Research on the Forming Detection Technology of Shell Plates Based on Laser Scanning

Abstract

In order to solve the problems of low efficiency and insufficient accuracy of the traditional manual template method in the forming detection of shell plates, a digital solution based on laser scanning detection system was proposed. By introducing a six-degree-of-freedom robotic arm and a high-precision line laser sensor to build a three-dimensional detection platform, a digital template method framework including data acquisition, point cloud registration, surface reconstruction, and deviation analysis was innovatively constructed. A point cloud non-penetration registration algorithm fused with boundary geometric information was proposed. Based on the improved Delaunay triangulation algorithm, the surface is reconstructed and the digital template is extracted. Experimental verification shows that the method achieves an accuracy of less than 1 mm of error in the detection of outer plates, shortens the single detection time to less than 10 min, and improves the detection efficiency by more than 75% compared with the traditional method.

1. Introduction

The forming quality of hull shell plates is a key factor in the precision control of ship construction. The forming of complex curved outer plates of a hull requires cold bending and water-fire bending. Forming detection is an important process in the water-fire bending plate process, which is used to judge the forming situation and guide the prediction of secondary processing parameters, which directly affects the forming quality of the outer plate. At present, this process is usually carried out by skilled workers using templates and template boxes, which has problems such as a strong empirical element, low degree of automation, low detection efficiency and low accuracy [1].

The mainstream research direction is the research of forming detection technology based on 3D laser scanners and vision sensors.

In the study based on a 3D laser scanner, Hiekata et al. [2] at the University of Tokyo developed a 3D laser scanner-based curvature deviation evaluation system for an outer plate. The system obtained the point cloud data of the target outer plate and compared it with the CAD model to obtain the rib line curvature deviation, which was used to guide workers in secondary processing. It realized non-contact detection and got rid of the problem of insufficient versatility of wooden samples. However, the curvature deviation obtained is difficult for workers to apply to actual work scenarios. Based on the 3D laser scanner, Deng Guiyang et al. [3] established a data matching evaluation system for the detection board based on a 3D laser scanner combined with a gantry-type five-axis mechanical structure and expert system software. The problem of automatic detection of the machine after the plate forming was solved, but the method of using the gray value for point cloud segmentation could easily lead to an incomplete edge in the obtained measurement data. Also, only the rib line data of the target plate is used to match the rib line data of the measuring plate when the point cloud data is registered. Thus, the overall forming of the outer plate cannot be characterized. Furthermore, there is the problem that the positioning of the detection line is inaccurate. Yamauchi et al. [4] of the Japan Society of Marine Architecture and Marine Engineering developed a curved plate measurement and evaluation system, which can be successfully applied to the accuracy measurement of the curved plates of large container ships by setting up a 3D laser scanner on the crossbeam of the factory to obtain a point cloud of curved plates and importing them into industrial software for the detection of rib lines. The registration process adopts PCA and ICP algorithms, and the authors found that the point cloud penetration phenomenon occurs when the method is applied to curved plates with large curvatures, which is not in line with the actual situation of the project.

In a study based on vision sensors, Shin [5] of Seoul National University in South Korea used stereo vision technology to measure the curved plate of a formed ship. The linear structured light visual measurement sensor was installed on a displacement mechanism with three translational degrees of freedom to realize the in-situ measurement of the curved plate of the formed ship, but there was also the problem that the forming detection results were difficult to directly apply. Zhen Xijin et al. [6] from the Shanghai Institute of Shipbuilding Technology combined CCD and a laser emitter to establish a detection system and display the rib deviation of the measurement model in real time. The in-place detection in the bending process was realized, but the detection results could not be applied to the secondary processing. Zhao Liang et al. [7] from the Wuhan University of Technology proposed a computer binocular vision measurement method based on structured light-assisted scanning, which obtains point cloud data on the surface of a saddle plate by extracting the laser centerline coordinates in the image. The stability and accuracy of the measurement system are high, but it is limited by the laser scanning range and is not suitable for measuring large and complex curved plates. Shi Dan [8] of the Jiangsu University of Science and Technology proposed a 3D measurement technology based on binocular vision which improved the detection accuracy by improving on Zhang’s calibration method and significantly reduced the amount of noise by using the multi-threshold segmentation processing method, but the image preprocessing time of this technology was long and could not complete real-time and rapid detection work. Chen studied the application of visual technology in marine ship inspection and proposed an image restoration technology which can effectively deal with rain and fog in images, which has certain reference significance for the research content of this paper [9].

Current research focuses on the use of 3D laser scanners for forming inspection, which is time-consuming and requires a high working environment [10,11]. In addition, the existing detection methods have the problems of insufficient versatility and low efficiency of formability evaluation, which are difficult to adapt to the complex application scenarios of shipyards.

In this paper, a six-degree-of-freedom robotic arm equipped with a line laser sensor is introduced for measurement. This arrangement can achieve high-precision and large-range flexible scanning [12], and the measurement accuracy can reach up to 0.04 mm. Based on the advanced feature point recognition algorithm, the boundary information of the outer plate is obtained. The point cloud data of the outer plate was processed by ICP registration with boundary geometric information constraints. According to the actual situation of the project, the non-penetrating registration technology was studied based on the nonlinear constraint optimization problem. The rib point data was obtained from the measurement model and the design model by extracting the digital template after the reconstruction of the Delaunay triangular surface. Based on the mathematical method of spline interpolation, the formability was evaluated and the inspection report generated. The framework of the method is shown in Figure 1. In this paper, I use Matlab R2022a, Robot Studio 6.07 and Python 3.10.

Compared with the traditional manual detection method, the laser scanning system used in this paper has significant advantages. From the point of view of detection efficiency, the time of manual detection of the sail plate used in this experiment is about 1~2 times the detection time of the laser scanning system, and the automatic laser scanning system will be more efficient than the manual detection of large plates in actual engineering. From the perspective of detection accuracy, the laser scanning system can achieve sub-millimeter measurement, which has higher accuracy than visual technologies such as photogrammetry. In addition, in the complex construction environment of the shipyard, the laser sensor scanning through the high-degree of freedom industrial robot has higher flexibility and adaptability than the structured light scanning method. Compared to the detection method of the 3D laser scanner, the scanning system in this paper can adapt to harsh environments such as strong light and high temperature. The robot and sensor can also adapt to the complex work in the welding environment after inspection.

This method improves the detection accuracy to less than 1 mm, which is a significant improvement of the detection efficiency compared with the traditional method. In an actual project, the amount of bending in a plate accounts for about 10~18% of the hull steel processing workload, so it is very important to improve the efficiency of the outer plate forming detection. In order to evaluate the superiority of the method in this paper, comparative experiments were carried out on the outer plate of the same hull: for a sail plate with a size of 1200 mm × 800 mm × 15 mm, the traditional detection method using a 3D laser scanner takes about 40 min on average and the average accuracy is 2~3 mm. The method used in this paper takes an average time of 10 min and has an average accuracy of 0.5~1 mm, which proves the efficiency and accuracy of the proposed method.

2. Methods

2.1. Research on the Point Cloud Data Preprocessing

2.1.1. Surface Point Cloud Data Preprocessing

In the actual scanning process, due to the existence of other objects around the outer plate of the hull, the obtained 3D point cloud data contains the surface information of the object outside the surface, and due to the physical characteristics of the line laser, the obtained point cloud data also contains the point cloud information along the plate’s thickness direction, so the surface point cloud data need to be extracted.

The shape of the outer plate surface of the hull is characterized by smooth changes in the curvature of the plate surface and obvious changes in the curvature of the plate edge, so the use of curvature and normal vector for point cloud segmentation is considered. The regional growth method is a commonly used segmentation algorithm for processing images, and its principle is to first find a seed point for each area to be segmented as the starting point of growth, then merge the points with the same or similar properties as the seed in the neighborhood around the seed point into the area where the seed pixel is located. New points continue to grow as seeds in all directions until there are no more points that meet the conditions to be included, and the area is grown.

Due to the particularity of the point cloud data obtained by line laser scanning, the partitioning of discrete points by a regional growth method will produce a large number of clusters. In order to obtain the point cloud data of the board’s surface, it is necessary to find and merge the clusters of points located on the board surface according to their characteristics, distinguishing them from tens of thousands of clusters. However, because the point features inside the board surface are not obvious, it is difficult to segment them by conventional methods, and the expected effect cannot be achieved.

Because the point cloud data of line laser scanning is a segmented structure and the overall characteristics are not obvious, the point cloud needs to be triangulated first. This way, the area growth based on the normal of the triangular patch will be more targeted, and the point cloud segmentation effect will be better. The process of regional growth method based on triangulation is shown in Figure 2.

The segmentation result is shown in Figure 3. It shows that the method of extracting the point cloud data of a curved plate by using the regional growth method based on triangulation can effectively segment the point cloud data along the direction of plate thickness, which lays a good foundation for follow-up research work, such as registration.

2.1.2. Boundary Point Cloud Data Preprocessing

The boundary information of the outer board can help improve the registration accuracy of the point cloud data of the design model and the measurement model and realize the positioning of the inspection line. Compared with the traditional 3D detection method, the measurement method adopted in this paper has higher flexibility and thus can accurately obtain the point cloud information at the boundary of the outer plate.

The advanced boundary feature point recognition algorithm is used to calculate and output the feature points of each scan line, and the point cloud dataset on the outer plate boundary can be obtained, as shown in Figure 4. Due to the different sampling frequencies of the line laser sensor and the robotic arm, the point cloud information at the corner position is incomplete, so it is necessary to design an algorithm to obtain the corner coordinates. The algorithm steps are as follows:

(1)

External point reduction processing on the boundary point cloud dataset: for the external point that deviates from the original boundary, the external point is filtered based on the curvature value, and the point is removed at the location where the curvature changes greatly.

(2)

Spline interpolation: The boundary is expressed mathematically based on the cubic spline interpolation curve.

(3)

Base plane projection processing: The functional equations of the four boundaries are projected to the XY base plane, which is the plane where the outer plate is placed horizontally.

(4)

Finding the intersection point: The intersection point of each boundary projection is taken as the parameter of the input interpolation equation, and the average value of the function is used as the corner point.

2.2. Research on the Point Cloud Non-Penetrating Registration Method

2.2.1. Coarse Registration of Point Clouds Based on Boundary Information

Based on the boundary geometric information, the coarse registration algorithm is designed, and the steps are as follows:

(1)

Center of gravity registration: Based on the average value of the coordinates of each dimension of the point cloud data, the position of the center of gravity of the outer plate is obtained, and the translation makes the center of gravity of the point cloud of the design model coincide with the center of gravity of the point cloud of the measurement model.

(2)

Diagonal normal vector registration: According to the coordinates of the four corner points, find the cross product of the two diagonal vectors, that is, the normal vector of the geometric center of the surface, and rotate and align the two normal vectors.

(3)

Midpoint registration: Based on the calculated corner coordinates, calculate the theoretical midpoint position of the upper and lower left and right sides and find the actual midpoint position according to the minimum distance from the point cloud data to the theoretical midpoint. According to the midpoint set of the design model and the measurement model, the position of the centroid of the point set is calculated and the covariance matrix is established. The optimal rotation matrix from the design model to the measurement model is obtained by using the Singular Value Decomposition (SVD) method [13]. After rotation, the difference between the centroid of the design model point set and the centroid of the measurement model is calculated as the translation vector. Finally, the point set obtained in step 2 is rotated and translated. A set of point sets and outer plate boundary point cloud data is constructed for coarse registration, and the registration results are shown in Figure 5. The blue dots are the design values used for coarse registration algorithm experiments.

The coarse registration algorithm based on boundary feature points is compared with two commonly used coarse registration algorithms for registration accuracy and speed, as shown in the

Table 1. Comparison of coarse registration methods.

Coarse Registration Method	The Average Distance Between the Nearest Points (mm)	Average Run Time (s)
Boundary feature point registration	21.3814	5.76
SAC_IA	111.4346	46.85
RANSAC	147.4253	55.87

. In the accuracy comparison, this paper adopts the method of closest point comparison, that is, the points with the closest distance between two sets of points are found for the calculation of the average spacing, so as to achieve a more accurate measurement of the reaction registration. According to the registration results, the coarse registration method of boundary feature points proposed in this paper is more accurate and efficient than the traditional method.

2.2.2. Precise Registration of Point Clouds Based on ICP

After coarse registration, the design model and the measurement model are in a relatively close initial position, and higher accuracy registration is required in order to maximize the degree of conformity between the two. The Iterative Closest Point (ICP) registration algorithm is an algorithm that calculates the optimal transformation relationship between two sets of point clouds’ coarse iterative optimization. The ICP algorithm can be divided into point-to-point ICP and point-to-area ICP [14]. The point-to-point ICP algorithm finds the closest point in the Q point set in P, pairs it according to the two point sets P and Q, takes the sum of the Euclidean distances between all the paired points as the objective function, and solves the rotation and translation matrix corresponding to the smallest objective function, which has the characteristics of high precision, strong versatility, and a simple and intuitive procedure. The objective function can be expressed as a least squares problem as follows:

$$E(R,T)=\underset{R\in SO(D),T\in {ℝ}^D}{\arg \min }{\displaystyle \sum _{i=1}^n{W}_i}{‖R{q}_i+T-p_i‖}^2$$

(1)

R and T represent the rotation matrix and translation vector, respectively, while W represents the weight and p and q correspond to the points in two point sets. The solution to this equation commonly uses Singular Value Decomposition (SVD), where R and T are treated as independent variables for computation.

The basic principle of the point-to-plane ICP algorithm is the same as the point-to- point ICP algorithm, but when calculating the distance, the point-to-plane distance between the point and the matched point’s plane is used. Compared to the point-to-point ICP algorithm, the point-to-plane ICP algorithm takes into account the local features of the point cloud data, helping to filter out incorrect matching point pairs, making it more suitable for point cloud registration of complex curvature surfaces [15]. Point-to-plane ICP is slower than point-to-point ICP in the initial matching stage. However, because it considers local features, it helps avoid local optimal solutions during the registration process, reducing the number of iterations and speeding up convergence when handling large point cloud datasets. The nearest point pairing is performed on the new point set to generate the paired points, and iterative calculations are carried out until the distance between the generated point set and the reference point set meets the error requirement. The error threshold is set to 0.1 mm, as shown in Figure 6.

2.2.3. Non-Penetrating Registration Based on Nonlinear Constraint
Optimization Problems

In order to simulate the workflow of the shipyard workers’ manual card template [16], the design model is rotated and translated on the measurement model and the coordinate value of the design model in the deflection direction is greater than or equal to the measurement model. The optimization problem is then constructed based on the distance and minimum registration principle, which is essentially a nonlinear constraint optimization problem [17]. The general form of a nonlinear constrained optimization problem is:

$$\begin{array}{l} \min f(x)\\ s.t. g_i(x)\le 0, i=1,\cdots ,p\\ h_j(x)=0, j=1,\cdots ,q \end{array}$$

(2)

For the non-penetrating registration problem, the point cloud of the design model is Q, the point cloud of the measurement model is P, and the objective function can be expressed as:

$$f(\overrightarrow{x})={\displaystyle \sum _{i=1}^n(Q_i(\alpha ,\beta ,\gamma ,T_1,T_2,T_3)}-P_i{)}^2$$

(3)

where $\overrightarrow{x}$ represents the vector composed of the rotating Euler angle and translation components, n represents the number of points of the point cloud Q, and $Q_i(\alpha ,\beta ,\gamma ,T_1,T_2,T_3)$ represents the coordinate value of the ith point of the point cloud Q after coordinate transformation.

If the Z-direction coordinate value of the point cloud Q is greater than or equal to the coordinate value of the point cloud P, the constraint can be expressed as:

$$P_{zi}-Q_{zi}\le 0 i=1,\cdots ,n$$

(4)

The optimization problem in this paper meets the requirements of using the interior point method, i.e., the obstacle function method, which is a kind of penalty function method that can be used to solve linear programming or nonlinear convex optimization problems. The basic principle is to transform the inequality constraint problem into an equality constraint problem by introducing an obstacle function, which tends to be zero in the feasible domain but infinite outside the feasible domain, thus effectively limiting the scope of iteration. Since the optimization problem in this article contains only inequality constraints, the objective function can be written as:

$$\begin{array}{l} \min f(x)\\ s.t. g_i(x)\le 0, i=1,\cdots ,p\end{array}$$

(5)

When x moves from the inside of the feasible domain to the boundary of the feasible domain and $g_i(x)$ approaches zero, then the barrier function is constructed as:

$$B_{{\mu }_k}(x)=-{\mu }_k{\displaystyle \sum _{i=1}^n\ln \left[-g_i(x)\right]} x\in D_I k=1,\cdots ,n$$

(6)

where ${\mu }_k$ is a monotonically decreasing sequence of positive penalty factors that tends to zero and $D_I$ represents the interior of the feasible domain. An augmentation function defined inside a feasible domain can be expressed as:

$$F_{{\mu }_k}(x)=f(x)+B_{{\mu }_k}(x) x\in D_I$$

(7)

The iterative termination condition can take the value of the barrier function or the value of the constraint function:

$$B_{{\mu }_k}(x^k) \le \epsilon \mathrm{or} \underset{1\le i\le n}{ \min }\left|g_i(x^k)\right|\le \epsilon$$

(8)

At this point, the inequality constraint problem has been transformed into an equality constraint problem. When x approaches zero in the feasible domain, the value of $F_{{\mu }_k}(x)$ approaches infinity, ensuring that the optimal solution falls inside the feasible domain. The advantage of the interior point method is that it can deal with large-scale, high-dimensional constraint problems, has a fast convergence speed, and can quickly find the optimal solution through the center path when there are many constraints.

Compared with the interior point method, the Sequential Quadratic Programming (SQP) method is suitable for the optimization problem of smooth nonlinear constraints, has better performance in fitting and nonlinear regression model construction, and the optimization results are more accurate, which makes it suitable for medium-scale optimization problems. The basic principle of the SQP method is to transform the complex nonlinear constrained optimization problem into a quadratic programming problem for solving. The basic idea of quadratic programming is to use Taylor series to expand the objective function of the nonlinear constraint problem, simplify the constraints to a linear function, let the current iteration point be x^k+1, and simplify the constraints:

$$\begin{array}{l} \min f(x^k) + \nabla f(x^k{)}^{T}d+{\displaystyle \frac{1}{2}}{d}^T{\nabla }^{2}f(x^k)d\\ s.t. g_i(x^k)+\nabla g_i(x^k{)}^{T}d\le 0, i=1,\cdots ,n\end{array}$$

(9)

where $d=x-x^k$ represents the step size.

The iterative solution steps are as follows: First, the initial point is given, the convergence precision is set, and the Heisen array is initialized as the identity matrix. Then, the original objective function is simplified to a quadratic programming problem. The quadratic programming problem is solved and the optimal solution is taken as the next search direction of the original problem. In this direction, the constrained one-dimensional search of the original objective function is carried out to obtain the next iteration point x^k+1. Finally, if x^k+1 satisfies the termination condition of the given precision, the iteration is stopped and f(x^k+1) is the optimal solution of the most objective function. Otherwise, the Heisen array is modified and the iteration continues from the simplified objective function.

When the SQP method is used to process the 3D point cloud data, the initial position vector is (0, 0, 0.0055, −10, −5.0208, 50.3894), the iterative solution is (0.0044, 0, 0, 0, −10.0006, 0.5954, 10.2116), and the solution obtained by the interior point method is (0.00189, 2.6726, 3.0012, −9.9998, 25.9349, 30.0364). The first three terms of the vector represent the rotation angle and the last three represent the translational amount.

It is found that the non-penetrating registration of 3D point cloud data has high requirements for the initial point position, and due to the more variable parameters and fewer constraints the iterative process can easily to fall into the local optimal solution. Two sets of 3D point cloud data are inputted. The number of point clouds in the design model Q is 158, the number of point clouds in the measurement model P is 175, and the initial relative position is shown in the figure. The number of penetration points is 44. The penetration point is the point where the deflection difference between the design model point cloud and the measurement model point cloud is negative. The SQP method and the interior point method were used for iterative calculation.

Compare the number of iterations, convergence time, and the optimal solution of the objective function as shown in the

Table 2. The performance of the SQP method and the interior point method in 3D and 2D point cloud data processing was compared.

	Evaluation Parameters	3D Point Cloud	2D Point Cloud
SQP	Iteration duration (s)	3.19	1.49
	The number of iterations (times)	24	6
	Optimum (mm2)	1864.018	1868.359
Interior point method	Iteration duration (s)	2.22	1.28
	The number of iterations (times)	118	17
	Optimum (mm2)	2020.751	1893.478

.

Based on the accuracy and efficiency of the two methods, it can be considered that SQP is more suitable for solving the point cloud penetration problem in this paper.

2.3. Research on Digital Template Detection Technology

2.3.1. Digital Template and Detection Line Extraction

In this paper, the registered point cloud model is processed based on the Delaunay triangulation surface reconstruction method, and the digital template in the design model is extracted to detect the position to be detected in the measurement model.

Delaunay triangulation is a key technical method for point cloud data reconstruction. For a discrete set of points in a given three-dimensional space, an improved divide and conquer algorithm is used to construct the topological relationship according to the properties of the space and the maximum nature of the minimum angle, as shown in Figure 7.

Compared with the method of directly processing discrete point cloud data, this method is more in line with the mathematical characteristics of the outer plate surface, which makes the point cloud data obtained by cutting more accurate and smooth. According to the input cross-section position and cross-section normal vector, the point cloud coordinates of the digital template of the design model are obtained. The procedure steps are as follows:

(1)

Enter the cross-section position information, including the three-dimensional coordinates of the cross-section origin and the normal vector of the cross-section.

(2)

Obtain the triangle patch and vertex index. Obtain the vertex index from the patch index, then obtain the vertex XYZ coordinate index.

(3)

Filter the triangular patch according to the cross-section information. First, the projection of each triangular patch perpendicular to the cross-sectional direction is obtained, according to the cross-sectional position such as X = X₀. Then, the three coordinates X₁, X₂, X₃ of each triangle patch in the X direction are sorted and the triangular patch through the cross-section is obtained according to the relationship X₀ ≤ Max(X_i) & X₀ ≥ Min(X_i). Then, the triangle patch is divided into a right triangle and non-right triangle, and the position relationship between the triangle patch and the cross-section is divided into three categories according to the sorting results, namely, the cross-section located between the left vertex and the middle vertex (X_L < X₀ < X_M), the cross-section located at the middle vertex (X_M = X₀), and the cross-section located between the right vertex and the middle vertex (X_M < X₀ < X_R).

(4)

Calculate the intersection point of the cross-section and the triangle patch. For the right-angled triangle, if the cross-section is tangent to the right-angled edge, the two vertex coordinates corresponding to the right-angled edge are output. Otherwise, the intersection is calculated according to the law of similar triangles. For non-right-angled triangles, the intersection coordinates are calculated according to the spatial coordinate calculation method according to the classification results. The classification of triangular patches is shown in Figure 8.

Through the design program form, the user can observe the position relationship between the cross-section point coordinate information and the triangular patch in real time by inputting the point coordinates and normal vectors, as shown in Figure 9. In a departure from the traditional method, before solving the cross-section normal, this paper first locates the key point of the position to be cut through the boundary feature point and obtains the normal vector of the position through the PCA method.

2.3.2. Standard of Detection

The existing inspection standard for the forming of the outer hull plate refers to the China Shipbuilding Quality Standard (CSQS) issued in 2016 [18]. For a hyperbolic plate, the allowable limit of the deviation between the cable and the reference line on the template is 3 mm, the standard range is 2 mm, the allowable limit of the gap between the rib direction and the template box is 5 mm, and the standard range is less than or equal to 4 mm. The allowable limit of the gap between the length and the template box is 5 mm, and the standard range is less than or equal to 3 mm.

In order to solve the problems of the single index of the traditional detection standard and ignoring the smoothness of the outer plate, this paper proposes a digital forming detection standard for the complex curvature of an outer plate which is based on the transverse detection line, the longitudinal detection line, and the diagonal detection line and adopts the evaluation index that integrates the deflection deviation and bending degree deviation to carry out the forming detection of the complex curvature outer plate. The evaluation indicators are as follows:

(1)

Deflection deviation integral

The data points are interpolated to the function equation of the two sets of curves, and the difference between the equations in the deflection direction is integrated according to the coordinate range of the two sets of data points and in the common definition domain of the curve equation:

$$S_{\mathrm{Z}}={\displaystyle {\int }_a^b\left|f_{\mathrm{design}}(x)-f_{\mathrm{measure}}(x)\right|}dt$$

(10)

where ƒ_design and ƒ_measure are the detection line function equations corresponding to the design model and the measurement model, respectively.

(2)

The average value of the deflection deviation

The integrals of the difference in the deflection direction of the parametric equation divided by the length of the integration interval are obtained:

$$S_{\mathrm{Z}\text{-}\mathrm{average}}={\displaystyle \frac{1}{b-a}}{\displaystyle {\int }_a^b\left|f_{\mathrm{design}}(x)-f_{\mathrm{measure}}(x)\right|}dt$$

(11)

(3)

The maximum value of deflection deviation

Referring to the allowable limit in the manual testing standard, the maximum value of the difference in the direction of the deflection of the parametric equation is calculated:

$$Z_{\max }=\max \left|f_{\mathrm{design}}(x)-f_{\mathrm{measure}}(x)\right|$$

(12)

The smoothness of the surface is an important factor in the forming detection of the outer plate of the hull, and one of the important bases for evaluating the smoothness of the outer plate is the change of the bending degree of the outer plate, so in order to measure the change of the bending degree of the detection line, the following indicators are used for evaluation:

(4)

Curvature deviation integral

The curvature integral is a cumulative value used to express the difference in the degree of curvature of the design surface and the measurement surface at the test line. The formula is as follows:

$$CI={\displaystyle {\int }_L\left|K_{\mathrm{design}}(s)-K_{\mathrm{measure}}(s)\right|}ds$$

(13)

where K_design and K_measure are the curvature equations of the design model and the measurement model detection line, respectively. L is the total arc length of the detection line.

(5)

The average value of curvature deviation

The average curvature deviation is calculated by dividing the curvature deviation integral by the total arc length of the detection line:

$$S_{\mathrm{C}\text{-}\mathrm{average}}={\displaystyle \frac{1}{L}}{\displaystyle {\int }_L\left|K_{\mathrm{design}}(s)-K_{\mathrm{measure}}(s)\right|}ds$$

(14)

The arc length is calculated using an approximate formula:

$$L={\displaystyle {\int }_a^b\sqrt{1+[f^{\prime }(x){]}^2}}dx$$

(15)

(6)

The maximum deviation of curvature

Refer to the allowable limit standard for deflection deviation and set the maximum curvature deviation:

$$C_{\max }=\max \left|K_{\mathrm{design}}(s)-K_{\mathrm{measure}}(s)\right|$$

(16)

3. Results

The laser scanning system used in this paper consists of: ABB robot, robot control cabinet, host computer and operating software, switch, line laser sensor, and gripper. The robot model is IRB 4600-40/2.55 produced by ABB (Oerlikon, Switzerland) and the line laser sensor model is HD8-0050W produced by Suzhou Bozhida Laser Technology Ltd. (Suzhou, China).

The conversion method of the robot coordinate system and the sensor acquisition data is as follows: Based on the robot’s kinematics, the robot’s DH parameters are derived, the conversion matrix of the laser sensor coordinate system and the robot base coordinate system is obtained, the C# development environment is combined with the Robot Studio online programming function, the robot communication software is written, and the host computer program is designed to interact with the coordinate information obtained by the robot and the sensor in real time. Finally, the acquisition and storage of the three-dimensional point cloud data on the surface of the curved plate are realized in the actual measurement. The coordinate system used in this paper is the coordinate system of the robot base, the origin is the robot base, the X-axis direction is the guide rail direction, the Y-axis direction is perpendicular to the guide rail direction, and the Z-axis direction is perpendicular to the ground.

The accuracy of the laser scanning system itself is 0.1~0.5 mm. After data processing, the accuracy can be controlled to within 1 mm, assuming that due to the influence of vibration and other factors, the scanning system will produce a large number of external points. However, based on the point cloud preprocessing method in this paper, these external points can be effectively filtered. Assuming that the laser scanning system performs perfectly and achieves a measurement accuracy of 0.1 mm, the detection accuracy can be controlled to within 0.5 mm.

The object of this paper was a sail-shaped plate with a length of 1200 mm, a width of 800 mm, and a thickness of 15 mm. According to the size and shape of the sail board, 14 equally spaced transverse detection lines with a spacing of 80 mm, 7 equally spaced longitudinal detection lines with a spacing of 100 mm, and two diagonal detection lines were taken, as shown in Figure 10.

After the measurement model and the design model were registered, the digital template was extracted from the design model and the detection line was extracted from the measurement model through the digital template extraction technology, as shown in Figure 11.

In order to carry out comparative experiments, a traditional 3D laser scanner was used to inspect the same shell panel, and the time from scanning to the production of inspection results was recorded. The experiments showed that the average scanning time of the traditional method was about 25 min, while the average scanning time of the method used in this paper is about 4 min. The data processing time of the traditional method is about 15 min; the data processing time of the method used in this paper is about 6 min.

The method process adopted in this paper is as follows: First, based on the offline programming simulation technology, the laser scanning system is imported into the simulation environment, the scanning path is set, and the pre-written robot motion instructions are transmitted to the physical teaching pendant through the synchronization to the teach pendant function of Robot Studio. The position status of the robot can be monitored in real time in the software. Then, the scanning system is manipulated by motion commands to automatically scan the board’s surface and edges. The point cloud data obtained by scanning are fed into the host computer program in real time. After the scanning, based on the point cloud data preprocessing method described above, the point cloud data outside the curved plate are segmented to obtain the target point cloud on the surface of the curved plate. Then, the coarse registration based on boundary feature points and the fine registration based on ICP is carried out, so that the measurement surface model and the design surface model are in the same coordinate system and fit to the greatest extent.

In order to make the relative position of the two conform to the actual situation of the project, the position optimization of the design model and the measurement model is carried out based on non-penetrating registration technology so that the design model is located above the measurement model and tends to fit. Finally, based on the digital forming detection standard, the transverse, longitudinal, and diagonal detection lines were detected, the method of combining numerical integration and sampling points was used to calculate the deviation value of the detection line, and the average spacing of the sampling points was 1 mm, which ensured the distribution density of the sampling points and avoided the limitations of insufficient accuracy of the traditional detection methods. The generated detection report is shown in

Table 3. Transverse forming detection report.

Detection Line Serial Number	SZ (mm2)	SZ-average (mm)	Zmax (mm)	CI	SC-average (mm−1)	Cmax (mm−1)
1	375.8725	0.4764	0.8807	1.8057	0.0023	0.1582
2	356.7683	0.4551	0.8964	3.6764	0.0047	0.0155
3	422.3746	0.5385	0.8770	4.8851	0.0062	0.0649
4	340.2731	0.4327	0.7411	1.5328	0.0019	0.0523
5	411.7066	0.5235	0.8595	2.1162	0.0027	0.2678
6	411.9992	0.5239	0.8627	1.5823	0.0020	0.2822
7	448.9025	0.5691	0.9254	3.7085	0.0047	0.7123
8	408.2246	0.5155	2.2003	3.3758	0.0042	0.1708
9	448.0905	0.5657	1.0755	5.1318	0.0064	0.5579
10	557.3721	0.7092	1.2392	2.7188	0.0034	0.2987
11	512.2590	0.6523	1.2647	4.3445	0.0052	1.2255
12	548.2442	0.7024	1.2141	2.9446	0.0037	0.2882
13	358.0468	0.4527	0.7706	2.2842	0.0029	0.0409
14	443.7020	0.5620	1.0194	1.3883	0.0017	0.0395

,

Table 4. Longitudinal forming detection report.

Detection Line Serial Number	SZ (mm2)	SZ-average (mm)	Zmax (mm)	CI	SC-average (mm−1)	Cmax (mm−1)
1	788.9903	0.6564	1.1554	9.7685	0.0070	0.1417
2	764.9880	0.6380	1.4043	9.9301	0.0083	0.3391
3	770.4791	0.6399	1.2152	8.7202	0.0072	0.4177
4	637.4193	0.5324	1.1445	11.9461	0.0100	0.0669
5	656.6828	0.5470	1.0935	7.8518	0.0065	0.1204
6	1462.5702	1.2106	1.7366	18.7312	0.0143	0.1882
7	570.4808	0.4747	1.2329	13.5704	0.0113	0.1998

and

Table 5. Diagonal forming detection report.

Detection Line Serial Number	SZ (mm2)	SZ-average (mm)	Zmax (mm)	CI	SC-average (mm−1)	Cmax (mm−1)
1	1571.0582	0.9338	1.8061	8.7122	0.0052	0.0194
2	1991.5769	1.1651	2.0169	8.6182	0.0050	0.0905

.

4. Conclusions

In order to solve the problems of low efficiency and insufficient accuracy of the traditional manual card template method in the forming inspection of shell panels, a digital forming detection method based on a laser scanning system is proposed, and the conclusions are as follows:

(1)

The coarse registration method based on boundary information and the point cloud non-penetration registration method effectively solve the problems of low matching efficiency and point cloud penetration of complex surfaces, so that the detection results are in line with the actual engineering conditions.

(2)

Based on the digital template and detection line extraction technology of Delaunay triangulation, any number of detection point data can be quickly extracted with a detection accuracy better than 1 mm.

(3)

Based on the digital forming inspection standard, the visual inspection report can be quickly generated. The detection efficiency is increased by more than 75% compared with the traditional method, and the detection results can directly guide the prediction of secondary processing parameters, which provide a reliable technical solution for the digital inspection of the outer hull panel.

The innovations of this paper are as follows:

(1)

Aiming at the point cloud penetration problem that occurs after ICP registration of the design surface model and the measurement surface model, a point cloud non-penetration registration method based on the nonlinear constraint optimization problem is proposed. The registration algorithm is designed based on the SQP method, which effectively solves the point cloud penetration problem and makes the digital sample detection method in this paper conform to the actual situation of the project.

(2)

A digital forming detection method of a curved hull plate combined with deflection deviation and bending degree deviation was proposed, and the detection object was divided into transverse, longitudinal, and diagonal detection lines. The overall forming situation of the curved plate was effectively reflected through the indicators of deviation integral, average value, and maximum value, and the accuracy of the digital forming detection method was improved.