Proceedings of the Fifth International Conference on Machine Learning and Cybernetics, Dalian, 13-16 August 2006

1-4244-0060-0/06/$20.00 ©2006 IEEE

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MULTI-ROBOT COOPERATION COALITION FORMATION BASED ON GENETIC ALGORITHM

HUI-YI LIU, JIN-FENG CHEN

Computer & Information Engineering College, Hohai University Nanjing, 210098,China E-MAIL: hyliu@hhu.edu.cn,hhuchen@163.com

Abstract: Coalition formation is a key problem in multi-robot

cooperation. The formation of multi-robot cooperation for single or multi-task is settled via finding the coalition maximum value or the coalition structure with the largest total coalition value. Traditionally, exhaustive method is used to get the optimal coalition or coalition structure with huge consumption in both computation time and communication overhead, which possibly results in search combination explosion. In this paper, agent coalition cooperation in MAS (Multi-Agent System) is applied in multi-robot system with genetic algorithm for the formation of multi-robot coalition and coalition structure in order to gain the possible maximum coalition value during task execution through finding the optimal solution or feasible solution for multi-robot coalition and coalition structure.

Keywords: Multi-robot Cooperation; Agent; Coalition Formation;

Genetic Algorithm

1.ntroduction

If a single robot can not or not effectively complete a given task in multi-robot systems, robots form a certain combination (such as coalition) to complete it and reach global optimization[1]. Different robot coalitions execute tasks at different efficiency and cost due to the different number and ability of their members. It is a priority to form a best robot coalition and assign the task reasonably with maximum benefit at lest cost during task execution in multi-robot cooperation.

In multi-robot systems, the formation of robot coalition is an optimization problem. The same with other optimization ones, it requires the optimal or satisfying solution in a complex and large searching space [2,3]. The strategy for the coalition formation is looking for a global optimization plan for task assignment. To reach that, the task assignment process exhaustively enumerates all possible coalitions, and tries to assign each task, and finally

reaches the global optimization. For a system with n robots,

the number for all possible coalitions is 121

−=∑=

nn

i

inC .

When n is comparatively large, this method will consume a lot in computation time and communication overhead and may lead to possible search combination explosion. Many researchers have studied on how to reduce the complexity of exhaustive method. Shehory[4] introduces a constant K to limit the number of agents in agent coalition, which reduces that number effectively, communication overhead and computation load to save cooperation time and increase cooperation efficiency. However, there is some blindness in the determination of K and coalition formation. Sinclair [5] and Billionnet[6] introduce low estimation method, reducing search space via cutting down search tree with given restriction requirements. Those two methods have done much on reducing the complexity with a high cost on the optimality, and the second optimal solution is usually gained. In this paper, agent coalition cooperation in MAS (Multi-Agent System) is applied to multi-robot system with genetic algorithm[7] to settle the formation of robot coalition. In that way, the cost is greatly cut for finding the optimal or feasible solution for robot coalition or coalition structure, and the coalition value will be possibly maximized during task execution with the solution (of robot coalition or coalition structure), to sufficiently increase the efficiency of the whole multi-robot system.

2. Formation of Robot Coalition

2.1. Coalition and Coalition Structure

Each robot can be regarded an agent, and one robot coalition C is a set for one or more robot agents to complete a task a single one cannot in systems. A robot CS (Coalition Structure) is a combination for finite numbers and non-intersection coalition in set A of robot agents, to complete task lists that a single one cannot. In one robot CS,

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each robot agent only belongs to one coalition, and there is only one agent in some robot coalitions. For instance, in set S = {a, b, c} (a, b and c represents a robot agent respectively), there are seven possible robot coalitions: {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}, and five possible robot CS: {{a}, {b}, {c}}, {{a}, {b, c}}, {{b}, {a, c)}, {{a, b}, {c}}, {{a, b, c}}.

2.2. Models of Coalition and Coalition Structure Formations

Suppose that set },,,{ 21 nAAAA K= is for n robot agents and that each Agent iA has r-dimension capability

vector )1,1(,0,,,, 21 rjnibbbbB ji

riiii ≤≤≤≤≥>=< K . Individual

capability jib is a property of iA to quantitatively

describe the capability of iA to take a certain action or complete an unseparate task . Suppose that set

},,{ 21 mtttT K= is for m tasks, and that a set of corresponding capability vector

)1,1(,0,,,, 21 rkmibbbbB rt

rtttt

iiiii≤≤≤≤≥>=< K is required to

complete each task it in T. After it is completed, Benefit

itofitPr is acquired. There is an capability vector

>=< rCCCC BBBB K,, 21 in coalition C, and CB is the total

of capability vectors of all agents. To execute task t, robot agent coalition C must satisfy )1( riBB i

Cit ≤≤≤ . For each

C, there exist coalition cost CCost and coalition value CValue . CCost is the total cost for members to complete t in C. And the coalition value is determined by

CCost and tofitPr upon the completeness of t. If the

coalition satisfies )1( riBB iC

it ≤≤≤ , CValue is positive,

otherwise it is 0. For a given task, the more CCost , the less

CValue . Formal description of the problem: For a given set

},,{ 21 mtttT K= with m sub-tasks, each lt has a set for required capability vector set lB , and each robot agent

iA has its own capability vector iB . The problem is how to assign each single task )( Ttt ii ∈ or task list

),1,1,,(,, jimjmiTtttt jiji ≠≤≤≤≤∈K to some robot agent C or robot CS while CValue or CSValue can be maximized.

3. Algorithm Design

Suppose the environment for robot coalition formation

is non-superadditive [8],which means it will cost for adding additional robots into the coalition. The increase of coalition members does not always mean benefit.

3.1. Algorithm Design of Coalition Formation

(1) Genetic Encoding Suppose that },,,{ 21 nAAAA K= is a set for robot agents and that at a given time, the number of free agents is

)0( nmm ≤≤ . Make coding with chromosome whose length can be changed and binary code {0，1} for encoding. Define an individual }),,,{( 21 maaaCC K= as a coalition result, and a gene )1( miai ≤≤ represents whether agent

iA exists in the coalition ( ia ＝1 for true, and ia ＝0 for false). (2) Fitness Function Design Suppose cCost for the cost of coalition C, ｔ for task, and tofitPr for benefit. If C cannot complete t, the coalition value 0=cValue ，otherwise it is positive. And the more the cost, the less the value. The fitness function is defined as CtC CostofitValueCF /Pr)( == , where C cannot complete ｔ, tofitPr ＝0. (3) Initialization Choose an integer p（20-500 commonly）for the population scale. (4) Selection For individuals with higher fitness values to have more chance to survive, roulette wheel selection or Monte Carlo method is introduced with optimization reservation. The purpose for the integration is to constantly increase the average fitness value during genetic operation, and to ensure the value of the best individuals will not drop. (5) Crossover Random one-point crossover is to intersect the individuals with the highest fitness value with those with the second highest. In the same way, the offspring is guaranteed to be better than their father individuals. And after mate, the population scale must not be less than p. (6) Mutation For concrete algorithm, the mutation probability is determined between 1-5% with dynamically change. During the starting phase of the algorithm, the probability is used with comparatively small values, and increases with the execution of the algorithm. In this way, the algorithm has the ability to partly search randomly, and premature convergence is prevented. (7) Replacement For better solution, optimization reservation is used.

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That means to replace worstC with the lest fitness value in the coming sub-population with the best individual bestC of all generations. (8) Termination Requirement Regulate genetic generation is used.

3.2. Algorithm Design of Coalition Structure Formation

Compared to that of coalition formation, the algorithm design of coalition structure formation is different in genetic code and fitness function.

(1) Genetic Code Suppose set },,,{ 21 nAAAA K= for robot agents. At a

given time, the number of free agents is )0( nmm ≤≤ and },,{ 21 ltttT K= for task. Make coding with chromosome

whose length can be changed and decimal symbol set },,0{ lK for encoding, where l is the number of T. Define an individual }),,,{( 21 maaaCSCS K= as a result

of a coalition structure, and a gene )1( miai ≤≤ determining which sub-task coalition to which Agent iA belongs, and )0(, lkkai ≤≤= determining No. K sub-task to which iA belongs. When 0=ia , iA does not belong to any coalition.

(2) Fitness Function Design For one coalition structure CS and task list

},,{ 21 ltttT K= , to put it simple, CS can be formed by l coalitions, every coalition iC is in correspondence with

it , and iC includes several agents or none. If iC does not include agent, sub-task it is not chosen. Therefore, the fitness function is defined as

∑∑==

−==l

iCt

l

iC iii

CostofitValueCSF00

Pr)( .

4. Computation Examples

(1)Experiment I: Coalition Formation Suppose task l enters the system. The task requires

four different capabilities of robots: },,,{ 4321lllll bbbbB = . The

benefit for l is 4321 432Pr iiiil bbbbofit ⋅+⋅+⋅+= . If the

coalition cannot complete the task, 0Pr =lofit . The fitness function is CtC CostofitValueCF /Pr)( == . And this evaluation function can be regarded as the utilization rate of

coalition resources. Suppose there ten free robots in the system with their

capability vectors indicated in Table 1 (agent in column, and capability value in row).

Select l , lB ={185，135，136，162}， ofitPr ＝1511.

With the above genetic iterative, the optimal solution is }0000111100{=C with the fitness

value 0.9402. That means 6543 ,,, AAAA agents are chosen to execute the task for the maximum benefit.

(2)Experiment II: Coalition Structure Formation Suppose task list },,{ 1 mttT K= enters the system

dynamically. Tasks have no priority and require robots with four different capabilities: },,,{ 4321

lllll bbbbB = , and the benefit for l is )432(3Pr 4321

iiiil bbbbofit ⋅+⋅+⋅+= . The fitness function is

∑∑==

−==l

iCt

l

iC iii

CostofitValueCSF00

Pr)(, and this

evaluation function can be regarded as the benefit value for CS to execute },,{ 1 mttT K= .

Suppose there are 10 free robots in collaborative virtual environment with the same capability value in Experiment I. In task list },,{ 321 tttT = , 1t requires the capability value }112130105110{

1=tB and

1Pr tofit

＝ 1158, 2t requires }105147135144{2

=tB and

2Pr tofit ＝1275, and 3t requires

}126104121141{3

=tB and 3

Pr tofit ＝ 1250. With the above genetic iterative, the optimal solution is

}2313313122{=C with the fitness value 6134.0. That means 1t , 2t and 3t are executed with 853 AAA , 1021 AAA and 9764 AAAA respectively.

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5. Conclusions

To find a proper robot coalition is an optimizing combination problem. For comparatively large robot systems, exhaust method takes a lot in time with low efficiency due to the large number of robots. Genetic algorithm has advantage in search performance and quick convergence, greatly reducing the cost for finding the optimal or feasible solution of robot coalition or coalition structure, and the found robot coalition or coalition structure executes the task with a lower cost, and is increases the efficiency of the system. However, the genetic algorithm is only suitable for comparatively large multi-robot systems. The advantage can not be seen for systems with robot agents in fewer numbers while exhaust method is much more proper.

References

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[3] Yong Gong, Li Yao and Wei Zhang,et al., “A Survey of Coalition Formation Technology in MASs”, Computer Engineering&Science, Vol 26, No.6, pp. 100-104, 2004.

[4] Shehory O. and Kraus S., “Methods for task allocation via agent coalition formation”, Artificial Intelligence, Vol 101, No. 1-2, pp. 165-200, 1998.

[5] Billionnet A., Costa M. C. and Sutter A., “An efficient algorithm for a task allocation problem”, Journal of ACM, Vol 39, No. 2, pp. 502-518,1992.

[6] David Todd, Pratyush Sen., “Distributed task scheduling and allocation using genetic algorithms”, Computer and Industrial Engineering, Vol 37, pp. 47-50, 1999.

[7] Ming Zhou,Shudong Sun., “Principle and Application of Genetic Algorithm”, National Defense Industry Press,Beijing,1999.

[8] Sandholm T. W. and Lesser V. R., “Coalition among computationally bounded Agents”, Artificial Intelligence, Vol 94, No. 1, pp. 99-137, 1997