Dynamic performance modelling and measuring for machine tools with continuous-state wear
processes
Bi-Min Hsua, Ming-Hung Shub* and Lang Wuc
aDepartment of Industrial Engineering & Management, Cheng Shiu University, Kaohsiung, Taiwan;bDepartment of Industrial Engineering & Management, National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan;cDepartment of Statistics,
University of British Columbia, Vancouver, BC, Canada (Received 5 November 2012;final version received 3 April 2013)
In this paper, we consider the problem of using empirical continuous-state wear data of machine tools to estimate the dynamic lifetime distribution and to measure the performance of a machining process subject to stochastic tool-wear evolution. Machining systems are dynamic processes whose performance variable is usually characterised by the amount of tool wear that advances gradually with a continuous range of values. To accurately capture the performance of these continuous-state wear processes, neither traditional models such as the binary-state models nor multi-state models are suitable. In this paper, an exponential mixed-effects (EME) model is first developed. The EME model is subsequently transformed into a linear mixed-effects (LME) model to enhance thefit and predictability of the wear process data. The LME models take into consideration the correlations among repeated wear measurements collected at different time points within each subject. We then implement the expectation-maximisation (EM) algorithm to obtain the full maximum likelihood estimates (MLEs) of the parameters of the LME models whose asymptotic normal distributions can be used to acquire approximate confidence intervals and a testing hypothesis for the parameters. In addition, to measure the dynamic performance of tools, the amount of wear over time estimated from LME models is compared with a given tool-failure threshold. Consequently, we obtain the reliability of the tool and the estimation of its residual-lifetime distri-bution, which is critical information for the tool replacement or maintenance strategy. Finally, the lower and upper wear prediction limits of the 95% confidence level are presented. A practical application of the proposed methodology is illus-trated throughout the paper.
Keywords: manufacturing systems engineering; process modelling; production control; quality measurement; performance measures; continuous-wear processes; tool reliability
1. Introduction
Machine tools play a major role in most areas of industrial fabrication for the production of quality goods in modern manufacturing industries. To manufacture simple as well as complex components, machine tools carry out tasks such as turning, boring, milling, drilling, and tapping in many materials, ranging from alloys to plastics. Wear-related costs make up a significant portion of the expense of parts produced, accounting for 25–30% of the manufacturing operating costs (Veeramani, Upton, and Barash 1992). Nowadays, continuous advancements in automation and computer-aided manu-facturing techniques have all emphasised the importance of the modelling and measurement of tool performance to assist manufacturers in their worldwide competition (Heinemann and Hinduja 2012).
Many machining failure mechanisms can be traced to an underlying wear process of the machine tools. Understand-ing how wear levels increase over time is the key element for evaluatUnderstand-ing the performance of machinUnderstand-ing systems or pro-cesses. In most machining systems, the tool condition starts from being sharp, deteriorating to dull, and eventually tool breakage (Shaw 2005, Yesilyurt and Ozturk 2007). This wear process information can be viewed as decisive evidence for product quality and reliability. For example, surface roughness is a key quality characteristic of metallic materials that changes in the machining process as tool wear develops. In general, when tool wear increases, the surface rough-ness quality decreases (Ozcelik and Bayramoglu 2006). Therefore, providing an accurate model and effective perfor-mance measures for this progressive wear process can greatly benefit modern manufacturing industry for the purposes of producing reliable products of high quality.
*Corresponding author. Email: [email protected]
Vol. 51, No. 15, 4718–4731, http://dx.doi.org/10.1080/00207543.2013.793858
Ó 2013 Taylor & Francis
When modelling and measuring the performance of machine tools, traditional studies either identify and analyse binary state models (fresh (success) and worn (failure)) in terms of reliability (Sun et al. 2004, Liu and Kapur 2006), or multi-state models with limited intermediate states between a functioning state and a failure state (Xue and Yang 1995, Sun et al. 2005, Hsu and Shu 2010, Shu, Hsu, and Kapur 2010). However, the machine wear process is a dynamic pro-cess whose wear states deteriorate through a continuum of values, as shown in Figure 1. Thus, it is required to develop a continuous-state wear model, an extension of traditional models, to better capture this progressive wear process.
Li and Kapur (2012) used the fuzzy set theory to model continuous-state gamma wear processes with assumed known parameters over time. However, in practice, the parameters are generally unknown. They can only be estimated from observations taken over time. The use of continuous-state wear observations to measure component performance was addressed in the work of Lu and Meeker (1993). They proposed path models with mixed-effects parameters for fit-ting observed continuous-state wear measurements. Instead of using full maximum likelihood estimation (MLE) for the parametric statistical model, a two-stage estimation method on the basis of ordinary least square (OLS) estimates is pro-posed to simplify the estimation computation of the random-effect parameters. This OLS-based two-stage method, how-ever, requires appropriate re-parameterization techniques to transform first-stage estimates for the random-effects parameters to be modelled as a normal distribution in the second stage. Zuo, Jiang, and Yam (1999) introduced regres-sion-like approaches that model continuous-state wear components with balanced and unbalanced data. Their approaches consider that within- and between-subject wear data are all independent. Nevertheless, continuous-state wear data within the same subject may be correlated. In the analysis of correlated data, the correlation should be incorporated in the analysis in order to avoid potential evaluation bias and loss of efficiency.
A mixed-effects model viewed as an extension of the regression model introduces random effects in the model accounting for correlations within subjects and variations between subjects (Wu 2009). Moreover, in the mixed-effects model, the repeated measurements, fsi1; si2; . . . ; sinig, of the response within each individual can be taken at different
time points for different individuals, and the number of measurements, ni, may also vary across individuals. In other
words, the mixed-effects model allows unbalanced data in the response. This is also an advantage of the mixed-effects model. Motivated by prognostic applications in dynamic autonomous machining processes, with physical experiments this research provides a continuous-state statistical model and reliability measures to assess and capture the current and future performance of machining processes undergoing tool wear so as to prevent catastrophic failure or sub-optimal replacement or maintenance activities for the machining processes.
The paper is organised as follows. An exponential mixed-effects (EME) model isfirst developed based on observed measurements of tool-wear processes. To enhance the fit and predictability, the EME model is then transformed into a linear mixed-effects (LME) model. The LME models take into consideration the correlations among repeated wear mea-surements taken at different time points within each subject. Full maximum likelihood estimates (MLEs) of the parame-ters using the expectation–maximisation (EM) algorithm are presented. The asymptotic normal distributions of the MLEs are used to approximate confidence intervals and perform hypothesis tests for the parameters. Moreover, by com-paring the quantities between the tool failure threshold and the amount of wear over time acquired from LME models
0 50 100 150 200 250 0 102 03 04 05 0 Cycles (102) W ear (10 −2 mm)
Figure 1. The wear of machine tools in a friction drilling process.
enables us to estimate the wear evolution and lifetime of the tool. Estimation of the distribution of the residual life up to the tool wear threshold is also presented. This information can be used to estimate the lifetime at a specific tool wear level or estimate the tool wear level at a specific time. Finally, we present the lower and upper wear prediction limits of the 95% confidence level. For practical applicability, the proposed methodology is illustrated by a friction drilling manu-facturing process with empirical continuous-wear data collected from physical experiments. In contrast to the proposed LME model, a linear regression model, a primitive fit that overlooks the correlation among repeated measurements, is included to serve as a baseline method due to its popularity.
2. Tool-wear model development
Before constructing the tool-wear model, we introduce the notation used in this paper. We use lower-case letters (e.g., s, x, and r) to denote scalar quantities, either fixed or random. Occasionally, we also use upper-case letters (e.g., X and Y ) to denote random variables. Lower-case bold letters (e.g., x and y) will be used for vectors, and uppercase bold letters (e.g., X and Z) will be used for matrices. Any vector is assumed to be a column vector. The transpose of a vector x and a matrix X is denoted as xT and XT, respectively. Thus, a row vector is denoted as xT.
2.1 Exponential mixed-effects models
Since tool wear is a continuous stochastic process, continuous states with respect to a regressor x, let sij denote the
exponential wear signals for the jth of ni observation times in the ith of n subjects shown in Figure 1. Therefore, we
can module these exponential tool-wear processes shown in Figure 6 as an exponential mixed-effects (EME) model,
sij¼ b00 exp½b0iþ (b1þ b1i)xij ij; b0i N (0; r20); b1i N (0; r21); Cov(b0i; b1i) ¼ r01;
ij LogN (0; r2); ð1Þ
where b00 andb1 are the fixed-effect coefficients, which are identical for all subjects, xij is the fixed-effect regressor for
observation j of subject i, b0i and b1i are the random-effect coefficients for subject i, assumed to be bivariate normally
distributed, and ij is a log-normal random variable. The structural and stochastic properties describe the deterministic
and probabilistic relationships between the population state and the states of individual subjects at a fixed moment in time. It should be noted that b0i and b1i are thought of as random variables, not as parameters, and are similar in this
respect to the error ij. The random effects b0i and b1i and the error ij are assumed to be independent for different
subjects and to be independent of each other for the same subject.
2.2 Linear mixed-effects models
The EME model shown by Equation (1) will be advantageous, after transformation, in becoming a linear mixed-effects (LME) model that may enhance the fit and predictability of the data (Venables and Ripley 2002), that is
yij¼ (b0þ b1xij) þ (b0iþ b1ixij) þ eij; i ¼ 1; . . . ; n; j ¼ 1; . . . ; ni; (2) where yij¼ ln (sij), b0¼ ln (b00), and eij¼ ln (ij) N (0; r2).
Equivalently, in general matrix form,
yi¼ Xib þ Zibiþ ei; bi N2(0; R); ei Nni(0; r
2I
ni); (3)
where, in this case,
yi¼ yi1 yi2 ... yini 0 B B B @ 1 C C C A; Xi¼ Zi¼ 1 xi1 1 xi2 ... ... 1 xini 0 B B B @ 1 C C C A; b ¼ bb01 ; bi¼ b0i b1i ; R ¼ r20 r01 r01 r21 :
Therefore, the marginal distribution of the response yiis given by
yi N (Xib; Vi); (4) where Vi¼ ZiRZTi þ r2Ini.
3. Inference based on likelihood methods 3.1 Estimation
Statistical inference for Equation (2) is typically based on the maximum likelihood (ML) or the restricted maximum likelihood (REML) method (Laird and Ware 1982, Lindstrom and Bates 1988). Let h denote the vector of all distinct parameters in the variance–covariance matrices R and r2, and let p ¼ (b; h) denote all parameters in Equation (2). The
likelihood for the observed data y¼ ½yT
1; . . . ; yTn T is given by L(p j y) ¼Yn i¼1 f(yijb; h) ¼ Yn i¼1 Z f(yij b; bi; r2) f (bijR)dbi; (5) where f(yij b; bi; r2) ¼ 1 r (2)ni=2exp½r 2(y i Xib Zibi)T(yi Xib Zibi); f(bijR) ¼ 1 2pffiffiffiffiffiffijRjexp( biR 1b i):
For the case given the variance–covariance R, the values of b and r2 that maximise the likelihood are
^b ¼ (XT iV 1 i Xi)1XTiV 1 i yi; (6) ^r2¼ 1 ni (yi Xi^b)TV1i (yi Xi^b): (7)
In the tool-wear process, the variance–covariance parameters are usually unknown. The maximum likelihood estimates (MLEs) of all the unknown parameters p in Equation (2) whose values maximise the probability of observing the spe-cific pattern can be obtained using an iterative numeric algorithm such as an expectation–maximisation (EM) algorithm or a Newton–Raphson method (Laird and Ware 1982, Lindstrom and Bates 1988).
3.2 EM algorithm
The EM algorithm is a popular iterative algorithm typically used to compute MLEs in the presence of missing data or unobservables (Dampster, Laird, and Rubin 1977). It is general, simple, and reliable. Given starting values of unknown parameters, the EM algorithm iterates between an E-step, which computes the conditional expectation of the log-likelihood of the complete data, the observed data and current parameter estimates, and an M-step, which maximises the conditional expectation in the E-step to obtain updated parameter estimates until convergence.
For the model given by Equation (2), we can treat the random effects as missing data, so we have complete data (y; b) ¼ f(yi; bi); i ¼ 1; 2; . . . ; ng. Let k, k ¼ 0; 1; 2; . . ., denote the iteration number. The E-step of the EM algorithm at
iteration k computes Q(p j p(k)) ¼ E½ln L(p j y; b) j y; p(k) ¼ E X n i¼1 f(ln f (yij b; bi; r2) þ ln f (bij R)) j yi; p(k)g " # : (8)
The M-step finds and updates parameter estimates p(kþ1) that maximise p(k). By iterating the above procedure until convergence, we obtain the MLE ofp.
Note that the variance components are underestimated since the degrees of freedom lost in the estimation of the fixed effects b are not incorporated in the estimation of R. Less-biased estimates of the variance components R can be obtained using the REML estimates (Harville 1977). Pinheiro and Bates (2002) provide computational details.
3.3 Inference
Standard errors of the MLEs are not directly produced in the EM algorithm, but they can be obtained separately. Confi-dence intervals and hypothesis tests for the parameters are often based on asymptotic results (Verbeke and Molenberghs 2001), since the exact finite sample distribution of the MLEs cannot be explicitly derived. That is, we can use the asymptotic normal distribution of the MLE to obtain approximate confidence intervals and hypothesis tests for the parameters. From the MLEs’ properties, ^yi can hence be regarded as normally distributed,
^yi N (Xi^b; ^Vi); (9)
where ^Vi¼ Zi^RZTi þ^r2Ini.
Finally, given a data set, the random effects can be estimated by empirical Bayesian estimators (Wu 2009) ^bi¼ E(bijyi; ^p) ¼ ^RZ
T
i^Vi(yi Xi^b): (10)
These random-effects estimates can then be used for subject-specific inference. 4. A friction drilling process
Empirical continuous-wear data collected from a friction drilling manufacturing process is illustrated in this section. Using the above methodologies, the LME model is developed to better capture the performance of machine tools with continuous-state wear behaviour. Moreover, in addition to the presented LME model, a linear regression model, a funda-mental fit disregarding the correlation among repeated measurements that potentially results in evaluation bias and loss of efficiency, is presented to serve as a baseline method due to its wide usage in real applications.
Example 4.1 Friction drilling is a modern machining process that utilises frictional heat between a rotating conical tool and the workpiece to increase the ductility of the work material. It is extruded onto both the front and back sides of the material being drilled, and penetrates the workpiece to form a hole (Miller, Blau, and Shih 2007). Figure 2 shows a
Figure 2. A friction drilling tool.
where U1Lognormal(l; r; P) is the inverse of the log-normal CDF with parameters l and r at a corresponding probability P.
Case study 5.3 (continued from Example 4.1): Based on Equations (18), (19), and (20), we can plot Figure 11. From the figure, we can first predict the change in wear length for a given number of cycles or, conversely, the number of applied cycles for a given change in wear length, and then obtain the corresponding lower and upper wear prediction limits of the 95% confidence level.
6. Conclusions
In modern manufacturing industries, machining tools play a major role in most applications of industrial fabrication for the production of quality goods. Continuous development in automation, computer-aided manufacturing techniques, and prognostic applications in autonomous machining processes has emphasised the significance of tool performance model-ling and measuremant in reducing the expense of the parts produced so as to assist manufacturers' in their world-wide competition. Machining systems are dynamic processes whose performance variable is usually characterised by the amount of tool wear, which proceeds gradually with a continuous range of values. To effectively monitor the perfor-mance of these continuous-state wear processes, simplified traditional models such as the binary-state (success and fail-ure) models and multi-state (limited classification) models may not be entirely satisfactory. This paper provides continuous-state statistical models and reliability measures to evaluate their current and future performance, which is useful for eliminating catastrophic failures as well as avoiding excessive replacement or maintenance activities. In this research, an exponential mixed-effects (EME) model is first introduced. The EME model is then transformed into a lin-ear mixed-effects (LME) model to take advantage of the fit and predictability for the observed wear process measure-ments. The LME model is capable of considering correlations among the repeated wear data taken at equal or unequal time points within each subject. We subsequently employ the expectation–maximisation (EM) algorithm to obtain full maximum likelihood estimates (MLEs) of the parameters of the LME model. The MLEs have the property of asymp-totic normal distributions that allow approximate confidence intervals and testing hypotheses for the parameters. When a pre-determined failure threshold h is specified, we obtain the reliability of the tool. The cumulative distribution function (CDF) of wear over time is suitable for optimising a maintenance strategy. The residual-lifetime distribution and the mean residual lifetime are useful information for the tool replacement strategy in the job shop. The prediction of the change in wear length for a given time or, conversely, the time for a given change in wear length, and the corresponding lower and upper wear prediction limits of the 95% confidence level are also presented. Throughout the paper, the proposed methodology is demonstrated using a friction drilling manufacturing process.
Acknowledgements
The authors acknowledge the helpful comments made by four anonymous referees. The work is partially supported by the National Science Council in Taiwan under grant 101-2628-E-151-002-MY3.
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