The models presented in Section 2.2 serve as an aid to understand the propagation of variation during the assembly process, while the techniques presented in Section 2.5 illustrate the operations that can be used to control the variability. However, representing and interpreting this variability is also of significant importance as it is often highly correlated and has a significant influence on the ability to monitor and diagnose faults, as discussed in Section 2.3. With the increasing use of three dimensional point cloud measurement devices, understanding the process variability across the complete structure, not just at discrete points, is required. This section provides a review of methods for interpreting and visualising variability in both measurement data and model-based estimations.
Variations in the production of sheet metal components can result in a variety of errors to the final component output. These errors can be: localised to a specific
§2.6 Characterising shape variations 39
region, such as a ripple in the surface; specific to a feature, such as spring back in a flange; or distributed across the component, such as component twist. Further errors can present themselves during the assembly stage. Camelioet al.(2004b) have identified three main sources of variation during the assembly process: fixture variation, component variation and joining tool variation. When measurements are made, the component and assembly errors result in highly correlated data due to the geometric covariance of the continuous sheet metal surface (Merkley, 1998). Using pattern recognition techniques, correlated data can be analysed and the uncorrelated variation patterns extracted. A common approach to identifying these patterns is known as Principal Component Analysis (PCA), which identifies and ranks the contribution of each pattern in a data set. The largest variations in a measurement set can then be identified. Utilising PCA, Hu and Wu (1992) analysed mean shifts in panel placement and successfully identified the dominant variations.
The field of computer vision has seen extensive research in the area of shape and variation representation. One particular method that has seen widespread applications is the Point Distribution Model (PDM) presented by Cooteset al.(1992, 1995). The technique is based on the application of PCA to generate an initial training model. PCA linearly separates the correlated patterns in the original training data and allows a new geometric measurement to be mapped onto this reduced dimensional space for evaluation. The PDM technique has successfully been applied to areas involving medical imaging (Hill et al., 1993), sheet metal stamping variation (Rolfe et al., 2003a,b), and sheet metal stamping tonnage signatures for component failure identification (Doolanet al., 2003).
Since PCA is a linear technique the method can struggle with representing non- linear correlations. This is an issue that can be circumvented by the application of a kernel based PCA approach. Unlike traditional PCA, which creates a linear mapping, kernel PCA approaches are non-linear. This is achieved by an initial data transform to a feature space with a different dimension prior to applying PCA. Then, when
PCA is performed, the linear mapping is based on the feature space, not the initial data space. Therefore, in relation to the original data space the mapping becomes non-linear. However, it can be difficult to determine an appropriate kernel because the kernel needs to represent a mapping of the non-linear information to a linear representation, and as a result it is highly data dependent.
Other shape representation techniques that have been proposed include the use of fractals (Liao and Wang, 2005a) and wavelets (Liao and Wang, 2005b). The primary focus of these works has been on representing smaller scale levels of variability that would otherwise be missed with techniques such as PCA which specifically identifies the most dominant variations.
While the PDM is a powerful technique, it is also completely data-driven. Consequently, any representation will not be based on the properties of the underlying geometry but purely on the contents of the training data and is therefore sample specific, which can result in the production of inconsistent models (Huang
et al., 2013). Although the training data can be established to relate to meaningful properties, it must be orthogonal and independent prior to training. PCA generally results in a loss in the physical significance in the resulting principal components, as two or more potential variations that contain correlated information are separated. Alternative approaches utilising model-based data have been developed as methods for maintaining underlying geometric information. Specifically, the use of geometric stiffness properties and modal shapes as a representation of variations (Pentland and Sclaroff, 1991; Terzopoulos and Metaxas, 1991; Nastar and Ayache, 1993).
To represent part form error for compliant components using a model-based technique, Huang and Ceglarek (2002) presented the Discrete Cosine Transform (DCT) as a method for geometric representation. A significant advantage of this representation is the ability to relate form error directly to interpretable representations of variability, such as twist or bow of the component. Furthermore, any geometric form can be expressed by using a larger number of representative
§2.6 Characterising shape variations 41
components. A limitation of the DCT method is the height field assumption where
z = f(x,y), which limits the technique to single panels without bent vertical flanges. Consequently, application of the method to a three dimensional assembly is difficult, unless done on a part by part basis. Huang et al. (2013) more recently presented Statistical Modal Analysis (SMA) as a method for quality control of compliant components. In this work DCT was also used for representation, although it was applied as just an example of one of many possible modal analysis based shape representations. Huang and Kong (2008) then further developed this approach by combining it with methods to also monitor rigid body movements during the assembly process.
Once a representation has been obtained, either data or model-based, the next step is utilising the information for visualisation, quality control or diagnosis. Based on PCA, Wells et al. (2011) presented a framework for variation visualisation that allowed for complete visualisation monitoring of variation patterns within the data by morphing the underlying geometry in the virtual world. In doing so, the approach allows for an interpretable understanding of the principal component vectors. Lindau et al. (2013) also presented an application of using PCA based models in the virtual environment; however, with the aim of simulating a virtual assembly process. An important aspect here is understanding the variability of the shape representation. Cootes and Taylor (1999) developed an extension to the PDM using a mixture model approach for representing the distributions within the shape variations. This was extended utilising kernel density estimation in the application to the Kernel Density Estimate - Point Distribution Model (KDE-PDM) presented by Matuszyk et al. (2010) for shape variation representation in manufacturing quality control.
Off-line methods of interpreting variability have been investigated by Söderberg
et al. (2008) using a virtual environment. Further development has also been completed by studying the human interpretation of the computer based
visualisation (Forslund et al., 2011). This is an important aspect to understanding variability relationships between the pre-production design phase and the perception of quality in the final production output.
All of the techniques mentioned thus far, particularly the data-based models, rely on careful selection of measurement information. Given the increasing use of three dimensional point cloud acquisition technologies, more data are obtained than is required. This poses two problems: the time involved in acquisition, and the necessary processing requirements. Wells et al. (2012) discussed utilising high-density point cloud data to reduce the required measurements when using multiple acquisition technologies, while Wärmefjord et al. (2009) investigated the loss of information when reducing measurement points. The influence on sample size with reduced inspection point data has also been considered (Wärmefjordet al., 2010a). While reducing the number of points is an option, selecting which points to remove and which will remain creates other issues. When specific shape variations are known, there are a variety of techniques that can be applied, including the Effective independence (Efi) method (Kammer, 1991), genetic algorithms (Yao et al., 1993) and average mutual information (Trendafilovaet al., 2001), amongst others.
This section has presented a review of a variety of shape representation methods for compliant components. Two distinct segmentations have formed in the design-to-manufacture product life cycle regarding shape representation. They are: methods for variation simulation and tolerance analysis of compliant components, and methods for process control and root cause diagnosis of variation for the components. However, recent efforts have seen attempts at utilising heavily data-based methods, from the later category, for the application of variation simulation and tolerance analysis earlier on in the design phase. In these situations the required information to generate the appropriate model has usually not been obtained. Huang et al. (2013) also noted this disconnect when presenting the SMA, intended as a holistic approach to variation analysis. Furthermore, these techniques