# Three-Dimensional Mapping Technology for Structural Deformation during Aircraft Assembly Process

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## Abstract

**:**

## 1. Introduction

## 2. General Scheme

## 3. Methods

#### 3.1. Curvature Conversion Model of Optical Fiber Monitoring

#### 3.1.1. Curvature Conversion Model

#### 3.1.2. Coordinate Value Derivation Based on Curvature

#### 3.2. Grid Distortion Optimization Algorithm for Base Point-Cloud Optimization

- Based on the optimized half-edge folding method, the original mesh model was simplified to obtain an optimized mesh with sparse point clouds but no effect on surface quality.
- The absolute coordinates of the mesh model were converted into differential coordinates.
- The optical fiber monitoring point was set as the control point, and the mapping relationship between the original model and the control point was established.
- The absolute coordinate values of control points after distortion were obtained according to optical fiber monitoring data.
- The absolute coordinates of the other vertices of the distorted model were inverted based on the known differential coordinates and the absolute coordinates of the control points.

#### 3.2.1. Model Simplification

- Calculation of the average area of a vertex neighborhood triangle $\overline{S}$

- 2.
- Calculation of the discrete curvature $Q$ of vertices

- 3.
- Determination of a boundary point

- 4.
- Calculation of distance between vertices $d\left(u,v\right)$

- 5.
- Fold cost calculation

#### 3.2.2. Mesh Distortion Optimization Algorithm

- Laplacian mesh distortion

- 2.
- Laplacian matrix

- 3.
- Addition of constraints

- 4.
- Establishment of deformation constraints

#### 3.2.3. Pseudocode

Algorithm 1. Pseudocode for HEC-Laplace
| |

1: | Input: Original grid control points before and after deformation simplification threshold |

2: | Output: Deformed grid runtime |

3: | Initialization: |

4: | Compute folding times based on simplification threshold |

5: | Model simplification: (Section 3.2.1) |

6: | Compute folding cost by Equation (15); |

7: | Sort vertices according to collapse cost; |

8: | Folding edges with the least cost by priority; |

9: | Grid Deformation: (Section 3.2.2) |

10: | Verify the set of one-ring neighborhood points for each vertex; |

11: | According to Equation (16) define the Laplace coordinates of the vertices; |

12: | According to Equation (17) compute cotangent weights; |

13: | Introduce a Laplacian matrix based on Equation (19) convert Cartesian coordinates to differential coordinates; |

14: | According to Equation (23) establishment of statically indeterminate systems of linear equations; |

15: | Solve equations to obtain the new position of the vertex under the Cartesian coordinate system. |

## 4. Results and Discussion

#### 4.1. Comparison of Deformation Algorithm through Simulations

#### 4.1.1. Model Evaluation

#### 4.1.2. Simulation Experiments

#### 4.2. Application of Structural Deformation Mapping

#### 4.2.1. Establishment of Experimental System

^{3}) and a resolution of 0.05mm. The binocular vision scanning system was based on the binocular stereo vision measurement principle and the structured light 3D vision measurement principle to quickly obtain the high-density point cloud data of the product. The results are shown in Figure 13; the number of triangular patches in the scanning data was 868,286, and the number of vertices was 434,152, which provides true and reliable data for subsequent 3D mesh deformation.

#### 4.2.2. Experimental Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 10.**Comparison of deviation between the deformed mesh algorithm and actual deformed model: (

**a**) Interpolation algorithm; (

**b**) Gaussian fusion; (

**c**) Laplace; and (

**d**) HEC−Laplace.

**Figure 15.**Wing deformation/reconstruction deviation diagrams or the deviation distribution diagrams under different loading modes. (

**a1**–

**e1**) Results of wing model shape variables under five load modes; (

**a2**–

**e2**) The deviation consistency test results between the reconstructed model and the binocular measured point cloud data under the five loading modes; (

**a3**–

**e3**) Percentage of deviation result point cloud distribution under five loading modes.

Model | Evaluation Index | |||
---|---|---|---|---|

MAE | RE | RMSE | Time (s) | |

Interpolation algorithm | 0.388 | 19.77% | 0.609 | 10.47 |

Gaussian fusion | 0.064 | 5.28% | 0.090 | 50.54 |

Laplace | 0.058 | 3.47% | 0.075 | 591.22 |

HEC−Laplace | 0.046 | 2.92% | 0.062 | 42.47 |

Node Number | Loading Mode | Load Size | Load Application Position |
---|---|---|---|

1 | Single point | 2 N | Point A |

2 | Single point | 5 N | Point A |

3 | Single point | 5 N | Point B |

4 | Multipoint loading | 2 N 5 N | Point A Point C |

5 | Multipoint loading | 2 × 5 N | Points B and C |

Node Number | Maximum Deformation (mm) | Evaluation Index | ||
---|---|---|---|---|

MAE | RE | RMSE | ||

1 | 29.94 | 0.104 | 3.78% | 0.146 |

2 | 73.68 | 0.210 | 3.02% | 0.307 |

3 | 28.06 | 0.094 | 2.62% | 0.111 |

4 | 64.03 | 0.200 | 2.82% | 0.272 |

5 | 63.70 | 0.247 | 3.50% | 0.325 |

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## Share and Cite

**MDPI and ACS Style**

Liu, Y.; Yan, D.; Li, L.; Lin, X.; Guo, L.
Three-Dimensional Mapping Technology for Structural Deformation during Aircraft Assembly Process. *Photonics* **2023**, *10*, 318.
https://doi.org/10.3390/photonics10030318

**AMA Style**

Liu Y, Yan D, Li L, Lin X, Guo L.
Three-Dimensional Mapping Technology for Structural Deformation during Aircraft Assembly Process. *Photonics*. 2023; 10(3):318.
https://doi.org/10.3390/photonics10030318

**Chicago/Turabian Style**

Liu, Yue, Dongming Yan, Lijuan Li, Xuezhu Lin, and Lili Guo.
2023. "Three-Dimensional Mapping Technology for Structural Deformation during Aircraft Assembly Process" *Photonics* 10, no. 3: 318.
https://doi.org/10.3390/photonics10030318