Sampling Point Planning for Complex Surface Inspection based on Feature Points under Area Division

Abstract

Traditional sampling point planning methods usually apply the same sampling strategy to all areas of the complex surface. It is probable that it misses the extreme points in areas with large curvature variations, resulting in low fitting accuracy. This paper proposes a new sampling method for complex surfaces based on feature points under area division. First, the curvature characteristics of the complex surface is analyzed, and the complex surface is divided into flat and sharply-edged areas. Then, for the flat areas the uniform distribution method is used, and for the sharply-edged areas, the feature points are defined and searched, and the curvature adaptive planning method based on the feature points is adopted. Finally, the repeated and redundant points are optimized and adjusted. The experimental verification results show that compared with four traditional sampling methods, the maximum and mean fitting errors of the proposed method are significantly reduced and the measuring accuracy is efficiently improved.

Appendices

Appendix 1

→ Forward reshape operation

$${\textbf{A}}_{p\times q}=\left[\begin{array}{ccc}{a}_{11}& \cdots & {a}_{1q}\\ {}\vdots & \ddots & \vdots \\ {}{a}_{p1}& \cdots & {a}_{pq}\end{array}\right],$$

$${\overrightarrow{\textbf{A}}}_{pq\times 1}={\left[\begin{array}{ccc}{a}_{11}& \cdots & \begin{array}{cccccccc}{a}_{p1}& {a}_{12}& \cdots & {a}_{p2}& \cdots & {a}_{1q}& \cdots & {a}_{pq}\end{array}\end{array}\right]}^T$$

← Reverse reshape operation

$${\textbf{A}}_{pq\times 1}={\left[\begin{array}{ccc}{a}_{11}& \cdots & \begin{array}{cccccccc}{a}_{p1}& {a}_{12}& \cdots & {a}_{p2}& \cdots & {a}_{1q}& \cdots & {a}_{pq}\end{array}\end{array}\right]}^T,$$

$${\overleftarrow{\textbf{A}}}_{p\times q}=\left[\begin{array}{ccc}{a}_{11}& \cdots & {a}_{1q}\\ {}\vdots & \ddots & \vdots \\ {}{a}_{p1}& \cdots & {a}_{pq}\end{array}\right]$$

Appendix 2

⋅ Dot product operation

${\textbf{A}}_{p\times q}=\left[\begin{array}{ccc}{a}_{11}& \cdots & {a}_{1q}\\ {}\vdots & \ddots & \vdots \\ {}{a}_{p1}& \cdots & {a}_{pq}\end{array}\right]$,${\textbf{B}}_{p\times q}=\left[\begin{array}{ccc}{b}_{11}& \cdots & {b}_{1q}\\ {}\vdots & \ddots & \vdots \\ {}{b}_{p1}& \cdots & {b}_{pq}\end{array}\right]$,

Then ${\textbf{C}}_{p\times 1}=\textbf{A}\cdot \textbf{B}=\left[\begin{array}{c}{a}_{11}{b}_{11}+{a}_{12}{b}_{12}+\cdots \cdots +{a}_{1q}{b}_{1q}\\ {}\vdots \\ {}{a}_{p1}{b}_{p1}+{a}_{p2}{b}_{p2}+\cdots \cdots +{a}_{pq}{b}_{pq}\end{array}\right]$

Appendix 3

× Cross product operation

${\textbf{A}}_{n\times 3}=\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {}\vdots & \ddots & \vdots \\ {}{a}_{n1}& {a}_{n2}& {a}_{n3}\end{array}\right]$,${\textbf{B}}_{n\times 3}=\left[\begin{array}{ccc}{b}_{11}& {b}_{12}& {b}_{13}\\ {}\vdots & \ddots & \vdots \\ {}{b}_{n1}& {b}_{n2}& {b}_{n3}\end{array}\right]$,

Then ${\textbf{C}}_{n\times 3}=\textbf{A}\times \textbf{B}=\left[\begin{array}{ccc}{c}_{11}& {c}_{12}& {c}_{13}\\ {}\vdots & \ddots & \vdots \\ {}{c}_{n1}& {c}_{n2}& {c}_{n3}\end{array}\right]$

Where$\left\{\begin{array}{c}{a}_{11}{c}_{11}+{a}_{12}{c}_{12}+{a}_{13}{c}_{13}=0\\ {}\cdots \cdots \\ {}{a}_{n1}{c}_{n1}+{a}_{n2}{c}_{n2}+{a}_{n3}{c}_{n3}=0\end{array}\right.$,$\left\{\begin{array}{c}{b}_{11}{c}_{11}+{b}_{12}{c}_{12}+{b}_{13}{c}_{13}=0\\ {}\cdots \cdots \\ {}{b}_{n1}{c}_{n1}+{b}_{n2}{c}_{n2}+{b}_{n3}{c}_{n3}=0\end{array}\right.$

Appendix 4

Table 2 Fitting error of UDM in sharply-edged and flat areas

Appendix 5

Table 3 Fitting error of CADM in sharply-edged and flat areas

Appendix 6

Table 4 Fitting error of SCM in sharply-edged and flat areas

Appendix 7

Table 5 Fitting error of MGCAM in sharply-edged and flat areas

Appendix 8

Table 6 Fitting error of THE PROPOSED METHOD in sharply-edged and flat areas