Geometric deviations inevitably occur in product manufacturing and seriously affect the assembly quality and product functionality. Assembly simulations on the basis of computer-aided design (CAD) package could imitate the assembly process and thus find out the design deficiencies and detect the assemblability of the components. Although lots of researches have been done on the prediction of assembly variation considering the geometric errors, most of them only simplify the geometric variation as orientation and position deviation rather than the manufacturing deformation. However, in machinery manufacturing, even if the manufacturing defects are limited, they could propagate and accumulate through components and lead to a noncompliant assembly. Recently, many point-based models have been applied to assembly simulation; however they are mainly interested in simulating the resulting positions of the assembled parts and lack the consideration of the postprocessing after positioning. This paper enriches the complete assembly simulation process based on skin model and presents a simple and effective posture evaluation and optimization method. The studied approach includes a software algorithm applied to evaluate the contact state of the assembly parts and a mathematical model based on the particle swarm optimization to acquire the optimal assembly posture. To verify the efficiency and feasibility of the proposed method, a case study on the aircraft wing box scaling model assembly is performed.
Traditional assembly simulation aims at evaluating the assemblability of the components and to predict the interference between assembly parts, which is normally based on the nominal CAD models. The virtual assemblies so obtained have the limitation of considering the nominal geometry only. In order to calculate the assembly variation more accurately, many researchers have devoted their efforts to modeling and analyzing the assembly variation on the basis of geometric deviations. Monte Carlo simulations are commonly used in tolerance analysis [
The skin model is a basic concept within Geometrical Product Specification (GPS) and verification and attracts more and more attention in computer-aided tolerance (CAT) analysis [
The difference between theoretical model and skin model.
Posture evaluation is a necessary task in large structure assembly, which is aimed at calculating the posture deviations of the assembled part and using the calculated results to guide the position and orientation adjustment of the components. B. Zhang et al. [
This paper enriched the complete assembly simulation process based on skin model. Not only was the numerical solution of the posture optimization problem presented in detail, but also the contact states of the assembled parts as well as the assembly gaps are evaluated quantitatively, which could be applied as a guide in the assembly postprocessing (shimming or fettling) to improve the assembly efficiency and accuracy. The profiles of mating surfaces are measured and represented by point clouds. The assembly deviation distribution between multiple components is investigated quantitatively. The structure of this paper is organized as follows. Section
In this section, the skin models of the manufactured parts are firstly positioned according to their assembly process. Then a constrained optimization algorithm on the basis of particle swarm optimization (PSO) is presented to adjust the part posture and the transformation matrices of adjustment are obtained. In order to describe the algorithm conveniently, some terms are defined as follows: Assembly coordinate system (ACS) is termed as Measurement coordinate system (MCS) is the default coordinate system of optical facilities and can be defined as Local coordinate system (LCS) for the movable components during assembly is known as
Classification of part postures: Theoretical posture is the position and orientation of the digital model in the ACS Measurement posture is acquired through the three-dimensional laser scanner in the MCS Target posture, i.e., the final state of the panel, is set according to the adjusting result
The complete procedure of assembly simulation and posture optimization for mechanical assemblies consists of three main steps, i.e., the skin model positioning of the assembled parts, the software algorithm of the posture optimization, and the assembly postprocessing, as demonstrated in Figure
The assembly simulation and posture optimization process.
The assembly simulation conducted in various CAD packages such as CATIA, UG, and SolidWorks is usually worked as a detection tool of product design, which is normally based on a nominal CAD model measured in ACS. They are not capable of dealing with discrete geometry model and do not allow the consideration of form errors. The skin model could however reflect the real surface of the manufactured part but the application in assembly simulation is still a challenge.
The positioning of skin models is based on the registration of their point clouds. The iterative closest point (ICP) method introduced by P. Besl et al. [
Skin model positioning based on assembly features.
Due to the form errors on the component surfaces, the interference or gaps inevitably occur between the mating surfaces. Although special clearance gauges are used to quantitatively measure the gaps, the handling process is relatively of low efficiency. Not only are highly skilled engineers required, but also multiple trials are needed to achieve the desired shape. To eliminate the gaps between the mating surfaces, forces are applied to deform the part, which could induce assembly stresses around the structure and impair the reliability. Alternatively, shims could be used when the gaps are relatively large [
Wing box panel assembly.
Since part
In order to consider the assembly process and the actual assembly situation, constraints need to be added onto the transformation matrices
The point clouds of two assembled parts.
In order to check the contact status of the mating surfaces, the signed weighted distance
In this problem, we define the unsolved posture as a particle and the six parameters (
Another key issue that needs to be solved in this optimization problem is that the objective function has an inequality constraint which was imposed to avoid interference between assembled parts. Since PSO is primarily applied on unconstraint optimization problems, a constraint-handling technique was added. There are several approaches that can handle constraints within the framework of PSO algorithm. For example, many penalty-based PSO approaches have been developed and examined in literatures [
The procedure of the PSO algorithm is the particle searching their position and velocity according to the fitness of the corresponding objective function and the constraints to find the optimal solution. The calculating procedure of the PSO algorithm is as follows.
After random initialization, calculate the fitness so as to find the optimal position
Set evolution times
Update the velocity vector of every particle according to (
Update the position vector of every particle according to (
Update
If the solution satisfies accuracy requirement or the iteration time satisfies
After every iteration, the points on part
Generally, the gaps between mating surfaces could be assessed using special gauges or other manual inspection techniques; then, based on the detection results, components need to be shimmed or fettled to maintain the assembly tolerances. However, this detection process is relatively time-consuming and of low efficiency. Based on the aforementioned iteration algorithm in Section
After the posture optimization process, all the assembled parts are positioned on the optimal position. Then, the assembly deviations between the multiple parts could be evaluated. Parts can only generate assembly deviations between mating surfaces, which means other surfaces are not involved in the assembly and do not affect the deviation results. Hence, in order to simplify the calculation, only the mating surfaces are extracted. Figure
Mating surfaces extraction.
After extracting the mating surfaces of the assembled parts, an interference calculation method based on the point cloud slicing is proposed. As skin models are dense and discrete points, it is hard to get a complete section curve on one slice plane. Thus, a section step size
Point cloud slicing diagram.
Assembly deviation calculation procedures.
The proposed posture optimization algorithm and assembly deviation calculating method has been applied to a real aircraft wing box scaling model assembly process to validate the feasibility and efficiency. The structure of the wing box consists of two main parts, i.e., the skeleton, which contains ribs and spars, and the panel. The posture optimization method proposed in this paper is applied on the adjustment of the panel. The main steps of the assembly simulation and posture optimization experiment consist of the positioning of the assembled parts, the posture optimization, and the assembly postprocessing, which have been discussed in Section
The skin models of the manufactured components were obtained by the FARO Edge 2.7 optical measurement system in this experiment. Since scanned points are dense and noisy, some preprocesses of the point clouds need to be conducted. Distorted point removal: Since the data point clouds were obtained through optical devices, the background points are inevitably scanned and cause some noise. These points must be filtered out before the analysis Data reduction: The number of scanned points could be up to millions and huge computing cost will be needed to handle the mass points. However, the more points do not mean the higher computing accuracy, so appropriately simplified point clouds could maintain the calculating precision. Furthermore, data reduction improves the computing efficiency and hence saves cost Multiview point cloud registration: Since the manufactured parts are large and complex, we could not obtain the whole surfaces of the model by one scan. Thus, the multiview registration is implemented to get a complete model, which could be realized by the common characteristic points registration on different point cloud pieces
After the preprocess of the point clouds, the skin models of the manufactured parts could be obtained. Then the multiple parts are located according to the assembly process and the coordinate systems are transformed from MCS to ACS. Figure
The skin model acquisition: (a) scanning process, (b) the panel, (c) the rib, and (d) the spar.
During the wing box assembly process, the skeleton is set as assembly datum and the panel is moved and adjusted to fit the skeleton. The signed weighted distance
According to Garg [
and the global optimal position
Based on the improved PSO algorithm, an optimization mathematical model under certain constraints is solved by Matlab®. Since PSO is a stochastic algorithm, statistical method was applied in the experiment to verify the reliability of the results, as adopted in literatures [
Statistical comparisons for the experiment.
Best | Median | Worst | Mean | Standard deviation |
---|---|---|---|---|
| 85.505687 | 87.378064 | 85.953390 | 1.060213 |
| 83.258915 | 84.182042 | 83.472345 | 0.532362 |
The posture adjustment parameters.
| | | | | |
---|---|---|---|---|---|
| 7.042101 | -5.130164 | -0.003009 | 0.002831 | -0.003804 |
| 8.697785 | -7.096523 | 0.000597 | -0.001751 | -0.001828 |
Wilcoxon rank sum test: Ranks of two methods
Method | N | Mean Rank | Sum of Ranks |
---|---|---|---|
Improved Method | 20 | 10.50 | 210.00 |
Previous Method | 20 | 30.50 | 610.00 |
Wilcoxon rank sum test: Test statistics of two methods
Mann-Whitney U | Wilcoxon W | Z | Asymp. Sig. (2-tailed) | Exact Sig. [2 |
---|---|---|---|---|
.000 | 210.000 | -5.410 | 6.3018E-8 | 1.4509E-11 |
After that, the virtual assembly process of the whole structure is simulated by CATIA®, as shown in Figure
Assembly simulation: (a) virtual assembly of the wing box and (b) assembly deviation calculation.
The assembly deviations of the panel and the skeleton before and after adjustment are calculated, respectively, by the improved algorithm, and the percentage of deviation points is counted, as illustrated in Figure
The percentage of deviation points: (a) nominal position and (b) optimal position.
Measurement points on the actual part.
Geometric deviations inevitably exist in every manufactured part and even a tiny form error may induce noncompliant assembly. However, only the orientation and position deviations are considered in most models in analysis of geometric variation rather than the form errors. This paper developed an approach for the assembly simulation and posture adjustment based on the skin model, which enables the prediction of the assembly deviation caused by part geometric errors.
In this method, the mechanical assembly of rigid parts with manufacture deviation is investigated and a numerical optimization method for the posture adjustment is employed. For the positioning of the skin model, an assembly feature based registration method is applied to improve the accuracy and efficiency of point cloud registration. Then the PSO algorithm is employed to solve an optimization mathematical model so as to search the optimal posture of the assembled part. The numerical method is validated with experimental data obtained from a real wing box assembly in a production floor. Afterward, the assembly deviation on the mating surfaces is calculated based on the point cloud slicing method, which could quantitatively evaluate the contact quality of the multiple parts. Before computing the deviation, the mating surfaces of the assembled parts are extracted according to the change of the normal vector at the boundary regions. Finally, a case study of the aircraft wing box scaling model is conducted. The panel is adjusted and located according to the final transformation matrices, and the assembly deviations of the skeleton and the panel are calculated. It can be seen that, after adjusting, the overall gaps between the assembled parts become uniform, which could facilitate later assembly process, and the final shape of the structure is within the tolerance. This paper discussed the assembly simulation and part posture adjustment on the basis of point clouds, which is effective in reducing the handling work (such as fettling or shimming) to maintain the interface tolerances. However, this simulation model simplified the posture adjustment as rigid transformation; more complex assembly processes considering the bolt fastening force and part deformation need to be combined with the skin model representation in future research works. The authors are already working on the extension of the methodology employed in this paper to more complex nonrigid assembly situations. In future work, the issues of more different assembly deviation calculation methods will also be addressed.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work is supported by the National Science and Technology Major Project of the Ministry of Science and Technology of China (Grant No. 2014ZX04001071).