A posture evaluation method for aircraft wing based on a simple measurement system

Abstract

Posture alignment is a basic process for the joining of large-size components of aircraft, while posture evaluation is the first problem to be solved for posture alignment. The current posture evaluation method is dependent on advanced measurement equipments, which is not favorable for the wide application. Aimed at the shortcomings, this article puts forward a kind of posture evaluation method for aircraft wing. In the method, a simple measurement system composed of several linear displacement sensors is constructed, which can realize the rapid measurement of the wing posture. Then, the wing posture evaluation problem is transformed into a registration problem between space points and surface. Based on the iterative closest point algorithm, the wing posture evaluation algorithm is presented. An industrial example is presented to validate the proposed method, and the results suggest its feasibility.

Introduction

As the critical component for aircraft with aerodynamic configuration requirements, the wing’s posture accuracy plays an important role in the manufacturing quality of aircraft. In order to ensure the assembly accuracy requirements of the wing, it needs to accurately adjust the wing posture during the joining between wing and fuselage, while the measurement and evaluation for the wing posture are the foundation to realize the process. The general process of posture evaluation can be divided into two aspects: (1) getting measurement data for the target components through specific measurement systems and (2) calculating the posture parameters with an algorithm. There are many research works on these two aspects, and we survey them as follows.

As for the former, Benoit and Bernard ¹ introduced a kind of measurement-assisted assembly (MAA) system to complete the measurement and adjustment of wing posture, and the system has been successfully applied in the airbus assembly. Maisano et al. ² discussed two distributed large-volume measurement systems, the mobile spatial coordinate measuring system and the indoor global positioning system (GPS), to give some helpful guides for the application of these systems. Jamshidi et al. ³ reviewed the application of large-volume metrology in the aircraft industry and used a large-size aircraft wing to illustrate the automatic metrology-assisted assembly process. Muelaner et al. ⁴ provided an overview of the important selection criteria for typical measurement processes and presented some novel selection strategies, which are helpful in actual application. Jayaweera et al., ⁵ Jayaweera and Webb ⁶ and Muelaner et al. ⁷ described a measurement-assisted robotic assembly for aerospace applications. Galetto et al. ⁸ and Franceschini et al. ⁹ proposed several novel approaches for large-scale metrology.

As for the latter, Frederik and Hanebeck ¹⁰ proposed a new closed-form solution to posture evaluation based on range measurements and decoupling position and orientation. The proposed method can improve the accuracy of calculation. Fischler and Bolls ¹¹ applied perspective-n-point to solve the posture parameters of targets relative to the camera. Lowe ¹² proposed the least-squares model to calculate the posture parameter. Vincze et al. ¹³ systematically discussed the posture calculation method based on laser tracking, which has enhanced the motion accuracy of robot end effectors. Edouard and Shingo ¹⁴ proposed dynamical models for position measurement with global shutter and rolling shutter cameras, which can timely and accurately capture the target posture. Hiroyuki and Toshihiko ¹⁵ adopted two sets of electronic cameras with high performance to compose the measurement system and calculated the target posture with the principle of spatial angle intersection.

Based on the analysis of the literature, we can see that large-volume metrology has been widely used in aircraft manufacturing, and these posture evaluation methods can achieve a high accuracy. A metrology-assisted assembly system plays an important role in improving the level of automation and flexibility of aircraft assembly. Nevertheless, there are still a few issues that need to be addressed. For example, although the currently used metrology systems, such as mobile spatial coordinate measuring system and indoor GPS, can achieve a high accuracy, the cost of such systems is relatively high for common use and the process of using such systems is complex. It will benefit from developing a simple measurement system. In this article, a simple measurement system that is composed of several linear displacement sensors is constructed for the measurement of aircraft wing. Then, the wing posture evaluation problem is transformed into a registration problem between space points and free surface. Finally, it is realized using iterative closest point (ICP) algorithm.

Problem formulation

As shown in Figure 1, $\Omega$ and $\Omega \prime$ are the wing with theoretical posture and the wing with actual posture, respectively. $\mathrm{OXYZ}$ is the global coordinate system; $O_1{X}_1{Y}_1{Z}_1$ and ${O\prime }_1{X\prime }_1{Y\prime }_1{Z\prime }_1$ are the wing’s theoretical posture and actual posture, respectively. A certain number of measurement points are selected on the skin to represent the posture of the wing, such as $A_i$ , $i=1,2,\dots ,N$ , where $N$ is the number of measurement points. The linear displacement sensor sets under $A_i$ and moves along z direction. When touching the skin, z value of the measurement point can be obtained. Combined with x and y coordinates of linear displacement sensor, three coordinate values of the measurement point can be obtained. With the existence of error, measured z value at linear displacement sensor is not that of $A_i$ , rather than it is another point ${B\prime }_i$ on $\Omega \prime$ . The set of measurement points can be shown as $B\prime =\{{B\prime }_1,{B\prime }_2,\dots ,{B\prime }_N\}$ . Without the consideration of wing deformation and measurement error, $B\prime$ is obtained after the rotation and translation of a certain point set $B=\{B_1,B_2,\dots ,B_N\}$ on $\Omega$ . $B$ can be called the matching point set of $B\prime$ , and the rotation and translation relationship between $B$ and $B\prime$ can express the wing posture.

Figure 1.

Illustrative diagram of the measurement system.

The wing posture is expressed by six parameters, i.e., $X=[\begin{array}{cccccc}\hfill x_1\hfill & \hfill x_2\hfill & \hfill x_3\hfill & \hfill x_4\hfill & \hfill x_5\hfill & \hfill x_6\hfill \end{array}{]}^T$ , where $x_1$ , $x_2$ and $x_3$ are Euler angles; $x_4$ , $x_5$ and $x_6$ are translation parameters. Let ${B\prime }_i=(B{\prime }_{\mathrm{ix}},{B\prime }_{\mathrm{iy}},{B\prime }_{\mathrm{iz}}{)}^T$ and $B_i=(B_{\mathrm{ix}},B_{\mathrm{iy}},B_{\mathrm{iz}}{)}^T$ be the coordinates of $B{\prime }_i$ and $B_i$ , respectively. Then, the relationship between these two points can be expressed as

$$B{\prime }_i=R{B}_i+T$$

(1)

where $R$ is the rotation matrix, and the expression of $R$ can be shown in equation (2), where $\text{c}\theta =\cos \theta$ and $\text{s}\theta =\sin \theta$

$$R(X)=\left[\begin{array}{ccc}\hfill \text{c}{x}_1\text{c}{x}_2\hfill & \hfill \text{c}{x}_1\text{s}{x}_2\text{s}{x}_3-\text{s}{x}_1\text{c}{x}_3\hfill & \hfill \text{c}{x}_1\text{s}{x}_2\text{c}{x}_3+\text{s}{x}_1\text{s}{x}_3\hfill \\ \multicolumn{1}{c}{\text{s}{x}_1\text{c}{x}_2}& \hfill \text{s}{x}_1\text{s}{x}_2\text{s}{x}_3+\text{c}{x}_1\text{c}{x}_3\hfill & \hfill \text{s}{x}_1\text{s}{x}_2\text{c}{x}_3-\text{c}{x}_1\text{s}{x}_3\hfill \\ \multicolumn{1}{c}{-\text{s}{x}_2}& \hfill \text{c}{x}_2\text{s}{x}_3\hfill & \hfill \text{c}{x}_2\text{c}{x}_3\hfill \end{array}\right]$$

(2)

$T=[\begin{array}{ccc}\hfill x_4\hfill & \hfill x_5\hfill & \hfill x_6\hfill \end{array}{]}^T$ is the translation vector. The six posture parameters can be obtained by solving formula (1).

The posture evaluation algorithm

The process of solution formula (1) can be transformed into two steps: (1) searching for $B_i$ and (2) solving the six posture parameters. $B_i$ can be searched with ICP algorithm.^16,17 Let $B{\prime }_{i,0}$ and $B_{i,0}$ be the measurement values of ${B\prime }_i$ and the nearest point to surface $\Omega$ , respectively, where $i=1,2,\dots ,N$ and $N$ is the number of measurement points. Rotating or translating $B{\prime }_{i,0}$ can obtain a new point ${B\prime }_{i,k}$ , where $k=0,1,\dots$ shows the times of rotation or translation. If $B_{i,k}$ is the nearest point for ${B\prime }_{i,k}$ to surface $\Omega$ , then the sum of squared distance between $B_{i,k}$ and ${B\prime }_{i,k}$ can be expressed as

$$d_k=\sum _{i=1}^N{\Vert {B\prime }_{i,k}-B_{i,k}\Vert }^2$$

(3)

When $d_k$ is the smallest, $B_{i,k}$ is the $B_i$ point to be solved. Having $B_i$ point, the posture parameters can be obtained with singular value decomposition (SVD) method. ¹⁸ The posture evaluation algorithm can be expressed as follows:

• Step 1: Assume $k=0$ , solve $B_{i,0}$ and $d_0$ ; Given accuracy $\epsilon >0$ ;

• Step 2: $k=k+1$ ;

• Step 3: Find $R_k\left(X\right)$ and $T_k$ which minimizes $d_k=\sum _{i=1}^N{\Vert R_k{B\prime }_{i,k-1}+T_k-B_{i,k-1}\Vert }^2$ with SVD.

• Step 4: Calculate ${B\prime }_{i,k}=R_k{B\prime }_{i,k-1}+T_k$ ;

• Step 5: Search the nearest point $B_{i,k}$ for ${B\prime }_{i,k}$ to surface $\Omega$ ;

• Step 6: Calculate $d_k=\sum _{i=1}^N{\Vert {B\prime }_{i,k}-B_{i,k}\Vert }^2$ ;

• Step 7: Assume $\delta =1-d_k/d_{k-1}$ , if $\delta \ge \text{ε}$ , go to Step 2. If not, go to Step 8;

• Step 8: Find $X$ which minimizes $\sum _{i=1}^N{\Vert R(X){B\prime }_{i,k}-B_{i,0}\Vert }^2$ .

Case study

The method proposed in this article has been integrated in CATIA&DELMIA R18 and successfully applied in the assembly of a wing. With the background of this kind of wing, this article has designed a simulation model to verify the validity of the algorithm. Figure 2 shows the simulation model including the wing and fixture. The wing can be divided into three components—XX_1, XX_2 and XX_3—which will be joined together in the fixture. The fixture includes several linear displacement sensors and adjustment mechanisms. The linear displacement sensors are used to measure the posture, while the adjustment mechanisms are used to adjust the wing posture. Take XX_1 as an example; the theoretical posture of XX_1 is $X={\left[\begin{array}{cccccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]}^T$ . Drag the component in the system and change the posture as $X={\left[\begin{array}{cccccc}\hfill 0.003\hfill & \hfill 0.003\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]}^T$ . All measurement points and their matching points are shown in Table 1, where $B{\prime }_i$ and $B_i$ are the measurement point and the matching point, respectively. Posture evaluation results are shown in Table 2. From the results, it can be seen that the errors between the calculated values and the truth values of the posture are small, which can meet the requirements of industrial application.

Figure 2.

Verification system of the proposed method.

Table 1.

Measurement points and their matching points.

$i$	${B\prime }_i$			$B_i$
	$x$	$y$	$z$	$x$	$y$	$z$
1	1950	788.675	−163.489	1949.619	789.090	−163.494
2	1950	1596.965	−161.221	1949.626	1597.378	−161.138
3	3151.969	1482.632	−164.969	3151.589	1483.044	−164.975
4	3151.969	2290.923	−162.702	3151.596	2291.332	−162.619
…				…
11	−1950	788.675	−152.153	−1950.365	789.090	−152.127
12	−1950	1596.965	−149.822	−1950.358	1597.377	−149.771
13	−3151.969	1482.632	−146.627	−3152.323	1483.044	−146.601
14	−3151.969	2290.923	−144.296	−3152.316	2291.332	−144.245

Table 2.

Results.

	$x_1$	$x_2$	$x_3$	$x_4$	$x_5$	$x_6$
Truth value	0.0030	0.0030	0	0	0	0
Calculated value	0.0029	0.0029	0	0.0871	−0.0483	0.0372
Calculation error	0.0001	0.0001	0	0.0871	0.0483	0.0372

Conclusion

With the construction of a simple measurement system, this article adopts ICP algorithm to realize the wing posture evaluation. Integrating the method into CATIA&DELMIA can realize the application in the enterprise. Compared with existing methods, the method proposed in this article has the following advantages:

1. The measurement system is composed of several linear displacement sensors with simple design, convenient use and low cost. In the actual application, it can increase or decrease the number of linear displacement sensors according to the wing size.

2. ICP algorithm is used to realize the posture evaluation, which is a mature algorithm and easy to realize. The accuracy can satisfy the application requirements.