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Thermal error optimization modeling and real-time

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j o u r n a l h o m e p a g e :w w w.e l s e v i e r.c o m /l o c a t e /j m a t p r o t e

Thermal error optimization modeling and real-time compensation on a CNC turning center

Wu Hao ?,Zhang Hongtao,Guo Qianjian,Wang Xiushan,Yang Jianguo

School of Mechanical Engineering,Shanghai Jiaotong University,Shanghai 200240,PR China

a r t i c l e

i n f o Article history:

Received 22May 2007Received in revised form 24October 2007

Accepted 15December 2007

Keywords:Thermal error

Optimization modeling Genetic algorithm Arti?cial neural networks NC machine tool

a b s t r a c t

Thermal errors are the largest contributor to the dimensional errors of a workpiece in preci-sion machining.The error compensation technique is an effective way of reducing thermal errors.Accurate modeling of errors is a key part of error compensation.The thermal errors of a machine tool can be treated as the superposition of a series of thermal error modes.In this paper,?ve key temperature points of a turning center were obtained based on the ther-mal error mode analysis.A thermal error model based on the ?ve key temperature points was proposed by using genetic algorithm-based back propagation neural network (GA-BPN).

The GA-BPN method improves the accuracy and reduces computational cost for the predic-tion of thermal deformation in the turning center.A thermal error real-time compensation system was developed based on the proposed model.An experiment was carried out to ver-ify the performance of the compensation system.The experimental results show that the diameter error of the workpiece reduced from about 27–10?m after implementation of the compensation.

1.Introduction

The inherent inaccuracy of machine tools is a major con-tributor to workpiece errors.Among many various sources of machine tool errors,thermal errors are the largest contrib-utor,accounting for as much as 70%of workpiece errors in precision machining (Weck et al.,1995).Researchers have con-sidered many ways of reducing thermal errors,including the thermally symmetric design of a structure,separation of the heat sources from the main body of a machine tool,installa-tion of a cooling unit,and so on.However,the manufacturing costs associated with the above-mentioned approaches are usually very high.In addition,there are many physical lim-itations in implementing process,which cannot be overcome solely by design techniques.As a result,error compensation technique used to improve machine accuracy cost-effectively has received signi?cant attention in recent years (Yang et al.,1996a ).

Corresponding author .Tel.:+86138********.

E-mail addresses:wuhaoer@https://www.sodocs.net/doc/2712121773.html, (W .Hao),jgyang@https://www.sodocs.net/doc/2712121773.html, (Y .Jianguo).Accurate modeling of errors is a key part of error compen-sation.The thermal errors of a machine tool origin from the non-linear and time-varying thermal deformations caused by the non-uniform temperature variations in the machine structure.The temperature variations are related to the heat source location,heat source intensity,thermal resistance coef?cient and the machine system con?guration.Therefore,the thermal error model is usually achieved by non-linear empirical modeling approaches which correlate machine thermal errors to temperature measurements of a machine.In recent years,it has been shown that thermal error map of a machine tool can be successfully approximated by empirical modeling approaches such as multiple regression analysis techniques (Yang et al.,1996b,1999,2002;Lee and Yang,2002),types of arti?cial neural networks (Yang et al.,1996c;Yang and Lee,1998;Mize and Ziegert,2000;Lee et al.,2003;Yang and Ni,2005),grey system theory (Wang et al.,1998),genetic algorithm (Choi and Lee,2002),rigid body kinematics (Okafor

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Fig.1–T urning center and locations of sensors.

and Ertekin,2000;Delbressine et al.,2006),or a combination of several different modeling methods(Attia et al.,1999; Barakat et al.,2000;Kang et al.,2007).A genetic algorithm-based back propagation neural network(GA-BPN)technique is developed in this paper.GA-BPN is used to construct the model of thermal errors,which improves the accuracy of the thermal error model and reduces the computational cost for prediction of thermal deformation.Also,an accuracy and low-cost thermal error compensation system based on GA-BPN was developed to reduce the thermal drift errors on the machine tool effectively.

2.Experimental setup

This study was carried out on an INDEX-G200turning center(Fig.1(a))with high precision.The turning center has the high geometric accuracy and positioning accuracy. Therefore,the thermal errors are the important factors sig-ni?cantly affecting the accuracy of the machine tool.In the experiment,detecting of the temperature?eld of the machine and thermal deformation characteristics was accom-plished by the installation of a total of16thermistors on the turning center as shown in Fig.1.These thermistors are divided into six groups in accordance with their loca-tions:

(1)two thermistors(No.1and2)for measuring the tempera-

tures of the spindle fore-end(Fig.1(b));

(2)four thermistors(No.3–6)for measuring the temperatures

of the spindle rear-end(Fig.1(c));

(3)three thermistors(No.7–9)for measuring the tempera-

tures of the headstock(Fig.1(d));

(4)two thermistors(No.10and11)for measuring the temper-

atures of the X-axis lead screw(Fig.1(e));

(5)four thermistors(No.12–15)for measuring the tempera-

tures of the machine tool bed(they are not shown in the ?gure);

(6)one thermistor(No.16)for measuring the ambient tem-

perature(they are not shown in the?gure).

The machining accuracy is decided by the accuracy of the relative movement between the cutting tool and the part.As

174

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Fig.2–Temperature variations of the six groups of thermistors.

shown in Fig.1(f),a displacement sensor mounted on the cutter rest was used for measuring the thermal drifts of the spindle in the x direction,which is also the thermal errors of the spindle in the radial direction.The thermal drifts of the spindle in the z direction were neglected because the ther-mal deformations of the turning center in the z direction are in?nitesimal.

First,an experiment was carried out by simulating the machine working without implementing real cutting process,in which the machine spindle is rotating,the carriage is mov-ing and the coolant is ?owing.Initially,the machine kept running for 50min.Then,the machine paused for 10min.

After that,the machine ran for another 1h and stopped for 20min.In the simulation,the spindle speed was set at a high speed of 4500rpm.Fig.2(a)–(f)show the temperature variations of the six groups of thermistors over time series,respectively.Fig.3shows the thermal errors in the radial direction over time series.The conclusions are drawn from Figs.2and 3:

(1)The part radius decreases with the increase of machine

temperatures but having some time-lag.

(2)The thermal error range in the radial direction is about

27?m,which is larger than the expected value.

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Fig.3–Thermal error in the radial direction.

Modeling of the thermal errors

3.1.

Choosing of the temperature variables

The number of the temperature variables is a key factor to the accuracy of the thermal error model.The accuracy of the thermal error model could be decreased if the number is too small.At the same time,the calculating time and the rel-evant cost increase if the number is too big.Therefore,the temperature variables should be determined before the con-struction of the thermal error model.The thermal errors of the machine can be treated as the superposition of a series of thermal error modes with corresponding modes’shapes and time constants.In this study,the temperature variables are determined through the thermal error mode analysis of the machine (Yang et al.,1999).There are two basic thermal error modes,namely,thermal expansion and thermal bend-ing.Through investigation and analysis of the mechanical structure,working conditions,heat sources and thermal dis-tortion of the turning center,?ve key thermal error modes,which cause the thermal errors in the radial direction,were identi?ed.

(1)Expansion mode of the spindle column .Due to the rotation of

the spindle,the friction heat arises at the spindle front

bearing (i.e.,thrust roller bearing)and the spindle rear bearing (i.e.,angular contact ball bearing).On one hand,the friction heat causes the spindle column expansion in the vertical.On the other hand,the spindle inclines due to the temperature difference between the front bearing and the rear bearing.This thermal error mode can be effec-tively estimated by using two sensors:one is on the spindle fore-end (sensor 2)and the other on the spindle rear-end (sensor 4).

(2)

Expansion mode of the spindle .For the INDEX-G200,the front bearing of spindle cannot move along the axial direction.Therefore,the spindle fore-end will expand and produce machining errors in the radial direction when the friction heat arises.The thermal expansion of the spindle rear-end in the axial direction can be neglected since the rear bearing (i.e.,angular contact ball bearing)of the spindle is allowed to move a little along the axial direction.This thermal error can be effectively estimated by using the temperature of sensor 2.

(3)

Expansion mode of the X-axis screw .When the X -axis slide moves,the temperature of the X -axis ball screw rises.The expansion of the ball screw pushes the screw nut to the back side since the bearing at the front end is a thrust bearing,which results in increase of the part radius.Sim-ilarly,the temperature of the ball screw can be estimated using sensor 10.

(4)

Bending mode of the base .The tank inside the machine base is used as a reservoir for storing the liquid coolant,so the temperature variation of the coolant will directly cause a temperature change of the bottom plate of the base.The distance between the cutting tool and workpiece decreases when the temperature of the coolant increases.This thermal error mode can be effectively estimated by using the temperature difference between two sensors:one is on the upper plate of the base (sensor 12)and the other sinks in the coolant close to the bottom plate (sensor 15).

(5)

Expansion mode of the base .Due to the heat conduc-tion effect,the temperature of the upper plate of the base increases gradually along with the temperature of the lower plate.The thermal expansion mode of the base makes the part size larger when the base expands.This thermal error can be effectively esti-mated using the average temperature of sensor 12and

15.

Fig.4–Locations of the key temperature points on the turning center.

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Through thermal error mode analysis,it is clear that just ?ve thermistors are enough to estimate the thermal errors of the machine as shown in Fig.4.

3.2.Optimization modeling

Arti?cial neural network(ANN)composed of lots of neurons with non-linear mapping ability is a non-linear dynamic sys-tem.ANN has been applied successfully in many?elds and solved many complicated problems that could not be solved using traditional methods.Back propagation neural network (BPN)is used more widely than the other ANNs.BPN contains one or more hidden layers besides the input and output layers. The nodes in the same layer have no coupling and the transfer function of neuron of BPN usually is sigmoid function.The out-puts of BPN vary continuously but not linearly as the inputs change.Therefore,BPN is very appropriate for modeling the complicated and non-linear relationship between input space and output space.

The learning process of the BP algorithm is composed of two phases:the forward propagation phase and the back prop-agation phase.During the forward propagation phase,input signals are dealt with from the input layer across the hidden layer and transferred to the output layer.If the expected output cannot be obtained in the output layer,an error value is then propagated backwards through the network,and changes are made to the weights in each layer.The weights are repeat-edly modi?ed until the overall error value drops below a pre-determined threshold.BP algorithm is simple and easy to realize,but it has some shortcomings such as low convergence rate,bad stability and proneness to yield to local minimum.

Genetic algorithm(GA)was?rstly put forward by Holland (1975).GA is a stochastic optimization method based on the mechanics of natural evolution and natural genetics.The pop-ulation of GA with possible solutions is treated time after time by using genetic operations.New populations are created in accordance with the Darwinian evolution theory of the natural selection or the survival of the?ttest.The optimum individ-uals are searched by using global parallel mode to seek the optimum solution at the same time.Due to its strong abil-ity of global search,GA is appropriate for solving complicated problems such as control,function optimization and machine learning,etc.(Sahoo and Ray,2006;Hao et al.,2006).GA-BPN uses GA to train the BPN in this study.The convergence rate and the prediction precision of the BPN are improved.On this basis,a model of thermal errors with strong robustness is built.

GA has the following components:encoding mechanism, control parameters,?tness function and genetic operators. Fundamental to any genetic algorithm structure is the encod-ing mechanism for representing a solution of the problem to be solved.The encoding mechanism depends on the nature of the problem variables and maps each solution to a unique binary string.The?tness function evaluates each solution to decide whether it will contribute to the next generation of solutions.The?tness function is also the objective function in the optimization problem.Highly?t solutions are given more opportunities to reproduce,so that the offspring inherit good characteristics from their parents.The control parameters operation determines the population size,crossover probabil-ity and mutation probability,etc.

Selection,crossover and mutation are three main genetic operators.The individuals producing offspring are chosen in accordance with the?tness values of the individuals through the selection operator.Selection gives a direction to evolution and conserves successful states,but it reduces the diversity of the population at the same time.If only the selection operator exists,the GA will lose progress and the offspring population cannot excel the parent population.Hence,the other oper-ators are required.The most widely used operators include crossover operator and mutation operator.Crossover repre-sents mating between individuals.That is to say:?rstly,two individuals are chosen from the population using the selection operator and a crossover site along the bit strings is randomly chosen.Then,the values of the two strings are exchanged up to this https://www.sodocs.net/doc/2712121773.html,stly,the two new offspring created from mating are put into the next generation of the population and the bet-ter individuals are likely to be created.Mutation prevents the premature stopping of the algorithm in a local solution.The mutation operator is de?ned by a random bit value change in a chosen string with a low probability.Mutation adds a ran-dom search character to the GA,and it is necessary to avoid that,after some generations,all possible solutions were very similar ones.

In this study the connection weights and threshold values of the BPN are optimized by using the GA.The process includes the following steps:

(1)Encoding:encode the initial weights and threshold values

of the BPN into binary string and produce the initial pop-ulation.

(2)Fitness evaluation:the?tness value of each individual

among the present population is calculated in accor-dance with the?tness function.The?tness function is the inverse of the error function in this study.The smaller the error,the larger is the?tness value.

(3)Selection:abandon the low?tness individuals and select

the high?tness individuals.Hence the successive off-spring inherit good characteristics.

(4)Genetic operation:treat the present generation using the

crossover operator and mutation operator and then pro-duce new generation.

(5)Make the weights and threshold values ceaselessly evolve

until the search goal is achieved;otherwise go to step(2).

A BPN with three layers was adopted in this study.Accord-ing to the above-mentioned analysis,the input layer has?ve nodes:the temperature variations of the?ve key temperature points T2, T4, T10, T12and T15.And the output layer has one node:thermal error in the radial direction?T r.

The number of the hidden nodes is closely related to the convergence rate of the BPN and is a key factor to the suc-cess of the network.It is impossible that the learning process of the BPN is convergent if the number of the hidden nodes is too small.The BPN will overwork and weaken its anti-interference capability if the number of the hidden nodes is too large.So far,there are no perfect theories that can be used to direct us for determining the number of the hidden nodes. Therefore,the general procedure is by tentatively choosing the number of the hidden nodes in accordance with the actual condition,and then the number is gradually modi?ed to an

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Fig.5–Structural chart of the BPN.

optimal value.Considering the error compensation system in this study,the thermal errors are continuous functions of the temperature variations.Therefore,the number of the hid-den nodes is determined according to Kolmogorov theorem (Hecht-Nielson,1987).It is 2M +1=2×5+1=11nodes (where M is the number of the input layer nodes)and the structure of the BPN in this system is shown in Fig.5.

The training data set with 70examples were obtained from the temperature sensors and the displacement sensor to train the BPN.The selected parameters and their corre-sponding values of GA and the BPN are listed in Tables 1and 2,respectively.According to the above-mentioned research,the topology structure of the BPN in this study is 5-11-1.So,a total of 78connection weights and threshold values need to be opti-mized.The crossover and mutation operators are carried out in GA-BPN until the stopping condition is satis?ed.For the controlling parameters of the GA search,the population size

Table 1–GA parameters and their values GA parameters

Value

Population (P )

100Crossover probability (p c )0.05Mutation probability (p m )0.01Maximum iterations

1000

P was set to 100organisms.The strings used in the hybrid algorithm for this study were encoded as following.The ?rst 66bits represent the connection weights between the input layer and the hidden layer,as well as the hidden layer and the output layer.The following 12bits are the threshold values for error prediction.The effect of the following two main training parameters on the training error convergence was also inves-tigated.It includes the learning rate l r that controls the speed of the adaptation of the connection weights between the neu-rons,and the momentum rate m that takes into account the rate of the last change of the connection weights.The training process is supervised by the desired outputs of the network.All of the training sample values are scaled to ?t into the nor-malized range of (0,1).

Table

2–BPN parameters and their values BP networks parameters

Value

Networks topology structure 5-11-1

Initial weights

Random of (0,1)Momentum rate (m )0.9Learning rate (l r )

0.03Learning adjustment coef?cient (l a )

0.85

Fig.6–Flowchart of the GA-BPN model.

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Fig.7–Schematic diagram of the thermal error compensation system.

The ?tness value calculated by the ?tness function is a cri-terion for evaluating of each solution.The ?tness function in this paper is set as f i (x )=

where E is the sum squared error of the BPN:E = n

i =1(y i

??y i )2

,y i :the actual measured value of the output of the BPN,?y

i :the predicted value of the output of the BPN.

The genetic optimization does not stop until the sum squared error of the BPN is less than the value ε.Here ε=0.1,which is set in advance.

The ?owchart of the optimization is shown in Fig.6.

4.Real-time compensation of the thermal errors

4.1.

Implementation of the error compensation

The schematic diagram of the thermal error compensation system is shown in Fig.7.During machining process,the tem-perature variations of the ?ve key points and the thermal errors in the radial direction are measured with the tempera-ture sensors and the displacement sensor,respectively.Then the data are sent to the database through an A/D board to con-struct the model of the thermal errors by using the GA-BPN technique.Finally,the thermal errors model is stored in a DSP to implement the error real-time compensation.

When compensating,the model of thermal errors is used to calculate the values of error compensation in accordance with the temperature variations on the key points.Then the feedback of the compensation value is sent to the CNC of the machine tool.The real-time compensation is realized after the feedback is added to the control signal of the servo loop.

https://www.sodocs.net/doc/2712121773.html,pensation results

In order to evaluate the performance of the GA-BPN model,another experiment is carried out.The experimental setting is similar with the experimental setting described in Sec-tion 2,in which only the spindle the machine tool is rotating

with no real cutting process,the carriage is moving,and the coolant is ?owing.The ?ve temperature sensors (No.2,4,10,12and 14)and the displacement sensor are only used in this experiment.Initially,the machine tool kept running for 1h.Then the machine was cooled down for 1h to simu-late as nooning.After that,the machine continued to run for 40min and paused for 10min.At last,the machine ran for another 20min and then stopped.The spindle speed was set at 4500rpm.

A comparison between the measured error data and the predicted values gotten by using the error model is repre-sented in Fig.8.It can be observed that the model predicts the error very well,and the residual error range of the model is smaller than 8?m,depicting that an accurate thermal model has been obtained.

Based on the proposed model,a compensation system of the thermal errors was developed.In order to evaluate the per-formance of the compensation system,a cutting experiment was carried out.Forty shafts were machined on the INDEX-G200turning center after using the compensation system of thermal errors.The material of the workpieces is 40Cr steel.The machining was done with spindle speed at 1200rpm and the axial feed at 4

mm/r.

Fig.8–Predicted value from model and experimental data.

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Fig.9–Machining errors of the turning center with compensation and without compensation.

As shown in Fig.9,the size variations of the workpieces were reduced from27?m(before compensation)to10?m (after compensation),which demonstrates the compensation system is effective.Thus,the machining accuracy of the turn-ing center is improved signi?cantly.

5.Conclusion

In this paper,a GA-BPN-based thermal error model was proposed for on-line prediction of the dynamic and highly non-linear thermal errors on an INDEX-G200turning center. The proposed model not only enhances the prediction accu-racy of the thermal errors but also reduces the computational cost of the BPN.A real-time compensation system of thermal errors based on the model has been developed to effectively compensate for the thermal drift errors.The experimental results demonstrated that the machining accuracy of the turn-ing center was improved signi?cantly after implementation of the compensation of the thermal errors.

Acknowledgement

The project is supported by the Foundation for the Author of National Excellent Doctoral Dissertation of PR China(Project No.200131).

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Thermal error optimization modeling and real-time

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