Stencil Printing Process Modeling and Control ... - Magnus Egerstedt

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IEEE TRANSACTIONS ON ELECTRONICS PACKAGING MANUFACTURING, VOL. 31, NO. 1, JANUARY 2008

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Stencil Printing Process Modeling and Control Using Statistical Neural Networks Leandro G. Barajas, Senior Member, IEEE, Magnus B. Egerstedt, Senior Member, IEEE, Edward W. Kamen, Fellow, IEEE, and Alex Goldstein, Senior Member, IEEE

Abstract—This paper presents a neural network model for the stencil printing process (SPP) in surface-mount technology (SMT) manufacturing of printed circuit boards (PCBs). A practical model description that decomposes the overall steady-state process in independently modeled subspaces is provided. The neural network model can be updated in real-time procuring a method to control the process by dynamically searching the optimal set point of the control variables. The optimization is performed by minimizing the weighted mean squared error with respect to the desired solder brick height or volume; furthermore, in the case when multiple solutions exist, the set point that yields the lowest variance is used. The process simulator is mainly suitable for offline testing and debugging of more complex closed-loop control algorithms for the SPP optimization providing a common and realistic framework for algorithm performance evaluation. An important consideration in this paper is based on the fact that the estimation of the sampled moments of the probability distributions is made using a statistically significant number of data samples from each board, for each component type, for each printing direction, and for each pad orientation. Index Terms—Printed circuits, soldering, stencil printing process (SPP) model, statistical neural networks, surface-mount technology (SMT), closed-loop control.

I. INTRODUCTION HE MAIN limitation in the optimization of the stencil printing process (SPP) is the elevated evaluation cost of the system output function. In order to obtain a single sample output value of the process an individual printed circuit board (PCB) has to be printed and inspected. Such a fact makes the necessity of finding a method for economically and adequately simulate and study the SPP a quite compelling industrial challenge. Currently, there are no comprehensive practical models for the SPP; however, interesting approaches toward the process understanding and offline optimization include the works by

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Manuscript received October 3, 2004; revised September 11, 2006 and January 17, 2007. This work was recommended for publication by Associate Editor J. Fowler upon evaluation of the reviewers comments. L. G. Barajas was with the Center for Board Assembly Research (CBAR), Manufacturing Research Center (MARC) and with the Department of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA. He is now with the Manufacturing Systems Research Laboratory, General Motors R&D Center, Warren, MI 48090-9055 USA (e-mail: l.g.barajas@ieee. org). M. B. Egerstedt, E. W. Kamen, and A. Goldstein are with the Department of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TEPM.2007.914236

[1]–[12]. For example, Li and Mahajan [1] studied the effects of PCB design in the surface-mount technology (SMT) manufacturing yield by identifying critical parameters through the use of analysis of variance (ANOVA) and using statistical regression and neural networks models for assembly yield prediction. Later, Mahajan [2] introduced a statistical design of experiment (DOE)-based neural network model. This model is based on a Taguchi L27 orthogonal array that is used to statically optimize equipment settings in order to minimize solder paste height variation; in the same study techniques for transferring artificial neural network (ANN) process models between different stencil printers are proposed. Posteriorly, the same author [3] developed the concepts of physical-neural network models and models transfer, where he demonstrated that they are appropriate in the procurement of low-complexity accurate ANN models. The same paper also provides additional developments in novel process control strategies using statistical ANN. A knowledge-based approach to SPP controller design was developed by Barajas et al. [4] by using a combination of neural networks and fuzzy logic. The neural networks were used to determine the membership functions for a fuzzy logic controller and to simulate the process for control signal verification. The fuzzy logic controller was designed in such a way that heuristic knowledge acquired from machine operators was incorporated in the control signal generation. The same authors also presented in [5] a hybrid closed-loop control for the SPP where the optimization objective was to maintain constant the solderpaste-volume deposition while minimizing the response time. The merit of this control is that it minimizes the variance and the steady-state error of the weighted sample mean versus the desired height. It also considers print direction and different component types independently. A generalization of this previous work is also presented by Barajas et al. in [6]; this control scheme is based on a weak-search algorithm that can be applied in the presence of large amounts of noise and when minimal information is known about the process. The controller employs an affine estimator implemented with a modified version of a constrained conjugated gradient method transitioned into a windowed smoothed block-form of the least-squares algorithm [13]. Recently, Coit et al. [7] designed a set of hierarchically connected neural networks. The networks were used to predict thermal behavior of PCBs utilizing board design parameters and process settings. Ultimately, such predictions were used to estimate the quality of solder connections. In addition, Ho and Xie [8] showed that ANN modeling with confidence bounds is an effective methodology that complements standard statistical process control (SPC) techniques for online process

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IEEE TRANSACTIONS ON ELECTRONICS PACKAGING MANUFACTURING, VOL. 31, NO. 1, JANUARY 2008

Fig. 2. SMT manufacturing line. Fig. 1. Stencil printing subprocesses.

optimization; they used a multilayer feed-forward ANN for SPP performance monitoring. In a closely related paradigm, Morad et al. [9] proposed the utilization of a genetic algorithm for process optimization of an ANN process model; the random search algorithms presented were suitable to be used with different optimization criteria of the process capability index. Other related approaches to the implementation of SPP practical models include the work by Ekere et al. [10], where a process modeling map with six main subprocesses for the SPP is presented. These subprocesses are depicted in Fig. 1. Such a study shows that the solder paste physical characteristics are the main factors in the solder paste print quality as well as in the interaction among processes. An alternative approach is the use of advanced computational fluid dynamics (CFD) methods as the ones proposed by Glinski et al. [11]; they provide realistic simulations at macro and microscopic scales. A unique approach is the one offered by Lotfi and Howarth [12] where a generic estimation of the solder-paste-volume deposition is made by the use of a set of orthogonal fuzzy rules. The main contribution of this paper is the development of a realistic and practical process model for the SPP. The underlying assumption is that the process can be modeled as a mixture of Gaussian (MoG) probability distributions; this assumption has been extensively tested in several experiments performed by the authors and it is substantiated in detail in Section II-B. The model is suitable for online and offline process optimization. During online optimization, the training of the ANN has to be done concurrently with the data acquisition of recently inspected PCBs. In the offline case, this model can serve as a test bed for optimization and/or tuning of more complex process control algorithms, like the one presented in [6]. The outline of this paper is as follows: In Section II a description of the SPP is given and in particular the performance objectives and process constraints are presented; this is followed by an ANN model analysis for the system in Section III. In Section IV a basic closed-loop control algorithm is described. As conclusion, simulations, and experimental results are presented in Section V. II. STENCIL PRINTING PROCESS The goal of the SPP in SMT manufacturing is to apply an accurate and repeatable volume of solder paste deposits at precise locations [14]–[16]. Given that most of the defects in SMT manufacturing can be attributed to the SPP [4], [17], [18]; this makes it the most critical step in the process. A. Process Description A simplified version of the SMT manufacturing process is illustrated in Fig. 2. In order to solder components to a PCB, it is necessary to print solder bricks over the metallic contact

pads on the PCB. Once this is achieved, the components are dispensed on top of the solder bricks, pushing their leads into the paste. After the components have been placed, the solder paste is melted using either reflow soldering or vapor-phase soldering in order to create the electromechanical junctures. Finally, the manufactured PCBs are inspected and tested. B. Problem Definition The SPP is characterized by having high process-noise levels and by requiring constant solder-paste-volume deposition at all times. This process has particular characteristics that make it difficult to control [4], [18]–[20]; some of which are the following: • poorly understood process physics; • difficulty in measuring key variables; • high-noise environment; • limited number of measurements; • software/hardware implementation limitations. In order to gain an better understanding of the overall SPP through its descriptive statistics [21], a typical histogram, normality test [22], and confidence intervals of the 3-D laser measurements for the solder brick heights in a single board are shown in Fig. 3. Additionally, Table I shows the results of two normality tests performed for each solder brick type and for the overall process. The Anderson–Darling (AD) [22] and Kolmogorov–Smirnov (KS) [23] tests are based on the empirical cumulative distribution function, and even when the KS test has lesser statistical power it is included here for reference purposes. : data follows The common null hypothesis for these tests is a normal distribution. If the P-value of the test is less than the level, reject . This demonstrates that the normality assumption holds at a solder brick type level P-values but not . Note that the metric for the overall process P-values units for Fig. 3 are given in millimeters. A more detailed description of the process complexity is depicted in Fig. 4. Here, the main factors of the SPP are classified in one of six main categories and by their controllability and observability properties from an online optimization perare directly or spective. The factors marked with a dagger indirectly measurable; however, they should be kept constant; for some of them, their value may be known. The ones with an asterisk (*) are factors that can be used to dynamically control the solder-paste-volume deposition and can be considered also as observable. Finally, all other factors are either static or not accessible during online optimization; such parameters are usually optimized offline via a design of experiment (DOE) methodology [2]. This classification has been made based on extensive experimentation performed at the Center for Board Assembly Research (CBAR) at the Georgia Institute of Technology, complemented with knowledge available in the existing literature. It should be noted that this classification assumes that state-of-the-art technology is available, and it may depend upon

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Fig. 3. Stencil printing process descriptive statistics for a single board. TABLE I NORMALITY TESTS BY PAD TYPE ( = 0:05)

the sophistication of the measurement and control equipment available. The observability of the system is limited because key variables like solder paste viscosity and hydraulic pressure cannot be directly measured or estimated in most of the existing industrial production equipment. The process is furthermore corrupted by two types of noise, respectively caused by inaccuracies in the measurement and by the internal system variability. The former can be disregarded in the case when 3-D laser measurement techniques are used. The latter has a six-sigma interval of approximately 30% of the mean of the probability distribution function of the signal [5]. This makes the process outputs extremely variable even under constant conditions. The associated cost of taking a measurement of the system output is high since for this it is necessary to print solder paste bricks on a PCB. It should be noted that the measurements are multivariate. The number of such outputs is related to the total number of solder bricks printed and inspected in each board. This set of measurements as a whole is considered as a single realization of the system, and it is desirable to minimize the

number of such evaluations to be able to generate control values while printing as few PCBs as possible. The industry standard for measuring the quality characteristics of the process is solder-paste-volume deposition. However, as of yet, there are no machines that can directly measure the solder brick volume. Instead, algorithms are used to estimate the effective area and mean height of the solder paste deposit, and therefore their product becomes the estimated volume. Given that under normal conditions, the area of the solder brick deposits do not change significantly with the modification of the control parameters, the height of the deposits becomes the measure of interest. Commonly, a direct sample mean of such values is used as quality characteristics; while a more sophisticated approach would be to assign different weights to each solder brick type so that problematic components can be given more importance in the quality characteristics generation process. Such a weighted scheme can be represented by

(1)

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IEEE TRANSACTIONS ON ELECTRONICS PACKAGING MANUFACTURING, VOL. 31, NO. 1, JANUARY 2008

Fig. 4. Stencil printing process relevant factors.

where is the number of solder bricks present in the th board, is the weight assigned to the th solder brick. Extensive and treatment on how to automatically select the ’s can be found in [13]. For steady-state performance evaluation, the weighted meanand squared error between the mean weighted height in board-by-board basis can be used. This the desired height representation, as shown in (3), is appropriate since it addresses two of the key factors in the process; i.e., the mean squared error stands for the expected and the variance of the process. value of the random variable . MSE

(2)

(3)

C. Convergence of Sampled Moments In order to produce statistical models of the SPP, it is vitally important that a statistically significant number of samples are used. In the SPP, the height measurements of specific solder bricks placed on the same orientation and printed under the same printing conditions can be approximated as a Gaussian distribution. The results depicted in Fig. 5 are based on 100 000 000 realizations of 10 000 sample series drawn from a Gaussian process, . The average error across realizations of the sampled mean and the sampled standard deviation are evaluated. On average, the higher the number of samples used, the lower the estimation error of the real mean and standard deviation.

Fig. 5. Average convergence for sampled mean and sampled standard deviation.

For an average 5% error in the sample mean, over 250 measurements are required, but for only 10% accuracy, less than 65 will suffice. This is the mean for the worse case scenario when there is no correlation among samples. Therefore, for the SPP the results will be more optimistic. In the case of the sample standard deviation in Fig. 5, the results are similar. For a 5% error in the sample variance, less than 520 measurements are required, and for only 10% accuracy, about 135 will be enough. These numbers will provide guidelines for producing accurate statistical process models. III. NEURAL NETWORK MODEL One of the major limitations for the optimization of the SPP is the extreme difficulty for extracting reliable information for

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and be the output space space the variables are as follows:

where

pdf (P)arameters (D)imension (d)irection pad (T)ype (O)rientation Machine (S)ettings Fig. 6. Generic process model for random variables generation.

where mean;

noisy data. This also translates to the need of a complex simulator that can generate adequate and coherent random data for process optimization purposes. The solder brick measurement from the PCBs can be considered locally Gaussian under the assumptions to be discussed in this section. An important consideration is the selection of the quality characteristic for the model. Traditionally, solder-paste-volume deposition has been widely accepted in the production environment. The main reason for using volumetric measurements is the fact that ultimately what is relevant for obtaining an adequate electromechanical junction between the PCB and the component is the amount of solder paste present in the juncture. However, there are two issues to consider. First, for defect detection-related issues, measurement of solder brick effective area and height separately will provide more information than their product alone. Second, for process control purposes, solder paste brick height is the relevant quality characteristic of the process. In conclusion, in order to provide a more complete model of the SPP, both solder paste brick height and effective area must be modeled independently. Several authors [2], [8], [24], [25] have used ANN models based on Taguchi orthogonal arrays or time series for SPP optimization. However, such implementations consider mostly static control factors that cannot be changed online. The factors include squeegee material, pad material, and squeegee angle, and aperture size and shape. These approaches also investigate semidynamic factors like temperature, snap-off distance and separation speed. These set of factors can only be optimized to minimize the occurrence of defects in the process but cannot be efficiently used for dynamically control of the solder-paste-volume deposition. In addition, it is common practice to consider printing direction as control or noise factor. In this paper, only the two control factors that allow efficient dynamic modification of the solder brick height are used as inputs in the ANN, namely squeegee speed and pressure. All static factors have been fixed to nominal values and any attempt to optimize them must be done offline. A relevant consideration is not to use printing direction as an input of the ANN but rather to construct independent models for backward and forward directions; this increases the accuracy of the model and also simplifies the optimization procedure because it removes the only discrete input type to the model. Besides, it is pointless to find the optimal printing direction given that in real industrial production, bidirectional printing is a process requirement and cannot be avoided in practice. As a conclusion, the SPP data should be analyzed (or simulated) as sets of different partitions. Formally, let be the input

variance; height; area; forward print direction; backward print direction; pad type ; perpendicular to print direction; parallel to print direction; 45 w.r.t. print direction; speed; pressure. In this case, is a output feature-space that accurately defines the main characteristics of the SPP, where is the number of pad types, and is a two-dimensional input space given by the stencil printer squeegee speed and pressure. Any attempt to recreate the behavior of the process should consider at least all these aspects. Fig. 6 illustrates a generic process model, which is independent of the method used to estimate the probability density function (pdf) parameters. In this formulation, the pdf parameter estimation is performed by the ANN that has been trained using sample data from experimental runs of the SPP. This steady-state model was then validated by additional and independent board runs. A. Model Description The SPP cannot be characterized with a simple input-output relationship, in fact, it is a multiple-input multiple-output time variant nonlinear system. If in a PCB there are enough pads of the same type for each pad type, then their distributions can be approximated individually by Gaussian distributions. It should be noted that pads of the same type but with different orientation with respect to the printing direction should be considered as different pad types; the reason for this is that there exists a noticeable difference in solder-paste-volume deposition depending on the orientation of the pad. This difference in printing direction quality is a well-known and documented effect on stencil printing [26]. Gaussian distributions can be characterized in terms of their mean and variance. For the generation of a process model, this fact can be useful because it will simplify the generation of simulated data. In fact, it will suffice to draw samples from a

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Fig. 7. Neural network convergence for mean height model.

Gaussian distribution parameterized by its mean and variance. The pressure and the speed are the inputs of the model. Additional parameters can be considered but for simplicity, only the two main ones are used. There will be independent outputs for area and height, constructing the volume by their arithmetic product in sample-by-sample basis. It is also necessary to construct independent models for each printing direction; they will be identical in structure; however, their values of mean and variance may differ. Finally, the Cartesian product of the ordinate pairs given by the pad type and the pad orientation will define the number of outputs of the model. In this way, the sample realizations of each combination of pad type and orientation will be simulated independently. The inputs for each individual model will be the mean and variance of the respective pad type in a particular orientation for each printing direction, and the current printing speed and squeegee pressure. The orientation of the pad may be perpendicular, parallel, or with respect to the printing direction. The pad type depends on the geometry of the electronic component to be soldered on the PCB. Fig. 6 illustrates the RV generation process and complements the summary of the input and output spaces and previously introduced. Using the partitioning described in Section III-A, a nonlinear model was implemented by the use of feed-forward neural networks. The training sequences are taken from real data acquired from several experiments. In order to assure continuity between consequent data values at adjacent points in the speed-pressure input space, polynomial interpolation was used. It should be noted, however, that such interpolation was done in pad-to-pad basis across different boards, such that the information contained in each solder brick measurement was preserved. The noise-filtering characteristic of the neural network took care of the issue of having large levels of noise in the process. There are actually two complementary models for the process. The first model estimates the means of the height and area of each individual solder brick. Similarly, the second model estimates the variances. However, the internal structures of the networks are completely different as well as their convergence characteristics. Fig. 7 shows the convergence for the

Fig. 8. Neural network convergence for standard deviation model.

model that estimates the mean of the height. The performance measurement used is based on the error across all pad types in different orientations but in a single printing direction. Validation, test, and training data are used in the learning procedure to avoid over-fitting. Fig. 8 shows the convergence for the model that estimates the variance of the height. Similar estimates for the mean and variance of the solder brick area are also implemented in an analogous way. This process model is used in the framework depicted in Fig. 6. B. Neural Network Structure The topology of the network used for the proposed model is now explained. The network structure for mean and standard deviation models is , meaning that three layers are used; two layers with neurons each, having one neuron for each pad type, and one single-neuron output layer. However, even when the activation functions for the mean and standard deviation are the same for the first two layers (tangent sigmoid), using a linear transfer function for the output layer for the standard deviation yields a remarkable improvement over the sigmoid used in the mean model. The function used in the ANN back-propagation algorithm is based on a Levenberg–Marquardt optimization method [27]. The main advantage of this method is its speed of convergence; usually only five to ten iterations are necessary for convergence; however, the large amount of memory storage required for the method may become a practical issue for high-dimensional applications like in the SPP. However, by training the different ANN in independent batches (height, area, mean, and standard deviation) it is possible to reduce the memory requirements and the training time. The training data sets were obtained by a full factorial DOE methodology. The DOE used speed and pressure as main factors with four levels each of (0.5, 1, 2, 3)in/s and (0.5, 1, 2, 3)lb/in, respectively. Four repetitions were made for each combination of the control factors, two in each individual printing direction. All other control parameters in the SPP were kept constant (e.g., zero snap-off distance, i.e., contact printing; snap-off

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Fig. 9. Neural network structure.

speed, 0.05 in/s; squeegee angle, 60 ). In this study, printing direction was not considered as a noise factor as it has been previously done in [3]. However, the dominant printing direction effects were compensated by having independent SPP models for forward and backward directions. The data sets were divided in training, validation, and test vectors. The validation set is used to perform an early termination of the ANN training process if the ANN response is not improved for an specific number of iterations. The test set has no effect in the training procedure; however, it is used to monitor the network generalizing properties, e.g., in order to verify that over-fitting has not occurred. A noticeable difference in the precision and reliability of the model presented in this paper compared with previous approaches considered [1]–[3], [7]–[9], [11], [12], [14], [15] is that the number of sample measurements used per PCB is statistically significant. Usually, due to technical/physical/temporal limitations, similar approaches have only used less than ten training samples per PCB. Such samples are normally selected at random from each PCB, sometimes also disregarding pad type, printing direction, and component orientation. The number of solder bricks measured from each pad type for each PCB is shown in Table I. Correlating this information with the explanation previously given in Section II-C, it can be inferred that, on average, the accuracy of the proposed model is improved by at least one order in magnitude (when using sample mean and variance) by just considering the larger number of samples used for the surface fitting model. In fact, the error percentage for the estimation of the mean is less than 8% for individual solder brick types and about 3% for the entire board. In the same manner, as shown in Fig. 5, the estimates accuracy for the standard deviation are just 1.5 dB above these values ( 12% and 4.2%, respectively). A graphical representation of the internal structure of the ANN used for the model of the solder brick standard deviation and LW are the input and is shown in Fig. 9, where IW are the bias vectors. The struclayer weight matrices and ture for the model of the solder brick mean only differs in the final output activation function; in this case, it uses a tangent sigmoid rather than a linear one. Empirically it was found that, for the training data available, using only one hidden layer in the ANN it is possible to create a working model of the process; however, using a second hidden layer drastically improves the precision and speed of convergence of the estimates; the use of a third hidden layer duplicates the training time and not always yields better results than the two hidden-layer model. It was also found that for the number of neurons in each hidden layer the best tradeoff between training time and model precision always lies near the respective number of different solder brick types that are being estimated.

Fig. 10. Neural network response surface for the pdf mean of the height.

This same structure is replicated to independently estimate different orientations inside the PCB, different printing directions across PCBs, and height and area independently. Complete information on the MATLAB Neural Network Toolbox and the specifics of the functions and parameters used in the construction, training, and testing of the ANN can be found in [27]. IV. CLOSED-LOOP CONTROLLER The main application of having a working SPP model for the steady-state of the process is that it is possible to find the desired set point by performing multiobjective optimization over the response surfaces of the different pdfs that are being estimated by the ANN. Such optimization should consider the number of solder bricks of each pad type and their orientation, as well as solder volume deposition as estimated by the height and area pdfs. A. Search Algorithm A common formulation for multiobjective optimization is the formulation of an aggregation function to represent the total cost function for the overall problem. An important consideration is the dimensionality of the different objectives of the problem and therefore it is common practice to normalize each individual objective by a nominal value. These two considerations are used to generate the total cost function shown in (4), where is the number of objectives. A complete description of this formulation is found in [24]. For the SPP, the functions are the individual output surfaces of the sample mean and standard deviation each solder brick type height and area, printed in a specific direction and with the same orientation; typical examples of s are depicted in Figs. 10 and 11: (4) For the SPP, the multiobjective problem is subdivided in two main types, namely solder brick type mean and variance. The variance estimation can be simplified by considering the standard deviation instead. Several ANN training experiments

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composed from historical and real-time sample measurements; in each adaptation, the ANN output, error, and network adjustment will be calculated for each given input data set; this whole procedure has to be iterated until the output error is reduced under a permissible level or until one of the additional stopping samfactors is triggered. For a specific input sequence with ples, the network is updated as follows [27]. Each sample in the sequence of inputs is presented to the network one at a time. The network’s weight and bias values are updated after each step, before the next step in the sequence is presented. Thus, the times. network is updated V. SIMULATION AND EXPERIMENTAL RESULTS

Fig. 11. Neural network response surface for the pdf mean of the area.

Fig. 12. Neural network closed-loop controller.

showed a large increase in accuracy and therefore MSE reduction by the use of this simple technique. Fig. 10 shows the output of the simulator for the mean height of a specific pad type for only one printing direction and orientation. Furthermore, by using the search scheme depicted in Fig. 12, it is possible to find an optimum value for the SPP control parameters. It may be the case that the problem does have nonunique solutions. However, Once the ANN has been trained, its evaluation can be done at a minimal cost given the low complexity of the resulting input–output relationship. Therefore, the application of standard numerical search techniques over the entire function domain provides an optimal solution for practically any proposed cost function. For our controller implementation, the Nelder–Mead simplex (direct search) method was used [28], [29]. If minimization of the bias error and variance does not yield a unique mathematical solution, a practical consideration can be used in the selection of the most adequate values for the control variables. Due to solder paste rheological considerations, for most SPP cases, the lowest printing speed can be selected as the unique and optimal solution. B. Online Network Update In order to increase the adaptability of the model, it is possible to perform online training of the neural network. The function adapt from the MATLAB Neural Network Toolbox allows the dynamic updating of the network during operational time. By iteratively applying the simulation and learning functions over the ANN training set, adapt changes the weights and biases according to the error. The data used to perform this procedure are

Using the neural network models previously described, simulations were performed and later verified by real board run manufacturing. In Figs. 10 and 11, typical response surfaces for a specific solder brick type distribution are shown. In this case, only mean height and area are depicted; however, the standard deviation is also considered in the ANN complete model for these two dimensions. The assembly of all solder brick partitions defined in Section III-A is used to construct the overall model for the SPP. This embodiment provides a suitable alternative for solder-paste-volume deposition simulation. The experimental setup used for this paper includes a Speedline MPM-3000 stencil printer and a Cyberoptics Sentry-2000 3D-Laser inspection system which are part of one of the Surface Mount Technology (SMT) manufacturing lines located in the Center for Board Assembly Research (CBAR) that is part of the Manufacturing Research Center (MARC) at the Georgia Institute of Technology. A 12 in metallic squeegee was used to perform the experiments over a laser-cut 5-mil (127- m) stencil, using nonclean 63%Sn–37%Pb solder paste Type IV. The pad distribution of the test board is also shown in Table I. Given the large amount of solder bricks inspected for each pad type, it is reasonable to assure that the sample is statistically and practically significant [30], and that the estimated parameters will adequately represent the distribution of the samples in the SPP. Using experimental data from a 64 board full-factorial DOE run (with two repetitions plus eight more control boards), an of the data. Later on, the ANN valANN was trained using idation and testing was performed with of the data each. Similar experiments were performed for several solder paste types. The control inputs of the ANN are the squeegee speed and printing pressure of the stencil printer. All other control parameters were fixed to optimized values that from previous and more extensive DOE analysis yielded the best results. The outputs of the network estimate the pdf parameters of the Gaussian distri. This is done independently for the solder bution, bricks for each individual solder brick type, in each printing , in each printing direction (forward, backorientation ward), and for each physical dimension (height, area). The optimality criteria used on the solution of the problem is based in (1) and (4) for the case of equal weights, i.e., . The rationale for the equal value assumption is based on the fact that the area and height values have been normalized with respect to their historical optimal averages; this is particularity important in order to

BARAJAS et al.: STENCIL PRINTING PROCESS MODELING AND CONTROL USING STATISTICAL NEURAL NETWORKS

be able to compare large and small area components in an unbiased manner. The numerical algorithm used is a complex mathematical engine provided by the fminsearch algorithm of the MATLAB Optimization Toolbox [29, p. 6–61]; the function fminsearch minimizes a multivariate scalar function given an initial condition; this is also known as unconstrained nonlinear optimization. This numerical routine specifically uses the Nelder–Mead simplex (direct search) method [28], [29]. Local and global simulated annealing [31] was used to minimize the effects of possible suboptimal solutions caused by local minima. In the global simulated annealing procedure, when restarting the search algorithm from different random initial conditions, it was observed that, on average, the benefit became marginal when more than 10–15 replications were used. For this specific data set, it was found that the control values for speed and pressure that minimize the variance and the MSE with respect to the desired mean solder paste deposition height of 5.5 mil ( 139.7 m) are 1.8 in/s (4.572 cm/s) and 2.1lb/in (123.037 kg/m), respectively. It should be clear that these values are optimal only for the test PCB used, and under the conditions specified at the beginning of this section. Further experimental corroboration of the precision of the model showed that, in steady-state, during 20 board runs, the predicted pdf mean and standard deviation for all component types were inside the confidence bounds as proposed in [8]. VI. CONCLUSION This paper has presented an SPP process simulator that allows the testing and refining of complex algorithms for SMT manufacturing yield improvement. A basic numerical controller that allows online training is also described and tested in a real manufacturing line. The results show that, in practice, it is possible to control the SPP in order to achieve the desired solder-pastevolume deposition and to minimize the variance of the distribution. A practical and mathematically sound cost function for SPP optimization was also proposed and validated. By the use of a statistically significant number of samples of each pad type, it was guaranteed that the model was trained with sufficient data such that it provides a total error percentage in the estimation of the sample moments of about 5% for the mean and 8% for the standard deviation. REFERENCES [1] Y. Li, R. Mahajan, and J. Tong, “Design factors and their effect on pcb assembly yield-statistical and neural network predictive models,” IEEE Trans. Compon., Packag., Manuf. Technol., Part A, vol. 17, no. 2, pp. 183–191, Jun. 1994. [2] R. L. Mahajan, “Statistical neural network modeling for stencil printing,” in Surface Mount Int., 1996, pp. 573–578. [3] R. L. Mahajan, “Neural nets for modeling, optimization and control in semiconductor manufacturing,” Proc. SPIE, vol. 3812, pp. 176–187, 1999. [4] L. G. Barajas, E. W. Kamen, and A. Goldstein, “On-line enhancement of the stencil printing process,” Circuits Assembly, pp. 32–36, Mar. 2001. [5] L. G. Barajas, E. W. Kamen, A. Goldstein, M. Egerstedt, and B. Small, “A closed-loop hybrid control algorithm for stencil printing,” in Proc. Surface Mount Technol. Assoc. Int. Conf. (SMTA’02), Boston, MA, 2002, pp. 51–58. [6] L. G. Barajas, M. Egerstedt, E. W. Kamen, and A. Goldstein, “Process control in a high-noise environment with limited number of measurements,” in Proc. Amer. Control Conf., Denver, CO, 2003, pp. 597–602.

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[7] D. W. Coit, B. T. Jackson, and A. E. Smith, “Neural network open loop control system for wave soldering,” J. Electron. Manuf., vol. 11, no. 1, pp. 95–105, 2002. [8] S. Ho, M. Xie, L. Tang, K. Xu, and T. Goh, “Neural network modeling with confidence bounds: A case study on the solder paste deposition process,” IEEE Trans. Electron. Packag. Manuf., vol. 24, no. 4, pp. 323–332, Oct. 2001. [9] N. Morad, H. K. Yii, M. Hitam, and C. P. Lim, “Development of an intelligent system for the solder paste printing process,” in Proc. TENCON 2000. , Penang, Malaysia, 2000, vol. 3, pp. 479–483, School Ind. Technol., Univ. Sains Malaysia. [10] N. Ekere, S. Mannan, and M. Currie, “Solder paste printing process modelling map,” in Proc. Electron. Manuf. Technol. Symp., Proc. 1995 Japan Int., 18th IEEE/CPMT Int., UK, 1995, pp. 137–141, Dept. Aeronaut., Mech. Manuf. Eng., Salford Univ., U.K. [11] G. Glinski, C. Bailey, and K. Pericleous, “Simulation of the stencil printing process [solder pastes],” in Proc. Int. Symp. Electron. Mater. Packag. (EMAP’00)., J. Kim, A. Teng, and S.-W. R. Lee, Eds., London, U.K., 2000, pp. 364–370. [12] A. Lotfi, M. Howarth, and P. Thomas, “Orthogonal fuzzy model of the solder paste printing stage of surface mount technology,” in Proc. 6th IEEE Int. Conf. Fuzzy Syst., 1997, vol. 3, pp. 1433–1437. [13] L. G. Barajas, Process control in high-noise environments using a limited number of measurements Ph.D. dissertation, Georgia Inst. Technol.. Atlanta, 2003. [14] S. Fujiuchi, “Fundamental study on solder paste for fine pitch soldering,” in Proc. 11th IEEE/CHMT Int. Japan IBM Electron. Manuf. Technol. Symp., Shiga, Japan, 1991, pp. 163–165. [15] S. Venkateswaran, K. Srihari, J. Adriance, and G. Westby, “A realtime process control system for solder paste stencil printing,” in Proc. 21st IEEE/CPMT Int. Electron. Manuf. Technol. Symp., Binghamton, NY, 1997, pp. 62–67. [16] L. Gopalakrishnan and K. Srihari, “Solder paste deposition through high speed stencil printing for a contract assembly environment,” J. Electron. Manuf., vol. 8, no. 2, pp. 89–101, 1998. [17] J. Pan, G. Tonkay, R. Storer, R. Sallade, and D. Leandri, “Critical variables of solder paste stencil printing for micro-bga and fine pitch qfp,” in Proc. 24th IEEE/CPMT Electron. Manuf. Technol. Symp., Bethlehem, PA, 1999, pp. 94–101. [18] F. K. H. Lau and V. W. S. Yeung, “A hierarchical evaluation of the solder paste printing process,” J. Mater. Process. Technol., vol. 69, no. 1–3, pp. 79–89, 1997. [19] A. Johnson and A. Flori, “High density/fine feature solder paste printing,” in Proc. APEX, 2001, pp. MP2-3-1–MP2-3-8. [20] R. Durairaj, T. A. Nguty, and N. N. Ekere, “Critical factors affecting paste flow during the stencil printing of solder paste,” Soldering Surface Mount Technol., vol. 13, no. 2, pp. 30–34, 2001. [21] Minitab, “One-sample T-test,” 2003 [Online]. Available: http://www. minitab.com/training/t-Test.pdf [22] T. W. Anderson and D. A. Darling, “Asymptotic theory of certain “goodness of fit” criteria based on stochastic processes,” Ann. Math. Statist., no. 23, pp. 193–212, 1952. [23] H. W. Lilliefors, “On the kolmogorov-smirnov test for normality with mean and variance unknown,” J. Amer. Statist. Assoc., vol. 62, pp. 399–402, 1967. [24] H. K. Yii, N. Morad, and M. S. Hitam, “Optimisation of a solder paste printing process parameters using a hybrid intelligent approach,” Neural Netw. World, vol. 11, no. 2, pp. 109–127, 2001. [25] Y. Li, “Yield improvement, reliability modeling and design optimization for solder interconnection (joints),” Ph.D. dissertation, Univ. Colorado, Boulder, 1996. [26] R. P. Prasad, Surface Mount Technology: Principles and Practice, version 4, 2nd ed. New York: Chapman & Hall, 1997. [27] H. Demuth and M. Beale, Neural Network Toolbox User’s Guide: for use with MATLAB. Natick, MA: MathWorks, 2000 [Online]. Available: http://www.mathworks.com/products/neuralnet/ [28] J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optimization, vol. 9, no. 1, pp. 112–147, 1998. [29] Optimization Toolbox User’s Guide: for Use with MATLAB, version 3.0.3.. Natick, MA: Mathworks, 2005 [Online]. Available: http://www.mathworks.com/access/helpdesk/help/toolbox/optim/ [30] L. Tenorio, “Statistical regularization of inverse problems,” SIAM Rev., vol. 43, no. 2, pp. 347–366, 2001. [31] S. Kirkpatrick, C. D. Gelatt, Jr., and M. Vecchi, “Optimization by simulated annealing,” Science, vol. 220, no. 4598, pp. 671–680, May 13, 1983.

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IEEE TRANSACTIONS ON ELECTRONICS PACKAGING MANUFACTURING, VOL. 31, NO. 1, JANUARY 2008

Leandro G. Barajas (S’95–M’99–SM’06) was born in Bogotá, Colombia, in 1973. He received the Honor degree in electronics engineering as Valedictorian from the Universidad Distrital F.J.C., Bogotá, Colombia, in 1998, and the M.S. and Ph.D. degrees in electrical and computer engineering from the Georgia Institute of Technology (Georgia Tech), Atlanta, in 2000 and 2003, respectively. He is currently a Senior Research Engineer at the General Motors R&D Center, Manufacturing Systems Research Laboratory, Warren, MI, where he focuses on the area of Plant Floor Systems and Controls. During his graduate studies, he worked at the Center for Board Assembly Research (CBAR), Manufacturing Research Center (MARC), Georgia Tech. Dr. Barajas is a Senior Member of the Society of Manufacturing Engineers (SME), Elected Full Member of SIGMA XI (The Scientific Research Society), member of the societies Automotive Engineers (SAE), Hispanic Professional Engineers (SHPE), and the Surface Mount Technology Association (SMTA). In 2000 and 2003, he received M.S. and Ph.D. OMED Tower Awards from Georgia Tech. During his tenure at GM, he has been distinguished with the 2005 GM R&D “Spark-Plug” Award, the 2006 GM Chairman’s Honors Award, the 2006 GM R&D Charles L. McCuen Special Achievement Innovation Award, and the 2007 SME Kuo K. Wang Outstanding Young Manufacturing Engineer Award.

Magnus B. Egerstedt (S’99–M’00–SM’05) was born in Stockholm, Sweden. He received the B.A. degree in philosophy from Stockholm University in 1996, and the M.S. degree in engineering physics and the Ph.D. degree in applied mathematics, both from the Royal Institute of Technology, Stockholm, in 1996 and 2000, respectively. He spent 2000 to 2001 as a Postdoctoral Fellow with the Division of Engineering and Applied Science, Harvard University, Cambridge, MA. In 1998, he was a Visiting Scholar at the Robotics Laboratory, University of California, Berkeley. He is currently an Associate Professor with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta. His research interests include optimal control as well as modeling and analysis of hybrid and discrete-event systems, with emphasis on motion planning and control of (teams of) mobile robots. He has authored over 100 articles in the areas of robotics and controls. Dr. Egerstedt received the CAREER Award from the National Science Foundation in 2003.

Edward W. Kamen (S’71–M’71–SM’93–F’94) received the B.E.E. degree from the Georgia Institute of Technology (Georgia Tech) in 1967 and the M.S. and Ph.D. degrees in electrical engineering from Stanford University, Stanford, CA, in 1969 and 1971. He is Professor Emeritus in the School of Electrical and Computer Engineering, Georgia Tech. From 1980 to 1986, he was Professor of Electrical Engineering at the University of Florida. From 1986 to 1990, he was Professor and Chair of Electrical Engineering at the University of Pittsburgh. He is the author/coauthor of over 100 journal research publications and six textbooks in the areas of signals, systems, controls, and manufacturing. Dr. Kamen is the recipient of several teaching and research awards. He was General Chairman of the 1999 IEEE Conference on Decision and Control.

Alex Goldstein (M’91–SM’99) received the M.S. degree (with honors) in electrical engineering from the Leningrad Electrotechnical Institute, Leningrad, USSR. He is Director of Operations of the Center for Board Assembly Research, Georgia Institute of Technology, Atlanta. His current research interests include hybrid data-driven closed-loop control for high-density interconnect technologies. With over 30 years of experience in systems and controls, he has lead engineering teams in development and integration of novel supervisory control and data acquisition systems and distributed control systems for various industrial and scientific applications. Mr. Goldstein is a Senior Member of the Society of Manufacturing Engineers. He served as Registration Chair of the 38th IEEE Conference on Decision and Control.