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Article Excerpt
1. Introduction
The extensive adoption of sensor technology in manufacturing processes, for example the use of in-line Coordinate Measuring Machines (CMMs) in automotive body assembly processes, has resulted in extensive databases on product quality characteristics becoming available to manufacturers. The analysis of this sensor measurement data can provide valuable information about the state of the manufacturing process allowing the identification of the root causes of abnormal sensor observations. One cause of abnormal observations is a process fault, such as the malfunction of tooling elements (such as fixture, cutting tool and welding gun) used in the manufacturing process. Another cause is the malfunction of the sensors used to take the measurements, which is referred to as a sensor fault in this paper.
Statistical Process Control (SPC) (Woodall and Montgomery, 1999) has been widely used to detect out-of-control conditions in manufacturing processes thereby improving process capacity through variation reduction. By monitoring the variability in key quality characteristics, SPC can be used to detect the presence of assignable causes. However, SPC techniques are not able to differentiate between sensor faults and process faults, the two major root causes of the abnormal observations taken by sensors.
In many manufacturing processes, a linear model can explain how process faults and sensor measurement noise affect the observations of the product quality characteristics. Let an n x 1 vector y = [[y.sub.1], [y.sub.2], ..., [y.sub.n].sup.T] denote the sensor measurements on a set of n product quality characteristics. Suppose we have a sample of N products, whose product quality characteristics are denoted by y(j), j = 1, 2, ..., N. A fault quality model can be written as
y(j) = Cu + w(j), j = 1, ..., N, (1)
where the p x 1 vector u = [[[u.sub.1], [u.sub.2], ..., [u.sub.p]].sup.T] represents p potential process faults. The process faults u will affect the quality characteristics y(j) linearly through an n x p matrix C, which can be usually obtained through engineering analysis (for details, see Ceglarek and Shi (1996), Jin and Shi (1999) and Zhou et al. (2003)). If no process fault is present, then u = 0. The n x 1 vector w(j) denotes the aggregated effects of sensor noise and any inherent unmodeled variation in the manufacturing process for product j. We assume the columns of C are linearly independent. In addition, w(j) is a Gaussian random variable with mean [[mu].sub.w] and covariance matrix [K.sub.w] = [[sigma].sup.2] [I.sub.w], i.e., w(j) ~ N([[mu].sub.w], [[sigma].sup.2] [I.sub.w]. The variance [[sigma].sup.2] includes the variances of the sensors and other process noises. [I.sub.w] is an n x n identity matrix, indicating that the noises affecting each measurement point are uncorrelated with equal variance. In Section 4, we will discuss the more general situation when the covariance matrix is of a general form [K.sub.w].
The above linear fault quality model has been extensively applied in process fault diagnosis for discrete-part manufacturing processes. Ceglarek and Shi (1996) developed a Principal Component Analysis (PCA)-based mapping procedure to diagnose a single fixture-fault in automotive body assembly processes. Ceglarek and Shi (1999) improved their previous method by considering the impact of measurement noises on the diagnostic results. They presented a diagnostic index as a function of noise, fixture geometry and sensor location, which helped to identify a single fixture-fault in sheet metal assembly processes. Apley and Shi (1998) presented an algorithm using least squares estimation to identify multiple fixture-faults in panel assembly processes based on the linear fault quality model. Jin and Shi (1999) extended the single-stage fault quality model to a more general state space model for multi-stage manufacturing processes. Ding et al. (2002) applied PCA and pattern recognition methods to map the process faults for multi-stage processes. Ding et al. (2005) studied the properties of selected statistical estimators of process variation sources and compared their performances with respect to computational and statistical efficiencies.
In the recent literature on linear profile monitoring there is a trend for SPC techniques to be combined with the linear fault quality model of Equation (1). Kang and Albin (2000) and Kim et al. (2003) studied a simple linear model that is a special case of model (1) in which the matrix C only has two columns with the first column equal to 1, which corresponds to the intercept term. They applied least squares regression and a multivariate [T.sup.2] control chart to monitor mean shifts in the intercept and slope parameters. Zantek et al. (2007) recently studied a MEWMA control chart to detect and identify mean shifts caused by process faults based on a more general linear fault quality model.
All the above fault monitoring and diagnosis papers have assumed that the sensors are functioning normally and also that the mean of the noise term w(j) in model (1) is 0. In practice, however, this is not always the case. The malfunction of sensors may occur, which results in mean shifts in w(j). Thus, from model (1) we can see that both process faults and sensor faults may cause mean shifts in sensor observations. If a sensor fault is overlooked and confused with a process fault, resources will be wasted in trying to identify and eliminate process faults which may not really exist. Therefore, it is crucial to distinguish sensor faults from process faults in manufacturing quality control applications. Process fault detection based on sensor observations is valid only when it is assured that all sensors are working properly.
Some studies in detection and identification of faulty sensors have been performed for dynamic processes, especially chemical processes. Fantoni and Mazzola (1996) applied a non-linear PCA method to detect and identify faulty sensors in nuclear power plants. Chou and Varhaegen (1997) developed a subspace algorithm to identify multivariable finite dimensional linear time-invariant systems assuming that the input and output measurements are contaminated by a white noise. Based on the subspace identification model, Qin and Li (2001) proposed a dynamic structured residual approach to identify faulty sensors in dynamic processes. These studies only focus on the occurrence of sensor faults with no process fault involved.
In this paper, we will focus on the detection of sensor faults that cause mean shifts of one or more elements of the vector w(j) using a so-called W control chart, while effectively distinguishing sensor faults from process faults. In Section 2 of this paper, we provide the procedure to derive the W control chart from the fault quality model. The sensitivity of the W chart to single, double or multiple sensor faults is investigated in Section 3. In Section 4, the W chart is extended to sensors with unequal measurement precisions. Section 5 presents an example from automotive body assembly processes to demonstrate the performance of...
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