# A New Method of Distribution of Measurement Points on Curvilinear Surfaces of Products

## Abstract

**:**

## 1. Introduction

## 2. State of the Art in the Area of Determining the Location of Measurement Points

## 3. New Method of the Distribution of Measurement Points on a Free-Form Surface

#### 3.1. Algorithm of the Distribution of Scanning Lines

- a free-form surface is divided into intervals in two directions;
- only one point in each row and column of a surface is selected.

- average curvatures for the generated cross-sections based on the values of curvatures at points uniformly distributed along the considered cross-sections;
- lengths of the created cross-sections of a measured free-form surface of a product;
- deviations of the analyzed cross-sections from the 2D curve.

#### 3.2. Algorithm of the Distribution of Measurement Points along Selected Scanning Lines

- process of the probe radius correction carried out with the use of selected compensation methods;
- substitute curves fitted to the groups of measurement points generated along the scanning lines and representing a measured free-form surface.

## 4. Fuzzy-Logic-Based System

#### 4.1. Part of the System Used for the Selection of Scanning Lines

#### 4.2. Part of the System Used for the Selection of Number of Measurement Points

## 5. Analyzed Workpieces Used during Simulation Investigations

## 6. Simulation Investigations

## 7. Experimental Investigations

- ${E}_{L,MPE}=(1.6+L/333)\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$;
- ${P}_{FTU,MPE}=1.7\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$;
- $MP{E}_{Tij}=2.5\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$;
- $MP{T}_{\tau ij}=50.0\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$;

- the method that enables the distribution of measurement points between two points located on a measured free-form surface and chosen by the operator of a CMM;
- the method of single measurement points, which are randomly selected on a considered curvilinear surface of a measured object by the user of a CMM.

## 8. Implementation of the System Based on the Fuzzy Logic in the Commercial Measurement Software

## 9. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**The calculation of the deviation of the selected cross-section from the 2D curve based on the arithmetic mean of the angles between normal vectors.

**Figure 6.**The membership functions for the input parameters in the case of the first free-form surface.

**Figure 7.**The membership functions for the output parameter in the case of the first free-form surface.

**Figure 9.**The membership functions for the input parameters in the case of both free-form surfaces and the second part of the expert system.

**Figure 10.**The membership functions for the output parameter in the case of both free-form surfaces and the second part of the expert system.

**Figure 12.**The generated cross-sections for two investigated free-form surfaces and the non-measurement points obtaind using the LHS method.

**Figure 15.**The distribution of form deviations on the measured free-form surface of the product together with the generated scanning lines and the selected value of the tolerance.

**Figure 16.**The selected methods of the distribution of measurement points available in the Calypso software.

**Figure 17.**The most important stages of implementation of the created method in the commercial measurement software.

Curvature | Length | Deviation | Difficulty | Result |
---|---|---|---|---|

small | small | small | difficult | 1 |

small | small | small | medium | 2 |

small | small | small | easy | 3 |

small | small | medium | difficult | 0 |

small | small | medium | medium | 1 |

small | small | medium | easy | 2 |

small | small | large | difficult | 0 |

small | small | large | medium | 1 |

small | small | large | easy | 2 |

small | medium | small | difficult | 2 |

small | medium | small | medium | 3 |

small | medium | small | easy | 4 |

small | medium | medium | difficult | 1 |

small | medium | medium | medium | 2 |

small | medium | medium | easy | 3 |

small | medium | large | difficult | 0 |

small | medium | large | medium | 1 |

small | medium | large | easy | 2 |

small | large | small | difficult | 3 |

small | large | small | medium | 4 |

small | large | small | easy | 5 |

small | large | medium | difficult | 2 |

small | large | medium | medium | 3 |

Probe Radius Correction—1st Method | Probe Radius Correction—2nd Method | Substitute Model | Result |
---|---|---|---|

small | small | small | decrease |

small | small | medium | decrease |

small | small | large | no change |

small | medium | small | decrease |

small | medium | medium | decrease |

small | medium | large | no change |

small | large | small | no change |

small | large | medium | no change |

small | large | large | increase |

medium | small | small | decrease |

medium | small | medium | decrease |

medium | small | large | no change |

medium | medium | small | decrease |

medium | medium | medium | no change |

medium | medium | large | no change |

medium | large | small | no change |

medium | large | medium | no change |

medium | large | large | increase |

large | small | small | no change |

large | small | medium | no change |

large | small | large | increase |

large | medium | small | no change |

large | medium | medium | no change |

**Table 3.**The selected results of simulation investigations being the basis for selecting the scanning lines for the first free-form surface.

Point | Cross-Section | Curvature, mm${}^{-1}$ | Length, mm | Deviation, degree | Difficulty | Result |
---|---|---|---|---|---|---|

1 | 1 | $0.022$ | $247.545$ | $3.857$ | $0.5$ | 21.8 |

2 | $0.008$ | $208.244$ | $22.423$ | $0.5$ | $11.2$ | |

3 | $0.013$ | $204.599$ | $26.974$ | $0.5$ | $12.0$ | |

4 | $0.015$ | $298.126$ | $17.423$ | $0.5$ | $\mathbf{17.8}$ | |

2 | 5 | $0.023$ | $247.657$ | $5.553$ | $0.5$ | $\mathbf{21.6}$ |

6 | $0.018$ | $172.604$ | $23.316$ | $0.5$ | $14.0$ | |

7 | $0.020$ | $208.571$ | $32.389$ | $0.5$ | $13.4$ | |

8 | $0.016$ | $262.617$ | $17.467$ | $0.5$ | $\mathbf{16.1}$ | |

3 | 9 | $0.021$ | $246.971$ | $4.244$ | $0.5$ | $\mathbf{21.4}$ |

10 | $0.016$ | $299.152$ | $18.525$ | $0.5$ | $\mathbf{17.9}$ | |

11 | $0.013$ | $204.397$ | $14.194$ | $0.5$ | $14.0$ | |

12 | $0.010$ | $176.200$ | $16.536$ | $0.5$ | $12.5$ | |

4 | 13 | $0.020$ | $246.746$ | $4.899$ | $0.5$ | $\mathbf{20.8}$ |

14 | $0.015$ | $252.293$ | $18.580$ | $0.5$ | $15.1$ | |

15 | $0.003$ | $201.231$ | $6.238$ | $0.5$ | $12.8$ | |

16 | $0.016$ | $269.904$ | $16.918$ | $0.5$ | $\mathbf{16.6}$ |

**Table 4.**The selected results of simulation investigations being the basis for selecting the scanning lines for the second free-form surface.

Point | Cross-Section | Curvature, mm${}^{-1}$ | Length, mm | Deviation, degree | Difficulty | Result |
---|---|---|---|---|---|---|

1 | 1 | $0.014$ | $222.527$ | $14.010$ | $0.5$ | $\mathbf{15.2}$ |

2 | $0.012$ | $206.815$ | $16.586$ | $0.5$ | $13.8$ | |

3 | $0.014$ | $247.648$ | $11.675$ | $0.5$ | $\mathbf{17.7}$ | |

4 | $0.007$ | $279.070$ | $23.548$ | $0.5$ | $12.4$ | |

2 | 5 | $0.017$ | $229.304$ | $25.655$ | $0.5$ | $15.1$ |

6 | $0.018$ | $188.269$ | $25.37$ | $0.5$ | $14.6$ | |

7 | $0.019$ | $232.346$ | $3.421$ | $0.5$ | $\mathbf{22.2}$ | |

8 | $0.010$ | $263.425$ | $20.638$ | $0.5$ | $\mathbf{15.4}$ | |

3 | 9 | $0.013$ | $212.116$ | $12.813$ | $0.5$ | $15.1$ |

10 | $0.011$ | $298.818$ | $16.755$ | $0.5$ | $\mathbf{17.5}$ | |

11 | $0.015$ | $241.833$ | $20.631$ | $0.5$ | $\mathbf{16.1}$ | |

12 | $0.014$ | $184.554$ | $20.349$ | $0.5$ | $13.8$ | |

4 | 13 | $0.017$ | $233.728$ | $19.435$ | $0.5$ | $18.0$ |

14 | $0.013$ | $278.197$ | $9.091$ | $0.5$ | $\mathbf{20.5}$ | |

15 | $0.019$ | $239.164$ | $19.862$ | $0.5$ | $\mathbf{19.1}$ | |

16 | $0.010$ | $267.040$ | $21.276$ | $0.5$ | $15.0$ |

**Table 5.**The results of the second part of the simulation investigations and the recommendations regarding the number of measurement points located along the selected measured cross-sections for the first free-form surface.

Point | Cross-Section | Deviation—1st Method of the Probe Radius Correction, mm | Deviation—2nd Method of the Probe Radius Correction, mm | Deviation—Substitute Model, mm | Result | Recommendation |
---|---|---|---|---|---|---|

1 | 1 | $0.002011$ | $0.000705$ | $0.260878$ | $3.7$ | increase |

4 | $0.001353$ | $0.000668$ | $0.296815$ | $5.2$ | no change | |

2 | 5 | $0.002174$ | $0.000879$ | $0.287335$ | $2.4$ | increase |

8 | $0.000634$ | $0.000131$ | $0.055623$ | $7.9$ | decrease | |

3 | 9 | $0.001378$ | $0.000242$ | $0.125011$ | $7.8$ | decrease |

10 | $0.001360$ | $0.000818$ | $0.328597$ | $5.1$ | no change | |

4 | 13 | $0.001396$ | $0.000357$ | $0.151204$ | $7.6$ | decrease |

16 | $0.000647$ | $0.000134$ | $0.045928$ | $7.9$ | decrease | |

5 | 17 | $0.001347$ | $0.000220$ | $0.125403$ | $7.9$ | decrease |

20 | $0.001730$ | $0.000944$ | $0.437777$ | $4.6$ | no change | |

6 | 21 | $0.001605$ | $0.000394$ | $0.177740$ | $7.4$ | decrease |

24 | $0.001103$ | $0.000316$ | $0.145900$ | $7.7$ | decrease | |

7 | 25 | $0.001471$ | $0.000293$ | $0.124325$ | $7.8$ | decrease |

27 | $0.000524$ | $0.000095$ | $0.077839$ | $7.9$ | decrease | |

8 | 29 | $0.002449$ | $0.001109$ | $0.332622$ | $2.1$ | increase |

32 | $0.000228$ | $0.000095$ | $0.027453$ | $7.9$ | decrease | |

9 | 33 | $0.001360$ | $0.000358$ | $0.149707$ | $7.7$ | decrease |

36 | $0.000947$ | $0.000182$ | $0.144307$ | $7.9$ | decrease | |

10 | 37 | $0.002240$ | $0.000923$ | $0.306250$ | $2.3$ | increase |

38 | $0.000517$ | $0.000144$ | $0.081611$ | $7.8$ | decrease |

**Table 6.**The results of the second part of the simulation investigations and the recommendations regarding the number of measurement points located along the selected measured cross-sections for the second free-form surface.

Point | Cross-Section | Deviation—1st Method of the Probe Radius Correction, mm | Deviation—2nd Method of the Probe Radius Correction, mm | Deviation—Substitute Model, mm | Result | Recommendation |
---|---|---|---|---|---|---|

1 | 1 | $0.000191$ | $0.000143$ | $0.073943$ | $7.85$ | decrease |

3 | $0.000342$ | $0.000122$ | $0.057719$ | $7.9$ | decrease | |

2 | 7 | $0.000365$ | $0.000049$ | $0.051129$ | $7.94$ | decrease |

8 | $0.000149$ | $0.000095$ | $0.042674$ | $7.93$ | decrease | |

3 | 10 | $0.000304$ | $0.000097$ | $0.113102$ | $7.93$ | decrease |

11 | $0.000296$ | $0.000116$ | $0.040633$ | $7.92$ | decrease | |

4 | 14 | $0.000164$ | $0.000058$ | $0.055943$ | $7.94$ | decrease |

15 | $0.000410$ | $0.000101$ | $0.050415$ | $7.9$ | decrease | |

5 | 19 | $0.000351$ | $0.000090$ | $0.0485772$ | $7.92$ | decrease |

20 | $0.000147$ | $0.000102$ | $0.070310$ | $7.89$ | decrease | |

6 | 21 | $0.000105$ | $0.000075$ | $0.069746$ | $7.91$ | decrease |

23 | $0.000216$ | $0.000062$ | $0.036623$ | $7.95$ | decrease | |

7 | 25 | $0.000213$ | $0.000113$ | $0.057528$ | $7.91$ | decrease |

27 | $0.000286$ | $0.000126$ | $0.036728$ | $7.92$ | decrease | |

8 | 31 | $0.000390$ | $0.000069$ | $0.051482$ | $7.93$ | decrease |

32 | $0.000095$ | $0.000046$ | $0.012794$ | $7.96$ | decrease | |

9 | 33 | $0.000252$ | $0.000049$ | $0.048382$ | $7.95$ | decrease |

35 | $0.000387$ | $0.000127$ | $0.062302$ | $7.89$ | decrease | |

10 | 38 | $0.000173$ | $0.000062$ | $0.046361$ | $7.94$ | decrease |

39 | $0.000352$ | $0.000123$ | $0.051950$ | $7.91$ | decrease |

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Magdziak, M.
A New Method of Distribution of Measurement Points on Curvilinear Surfaces of Products. *Sensors* **2019**, *19*, 2667.
https://doi.org/10.3390/s19122667

**AMA Style**

Magdziak M.
A New Method of Distribution of Measurement Points on Curvilinear Surfaces of Products. *Sensors*. 2019; 19(12):2667.
https://doi.org/10.3390/s19122667

**Chicago/Turabian Style**

Magdziak, Marek.
2019. "A New Method of Distribution of Measurement Points on Curvilinear Surfaces of Products" *Sensors* 19, no. 12: 2667.
https://doi.org/10.3390/s19122667