# Evaluation and Optimization of Task-oriented Measurement Uncertainty for Coordinate Measuring Machines Based on Geometrical Product Specifications

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## Abstract

**:**

## 1. Introduction

## 2. Evaluation Method

#### 2.1. Source Analysis of Uncertainty

_{L, MPE}) is mainly related to the errors of the distance and other dimensional elements, and the MPE

_{P}indicates the error of the whole measurement system in a very small test space, generally affecting the form measurement.

#### 2.2. Uncertainty Evaluation Model

## 3. Modeling for Typical Task Uncertainty Evaluation

#### 3.1. Uncertainty Model for Dimensional Measurement Task

_{L,MPE}is used to express the CMM ability for dimensional measurement. Therefore, the uncertainty component ${u}_{\mathrm{E}}$ caused by the bias and linearity of the CMM dimensional measurement tasks is as follows:

#### 3.2. Uncertainty Model for Form Error Measurement Task

_{P}and is calibrated by the sphericity of the standard ball, which essentially reflects the comprehensive impact of the residual system errors on the form measurement results in different directions and at different positions. It is more reliable for using MPE

_{P}to evaluate the uncertainty component caused by the indication errors of the CMM form error measurement; the formula for qualification is as follows:

#### 3.3. Uncertainty Model for Location and Orientation Errors Measurement Tasks

_{L, MPE}=A+B·L may be used to represent the influence of its indication error. However, the location and orientation errors are still greatly different from dimensional measurement due to the impacts of the instrument offset and linearity; different types of position errors are differently affected by the offset and linearity of the measuring instrument. Here follows the analysis and discussion of the uncertainty components for position errors caused by the indication errors of the measuring instrument.

_{L, MPE}and the error range of the overestimated parallelism tolerance t to get:

_{1}and L

_{2}indicate the distances from the measured elements to the fitting elements.

_{1}and L

_{2}both are miniature dimensions, not considering the influence of linearity but only the constant term of the measuring instrument deviation. The uncertainty ${u}_{{\mathrm{EW}}_{2}}$ of angularity and perpendicularity measurement tasks caused by the indication errors is as follows according to uncertainty type B evaluation method:

_{L, MPE}=A+B·L.

## 4. Method for Optimizing Measurement Uncertainty

#### 4.1. Secondary Optimal Evaluation of Uncertainty Components

_{L = 60, MPE}and MPE

_{P}calibration results by using standard gauge blocks and master balls with calibration uncertainty able to be neglected. It is shown that the uncertainty component introduced by the indication errors after secondary evaluation has been significantly reduced compared with the initial evaluation.

#### 4.2. Real-Time Updating of Repeatability Uncertainty Component

## 5. Experimental Analysis

#### 5.1. Example for Evaluating Uncertainty of Diameter Measurement

#### 5.2. Example for Evaluating Uncertainty of Flatness Measurement

_{r}= 0.618 μm; if the average value of three times’ measurements is taken as the best estimate of the flatness measurement, then the standard uncertainty caused by measurement repeatability is as follows:

#### 5.3. Example for Evaluating Uncertainty of Perpendicularity Measurement

_{r}= 0.682 μm; if the average value of three times’ measurements is taken as the best estimate of the perpendicularity measurement, then the standard uncertainty caused by measurement repeatability is as follows:

#### 5.4. Result Analysis and Optimizing Uncertainty

_{o}cannot be satisfied; however, the extended uncertainty of the flatness measurement tasks for three workpieces to be measured after optimal evaluation can meet the accuracy requirements of the measurement tasks, but the sacrifice will be relatively high accordingly. Therefore, according to the basic principle of the task-oriented uncertainty optimal evaluation described in this section, the surveyors should make choices according to their own measurement conditions.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The uncertainty sources of the measurement system. EN: European Norm; ASME: American Society of Mechanical Engineers; GB: Chinese National Standards.

Error | Error Limit | Uncertainty Component |
---|---|---|

E_{L = 60, MPE} | 3.24 μm (L = 60 mm) | 1.87 μm |

E_{L = 60} | 1.4 μm (L = 60 mm) | 0.81 μm |

MPE_{P} | 3.5 μm | 2.02 μm |

P | 1.2 μm | 0.69 μm |

No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

measured value d_{i} (mm) | 62.0010 | 62.0011 | 62.0010 | 61.9998 | 62.0011 | 62.0002 | 62.0008 | 61.9998 | 62.0007 | 62.0008 |

The three times measurement mean of surveyor A ${\overline{d}}_{j}$ (mm) | Group Ⅰ | Group Ⅱ | Group Ⅲ |

62.0012 | 61.9998 | 62.0013 | |

The three times measurement mean of surveyor B ${\overline{d}}_{j}$ (mm) | Group Ⅳ | Group Ⅴ | Group Ⅵ |

61.9998 | 62.0016 | 62.0001 | |

The three times measurement mean of surveyor C ${\overline{d}}_{j}$ (mm) | Group Ⅶ | Group Ⅷ | Group Ⅸ |

62.0013 | 61.9996 | 62.0016 |

Standard Uncertainty | Source of Uncertainty | Evaluation Result |
---|---|---|

${u}_{\mathrm{E}}$ | Indication error | 1.875 μm |

${u}_{\mathrm{r}}$ | Repeatability | 0.294 μm |

${u}_{\mathrm{R}}$ | Reproducibility | 0.850 μm |

Standard Uncertainty | Source of Uncertainty | Evaluation Result |
---|---|---|

${u}_{\mathrm{E}}$ | Indication error | 2.021 μm |

${u}_{\mathrm{r}}$ | Repeatability | 0.357 μm |

${u}_{\mathrm{R}}$ | Reproducibility | 0.915 μm |

Standard Uncertainty | Source of Uncertainty | Evaluation Result |
---|---|---|

${u}_{\mathrm{E}}$ | Indication error | 2.449 μm |

${u}_{\mathrm{r}}$ | Repeatability | 0.394 μm |

${u}_{\mathrm{R}}$ | Reproducibility | 1.060 μm |

Tolerance Grade | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Proportion of Uncertainty to Tolerance | 33% | 25% | 20% | 16% | 12.5% | 10% |

The three times measurement mean of surveyor A ${\overline{t}}_{j}$ (mm) | Group Ⅰ | Group Ⅱ | Group Ⅲ |

0.0053 | 0.0062 | 0.0045 | |

The three times measurement mean of surveyor B ${\overline{t}}_{j}$ (mm) | Group Ⅳ | Group Ⅴ | Group Ⅵ |

0.0041 | 0.0058 | 0.0065 | |

The three times measurement mean of surveyor C ${\overline{t}}_{j}$ (mm) | Group Ⅶ | Group Ⅷ | Group Ⅸ |

0.0047 | 0.0066 | 0.0049 |

The three times measurement mean of surveyor A ${\overline{t}}_{j}$(mm) | Group Ⅰ | Group Ⅱ | Group Ⅲ |

0.0051 | 0.0058 | 0.0048 | |

The three times measurement mean of surveyor B ${\overline{t}}_{j}$(mm) | Group Ⅳ | Group Ⅴ | Group Ⅵ |

0.0047 | 0.0058 | 0.0055 | |

The three times measurement mean of surveyor C ${\overline{t}}_{j}$(mm) | Group Ⅶ | Group Ⅷ | Group Ⅸ |

0.0049 | 0.0061 | 0.0049 |

Measured Value | Workpiece A/t_{1j} | Workpiece B/t_{2j} | Workpiece C/t_{3j} |
---|---|---|---|

t_{i}_{1} | 0.0057 mm | 0.0051 mm | 0.0060 mm |

t_{i}_{2} | 0.0048 mm | 0.0044 mm | 0.0049 mm |

t_{i}_{3} | 0.0051 mm | 0.0054 mm | 0.0053 mm |

Mean value ${\overline{t}}_{i}$ | 0.0052 mm | 0.0050 mm | 0.0054 mm |

n_{i} | 3 | 3 | 3 |

Standard deviation | 0.458 μm | 0.513 μm | 0.556 μm |

u_{r} | 0.265 μm | 0.296 μm | 0.321 μm |

Repeatability | Workpiece A | Workpiece B | Workpiece C |
---|---|---|---|

Repeatability of prediction | 0.357 μm | 0.357 μm | 0.357 μm |

Repeatability of sample data | 0.265 μm | 0.296 μm | 0.321 μm |

Repeatability of real-time updates | 0.339 μm | 0.323 μm | 0.309 μm |

Uncertainty Components | Initial Evaluation Results | Optimized Evaluation Results | |||
---|---|---|---|---|---|

Symbol | Sources | Workpiece A | Workpiece B | Workpiece C | |

${u}_{\mathrm{E}}$ | Indication error | 2.021 μm | 0.69 μm | 0.69 μm | 0.69 μm |

${u}_{\mathrm{r}}$ | Repeatability | 0.357 μm | 0.339 μm | 0.323 μm | 0.309 μm |

${u}_{\mathrm{R}}$ | Reproducibility | 0.915 μm | 0.518 μm | 0.518 μm | 0.518 μm |

Standard uncertainty u_{c} | 2.3 μm | 0.9 μm | 0.9 μm | 0.9 μm | |

Expanded uncertainty U (p = 95%) | 4.6 μm | 1.8 μm | 1.8 μm | 1.8 μm | |

Comparison with target uncertainty U_{o} = 2.0 μm | Excess | Less | Less | Less |

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**MDPI and ACS Style**

Cheng, Y.; Wang, Z.; Chen, X.; Li, Y.; Li, H.; Li, H.; Wang, H. Evaluation and Optimization of Task-oriented Measurement Uncertainty for Coordinate Measuring Machines Based on Geometrical Product Specifications. *Appl. Sci.* **2019**, *9*, 6.
https://doi.org/10.3390/app9010006

**AMA Style**

Cheng Y, Wang Z, Chen X, Li Y, Li H, Li H, Wang H. Evaluation and Optimization of Task-oriented Measurement Uncertainty for Coordinate Measuring Machines Based on Geometrical Product Specifications. *Applied Sciences*. 2019; 9(1):6.
https://doi.org/10.3390/app9010006

**Chicago/Turabian Style**

Cheng, Yinbao, Zhongyu Wang, Xiaohuai Chen, Yaru Li, Hongyang Li, Hongli Li, and Hanbin Wang. 2019. "Evaluation and Optimization of Task-oriented Measurement Uncertainty for Coordinate Measuring Machines Based on Geometrical Product Specifications" *Applied Sciences* 9, no. 1: 6.
https://doi.org/10.3390/app9010006