Prediction of Surface Roughness in Ultraprecision Turning Using Fuzzy Logic
A.K. Nandi Central Mechanical Engineering Research Institute Durgapur-713209, WB, India. Email:
[email protected]
Abstract Ultraprecision turning is a manufacturing process used to generate a high surface roughness in precision components, and its input-output relationships are highly nonlinear. Surface roughness of a turned surface depends on the selection of cutting variables, such as cutting speed, feed and depth of cut. Realizing the fact that fuzzy logic controller (FLC) is a powerful tool for dealing with impression and uncertainty, in this paper two approaches are developed to model the Ultraprecision turning operation using fuzzy logic. In the first approach a combined Genetic-fuzzy (GA-Fuzzy) system is used, where as a Genetic-fuzzy basis function network (GA-FBFN) is utilized in the second approach. The results of these two approaches are compared with those of the empirical expression for making prediction of surface roughness in Ultraprecision turning. It has been found that the performance of the second approach is much better than the previous one. It may happen because, in the GAFuzzy system, the main draw back lies in the fact that it does not keep track on the number of times a particular rule is getting fired during training. Keywords: Turning, Fuzzy logic, FBFN, GA, Surface roughness.
1
Introduction
Now-a-days, due to the increasing demand of higher precision components for its functional aspect, surface roughness of a machined part plays an important role in the modern manufacturing process. Turning is a machining operation, which is carried out on lathe. Ultraprecision turning is an important technology used to generate a high surface finish of machined components in the order of nano-meter. In the recent past, computerized numerically controlled (CNC) machines have attracted much attention of the researchers. Development of a reliable strategy for predicting surface finish of a machined component before carrying out the actual machining is a challenging task [1]. Due to non-linearity of the cutting parameters, toolwork combination and rigidity of machine tool, soft computing (Neural network, Fuzzy logic, Genetic algorithm and their hybridizations) is used for prediction of important parameters in manufacturing [3, 4, 10]. Fuzzy logic controller, a successful application of fuzzy set theory [13] is a potential tool for dealing with imprecision and uncertainty [6]. Realizing this fact, an attempt is made by the author to model the Ultraprecision turning operation using a combined GA-Fuzzy and GA-FBFN approaches separately. Genetic algorithm (GA) is a populationbased search and optimization technique [2, 7]. In the GA-Fuzzy system, the knowledge base (KB), which consists of Rule base (RB) and membership function distributions of FLC, is designed using a GA-based tuning. In the GA-FBFN system, the membership function distributions, RB and the weight factors (for each rule) of fuzzy basis function network (FBFN) are designed based on a GA. In
both the approaches, the GA-based tuning is made based on error reduction measure using the experimental data [5]. The performance of the both approaches is compared with the empirical expression for making prediction of surface roughness in ultraprecision turning operation.
2
Mathematical formulation of the problem
The functional relationship between surface roughness and the cutting parameters in ultra precision turning can be expressed by the following empirical expression [5] for a job-tool combination of aluminum alloy and diamond tool.
R = 13.635Vc − 0.102F0.5123Dc − 0.0382 ,
(1)
where R is the surface finish in nano-meter, Vc is the cutting speed in meter per min, F is the feed rate in micro-meter per revolution and Dc is the depth of cut in micro-meter. The above empirical expression was established based on the regression analysis for predicting surface roughness of ultraprecision turned surface with a single crystal diamond tool. But, the main drawbacks of the above model are that, it is limited to use for a large range of the cutting variables and the predicted surface roughness of a machined surface may not be accurate.
3
Mathematically, the second order polynomial function used here, can be represented as follows: Y= c0X + c1X2 ,
(2)
where X is the distance measured along the basewidth of the membership function distribution, Y is the fuzzy membership function value whose numerical value varies in the range of (0, 1), c0 and c1 are the coefficients, which can be obtained based on some specified conditions. The values of Y are assumed to be equal to 0 and 1 at X=0 and X=b (the half base-width), respectively. Moreover, the rate of change of Y with respect to X is assumed to be equal to 0 at X=b, to have a feasible solution of equation (2). The variables `b’ (i.e., b1, b2, b3 and b4 of Figures 2) are optimized using a GA for fine-tuning. In the Figure 2, b1 is set equal to 99 corresponding to a scaling factor for cutting speed, SFc = 297, b2 is made equal to 1.6 corresponding to a scaling factor for feed, SFf = 4.8, b3 is set equal to 2.3 corresponding to a scaling factor for depth of cut, SFd = 6.9, and b4 is set equal to 2.35 corresponding to a scaling factor for surface roughness, SFs = 7.05. Approach 1:
GA-Fuzzy system
Figure 1 shows the schematic diagram of a geneticfuzzy system in which a GA-based tuning is adopted, off-line, to improve the performance (online) of an FLC.
Proposed approaches
In the present work, two different approaches are developed, as explained below. The input as well as output variables of the both approaches are considered to have second order polynomial membership function distributions [11]. Each variable is assumed to have four different values, which characterizes the linguistic terms (L - Low, M - Medium, H - High and VH - Very high) within its range. As there are three input variables of both the FLC and FBFN (i.e., Cutting Speed, Feed and Depth of cut) and each of these inputs have four different values within its range, there could be a maximum of 4 ×4 ×4 = 64 rules in the RB of the FLC and FBFN. A typical fuzzy rule will look as follows: IF Cutting Speed is L AND Feed is L AND Depth of cut is H THEN Surface roughness in Ultraprecision turning will be H.
Figure 1: Schematic diagram of the GA-Fuzzy system A GA is used to design the membership function distributions as well as RB of the FLC automatically based on the experimental values. As a GA [2] is computationally expensive, GA-based tuning is carried out off-line. A binary coded GA with 212-
bits long string is used during optimization. The first 64 bits will indicate the existence of the rules (1 and 0 for the presence and absence of a rule, respectively) and the next 20 bits will carry information of the four continuous variables (5 bits of each of the variables namely, Cutting Speed, Feed, Depth of cut and surface roughness). Among the rest 128 bits, every two bits will be used to determine the output of a particular rule (00 for L, 01 for M, 10 for H and 11 for VH). Thus, if the first rule is selected to be present in the RB, its output will be decided by the two bits (85th and 86th) and so on. The initial settings of membership function distributions for the input and output variables as shown in Figure 1 are decided based on the author’s knowledge of the process to be controlled. During
GA-based tuning, the ranges of variation for the half base-widths of the input and output variables are decided after a thorough careful study. The ranges of variation for b1 (cutting speed), b2 (feed rate), b3 (depth of cut) and b4 (surface roughness) are assumed to be equal to (75, 160), (0.85, 2.9) (1.5, 2.8) and (2.8, 4.0), respectively.
Figure 2: Author-defined membership function distribution The objective of this study is to find optimal size of the membership function distributions for condition and action variables of the FLC and to obtain an appropriate RB automatically using a GA based on error reduction measures during training, so that, it can predict surface roughness in Ultraprecision turning with minimum error.
Approach 2:
GA-FBFN system
The way of representing fuzzy inference systems was provided using fuzzy basis function network by Wang and Mandel [12], where the FBFN was represented in a simple structure, which is similar to those of radial basis function network [9]. The fuzzy basis function (FBF) can be defined as [8] P
j (I )=
j ∏ iN= 1 Y i (I i ) , S j N ∑ ∏ i = 1 Y i (I i ) j= 1
(3)
where S is the number of rules and N is the number of input variables (Ii). Y is the membership function value. The weight factor of the jth rule is defined by Wj. The value of the weight factors are assumed to lie within the range of 0.0 to 1.0. Figure 3 shows the architecture of FBFN with three inputs (cutting speed, feed and depth of cut) and single output (i.e., surface roughness). Since, each of the input variables is assumed to have two different values, L and M, within its range, a maximum of 2 ×2 ×2 = 8 rules and the weight factors (Wj) for each rule are depicted in the Figure 3. In the present work, each variable is considered to have four different values in its range and the methods of product inference and centroid defuzzification have been adopted. A method of automatic design of membership function distributions of the input and output variables, RB and WFs (for determining the strength of the rules) of the FBFN is adopted, off-line, based on error reduction measure. A 532 bits long string, binary coded GA is used for both data base tuning and design of RB as well as WFs (for each rule). The first 64 bits are used to indicate the existence of the rules, next 20 bits will carry information of the four continuous variables (5 bits for each variable), next 128 bits are utilized for designing the RB (in the manner, as discussed in Approach 1), and rest 320 bits are used for WFs (5 bits for each weight factors). The initial settings of membership function distributions of the input and output variables are considered same as in Approach 1. During GAbased tuning, the ranges of variation of both the input as well as output variables are kept as equal to Approach 1 and the weight factors are varied in the range of 0.0 to 1.0. The objective of this study is to find the optimal size of the membership function distributions for condition and action variables of the FBFN and to
obtain an appropriate RB and WFs for each selected rule based on error reduction measures during training, so that it can predict surface finish in ultraprecision turning with less error. Layer 0
Layer 1
L
Layer 2
P1
M
P2
M
P3
Table 1. Experimental values of surface roughness
Layer 4
Layer 3
L
as well as FBFN is to reduce the average error based on the 15 training sets.
Cutting Feed rate Depth of Surface speed (micro- cut (micro- roughness (m/min) meter/rev) meter) (nanometer)
W1
Cutting speed
W2 M
W3
L
L
P4
W4
W5
Feed L
W6
M
1.8
1.5
11.16
314
1.8
1.5
9.94
113
5.0
1.5
20.65
314
5.0
1.5
16.29
113
1.8
6.0
9.9
314
1.8
6.0
10.72
113
5.0
6.0
18.15
314
5.0
6.0
15.67
91
3.0
3.0
15.04
188
1.5
3.0
9.88
188
6.2
3.0
23.77
188
3.0
1.13
13.64
188
3.0
8.0
12.23
188
3.0
3.0
14.02
388
3.0
3.0
14.1
P5
M
L
Surface roughness
113
P6 W7
Depth of cut M
L
P7
M
P8
W8
Figure 3: The architecture of FBFN with 8 rules
3.1 Fitness evaluation 15 different sets of input parameters (i.e., cutting speed, feed and depth of cut) are considered, during the GA-based tuning, as shown in Table 1. In Approach 1, for each of the training cases, the FLC will determine the output, i.e., surface roughness in ultraprecision turning. Moreover, for each set of input parameters, surface roughness in turning is measured by experimentation [5] as depicted in Table 1. The difference in surface roughness, as determined by the FLC and that of the experimental value is the error in prediction of surface roughness of the FLC. The absolute value of the error is considered, since the error can be either positive or negative. The average error of the 15 training cases is taken as the fitness of a GA- solution. Similarly, in Approach 2, the fitness value is obtained by determining the average error in prediction of surface roughness using the FBFN based on the experimental values of the same training cases as in Approach 1. The main objective of training the FLC
4
Results and Discussion
A sub-micro CNC ultraprecision turning machine, developed by Harbin Institute of Technology, China, is employed to perform the turning operation [5]. The schematic layout of the sub-micro CNC ultraprecision turning machine is shown in Figure 4. An AFM (Atomic Force Microscopy) is used to measure the surface roughness of the machined part made of annealed aluminium alloy (LY 12). A natural single crystal diamond tool produced by Contour Fine Tooling of England is utilized for the cutting test. The tool rake angle, the clearance angle, the corner radius, the rounded cutting edge radius are kept fixed as 00, 70, 1.5 mm, 175 nano-meter, respectively.
It has been found that the GA has selected 36 rules using Approach 1 and 30 rules using Approach 2, out of a total of 64 rules, respectively, during GAbased tuning. The results of 10 test cases (considered at random) for making prediction of surface roughness in Ultraprecision turning, using Approach 1 and Approach 2, are shown in Table 2. R1 and R2 are the surface roughness determined using Approach 1 and Approach 2 respectively, and R3 is the surface roughness calculated using Equation (1). A comparison of results of both the approaches with the mathematical expression (as explained in Section 2) is also made in Table 2.
Figure 4: Schematic layout of the sub-micro CNC ultra-precision turning machine The optimal GA parameters determined through a parametric study, are kept as same for both the Approaches, as follows: Population size =250; Number of generation =500; Crossover Probability =0.98; Mutation probability =0.012 . Table 2. Results of the test cases – surface finish prediction Vc
F
Dc
R1
R2
R3
% % Error Error I II
170
4.8
5.72
15.4
16.7
16.8
8.33
150
5.2
6.28
15.9
16.9
17.7 10.17 4.52
120
5.8
7.20
17.1
19.0
19.1 10.47 0.52
100
6.1
7.60
16.7
18.8
19.9 16.08 5.53
185
4.5
3.48
14.7
17.8
16.5 10.91 7.88
285
5.5
6.70
17.9
16.8
17.0
5.29
1.18
260
4.7
5.58
16.4
16.1
16.0
2.50
1.63
310
6.1
1.24
15.6
19.2
19.0 17.89 1.05
278
5.2
2.50
15.5
17.1
17.2
9.88
0.58
270
4.9
2.92
18.1
16.8
16.6
9.04
1.20
0.60
Error I and Error II are the deviations of surface roughness predicted using Approach 1 and Approach 2 respectively, from the value obtained using the empirical expression. It is interesting to note that, for all the cases, Approach 2 is performed better than Approach 1. This is obvious, because, in the GA-fuzzy approach, the main draw back lies in the fact that it does not keep track on the number of times a particular rule is getting fired during training. Thus, there may be a chance to presence some of the redundant rule in the optimized rule base of the FLC. This problem is taken care using FBFN as discussed in Approach 2.
5
Concluding remarks
In this paper, two approaches are used to model the Ultraprecision turning operation. In Approach 1, a combined GA-Fuzzy system is utilized, where as a GA-FBFN system is employed in the Approach 2. A method of automatic design of KB of the FLC as well as FBFN using a GA is adopted based on error reduction measure with experimental data. The results of these two approaches are compared with those of the empirical expression for making prediction of surface roughness in Ultraprecision turning. It has been found, that the Approach 2 outperforms than Approach 1. It is obvious because, in the GA-Fuzzy system, the main draw is that it does not keep track on the number of times a particular rule is getting fired during training. Comparison of the results of both the approaches with those of the empirical expression indicates that, using fuzzy logic, it is possible to predict surface roughness in Ultraprecision turning with quite reasonable accuracy.
Acknowledgements The author would like to thanks to the Director of CMERI, Durgapur, India for his support to carry out this research work. The author would also express his gratitude to Professor F. Klawonn for his encouragement to attain EUSFLAT2003.
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