Applications of GMDH-type modeling in manufacturing

Applications of GMDH-type Modeling in Manu am'ing P.Y. Chao, Purdue University, West Lafayette, Indiana P.M. Ferreira, University of Illinois at Urbana-Champaign, Illinois C.R. Liu, Purdue University, West Lafayette, Indiana

Abstract

Introduction

This paper introduces the group method of data h a n d l i n g (GMDH) a l g o r i t h m w h i c h addresses the problem of modeling complex systems. It is a heuristic self-organizing modeling approach for modeling different processes/ phenomena on a metalworking shop floor. These models which can be used in monitoring/ prediction of the processes also present information in a concise and accurate form to the planning functions, such as process planning and scheduling. This helps in making these functions more dynamic and the decisions more realistic. Most shop floor processes are difficult to model because of the large number of influencing factors and the complex interactions between them. The GMDH is particularly well suited for the modeling and control of manufacturing processes and shop floor phenomena. In this paper the GMDH algorithm is described and its application in three problems, i.e., modeling of positioning error of a machine under varying thermal conditions, modeling of tool wear and tool wear rate, and modeling of cutting forces is presented. These initial applications of the method suggest that the GMDH algorithm has good potential for use in the abstraction and computer control of such processes.

The concept of flexibility in manufacturing systems has opened up the possibility of efficient automated manufacture of small batch workpieces. However, in order to achieve the full benefit of this flexibility, a very complex control problem has to be solved. For a manufacturing system to function flexibly, it must possess certain basic features. First, it must be able to sense changes and then it should be able to adapt to them. Second, it should be able to predict certain changes and be able to take the required steps either to accommodate or avert them. Finally, the two preceding requirements must be met using an automated approach. In the structural setup of an automated manufacturing facility, one finds information on the state of the system moving upwards from the shop floor to different levels of the planning elements of the system, while commands for change move downwards to the shop floor. Various structures for the control of such systems have been proposed. Of these, the five-level hierarchy proposed and implemented by the National Bureau of Standards ~(NBS) is probably the most realistic. This hierarchy consists of layers ranging from primitive servo commands to machine commands to higher task-level c o m m a n d specification to cell level c o m m a n d specification to, finally, the facility controller. A c o m m a n d from the higher level gets decomposed as it moves down the hierarchy, the effects of

Keywords." Modeling, Adaptive Control, Positioning Errors, Tool Life, Metal Cutting, FMS, CIM.

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response at each level is fed back upwards in the form of changes to the 'world model'. The response time of the layers in this hierarchy increases as one goes up the hierarchy. The machine c o m m a n d level might have a response time of a few microseconds, while that at the top might respond in a matter of days or weeks. In the information flow upwards, one observes the volume of information increasing tremendously. The sheer volume of the information needed to characterize the state of the shop floor and changes in it makes it impossible to carry all the information in its raw state. Processing this information at the higher levels would also pose a serious problem. Simpson, Hocken and Albus' identify this problem and suggest that the level of abstraction of the data increases as it is fed back to the upper layers of the hierarchy. In this paper we investigate the possibility of using the GMDH algorithm to perform this abstraction for data moving up the hierarchy of such a control system. More specifically, we address the problem of making floor level information available to process planning modules so that the plans developed in this planning function are more realistic. It also allows for the higher levels of planning becoming more adaptive to dynamic changes on the shop floor.

mentioned above, it must be provided in a concise, efficient, and easily retrievable form. Further, it should be easy to update the data so that it is relevant and realistic. The type of problems we address in this research have certain c o m m o n features: 1. The data is not readily observable 2. It depends on a large number of variables 3. There are no analytical models relating to observable variables 4. Data available in handbooks, while giving indication of the general behavior, does not generally apply to the machine shop floor. In such a situation, we propose the use of a self-organizing modeling approach to represent such data as functions of easily observable variables. The use of a self-organizing modeling approach affords the following: 1. The models are data-dependent and hence apply directly to control of the processes which generated them 2. The model development and periodic updating can be done automatically 3. The representation of data in the form of models thus obtained provides a concise representation for the data. In the light of these characteristics, we propose the use of the GMDH algorithm. This is a heuristic self-organizing model building procedure which develops polynomial models relating the dependent variable in terms of a set of given independent variables. It is capable of working with moderate amounts of data and in the presence of noise. Further, it can be completely automated. The next section discusses in detail the GMDH algorithm.

Feedback in a Metal Cutting Facility We identify three broad categories of information in a metal cutting facility which have to be fed back: the machine state, process information, and the tool state. While most of the information to characterize these states might be readily available through observations or from storage areas, some information, such as machine accuracies, expected tool life, and machining forces, is not readily available. The importance of such information at planning stages is quite obvious. The machining errors and tool wear information can be used to adjust the cutting parameters of the process plan. Thus, the precision of the machine/tool can be controlled. Using predicted tool life information, we can schedule tool changes which is another important activity in the shop floor. The modeling of cutting forces can be used to select between processes, machines and fixturing devices in the process plan. In order to use such data in the planning stages

GMDH Algorithm This section presents a heuristic self-organizing model building algorithm called the GMDH (group method of data handling) algorithm. It is used in system/plant identification and models the system/ plant as a high-order polynomial function of its inputs. It has the advantages of being able to deal with a large number of predictors (it sifts through them retaining only the significant ones) and still using high-order polynomials. These features make it highly amenable to the problem of modeling processes on a manufacturing shop floor.

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T h i s s e c t i o n will i n t r o d u c e t h e GMDH algorithm, describe it, and survey some of its applications. T h e G M D H A I g o r i t h m . The GMDH algorithm was proposed by a Soviet cybernetics expert, A.G. Ivakhnenko in 19683 Since then, it has been widely used by Soviet controls and systems engineers in plant identification. The algorithm has also found widespread applications in the modeling of ecological and economic systems. In the U.S. a similar idea called the perceptron was proposed and used by RosenblaW in 1958 to model the information storage and organization in the h u m a n brain. This concept did have engineering application and the Department of Defense and the Air Force sponsored further research into the application of such an approach in self-organizing flight controllers. The approach came to be known as the 'Adaptive Learning Network' approach. However, because of its heuristic nature, the approach soon lost the interest of academicians.

Yo = A + Bx, + Cx i + Dx, z + Exiz + Fx, x i

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This results in m ( m - l ) / 2 second generation variables for predicting Ye instead of the original m variables. F r o m these, the m, best polynomials (based on some criteria which are discussed later) are retained. These are used as independent variables for the next layer, to generate new polynomials (of degree 4 in terms of the original variables) which are used to predictyo. The process is continued until the regression equations begin to have poorer predictability power than those at the previous level. Figure 2 shows the scheme for the generation of higher-order variables and Figure 3 shows how the variables are propagated mathematically. Thus, an estimate o f y o is obtained as a quadratic of two variables which are themselves quadratics of some other variables ..... which are quadratics of the original variables. In other words, a very complicated polynomial of the form:

Overview o f the A lgorithm. The GMDH algorithm is implemented in a multilayered perceptrontype network structure (see Figure 1). At each layer, the elements of the network implement a nonlinear transformation of its inputs. This transformation generally takes the form of a second order polynomial with two variables, the coefficients of which are determined by regression. Suppose a relation was required between one output variable, say yo and m input variables x z,x z..... x , . For each pair of variables, x~, and x~, the regression equation with Yo can be obtained as

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Journal of ,~lanufacturing Systems Volume 7,' No. 3

m

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which is known as the Ivakhnenko polynomial. Thus, if the process was started with 10 variables and went through 10 layers, then a polynomial of degree 256 would be obtained. The following are the steps in which the algorithm is implemented. 1. Division o f Data: The data is divided into two sets, the training set and the checking set. 2. Construction o f N e w Variables: All the independent variables are taken two at a time and regression polynomials are constructed using only the checking data set. These new polynomials are evaluated at all the data points. 3. Screening Out o f the Least Effective Variables: In this step the old independent variables are replaced by the new independent variable (the second degree polynomials) that best estimate the dependent variable over the checking set. This is done with the regularity criterion which is defined as

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RMIN of the layer at which analysis is being done is less than that at the previous design, then the procedures are carried out to the next layer, otherwise the optimal Ivakhnenko polynomial has been obtained. The graph of RMIN vs. iterations generally looks like that shown in Figure 4. The algorithm described above is a very basic version of the GMDH approach to model building. A tool life model is shown as an example in Appendix A which shows the procedures in some detail. A number of improvements and modifications have been made to the algorithm to make it less heuristic and more applicable to particular situations. However, the following salient features are obvious. 1. The algorithm is capable of "discovering" complicated relationships between the dependent and independent variables. In other modeling procedures the modeler generally constrains the search by proposing a model structure and using the data to find out the model coefficients. In this case the model structure and coefficients are obtained from the data. This makes the entire modeling process more objective. 2. High-order models with a large number of variables are made possible by the way in which the model selects only the relevant elements to be passed on to the next level. 3. The algorithm is capable of functioning under conditions where noise is present in the input data. 4. The a m o u n t of data required in order to use this procedure is not very large. A small a m o u n t of data well distributed over the range

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of values the independent variables take, is enough to get accurate results. 5. Once the "optimal" model structure is obtained, updating the model to account for drifts in the system is a relatively easy procedure.

where a number of prediction problems, like population forecasting, concentration of dissolved oxygen in river water and other such problems are solved. Of all the examples surveyed, only one was obtained in the field of manufacturing. Paterau t2 used the GMDH algorithm to investigate the possibility of predicting tool life. An extremely complex model with GMDH was developed which involved about 14 different parameters and were able to get about 86% accuracy in predicting the tool life when their experimental error itself was around 10%.

The above features make the GMDH algorithm a very attractive method for modeling systems in the field of manufacturing sciences where, typically, the problems are ill-structured, the environments are complicated and the physical laws governing most of the systems are only partially known. In this paper, we think that the application of the GMDH algorithm of modeling the manufacturing processes could compliment the dynamic data system (DDS) approach. 5 The GMDH approach might be more useful when the number of variables influencing the observed variable is large (multivariate) and the models are nonlinear.

Modeling of Positioning Error In this section, w e describe how the GMDH algorithm is used to develop models for predicting positioning errors of a machine at a given point given its thermal condition. The errors of a machine tool might broadly be divided into the quasistatic error and the dynamic errors. The latter includes errors like the spindle error motion and controller errors and, by their very nature, vary with a relatively high frequency. Hence, they require real-time compensation. The quasistatic errors, as the name suggests, are slowly varying errors of relative positioning and are generally related to the s t r u c t u r e of the machine, z3 These errors might account for more than 70% of the total machine tool error. '4 The quasistatic errors consist of errors arising from inaccuracies in the geometry and alignment of the positioning elements (or axes) of the machine called geometric errors. These errors, resulting from inaccuracies in the manufacture of the elements and their assembly into the machine, are influenced by the slowly varying thermal changes and the static loading on the machine. Hence, the name quasistatic errors. A m o n g the various components contributing to the geometric error, the displacement, or positioning error of each of the positioning elements is undoubtedly the major contributor. This component alone, as experiments indicate, might contribute more than 0.1 m m to the total quasistatic error of the machine tool. Of the two main influencing factors, the thermal effects are known to have very strong influences on the machine's errors, t5

Some Application of GMDH. The GMDH algorithm has been used for purposes varying from modeling the dynamics of complex plants to discovering physical laws that govern systems to modeling economic and ecological systems. Farlow 4 gives a detailed survey of the entire spectrum of applications of this algorithm. It also gives the variations and different optimization criteria that are used in the algorithm. The Soviet control and cybernetics journal called Soviet Automatic Control has papers on the theoretical and application aspects of the algorithm in almost every issue. Some of these important papers are by Ivakhnenko et.al., 64 on the development of a polynomial and logical theory of dynamic systems. In these papers new polynomial and logical theories are introduced, meant to replace, in complex systems, the existing theory based on differential equations. The synthesis of polynomial representations of typical components of automatic control systems in different circuit configurations (linear and nonlinear) are worked out. The inflation in the British economy is worked out as an example. The capability of GMDH in discovering physical laws is demonstrated by Ivakhnenko 9 in his paper in which physical laws for calculation of the power in a DC circuit, Onsagers law and governing equation for the action of pesticides are verified. Finally, the GMDH algorithm performance in prediction problems is demonstrated by Ivakhnenko ~*.t'

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The m a x i m u m temperature rise on well designed machines is usually held to about 5-10 degrees and, for a positioning element which is one meter long (not unusual in a mid-size machining center), this could result in a 0.05 to 0.1 mm change in length, t5 Thus, by modeling the positioning error under varying thermal conditions and compensating this error, at least an order-of-magnitude improvement in accuracy can be expected. Further, in order that a machine can be properly selected to match a particular job and for computing the location of the workpiece in the machine's workpiece in order to maintain the specified accuracy, this information must be available to the (process) planning function. The first problem encountered when trying to model the positioning error is its observability. Unless, the error is measured at the axis of movement of the positioning element, the measured error, due to the Abbe's offset error, 16 is influenced by the angular errors (undesired angular motions and misalignments between the machine's structural members). Otherwise, two measurements along different straight lines are required to calculate and compensate for the angular movement. In this experiment, the effect of the angular errors was made as small as possible by making the measurements as close to the axis of the positioning element as possible. The next important alignment problem is that of recognizing the thermal state of the positioning element. Once again, it is possible to observe the temperatures only at a finite number of points on the surface of the element. Because of the varying intensity (depending on the cutting forces and the speed) and positions of the heat sources (heat is generated by friction at points on the machine slide and the lead screw which depend on the position of the carriage), this requires a large number of measurement points to completely characterize the thermal state of the element to avoid the possibility of having two different thermal states with the same temperatures at these points. In the other method, a smaller number of measurement points are used, but the thermal history at these points is also taken into consideration. The latter was selected because, besides requiring less components and being more reliable, it is capable of recognizing the thermal trends, i.e., whether the machine is heating up or cooling down.

An NC, three axes, horizontal machining center with an index table was used in the investigation. The machine had a resolution o f + 2 . 0 microns and a repeatability of about +7.0 microns. The position feedback for each axis of the machine was through a resolver. Therefore, the machine was very sensitive to any changes in the lead screw system. In order to characterize the thermal state of each positioning element, six strategic points were chosen. One point was located at the bearing of the lead screw, another on its nut, the third on the axis motor, and the remaining three on the structural part of the element. At any instant, the instantaneous temperatures at these points along with two previous sets of temperatures taken at intervals of 15 minutes, were considered representative of the thermal state of the element. 8o.oo

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Journal of Manufacturing Systems V o l u m e 7"/No. 3

GMDH p r o g r a m in the order in which it was collected. The performance of the algorithm, in this case, indicated its capability to obtain a model which fit the data. The results for each of the axes are shown in Figure 5. In these cases, the algorithm was able to obtain models with a correlation coefficient of a b o v e 0.95. The m a x i m u m error of prediction encountered was a b o u t 12 microns and the average absolute errors for each of the axes were a r o u n d 3 microns. The algorithm t o o k a b o u t six iterations to converge, suggesting a very complicated polynomial model. In the second case, the data sequence was randomized and some data was excluded from the model building procedure and held to later test the prediction capabilities of the model. Figure 6 gives the results of the model developed with the data

The positioning error along each axis was obtained by comparing the position recorded by the machine controller with a displacement reference o b t a i n e d by a laser m e a s u r e m e n t system (HP 5526A). The error was observed at steps of 100mm along the axis of the positioning element. This was repeated at intervals of 30 minutes while the machine was taken through an 8-hour cycle of heating and cooling, so as to take the positioning element through most of its range of thermal states. D a t a for each of the three axes was obtained and used to develop models using the GMDH algorithm (Fortran code modified from Farlow4). F o r relating positioning error to the position along the axis and the temperature readings of the thermocouples, two different cases were analyzed. In the first case, all the data was fed into the 80.00

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Journal of Manufacturing Systems Volume 7 No. 3

sequence randomized. In Figure 7, the errors predicted by this model for thermal states of the machine which were not used in model building are shown. In these figures the "observation point number" on the abscissa correspond to vectors of temperature readings representing the thermal state of the machine and the position of the axes when the position error was observed. Once again, the models were able to explain more than 95% of the observed error. The m a x i m u m error encountered in these models was around 13 microns and the mean absolute error was around 3 microns. It might be observed that, when the observed positioning error is small, the percentage errors in prediction are large. This is due to the fact that when the observed positioning error is of the order of magnitude of the repeatability of the machine, the signal-to-noise ratio is small. Table 1 gives a comparison of the performances of the algorithm for the two cases mentioned above. The table suggests that the performance of the algorithm is not influenced by the sequence of the data. Also, the prediction power of the model over unseen data is not significantly different from its fitting over the data used to generate it. Finally, considering the ratio of the mean to standard devia-

tion of the residuals, the models might be considered to be unbiased.

Modeling of Tool Life and Tool Wear Tool wear, the reduction in longitudinal (axial, for rotating tools) or transverse (radial) dimensions of the cutting tool during cutting, could result in an error of about 0.05mm on the finished workpiece. If accuracy is to be maintained, it becomes necessary to be able to accurately predict the tool life and the rate at which the tool wears. The wear of the tool can be related to a n u m b e r of influencing factors; cutting parameters such as cutting speed, the undeformed chip thickness, undeformed chip width, the tool geometry and the workpiece and tool material properties. Since there are so m a n y influencing factors, it becomes particularly difficult to predict tool wear.

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undeformed chip width, mm milling width, m m cutter diameter, m m l e n g t h of m a c h i n e d s u r f a c e , m m f e e d p e r tooth, m m depth of cut, m m plan approach angle

Figure 8 Machining Parameters and Elements of Undeformed Chip in Symmetrical Face Milling 17

(All values in microns)

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Journal of Manufacturing Systems Volume

For the purpose of demonstrating the capability of GMDH to identify models relating tool wear to process parameters, data on tool wear reported by Pandey and Mehta" was used. In this study '~ of face milling, the following process parameters were selected as independent variables: cutting speed, undeformed chip width, undeformed contact arc length and mean undeformed chip thickness. These parameters are shown in Figure 8. The criterion chosen for tool life was a flank width of 0 . 8 m m and tool wear rate was defined as the width of the flank wear land in mm corresponding to a 1000mm length of the machined surface on the basis of a full-factorial experiment design. Sixteen combinations of the process parameters (mentioned above) were selected for their study. The data generated by their experiment (i.e., tool life and wear rate corresponding to each combination of process parameters) was used (by us) as input to the GMDH algorithm to identify relationships of tool life and wear rate to the process parameters. The training set for the algorithm consisted of the 16 observation vectors from Pandey and Mehta. '~

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32

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No. 3

Since we were essentially fitting a model to the observed data, the 16 observations were duplicated in the checking set. The data and the results obtained are shown in Figures 9-11. Figure 12 shows the result of the tool wear rate model. Once again, each observation number on the abcissa in Figures l l and 12 corresponds to a vector of some combination of independent variables.

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Figure 10

Input Data to the GMDH Tool Life Model

Sample Output Obtained While Modeling Tool Life

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Journal of Manufacturing Systems Volume 7 No. 3

150.0

v

maintained on the workpiece, the residual stresses, surface finish and properties on the finished surface, the cutting temperatures and tool wear and life, and the stability of the cutting process itself. It therefore becomes necessary to properly model and predict the cutting forces and to use this in the selection of a process and its parameters.

Observed /, Predicted

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Much research has been performed on cutting mechanisms in order to develop general models to predict cutting forces during actual operations. This has provided the basic understanding of phenomena involved in metal cutting. '~2° However, it usually cannot be used for the purpose of predictions because of the large number of influencing factors, most of which are specific to the particular set of circumstances. Therefore, most of the data used today is experimental.

16.00

No.

Figure 11 Modeling of Tool Life

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Once again, this data helps provide a good starting point for parameter selection and rough force calculations, but in many cases turns out to be inadequate. Further, most of the experimental models characterize the variations of forces with respect to a single (or few) parameter(s) for a certain range of variations. Thus, they cannot fully model the cutting forces for the entire range of conditions encountered in practice. For example, single point, orthogonal cutting is the simplest case of metal cutting processes and has been studied for decades. Most of the work on this case considers only changes of one aspect, e.g., change of cutting parameters, tool geometry, or workpiece material. Not much research has been done in developing predictive models which take the simultaneous variation of all the above parameters into considerations. Hence, usually the models developed are applicable to specific setups and very limited conditions.

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Appendix A explains how the GMDH model is built. The models are capable of explaining 98% and 87% of the variations in tool life and wear rate, respectively. In this example, we see how GMDH can be used to build empirical models to relate wear phenomena to machining parameters over which we have some control. One could conceive of more complex models involving material properties, tool geometry and other such quantities. In the models given above, knowing the parameters used on the right-hand side of the equation, the tool life and wear rate can be estimated. The computed tool life can be used to schedule tool changes. The wear rate may be used for computing periodic compensations on the machine.

Since we are dealing with a system with many influencing variables and almost no specific knowledge as to the manner in which they influence the observations, the GMDH algorithm turns out to be a very useful approach to modeling the system. A relatively small number of observations with a large amount of variables on input/output data can be used to construct a model. Further, as more data is obtained, it can be used by the system for the adjustment of the model to generate more accurate predictions.

Prediction of Cutting Forces Modeling of forces in a metal cutting process finds important application in process selection and control. The cutting forces decide the extent of workpiece/tool deflection, and hence the accuracy

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Journal of Manufacturing Systems Volume 7/No. 3

The initial data input to the system includes the variables which may have influence on the cutting force model. Therefore, variables such as speed, feed, tool geometry, property ofworkpiece material are taken into account. In the first test case, the cutting forces generated in single point orthogonal cutting were modeled. Hasting's work shows that cutting forces are affected by workpiece material, tool geometry, and cutting parameters:' Their work showed the effect of each individual variable. In this paper we used Hasting's result, but concentrated on simultaneously including all the influencing effects in the model. The material property is represented by carbon content. Two different types of material are used. The tool geometry includes rake angle, clearance angle, and nose radius. The cutting parameters are speed, feed, and depth of cut. There are four variables in this example: carbon content of workpiece, cutting speed, feed, and rake angle. In this case, the algorithm selected only two variables, speed and feed, to explain the variability in the cutting force. It was able to explain the 96% of the variation of the given

data, predicting the cutting force with an average error of 10.11%. Figure 13 shows the predicted and the original values as well as their prediction errors. The fluctuation shown on the horizontal axis is the prediction error at each observation point. Take two observation points as examples, point 28 is the case where carbon content is 0.2%, speed is 400 m / m i n , feed rate is 0.125 m m / m i n , and rake angle is -5 degrees. The measured force is 1200N. The predicted force is 1234.9N and the error is 34.9N. Point 29 is the case with carbon content 0.38%, cutting speed 25 m / m i n , feed rate 0.125, rake angle is -5 degrees, with measured 1670N. The predicted force is 1550.8N with error of l19N. The errors are shown on points 28 and 29 of the horizontal axis. These two points do not relate to each other. In this paper, we considered the simultaneous changes of all variables. The second example is the single point, oblique cutting operation. Hasting's work has been extended to oblique cutting operation, n This extension shows that the variables which need to be taken into account are: workpiece material, tool geometry, oblique angle, and cutting parameters. In the input data set, only carbon content, speed, cutting edge angle, and oblique angle are changed and have been used in the model. The result from the GMDH algorithm is shown in Figure 14. In this example, the algorithm reached the second level, using all the variables at the first. This indicated a much more complex interaction between the variables, the average prediction error in this case, however, was less than 3.0%.

Observed _~, Predicted

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Concluding Remarks

Modeling of Orthogonal Cutting Force

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The GMDH algorithm provides an easy procedure of automatically and accurately modeling complex systems. The simplicity of use, generality, the self-organizing ability, as well as the robust nature are the major advantages of the GMDH algorithm over the tradition approaches. In order to increase the capabilities of a computerized manufacturing system like an FMS, a GMDH-type selforganizing modeling and compensating scheme can easily be set up for individual machining centers and tool families. The control structure in such systems greatly facilitate the data acquisition, storage and m a n i p u l a t i o n that such c o m p e n s a t i o n systems

Observed Predicted

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~=

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Figure 14 Modeling of Oblique Cutting Force

251

Journal of Manufacturing Systems

Volume 7/' No. 3

w o u l d require, F u r t h e r , the entire w o r k i n g o f the G M D H a l g o r i t h m c a n be easily a u t o m a t e d . This is a p a r t i c u l a r l y a d v a n t a g e o u s feature o f the a p p r o a c h . T h e c o m p l e x p o l y n o m i a l identified can be stored a n d e v a l u a t e d u s i n g a n efficient d a t a s t r u c t u r e a n d s t o r i n g the coefficients in a m a t r i x f o r m . This allows for the q u i c k e v a l u a t i o n / r e t r i e v a l o f the d a t a corr e s p o n d i n g to p a r t i c u l a r process states t h a t are m o n i t o r e d o n the s h o p floor,

( 101 103) f r o m the p r e v i o u s level (i.e., zl a n d zz). T h e coefficients to the e q u a t i o n are listed in the first r o w o f the tree m a t r i x o f level 2. T h e first e q u a t i o n at this level has the best p r e d i c t i o n p o w e r a n d we use it in the final m o d e l . T h e final f o r m o f e q u a t i o n for t o o l life is

Acknowledgement

References

This research was s u p p o r t e d by the Office o f N a v a l R e s e a r c h u n d e r c o n t r a c t # N 8 3 K 0 3 5 8 a n d the E n g i n e e r i n g R e s e a r c h C e n t e r at P u r d u e University.

1. J.A. Simpson, R.J, Hocken,J.S. Albus. "The Automated Manufacturing Research Facilityof the National Bureau of Standards", Journal of Manufacturing Systems, Vol. I/I, 1982, pp. 17-32. 2. I.G. Ivakhnenko."The Group Method of Data Handling- A Rival Method to StochasticApproximation",Automation and Remote Control Vol. 1, No. 3, 1968, p. 43. 3. F. Rosenblatt. "The Perceptron: A Probabilistic Model for Information Storage and Organization in the Brain", Psychology Rev., Vol. 65, 1958, pp. 386-408. 4. S.J. Farlow. Self-Organizing Methods in Modeling - GMDH-type Algorithms, Marcel Dekker, Inc.. 1984. 5. S.M. Pandit, S.M. Wu. Time Series and System Analysis with Application Approach, John Wileyand Sons, 1983. 6. A.G. Ivakhnenko, Yu. V. Koppa, W.S. Min. "Polynomial and Logical Theory of Dynamic Systems (Part 1)". Automation and Remote Control, Vol. 15, No. 3, 1970, pp.l-13. 7. A.G. Ivakhnenko, Yu. V. Koppa, W.S. Min. "'Polynomial and Logical Theory of Dynamic Systems (Part 2)", Automation and Remote Control, Vol. 14, No. 4, 1970, pp. 11-30. g.A.G. Ivakhnenko. "Polynomial Theory of Complex Systems", IEEE Trans. Svst., Man, Cybern., Vol. I, 1971, pp. 364-378. 9. A.G. Ivaki:anenko."Discovery of Physical Laws by the GMDH Method Using the Absence-of-Bias Criterion", Automation and Remote Control, Vol. 6, No. 6, 1973. 10. A.G. lvakhnenko, N.A. Ivakhnenko. "Long-term Predictions by the G MDH Algorithm Using the Unbiased Criterion and the Balanceof-Variables Criterion (pan I)", Automation and Remote Control, Vol. 7, No. 4, 1974, pp.40-.45. 11. A.G. Ivakhnenko, N.A. Ivakhnenko. "Long-term Predictions by the GMDH Algorithm Usingthe Unbiased Criterion and the Balanceof-Variables Criterion (part 2)",Automation and Remote Control Vol, 8, No. 4, 1975, pp. 24-38. 12. S.G. Patereu, N.S. Ravs'ka, V.M. Kul'chiy. "A Mathematical Model of the Metal Cutting Process", Automation and Remote Control, Vol. 8, 1975. 13. R.J. Hocken and Machine Tool Task Force. "Technology of Machine Tools, Vol. 5: Machine Tool Accuracy", UCRL-52960-5, Lawrence Livermore Laboratory, Universityof California, 1980. 14. R. McClure, M. Week, G. Petuelli. "Thermally Induced Errors", in

y - 20,37 + 0.9180z t - 0.9868z 2 - 0.02876z~ z 0.02114zz 2 + 0,05820zlz z

Appendix A I n this A p p e n d i x , we s h o w h o w the G M D H a l g o r i t h m w o r k s u s i n g the t o o l life m o d e l as an e x a m p l e , T h e input d a t a to the t o o l life m o d e l is s h o w n in F i g u r e 9. Line 11 s h o w s the n u m b e r o f variables, the n u m b e r o f o b s e r v a t i o n s , the n u m b e r o f d a t a in the t r a i n i n g set, the c o d e to print o u t the i n p u t d a t a , the c o d e f o r c o n t r o l l i n g the level o f i t e r a t i o n (0 for the G M D H to c o n t r o l by itself), a n d the f r a c t i o n a l ratio increase in the n u m b e r o f v a r i a bles at e a c h iteration. T h e r e are 60% m o r e new variables g e n e r a t e d at e a c h iteration level. T h i r t y t w o o b s e r v a t i o n points are listed below with the d e p e n d e n t variable f o l l o w e d by f o u r i n d e p e n d e n t variables. S i x t e e n o b s e r v a t i o n p o i n t s are put in the t r a i n i n g set. F i g u r e 10 s h o w s the s a m p l e o u t p u t o b t a i n e d while m o d e l i n g t o o l life. A t the first level, t h e / t r e e m a t r i x s h o w s the variables xt a n d x z ( r e p r e s e n t e d by 101102 in the print out) c o n s t r u c t the best new indep e n d e n t v a r i a b l e o f the s e c o n d level u s i n g Eq. (1). Coefficients to the e q u a t i o n are listed in the first r o w o f the tree m a t r i x o f level 1. By the s a m e t o k e n , v a r i a b l e s x l a n d x3 ( r e p r e s e n t e d by 101103 afterw a r d ) c o n s t r u c t the s e c o n d best i n d e p e n d e n t v a r i a ble o f the s e c o n d level. T h e s e t w o new variables at the s e c o n d level are given by the f o l l o w i n g e q u a t i o n s .

Technology of Machine Tools, M. T. T.F., Vol. 5: Machine Tool Accuracy, Lawrence LivermoreLaboratory, University of California, 1980.

15. C.P. Hemingray. "Some Aspects of the Accuracy Evaluation of Machine Tools", M. T. D. R. Conference Proceedings, 1973,pp. 281-284. 16. J.B. Bryan."The Abbe Principle Revisited: An Updated Interpretation", Precision Engineering, Vol. I, 1979, pp. 129-132. 17. P.C. Pandey, N.K. Mehta. "Application of Design of Experiment Technique to Milling Operation Studies", Indian Journal of Technology, Vol. 22, 1984, pp. 93-98. 18. E.J. Armarego, R.H. Brown. The Machining of Metals, PrenticeHall, EnglewoodCliffs, New Jersey, 1968. 19. G. Boothroyd. Fundamentals of Metal Machining and Machine Tools, McGraw-Hill, New York, 1975. 20. M.M. Nigm, M.M. Sadek, S.A. Tobias. "Dimensional Analysisof the SteadyState Onhogonal Cutting Process",lnternationalJournalof Machine Tool Design Research, Vol. 17, 1977. 21. W.F. Hastings, P. Mathew, P.L.B. Oxley. "A Machining Theory for PredictingChip Geometry,Cutting Forcesetc. from Work Material Properties and Cutting Conditions", Proceedings of Royal Society of London, 1980. 22. G.C. Lin, P. Mathew, P.L.B. Oxley, A.R. Watson. "Predicting Cutting Forces for Oblique MachiningConditions", Proceedings of the Institution of Mechanical Engineers, Vol. 196, 1982.

z I = 141.3 + 0.1951x~ + 4.842x 2 - 0.003066xt 2 - 7.912x~: + 0 . 1 0 4 5 x j x z z 2 - - 37.67 + 0.4824x t ÷ 3.096x 3 - 0.003382x~Z - 0.01582xa ~ + 0.0004067x~ xa A t level 2, the best variable g e n e r a t e d in this level is m a d e up o f v a r i a b l e s 1 (101102) a n d 2

252

Journal of Manufacturing Systems Volume 7/No. 3

Author(s) Biography Ping Yi Chao graduated from the National Chung Hsing University, Taichung, Taiwan, with a B.S. in Mechanical Engineering in 1979. He obtained a M.S.M.E. from Auburn University, Alabama, in 1984. Mr. Chao is currently a student at Purdue University, working toward his Ph.D. in the School of Industrial Engineering. His research interests include manufacturing processes, manufacturing systems engineering, and the application of artificial intelligence to manufacturing. Placid M. Ferreira obtained his Bachelor's degree in Mechanical Engineering from the University of Bombay in 1980 and a Master's degree (1982) in Mechanical Engineering from the Indian Institute of Technology, Bombay, with specialization in Manufacturing Engineering. He received his Ph.D. from the School of Industrial Engineering, Purdue University in 1987. Dr. Ferreira is now an assistant professor in the Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign. His research interests include manufacturing processes, computer aided manufacturing and artificial intelligence. C.R. Liu started his research and teaching career with Whirlpool Cooperation and Stanford University from 1973 to 1978. He has been affiliated with Purdue University since 1978 and is now a Professor of Industrial Engineering. His current research activities include various aspects of computer integrated manufacturing, such as precision engineering, representation, classification and integrated planning. Dr. Liu is a recipient of the 1981 Outstanding Young Manufacturing Engineer Award of the Society of Manufacturing Engineers, and a recipient of the 1984 Blackall Machine Tool and Gage Award, ASME. Dr. Liu and his Ph.D. student, Mr. A. Donmez, are among a group of scientists at NBS awarded the 1985 IRI00 Award. He was also awarded a certificate of appreciation by the Governor of the State of Indiana. Dr. Liu has been active in research in the emerging area of intelligent and integrated manufacturing (IIM). His activities include serving as Organizer and Chairman of the ASME Symposia related to IIM in 1985, 1986, and 1987.

253