Hui Wang Qiang Huang1 e-mail:
[email protected] Department of Industrial and Management Systems Engineering, University of South Florida, Tampa, FL 33620
1
Using Error Equivalence Concept to Automatically Adjust Discrete Manufacturing Processes for Dimensional Variation Control Traditional statistical process control (SPC) has been widely employed for the process monitoring in discrete part manufacturing. However, SPC does not consider any adjustment preventing the process drift. Furthermore, many in-line adjustment approaches, such as thermal error compensation and avoidance, are designed only for machine tool error reduction. This paper intends to fully utilize the engineering process information and to propose an alternative compensation strategy based on equivalent fixture error (EFE) concept that could reduce overall effect of the process errors. Considering three types of error sources in a machining process, we propose to adjust fixture locators to compensate errors using the EFE model. The dynamic property of EFE is investigated for the feedback adjustment of both static and quasi-static errors in machining processes. A minimum-mean-square-error controller is designed based on the dynamic EFE model. We then evaluate the performance of the controller such as stability and sensitivity. A selfupdating algorithm for the controller will track the latest process information to stabilize the controller output. Finally, we simulate this process adjustment using the data collected from a real machining process. The results show that this algorithm can improve the machining quality. 关DOI: 10.1115/1.2714581兴
Introduction
The objective of process control is to keep the output as close as possible to the target all the time. Statistical process control 共SPC兲 focuses mainly on process change detection. It does not apply an adjustment to prevent process drift. Automatic process control 共APC兲 uses feedback or feed forward control to counteract the effects of root causes and to reduce the process variation. It has been applied mainly in continuous process industries where the process output has a tendency to drift away. Early research on APC can be tracked back to Box’s early work 关1,2兴. Less work has been done on the application of APC in a discrete part manufacturing process due to the conventional perceptions: 共1兲 feed materials of machining process such as raw workpiece can be reasonably controlled; 共2兲 the cost of process adjustment and frequent monitoring the process is substantial; and 共3兲 there are few maneuverable variables that can be easily controlled to adjust the machining process when multiple errors occur 关3兴. Therefore, the machining process control relies mainly on control of servo motor, interpolator, and adaptive loop in machine tools to reduce machine tool errors. 共A complete review of monitoring and control of machining can be founded in 关4,5兴.兲 Chen et al. 关6兴 considered thermal error reduction through real time machine tool error compensation, in which the thermal error is modeled as a function of machine temperatures collected by thermal sensors. Yang and Ni 关7兴 applied system identification theory into machine tool thermal error modeling. Donmez et al. 关8兴 and Mou et al. 关9兴 developed and experimentally verified a method that can compensate thermal errors by adjusting computer numerical control 共CNC兲 program. This line of research has been successful in reducing machine tool errors. However, the error com1
Correspondence author. Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received August 26, 2005; final manuscript received November 13, 2006. Review conducted by S. Jack Hu. Paper presented at the 2005 ASME International Mechanical Engineering Congress 共IMECE2005兲, November 5–11, 2005, Orlando, FL.
644 / Vol. 129, JUNE 2007
pensation approaches normally require the change and reload of the NC program, which may interrupt production. From the process point of view, there is also a lack of strategy to simultaneously compensate all error sources in machining processes. This study aims to develop an alternative compensation strategy and to apply APC in discrete manufacturing processes. Unlike the traditional approach, by which multiple process errors are compensated separately, our method intends to treat all the errors as one “system” and to use one type of error to compensate others. The method is based on the concept of “error equivalence,” which describes the phenomenon where by different error sources may result in identical variation patterns on part features. We have developed an equivalent fixture error 共EFE兲 model in the machining processes 关10,11兴, which transforms datum errors and machine tool errors into the equivalent amount of fixture locator deviations. This model suggests that by properly adjusting the fixture locator length, process errors can be compensated. This idea has been demonstrated through static error compensation in 关11兴. In this study we use the EFE model to compensate both static errors as well as quasi-static errors caused by the thermal effect of machine tools. Uninterrupted production can be achieved by automatically adjusting fixture locators during the period of changing the workpiece. In Sec. 2, we investigate the impact of errors on the process output and build a process variation model considering the quasistationarity in the process. In Sec. 3, an minimum-mean-squareerror 共MMSE兲 control rule is derived to counteract process variation. The control algorithm is implemented via a case study. In Sec. 4, the performance of the controller, such as stability and sensitivity when a change in the dynamics of the process occurs, is evaluated. Conclusions are drawn in Sec. 5.
2 Dynamic Equivalent Fixture Error Model for Feedback Adjustment A properly designed process adjustment algorithm should minimize process output variation. To achieve this objective, process information should be effectively utilized to predict the output
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deviation. Two tasks are involved: 共1兲 to predict the impact of process errors on the part feature deviation, and 共2兲 to model the process errors, especially the dynamic property of process errors. These two tasks have been accomplished separately; e.g., in 关10,11兴. In this paper, we aim to investigate the dynamic EFE model for feedback adjustment. The EFE model was first proposed in 关10兴 by observing the fact that part feature deviations can be affected by fixture, datum, and machine tool errors, and that these three types of errors could generate the same error pattern on machined features. The advantages of applying the EFE model in feedback adjustment are that: 共1兲 the EFE model transforms datum and machine tool errors into the equivalent amount of fixture locator errors, and 共2兲 feature deviations are predicted in terms of EFE corresponding to all three types of process errors. The detailed procedures are presented in 关10兴. Therefore, it provides a mathematical model to use fixture locators to compensate feature deviations without distinguishing the exact amount of three process errors. For a machining process, the kinematic fixture, datum, and machine tool kinematic errors can reasonably be regarded as static after the machining system has been running in a steady state. However, the thermal error of machine tool can be quasi-static in that thermal effect may cause the error to vary with time. The accumulated heat by previous machining may affect thermal error at the current time. Therefore, it is also reasonable to expect observation of machine tool error caused by thermal effect to exhibit time-varying characteristic. Under a general 3-2-1 locating scheme, the EFE due to thermal error or ⌬m is also time varying and determined by 关10兴 ⌬m共n兲 = K␦qm共n兲
冢
˜ = 关T ˜ 共1,n − l + 1兲 T ˜ 共2,n − l + 2兲 ¯ T ˜ 共l,n兲 Q ˜ 共1,n − l X m ˜ 共2,n − l + 2兲 ¯ Q ˜ 共l − 1,n − 1兲兴 + 1兲 Q m m
共1兲
where
K=
model fitting when the sensing information is limited and data are highly correlated. The temperature can be measured by placing the thermal sensors on the machine and thermal errors are obtained using in-line probes. The sensor readings for the temperatures t共n兲 are denoted by s共n兲 = 关s1共n兲s2共n兲 ¯ sr共n兲兴T, where si共n兲 is the reading from the ith thermal sensor. r is the number of sensors. The measured readings for thermal error ␦qm共n兲 are denoted by sm共n兲6⫻1. In order to fit the model to non-stationary data using LVM, the common treatment is to take first- or second-order difference on original data and check the first two moments for adequacy test 关16兴. If the first-order difference of data is adequate to yield stationary sequence, denote T共i , j兲 = 关s共i兲 − s共i − 1兲 s共i + 1兲 T − s共i兲 ¯ s共j兲 − s共j − 1兲兴共j−i+1兲⫻r and Qm共i , j兲 = 关sm共i兲 − sm共i − 1兲 sm共i T and i ⬍ j. Data should be + 1兲 − sm共i兲 ¯ sm共j兲 − sm共j − 1兲兴共j−i+1兲⫻6 mean-centered and scaled to the unit variance before the modeling. In this paper, we add operator “⬃” on the top of the notation to represent the scaled variable or data matrix 共scaled for each column兲. Suppose the speculated time lag is l, which can be chosen as a number large initially, and total n observations are col˜ and output vector ⌼ ˜ lected as training data. The input vector X can be represented as
0
0
− 1 − f 1y
f 1x
0
0
0
− 1 − f 2y
f 2x
0
0
0
− 1 − f 3y
f 3x
0
0
−1
0
f 4z
0
− f 4x
0
−1
0
f 5z
0
− f 5x
−1
0
0
0
− f 6z
f 6y
冣
˜ =Q ˜ 共l,n兲 Y m
共3兲
˜ is an n − l + 1 by rl + 6共l − 1兲 matrix consisting of the data X collection of temperatures and thermal errors. Here, the block ma˜ 共i , n − l + i兲 contains n − l + 1 scaled temperature data vectors trix T for 兵s共i兲 ⬃ s共n − l + i兲其i=1,2,. . .,l and can be regarded as the data collection of the variable difference ˜t共i兲 −˜t共i − 1兲, i = 1 , 2 . . . , l, over a ˜ 共i , n − l + i兲 includes period from i to n − l + i. Similarly, matrix Q
,
m
thermal error ␦qm = 共x y z ␣  ␥兲T and discrete variable n is the index for the time period. Each period can be a fixed amount of minutes or the time period for machining a batch of parts. The quasi-static error ⌬m共n兲 is considered to be relatively constant within the time period between the adjacent adjustments. x, y, z are translations along three directions of fixture coordinate while ␣, , ␥ are the rotational angles around these directions. Notations f 1 f 2 , . . . , f 6 represent coordinates of six fixture locators. We have derived and experimentally validated 关11兴 the predictive model for part feature deviation x共n兲 at time period n
n − l + 1 scaled thermal error vectors for 兵sm共i兲 ⬃ sm共n − l + i兲其i=1,2,. . .,l and is an 共n − l + 1兲-period 共from i to n − l + i兲 data collection of the variable difference ␦˜qm共i兲 − ␦˜qm共i − 1兲. Temperatures will be used for input and thermal errors will be for autoregressive ˜ includes the data collection of thermal terms in the model. ⌼ errors ␦˜qm共n兲. By LVM fitting procedure, we fit the regression coefficient G in Eq. 共A2兲 to the data in Eq. 共3兲. Hence, the firstorder differences of errors at time period n can be represented as the function of error sources in the previous periods: p1
␦˜qˆ m共n兲 − ␦˜qˆ m共n − 1兲 = −
兺 A˜ 共l兲关␦˜qˆ
m共n
− l兲 − ␦˜qˆ m共n − l − 1兲兴
l=1
x共n兲 = ⌫关⌬u + ⌬m共n兲兴 + 共n兲
共2兲
where ⌫ is the coefficient matrix and 共n兲 is the process noise vector whose entries are all assumed to follow zero-mean normal distribution. By vectorial surface model 关12–14兴, x is a vector consisting of surface orientation deviation and position deviation. ⌬u is the total EFE caused by static kinematic error of the machining process. ⌬u and ⌬m all compose of six locator-length deviations, e.g., under 3-2-1 locating scheme, ⌬m = 共⌬m1 , ⌬m2 , . . . , ⌬m6兲T. Next is to set up the dynamic model for ⌬m共n兲 based on the historical information of thermal errors 共␦qm or ⌬m兲 and temperatures t. The latent variable method 共LVM兲 关15兴 is adopted in this paper to fit the model ⌬m共n兲. Since LVM captures the underlying structure of input 共temperature兲 and output 共errors兲, rather than the impact of input on the output, it is especially appropriate for Journal of Manufacturing Science and Engineering
p2
+
兺 B˜ 共l兲关t˜共n − l兲 − ˜t共n − l − 1兲兴
共4兲
l=1
˜ 共l兲 is a 6 ⫻ 6 square coefficient matrix and its non-zero where A entries come from the entries in G corresponding to auto˜ 共l兲 is a 6 ⫻ 11 coefficient matrix and its nonregressive terms. B zero entries come from the entries in G corresponding to the temperature variables 共see the example of the coefficient matrices in the case study兲. p1 and p2 represent the maximum time lags for temperature and thermal error in the model, respectively. Time lag ˜ and p for B ˜ , n 艌 n . n is the starting period when the p1 is for A 2 0 0 adjustment applies. Scaling the data back with the mean and variance from the training dataset, we have JUNE 2007, Vol. 129 / 645
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p1
␦qˆ m共n兲 − ␦qˆ m共n − 1兲 = −
兺 A共l兲关␦qˆ
m共n
− l兲 − ␦qˆ m共n − l − 1兲兴
l=1 p2
+
兺 B共l兲关t共n − l兲 − t共n − l − 1兲兴 + D 共n兲 0
l=1
− D0共n − 1兲
共5兲
where D0共n兲 is the intercept term that is the linear combination of the means of the original data. A共l兲 and B共l兲 are the coefficient matrices after scaling back the data. Considering Eq. 共1兲, we get Fig. 1 Process control and monitoring
p1
⌬m共n兲 − ⌬m共n − 1兲 = − K
兺
A共l兲K−1关⌬m共n − l兲 − ⌬m共n − l
l=1
p2
− 1兲兴 + K
兺 B共l兲关t共n − l兲 − t共n − l − 1兲兴 l=1
+ K关D0共n兲 − D0共n − 1兲兴
共6兲
Denote q as the backward operator; e.g., q ⌬m共n兲 represents ⌬m共n − 1兲. Canceling 共1 − q−1兲 on both sides of Eq. 共6兲 leads to −1
−1
p1
⌬m共n兲 = − K
兺 A共l兲K
p2
−1
⌬m共n − l兲 + K
l=1
兺 B共l兲t共n − l兲 + K 共n兲 0
l=1
共7兲 where K0共n兲 is a matrix that is related to the initial condition t共n0兲, ⌬m共n0兲, and intercept term D0共n兲. Equation 共7兲 is the fitted model for the quasi-static EFE thermal error. It will predict the thermal error at the next period based on all the previous information such as the temperatures and thermal errors collected.
3 Feedback Adjustment Based on a Dynamic Error Equivalence Model 3.1 Overview of the Methodology. The proposed compensation in this paper is an APC methodology that can adjust the machining process. The most widely discussed control rule in APC is the MMSE control. It is based on the stochastic control theory 关17兴 to find out the optimal control rule to minimize the MME of the process output. More recently, more research efforts are directed towards the approach combining SPC and APC to secure both the process optimization and quality improvement. MacGregor 关18兴 was among the first to suggest SPC charts to monitor the controlled process. The similarities and overlap between SPC and APC were described. The integration of APC and SPC has been reviewed in 关3兴. In these early papers, a minimumcost strategy is suggested to adjust the process and SPC charts are used as dead-bands or filtering devices 关19兴 for the feedback controlled process. This dead-band concept was extended for multivariate problems in 关20兴. Vander Wiel et al. 关21兴 proposed an algorithmic statistical process control 共ASPC兲, which reduces the process variation by APC and then monitors the process to detect and remove root cause of variation using SPC. Tucker, Faultin, and Vander Wiel 关22兴 elaborated on the ASPC by giving an overall philosophy, guidelines, justification, and indicating related re-
search issues. SPC integrated APC in a discrete machining process can be represented by Fig. 1. One can see that a nominal machining process is disturbed by errors ⌬u + ⌬m共n兲 共represented by EFE兲 and the observation noise 共n兲. Errors ⌬u + ⌬m共n兲, noise 共n兲, and the machining process constitute a disturbed process, as marked in the dashed line block. Using the observed feature deviation x共n兲 as input, a controller is introduced to generate signal c共n兲 to manipulate adjustable fixture locators to counteract the root cause ⌬u + ⌬m共n + 1兲 for the 共n + 1兲th period, where c共n兲 represents the cumulative amount of adjustment for period n + 1 after completing period n. The measurement on features x共n兲 is only applied before each adjustment. The adjustment for machining period n + 1 should be c共n兲 − c共n − 1兲. Then, the measured feature deviation x共n + 1兲 is x共n + 1兲 = ⌫c共n兲 + N共n + 1兲 N共n + 1兲 = ⌫关⌬u + ⌬m共n + 1兲兴 + 共n + 1兲
共8兲
where N共n兲 is the disturbance that describes how the process drifts away from the target without an adjustment being made. It represents the varying nature of the disturbed process in Fig. 1. Due to the variability of the controlled system such as deformation and looseness of fixture locators, the potential failure of sensors and probes, it might be necessary to apply the control chart to the control error x and the control signal c. The observations outside the control limits on the control charts may indicate periods where large variation of the control signal is generated. For the example of EFE adjustment, out-of-control on EWMA chart may indicate a large variation of adjustable fixture locators. Therefore, the proposed approach should include the tasks of statistical model training, controller design, controller performance evaluation, and process monitoring. The adjustment using EFE can be illustrated with an example in Fig. 2, where a prismatic part is set up in a fixture with locators f1, f2, and f3. We expect to perform a parallel cutting on the top plane of the part. If the tool path tilts due to the thermal effect, the yielded top plane will also tilt the same angle. However, under the fixture where the length of locator pin is adjustable, we may find out the adjustment amount 共black bar in right panel of Fig. 2兲 for f1, f2, and f3 such that the part tilts the same angle as the deviated tool path. Obviously, a conforming part can still be obtained.
Fig. 2 Process adjustment using EFE concept
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Fig. 3 Raw workpiece setup for a milling process †11‡
Similarly, we can also adjust fixture locators to compensate the datum error. The amount of adjustment can be determined by EFE using Eq. 共1兲. With this concept, the feature deviation caused by machine tool thermal error 共tilted tool path兲 can also be generated by EFE 共⌬m1 , ⌬m2 , ⌬m3兲 alone. In order to compensate this error, we must apply the amount of adjustment 共−⌬m1 , −⌬m2 , −⌬m3兲 to three locating pins.
the disturbed process. Substituting process model 共7兲 into Eq. 共8兲, the prediction for feature deviation at period n + 1 is
3.2 EFE Controller. The design of controller based on the EFE model 共namely, EFE controller兲 relies on the knowledge of
The control algorithm can be designed to minimize the meansquared deviation of product feature; i.e., min E关x2共n + 1兲兴. As
Fig. 4
p2
xˆ 共n + 1兲 = ⌫c共n兲 − ⌫K
兺
p2
A共l兲K−1⌬m共n − l兲 + ⌫K
l=1
兺 B共l兲t共n − l兲 l=1
+ ⌫关⌬u + K0共n兲兴
共9兲
„a… Machine tool temperature and error data †15‡. „b… Thermal error measurements.
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pointed out by 关23兴, we can treat a simpler problem of minimizing an instantaneous performance index: min关xˆ 2共n + 1兲兴. Taking the first derivative of xˆ 2共n + 1兲 in Eq. 共9兲 and setting it equal to zero, we find out that control signal c共n兲 must be able to cancel the predicted errors in period n + 1, i.e.
p1
c共n兲 = K
兺 A共l兲K
p2
−1
⌬m共n − l兲 − K
l=1
− K0共n兲
兺 B共l兲t共n − l兲 − ⌬u l=1
c共n0兲 = 0
ˆ 共n − l兲 = − c共n − l − 1兲 + 共⌫T⌫兲−1⌫Tx共n − l兲 − ⌬u ⌬m
n 艌 n0 共10兲
Equation 共10兲, the control rule for the machining adjustment, shows that, at the end of machining period n, this algorithm predicts the errors at period n + 1 and uses negative value of this prediction to implement compensation. The static errors ⌬u can be determined by subtracting quasi-static thermal errors from 共⌫T⌫兲−1⌫Ty共1兲, since they are invariant with period n; i.e.
⌬u = 共⌫T⌫兲−1⌫Ty共1兲 − ⌬m共1兲
共11兲
In practice, the accuracy that the adjustable locator can achieve must be considered. Suppose the standard deviation of locator’s movement is f . We use a dead-band adjustment 关2兴 scheme by which the locators are adjusted only when the predicted EFE fall outside the dead-band 共or stopping region兲. The stopping region for applying error compensation with 99.73% confidence is set to be
− 3 f 艋 c共n兲 − c共n − 1兲 艋 3 f
共12兲
As an alternative strategy to the conventional compensation, the proposed method has its application conditions. First, the adjustment could introduce a new random “error source” because of the variability in the actuator. Therefore, the adjusted total process error ua has uˆ a = c共n兲 + ⌬uˆ + ⌬mˆ 共n兲 and ⌺uˆ a = ⌺c共n兲 + ⌺⌬uˆ + ⌺⌬mˆ 共n兲, where and ⌺ represent the expectation and variancecovariance matrices of the variable in the subscript, respectively. However, the generalized variance of error ua or 兩⌺uˆ a兩 共兩·兩 stands for the determinant of a matrix兲 is not necessary to be smaller than the one without adjustment. The method is effective only when 兩⌺c共n兲兩 艋 兩⌺⌬fˆ兩, where ⌬f is the fixture error included in the static error ⌬u. • 兩⌺c共n兲兩 ⬎ 兩⌺⌬fˆ兩, but the increase of total process variation 共兩⌺uˆ a兩 − 兩⌺共⌬uˆ +⌬mˆ 共n兲兲兩兲 / 兩⌺关⌬uˆ +⌬mˆ 共n兲兴兩 is insignificant.
•
Second, it should be pointed out that the two compensation strategies can be applied complementarily. The conventional compensation strategy aims to offset ⌬uˆ and ⌬mˆ 共n兲 and reduce their corresponding variance individually. The error sources with the largest variations can be compensated using conventional methods to reduce ⌺关⌬uˆ +⌬mˆ 共n兲兴. The new compensation strategy is to cancel the mean-shift of the process outcome. 3.3 Self-Updating EFE Control. Due to the change of process conditions or occurrence of unexpected errors, the fitted thermal model may show a large prediction error. When SPC signals 648 / Vol. 129, JUNE 2007
an alert, the model might need to be updated so that it can catch up the latest information of the process. Suppose we measure the temperature and thermal error every period, and the measurement data are available at the period 1 ⬃ n0. The updating control procedure is proposed as follows: 1. At the beginning of period n0 + k, the data, including part features 共measured by a coordinate measuring machine p1 共CMM兲兲 兵x共n0 + k − l兲其l=1 , thermal errors 共measured by inp1 line probes兲 兵␦qm共n0 + k − l兲其l=1 , and temperatures 共measured p1 by thermal sensors兲 兵t共n0 + k − l兲其l=1 , are collected to compute the locator adjustment c共n0 + k − 1兲 − c共n0 + k − 2兲. k is the period when SPC identifies significant errors in the controlled process 共Eqs. 共10兲 and 共A2兲兲. Then cut the parts after the adjustment. With the updating scheme, the fitted coefficient p1 p2 and 兵B共l兲其l=1 in Eq. 共10兲 also change with matrices 兵A共l兲其l=1 period n 共or equivalently, updating iteration兲. Thus, it is reap1 p2 and 兵Bn共l兲其l=1 . sonable to denote them as 兵An共l兲其l=1 2. At the end of period n0 + k, measure the parts and take the average of measurement results to estimate x共n0 + k兲. The updating scheme can enhance the robustness of the EFE controller to the process change. 3.4 Case Study. In this paper, we use a single-stage milling process to implement the process adjustment. The process performs cutting on two planes X1 and X2 as shown in Fig. 3. Thickness along the z direction 共lz兲 and y direction 共ly兲 are the part features to be controlled 共the nominal thickness of the finished part is lz = 15.24± 0.1 mm and ly = 96.5± 0.1 mm兲. In this simulation, we use the data 共Fig. 4共a兲兲 from the experiment in 关15兴. There are 11 thermal sensors mounted on the CNC milling machine to collect data 共r = 11兲. Figure 4共b兲 shows the thermal deviation along two directions: the angular deviation ␣ around the x axis and translational deformation along the z direction of the tool head. The upper left panel of Fig. 4共a兲 shows the readings from 11 thermal sensors. The upper middle, right, and lower panels show the measurement of thermal errors. The data are collected in every period. We have derived ⌫ in Eq. 共2兲 to be 关11兴
⌫=
冢
0
0
0
− 0.0158
0
0.0079
0.0079
0
0
0
0
0
0
0
0
0
0 − 0.0828
− 0.0263 0.0263
− 0.1379 0.1379 1.3368 − 1.3368 − 1 0.0414
0.0414
− 1.3033 − 0.8483 1.1517 0
0
0
0
0
0
− 1.5
0.5
0
0
− 0.0263 0.0263
0.0158 − 0.0079 − 0.0079
0 0 0
0
0
0
0
0
0
0
0.2632
− 0.2632 − 1.2026 1.2026 − 1
0.158
− 0.079
− 0.079
− 1.5
0.5
0
0.2212 − 1.6106
0.3894
0
0
0
冣
共13兲
Suppose the maximum time lag in the model is 5, and n = 95. It can be shown that the first-order difference is sufficient in resulting stationary time sequences for the temperature and thermal deformation data. Then, the fitted coefficient matrix G is Transactions of the ASME
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Fig. 5 EFE adjustment
G=
冢
0 ¯ a1共1兲 0 v1共4兲 ¯ v11共4兲 ¯ v1共0兲 ¯ v11共0兲 a1共4兲 0 0 a2共4兲 ¯ a2共1兲 w1共4兲 ¯ w11共4兲 ¯ w1共0兲 ¯ w11共0兲
where vi共l兲 and ai共l兲 are fitted coefficients. Then, coefficient ma˜ 共l兲 and B ˜ 共l兲 are trices A
冢 冢
˜ 共l兲 = A
˜ 共l兲 = B
0 0
0
0
0 0
0 0
0
0
0 0
0
0 0
0 0 − a1共l兲
− a2共l兲 0 0
0 0
0
0 0
0
0
0 0
0 0
0
0
0 0
0
0
0
¯
0
0
0
0
¯
0
冣
v1共l兲 v2共l兲 v3共l兲 ¯ v11共l兲 w1共l兲 w2共l兲 w3共l兲 ¯ w11共l兲
0
0
0
¯
0
0
0
0
¯
0
and
冣
共14兲
6⫻11
Both static kinematic errors and machine tool thermal errors have been introduced to the milling process. The static kinematic errors, after being transformed to EFE, are assumed to be ⌬u = 关0.4 0 0.35 0 0 0兴T mm. The measurement noise 共n兲 is assumed to follow N共0 , 共0.002 mm兲2兲 for displacement and N共0 , 共0.001 rad兲2兲 for orientation. For each period, five parts go through the cutting operation. We use the average of five measurements to estimate the real feature deviation for each period. Thermal error and temperature for 95 periods 共thus, n0 = 95兲 are available before the adjustment is applied. The measurements of temperature from i ⬃ 95+ i periods and thermal error from i ⬃ 94 + i are used to estimate the adjustment of locator pins for the 95 + ith period, i = 1 , 2 , . . . , 20. The controller is updated after measuring the parts at the 95+ ith period. Journal of Manufacturing Science and Engineering
冣
T
The accuracy of the locator movement is assumed to be f = 0.003 mm and the criterion for stopping the compensation is −0.01艋 c共n兲 − c共n − 1兲 艋 0.01 mm. The values of adjustments for six locators are given by the solid line in the Fig. 5. The dash dot line represents the value of ±3 f . The adjustments for locators 4, 5, and 6 are all zero since no EFEs are introduced on these locators in this example. The effect of the automatic process adjustment can be evaluated by monitoring the thicknesses of the parts ly and lz. The mean of such distance 共in each period兲 is estimated by the average of four edge lengths along the y and z directions. The variability of four edge lengths in individual parts is estimated by the variance of the four edge lengths. Figure 6 shows the mean and standard deviation of the thickness for 20 periods 共period 95–114兲. There is no adjustment applied in period 95. We can see that, after the process adjustment, the mean of the thickness is within the specification limit 共±0.01 mm兲 and the variance is greatly reduced. We conclude that the proposed control algorithm can significantly increase the product quality. It should be noticed that the thickness ly has less mean-shift than lz. This is because plane X2 tilts around the x axis and the distances between edges ly are smaller along z direction. Such edge layout leads to edge length with less variance and mean-shift.
4
Stability and Sensitivity of EFE Controller
MMSE control has unstable modes 关17兴. In some occasions, it causes the process to adapt to the disturbance changes and causes larger output response. In this paper, the stability of a controller means that an error in the output can be canceled by an adjustment sequence that converges to zero 关23兴. Introducing backward operator q−1, Eq. 共10兲 can be represented as JUNE 2007, Vol. 129 / 649
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Fig. 6 Monitor for thickness and standard deviation of edge length
冋
p1
I+K
兺 l=1
册
p1
A共l兲K−1q−l−1 c共n兲 = K
兺
p1
A共l兲K−1共⌫T⌫兲−1⌫Tq−lx共n兲
c j共n兲 = −
l=1
兺 B共l兲q
兺兺
p2
关a1共l兲 + a2共l兲f jy兴hi⌬mi共n − l兲 −
l=1 i=1
−l
11
兺 兺 关v 共l兲 i
l=1 i=1
+ wi共l兲f 1y兴ti共n − l兲 − ⌬ui − k0i共n兲,
p2
−K
3
j = 1,2,3
t共n兲
l=1
p1
p1
−K
兺 A共l兲K
−1
⌬u − ⌬u − K0共n兲
l=1
共15兲 The stability of the controller is governed by the entries in 6 ⫻ 6 p1 matrix 关I + K 兺l=1 A共l兲K−1q−l−1兴−1. If the roots 共poles兲 of denominator of each entry in this matrix are inside the unit circle in q plane, the controller is stable. It is clear that the controller is always stable if the thermal error model does not contain an autoregressive term, i.e., A共l兲 = 0. When the auto-regressive terms are included in the model, the controller may be unstable. The designed MMSE controller at certain periods may contain unstable poles 共poles outside unit circle兲. This may cause the MMSE controller to exhibit fluctuation and large output if the controller parameters A共l兲 and B共l兲 were unchanged as n increased. One solution for the unstable output is to use a thermal model 共Eq. 共7兲兲 without auto-regressive term since the derived controller is always stable. In our adjustment algorithm, we introduce the updating scheme, which makes the controller output capture the latest process information. In this case, Eq. 共15兲 is not strictly proper to evaluate the p1 p2 stability because the coefficient matrices 兵A共l兲其l=1 and 兵B共l兲其l=1 p1 are varying with the adjustment period n 共denoted as 兵An共l兲其l=1 p2 and 兵Bn共l兲其l=1 兲. In practice, the proposed algorithm can achieve satisfactory results. This has been validated by the results from the simulation study. Another important issue is the sensitivity of the controllers to the modeling errors that can feasibly occur. If there are moderate changes of modeling parameters 共entries in matrices An共l兲 and Bn共l兲兲, we are more interested in how the quality of the product could be affected. Such change may be due to several reasons, including sensor reading errors and change of lubrication condition. To study the sensitivity, expand Eq. 共10兲 as 650 / Vol. 129, JUNE 2007
c j共n兲 = −
3
兺兺
p2
a2共l兲f jzhi⌬mi共n − l兲 −
l=1 i=1
− k0i共n兲,
11
兺 兺 w 共l兲f i
jzti共n
− l兲 − ⌬ui
l=1 i=1
j = 4,5 c6共n兲 = 0
共16兲
where hi is the function of fixture coordinates f 1 , . . . , f 6. Differentiating both hand sides of Eq. 共16兲 leads to p1
⌬c j共n兲 = −
3
兺兺
p1
hi⌬mi共n − l兲⌬a1共l兲 −
l=1 i=1
3
兺 兺 h ⌬m 共n
p2
− l兲f jy⌬a2共l兲 −
11
− l兲⌬wi共l兲,
⌬c j共n兲 = −
i
p2
11
兺 兺 t 共n − l兲⌬v 共l兲 − 兺 兺 f i
i
l=1 i=1
p1
i
l=1 i=1
j = 1,2,3
3
兺兺f
1y ti共n
l=1 i=1
p2
jzhi⌬mi共n
− l兲⌬a2共l兲 −
l=1 i=1
− l兲⌬wi共l兲,
11
兺兺f
jzti共n
l=1 i=1
j = 4,5 ⌬c6共n兲 = 0
共17兲
⌬m共n − l兲 is only related to the previously fitted model and is not affected by the fitting error of An共l兲 and Bn共l兲. It can be considered as a constant when we conduct the sensitivity analysis. For the example in Sec. 3.4, substituting the values of coordinates yields Transactions of the ASME
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Fig. 7 Effect of controller parameters change
p1
⌬c1共n兲 =
兺 共25⌬m
p1
1
+ 16.3⌬m2 − 22.1⌬m3兲⌬a2 +
l=1
兺 共1.3⌬m l=1
p2
+ 0.8⌬m2 − 1.2⌬m3兲⌬a1 −
11
兺 兺 t ⌬v i
i
l=1 i=1
p2
− 19.2
11
兺 兺 t ⌬w i
i
l=1 i=1
p1
⌬c j共n兲 =
兺 共107.5⌬m
p1
1
+ 70⌬m2 − 95⌬m3兲⌬a2 +
l=1
兺 共1.3⌬m
1
l=1
p2
+ 0.8⌬m2 − 1.2⌬m3兲⌬a1 − p2
兺 兺 t ⌬v i
i
5
11
兺 兺 t ⌬w , i
j = 2,3
i
l=1 i=1
p1
⌬c j共n兲 =
兺 共− 13.3⌬m
1
− 8.4⌬m2 + 11.5⌬m3兲⌬a2
l=1
p2
− 10
11
兺 兺 t ⌬w , i
i
j = 4,5
l=1 i=1
⌬c6共n兲 = 0
共18兲
To simplify the representation, time indices 共n − l兲 and l are dropped in this equation. We can conclude the following about the designed controller at time period n: • •
•
The updating scheme can effectively increase the sensitivity robustness of the controller. We have simulated the feature deviation when there are changes of 50%, 200%, 350%, and 500% in the coefficients v6共0兲 and w6共0兲 in matrix B105共0兲. Figure 7 shows an example when there are changes up to 500% in the coefficients. We can notice a large variation of feature lz at period 104 and 105. Feature ly is not too much affected. After period 105, the feature lz falls within the specification limit since the fitting error has been counteracted by the updated model.
11
l=1 i=1
− 82.5
error occurs only around z and along x directions. The EFEs on locators 1, 2, and 3 have more impact on the feature deviation than locators 4 and 5. Locator 6 never affects feature deviation along these two directions.
1
There is no adjustment on the locator 6; Deviation of coefficients a1共n − l兲 and vi共n − l兲 does not affect the adjustment c4共n兲 and c5共n兲; a1共n − l兲 has the same effect on the adjustment of c1共n兲, c2共n兲, and c3共n兲; The adjustment for the locators 2 and 3 are most likely to be affected by the fitting errors. Locators 4 and 5 are least sensitive to the fitting error. This is because the thermal
Journal of Manufacturing Science and Engineering
Conclusion
Regarding error compensation, the conventional method in machining processes is to compensate the multiple errors individually. However, APC and its integration with traditional SPC have not been fully addressed in error compensation in discrete part machining processes. This paper introduced APC to discrete manufacturing process as an alternative compensation methodology using error equivalence concept. This new strategy is to use one type of error to compensate the overall effect of multiple error sources, rather than to reduce the errors individually. An EFE controller design is outlined based on the engineering process fault model and statistical disturbance model. It uses model prediction to compensate the errors in the future periods. SPC is applied to the controlled process to identify the unexpected process errors. When SPC signals an alert, the fitted model is updated to obtain the latest information of the dynamic process. The control algorithm is implemented using the data collected from a milling process. We have shown that the EFE controller can effectively improve the machining accuracy and reduce the variation. The applicable condition of this new compensation method is discussed as well. We pointed out that the conventional thermal error compensation will be preferable if this condition does not hold. Therefore, this strategy can be implemented with conventional ones complimentarily. JUNE 2007, Vol. 129 / 651
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The performance of designed controller is analyzed. Without controller updating, stability of EFE controller is sensitive to the process data. We have demonstrated that the proposed updating scheme is effective to tune the EFE controller parameters and stabilize its output. We also studied the sensitivity of controller output to the change of controller parameters. It helps to find out parameters that contribute most to the controller output deviations.
Acknowledgment The authors are grateful to the comments and suggestions from the anonymous reviewers.
Appendix: Latent Variable Modeling Latent variable modeling is a method for establishing predictive model when the data available are potentially collinear. By this method, the data matrices X and Y are represented by linear combination of underlying latent variables in L X = LPT + E T
Y = LQ + F
关4兴 关5兴 关6兴 关7兴 关8兴 关9兴
关10兴 关11兴
共A1兲
where each column of input data set X consists of process measurements at certain time, and each row of output matrix Y consists of quality variable or process output. Each column of X and Y is centered and scaled to the unit variance. P and Q are loading matrices for X and Y, respectively. Score matrix L = XW共PTW兲−1, each of which is one realization of latent variable. W is the weights matrix. E and F are un-modeled noise terms. Equation 共A1兲 projects X and Y spaces into lower dimensional subspaces spanned by the columns of L. The subspaces capture the most relevant structures of X and Y spaces. Y can be expressed in a regression form as Y = XG + F
关3兴
共A2兲
关12兴 关13兴 关14兴 关15兴 关16兴 关17兴 关18兴 关19兴
where G = W共PTW兲−1QT. Partial least-squares estimation determines the latent variable by maximizing the covariance between X and Y. It is widely applied in model fitting and adopted in this paper.
关21兴
References
关22兴
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652 / Vol. 129, JUNE 2007
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Transactions of the ASME
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