# Selection of the Best Model of Distribution of Measurement Points in Contact Coordinate Measurements of Free-Form Surfaces of Products

## Abstract

**:**

## 1. Introduction

## 2. Proposed Method of the Distribution of Measurement Points

## 3. AHP Hierarchical Structure

- form deviations calculated by using the applied model of the distribution of measurement points,
- number of measurement points belonging to the considered model of the distribution,
- accuracy of substitute models created based on the measurement points located on a free-form surface,
- time of contact coordinate measurements,
- areas of 3D substitute models representing the models of the distribution of measurement points.

## 4. Simulation and Experimental Investigations

#### 4.1. Analyzed Curvilinear Surface

#### 4.2. The Considered Models of the Distribution of Measurement Points

#### 4.3. Results of Simulation Investigations

#### 4.4. Results of Experimental Research

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

## 5. AHP Analysis of the Considered Models of the Distribution

## 6. The Time of Coordinate Measurements Carried Out by Using the New Method of the Distribution of Measurement Points

## 7. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The stages of measurements of a series of products when using the new method of determining the location of measurement points.

**Figure 2.**AHP hierarchical structure applied to select the best model of the distribution of measurement points.

**Figure 3.**The view of the investigated free-form surface for which the best distribution of measurement points was searched and the reference model of the distribution of measurement points.

**Figure 7.**The consolidated weights of the analyzed models of the distribution of measurement points.

Substitute Model | Surface Area, ${\mathbf{mm}}^{2}$ |
---|---|

Substitute model 1 | 7046 |

Substitute model 2 | 13,935 |

Substitute model 3 | 10,364 |

Substitute model 4 | 14,292 |

Substitute model 5 | 10,624 |

Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | |
---|---|---|---|---|---|

Model 1 | 1 | 2.3 | 3.7 | 2.7 | 1.7 |

Model 2 | 0.4 | 1 | 1.6 | 1.1 | 0.7 |

Model 3 | 0.3 | 0.6 | 1 | 0.7 | 0.5 |

Model 4 | 0.4 | 0.9 | 1.4 | 1 | 0.6 |

Model 5 | 0.6 | 1.4 | 2.2 | 1.6 | 1 |

Model | Deviation from the Reference Model, mm | Time, s |
---|---|---|

Model 1 | 0.0570 | 99 |

Model 2 | 0.0058 | 63 |

Model 3 | 0.0721 | 66 |

Model 4 | 0.0056 | 61 |

Model 5 | 0.0653 | 82 |

**Table 4.**The scale for performing pairwise comparison [19].

Intensity of Importance | Definition | Explanation |
---|---|---|

1 | Equal importance | Two factors (A and B) are equally important |

3 | Moderate importance | Experience and judgment favor A over B |

5 | Strong importance | A is strongly favored over B |

7 | Very strong importance | A is very strongly favored over B |

9 | Extreme importance | The evidence favoring A over B is of the highest possible validity |

2, 4, 6, 8 | Intermediate values between judgments | A compromise is required |

Form Deviation | Time | Number of Points | Substitute Model | Area | |
---|---|---|---|---|---|

Form deviation | 1 | 5 | 6 | 5 | 5 |

Time | 0.2 | 1 | 3 | 0.33 | 3 |

Number of points | 0.17 | 0.33 | 1 | 0.25 | 2 |

Substitute model | 0.2 | 3 | 4 | 1 | 4 |

Area | 0.2 | 0.33 | 0.5 | 0.25 | 1 |

Category | Priority | Rank |
---|---|---|

Form deviation | 53.9% | 1 |

Time | 12.3% | 3 |

Number of points | 6.7% | 4 |

Substitute model | 21.8% | 2 |

Area | 5.4% | 5 |

Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | |
---|---|---|---|---|---|

Model 1 | 1 | 0.14 | 1 | 0.14 | 1 |

Model 2 | 7 | 1 | 9 | 1 | 8 |

Model 3 | 1 | 0.11 | 1 | 0.11 | 1 |

Model 4 | 7 | 1 | 9 | 1 | 8 |

Model 5 | 1 | 0.12 | 1 | 0.12 | 1 |

Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | |
---|---|---|---|---|---|

Model 1 | 1 | 0.12 | 0.14 | 0.11 | 0.33 |

Model 2 | 8 | 1 | 1 | 1 | 5 |

Model 3 | 7 | 1 | 1 | 0.5 | 4 |

Model 4 | 9 | 1 | 2 | 1 | 5 |

Model 5 | 3 | 0.2 | 0.25 | 0.2 | 1 |

Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | |
---|---|---|---|---|---|

Model 1 | 1 | 9 | 9 | 9 | 4 |

Model 2 | 0.11 | 1 | 1 | 1 | 0.2 |

Model 3 | 0.11 | 1 | 1 | 1 | 0.2 |

Model 4 | 0.11 | 1 | 1 | 1 | 0.2 |

Model 5 | 0.25 | 5 | 5 | 5 | 1 |

Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | |
---|---|---|---|---|---|

Model 1 | 1 | 0.11 | 0.2 | 0.11 | 0.2 |

Model 2 | 9 | 1 | 4 | 1 | 3 |

Model 3 | 5 | 0.25 | 1 | 0.25 | 1 |

Model 4 | 9 | 1 | 4 | 1 | 4 |

Model 5 | 5 | 0.33 | 1 | 0.25 | 1 |

Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | |
---|---|---|---|---|---|

Model 1 | 1 | 5 | 9 | 6 | 3 |

Model 2 | 0.2 | 1 | 2 | 1 | 0.5 |

Model 3 | 0.11 | 0.5 | 1 | 0.5 | 0.25 |

Model 4 | 0.17 | 1 | 2 | 1 | 0.5 |

Model 5 | 0.33 | 2 | 4 | 2 | 1 |

Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | |
---|---|---|---|---|---|

Form deviation | 0.056 | 0.421 | 0.050 | 0.421 | 0.053 |

Time | 0.033 | 0.299 | 0.243 | 0.353 | 0.071 |

Number of points | 0.597 | 0.055 | 0.055 | 0.055 | 0.238 |

Substitute model | 0.545 | 0.103 | 0.053 | 0.100 | 0.199 |

Area | 0.031 | 0.358 | 0.112 | 0.381 | 0.118 |

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**MDPI and ACS Style**

Magdziak, M.
Selection of the Best Model of Distribution of Measurement Points in Contact Coordinate Measurements of Free-Form Surfaces of Products. *Sensors* **2019**, *19*, 5346.
https://doi.org/10.3390/s19245346

**AMA Style**

Magdziak M.
Selection of the Best Model of Distribution of Measurement Points in Contact Coordinate Measurements of Free-Form Surfaces of Products. *Sensors*. 2019; 19(24):5346.
https://doi.org/10.3390/s19245346

**Chicago/Turabian Style**

Magdziak, Marek.
2019. "Selection of the Best Model of Distribution of Measurement Points in Contact Coordinate Measurements of Free-Form Surfaces of Products" *Sensors* 19, no. 24: 5346.
https://doi.org/10.3390/s19245346