Int J Adv Manuf Technol (2016) 87:2363–2374 DOI 10.1007/s00170-016-8585-5
ORIGINAL ARTICLE
Iterative optimization of tool path planning in 5-axis flank milling of ruled surfaces by integrating sampling techniques Chih-Hsing Chu 1 & Chi-Lung Kuo 1
Received: 28 October 2015 / Accepted: 2 March 2016 / Published online: 28 March 2016 # Springer-Verlag London 2016
Abstract Simultaneous adjustment of all cutter locations in a tool path using meta-heuristic algorithms provides a systematic approach to reducing machining errors in 5-axis flank machining of ruled surfaces. However, these algorithms experienced unsatisfactory quality of optimal solutions and lengthy search time in high-dimensional search space. To solve these problems, we propose an iterative optimization scheme that progressively simplifies solution space by sampling techniques. Akaike information criterion (AIC) is used to screen significant factors from sampling data. An electromagnetismlike mechanism (EM) algorithm searches through a simplified solution space constructed only using these factors. An iteration process consisting of such sampling, screening, and searching steps repeats several times until final optimal solutions are obtained. Test results of representative surfaces validate the effectiveness of the proposed scheme. Both solution quality and search efficiency are improved comparing to those produced by previous studies. This work enhances the practicality of optimization-driven tool path planning in 5-axis flank machining.
Keywords Sampling . Optimization . Five-axis machining . Flank milling . Tool path planning
* Chih-Hsing Chu
[email protected]
1
Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu, Taiwan
1 Introduction Five-axis CNC machining has received much attention in various industries since the late 1990s. This advanced machining operation is particularly suitable in manufacturing complex components such as turbine blades, compressors, molds, and automobile as well as aerospace structure parts. With two rotational degrees of freedom in the cutter motion, 5axis CNC machining offers superior shaping capability and reduces part handling tasks compared to traditional 3-axis machining. However, tool path planning becomes highly complicated because the cutter is likely to collide with objects in the machining environment. Most current CAD/CAM tools provide graphic simulation functions for detecting such unexpected collisions prior to real machining. The planning task inevitably requires intensive human involvements and thus becomes a bottleneck in computer-aided process planning [1]. Five-axis CNC milling is usually categorized into end and flank milling by the location of the cutter removing the work material. The cutting edges near the end of a cutter perform material removal in end milling. The peripheral part of a cutter mainly does the cutting in flank milling. Five-axis flank milling is commonly used to machine-ruled geometries, which can be swept out by moving a line in 3D space. Except for simple geometry such as a cylindrical and conical surface, a cylindrical cutter cannot create an exact ruled surface without producing tool overcut or undercut [2]. The cutter induces substantial errors near twisted surface regions, or in a mathematical term, not locally developable [3]. In practice, the machined surface is considered acceptable if the amount of the accumulated errors is restricted within a given tolerance. Automatic precise control of the machining errors is still lacking in previous tool path planning methods developed in 5-axis flank milling.
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Various local adjustment methods [4–7] have been proposed to reduce the geometrical errors in 5-axis flank milling of ruled surfaces. These methods analyzed tool engagement conditions locally at a cutter location and applied geometric algorithms to separately adjust individual cutter locations, without considering the errors induced by the tool motion between cutter locations. Based on both theoretical and experimental results, our previous study [8] has demonstrated that such a local adjustment approach fails to produce minimal errors accumulated on the entire machined surface. A global approach is needed to adjust individual cutter locations simultaneously, as the tool motion linearly interpolated from consecutive cutter locations might yield a greater amount of errors than that induced at a cutter location. Various optimization schemes based on meta-heuristic algorithms were developed to conduct the adjustment in this global approach. Wu et al. [9] transformed the tool path planning task, originally a geometric problem, into an optimal curve matching problem. They applied an Ant Colony Systems (ACS) algorithm to calculate the optimal matching with the accumulated geometrical errors on the machined surface as the objective function. In this study, the cutter was only allowed to contact the design surface at predefined discrete points on its boundary curves. Hsieh and Chu [10] relaxed this constraint by allowing the cutter to freely move along the surface. A particle swarm optimization (PSO) algorithm was used to search for optimal solutions. The results were improved compared to the solutions produced by [9]. Their later study [11] compared the performance of different particle swarm algorithms including PSO, Advanced Particle Swarm Optimization, and Fully Informed Particle Swarm Optimization on the tool path planning problem. Test results showed that FIPS produces best solutions in most test cases, particularly with a large number of cutter locations. All the search processes based on those algorithms were time-consuming and easily converged to local optima of poor solution quality. The above reviews have shown that simultaneous adjustment of all cutter locations comprising of a tool path through optimization provides a systematic approach to controlling and reducing the geometrical errors in 5-axis flank finishing cut of ruled surfaces. This approach automates tool path planning while ensuring good machining quality on the finished surface. However, the optimization schemes used in previous studies were not effective. The search process is often lengthy and fails to yield good optimal solutions due to high dimensionality of the solution space and nonlinearity of the objective function. The practical value of implementing tool path planning based on the approach is thus limited. In this study, we propose a new optimization scheme to solve the problem. This scheme integrates statistical techniques with meta-heuristic algorithms for accelerating the search process in a simplified solution space. First, cutter locations are generated according to surface twist. Such a cutter
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location distribution is more efficient than that produced by equal parameter intervals used by previous studies. Next, sampling methods are applied to identify significant factors from a large set of optimization variables using Akaike information criterion (AIC). Electromagnetism-like mechanism (EM) algorithms are applied to search through the solution space constructed only using those significant factors. The above steps of sampling, screening, and searching repeat for a number of iterations until optimal solutions are attained. The proposed scheme is tested against representative surfaces to validate its effectiveness. The test results show an improvement on both search efficiency and solution quality in the optimization of tool path planning compared with those produced by previous methods. Detailed discussions highlight the influence of different factors involved at each step and trade-offs among them. This work enhances the practicality of the optimization-driven tool path planning in 5-axis flank milling of ruled surfaces.
2 Background A CNC tool path is defined by a set of cutter locations (CL). A cutter location specifies the cutter center point and the cutter axis in 5-axis CNC machining. The cutter normally follows a linearly interpolated tool motion between consecutive cutter locations. In 5-axis flank finishing cut of a ruled surface, the simplest method of tool path planning is to let the cutter move along the surface rulings. Despite its simplicity, the resultant path may produce excessive machining deviations in twisted surface regions. A ruled surface is constructed by linearly interpolating two boundary curves given in 3D space. The line segments connecting the curves are referred to as the ruling line (or the surface ruling). Except for simple geometries like cylindrical and conical surfaces, machining errors usually arises when a cylindrical cutter contacts a surface ruling. As shown in Fig. 1, the cutter cannot contact tangentially with both boundary curves at the same time [2]. Mathematically speaking, the surface is locally non-developable around the surface ruling. The errors thus induced can be reduced by
cutter R B’(w) A
max overcut = R(1 − cosθ) B
ruling
A’(w)
Fig. 1 Machining errors occur around a surface ruling due to local nondevelopability [2]
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properly adjusting the cutter position and the cutter axis, but never vanish. In practice, the machined quality is considered acceptable as long as the amount of the errors is smaller than a given tolerance. Figure 2 shows how to specify a cutter location from a surface ruling. pA and pB denote the points corresponding to the curve parameters uA and uB on the two boundary curves (A and B), respectively. Offsetting pA and pB along the surface normal with a distance of the cutter radius produces two cutter center points. The cutter axis is determined by connecting the two center points. To reduce machining errors, we allow the cutter center points to deviate from the current positions in the surface normal, tangent, and bi-normal directions, as shown in Fig. 2: cA ¼ pA þ N A nA þ T A tA þ BA bA
Assume that a tool path consists of n cutter locations. The tool path planning is formulated as an optimization problem [12]: X n−2 M ϵ CL → CL Min ð3Þ i iþ1 0 where ϵ is the geometrical errors induced by the cutter motion M from CLi to CLi + 1. A cutter location CLi can be optimally determined by varying a set of parameter values: fuA ; uB ; N A ; N B ; T A ; T B ; BA ; BB g
Previous methods [4–7] adjusting individual cutter locations independently treat the tool path planning as:
ð1Þ Min
cB ¼ pB þ N B nB þ T B tB þ BB bB
ð2Þ
where cA and cB nA and nB NA and NB tA and tB TA and TB bA and bB BA and BB
are new cutter center points are the unit surface normal vector at pA and pB, respectively are the adjustment magnitude in nA and nB, respectively are the unit tangent vector at pA and pB, respectively are the adjustment magnitude in tA and tB, respectively are the unit bi-normal vector at pA and pB, respectively are the adjustment magnitude in bA and bB, respectively
X n−1 0
ϵ ðCLi Þ
ð5Þ
which ignores the cutter motion interpolated between consecutive cutter locations. The optimal solutions thus obtained do not minimize the accumulated errors on the surface machined by a CNC tool path. To obtain an optimal tool path able to yield minimized errors requires adjustment of all cutter locations simultaneously. This involves a search for optimal solutions in a high-dimensional solution space (8n dimensions in this case). Exact estimation of the machining errors is highly difficult and might not be required. The stock height method has been developed to compute the errors approximately [9]. As shown in Fig. 3, the stock height method contains four calculation steps. The design surface is first discretized into a set of sample points. Two straight lines are extended along the positive and negative normal directions at each sampling point. The cutter sweeps across these lines along a given tool path. The cutter motion between cutter locations is approximated by linearly interpolating a finite number of intermediate cutter positions. Those lines are trimmed by intersecting with the peripheral surface of the cutter at each position. The geometrical errors are calculated as the sum of the lengths of the trimmed straight lines. The problem described in Eq. (3) has thus become: Min
Fig. 2 Determine a cutter location from a ruled surface
ð4Þ
X n−2 interpolation * ϵ CL → CL i iþ1 i¼0
ð6Þ
where ϵ* represents the approximate errors estimated by the stock height method. A set of intermediate cutter locations linear interpolated from CLi and CLi + 1 replaces continuous tool motion. This mimics the linear interpolation normally conducted by a CNC controller. The objective function calculates the accumulated errors on the finished surface using the stock height method and CNC linear interpolation.
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1. Generate sample points
2. Extend line segments at each sample point
3. Intersect the lines with the cutter
4. Calculate the accumulated errors
3 Optimization scheme Choosing sample points
As discussed previously, simultaneous adjustment of all cutter locations in a tool path is theoretically a feasible approach to reducing the geometrical errors accumulated on the finished surface. However, the tool path planning problem described in Eq. (6) involves a search for solutions in an extremely highdimension solution space. For instance, a regular CNC tool path in 5-axis flank milling of a ruled surface contains a few dozen cutter locations, sometimes even up to hundreds, producing a several-hundred dimension solution space. Optimization schemes performing a search under this circumstance may suffer from low computational efficiency and/or poor solution quality. A possible solution is to reduce the complexity of the solution space. For those 8n variables involving in Eq. (3) or (6), some of them are more significant than the others and thus have a greater influence on optimal solutions. An approximate solution space of a lower dimension can be constructed only from those significant factors. The search process may be accelerated in such a simplified space but inevitably create approximation errors to optimal solutions. Changing the number of variables constructing the solution space provides trade-offs between the search efficiency in the optimization process and the final solution quality. As shown in Fig. 4, an optimization scheme is proposed to realize this idea. This scheme integrates sampling techniques with a meta-heuristic search algorithm. Optimal solutions are incrementally obtained in iterations.
Sampling
Screening factors N iterations Approximating Solution Space
Searching
Results Fig. 4 The proposed optimization scheme
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Parameter setting in the EM-based algorithms
Parameter
Value
Number of particles
50
Maximum number of iterations
50
Maximum number of iterations in local search Upper bound of parameters (NA, NB, TA, TB, BA, BB) Lower bound of parameters (NA, NB, TA, TB, BA, BB) Upper bound of parameter change in (ua, ub) Lower bound of parameter change in (ua, ub)
1 0.05 mm −0.05 mm 0.025 0.0001
Number of samples
1000
Fig. 5 A schematic of stratified sampling
Each iteration undergoes sampling, screening, constructing solution space, and searching steps. Details of each step in the scheme will be discussed as follows. 3.1 Sampling Factorial experiment is a commonly used technique in statistics to identify significant factors in a model. A full factorial experiment is not suitable in this research due to a large number of variables involved. The proposed scheme adopts sampling to estimate the impact of various variables on the objective function and to identify significant factors from the sampling result. Two sampling methods, simple random sampling (SRS) and stratified sampling (STS), were tested for this purpose. Both generate a finite number of samples from different settings of variables. A sample contains a tool path specified by a set of cutter locations and the machining errors estimated by the stock height method. In statistics, a simple random sample is a subset of individuals (a sample) chosen from a larger set (a population). Each individual is chosen randomly and entirely by chance, meaning each individual has the same probability of being chosen at any stage during the sampling process. Each subset of k individuals has the same probability of being chosen for the sample as any other subset of k individuals. This technique is known as simple random sampling. It is a basic sampling technique and sometimes combines with other techniques in Table 1
Setting of machining parameters
Parameter
Value
No. of sample points in the u direction No. of sample points in the v direction No. of cutter locations (CL) Cutter length Cutter radius No. of interpolations between cutter locations
200 10 40 30 mm 2 mm 10
complex sampling methods. The value of each factor (variable) is randomly generated between predetermined maximum and minimum values. In statistical surveys, when subpopulations within an overall population vary, it is advantageous to sample each subpopulation (stratum) independently. Stratification is the process of dividing members of the population into homogeneous subgroups before sampling. The strata should be mutually exclusive: Every element in the population must be assigned to only one stratum. The strata should also be collectively exhaustive: No population element can be excluded. The idea of stratified sampling is shown in Fig. 5. The range of each factor is divided into three stratums. The number of sampling in each stratum remains the same.
3.2 Screening factors The goal of this step is to identify significant factors that sufficiently explain variation of the sampling results. The Akaike information criterion (AIC) [13] is a measure of the relative quality of a statistical model for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model relative to each of the other models. AIC provides an indication of suitability to fit model selection,
Table 3 Machining errors (in mm) obtained after different numbers of function calls
Surface
Surf1 Surf2 Surf3 Surf4 Surf5 Surf6 Surf7 Surf8
No. of function calls 120 k
480 k
840 k
4.14 8.00 3.71 7.13 4.46 4.79 6.18 17.59
3.67 7.89 3.41 4.61 4.14 4.25 5.02 17.03
3.11 7.72 3.31 4.45 4.05 4.09 4.39 16.99
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Int J Adv Manuf Technol (2016) 87:2363–2374 Comparisons with the results produced by EM
Table 5
Comparison of different distributions of cutting locations
Surface
Error (mm) EM Sampling + EM (120 k)
Difference (%)
Surface
Surf1 Surf2
3.30 8.70
25.52 % −8.06 %
Surf1
5.65
4.14
26.73 %
Surf2 Surf3
10.91 7.68
8.00 3.71
26.67 % 51.69 %
Surf4
10.17
7.13
29.89 %
Surf5
8.39
4.46
46.84 %
Surf6 Surf7
10.48 9.77
4.79 6.18
54.29 % 36.75 %
Surf8
22.86
17.59
23.05 %
4.14 8.00
Surf3 Surf4
3.64 4.58
3.71 7.13
2.05 % 55.57 %
Surf5
4.32
4.46
3.14 %
Surf6 Surf7
4.25 4.87
4.79 6.18
12.66 % 26.80 %
Surf8
17.85
17.59
Surface
Difference (%)
Surf1 Surf2
Error (mm) EM Sampling + EM (480 k) 3.30 3.67 8.70 7.89
Surf3
3.64
3.41
−6.30 %
Surf4 Surf5 Surf6
4.58 4.32 4.25
4.61 4.14 4.25
0.60 % −4.12 % −0.08 %
Surf7
4.87
5.02
2.91 %
Surf8 Surface
17.03
−4.57 % Differe nce (%)
Surf1 Surf2
17.85 Error (mm) EM 3.30 8.70
Sampling + EM (840 k) 3.11 7.72
−5.76 % −11.22 %
Surf3 Surf4 Surf5 Surf6
3.64 4.58 4.32 4.25
3.31 4.45 4.05 4.09
−9.18 −2.93 −6.32 −3.66
Surf7 Surf8
4.87 17.85
4.39 16.99
−9.88 % −4.84 %
−1.43 %
5.15 % −9.26 %
RSS ¼
Xm i¼1
Twist-based distribution
Difference (%)
3.3 Searching
% % % %
especially in the situation with a lot of factors. The general formula of AIC is shown as follows: RSS m
ð7Þ
½yi −f ðxi Þ2
ð8Þ
AIC ¼ 2k þ m ln
Curve parameter-based distribution
where k is the number of factors, m is the number of samples, yi is the real error amount, f(xi) is the error estimated from the sampling results, and RSS is the difference between the real and estimated error amounts. The initial AIC can be obtained from Eq. (7). Then, choose one factor and remove it from the sampling results. A new AIC can be recalculated. Comparing the two values, if the initial value is smaller, keep the corresponding model and choose another factor. Otherwise, use the modified model and select another factor. Finally, the factors remaining in the model are significant factors.
Electromagnetism-like mechanism (EM) is a stochastic optimization method based on electromagnetism [14]. It is a search algorithm based on random population. The original EM algorithm imitates the attraction-repulsion mechanism of the electromagnetism theory. In the algorithm, a solution is a charged particle in search space and its charge relates to the objective function value. Due to the electromagnetic force between two particles, a particle with more charge attracts the other, while the other one repulses the former. The better the objective function value, the higher the magnitude of attraction or repulsion between particles, determined by the particle charge. The original EM algorithm consists of four phases: initialization of the algorithm (Initialize), application of a neighborhood search to find the local (Local), calculation of the total force (CalcF) exerted on each particle, and movement along the force direction (Move). The technical detail of each phase is omitted here (please refer to [14]). In this work, a particle in the EM algorithm represents a tool path consisting of n cutter locations. Each cutter location is defined by eight decision variables (four from each boundary). The previous study [12] has recognized the EM algorithm as the most efficient method among all meta-heuristics in comparison. It is thus used in this research.
3.4 Iteration Iteration in mathematics refers to the process of iterating a function using the output from one iteration as the input to the next. Iterative algorithms are often used to produce approximate numerical solutions to certain mathematical problems, particularly of many unknown variables or uncertain property [15]. As indicated by the arrow in Fig. 4, the proposed scheme is an iterative process. A complete iteration consisting of the steps of sampling, screening, constructing solution space, and searching repeats for a given times
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Table 6 Statistical data of different distributions of cutting locations
Curve parameter-based distribution
Twist-based distribution
R-Squared percentage Number of significant factors
71.67 % 81
82.33 % 43
Average percentage per factor
0.88 %
1.92 %
specified by users. The final solution obtained by the current iteration works as the initial solution in the search process of the next. The significant factors identified may change in each iteration during the process, so does the solution space constructed.
4 Results and discussions 4.1 Comparisons with EM-based search The past study has constructed eight representative ruled surfaces for comparing the efficiency of different tool path planning methods in 5-axis flank machining [11]. These surfaces are formed according to combinations of three different properties: length difference between boundary curves, twist, and surface curvature, all considered to be highly related to cutter interference. As shown in Appendix 1, those surfaces were created by alternatively changing each property from a lower to an upper bound. Each surface is defined by two cubic Bezier curves. Table 1 lists the setting of basic machining parameters in this study. The numbers of sample points control the precision of estimating machining errors using the stock height method. The number of interpolations between cutter locations has a similar effect. The setting is derived from the previous studies [10, 11] for a fair comparison with their results. Table 2 lists the setting of parameters used in the EM algorithm. The previous study [12] has shown that such a parameter setting produces best search result in the tool path planning under investigation. Note that the range of curve parameters is different from that of the other Table 7
Error comparison of different sampling methods Errors (mm)
Number of significant factors
Iteration
SRS
STS
SRS
STS
1 2 3 4 5
10.84 8.83 7.35 6.87 5.86
8.99 7.11 6.22 5.71 4.14
174 132 99 76 65
115 89 77 58 43
parameters. The number of samples is chosen as 1000 in the initial setting. This number controls the amount of data generated for identifying significant factors. Table 3 shows the optimal solutions obtained by the scheme in different numbers of function calls (or estimating the objective function). We control those numbers by different numbers of iteration cycles. Each search was conducted three times to reduce the influence of randomness in the EM-based search step. A function call estimates the amount of machining errors induced by a given tool path. The number of function calls in the table is the sum of the number of sampling and the number of function calls made in the search process. Each sampling requires executing the objective function exactly once. As shown in the table, the error amount decreases with increasing number of function calls, but the change becomes diminishing. Table 4 compares the results with those produced by the EM algorithm-based search in the original solution space. The total number of function calls made by the EM algorithm on this problem is approximately 840,000 [12]. The proposed scheme (sampling + EM) after 120,000 function calls performs worse than the EM algorithm in most surfaces, except Surf2 and Surf8. The results produced after 480,000 function calls are comparable to those of EM for most surfaces, whereas the function calls required are significantly reduced by 42.9 % from 840,000. Indicated in the third table, the proposed scheme outperforms the EM algorithm by producing smaller errors on all test surfaces with the same number of function calls at 840,000. The error reduction amount is 6.72 % in average. We conclude that reducing the dimensionality of solution space by sampling techniques improves the search efficiency and provides satisfactory solution quality at the same time. The results validate the effectiveness of the proposed optimization scheme on accelerating the search process.
5 Discussions 5.1 Distribution of cutter locations Most previous methods generated cutter locations from the boundary curves of the design surface by varying curve parameter at a fixed interval. The corresponding distribution of
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Table 8
Error comparison between AIC and ANOVA
Iteration
Errors (mm)
words, the factors identified from the twist-based distribution are more relevant in determining the machining errors than those of the other distribution.
Number of significant factors
ANOVA
AIC
ANOVA
AIC
1 2
10.76 8.68
8.99 7.11
286 247
115 89
3
7.29
6.22
215
77
4 5
6.82 6.53
5.71 4.14
199 192
58 43
5.2 Screening criteria
cutter locations may not be most effective in reducing machining errors. An intuitive thought is to scatter more cutter locations in the surface regions potentially producing greater errors. Hence, we propose to allocate initial cutter locations by surface twist along the surface boundary, as this property is directly related to the machining errors induced by cutter interference (see Fig. 1). The twist is estimated as the angle θj extended by the two tangent vectors projected along the jth surface ruling. The sum of the extended angle θ along a list of M surface rulings is calculated as the total twist Γ. An average twist interval γ is expressed as γ ¼ n⋅
Γ M
ð9Þ
where n is the number of cutter locations in a CNC tool path. M is much greater than n in this study. A cutter location is generated from the surface when X te j¼t s
θj ≥γ
ð10Þ
where 1 ≤ j ≤ M, and ts and te are the start and end index in the ruling list, respectively. The above equation implies that a cutter location is generated when the twist amount accumulated from the previous cutter location reaches γ. The test results with the two distributions are shown in Table 5. They were produced by the proposed scheme after 120,000 function calls. The performance of the twist-based distribution is notably better than that of the curve parameter-based distribution in every test surface. A possible reason is that computation resources are committed more efficiently in the search process with the twistbased distribution. This can be reflected by the capability of the significant factors that explains data variations in reality. As shown in Table 6, the R-squared value indicates how well data fits a statistical model. The ability to explain variations under the twist-based situation is 13 % higher than that under the curve parameter-based situation, while the number of significant factors is 47 % fewer (decreasing from 81 to 43). The explanatory capability of each significant factor between two circumstances is about 2.2 times different (1.92 vs. 0.88 %). In other
Sampling is a crucial step in the proposed scheme. The goal is to select representative data that helps identify significant factors. The test results produced by SRS and STS show the difference of their effectiveness in this regard. Table 7 lists the machining errors of Surf1 produced by these two sampling methods in each iteration. The performance of STS is superior to that of SRS in this case. A possible reason is that to choose an appropriate range for each parameter in the optimization of tool path planning without any prior knowledge is lack of systematic methods. If the parameter values are randomly generated like SRS, the result does not guarantee to be representative of data or sufficiently close to optima in solution space [16, 17]. STS may avoid such ill conditions with sampling data more evenly covering the solution space. As shown in Table 7, the number of significant factors obtained by STS is lower than that of SRS in every iteration. The solution space constructed by the factors identified by STS presumably contains more relevant information than the one created by SRS. Besides AIC, other criteria can be used in statistical hypothesis testing. For instance, the analysis of variance (AVONA) is commonly used to explain observations. AIC gives a relative estimate of the information lost when a given model is used to represent the process that generates the data. The principle of AIC is stepwise regression. It is advantageous to compare the performance difference of using AIC and ANOVA in screening factors. As shown in Table 8, the machining errors produced by AIC-based screening are smaller than those of ANOVA despite of fewer factors identified. The screening effectiveness of AIC is better than that of ANOVA in this case. ANOVA detects significant factors only once based on the current model. Theoretically, it requires more sampling data to
Fig. 6 Error change in 15 iterations for 8 test surfaces
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improve the screening performance as iterations proceed. However, the sample size was not increased in the tests. Consequently, AIC can identify factors more efficiently than ANOVA with the same amount of information obtained. 5.3 Trade-offs As described previously, the proposed optimization scheme is an iterative procedure. If the scheme is effective, the optimal solutions should keep improving as the iteration continues. Figure 6 shows the test results in 10 iterations. The errors are decreased with increasing iterations for all test surfaces. Significant factors are also decreased (see Tables 7 and 8) as the iteration proceeds. Initial search was conducted in a solution space approximated by a large number of significant factors. The search result may be sufficiently close to optima after several iterations. Search can be conducted in a relatively small region under this circumstance, and thus, fewer factors are needed to construct the solution space. The above results validate the effectiveness of the ideas of incremental modeling and iterative search suggested by this study in the tool path planning under investigation. Both increasing iterations and sample sizes help generate better solutions. Using large sample sizes improves the screening result from sampling data and precision of the solution space thus constructed. The computational time increases accordingly, though. Figures 7 and 8 show the trade-off curve of every test surface with different sampling numbers after five iteration cycles. With the sampling number increased from 1000 to 10,000, the error amount continues to decrease but shows diminishing improvement. This provides a useful reference for selecting proper sampling numbers and solution quality in practice. The test results shown above were generated from C++ programs for validation purpose. All the statistical functions used were included as C++ library in the programs. CAM commercial packages can implement the optimization scheme proposed by this work as a specialized tool path planning module as long as they provide C++ APIs for customized development.
Fig. 7 Trade-offs between sample sizes and solution quality (except Surf6 and Surf8)
problem. SRS and STS methods were applied to identify significant factors from a given number of samples. A simplified solution space was constructed only by those factors. An electromagnetism-like mechanism-based algorithm was developed to search for optimal solutions in the simplified space. One iteration process consists of such sampling, screening factors, approximating solution space, and searching steps. The test results of representative surfaces reveal the following findings (Table 9). Optimal solutions can be obtained after a given number of iterations more efficiently than those produced by previous meta-heuristic search methods, while the solution quality is improved. Distributing initial cutter locations by the surface twist produces better results than that of a fixed parameter interval. The solution space constructed with the factors identified by STS contains more precise information than the one created by SRS. AIC outperforms ANOVA in screening significant factors from sampling data. Finally, trade-off curves between the number of computations and the solution quality help satisfy various requirements in practice. This study has demonstrated the feasibility of integrating sampling techniques with meta-heuristic algorithms and progressive search space reduction on optimization of tool path planning in 5-axis flank machining of ruled surfaces. Future work is to incorporate physical constraints such as cutting forces and avoidance of tool chatter in the optimization scheme.
6 Conclusions Most previous studies related to optimization of the tool path planning in 5-axis flank milling of ruled surfaces use metaheuristic algorithms to search for solutions. To obtain satisfactory solution quality normally requires lengthy computational time and thus decreases the applicability of the tool path planning in practice. It is necessary to reduce the complexity of the solution space involved in the optimization process. This study develops an iterative optimization scheme to solve this
Fig. 8 Trade-offs between sample sizes and solution quality (Surf6 and Surf8)
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Appendix 1
Table 9
Data of test surfaces
Surface
Figure
Length difference
Curvature
Twist
Control points of A and B boundary curves A1(70, 2, -10) A2(81.545, 6, -10) A3(92.157, 0, -10)
Surf1
Small
Small
Small
A4(109.3, 4, -10) B1(70, 5, 0) B2(81.545, 1, 0) B3(92.157, 5, 0) B4(109.3, 1, 0) A1(79.125, 2, -10) A2(84.898, 6, -10) A3(90.204, 0, -10)
Surf2
Large
Small
Small
A4(98.775, 4, -10) B1(70, 5, 0) B2(81.545, 1, 0) B3(92.157, 5, 0) B4(109.3, 1, 0) A1(79.125, 4, -10) A2(84.898, 12, -10) A3(90.204, 0, -10)
Surf3
Small
Large
Small
A4(98.775, 8, -10) B1(79.125, 2, 0) B2(84.898, 10, 0) B3(90.204, 2, 0) B4(98.775, 10, 0) A1(79.125, 4, -10) A2(84.898, 12, -10) A3(90.204, 0, -10)
Surf4
Large
Large
Small
A4(98.775, 8, -10) B1(70, 4, 0) B2(81.545, 20, 0) B3(92.157, 4, 0) B4(109.3, 20, 0)
Int J Adv Manuf Technol (2016) 87:2363–2374 Table 9
2373
(continued)
A1(70, 2, -10) A2(81.545, 6, -10) A3(92.157, 0, -10) Surf5
Small
Small
Large
A4(109.3, 4, -10) B1(73.4, 13.86, 0) B2(81.4, 4.62, 0) B3(92.63, 2.78, 0) B4(105.48, -9.26, 0) A1(79.125, 2, -10) A2(84.898, 6, -10) A3(90.204, 0, -10)
Surf6
Large
Small
Large
A4(98.775, 4, -10) B1(73.4, 13.86, 0) B2(81.4, 4.62, 0) B3(92.63, 2.78, 0) B4(105.48, -9.26, 0) A1(79.125, 4, -10) A2(84.898, 12, -10) A3(90.204, 0, -10)
Surf7
Small
Large
Large
A4(98.775, 8, -10) B1(82.35, -2.027, 0) B2(83.35, 7.79, 0) B3(91.94, 3.51, 0) B4(95.36, 14.73, 0) A1(79.125, 4, -10) A2(84.898, 12, -10) A3(90.204, 0, -10)
Surf8
Large
Large
Large
A4(98.775, 8, -10) B1(76.44, -4.05, 0) B2(78.44, 15.58, 0) B3(95.63, 7.03, 0) B4(102.48, 29.45, 0)
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