Comparison and Research on Different Cutting Force Prediction Modeling Methods

1 INTRODUCTION In cutting technology research and actual cutting, the data about cutting force is an important basis for calculating cutting power, designing and using machine tools, tools and fixtures, developing cutting database, and realizing cutting force control in machining. In actual production, in order to make full use of the power of the machine tool during rough machining and effectively ensure the quality of the workpiece during finishing, it is necessary to reasonably select the cutting conditions and predict the cutting force under the selected cutting conditions. The empirical model for predicting cutting forces is mainly based on the least-squares regression method. In recent years, the application of artificial neural network methods and gray theoretical modeling methods has become more and more. These modeling methods have different characteristics and conditions of use, and have advantages and disadvantages. In this paper, the modeling characteristics and advantages and disadvantages of the artificial neural network method and the grey theory modeling method are analyzed in depth and compared with the commonly used least squares regression method to provide a reasonable selection modeling method. Reference. 2 Prediction model of cutting force based on radial basis neural network Three-layer BP neural network based on Kolmogorov theorem can more accurately fit any continuous function. When the number of input nodes is n, the number of hidden layer nodes is (2n+1). And often choose Sigmoid transfer function. In practical applications, a large number of BP hidden layer nodes are often needed. By increasing the number of hidden layers, the number of nodes on each hidden layer can be reduced. However, there has been no unified method for selecting hidden layers and their number of nodes in BP networks. In addition, the standard BP and various improved BP algorithms all have problems of local minima and convergence speed. The ability of radial basis neural network (RBF) to accurately fit any continuous (or discontinuous) objective function and learning speed is better than BP network. The hidden layer node of RBF adopts the radial basis transfer function. The number of nodes does not need to be set in advance as in the BP network, but increases continuously during the learning process until the error indicator is met. According to the characteristics of cutting force and its influencing factors, design the RBF network as shown in the figure below. As can be seen from the figure, the RBF network includes an input layer, an RBF hidden layer, and an output layer. The output layer contains a linear node for outputting predicted cutting forces. The hidden layer contains S1 RBF nodes and the S1 value dynamically increases during the learning process. The R × Q order input vector matrix P of the input layer indicates that there are R input nodes, and Q samples are input at each node (Q is equal to the test group number m). Each input node represents an influence factor of the cutting force, and all quantifiable influence factors of the cutting force can be abstracted as one input node. Considering cutting depth, feed rate, cutting speed, shear yield stress of workpiece material, tool material, tool's negative chamfer width, leading angle, edge angle, tip radius, tool wear, cutting fluid, etc. Influencing factors can have multiple input nodes. According to the actual modeling experience, the influence of depth of cut and feed can be mainly considered. At this time, the number of input nodes is R=2.

Cutting force prediction of radial basis neural network structure diagram
(S2=1, S1 dynamic determination)

The test data set under the cutting conditions of group m is [P,T], the target output is T=[Fz1,...,Fzm], and the R×Q order input vector matrix P is represented by each group in the Q=m group test. Consider R cutting force influence factors. After selecting the relevant design control parameters (such as expected network output error squared index, etc.), based on the experimental data [P, T], the RBF design algorithm can be used to determine the hidden layer and output layer of the RBF network in a relatively short time. The weight matrix W, the deviation matrix b, and the number of nodes S1 on the hidden layer complete the neural modeling of the cutting force prediction. 3 Grey Modeling of Cutting Force Prediction Grey Set Theory is good at dealing with uncertain objects that have the characteristics of “partial information is known, part of information is not known, small samples, and lean information”. It generates and identifies “partially known information”. Development, extraction of useful information, and ultimately the effective description of the internal laws of the research object. Cutting practice shows that due to various factors, cutting forces often exhibit uncertainties. The GM(1,1) gray modeling principle is described as follows: Let the measured raw data sequence be Y0={Y0(1),Y0(2),...,Y0(j),...,Y0(n)} Accumulated generation sequence is defined as Y1={Y1(1),Y1(2),...,Y1(j),...,Y1(n)} Where Y1(j) is Y1(j)=j Y0(i) Σ i=1 (1) For the sequence Y1, the adjacent average generation is defined as Z1={Z1(2),..., Z1(i),..., Z1(n)} where Z1(i) can be expressed as Z1 ( i) = 0.5Y1(i) + 0.5Y1(i-1) Assume that the column vector Y = Y0(2), Y0(3), ..., Y0(n)]T, and the matrix B is defined as B = [-Z1 (2),1;-Z1(3),1;...;-Z1(n),1] In the grey differential equation dY1/dt+aY1(t)=b, the estimated values ​​of parameters a and b are determined as [ a,b]T=(BTB)Z-1BTY According to formula (1), there is Y1(0)=Y0(1). The solution of the gray differential equation is Ŷ1(1)=Y0(1) (2) Ŷ1(i)=[Y0(1)-(b/a)]exp[-a(i-1)]+(b/a (3) where: i=2,3,...,n. Equations (2) and (3) can be used to find the simulated sequence Y1 of Y1. Therefore, Y0's simulation sequence Y0 can be determined as Ŷ0(1)=Y0(1) (4) Ŷ0(i)=Ŷ1(i)-Ŷ1(i-1) (5) where: i=2,3,... ,n. Using equations (4), (5), i ≤ n is used for the simulation of the original sequence Y0; i>n is used for prediction of the cutting force uncertainty. 4 Model verification and analysis In order to verify the validity and accuracy of the modeling method, it is necessary to obtain modeling data and evaluation data. Table 1 shows the cutting force data for the turning test. Test conditions: Workpiece material 45 steel (normalizing, HB=187), workpiece diameter 81mm; YT15 external turning tool (416A), rake angle 15°, relief angle 68°, auxiliary relief angle 4° to 6°, main deviation Angular 75°, sub-deflection angle 10°～12°, cutting edge angle 0°, tip radius R0.2mm, negative chamfer width 0; spindle speed n=380r/min, cutting speed v=96m/min. Table 1 Cutting force measurement data No. 1 2 3 4 5 6 7 8 9 ap (mm) 2 2 2 2 3 3 3 3 f (mm/r) 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 Main cutting force Fz ( N) Validation and analysis of neural network model Based on the model shown in Figure 1 and the data in Table 1, ap and f are selected as input nodes, and the cutting force Fz is measured as the target output. Data samples Nos. 1 to 8 in Table 1 were selected for modeling, and data sample No. 9 was used for model evaluation. When the radial basis learning algorithm is used to design the specific model shown in Fig. 1, the learning control parameters are as follows: network output error squared and expected value e=0.01, radial base scattering value sp=1.0, hidden layer maximum node number nr=1000, display frequency Df=25. Through programming and calculation, the specific model parameters of the cutting force neural network shown in Table 2 are obtained. The calculation results of the cutting force neural network based on the Table 2 model and the cutting conditions corresponding to Table 1 are listed in Table 3. For comparison, Table 3 also lists the measured values ​​of cutting forces and the calculation results using the ordinary least-squares multiple linear regression model. Least squares regression modeling is based on data samples Nos. 1 to 8 in Table 1, and data sample No. 9 is used for model evaluation. The linear form of the regression model is Yp = 5.135186 + 0.9719143 [ln (ap)] p + 0.862146 [ln (f)] p (6a) Fz = exp (Yp) (6b) Table 2 Specific model parameters of the cutting force neural network on the hidden layer Weight W Hidden layer deviation bh Output layer Wo (×106) W11=0.4, W12=3.0 b1=0.8326 W11=0.4487 W21=0.3, W22=3.0 b2=0.8326 W12=-0.3428 W31=0.5, W32=2.0 b3=0.8326 W13=0.6923 W41=0.4, W42=2.0 b4=0.8326 W14=-1.0032 W51=0.1, W52=3.0 b5=0.8326 W15=0.5320 W61=0.2, W62=2.0 b6=0.8326 W16=0.3526 W71=0.2, W72=3.0 B7=0.8326 W17=-0.5968 Other structural parameters: two input nodes ap,f; hidden layer node number S1=7 (determined through learning); hidden layer node transfer function is radbas; output layer node number S1=1 (preselection The output node transfer function is a linear function; Output layer deviation b0 = -4.0992 × 104 Table 3 The neural network predicted value of the cutting force Fz, the comparison of the least square estimate with the measured value No. Cutting force Fz (N) Actual measured value Neural network predictive value relative error B% least square estimate relative error B% 1 878 878 0 815 -7.2 2 1129 1129 0 1157 +2.5 3 1443 1443 0 1482 +2.7 4 1756 1756 0 1797 +2.3 5 627 627 0 66 5 +6.1 6 1255 1255 0 1209 -3.7 7 1756 1756 0 1715 -2.3 8 2195 2195 0 2198 +0.14 9 2760 2415 -12.5 2665 -3.4 In Table 3, the relative error is defined as: B% = [(predicted value - measured Value) / measured value] x 100%. From Table 3, we can see that radial basis neural network (RBF) modeling method has the following characteristics: 1 can be accurately fitted to any continuous or non-continuous function (such as data samples No. 1 ~ 8), and its fitting accuracy is higher than commonly used Least squares regression method; because the nodes on the RBF hidden layer adopt the radial basis transfer function, the number of nodes does not need to be set in advance, but increases continuously during the learning process until the error indicator is met. 2 When the selected cutting conditions are outside the upper or lower limit of the cutting conditions of the modeled test samples, the radial basis neural network has a poor prediction effect, and the fitting accuracy is lower than the least squares regression method (such as data sample No. 9). . 3 In order to make the prediction range of neural network method wider and the prediction result more accurate, the difference in cutting conditions between each test sample selected for modeling should not be too large, and as many test samples as possible should be collected, but RBF at this time The number of hidden nodes in the network will increase. Verification and Analysis of Grey Prediction Model According to the aforementioned GM (1,1) gray modeling principle, data samples Nos. 1 to 8 in Table 1 are selected for modeling, and data sample No. 9 is used for model evaluation. The gray model prediction values ​​of the cutting force Fz shown in Table 4 and the comparison results with the least squares estimated values ​​and the actual measured values ​​can be obtained. From Table 4, we can see that the fitting and prediction accuracy of the gray model is lower than the commonly used least squares regression method (ie, the relative error is relatively large), because the distribution of the model data of the gray model can not be better obeyed by the regular distribution of the e index. . Table 4 Comparison of gray model predictions, least squares estimates, and measured values ​​of cutting force Fz No. Cutting force Fz(N) Measured value Grey model Predicted value Relative error B% Least square estimate Relative error B% 1
2
3
4
5
6
7
8
9 878
1129
1443
1756
627
1255
1756
2195
2760 878.00
1075.2
1179.2
1293.3
1418.4
1555.5
1706.0
1871.0
2051.9 0.0
-4.8
-18.3
-26.3
+126
+23.9
-2.8
-14.8
-25.7 815
1157
1482
1797
665
1209
1715
2198
665 -7.2
+2.5
+2.7
+2.3
+6.1
-3.7
-2.3
+0.14
-3.4 Parameter estimation of gray differential equations: [a, b] = [-0.0923, 945.2975] Data samples Nos. 4 to 6 are removed from Table 1, and data samples Nos. 1 to 3 and Nos. 7 to 8 are used Modeling, No.9 is still used for model evaluation, so that the modeling data is more similar to the e-exponential distribution, and the resulting gray model prediction results are shown in Table 5. Obviously, compared with Table 4, the gray model fitting and forecasting accuracy of Table 5 has been significantly improved, and there are fewer sample data for the model. This is one of the salient features of the gray modeling method. Table 5 Grey Model Prediction of Cutting Force Fz No. Cutting Force Fz(N) Measured Value Grey Model Predicted Value Relative Error B% 1
2
3
7
8
9 878
1129
1443
1756
2195
2760 878
1138
1413.3
1755.3
2179.9
2707.3 0.0
+0.79716
-2.05821
-0.03986
-0.68792
-1.90942 parameter estimation of grey differential equations: [a,b]=[-0.2167,828.9403] The grey model prediction method is suitable for "small sample, poor information" modeling, and can obtain high model fitting and prediction accuracy. However, the precondition for this is that the model data sample must obey the distribution law of e-index. In order to further verify its modeling characteristics, Table 6 lists another calculation example that uses the gray model to predict the normal grinding force Fn in cylindrical grinding. Among them, data samples Nos. 1 to 5 are used for modeling, data samples No. 6 are used for model evaluation, and measured values ​​of the normal grinding force Fn are cited from the references. As can be seen from Table 6, as long as the modeling data samples are better obeyed by the e-exponential distribution law, the fitting and prediction accuracy of the grey model prediction method is better than the least-squares regression method. Table 6 Grey Model Prediction of Normal Grinding Force Fn in Cylindrical Grinding Process No. Cutting Force Fz(N) Measured Value Grey Model Predicted Value Relative Error B% Least Square Estimate Relative Error B% 1
2
3
4
5
6 7.84
8.10
8.38
8.75
9.32
9.93 7.84
8.04
8.42
8.83
9.25
9.69 0
-0.77
+0.52
+0.89
-0.72
-2.3 7.69
8.24
8.58
8.83
9.09
9.19 -1.865
+1.76
+2.406
+0.92
-3.13
-7.415 Parameter estimates for grey differential equations: [a,b]=[-0.04695,7.48196] 5 CONCLUSION For prediction of cutting force, least-squares regression and artificial neural network methods are effective modeling methods. Both of these modeling methods require the provision of as many data samples as possible to ensure a high degree of fitting accuracy and applicability. The accuracy of the modeling and fitting of the neural network method is better than that of the least squares regression method, but the least squares regression method is more advantageous in the data prediction than the modeling data. Under the premise that the modeling data sample obeys the e-distribution distribution rule, the fitting and prediction accuracy of the grey model prediction method is better than the least squares regression method, and can be realized under the condition of “small sample, poor information” data. mold. If the data sample does not obey the e-index distribution law, the least squares regression method is better for modeling.