

3 ∑ 𝑚=1

70

[ ( )( ) ( ) ] 𝑐𝑚 ℎ 𝑛 + 𝑝𝑚 1 + 𝛿𝑚 − ℎ 𝑛 + 𝑝𝑚 − 1 𝛿𝑚

(33)

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Fig. 3. (A) Roundness profile with harmonic components from 2 to 30 undulations per revolution. (B) The amplitudes of the harmonic components were distributed between 5 μm and 10 μm.

Fig. 4. Accuracy of error separation under the G3P method. Fig. 6. Accuracy of error separation under the II-D3P method.

derived: ( ) ( ) ( ) ℎ̂ 1 (𝑛) = 1 + 𝛿1 ℎ̂ 𝑛 + 𝑝1 − 𝛿1 ℎ̂ 1 𝑛 + 𝑝1 − 1 ( ) ( ) ( ) ℎ̂ 2 (𝑛) = 1 + 𝛿2 ℎ̂ 𝑛 + 𝑝2 − 𝛿2 ℎ̂ 1 𝑛 + 𝑝2 − 1

(35) (36)

Finally, the motion error estimated values 𝑔̂𝑥 (𝑛) and 𝑔̂𝑦 (𝑛) can be obtained by substituting Eqs. (35) and (36) into Eqs. (16) and (17).

4.2. Three-point method based on the backward second-order difference equation

In the II-D3P method, the approximate decomposition of ℎ(𝑛 + 𝑝𝑚 + 𝛿𝑚 ) is carried out by using the backward second-order difference equation; the following equation can be derived:

Fig. 5. Accuracy of error separation under the I-D3P method.

( ) ( ) ( ) ( ) ℎ 𝑛 + 𝑝𝑚 + 𝛿𝑚 =ℎ 𝑛 + 𝑝𝑚 + 𝛿𝑚 ℎ′ 𝑛 + 𝑝𝑚 + 0.5𝛿𝑚2 ℎ′′ 𝑛 + 𝑝𝑚 ( ) ( ) ( ) =ℎ 𝑛 + 𝑝𝑚 + 𝛿𝑚 + 0.5𝛿𝑚2 ℎ′ 𝑛 + 𝑝𝑚 ( ) − 0.5𝛿𝑚2 ℎ′ 𝑛 + 𝑝𝑚 − 1 ( ) ( ) ( ) ( ) = 1 + 𝛿𝑚 + 0.5𝛿𝑚2 ℎ 𝑛 + 𝑝𝑚 − 𝛿𝑚 + 𝛿𝑚2 ℎ 𝑛 + 𝑝𝑚 − 1 ( ) + 0.5𝛿𝑚2 ℎ 𝑛 + 𝑝𝑚 − 2

By using the discrete Fourier transform, a new transfer function 𝑊 (𝑘) can be derived: 𝑊 (𝑘) =

3 ∑

2𝜋i [( ) 2𝜋i ] 𝑐𝑚 1 + 𝛿𝑚 𝑒 𝑁 𝑘𝑝𝑚 − 𝛿𝑚 𝑒 𝑁 𝑘(𝑝𝑚 −1)

(34)

𝑚=1

The new roundness error estimate values of ℎ̂ can be obtained by substituting Eq. (34) into Eq. (15); then, the following equation can be

(37) 71

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Substituting Eq. (37) into Eq. (10), the following equation can be derived: 𝑟 (𝑛) =

3 ∑

𝑐𝑚

[

) ) ( )( ( )( ℎ 𝑛 + 𝑝𝑚 1 + 𝛿𝑚 + 0.5𝛿𝑚2 − ℎ 𝑛 + 𝑝𝑚 − 1 𝛿𝑚 + 𝛿𝑚2

𝑚=1

( ) ] + 0.5ℎ 𝑛 + 𝑝𝑚 − 2 𝛿𝑚2 (38)

By using the discrete Fourier transform, a new transfer function 𝑊 (𝑘) can be derived: [ 3 ∑ ) 2𝜋i ( ) 2𝜋i ( 𝑊 (𝑘) = 𝑐𝑚 1 + 𝛿𝑚 + 0.5𝛿𝑚2 𝑒 𝑁 𝑘𝑝𝑚 − 𝛿𝑚 + 𝛿𝑚2 𝑒 𝑁 𝑘(𝑝𝑚 −1) 𝑚=1 (39) ] 2𝜋i + 0.5𝛿 2 𝑒 𝑁 𝑘(𝑝𝑚 −2) 𝑚

The new roundness error estimated values of ℎ̂ can be obtained by substituting Eq. (39) into Eq. (15); then, the following equation can be derived: ) ( ) ) ( ) ( ( ℎ̂ 1 (𝑛) = 1 + 𝛿1 + 0.5𝛿12 ℎ̂ 𝑛 + 𝑝1 − 𝛿1 + 𝛿12 ℎ̂ 𝑛 + 𝑝1 − 1 ( ) + 0.5𝛿12 ℎ̂ 𝑛 + 𝑝1 − 2 (40) ( ) ( ) ( ) ( ) ̂ℎ2 (𝑛) = 1 + 𝛿2 + 0.5𝛿 2 ℎ̂ 𝑛 + 𝑝2 − 𝛿2 + 𝛿 2 ℎ̂ 𝑛 + 𝑝2 − 1 2 2 ( ) + 0.5𝛿22 ℎ̂ 𝑛 + 𝑝2 − 2 (41)

Fig. 7. Comparison between the G3P method and the I-D3P method.

Finally, the motion error estimated values 𝑔̂𝑥 (𝑛) and 𝑔̂𝑦 (𝑛) can be obtained by substituting Eqs. (40) and (41) into Eqs. (16) and (17). 5. Simulations and experiment 5.1. Computer simulation Suppose that 𝑆 represents the original value sequence and that 𝑆̂ represents the estimated value sequence of 𝑆. Use the ratio of the root of mean square error of the 𝑆̂ and the highest peak of 𝑆 (RRMSE) to indicate the overall difference between the estimated values and the original values, which is defined as follows: √ √ 𝑁 √1 ∑ ( )2 1 √ 𝑆̂ − 𝑆𝑖 ⋅ 100% (42) RRMSE (𝑆) = max (𝑆) − min (𝑆) 𝑁 𝑖=1 𝑖

Fig. 8. Comparison between the G3P method and the II-D3P method.

The maximum difference between the estimated values and the original values is expressed by using the ratio of the peak error of 𝑆̂ and the highest peak of 𝑆 (RPPE), which is defined as follows: ( ( ) ( ))| | |max (𝑆) − min (𝑆) − max 𝑆̂ − min 𝑆̂ | | ⋅ 100% RPPE (𝑆) = | (43) max (𝑆) − min (𝑆) In this paper, 𝑆 represents ℎ(𝑛). The parameters of the simulation are given as follows: 𝜑1 = 0.00◦ 𝜑2 = 60.29◦ 𝜑3 = 180.17◦ 𝑔𝑥 (𝜃) = 20 cos 𝜃 𝑔𝑦 (𝜑) = 20 sin 𝜃 The harmonic components are randomly distributed between 5 μm and 10 μm, and the phase of each harmonic component is optimized by computer to minimize the peak of the resulting roundness error. The roundness error and its spectrum diagram are shown in Fig. 3. For 𝑁 = 2𝑘 (𝑘 ∈ [8, 15]), the simulation is carried out by using the G3P method, I-D3P method and II-D3P method. The accuracy of error separation under the G3P method, I-D3P method and II-D3P method is shown in Fig. 4, Fig. 5 and Fig. 6, respectively. From the figures, it can be observed that the error separation accuracy of the three methods increases with increasing sampling period 𝑁.

Fig. 9. Comparison between the I-D3P method and the II-D3P method.

On the one hand, Fig. 4 shows that when (𝑁 ≤ 2048), the RRMSE and RPPE obtained by the G3P method exceed 100%, and when 𝑁 = 4096, although the RRMSE is less than 15%, the RPPE exceeds 50%, and it is clear that the error separation fails in the case of 𝑁 ≤ 4096. On the other hand, as shown in Figs. 5 and 6, when 𝑁 = 256, the RRMSE obtained by the I-D3P method is less than 5%, and the RPPE is less than 10%; in addition, the RRMSE obtained by the II-D3P method 72

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Fig. 10. Global scope error separation accuracy of the G3P method.

Fig. 11. Global scope error separation accuracy of the I-D3P method.

is less than 4%, and the RPPE is less than 9%. Clearly, error separation can be accomplished well by the D3P method when the error separation of the G3P method fails.

I-D3P method, and RRMSE2 and RPPE2 represent the results obtained by the II-D3P method. Figs. 7 and 8 show that for different values of 𝑘, RRMSE0 ∕RRMSE1 and RPPE0 ∕RPPE1 are greater than 66, and RRMSE0 ∕RRMSE2 and RPPE0 ∕RPPE2 are greater than 74, which means that the accuracy of the results obtained by the D3P method is at least 66 times higher than that of the G3P method which is a substantial boost. In addition, note that RRMSE0 ∕RRMSE1 and RRMSE0 ∕RRMSE2 with increasing 𝑁(𝑁 ≥ 4096) have a relatively obvious increasing trend, which means that the advantage of the D3P method becomes more prominent NN

Further, a comparison of Figs. 4–6 indicates that the accuracy of the I-D3P method and the II-D3P method in 𝑁 = 4096 is better than that of the G3P method in 𝑁 = 32 768. A comparison of the accuracy between the G3P method, I-D3P method and II-D3P method is shown in Fig. 7, Fig. 8 and Fig. 9, respectively. RRMSE0 and RPPE0 represent the results obtained by the G3P method, RRMSE1 and RPPE1 represent the results obtained by the 73

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Fig. 12. Global scope error separation accuracy of the II-D3P method.

Fig. 13. Error separation experiment platform.

Table 1 Error separation accuracy statistics.

Fig. 9 shows that RRMSE1 ∕RRMSE2 is greater than 1, which means that the accuracy of the results obtained by the II-D3P method is better than that of the II-D3P method. It is foreseeable that the accuracy of the results obtained by a higher-order-difference three-point method would be better. For the above simulation analysis and comparison of the error separation accuracy between the D3P method and the G3P method when the sensor angles are 0◦ , 60.29◦ and 180.17◦ , although the simulation results show that the D3P method has higher precision than the G3P method, they do not explain the superiority of the D3P method in the global scope. Thus, more sensor angles are needed to verify the error separation accuracy of the D3P method in the global scope. To further study the error separation accuracy of the D3P method, take the following sensor angle set: 𝜑1 = 0◦ , 𝜑2 ∈ [30.29◦ , 320.29◦ ], 𝜑3 ∈ [50.17◦ , 350.17◦ ] and 𝜑3 ≥ 𝜑2 + 20◦ . Calculate the error separation accuracy of the G3P method and the I-D3P method in 𝑁 = 1024. The results are shown in Figs. 10–12. Interval statistics are performed on the

>20% 20%∼10% 10%∼5% 5%∼1% <1%

G3P method (%)

I-D3P method (%)

II-D3P method (%)

RRMSE

RPPE

RRMSE

RPPE

RRMSE

RPPE

7.59 4.45 6.08 71.29 10.59

13.86 8.09 25.14 46.21 6.70

0.00 0.06 0.44 10.16 89.34

0.06 0.13 1.44 15,55 82.82

0.00 0.00 0.00 2.38 97.62

0.00 0.00 0.13 9.28 90.59

results, and we calculate the proportion of the values in each interval; the statistical results are shown in Table 1. A comparison of Figs. 10–12 indicates that the number of error separation failures of the G3P method is significantly higher than that of the D3P method. As the statistical results of Table 1 show, on the 74

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Fig. 14. Sensor sampling data.

Fig. 17. Comparison of the result of three methods (G3P, I-D3P and II-D3P) and reference roundness.

improved clearly, which also shows that the D3P method has stronger robustness. 5.2. Experiment An experimental platform for error separation is set up as shown in Fig. 13. In the actual test, due to installation error of the sensor, machining and installation errors of its bracket, and other reasons, the position angle of the sensor does not match the ideal value, so it is necessary to identify the installation position of the sensor. In the rotational motion of the spindle, the sampling data of three sensors are always related, so the phase relation of the sampled data of each sensor can be calculated by using the correlation analysis method. Assume that the sampling data for three sensors are S1, S2, and S3. Calculate the correlation functions between S2 and S1 and between S3 and S1 as follows:

Fig. 15. Sensor sampling data (Processed).

𝑅1 (𝑚) =

𝑁−1 1 ∑ 𝑆 (𝑛) 𝑆2 (𝑛 + 𝑚) 𝑁 𝑛=0 1

(44)

𝑅2 (𝑚) =

𝑁−1 1 ∑ 𝑆 (𝑛) 𝑆3 (𝑛 + 𝑚) 𝑁 𝑛=0 1

(45)

one hand, the value of RRMSE and RPPE less than 5% under the ID3P method and the II-D3P method are more than 98%, while the proportion of results under the G3P method is less than 82%. On the other hand, the value of RPPE greater than 20% under the I-D3P method and II-D3P method are less than 1%, while the proportion of

Since the 1st sensor angle is considered to be 0◦ , the 2nd and the 3rd sensor angles can be determined by

results under the G3P method exceeds 13%. Furthermore, the error separation effect of the D3P method is better than that of the G3P

𝜑2 =

method in the global range, and the error separation effect has been

360◦ max 𝑅1 (𝑚) 𝑚 𝑁

Fig. 16. Roundness measured. 75

(46)

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Fig. 18. Comparison of the calculated harmonics of three methods(G3P, I-D3P and II-D3P) results and reference roundness result.

Fig. 19. Output from Monte Carlo simulation where the standard uncertainties are shown as error bars for the harmonics. Table 2 Sensor position angles measured in five tests. Test

1

2

3

4

5

𝜑2 𝜑3

46.94 155.72

46.92 155.69

46.99 155.69

46.99 155.72

46.99 155.72

𝜑3 =

360◦ max 𝑅2 (𝑚) 𝑚 𝑁

The original signal of the sensors is processed by DC removal and noise reduction and is multiplied by the calibration coefficient; the results are shown in Fig. 15. Sensor sampling is carried out at 𝑁 = 65 536, 𝑁 = 4096 and 𝑁 = 1024 respectively, and error separation is carried out using the G3P method and the I-D3P method and the II-D3P method. The results are shown in Fig. 16. To quantify the deviation between the two results 𝑅1 and 𝑅2 , a criterion dv is defined: √ √ 𝑁 √1 ∑ ]2 [ (48) dv = √ 𝑅 (𝑖) − 𝑅2 (𝑖) 𝑁 𝑖=1 1

(47)

To reduce the effect of the harmonic factor on the correlation function in the sampled data, a piece of copper foil is added to the spindle. The resulting sensor sampling data are shown in Fig. 14.

The deviation under different error separation methods and different sampling cycles is shown in Table 3. The deviation between the results of the G3P method, I-D3P method and II-D3P method are less than 0.015 μm at 𝑁 = 65 536, which is a small value, indicating that the accuracy of the two methods is similar under these conditions. Comparing the dv values obtained by three methods under different 𝑁 values, it can be found that the deviation between the results of the I-D3P method and the II-D3P method is the least, and significantly less

Five tests (𝑁 = 65 536 and 𝜑1 = 270◦ ) were conducted using this method to estimate the sensor angles. The results are given in Table 2. Table 2 shows that the sensor angle estimation has satisfactory stability, and its limit difference is lower than 0.07◦ . Take the mean value of 5 tests as the final sensor angle: 𝜑2 = 46.97◦ , and 𝜑3 = 155.71◦ . 76

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Fig. 20. Outputs from Monte Carlo simulations for sensor error, with the standard uncertainties shown as error bars. These simulations were run with sensor S1 scale error (𝜎 = 1 μm).

Fig. 21. Outputs from Monte Carlo simulations for sensor error, with the standard uncertainties shown as error bars. These simulations were run with sensor S2 scale error (𝜎 = 1 μm).

than the deviation between the results of the G3P method and the ID3P method and the II-D3P method. This shows that the results of the I-D3P method are more similar to that of the II-D3P method, and the deviation between the two methods decreases as the 𝑁 value increases. Take the results obtained by the II-D3P method with 𝑁 = 65 536 as a reference, it can be found that with the increase of 𝑁, the dv value obtained by the same method is decreasing, which indicates that the increase of sampling period in a certain range can help to improve the accuracy of measurement.

often unknown, the error cannot be obtained accurately. So the error measurement is uncertain. Measurement uncertainty can be estimated with the GUM (Guide to the expression of uncertainty in measurement) method which is fairly unambiguous and used commonly if the measurement method and model is simple, linear and well defined [19]. However, once the measurement model becomes complex, it is challenging to establish the sensitivity coefficients. In 2008, ‘‘Supplements to the GUM’’ were published describing the use of the Monte Carlo method for uncertainty evaluation [20]. The Monte Carlo method according to the probability distribution of a given amount of input in the model to simulate random sampling, by statistical calculation, will receive the output of the arithmetic mean and standard deviation of as the average and the standard uncertainty of the best estimates [21,22].

5.3. Uncertainty evaluation by simulation For a long time, the accuracy of the measurement results has been expressed by measurement errors. The measurement error is the measured result minus the true value measured, which contains random error and system error. However, since the true value measured is 77

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Fig. 22. Outputs from Monte Carlo simulations for sensor error, with the standard uncertainties shown as error bars. These simulations were run with sensor S3 scale error (𝜎 = 1 μm).

Fig. 23. Outputs from Monte Carlo simulations for alignment error, with standard uncertainties shown as error bars. These simulations were run with sensor S1 angular position error (𝜎 = 0.5◦ ). Table 3 Deviation values of the results obtained by different methods under different sampling cycles. dv (μm)

𝑁 = 65 536

𝑁 = 4096

𝑁 = 1024

G3P

I-D3P

II-D3P

G3P

I-D3P

II-D3P

G3P

I-D3P

II-D3P

𝑁 = 65 536

G3P I-D3P II-D3P

0 0.013 0.012

– 0 0.003

– – 0

– – –

– – –

– – –

– – –

– – –

– – –

𝑁 = 4096

G3P I-D3P II-D3P

0.192 0.181 0.180

0.190 0.183 0.179

0.190 0.183 0.179

0 0.049 0.049

– 0 0.009

– – 0

– – –

– – –

– – –

𝑁 = 1024

G3P I-D3P II-D3P

0.585 0.572 0.535

0.584 0.572 0.535

0.584 0.572 0.535

0.545 0.522 0.515

0.541 0.528 0.521

0.541 0.538 0.521

0 0.181 0.186

– 0 0.050

– – 0

155.71◦ respectively, and the surface profile of the workpiece was

Table 4 presents the error contributions. In the Monte Carlo simulations, the angles of sensor S1, S2 and S3 were

270.00◦ ,

46.97◦

and

shown in Fig. 3. 78

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Fig. 24. Outputs from Monte Carlo simulations for alignment error, with standard uncertainties shown as error bars. These simulations were run with sensor S2 angular position error (𝜎 = 0.5◦ ).

Fig. 25. Outputs from Monte Carlo simulations for alignment error, with standard uncertainties shown as error bars. These simulations were run with sensor S3 angular position error (𝜎 = 0.5◦ ). Table 4 Error sources and their statistical distributions. Error source

Scale error of the sensor Angular position error of the sensor Vertical position error of the frame Horizontal position error of the frame Temperature change

PDF

Normal(𝜇, 𝜎 2 ) Normal(𝜇, 𝜎 2 ) Normal(𝜇, 𝜎 2 ) Normal(𝜇, 𝜎 2 ) Normal(𝜇, 𝜎 2 )

Table 5 Maximum deviation under scale error of the sensors. Parameters

Error source

𝜇

𝜎

0 μm 0◦ 0 mm 0 mm 20 ◦ C

1 μm 0.5◦ 0.25 mm 0.25 mm 0.5 ◦ C

Sensor S1 Sensor S2 Sensor S3

Maximum deviation (μm) G3P

I-D3P

II-D3P

1.226 1.238 1.240

0.387 0.399 0.402

0.213 0.225 0.228

methods in Fig. 18 and the harmonic components of the reference also indicates this point. The maximum harmonic amplitude deviations for

Under the above parameters, Figs. 17 and 18 compares the difference between the simulation results of three methods (G3P, I-D3P and II-D3P) and the reference results. The difference between the roundness plots in Fig. 17 indicates the presence of harmonic suppression, the difference between the harmonic components obtained by the three

the G3P method, I-D3P method, and II-D3P method results were 1.231 μm (30th), 0.392 μm (30th), and 0.218 μm (30th), respectively. Obviously, the I-D3P and II-D3P methods are more resistant to harmonic suppression than the G3P method. 79

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Fig. 26. Outputs from Monte Carlo simulations for horizontal position error of the frame (𝜎 = 0.25 mm), with standard uncertainties shown as error bars.

Fig. 27. Outputs from Monte Carlo simulations for vertical position error of the frame (𝜎 = 0.25 mm), with standard uncertainties shown as error bars. Table 6 Maximum deviation under angular position error of the sensors.

The output from the Monte Carlo simulations with 100,000 runs is shown in Figs. 19–28, where the different standard uncertainties are shown as error bars. And, to make these illustrations more intuitive and easy to analyse, the amplitude of each harmonic is removed and only the deviation is retained.

Error source

Sensor S1 Sensor S2 Sensor S3

Fig. 19 shows the result of the simulation with all the error sources specified in Table 4. The maximum standard uncertainties for the G3P method, the I-D3P method and the II-D3P method were 1.858 μm (30th), 1.081 μm (30th) and 0.919 μm (30th), respectively.

Maximum deviation (μm) G3P

I-D3P

II-D3P

0.954 1.550 1.440

0.325 0.743 0.622

0.291 0.575 0.452

deviation of different methods under different angular position error of the sensors . The sensitivity of the three measurement method to the horizontal position error of the frame is presented in Fig. 26. With the maximum deviation for G3P, I-D3P and II-D3P methods were 1.894 μm (30th), 1.121 μm (30th) and 0.960 μm (30th), respectively The sensitivity of the three measurement methods to the vertical position error of the frame is presented in Fig. 27. And the maximum

The sensitivity of the three method to the scale error of the sensors is presented in Figs. 20–22. As can be seen from the figures, the maximum deviation of the results of each method occurs at the 30th harmonic. Table 5 shows the maximum deviation of different methods under different sensor scale errors. The sensitivity of the three method to the angular position error of the sensors is presented in Figs. 23–25. Table 6 shows the maximum 80

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Fig. 28. Outputs from Monte Carlo simulations for temperature change (𝜎 = 0.5◦ C), with standard uncertainties shown as error bars.

Declaration of competing interest

deviation for G3P, I-D3P and II-D3P methods were 1.639 μm (30th), 0.841 μm (30th) and 0.675 μm (30th), respectively Fig. 28 presents the sensitivity of the three method to the temperature change. For G3P, I-D3P and II-D3P methods, the maximum deviation were 1.231 μm (30th), 0.392 μm (30th) and 0.218 μm (30th), respectively. However, it is important to note that the results at this time are almost equal to the results when there is no error source. It can be considered that the effect of temperature changes on the measurement uncertainty is weak. Throughout Figs. 19–28, it can be found that for different error sources, the uncertainty of the D3P method is always smaller than the G3P method, the II-D3P method is the least sensitive, the I-D3P method is the second and the G3P method is the most sensitive, thus verifying that the D3P method is more robust than the G3P method.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This research is supported by the National Natural Science Foundation of China (No. 51375275) and Natural Science Foundation of Shandong Province of China (No. ZR2019MEE064), which is highly appreciated by the authors. References

6. Conclusion

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1. The harmonic suppression of the G3P method is analysed, as is the rounding error caused by the inconsistent measurement position of each sensor under the condition of a restricted sampling period. 2. A three-point method based on a difference algorithm is proposed, which implements differential decomposition of the roundness error and then reconstructs the transfer function 𝑊 (𝑘), thus reducing the calculation error caused by the rounding error. 3. The error separation accuracy under the G3P method, I-D3P method and II-D3P method in the sensor position 𝜑𝑚 = {0.00◦ , 60.29◦ , 180.17◦ } and sampling period 𝑁 = 2𝑘 (𝑘 ∈ [8, 15]) is compared and analysed by simulation, and the results show that the error separation accuracy of the three methods increases with increasing sampling period. The error separation accuracy of the I-D3P method is at least 66 times that of the G3P method, and the error separation accuracy of the II-D3P method is at least 77 times that of the G3P method for 256 ≤ 𝑁 ≤ 32 768. Moreover, the I-D3P method and the II-D3P method maintain a high error separation accuracy under a condition (𝑁 ≤ 512) in which the G3P method fails. 4. Simulations are used to compare the accuracy of the D3P method and the G3P method in the global scope, and the results show that the D3P method has higher accuracy and higher robustness. 5. Experiments verify the effectiveness of the D3P method and confirm that it is more accurate than the G3P method. 81

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[20] BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, OIML. Evaluation of measurement data—supplement 1 to the Guide to the expression of uncertainty in measurement—propagation of distributions using a Monte Carlo method. International Organization for Standardization, JCGM; 2008. [21] Viitala R, Widmaier T, Hemming B, et al. Uncertainty analysis of phase and amplitude of harmonic components of bearing inner ring four-point roundness measurement. Precis Eng 2018;54:118–30. [22] Widmaier T, Hemming B, Juhanko J, et al. Application of Monte Carlo simulation for estimation of uncertainty of four-point roundness measurements of rolls. Precis Eng 2017;48:181–90.

[16] Muralikrishnan B, Venkatachalam S, Raja J, et al. A note on the three-point method for roundness measurement. Precis Eng 2005;29(2):257–60. [17] Ping M, Huang JJ, Li DN. The comparative analysis about rotational error separation with three-point method and approximate three-point method. In: 2011 International conference on consumer electronics, communications and networks (CECNet). XianNing; 2011. p. 1–5. [18] Zhang Y, Wang X, Zhang G, et al. A study on improving the accuracy of (for roundness and spindle motion error measurements with three-point method). J Beijing Inst Technol 1999;19(2):157–62. [19] BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, OIML. Guide to the expression of uncertainty in measurement (GUM 1995 with minor corrections). International Organization for Standardization, JCGM; 2010.

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