Owing to eccentricity and inclination, circularity of a cylindrical workpiece cannot be measured precisely by a circularity measuring machine when the workpiece has a small dimension (diameter ≤ 3 mm). In this paper, with the aim of solving this problem, circularity metrology of a small cylindrical workpiece using a segmenting scanning method is analyzed. The cross-sectional circle of the cylinder is segmented into several equivalent arcs for measurement by a two-dimensional coordinate measuring machine (profilometer). The circularity contour is obtained by stitching together arc contours obtained by data processing of the coordinates. Different segmenting patterns for coordinate scanning are considered. Measurement results are presented for three segmentation patterns, with 8, 10, and 12 equal segments, respectively. These results are evaluated in terms of the matching coefficient between neighboring arc contours on circumferential stitching, the Euclidean distance between neighboring arc contours on radial stitching, and the curvature of the arcs. From these evaluations, it is found that as the number of segments is increased, the matching coefficient increases from 0.14 to 0.50, the Euclidean distance decreases from 32 nm to 26 nm, and the curvature becomes close to the standard value.

Topics
Measuring instruments, Metrology, Quality assurance, Rotational dynamics, Linear filters, Data processing, Precision engineering

  • A segmenting scan method is used for precision circularity metrology of small cylindrical parts of diameter ≤ 3 mm.

  • A cross-sectional circle of the measured cylinder is segmented into 8, 10, or 12 equal parts to be scanned by a profilometer.

  • Circularity contours are characterized and reconstructed on the basis of the obtained coordinate data.

Topics
Measuring instruments, Metrology, Quality assurance, Rotational dynamics, Linear filters, Data processing, Precision engineering

Cylindrical parts are important components of many pieces of machinery used in precision manufacturing, such as the rollers of a rotate vector (RV) reducer, the needle rollers of a needle roller bearing, and pin gauges for precision machining.1–3 To provide the required equipment performance and lifespan, these parts must have high accuracy and small dimensions.4–7 Accurate measurements are essential for process and quality control in precision manufacturing, not only to determine whether manufactured parts meet assigned tolerances, but also, in many cases, to reduce deviations of these parts from designed parameters through improvements in manufacturing techniques based on the measurement results.8–11 Circularity is an important parameter of a cylindrical part. Traditionally, circularity is measured by the rotary scan technique, in which the cylindrical part is aligned on the center of rotation of the measuring machine, with the stylus of the machine kept in contact with the part.12–14 The circularity contour and value can be obtained by calculating the distance between the stylus tip and the rotational datum axis at each angular position of the rotation stage.15 However, this technique is not capable of high-accuracy measurements when the dimension of a cylindrical part is too small, since eccentricity and inclination have significant effects on the measurement accuracy.16–18 Normally, the alignment of the eccentricity for circularity measurement by the rotary scan technique should be less than 1 μm, although residual eccentricity can be further reduced by software compensation. Therefore, achieving sufficient alignment for cylindrical parts with small dimensions is challenging in the traditional rotary scan technique. In recent years, an orthogonal mixed technique employing a pair of displacement and angle sensors for error separation has been proposed for circularity measurement.19,20 On the basis of this approach, a noncontact circularity measurement method using three chromatic confocal sensors for error separation has been proposed,21 as has an online method for circularity and diameter measurement.22 However, none of these circularity measurement methods are suitable for small cylindrical workpieces. An alternative stitching linear scan technique has been proposed for circularity metrology of small cylindrical parts,23–25 although this also encounters difficulties in the case of cylinders of small dimensions.

To solve this problem, an analysis of circularity metrology of small cylindrical workpieces using a segmenting scanning method is carried out in this paper. Precision coordinate measurement of the measured cylinder’s cross-sectional circle is segmented into several equivalent arcs to be carried out by a two-dimensional coordinate measuring machine (profilometer). The arc contours of these equivalent arcs are generated by data processing. Thus, the entire circularity contour can be obtained by stitching these arc contours together. The proposed method does not require alignments of eccentricity and inclination in the rotary scan technique, since the measured workpiece is set on a V-groove device and linearly scanned.

As shown in Figs. 1(a) and 2(a), the small cylindrical workpiece is placed on a V-block and scanned by the diamond stylus of a profilometer to obtain the two-dimensional coordinates of its surface. The end of the workpiece is held in a round magnetic jig, on which there are n marks for equal segmenting. The stylus is set to scan along the direction perpendicular to the axis of the workpiece and then return to its initial position. The workpiece is then rotated to a new angular position manually by rotating the jig anticlockwise through an equal angular displacement 360°/n. The stylus is set to scan and return again. This operation is repeated n − 1 times, and the cross-sectional circle of the workpiece is thus segmented n times for scanning to obtain n groups of arc coordinate data, as shown in Fig. 1(b). The radius ri and center coordinates xi,zi of each arc are calculated by fitting with the least squares method.26 The mean value of these arc radii is taken as the measured workpiece radius:
(1)
FIG. 1.
Principle of coordinate modeling–segmenting method: (a) measuring device; (b) measured arc data; (c) arc contour obtained by data processing; (d) stitching of arc contours; (e) matching for angle error calculation; (f) accurate stitching of neighboring arc contours; (g) combining of overlapping parts of neighboring arc contours; (h) final circularity contour.
View largeDownload slide

Principle of coordinate modeling–segmenting method: (a) measuring device; (b) measured arc data; (c) arc contour obtained by data processing; (d) stitching of arc contours; (e) matching for angle error calculation; (f) accurate stitching of neighboring arc contours; (g) combining of overlapping parts of neighboring arc contours; (h) final circularity contour.

FIG. 1.
Principle of coordinate modeling–segmenting method: (a) measuring device; (b) measured arc data; (c) arc contour obtained by data processing; (d) stitching of arc contours; (e) matching for angle error calculation; (f) accurate stitching of neighboring arc contours; (g) combining of overlapping parts of neighboring arc contours; (h) final circularity contour.
View largeDownload slide

Principle of coordinate modeling–segmenting method: (a) measuring device; (b) measured arc data; (c) arc contour obtained by data processing; (d) stitching of arc contours; (e) matching for angle error calculation; (f) accurate stitching of neighboring arc contours; (g) combining of overlapping parts of neighboring arc contours; (h) final circularity contour.

Close modal
FIG. 2.
Small cylindrical workpiece scanned by profilometer stylus: (a) photograph of scanning setup; (b) diagram of scanning process.
View largeDownload slide

Small cylindrical workpiece scanned by profilometer stylus: (a) photograph of scanning setup; (b) diagram of scanning process.

FIG. 2.
Small cylindrical workpiece scanned by profilometer stylus: (a) photograph of scanning setup; (b) diagram of scanning process.
View largeDownload slide

Small cylindrical workpiece scanned by profilometer stylus: (a) photograph of scanning setup; (b) diagram of scanning process.

Close modal
The radius of the jth coordinate point on the ith arc is calculated as
(2)
As shown in Fig. 3(c), the contour of each arc, which consists of numerous coordinates, can be characterized by the following equations, where (Δxi,j, Δzi,j) are the coordinate of each arc contour, and θi,j are the angular positions of each coordinate point on each arc contour:
(3)
(4)
FIG. 3.
Segmenting using a round magnetic jig: (a) round magnetic jig with segmenting tool; (b) initial segmenting; (c) segmenting after first rotation; (d) segmenting after second rotation; (e) segmenting after third rotation.
View largeDownload slide

Segmenting using a round magnetic jig: (a) round magnetic jig with segmenting tool; (b) initial segmenting; (c) segmenting after first rotation; (d) segmenting after second rotation; (e) segmenting after third rotation.

FIG. 3.
Segmenting using a round magnetic jig: (a) round magnetic jig with segmenting tool; (b) initial segmenting; (c) segmenting after first rotation; (d) segmenting after second rotation; (e) segmenting after third rotation.
View largeDownload slide

Segmenting using a round magnetic jig: (a) round magnetic jig with segmenting tool; (b) initial segmenting; (c) segmenting after first rotation; (d) segmenting after second rotation; (e) segmenting after third rotation.

Close modal

As shown in Fig. 1(d), the first arc contour (in blue) is kept static as contour 1, and contour 2 (in red) is rotated from the position of contour 1 clockwise by 360°/n. Similarly, the third arc contour (in green) is rotated from the second arc contour clockwise by 360°/n. This procedure is repeated n − 1 times, such that all the arc contours are stitched into a circularity contour. However, this stitched contour is inaccurate, since the manual rotation angle in the scanning procedure is not always 360°/n, which means that the real stitching angle is not always 360°/n. Therefore, stitching angle error compensation is necessary for accurate stitching.

For the real stitching angle, as shown in Fig. 1(e), the contour data in the X–Z coordinate system must first be transformed into the θ −ΔR coordinate system. It should be noted that both of these are rectangular coordinate systems. The stitched arc contour in the θ −ΔR coordinate system is processed by a low-pass filter with 50 UPR. It is found that the overlapping parts of neighboring arc contours are mismatched, although theoretically they should coincide since they are at the same measured positions on the workpiece. Obviously, the mismatch corresponds to the stitching angle error, which can be calculated by the matching coefficient Ci∼i+1 of the overlapping parts of neighboring arc contours, which is given by the following cross-correlation function:27–29 
(5)
where fi(θ) is the ith arc contour, fi+1θ is the i+1th arc contour, θ is the angular position, and Δθi∼i+1 is the stitching angle error of the overlapping parts. The overlapping parts of neighboring arc contours can be matched at the correction position when the matching coefficient Ci∼i+1 reaches its maximum value, and the stitching angle error Δθi∼i+1 can then be calculated. The first arc contour in the θ −ΔR coordinate system is kept static, and the second arc contour is moved to match at the correct position. The angular displacement Δθ1∼2 of the second arc contour is then the stitching angle error. On returning to the X–Z coordinate system, the first arc contour is kept static and the second arc contour is rotated by Δθ1∼2.
Subsequently, the second arc contour is kept static and the third arc contour is rotated by Δθ2∼3. As shown in Fig. 1(f), the entire stitching angle error compensation of neighboring arc contours can be completed after repeating this procedure n − 1 times. To obtain a continuous circularity contour, the overlapping parts of neighboring arc contours are integrated into a single contour, as shown in Fig. 1(g). As shown in Fig. 1(h), an integral, accurate, continuous, and smooth circularity contour can be obtained after low-pass filtering, which can be taken as the final circularity contour result. The root-mean-square circularity deviation ΔRrms is used in this paper for evaluating the circularity. ΔRrms is calculated as follows:
(6)
where ri,j (max) and ri,jmin are the maximum and minimum values, respectively, among all the ri,j. With increasing number of segment of the cross-sectional circle of the measured small cylindrical workpiece, the final circularity value will become more accurate.

As shown in Fig. 2(a), a small cylindrical workpiece of diameter 3 mm and length 50 mm is placed in the V-groove and scanned linearly from left to right by the profilometer stylus. As shown in Fig. 2(b), the maximum measurable range of the stylus is 90°, which is the limit of the machine. When there are fewer than four equal segments, the cross-sectional circle of the workpiece cannot be scanned completely. Although the cross-sectional circle is segmented into four equivalent parts to be scanned, the stitching of neighboring arc contours cannot be carried out in this case, since there are no overlapping parts between neighboring arc contours for the stitching mark. Therefore, the segmenting should be considered first.

Figure 3(a) shows the round magnetic jig that holds the measured cylindrical workpiece and rotates it while it is segmented by the segmenting tool. As shown in Fig. 3(b), the workpiece can be segmented into two equal parts by carving the tool across the center of the jig. It can be segmented into two equal parts in a different way by rotating the jig through an angle α1 and carving again, as shown in Fig. 3(c). By repeating this procedure, three and four different ways of segmenting the workpiece into two equal parts can be achieved, as shown in Figs. 3(d) and 3(e), respectively. Therefore, the overall segmenting procedure can be summarized as follows:
(7)
To simplify the carving process, it is assumed that there are only two ways to segment the workpiece into two equal segments, and Eq. (7) then simplifies to
(8)
It should be noted that β and γ should be less than 60°, since sufficient overlapping of neighboring arc contours is required for correct matching. The carving procedure can be further simplified by taking β = γ, whereupon Eq. (8) simplifies further to
(9)
where θr is the rotation angle.
According to the discussion above, the segmenting patterns are as shown in Fig. 4. However, not all of these can be used for the following experiment. Figure 5 presents an analysis of the stitching of neighboring arc contours for different segmenting patterns. From the analysis of stitching of neighboring arc contours presented above, the relationship between the rotation angle θr, the overlap angle θo between neighboring arc contours, and the angle θe extracted from the scanning data can be written as
(10)
where no is the overlap number between the arc contours. As shown in Fig. 5(a), if the rotation angle θr = 60°, the overlap angle between neighboring arc contours is θo = 25°, which is not suitable for use for stitching, since there are not enough similar features available for matching by the cross-correlation function. If θr = 20°, then θo = 65°, which appears to be suitable for stitching. However, the overlapping of neighboring arc contours with this pattern is too complicated, as shown in Fig. 5(e). Therefore, the rotation angle θr should be set in an appropriate range that provides sufficient but not too complicated overlapping. Thus, finally, the three patterns shown in Figs. 5(b)–5(d) are selected for the proposed coordinate modeling segmenting method for circularity metrology.
FIG. 4.
Different segmenting patterns, with (a) 6, (b) 8, (c) 10, (d) 12, (e) 20, and (f) 24 equal segments.
View largeDownload slide

Different segmenting patterns, with (a) 6, (b) 8, (c) 10, (d) 12, (e) 20, and (f) 24 equal segments.

FIG. 4.
Different segmenting patterns, with (a) 6, (b) 8, (c) 10, (d) 12, (e) 20, and (f) 24 equal segments.
View largeDownload slide

Different segmenting patterns, with (a) 6, (b) 8, (c) 10, (d) 12, (e) 20, and (f) 24 equal segments.

Close modal
FIG. 5.
Analyses of stitching for stitching of (a) 6, (b) 8, (c) 10, (d), 12, and (e) 18 equal segments.
View largeDownload slide

Analyses of stitching for stitching of (a) 6, (b) 8, (c) 10, (d), 12, and (e) 18 equal segments.

FIG. 5.
Analyses of stitching for stitching of (a) 6, (b) 8, (c) 10, (d), 12, and (e) 18 equal segments.
View largeDownload slide

Analyses of stitching for stitching of (a) 6, (b) 8, (c) 10, (d), 12, and (e) 18 equal segments.

Close modal
The matching results for the three segmenting patterns are presented in Fig. 6. The stitched circularity contour, the circularity contour after integration of overlapping parts, and the circularity contour after low-pass filtering with 50 UPR are presented in Fig. 7. Three methods are adopted for evaluating the measurement quality of the three segmenting patterns: as shown in Fig. 8(a), the circumferential deviation of the overlapping parts of neighboring arc contours can be evaluated by the matching coefficient Ci∼i+1 between these overlapping parts given by Eq. (5); as shown in Fig. 8(b), the radial deviation of overlapping parts of neighboring arc contours can be evaluated by the Euclidean distance30 
(11)
where nc is the coordinate number of the scanned arc, θi,j is the angular position of the arc contour coordinate point, and Δri,j is the arc contour; and as shown in Fig. 8(c), the curvature of the measured arcs, calculated as
(12)
can be compared with the standard value. The results of these evaluations for the three segmenting patterns are shown in Table I. Under ideal circumstances, the overlapping parts of neighboring arc contours should overlap completely. In practice, however, there are some deviations between the overlapping parts, owing to measurement errors. In addition, the curvatures of these arc contours differ from the standard value. The closer the curvature is to the standard value, the better is the measurement quality.
FIG. 6.
Matching of neighboring arc contours for three segmenting patterns with (a) 8, (b) 10, and (c) 12 equal arc contours.
View largeDownload slide

Matching of neighboring arc contours for three segmenting patterns with (a) 8, (b) 10, and (c) 12 equal arc contours.

FIG. 6.
Matching of neighboring arc contours for three segmenting patterns with (a) 8, (b) 10, and (c) 12 equal arc contours.
View largeDownload slide

Matching of neighboring arc contours for three segmenting patterns with (a) 8, (b) 10, and (c) 12 equal arc contours.

Close modal
FIG. 7.
Circularity contours for three segmenting patterns with (a) 8, (b) 10, and (c) 12 equal segments.
View largeDownload slide

Circularity contours for three segmenting patterns with (a) 8, (b) 10, and (c) 12 equal segments.

FIG. 7.
Circularity contours for three segmenting patterns with (a) 8, (b) 10, and (c) 12 equal segments.
View largeDownload slide

Circularity contours for three segmenting patterns with (a) 8, (b) 10, and (c) 12 equal segments.

Close modal
FIG. 8.
Evaluation of measurement quality: (a) circumferential deviation between neighboring arc contours; (b) radial deviation of neighboring arc contours; (c) curvature of an arc contour.
View largeDownload slide

Evaluation of measurement quality: (a) circumferential deviation between neighboring arc contours; (b) radial deviation of neighboring arc contours; (c) curvature of an arc contour.

FIG. 8.
Evaluation of measurement quality: (a) circumferential deviation between neighboring arc contours; (b) radial deviation of neighboring arc contours; (c) curvature of an arc contour.
View largeDownload slide

Evaluation of measurement quality: (a) circumferential deviation between neighboring arc contours; (b) radial deviation of neighboring arc contours; (c) curvature of an arc contour.

Close modal
TABLE I.

Evaluation results for three segmenting patterns.

Segmenting patternAverage matching coefficientAverage Euclidean distance (µm)Average curvatureDiameter (mm)Circularity (µm)
8 equal parts 0.14 0.032 0.665 883 3.003 53 0.16 
10 equal parts 0.41 0.029 0.666 275 3.001 76 0.14 
12 equal parts 0.50 0.026 0.666 298 3.001 66 0.12 
Segmenting patternAverage matching coefficientAverage Euclidean distance (µm)Average curvatureDiameter (mm)Circularity (µm)
8 equal parts 0.14 0.032 0.665 883 3.003 53 0.16 
10 equal parts 0.41 0.029 0.666 275 3.001 76 0.14 
12 equal parts 0.50 0.026 0.666 298 3.001 66 0.12 
View Large

It can be seen from these results that with increasing number of segments, the average matching coefficient increases, the average Euclidean distance decreases, and the mean value of the curvature of the arc contours becomes closer to the standard value. This means that the circumferential and radial deviations can be reduced by increasing the number of segments of the arc. The mean value of the curvature can also be improved by increasing the number of segments. However, measurement efficiency needs to be taken into account here, since the greater the number of segments, the more time-consuming is the measurement procedure. Therefore, a comprehensive consideration is necessary to obtain an appropriate segmenting pattern for the circularity and diameter metrology of small cylindrical workpieces.

A coordinate modeling–segmenting method for circularity and diameter metrology of small cylindrical workpieces has been proposed that overcomes the alignment problem. The cross-sectional circle of the cylinder is segmented into several equivalent arcs for precision coordinate measurement by a two-dimensional coordinate measuring machine (profilometer). The contour and diameter of the obtained arcs are calculated by a data processing process. The circularity contour is formed by stitching these arc contours together. The arc segmenting by this method has been analyzed, and the measurement quality of three segmenting patterns with 8, 10, and 12 equal segments, respectively, have been experimentally evaluated in terms of the circumferential deviation, the radial deviation, and the curvature of the obtained arc contour. The results show that the measuring quality can be improved by increasing the number of arc segments, although it is also necessary to take account of measurement efficiency, since the greater the number of segments, the more time-consuming is the measurement procedure. Although cylindrical workpieces of diameter ≤ 3 mm can be measured precisely by the proposed method, which avoids the problems arise with the traditional rotary scan technique, the segmenting process needs to be analyzed in greater detail in the future work, since both the measurement quality and efficiency are affected by the number of arc segments used.

This work was supported by the National Defense Basic Scientific Research Program of China (Grant No. JCKY2019427D002).

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
Xie
YH
,
Xu
LX
,
Deng
YQ
.
A dynamic approach for evaluating the moment rigidity and rotation precision of a bearing-planetary frame rotor system used in RV reducer
.
Mech Mach Theory
2022
;
173
:
104851
.
https://doi.org/10.1016/j.mechmachtheory.2022.104851
Google Scholar
Crossref
Search ADS
 
2.
Liu
J
,
Yan
Z
,
Shao
Y
.
An investigation for the friction torque of a needle roller bearing with the roundness error
.
Mech Mach Theory
2018
;
121
:
259
–
272
.
https://doi.org/10.1016/j.mechmachtheory.2017.10.028
Google Scholar
Crossref
Search ADS
 
3.
Waghole
V
,
Tiwari
R
.
Optimization of needle roller bearing design using novel hybrid methods
.
Mech Mach Theory
2014
;
72
:
71
–
85
.
https://doi.org/10.1016/j.mechmachtheory.2013.10.001
Google Scholar
Crossref
Search ADS
 
4.
Weckenmann
A
,
Bruning
J
,
Patterson
S
,
Knight
P
,
Bryan
J
.
Grazing incidence interferometry for high precision measurements of cylindrical form deviations
.
CIRP Ann
2001
;
50
(
1
):
381
–
384
.
https://doi.org/10.1016/s0007-8506(07)62145-3
Google Scholar
Crossref
Search ADS
 
5.
Adamczak
S
,
Zmarzły
P
,
Janecki
D
.
Theoretical and practical investigations of V-block waviness measurement of cylindrical parts
.
Metrol Meas Syst
2015
;
22
(
2
):
181
–
192
.
https://doi.org/10.1515/mms-2015-0023
Google Scholar
Crossref
Search ADS
 
6.
Shimizu
Y
,
Li
Q
,
Kogure
M
, et al.
On-machine diameter measurement of a cylindrical workpiece with a reference artefact
.
Meas Sci Technol
2021
;
32
(
10
):
105012
.
https://doi.org/10.1088/1361-6501/ac0ace
Google Scholar
Crossref
Search ADS
 
7.
Zhang
L
,
Jiang
B
,
Zhang
P
, et al.
Methods for fatigue-life estimation: A review of the current status and future trends
.
Nanotechnol Precis Eng
2023
;
6
(
2
):
025001
.
https://doi.org/10.1063/10.0017255
Google Scholar
Crossref
Search ADS
 
8.
Gao
W
. Precision nanometrology.
London
:
Springer
.
2010
.
9.
Gao
W
. Metrology.
Gao
W
, ed.
Singapore
:
Springer
.
2019
.
10.
Gao
W
. Surface metrology for micro- and nanofabrication.
Elsevier
.
2020
.
11.
Gao
W
.
Precision nanometrology and its applications to precision nanosystems
.
Int J Precis Eng Manuf
2005
;
6
(
4
):
14
–
20
.
Google Scholar
 
12.
ISO 12181-1 2003
. Geometrical Product Specifications (GPS)-Roundness-Part 1: Vocabulary and Parameters of Roundness.
13.
Taylor Hobson Ltd.
A guide to the Measurement of Roundness Introduction to roundness.
14.
Taylor Hobson Ltd.
Roundness Measurement Equipment|Form Measurement|Cylindricity Measuring Instrument|Roundness Tester.
15.
Kühnel
M
,
Ullmann
V
,
Gerhardt
U
,
Manske
E
.
Automated setup for non-tactile high-precision measurements of roundness and cylindricity using two laser interferometers
.
Meas Sci Technol
2012
;
23
(
7
):
074016
.
https://doi.org/10.1088/0957-0233/23/7/074016
Google Scholar
Crossref
Search ADS
 
16.
Thalmann
R
. Basics of highest accuracy roundness measurement.
Queretaro, Mexico
:
Simposio de Metrologia
.
2006
.
17.
Surtronic R50-R80
, Taylor Hobson, www.taylor-hobson.com.
18.
Li
Q
,
Shimizu
Y
,
Wang
X
,
Li
X
,
Gao
W
.
High-accuracy roundness measurement of small cylindrical workpieces by a high-frequency filtering method
.
Precis Eng
2024
;
85
:
241
–
246
.
https://doi.org/10.1016/j.precisioneng.2023.10.010
Google Scholar
Crossref
Search ADS
 
19.
Chen
Y
,
Zhao
X
,
Gao
W
,
Hu
G
,
Zhang
S
,
Zhang
D
.
A novel multi-probe method for separating spindle radial error from artifact roundness error
.
Int J Adv Des Manuf Technol
2017
;
93
(
1-4
):
623
.
https://doi.org/10.1007/s00170-017-0533-5
Google Scholar
Crossref
Search ADS
 
20.
Gao
W
,
Haitjema
H
,
Fang
FZ
, et al.
On-machine and in-process surface metrology for precision manufacturing
.
CIRP Ann
2019
;
68
(
2
):
843
–
866
.
https://doi.org/10.1016/j.cirp.2019.05.005
Google Scholar
Crossref
Search ADS
 
21.
Bai
J
,
Wang
Y
,
Wang
X
, et al.
Three-probe error separation with chromatic confocal sensors for roundness measurement
.
Nanomanuf Metrol
2021
;
4
(
4
):
247
–
255
.
https://doi.org/10.1007/s41871-021-00120-8
Google Scholar
Crossref
Search ADS
 
22.
Cai
Y
,
Xie
B
,
Ling
S
,
Fan
KC
.
On-line measurement method for diameter and roundness error of balls
.
Nanomanuf Metrol
2020
;
3
(
3
):
218
–
227
.
https://doi.org/10.1007/s41871-020-00071-6
Google Scholar
Crossref
Search ADS
 
23.
Chen
YL
,
Machida
Y
,
Shimizu
Y
,
Matsukuma
H
,
Gao
W
.
A stitching linear-scan method for roundness measurement of small cylinders
.
CIRP Ann
2018
;
67
(
1
):
535
–
538
.
https://doi.org/10.1016/j.cirp.2018.04.009
Google Scholar
Crossref
Search ADS
 
24.
Li
Q
,
Shimizu
Y
,
Saito
T
,
Matsukuma
H
,
Gao
W
.
Measurement uncertainty analysis of a stitching linear-scan method for the evaluation of roundness of small cylinders
.
Appl Sci
2020
;
10
(
14
):
4750
.
https://doi.org/10.3390/app10144750
Google Scholar
Crossref
Search ADS
 
25.
Li
Q
,
Shimizu
Y
,
Saito
T
,
Matsukuma
H
,
Cai
Y
,
Gao
W
.
Improvement of a stitching operation in the stitching linear-scan method for measurement of cylinders in a small dimension
.
Appl Sci
2021
;
11
(
10
):
4705
.
https://doi.org/10.3390/app11104705
Google Scholar
Crossref
Search ADS
 
26.
Chernov
NLC
,
Lesort
C
.
Least squares fitting of circles
.
J Math Imaging Vis
2005
;
23
:
239
–
252
.
https://doi.org/10.1007/s10851-005-0482-8
Google Scholar
Crossref
Search ADS
 
27.
Konecny
J
,
Kromer
P
,
Prauzek
M
,
Musilek
P
.
Scan matching by cross-correlation and differential evolution
.
Electronics
2019
;
8
(
8
):
856
.
https://doi.org/10.3390/electronics8080856
Google Scholar
Crossref
Search ADS
 
28.
Gomez-Perez
D
,
Winterhof
A
.
A note on the cross-correlation of costas permutations
.
IEEE Trans Inf Theory
2020
;
66
(
12
):
7724
–
7727
.
https://doi.org/10.1109/TIT.2020.3009880
Google Scholar
Crossref
Search ADS
 
29.
Yoon
H
,
Hoskere
V
,
Park
JW
,
Spencer
BF
.
Cross-correlation-based structural system identification using unmanned aerial vehicles
.
Sensors
2017
;
17
(
9
):
2075
.
https://doi.org/10.3390/s17092075
Google Scholar
Crossref
Search ADS
PubMed
 
30.
Toledo
FJ
,
Galiano
V
,
Blanes
JM
,
Herranz
V
,
Batzelis
E
.
Photovoltaic single-diode model parametrization. An application to the calculus of the Euclidean distance to an I–V curve
.
Math Comput Simul
2024
;
225
:
794
.
https://doi.org/10.1016/j.matcom.2023.01.005
Google Scholar
Crossref
Search ADS
 

Qiaolin Li received his Ph.D. degree from Tohoku University in 2022. He is currently a postdoctoral researcher at Tsinghua University. His main research interest include precision metrology of fine mechanics, measurement uncertainty, and precision instruments.

Chuang Zeng received his Bachelor’s degree in Measurement and Control Technology and Instruments from Tianjin University in 2021, and is now pursuing a Master’s degree in Electronic and Information Engineering at Tsinghua University. His main research interest is precise measurement.

Jiali Zhao received his Ph.D. from Tianjin University of Machinery, Manufacturing, and Automation in 2007. He is currently a Professor and postgraduate supervisor at Lanzhou University of Technology. His main research interest include numerical control technology, precision measurement, and complex manufacturing process quality control.

Dan Wu received the B.S. degree in Financial Management from Lyuliang University, China, in 2021. She is currently a graduate student at Lanzhou University of Technology. Her main research interest is precision metrology, instrumentation and engineering technology.

Liang Zhang received the B.S. degree in Information Management and Information Systems from Hebei University of Science and Technology, China, in 2021. She is currently a graduate student at Lanzhou University of Technology. Her main research interest are uncertainty analysis, instrumentation, and engineering technology.

© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).