Measurement Uncertainty Analysis of the Stitching Linear-Scan Method for the Measurable Dimension of Small Cylinders

Abstract

A stitching linear scan method is proposed for roundness and diameter measurement of small cylindrical workpieces instead of the conventional rotary scan method due to the crucial alignments of eccentricity and inclination. To verify the reliability of the proposed method, by which the coordinates of the cross-sectional circle of a small cylinder is divided into several equal parts to be obtained and reconstructed, the diameter and roundness measurement uncertainties of the small cylinders with a diameter 1.5 mm are evaluated to be 0.047 μm and 0.095 μm, respectively, which can meet the uncertainty target of 0.1 μm. To investigate the measurable dimension by the proposed method, measurement uncertainty analysis of the small cylinders with various dimensions has been conducted according to the previous evaluation, since all the procedures are the same except for the uncertainty of X coordinates, which changes with the measured dimension’s change. The results show that the small cylinders with a diameter range from 0.01 mm to 50 mm can be measured by the proposed method when the position error θ_Z is reduced to the corresponding value. There is no measuring limitation set by the proposed model theoretically in the case of θ_X = θ_Z = 0.1°, while the machine has a measuring limitation.

1. Introduction

Fine mechanics, such as RV reducers, play a significant role in industrial robotics and machine tool spindles. There are many cylindrical parts with various diameters employed in RV reducers. The performance and life of an RV reducer are affected by the quality of these cylindrical parts very much [1,2,3,4,5]. The parameters of roundness and diameter are always used to evaluate the quality of the cylindrical parts. The more precise the parameter is, the better the quality is. To enhance the quality of the cylindrical parts, precision measurement is necessary [6,7,8,9,10,11]. Conventionally, roundness measurement can be carried out by the rotary scan method with a roundness measuring machine [12,13,14,15]. In recent years, some alternative roundness measurement methods have been proposed, such as the multiple probe method and online method [16,17,18,19]. However, due to the crucial alignments (inclination and eccentricity) of the rotary scan method, precision roundness measurement become difficult when the diameter of the measured workpiece becomes smaller. To address this issue, the stitching linear scan method is proposed [20,21,22]. In this method, the coordinates of the cross-sectional circle of the cylindrical workpiece, which is attached to a round magnetic jig and mounted on a V-groove, are divided into several equal arcs to be scanned linearly by a profilometer. A set of arc coordinate data can be obtained after one linear scan on the surface of the cylindrical workpiece. The measured cylindrical workpiece is rotated by an equal angle to a new measuring position by rotating the magnetic jig and then scanned again. Another set of arc data can be obtained. We then repeat the procedure several times till the entire cross-sectional circle of the cylindrical workpiece is scanned completely, then a series of arcs can be obtained. The arc profiles, which can be characterized according to the obtained arcs, are used to stitch into a roundness profile. Through the stitching angle error compensation, overlap parts integration, and filter processing, an accurate, integral, smooth, continuous roundness profile can be obtained [20,21,22].

Although small cylindrical parts can be measured by the stitching linear scan method, the measurable range of this method has not been analyzed. In this paper, measurement uncertainty is used to verify the reliability of the stitching linear scan method for the roundness measurement of small cylindrical parts with various diameters. Due to ambiguous random error and system error, the concept of measurement uncertainty was proposed in 1993. The measured result can be reasonably and quantitively evaluated according to the GUM (Guide to the Expression of Uncertainty in Measurement), which is actually an error propagation evaluation [23,24,25]. Measurement uncertainty is necessary for multiple disciplines, such as air gauge back-pressure uncertainty estimation for advanced test rigs and precision positioning measurement uncertainty [26,27]. In this paper, the analysis of measurement uncertainty is presented and the measurable range of the cylindrical workpiece can be confirmed.

2. Principles and Experiment

As is known to all, the surface form of a measured workpiece can be modeled and reconstructed by coordinates. In this paper, the coordinates of the cross-sectional circle of a small cylindrical workpiece are divided into several equal arcs to be scanned by a profilometer. The diameter and center coordinates of these arcs can be fitted by the least square method. The arc profiles can be characterized according to the arc data. The roundness profile of the small cylindrical part can be obtained by stitching these arc profiles one by one in sequence.

As shown in Figure 1a, a small cylindrical workpiece with a diameter of 1.5 mm and length of 7.8 mm is attached to the circular magnetic jig, which is marked by eight equal lines on the surface. This kind of combination is mounted on the V-groove. As shown in Figure 1b, the stylus of the profilometer is brought to scan on the surface of the small cylindrical workpiece and returned to the initial position, after which the first arc coordinate data can be obtained. The small cylindrical workpiece is rotated by 45° by rotating the circular magnetic jig by one equal part. The stylus scans on the surface of the workpiece again and returns, then the second set of arc coordinate data can be obtained. We repeat the procedure seven times, then eight sets of arc coordinate data can be obtained. Since the maximum measuring inclination of the stylus is ±45°, only 85° arc coordinate data are extracted for the following data processing.

The eight arc profiles with respect to the arc coordinate can be characterized by a series of data processes. An inaccurate roundness profile can be formed by stitching these eight arc profiles together. Regarding the stitching procedure, as shown in Figure 2a,b, arc profiles 1 and 2 are at the initial position. As shown in Figure 2c, the first arc profile is kept static and the second arc profile is rotated by 45°, since there is a 45° angular displacement between adjacent arc profiles according to the measuring procedure. As shown in Figure 2d, the second arc profile is kept static, and the third arc profile is rotated by 45°, while the rest of the arc profiles are stitched in this manner. However, there are some stitching angle errors between adjacent arc profiles, since the small cylindrical workpiece is rotated manually, namely, the angular displacement between adjacent arc profiles is not always 45°. Therefore, stitching angle error compensation is necessary. An accurate, integral, smooth, continuous roundness profile can be obtained after stitching error compensation, overlap part combination, and low-pass filtering.

3. Measurement Uncertainty Analysis of the Linear Scan Method

3.1. Mathematical Modeling

The uncertainty of roundness and diameter measurements of a small cylinder with a diameter of 1.5 mm and length of 7.8 mm by the stitching linear scan method is carried out at first. Mathematical modelling was conducted to calculate the measurement uncertainty of the diameter and roundness. The arc coordinate (x_i, z_i) in the rectangular coordinate system can be obtained by a profilometer with its stylus. The mathematical model used for radius calculation and the uncertainty factors deriving from the measurements are shown in Figure 3a. The radius of each arc can be fitted by the least square method. Radius R_i of an arbitrary measuring point can be expressed by Equation (1), where (x_i, z_i) is the arc coordinate.

$$R_i=\sqrt{{x_i}^2+{z_i}^2}$$

(1)

As derived from Equation (1), the combined standard uncertainty u(R_i) of the measured arc radius can be expressed as follows:

$$u^2\left(R_i\right)=\left(\frac{\partial R_i}{\partial x_i}\right)u^2\left(x_i\right)+\left(\frac{\partial R_i}{\partial z_i}\right)u^2\left(z_i\right)=\frac{{x_i}^2}{{z_i}^2+{x_i}^2}{u}^2\left(x_i\right)+\frac{{z_i}^2}{{z_i}^2+{x_i}^2}{u}^2(z_i)$$

(2)

The uncertainties in the figure are as follows.

(1)

Uncertainty of output z_i in the Z-axis direction.

u(e_{calibration_Z}): Uncertainty of stylus calibration in the Z-axis direction.

u(e_resolution): Uncertainty due to stylus resolution.

u(e_repeat): Uncertainty of repeatability.

(2)

Uncertainty of output x_i in the X-axis direction

u(e_{calibration_X}): Uncertainty of stylus calibration in the X-axis direction.

u(e_{alignment_Z}): Uncertainty due to position error around the Z-axis of the workpiece.

u(e_{alignment_X}): Uncertainty due to position error around the X-axis of the workpiece.

In summary, the combined standard uncertainty z_i of the Z-axis output and the combined standard uncertainty of the X-axis coordinate x_i are expressed by Equations (3) and (4), respectively.

$$u^2\left(z_i\right)=u^2\left(e_{calibratio{n\_}_z}\right)+u^2\left(e_{resolution}\right){+u}^2\left(e_{repeat}\right)$$

(3)

$$u^2\left(x_i\right)=u^2\left(e_{calibation\_X}\right)+u^2\left(e_{alignment\_Z}\right)+u^2\left(e_{alignment\_X}\right)$$

(4)

For the calculation of roundness, it is assumed that there is a deviation from the approximate center of the circle, rather than the radian of a perfect circle, and the approximate center of the circle is set as the origin. The output of z_i can be obtained by Equation (5), where R_i is the radius of the arbitrary point of the measured arc and R₀ is the peak radius.

$$z_i=R_0-\sqrt{{R_i}^2-{x_i}^2}$$

(5)

The radius R_i after the stitching process is calculated according to Equation (6).

$$u\left(∆zq\right)=u\left({∆r\left(\theta \right)}_{max}\right)+u\left({∆r\left(\theta \right)}_{min}\right)$$

(6)

Δr_i is the radial deviation at any measured point and Δr₀ is the radial deviation at the arc apex.

$$R_i=\overline{R}+∆r_i(\overline{R}\gg ∆r_i)$$

(7)

$$R_0=\overline{R}+∆r_0(\overline{R}\gg ∆r_0)$$

(8)

We square both sides of Equation (5), substitute this into Equations (7) and (8), and rearrange.

$$∆r_i=\frac{2\overline{R}{z}_i-{z_i}^2-{x_i}^2}{2\overline{R}}$$

(9)

From Equation (9), the combined standard uncertainty u(Δr_i) of the radial deviation of any measurement point can be expressed as:

$$\begin{gathered} u^2\left(∆r_i\right)={\left(\frac{\partial ∆r_i}{\partial \overline{R}}\right)}^2{u}^2\left(\overline{R}\right)+{\left(\frac{\partial ∆r_i}{\partial x_i}\right)}^2{u}^2\left(x_i\right)+{\left(\frac{\partial ∆r_i}{\partial z_i}\right)}^2{u}^2\left(z_i\right) \\ ={\left(\frac{{x_i}^2+{z_i}^2}{2{\overline{R}}^2}\right)}^2{u}^2\left(\overline{R}\right)+{\left(-\frac{x_i}{\overline{R}}\right)}^2{u}^2(x_i)+{\left(1-\frac{z_i}{R}\right)}^2{u}^2(z_i) \end{gathered}$$

(10)

The uncertainty factors shown in Figure 3b are similar to the radius measurement model stated above. Therefore, the combined standard uncertainty z_i for the Z-axis output and the combined standard uncertainty for the X-axis coordinate x_i are expressed by Equations (6) and (7), respectively. The uncertainty of the radius after the stitching process is expressed by Equation (11), where u(R_i) is the combined standard uncertainty of the radius obtained from Equation (6) and N is the dividing number.

$$u\left(\overline{R}\right)=\frac{u(R_i)}{\sqrt{N}}=\frac{u(R_i)}{\sqrt{8}}$$

(11)

3.2. Measurement Uncertainty Evaluation

The standard uncertainty of each uncertainty factor can be analyzed and obtained according to GUM.

• Uncertainty coefficient of the Z-axis coordinate z_i.

(1)

Uncertainty of stylus calibration in Z-axis direction u(e_{calibration_Z}).

The calibration of the employed Form Talysurf PGI420 is conducted with the dedicated master sphere shown in Figure 4a. The Pt value, which is the minimum and maximum width after linear expansion of the arc measured at the known radius values shown in Figure 4b, includes the shape error of the master ball, the shape error of the stylus tip diameter, the electrical noise of the system, and mechanical and environmental disturbances. The Pt value = 0.0337 μm exists in the radial direction, but assuming that it exists in the Z-axis direction and the X-axis direction, respectively, the uncertainty of the probe calibration in the Z-axis direction can be expressed by the following equation.

$$u\left(e_{calibration\_Z}\right)=0.0337 \mathsf{\mu }\mathrm{m}=33.7\mathrm{n}\mathrm{m}$$

(12)

(2)

Uncertainty due to stylus resolution u(e_resolution).

The stylus of the Form Talysurf PGI420 has a vertical resolution of 3.2 nm. Since the uncertainty due to the resolution of the probe is considered to be a square distribution of ±1.6 nm, it is obtained by the following equation using B-type evaluation.

$$u\left(e_{resolution}\right)=\frac{3.2/2}{\sqrt{3}}=0.92 \mathrm{n}\mathrm{m}$$

(13)

(3)

Uncertainty of repeated measurements u(e_repeat).

The uncertainty due to the measurement error can be evaluated by the results of the repeated measurements of the small cylinder with a diameter 1.5 mm. Since it has a repeatability of 59.93 nm from the results of 10 repeated measurements, the uncertainty of repeated measurements can be obtained by the following formula using A type evaluation.

$$u\left(e_{repeat}\right)=\frac{59.93}{\sqrt{10}}=18.95 \mathrm{n}\mathrm{m}$$

(14)

2.

Uncertainty coefficient of the X-axis coordinate x_i.

(1)

Uncertainty of stylus calibration in the X-axis direction u(e_{calibration_X}).

When picking up the output z_i in the Z-axis direction as the uncertainty coefficient, it is assumed that Pt values exist in the Z-axis and X-axis directions, respectively, so the uncertainty of the stylus calibration in the X-axis direction is the same as that in the X-axis direction, expressed by the following equation.

$$u\left(e_{calibration\_X}\right)=33.7 \mathrm{n}\mathrm{m}$$

(15)

(2)

Uncertainty due to position error around the Z-axis of the workpiece u(e_{alignment_Z}).

The uncertainty due to the workpiece position error can be obtained according to B type evaluation by using the value of the calculated measurement error, which can be obtained from the mathematical modeling. The influence of the position error around the Z-axis of the measured cylinder is shown in Figure 5a, in which D is the diameter of the workpiece, R is the radius, (x_edge, z_edge) is the end point coordinate of the obtained arc, and θ_Z is the angle of the position error around the Z-axis. The position error around the Z-axis causes the measurement result of the geometric circle of the geometric cylinder to appear as an ellipse, with its main axis in the X-axis direction. Since the arc of φ = 85°, that is, the arc of height h from the vertex shown in Equation (17), is extracted regardless of the size of the workpiece diameter, an error occurs in the X-axis coordinate. If the error angle is ±θ_Z° and this error has a rectangular distribution, the uncertainty due to the position error around the Z-axis can be represented by the following equation.

$$h=R\times (1-\cos \frac{\phi }{2})$$

(16)

$$u\left(e_{alignment\_Z}\right)=\frac{x_{edge}}{\cos \frac{{\theta }_Z}{\sqrt{3}}}-x_{edge}=R\sin \frac{\phi }{2}(\frac{1}{\cos \frac{{\theta }_Z}{\sqrt{3}}}-1)$$

(17)

In the actual measurement, the alignment of the rotating stage and the scanning of the stylus were repeated to converge the obtained arc length to the minimum, and the measurement was performed with the error angle θ_Z being as small as possible. Assuming that the error angle at this time is ±1°, the uncertainty due to the attitude error around the Z-axis can be obtained as follows, by substituting the radius after stitching into Equation (18).

$$u\left(e_{alignment\_Z}\right)=0.74940\times \sin \frac{85°}{2}\left(\frac{1}{\cos \frac{1°}{\sqrt{3}}}-1\right)=0.00002568 \mathrm{m}\mathrm{m}=25.68 \mathrm{n}\mathrm{m}$$

(18)

(3)

Uncertainty due to position error around the X-axis of the workpiece u(e_{alignment_Z}).

The influence of the position error around the X-axis of the measured cylinder is shown in Figure 5b, in which D is the workpiece diameter, R is the radius, the coordinates (x_edge, z_edge) are the end point of the obtained arc, and θ_X is the angle of position error, which causes measurements of geometric circles to geometric cylinders to appear as ellipses, with the major axis along the Z-axis. Since the arc with height h from the vertex is extracted in the same way as when there is a posture error around the Z-axis, an error occurs in the X-axis coordinate. Assuming that the error angle is θ_X, which forms a rectangular distribution, the uncertainty due to the position error around the X-axis is expressed by the following equation.

$$u\left(e_{alignment\_X}\right)=x_{edge}-x_{edge}\cos \frac{{\theta }_X}{\sqrt{3}}=R\sin \frac{\phi }{2}(1-\cos \frac{{\theta }_X}{\sqrt{3}})$$

(19)

In the actual measurement, the manual stage was moved in the Y-axis direction while the stylus was in contact with the flat surface of the mounting table, and alignment was performed using the tilting stage so that the change in displacement output was minimized. As a result, the error angle θ_X was measured as small as possible. Assuming that the error angle at this time is ±0.1°, the uncertainty due to the attitude error around the X-axis is obtained as follows, by substituting the radius after stitching processing into Equation (20).

$$u\left(e_{alignment\_X}\right)=0.74940\times \sin \frac{85°}{2}\left(1-\cos \frac{1°}{\sqrt{3}}\right)=0.00000257 \mathrm{m}\mathrm{m}=0.26 \mathrm{n}\mathrm{m}$$

(20)

Table 1. Uncertainty budget of the output in the Z-axis direction.

Source of Uncertainty	Symbol	Type	Coverage Factor	Standard Uncertainty	Sensitivity Coefficient	|ci| × u(xi) nm
Calibration of probe	u(ecalibration_Z)	A	―	33.7	1	33.7
Resolution	u(eresolution)	B	$\sqrt{3}$	0.92	1	0.92
Repeatability	u(erepeat)	A	―	18.95	1	18.95
Combined standard uncertainty	u(zi)	―				38.45

and

Table 2. Uncertainty budget of the X-axis coordinate.

Source of Uncertainty	Symbol	Type	Coverage Factor	Standard Uncertainty	Sensitivity Coefficient	|ci| × u(xi) nm
Calibration of probe	u(ecalibration_X)	A	―	33.7	1	33.7
Attitude error around Z-axis	u(ealignment_Z)	B	$\sqrt{3}$	25.68	1	25.68
Attitude error around X-axis	u(ealignment_X)	B	$\sqrt{3}$	0.26	1	0.26
Combined standard uncertainty	u(xi)	―				42.37

show a summary of the calculation results for each uncertainty. From these results, the combined standard uncertainty of the Z-axis direction output z_i and the combined standard uncertainty of the X-axis coordinate x_i are expressed by Equation (21).

$$\begin{gathered} u\left(z_i\right)=\sqrt{u^2\left(e_{calibratio{n}_{\_Z}}\right)+u^2\left(e_{resolution}\right)+u^2\left(e_{repeat}\right)} \\ =\sqrt{{33.7}^2+{0.92}^2+{18.95}^2}=38.45 \mathrm{n}\mathrm{m} \end{gathered}$$

(21)

$$\begin{gathered} u\left(x_i\right)=\sqrt{u^2\left(e_{calibration\_X}\right)+u^2\left(e_{alignment\_Z}\right)+u^2\left(e_{alignment\_X}\right)} \\ =\sqrt{{33.7}^2+{25.68}^2+{0.26}^2}=42.37 \mathrm{n}\mathrm{m} \end{gathered}$$

(22)

Using the calculated uncertainties, the combined standard uncertainty u(R_i) of the radius of the measured arc can be obtained from Equation (23). u(R_i) can be obtained by Equation (24).

$$u\left(∆r\left(\theta \right)\right)=u\left(m\left(\theta \right)\right)=26.92 \mathrm{n}\mathrm{m}$$

(23)

$$\begin{array}{l}u\left(R_i\right)=\sqrt{\frac{{x_i}^2}{{z_i}^2+x_i^2}{u}^2\left(x_i\right)+\frac{{z_i}^2}{{z_i}^2+x_i^2}{u}^2\left(z_i\right)}\\ =\sqrt{\frac{{0.50575}^2}{{0.55280}^2+{0.50575}^2}\times {42.37}^2+\frac{{0.55280}^2}{{0.55280}^2+{0.50575}^2}\times {38.45}^2}\\ =40.28 \mathrm{n}\mathrm{m}\end{array}$$

(24)

The uncertainty of the radius after the stitching process can be obtained by substituting the values obtained in Equation (13) into Equation (11) to obtain the following equation.

$$u\left(\overline{R}\right)=\frac{u(R_i)}{\sqrt{8}}=\frac{40.28}{\sqrt{8}}=14.24 \mathrm{n}\mathrm{m}$$

(25)

The uncertainty of the diameter measurement is twice the uncertainty of the radius.

$$u\left(D\right)=2\times u\left(R_i\right)=2\times 14.24=28.48 \mathrm{n}\mathrm{m}$$

(26)

The expanded uncertainty can be calculated when k = 2.

$$U\left(D\right)=ku\left(D\right)=2\times 28.48=56.96 \mathrm{n}\mathrm{m}$$

(27)

As can be seen above, the uncertainty of the diameter measurement of the 1.5 mm small cylindrical workpiece using the linear scanning method was estimated to be ±47.24 nm, achieving a target measurement uncertainty within ±0.1 μm.

Table 3. Uncertainty budget of diameter measurement by the proposed method.

Source of Uncertainty	Symbol	Type	Coverage Factor	Standard Uncertainty	Sensitivity Coefficient	|ci| × u(xi) nm
Output in Z-axis direction	u(zi)	―	―	38.45	1	38.45
Coordinate of X-axis	u(xi)	―	―	42.37	1	42.37
Combined standard uncertainty	u(Ri)	―				40.28
Radius after stitching process	$u(\overline{R})$	―				14.24
Diameter	u(D)	―				28.48
Expanded uncertainty (k = 2)	U(D)	―				56.96

summarizes the results of the standard uncertainty.

Using the calculated uncertainty, the combined standard uncertainty u(Δr_i) of the radial deviation of an arbitrary measured point on the arc can be obtained from Equation (10). Substituting the combination of (x_i, m_i), the coordinates of the endpoint of the arc (x_edge, z_edge) = (0.50575, 0.55280) and the radius after the stitching process, for which u obtained from the measurement u(Δr_i) is maximum, the following equation is obtained.

$$\begin{array}{l}u\left({∆r}_i\right)=\sqrt{{\left(\frac{x_i^2+z_i^2}{{2\overline{R}}^2}\right)}^2{u}^2\left(\overline{R}\right)+{\left(-\frac{x_i}{\overline{R}}\right)}^2{u}^2\left(x_i\right)+{\left(1-\frac{z_i}{\overline{R}}\right)}^2{u}^2\left(z_i\right)}\\ =\sqrt{{\left(\frac{{0.50575}^2+{0.55280}^2}{{2\times 0.74940}^2}\right)}^2\times {14.24}^2+{\left(-\frac{0.50575}{0.74940}\right)}^2\times {42.37}^2+{\left(1-\frac{0.55280}{0.74940}\right)}^2\times {38.45}^2}\\ =34.84 \mathrm{n}\mathrm{m}\end{array}$$

(28)

Using the same assumptions as in Equations (6) and (29), the combined standard uncertainty u(Δzq) of roundness can be expressed as follows.

$$u\left(∆zq\right)=2u(∆r(\theta ))$$

(29)

$$u\left({∆z}_q\right)=2u(∆r_i)$$

(30)

Substituting the uncertainty calculated in Equation (26), the combined standard uncertainty u(Δzq) is obtained from Equation (27), as shown in Equation (31) below.

$$u\left({∆z}_q\right)=2u\left(∆r_i\right)=2\times 34.84=69.68 \mathrm{n}\mathrm{m}$$

(31)

The expanded uncertainty can be obtained when k = 2 shown in Equation (32).

$$U\left(∆z_q\right)=ku\left({∆z}_q\right)=2\times 69.68=139.36 \mathrm{n}\mathrm{m}$$

(32)

As can be seen above, the uncertainty in the roundness measurement of a small cylindrical workpiece with a diameter of 1.5 mm was estimated to be ±95.44 nm using the linear scanning method, achieving a target measurement uncertainty of ±0.1 μm or less. Table 3 summarizes the results of the standard uncertainty.

3.3. Variation of Uncertainty Due to Change in Workpiece Diameter

For the purpose of examining the cylindrical work diameter suitable for roundness measurement using the profilometer, which is the stitching linear scan method, a simulation is performed to calculate the measurement uncertainty when the work diameter is changed. Of the uncertainty factors in (1) to (3) of 1 and (1) to (3) of 2, shown in the previous section, the ones that change significantly depending on the workpiece diameter are the uncertainty u(e_{alignment_Z}) due to the workpiece orientation error around the Z-axis and the uncertainty u(e_{alignment_X}) due to the workpiece orientation error around the X-axis. Figure 6a,b show the graphs of the uncertainty u(x_i) of the X-axis coordinate when the workpiece diameter is varied from 0.01 mm to 50.00 mm using Equations (13) and (14). From Figure 6a, as the diameter increases, u(e_{alignment_Z}) increases and becomes almost the same value as u(x_i), but the amount of change in u(e_{alignment_X}) is small. In other words, u(e_{alignment_Z}) is the dominant factor for changes in u(x_i). This is because the attitude error angles θ_Z = ±1° and θ_X = ±0.1° are set in consideration of the alignment method and stylus resolution, and θ_Z is estimated to be larger than θ_X. Figure 7a shows a graph showing changes in diameter expanded uncertainty U(D) and roundness expanded uncertainty U(Δzq) created by substituting u(x_i), which varies depending on the workpiece diameter, and other standard uncertainties. It can be seen that both U(D) and U(Δzq) increase as the workpiece diameter increases due to the influence of u(x_i). Figure 7b shows an enlarged graph of the work diameter range from 0.01 mm to 10 mm. From this graph, when the attitude error angle θ_Z = ±1° and θ_X = ±0.1°, it can be read that the conditions for the workpiece diameter that can theoretically achieve the target measurement uncertainty within ±0.1 μm are a U(D) of 5.58 mm or less and a U(Δzq) of 2.11 mm or less.

In the previous simulation, the position error angles were set to θ_Z = ±1° and θ_X = ±0.1°, but improving the alignment increases the upper limit of the workpiece diameter that satisfies the target measurement uncertainty. There is room for improvement in the position error angle θ_Z around the Z-axis. Therefore, Figure 8a,b show the changes in the expanded uncertainty of diameter U(D) and the expanded uncertainty of roundness U(Δzq) when the attitude angle error occurs in a rectangular distribution of θ_Z = ±0.1°, ±0.3°, ±0.5°, ±0.6°, ±0.7°, ±0.8°, ±0.9°, ±1°, ±1.5° and ±2°.

Table 4. Limits of diameter that can be achieved.

Attitude Error Angle θZ deg.	0.1	0.3	0.5	0.6	0.7	0.8	0.9	1	1.5	2
Work Diameter Upper Limit mm (U(D))	-	61.65	22.31	15.50	11.39	8.72	6.89	5.58	2.48	1.39
Work Diameter Upper Limit mm (U(Δzq))	-	23.34	8.44	5.86	4.31	3.30	2.60	2.11	0.93	0.52

summarizes the upper limit of the workpiece diameter that can achieve the target uncertainty of diameter and roundness within ±0.1 μm at each attitude error angle θ_Z. From Table 4, we see that it is desirable to converge θ_Z as small as possible by more accurate alignment when measuring cylindrical workpieces with a large diameter. It can be confirmed that the proposed method can cover all diameters of less than ϕ 3 mm, which are difficult to measure using the rotational scanning measurement method, as long as θ_Z is kept smaller than ±0.8°. Furthermore, if θ_Z can be made smaller, the proposed method can be used for workpieces with a larger diameter. If alignment is possible up to θ_Z = θ_X = ±0.1°, there is no limitation on the workpiece diameter theoretically. However, as the workpiece diameter increases, the measurement range in the Z-axis and X-axis directions expands, so it is thought that the upper limit of the workpiece diameter will be reached due to the increase in uncertainty and the limitation of the measurement range of the measuring equipment used.

3.4. Discussion

For the precision roundness measurement of cylindrical parts, the conventional rotary scan method cannot meet the requirements. Therefore, the stitching linear scan method has been proposed. Since the repeatability of measurement is very important, large numbers of experiments are necessary [28]. For practical applications, some cylindrical parts, such as the motor cylinders [29], can be measured by the stitching linear scan method. Meanwhile, the reliability of the proposed method can be verified in practical application. Contact measurements with a stylus or probe may not be accurate due to the measuring force and deformations, therefor it is necessary to develop non-contact measurements with optical linear encoder technology [30,31]. Furthermore, the proposed method and the verification manner can be applied for the measurement or testing of cylindrical parts or round parts made of other materials, such as functionally graded plates, which play a significant role in building materials [32,33]. Measurement uncertainty is necessary for testing, inspection and precision measurement. For the verification of the reliability of the proposed method, the measurement of uncertainty is carried out in this paper. The upper limitation of the proposed method can be investigated by the analysis and applications in a variety of fields.

4. Conclusions

The roundness profile of the small cylindrical parts can be reconstructed by coordinate modeling in the stitching linear scan method. To verify the reliability of the method, the measurement uncertainty of a small cylinder with a diameter 1.5 mm and length 7.8 mm was evaluated to be 0.095 μm, which satisfies the target uncertainty of 0.1 μm. According to the analysis, it can be determined that the uncertainty due to the position error around the Z-axis is the main uncertainty source of the X coordinate, after which the measurement uncertainties of the workpieces with a diameter from 0.01 mm to 50 mm can be analyzed. As a result, all the small cylinders with the dimensions mentioned above can be measured with a measurement uncertainty less than 0.1 μm when the alignment is improved properly. Theoretically, there is no limitation to the stitching linear scan method, while the employed profilometer has a measuring limitation. Therefore, a machine with a large measuring range will be used for investigating the limitation of the stitching linear scan method for roundness measurements of cylindrical workpieces. In addition, the measuring efficiency by the stitching linear scan method should be enhanced as well.