ICA based thermal source extraction and thermal distortion compensation method for a machine tool

International Journal of Machine Tools & Manufacture 43 (2003) 589–597

ICA based thermal source extraction and thermal distortion compensation method for a machine tool D.S. Lee a, J.Y. Choi a,∗, D.-H. Choi b b

a School of Electrical Engineering, Seoul National University, Kwanak, P.O. Box 34, Seoul 151-742, South Korea School of Electrical Engineering and Computer Science, Kyungpook National University, Daegu 702-701, South Korea

Received 15 July 2002; received in revised form 9 December 2002; accepted 3 January 2003

Abstract Thermal distortion in machine tools is one of the most significant causes of machining errors. One of the difficult issues in developing a system to compensate for thermal distortion is to select the appropriate temperature variables and to obtain an accurate thermal distortion model. This paper presents a new thermal distortion compensation method based on the Independent Component Analysis (ICA) method. The ICA method was used to extract the thermal sources from the temperature variables. The Optimal Brain Surgeon (OBS) algorithm was used to reduce the temperature variables with insignificant information. Using the extracted sources, a new thermal distortion model and a compensation method is proposed and is implemented in real-time hardware. In these experiments, the proposed method was shown to be capable of compensating for thermal distortions to a few micrometers.  2003 Elsevier Science Ltd. All rights reserved. Keywords: Machine tool; Thermal distortion compensation; Independent component analysis (ICA); Thermal source extraction; Variable reduction; Real-time compensation

1. Introduction In order to produce products with a high accuracy and good quality, errors in the manufacturing process should be minimized. Computerized Numerical Control (CNC) manufacturing inaccuracies are caused by the following errors: (1) geometric errors of the machine components and structures, (2) errors induced by thermal distortions, (3) deflections caused by the cutting forces, and (4) the others factors such as servo errors of the machine axis and NC interpolation algorithmic errors [1]. Thermal distortion in machine tools is one of the most significant causes of machining errors. Thermal distortion comprises more than 50% of distortions for kinematically well-aligned machines [2,3]. Recent technology to reduce thermal distortion compensates for thermal distortion by using a proper thermal distortion model. Several thermal distortion models with linear regression

Corresponding author. Tel.: +82-2-880-8372; fax: +82-2-8884459. E-mail address: [email protected] (J.Y. Choi). ∗

analysis have been suggested [4,5]. Other researchers have attempted to model the thermal distortions of machine tools using neural networks [6,7]. One of the difficult problems in a thermal distortion compensation system for machine tools is to select the appropriate temperature variables and to obtain an accurate thermal distortion model. In a recent study [4], correlation grouping, representative searching, group searching and variable searching were suggested to find the optimal temperature variables and the optimal model. However, the procedure required a huge amount of computation time. This paper presents a new thermal distortion compensation method based on independent component analysis (ICA) [8]. In the proposed ICA method based on a thermal distortion compensation system, the thermal sources were extracted from the measured temperature variables and insignificant thermal sources were eliminated. The dimensions of the temperature variables were reduced by the OBS (Optimal Brain Surgeon) pruning algorithm [9]. A compensation model using significant thermal sources is proposed. The proposed method is shown to be able to compensate for thermal distortions to a few micrometers.

0890-6955/03/$ - see front matter  2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0890-6955(03)00017-8

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Section 2 presents the proposed thermal source extraction method. Section 3 presents the proposed thermal compensation model. Section 4 describes the experiment settings and evaluates the performance of the proposed real-time compensation system. Conclusions follow in Section 5.

2. Thermal source extraction It is difficult to determine the optimum location of the temperature sensors and to select the number of sensors in machine tools. In general, the heating elements in machine tools are bearings, gears, hydraulic oil, drives, clutches, pumps, motors, guide-ways, cutting action and swarf, and the external heat elements. Many temperature sensors are needed to measure all the possible heat elements. However, it is impossible to measure some of these heat elements directly and some or perhaps all temperature sensors are correlated with each other. Fig. 1 shows the test machine (MCH-10, Tongil Heavy Industry, Korea) and the CNC Controller (FANUC 15M, FANUC, JAPAN) used to compensate for thermal distortion. Fig. 2 shows the structure of the test machine and the attached temperature sensor positions. Ti denotes the position of the temperature sensor variable. Sixteen temperature sensors were attached to the spindle housing, the machine body, the guide-way, the worktable, and the machine housing. If thermal sources can be found, a thermal distortion model can be constructed by the thermal sources instead of the temperature variables. Fig. 3 shows conceptually the relationship of the thermal sources and the measured temperature variables. xi and si indicate the temperature variable and thermal source, respectively. Each xi is influenced by all the thermal sources, therefore, they are closely related to each other. The goal to extract the thermal sources from the temperature variables is the same as the goal of the ICA. x 1, x 2,…, xN are expressed by a linear combination of unknown sources, which are denoted by s 1, s 2,…, sM. The sources are assumed to be

Fig. 1.

Fig. 2. The structure of the test machine (MCH-10) and the positions of the temperature sensors.

mutually statistically independent from each other. Define x(t) =[x1(t),%, xM(t)]T and s(t) = [s1(t), %, sM(t)]T at each time point t. Then x(t) ⫽ As(t),

(1)

where A is an unknown mixing matrix. The goal of the ICA is to recover the independent sources from the sensor observations that are unknown mixtures of independent source signals. In general, an estimate of the thermal source is sought as a linear transformation of the measured variables, i.e., u(t) ⫽ Wx(t) ⫽ WAs(t),

(a) Test machining center (MCH-10). (b) CNC controller (FANUC-15M).

(2)

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cant thermal sources) and the insignificant thermal sources (M - m 0; the number of insignificant thermal sources). As the significant thermal sources are relevant to the thermal distortions in machine tools, the insignificant thermal sources, which are irrelevant to the thermal distortions, can be eliminated. The waveform of the insignificant thermal sources is similar to the white noise signal. 3. Thermal distortion modeling Fig. 3. The conceptual relationship of the thermal source (si) and the temperature variable (xi).

where W is a matrix to be determined and u is an estimate. The sources are recovered exactly when W is the inverse of A up to a permutation and scale change. P ⫽ RS ⫽ WA,

(3)

where R is the permutation matrix and S is the scaling matrix. The two matrices define the performance matrix, P, so that a perfect separation leads to the identity matrix if P is normalized and reordered. The ICA procedure is illustrated in Fig. 4. si can be considered as the thermal source of a machine tool, xi can be considered as the temperature measurement, and ui can be considered to be the estimate of the thermal source. In this paper, the extended infomax algorithm [8] to determine W, ⌬W⬀[I⫺Ktanh(u)uT⫺uuT]W

再

ki ⫽ 1: for supergaussian data ,

(4)

ki ⫽ ⫺1: for subgaussian data

where ki is a diagonal element of the diagonal matrix K and the other element is 0. The switching parameter, ki, is determined by ki ⫽ sign[E{sech2(ui)}E{u2i }⫺E{tanh(ui)ui}].

(5)

The estimated thermal sources are comprised of the significant thermal sources (m 0; the number of signifi-

Independent thermal sources were determined by the ICA method in the previous section. If some of the temperature variables are unrelated to the extracted sources, they can be eliminated. Thus, the cost of temperature sensors can be reduced by reducing the dimension of the temperature variables. In this paper, the Optimal Brain Surgeon [9] method was used to prune the temperature variables. In Fig. 4, a de-mixture matrix, W, can be considered as a set of network weights. Let W be a network that is trained to a local minimum in error. The mean square error on the training set is defined as

冘 P

E⫽

1 (u[k]⫺u¯[k])T(u[k]⫺u¯[k]), 2Pk ⫽ 1

(6)

where P is the number of training patterns, and u[k] and u¯[k] are the original network output and the network output after pruning a weight for the k-th training pattern, respectively. The functional Taylor series of the error with respect to the weights is described by dE ⫽ (

1 ∂E T ) ·dw ⫹ dwT·H·dw ⫹ O(|dw|3), ∂w 2

(7)

∂2E is the Hessian matrix, w = ∂w2 [w11 … w1N w21 … w2n … wm01 … wm0N] T is a weight vector, m 0 is the number of the significant thermal sources, N is the number of the temperature variables, and the superscript T denotes the vector transpose. Our goal is then to set one of the weights to zero (which is called wq) in order to minimize the increase in error given by Eq. (7). For a network trained to a local minimum error, the first term in Eq. (7) vanishes. In addition, the third and all higher order terms are ignored. Hence, only the second term in Eq. (7) is considered. Eliminating wq is described as dwq + wq = 0 or more generally, where H ⬅

eTq·dw ⫹ wq ⫽ 0,

(8)

where eq is the unit vector in weight space corresponding to the weight, wq. The objective function used to solve this problem is defined as 1 min{min( dwT·H·dw), q dw 2 Fig. 4.

The ICA structure for thermal source extraction.

subject to e ·dw ⫹ wq ⫽ 0}. T q

(9)

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To solve Eq. (9), a Lagrangian is formulized from Eqs. (6) and (8). 1 L ⫽ dwT·H·dw ⫹ l(eTq·dw ⫹ wq) 2

(10)

where l is a Lagrange multiplier. The functional derivatives are taken, the constraints of Eq. (8) are employed, and a matrix inversion used to determine the optimal weight change, dw. The resulting changes, Lq, in error are as follows: dw ⫽ ⫺ Lq ⫽

1 wq H⫺1·eq, 2[H⫺1]qq 2 q ⫺1

w , (2[H ]qq)

(11)

-1

1. Compute H . 2. Find the q that has the lowest saliency Lq = w2q / (2[H⫺1]qq). If this candidate error increase is much smaller than a specific value ⑀, then the qth weight should be deleted, and move to step 3. Otherwise stop the pruning procedure. 3. Use the q from step 2 to update all weights. Go to step 1. The network pruning of weights could be interpreted as the reduction in the number of temperature sensor variables. Fig. 4 shows a pruning example. Two estimated significant sources will be obtained by the ICA method if there are two significant sources and M measured variables. As a result, the insignificant temperature variables can be removed by the OBS algorithm and the minimum significant temperature variables will remain. The thick solid lines in Fig. 4 denote the significant signal flow and the thin solid lines denote the insignificant signal flow. In the de-mixture matrix, W, the thick lines denote the signal between the extracted thermal sources and the reduced temperature variables. In general, the thermal distortion model can be designed to the linear regression model using a temperature variation, ⌬xi(t), as follows [1],

冘 M

xi(t) ⫽

aijsj(t),

where M is the number of thermal sources. Eq. (13) can be represented by Eq. (14), as follows:

冘 冘 冘冘 冘 N

冘冘 N

⫹

B1i⌬xi(t)

B2ik⌬xi(t)⌬xk(t) ⫹ % ⫹ ek,

B1i[

i⫽1

N

⫹

N

aijsj(t)⫺xi(0)]

j⫽1

M

B2ik[

i ⫽ 1k ⫽ 1

i ⫽ 1k ⫽ 1

∀k苸{x,y,z}, where N is the number of temperature sensors and

akjsj(t)⫺xk(0)]

j⫽1

⫹ % ⫹ ek. (15) Since, si is assumed to be independent,

冘冘 N

N

sisj⬇0.

(16)

i ⫽ 1j ⫽ 1

Therefore Eq. (15) becomes

冘 M

ek(t) ⫽ C0 ⫹

冘 M

C1jsj(t) ⫹

j⫽1

C2js2j (t) ⫹ ek.

(17)

j⫽1

Where

冘 冘冘 冘冘 N

C0 ⫽ B0⫺

i⫽1

N

冘冘 N

B1ixi(0) ⫹

N

B2ikxi(0)xk(0) ,

i ⫽ 1k ⫽ 1

N

C1j ⫽ ⫺

B2ik(aijxk(0) ⫹ akjxi(0)) ,

i ⫽ 1k ⫽ 1

N

C2j ⫽

N

B2ikaijakj .

i ⫽ 1k ⫽ 1

In this paper, a new thermal distortion model is proposed using the estimate, uj, of the unmixed significant thermal source signals, sj, that are extracted by the ICA method and significant temperature variables. In addition, the estimated thermal distortion eˆ k(t) is modeled as follows,

冘 m0

(13)

冘 M

aijsj(t)⫺xi(0)][

j⫽1

eˆ k(t) ⫽ Cˆ 0 ⫹

i⫽1

N

M

ek(t) ⫽ B0 ⫹

N

ek(t) ⫽ B0 ⫹

(14)

j⫽1

(12)

where Lq is referred to as the saliency of the weight q, which is the increase in error that results when the weight is eliminated. The Optimal Brain Surgeon Procedure used in this study is briefly explained as follows:

冘

⌬xi(t) denotes a temperature variation. ex(t) denotes the thermal distortion in the k-axis. The teacher values of ek(t) are obtained by a gab sensor that is off-line. The temperature variables can be described by the thermal sources.

冘 m0

Cˆ 1juj(t) ⫹

j⫽1

Cˆ 2ju2j (t) ⫹ ek,

(18)

j⫽1

∀k苸{x,y,z}

冘 n0

uj(t) ⫽

i⫽1

wjixi(t),

(19)

D.S. Lee et al. / International Journal of Machine Tools & Manufacture 43 (2003) 589–597

Fig. 5.

Experimental settings.

where m 0 is the number of selected significant thermal sources, uj(t) denotes the unmixed independent thermal source signal and n 0 is the number of selected significant temperature variables. The coefficient in Eq. (18) can be fitted by the Recursive Least-Squares (RLS) algorithm [10]. Consider the regressor, f(t) = [1 u1(t) … um0(t) u21(t) … 2 um0(t)]T, and the distortion observations, y(t). The recursive least-squares algorithm is as follows: q(t) ⫽ q(t⫺1) ⫹ P(t⫺1)f(t)j(t), j(t) ⫽ y(t)⫺f(t)Tq(t⫺1), P(t) ⫽ P(t⫺1)⫺

P(t⫺1)f(t)f(t)TP(t⫺1) , 1 ⫹ f(t)TP(t⫺1)f(t)

593

(20)

P0 given, where q(t) = [Cˆ 0 Cˆ 11 … Cˆ 1m0 Cˆ 21 … Cˆ 2m0] is the parameter estimate, j(t) is predicted error. Except for a factor s 2, the matrix, P(t), constitutes an estimate of the parameter covariance at the recursion number, t.

4. Experiments 4.1. Experiment settings Using the proposed method, a compensation system was implemented with real-time hardware. Fig. 5 shows the architecture of the overall compensation system. The compensation system consists of three modules: A temperature measurement (TM), thermal distortion modeling (TDM), and thermal distortion compensation (TDC). The dashed lines in the Fig. 5 denote the signal flow for modeling in the off-line condition and the solid lines denote the signal flow for compensation in the online condition. After the modeling is complete, the signal flow for the dashed line is deactivated and the compensation module sends the compensation data to the PMC (Programmable Machine Controller). 4.1.1. Temperature measurement (TM) module Fig. 6 shows the block diagram of the temperature measurement module. Thermocouples were used to measure the temperatures at their installed position inside and around the machine tool. Standard T-type (copper-

constantan) or K-type (chromel-alumel) thermocouples were connected to a thermal isolation block. The shielded thermocouples were used to remove noise. A lowpass filter was designed to reduce the high frequency noise [11]. Using the analog multiplexer, the input channels can be extended to 32 channels. Two amplifiers were used to suppress the noise and enhance the sensitivity. The first Amplifier was a monolithic thermocouple amplifier (AD595) with cold junction compensation. The common-mode noise voltage is rejected by the differential inputs of the circuit. The second amplifier (AD627) improves the sensitivity of the temperature data to 100 mV/°C. The analog signals were converted to 16 bit digital data by an A/D converter (AD1674). The operating temperature range of the constructed TM module was ⫺55–125 °C and the resolution was 0.05 °C. The converted digital data was sent to thermal distortion modeling (TDM) module. This module has a calibration circuit to improve the accuracy of the temperature measurement. The thermal compensation system may generate wrong compensation data when there is a failure in the temperature measurement. The TM module detects sensor failure such as a short circuit and an open circuit in the hardware. When failure occurs, the TM module attempts to recover the failed sensor data using the other correlated sensor data and generates an alarm signal.

Fig. 6.

Temperature measurement module.

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Fig. 7. (a) Three Non-contact capacitance probes are mounted on a machining table, (b) A measurement system of the thermal deformation of a machine tool.

4.1.2. Thermal distortion modeling (TDM) module This module produced a thermal distortion model from the observed temperature and distortion data. Data on the temperature and thermal distortions were collected in order to model the thermal distortions of a machine tool itself. A precision capacitance gage (MS3 three probe system) made by Lion Precision was used to measure the thermal distortions [12]. Three Non-contact capacitance probes were mounted on a machining table

Fig. 8.

at 90° to each other. Typically, the three probes were aligned with the x, y, and z axes of the machine tool. Although the maximum resolution of the gauge is 2.5 nm, 0.01 µm resolutions were used in this paper due to the noise level. Fig. 7 shows a photograph of the machine equipped with a deformation measuring system. In the proposed compensation system, the thermal sources were extracted from the temperature variables using the ICA approach described in section 2. The

(a) The flowchart of Thermal Distortion Compensation (TDC) Module, (b) The compensation data format.

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axis of a machine tool is sent to the CNC controller. The calculated value calculated in the TDM module is in a 12-bit data format. The most significant bit is the compensation enable/disable signal. The following 3-bits is the axis selection bit. Only one bit is enabled at one time. The following 8-bits is the compensation value, ranging from ⫺128–127 µm with a 1 µm resolution. These bits can be extended to 20-bit data because the compensation hardware is designed to control a 24-bit digital output. In this case, if one uses all 24 bits, the compensation range is ⫺524288–524287 µm in a machine with a least setting unit of 1 µm. The compensation data is sent to the CNC controller to correct the thermal distortions. Fig. 9. The ICA component error due to the number of temperature sensors.

extracted sources were used to model the thermal distortion of the machine tools, as mentioned in section 3. The temperature variables were reduced using the pruning algorithm described in section 3. The final thermal compensation model used a minimal number of temperature sensors. 4.1.3. Thermal distortion compensation (TDC) module Several methods of feeding the compensation data into the CNC controller have been reported [6]. In the proposed compensation system, the real-time error compensation data was sent to the PMC or PLC using digital I/O communication. The advantage of digital communication is that it can be applied to most modern CNC controllers easily without the need for hardware modification. The ladder program in the proposed compensation system supports the FANUC series, and the KSNC (Korean Standard Numerical Controller) with little modification. The temperature measurements, compensation value calculation, and compensation data feeding were carried out at each compensation period. Fig. 8 shows the flowchart of the thermal distortion compensation module. The period of compensation should be greater than the control period of the CNC controller. The period can be chosen according to the magnitude of the thermal distortions. 10 s to 60 s was suitable in our experiments. In the TDC Module, a compensation value for each

4.2. Performance evaluation 4.2.1. Thermal distortion modeling performance Two operation scenarios were carried out to collect the data sets. In the first scenario, the spindle ran on its maximum speed of 4000 rpm for 10 h and stopped for 14 h. The 14 h stoppage guaranteed that the temperatures and thermal distortions approached the nominal state of thermal equilibrium. From this run, the largest amount of thermal distortion and the maximum temperatures can be measured. The same operating scenario was applied at a speed of 1000, 2000, and 3000 rpm. In the second scenario, the spindle speed was changed every 30 min. These scenarios provide diverse data for the various running conditions. The data on the temperatures and distortions were gathered during the run. These were used to extract the thermal sources, select the optimal temperature variables, model the thermal distortions, and verify the model. In the case of the particular test machine (MCH-10, horizontal type machining center) and the CNC (FANUC 15 M) under consideration, only one independent thermal source, that is m = 1, was sufficient to obtain a thermal distortion model. The reason can be explained from Fig. 3. The thermal sources can be contained in an envelope (thin solid line). The envelope itself can be considered as a single thermal source. Fig. 9 shows the ICA component error due to the number of temperature sensor variables after the pruning weights. The figure shows that the reasonable number of temperature sensor variables is three or four. Table 1 denotes all the fitted parameters in the thermal

Table 1 Parameters of the proposed model based on a virtual thermal source Axis

x

y

z

Parameters Startup from relaxed state Nominal operation Temporary stoppage

C0 0.138

C11 3.059

C21 0

C0 –3.0

C11 7.431

C21 0

C0 3.436

C11 –5.311

C21 0

22.622 4.205

0.036 –0.076

–0.794 1.770

26.015 –23.992

0.259 0.125

⫺0.709 –8.281

171.459 19.837

–0.188 –8.387

⫺9.803 1.770

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distortion model. The thermal distortion model for each axis consisted of three sub-models according to the operating conditions. In addition, the time interval was divided into three according to the state of the operating conditions (start up, running, or temporary stoppage) of a particular machine tool. Fig. 10 shows the modeling

error of each axis. The average error for each axis was ⬍ 5 µm. Table 2 shows the computational time of the temperature sensor variables selection as a comparison of the proposed ICA with the pruning algorithm with the conventional methods [4]. The proposed method significantly reduces the computational burden compared to the conventional ones. 4.2.2. Thermal distortion compensation performance A horizontal-type machining center (MCH-10, Tongil Heavy Industry, Korea) and FANUC-15M CNC controller were used as a test system to evaluate the developed thermal error model. Fig. 11 shows the job used to demonstrate the quality of the proposed thermal error modeling and the real-time compensation. It consisted of 16 equally spaced holes drilled into a steel block. The material used for the work-piece was SS41C (carbon steel). It was manufactured at a feedrate of 20 mm/min, and a spindle speed of 4000 rpm. The tool diameter was 10 mm. The depth of the holes was 10 mm and the interval between holes was 20 mm. The directions of tool movement are also shown in Fig. 11. Two products were manufactured. One was manufactured without thermal compensation and the other was manufactured using the proposed thermal distortion compensation model. The period of compensation was set to 60 s. Each hole was milled at an interval of 30 min to examine the thermal behavior. Therefore, the total operating time was 480 min. The position and depth of each hole in the two products were measured using a CMM (Coordinate Measuring Machine) with 0.1 µm precision. Table 3 compares the average distortions during the test operations. The average error for the z axis without compensation was approximately 155.5 µm. However, the error was reduced to 3.5 µm using the suggested compensation model. For the other axes, the performance improvement was obvious. From the result of the test operations, the proposed thermal source extraction and modeling method is useful for compensating for real-time thermal distortion. Using the proposed method, a machining center with a hundred-micron precision can be converted into a center with a ten-micron precision.

5. Conclusion

Fig. 10. Thermal error for each axis. (Legend: Solid: thermal error, Dotted: estimated thermal error).

This paper presented a new thermal distortion modeling method based on Independent Component Analysis (ICA). The proposed algorithm extracted the independent thermal sources from the measured temperature variables and eliminated the insignificant thermal sources. In addition, the temperature variables with insignificant information were reduced using the Optimal Brain Surgeon (OBS) algorithm. The thermal distortion model was constructed using significant thermal sources

D.S. Lee et al. / International Journal of Machine Tools & Manufacture 43 (2003) 589–597

Table 2 Computation Time (P4-1.7Ghz)

597

Acknowledgements

Selection method

Computation time (s)

Full search Grouping + Search [4] ICA + Sensor pruning (proposed)

98959 2931 195

This Project was supported by the Brain Korea 21 Program of the Korea Ministry of Education.

References

Fig. 11.

A test product and its machining procedure.

Table 3 Average distortion [µm] for each axis Axis

x

y

z

Without compensation With compensation

19.4 3.3

148.3 3.7

155.5 3.5

and temperature variables and was used as a part of the compensation system. In the experiments, the proposed method was capable of compensating for thermal distortions within a high precision range. The proposed compensation system provides many desirable properties; robustness to some kinds of sensor failure, a reduced computational burden, prevention of parameter overestimation, and cost reduction.

[1] J. Ni, CNC machine accuracy enhancement through real-time error compensation, J Manufact Sci Eng 119 (11) (1997) 717– 727. [2] M. Weck, P. Mckeown, R. Bonse, U. Herbst, Reduction and compensation of thermal errors in machine tools, Annals of the CIRP 44 (2) (1995) 589–598. [3] R. Ramesh, M.A. Mannan, A.N. Poo, Error compensation in machine tools—a review Part II: thermal errors’, International Journal of Machinery Tools Manufacturing 40 (2000) 1257– 1284. [4] C.-H. Lo, J. Yuan, J. Ni, Optimal temperature variable selection by grouping approach for thermal error modeling and compensation, International Journal of Machinery Tools Manufacturing 39 (1999) 1383–1396. [5] J. Yang, J. Yuan, J. Ni, Thermal error mode analysis and robust modeling for error compensation on a CNC turning center, International Journal of Machinery Tools Manufacturing 39 (1999) 1367–1381. [6] J.S. Chen, G. Chiou, Quick testing and modeling of thermally induced errors of CNC machine tools, International Journal of Machinery Tools Manufacturing 35 (7) (1995) 1063–1074. [7] M. Weck, U. Herbst. Compensation for the thermal errors in machine tools with a minimum number of temperature probes based on neural networks, ASME Annual Meeting (1998) 45–53. [8] T.W. Lee, M. Girolami, T.J. Sejnowski, Independent component analysis using extended infomax algorithm for mixed subgaussian and supergaussian sources, Neural Computation 11 (1999) 417–441. [9] B. Hassibi, D.G. Stock, G.J. Wolff, Optimal brain surgeon and general network pruning, IEEE International Conference on Neural Networks, Vol. 1, San Francisco, CA 1993, pp. 293-299. [10] R. Johansson, System modeling and Identification, PrenticeHall, 1993. [11] C. Johnson, Process Control Instrumentation Technology, fourth ed., Prentice-Hall, 1993. [12] D.L. Martin, A.N. Tabenkin, F.G. Parsons, Precision spindle and bearing error analysis, International Journal of Machinery Tools Manufacturing 35 (2) (1995) 187–193.