INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 13, No. 1, pp. 77-84
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DOI: 10.1007/s12541-012-0011-9
Optimization of Tool Path Planning in 5-Axis Flank Milling of Ruled Surfaces with Improved PSO Hsin-Ta Hsieh1 and Chih-Hsing Chu1,# 1 School of Industrial Engineering and Engineering Management, National Tsing-Hua University, No. 101, Section 2, Kuang-Fu Road, Hsinchu, Taiwan, 30013 # Corresponding Author / E-mail:
[email protected], TEL: +886-3-574-2698, FAX: +886-3-572-2204 KEYWORDS: 5-Axis machining, Flank milling, Particle swarm optimization, Ruled surface
Previous studies have shown that machining error in 5-axis flank milling can be systematically reduced by optimization of tool path planning. However, the solution quality of the optimization methods adopted by those studies was not satisfactory, due to the constraint that the cutter must contact the boundary curves of the ruled surface to be machined. This work proposes an improved tool path planning method based on Particle Swarm Optimization (PSO) algorithms without this constraint. The method enlarges the feasible region in the optimization by allowing cutter location to deviate the boundary curves along the normal, tangent, and bi-normal directions. Design of experiment techniques are applied to determine better parameter setting in the PSO. A new objective function is formulated as the weighted sum of the errors induced by overcut and undercut. We can choose to minimize the total machining error, the overcut, or the undercut by properly adjusting the weight value. Compared to previous methods, the improved path planning method produces smaller machining error and offers better planning flexibility. Manuscript received: March 7, 2011 / Accepted: July 11, 2011
1. Introduction 5-Axis machining has been widely used in manufacturing of complex ruled geometries in automobile, aerospace, energy, and mould industries. Compared with traditional 3-axis machining, this machining operation offers better shaping capability and productivity with additional degrees of freedom in the tool motion. Tool path planning is a challenging task in most 5-axis machining operations, with tool collision and machining error control as two major concerns.1 It is highly difficult, if not impossible, to completely avoid tool overcut and undercut when a cylindrical cutter is used to create complex shapes. The machined surface is considered acceptable in practice as long as the amount of machining error is limited within a given tolerance. Previous studies2-4 have shown that the machining error in 5axis flank milling can be effectively reduced by means of global (or near global) optimization of tool path planning. The optimization approach also provides a systematic mechanism of machining error control. Wu and Chu2 proposed the first attempt to transform the tool path planning into a curve matching problem. Discrete dynamic programming techniques were then applied to generate an optimal matching with the total error on the machined surface as the objective function. Chu et al.3 solved the similar problem with Ant © KSPE and Springer 2012
Colony Systems (ACS) algorithms Simulation results showed that the computation time can be thus significantly reduced with a minor sacrifice at the final solution. Hsieh and Chu4 allowed the cutter to contact at any point on the boundary curves of the ruled surface. This relaxes the constraint in the two previous studies that the cutter can only contact at pre-defined discrete points on the boundary curves. They also adopted GPU computing technologies to accelerate the optimization process,5 which is usually computationintensive and limits the practicality of optimization-based tool path planning. All the previous methods mentioned above suffer from unsatisfactory solution quality of the optimization due to the assumption that the cutter must make contact with the boundary curves. This assumption greatly restricts the solution space in search for optima, resulting in worse tool paths. To overcome this deficiency, we propose to use a new scheme for encoding cutter location in the PSO-based tool path planning. The cutter can deviate from the surface to be machined along the normal, tangent, and binormal directions on the points of the boundary curves. The solution space in the optimization is thus significantly enlarged. Factorial experiment techniques are applied to systematically determine the parameter setting in the PSO algorithms. In addition, the objective function is re-formulated as a weighted sum of the
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errors induced by undercut and overcut respectively. It can be chosen to minimize the total machining error, the undercut, or the overcut by properly specifying the weight value. This offers a new planning flexibility in 5-axis flank milling, which helps meet various machining requirements in practice. This work improves the previous methods of tool path planning in 5-axis flank milling by offering smaller machining error and better planning flexibility.
2. Tool Path Planning based on PSO 2.1 PSO Encoding We first explain how to encode the tool path planning in PSO. In the previous studies,4,5 a tool path is represented by the connection between the boundary curves of the ruled surface. A series of curve parameter pairs is the actual variables to be optimized. This encoding can produce a smaller error on the machined surface, as compared to the result generated by traditional methods, e.g. the cutter follows the rulings or it moves at a tilt angle while moving along the rulings. However, it is not necessary that the cutter has to contact with the boundary curves. This constraint seriously restricts the search space in the optimization and produces a worse solution for the optimized tool path. We propose a new scheme of encoding cutter location that relaxes the constraint. A tool path consists of n cutter locations. Each cutter location is defined with a point pairs generated from the first and second boundary curves respectively. It contains the following variables to be determined: 1. A pair of curve parameters uij on the two boundaries. They act as reference positions for the other variables. 2. The movement tij along the tangent of the boundary curves at uij. 3. The movement nij along the surface normal at uij. 4. The movement bij along the bi-normal of the surface at uij. Note that the tangent, normal, and bi-normal vectors are mutually orthogonal. Where i=1,2 and j=1~n. We set i as the index of the reference position on the boundary curves and j represents the index of cutter location. n is the number of cutter locations. For a reference point on the boundary, a direction vector can be generated with three unit
Fig. 1 Variables to be varied in the encoding scheme
INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 13, No. 1
vectors and scalars, as shown in Fig. 1. It is expressed as (tT, nN, bB). The two center points of the cutter are translated with the corresponding direction vectors. Our task is to find an optimal set of scalars that minimizes the machining error.
2.2 Machining Errors Estimation Exact estimation of the machined geometry is highly difficult, if not impossible. The machining error is approximately estimated with the z-buffer method developed in the previous work.2 The estimation of the machining error consists of four steps as shown in Fig. 2. The design surface is first sampled in a discrete manner. At each sampling point, two straight lines are extended along the positive and negative normal directions with a distance of the cutter radius. The lengths of these lines will get updated after the cutter sweeps across them along a given tool path. The lines shown above/below the surface indicate the amount of undercut/overcut respectively. We approximate the tool swept surface by generating a finite number of tool positions interpolated by two consecutive cutter locations. This approximation is similar to the actual tool motion generated by CNC interpolation. The next step is to intersect the lines with the peripheral surface of the cutter. The machining error is calculated as the sum of the lengths of the trimmed straight lines. In practice undercut and overcut may correspond to machining deviation/imperfection of different degrees. For example, manual polishing sometimes can be applied to remove uncut materials left from previous finish cut. In contrast, it is more difficult to restore the material mistakenly removed by overcut. It is advantageous to control the distribution of overcut versus undercut while minimizing the total machining error. To achieve this goal, the objective function in the PSO is defined as a weighted sum of the error amounts induced by undercut and overcut, respectively. By properly choosing the weight values, we can choose to minimize the total error, overcut, or undercut induced by the optimized tool path. The objective function is expressed as:
Fig. 2 Machining error estimation with the stock height method
INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 13, No. 1
Deviation = Wg*sumGouge + We*sumExcess
(1)
Where Wg: weight of the undercut error, We: weight of the overcut error, sumGouge: sum of the undercut error, and sumExcess: sum of the overcut error.
2.3 PSO Algorithm Particle swarm optimization (PSO) is a stochastic optimization technique inspired by social behavior of bird flocking or fish schooling.6 A PSO algorithm is initialized with a population of random solutions and searches for optima by updating generations. The potential solutions in PSO, referred to as particles, explore the solution space by following the current optimum particles. Each particle keeps track of its positions in the solution space associated with the best solution it has achieved so far, noted as pbest. Another value recorded during the optimization process is the best value obtained so far by the neighbors of the particles. This location is called lbest. When a particle takes all the population as its neighbors, the value becomes a global best gbest. The PSO process changes the velocity of each particle at each iteration toward its pbest and lbest positions. The variables in the PSO algorithm are summarized as follows: Xgb(t): global optimal location; fgb(t): the objective value of Xgb(t); Xib(t): the optimal location of particle i in one iteration; fib(t): the objective value of Xib(t); Xi(t): the location of particle i at time t; fi(t): the value of Xi(t); Vi(t): the velocity of Xi(t); W: weight; C1, C2: learning factors; rand1, rand2: random numbers generated from the probability distribution of U(0,1); N: the population of particles; T: the number of iterations. The location of a particle Xi(t) corresponds to a tool path consisting of n cutter locations. The encoding of each cutter
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location contains four pairs of variables: uij, tij, nij, and bij. (see Fig. 3) Each tool path produces an objective value, i.e. the machining error fi(t). The initial velocity of particle i is null. The search process of PSO is described as follows: Vi(t+1)=W×Vi(t)+C1×rand1×(Xib(t)-Xi(t)) +C2×rand2×(Xgb(t)-Xi(t))
(2)
Xi(t+1)=Xi(t)+Vi(t) 0 ≤ t ≤ (T-1), rand1 ~ U(0,1), rand2 ~ U(0,1), i=1,2,…N
Where W, C1, and C2 are constants to be chosen by the user. The algorithm consists of the following steps: Step 1: N sets of tool path are randomly generated from uniform distribution. Each path corresponds to the location of a particle Xi. The initial velocity Vi(0) is null. We obtain the minimal value Xgb(0) by computing the machining error produced by each particle. Step 2: Based on the Eqs. (2) and (3), we compute Xi(t), Vi(t), and fi(t) for each particle. fib(t) is replaced with the smaller between fib(t) and the error. The same update is applied to fgb(t), too. Step 3: The process terminates after T times of iteration; otherwise it repeats Step 2.
3. Test Results 3.1 Factorial experiment The encoding length of the proposed PSO algorithm is increased by adding the three movements along the tangent, normal, and bi-normal directions. As a result, the parameter setting in the algorithm becomes more complex than the original one in previous studies. A systematic approach is needed for choosing the parameter values that lead to a good performance of PSO. We first analyze the influences of different parameters on the machining error with design of experiments techniques. The parameters under investigation include the limits of n, t, b, the division numbers of n, t, b, and the weight value of undercut. The weight value of overcut remains 1 throughout the experiment. The machining parameters are summarized in Table 1. The parameters in PSO are listed in Table 2. The test surface is defined with two cubic Bézier curves with control points shown in Fig. 4. All encoding variables in PSO Table 1 Comparison of measured roughness data Number of cutter locations Length of cutting edge Cutter radius Number of interpolations between cutter locations Table 2 Parameters in the PSO algorithm
Fig. 3 Encoding of a tool path
(3)
X C1 C2 Number of particles Number of iterations
0.5 0.5 0.5 100 100
40 30 mm 2 mm 10
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contain three levels in the experiment, as shown in Table 3. NR, TR, BR indicate the limits of cutter movement in the directions of n, t, b, respectively. Nd, Td, Bd represents the division numbers of the movement in the directions of n, t, b, respectively. The cutter is restricted with these parameters so that it can only move along these directions in pre-defined discrete positions. This helps reduce the encoding length and the corresponding solution space. The weights of Wg also contain three different values in the experiment. The values of NR, TR, BR are determined by the length of the boundary curve L and the number of cutter locations N. The levels of TR and BR were first chosen as L/N, 2L/N, and 3L/N. According to the finding of our previous work, the machining error is more sensitive to the cutter movement in the surface direction (NR) compared to the other directions (TR and BR). To reflect this sensitivity, the levels of NR are reduced to one tenth of the original values, i.e. L/(10N), 2L/(10N), and 3L/(10N). In Table 3, the boundary length of the test surface is 13.18 mm and the number of cutter locations is chosen as 40. Substituting these values into L/(10N) gives 0.0329475, which in turn determines the values of L/N, 2L/N, 3L/N, 2L/(10N), and 3L/(10N). A L2737 orthogonal array is constructed to reduce the number of trial runs in the experiment. Each trial is repeated 12 times with a different combination of factors. The experimental results are evaluated based on the errors induced by undercut and overcut. The ANOVA result and main effect plot are shown in Figs. 5, 6, and 7. The ANOVA result shown in Fig. 5 indicates that the factors most related to the error induced by undercut include NR, Td, Bd, and the weight. Similarly, Fig. 6 shows that the factors most related to the error induced by overcut are NR, TR, Nd, and Td. Note that the weight is not a significant factor, since it remains constant in the experiment. The analysis results also show that the determination coefficient of overcut (78.1%) is higher than that of undercut (18.99%). For this reason, we select the factor levels mainly based on the overcut. The final parameter setting is listed in Table 4.
Table 5 shows the results calculated from all 27 trial runs in the factorial experiment. Each run is repeated 12 times to reduce the effect of randomness in PSO and the minimized error is determined as the average value. The largest error is 122.8778 mm produced in
Fig. 5 The ANOVA result for undercut
Fig. 6 The ANOVA result for overcut Main Effects Plot (data means) for gauge Normal_range
0.0008
30.7, 0, 0.309 42.245, 0, -0.449 52.857, 0, -2 70, 0, 2
P5 P6 P7 P8
Bi-normal_range
0.0004 0.0000 e g u a g f o
0.0008
n a e M
0.0329475 0.0658950 0.0988425 0.329475 0.658950 0.988425 Normal_div ide Tangent_div ide
30.7, 10, -0.309 42.245, 10, 0.449 52.857, 10, 2 70, 10, -2
0.329475 0.658950 0.988425 Bi-normal_div ide
0.0004 0.0000
3
0.0008
P1 P2 P3 P4
Tangent_range
5 penalty
7
1000
1500
3
5
7
3
5
7
0.0004 0.0000
500
Main Effects Plot (data means) for excess Normal_range
110
Tangent_range
Bi-normal_range
90 70 s s e c x e f o
Fig. 4 Test ruled surface and its control points
110
0.0329475 0.0658950 0.0988425 Normal_div ide
0.658950 0.988425 Tangent_div ide
0.329475
0.658950 0.988425 Bi-normal_div ide
90
n a e M
70
3
5 penalty
7
500
1000
1500
110
Table 3 Parameter setting in factorial experiment Factor Level NR 0.0329475 mm 0.065895 mm 0.0988425 mm TR 0.329475 mm 0.65895 mm 0.988425 mm BR 0.329475 mm 0.65895 mm 0.988425 mm Nd 3 5 7 Td 3 5 7 Bd 3 5 7 Wg 500 1000 1500
0.329475
3
5
7
3
5
7
90 70
Fig. 7 Main effect plot of the experiment Table 4 Final parameter setting based on factorial experiment TR BR Nd Td Bd Wg NR 0.03295 0.3295 0.6590 5 7 5 1500
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Run #25. The best result is 61.0058 mm produced in Run #2. In this case, with a proper parameter setting like Run #2, the optimized value can be further reduced more than half. This indicates the necessity of determining best parameter setting by factorial experiment, although the experiment involves extra computational time.
3.2 Test Results and Discussion This section describes the test results of the proposed method and compares them with two previous methods. The purpose is to show the improvement of our approach over the previous ones. The ruled surface shown in Fig. 4 is used in the test. The machining parameters and the parameters in the PSO algorithm remain the same as Tables 1 and 2, respectively. The optimal setting determined by factorial experiment (Table 4) is also used. A similar PSO algorithm is applied to find the optimal tool path with the cutter contacting the boundary curves, i.e. in a restricted solution space. The encoding length of PSO is shorter in this case. Each cutter location is determined by a pair of the curve parameters. The number of particles is reduced to 40 accordingly, while the number is 100 in the enlarged solution space. A heuristic method is also implemented as a greedy approach. The result demonstrates the necessity of global optimization as opposed to local search. The heuristic method was proposed by Tsay et al..7 It first generates a set of ruling lines with equal parametric intervals from the design surface. These ruling lines determine the initial tool orientations. The next step is to calculate the average direction m between the
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surface normal vectors (nA and nB) at the end points of each ruling (see Fig. 8). The two normal vectors are then rotated about the direction m, producing a family of tool axes. The rotation angle α is limited by an upper bound α*. A one-dimensional search is then performed to find the position at which the machining error around the ruling line being considered is minimal. The search process of the improved PSO algorithm is shown in Fig. 9. It can be seen that the optimal solution remains almost identical after 30 iterations. The search process in the restricted solution space is shown in Fig. 10. The similar convergence can be
Fig. 8 Schematic of the heuristic method for local adjustment of tool axis
Table 5 The results generated from all trial runs in the factorial experiment # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
NR 0.032948 0.032948 0.032948 0.032948 0.032948 0.032948 0.032948 0.032948 0.032948 0.065895 0.065895 0.065895 0.065895 0.065895 0.065895 0.065895 0.065895 0.065895 0.098843 0.098843 0.098843 0.098843 0.098843 0.098843 0.098843 0.098843 0.098843
TR 0.329475 0.329475 0.329475 0.65895 0.65895 0.65895 0.988425 0.988425 0.988425 0.329475 0.329475 0.329475 0.65895 0.65895 0.65895 0.988425 0.988425 0.988425 0.329475 0.329475 0.329475 0.65895 0.65895 0.65895 0.988425 0.988425 0.988425
BR 0.329475 0.329475 0.329475 0.65895 0.65895 0.65895 0.988425 0.988425 0.988425 0.65895 0.65895 0.65895 0.988425 0.988425 0.988425 0.329475 0.329475 0.329475 0.988425 0.988425 0.988425 0.329475 0.329475 0.329475 0.65895 0.65895 0.65895
Nd 3 3 3 5 5 5 7 7 7 7 7 7 3 3 3 5 5 5 5 5 5 7 7 7 3 3 3
Td 3 5 7 3 5 7 3 5 7 3 5 7 3 5 7 3 5 7 3 5 7 3 5 7 3 5 7
Bd 3 5 7 3 5 7 3 5 7 5 7 3 5 7 3 5 7 3 7 3 5 7 3 5 7 3 5
Wg 500 1000 1500 500 1000 1500 500 1000 1500 1500 500 1000 1500 500 1000 1500 500 1000 1000 1500 500 1000 1500 500 1000 1500 500
Error 71.78033 61.00575 67.01592 68.043 65.22042 62.66692 84.38525 73.8985 74.53308 83.7525 80.088 76.88942 91.00858 86.90958 87.34275 97.85283 92.16942 90.05525 105.3948 100.6089 107.0556 106.6848 109.5723 106.0693 122.8778 116.0002 115.1598
Fig. 9 The search process of the improved PSO algorithm
Fig. 10 The search process in the restricted solution space
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observed after 30 iterations. The greedy approach applied a onedimensional research at each cutter location. The results of the three methods are summarized in Table 6. The improved PSO algorithm produces the smallest machining error (9.30562 mm). It outperforms the original PSO method (11.8697 mm) because of its enlarged solution space. Both PSO methods perform better than the greedy approach. The result reinforces the finding obtained by the previous study.8 The computation time of the improved PSO is 496.76 seconds compared to 490.52 seconds of the original PSO. The search process in the enlarged solution space is not much lengthened. In addition, the weight value of undercut is varied to compute the optimal tool path in the enlarged solution space. The purpose is to show how to control the machining result by the weight to meet different machining requirements. The machining parameters and the setting in the PSO remain the same as the previous cases. A new test surface shown in Fig. 11 is used. Three different conditions shown as follows are tested.
total machining error (16.6734 mm). These results demonstrate that the new objective function offers a good planning flexibility. The users can choose to minimize the total error, undercut, or overcut by properly adjusting the weight value. A commercial NC package NCVericut is used to simulate the tool paths generated with different weights. It also displays the error distribution on the machined surface. The stock material consisting of three test surfaces is shown in Fig. 12. The machined surfaces generated by minimizing undercut, overcut, and the total error are shown in Figs. 13(a), (b), and (c) respectively. The region of a lighter color indicates occurrence of overcut on the machined stock and the one of a darker color indicates occurrence of undercut. The simulation results of Figs. 13(a) and 13(b) verify the effectiveness of controlling the error distribution by the weight. The dispersed dark areas in Fig. 13(a) may be produced as a result of approximation error in the simulation, so do the light areas in Fig. 13(b). Both overcut and undercut randomly scatter in Fig. 13(c).
Minimize the undercut error Objective function = 1500*sumGouge + sumExcess Minimize the overcut error Objective function = sumGouge + 1500*sumExcess Minimize the total machining error Objective function = sumGouge + sumExcess The results obtained with different weights are shown in Table 7. The error amount induced by undercut is null when its weight is set as 1500. A large amount of overcut (43.6346 mm) is produced at the same time. The similar result can be obtained for overcut. The overcut error can be completely eliminated, but the corresponding undercut becomes excessive (41.8689 mm). The total errors in both cases are much greater than the result generated by minimizing the Table 6 Comparison of the machining errors for three methods Method Improved PSO Original PSO Heuristic Error (mm) 9.30562 11.8697 12.2639 P1 P2 P3 P4
30.7, 0, 1 42.245, 0, -3 52.857, 0, 3 70, 0, -1
P5 P6 P7 P8
Fig. 12 Machining simulation using NCVericut
(a) Undercut
30.7, 10, -2 42.245, 10, 2 52.857, 10, -2 70, 10, 2
(b) Overcut
Fig. 11 Test ruled surface for testing different objective functions Table 7 Test results with different weights Method Min undercut Min overcut Undercut 0.0000 41.8689 Overcut 43.6346 0.0000 Total error 43.6346 41.8689
Min total error 6.5969 10.0765 16.6734
(c) Total machining error Fig. 13 Machined surfaces generated by minimizing
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3.3 Machining test A machining test is conducted to demonstrate the effectiveness of the tool path planning method proposed by this work. A DeckelMaho 5-axis CNC machine tool with a Heidenhain ITNC 530 controller is used. The stock is made of Epoxy as shown in Fig. 14(a). The test consists of two machining operations. A roughing cut is first conducted to remove the majority of the stock material using a φ-4 cylindrical cutter with 30-mm tool length and 20-mm cutting edge length. A new cutter of the same type is used in the subsequent finishing cut. The spindle speed and the feedrate are chosen as 15000 rpm and 3000 mm/min, respectively in both cuts. The finished part shown in Fig. 14(b) contains three machined surfaces produced with different paths: heuristic, original PSO and improved PSO methods respectively. The design surface is the one shown in Fig. 11. The machined surfaces are measured using a ZEISS UMC 850 CMM. Each surface is sampled along the u and v directions with equal parametric intervals. The total machining error is determined as the sum of the error amount measured on each sample point. Table 8 summarizes the measured errors of the three surfaces. The tool path generated from improved PSO gives the smallest error.
(a)
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Both tool paths generated with PSO methods outperform that of the heuristic method. The test result demonstrates the advantage of enlarging the solution space in the PSO process. Notice that the measured errors do not numerically match the simulated ones as shown in Table 6. This is because that the way the errors were measured on the CMM does not exactly correspond to the error estimation in the optimization process. Due to the restriction of the machine resolution, the sampling density of the CMM is lower than that of the optimization. The sampling locations are also different on the surface, as there is a 1-mm safety margin along the surface boundary during the measurement. However, the estimated values match well with the tendency of the actual ones in all three cuts.
4. Conclusions Machining error control is a critical issue in 5-axis flank milling of ruled surfaces. Previous studies showed that the machining error can be reduced by optimization of tool path planning. However, the optimization methods adopted by these studies failed to produce good solution quality, due to the constraint that the cutter must contact with the surface to be machined. This paper proposed an improved PSO method for the tool path planning. It offers several advantages over the PSO method originally developed by the previous study. The constraint of following the rulings has been relaxed by allowing the cutter to deviate the surface along the normal, tangent, and bi-normal directions. The new method encoded the movement steps in the three directions and searched for their optima. The parameter setting was determined by the systematic analysis based on design of experiments. Besides, the objective function was re-defined as a weighted sum of the errors induced by overcut and undercut. By properly adjusting the weight value, the user can choose to minimize the total machining error, undercut, or overcut in the optimization of tool path. This novelty offers a planning flexibility required by various machining requirements in practice. The implementation results verified the effectiveness of the proposed method and demonstrated its practical values. This work enhances the capability of 5-axis flank milling by providing a precise, flexible, and systematic mechanism of the machining error control.
REFERENCES 1. Park, J. W., Lee, J. G. and Jun, C. S., “Near Net-Shape FiveAxis Face Milling of Marine Propellers,” Int. J. Precis. Eng. Manuf., Vol. 10, No. 4, pp. 5-12, 2009. (b) Fig. 14 (a) Stock design and (b) Three machined surfaces Table 8 Measured errors of three surfaces machined with different tool paths Method Improved PSO Original PSO Heuristic Error (mm) 6.1058 7.2637 7.9639
2. Wu, P. H., Li, Y. W. and Chu, C. H., “Optimized tool path generation based on dynamic programming for five-axis flank milling of rule surface,” International Journal of Machine Tools & Manufacture, Vol. 48, No. 11, pp. 1224-1233, 2008. 3. Chu, C. H., Lee, C. T., Tien, K. W. and Ting, C. J., “Efficient Tool Path Planning for 5-Axis Flank Milling of Ruled Surfaces using Ant Colony System Algorithms,” International Journal of
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Production Research, Vol. 49, No. 6, pp. 1557-1574, 2011. 4. Chu, C. H. and Hsieh, H. T., “Generation of Reciprocating Tool Motion in 5-Axis Flank Milling based on Particle Swarm Optimization,” Journal of Intelligent Manufacturing, DOI: 10.1007/s10845-010-0450-z, 2010. 5. Hsieh, H. T. and Chu, C. H., “Particle Swarm Optimization based Tool Path Planning for 5-Axis Flank Milling Accelerated by Graphics Processing Unit,” International Journal of Computer Integrated Manufacturing, Vol. 24, No. 7, pp. 676687, 2011. 6. Kennedy, J. and Eberhart, R. C., “Particle swarm optimization,” Proc. IEEE International Conference on Neural Networks, Vol. 4, pp. 1942-1948, 1995. 7. Tsay, D. M. and Her, M. J., “Accurate 5-Axis Machining of Twisted Ruled Surfaces,” ASME Journal of Manufacturing Science and Engineering, Vol. 123, No. 4, pp. 731-738, 2001.
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