A genetic-algorithm-based approach to the generation of robotic assembly sequences

Control Engineering Practice 7 (1999) 151—159

A genetic-algorithm-based approach to the generationof robotic assembly sequences

D.S. Hong!,*, H.S. Cho"

!Department of Mechanical Design and Manufacturing, Changwon National University, 9, Sarim-dong, Changwon, Kyeongnam, 641-773, South Korea"Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1, Kusong-dong,

Yusong-gu, Taejon, 305-701, South Korea

Received 29 July 1997; accepted 15 September 1998

Abstract

An assembly sequence is considered to be optimal when it minimizes the assembly cost while satisfying assembly constraints. Togenerate such sequences for robotic assembly, this paper proposes a method using a genetic algorithm (GA). This method denotes anassembly sequence as an individual, which is assigned a fitness related to the assembly cost. Then, a population consisting of a numberof individuals evolves to the next generation through the genetic operations of crossover and mutation, based upon the fitness of theindividuals. The population continues to evolve repetitively, and finally the fittest individual with its corresponding assemblysequence is found. Through case studies for industrial products such as an electrical relay and an automobile alternator, theeffectiveness of the proposed method is demonstrated, and the performance is analyzed. ( 1999 Elsevier Science Ltd. All rightsreserved.

Keywords: Assembly sequence generation; genetic algorithms; robotic assembly; assembly cost; assembly constraints

1. Introduction

It has been reported that, on average, assembly costsaccount for 10—30% of the total cost of most industrialproducts (Nevins and Whitney, 1980). To reduce suchassembly costs, much research effort has been put intoassembly sequence generation. As has been pointed outin other research work, assembly sequence planning hasdirect consequences for productivity, product qualityand the fixed costs that are involved with the assemblymachines and other equipment.

Up to now, various methods for generating assemblysequences have been reported: tree-search or graph-search methods (Cho, 1992; Cho and Cho, 1993; Choet al., 1993; Fazio and Whitney, 1987; Mello and Sander-son, 1990), disassembly methods (Lee, 1989; Shin andCho, 1994; Shin et al., 1995), neural-network-based ap-proaches (Chen, 1990; Hong and Cho, 1995) and thesimulated annealing (SA) method (Hong and Cho, 1998).The search methods can find the optimal solutions, but

*Corresponding author. Tel.: #81-551-279-7577; fax: #81-551-263-5221; e-mail: [email protected].

have a problem of the search space explosion for anincreased number of parts. On the other hand, the dis-assembly methods and the neural-network-based ap-proaches can cope with larger parts, but usually findlocal optimal solutions. The simulated annealingmethod, however, can find an optimal solution with highsuccess rates even for an increased number of parts.

As an alternative search method, genetic algorithms(GAs) have been used for solving various optimizationproblems (Caponetto et al., 1993; Goldberg, 1989;Holland, 1971; Holsapple et al., 1993; Winston, 1992). Tosearch for a solution, the GA starts from a set of points ofan objective function, whereas the SA starts from a singlepoint of the objective function. It has been reported fromprevious works that the GA can find an optimal solutionfor relatively large optimization problems (Caponettoet al., 1993; Goldberg, 1989). In general, GAs regard thefittest individual resulting from the natural evolution ofa population as a solution. The population consists ofa number of individuals, each of which has a number ofgenes. The natural law, which operates on the genes, ischaracterized by natural selection, which produces thenext generation through crossover and mutation.

0967-0661/99/$ — see front matter ( 1999 Elsevier Science Ltd. All rights reserved.PII: S 0 9 6 7 - 0 6 6 1 ( 9 8 ) 0 0 1 7 7 - 4

Using such evolutionary characteristics, this paperproposes a new method for generating robotic assemblysequences by using a genetic algorithm. In this method,an assembly sequence is represented as an individual,which is assigned a fitness related to the assembly cost.Initially, a population consisting of a number of indi-viduals is created. Then, the population evolves to thenext generation through crossover and mutation, basedupon the fitness of the individuals. As a result of suchrepetitive evolution, assembly sequences, represented asthe fittest individuals, are finally found.

To show the effectiveness of the proposed approach,case studies for industrial products such as an electricrelay and an automobile alternator have been carriedout. The results show that the proposed approach suc-cessfully generates robotic assembly sequences, whileminimizing the assembly cost. This paper is organized asfollows: assembly sequence description, generation of as-sembly sequences using a genetic algorithm, and casestudies and discussion. Finally, conclusions are drawnfrom the results of the case studies.

2. Assembly sequence description

As a prerequisite to the subsequent assembly planning,the assembled states of parts in a product must be ad-equately described. Such a description, called the productmodeling, will be utilized to infer the assembly con-straints, and to evaluate the assembly cost. This studyadopts the modeling method previously proposed byCho and Cho (1993), in which products are modeled bya set of liaison data between parts. This section describesthe modeling method of products, and the concept ofassembly sequences.

2.1. Product modeling

A product is assumed to be suitable for robotic assem-bly; it is composed of rigid parts, interconnected witheach other in mutually orthogonal directions. Each partcan be assembled by a simple insertion or fasteningaction such as screwing. Fig. 1 shows such a product. Leta product A"(P, ¸) consist of n parts

P"Mpa Da"1, 2,2, nN (1)

interconnected by r liaisons

¸"Mlab Da, b"1, 2,2, n. aObN, (2)

where r"D¸ D and (n!1))r)n (n!1)/2. A liaisonlab represents the connective relations between a pair ofparts pa and pb . The connective relations can be dividedinto contact-type connections and fit-type ones. Theassembly directions shown in the figure are definedwith respect to x, y, z, xN , yN , zN . Then, a liaison lab is

Fig. 1. An example of a product: ten-part electrical relay.

expressed by a predicate

lab"liaison(pa , Cab , Fab , pb). (3)

The Cab is called the contact-type connection matrix, andthe Fab is the fit-type connection matrix. Each matrixconsists of 2]3 elements, and is expressed by

Cab"Acx

cy

cz

cxN

cyN

czNB , Fab"A

fx

fy

fz

fxN

fyN

fzNB . (4)

The element cd

is given 0 for no contact, rc for realcontact, and vc for virtual contact in the d directionbetween two parts pa and pb , where d3Mx, y, z, xN , yN , zN N.While, the element f

dis given 0 for no fit, sw for

screwing, r f for round peg-in-hole, and mp formultiple round pegs-in-holes. A more-detailed descrip-tion on the connective relations can be found in Cho andCho (1993).

2.2. Feasible, stable and optimal sequences

Generating an assembly sequence is equivalent tofinding a series of operations by which n parts are

152 D.S. Hong, H.S. Cho/Control Engineering Practice 7 (1999) 151—159

sequentially assembled to form an end-product. Impor-tant factors in the generation of assembly sequences areassembly constraints, which can be classified intoprecedence constraints and connectivity constraints.These assembly constraints can be inferred from theliaisons of the product (Cho and Cho, 1993).

The precedence constraints of a liaison are representedby a set of parts that must be connected before a pair ofparts are interconnected. These constraints result fromthe geometrical relationships between parts. The otherconstraints, the connectivity constraints, state that a partto be assembled onto an in-process subassembly musthave at least one real connection, i.e., a real contact ora fit, with some part belonging to the in-process sub-assembly. The connectivity constraints play the role ofpreserving only one in-process subassembly throughoutthe assembly of a product. It is noted that such assemblyconstraints are directly derived from the physical rela-tionships between parts in a real product.

Once the assembly constraints have been inferred, as-sembly sequences that satisfy the assembly constraintscan be inferred. Such assembly sequences are called thefeasible assembly sequences.

The feasible assembly sequences, however, do notalways guarantee the parts to fix onto an in-processsubassembly; parts may be loosely connected, and maycome apart when the subassembly is turned or moved.Such assembly sequences that maintain the stability ofin-process subassembly movement are called the stablesequences, by means of which the parts can be success-fully assembled to form an end-product. As can be ex-pected, if a large-degree-of-freedom robot is used insteadof a smaller one, an increased number of stable sequencescan be found, since the robot motion may eliminate someotherwise necessary subassembly movements. Stablesequences, therefore, are related not only to the sub-assembly state, but also to the degrees of freedom of therobot motion.

Among the stable sequences, optimal sequences areselected as the ones having the minimum assembly cost.The assembly cost is defined to reflect the normalizeddegree of motion instability C

as(0)C

as)1) and the

normalized number of assembly direction changes Cnt

(0)Cnt)1) when the degree of freedom of robot

motion is chosen (Hong and Cho, 1995). It should benoted that the C

asstrongly affects the complexity of jigs

and fixtures, while the Cnt

is closely related to the numberof turning devices. Taking into consideration theseC

asand C

nt, the assembly cost J (0)J)1) can then be

expressed by

J"G1 if an assembly sequence

violates assembly constraints,

or it is unstable,

osC

as#o

tC

ntotherwise,

(5)

where osand o

tdenote weighting factors determined by

the assembly system and the assembly cycle time (Cho,1992) and o

s#o

t"1.

The above equation will be considered when theenergy function and fitness associated with assemblysequence are to be derived.

3. Generation of assembly sequences using a geneticalgorithm

This section describes a genetic algorithm that findsthe fittest individual from the repetitive evolution ofa population. For such purposes, an assembly sequence isrepresented as an individual, which is assigned a fitnessrelated to the assembly cost. As a result of such evolution,the assembly sequences represented by the fittest indi-viduals are finally found.

3.1. Population, individual and fitness of individuals

A population consists of a number of individuals.Here, the number of individuals, denoted by n

pop, is

called the population size. Each individual represents anassembly sequence of an n-part product. Accordingly, anindividual consists of n genes. Each gene is attributed aninteger lying between 1 and n, such that the ith genenumber represents the part number to be assembled atthe ith assembly step. For example, the individual atthe top of Fig. 2 represents an assembly sequenceM1, 8, 9, 3, 2, 4, 7, 5, 6N of a nine-part product.

Each individual is assigned a fitness, which means theprobability that the individual will survive and createoffspring in the next generation (Goldberg, 1989; Win-ston 1992). It can be easily deduced from the concept of

Fig. 2. An example of crossover operation: partially matched crossover(PMX). The marks . represent the crossover points.

D.S. Hong, H.S. Cho/Control Engineering Practice 7 (1999) 151—159 153

fitness that a sequence having a low cost is regarded asthe one having a high fitness. As an auxiliary measure ofthe fitness, this study introduces an energy function asso-ciated with an assembly sequence by reflecting the assem-bly cost and assembly constraints. The energy functionE is defined as

E"CJJ#

n+i/1

(CPki#C

Sji), (6)

where J denotes the assembly cost in Eq. (5), kiis the

precedence index of the ith part, i.e., the ith gene, and jiis

the connectivity index. The CJ, C

Pand C

Sare positive

constants. The precedence index kiis assigned a value of

zero if the ith gene of an individual satisfies the preced-ence constraints. Otherwise, it is assigned a value of one.In a similar manner, the connectivity index j

iis assigned

a value of either zero or one.To determine the fitness, various methods have been

reported. This study adopts two fitness functions: thestandard method and the ranking method (Winston,1992). The standard method is a common one, whichequates the fitness of an individual with the relativequality of the individual. In assembly planning, the reci-procal of the energy can be attributed to the relativequality. Since the fitness means the probability, this studydefines the fitness such that the sum of the fitnesses for allindividuals in a population becomes unity. Accordingly,the fitness F

i(i"1, 2,2, n

pop) of the ith individual is

expressed by

Fi"

(1/Ei)

+npop

j/1(1/E

j), (7)

where Eimeans the energy of the ith individual in Eq. (6).

As an alternative, the ranking method that links thefitness to the quality ranking was reported (Winston,1992). It has been known that the ranking method notonly offers a way of controlling the bias toward the bestindividual, but also eliminates any unfortunate choice ofmeasurement scale that might do harm. In the rankingmethod, the fitness of the highest-quality individual isassigned a fixed probability p

rk, while the fitness sum of

the rest of the individuals is 1!prk

. In a similar manner,the second-ranked individual has the fitness p

rk(1!p

rk),

and so on. Accordingly, the fitness of the ith-rankedindividual F

i(i"1, 2,2, n

pop) is expressed by

Fi"G

prk

(1!prk)(i~1) for i(n

pop,

(1!prk)(i~1) for i"n

pop.

(8)

The performance of the two methods will be demon-strated and compared in the next section.

3.2. Genetic operations

Initially, a set of individuals are randomly generated.Then, the population is evolved by natural selection,

which produces the next generation through crossoverand mutation. Crossover creates two offspring as newindividuals from two parents among the current indi-viduals, whereas mutation operates only on one parent,and creates an offspring that is different from the parent.These crossover and mutation operations are describedbelow.

3.2.1. CrossoverThe genetic operation of crossover acts on two indi-

viduals, and allows two new individuals to be created.Various methods of the crossover operation have beenreported. In this study, the partially matched crossover(PMX) is adopted (Goldberg, 1992; Holsapple et al.,1993).

Fig. 2 shows an example of the PMX operation. Ini-tially, two individuals are selected as parents, based upontheir fitness, and two crossover points are randomlypicked along the individuals, as shown in Fig. 2a. Then,the shaded middle segments of the individuals are ex-changed to create new individuals, as shown in Fig. 2b.However, it can be seen from the figure that each newindividual will not always represent a meaningful assem-bly sequence; the first individual has two duplicate parts,8 and 5. To eliminate this conflict, the duplicate numbersoutside the middle segments are exchanged with the onesthat have been matched in the middle segments. Fig. 2cshows the offspring resulting from this partially matchedcrossover.

3.2.2. MutationMutation operates on one parent, and creates an off-

spring that is different from the parent. Fig. 3 illustratesan example of the swap operation for mutation (Hol-sapple et al., 1993). Initially, a gene is randomly selected,and is to be changed with another number, as shown inFig. 3a. This operation results in the creation of a newindividual, shown in Fig. 3b, but this individual does notnormally represent a meaningful assembly sequence; theindividual has a duplicate part 4. To eliminate this con-flict, the selected gene and the one having the mutatednumber are swapped with each other. Fig. 3c shows theoffspring resulting from this swap operation.

3.3. Genetic algorithm for generating assembly sequence

To simulate a GA, parameters for controlling the algo-rithm should be determined. The parameters include thecrossover probability p

cr, the mutation probability p

mu,

the ranking probability prk

and the population size npop

.Fig. 4 shows the simulation flow of the GA. The cross-over and mutation operations are summarized as follows:

Crossover. Generate a random number NR. If N

R)p

cr,

perform crossover to create two new individuals. Other-wise, reproduce the two individuals as new ones.


Fig. 3. An example of mutation operation: swap.

Fig. 4. The flowchart of the genetic algorithm for generating assemblysequences.

Mutation. Generate a random number NR. If N

R)p

mu,

perform mutation for each gene of the parents. Other-wise, reproduce the parents as new individuals.

The procedure shown in the figure terminates when thenumber of generations reaches a specified number, orwhen no change in the maximum fitness occurs for aninterval of a specified number of generations. In the latter

case, the individuals with the maximum fitness areobtained as the solution of the assembly sequences. Inthe next section, case studies are carried out using thisalgorithm.

4. Case studies and discussion

4.1. Simulation conditions

To show the performance of the proposed approach, itwas applied to the generation of assembly sequences forindustrial products such as an electrical relay (Fig. 1) andan automobile alternator (Fig. 5). As presented in theprevious section, the simulation flow is shown in Fig. 4.A simulation run terminates when the number of genera-tions reaches a specified number, or when no change inthe maximum fitness occurs during a specified interval ofa number of generations. In this study, the specifiednumber was assigned as 1000, and the interval for check-ing the termination was given as 20 generations. Theweighting factors in Eq. (5) were assumed to be o

s"0.5

and ot"0.5, while the energy constants were determined

to be CJ"1, C

P"1, and C

S"1. The simulation pro-

gram was written in Turbo Prolog and executed on anIBM PC 486(66 MHz).

4.2. Simulation results and discussion

Fig. 6 shows an example of the evolution of a popula-tion consisting of sixteen individuals for the ten-part

Fig. 5. An exploded view of a 13-part automobile alternator.


Fig. 6. Example of evolution of population for a relay (npop

"16, pcr"0.5, p

mu"0.08, p

rk"0.2).

relay. Each individual represents an assembly sequence,and is assigned an assembly cost. In terms of the geneticalgorithm, a lower energy means a higher fitness, asdescribed in Section 3.1. Fig. 6a shows an initial popula-tion in which individuals have relatively high energy. Asthe population evolves, assembly sequences with lowenergy are created; the first individual of the 61st genera-tion in Fig. 6d yields a low energy 0.362, and so on. At the140th generation, shown in Fig. 6e, an individual with thelowest energy 0.329 is created. As the population evolves,no individual whose energy is lower than 0.329 is created.As a result, assembly sequences whose assembly cost is0.329 in Fig 6f are found as the solution in this example.In this case, energy lower than the unity means theassembly cost itself. It can be seen from this example thatthe assembly cost is minimized by using the proposedmethod.

The results for the 10-part relay are shown inFigs. 7—10. Each result under a simulation condition wasobtained from twenty simulation runs, and is expressedin terms of both the rate of yielding an optimal sequence

Fig. 7. The results from 20 simulation runs at various mutation prob-abilities for a 10-part relay (n

pop"20, p

cr"0.5, p

rk"0.2).

out of 20 simulation runs, and the average number ofgenerations to complete a run. Here, the optimal se-quences were analytically obtained from the expert sys-tem (Cho, 1992). The procedure of assembly-sequencegeneration from the expert system can be briefly ex-plained as follows. Firstly, the assembly constraints areinferred from the liaison data of a product, as explained


Fig. 8. The results from 20 simulation runs at various crossover prob-abilities for a 10-part relay (n

pop"20, p

mu"0.08, p

rk"0.2).

Fig. 9. The results from 20 simulation runs at various ranking prob-abilities for a 10-part relay (n

pop"20, p

cr"0.05, p

mu"0.08).

Fig. 10. The results from 20 simulation runs at various population sizesfor a 10-part relay (p

cr"0.5, p

mu"0.08, p

rk"0.2).

in Section 2. Secondly, feasible assembly sequences sat-isfying the assembly constraints are generated. Thirdly,stable sequences that maintain the stability of the in-process subassembly movements are inferred from thefeasible sequences. Lastly, by the evaluation of assemblycost of the stable sequences, optimal sequences, yieldingthe minimum cost, are generated.

On the other hand, the proposed approach obtainsassembly sequences by using a genetic algorithm. Thesequences obtained are then compared with those of theexpert system to verify their optimality. Through casestudies, it was found that the proposed approach can find

the optimal solution, the sequence both satisfying assem-bly constraints and yielding the minimum assembly cost.Then, as a measure of performance, the rate of yield ofoptimal sequences is evaluated in this study.

Fig. 7 shows the results at various mutation probabilit-ies between zero and 0.13. The figure shows that the rateincreases until the mutation probability reaches 0.12.However, when the mutation probability is larger than0.14, the solution was observed to be unstable; the max-imum fitness fails to converge due to its irregular oscilla-tion during the evolution. The number of generationsrequired to complete a run is shown to be a minimumaround the mutation probability 0.07. It can be seen fromthese results that the mutation probability strongly af-fects the performance of the GA in this study.

Fig. 8 shows the results at various crossover probabil-ities from zero to unity. The figure shows that thecrossover probability barely affects the performance ofthe GA.

Fig. 9 shows the results for various ranking probabilit-ies. The result of the standard method is shown at thezero ranking probability for simplicity, while the result ofranking methods are shown at probabilities of more than0.1. The figure shows that the ranking method givessuperior performance to the standard method. Theranking probability 0.2 gives a relatively good perfor-mance both in the rate of yielding optimal sequences, andin the number of generations required to completea simulation run.

Fig. 10 shows the results at various population sizes. Itcan be seen from the figure that population sizes of morethan 16 give good performance in yielding optimal se-quences. As the population size increases, the executiontime to complete a simulation run becomes longer, al-though the number of generations becomes smaller.Thus, it is recommended that the population should bechosen to be as small as possible, provided that goodsolutions are guaranteed.

The results for the 13-part alternator are shown inFigs. 11—14. Each result under simulation conditions wasalso obtained from 20 simulation runs. Unlike the case ofthe 10-part relay, this case shows relatively poor perfor-mance; at best the rate of yielding optimal sequence 0.15was found as can be seen from Fig. 11a. To clarify theperformance, new measures are adopted instead of therate of yielding optimal sequences: the rate of yieldingstable sequences and the solution quality. The solutionquality q

#045for the stable sequences obtained from 20

simulation runs is defined by

q#045

"

J3%&!JM

J3%&!J

015

, (9)

where J015

represents the optimal assembly cost, JM is theaverage cost for the stable sequences obtained, and J

3%&is

a reference cost. In this study, the maximum cost of the


Fig. 11. The results from 20 simulation runs at various mutation prob-abilities for a 13-part alternator (n

pop"30, p

cr"0.04, p

rk"0.2).

Fig. 12. The results from 20 simulation runs at various crossover prob-abilities for a 13-part alternator (n

pop"30, p

mu"0.06, p

rk"0.2).

Fig. 13. The results from 20 simulation runs at various ranking prob-abilities for a 13-part alternator (n

pop"30, p

cr"0.04, p

mu"0.06).

Fig. 14. The results from 20 simulation runs at various population sizesfor a 13-part alternator (p

cr"0.04, p

mu"0.06, p

rk"0.2).

assembly sequence was assigned a value of unity; thus,the J

3%&is chosen as unity. The above equation states that

the solution quality for the optimal sequences becomesone, and 0)q

#045)1.

Fig. 11a and b show the results at various mutationprobabilities. The figures show that the mutation prob-abilities between 0.04 and 0.07 give relatively goodsolutions; the rate of yielding stable sequences and thesolution quality are high, while the number of genera-tions required to complete a simulation run is low.

Fig. 12 shows the results at various crossover probabil-ities between zero and one. The results show a tendencyquite similar to those of the relay; the crossover probabil-ity barely affects the performance of the GA.

Fig. 13 shows the results at various ranking probabilit-ies. As can be seen in the figure, the standard methodwhose result is depicted at the probability of zero cannotgive any stable sequences. In the ranking method, theranking probabilities of more than 0.2 are shown to giverelatively good solutions. This tendency is the same asthe case study for the relay.

Fig. 14 shows the results at various population sizes.The figure shows a tendency similar to the case of therelay. It can be seen that the population size of 20 givesgood performance, both in yielding stable sequences andin the execution times. The figure also implies that thepopulation should be chosen to be as small as possible,provided that good solutions are guaranteed.

From the above results, it is concluded that the pro-posed approach successfully generates robotic assemblysequences, while minimizing the assembly cost. The per-formance of the GA is shown to be strongly affected bythe fitness function and the mutation probability.

5. Conclusions

In designing assembly systems it is essential to deter-mine optimized assembly sequences for increasing pro-ductivity and reducing assembly costs. In this study,a genetic-algorithmic-base approach was proposed togenerate assembly sequences for robotic assembly.


The study denotes an assembly sequence as anindividual. The individual consists of a set of genes,and is assigned a fitness related to the assembly cost.Then, assembly sequences are found as the fittestindividuals resulting from the evolution of the popula-tion. Here, the population consists of a number ofindividuals, and the evolution is characterized by naturalselection, which produces the next generation of thepopulation through the genetic operations of crossoverand mutation.

Through case studies for industrial products, theeffectiveness was demonstrated, and the results wereanalyzed. The performance was shown to be stronglyaffected by the fitness function and the mutationprobability. From the results of the case studies, itis concluded that the proposed approach successfullygenerates robotic assembly sequences, while minimizingthe assembly cost.

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A genetic-algorithm-based approach to the generation of robotic assembly sequences