a study on torch path planning in laser cutting processes part 2: cutting path optimization using...

~ Journal of Manu/acturing Pnwesses , V o l . I / N o . 1

1999

A Study on Torch Path Planning in Laser Cutting Processes Part 2: Cutting Path Optimization Using Simulated Annealing Guk-chan Han, Production Engineering Center, Samsung Electronics Co., Suwon, Kyungki-do, South Korea E-mail: [email protected] Suck-joo Na, Dept. of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Taejon, South Korea. E-mail: [email protected]

Abstract In Part 1 of this paper [beginning on page 54], the heat

flow in contour laser beam cutting was calculated by using the finite difference model, and a modified analytic model was developed based on the numerical experiments. Part 2 addresses the problem of optimal torch path planning for the 2D laser cutting of a stock plate nested with irregular parts. Under the constraint of the relative positions of parts enforced by nesting, the optimization algorithm generates a feasible cutting path. The simulated annealing technique is adopted for solving the torch path optimization problem to minimize a specified cost function. The objective is to traverse the cutting contours with a minimum path length and, at the same time, to minimize the effect of heat on the cutting path sequence. To minimize the heat effect and avoid overheating, the critical temperature that should be avoided during the whole cutting sequence is considered. In this way, a global solution can be obtained in a reasonable time. Several examples are presented to illustrate the effectiveness of the proposed method.

Keywords: Laser Beam Cutting, Torch Path Planning, Simulated Annealing, Cost Function, Critical Temperature

Introduction The productivity of laser cutting machines can be

significantly improved by using microcomputer- based CAD/CAM systems for CAD to CNC program generation. Furthermore, optimal torch path generation has a considerable impact on production time. Currently, a number of commercial CAD/CAM packages that provide automatic torch path sequencing have been developed and applied to various cutting processes, such as flame, plasma, and laser cutting.

There are many constraints in the laser cutting process that are usually complicated and not feasible to be linked with an optimization algorithm. In the

practical laser cutting of thin plates, cut quality should be considered for complex part geometries. In the case of acute opening angles, an undesirable reduction in cutting quality may result due to excessive workpiece heating. For this reason, an ideal path-generation algorithm should strive to find an optimal cutting path satisfying user demand for cutting quality. Unfortunately, the effect of workpiece heating on the cutting path sequence is difficult to be incorporated into an optimization algorithm; therefore, most of the investigations on the cutting path problem deal with the cost associated only with the cutting path length.~-3

Manber and Isran? solved a problem of finding a sequence of torch paths to cut a nested stock sheet with a minimum number of piercing points. They did not take into account additional manufacturing constraints such as inside hole cutting. Raggenbass and Reissner 2 presented an expert system that generates a manufacturing plan containing the stamping-laser combination. They relied on the relation- ship between the optimal laser cutting speed and the radius of the curved section but not on the optimal path sequence. Jackson and Mittal 3 introduced a cutter path planning algorithm for the automatic generation of NC machine programs. The method extracts the entity information from CAD data and creates a torch path with the proper sequence and direction of cut for internal and external part contours.

Torch path problems can be formulated as a spe- cial case of the traveling salesman problem (TSP), which is one of the most widely studied combinatorial optimization problems. 4-7 The most common practical interpretation of the TSP is that of a sales-

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man seeking the shortest tour through N cities. The problem statement is deceptively simple, but it still remains as one of the most challenging problems in operational research. Due to this reason, the torch path algorithm can be classified into a hard combinatorial optimization problem, termed an NP-hard problem with the part nesting algorithm. A direct consequence of the property of NP-completeness is that optimal solutions cannot be obtained in a reasonable amount of computation time. Therefore, until now, a number of researchers have investigated various heuristic search methods that do not use the optimizing technique, but instead use the heuristic information obtained through experiences and near- optimizing techniques.

In relation to the path-optimization problem, a number of optimization algorithms have been developed. Since the optimization algorithms are capable of only solving small-sized problems, 4,s a number of researchers have developed a wide variety of heuristic algorithms for solving medium-sized and large-sized problems. Recently, impressive results were obtained in the large-scale optimization problem by applying simulated annealing (SA)3 '9 tabu search (TS)J ° and neural networks (NN). u,j2 Ever since Kirkpatrick, Gelatt, and Vecchi ~3 introduced the concept of annealing into the TSP in 1983, interest in the technique of obtaining approximation solutions for problems with combinatorial optimization has grown rapidly.

This paper presents an efficient laser cutting path planning algorithm based on simulated annealing that is an improved version of previously suggested TSP models. One of the main features of the proposed algorithm is that the heat effect is incorporated into the cost function. The cost function adopted in this paper includes two conflicting goals in the minimization: minimum cutting path length and minimum heat effect. The requirement of any sequence of the path is that all the edges of nested parts should be traversed exactly once by the laser torch.

Brief Review of Simulated Annealing The standard iterative improvement is a downhill-

only style that can be trapped in local minima, because each new perturbation moves to a configuration downhill from the previous one. To overcome this, a number of random starts are needed, but this is still not guaranteed to find a good solution.

The SA algorithm offers a strategy very similar to

the iterative improvement, with one major difference: the annealing allows the perturbations to move uphill in a controlled fashion. Because each movement can now change the configuration into a worse one, it is possible to jump out of local minima and potentially fall into a more promising downhill path. The relevant analogy here is the physical annealing of a solid. To coerce a material into a low energy state, the solid is first heated and then cooled very slowly while it is allowed to come to the thermal equilibrium at each temperature. The way in which the temperature decreases is called the cooling schedule, and the probability of energy states is determined by Boltzmann's law.

The annealing algorithm is based on the Monte Carlo technique proposed by Metropolis et al) 4 and can be mathematically modeled using the concept of the theory of Markov chains. 9 The technique of SA and its theoretical background is fully described by Arts and Korst) s

The SA process starts with an arbitrary initial configuration Xo and the initial control parameter Co. The control parameter c is analogous to the temperature in a physical system and decreases as the algorithm progresses. At each control parameter c, the algorithm generates a new configuration X', cal- culates the cost difference AC = C(X') - C(X), and determines the acceptance probability P of the new configuration given by the following:

l (lAc fAc<_o P=

t e x p ~ - v otherwise (1)

This Metropolis acceptance criterion can be described in the following manner. When AC _< 0, the system configuration is improved and the new configuration is accepted with the probability of 1. When AC > 0, the new configuration is accepted with the limited probability. The latter half of tile acceptance scheme is implemented by picking a random number R from a uniform distribution between zero and one and comparing it with P = exp(-AC/c). IfR is small- er than P, the new configuration X' is accepted) 7 The probability of this limited acceptance is 1 when c = ~, and decreases to zero as the control parameter is lowered down to c = 0.

The implementation of the SA algorithm can be realized by generating the homogeneous Markov

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chains of finite length for a finite sequence of descending values of the control parameter. To achieve this, a set of parameters that governs the con- vergence of the algorithm must be specified. These parameters are combined into a so-called cooling schedule: initial value of the control parameter, cooling rule for decreasing the control parameter, final value of the control parameter specified by a stop criterion, and finite length of each homogeneous Markov chain (finite number of transitions at each control parameter). Since the quality of solutions is sensitive to a number of different parameters, the best schedule will vary from one problem type to another.

Problem Definition Geometric Representation

In this paper, the basic object is a polygon (a many-sided figure) with holes. A part may be represented as a number of line segments connected end to end in counterclockwise order and formed into a closed contour as required for cutting paths (see Figure 1). The line segments that make up the part boundary are called edges, and the endpoints of edges are called the polygon vertices. The information required to specify the part is the number of edges and the coordinates of the vertex points. A part may include the inside holes to be cut, which also can be represented by a polygon in the same way as the outer contour. The geometry of parts is entered by means of the simple drafting routine.

~,~ Hole 1

I t1~,

V°,~

. . . . I Pad n

V.,r

Figure 1 Definition of Irregular Part C on to u r s

Formulation of Torch Path Problem This section first introduces the notations.

Considered is the torch path problem for N parts with V(n) vertices and h(n) inside holes with respect to each part n (1 -< n --< AD. At this moment, each hole k (1 <- k <- h(n)) on part n has v(n, k) vertices. In Figure 1, Vn,t denotes the l-th (1 <- l <- V(n)) vertex of part n and v,,k.,, denotes the m-th (1 -< m -< v(n, k)) vertex of hole k on part n. The piercing point is assigned to the starting point on each cutting contour; for example, V, a is the outer contour's piercing point and v,,aa and v,,, a are the inner holes' piercing points, respectively, as shown in Figure 1. Since all piercing points in contours can be replaced by any other vertices during the optimization process of algorithm, the total number of vertices to be considered is ~ V(n) + Y~W_v(n, k). However, V(n) and v(n, k) are defined as predefined closed paths visiting each vertex exactly once. Therefore, only the N + ~h(n) vertices are selected as piercing points at each iteration during the optimization process of this algorithm.

The effectiveness of a given optimization algorithm and the quality of the resulting cutting path strongly depend on the cost function used. The cost function used in this paper consists of two conflicting terms: the cutting path length to be traversed and the penalty of the heat effect. The first cost of the torch path sequence, between two piercing points of neighboring contours, is to make the path length as short as possible. Mathematically this can be described as follows:

minimize h,,+l • Pa,,,,+l + (1 - h,+l) • Pb,.,+l (2)

where

n n L ~r

Pb,,,,,+ I = Z g,,i V,+] /

The first vertex, V.,I, is the piercing point in the outer contour of part n, v.+L1 a is the piercing point in the first hole of part n+l, and v.+,,~ a and v.+~,k+~a are the piercing points in the k-th and k+l-th holes of part n+l, respectively. If part n+l includes the inner hole, then h.+~ = 1, otherwise h.+l = 0. Therefore, Pa.,.+j is the total cutting path length when the part n+l includes inner holes in it, while Pb,,.,,+~ is for the case of no holes.

Another cost of path sequence is to make the heat effect as small as possible and, hence, reduce the

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thermal distortion or burnoff caused by the overheating. A threshold function is adopted as a penalty to implement this heat effect consideration. The cost can be then described as follows:

minimize ~t • h,+~ • Ta,.,+l + ~ " (1 - h,+0 " Tb,,,+l (3)

where

f l ) t l , I t + ]

} zz[ ) 1 - T. + 0 T v,,+m~+ u - T,

+ ] pt

-- q tl

For this objective to be satisfied, the initial temperature (T) of any piercing point should be below the critical temperature (T,.). The threshold function q~[ T - T,] - 0 when T - T,. < 0, while q)[T - T,] - I when T - T,, >-- O. The initial temperature at the piercing point o f nested parts is calculated repeti- tively during the optimization process. This initial temperature that is formed during the cutting of contours with the laser beam can be calculated by using the modif ied analytical solution suggested in Part 1 of this paper. The algorithm now finds the shortest path sequence that avoids the occurrence of critical temperature. The relative importance that it assigns to the path length versus the heat effect is determined by the value of It.

In practical manufacturing situations, a resource sheet is usually placed on a flat bed on the work- table. The cut parts do not actually fall, but they are removed after the completion of the entire cutting process. Therefore, i fa cut part needs further cuts on the inside, this is theoretically possible. However, by reason of thermal distortion and movement, it is generally advisable to complete all inside cuts of a part before dropping it; this is adopted as a constraint in this research. The closed path of V(n) vertices should be visited just after visiting the closed path of h(n) vertices. Figure 2 shows the way of finding the optimal piercing point to minimize the cost for a selected part. The cost for the torch path sequence between two outer contours and an inner contour can be obtained by the sum of two components: the cost between vertex V;,~ and vertex vi.u, and the cost between vertex vj,~,t and vertcx ~.j.

In addition, there are other practical manufacturing constraints to be considered. To prevent excessive

burning of the part material at the initial piercing of the plate, lead-in cuts are often used. The torch pierces the material away from the part and enters the part periphery with a relatively gradual increase in material removal. Furthermore, when cutting out parts, torch characteristics often dictate the torch direction relative to the part edge for the best cut quality as well as kerf width. In this study, however, lead- in cuts are not considered. It is also assumed that the torch directions of all nested parts and their inside holes are known as counterclockwise in advance.

Initial Temperature of Piercing Point Since the primary purpose of the heat effect con-

sideration is to avoid the critical temperature that can cause plate overheating, a simplified analytic equation is adopted in Part 1 of this paper to predict the overheating, as follows:

T'(~, t) = T, -~ q expl ~2 I Vdpc \ / 4 n m - ~ m ( 4 )

where T'(~, t) is the nonstationary state heat conduction equation for the opening angle of [3=180 ° and ~ is the shortest distance between the edge and piercing point.

From the fact that the initial temperature at a piercing point strongly depends on the heat flow s temming from the previous cut contour, it is assumed that the heat flow that affects the initial temperature at a piercing point (V,+~,) is conducted from only an edge connected to the nearest vertex (C,) from V,+ u. As shown in Figure 3a, the edges effective in considering the initial temperature are

Initial path / 4 - ~t Intal p erc ng \

r / I New piercing point

Optimized path '-~ ~T /

,,

Figure 2 Illustration of Finding the Best Piercing Point to Minimize Cost

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Journal of Manl{facturing Processc~" Vol. I/No. 1 [ 999

/.tto,-edge --j'

v. / k~eg,o , , ,~ v.., \

. , " ~ . /

(a) "---./

~ j . l . ~ ¸~ " J ~

~ / / latter-edge ax+by+c=O )

~n So..f Part n . ~ h / ~

: former-edge N~ ] P rt O 1 5

V..,¢~ ....................... / / 7 a +/// ( b ) / ~ ~

Figure 3 Two Different Types of Contour Cutting

(a) Convex contour cutting (b) Concave contour cutting

than 120 °, therefore, the nonstationary state heat conduction equation can be used without any modification. In this case, the initial temperature of V.+~ a at a time t, T(V.+~a) is given by T'(t~, t) from Eq. (4).

As shown in Figure 3a, the piercing point is denoted by V.+la(X.+b Y.+1) and the edge by ax + by + c = 0, where a, b, and c are the coefficients of the equation for the line. A line that starts from the piercing point V.+L~ and is perpendicular to the given edge intersects that edge at the point S.(x.s, y.,). The coordinates of the intersection point are defined as (x.~., y.,,) = (b(bx.+l - ay..O - ac, - b c - a(bx.+~ - ay.+O). Based on this information, the distance ~ between the piercing point V.+~a and the intersection point S.(x.~, y.,) is formulated as follows: 17

I"'x,,+, +"- vn+, +el = " (5)

\ / a 2 + b 2

The instantaneous line heat source at S. affects the initial temperature of piercing point V.+~j, which changes during the torch traverse from S. to V.,~ with beam-on, and from V. a to V.+~,~ with beam-off. The time t required for the torch to traverse the path sequence from 5'. to V.+ u is given by the following:

divided into two types: former-edge and latter-edge. And V.+la may be located in three different regions: region 1, region If, and region lII. If V.+~ a is in region 1I or region lII, the intersection point S., which is the nearest point from V.+j,1 is on the latter- edge; otherwise, the intersection point S. is on the former-edge. During the concave contour (small opening angle) cutting as shown in Figure 3b, however, the point S. should be located on the latter- edge for a margin of safety. In this research, the pre- heating of the workpiece is initially neglected in every calculation of temperature distributions. Accordingly, the initial temperature is determined by solving the analytic heat conduction equation [Eq. (4)].

When the laser beam changes its direction at a corner during the cutting operation, two different heat flow features are expected. At small opening angles, burnoff of the corner may take place. This is due to the reduced heat dissipation caused by two adiabatic sides of the kerr. But, at wide opening angles, there is no reduction in the cutting quality along the edge because the heat applied can be eas- ily dispersed in the plate. For opening angles greater

SUM(S,, V,,,~) V IV,,+, 1 t = + ' ' (6)

v v,

where SUM(S., V.j) is the sum of the path length from point S. to point V.j on the cutting path, V is the cutting speed with beam-on, and Vs is the rapid traverse speed between two piercing points with beam-off.

In the case of small opening angles ([3 - 120°), a heat-affected zone broader than at wide opening angles may result. The initial temperature of V.+~ a at a time t, T(V.+u) , is given by the following:

T(V.+,,,, 13) = T'(~, t ) . m([3, T'(~, t)) (8)

where 535 F ¢ -72 - '

L m(~, T'(~, t)) is the modification factor varying with the opening angle and temperature and T' (~, t) is given by Eq. (4). The correlation between temperature T and opening angle [3 was examined in Part 1 of this paper.

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Implementation of SA to Torch Path Planning

The sequence of probabilistic transitions is exe- cuted at each control parameter until the quasi-equilibrium is attained (inner-loop). Then the control parameter c is lowered according to the cooling rule, and the sequence of probabilistic transitions is repeated. The whole process continues until the stop criterion is satisfied (outer-loop).

The annealing algorithm for this task needs some basic components, such as neighborhood structure, cost function, Metropolis criterion, and cooling schedule. The SA algorithm implemented for the pattern nesting is represented as tbllows.

generate random path, X ; whi le( stopping_criterion( loop_count )<i TE MP and c > e,) {

while(inner_loop_criterion<nOVER and nsucc <nLIMIT) {

do a shifted pathing, X, ; _A C=cost(X)-cost(X,) ; if(AC<0 or exp(AC/c) > R) accept(X=X~) ; else {

do a reversed pathing, Xr ; AC=cost(X)-cost(Xr) ; if (AC<0 or exp(AC/c) > R) accept(X=Xr) :

choose a new pierc ing po in t in a outer part contour, X r ; AC=cost(X)-cost(Xp) ; if(AC<0 or exp(AC/c) > R) accept(X=X,,) ; elsel

choose a new pierc ing po in t in a inner hole contour, Xh ; AC=cost(X)-cost(Xh) ; if (AC<0 or exp(AC/c) > R) accept(X=Xh) :

} t f

update c;

where iTEMP is the number of times by which the annealing is performed, e, is a small positive number as the stop criterion, nOVER is the maximum number of solutions evaluated at each control parameter, and nLIMIT is the maximum number of new solutions to be accepted at each control parameter. The

inner_loop is repeated nOVER times or until the number of new solutions accepted is equal to nL1MIT, whichever occurs first. Press et al. ~6 suggested the guidelines for setting the values of the parameters in the traveling salesman problem (TSP). Based on the experiments, nOVER was set at 100N and nLIMIT was set at 1 ON, where N is the number of parts to be cut in the nested stock plate.

in addition to the annealing schedule, the quality of the final solution obtained by SA based on the Monte Carlo technique usually strongly depends on the neighborhood structure and its generation mechanism. The neighborhood structure in this torch path problem is obtained by rearranging a section of torch path to any order. In this paper, the neighborhood structure is obtained by three types of generation mechanisms: (1) a section of path is removed and then replaced with the same parts running in the opposite order (Figure 4a); (2) a section of path is removed and then placed between two parts randomly chosen (I~Tgure 4b); and (3) a piercing point of an outer part contour and inner hole contour is newly chosen (Figure 5). The type (1) and (2) are the well-known local search algorithms for the TSP using 2-changes, which has been suggested by Lin. j8 This paper calls these two types of rearrangement strategy the reversed pathing and the shifiedpathing, respectively. In Figure 4, the number of parts k that should be removed and rearranged is randomly selected between 2 and N - 2 at each iteration.

During a torch traverse on a nested stock plate to cut arbitrarily shaped part contours, the induced laser beam initially pierces the plate and then enters the part contours with material removal. These piercing points in the torch path problem correspond to the cities in the TSP problem. While most of the early lit- erature addressed the torch path problem with fixed piercing points to simplify this hard optimization problem, the piercing points in this paper are uncon- strained on the contours and accordingly can move during the optimization process. Therefore, the rearrangement of the vertices for cutting parts is per- tbrmed at each trial as shown in Figure 5. In implementation of the SA algorithm, cost differences are calculated incrementally at each iteration for speed- ing up the algorithm. The new configuration after the move is accepted according to the given probability function (Metropolis acceptance criterion), which decreases as the algorithm progresses.

Generally, the optimization problem can be solved

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Journal ~?f Manu[acturing Processes Vol. 1,'No. 1 1999

. . . . . . . . . . q - 1 . . . .

{a)

• . . ; 2 ! - . . k - . I ! [ ] . . . . [ i4 _ ' k . . . . . . .

• . . • . . . . [ t . . . . .

(b)

Figure 4 Cutting Path Generation Mechanism

(a) Shifted pathing, (b) Reversed pathing

from two different starting solutions, which are gener- ated totally randomly or from a fast heuristic method. The former may take a long time to converge toward the useful areas of the search, but the latter may lead to the algorithm becoming stuck near the solution at an early stage. In general, at large values of c in SA, all the proposed transitions accepted differ from the general local search algorithms. This is one of the strong features of the SA that do not strongly depend on the choice of the initial solution. Accordingly, the starting solution is given by a random sequence of the N parts, which is far from an optimal solution. After all parts are visited, the loop is completed by returning to the starting point. In the present study, a constant starting control parameter was determined through a number of experimental calculations.

Experimental Results and Discussion Figure 6 shows the four intermediate results of

the optimization process carried out by SA for torch path planning without heat effect consideration. The initial solution, shown in Figure 6a, is given by a random sequence of 35 identical parts, which is far from an optimal solution. The solution looks very chaotic and the cost is large, in the course of the optimization process, the solution becomes less and less chaotic (Figures 6b and 6c), and the path length decreases. Finally, the near-optimal solution shown in Figure 6d is obtained. The cooling rate employed for this problem used an initial control parameter of 1.5 with 100N iteration at each temperature. In the

Part n

a o o * l [ [ } ~" o o e i i i i

_ ~ s t edge New plercing point

. . . . . . . . .

Last edge

Part n

First edge

[ ~ -t" * • t ...... I l l o

Figure 5 Rearranging of Piercing Point

torch path problem, the control parameter is reduced by a factor of 0.9 in the outer-loop; that is, c,+~ = 0.9 × c,. The algorithm is stopped when the outer-loop has reached iTEMP or the control parameter has reached a given value e,.. These parameters deter- mine the number of times that the SA schedule is to be performed. In this study, the parameters iTEMP and e, were set at 150 and 0.00001, respectively.

Figure 7 is the result of the SA process incorporat- ing the heat effect consideration for the same cutting problem as in Fignu'e 6. The critical temperature adopted in the computation was set at 240°C, which is equivalent to the allowable kerf width of 113%. The correlation between the workpiece temperature and the kerr width was examined in Part I of this paper. The total torch path length of Figure 7 is larger than that of Figure 6 because the specified critical temperature for given material and cutting parameters should be avoided on the entire path sequence. The plot of the cost function versus control parameter is shown in Figure 8. (Note that the plot is read from right to left because the annealing is processed from hot to cold temperature.) During the first portion of the run, there can be much random perturbation so that the cost even increases, which is necessary to avoid an entrap- ment at local minima. As the process goes on, however, the total cost decreases, and at the end of the run the heat effect is entirely diminished and only the path length term of the cost function remains.

Figures 9 and 10 show other examples of the optimized cutting path sequence obtained by SA with and without heat effect consideration, respectively.

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Figure 6 Intermediate Results of Simulated Annealing (SA)

Optimization Process with Increasing Time from (a) to (d) Without Heat Effect Consideration

Since several irregular parts with holes are included in the workpiece, the constraint in the cutting path sequencing between the outer contour and inner hole was considered in these examples. The adopted cutting sequencing constraint was that the outer contour should be cut just after cutting the inner holes. In this research, the cutting parameters used for the laser beam cutting of type 304 stainless steel with 1 mm thickness are as follows: laser power 250 W, cutting speed 2 m/rain., and rapid traverse speed 5 m/rain. Since the initial temperatures at all the piercing points are kept less than the specified critical temperature, uniform cut quality may be expected along the optimized cutting path sequence.

Concluding Remarks A new' solution to the optimal torch path planning

problem for a 2D laser cutting process was proposed with the help of heat conduction analysis introduced in Part 1 of this paper. The objective is to traverse the cutting contours with a minimum path length and to avoid the critical heat effect of the workpiece from the laser beam. One of the main f~atures of the proposed algorithm is that, to optimize cutting path planning, heat effect is incorporated into the cost function based on the modified analytical model of

Figure 7 Example of Torch Path Planning for 35 Identical Parts

with Heat Effect Consideration

Cost 25.0

200

15.0

10.0

50

0.0

optimization t ~

--... End of

optimization

i I I 1

0,0 0.5 1.0 1.5 20

Control parameter, c

Figure 8 Optimization Curve: Cost vs. Control Parameter of Figure 7

heat conduction. The proposed torch path algorithm adopted the SA-based stochastic process, which has a much higher possibility to obtain a global solution than a deterministic searching technique. And an efficient generation mechanism of neighborhood structure was introduced to solve the combinatorial optimization problem. From the simulation results,

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Journal of Mantgfacturing Processes Vol. 1/'No. 1 1999

260 mm

It

I / [

)

Figure 9 Example of Torch Path Optimization for Irregular Parts

Without Heat Effect Consideration

it can be concluded that the proposed SA model is both efficient and effective in solving the large-sized optimization problem such as cutting path planning. The method can effectively automate CNC pro- gramming for various cutting processes, such as flame, plasma, and laser cutting; considerably improve the productivity; and consequently lower production costs.

References I. U. Manber and S. lsrani, "Pierce Point Minimization and Optimal

Torch Path Determination in Flame Cutting," .Journal of Mam~/acturing Systems (v3, n I, 1984), pp81-89.

2. A. Raggenbass and J. Reissner, "Stamping-Laser Combination in Sheet Processing," Annals ~flthe ('IRP (v38, nl, 1989), pp291-294.

3. S.D. Jackson and R.O. Mittal, "'Automatic Generation of 2-Axis Laser Cutter NC Machine Program and Path Planning from CAD," ('omputers in huhesttlv (v21, 1993), pp223-23 I.

4. G. Laportc, "'The Traveling Salesman Problem: An Overview of Exact and Approximate Algorithms," European Journal o['Operations Rexeareh (v59, 1992), pp231-247.

5. A. Langcvin and E Soumis, "Classification of Traveling Salesman Problem Formulations," Operations Research Letters (vg, 1990), pp 127-132.

6. J.K. Lenstra and A.H.G. Rinnooy Kan, "Some Simple Applications of the Traveling Salesman Problem," Operational Re,'earth Quarterly 1v26, n4, 1975), pp717-733.

7. R.S. Garfinkel. "Minmfizing Wallpaper Waste, Part I: A Class of Traveling Salesman Problems," Operational Research (v25, n5, 1977), pp741-751.

8. R.A. Rutenbar, "Simulated Annealing Algorithms: An Overview," IEEE Circuits and Devices Magazine (1989), pp 19-26.

9. V. Cenrny, "Thermodynamical Approach to the Traveling Salesman Problem: An Efficient Simulation Algorithm," Journal ~4f Optimization Theoo~ and Applications (v45, hi, 19851, pp41-51. 10. J.A. Bland and G.E Dawson, "Tabu Search and Design Optimization," Computer-Aided Design (v23, n3, 1991 ), pp195-201. I 1. JJ. Hopfield and D.W. Tank, "Neural ('omputation of Decisions in Optimization Problems," Biological Cvhernetics ( v52, 1985), pp 141 - 152.

/

260 turn ,

L _ ~ ~ ' ~ ' ~ 185mm

. . . . . . i r

Figure I0 Torch Path Optimization with Heat Effect Consideration

(critical temperature = 242°C)

12. M.K. Mehmet All and E Kamoum "Neural Networks for Shortest Path Computation and Routing m Computer Networks," IEEE Trans. on Neural Nem'orks (v4, n6, No','. 1993), pp941-954. 13. S. Kirkpatrick, C.D. Gelatt, and M.E Vecchi, "Optimization by Simulated Annealing," Science (v220, n4598, 1983 ), pp671-680. 14. N. Metropolis, A.~: Rosenbluth, M.N. Rosenbluth, and A.H. "Feller, "Equation of State Calculations by Fast Computing Machines," Journal ~?1 Chemkal Physics (v20, n1334, 1953), pp1087-1092. 15. I';.Hi. Arts and J.II.M. Korst, 37mulatedAnnealing and Bohzmann Machines (New York: John Wiley & Sons, 1989), ppl3-114 , 16. WH. Press, B.R Flannery, S.A Teukolsky, and W,T. Vetterling, Numerical Reeipes in (': The Art 0[' Seienti[ic ('omputing (New York: ('ambridgc Univ, Press, 19901. 17. S. Harrington, Computer Graphics' (New York: McGraw-Hill, 1987). 18. S, Lin, "Computer Solutions of the Traveling Salesman Problem," The Bell Svxtem Technical Journal (v44, 1965), pp2245-2269.

Authors' Biographies Guk-Chan Han received his BS in mechanical design from Sung-Kyun-

Kwan University (Korea) in 1989 and his MS in precision engineering and mechatronics in 1992 and PhD in mechanical engineering in 1996 from the Korea Advanced Institute of Science and Technology (KAIST). In 1996, he joined Samsung Electronics as a senior researcher, tfe is currently doing research in computerized numerical control systems, especially the development of CNC control software and its applications. His research interests include laser materials processing, combinatorial and stochastic optimization techniques, simulated annealing algorithms and their application, part nesting problems, open ('NC architecture, and high-speed motion control algorithms.

Suck-Joo Na received his BS in mechanical engineering from Scoul National University (Korea) in 1975, his MS in mechanical engineering fi'om the Korea Advanced Institute of Science and Technology (KA1ST) in 1977, and his Dr.lng. in welding engineering from T[J Braunschweig (Germany) in 1983. In 1983, he .joined KAIST to research and lecture on welding and other thermal processes. He is now working mostly in the field of heat and mass flow in the welding process, laser materials processing. vacuum brazing of metals and ceramics, and automation of welding and cutting processes for development of seam tracking sensors and automatic nesting systems. Dr. Na is a member of the Korean Society of Mechanical Engineers, the Korean Welding Society. the American Wekting Society, and the German Welding Society.

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a study on torch path planning in laser cutting processes part 2: cutting path optimization using...

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