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Article Excerpt
Notation
[t.sub.i] = coordinates of the ith locator in a reference co-ordinate system;
[v.sub.i] = small deviations of the ith locator from its nominal positions;
U = the small tolerance on the variation of the locator position;
q = location and orientation deviations of the parts;
[q.sub.o] = nominal location and orientation of the part;
[s.sub.i] = process design parameters;
[gamma] = parameters of the surrogate model for a single part;
[n.sub.i] = outgoing norm vector at the ith locator nominal position;
[beta] = parameters of the a single rational function;
[^.[beta]] = least square estimates of parameter [beta];
J = Jacobian matrix of the constraints of the locators;
A = coordinate transformation matrix from global coordinate to reference coordinate;
[H.sub.q] = homogeneous transformation matrix of reference system from deviated position to nominal position;
[(A).sub.[i,j]] = the element at the ith row and jth column of matrix A;
[N.sup.k](t) = kth-order polynomial function of variable t;
[q.sub.i] = the ith element of q;
[a.sub.i.sup.j] = the jth rational components of [q.sub.i];
d[P.sub.i] = dimensional deviation of mating point i propagated from previous stage.
1. Introduction
In recent years, dimensional variation reduction has become a crucial engineering objective during all stages of a product life cycle to maintain high-quality products while achieving ever shorter time-to-market requirements. Consequently, the modeling of dimensional variation propagation in multistage manufacturing processes has drawn significant attention. The dimensional variation propagation models are mathematical descriptions of the relationship between the dimensional quality of the final product and the various process parameters (e.g., the fixture layout, locator position deviation and the inaccuracy in machine geometry) and provide a quantitative basis for process design optimization and process monitoring and diagnosis.
The existing approaches for variation modeling in multistage manufacturing processes can be classified into two categories: analytical approaches and numerical approaches. In the analytical approaches, the modeling of the variation propagation is based on a physical analysis of the basic underlying operations of complicated manufacturing processes. A set of closed-form equations describing the relationship between the process variation sources (e.g., the variation of the positions of fixture locators and the variation of the incoming raw parts) and the product dimensional quality for each manufacturing stage is first derived; then, these equations are linked together to form the overall model for the complete multistage process. To make the derivation analytically tractable, the higher-order terms are often ignored in the derivation through a linearization procedure. Analytical variation propagation model shave been successfully derived for assembly processes (Shiu et al, 1996; Jin and Shi, 1999; Ding et al., 2000; Ceglarek et al., 2004) and machining processes (Huang et al., 2000; Djurdjanovic and Ni. 2001; Zhong et al., 2002; Zhou et al, 2003: Loose et al., 2007). These analytical models provide a theoretical foundation for process monitoring and diagnosis to identify the major variation sources in the process (Ceglarek and Shi, 1996, 1999; Ding et al. 2002a; Ding, Zhou and Chen, 2005; Li and Zhou, 2006; Li et al., 2007), process-oriented tolerance allocation (Ding et al., 2002b; Ding, Jin, Ceglarek and Shi, 2005) and sensor distribution optimization (Khan and Ceglarek, 2000; Ding et al., 2003). In the numerical approaches, practitioners rely on computer simulation to describe the underlying physical relationships between the product quality and process parameters. Several software packages, such as Tecnomatix, Sigmetric and Dimensional Control Systems (DCS), arc available to simulate the variation propagation in multistage assembly processes, particularly in automotive assembly processes. These simulation models can often describe very complicated interactions within the manufacturing process and provide realistic results for large-scale systems. However, given the very large number of parameters typically included in an analysis, it is often very time-consuming to establish a simulation model and for certain physical processes, it is also very time-consuming to finish one simulation run. Thus, these models are currently used most in process design validation.
From a brief review of the existing modeling approaches for variation propagation, it can be seen that a common shortcoming of existing approaches is that they only describe the variation propagation under fixed process design settings, such as fixture layout. In other words, many very important process design parameters (e.g., the nominal positions of the fixture locators) are treated as constant values in these models. Thus, these models cannot be used in a process design optimization on these parameters. Although, in theory, all design parameters can be treated as free variables instead of known constant values in the analytical approaches, the extremely large number of design parameters and their complex interactions in multistage manufacturing processes make it impractical to analytically derive useful closed-form equations describing the variation propagation and include all design parameters as free variables. Similarly, it is impractical to use the simulation models in process design optimization because it is often time-consuming to change the process design settings in the simulation and also time-consuming to run the simulation. Consequently, experience-based trial-and-error methods are still commonly used in practice for design optimization of manufacturing processes. Clearly, these methods are costly and error-prone. There is an urgent need to develop a modeling technique for dimensional variation propagation that can take the large number of process design parameters into consideration.
The surrogate modeling technique which is based on computer simulation has recently become a popular method in engineering design. The basic idea of surrogate modeling is to first run a set of controlled computer simulation experiments. In the second step, based on the simulation results, a statistical model is established to describe the relation between inputs and outputs. Different types of models have been proposed as surrogate models (also referred to as metamodels in the literature). The most common one is based on polynomial functions to represent linear response surfaces (Myers and Montgomery, 1995). Its main limitation is that only a small number of parameters can be typically included in the analyses. Sacks et al (1989) proposed the use of a stochastic model borrowed from spatial statistics called a Kriging model. In this model, the deterministic response from a simulated experiment is determined as the sum of a regression function, acting as global approximant to the data and a random process acting as local perturbation to interpolate the data. Kriging models have been successfully applied to fields as diverse as aerospace engineering design (Simpson el al., 1998) and electrical engineering (Sacks et al., 1989). Additional models, such as the radial basis functions, multivariate adaptive regression splines and also Neural Networks (NNs) have been employed successfully as metamodels. For a complete review of these models, the authors refer the readers to a set of review papers (Jin et al. 2001; Simpson et al, 2001; Chen et al, 2006; Wang and Shan, 2007). The aforementioned models can then be used within an optimization routine in place of the complicated simulation model to find the best combination of input parameters within a prespecified design space leading to an acceptable set of output parameters. The obtained results are then validated using the simulation package and the decision is made to execute further runs if necessary. Although generic, the existing surrogate modeling techniques cannot be directly applied to the problem of variation propagation modeling of multistage manufacturing processes. The first issue is that surrogate models are often chosen based on their interpolative capability and may not be good predictive models. Therefore, the model might not be accurate if the optimization design space is outside the model training space. In practice, people often use a validation step to address this issue. A comparative study of different model validation techniques by Jin et al., (2001) recommends the determination of the[R.sup.2] error, relative average absolute error, or relative maximum absolute error at untried design sites as a basis for validating the surrogate model. The second issue comes from the fact that a typical manufacturing process (even a single-stage process) has a very large number of parameters to be simultaneously optimized. During the design of a single assembly stage for instance, decisions will be made on the location of numerous locators, leading to the simultaneous optimization of up to 36 parameters(the three coordinates of the three-dimensions of up to 12 locators) with complicated interactions. According to...
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