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

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## Abstract

**:**

_{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

## 2. Principles and Experiment

## 3. Measurement Uncertainty Analysis of the Linear Scan Method

#### 3.1. Mathematical Modeling

_{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.

_{i}) of the measured arc radius can be expressed 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 directionu(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.

_{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.

_{i}can be obtained by Equation (5), where R

_{i}is the radius of the arbitrary point of the measured arc and R

_{0}is the peak radius.

_{i}after the stitching process is calculated according to Equation (6).

_{i}is the radial deviation at any measured point and Δr

_{0}is the radial deviation at the arc apex.

_{i}) of the radial deviation of any measurement point can be expressed as:

_{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.

#### 3.2. Measurement Uncertainty Evaluation

- Uncertainty coefficient of the Z-axis coordinate z
_{i}.

- (1)
- Uncertainty of stylus calibration in Z-axis direction u(e
_{calibration_Z}).

- (2)
- Uncertainty due to stylus resolution u(e
_{resolution}).

- (3)
- Uncertainty of repeated measurements u(e
_{repeat}).

- 2.
- Uncertainty coefficient of the X-axis coordinate x
_{i}. - (1)
- Uncertainty of stylus calibration in the X-axis direction u(e
_{calibration_X}).

_{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.

- (2)
- Uncertainty due to position error around the Z-axis of the workpiece u(e
_{alignment_Z}).

_{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.

_{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).

- (3)
- Uncertainty due to position error around the X-axis of the workpiece u(e
_{alignment_Z}).

_{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.

_{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).

_{i}and the combined standard uncertainty of the X-axis coordinate x

_{i}are expressed by Equation (21).

_{i}) of the radius of the measured arc can be obtained from Equation (23). u(R

_{i}) can be obtained by Equation (24).

_{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.

#### 3.3. Variation of Uncertainty Due to Change in Workpiece Diameter

_{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.

_{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 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

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Linear scan with a stylus; (

**b**) small cylinder mounted on the V-groove and scanned linearly.

**Figure 2.**The stitching procedure: (

**a**) profile 1; (

**b**) profile 2; (

**c**) the stitching of profiles 1 and 2; (

**d**) the stitching of the rest of the arc profiles.

**Figure 3.**(

**a**) Schematic of the radius measurement and the uncertainty components; (

**b**) schematic of the roundness measurement and the uncertainty components.

**Figure 5.**(

**a**) Influence of attitude error of roll workpiece around Z-axis; (

**b**) influence of attitude error of roll workpiece around X-axis.

**Figure 6.**Variation of u(x

_{i}) in accordance with the diameter of workpiece: (

**a**) influence of u(Δe

_{alignment_Z}); (

**b**) influence of u(Δe

_{alignment_X}).

**Figure 7.**Variation of U(D) and U(Δzq) in accordance with diameter of workpiece: (

**a**) diameter ф 0.01–50 mm; (

**b**) diameter ф 0.01–10 mm.

**Figure 8.**(

**a**) Variation in U(D) with the diameter of the workpiece; (

**b**) variation in U(Δz

_{q}) in accordance with the diameter of workpiece.

Source of Uncertainty | Symbol | Type | Coverage Factor | Standard Uncertainty | Sensitivity Coefficient | |c_{i}| × u(x_{i})nm |
---|---|---|---|---|---|---|

Calibration of probe | u(e_{calibration_Z}) | A | ― | 33.7 | 1 | 33.7 |

Resolution | u(e_{resolution}) | B | $\sqrt{3}$ | 0.92 | 1 | 0.92 |

Repeatability | u(e_{repeat}) | A | ― | 18.95 | 1 | 18.95 |

Combined standard uncertainty | u(z_{i}) | ― | 38.45 |

Source of Uncertainty | Symbol | Type | Coverage Factor | Standard Uncertainty | Sensitivity Coefficient | |c_{i}| × u(x_{i})nm |
---|---|---|---|---|---|---|

Calibration of probe | u(e_{calibration_}_{X}) | A | ― | 33.7 | 1 | 33.7 |

Attitude error around Z-axis | u(e_{alignment_Z}) | B | $\sqrt{3}$ | 25.68 | 1 | 25.68 |

Attitude error around X-axis | u(e_{alignment_}_{X}) | B | $\sqrt{3}$ | 0.26 | 1 | 0.26 |

Combined standard uncertainty | u(x_{i}) | ― | 42.37 |

Source of Uncertainty | Symbol | Type | Coverage Factor | Standard Uncertainty | Sensitivity Coefficient | |c_{i}| × u(x_{i})nm |
---|---|---|---|---|---|---|

Output in Z-axis direction | u(z_{i}) | ― | ― | 38.45 | 1 | 38.45 |

Coordinate of X-axis | u(x_{i}) | ― | ― | 42.37 | 1 | 42.37 |

Combined standard uncertainty | u(R_{i}) | ― | 40.28 | |||

Radius after stitching process | $u(\overline{R})$ | ― | 14.24 | |||

Diameter | u(D) | ― | 28.48 | |||

Expanded uncertainty (k = 2) | U(D) | ― | 56.96 |

Attitude Error Angle θ _{Z} 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(Δz_{q})) | - | 23.34 | 8.44 | 5.86 | 4.31 | 3.30 | 2.60 | 2.11 | 0.93 | 0.52 |

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## Share and Cite

**MDPI and ACS Style**

Zhao, J.; Zhang, L.; Wu, D.; Shen, B.; Li, Q.
Measurement Uncertainty Analysis of the Stitching Linear-Scan Method for the Measurable Dimension of Small Cylinders. *Appl. Sci.* **2023**, *13*, 9091.
https://doi.org/10.3390/app13169091

**AMA Style**

Zhao J, Zhang L, Wu D, Shen B, Li Q.
Measurement Uncertainty Analysis of the Stitching Linear-Scan Method for the Measurable Dimension of Small Cylinders. *Applied Sciences*. 2023; 13(16):9091.
https://doi.org/10.3390/app13169091

**Chicago/Turabian Style**

Zhao, Jiali, Liang Zhang, Dan Wu, Bobo Shen, and Qiaolin Li.
2023. "Measurement Uncertainty Analysis of the Stitching Linear-Scan Method for the Measurable Dimension of Small Cylinders" *Applied Sciences* 13, no. 16: 9091.
https://doi.org/10.3390/app13169091