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Electromagnetism-like algorithms for optimized tool path planning in 5-axis flank machining
Accepted Manuscript Electromagnetism-like Algorithms for Optimised Tool Path Planning in 5-Axis Flank Machining Chi-Lung Kuo, Chih-Hsing Chu, Ying Li, Xinyu Li, Liang Gao PII: DOI: Reference:
S0360-8352(14)00411-2 http://dx.doi.org/10.1016/j.cie.2014.11.023 CAIE 3876
To appear in:
Computers & Industrial Engineering
Please cite this article as: Kuo, C-L., Chu, C-H., Li, Y., Li, X., Gao, L., Electromagnetism-like Algorithms for Optimised Tool Path Planning in 5-Axis Flank Machining, Computers & Industrial Engineering (2014), doi: http:// dx.doi.org/10.1016/j.cie.2014.11.023
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Electromagnetism-like Algorithms for Optimised Tool Path Planning in 5-Axis Flank Machining Chi-Lung Kuo, Chih-Hsing Chu* Department of Industrial Engineering and Engineering Management National Tsing-Hua University, Hsinchu, Taiwan Ying Li, Xinyu Li, Liang Gao State Key Lab of Digital Manufacturing Equipment & Technology Huazhong University of Science & Technology, Wuhan, China
*[email protected] Abstract Optimisation of tool path planning using metaheuristic algorithms such as ant colony systems (ACS) and particle swarm optimisation (PSO) provides a feasible approach to reduce geometrical machining errors in 5-axis flank machining of ruled surfaces. The optimal solutions of these algorithms exhibit an unsatisfactory quality in a high-dimensional search space. In this study, various algorithms derived from the electromagnetism-like mechanism (EM) were applied. The test results of representative surfaces showed that all EM-based methods yield more effective optimal solutions than does PSO, despite a longer search time. A new EM-MSS (electromagnetism-like mechanism with move solution screening) algorithm produces the most favourable results by ensuring the continuous improvement of new searches. Incorporating an SPSA (simultaneous perturbation stochastic approximation) technique further improves the search results with effective initial solutions. This work enhances the practical values of tool path planning by providing a satisfactory machining quality.
Keywords: electromagnetism-like mechanism algorithm, 5-axis machining, flank milling, tool path planning 1
1.
Introduction 5-axis CNC machining has been commonly used in manufacturing of complex
geometries in automobile, aerospace, energy, and mold industries. This advanced machining operation provides better shaping capability and higher productivity compared to traditional 3-axis machining. Tool path planning becomes a challenging task in most 5-axis machining operations due to additional degrees of freedom in the tool motion. A primary goal in planning the tool path is to avoid tool collision. In 5-axis flank machining, the flank part of a cutter is used to remove stock materials and finish the design surface. To completely eliminate geometrical deviations when creating complex shapes with a cylindrical cutter is highly difficult, if not impossible. The cutter will induce substantial deviations near twisted surface regions, or in a mathematical term, not locally developable [1]. Precise control of the machining deviations is lack of solutions. The geometrical deviations in 5-axis flank finishing of ruled surfaces can be reduced by adjusting all cutter locations of a tool path simultaneously. Previous studies [2-4] have proposed various optimization schemes based on meta-heuristic algorithms to conduct the adjustment. Chu et al. [2] transformed the tool path planning of a ruled surface into a 2D curve matching problem. An Ant Colony Systems (ACS) algorithm was applied to calculate an optimal matching with the accumulated geometrical deviations on the machined surface as the objective function. Their method restricted that the cutter contacts the surface at pre-defined discrete points on its boundary curves. Hsieh and Chu [3] relaxed this constraint by allowing the cutter freely contact the surface and applied the particle swarm optimization (PSO) algorithm to search for optimal solutions. They adopted GPU computing techniques to accelerate the search process. This approach still suffered from unsatisfactory quality of the search results, as the cutter had to make contact with the boundary curves of the surface to be machined. To overcome this problem, their later study [4] proposed a new encoding scheme of 2
cutter locations in the tool path planning driven by PSO. The cutter could deviate from the surface along the normal, tangent, and bi-normal directions at the end points of the surface rulings. Such additional freedoms in the tool motion enlarge the solution space in the optimization, thus yielding better search results than those of previous studies. Hsieh and Chu 5] compared the performance of various particle swarm algorithms such PSO, Advanced Particle Swarm Optimization (APSO), and Fully Informed Particle Swarm Optimization (FIPS) algorithms on the tool path planning problem. They constructed a set of representative test surfaces by systematic variations of three surface properties. The test results showed that FIPS perform best in all trials with a large number of cutter locations, but the improvement was not significant when the number is small. Besides, the search process easily converged to local optima and results in poor solutions. The above reviews have shown that meta-heuristic algorithms provide a systematic approach to controlling and reducing the geometrical deviations in 5-axis flank finishing cut of ruled surfaces through optimization of tool path planning. However, the optimization schemes employed by previous studies including ACS and various PSO algorithms failed to produce good search results due to high dimensionality of the solution space. This study applied optimization methods based on the electromagnetism-like mechanism (EM) to further enhance the solution quality. Two new algorithms were developed from the original EM: a simplified electromagnetism-like mechanism (SEM) algorithm and an electromagnetism-like mechanism with move solution screening (EM-MSS) algorithm. The SEM algorithm simplifies calculation of the interaction forces between particles. The EM-MSS algorithm guarantees continuous improvement of search results by adding a solution screening step. These algorithms were used to optimize the tool path planning with a set of representative test surfaces. The test results were compared with those produced by PSO on computational time and solution quality. In addition, the EM-MSS algorithm incorporates a simultaneous perturbation stochastic approximation (SPSA) procedure to start search with good initial 3
solutions. Conclusion remarks were given to summarize the effectiveness of the EM-based methods on reducing the geometrical deviations in 5-axis flank finishing cut of ruled surfaces.
2.
Tool Path Planning in 5-axis Flank Machining Tool Path Encoding A CNC tool path is defined by a set of cutter locations (CL). 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 for tool path planning is to allow the cutter to move along the surface rulings. The resultant path produces excessive machining deviations in twisted surface regions, though.
Figure 1 Defining a cutter location with respect to a ruled surface
A cutter location is specified with respect to a ruled surface as shown in Figure 1. The cutter contact points pA and pB are first generated from the two boundary curves (A and B) and each corresponds to the parameter values uA and uB, respectively. The cutter center point is then determined by offsetting the contact point along the surface normal with a distance of the cutter radius. The cutter orientation lies in the direction connecting the two cutter center 4
points. In this study, we allow the cutter center point to deviate from the cutter contact point in the surface normal, tangent, and bi-normal directions. They are expressed as: (1)
where: cA and cB are cutter center points; pA and pB are cutter contact points corresponding to curve parameter values uA and uB, respectively; nA and nB are the unit surface normal vector at pA and pB, respectively; NA and NB are the magnitude of deviation in nA and nB, respectively; tA and tB are the unit tangent vector at pA and pB, respectively; TA and TB are the magnitude of deviation in tA and tB, respectively; bA and bB are the unit bi-normal vector at pA and pB, respectively; BA and BB are the magnitude of deviation in bA and bB, respectively. When the objective function is to minimize the accumulated deviations on the machined surface, optimization of the tool path planning is expressed as:
(2) Subject to where
0 i (N 1), 0 uA 1, 0 uB 1;
is the geometrical deviation induced by the cutter motion
A tool path consists of N cutter locations. A cutter location
from
to
.
can be adjusted in 3D space
by varying a set of parameter values {uA, uB, VA, VB, TA, TB, BA, BB}. To obtain an optimal tool path yielding minimized geometric deviations requires adjustment of all cutter locations simultaneously. This involves search for optimal solutions in a high-dimensional solution space (8N dimensions in this case). 5
1. Generate sample points
2. Produce line segments at each sample point
3. Intersect the lines with the cutter
4. Calculate the accumulated deviations
Figure 2 Estimating machining deviations by the stock height method [6]
Estimation of Geometrical Machining Deviations In this study, the objective function applied in the optimization process is the accumulated geometrical deviations on the finished surface. Exact estimation of the geometrical deviations is highly difficult and might not be required. The stock height method developed in the previous study [6] estimates the deviations approximately. As shown in Figure 2, the estimation process contains four steps. First, the design surface is discretized into 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 tool swept surface is approximated by interpolating a finite number of tool positions between consecutive cutter locations. Next, those lines are trimmed by intersecting with the peripheral surface of the cutter at each position. The geometrical deviations are calculated as 6
the sum of the lengths of the trimmed straight lines. The sampling density in the method controls the estimation precision of the deviations and the computational time required in the optimization process. The optimization problem described in Eq.(2) has thus become:
(3) where
is the approximate deviations estimated by the stock height method. A set of
intermediate cutter locations linear interpolated from
and
replaces the
continuous tool motion. This mimics the linear interpolation normally conducted by a CNC controller. Thus, the objective function is the accumulated machining deviations on the finished surface, which is estimated by the stock height method using CNC linear interpolation. The optimization variables in Eq.(3) are {uiA, uiB, NiA, NiB, TiA, TiB, BiA, BiB}, where 1 i N and N is the number of cutter locations in a tool path.
3.
Optimization Schemes Electromagnetism-like Mechanism (EM) Electromagnetism-like mechanism (EM) is a stochastic optimization method based on
electromagnetism [7]. It is a population-based random search algorithm similar to GA. 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 and 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 [8]: initialization of the algorithm (Initialize), application of neighborhood search to exploit the local (Local), calculation of the total force (CalcF) exerted on each particle, and movement along the force direction (Move). 7
A general framework is described as follows. Note that a particle in the EM algorithms represents a tool path consisting of N cutter locations. Each cutter location is defined by 8 decision variables (4 from each boundary curve). Thus a particle corresponds to an 8xN matrix and is encoded as a one dimensional matrix during the optimization process. In EM algorithm, the particle vector is defined as
, superscript of i represent the particle index and
subscript of k represent the particle dimension.
EM Framework EM(m, MAXITER, LSITER, δ) m: particle number MAXITER: maximum number of iterations LSITER: maximum number of local search iterations δ: local search parameter，δ∈[0,1] 1: 2:
Initialize () iteration
3: 4:
while iteration < MAXITER do Local (LSITER, δ)
5: 6: 7:
F CalcF () Move (F) iteration iteration + 1
8:
1
end while
Initialize The procedure Initialize randomly samples m points in an n-dimensional search space. Assume that the coordinates of a sampling point follow a uniform distribution U(0, 1) between given upper and lower bounds, uk and lk. Each sampling point corresponds to a particle in the EM algorithm. Sampling points follow a uniform distribution between given upper and lower bounds, uk and lk. The particle with the best value is stored as
xbest .
The
particle dimensions equal to 8 times number of cutter locations, i.e. n = 8 N.
8
Initialize () 1: 2: 3: 4:
for i = 1 to m do for k = 1 to n do U (0,1) xki
5: 6: 7: 8:
lk
(uk lk )
end for Calculate
f ( xi )
end for xbest
arg min{ f ( xi ), i}
Local Most random search algorithms reply on local search to attain good optimal solutions. The procedure Local is applied to improve the quality of the initial solutions obtained by Initialize. It is a simple random line search determined by two parameters – LSITER and δ, representing the number of iterations and a multiplier for the neighborhood search, respectively.
Local (LSITER, δ) ___________________________________________________________________________ 1: counter 1 2: Length (max k {uk lk }) 3: 4: 5: 6: 7: 8:
for i = 1 to m do for k = 1 to n do U (0,1)
1
while counter < LSITER do xi
y
U (0,1)
2
9: 10:
if
11: 12:
else
13: 14:
end if if f ( y)
1
0.5
yk
yk
then
yk
2 ( Length)
yk
2 ( Length)
f ( xi )
then 9
15:
xi
16:
counter
17: 18: 19: 20: 21: 22:
y
LSITER 1
end if counter counter 1 end while
end for end for xbest
arg min{ f ( xi ), i}
CalcF The procedure CalcF integrates local and global search results by calculating the electromagnetic-like force between two particles. The force exerted on a particle by the other is inversely proportional to the distance between them and directly proportional to the product of their charges (see Eq.(4)). Note that the charge carried by particles can change in the EM algorithm, but this change is not possible in reality. The charge of particle i is determined as:
q
i
exp
n
f xi m k 1
f xk
f xbest f xbest
, i.
(4)
This equation indicates that higher charges produce better objective function values. The number of dimensions n multiplied in the equation avoids overflow in the exponential function when the denominator becomes too large. Experimental results obtained by the previous study [9] have shown that determining the particle charge by Eq. (4) normally gives satisfactory search results. The total force Fi exerted on particle i is computed as:
(5)
The charge calculated by Eq. (4) does not carry a positive or negative sign. The direction of the electromagnetic-like force is determined by Eq. (5). As shown in the procedure CalcF, a 10
particle with a better objective function value attracts the other particle (lines 7 to 8); the one with a worse value repels the other (lines 9 to 10). Since xbest has the best objective function value, it attracts all other points in the population.
CalcF(): F ___________________________________________________________________________ 1: 2:
for i = 1 to m do qi
f
exp
x
m k 1 f
3: 4: 5: 6: 7:
Fi
i
f
n x
k
x f
best
x
best
0
end for for i = 1 to m do for j = 1 to m do if f ( x j ) f ( xi ) then
8:
{attraction}
9:
else
10: 11: 12: 13:
{repulsion} end if end for end for
Move In this procedure, particle i will be moved in the direction of Fi by a random step length λgiven by: x
i
x
i
Fi ( RNG ) || F i ||
i 1, 2,
m
(6)
We assume that λ obeys a uniform distribution between 0 and 1. Distributions of other types can also be used in calculating the step length. A step length randomly determined guarantees a nonzero probability of moving toward unvisited search space. 11
Move (F) ___________________________________________________________________________ 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13:
for i = 1 to m do if i ≠ best then U (0,1) F
Fi
i
i || F ||
for k = 1 to n do if Fki 0 then xki
xki
Fki (uk
xki )
xki
Fki ( xki
lk )
else xki
end if end for end if end for
Simplified Electromagnetism-like Algorithm (SEM) In the EM algorithm, estimation of the particle charge using Eq.(4) and the resultant force using Eq.(5) involves a large number of algebraic operations. The optimization process may require lengthy computational time when solving complex problems. A greater value of the denominator in Eq. (5) produces smaller forces between particles. As a result, the step length in the procedure Move tends to be too short and the search result is thus likely to be restricted within a small neighborhood in the solution space. To avoid this problem, a simplified formula is proposed to calculate the particle charge: (7) The force is written as:
(8) 12
The algorithm adopting this new procedure is named as Simplified Electromagnetism-like Algorithm (SEM).
Electromagnetism-like Algorithm with Move Solution Screening (EM-MSS) SEM does not guarantee that the solution generated by the Move procedure is better than the original one. This may lengthen the computational time during the optimization process and deteriorate the solution quality. A screening mechanism is proposed to assure generation of better solutions after the Move procedure. This procedure repeats until the new solution is of a better objective function value than the previous one. The EM algorithm using this screening mechanism is referred to as Electromagnetism-like Algorithm with Move Solution Screening (EM-MSS).
4.
Test Results The algorithms described above have been applied to generate optimized tool paths with
various test surfaces. The previous study [5] proposed a set of representative ruled surfaces for comparing the effectiveness of different tool path planning methods. These surfaces were constructed by varying three intrinsic properties: curvature (C), twist (T), and the length difference (L) between two boundary curves defining a ruled surface. The surface curvature refers to the maximum normal curvature of a C2 surface and measures the maximum degree of bending in a surface point-wise. Twist indicates the deviation between the tangent directions at the end points of a ruling connecting the boundary curves. It is usually estimated as the angle extended by the projections of the directions along the ruling onto a plane. In this study, twist refers to as the sum of the angle for all surface rulings.
13
Surf1: small C, small T, small L
Surf2: small C, small T, large L
Surf3: large C, small T, small L
Surf4: large C, small T, large L
Surf5: small C, large T, small L
Surf6: small C, large T, large L
Surf7: large C, large T, small L
Surf8: large C, large T, large L
C: curvature; T: twist; L: length difference Figure 3 Eight different surfaces generated by varying three intrinsic properties [5]
Eight types of ruled surfaces can be constructed by varying those three properties as follows. Connecting two boundary curves of large normal curvatures produces a surface of a large curvature. Rotating a boundary curve with respect to the other yields a twisted surface. Placing the control points of a free-form curve in space controls its arc length. The surfaces thus generated are shown in Figure 3 and their control points can be referred to [5]. Table 1 Setting of cutting parameter # of sample points in the u direction
200 14
# of sample points in the v direction
10
# of cutter locations (CL)
40
cutter length cutter radius # of interpolations between cutter locations
30 mm 2 mm 10
Table 1 lists the setting of cutting parameters in the subsequent tests. The numbers of sample points determine the precision of estimating the objective function, i.e. the geometrical deviations estimated using the stock height method. The number of interpolation between consecutive cutter locations also relates to the precision. The values of those parameters were chosen as recommended by the previous studies [3-5]. PSO-based tool path planning was also tested for comparison with EM-based approaches. Our previous studies [3, 4] described the details of the path planning and the settings in the PSO algorithm (thus omitted in this paper). The parameter setting in the EM-based algorithms is shown in Table 2. The parameter values are chosen as the ones suggested by [7]. Our previous work [9] has shown that the EM algorithms have a satisfactory performance under such a parameter setting. Each algorithm was conducted three times to reduce the influence of randomness. An initial tool path was generated by allowing the cutter to contact discrete surface rulings determined by varying the curve parameter u with a constant increment of 0.025, as the number of cutter locations is 40 in the tests. The PSO algorithm adopts the same initial solution for fair comparison. ACS was applied to solve the similar tool path planning problem in the previous work [10]. A major limitation of using the ACS algorithms is that the cutter is only allowed to move among pre-defined discrete points on the boundary curves of the surface to be machined. This limitation results in a very restricted solution space and consequently suboptimal solutions. On the other hand, to search for optimal solutions among pre-defined cutter locations requires less computation time. A fair comparison with EM or PSO is not feasible and thus not included in this paper. 15
Table 2 Parameter setting in the EM-based algorithms Parameter
Notation
Value
# of particles
m
50
maximum number of iterations
MAXITER
50
maximum number of iterations in local search upper bound of variable change in one iteration lower bound of variable change in one iteration upper bound of variable setting lower bound of variable setting
LSITER uk lk umax lmax
1 0.05 -0.05 1 -1
Table 3 Test results of different algorithms Surface Surf1 Surf2 Surf3 Surf4 Surf5 Surf6 Surf7 Surf8
PSO Best Ave. 8.534 12.119 15.812 20.562 7.315 11.691 7.601 12.321 25.753 32.449 27.969 30.953 21.364 23.451 25.352 28.087
EM Best Ave. 6.334 7.845 9.456 11.170 9.434 11.922 11.879 15.046 10.101 12.652 20.049 22.124 13.424 14.231 24.435 26.657
SEM Best Ave. 6.486 7.737 9.196 11.155 9.325 11.876 11.235 14.757 11.437 12.121 20.135 22.121 12.676 14.206 22.365 25.829
EM-MSS Best Ave. 4.642 6.323 9.275 10.234 4.678 7.244 6.539 8.431 7.919 11.257 4.831 5.267 7.602 12.367 22.288 24.125
Table 3 lists the results, the best and average values in three runs, generated by PSO, EM, SEM, and EM-MSS algorithms for eight test surfaces. Figure 3 shows that the search processes produced by all gradually converge to the final solutions. Figures 4 and 5 compare the test results in bar charts. The EM-MSS algorithm yields the best solutions in all test cases. The EM and SEM algorithms perform better than PSO in most tests except Surf3 and Surf4. Large surface twist normally results in an excessive amount of the machining deviations. The test results (Surfs 5, 6, 7, and 8) indicate that the solution quality of the PSO algorithm is worse than those of the EM-based algorithms in this case.
16
Figure 3 The convergence processes of all algorithms
Figure 4 Comparison of optimal solutions (best value) in bar charts
17
Figure 5 Comparison of optimal solutions (average value) in bar charts EM-MSS yields better solutions than EM and SEM in all test surfaces. We thus conclude that the screening mechanism works well by assuring good search results every time after applying the Move procedure. The solution quality of SEM is slightly better than that of EM in all tests. Improvement of the search results is fairly limited using the new force calculation Eq.(5). However, SEM indeed requires shorter computational time in the optimization process with this simplified formula (see Table 4). The number of evaluating the objective function in EM is four-time greater than that of PSO, which shows that the effectiveness of the original EM is not satisfactory. The optimal solutions obtained by those advanced EM methods perform significantly better than those of PSO. SME and EM-MSS show in average 24.9% and 47.2% improvement respectively based on the eight test surfaces (see Table 3). In addition, the PSO algorithm simply stops or gets trapped in local optima in the tests. The optimal solutions obtained by PSO, if not with advanced search mechanisms, cannot be further improved even with a greater number of evaluating the objective function.
18
Table 4 The computational time of different algorithms (units: second) Surface Surf1 Surf2 Surf3 Surf4 Surf5 Surf6 Surf7 Surf8
PSO Best 9634 9782 9843 9756 10156 10537 10342 10432
Ave. 10123 10243 10356 10342 10643 10789 10974 10622
EM Best 33231 32796 34288 34531 33998 35413 32168 34468
Ave. 34592 35693 37119 37654 38852 38752 35575 35857
SEM Best Ave. 29078 29806 31044 32018 33462 35961 32344 32845 31137 31732 30560 31440 31727 34005 33038 34677
EM-MSS Best Ave. 36257 37369 42396 45056 44199 47264 41056 42965 38921 42337 38200 41463 42695 44223 41095 44747
Initial solutions play an important role in the global optimization based on meta-heuristic algorithms such as EM. Starting with good initial solutions may accelerate the convergence of the search process and/or improve the final solution quality. Random number generation is commonly used in most optimization problems without prior knowledge. The solutions thus generated are not guaranteed to good ones. In this study, the simultaneous perturbation stochastic approximation method was applied to produce an initial set of particles with satisfactory solution quality. The focus was to demonstrate to what extent such an initial condition would influence optimal solutions. It is also insightful to examine how the ratio of good particles in the initial set affects the search results. Simultaneous perturbation stochastic approximation (SPSA) is an optimization scheme that uses only objective function measurements, instead of the gradient of the objective function, in finding optimal solutions [11]. SPSA is especially efficient in high-dimensional problems in terms of generating good solutions with a relatively small number of estimations of the objective function [12]. In this study, the optimization variables NA, NB, TA, TB, BA, and BB were randomly adjusted using SPSA. The adjustment process terminates when the objective function value corresponding to those variables is better than the value generated by all null values, i.e. the cutter follows the surface rulings. In other words, here a “good” initial solution means that the resultant tool path yields smaller geometrical deviations than that of the path following the surface rulings. 19
Table 5 Search results with good initial solutions generated by SPSA EM-MSS Best 4.642 9.275 4.678 6.539 7.919 4.831 7.602 22.288
Surface Surf1 Surf2 Surf3 Surf4 Surf5 Surf6 Surf7 Surf8
EM-MSS+SPSA Best 3.295 8.702 3.638 4.582 4.321 4.253 4.866 17.852
Table 5 shows the test results with initial solutions generated by SPSA. The best value of the objective function was improved in all surfaces, thus validating the necessity of good initial particles in the EM algorithms. This conclusion demonstrates the effectiveness of the SPSA procedure. Table 6 shows the test results for different numbers of good initial solutions produced by the same method. Generally speaking, the objective function value decreases with the increasing number of good initial solutions. Figure 6 shows evidence to support this statement. As shown in Table 7, the computational time of the optimization process increases correspondingly, as additional computations are required by the SPSA procedure. All EM-based algorithms involve a greater number of estimating the objective function than the PSO algorithm does (see Table 8). As anticipated, they yield better optimal solutions with more search attempts. The EM-MSS algorithm proposed by this work performs better than the EM and SEM algorithms in the tool path planning problem, with a small amount of additional computations. The improvement becomes even more significant when searching with good initial solutions. Table 6 Search results for different numbers of good initial solutions generated by SPSA Surface Surf1 Surf2 Surf3 Surf4
# of good initial solutions 5
10
15
20
25
30
35
40
45
50
3.780 14.22 4.691 5.388
3.898 13.51 4.178 5.279
3.295 11.93 4.058 5.226
3.579 11.78 4.094 5.174
3.674 11.55 4.034 5.122
3.687 11.31 4.009 5.119
3.598 10.16 3.877 5.074
3.457 12.24 3.805 4.660
3.244 9.946 3.685 4.719
3.799 8.702 3.638 4.582 20
Surf5 Surf6 Surf7 Surf8
6.708 7.286 7.259 20.07
5.181 6.234 6.871 17.85
5.355 5.682 6.714 19.34
5.302 5.528 6.260 18.61
5.158 5.226 6.284 19.06
5.123 5.376 6.493 18.78
5.000 5.819 5.800 18.90
4.772 4.871 5.625 18.93
4.562 4.814 5.649 18.59
4.321 4.253 4.866 18.21
Table 7 The computational time with different numbers of good initial solutions (units: second) Surface Surf1 Surf2 Surf3 Surf4 Surf5 Surf6 Surf7 Surf8
# of good initial solutions 5
10
15
20
25
30
35
40
45
50
36348 45055 46828 40430 38921 38200 44659 46291
36257 46273 44951 41056 39665 39300 46256 44596
38397 45721 46438 42143 41677 40700 43432 44347
37323 45234 45236 43963 41802 40943 43695 46935
36033 44527 44199 42857 42984 41613 42695 46411
37413 46238 47771 42213 43896 41011 44998 44449
37980 46190 48072 43013 43536 41372 43896 44792
38220 42396 49345 43338 43235 42047 43586 43609
36840 44987 49932 44213 44620 42637 43320 41095
38880 43938 49872 46424 43032 46804 45689 44942
Figure 6 Optimal solutions vary with the number of initial solutions generated by SPSA Table 8 The number of estimating the objective function for all test algorithms Surface
PSO
EM
SEM
EM-MSS
EM-MSS+SPSA
Surf1 Surf2 Surf3
146240 157420 178210
822500 822500 822500
822500 822500 822500
830124 824521 826573
860823 827485 831700 21
Surf4 Surf5 Surf6 Surf7 Surf8 5.
172240 179810 186840 195210 201450
822500 822500 822500 822500 822500
822500 822500 822500 822500 822500
827549 829582 823565 823070 823507
838385 847224 825240 824043 825163
Conclusion Previous studies have shown that optimized tool path planning driven by meta-heuristic
algorithms provides a systematic approach to reducing the geometrical deviations in 5-axis flank finishing cut of ruled surfaces. For example, PSO was applied to simultaneously adjust cutter locations comprising of a tool path so as to minimize the accumulated deviations on the finished surface. The search results thus generated seem to quickly converge to local optima and are difficult to be further improved. It is advantageous to study whether or not other algorithms can enhance the optimal solutions obtained by PSO. Therefore, this study developed various EM-based algorithms and applied them to the tool path planning problem. The SEM algorithm simplifies calculation of the particle charge in EM to increase the step length in determining new search result. The EM-MSS algorithm includes a screening step to guarantee continuous improvement in the Move procedure. Test results with representative ruled surfaces have shown that the EM algorithms normally yield smaller optimal solutions than those of PSO, but require a longer computational time. The improvement of using simplified particle charge was not significant. The EM-MSS algorithm produced best solutions in all test surfaces, with a small increase in search times. In addition, a SPSA procedure was proposed to ensure that the initial particles in the EM algorithms are of satisfactory solution quality. This procedure significantly improves the search results over the initial solutions randomly generated. The degree of the improvement increases with the number of good initial solutions. This study has demonstrated the feasibility of the global optimization algorithm EM on tool path planning in 5-axis flank machining of ruled surfaces. Although the EM algorithms 22
are effective in finding better solutions, they require a greater number of search trials than PSO. A potential enhancement is to reduce the range of local search in those algorithms. Sampling techniques can be incorporated to reduce the number of optimization variables at various iteration stages during the search process. Our future work will be focused on this area.
23
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Chu, C. H. and Chen, J. T., “Tool path planning for 5-axis flank milling with developable surface approximation,” International Journal of Advanced Manufacturing Technology, Vol. 29, No. 7-8, pp. 707-713, 2006.
[2]
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 Production Research, Vol. 49, No. 6, pp. 1557-1574, 2011.
[3]
Hsieh, H. T. and Chu C. H., “PSO-based tool path planning for 5-axis flank milling accelerated by GPU,” International Journal of Computer Integrated Manufacturing, Vol. 24, pp. 676-687, 2010.
[4]
Hsieh, H. T. and Chu, C. H., “Optimization of tool path planning in 5-axis flank milling of ruled surfaces with improved PSO,” International Journal of Precision Engineering and Manufacturing, Vol. 13, No. 1, pp. 77-84, 2012.
[5]
Hsieh, H. T. and Chu, C. H., “Improving tool path planning in 5-axis flank milling of ruled surfaces using advanced PSO algorithms,” Robotics & Computer Integrated Manufacturing, Vol. 29, No. 3, pp. 3-11, 2013.
[6]
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 and Manufacture, Vol. 48, No. 11, pp.1224-1233, 2008.
[7]
Birbil, Ş. İ. and Fang, S. C., “An electromagnetism-like mechanism for global optimization,” Journal of Global Optimization, Vol. 25, No. 3, pp. 263-282, 2003.
[8]
Zhang, C., Li, X., Gao, L., and Wu, Q., “An improved electromagnetism-like mechanism algorithm for constrained optimization,” Expert Systems with Applications, Vol. 40, No. 14, pp. 5621-5634, 2013. [9] Wu, Q., Li, X., Gao, L. and Li, Y. “A simplified electromagnetism-like mechanism algorithm for tool path planning in 5-axis flank milling,” IEEE 17th International Conference on Computer Supported Cooperative Work in Design, Whistler, Canada, 2013. [10] Chu, C.H., Tien, K.W., Lee, C.T. and Ting, C.J. “Efficient tool path planning in 5-axis milling of ruled surfaces using ant colony system algorithms,” International Journal of Production Research, Vol. 49, No. 6, pp. 1557–1574, 2011.
[11] Spall, J. C., “Simultaneous Perturbation Stochastic Approximation,” IEEE Transactions on Automatic Control, Vol. 37, pp. 332-341, 1992. [12] Cheng, S., Qin, T. Y., and Jing, L. B., “Multi-objective optimization of accommodating distributed generation considering power loss, power quality, and system stability,” Journal of Industrial and Production Engineering, Vol. 31, No.3, pp. 1-9, 2014.
1
Enhances the practical values of tool path planning in 5-axis flank milling.
New algorithms derived from the electromagnetism-like (EM) mechanism.
Those EM-based algorithms yield more effective optimal solutions than does PSO.
Incorporate an SPSA technique to improve the search results.
S0360-8352(14)00411-2 http://dx.doi.org/10.1016/j.cie.2014.11.023 CAIE 3876
To appear in:
Computers & Industrial Engineering
Please cite this article as: Kuo, C-L., Chu, C-H., Li, Y., Li, X., Gao, L., Electromagnetism-like Algorithms for Optimised Tool Path Planning in 5-Axis Flank Machining, Computers & Industrial Engineering (2014), doi: http:// dx.doi.org/10.1016/j.cie.2014.11.023
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Electromagnetism-like Algorithms for Optimised Tool Path Planning in 5-Axis Flank Machining Chi-Lung Kuo, Chih-Hsing Chu* Department of Industrial Engineering and Engineering Management National Tsing-Hua University, Hsinchu, Taiwan Ying Li, Xinyu Li, Liang Gao State Key Lab of Digital Manufacturing Equipment & Technology Huazhong University of Science & Technology, Wuhan, China
*[email protected] Abstract Optimisation of tool path planning using metaheuristic algorithms such as ant colony systems (ACS) and particle swarm optimisation (PSO) provides a feasible approach to reduce geometrical machining errors in 5-axis flank machining of ruled surfaces. The optimal solutions of these algorithms exhibit an unsatisfactory quality in a high-dimensional search space. In this study, various algorithms derived from the electromagnetism-like mechanism (EM) were applied. The test results of representative surfaces showed that all EM-based methods yield more effective optimal solutions than does PSO, despite a longer search time. A new EM-MSS (electromagnetism-like mechanism with move solution screening) algorithm produces the most favourable results by ensuring the continuous improvement of new searches. Incorporating an SPSA (simultaneous perturbation stochastic approximation) technique further improves the search results with effective initial solutions. This work enhances the practical values of tool path planning by providing a satisfactory machining quality.
Keywords: electromagnetism-like mechanism algorithm, 5-axis machining, flank milling, tool path planning 1
1.
Introduction 5-axis CNC machining has been commonly used in manufacturing of complex
geometries in automobile, aerospace, energy, and mold industries. This advanced machining operation provides better shaping capability and higher productivity compared to traditional 3-axis machining. Tool path planning becomes a challenging task in most 5-axis machining operations due to additional degrees of freedom in the tool motion. A primary goal in planning the tool path is to avoid tool collision. In 5-axis flank machining, the flank part of a cutter is used to remove stock materials and finish the design surface. To completely eliminate geometrical deviations when creating complex shapes with a cylindrical cutter is highly difficult, if not impossible. The cutter will induce substantial deviations near twisted surface regions, or in a mathematical term, not locally developable [1]. Precise control of the machining deviations is lack of solutions. The geometrical deviations in 5-axis flank finishing of ruled surfaces can be reduced by adjusting all cutter locations of a tool path simultaneously. Previous studies [2-4] have proposed various optimization schemes based on meta-heuristic algorithms to conduct the adjustment. Chu et al. [2] transformed the tool path planning of a ruled surface into a 2D curve matching problem. An Ant Colony Systems (ACS) algorithm was applied to calculate an optimal matching with the accumulated geometrical deviations on the machined surface as the objective function. Their method restricted that the cutter contacts the surface at pre-defined discrete points on its boundary curves. Hsieh and Chu [3] relaxed this constraint by allowing the cutter freely contact the surface and applied the particle swarm optimization (PSO) algorithm to search for optimal solutions. They adopted GPU computing techniques to accelerate the search process. This approach still suffered from unsatisfactory quality of the search results, as the cutter had to make contact with the boundary curves of the surface to be machined. To overcome this problem, their later study [4] proposed a new encoding scheme of 2
cutter locations in the tool path planning driven by PSO. The cutter could deviate from the surface along the normal, tangent, and bi-normal directions at the end points of the surface rulings. Such additional freedoms in the tool motion enlarge the solution space in the optimization, thus yielding better search results than those of previous studies. Hsieh and Chu 5] compared the performance of various particle swarm algorithms such PSO, Advanced Particle Swarm Optimization (APSO), and Fully Informed Particle Swarm Optimization (FIPS) algorithms on the tool path planning problem. They constructed a set of representative test surfaces by systematic variations of three surface properties. The test results showed that FIPS perform best in all trials with a large number of cutter locations, but the improvement was not significant when the number is small. Besides, the search process easily converged to local optima and results in poor solutions. The above reviews have shown that meta-heuristic algorithms provide a systematic approach to controlling and reducing the geometrical deviations in 5-axis flank finishing cut of ruled surfaces through optimization of tool path planning. However, the optimization schemes employed by previous studies including ACS and various PSO algorithms failed to produce good search results due to high dimensionality of the solution space. This study applied optimization methods based on the electromagnetism-like mechanism (EM) to further enhance the solution quality. Two new algorithms were developed from the original EM: a simplified electromagnetism-like mechanism (SEM) algorithm and an electromagnetism-like mechanism with move solution screening (EM-MSS) algorithm. The SEM algorithm simplifies calculation of the interaction forces between particles. The EM-MSS algorithm guarantees continuous improvement of search results by adding a solution screening step. These algorithms were used to optimize the tool path planning with a set of representative test surfaces. The test results were compared with those produced by PSO on computational time and solution quality. In addition, the EM-MSS algorithm incorporates a simultaneous perturbation stochastic approximation (SPSA) procedure to start search with good initial 3
solutions. Conclusion remarks were given to summarize the effectiveness of the EM-based methods on reducing the geometrical deviations in 5-axis flank finishing cut of ruled surfaces.
2.
Tool Path Planning in 5-axis Flank Machining Tool Path Encoding A CNC tool path is defined by a set of cutter locations (CL). 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 for tool path planning is to allow the cutter to move along the surface rulings. The resultant path produces excessive machining deviations in twisted surface regions, though.
Figure 1 Defining a cutter location with respect to a ruled surface
A cutter location is specified with respect to a ruled surface as shown in Figure 1. The cutter contact points pA and pB are first generated from the two boundary curves (A and B) and each corresponds to the parameter values uA and uB, respectively. The cutter center point is then determined by offsetting the contact point along the surface normal with a distance of the cutter radius. The cutter orientation lies in the direction connecting the two cutter center 4
points. In this study, we allow the cutter center point to deviate from the cutter contact point in the surface normal, tangent, and bi-normal directions. They are expressed as: (1)
where: cA and cB are cutter center points; pA and pB are cutter contact points corresponding to curve parameter values uA and uB, respectively; nA and nB are the unit surface normal vector at pA and pB, respectively; NA and NB are the magnitude of deviation in nA and nB, respectively; tA and tB are the unit tangent vector at pA and pB, respectively; TA and TB are the magnitude of deviation in tA and tB, respectively; bA and bB are the unit bi-normal vector at pA and pB, respectively; BA and BB are the magnitude of deviation in bA and bB, respectively. When the objective function is to minimize the accumulated deviations on the machined surface, optimization of the tool path planning is expressed as:
(2) Subject to where
0 i (N 1), 0 uA 1, 0 uB 1;
is the geometrical deviation induced by the cutter motion
A tool path consists of N cutter locations. A cutter location
from
to
.
can be adjusted in 3D space
by varying a set of parameter values {uA, uB, VA, VB, TA, TB, BA, BB}. To obtain an optimal tool path yielding minimized geometric deviations requires adjustment of all cutter locations simultaneously. This involves search for optimal solutions in a high-dimensional solution space (8N dimensions in this case). 5
1. Generate sample points
2. Produce line segments at each sample point
3. Intersect the lines with the cutter
4. Calculate the accumulated deviations
Figure 2 Estimating machining deviations by the stock height method [6]
Estimation of Geometrical Machining Deviations In this study, the objective function applied in the optimization process is the accumulated geometrical deviations on the finished surface. Exact estimation of the geometrical deviations is highly difficult and might not be required. The stock height method developed in the previous study [6] estimates the deviations approximately. As shown in Figure 2, the estimation process contains four steps. First, the design surface is discretized into 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 tool swept surface is approximated by interpolating a finite number of tool positions between consecutive cutter locations. Next, those lines are trimmed by intersecting with the peripheral surface of the cutter at each position. The geometrical deviations are calculated as 6
the sum of the lengths of the trimmed straight lines. The sampling density in the method controls the estimation precision of the deviations and the computational time required in the optimization process. The optimization problem described in Eq.(2) has thus become:
(3) where
is the approximate deviations estimated by the stock height method. A set of
intermediate cutter locations linear interpolated from
and
replaces the
continuous tool motion. This mimics the linear interpolation normally conducted by a CNC controller. Thus, the objective function is the accumulated machining deviations on the finished surface, which is estimated by the stock height method using CNC linear interpolation. The optimization variables in Eq.(3) are {uiA, uiB, NiA, NiB, TiA, TiB, BiA, BiB}, where 1 i N and N is the number of cutter locations in a tool path.
3.
Optimization Schemes Electromagnetism-like Mechanism (EM) Electromagnetism-like mechanism (EM) is a stochastic optimization method based on
electromagnetism [7]. It is a population-based random search algorithm similar to GA. 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 and 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 [8]: initialization of the algorithm (Initialize), application of neighborhood search to exploit the local (Local), calculation of the total force (CalcF) exerted on each particle, and movement along the force direction (Move). 7
A general framework is described as follows. Note that a particle in the EM algorithms represents a tool path consisting of N cutter locations. Each cutter location is defined by 8 decision variables (4 from each boundary curve). Thus a particle corresponds to an 8xN matrix and is encoded as a one dimensional matrix during the optimization process. In EM algorithm, the particle vector is defined as
, superscript of i represent the particle index and
subscript of k represent the particle dimension.
EM Framework EM(m, MAXITER, LSITER, δ) m: particle number MAXITER: maximum number of iterations LSITER: maximum number of local search iterations δ: local search parameter，δ∈[0,1] 1: 2:
Initialize () iteration
3: 4:
while iteration < MAXITER do Local (LSITER, δ)
5: 6: 7:
F CalcF () Move (F) iteration iteration + 1
8:
1
end while
Initialize The procedure Initialize randomly samples m points in an n-dimensional search space. Assume that the coordinates of a sampling point follow a uniform distribution U(0, 1) between given upper and lower bounds, uk and lk. Each sampling point corresponds to a particle in the EM algorithm. Sampling points follow a uniform distribution between given upper and lower bounds, uk and lk. The particle with the best value is stored as
xbest .
The
particle dimensions equal to 8 times number of cutter locations, i.e. n = 8 N.
8
Initialize () 1: 2: 3: 4:
for i = 1 to m do for k = 1 to n do U (0,1) xki
5: 6: 7: 8:
lk
(uk lk )
end for Calculate
f ( xi )
end for xbest
arg min{ f ( xi ), i}
Local Most random search algorithms reply on local search to attain good optimal solutions. The procedure Local is applied to improve the quality of the initial solutions obtained by Initialize. It is a simple random line search determined by two parameters – LSITER and δ, representing the number of iterations and a multiplier for the neighborhood search, respectively.
Local (LSITER, δ) ___________________________________________________________________________ 1: counter 1 2: Length (max k {uk lk }) 3: 4: 5: 6: 7: 8:
for i = 1 to m do for k = 1 to n do U (0,1)
1
while counter < LSITER do xi
y
U (0,1)
2
9: 10:
if
11: 12:
else
13: 14:
end if if f ( y)
1
0.5
yk
yk
then
yk
2 ( Length)
yk
2 ( Length)
f ( xi )
then 9
15:
xi
16:
counter
17: 18: 19: 20: 21: 22:
y
LSITER 1
end if counter counter 1 end while
end for end for xbest
arg min{ f ( xi ), i}
CalcF The procedure CalcF integrates local and global search results by calculating the electromagnetic-like force between two particles. The force exerted on a particle by the other is inversely proportional to the distance between them and directly proportional to the product of their charges (see Eq.(4)). Note that the charge carried by particles can change in the EM algorithm, but this change is not possible in reality. The charge of particle i is determined as:
q
i
exp
n
f xi m k 1
f xk
f xbest f xbest
, i.
(4)
This equation indicates that higher charges produce better objective function values. The number of dimensions n multiplied in the equation avoids overflow in the exponential function when the denominator becomes too large. Experimental results obtained by the previous study [9] have shown that determining the particle charge by Eq. (4) normally gives satisfactory search results. The total force Fi exerted on particle i is computed as:
(5)
The charge calculated by Eq. (4) does not carry a positive or negative sign. The direction of the electromagnetic-like force is determined by Eq. (5). As shown in the procedure CalcF, a 10
particle with a better objective function value attracts the other particle (lines 7 to 8); the one with a worse value repels the other (lines 9 to 10). Since xbest has the best objective function value, it attracts all other points in the population.
CalcF(): F ___________________________________________________________________________ 1: 2:
for i = 1 to m do qi
f
exp
x
m k 1 f
3: 4: 5: 6: 7:
Fi
i
f
n x
k
x f
best
x
best
0
end for for i = 1 to m do for j = 1 to m do if f ( x j ) f ( xi ) then
8:
{attraction}
9:
else
10: 11: 12: 13:
{repulsion} end if end for end for
Move In this procedure, particle i will be moved in the direction of Fi by a random step length λgiven by: x
i
x
i
Fi ( RNG ) || F i ||
i 1, 2,
m
(6)
We assume that λ obeys a uniform distribution between 0 and 1. Distributions of other types can also be used in calculating the step length. A step length randomly determined guarantees a nonzero probability of moving toward unvisited search space. 11
Move (F) ___________________________________________________________________________ 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13:
for i = 1 to m do if i ≠ best then U (0,1) F
Fi
i
i || F ||
for k = 1 to n do if Fki 0 then xki
xki
Fki (uk
xki )
xki
Fki ( xki
lk )
else xki
end if end for end if end for
Simplified Electromagnetism-like Algorithm (SEM) In the EM algorithm, estimation of the particle charge using Eq.(4) and the resultant force using Eq.(5) involves a large number of algebraic operations. The optimization process may require lengthy computational time when solving complex problems. A greater value of the denominator in Eq. (5) produces smaller forces between particles. As a result, the step length in the procedure Move tends to be too short and the search result is thus likely to be restricted within a small neighborhood in the solution space. To avoid this problem, a simplified formula is proposed to calculate the particle charge: (7) The force is written as:
(8) 12
The algorithm adopting this new procedure is named as Simplified Electromagnetism-like Algorithm (SEM).
Electromagnetism-like Algorithm with Move Solution Screening (EM-MSS) SEM does not guarantee that the solution generated by the Move procedure is better than the original one. This may lengthen the computational time during the optimization process and deteriorate the solution quality. A screening mechanism is proposed to assure generation of better solutions after the Move procedure. This procedure repeats until the new solution is of a better objective function value than the previous one. The EM algorithm using this screening mechanism is referred to as Electromagnetism-like Algorithm with Move Solution Screening (EM-MSS).
4.
Test Results The algorithms described above have been applied to generate optimized tool paths with
various test surfaces. The previous study [5] proposed a set of representative ruled surfaces for comparing the effectiveness of different tool path planning methods. These surfaces were constructed by varying three intrinsic properties: curvature (C), twist (T), and the length difference (L) between two boundary curves defining a ruled surface. The surface curvature refers to the maximum normal curvature of a C2 surface and measures the maximum degree of bending in a surface point-wise. Twist indicates the deviation between the tangent directions at the end points of a ruling connecting the boundary curves. It is usually estimated as the angle extended by the projections of the directions along the ruling onto a plane. In this study, twist refers to as the sum of the angle for all surface rulings.
13
Surf1: small C, small T, small L
Surf2: small C, small T, large L
Surf3: large C, small T, small L
Surf4: large C, small T, large L
Surf5: small C, large T, small L
Surf6: small C, large T, large L
Surf7: large C, large T, small L
Surf8: large C, large T, large L
C: curvature; T: twist; L: length difference Figure 3 Eight different surfaces generated by varying three intrinsic properties [5]
Eight types of ruled surfaces can be constructed by varying those three properties as follows. Connecting two boundary curves of large normal curvatures produces a surface of a large curvature. Rotating a boundary curve with respect to the other yields a twisted surface. Placing the control points of a free-form curve in space controls its arc length. The surfaces thus generated are shown in Figure 3 and their control points can be referred to [5]. Table 1 Setting of cutting parameter # of sample points in the u direction
200 14
# of sample points in the v direction
10
# of cutter locations (CL)
40
cutter length cutter radius # of interpolations between cutter locations
30 mm 2 mm 10
Table 1 lists the setting of cutting parameters in the subsequent tests. The numbers of sample points determine the precision of estimating the objective function, i.e. the geometrical deviations estimated using the stock height method. The number of interpolation between consecutive cutter locations also relates to the precision. The values of those parameters were chosen as recommended by the previous studies [3-5]. PSO-based tool path planning was also tested for comparison with EM-based approaches. Our previous studies [3, 4] described the details of the path planning and the settings in the PSO algorithm (thus omitted in this paper). The parameter setting in the EM-based algorithms is shown in Table 2. The parameter values are chosen as the ones suggested by [7]. Our previous work [9] has shown that the EM algorithms have a satisfactory performance under such a parameter setting. Each algorithm was conducted three times to reduce the influence of randomness. An initial tool path was generated by allowing the cutter to contact discrete surface rulings determined by varying the curve parameter u with a constant increment of 0.025, as the number of cutter locations is 40 in the tests. The PSO algorithm adopts the same initial solution for fair comparison. ACS was applied to solve the similar tool path planning problem in the previous work [10]. A major limitation of using the ACS algorithms is that the cutter is only allowed to move among pre-defined discrete points on the boundary curves of the surface to be machined. This limitation results in a very restricted solution space and consequently suboptimal solutions. On the other hand, to search for optimal solutions among pre-defined cutter locations requires less computation time. A fair comparison with EM or PSO is not feasible and thus not included in this paper. 15
Table 2 Parameter setting in the EM-based algorithms Parameter
Notation
Value
# of particles
m
50
maximum number of iterations
MAXITER
50
maximum number of iterations in local search upper bound of variable change in one iteration lower bound of variable change in one iteration upper bound of variable setting lower bound of variable setting
LSITER uk lk umax lmax
1 0.05 -0.05 1 -1
Table 3 Test results of different algorithms Surface Surf1 Surf2 Surf3 Surf4 Surf5 Surf6 Surf7 Surf8
PSO Best Ave. 8.534 12.119 15.812 20.562 7.315 11.691 7.601 12.321 25.753 32.449 27.969 30.953 21.364 23.451 25.352 28.087
EM Best Ave. 6.334 7.845 9.456 11.170 9.434 11.922 11.879 15.046 10.101 12.652 20.049 22.124 13.424 14.231 24.435 26.657
SEM Best Ave. 6.486 7.737 9.196 11.155 9.325 11.876 11.235 14.757 11.437 12.121 20.135 22.121 12.676 14.206 22.365 25.829
EM-MSS Best Ave. 4.642 6.323 9.275 10.234 4.678 7.244 6.539 8.431 7.919 11.257 4.831 5.267 7.602 12.367 22.288 24.125
Table 3 lists the results, the best and average values in three runs, generated by PSO, EM, SEM, and EM-MSS algorithms for eight test surfaces. Figure 3 shows that the search processes produced by all gradually converge to the final solutions. Figures 4 and 5 compare the test results in bar charts. The EM-MSS algorithm yields the best solutions in all test cases. The EM and SEM algorithms perform better than PSO in most tests except Surf3 and Surf4. Large surface twist normally results in an excessive amount of the machining deviations. The test results (Surfs 5, 6, 7, and 8) indicate that the solution quality of the PSO algorithm is worse than those of the EM-based algorithms in this case.
16
Figure 3 The convergence processes of all algorithms
Figure 4 Comparison of optimal solutions (best value) in bar charts
17
Figure 5 Comparison of optimal solutions (average value) in bar charts EM-MSS yields better solutions than EM and SEM in all test surfaces. We thus conclude that the screening mechanism works well by assuring good search results every time after applying the Move procedure. The solution quality of SEM is slightly better than that of EM in all tests. Improvement of the search results is fairly limited using the new force calculation Eq.(5). However, SEM indeed requires shorter computational time in the optimization process with this simplified formula (see Table 4). The number of evaluating the objective function in EM is four-time greater than that of PSO, which shows that the effectiveness of the original EM is not satisfactory. The optimal solutions obtained by those advanced EM methods perform significantly better than those of PSO. SME and EM-MSS show in average 24.9% and 47.2% improvement respectively based on the eight test surfaces (see Table 3). In addition, the PSO algorithm simply stops or gets trapped in local optima in the tests. The optimal solutions obtained by PSO, if not with advanced search mechanisms, cannot be further improved even with a greater number of evaluating the objective function.
18
Table 4 The computational time of different algorithms (units: second) Surface Surf1 Surf2 Surf3 Surf4 Surf5 Surf6 Surf7 Surf8
PSO Best 9634 9782 9843 9756 10156 10537 10342 10432
Ave. 10123 10243 10356 10342 10643 10789 10974 10622
EM Best 33231 32796 34288 34531 33998 35413 32168 34468
Ave. 34592 35693 37119 37654 38852 38752 35575 35857
SEM Best Ave. 29078 29806 31044 32018 33462 35961 32344 32845 31137 31732 30560 31440 31727 34005 33038 34677
EM-MSS Best Ave. 36257 37369 42396 45056 44199 47264 41056 42965 38921 42337 38200 41463 42695 44223 41095 44747
Initial solutions play an important role in the global optimization based on meta-heuristic algorithms such as EM. Starting with good initial solutions may accelerate the convergence of the search process and/or improve the final solution quality. Random number generation is commonly used in most optimization problems without prior knowledge. The solutions thus generated are not guaranteed to good ones. In this study, the simultaneous perturbation stochastic approximation method was applied to produce an initial set of particles with satisfactory solution quality. The focus was to demonstrate to what extent such an initial condition would influence optimal solutions. It is also insightful to examine how the ratio of good particles in the initial set affects the search results. Simultaneous perturbation stochastic approximation (SPSA) is an optimization scheme that uses only objective function measurements, instead of the gradient of the objective function, in finding optimal solutions [11]. SPSA is especially efficient in high-dimensional problems in terms of generating good solutions with a relatively small number of estimations of the objective function [12]. In this study, the optimization variables NA, NB, TA, TB, BA, and BB were randomly adjusted using SPSA. The adjustment process terminates when the objective function value corresponding to those variables is better than the value generated by all null values, i.e. the cutter follows the surface rulings. In other words, here a “good” initial solution means that the resultant tool path yields smaller geometrical deviations than that of the path following the surface rulings. 19
Table 5 Search results with good initial solutions generated by SPSA EM-MSS Best 4.642 9.275 4.678 6.539 7.919 4.831 7.602 22.288
Surface Surf1 Surf2 Surf3 Surf4 Surf5 Surf6 Surf7 Surf8
EM-MSS+SPSA Best 3.295 8.702 3.638 4.582 4.321 4.253 4.866 17.852
Table 5 shows the test results with initial solutions generated by SPSA. The best value of the objective function was improved in all surfaces, thus validating the necessity of good initial particles in the EM algorithms. This conclusion demonstrates the effectiveness of the SPSA procedure. Table 6 shows the test results for different numbers of good initial solutions produced by the same method. Generally speaking, the objective function value decreases with the increasing number of good initial solutions. Figure 6 shows evidence to support this statement. As shown in Table 7, the computational time of the optimization process increases correspondingly, as additional computations are required by the SPSA procedure. All EM-based algorithms involve a greater number of estimating the objective function than the PSO algorithm does (see Table 8). As anticipated, they yield better optimal solutions with more search attempts. The EM-MSS algorithm proposed by this work performs better than the EM and SEM algorithms in the tool path planning problem, with a small amount of additional computations. The improvement becomes even more significant when searching with good initial solutions. Table 6 Search results for different numbers of good initial solutions generated by SPSA Surface Surf1 Surf2 Surf3 Surf4
# of good initial solutions 5
10
15
20
25
30
35
40
45
50
3.780 14.22 4.691 5.388
3.898 13.51 4.178 5.279
3.295 11.93 4.058 5.226
3.579 11.78 4.094 5.174
3.674 11.55 4.034 5.122
3.687 11.31 4.009 5.119
3.598 10.16 3.877 5.074
3.457 12.24 3.805 4.660
3.244 9.946 3.685 4.719
3.799 8.702 3.638 4.582 20
Surf5 Surf6 Surf7 Surf8
6.708 7.286 7.259 20.07
5.181 6.234 6.871 17.85
5.355 5.682 6.714 19.34
5.302 5.528 6.260 18.61
5.158 5.226 6.284 19.06
5.123 5.376 6.493 18.78
5.000 5.819 5.800 18.90
4.772 4.871 5.625 18.93
4.562 4.814 5.649 18.59
4.321 4.253 4.866 18.21
Table 7 The computational time with different numbers of good initial solutions (units: second) Surface Surf1 Surf2 Surf3 Surf4 Surf5 Surf6 Surf7 Surf8
# of good initial solutions 5
10
15
20
25
30
35
40
45
50
36348 45055 46828 40430 38921 38200 44659 46291
36257 46273 44951 41056 39665 39300 46256 44596
38397 45721 46438 42143 41677 40700 43432 44347
37323 45234 45236 43963 41802 40943 43695 46935
36033 44527 44199 42857 42984 41613 42695 46411
37413 46238 47771 42213 43896 41011 44998 44449
37980 46190 48072 43013 43536 41372 43896 44792
38220 42396 49345 43338 43235 42047 43586 43609
36840 44987 49932 44213 44620 42637 43320 41095
38880 43938 49872 46424 43032 46804 45689 44942
Figure 6 Optimal solutions vary with the number of initial solutions generated by SPSA Table 8 The number of estimating the objective function for all test algorithms Surface
PSO
EM
SEM
EM-MSS
EM-MSS+SPSA
Surf1 Surf2 Surf3
146240 157420 178210
822500 822500 822500
822500 822500 822500
830124 824521 826573
860823 827485 831700 21
Surf4 Surf5 Surf6 Surf7 Surf8 5.
172240 179810 186840 195210 201450
822500 822500 822500 822500 822500
822500 822500 822500 822500 822500
827549 829582 823565 823070 823507
838385 847224 825240 824043 825163
Conclusion Previous studies have shown that optimized tool path planning driven by meta-heuristic
algorithms provides a systematic approach to reducing the geometrical deviations in 5-axis flank finishing cut of ruled surfaces. For example, PSO was applied to simultaneously adjust cutter locations comprising of a tool path so as to minimize the accumulated deviations on the finished surface. The search results thus generated seem to quickly converge to local optima and are difficult to be further improved. It is advantageous to study whether or not other algorithms can enhance the optimal solutions obtained by PSO. Therefore, this study developed various EM-based algorithms and applied them to the tool path planning problem. The SEM algorithm simplifies calculation of the particle charge in EM to increase the step length in determining new search result. The EM-MSS algorithm includes a screening step to guarantee continuous improvement in the Move procedure. Test results with representative ruled surfaces have shown that the EM algorithms normally yield smaller optimal solutions than those of PSO, but require a longer computational time. The improvement of using simplified particle charge was not significant. The EM-MSS algorithm produced best solutions in all test surfaces, with a small increase in search times. In addition, a SPSA procedure was proposed to ensure that the initial particles in the EM algorithms are of satisfactory solution quality. This procedure significantly improves the search results over the initial solutions randomly generated. The degree of the improvement increases with the number of good initial solutions. This study has demonstrated the feasibility of the global optimization algorithm EM on tool path planning in 5-axis flank machining of ruled surfaces. Although the EM algorithms 22
are effective in finding better solutions, they require a greater number of search trials than PSO. A potential enhancement is to reduce the range of local search in those algorithms. Sampling techniques can be incorporated to reduce the number of optimization variables at various iteration stages during the search process. Our future work will be focused on this area.
23
Reference [1]
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1
Enhances the practical values of tool path planning in 5-axis flank milling.
New algorithms derived from the electromagnetism-like (EM) mechanism.
Those EM-based algorithms yield more effective optimal solutions than does PSO.
Incorporate an SPSA technique to improve the search results.