# Modeling Vehicle Collision Angle in Traffic Crashes Based on Three-Dimensional Laser Scanning Data

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## Abstract

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## 1. Introduction

## 2. Issue Description and Modeling

#### 2.1. Vehicle Collision and the Relevant Hypothesis

#### 2.2. Modeling

## 3. Reconstruction of the Crashed Surface

#### 3.1. Delaunay Triangulation and Improvement Based on the Mapping Method

_{i}, with the distance l to the point P, to the plane π, the point P

_{i}is changed to point ${P}_{i}^{\prime}$ on the plane π, and the distance between P and ${P}_{i}^{\prime}$ is changed to ${l}^{\prime}$. Generally, l is not equal to ${l}^{\prime}$ , and so the position of ${P}_{i}^{\prime}$ should be adjusted after mapping.

_{P}with radius l centered at point P is drawn. The next adjustment position of P

_{i}is on O

_{P}. In addition, the next adjustment position of P

_{i}should meet the distance of another mapping point P

_{j}, as shown in Figure 1c.

_{j}after the first mapping is located at ${P}_{j}^{\u2033}$. To meet the original distance of $\overline{{P}_{i}{P}_{j}}$, another circle is drawn with radius $\overline{{P}_{i}{P}_{j}}$ centered at point ${P}_{i}^{\u2033}$. The circle will intersect circle O

_{P}at point ${P}_{i}^{\u2034}$, which is the second adjusted point of P

_{i}. After the mapping process, the mapped point of P

_{i}, namely ${P}_{i}^{\u2034}$, meets the distance of the seed and another point. Then, the mapped point of P

_{i}is adjusted k times (kth-nearest neighbors, chosen as 5 in this paper) in the same way. After these mappings are done, the mapping position of P

_{i}can be determined by the following formula based on the least-squares method:

_{i}, ${d}_{{P}_{it}{P}_{n}}$ is the distance between ${P}_{it}$ and ${P}_{n}$ in 3D space, and ${d}_{{P}_{it}{P}_{n}}$ is the distance between P

_{i}and P

_{n}.

#### 3.2. Analysis of Delaunay Triangulation Based on Improved Mapping Method

#### 3.3. Impact of Noise on the Triangulation Result

^{−11}. From the numerical point of view, the triangles formed by triangulation are quite uniform and the space structure is also very good.

^{−11}. It shows that the size of the triangle formed by the triangulation is almost the same, and this result shows that the triangulation result meets the requirements of this paper when the measurement noise is ±0.5 mm.

^{−11}, and there is no obvious regularity. Therefore, in the triangulation the size of the triangle in the triangular mesh is not sensitive to the measurement error within ±2 mm. In using the mapping method proposed in this paper, in the Delaunay triangulation, the influence of the measurement error within ±2 mm cannot be underestimated.

#### 3.4. Analysis of Influence of Projection Surface Selection on the Triangulation Result

^{−5}. Orders of magnitude have been increased from 10

^{−5}to 10

^{−7}in Figure 3b, which shows that the triangulation result is a sensitive choice of the projection surface. In order to study the sensitivity of different projections relative to triangulation, in this paper the projection plane X = 0 is rotated along the Z axis to the Y = 0 plane. We also select a projection surface at 15° per interval and calculate the variance of the triangular mesh generated on these projected surfaces. The variance results are shown in Table 2.

## 4. Model Validation and Results Analysis

#### 4.1. Experimental Verification

#### 4.2. Verification of Real Vehicle Collision Accident

## 5. Conclusions

- Study the angle of impact from different body material stiffnesses, and have an accurately assignment about the collision point;
- Propose the correct factors aiming at different deformation surfaces, and make the model suitable for different collision angles’ solving;
- When it comes to an elongated, groove shape of a collision surface, the collision area can be partitioned for calculation and the research of the collision angle.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Projection approach with distance adjustment. (

**a**) Direct projection; (

**b**) First distance adjustment; (

**c**) Second distance adjustment.

**Figure 4.**Triangulation and collision points’ normal vectors results. (

**a**) Triangulation results; (

**b**) Calculation of the collision points’ normal vectors.

Error size | 0 (no error) | 0.5 mm | 1 mm | 1.5 mm | 2 mm |

Variance | 8.486 × 10^{−11} | 7.271 × 10^{−11} | 7.483 × 10^{−11} | 8.650 × 10^{−11} | 9.872 × 10^{−11} |

Plane | YOZ Plane | Rotated 15° | Rotated 30° | Rotated 45° | Rotated 60° | Rotated 75° | XOZ Plane |

Variance | 1.529 × 10^{−7} | 3.834 × 10^{−7} | 9.115 × 10^{−7} | 2.250 × 10^{−6} | 5.225 × 10^{−6} | 1.187 × 10^{−6} | 3.153 × 10^{−5} |

Angle (°) | Plane Normal $\overrightarrow{\mathit{n}}$ | Normal after Collision $\overrightarrow{{\mathit{\lambda}}_{\mathit{c}}}$ | Collision Angle (°) | Error (°) |
---|---|---|---|---|

90 | (0.8461, 0.5301, 0.0551) | (0.8195, 0.5670, 0.0594) | 85.7 | 4.3 |

75 | (0.7981, 0.4862, 0.3753) | 71.1 | 3.9 | |

60 | (0.6985, 0.5389, 0.4709) | 64.5 | 4.5 | |

45 | (0.5663, 0.2106, 0.7968) | 39.4 | 5.6 | |

30 | (0.3975, 0.2252, −0.8895) | 24 | 6.0 |

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**MDPI and ACS Style**

Lyu, N.; Huang, G.; Wu, C.; Duan, Z.; Li, P.
Modeling Vehicle Collision Angle in Traffic Crashes Based on Three-Dimensional Laser Scanning Data. *Sensors* **2017**, *17*, 482.
https://doi.org/10.3390/s17030482

**AMA Style**

Lyu N, Huang G, Wu C, Duan Z, Li P.
Modeling Vehicle Collision Angle in Traffic Crashes Based on Three-Dimensional Laser Scanning Data. *Sensors*. 2017; 17(3):482.
https://doi.org/10.3390/s17030482

**Chicago/Turabian Style**

Lyu, Nengchao, Gang Huang, Chaozhong Wu, Zhicheng Duan, and Pingfan Li.
2017. "Modeling Vehicle Collision Angle in Traffic Crashes Based on Three-Dimensional Laser Scanning Data" *Sensors* 17, no. 3: 482.
https://doi.org/10.3390/s17030482