Neuro-fuzzy Drilling Modeling

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with a PC-based controller is used. The sampling frequency is

200 Hz. The machining center is a three-axis system, and the po-

sitioning resolution is 1 m. The voltage command sent to theservo pack is equivalent to an analog velocity reference for thevelocity loop. The allowable range for the velocity reference is

from 10 to +10 V, with +10 V corresponding to a command

feed rate of 400 mm/ s. The composite laminate specimens areheld in a rigid fixture attached to a force-torque Kistler 9271Adynamometer during drilling. The experimental setup is shown inFig. 1.

3 Neural Model with Fuzzy SwitchesA Global Mod-eling Strategy for Drilling Process

Neural networks have the ability to learn static/dynamic andlinear/non-linear characteristics of the controlled plant. In this sec-tion, neural networks will be utilized to model the thrust force.

3.1 Structure of Neural Network Model. The most commonlinear model structures are autoregressive with exogenous input

ARX , autoregressive moving average with exogenous input ARMAX and output error OE models 14 . Their non-linearcounterparts are non-linear ARX NARX , Non-linear ARMAX

Fig. 1 Experimental setup

Fig. 2 Architecture of NN model with one hidden layer: a NNARX model; b NNOEmodel note that the bias nodes are not shown

Journal of Dynamic Systems, Measurement, and Control DECEMBER 2006, Vol. 128 / 847

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NARMAX and non-linear OE NOE . When NNs are applied torepresent the non-linear process, we can get the NN counterpartsof ARX, ARMAX and OE as NNARX, NNARMAX and NNOE,respectively. Since for non-linear problems the complexity in-creases strongly with the input space dimensionality, the applica-tion of lower-dimensional NNARX or NNOE models is morewidespread. The architecture of NNARX and NNOE models withone hidden layer is shown in Figs. 2 a and 2 b , respectively.NNARX model can be represented as

F k = f F k 1 , . . . , F k na ,u k nk , . . . , u k nk nb + 1

1NNOE model can be represented as

F k = f F k 1 , . . . , F k na ,u k nk , . . . , u k nk nb + 1

2

where F is the measured force, F is the estimated force, u is the

command feed rate, nk is the time delay and na and nb definesystem dynamic orders.

In 9 RNNs were selected as the NN model and NN controller.RNNs are neural networks that use outputs of network units at

time instant k as the input to other units at time k+1. In this way,they support a form of directed cycles in the network and thushave the potential for better approximation ability due to theirdynamic nature

15,16

. In the NNOE model discussed above,

predictions at time instant k, k1,..., can be used as inputs to the

network to predict the output at instant k+1. In this way, there isa form of indirect cycles in the NNOE networks which providesNNOE an dynamic nature. Due to the popularity of feedforwardneural networks and the dynamic nature similar to RNN inNNOE, NNOE model is selected in this paper. Compared toNNARX model, the NNOE model is difficult to determine butonce it is obtained, it can provide a true representation of the plantbehavior. For each neuron, the activation function can be eitherlinear or hyperbolic tangent tanh or logistic sigmoid function.Empirically, it is often found that tanh activation functions giverise to faster convergence of training algorithm than logistic sig-moid functions. Thus, in this paper, the hidden neurons have atanh activation function, i.e.

tanh x ex

ex

ex+ ex 3

We use a linear neuron for the output layer so that it has anunlimited range. The NNOE model in Eq. 2 can be further rep-resented as

F k =j=1

M2

Wjtanhi=1

M1

wjii k + wj0 + W0 f k ,

k = F k 1 , . . . , F k na , u k nk , . . . , u k nk nb + 1T

4

where wj0 and W0 are weights related to biases at input and hidden

layers, respectively. = WjwjiT, j = 0 , 1 , . . . ,M2, i = 0 , 1 , . . . ,M1,

is a vector with length of p = M1 + 2 M2 +1, which combines the

hidden-to-output layer and input-to-hidden layer weights of the

neural network. M1 and M2 are the number of inputs to the net-work and the number of hidden neurons of the network, respec-tively. Initial weights were selected randomly and they were uni-

formly distributed between 0.5 and +0.5. The number of neuronsof hidden layer is six for the NN model unless other remarks areprovided.

In order to be able to achieve a good mapping f , the dynamic

order na and nb and the time delay nk of the system must beknown a priori or be estimated from experimental data. The moreaccurate information of these parameters, the better performance

of the model. In this paper, they are determined through trial and

error. Different parameters of na, nb, and nk are used for the NNmodels. Predicted force responses from NN models are compared

to measured forces. It is found that nk=1 can provide the best

prediction accuracy and will be used in this paper. Large na and nbwill cause long NN calculation time. It is found that no obviousimprovement in prediction accuracy can be obtained when higher

orders of na and nb than na = 2, nb =2 are used. Thereby, na = 2,

nb =2, and nk=1 are selected in this paper.

3.2 Training Signal. After selecting the model a proper train-

ing set u k , F k , k= 1 , . . . ,N has to be selected. To obtain goodidentification results the input excitation signal command fee-drate u k must be chosen properly. In 9 , excitation signal in

triangle profile was used by Stone and Krishnamurthy to traintheir NN models. Pseudo random binary signal PRBS can pro-vide more excitations than triangle profile. For a non-linear sys-tem, amplitude of PRBS can be modulated to different levels inorder to excite non-linear system behaviors in different operationregions. Thereby, amplitude modulated pseudo random binary sig-nal APRBS is selected in this paper. One example of APRBS isshown in Fig. 3.

The minimum hold time MHT of APRBS plays an importantrole for the successful training of the neural model. It should besufficiently long so that the system output has time to approach tothe new set point. For a non-linear system, system behaviors, suchas process gain, time constant, et al., vary at different operation

Fig. 3 APRBS signal; AL specimen, CT drill, open loop: aforce response; b command feed rate

848 / Vol. 128, DECEMBER 2006 Transactions of the ASME



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regions. It takes different time for the system output to reach acertain level if the process gains and/or time constant are different.In addition, the system output will reach different levels when theinput signal is kept at one level for different durations. It is noteasy to distinguish the system at which the operation point if theMHT is too short, e.g., shorter than its maximum time constant forthe whole operation range. But the MHT cannot be too long be-cause it will lead to quasi-static excitation 17 . In this paper, theMHT is chosen equal to the process time constant. Since the sys-tem is non-linear, this value varies for different operational con-

ditions. For example, it is about 90 ms for the drilling of AL

around 200 N and about 40 ms for the drilling of CM around

110 N. While CM is more difficult to drill properly, the dynamicsfrom command feedrate to thrust force are the same between drill-

ing of CM and AL specimens. In this paper, AL specimens will beused to investigate the modeling of thrust force in drilling process

together with CM specimens. MHT=100 ms is chosen for drillingexperiments of AL specimens. Furthermore, it is important tomake the excitation signal cover the operating range. The neuralmodel will be more accurate in the range of the training data.Outside the training range the neural model will be less accurate.So it is of great importance to know the working range of the realsystem.

Six sets of APRBS signals labled as A, B, C, D, E, F aregenerated and used for command feed rates. Every set of APRBSsignal is used to drill six holes in AL specimen in a row using anew CT drill. The sequence number SN of each hole is recorded.Different experimental data sets are used in training and validationof the NN model. For example, for the six experimental data sets

with SN=5, five of them are used for training and the remainingone is used for validation and testing.

3.3 Data Scaling. In order to avoid saturation of activationfunctions in NN, input and output data are scaled as

u k scaled =u k

UMAX, F k scaled =

F k

FMAX 5

where UMAX=6.0 V and FMAX=1800 N.

Table 1 Comparison of max and for middle drilling stagebetween Test I using data from entire drilling process andTest II using data from middle drilling stage only

Test max N N

I 55.2 18.4II 30.1 11.3

Fig. 4 Validation results of NN models trained with APRBS signal; AL specimen, CT drill, open loop; solid lines representthe measured forces and dashed lines represent the simulated forces in a and c: a force responses of the entire drilling

process; b command feed rate of the entire drilling process; c force responses of the middle stage of the drillingprocess; d command feed rate of the middle stage of the drilling process




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3.4 Training Algorithm of Neural Model. Normally, the cal-

culation of weights is done by minimizing the error function

E =1

2k=1

N

F k F k 2 =1

2k=1

N

F k f k , 2

=1

2k=1

N

k2 =

1

22 6

where k= F k F k and is a vector with elements k. Usingthe first-order Taylor expansion of f

k

,

around the initial ,

we obtain

f k ,+ = f k , + J1 kT+ O f k ,

+ J1 kT, for small 7

where J1 k =f k ,

1

f k ,

2

f k ,

p T, is a vector with size

of p. Thus E + can be represented as

E + E + gT+1

2TJTJ 8

where J= J1 1 J1 2 . . .J1 NT is the Jacobian matrix with size

of Np. And g = E =JT. LettingE +

= 0, becomes

= JTJ 1JT 9

In principle, the update formula in Eq. 9 could be applied itera-tively to minimize the error function. The problem with suchmethod is that the step size given by Eq. 9 could be relativelylarge, in which case the linear approximation in Eq. 7 would nolonger be valid. In the Levenberg-Marquardt algorithm 1820 , amodified error function is considered to solve this problem whichis of the form

E + E + gT+1

2TJTJ+

1

22 10

where the parameter governs the step size. Through this strat-egy, we can minimize the error function while at the same timekeeping the step size small so as to ensure that the linear approxi-

mation remains valid 20 . LettingE +

=0, we obtain as

= JTJ+ I 1JT 11

This algorithm is implemented in MATLAB Toolbox 21 and isused to train the neural models in this paper.

3.5 Validation of Neural Model and Discussion. Maximum

error max and mean square error defined as below are used toevaluate the model performance in this paper.

max = maxk=1 N

F k F k 12

= k=1N F k F k 2N 1

13

where N is the number of samples. Recall that sequence numbers SN of holes from the first hole drilled with the same drill arerecorded. Two types of data were tried for training and validation:one using the data from the entire drilling process Test I and theother using the data from the middle stage of the drilling process

Test II . Validation results of neural models trained and tested bydata sets with SN=5 are shown in Fig. 4. The comparisons ofmaxand between the two tests are listed in Table 1.

Comparing Figs. 4 a and 4 c and considering the results listedin Table 1, the middle stage modeling accuracy is better if we trainthe NN model using only the middle part of corresponding train-ing data. The reason is that the process gain is different for en-trance, middle, and exit drilling stages. The effective part of the

drill flute which really takes part in drilling varies in the three

stages as shown in Fig. 5, where H is the thickness of the speci-

men and y is the distance from the drill tip to the upper surface ofthe specimen. In order to capture such changes, the information onthe drill head position needs to be included in the NNOE model at

the entrance and exit stages of the drilling process. The term y isa good choice to represent such information for the entrance stage.The difficulty is to determine the position of the upper surface ofthe workpiece. In this paper, drilling is not considered to takeplace unless the measured thrust force exceeds a preset threshold.Furthermore, the drill tip position at this critical instant is re-corded and considered as the reference position of the upper sur-

face of the workpiece. It is then straightforward to obtain y bycomparing current drill position and the reference position of theupper surface. The extended NNOE model for the entrance stagecan be represented as

F k = f F k 1 , .. . ,F k na , u k nk1 , . . . ,

Fig. 5 Different drill-specimen relationships, the shadowedarea of the drill head is the part which really takes part in drill-ing: a entrance stage; b middle stage; c exit stage




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u k nk1 nb1 + 1 ,y k nk2 , .. . ,y k nk2 nb2 + 1 14

where, as stated before, y k is the drill tip position relative to theupper surface of the specimen. The number of hidden neurons is

12 and the parameter values of na , nb1 , nb2 , nk1 , nk2 are 2 , 2 , 2 , 1 , 1 . The validation results of neural models trained andtested by the entrance stage data with SN=5 are shown in Fig. 6.

The comparisons of max and between the two tests without yand with y information are listed in Table 2.

Comparing Figs. 6 a and 6 c and considering the results listedin Table 2, the modeling accuracy is improved when the informa-tion on the drill head position is introduced to the network. In

order to avoid the saturation problem of the activation function inneural networks, y k is scaled as

y k scaled =y k

YMAX 15

where YMAX=2.2 mm, the average conical length of the used

drills. Similar strategy can be applied to the exit stage. YMAX

y k H is used instead of using y k directly in order to accu-rately represent the effect of the variation of the cutting part of thedrill during exiting on the thrust force.

Another observation shows that the accuracy of the NN modelfor the entrance stage becomes worse when the drill approachesthe transition region from entrance stage to middle stage, asshown in Fig. 6

c

. One reason for this is the different wearamount of each drill due to all kinds of uncertainties and/or dis-turbances in the drilling process. As stated before, drilling is notconsidered to take place unless the measured thrust force exceeds

a preset threshold which is 10 N in this paper. Thus the distancedrilled before the thrust force reaches the threshold will be differ-ent. This will cause the selected upper surface position to be dif-ferent each time and make the drill tip position different since it ismeasured based on the reference position of the upper surface ofthe specimen. An extra sensor such as a linear variable displace-ment transducer mounted to measure the actual position of drillhead is only a partial solution to this problem. One importantfactor in this problem is the variation of the conical length of

Table 2 Comparison of max and for entrance drilling stagebetween Test I without y information and Test II with yinformation

Test max N N

I 67.5 22.3II 37.0 7.8

Fig. 6 Validation results of NN models; entrance stage of the drilling process; AL specimen, CT drill, open loop, solidlines represent the measured forces and dashed lines represent the simulated forces in a and c: a force responses

without the information of the drill head position; b command feed rate; c force responses with the information of thedrill head position; d command feed rates




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different drills. Therefore, it is hard to determine when the en-trance stage finishes. Similar problems exist for the transitionfrom the middle stage to the exit stage.

3.6 Fuzzy Switching Strategy. In this section, a fuzzyswitching strategy is proposed to solve the modeling problem dueto the transition between different drilling stages. Fuzzy modelscan be viewed as a class of local modeling approaches, whichattempt to solve a complex modeling problem by decomposing itinto a number of simpler subproblems. The theory of fuzzy setsoffers an excellent tool for representing the uncertainty associatedwith the decomposition task, for providing smooth transitions be-tween the individual local submodels, and for integrating varioustypes of knowledge within one common framework. The deliber-ate overlap of the membership functions allows to represent situ-ations not completely captured by one set of rules. In mathemati-cal terms, the inference process in fuzzy models can be regardedas an interpolation between the outcomes of the individual rules.

3.6.1 Basic Idea. The entire drilling range is divided into threesubranges, i.e., entrance stage ENTR , middle stage MID , andexit stage EXIT . Three non-linear neural models are developedfor each subrange, respectively. The rules considered in this paperare

R1: IF y k is ENTR THEN F1 k+ 1 =NN1 R2: IF y k is MID THEN F2 k+ 1 =NN2 R3: IF y k is EXIT THEN F3 k+ 1 =NN3

where NN1, NN2, and NN3 stand for the NNOE models for ENTR,

MID , and EXIT stages, respectively. The membership function is

shown in Fig. 7, where ENTR is defined from US Upper Surfaceof the specimen, the position where the drill touches the speci-

men

to EMR , MID is defined from EML to MXR and EXIT is

defined from MXL to BS

. BS

is BS Bottom Surface of the speci-

men plus the drill conical length, i.e., BS+2.2 MM. EMR , EML ,

MXR , and MXL are as defined in Fig. 7. Then the weighted aver-age representing the combined consequents is

F k+ 1 =i=1

Mi F

i k+ 1

i=1M

i

16

where i is the membership degree of the fuzzy model, as shownin Fig. 7, M=3 is the number of fuzzy rules. In this way, thecomposition of all consequents is a crisp number.

Validation and Discussion. Validation results of NN modelswith the proposed fuzzy switching strategy for the entrance andentire drilling process are shown in Fig. 8. The comparisons of

max and between the two tests without/with fuzzy switchingfor entrance drilling stage are listed in Table 3.

From the force responses in Fig. 8 a and considering the re-sults listed in Table 3, the model accuracy is kept for the wholeactual entrance stage when the proposed fuzzy switching strategyis used. For comparison, from the force responses in Fig. 6 c ,there is a non-trivial accuracy loss at the end of entrance stage

from 15.62 to 15.66 s without the fuzzy switching strategy time15.66 s corresponds to 2.2 mm or EM

. It can be concluded that

the proposed fuzzy switching technique makes the transition be-tween different drilling stages smooth.

Table 3 Comparison of max and for entrance drilling stagebetween Test I without fuzzy switching and Test II with fuzzyswitching

Test max N N

I 37.0 7.8II 19.9 5.9

Fig. 7 Fuzzy membership function for drill depth, AL speci-

men, H=9.3 mm, EM-US=BSMX=2.2 mm, EM-EML=EMR-EM=MX-MXL=MXR-MX=0.8 mm

Fig. 8 Validation results of NN models; entrance stage of thedrilling process; AL specimen, CT drill, open loop, solid linesrepresent the measured forces and dashed lines represent the

simulated forces in a: a force responses with the fuzzyswitching strategy; b command feed rate




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4 Neural Model With SN SwitchesA Solution for

Gain Variation by Drill Wear

Drill wear will also cause gain variation. In order to examinethis problem explicitly, a set of consecutive drilling experimentsfor CM specimens are conducted to get the open loop responses of

thrust force. In the experiments, a constant command feed rate u

=3.0 V is applied, a new HSS drill is repeatedly used, and thecorresponding SN is recorded. In drilling of the CM specimen,delamination occurs at any instant when the thrust force exceeds acritical value. Thereby, the maximum force is more important thanthe mean thrust force and needs more investigation. The evolution

of the maximum force is shown in Fig. 9 where SN varies from 1

to 10. The standard deviation STD is 40.3 N. From the results, itis found that the thrust force varies by almost 57% even under thesame command feed rate when the same drill is used repeatedly.One reason for this variation is the drill wear, which usually re-sults in the increase of thrust force. This is an important factor

which will affect the fixed gain controller such as PI controller forregulation of the thrust force. Drill wear will be slower if a CTdrill is used instead of a HSS drill. The evolution of the averagedmaximum thrust force three duplicated sets of drilling in drillingAL specimen with a CT drill is shown in Fig. 10 where SN varies

from 1 to 6. The standard deviation STD is 7.5 N. It is observedthat the maximum thrust force rises at a much lower rate.

SN can be used to switch between different NN models duringboth the training and application phases. The strategy is shown in

Fig. 11. During the training phase, only ith drilling data by differ-

ent drills is used to train the neural model with SN= i, and so on.Then the corresponding model will be used to predict the forceresponse at any drilling sequence during the application phase. Inthis way, we can partially solve the process gain variation problem

Fig. 9 Evolution of maximum force under constant commandfeed rate U=3.0 V, open loop, CM specimen, HSS drill

Fig. 10 Evolution of maximum force under constant command feed rate U=1.8 V, open loop, AL specimen, CT drill

Fig. 11 Neural models with switching strategy based on SN:switches S1-1 and S1-2 will switch to the same SN channel, N isthe maximum number of holes satisfying certain specificationsa drill can make




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caused by drill wear. We can also avoid real-time adaptation of theNN model weights which is not easy to analyze and implement.Validation results for NN models with fuzzy and SN switches are

shown in Fig. 12. The comparison ofmax and between the fourtests is listed in Table 4.

Recall that the maximum thrust force rises at a much lower rate

in drilling an Al specimen using a CT drill than in drilling an CMspecimen using a HSS drill. The validation of the proposed SNswitching method will be based on drilling of an AL specimenusing a HSS drill which will be better in showing the usefulness

Fig. 12 Validation results of NN models; entire drilling process; AL specimen, CT drill, open loop; solid lines repre-sent the measured forces and dashed lines represent the simulated forces in cf: a command feed rate withSN=1; b command feed rate with SN=6; c force responses, SN=1 for training and SN=1 for testing; d forceresponses, SN=6 for training and SN=6 for testing; e force responses, SN=1 for training and SN=6 for testing; fforce responses, SN=6 for training and SN=1 for testing




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and effectiveness of the method. Error of maximum force maxdefined as below is added to evaluate the model performance

since maximum force Fmax is directly related to the delaminationin drilling of CM specimen.

max = maxk=1. . .N

F k maxk=1 N

F k 17

where N is the number of samples. The results shown in Fig. 12and listed in Table 4 support that the proposed modeling strategy

may be effective. Specificaly, max dropped from 35 or 38 N to 6

or 10 N if the SN switching method is used. Note that all data setsused for testing are not included in those ones which are used fortraining. If the time-varying nature is too dependent on the drilland workpiece, real-time adaptation will have to be performed.

5 Conclusions

Intelligent modeling of thrust force in drilling process was in-vestigated in this paper. Main contributions and conclusions ofthis paper are: 1 A set of NN models and the correspondingfuzzy switching strategy were introduced to solve the gain varia-tion problem due to the drill transition between different drillingstages; 2 SN and the corresponding switching strategy were pro-posed to compensate for gain variations caused by drill wear; 3The neural models trained by APRBS signal with suitable MHTcan represent the drilling dynamics well and 4 The modelingstrategy presented in this paper is not limited to the drilling pro-cess. Similar strategy can be applied to complex non-linear plantswith time-varying parameters. In general, for a plant with non-linear dynamics which is dependent on the operation point, NNmodel/models may be chosen to capture the non-linear dynamics

around each operation point and the fuzzy switches may be usedto obtain smooth transition between them. In addition, if this planthas some time-varying characteristics due to tool wear, etc. , theSN method may be a possible solution for it.

Acknowledgment

The authors thank Dr. Oliver Nelles for constructive commentsand discussions. This research was funded in part by grant fromthe U.S. National Science Foundation DMI-9713751 . Composite

prepreg materials are furnished by Space Systems/Loral Corpora-tion, Palo Alto, CA.

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Table 4 Comparison of max and for entire drilling processamong four tests with different SN

Test Training SN Testing SN max N N max N

I 1 1 31.2 7.1 6II 6 6 29.8 6.8 10III 1 6 50.0 10.9 38IV 6 1 48.4 9.7 35


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Neuro-fuzzy Drilling Modeling