# 基于三维激光点云的钢架螺栓球体节点要素提取研究Research on Geometric Element Extraction of Steel Frame Bolts Based on 3D Laser Point Cloud

DOI: 10.12677/GST.2019.74026, PDF, HTML, XML, 下载: 495  浏览: 724  科研立项经费支持

Abstract: At present, in the construction of large-span and complex buildings such as new airports and high-speed railway stations, the main frame is usually constructed by steel frame structure, and the internal construction of such steel frame structures needs to obtain the geometric elements of the center point coordinates and diameter of the bolt sphere. In order to extract the geometrical elements of the bolt sphere efficiently and accurately, this paper obtains the point cloud data based on the 3D laser scanning technology, and proposes to extract the parameters by using the random sampling consensus method. The results of the random sampling consistency method and the least squares fitting result are compared. The random sampling consensus algorithm extracts the sphere information better and has higher precision.

1. 引言

2. 原理与方法

2.1. 最小二乘法

${\left(x-a\right)}^{2}+{\left(y-b\right)}^{2}+{\left(z-c\right)}^{2}={r}^{2}$ (1)

${x}^{2}+{y}^{2}+{z}^{2}-2ax-2by-2cz+{a}^{2}+{b}^{2}+{c}^{2}={r}^{2}$ (2)

${x}^{2}+{y}^{2}+{z}^{2}-Ax-By-Cz+D=0$ (3)

$a=\frac{A}{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}b=\frac{B}{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}c=\frac{C}{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}r=\sqrt{{a}^{2}+{b}^{2}+{c}^{2}-D}$ (4)

2.2. 随机采样算法

Table 1. Random sampling algorithm parameters

N维球体被定义为满足以下公式的一组点：

${\left(p-c\right)}^{\text{T}}\left(p-c\right)={r}^{2}$ (5)

$\forall i,j\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left({p}_{i}-c\right)}^{\text{T}}\left({p}_{i}-c\right)={\left({p}_{j}-c\right)}^{\text{T}}\left({p}_{j}-c\right)={r}^{2}$ (6)

${\left({p}_{i}-{p}_{j}\right)}^{\text{T}}c=0.5\left({p}_{i}^{\text{T}}{p}_{i}-{p}_{j}^{\text{T}}{p}_{j}\right)$ (7)

$\left[\begin{array}{cc}\left({p}_{1x}-{p}_{2x}\right)& \left({p}_{1y}-{p}_{2y}\right)\\ \left({p}_{1x}-{p}_{3x}\right)& \left({p}_{1y}-{p}_{3y}\right)\end{array}\right]\left[\begin{array}{c}{c}_{x}\\ {c}_{y}\end{array}\right]=0.5\left[\begin{array}{c}\left({p}_{1x}^{2}-{p}_{2x}^{2}\right)+\left({p}_{1x}^{2}-{p}_{2y}^{2}\right)\\ \left({p}_{1x}^{2}-{p}_{3x}^{2}\right)+\left({p}_{1x}^{2}-{p}_{3y}^{2}\right)\end{array}\right]$ (8)

$A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ , ${A}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$ (9)

(10)

$A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}{A}^{-1}=\frac{1}{a\left(ei-fh\right)-b\left(di-gh\right)+c\left(dh-eg\right)}\left[\begin{array}{ccc}ei-fh& ch-bi& bf-ce\\ fg-di& ai-cg& cd-af\\ dh-eg& bg-ah& ae-ah\end{array}\right]$ (11)

${\delta }_{i}={p}_{i}^{\text{T}}{p}_{i}-2{p}_{i}^{\text{T}}c+{c}^{\text{T}}c-{r}^{2}$ (12)

$\left[{c}^{*},{r}^{*}\right]=\mathrm{arg}{\mathrm{min}}_{c,r}{\sum }_{i=1}^{m}{\delta }_{i}{}^{2}$ (13)

$m={c}^{\text{T}}c-{r}^{2}$，得到以下线性方程组 $\left(Ax=b\right)$

(14)

${\sum }_{i=1}^{m}{\delta }_{i}{}^{2}={‖Ax-b‖}^{2}$ (15)

3. 实验与分析

“全站仪 + 最小二乘法”是利用全站仪采集螺栓球体表面上的4个随机点，然后通过MATLAB编写的最小二乘法程序对数据进行计算，最终得到螺栓球的半径和球体中心坐标。

Figure 1. Experimental flow

Table 2. RANSAC processing results (unit: m)

Table 3. Least squares processing results (unit: m)

Figure 2. Error statistics for two experimental methods

4. 实际应用

Figure 3. International airport point cloud

Figure 4. Bolt ball lowest point coordinates

Figure 5. Construction site before ceiling installation

Figure 6. Site after completion of ceiling ceiling work

5. 结论

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